Sir 78440 N E 2 0 N _ Two TREAT 1 SES "OF THE Quadrature of CURVES, AND | ANALYSIS by Equations of an infinite Number of Terms * explained: CONTAINING * The 'Freatiſes themſelves, tranſlated into Engliſh, with a large Com- mentary; in which the Demonſtrations are ſupplied where wanting, the Doctrine illuſtrated, and the whole accommodated to the Capa- cities of Beginners, for whom it is Ber deſigned. By JOHN STEWART, A. M. Profeſſor of Mathematicks in the Mariſhal College and Univerſity of An 1 10 0 % I — — — — — 2 — — — — — — — — — — "> — — - LANDON: | Printed by JaMzs BzTTENHAM, At the Expence of the SocizTy rox THE ENCOURAGEMENT OF LEARNING; And ſold by Jonx Nouxsx atTemple-bar and Joun WrisToON in Fleet-/ reet, | Bookſellers to the ſaid SOCIETY, M. DCC. XLV. [Price twelve Shillings in Sheets.] Y 4 . = 4 a 9 2 * „ £ * Ya 74 | «, - are. © % 1 — # 4 It - ” 2 *; * THOMAS Duke of LEE DS MY LORD, HEN a ſtranger implores your GRACE's protection to a mathematical work, he is incouraged to it from the patronage, with which you are pleaſed to favour all the liberal ſciences ; and not from any peculiar regard vouchſafed to this more than others. For among the many vir- tues ſo univerſally acknowledged in your GRace's character, your zeal for the general advancement of learning 1s eminently conſpicuous, as being the Prefident and life of a Society founded ſolely for its incouragement. A Society ſo generous and diſintereſted in its inſtitution, that neither Greece nor Rome could ever boaſt of a nobler,defign; and ſo applauded by foreign nations, that they would gladly imitate it, were they bleſſed with ſuch a Maecenas to inſpire them with a like ardour for the promoting of ſcience. For whatever enter- tainment and delight knowledge affords the hu- man mind, and how far ſoever it elevates thoſe, who are poſſeſſed of it, above thereſtof their ſpecies; | IG A 2 yet DEDICATION. yet they only, who communicate it to others, have the merit of rendering it beneficial to mankind. + THE great regard, which your honoured Society retain for the works of the illuſtrious Newton, has ingaged them to condeſcend fo far, as to coun- 5 tenance his Commentator. The ſame motive ani- mated my indeavours to illuſtrate and explain thoſe principles, which are the foundation of the greateſt and moſt ſurpriſing diſcoveries, either of the pre- ſent, or any preceding age. And this being the firſt book of the kind, which the Society have taken under their care, as the dignity of the author juſtifies their choice; ſo, I would hope, it may ren- | der the public more favourable to my performance, ; I ſhall however have this ſatisfaction, that while I | offer a ſmall tribute to the memory of Sir 1/aac | Newton; I have the honour to pay my ſincereſt acknowledgements of gratitude toyour GRACE, and with the higheſt eſteem and veneration to ſubſcribe myſelf, May it pleaſe your GRACE, J Your GRACE's moſt obliged, 3 MfLſt devoted, and Z J Moſt obedient humble Servant, =—_ JOHN STEWART. The PREFACE. UCH as have duly read and conſidered Sir TJaac 8 Newton's Treatiſe of the Quadrature of Curves, and his other Yorks allied to it, muſt be ſenſible, that he has greatly extended the Limits of Geometry, by diſcovering a Method of inveſtigating Truth, ifferens from all thoſe Methods, which Geometricians had made uſe of before his Time; and which leads to the Diſcovery of the moſt hidden and remote Truths in Mathematicks, and Philoſophy. But his. invaluable Diſcoveries, eſpecially ſuch of them as are contained in this Treatiſe, are delivered by himſelf in ſuch a conciſe Way (the Effect of his vaſt Genius), that they ſtill lye beyond the Reach of not a few, who have made ſome conſiderable Progreſs in the Mathematical Sciences: or at leaſt they find ſuch Stops and Difficul- ties, before they are well entered into this Work, that they give it over; either apprehending that thoſe Difficultics. are ſo great, that they cannot ſurmount them at all; or, if they are able, that it will require more Time and Pains, than they can afford. to beſtow upon. ſuch abſtruſe and: deep Speculations. Yet the ſame Perſons, moved. by. the Character of the Author and the Work, would willingly beftow: L \ PREFACE. beſtow any reaſonable Time and Pains to underſtand theſe Diſcoveries, were they furniſhed with proper and ſufficient Helps for that Purpoſe. This Conſideration may juſtly make one wonder, why among the vaſt Number of Books and Performances, which have been ſent abroad into the World, ſince the Quadrature of Curves was farſt publiſhed, in the year 1704, no one has ever obliged the publick (ſo far as I know) with a Commentary upon this Work, or any Explication of it; by which Means it happens, that it's great Worth and Excellency ſtill lye in a great Meaſure concealed from the Bulk even of thoſe, who have made ſome tolerable Proficiency in the Study of Ma- thematicks. I do not pretend to aſſign all the Cauſes, that may have concurred to produce this Effect. It certainly cannot be owing to this, that learned Men are not ſenſible of the Neceſſity and Uſefulneſs of ſuch an Undertaking. One Reaſon perhaps may be, that they, who are beſt qualified for it, and could execute it to beſt Purpoſe, having once engaged ſo deep in the Study of Geometry, as the full underſtanding. of our Author's Treatiſe requires, forth- with affect to be Authors themſelves, rather than labour in the lower Office of explaining and illuſtrating the original Diſcoveries of others. Notwithſtanding it muſt be owned, that one may be as uſefully employed, to ſay no more, in this latter Office, as in the former, which is efteemed higher. Since it has been juſtly enough obſerved, that this Science is now brought to ſuch a Degree of Perfection, that the Application of theſe abſtruſe and ſublime Specu- lations to ſome uſeful Purpoſes, either in common Life or Philoſophy, is much more wanted, and would tend more to the Good of Mankind; than the carrying them to yet a greater - * * g P Ig * * L © 1 — _ 7 hy EGS. 2 * 5 7 4 LETS AE 4 q " 2 3 1 r r 2 +. - * * | "PREFACE. reater Height, without making any Application of them 2. Purpoſes mentioned, by which the Good of Society might be promoted, and the wi/e and excellent Conſtitu- ane Things in this material Werte better underſtood. This naturally enough leads me to anſwer an Objection, that ſome perhaps may be ready to ſtart, both againſt our Author's Fears and this Attempt to make it more uni- verſally known. Cui uſui is the Word: Why ſpend ſo much Time and Labour upon ſuch abſtra& geometrical Speculations? We acknowledge, fay ſuch, the Purſuit and Acquiſition of Knowledge is an ble Exerciſe to the Mind of Man; but this alone is not ſufficient to account for waſting away ſo much Time, which our Nature and Circum- | ſtances, as reaſonable Beings and Members of Society, re- quire to be otherwiſe employed. As ſuch, others have an Intereſt in Us, and may juſtly demand a very conſiderable Share of our Time and Labour. Mon nobis ſolum nati ſumus, ortuſque noftri partem patria vindicat, partem amici. Cic. In anſwer to which, it is readily acknowledged, that other rational Beings have a very conſiderable .Intereft in our Time and Labours. But if the Matter be rightly con- ſidered, it will appear, that this Intereſt and Concern of theirs is by no Means overlooked or neglected; on the contrary, it is conſulted and regarded. The natural De- ſire of Knowledge found in all Mankind is a great Ex- citement to the Purſuit and Acquiſition of it, and at the ſame Time a ſufficient Indication, that Mature deſigned it for a Part of our Employment ; which is ſtill confirmed by the great Satisfaction and Delight, the Mind of Man enjoys in the Diſcovery and Poſſeſſion of Truth. Ex guatuor autem locis, in quot honefti naturam vimque 1 | diviſimus, - K - - : * = w = : A - on. 0 * - * 1 4 * © * | | | "WEIS "Ws" : 7 174 Irre | AIRES on % x ane „ > * „ Wk Y * * mn N >> } — * * * \ PREFACE. diviſnaus, primus ille, qui in veri cognitions confiftie, enaxime naturam attingit humanam, OQmnes enim tra- himur & ducimur ad meer et ſcientiæ cupidita- tem, in qua excellere pulcbrum putamus. Cic. Who- ever therefore contributes to the Advancement of Know- ledge in others, by beſtowing ſome Time and Pains to make way for the Entrance of it into their Minds; ſo far adminiſters to their Happineſs, and conſults: their Au- vantage. And this holds eſpecially in the preſent Caſe, if the natural Tendency and genuine Effect of mathema- tical Knowledge purſued and acquired be rightly conſi- dered, and attended to; as it ſtrengthens our intellectual Powers, by forming in the Mind an Habit of thinking cloſely, and reaſoning accurately, even upon other Subjects But further, every one, who 1s the leaſt converſant in theſe Matters, knows, that natural Philoſophy never was, nor can be ſucceſsfully proſecuted or advanced, but by the 1 Would you have a Man reaſon well, you mult uſe him to it betimes, exerciſe his Mind in obſerving the Connection of Ideas, and following them in train. Nothing does this better than Mathematicks ; which therefore, I think, ſhould be taught all thoſe, who have the Time and Opportunity, not ſo much to make them Mathematicians, as to make them reaſonable Creatures. Locke, On the Conduct of the Underſlanding. And a little after: I have mentioned Mathematicks as a way to ſettle in the Mind an Habit of rea- ſoning cloſely, and in Train; not that I think it neceſſary, that all Men ſhould be deep Mathematicians ; but that having got the way of reaſoning, which that Study neceſſa- rily brings the Mind to, they might be able to transfer it to other Parts of Knowledge, as they ſhall have occaſion. I think the Study of Mathematicks of infinite Uſe even to grown Men; firſt by experimentally convincing them, that to make any one reaſon well, it is not enough to have Parts, wherewith he is ſatisfied, and that ſerve him well enough in his ordinary Courſe. A Man in thoſe Studies will ſee, that however good he may think his Underftanding ; yet in many Things, and thoſe very viſible, it may fail him, This would take off that Preſumption, which moſt Men have in that Part. The Study of Mathematicks would ſhew them the Neceſſity there is in reaſoning to ſeparate all the diftin& Ideas, and ſee the Habitudes that all thoſe concerned in the preſent Enquiry have to one another, and to lay by thoſe which relate not to the. Pro- poſition in Hand, and wholly to leave them out of the reckoning. This is that, which in other Subjects beſides Quantity, is abſolutely neceſſary to juſt reaſoning. In geometria partem fatentur eſſe utilem teneris etatibus : agitari namque animes, atgue acui ingenia, & celeritatem percipiendi inds venire cancedunt, Quintil. | Help 1 PFREPF AGE. World, and all Things in it, are ſo wiſely framed, main- tained and preſerved, in that admirable and beautiful Or- der and Harmony, wherein we behold them: which is the true Notion and Deſign of natural Philoſophy. According - ly you will find, that the greateſt Maſters in theſe Sciences fecting of natural Knowledge. Witneſs our noble Author, who in that admirable Petformance, his Mathematical Prin- ciples of naturul P biloſophy, ' hath: happily ſhewn us, of what great Uſe/abſtra& Mathematical Knowledge' may be, for inveſtigating the Forces of natural Bodies; and thence explaining and demonſtrating the Phænomena and Laws, which Nature obſerves in her Operations, both in the Solar the Diſcoveries contained in that Book, is owing, to, , and founded upon, the Doctrine of Fluxions (the Invention of the ſame happy Genius) as was obſerved ſome Time ago by the noble and learned Marquis HN Hoſpital ; ſo we find in particular, chat he often proceeds upon the Qua- drature of Curves as a Poſtulatum, or Principle already known and granted. See Propo/..'46,-1;53 54, 56, 81, Book I. and many other Places. By which he hath ſhewn, that the moſt ſublime Parts of Geometry, and particularly the Doctrine of Fluxions, and the Quadrature of Curves, are of infinite Uſe in true Philoſophy, 2 Ceſt encore une Juſtice due au ſcavant Monſieur Newton, & que Monſieur Leibnitz lui a rendue lui-mEeme : qu'il avoit auſſi trouve quelque choſe de ſemblable au Calcul 3 differentiel, comme il paroit par Pexcellent livre intitule, Philaſopbiæ naturalis Prin- 4 cipia Mathematica, * nous donna en 1687: lequel eſt preſque tout de ce calcul. Anal, == des infiniment Petits. Pref, 4 | ; L a Wherefore * 1 9 v of theſe Sciences) that we, diſcover the Gonfituzlong and inveſtigate the Laus, decerding to Which this material have moſt ſucceſsfully laboured in the promoting and per- Syſtern, and upon this our Farth. And as a great Part of 1X I IT — — * N it } | — > _ - p — = 4 5 ad ae wg — — ͤ —— -—- - 4 — — * —— — — — PREFACE. Whetefote by cultivating Geometry, and ſtudying even it's Mebelt Pare, ſevefal noble” and uſeful rw ke are ſerved and promoted. Our rational and intellectual Powers are greatly enlarged and ſtreripthened, Many Things uſeful and eonvenient in common Liſe are diſcovered. Natural Philoſophy in all it's Branches is happily advanc- ed. And what is moſt cotifiderable of all,” the Mind is thereby led to perceive and obſerve the infinite Knowledge and Wiſdom, Power and Goodneſs of the Aumgbty Creator, and bountiful Preſerver and Governor of this Univerſe. The Knowledge and Contemplation of whoſe Works and Perfections. is the moſt noble and delightful Exerciſe of our reaſonable Powers, and naturally leads the Mind to thank and praiſe, reverence and admire, love and imitate this ſupreme Being; which is the greateſt Happineſs and Perfection of any reaſonable Creature, and the true End of all our Knowledge, and of bur very Exiſtence itſelf, It is chiefly with a Deſign of promoting theſe noble Ends, that I publiſh to the World this Commentary upon Sir Iſaac Meuton's Quadraturt of Curver, and his Analyſis by Equations of an infinite Number of Terms : And though my Attempt ſhould fail of the defigtied Effect, I ſhall at leaſt have the Pleafure of defigning well, which is ſome- thing to a virtuous Mind. | not Doh T- . * * * @%@ by - (14% 7 , PE * . 4 u 2,017 mus tives |: ; 31 1 * Le» 984 19 I * — e a 8 o cb *%.% . * ©. s 0 — 0 X AS — 7 . * % 1 = 4 1 - = — 4 1 l 4 E 2 7 * 2 * * * 2 ME WSN” =O 6 >a” > wa = * 1 4 1 e 2 ' 2 cy N IB en N 1598 Y = * e! 55201 il JLILS 1 * ©. * 4 | 4 be 14 1 1 Try * \ 122 — 15 _ N - . 1 ; ry * * 1 * * S8 20 ane | | \ WW _—_ LO N — — _ 5 urn eser 208 unden arne 1 3 W. +» Kd Contents of Sir Jac Newton's Treatiſe 7 hs ture ? RF « MRR e 1.2 5 of Curves... T. al; * 5 51 is 4 2 ww ohm 9h eee * 4 al wi LY E N E SS -of Mathematical Quantises, fr. xy (77 71! fi * VP he original, erer e O ad: 090 Flyxions very nearly as the Augments of the Flowing Quantities, gene- rated ee Bl but very ſmall Particles of Time: ee adatelhz- in- che firſt Ratio of the naſcent A ts, Art. 3. T his explained and illuſtrated by the Dekerpetnofplane Figure whoſe Fluxions ate as their Qrdinates, Art. 14. The Proportion of 7 Flpxions of the Abſcis, Wee an nc Cane Line, Gn Cury dad Art. . Dl Fiukions in the la Rat? of th the ev 8 Parts, Art, " hs 'FheProportion of the Fluxions of Solids ; ang their Sürfaces, FRY Examples of che Method of Fluxions, ; Art. 8. wins 2 The Quantity y flowing uniformly, to fin d the Fluxion of , Ast. JI. The Fluxions of all. Sorts of Quantities 8 be collected by the Method 7 prime and ultimate Ratios: and no Jeceſſity of introducing Fi- VOY fmall into Geometry, Art. 12. 5 | "Cc 4 $45 , n a 5 , "The. Quadratur of Curves, n AT 8. o The Not Netz e ang . dens 220 ap Ac La of chr Gr ra- 4/13. * * a 2 Prop. a I *ii CONTENTS. Prop. 1. Prob. An Equation being given involving any Number of flowing Quantities, to find the Fluxions, Art. 15. A fuller Explication of this Prop. with the Way of finding ſecond, third, and ſubſequent, Orders of Fluxions, w 18. Prop. 2. Prob. To find, ſich Curves as cun be ſquartd, Art. 22 Prop. 3. Th. When Ye Area of a urve is 2. tt being the Abſciſ5, and RD e + fz + gz" 3 8 0 what it's ra is equal to: demonſtrated, Art. 23. Prop. 4. Th. When the Area of a Curve i AR. g* -(2andR, being the fame as bifore ; and S k FI A m &c.)/what the Ordinate 7s equal to, Art. 25. Prop. 5. Th. When the Ordinate of a Curve is 2 R 1 a + bz” + 025 &c. what the Area Lad? be equal to: demonſtrated, Art. 26,, O Saz 10. Mie L nr rat e „ 1 00 Every Ordinate of a Curve is reſolved into a Serie two different Ways : with the Uſes of this for 'findihg the ' e of Curves; and when 1 they may be ſquared, or not, Art. 28. 1 | ; When the Ordinate is 4 Product contained uhr a Fatidhal Factor and an irreducible ſurd Factor, what to be done, Art. 2999 When the Ordinate is an irreducible rational Fraction, whoſe Deno nator is made up of two ot more Terms, what to be done, Art. 30. When the Ordinate is an irreducible F raction, whoſe Denominator i 18 a Product contained under a rational Factor, artian irreducible furd * * . * 4 = _ 1 5 © oP . " 4 > 3 # 7% 5 23 — 2 2 - - bs r 1 3 Rn = ("Eh 3a herd n 83 * 8 abs, * . ks * W < rat ma p TN N 2 Ps ? . . * ET : * = - = « + 4 — 8 * * J «4 — * * e, f, g, &c. remain ' the La and for ꝙ and r be wrote ſurceſſuely 2 integral. Numbers: and if there be given the Areas 'of : one, _ three, Cc. of the innumerable Curves whojs Ordinates are ſo drfigned Ge. when the Ordinates are Binomials, . Trinamiali, Quadrinemia reſpectively, under the V inculum of the Root ; I Areas of all theſe 3 Curves ſhall be given, Art. 33. 1 Prop. 8. Th. Vin "the Ordinate of any Curve 2 Roe pt” SE", the - Quantities 6, nA fe, e, f. g, k. I, m, &c. remaining 5 ag. of 6, T, v, be wrote * whole Numbers facceſroely : and if the Areas * * * IC FE { : GOMTEMNTS, "KH; the. Curves be given, which are defig ned by the Ordinates bus g, when R 2508 are Binomi als: or 7 three, four; &c. of F theſe At eas be gi ven, awhen'R and 8 taken together confiſt of five Nomes, fix Nomes, &c. W z the Areas of all the| Curves ſhall be given,.. Art, 27 Prop. 9. Th. The Areas 0 all thoſ Curves are aus], whoſe Ord: nates are reciprocally as the Fluxions of their Abſciſſes, Art. 40. Cor. 1. Innumerable Carved found whoſe Areas are equal among chem- ſelves, Art. 41. dh Rt Cor. 2. How a Curve whoſe Ordinate is 0-1 K e + - for” + 2 Sc is transformed into another equal Curve. Art. 2 Cor. 3. How a Curve waeks Ordinate is 2%" x a + bg” c &c. A J2" + 857 G, is transformed into another equal Gueve, Art. 43. | Cor. 4 How « Cura whoſe Ordioae, is 21 12 pn T cz" Or. Xe + /2" + g= Fc! x þ + be" + mz*" Ge. * oi transformed 5 into another equal Curve, Arte 44. Cor. 5. The Curve whoſe Ordinate is 2%") Xe ＋ f2" g Ser, | transformed by putting L =x, Art. 45. Cor. 6. The Curve . whoſe Ordinate i is 27 xe + fe" + gz* &c.\" „IE + mz" Scl, transformed the ſame Way, Arr. 46. Cor. 7. A Curve the Relation of whoſe Abſciſs and Ordinate is de- | fined by a certain affected Equation, made to paſs into another equal toit, where that Relation is expreſt by an Equation not affected, Art. 48. Cor. 8. A Curve having the Relation of the Abſciſs and Ordinate defined by an affected Equation of a certain Kind, transformed into another, where that Relation is expreſt by an Equation leſs affected, Art. 49. Cor. 9. How a Curve whoſe Ordinate is a" xw+rÞqxfs +5 + 21 x gz” Gel 1 1 + 3 x ez? ＋ fo + g N Se, 5 cransformed; into another whoſe Ordinate is & X a N Art. * « 0 > Xa, © 44 Y 9 . * N * 8 # 8 4 2 * ks . . $ ”'Y n 4 * — * - »h 4 "1 4 Pe". OO -- 4, 4 Sts Rod os. Ads _ ” \ * — . bed % "AE A” * 2 7 5 2 "INE «+ G ; 23 * * CNET. K

* 2 — U 1 3 * 208 6 * 2 Jr , "VI. "4 - Ta BR OTIS © * 2 2 2 3 - P ** 0 —__ gs: > x8 —_— 2 Ao. 88 r © 2 5 e | a . ett” V2 ©. ods —— . + + % CC 4 < "On. > kT -— - 2 . * a 2 — Po - 4 . A Mc; A 1 ö , x "y ” . * LE. re Ag” WY PIP LY OW 7 2 1 l 1 r AI # * - * 8 xi CONTENTS. Cor. 10. How a Curve baving Seki EE JR . for it's 8 is travslormeg. 30th. erh e e We AT, rt. 51. * - . d XK Prop. 10. Prob. 7 find the moſt imple » with which c. may be geometrically compared, whoſe Ordinate, Mak Fhe e Ts, is determined by an Equation 1 not affected, Art. 52. Caſe 1. When the Ordinate is 22, Art, 53. "Fa Lb Caſe a. When the Ordinate is 62%" x ce go * Art. 54. St z. When the 2 is. 2 . t — Ac. X e Lag” Sch „Art. 55. ; — 12 K WO a 7 — Caſe 4. When the Ordinat s © x FITS by” + . * 2" ＋ g Se. x & 4-12" + mz*" Sc.! * 56. Cale 5. When Tei Ordicate conſiſts of different Parts: what to be done, Art. 57. e Cor. 1. Every Curve whoſe Ordinate is. the aT S Roanof | it's own Equation, may be e with the moſt Ample 1 Fi "igurss, rectilinear, or curvilinear, Art. Cor. 2. Every Curve, whoſe — > bo being defined by an affected Equation, paſſes into an Equation not affected, is either ſquared, or compared with the moſt fimple Figures: and by this Means every Curve-whoſe (Equation conſiſts of three Tecnas, is.compared with the moſt ſimple Figures, Art. Cor. 3. Every Curve, whoſe Ouiinate 3 is Wes by an affecbell Equa- tion, which can be made to paſs into an affected. quadratic Equation, s either ſquared, or compared with the moſt ſimple Figures, Ant. 60. Schol. When Curves: are to he:ſquared, it is moſt cohV⁰en lor Nractioe to have Tables ef -Quadratures, , Art. 61. . A Table of the more Gmaple Kind of Cadets that may be TN Y = =, is quadrable or not; and when it is doubly EE cable. Art. 150. Rule for finding when Curves of this Claſs 2—1 x eÞ 7 * 2 2 TE =y, are quadrable or not; and when chey are doubly quadrable, Art. 162. Rule for finding the ſame Things in Curves of this cus 1 — — DME Fiz” =y, Art. 163. Rule for finding the ſame Things in Curves of this Olaf 2: „1 YT x a+ be Per ce. =, Art. 164. Rule for finding the ſame Things in Carves of this Claſs 2 X 7 PSN „ 2 +” F. =y, Art, 165. A Rule for finding when any Curve of this Form Z x = * 2 bz" - gz" Sc. =, may be ſquared, Arr. 166. | LOS 3 680 CONTENTS. | "FACT vY. 0 6 Of the Poſition and Limits of the Areas of Curves, Page 133. A general Rule for determining the, initial Limits of curvilinear Areas: - with their Poſition and Extent, Art. 168. A general Rule for determining the Points of the Abſciſs in which the Curve meets with it, Art. 170, | A general Rule for finding the Points where the Curve decuſſates the Baſe or Abſciſs, Art. 173. How the Area of a Curve muſt be eſtimated, when it lies partly above and partly below the Baſe, Art. 174. | That there are ſometimes different initial Limits of the Area: and what to be done in that Caſe, Art. 177. That the Area found by the Method of Quadratures, ſometimes lies partly upon one Side, and partly upon the other Side of the Ordinate: when this happens: and what to be done in that Caſe, Art. 18 1. Sometimes no initial Limit of the Area can be found: in hat Caſe this happens, Art. 184. What to be done when that happens, Art. 18 5. How to find the Area adjacent to any given Part of the Baſe, Art. 187. The Caſe of the hyperbolical Space adjacent to the Aſſymptote in the Apollonian Hyperbola, conſidered and explained, Art. 190. Of changing the Beginning of the Abſciſs; with its Uſe; and Effect, Art. 192. | f SECT. VI. Notes on Article 32 of the Quadrature of Curves. Demonſtration of Prop. 6. Quad. of Curves, Page 194. SECT. VI. Notes on Articles 33—39 of the Quadrature of Curves, Explication of Prop. 7. Quad. of Curves, Page 1 56. Demonſtration of Caſe 1. of that Prop. when R = e+fz" a Binomial, Art. 190. Two general Theorems for deriving the Areas of all ſuch Curves, when the Exponent of 2" is to be increaſed or diminiſhed by », Art. 202. b 2 Explication CONTENTS. Explication of Caſe 1. Prop. 7. when R =e . + gz" a Trino- mial, Art. 204. Three general Throremf Nr deriving the Areas of all ſuch Curves, when the Index of 2 =" is to be increaſed or diminiſhed by 3, Art. 206. A Caſe ſuppoſed by Sir Iſaac, not ſufficiently underſtood : with ſome general Obſervations upon it, Art. 20 Demonſtration of Caſe 2. Prop. 7. accommodated to Curves i in which it is R—e+-/z” a Binomial, Art. 2 10. Two general Theorems for deriving the Areas of all ach Curves, when the Index of R'E” is increaſed or diminiſhed by Units, Art. 211. | Caſes firſt and ſecond of Prop. 7. illuſtrated by an Example, Art. 213. Two general Theorems for deriving the Areas of all thoſe Curves in which R=e-+fz" + g a Trinomial; and the Index. of RE” is increaſed or diminiſhed by Units, Art. 215. Caſes third and fourth of Prop. 7. illuſtrated, 218. ; N Demonſtration of Prop. 8. Quad. of Curves : e into = different Caſes, Art. 219. SECT. VIII. Notes on Articles 40——51 Quadrature of Curves, Page 174. The Demonſtration of Prop. 9. Quad. of Curves, referred to Art. 38. of the Explication, Art. 224. | Cor. 1. Prop. 9. illuſtrated by Examples, in the Transformation of 8 Art. 225. To find any Number of Curves whoſe Areas bear any aſſignable Re- lation to the Area of any known Curve, Art. 226. Cor. 2. Prop. g. demonſtrated, Art. 227. Cor. 3. demonſtrated, Art. 228. Cor. 4. demonſtrated, Art. 229. Cor. 5. demonſtrated, Art. 230. Cor. 6. demonſtrated, Art. 231. Cor. 7. demonſtrated, Art. 233. Cor. 8, demonſtrated, Art. 234. Cor. g. demonſtrated, Art. 235. f Cor. 10. corrected, and demonſtrated, Art. 247. The Neceſſity of the Correction evinced, Art. 240. r — CONTENTS. OY e EN? ' Notes on Articles 52 —64 Quadrature of Curves, Page 18. Prop. 10. Quad. of Curves, and its Corollaries explained and illuſtrated, Art. 242. | | | The five different Caſes of that Prop. explained, Art. 243. Cor. 1. explained, Art. 2 52. What to be done when the Ordinate of the Curve has different Values, Ari. 253. | Cor. 2. demonſtrated, Art. 2 54. That every Curve whoſe Equation conſiſts of three Terms however affected, is either quadrable, or may be compared with the moſt ſimple Curves, illuſtrated by an Example, Art. 255. What to be done when the Equation defining the Relation of th: Abſciſs and Ordinate is ſuch that none of the preceding Methods are ſufficient for finding its Area, Art. 258. A general Account of the Deſign and Uſe of the Author's two Tables of Quadratures, Art. 259. Explication of Tab. 1. Art. 260. Demonſtratiog of the Quadrature of Curves belonging to Form 1. Tab. . Art. 261. | Demonſtration of the Quadrature of Curves belonging to Form 2. Tab. 1. and that they are doubly quadrable, Art. 262. | Demonſtration of the Quadrature of Curves belonging to Form 3. Tab. 1. Art. 264. ITE? A general Theorem for finding the Areas of all Curves of Form 3. with the Demonſtration of Sir 1/aac's Rule for carrying on the Series of Curves of this Form in inf. Art. 265. | Art. 266. How the Quadrature of Curves of Forms 3 and 4. Tab. 1. may be con- tinued forward in inf. by Means of the Theor. Art. 202, but that they cannot be continued from the firſt Species backwards, Art. 268. A general Theorem for expreſſing the Areas of all Curves of Form 4, by which Sir aac Newton's Obſervation (Art. 62. Quad. of Curves) for continuing the Progreſſion of Curves of this Form is demonſtrated, Art. 269. When the ; rn of the Radical in Form 3 is +, 2, +, &c. and in Form 4 is — 2, — 2, — , Gc. what to be done, Art. 271. Explication Demonftration of the Quadrature of Curves belonging to Form 4. Tab. 1. | xxi — * CONTENTS. Explication of Tab. 2. Art. 272. Demonſtration of the Quadrature of Curves belonging to Species E Form 1. Tab. 2. Art. 274. How the Series of Curves belonging to Form 1. may be carried on in inf. by the Canon, Art. 211. Explication, or by Diviſion: or by a general Expreſſion exhibiting all their Areas, Art. 275. Demonſtration of the Quadrature Species 1. Form 2. Tab. 2. with the Conſtruction of the conic Sections, Art. 276. The Matter farther illuſtrated and explained by an Example, Art. 279. How the Areas of the Curves belonging to Species 1. Form 4. are found, by ſuppoſing a different Relation of Abſciſſes from that the former: likewite how the Areas of all the Curves of that Form may be found, Art. 280. Demonſtration of the Quadrature of Curves belonging to Species 2. Form 5. Tab. 2. Art. 281. How the Series of Curves belonging to that Form may be continued in inf. Art, 282. Demonſtration of the Quadrature of Curves belonging to Form 7. . Species 1 Form 11. Tab. 2. Art. 283. General Remarks with Reſpect to the Die and Application of the Author's Tables of Quadratures, Page 212. | 1* How the Series of Curves belonging to all the Forms are to be con- tinued in inf. both forwards and backwards, under ſome Limitations, Art. 285. 2 What to be done when the Signs of the Quantities e, J. g. &c are changed, Art. 286. 3* With Reſpect to the Proportions and Limitations of theſe Quantities, Art. 287. 4* With Reſpect to the Conſtruction of the conic Sections requiſite for the Quadrature of the Curves in Tab. 2. Art. 288. 5 What to be done when the numeral Indexes 3, 6, become negative, Art. 314. 6 What to be done when the defining Equation of any Curve is com- pounded of ſeveral Equations, Art. 315. 7 With Reſpect to the initial Limits of the Areas, Art. 3 16. 8e How any propoſed Curve is to be managed, that its Area may be found by the Tables, Art. 317. g* With Reſpect to the Continuation of the Progreſſions of the Curves of the ſeveral Forms of both Tables; without the Limitations in Remark 1. Art. 318. 2 The . * x BY, OTE " 4 ” 8 f1 « "_ 1 * 1 4 1 — * 8 __ 2 KC 4 ren = "2 - * R * "= . 4 >» 4 . * 2 3 +a ww» N 1 _ 4 . Fo 2 28 2 - 4 — g na Wa , ws _ D F _ _— 7 e 7 * a Sy" *% 5 1 1 ” 40 » +4 2 — r P I 4 — % 8 _ — * 1 2 * 1 2 a) — —_ 2 2 2 * E 2 Wee 7 - th. gd - » = * * E +» 3 - Wo — * + x4 4 4 = *»* 2 KY 4 a * F — 9 * - 2 Y + ee ID —— 1 Ie. n 5 4 E 8 1 % OF N . b ba * - CONTENT &. Examples, Art. 3 19. | The Fig. of Subtangents in any Curve, when the Subtangents are ordi- nately applied to the right Series of the Arches, demonſtrated to be ual to the Area of the Curve, Art. 326. What the Fig. of Secants in a Circle is equal to, when the Secant is ordinately applied to the right Sine of the Arch, Art. 329. What the Figure of Secants in a Circle, Ellipſe, or Hyperbola, is equal to, when the Secant of the Arch is ordinately applied to the verſed Sine, Art. 328. | | That the Meridian Line in Mercator's Projection, is equal to the Dif- ference betwixt the logarithmical Radius and logarithmical Tangent of the half Complement of the Latitude, multiplied by 2. 302 58 5 &c. demonſtrated, Art. 329. The ſame Thing demonſtrated in another Way, Art. 330. To find the Areas of Mechanical Curves, Art. 331. How the Areas of Spirals may be found, Art. 335. To find the Areas of Curves generated by the Reyolution of a Radius about a fixt Point, Art. 336. SECT. X. Of the Reduction of the Hyperbola and Ellipſe ; together with all curvi- linear Areas, and Fluents, which depend upon theſe conic Sections, to The Application of the Author's Tables of Quadratures in a Variety of the Meaſures of Ratios, and Angles; or Logarithms, and circular Arebes, Page 240. The Definition and fundamental Property of the logiſtical or logarith- mical Curve, Art. 3 14 1 The Analogy betwixt 4 Abſciſſes of the logarithmical Curve, and the hyperbolical Spaces adjacent to the Aſſy mptote; as both are the Meaſures of Ratios, Art. 338. | N The Principles of logarithmical Aruhmetic demonſtrated, Art. 340. To find the Logarithm of any given Ratio: and conſtruct a Table of Logarithms, Art. 341. 1 Different Expreſſions of the Ratios of which hyperbolical Sectors, or hyperbolical Spaces adjacent to the Aſſymptote, are the Meaſures or Logarithms, Art. 344. | | What Proportion the Logarithms or Meaſures of the ſame Ratio in different Syſtems bear to one another, Art. 346. Definition of the Modulus of a Syſtem of Logarithms, Art. 348. ” „ a The xxiv CONTENTS. | The Modulus of the common Syſtem of Logarithms, called Briggs's Logarithms, Art. 349. N r The hyperbolical Logarithm of any Ratio multiplied by the Modulus of any other Syſtem, makes the Logarithm of that Ratio in that Syſtem, Art. 3 50. | | Defini:ion of the Ratio modularis : that it is the ſame in every Syſtem : and how to be found, Art. 351. | | A Logarithm being given, to find the Number or Ratio to which it belongs, Art. 352. How any hyperbolical Spaces may be found by Means of a logarithmi- cal Canon, Art. 355. | a. The Areas of all Curves capable of being compared with the Hyperbola; and all Fluents whoſe Fluxions are analogous to the Fluxions of thoſe Areas, found and expreſſed by Logarithms or Meaſures of Ratios, Art. 357. e | > 1 How LIP Area may be reduced to a circular Area: and fo be found by Means of the trigonometrical Canon, and the known Pro- portion of the Radius of a Circle to its Circumference, Art. 3 58. Elliptical Sectors one Syſtem of Meaſures of Angles, Art. 361. The Modulus of that Syſtem, what, Art. 362. The Areas of elliptical Spaces found by the trigonometrical Canon: il- luſtrated by an Example, Art. 304: | | 7 The Areas of all Curves capable of being compared with the Ellipſe; and all Fluents whoſe Fluxions are analogous to the Fluxions of thoſe Areas, found and expreſſed by the Meaſures of Angles, Art. 36 5. The Analogy betwixt the Meaſures of Angles, and Meaſures of Ratios illuſtrated, Art. 366. | | The Connection betwixt Sir Jaac Newton's Method of expreſſing Flu- ents in his Quadrature of Curves, and Mr. Cotes's Method in his Harmonia Menſurarum, explained by an Example, Art. 372. SECT; XI. Demonſtration of Prop. 11. of the Quyadrature of Curves, Page 268, SECT. XII. \ Explication of the laſt Schol. of the Qyadrature of Curves : with the Ap- plication of the preceding Doctrine to the Solution of ſome Problems, Page 272. | | | The Relation and Analogy betwixt the various Orders of Differences and Fluxions, enquired into: upon which Subject there ſeems x be ome CONTENTS. Dr. Keil's Correction, Art. 382. The Foundation of the Agreement betwixt the differential Method of Mr. Leibnitz and Sir Jſaac Newton's Method of Fluxions, as to their Concluſions, Art. 383. Some Remarks with Reſpect to the two general problems including the direct and inverſe Method of Fluxions, Art. 38 5. Fluents pure or mixt: with the Difference betwixt them: and how the true Fluent, although mixt, may, in certain Circumſtances, be found, Art. 387. The fame th fluxionary Expreſſion 12 have two different pure Fluents: how theſe Fluents may be found by the Rules formerly delivered for finding the Areas of Tak Art. 388 | The Relation of the flowin % Quantities the ſame; whether one, or other; or none of them be Fluxion, Art. 289. by the Quadrature of Curves, is founded, Art. 390. Prob. To determine the Maxima and Minima of Lyantitier, Art. 391. To find whether the Ordinate of 4 Curve arrives at a Maximum, or Minimum; or at a Point of contrary Flexure, or Retrogrefſion, when the Fluxion of the Ordinate vaniſhes, or is infinite, Art. 394. This Matter illuſtrated by ſeveral Examples, Art. 395. To find ſuch a Part of a ſpherical Superficies as can be illuminated in its farther Part; by Light coming from a great e and refracted : the nearer Hemiſphere, Art. 398. Prof. To draw Tangents to all forts of Curves, Art. 460. Examples hereof in 3 Curves, Art. 401. The Points of contrary Flexure how determined, Art. 402. How Tangents are drawn to mechanical Curves, and Spiras with Examples, Art. 403. Prob. To find the Length of Curve Lines: with Examples in geometri- | cal and mechanical Curves, Art. 406. To find as many Curves as you pleaſe, whoſe Lengths are equal to known Right Lines, Art. 415. Prob. To find the Areas of Curve Surfaces :. with Examples, Art. 4.16. Prob. To find the Contents o, Solids generated by the — Plane Figures: with Examples, Art. 432. What Fluents may be found by the Quadrature of Curves, Art. 43 5. poſed to flow with a conſtant or uniform The Principle upon which — finding the Fluents from the Fluxions, XXY. ſome Error or Miſtake in, the Author's Schol. with an Account of . XXVI CONTENTS. The various Caſes of fluxi Equations, from which the Relations of the Fluents can be found by Mags. of the Quadrature ob Curves, explained and illuſtrated, pic 436. 5 Ia finding the Relation of the Fluents, one of them is ſuppoſed to flow with an uniform Fluxion, which Fluxion is put equal to Unity: and that flowing Quantity is ſuppoſed to be imed-in- order- to- de- termine the other, Art. 444. How the F luents found by the preceding 3 be-alcertained, Art. The Author 's Concluſion expreſſive of the Greatneſs of his, Modeiy, and Vaſtneſs of his Underſtanding, Art. 440. Contents of the Apalyhis by F normal of an infinite Number Ter S. * (} The Author's own n Account of n Deſign and Contents ofthis read, Ali. 1 T The Quadrature of fimple- curves, Rule: 1, — 21. aus on The. TE compound:Gurwues, Rule 2. Pig. 322. mples of this Rule: in which the Brponent of the Powers of x are all poſitive, Art. 5. Second Examples: in which theſe ebene are all negative; Ave: 6. Third — | in which, theſe. Exponents are Ry ret partly. negative, Art. | The. Quadrature of al other. Curves ;; Rule 3. Page 224. Examples of this Rule, where you divide, Art. 12 Ar n y 2 Examples where the Square Root muſt be extractod, Art. 18 Examples by the Reſolution of Equations, Page 328. The numeral Reſolution of affected Equations, Art. a1 1 any Terms, or not; * with the 0 F. 28. How one kind of Quantities. may bo moſt eommodioully ſubſtituted for another, in an Equation, Art. 29. The literal Reſolution. affected ori, Page 3305 | Another Way of reſolving them; when. the Area is ſo -the-nearer the Truth the greater that x is; 332. Senn Remarks with Reſpect to what is to be done when the ecedi 4 Methods fail, Ar. = 5 W All Problems concerning the Lag of Curves 3 ths, Quantity y, and · Surface of Solids; and the Centers of Gravity, ane to the Quacdrature of Curves, Art. 36. The zh in 'S A F PA 3 p- 7 kk CONTENTS. Ixxvii 2 The Applicatibn of what ar ns vera Problems $07 IE Page 3: Lee * 0 18 To To find | the Logins of Gorey theſe Things, Art. 42. To find the Baſe from the Area given, Art. 43. To find the Baſe from the Lengh of the Curve given, Art. 4 5. 2 Concerning the Continuation of the Series of the Progreſſions, Art. 47. | The Application of what has been ſaid to mechanical Curves, Arr. 48. ade Demonſtration of the Quadrature of fimple Curves belmging to Rate 1 1. Page fs 7 hop gfe Curves which can be ſquared, Art. 56. My * — of the Reſolution of affected Equations, Art. 57, VO of Analyſis by Equations De. n. SECT, I. IntroduBion : containing an Account of the Riſe and Progreſs of the .Contemplation of Infinites ; and the Quadrature of Curves: t ther with ſome general Actount of the Contents of Sir Tſaac Newton's 8 two Treatiſes of the Quadrature of Curves; and Analyfis by infinite 2 age 444. Ts oundation of the Method of Exhauſtions laid down by Euclid, tt. 2. The Method of Exhauſtions made Uſe of by Euclid and Ardhi medes, Art. This Mithod 3 illuſtrated by an Example taken from Archimedes, Art. 4. The Gregorian and Netionian Methods of infinite converging Series, lying in Embryo in the Ancients Method of Exhauſtions, Art. 6. The Analogy, Gradation, and Difference of the Method of Exhauſtions; the Gregorian, and Newtonian Forms of converging Series, Art. 7. Some Account of the Method of Indiviſibles, introduced by Binaventura -___ Cavallerius, Art. 8. An Account of Dr. Wallis's Arithmetic of Infinites, Art, 9. Where Dr. FW allis's Method ſtopt, Art. 11. A large Account of the Original of Sir 1/aac Newton's Method of infinite Series taken from one of hisLetters to Mr. Oldenburg : and delivered in his own Words. Where alſo ſome Account is given of Mr. James Gregory's Diſcoveries, Art, 14. _ An Acconnt of the Contents of the Analyſis by Equations of an infinite Number of Terms, Art. 16. 2 : | The XXVII CONTENTS. The Treatiſe of the Quadrature of Curves a further Improvement : : with an Account of what it contains, Art. 1 | When the Methods laid down for finding the Areas of Curves in the Quadrature of Curves fails, Recourſe muſt be had to the Methods laid down in this Analyfis, Art. 20. SECT. ML; / % W of the firſt Part of the Author's third Rule, which ſhews the Method o le finding the Areas of thoſe Curves, which require the Reduction of complicate Fractions and _ cals.: contained in Art. 12— 20. Page 70 The Author's firſt Example upon Redu@tion of complicate Fractions by Diviſion, explained, Art. 2 5. The Author's ſecond Example illuſtrated, Art. 27. How the Diviſion in his laſt Example upon this Head is performed by two different Ways: with the Effects of both as to the Area of the Curve, Art. 28. That the Reduction of ſpecious Quantities into infinite Sten reſembles certain Operations in Decimal Arithmetic : with an Account of the Agreement and Difference betwixt numeral and Nen converging Series's, Art. 2 The Reduction of R Radicals into infnite Series, by extraction of Roots, reſembles the like Operation in Decimal Arithmetic, Art. 32. The Operation in the Author's — Art. 15. repreſented and: ex- plained more fully, Art. 33 A How the Law of Progreſſion of Series's is found, Art. 37. Circular and Hyperbolical Segments compared, Art. 38. The Computation of the Area of a Circle, Art. 40. The Rectification of an elliptical Arch; being an Explication of the Author's Example at Art. 18. Art. 42. What Uſes the Author's Binomial Theorem ſerves for, Art. 44. This famous Theorem exhibited, Art. 45. How it was fiſt found out, Art. 46. The Application of it in a Variety of Examples, Art. 47. Mr. James Gregory's Series for the Length of an elliptical Arch inveſti- gated, and compared with one of Sir 1/aac Newton's for the lame Purpofe, Art. 52. The Binomial Theorem ſerves for reducing complex FraRions into infinite Series; and for raiſing to any perfect Powers: as well as extracting Roots: with N Art. 54. The — "y CONTEMNTS. xxix ROY ries- for the curvilinear Area; or Fluent, found by Help of the | Binomial Theorem, may run out into an infinite Series, although the Curve be geometrically quadrable; i. e. the Area of the Curve; or Fluent, aſſignable in finite Terms, by oth 5 and 6. Quad. Curv. Art. 56. BL How the ſame Thing may be diſcovered by the Tranſmutation of Curyes, 8 Ari. The fame Thing found by Cor. g and 10. Prop. g. Quad. Curv. Art. 59. SECT. III. Expli a on of the numeral Reſolution of affetted Equations contain in * Art. 21— 2, Page 388. An Replanation of ſome Things with Reſpect to Decimal Fractions, Art. 60. . — The Operation for extrating the Roots of affected numeral Equations, with the Abridgments made uſe of therein by the Author fully e- plained, Art. 62. Dr. Halley's Rational Formula for the Reſolution of Equations deduced and demonſtrated, Art. 6 5. Many other Methods for approximating to the Roots of Equations, may be inveſtigated, Art. 66. When there is any Doubt about the Approximate Value of the Supple- ment being near enough the Truth, which is found by the former Method: what to be done, Art. 67. | A Rule Le down by Sir 1/aac with Reſpect to that Caſe, demonſtrated, a Art. 6 Dr. Halley's irrational Formula; and Mr. Raphſon's Method of reſolving Equations, virtually included in our Author's Method, Art. 69. This Method applicable to all Sorts of numeral Equations, whethes the Terms be complete or not: and conſequently to the extracting any Roots of Numbers, Art. 70. Although the ſupplementary Equation have as many Roots as the original = Equation, that yet the leaſt Root is the Supplement: and that the preceding Methods diſcover this nearly: demonſtrated, Art. 71. Some further Remarks with reſpect to this Method of > reſolving Equa- tions, Art. 72. 8 E C T. IV. | Explication of the Reſolution of affected, ſpecious or literal Equations, if contained in Art. 30o—34. 57. In ſpecious Equations involving two unknown Quantities, the one as x 4. 4A is conſidered as the Abſciſs, and the other y, as the Ordinate of a I Curve ; . CONTENTS Curve: and in their Reſolution x is conſidered as very ſmall; or very great; or differing very little from ſome given Quantity, Art. 73. The firſt Caſe, in which x is ſuppoſed very ſmall, chiefly inſiſted upon 1 by the Author: with the Manner of proceeding, Art. 74, Some Remarks ſerving to explain and demonſtrate the Author's Proceſs for extracting the Roots of literal Equations, when the ſpecies x is | ſuppoſed to be very little, Art. 785 : | An Abridgment for ſhortening the Proceſs in extracting the Roots of theſe Equations, mentioned by Sir 1/aac, Art. 3 1. explained, Art. 76. The Proceſs of the Extraction exhibited, according to the preceding Rules and Remarks, Page 406. : The ſame further illuſtrated by another Example, Art. 79. 0 Explication of — 3 * of * ing the Roots of literal uations, when the Species x is ſuppoſed to be very great, Art. 80. The Fg of the Extraction exhibited. Art. 81. I An Explication of the Steps of that Proceſs : with a Rule for determin- ing to what Length the Diviſion, for obtaining the laſt Terms of the Series, may be continued, Art. 82. How the Value of y may be found otherwiſe, in the Caſe of & being very great, Art. 83. 6 | wo The Value of the Ordinate y being extracted, how the Area of the Curve, and it's other Symptoms are determined, Art. 84. | How the Root y is extracted, when it is known or ſuppoſed that it dif- fers but little from ſome given Quantity, Art. 8 . How the three different Methods of extracting the Root y, may be re- duced to one, Art, 86. . 3 er. v. Of the various Methods of extracting and expreſſing the Roots of affected literal Equations, including two unknown Quantities, by Means of in- finite Seriess: in which the Author's 35, 42 —46 Articles are ex- plai ned, Page 416. Various Series 's capable of being deduced from the ſame Equation, Art. 87. Some Lemmas, ſerving to demonſtrate the Methods for diſcovering the firſt Terms of converging Series's, Art. 88. . Method 1. For finding the firſt Terms of converging Series's ; to be ap- plied when there are two or more Terms of the propoſed Equation, which involve not the Quantity x, and one or more of which in- volve the Quantity y, Page 423. <4 Method or very great, P CONTENT & Method 2. By which the firſt Terms of all the con verging Series s may be diſcovered; which are the Roots of the propoſed Equation, either when x is very ſtall, or very great, Page 424. 1353 277 Method 3. By which, for the moſt Part, the firſt Terms of two con- verging Series's. may be: diſcovered, one of them for the Caſe of x being very ſmall; the other for the Caſe of x being very great, Page 428. Method 4. By the Help of a rectangular Parallelogram: by Means of which the: firſt Terms of all the converging Series's may be found, which are Roots of the given Equation, whether & be very ſmall, 3 431. | 2X Of the different n. ts finding the ſubſequent Terms of the Quo- tient or converging Series for the Value of y, Page 436. . The firſt Method; by Means of the Parallelogram, Art. 11 5. In what Manner certain Equations muſt be prepared, Art. 117. 4 Another Method for diſcovering tlie ſubſequent Terms, being that made uſe of by. Sir 1ſaac. in. this Treatiſe : which is here fully ex- 0 Xx plained and demonſtrated, Art. 120. N I Prob.. To find whether the Exponents of the Powers of , in the Terms of the converging. Series, go on in an arkhmetical Progreſſion, or not: and if ſo, what the common Difference, of that Progreſſion ig, 2 Page 142. | = A LN aied, for extracting the Roots of affected Equations, includ- ing two unknown Quantities, by converging Series: vi. by tho Aſ- ſumption of a Series of a proper Form with indeterminate Coeffici- ents, Art. 126. ; *. The Reſolution of Equations by infinite Series, includes virtually in it; the Reduction of complex Fractions and Radicals into infinite Series, Art. 134. | : . vi The preceding Doctrine Reverfion: f Series, Art. 135. How Equations inſinitely affected are reſolved rity Art. 137. = How the famous Analyſt Mr. Abraham de Morvre's Theorem for reſolv- ing an infinite Equation ; together with other ſtill more general 8 may be inveſtigated by Help of the foregoing Doctrine, Art. 138. | The Terms of any Power or Root whatſoever of an infinite univerſal Series, diſtinctly exhibited, to the Length of ſeven Terms, Art. 139. What to be done when the Root of the fictitious Equation by Means of which the firſt Term of the converging Series ought to be found, is 2X Adfurd; or wholly unknown, Art. 140. What to be done when the fictitious Equation from whence the firſt X Term of the converging Series ought to be derived, hath no real or poſſible Root, Art. 142. That xxxil CONTENTS. That the Roots of Equations including two unknown variable Quanti- ties, may be extracted and expreſſed by infinite Ways, Art. 145. The Application of the Extraction of Roots of affected literal Equations to the Quadrature of Curves, illuſtrated by Example, Art. 140. Some Directions with reſpect to the finding of the Curvilinear Area L adjacent to any given Part of the Baſe or Abſciſs: and the changing 1 the Beginning of the Abſciſs, Art. 148. i Some Advantages in taking the Beginning of the Abſciſs exactly i inthe | Middle of the given Part of it, Art. 150. x [ | Some Obſervations for facilitating the Buſineſs of Quadratures, Art. 1 52. An infinite Series for the Root y, of an affected Equation, being once in- veſtigated, by ſuppoſing x to be indefinitely ſmall or ome”: will expreſs the Root, whatever be the Value of x. Art. 1 15 fo 8 E C T. VI. M 1 i ( The Iroefligation and Demonſtration of the Binomial Theorem, Page 470. £ Caſe 1. When the Exponent of the Power is a poſitive mann Art. 15 Caſe 2. When it is a poſitive Fraction, Art. 158. E Caſe 3. When the Exponent is any negative Number, integral or fraQti- iN onal, Art. 160. 1 This Theorem more eaſily demonſtrated by the Help of Fluxions: but in order to this the Rule for finding the Fluxion of any Power or | Root, muſt be demonſtrated from other Kere 6 N is bn done, Art. 161. Demonſtration of it upon that Foundation, Art. 16: $9 do What further remains, with reſpec to the Application of this Analyſis to other Subjects, ſufficiently a; Ing 1 General Scholium, by way of Concluſion to the n Page 478. Sir 1 = - 1 1 A - * - " . * =” £ 4 * « o - « * : l [ Tr. 1 © * * * * * — * , a | 4 — . 4 12 * .- _ — 1 — —_— * * * + - * 4 Sir ISAAC NEWTON'S "ITREATISE 2, u- 1 OF THE Quadrature of CURVES: 0. | 8 5s Þ 4 = INTRODUCTION to the Quadrature of Curves. 1 4 : 4 1. Jy Conſider mathematical Quantities in this Place not as conſiſt- ue ing of very ſmall Parts ; but as deſcrib'd by a continued Mo- * tion. Lines are deſcrib'd, and thereby generated not by the os mn Appoſition of Parts, but by the continued Motion of Points; Superficies's by the Motion of Lines ; Solids by the Motion of Super- ; ficies's ; Angles by the Rotation of the Sides; Portions of Time by a is cContinual Flux: and ſo in other Quantities. Theſe Geneſes really take Place in the Nature of Things, and are daily ſeen in the Motion of 4 Bodies. And after this Manner the Ancients, by drawing moveable $56 3 right Lines along immoveable right Lines, taught the Geneſis of Re&- angles. . = 3 4 Therefore conſidering that Quantities, which increaſe in equal Times, and by increaſing are generated, become greater or leſs accord- ing to the greater or leſs Velocity with which they increaſe and are generated; I ſought a Method of determining Wr from the Ve- = locities of the Motions or Increments, with which they are generated; and calling theſe Velocities of the Motions or Increments Fluxions, and 2* the generated Quantities Fluents, I fell by degrees upon the Method of * Fluxions, which I have made uſe of here in the Quadrature of Curves, „ in che 665 and 1666.1 ir H m NE Lears 1 5 And 1 . | ; : . 93 5 ih * — \ 1 5 ; 8 | <= 4 - To. Introduction to the | 3. Fluxions are very nearly as the Augments of the Fluents gene- rated in equal but very ſmall Particles of Time, and, to ſpeak accu- rately, they are in the firff Ratio of the naſcent Augments ; but they may be expounded by any Lines which are proportional to them. 4. Thus if the Area's ABC, ABDG. be deſcribed by the Ordinates BC, BD moving along the Baſe AB with an uniform Motion, the Fluxions of theſe Area's ſhall be to one another as the deſcribing Or- H now generated of the Abſcifs AB, the Ordinate BC and the Curve Line p ACc+; and the Sides of the Triangle 2 CET are in the ff} Ratio of theſe Augmentsconſidered as naſcent, there- fote the Fluxions of AB, BC and AC are as the Sides CE, ET and CT of pounded by theſe fame Sides, or, which is the ſame thing, by the Sides of the Triangle VBC, which is fimi- lar to the Triangle CET. | ſe to take the Fluxions in the 11ti- 6. It comes to the ſame Pu mate Ratio of the evaneſcent Parts. Draw the right Line Cc, and produce it to K. Let the Ordinate bc return into it's former Place BC, and when the Points C and c coaleſce, the tight Line CK will coincide with the Tangent CH, and the evaneſcent Triangle CE: in it's ulti- mate Form will become ſimilar to the Triangle CET, and it's evaneſ- cent Sides CE, Ec and Cc will be ultimately among themſelves as the Sides CE, ET and CT of the other Triangle CET, are, and' therefore the Fluxions of the Lines AB, BC and AC are in this ſame Ratio. If the Points C and c are diſtant from one another by any ſmall Diſtance, the right Line CK will likewiſe be diſtant from the Tangent CH. by a ſmall Diſtance. That the right Line CK may coincide with the Tangent CH, and the ultimate Ratios of the Lines CE, Ec and Cc may be found, the Points C and c ought to coaleſce and exactly coin- cide. The very. ſmalleſt Errors in mathematical Matters ate not to be neglected, | 7. By that Triangle CET, and may be ex, „ 6h = > © i. a a ww . YE W gy—_ Yy . 9 ultimate Ratio of Ph to YQuadrature of Cunves. 7. By the like way of reaſoning, if a Circle deſcrib'd with the Cen- ter B and Radius BC be drawn at right Angles along the Abſciſs AB, with an uniform Motion, the Flaxion of the generated Solid ABC will be as that generating Circle, and the Fluxion of it's Superficies will be as the Perimeter of that Circle and the Fluxion of the Curve Line AC jointly. For in whatever Time the Solid ABC is generated by draw- ing that Circle along the of the Abſciſs, in the ſame Time it's Superficies is generated by drawing the Perimeter of that Circle along the Length of the Curve AC. You may likewiſe take the following Examples of this Method. | 8. Let the right Line PB, revolving about the given Pole P, cut an- other right Line AB given in Poſition : it is required to find the Propor- tion of the Fluxions of theſe right Lines AB and PB. | | Let the Line PB move forward from it's Place PB into the new Place PB. In Ps take PC equal to PB, and draw PD to AB in ſuch manner that the Angle PD may be equal to the Angle BC; and becauſe the Triangles BC, PD are ſimilar, the Augment Bb will be to the Augment Ch as Pb to DG. Now let P5 return into it's former Place PB, that theſe Augments may evaniſh, then the ultimate Ra- tio of theſe evaneſcent A ts, that is the , ſhall be the fame with that of PB to DB, PDB being then a _— Angle, and therefore the Fluxion of is to the Fluxion of PB in that ſame Ratio. 9. Let the right Line PB, revolving about the given Pole P, cut other two right Lines given in Poſition, viz. AB and AE in B and E: the . Proportion of the Fluxions of theſe right Lines AB and AE is fought. Let the revolving right Line PB move forward from it's PB into the new Place PB, ſo as to cut the Lines AB, AE in the Points 6 and e: and draw BC parallel to AE meeting PG. in C, and it will be BS: BC :: Ab: Ae, and BC: Ee :: PB: PE, and by joining the Ratios, BS: Ee :: Ab PB: Ae x PE. Now let Pb return into it's — „ W mY former Place PB, and the evaneſcent Augment B& will be to the evaneſcent Augment Ee as AB x PB to AE x PE ; and therefore the Fluxion of the right Line AB is to the Fluxion of the right Line AE in the fame Ratio. B 2 10. Hence 2 FTW FS 7 = of * ” d * > „ — TY > : 2 # by = . ” . ; *. 4 bane & i f 9 E * * CE " * : N : * 4 2 2 Introduction, tc; > | $ 10. Hence if the revolving tight Line PB cut any curve Ling given Wl in Poſition in the Points B and E, and the right Lines AB, AE now ‚ becoming moveable, touch theſe Curves in the Points of Section Band RF = E: the Fluxion of the Curve, which the right Line AB touches, ſhall be to the Fluxion of the Curve, which the right Line AE touches, as ü AB * PB to AE x PE. The fame thing would happen if the right 3 Line PB perpetually touch'd any Curve given in Poſition in the move. able Point P, | üer 2s : 2 1541 1 11. Let the Quantity x flow uniformly,” and let it be propoſed to find I : the ion of x". | | "Ws OT 4 la the ſame Time that the Quantity x, by flowing, becomes x o the Quantity. x": will become ＋ ol”, that is, by the Method of in- ] finite Series's, x"'-+- no þ 000%? + &c. And the Au- „ a ments o and noπ . + 00x? + Gc. are to one another as x 1 3» and * + o + &c. | . 1 No let theſe Augments vaniſh, and their ultimate Ratio will be 1 . wa -. 8 | | = -: 12, By like fo of reaſoning, the Fluxions of Lines, whether right , or curve in all Caſes, as likewiſe the Fluxions of Superficies's, Angles MF : and other Quantities, may be collected by the Method of prime and -: ultimate Ratios. Now to inſtitute an Analyſis after this manner in \, finite Quantities and inveſtigate the prime or ultimate Ratios of theſe . - finite Quantities when in their naſcent orevaneſcent State, is conſonant = | 1 | to the Geometry of the Ancients: and I was willing to ſhow that, in 9 1 the Method of Fluxions, there is no neceſſity of introducing Figures | infinitely ſmall into Geometry. Yet the Analyſis may be performed in any kind of Figures, whether finite or infinitely ſmall, which are imagin'd fimilar to the evaneſcent Figures; as likewiſe in theſe Figures, x which, by the Method of Indiviſibles, uſe to be reckoned as infinitely WM ſmall, provided you proceed with due Caution. = From the Fluxions to find the Fluents,. is a much more difficult Problem, and the firſt Step of the Solution is equivalent to the Quadra- i ture of Curves ; concerning which I wrote what follows ſome confi * derable Time ago. * 4 7 * ; ; | F * « ” 4 * - + 2 * * 5 : , : o o , - N = p ” * P 1 TY Y g a by — 6 : a * 4 * S z . : 4, «% 2 ” . # F 4 . | 7 - " 1 5 o 2 , . P 1 . 1 . ” » „ l . * g P e 5 4 $ | * „ * . : * A E ? y * . mn, A " * U N Lt 7 Y * F 8 Y > U TY . * N * — / * 8 „ - ah . I * 2 aw N N # © \ : , a . * a "4 | b a 3 , 99 © * . * 1 * „ . * a : © * * X ba e 9 4 * 0 8 . 9 , . 4 9 * * X * » 8 - » = « — * 0" 7 n + «4 vb 3 n 1 ? — . 9 , * — "_ 4 * and. * * . 70 1 9 . +4 4 * * 77 4 SS. * - « « © £ -, FW | 1 — . 5 . 4 * * * «< oj 4 * * + * * * — * dF Sod l * n a 2 * N wo : L . 2 _ 2 af 4 « ®; ah Kar A - "Y a * * - * + » "1 ES 1 3 „ 3 „„ © +k — A 0 7 * : - : . S * * # - . ut s * , *. # , 9 8 ; O F T H E, : , * 8 * . - 5 4 0 0 „ 40 N = k . g —— 9 © . . . ” 9 — * 1 _ 6 6 QUADRATURE of CURVES. . » "IM « K «. 4 21 4 — => * * 8 A — 4 3 * = * » 5 _ * 8 42 5 13. IN what follows I conſider indeterminate Quantities as increaſing IN or decreaſing by a continued Motion, that is, as flowing forwards, or backwards, and I deſign them by the Letters z, y, x, v, and their . Fluxions or Celerities of increaſing I denote by the ſame Letters point 1 2, 5, X, V. There are likewiſe Fluxions or Mutations more or | 7 ſwift of theſe Fluxions, which may be call'd the ſecond Fluxions of the ſame Quantities z, y, x, v, and may be thus deſign'd x, y, x, v : .\ and the firſt Fluxions of theſe laſt, or the third Fluxions of z; y, 4 N F | | 22 88 | x, v are thus denoted 2, y, x, v: and the fourth Fluxions thus 2,9, 1 14 x, v. And after the ſame manner that 2, 3, x, v are the Fluxions of 1 the Quantities 2, y. x, b, and theſe the Fluxions of the Quantities 2, 5, æ, U; and theſe laſt the Fluxions of the Quantities 2, y, x, v: 2X fo the Quantities, y, x, v may be conſidered as the Fluxions of others, Wich I ſhall defign thus , J, &, b; and theſe as the Fluxions of | others 2, y, x v; and theſe laſt ſtill as the Fluxions of others 34 : Therefore 25 25 2, x, S, 2, 2, 2, &c. deſign a Series of Quantities whereof every one that follows is the Fluxion of the one immedi- ately preceding, and every one that goes before, is a flowing Quanti hw that which immediately ſucceeds, for it's Fluxion. The like | is the Series * 22 — Zz, Va — 22, * az — 22, Naz Z, Naz — Sz, s ——— x 5 8 az o+ — az T IE az LY az + tz: TOO. | V2z—22; as likewiſe the Series ==, =, 21, == += ara go $20 a —- % ? . 1 14. And it is to be remarked that any preceding Quantity in theſe x Scries's is as the Area of a Curvilinear Figure of which the ſucceeding z the rectangular Ordinate, and the Abſciſs is z : as Va = the Area z a 3 | þ ; | '. The Quadrature of Curves. of a Curve, whoſe Ordinate is Vaz — z, and Abſciſs 2. The Deſign of all theſe things will appear in the following Propopoſitions, PROF L PAGQS.L 9 pray” being given involving any Number of flowing Quan- titties, to find the Fluxtons. . SOLUTION, Let ney Term of the Equation be multiplied by the Index of the Power & of every flowing Quantity that it involves, and in every Mul- tiplication c the Side or Root of the Power into it's Fluxion, and the Aggregate of all the Products with their proper Signs, will be the new Equation. EXPLICATION, 16. Let a, 6, c, d, &c. be determinate and invariable Quantities, and let any Equation be propoſed involving the flowing Quantities 2, y, x, &c. as & — xy* -þ- &*z — o. Let the Terms be firſt multiplied by the Indexes of the Powers of x, and in every Multiplication for the Root, or x of one Dimenfion write x, and the Sum of the Factors will de zx — xy*. Do the fame in y, and there ariſes — 2xy9. Do the ſame in 2, g r . — N — Sum of theſe &s be put equal to nothing, you'll have the Equation 3xx* — xy* — ol + aa = ©, I fay the Relation of the Fluxions is defin'd by this Equation, | DEMONSTRATION. 17. For let o be a very ſmall Quantity, and let o, oy, ox be the Moments, that is the momentaneous ſynchronal Increments of the Quantities 2, y, x. And if the flowing Quantities are juſt now 2, y, x, then after a Moment of Time, being increas'd by their Incre- ments 02, oy, ox, theſe Quantities ſhall become x 02, yo, x+ox : which being wrote in the firſt Equation for z, y and x, give this * The Word tranſlated here Power is 1 Dignity, by which muſt be underſtood not only perfect, but alſo imperſect Powers or ſurd Roots, which are expreſs d in the Manner of perfect Powers, as is well known, by fractional Indexes. In which Senſe 4, x3, &. are Powers; + and 3 their Indexes, and x the Side or Root. I uſe the Word Power, becauſe Dignity is ſeldom us'd in Fagliſb in this Senſe, | | | Equation OH 1 d 5 d le rs 0 ts 13 and for it's firſt Fluxion to write Unity, ones, nothing. Let there be given the Equation zy3 — 2+ ＋ at =o, as above; and let z flow umformly, and let it's Fluxion be Unity: then by the firſt Operation it ſhall be y 32y5* 48. = ©; by the = ſecond 65y* + 32 + G — 122* = © ; by the third gyy - = 185*y + 329 ＋ 18 + 6 — 242 = 0. The Quadrature of Cutvis. tion x3 ＋ 2x%0x + 3x00*x 4- 93z3: = xy* — 0xy* = 2K — 22 — 29 = o + 4 + &02— b3—= O. Su tract the former Equation from the latter, divide the remaining Equation by o, and it will be zar ＋ 33x0x + #30* — % — 29 — 2xo0jy — x0jy *. iy + az — ©, Let the Quantity o be dimi- niſhed infinitely, and negſecting the Terms which vaniſh, there will remain 30 — % — 29 A O. Q. E. D. S A fuller Explication. 18. After the fame manner if the Equation were x3 — * + q aa — y* — þ3 == ©, thence would be produced 3x*%* — xy* — 2xyy + a — * = 0. Where if you would take away the Fluxion b , put ax - , and it will be ax — y* , and by that is 2 23 —Tx —yy. And thence 37 — % 2 os. 2V AX — 19. And by repeating the Operation, you proceed to ſecond, third | | and ſubſequent Fluxions. Let 2y3 — 2* +- A o be an Equation propos d, and by the firſt Operation it becomes 2y3 ＋ 327 — 4223 — 03 by the ſecond 2y3 + 62yy*+ 32 + 625*y9 — 4.423 — 222 == 0, by the third, 255 4 9259 * 18577 + 3 + 182 =_ 3 Aa > IE ? 0 62) — 422) — 36222 — 245 =0O, 20. But when one proceeds thus to ſecond, third and following Fluxions, it is proper to conſider ſome N as flowing uniformly, r the fecond and ſubſequent But in Equations of this Kind it muſt be conceived that the Fluxions 1 in all the Terms are of the fame Order, 7. e, either all of the firſt Or- 1 der , 2; or all of the ſecond y, y*, yz, 2*; or all of the third. , 9, J, 53, j*s, jb, 23, &c. And where the Caſe is otherwiſe the Order is to be completed by means of the Fluxions of a Quantity that 1 flows uniformly, which Fluxions are underſtood. Thus the laſt Equa- tion, 3 De Np-atere of donend tion, by completing the third Order, becomes 929 + 899 + 325 T 18zyy + 6253 — 24283 o. PROP. IL PROB. I. 22. To find ſuch Curves as can be ſquared, Let ABC be the Figure to be found, BC the rectanglar Ordinate, and AB the Abſciſs. Produce CB to E, fo that BE == 1, and complete the Parallelogram ABED: and the Fluxions of the Areas ABC, ABED ſhall be as BC and BE. Aſſume therefore any Equa- A 2 tion, by which the Relation of the Areas may be | 4 defined, and thence the Relation of the Ordinates | Wy and BE will be given by the firſt Fropolition, D E 7 I. The two following Propoſitions afford Examples of this. PR OP. III. TH E O R. I. 8 If for the Abſciſs AB and Area AE or AB x 1 you write 2 pro- miſcuouſly, and if for e + jfz" + ga + hb23" + &c. you write R: let the Area of the Curve be 2*R*, the Ordinate BC ſhall be equal to . Ty . , ba. HR x hav + Kc. x2! N. DEMONSTRATION. 24. For let R v, it will be (by Prop. 1.) 6 R RR“ b. For R in the firſt Term of the Equation and 2“ in the ſecond write RR*-" and 22%", and it will become 0zR + R x -R. =. But it was Re + g2*" + * + &c: and thence (by Prop. I.) it becomes R = f , + 2 25 + 3,2 + W &c. which being ſubſtituted, and BE Þ&ay * fe X £27" Fo ay x hz3" +&c. x 2%" -U. PROP. rote for Ma it becomes le J 818 3 TY ) — „ » FT W 1 * „ The r of Curves. 9 PROP. IV. 1 H E O R. Il 2 5. If AB the Abſciſs of a Curve be z, and if for e 72 + wy ＋ &c. you write R; and for & + 4 ＋ mz" ＋ &c. you write 8: 1 let the Area of the Curve be 2 R, the Ordinate BC will be 25 to WT, 455 eee nay g +0 2y +0 * — a 7. le Jon x f'lz * eee - +1 fan 9 +0 +0 5 I. , Kms 72 Anz; Lada x gmz#" ＋2¹¹ 2 This Propoſition is ene after the Manner of the pre- ceding. PR OP. V. T H E O R. III. 8 26. If the Abſciſs AB of a Curve be 2, and for e + fz» + g= * 7 hz" + &c. be wrote R: Let the Ordinate n a + bg" + ca + dz3" + &c: and put — =, r ＋ A= „ S$+A.- 7 N The Area ſball be equal to +3 Le -T TFI XB 27 r+2xe 247 x FC—? FT „2B A — 3" hs r+3xe ” =i+3x/D=t+2 x 1x4B * — Ss. .* 114 *. | — 10 "The Bundrature of Cunyvss, Where A, B, C, D, &c. denote the whole given 2 2 12 ven Terms 3 m the Series with their 8 Signs + and — „ eee C oF = ofA 8 B tha Coefficient of the frond SEO: == £ —C— = #B.- 6 | Che Coefficient of the third Term 2 — + 4,29 F 3 r 2 Xx „ | Ad0S ine 0 te wane ö ENS | DEMONSTRATION 5 27. According to the third Propofition, let the following Expreſſions - Ordinates of Curves, - + Their Areas. £ o +4. AF. +0 a} „ J 14 i ; I ben 0 „R ＋ *g +3aq x AZ &c. * ALR : * ene ta, 1 * ＋ e e, XgB2z3" Nc. by Bt R | 1% Ts * 5 25 . Teh 4® — . (Patt | And if the Sum of the Ordinates be put equal to the Ordinate a ＋ b + 2" + dz + &c. into 2 Rα⁰., the Sum of the Areas 2 R into A B + CZ + D* + &c. ſhall be equal to the Area of the Curve of which that is the Ordinate. Therefore let the Correſpondent Terms of the Ordinates be put equal, and it will be a — beA b = TFanx/A+TF5 xeB * | em "+ eee ee TF 2aXeC Co And thence A= ; i — == 8 r $5. Tex e _ And ſo on in infinttum. RW Now 0 we w The Quadrature of Cutvis, Now put . r, r + a =5, r, Ge. and in the Area tn Mito. 4 * . 2h „X Ba + Cz?” + Dz3" + &c. write the Values F A, B. Sc. juſt now found, and the Series propoſed will come out. Q. E. D. K 28. And it muſt be obſerved that every Ordinate is reſolved into a Series two different ways. For the Index y may either be affirmative or negative. bh | : 8 rr n Let the Ordinate — f= be propoſed : it may be wrote E bn) TA P either thus 2=#x 3 E- E + mz3|-* or thus 2=* x — [+-3k2—* x m—lg—+k2—3\-t, 5 In the former Cafe you have a= 3h, B= o, c=—/: e=k, f=0, g=— l, hb =m-:. A == 1=1, DJ —I=— = — =, $=—1, r - Y=0 _ 15 | In the latter it is a= — , So, cz: n, f=—l, go, b=k: XA, n=—1, (—1=—2, j=—1, r=1, =, £8=2, vs. 8 Both Cafes muſt be tried. And if either of the Series s, by means of the Terms at length failing, break off and terminate, = Area of the Curve will be had in finite Ferms. Thus in the firſt Caſe of this Example, by writing in the Series the Values of @, 6, c, e, ,, g, b, X, 0, r, 5, t, v, all the Terms after the firſt vaniſh in inf. and the Area of the Curve comes out — 2/ — ==, and this Area, becauſe of the negative Sign, is adjacent to the Abſciſs produced beyond the Ordinate. For every affirmative Area is adj both to Abſeiſs and Ordinate; but a negative Area falls upon the oppoſite Side of the Ordinate, and is adjacent to the Abſciſs produced, the Sign of the Ordinate being un- derftood to remain, By this means one of the Series s, and ſometimes both of them, always terminates and becomes finite, if the Curve can be ſquared geometrically. But if the Curve don't admit of being thus ſquared, both Series will be continued infinitely, and one of them will converge and give the Area by Approximation, except when 7 (becauſe of the Area being infinite) is either nothing or a negative In- teger ; or when 2 is equal to Unity. If = be leſs than Unity, that. Series will converge in which the Index 3 is affirmative: But if - 2 than Unity, the other Series will converge. If in the one Caſe the Area be adjacent to the Abſciſs drawn as far as the Ordmate, in the other it is adjacent to the W produced beyond the Ordinate. N | 2 e 29. Be- 12 The Quadrature of Cunvas. 29. Beſides you are to obſerve, that if the Ordinate be a Product contain'd under a rational Factor ond an, irreducible ſurd Factor R-, and the Side R of the ſurd Factor don't divide the rational Factor Q; it will be a —1= and R. == R. But if the Side R of the ſurd Factor divide the rational Factor once, then it will be a — 1 =7 +1, and R- — RI: if it divide it twice, it will be a — 1 ra, and Ri = RH: if thrice, then A — 12 ＋ 3, and R- — R.;: and ſo on. # 12 | 30. If the Ordinate be an irreducible rational Fractien having it's. Denominator made up of two or more Terms: the Denominator muſt be reſolved into all it's prime Diviſors. And if there be any Diviſor that has no equal, the Curve cannot be ſquared : but if there be two or more equal Diviſors, one of them is to be rejected, and if there be yet other two or more equal to each other and unequal to the former, one of them is likewiſe to be 11 and ſo you are to do in all the other ual Diviſors, if more ſtill remain: afterwards the Diviſor which is. left; or the Product contain'd under all the Diviſors which are left (if there are more of them) muſt be put for R, and the Reciprocal of it's Square R- for R-, unleſs when that Product is a Square, or a Cube, or a Biquadrate, &c, in which Caſe it's Side or Root muſt be put for R, and the Index of the Power, viz. 2, or 3, or 4, taken ne- gatively, for x: and the Ordinate is to be reduced to the Denominator Ra, or Rs, or R., or RS, Oc. 285 + 24 = $23 Thus if the Ordinate were Tec: . becauſe this Fraction is irreducible, and the Diviſors of the Denominator have all of them Equals; for they are 8 — 1, 2 — 1, — 3 and 2 +2, 2 T2, I reject one Diviſor of each Magnitude, and the Product of the remaining Diviſors z — 1, 2 — 1, 2 +2, vis. 23 — 33 ＋ 2, 1 put for R, and the ·Reciprocal of it's Square N or R, for R.. Then I reduce the Ordinate to the Denominator R or R, and it 25 = 02+ + $823 5 48 | : 1 * becomes r 1, e. BI X * 98 + 23 X 2 32 + 23\ And thence it is a=8, þ— 9, c o, d = 1, Cc. e=2,f=—3, g=0, b=1, A—1=—2, A=—1, 9=1, 8 =1=3, 0 4 Dr, z. Da, v=1. And theſe Values being inſerted in the . . 24 : ; . Series, the Area comes out -f, all the Terms of the Series throughout, after the firſt, vaniſhing, | 31. Finally, F + MM, » A TY The Quadrature of Curves. 31. Finally, if the Ordinate be an irreducible Fraction,” and it's De- nominator be a Product contained under a rational Factor Q and an irreducible ſurd Factor R., you muſt find all the prime Diviſors of the Side R, and reject one Diviſor of each Magnitude, and then multiply the rational Factor Q by the remaining Diviſors, if there be any ſuch; and if the Product thence ariſing be equal to the Side R, or any Power of it whoſe Index is a whole Number, let that Index be , and it will be X- i - - n and R. —= R=. b | As if the Ordinate was EZ = 5 becauſe R the = „e eee Side of the ſurd Factor or 5 + 9 — gx* — x3 has theſe Diviſors 7 + x, Nx, q — x, which are of two different Magnitudes, I re- je& one Diviſor of each Magnitude, and I multiply the rational Factor * x* by the Diviſor which remains g + x. And becauſe the Pro- duct 9g? + g*x — g — x3 is equal to the Side R, I put m=1, whence, ſince 7 =, it becomes A — I =— . Therefore I reduce the Ordinate to the Denominator RA, and it becomes x* x 37 Pai N T -, eee! Whence a = 34, b 295, Cc. == , f=4q*, c. o, lt Ar r =, 4=%, f=575, vo. And 10 1 ing wrote in the Series, the Area comes out — f, n e the Terms through the whole Series after the third vaniſhing, P R O P. VI. T H E O R. IV. 32. If the Abſciſs AB of a Curve be z, and R be put for e + /z" + g2*" + bz3" + &c. and 8 for & + Iz” + mz*" + nz" + &c. and if the Ordinate be 2 R*—' $%=7 into a + bn + cg + dE + Se; and the Rectangles of the Terms e, /, g, b, Sc. and &, I, m, n, &c. be „ . em fm gm hm Ec. . In the former Impreſſions there was an Error in the Expreſſion of this Ordinate, which. 295 — , + 99*x* — g*x3 — 6x3 | ſtood thus — 3 is eaſily collected from what follows. See: een | the Notes Art. 136. + Here alſo was an Error in the former Impreſſions, the Numerator of the Fraction being. according to them 39*x ＋ 3x3. See the Notes. | * | And 13 14 &c. remain the ſame; and for c and 7 be wrote fac The Ruadrature of Curves. And if the numeral Coefficients of theſe Rectangles reſpectively be 2 Ar. r+A=5s TAD. t+a=v, Ge. 5+ pt. + «= v. e m. w p A, Ge. T ＋＋ b. 0+ p=w0v.wHb w=x x + « =y, Se. The Area of the Curve will be Where A denotes the given Coefficient of the firſt Term 27 with it's Sign T or —, B the given Coefficient of the ſecond Term, C the given Coefficient of the * and ſo on. And one or more of the — a, b, c, Sc. e, J. g. Sc. k, I, n, &c. may be wanting. The Propoſition i is demi ned after the Manner of the preceding, and what Things are there remarked, likewiſe obtain here, More- over the Series of ſuch Propoſitions goes on e and the Pro- greſſion of the Series is manifeſt. PROP. VII THEOR.V. 33. If for e L + g2*" ＋ Ge. be wrote R as above, and in the Ordinate of any Curve e RAE", the given Quantities. 0, 2, 2, e, g. ly any in yo - Numbers: The Yuadrature of Cunvrs. Numbers: and if there be given the Area of one of theſe Curves which are deſign'd by innumerable Ordinates thus arifing, when the Ordinates are Binomials under the Vinculam of the Root; or if the Areas of two of the Curves be given when the Ordinates are Trinomials under the Vinculam of the Root ; or the Areas of three of the Curves, when the Ordinates are Quadrinomials under the Vinculum of the Root, and ſo on in infinitum : I ſay the Areas of all theſe Curves ſhall be given. 34. I reckon here for Names all the Terms under the Vinculum of the Root as well them that are deficient, as them that are full, the Ordinate Va*— ax? £ x+, becauſe of two deficient Terms be- twixt a+ and — 4x3 gught to be eſteemed a Quinquinomial. But = is a Binomial and t L — © a Trinomial, fince the Progreſſion goes on now by greater Differences. Now the Propoſition is thus demonſtrated. 81 Sy * 35. Let the Ordinates of two Curves be pz*—"R*-" and q and their Areas pA and gB, R being a Trinomial e + fz" g. And fince by Prop. 3. R is the Area of a Curve, whoſe Ordinate is be TOT - 8+ 2 x g2% into R, ſubtract the for- T —4— indy vale y from the lower Ordinate and Area, and there remains de 5 4 d „* f—q * =Þ of x ga x r the Ordi- nate of a new Curve, and R — pA — gB it's Area. Put de =p, and f f f = q, and the Ordinate becomes 6 ＋ 2 x g x 2—R*-", and the Area 2 — rA — /B -A B. Divide both by (g + 2, and call the Area thence ariſing C, and aſſuming any Quantity r, C will be the Area of a Curve whoſe Ordinate is rata" Ri And C correſponding to the Ordinate r2*+2»—R*=* from the Areas pA and gB, you may from the Areas 9B and C find a fourth Area, viz. D correſponding to the Ordinate 52%+3—"R®-", and fo on in inf. And the like is the Rate of the Progreflion which proceeds the con- by the ſame Method we have found the Area trary way from the Areas B and A. If any of the Terms 0, Q, and d C az be wanting and break off the Series, let the Area pA be aſſumed in the Beginning of the one Progreffion, and the Area B in the Beginning of the other, and from theſe two Areas, all the Areas will. be given in both Progreffions. And contrarily, from any other two Areas 13 15 The Ruadrature of Curves. Areas aſſumed you can return by Analyſis to the Areas A and B, ſo that from two Areas given all the reſt are given. Q. E. O. This is the Caſe of Curves in which 0 the Index ot z is increaſed or diminiſhed by the perpetual Addition or Subtraction of the Quantity . The other Caſe reſpects Curves where the Index A is increaſed or diminiſhed by Units. CASE II. 36. If the Ordinates p.. R and gi, to which the Areas 2A and qB may now be ſuppoſed to anſwer, be multiplied by R or e + for gz, and then again be divided by R, they become 1 * r and 2 ＋ 9 f2*" ＋ 9925” x SIR, And (b . 3.) R is the Area of a Curve, whoſe Ordinate is bae +8 + an x afz" +0 + 2ay N N R-, and HR the Area of a Curve whoſe Ordinate is 8 + / x bez" +8 + n + an x b/z*" + 8 + 9 + 2an Ng x 2 RI. And the Sum of theſe four Areas is þA + 9B + az RP + bz Rd and the Sum of the correſponding Ordinates Fer- f- Te +8+ ze TTT TI xgg T r + 1 xge + 1 * If the firſt Term, the third and fourth be put equal to nothing ſe- parately, by the firſt it will be hae + pe = o, or — da p, by the fourth — 85 — 26 — 2 , and by the third (exterminating p and ) 7 Sum of the four Ordinates is — 2 TR, and the Sum of as many correſpondingAreas is _ DR A —agB. Divide theſe Sums by eke, 15 if the laſt Quote be call'd D, then ſhall D be the Area of a Curve whoſe Ordinate is the firſt 33 2 ＋ 1IRN—1. And by the ſame Method, if you put all the Terms of the Ordinate fave the firſt equal cach to nothing, you may find the Area of a Curve whoſe Ordinate is 2 R*-', Let that Area be called C, and by the fame Means that the Areas C and D have been found from the Areas A and / B. Whence the ſecond becomes . and therefore the Fr . r . 0 --- . © Die Quadrature of Curves, A and B, you may from theſe Areas C and D find other two Areas E and F anſwering to the Ordinates 2 R and 28+" R*-?, and ſo on continually, And by a contrary Analyſis ou may return from the Areas E and F to the Areas C and D, and thence to the Areas A and B, and likewiſe to the other Areas which follow in the Progref- fion, Therefore if the Index > be increas'd or diminiſh'd by the con- tinual Addition or Subtraction of Units, and of the Areas which an- ſwer to the Ordinates thus ariſing two of the moſt fimple be given; all the others in inf. are given. CASE III. | 37. And by joining theſe two Caſes together, if both the Index 0 any how increas'd or diminiſh'd by the continual Addition or Sub- traction of 3, and the Index a by the continual Addition or Subtrac- tion of Unity, the Areas correſponding to the ſeveral Ordinates thus ariſing, ſhall be given. Q. E. O. CAS E IV. 38. And by a like way of reaſoning, if the Ordinate conſiſt of a Quadrinomial under the radical Sign, and there be given three of the Areas, or if it conſiſt of a Quinquinomial, and there be given four of the Areas, and ſo on: all the Areas ſhall be given which can be pro- duced by the Addition or Subtraction of the Number » to the Index 0, or of Unity to the Index a. And the like is the Reaſoning with re- ſpe to the Curves, whoſe Ordinates are made up of the Binomials, and one Area of thoſe which are not geometrically quadrable is given. n P R O P. vin. T H E o R. VI. 39. If for e + g2” + Sc. and x + Iz" + mz*" + &c. be wrote R and S as above, and in the Ordinate of any Curve 2%£»-R*i&rgut+» » the given Quantities 0, 1, A, f, e, /, g, K, I, m, &c. remain the ſame ; and for c, 7 and v be wrote any whole Numbers ſucceflively : and if the Areas of two of the Curves be given which are deſign d by the Ordinates thus ariſing, when the Quantities Rand 8 are Binomials, or if the Areas of three of the Curves be given, when R and 8 together conſiſt of five Nomes or Terms, or the Areas of four of the Curves when Rand 8 both taken together conſiſt of ſix Nomes or Terms, and fo on in inſinitum: 1 ſay the Areas of all the Curves ſhall be given. It is demonſtrated after the Manner of the preceding Propoſition. D PROP. 1 7 18 EUR TT fe gr TEN FETa fer The Quadrature of Counvs 8: p R O P. . THE OR. vn. 40. The Areas of thoſe Curves are equal among themſelves, whoſe Ordinates are reci as the Fluxions of their Abſciſſes. For the Rectangles pn a under the Ordinates, and the Fluxions of the Abſciſſes will be equal, and the Fluxions of the Areas are as theſe Rectangles. — CO RO L. I. 41. If there be aſſum'd any Relation betwixt the Abſciſſes of two Curves, and thence by Prop. I. you ſeck the Relation of the Fluxions of the Abſciſles, and the Ordinates be made reciprocally proportional to theſe Fluxions, you may find innumerable Curves, whoſe Areas are equal among themſelves. c OROL, u. 42. For every Curve whoſe Ordinate is 2 U x Ad Taz" Ton by aſſuming any. Quantity for , and putting and z'= x, paſſes into another Curve equal to itſelf whoſe Ordinate is Be —̃ *r f + ge” + Eel” 0040 L WW .:-;......; 43. Andevery'Curve whoſe Ordinate is 2% x PU . Re + fe" + g2” Selb, if you aſſume any Quantity for v, and put i, and a = x, paſſes into another Curve equal to itſelf, whoſe Ordinate is * * 2 c ＋ Sc. '% elf +-gx* TE COROL. IV. 44. And every Curve whoſe Ordinate is 2% X C c + Ex. xe+jz + T Ce. xk 4. E + ma ＋ Cc. by affuming any Quantity for , and putting , and &' =, paſſes i into an- other Curve equal to it, whoſe Ordinate is | Y MH * . < J 2 7 A} + EL 2, o . MY .M 2 „ ov * ig 4 4 f/ /f n l r EEE 8 . n A " 6h 1 2:58 = -Y is 3 "FJ A . 2 Dd By. 92 *8 - * ©] 1 2 „ Ko. 2 7. Tie 5 w__ - Py WF 2 ad n FLY Is, The Quadrature of Curvas. COROL.V. 45. And every Curve whoſe Ordinate is 2 x . +g229 Tc. P, by putting —== x, paſſes into another Curve equal to itſelf, whoſe Or- dinate is =: Ke fx gx=tt + &c.\*; : that is Ir F ex“ Jig if the Quantity under the Vinculum of the Root be a Binomial ; or * IT fx __ ex" 2n p, if it be « Trinomial: and fo for others. MW. | C 0 R O L. VI. 46. Andevery Curve whoſe Ordinate is 2 — x g + &c.Þ xk + Is + mz*" + c.“, by putting == x, paſſes into another equal Curve, whoſe Ordinate is TTX? + fx” + gx0® + Fol“ X IH Sell“, that i 1 prfarz „Ter * LAN if the Quan nantities under the Vinculums of the Roots be Binomials, or xs D Far x FEA if the Quantity under the Vinculum of the firſt Root be a Trinomial, and that under the ſecond a Binomial : and fo for others. 47. And obſerve, that the two equal Areas in theſe two laſt Co- rollaries, lye towards contrary Parts of the Ordinates. If the Area in the one Curve be adjacent to the Abſciſs, the Arca equal to it in the other Curve, is aGacent to the Abſciſs produced. COR OL. VII. 48. If the Relation betwixt the Ordinate y of any curve and the Abſciſs x be defin'd by any affected Equation of this Form, Nee eee. ee Sc. this Figure, by aſſuming — — x = = and A = —— paſſes into another equal to itſelf, whoſe Abſciſs x, from the Dedinate v given, is determined by an Equation not affected, iz. een aa 9 xt TY T ＋ e.. . COROL. 19 The Quadrature of CRV IG. COROL. VIIL 49. If the Relation betwixt y the Ordinate of any Curve, and z its Abſciſs be defined by any affected Equation of this Form y* x e 2 Sc. = t x 2 ly2*+my*""22 + &c. D f D &. This Figure, by taking s = = * , þ 2 and v —4 2, paſſes into another equal to itſelf, IH Abſciſs x is deter- mined from the Ordinate v given, by this Equation leſs affected v* x e + fo" g + &c. = Alu b-mv*" + Fe. + 5x e. COROL. N. 30. Every Cu Curve whoſe Ordinate is d x7 fe gen + Te! * TBN + fer +getn + &cll i 0 = ay, and you aſſume x = ez + fzt" + gy'+2* Þ To”, = = and $ = ==, paſſes into another equal to itfelf, whoſe Ordinate, is xa P. And you may obſerve that the former Ordinate, in this Corollary, becomes more ſimple by putting A = 1, or by puttin += 1, and ſo ordering it that the Root of the Power may be extract whoſe Index is o; or yet by putting @w =— 1, and A= l ===, to paſs other Curves. COROL. X. = g1.For ez+fzt"+g2 ee vo ee Tango Ts, A m + &c, and 2 + a + Sc. write R, r,S and 5 reſpeQively, and every Curve whoſe Ordinate is 28 + PRs x RUN FRY, if it be = == . r === 9, and RS = x, paſſes into e dee to itſelf, whoſe Ordinate There is in this Place an Error in the printed Copies, which I have corrected. See the Demonſtration in the Notes, Art. 332-335. 18 _ - as a and a a = The Quadrature of CUR VES. is x? X a PY. And obſerve that the former Ordinate becomes more fimple, by putting Unities for r, v, and A or fa, and fo ordering it that the Root of that Power may be extracted whoſe Index is , or by putting « n—— LM R=. | PR O FP. X. PROB. II. 2. To the moſt fumple Figures with which any Curve be Dk an whoſe Ordinate y, from the A0 2 os is determined by an Equation not affected. CASE TI. | | 53. Let the Ordinate be a2, and the Area ſhall be Zarb, as is eaſily collected from Prop. 5. by putting 6 =o =c=d=f=g=b and e=1. CASE II. £4. Let the Ordinate be 42 Xe + fx" + = + Se. Pi, and if the Figure can be geometrically compared with rectilinear Figures, it will be ſquared by Prop. 5, by putting & = d. But if otherwiſe, you muſt change it into another Curve equal to it, whoſe Ordinate is * Xe + fx + g + Sci, by Cor. 2. Prop. . Afterwards if you reject the Units from the Indexes of the Powers, viz.. — and x — 1 by the ſeventh Prop. till theſe Powers become as low as poſſible, you ſhall have arrived at the moſt ſimple Figures, which can be diſcovered in this way. Then every one of them gives an- other which is ſometimes more ſimple by Cor. 5. Prop. g. And from theſe compared together by Prop. 3. and Cor. g. and 10. Prop. g, more ſimple Figures as yet ſometimes diſcover themſelves. Finally, from the Aſſumption of the moſt ſimple Figures the Area ſought may de computed by coming back through the Tom Steps, CASE III. e+ fs + gz Cc, and this Figure, if it can be ſquared, will be ſquared by Prop. 5. If not, it's Ordinate muſt be diſtinguiſhed into the Parts zi xa xe + + gz” + Cc. l., -N H x e Tg + c. „c. and then the moſt ſimple Figures muſt be found, with which the Figures anſwering to theſe Parts can 2˙ 22 The Quadrature of CV Rvxs. can be compared, by Caſe 2. For the Areas of the Figures anſwer- ing to theſe Parts, join d together with their make up the whole Area required. proper Signs —— and — CASE NV. 56. Let the Ordinate be I „ A ＋ 5 er + &. x e T ga ＋ Sc. KETTE + mz" + C.; and if the Curve can be ſquared, it will be ſquared by Prop. 6. If otherwiſe, you muſt convert it into a more ſimple one by Cor: 4. Prop. g. and thence you may compare it with the moſt ſimple Figures by Prop. 8. and Cor, 6, 9 and 10. Prop. 9. as is done in Caſes ſecond and third, CASE V. 57. If the Ordinate conſiſt of. different Parts, all the Parts muſt be accounted as Ordinates of ſo many Curves ; and theſe Curves, as many of them as can be ſquared, are to be ſquared ſingly, and their Ordi- nates to be taken away from the whole Ordinate. Then the Curve, which the remaining Part of the Ordinate defigns, muſt be compared ſeparately (as in the ſecond, third and fourth Caſes) with the moſt ſim- ple Figures it can be compared with, And the Sum of all the Areas muſt be accounted the Area of the Curve propos'd. COR OL. I. 38. Hence alſo, every Curve whoſe Ordinate is the affected ſquare Root of it's own Equation, may be compared with the moſt ſimple Figures, either rectilinear, or curvilinear. For that Root always con- ſiſts of two Parts, which conſidered ſeparately, are not the affected Roots of Equations. | eres, RD Let the Equation a*y* ＋ 2*y* = 243y + 223y — be propoſed ; and the Root being extracted will be y „, EO. whoſe rational Part . and irrational Part — 2 IS == are the Ordinates of Curves, which, according to this Propoſition, may either be ſquared; or elſe compared with the moſt ſimple Figures, with which they admit of a geometrical Compariſon. COROL, * » *M 1 2 * 4 0 RR” 4 N » 4 Py. a — wy 38 * F 79 19 6 Ps. 4 4 v8 => , . 2 Fo „ . 1X - 3 wy * BS - "=. * Ky * : » 1 LO - - * a4 WEE. 5 * 3 NaF i a 7 2. _ ſl 2 2 2 0 7 1 1 4 a — Ga l af # F oy 7 +78 Ky i Is * . 3 * * 5 * + 1 "bb bo i p 2 «x * "I 1 2 — 1 1 * —— . 8 8 * "4b SEzE Mg 155 24 3 x RES. 8 bat : 2 8 , he” 4 - n MY P : S e 8 2 r EAR IX 3 fi 7 2 as 2 , G . PETS d ; 4 * : 2 8 ** ** . ; . 1 Wage 4% RI IS B. SF . 14 . D F e > * 8 Fe IT „* NO" wy; — * be hd ; , * . 2 3 5 e n e . 4 be 6d. ef 9 ” e 1 3 8 7 : k — W Ps a D the P A 2 we d : „ FI " M. 4 The Quadratur of Cux vgs. $59. And every Curve whole. Ordinate is gdefin'd by any affected ation, which, by Cor. 7. Prop. , paſſes into an Equation not ected, is either ſquared by this Propofition, if it can be ſquared ; or is compared with the moſt ſimple Figures with which it may be compared. And by this means every Cprve 1s ſquared whoſe Equa- tion conſiſts of three Terms. For th Equation, if it be affected, is changed into one not affected by Cor. 7. Prop. 9: and then, by Cor. 2% and 5", Prop. g, by paſſing into moſt ſimple Curve, it either gives the Quadrature of the Figure, if it admit of one; or gives the moſt ſimple Figure, with which it may be compared. COR OL. III. 60. And every Curve, whoſe Ordinate is defined by any affected Equation, which, by Cor. 8. Prop. , paſſes into an affected quadra- tic Equation, is either ſquared by this Prop. and Cor. 1, if it can be ſquaned ; or is compared with the moſt ſimple Figures with which it admits of a geometrical Compariſon. | | SCHOLIUM, 61: When Figures are to be ſquared, it would be too troubleſom a Work to have recourſe always to theſe general Rules: it's much better once to ſquare the more ſimple and uſeful Kind of Figures, and record the Quadratures in a Table; afterwards conſult the Table, as often as you have any ſuch Curve to ſquare. And the two follow- ing Tables are of this Kind: in which 2 denotes the Abſciſs, y the | Ordinate, and t the Area of the Curve to be ſquared ; and d, e, if, g, h, n; are given Quantities with their proper Signs. - and —, A TABLE „ The Ruadrature of Curves: —_—— —— . A TABLE - Of the more fimple kind of Curves which may be ſquared. | Forms of Curves. Areas of the Curves. 1 4 — £21 = f. | NA | I 4 * 2 x 1 II f 2 4E Or —4 8233 T- A T* - D [4 1% Cf yR t. Where R= V. 2E IL -I Ls bl 28 | e e R = | | 3 » — ww" | — 25 | | 1 eee eee e-, | 1 LT . Sof b —_ | [2 f__ xa ==5 ER = 7. | * * | fr? . * 7 . | N 37 1 F 93 IV 4 8 3—1 n 8 | LN” , =) LOSES ER =? Ac 3 gy” y G48 fr" —36ef*2*" + 30/327" 8 657 Sony q — — i —— > oa 4 0 (po + * 2 ; * 4 c MH Sy « EA 3 FI ne * PPP A Wh { a _ —-— — = + EE wn | os 1 ee | We” - , a 1 % D = = a= n m Jo amy | a W x. SR 4 HE SY % &_-_ / "Dai 2 2 | | yy : , 203 d 9m wnog q aq ut puy uE 243 mon du, q 20 omg q mog payengry aq pinoyy anf 273 12typaym ure oun sf 37 um Senn -Uend o zo οοαj,ẽicnn d 21. 20up — f puy D ry aq pur I ajeurpig on 7 ln aq zo Sον 21 1122 *pajodoad ain qi 30 22nzezpeey q 02 Paarnboa aq suorog 2000) 0443 JO srory tn oνfττpν jr 'S puy *payjdwoo 2q Odꝗ v weidoppjjereq 213 39] pur? I. uf qy $qy q Junzaw Id agu I od Me * poufof g do av dN I med q 32 axrurpag ur Zuse HY = Odd 0 OUg? 10 ꝗuꝗ roy aq pur a qq 2eupi0 §ô en uryad q & = gu 10 gy gy g12jqy 2y3 10 © 10 fy grojqy sq; 30 SunuurBag uaard amp ꝗqv xy 2anfucg fey Y x24 oy xy fy 2q anus sn 29] pur pajodoid 2410 213 Jo 2ameapen aq oi panne 81 r 2Joya uongag 20g r gd 10 00d 10 002 1] D ag dou q21qas 520m fo pury o1duy 340m 291 {O. A 1d VI. v Ruadrature of CUxvrs. S aa * - = 42 — \ p — 7 al — — — _— | Ir Sr et ¼—— —— ar +4 ig vod nf =0=1 2 ef 5 my) ps x = [mp 10 IRE © 2 ER 14 —— OO Dec | Al 4 +4 n vd on > = 7 = 1 «x * 24 5 4 5 N Wy Yan o 10 ayaoun + =;=1 > ax} 2 — 12 o+// ap ee ; D S = 1 Aer i x + ig Ladd q = — | eee B als Hig 9099 5; =1= Smart —ex — x = mg 20 E 1 . +442 Hig 099.40 ad „ 1224 22 = - 23 — — | ter Hg AC.LEACY o 140 109” OJul Ys * = xi + //) * mw (= {+ 2 — — - — 1 i F = * = 22 7 n 243 30 crery *23xurPIC) u werpeg Aug 2 Md — — * *'SDAINJ JO aui >” _þ FEE LZT, 7 = EEE — Pre 7 22 90 dd l . = x3 + 2 = 214 Is 4 (=n3+2/+1/;- 2 R * Dee. , — g e — — 4 i — if — @x7 _——__ r wa ” _— NY * — * uzg +7 | iert S 7 == ä 1 2 | = 8 — 2 an 4 ROE" os. 12 — dw.” — ſp} - 2 SEE. | 7 == = 2 > 7 oh — opt — e e oÞ+ preg" x =aq +I +? J '& es e r I Au 8 1 e 2. e OE = a. | | _— 5 a ——X WS a _ fs — — = LT c. £02 7th 8 ens 7 . "3 = 2 hl 9 7 2 124 8 ;= 2 TE 1 7 7 4 +3 | 4 +2 x x _ —_ = (= = : = ee | = 2 += a2 0+2 | — | (>) * D.. — == C=—. & SYS T e * IJ. fag ez | E 2T ++ 72 = 42 UE wed reds =—— rms .. x = Ka S n 4 ae per + hcl — Toe ; —_— > E 1=uþ?P F | 1 — — xIouh wn | EESRTTT = 4 (1 A Spy | => . 3 x & = = — — | VV art 6 fr ul i — — 4 . ö \ | =5 _ 24 "GD, . * | 5 a= C== TITER: | E | — — : aopy o+ apt + py —| = CHE of 1-22 ; e Dr 22 'V4dV+ga9” = wi +/+ = — 1 — wu fIJqy | dong mn Jo suẽjꝗ The Quadratur of Cufves. 62, In theſe Tables the Series's of Curves of each Form may be continued in infinitum towards both Parts. For in the firſt Table, with ref; to the Numerators of the Areas of the third and fourth Forms, the numeral Coefficients of the initial Terms (2, — 4, 16, — 96, 868, &c.) are found by multiplying the Numbers — 2, — 4,— 6, —8, — 10, Sc. continually into one another: and the Coefficients of the ſub- ſequent Terms are derived from thoſe of the 1 Terms, by multiply- ing them orderly, in the third Form, WM — 2, —£, —+, —- +— LY Sc. in the fourth Form, by — 2, — +, — "> — . — , Cc. And the Coefficients of the . 3,15, 105, &c. ariſe by mul- tiplying the Numbers 1, 3, 5 FE „Sc., continually into one another. 63. And in the ſecond Table, the Series's of Curves of the firſt, ſe- cond, fiſth, ſixth, ninth and tenth Forms, by the Help of Diviſion alone; and the other Forms by the Help of the thitd and fourth Pro- poſitions, are drawn out both Ways in inſinitum. 64. Moreover theſe Series's, by changing the Sign of the Number 3, uſe to vary. For thus, to take an Example, the Curve SN J2"=y becomes F,. | PROP. XI. T HE O R. VIII. 65. Let ADIC be any Curve, having the Abſciſs AB ==, the Ordinate BD y; and let AEK C be another Curve, whoſe Ordinate BE is equal to the Area of the for- mer ADB applied to Unity; and AFLC a third Curve, * | whoſe Ordinate BF is equal to Area of the ſecond AEB applied to Unity; and AGM G a fourth Curve whoſe Or- 42250 BG is equal to the 4 Area of the third Curve E AFB applied to Unity; and AHNC a fifth Curve, whoſe ed Ordinate BH is _ ” D AGB the Area of the fourt Z applied to Unity: and fo on — | in infinitum. And let A.B, 4== Pr C. D. E. Sc. be the Areas of A Trans | 2 29 The Quadrature of Cunves. | er Ordinates y, ay, 2, zy, xy, &c. and the common 606. Let there be any given Abſciſs AC== f, and let C 2 =x- and let P, Q. R, 8, T be the Areas of Curves having the Or- dinates y, xy, **, By, My, &c. and the common Abſciſs x. 1 _ — = r at the whole given Ab- AC, and likewiſe at inate CI given in Poſition or OS 1 of the Areas mentioned at the Beginning: The 1, ADIC = A = P IF 2%. AEKC=/A —HB=Q_ | FA — 2B +C_. iD 7. AFLC= HER 4, AGMC= == H= = 8 50, AHNC= HA — 4B + 6#4C —4D+E __ uo nA (167 Ih ef fs TN: COROL. 68. Whence if the Curves, whoſe Ordinates are y, zy, 2*y, 23y, &c. or y, xy, x*y, 5 &c. can be ſquared ; the Curves ADI C, AEKC, AFLC, AGMC, &c. may likewiſe be ſquared ; and we ſhall have the Ordinates BE, BF, BG, BH, which are proportional to the Areas of the Curves. SCHOLIUM. | 69. We ſaid formerly that there were firſt, ſecond, third, fourth, Ge. Fluxions of flowing Quantities. Theſe Flaxions are as the Terms of infinite converging Series. 951 04 11 70. Thus if 2" be the flowing Quantity, and by flowing become z Tob', and afterwards be reſolved into the converging Series 2” + nos A. oo þ E239 ©2%932%3 + &c. The firſt Term of this Series 2” will be that flowing Quantity ; the ſecond will be the firſt Increment or Difference, to which confider'd as naſcent, it's farſt Fluxion is proportional : the third 2002" will be * it's ſecond Increment or Difference, to which confider'd as naſcent the ſecond * . . — 2 . Fluxion is proportional: the fourth Term 2 on will be * it's third Increment or Difference, to which conſidered as naſcent the third Fluxion is proportional: and fo on in infinitum. | 71. But theſe Fluxions may be expounded by the Ordinates of Curves BD, BE, BF, BG, BH, &c. See the Fig. See the Notes upon the Place. 72. Thus The Pucdrature of Cunves. 72, Thus if the Ordinate BE (==>) be the flowing Quantity, it's firſt Fluxion nne 8 73. If BF (==) be the flowing Quantity, it's firſt Fluxion will be as the Ordinate BE; and it's ſecond Fluxion as the Ordinate BD, 74. If BH (= AGB, be the flowing Quantity, it's firſt, ſecond, third and fourth Fluxions will be as the Ordinates BG, BF, BE and BD reſpectively. 75. And AP in ſuch Equations as involve only two unknown Quantities, of which one is a Quantity flowing uniformly, and the other any Fluxion of the other flowing — that other Fluent may be found by the Quadrature of Curves. For let its Fluxion be expounded by BD, and if it be the firſt Fluxion, ſeek the Area ADB =—=BExX1; if it be the ſecond Fluxion, ſeek the Area AEB = BF x 1; if the third, ſeek the Area AFB — BG x1; Sc. and the Area fo found will be the Exponent of the Fluent ſought. 76. But moreover in ions which involve a Fluent and it's firſt Fluxion without the other Fluent ; or two Fluxions of the ſame Fluent, the firſt and ſecond, or the ſecond and third, or the third and fourth, Sc. without either of the two Fluents : the Fluents may be found by the Quadrature of Curves, Let there be given the Equation aa av + vv, while it is v = BE, v=BD, z—= AB and z= 1, and that Equation, by completing the Dimenfions of the F Juxions, will became aav = avs T- vos, or and Abſciſs v, you'll have the Fluent z. 78. Moreover let the Equation be aav = av 4+ vv ; while it is v=BF, BE, ö = BD, and z— AB; and by means of the Relation betwixt © and v or BD and BE, you'll find the Relation be- twixt AB and BE as in the immediately preceding Example. Afffter- wards by means of this Relation, you'll find the Relation betwixt AB and BF, by ſquaring the Curve AEB. 79. Such Equations as involve three unknown Quantities may ſome- times be reduced to Equations which involve two only: and in theſe Caſes the Fluents will be found from the Fluxions as above. Let the Equation @ — bx" = cxy'y 4 dy**3y be propoſed : put y = v, and it will be a — bx" = cxv + dv*, This Equation, by ſquaring the Curve, whoſe Abſciſs is x and Ordinate v, gives the Area v; and yo otner 2 * 32 other Equation, viz. yy , by returning back to the Fluents gives Te Quadrature of Cunves: x 77+! =v: whence the Fluent y is obtain d. 80. But beſides, in Equations which involve three unknown Quan- tities, although they can't be reduced to Equations which involve only two, the Fluents will be ſometimes found by the Quadrature of ' Curves, Let the Equation ax” + bx? rer y + wy - be given, while it is & I; and the latter Part re + oxy? — fyy, by going back to the Fluents, becomes ex — , which therefore is as the Area of a Curve whoſe Abſciſs is x, and Or- dinate ax” + bx"V : and thence the Fluent y is given. 81. Let the Equation & x ax” + N be given, and the = Ver) Fluent whoſe Fluxion is x X ax” ＋ IK, will be as the Area of a Curve, whoſe Abſciſs is x; and it's Ordinate ax” ＋ & V. Likewiſe the Fluent whoſe Fluxion is dy will be as the Area of a Curve Vip whoſe Abſciſs is 5 and it's Ordinate . , that is (by Caſe 1. Form ef" 4th, Tab. 1.) as the Area =ve + 5”. Therefore I put 7 + F/ equal to the Area of the Curve whoſe Abſciſs is x and Ordinate ax” ＋ bx"\?, and fo the Fluent y will be had. 82. And let it be remarked that every Fluent which is collected from a firſt Fluxion, may be augmented or diminiſhed by any Quantity which don't flow at all. That which is collected from a ſecond Fluxion may be augmented or diminiſhed by any Quantity whoſe ſecond Fluxion is no- thing. A Fluent collected from a third Fluxion may be increaſed or di- miniſhed by any Quantity that has no third Fluxion. And fo on in inf. 83. After the Fluents are collected from the Fluxions, if there ariſe any Doubt about the Truth of the Concluſion, then the Fluxions of the Fluents found are again to be collected, and compared with the Fluxions propoſed at the Beginning. For if they come out equal, the Concluſion is right : if otherwiſe, the Fluents muſt be fo corrected, that their Fluxions may be equal to the Fluxions at firſt propos'd, For a Fluent may be aſſumed at pleaſure, and the Aſſumption be corrected, by putting the Fluxion of the aſſumed Fluent equal to the Fluxion propoſed, and comparing the homologous Terms with one another. And thus, by theſe Beginnings, we are led to yet more important Truths. Sir Ka ER R A "YE L331 * —_— at od 1 ä * —— 8 A a 1 Sr ISAAC WE VO 's. ITREATISE 4 * 2 1 | 1 * LC e nne - OF THE Quadrature of Curves explained. SECT: L Containing Notes on the Introduction. I, HE incomparable Sir I/aac Newton, having not only en- riched Mathematics and Philoſophy with many new and uſeful Diſcoveries in their various Parts ; but alſo extended the Limits of Geometry, by the noble and ſublime In- vention of Fluxions, takes care firſt of all, in this Introduction to the uadrature of Curves, to give us juſt Notions of his Method of Fluxions : becauſe it is a Method entirely new, by which the moſt remote and uſeful Properties and Relations of Quantities are diſcovered, and de- monſtrated, which were in vain attempted by the geometrical Analyſis of the Ancients; or the ſpecious Analyſis of the Moderns: and is the Method he made uſe of in the Doctrine of the Quadrature of Curves contained in this Treatiſe : as he himſelf informs us in this Place. 2. The Method of Fluxions is twofold : or rather, it may be. re- duced to two very general and comprehenſive Problems. 19. From the Fluents given to find the Fluxions. 2%. From the Fluxions given to find the Fluents. Which two Problems may be conceived as equi- valent to theſe, 1%. The Length of the * deſcribed by a 8 g 34 The Ruadrature of Curves explained. Body being continually or at all Times given, to find the Velocity of the Motion at any Time propoſed. 2%. The Velocity of the Motion being at all Times given, to find the Length of the Space deſcribed at any Time propoſed. The firſt of theſe Problems includes the direct Method of Fluxions; the other, the inverſe Method. 3. Both theſe Methods of Fluxions are capable of being applied ; and accordingly have been applicd to a vaſt Number of uſeful Purpoſes, by ſuch as have treated of this Subject: of all which one of the moſt noble and uſeful is the Doctrine of the Quadrature of Curves delivered in this Treatiſe, The Deſign of which is to ſhew how infinite Num- bers of various Kinds of geometrical Curves may be ſquared, 7z. e. their Areas my determined, by comparing. them with right-lined Fi- gures: or, if that cannot be done, it ſhews what way their Areas ma be nearly determined, by an infinite converging Series ; and the Curves eometrically compared, either with the Circle, Ellipſe and Hyper- la; or elſe, with other the moſt ſimple Curves, with which they admit of a geometrical Compariſon. So that by what our Author de- livers in this Treatiſe, and in his Analyſis by Equations of an infinite Number of Terms, (which I have ſubpoined to this Performance) all geometrical Curves, i. e. ſuch, the Relation of whoſe Abſciſs to the Ordinate is defined by an algebraical Equation of any Degree, can be either exactly ſquared; or, if that cannot be done, the Area may be expreſſed by an infinite converging Series, of which a finite Number of Terms ſhall make ſuch a near Approach to the true Area, as to differ from it by leſs than any given Difference. And even in mechani- cal Curves, i. e. ſuch Curves, whoſe Property cannot be defined by any fintte Equation of what Degree ſoever, de ſhews how to exhibit ap- proximate Values of their Areas; or to reduce them to geometrical Curves. | 4. Butalthough this be the primary Deſign of this Treatiſe of our An- thor's; yet it may be conſidered in a more general and extenſive View, as containing the Doctrine of Fluxions, both direct and inverſe, whatever the Nature of the flowing Quantities be. For the firſt Prop. contains * Art. 2. the Solution of the firſt General Problem mentioned above v. And by of this Ex. means of the Quadrature of Curves, the ſecond General Problem, con- * taining the inverſe Method of Fluxions, may be ſolved in a great many Caſes, although not univerſally: as we ſhall have Occaſion to ſhew, when we come to explain our Author's laſt Scholium. Art. 3. 5. After the true Meaning of Fluents and Fluxions is laid down by of the = our Author, he proceeds next, in this Article, to ſhew what Propor- ture of fion the Fluxions of flowing Quantities bear to one another. Accord- Curves, | 1 | | ingly Kt 4 = ® 3 2 * g > 4A 3 N * 1 1 1 5. Ou . _ * 2 * 8 +a 1 * 1 -20 2 * * o +; 3 by” ö. _ - 9 * . JF * i 4 1 ”» _— —— * : * * 1 8 | is e A R 4 br "+ wA©_ — mu ＋ÿdJI²y . . aa att «an Ea wo r "XV 3 VV 5” WTR” I» Fy F*** —— — F by ” n m—_ — mad hath, a ed SEC a. * CS : U a — 0 2 8 2 EA e EOS Ra * - r mn 2% 2 1 * - 3 <4 © a8 4 8 4 TI Ad G MER "x of . . GGG Ls. eee We bz. £3 c n 6 p N * 1 — e. - » — - bs 1 7 Fi RET, Ws K The Ruadrature of Curvas explained. 35 ingly he tells us, They are very nearly to one another as the Aug- « ments of the Fluents generated in equal, but very ſmall Times: and te that if we would ſpeak accurately, they are in the fir Ratio of the e naſcent Augments, but may be expounded by any Lines, which are « proportional to them.” He ſays afterwards *, that it comes to * Art, 6. the ſame Purpoſe if the Fluxions be taken in the 4% Ratio of the « evaneſcent Parts.” Now, becauſe the Doctrine of prime and ulti- mate Ratios has been ſo much controverted of late *, I ſhall here en- quire whether we have any diſtin& Idea thereof: which I reckon to be the more neceſſary, becauſe the noble Inventor has laid down this Notion of Fluxions in this Place, as that by which we are to be di- rected in our Conceptions of them. 6. Before we enter upon the Diſcuſſion of this Point, it will be pro- per to obſerve, 15, That the Fluxions of flowing Quantities, being the Celerities with which they are ſuppoſed to flow ; and by which they are generated : the Fluxion of any variable Quantity, ought never to be conſidered ab/olutely, or by itſelf ; but relatively, or with Relation to the Fluxion of ſome other flowing Quantity of the ſame Kind. 80 that if at any Time, for Brevity's ſake, the Fluxion of a Quantity be mentioned abſolutely ; yet there is always a ſuppoſed Relation to the Fluxion of ſome other Quantity with which it is underſtood to be com- pared : and ordinarily in the Compariſon, one of the flowing Quantities is ſuppoſed to flow equably and * ; ſo that it's Fluxion, being con- ſtant and invariable, is conſidered as a Standard or Meaſure, to which the Fluxions of the other Quantities are referred. 2% That if the Fluxion of any flowing Quantity, or it's Celerity of flowing, be con- tinually varying, i. e. continually accelerated or retarded, for any Time, the Fluxion in any one Place or at any one Inſtant of that Time, is different from the Fluxion of that Quantity in any other Place, or at any other Inſtant of Time. For whatever is continually varying, by Increaſe or Decreaſe, by the very Suppoſition, muſt be of a different Value in every different Place, or at every different Inſtant of Time, otherwiſe it would not be continually varying. To illuſtrate which in the Caſe of a falling Body : the Velocity with which it falls in any one Place, or at any one Inſtant of Time, during it's Fall, is different from the Velocity it has in any other Place, or at any other Inſtant of Time: and it is all one, whether the Motion be uniformly accelerated, or re- 2 or whether it be accelerated or retarded according to any other W. 1 See the Analyſt publiſhed by the Biſhop of Cloyne, with an Anſwer to it by Philalethes Can- tabrigienfis : likewiſe the Biſhop's Defence, and Philalethes's Reply. | F 2 7. Before 36 The Quadrature of CuxvAS explained. 7. Before we proſecute this Affair any further, we ſhall here bor- row from Sir 1, Newton the following Lemma. * Quantities, as likewiſe the Ratios of Quantities, which conſtant! * tend to Equality during any finite Time, and before the End of that * Time conie nearer to one another than by any given Difference, at « laſt become equal, | * If you deny it; let them be at laſt unequal, and let their laſt * Difference be D. Therefore they cannot come nearer to Equality « than by the given Difference D: contrary to the Hypotheſis.” 8. If it ſhould be alledged by any, that, notwithſtanding this De- monſtration, the Quantities and Ratios mentioned in the Lemma may differ at laſt, 1 that Difference be leſs than any given or aſſign- able Difference: ſuch would do well to conſider, that ſuch a way of reaſoning, being admitted, would overturn ſome of the fineſt Demon- ſtrations of the moſt accurate Geometricians among the Ancients them- ſelves, ſuch as Euclid and Archimedes, whoſe Works have undergone the ſtricteſt Scrutiny and Examination of the beſt Geometricians, ſince their Time to this very Day. For Example, How does Archimedes demonſtrate that a Circle is equal to a rectangular Triangle, having it's © Baſe equal to the Circumference, and it's perpendicular Altitude equal 2 | to the Radius of the Circle *? He does it by ſhewing that a Circle can J | neither be greater nor leſs than ſuch a Triangle. But how does he prove this? By ſhewing that the Circle is neither greater nor leſs than the Triangle by any given Space, Again, Euclid demonſtrates 3 that Circles are to each other as the Squares of their Diameters, by ſhewing that the Square of the Diameter of the one Circle is to the Square of the Diameter of the other, neither as the firſt Circle is to a Space greater, nor yet to a Space leſs than the other Circle. But let us ſee what he means by a Space greater or leſs than the other Circle. Why, he means a Space differing from it by a given or aſſignable Difference; i. e. ac- cording to the Lemma premiſed to that Propoſition, ſuch a Difference, as, repeated a certain Number of Times, may exceed that Circle, as appears to any one that reads that Prop. and Lemma: which Lemma is the Foundation of the Method of Exhauſtions, made uſe of by Euclid and Archimedes in theſe and many other Propoſitions. 9. Now if any one ſhould object, that, notwithſtanding of what Archimedes and Euclid have demonſtrated in theſe Propoſitions, the Circle may be greater or leſs than the rectangular Triangle; and the Ratio of the Squares of the Diameters may be greater or leſs than the s Lem. 1. of his Principles. . 2 See his Kuxas MiTenoG. 3 See Prop. 2. B. 12. of his Elements. 4 I | Ratio ak — gm hs. i a. a. The Ruadrature of CUR VIS explained. Ratio of the Circles, although not by any given or aſſignable Difference; yet by a Difference leſs than any given Difference : is not this the very fame Objection as that raiſed againſt Sir J. Newton's Lemma? And therefore, if it be of no Weight againſt Euchd and Archimedes ; no more is it againſt our Author. But the Truth of the Matter is, that a2 Difference leſs than any Thing „ is the ſame Thing as no Difference at all: for repeat it as often as you pleaſe, it can never be equal to any finite Quantity: and therefore can bear no Ratio to it, by Def. 4. B. 5. Elements, conſequently it can be of no Importance to make the Thing greater or leſs ?. 10. Def. 1. The ultimate Ratio of Quantities is the Limit to which the Ratio of variable Quantities continually approaches during any finite Time, and juſt attains to at the End of the Time. { 11. Def. 2. The ultimate Ratio of evaneſcent Quantities is the Li- mit to which the Ratio of variable Quantities diminiſhing without Bound, continually approaches, to come nearer to it than by any given Difference ; but which it never goes beyond : yet no ſooner attains to, than the Quantities being diminiſhed infinitely, vaniſh. 12. Def. 3. The prime Ratio of Quantities is the Limit from which the Ratio of variable Quantities continually recedes ; during ſome finite Time, and with which it coincides at the very Beginning of the Time: and is only a different Way of conceiving the ſame Thing as is con- tain'd in Def. 1. 13. Def. 4. The prime Ratio of naſcent Quantities is the Ratio with which they ariſe; or it is the Limit from which their varying Ratio con- tinually recedes ; after the like Manner as the ultimate Ratio of evane- ſcent Quantities was defined a Limit, to which their varying Ratio con- tinually approaches in Def. 2. being only a different Way of conceiving the ſame Thing; although it cannot be ſo eaſily expreſs'd. The Whole is illuſtrated by what follows, 1 Archimedes tells us in the Preface to his Treatiſe of the Sphere and Cylinder, that he aſſumes this for a Principle or Poſtulate. "ET; N Tov &r10 wy Ypapppor, Y TOY Grows Irs Qartiur, r dien Frewwr, To fila Ta Id 2 cer drifð&kè ro, © vill you da illo ia i, quꝛalo iu ùrięi xi mails d vie. vd ee anna Aryopiruy. | And he mentions the ſame in the Preface to his Treatiſe. of Spiral Lines. Likewiſe in the Preface to his Quad. of the Parab. he tells us, that in order to demonſtrate that a Segment of the Parabola is 4 of a Triangle having the ſame Baſe and Altitude, he aſſumes this for a Foun- dation, that the Exceſs of unequal Spaces, whereby the greater exceeds the leſs, may be ſo often added to itſelf, as to exceed any propos'd finite Space. And he tells us, that Geometers before his Time had made uſe of the ſame Principle in demonſtrating, that Circles are to one another in the duplicate Ratio of their Diameters ; that Spheres are in the triplicate Ratio of their Diameters ; that a Pyramid is a third Part of a Priſm of the ſame Baſe and Altitude, &c. 14. Let 37 ) 38 The Quadrature of Curvns explained. 14. Let the two right Lines AB, AC given in Poſition, and meet- ing in the Point A be interſected by a third right Line DCB, alſo given in Poſition, in the Points C and B: and let another right Line DEF revolve about the given Point D as a Pole, always cutting AC, AB in the Points E and F; and ſuppoſe it to approach DCB continually, until it at length fall in with it : then the Ratio of DC to CB is t ultirnate Ratio of DE to EF. For the varying Ratio of DE to EF continually approaches to the given or invariable Ratio of DC to CB ; ſo as that before DEF fall in with DCB, the Ratio of DE to EF comes nearer to the Ratio of DC to CB than by any given Difference ; or, which is the ſame thing, than any other given or aſſignable Ratio does. For, produce DB unto G, ſo as that the Ratio of DC to CG be as near the Ratio of DC to CB, as that of * wa can ever — rive to, if poſſible: through G draw GH parallel to meeting * gh : in HH: let DEF by | revolving, come into the Poſition Def, ſo that / lye betwixt B and H: draw / pa- rallel to AC, meeting 8 DB in I. Then, from parallel Lines, De : ef :: DC: CI; but the Ratio of DC to W |. — CI is nearer to the | Ratio of DC to CB, than that of DC to is; wherefore alſo | the Ratio of De to ef is nearer to the Ratio of DC to CB than that of DC to CG is, contrary to the Hyp. Whence it appears that the Ratio of DC to CB is the Limit to which the __ Ratio of DE to EF continually approaches, and comes nearer to than by any given Difference, and no ſooner ar- rives at, than the Line DEF falls in with DCB; or, which is the * Art. 10. fame, ceaſes to move; therefore the Ratio of DC to CB, is * the ul- of this Ex- timate Ratio of the variable Lines DE, EF. plication. x 5 If the right Line DEF be ſuppos d to move back again from DCB, where it is ſuppos'd to begin to move; then the Ratio of DC * Art. 12. to CB is the * prime Ratio of DE to EF. e | ofthis Ex 16, Suppoſe the ſame Things as before: and let DEF, revolving E upon D as a Pole, move towards DCB until it coincide with it, and the "CAO © —_— ** e The Quadrature of CVRvAS explained. the Points E and F coincide with the Points C and B, then I ſay the Ratio of the variable Lines CE, BF is continually changing: the com- pounded Ratio of DC to DB and AC to AB, is a Limit to which their Ratio continually approaches, and comes nearer to than by any given Difference ; or than any other given Ratio doth ; and that before the Points E and F fall in with C and B; but no ſooner arrives at, than the variable Lines CE, BF being infinitely duniniſhed, vaniſh ; and the Points E and F fall in with C and B. | Through B draw BL parallel to AC meeting DF in L: then CE : BF *:: CE: BL BL: BF. Now CE: BL:: DC: DB, and © Def. 5 therefore conſtant or given; but BL: BF * is continually varying: Pop. 4. for when DLF comes into any other Poſition DF, if BL: BF :: B/: B.6.Elem. Bf, LF would be parallel to /f *, which it never can be; and it's evi- p dent alſo that B/: B/ is greater than BL: BF; wherefore CE: BF is B.6 continually varying and increaſing until DEF fall in with DCB. 25. If any given Ratio can be as near to the compounded Ratio of DC to DB and AC to AB as CE : BF comes to: let it be DC x AC: DB x AH a leſs Ratio than DC x AC: DB x AB, becauſe it has been ſhewn that CE: BF is always increaſing ; ſo that AH > AB. Afume the Point M any where in the Line AB, betwixt B and H, join CM: let DEF by revolving come into the Poſition Def parallel to CM. Then Ce: 135 2. . em. Bf :: Ce: Bl-þ+ Bl: Bf :: DC: DB AC: Au *; or Ce: Bf :: * Prop. 4. / DCx AC: DBR AM; where for Ce: Bf > DC x AC: DB x AH, B.. Hlem. (becauſe DC x AC: DB x AM DC x AC: DB x AH.) contrary to the Hyp. 3*. CE: BF never goes beyond DC x AC: DB x AB; for it has been demonſtrated that Ce: Bf:: DC x AC: DB AM SDC Xx AC: DB AB: But it approaches nearer to it than by any given Difference; and that, while CE and BF are yet of a finite Magnitude: therefore DC x AC: DB X AB is the Limit at which CE : BF no ſooner arrives, than CE and BF being infinitely diminiſhed vaniſh ; juſt while the Points E, F coincide with the Points C. B. So that DC x AC: DB x AB is the ultimate Ratio of the continually diminiſh- ing and at laſt evaneſcent Lines CE, BF . * Art. 11. If the right Line DEF be ſuppos d to turn back again from this Ex- 17. pc6, where it's Motion begins, then the ſame Ratio, viz, DC x AC: plication. DB x AB, is the prime Ratio of CE and BF confidered as naſcent *, » art 13. 18. If any one ſhould object that there can be no ultimate Ratio of this Ex- of contin By this way of Notation I mean a Ratio compounded of the two Ratios of CE to BL and of BL to BF, the ſame as that of CE x BL to BL x BF. | By this Notation I mean the Ratio of BL to BF. | before y diminiſhing and at laſt evaneſcent Quantities : becauſe *"*** 40 The Quadrature of Cu xvxs explained. before they vaniſh it is not the laſt ; and after they vaniſh, they have no Ratio. The Anſwer is, that the ultimate Ratio is neither the Ratio of them before they vaniſh; nor after they vaniſh ; but the Ratio wherewith they vaniſh, or the Limit to which their varying Ratio no ſooner arrives, than they vaniſh. If there was any Thing in this Ob- jection, it would infer that when a falling Body is ſtopt in it's Motion, Velocity before it was ſtopt it has no laſt or ultimate Velocity: for is not the laſt, and the Velocity, after it is ſtopt, is none at all. But every one may ſee, that by the laſt Velocity is meant neither the one nor the other of theſe ; but the Velocity it has at that very Inſtant it ſtops, which. it does not arrive at before it is ſtopt ; and no ſooner ar- rives at but it is ſtopt. For ſince it moves with a continually acce- lerated Velocity, it has a different Velocity for every different Inſtant of Time, and therefore at that Inſtant it ſtops, it has acquired a Ve- locity different from the Velocity it had at any other Inſtant of Time. And the ſame is the Caſe of the ultimate Ratio of evaneſcent Quanti- ties, whoſe Ratio is continually varying ; it is that Ratio they have that very Inſtant they vaniſh : for, ſince they are ſuppoſed to have a different Ratio for every different Inſtant of Time, they muſt have a certain determinate Ratio, with which they vaniſh, otherwiſe th never can vaniſh, contrary to the Hypotheſis : and that is the ultimate Ratio of the evaneſcent Quantities. * Art. 16. of this Ex- plication. . of the Quadra- ture of Curves. Art. 5. of this Ex- plication. 19. It ſignifies nothing to ſay ultimate Quantities cannot be aſſigned, in regard Quantity is diviſible without End : for it is not the Quanti- ties themſelves that are hereby determined, but only their Ratio: which is capable of being determined, as has been already ſhewn “. 20. As theſe are the only Objections that can be: rais'd againſt this Doctrine of prime and ultimate Ratios, the noble Author was too clear ſighted not to perceive them : and accordingly he has fully anſwered them elſewhere , to ſo great Satisfaction to every intelligent and un- prejudiced Reader, that the great Duſt which has been raiſed of late about the Whole of this Doctrine, muſt be owing to Weakneſs, or ſome worſe Principle. | 21. Having endeavoured to clear up the Meaning of prime and ul- timate Ratios, and remove the Objections that are brought againſt that Doctrine: that I may clearly demonſtrate the fundamental Propoſi- tion concerning the Proportions of Fluxions, laid down by our Au- thor, and formerly mentioned *, I ſhall premiſe ſome Lemmas re- ſpecting Motion. See Schol. to Sect. 1. of his Principles; and Lem. 2. B. 2. L EMMA R . Ä -— ST VV” mg” R8R = "The, Quadrature of 'Curyns explained. L EMMA l. 22. If the right Line AC be deſcribed by the Motion of the Point B, with a Velocity continually accelerated, the Spaces deſcribed in equal Times will {till increaſe, as the Time from the Beginning of it's Motion increaſes, For if the Motion = B 3 0 was equable or uniform, the X — rr Spaces deſcrib'd in equal Times would be equal, therefore when it is continually accelerated, the Spaces deſcrib'd in equal Times muſt till increaſe, the further the Time is from the Beginning of the Motion. LEMMA II 23. If the Motion of the Point B be continually retarded, the Spaces deſcribed in equal Times will till diminiſh. This is alſo evident by the ſame way of reaſoning. LEMMA III. 24. If the Point B move with a Motion continually accelerated or continually retarded, it's Velocity at any one Inſtant of Time is dif- ferent from what it is at any other Inſtant of Time. This is evident from the very Notion of a Motion continually ac- celerated, or continually retarded : for ſuch Motion cannot continue the fame for any the leaſt Time. X LEMMA IV. 25. If the Motion of the Point B be continually accelerated or con- tinually retarded, in ſuch manner, as that the variable Velocity receive finite Additions or Diminutions, only in finite Times; and the Velo- cities at any two Inſtants of Time be taken, during which the Point B moves, it muſt paſs through all poſſible intermediate Degrees of Ve- locity lying betwixt the two Velocities aſſumed, in the Space of Time lying betwixt the two Inſtants aſſumed. For let V denote any inter- mediate Velocity whatſoever, through which, if poſſible, the variable Velocity mentioned in this Lemma doth not paſs ; let M and N de- note two Velocities, one on each Side of V, and the nigheſt to it which the variable Velocity mentioned in the Lemma paſles through, then the Difference betwixt M and N: is a finite Difference, (ſince there is a Velocity V aſſigned betwixt them) conſequently the variable Velocity receives a finite Addition or Diminution in a Time leſs than any aſſign- able, contrary to the Hyp, a &c, LEMMA "4 k * "Ti Ne Boadrature of Cunvis explained. | r ern 26. If the Space or Augment Bb be deſcribed by the Point B, moy- ing with a Velocity continually accelerated or continually retarded, in any Time, an uniform Velocity capable of deſcribing the Space Bb in the ſame Time will fall in betwixt the two extreme Velocities, where- with the Point B moves at the Beginning and End of the Space or Aug- ment Bb. For when the Motion is accelerated, the Velocity with which the Point B moves at the Beginning of the Space Bb, continued the fame, would not be capable of deſcribing ſo great a Space as Bb, in the ſame Time it is deſcribed by the accelerated Motion : but the Velocity with which B moves at the End of the Space Bi, would produce a greater Space than Bb in that Time. Therefore an equable ot uniform Velo- city capable of deſcribing Bb in the fame Time it is defcribed by an accelerated Velocity, muſt fall in betwixt the two, i. e. it muſt be greater than the firſt and leſs than the ſecond, ſince, when uniform Velocities or Motions are compared, and the Time given, the Velocitics are as the Spaces deſcribed directly. And it is evident, that” the like way of reaſoning may be applied to the Caſe of the continually retard- ed Motion: whence the Truth of the Lemma appears. LEMMA VI. a. ac 27. If the Point B move with an uniform Motion, along the Line AC, and the Point E move along the Line DF with a Motion conti- nually accelerated or continually retarded : and Bb and Ee be Spaces deſcribed or Augments generated by the Motion of thefe Points in the fame or equal Times : the leſs the Spaces or Augments Bb, Ee be taken, or, which is the fame thing, the lefs the Time be in which they are de- ſcribed, the nearer will the Ratio of Bb, Ee approach to the Ratio of the Velocities with which the Points B and E move at the Beginning of the Spaces or Augments : and by diminifhing Bb and Ee more and more, their Ratio ſhall come nearer to the Ratio of theſe Velocities than by any given or aſſignable Difference: and therefore the ultimate Ratio of theſe Spaces or Augments Bb, Ee as they vaniſh (which is the ſame with their prime Ratio, confidered as naſcent) will be the Ratio of the Velocities mentioned. The leſs the Space or Augment Ee is, and conſequently, the leſs the Time is, in which it is deſcribed, the nearer will the Velocities, with which the Point E moves at the Beginning and End of that Space, | approach . f ooo ocXACcC_uocoaa wÞ.Þ oo xr. co coacc.£am_m2mGCC L_©RTDL A == - 32 wy ©” Þ- @ 4 FFP tv —_ 0 0 © » ogy w CO The Quadrature of Curvss explained. approach to one another: and by diminiſhing the Augment Ee, or the Time continually, theſe a 3 10 & extreme Velocities may be ͤ 2 made to approach nearer to one another than by 5 | any given Difference, as — — E . appears by Lem. 3 ; there- | fore much more, the uniform Velocity capable of producing Ee in the ſame Time, will ſtill approach nearer and nearer to the Velocity with which the Point E moves at the Beginning of the Space Ee, the leſs that Space, or the Time wherein it is deſcribed, is: and by continually leflening that Space or Time, the uniform Velocity ſhall differ from the Velocity, with which the Point E moves at the Beginning of the Aug- ment Ee, by leſs than any given Difference ; fince the uniform Velo- city falls in betwixt the two extreme Velocities v. If then we put B * Art. 26. for the Velocity with which the Point B moves, which is ſuppoſed to of thi Ex- be conſtant or invariable ; E for the Velocity with which Point E moves 2 98 at the Beginning of the Space Ee; and V for an uniform Velocity ca- pable of deſcribing Ee in the ſame Time it is deſcribed by the variable Velocity E: then from the Nature of uniform Motion, Bb: Ee: : B: V; but the leſs that Ee is taken, the leſs is the Difference betwixt V and plication, ſmall, or ſuch as are defcrib'd in very ſmall and equal Times, it will! be nearly Bb : Ee:: B: E. Finally, fince by continually diminiſhing Ee and conſequently BS, V approaches nearer to E than by any given Difference, it follows that Bb : Ee approaches nearer to B: E than by any given Difference : and therefore the ultimate Ratio of the evane- ſcent Augments or Spaces Bb, Ee, or, which is the ſame, the prime Ratio of them conſidered as naſcent, is the Ratio of B to E. Which * Art. 7. Things were to be demonſtrated, | hae: wang 28. Theſe Things being demonſtrated, the Truth of the fundamental * Propofition laid down by our Author *, will eaſily appear. For, - Art. z. Fluxions being defined the YVelocities of the Motions or Increments by of the which the flowing Quantities are generated, ſuppoſe any one Quantity _ to flow uniformly, i. e. to have a conſtant or invariable Fluxion at all Curves. Times; and another Quantity of the ſame kind to flow according to any Law whatſoever, whether by an uniform Velocity, or a Velocity continually varying ; and that at any rate, whether continually acce- lerated, or continually retarded : and 10 there be taken * * | 2 theſe 44 * Prop. 1. B. 6. Elem. The. Quadrature of C VVG explained. theſe flowing Quantities, generated in the ſame or equal Terms: then in the preceding Fig. AB and DE repreſent the flowing Quantities, whatever the Nature of theſe Quantities be (juſt as all kind of Quanti- ties are repreſented by right Lines in the fifth Book of Euclid's Elem.) Bb, Ee the cotemporaneous Increments. And therefore it appears from the Nature of Motion, 15%, That if the Fluents flow with conſtant or invariable Fluxions, the Fluxions are to one another as the Increments nerated in any equal Times. And from what has been demonſtrat- ed, 2%, That if the one Fluent have a conſtant Fluxion, and the other a Fluxion continually varying, and their Augments be taken ſuch as are produced in very ſmall and equal Times, the Fluxions will be nearly as theſe Augments. 3%, In every Caſe accurately, the Fluxions are in the prime Ratio of the naſcent Increments, or the ultimate Ratio of the evaneſcent Decrements, according as the flowing ＋ are ſup- poſed to increaſe or decreaſe. And further, becauſe this is true when one of the Fluxions is given or conſtant, it follows that the ſame is true, when the Fluxions are all variable. For let P, Q denote any two homogeneous flowing Quantities, whoſe Fluxions are variable: A, any other Fluent of the ſame kind, which has an invariable Fluxion : let theſe Fluxions be denoted by the Symbols P, Q, A reſpeCtively ; and let p, 9, a denote the prime Ratios of their naſcent Augments. Then, by what has been demonſtrated P: A:: p: 4 and A: Q.: a: g, there- fore by Equality P: Q :: p: 9, as was to be ſhewn. 29. After laying down what we have been about demonſtrating, touching the Proportion of Fluxions, our Author proceeds to ſhew the Application thereof, in determining the Fluxions of curvilinear Areas, in this Art.; and the Fluxions of the Abſciſſes, Ordinates and Curve Lines in the two ſucceeding Art. Now, what is laid down in Art. 4. may be thus demonſtrated. Suppoſing the ſame Things as are there mentioned, I fay F.ABC * : F.ABDG : : BC: BD, when ABDG is a Parallelogram. To demonſtrate which, we need only ſhew. that the prime Ratio of the naſcent Increments, when the Areas are ſuppoſed to flow by increaſe ; or the ultimate Ratio of the evaneſcent Decrements, when they are ſuppoſed to flow by decreaſe, is the Ratio of BC to BD. Let BC, BD, conſtituting one right Line, move forward, and come into the new Poſition bc, d; ſo that the Increments generated in the tame Time be B4cC and BD. Then it is evident, that, when cb4 has got to any finite Diſtance from CBD, BC: B4dD > BC: BD; for by completing the Parallelogram BCE5, it is BC: BD :: BCE5 : BoD * S BC: BD: but I fay BC: BD may be nearer to BC: F. ABC denotes the Fluxion of the Area ABC: and ſo of other like Notations. BD 3 — wiwI wi) = SH Wears ww ics — „ „ Ot fad K „ . © 6 py _— m— —_— The Quadrature of Curves explained. BD, than by any given Difference; or than any given or aſſignable Ratio, by ſuppoſing ic to be ſtill! nearer and nearer to BC. For if poſ- ſible let BF: BD be as near to BC: | | ** BD as Bic C: BbdD can be: draw Fx nk E KN 45 — F. . arallel to AB meeting the Curve in 2 - wk 8 complete the Parallelograms BBF 2 0 i and BD. Then BGF: BD 3 BSC: BSD, but BS＋ F: BSD :: | | BF: BD, therefore BF: BD BGH C: U A Bi 70 TI BID, wherefore BOC: BSD is [> g nearer to BC: BD than BF : BD is G - > (fince B&xC : BBD can never be leſs than BC : BD, as was ſhewn) con- trary to the Hyp. Therefore BC : BD is the prime Ratio of the naſcent * Augments BC and BAD *. And it is the ſame thing to conſider * Art. 13m, Bbc, BbdD as Decrements, by ſuppoſing chd to return back again into thb Ex- the Poſition CBD : for then it is demonſtrated the ſame way that BC: * BD is the ultimate Ratio of BC and BAD when vaniſhing. But the Fluxions of the Areas ABC and ABDG are to one another in the prime Ratio of the naſcent Increments , or ultimate Ratio of the * Art. 3. evaneſcent Decrements BC, BbdD *. Therefore F. ABC: F.ABDG :: * BC: BD, when ABDG is a Parallelogram, as was to be ſhewn, Quadra- 30. But further, although both Fluents were curvilinear Areas, I ſay due of their Fluxions ſhall be as the Ordinates, when the Ordinates do move Art 2 with the ſame uniform Motion along the Abſciſs AB. For let ABL of this Ex- be another curvilinear Area deſcribed by the Ordinate BL, then, by * what has been demonſtrated, F. ABC: F. ABD G:: BC: BD and F. ABD G: F. ABL :: BD: BL, therefore by Equality, F. ABC: F. ABL:: BC: BL. 31. Although this Method of inveſtigating and demonſtrating the Proportions of the Fluxions of plain Figures, be laid down by Sir J. Newton in this Place; and, as I have ſhewn, is both certain and geo- metrical: yet ſuch perhaps may be the Scruples of ſome; and ſuch the Obſtinacy of others, againſt the whole of this Doctrine of prime and ultimate Ratios of naſcent and evaneſcent Quantities, that I hope it will not be unacceptable to the Reader, if I ſhall ſhew that the Proportions of the Fluxions of plain Figures (and conſequently by Analogy of all other flowing Quantities) may be demonſtrated in another manner, upon the moſt indiſputable Principles of Geometry, without introducing either infinitely little Quantities, as has been done, without ſufficient Caution, 3 2 by 16 De Nadrature of Cunvas explained. by ſome z ws avoided by Sir J Neuron in this Treatiſe ; or et naſcent and evaneſcent Quantities, with their prime and ultimate tios, an which Foundation he has built this Doctrine. | Let ABC and ABDG — wa be a curvilinear Area and Þ E © Parallelogram deſcribed as Co Rn 4 | 1 | ing ſuppos d as formerly * fx 24 | 09 and farther let ABCF be Quadra- tn is another Paralle de- _ A — — . 9ſcribd at the — and Art. N by a given Ordinate equal of ihis Px to AF, fo that all the plication. | diet three Areas ABC, ABDG 1 34 and DE 4 deſcribed — fo — — uniform Motion 9 N. A. of their reſpective Ordi- nates, lying always in the ſame right Line: then I ſay F. ABC: F.ABDG :: BC: BD. For fince the Ordinates of the Parallelograms ABDG, ABCF are given or invariable right Lines, and theſe Lines are ſuppoſed to move | uniformly along the common Abſciſs AB, the Fluxions or Velocities of flowing of the Parallelograms muſt be conſtant and invariable, for 5 the Spaces generated in equal Terms are evidently equal in the reſpec- tive Parallelograms; and the Fluxions or Velocities of flowin 2 Te * Art. 28, Caſe are as the Increments generated in any the ſame times denten. Compoſition as the whole Areas generated, hat is FABCP: F.ABDG . pe * Wy ABDG ::) BC: BD *, BC being conſidered as the invariable B.6.Elem. Ordinate of ABCF. But the Velocity with which the curvilinear Area ABC is deſcribed muſt be leſs than the Velocity with which the Pa- rallelogram ABCF is deſcribed at any Time before the Ordinate BC arrive to it's preſent Poſition, as when it is in G, becauſe G x Bp : again, the Velocity with which ABC flows,. at any time after BC has Ty it's preſent Poſition, as when it is in Ec, is greater than the * fox ity with which ABCF flows, ſince bc > bf : — theſe things ar —ç let He and fe be as near to BC as you pleaſe, becauſe the heat AC and right Line FC interſe& each other in one Point only: but the Velocity with which ABC flows, is continually or inceſſantly vary- ing, ſince it's Ordinate BC never continues of the ſame Length, for any the leaſt Time. Wherefore, betwixt the two Inſtants of Time that the Ordinate BC has the Poſitions for and bc, the variable gs» 4 | — WI e 1 nr re = e 7 We * a 18 — er TV i \/4 #® © ,, x WY © 7 The Quadrature of Cunvns. evplaintd. 47 with which the Area ABC flows muſt paſs through all poſſible inter- mediate Degrees ®, therefore alſo through that Degree of Velocity · Art. 2g. with which ABCF flows, which is invariable and intermediate: Bi of this Ea. it has been demonſtrated, that this cannot happen at any Time, when 3 the Ordinate BC is out of the preſent Poſition, therefore it muſt when the Ordinate BC obtains the preſent Poſition : But the Flurions of flowing Quantities are the Velocities with which they flow or increaſe *: therefore, when the Ordinate BC comes to the-preſent » Art. 2. Poſition, the Fluxions of the curvilinear and rectilinear Areas ABC and of of ABCF are equal: but, as was already ſhewn, F.ABCF : F. BDO cape eel BC: BD, therefore F. ABC: F.ABDG :: Ord. BC: BD. +1] Curves. 32. From this, we may conclude as formerly *, that when the flow- v Art, 30. ing Areas are both Curvilinear, their Fluxions are to one another & 8 lic 4 the Ordinates. * 33. And thus we have demonſtrated that the Fluxions of all Figures, deſcribed by the equable Motion of their Ordinates 1 lying in the ſame variable right Line, and moving along a common Abſciſs, are every where as their relpective Ordinates; and that with- out introduci Tide infinitely ſmall Quantities, or yet the Doctrine of prime and ol imate Ratios of naſcent and evaneſcent Quantities. And 4 ney the Foundation which the Doctrine of Fluxions is built, would ſtill remain unſhaken, although the Doctrine of prime and ul- timate Ratios could be ſhewn to — uncertain and precarious; which yet t all Sir 7. Newton's Adverſaries ſhall never be able to do. From what has been faid the * — Corollaries may be deduced. 34. Cor. 1. The Fluxion of any ure, generated by the uni- form Motion of it's Ordinate ode 22 Abiciſs, is every where propor- tional to the Ordinate. 5. Cor. 2. Although our Author has hitherto mentioned no Motion of the Ordinates but what is uniform and the ſame along the Abſciſſes, or rather common Abſcits : yet the Motion of the Ordinates along the Abſciſſes of different Curves, i. e. the Fluxions of the Abſciſſes of dif. ferent Curves, may be ſuppoſed different ; yea and the Fluxion of the Abſciſs of the ſame Curve may be ſuppoſed variable, by being accele- rated or retarded, and that according to any Law. But I fay in every Caſe, the Fluxions of the Areas of plain Figures, deſcribed by the pa- rallel Motion of their Ordinates along the Abſciſſes, are always to one another as the Rectangles contained under the Ordinates and the Fluxions of the teſpective Abſciſſes. For by what has been demonſtrated, when the Fluxions of the Abſciſſes are uniform and the ſame in all, the Flux- ions of the Areas are as the Ordinates directly: but if the Fluxion 14 the Abſciſs be increaſed or diminiſhed in any Proportion, the Velocity with which the Area increaſes, i. e. it's Fluxion muſt be increaſed or De Nuadrature of CunvxG explained. — diminiſhed in the ſame Proportion; fince the Space that would be de- ſeribed with that Velocity in any given Time, would be increaſed or diminiſhed in that Proportion: and it is no matter, whether the Fluxion of the Abſciſs of the ſame Figure continue invariably the ſame, or vary according to any Law, by being accelerated or retarded; for it is only the preſent Fluxion of the Area: therefore it follows that the Fluxions of any plain Figures, are always as the Rectangles contained under the Ordinates of theſe Figures, and the Fluxions of their Abſciſſes, or right Lines proportional to and expounding the Fluxions of theſe Abſciſſes. 36. The fame thing may be more ſhortly demonſtrated by means of Symbols thus, as is done by our Author elſewhere *, in the foregoing * Art, 25 of tins Ex- plication. Figure, let the Ordinates BC, BD move at right Angles along the Ab- iis AB; let AG or BD be equal to Unity; the common Abſciſs AB == x, the curvilinear Area ABC applied to BC , or ABC XI , then is ABDG = (xx BD =xxX 1 =) x. Now putting & and æ to ſtand for the Fluxions of x and z reſpeively, by what has been demonſtrated, z : x :: BC: BD= 1 : for it is all one whether the Fluxion of AB, i. e. * continue invariably the ſame, or vary accord- ing to any Law: therefore by reducing the Analogy to an Equation 2z— EX TC —EXE) x x BC, i. e. F. ABC F. AB x BC, vis. the Flux- ion of the Abſciſs multiplied into the Ordinate, i. e. in other Words, if the Ordinate of any plain Figure, continuing the ſame in Length, that it is in any given Place, were ſuppoſed: to be carried perpendicular] along the Abſciſs with an uniform Velocity, equal to the Ar ves. wi which the Abſciſs increaſes at that particular Place, the ſuperficial Space thereby produced in any given Time, would be equal to the ſuperficial Space produced or deſcribed in that ſame Time, by the Velocity with ay the Area flows or increaſes at that Place, continuing uniformly the ſame. | 4 37. But ſuppoſe the Ordinates not to move perpendicularly along their Abſciſſes; but in any given Angle, provided it be the ſame in all, the Fluxions of the flowing Areas will {till be proportional to the Rectangles contained under the Ordinates and Fluxions of the Abſciſſes. For in the Demonſtration of our Author's fourth Art. the Ordinates BC, BD may make any given Angle with the Abſciſs AB *: thereſore it's eaſy to apply the Reaſoning at Art. 3 5. to this Caſe. » See his Method of Fluxions and infinite Series, publiſhed by J. Colſon, pag, 23, 238. Cor. . FLY CIS / ( ab. 5 ID wy 0H, . The Ryadrature of Curves explained. 38. Cor. 3. If two or more plain Figures begin to be deſcribed at the ſame Time, by their Ordinates moving along their Abſciſſes in a given Angle, and fo be deſcribed in the ſame Time; or, which comes to the ſame purpoſe, if be deſcribed in equal Times, and the Rectangles contain'd under the Ordinates and the Fluxions of the Ab- ſciſſes or right Lines proportional to and expounding the Fluxions of the Abſciſſes, be equal in theſe Figures, at every the ſame or cor- reſpondent Inſtant of their flowing or increaſing, throughout the whole Time of their Deſcription, the Areas deſcribed ſhall be equal. For theſe Rectangles are proportional to the Fluxions of the Areas “, and * Art. 35. therefore the Fluxions of the Areas are always equal to each other, 39 37. i. e. the Velocities with which the Figures are deſcribed, are every — where equal to each other: and therefore the Areas produced or de- ſcribed with theſe Velocities in the ſame or equal Times muſt be equal, from the Nature of Motion. FFF It would be the ſame thing, if the Ordinates of the Curves and the Fluxions of their Abſciſſes were reciprocally proportional: for then would the Rectangles mentioned be equal *. Prop. 14. 39. Cor. 4. Converſely if the flowing Areas be every where equal B. 6. Elem. to each other, the Rectangles contain'd under the Ordinates and the Fluxions of the Abſciſſes, ſhall be every where equal to each other: for, if not, the Fluxions of the plain Figures, 7. e. the Velocities with which they flow or increaſe, would not be equal: and if the Velocities of Deſcription were not every where equal, the Areas deſcribed could not be every where equal. | | 40. Schol. It may be here remarked, with reſpe& to variable or flowing Quantities of any kind whatſoever, that, if they be at all times equal to each other, the Velocities of their Mutation, i. e. the Fluxions with which they flow at any the ſame Time, muſt alſo be equal: converſely, if the Velocities of Mutation or Fluxions of any variable Quantities be always equal, and moreover theſe Quantities ariſe toze- ther, when they flow by increaſe; or vaniſh together, when they flow by decreaſe, the variable Quantities ſhall always be equal at the ſame Time. For theſe variable or flowing Quantities may be repreſented and expounded by the flowing plain Figures; and their Velocities of Mutation, by the Fluxions of theſe Figures, mentioned in Cor. 3. and 4. Wherefore, ſince the Fluxions of theſe equal Figures are always equal; and converſely the Fluxions being equal, the flowing plain Figures are equal; hence the variable Quantities of any kind, and their Velocities of Mutation or Fluxions muſt be equal. Which is likewile evident in itſelf from the Nature of the Thing, H More- 50 ments Bb and Ec The Quadrature of CN explained. | Moreover any variable or flowing Quantities, which are in a. given or conſtant Ratio, muſt have their Fluxions in the fame given Ratio: and converſely variable Quantities, whoſe Fluxions are always in a given Ratio; if ſo be the Quantities ariſe together, or vaniſh together as be- fore, are in the ſame given Ratio themſelves: as is nen evident from what hath been ſaid. | 41. Suppoſing the ſame Things as are ſuppoſed by our Author, *] day the prime Ratio of the naſcent Augments Bb, Ec and Ce, viz. of * the Abſcib, Ordinate and Curve-line AB, BC and AC; or the ultimate Ratio of theſe Lines conſidered as evaneſcent, is that of the Sides of the Triangle CET; or of the Triangle VBC ſimilar to it, viz. CE, ET and CT, or VB, BC and VC the Subtangent, Ordinate and Tangent. For chrough any Point as m of the Subtenſe Ce, lying within the Curve, draw the Ordinate, G parallel to BC or bc, meeting CE in e; and produce it, till it meet the Tangent VC in M: then Ce: ex > (Ce: M :) CE: ET; but Ce: ex (Ce: em'::) CE: Ec: therefore Ce: e falls in betwixt CE: ET and CE: Ec; conſequently Ce: ex is nearer to CE: ET than CE: Ec is 1 W hereforethenear- | K er bc is to BC, the | 5 ＋ nearer doth the Ra- | 2 G tio of the Incre- approach to the Ratio of CE to ET or of VB to BC, but fo as that, when the Points B and 3 are at any finite | : Diſtance from one LT 6 $4 Wi I e eee another CE: Ec > CE: Er. | Yet the Ratio of theſe Augments may be nearer to the Ratio of CE to ET, than any other aſſignable Ratio. For let the Point L be aſſumed in AB produced beyond V, fach as that LB : BC be a Limit, if poſſible, nearer than which the variable Ratio of the Augments Bb, Ec, can never approach to CE: ET or VB: BC: where LB muſt be greater than VB, becauſe CE: Ec > (CE: ET ::) VB: BC, while Fa 41 | there is yet any Diftance betwixt BC and bc. Draw LC and produce it, beyond C; then muſt it enter within the Curve, otherwiſe a right Line would fall in betwixt the Carve and it's Tangent, which is im- yy. - rough any Point n, of the Line LC produced, lying within © 1 3 n % p * @ Q @ d 5 1e 7 5 > 4 * «i I" K . _ b — : J * . vr * „ ö . * 9 ; ? a * SE OTE Ion dr 4 he = . {A god GA r . 5 a ae, TG , : « SIS TOS OT. A PO I ER I IO WA 3 e oor rs 24s Crn VAs 1 ' Fwy . D 5/4 — "a 2 Faw © 1 Win | VEL FRA, 7 40% > vu - FR A, 0 ge EIENCS. ba dt I 4 - WY; r © 4 4 D e FLY ys 1 3 d 2 8 . 2 d - k \ 95 F q , by ; L A. e I» 4 p 8 2 N PRES 3 1 8 = IE", % 25 8 Sy q WEL ' 8 N 15 "Hye r 5 2 L De en ee . een "WE "Wa" £5 4 '% 4 7 „ Wen Far 2 22 > - — — and . bd Yy. * ' 4 £2 , " * ; Bats £ 2 — / 7 2 8 1 * "_ : = > Fs 1 4 154. ff * b a R Ps” 42 - - 5 WE PP nrg r rere 2 8 4 PP S 8 * * . td * 7. \ - > 14 2 75 » 0 y 1 SYS hag” 3 „. ; 1 2 4 * * 5 9 * on . N * * »% - "_—_ | « * 2 * 1 0 D 1 FE 5 82 1 W * 7 G 8 IIS: of; ag 54 Fe. 3 v2 N * 2 * — 2 WA AS», vn p* Mr < OE. — ren r an OS ; | pg N The Quadrature of CURVE S explained. 1 the Radius's of their circular Baſes. Now 'tis evident that when BZ is of any finite Length, BC: BdͤD > (BEC: B4dD: :) * BC: BD; Rg. therefore J have ſuppoſed BF greater than BC. But ſince BEG S Cyl. BSF, we have BBxC: BSD S (BGF: BSD : :) * BF!: BD? ; Therefore BBH C: BSD is leſs than BF? : BD; but it was ſhewn that it cannot be leſs than BC7: BD, wherefore it comes nearer to BC: BD! than BF?: BD? is, contrary to the Hyp, Therefore the prime Ratio of the naſcent Augments B&C and BAD is that of BC? to. 47 11. BD?! *; or, which is the ſame, that of the Circles having BC and BD and 13. for their Radius'ss Wherefore the Fluxions of the Solid ABC and Cy- „& lication. linder ABDG are in the ſame Ratio . Whence it's evident (by con- L Art. 28. ſidering what was faid in Art. 30.) that the Fluxions of all ſolid Figures, * generated after this manner, the Fluxions of whoſe Abſciſſes ate every 4 where equal and uniform, are to one another in every Place, as the Circles, by whoſe Motion they are generated, are in that Place; or as the Squares of their Radius's: as our Author affirms, 44. Cor. Hence it will follow, that, although the Motion of the generating Circles along the Abſciſſes be different and unequal, z. e. although the Fluxions of the Abſciſſes be different in different Solids ; and even variable in the ſame Solid, yet univerſally, the Fluxions of Solids, generated by drawing the Circles along the Abſciſſes, are in a compounded Ratio of the generating Circles, and the Fluxions of the Abſciſſes: as may eaſily be demonſtrated, by conſidering what was ſaid at Art, 35. * | of this 45. The other Thing to be demonſtrated in this Art. is, that the _—_ Fluxion of the Curve-ſurface is as the Periphery of the generating Circle, having it's Radius BC, and the Fluxion of the Curve-line AC conjunct- ly; or in a compounded Ratio of theſe two. To illuſtrate and de- monſtrate which: ſuppoſe the Curve-line AC, and the Circumference of the Circle of which BC is Radius, to be both extended into right Lines; ſo as the latter, being made to move at right Angles along the former, may thereby generate or deſcribe a plain Figure having the right Line equal to AC for it's Abſciſs; and the other, equal to the Perimeter of the Circle, whoſe Radius is BC, for it's Ordinate: and ſuppoſe the Velocity with which the rectilinear Abſciſs flows, or it's Fluxion, to be always the ſame with the Velocity or Fluxion of the Curve-line AC; and it's Ordinate in every Place, equal to the Circum- ference of the Circle, having for it's Radius BC, in the correſponding Place; then it's evident that the Curve and plain Surfaces, generated in the ame time, will be equal; and that therefore their Fluxions, at the The Quadraturs of Curves explained. * Art. 40. the ſame Time, will be equal . But the Fluxion of the plain Surface Ex- js always as the Ordinate and Fluxion of the Abſciſs jointly &: therefore alſo the Fluxion of the Curve-ſurface is as the Perimeter of the Circle, of this Ex- having the Line BC for Radius, and the Fluxion of the Curve-line AC jointly. The plain meaning of which is this: that if there were any Number of Curve-ſurfaces, N Solids, generated after the man- ner that has been explained, the Velocities with which the ſeveral Curve-ſurfaces flow, are ſuch, that the ſuperficial Spaces, which would be deſcribed by theſe Velocities in any given time, ſuppoſing theſe Ve- locities to continue invariable for that time, are as the Rectangles con- tain'd under the Perimeters of the ſeveral generating Circles, and right Lines > Ahn to the Fluxions of the ſeveral Curve-lines, correſpond- ing to AC. ; 46. Schol. Upon theſe Properties of the Fluxions of Solids; and their Carve-ſurfaces, are founded two principal Branches of the Doctrine of Fluxions: viz. the Cubature of Solids; or the Method of finding their ſolid Content: and the Plaining of Curve-ſurfaces ; or the Method of finding the ſuperficial Content : both belonging to the inverſe Method of Fluxions. | * | 47. Theſe three Articles contain fome more Examples, for the fur. ther Illuſtration of the Method of prime and ultimate Ratio's, both in \. | and Curve Lines, \ conſidered as flow- * ing Quantities. The * Example contain- — ä e oe > „ - he." ain as to ne — 2 * bs further Illuſtra- — * tion: that in Art. q. A | TI 1 is coincident with ney the one formerly . B e adduced and ex- * plained Art. 16. * From which Ex- ample our Author's 10'* Art. is deduced by way of Corollary: and is the only thing remaining in theſe Articles, that needs any further Explica- tion. In order to which, let the right Line PB, revolving about the given Point P as a Pole, cut the Curves GB, GE given in poſition, in the Points B and E, and let the moveable right Lines AB and AE touch theſe Curves always in the moveable Points of Interſection B and E, I I fay Aa %@ * O< e I. «7 wt a+» AA ha = De Qgadrature of CVnvRsSs explained. 85 I fay the Fluxion of the Curve GB is to the Fluxion of the Curve GE in a Ratio compounded of AB : AE and PB: PE; that is F. GB: F.GE:: AB x PB: AE x PE. For let the revolving Line PB move forward from it's Place PB into the new Place Pb, while at the ſame time the Tangent-lines AB and AE, till touching the Curves come into the new Poſition Ah and Ae, ſo as þ and e be the new Points of Contact and Interſection: and let the Line Ph cut AB and AE in the moveable Points H and I. Now ſuppoſe the Line P6 to return to it's former Place PB; by which means the Lines Ab and Ae will return to their former Places AB and AE, and the Points A and a, B, & and H; E, eand1I at laſt coincide. In - which Caſe, the ultimate Ratio of the evaneſcent Augments Bb and BH; Ee and El is a Ratio of Equality, as may eaſily be collected from what was demonſtrated the latter Part of Art. 41. *: for the ultimate * of this Ratio of the Arch and Tangent, when vaniſhing, is a Ratio of Equality, Explica- whatever be the Angle, which the ſecant Line makes with the Curve or Tangent. Wherefore B45: Ee:: BH: EI; when vaniſhing : but in that Caſe, BH : EI :: AB IPB: AExXPE®*, Therefore alſo the ul- * Art. 16. timate Ratio of the evaneſcent Augments B& and Ee is AB Xx PB: 6 AE x PE. Therefore F. GB: F. GE :: AB X PB: AEX PE. As E Art 26. was to be proved. | of this Ex- 48. Moreover if the Line PB, inſtead of revolving about the im- "=== movable Point P as a Pole, ſhould perpetually touch a Curve given in Poſition in the movable Point P, the Ratio of the Fluxions of the Curves GB and GE, will be the fame as before: fince the ultimate Ratio of the evaneſcent Augments will be the tame as before ; as will eaſily ap- pear by the ſame way of reaſoning : ſo that there is not any neceſſity to inſiſt particularly upon it. 49. This Article teaches us how to find the Fluxion of any Power Art. rr. of a flowing Quantity, i. e. to find the Relation betwixt the Fluxion 0 the of any Power of a flowing Quantity, and the Fluxion of the Root or _— Quantity itſelf *. | Curves. Let x repreſent any flowing Quantity, ſuppoſed to flow uniformly, et. if and xn any Power of that Quantity: let x be increaſed by the ſmall plication. Augment o, ſo that in any ſhort Space of Time, it may become x o: then in the ſame Time, x will become x TO“; for into whatever new State or Value x comes, x” muſt come into ſuch a new State or Value, as that ſtill, in every Place, or at every Inſtant of Time, it mult be the nth Power of the Value of x in that Place or at that Inſtant of Time. Now by Sir Iſaac Newton's binomial Theorem, x T of = 4 » * . / - . 5 __—_ a 4 4 o 6 — het - 29447 1 Sa * 56 De Quadrature of CV RxvES explained. e Sis fore, in the ſame Time that x becomes x 4- o, x* becomes x*--zx*—19 + 1X — x*—200 + n X n 2 * 0 + &c. From which two new Values ſubtracting the former Values x and x” reſpeCtively, the Remainders, viz. o and nx"—o + n x — x"=200 + AXN "= * Di- + Cc. are the ſynchronal Increments: and if both be di- vided by o, we ſhall have 1 to mx" + 7 x Ny 2— 2 * 3 X x*=20 & *x%—300 ＋ &c. for the Ratio of the Augments generated in any the ſame Time, and that whether the Increment © be big or ſmall. Now the Fluxions of Quantities are to one another in the prime Ratio of the naſcent or ultimate Ratio of the diminiſhing, * Art. 28. and at length evaneſcent Augments “. But by diminiſhing o more of this Ex- plication. and more continually, the prime Ratio of the naſcent, or ultimate Ratio of the evaneſcent Augments will be that of 1 to nx"-!, be- Ho cauſe the varying Ratio of 1 to nx"! + nx —-x*=20 , D x 2 2 200 + c. (which is the Ratio of the Augments) by diminiſhing o continually; before o vaniſh, comes nearer to the Ratio of 1 to *. than by any given Difference. Therefore F. x: F. x:: 1: nx*%". A 24 2 0 \ 1) 7 G7 1 50. The ſame thing might be demon- ſtrated with greater Form, according to the ſame Principles: but waving that at this Time, I ſhall ſhew how it may be demonſtrated by a geometrical Conſtruc- tion: in which the Quantities x and x” are repreſented or expounded by plain Fi- gures. | Suppoſe then the plain Figure ABC and Rectangle AG to be deſcribed in the ſame Time, by the uniform Motion of the right Line BCG along the common Abſciſs AB. Let AB = x, and BG (= AD) be the Ordinate of the flowing Rectangle AG, and let the given right Line AD or BG : then is AG= (1 Xxx H) x, Let BC be the Ordinate of the Figure ABC, whoſe Area is ſuppoſed equal to x”. Now let the plain Figures ABC and AG be ſuppoſed firſt to flow by Increaſe, I ſo wa... be r Ms... b F % A” a. _ at A... — . The Ruadrature of Curves explained. 57 ſo that after a ſhort Time, the right Line BCG may come into the new Place bcg: again let them be ſuppoſed to flow backwards, by decreaſ- ing, BCG come into the new Place Gxy. Then let beg and ey return again to BCG, and fall in with it at the ſame Time, fo that Bb and BB may be always equal. And, by what was formerly de- monſtrated , the ultimate Ratio of the evaneſcent Augments Bic and * Art. 29. is BC: BG; and, for the ſame Reaſon, the ultimate Ratio of the — evaneſcent Decrements BH. C and B85, is alſo BC: BG. Therefore, when theſe Increments and Decrements are juſt vaniſhing, we have BbcC : :: BBH C: BSG: but becauſe BB is always equal to BS, and re alſo when vaniſhing, hence BHG = B, therefore alſo the ultimate Ratio of BicC and BBxC is a Ratio of Equality, Put then BS = O BB, and conſequently Ab—x+0o and AB—=x—0: and from the Property of the Figure ABC, we have Abc = x TO = (by Sir 1/aac Newton's bmomial Theorem) x"+nx%—!0+n x Deo An X = X — + &c; and Ar -O -i x Oo X 7 + &c.. From the former ſubtract ABC — x”; and ſubtract the latter from ABC x”, and you'll have nx"=l0+7 X DT i½οοm X — X J + Sc. the Augment BbcC; and nx"='o — 2 X x%=200+1 X — * Y-: — Cc. = the Decrement BBC. Now F. ABC: F.AG :: BbcC : BhbgG when vaniſhing *: and likewiſe F.ABC: F. AG:: BB«C: BSD, when va- * Art. 28. niſhing. Wherefore, I fay, it follows that F. ABC: F. AG:: A o: 1 o:: nx%1:; 1. For if F. ABC: F. AG < nx*%'o : o, then by diminiſhing of o, io +1 Xx —x%*00-þn x — X —x*—0" - Gc. : o might become at length leſs than nx"—'o : o, which can never be: again, if F. ABC: F.AG > nx"='0: o, then nx"-!10—n x S -:0. + 2X — * IEF x io — &c.: o, by diminiſhing o continually, might at length become greater than nx"—'o: o, which can never be either, becauſe in the Series nx%—'0— 1 X —-**=?00-nx—= x rio — Cc. (where the Terms are the ſame with thoſe of the 11 Power of the Reſidual x — ©, wanting the firſt Term, with the Signs changed) by diminiſh- ing o ſufficiently, each Term will be greater than the ſucceeding one, and therefore the Terms being affected with the Signs 4+ and — alter- nately, the whole Series, by diminiſhing o in inſinitum, can never be- I come 55 Art. 29. The Quadrature of Cunvns explained. come greater than nx%-'0. Wherefore, ſince it is neither F. ABC: F. AG xnx*-'0: o; nor yet F. ABC: F. AG > nx*—o : o, it follows | that F. ABC: F. AG:: nx—"0:0::nxw=": 1. 51. Schol. In this Demonſtration, the Index n of x* is poſitive; but ſuppoſing it to be negative, we can prove, by the ſame way of Reaſon- ing, that the Fluxion of x is to the Fluxion of & as 1 to — Xi. Which ſhews that when x flows by Increaſe, x” flows by Increaſe alſo; but x flows by Decreaſe, it's Velocity or Fluxion being negative. But farther, it appears, 1, That, when x flows uniformly by Increaſe, having it's Fluxion conſtant and invariable, if 7: be poſitive and greater than Unity, the Velocity with which x” increaſes, or it's Fluxion is con- tinually accelerated: for ſuch a Number being ſubſtituted for , in the Expreſſion nx, makes the Index z—1 poſitive: and therefore it is evident the greater x is. the greater will the Quantity x*- be, which expreſſes the Fluxion of x", when 1 expreſſes the Fluxion of x. 29. If n be poſitive, but leſs than Unity, the Index »—1, in the Expreſſion of the Fluxion a , becomes negative in the Numerator; or poſitive in the Denominator, therefore the greater x is, the leſs will the Value of the Expreſſion nx i be: wherefore the Velocity with which x” in- creaſes, or it's Fluxion is continually lefſened or retarded. 3%. If u be negative, whatever be it's Value, the Expreſſion — nx i or — — having the Index of x negative in the Numerator; or poſitive in the Denominator, ſhews the greater x is, the leſs will the Value of the ne- gative Expreſſion — ux i or — = be. Therefore it appears from thence, that when x flows uniformly by Increaſe, the Quantity x— flows backward or by Decreaſe, with a Velocity continually leſſened or retarded, 52. Cor. By what has been demonſtrated F.AG : F.ABC :: 1: nx; likewiſe F. AG: F. ABC:: BG: BC *; hence BG: BC :: 1: nxu- i: but BG =1, therefore BC — ne i Whence it follows, that jon. if the Abſciſs, perpendicular Ordinate and Area of a Curve be called x, , and 2 reſpectively; and æ then y ne u: and converſely if Y nx expreſs the Relation betwixt the Abſciſs and Ordinate, then 2 expreſſes the Relation betwixt the Area applied to Unity and the Abſciſs: or, if the Value of x be taken from the Equation y =nx"-", and inſert in place of it, in the Equation æ x", it becomes z = 8 — * = i for the Relation betwixt the Area applied to Unity and the Ordi- nate: or laſtly by taking the Value of x", we will have * x for the Relation betwixt the Area of the Curve, and the Rectangle contain- I 5 ed W. * is The Quadrature of Curves explained. 59 ed under the Abſciſs and Ordinate. And this is the Foundation of the Quadrature of fimple Curyes ?. From this it appears that if navi y expreſs the Relation of the Abſciſs and Ordinate of a Curve; where @ denotes any given Quantity: then ax"==2 ſhall exhibit the Relation betwixt the Abſciſs and Area ap- plied to Unity: and converſely, if gr, then y==nax"-?, For ſince nx"-" and navi are in a given or conſtant Ratio, v:z. that of 1 to a: it follows, by Compoſition, that the Areas of the Curves which have & i and nax”-" for their Ordinates, and x for their com- mon Abſciſs, will be in the ſame given Ratio: but when y==nx*", it is 2 x (as has been ſhewn) therefore when y = navi, it will be zar. Wherefore alſo converſely, if z=ax", it follows that y= navi. The fame Thing may be eaſily demonſtrated by the Help * Of this of Art. 38, 39, 40 *. —_— 53. After our Author has obſerved in this Article, that by like ways Art. 12. of reaſoning, in the Method of prime and ultimate Ratio's, the ef the Fluxions of right and curve Lines in all Caſes whatſoever; the Fluxions ture of likewiſe of Surfaces, Angles and other Quantities may be inveſtigated ; Curves. he deſires it may be remarked, that the manner of proceeding in the Doctrine of Fluxions, by means of prime and ultimate Ratio's, is per- fectly agreeable to the Analyſis of the Ancients: and that there is not any neceſſity of introducing into Geometry, by the Method of Fluxions, Figures infinitely ſmall. This is carefully to be obſerved : and the Truth of it will appear by conſidering, that the Method which he takes in the ſeveral Examples, contained in this Introduction to his Treatiſe of the Quadrature of Curves, is this: firſt he determines the Ratio's of the cotemporaneous or ſynchronal Increments, by ſuch general Expreſſions, as are capable of being applied to theſe Increments whether taken great or ſmall; or whether generated in a greater or leſs, only the ſame Time: then, from theſe general Expreſſions, he finds in particular, what muſt be the prime or ultimate Ratio's of theſe Increments, conſidered as juſt ariſing or vaniſhing. Which is certainly a juſt and geometrical wa of proceeding; to give an univerſal Solution of a Problem, applicable to all poſſible Caſes, and then apply it to one particular determinate Caſe. This Method of Solution is the moſt elegant; as well as moſt uſeful of any other, and what is commonly made uſe of both in the Analyſis of the Ancients; and that of the Moderns. Thus in deter- 1 See our Author's admirable Treatiſe, Analyſis by Equations of an infinite Number of Terms, Rule firſt, and it's Demonſtration towards the End of that Treatiſe : likewiſe his Method of Fluxions and infinite Series, Prob. . * I 2 mining 60 The Quadrature of Cunvzs explained. | mining the Fluxion of x”, or rather the Ratio of the Fluxion of x* to the Fluxion of x: he firſt ſhews that the Ratio of their ſynchronal In- crements, whether theſe Increments be great or ſmall, or, which is the ſame, whatever the Time be, wherein theſe cotemporaneous Incre- ments are generated, is thatof nx»! + ZZxn-20 4 © F2%,9-100 + Cc. to 1, Which expreſſes the Ratio of theſe Increments at all Times and in every Caſe: and therefore alſo, when theſe Increments are juſt ariſing ; or juſt vaniſhing, as well as at any other Time: and then he ſhews that the foreſaid Ratio, in that particular Caſe, when the Increments are juſt ariſing or vaniſhing, becomes the ſame with Ax"=! : 15. The ſame way of reaſoning may be obſerved, in the other Examples propoſed in this Introduction: thus in the Example at Art. q, in which our Author propoſes to determine the Ratio of the Fluxions of the flowing Lines AB and AE: he firſt demonſtrates, that their ſyn. chronal Increments Bb and Ee, in all Caſes, whether theſe Increments be great or ſmall, are as Ab x PB to Ae x PE: then he ſhews what that variable Ratio, ſtill expreſſed by finite Lines, will be, in that par- ticular Caſe, when the Augments are juſt ariſing; or juſt vaniſhing: and he demonſtrates that it will, in that Cafe, become the fame with the Ratio of AB x PB to AE x PE, and therefore he infers, in conſe- quence of what was formerly demonſtrated, that the ſame is the Ratio of the Fluxions of the Lines AB and AE. And thus it is evident, by this way of proceeding, as our Author obſerves, that the Analyſis is carried on, and the Fluxions of flowing or variable Quantities are ex- unded by finite Quantities: and although, as he obſerves, the Ana- yſis may be performed by means of Figures either finite or infinitely ſmall, which are ſuppoſed, fimilar to the evaneſcent Figures; or yet by Figures, which, by the Method of Indiviſibles, are accounted as in- finitely ſmall; yet ſince the Method of Indiviſibles is reckoned not ſo agreeable to the Principles of ſtrict Geometry; and our Ideas of infinite- : Hence it appears that there is no Fallacy or Paralogiſm in this Method of Reaſoning, made uſe of by the ingenious Inventer, for finding the Fluxion of &, as is alledged by the Author of the Analy/t. See Art. 13, 14, 15, 16 of that Piece. For it is moſt evident that 2 2 ＋ o &: 1, expreſſes the Ratio of the ſynchronal Increments of x” and x in every Caſe : but when theſe Increments are pain to he diminiſhed continually until they vaniſh, there muſt be ſome laſt Ratio, with which they vaniſh, otherwiſe they never could vaniſh: and fince it is demonſtrated that the ultimate Ratio with which the Augments vaniſh can neither be leſs nor greater than the Ratio of * to 1, it follows that the ultimate Ra- tio of the evaneſcent Increments of x and x, and conſequently the Ratio of their Fluxions is | that of & to 1. For a further Anſwer, ſee a Letter to the Author of the Anal by Pbilalethes Cantabrigienſu, pag. 54+ k y 2 . ey SF : 2 ” 9 of G5 2 . . A at's 8 32 9 R 23 = 4a - SE K LL By. 5 IDES n Vanda a 2 . r : ' VG r © NOM mM 6 £85.15” a Fa. <©5H Con — 4 3 4, 7 — * * . 3 * ak . * "WF; at 85 e 2 , "—_— gh N mY * Nd Eo 1 - nm * . ” T my & oe „ AO. IS cao0234 +4, ama gdb DE nmOOorpm ku. Tr . = AS % * * — * 5 1 "il » rd 0 , — > 1 5 9 5 4 7 . — - 6” * . A 1 3 U * = S SD PF, * an 4 q 1 * 1 1 1 2 * z «4 3 _— 4 \ % TI 1 * of * 4 . 3 - = 7 p 4 *%* * <4 P BY " - *48 = "6s e 8 & * 0 „ hy 3 "YI * = 1 . « A . - l 2 3 % - i pe x ” * - my * 0 *. * W 7 2 * * F * * FS 2 4 7 * \ - — * ** „ N 99 T2 Þ . = by pb * 28 . 22 bo - The Quadrature of Curxv'ss explained. 60. Now as the Velocity with which any ical Quantity flows, by augmenting or diminiſhing, includes Time and Space in the Notion of it, whereby it may be meaſured and determined; it will be per to obſerve, that, when the variable or flowing Magnitude is a 2 it is ſuppoſed to be generated or produced by the continual Mo- tion or Flowing of a Point: and then, the Fluxion of the flowing Line, at any Time, is moſt naturally meaſured by the Length which the moving Point would run over in a certain given Time, ſuppoſing the Velocity, with which the Point moves, continued invariably the ſame, during the given Time, that it was at that Inſtant, or at that particular Place, when and where it's Fluxion is ſpoke of and enquired after. If the flowing 52 be a Superficies, it is ſuppoſed to be generated or produced by the continual Motion of a Line: and the Fluxion of the flowing ſuperficial Space at any Place, or Inſtant of Time, during the flowing, is naturally meaſured by the Quantity of ſuperficial Space, that would be deſcribed in a certain given Time, by ſuppoſing the ge- nerating Line, continuing invariably the fame, to be carried along an- other Line, with an uniform Motion, and thereby deſcribe a Space, which increaſes equably, and juſt as faſt through the whole given Time, as the flowing ſuperficial Space increaſed at that Place or Inſtant of Time. Laſtly, if the flowing Quantity be a Solid, it is ſuppoſed to be generated by the Motion of a plain Figure: and the Fluxion of the Solid at any Time or Place, is naturally meaſured by the Quantity of ſolid Space that would be deſcribed by the generating plain Figure con- tinuing invariably the ſame, and drawn along a right Line, with an uniform Motion, ſo as to make the ſolid Space increaſe, through the whole given Time, juſt as faſt as the flowing Quantity increaſes at that particular Place or Inſtant of Time: when it's Fluxion is ſought." 61. This is the moſt eaſy and natural Way to conceive the Meaſure of the Fluxion of flowing Magnitudes: but notwithſtanding of this, the Fluxion of any Magnitude may be repreſented or expounded, by a Magnitude of any kind, provided it be always proportional to that Fluxion : thus the Fluxions of Superficies's and Solids; as well as of Lines, may be expounded by Lines, which are the moſt ſimple kind of Magnitudes, provided theſe Lines, by which they are expounded and repreſented, be always proportional to theſe Fluxions: juſt as Euclid. has repreſented all Magnitudes by right Lines, in the 55 Book of the Elements. Thus, as has been ſhewn already, when a plain Figure is generated by the uniform Motion of the Ordinate along the Abſciſs, the Fluxion of the plain Figure is always as the Ordinate: and there- fore may. be expounded thereby. | 62. When 64 Art. 31. of this Ex- plication. Art. 47. of this Ex- plication. - * Art. 46. of this Ex- plication, The Quadrature of CV RvxS explained. 62. When Fluxion or Velocity of flowing is applied to other Quan- tities, it muſt be underſtood in an analogical Senſe, it being applied to them, becauſe of the Similitude there is betwixt the increaſing and diminiſhing of any Quantity whatſoever, which is ſuppoſed to be con- tinually paſſing from one State or Value to another; and the increaſing and diminiſhing of flowing or variable Magnitudes. Now. the whole Buſineſs of the different Orders of Fluxions may be explained in the following manner. 63. Let ABC be a plain Figure generated or deſcribed, by the uni- form or equable Motion of it's Ordinate BC, along the Baſe or Abſciſs AB, always at right Angles to each other: and putting the Abſciſs AB=x, let the Nature of the Figure be ſuch, that it's Area ABC==ax"; where à is any given or invariable Quantity; and x any indefinite Index to x: and let AG==1. Now ſuppoſe the Fluxion of the Abſciſs AB, i. e. the Velocity with which the Point B moves, which is equable and uniform, to be ſuch that in a given Time, ſuppoſe a Second of Time, it moves through a Space equal to AG= 1, Draw AH perpendicular to AG and equal to it: and let the Rectangle HB be deſcribed, at the fame Time with ABC, by it's invariable Ordinate equal to AH or Unity. Then tis evident, that in the ſame Time, the Point B moves through a linear Space equal to AG or 1, the Ordinate of the Rectangle BH will move through or deſcribe a ſuperficial Space equal to AG x AH or the Square of 1: that is, it will do it in a Second of Time. But, if the Velocity with which the Area ABC increaſes, were continued invariably the ſame, for a ſecond of Time, or which is the ſame *, if the Ordinate BC continued invariably the ſame, moving along AB, the ſuperficial Space deſcribed by BC, in a Second of Time, would be BCXAG—BCxX 1. Therefore AG x AH or a ſuperficial Unite, be- ing the Meaſure of the Fluxion of the Rectangle HB ABN 1; BC * AGS BC 1 will be the Meaſure of the Fluxion of the flowing Area ABC. But ſince ABC = ax”, it appears from what was for- merly demonſtrated , that the Ordinate BC n, therefore the ſaperficial Space, which is the Meaſure of the Fluxion of ABC, may likewiſe be expreſſed by nax x 1 or naaxx. 64. Now the Expreſſion nax , when AB — x flows, denotes a variable or flowing Quantity, unleſs when : and a Quantity flowing by Increaſe, when i is poſitive *, and x increaſes: there- fore the Fluxion of ABC is itſelf a variable or flowing Quantity, z. e. the Celerity with which ABC flows, is continually flowing or varying as to it's Value, which, in this Caſe, grows (till greater and greater. Sup- * poſe The Nuadrature of Curnvus explained. 65 poſe then another plain Figure ABD, whoſe Area ABD= BC x AG or na N = nax""", G | | deſcribed by it's Ordinate BD, moving along AB, at the fame Time ABC is deſcribed by it's Ordinate BC: then the Fluxi- on of ABC is meaſured and expounded by ABD, which continually flows by Increaſe. But the Fluxion of ABD, or the Velocity with which it en- creaſes, is ſuch, as would be ſufficient to deſcribe a ſuper- ficial Space equal to BD * AG = BD x 1, in the ſame Time the Point B deſcribes the linear Unite ; the Ordinate of the Rectangle HB deſcribes the ſuperficial Unite; or the Or- dinate BC of the Figure ABC, continuing invariably the ſame, would deſcribe the ſuperficial Space BC x 1 ; viz. in a Se- cond of Time; as eaſily ap- 3 from what has been ſaid þ 3 with reſpect to the flowing of © = ABC: therefore BD x 12 BD XAG is the Meaſure of the Velocity, with which ABD flows or increaſes: conſequently BD AG or BD X 1, is proportional to, and therefore may be the e of the Celerity, with which the Fluxion of ABC changes, 7. e. it is pro- portional to, or the Exponent of the Fluxion of the Fluxion of ABC, which is called the ſecond Fluxion of ABC. But, ſince the Area ABD na x 1 or nax"”"', the Ordinate BD N Xx n—1 X ax"* or n- x ax"—* *, therefore - X ax"—* x 1 =n*—n XK is - Art. 52. the Exponent of the ſecond Fluxion of ABC. | | of this Ex- wb 31 — L lication, 67 Again if 1 2, then 1 <7 x ax*—=TJ 2 X A 24 ib conſtant or invariable Quantity: in which Caſe the Fluxion of ABD, or the ſecond Fluxion of ABC is conſtant and invariable, ſince it's Ex- ponent is ſo; conſequently ABC will have. no third Fluxion, which | | _— denotes +. * 66 Art. 51. of this Ex- plication, Art. c2. of this Ba. plication, variable: and conſequently no Fluxion of any Order, in chat Caſe, can The Quadrature of Curxyss explained. denotes the Celerity of the Mutation of the cond Fluxion. This will happen, when ABC is a right angled Triangle, having it's Baſe equal to 2ax: and then ABD is a Rectangle, having the Abſciſs AB. for it's variable Altitude, and the invariable Line denoted by £# for it's Baſe. But if u be not equal to 2, the Expteſſion * M is vatiable : and if 1 2, it is always increaſing, as x increaſes *. In which Cafe; ſuppoſe ABE another plain Figure, having it's Area equal to BI x AG or BDI H - N N or -N : and let it be de- ſcribed by the Motion of it's Ordinate BE along the common Ab. ſciſs AB, at the ſame Time ABC and ABD are deſcribed by their Ocdinates BC and BD. Then it appears, by the faite way of rea- ſoning as before, that the Area ABE is always proportional to, or the Exponent of the ſecond Fluxion of ABC: and farther, that the Velocity with which the flowing Area ABE increaſes is meaſured by BE Xx AG=BE x I: Therefore the Velocity with which ABE in- creaſes of it's Fluxion will be always tional to, or the Exponent of the ſecond Pluxion of ABD; and of the third Fluxion ef ABC; or, which is the ſame thing BE x AG — BE x 1 is always proportional to and conſequently the Exponent of the third Fluxion of ABC, More- over ſince ABE = E x ax”, the Ordinate BE IN <2 a * —n3.— 3n* ＋ 21x ax*-3, therefore 13 — 3 T 27 ax 3 is likewife the Exponent of the firſt Fluxion of ABE; of the ſe- cond Fluxion of ABD; and of the third Fluxion of the f ſt flowing Area ABC. Now if, in the Expreſſion ax”, which is the Value of the Area ABC, we ſuppoſe a=; and =, we ſhall have BE= 105 (n- gu zA x ) x*: which ſhews that ABE is the Apollonian or common Parabola, having AG = 1 for the Parameter of the Axis. Again if ABF be a plain Figure, deſcribed by the uniform Motion of it's Ordinate BF along AB, together with the other plain Figures; fuch, that it's Area ABF = BE x AG = BE x 1 =n3— 35 ＋ 2 „ a x 1, then the Area ABFa, will alſo be proportional to, or the Exponent of the firſt Fluxion of ABE; of the ſecond Fluxion of ABD; and of the third Fluxion of the firſt flowing Area ABC. 66. And thus we might proceed to ſubſequent Orders of Fluxions; which will run on infinitely, if the indefinite Index of x; viz. M being poſitive, be not an Integer; or whatever it be, if negative: becauſe in every ſuch Caſe, the Exponent of x in the Expreſſion of any of the flowing Areas, in the Series of plain Figures, can never become no- thing: and therefore the Exponents of their Fluxions will always be — ever * - * 1 4 i 1 ; * » > 0 * 7 a N 1 a 2 1 N abr * by „ 8 *. ** * * [ # N Payee N A 2 5 g. S .es 2 8 082 — » >. 2000 „% wad os A. = 5 me reer r . ws. R . below the Ordinate, for according to the Order of the Progteſſion, that 3 The Quadrature of Cunvas explained. 67 over he a conſtant Quantity. Only it may be obſerved, that when the Exponent of the Power of x, in the Expreſſion of the Area of any Curve, in the Progreſſion of Curvilinear Areas, becomes negative, the Fluxion of that Area muſt likewiſe be negative: for that Area di- - Art. 5m. miniſhes as & increaſes; and therefore is poſited below the Ordinate; of this Bux conſequently it is ſo far from having any Velocity of Increaſe, thay F 10h. upon the contrary, it deereafes, or diminiſhes, and ſo the Velocity with which it diminiſhes, is call'd a negative Fluxion. Thus, if we ſyppoſe a = r and n== +, as formerly, and conſequently ABE a Parabola, then the Curve aF will be of the hyperbolical Kind, approaching the Aſſymptotes AB and AH continually: and it's Area ABE a being equal to BE IN XK, it's Ordinate BF = or 2 and · An. c2 therefare it's Fluxion will be expounded by a Curvilinear Area lying of this Ex- plication. Area muſt be ro XI Or ze, the Fluxion of which muſt be negative. 67. The Progreſſion of Curvilinear Areas, as Exponents of the various Orders of Fluxions and Fluents, may be ſuppoſed to be carried on like- wiſe the contrary Way. Thus if ABA be a Curvilinear Figure havin it's Ordinate BA equal to the Area ABC a plied to AG or nity : and i there were another Curve, whoſe Ordinate 1s equal to ABA applied to AG, and ſo on; that would conſtitute a Series of Curves running the contrary way, ſuch, that every Curvilinear Area in this Series will be a Fluent, of which the immediately preceding is the Exponent of it's Fluxion; that two Steps before it, is the Exponent of it's ſecond Fluxion, Fr. And thus the Progreſſion of Curves, reckoning from ABC, may be ſuppoſed to be carried on both ways in inf. ; 68. And thus, I have endeavoured to explain and illuſtrate the whole Affair of the different Orders of Fluxions, ſo as to ſhew and de- monſtrate that they have a real Foundation in Nature; and may be iſti conceived. ; 69. Since the Cuzvilinear Areas ABD; ABE, ABFa, Cr. are equal to the Ordinates BC, BD, RE, &c, multiplied by AG or 1 reſpective- ly: therefore the Ordinates BC, BD, BE, Sc. will alſo be the Expo- nents of the firſt, ſecond, third, c. Fluxions of the flowing Area ABC. 70. But it may be proper here to caution the Beginner, that he don't underſtand me, as if I ſaid, that the Areas ABD, ABE, ABFa, &c- or the Ordinates BC, BD, BE, Ce. were really the firſt, ſecond, third, &c. Fluxions of the flowing Area ABC. They are only Expo- nents of theſe Fluxions, as being AW to. them : for- ans, | K 2 w 68 The Quadrature of Curves explained. which denotes the Celerity of Flowing or Mutation, is one thing; and a Surface, or a Line is another: although a Surface or a Line may be proportional to, and therefore the Exponent of a Celerity. So that ox, the true Meaning of the thing is this: Suppoſe BCFDE, and Scde, to pro be two different Poſitions of the Ordinates of the Curves already men- Vel tioned, generated and related to one another as above, the firſt Fluxion beg of ABC is to the firſt Fluxion of Abc as ABD to Abd; or as BC to o- the the ſecond, third, &c. Fluxions of ABC are to the ſecond, third, &:, MF the Fluxions of Abc reſpectively, as ABE, ABFa, &c. to Abe, Abſa, GS-o. are or as BD, BE, Sc. to bd, be, Sc. reſpectively. me 71. From what has been ſaid, the Truth of what our Author ad- — | | bt ns WH 9 Ip | the vances in Art. 14. will eafily appear, viz. If z. Z. z. Z. 2.2. &c. | the Anz — 22. Nat — 22. Yaz — 23. Naz — 22. Yaz — 22, Cc. Ti Eh = —.— : — 2 . == Sc. be Series's of Quantities, fo |: = related to each other, that every ſubſequent Quantity in any of theſe Se- ms rics's, is the Fluxion of the preceding Quantity; and converſely every pre- 0% ceding Quantity being conſidered as Fluent have the ſucceeding one it's ſuc Fluxion: I ſay if this be the Relation of the Quantities to each other, thgen 1, every preceding Quantity is as the Area of a Curvilinear Figure, whereof FY ho the immediately following is the rectangular Ordinate and ⁊ the Abſciſs. FR ag Art. 29, For it has been demonſtrated * that the Fluxion of any plain Figure, tat 31, 63: as ABC (ſee the preceding Fig.) is ſuch, that when it is compared I &@ plication. With the Fluxion of AB the Abſciſs; or rather with the Fluxion of the or Rectangle AB x 1, there ariſes this Analogy F. AB x 1: F. ABC:: 1: FR tr BC. Wherefore if the Fluxion of AB x 1 or AB be denoted by x th (which is ſuppoſed) then the Fluxion of ABC will be denoted by BC, 1 1. e. the rectangular Ordinate. Conſequently if any Quantity ſuch as . . . ne Naz — 22 expreſs the Area of a Curve, whoſe Abſciſs is z:. and 2== 1, {h as is ſuppoſed, then F. Va 22 i. e. Vaz—2z = the rectangular al Ordinate, as our Author affirms : and ſo of others. ye 8 K & F. im. ſe Containing Motes en Art. 1 5 24. - | Explication of Prop. I. | CC 72. N Difficulty there is in our Author's Demonſtration of this as Propoſition, arifes from aſſuming ox, oy and oz. for the ſyn- 7 chronal Increments of the flowing Quantities x, y and g: and neglect- ing theſe Terms, which involve © at the Reſult of the Operation. 8 ce 73+ De Quadrature of CVUxvxRS explained. 73. If the ſynchronal Increments of æ, y and z, be ſuppoſed to have already acquired any finite Magnitude, it muſt be acknowledged that ox, oy and oa cannot expreſs ſuch Increments : becauſe ox, oy and oz are proportional to x, y and 2, viz. to the Fluxions of x, y and æ; or the Velocities with which they flow or increaſe, juſt as the Increments are beginning to ariſe: but finite Increments are not always proportional to the Fluxions of the flowing Quantities, fince the Velocities, with which theſe finite Increments are generated, may vary during the Time they are generated: therefore ox, oy, oa cannot properly expreſs finite Incre- 69 ments. But it was ſhewn formerly“, that the leſs the ſynchronal In- · Art 27. crements of flowing Quantities are; or the leſs the Time be, in which of this Ex- they are produced, the nearer will their Ratio approach to the Ratio of "=" the Fluxions: ſo as that, by diminiſhing theſe Increments; or the Time in which they are generated, continually more and more, the Ratio of the Increments, ſhall approach nearer to the Ratio of the Fluxions, than by any given Difference; and conſequently their ulti- mate Ratio be the ſame with the Ratio of the Fluxions. Therefore by 0x, 09,02 we muſt underſtand ſynehronal Increments in this Senſe, wiz. ſuch as are indefinitely ſmall, called by our Author Moments. or fyrchro- nal momentaneous Intrements. And ſurely, if there can be no Increments = how ſmall ſoever aſſigned, but there are Increments yet leſs and lefs, aſſign as ſmall ones as you pleaſe; he may be allowed to aſſume a No- tation denoting naſcent or evaneſcent Increments, which are indefinitely ſmall, although we have no poſitive adequate Idea of ſuch Increments, or Moments. For a negative Idea is ſufficient, in many Cales, for de- termining the Properties and Proportions of Quantities. What other than a negative Idea have I of the infinite Decimal ,6666, Sc. Yet I can demonſtrate that it is exactly equal to . What other than a negative and very imperfect Idea has any one of 4/2? Yet it can be ſhewn that it is a mean Proportional betwixt 1 and 2. And although all Surds are negative Ideas; and involve fome kind of Infinity in them, yet they admit of arithmetical Operations: and can be compared with one another, Thus though none can tell how much 4/2 and V8 are ſeparately : yet we know that their Product is 4; and that the latter is double the former. After the ſame manner, though the momentaneous Increments of flowing Quantities cannot be diſtinctly and adequately conceived by the Mind, yet. we may make uſe of ſome kind of Nota- tion, by way of Symbol or Repreſentation of them : and their Relations may be determined and expreſs'd by finite Quantities, which are di- ſtinctly conceived: and this is all this Propoſition requires or propoſes. 74. The * The Quadratur of Cunvis explained. 74. The other thing I mentioned, as having any Difficulty, in the Demonſtration given by Sir Jaac of this iſt Prop. is the neglecting the Terms of the Equation, at the Reſult of the Operation, whercin the Quantity o is found. The Reaſon of it is aſſigned briefly by himſelf: viz. becauſe the Quantity o is infinitely diminiſhed and vaniſhes, For the N expreſſing the Relation of the ſynchronal Augments, will only then expreſs the Relation of the Fluxions, when the Quantity o is juſt vaniſhing: but when that Quantity vaniſhes, all the Terms where- in it is found muſt vaniſh likewiſe; which is the Reaſon of their being neglected and thrown out. jo | 75. But in regard this 3 is a fundamental one and con- tains. the Subſtance of the direct Method of Fluxions, I ſhall ſhew how it may be demonſtrated at fuller Length. In order to which, it will be neceflary to ſhew. how the Fluxion of any fimple Quantity, which contains the Product of any Number of flowing Quantities may be in- veſtigated, | 1 vo | * Caſe 1. When the flowing Quantity is the Rectangle or Product of two flowing Quantities, as xy; ſuppoſe & and y to denote the Fluxions of the Factors æ add y, Tay PE. ur Put æ ; and let x, y, 4 denote any ſynchronal Increments, great or ſmall of the flowing Quantities x, y, 2: fo that in the fame Time that x becomes x + x, y and æ may become yy. and'z + z reſpectively. Then becauſe the Equation xy —=z exhibits the, Relation, of the flowing Quantities x, y and 2 indifferently, or at all times, into whatever new Values they, paſs, therefore z ＋ 3=x+5 x37 4r5= * Art. 28. of this Ex- "plication, ; Axt. 40. of this Ex- Plication. xy + ay + xy b ay: take away = from the one. Side of the Equation and xy equal to it, from the other; and; we have π - xy an Equation defining the Relation of the Increments generated in the ſame Time, great or ſmall; and by, dividing. by &, it is Eg + . Now ſuppoſe the Increments x, y and:z to. be diminiſhed in- finitely, ſo as to be juſt vaniſhing, and in their ultimate Ratio: in that Caſe the Ratio of the evaneſcent Increments, is the ſame with the . EE IL EO y 8 . Ratio of the Fluxions &, 7, e. === and = : Which Values be- ing inſerted the Equation becomes === y 41x, where the Term 2. being infinitely diminiſhed, vaniſhes and, goes ont; conſequently we have = == &, for the Relation of the Fluxions, i. e. 2=xy-þ-xp. But fince z= xy, F. xy , therefore F.xy == xy + xy. Fl 77.Cor. . 8 , _ | Wa 2 - 2 2 . * oo % As. * * * 2 3 TS * * b L 2 y 1 41 P — 2 Y 8 = * * * q * f * . RN g 4 — 5 * * = 4 2 l 8 R 7 r N 1 ys A p n By © K 4 4 N o L CL ed. wo a>; Do oft id. My. 5 22.0 = ; e a. . . N : | 22 5 * 1 „ - - * N 2 p * . Y ; ACS» | _ - WL WEIL C4 . 2 of R - a 8 f N 8 phe. >. at r . 2 , * * 3 . * by Se . « l . == b . : 3 2 2 . ; * cafily appears from Art.4o ® . 8 of this * i b. Gor 2. F.xx=2xx: for it is xx - xX=2xx. Explica- 59. Caſe 2. When the Floent is the Produck of three flowing Quin- tities as yx, I fay F.xyz = xyz + xz + H. For put y n, then = #3, therefore F. u = F. uE * = +45 *: but fincs Y n, * Art. 40. 5 + xj =i. Wherefote if we infert xy + xy for i and reſtore xy „ch He- for u, in the Equation F.xyz u + us, we have Fama R 1 REY . 80. Cor. F.axz == ax2 ＋ ax. F.ax* = 24x. For the flowing Quaiitities x, a; and &“, ax*, are in a given or conſtant Ratio, viz. that of 1 to 4: therefore their Fluxions are in the ſame given Ratio“: * Art. 40. but F.xz==x2 + xa, and F.x* xx, therefore F.axz = @x% + ax plication. and F.ax* = 2axx. Likewiſe F.x3== zx: for it is xxx + xxx + XXX = xa. PISS 4 bs 4 1 Dee a 48 0 . 81. Gale 3. When the Fluent is the Product of any Number of flowing Quantities, the 3 Rule for finding the Fluxion in every Caſe is this. Multiply Fluent by the Index of the Power of every flowing Quantity it involves; and in every Multiplication, change the Root of the Power into it's Fluxion: and the Aggregate of all theſe Products is the Fluxion required. | 2 6 For when the Fluent conſiſts of four Dimenſions, and as many dif- ferent Letters, as æyau, it's Fluxion is 0 + ay2u þ -T xyzu : which is demonſtrated by reducing it to three, vzz. by putting f=xy,, i. e. making ? a 4 Proportional to 1, x and y: and then xyzu= tu, therefore F. xy = F./zu == , F x 2u4- ,. But ſince = , Art 59. t + xj *, therefore by Reſtitution, F.xyzu = xy Þ xy x 21 ay oc . | plication. XT = #y2u + xj2u + xy2u + xy2zu. And thus by means of AE. 0s a 3 of four Dimenſions, you may diſcover the Fluxion of r Fluent of five Dimenſions, and ſo on: in ſuch manner, that he who rightly conſiders the Proceſs, will ſee this Truth demonſtrated, viz. That the Fluxion of any Fluent, which is itſelf a fimple Quantity, i. e. a Quantity of one Term, be it's Dimenſions what they will, is found by ſetting down the Quantity as often as there are flowing prime Diviſors nn = plication. 72 The Quadrature of Curves explained. in it, only changing each ptime Diviſor into it's Fluxion, in order, one after another, and taking the Sum of all, after the Manner of the preceding Examples. Which is coincident with the general Rule pre- ſcribed under this 34 Caſe: or at leaſt manifeſtly included in it. However that the general Rule may be more clearly demonſtrated, take the Quantity x"y"z? with indefinite Exponents: and I ſay F.x"y"z? == nxx"—"y" 2d 4 mx" +px"y"22=", For let it be 1:x":r 11, 1:9":5:!, and 1:7: f , or r, and #==7: then F. x = F. rot Fit +rit ri: but becauſe x" t; = and 2 =, therefore F. , α ==# : F.,y" = =5: and F. 27 = pe . Subſtitute for 7, 5 and 7 theſe their Values, and re- place x", y” and 2, for r, 5 and t in the Equation F.x"y"z*== r5/ Tru Tra, and we have F. = nx""xy"28 F Ny + * Art. 40. of this Ex- plication. px _ 'z, according to the general Rule, f the Fluent have any conſtant or invariable Quantity among it's Factors, as 4a*xy32+ the general Rule ſtill holds. For F. a * = a*xy32# + 3a*xyy*2+ | 49*xy3223, as appears from what has been ſaid already, fince xy32+ and a*xyiz+ are in a given Ratio, and therefore their Fluxions in that ſame given Ratio“: But that likewiſe is the Fluxion of a*xy3z+ according to the general Rule above, ſince a =o, and therefore the Term which involves à equal to nothing likewiſe, being 26 _ 3 F 82. And now theſe things being underſtood, the Solution of our Author's firſt Prop. will be very plain and eaſy, vig. thus: An Equa- tion being given involving any number of flowing Quantities, tranſpoſe all the negative Terms to the oppoſite Side, that all may become poſitive, then find the Fluxions of both Sides, by taking the Aggregate of the Fluxions of all the Terms, found by the general Rule above: put the Fluxions of both Sides equal to each other: and ſo you ſhall — an Equation exhibiting the Relation of the Fluxions. Thus if the firſt Equation mentioned in this Prop. were given, vig. x3 — xy* + a*2z — 63 Do, make it by Tranſpoſition ſtand thus x3 + @*z = xy* 4- 43, Then take the Fluxions of both Sides, which, being the Sum of the Fluxions of all the Terms, I do, by taking the Fluxion of each Term, and add- ing them together: whereby we will have gxx* + a*2z = xy* + 2x5; or by tranſpoſing all to one Side, 3xx* — xy* — 2xyy + 4˙ =o, for the Fluxion of 43 is nothing. Which is the fame with that diſcovered by our Author: and defines the Relation of the Fluxions, in as far as that Relation can be determined from one Equation only including the flowing Quantities. OP . U The 4A L Wi p a — es 2 r THO, — A V ee te EN ee al x S „ oa aA Aa4_ a RR. a___= .w_ww=—r 22 2 — 7 4 Wo - —_ WY Xw >” SY WW %- ETA eto cn JET fl Vs FO ES tl ² ˙ K $A" N De Quadrature of CU RVS explained. 73 The Demonſtration of this Solution is founded upon Art. 40 “. Where it was ſhewn that the Fluxions of equal Quantities are equll. barbie 83. It is evident that the Solution will be the ſame, if all the Quan- tities being brought to one Side, and made equal to nothing, the Fluxion of the whole be taken and put equal to nothing, it being un- derſtood that every Part of the Fluxion of a negative Term, muſt have an oppoſite Sign to what it would have, if the Term were poſitive. And ſo the Solution will be the ſame with our Author's, 0 8 84. But becauſe fractional and ſurd Quantities may often occur in the fluential * Equation propoſed, as in our Author's 2 Example &, . Art. 18. it will be proper to ſhew more particularly how the Fluxions of ſuch of the Quantities are to be found. And firſt with Reſpect to fractional Quan- Beg tities, Let 7 repreſent any fractional Quantity, having both Num. Curves. and Den“, flowing Quantities. Put y: & :: 1: v; or v or yet x=vy: then x=%y +-v*: and by Tranſpoſition and Diviſion » Art. 40. . * X—©y . | | 2 Xn —y — 2 2 of this Ex- —v. Reſtore — for v, and it is 10 Sa, i. e. (be- plication. , bam; ha” * * cauſe v e 2. If either Numerator or Denominator was a conſtant Quantity, it is only putting & or y equal to nothing, and the Fluxion will be inveſtigated the ſame way. = which way of reaſoning, we deduce this general Rule for finding the Fluxion of a Fraction, viz. Multiply the Fluxion of the Numerator into the Deno- minator ; ſubtra& from that Product the Fluxion of the Denominator multiplied into the Numerator: and divide the Difference by the Square 2 — a3 O=m-gatzad 2a xed- 34 or — 3a. u Which Expreſſions of the Fluxion are all the fame in effect: the firſt being the Notation ariſing from the Rule for finding the Fluxion of a Fraction; the laſt Notation ariſes immediately from the Application of the general Rule above “: for - SN 3. Likewiſe * Art. * — * N . 33 of this Ex- GEESE K* r By the ſame plication, way of reaſoning 1 — 7x-*-'x which is a further De- monſtration of Art. 51 x. of this 1 By a fluential Equation is meant an 1 containing flowing Quantities, whereby their Explica- Relation is determined : the Equation thence deduced according to this firſt Prop.; or even tion, any other propoſed, which contains Fluxions, and thereby determines the Relation of the Fluxions, is called a fluxional Equation. . L 85. Now 74 The Bruadrature of Cunvns explained. | 85. Now with reſpect to the Fluxions of Surds: let /ax = or G7 <7N be aSurd with an indefinite Index. In order to findit'sFluxion, +7 þ 19 f ſuppoſe ax — x* » D ax—x*=w". Take the Fluxions of both, and we have ax 2 =1nvv*-" : or, by dividing both by avi, ED b. Whence, by inſerting the Value of v in place of it, we have * == — or — Xax — xx)" xax — 2x, Which is in ne Ce wihichet toni es; he; accnting to. the Bale fir ding. . Of this ing the Fluxion of any Power of a flowing Quantity, Art. 49 and go * — 180 or by the general Rule at Art. 9 1. Likewiſe F. x* e © '#. For — , whoſe Fluxion, by what has been now ſaid, 3 Lhe 2 X nxt = =x" „ nx" = = x. And F. x = 2 Fes "= 2 whoſe Fluxion, by the laſt Art. and by Bn | PL n + | | Tn TA EEE what has been juſt now ſhewn, is — =— . 22 1 * was | Wherefore univerſally the Rule for finding the Fluxion of any Power of a flowing Quantity, be it's Exponent. poſitive or negative; integral or fractional, and whether the Quantity affected by the Exponent be fimple or compound, is this, multiply the flowing Quantity by the Exponent of the Power, diminiſh the Exponent by a Unity, and | multiply further by the Fluxion of the Quantity or Root : agreeably- to * Of this Art. 50 and 817. l Explic- $6. Suppoſe now the Equation & — xy* + aavax—y — o, our Author's 2 Ex. was propoſed, to find the Fluxions: By taking the Fluxions of all the Terms, according to the preceding Rules, it gives 3 — 3y* —2x05y + aaa X ax — py = Y o, or, which is the ſame thing, 33x* — . — 2% + FIZ o. Which exhibits the Relation of x and 7. And fo of others. 87. Schol. When FraQtions and Radicals, involving the flowing Quantities, enter the fluential Equation, you may inveſtigate the Equa- tion exhibiting the Relation of the Fluxions, by a Subſtitution ; thus, let the Equation x* — ay* + A —xxvVay + © =o expreſs the Rela- I tion 5 | 9 2 ; | 4 The Ruadrature of Cunxves explained. 7g tion of x and y: put I =z and xxVay Þ x* =v: and fo you will have three Equations, vis. x3 — 4 + 5 —v=o, az + 5 — by o, and axty + x* — o: the firſt of which gives 3xx* — 24% + z — V=0, the ſecond gives mA AF T1 on, and the third, 4axx*y 2 + 6xx5 — 2vv==0, for the Relations of x, 5, 2, ö. Now by Reduction of Equations from the two laſt, you may find æ — and d= LEFT”: which Values of & and 4 being ſubſtituted in place of them, in the firſt of the three fluxionary Equa- cows, ther ite gx — 9p + SES ENDLESS he 2 by reftoring the Values of 2 and v, 3xx* — 2ajy + 5 — TS : which expreſſes the Relation of x and . And after the ame manner you may proceed, when there are other more com- plicate Terms, in the fluential Equation propoſed. 88. The Fluxions we have been ſpeaking of hitherto, under this firſt Prop. are firſt Fluxions. But our Author proceeds at Art. 19. * to * Of the ſhew how ſecond, third and following Fluxions may be inveſtigated. _—_ And he tells us, it is only by repeating the ſame Operation, by which Curves. the firſt Fluxions are found. | | Thus if we take the fluential Equation zy3—2+*4-a#=0 of Art.19.*: * Of the from it we deduce this fluxional Equation zy3 ＋ 32 — 4223 So, Wader | | for the Relation of the firſt Fluxions z and y, Which Equation may Curves. | 329 be reduced to an Analogy 7 = or 55 1: 32)* : 423 — 55. Again from the Equation 2y3 + 325 — 4223 =o, by conſidering 2 and y as flowing Quantities as well as z and y, we deduce this other | Equation Zy3 ＋ 32yy* + 32 + 32jy* + 62yyy — 4223 — 12222 =0, or 2y3 ＋ 62yy* ＋ 32% + 625% — 4223 — 122*2* o, con- taining the ſecond Fluxions of z and y. After the ſame manner, by conſidering Z and 5, as well as 2, y, 2 and y, as flowing Quantities and applying the general Rule contained in this Prop. we ſhall have 233 ＋ 39% + 6339 + 625y* ＋ 12% ＋ 3% + 325y* + 6zjyy + 623%» + 122yjy + 623% — 4223 — 12222* — 24222* — 24232 Do, or, by adding ſimilar Terms 2y3 + 92yy* 4 g2jy* + 1887 + 32jy* + 18zjyy + 6233 — 4223 — 36822 — 24232 =o, an E- quation containing the third Fluxions of the flowing Quantities x and y. And ſo we might proceed to Equations containing the ſubſequent Orders of Fluxions. The Reaſon of which Operations is plainly _—_— L 2 what 76 The Quadrature of Curves explained. what has been ſaid already, with reſpect to the finding an, Equation. containing the Relation of the firſt Fluxions from a fluential Equation ropoſed. For N the fluential Equation contain three or more wing Quantities, a fluxional Equation, may notwithſtanding, be de- duced from it, exhibiting the Relation of the Fluxions: by proceeding according to the general Rule. | 89. If we reſume the fluential Equation at the Beginning of the laſt Art. viz. 2y3 — 2* + a* o, and the Relation of z and y thence de- rived, viz. 5 =, the firſt expreſſes the Relation of the flowing Quantities x and y at all Times, or in every State; and the other, the Relation of their Fluxions, at all Times, in Terms made up of z and y. Wherefore if we aſſume a particular determinate Value for z or y, the correſponding Value of the other may be found, from the firſt Equa- tion and thereby the Ratio of à and y or — will be wholly known. Thus ſuppoſe that at any Time z = 22: then by ſubſtituting 2a for ⁊, in the Equation 2) — 2+ ＋ at oO, we have 2453 — 1 fa. o, whence y3 = or — Subſtitute theſe Values of z and y in the Expreſſion of the Ratio of à and y, and it becomes 7 — C | 3 = wholly a known Quantity. Which ſhews y/ 117649 | that, at what Time or Place, the variable Quantity z becomes equal to the known determinate Quantity 24, the Velocity with which 2 flows, at that Time, is to the Velocity with which y flows at the ſame Time, in the ſubtriplicate Ratio of 97200 to 117649; or as the Cube Roots of theſe Numbers. And if z be taken of any other Value, an- other known Relation of 2 and y will ariſe. go. Hence it appears, that when a fluential Equation is propoſed, containing two variable or flowing Quantities only, the Relation of the firſt Fluxions of theſe two flowing Quantities, may always be had in Terms containing the two variable Quantities and known Quantities only: and therefore by aſſuming one of the flowing Quantities at plea- ſure, the fluential Equation, by the Rules of Algebra, will give a cor- 3 Ye 8 reſponding known Value of the other flowing Quantity: wherefore the Relation which the Fluxions of theſe two Quantities, bear to one another at that Time, will be fully known, and determined, by ſubſtituting the particular determined Values of the flowing Quantities, 1n place of them, in the fluxionary Equation, . : 91. Take AS. T A ve.” The Quadrature of Curves —_ 91. Take another Example. Let the fluential Equation px =y*—=0 be propoſed: which belongs to the common Parabola: x being the Abſciſs; y, the Ordinate; and p, the Parameter of the Diameter. The fluxionary Equation thence ariſing, is px — 2yy So, whence &: :: 2y : P. i. e. the variable Ratio of x and , will be at all Times, and in every Place as twice the Ordinate in that Place to the Parameter. Now ſuppoſe I would know, what that Ratio would be when x— p, I in- ſert p for x, in the Equation px —y*—0, and it becomes p* — y*=0 or p=y: ſo that when the Abſciſs is equal to the Parameter; ſo is the Ordinate alſo, Wherefore inſert p for y, in the Value of the Ratio of x to y, and it is &: :: 2p: p:: 2: 1, ſo that, at that Time, the Velocity, with which the Abſciſs increaſes, is double the Velocity with which the Ordinate increaſes. And ſo by aſſuming other Values of x; or of y, other known Relations of x and y will ariſe, | 92. But if there be three variable or lowing Quantities in a fluential Equation propoſed, another Equation including at leaſt two of them, ought alſo to be given, that the Relation of their Fluxions may be fully determined; and alſo their Relation among themſelves. Thus let the fluential Equation ax + by* — cxz=—=0 including three flowing Quan- tities x, y and z be given. The fluxional Equation thence deduced is ax + 2byy — c - cx o, Which gives the Relation of the Fluxions x, y and >, as far as that Relation can be determined from one fluential Equation only. But the Relation of the Fluents in the fluential Equa- tion, and of their Fluxions in the fluxionary Equation, will not be fully determined, unleſs another fluential Equation be given. As if it were ſuppoſed that x — ay + z =o. From whence we deduce x—ay + o, for another Relation of the Fluxions, beſides the former. Therefore by comparing the two fluential Equations together; and the two fluxional Equations, thence deduced, together, you may extermi- nate any one of the flowing Quantities; and alſo any one of the Fluxions, and thereby you may obtain an Equation, which will entirely determine the Relation of the other two, whether flowing Quantities or Fluxions. Thus if I wanted to have the Relation of x and y, fully determined : The firſt fluxional Equation gives à =< 2 ==, the other, which is deduced from the 2* fluential Equation, &c. either given; or aſſumed, if not given, will give Zz==ay — x: therefore, by equating theſe Values of z, we have — 2 == = ay —x, Again by means of one of the fluential Equations, as of x — ay ＋ 2 =0 take a Value of 2, viz. 2 S ay -x, and put that in place of z in the Equation =O —.— C 77 Art. 6. of this Ex- plication. The Quadrature of Curves explained. Say &, and it becomes ax + 2byy — acxy — acxy + 2cxx o, or x * acx — aby: e Where if you take x at pleaſure, the fluential Equations will give y; and ſo the Relation of x and y will be entirely determined. More ſhortly thus. From the fluential Equation ax + by* — cx2z= 0, propoſed; and the other x — ay f A =o, either given or aſſumed, find another Equation free of 2; which will be ax + by* =acxy — cx®; and from that you'll have ax + 20% — acxy — @cxy — 2Cxx, as formerly. . 93. It may be obſerved here, that, when there are three flowing Quantities, and but one Equation to determine their Relation, the Fluxions, as well as the Quantities themſelves, admit of an infinite Variety of different Relations: and therefore when in that Caſe, I aſſume another Equation, as above, I only thereby determine one of theſe Relations: which, from the Nature of the Thing, are indefinitely many. = If there are four, five or any other Number of flowing Quanti- ties, there ought to be given as many Equations ſave one, as there are 2 in order to the full Determination of the Relations of the firſt Fluxions of theſe flowing Quantities: as plainly appears from what has been ſaid. $3> 24. 95. It was obſerved formerly *, that the Fluxions of homogeneous Quantities are ſtill to be conſidered in relation to one another: for without ſuch a Conſideration, we can make nothing of the Doctrine of Fluxions. And therefore, ſince every fluxionary Equation contains the Fluxions of two flowing Quantities at leaſt, either expreſs'd or under- ſtood ; and thereby determines only the Relation of theſe Fluxions, it is left at liberty, to ſuppoſe one of theſe flowing Quantities to flow or change at any rate, either equably and uniforraly; or according to any Law of Acceleration or Retardation, we think fit to frame or ſuppoſe: becauſe ſuch a Suppoſition never alters the Relation of the Fluxions or Velocities of flowing. Hence it appears, that our Conception of the Relation of the Fluxions, will then be moſt clear and diſtin, when we ſuppoſe one of the flowing Quantities, to flow with an uniform and invariable Velocity, and call it Unity, For thereby that uniform Fluxion is made a common Standard, by which to meaſure the reſt. Thus in the Equation px -— y* = 0, mentioned above, to a Parabola, from which the fluxionary Equation px — 2j o is deduced, I ſup- poſe the Abſciſs x to flow Ars aumray and put it's Fluxion x = 1, By which means the fluxional Equation becomes p — 2) So, i. e. J= E fo that the Fluxion of the Abſciſs being always 1; the Fluxion of r e YE 1 * k . 4 * _ n CO 4 8 * 8 1 2 c 4 FA n of 99 . 1 a Wr "mY * 5 4 ty. web SI ENS SER IN! ts 23 35 to — 2 E A TI - * < The Ruadrature of Cuxvss explained. of the Ordinate, will always be expreſs'd by the Quantity 2. I might have likewiſe m_—_— the Parabola to be ſo deſcribed, by the Motion 0 of the Ordinate along the Axis, that the Velocity, with which the Or- dinate increaſes, is always invariably the ſame, and call it Unity: then 3 2j o becomes p — 2y =o, or =>: i. e. in ſuch a Caſe, the Velocity, with which the Abſciſs flows will be always expreſs d by 2 And fo of others. 96. This Suppoſition is of further Uſe, when we have occaſion to roceed to 20, 3 and other ſuperior Orders of Fluxions. For in all the E Boca, for determining the Relations of theſe ſuperior Orders of Fluxions, the Symbol, for expreſſing the firſt Fluxion of the uniformly flowing Quantity, may be daſhed out, wherever it is found, being Unity; and all theſe Terms thrown out, in which a ſecond, third or any ſubſequent Fluxion of the uniformly flowing Quantity, would be involved : for ſince it's firſt Fluxion is Unity, a conſtant and invariable Quantity; every one of the ſubſequent Fluxions is nothing. Which our Author obferves in Art. 20.“ and by this means the Equations, for * Of the determining the Relations of the Fluxions, are conſiderably abridg'd, as Quade: is there to be ſeen. 3 97. But hence it muſt happen, that a fluxionary Equation, will ſometimes appear not to have the Symbols of the Fluxions of the ſame Dimenſions, through all the Terms: which yet it ought to have, if we conſider the Manner of forming any fluxional Equation from it's fluential. Fluxions of the ſame Order, or whoſe Symbols are of the fame Dimenſions, are ſuch as theſe, x. y. S of one Dimenſion ; &. &. xy, of two Dimenſions; &. xy. x3. K. xyz of three Dimenſions, &c. That the Symbols of the Fluxions, in each Term of any fluxionary Equation, as it is formed from the correſponding Fluential, whether it be a fluxio- nary Equation of the 1, 24, 3*, or any other Order, muſt always be of the ſame Dimenſions, will eaſily appear to any one that conſiders, how the ſeveral Orders of fluxional Equations are formed above *. And * Art. 88. it is neceſſary it ſhould be ſo, ſince Fluxions are Quantities of a different — 3g kind from the Quantities, whereof they are the Fluxions. Thus taking and Art. the preceding Example px — o: hence ariſes px — 2 o, where and. the Symbols of the Fluxions, in both Times, are of the ſame Dimen- Ouadra- ſion. But if we put v 1, it becomes p — 2 S a, where one Term ture of contains no Symbol of a Fluxion at all. Thus alſo, when we have the Curved. Equation zy3 — 2* + a = © mentioned Art, 88 *, the third fluxional * Of this 4, Expl ica- Equation, thence deduced, was ſhewn to be 2y3 ＋ g#yy* 4- gz I Fon. Art. 2. of this Ex- plication. The Quadrature of Curvss explained. 185 2575 + 1827 + 629% — 4223 — 36222* H 2412 — 15 25 ail he 3 Symbols of Fluxions of three Di- menſions. But if we ſuppoſe 2 = 1, it appears in another Form, thus M + 185% ＋ 3 + 18zjyy ＋ 6293 — 242 . 98. Therefore, whenever a fluxionary Equation is propoſed, where the Symbols of the Fluxions, in all the Terms, are not of the fame Dimenſions, theſe Dimenſions muſt be completed, or ſuppoſed to be completed, by multiplying the lower Terms, by the Fluxion of ſome flowing Quantity, whoſe Fluxion is underſtood to be Unity, until the Symbols of the Fluxions ariſe to the ſame Number of Dimenſions, in all the Terms. Thus the fluxionary Equation, juſt now mentioned, is to be completed, by multiplying the two firſt Terms gjy* and 187 by z; and the laſt Term — 242, by 23: 2 being ſuppoſed equal to Unity, And fo the Equation will appear thus g2jy* + 1825*y + 325 + 182jyy + 6257 — 24232 =0. here the Dimenſions are com- pleted, by multiplying the lower Terms as often as is neceſſary by æ, which is the Fluxion of a flowing Quantity z, viz. ſuch a one as is found in the given fluxional Equation, while in the mean Time, it's Fluxion is not found, | 99. But if there be no variable or flowing Quantity in the given fluxionary Equation, but what has it's Fluxion in it likewiſe, then the Order of the Fluxions in the lower Terms, muſt be completed, by ſup- oſing ſome other variable or flowing Quantity, which flows uniform- f , Whoſe Fluxion muſt be underſtood to be Unity, and multiplyin the lower Terms by that Fluxion, ſo often, until the Dimenſions o the fluxionary Symbols be the ſame through all the Terms. Thus if the Equation x + xyx — ax*— © were propoſed, we are to ſuppoſe ſome third Quantity as z, which flows uniformly, having it's Fluxion 23= 1: and then multiply the firſt Term by z, and the laſt Term by 22. And ſo we muſt conceive of the given fluxional Equation x + xyx- — 4 So, as if it had been derived from this one x2 + xyx — azzx* Do, by putting 2= 1. Likewiſe if aav — av + vg were given, I complete the Dimenſions, by ſuppoſing another flowing Quantity z, whoſe Fluxion z= 1. As if the Equation had been originally aa — au — vis oo. And ſo of others. | 100. This firſt Prop. of our Author's, which I have been explaining, lays the Foundation of the whole Doctrine of Fluxions: particularly it contains the Solution of the firſt of the two general Problems mentioned at the Beginning *, viz. the og? of the Space deſcribed, being conti- nually or at all Times given, to determine the Velocity of the Motion at any Time propoſed : which in effect amounts to the ſame thing, with 1 I the Ke 8 mee NE BLOG IN O * FE — * 4 4 — 3 N F ä ” : K a F TRE 222 : 8 7 hr IK n r 8 apr oe * bs - - L 4 % * * . De Quadrature of Cuxves explained. the Propoſition as here expreſs d: and is called the direct Method of Fluxions. The Application of theſe two general Problems is very ex- tenſive: particularly by them we are enabled to ſolve the following uſeſul Problems, v:z. 1 1. To determine the Maxima and Minima of Quantities. 20. To draw Tangents to all Sorts of Curves, whether geometrical or mechanical. | z. To determine the Points of contrary Flexure, and Retrogteſſion in Curves. 4. To determine the Quantity of Curvature at a given Point of a given Curve: i. e. to find the Length of the Radius of an equicurve Circle, Under this may be comprehended the finding the Points, where a Curve has any given Degree of Curvature : where it has the greateſt or leaſt Curvature: and the Determination of the Locus of the Center of Curvature. ; 5. To determihe the Quality of Curvature at a given Point of a given Curve: z.e. it's Form, as it is more or leſs inequable; or as it varies more or leſs in it's Progreſs through different Parts of the Curve. 6. To find as many Curves as we pleaſe, which may be ſquared: 7.e, to find the Nature an Property of all ſuch Curves, whoſe Areas are exhibited by finite Equations *. 7. To find as many Curves as we pleaſe, the Relation of whoſe Areas to the Area of any given Curve is aſſignable by finite Equations. 8. To find as many Curves as we pleaſe, whoſe Lengths may be ex- preſs d by finite Equations. To this Problem and the 4 belongs the determining the Evolutes of Curves. | 9e. To find as many Curves as you pleaſe, whoſe Lengths may be compared with the Length of any Curve propoſed ; or with it's Area ap- plied to a given Line. | 10%. To find the Cauſtics to all Sorts of Curves, whether by Reflec- tion or Refraction. 3 11. The Quadrature of Curves; or finding their Areas. 12%, The Rec iſication of Curve-lines; or finding their Lengths. 13%. The Plaining of Curve- ſurfaces; or reducing them to plain Surfaces. | 14. The Cubature of Solids; or finding their ſolid Contents. 15. The finding the Centers of Gravity: and of Oſcillation or Per- cuſſion of all Sorts of Lines, Superficies's and Bodies. | 101. Theſe Problems for moſt part belong immediately to the Geo- metry of Curves; but by their means, ſerve to other excellent Purpoſes M in 2 4.0 The Buadrature of Curyss explained. in Mathematics and natural Philoſophy : witneſs that inimitable Work, the mathematical Principles of natural Philoſophy: a great Part of which is founded upon the Geometry of Curves: and Principles of the | | fame or like. Nature with thoſe of Fluxions. 102. To go through all theſe Problems would take up a great deal of Time; and be foreign to our Deſign: however ſome of them that come in moſt naturally, ſhall be explain'd and reſolved in the Pr of this Work: particularly the 17, 2, 30, 6®, 70 95,2; 24> 14, The 11 is that whoſe Explication and Solution takes up the great Part of this Work. 103. Among geometrical Problems, one of the moſt noble and uſeful is the Quadrature of Curves: and accordingly the greateſt Geo- metricians, in all Ages, have laboured this Point with all the Force of Invention; and exerted themſelves to the utmoſt upon this Head. Yet there was very little done in that way, till after the Middle of the laſt Century, that the Doctrine of infinite Series was introduced; found- ed upon the Arithmetic of Infinites, made uſe of by Dr. Wallis. Which Method of infinite Series was improved and perfected by the Diſcovery of the Method of Fluxions by our celebrated Author about the Year 1666 * : whereby the Quadrature of Curves is rendered univerſal. 104. This is a new kind of Analyfis: by means of which the Areas of all Sorts of Curves are inveſtigated and determined, either to-a per- fect Exactneſs; or, when the Nature of the Curve will not admit of that, to any Degree of Exactneſs, one can poſſibly require. And ſuch is the Subject of this Treatiſe. 105. But if any one ſhould think that the Quadrature of Curves, being accounted but one Branch of the Application of the Doctrine of Fluxions, ſcarcely deſerves to be infiſted upon at ſo-great length: it muſt be known, that, the Quadrature of Curves, is not only among the moſt ſublime and uſeful Diſcoveries in Geometry, confidered in it- ſelf: but in effect includes under it, the finding the Length of Curve- lines; the Solidity of Bodies, the Contents of their Surfaces, and Innu- merable other Problems. Add to this that the original Prindiples, and See Dr. Wallii's Treatiſe of Algebra, Chap. 82, 85, 91, and the Commorcium Epiftolicum. de varia re Mathematica, particularly a Letter from Sir J. Newton to Mr. Oldenburg,, dated ©Xober 24, 1676. Some time after Sir J. Newton had diſcovered his Doctrine of infinite Series and Fluxions, Mr. James Gregory Profeſſor of Mathematics at St. Andreaus, about the Rnd of "the Year 1670, fell upon a like general Method, ſerving for the ſame Purpoſes; having diſ- covered many particular Series's long before that Time: as, appears from. the Gommercium -£piſt. See Mr. Gregory's Letters to Mr. Colins, dated Sept. 5. Now. 23. Dec. 24. Anno 1670. Feb. 15. 5 — 1572, likewiſe Mr. Olias's Letter to Dr. David Gregory : and Sir J. Neæcotan 's ſecond Vetter to Mr. Olaenburg already mentioned. f ” juſteſt 2 a >... 0A a. ae. ©. ad os 8 ewe wu = pF —. NAFTA M E oO me. The Quadrature of CVRvRS explained. juſteſt Conceptions of the Methad of Fluxions, are deduced from the Conſideration: of the Deſcription of curvilinear Areas, after the Manner already explained. Upon which account, the Quadrature of Curves, here laid down by our Author, ought to be fully underſtood by all ſuch as would bave a juſt and comprehenſive Notion of Fluxions. TS Explication of Prop. 2. | 106. By this Propoſition, the Relation betwixt the Area of any Curve and it's Abſciſs or rather the Rectangle contain'd under the Ab- ſciſs and an invariable right Line, ſuppoſed to be Unity, being exhibit- ed, you can always determine the Nature of the Curve, i. e. the Rela- tion betwixt the Abſciſs and Ordinate: or you may aſſume or feign any Relation you pleaſe, betwixt the curvilinear and rectilinear Areas ABC and ABED *, and thence determine the Relation of the cor- reſponding Ordinates AB and BC. TK | 107. Although the two following Prop, be Examples of this, yet, for the ſake of a further Jluſtration of this Prop. I ſhall ſhew it's Ap- plication, in ſome Examples, which are more ſimple and eaſy: hy which, at the ſame Time, the Demonſtration df it will be made more evident, „ Ex. 1. Calling the Abſciſs AB ; the Area ABC :?; the Ordi- nate BC y, and the Ordinate BE=(AD =) 1: and conſequently the Rectangle ABED D A x IS: Let the Relation of the two Areas ABC and ABED, i. e. of f and z, be expreſsd by this Equation 83 2*==f:; then taking the Fluxions by Prop. 1, we have 255 =]: i. e. . Art. 40. by reducing to an Analogy, 1: 22 :: 2: f: but 3: 1:: EB: BC®: of this Ex- therefore 0 Equality, 1:22z:: (EB: BC ::) 1: 5: whence 22 ==7: P lication. Art. 2 Which defines the Relation of the Abſciſs and Ordinate of the Figure of this Ex- ABC: and ſhews it to be a right-angled Triangle, having the Baſe BE plication» double the Altitude AB. Ex. 2. Suppoſe we aſſume 12 = t, or zal t, for the Relation of the Areas ABED and ABC: then, by taking the Fluxions, we have @izz* =&, i. e. I: a:: (S: f: :) EB=1: BC=y: Whence /az —=y or az = y*: which ſhews the Curve to be the com- mon Parabola. Ex. 3. Let 423 = 365% + 36bcz3t exhibit the Relation of the Areas: thence, by taking the Fluxions, we have 3a = 3bb# + bet + 36c23t; or, by dividing by 3 and tranſpoſing az* — 3bcz*t x & =bb + be23 xt, i. e. W + KR: a2* —2bezt ii (: 1 ::) 159; 1 See the Fig. belonging to this Prop. = M 2 whence, 84 The Quadrature of CurvE's explained 7. whence, by multiplying the Extremes and Means, we have hy + bezry = 4a2* — 3bc2*t for the Relation of z, y and f. Now that # may be exterminated : from the Equation expreſſing the Relation of the Areas, + = 36*t + 36bcz3!, find the Value of t, which is = TS; ſubſtitute this Value in the Equation juſt now found, viz. þ*y 4 bci =42* — 3bc2*t, and it becomes b*y + bee . — 3bcz* x 7 _ =_—_ i. e. by Reduction, y FR —z; which ſhews the Relation of 2 and y or AB and BC, and ſo determines the Nature of the Curve ABC ſought. ; Ex. 4. Let the Equation 24 — 42 — 4 =? be aſſumed for the Relation of z and 7: by taking the Fluxions, we have 24** — 427 — 4 ait; or 4˙ — 22 —!t==22t, i. e. 4 —22 — : 2t:; (31 ::) 1:5 whence y = ==: and in order to exterminate 7, let it's Value be found from the Equation 24 — 4 — a* = if, ex- preſſing the Relation of the Areas: which will be T= — 22 + 2%. — 4: inſert this Value of ? in place of it, in the Equation 2 a 24% — + * — 5 = —, and we ſhall have A = — =+ 9 2 for defining the Nature of the Ex. 5. Let the Equation — 51 = /} et be given or aſſum- ed for expreſſing the Relation of z and f ef then by taking the Fluxions, | nan} — cæx — 2ab + acz* . vg WP * 7 Tc + 7 X — z; or by due Reduction > <7 3 az3 2-24 rr =I: Vhence it is 1: = G.:) 1:5; or y= pe- 4 | HEY. I 3 * Ar 29, 108. Schol, Since we have always : 2: :: 17, thence Zy r, is 31. of this - og Equation, if we ſuppoſe 2 1, as we may do“, it becomes y —7. * Art. g. Whence the Solution of this Problem may be a little more expeditious, of this Ex- . , , . vlication, thus: From the Equation propos'd or aſſum'd, exprefling the Relation of f and 2, find the Equation expreſſing the Relation of the Fluxions and 2: in which Equation, = 1 for 2; or dafh out > every where, and then the reſulting Value of f, will be the Value alſo of y: therefore : fubftitute y for : and ſo you ſhall have an Equation expreſſing the Nature of the Curve. Thus in the 2 Ex. za =7, the Relation of the an Equation expreſſing the Relation of z and ? in every Caſe : in which Z' | 9 De Quadrature of COR VES explained. 85 the Fluxions is ae , i. e. by ſubſtituting 1 for z and y for ?, Jaz=y. Likewiſe in the 4 Ex. viz. zart — 42t — a+ tt, the Relation of the Fluxions is 4. — 2 — 227 =, i. e. a — 2f — 22y 5 — yt, by means of which Equation, and the propoſed Equation, # be- . y ing exterminated, we obtain, as formerly 72 45 2 2 — 2 for de- | ? fining the Nature of the Curve. And ſo in all other Caſes. R Our Author's 34 Prop. is an Example of the fame thing: and is ſo plainly demonſtrated by himſelf, that, if what hath been ſaid in the e foregoing Schol. be conſidered, there cannot remain any Difficulty in it: and therefore I ſhall paſs on to the next Propoſition. Demonſtration of Prop. 4. = The Demonſtratian of this Prop. is omitted by our Author, k becauſe it may be * ſupplied: if the Demonſtration of Prop. 3? de duly conſidered: and it is thus. Let us ſuppoſe the Area S/ Sui: then by taking the Fluxions according to Prop. x, we have I _6G2= RS bar RRS + R S8“ =0, Inſtead of R, in the firſt Term of the Equation, ſubſtitute RRa = 88 1; inſtead of 208“, in the ſecond Term, 22%'S$*=" : and inſtead e of 2B), in the third Term, ſubſtitute z22-"RR>=": which Expreſſions = are cvidently equal to the others: and it becomes b - iRR.-188“/ + N- IRR Y 188 + u -iRR.-— 188711 I] Or RS TRS + ARS - RS — 0. Bat,'by Hyp. R=e r + gz" + Ge. and Sr A TIE -þ mz; A From which two laſt Equations, find the Relations of the Fluxions by Prop. 1, and you'll have R= + 219222=! + Cc. And S N⁰N -! + 22π¼ mm i CG. Subſtitute theſe Values of R and 8, in place of them in the preced- ing fluxional Equation; and likewiſe the Values of R and S in place of them: and put Zz=1. And fo you'l have 8 xe + far + g + e. Xx ENTE + nE + Cc. + N nfs”! + 2109291 +. c. x k + bn + mz" + &c. + Ne + fs + gz" Sc. x wg! + 2nm2z%=! + Sc. into 2—R-ISH = 5 = % =) BO® vartios. That is, if you actually multiply the ſeveral Factors; and obſerve to of this Ex- rank the Terms according to the Dimenſions of tho variable Quantity 2; plication. y ig ſuppoſe.l to. ſtand for the Ordinate of the Curve. | beke The Quadratare of Cunvis epa 02k-\-0 x ft +90 | 1.4 GS-4 Sos „ 44 of + < © +a +0 wo a ww * I-27 T2 Sk fo - Jed T +0 * +0 » 24 +9 Ws '= — - 8 Lag x +50 x flz 255 Vgl * —— 7 ＋¹ 1 4 P „ Kenz, J. xfmar % N gt n 2 u Ay 2A [ 2pn + 2pe1 18 110. Cor. 1. Since f and x, the Exponents of the Powers of z and R, in the 3* Prop. and 0, A, , the Exponents of the Powers of 2, Rand 8, in the 4 Prop. may be any Numibers poſitive or negative; integral or fractional: for the reaſoning is the ſame, whatever they be: hence it follows, that, when any of - theſe Exponents is negative, in the Ex- preſſion of the Area of the Curve, all the Change that it brings apon the Expreſſion of the correſponding Ordinate, is this, that, wherever that negative Exponent, or theſe negative Exponents occur, it, or they muſt be affected with the negative Sign. Thus: in. the g* Prop. the following Areas and Ordinates correſpond. e Areas. ey” Ordinates 111. R — 62 — 6 — 0 * 8 r — Wee” 0 few To xz Of, x R — Ay — 2K KY * In mo — 3 113. SUR e == * 280 X ge- c - iR And the like are the Changes that will ariſe, in the Expreſſion of the Ordinate in Prop. 4. Moreover it is evident that, if the Exponent of the Power of z with- out the Vinculum, in the Expreſſion of the Areas of the Curves belong- ing to theſe two Prop. were made n; ſo that the Area in Prop. 3, ſtood thus 2 R): and that in Prop. 4. ſtood thus at NN, where vis ſuppoſed to denote any integer Number: all the Change, thereby brought, upon the Form of the Expreſſion of the correſponding Ordi- . nates, is this, that 4 + ys muſt be ſubſtituted in place of 9, wherever it is found. So that the following Areas and Ordinates will correſpond. Areas, . 0 * 8 3 ets N a. tub. beds 5 a. eye: a n < ee RIO »% Vn _ wie > wov XE ö =» x lf 24 * The Ryadrature of Curyts explained. 87 Areas. - 2 Ordinates. | — . — ——— — : 1e -e 4 Iz" fo . r +&c. - 1-1 rr rr Or oe u * „ 1＋ 123 Ken + be. x IT A An Ton Mo of Ye Fal Wag LETT 7 INS F T8 l 1 | "ed LT. | '7 11g. N - - Eonxck a, Y f 4 — 3x“. =. A2 —1 Do, m— — 3. Inſert theſe gb3 * + x — 4A? for the Ordinate of the Curve whoſe Area is * | FER v/05 + cox — dx* 5 | The laſt Factor, correſponding to 8, is repreſented as wanting the 2 Term, viz.. thus 34 * — 3x*, becauſe, ſince the Powers of 2 be- Kr | longing 0 A. — 0 of 4 , «© *4 7 "As 88 "The Quadrature of Curvzs explained. longing to R and 8, in the general Form, are ſuppos'd to have their Indexes going on by the ſame Differences, being 3, 2, 3y, &c. and ſince the Value of 3, in this particular Caſe, muſt be Unity: therefore it is evident why the ſecond Term of the Value of 8 muſt be nothing, and therefore o. 5 | And moreover by the ſame way of reaſoning it appears that, if it be R —e + jz-" + gg-*” + bz-3" + &c. in Prop. 3: and R =e + fe» +2g2-2-+ &c. and likewiſe S + [/z-" + mz-*" + nz=3" + Ge. in Prop. 4: where the Indexes of the Powers of z under the Vinculums are all negative: then the Sign of y muſt be chang'd every where into k it's Oppoſite in the Expreſſions of the Ordinates of the Curves, belong- ing to theſe 1 reſpectively. : | To 115. It may likewiſe be obſerved that, if the general Forms of the Areas, viz. RD and RS, be multiplied by any conſtant or given Quantity, as A, and ſo become Az/R>® and Az*R*S*, the correfponding Ordinates of the Curves muſt be multiplied by the ſame given Quantity A. For : the Fluents being increaſed or diminiſhed in a given Ratio, the Fluxions Arx. 40. are increaſed or diminiſhed in the ſame given Ratio“: and the Areas l of thsEx- being the Fluents, the Ordinates are as the Fluxions in this Caſe &, be- EA. . cauſe the Fluxions of the Abſciſſes are the ſame. | 3 i of this Re 116. By the ſame way of reaſoning as that made uſe of in theſe two C plication. prop. it appears that, if the Area of a Curve be 2*R'S#T", where v is any given Number, and T —= pa qu ra 4- Fc. and the other F Symbols ſtand for the ſame things as before, the Ordinate of ſuch Curve may be found: and ſo for other Areas more complex. 117. It follows from Prop. 3, and what has been ſaid that if ( Fa Fon AE hon, fag" Ker Fer 6 cath ko 5 a; Te. e. A“-1 - q be the Ordinate of a Curve, the Area of the Curve is Aa t R: for 1 if not, the Area Az R would belong to a Curve having a diffe- f rent Ordinate, contrary to what has been demonſtrated. And the like is to be underſtood with reſpect to the Area and Ordinate contain d in the general Expreſſions in Prop. 4. | , t t SECT, E ee, of Cuxvzs explained. SECT. Tv. Containing Motes on Art. 27-—31. Some Lemmas ſerving to demonſtrate ſeveral Things contain'd in the 5 Author's Remarks upon Prop. 5. >, LEMMA. I. 119. An infinite Series of Powers of a proper FraCtion, whoſe Indexes conſtitute an arith \ Progreſſion aſcending, having finite Coefficients *, will converge : i. e. a finite Number of Terms of the Series, will differ from * Sum of the whole Series by leſs than nt given Difference. | Let 5 © repreſent any proper Fraction, 4 J.. 3 Oc. in inf an infinite Series of Powers of that FraQtion, wich Expo- nents are n,n+p.n+2p.n+ 3þ, &c. then ah, „ FFAs is an infinite geometrical Progreſſion deſcending, having J for the common Multiplier: wherefore, aa to the Nature of ſuch a Progreſſion the Sum of the whole is — Let 9 be any given n+ 2 N N. ＋ 9/ Quantity how ſmall ſoever; and T a Term of the Series leſs than * 1 A (fr, it is evident, ſuch a Term may be found) ſo that T=xqXI1=—H N Now, the Sum of all the Terms he T incluſive, from the Property of a geometrical Progreſſion, is _ therefore that Sum is leſs than — = 7. Now, ſuppoſing the ſeveral Terms "on Wh to have any y finite Coefficients prefix d, ſo that none of them be greater than A: then you need only aſſume any Term of the Series, which * By finite Coefficients in this Lem. I mean ſuch as the greateſt of them is not greater than any abgnable Quantity. | N without Net” = with different Sow: — De Quadrature Yeu: eee without it's Coefficient, is leſs than 1 11 : and then all the ſubſequent Terms from m that Term incluſive, We * Coefficients, £ 1722 A vl be les than —— —.— 2. as has been ſhea. Aber, Gnee no Coefficient is greater than A, the Sum of we theſe Terms with their Coefficients prefixt, cannot be ſo great as - XA = 7, e. the dum of them all is leſs than any given Quantity. 120. Schol. Hence it appears that an infinite Series of Powers of any Quantity greater than Unity, whoſe Indexes decreaſe in an arithmetical Progreſſion, without Coefficients, or with finite Coeffi- cients, will converge. For if Ax N*.BxN*7?.Cx N &c, in inf. repreſent _ * a Series: it is equivalent to this other AN“ x a BN" N, CN" A* Ge. 1 5 15 the foregoing n | LEMMA. n i121. An infinite Series, ſuch as theſe rſt now mentioned, which converges when all the Terms are affected with the fame Sign, will alſo converge when the Terms are affected with different Signs. For ſince a Term may be found, when all are b e or all nega- 1 tity, when he pen affect- therefore the Series coriyerges: fince the Amount of the whole Series is ſuppoſed not to be leſs than a finite Quantity. oy D. LEMMA IL 122, If any rational integral Fluent, including only one indetermi- nate or flowing Quantity x: and it's F luxion, have any common prime Diviſor, which includes x : I ſay the Fluent ſhall contain ſuch com- m_ prime Diviſor oftener by once and no more, than it's Fluxion dot Let @ + bx" + ox + du + Ge. be a Fluent, where the Quan- tities a, &, c, d, Fc. are conſtant or invariable; and any of them may be equal to nothing. It's Fluxion is nbxx"" Anαπi, + — EC ͤ ˙ A The. Rgadrateire of Cunryys explained; 9 + G Now it, is demonſtratad by Writers upon Algebra ie that, * Art. 82. of this Ex if the ſeveral Terms of the Quantity a. N- bx" A c. g de 4+ G plieadon. — multiplied by the of the Powers of x in theſe Tons wes reſpectively, and the Products thence ariſing be divided by x: and if the Quantity which reſults, viz. nb“. ＋ an‚& + gndesn=* 4 Sc. have any common prime Diviſor with the Quantity . propoſed 4 + bx* c + dx” 1 & Sc. that prime Diviſor ſhall be contain'd. juſt once oftener in the Quantity + bx" * þ dx + Ge. than in the Quantity nb] . + 2ncx*"=" + ud | Cc: therefore it the flowing Quantity propoſed a ＋ bx" + cx -| dx3" + Cc. and it's Fluxion nbx x * a2ucxx * Zzudæx 31 + Se. have any com- mon prime Diviſor, including the variable Quantity x, ſuch Diviſor muſt be contain d juſt once oſtener in the Fluent than in the F luxion. QE. D. Explication of Prop. 5. and of the Author's Retnarks upon it. Although our Author's Demonſtration of this Propoſition may be evident enough to any one that carefully conſiders it, and-underftands what hath been already taught: yet for the fake of a farther Illuſtra- tion; and ſome Obſervations to be made upon it, I ſhall. exhibit the Demonſtration at ſome further Length. 123. Wherefore, ſuppoſing the ſame things as are ſuppoſed by Sir [aac ; and moreover that the Factor a+ bz” + c + dz" Ce. continued one Term further, is a + bz” + ca + d=“ + 8 4 &c, fo that the Ordinate of the Curve may be R x a TN c ＋ d ＋ e oþ+ Ge; likewiſe that R= L * ＋ 525 ＋ i + Se; as alſo that v＋ x == + then by what been demonſtrated *, the following Ordinates and Areas ſet ſet oppo- ſite to them, will belong to the ſame Curves. nos of this Ex- Areas. plication. f TE) of | | 5 TEE" By, 1. 2 i WIEN 2. 4 222 eka Kc. See Reyneau's Analyſe demontrie, B. 4. Sect. 4. | N 2 Therefore n 3 * 5 92 De Quadrature'of Curves — - Therefore if the Sum of theſe Ordinates be ſuppos d equal to the given Ordinate a+62"+r2"+dz"+ux# IG X- R-, the Sum of the correſponding Areas, viz, ABZ TCE +Dz%%+Ez#" + G. x 2*R> ſhall be — to the Area of the Curve, whereof the Ordi- nate propos d, is the Ordinate. Wherefore that the Values of A, B, C, D, E, &c. which are deter- minate Quantities, may be diſcovered, let the correſponding Terms of the two Ordinates be compared, by 1 them e and thence ariſe the following 1 * 10. A 2 20. N Y wy (FaxgA4STFF f xs =Ee 4. (pax bA+FFinxgt FF mx . = Y 5e. N i ene, CN οο Y x & * =«. From which, by Reduction of Equations, we obtain che following Equations, expreſſing the Values of A, B, C, &c. viz. 1. A= 2%. > 3 z. F SOR o2 - 08 Þ2nxe ö D= .. + 2 2 — x FC wo 7 E . eee un- 2 rr Sc. in inf. Hence it appears, that, if the given Ordinate a + bz" + c2** þ E + 12% + &c, KR. be ſuppos d equal to the Sum of the Ordinates above; and conſequently the Area of the Curve, whoſe Ordinate is given, equal to the Sum of the correſpond- ing Areas, the Values of A, B, C, &c. muſt be thoſe expreſs'd by the laſt Equations juſt now mentioned. But the Sum of the Areas is ATB + Cz** + D* * Ex h + &c. „N. Wherefore, by inſerting the Values of A, B, C, Cc. found above, we ſhall have R 19 4 — — . f -A- 11 be Te x bA — FT Fan x gB -N xfC _ 3 TT. „ wh 4 . .,, pa A C. for the Value of the Arca of the Curve, | having the propos'd Ordinate Rx a+ b2* + 2 d + ez + &c, 124. Which Arca, by putting = f or = r + a=s, 5+ l, N=, N=; and inverting the Order of all the Terms ſave the firſt, in the ſeveral Numerators, ſtands thus | _ SR x4" 2 re 1 —b if A n * T = Ixe I — | — 2 r+2xe 1 41 PI B . | rÞ+ 3xe | ar FL 2 „CTI xbB- wA * . — , | p , — — * r+4x0e | + Ge. 3 The ſame with the Expreſſion of the Area delivered by our Author; ſave only that he hath not ſet down the firſt and laſt Terms of the laſt Numerator, viz. —s and — ui; which I have inſerted. And the Law of Progreſſion of the Series is manifeſt. 12 5. But that the Properties of it; and the Laws or Conditions, ac- cording to which it cither terminates; or is continued infinitely, =P —— 94 — F Cu nas explained. the more el: appear: and that the Expreſſion of the Area may 2 wholly in known Terms, let us inſert inſtead of the Capitals A, B, C, D, Sc. their Values at length, which are eaſily deduced from the foregoing 1 5 . . done, the Series e at full dy * 2 11 22 Fo iNN ee — — BAT i S2 — 2.2 2 xe PCI vr TZ xe 6: e e ge = C= g N bs 2011 155 657 IE F — . I —＋ 2 | — " 2:3" 25 23” 12520 2 25 PR 425 Tx th x La 2 1 2 * F ＋ * : 72 ox oF 3 Pry | t+1 Yen : i T xiſp xa wh x _ r+-1 — uri xr43x3 „u 3 * ev -- —Dz3" | I I — +3 x fx —4 4 25 „ 8 — — 2 + «+3 J 2 7 4 2:4" Ax LA F ] xf1x=b PH "bo eee 5 | . e N 2 „ 7 ya wry x fp x—b 1 — ES Pi x1 F3 x _ my e „ er 7 e rxr+3 n N g x = 7 Yi x7 x —=b — 1 Horner 2 & 7 4 x e 2 * I/ . . he . . NIIEy's e eee Fe W = + 3 © hn . 8.1 ꝗ——. EB | Fo: . H * ” | Ex 3 + &c. | The 'The Buodrature of Cvavanerplained. The fame with the: former-Series, ee Values of the Capitals A, B, C. *. And that it may b red to_regalar, Series 85 I begin at the firſt Term r and running * all the reſt in order, ſet down in one Serien, all theſe Parts of each complex Term, which will conſtitute a Series of geometrical Proportionals going on by the common Multiplier = 77 with their proper < Ceefficictits: then begin- ing with the firſt Part of the ſecond complex Term, viz. . * and running through all the ſubſ; ques Terms in order, ſetting down in another Series, all thoſe Parts of each complex Term that will conſtitute a ſecond geometrical Progreſſion, going on by the ſame common Multi- plier as before —, with their proper Ooefficients 28 5 then beginning with the firſt Part of the third complex Term, via. __ * — take out of the ſubſequent Terms, as before, ſuch Parts as conſtitute a ind grome- tricel Progreſſion proceeding by the ſame common Multiplier VIZ * with their proper Coefficients : and ſo do contintally. And thence | the Area of the Curve, having the propos'd Ordinate 2 Ri x a + bz" oh ca + dz" + «24 Cc. ſhall be expreſs'd as follows, W N L * 2 — 3 I ne Y = x of) xy rx r+1 TY xr + 2. TAT * — T TNF == 1 Na = Ci © i —= — * f g* | s 2 * x —b 23% 2 5+3 Exam aft en Yn — —— r ————rꝘꝛ EL ta „ e FF = r 1 &'1 14 E 2 = * A z * 2 x if *g x—a — FT, | ——ů—— Rau A e r* 2 x 3 x r+4 Tx fp #—a 5 | DDI e N fz Mr A x r+4 i . | WA; | = =I ef EO ia — —— — — ' ¶ ͤUñun — — — — — To Tin Fx The Ruadrature of Convus explained. — * es 0 F. wo | . \ | Co Fingers N. N, A n r Y rern lar , "I. FI x of heme * 1 T * =—==———_ nh xr Fa x mts F ga FOOT | | , 2 — 1 T 4x rpl xrÞ+4 Ge. in inf. Which Series exhibits the Value of the Area; and is the ſame with the immediately preceding one; only that it is reduced into Order; and divided into ſo many diſtinct Series's: the Progreſſion of which, when duly conſidered, will be manifeſt. For the Coefficients of the firſt Terms of each diſtinct rs conſtituting the firſt perpendicular Row, are — . . © &c. then the Coefficients of the ſubſequent r+1 = 9 perpendicular Rows, are all formed from the preceding in order: the ſecond U +» 464 nd perpen d on from the third, &c. after ſuch an eaſy and natural way, as will eaſily appear to any one that conſiders it. Moreover the Pro- reflion of the uppermoſt horizontal Rows, in each particular Series is evident. | | dr ©, ; 126. And now by confidering theſe two Forms of the Expreſſion of the Area, v/z. that exhibited by our Author; and the immediately receding one, we ſhall be enabled to diſcover ſeveral Properties, and E or Conditions, according to which, the Series will terminate or ſtop, after a certain Number of Terms from the Beginning; and when it will run out infinitely: and in what Caſes it may or may not con- verge. Of all which by and by. _ - | 127. Our Author obſerves * that eve ndicular. Row. from the firſt; the third from the ſreond; £ ry Ordinate of a Curve is ca- Art. 28. ble of being reſolvd into a Series two different Ways: viz. by mak- ef te ing the Index q either poſitive, or negative. Thus let the Ordinate be propoſed: it is reduced to the general Form men- * = F 2 tioned in this Prop. 19. by bringing the Quantity 2*/kz — [23 Þ+ mz+ or 2* * kz — 123 + niz*'* out of the Denominator into the Nume- rator : which is done by — the Signs of the Indexes of the Factors 2* and kz — E T mz#\* into their Oppoſites: by which the Ordinate is expreſs d thus z x kz — [23 + mz+*\”* x 3E —1Zʃ. 2. By purging the radical Expreſſion of 22: which is done by di- viding it by 2, and multiplying the other Factor 2—— by 2-7, which two Operations compenſate each other: for the radical Ex- preſſion muſt be made as ſimple as may be: and ſo the whole Expreſ- ſion of the Ordinate propos d, is reduc'd to this a x k—/z*+mz3]-+ * 34 — E=; or rather 2 ⁷ ⁰ - X —12* + 3-7, capa- ble of being compared with the general Form IR- X a+bz" +c27" +d23" + &c. i. e 25-1 x e+f2" +g2% S+hz3" + &c.| 1 Xa+b2"+ i" d + &c. This is one Form of the Ordinate, where the Indexes of z, in the two Factors k — E + mz3\—% and 3*— E. are poſitive. But it may be brought to another Form, in which theſe Indexes are negative, viz. thus: divide the radical or ſurd Factor & — [z2* + mz3\"* by zi, and multiply the other Factor 2 by the ſame 2)—* or 2, and then the Ordinate ſtands thus z Xx ETZ 3} — E=: further, divide the Factor 5 ED by z*, and multiply 2. by 8 ſame 2˙, and ſo youll | ave Quadra- ture of Curves. 1 * bn 4 , 4 = N 1 A hens . * 1 a 1 . * 4 k * 4 1 : Z ” o * ** © p > * * * * 4 * 8 % 3» : Iz bi ; X : »” 4 — Y 12 — & \ 4 * e Cc N n N ; e 9 * N i N r ee Y " + * 82 mne * 1 ** N FDI $4Y eren £634 K J 3 b ; d . 8 4 . *% * — - * — %% SL g 7 2 * ** * 4 N * 7 N : ** 1 7% Y 4 Ly 6 1 o a 4 : * of; N - "*, 2 7: — N 9 , 1 we U 0 2 — 7 * > > * ? 1 » . | "IS. | - 1 & 4 p a — Quadratur Cuꝝxv | nee. | x rature of "Curves explained. 97 & FUSES 6 * C4 * % > 4 þ % 5 0 98 Art. 127. of this Ex. plication-. The Quadrature of Cunvns explained. have 2— Xx m — E + kz—*\-* x/ — 3E —, or, which is the ſame, 2——" Xx m — e + &2—3}*—"* 3E, where the In- dexes of the Powers of 2 in the two laſt Factors, which were for- merly poſitive, are made negative, And after the like manner, every Ordinate of a Curve may be expreſs'd after theſe two different ways, Accordingly, when any Ordinate is propoſed, it is to be reduced to theſe two different Forms: and both Caſes are to be tried. Which is done by applying the general Form of the Area, exhibited in this Prop. to the particular Caſe propoſed, in ſuch manner as our Au- thor ſhews, viz. by ſubſtituting the particular Values inſtead of the correſponding general Expreſſions. By doing of which, the Series ex- prefling the Area, puts on two different Forms: one or both of which will terminate or become finite, by the Terms vaniſhing at length; or elſe both of them will run out into infinite Series's. if one or both of the Expreſſions of the Area terminate, you have the Area exactly, and ſo the Curve is quadrable: but if both the Expreſſions of the Area run out infinitely, one of them will converge, and give the Area by A proximation, when the Curve is not quadrable. * 1 128. Let DCs be a Curve, deſcrib'd by the Ordinate BC movin 8 | ndicularly along the Abſciſs AB, A | Ft AB =, BC=y, and 3, a -Bl.. i — given Line: and let the Equation to 8 2 the Curve be y= = This Or- dinate brought to due Form is 2* | NEED" Roar En Xx a, or yet, by the Method ſhewn above *, 2—=2 x 1+2—"\—* A, i.e, $1 x 1 +1" xe. By comparing the firſt Form of the Or- dinate, viz. zii X 1+2 I with the general Form 2%R*— X @ + bz" + &c. we find 8=1, | | 0 | 1=1.a=—1.r=(—=)1.4=(r+a=)0.c=1.f=1, go, Gc. a=a.b=0=c, Cc. Whence, by Subſtitution, 64 ofa of | the general Expreſſion for the Area, R Ti + ==" + Ge. becomes #x F N or 5555 All the Terms except the firſt va niſhing. The Quadrature of Curves explained. 99 niſhing. But the other Form of the Ordinate propoſed is x11 x IFK: which compar'd with the general Form, gives f=—I.q=—I, A2 —1 = I.s=(r Fa=)o, the reſt as formerly. Whence the Area, correſponding to this Form of the Ordinate, will be found, by a due Subſtitution of Values, in the general Expreſſion for the Area, 2 1 Xx — a, or I R all the Terms fave the ficſt vaniſhing in this Caſe alſo. 129. Hence it appears that we obtain two different Areas, both be- longing to one and the ſame Curve DCa. But the firſt © is the Area ABCD lying above the Ordinate BC, and comprehended betwixt it and the fame Line AD drawn parallel to BC through the Beginning of the Abſciſs A: whereas the other Area, viz. r is repreſented by the S «BC, lying below the Ordinate BC, and adjacent to the Abſciſs infinitely produced towards x. Now that there is nothing ſtrange in this; but on the contrary, that it is what ought naturally to happen, will appear, if we conſider that the two Areas ABCD and BC have one and the ſame common Fluxion : becauſe the Velocity with which the one flows by Increaſe, is the fame with that whereb the other flows by Decreaſe: or the poſitive Fluxion of the one is the negative Fluxion of the other ; accordingly while the one increaſes, the other decreaſes, and contrarily. And ſince a poſitive Fluxion produces a poſitive Fluent ; and a negative Fluxion, a negative Fluent, hence AB (=2) and conſequently the Area ABCD, being both ſuppoſed to flow by Increaſe, that Arca is repreſented poſitive, being f; but the other aBC is repreſented negative, being ==: And ſo it always happens, that two Areas, terminating at the ſame Ordinate of a Curve, 3 and ſituate upon different Sides of that Ordinate; but upon the ſame Side of the Abſciſs or Baſe, are affected with oppoſite Signs. And therefore fince the Area above the Ordinate, becauſe of it's poſitive Fluxion is poſitive; the Area below the Ordinate, if it lie upon the fame Side of the Abſciſs produc'd, muſt be negative. 130. There are ſeveral other Things reſpecting the Poſition and true Values of curvilinear Areas, of which more afterwards. Let it ſuffice juſt now to ſhew how one may determine the initial Limit of any Area, or the Term whence it commences. To do this, all that is neceſſary, 1s to ſuppoſe the Expreſſion of the Area equal to nothing, and the Value of the Abſciſs z, agreeing to that r will ſhew the 5 21 | 2 Imit. 100 25. Golla 1 co URVES. TY C Limit. Thus in the Example above, the firſt of the two Areas was 1 = whetelbreeput r 0 hence az o, 1. e. & o, ſince 4 is a given Quantity: therefore, when the Area 22 is nothing, the Ab- ſciſs z is nothing; and in no other Circumſtances can the Expreſſion 1 = be nothing, but when the Abſciſs z vaniſhes. And this ſhews that the initial Limit of the Area is at the Beginning of the Abſciſs, or is to be computed from the Line AD, drawn through A, parallel to BC: whence it follows that it is the. Area ABCD. Again put the other Expreſſion of the Area, viz. = o: and in that Caſe, z muſt be infinitely great, ſince otherwiſe the Expreſſion — could not be equal to nothing: which ſhews that the Area deſigned thereby, lies along the Abſciſs infinitely produced below the Ordinate BC: and therefore de- notes the Area aBC, as was faid. 131. Now, although the initial Limit, from whence any curvilinear Area commences, may be different from either of theſe; and fall in any other Place; yet wherever it falls, it may be determin'd the ſame way. Thus taking the Area — * = , Which is ſhewn by our 3 34— bz* Author to belong to the Curve, whoſe Ordinate Is ETD EEE — —2 — =0,and thence you'll have this Equation k — Iz + o mæs o, by the Conſtruction of which cubical Equation the Length of 2 the Abſciſs is determined; through the Extremity of which an Ordi- nate being drawn gives the initial Limit of the Area. And thus far with reſpect to the ffect of the two-fold Expreſſion or Form of the Ordinate, in determining the initial Limit and Poſition of the Area of the Curve, in order to * 10 and illuſtrate what ! is ſaid by our Author of te in Art. 28 *, . 4.7K herefore when an Ordinate of any Curve is 3 both 4 . Expreſion or Forms of the Ordinate muſt be tried : and ſometimes, the correſponding Areas will be finite, and exactly determined, as „Art 128. in the Example above *: ſometimes, one of them only can be exactly of this Ex- found, as in a e Example adduced by Sir 1/aac . One or other of theſe 1 will happen, if the Curve be quadrable, i. e. admit of a geometrical Com- of the pariſon with a rectilinear Figure. But if the Curve be not quadrable, Quadr?- both Expreſſions of the Area will run out into an infinite Series: yet Carte. one or other of them will converge, and thereby approximate to = 3 8 Value - put Ain 5 * hs ape, * — > - * , | * F = 9 , 2 — * *. #4 \ * x» = 7 "A | 5 1 1 x X * ; Pe r _— i A X. 0 De Quadrature of Curves explained. tions. 1%, When the Value of * in both Series's expreſſing the Area is O or a negative Integer; although it be not o in both; nor a negative Integer in both at the ſame Time. For ſince 7, r +1, T2. +3, Se. are Factors in the Denominators of the Terms of the Series ex- 0b value of the Area. To which notwithſtanding there are ſome Excep- — the Area of the Curve, according to the Author's Form of it ®, * Art. 26. ence it follows that if T o or a negative Integer, ſome Term muſt of the become - infinitely great, the Denominator vaniſhing :' and therefore if ture of this be the Caſe with reſpe& to both Forms of the Series, both that in Curves. which the Powers of z have poſitive Exponents; and that alſo in which they have negative Exponents, the Value of the Area in both Caſes is infinitely great, i. e. the Area lying on both Sides of the Ordinate. Whereas if r be o or a negative Integer in one of the Caſes only, you conclude from this that the Area upon the one Side'of the Ordinate is infinite; but finite upon the other: which muſt be obſerved, left the Ambiguity in the Manner it is expreſs d by Sir Iſaac, ſhould lead one into a Miſtake. Thus taking the Equation to the equilateral Hyperbola y: which reduced to Form is either 2*—" x 142-7 = y, 1＋ | or IZ —y; by comparing the laſt of which with the general Form of Ordinates 2%" X e + 2"! , you have /=0==a, and therefore r —= (= —) o: whence you conclude that the Area lying along the Abſciſs infinitely produced beyond the Ordinate, is infinitely great. Whereas, ſince in the former Caſe it is en, hence r— (= — 1, and therefore the Area adjacent to the Abſciſs and terminated at the Ordinate is finite, and ſo, according to this Prop. is thus expreſs'd z x 1 — 22 ＋ s — 23 + c. in inf. or & — 12 + i233 — 124 Cc. which converges, and approximates to the Value of the Area, if z do not exceed 1; otherwiſe not. But it muſt be obſerved that this Ex- ception muſt be taken with this Limitation, viz, That the Term whoſe *Denominator vaniſhes, have not it's Numerator equal to o at the ſame Time: for if that ſhould happen, the whole Term vaniſhes, and the Area is finite. There is another Exception mentioned by our Author againſt the general Rule about the Convergency of the Series by which the Area is expreſs'd : which is this, if - =1 the Series does not con- werge. In this Caſe the Powers of - are all the ſame, viz. 1, and therefore by obſerving the Form and Progreſſion of the Series inArt. 12 5 *, * of chip where © is the common Multiplier, in producing the laſt Factor agg each li n 102 De Quadrature of Curvns explained. each Term from that of the preceding, it appears that the Series can. not converge, unleſs it be by means of the Coefficients, which ſeldom happens : but that this ſometimes happens in the preſent Caſe, we ſhall demonftrate juſt now : and therefore this Exception muſt be taken with ſome Limitation : as likewiſe muſt that which follows in this Art. viz. That if = be leſs than 1, the Series will converge in which the Index y is poſitive ; but if = be greater than 1, the other Series will converge, For, that this will not always hold, will appear by the following Ex- ample. Let there be given this Equation y— 1 =2—! x e+fz%"; by comparing which with the general Expreſſion. 2%" x f, you have f=1, q=1 , A=0.r=(-=)1, 5=1 t whence the Series for the Area is Zu I , to = —Gc.im inf +z Eo F where, if we ſuppoſe E to be any thing greater than Unity, the Series will not converge, whatever the Value of = be. Thus ſuppoſe f= 3, =2, 2= 1, and therefore =— 2: then the Series for the Area will be 4X1 — x4 TIN - N + 4x24/*— Sc. in 2 where, it's evident, the Value of the Terms after the third, continues to increaſe ſtill more and more, and therefore the Series doth not con- 2 verge although - = —==+ be leſs than 1. 2. If it be 1, yet if F be any thing leſs than 1, the Series will converge by Art. 119, * Of this 121 *, for in that Caſe S I 1. And fo having ſhewn by this Exam- Explica- tion, ple that the Exception, and Rules mentioned by our Author, with re- ſpect to the Value of - in determining the Convergency or Non-con- vergency of the Series, muſt be taken with ſome Limitation, I ſhall here ſhew how you may determine when the Series for the Area of a binomial Curve will not converge ; and when it will converge. 133. In the Series for the Area of a binomial Curve, if = be greater than 1, the Series in which the Exponents of the Powers of z are poſitive cannot converge, whatever be the Value of . For the Series for binomial Curves, putting A, B, C, D, Ge. for the firſt, ſecond, third, fourth, &c. Terms ( and not for their Coefficients) with their proper Signs, ſtands thus z*R* x The Quadrature of Cv ITT] explained. 103 —— _—_— 2. „ fg. ihr fin ha fil 1s . Now, if you ſuppoſe s to be leſs than 1 1, then it's evident, that, ſince the Cofficients 2 I, 5 Cc. go on continually by the Addition of Units to the Numerator and Denominator, ſome Coeffi- cient muſt at length become a N poſitive Fraction, and always continue after that to increaſe, ſince 5 is leſs than 2 ＋ x ( where a negative Quantity is conſidered as leſs than o, or any poſitive Quan- oy, or yet a leſs negative) becauſe the Difference continuing the e the Ratio of the leſs to the greater will increaſe : therefore it's evident that if > 1, the Coefficient multiplied into a will at length ex- ceed 1; and fo the Terms, after that, muſt increaſe : but if s be greater than r ＋ 1, then it's evident that whether = be greater than 1; or equal to 1, the Value of the Terms muſt increaſe as the Series proceeds ; and therefore whatever be the Values of r and s, the Series in which the Exponents of the Powers of z are poſitive cannot con- verge, if it be 2 > I. 134. In any binomial Curve thus generally expreſs'd ai x] F/ == y, if the Quantity 2 be leſs than 1, the Series for the Area in which the Exponent is affirmative, will converge: but if = be greater than 1, the other Series, in which 3 is negative. For the Series for the Areas of Curves, by applying it to the Caſe of bino- racer Smog , nnr eta ION. n = * „ $ . » az? * * Yi als mial Curves, may ſtand thus R x 1 — 2 x Ls x 1x Fi x oft — == * r * — Cc. where, if ; r+1 «TZ Pz 7 in zA A... it be s xr +1, in the Senſe already mentioned, the Coefficients, after a certain Number of Terms from the Beginning, muſt continu- ally diminiſh (as is evident) and therefore the Series, in this Caſe, will rs WW converge by Lemmas 2. and 4. preceding. Again, if it be 5 >r +1, 5. and you take 7 an integral Number not leſs than dT TEES the c. Terms of the Series after the n ＋ 2 Place ſhall continually diminiſh or WW faſter and faſter, or in a greater Proportion continually ; and there- x, WF fore the Series ſhall converge, For calling P the Term which imme- ; we diately I. Quadrature' of on URVES ee diately precedes the n T 2 Term, and 2 next: from the Natute of the Series, you'll have Q= << =p; where, by inſerting for n, the Quantity 2 LAT ALY . and reducing, you'll have a= ( r —) P; conſequently if 7 be taken greater, -Q will be S——1 x offs" leſs than P: fo that hence it appears that the Value of the Terms after this conſtantly diminiſhes : and it is moreover evident that they muſt diminiſh Mill in a greater Ratio, ſince I > 2: wWhere- fore, in this Caſe alſo, the Series will converge. But finally if it be sS=r +1, then the Curve is quadrable : for A = 7, fince S$= N. Again, if it be © >1, I fay the Serics will converge in which the Powers of 2 are negative ; for ſince 2 >1, it muſt be = <1, and therefore, ſince the Powers of —— ra =" which i is leſs than I, run through the Terms of the Series, it appears by Lemma 2, and 4. above, and what has been juſt now demonſtrated, that the Series muſt erer Therefore in binomial Curves, which are not quadrable, whether 2 - be leſs than nothin g or greater than nothing, one or other of the Series's for the Area will converge : and if it be = — I 1, the Convergence will be the quicker (ceteris paribus) the leſs & is; but if = > I, the quicker, the greater — E. — 1 There is a third Caſe, viz. when & — 1. In which Caſe the Convergency depends entirely upon the Coefficients : therefore if 5 > r + 1, or, which is the ſame, if A> 1, neither of the Series's for the Area will converge, becauſe the Quantity "= II by which the pre- r. Term of the Series is multiplied, becomes greater than Unity. But if s xr + 1, or which is the ſame, a I, both the Series's for the Area will converge ; for the Quantity — 3+1 x 3+ 2 x 5+ 3, . r +1 x r +2 * r +3 x r+4, He. mn continually decreaſe within a finite Number of Terms, and be- come Jeſs than any given Quantity, and that in both Forms 85 the | | ries; de Quadrature of CunxvES explained. _—_ Ire Series: and the Terms being affected with the Signs F and! alter nately, each Term will be greater than all the ſubſequent Part of the _ "S Series. For let a—b4+r—d+e—f, &c. repreſent a Series of an oy infinite Number of Terms, which continually diminiſh, and are alter- * nate, the Amount of the Series is a 4- e, &c. — 6 - 4 —½ &c. be now a+ c + e, &. > b -|- d + f, &c. but b+ 4 ＋ V &c. ce, &c. therefore a > a—b +c— d+e—f, &c. in inf. Laſtly, if * x —= 1, the Curve is quadrable. Ac * 13 5. In Art. 29. our Author deſires to remark, that, if the Ordi- Art. 29— nate of a Curve be contain d under a rational Factor, which he de- —— = notes by Q, and another irreducible ſurd Factor, denoted by R“; and ture of the Quantity under the radical Sign, viz. R, do not divide the rational Curves. Factor Q, then the Expreſſion of the Ordinate, remaining as it ſtands; r, the Index of R in the general Form of Ordinates, viz. 2 Rt x a + bz" + c2*» + &c.) muſt be put equal to : fo that R= R“. For in this Caſe, the Expreſſion of the given Ordinate cannot be ren- dered more ſimple. And fo in the Application of the grvenl Theo- fem, we'll have a= + 1. But if Q can be divided by R, let it be divided, and then put a — l , or = R*+ ſo that Jr 2. Again, Q x R expreſſing any Ordinate of a Curve, if R divide Keie ie Ge. aller Q's divided twice, thrice, &c. reſpectively by R, you muſt put x — I 2, a - l +3, Sc. reſpectively; or R.) — RH; R. — R*T3, &c. This Ope- ration reduces the Expreſſion of the Ordinate to due Form; but it is evident it alters not it's Value: for Q R Ax RF — 8 * R == Sv R*T3, &c. 7 A Thus, y and > ſtanding for the Ordinate and Abſciſs, let it be y A = a FTI; here Q- , R=ao+2, 2 — 2 and R. A + Alt: and in regard a* — 2* can be divided by a + 2 once, I divide a* — 2* by a+ 2, and it quotes @ — 2, but then to compenſate that, IJ increaſe the Index of the Surd à , by Unity: and ſo the Expreſſion y = a* — 2* Xa lis reduced to this A—Z X a+3Z * F ==F, where R- = a+2Z 18 a2 2 and fo x +, Which Alteration being made upon the Form of the 8 the Area is found by this Propoſition. And ſo in other ſes. | | nie FP 136. At AO 106 The Quadrature of Cunvns explained. * Of the 136. At Art. 30 *. our Author puts another Caſe, viz. that the Or- re of dinate of a Curve propoſed, is a rational irreducible Fraction, having Curves, it's Denominator compounded of two or more Terms. In which Caſe, the Denominator of the Fraction muſt be reſolved into it's prime or in- compoſite Divifors-* : this done, either there muſt be ſome ptime Di- viſor that has no Fellow, the ſame with it; or then each prime Diviſor muſt. have at leaſt one Fellow equal to it. If the firſt be the Caſe, it's an Evidence the Curve cannot be ſquared (the Demonſtration of which *Art.139. ſhall be given by and by *:) but when the laſt happens, you muſt re- of this Ex-; pli cation, je& one prime Diviſor of each Set or Kind: this done, the Diviſor, which remains; or the Product contained under all the remaining Di- viſors, when there are more left, is to be put for R, in the general Expreſſion of the Ordinate, and —2 for a—2 or R for Rt, and the Ordinate propoſed, muſt be reduced to a Fraction having R- for it's Denominator : except it ſhould happen, that the Product contained under the fare prime Diviſors, be a true Square, or Cube, or Biqua- drate, Cc. in which Caſe, the correſponding Square-root, or Cube-root, or Biquadratic- root, is to be put for R, id the Indexes made negative, wiz. — 2, or — 3, or —4, &c. put for x reſpectively; or R- — R-, R= RC, R5— RR, Oc. and then the Ordinate is to be reduced to a Fraction having RA, or R., or Rs, Sc. for it's Denomi- nator. All which, with the Reaſon of it will be very plain by the following Examples. * e 137. Let © R be an Ordinate of a Curve, which is a rational irreducible Fraction; and the Example propoſed by the Author. The prime Diviſors of the Denominator are z — 1, z — 1, 2 — 1, one Set; 2 ＋ 2, 2 ＋ 2 another: here there is no prime Diviſor wanting a Fellow. Now tis evident that the Ordinate propoſed: is | 2 vhere the Denomina- | Z—1 x 2+ Z—1X&—1I XxX Z Þ 2 | tor of the firſt Factor is made up of the: ſeveral prime Diviſors, which, according to the Rule, are to be rejected: and the Denominator of the ſecond Factor, viz. $—1X2Z—1X2S+2, i. e. 21— 32 +2 is | I 2 1 1 to vo put for R, and ST OT EN — = or R for R, i. e. Z — Z TZ for R. ＋“ . Wherefore, the propoſed i | | F „„ Ordinate muſt be reduced to the Denominator 2 — 1 X 2—1 Xx 22 For doing this ſee 4rith. Univerſalis, cap. de Inventione Diviſorum. 1 . — — — ff S 00 The Quadrature of Curves explained. = zZ Þ 2. To do which, without altering the Value of the Expreſſion a - ta Ps =» you muſt multiply the Nu- t + 2+ — 523 — 2* + 83 — 4 5 | . merator 25 L S. — $23 by ſuch Prior (or the Product of ſuch Di- viſors, if chere were more than one) as is ſupernumerary in the De- nominator 2 — 1 K — 1X2+2 or 23 — 32 +2 above thoſe in the Denominator 2 — 1 X 2 7, which were to be rejected: here there is only one ſuch ſupernumerary Diviſor, viz. & — 1 : and ſo 88 . | N A 2 25 — 982+ + 323 ' the Ordinage 1 in this Form ===, or 23 x 4 25 — 3. . ; ME toon 2 9 1 1 * + 2+ — $23 . 4 21 . | was. 69 & ＋ 24 — $23 i x =o} 5 xt + $1 e Owe = ge f = z FI , x2—jz +2) *x 92 -+ 23, or, to bring it exactly to the general Form of Ordinates, 2—"xX2 =3z + 2x8 — 92 + 23, And after the fame Manner, may any other Ordinate, having the Properties mentioned, be reduced to due Form, and ſo be compared with the general Form of Ordinates, viz. 2—"R*—1 x a + bz» + 2» + Xe. or 1 x e + f2" + g + Kc. -i a + bz» + c2*» + &c. and by a due Subſtitution of Values, the Area found. Particularly, in the preſent Example, we will have (=4.n=1,a=—1.r=(==)4.s =(r+a=)3.i=(s+a=)2.v=(t+a=)1.c=2.f= —3}.-g=0.b=1.a=8,b=—g.c=0.d=1, Which Values being inſerted in the Series for finding the Areas of Curves, viz. I I I — —b — if A De - TI KIZ -A | 2R* x — _ —2 — —2 29 1 TY 787 + 2 ＋ &c, 3 1 oY * = ===: for ſince the ſecond, third and fourth Terms are each equal to nothing, i.e. BO C= D, and the Factors 2—38-＋r- 11 and 8 — 92 + 23 only Quadrinomials, it will be evident, by con- ſidering — zz, 0, Ti- out 2x2 — 37 + 2 xp + Ln + . Progreſſion of the Series, 8 all the ſubſequent Terms | | 2 zh — _ = — ͤ—w TI ‚—— I rn — — * 108 | zr 2 — 33 ＋＋ 2 == and 8 — gz +23, belonging to the Expreſſion of = 5 —_— rational irreducible Fraction: the prime Diviſors and bringing the propoſed Ordinate to due Form 3x | to the propoſed Ordinate +. _ > * De Quadrature of Curvns explained. in infinitum, muſt vaniſh, and therefore the Area of the Curve is VB. It is to be obſerved in this Example that the Factors the Ordinate, are to be conſidered as Quadrinomials, and not Trino- mials: for. the Name of the Quantity under the Vinculum, depends upon the Number of Terms, in which the Indexes of the Powers of 2, conſtitute an arithmetical Progreſſion : and thus theſe two F actors are to be conſidered as ſtanding in this Form 2 — 3z + o2* + IN and 8B — 92 + oz2* + 23 : for the Exponents of the Powers of x, vis. o, 1. 2. 3, &c. in the general Form of the Ordinate 2*— x e Tg + bz31 + Cc. x a+ bz" + cx + de + &c. conſtitute an arithmetical Progreſſion, _ 8 138. But let us next ſuppoſe the Ordinate of a Curve to be 2 + 3 of whoſe Denominator are 1+x.1+x.1+x one ſort; and *. X. *. «. x. another, and therefore no prime Diviſor without a Fellow: reje& one prime Divifor of each fort, and the Quotient is ITX I +xXxXxXxXx=x*++ 2x5 + x a true Square, hay- ing x*-+-x3 for it's Root; therefore, I put & &= for R, x*Þx3\-: for Ro; but * + x\-3 for R—17, vis. by reducing the Ordinate to the Denominator x* + x3 . The Manner of which Reduction, „ appears thus. . 2+ — 243 — a * * ＋ 3 z oe nf PITF IFT NN NN Terz — 2 + 3x Woe 1 GE 2+ 3x . i+xXx1+FXXXXXXXXX 1+x * K FE * I Tax iN 2 + 3x x x 2 + 3x & x 2 ＋ 2x * —_—_—_ ==) - = == x5 — XT NN TF T x x*| x* x 1 + a X x 1 x2 + 3x or —+—x1+x X 2 + 3x, which is equal in Value x5 32 + 3x7 Y 3 but brought to due Form, ſo as to be capable of being compared with the general Form of Ordinates, viz. 2 R x a+bz»Þc2**+ Tt : by comparing of which, we ſhall have 0 = — 4. 7228 1 1 Here we need proceed no further in the Compariſon, ſince 7 is a negative =... The Ruadrature of Counyas expleined. 209. egative In for this is an Evidence. the ariſing from thi 3 Horn of the On rr is infinite 4 r e to w what Art. 133, ” } 3 - was ſhe denn at Art. 427 1 convert. Saas: Wire * ET 1144 121.2. dry $=4.t=2. 9.0 WES #2 5 go, Kc. a 2 3. b . c o, &c. Hence, by inſerting cheſe Values in the Series for Areas, we find x>* Xx Ir NY“ Xx X — N= for the Area of the Curve whoſe Ordinate was. propos d. all the Terms ſave the firſt vaniſhing. And * 1 in all other Caſes of the like Nature. ele 139. But it was ſaid above 6. that, — the Demio of à ra- * Art 30. tional irreducible Fraction, expreffing the Ordinate of à Cutve, vis. f = in Terms made up of the Abſeiſs and known Quantities,” is reſolved ww into it's cant prime Diviſors *, if there be any of theſe prime Curves. Diviſors without a Fellow equal to it, the Curve is not tquadrable : EIS: Demonſtration of which 1 Ek! 2 to this Place. And now in order plication. to demonſtrate it: I ſuppoſe it is eaſily underſtood, that, if any Fluent, be a rational Fraction, * one * Quanti only x, having it's Fluxion x = 1, the Fluxion will be a rational Fraction : and if any fluxionary Expreſſion, including only one flowing Quantity x and it's Fluxion x = 1, be a rational Fraction, it's Fluent, i finite, muſt be a rational Fraction, alſo ; for the Fluxion of a Surd can never, be a rational Quantity; nor the Fluxion of a rational Integer, be a Fraction. Which Things are e cally deduced from what was ſhewn on Prop. 1. Wherefore let 75 5 be a rational irreducible Fraction expreſſing he Area of a Curve, which therefore is quadrable : then — is the Fraction of the curvilinear Area N, and conſequently the Value of the Ordinate when the Fluxion of the Abſciſs is Unity; and muſt be a rational fractional Quantity. Now in the Expreſſion 2 , every prime Diviſor of the Denominator D* muſt be capable of dividing it \ e Due, Kaz an in what den in un At. is meant fach as include the ſome 5 F „ . i, . * 0 * + + V - The "Ruadrature" of Cynvzs explained. - rg Number of Times, fine it a Square, and of dividing D ; > . t Number of Times, apd D juſt once leſs chan half the Num. mes, by Lans SHAH, Let P repreſerit any ſuch prime Di. FFF erator dt all: becauſe P divides ND the firſt Part of the Nl ſince it divides D once; but it n't divide the other Part of the Numerz- tor, fince it can t divide O, by Lemma 3, nor N. by Hyp.z N and D being prime 10 each other, and 40 not ND ; therefore it's plain, that P won't divide the Numerator ND — NH at all, when it divide the Denominator Dꝰ but twice. Again, if P divide D- four Times, it can divide ND -N only once: for it divides ND twice and | ND but once, for the fame Reaſon ay before, fherefore/it can divide ND =N only beg, "when it dice P. four Times: and thus che lame way of reaſoning, whatever Nuinber of Times any jerks Diviſor divides the Denaminator D, it will divide the Numerator ND — ND Juſt once leſs than half that Number of Times. Whence it plajnly follows that when n 1. It's loweſt Ter ms, theroca he no prime Divifor in the Deoralnater without a Fellow | to it, Therefore if the Ordinate of a Curve be a rational irreducible Frac- tion, which contains any prime Diviſfor-in it's Denominator, without Ane equal to it, the Curve cannot be ſquared. Q. E. 6. 40. If the Ordinate of a Curve be a rational irreducible FraQtion, whoſe & Depoiminatet: confiſtirg of ſeveral Terms, is not a true Square; and ſome prime Diviſor of that Denominator, be not contain'd at leaſt three Times in it, the Curve is nat quadrable : ag eaſily appears from what has been ſaid. Which is ſtill a further Limitation, beyond that mentioned by Sir J@ec. Moreover, let 4, J, c, d, &c. be the prime N contained in the Denominator thrice above ; and let a, 6 , &c. be the higheſt Powers of a, 5, c, d. &c. which divide the | Ne the Fraction in it's loweſt Terms, then the Curve is not quadrable, unleſs when the Denominator of the Fraftion in it's loweſt Terms being multiplied by a*=2b5-2c2=2>2, &c. makes a com- plete Square: for the. Multiplication, of the Numerator and Denomi- nator by this laſt Quantity, muſt bring back the — of the Ordinate to it's original Form: as may eaſily a from what has been ſaid: and therefore if the Curve be q , the Ordinate vil T oO ni en- r HEAL ger nn Kno S8 8 5 The Ruadrature of Cunyns, explained. 122 de brought dack to this Form n is dice Ros manage, when * Of the a true Square: ene ITY f the Ordinate is an irreducible Fractionm, \ of a rational, Fader, Which he. calls actor, denoted by R, t ! to due Form. Operation contains no D. 1-1 amounts to this. After you have thrown by one Pale Diviſor ot each Magnitude ot Sort, contained in the Root N, fee whether the Quantity Q, muſtipſied by the Diviſors of R. which : remain, either make up R, or any poſitive Power of R, whoſe Inden is an Integer, denoted by m then, if that be the Caſe, e general Ex- preſſion of the Ordinate, becomes R. viz. poſed fractional Ordinate, to the Denominator duction will be always made thus: multiply propoſed Fraction, by the Diviſors of the Root R, which 7 after one of carts — or Kind is thrown. by; and the Denominator Q R“, write R fe, and it is done. That the Thing may de perfeQtly evident, let 2 repreſent the propoſed fu tional Ordinate, and ſuppoſe 4. 5. e and . 6 S. c. ec to denote the component prime Diviſors of Qand R relpedinly, ſo. that the Ordinate may be repreſented _— where, after throwing by one prime Diviſor of each Sort, of R, viz. 4. 6 c and multipl ing Q== abe by all the remaining Diviſors of R, US. a. e. c, 15 Product is A ages = R. Therefore, it's evident, if N be multi- plied, alſo: by the remaining prime Diviſors a. c . o, the Fraction a Lo will be converted into this equivalent one. MI, abc x «\ cc * ec acxN acc x N o r = l After the ſame manner, if we ſuppoſe — == where; after throwing by 4. 6. c, oof a Cs prime Diviſors of R, and multiplying Oase, by the Product of the remaining ones, via. ac“, ae ar, which is the e of R 5 it will appear that —— = QxR” 4 Thee X & Fo — zo x a*be og > =) TRY And fo of others, | N ˖ — S #4 o - * 112 N Vadrature of Conv \eaplained. Thus let = EIS be the Ordinate of a Curve; Ir — gx* — x , oy prime Pride 15 EE ITT. are Y ＋ K* g+x, x after" thro ad 7 . multiply the rational Facto ee y 15 85 prime Driſor , 4+-x, the Product is 7% Wigg® = 23,5 Fs e you” tfan ane the Ordinate by multiplying the Nutticjator by! 4 K x and making the Denominator PEPE Dee, 4 Nands thus Win FTT 4 5 Pn fo fol wt EET DEE * ___— PF Agee i * 37 P e gr by comparing which, with the general Otlinates , Ra—rin, a PH cz” TE; and then ſübſtituting the particular Values in the Series ſor Areas, b . of the Curve 1 1 "_”__ 18 be 1 EE 5g 1 8 | | »b] — 7700 21 15 4+ f 10 1 "Our Author, in this Propoſition, exhibits a na Theorem or Canon, by which the Atea of any Curve, whoſe Ordinate is of this Form 2 * Y Þ 295 = Rc, may be deter- mined ; and that either perfectly true, viz. when the — — Area admits of a Nes b Compariſon with right-lined Figures; or elſe ſo near the Truth, that the Difference may ſtill be made leſs and leſs without End, viæ. when the Curve cannot be ſquared. But where- as, it may be of conſiderable Uſe, to know when a Curve may. or may not be ſquared:: i. e. when it's Area may be expreſſed by a finite Number of Terms, or cannot be ſo expreſſed: therefore I ſhall here lay down ſome general Rules reſpecting this Matter: and for that End ſhall diſtinguiſh ſome Claſſes of Curves that come under the general Form mentioned in this Propoſition ; or may be eaſily reduced to it, And they are theſe following : . ax! g. 2%. a7 + bt + car + det, Nec. . z“, az * DDs. 4. an* f E= Fe. at * N= x a+bz"=y. 69, 25-1 * X@felt- on 7 2-1 „ - f% "x 2 Ba" o cn Gee, = 8. 25-1 x - f2 SF 2 Pz Kc. = 5. Where the Sym- bols are to be underſtood in the ſame Senſe, they are taken by our 1 vix. 2 denotes the Abſciſs, and y the cerreſponding Ordi- 3 Nate + ern . . ³ - ĩ³ . ' j | N 5 — c—_— n — — ůNU—U—U—U—̃ ꝛ Ü ꝛ x ꝛ — ——— ——— — = =_ = — — « JT p 7 * 25 FX Wo 7 8 44 . Sat AST © *-- The” Quadrature of Curves: explained. nate: „e. de F. g. 0. . X alſo p.. are conſtant ter integral or nend poſitive or — 1 143. All Curves of this Form ate = may be ſquared, unleſs when it is 0 o: and the Area is 720. By For if the Ordinate a or 2e «a be compared with the general Frm of Ordinates 2*!R>=1x2 Y Te &c. you find o Fg, Kc. G O c, &c. R (e =!) 1 =, &. Therefore inſert this fifth Propoſition: and it will cafily appear, that all the Terms of - 1B | &c. 7 85 h, &c. being equal to nothing: we Ares mY * 2 Da becomes re 22. Which ede * exhibits the Area exactly, and gives it finite, unleſs when o; for then it becomes © > infinite. WB O. See alſo Art. 52 9. ele rebel are ſquared, | ew Am 903 Ex. 1. Suppoſe it be p , or PD =. ts 8 and b — 1 . or 0=2, the Area is pr: or by inſerting y for pz, = which gives the Area of the common. Parabola : being 5 2 of the Rectangle under the Abſciſs and Ordinate. Ex. 2. Let it be y2* —@=0 or az=*=y: 1 . or b = 1, therefore the Area is — 42 or — =; which denotes an hyperbolical Space lying betwixt the Hyperbola and Aſſymptote infinitely produced below the Ordinate. But if it had been 25 = 6 of ang, where - 1 = — or o, the Area comes out finitely great: which is the Caſe of the common Hyperbola : p 400 what muſt be done here, ſhall be ſhewn afterwards *. 145. All Curves of this Form a + bz! + c2r + dz', &c be ſquared, unleſs ſome of the n p. 9 r, &c. be — 1: and the Area is ff. "T7 1 z i &c. Q | For theſe Values into the general Expreſſion for the Area, 0 the Series after the firſt 2 . ", vaniſhing, becauſe. of the Letters , c, d, * > p 6 11 | r when the Number of Terms of the Value of Y, is 1 may . 11.4 The: Quedrature:of Cunrvis explained. For the complex Ordinate az? + ba, &c. is to be conſidered as compounded of the feveral ſimple Ordinates 42, bat, c, &: and therefore the Area of the Curve is cor ded of the ſeveral. Areas belonging to the ſimple Ordinates, which being aſſignable ; by. the l Rule, che Ares according thereto, will be ff oþ gt © +4 2 2. 22-3. 11 therefore by a proper Subſtitution of Values the Area will be 21 — 22 + 23. | 3 146. Cor. Hence it follows, if 2 + 2» + c2*» + d, &c. in inf. be a converging Series, approaching continually to the Value of che Ordinate of a Curve, then az + At + anti | 1 + ri — -23"+1 &c. in inf. will be an infinite converging Series, approach- ing continually to the Value of the correſponding Area, Caves the Relation And by n of whoſe: Abſciſs and Ordinate is expreſſed by any -algebraical Equa- tion howſoever affected, may be — by * A ing Series: when they cannot otherwiſe be ſquared,” by the Methods ſhewn in this Treatiſe. of Quadratures. For by means of ſuch an Bean, the Value of the Ordinate may always be expteſſed by an infinite Series of fimple Terms, containing the Powers of the Abſcis z and known Quantities only, when it cannot be expreſſed in finite Terms: and that by the Methods taught by our Author elſewhere *: and thence by what has been juſt now ſaid, the Area may be deter- mined by an Approximation. Thus if y3-+ ay ++ azy —24%23=0 be an Equation expreſſing the Relation of the Abſciſs and Ordinate of a Curve: by reſolving the Equation according to his Method, by an infinite Series, it is .y == @ — 7 ＋ + == &c. in inf. whence, See his Treatiſe of Analy/fs by Equations of an infinite Number of Term : likewiſe hi Method of Fluxions and infinite Series, Chap. 1. publiſhed Anno 1736. 8 by Ne Quodrature of Cunvge explained. 215 by what be tas, been juſt. nom ſeid, we Lal, have ar — fe. 7 — £ + == zocker: Nc. an approximate Value of che Are. Likewiſe if we have vr — 22 =}, an{Equation. to the Circle: by | 2 extracting the ſquare Root of + fon * according to his binomial The- orem, it bs — 2 — . — ig. — , — Ke, in inf. : where- | fore the circular Area adjacent to the Abel 2, which begins at the K* <5. Center, is * — - Tr = Ge. 5 being the Re- D C dius. Thus let ADE be a Quadrant of a Circle (ſee Fig. AES . ABS, and let BC be perpendicular to AB. Put r =1.8S==0;1. then the Arca ABCD is ES 8 „ ic 100 MM: — gy Kg W e Tin er eee: But 199 22 1, the Area e is 1 —7- _ a 10 On 11 — at nearly. 147. All Curves of thisForm agt=" x Xe wor =y, Lay: . be ſquared, when 3, that is ＋ is either equal to nothing, or a negative Integer, as — 1, — 2, — 3, &c. unleſs when r, that is =, is either nothing, or a negative Interger, not: greater than 5, at the ſame Time: in which Case de Area is infi- nitely great And the Area of the Curye is = * 6+ Iv * I ns of | - 7 x f* 2 „z js „ nix 8 * * PK Z 7 + . Wbere the Number of Terms of the Series is one, when-s = ©: 2, when s =— 1: 3, when s—= — 2, Sc. i. e. one more than the negative Units contain d in 3. In other Caſes the Curve is not quadrable. But it muſt be obſerved, that both Expreſ- ions of the Ordinate muſt be tried by the Rule; as well that | which has the Index of 2; under the Vinculum negative; as that, . ſitive. For b ering the Series for Areas in general, exhibited by , | our Au Ky in this Propoſition *: and applying it to the Caſe of a f the * f the imple: binomial Curve, ſuch as we are now ſpeaking of: where 5 it is b=0 c == d, &c. likewiſe g=0==hb=1, &c. in ur” c | ved 2 un, 116 tum, it —— . reduced to this Shape 20 X 7 1 4 —a * / — E ee . = *. 23 5 7 & De Vadrature of ComuRalky. * NR n n Pe aw F FA — . — F 2 * : { "| jp e, pr 9 5 r*. f 5 1 ei * 1 | Or, which comes to By ſame, * * e + far * wg * * £3.00 * 2 4: * Of this AS appears by Art. ie: for the w bee De, in this Caſe, is Explica- tion. or more of the precedi greſſion o reduc'd to the firſt horizontal Row, as it is expreſſed towards he End of that Art. Which, by taking the common Factor a, out of all the Terms, and prefixing it to- Fug common Mul- lice 25 makes the Series mentioned, VIS, 2 „ e + fan 420 13 3 £ r : fees, 4 A . 1 From which Series, expreſſing the Area of a ſimple binomial Curve in general, it appears that if it be's == ©, the ſecond and all the ſub- ſequent Terms will vaniſh, - becauſe of the common Factor 5 = 0 "Found in them all: if it be s+ I=0 or S=— 1, all the Terms "after the ſecond muſt vaniſh ; becauſe of the common Factor 5s + 1 Do: if it be s=—2 ora , all the Terms after the third _- muſt; vaniſh, and ſo on continualhy: wherefore if s be nothing, or a negative Integer, the Series will terminate; and the Nymbef of Terms, will be one more than the negative Units contain'd in = A: but if r be equal to nothing, or any negative Integer not greater than 5, the Denominators of ſome one or more of the Terms preceding that Term, which ſhould be the laſt, will vaniſh, and ſo make one or more Terms infinite: thus e. g. if it were s = 2 or S$+2=0, there would be three Terms; * t if, at the ſame Time, it be 12, or r——1, or r So, i. e. „ oh 2 , orr +1 So, or = o, one ng Denominators muſt contain a Factor equal to nothing, and thereby make the Term or Terms, to which it or they belon ng, infinite: and the Reaſoning, from the Nature and Pro- t he Series, is evidently the ſame, whenever 7 is nothing, or a negative Integer not greater than s. Moreover, it is evident, that in no other Caſe, except when 5 1s . Either nothing or a negative Integer, can the Series terminate: 1. e. in no . The Nuadrature of Cuxvas platens Caſe; can the Curve be ſquared :_theſe things being ſuppoſed | . to both Expreſſions of the Ordinate; as well that — 2 in the Index 3 is . 3 as that wherein it is affirmative ; which will appear by the i Therefore, &c. 265 27 — 108. + 1442* — 0423 421 Sign in the Denominator, is a true ny, whoſe cube Root is 3 — 42, thereſore I reduce it to this Tape =y, which is a ſim- ple binomial | Curve: When it is brought 2 Form, it is either 2211 3 — AE or 22 - * T == 1. In this laſt Caſe, by Lenne ing with the general Form of Ordinates for binomial Curves a2%-! -i, we have t=0.y=—1. r=(==) o, whence, without proceeding further, I conclude the Area, arifing from the ſecond Form of the Ordinate, to be infinite: which is al- ways the Caſe when o. But I cannot thence conclude that the propos d Curve cannot be ſquared, until I apply the Rule to the firſt Form of the Ordinate, viz. 28 * 3 — 42)” ; where we find. L. d r. A-. r=4.s=2z, hr being poſitive, ſhews that the Series for the Area, ariſing from this Form of the Ordinate, runs out into an infinite Series: and now I conclude chat the propoſed Curve cannot be quared. Ex. 24. Suppoſe it to be 28 Nr i= =. * Ordinate may, when brought to due Form, * — 22 * JA —+ —1 or 22"! x —4 +3214", From the firſt, we have o = . I. A -A r. o: whence I conclude the Curve may | be ſquared: and fince moreover it is e=3 .f=—4.4=2; tho Area, which according to the Rule, is thus generally expreſſed -— 1 * 28 will be, by Subſtitution of Values, 22* x 3—421-* x S: although, if you try the other Form of the Ordinate — * 7 38 E, you would find the nN for the Area to run out into an infinite Series. Ex. Ex. 1. Let — —— expreſs the ue ofthe Abſciſs and Ordinat of a Curve: by extracting the ſquare Root, it is ==: and becauſe the Quantity under the ra- 0 8 we Nada of Coutves plain. Ex. 4%. Let ch 4 — ac 4-y2%=0 be an Equation, con. talning the Relation of the Ab an Ordinate of a Curve: by taking the Value of y, it is y = = De where the Denominator being . Art. a true Square, I make it ſtand thus * =, or a2 N c -: 130, 137. | fk A Fx | dan, okck, being reduced to due Form, is either 1 x Cc or yet 2—*—"X I Tecz i. From the firſt Manner of ex- preſſing, we have 0 S . y—2.,Aa=—1.r7=1.,s=0 : whence I conclude the Area ariſing from this Form of the Ordinate to be finite and aſſignable: and ſince it is moreover em=cc . f = -l. a g=: the Area will be . Again, by comparing the ſecond Form of zz the propos d Ordinate, vis. n,. - Iren, with the general Form, we have 8 = — 2. 1 28 —2. A2 — 1. ri. $=0: whence it appears that the Area, ariſing from this other Form of the Ordinate, may be exactly determined likewiſe : and ſince it is moreover e -= cc. a ==; by a proper Subſtitution and Reduction, the Area will come out : By this means we obtain two Areas, perfectly aſſignable, and both belonging to the ſame Curve: 4 * art, Of which we gave an Example formerly ?. | * eiche. Ex. 46. Let az*"=1v/e 72" be the Ordinate of .a Curve: it may plication. ſtand thus aπαπꝰ/¾ x e+ fant or thus e K f Tei-. From the firſt, it is 0 2 2. =. AZ. r 2. : whence the Series for the Area, ariſing from this Form of the Ordinate runs Gat "infinitely. Wherefore I try the other Form azi=" = -: here I find g . N = -n. A=. r == — + , f==— 1: wheneer I conclude that the Curve may be ſquared : and ſince it is moreover e. F Se. a Sa, the Area by a proper Subſtitution of Values in the Series for the Areas of binomial Curves, will be = * = — — * = — IF -» = (by a proper Reduction) 4 X e+ felt x — en N 2 1577 ; 2 148. Cor. From what has been ſaid it appears, that, when we conſider ſimple binomial Curves that are capable of being ſquared: ſome are of that Nature that one Area only can be exactly found, as in the ſecond and fourth Examples: which Area is ſometimes tags The Qyadrature of Cunyas exploined. 319 from. that Form of the Ordinate, where , the Index of 2, is poſi: tive, as in the ſecond Example; Sumerer fg the ather Form of the Ordinate, where. y is negative, as in the fourth Example. But there are other Curves of ſuch a Nature that both Areas may be ex- ally found, as in Example third. This la Sort. we may call doubly Quadrable. Therefore it may be of uſe to have a Rule ready, at hand whereby. we may at fuſt View determine, when any propos d bino- mial Curve may, or may not be ſquared at all: and when it is doubly To find when a ſimple binomial Curve may, or may not be ſquared at all: and when it is doubly Quadrable. TA 149. Having brought the Ordinate of the Curve propoſed, to ſtand in due Form, viz. thus az%3 x e ;: being that Form, where and & being either affirmative or negative. { | If — + > be nothing, or a negative Integer ; and at the fame Time = be nothing, nor a negative Integer as ſmall or ſmaller than it; or if — 2 + 1 be nothing, er 4 negative Integer; and at the Game Time — = + I —A be not nothing, nor a negative Integer as fmall or ſmaller than it: then if either of theſe happen, the Curve may be ſquared: if neither of theſe happen, the Curve cannot be ſquared: if both of them happen, the Curve is doubly quadrable. The onſtration of this will be evident from what has been for- merly fail in Art. 147 * if it be confidered that the Ordinate a- x 5 Of this TDi may be alſo exprefied otherwiſe thus a == i x tion. Per-; for from the firſt, we have r=. ==>+a; and from the ſecond, = A +1 = + 1. 150. Hitherto we have been capable of laying down general Rules, by which, we may always determine, when any Curve, of the Kinds mentioned, is propoſed ; whether it may be ſquared, or not, and that immediately from the Properties of the Exponents. But with reſpect to trinomial Curves, viz. whoſe Ordinates come under this Form an- x e + fz" + gz, all we can do by means of the Proper- ties and Relations of the Symbols 6, y, x, which belong to the Ex- ponents, is to ſhew ſome Limitations, without which no ſuch Curve can be ſquared. And whereas there are infinite different ag” whic . 1 ee er es B e.3 .. & 120 De Quadrature of Cunvss explained. which ſubſiſting among all the Symbols 0. . X. . . g taken together make the 525 ö Ay we Zen ſhew how ſome of theſe Relation, may be inveſtigated. Finally, I ſhall lay down a Method or Rule, by which we can find, without much trouble, whether any propoſed ſimple'trinomial Curve may, or may not be ſquared at all. In order to which it will eaſily appear, by conſidering the general Series. for * Art. 26. Areas, exhibited. by our Author in this Propoſition *, and applying it - the of to the Caſe of a ſimple trinomial Curve, that it muſt put on this Shape ture of I | | I 2 D g x nt erz. TATA of this Ex- 1 re r+1xe 7 ＋ 2 * * pli cation. — — — — 12 Nen. de other 4 r ＋ 4 * r # {> = Parts of the Numeratots of each complex Term, vaniſhing, becauſe S oO , &c. and B= O ==. &c. in infinitum. From which we may deduce what follo woass s. 151. Any Curve of this Form az! x ez" + g2**'”' = yp, being propoſed, ſet down the ſeveral Terms in order, of which it's Area is compoſed, by ſubſtituting the particular Values, in ../, place of the Symbols, in the general Theorem or Series for the Area of trinomial Curves, expreſſed in the preceding Article: and if you come to two Terms *, women ſucceeding one an- other, both ſeparately taken, equal to nothing, the Curve may de ſquared: And the Area is juſt equal to the preceding Part of the Series: provided always that none of the preceding Terms become infinitely great, by r becoming o or any negative In- teger leſs than the Number of ſignificant Terms: which would make the Area infinitely great. If there be but two -fignih- cant Terms, it muſt neither be o, nor r ——1: if there be three Terms preceding the two Terms which break off the Series, + muſt not be o, — 1 nor — 2, and fo on: for there is no quadrable Curve of this Form, which can have but one Term, the Area being neceſſarily infinite. The Demonſtration of this will be evident, if it be conſidered, that there are only two capital Letters, in each complex - Term of the fore- 1 | — : By Terms here are meant Terms of the laſt Factor, conſtituting the Series — * * 2A, &c. N HE going De Quadrature' of Cx v explained. going Series for the Area of a trinomial Curve, ' after you paſs the ſe⸗ cond Term, in inf. which two capital Letters always denotè the CO. efficients of the two immediately preceding Terms: and theſe Capitals are each a Factor of one of the two Parts of which the Numerator of the Term conſiſts: therefore, if any two Terms immediately ſucceed- ing one another, vaniſh; and conſequently their Coefficients, denoted by the two capital Letters, contain'd in the Term immediately fol- lowing them, that Term muſt alſo vaniſh : and fo alſo the next Term after it, for the like Reaſon : and therefore all the ſubſequent Terms in inf. by the ſame way of reaſoning, Whence if two immediately S ſucceeding Terms vaniſh, all the ſubſequent Terms muſt alſo vaniſh : nnd therefore the Series muſt terminate with the Term immediately preceding them; and ſo exhibit the full Area of the Curve, and ſhew it to be quadrable : unleſs v by being equal to nothing; or to any ne- gative Integer, leſs than the Number of ſignificant Terms do make the Denominator of ſome Term to vaniſh: it's Numerator being ſup- ſed not to vaniſh : for in ſuch a Caſe one or more of the precedin ferme muſt be infinitely great: as eaſily appears by conſidering the Series, and what was formerly ſaid. Moreover, that there can be no quadrable Curve of this Form, 527 110 37 b OPEN IG whoſe Area conſiſts of no more but the firſt Term, via. — x 2! x Pg, will appear thus. When all the Terms, but the firſt vaniſh, we will have the third Term — fn ub RF Sdn oh gt o, that is, becauſe the ſecond Term alſo, and conſequently it's Coefficient de- noted by B, vaniſhes, gA —= o: which can only happen here, when | if A it is f=0: but ſeeing the ſecond Term —2 . 2%" = © : Which r+IxX#& cannot be in this Caſe, but when it is o: therefore when all the Terms of the Series except the firſt vaniſh, it muſt be = (o =)s : but * =s — a, therefore it is alſo Aa— ©: again, -A, there- fore it is r—=© : conſequently the firſt Term of the Series, and there- my the Area of the Curve is infinitely great. Which Things were to N. B. The Rule contain'd in this Article, may be applied to both Series's, which denote the Areas of the Curve ; both to kat where the Index ij is affirmative ; and that where it is negative. And by ſo doing, if you come at two immediately ſucceeding Terms, equal each to no- thing, in both Series's, the Curve is doubly quadrable. Moreover : R mu 121 122 Art. 18 1. of this Ex- plication. The Ruaarature of Curves explained. muſt be remarked: that r may, poſſibly be equal to a negative Integer leßß than the Number of ſignificant Terms, and yet the Area be eule and tho Curve quadrable: if ſo be the Numerator of that Term vaniſh, whoſe Denominator vaniſhes. Thus if it be r =— 2, the Denomi- nator of the third Term, vis. # + 2 xe vaniſhes, which makes that Term and conſequently the Area infinitely great, provided the Nume- rator TPI xfB THA don't. yaniſh at the fame Time: but if that - Numerator be nothing, then the third Term becomes nothing, and therefore the Area is finite, and the Curve will be quadrable upon the Conditions mentioned in this Art, And it muſt be obſerved, that the like Exception reſpecting the Infinity of Areas ariſing from the Value of r being nothing or a negative Integer, muſt be underſtood with the like Limitation in all the ſubſequent Rules contained in this Section. 152. In the preceding Article, I have demonſtrated that, when any two immediately ſucceeding Terms of either of the two Series's, for the Area of a ſimple trinomial Curve, vaniſh, and thereby break off the Series, the Curve is quadrable : and the Area made up of all the Terms preceding theſe. 1 ſhall next ſhew how one may inveſtigate certain Laws or Conditions, arifing from the Properties and Relations of the Quantities, which enter the Exprellion of * Ordinate: which Laws or Conditions obtaining, the Curve may be ſquared. 153. Put any two next adjoining Terms, beginning with the third and fourth *, of the Series expreffing the Area of any ſimple trinomial Curve, each equal to nothing: and thence will ariſe a certain Rela- tion, or certain Relations among the Quantities : which ſubſiſting, the Curve may lquoeretl:-. ids fs 1 rein 5 F | Suppoſe the third and fourth Terms equal to nothing: i. e, — tg Dus n, o and — R <2" o : then, r +2xe r+3xe becauſe C = (— D 22 ) o, therefore, from the ſecond r Xx Equation, by inſerting o for C, and diyiding by —_ we will have | | | | ze | #+ 1xgB=0: i. e. becauſe neither g nor B is nothing, #-þ 1=0 or r =— 1: whence s=(f—A=)—1—Aandr=(5s—A=) — 1 — 2A: inſert theſe Values of s and ? in place of them, in the firſt Equation — EARS; s o, and divide it br r 2 Xe | 2 Xe and it becomes — JB — gA = o: inſert for B its Value, vis. — E es a Mapper 1 123 . 8 7 1 \ 4 % — * — 1 — A = 1 4 * , WW w% : - * 2AXe . ; , , Whence it follows, that, bo ps "Lag 44 2 3 the third and fourth Terms of the Series, expreſſing the Value of the Area, both vaniſh: and * the Br that has ſuch an Ordi- nate, as that A = — I, and — 1 1 may be ſquared : pro- vided that z — #, which is the Value of f or 1, bs neither o: nor — 1 : as appears by the laſt Article, e ee a, Example. Let the Ordinate of a Curve be 2 = Zz N : duce it to Form and it ſtands thus, apt! x2 + 32) + 32M; where A = 7 2. —_— 123 Hera T3 3823 Wine DA in ( A. where fore the Curve may be ſquared, and be in y, by 2 the particular Values of the indefinite Symbok, in the Series which expreſſes the Value of the Area "of a Triamia , you'll find the Area "Art age. of the Curve propoſed, = 5 — 2 * g il enn — after the firſt two, of the geties vaniſhing + for {=i(r+ 2x=) —1, 154. Now, by the very ſame way of teaſoning, you may inveſtigate and diſcover other Properties and Relations of the ſeveral Quanties ( Y. *. e. F. and g: which ſubſiſting, a ſimple trinomial Curve, whoſe OP] is of this Form ag x , may beſquared: viz. by putting the fourth and fifth Terms; the fifth and fixth Terms; the fixth and ſeventh Terms, and ſo on, equal to nothing: bat the In- —.— becomes ſtill more and more tedious, and perplex d: be- cauſe it involves you ſtill in higher Equations: and then after all, you ſhall not be able to diſcover a general Rule, by which you might al- ways know when any ſimple trinomial Curve, may or may not be ſquared: for tis evident that all the Rules thus diſcovered, extend only to particular Caſes; by ſhewing ſome of the infinitely many Re- lations, which theſe Quantities bear, in order to make the Curve qua- drable. And I ſuppoſe it may be eafily gathered from what was ſaid R 2 formerly, — ——V—ñj— —— 124 We \Quadrature of 'Conves: e formerly, that this Method of Inveſtigation may be a ied to both *Art 147. Expreſfions of the Ordinate © * 11 _ IVC 3 what has been ſaid in the four preceding Articles, we may eaſily deduce ſeveral things, ſerving as Limitations or Conditions, with- out which no trinomial Curve can be ſquared. at all. Having therefore brought any propos d Ordinate of a Curve of this Sort, to ſtand in this Form a x HDD 1,” where 15 the Exponent of 2 under the Vincalam, muſt be ene, Then, 1 55. 19. If both — — + 2X and ·— — + 2 be nothing ; Fractions of any Kind, or poſitive Integers, the Curve cannot be ſquared : for theſe are the two Values of the Symbol 2, according to the two different Ways of expreſſing the Ordinate: and it is evident from what has been demonſtrated, that the Series expreſſing the Area cannot terminate, unleſs ? be a negative Integer: becauſe no two Terms immetlately ſucceeding one another can vaniſh, but ® Of this when 7 is a * re as appears from Art. 153 * — — go." L If both and — . — 2 + 2 . I, the Curve cannot be ſquared : for theſe are the two Values of 7, in the two different Expreſſions of the Ordinate: therefore, in this Caſe the (rage cannot be ſquared, we Area being infinitely yu Art. 151 * "FR 35 1 be Ao, the Curve cannot be ſquared : for in that -Caſe HAS . — 2x ==)t + but the Curve cannot be ſquared unleſs t be a negative Integer by Art. 155 : now if ? being a negative Integer, the Series terminate, the laſt fignificant Term of the Se- . Ties, becomes infinitely great: becauſe it's Denominator vaniſhes, when 7 t, and 7a negative Integer: and therefore the Area is infinitely great: which will appear by conſidering the Series at * Of this Art. 150 *; and what has been demonſtrating in Art. 151 *. — And now, from the Conſideration of all that has been ſaid about Of this trinomial Curves, we deduce this Explica- N | | RULE | 158. To know when any ſimple trinomial Curve, whoſe Ordinate is of this Form az! * e- NAT gz Pt, may, or may not be ſquared; and when it is doubly quadrable. 15 ſuch Curve being propoſed, ſet down it's Ordinate in that Form where 1 is affirmative : apply to it the Limitations . in rt. The Qeadrature of Convns exloin . 24s Art. 155. 156, 157 “: and if any of theſe Limitations ſhew that it? Of this cannot be ſquared, you proceed no further. But if none of them de- 3 termine this: then ſet down likewiſe the other Form of the Ordinate viz. that where y is negative: ſeek the two Values of t, belonging to the two different Expreſſions of the Ordinate ; one or other, or both of them muſt be negative Integers, from the Limitations already men- tioned: if only one of the Values of # be a negative Integer, you need only ſet down the Series for the Area, belonging to that Form of the Ordinate, according to which F is a negative Integer: but if both Values of # be negative Integers, you muſt ſet down both Series's for the Value of the Area: that where y is poſitive, and the other alſo where it is negative, Then, if one of the Values of ? only be a nega- tive Integer, it muſt be? + == o, or f ＋ 2 o, orf+3=6, &c. ſubſtitute the particular Values of the Letters, through all the different Terms of the correſponding Series for the Area, in order from the firſt, until * op come to that Term of the Series, which immedi- ately precedes Term, where you have f - 1 o, or f +2 Do, or t +3 =o, &c. and if that Term be not nothing, the Curve cannot be ſquared at all: if it be nothing the Curve is quadrable : (provided ſtill be not equal to o nor any negative Integer ſo ſmall or ſmaller than t) and the Area is equal to the preceding Terme. | But if both Values of ? be negative Integers, you muſt try both Series's for the Area, after the Manner juſt now mentioned: and if the Term of the Series's immediately preceding that Term, where you have f ＋ IS o, orf + 2=0, or ft + 3 =o, &c. be equal to nothing, in neither Series, the Curve is not quadrable : but if that Term be nothing, in either of the Series's, the Curve is quadrable, and you have the Area: if it be ſo in them both, the Curve is doubly 2 and you have both Areas: it being ſuppos'd ſtill as before, at the Value or Values of r be not o; nor any negative Integer or Integers fo ſmall or ſmaller than the Value or Values of f reſpectively. There needs nothing be ſaid for demonſtrating the Juſtneſs of this Rule or Solution, after what has been already -ſhewn : unleſs perhaps it ſhould be thought neceſſary to demonſtrate, that if the Term of the Se- ries for the Area, which immediately precedes that Term where it is t+1=0, orf+2=0, or f +3 Do, Cc. be not nothing, the Terms of the Series will go on in in,. ſo as that it can never terminate: nor conſequently the Area of the Curve, denoted by that Series, be exactly found. Which appears thus. 1 3 159. Suppoſe it were f=—2 or ＋ 2 =0: (See the Series at * of this Art. 150 *.) the Term where 7 ＋ 2 is found, is the fifth. Your 1 y . 126 We Nadruture of Cunyns explained. fay if the fourth Term, VIS, — thin CLE be not equal to nothing, the Series will never terminate: for if it terminate the fourth Term, or ſome Term after the fourth muſt be the laſt ſignificant Term of the Series: it cannot be the fourth Term; for then, ſince the next two Terms vaniſh, vis. D E and — Dr we have — LANE o b „ rÞ+5 | = ; * is alſo E o: for E denotes the Coefficient of the preceding Term, which is nothing: therefore we will have — 27 o, conſequent- rop5 xe , ly £:+ 3X2D =o: but g is not nothing, from the Nature of the frinom al; and D is not nothing by the Hyp. , for it is the Coeffi- cient of the fourth Term, therefore it muſt be f -- 3 — ©: but that is contrary to the Hyp., for it was ſuppos d that ? ＋ 2 =o. Now by the very ſame way of reaſoning the fifth cannot be the laſt ſignifi- cant Term, nor the ſixth, not any ſubſequent Term whatſoever in the Seties, ſtill for the ſame Reaſon, therefore the Series will never termi- nate. And it's evident, the Reaſoning is exactly the ſame if we ſup- poſe /=— 1, of f=—4, or 1 = — 4, Sc. i. e. univerſally, what- ever negative Integer be the Value of 7. Therefore, Sc. Q. E. D. 160. Cor, If the Series exhibiting the Value of the Arearof a ſim- ple trinomial Curve, do terminate, the Area will at leaſt contain two Terms of the Series, multiplied into 2 Rꝰ, vis. when it is fx 1: the Area will be exp by 3, 4 5, Sc. Terms multiplied into 2R*, when it is = — 2, 7=—3, f=—4, &c. reſpectively, i. e. the Number of Terms will be one more than the Number of ne- gative Units contain'd in the Value of 7. | Now I ſhall ſubjoin a few Examples of the Application of the pte- ceding Rule. 2 | Ex. 1. Let *＋νe= 3 42+ y expreſs the Relation of the Ab- cis and Ordinate of a Curve: when it is reduced to Form, accord- ing to the Rule, it ſtands thus 4 x 2-38 442+" : here Oo. . A. therefore - So; and 2 + 2x=23 ; and — Lþ2=2: therefore the Curve is of that Number that cannot be ſquared by 11 Art. 155 . The ſame may be concluded from Art. 1 56 *, ſince it is tion. LIL and = = ew 2A þ 2 == — . | Ex. The Quadrature of Curva explained. Ex. 2, Suppoſe a x 1-22-32") y 7 reduce it to Form according to the Rule, and it ſtands thus az"! > IT-: where we have 22 1 and —=+2=2. further = = and —- 2 2a +80: and fince it is not A==0 hence it ap- pears that this Curve comes not under the Limitations contain'd in Art. 155, 156, 157 “, ſo as that we can conclude it not to be quadra- * of this ble. Therefore ſince it is LINE 2A = — I, and 21. which are Pha the Values of f and r belonging to that Form of the Ordinate, where 4 is poſitive : I ſet down the Series for the Area where the Value of » js poſitive ; which by a proper Subſtitution of Values (ce Art. 150 5) * Of this ſtands thus af x 1—22-321"t 2 — az, Ge. Hence then tion. I conclude the Curve cannot be ſquared : for fince it is f = — 1, the third Term muſt have been nothing, in order to make the Series ter- minate : and the other Value of ? ariſing from the other Expreſſion of the Ordinate being +, the other Form of the Series for the Area, muſt run out infinitely, | N. B. The fame thing might have been concluded from Art. 1 53 *. Of this . Explica- Ex. 3. Let it be - 3 . a ä 3 1 : =: reduce it to the Form, where tion. the Value of j is poſitive: and it ſtands thus a2 — 1—2z+3z]*"" : *, » 3D 2 . Ly 2 6 0 where it is 0 . — 2. 2. A: whence = ＋ 2 =—1; — 5 ＋2 4, and . — 2: therefore I cannot conclude from Art. 1 —157 *, that the Curve cannot be ſquared, only I know it is not Rack. doubly quadrable. Wherefore, agreeably to the Rule, ſince the Va- tion. lue of t, belonging to the Form of the Ordinate, having » poſitive, is — I, and that the correſponding Value of r is — 2, a negative Integer not ſo fmall as /; I ſet down the ſeveral Terms of the corre- ſponding Series for the Area, according to the proper Values of the ge- neral Symbols: and thence I obtain 2=* x 122+ 32*)*x —tg9—E2—0 where, becauſe the third Term, which immediately precedes that one, where we have f ＋ 1 o, vaniſhes, the Curve is quadrable : and — — the full Area is, what I have juſt now ſet down; or - TA 6 — 22 + 32%. . | | 17 161. Schol. 128 De Quadrature of Curves explained. 158561. Schol. It appears from what has been faid about ſimple trinomial Curves, that Things of a like Nature might be demonſtrated with re. ſpe to ſimple Quadrinomials, viz. whoſe Ordinates are thus generally expreſſed az! e- CFA =; that is we might ſhew certain Conditions, without which no quadrinomial Curve can be ſquared at all, ariſing from the Relations and Properties of the Expo- nents 0. . A, analogous to what has been laid down in Art. 155— -» Of this 157 * concerning Trinomials: likewiſe, how to inveſtigate. certain Explica- Relations of all the Symbols . 1. X. e / g. h: which obtaining, the tion. * Of this Curve is quadrable ; like to what was done in Art. 153 *: and finally, | Explica- how a general Rule might be laid down; by which it might be known, non. whenany ſimple quadrinomial Curve may or may not be ſquared at all: * Of this analogous to that laid down in Art. 158 *. And ſo likewiſe with re- Explica- ſpect to Curves, whoſe Ordinates contain ſtill more complicate radi. cal Expreſſions. But this would lead us beyond our Deſign: it being ſufficient to have ſhewn theſe ſeveral things with reſpect to Trinomi. als: and thereby pointed out the Way for doing the ſame in theſe others, whoſe Ordinates are more complex. N 162. A Curve of this Form 2%! x e + f="V' x a+bz»=y may be ſquared, _. | 1, When it is F — 1 and the firſt Term of the Series for th Area multiplied into the common Factor RO, i. e. 2#x4fen\'% E ARE | | | | 2— exhibits the full Ares. RW | | 2% When s (==+ x) is any negative Integer, as — I, — 2, — 3, Sc. unleſs x ( ==) be either o or a negative, Integer not greater than s (in which Caſe the Area is infinitely great) and the Area of the Curve conſiſts of ſuch a Number of Terms of the Series, multiplied into the common Factor 2 R, as exceeds the . Number of negative Units contain'd in s, by one. Which two Rules are to be applied to both Forms of the Ordinate: and if they agree to either, the Curve is quadrable . if to nei- ther, the Curve cannot be ſquared : and if they agree to both, the Curve is doubly quadrable. | Theſe two Rules will eaſily be demonſtrated, from the Nature of the Series expreſſing the Area of ſuch a Curve: which Series, by re- ducing the general one, contain'd in our Author's fifth Propoſition, , | | ag M +; as . .. Fm. The Auadrature of Curves explained. to the preſent Caſe, will be thus expreſſed a x e +fe VP = — ine 2 * 4 „„ NA. tr 5 . Nature of the Progreſſion of this Series, it is evident, that, if any one Term, after you the firſt, vaniſh, all that follow it in inf. muſt vaniſh like wiſe: uſe the capital Letter, which is a Factor in each Term, denotes the Coefficient of the immediately pre- ceding Term, 5 | OP re abt os Now let us ugpols tte fecond Term —— = o: thence we have 25 — R =0: from which by inſerting = + 2 for s, and I > for A, and reducing, we obtain this Equation 2 = —1, There- fore converſely if it be 7 = — 1, the ſecond Term, and conſe- quently all the ſubſequent ones, of the Series, vaniſh. 2. If it bes=—1'ors+ 1=0o, the third Term, and conſe- quently all the ſubſequent Terms, vaniſh : if it be «= — 2, or 5s + 2 = 0, the fourth Term, and all following it, vaniſh ; and ſo on. There- fore fince no Term after the ſecond can vaniſh, but upon ſuppoſing s5--1=0 or 5+ 2 =o, Sc. and ſince the other things, with re- ſpect to the Values of 2, which make the Area infinitely great, eaſily appear from what has been ſaid formerly, the ſeveral things that were to be demonſtrated, are evident. ba» Ex. 1. Suppoſe —=< = Sy: reduce it to Form, and it may ſtand 2 thus, 82 — x 3—42Þt- "X2—32* :; which Form falls under the Conſideration of this Article : and by comparing it with the general Form of Ordinates of this Claſs, 2%! x e + fz"Þ”' x a + bz, we find P=2.,q=+.a=t.r=(5=)4.5s = 4%: whence it ap- pears that we cannot determine the Curve to be quadrable, by the ſecond Part of the Rule; from this Expreſſion of the Ordinate, Therefore I try it by the firſt Part of the Rule, to ſee whether it e — 1: and ſince it is a =2 5 23.283 f==4 8 I find The Quadrature of Cunxves explained. 1 find Fu x a=(p— 1=4=)%, therefore I 'cohclide the Curr, may be r the Terms of the Series after the firſt vaniſhing, Accordingly, by a due Subſtitution, the Area will be found to be * „ Ex. 2. Let it be = =, this E ſtands thus, 2— „ 4 a. 45% + 2; or thus, 21 3a + 4% 24 + 4632—*, In the firſt Form we have 2 D2 . N. I. . e = 4 = 34. a. 3 aa: where ſnce neither 5 is a negative Integer nor 7 . — 1: I proceed to the other Form where y is negative, there I find = 3. y =—2, N=. -. 1, thence I conclude the Curve is quadra- ble, and the Area conſiſts of two Terms of the Series multiplied into the common Factor R: and ſince in this Caſe it is e 34 f=443, a 2. U 405, by ſubſtituting theſe Values into the general The- 8 Ss REL orem for the Area, viz. 23x e + fz x 2. 4- De NS WE A SE PPP Iu * ri r azuxe „1 * 465 - oa 22 | a Ge. you'll obtain F 93 ＋3 * for the full Area, after a ptoper Reduction. E 1563 . A Curve of this Form ae & a + ba» c may be ſquared. od i aa | 1. When it is 2 = — :- provided it: be not 7 o, nor yet == — 1 (in which Caſe the Area will be infinitely great) and the Series for the Area terminates with the ſecond Term, 2. When is equal to any negative Integer greater than — 1 : pro- vided r be not © nor any negative Integer ſo {mall or ſmaller than : (in which Caſes the Area becomes infinitely great) and the de- tiles expreſſing the Area, will conſiſt of as many Tertns and one more, multiplied into R, as g contains negative Units. The Curve is quadrable, if theſe Conditions or Properties agree to the Ordinate in either of it's Forms : it is doubly quadrable, if they agtee to both: but if none of them agree to either, the Curve cannot be ſquared at all. | Ping Theſe Things will eaſily appear, by conſidering the Series for the Areas of Curves, exhibited by our Author: when it is reduced to ay '$ | | preſent ri 8 B44 The Reuanrature of "Cunvss. explained. 25 131 48 | I "$213: 235 1 e een PPR preſent Caſe : for it will ſtand thus AxeFfo x2 +? _—# „ Y "Xo j ? * + . ule $248 „ . Sc. where xr +2 * r+3Xe r * e it is evident, if any Term aſter the ſecond, become nothing, all the ſubſequent Terms in inf. muſt vaniſh : as appears by what was ſaid in 53 | ett qc—i+1xf/B the preceding Art. Now if the third Term - = . De be put equal to nothing, you'll thence obtain 22 = A L. whence the firſt Part of the Rule appears. 4 Again, if it be = - 2 or $2 =o, the fourth Term vaniſhes: if it be = — 3, or 5s 3 , the fifth Term vaniſhes, and fo on. But it is evident, from what has been juſt now ſaid, that the third Term of the Series cannot vaniſh, unleſs it be = =X% md ce> if it don't vaniſh, none of the ſubſequent Terms can vaniſh, unleſs s be a negative Integer greater than — 1 : therefore if none of theſe hap pen, the Series will run out infinitely : conſequently, fince the other things, reſpecting the Value of 7, are plain by Inſpection of the Series: the ſeveral things belonging to this Rule, are abundantly evident. 164. When a Curve is of this Form zi x - . a+bz2"4-c2" + dz*" + &c. = y, where the laſt Factor à + bzn + c2*", &c. conſiſts of four, or five, or more Terms: it is quadrable 1%, When —d= s+2xfC; or g = f 3»fD, &c. reſpectively. 2. When 5 is equal to a negative Integer not leſs than 3 or 4, Ee. reſpectively : provided till that no Term become infinitely great by rr being equal to nothing, or any negative Integer ſo ſmall or ſmaller than 3. It being ſuppos'd, you apply theſe two Rules to men And if neither of theſe happen the Curve is not quadrable. Which things are evident enough from what hath been already ſaid. 165. Finally, when a Curve of this Form 2%: x e+fzr+gz2- =" X a - bz" + c2*"» + c. y is propoſed, where the Quantity under the Vinculum of the Root is a Trinomial ; and the laſt Factor a tie, &c, conſiſts of two, or three, or more Terms: proceed thus. S 2 Set 132 pe - *%. 4 3 * — *. * b 4 „ 0 6 8 * as. | * * o "The Quadrature of Co xVis explained. © Set down in order the ſeveral Terms of the Series expteſſing the Area, by ſubſtituting the particular Values anſwering to the general Symbols in the Theorem exhibited by our Author in this fifth Prop.; and after you have ſet down as many Terms of that Series as there are Terms in 0% Factor of the Expreſſion of the Ordinate, viz. 4 -þ bz" + c2*", Cc. 8 | 111 the laſt two of theſe Terms be each equal to nothing, the Curve is quadrable: and the Terms preceding theſe laſt two, multiplied into the common Factor 2%R*, make up the Value of the Area. 2. If the haſt one of theſe Terms only be found equal to nothing, and the Symbol ? be any negative Integer (among which o is included) equal to the Number of Terms contained in the Factor a ＋ 624. c22* T c. diminiſhed by 2, the Curve is quadrable; and the Area is equal to the preceding Terms multiplied into the common Factor R. | | 1% | 3*. If neither of the two preceding Caſes happen, then ſee whether t be a negative Integer as great or greater than the Number of Terms leſs one, contain'd in the Factor a + bz» + c2*", Sc. for if it be not, the Curve cannot 'be ſquared : if it be, continue to write down the Terms of the Series for the Area, until you atrive at that Term which immediately precedes the one where it is ? + 1 =o, or f +2 D, or r ＋ 3 Do, &c. and if ſuch Term be not nothing the Curve is not quadrable : but if it be "_ the Curve is quadrable ; and you have got the Value of the Area: for all the ſubſequent Terms in inf. va- niſh : it being ſtill ſuppoſed in the Caſes where I have determined the Curve to be quadrable, that the Symbol r be not mary » nor any ne- gative Integer ſo ſmall as to render any Term from the firſt to the laſt mentioned, infinitely great, in the Manner formerly explained ; which will appear as you proceed in writing down the Forms of the Series : and moreover, that the above Rules be applied to both Terms of the Series; both to that in which the Powers of z are poſitive ; and alſo to that in which they are negative. | Theſe things will eaſily appear to any one that conſiders the Pro- greſſion of the Series for expreſſing the Areas of Curves; together with what hath been ſaid already upon this Subject. And this is all I ſhall ſay with reſpect to ſuch Curves, to determine when they are quadra- ble, and when they are not : becauſe although there might be other * particular Rules given for this Purpoſe, yet they are too particular and 1ntricate to be of any conſiderable Uſe. hehe 166. Cor. It eaſily follows from what has been ſaid with reſpect to binomial Curves, thus univerſally expreſſed - x e F x n 4 + — g ccc . - 74 ” 9 4%. *T a be! 1d bp 2 , 4 4 7 * 1 8 de Quadrature of Cunvss explained. 133 oF ban + ＋ G. =y, that they may be ſquared when s (= L) i cqual to any negative Integer (among which © is in- * f | THI | . Auded) not leſs than the greateſt Repetition of y among the Expo- Kern 95 Z in the laſt Factor a + bz» ca + &c. which is always one leſs than the Number of regular Terms contain'd in that Factor: # e. if there be none but the firſt Term a; the Curve may always be uared, when ; is o, or — 1, or — 2, &c. : when there are two erms, viz. 4 + bz", it may be done if 5s be — 1, or — 2, or — z, Gr.: if there are three Terms, as a + bz" cz, or a # + 2": it can be done when is 2, or — 3, or — 4, &c.and ſo for others: provided till that 1 ( = 7) be not o, nor any negative Integer fo ſmall, or ſmaller than . | Gr In fimple Binomials, where it is 2%" x e. ν x az®", or 421 * Ti, the Curve can never be ſquaredin any other Caſe *: 1 and in others, it very ſeldom happens * : ſuppoſing till that both gitentlen. Forms or Expreſſions of the Ordinate be tried by this Rule. . 16g. | of this Ex- ; | | plication, SECT. V. Of the Poſition and Limits of the Areas of Curves. 167, & hp Areas of Curves, diſcovered by the Method of Qua- dratures, laid down by our Author in this Treatiſe, may have various Poſitions and Limits, according to the different Natures and Properties of the Curves, whoſe Areas are thus determined. There- fore that the young Geometrician may not be at a loſs with reſpe& to this Affair, I ſhall explain by ſome Examples, how this Variety hap- pens: and then ſhew, by what means, the Area of a Curve, adjacent to any Portion of the Abſciſs, may always be found : when once the Area of the Curve in general is diſcovered, by the Methods taught by our Author. | | 1 2 168. In order to which, ſuppoſe aa ? = y be an Equation, expreſ- fing the Relation of the Abſciſs, and perpendicular Ordinate of a 4d - | | 11 Curve, The Area, found by the Method of Quadratures, is _ S of W And this Area, as likewiſe, every other curvilinear Area, found by plication. the ſame Method, is bounded by the Ordinate drawn through the Ex- tremity of the Abſciſs, and more or leſs of the Abſciſs and Curve-line, CRETE? | proceeding 134 The Quadratute of Cunvis explained. - proceeding from the Extremities of that Ordinate: but this Ares ſometimes lyes upon the. nearer Side of the Ordinate; ( which? mean the ſame Side the Beginning of the Abſciſs lyes upon) at other times, it lyes upon the further Side of the Ordinate: it may ha ben likewiſe, that an Ne may lye partly upon one Side, and partly u the other, in ſome Curves, according to the different Terms, by w ich 1 0 Thus in the Example e juſt now mentioned, | where the Area is —— that Area is expreſſed. Now that you may know how far the Area extends, and upon what Side of the Ordinate it is ſituate, and conſe. quently irs initial Limit, from whence it is to be computed, put the — ion of the Area equal to nothing, and the Value of z thence ng, or which agrees to that Suppoſition, will give you this initial Limit: 4. e. an Ordinate drawn through the Extremity of &, taken of that Length; which is thus determin'd, will be the initial Limit of the Area : and ſo: you'lt have the Limits of the Ares upon/evety Side. r ) * | put or ſuppoſe 2 Ar o: * i the Index j + 1 be poſitive, it is evident, this will happen when 2 o: and cannot happen in any other Caſe. But if the Index 14 + 1 be negative, the Area denoted RET | by the Expreſſion above, viz. ' IT" can never vaniſh, but when z is infinitely great. Hence you conclude, that in the former Caſe, the * commences at the Beginning of the Abſciſs, where alſo the Curve interſects it, and ſo reaches from that, along the Abſciſs, all the way to the Ordinate. Whereas in the other Caſe, it reaches alopg the Ab- | ſciſs infinitely produced beyond the Ordinate. Let AB be the Abſcis, and BC the perpendicular Ordinate, belonging to the Curve AC and take BD = a: ſuppoſe likewiſe, in the preceding Example, y— : ſo that ES I © Property of e Curve be 2 | FEET ant by the Equation az* y; by inſerting + for 7, the Area is NY = 2BD x AB or AB x BC, which is the Area ABC: fince the Ex- preflion 2a2* can never be equal to nothing, but when z = AB is nothing. But, if the Relation of AB toBC be exprefled * this Equation 85 = or „r- (ſee the Fig. in the next Page) — wha by * —2 for y, becomes * 4 S. S Sergei gaga gr ſo — — . wy e. F =, 2 —= ' 1 A Ar ichn ann aal tzecomes — a of — : which denotes the Arek BCa lying por K . S. EN N USS 3842 The Raasrature of Cur yRsaflointd. the forthbr Side of the Orfinare B and extendin along the Line Ba „ A F 2 10 1A alt: f infinitely, pr uced : for if you put 114 Der yo ene ten don 7603 bah great, finee u 18 a glven Quantity: $9 | [4% 4 | which fhews that the initial Limite 5 of the Area removes to un infnite Diſtance. This Area comes out negative, in regard it is fitunte upon 1 1 the further 'Side of the Ordinate (c BC, and wholly above the Abſciſs © * Pd. | AB produc'd : for ſince the Area c | ABC upon the nearer Side of the Ordinate, and above the Abſcifs is gh NE poſitive, the Area BCa ought to be 3 f repreſented negative, from the oppoſite Nature of poſitive and n tive Quantities, Thus if we ſuppoſe, g== 1, and æ = AB=1. allo, we have — = —— 1 for the Area BC: whoſe poſitive Value there- fore is 1 = AB! or the 5 — AC, for in this Caſe AB = BC. 169. After the ſame Manner, and for the like Reaſon, it will be found, that the Area of every Curve, which conſiſts of any Number of ſimple Terms, in every one of whoſe Numerators there is contain d ſome poſitive Power of z, will have it's initial Limit, where the Ab- ſeiſs begins: And every Area whoſe Expreſſion contains ſome poſitive Power of x in the Denominator of every Tetm, will be ſituate upon the further Side of the Ordinate, and extend itſelf along the Abſciſs in- finitely produc'd beyond the Ordinate. UE Let a + bz + ca y be an Equation, containing the Relation of the Abſciſs and Ordinate of a Curve. The Area, by Art. 145 “, is ® Of this oz + 3b2* 4- gc: where the poſitive Powers of z run through the garen. Numerators of all the Terms: therefore the initial Limit of that Area is at the Beginning of the Abſciſs: and the Area is bounded by an Or- dinate drawn-through the Beginning of the Abſciſs, the Ordinate be- longing to the Abſciſs z, and the Abſciſs and Curve-line intercepted betwixt theſe' two Ordinates. | EIT | Fx by 170. I have ſaid, yon muſt draw an Ordinate to the Curve at the Beginning of the Abſciſs in this Example, in order to have the initial Limit of the Area: becauſe the Curve does not meet with the Abſciſs at the Beginning, as is evident: for ſince y=a + bz + *; wed 136 the Roots of the Equation c2* + bz + @ = o, are imaginary : and t - \ . Ne Buadrature of Curves explaintd: the Abſciſe 2 o, we have y : which ſhews the Ordinate at the Beginning of the Abſciſs, to be to the given Quantity a, as is te- preſented in the Fig. annex'd : w if you draw AD perpendicular to the Abſciſs AB ; meeting the Curve in the Point D, the Area ABCD is that denoted by the Expreſſion az + 5bz* + 3c2%; and AD a. And that you may know in Caſe, where the Curve meets with the or Abſciſs produc'd, if need be, towards both Hands, ſuppoſe the Value of y equal to nothing, and the Value or Values of | 2, which ariſe from that Suppoſition, will | fhew you the Point or Points of the Abſciſs, in which the Curve- line meets with it. Thus in the Example juſt now mentioned, where y = @ + bz Tc, put c2* + bz + @=0; reſolve the Equation, and you'll find a double Value of 2, viz, = EL, both which Values are negative, and ſhew that if the Abſciſs AB be produced backwards, and the Points H and I be taken ſuch, that AH= 1 and AI — ——— the Curve CD continued, will meet with the Abſciſs produc d, in the Points H and I: and in no other Points, in regard z has no other Value in the Equation @ + bz + c2* =o, i. e. when y is equal to nothing. But if the given Quantities a, ô and c be ſuch that it be * = 4c, then the two Points H and I fall into one, becauſe V — gac be- comes nothing: and ſo there will be but one Point of the Abſciſs, in which the Curve will meet with it, diſtant from the Beginning of the Abſciſs, produc'd backwards, by the given Quantity 2. | Again, if 4ac be greater than 4*, the Expreſſion /F— Jac is im- poſſible or imaginary, and ſo there will be no poſſible Value of 2, or this ſhews that the Curve can no where meet with the Abſciſs, in this Caſe. And by the ſame Method, you may always find whether any Curve propoſed meets with it's Abſciſs or not; and if it do, in what Point or Points of the Abſciſs it meets with it. The Rule is this. 171. Put the Value of the Ordinate, expreſſed by given Quantities and the Abſciſs z, equal to nothing: and the Values of x thence ari- ſing, will be the Diſtances of theſe Points of Concurrence from the * erg. of the Abſciſs : and if all the Roots of the Equation be im- poſſible, there is no Point of Concurrence. 172, But BE 8. 4 N 3 889 ff *© = my 75 9, @ 55 FS.. „ r e 8 & © 5 5 The Quadrature of Curves explained. 172. But to return to the Limits of Areas, If the poſitive Powers of z run through the Denominators of all the Terms of the Expreſ- fon of the Area, the initial Limit runs off to an infinite Diſtance and ſo the Area extends itſelf from the Ordinate, on the further Side, all along the Abſciſs infinitely produc'd beyond the Ordinate. Thus in the Fig · p· 13 5, if the Relation of AB to BC were defin'd by this Equation 2—* + 2—+ =y: the Area thence ariſing, Is — 2 — — 22-4, of — . a g © _ = b E . * 137 Los - ; where the poſitive Powers of z are found in the Denomi- *Art.145. 3 0 nators of the ſeveral Terms; and both Terms are negative: therefore — it is the Area BCa, infinitely extended upon the further Side of the Ordinate: for if you put — — — I o, you find the Value of 2, infinite : which ſhews that that Area can never be nothing, while the Abſciſs AB is of a finite Length : but it is evident from the very Nature of the Expreſſion — — — 4, that, as z or AB diminiſhes, it increaſes ; and as AB increaſes, it continually diminiſhes : therefore, it muſt lye beyond the Ordinate BC: and it alſo lyes wholly above the Abſciſs, as will appear by the preceding Art. For if you put + 4 (=y) So, 2 can have no finite Value. From both which Conſiderations and Circumſtances it appears, why it is repreſented ne- gative. Let AB () , then — — — - 144 . — — 4 1 I7, which being made poſitive, we have 1; for the Value of the Area BC. 173 But ſometimes the Area of a Curve will lye partly above, and partly below the Abſciſs or Baſe : which happens, when the Curve decuſſates or croſſes the Baſe, and paſſes to the oppoſite Side, as is re- preſented by the Figures in p. 136, 140 and 141. In order to diſcover whether there be any Points of Decuſſation; and if they be, where the are ; do this, Find the Points of the Abſciſs, where the Curve meets wit it, by Art. 171 *, And if there be no Point of Concurrence, it is. Of evident, there can be no Point of Decuſſation: in which Caſe the Explica- Area muſt lye wholly upon one Side of the Abſciſs. But if you find tion. Points of Concurrence, then you may know whether any of theſe be a Point of Decuſſation, thus: add to, and ſubtract from the given Length of the Abſciſs at the Point of Concurrence, a very ſmall Quan- tity, which call p: and inſtead of 2, in the Value of y, inſert z + p, and then z =: then neglecting all the Terms, where p riſes higher than the Root, or other loweſt Power ; obſerve whether the Values of | T . f this Ex- this 138 Of the Quadra- ture of Curves. 3, which remain, be, the one affirmative, the other negative: if they you obtain 15 De Quadrature of Cunvns explained. be, there is a Point of Decuſſation, at that Point of Concurrence : if not, but that both Values be affirmative, or both negative, there is no Point of Decuſſation, in that Place: and if one of the Values involve any impoſſible Suppoſition, the Curve is reflected back towards the Parts from whence it came, The Reaſon of this Rule will eaſily appear, by conſidering, that, when there is a Point of Decuſſation, the Values of the Ordinate to- wards either hand of that Point, muſt be affected with oppoſite Signs: and the Powers of p above the Root, or above the loweſt Power, may be neglected, becauſe p is ſuppos d to be diminiſhed infinitely. (See Art, 17 *. and Note 450 it). 5 The Thing will be further illuſtrated by Examples. Ex. 1. Let AB=2z, and BC=y, be the Abſciſs and Ordinate of the Curve CDE (fee the Fig. in p. 141.) and ſuppoſe - — n expreſs the — > > FO 2 2 Relation of the Abſciſs and Ordinate : then putting — _ — 3 Do, we have 2 = infinite; and æ f: ſo that at the Extremity of the Abſciſs 1, there is a Point of Concurrence, wherefore I take z P br and then 2 — p, and inſert theſe for z, in the Value of y, v:z. by — 4 : thence =, Le. by inſerting the Value of 2 at I Y N that Point, viz. 1, — 1 N plainly negative: ſince 1 + is greater than 1. _ by inſerting 1 — p for 2 you'll have 2* 18 == e. & = 2 11 9 '* ==; Poſitive, becauſe 1 — is leſs than 1: wherefore if you take Mp 1 the Point D is a Point of 0 Ex. 24, In the Curve CDH I, (ſee the Fig. in p. 136.) whoſe Abſcif is ABD, and Ordinate BC==y : let it be e by putting a7 a+ bz-+c2*=0, you. obtain & = = — and 2== — — —, denoting two Points of Coturitnce as before. For the firſt, take N = +, and then AL CET p, or by ſubſtituting 4 for + EEE „ (for Brevity's fake) q ＋ p and 9 — p: inſert theſe - 1 4 M37 ” — oo The Qtadrature of CV RVS explained. — Neglect or throw away what is common in theſe two Values of y; what remains is 1 Wig bp in the firſt; and — 2cqp — y in the ſe- cond: which are, the one affirmative ; the other negative: therefore if you produce the Abſciſs AB backwards to the Point H, and take AH === 4 the Point H will be a Point of Decuſſation. And after the ſame Manner, if you produce it ſtill further, in the fame Direction, to the Point I, ſo that it be AT = , the Point I would be found to be another Point of Decuſſation. Ex. 3. Suppoſe 42* — 23 „ expreſs the Relation of the Abſciſs AB=z2 and Ordinate BC, in the Curve ACD (fee the Fig. in p. 140.) put the Value of y equal to nothing, or 42*— 23 =o: hence you ob- tain two Values of z, viz. a ,; and z= 4: wherefore taking AD = 4, the Curve meets with the Abſciſs in the Points A and D. There- fore to find whether the Curve decuſſates the Baſe in the Point A, where 2 = ©, put p, and then —p inſtead of z in one and the ſame Value of y; ſuppoſe the poſitive Value of y, viz. +-v/42*—23 or 2V4—2: (for it is y=+4/42* — 23 =+24/4—2) hence you'll have + pv/4 p and — p1/4 + for the two Values of y; of which the firſt being poſitive, the other negative, ſhews the Curve decuſſates the Baſe at the Point A. Again, for the other Point of Concurrence, viz. D; where you have æ == 4, by inſerting 4 + p and then 4 — p in place of & in the Equation y =24/4 2, you have 1*y = 4 + pv ——p, and 2% 4 — pp: the firſt of which Values being impoſſible, hence you con- clude that the Curve proceeds no further that way, but returns back towards the Beginning of the Abſciſs, making either a Cuſpis, or con- tinued Arch: which laſt it doth in this Caſe, forming another half upon the lower Side of the Baſe AB, exactly ſimilar and equal to the half above: fince the Property of the Curve is repreſented by the Equa- tion y* = 42* — 23, and conſequently y = +4/4z* — 23 : which 2 ſhews * 139 140 The Quadrature of Curves explained. ſhews that the Ordinate, at any Point of the Abſciſs, meets the Curye on both Sides, at the ſame Diſtance, as is repreſented by the Fig. annex d, bein g the modated Parabola, which conſtitutes the fixty-eighth Species of Curves of the ſecond Order, accordi to our Author's Enumeration of Lines of the third Order. | And after the like Manner, in any other Caſe, it will appear whether the Point of Concurrence with the Ab. ſciſs : as for other Circumſtances, which may occur, with reſpect to the Points of Concurrence and Decuſla- tion, the Reflection and Continuation of Curves, they muſt be left to the young Geometrician's own Sagacity : uſe the inſiſting upon theſe thin in this Place would carry us too far away from our original Deſign *, 174. If the Curve decuſſate the Baſe or Abſciſs betwixt the initial Li- mit of the Area and the Ordinate : whereby one Part of the curvilinear Area lyes above the Baſe, and another Part below, the Area which ariſes by the Method of Quadratures, will exhibit the Difference of the two Parts of the Area above and below the Baſe, lying betwixt the initial Limit and the Ordinate belonging to any given Abſciſs. | Thus in the Curve CDE (ſee the Fig. in the oppoſite Page) whoſe Abſciſs is AB, and perpendicular Ordinate BC, ſuppoſe it be 2— — 2 *Art.173. Sy: it was ſhewn above *, that the Curve will decuſſate it's Baſe AB, in —— Ex one Point only, at the Diſtance 1 from A the Beginning of the Abſciſs: 3 therefore, if we take AD= 1, D is the Point, at which the Curve eroſſes the Baſe, and afterwards extends itſelf infinitely below it: Now t. 145. the Area of the Curve is — A*: which Area extends itſelf, of this Ex- * plication. upon the further Side of the Ordinate BC, along the Abſciſs infinitely produc'd towards *; becauſe of the poſitive Powers of z, in the De- ? ok. nominators of the Terms *: but that Expreſſion denotes the Diffe- 168, 169. | of this Ex- _ ; 8 | , | Wr. Some Things of this ſort, reſpecting the Symptoms of Curves, wilt occur afterwards in 22 the Notes upon the laſt Scholium. 4 | . rence Curve proceed forward or not, at any 09 ni vw" OD mos WW © 1 Py © oy» mw CAC ® oo | @u : WW 83@© The Quadrature of Curve $ explained. rence betwixt the Area BCD, above the Baſe, and the Area DE, extended infinitely below the Baſe. | 175. The thing is f explain d thus. Sup- G poſe the Curve GF to be deſcrib'd by the Ordinate = - : then — 2, which is the firſt Term of the Expreſſion = 7 + | = denotes the Area GBa, infinitely ex- tended along Ba and lying wholly above it . Again, let GC be always equal to 1 or =, the other A z « Part of the Expreſſion of the propoſed Ordinate, fo that the Points G, C always lye in the Curve-lines GF and CDE: then, as eaſily a pears from what has been formerly ſaid, the Area belonging to the Ordi- nate 2=* = CG, vis, — = muſt be that which reaches from CG, all the way betwixt the two Curves CDE, GF infinitely produced : there- fore, ſince the Ordinate BC BG — BC, the Area of the Curve hav- ing BC for it's Ordinate, will be made up of the Difference of the former two Areas by ſubtracting the latter, vis. — + from the for- 2} mer, vi. — =: wherefore — — + 4 denotes the Area GBaF — GCDEF, i. e. ſince GCDaF is common to both, CBD — DE. And fince the Part CBD, lying beyond the Ordinate and above the Baſe, is always repreſented negative, in the algebraical Expreſſion, for the Rea- ſon already aſſign'd “: the Part DEæ lying below the Baſe and be- . Art. 268. yond the Ordinate will be poſitive, as being oppoſite in it's Nature to of this Ex- the other: therefore if the Expreſſion — — + 4 de in the whole ne- gative, it gives the Exceſs of CBD above DEa; but if, on the whole, | it 142 . Thus when B and D coincide AB =I: therefore the Area tx DEA =( 2 + 3; =) 1: but if AB () z; as formerly, then — The Quadrature of Curvss explained. it be poſitive, it denotes the Exceſs of DEa above CBD; but if it amount to nothing, it denotes that the Areas CBD, DEa are equal. Suppoſe AB = , then — 2 ＋ = 9 ＋ 6 = — 3, therefore in that Caſe CBD would exceed DEz by 3, but if it be AB D, then — — + == 2 + 3 = poſitive, which ſhews that DEa exceeds CBD by ſo much. Laſtly, if it were AB , then — 24 2 =— 4 + 4 = 0: in which Caſe CBD and DEs are exactly equ al. 176. But if you would have the Sum of both Areas CBD and DEa; find the Value of the Area DEa by itſelf, v:z. by ſuppoſing the Point B the Foot of the Ordinate, to fall in with the Point of Decuſ. ſation D: and ſo having obtain'd the Area DEa, lying all upon one Side of the Baſe: and having formerly found the Difference of the two Areas CBD and DEa, above and below the Baſe, you eaſily ob- y tain the Area CBD alſo by itſelf, and conſequently the Sum of both, —+ = = — 3, which is the Difference of the two Areas, the Dif. ference then being made poſitive is 3: which is the Exceſs of the Part lying above the Baſe, above the Part lying below. Therefore the Part lying above is 3 ＋ 1 = 4, and the Sum of both is 4 +1=—=5: and ſo in other Caſes, 177. Cor. Hence it follows, that, if the Abſciſs AB, be of ſuch a Length, as to make the Areas CBD and DE above and below the Baſe, equal, and you take the Abſciſs Ab leſs than AB, and draw the Ordinate c: and then ſuppoſe Ab = 2, the Expreſſion — — +5 will exhibit the Value of the Area BCcb, for that is the ſame with cDb — DEas. And thus the Ordinate BC, fo drawn, will be an ini- tial Limit, from whence the Area is computed, when the Abſciſs Ab is leſs than AB, And thus there are two initial Limits in this Curve: the one at an infinite Diſtance from A, the Beginning of the Abſciſs, towards ; the other is an Ordinate drawn from that Point of the Abſciſs; which is diſtant from A, by JAD —= 5. Theſe initial Limits, both ariſe, by putting the Expreſſion of the Area, vis. — ; +5 20 & 7 e * 2 4. EY I Ry _* R ** * va" Ins * ry ar 2 * . Cy — 2 % ent thy ex kt dds * 2 * % J an — % S & 4 PI ed. tha YZ * „ * 4 1 * > f * 9 A ms» ſc M4 > 5 4 1 s 8 [1 V #1 4 a g 4 6 4 F- ; 1 K i I 5 4 4 a * > 4 2 14 8 * * * The Quadrature of CURA VES explained. — 0: for that will happen, both when 2 is infinitely great, and alſo when 2 r: as will appear by reſolving the Equation. 178. By conſidering of this, you'll know how to apply any like Caſe, that may happen: where, by putting the Expreſſion for the Area equal to nothing, as directed above, in order to find it's initial Limit, you obtain different Values of 2, or Roots of the Equation : for theſe different Values of , give you ſo many different initial Limits, from any one of which, the Area may be computed. The Reaſon of which is evident: becaufe the curvilinear Area lying betwixt any two initial Limits amounts to nothing, in the algebraical Notation ; the Parts above and below the Baſe, which are oppoſite Quantities, exactly compenſating one another. | But although the curvilinear Area may be thus computed from any one of the initial Limits mentioned : yet it is moſt ſimple and natural to compute the Area from the nigheſt initial Limit. Thus when BC (ſee the Fig. in p. 14 1.) is an initial Limit : if you take Ab. leſs than AB, e.g. : the Area thence ariſing, viz, — — + , = — 3, is the Area BCœb; as well as the Difference betwixt Doc and DE. 179. Let it be y=a+bz-+c2? (ſe the Fig. annexed) as before, the Area is az ＋ ib2* + cz: put that Expreſſion of the Area az + 2622 + ic233 o: thence you obtain, 1% 2 — 3b + N — 48ac 4c er =20. 2*- 2 — 3b — / gÞ* — 4Bac 35.8 2 Whence you have three initial Limits: one at the Point A, where the Abiciſs begins: the other two beyond the Beginning of the Abſciſs AB, viz. produced towards H, as is denoted by the negative Values of z. The two laſt evidently fall into one, if ſo be 16ac = 36: for then the | N Op 1 common radical Part /94* — 48ac =o, in — the two negative Values of 2: ſo that, in that Caſe, they both be- come 2 = — 25 Let us ſuppoſe a 2 1. 6 4. c 3: then 16ac = 48, = 30, therefore gh —48ac=0: and ſo æ —=3b+\/ 9:*—38ac 4c tho Za == aac _ pr £ =— likewiſe. In this Caſe the =-— 1; and 2 143 144 The Ruadrature of Cu RVES explained. the Points of Decuſſation H and I will be ſuch, that AH = — and Al = — 1: as will eaſily appear, by ſubſtituting theſe Values of the Symbols a, b, and c in place of them, in the Equations above, for de. * Of this termining theſe Points (ſee Ex. 2. Art. 173). Hence if you take _— Al=— 71, i. e. produce the Abſciſs AB backwards to I, fo that Al Di, the Points A and I will be the only initial Limits: and the Area *Art.178. ADH == HKI“. To find the Area ADH, ſubſtitute — + for 2 in 0:15 =x- the general Expreſſion of the Area, viz. as ＋ £b2* 4- £23, and! plication. 8 P N , for a, b, e; and it becomes — 5 + + — 5 =— +. Again, to find the Difference betwixt the two Areas ADH and HKI, ſubſti- tte — 1 for 2 in the Expreſſion for the Area, vis. as + 2 ＋ Ic, and you'll have — 1 + 2 — 1 = for the Difference. 180. Reſuming Ex.3*. of Art. 173 *,viz.42*—2I=:, belonging to the Curve repreſented by the Fig. in p. 140, by extracting the ſquare Root, it is * Art. y=24/4—2: therefore the Area is x4 — 2#+#*, To find the 147, 149. 15 of hisEx- initial Limit, put © =D : and you find this to hap- N pen in two Caſes, and no more, viz. 19. when 2 4; 29. when 2 = — +; therefore if you take in the Abſciſs AB, produc'd backwards, Ag =, and draw the perpendicular Ordinate ag: the two initial Limits will be at a, and at the Point D, where the Curve meets with *Art.173. the Abſciſs : for it was demonſtrated already that AD —= 4 *. Where ben Er fore take the Abſciſs AB—=2 : and the general Expreſſion of the plication. Area, viz. x4 — 2 becomes © — X 2 = — 224/$; which being reckoned from the initial Limit D, denotes the Area DBC; or being reckoned from the initial Limit af denotes the Difference of the two Areas ABC and Ag lying above and below the Baſe. 181. Sometimes the Area, which ariſes by the Method of Quadra- tures, is ſituate partly upon one Side, and partly upon the other Side of the ſame Ordinate : which happens when the Value of y is made up of ſimple Terms, whereof ſome have the poſitive Powers of z in the Numerators ; and others have them in the Denominators. | Thus let the Relation of the Ab- ſciſs AB— 2, and Ordinate BC), be defin'd by this Equation 2 + 2 Sy (ſee the Fig. annex d.) The Area, by the Method of Quadratures, will be;2%— — *: the firſt Part of which, 25 de- notes an Area lying betwixt the Be- ginning Art. 145. of this Ex- plication. © *»*. ee -— I © 9 - . - " bn: The RNuadrature of Curves explained. | 145 ginning of the Abſciſs and the Ordinate; the other Part of the Ex- preſſion, v. — — denotes an Area ſituate upon the further Side of the Ordinate, and extends itſelf to an infinite Diſtance upon that Side *. t. 189. The * is explained after the ſame Manner as the Example in —́ Art. 175. For let Ad) be another Curve, whoſe Ordinate BD =2*, * — this the firſt Part of the Expreſſion 2* + * C; then, it is evident, the re · tion. maining Part of the Ordinate BC, v/z. DCA, the other Part of the Ex- preſſion 2 + = Now the Area belonging to the Ordinate BD=22, viz. £23, is ABD ; and the Area belonging to the other Ordinate DC= 5 Vis, — —, is Oa, lying upon the further Side of the Ordinate BC, and extending itſelf infinitely betwixt the two Curve-lines c Ca and AdDa, after the Manner ＋ ＋ explained in that Art. 175 *. Of this 0 182. If we ſhall ſeek for the initial Limit of the Area in this Caſe, - "ak viz. by putting the Area {23 — = o, we find only one Value of 2 viz. 2 J: Take the Abſcifs AB Vn, draw through the Point B, the Ordinate BDC, ſubſtitute V/; for 2 in the Expreſſion of the Area, and you'll have the Area £23 — —= 62 7— =/it— * o. This ſhews that when the Ordinate BDC is 13 at the Diſtance 7/3 from A, the Areas ABD and CDe are equal. 183. If * given Abſciſs AB be greater than the Abſciſs AB, which reaches to the Limit, the Expreſſion {23— — is poſitive and denotes the Area BOC: but when the Abſciſs Ab is leſs than AB, the Expreſſion 123 —— is negative : but being changed into it's oppoſite, will de- note the Area BbcC. For when 2= AP, then 42) — = = A0 — ade = ABD + BSD — CDs -+ CDG — BHD -+ CDde or BBC for ABD=CDa. Again, when z = Ab, then {23 — = = Abd— ade : but Abd == ABD — 4bBD4, and ade = «DC + DCcad: ſubtract the latter from the former, and you have £23 — — = ABD — BDA — «DC — DCed = — bBDd — DCed = — BHC. Let it be AB = 2, then 125 — = = — += 25, the Value of the Area BHC lying betwixt the Ordinate G and initial Limit BC. Again U ſuppoſe = — 146 "= Qradrature of Curves explained. ſuppoſe Ab = 4, then 423 — == —4 == 1:5; : which being made poſitive, is the Value of the Area BGC. OY 184. Sometimes no initial Limit of the Area can be found at all, This will always happen, when, putting the Expreſſion for the Area equal to nothing, the Equation has no real Root or Value of z. Thug if the Curve be ſuch that z2* — : y: then the Area, by the Me. 1 145 thod of Quadratures, Is +23 += *: if we put 423 + = == ©, thence of this Ex- _ | plication. e have 24 ＋ 3=0 of 2 3, an impoſſible Expreſſion : and thus, there ĩs not any poſſible or real Value of z, in the Equation 3234 Do, or * + 3 o. Therefore no initial Limit, from whence the curvilinear Area ought to be computed, can be found. Again, ſuppoſe the Curve to be defin'd by this Equation zv/a* + 2* =: *Art147. The Area, by the Method of Quadratures, is "IZA F# *. Put O S Ex- Le 22 ä—Xßç2ł‚ðw!˖•m— | | . plication. it equal to nothing, 2. e. — a o, and you find this can never happen, unleſs it could be a* + 2* O or z= - a*, which is impoſſible. Therefore no initial Limit can be found: and fo in other Caſes. | | 185. When this happens, you may, notwithſtanding, find the Area which is adjacent to any given Part of the Abſciſs, and lying betwixt the two Ordinates at the two Extremities of that given Part of the Abſciſs, viz. thus: ſubtract the Area belonging to the leſſer Abſcifs, arifing by the Method of Quadratures, from the Area belonging to the . — er Abſciſs, the Difference is the Value of the Area adjacent to the ifference of the. Abſciſſes. Let the Curve AC (ſee Fig. annexed) whoſe Baſe is AB, be defin'd by the Equation z/a* + 2* = y, mention- ed juſk now: the Area, by the Doctrine of Quadra- tures is —— az. Wherefore, to find the cur- vilinear Area adjacent to any given Part of the Abſciis as Bb: let AB = 2, Ab— 1: ſubſtitute 2 = AB fos 2 in the general Expreſſion of the Area IEF, and it becomes —— again ſubſtitute 12 A Ab for &, and you'll have 2 U F 1 : ſubtiad this 4 The Quadrature of CV explained. this laſt from the other, and it gives EY ee a a „ ſor the juſt Value of the Area BbcC adjacent to the given Part of the Abſciſs Bb. 186. Univerſally : call the greater Abſciſs AB ; the leſſer Ah 8 and then the Area BbcC is T&F — e + ©, Ace then © f f — =, for the Value of the Area ABC, Een from . Beginning of the Abſciſs. It _ to the ſame Purpoſe to ſuppoſe z= © in the —_ Expreſſion f L + 23, and then ſubtract what NY vie, ; 2 = - from Ka general Expreſſion, giving _ 2 * — = for the Value of the Area ABC adjacent to the whole hah AB. La it be a="1 = AB; then the Arca ABC = (ET nt — =) 24/2 — +. NL. Ta The FASO Method may be uſed in the Caſe of any othet Cures whether it admit of an initial Limit for the Area, according to the Rule delivered above, or not. Yea in regard in ſome Curves, one may be perplex'd, and ready to miſtake, you do belt to follow os fame Method, in the Caſe of any Curve propoſed. Thus in the Curve ACD, repreſented by the Fig. p. 140. mh, We have y=2v/4—2: let it be required to find the Area Bic C adjacent to 35 a given Part of the Abſciſs. The general Expreſſion of the Area is wat bo 7 = X 4 — 2)z, Calling therefore the leſſer Abſciſs AB; and the greater Abſciſs Ab &, ſubtract the Area belonging to z from the Area be- longing to C and you'll have — _ X 4 —T N * — —5 X 4—2}, for the Value of the Area B4cC. And if 2 (= ay == © you'll have 1 — . 1 ++ *. DT, for the true Va- lue of the Area Abc. Lat it be AB=2 =2,and Ab = z, then —.— N I x4 —=2t=— 34 + 3234/8 = Area BC. Again, putting 2 = AB Do, r have (by retaining the ame Value of Ab —= 07) DNN ee 27 for 3 15 U 2 | the — — —— —— — . — — *Art 181. to be 323 — 2 * Call Ag D z and Ab = &, then the Area bGuc = 248 The Quadrature of Curves explained. the Value of the Area Abc, reckoning from the Beginning of the Ab. ſciſs at A. 188. Thes likewiſe, when the Curve is doubly quadrable : e. g let = it be y = == (lee the Fig. at Art. 128 *.) It was there demonſtrat- ed that . bs of the Curve was either od - or yet == : Either of which Expreſſions will ſerve to find the Area BSC adj cent to any given Part of the Abſciſs as * Calling the leſſer Abſciſs Ab = . the firſt Form will give you —— — Te: the ſecond Form will give you , = 2 ＋ PER: either of which exhibits the Value of the Area BGC. A A Let it be AB =z=2; A = — * N a for the Area BbcC. Again —- + == + ==4a, the rn iR 7 : i ſame as before. In the firſt 2 if you put F=Ahb =o, in order * find 1 Area ABC a n to the whole Abſciſs AB = 2, then * 1 becomes .= = AC. Again, the ſecond 189. Let the Nature of the Curve be defin'd by this Equation 2.4. ==, ſeethe Fig. p. 144. The general Expreſſion of the Area was found of this Ex- | plication. 4 — 2 — 0 +>: ſuppoſe 22. As = =: then Ig) — 2 18 ＋ 7 =} r = 311 the Value of the Area bBxc. Again, ſuppoſe C — Vz, and à 2, as before: then V7 1 I ＋ : 12) — 7 rr = - T = —T=27 3 *Art.183. was found by means of the initial Limit before “. of thisEx- But, if you either ſuppole 4 EAA 0 or yet Ag e infinite: plication. then the Expreſſion 123 — = — 4G + - 7» in both Caſes, becomes infinitely great: for when =o, the Term I is infinite: again when 2 = infinite, the Term Zs is infinite. The firſt ſhews that the Area upon the nearer Side of any Ordinate, comprehended betwixt it and AG per- pendicular — 3 I — h e The Ruadrature-of CURvRS explained. icular to AB, is infinitely great: the other ſignifies that the Area tuate upon the further Side of the Ordinate, and extending along the Abſciſs infinitely produc'd, is alſo infinitely great. Therefore in this and the like Caſes, we muſt be contented to find the curvilinear Area, adjacent to any middle Part of the Abſciſs, as has been juſt now ſhewn. COOL 190. And this naturally leads us to ſpeak of the hyperbolical Space contain'd betwixt the Curve and Aſſy mptote in the Caſe of the Apol- lian Hyperbola, whoſe Aſſymptotes are at right Angles to each other: the Conſideration of which was put off to this Place. Suppoſe cCxa to be ſuch an Hyperbola; (fee the Fig. p.13 5.) AB, AG irsAſſymptotes: the Equation to it is zy=a, or y =, a denoting a given ſuperficial Space. The Area, by the Method of Quadratures, is —— 1 — infinite : whence it is evident that we can neither find the Area plication. on the nearer Side of any Ordinate BC, lying between it and the Aſ- ſymptote AG ; nor the Area CBa, lying on the further Side of the Ordinate BC, along Ba infinitely produced; theſe Spaces, in both Caſes being infinite : nor yet can we, by the foregoing Method, find the Value of any intermediate Area, as BôcC, adjacent to any middle Part of the Abſciſs Bb, as long as the Point A continues to be the Beginning of the Abſciſs. Therefore, to remedy this, you muſt alter the Beginning of the Abſciſs: and ſuppoſe it to be at ſome Point in the Aſſymptote, as at the Point B, at a given Diſtance from A: which Diſtance, call 4: draw the Ordinate BC; likewiſe the Ordi- nates Ox, bc, on each Side of BC. Then you'll have BB = , Bb =—2, A 2 d-, and Ab—d—2z: and ſo the Ordinate By —= To and bc = 5—. Reduce the Ordinate 7 to Form, and it ſtands thus 2 x d ＋ Zl! x a: which belongs to a Curve that is not quadrable, as appears by Art. 149. Wherefore we muſt con- Of this tent ourſelves with an Approximation. Accordingly, by ſubſtituting Flic 1 for 0, y, , s and /; and d, à fore, a reſpectively, in the general 225 Series for the Area of a fimple Binomial, viz. & x e + fz»Þ * r ne 1 r+2xe 1+, xe r+4xe 4 you'll have zx4 +2 x5 — ,;3+ 17 2 — 23 4 7 — Ce. S XI T- Ge. in inf. which denotes the | | Area un 1 50 Art. 169. of this Ex- plication. * Of this Explica- tion. * Art. 28. of the Quadra- ture of Curves. *Art.144. of this Ex- plication. De Quadrature of Curves explained. Area BBxC, becauſe of the poſitive Powers of z contain'd in the Nu. merators of the ſeveral Terms “. After the ſame manner, if you take y = En, you'll find, by making the like Subſtitution in the general Theorem, expreſſing the Areas of binomial Curves, 7 X 1+ = * = + 75 * = + &c. in inf for the Value of the Area BbcC, for the like Reaſon. 191. Whence it follows, that, if BB = Bb, then by adding both Series's together, you'll have X 1 + = + = + _ + Oc. in inf. for the Value of the Area u , made up of both the preceding Areas: which Series converges double as faft as the preceding ones. And thus you may, moſt conveniently, find an approximate Value of any hy. perboligal Space, ſuch as Ou, adjoining to any intermediate Part 30 of the Aſſymptote, viz. by ſuppoſing the Beginning of the Abſciſs 2 to be at the Point B, exactly in the Middle betwixt & and H: for by this means the Series made up of the two others, as above, conver faſter, than by ſuppoſing the Beginning of the Abſciſs to be at one of the Extremities of the given Part of the Aſſymptote; or by throwing the Beginning of the Abſciſs without the Part , and working as in Art. 186 *: which may likewiſe be done here. Ex. Let As o, g, AB= 1,1: and you would have the Area bc. Take bþB — o, 1 = BGH = 2, and conſequently AB d= 1 : then ſubſtitute 0,1 for 2 and 1 for d, in the preceding Series, and you'll have o. 2a X r= w_ — =_ — + Sc. = @ x 0.2 + 0.000666666, &c. ＋ 0.000004 ＋ 0.000000028, Sc. + &c. o. 20067069, &c. x BC for an approximate Value of the Area xc: and ſince the Progreſſion of the Series is manifeſt, the Value of the Area, may be carried to any Degree of Exactneſs you pleaſe. Now although the Series's above, would 'converge very flowly, if it were 2 ; and not at all, if it were z greater than 4 *, yet it is evident, by the above expedient, z muſt always be leſs than d: for Bb BG = muſt al- ways be leſs than AB — d. And thus we have ſhewn at full Length, what is to be done in or- der to find the curvilinear Area adjacent to any middle Part of the Aſſymptote, in the Caſe of the Apotlonran Hyperbola, as we promis'd above *. | | 192. And as it has been ſhewn, in the Caſe of the Hyperbola, how you may change the Beginning of the Abſciſs; it may not be impro- wh per ? Mt ON my, > e 5 03 co» = bt o* 1 P * The Quadrature of Curxvus explained. here to obſerve, that the ſame Expedient may be uſed, in the Caſe of any other Curve whatſoever : which may be of conſiderable Uſe, upon Occaſions, for extricating one out of ſeveral Difficulties that may occur in this Doctrine of Curve-lines : particularly, by this means, you may expreſs one and the ſame curvilinear Area, by infinite dif- ferent Ways: ſince every different Beginning of an Abſciſs, which may be infinitely varied, will afford you a different Expreſſion for the Area. And thus likewiſe you may get free of an hy Heal Term, which may enter at any Time into the Expreſſion of the Ordinate of a Cutve. 1 call an hyperbolical Term, that which contains the firſt Power ot Root of the Abſciſs 2 in it's Denominator : which always denotes an hyperbolical Ordinate, applied to the Aſſymptote. Let z + - = be an Equation to a Curve: then becauſe of the hyperbolical Term — the Area becomes infinite, viz. 22 — but if you ſuppoſe the Abſciſs z to be increas d or diminiſh d by any given Length, e. g. to be diminiſh d by 1, by taking a new Abfcifs & —=2— 1 or ITF =-: and ſubſtituting 1 + & for z in the Equation to the Curve: hence you obtain y 1+ F = and therefore ſince the Area correſpond- ing to 1 4 F is CT 1, and that correſponding to the Part A is 25 _ 2 + = _ 2 ＋ &c. in inf. after the Manner already demonſtrat- ed: hence by joining theſe Areas together, you'll have 37 —* + 207 — ft + 4H — &c. in inf. for an approximate Value of the Area of the r Je 193. And fo I ſhall finiſh this Section, reſpecting the Poſition and Limits of the Areas of Curves, being afraid I have inſiſted too long upon it already, when I ſhall have remarked that, if the Curve decuſ- fate it's Baſe, betwixt the two Extremities of that Part of it, to which the Area to be found, adjoins, then, by obferving the Rule delivered above, viz. by ſubtracting the Area belonging to the ſhorter Abſciſs from the Area belonging to the longer Abſciſs, found by the Method of Quadratures, the thing which ariſes is the Difference between that Part of the intervening Area, which lyes above the Baſe ; and that Part, which lyes below, the latter being ſubtracted from the former. Thus it it be AB z and Ag & (ſee the Fig. p. 14 1.) : the Area belonging 5 larger to the leſſer Abſciſs AB = 2, is — + +-;: and that belonging to the 151 152 . FR The — of Curves — larger Abſciſ Ag 5 is — 4 ＋ 2 nt — therefore — + 7 +5 $5 14 == 3 denotes the Area DBC — DSG. Let ABD == and AB r, then — + = I —4 14 S i the Difference Me by fubrraing PH De from Dbc. Again, let it be AB = z = 25, and AB = T=4: then — TEASE EIS TIENTS = — 745 06 Dit rence ariſing by ſubtracting the Area DG below the Baſe from the Area DBC above the Baſe : which comes out negative, in regard, DB is, in this Caſe, greater than DBC. Finally, let it be AB = = , and Ag 4: ret -A =-- 3 which ſhews that the Parts o the Area above and below the Baſe are equal. = SECT. VI. Containing Notes on Art. 32. DEMONSTRATION of PRop. vi. HE Demonſtration of this Prop. which our Author has omit- ted, is as follows. 194. By Prop. 4. and Art. III, the following Ordinates and Area of Curves correſpond. Ordinates. Areas. * E 0 : [Jn Ae] 1 0 29 4-0 ad SO 26 An AEZ | | 2 6 * — — — —— — „ -4 a” - The Quadratur of C URVES ads Ordinates. 7 ry 12 TT : 46.2 fiBa*" T5 ghBe3" | 0Fr, m..21 — N {47 elBæ un fIBz?" 9—1 pa—1 g—1 Bx NR b 1 „„ oc node dJ ⅛ ͤͤůA’“u - Abe re 2 Tv 21 ; 50 Ae. „ . . | * iy eFnckDza?? oc. xt OD DSZ Now if the Sum of theſe Ordinates be ſuppos'd equal to the Ordi- nate Fc + d= + Sc. x2 - I propoſed, then the Sum of all the Areas, correſponding to the ſeveral Ordinates, i. e. ARS? + Bz CRS. + Co R*S+ + Dx LRS + Ge. or TBZ + Cz*» DZ + &c. RS., ſhall be equal to the Area of the Curve, whoſe Ordinate is A + dz3n + &c. x 21 R181. Wherefore, let the correſpondent Terms of the Ordinates be put equal, i. e. ſuch Terms as have the ſame Power of Z in them: that b that means, we may diſcover the Values of A, B, C, D, &c. w are of certain Values, although not yet determined. Hence we have | theſe Equations, 1%. a =OekA. 2 = NN, 6+pn xelA + TFn xeB. z. c AN xgtA + SPN) LY HA + 12 X em + Mena xftB + xe + Rel. 4. dN A + NN xglA + HA x fmA TO TZ X enA + 0+n-+2ay x g&B + 08+n+an+ wy x FIB + 0+1+2pn x emB UTA xfkC+ a xe + Tx . &c, X In 153 The Quadrature of CV RVS explained. In which Equations, if you put or ſuppoſe r. r +a; 5s + AS. t TNS. &c. And again r ＋πH =. A as they are expreſſed by the Author himſelf: they will appear 1 15. a== rnkA. | 2. „ ] - vel + r + 1 XeB. 3% c = tnghA A A tnemA + PIX VIB + 5+ 1 xl + r+2 X yekC. | op 4*. df + N + fnR + vnenA + TPI x NAB þ t+1 BTI x nemB + a x of x NIN. Sc. From which Equations, by a due Reduction, you may find the Values of A, B, C, D, &c. which are the following, viz. — 0 —— i =" — * 1 | = S+2 x e — 1 x// „ n — t-1 xm A n D — — — — = I & | | r+3 x — tgk | —C ww I K k / een 8 — fl A — $1 x & C= — 2 x et Wherefore, fince, by putting the Sum of the Ordinates above men- tioned, equal to the Ordinate @ ++ 5&2" + c + dg3 + Se. xz R. S. and the correſponding Terms of the one and of the other equal liKewiſc, theſe Values of the Coeſficients A, B, C, D, &c. do ariſe, it follows that, if you infert theſe Values of A, B, C, D, &c. ſo determin'd, in place of them, in the Sum of the Areas, vis. A+ Bz" + C2" + Dei" + Sc. XR; what reſults will be the Area of the Curve, whoſe Ordinate was propos d, which will be thereby determined: and accordingly it will be expreſſed by the fol lowing Series. | 2 2 R The Quadrature of Cunves explained. MWEx+L Ip ft KEY” wy + Fi Tm.” ** * 1 + i x of — th v6 | » £2 x & » —ATizx BA 5 — 2x) — t+t en = nes Holes Bcd... — r +3 * e + &c. In which Series, although the Capitals A, B, C, &c. are retain'd, yet 1 it is evident, they are now fully known: for A= the firſt Term, ret which is wholly known : conſequently the Coefficient of the ſecond Term is wholly known, i. e. B is wholly known: and fo on for all the reſt. And as A, B, C, Cc. are the Values of the Coefficients of the firſt, ſecond, third, &c. Terms; ſo, when any of theſe Coefficients is negative, the capital Letter, which ſtands for it, muſt be negative likewiſe. And therefore if a ＋ bz» + cz» + d= + Sc. x R Dube the Ordinate of a Curve, the Series expreſſed above, will ex- hibit it's Area: and that either perfectly, viz. when the Series ter- minates ; or elſe by Approximation, vi. when the Series runs out infinitely. But it muſt ſtill be remember'd, that every Ordinate of a Curve may be expreſſed in two different Forms: for either the Ex- ponents of the Powers of 2, contain'd in R, 8 and a+#2»+c27* + &c. may be poſitive, or negative: which was ſufficiently explain'd upon the foregoing Propoſition : and accordingly both Forms of the Ordi- nate muſt be tried, in the Application of this Theorem: and if the 135 Curve be quadrable, the Series will terminate in one or other of the Caſes: but if otherwiſe, the Series will converge, either when » is po- itive, or elſe when it is negative; and fo it will exhibit an approxi- mate Value of the Area, X 2 195. What - -— \ © 156 The Quadrature of Curves explained. | 195. Whatever Obſervations or Remarks were made upon the fore. going Prop. the like may be made upon this, with reſpect to the Laws or Conditions of the Termination, Continuation and Conver. gency of the Series: and the bringing Ordinates of Curves propor'd, which fall under this Prop. into due Form, as likewiſe wi reſpe& to the Poſition and Limits of Areas, and other things relating thereto, But that would be too large a Subje to enter upon : and therefore 1 ſhall let it alone. 171 The Series exhibiting the Areas of Curves falling under this Prop, may be eaſily continued, by conſidering the Law of the Progreſſion. It is evident likewiſe that the Series of ſuch Propoſitions, as the fifth and ſixth, is infinite ; as well as that of the third and fourth, which was *Art.117, mentioned already , 7. e. we may ſuppoſe T to ſtand for «+-f2»4-y2 — + Cc. and » for any numeral Index: and then if the Ordinate of a Curve be of this Form z R- Tm? xx a+ bzn+cptrno- gun + you may exhibit a Series for it's Area, analogous to theſe in Prop. fifth * Of this and fixth, by conſidering what was faid in Art. 117 *, and applying Explica- that to this Caſe, as Prop. third and fourth have been applied for de- den. monſtrating Prop. fifth and ſixth: and fo likewiſe for other Ordinates ſtill more complex. SECT. VII. Containing Notes on Art. 33———39. | ExPLICATION of PROP. vii. CASE LL 196. CUppoſing the ſame things mentioned by the Author in this Prop. Let Re + fz" a Binomial : I ſay if the Area of any one of the infinitely many Curves, whoſe Ordinates are thus ge- nerally expreſs'd a A, be given, the Areas of all theſe Curves will be given. | The Demonſtration of this Prop. as far as it reſpects binomial Curves, is left by our Author to be gathered from his Demonſtration of it in the Caſe of Trinomials : which therefore I ſhall here fupply. The firſt of the four Caſes into which this Prop. is divided by him, reſpects the Increaſe or Diminution of the Index of z, viz. 8+ on, by any Repetition of 2 197. Let The Quadrature of Co RVS explained. 157 197. Let the Area of the Curve, whoſe Ordinate is R., be : and call it A, i. e. multiplying both by the Quantity p, whoſe alue is afterwards to be determined, we ſhall have pa- Ri the Ordinate of a Curve, whoſe Area is pA. But, by Prop. 3, 8&&+04-2n x yu g9=1R>-", is the Ordinate of a Curve, whoſe Area is 2 R: ſubtract the former Ordinate and Area from the latter Ordinate and Area re- pectivel and you'll have de — p +8 +39 = N -R! and . the N the firſt the Ordinate of a Curve, whereof the laſt is the Area: and that whatever be the Value of p, which is the Quantity by which the given Ordinate and Area, vi. 2%1R>—1 and A, were multiplied, Now put p == be a known Quantity, and ſubſtitute the one for the other in the laſt mentioned Area, and thence you'll have 0 + ay * F x 24—-1R>-1 and 2R> — be, for correſpon- dent Ordinate and Area, Again, dividing both by the given Quantity Faq *,, we have 2*+—1R>- the Ordinate of a Curve, whoſe Area is — which is wholly known, . That the Proceſs may appear at one View, I ſhall ſet down the ſeyeral Steps in Order. Ordinates Areas, rp * Xx RN g PA 2. be + NN x i t UR 3˙ be—p+0-+ ay X fon xX 25—1R>z=1 — — R — pA. 4 Pu xf X iR RN A. OE TY” I- IR! 222 e. (Fon xf ; 198. By the fame way of reaſoning, if you call this Area A 1. —— = B, whoſe Ordinate is z&»—1R>-1, you may find the Area of another Curve, whoſe Ordinate is z*+2—1R>— ;. by conſider- | ing what is expreſſed in Art. 114%. Which is done by ſubtracting » of t the Ordinate g2tra—1Ra-! Or 92" x ü- iR. 1, and correſpondent Area Explica- — — — b tion. B from the Ordinate 6 + » « ez» + 8 + ＋M x f221 » g-iR -i, and correſpondent Area 2*+"R reſpectively: which laſt Ordinate and Area The'Quadrature of Cunvss explained. Area belong to the ſame Curve by that Art. and then putting gg%— PY e, or = e: and ſubſtituting this Value in place of it: for thence you'll obtain 8 + y + an x f2" Ri and 2*+R1 neh; or, by dividing both by 8 9 4 xf, -N + — x 4B = YI for Ordinate and Area belonging to one and the ſame Curve: "hich Ordinate has the Exponent of x, v/z. 9 -þ 25 — 1 greater than the Exponent of z in the Ordinate 2 Ni be- longing to the given Area A, by 29: and fince B is known, being formerly found, the Area : —.— V a5 wholly known, Fn Finxf And thus it is evident, that, by the like way. of proceeding, you may find the Areas of all the Curves, whoſe. Ordinates are thus ge- ty expreſſed a - iR, & being a poſitive Integer. For if 25+o1-—1R>=1 be he Ordinate of a Curve, and A denote the Area of that Curve, then * mom . + 2 —. will be the Area of another Curve, belonging to that ſame Progreſſion of Curves, which has the Index of z in it's Ordinate greater by once ; than it is in the Ordinate * Of this of the other: as appears from Art. 114 *, compared with what hath fob. been already faid. tion, 199. Again, by the like way of reaſoning, it will appear, that if the Area of the Curve be given, whoſe Ordinate is 29-1R>-1 as for- merly, which call A, then the Area of the Curve, > whaſe Ordinate is z R-, may be thence found, and will be . — EXE) For you u ll have Ordinates and Areas r to ach other as follows. Ordinates. | Areas. 19. —— — —— þ * 2 —-1N—1 DA. echt. . JO 1e = N E - 2. N + plication. Subtract firſt Step f from ſecond : put Pf 6—1+ e and divide by z Xe. R 1-1 en - e This laſt Area being found, call it P, and then from the known Area * you proceed to find the Area of the Curye next in order 5 t - 6 = (jo . 088 = 5 F 8892S yp =» my *r 8 N X . 133 7A in us The Quadratur of Cuny ns explained. this n af Curves, viz. the Area of the Curve, whoſe Ordinate is S- R-: which by the ſame way of reaſoning, will be found obe : and ſo you may proceed from this laſt Area, now given, to determine the Area of that Curve which comes next in the Order of the P of Curves, viz. that whoſe Ordinate is 23 -R i: and fo on continually. So that univerſally if you put P for the Area of any Curve whoſe Ordinate is 2 e- Ni, * . = x fÞP | | | chen —ſhall be the Area of a Curve whoſe 9 -- Ordinate is 2˙—⁰ -R I: as eafil by ſubſtituting 9 for 0 in the firſt Part of this Art. N n 200. And thus it appears that the Area of a Curve, whoſe Ordi- nate is 2%—'R?-"?, being given, the Area of every other Curve, whoſe Ordinate is thus generally expreſſed z%+9—!R>—", is likewiſe given. Wherefore by a contrary Analyſts, you may come back from the Area of any Curve, whoſe Ordinate is 2%+-1—=1R>=1, to the Area of the Curve, whoſe Ordinate is 2%!R>-!, For, if you put A for the Area of any Curve, whoſe Ordinate is z&+-—1R>=1 : and B for the Area of the Curve whoſe Ordinate is z%#+++—1R>=1 2: then, by what has been demonſtrated, B = ——_ — — : therefore it follows con- Tn An N trarily by reſolving the Equation, > at if 1 B be given and A ſought, it wn A. will be A. And ogain, in the other Pro- Ko xe | greſſion of Areas ding the contrary way, if P ſtand for the Area of a Curve, hole Ordinate is - -iR 1; and Q for the Area of a Curve, whoſe Ordinate is 2 ⁹π⏑⁹0 e RI: it has been demonſtrated 002 — | that Or. Wherefore, ſuppoſing Q fg —on—nxe | given, and P ſought, by refolving the Equation, we have P * N ——n—n x eQ ee ED | : * From both which it plainly follows, that, — 07 - t An X | if the Area of any one of the infinitely many Curves, whoſe Ordinates are generally expreſſed thus 2*+91—"R?=1, be given, the Areas of all the reſt are alſo given, viz. when R = e +2 a Binomial. N. B. It's the ſame thing whether the general Form of the Ordi- nates be expreſſed thus 2 Ri; or thus 2 R, as it is by the Author himſelf in the Prop. you need only ſuppoſe that d in the laſt is equivalent to 8 — 1 in the fuſt: and that a+T in the laſt js | | equivalent: * * * * D ©. » 160 The. Quadrature of Cunves explained. | equivalent to x — 1 in the firſt; ora+r+1 in the laſt, the ſame with x in the firſt, 1 i Of 3 201. It appears from what has been ſhewn in the three preceding Art. that if A and B be made to ſtand for the Areas of any two Curvez next to one another, whether they be in the Progeſſion running to- wards the one hand, or towards the other: ſo that the Exponent of 2 in the Expreſſion of the Ordinate anſwering to the Area B, exceed the Exponent of z in the Ordinate correſponding to A, by + : then, we may make 2 -R and 2%+1—1R>-1 to repreſent any two ſuch Ordinates univerſally, ſince 0 may repreſent any Index poſitive or ne. gative, integral or fractional. From whence, by conſidering what hath been ſaid, we ſhall have the two following Theorems for determining the Areas of all binomial Curves, from any one Area being given. 0A 202, Th. 1. B 2 2 2 * . » „ ao DH a a as oe an oa * x f 5 A | 203. Th. 2. A —5 . 204. And thus having demonſtrated the firſt Caſe of the P when Rr e , a Binomial; the Demonſtration of the firſt 805 of the Prop. accommodated to the Suppoſition of R = e + fz» + g2*", a Trinomial, ſet down by the Author himſelf, includes no new Difficulty. The way of finding the Area of the Curve whoſe Ordi- nate is zt. -i i, from the Areas of the two Curves, whoſe Or- dinates are 2%-!R*-! and z —1R>-?!, being given, he ſhews himſelf at length: and ſo you have the Method of determining the Area of any Curve whoſe Ordinate is 2%F-+—R>-!, when the Areas of the two Curves are given, whoſe Ordinates are 2 N and 2 - iR. And he tells you that the Method of proceeding, in order to find the Areas of Curves, in the Progreſſion of Curves, which goes the con- trary way, via. thoſe Curves whoſe Ordinates are thus generally ex- preſſed 2&—-—!R>-", is like to that ſame, which he ſhews for the Areas of Curves in the other Progreſſion. That the Truth of this ma appear I ſhall ſet down the Proceſs of the Steps in order. Ordinates. Areas. 10. 9528595 o'>'s e LT om. —— OO I 2 a gn xn Rt! 75 Art. 114. 3 xe NR dna ® — Brͤubtract the firſt and ſecond Steps from the third: put == f xF and 12 0 —» + 2an x g: and divide by & —» x e. 18 nr 8 fb xgÞ 2 $ - TUE EEEEEEEESY - — Which 9— 2 * 4 The Ruadrature: of Curves explained. Which laſt Area is that which was to be found, and is now wholly known, fince A and B were given. And after the-ſame Manner that our Author ſhews how to find the Area of the Curve, whoſe Ordinate is f -i R—1, from the Areas of two Curves being given, whoſe Ordinates are 29—R*—1 and 2&+—1R» ; and that I have juſt now ſnewn, bow from theſe fame Areas given, to find the Area of the Curve; whoſe Ordinate is 2 - Ri, the Areas of all the Curves in both Progreſſions may be found. For, univerfally if A and B ſtand for the Areas of two Curves next one another in the firſt Progref- fion, any where, vig. ſuch whoſe Ordinates are z%+-—1R>=1 and iR, then An. 9 + on x A —d port x /B will be the | | 9 9 2anxg | Area of the Curve following next in that Progteſfion, pix. whoſe Or- dinate is -r iR I: as eafily appears by inſerting 8+ oy in- ſtead of 8 in our Author's Demonſtration, i. e. by ſuppoſing 8 + oy here, to be equivalent to 0 there. | Again, if P and Q ſtand for the Areas of any two Curves next to each other in the Progreſſion bf Curves going the contrary way, f. e. if they be the Areas of any two Curves, whoſe Ordinates are 2%--1=1R>=: and go=0n—mn—1 RN 1, then — — erer | 9 — on—21nxe a muſt be the Area of the Curve following next in order in this Progreſſion, viz, the Curve whoſe Ordinate is 2%-»=2*R>-1 : which eaſily appears by inſerting # — oq— y inſtead of 8 in the Demonſtration juſt now oven; 7. e. by ſuppoſing 8— ox — here, to be equivalent to h there. And therefore it hence appears that, if the Areas of the two Curves, whoſe Ordinates are -R and 23*+-!R*-!, be given, the Areas of all the Curves in both Progreſſions, are thence alſo given. And therefore by a contrary Analyſis, if the Areas of any two Curves, whe- ther both in the Progreſſion of Curves proceeding the ſame way, or one in the Progreſſion proceeding the one way; and the other in the Progreſſion that proceeds the contrary way, be given, the Areas of the two Curves whoſe Ordinates are Zů⁹ Ri and z - ⁰m—- IR.; and: alſo of all the others whoſe Ordinates are thus generally exprefled, z -R, may thence be found. For let A, B, C, D, &c. denote the Areas of the Curves belonging to the Progreſſion, whoſe Qrdinates: are thus expreſſed a . - R-, where on is poſitive, and A, the Area of the Curve whoſe Ordinate is 29 Rπ]] .: and again let A, P, QR, &c. denote the Areas of the Curves belonging to the Progreſſion proceeding the contrary way, viz: thoſe whoſe Ordinates are thus ex- preſſed 24-0—1R>1, A being as before the Area belonging to the TY | N Y Curve, 161 . k . * 162 Iy if any two Areas be given, all the reſt may be found. The Nuadrature of © unvnss: explained. Curve, whoſe Ordinate is 2. -: which Curve ſtands in the Middle betwirt the two Progreſſions, or rather js the firſt of both. Then by what has been ſhewn, we have an Equation defining the Relation of A, B and C; another defining the Relation of B, C and D; a third defining the Relation of C, D and E, and fo on: whereby we have a new Equation for every new Area. Again, we have an Equation defining the Relation of B, A and P, when we proceed now the contrary way: we have another, exhibiting the Relation of A, P and Q: another that of P, Q and R, and ſo on: therefore, it is evident, there will be as many diſtinct Equations except two, as there are Areas: conſequent- 205. Although in the preceding Art. we have confidered the Curves whoſe Ordinates are generally expreſſed thus 2%-»=1R*?, and thoſe whole Ordinates are expreſſed thus g%--— Ra, diſtinctly: and ac- cordingly exhibited two diſtin& Equations, for determining the Areas of the Curves belonging to the one, and the other : yet they are both in effect one and the ſame, and both virtually included in that one Equation or Theorem mentioned by the Author himſelf. For if A, B and C repreſent the Areas of three different Curves next to one an- other, fo that A be the Area of a Curve any where belonging to the Claſs of Curves, whoſe Ordinates are thus generally expreſſod SNR; B the Area of a Curve, in whoſe Ordinate the Index of 2 is greater by -- A; and C the Area of a third, in whoſe Ordinate the Index of z is greater by ＋ 23, than it is in the Ordinate corre- ſponding to the Area A; then we may make 2 ROA to repreſent univerſally the Ordinate of the firſt ; a R the Ordinate of the ſecond; and zi+2»—!R>=1 to repreſent the Ordinate of the third, cor- reſponding to the three Areas A, B, C: fo that the Equation 2*R>— deR — 0 + ay x fB= 0+ 2aq gC mentioned by him, may ſerv for finding all the Areas in this whole Claſs of Curves, whoſe Ord- nates are thus generally expreſſed z*r-o—R>-', any two of them be- ing given : which contains theſe three 'Theorems following. D 2 X 2 207. Th. 2. B = ER e Den x f 208. Th. 3. A = LD. 209. But, ſays our Author, if any of the Terms 0 | 7 6 +3 6 + 2ay be wanting or vaniſh, or break off the Series: The Ruadrature of CV RVES explained. pA be aſſum'd in the Beginning of the one Progreſſion, and the Area qB in the Beginning of the other, and from theſe tuo Areas, will be given all the Areas in both Progreſſions. | = 3 oh From the preceding Theorems it is evident, that if 8 or 8 ＋ xy or $ + 2*q be equal to nothing, the Series of Curves is broke off or in- terrupted : thus in the general Equation or Theorem 2%R* — 82A — Tan xfB=0+2a1 xg0, if it be 9 = o, A diſappears; if it be Mu s, B diſappears; and if it be 8 + 2 —o, C oo Foie But the Remedy our Author propoſes in this Caſe, I acknowledge I don't ſufficiently underſtand, according as he expreſſes it above. For ſuppoſe we call A, B, C, D, &c. the aſcending Progreſſion ; and A, P, Q, R, Sc. the deſcending Progreſſion, fignifying as above: the Meaning of what he ſays, would ſeem to be this, that when 8 So, from the Aſſumption of B and P, all the Areas in both Progreſſions on both Sides of A may be found: that if it were 8 + ay — o, all the Areas on both Sides of B may be found from the Aſſumption of A and C; and that if it be 8 + 2&4 =o, all the Areas on both Sides of C may be found from the Aſſumption of B and D. But this is not true: for when 8 + ay — o, from the Aſſumption of A and C, you cannot find the Area of any other Curve in either Progreſſion. For in this Caſe A and C are not ſufficient for determining B: ſince the Equation above z*R*— dA — 8 + ay xXfB= 6+ 2 x gC becomes 2R> — beA —= an x gC, where B diſappears: but without B and C, neither D, nor any of the Areas above C can be found ; and without Band A both, neither P, nor any of the Areas in the deſcending Pro- greflion can be found : as may eaſily appear from the Conſideration of the three preceding Theorems, and what has been ſaid formerly. How- ever in the Caſe propoſed, viz. when any of the Terms d, 8+ ay, 0 + 2ay is wanting, 1*, If it be 6 = o, from the Aſſumption of A and B, all the other Areas in both Progreſſions may be found. 2. If it be G ＋ xy =o, from the Aſſumption of the ſame two Areas, all the others may be found. 3. If it be 6 + 2 =o, from the Aſ, ſumption of A and B, 'the Areas of the Curves in the aſcending Pro- greſſion cannot be found: but if you aſſume A and C, all the Areas in both Progreſſions will be found, as appears from the foregoing The- orems. Yet it will not thence follow univerſally, that any two Areas being given, all the Areas can thence be found. For, 1*%. when it is So, if the two given Areas be both in the aſcending Progreſſion beyond A, none of the Areas in the defcending Progreſſion can be thence found: ſince the Equation that includes B and C, Ann, in L 2 clude: — S 7 F — — The was - — 2 . * — I - TIE...” 4 A | 164 The' Ruadrature of CVR VES explained. | | clude A, the Term. 8A. vaniſhing : and this evidently prevents the paſling from the aſcending to the deſcending Progreſſion. 20. If it be 0+ a1 =o, from A and C given, none of the other Areas can be found, 38 was ſhewn already. 3*. If it be 8 ＋ 2ay o, from the Areas A and B given, or any two Areas in the deſcending Progre none of the Areas in the aſcending Progreſſion beyond B, can-be thence found: for the Equation including A and B, doth not include C, and therefore this interrupts the Paſſage from A and B to the other Areas in the aſcending Progreſſion. But if you except theſe Caſes, any two Areas being given, either both in the aſcending, or both in the de- ſcending; or one in the one, and another in the other, all the other Areas in both Progreſſions may be found by help of the preceding o Theorems. : . 2 Schol. 1. In the preceding Caſes, from one Area only given, in the Caſe of Trinomials, other Areas are found. Thus 1“, when it is Do, one Area above A in the aſcending Progreſſion, ſuch as B, C, Sc. being given, all the others in that Progreſſion, except A, may be found. 22. If it be 8 + Ay o, the Area C may be Pund from A, or A from C. 3. If it be 0 + 241 =o, from B or A, or any one Area in the deſcending Progreſſion given, all the other Areas in that | Progreſſion (including alſo B) may be found: as may eaſily appear from the preceding Theorems, and what has been ſaid. © Schol. 2. There are analogous Obſervations which hold with re- ſpect to other more complex Curves, ſuch as Quadrinomials, Quin- quinomials, &c. And particularly with reſpe& to binomial Curves, it * Of this appears from Art. 202, 203 *, that when 6 = o, from B, or any 8 one of the Areas in the aſcending Progreſſion above B, given, neither A4, nor any of the Areas in the deſcending Progreſſion can be found. 2% When 8 + a1 =o, from the Area A, or any one in the deſcending Progreſſion, given, neither B, nor any other Area in the aſcending Progreſſion can be thence found. But that in the firſt Caſe, the Curye whoſe Ordinate is az" Xe = Jv" me A is quadrable, it's Area being * + fz"\*: and in the ſecond Caſe, the Curve whoſe Ordinate is PAS hö 2 JU 0 + 7. d | a2)! x , is quadrable ; and it's Area — — -: as ap- * pears alſo by Art. 149 *. | | | 210. The ſecond Caſe of the Prop. is where >, the Exponent of R, in the Expreſſion of the Ordinate, is to be increaſed, or diminiſh- — —— w— © + ——__ De Quadrature of Cunyss explained. ed, by the continual Addition, or Subtraction, of Units. This Caſe, as far as it reſpects binomial Curves, ag. when R= e , is, left out by our Author: and is ſupplied in what follows. * Suppoſe the Area of one ſuch binomial Curve to be given, whoſe - Ordinate is 2%-'R?, and that Area to be called A. Then p iR pA are correſpondent Ordinate and Area, 33 any conſtant Quan- tity *. Fonker, divide, and then multiply the Ordinate p N by "ne 1 6. R or e *, and it appears thus pe + pfz" x 2%—R>= different in — —.— but the ſame in Effect, EI it was before, ſince the Multiplication and Diviſion by the ſame thing, mutually compenſate each other. Then, fince the Ordinate he ＋ 0 + an X f2" X -R. and Area 2 RN anſwer to each other, by Prop. 3, add the former Or- dinate and Area to the latter Ordinate and Area, reſpectively: thence you'll have ÞFT Ne x fe" x N-. for the Ordinate be: longing to the Area pA + R: ſuppoſe the ſecond Term in the Va- lue of the Ordinate equal to nothing, that is p 4-08 + ay N= o, or þ +0 +24 =0 : thence p -x: inſert this Value of þ in place of it, in the firſt Term of the Ordinate, and in the corre- ſponding Area, and you'll have — ye iR and — Q N XA + 2#R3, or, by dividing both by — aye, 21Ra1 and DR | ; mos 7 *: [16% an Ordinate and Area belonging to the ſame Curve : which. Area is wholly known, ſince A is given; and the Ordinate has the Index of R leſs by Unity, than the Index of R in the Ordinate of the given Curve. The Steps of the Proceſs, ſet down diſtinctly, follow, 265 Ordinates. | Areas, 1%. FE 7 S- INR: - pA. in | 2% T ... opp Put the Sum of the ſecond Terms equal to nothing, or p=—0 + an. | 1 3. e X - i˖if - DX ANR. Divide both by — ave. 12 E . . - - - S—1Rm=t.,c.- B. K 2 At ic TT ae. . Now, | 165 # Art. 202, 203. of this Ex- plication. = — 3 , * — — * . * * 5 The Qadrature of Curves explained. Now, after the ſame Manner, that, from the Area A belonging to the Ordinate 2*—'R>, given, the Area belonging to the Ordinate z 1R—17, was juſt now found, the Area belonging to this laſt Or. dinate, being given, you may find the Area belonging to another Curve, whoſe Ordinate is 29—'R*—2: the Method of doing it and the De- monſtration differs in nothing, but inſerting a — 1, every where, for A in the foregoing : ſo that if the Area juſt now found be called B; and the Area belonging to the Ordinate 2 Ra, be denoted by C, then C = . And ſo you may proceed to find An n Xe the Area belonging to the Ordinate 2*—"R*—3, by a like Proceſs, viz. by inſerting a — 2 for x, and fo on continually. For in effect, the Theorem, which makes the Concluſion of the firſt Demonſtration, vis. B= DEA, includes all poſſible Caſes of binomil Ane | Curves, in which the Area of any one being given, the Area of an- other is ſought, in whoſe Ordinate, the Index of R, is leſſened by Unity: becauſe whatever be the Index of R, in the Ordinate be- longing to the given Area; whether it be affirmative, or negative, you may ſuppoſe it to be repreſented by A : and again, whatever be the Index of 2 without the Vinculum, it may be always repreſented by 6— 1. Therefore alſo converſely, if A and B repreſent the Areas of any two binomial Curves, ſuch, that the Index of R be greater by a poſitive Unit, in the Ordinate belonging to- A than in the Ordinate longing to B, by reſolving the Equation, and conſidering B as given, 0A and A as ſought, you'll have A — 2 Which furniſhes 1 | | an X Ee | with the two following Theorems, ſerving univerſally for carrying on the Progreſſion of binomial Curves, from the Area of any one being given, in whoſe Ordinates the Index of R is to be diminiſh'd or in- creas'd by the continual Subtraction or Addition of Units. © | 5 2 A 1 211. Th. 1. B — * F 212. Th. 2. A = EEE, 213. I ſhall illuſtrate what concerns the Increaſe and Diminu- tion of the Index of R by Units; and of the Index of 2 without the Vinculum by »y: for which I gave two Theorems formerly * by an Example, in the Caſe of a binomial Curve, which will make all evident. | Let 1 „„ 0 re tt ©. De Quadratur of Curys explained. 167 Let e /, define the Relation of the Abſciſs and Ordinate of a Curve: and ſuppoſe I would compare the Area of this Curve, with the Area of another Curve, whoſe Ordinate is Ya — 27, 2 be- ing the common Abſciſs to both : which laſt Ordinate plainly belongs to a Circle, in which à is the Radius, and the Abſciſs x reaching from the Center to the Ordinate : and therefore the circular Area, with which the Area of the Curve propoſed, is to be compared, is that lying betwixt the Ordinate and the Radius of the Circle parallel to 2 * Compariſon, let the given Ordinate, vis. 2 be brought to the Form 23” X a* p. :; call the Area , and from the Aſſumption of B, let us firſt find the Area belonging to an Ordinate expreſſed thus 23" Xa* — g where the Exponent of the Quantity under the Vin- culum, vi. a*— 2“, anſwering to R, is greater by Unity. Call the Area belonging to this laſt Ordinate, « : then, by comparing this Ex- ample with the two general Theorems juſt now laid down, you'll find A=a.B=B.1=;.n=2.aA=;.ec=as. Therefore by the | ; : 3 ſecond Theorem you have a = = 4 — —, for the Relation of the Areas à and B. 45 3 Again from the Area a, belonging to the Ordinate 2 x a*—2\t, I find the Area belonging to the Ordinate a e - H or /a*—2*: which is the former Ordinate, having the Index of z without the Vinculum, diminiſh'd by 2 =. This is done by Caſe 1. of this Prop. For calling the Area correſponding to this laſt circular Ordi- nate, viz. Va — 2*, ; and reducing the two laſt Ordinates to due Form, you have 23—" x a* — ii, anſwering to the Area &; and z x a* — 21" anſwering to the Area y: wherefore, by compar- IDE by Art, 203 * you'll have y = =—— 44. — 3 Value of « ſet down above) ESPEt Ss di, woe. Wat — 2* + g. Whence G == Y — 2e — , i. e. the Area I. e it, as appears by Art. 146 *: which Area is repreſented by ABCD in * Of this the Fig. belonging to that Art. when AB S. Now, to make this ca. ing this Example with Art. 201, 202, 203 *, you'll find B—=e@ and - Of this Azy.0=1.n=2.A=+,e=as.f= — 1. And therefore Explics- — (by inſerting the » of this the Curve pr . . belongin Zz - iR and ture of Ordinate 2 f - iR: de Quadrature of Curves explained. is equal to the circular Area, leſs the R contain d under the 3 Abſciſs z and the Ordinate of the Circ ® Of this 1. e. Are 146 , the Area 214. Our Author ſhews at full to the —5 trons 1R>, 403 * Fy pA +gB 1 Tg“ gam of the Areas, Then ſuppoſe the Coefficients of all the Terms, 17 the firſt, of the complex Ordinate, equal, each to nothing: by which you'll have From the laſt of which, viz. 8 + » + 2 x bg + 9g = ©, find a Value of 9, which is q=— b + y +231 xt. AC feet this Value of ꝙ in place of it, 7 the ſecond which ariſes from the third Term, v7z. 8 + 2a1 x ag +6 Fn +4 x bf o+ pg + + gf = o, thence you have a Value of p, viz. px b — 6 8 + 2a 2A71 X 0. three diſtinct Equations. „how, from the two Area Curves, whoſe Ordinates are 1 the Area correſponding to the ſays by the like Means you may Curves. find the Area belonging to the Dave whoſ: Ordinate is 2 —1R —i. The Manner of doing it is thus. Let the four Ordinates, and four correſponding Areas be ſet down, as he hath done: 1 - D ——— ini gum of the Ord- * xbe eee + Equation, Again inſert theſe Values of p and q now found, In the firſt Equation, ariſing from the ſecond Term, which is 6 +1 x af + 0+ nxbetpf + ge So: and fo you'll have an Equation free of p and 9, vis. N — an x be N NN by a proper Reduction, gives you the Value of b; viz. 5 =z Next, ſubſtitute this Value of b in place of it, in the vent Equa- =0Q: which Mc — 2g. tions, e the Values of 9 and p, viz. ꝙ —= : = 2A N 8 An xL—0 + 221 Xa: and fo you'll have 9 201 X a. And ſo having found theſe. Values df 75 The Quadratur of Curvy 28 explained. 158069 9, F and 6, , from the Suppoſition of the Coefficients of the | three laſt Terms of the complex Ordinate being equal to nothing : put theſe Values of q, p and 5 in place of them in the Sum of the Ordi- nates and Areas, and you'll obtain dae + 27" — 0+ 2a1 X ae x Re- for the Sum of the Ordinates. And Lz „A- A= AB + az + Fi — R Sum of the Areas. oo > — 2 Multiply both by the given Quantity /* — aeg, and then divide both by the given Quantity 4ayaceg — Anaef or — Anza x f*— A, and you'll obtain the Ordinate 2—R*—1, and correſpondent Area FEonx = x: x — FFD x/j1 x Bb gt PRO + Fine x Ad . | — Ante Xx /* — 4 | b Which therefore is the Area ſought, and is wholly known, ſince A and B were given. Wherefore, if you call this Area C, and call A, B, and D the Areas correſponding to the Ordinates 2*—"R>, 2%+»—'Ra> and 2 iR i reſpectively, as they are uſed by the Author: by con- fidering what he hath here demonſtrated in this 36th Art “. and what * Of the is demonſtrated above, you'll have the two following general Theorems _ ſerving to find the Areas of all trinomial Curves whoſe Ordinates are Cave. thus generally expreſſed Rr, when the Areas of any two of them are given, where the Index of z without the V inculum is greater by + » in the Ordinates of the Curves whoſe Areas are repreſented by B and D, than in the Ordinates of thoſe repreſented by A and C: but the Index of R in the Ordinates correſponding to the Areas A and B greater by Unity than the Index of R in the Ordinates of thoſe whoſe Areas are C and D. +» 415 Tb. . PE- Fd TETEIS 8 —ͤ — an x f* — 40 | 216. Th. 2 OC 2 — 8425 X 84 x 2 —— xP*x A-b+24+22n x/z xB — Aue x f'* — 217. 2 theſe two Theorems, if What Was aid formerly *, be duly Dry conſidered, and the Areas of two of the moſt ſimple trinomial Curves, plication. be given, all others, in whoſe Ordinates, the Index of R under the Vinculum, is increas'd or diminiſh'd by any Number of Units, may be found, | CASES 3, 4. 3 218. And now it appears, by what means, if both the Index of z of the without the Vinculum is increas'd or diminiſh'd by the continual Ad- _— 7 dition Curves. 170 The Quadrature of CUuxvES explained. dition or Subtraction of y ; and the Index of R under the Vinculum by the continual Addition or Subtraction of Unity: the Areas anſwer. ing to all the Ordinates will be given, when the Area of any one of the binomial Curves is given: or the Areas of two of the trinomial Curyes are given. For this is evidently done by the firſt and ſecond Caſe, conjoin d. An Example of this when the Curves are Binomials, was Aut. 213. given already . And the like may be done in the Caſe of Trinomi | — when the Areas of any two of them are given. For the Ordinates of any two trinomial Curves may be repreſented thus, 2 Rꝰ and 2 Rg; ſuppoſing then the Areas of the two Curves, to which theſe Ordi. nates. belong, to be given: aſſume the Area belonging to the Ordinate a R, as if it were given: then from the two Areas correſpondin to the two Ordinates a and z'+»R>, you can, by Caſe firſt, find the Areas correſponding to the Ordinates 2##-"R* and a]. R from | the two laſt, you may find the Area belonging to the Ordinate l aA RA by Caſe ſecond : and this Area being the other one which | was given, it follows converſely that from the two Areas, belonging 1 to the Ordinates 2%R* and 2 R, given, you may find the Area I belonging to the Ordinate S R, which was formerly aſſumed. 1 | And therefore the two Areas, correſponding to the two Ordinates 2%R* and 2*-*R*, being now both given, it is evident you can thence find the Area of any other trinomial Curve, whoſe Ordinate is thus indefinitely expreſſed z%t-1R>tr, As to the fourth Caſe of the Prop. when R is a Quadrinomial, or any other Multinomial : the Demonſtration of it is made out after the like Manner as when R was a Trinomial. DEMONSTRATION of PRoP. vil. CASE 1. 1 The firſt Caſe of the Prop. is when the Index of 2 without the Vin- | culum, is increas'd or diminiſh'd by the continual Addition or Sub- | traction of 9. 219. Let the Ordinates of two. Curves be pz ROSH! and . and the correſponding Areas pA and 45 when De f and S 2 & + , both:Binomials : then you'll have as follows, The Ryadrature of Cunvis txplaind. 11 Ordinates 1 1 : — | - - gB. 2 ; e et Ig. ep. — - AN- Ye: and then take the Sum of the Ordinates and ee: eee -A- Divide both by the given Quantity N N. 996 Ur— „„ Ir! „ -d i | nent? = NEX. Which laſt Area belongs to the Ordinate 2˙T iR i, and is wholly known when A and B are known. And by the like way of proceeding, if you call this laſt Area C; from the Areas B and C you may find a fourth Area correſponding to the Ordinate 2 R181; and ſo on. And the Method of proceeding in the Progreſſion going the contrary way, viz. from A and B to find the Areas correſponding 1 to the Ordinates Rs, z iN, Ge. in inf. 10 | after the ſame Sort . And contrarily, from any other two Areas, . f _ | you may, by Analyſis, return back to the Areas A and B: as will ofthis E- eaſily appear by conſidering what was faid upon the foregoing Prop. plication. ſo that from any two of theſe Areas given, all the others will alſo 2 — And when R and S taken together conſiſt of five Terms or Members, then there muſt be three Areas given: if R and 8 taken t conſiſt of ſix Members, the Areas of four Curves muſt be given, Gc. and then the Areas of all the reſt may be found, after the like Manner, as hath been done, when R and 8 are each Binomials, and the Areas of two Curves given. CASE 2. The ſecond: Caſe of the Prop. is when >, the Index of R, is in- creas'd ar diminiſh'd by the continual Addition or Subtraction of Unity. T 2 | 221, Let 172 The NQuadrature of Curves explained. 221, Let the Ordinates pa. R“ and g -i RS“ have pA and 4B for their correſponding Areas: and let the Ordinates be firſt myl. tiplied, and then divided by R — e + fz", as in Caſe ſecond of the laft Prop. by which they ſtand thus pe + pz" x 2%1R>-19% and ge” ＋ % x R718“: again, by Prop. 4. the Ordinate — | _ — bet +6 +anxft+8+ pn Xe + 6+ an + py +1 x flen x 28—1R>-18* and Area , belong to one and the ſame Curve Therefore take the Sum of the,three Ordinates and the Sum of the three Areas, which will make an Ordinate and Area correſponding to each other : then ſuppoſing the Coefficients of the' firſt and third Terms of the complex Ordinate, each equal to nothing, viz. pe ＋ bel —o and 6 + ay N to; or, by reducing, þ = — K and g = — 8 +31 + pn + 1 x1: which Values ſubſtitute in place of - and g: divide by ay x f# - el: and thence you'll obtain the Area RS“ — MA D wondin ; * A , 9 ponding to the Ordinate zT 1—1R>=1S*, as follows. | Ordinates. Areas, 1. pe + pf" ALES TINETEDLS, . 74 2%, ge + ofs" | iN 75 3e. 6 =y Mr Ng - - NS _ ge 5 Atm RAI.» SL- An x fa — 4 | If, inſtead of putting the Coefficients of the firſt and third Terms equal to nothing, in the complex Ordinate, you had put the Coeffi- cients of the ſecond and third Terms equal to nothing, you would have . thence obtained the Area correſponding to the Ordinate 2%-R>-S : and calling theſe two Areas C and D, after the ſame Manner that from A and B given, the Areas C and D are found, you may from the Areas C and D, proceed to find the Areas correſponding to the Ordinates 29-"R>=2S* and zt iR“, and ſo on continually. And converſely if C and D be given, by Analyſis, you find A and B; and thence the Areas of the Curves belonging to the Progreſſion going 5 The Ruadrature of CURvES explained. ing the contrary way, viz. where A is continually increaſed by nity : and the reſt follows as formerly, v Moreover the ſame may be done, by the like Means, when R or 8, or both of them are more than Binomials: provided ſtill the Number of given Areas be but two leſs than the Number of Members or Terms of which R and S, taken together, do conſiſt, | | CASE 3. 222, The third Caſe of the Prop. is when the Index of 8 is in- creas d or diminiſhed by the continual Addition or Subtraction of Unity. Let pA and gB be the Areas to which the Ordinates pz%!R>$* and 1R*S* correſpond: when R and S are Binomials. Then from theſe two Areas pA and B given, you may find other two Areas belonging to the Curves whoſe Ordinates are 2 R and RS, after the like Manner that, by Caſe ſecond, you found the Areas correſponding to the Ordinates 2 R S“ and z iR ig: all the Difference is, that the Ordinates of the two Curves whoſe Areas are given, are firſt to be multiplied, and then divided by 8 þ + Iz", inſtead of being multiplied and divided by R = e + fer: and then all the reſt goes on as formerly. CASE 4. 223. The fourth Caſe of the Prop. is when the Index of > is in- 3 or diminiſh'd by the continual Addition or Subtraction of 3, and likewiſe the Exponents of R and 8 by the continual Addition or Subtration of Unity. In which Caſe the Areas anſwering to each of the Ordinates ſhall be given, upon the Conditions mentioned in the Prop. and it is done by the three former Caſes jointly. Let 2*"'R*S* and zπ＋ -R repreſent the Ordinates of any two Curves whatſoever, whoſe Areas are given, R and 8 being Bi- nomials : then aſſume, or 1 to be given, the Area anſwering to the Ordinate 2 g“ From the Areas correſponding to the Or- dinates 9—RIS* and z - R, you may, by Caſe firſt, find the Areas anſwering to the Ordinates 2 iR“ and . R: | from theſe laſt two Areas you can find, by Caſe ſecond, other two Areas, correſponding to the Ordinates 2 . HRA and 2 Tete iR AT: again from theſe laſt two Areas, you may, by Caſe third, find the Area anſwering to the Ordinate 2 i a R which was one of the Areas originally given: therefore converſely, by | Analyſis, TD Nadrature of CVuvxG explained. Analyſts, when the Areas belonging to the Ordinates 2 N and Te- RAe are given, as at firſt, thence may be Wund the Are, $7: which anfwers to the Ordinate #4#+—'eS$* ; which — firſt af. famed: and conſequently it being now given, together with the Are, correſponding to the Ordinate 2%— RS", from theſe two Areas, now given, the Area of any other Curve in general belonging to the Claſs of Curves, mentioned in this Prop. may be found, when R and 8 are Binomials. And the like is the Reaſoning in all other Caſes, provided the Number of given Areas be ſuch, as is mentioned in the Prop, SECT. VIIL Containing Notes on Art. 40—— 51. | PROP. IX. Art. 40. 224. HE Truth contained in this Prop. vig. That the Areas of 2 ' theſe Curves are equal whoſe Ordinates are reciprocally ture of Proportional to the Fluxions of their Abſciſſes, was at large inſiſted 3 , upon towards the Beginning, and was there demonſtrated “. of this Ex- plication, COR. 1. 225. By means of this Corollary, we may find innumerable Curves, whoſe Areas ſhall all be equal : of which take the following Examples, Ex. 1. Let dx — x* =0* be an Equation given, belonging to a Circle: and it is required to find other Curves, whoſe Areas may be equal to that of the Circle. | | Let z and y be the Abſcifs and Ordinate of another Curve, having | | it's Area equal to that of the Circle: and ſuppoſe dx = 27 expreſs the | Relation of the Abfeiſſes. From this Equation 4x — , find the Relation of their Fluxions, by Prop. 1, which is 4x = 222, that is by ſuppofing x 1, {= 2zz or 2 2: . wherefore by this Prop. 32 2 I:: v A: y: whence y=—Jv/dx x, but , by Hyp. or 1 =: which Value being inſerted. for x, you have y == 4.— 2* for the Pro- perty of the Curve ſought, whoſe Area is equal to that of the Circle. „Prop 1. Again let it be # u: then 32 Q =0*, or z=—— : where- fore ſince the Ordinates and Fluxions of the Abſciſſes are reciprocally pro- 5 e w a 0 N= wins FP * OY Dm ' 2 The Ruadrature of Cuxvxs explained. proportional, you'll have — : 1 :: de — : 5, that is y=— = ; or (by inſerting a for x, from the given Equation dd y== * — 4. dz, the Equation to the Curve ſought : here the curvilinear Area is ſituate beyond the Ordinate, extending itſelf along the Abſciſs infinitely produc d: for à is negative, 2 diminiſhing as x increaſes, and contrarily : and when x o, 2 is infinite, but when x d. a: which things eaſily appear from the Equations above. Thus let ABCD be a Semiciccle, Diameter AD= d, the Abſciſs AB — x, and Ordinate BC A —v: produce AD indefinitely towards G, take AE a third proportional to AB and AD, which 4 call z : whence it is z =: draw EP perpendi- cular to AE, and take it equal to 2 f r =. Then the Ordinate EF ſhall deſcribe a cur- vilinear Area extending itfelf infinitely along the right Line DG, which is the Aſſymptote to the Curve: ſo that the Area EFG adjacent to the Ab- ſciſs infinitely produced beyond the Ordinate, ſhall be equal to the circular Area ABC adjacent to the Abſciſs AB; and the remaining Area DEF, lying EHF above the Ordinate EF, and adjacent to the Part of the Abſciſs DE, is equal to the remaining cir- — cular Area BCD: and the whole curvilinear Area DGP D — the Se- micircle ACD A. And if you take DE AD, you ſhall have DEFD equal to the Quadrant of the Circle. And thus you may, by aſſuming new Relations of & and x, find infinite other Curves, whoſe Areas ſhall be equal to the Area of the Circle. 3 | Ex. 2. Let a v be an Equation. to the Hyperbola : and you would find another Curve, whoſe Area is equal to the hyperbolical Area: call it's Abſciſs and Ordinate z and y, as formerly: then take any Relation of the Abſciſſes, as thus, bx =2® : thence by Prop. 1, 56222 or ==: therefore =; 12 = , whence y n = (by inſerting the Value of *, viz. 5) Ig which ſhews the Property of the Curve, whoſe Area. is equal to the hyperbolical Area. After the like manner you may find infinite other Curves, having their 7 | Areas 175 176 Art. 108. of this Ex- plication, De Quadrature of Cunvaes explained. Areas equal to the hyperbolical Area. And likewiſe any known Curve being aſſign d, you may, by the like means, find as many Curves a you pleaſe, having their Areas, each, equal to the Area of the known CREED. . | | N. B. This Operation may be called Transformation of Curves : be- cauſe by it you are taught a general Method of transforming any given Curve into as many other equivalent Curves, as you pleaſe: which may be of great Uſe in the Solution of Problems in Geometry, 226. It would be much the ſame, if it were required to find any Number of Curves, each of whoſe Areas bears any aſſignable Relation to the Area of any known Curve, | Thus let x, v; 2, b repreſent the Abſciſſes and Ordinates, of the given Curve, and of the Curve ſought, as before: moreover let 3 de. note the circular, elliptical, hyperbolical, or any other Area of a known Curve; and ? the Area of the other Curve. And ſuppoſe the given Curve be defined by the Equation a* — px = av* o a . —v: which is to the Ellipſe, having @ the half Axe, and p it's half Parameter: let 25 + - xv=0o, be the aſſigned Rel. tion of the Areas. Then, you aſſume any Relation of the Abſciſſes x and 2, as px + az*x* =pa*. And by means of theſe three Equa- tions you may find the Value of y, i. e. the Property of the Curve, whoſe Area bears the Relation aſſigned to the elliptical Area, The three given Equations are, 1“. of — px* a', being to the given Curve: 29. p + az*x* = pa*, defining the Relation of the Abſciſſes. 3. 25 + f — xv = ©, containing the given Relation of the Areas, from which y is ſought. Ek Now ſince * = yz or y, you muſt find the Ratio of 7 to z or 2 X 1 in Terms made up of z and known Quantities, which can always be done by means of the three given Equations. In order to which, from the ſecond Equation px* - a2*x* = paa, take the Re- lation of the Fluxions, viz. 2pxx + 2422x* ＋ 2a2*xx = o, or 2 — x — px . 1 e. by ſuppoſing + = 1 (for we may always ſuppoſe one of the flowing Quantities to flow uniformly) à = ZE: which Value of z you ſubſtitute for it in the Equation y = 25 and it be- comes y = — having one fluxionary Letter only: that you may get free of it likewiſe, from the third given Equation 25 + # — xv Do, take the Relation of the Fluxions, viz. 474 1— 1 xV=0, | OD whence as ©. % . a” a. Mit. «a = 7 Conver\iand, 5) Abende you have #==(xw "xv A= #6 23= (becauſe 3 * — ur bot from che firſt df the Sven Equations, vd. 4% Pe = , we have — 2 = z2a v. or Hh = : : fubſtitute this for o in the Equation # = x% — v, and you have 2 — ie. by inſerting H. WIE for v. from the firſt of the three given At 7-5 * 1012 2 40! 27 in LEE” od. 4 8. Ff je” 5 y= ———— Filly ede. of the three given Equations, viz. fx* ＋ az*x* = a*p, find the Value of æ, which is SF — "inſert it N run and reduce, and you'll obtain, at length, e the Equation de. fining the Pro} rty of the Curve ſoug ht: whoſe Area is to the Area | of the Ellipſe in the Relation Rai And after, the ſame 2 , if other Relations of the Abſciſſes of x and 2, different. from the fe 1 going, weile afſumed, the Relation of the Areas continuing the fame, you may find other Curves, each of whoſe Areas ſhall be in lame given Relation to the Area of the given Ellipſe. And by the lame way of proceeding, when an — Curve is aſſigned, you may find infinite other Curves, whole dae dn have any given — corn 92 eee, ere > © rh 227. The Truth 1 rs thus. Since by the Hyp: x, thence by taking the Fluxions, y ou have & 2. e. F =>» 4 by ſubſtituting = for 5, x for 2 and & 7 for or for 2—1, which is deduced from the ion en 2. = D = X * hence deu obtain the Ratio of the Fluxions of x and 2, viz. 4 :* —7 * . But, by this Prop, theſe Curves have equal Aras. vboſe Ocdivates are reciprocally proportional to the Fluxions of their Ablcſies: yoherofore, if to. the three Quantitcs 2x5 1 nnd Me 2-1 - - - } 3? 2 ©» CH ww *- * 14 | | | | | 178 The Ruadrature of Curves explaintd. 29-1 = gz Sc.“, find ſourth Proportional ſhall be the ed. A Carry 8 is equal to the - Dog the Curve propoſed, it's Abſciſs being &: but a fourth e to 1 2n A theſe three Quantities is e — ſtitute . for 2", and . . br =!, from the Equation x' = x given at the Beginning, you ſhall have 2 + G xenon for the Ordinate of a Curve, which having x for it's Abſciſß, has it's Area | equal to the Area of the —— Curve, erb being z has by Oe, ie en Lb - * 99 : « 7 w 1 * « * | c 0 3 3. Art. 43: 228, This Corollary appears thus: ſince by 8 Hyp. it is x=x — +, thence by Prop. 1. $=2z 7, 7. | wherefore if you put 2 1 N a e conſequently, if to the three Terms 25 , 1, wi xa Do teeÞ cnn f gan TeV; you fn ra Nan — TS be the Ordinate of the Oarve whok Abſciſs is #, and it's Area equal to the Area of the Curye propoſed: , a+be' e er Ge. the fourth Proportional is rer | 7. e. (by ſubſtituting & for 25, x * Tes and x» _— from the Equation #= 2", and reducing) . © xXIFF Fan Fe. co R. ai Art. 44” 224. | Since the Relation of the Abſciſſs, and conſequently of thi oY Fluxions is the fame as before ; to the three Quantities 2 , 1, and = Curves. -i, Fer +a Tc ce Deg Del! HEA, — ol br 4 b ar RG wks. and 1 24'S" e x JOE On b being . ſhall have it's Area equal & to o that of the Curve | COR. x. 5 14 230. gince * == —, thence 2 dee eg the anc 0 Ocdinate of the Curve, whoſe AbſciG is. x, u, you'll have this Prop. Quites — 2: r iK of fr penn: TV: v, i. . vr ati x P/ Þ ge” TS. or, by ſubſtituting x—» for 2 and & —1 for 2+" from the gyen tors yaw} ling | Vn Je x TDA Dh; 7. ©, wher the Quantity under the Vin- ah e Nieht eee and then 8 ee == er ; but if the Quantity under the Vincu- um be a Trinomial, then by firſt dividing, and then multiplying by xd, un aaa: which are the 'Ordi- nates of Oytves, having x = 2 for their Abſcis, and their Areas equal to the Areas of the Curves whoſe Abſcifſes being z have their Ordinates g-1 xe T, and 2-1 e n 18 | COR. 6. k . — app the fn we * 46. 2111 1: , youll find that 2-1 * PDD F x 1 rr IF x == + g-= FE i; [FF FE :; — 2:1, by confidering that x = =, And when the Quantities under the Vineulums are Binomials, * the Ordinate — ruf HD- ee *πνi e = Fm Fee, by dividing the firſt Factor by , and multiplying the ſecond Factor by x» and the laſt Factor by x%, Which ns compenſate each other, becomes iN Ken x[E- RV: and if there be re- Members Aa 2 under — Curves. The: Ryadrature Cu — under the former. Vincu two under the latter; by dividing the bald Faber by N multiplying 1 the ſecond an by py and the laſt by x, the Ordinate appears thus g +2 + ex» x1 + tz"\*, And ſo. in other Caſes. Art. 47. 15 2 32. The Ordinates belonging to the Curves whoſe Abſciſs is x, — the two laſt Corollaries, come out negative in the Operation, where. — by thay are diſooveted: although the negative Sign be not prefixt by Curves. the Author : which. ariſes from this, that the Fluxion of x is nega tive; or x diminiſhes as 2 inereaſes, and contrarily ; ; whence it hap- pens, that, if the Area of the Curve havipg z for it's Abſciſs, lye —4 the nearer-Bide of 7 * Ordinate, and adjacent to the Abſcifs, the Area of ofthe other Curve, having x for it's Abſciſs, and whoſe Area is equal t of tlie ferner, weill be fituate along the Abſeiſs produc'd u thi further Side Vf it's Ordinate, and contrarily, 4s is obſerved by him in this Att. For if the Area of the Cut ve, having 2⁊ for it's Abſeil, be adjacent to the Abſciſs, and ſo. increaſe as the Abſciſs z increaſes; the Area of the other Curve, that it way increaſe at the ſame Time, muſt) be ſituate: beyond it's Ordinate, for the Increafe of z is the Di- minution of x. But if che Area of the Curve, whoſe Abſciſs is 2. ye upon the further Side of the Ordihate, the equal Area of the other will ſye upon the nearer Side of the Ordinate for "the ſame e Reaſon, * Of this See 2 2255 AF ITT X Explica | -; Wo. c 0 R. 6 2 33. — Truth of this Coxollary Wees. It eke that | pon Ho K g AHA T Ar FB ind the Relation of the Fluxions x 4 n 7 heh of thee —ü— — „C „% „% =... Equations, vi. & 1 Ff. (i D:: : : 2 X. Therefore, fince the Areas 0 of the _Cyryes are equal by the Hyp. It is 57 22 K* 52: So I 2 —1 9 TH whence . you have y = # ©; x;57v 2. from thence ” = s 2 Fa. e, and Ni Ge. gain f f he, and = 7, Ge. Wherefore i in the Equation to the given Curve, viz. | ya xe ee Dc. = b ND Tc. by 25 erting in place of *, * 88, al, theſe Values juſt now found, it will ſtand thus al; pd. SET, GET | — 2 JIG ay == MM * 7 T 5 2 2 2 — 7 20 20 ee - 1 x ec The Quadrature of Curves — 167 TE F e „ het .#*, e TT. But = — LY from the firſt of the original: dn, aberefore, by throwing out repugnant Expreſſions, the Equation is on I ex ND e. x FTI N . Divide both ies of the Equizzon by = XTR 1 . . : * : 231 A — 14 — >a 1 2 and moreover by x and you'll have i it ſtanding thus © 7 =p > Ear e Foe x FFT Fm" r , . 5. B eee from the firſt and third of td original Equations, Wherefore by inſerting theſe Values, and raiſing 15 Sides of the Equation to the Power whoſe Exponent is , We ſhall have the Equation fully. reduced, ſtanding thus 2 * T Fee e Gl TTD b DF D Which expreſſes the Relation betwixt the Abſciſs wid Ordinate, in My: Curve having x and v for it's Abſciſs and Ordinate, and that by an ye — not affected: and whoſe Area is equal to the Area of the Curve d, whoſe ' Abſciſs and Ordinate are z and 5. By means of w the Abſciſs x is determin d, when the Ordinate v 1s 9 ub 4k 1011 31 C O R. 8. | . ne | 50 10 this Corollary it is ſuppoſed that, 1®, 5 = = 2%. .x = Art. 4 — +8 e —— = ent: = - of the 3e * a — 3, 4 + —— 1 ture of 3 the Relation of the Ordinates 7 and v, is the me as in Curves. * the ws Saga : ſince, by the ſecond Equation, the Relation of the Abſci and conſequently of their Fluxions, is the ſame : ſo that the Values of ye, * as, 2? may be thenee transferred. Wee have moreover in this Corollary < | Wherefore, by ingen. ing theſe Values in the given Equation — DN + &c. = 3 3 TS; - + 27 * | tr gz? + 9 T Oc. we ſhall have u — Sth —-— I — 1 2 # # S foes wink Ga = lx 4 Pl a EE a; x7 7+&e. b5rx7 pe u + Ce. 183 V=— . —— —— The Quadrature of Co nvas explained. But, as in the laſt Corollary, ZZ = o: and moreover, in this = and : =»; as appears from the firſt and fourth original Equations compared, therefore, by throwing out Repugnan- cies, dividing both Sides of the Equation by s = * — and ſubſtitut- ing þ and » for their Values, we ſhall have — = xk TVT] + &c. re u + &c. Li . ce Raise of of Ge Ab Abſciſs RN Or. dinate x and v, "ne to a Curve whoſe Area is equal to the Area of the other Curve propoſed, whoſe Abſciſs and — are x and 9: and * leſs Abe. I COR. 9. | 235. This Corllry is Ligen ger after the ſame Manner a aries fifth and Here it is ſuppoſed has, 1. 0 =, — * eee. 3. % . f. 4. ==, From the ſecond, by Prop. 1, we have the Relation of the Fluxion of x and . which is expreſſed by this Proportion & : 2 :: mx eee en- i- K . +&c : but &: :: 5: v, by this Prop. Therefore the two laſt Ratios of theſe two Proportions are equal : conſequent tly, if you reduce them into an Equation, and in place of y, infert it's Value from the Co- rollary itſelf, you'll have PPP Which, by dividing the firſt Factor of the Numerator by the firſt Factor r of the Denominator, the ſecond by the third, and the third by the ſecond, will ſtand thus v ei x 21 „ 2 X 55 e Ga) That is by multiplying the Factors, and inferting x for . Fat OO v=o xam xt” xa+be-”. That is, (becauſe ay = b and r =, and "=" = from the oi ginal Equations) v = a +be\, Which The. Quadrature. of Cunv 28 avplained. 193 Which mates exhibiu the Property of a Curve having fbr it's Abſciſs, and in Area to the Curve propoſed, whole Abſciſ i is 2; the Relation equa and z being that mentioned above. has Sir Jſaac deſires to remark three different Caſes, .or Suppoki- | tions, upon which the former of the two Ordinates in this Corollary ue more ſimple. The firſt is When x == 1 : for then the Or- is Ke F +7Fa x gn 4 Ge x EE. e Bapaiten if . and the Root denoted by the Index « be capable of being extracted: which would evidently 4 the Expreſſion more ſimp 15 | te third Caſe unge is when = —1 and 1 r Sr: for then the former Ordinate will put on this Form — and the latter 0: . eee + ac And beſides theſe three Caſes, there are a Variety of others, "pow Suppoſition of which, the propoſed Ordinate will become more ſimple, COR, 10. vii 1510 237. By ſome means or another an Error has crept into this Corol- lary, both I -o the Edition publiſhed by the Author himſelf, at the End of z and that publiſhed by Mr. Jones, with other original Pieces 0 Sir Laar n The * fimple Correction is by by .cheriog the Sign of the Index v, belonging to Sin the laſt Factor of the 58 of the Ordinate pro- poſed: ſo that, inſtead of Fe R. FN, it be made Sr R R-. = , and the other things remain the ſame, Or the Correction may be [made thus: fu the Expeſion of the given Ordinate to be made the ſame, inftead of —— _——— =, put 22 =, that is, you change the of » in theſe Equations, into it's oppoſite. But without one or of theſe Corrections, the Curve whoſe Ordinate is d, can- not be transform'd into the other that is mentioned. T ore I ſhall firſt demanſtrate the Truth of the Co ſu the Corrections, which I have mentioned to be made: then ſhew that the Corol- lary will not hold as the things are exprefiee i in the printed Copies. 238. For — of co URVES — 21 238. ret e eee Ge. 61 + Fr x fart =: + vo 270 K gain! 2 Sc. k E. -+ mz" + Cc. and warm 4. 2uymgir—! + Cc. write R, , 8, and 7 reſpectively, and every Curve whoſe Ordinate is HR 18. — * RN N, if it be , 122 6, : 2 == 9, and RS x, paſſes int another one equal to it in Ares, whoſe Ordinate is c * . For fince x = RS, we have by Prop. 1, 8 ese: but it's-evident that . A and 12 or R | Dis: by inſerting of which Values of R and 8 in the, preceding 4 you'll have x AR + oR*S*7! 175, :Whence & 3: 2K 78 + GR"; ; 1. Wherefore, by — if the given Ordinate 187 + HN x RCS“ x 4 + bR']", be divided by Re + SR-Se- 1j, we ſhall have RN e Y N, « the Ordinate * another Curve, equal 1 in Area to the former, jo 7" ing * for it's Abſciſs. But, becauſe — 6 or RN 85 ha the Ordinate may and thus," R-"8"S? eee , or, be- cauſe - =>, i. e. S 8 it wil ſtand thus Rn 184 - a+ has : Where if, inſtead of N — vs you put in = =; and in- ſtead of =, r, deduced from the original Equations you'll have K 1 . Nr 2 Gui is, becauſe RS? =x, and | 1-0 Sar x2X'aF-bxA*:; which is that mentioned by our Author, and is the Ordinate of a Curve equal | in Area to the Curve ſed ; and hay- * bot irs Abſcifs, - ral 239. Again, if the Values of R. 7, S and 5 be the fame as foe merly, and the Ordinate of a Curve be 78 JoR + @Rs X RS" TRA ; and moreover —_ e -— ry RS = x, the Curve will n into another equal to it, whoſe Ordi- nate is Le 4.2/4 p68 | {lt : For De Quadrature of Convss explained. | For we prove, as formerly, that the Ordinate of the new Curve is N87 48. T . e. becauſe — 2 2 us Pf = G KT lr 1 which, by inſerting 2 for p + uw and & 4 N- for S, from the original Equations, ſtands thus, R 8 xa +bx1*, that is x x a4-bx*\* as formerly. | 240. But if the Signs of all the Quantities be retained the ſame as in the printed Copies, then it may be demonſtrated, as above, that the Ordinate of the Curve will be RN x aS*+bR"1" or Re BUG T a — * +-bx'nS” = „ fe. WY YR aS*4-bx"S—*1" or R*-"S* „ 2 + l = x? x 48%» + bx-\*; whereas it ought to be * x TERA. 241. Our Author takes notice of certain Caſes, among many others, | in which the former of the two Ordinates becomes more ſimple, SE CT. N. Containing Notes on Art. 52——64. PROP. 10. and it's Corollaries explained and illuſtrated. 242. HE Deſign of this Propoſition is to diſcover the moſt 5 ſimple Figures, with which any propoſed Curve may be geometrically compared ; which is done by the Help of the foregoing Propoſitions. When any propoſed Curve 1s capable of being compared with a rectilinear Figure, ſo as the Relation betwixt them may be exactly determined, then ſuch a Curve is ſtrictly guadrable : becauſe we can find a Square equal to it by Prop. 14. B. 24. Elem. When this can't be done, the Curve, properly ſpeaking, is not quadrable: yet a Curve may be ſuch, that, though it's Area be not capable of a geometrical Compariſon with a right-lined Figure, it may be compared with the Area of another Curve, or Curves more ſimple than it: by which means we may have a more clear and diſtinct Conception of it. Thus if a Curve be ſach, that it is capable of being compared geometrically with a Circle, one ſhall have a more diſtinct and ſatisfactory Notion of it upon that very account: becauſe the Circle is a Figure ſo fami- B b liar 165 * Of this — theſe = conſult Art. 147 —161 *. The Quadrature of Curyps explained. liar and well underſtood in Geometry, and ſo N deſcribed, 8 likewiſe with reſpect to = „ res as may be compared with 4 Ellipſe and 13 be better — of, upon that very account, than if they — 5 not capable of ſuch Compariſon, and their Areas more eaſily exhibited. 80 that, . how well — Conic Sections are known, any Curve that admits o Compariſon with them, may, in ſome Senſe, be 5 Accordingly our Author in the ſecond of the two following Tables has given us a Catalogue of the more ſimple Kind of Curves that ad. mit of a geometrical Compariſon with the Ellipſe (including the Cir- cle) and Hyperbola : even as in the firſt, we have a Catalogue of the more In Curves, that are ſtrictly quadrable. CASE I:. 243. This Caſe is plain: every Curve of this Sort bein n 's See Art. 52. and 143 * 9 CAS E 2. 244. If azb-1 xe e + fe + 2: | chi be the Ordinate of : Curve, the Curve may admit of a Compariſon with a rectilinear Figure or not. If it do, you'll have it's Area exactly by Prop, 5 and that you may know when this will hap Nah in the Caſe of the more ſimple Curves, belonging to this general and have the Canons at hand for de- But if the propoſed Curve cannot be compared with a . lin'd Pig gure, the next thing is to find ont the moſt ſimple Curve or Curves, ich which it may be compared, whether it be Conic Section or any others. To which parpoe, 1 ou firſt make uſe of Cor. 2. Prop. 9. which ſee. bs -4 it, if you aſſume y any Quantity you pleaſe, and put x = 7 7, the Curve whoſe Ordinate is 142)! x e+ fan + gamn + Seh- may be changed into another of an equi Area, whoſe Ordinate is r X / + gz” | &c\*": and therefore if you put y= 1, the Curve 48, cunt will be changed into one more ſimple whoſe Ordinate is * XK N 2x*--&c Se-, where the Indexes of x under the Vinculum 80 on in the natural Order of the Numbers, * The Quadrature of Curves explained. 18 7 Ex. Let = 42" X 2 + 323*—-"' be the Ordinate of a curve: I perceive it cannot be compared with a rectilinear Figure, by Art. 139, of 149 *, Wherefore, by comparing this Ordinate with * Of this © the general Ordinate a- e + fn e-, I find 8=3. „;. Epic 1228 4 =. . =: therefore the Curve propoſed is con- 25 verted into another equal to it, whoſe Ordinate is 4 x 2 + z or 8 = whoſe Abſciſs is x = 23: and which Ordinate belongs to the rectangular Hyperbola, as it is applied to the Aſſymptote: and there- fore the Area of the Curve. propos'd, whoſe Ordinate is JT is re- ducible to an hyperbolical Area. 6: 3 The Compariſon is thus. Reduce the Ordinate —+— to , where | 3 NN | 9 0 7] the numeral Coefficient of x is Unity : wherefore, having drawn the right Line ABN for one of the Aſſymptotes, (fee Fig. 1. Tab. 2. p. 25.) in it take Au =, (linear Unity being ſuppos'd to be already determined) and aB — x : likewiſe having drawn aG perpendicular to AB, and equal to g;: if with the Center A and Aſſymptote ABN you deſcribe a rectangular Hyperbola GDS through the Point G; and then draw the perpendicular Ordinate BD, the Area «BDG ſhall be equal to the Area of the Curve propoſed, For from the Property of the Hyperbola AB x: Ag :: =: BD, which there- fore is equal to 3 or #—: ſo chat DG is equal to the Area of the Curve propoſed: for that is the Area belonging to the Ordinate BD and Abſciſs BB =x ®, : *Art.190. N. B. It would have come to the fame, if you had expreſſed the 1mm” Ordinate - 5 thus $X =. only in that Caſe you muſt take a =2, and oi the Area of the Curve will be to the hyperbolical Area 246. But if the given Ordinate agb-i x eþ f2n-k gan c. being reduced to the Ordinate Ex" * e+ fx + gx*+ c. P., this laſt one is not yet ſo ſimple as may be, you muſt then diminiſh the Indexes - — 1 and A — 1 by Unity always, by Prop. 7. until they be as low as poſſible, and ſo you ſhall have the moſt ſimple Figures, which can be diſcovered by this Method, with which the Curve pro- poſed may be geometrically compared. Bb 2 Ex. 188 ene E Wa | | Ex, 1. Let the Ordinate . or dev! x7 FJEN=" be pro bf Edi poſed. By putting x =2”, I reduce i it by the Method * ſhewn, to this other Ordinate _—_ x e + fo Jar - - belonging to a Curve of an equal Area. Then, by rejectin the Units from the Index of x without the Vinculum, by Prop. 7. Caſe 1, I reduce the Ordinate further, viz. thus : call the Ave bang to the Curve propoſed, f, and con. ſequently the Area belonging to the Ordinate =, 2. you'll Of this find, by applying the general Theorem at Art. 203 », viz. A dN n. R LET x. fB that — is the Area of a Curve whoſe Ordinate is - Again, by repeating the Application of che ſame Theorem, you'll find the Arca belonging to the Or- dinate —- 2 which is to o the Hyperbola as before, Therefore if we call this hy perbolical Area, , we will have == == EP =; whence, by proper Reduction, 55 e 2 A £115 On 2 lit and therefore, if the hyperbolical Area 3 be given, "as Area of 4 Curve propoſed is given likewiſe. Ex. 2. Let the Ordinate 3234/1 ＋ 42* 2 32+ be propoſed : which, reduced to Form, is 32%" x ITA Tann. By Cor. 2. Prop. g, if you put 2*= x, you'll have & VI + 4x + 3* for the Ordinate of another more ſimple but equal Curve. This done, I next reject Unity from the Index of « without the Vinculum, twice. To do which, bring it to this Form Zi 14x + 3x2)", and call the Area belonging to it, z that i 6 the Area belonging to the Ordinate x?—! x IAT FED tal is 5. Now ſuppoſe s and & denote the Areas belonging to the Ordinates ITA TZ and ũ— 701 14 zx“: then, here are three Ordinates K 1 IAT EX 1, and K 1 —1 x ITA TK- and = „ ITA Tzx t-, whoſe Areas are denoted by e, s and 2 — reſpec- * Of this tively. Therefore, by applying the Theor, i in Art. 206“, * C= Explica- On ANN, /B . cation. „ you'll have 9 O. 12821 — Ii. 8 b ＋ 2 X 4 2 | f=4 De Qgadrature f Cuny. 0 explained. 2 Sue A=r. B=s. .C = . a = \ \ | F= . ber fines 8=0, o goes out. Wherefore if you adh; by x you have f= — —+ theArca of the Curve pro- : ſo that when tis Ares rvidbngh to the Ordinate TIT, is given, the Area r will likewiſe . given. , Now the Ordinate STTax4+3x* is an Ordinate to the Hy perbola : which is thus con- ſtructed. Cee Fig. 30. Tab. 20. p 51889. Wich the * Axis and Latus Rectum 2, deſcribe the Hy la aGDS, hav- ak it's Center: take a = Aa, 1 and draw the Ordinates O, BD: then you'll have BD = N14 4x+3x* and aBDG —s, and conſequently 35D — aBDG = ?' the Area of the Curve whoſe Ordinate was mr pn viz. 3234/1 +42* + 32+: for, from the the Property of the Hyperbola, putting L for the Latus Rectum, it is Ka + aB x aB: BD? :: Ka: L, that is in the preſent Caſe (where | we have Ka = 2, aB = + +x and L=2) I+xXx5 Tr: B BD! ;: 2: whence BD V a N 1e and the initial Limit of the B being aG, as may be collected from Sect. 5. concerning = Poſition and Limits of curvilinear Areas compared with Prop. 5. it muſt be 9 and therefore 23D — «BDG is the Arca of the Curve pro 45 x was "0 247. But although in | the laſt Example, 412 ma of a inomial- Curve is compared geometrically with the Area of one Curve only, which is more ſimple than itſelf ; yet it moſt frequently happens, that the Areas of two Curves muſt be ſuppoſed to be given, that you may have the Area of a trinomial Curve. See Art. 209 *. Schol. 1. Thus if Yikes +3 was the Ordinate of a Curve propoſed, in order d ton. to find the moſt ſimple Curves with which it may be compared, you will find that it requires two Conic Sections to it's Quadrature. The Method of proceeding is thus. The propoſed Ordinate 2%—* x 1+42*+32+1* by putting 2* = x, according to Cor. 2. Prop. g, is changed into the Ordinate $x—"x 1 Ez: where 14+-4x+ 33: is the Ordinate of an Hyperbola as was ſhewn formerly: fo that if we ſhould call the Area of the Curve propoſed, ?; and the Area of the Hyperbola, s, we would have 27 and 5, for the Areas of two Curve 190 We Nuala of Curves pe Curves whoſe Ordinates are * * 1 TN and x x 14x31: from one of which being given, the other cannot be found: as you will find by trial, by a lying any of the Canons con. 2 tain d in Art. 206, 20% or-208 *. __Wherefore,- as our Author next ton. directs, I transform the Ordinate x TTA Tinto this other — EXIT TE, which belongs to an equal Curve by Cor. $. Prop. 9, by putting =: where the radical Part T is the Ordinate of an Hyperbola, Suppoſe you call the hyperbolical Area correſponding. to the Ordinate 3+4Z+2* U, „: then by means z of the two Areas s, and g you may find t the Area ſought, For by | Cor. 5. Prop. 9. the Ordinate rex Fat i is transform'd into the Ordinate — f N 3-+4E+E*F, whoſe correſponding Area is s : where- fore there ay three Ordinates, via. 15, 175 n z FA * . afp, ing to the Ordinate pom X 5 + 4Z NEN and E 1 * Of this 3+ Fatt |—!, you'll find, by applying the Canon Art. 206 *, viz. — e RN. ——P h 2 N E that k X 3F4EFEN — 6 — 47, is 2an X R | the Area of Curve next in order, vi2, that whoſe Ordinate is 5 * 3+aZ PER i—1, Renew the Application of the ſame Canon o find the Relation of the Areas belonging to the three Ordinates er TE EI), g- x PIE PA and £ ee which Areas are — zt, E X 3 TAE TFN — 6s — 4 and 75 I you will obtain this Equation & = = J AL — JET 2t + nf that 1 is, by reducing, f— 6 — 6s + 3&=*+4 323 * 3 ＋4 FTE, which, by ſubſti- tuting T for 3 + 4£ El, and v for E x 3 +42 + ZN: = I Ne will give Te 6s + 3xv + {v — T — r, the Area ſought. And the . Deſcription of he Conic Sections may be eaſily — from what was ſaid formerly. 248, Now, "The Nuadrature of CRV explained. 1 248. Now, after you have found out the moſt ſimple Curyes with which any propoſed Curve-may be compared, by the preceding Me- ' thods: you may ſometimes find Curves yet more” fimple; 2 the former ones can be ſo compared and joined together, by the Help of Prop. 3, as to conſtitute a new Curve, either of the Form men- tioned in Cor. g ; or of that mentioned Cor. 10. Prop. 9: which will ſometimes happen. 249. When any propoſed Ordinate is of this Form 2% x a+ bz» + 2%» + Sc. Xe + fz" + g2** + Sc. V;: and the Curve cannot be ſquared by Prop. 5, then you are to diſtinguiſh it's Ordinate into it's conſtituent Parts, which are to be conſidered as ſo many di- ſtint Ordinates: every one of which muſt be handled, as the Ordi- nate in Caſe 2: and having by that means diſcovered the Areas be- longing to theſe ſeveral Ordinates, join them all together by their pro- per Signs, and you'll have the Area required. See Art. 1659. Yet . of this it may be proper here to obſerve, in order to prevent Miſtakes, that Explia- ſuch a Curve may be quadrable, although all the Parts of which the ©” Ordinate is compos d are not quadrable,, as appears by comparing Art. 147 with Art. 162 and 163 * : ſuch is the Ordinate za. * Of this . g | C A 8 E 4. | tion, | 250. This Caſe reſpects Curves of the Form mentioned in Prop. 4 and 6: which are to be treated much after the fame Manner as thoſe mentioned in Caſe 2*: according as our Author directs. 25 F417 1 We 251. The Meaning of what is affirmed by our Author in this Caſe, together with the Reaſon of it, appears from what has been ſaid for- merly in Caſe 2 and Art. 164 and 16 5 *: and will be further illuſ- Of this trated by what is ſaid in the Cor. ſucceeding. Ekxgplica- lf few: ; tion. CORK. I. 252, It is evident from what has been faid, that, if the Relation of the Abſciſs and Ordinate be expreſſed by $a Equation in which the Ordinate y aſcends not above the Square, ſuch urve may be compared with the moſt ſimple Figures with which it admits of a geometrical Compariſon, whether rectilinear or curvilinear. Thus in the 3 adduced nd addoced by cur Rüber: r er % + 2 = 24% + of y i made up of two Parts, which are to be conſidered as two di- ſtinct Ordinates, viz. = and SLES . which muſt be. y has different Values (as in the above Example there are two different Values, according as the irrational Part is affected with the politiye fin'd by the Equation - Lo x e&fv +g0*+&cV NH nee 23. Yaadrature of Convs explain. Sr where the Value a ＋ * long to Curves that may either be ſquared, or elſe compared with the moſt ſimple Curves, bo the Methods already delivered, 253. It may be proper here. to obſerve, that, when the Ordinate —— 1} 1-1 K+ 8 * "ho or negative Sign) there will be as many diſtin Values of the Are, as there are Values of y: for in ſo man ny different Points will the Or- dinate meet with the Gres: and ſo each particular Area muſt be con- ſidered as extending along that Part of the Curve, to which the cor- reſponding Value of the Ordinate belongs: by obſerving of which, the &reral Areas may eaſily be diſtinguiſhed from each other, — nm * » COR. 2. 254. By means of this Corollary, any Curve whoſe Ordinate is defin'd by an affected Equation of that Sort which is mentioned in Cor. 7. Prop. g. viz. ſuch a Curve as may be made to paſs into an- other equal to itſelf, whoſe Abſciſs may be determined from the Or- dinate given, by an Equation not affected, may either be ſquared, ot elſe compared with the moſt ſimple Curves it can be compared with. For the general Equation to that Sort of Curves is 7 Xe + jy"27 PS __— Fe. = — 28 X 2 þ + Iys)? - + myꝛ¾egaꝭ + &c. which by putting = =. x= —2 and A 2 ＋12= FS. paſſes into an- other equal Curve, inwhich the Relation of the Abſciſs and Ordinate is de- Dx; by which, from the Ordinate v given, the Abſciſs x is deter- mined, and that by an Equation not affected. T en. by taking the Fluxions, according to Prop. 1, we have & = Yu α x fo FE x if iv Go n+ xv oranges +6, «fo FO! FIG xt ago oF anmn—10 46 * Hp -i eFfoF&c.Þ x —v*. And Wo As 7 Tones xs adit | And if-you ang apt both sides of the Equation. by v, it becomes aN N ν for + &.YxiFhb D R x Fu 22 2, Cc. * + Oc. l—1 „ R u + Se 0 „ : 209 x ahi amr + &c. xk+hv + Sc. i e+fo" DD uh . > n þecauſe. xv and y are the Fluxions of equal Curves: N > | | ſequently, by Prop. 10, a Curve having v for it's Abſciſs, and Dos * Ae + Se k+hv F Sc 7 Ax wo + 2ngu*" + Sc. c K 1 Tem! X Ele. 4 Tel 48 r X 7277 mus Fe Tc. ctr Fen = x fo + &cl for it's Ordinate, is equal 4 to the Curve propoſed, having ⁊ and y for it's Abſciſs and Ordinate. 7 Now the Ordinate of the Curve having v for it's Abſciſs, is made 3 8 of theſe. Parts. 1% Kr DA AI i | -. 5 *g Ge. X e Sc. EF FG). CA 2 xIF2mr Foc xe+frbocy x Iv. A t Curves having theſe ſeveral Parts of their Ordinates, and v for their common Abſciſs, may either be ſquared by Prop. fifth and 2 fixth, or compared with the moſt ſimple Curves, they can be com- 9 pared with by Caſes third and fourth of this Prop. and conſequently * the Curve propoſed is either ſquared; or compared with the moit {imple Curyes, it admits of a geomettical Compariſon with. | . « Accotgin gly, as our Author affirms, every Curve whoſe Pro- perty 3 is Jehn'd by any | Equation conſiſting of three Terms only, how- | ever, . BAY either be ſquared by-this Corollary ; or compared l with dne moſt. ſimple C rves poſlible. F 1 W Tad be ſack an Equation : by reducing it 4 to the Ferm 3 Cor. 7. Prop: g, it may ſtand thus y3 x 1 = 23 x 2 —1+ : which and with the general Form, gives a = 3 =Þ, J=—2, I. e=1,k=—1.,.1=a. s =(—= =) 3. = 3), x = — 1: whence by a proper Subſtitution you'll obtain — EF = * x for the Relation of the Abſciſs and Ordinate of an | 5 qua ne: and by taking the Fluxions ; 2 — 72 ; and fur- 8 N : Cc ther, : 7 . . x . 0 ; * * A 2 * c Nt | * 5 * i» ' 4 , * " T4 » "ANTS. 7 „ Py b 4 A 9 * 2 4 2 Yi - 4 >. : +. 3X \ £. its YA _ 8 * 7212 . be, * 94 9 . - J * * "= * 5 7 » 4 8 * * % ae” a Ay »\ « * wy & . d * - . bp 6 1 by ; / 5 * * a, ff f 3 _ .. * 4 * 1 7 aa 4) * 0 4 5 0 Y >. s : 4 1 * - * . - 44 x * of 9 | 5 * n 9 « % 7 ; £ 2 34 1 * 1 * uy - 4 ” © 6 * 0 1 : z& 4 * * 3 2.& 1 &Y . 2 os > * of . - 44 # 4 5 {8 7 * 4 * 4 - bo * a * 1 = F * 4 * * * * "WA | 1 FY ” a SF - - X - 1 2 > ' 5 a 8 * . . * 33 I . © „Art. 145. — =; educe this to an equivalent Expreſſion i of this 85. 20% 3 Now to r oy P In Term plication. of 2 and y: you have &: :: y: v, by what has been ſhewn, and Art. 169. puted from the Beginning of the Abſciſs *; the firſt two upon the one of this Ex · ide of the Abſciſs; and the laſt, which is negative, upon the other plication. Side of the Abſciſs, becauſe of the negative Value of the Ordinate. The Quadrature of Curvus explained. ther, by multiplying both Sides of the Equation by v, it is 2 12 _ = 3 = zy : wherefore the Area of the Curve, which has v for it Abſciſs, and —— 75 for it's Ordinate, is equal to the Area of the 93 Curve propoſed : which therefore, by the Method of Quadratures, is ©. again from the Equation 423 = x, by taking the Fluxions, you, haye 2 * : 2 :: 22: 1: therefore y: v:: 2* : 13 whence v = Subſtitute I 2a this Value of v in the Expreſſion — —, 7 and you'll obtain = _ = for the Value of the Area required in Terms of zand y. Therefore, if you aſſume 2 or y of any determined Value, and ſub. ſtitute that Value of z or y, in the Equation to the Curve, you'll, by the Reſolution of the Equation, find the correſponding Value or Value; of the other (for there may be as many Values of the unknown Quan- tity, retain'd in the Equation, as it has Dimenſions) and fo by ſub- „ theſe Values of z and y in the Expreſſion of the Area, you'll obtain ſo many Values of the Area, as the Nature of the Curve requires. Thus in the foregoing Example, if you take the Abſciſs z , then by ſubſtituting this Value for z in the Equation to the Curve, via. 23> azy + yi3= o, it becomes 443— 7aay + y3i=o; whence you'll obtain three Values of y correſponding to that Value of 2, viz. 1*.y DA: 2 y = „. „ = -N: the firſt t f. =: 2% 9 4. 3% y= * wo po tive; the other negative. Wherefore if you ſubſtitute theſe Values of y for it, and z for z in the Expreſſion 7 — 255 you'll have 15. aa. © 27 —11y/5 oe 27 +114/5 n 27. N 30% — 5077. for the three correſponding Values of the Area, according to the different Parts of the Curve, to which the different Values of y belong: which Areas are to be com- 2 56. The Reſult will be the ſame, if we proceed as follows. After having found, as above, that x —= = — —, inſtead of conſidering 343? x as the Abſciſs and v as it's correſponding Ordinate, you may, on the contrary, conſider v as the Abſciſs, and x as it's Ordinate: and then | multiply- The Ruadrature of Curves explained. 195 multiplying both by v, it is xv = - — =, which is the Fluxion of 9 the curvilinear Space, that, together with the curvilinear Area, belong- ing to the Abſciſs x and Ordinate v, makes up the circumſcribed Red. angle xv : Wherefore find the Area of the Curve having v for it's Ab- ſciſs and — — = for it's Ordinate : which is — = —+ — ſubtract it from xv == — IS and there remains = — for the Area required, to be converted into Terms of z and y as formerly, COR. 3. 257. What our Author lays down in this Corollary is evident, viz. that any Curve whoſe Ordinate is defin'd by any affected Equation whatſoever, that by the Application of Cor. 8. Prop. g. can be made to paſs into an affected quadratic Equation, may either be ſquared; or elſe compared with the moſt fimple Figures, that it can be com- pared with, Which appears ſufficiently from what has been ſaid. 258. Schol. But if the Equation defining the Relation of the Ab- ſci and Ordinate of any Curve propoſed, be ſuch, that none of the preceding Methods are ſufficient for finding it's Area : then we muſt have recourſe to the Method of infinite Series, laid down and explain'd by our Author in his Analyſis by Equations of an infinite Number of Terms: by means of which, any Equation how high or affected ſo- ever it be, that defines the Relation of the Abſciſs and Ordinate of a Curve, may be reduced into an infinite converging Series, which ex- hibits the Value of the Ordinate, by an infinite Number of ſimple Terms, made up of the Abſciſs and known Quantities: each of which Terms is quadrable by Art. 145 *: and therefore we thence find an » of this approximate Value of the curvilinear Area, An Example of this we Explica- gave already Art: 146 *: more of which, and the Method of Reſo- v* * Of this lution, may be ſeen in the Treatiſe itſelf : which, becauſe of the Af- Explica- finity of the Subject, I have here annexed. | hon. 259. And thus far I have endeavoured to explain the Principles upon which the Quadrature of Curves, and the Doctrine of Fluxions in ge- neral, are founded; and to illuſtrate the foregoing Propoſitions con- tained in this Treatiſe, in which, the general Methods for finding the Areas of Curves are laid down. Theſe the ſkilful and cautious Geo- metrician may have recourſe to and conſult, as often as Occaſion. re- quires: but in regard it would be a troubleſom and tedious Affair to be obliged always to have recourſe to thoſe general Methods, when the Area of any Curve is required: therefore our Author has furniſhed _.. Cc 2 us Form * Of this is that ſet down in the Table as appears by Art. 143 *. But if it be; neh o, appearing thus d = y, it belongs to the Hyperbola : for tion. * Of tion. Farm — this Author directs Art. 30 +: that is to ſay, you muſt bring the Expreſ- Ch y— I y— tion. Curvy *Art. of this Ex. to the firſt Form of the Ordinate belongs the Area — — ; totheſe- plication. | ne -v 1 Of this cond the Area — , as may be eaſily collected from Art. 147 *; xplica- N tion. in which the third Example belongs to this Form. The firſt Are tion. Explica- 262, All Curves belonging to the ſecond Form are doubly quadra- + Of the fion of the Ordinate, viz. - to ſtand thus, =" which joan id is expreſſed thus dan" x e AN n thus 2 —1 ez —2 *. : ＋ b. Nen. gative, are not quadrable, as appears by Art. 147 and 149 *. . a * * ee . "0 > ol - l > {” & * 5 JP 4. A. Li Mot . WR Gt ; * » it 2 2 + Ws © - ; e 4 - ** * * L773 WY) . JJ POS Ws 3 ODS A's n ” . s _ * — 1 - 4 = = - — - -- oa 4. - A "The Quadrature of Curves explained. us with the two following Tables. The firſt of Which exhibits im- mediately, the Areas of the more fimple Kinds of Curves, that ate capable of being ſquared geometrically. The other contains the more ſimple Kinds of Curves, that are . agg of a geometrical Compariſon with the Ellipſe and Hyperbola, under the firſt of which the Circle B comprehended : in which the Relation of the curvilinear Areas, to the Areas of the Conic Sections, is ſet before our View, ſo as that it may be ſeen at once. They are both divided into certain Forms, Orders, or Clafles, which contain certain Species's of Curves under them: which Tables are a fort of Speculum, exhibiting the Areas of infinite Numbers and Varieties of Curves. ExPLICATIon of TAB L E I. 260. This Table contains four different Forms or Orders of Curves: the firſt two Orders contain only one Species each; the other two Forms contain under them innumerable Species's of Curves, although the firſt four Species's, which are the moſt ſimple, be only inſerted in the Table. iſt, 261, The Curves of the firſt Form are quadrable : and the Area this which ſee Art. 190, 191 *.. 24. ble, as appears by Art. 149 *: and the Area is to be found as our Ef 127. 1 * commences at the Beginning of the Abſciſs; the other lyes along the Abſciſs infinitely produced beyond the Ordinate. See Sect. 5. concern- ing the Poſition and Limits of Areas. | 263. The other Species of Curves under this Form, whoſe Ordi- 1—1 nates are denoted thus =, where p is any Integer, poſitive or ne- 264. The Ne Quadrature f Curvns explained. 199 264. The Curves belonging to Form third are ,quadrable by Art, Form 34. 147 and 149 “: which may be thus generally expreſſed def t- 18 1 r = y : p being a poſitive Integer: but by them it appears that tion. 9 th alüng from the Ordinate retained in this Form, runs out into an infinite Series: and therefore it muſt be converted into the her Form, viz. d fr 1% T= and chence you'll find the Area by Prop. 5. according to Art. 14. The fourth Example in 2250 4 that Art. is the ſame: with the Curve of Species ſecond. of this Order: tion. which you may ſee. | alt at une 0) 265. Now that we may have a general Theorem for the Areas of all Curves belonging to this Form; and know how theſe Areas may de continued through all the different Species's in inf. let the general 1 i Ordinate d.. + fz", turned into this Shape 3 © © x 7 Tez 1—1 be compared with the Ordinate for binomial Curves Art. 147 *, and you'll find by ſubſtituting B23 for 0, — 1 for 3, * this lica- 1 for A, — = a a for 7, and —p-+ 1 for 6, that the Series for the don. van 2 — ill be dz W „Teri into = - Area w SO. ea r e | 2x2X2XÞ—1 xÞ=2Xxe* | 1 2X2X2%X 2xpþ—=1xþ—=2XfS=—-3xe 8 1 f e e e where you may 2þ4-1 x 2p—1 x 2p—3 x 2p—5 x 2p—7 x nfs obſerve, | ; 1*, That the Number of Terms muſt be equal to Pp. 2*, Suppoſing p to be 1.2.3.4. 5, &c. ſucceſſively, the nume- ral Coefficient of the Numerator of the laſt Term will be 2, — 4, + 16, — 96, ＋ 868, Cc. reſpectively : which are produced by mul- tiplying by — 2, — 4, — 6, — 8, Sc. continually, i. e. multiply the firſt, viz. 2 by — 2 that produces the 2%, — 4 ; multiply this by — 4, it produces - 16 ; this by — 6, produces — 96, Cc. as eaſily appears from the Progreſſion of the Terms. * U e 3*. P ſtanding for any poſitive Integer whatſoever, it will eaſily appear, by conſidering the Progreſſion of the Series, that the numeral Coefficients of the ee the ſeveral Terms, beginning with the laſt and going backwards, are formed by dividing by the Terms of this Series, 2 , 4+ 6. 8, Sc. and the Denominators of the Terms, ”. - 2 - _—_— wy p a . , , * N K = . by 4 = A l « z Y ? l . _ * er Las - wt " 2 FY 2 Þ» IR r p . 4 T4 . o \ » * - by 9 «\ Me a. * 1 : : 8 % 2 WM 1 - \ 4% 9 * * 7 Ar Yo 4% - 7" \ © 4 L K. 1 l 5 * %, a * - | 7 a2 "9% | f e 2 7" „ 4 . U n > N * 1% 4 p C7 x 9 A * 5 Lo 3 * a C : Y 5 — | | | | 198 in the ſame Order, are formed by dividing by the Terms: of this Serie, 3.5.7 9, Gc. Wherefore, the numeral Coefficients. of the ſeveri 7503 0 Terms, beginning with the laſt and going backwards, if vou reſpeg the Signs, are formed by multiplying by the Terms of this Series — | . 2 — +, — . &c. | g Grder of the Terms be inverted, the Relation of the numeral Coef. ficients of the ſeveral Terms will remain the fame as before, the fitſt 7. 9, Fc. into one another continually. Form 4th. * Of this Explica- - don. The Quadrature of Curves explained. Therefore, if the foregoing Expreſſion of the Area be duly reduced, inging all the Terms under one common Denominator ; and the Term now coming in place of the laſt formerly ; the ſecond in place of the laſt ſave one, Gc. and ſince that which becomes the common Denominator, is the ſame with the Denominator of the laſt Term for. merly, as in the Series above: hence it appears, that, after the Ex. preſſions of the Areas are. reduced to the ſame Denominator, if þ be 1. 2. 3. 4. Cc. ſucceſſively, the Denominator will be 3. 1 5 10g, 945, &c. reſpectively, formed by multiplying the Numbers 1. 3. 5. , . And ſo you have the Demonſtration of what Sir Iſaac Newton ſays at the End of the ſecond Table, with reſpe& to the Continuation 10 the Series of Curves belonging to this third Form. Only you may obſerve here, that the firſt Factor, viz. dz N Te-, being multiplied by the negative Power of z contain d in the laſt Term of the Series for the Area above, viz. —p — IX, will always become d x e+f27\% or 'dR3 : to compenſate which Multiplication, all the other Ferms are divided by the ſame negative Power of 2; and fo the Ex- preſſion for the Area ſhall put on the ſame Form as in the Table. 266. The Specics's of Curves belonging to Form fourth are like- Y 1 «013 VIVID | os aa , atn—1. ... wiſe infinite, and may be thus generally expreſſed T, The Area Alun go e d 025 0 N 4 NT. © arifing from the Form of the Ordinate d“ x e+f2"\-3, runs out into an infinite Series, as appears from Art. 14 and 149 “: but, when the Ordinate is turned into this other Form, viz d IX Tea -i, you'll find, by theſe ſame Art. that the Expreſſion for the Area ter. minates: and therefore all the Curves. belonging to this Form in inf. are quadrable, grey | 267. Let the Curve of the firſt Species be propoſed, viz. dz"-' x eZ = y I turn the Ordinate into this other Form, viz. dz*""* wfez-1\-3: then by comparing it with the general Form of Ordi- 5 nates e 155 Qadrature f Curvss explaintl. nates of bilghpial Curves Art. 24) », I find $==4y.1 y=—9 /A==2 .* Of thi -f oO. ad. e Fe. Therefore by a due Subſti- gn tution, the Area comes out t. * | Tesa = e FI. 3 ain the T... e aids e ee And after the ſame manner, the Quadratures of all the other Spe- cies's of Curves, belonging to this Form, my be demonſtrated, vis. the direct and immediate Application of Art. 1477. * Of this 268. But the Areas of all the Curves in this Species, and the * 1 ing in inf. may be found, and the Series of Areas continued after be Area of the firſt is knovrn, by the Help of the Theorem Art. 202 , * Of this * | Ming; "> Explica- VIZ. n= thus to find the Area of the ſecond Species of tion. this Form, whoſe Ordinate is d i = e4-fz1\—+, you'll have 84. © A1. A — e + fz": therefore, by a due Subſtitution of Values, V. — 7 e ; AM. aan multiplied by 4, gives om T e for the Area of the Curve conſtituting the ſecond Species of this Order: and fo for the others. But the Species's of Curves belonging to the Progreſſion, which runs backward above the firſt Species, both in this and the preceding Form, 3 r rn expreſſed thus de /e + e and — == y, cannot be found by the Help of the Theor. Art. 203 *: See Schol. 2. Art. 209 Þ: · of this neither are they quadrable, as appears by Art. 149 f; and they con- Explica- ſtitute Forms third and fourth, Tab. 2. | 70. this 269. You may likewiſe, by Prop. 5, and Art. 147 *, find a gene- Explica- ral Area correſponding to the general Ordinate dz?" x e4-fzn\=t, t Of this as was ſhewn in the Curves belonging to the third Order: which ge- Explica- any 1 tion. neral Expreſſion will ſtand thus dæ = »« f Þ 22 1+ into ©, Of this — ; c— cw | Explica- 2 —| Xe 7 —1 —2 Xx e | F 1 apt, Vii 265 nee — el cat —— 221 3 2þ=1x of 2p 1 x 2p=3 x »f* 2p—I x 2 — x „ x N 16 xp=1 x p=2 xÞ—3 x& . 46H n 25—1 7 5 1 27—7 a TR * Ec. Which Series will ſerve to find the Area of any Curve belonging to this Order, ſince the Progreſſion of the Series is evident. And by con- | ſidering eee it in the ſame Light, in which the general of the Curves of the — ng Order, as conſider eaſily ſee the Demonſtration ol what Sir Hanse ning the Continuation of the Series of Curves of this Order in "exo at the End of the ſecond Tab. which jt would be needleſs to inſiſt dpDD part. - | "cularly, after what has deen ſhewn at length, with reſpe@ | ta thi "= Matter, u Order third: | 5 1 0 270, 1 o ſhew the Application of the preceding general eine put F 4 5 then you'll have 4a « Fe. into - = | ar 1 28 SX ZX2%6 1 16x3x2X1X8 eee eee. which, by red all tt ator of the laſt Term, and "I the: Expteflioh.of the nege . 1 becomes dE. far FE s 2. — e ee —¹ 10 + : the ſame as in the Tabw. SE ROY 271. If in the Ordinates of Curves telonging to the laſt two 955 ral Forms, the Index of the Radical was +, +, 2, &c. in the third - Form; or — 2, — f, 2, &c. in the fourth, they may eaſily he =: reduced to the Curves-3 in the Tab. by Caſe ſecond of Prop. 7, Vis, | by Jitniniſhing or increaſing that Index by Unity, till it become * Of chis or — 2 according to the Canons i in Art, 21 I and DIV ** tion. . Het x -4 S 7. * - N 4 3. „ 2 BxPLICATION of TAL E 3B 2. When we enquite into the Area of any Curve propoſed, that 2+ * | Are may be exhibited either arithmeti ically : which 3 1s done by Prop, faſt Ane * N or cometrically by finding and. deſcribing other more e Curves; with which it may be compared. Now by the Hel oy Prop* - ninth and tenth, our Author has conſtructed this Table or J e of Curves, that are, capable. of being compared *geometri- wage the- Ellipſe and Hyperbola; ſo that 12 Areas may be ex- dib by the: Deſcription of theſe 8 and wonſequently given, when theſe _Eigures are given, . 273. It is not my Deſign here to 'go through all the Orders '< Curyes contained in this Tab. and demonſtrate: the Quadratures, of the ſeveral Species of -Curves, belonging to each Order. All T ſhall do, is to ſhew'by a few Exainples, the Manner. of demoniſtrating all the reſt : then make ſome general Remarks upon the Conſtruction and Uſe of this Table: and finally ſhew the Application of it in ſome Ex- "_— moſtly taken from the Author himſelf, - _ JOIN | 5 274. The f * a Ds — a | j * | ” ©, _— As. I . 8 2 ya „ „ 3 4 , Ne Buadrature of Curves explained. 201 274. The Curve of the firſt Species of Form firſt, is expreſſed thus — —y: which is not capable of a geometrical Compariſon with 2 Figure, as appears by Art. 147 and 1 49 * Wherefore, * Of this putting 2" =, (by Cor. 4: Prop. 9.) it paſſes into another equal bes Curve, whoſe Ordinate is x : being + the Abſciſs. Thete- fore, if the Area of the G which hath 7 for it's Ordinate, be called s, the Arca of * Curve propoſed ſhall be —s, fince y is a given Quantity. Now —<— —_— Fe = V is an Equation to the redtangular Hyperbola, as was ſhewn formerly *. Art 245. The Conſtruction of the Hyperbola, which ſerves a for all of ti the Species's of Curves under this general Form, is thus. See Fig. 1. plication. Weng ing to this Table, p. 2 5. Draw the right Line ABN for one of . A ymptotes, make F linear Unity, take A; De, and aB=x: draw aG perpendicular to ABN, and equal to LY or a third Pro- portional to the three Lines e, / and 4: throu gh G, with'the Center A and Aſſymptote ABN, deſcribe the — Hyperbola GDS, draw the Ordinate BD: then you have v» BD . d= «GDB: therefore it is <2? = the Area required . 3 *Art.199. 27 5. The Quadratures of the other Species's of this onder, are de- plication. , monitrated by the Help of the Canon in Art. 211 1 : the Area of the of this firſt Species being once found. - Explica- The Series of Curves belonging to this Order, may be carried on likewiſe in inf. by Diviſion, Thus let the fourth Species be propoſed Wh ==; divide dz -I by Fe, and the N 25 Is SUI — e N | Z 42 n- + I 2 which Diviſion is to be carried on, till you have as many Terms in the Quote, as the Number prefixt to ; in the Index of ⁊ without the Vinculum, leſs one: to which annex the Remainder with it's proper Sign, and e + under it for it's De- nominator: and then all the Terms fave the laſt, denote Ordinates of Curves belonging to Order firſt, Table firſt, which are thence quadra- ble; and the laſt Term is the Ordinate of a Curve, belonging to Spe- cies firſt of this Order : therefore all the Areas, , joined with their pro- per Form 3d. Art. 42. of the Quadra- ture of Curves, Op A * Of this you do by the Theorem in Art. 202 ®, viz. 3892 . For by _— The Quadrature of Curvss' en per Signs, make the Area required, vz. Dan — Ent Le 377 27 | 5 And ſo o 2/988 = it is eaſy to ſee what is the Progreſlion of the gene Sees for the Ars, vi. —<— — + —< p=—1 x of p—2x 7 £—3x f 3 276. The Quadrature at Species firſt, Order third, is thus demon- ſtrated. Since the Property of the Curve is exprefled thus Ve =y: by putti = *, an equal Curve, whoſe Abſciſs is x, wil have it's Ordinate v = — FR. ex*, Wherefore, from the Af. ſumption of the Area 7, belonging to the Ordinate — EFF or of the Area — = — belonging to the Ordinate PV, + ex*, find (by Prop. 7.) he An belonging to the Ordinate : which Y 3 | aſſuming the Area — * — agreeing to the Ordinate x x Ter-, 1 bind the Area Ok x2 to the Ordinate x'— „T= by in 9 1. 2. A= T. e . 7 dene 8 Fan. 2 which accordingly is — : call this Area s, then you 3 of have this Equation s =—J-—— 7: by reducing of which, you obtain 7 75 WP — 1 * — the Area of the Curve propoſed: wm therefore is given, when s 18 given. Now 5 denotes the Area of a Conic Section, for it belongs to o the Curve whoſe Ordinate is „/ + ex*, which, when f and e are both poſitive, is an Hyperbola, VIZ. that F be negative, it belongs to the Hyperbola at Fig. 3. p. 25. be negative, it belongs to the Ellipſe at Fig. 4. p. 25. The repreſented at Fig. 2. p. 21 or ly, if De Ruadrature of Curyrs explain. The ConſtruQtion for the firſt Caſe is thus: with the Center A, the half tranſverſe Axis AP = /f (that is a mean Proportional betwixt Unity and 7) and Parameter of the Axis I (that is a fourth Pro- portional to the Lines e, 2 and AP) deſcribe the Hyperbola PGD: draw the indeterminate Axis AN, in which take AB==ox A. to which draw the Ordinate BD: fo ſhall it be BD =v =D ex*, and s = APDB. For from the Property of the Hyperbola, calling the Lows Tranſverſum t, and it's Parameter p, it is BD+AP x BD—AP: t :p or BD'— AP?: AB:; : p, that is, by m the — according to the Conſtruction, BD! — f: :: af; : — BOue47 FT, as it ought to be; and fo you'll have — Now in order to find what the Atea of the Curve in qual to, in the geometrical Deſcription; we muſt find Value of the Expteffion T , or which is the ſame, =, For this end, draw the Tangent DT alike AB in T: then from the Property of the Tangent, you'll have AT = . that is, by ſubſtituting the Symbols, AT gg: add AB=x to each, and it is BY =*XX, that is, becauſe * = =f, BT — === 21 Wege multiply- ing by 2 = 2, you obtain L. , ;. . TDB = Z Hence the Area of the Curve propoſed, | whole Ordinate is 2 „which was ſhewn to renz into TDB ApDB: which is ſet down in the Table thus © - into APDB — TDB, denoting that it is 3 whether the ſecond. is to be ſubtracted from - firſt, or the firſt from the ſecond : for ſometimes TDB may exceed APDB, ſometimes be leſs: but the Area of the 8 opofed is equal to that Difference: if TDB — APDB be poſitive, the Aten quired being poſitive, lyes above the Ordinate; if TDB - APDB be bays. 4 Area is fituate upon the othet Side of the Ordinate; and — to the Abſciſs produced beyond the Ordinate. 277 If the Quantity F be negative in the Value of y, ſo that it be 21 them the Arta of the Curve is found by the Deſcrip- Dd 2 tion 204 420 DB, you'll have = X 4 — 55 == into aGDB — TDB or - De Quadrature of Curves explained. tion of the H la likewiſe, but according to a different Conſtrye. tion, viz. at Fig. 3. p.25. For now the Curve propoſed, will be equal to that whoſe Ordinate is — Ex—/=7 + ex* 2 and ſo the 'Conic Section muſt be ſuch, as to have it's Ordinate Per The Conſtruction is thus: with the Center A, tranſverſe Axe Ka = 20 and Parameter 2%, deſcribe the Hyperbola aGDS : take AB ==x, draw the Ordinate BD and the Tangent DT: and the Area, which now by changing the Sign of /, becomes — © x = — or — - 2 4 7 X — =, will be, according to this ConſtruQion, = into aGDB IDB or & x -D: which ſhews the Arca to be on the fr ther Side of the Ordinate. | For from the Property of the H la, (uſing the Symbols: and p in the ſame Senſe as formerly) KB x BA: BD? :: f: p; or att: v:: fp; that is, by inſerting the Values of ? and p from the Con- ſtruction, * — 7 2 22 20 2% f: whence we have v=v—f Ter as it ought to be. Again, from the Property of the Tangent, the 1. K* — Subtangent BT =x — === = (by inſerting 2 for x* — 44, as above) 2 or vs — = = = BT: therefore, multiplying by >. Of +l © rs — =» It 8 — =——— = ATDB. Wherefore, . fince i= into — aGDT. Which negative Value ſhews that the Area of the Curve is adjacent to the Abicils produged beyond the Ordinate. 278. If e be negative, the Conſtruction is by means of the Ellipſe at Fig. 4. p. 2 5. for now it is v= \/f— ex*. The Conſtruction is thus: upon the Center A, with the tranſverſe Axis Ka — 2wC, and half conjugate Axis AP = /f; deſcribe the Ellipſe aDPSK + take AB = x, draw the Ordinate DB, then it will be DB (V =) /f— ex, and APDB=s: conſequently the Area, which now, by changing the Sign of e, becomes — © x = — 5, or * s, will be © into TDB-+APDB ; or elſe © into TDB—dGDB, that is © x aGDT. | | For The Qgadrature f CukvRSs explained. For from the Property of the Ellipſe, KB x BA: BD? :: KAY : AP,, that is by inſerting the Symbols and Values from the Conſtruction, e ff if or -b: C ; whence vers, —BD as it ought to be. | 279. As to the two-fold Value of the Area, viz. 5 x TDB-+APD and x TDB — 2GDB, it is to be obſerved, that — s, in the Ex- n denoting the Area of the Curve, may ſignify, either that the Area of the Conic Section, which is adjacent to the Abſciſs, is ſub- tracted from the other Quantities contain d in the Expreſſion; or that the Area of the Conic Section, lying upon the further Side of the Or- dinate and along the Abſciſs produced, is added to the other Quan- tities: even ſo, + denotes indifferently either that the firſt is to be added, or the ſecond ſubtracted, as it may ſeem convenient; or the caſe require. Wherefore in the Expreſſion 5 * = + 5s, the Term + 5 denotes either that the Area APP; is to be added to _— TDB; or that a DB is to be ſubtracted from TDB. And this laſt Value is that which is exhibited in the Table by Sir Jaac: becauſe it is the proper Value in this Caſe, For if you expreſs the Area of the Curve, whoſe Ordinate is e T = dzb»=1/f—72—5, analytically by a converging Series, according to Prop. 5, you'll find by the Method of determining the Limits of Areas (Sect. 5.) that the Area vaniſhes when it is 2» —= =, that is when x=IS/FL : becauſe "==: but according to the Conſtruction WE NY therefore the Area of the Curve vaniſhes, when AB (= x) becomes equal to Aa: which can- — 3 not happen to the Expreſſion - X— ++ but when - ſignifies that aGDB is to be ſubtracted from 2 = TDB. And thus I have ſhewn at full length, what way the Conic Sections ae to be conſtructed, with which the Curves belonging to Species firſt, Order third, are geometrically compared, in all poſſible Caſes: and what the Value of the Area in each Caſe will be. How=- 205 The Quadratur of Cunves explained. Fig. 1 ? * Fig. Pit 251116% i | D 1:1 Fig. 2, U 7 ad a. aa ara S a what hath been faid in theſe laſt three Articles by Let HL be a Curve having the Abſciſs AT , and | == EE = Fig. 1. or ALT Fig. 2. 222 . e=i=f=d<, va, *=>, == . and v ( =) V1+ x*, Fig. 1; or /—1-+-x*, Fig. 2. or /1—x*, Fig. 3. ſo that the Axes and Parameters of the Conic Sections being become equilateral Hyperbolas in the firſt two Caſes, and the Circle in the other Caſe: in all which the Semiaxes is 1. 1 | Wherefore, when it is y =, upon the Center A with the ſemitranfverſe Axis AP — 1, and indeterminate Axis AT (Fig. 1.) defcribe the equilateral Hyperbola PDa, to. which draw the Tan- gent TD through the Point T, and through D the Ordinate DB: Mg then from the Property of the Tangent (as in Art. 276 *.) z =(AT= 2 AB x AP? AP? I — _— Barf = AB . whence AB = x and BD = VNL —— — — 1 207 F ex“; = and ATDB ==; like - wiſe APDB = 5 : therefore the Area of the Curve, viz. 1 — — 7 is in this Caſe twice AT DB — APDB, Whence (by Sed. 5.) if the Point ? be ſuch, that Arab — AP4þ — o, the curvilinear Area is no- thing, and the Ordinate 7h is the initial Limit, from whence the Area muſt be reckoned : ſo that if TDB — APDB be poſitive, the curvilinear Area THbt lyes upon the fame Side of the Ordinate TH with the Point A the Beginning of the Abſciſs: but if TDB—APDB be negative, it lyes upon the oppoſite Side of the Ordinate. Now although this initial Limit cannot be found in this Caſe, without draw- ing ta ſo as to make dh = APDB, which is equivalent to the 2 drature of the Hyperbola, yet nevertheleſs the curvilinear Area TH lying betwixt any two Ordinates TH, ö, is equal to the hyperboli- cal Area TD4. For the Area correſponding to the Ordinate TH having the greateſt Abſciſs AT, is TDB — APDB; and that corre- nding to the Ordinate 7h, having the leſſer Abſciſs At, is b — pab; therefore it will appear from the Conſideration of what hath been ſaid, that, whatever way the Areas lye, the Area TH incloſed betwixt the two Ordinates TH, 7h, is IDB — APDB — #4b APA I Dat, the Difference betwixt the other two: agreeably to what - was ſhewn Art. 185—187 *. | * Of this If through the oppoſite Vertex of the Hyperbola PDg, viz. Q, you — draw QR. perpendigukr to PQ, then-PQ, and M are two Am- | ptotes to the Curve HAL. ee ee e ker ab But, 2, If it be y = (ee Fig. a. p.206,) with the Center A. and ſemitranſverſe Axis Aa = 1, deſcribe the equilateral Hyperbola 4D: from the Point T draw the Line TD touching the Hyperbola in D; and from D, the Ordinate DB: then it is AB = x, ſince AT =2, Aa = 1 and AB are proportional, from the Property of the Tangent. Moreover the Curve HL paſſes through the Point a: fot lince TH === —=, put od, and thence you have & =1=* Ag, Now it may be cafily ſhewn here, as in the pre- Art. 12 r. ceding Caſe, that ATDB —_— = ; and DB =s : therefore ſince 41 plication. TY! a | ſe ' | 3 - <1 5 the curvilinear Area is 7 * — =, as was ſhewn above, it becomes in this Caſe 2 5 DB — TDB=— 2aDT negative: wherefore, ſince DT 208 De Quadrature of CURVvAHS explained. aDT — o when AT = Aa, the curvilinear Area commences at , being TH, which lyes on chat Side of the Ordinate, which is oppo ſite to A. The indeterminate Axis of the Hyperbola, v2. AL i an Aſſymptote to the Curve AHL : and if AT be taken = 0,816, Sc. of Aa, the Ordinate TH will paſs ooh a Point of Tage Flexure. Again, ze If it be y = I 4 (ce Fig. ; ) with the Radius At — 1, deſcribe the Semicircle ADPK; through T draw the Tangent 'TD, and through D the Ordinate DB: then ſince AT=2, Aa=; and AB are proportional, you have AB=x : and as formerly aD ; and now it is APDB = and DB — —-s : wherefore the nes Area, as was ſhewn above, muſt either be 2 « TDB4-APD} or 2 TDB —ADB=—24DT : the laſt of which is equal to the cur- vilinear Area aTH : for ſince the Curve HL paſſes through the Point a, the Areas aTH and 2D T both commence at the Point @ : and the Fluxion of ATH is double the Fluxion of aDT, ſince HT = (y= Sor if ===v=) DB, and the Fluxion of AHT is to the Fluxion of DT, as HT « Fluxion of aT to DB « Fluxion of 41 * Of this (by Art. 48 *.) that is as 2 to 1: therefore (by Art. 40 ®.) aTH= Explica- > DT. Draw AP perpendicular to Aa; and the Tangent PL is an tion. » Of this Aſſymptote to the Curve HL. © Explica- 280. But by ſuppoſing a different Relation of the Abſeiſſes 2 ands , ti Fr VIS, that - x, as in the Table, you may obtain the Area of the Curve alter a different Manner. For upon that Suppoſition, you'll find the Curve propoſed to paſs into another equal Curve having — Per fx Tex it's Ordinate: ſo that, by Prop. 7, you may find the Relation of the Curve propoſed, to the Curve whoſe Ordinate is x/fx- FP which you do by turning the Ordinate — 2 Fer into — 2 Hex, then increaſing the Exponent of x, viz. — by 2: and ſo you may conſtru& the Conic Section, with which the Curve may be compared. The other Species's of Curves of this Form are deduced from the firſt by Prop. 7. Form 5th, 281. I ſhall next ſhew x Demonſtration of the Quadrature of the Curve, conſtituting Species ſecond of Form fifth. which requires 90 Conic Sections. The The Readratore of Convzs explained. The Curve is expreſſed thus - — . Y. and the Relation of a gx the Abſciſs 2 to the Abſciſſes x and E, which are theſe of the two Conic Sections, is expreſſed thus eee. T and q ,- gan k. Wherefore, I firſt feek for the Ordinate of a Curve, which having it's Abſciſs x, ſhall be equal to the Curve propoſed, viz. thus. The Ordinate fought (by Prop. 9.) is equal to 2, that is, by ſuppo- | 2y—1 fing Z=1, 7 or by r in the Value of y, = — —-— * * fe" þ gu” wherelore rom. the Equation a ror 5 bud the Relation e g⁊ yl | 2-1 _. Wi: | the Fluzion which is += * : inſert this in of % 8 Fee _ ; i Gp ale of x, ind the Ordinate ſought will be == , : where, if you 2 ID 522 put for 25. it's Value, V, EA f 2, from the preceding Equation, and them reduce, you'll hin mien TORR. for the Value. of the Ordinate of a Carve, Fd hes br it's Abſciſs, and is equal to the Curve propoſed at firſt : which Ordinate is made up of two diſtinct Parts. The Area belonging to the firſt Part of the Ordinate, may be compared with the Area . the Conic Section whoſe Ordinate i is v = = + x*, by help of the Theorem in Art, 22 — * 211 . viz. B= — _ 1.4 : by applying of which, you'll obtain he this lica- n the Area belonging > WE Or — 1525 e which might atſo be deduced. from the Quadrature of the Cut e of the firſt Species of this Order. The other Part of the Ordinate 4s — 2 and in. order to find the Arca belonging! to. this Part of the Ordinate, take x = A for the Relation of the Aþſciſles. . of; 10 Canic eee dis 1 * e, : 2 * P * Ting _— : 1 4 V 4 * = N . 11 * C 1 b ge eau, - os 210 The Quadrature of Cunvss explained. Table, whoſe. Areas are equal ; the Ordinate of the firſt being =. : from whence find the Ordinate of the other. It will be, by Prop g, > or — = by putting #=1 : inſtead of x inſert it's Value V 543 and inſtead of } it's Value -, deduced from the 4 — Equation N 5; „and fo you'll obtain 2 x l for the Ord. nate of a Curve, wich having E for - it's Abſciſs, is equal to the Curye which has x for it's Abſciſs and — — = bor irs Ordinate. Now, according to the Table, * = 2 : which denotes the Ordi. dee nate to the Aſſymptote in a rectangular Hyperbola “, and to which — tho een Area is 7: that is the Area belonging to the Ordinate TIE, is — : which therefore is RAR TED CoD ® Of ths =, & hs 190192 ®, Explica- ** Whence by putting the two Areas together, you'll have == _— At the Area of the Curve propoſed, expreſſible by the Deſcription of two Conic Sections. pecies s of Curves of this Order in inf. may be found by Diviſion, the Areas of the two firſt being once found. Thus for the third Species, where it-is | — =y, divide dani by . NE — . | gu" * +6 and the Quoticnt is Ne — — 9 0 r hy — r 2 which is the Value no it the Gd. | 63+" Akoting the third of this Form : the firſt Part of which is the Ordinate of a Ouree quarbe by by Theorem firſt, Table firſt ; the other two belong to the firſt and ſecond Species of this Form fifth: whence — orms 7th 2 ve ſaid as m Example 20, Caſe 2*, Prop. * nd 10, as may ſerve to illuſtrate the — — of the Curves belong to Form 7. Therefore I ſhall 5 abe one Inſtance more: w ſhall be the Curve conſtituting the Specirof the kaif Form, whoſe Property ——— De Quadrature of Curyzs explaintd. Property is ths exprefied de—=\/ s: and two Conle Sec- tons are required to it's Quadrature, In order to which, I find the” Ordinate of a Curve, which having & for it's Abſciſs, ſhall be equal to the Curve propoſed, ſuppoſing x = /g 4 hz» : the Ordinate ſought is 2 of Z, putting à = 1. Wherefore, if you put in the Value of y, is, de U, Alas, and the Value of &, vis. =, deduced from the Equation x=v/g + hz", the Ordinate ſought will be — — 3 . ae which, by inſerting —* for 2" and x for 4/g + bz", and bringing to Order, gives = 3 + £x* x FE for the Ordinate of a Curve equal to the Curve at firſt propoſed, I next find the Ordinate of a Curve, whoſe Abſciſs, as before, ing x, ſhall have it's Area equal to , which is the Area of the fe- Conic Section, whoſe Abſciſs being E, has it's Ordinate TY = = + , the Relation of the Abſcifſes being I= which laſt is eaſily deduced from the Values of x and # in the Tab. And by proceeding after the ſame manner, as in the preceding Part of this Art. you'll find that Ordinate to be — g. I + Zan x FIN. So that now you have three Ordinates, vis. 1*, S* AE + fi x=gt xl. 2. * = I, 35. — fox=eF & , to which the three correſponding Areas are , = and _ I: ſo that from any two of them given, the third may be found by Prop. 8 : by applying of which to the preſent Caſe, —c you'll obtain £ = 2 — bx x = + A- xt, acc x =; or by a due Subſtitution of xz=" for bx x FH, — Ai 2d r = 4df; — 4deo . and v3 for -x. ; — = E, the Area required. 284. And thus I have ſhewn in a Variety of Caſes the Demon- ſtration of the Quadratures of Curves "> to ſeveral different 2 3 — 212 De Qadraturę f CU nv iv explained. Farqt i tn Table: by obſervide of which carefully, all the other may be. demonſtrated in like Manner : wheretfore, I now proceed to make ſome. general Obſervations upon this Affair, taken moſtly from 285. 1e, The seties of Cut ves in each Form may be continued i inf towards both Hands, by the Help of Propoſition third, fourth, ſeventh and eighth: that is, the Progreſſion of Curves may be carried backwards by diminiſhing the Index of 2 continually. by y ; as well a5 forwards. according to the Order in the Table, by increaſing the Index of & © The Ruadrature of Curves explained. -229 4raw CB perpendicular to AF, it ſhall always be AB: BD :: AF: AE, the Point C touches a Curve, 1 whoſe Nature and Area are re- 3 | | Call AB 2, BC=y, AE d, AF = 6: then becauſe AF —b: AE Sg a:: AB =z: BD, | hence we have BD =: and be- F B D A | cauſe of the right-angled Triangle BCD, BC = VN — 57, that is y = 2 — 22 : By comparing this with Art. 306 , where 4: you have v 4 t Ex=, it appears to belong to an Ellipſe: and — by comparing the homologous Terms, you have {pf = aa and = ; whence ariſes *= 25 = 2AP, and p=—YE, that is the tranſverſe Axis is 2AF and it's Parameter a third Proportional to 2AF and 2AE ; fo that it's conjugate Axis is 2AE, and A the Center. 325. Ex. 7. Let AB be a given right Line, CAD a rectangular Norma, revolving upon the Point A as a Pole, of which one Leg AC n L. ; : i F N = | © — CO N [ | D — AB 230 * Of this fought. By comparing which with Ex. 2. Art. 320 , it appears ty Explica- tion. = 46: Subtangent, you have : S:: y: 3, or ys = 2y, the firſt of which plication, | 1 Az; the other AD indefinite: and while the Norma revolves, 1, The Quadrature of Cunvns explained. the indefinite right Line CE drawn through C and cutting AB x right Angles in G, always interſect AD in the Point F: the Point deſcribes a Cutve, whoſe Area is ſought. Call AG= 2, GF — y, AB = = AC: then CG = (VAC? — AG! =) aa — 2* and becauſe CG — /a* =, AG—2, and GF y are proportional, hence it is y == —— Equation to the Curve whoſe Area i; ; 85 = od b be the ſame Curve: and therefore it's Area is found, as is there ſhewn, by comparing it with Species ſecond, Form fourth, Table ſecond. But that you may ſee how one and the ſame Curve may belong to diff. rent Orders and Species's in the Table, you may compare it with Spe. | 331 | cies third, Form eight, Table ſecond, vi. === , by put- | SG ting y— 1=d and f=0: and moreover e = aa, g=—1, 2 x, Ja — * u, and 5s — xv ==f, as formerly. Now fince it iSx=2, and vg NV — x* =x/a* — 2, hence it appears that the Conic Section with which the Curve is compared is a Circle, whereof the Radius is a = AB, and the Abſciſs x, beginning at the Center, equal to = AG. Wherefore, if with the Center A and Radius AB, the Quadrant BCH be deſcribed, and CI drawn parallel to AB, then GC v, the Rectangle GI = xv, and the Area AGCH = 5: conſequent- ly AGCH — GI —= CHI =s — xv = # = curvilinear Area AGF required, And if Bæ be drawn perpendicular to AB, it is evident it will be Aſſymptote to the Curve; and that the remaining Part of the curvilinear Area, lying below GF, are infinitely extended along the Aſſymptote, viz. BGF = Area ABCI : and therefore the whole cur- vilinear Area ABæ = the Quadrant ABCH. 326. Schol. 1. If you draw the Tangent CK, meeting AH pro- duced in the Point K, it appears that the two Triangles AGF, CIK are equal and ſimilar: and therefore the Ordinate GF — IK the Sub- tangent of the Arch CH ; and AF CK it's Tangent: ſo that ABaFA is a Figure of Subtangents in the Circle, lay applied to the right Sines of the Arches as their Abſcifles: any Part of which Figure of Subtangents, as AGF is equal to the correſpondent circular Area CHI, and the whole equal to the Quadrant. And univerſally, fince in any Curve whatſoever, if you call z the Abſciſs, y the Ordinate, and s the The Ruadrature of COR VES explained. 231 is the Fluxion of the Figure of Subtangents applied to the Ordinates or right Sines as their Abſciſſes; and the other, the Fluxion of the curvilinear Area * : * hence the Fluents are equal, ſince they both , 4, begin to be generated at the ſame time *, 7, e. the Figure of Subtan- of this E- gents applied to the Ordinates of any Curve for their Abſciſſes, is equal plication. to the Area of the Curve. 9 1 wa 327. Schol. 2. Moreover, it appears, that the Line CGF, in the plication. preceding Figure, is equal to the Secant of the Arch CH : and there- tore AH BAFA is equal to a Figure of Secants ordinately applied to the right Sines as their Abſcifſes : which Figure of Secants, therefore, ad- jacent to the whole Radius, is equal to the Semicircle. It's Deſcrip- tion may be thus: produce HA to L, fo that AL = AH, and take every where upon GF produced, FM = CG, then the Curve LM, which M touches, belongs to a Figure of Secants, ſuch as has been mentioned, having AG IC for the Abſciſs, and GM = AK for it's Ordinate. And fo the Area AGML = (AGCHA + CHI =) twice Sector ACH : and the whole Area ABM infinitely extended along the Aſſymptote Ba, is equal to twice the Quadrant AHB: and when the Arch HC 45, the Ordinate GM biſects the curvilinear | Area, making each half equal to the Quadrant. Accordingly GM — (GF + C —=— +vVZ—2 =) —=— of —=—:* 1 Mat z* X ITA which belongs to Species 1. Order 4. Table 2, vis. 7 = = =) 3 | as appears by putting d= a, e=—1, =, y=—2 : ſo that it 1 * = (2 =) 2*, or x = =AG, v= (Va*— XK N - 2 =) CG, and * == 2 K — 7x0 =2ACH = AGML re- quited. The ſame may be found by comparing the given Ordinate a . . . * * Of this TE with Species 1. Order 8. as in Art. 325 *. 1 328. Schol. 3. But if the Secants be ordinately applied to the verſed tion. Sines of their Arches, that is, if the rectangular Ordinate IO be always equal to the Secant AK, then AI, AH, AO are proportional: there- | fore if you complete the Square AN, it is evident that the Curve When I ſay here that js and zy are the Fluxions of the Figure of Subtangents and Area of the Curve, I hope the Reader won't miſtake me, or alledge that I have departed from the true Notion of Fluxions laid down in the Beginning of this Work : as if E meant that theſe Ex- os were in ſtrict Propriety Fluxions or Velocities of flowing: it is enough that they are preſentations or the ents of the Fluxions and proportional to them, which is all that the Reaſoning requires. This I thought proper to obſerve in this Place once for all: that ſo the Reader may not miſtake me in any Place, where he meets with the like way of ſpeaking. NOP, 232 1 * 188 The Quadrature of CV AVvEs explained. NOP, deſcribed by the Point O, is the rectangular Hyperbola haying A for it's Center and N for it's Vertex Principalis. Which is the cr. thographical Projection of the Figure of Secants applied ordinately to the Arches as their Abſciſſes, theſe Secants, made Ordinates, being ſuppoſed to be perpendicular to the Plane of the Circle, which is itſel perpendicular to the Plane of Projection. Thus likewiſe in the Ellipke and Hyperbola, if you put à for the half Axis and 2 for the Abſcib commencing at the Center, you'll have - for the Secant : which ſhey; that in theſe Curves likewiſe the Secants applied as Ordinates to the Axis, will touch with their Extremities rectangular Hyperbolas, whoſe Aſſymptotes are the tranſverſe and conjugate Axes of the Conic Sec. tions, and the Axes of thoſe are to the Axes of theſe in the ſubduplicate Ratio of 2 to 1: which rectangular Hyperbolas are likewiſe the or. thographic Projections of the Figures of Secants applied as Ordinates to the Arches for their Abſciſſes in the Conic Sections, in the ſame Senſe as in the Caſe of the Circle already mentioned. And the Hyperbol, which is formed by the Secants, in the Caſe of the Ellipſe, ordinateh applied to the verſed Sines or Sagittas as their Abſciſſes, is the ſame with the Hyperbola which is formed by the Secants in the Circle ſimi- larly applied, when the Diameter of the Circle is the ſame with the Axis of the Ellipſe: becauſe when a Circle is circumſcribed about an Ellipſe, or inſcribed in it, the Secants of the Arches, whoſe verſed Sines are equal, are themſelves equal: as is evident from the common Property of the Tangent in both. 329. Schol. 4. Let the Hyperbola NOP remain as before, viz, a Figure of Secants ordinately applied to the verſed Sines of their Arches as the Abſciſſes: and ſuppoſe now that the Quadrant ABH is the fourth Part of the primitive Circle, upon whoſe Plan the Sphere is ſtereogra- R phically projected, that primitive Circle ing the Equator : in which A is the Repreſentation of the Pole, and AH that of the Half of a meridian Circle, According to the known Law of this Sort of Projection, the right Line Al is the Tangent of one half of the me- ridional Arch of which it is the Repre- ſentation. For in the Figure adjoined, let the Eye be ſuppoſed to be at Q, one of the Poles; R being the other, QR the Axis, AH the Line of Meaſures, lying in the Plane of the Equator : . et „ M .43> IMC oe ecSn 2 RORSSz - > * = o -= T 4 = 1 - , - ! "IF : * 4 = - ” 7 5 „ F * , - De Quadrature of Curvus explained. 233 let AH in this Figure, be equal to AH in the preceding: then the Quadrant HR being deſcribed is a fourth Part of a meridian Circle: draw Ql8, cutting AH in I, and HR in 8, then it is evident that Al is the Tangent of the Half of RS. Now ſuppoſe the Line QIS to revolve upon Q as a Pole, while IO in the preceding Figure moves rallel to itſelf, ſo that AI and HI in both Figures be always of the ame Length: and let QIS move into the new Place Qzs, the two Triangles QSs, Qli, in their naſcent State, are ſimilar: whence Ss : Ii :: QS : Qi or QI; but, if you draw ST perpendicular to AH at the Point T, from the ſimilar Triangles AIQ, IST, it is QS : QI :: AT: Al, therefore Ss: Ii :: AT: AI; but Ss: I; :: Fluxion of HS: Fluxion of HI“, that is, calling HS=z and HI=x, z:x::(AT:,,, 2g Al ::) Coſ. HS: T. 2 comp. HS. Again, in the preceding Figure, ofthis Ex- F. IN: F.IOHN (:: HN :IO:: AI: AH) :: T. comp. : R; but plication. F. IN: F.IOHN :: 11 K 2 . therefore &: E:: N comp. 2 . R. But it was juſt now demonſtrated that &: x :: Col. z: T. comp. æ; therefore by Equality = : 2 COLOR! R > Sec: 2, But ſuppoſing p to expreſs the Length of the meridional Arch HS or , as |, in the nautical Chart, according to Mer- cator's Projection, from the known Property of that Projection, it is * | hb e 3:5 : R: Sec. 2; wherefore, by taking equal Ratios, it is 2 : HN 3 F. IONII | | IONH __ 1 : 3:5; whence H =P, and therefore HN = þ * the pro- . Art. 40. tracted meridional Arch HS. Wherefore the hyperbolical Area IONH — is equal to the protracted meridional Arch multiplied into the Ra- dius. Now from the known Property of the Hyperbola, the Area IO NH is the Logarithm or Meaſure of the Ratio of AH $0 AI, that is, calling the Rad. AH, 1, the Logarithm of —- = and x PT. x comp. < | whereas the hyperbolical Logarithm of any Ratio, is to the tabular Logarithm of that Ratio, taken from Briggs or the common Tables as 2.302585, Sc. to 1: therefore the Logarithm of TFT — * taken from the common Tables, being multiplied by 2. 302 58 5, &c. will give the hyperbolical Area IONH, or the Length of the pro- tracted meridional Arch, the Radius being Unity: ſo that the Rule is this; ſubtract logarithmical Tangent of : comp. lat. from logarithmical Rad, and multiply the Difference by 2.302585, &c. Moreover, if you put A and à for a greater and leſſer Arch of the Meridian, reck- oning from the Equator, and / for an hyperbolical Logarithm : then H h it The Nuadnatun of CA tuflointd.. it appears from hat bas been ſaid, that the protraGed Length of th meridional Arch'A — & L. Trag A I Fa- . T. comp. a — J. T. 4. mp · A, that 6, Jubtract the Log. f .the L gent of the half Complement .of the greater Arch 1 dhe * the Tangent of the half Complement of the leaſt Arch, and the Dif. ference is the protracted Length of the Arch A4: meaning by Ly. garitbm the hyperbolical Logarithm. Wherefore when you jake the logarithmical or artificial Tangents from the common teſgonometrioil Tables, the Difference mentioned above muſt he multiplied by 2.392 585 Ec. to have che protracted meridjonal Arch lying -betwixt the two Latitudes A and a, the Rad. of the Globe being 1. But if you would have this meridional Argh expreſſed in the common way by Miles ot Minutes, fo that each Minute of Latitude be called 1, then the Ex. preſſion already found muſt be multiplied by 3437-74977, Ge. he cauſe ſo many times is one Minute of a great Circle contained in the Radius. Or, by one Operation, multiply the Difference of the ani. ficial Tangents of the half Complements .of the Latitudes taken from the common Tableg, by 791,5,70446789, Sc. = 2.302 5B 50g, G. X 3437,74677078, Sc. e. g. Let it he required to find the Length of the meridional Arch in Mercgtor's Chart, lying betwixt the "Tropic in Lat. 23* 30“, and the polar Circle in 66 30“: which being Com- plements to each other, therefore the Halves of their Complement are 33* 15, and 11* 4 55 whoſe artificial Tangents are 9.816658, and 9.3 180640, whoſe Difference is 0.498 594, which multiplied by 7915, 704, Gc. gives 3946 2, Gc. Miles or Minutes of a great Cu- cle, contained in that meridional Arch protracted. ; 330. Schol, 4. Becauſe this Affair is under Conſideration, and the Figures ſuited to the Purpoſe, I ſhall ſhew another Method of inveſ- tigating and demonſtrating this Property of the meridian Line in Mer- cator's Projection. In the laſt Figure draw IC perpendicular to AH, meeting the Circumference of the Circle in C: at C and 8 draw the Tangents CK, SV, meeting AH produced in K and V. Then, uſing the Symbols x, z and p as formerly, and ſuppoſing the ſame things as before, from the Prgperty of the Tangent, you have AV: AH: AT — and X: AH: AI , whence AV : AK:: (AI: AT:: Q: Q:: L: S.:: ) *: &, that is AV x = AK x but the firſt is the Fluxion of the Figure of Secants applied to their Arches as Abſciſſes; the other is the Fluxion of a Figure of Secants applied to the verſed Sines as their Abſciſſes; the correſponding or ſynchronal Abſciſſes be- cre: bg HS and BIL: therefore the Axeas of theſe Figures are equal , thi — eo od wad feds a as. Be LS Tha Ryndrature of CU v G ain. 1 e ds Fut HION (im the prededing Figure) is Area of the Figurè of Secants-ordinately applied: to the — no Hig for the Abſtiſs, in — Rigure. But the Area laſt men- tioned is to a Rectangle having it's Baſe — HSors, and Altituds, the Nadius, as" the of the Arch HS, to-it's natural Length (for the Fluxions of theſe Figures are as the Secant of HS to Radius) _—_ HION : 2xR :: g: 2; that is HIONX2#=2 x Rx, or fg. ſo that when Rr, HION =p the protracted meridional Arch HS: and therefore the reſt follows as before. And here it may be obſerved that a Method is ſuggeſted of finding 2 geometrical ( Curve having it's: Area equal to that of a mechanical Curve': ſince it has been demenſtrated that the hyperbolical Area RION' 18 8 1 to the Area of the Curve which has an Arch of a Cir- cle; vis. ſor it's Abſcifs,. and the Secant AV for it's 9 of which more immediately). PROBLEM. 331. Jo, M the Ateas'of mechanical Curves. A Curve, whoſe Nature or Property may be expreſſed by an alge- braicdl Equation of any finite Order or Degree, which defines the Re- lation of the Abſciſs and Ordinate to each other, is commonly called an atzebraical or trical Curve: ſo that, according to the Num- ber of Dimenſions of the Equation by which ſuch Relation is defined, Curves are diſtinguiſhed" into different Orders: when the Equation: is of two Dimenſions, the Curve is of the firſt Order, the Circle and Conis Sections are the only Curves of this Order: when the Equation is of three Dimenſſons, the Curves ate of the ſecond Order, ſuch are the cubical Parabola; the Ciſſoid of Diocles; and others; an Enume- ration of which our learned Author hath made in + ſeparate Treatiſe by itſelf; in wHich' he reduces them all to ſeventy-two Kinds. And after the like manner, Curves of the third, fourth and higher Orders are ſuch; the Relation of whoſe Ordinates to their Abſciſſes, is defined by Equations of four, five or more Dimenſions reſpectively. It amounts to the ſame to ſay that Curves of the firſt, ſecond, third, Cc. Orders ate ſuch as may be cut by a right Line in two, three, four, &c. Points reſpectively. And a Curve of the firſt Order is eſteemed a Line of the ſecond Order; a Curve of the ſecond Order, a Line of the third, Sc. the right Line being the firſt Order of Lines; which can be in- terſected by another, only in one * Accordingly a Curve which h 2 may * to have the Abſciſs or Ordinate equal to ſome known Curve, as a Cir. ' The Quadrature of Curves explained. may be interſected by a right Line in an infinite Number of Point, E] is ſaĩd to be/ of an infiniteſimal Order: theſe Curves, or ſuch, the Re. Pc lation of whoſe Abſciſs to the Ordinate cannot be expreſſed by an alge. th braical uation of any finite Order or Dimenſions; are termed mecha. nical or tranſcendental, ſuch as the common Cycloid, Quadratrix, Spi ral and infinite others *, | Theſe mechanical Curves (at leaſt the Order of them) are ſuppoſed cle or Conic Section; or even one of ſome higher Order; or elſe to bear ſome known Relation to ſuch Curve-line ; or curvilinear Area a plied to Unity. . Now although the Relation of the Abſciſs and Or. dinate in theſe mechanical or tranſcendental Curves, cannot be expreſſed by any algebraical Equation of finite Dimenſions : yet the Relation of the Fluxions of the Abſciſs and Ordinate, may be expreſſed by an al. gebraical Equation ; by means of which we may find a geometrical Curve which has the ſame Fluxion : and thence the Area of the me- chanical Curve may be found, by the Quadrature of Curves : of which take the following Examples. 332. Ex. 1. Let the common Cycloid or Trochoid ABCD be pro- poſed, generated by the Rolling of the Circle AHGE upon the Baſe 1 Des Cartes calls thoſe Curves, in which the Relation of the Abſciſs and Ordinate is ex- reſſed by an Equation of two Dimenſions, Curves of the firſt Order or Kind, vix. the Circle, llipſe, Parabola and Hyperbola : but under Curves of the ſecond Order, he reckons both ſuch, the Relation of whoſe Abſciſs and Ordinate is expreſſed by an Equation of three Di- menſions; and likewiſe thoſe, in which it is expreſſed by an Equation of ſour Dimenſions, And under Curves of the third Kind or Order, he reckons all thoſe in which the Relation of the Abſciſs and Ordinate is defined either by an Equation of four or of five Dimenſions, and ſo on. See his Geometry, Book 2. But the Diviſion above, which is Sir 1/aac Newton's, is the moſt ſimple and natural. See his Enumeratio Linearum tertii Ordinis at the Beginning, And whereas there have been ſome Differences among learned Men, with reſpe& to what Curves ought to be called geometrical ; and as to their Preference in geometrical Conſtrue- tions of Problems. See our Author's Opinion of this Matter in the Appendix to his Arith, Univ. de Conſtructione Lineari, Where he ſhews that the Rule by which to determine the Preference of one Curve to another in Geometry, is the Eaſineſs of the Deſcription ; and not the greater Simplicity of the Equation by which it is defined: which laſt is a Conſideration entirely algebraical, otherwiſe the Parabola ought to be preferred to the Circle in geomett- cal Conftructions : which no one will affirm. | The general Diviſion of Curves by our Author is into. geometrically rational, and geometri- cally irrational. Curwvas geometricè rationales appello, quarum puncta omnia per longitudines & equationibus defnitas, i. e. per longitudinum rationes complicatas, determinari poſſunt, ; cates * rafque (ut Spirales, Quadratrices, Trochoides) geometrice irrationales, Nam longitudines que « ſunt vel non ſunt ut Numerus ad Numerum ( quemadmodum in decino Elementorum unt arith- « metice rationales vel irrationales.” Phil. Nat. Prin. Lib. 1. Cor. ad Lem\g8. By which it appears that theſe Curves which he calls geometrically rational, are the ſame with ſuch as we. (after others) have called geometrical or algebraical : and the geometrically irrational, the fame, we have termed mechanical or tranſcendental : which include thoſe called exporential by Mr. Zeibnitæ, wiz. ſuch whoſe Equations involve ſome Power of the Abſciſs or Ordinate, that has a variable Quantity for it's Exponent as x, x* K. ED | | | al | The Ruaarature of. C URVES explained. 2437 ED, of which the Diameter AE is the Axis of the Cycloid: from an Point of which as B, having drawn | | the rectangular Ordinate BGC, meet- ne the Circumference of the Circle 17 n 8, and the Curve in C: complete . E . ee. A the Parallelograms ABCF, AEDI ; 7 . draw the Chord AG ; and ſuppoſe 8 — CT a Tangent to the Curve at C. F 0 The Area ABC 1s required. Call AB , BC =y, BG = v, AE d: then from the Nature of | the Curve (as ſhall likewiſe be ſhewn 5 — E afterwards) CT is parallel to AG, fo that the two Triangles BCT and BGA are ſimilar, whence &: :: (TB: BC® ::)z: v, that is y == == (by inſerting Vdæ — 25 in- Art. 47. — of this Ex- ſtead of y, from the Property of the Circle) = —.— — : whence you Plication. have the Relation of the Fluxions of the Abſciſs and Ordinate. There- fore, while AB flows with the Fluxion 3, BC or AF flows with the Fluxion YEE: confequently, if you multiply this laſt Fluxion | into CF= 2, you'll have zVdz — 22, for the Fluxion of the exter- nal Area AFC, which, with ABC, completes the Parallelogram ABCF. Wherefore (by Prop. 9.) the Area AFC is equal to the Area of a geo- metrical Curve having 2 for it's Abſciſs, and /dz — zz for it's Ordi- nate: which is the circular Area ABG deſcribed by the Ordinate BG x — Zz and Abſciſs AB S g: ſo that the complemental Area AFC is equal to the circular Area AH GB; and the whole Area AIDCA equal to the Semicircle AGEA : therefore the Area ABC = ABCF — AHB; and the Semicycloid AED — AEDI — AHGE, f. e. (be- cauſe the Baſe ED is equal to the Semicircumference AHGE, and therefore the circumſcribed Rectangle AD AE x ED=AE x AHGE =4AHGEA) the Semicycloid AED= 3 Semicircles AHGEA; and ſo the whole Cycloid is equal to three times the generating Circle, and ACDEGHA equal to that Circle. 333. Ex. 2. After the ſame manner, if ACDE was a Figure of Arches of any geometrical Curve, as the Circle,. Ellipſe or any other known Curve: fo that the Ordinate BC applied to the verſed Sine AB, were equal to the Arch AHG: you may find the complemental Area AFC, and thence the Area AHGB as formerly. For, fince BC — ; AHG, 238 De Luuurururt of CV EV explained. ut. 4. AHG;- their Flitxions ate equal : therefore, from the known PG — perty of the Curve ABG, you may find the Relation of the Fluxioh, of AB and AHG or BC, that is AF; and then you proceed as in the laſt Article. | | Thus if AGE be a Circle, uſing the fame Symbols as before, the Lit not BY or AG, i g ht Gy plication. tuting for & it's Value, viz. 2* , dedueed from the Eq. tion to the Circle, v = d —2*, and redueing) 12 =: — 22 multiply this into FC=2, and ydu have 2 72 : where. fore a geometrical Curve, whoſe! Abſciſs is 2 τ AB; and Ordinate * Prop. 9. —_ is equal to the complemental Area AFC *: which may be reduced to Species ſecond, Order eight, Table ſecond, expreſſed thus 2y—1 | — = y, by putting 1 1, d= d, e o, f= d. g= V | — I, == (=) = AB; ve+ fr + gf = = N, = BG, and f= 25 — xv= 2ABGHA — 2AABG=2AGHA: which therefore is equal to the 2 Area AFC; and ſo the whole Area AID is equal to twice the Setnicircle; Whenoe, by ſubtracting twice the Segment AG HA ftom the given Rectangle AC, there re- mains the Area ABC required: and ſince ED== AHGE, and conſe- yk the Rectangle AD'=2 4AHGEA;,and ADCA == 2AHGEA,, therefore the Figure of Arches AEDCA is juſt equal to the Circle whoſe Diameter is AE: fo that the Rectangle AD is biſſected by the Curve- line ACD, and the Area AED by the Semicircumference AHGE. The ſame things may be found by comparing the Ordinate, aftet it is reduced to this Form y = with Cale ſecond, Species =» , onnr_rs serer r ; | — 1 ＋ = 5 ſecond, Order fourth, Table ſecond;. by putting y — — 1, &c. And thus you might find the Areas of Figures of Arches in the Ellipſe and Hyperbola, and compare them with geometrical Curves: only in this Caſe, it will be proper to compute the Beginning of the Abſciſs from the Centers of-the Conie Sections, rather than from the Extremities of the Diameters: as we have done in the Caſe of the Circle. 334. Ex. 3. Let GH I be a Circle, which the right Line HK touches in the Point H, and ſuppoſe the right Line GK to revolve about the Center G as a Pole, and always interſect the Tangent HK in the N , * 0 +» 1 uadnatume of Cunves evflained. E. and the Circumference. of the-Eircle.jn I: and let FCD be a Curve of ſuch a Nature | F ; that-it's Abſciſs A . B be equal to the Arob HI, and it's rectangular Ordi- nate BC equal to the Secant GK. o as. to conſtitute ſuch a Figure of We 239 a ö | 1 N Join GH, and ket -GIK move into the new Place Git ; draw KL perpendicular to Gt: call HI r AB , and GK or BC then Z'is to 5 in the prime Ratio of the naſcent Augments Ii and Lk , that is, (becauſe the Triangles Gli, -GKL, and LK#, GHK *&r.z,rz xe in that Caſe fimilar) 8:9 1: L: KLA-KL:Lhors: j .. Gli ome, 0K +GH HR, that is, if you call the Radius ; 2 25 ::: ture of Wir, or e ir, which defines the Relation of tbe Fluxiqns of . — _ | 3 Value of 5 is mint, thav- ing ) ip it's Compaſition, otherwiſe than happened in the preceding Examples, I. cant find the complemental Area as in them; but I take the Value of , ws. 2 = —=—, and multiply it into BC=y, ; ISS > rt which produces e = i che Pluxion af the curvilinear Area 511-1 ABED : and therefore that curvilincar Area is equal to the Area of a geometrical: Curve having y = BC or GK for it's Abſciſs, and | | | | * 7*— rr for its Ordinate *, that is, if yau draw GO perpeadicular to GK, Prop. 9. meeting the Tangent KH produced, in O, the Line HO, which is the Cotangent of HI. Now this. Ordinate = being made to ſtand thus = Z = belongs to Species 1. Order 4. Table 2, expreſſed thus ——— = y, „ bee we 8 by putting d = =I, =-, ==), 1 = 2: in which Species, 240 * Of this tion. + Of this Explica- tion. 1 Of this Explica- tion. Art. 40. of this Ex- plication. De Quadrature of Curves explained. Species, becauſe 5 —=Xx*, ff +ex* u, and — 2 making a proper Subſtitution of Values you'll find that the Conic "aq tion, with which the Curve is compared, has it's Abſciſs x = y= BC, or GK, and it's Ordinate v = — r* + x*—= BC! — ADi— HK; and that the Area of the Curve ſought is 2 X 4xv - te Which points out this Conſtruction: with the Center A and hal tranſverſe Axis AD, deſcribe the rectangular Hyperbola DPQ, through C draw PCR parallel to AB meeting the Hyperbola in P, and the tranſverſe Axis in R; join AP, then the curvilinear Area ABCD — 2APDA.. For AR =x, RP = AR! — AD' = HK = VN v. ARP D and DRP . And this hyperbolical Sector APDAa is equal to one half of the hyperbolical Area IO NH in the Figur bis Art. 32 5 *, according to the Meaning of Art. 329 +, when the Arch HS in the Figure Art. 329 f, and the Arch HI in this Figure are equa, the Radius's of both Circles being ſuppoſed equal: as might be eafily demonſtrated. ces | 335. The Areas of Spirals may be determined in the following manner. Let FC be a Spiral of any Kind deſcribed from the Point A as a Pole; BC an Arch of a Circle deſcribed from the ſame Point A as a Center, moving uniformly along the indefinite right Line AB given in Poſition, as a Baſe or Abſciſs: and let the Arch BC, while it moves, always touch the Spiral with it's Extremity C: then it is evi- dent that the ſpiral Space terminated at * BC, is deſcribed with the ſame Fluxion I as a Curve having AB for the Abſciß, and a right Line equal to BC for the S/ rectangular Ordinate; and therefore the ,; ſpiral Area and curvilinear Area adjacent to the ſame Part of the Abſciſs, muſt be equal *, Thus, if at the Point B of the Baſe AB, you draw the right Line BD perpendicular to AB, and equal to the Arch BC, ſo that while BC moves along AB, BD may move together with it, and be always equal to it, ſo as to deſcribe the Curve JED, the Fluxions of the ſpiral and curvilinear Areas are equal: therefore if from the Point þ you draw the Perpendicular þ4 = bc, the 2 | bBD =, => og => The RQuadrature of Cuxvss explained. 241 3BD4 and bBCcb are equal. Wherefore calling AB , BC or * Art 40. BD = y, the Relation of z and y are defined by an Equation expreſ- — ſing the Nature of the Spiral; and thence the Area may be found, as formerly, from the Quadrature of Curves. Thus let * = 42” be an Equation defining the Relation of z and y; or AB and BC or BD; which may denote an infinite Number of different Kinds of Spirals : thence we have y TT for the Value of the Ordinate BD of the Curve JED : therefore the Area is 2 nn | ; 22 denoting the curvilinear Area ABD or ſpiral Space ABCA, N beginning at the Pole A, when the Exponent * is poſitive; and ; adjacent to the Abſciſs AB and bounded by BD and DA, or BC and CA“: but if == be negative, the Area lyes upon the other Side - Art. 168. of BD or BC : and in every Caſe, the Area adjacent to any Part of —— 5 the Abſciſs Bb, viz. BbcCB is found, by ſubtracting the Area belong- + Ibid. ing to the leſſer Abſciſs from that belonging to the greater “. *Art.187. Ex. 1. Let the Equation be y = a="z*, where we have m = 1, — 4 p=—1 and n=2: therefore the Area is 17 (by ſubſtituting y for — . 22 = ABCA = ABDA. In which Caſe it appears that the Curve AED is the common Parabola having it's principal Vertex in A, where AB touches it, and BD a Diameter; and à the Latus Rectum to the Axis: for it is a x BD= ABI. And the Spiral is that of Archimedes. For ſuppoſe ACFM to be his firſt Spiral, AM the Line of the firſt Rotation ; draw ACG cutting the Circle MEGM, which bounds the Spiral, in the Point G: call AM = = AG, MEGM c, MEG Do, and retain z and y as formerly. Then from the Deſcription of Archimedes's Spiral : :: v: c, but Z: :: : v, therefore, by equating the two Values of v, deduced from theſe two Proportions, you have ===>, or ＋ y: Which is the fame with the aſſumed Equation _ „by ſuppoſing a ==, that is a third Proportional to the Circumference and Radius of the Circle MEGM bounding the firſt Spiral: which therefore is the Latus Rectum of the Parabola ED, whoſe external Area ABD is equal to the external ſpiral Space S. CO IE Moreover, if you ſuppoſe the right Line AD drawn, the parabo- lical Segment AEDA is equal to the internal ſpiral Area AcCA : for I1 the — — 242 The Ruadrature of Curves explained. the whole circular Sector ABCA == A BD. BD: by 2 the external ſpiral Space ABCcA — ABDEA, therefore the interny Space AcCA = AEDA. From which, as likewiſe from ay deng. ing the ſpiral Space ABCcA, as was demonftrated, it follows that ABCA = E or ABCA, and AcCA — + of the ſame: that the whole internal ſpiral Area ACCMA =— + Circle MECM, and the ex. ternal Part lying without that, and within the QCircumference of the Circle MEGM is + of the ſame Circle. And by a like way of tes. ſoning it may be ſhewn, that if the right Line AM be ſuppoſed to be produced indefinitely, and by revolving to deſcribe Spirals of the {. cond, third, fourth, &c. Revolutions the Proportions of the ſeveral ſpiral Spaces, belonging to the ſeveral Revolutions, may be ſhewn to be what Arobi medes has demonſtrated in his Treatiſe de Liners Spire. libus. Moreover, the ſpiral AcCA is to the whole ſpiral 8 AcCMA, as the circular Sector ABCA, to the whole Circle MEGM, that is in a Ratio compounded of the Sector ABCA to the Set AMEGA, and the Sector AMEGA to the whole Circle MEGM; or of AC? to AG? and MEG to MEGM, or AC to AG, that is the Ratio of AC to AM, as Pappus has demonſtrated 7. Ex. 2, Let the Nature of the Spiral be expreſſed by this Equation 2 23 _ me P; — Sy: whence the ſpiral Area AcCBA — 15 2 2 = ABDEA: where the Curve AED to whoſe Area the ſpiral Space is equal, is a Parabola of the ſecond Kind: and the external ſpiral Space AcCBA is + of the circular Sector ABCA ; and therefore the internal AcCA is +'thereof : and the whole CMA is f of the whole Circk MEGM. * 5 5 I 2 x. 3. Let AC be a Spital — : IT. deſcribed from the Pale A, of ſuch a Nature that if the right Line AB (called the firſt Radius) be drawn, and from the Center A, with any Di- tance AB, the Arch BC of a given Length, be deſcribed, it's Extremity C may always touch the Spiral CcA. In this Caſe calling AB = z and BC=y as before; the general Equz- tion * u, becomes y = a2® or y = a, and therefore the Area is 1 Pappi Collect. Mathem. Prop. 22. Lib. 4. az = 2, wee wane wt > ws 111 * i... The Ruadrature of Curves explained. 4% . erefore if the right Line HA be drawn perpendicular to AB, ſo that AH be equal to BC; and through H, HI parallel to AB; HI ſhall be an 9 to the Spiral: and if through B, 6, you draw BD, bd, perpe ſpiral 8 BCch = Rectangle BDab. Moreover let AC interſe& bc in the Point 7, then, if from BCcb = Bb x be, you take away BCf6 BN, there remains Cfe = Bb x £, to which add Afe = AE, and you have the ſpiral Setor AC: , therefore when the Abſciſſes AB, A6 are given, the ſpiral Sector ACc is found. This Spiral is called the . gy? Spiral, with reſpect to the common Spiral of Archimedes: becauſe, as in that of Archimedes, the Radius BC is pro- portional to the Angle BAC, which it makes with the firſt Radius, called the Beginning of the Circulation ; ſo in this other, BC is reci- procally proportional to the Angle BAC, as eaſily appears. Ex. 4. But if the Spiral CcA be of ſuch a Nature, that the Arch BC be reciprocally proportional to AB, or BC: :: A: AB: then the Equation to it is y , which plainly appears to be an Equa- tion to the Hyperbola, reckoning the Abſciſs z from the Center of the Hyperbola upon one of the Aſſymptotes, and the Ordinate drawn from the 1 upon the Aſſymptote, parallel to the other Aſ- ſymptote. Therefore, ſuppoſe the Point 6 ſuch, that bc = Ab; com- plete the Square Ad; and with the Center A and principal Vertex 4, deſcribe an equilateral Hyperbola, to which AB, AH are the Aſſym- totes: draw the Ordinates BD, 6d, then it is evident, the ſpiral Area BC is equal to the hyperbolical Area BD4b, the right Line Dl in the former Example, becoming an Hyperbola in this. This Spiral continually approaches to AB as an Aſſymptote. Moreover, fince it is AB: Ab:: bc : BC, the Sectors ABC, Abc are always equal, hence it appears that the ſpiral Sector Ace == BC. Whence it follows that BCch and AC: are as the Logarithm of the Ratio of AC to Ac. And theſe Examples ſhall ſuffice to ſhew after what manner me- chanical Curves may be reduced to geometrical Figures, and thereb their Areas determined : which was what we propoſed to do in this Problem. I ſhall next ſhew after what manner the Areas of Curves may be determined, when they are ſuppoſed to be generated by the Revolution of a Radius round ſome immoveable Point as a Pole or A 1i 2 . PROBLEM. 243 icular to AB, . HI in B, 3, the 244 * Of this Explica- tion .* | The Quadrature of Curves explained. PROBLEM. 336. To find the Areas of Curves ſuppoſed to be generated by the Reg. lution of a Radius about a fixt Point. Although in this Treatiſe of the Quadrature of Curves, the 7/luftr;. ous Author conſiders curvilinear Areas as generated by the Motion of x rectangular Ordinate along a Baſe or Abſciſs: which is certainly the moſt natural and convenient Conception of the Production of geome- trical Figures; and likewiſe the moſt Part of mechanical Ones; in order to determine their Areas by his Method of Fluxions : yet the Method of Fluxions, as applied to the Quadrature of Curves, is not limited to this manner of conceiving the Production of curvilinear Areas; but may be applied to other manners of conceiving their Pro- duction or Generation, by Motion variouſly modified: particular when they are produced, or ſuppoſed to be produced, by the Revoly- tion of a right Line about an immoveable Point. In which Caſe, even as in the other, to which our Author confines himſelf, there is ſome certain Line ſuppoſed to flow with an uniform or conſtant Fluxion; and ſome Area likewiſe to flow with a conſtant Fluxion, which ariſes from the Multiplication of the Fluxion of the uniformly flowing Line into a given Line : then, the Relation of the Fluxion of the curvili- near Area, deſcribed by the Revolution of a Radius, to the Fluxion of the uniformly flowing Area, is inveſtigated, and thence by means of the Equation to the Curve, the Relation of the Fluents is found as in the other Method. But whereas this way of conceiving the Deſcrip- tion of Areas, is of very little uſe, unleſs in the Caſe of Spirals, 1 ſhall juſt ſhew in an Example or two, the Application of it to this ſort of Curves. | Ex. 1. Let ACM be the Spiral of Archimedes, (fee the Fig. at Art. 335 *. p. 240.) the Lines AM, ACG as before : fo that the Radius ACG revolving about the Center A with an uniform or equable Mo- tian, may with the Point G deſcribe the Circumference of the Circle MEGM, and with the Point C the ſpiral Line AcCFM. In this Caſe, it is evident, that both the Circumference and Area of the cir- cular Sector AMEGM flow with a conſtant Fluxion : and moreover that the Fluxion of the ſpiral Sector AcCA is to the Fluxion of the circular Sector AMEGM as AC! to AG? ; for this is the prime Ratio of the naſcent Augments of theſe Sectors. Wherefore, call AG or AM Sa, MEGM=c, MEG —z, AC y, the circular w—_— "” pira The Ruadrature of CRV ES explained. ſpiral Sector = 5, and you'll have S: f:: 4*: y*, or 5 ass. But gnce the Fluxion of MEG or ⁊ is 2; or of 1X2 is 1 *, thence the Fluxion of the circular Sector is — *: ſubſtitute this for S in the pre- *Art.3.12 245 of the ceding Equation 1 , and you'll have 5 ===, which is the gene- _— al Formula for the Fluxion of all Spirals. But from the Property of 3 the Spiral, it is : 4 :: : c, or y== 3 by ſubſtituting of which azz Value of y in place of it, you have ===: wherefore it is :e , „ange. ac —— the Value of the ſpiral Sector AcCA: and when z=6, it is + = atten, tx MEGM ; of the Circle MEGM = AcCFMA, as before. Since it is 5 =E , if you ſubſtitute for z it's Value, viz. =, the ſpiral Sec- tor AcCA will be otherwiſe expreſſed thus : which includes the ſame things that were demonſtrated above. Ex. 2. Let MFC (ſee the ſame Fig. p. 240.) be ſuppoſed a Spiral of ſuch a ſort, that if from it's Pole A with the given Diſtance AM, the Circle MGEM be deſcribed, and any Radius ACG be drawn, the Sum of the Squares of AC and MG be equal to a given Area, e. g. to the Square of AM. Then putting AM = a, MG =, and AC y, it is * + 2* =, or *=4* — : but the Fluxion of the ſpiral Sector is — , as was ſhewn in the preceding Example, in which fluxionary Expreſſion inſert a — 2* for *, and it becomes 22 — 0 2a ? the Fluent of which, v:z. _ —5 is the ſpiral Area ſought, viz. MFC A: which Expreſſion conſiſts of two Parts; the firſt = de- notes the circular Sector AMG A; the other =, the external ſpiral Space MFCGM. And when z—=@ or MG— AM, the Point C falls os and the whole internal ſpiral Space is 24 — 44* . And thus you may feign any Relations of z and y, you pleaſe, b which an infinite Vai of Spirals will ariſe, £2 + 7 — 3 be determined after the like manner as in theſe two Examples; or elie may be compared with the Areas of Conic Sections, or other the moſt ple geometrical Curves, they can be compared with. p SECT. The Quadrature of Cuxvxs explained. SECT. X. Of the Reduttion of the Hyperbola ond Ellipfe ; taguks _ with all curvilinear Areas, and Fluents, which depend upon theſe Conic Sections, to the Meaſures of Ratiq, and Angles ; or Logarithms and circular Arches, 337- EF. Let the right Line BC move along the right Line Ag ina Inge Angle, ſo that while AB increaſes or de- creaſes with an equable or uniform Velocity, the Ordinate BC in- ' creaſes or decreaſes with a Velocity always proportional to itſelf : then AB is called the Logarithm of the Ratio ef BC to AD; AD being the ſame with BC when the Points B and A coincide : and the Curve DC which the Point C deſcribes is called the Agaritbmical or hogifi- cal Curve: becauſe, if the Abſciſs AB be divided into very ſmall and equal Parts, and Ordinates drawn thro all the Points of Di. | viſion, they will con- ſtitute a Series of con portionals, falling in ſo that if the Ratio of ö x | BC to AD be conſi- | dered as compound. 1 . _ ed or mdde op of (© 35 H A.-B equal Ratios of all the adjoining Ordinates beginning with BC and ending with AD, the Abſciſs AB will expreſs the Number of the equal compounding Ra- tios, which make up the Ratio of BC to AD: which is the true and original Notion of the Word Logarithm. The right Line AB is like- wiſe called the Meaſure of the Ratio of BC to AD. According to which Meaning of the Word Ratio, the Meaſure of the Duplicate, Triplicate, Quadruplicate, &c. of any Ratio is double, triple, qua- druple, &c. and the Meaſure of the Subduplicate, Subtriplicate, &c. is 2, , Cc. of the Meaſure of the fimple Ratio: the Meaſure of the Ratio of Equality is o; becauſe the Duplicate-triplicate, _ 3 Sub- | | uplicate- a 2 The Qtadrature of CURVES explained. duplicate-ſubtriphcate, &c. of a Ratio of Equality is ſtill the fame with itſelf, and therefore according to the former Notion of Ratio, it's Meaſure muſt be o: accordingly when BC coincides with AD, the Abſciſs AB vaniſhes. And when the Meaſure of the Ratio of greater to leſs is contidered as poſitive ; upon the contrary the Meaſure of the Ratio of leſs to greater, is negative: for when BC by moving towards AD, and getting npon the oppolite Side of it, comes into the Poſition &, the Abſciſs AB by diminiſhing firſt becomes nothing, from being poſttive ; and afterwards becomes negative, repreſented by Ab, which is negative with reſpect to AB, and denotes the Meaſure of the Ratio of bc to AD, or it's Logarithm. | 338. Let the Parallelogram ADFH be completed; and with the Aſſymptotes AD, AH, and principal Vertex F, deſcribe the Hyper- bola FG: through C draw CEG parallel to AB, cutting AD in E, and the Hyperbola in G, the hyperbolical Area DEGF is proportional to the right Line AB, For, from the known Property of the Hyper- bola, EG is reciprocally proportional to AE, that is to the Velocity or Fluxion of AE — BC, fince BC was ſuppoſed to increaſe, or de- creaſe with a Velocity proportional to itſelf : in which Caſe the · Art. 33. Fluxion of the hyperbolical Area DEGF is conſtant and invariable (as ofthis Ex- eaſily follows from Art. 37 *.) but fo is the Fluxion of AB alſo; therefore the Fluents being as the Time in which they are deſcribed, Explica- theſe Flnents, v:z. the right Line AB and hyperbolical Area DEGF tion. ae always proportional. Whence it follows that the Area DEGF is the Logarithm, or Meaſure, of the Ratio of AE to AD, or DF to EG. And in this Caſe likewiſe, as in the preceding, the Meaſure of the Ratio of Equality is o, for DEGF vaniſhes, when the Point E falls in with D; and the Logarithm or Meaſure of the Ratio of leſs to greater is negative, for DegF, the Meaſure of the Ratio of Ae to AD, is negative, v/z. when the Point E falls in e betwixt A and D. 339. If through any two Points K and M of the Hyperbola, the Ordinates KI, ML be drawn, parallel to the Aſſymptote AH, the Area IRML, will be the Logarithm or Meaſure of the Ratio of AL to AI or IK to LM: for if you take AE: AD :: AL: Al, and ſup- poſe the Fluxions of AE, AL, as formerly to be proportional to theſe Lines, the Lines DE, IL ; and conſequently the Areas DFGE, IKML, are generated in equal Times ; and therefore, fince the Fluxions with which the Augments DFGE, IKML are deſcribed, are likewiſe con- ſtantly the fame * theſe Augments are equal: but DFGE is the Lo- PArt.338. garithm or Meaſure of the Ratio of AE to AD, therefore IKLM is likewiſe the Meaſure of that Ratio, i. e. of AL to Al or IK to LM. 340.. Hence 248 De Quadrature of Curves explained. 340. Hence follows the fundamental Property upon which loga- rithmical Arithmetic is founded, viz. that the Logarithm of any Ratio made up or compounded of other Ratios is equal to the Sum of the Logarithms of the compounding Ratios: and the Logarithm of a2 Ratio, which ariſes by ſubducting one Ratio from another is equal to the Difference of their Logarithms, For let the Ratio of AR to AD be compounded of the Ratios of AP to AN and AL to AI; through R, P, N draw the Ordinates RS, PQ, NO. Take the Point E ſuch that it may be AP: AN :: AR: AE; then muſt it be AL: A]. AE : AD, otherwiſe the Ratio of AR to AD would not be compound. ed of the Ratios of AP to AN, and AL to AI. Wherefore it a pears, by the laſt Article, that the Sum of the two Areas NOPQ, IKML is equal to the Area DFSR : the firſt of which is the Sum of the N of the compounding Ratios, and the laſt is the Lo- rithm of the compounded Ratio, and the reſt eaſily follows. Whence it follows that the Sum of the Logarithms of two Numbers is equal to the Logarithm of their Product, and the Difference of the Lo- garithms of two Numbers is the Logarithm of their Quotient, Ge. For what is ordinarily called the Logarithm of any Number, is, pro- perly ſpeaking, the Logarithm of the Ratio of the Number to Unity. 341. If BC, and conſequently AD, be perpendicular to AB, the Hy- perbola FG is rectangular: in which Caſe, if you call AD 4, 2 conſtant Quantity, DE x, and EG = y, two variable Quantities, then, from the Property of the Hyperbola a + x xy = @*, or y = ar but the Fluxion of DFGE, that is of the Logarithm, or Mes- ſure of the Ratio of AE to AD, or à ＋T x to a, is * = <= { that if AD = (a =) I, the Logarithm or Meaſure of the Ratio of 1 to is = (by Diviſion) x — & ＋ xx* — K ＋ A — Cc. in inf. all the Terms of which being ſquared, you'll have * — 2 ＋ 3x3 — 4 + ix5 — &c. in inf. for the Value of the hyperbolical Area DFGE ; and conſequently of the Logarithm, or x By ſubducting of one Ratio from another, I mean the Reverſe of that which is called of Ratios, agreeable to the Way in which we conſider the Doctrine of Ratios in this Place. Thus, if the Ratio of A to B were compounded or made up of the Ratio of C to D, and of E to F; which is when A: B:: CX E: DXF; fo contrarily the Ratio of * F ſubdued from the Ratio of A to B, leaves the Ratio of C to D, viz. when A | F F. C: D. This laſt Operation might be called Reſolution of Ratios, being the op- poſite of Compoſition of Ratios. But it is all one how it be termed, if the thing itſelf be underſtood, Meaſure The Ruadrature of Curves explained. 249 Meaſure of the Ratio of 1 ＋ * to 1: which is commonly called it's hyperbolical Logarithm, to diſtinguiſh it from the common or tabu- lar Logarithm of that Ratio, commonly called ſimply the Logarithm of 1+ x *. By the ſame way of reaſoning, the Area DFge will be found to be — x — * — 3x3 — 4 — Fc. in inf. by ſuppoſing De — — x *; which therefore is the Logarithm of the Ratio of 1 — x * Art. to 1: Or, which is the ſame, x + ix* + K* =_ Tot =_ Sc. in inf. 190, 191. is the Logarithm of the Ratio of AD to Ae, or 1 to 1 — x. Conſe- — quently if it be De = DE, the Logarithm of the Ratio of AE to Ae or 1+ x to 1 —x, viz, EGge, is equal to the Sum of x — * ＋ £x3 - + Sc. and x A , 3x34 ix+4- c. that is 2x + * *Art,340. 2x5 ＋ 3x), Gc. in inf. = 2 Xx + 5x3 + 3x5 + 5x? + Cc. in inf. TR which Series converges twice as faſt as the former ones: and affords one of the moſt convenient and expeditious Methods of conſtructing and computing a Table of Logarithms. If the Difference betwixt the two Areas DFGE and DFge be ſought, ſubtract the leſſer DFGE — x — * + 4x3 — * + c. from the greater DFge = x + ix* + * + 4* + Sc. and there remains x* + A* + ix* + ix* + Ge. = DFge — DFGE. - Now if you ſuppoſe DG =Dg = o. 1, that is AG=1.1 and Ag 0.9 : by ſubſtituting o. 1 for x in the Series 2 * C- + &c. you'll find the Area EGge = 0.20067069 5462, Cc *: which is there- Art. 91. fore the hyperbolical Logarithm, or Meaſure of the Ratio of 1.1 to 0.9 1 or 11 to 9, in this particular Syſtem. And by ſubſtituting o. 1 for x Pen. in the other Series x* + Z + ix* + ix* + Cc. you'll obtain 0.01005033 58, &c. for the Value of DFge — DFGE : wherefore by adding this Number to and ſubtracting it from the former, the Half of the Sum, viz. 0.1053605156 = DFyge ; and the Half of the Dif- ference, vi. 0.0953101798, Sc. = DFGE ; the firſt of which is the Logarithm of the Ratio of 1.1 to 1; and the other the Logarithm of the Ratio of 1 to 0.9. After the ſame manner by fuppoſing DG — 0.2 = Dg, you'll find the Logarithm of the Ratio of 1.2 to 0.8, or 12 to 8, or 3 to 2 to be 0.4054651081, Sc. of the Ratio 1.2 to 1 to be o. 22314355 13, Cc. and of 1 to 0.8 to be o. 18232 15567, &c. And thus by ſuppoſing DE = De = 0.01, 0.02, &c. 0.001, 0.002, Ec. you may inveſtigate the Logarithms of other Ratios, in the ſame The Tables of Logarithms, which are in common uſe, are called Briggs's Logarithms, to diſtinguiſh them from that Form of Logarithms firſt invented by Lord Neper : which laſt are the ſame with thoſe which I have mentioned under the Name of hyperbolical Logarithms. The Difference betwixt the two Forms will appear in what follows. | manner: 250 Art. 338. of this Ex- plication. The Quadrature of Curves explained. manner: by the Addition of which Logarithms, their SubtraQion Multiplication and Diviſion, all other Logarithms belonging to this Syſtem may be found moſt conveniently and expeditiouſly : particular. ly fince = Xx —=+ or 2, and & = = 3, and * x = H;, and 4X += = 10: you'll find by Addition of the Logarithms formerly found, agreeably to what was ſaid in the preceding Article, the Logarithms of the Numbers 2, 3, 5, 10; rather of their Ratios to Unity, to be 0.693147 1805,&c.1 6980 122886, Sc. 1. 6094379124, Cc. and 2.302 58 509299, &c. And fince the Logarithm of 10, or of the decuple Ratio in the common or Briggs Tables is i, hence you ſee what Proportion the Logarithm or Meaſure of the decuple Ratio in the hyperbolical Logarithms, bears to the Lo- garithm of that ſame Ratio in the common Syſtem of Logarithms, uiz that of 2.302 58509299, Sc. to 1: and the fame is the Proportion that the hyperbolical Logarithm of any other Ratio bears to the tabu. lar — 46 m of that ſame Ratio: whence the hyperbolical Logarithms being found, in the Manner juſt now ſhewn, the tabular Logarithm may thence be-found, and contrarily : more of which by and by. 342. Schol. As there may be an infinite Variety of different Hy. may” hence it appears that there may be an infinite Variety of dit. erent Syſtems or Scales of Logarithms, or Meaſures of Ratios : and what Proportion the Logarithms or Meaſures of the fame Ratio, in the different Syſtems, bear to each other, will appear by what follows, FXOF. 343. If from any two Points K and S of an Hyperbola FSK (ſe the foregoing Fig. p. 246.) the Semi-diameters KA, SA be drawn, and the two Ordinates KI, SR, as before, the hyperbolical Sector KAS is equal to the Area KIRS; and conſequently, is the Logarithm or Mex ſure of the Ratio of AI to AR, or RS to IK, in that particular Syſtem, For the two Triangles ARS, AIK are qual, ſince the Angles at R and I are equal, and the Sides about theſe equal Angles reciprocally proportional : wherefore, if from the Area ASKIA, theſe Triangles be ſubtracted ſeparately, there remains KIRS = KAS. Wherefare allo KAS is the Logarithm or Meaſure of the Ratio of AI to AR, or R to IK X. PK OP. 344- Let aC be an Hyperbola deſcrib'd with the Centre A, ſemi- tranſverſe Aa, and Aſſymptotes AD, AE: through any Point of the ; | Hypetbola . — The Ruadrature of Cunvzs explained. 251 Hyperbola C draw the Ordinate CBE to the tranſyerſe Axis, cutting it in B, the Curve in C and F, and the Aſſymptotes in D and E; draw likewiſe CLK an Ordinate to the conjugate AK, meeting the i Aſſymptote AD in L; draw | bo alſo the Semi-diameter AC: let MG be a Tangent at a K meeting AC, AD in M and Shs G: and through @ and C TENG day oH, CI parallel to the — Aſſymptote AE, and meet- N N ing the other Aſſymptote A) A 1 in H and I: then I fay the | 2 Ares aHIC, or Sector ACa, | is the Meaſure of the Ratio | of FD to aG or aG to CD; . or KC + KL to Aa or Aa F to KC — KL; or BD + BC to /BD#? — BCY or 48 | PLD, + M to JG — aMr; or KC+KL to VR — KLF. For 4 from parallel Lines, aH: CI : : aG : CD. Again, from the Pro- "2 perty of the Hyperbola, aG : CD:: CE DF: 2G; Likewiſe, from fimilar Triangles AaH, LCI, it is aH: CI :: Az: LC; and from the Property of the Hyperbola Aa: LC :: KC + KL: Ag. Since then all theſe Ratios are equal to the Ratio of 4H to CI, which the Area HIC or Sector AC# is the Meaſure of: hence it appears that HIC or Aa is the Meaſure of any of theſe Ratios, which have been men- tioned, in the Syſtem belonging to the given Hyperbola. And as to the other Ratios mentioned in the Prop. viz. theſe of BD BC to BD! — BC? &c. ſince it is BD ＋ BC, 2G, and BD — BC, propor- tional; hence BD + BC FD: 4G :: /BD + BC: /BD — BC ::/ BD ＋ BC: /BD# — BC? : hence it appears that the Ratio of BD - BC to /BD#s — BC? is the fame with the Ratio of FD to aG. And by the ſame way it may be ſhewn that the Ratio of KC + KL to RO RL is the ſame with that of KC + KL to Aa: Moreover it is BD: BC::aG: aM, therefore it follows that BD -+ BC: D- BV:: aG + aM : /aG! — aN. Whence it follows, that HIC or AgC is likewiſe the Meaſure of the Ratio of BD + BC to D LAE! C K k 2 y/ BD! — BC? The Quadrature of CVnvVxs explained. BD D or + M to V= N, or of KC A KIL. 1 RO — KLY in the ſame Syſtem. | 345. Schol. It would be the fame thing if Aa and AK were any two conjugate Diameters, for the Demonſtration is the ſame, as when they are the two Axes, PROP. 346. Let al and BC be * two Hyper. bolas deſcribed from a common Centre A, with the ſemi-tranf. verſe Axes Aa, AB, and Aſſymptotes AF, AD reſpectively: draw - | — MN, BE Tangents to A H N P no? the Curves at g and B, meeting their Aſſymptotes in N and E: then I ſay the Logarithm, or Meaſure of any Ratio, in the Syſtem belonging to the Hyperbola al; is to the Logarithm, or Meaſure of the ſame Ratio, in the Syſtem be- longing to the Hyperbola BC, as the Triangle AaN to the Triangle For with the Center A, principal Vertex a, and Aſſymptote AD, deſcribe another Hyperbola aK, which therefore is ſimilar to the Hy- perbola BC. From any Point O in the Axis produced, draw t Ordinate OP, cutting the Aſſymptotes AF, AD, in P and T, and the Hyperbola's al, K in Q and M: join AQ, AM, interſecting aN in ai 5: and let AM, produced if need be, meet the Hyper- bola BC in the Point C: through C, draw the Ordinate RCS to the Axis, meeting it in R, and the Aſſymptote AD in 8: moreover let the Aſſymptote AD and Tangent aN interſect each other in the Point c. Then becauſe the Hyperbolas BC and aK are ſimilar, their ſimilar Sectors ABC, Aa M are to one another, as the Triangle ABE to the Triangle Aac (as is demonſtrated in the Lemma annexed.) Again, Sector Aa M: Sector Aa Q:: AAac : AAN (as is there demonſtrated alſo) therefore ex gu Sector ABC : Sector AaQ :: AABE : AAN. Now it appears, by the Lemma, that AC: AM:: AB: Aa :: BE: ac; but AC: AM:: CS: MT, therefore CS: MT :: BE: ac ; or by Alternation, CS: BE:: MT : ac. Again, The Quadrature of Cuxvxs explained. in, in comparing the two Hyperbolas aK, al together, you'll the Property of the Hyperbola, OT! — OMr : OP! — OO: (act: aN! ::) OTY : OP?; and by Alternation, Inverſion, and Diviſion, OT? : OM? :: OPT: OO; or OT: OM :: OP: OQ; by Diviſion, OT: OP:: MT: QP; but OT: OP:: ac: aN, whence Mr: OP:: ac: aN, or MT: ac:: Q: aN; but it was ſhewn at the End of the laſt Paragraph, that MT: ac : : CS: BE, therefore U: N:: CS: BE: wheretore, by the laſt Prop. the Sectors Aa reſ ABC are Logarithms or Meaſures of the ſame Ratio, in their pective Syſtems, belonging to the Hyperbolas al and BC : but, as has been ſaid, and is demonſtrated in the Lemma ſubjoined, theſe Sectors are as the Triangles AaN, Aac: therefore the Logarithms or Meaſures of the ſame Ratio, in the different Syſtems, are as the Tri- angles AaN, Aac. Q. E. D. | LEMMA. 247. It was faid that Sector ABC: Sector AaM :: AABE : AAac; and that Sector Aa M: Sector AaQ_:: AAac : AAaN; and conſequent- ly, by Equality, that Sector ABC: Sector Aa Q:: AABE : AAN. Which things appear thus. pal Becauſe the tranſverſe and conjugate Axes of the two Hyperbolas BC and aK are in the fame Ratio, therefore, from the Property of the Ordinates, AR! — AB? : AOT: — Ag? :: (RO: OM?r::) AR? ; AO:; or by Diviſion, AB“: Aa? :: AR? : AO:: AC: AM”; but F. ABC: F. AaM :: AC? : AM“, fince this is the Ratio of their naſcent Augments, and ACs : AM? :: AB: Ag! :: AABE.: AAac; therefore 253 F.ABC : F.AaM :: AABE : AAac, which is a given or conſtant Ra- tio, hence it follows that Sector ABC: Sector AaM * :: ABE: AAac:'® Art. 403 which was the firſt thing to be ſhewn. of this Rx- Again it is demonſtrated, as in the Prop. itſelf, that OM: OQ : :. plication. ac:aN :: AAac : AAaN, which is a given Ratio; and that being the Ratio of the Fluxions of the hyperbolical Areas, or half Segments OM, aOQ, hence theſe Areas themſelves are in the ſame Ratio: but the Triangles AOM, AOQ are in that Ratio, conſequently the hy- perbolical Sectors AaM, AaQ, which are their Differences, are in the lame Ratio, viz. of the Triangles Aac, AaN. By comparing of which with the Proportion at the End of the laſt Paragraph, viz. Sector ABC: Sector AaM :: AABE : AAac, it is, by Equality, Sector ABC: Sector AaQ ::: AABE : AAac, Q. E. D. Ro 9 348. Def. The Triangle Aa N is called by the learned and inge- nious Mr. Cotes, the Modulus of the Syſtem, belonging to the H 1 TY | Ola: 254 The Quadrature of Curves explained. bola al; and the like in other Hyperbolas. It is the ſame thing to take the Parallelogram AHG, inſcribed betwixt the Hyperbola, and it's Aſſymptotes AF, AG; for AN = 2AH : yea, and although Az be not the Axis, but any other Diameter, the Triangle Aa N and Pa. rallelogram AHaG are always equal to one anotlier, and to the Pa. rallelogram AHG inſcribed at the Vertex Principalis: therefore any of them will be the Modulus. Whence it follows that the Logarithm or Meaſure of any Ratio, in one Syſtem, or Hyperbola, is to the Logarithm or Meaſure of the ſame Ratio in another Syſtem; as the Modulus of the firſt, to the Modulus of the other. * Of ths 349. Wherefore, by the Help of what was demonſtrated Art. 3418, Explice- the Modulus of the common Syſtem of Logarithms, is eaſily nd for ſuppoſing Aal to be an equilateral Hyperbola, in which AH! or Aa—+/2 —aN, the Modulus is 1, and the Meaſure of the decu- ple Ratio, as was there ſhewn, is 2. 302 585092994, &c. and the Lo- garithm of the Number 10, f. e. of the decuple Ratio in Briggrs Logarithms, is 1: therefore a third Porportional to 2.30258, c. and 7, that is — — — = 0.434294481903, &c. is the Modulus of the common Syſtem of Logarithms. 350. From what has been ſaid, it likewiſe appears, that the hy- * Of this perbolical Logarithm of any Ratio being found by Art. 341 “, in nere which the Modulus is 1, that Logarithm multiplied by the Modulus of any other Syſtem, will give the a of that ſame Ratio, in that other Syſtem. Moreover, that, if the tabular Logarithm of any given Ratio be called /; and M denote. the Modulus of any other Syſtem, and in the Meaſure of the given Ratio in that Syſtem, then it 43420, Kc. M x 7 | will be M = =: and n 5.477429, E. or M x1 x 2.302 58 og, Sc. ſince the Meaſure of any given Ratio is always as the Modulus of the Syſtem to which it belongs. 351. Def. The Ratio, of which the Modulus in any Syſtem, is the Meaſure, is called, by the ſame ingenious Author, the Ratio Mo- dularis. And it is evident, that this Ratio is the ſame in every Sy- ſtem: for the Meaſure of any given Ratio is as the Modulus of the *Art 345. Syſtem *, Therefore, if in the forementioned equilateral Hyperbola, _ you find the Ratio whoſe Meaſure is 1, that ſhall be the Ratio Mo- * Of this dularis. In order to which (ſee Fig. Art. 337 *. p. 246.) if you put _— AD = 1, and DE = x = De, then by Art. 341 + you have the + Of this Logarithm or Meaſure of the Ratio of AE to Ae = 2x + £xi + Explica- 2x5 + 2x?, &c. which Series expreſſes the Value of the Area EGge. tion. Wh ere- — ww. 1e JAMA. The Quadrature of Curves explained. Wherefore, if you put that Series equal to Unity, and by the Method of Reverſion of Series (taught by our Author ne the Value of 'x — DE or De, by a converging Series, and add that Value of x to AD= 1, and ſubtra& it from AD; the firſt will give AE the An- tecedent, and the ſecond Ae the Conſequent of the Ratio Modularis, it's Meaſure being 1, which is here the Modulus. You might find the fame Ratio by putting x — zu + 4x3 — 4x4, Sc. I; or x + 2x* + 333 + 4x+, Cc. Iz and the Value of x, reſulting from the firſt added to AD = 1, will give the Antecedent of the Ratio Modularis to AD or 1, the Confequent ; and the Value of x reſult- ing from the other, ſubtracted from 1, gives the Conſequent of that Ratio, to the Antecedent 1: as appears from Art. 341, and what » of this hath been faid : but the former Series converging twice as faſt as theſe Explica- two laſt, is to be preferred. By which means it will be found, that the Ratio Modularis is that of 2.71828181828459, Sc. to 1, or 1 to 0.367879441171, &c, the firſt of which Numbers has 0.43429448, Cc. it's Log. in the common Tables. | 352. And here by the by you ſee the Solution of this Problem, From the Logarithm of a Number or Ratio given, to find the Number or Ratio: which is the Converſe of the Prob. treated of Art. 341 *. * Of this 353: According to the Notation made uſe of by Mr. Cotes, let ore M tion. 5 denote the Meaſure of the Ratio of A to C, in the Syſtem whoſe Modulus is M; A being the Antecedent, and C the Conſequent: then if n denote any Number, integral or fractional, you ſhall have M * = M 4 A MIA 14 A — and M= ==. For M|= : M 2 :: 1:2 * and Art. 348. c * Oe C + os. : of this kx. plication. "M 8 nM 8 n: 1% Therefore ex £quo * 2 : AM A. 11. 4337 C = C 5 * . . q Cc - N * * 121. of this Ex- And the ſame is the Reaſoning in the other Caſe. | On 354. Hence it follows (ſee Fig. Art. 344. p. 251.) that the Area * of this eHICs, or Sector AC = A K or A IB TBS: likewiſe tion. MHlca, or AoC = AG | FIR. For EC: 48: CD and alſo KC KL: Aa: KC - KL ==, from the Property of the Hy- perbola : hence the thing is evident by comparing the laſt Article with Art. 348 *. * Of this: Explica- See Analyſis by means of Equations infinite in Number of Terms, Sect. Invention of the os og Baſe from the Area given, likewife our Author's Letter to Mr, Oldenburg, 24 O06. 1676. I 355. Schol. 256 De Nuadrature of Curves explained. +. 355. Schol. What is principally to be remarked. here, is, that means of a logarithmical Canon, 1 Space may be found. For let aGDS be any conical Hyperbola, deſcribed with the Center A tranſverſe and conjugate Axes Ka, PQ; and Aſſymptote AC. Draw the Ordinates DB, DO, meeting the Axes in B and O, and the Af. any other Point G the Ordinates Ga, D _ GF, meeting the tranſverſe and con. | — jugate Axes in the Points à and P: F at a draw the Tangent 4H, meet. | ha i | ing the Aſſymptote in H: join AG ic N AD, KD; and let DT? be a Tan- Q gent at D, meeting the tranſyerſ Axis in T, and Conjugate in . *. as Then it appears from what has been Aid, that the hyperbolical Spaces ADGa, aGDB, GDB, 400, 40 DT, aGDK, AODGa, FODG, &c. may be found by logarithmi- cal Canon. For the Sector ADGa is the Meaſure of the Ratio of Of this aH to DC, as it is variouſly expreſſed Art. 344 *, in the Syſtem whoſe — Modulus is the Triangle Aa H, and therefore if the Logatithm of that Ratio be taken from the logarithmical Canon, the Modulus of the lo- garithmical Canon is the Logarithm of the Ratio of aH to DC; 2 Art. 350. the Triangle Aa H is to the Sector ADGa ®, which therefore is found. ofthis Ex- The ſame way is the Sector AG found. And therefore, fince al pt” the other hyperbolical Areas are made up of theſe and known right- lined Figures, they are found. 356. Wherefore, if the Lines belonging to the hyperbolical Areas, be expreſſed by Numbers, theſe Areas are moſt conveniently found by a common Table of Logarithms : which is a Syſtem of the Mea- fures of Ratios expreſſed by Numbers (for there may be a great Va- riety of Syſtems, in which theſe Meaſures may be expreſſed by any other Kind of Quantities). Thus in this Figure, ſuppoſe Ka —=1. aH — 0,4 . aB o, 5: ſo that the Triangle AdH = 0.1, the Mo- dulus of the Syſtem. And becauſe Aa o. 5: aH—0.4 :: AB=1: BC, therefore BC — 0.8 : again, ſince aH! —= BC! — BDY, there. — Wh fore BD = (VBC! — aH! — 0.64 — 0.16 =) 0.6928203, &c. and therefore DC = (BC — BD = 0.8 — 0.6928203 =) o. 1071796, Sc. Wherefore the Sector A2GD = o.1 x 2.302 58 50g x Log. Ratio of 0.4 to o. 1071796. Now the Logarithm of any Ratio taken from a Table of Logarithms, is found by ſubtracting the Logarithm of the Conſequent from the Logarithm of the Antecedent : for if N and ' denote * 0 b aa «a. l4 VA au aaa _— % ſymptote in C and I: likewiſe from - S The. Ruadrature of Co RVS explained. denote. any two Numbers, then Ny: :: N: 141: 2; but the Lo- ihm of N to 1, is that which is called the Logarithm of N; and Logarithm of 1 to » is the oppoſite of the Logarithm of u to 1, that is — Log. u: ſo that the Logarithm of N to u, is Log, . Therefore Sector AGD = o. 1 x 2.30258509 x gubtract this from AABD = 2 = 0.346410, &c. remains 0.214714 — = DB. Subtract the ſame Seftor from AADa — Ae x UB . 17320 fo, remains 0.041 509 — == Segment aGDa. Again, to the Sector AaGD add that AADa = AADK, the Sum is 0.304901'== DK. Then for the Space aGDT, from the Pro- perty of the Tangent, you have AB=1 : Aa =o.5:: Aa: AT 0.25. | Therefore AADT => >= 0.0866020 ; which ſubtract- 2 ing from the Sector AaGD = 0.13 1696, remains 0.04 5094 = aGDT. Farther to the Seftor AGD add AAOD = 2 — 0.346410, the Sum is 0.478106 = Area AODGa, And it is abundantly evident, how, by the ſame Method, the Sector aGA may be found, and from thence the Areas aGa and AF Ga: which being ſubtracted from 328DB and AODGa, you ſhall thereby find aG DB and FGGDO. Which were the things I propoſed to ſhew. 357. Hence it appears that the Areas of all thoſe Curves, which ue capable of being compared with the Hyperbola, may be found, by the Meaſures of Ratios, or by the Help of a Table of Logarithms : and therefore alſo, all ſach Fluents of any kind, whoſe Fluxions are analogous to the Fluxions of theſe Curves. More of which afterwards. . PROP. 8. Every elliptical Area may be reduced to a circular Area: and fo 2 Sand by meats of the trigonometrical Canon, and the known Proportion of the Radius of a Circle to the Circumference. Let 488 be any given Ellipſe, deſcribed with the Center A, tranſ- verſe and conjugate Axes Ka, PH : let AD be any Radius of the El- lipſe and 2GDA a Sector adjacent to the tranſverſe Axe, and DPA a Stor adjacent to the ſhorter Axe: with the Radius Az, deſcribe the antal Arch aQR, meeting AP produced, in R: through D draw he Ordinate DB, meeting the tranſverſe Axis in B, and produce it ll it cut the Circle in Q. join AQ. Ifay N Sector _ circular 2 2 . 0.4 *Art.340, Log. 0.1071796 = 0.2302585, &c. x 1.6020600 — 1.0301121 ofthis Ex, —0.2302585 x 0.57 19479 = 0.13 1696, &c, = Sector AGD nearly. 9 | | 1 258 Ne Quadratur: of Cunyv us explained. circular Se&or QA BD B ' AP As 3 and "therefore Set 1 10 an r c Nee | 1 21 a Fr ; 7 * aQ, t 02 8 to ili ig F 1 Mogg elt ef And Sector BPA * * * 4 100 on 7” | ' 45 80 e a 11 8 = . : p . 7 1 4 N d, , 18 Q * o. 5D ,3 07 SO 8.0 rag! 350 2: AP: N 30 1102 OO) AR, therefore 26D and. ee: AP AR, fine 2 tteix Fluxions are in that given A2 but AABD: ul; C4 08880 0 = 1. 4 'H 11 2 . =4AP. x ot And une i follows that C, rA AT nen 01 19(111u7 dnpbies x PROF 54.0 7 1259. aa the Radius AP (ce 8 1 l deſcribe the quadrantabArch Per, meeting the longer Axe in 7, and the Or- dinate DB, belonging to the ſhorter Axe, in the Point the Sector DPA = Ax Pg. and the Sector 0A e II js demon- ſtrated the ſame way as the preceding 360. Since it is 5D : 159 (Ae *. 0 BQ.: BD. A by [Diviſion D = AB: D:: DB: Q. pepe it appears that the Points A, 4, Q lye in the ſame right Line: ſo that the circular gectors RQA a Pe; aQA and rgA; and Arches RQ and Pg, aQ and 79, in thi and the receding opoſition, are ſmilar. Moreover, if the Tangent be drawn at the Point =” and AgQ 1 noduced till. it meet jt in kt th ae Ao for i Rai, 40 for its "Tangent and AC for 1 it's Secant : whic 1 1 rtional to and AQ or Ag. And it is evident a ike. Cor nſtrycion may Made for the Circle Pgr ;, and that the Arghes R uch | ius, reden 2 Secant af that. A5, en as the theſe Arches. he! 361. Hence it appears that as the byperbolieal Sectors are Ma- faxes of. Ratios, ſo the elliptical Sectors are Meaſures of Angles : thus the; elliptical Sector 2GDA is the Meaſure, of the Angle AQ; for fince aGDA — AP x aQ, it appears that aGDA increaſes m min | + YO 2» * N 516 15 ?! {1436 91} tg, 4+ . Quadrature of CUunvHE explained. miniſhes in;the ſame Ratio the Arch ode chat is in the ſame, Ratio the Angle n But as the Meaſure, of any given Ratio is not ſufficiently. is „ unleſs ou know to what Syſt ſte it belon which may known from Modulus (s 4 xplained at "Fall Length) ſo neither can t e Mealufe of a given Angle be known, un- Therefore it is neceſſary to conſider the Meaſure of ſome, certain de- terminate Angle, as a Modulus, to which the Meaſures of all other An- gles in the Syſtem,” may be referred and compared, and thereby de- termined. Now there may be conceived various Kinds of Meaſures of Angles : Arches of Circles, the angular Point being at the Center, make Gn Syſtera of Meaſures: circular Sectors make another. fates ; but b6th increaſe and decreaſe in the fame Ratio wi the An- leſs you know to what 'Syſtem of Meaſtres. of Anples it belongs. | The Meaſures in the firſt Caſe being Lines, in the other 6510p t 239 gles “. There may be different Modulus's: but the moſt convenient * Eucl. in the Syſtems of Arches, upon ſeveral Accounts, particularly becauſe 5 . = of the Analogy betwixt the Meaſures of Ratios and Angles, and the x; 5. natural Paſſage from the one to the other, is the Radius. Which be- eſtabliſhed for the Modulus in this Kind, the Half of the Square f the Radius will be the Modulus in the Sy ſtems: of circular Sectors: gem — Radius bears the ſame Ratio to an Arch, which ſubtends any „as half the Square of the Radius bears to the circular ch has the — Arch for it's Baſe; and is the Meaſure of and Notion of a Modulus. 362. Hence it — that in a Syſtem of elliptical Sector, con- Gdered as the Meaſures of Angles, the Modulus is half the Rectangle contained under the ſemitranſverſe and ſemiconjugate Axes, i. e. (in the foregoing Fig.) if you ſuppoſe the right Line Pa joined, the Tri- angle AaP : which is the ſame with the Modulus of the Syſtem of Meaſures 'of Ratios, 0. K. to an er. having the ſame Axes with the Ellipſe. For =: :: DA: 20A; which two laſt are the Meaſures of the fame Angle aAQ in the reſpective Syſtems: and therefore ſince zs che Modulus of the one Syſtem, e Hill be the Modulus of ip other Syſtem. 36 3. From what has been ſaid it appears, that, in a given Ellipſe aP, whoſe tranſverſe and conjugate Axes are known, if moreover the Ratio of Aa to aC or to AC, which are the Tangent and Secant 'of the Angle aAQ, in the Circle whoſe Radius is Aa, be given, the Arch l | L1 2 | "=" _ 155 9 given Angle in the Syſtem of circular Sectors: which i is the 260 Ne Quadrature of Curve s explained. 20, and elliptical Sector DA may be found. Now the Ratio of Aa to aC, or, which is the fame, of AB to BQ is given, when Ap is given, for BQ= V — A: fo that the Sector aGDA is the Meaſure of an Angle whereof the Radius, Tangent and Secant ar, Aa, ac and AC; or AB, BQ and Aa, the Modulus being —*4 And it is evident from Ab given, by the like means the other elſy. tical Sector PDA is found, by ſuppoſing a Tangent to the Circle Pg drawn at the Point P, and 177. till it meet For it is d Meaſure of an Angle, whoſe Radius, Tangent, Secant are as A, by and AP, the Modulus being the fame as before, vis, = * 64. Hence it appears, that, when the Lengths of the given Ling 8 by Numbers, the Number of Degrees, Minutes, &c. 2 pr recon 49-7Yy res found, with great eaſe, by means of the trigonometrical ng tical Sectors will be found, from the known Ratio of the Radius of Circle to it's Circumference, whereby the Length of an Arch of a known Number of Minutes, &c. with reſpect to the t this of the Radius is determin'd ; thus, in the preceding Fig. p. 258 #, ca gGDA= AP xaQ®, and PDA = ZAP XK RQ. And 4.268. in this Caſe, the Areas of any other elliptical Spaces may be found, .cchisBx- That is, if you join KD, draw the Tangent DT meeting the Axis in plication. T, and the Ordinate Ga, you may find the Areas of the ellipti Spaces aGDB, aDGa, aGDT, aGDK, AbDGa, PSD, ABDP, «GDB, Sc. after N n formerly, in the Caſe of the Hyperbola : ſince ellipti ces are made up of ellipt Sectors and known right-lined Figu 2 Addition or Lim in. Ex. Let us ſuppoſe Ka=1, HP =0.8, AB o. 25. Then AB So. 25: AQr o. :: Rad.: Sec. . aA Q; whence Sec. L AQ 2 X Rad. wherefore add the Log. of 2 to the logarithmical Rad. 1. . o. 30 10300 to 10.0000000, and the Sum 10. 3e 10300 will be found in the Table of artificial Secants, oppoſite to an Angle of 60: therefore the Arch is an Arch of 609, i. e. 4 of the whole Circumference, But the Circumference of a Circle is to the Rad. as 6.283 18 5307179, &c. to 1: therefore that Number divided by 6, quotes 1.047197 551196, &c. the Length of + of the Periphery of a Circle having it's Radius 1: fince it is Ag=4, if you divide the former Number by 2, you'l 2 When in this Place, or any other, I mention the Radius of an Angle, in conjunction with the T AB zus, T and Secant of 2 circular Arch which meaſares that Angie. 0 ha ve tbe. Au i,,,ʒ of, Curvy 8 explained. 0523598775598 the Length of oQ; which multiplied by 2 N 9.4047 197551196 = aGDA nearly. N \ "Again K 1: HP/=0.64 :: KB x Ba=0.1875 : Dv; there- fore BD = +/0.64 X ©. 175 o. 346410161513, &c. and therefore the Triangle ABD = _ = o. o4330 1270189, which ſubtraQted om DA = 0104719755119, leaves o. o6 14184849 = «GDB 257 And we w_— not Tant particularly, in 1 how all the other elliptical Areas mentioned, are deduced after the fame Manner: after what has been done already, in the like Caſe of the hypetboli- appears, that the Arcas of all ſuch Curves as are capable of being com- — ot the Meaſures of Angles, i. e, b in a linear Deſcription: ( 7 metica] P. R 1 K juga draw the right Line AcC, C, c, the Lines CB, ch, perpen- Marta * A, os tha n ny hm a, 1 angent apg, cutting AC, AE in g and p. yperholical tor AC = EEE and the elliptical Sector Aur = r by the lan of which two Notation, 1 mean the Meaſure of an Angle, whoſe Radius, Tangent, and Sccant are AP, ay, and MNT, to the Modulus ; where it muſt be obſerved that of the two Quantities as AP. ag . the firſt always defigns the Radius 262 The Bittidratire bf Cu vis anptinad; ard the ſecond the Tangent, For if with the Center A; and Aa, the eqyilatergl'Hyperbold 209, drid'quadrapfat Arch AR be 2 feribed 3 ABymiptote ANF to the Hy 'aD be drawn: © and 1 B be produced till it meet; the H R Din D, and the Al. mptqtes „AF, in E. and F; and bc be prodpeed till it mer th 745 © in the int and moreover aqp, Pt if need be, till i er the Ally ptote AF in N;: then, becauſe it is b: 12 :: (AP— Ts 1 . : BF: :) BC: : BD * 1 | ofthis Ex- D, lye in che fame right Line, Let that right Line AdD be joined, and interſe&t aN, in the Point o. ae 1 N me byperbolical Sector AaC —= W yt" k for AP ; 01 AP A= as. As Arabs = c (Again the elliptical Sector 13 I —_ — — = (becauſe N:; a: BF: : BD':: BE * 'BC:: 2 70 LAP. 248 0 2 i | | "I = as was to be ſbevp- 10 1 13150 a yr Cor: 1. Let Pp- be joined and produced indefininly: towards r, and AcC produc'd, if need be, meet it in 1. then, from ſimilar Tlapgles, it is AP : aq :: Pr: Aa; therefore it follows that the 9 as r by approaching to the Point 5 n Wi _ coincide with it, the hyperbolical Sector Fad AaC becomes infinite, and ſo doth the Meaſure of the Ratio; for the Conſequent of the Ratio vaniſhes, But the, elliptical Sector Aar becomes the Meaſure of an Angle of 45, be- cauſe Ap and aq or Pr and Aa, that is the Radius and Tangent be- _ equal. And if g and 7 appear upon the oppokite e Side of p, wiz. in Q and R, the hyperbolical Seftor AC becomes . A - or == 2 — T. both impoſſible, as including” negative Squares. ann 2 12 AP. 5 ms Hah cli ben nw ba pL. tas PR. aa | vil . t N, TA 2s 2:4: es 368. Cor. INE fame 5 lical Sector A 0 2 Aa x — Bk pe | y » Sg ; and thellpiclSeforAc= or | | 'F | | LE 61 8 22 2 As And when the fan P | | Fan DW, * s Da. Ca. lk The-Quadrature-of Qunvss explanied. 263 368. Cotti 20 In this laſt Caſe; viz; when'ths Points ꝙ and r fall in und R, with-the-ſemitranſverſe. and ſamicem jugate Axes AP, Aa, ow PR a wp Hyperbola to dhe ferner n then, by the N E the hyper reer r APK. = — — 75 e [a= ad theelliptical Sector AA AAP 4 917: u ad . r Xi. abogolsng af DT . Dorle . 1 5 9 I L Arif 2 0 al. % brit} KI 0 3690 Sd dec f t it „Me that, .t the 1012 eee eee | e . And APK = AKA Age =; , And LIE 3 Ul! U Dein A n e - oily av $19 i N. hw) 7 99 glodioqrt ot vdingtob boy 1 191216168 Jo, Schol. — the Meaſure of any Ratio or Angle and 2 7 „ W oY 2825 51. the + a 5 a e tn | e Expreſſions Frogs ind e, e f ; : V ITE plication. Crt End rien eds in which laſt two Expreſſions,” AP. on; fis ;confideredoas the-Methh- lus, and the Meaſure of the Ratio, or FIN to that Modulus, is multiplied-by,*As ;,a9d © in other like Caſed. , Moreover! it may be obſerved, 457.5 Bd AP. ag. RN aq? may be ankiplied ar divided þy , withaut pheriog the Ne off RG or Angle SH FA N45 71. ty am inſiſt e longer upon this Affair of the Meaſures of atio and 0 it e with | hyperbotical and elbptical Areas, that 1 ght ES ſhew the Connection there is betwixt our Author Mabel dating curvilinear Aregs, ahd conſtquenty-other analogous Fhiehits,' to hyperbolical, elli tical and circular Areas; and the Method of the our! ME) Co £3 y Twhitl tHe red bb ſuch he Fon Ne with many be Men, to the Meaſures of-Ratios and An- publiſhed -and-emtirged by the - med Dr.-Smzth. So that what have ſaid, 41 an In- trodudtion-t young eee etrieizn; reading and 1 ing Mr. Coters Method, in the Book juſt now mentioned. For a fur- | ” ther Commentary upon which, the Reader may oonſult a Performanec entitled Epiſtola ad Amicum de Catefii Inventis. | 372. And © De Nuadrature of Cunvxs explained. 372. And now I ſhall illuſtrate this Connection of the two dif. rent Methods by an Example. Take the Curve belonging to Form third Species firſt of Sir aac Newton's ſecond Table, defined by thi Equation Zi =. the Quadrature of which was demonſtrat ed, and explained at full Length already, according to our Author, Art. 256 Method *. The analogous Fluxion and Fluent. are found in Form elek- third of Mr. Caters firſt Table, viz. dür Y Þ fe", and dp —_ where the Values of P. R. T. s. ae PN PN 264 *, Of this A Art. 256 *, if (ſee Fig. 2. Tab, 2 — Babe Nl, and half tranſverſe Axis AP = M and half 3 YL, you deſcribe the Hyperbola PGDS ; and, upon the conjugate Axis AN, you take AB =(x=) -- e, draw the Ordinat BD, ; Ind this Tangent DE aneting A in T: then the Area of the Curve ſought is = x TDB — APD DB. Join AD, and + the conju. git Axis is U., becher if 0 AO Ter. OD=x, and X AP SV, you find a fourth Proportional, viz. Se (Wi ſting 75 for u) H fn then wenn hay by what fas ha oh 366. demonſtrated , che Sector APGD = 5 = (by chi, Ex. _— 275 - 7 YE : to which if you ch Th ADB= WEE = DEE, hn FE + t., KO 2% ee * — ADB. But the Triangle TDB 3 to be equal to 2 22 5 EZ TIE: — — . =: therefore TDB—APDB D LE - = Ap 228 92 * Le + fe 72 At ve + fe —=Lp—[R v fe" 175 9 277 \ E The Nadraturs of Cunvrs — 265 727 ; wherefore x TDB—APDB= =P — R : u it is in Mr. Cotes's Table. 373. But if F be negative, fo that it be N= n then by what was ſhewn at Art. 277 ®, (ſee Fig. 3. Tab. 2. p. 25.) if, with * Of the Center A, half Tranſverſe A2 = A, and half eats = , tion. you deſcribe the Hyperbola aGDS; take AB= (x =) —, draw the Ordinate BD, the Line AD, > ap Tangent DT, then ho 4. of the Curve ſought is equal to — — Sx aGDT. But, by what has been de- monſtrated Art. 366 *, if to AB BD = T, and * Of this A 422 con, 4 A. C, you find a fourth Proportional, via. AA, then you ſhall have the hyperbolical Sector 4 2 * * 2. 1 = Le LTZ: : from which ſubtract the FE CaO — A — 2 Xx c Triangle ADT = ( 2 32 1.0.2 pn Ink _ += x GH =) Le + 7z", there remains aGDT — MULES g. HR — gr, ve E xaGDT = 2d = =dR E. the fame as in Mr. Cotess Table, 374. Finally, if it be e + fz" —=y : then, by what was de- monſtrated Art. 278 * (ſee Fig. 4. Tab. 2. p. 25.) if with the Center · of this A, Semitranſyerſe Aa = A, L, and Semiconjugate AP = If (che noblen tion. lame as in the preceding Figure) the Ellipſe aGDS be deſcribed ; the Reading AD, and Tangent DT be drawn; and you take AB (x= 7 the Area of the Curve ſought is rer. But (by what has been demonſtrated Art. 366 *.) o AB —=x, BD -e, Of this Mm and Explica- tion. 266 The Quadrature of Curves explained. VE and A = A. C, you find a fourth Proportional, via. — * | | Fe A 2 T4 — the elliptical Sector aGDA = ＋＋ 2 . . wr ve which being ſubtradted from the Triangle AD 3 TD A = N — — (= - = — - =) Ly ee, there remains Ly —etf2 Cp. LN e -R 24P — RN: the ſame as in Mr. Cers Table. 375. Schol. It may be obſerved that the ſame Notation, making allowance for the Difference of the Signs, ſerves all the Cafes, in thi Example: and the ſame happens in all others: when the Fluent are expreſſed by the Meaſures of Ratios, and Angles. In the lat Caſe, where the Area is expreſſed by the elliptical Area ; and the Fluent by the Meaſure of an Angle, we might have taken AB for the Radius, and MA — AP?, for the Tangent of the Angle to the ſame . A rd ———= _ 77 LAST, for the Value of the Sector CDA, as be fore. It may likewiſe be obſerved that the two Hyperbolas are con- jugate to one another, the tranſverſe Axis of the one being the conju- gate Axis of the other; and the conjugate Axis the ſame with the tranſverſe : and the Ellipſe has the fame Axes with the Hyperbolas; ſo as to be that which is inſcribed within the conjugate Hyperbolas, as of this repreſented by the Fig. at Art. 367 ®. Moreover the Expreſſions for Explica- the Meaſures of the Ratios and Angles might have been deduced * Of this Otherwiſe, by conſidering what was ſhewn Art. 367, 368, 3699. 2 376. If the Area of the Curve whoſe Equation is LD e + fy —y; or the analogous Fluent of the Fluxion d&z—"/—e + fz", be required in a linear Deſcription by means of a Circle, it may be thus. Take ſome Quantity for linear Unity in the Conſtruction, by * x] : Wherefore 27 x aGDT = Modulus as before: and ſo it would have been Le Quadrature of CunvzG explained. let the Dimenſions of the ſeveral Terms be reduced to the juſt Num- ber, by Multiplication and | 2 Diviſion thereby: ſuppoſe * it be . Then take AP F — f, and _— r {2 ced beyond A, take | A; 8 2 — 7 and 2 E | N b $ having deſcribed upon PB and PC as Diameters, two | Semicircles; through A | draw ADE perpendicular to AP, meeting the Semi- circles in D and E: with the Radius AD deſcribe | the Quadrant DHF, draw E the Tangent DG; and with the Radius AE deſcribe the Arch EG cutting the Tangent in G, join AG: and the Area, or Fluent required is equal to = x DG — DN. 267 FT For it is evident by the Conſtruction, that AD = Ve and AE — AG, and therefore DG = e : whence it follows that the Arch DH = Ve [© Lt and DG — DHV ä de = PR ap — =dR Pr. as it ought to be *. W | 377- The Conſtruction may be made otherwiſe thus. \Suppoſi as before AP =f— 1, AC = 2», AB =e, AE = VfR = Na, and AD =we : produce EA towards L, upon which take AI a third Proportional to AE V and AP = 1, ſo is Al= ( 3 e x (ſee Table 2. Form 3. Quad. p. 25. and take Ag a third Proportional to AD=ve and AP=1, ſo is Aa 2. With the Radius Aa de- ſcribe the Quadrant aMO : through I draw IM perpendicular to Ag, cutting the Quadrant in M : draw the Radius AM, and the Tangent ML meeting Aa produced in L, then the Area of the Curve whoſe Ordinate is =o — e+ f2”, or the analogous Fluent whoſe Fluxion is 4 — - 20), —— ks iV—e+f2" is = X LM Xx AB — aM x AB. For with the Semi- a Mm 2 axcs === therefore * x DG — DH = 268 Art. 278. of this Ex- plication. The Quadrature of Cunvss explained. axes Aa, AP, deſcribe the Quadrant of the Ellipſe aNP, cutting III in the Point N; join NL, which is a Tangent to the Ellipſe at the Point N, then by what was formerly demonſtrated “, the Area or Fluent is equal to 7 X aNL ; but it is NL: ML:: (IN: IM 5 AF==1:' Ad= 75 , therefore aNL = AML x ve=LM <7 = xe = LM — M x 55; ** LM N x2: ſubſtitute thiz for aNL in the foregoing Expreſſion = X aNL, 1 for h, and AB for e, and it becomes *xLMxAB—aM x AB for the Area or Fluent required, 8 E C T. XI. Containing the Demonſtration of Prop. 1 f. 8. OUppoſing the ſame Things as are ſuppoſed by our Author a F (which ſee with the Figure belonging = it) only let us further ſuppoſe that the Areas of the Curyes ADB, AEB, AFB, Gc. are repreſented by a, G, y, &c. and conſequently that the Ordinates BE, BF, BG, &c. are F, =, 2, Ce. Then! fay, if the Areas a, Þ, y, &c. be terminated at the whole given Ab- ſciſs AC t, and at the Ordinate CI given in Pofition and infinitely produced, i. e. when they are the Areas ADIC, AEKC, AFLC, &c. and the Areas A, B, C, &c. be terminated at the ſame Abſciſs and Or- dinate, then it ſhall be ITN 19. ADIC = A. 6 2. AEKC= A — B. 3. AFLC= &. % 4A — 3*B+ 3C—D 4% AGMC= 2 -, 9. AHNC= = ee. &c. That is to ſay univerſally, if e repreſent the Area of any Curve in the Series a, G, Y, &c. at any Diſtance of Place from the firſt a; and ſuch indefinite Diſtance be called d, then, when e and A, B, C, &c. are terminated at the whole given Abſciſs AC, and infinite Ordinate CT, it ſhall be e E = The Ruadrature of Curxvzs explained. fA AB x ZAC — 2 3p + &c. —{ >. — ATP dx Ti x4 —2 &c. Where the numeral Coefficients of the Terms in the Numerator, viz. 1, — 4, dXð &c. are the ſame with thoſe which belong to the Terms of a Reſidual, ſuch as a — b, raiſed to the Power whoſe Ex- of the Power of 7 in the firſt Term is 4, in the ſecond, d — 1, in the third, 4 — 2, &c. the ſame as the Exponents of the Powers of a the firſt Member of the Reſidual, and A, B, C, &c. are Factors of the firſt, ſecond, third, &c. Terms reſpectively. And the Factors by whoſe Multiplication the Denominator is formed, diminiſh each by Unity from the firſt d, and are continued till the laſt be equal to- Unity; or which is the ſame, till their Number be equal to d. 379: To demonſtrate which, I ſhall prove that, when the Areas A, B, C, D, &c. and e, have any common Abſciſs z = AB, the Rela- tion of the Areas ſhall be expreſſed after the ſame manner, viz. thus. AA dB +4 x wy x = „ian ＋&c. 8 dx 0=1 x i= ie, ; For by Propoſition 1, the Fluxion of d—r dns & -A iB —4 x nn one MO + Kc. SO, eee d x d=—1 x d=—2 &c. 18 1 A N A 4—1 x 22 &c. A x d—1 x 2 &c. I xe "a 415 — 5 — 4—1 x d—2 &C. 4 x d—1 * d—2 &c. 2 2 * d—1 „ Lz &c. IX II d—2 &c. Gant Gow 4—1 4—2 2. i „ 85 n d x — x —— d—1 x d—2 &c. | d x d—1 x Lz &c. &c. &c. But A=zy . B = (22y =) 2A C S A. B= (22%! =) 23A, &c.. which Values of B, C, D, &c.. being put for them, in 2 269 ponent is d, 1. e. of @ — W when actually involved: the Exponent 270 N Quadrature of Cunvzs explained. in the Terms which conſtitute the ſecond Column, it become, Add 44x bt x ox Ab be. of; 7 = bee * 7 E — 4— 4—2 * 1-4 - IX x 1 + &c. where the laſt Fado 1A AN = — &c. being the numeral Coefficients of a Reſiduil raiſed to the Power whoſe Exponent is d, is therefore equal to 30. thing : conſequently all the Terms conſtituting the ſecond Column taken together are equal to nothing, Wherefore the Terms conſtitut. ing the firſt Column, make up the Fluxion of the Quantity be 41 4— 4—2 A -A. 'Bpdx e- = e Df ve. d x Ai x U &c. Which therefore is 4—2 7. ; Feed EY =" r J 30 — &c. 2 Xx — ws — | 4— 1 xd—2 &C. | Wherefore, if the firſt of theſe p__ the Area of a Curve whoſe Ab- ſciſs is 2, the ſecond will expreſs the Fluxion of that Area: and there- fore, if it be divided by 2, it will . the Value of the Ordinate of that Curve. In which laſt Expreſſion, the numeral Coefficients of the Terms in the Numerator are the ſame with thoſe of the Power of a Reſidual, whoſe Exponent is 4— 1: the Indexes of the Power of z are d—1, d—2, d— 3, &c. and A, B, C, &c. are Factor of the firſt, ſecond, third, &c. Terms reſpectively. Whence it ap- rs that the Number of Terms in this Expreſſion of the Ordinate, is one leſs than in the Expreſſion of the correſponding Area: and fur- ther, the Denominator of the laſt Expreſſion 1s equal to the Denomi- nator of the former divided by d. Moreover if d be 1; ſo that the Area of the Curve be zA — 3, the Fluxion is 2A + Az — B = (ZA + Ly — y =) SA, and conſequently the Ordinate is A, by conſidering of which things it ap- pears that in the Series 22A —2zB4+C 23A-32*B43zC-D 24A—4z3Bþ 6z2*C — 4zD+E A. 2A — B. - +» 7 2 Sc. in inf. If you take any two next adjoining Terms, the firſt of the two be- ing the Ordinate of a Curve whoſe Abſciſs is 2, the laſt of the "= N | W The Ruadrature of Curves explained. will be the Area, Or, which is the ſame, if you conſider all the Terms as Areas of Curves, having the common Abſciſs z; and ſap- ſezor K 1, then each Area will be the Fluxion, or, if you \ pleaſe, the Exponent of the Fluxion of the one immediately follow- ing. But a. . 1. 9 &c. is a Series of Areas having the ſame Abſcifs, and Relation as the former: and the firſt Area A is equal to the firſt Area a, fince they have the ſame Abſciſs z and Ordinate y: there- fore all the ſucceeding Terms of the one Series are equal to all the ſucceeding Terms of the other, each to each. Now ſuppoſe the Ab- ſci 2 to become equal to AC or f, then a, , , d, &c. become ADIC, AEKC, AFLC, &c. and therefore it will be | 20% AEKC== A — B. 3 AFLC 124 — 2tB + C . 2 . 4*. AGMC= = =. o& AHNC= = YH DART 24 &c. This was the firſt thing to be demonſtrated. 80. The ſecond thing to be demonſtrated is, that, if x =#— 2, and P. Q. R. 8, &c. be a Series of Areas having x for their common Abſciſs, and y, xy, xy, x, &c. for their Ordinates reſpectively, when the Areas P. Q. R. 8 &c. as well as A. B. C. D &c. are terminated: at the whole given Abſciſs AC , and infinite Ordinate CI, it ſhall be P — &. Q= A —B. R= A — 2B + C. S —=3A — 3B + 3iT—D. &c. in inf. Ne | Or univerfally, if 4 denote the Diſtance of Place of any Area from the firſt P, in the Series of Areas P. Q. R. S. &c. that Area ſhall: be equal to WA dB = νπ. xD, &c. the ſame Series as that mentioned in the preceding Part of the Pro- polition, only wanting the Denominator, For the Curve belonging to the Series P. Q. R. S &c. which is at 271 the Diſtance d from the firſt, has x*y for its Ordinate: but æ f -, va 272 The Quadrature of Cunvss explained. or of = , = =) tm d. + dx Dm = dui, 14-323 b+ &c. therefore xy = cy dig dx N= 2 —4* x Y + &c. Now the Ordinates , ay, 2*y, &c. belong to the Areas A, B. 0 &c. conſequently, ſince t is a given Quantity, the Area belonging w the compounded Ordinate #y — di + dx e- 4 * ee b Kc. will be HA — d#—B + Ne- dx Xx u + &c. the Abſciſs being 2 and the ſame i; WR. true, when the Abſciſs z grows into the whole given Abſciſs r. There. fore, ſince the Ordinate y at every Point of the given Abſcifs AC At, is equal to the compounded Ordinate, the Area deſcribed by the Ordinate x4 drawn along the whole Abſciſs AC, is equal to the for. mer compounded Area. Hence the ſecond Part of the Propoſition appears. * therefore by conſidering what has been demonſtrated in this and the preceding Article, the Truth of what our Author aſſerts in this Propoſition is manifeſt ; and the Corallary, thence deduced, plain. ly appears from the Propoſition itſelf, | $ECT. XII. 8 Containing an Explication of our Author's laſt Scboliun; with the Application of the preceding Doctrine to the Solution of ſome Problems. 381. HE Affair of the different Orders of Fluxions was fully explained at the Beginning of this Work in Sect. 2; and the Juſtneſs of the Conception vindicated from the Objections that have been raiſed againſt it by a late Author. In the fame Place | likewiſe ſhewed how theſe different Orders of Fluxions may be ex- pounded by the Ordinates of Curves, after ſuch manner as is men- of the tioned by our Author here in Art. 7 1—74 * : ſo that it would be to Quadr?- no Purpoſe to ſpend more Time in explaining that Matter. But what Curves. is contained in Art, 70 + muſt be illuftrated a little: and ſo much 44 the rather, that there ſeems to be ſome Miſtake in the Text. ture of Curves, | | 382, Let 382. Let ABC be a Curve deſcribed by ads the AbG& KB. Cal AB, S —2", Let BC, Ba Ca, B; Cz, Bꝗ Ca, be. be Bebtihes Poſitions of che Ordi- nate, fo that the Diſtances B Bz, 32 Bz, B3 BA, &c. be equal, any of Which Piſtances call o. Then the ſucceſſſve Values of the Abſciſs are 2, 8 , 2 +20, 4 +,39, &c : to which the ſucceſſive correſponding Values of the Or- dinate are 2%, Tol“, S Tao, 2+30Þ . | By throwing theſe Values of y ino Be: nis a, you'll, have them expreſſed thus. — e e | —_ 4 „ — —2 . & Puro NN F X Fier x | ANCE SO, Se. | 1. 2 Pag oA. oo F B R x" 208 4161 _ 1 — : 4. W o X 2500279 X— X*=2*=30348 14 you] yw=—=2 2 * 3 r o | Where the firſt Term of each Series is the flowing Quantity z" itſelf : the numeral Coefficients of the ſecond Terms are 1. 2. 3. 4, Sc; of the third Terms, 1. 4 . 9 . 16, Cc; of the fourth, 1. 8. 27. 64, &c of the fifth, 1. 16.81. 256, Gc. that is, the numeral Coefficients of the ſecond, third, fourth, &c. Terms are Series's of the firſt, ſecond, third, fourth, &c. Powers of the natural Numbers. . If you ſubtract the firſt Value of the Ordinate from the ſecond, the ſecond from the third, che third from the fourth, &c. you'll have the firſt Differences of the Ordinates BC, B2C2, B3C3, B4C4, &c. the firſt Term of which in them all is y2»—*o. Again, by ſubtracting the firſt Differences from each other, you have the Differences of the Differences, or ſecond Differences of the Ordinates, the firſt Term of ach of which is 29 X = 2%=*00 = 7X19 —12%*00, After the ſame Nen | manner, X x —25—0b, Ge. We Nalrarure of Cones ename. 273 Ordinate BC moving Art. 70: tt nie an ban A =7: and let the Curve be of that Nature A ture of Curves. 274 — the after the fame Manner as in Art. 11 *, and in the Notes upon it: ua — ture of Curves. manner, by ſubtracting the ſecond Differences from each other, you The Quadrature of Curves explained. find the third Differences of the Ordinates, the. firſt Term of which in all is GUN = X20 = Kö — 2X2%303, & likewiſe the firſt Term of the fourth Differences will be 247 x x N =zjzg Xu — IX — 2X $— 3X29; and fo on through the ſubfequent Differences in nf. as is deduced from the Differences of the Series's of Powers of the natural Number, to which the numeral Coefficients of the Terms in the ſucceſſive Value; of the Ordinate 2 correſpond. Now by conſidering theſe ſeveril Orders of Differences, it will appear that there are infinite Orders af them in every Caſe, unleſs when 3 is a poſitive Integer: in which hf Caſe the Number of Orders is the ſame with 3. And the firſt of the firſt Order is called the firſt Difference 2" the firſt of the ſecond Order is called the ſecond Difference of 22; the firſt of the third Or. der, the third Difference of 2, and ſo on. Only it may be obſerved that theſe various Orders of Differences of 2", are generally underſtood by Writers upon this Subject, as meant of them when infinitely dimi. niſhed : in which Caſe any one of the ſame Order of Differences of the Ordinates is the ſame with the firſt of that Order. And ſo it appear that the firſt Terms of the different Orders of Differences. of 2", arc equal to the ſecond Term; twiee the third Term; fix times the fourth Term; twenty-four times the fifth Term, &c. reſpectively, of the 3 . — 1 0 _ converging Series -C vox -O & = X gel NX rot, &c. = 2 +oV, | | 2 ; 1 5 Now when the Increment of z, viz. o, which denotes the Diſtan- ces of the Ordinates, is diminiſhed more and more in inf. the prime of ultimate Ratios of the firſt Terms of the Differences, to the whole Differences reſpectively, will be a Ratio of Equality, by reaſoning > .Q © @.. © 3 F rr •⁰ůãmἰìu ̃ nr Tr T7 WF * 4 + ati. wherefore, hence it appears, that, in the converging Series 2" -þ yzw'0 + nx =2—*0* + 1X x D 101-4 Sc. the firſt Term 2 D 2 being the Fluent, the ſecond, third, fourth, &c. Terms will be pro- tional to the firſt, ſecond, third, Sc. Differences of 2", it being underſtood” that the Quantities are taken in their prime or ultimate Ratio, v/z. when. theſe Differences are im their naſcent or evaneſcent State. But from what was demonſtrated in the Beginning of this Work, the Fluxion of z being uniform and conſtant, and put equal | | 4 The Qaadrature of Curves explained. to Unity, the firſt, ſecond, third, &c. Fluxions of 2» will be 22 —9, „IN., 1X15 1X — 2X23, Oc. Hence it appears, that theſe firſt, ſecond, third, &c. Fluxions are proportional to the ſe- cond, third, fourth, &c. Terms of the converging Series; and cohſe- quently to the firſt, ſecond, third, fourth, &c. ifferences of 2", when theſe Differences are in their naſcent or evaneſcent State, . . But here you muſt obſerve, that I don't mean that there is any Ra- tio ſubſiſts betwixt Fluxions, repreſented here by finite aſſignable Quan- tities, and evaneſcent Differences: but only that a Fluxion of any Order is to a Fluxion of that ſame Order, as a Difference of the like Order is to another Difference of that Order, taken in their prime Ratio, And this I take to be the —— of what our Author here advances, which ſeems not to be expreſſed with that Clearneſs and Accuracy, which he was Maſter of upon other Occaſions, For the ſecond, third, fourth, &c. Terms of the converging Series are ſaid to be equal to the firſt, ſecond, third,” Cc. Increments or Differences; which certainly is not true, Therefore Dr. Keil ( ſee Commercium Epiſt. pag. 41.) ſuppoſes that the Particle r has ſtood originally be- twixt the Word erit and Incrementum, ſo as to make the Text run, tertius Door erit ut ejus Incrementum ſecundum: and again, — quarfus O. erit ut ejus Incrementum tertium, &c. and that this Particle f by ſome Accident or another hath been left out. But even when this Correction is made, I don't think the Text clear d: becauſe the plain and natural Meaning would ſtill appear to be that the ſecond Term of the converging Series will be the firſt Increment or Difference abfolutely; and the third Term of the Series will be as the ſecond Increment or Difference abſolutely, &c. contrary to what has been ſhewn, But if we ſhall ſuppoſe that by firſt, ſecond, third, Ec. Increments or Differences of 2, Sir 1/aac Newton-meant the ſe- cond, third, fourth, Cc. Terms of the converging Series 2» ＋ yozr—* ——0021= + 3 * o-, Cc. (as he might have done) explained the Author's Meaning: and the Omiſſion, Inaccuracy, or Miſtake, no way affects the Subject. Thefe was no leſs than twenty- eight or thirty Years interven'd betwixt Sir 1/aac's writing the former Part of the Treatiſe, and this Scholium, which was annex d A®, 1 704. the Addition of cui naſcenti is ſuperfluous. However I ſuppoſe I have 273 383. And hence by the by it appears how the = e Method of Mr. Leibnitz, and the foreign Mathematicians, will agree with the Nn 2 Method 276 The. Quadrature of CUnvxS explained. Mlethad of Fluxions in it's Concluſions 3 becauſe the Ratios of the Pit ferences are the ſame with the Ratios of the Fluxions, although the Principles they reaſon upon are not ſo juſt and geometrical, | In order to underſtand the ſubſequent. Part of this, Scholium : the Application of which is of very great Extent, there are ſeveral things to be taken Notice of, and a little explain'd, with reſpect to the Na. ture of Fluents and Fluxions, and the Conſtitution. of, Equations de- fining the Relations of the one or the other: *. 384. It was obſerved in Section ſecond, that every flux ional Equa, tion muſt contain the Fluxions of two. Quantities. at leaſt, either ex. * Art.g;. preſſed or underſtood “: becauſe Fluxion or Velocity being a relative of this Ex- Term, there muſt always be the Fluxions of two fſowing Quantities IN at leaſt, in every ſuch Equation ; even as there muſt, be, at · leaſt two flowing Quantities , in every flaential Equation. Wherefore, if any fluxional Equation be propoſed, in which. the Fluxion of one, Quantity only is expreſſed, the Fluxion of ſome. other flowing Qgantity is under. ſtood, although it don't appear; by, which, all the Terms, that don't include the Fluxion of the. fixſk mentioned Quantity, muſt be ſup- ſed to be multiplied. Which other flowing Quantity is ſuppoſed to. Row uniformly, and to have it's Fluxion equal to Unity, by which means it diſappears in theſe” Terms, and muſt be ſupplied, after the Art. Manner taught in that Place “. „ 22 385. In the Doctrine of Fluxions, there are two different ways of plication. expreſſing each of the two general Problems, that contain the diref, and inuef e Method of Fluxions: which, although they may be te- duced to one manner of Expreſſion, yet that the, young Geom trician may. Gill preſerve clear and diftnét Ideas, I fili explain, a le n this Place, to make: the Way plainer to what follows. In the direct Method, an Equation is ſometimes given, which muſt. contain at leaſt two variable or flowing Quantities, and the Problem is expreſſed thus: From the given Equation expreſſing the Relation of \ the Fluents, to find un Equation” defining the Relation of the Fluxions Thus our Author expreſles it. in Prope1: and the Meaning of it was explained at full length, in the. Notes upon that Prop. which ſee. Again, the ſame Problem. is expreſſed; otherwiſe thus: Auy Flunt being given, to find it's Fluxiow. Thus the, flowing: Quantity xx be · ing. given, to find it's Fluxion,: the Fluent ax3 —4h⁰¹ 8 ab being given, to find it's Fluxion: likewiſe; the- flowing Quantity x3 — 3 ＋ 3% — 3 + ab being propoſed ta; find. ĩt 5 Fluxion: and ſo for other Fluents, including more variable Quantities than two. The Meaning of all which muſt be conceived to be this. The Flux- DOE TITLE) 10m %% The. Ruadrature of Curves explained. 277 :0ns or Velocitirs of: flowing of the vatiable Quantities, which com- poſe theſe Fluents, being ſuppoſed to be known, to find what are the Fluxions or Velocities with which the whole Fluents flow: or what Relation they bear to the Fluxion, or Fluxions of the flowing Quan- tity, or Quantities of Which they are compounded; Thus the Flux- jon of * is n; the Flaxion of ax3 — 4gabx* + 8bx* — ab, is zax*x — Salem A 16π]% of x3 — zx A. 30% = 93 ＋ ab is zue — bxxy e Z f 30 + 6 350%: the Meaning of all which is, that, when Fluxions of x and y are & and 5, the Fluxions of the Fluents mentioned, are ſuch, or bear ſuch Relation to them, as the fuxional- Expreſſions ſet down, import. Whence the two different Ways of expreſſing the ſecond genetal Problem, containing the in- 5 Method of Fluxions, may be eaſily underſtood. One of which is this : Any Equation being given, defining the Relation of the Fluxi- ons of any flowing Quantities; to find an Equation, defining © the Re- lation of. the Flurnts. The other runs thus: To find' the Fluent of” any fluxionary Expreſſjon propoſed. Which are the Converſe of the former two Problems. | | 386. Now although theſe be different Ways of —_— the two general Problems, yet in effect they mean the ſame thing. For any Fluent being propoſed, to find it's Fluxion, and converſely; you may put the flowing Quantity in the firſt Caſe, equal to ſome flowing Quantity expreffed by a Letter or Symbol: and the fluxionary Ex- preſſion in the ſecond Caſe, to the Fluxion of ſome flowing Quantity expreſſed by the proper Notation of a fluxionary Symbol. Thus | when it is propoſed; as above, to find the Fluxion of ax3 — 4abx* - dhe ab; ſappeſt the Fluent” equal to 2, fo that it ſtant thus ax3 the Relation of the Fluents x and-2z;. you find an Equation expreſſing the Relation of their Fluxions. Again, when the fluxionary Expreſ- bon 3x5 — bxaty — 34 ＋ 339% + 6x59 — 35% is ptopoſed, to find it's Fluent : you ſuppoſe it equal to the Fluxion of 25 ſo as to ſtand thus, zur — 6xxy — 3x% + 2ap>Þ bxyy — 35% = and (o it will amount to this, from the Equation juſt now mentioned, cxpreing | the Relation of the Fluxions x, y and/z, to find an Equation expreſ- ling the Relation of the Flyents : far the Import of which ſee Art. 89, go , ſo that the two different Ways of expreffing the two gene- · of this ral Problems, will in effect amount to one and the ſame thing. Explica- 387. But here ſomething occurs that muſt be carefully obſerved, v2. that the Fluent belonging to any Fluxion, is either pure or mixt. Thus in one of the Examples above, it was ſhewn that * abxx: 278 The Quadrature of Curves explained. Babxx + 16bxx was the Fluxion of a — 4abx* + 8bx* — 2b: but it is alſo the Fluxion of the Expreſſion ax3 — 4abx* + 8bx*, becauſe — ab is a conſtant Quantity, producing no Fluxion. The laſt, ! call a pure Fluent ;- becauſe it contains no Part but what flows, ang leaves ſome Veſtige of it in the Fluxion: whereas the other is called a mixt Fluent, as being made up of Parts, which don't all flow; there being a Part — ab which is conſtant, producing no Fluxion, and therefore leaving no Veſtige of itſelf in the fluxionary Expreſſion, by which one might know that it was in the Fluent, = Now the finding the pure Fluents from the Fluxions given, is 2 determinate-Problem, which in many Cafes may be ſolved by the Quadrature of Curves, or the Method of finding their Areas taught by our Author in this Treatiſe : whereas, to find a mixt Fluent be. longing to any propoſed Fluxion is indeterminate, ſince there may be an infinite Number of ſuch. However, there may be fuch Limita- tions either aſſigned, or ariſing from the Conditions and Circumſtances of the Caſe, as may make this laſt Problem determinate ; although -otherwiſe it would not. Thus if you know that the Fluent ought to be equal to a certain known Quantity in certain Circumſtances, you may thence find the mixt Fluent you want; provided you can firſt find the pure Fluent. RES Thus if the Fluxion —— mixt Fluent, to which that Fluxion belongs, ſo circumſtantiate or limited, as that, when x = 6, the Fluent may be o, or 5; or = or 5 az; or any other known Quantity whatſoever : then I firſt ſeek a pure Fluent, which will be the ſame Expreſſion with that, which ex- hibits the Area of a Curve, whoſe Abſciſs is x, and Ordinate =; CAX as will be ſhewn more clearly below: this Area or pure Fluent you'll - was propoſed, in order to find a 1 find (by Prop. 5. or Art. 147 *; or, more ſhortly, by comparing gation. ax = with the Ordinate of the Curve belonging to Form ſecond, Table firſt) to be a Tz Which found, ſuppoſe you take 9 for any given Quantity, to be juſt now determined: put >—=77 + So, or 6, or =, or b—a, or any other given Quantity, to which the mix d Fluent muſt be equal, when x = b, that is, (by Rees putting E Qaadrature of Curves explained. 279 putting x=#) T7 o, or b, or =; or 6 - a, &c. Whence you find 9 == Nr A orf er 1230 —4 — 2. &c. Therefore, in order to find the true Fluents way. to theſe feveral Caſes or Conditions: to the pure Fluent formerly found, you muſt annex the correſponding Values of % which will give 26 Fae 2 * „ on — K ax* be a ax” a A r — 7432 &c. for the true Fluents required, correſponding to the ſeveral Caſes, or Sy ſitions put. And the ſame may be done in other Caſes, when like imitations, or Conditions of the Fluent are aſſigned or known. 388. It may likewiſe be obſerved,. that there are fometimes two pure Fluents belonging to one and the ſame Fluxion: thus the fluxion- ary Expreſſion Ay mentioned above, has another pure Fluent, ws. T which is found by changing the Form of the given Fluxion —=;, into this other = 3 and finding the Arca of a = © =@ — 2x, Whence ariſes x = ia, i. e. AD = AB, in order to make the incribed Parallelogram a Maximum. And this coincides with Prop. 27. Lib. 6. Elem. And after the ſame manner 7 . determine the inſcrib'd Pa- rallelogram to a Maximum, it ABC had been a Segment of any known Curve : thus if it had been a Portion of a Parabola, having AB for the Diameter and BC an Ordinate bounding the Figure, then becauſe AB (a), AD (x), BC? (5*), DE? are proportional, hence DE : therefore put the Fluxion of DE x DB, i. e. 4a — * * V. - or, which is the ſame (by dividing by the given Quantity 75 of ax — x* equal to nothing, that is Zar — * * o: whence, by reducing you find x —= 4a : which ſhews that the in- ſcribed Parallelogram is in this caſe a Maximum, when AD is < of AB. And fo you proceed in other like Caſes. 398. Let it be propoſed to find ſuch a part of a ſpherical Superficies, as can be illuminated m it's farther Part, by Light coming from a great Diſtance, and refratted by the nearer Hemiſphere. To do which, let AB be a Ray of Light coming from a great Diſtance, falling upon the nearer Hemiſphere at B: let BEGH be a Section E „ i | g | 5 288 De Quadrature of Curves explained. | Section of the Sphere through the Center C: draw the Diameter ey | | JA ECG parallel to AB: let the Ray be refracted at 3 B into BHD, meeting the Circumference of the Circle in H, and the Diameter EG produced, if need be, in D. From B draw BF perpendicular to EG at F, join BE, BC, and GH. Then becauſe the Angles BCE and CBD are equal to the Angles of Incidence and Refraction, therefore in the Triangle BCD, the Ratio of Bb to DC is a given invariable Ratio, being that of the Sine of the Angle of Incidence to the Sine of the Angle of Refraction, whereſoever the Point B falls, for the Ray AB is always parallel to EG, fince the Light is ſuppos d to come from a great Diſtance, But it is evident that the right Line GH is the Chord of half the ſpherical Segment, the Circumference of whoſe Baſe is enlightened by all the Rays which fall upon the nearer Hemi- ſphere at the ſame Diſtance from the Point E as AB falls: when the Point Bis very near to E, GH is very ſmall ; when EB increaſes, GH in- creaſes with it for ſome time, but afterwards decreaſes : therefore the Problem will come to this to determine when GH is a Maximum, ? To do which call the Radius EC —1, EF =x, CD —=z : whence becauſe GE, EB, EF are proportional, you have EB = 2x: more- over, if à expreſs the Ratio of the Sine of the Angle of Incidence to the Sine of the Angle of Refraction, then BD = az : likewiſe from the ſimilar Triangles DBE, DGH, you have DB (az): BE (VZx) :: DG (z — 1) : GH, whence GH = ax: therefore —»/2x, or, which is the ſame (by dividing by the given Quantity ) x! — fr muſt be determined to a Maximum: wherefore by putting its Fluxion equal to nothing, you have T7 — 3 4 _ = 0, that is (by multiplying by 22*4/x) 2*x — 2x + 2x2 = 0. | Now in order to eliminate one of the unknown Quantities, and it's Fluxion, you muſt find an Equation expreſſing the Relation of x and z, which you may deduce from the right-angled Triangle DBF where DB! (a*2*) = DF? (2*+ 22 — 22x — 2x + x*+ 1) +BF; (2x — x*) that is 4˙ = 2* + 22 — 2x2 -|- 1: from which you have this fluxional Equation 24*z2 = 222 ＋ 23 — 2x2 — 2 whence — — ͥ — ͤ w — — — — q 5 <4 = | kh * A "a : «ns ih : At n 9 » 4 » GR . a , N 7 4 wm. 1. , , " * 5 * „ © 1 2 * - a © > w » 8 %* o , 4 n 25 * 9 we : = Z . " „ * * — . The Nuadrature of CV VES explained. whence Z = —— _ which Value of 2 being inſerted for it in the preceding fluxional Equation, viz. 2 — zx + 2x2 =o, and the Equation reduced, you'll have 2* — 4˙ — x2 + 4˙ ＋ 3x — 1 —0: by means of which, and the preceding Equation, biz. 42 = * + 25 — 2x2 Þ 1, if you throw out x, you'll obtain this cubi- cal Equation 23% — @*23 — 4˙ + 2* + 32 ＋ 3, whoſe Roots are 3 3 — . n * of which the firſt two being negative, can be of no uſe in this Affair, ſince it is evident the Point D muſt fall upon the ſame Side of the Center with G : therefore GH is a Maximum when CD — Ny == « by inſerting which Value of z for it in the preceding Equation 4a*2* = 2* ＋ 22 — 2xz + 1, and reducing, you'll get x =—— — ＋1— 444 —2 | w - EF. Thus, for Example, ſuppoſe the Globe was Glaſs, in which Caſe the Sine of the Angle of Incidence to the Sine of the Angle of Refraction is nearly as 3 to 2, therefore by inſerting + for a, you'll have D=/ x5; V. 4 = 1, 5492, &c. of CG; and EF = o. 3 545, &c. of CE. And ſince it was GH = e hence by ſubſtituting for z and x their Values, it is GH = 0.198, Fc. which is nearly the Subtenſe of an Arch 11% 22", in a Circle whoſe Rad. is i I : therefore the Part of the farther Hemiſphere which is enlightened, is a Segment, the Diameter of whoſe Baſe ſubtends an Arch of 22% 44, nearly: and in this Caſe EB is nearly an Arch of 49®, 48". | 399. Schol. By means of the above Equation a*z* — 2* — 22 + 2x2 — 1 o, defining the Relation of x = EF and a CD, you may determine ſeveral things in this Affair: e. g. by putting z= 1, you may find the Diſtance of thoſe Rays from E, which after Re- fraction will paſs through C. By ye x = 1 = Rad. you may find where thoſe Rays that touch the Globe after Refraction at their Entrance would cut CG, By putting x = © the Value of z will give the Focus of parallel Rays after Refraction at the firſt Surface, &c. thence find a, that is the Law of Refraction. And if any two correſponding Values of x and æ are given, you may * Of the pends is laid down by our Author Art. 4 * ; and was explain d at ful Quadra- ture of Curves. + Of this Explica- tion. The Quadrature of Curves explained. FROM: To draw Tangents to all forts of Curves. 400. The Principle upon which the Solution of this Problem de ngth Art. 41, 42 +. which amounts to this: if the Curve DC be deſcribed by the Ordinate moving along the Abſciſs AB; and CE be $46 Than ag a Tangent to the Curve x | the Extremity of the Ordi. nate, meeting the Abſciſs in E, the Sides BE, BC and CE of the Triangle BCE are proportional to the Flux. ions of AB, BC and DC, the Abſciſs, Ordinate and Curye- line; and it is the fame thing whether BC be a lar Ordinate; or an Ordi- nate applied in any other given Angle. Wherefor calling AB = , BC and BE =s (which laſt jy termed the Subtangent) you have this Proportion 5: x :: P35, OF y 7 =: which . | is the general Formula for finding the Subtangent BE. Therefore if an Equation be given de- fining the Relation of x and y, we may thence find the Relation of their Fluxions, and conſequently from BC or AB given you find the Point E, ſo that EC being join'd may be a Tangent at the Point C. 401. Hence it is evident, that if, while the Abſciſs increaſes, the Ordinate increaſe at the ſame time, then — is poſitive, and conſequent- ly the Subtangent is poſitive : in which Caſe BE lyes upon the ſame Side of the Ordinate that the Point A the Beginning of the Abſciſs does: if y become infinitely great in compariſon of æ, then 5 vaniſhes, and conſequently EB or , if y be finite, and the Tangent falls in with the Ordinate: if y vaniſh, 7 becomes infinitely great, conſe- quently EB is infinite; and the Tangent becomes parallel to _ 155 ut The Ruadrature of C URVES explained. | but if BC diminiſh while AB increaſes, and conſequently have a ne- tive Fluxion, then 7. and conſequently y x , is negative, in which Caſe BE lyes upon that Side of the Ordinate which is oppo- to A. N | h | | _ 1. Let y* == a*="x define the Nature of the Curve: then, by 1 n—1 | 1 þ * 1 = XxX ny x Prop. 1, ny*—"y == 4"—"x, Or FL therefore —— - = (* —) s the Length of the Subtangent: or if you would have the Sub- tangent oxpreſſed in Terms of the Abſciſs, inſert for y* it's Value a*—"x fom the Nature of the Curve, and you have = x : wherefore if you take BE: AB:: : 1, and join EC it is the Tangent required, Thus if it be 2 = 2, the Equation belongs to the common Parabola, and ſo EB= 2AB. And if » be 3, 4, 5, &c. which gives a Series of Parabolas of different Orders, you have EB equal 3AB, 4AB, 5AB, Fc, reſpectively. Again, if it be 21 - 1, then the Equation be- comes ) = a—*x or 3x , which is to the common Hyperbola, having the Center A, one of it's Aſſymptotes AB, and the other pa- rallel to BC: here you have 5s = — x, therefore you muſt take EB — AR upon the other Side of the Ordinate. After the ſame manner if you take 7 = — 2, — 3, -— 4, &c. you'll have different Orders of Hyperbolas, whoſe Subtangents are — 2x, — 3x, — 4x, Gc. i. e. 2AB, 3AB, 4AB, &c. taken upon the Abſciſs produced beyond the Ordinate. Ex. 2. Let abx — bx* — ay* o expreſs the Relation of AB and BC, in which caſe the Curve DC is an Ellipſe, A the Vertex of the Diameter, @ the Diameter, 5, its Parameter: then by taking the Flux- ions, it is abx — 2bxx — 24% So; whence = , therefore 20% 292 = T = (by ſubſtituting abx — bx* for ay* from the Equation to the Curve) . = EZ = BE : whence BE is a fourth 24 — * ciples. Hence it that, ſince the Value of BE the Subtangent, doth not include the Parameter 5, any two Diameters of any two Ellipſes, and their Abſciſſes, being equal, the Subtangents are equal : 2 31 jonal to 22 — x, 4 — Xx, and x, as is known from other Prin- and therefore if there be any Number of Ellipſes having a common Axis, including alſo the Circle, the Tangents drawn from the Extre- mities. of the Ordinates applied to the ſame Point of the common Axis, will meet that Axis in the ſame Point; If the Equation had becn.abx- Pp 2 0 _ 1 a : o N 1 —_— 4 . N mee o 4 4 6 4 292 De Quadrature of Cunvas' explained. | + btn o, it would have been to the Hyperbola, and bo — 3 == BE. 2 te ; * Ex. 3. Let AGE be the Ciſſoiĩd of Diocles (ſe Fig. Art. 3219 ot this belonging to the Circle AD Q, the fame things being ſuppoſed as in — that Article, call AQ = a, AC x, CE =y: and it is required to draw a Tangent to the Curve at the Point E. 3x» A The Equation to the Curve is y = — or, by ſquaring, a — xy*= x3, therefore, by taking the Fluxions, 2ayy — 2 * oY & & , | a x | 2ay — 2xy | * | | * — 2X X y* — 2217 — — a 3 — — — ps. = 3 whence r wherefore File 4 Bar = (by . - WW <- © Q 2 — inſerting er for 5% and redueing) 22 S ge which poim aut this Conſtruction: produce AQ to I, ſo that Q be equal to the Radius of the Circle ADQ, join ID, draw DK perpendicular to it, cutting the Abſciſs AC in K, join KE, and it is the Tangent required. Ex. 4. Let PEH be the Conchoid of Nicomedes (ſee the Fig, An. * Of this 322 *) and ſuppoſing the ſame things as there mentioned, only con- — der AB as the Abſciſs, and BE as the rectangular Ordinate, which call x and y, the other Symbols remaining the ſame ; it is required to draw a Tangent to the Curve at the Point K. | The Equation to the Curve is xy = 6 + yvcc — yy : therefore by „ Art,87. taking theFluxions, you have xyxy=jJvcc—yy+bÞy x ——2 » of this Ex- EE | 4 333 ö V -= f plic os. 1 „ 1 | : . ce — yy p * 3 whence > = — ee : therefore 5 = Wie — yy = 2 — £= CD — — AB= s the Subtangent: which Value being always negative, in regard AB always exceeds CD, ſhews that it is to be taken upon the Abſciſs AB produced beyond the Or- dinate : wherefore if you take the Point T in the Aſſymptote AB, upon the further Side of the Ordinate equal to AL + GL x AC nd = join TE, it ſhall be the Tangent required. , 402. This Curve affords us an Example of a Point of contrary Flexure, which ſeparates betwixt the Part of the Curve that is concave to the Aſſymptote AB and the Part that is convex; which may be thus determined from the Conſideration of the Tangent. When AB is no- thing, and the Point E coincident with P the Vertex of the Axis, the . Tangent The Quadrature of Curtis explained. Tangent ET is parallel to the Aſſymptote AT, in which caſe the cybtangent BT is infinite; and if you oppo ET produced till it meet the Axis AP in F, the Part PF vaniſhes, when AB and conſe- quently PC is of any finite Magnitude BC and PF are finite Magni- tudes; ſo that, ſuppoſing the Abſciſs AB ariſing from nothing to in- creaſe, the Line AT will decreaſe and AF increaſe, until ſuch time, as the Ordinate BE fall upon the Point of contrary Flexure : after «hich, AB till increaſing, AT will begin to increaſe and AF to di- g miniſn. Hence it is evident that AT and AF attain an extreme Va- 9 51 lue, the firſt becoming a Minimum, the other a Maximum, when LN the Ordinate falls upon the Point of contrary Flexure. Therefore if you take the Value of any of theſe Lines, and put it's Fluxion equal 0 nothing, the reſulting Equation yill determine the Point of con- | I LO 0M SET DEN ITT HO RIS 1 b FR - — VEIIIY * * 7 g " , | Now ſince we found — BT S ce =p LIL: . * hence - | * | * 21 De oe MS LF \ allng ATI. we ſhall have — cf —5y + ZE al From which find an Equation expreſſing the Relation of the Fluxions, in which ſuppoſe = ®; for x and + inſert their Values "Eee I Ae. ad t =. — — 5, from the Equation to the Curye, Pe. and reduce the reſulting Equation, and it will give you yi— 355 — 24c* So. Therefore if this Equation be conſtructed, the Length of BE will be thereby found : to which if you take AC equal, and draw CE parallel to AB, the Point E ſhall be the Point of contrary Flexure required, And after the ſame manner the Points of contrary Flexures, in other Curves where there are ſuch Points, may be inveſtizated.© - 403. Although the Curves which are propoſed, be mechanical, Tangents may be drawn to them by the like means as if they were geometrical, All that is neceſſary is to find the Relation of the Flux- ions of the Abſciſs and Ordinate, expreſſed by finite Lines: which may be done although there be no finite Equation that can define the Relation of the Abſciſs and Ordinate. 2 | Ex. Take the common Cycloid or Trochoid ACD {ſee the Fig. Of this Art. 332 ) and ſuppoſe the ſame things as are there mentioned, only - af call AB=x and AHG ; and let it be required to draw a Tan- gent to the Curve at the Point C. | To do which, from the Property of the Curve, you have BC — BG + AHG, that is y =v+2: therefore, by taking the Fluxions, you % % 294 The Ruadrature of Curxvss explained. =. Sy you have y—=v + z= (becauſe 5 = T n by Art. 42 8.) 3 tion, + Ad: but from the Property of the Circle, viz. v* = dx — xt by taking the Fluxions, you get v = 2 x : which Value of ; being ſubſtituted in the preceding Equation 5 =v + v® FT, and it duly reduced, you have = 2 whence 7 = = 2 =BCxgg=5 = BT. Whener it appears thu the Subtangent BT is a fourth Proportional to BG, AB and BC; 0 that the Tangent CT is parallel to the Chord AG: which Property we * Of this aſſumed at Art. 326 *, in order to abridge the Demonſtration. Explic-' = 404. And by the like Methods may Tangents be drawn to other emo mechanical Curves, which are related after any other manner to geo- metrical Curves : but whereas it is not ſo evi how to proceed in the Caſe of-Spirals, I ſhall ſhew the young Geometrician how Tan- gents may be drawn to them. EL A Shake - * See Fig · Let ACM bea Spiral “, MEGM a Circle deſcrib'd from the Center Art. 335- A with a given Radius AM: and let the Radius Ac beginning at — AM reyolve upon Aas a Pole, and in the mean time the Point C, mo- ing along AC, deſcribe the Spiral ACFM : draw ATS perpendicular to ACG, and let CF be a Tangent to the Spiral at C meeting ATS in T: through G draw G9 parallel to CT, meetidg AT'S'in S. Sup- poſe ACG to move forward into the new Poſition Azy, with the Cen- ter A and Diſtance AC draw the ſmall Arch C9 meeting Ax in d: then by ſuppoſing the Angle GAy to be juſt vaniſhing, the Triangle Ci * Art.q1, is ultimately fimilar to the Triangle ACT “, that it is d: :; of this Ex. AC: AT. Again from ſimilar Arches it is C: :: (AC: AG:: plication. AT: AS, therefore ex ue : G:: AC: AS; but & is to yG as * Art.28. the Fluxion of AC to the Fluxion of MEG *. Therefore an Equa- of this Ex- tion being given from the Nature of the Spiral, defining the Relation plication. of AC and MEG, thence find the Ratio of their Fluxions as for merly : and take AS to AC in the Ratio of the Fluxion of MEG to the Fluxion of AC, fo ſhall SG be parallel to the Tangent at C: therefore SE being join'd, through C draw CT parallel to it, and it ſhall be a Tangent ta the Spiral at C. Ex. 1. Call MEG = x, and AC y, which you may conſider as Abſciſs and Ordinate; and let it be x: :: 4: b, a given Ratio, ſo that the Spiral ſhall be that of Archimedes: then bx ay, whence bx=uy, ſo that j: & :: b: ::): F = AS: therefore take AS a fourth Propor- | tional 5 The Ruadrature of CUxvES explained. tional to 5, 4, and AC, join GS, and CT drawn parallel to it touches the Spiral at C. It the given Circle MEGM be that which bounds the firſt Spiral, then à is to b as the Circumference of a Circle to it's Radius : which therefore will be the Ratio of AS to AC : conſequent- y when the Points C and G coaleſce at M, and conſequently T with $, the Subtangent AT is equal to the Circumference of the Circle MEGM, as 3 has 1 * Ex. 2. Let the Equation to the Spiral be generally expreſſed thus, off = bx*: then by taking the Fluxions it is — whence 5: K 33 nhxv—" : may"-? :: y : AS, therefore AS — = 2 NPR ab 1 a — | - | PZ 4 = (by iaferting 2 for i, deduced from the given Equation and 4 by reducing) — — nb Ex. 3. Let the Equation x* — 24 + by*= © expreſs the Rela- 4 01 and y: then by taking the Fluxions, you have 2xx — 24 + 2% = ©, whence 24 — 2x : 2by :: 6 #33) y: ASS = (by ſubſtituting a — J — by* for x, deduced from the given Equation ) 2 therefore if you call AM r, then ſince : y:: AS: AF, it will be > = = AT, the Subtangent. And fo you may proceed in all other like Caſes. 405. Theſe two general Problems, which we have been explain- ing, are an Application of our Author's firſt Propoſition, and belong to the direct Method of Fluxions, becauſe they only require the find- ing the Fluxions from the Fluents given, But the far greater Num- ber of Problems to which the Doctrine of Fluxions is applied, require the finding the Fluents from the Fluxions given, in order to their So- lution, This cannot always be done by the Quadrature of Curves ; but may be done by it in many Caſes, of which theſe that follow are Examples. | 295 p RO B. 252865 | The. Ruadrature of Curves' explained. „ tte 6 95 lein oa TNG D 0 Jo find the Length of Curve: lines. 406. The Principle, upon which the Solution of this A Eagles. pends, was explained and demonſtrated in Art. 41 and 42 * (ſee the tion. Fig. belonging to that Place) which is this: if you put the Abſcig Ah x, the rectangular Ordinate BC y, and the Curve-line ter. minated at the Extremity of the Ordinate as AC = 2, then it is 3 — — — a/x* + y* : where you muſt obſerve that it is not neceſſary to fy the tes to begin at A or even to paſs through A, r do ſo in this Figure: but the initial Limit muſt be determined from the Nature of the Curve, as will appear afterwards. Therefore, if from the Nature of the Curve, an Equation be given, expreſſing the Relation of x and y, thence by Prop. 1. you may find the Relation of their Fluxions in another Equation : from which two Equations you may find the Value of the Fluxion of y, in Terms made up of the Fluxion of x, x itſelf, and knowr. Quantities : which Value of the Fluxion of y being ſubſtituted for it in the Equation z = /3* FF the ſecond Member ſhall contain no variable Quantity but x, and 10 Fluxion but that of x ; and ſo the Fluent z, that is the Length of the Curve-line, may be found by the Quadrature of Curves, as will ap- pear by what follows. 3 | 407. Let x" ae. be an Equation. to a Parabola of any Order, as AC, where n repreſents any poſitive Integer; and it is required to find the Length of the Curve, which call z. | ; Then you have 2 =vx* CT *: but from — — 3 — 2 e g , * ks * ER "3; > r * ——— — 2 2 — 2 >. — = - * — -———_> . as = * 18 —Y * - n the Equation to the Curve, viz. x" au-, 1 by taking the Fluxions, it is nx*—"s = | yl . 4 2 99 un — IX, whence 5 = — —5 Q : 1 4K 1 that is (by inſerting —— RP” | | . an —1 1 duced from the given Equation) J= — — or by ſquaring, n — IX a : now if this Value of y⸗ be put in place 1* — 22 + LX — of The Quadrature of Cunvss explaind. 297 of it in the fluxional Equation Zz == /x* + 5*, hence will ariſe 2 = | 2 * 1 | | i 1 — — — But it appears that this Expreſſion | n* — 2n + 1X4" 0 genotes the Fluxion of a curvilinear Area, whoſe Abſciſs is x, and rectangular Ordinate the radical if 1 +: —— Wt, n —2n-þ-1X 4" h therefore the Curve-line z, and ſuch curvilinear Area, flow according to the ſame Law; the one as a Line, the other as a plane Superfi- cies, conſequently the two Fluents will be analogous, ſuppoſing them to begin at the ſame time : wherefore the curviliner Area having ſuch Abſciſs and Ordinate, being found by the Quadrature of Curves, ap- ply it to linear Unity, and that ſhall exhibit the Length of z required; or the curvilinear Area being found in an algebraical Expreſſion, con- ider ſuch Expreſſion no longer as denoting a Superficies, but a Length, the ſame algebraical Expreſſion ſerving for both, ſince the Diviſion by Unity makes no Alteration upon it. "Wherefore if the Value of ꝝ be ſuch, that the Ordinate IF gh 1 + - - — * 57 belong to a nm —2n+1x0 | Curve that is quadrable, the Length of the Curve-line is found ex- aAly ; if not, the Length of the Curve-line is found approximately, by Propoſitions fifth and fixth : or in this laſt Caſe, may be exhibited by the Area of a Conic Section; or perhaps of ſome other Curve ap- plied to Unity. So that our Author's Tables of Quadratures ſerve 25 this Purpoſe, as well as for diſcovering the Areas of Curves : theſe Lengths likewiſe may ſometimes be found by Logarithms and circu- lar Arches, according to Sect. 10. For finding when theſe Curve- lines can be rectified exactly, or not conſult Art. 147—149 *. * Of this Thus if it be 2 — 3, ſo that the Equation to the Parabola ABC Explica- be x3 = ay*, which is a Parabola of the ſecond Kind: then 8 = ** i 1+ EL which is analogous to the Fluxion of the Area of the Curve whoſe Ordinate is IF 4 1+ 2 which may be ſquared by Art. 147149 , and belongs to Species 1. Form 3. Table 1. which. Of ti, ſtands thus dz — e y, by putting Y = 1, d= 1, 1, 1 * Q FA 298 The Quadrature of Curves explainad. f= bi: wherefore the Fluent 3, R. in the Tuble, is by a proge — Subſtitution of Values, 55 * 1+ 2 P or wt 14 2 which therefore denotes the Length of this Parabola. And after the fam. manner, the Legs of all theſe Parabolas may be found where is any poſitive odd Number greater than 3, as 5, 7, 9, &c. by hel of the ſecond, third, fourth, &c. Species's of Form third, Table fr _ PR... fame thing might be done by having recourſe to Prop. 3. % {thi Art, 39 f 1 ö bn. 408. But whereas it is not yet determined from what initial Limit, the Length of the Curve ſo found is to be computed, this may be done by putting the Quantity "= 1 J. E equal to nothing, and ür Value of x thence ariſing will determine the initial Limit; as was ſhewn formerly with reſpect to curvilinear Areas in Sect. 5. Thus if we put % /1+E o, hence ariſes 4, bro, or 3+ =, either of which gives x = — t@ : which Value being negative, ſhew that if you take upon the Abſciſs AB produced bęyond A, the Point P ſuch that it be AP ga, and draw the Ordinate PQ, the Part QAC= Ei If you would compute the Length of the Cure from the Beginning of the Abſciſs at A, then here is a known Con- dition or Limitation of the Fluent, viz, that it is nothing when z= o, * Of this therefore by Art. 387 *, K . 1+E — = More wi Explica- SZ | 32 OE, £0 A2 n | tion. yerſally : if you would have any Part of the Curve line as yC, h. ing betwixt any two given Ordinates Sy, BC, ſubtract the Value of z ariſing from putting x == AB the leſſer Ordinate, from the Value of z arifing by putting x = AB the greater Ordinate, the Difference *Art,187. is the Length o *: thus if you call AB = E and AB = x, then this Ex- | — g ; plication. you have © i + E — EXE 1 +E: where F vaniſh, the Computation is from the Beginning of the Abſciſs, or rather the Ordinate paſſing through it. ator 92 409. Suppoſe now that in the preceding general Equation x" 2 ay*—*, 1 is 4, ſo that the Property of the Parabola AC be expreſſed thus xt = ys; then it will bez = 1 ＋ . by comparing thi = . with —— Cunvzs explained. with 1 1 49 „ youll find the radical dinate of a whether it may —1* reduced to any Ordinate found in our Author's ſecond Table of Quadratures; which as it ſtands, it cannot; but if 299 Expreſſion denotes the Or- Ofthis Curve that cannot be ſquared: therefore I next conſider df A jou redues it to this other Form .f, 1, you find * 3 ge? NN. belong to Species third, Order third of Table ſecond, Ws, 5 = 3 plication. z, by piting f, d = 1, = 5, f=7 by "ga?" 3 comparing which with Art. 298 *, you'll find the Conic Section, Nacken weans of which the Arca of the Curve, whoſe Ordinate wv x we. may be compared to be an Hyperbols belonging to Fig. third Table frond, whoſe Latus Rectum is 1 and Lat: Tranfv. is =". And if t 4 1, or * = 193, for the Equation tothe Curie po m0; you'll have the Goring l ſecond, Quad. p. 2 5. Wi Ka = a deſeribe the aGDS, take 1 that is, take aB the ſecond of two middle jonals betwirt 4 and AB the Abſciſs of the given Parabola AC, draw the Ordinate BD, then the Area of the Curve whoſe Ordinate is V1 + „ is 240DB : conſequently, the Area 2 being applied tothe Lat. Rectum 4, 12 o the Line alifin 8 Parabola AC, of t 2 Ki Now if we would know from whence the . of the ine is to be computed, it is evident, that, ſince æ — <= 2 0 So, then x o: but aGDB is nothing when the Abſciſ aB is no- thing, and aB — AB}, therefore 2 is nothing when AB the Abſciſs of the een, Parabola AC is nothing: therefore the Length of the Curve-line found is to be computed from A, being AC. 410, If we would have the Length of the Curve-line expreſſed by Logarithms, according to what has been delivered in Sect. 10: then ſee Fig. Art. 355 *, which ſubſtitute in place of Fig. 3. Table 2. the Parameter a, and tranſverſe Axe Quad. ſuppoſing it deſcribed with the ſame Latus Tranſverſum, and EPs Neu, as Noe then 2 — — inn = e. g. ſuppoſe the Qq 2 Length Conſtruction. (dee Ter ine 12. * Of this tion. * Art. 5 370- 3 this Ex- plication. 4 a% 4 S ˙ 4 +252 Art. 349. — — => 2 2 300 We Nuadrature of Curves explained. Length of the Abſciſs AB = x = a = 1, then in the Hyperbola by making aB = (AB =) 1, you'll find H=4 = 0.6666, Gc. in inf 2 = 44'== 1.41666, Ge. in inf. BD = 4 = 1,25, whence DC . o. 16666, in inf. therefore it is & = 1.80569, &c, — 0.296296, G — wherefore, ſince the Modulus of the tabular Logarithn of thiekz, lis 0.434294, Sc. L. hence it follows that it is z = 1.80569, Cc. —. VVV of chie Ex. Cc. — 0.682247, &c. x Log. 4 = 1.80569, Cc. — 0.682247, &. plication. o. 6020600 = 1.80569, Sc. — 1975 . — the Length of the Curve- line AC required. | . The ſame thing may be done by reducing the Value of the Area into a converging Series by Prop. 5. And after the like manner you may find the Lengths of de Curve- lines belonging to the other Orders of Parabolas, when 7: is 6, 8, 10 or any other even Number. 411. Ex. 2. Let it be propoſed to find the Length of the Cycloid L, Of this (ſee the Fig. at Art. 332 5). Uſe the ſame Symbols as in that Article, nen. only call the Abſciſs AB , and the Curve- line 2. Then by what is there demonſtrated, you have 5 = *VE==, „ = inſert this Value of 5*' in the general Formula à = , and it ſtands thus & = V ＋ = == or d*x—*x : which belongs to Form firſt, Table firſt, Quad. viz. de" —= ye ſo that if you ſubſtitate'd? for 4, + for y, and & for 2, the Fluent is 2V/dx = 2 Chord AG : Which therefore denotes the Length of the Arch of the Cycloid AC : for it's initial Limit is at A, fince 2 dx is nothing when x == AB is nothing. Hence it follows that ACD = 2AE, twice the Diameter of the generating Circle. 412. Ex. 3. Let the common Parabola ABC be propoſed, and it is required to find the Length of the Part AC of the Curve-line, AB being the Axis, and BC it's Ordinate. Te C | To do which, uſing N the ſame Symbols for Abſciſs AB, Ordinate BC and Curve. line as formerly; call the Pa- rameter a : then ſince it Dis ax ; hence by taking * De Ruadrature of Curves explained. 301 taking the Fluxions you ax = 2yy, that is y = (= 2 2 af 1 Yo * inſert this Value of y in the general Formula z = /x* + , and 1 becomes, by a due Reduction, & = x X 34/4 +ax=* : which being the fame with the Fluxion of a Curve that is not quadrable (by Art. 147, 149 *.) I find it belongs to Species ſecond, Order third, Table Elle ſecond, VIS. * + 2" y, by putting » = 1, Cc. By which n. ns it may be found, by applying an hyperbolical Space to linear Vaity, as was ſhewn before in the Caſe of a Parabola of the third der. . 13. Moreover it may be obſerved, that as we arrive at the Lengths of Curves by ſubſtituting the Value of y* in Terms made up of x and it's Flux- jons, in the general Formula z==v/x* ral ſo you may ſubſtitute * it's Value made up of y and it's Fluxion, and by that means do the fame: thus becauſe it is in this Example, ax , ſubſtitute = for x, or . for 4, thence itwillbe Z=. LA ＋ ) 1 +=; which agrees with . # vix. Z= ® Ofthis the general Expreſſion Len all Parabolas, as in Art. 407 , viz. & * tion. _ Ne 4 1+ 4.4 , with this only Difference, that you | nm — 2 T X . | | have y here for x there, ariſing from this Circumſtance, that the Abſciſs in the former Caſe was taken upon a Line perpendicular to the Axis upon which the Abſciſs is taken in this Example, as is evident. Therefore now that it is 2 Nl + vo or Z= yy + + 5, which belongs to Species ſecond, Order third, Table ſecond, viz. ve + . = y, by putting q= — 2, z==y, &c. it will appear by conſulting Art. 295 *, and comparing it with the Expreſſions * of this 3 „ 4 FELL 24 5 0 . ; Explica- F ex =v, and — st, in the Table, that, aſſum- ion. ing any Line for linear Unity, if you take (ſee Fig. 2. Table 2. p- 25.) AP= 1, and with it for the half tranſverſe Axis, and 1 for the Latus Rectum, deſcribe the Hyperbola PGDS, take upon AN the indeterminate Axis, AB equal to BC the Ordinate of the pro- 3 poſed * Of 302 Erplica- tion. „of this See Art. 100, 225, 220 , e. g. let it be FR to find the Curve Explica- tion, ths i, it will follow by conſidering Art. 344 *, that the hy- The Quadrature of Cunrvss explained. ſed Parabola ABC, and draw the Ordinate BD, then you ſhall har APP Ac the parabolical Arch required. 414. Whence we learn to expreſs the Length of the ra | Arch AC by the Meaſures of Ratios thus. Draw the bene meeting the Axis in E, through A 8 AF perpendicular to AB, meeting the Tangent in E, likewiſe CD perpendicular to the Tan. gent meeting the Axis in G : then * FG the Meaſure of the Ratio of AE +EF to AF to the Modulus BD, fo ſhall it be CG = Arq AC: which according to the Notation in Sect. 10. is expreſſed thus; AC==CF D. y For ſince it has been ſhewn that AC== in in the preceding Ar, and ABDP = AABD -+ Sector APGD; and moreover that in the Hyperbola you have AP = 1, Lat. Rect. = id, AB =y and BD ws 4 WV WV, Fo 3 pou wy perl den APOD = to E = ( by the Conſtne: tion above in the Parabola) BD |, or (from fimilar Tri angles) BD BD [gp _— and the en ABD in the Hyperbola is qu) + =x* FeFY — NNE 2 =; ELD = (in the Parabola) 2 2x0 CF. Therefore the parabolical Arch AC= (CF ＋ 25D 1512 0 — CF + FG) = CG. As was to be ſhewn. And this ſhall ſuf- fice for wy and ſolving this general Problem: 20 find the Length f Curve- lines. 415. Schol. Here it is proper to obſerve that you may find as many Curves as you pleaſe, whoſe Lengths are equal to known right Lines; or may be compared to the Aloe of known * a olied. to Unity, like to what was ſhewn n; with reſpett he Afeas of Curves. whoſe Length z Kl. Therefore by taking the LS, it 1 S=2 4 and by ſquaring, * x=: but it is 2* K ＋ 37, therefore by Subſtitution, x#* = +2, 0 or & — 1 I Xx* , that is by m the Square Root ** The Nuadrature of Curves explained. the Area of a Curve whoſe Abſciſs is x and Ordinate /X — 4. — . +£x*, you will find that it denotes the Ordinate of an Ellipſe : in which, by comparing the homologous Terms The Ruadrature of CV RV EGS explained. 305 Terms by the Rules of Algebra, you'll get Iwo, f—þ He = p 20. þ = _ 3% 4 = th 7 21 or if — 1K. By conſider- ing which you may deduce the following Conſtruction. Let AGE be a Quadrant of the generating Ellipſe, AG half the longer and GE half the ſhorter Axis, $9 Iv ret, BC an Ordinate to the longer Axis, F Pe! L the Focus. Then upon GA produc'd . take GL a third Proportional to GF and GA : with the ſemitranſverſe GL and ſemiconjugate GE, deſcribe the d N Ellipſis LE, draw AN perpendicular NY to AG, meeting the Ellipſe LE in the v Point N, and produce BC till it meet r it in M: then the Area ABMN is to | the Surface of the ſpheroidical Seg- R 85 ment generated by the Rotation of *' ABC, as 7 to c, that is as Radius of a Circle to the Circumference, 1 ner D For ſince it is GA — 54, and half i + 3 Parameter = , hence it is GE —= Xx 2 therefore GF/=(GA!T—GE'=) - or GF=vVi41*—z, conſequently GL = (2 = 24. . =: but from the Property of the Ellipſis LGE, BW : GET VH) :: GL! — GB? (= — Tay): GL. ©): from whence, by a proper Reduction, you have BM AF V Ml + = . Wherefore the Fluxion of the elliptical Space ABMN is 2A. +l ＋ l = * but it was ſhewn above that 5 = A. Hl + STA — ＋ A, therefore the Fluxions are to one another as r to c, a given Ratio, and the Fluents have both the ſame initial Limit at AN, therefore ABMN : the Surface of the ſpheroidical Segment:: : c. | 419. Cor. 1, Hence it appears that the whole ſpheroidical Surface 5 to 4AGEN, as the Periphery of a Circle to it's Radius. | 420. Cor: 2. If GE remaining the ſame, GA grow ſhorter and ſhorter, GL will thereby grow longer and longer, until GA become R r equal 306 The Quadrature of CV RVR explained. equal to GE, when the revolving Ellipſe becomes a Circle, and the Spheroid, a Sphere, and the Focus F falls in with G; and then Gl. which is a third Proportional to GF and GA, turning infinite, EMN becomes a ſtrait Line parallel to GA, and fo AGEN and ABMN are Nectangles: whence it follows that the Surface of the ſpherical Se. ment is equal to the correſponding convex Surface of a right Cylinder having GE for the Radius of its Baſe : for the Rectangle AGEN, þy it's Rotation about AG, generates a right Cylinder, and EMN the convex cylindrical Sutface: where it is evident the Part of the cylin- drical Surface generated by MN is to the Rectangle ABMN as c tor, Whence appears the Proportion betwixt the Surface of the Sphere and that of the circumſcribed Cylinder ; and that the Surface of the Sphere is equal to four times the Area of a great Circle. 421. Cor. 3. But upon the contrary if the Elhpſe AGE be fp. poſed to grow more and more excentric, by the Center G removing, the Points A and F remaining, until G be infinitely. diftant, the El. lipſe becomes a Parabola, having A for it's principal Vertex and F for it's Focus. |Now if the Surface of any Segment of this Paraboloid was — you need only make ſuch a Change upon the former fluxionary Equation j=<&/ 2 + ZZ as the Suppo- ſition of & being infinite requires: which will turn it into this 3 = VI + { which being compared with Species 1. Form 3. Tab. . is found to denote the Fluxion of a quadrable Curve whoſe Area is, by a proper Subſtitution, 7 XIT NVU R, which will be found » Of this to be the ſame with the Expreſſion formerly found Art. 4 17 *. but to = agg have the true Fluent, it muſt be made 7 X 21 E — Ld : for which a geometrical Conſtruction may be eaſily made. 422. Cor. 4. If the revolving Axis AG be ſuppoſed to grow ſhorter than the Axis GE of the Ellipſis, as in the Figure adjoin'd : then, ſuppoſing & to denote the ſhorter Axis, and / it's Latus Rectum, the Expreſſion for the Fluxion of the Surface of the Segment of the Spheroid, now become oblate, will be the ſame as before, only / be- ing now greater than E, there will be a Difference in the Signs of the Terms of the radical Expreſſion ſo as to give 5 = FP Fo mos 2 2p 27 20 = UM "Las ae by The Reaadrature of Curves explained. by comparing of which with Art. 294 *, the radical Expreſſion will * Ot this be found to denote an Ordinate to the ſecond or conjugate Axis of an Hyperbola. And if the Terms of the radical Expreſſion WE al TT 2 + 2 be compared with the homologous Terms of the hyperbolical Ordinate in that Art. viz. . D TI, TT, 6 | F as formerly, you will obtain theſe Values of t, þ and a, vis. f = Vi, þ= 2 I and a . Whence ariſes this Conſtruction. F being the Focus of the Ellipſe, join AF, draw AO perpendicular to AF, meeting EG produced in the Point O: with half the tranſ- verſe Axis GE, and half the * 7 GL = GO, and the principal Vertex E, deſcribe the Hyperbola EMN ; draw AN perpendicular to AG meeting the Hyperbola in N; produce BC till it meet the Hy- perbola in M: then the hyperbolical Area ABMN is to the Surface of the Segment of the oblate Spheroid generated by the Revolution of ABC as r to c. | fror GF = (vAFT—AG!=) VIH—TÞF, and CI. (2. = | FG . but from the Property of the Hyperbola it is BM? — GE' : BG! :: GE? : GL, or by Alternation and Compoſition, BM : GE: (40 :: BG! OL: (FE=2 +) : L (i. whence BM = E! x BG! + GL? . * =h . eee =) by a due Reduion f, Kfer f e, whence it follows as in the 9 Conſtruction, that the hyper- bolical Space ABMN is to the Surface of the Segment of the oblate Spheroid, as Radius of a Circle to it's Circumference. 423. Schol. By comparing theſe two Caſes of the oblong and ob- late Spheroid together, and by ſuppoſing the one to paſs through the Sphere into the other, you ſee the gradual Progreſſion of the Values of the Surfaces, by the Line EMN paſſing from an Ellipfe, concave to AB, into a right Line parallel to it, and then into an Hyperbola convex to it : and that the revolving Axis GE is always one Axis of 2 Nr Section EMN, and the other GL a third Proportional to and GA. | In the oo. Inveſtigations of the Surfaces of the obl ong and oblate Spheroid, if you had ſuppoſed GB = , fo that the Abſciſs Rr 2 had N.. — — . — * X — 5 ( K — ¶ — — —— — — — — — 308 * Of this comparing this with Article 296 *, where it is v = XPUCA- tion, * . The Quadrature of Cun ves explained. had begun at the Center, the Solution would have been more eaſy and ſimple : for in that caſe you would have had 5—- * . _ 2 Bl —— I 424. Suppoſe now the revolving Conic Section to be an Hyper- bola ABC, G the Center, GA half the tranſverſe, GE half the conjugate Axis and F the Focus as before ; and ima. gine it to revolve about the tranſyerſe Axis GAB; BC being an Ordinate thereto as formerly: and the Surface of the Segment of the hyperbolical Co. Pg noeid is required. Suppoſe AB = z, ws 2 x | BC —y, and & and / to denote the F... 3 fore; then from the Property of this Conic Section it is y* = r= nd by taking the Fluxions 2yy = ==, whence you k . . . LIB + Ihe ** 4x 2— 222 ee obtain * — X his | Lys 3 mula 5 = v2? -+ , and bringing all to order as before, you'll by inſerting which in the general For. thereby obtain this Equation j = li + Fa - b 9 BID +ETY, +2, you may ſee that it belongs to an Hyperbola ; ſo that by comparing the homologous Terms as before, it will give te = i - = . = or 2 — tt. Whence ariſes the following Conſtruction. Take GL a third Proportional to GF and GA, then with the Cen- ter G, half tranſverſe GL, and half conjugate GE, deſcribe the Hy- perbola LNM, and finiſh the reſt of the Conſtruction as before: then the hyperbolical Area ABMN is to the Surface of the Segment of the hy perbolical Conoeid formed by the Revolution of ABC and AB, as Radius of a Circle to it's Circumference. The Demonſtration of which is the fame as before. And as GL was a third Proportional to GF and GA, and 2GE was a common Axis to both Conic Sections in the preceding Caſes ; it is the ſame in this Caſe. 425. Schol as J ada. - abs. tranſverſe Axis and it's Parameter as be- The Ruadrature of Curves explained. 425. Schol. If the Points A and F continuing, the Center G be ſuppoſed to remove to an infinite Diſtance, AC becomes a Parabola, and LNM another, having both the ſame Latus Rectum, and AL —AF : for by ſuppoſing E infinite, the fluxional Equation 5 — l + Iz + gs becomes 5 D N : where I denotes the Latus Rectum of the Axis of the revolving Parabola. Wherefore fince it is AL = (AF ==) +, you have LB =i/ +2, multiply this by J and it produces zl + E, which is the Square of the Ordinate "BM, therefore LNM is a Parabola having the fame Latus Rectum with the revolving Parameter AC: ſo that the Surface of the Seg- ment of the Paraboloid, generated by the Rotation of the Parabola ABC, is to the parabolical Space ABMN, as c to r. 426. Finally, if the Hyperbola AC be ſuppoſed to turn round upon the conjugate Axis GE, and thereby generate a Surface convex to- wards GE ; take upon GE, produced if need be, any Abſciſs Gg, and draw an Ordinate Þx : then calling GB = z, and G =, you may find as before another Hyperbola giving an hyperbolical Space, which ſhall be to the Surface generated by the Rotation of Ax, as r to c. This other Hyperbola has GA for it's half Tranſverſe, and a third Proportional to GF and GE for it's half Conjugate : the ſame Property as the Conic Sections formerly found. ; 427. If the Area ABMN in all the preceding Caſes be applied to the Abſciſs AB, and the Line reſulting be made the Altitude of a Rectangle whereof AB is the Baſe; and ſuch Rectangle, by revolv- ing round AB as an Axis, generate a right Cylinder, the convex Sur- face of the Cylinder is equal to the Curve-Surface of the Segment of the Spheroid or Conoeid 8 by the Rotation of ABC about AB. For th cylindrical Surface is to the Area ABMN as c tor: as is evident. Underſtand the fame of the Surface formed by the Rotation 2 of Ax about Ag, when the hyperbolical Area ariſing is applied to GS. 428. By conſidering what has been demonſtrated, and comparing, it with what was ſhewn formerly in Sect. 10. you may know how to expreſs theſe ſeveral things by the Meaſures of Angles and Ratios. Thus in the Caſe of the oblong Spheroid (ſee Fig. Art. 418 K.) if you * Of this imagine the two right Lines or Radius's GM, GN to be drawn, the _— Area ABMN —= LBMNL — LAN =— Sector LGM — ABGM — Sector LGN + AAGN : but the Sector LGM =— {GE x GL. = 10. and the Sector LGN GE x 61 Gene, At the ofthis Ex- "2 plication.. 309 — — — - > * — 3 f - — — W 22 _ 1 e — =_ , r a | > — — — — > —— CS — : : : . tH . , — — ” * — — — — — - — — 4 = * = -— — 2 IS 310 * Art. 363, 370. of this Ex- plication. The Quadrature of Curves explained. the BGM —= = - 2, and AAGN = = = : therefore the Are, ABMN = GE x or. [LED as GL' oa 2 — — — . wherefore if this Quantity 8 multiplied by < — 6.283 118 53, Sc. the Quantity ariſing will denote the Value of the Surface generated by the Rotation of AC about AB. 429. If the Surface generated by the Rotation of the Arch Ec were required, then it is equal to — x BGEM: but BGEM — (Sector EGM 4- ROK —) 2GE x GL |\ ares, GB +20xM, vu fore if the Arch which meaſures the Angle whoſe Radius, Tangent and Secant are to one another as L — GB?, GB, and GL, or (which is the fame) whoſe Radius is to it's Sine as GL to GB, be multiplied into GE, and the Triangle BGM added thereto, then that Sum increaſed in the Ratio of c to f ſhall be equal to the Curve. Surface required. Which may be conſtructed thus. With the Center G and Radius GL deſcribe the Quadrant LP0, produce BCM till it meet it in P: take QR: BG :: BM: GE; to QR add RS—OP, then the Surface generated by the Rotation of the Arch CE is to the Circle deſcribed with the Radius GE as Qs to GE. For the Surface generated by the Rotation of BC is {GE x GL — Zo. 2. 4 — n Mi. 4 „ — ** 7 — Gn. p c ler 1 2 ci, EOF TL, x <=GE x QS x _ but the Circle deſcribed with the Radius GE is :GEx < : therefore, dividing both by * ere, (they are to one another u Qs to GE. And this is the Con ruQtion given by Cotes, Harmonia Menſurarum, p. 28. And after the ſame manner, if you produce AN till it meet the Circle in T, take HI : GA :: AN: GE and HK : GB:: BM: GE; and upon HI produced take ID = PT, then the Surface produced by the Rotation of the Arch AC, ſhall be to the Circle deſcribed with the Radius GE, as KD to GE. Moreover, if you take HI the fame as before, and add to it Ix = OT, then the Surface generated by the Rotation of ACE, that 1s — 2d ORE ao = Tr Rv The Ruadrature of Curves explained. 312 half of the Surface of the oblong Spheroid, is to the Circle deſcribed with the Radius GE, as HX to GE. | All which things _ be ay found by the trigonometrical Canon, and the known Ratio of the Radius of a Circle to it's Circumference. i 43zo. With reſpect to the oblate Spheroid (fee Fig. Art. 422 *.) the * — this Surface generated by the Rotation of CE about BG is equal to don. x BGEM : now if you draw the Radius GM to the Hyperbola NEM, it appears that the Area BGEM = Sector GEM--ABGM : but if you draw the Aſſymptote GPQ, cutting BM and AN in Pand Q, the gector GEM may be variouſly exp by the Meaſure of a Ratio, | EM-A. BP BM+Vanrooet don. a ern * 2 BM BM'-GE xGE lp or LOGE rang &c.or *LG«xGE — : wherefore BGEM —= ABGM, together with the Meaſure of any of theſe Ratios to the Modulus GE multiplied into LG: for which it might be eaſy to make a Conſtruction, fimilar to that for the Surface of the oblong Spheroid, and like what Mr. Cotes has given in the Place already mentioned. And if the Lengths of the Lines GE, GA, 6B be expreſſed in Numbers, you may eaſily compute the Surface of any Part or the Whole of the oblate Spheroid by means of a Table . of Logas The ſame things are to be underſtood of the other Solids which have been mentioned. Which I ſhall leave to the young Geometrician for his own Exerciſe : and paſs on to the next Problem that I may haften to a Conclufion. PROB. To find the Contents of Solids generated by the Rotation of plane Figures. 431. The Principle upon which the Solution of this Problem de- pends, was fully explained Art. 43, 44 *, which amounts to this: if * Of dia 2 denote the Fluxion of a Rectangular parallelopiped, whoſe Breadth Explica- and Thickneſs. is 1, or having a ſquare Baſe whoſe Side is 1; and the 2 variable Length of the Parallelopiped, z, then the Fluxion of any Body, generated by the Rotation of any plane Figure, having æ for its Abſciſs and y for it's perpendicular Ordinate, will be equal to 2 multiplied into the Circle ſe Radius is y. Now if v and c fig- nify the fame thing as before, then the Area of a Circle whoſe Ra- Gus is y, will be deſigned by 2 : therefore if the ſolid Content of any : | ſucks — — 2 — ” — — — —— — 1 Þ 312 and retain a Value of 5 including only y and it's Fluxion : by either * Of this tent of the Solid. See Art. 390, 407 *. Explica- tion, * Of this Explica- tion, the Equation to the Curve is y* = —— , therefore by inſerting * Of this Explica- * tion. which is generated by the Revolution of a right-angled Triangle round The Quadrature of Cunvss explained. ſuch Body be called s, then you have this general Formula j 7 . Wherefore an Equation being given containing the Relation 27 of 2 and y, you may by help of it throw y out of the Equation 5— cy 2 . ; . . . =— or, if you find it more convenient, you may throw out 25 of which means you may find the Fluent by the Quadrature of Curye, by finding the Area of a Curve having an analogous Fluxion ; which Area being found, and ſuppoſed to be multiplied by 1, gives the Con. 432. Ex. 1. Let it be propoſed to find the Solidity of a right Cone; one of the Sides including the right Angle. Wherefore calling the Side about which the Triangle revolves a, and the other Side, which generates the circular Baſe, &; let z and y denote an Abſciſs and Or- dinate in that Triangle: then 2 : y:: 4: b, or y = : wherefore in the general Formula, VIS. = 2 , Inſert this Value of y, and 4 „ c*2'2 . N | - | it becomes 5 = ——=, which may denote the Fluxion of a quadrable Curve belonging to Form 1. Table 1. or the Fluents may be found by the Converſe of our Author's firſt Propoſition ; by either of which cb you'll have.s = — == (by inſerting ** for S) 2 the Value of the Cone having E for it's Altitude: which when z and y become a and b, is = i. e. a third Part of the Cylinder having the ſame Baſe and chꝛa ; Altitude, which is —. 433. Ex. 2. Let it be required to find the Solidity of any Segment of an oblong Spheroid, as that generated by the Rotation of ABC (ſee Fig. Art. 418 *.) about the Axis AG. Let @ and 6 denote the Halves of the longer and ſhorter Axis: call AB = z, BC =y, then 2b*z b*z* this Value of y* into the general Formula 5 — 2, it becomes V ch R chꝰ * : wherefore if you find (by Art. 14.5 *, or otherwiſe) the BD for it's perpendicular Ordinate; and call AB=x, BD==y, and let a, ö, e, &c. be given Quantities, and m and 7 whole Numbers. Then | | | =” "0 The Quadrature of Simple Curves, „ a - WL 4. Vf ax i it ſhall be „i Ates ABD, | The thing will be evident by an Example. 1. If x (=1x*) y, that is a=1=7, and m=2 ; it ſhall be {x3 ABD. T t 2. Suppoſe 322 ANALYSIS y Equations 2. Suppoſe 4x (Axt) ); it will be L (=34/x3) = ABD. 3. I Vo (=D) ; it will be 4x? (=3V i") = ABD. 4. If = x-*) =y, that if 22 in, and m=—2; It will be ExT=) — (= =) = «BD, infinitely extended towards &, which the Calculation places negative, becauſe it lyes upon the other ſide of the Line BD, * 5. If n (x) =; it will be (S = 2 — = BDa. Bw 4 A B 6. If ( =; it will be 2x? = * —txX1=2 —an infinite Quantity; ſuch as is the Area of the Hyperbola upon both Sides of the Line BD. The Quadrature of Curves compounded of ſimple ones, RULE ll. 4. If the Value of y be made up of ſeveral ſuch Terms, the Are: likewiſe ſhall be rhade up of the Areas which reſult from every one of the Terms, The firſt Examples. 5. If it be x*-þ- x} ==» ; it will be 2. Jr ABD. WO - For if it be always x* = BF and x: F, .” you will have by the preceding Rule {xi = Superficies AFB ; deſcribed by the Line BF; and 2x3: = AFD deſcribed by DF; where- fore * + ix: = the whole Area ABD. Thus if it be x*— x3 ; it will be 3x" | — ABD. And if it be 9x — 2K + x3 — gx+ = y; it 'willibe e -A! xt x5 ABD. | * Whatever is laid down by our Anthorwith reſpett to the Pofition df Areas of Carves ir 8 following Rules, is explained at full length in Se&. 5 of the preceding Treat. The The ſecond Examples. y-1 — 2 aD. Or if it be «K —x-+=y; it will be — x + 25: = aBD. And if you change the Signs of the Quantities, you will have the affirmative Value (x=! + 2x=5, or x — 2x-+) of the Superficies «BD, provided the whole of it fall above the Baſe AB. 7. But if any Part fall below (which happens when the Curve decuſſates or croſſes it's Baſe betwixt Band æ, x you ſee here in q) you are to ſubtract that Part from the Part above the Baſe; and fo you ſhall have the Value of the Difference : but if you would bave their Sum, ſeek both the Superficies's ſeparately, and add them. And the ame thing I would have obſerved in the D other Examples belonging to this Rule. | The third Examples. 8. If x* + x-?=y; it will be * - = the Superficies deſcribed. But here it muſt be remarked that the Parts of the ſaid Superficies ſo found, lye upon oppoſite Sides of the Line BD. That is, putting x* = BF, and x=2? =FD; it ſhall be * = ABF the Su- perficies deſcribed by BF, and — x=' —= DFa the Superficies deſcribed by DF. 9. And this always happens when the Indexes (==) of the Ratios of the Baſe x in the Value of the Superficies ſought, are affected with different Signs. In ſuch Caſes any middle part BD9P of the Superficies (which only can be given, when the Superficies is infinite upon both Sides) is thus found. Subtract the Superfieies belonging to the leſſer Baſe AB from the Superficies belonging to the greater Baſe AB, and you ſhall have BDA the Superficies inſiſting upon the difference of the Baſes, Thus in this Example (fe the preceding Fig.) WO 7 + nyo mm. - Tt2 If 324 V @. —— - * * = = — w „„ . — — * — , ie A - —— 2 —— . „ ͤ . „ * 324 ANALYSIS by Equations If AB=2, and AB—1; it will be BD Y = 7 : For the Superficies — — to AB (viz, ABF — DFa) will be - or ; and the Superficies belonging to AB (viz. ApB — Ia) will be — 1, or — : and their Difference (viz. ABF — DF AGB ＋ d — GBD9) will be 3 + + or . After the ſame manner, if AB I, and AB=x; it will be BD) = IN — N. Thus if 2x3 — 3x — 2x-4 + x-+==y, and AB==1; It will be BBD = Zx+ — 4x*% + 2x-3 + 4x5 — 42, | 10. Finally it may be obſerved, that if the Quantity æ-1 be found N 8 in the Value of y, that Term (ſince it ge- | 3 nerates an hyperbolical Surface) is to be conſidered apart from the reſt. As if it were x* + x-3 ＋ x-! =, let it be x=! BF, and x* + x— —FÞ, and AB—1; and it will be dFD 27 + "Woes 3x3 — xc, as being that which is gene- 5 —- Tated by the Terms x* EX. A | Whererefore if the remaining Superficies DFB, which is hyperbolical, be given by any Method of Computati- on, the whole BBD d will be given. The Quadrature of all other Curves. RULE III. x1. But if the Value of y, or any of it's Terms be more compound- ed than the foregoing, it muſt be reduced into more ſimple Terms; by performing the Operation in Letters, after the ſame Manner as Arithmeticians divide in Decimal Numbers, extract the Square Root, or reſolve affected Equations; and afterwards by the preceding Rules you will diſcover the Superficies of the Curve ſought. Examples, where you divide. 12, Let 18 y; Viz. where the Curve is an Hyperbola. X Now that that Equation may be freed from it's Denominator, I make the Diviſion thus. | | I > E b+x) of an infinite Number of Terms, 325 ehe = + Io = Ge 6 aa TT. e a — deen eee 15 — o + 5-TO aax* aax3 TIS. | X aax3 les: Gets aax3 aax* „ — —— n. o + 1 2 And thus in Place of this 5 N= a new Equation ariſes, viz. Y- += —= Ge. this Series being continued infinitely ; and therefore (by the ſecond Rule) The Area ſought ABDC will be equal to — . + 7 — 22 Ec. an infinite Series likewiſe, but yet ſuch, that a few of the initial Terms are exact enough for any Uſe, provided that þ be equal to x re- peated ſome few times. 13. After the ſame Manner if it be 1 =y, by dividing there ariſes | J=1—xx + x*—x* + x* &c. Whence (by the ſecond Rule) You will have ABDC= x — 4x3 + 4x5 — 2x7 + 249 Ce. Or if x* be made the firſt Term in the Diviſor, vig. thus: & A 10 there will ariſe x2 — K 14 + x=5 — x-8 &c. for the Value of ”; whence (by the ſecond Rule) It will be BDa = — Ki + 4-3 — 4-5 + 3x-7 &c. You muſt proceed in the firſt Way when x is ſmall enough, but the ſecond: Way, when it is ſuppoſed great enough.. | x4. Finally, ANALY $1 8 % Bevarions 14. Finally, if it be — — =); by dividing there ariſes . * 3x 2x% — 2x + 7x} — 13* + 34K &c. whence it will be ABDC = 4xi — x* + Ki — x3 &c, Examples, where the Square Root muſt be extracted. 15. If it be Vaa+xx = y, I extract the Root thus: | 2 x+ & x* . aa + xx (+5; = or + a = r. aa O 2 * 7 ee 1/2 * O — Jas C RF 3 8a 64 7 A * e _— 9 5 x6 x3 x10 K 15 + r 165 84 52 25 a 1 5 xÞ x10 4 x12 « ws 6445 - 54a 25610 Ge. Whence for by Lr Naa +, Xx = y, a new one is produced, vi. y += — + 2 — Sc. And (by the ſecond Rule) You will _ the Area ſought ABDC =ax + - = — — _ 15705 Sc. And this is the Quadrature of the Hyperbola. +; 16. After the fame Manner if it be Va — xx =, it's Root will be a — 5 — * — 2 — 55 Sc. and therefore the Area ſought ABDC x5 x7 will be equal to a — 7 — Joa — 1 — c. And this is the Quadrature of the Circle. 9 115247 17. . of an infinite Number of Ternit. 327 17, Or if you ſuppoſe Vx — xx = y, the Root will be equal to this infinite Series xt — 1g} — Lack — 1K — T Sc. And the Area fought ABD will be * — 5x* — — K* _ _— Se. Or x: XIX - TAU -f sA Ce. And this is the Quadrature of the Circle's Area. ; 18. If va = (whoſe Quadrature gives the Length of the Curve Elli ſe a extracting both Roots, there ariſes „ 1 * 2 1a — Ge. And by dividing as is done in the Caſe of Decimal Fractions, you il have g 1 +3 + xt + 5b) e + neh. GG. - T +34 +a al 1 - heb T + 75:68b And therefore the Area required x 4+ 5b x3 + 3-4 x5 &c. | +38 + 7:46 — 284 19. But it is to be obſerved that the Operation may be often ab- breviated by a due Preparation of the Equation, as in the Example juſt now adduced = —y. If you multiply hath Parts of the Ito 7. Fraction by Vi — , there will ariſe = — y; and the reſt of the work is performed by extracting the Root of the Nume- rator only, and then dividing by the Denominator. 20. From thefe Examples, I ſuppoſe, it will be ſufficiently evident after what Manner any Value of y may be reduced (with whatever Roots or Denominators it may be involyed, as you may ſee here 4 V==V/1== _— Vo+ 235 =? | Marx + x3 V — 3 imple Terms, from which, by the ſecond Rule, the Superficies re- quired may be known. For the Explication of this ſee Art. 341-343 of the preceding Treatiſe. Examples =y) into infinite Series of 328 near the Truth; therefore I write o, i in the Quotient, and then ſup- there are in the Quotient except one, inſtead of q I ſubſtitute — o, oog ANALYSIS by Equations Examples by the Reſolution of EqQuaTions, The Numeral Reſolution of Affetted Equations. 21. Becauſe the whole Difficulty lyes in the Reſolution, I ſhall firſt illuſtrate the Method I uſe in a numeral Equation. Let the Equation y} — 2y — 5 =o be propoſed to be reſolved: and let 2 be a number which differs from the Root ſought, by leſs than a tenth Part of itſelf. Then I put 2 +p =y, and I ſubſtitute thi; Value in Place of it in the Equation, and thence a new Equation ariſes, viz. p3 + 6p* + 10% — 1 = oO, whoſe Root p is to be ſought for, that it may be added to the Quotient: v/z. thus (neglecting y3 4. 6p* upon the Account of their ſmallneſs) 10% — 1 o, or p =o, i; poſe o, 1 p, and this it's value I ſubſtitute, as formerly, whence reſults 93 + 6,349* + 11,23q + 0,061 o. 22. And ſince 11, 239 + 0,061 o comes near to the Truth, ot ſince q is almoſt equal to — 0,0054. (viz. by dividing until as many Figures ariſe as there are places betwixt the firſt Figures of this and the principal Quotient) I write — o, oo 5̃4 in the lower Part of the Quotient, ſince it is negative. 23. And then ſuppoſing — o, oo 54 Kr ==, I ſubſtitute this as formerly, and thus the Operation is continued as far as you pleaſe. But if I deſire to continue the Work only to twice as many Figures as + into this 6,393 ＋ 11,239 + 0,061, vig. neglecting it's firſt Term () upon the Account of it's Smallneſs, and there ariſes 6, zr. + 11,16196r ＋ 0,000541708 S o almoſt, or (rejecting 6,37*) r= — 567 ED = — 0,00004853 almoſt, which I write in the negative Part of the Quotient. Finally, ſubducting the negative Part of i Quotient from the affirmative, I have 2,094 55147 the Quotient ſought, fe of an infinite Number of Terms. | 51 — 25 — 5 0 ＋2, 10000000 —=2005443853 | ＋2, 9455 147 = T2] —4— 25 — no \ 2 +p=y | TYATES > N 5 Sum | — 1 + 10% + 6p* + 55 + 0,00 ＋ 12 —+ 6,0 + 1, ＋ 10 — 1 — 1, „ T= 0 + 0,001 + o, o37 + 0,37 5 Pp Sum | + 0,061 + 11,234 + 6,39 + 93 — , 60642 T 11,23 + 11,23 0,061 + 0,061 ab 0,0054 +r=9q | + 0 To, ooo 183708 — 0,06804r ＋ 6, 37 N — 0,000048 54 +5 =7 | Sum ＋o, ooo 541708 + 1 1,16196r + 6,377 24. Equations of more Dimenſions are reſolved the ſame way, and you'll fave a good deal of Trouble, towards the End, as was done here, if you omit the firſt Terms gradually. 25. Moreover it is to be obſerved that in this Example, if T had any doubt whether o, 1 =p approached near enough to the Truth, inſtead of 10þ — 1 0, I had feigned 6p* + 10p —1 = o,* and writ the firſt Figure of it's Root in the Quotient; and indeed it's convenient to ſearch out the ſecond and third Figure of the Quotient after this Manner, when in the laſt reſulting Equation, the Square of the Coeſficient of the pe- nult Term is not ten times greater than the Product of the laſt Term and Coefficient of the antepenult Term. | | 26, Yea you will for the moſt part ſave Labour, eſpecially in Equa- tions of many Dimenſions, if you find out all the Figures to be ad- jolhed to the Quotient by the ſame Means, i. e. by extracting the lefler of the Roots out of the three laſt Terms of the laſt reſulting Equation : For in that way you will gain double as many Figures, at each Time, to be annexed to the Quotient. '# | 27. Whether 330 ANALYSIS by Eavartons 27. Whether this Method of reſolving Equations be vu practiſed I cannot tell, bit ſurely to me it ãppeats ſimple in Comp. riſon of others, and more accommodated to Practice. The Demon. ſtration of it appears from the very Manner of Operation, whence, x; Occaſion requires, you may eaſily call it into your Mind. 28. Equations wanting any of the Terms, '6r net, are managed al. moſt with the ſame Eaſe : And an Equation is always left, whoſe R together with the Quotient already found, is equal to the Root of the Equation at firſt propoſed. Whence :you may examine or prove the Work here as well as in other Parts of Arithmetic, viz. by takin away the. Quotient from the Root of the fitſt Equation (as is known to Analyſts) that fo the laſt Equation; or two or three of it's laſt Terms may be thence produced. Any Labour thete is in this Affair will be found to ariſe from the ſubſtiküting the one Kind of Quantities for the dther: Which you may do different Ways, but I think the follow. ing the moſt expeditious, eſpecially when the numeral Coefficients con- ſiſf of many Figures. , 29. et y 3 be'tobe'fabſtituted for y in this Equation 54 — 4% —+ 5 — 12y + 17 o: And ſinee it may be reſolved into this Form, by —4 * Y ＋FNYY— iz v iy Sd, the new Equation will be ge. nerated thus; p I xp +3 =p +2þ—3, and y. + 2p +2 xp+; bs 5þ* + Bp +6, and p3 f +8 3 =p + 2307 e = 18, and po f fe a. 1p 1 =0, mh vas lotight. The literal Reſolution of affected Equations. 30. Theſe things being thas ſhewn in Numbers: Let y3 + 2 — 245 ＋ axy — x D M be propoſed to be reſolved. 85 J firſt ſearch for the Value of y when x is nothing, that is, I ſeck the Root of this Equation y3 y — 243 , and I find it to be + a. Therefore I write ¶ꝙ à in the Quotient; and ſuppoſing -+ 2 -+p=y, I fubſtitate-for y it's Value, and put the Terms thence te- falting (p3 -|-34p* 4 4a*p Fc.) in the Margin; out of which I affume + 4a + a*x, viz. the Terms where p and x ſeparately are of the leaſt Dimenſions, and I ſuppoſe them equal almoſt to nothing, ot 't þ = — 2x-almoſt, or p =- e g. And writing — {x in the Quotient, I ſubſtitute — 2x ＋ for p; and the Terms thence reſulting I write again in the Margin, as you may fee in the Scheme annexed; al . Sb. r F an infinite. N umber of Terms. | 331 and thence I aſſumę the Quantities 44% — fear“, viz, thoſe in which 5 and „ ſeparately- are of the leaſt Dimenſions, and I feign, q = = almoſt, or q = 75 Tr; and ſubjoining + 555 to the Quotient, 644 | 44 4a [ ſubſtituts * for q; and thus I procced as far as you pleaſe. * + ay — 243 + axy — K* 0 2 131 x ogx+ y=4=7 d + fa © nr Ge. G12a* 1638483 F ＋ 34 + 3% N 5 ＋ a? 15 131 * 5ogs+ | + 4a — % + Ver-) FT 1 = 49g6a ( + 5124* * 163844 31. But if I deſire only double as many Terms ſave one to be fur- ther adjoined to the Quotient: I omit. the firſt Term (93) of the laſt reſulting Equation, and likewiſe that Part (— jxg*) of the ſecond Term, where x is of as many Dimenſions as in the penult Term of the Quo- dent; and ſubſtitate 77; +» forginto the other Terms (3091-427 Se) placed in the Margin as you ſee; and then from the laſt two Terms | Unu2 ( I5x+ 409 a S 4 . ANALYSIS by EON 2 8 4096a WE, * + For * Laxr + 40%) of the Equation thence reſulting, by dividing thus: E 131 x+ . 44 — 4x + Prat) + ! — 2 (1 bring out + == + 2 5124% hoon to be adjoined to the Quotient, 32. Finally that Quotient (a — + 855 Sc.) by the ſecond Rule, . . * 2 * * will give ax — + + 7; > 2. L 119255 Sc. for the Are ſought, which approaches ſo much nearer to the Truth, the leſs that *# " | Another Way of reſolving the ſame Equati ons, 33. But if the Value of the Area ought to approach the nearer to the Truth the greater that x is; of which let this be an Example 43 + axy + x*y — a3 — 2x3 =o. Therefore, being about to reſolye this, I chuſe out the Terms y3 + x*y — 2x3, in which x and y either ſeparately, or multiplied together are of the moſt and at the ſame Tine of equal Dimenſions every where; and from them as if equal to no- thing I diſcover the Root. This I find to be x, which I write in the Quotient. Or, which comes to the fame Purpoſe, I extract the Root out of 1 ＋ y - 2 =0 (ſubſtituting Unity fr x) which Root here comes out 1, and I multiply it by x, and write the Product (x) in the Quotient. Finally I put x + p =, and foI proceed as in the former Example until I obtain the Quotient x — © + Q_.z; 3165 , _gagal — — 64x 512* | 16384 — 42 4* + KITE 1 goga⸗ &c. and fo the Area ſought — : +| — 9 concerning which ſee the third Examples of the ſecond Rule. For the ſake of illuſtration J have given this Example the ſame in every Reſpe with the former, provided x and a be there ſubſtituted for one another, that there might not be any Neceſſity to adjoin here any other Example of this Reſolution. 34. But the Area - — 7 —- which extends itſelf infinitely along ſome Aſſymptote; and the initial 1 Sir 1/aac uſes this Notation to denote the curvilinear Area belonging to the Ordinate 8 | Gaz? which belongs to the Hyperbola, For underſtanding of which conſult Art. 190 — 19? of the pre:eding Treatiſe. 7 1 FE | | 577] &c. is terminated at a Curve Terms F an infinite Number of Terms. Terms x — 54) of the Value of y extracted, always terminate in that Aſſymptote; whence you may eaſily find the Portion of the Aſſymptote. The ſame thing is always to be obſerved when the Area is denoted by Terms divided more and more by x continually, only that in place of right-lined Aſſymptote, ſometimes you will have the conical Parabola, or perhaps one more compounded, | 35. But paſſing this Manner of Reſolution, as being particular, be- cauſe not applicable to Curves which return into themſelves like El- lipſes ; with 1 4 to the other Manner of Reſolution, which was ſhewn above in the Example y3 N- a*y + axy — 243 — x3 = O (viz, that Method in which the Dimenſions of x-perpetually increaſe in the Numerators of the Quotient) the following Things may be remarked. 1. If it happen at any Time that the Value of y, when x is ſuppoſed to be nothing, be a ſurd Quantity; or one entirely unknown, yet you may deſign it by ſome Letter. Thus in the Example y3 + a*y -þ axy — 245 — x3 o, if the Root of this Equation y3 ++ a*y— 243 = 0 had been a Surd, or unknown, I would have ſuppoſed any Quantity (5) put for it; and have performed the Operation as follows. Writing 6 in the Quotient, I ſuppoſe þ+p = y, and ſubſtitute that for y, as you may ſee; whence a new Equation p3 + 30% &c. = © ariſes, in which the Terms 6 ＋ a*h— 243 are to be rejected, as bein equal to nothing, becauſe 5 is ſuppoſed the Root of this Equation ay — 243 Soo. Then the Terms 36*þ + ap + abx give — a to be adjoined to the Quotient ; and — = + to be ſubſtituted for p, &c. : 333 3.34 ANALYSTS 5 EquaTroNs —— * 4- a'y * — 243 —» x3 o. Put ce 234 a®, | 5 a#*bx* x3 „ aibixz3} —- alba) aShisn z=b===+ —— + x Fa Da TH Tar 3 | TH 1 | 4-43 + 36 2 Þ j3 y A7 a 1 3 Ib | + aay | + aab + a*p | | 3 xt tab vx ＋ 3 re. 5 h | —abs + c | bGaber M athx* n . * ＋ ax — ——) —- + 8 + = (== + => + — — Ge. | The Work being finiſhed, I take any Number for a, and I reſolve this Equation y3 - a*y —243 = o, after the ſame Manner as was ſhewn above in the Caſe of Numeral Equations; and then I ſubſti- tute it's Root in Place of 5. | | 2. If it ſhould happen that the ſaid Value be nothing, that is to ſay, if in the Equation to be reſolved, there be no Term but what i multiplied by x or y, as in this y} —axy + x3 = o; then I ſelect the Terms (—axy A x3) in which x ſeparately, and alſo y ſeparately, if that can be had, otherwiſe multiplied by x, is of the feweſt Dimenſions. And theſe Terms give + — for the firſt Term of the Quotient; and = p to be ſubſtituted for y. In this Equation y3 — a*y + axy — x3 So, you may diſcover the firſt Term of the Quotient, either from — a*y 2 ; A from y} — afy. | 8 If that Value of y be imaginary, as in this y+ ＋ — 27 + " 9 *) — 2x + x* + x* =o, I increaſe or diminiſh the Quantity x until the ſaid Value become real. Thus of un infinite Number of Terme. Thus in the Figure annexed, when AC (x) is nothing, then CD ( ”) is\tmaginary. E= But if AC be diminiſhed by the given Line PR A AB, that ſo BC may become x; then ſuppoſing : | ; BC (c) to be nothing, CD (Y) has a fourfold C TT Value (CE, CF, CG or CH) each of them real ; any of which Roots (CE, CF, CG or CH) 2 1 may be the firſt Term of the Quotient, ac- cording as the Surface BEDC, 'BFDC, BGDC or BHDC is deſired. jn other Caſes likewiſe, where you find any Stop, you may extricate yourſelf by the ſame Means. | Finally if the Index of the Power of x or y be a Fraction, I reduce it to an Integer; as in this Example y3 — xy: + xt o putting == u, and x3 2, there ariſes v® — 239 + 2+ o, whoſe Root is y=2 + 23 &c. or (by reſtoring the Values) yz =-x3 -+ x &c..and by ſquaring, y = A 2&1 &c. | 36. And this much. ſhall ſuffice to have ſaid touching the Inveſtiga- tion of the Areas of Curves. Yea ſince all; Problems concerning the Length of Curves; the Quantity and Surface of Solids; and the Cen- ters of Gravity may at length be brought to this, viz. that the Quan- tity of ſome Plane Surface bounded by a Curve Line 1s ſought, there is no Neceſſity to ſay any thing about them here. However. ſhall ſhew very briefly after what Manner 1, perform the Operation in theſe Things. The Application f ꝛwbat has been ſaid to other Problems of that Kind. 37. Let ABD be any Curve, and AHKB D A Rectangle, whoſe Side AH or BK is Unity : _—_— And imagine the Right Line DBK to move uniformly from AH, fo as to deſcribe the Areas ABD and AK; and that BK (i) is the B Moment with which AK (x), and BD ()) 4 5 the Moment with which ABD is gradually encreaſed ; and that from the Moment B 7 — continually given, you can, by Means of the 1 K preceding Rules, inveſtigate the Area ABD deſcribed by it, or com- pare it with AK (x), which is decribed with the Moment 1. Now by the fame Means that the Superficies ABD from it's Moment being at all Times given, is diſcovered by the foregoing Rules, by the like Means may any other Quantity be inveſtigated from it's Moment given in like manner.. The Thing will be clearer by an Example. —— To 335 336 ANALYSIS h Eqvarions To find the Lengths of Curves. 38. Let ADLE be a Circle, the Length of whoſe Arch AD i, 0 be inveſtigated. Draw the Tangent DHT, and having complete the indefinitely ſmall Rectangle L HGBK, and put AE=1=2AC 1 D it ſhall be as BK or GH the Moment 7 of the Baſe AB (x) to HD the Mo- ; | ment of the Arch AD : : BT: Dr 2 3 Wow :: BD (V -x) : DC(<+)::; F ARI C E. (BK): =—(DH). And f 24/ — K or Y*—= is the Moment of the Arch AD. Which being 2 * X —— XX 2X — ZXX reduced makes * + A* ＋ xt + x5 + e b ! Ge, Wherefore by Rule the ſecond the Length of the Arch is x: + ix! + | xz —— Tr —— Tir * Tre, Se. Or Xi into I + * + =O ; + Tx TIN ix, Oc. | | 39. After the ſame Manner by ſuppoſing CB to be x, the Radius CA to be 1, you will find the Arch LD to be x + 4x3 + x5 - T7, Se. 40. But it is to be remarked that that Unity which is put for the Moment, is a Superficies, when the Queſtion is about Solids ; and a Line when about Superficies; and a Point when it is about Lines (as in this Example.) Neither am I afraid to ſpeak of Unity in Points, ct | | Lines infinitely ſmall, ſince Geometers are wont now to conſider Pro- portions even in ſuch a Caſe, when they make uſe of the Methods ot Indiviſibles. | 41. From theſe Things one may gueſs how one ought to proceed in inveſtigating the Superficies and Contents of Solids; and likewiſe the Centers of Gravity. To find the Converſe of theſe Things. 42. But if upon the contrary, from the Area, or Length, &c. ct any Curve being given, the Length of the Baſe AB be required, then you mult extract the Root x, out of the Equations which have been found by the preceding Rules, — G — — 7 Ty 8. Cats — —— — F an infinite. Number of Terms. To find the Baſe from the Area given. 43. Thus if from the Area ABDC of the Hy- p14 perbola ( 7 — =) given I wanted to inveſtigate the Baſe AB, calling the Area z, I extract the Root of this Equation 2 (ABCD) = x — AK + ix3 — 4+, &c. neglecting thoſe Terms in which x is of more Dimenſions than ⁊ is deſired in the Quotient. - As if I would have z to riſe to five Dimen- | E ſions only in the Quotient, I neglect all the Terms — z' + * — 1x3, Cc. and extract the Root of this only 4x5 — 4x+* + x3 — * +bx—2Z=0, x=2 +12 +12) + 42* + 51525 Ge. 2 ＋ = | + 4x5 | + 425 Ge. — 2 — 22 —- 2% Ge. T 3 | T5385 + 2Þ Tp Ge. ( —j—p — +x [TS + \ — 2 — 2 1 = T 4, Ge. + * 1 TE — 4 — 42 + 1993 + 223 1 — Ez) 223 — 12 + 5235 (69 + rt 8 — 44. I have laid the Steps of the Reſolution before you, as you ſee, upon the Account of the two following Remarks. I. That in the Subſtitution, I always omit thoſe Terms, which T foreſee will be of no Uſe afterwards, Concerning which this is the n X X Rule; 337 338 ANALYSIS by Eqvartrons Rule; That after the firſt Term reſulting from each Quantity that i; collateral to it, I add no more Terms upon the right Hand than the Index of the Dimenſion of that firſt Term wants Units of the In- dex of the greateſt Dimenſion. As in this Example, where the greateſt Dimenſion is 5, I neglect all the Terms after 25, J put one after 2+, and two only after 23, When the Root (x) to be extracted, is every where of even or odd Dimenfions, let this be the Rule: That after the firſt Term, reſulting from each Quantity which is collateral to it, you add no more Terms towards the Right Hand, than what the Index of the Dimenſion of that firſt Term, wants Pairs of Units of the Index of the higheſt Dimenſion; or no more than what it wants Ternaries of Units, when the Indexes of the Dimenſions of x differ by three Units ; and fo in others. 2. When I ſee that p, 9, or r, Cc. in the laſt reſulting Equation, is found of one Dimenſion only, I ſeek it's Value, that is to ſay the remaining Terms, which are ſtill to be added to the Quotient, by means of Diviſion ; as you ſee done here. To find the Baſe from the Lengtb of the Curve given. & 45. If from the Arch D given the Sine AB Sr was required; I extract the Root of the Equation E found above, vis. 2 =# + 4x3 + 3x5 + Taz? (it being ſuppoſed that AB x, aD=z, and Ag = 1) by which I find x 2 — ($23 + Ig Terz + 7 5 * o 5? Tc. 46. And moreover if the Coſine AB were A E required from that Arch given, make AB (= I- i — 2˙ + 272 — 542 + 72 0 1 8 | ; 4034 L Trrretsz , Se. Concerning the Continuation of the Series of the Progreſſions. 47. Let it be obſerved here, by the bye, that when 5 or 6 Terms of thoſe Roots are known, they -may be continued at Pleaſure for moſt Part, by obſerving the Analogy of the Progreſſion. Thus you may continue this x = 3 + 42* + 423 + £2* + 5552) &c. by dividing the laſt Term by the following Numbers in Order, 2, 3, 4, 5, 6, 7 Se. : And this x = 2 — 423 ＋ 125 — 5227, &c. by dividing by theſe Numbers 2 X 3, 4X5, 6X7, 8X9, 10 X11, &c, Aud F an infinite Mumber of Terms. And this x =1 — 2 + 2781 — ga“, Cc. by theſe 1 x 2 5x6, 7X8, 9X10, &c, 7 ; ö * And — — ＋ 2a * Ir Shack, &c. by multiplying by . * 5K 7X . theſe, * 225 axe? Gun? Bug? Sc. And ſo in other 8. The Application of what bas been ſaid to Mechanical Curves, 48. And what has been faid ſhall ſuffice with Reſpect to Geometri- cal Curves. But now, although the Curve be Mechanical, yet this Method of ours may be applied to it. For Example, let ADFG be the Trochoid whoſe Vertex is A, and Axis AH, and AKH G the Wheel or Circle with which it is deſcribed, © And the: Superficies ABD is required. In Order s/ | to which, putting AB x and BD Dy as for- merly; and AH = 1, I firſt ſeek the of BD. Now from the) Nature of the Trochoid, you have KD Arch AK: Wherefore the whole Line BD = BD + Arch AK. But BK (= Vx —xx) = x* — xt —qaxt— 4, Cc. T ABC HH And (from what was formerly ſhewn) the Arch AK = x + qxi + 2-x+ + xt &c. Therefore the whole BD = 2XT - IXI — Z-x* — f Cc. And (by Rule the ſecond) the Area ABD = 4xt — Fri — xt zx? &c. | Or more ſhortly thus: Since the Right Line AK is parallel to the Tangent TD, it will be AB to BK, as the Moment of the Line AB to the Moment of the Line BD, that is, x: /x — xx 2 1 2 & = X=+ = A* — xi — xt — er Cc. Wherefore (by Rule the ſecond) BD = 2x% — K* — xi — r — 53x? Cc. And the Superficies ABD = * — xt — xt — * — Fox? Cc. 49. Much after the ſame Manner (if C be b. the Center of the Circle and CB x) you may find the Area CBDF Cc, | 50. Let it be required to find the Area ABDU B of the Quadrature UDE (whoſe Vertex is U, and A the Center of the interior Circle UK to which it is fitted. | Having drawn any Line as AKD, I let fall the Perpendiculars DB, DC, KG. And fo it ſhall be KG: AG:: AB (x): BD (»), R x2 or 339 = — - — 3 — ùu—ͤ— 1 ANALYSIS by EqwvATTONSs x x AG or K =y. But from the Nature of the Quadratrix, you have BA (=DC) = Arch UK; or UK x. Wherefore putting AU =, it will be GK = & — 5x3 ex, Cc. from what was ſhewn above, and GA =1 — A* + Z#x* — -i-x* &c. IAG 1—i$x* + xt — A Ofc. And therefore y (= = EEE e., Or, by actual Diviſion y== 1 — {x* — n — -+-x* &c. and (by Rule the ſecond) the Area AUDB = x — 5x3 = e — 37x? Tc, 51. Thus alſo the Length of the Quadratrix UD may be determined, although the Calculation be ſomething more difficult, 52, Neither do I know any Thing of this Kind to which this Me- thod doth not extend, and that in various Ways. Yea Tangents may be drawn to Mechanical Curves by it, when it happens that it can be done by no other Means. And whatever the common Analyſis per. forms by Means of Equations of a finite Number of Terms (provided that can be done) this can always perform the ſame by Means of in- finite Equations : So that I have not made any Queſtion of giving this the Name of Analyſis likewiſe. For the Reaſonings in this are no leſs certain than in the other; nor the Equations leſs exact; albeit we Mor- tals whoſe reaſoning Powers are confined within narrow Limits, can neither expreſs, nor ſo conceive all the Terms of theſe Equations, as to know exactly from thence the Quantities we want: Even as the ſurd Roots of finite Equations can neither be ſo expreſt by Numbers, nor any analytical Contrivance, that the Quantity of any one of them can be ſo diſtinguiſhed from all the reſt, as to be underſtood exactly. 53. To conclude, we _ juſtly reckon that to belong to the Anahytic Art, by the Help of which the Areas and Lengths &c. of | Curves may be exactly and geometrically determined (when ſuch a thing is poſſible). But this is not a Place for inſiſting upon theſe Things. There are two Things eſpecially which an attentive Reader will ſe need to be demonſtrated. 1. The Demonſtration of the Quadrature of Simple Curves belonging to Rule the firſt. Preparation for demonſtrating the firſt Rule. 54. Let then AD9 be any Curve whoſe Baſe AB— x, the perpen- dicular Ordinate BD = y, and the Area ABD = 2, as at the Begin- ping. as ww ore he © Sg =D of an infinite Number of Terms. ning. Likewiſe pur BB = o, BK u; and — Ne Rectangle BEHK. (ov) equal to the Space — H JD D. a it is A x To, and AB 2 ov: Which Things being premiſed, aſſume any Relation betwixt x and z that you pleaſe, 3 and ſeek for b in the following Manner. C Take at Pleaſure 3x} =; or 4x3 = 27, Then x + o (AP) being ſubſtituted for x, and 2 + ov (AP) for 2, there ariſes + into x3 4- 3x0* gx + 03 (from the Nature of the Curve) 2* + 220 + o*v*, And taking away Equals (4x3 and 2*) and dividing the Remainders by o, there ariſes + into 3x* + 2x0 + 00 =22v + ovv. Now if we ſuppoſe BB to be diminiſhed infinitely and to vaniſh, or © to be nothing, v and y, in that Caſe will be equal, and the Terms which are multiplied by o will vaniſh : So that there will remain + x 3x* = 2zv, or 4x* (= zy) xh; or xi == y. Wherefore converſely if it be x ==y, it ſhall be 2xi g. The Demonſtration. m+n 55. Or univerſally, if * X ax ; or, putting — 2 c and m+ p, if 2 2; or x? =2": Then by ſubſtituting x+o for x, and 2 + ov (or which is the ſame z + oy) for a, there aries c into x + po, Cc. = 2" + noyz"”" T&s. the other Terms, which would at length vaniſh being neglected. Now taking away e and 2* which are equal, and dividing the Remainders by o, there temains px?" = n (= * = , or, by dividing by 2 2 cx Ox, it (hall be p = 2 z or ber d = or by reſtoring nn Cx» | wy for e, and m -þ 7: for p, that is m for p— , and na for pc, it becomes m m m- n as =y. Wherefore converſely, if ax* =y, it ſhall be 1 Ua n =. QE. D. * ON To 341 ANALYSIS by Equations 2 To find thoſe Curves which can be ſquared. Ml 56. Hence by the Way you may obſerve after , what Manner 2 1 many Curves as you pleaſe may be found, whoſe Areas are known * viz. aſſume any Equation you pleaſe for the Relation betwixt the Are, S and Baſe x, and thence let the Ordinate y be fought. Thus if you ſuppoſe /aa+xx = , by performing the Calculation you will find Qu . And ſo in other Caſes. 1 II. The Demonſtration of the Reſolution of Affeded p Equations, J. 5). The other Thing to be demonſtrated, is the literal Reſolution f of affected Equations. Vix. That the Quotient, when x is ſufficiently - ſmall, the further it is produced, approaches ſo much nearer to the - Truth, fo that the defect (p, 9, or r &c.) by which it differs from the full Value of y, at Length becomes leſs than any given Quantity, and that Quotient being produced infinitely is exactly equal to) Which will appear thus. | 1*, Becauſe that Quantity in which x is of the loweſt Dimenſion (that is to lay, — — the half of the laſt Term, provided ſuppoſe x ſmall enough) in every Operation is perpetually taken — of the laſt Term of the Equations, Ff-which A n 7 2 are the Roots: Therefore that laſt Term (by x. 10. Elem.) at length be. comes leſs than any given Quantity; and would entirely vaniſh, if the Operation were infinitely continued, | s (2210 Thus if it be x = 4, you have x the half of all theſe x + x* + x + x+, &c. and x* the half of all theſe x* + x3 + + + x5, Ge. Therefore if x , x ſhall be greater than the half of all theſe x + x* + x3, Sc. and x* greater than the half of all theſe x* + x3 ＋ x+, Cc. Thus if - 2, you ſhall have x 3 | 3 — 4 Þ more than the half of all theſe x + 4 = e 7 + , Sc. And the ſame Way of others. And as to the numeral Coefficients, for moſt part they perpetually decreaſe: Or if at any Time they increaſe, you need only ſuppoſe x ſome few Times leſs. 11 of an infinite Number of Terms. 29. If the laſt Term of any Equation be continually diminiſhed, ll it at length vaniſh, one of it's Roots ſhall likewiſe be diminiſhed, until the laſt Term vaniſhing, it vaniſh along with it. ze. Wherefore one Value of the Quantities p, g, r, &c. continually decreaſes until at length, when the Operation is infinitely produced, it qaniſh entirely. | . But the Values of theſe Roots p, 9, r, &c. together with the Quotient already extracted, are equal to the Roots of the propoſed Equation (thus in the Reſolution of the Equation yi + a*y + axy — 20 — * o, ſhewn above, you will obſerve that y=a + p= 4 ix e + ze +7, Ge.) Whence it is ſufficiently evident that the Quotient infinitely produced, is one of the Values of y: Which was what I propoſed to ſhew. _ Is | 58. The fame. Thing will appear by ſubſtituting the Quotient in- ſtead of y in the Equation propoſed. For you will perceive that thoſe Terms perpetually deſtroy one another in which x is of the leaſt Di- menſions. Sir 343 — 4 8 4 — — 1 . „ — n At — 1 — Sir ISAAC NME M O -; ANALYSIS Equations of an infinite Number of Terms, explained. A SEW Ii. L INTRODUCTION, AVING endeavoured, in what goes before, to explain and illuſtrate Sir Jaac Newton's Treatiſe of the Quadrature of Curves: In order to render this Doctrine, together with what depends upon it, as compleat as I can, for the Ute of the young Geometrician, I now defign to give him ſome Aſſiſtance, for his better underſtanding this other in Performance of our cele- brated Author, according as I promiſed. And therefore, as this Ana- lyſis treats of the Doctrine of Infinite Series and Equations, chief with a View to the Quadrature of Curves, and other Parts of the Geo- metry of Curve-Lines, I ſuppoſe it will neither be unſuitable to the Deſign, nor unacceptable to the Reader, to lay before him a brief Ac- count of the Riſe and Progreſs of the Contemplation of Infinites and the Quadrature of Curves, in Mathematics, — it's Infancy in the Method of Exhauſtions, uſed by the Ancients, to it's preſent State, as it has been perfected by Sir Jaac Newton into the Method of Infinite Series and Fluxions. «> 2. When I. 1 . a» a. a an es ANALYSIS 4y EuATIONS, &c. 2. When the Ancients began to indulge themſelves in the Study and Contemplation of the Properties and Relations of Magnitudes, they yery ſoon found it neceſſary to conſider and inveſtigate the fundamental Principles of the Doctrine of Proportion: And accordingly we find it tao Magnitudes of the SAME KIND to each other according to Quantity: And in the next Definition, he defines Magnitudes capable ot bearing Ratio or Proportion, to be ſuch, as being multiplied or repeated ſome cer- appears that Quantities that are of different Kinds, ſuch as Lines and carfaces, Surfaces and Bodies; any of theſe and Time; Time and Velocity, &c. cannot be ſaid to have Ratio or Proportion to one ano- ther: Both becauſe they are of different Kinds ; and likewiſe (which indeed follows from the other) becauſe no Multiplication or Repetition of the one can ever make it exceed the other; or even conſtitute any Quantity of that Kind : For no Multiplication of a Line can ever make any Surface; nor any Portion of Time, however multiplied, make any Degree of Velocity, Cc. But it is evident, by the ſecond of the preceding Definitions, that Euclid deſigned by it to limit Magnitudes Homogeneity : Otherwiſe his fourth Definition had been ſuperfluous. And by what comes afterwards, in the ſucceeding Parts of theſe Ele- ments, it appears, that he meant to lay a Foundation for the Doctrine of Exhauſtzons, as it is wont to be called; as well as that of Incom- menſurables. For being about to treat of the Doctrine of Incom- menſurables in the tenth Book, he lays down and demonſtrates this Propoſition by way of a Lemma, viz. Two unequal Magnitudes being propoſed, if from the greater you take away more than the half, and from the Remainder more than it's half; and ſo on continually, there will remain a Magnitude at length, by this continual Subduction, which is leſs than the leaſt of the two Magnitudes propoſed. In demonſtrating which he aſſumes this Principle or Poſtulate, that he leſſer of the two Magnitudes being multiplied, will at length exceed the greater: For if not, they can bear no Ratio to one another by Def. 3. B. 5. which yet they are ſuppoſed to do, that they may be capable of any Com- pariſon or Relation to one another. 3. Now this Propoſition or Lemma is the Foundation upon which the Method of Exbauſtions is built. Which Method is made uſe of by Euclid in demonſtrating ſome Propoſitions, as the ſecond, tenth, A/ ig Ibo pwryivav opoymar 1 Kala ννẽ0d ö d Told TXE0K. | Yy eleventh, delivered by Euclid in the fifth Book of his Elements. In the third Definition, he detines Ratio or Proportion to be * a certain Habitude 0 f tain Number of Times, may EXCEED one another, From whence it. 345 that bear Proportion to each other, to ſome other Idea than that of | 346 ANALYSIS by EqvarTions, &c. eleventh, twelfth, and eighteenth, B. 12. El. in which he proves that Circles are to one another as the Squares of their Diameters; thy a Cone is a third part of a Cylinder of the ſame Baſe and Altitude: that Cones and Cylinders of the ſame Altitude are to one another 8 their Baſes; that ſimilar Cones and Cylinders are to one another in triplicate Proportion of the Diameters of their Baſes; and that Sphere, are to one another in a triplicate Proportion of their Diameters. And Archimedes makes conſtant Uſe of it throughout all his Writings. For he, beginning where Euclid and other Geometricians before himſelf left off, found that this, or ſome Principle equivalent to it, was abſolutely neceſſary to make any Progreſs in the Contemplation of Plain Cumi. linear Figures; and Solid Figures generated by their Rotation; which were not capable of being divided into Parts bounded by Right Lines or Plain Surfaces, and therefore we find him, in ſeveral different Places, aſſuming this Poſtulate, that of rio unequal Magnitudes of the [any Kind, the Exceſs of the greater above the leſſer cannot be ſo ſmall but that it may be ſo multiplied as to exceed either of them; or any other Magnitude of the ſame Kind. Accordingly, by the Help of the Prin- ciple, he has demonſtrated, by the Method of Exhauſtion or continual Approach, the Quadrature of the Parabola; having ſhewn that any Segment of a Parabola cut off by a Right Line is to the inſcribed Tri. angle of the ſame Baſe and Altitude as 4 to 3; that a Circle is equal to a right-angled Triangle whereof the two Sides including the Right- angle, are equal the one to the Semi-diameter, the other to the Peri- phery of the Circle; and has ſhewn how to approach to the Quz- drature of the Circle and Ellipſe. He has demonſtrated the Ratio of the Cylinder, Hemiſphere and Cone having the ſame Baſe and Height, both with reſpect to their folid Contents, and Surfaces ; and ſhewn in general how to compare Spheres, Spheroids, Parabolical, and Hyper- bolical Conoeids, or any Portions of them, with Cones or Cylinders: And with Reſpect to the Surface of the Sphere, he has demonſtrated that it is equal to 4 times the Area of one of it's great Circles. And beſides many noble and uſeful Diſcoveries with Reſpect to theſe Figures mentioned, he wrote a moſt elaborate Treatiſe upon the Spiral Line, which commonly goes under the Name of Archimedes's Spiral, in which he demonſtrates the Proportion of the Spiral Spaces to the Areas of the circumſcribing Circles; and that of the Subtangents in ti:e Spiral, to the Perimeters of the Circles. All theſe Things are demonſtrated moſtly by the Method of Exhauſtions. Which may be conſidered as the firlt Step towards the general Method of Quadratures, and infinite converging Series lately introduced. This bath got the N my ys | | et EATPLAITINE D. Method of Exhauſtions, becauſe, in comparing two unequal Quantities with one another, by taking more than half the Difference, and addin it to the leaſt, or ſubducting it from the greateſt ; and again takin more than the half of the remaining Difference, and adding it to the leaſt, or ſubducting it from the greateſt ; and fo on continually, the Difference betwixt the two Quantities is gradually exhauſted, fo as to become leſs than any given Quantity, in virtue of what is demonſtrated by Euclid in that Lemma, which hath been mentioned : And it would be the ſame thing, if, inſtead of taking more than half the Difference, you take half the Difference, or even any given Part of the Difference, and add it continually to the leaſt ; or ſubduct it continually from the greateſt of the two Quantities: For this likewiſe will gradually exhauſt the Difference, ſo as to make it leſs than any given Quantity. | 4. That the Way of proceeding in this Method may be the bette underſtood, I ſhall ſhew how Archimedes demonſtrates by it, that a Circle is equal to a right-angled Triangle, whereof one of the Sides including the Right Angle, is equal to the Radius of the Circle, and the other to it's Circumference. In Order to which, let the Circle and Triangle be repreſented by the two Right Lines AB and hls E FGBEI II CD, as Symbols. Moreover let A. n AE, AF, AG, Sc. repreſent a Square, an eight ſided, a ſix- | teen fided Figure, &c. regularly A inſcribed in the Circle: The | Number of Sides being always doubled. Again let AH, AI, AK, Ge. repreſent a Square, a regular eight ſided, fixteen ſided, &c. Figure, circumſcribed about the Circle. Then I ſay AB = CD. For if not, then AB is either greater than CD; or leſs. If AB be greater than CD, let the Exceſs be x: Then fince EB, FB, GB, Sc. are the Differences betwixt the inſcribed Figures and the Circle, EF is greater than the half of EB; FG greater than the half of FB, &c. (as may eaſily be demonſtrated) conſequently (by Prop. 1. Elem. 10.) you muſt come at length to ſome Figure inſcribed as AG, whoſe Difference from the Circle AB, viz. GB is leſs than x : Wherefore AG muſt be greater than the Triangle CD, ſince AB was equal to CD . But AG, being an inſcribed Figure, is only equal to a right-angled Triangle having one of the ſides equal to the Perpendicular drawn from the Center upon one of the Sides of the Polygon ; and the other to the Sum of the Sides of the Polygon; the firſt of which is leſs than the Radius of the Circle; and the other leſs than the Circumference ; Yy2 to C 4D 347 348 Difference, but never to go beyond them, they concluded theſe Quanti- ANALYSIS by Equations, &c. to which the Sides of the Triangle CD are equal: Therefore AG;; leſs than CD, contrary to what has been demonſtrated. Wherefore it is not true that AB is greater than CD. I fay neither is AB leſs than CD. For if AB be leſs than CD; let the Defect be x, fo that it be AB ++ x = CD; then it is eaſy to ſhew that HI is greater than HB; that IK is greater than + IB, Sc. and that therefore you muſt (by the Propoſition already mentioned) at length arrive at ſome Difference a KB leſs than x: Wherefore AK is leſs than CD, ſince AB + x = Ch But AK, being a regular Polygon circumſcribed about the Circle, i; equal to a right-angled Triangle having one of it's Sides equal to the Radius, and the other equal to the Sum of all the Sides of the Polygon, i. e. greater than the Circumference of the Circle : Which Triangle therefore is greater than the Triangle CD : Wherefore AK is alſo greater than CD; contrary to what has been demonſtrated : For it was ſhewn to be leſs. Therefore the Circle AB is not leſs than the Triangle CD; and it was demonſtrated that it is not greater. Therefore they are equal. Q. E. D. 5. By this example it may appear in what Manner the ancient Geo. metricians proceeded, when they demonſtrated any Propoſition by the Method of Exhauſtions. The Characteriſtic of which is, that one Quantity makes a continual Approach to another, ſo as to exhauſt the Difference by degrees, until it become leſs than any given Quantity. And it is the ſame in Ratio's: When a Ratio, which is variable, con- tinually approaches to another conſtant Ratio, ſo as to differ from it at length by leſs than any given Ratio. Where they could diſcover any two Quantities, or two Ratios, which were both Limits to which the ſame variable Quantity, or Ratio, continu- ally approached, ſo as at length to come nearer them than by any given ties, or Ratios, which were the Limits, to be equal. And this they de- monſtrated by that kind of Demonſtration which is called ducens ad abſurdum: Which Demonſtrations were ſometimes Jong ; but always ſtrict and exact. To avoid the Tediouſneſs of ſuch Demonſtrations, it was that our Author invented his Method of prime and ultimate Ratios *, Which is founded upon a like Principle, as the Method of Exhauſtions; and the ſame Objections lye againſt the one as againſt the other, as we have ſhewn towards the Beginning of our Notes upon the Quadrature gf Curves. * 70% 6. As, by what has been ſaid, it appears in what Manner the An- cients entered into the Contemplation of Curvilinear Areas, and Solids 2 Phil. Nat. Prin. Sect. 1. Schcl. | generated —— ES LATNED. generated by their Revolution; ſo likewiſe we may obſerve the Doctrine of infinite converging Series, beginning to appear in Embryo, as it were, in their continual Approaches, by exhauſting the Difference. For the Steps of the Approach, may be ſuppoſed to be continued indefinitely : For although a finite number, or even a very few of them, is ſufficient to make out their Demonſtrations; yet it is evident that no finite Number of them can ever entirely exhauſt the Difference. Thus in the preceding Example taken from Archimedes, AE -|- EF + FG, Ge. in inf. repreſents an infinite Series of ſimple Terms, according to Sir Iſaac Newton's Method ; which is that generally uſed ; where the Ag- oregate of all the Terms, infinite in Number, is equal to AB; and the greater the Number of them, taken from the Beginning, is, the nearer doth the Aggregate of them approach to AB: So that a finite Number of Terms differs from AB by leſs than any given Difference. Again AH . AE, AI. AF, AK. AG, Gc. repreſents Mr. James Gregory's Method of converging Series in his excellent Treatiſe Vera Circuli & Eyperbolæ Quadratura, In which he ſuppoſes the Terms of the Series to proceed by Pairs, as A and B, C and D, E and E, Sc. ſuch, that the Difference betwixt A and B is greater than the Difference betwixt Cand D; and that after the ſame Manner that C and D are formed by an analytical Operation from A and B; after the like Manner, or by the like analytical Operation, E and F are formed from C and D; and ſo on continually : Every new Pair of Terms being produced al- ways in the ſame Manner from the Pair of Terms immediately preced- ing. By which the Difference continually leſſening, becomes leſs than any given Quantity; and the Series being ſuppoſed to be continued 7z infuitum, that Difference quite vaniſhes, and the two Terms become equal; either of which is the Quantity ſought ; and the other Terms approach ſo much nearer to it, in order, the further they are diſtant from the Beginning of the Series. He mentions another Form of con- verging Series conſiſting of ſimple Terms, to which the foregoing may be reduced; which he defines thus: Let A, B, C, D, E, Sc. be a Series of ſimple Terms, and let them be of ſuch a Nature that the third Term C is compounded after the fame Manner of A and B, the firſt and ſecond, that the fourth D is of B and C; and the fifth E is of O and D, and ſo on in injinitum : And let the Difference of the antecedent Terms A and B, be always greater than the Difference of the immediately ſubſequent B and C. This Series being ſuppoſed to be continued infinitely, each new Term makes a nearer Approach to the Quantity required, and you may continue it ſo far as to arrive at a Term, which differs from it by leſs than any given Difference: So et What 349 > bu ANALYSIS by Eq vATTONSs, &. what he calls the laſt Term (for want of another Word to expreſs it is the very Quantity required. Such Series appears in Embryo, like. wiſe, in the preceding Propoſition from Archimedes; where AB being the Quantity ſought, AE, AF, AG, &c. may repreſent the Terms of one ſuch Series; and AH, AI, AK, &c. the Terms of another, 7. But notwithſtanding there is ſome Analogy in all the three Me. thods, viz. that of Exhauſtions; that of Mr. Gregory's Forms ef Series ; and that of Sir //aac's; as they repreſent a continual Approach to ſome determined Limit; yet there is a manifeſt Gradation as to their Extent and Univer/ality. For the Approach uſed by the Ancients pro. ceeded by taking away or adding the half Difference, or more than the half Difference, continually. In Mr. Gregory's Form of converg- ing Series, it is ſufficient if the Terms by which the continual Ap. proach is made be ſo related, that the ſubſequent ones be always com- pounded of the preceding ones after the ſame Manner, by any analy. tical Operation whatſoever, including Addition, Subtraction, Mult plication, Diviſion, and Extraction of Roots. Sir Iſaac Newton's Form of Infinite Series doth not even require this; it being ſufficient that the Sum of the Terms from the Beginning, continually approach to the Quantity ſought ; ſo as a finite Number of them may differ from the thing ſought by leſs than any given Difference, without any fur- ther Limitation ; although the Law according to which the Approxi- mation is made, may often, if not always, be diſcovered. The Me- thod of inveſtigating theſe Series's being much more general, as well a; the Relation of the Terms of which they are compoſed, Mr. Gregs afterwards fell into the ſame Method; when he ſaw it to have fo Xs Advantage over the Method he had formerly uſed in his Quadrature of the Circle, Ellipſe and Hyperbola; as we ſhall ſee afterwards, 8. After Archimedes there was little or nothing done in the Geome- try of Curve Lines, till towards the Middle of the laſt Century. In the Year 1635 Bonaventura Cavallerius introduced his Method of I- diviſibles: By which he endeavoured to render the Method of Ex- hauſtions of the Ancients, ſomewhat eaſier and ſhorter. In this Me- thod Figures are conceived as made up of infinitely ſmall or nd:v:fibe Parts or Elements; a Line of an infinite Number of Points; a Surface of innumerable Lines, right or curve, as is moſt convenient ; or per- haps of other Elements, as infinitely ſmall Triangles, Sectors, &c. a Solid, of an infinite Number of Surfaces, plain or curve; or perhaps of other Elements; as a Sphere of innumerable ſmall Pyramids, all terminating in the Center, and conſtituting the ſpherical Surface with their Baſes, &c. In which Method, Points, Lines, and Surfaces, con. ſidered EXPLAIMNE D. ani fdered as the Elements of which Magnitudes are compoſed, are not to be conceived as geometrical Points, Lines and Surfaces ; but as geo- metrical Lines, Sur faces and Solids, of infinitely ſmall Length, Breadth and Thickneſs. By this Method many uſeful Truths were diſcovered; and demonſtrated in a much more conciſe Way than by the Method of Exhauſtions : Vet one had need proceed with great Caution, other- wiſe one may be very readily led, by it, into Errors, and falſe Conclu- ſions. It was found of very great Uſe for inveſtigating thoſe Proper- ties and Relations of Magnitudes ; which could not be eaſily diſcovered by the former Methods: Which being once diſcovered might always be aſcertained, by reducing the Demonſtrations to the Apagogical Form of the Ancients ; where there was any Scruple left as to their Truth ; which indeed was frequently the Caſe. For the Foundation, upon which the Geometry of Indiviſibles is built, ſeems not to be quite ſo ſatisfying, or ſtrictly geometrical; in Regard the Suppoſition of finite Magnitudes being made up of infinitely ſmall and iniviſible Parts, is a Notion hard to digeſt; and, to ſay the beſt of it, is not capable of being altogether freed from ſome Confuſion, which ſtill remains in the Mind, after all the Pains we can be at to comprehend it. - 9. After Cavallerius and his Scholar Torricellius, there aroſe a great Number of famous Men in the Mathematical Way, who have con- fiderably enlarged and promoted this Science, in it's higher Parts; ſuch were M. Des Cartes, Fermat, Huddenius, Gregory of St. Vincent, Mr. Huygens, Dr. Barrow, and many others: But the moſt conſiderable Step by far that was made in the Geometry of Curve Lines, and the Doctrine of Quadratures, was by Dr. Wallis in his Arithmetic of Inft- nites, publiſhed in the Year 1656. The Doctrine of infinite Series had been very little conſidered before this Time: But he perceiving the Analogy betwixt the Terms of certain Series's and the Otdinates of cer- tain Curves, ſought out Rules for finding the amount of all the Terms of theſe Series's: By Means of which he was capable of ſquaring all thoſe Curves, whoſe Ordinates were proportional to the Terms of ſuch Series'ss He began by conſidering an arithmetical Progreſſion : From that he proceeded to Progreſſions of Powers, that is whoſe Terms were as the Squares, Cubes, Biquadrates, &c. of the Terms of an Arith- metical Progreſſion as o, 1, 2, 3, 4, &c. then to Progreſſions of Roots, that is, whoſe Terms are as the Square Roots, Cube Roots, Biquadra- tic Roots, &c. of the Terms of an arithmetical Progreſſion. Afffter- wards he was led to conſider Progreſſions whoſe Terms are any Dimen- hon whatſoever of the Terms of the Arithmetical Progreſſion, 7z. e. the Indexes or Exponents of whoſe Dimenſions are any Numbers integral, fractional, 352 moſt general and comprehenſive Propoſition, That the Sum of all the | finites, and other mathematical Works. ANALYSIS by Equations, &c. fractional, or ſurd, whether poſitive, or negative *. He conſidered theſe Progreſſions as conſiſting of an infinite Number of Terms, the laſt Term, which repreſented the loweſt Ordinate of the Curve, being ſtill finite, and the intermediate Terms from o to the laſt, being in. finite in Number, repreſented Ordinatcs applied to the Axis at ink. nitely ſmall and equal Diſtances, betwixt the Vertex and loweſt Ordi- nate: Or perhaps theſe Terms repreſented any other Lines Right or Curve; or any Plain or Curve Surfaces, in the Caſe of Solids, which were proportional to them. And he found, in a Way of Induction, this Terms of any ſuch Series is to ſo many times the greateſt, as Unity is u the Index or Exfonent of the Power, Root, or Dimenſion whatſoever, encreaſed by 1. Which amounts to this: Suppoſing o, 1, 2, 3, 4, &. x to be an Arithmetical Progreſſion conſiſting of an infinite Number of Terms, in the natural Order of the Numbers, having the laſt Term x; and let o, 1”, 2”, 3”, &c. x” be a Progreſſion of Terms, which are any the ſame Power, Root, or Dimenſion whatſoever, of the formerTerm, whoſe Exponent is denoted by , then ſhall the Sum of this laſt Series ot | m+ Progreſſion be equal to = - + And this is in Subſtance the ſame with the firſt Rule of this Analyſis of Sir I/aac Newton's, which was dil- covered by an Induction of Particulars by Dr. Wallis; but is demon- ſtrated univerſally, by Means of an indefinite Index, by Sir 1/aac. 10. From thence Dr. Wallis proceeded to the Conſideration of ſuch Progreſſions as have their Terms made up of Combinations of the ſim- ple Terms of the forementioned Series's, by Addition and Subtraction; and likewiſe of the Powers of ſuch Sums and Differences: And thence he inveſtigated what is in Subſtance the ſame with our Author's ſecond Rule of his Analyſis, for the Quadrature of compound Curves. B this Means he was enabled to give the Quadrature of all ſorts of Parabo- las, of ſpiral ſpaces, and of many plain Figures which extend themſelves infinitely along an aſſymptote: Likewiſe the Contents of ſuch ſolids as are generated by the Revolution of theſe Parabolical and Hyperbolical Curves; and thence diſcovered other Things, which depended upon theſe Quadratures, ſuch as the rectifying of ſeveral Curve Lines, and plaining of Curve Surfaces; as he has ſhewn in his Arithmetic of In- 11. But he found his Method fail him in the Caſe of thoſe Progrefli ons, whoſe Terms were the Roots of the Sums or Differences of {imple 2 Dr. Wallis is reckoned the firſt Perſon who brought in the Uſe of fractional and negative Indexes; and likewiſe of indefinite and ſurd ones. Terms, EX PEAINVE D. Terms, called Roots univerſal, ſuch as /i o, V/ 1, VAT 2 N73 ibs, Sc. n-: Which he calls a Series of Terms in the ſubduplicate Ratio of a Series of Equals leſſened by a Series of and a for any of the infinitely ſmall and equal Diſtances of the Ordi- nates in the Quadrant of the. Circle, beginning at it's Center, ſuch Quadrant 1s equal to the Sum of all the Terms of this Progreſſion; as he ſhew¾s in Prop. 12 1. Arith. Inf. The fame Series with the Sine of the ſecond Term under the Vinculum, changed into it's Oppoſite, that is Jr + bn, / Þ 1a, V2 4, Oc. V, being ſummed up, would give the Quadrature of the equilateral Hyperbola. He like- wiſe ſhews other two Series's, by the ſuming up of whoſe Terms, the Quadrature of the Circle and Hyperbola would be found, but labouring under the ſame Inconveniency. Here Dr. Wallis ſtuck : But this Diffi- culty was removed afterwards by our incomparable Author, by his third Rule in this Analyſis, as we ſhall ſee anon. However Dr. Wallis being very ſolicitous to do ſomething towards the Quadrature of the Circle; which was his principal View when he engaged in the Proſe- cution of theſe Enquiries (as he tells us in the Preface to that Work) he thought u He another Method, which he calls Interpolation of Series. By which he means a Method of diſcovering certain inter- Properties of the Progreſſion, and the Relations of the Terms to 'each other. Of this he gives ſeveral Inſtances for finding the Area of the Circle, Of which this is one: In the Progreſſion 1, +, 5, , &c. whoſe Terms are produced by the continual Multiplication of 1 x + x + X, Cc. to find the intermediate Term betwixt 1 and 2. Such in- termediate Term he expreſſes by this Symbol ©, even as the Square Root of 2 is expreſſed thus, /2 ; and he ſhews that the Circle is to the Square of it's Diameter as 1 to q. But the Reſult of his Enquiry Way of Notation; which is nothing more ſtrange than that /a or any other ſurd Root is not capable of being adequately expreſſed, in that Way: But whereas the Value of /a, or any other Surd, may be ex- preſſed approximately by the common Notation, ſo likewiſe he found that the approximate Value of ©, was 1 * 25452526210 = . — or 1+ A+ZB+:C+#D &c. in inf. the Capitals A, B, C, &c. denoting the firſt, ſecond, third, &c. Terms. 12, When theſe Diſcoveries, which were indeed very noble and uſetul, and in Point of Generality and 5 far exceeding any thing 2 that mediate Terms of a regular Series or Progreſſion, by conſidering the 333 gecundans or Squares. Where if » ſtand for the Radius of a Circle; was, that the Value of © cannot be adequately expreſſed by any received ANALYSIS by Eqvarions, &c. that had been formerly done by others in the Geometry of Curye Lines, were made public, it was objected to him by Monfieur Fer nat: and others, that however valuable his Diſcoveries were, and true in themſelves, yet the Demonſtrations of them in the Way of Induction by Dr. Wallis, did not come up to that Accuracy which a Geome. trical Subject required, and the ancient Geometricians had all alon obſerved in all their Performances. To which Dr. Wallis made An. ſwer (even as he had remarked in the Arithmetic of Infinites itſelf that he did not ſo much deſign to demonſtrate his Diſcoveries, as tg lay open to others the Method he uſed in making them; which the Ancients purpoſely concealed ; that, notwithſtanding, he thought the Proof by way of Induction was ſatisfying and convincing ; that it would be an eaſy Matter for any Perſon moderately ſkilled in Geometry to demonſtrate thefe Things with all the Pomp and Apparatus made Uſe of by the Ancients : But that was a Labour he never deſigned to under. take, However, to give ſome SatisfaCtion in this Matter, he ſhews by ſome Examples in the 78th Chap. of his Algebra, how the Propoſi tions he had diſcovered, might be demonſtrated after the Manner of the Ancients, in imitation of what had been done by Archimedes in the 1oth and 11th Propoſitions of his Treatiſe of Spiral Lines: In which Archimedes demonſtrates, what is the Sum of all the Terms of a Pro- greſſion of Squares, whoſe Sides conſtitute an arithmetical Progreſſion of Lines, having the common Difference equal to the firſt Term; when compared with ſo many times the greateſt Square: And the Limits betwixt which the Sum of the Terms of ſuch a Progreſſion is contained, although the common Difference of their Sides be not the ſame with the firſt of them. Which he applies to the finding the Relation of the Spiral Spaces to the Circular Sectors; even as Dr. Wallis, by proſecuting this Affair to a much greater Length, ſhews how to find not only the Sum of all the Squares; but the Sum of any Powers, or Roots whatſoever, of an arithmetical Progreſſion ; and thereby to compare infinite Numbers of curvilinear Areas with right-lined Fi- gures, and with one another. And truly, when one attentively con- ſiders this elaborate Treatiſe of Archimedes, and the other Works of that ſubtile and penetrating Genius, one cannot help ſeeing Dr. Wallis Arithmetic of Infinites, Mr. Gregory's Method of Inſcription and Cir- cumſcription of Polygons; and even our ſecond Archimedes's Method of prime and ultimate Ratios, beginning to diſcover themſelves, as it were in Embryo, in order to be brought forth afterwards to open Light, : See Dr. Wallis Algebra, Chap. 79. 13. And EAXPLAINE D. 13. And now we are arrived at the laſt Step of this Progreſs, as it was perfected into the Method of infinite Series and Fluxions by our celebrated Author. By what Means this was done, I cannot better re- reſent to the Reader than in the Account he gives of it himſelf in his — to Mr. Oldenburg, Octob. 24, 1676. | | 14. About the Beginning of my mathematical Studies, as ſoon « as the Works of our celebrated Countryman Dr. Wallis fell into my « Hands, by conſidering the Series's, by the Intercalation of which, « he exhibits the Area of the Circle and Hyperbola ; viz. that in a « Serjes of Curves, whoſe common Baſe or Axis is x, and the Ordinates IK , I-, I—xxl, I—xxV, I:, T—-xx}, Sc. « if the Areas of every other of them, that is x, x — *, x — *x3 « þ 1x5, x — 2x3 + 4x5 — g, &c. could be interpolated, we « would have the Areas of the intermediate ones, of which the firſt * 1—xx} is the Circle: To interpolate theſe, I obſerved that the « firſt Term in all of them was x, and that the ſecond Terms 2x3, * 2x3, 2x3, 2x3, &c. were in arithmetical Progreffion ; and that there- « fore the two firſt Terms of the Series's to be intercalated, ſhould be r x; 3x3 For intercalating the reſt, I confidered that the Denominators 1, * 3, 5, 7, Cc. were in arithmetical Progreſſion, and that nothing *© remained but to inveſtigate the numeral Coefficients of the Numera- tors. Now theſe I ſaw, in the alternate Areas that were known, * to be the Figures which expreſs the Powers of the Number 1 r, viz. “11% 11 112, 113, 11+, Cc. that is the firſt was 1; the ſecond * 1, 1; the third 1, 2, 1; the fourth 1, 3, 3, 1; the fifth 1, 4, 6, W * Wherefore I enquired how, from the two firſt Figures given in * theſe Series's the reſt might be derived; and I found that if the * ſecond Figure be called , the reſt might be produced by a conti- * nua] Multiplication of the Terms of this Series —— X — Xx = 2 3 N n Sc. * For Example, let the ſecond Term m == 4, then 4 * — that * 18 6, will be the third Term; and 6 x = that is 4, the fourth * boi. © this Letter, which contains many valuable Diſcoveries of Sir Tac in the Com. Nes 55. 2 2 2 “ Term; 355 356 ANALYSIS by Equations, &c, © Term; and 4X==, that is 1, the fifth; and K that is o the ſixth Term: With which the Series, in this Caſe, terminates | Therefore I applied this Rule for intercalating the Series's, And * ſince, for the Circle, the ſecond Term was 5 2 1 put n , and * the Terms ariſing were „ , or — +; —+X Dor +; 42 1 | * bK 2, or — 2; and fo on in inf. Whence I came 00 © underſtand that the Area of the circular Segment, which I want Ce was x — = — = — 25 — =, Ge. And the like Method may be made uſe of to interpolate other e Series s, and that although they be deficient by the Intervals of two te or more Terms together. Such was my firſt Entry upon theſe Studies; © which had certainly eſcaped me, had I not accidentally caſt my Eye e upon ſome looſe Papers that came in my Way a few Weeks apo, <« But having arrived this Length, I immediately began to conſider that e the Terms I— h, I— n, I , Il, Ge. that i © to ſay 1, 1— &, 1 —2x* ＋ x., 1 — 3x. + 3x. — &&, &c. might te be intercalated after the ſame Manner as the Areas generated by « them; and that nothing was required for this Purpoſe but to omit the Denominators 1, 3, 5, 7, &c. which are in the Terms expreſſing de the Areas: That is to ſay that the Coefficients of the Terms of the « Quantity to be intercalated x— xx P, or T—xx1}; or in general of « T—xx|”, ariſe by the continual Multiplication of the Terms of © this Series mx * 27 * 2. Sc. And thus I perceived ee C. e that 1— xxl* was equivalent to 1— 4x? — 4x* — &'; and 1 xx}, « to 1— A* + * + Zzx5, Sc. and 1—xxÞ, to 1 — * — {xt 6 — , Se. « And thus I was poſſeſſed of a general Method of reducing Radicals © to infinite Series, by the Rule I mentioned towards the Beginning of « my former Letter *, and that before I was acquainted with the a Extractions 1 This refers to his Letter wrote to Mr. Oldenburg 13 June 1676, at the Beginning of be” > 8 which he had ſet down his famous Binomial Theorem thus, P Pa =P” + = AQ + — 25 3 — CQ, &c. which is what he underſtands by the Rule he mentioned. See Com, EXPLAIN EE D. « Extractions of Roots. But once this was known, I could not remain « Jong ignorant of that other. For in order to prove theſe Operations, « ] multiplied 1 — * — 4x+ — s, Cc. into itſelf, and I found « the Product 1—xx, the reſt of the Terms in inf. vaniſhing, by the « Continuation of the Series. And thus likewiſe the Series 1 — & « 1x4 — f, Oc. multiplied twice into itſelf produced 1 en! « Which, as it afforded a certain Proof of the Truth of theſe Con- « cluſions; ſo it led me to try whether, converſely, theſe Series's, « which thus appeared to be the Roots of the Quantity 1 — , « might not in an arithmetical Manner, be extracted out of it. « The Matter ſucceeded to my Mind. The Form of the Operation « in the Extraction was this. (See it under the Examples for extratting « Roots in this Analyſis. Art. 15. and following.) Having diſcovered this, I entirely neglected the Interpolation of « Series's, and made uſe of theſe Operations only, as a more genuine « Foundation. And as to Reduction into infinite Series by Diviſion, « jt did not lye long concealed, being a much eaſier Affair. Yea I « jmmediately attempted the Reſolution of affected Equations, and « obtained it likewiſe. Whence the Ordinates, the Segments of the « Axes, and any other Right Lines, were at once known, from the « Areas or Arches of the Curves being given. „About that Time the Plague breaking out *, obliged me to get * hence (from Cambridge) and think of ſomething elſe. Notwith- * ſtanding, I added to my other Diſcoveries, at that Time, a certain * Conſtruction of Logarithms deduced from the Area of the Hyperbola.” And after ſhewing how this Conſtruction was affected, he proceeds thus : % am aſhamed to tell to how many Places of Figures I carried * theſe Computations, having no other Buſineſs at that Time: For I * pleaſed myſelf too much with theſe Diſcoveries. But as ſoon as that ingenious Performance of Nicolaus Mercator appeared, I mean his Logarithmotechnia (who I doubt not, firſt diſcovered the Things * he publiſhed) I began to cool a little upon theſe Things, ſuſpecting * that either he underſtood the Extraction of Roots; as well as Diviſion * of Fractions; or at leaſt that others upon the Diſcovery of Diviſion, * would find out the reſt, before I could be of a ſufhcient Age for writing. Com. Epiſt. No 48. Theſe two Letters of Sir 1/aac Neauton to Mr. Oldenburg give an Account of many of his admirable Diſcoveries in the Buſineſs of the Quadrature of Curves, and infinite deries ; and upon that Account deſerve the mathematical Reader's careful Peruſal. * This happened in the Years 1665 and 1666, 1 357 358 ANALYSIS h Ev ATTOoNSs, &c, © Yet about the ſame Time that Performance appeared, I commu. nicated to Mr. Collins by the Intervention of our Friend Dr. Bax,,;, (then Profeſſor of Mathematics at Cambridge) a certain Compend « of theſe Series's3, In which I had ſignified that the Areas and * Lengths of all Curves; and the Surfaces and Contents of Solid, might be determined from Right Lines given ; and converſely from theſe given, the others might be found: And I illuſtrate e the Method I pointed out, by ſeveral Series's. After this a literary Correſpondence having ariſen betwixt us, Mr. Collins, a Perſon born for the Advancement of mathematical Learning, infiſted ſtrongly upon my publiſhing theſe other Thing; Accordingly five Years ago (1671) when at the Perſuaſion of m Friends, I had reſolved to publiſh a Treatiſe of the Refraction of Light; and of Colours, which I had then ready by me, I began to turn my Thoughts again to theſe Series's, and wrote likewiſe a Treatiſe upon them *, that I might publiſh both together, But having wrote you a Letter upon the Subject of the refleQing Teleſcope, in which I explained briefly my Notions of the Nature of Light, ſomething unexpected fell out, which made me imagine it concerned me to write to you ſpeedily about printing that Letter, Immediately upon the back of this I met with frequent Interru tions, from ſeveral Perſons letters, taken up with Objections, and other 'Things, which frighted me entirely from executing my De- ſign ; and made me find Fault with myſelf for my Imprudence, that by catching at a Shadow, I had been ſo far deprived of the Peace and Quiet of my own Mind, which is a thing of more ſubſtantill * Worth. * About that Time, James Gregory, from one certain Series of mine, which Mr. Collins had ſent him, after much Thought and Ap- plication (as he himſelf wrote back to Collins) lighted upon the ſame Method, and hath left a Treatiſe upon it, which we hope will be publiſhed by his Friends. Since, conſidering his extraordina- ry Genius, he muſt have added many new Diſcoveries of his own, which would be a great Loſs to mathematical Learning, ſhould they te be loſt . ; &« But 3 By which he means this Analyſis by infinite Equations. | 1 This Treatiſe is mentioned by Mr. Collins in two Letters publiſhed in the Com. Epil: No 22, 23; and it appears, by the Account given of it by him, and by what follows in thus Letter of the Author, ro be the ſame with that publiſhed by the ingenious Mr. Jon Colſon, now Profeſſor of Mathematics in the Univerſity of Cambridge, Ao 1736, under the Title of Thr Method cf Fluxions and infinite Series; with it's Application to the Geometry of Curve Lines. : Here we have Sir Jſaac Newton's own Teſtimony that Mr. James Gregory diſcovered à General Method of Quadratures and Series's, of the ſame Sort with his own, I his 1s ſufficient- ly cc cc cc cc cc cc cc nv Series, which proceeds by Pairs of Terms, of that Sort I have mentioned above, Art. 7. by which he ſhews how to approach to the Area of the Hyperbola, as well as that of the Circle, as near aa you pleaſe. In April next Year, Lord Viſcount Brounkey publiſhed an infinite Series (of that kind, which I have called above, Art. 6. the Newtonian) for the ſame Purpoſe : — 2 8 I I 1 1 2 he expreſſes thus * 728 THe Sc. in inf. That ſame Year Mr. Nicholas Mercator, an Holſatian by Birth, but who had ſpent a great Part of his Life in Eng/and, pub- of liked his Logarithmotechnia, in which he ſhewed how Lord Viſcount Brounker's Series might be found, by reducing a complex Fraction to an infinite Series of {imple Terms by Diviſion. in Which was but a Improvement upon what Dr. Valli had ſhewn in his Math, Univ. Chap. 33. with reſpe& to Diviſion (although the Doctor had not given any * — Example, he in which the Number of Terms in the Quotient was infinite, as ſome alledge) Mercator having reduced the complex Fraction to an infinite Series of fimple Terms, had no more ado after m that, but apply the ſame Perſon's Method of Quadratures in his 4rith. Ivf. for ſquaring tbe er ſeveral Terms. Upon the Publication of the Logarithmetechnia, Dr. Wallis illuſtrated the Diſ- he covery 3 ve another infinite Series for the ſame * in the Philo/oph. Tran/. for ug. 1668. Mercator and Wallir's Series's were in effect the with thoſe mentioned above, ea a*x a*x* a ab x+ | a*x , a*x* 4 l ax+ be Art, 25, wit. ＋ — Frier. ny Tg Ti YT Tr cal the former being Mercator's ; the latter Dr. Walks's. Towards the End of the ſame Year ors 1668, Mr. James Gregory publiſhed his Exercitationes Geometric : in which he promoted and Mercator's Diſcovery, and gave a geometrical Demonſtration of it, by means of ſum- ms ing up the Secants of a circular Arch. However it a from what we have taken out of his du aac Newton's Letter to Mr. O , recorded above, and other undoubted Evidences, ” that dir Iſaac was poſſeſt of his General Method of reducing Complicate Frattions, Radicals, and afeed Equations, inte infinite Strict i, above two Years before the Logarithmorechnia was publiſhed. : Bbb arc 370 ANALYSIS V EQ ATTToNd, Cc. are all poſitive Integers below 10; whereas in the latter, which is 6f 2 more general and indefinite Nature, the Scales or Progreſſions may be infinitely varied, fo as to decreaſe in a decuple, or any other · Proportion whatſoever ; and the numeral Coefficients may be any Numbers, integral or fractional; poſitive or negative. Thus we may conceive any integral Number as a Series of Powers with their proper Coefficients, i. e. the Number 86027 may be conceived as expreſſed thus; 8 x 710. K. 6 x ĩ0h + 0x10* + 2 x10!" + 7 x 101: and if the Number be continued into Places below the Place of Units, as thus 86027 . 4638, then it is equivalent to this 8 x 10 + * + 0x10 + 2 x10! + 7X 101 + 4* 101 | 6 X 101”? + 3x 103+ 8 x 160+: the fractional Part of which may be otherwiſe denoted thus = + = = | | * X | 10 + = + We may likewiſe ſuppoſe 0. 1 to be the Root inſtead r 10, and * the preceding Number 86027 . 4638 will ſtand > eee root = 0. 6x0.1* + 3x0.1* + 8x0.1): the fractional Part of which, : | . 4 6 1 VIZ. « 4638 may likewiſe ſtand thus E. + 238 + _ + 2 All which Series's would be infinite if the Decimal Fraction were infinite. And after the fame Manner a great Variety of other Scales, or Forms of Progreſſion, might be deviſed, with ſuitable Roots for different Kinds of Arithmetic : ſuch as the ſexagenary uſed in Aſtro- nomy ; in which the Root is 60 or it's Reciprocal. Further it ma be obſerved, that the tions in Decimal Arithmetic, by whi the Quotient, or Quantity ſought, is diſcoverdd, *correſpond to the rations relating to infinite Series in ſpecious Arithmetic, in this Reſpect that every new Step of the Operation, in both Caſes, by which the conſtituent Parts of the Quotient are found, makes as great an Ad- vance towards the Supply of what the Quotient is yet deficient, as the Nature of the Progreſſion will admit | 30. In thoſe Series's which belong to the literal Arithmetic, the Indexes or Exponents of the Powers or Dimenſions, may be fractional as well as integral; and are ſometimes left indefinite: but they ate al- ways ſuppoſed to proceed by equal Differences. The numeral _— £X Þ £5 ATNED.,! x * cients are mene ſometimes left indefinite. And then theſe indefinite Exponents and Coęfficients are expteſt hy Letteth. Suck a general and indefinite con verging Series may be repreſented thus, Ax" + B. + c + Dx"E3", ec. where x" denotes the Root of the Scaleo er Knits, A, B, C, D, &c. any Coefficients, poſitive or negative ; ; mand n any poſitive Numbers whatſoever : m may de nothing and ſo may one or more of the Quantities, A, B, C, SST is 31. But every Series in ſpecious Arithmetic, when it is to be applied in Practice, muſt have it's feveral ies and indefinite Characters, reduced to determinate and known Values, that ſo the Value of the ſeveral Terms, and conſequently of the Whole, may be. collected. Thus ſuppoſe we are going to make uſe of the Series 2 * abr. 4 24" | 55 + . Sc. [See above Art. 25.) for finding the hyperbolical Area ABDC (See the Fig: referred to in that Place) where AB as 4, abid b, the ſame as there mentioned: ſuppoſe a= I; and x—=0.1 or AB = o. 2, then you ou'll have ABDC (= =P EE 3, + + . 8 * .) e. þ 2g . 2er f age 2 of. 1 0 ,0006666 Cc. + 0.000004 + 0.000000028 Cc. So. 20067069 &c. I now proceed to explain the Reduction of complex Radicals into infinite Series: and the Quadratures depending thereow': of which 'our —_ treats in Art. 15—20. 32. In the Account 1 have given above of the Riſe and Progreſs of the Doctrine of Quadratures and Series's, I have ſhewn after what Man- ner, and by what Steps our Author diſcovered the Reduction of Radi- cals into infinite Series. As for the Manner of Operation, it reſembles the like Operation in Decimal Arithmetic for extracting the Roots of Numbers which are not true Powers of their Kind: In which Caſe, the Root or Quotient runs out into an infinite Decimal, Wherefore this being underſtood ; and the Reader being uf de to be acquainted with the Operations of literal Arithmetic ; the Fonndeden upon which the Extraction of Roots is built, I ſhall not detain him in ex- plaining this Operation, when applied to the Reduction of complex Surds into infinite Series, further than to repreſent the Parts of the Operation in the Example Art. 15. a little more fully. | 33. Let it be propoſed then to extract the Square Root of 44 -- xx; or to find the Value of y in an infinite Series, ſuppoſing that f= Ja + xx. The Manner of proceeding is as follows. Bbb 2 aa ＋ xx 37 37a ANALYSIS by Eqavations, Wc, „ x6 A. 0 4 CIT Ter- Te, aa 20+ )* + xx 1 * + xx + = 2 - - * EAT 2 - er "+9 iro +2... e * = _& = —— | K Os Sc. Where, having ſet down the Quantity ae T- xx, whoſe Square Root is to be extracted, you take the Square Root of 44 the firſt Term, viz. a, which you place in the Quotient: and then ſubtracting a“ from the propoſed Quantity, the Remainder + x* is divided by 29, which gives + = to be ſubjoined to a in the Quotient: You annex it likewiſe to the Diviſor 2, making 24 + 2, and then multiplying this augmented Diviſor by it, place the Product + xx + rr under the Reſolvend or Dividend + xx : ſubtract it from it, and the Remainder — Lr gives a new Reſolvend or Dividend. For forming the Divifor to which, take double the Part of the Root or Quotient already found, viz. 24 ＋ = : which place for the Diviſor as you ſee, leaving Room for another Term to be ſubjoined, Divide — =; by 24, the firſt Tem of this Diviſor ; which gives — Jer to be fubjoined to. the preceding - 2 EXPLAINED. 373 part of the Quotient; and likewiſe to the Diviſor: then multiply the wgmented Diviſor 2a + 2 — #5; by it, forming thereby the Able- FAIRE, ST. to be ſubtracted from the Reſolvend _ =: and fo you proceed, repeating always the ſame Operation, until you obtain as many Terms in the Root as you defire, Where you ma obſerve that thoſe Terms may be neglected in the Operation, which have no Influence upon the Quotient within the propoſed Limits. Thus you'll obſerve that if the Quotient is to be continued to the roth Power of x only, thoſe Terms are neglected, which can be of no Uſe within thefe Bounds : which I have accordingly marked with Aſteriſks. 34. The Operation being finiſhed as far as you intend, the Truth of it may be proved by multiplying the Root into itſelf : for that ought to produce a. + , the ſubſequent Terms in inf. deſtroying one another: which you will find to be ſo here, as far as the Product can be carried by the defective Root found, viz. to as many Terms as have been found in the Quotient, And after the ſame Manner you are to proceed in other Examples. 35. And thus having found y (V =) a+ — =+= _ — 17655 Sc. the Area of the Curve whoſe Property is defined Ny +x*, ory V e, is (by the fecond Rule) ax 4 = © + =; — © c. Now if CD (Ser the Fig. 40a 201 3 11245 115247 Art. 15.) be a rectangular Hyperbola, A it's Center, AC (Sa) the half of the determinate Axis; and conſequently of the Latus Rectum, AB(=x) any Part of the ſecond Axis, and BD ()] it's Ordinate: from the Property of the Hyperbola, you have BD + AC x BD—AT =ABy, or y + aXy— ax“, that is * — a*=x* or y* S x*; which is the Property required. Therefore ax + = — = G. = 40a3 ABDC the Area required: for the Area begins at. A-Q, fince it vaniſhes, when x = AB vaniſhes. And to make this Series con- verge, x ſhould be leſs than a: and the leſs x is, @ remaining, the; quicker is the Convergency. The ſame hyperbolical Area would be found by Prop. 5. of the Treatiſe of Quadratures, to be cnet XxX 4x3 , 8x5 04x7 | 128x? 2 "Io ad 4x5 £3 8x7 04x9 2 TE” er R. - 5% gk 36. If 374 ANALYSIS W Equations, &c, 36; It it wete y == =, the Equation would be to the Cid (See the Fig. Art. 16.) ABCD: where the Radius A=, AB—, D =y: for BD. = AC — ABY, that is y* =a* — x*, Now if the 3 Root be extracted out of * x*, after the Manner of the bu. FE ns, it will be found „-=- 2 A ry — &c. which differs from the Series for the Ordinate to the &. cond Axis of the Hyperbola, as above, in no other Reſpe& but that all the Terms after the firſt are negative; which in the Hyperbola arc | yes and n alternately, Whence the circular Area ABDC x7 x9 111 — 2 Ws _ "rn", Tiger — 2 Sc. differing from the Series for the Hyperbolical Area in the fame Reſpect thoſe for the Ordinates differ. The double of this Series will give the Area of the . Circular Zone adjacent to the Diameter, and having it's Right Sine or Altitude ul. to * wit, Hh ear >, wee ie als wn ren n 2RB — * — D — —— Ce. the ſame with that which was ſent — by Mr. Collins in one of his Letters to Mr. James Gregory, as a Speci- men of Sir Iſaac's Method of Series. See Com. Epiſt. No. 18. The ſame circular Area — — would be found by Prop. 5. of the Qui 'dratures to be) K ++ 7 © r +55” - £4 5 Se. the ſame T5 as with that for the e Area found by the ſame Prop. except only i in the Signs. The former Series ax +=:— = + = Ge. ud Fm Ai 4 — 11245 this one y x +4 77 S+ * 2 Ge. wil 8 ſound the fame, if for you inſert 0 Vau, vix. 44 2 * 24 = Ra Ge: : nnd hen malig 822 pi by * 2 : + = 72 ＋ 389 &c. 37. When a few of the initial Terms of a Bete are once found, it is of confiderable Uſe to know the Law of the. Progreſſion by which che Series proceeds, that thereby it may be continued at Pleaſure, This Law is found either by obſerving the Steps of the. general Theorem from whence the Series is deduced ; or by conſidering the Relations of n Terms wh ich are already found in the beginni ing of the 1 us * = S 8 3 PT = EXPLAINED, The the Sees ex EZ (og Eg —= + &c.forthe 40a — 11245 115 247 + 281609 or circular Area: found as above by the Method of in- finite — may be continued at Plcaſure, by obſerving that the nu- meral Coefficients . 7; . Tr. 55777 + &c. are formed by multiplying continually b y 251 . Joe + 227 12 . =, Sc. due Regard being had to the Signs. So likewiſe the other Series for the ſame Areas, derived from Prop. 5. of the Quadratures, viz. y x x T F 408 Se. may be continued, by obſerving that the W Coefficients = . - 2, 5 Oc. are formed by the continual Multiplication of the ing Fracti ctions . 5 . 42, &c. g Hence if CD, Cd; be ns and retangular Hyberbola, deſcribed with the Center A, and Tranſverſe Axis AC; and AB x the Baſe or Abſciſs, as before, upon the conjugate Axis; BD or Bd y the Ordinate: and the Ordinates DE, de, to the Tranſverſe Axis be drawn: then from ABDC=ax —= — = — = — oa 40a 11245 e ſubtracting the K. f 287855 | ; Rectangle ABDE — m__ „BD — En 4 — A B F 7 2 — ar — 2 — 2. — 18 — 5 2768 &c. there remains DCE == + = + 2 2 - + = 5 Sc. which ſerves for finding the * or half — DCE from the Right Sine AB or DE given. And after the like manner, from AB or de x given in the Hy- ——_ you'll find 2 Hyperbolical half Segment 4Ce equal to —— 22 —+X — 2 — + Gc. which two Series may be con- dba at Pleaſure by forming the numeral Coefficients of each Term by the continual Multiplication of fo many Terms of the following Series, * Ixl 1x3 . $x7 7x9 n u the Place of the Term denotes, viz. 13 » 96 — Hl BA 1 GO By adding the two Series = + 5 + 3 Ge. and Z = 376 ANALYSIS h Equations, Oc, =_ Es &c. together, you have = + + 4-2 Sc. = DCE 23a5 704aꝰ + dCe : and by ſubtracting the latter from the former you have a + - Sc. = DCE — dCe: the firſt is the Sum of the two half 7 Segments; the laſt their Difference: both which converge very faſt; ſo doth the Series 25 + — + 2 == the intermediate Are 140849 Ded the Difference betwixt AB DC and ABdC: and alſo 2ax— = — = Ce. equal to their Sum, | 4 57 | | 39. Our Author's next Example (Art. 17.) is y=%x — xx or VIxx—xx: Which is to the Circle ABD (See the Fig. belonging 1. that Art.) having it's Diameter equal to 1; and the Abſciſs AB x beginning at the extremity of the Diameter; in which Caſe you have JY ( = BLY = 1=—x x Xx) =X —Xx*; Or y D V-. Now if the Square Root of x — xx be extracted, you'll find it to be xt — 1 — * — * — {x7 — NG. = and therefore by ſquaring all the Terms, it is * — 4xt — K — z — er — 0g Sc. = Area ABD. The numeral Coefficients of which are produced by the continual Multiplication of:. == , +. . 22, Pig. by which the Series may be continued as far as you pleaſe. 40. Now by Means of the foregoing Series's contained in the four preceding Articles, you may eaſily find any hyperbolical, or circular Area, in any Caſe propoſed. For the Satisfaction of the young Geo- metrician, I ſhall lay before him the Computation of the circular Area from the laſt Series * — Ki — 7x3: &c, = ABD. Here the Dia- meter is 1: let us ſuppoſe AB (x) to be any Part of the Diameter, the leſs Part it is taken, the Series will converge the more quickly; but becauſe it is moſt convenient for the computing the Area of the whole Circle, let it be x —= + that is the verſed Sine of 60 Degrees, By what was ſhewn in the laſt Article, the firſt Term being *x' , the ſecond Term will be found by multiplying the firſt by — x =— 2x5 -xz the third, by multiplying the ſecond, vis. 5 * by 2x or x: and ſo on. But in ſuch numeral Calculations, you'll do beſt to reduce all to Decimals: and ſet down each Term, as you ſee in the Margin, after it is found. Now fince x = +, it appears that * = x TSO. 083333 Sc. Which I ſet down in the Margin with | ; its EXPLAIMNE D. | it's proper Sign. This multiplied by -- 0.0833333333333333 — 3-x = — Z;, makes — 0.0062 5. Which being placed duly under the 0.0062 5 former with it's proper Sign, you mul- 00279017857 1429 tiply it by #/5X = , and it produces 0.0000271207361111 — 0.0002790178571429 Cc. And ſo on © 0.000003467906605 1 for the reſt ; which will be ſuch as you ſee 0.0000005135169396 expreſſed in the Margin. Where the firſt 0.0000000834465027 Line only is poſitive ; and all the reſt negav 000001 44628917 tive. Wherefore ſubtract the Sum of all the © 2299000002628 53 54. Negatives, vi g. 0.000 560227 17028600. 0.00000000049 54581 from the firſt Term o. o83 3333333333333 © 0.0000000000961296 c. and there remains 0.036773 1061630473 © 0.000000000019094H5 Sc. = the half Segment ABD. The — $:2299000000038676 Radius CD being drawn, the Triangle — 0.000000Cc000007903 BC x CD — 0.0000000000001663 BCD = ——— = . 4 2 2 32 — 0.00000000000003 52 0.0541265877305275 Cc. which being — 0.000000000000007 5 added to ABD, makes 0.13089969389957 — 0.0000000000000016 Ec. = Sector ACD a ſixth Part of the — 0.0000000000000004. whole Circle : ” Hat multiplying it by — 0.00000c000000000 1 b, you have o. 816 n Ch of a Girde whole Damiter is1: — 2:200560227 1702860 true to 14 Places of Decimals. And there- + 0.076773 1061630473 fore the Diameter of any Circle being + 0.0541265877365275 en, multiply this Number by the Square - bf that Bar, and you — it's ; = 0.13089909339957 Moreover the Area being found divide by: Radius, and you'll have the Circumference : that is in the preſent Caſe, divide the preceding Number by + or multiply it by 4, and it gives 3.14 1592653 58979 &c. for the Circumference of a Circle whoſe Diameter is 1. 41. After the like Manner you may find the Area of the Hyperbola from the Equation y =v/x+ xx: and both that of the Circle and that of the Hyperbola, by Means of the other Series's which have been mentioned. And it appears with what Eaſe and Expedition this is per- formed: So that for Eaſe and Expedition it vaſtly ſurpaſſes any former Method for approaching the Area of the Circle, made uſe of before our Author's Time: not excepting Mr. James Gregory's, in his Vera Grculi & Hyperbolæ Quadratura (which was the beſt) as will eaſily appear by conſidering the Operation made uſe of for this Purpoſe in Prop. 29; and comparing it with the Method here ſhewn: which jet is capable of great Improvement. Cee 42. Our 377 378 ANALYSIS by EAT Tos, &c. 42. Our Author's next Example (Art. 18.) is — =y: the Qu. drature of which gives the Length of an Elliptical Arch. For fuppoſe the Figure ACF, belonging to Art. 38 above, to be an Ellipſe, havin AF the half Franſverſe and AC the half conjugate Axis; and DB à DE Ordinates to them reſpectively: call ABR x, BD v, AF =. AC r, and the Arch CD : then by what was demonſtrated in the preceding Freatiſe (See Notes upon Art. 5. of the Treatiſe of Qua- drathres) it is #= PF > : that is (by inſerting == for be; which is equal to it from the Property of the Curve, and reducing) — c * \/ c4 PE pe Z = # . Wherefore a Curve which has x for it's Abſciſs and F 7 — or it's Ordinate, flows with a Fluxion or Velocity analogous to that with which the Elliptical Arch CD flows: conſequently the Area generated by that Ordinate is analogous to the Elliptical Arch CD: So that the ſame Algebraical Expreſſion ſhall denote both agreeably to Art. 37. of this Analyſis; and what was ſhewn with Reſpect to the ReQification of Curves in the preceding Treatiſe. Therefore if it be de Quadrature of it will give the Elliptical Arch, p- ie! | poſing c=1, ft — cc a, and cc . 43. For this Purpofe you may extract the Square Roots of I ax and x— 6x*, and divide the former by the latter: but, as our Author obſerves, you may abridge the Operation by multiplying the Numerator andDeno- 8 | -x minator by YT — ; by which it becomes . .—: and you need only extract the Square Root of the Numerator, and divide by the Denominator, in order to have the Value of y, The Opetation follows. EXPLAINE D. 379 eee Wen VTi —1⁰⁰ el * . Which Square Root, now reduced into a Series of ſimple Terms, may be cafily divided, dy 1—6x* after the 14 ſhewn yg by - doing of which, you 'll obtain this Quotient HY 55 4 + Lag &c. And therefore, by Rule the ſecond, the Terms being ſquared, will give * 3 — bal? 3x7 Cc. Elliptical Arch CD T + for 72 2 thee an 7% for the initial Limit of the Arch is at the Point C: ſince the Series becomes nothing when x:== AB vaniſhes : that is when CD vaniſhes. 44. I ſhould next proceed to ſhew how the Cube Root; and other higher Roots, of complex Radicals may be extracted, and thereby re- duced-to infinite eries s: which ns are perſormed after the ſame Manner as they are in ſpecious Arithmetic, by conſidering the Geneſis of the ſeveral Powers of a Binomial; and petforming the Extraction or Reſolution in every Caſe, as the Compotition of the Power, whoſe Root is extracted, may require; and continuing the Operation, after the Manner you do in common Arithmetic, in Decimal Fractions: as may be cablly underſtood by the Method of reſolving quadratic Ra- dicals wa » Burg But I think it needleſs to detain the Reader wk Ccc2 this: 380 af ANALYSIS by EquaTions, &c. this: ſince our Author's famous Binomial Theorem, mentioned alread in the Hiſtory of his Diſcoveries, ſerves more readily and generally not only for extracting Roots properly ſo called; but for expreſſing by a Series of ſimple Terms, any Root, Power, or Dimenſion whatſoeyer of a compound Quantity, having any Index poſitive ; negative; inte. gral or fractional. Which, without all Peradventure, is one of the moſt general and uſeful Theorems, that ever were diſcovered. 45. Sir 1/aac exhibits the Theorem thus: PFPQ" + P* ＋ AQ + *BQ + COT D G. * Where P + PQrepee ſents any compound Quantity, whoſe Root or Dimenſion whatſoever is to be inveſtigated : P, the firſt Term of that Quantity; Q, the reg of the Terms divided by the firſt; and conſequently, PQ, all the reſt of the Terms: — the numeral Index of the Root, Power, or Di- menſion of P -- PQ; whether that numeral Index be integral, or fractional; poſitive, or negative. Moreover the Capitals A, B, C, D, Sc. ſtand for the firſt, ſecond, third, fourth, Cc. Terms: i.e. A=P"; B=ZAQ; C=*ZBQ: &c. The fame Theo- rem may ſtand otherwiſe thus: PTPQ' = (P" x IO — p err "Wha" hap m © 3n is 4 QHE Which is the ſame with the former, only differently expreſt : ſome- times the one Expreſſion being more commodious ; ſometimes the other. | 5 460. This celebrated Theorem was diſcovered at firſt by the Method of Trial and Induction, as may be ſeen in the foregoing Account of our Author's Diſcoveries, Art. 14. Since that Time ſeveral Demon- ſtrations of it have been given ; or alledged to be given. The Me- thod of Fluxions is moſt frequently made uſe of for this Purpoſe : vis. M— 2, ; += E m— = M— 8 — 3 this Prop: that the Fluxion of ax is a . Which therefore ought to be proved independently of the Binomial Theorem : other- wiſe the Proof goes in a Circle. But we ſhall ſuperſede the Demon- ſtration of it, in this Place, reſerving it till we have explained the See his Letter to Mr. Oldenburg June 13, 1676. Com. Epiſt. No 48. See the Account of our Author's Diſcoveries above, in Art. 14. e Reſolution EXPLAIMNE D. 387 Reſolution of affected Equations: and only ſhew, by ſome Examples, in what Manner it is applied. 47. Ex. 1. Let it be required to expreſs Ver in an infinite Series; or to find the Value of y in ſuch a Series, ſuppoſing y = cc — xx *. By comparing this Example with the General Theorem P 22 =P +ZAQ+*ZBQ + *ZCQ &c. You have P cc, Q= 3” — — M=1, 12: therefore alſo P cc =c=A, =AQ = B = ATN = -C. * ＋ fir =D, D — = -E, Sc. Therefore it is y (V =) n £ Sc. the fame as above, Art. 36, being the Sine of an Arch in a Circle, whoſe Radius is c, the Coſine being x. 48. Ex. 2. Let it be y a Pi, y is fought, Here Pa-, 422 75 m=1, z: therefore allo it is P = os = A, AQ - e SRL nA ee AT. I AAS AG a x 8 =D, — BQ = GX 5 * == nm — 22 *. * ox 3 mn — Jn 8 5 . 3" 9 933 5 1 8145 1 NE $145 14 3 =” it — 1 ISE, &c. Wherefore it is y = (ax* + xV =) | 2434 | 8 11 I 1 2 F T 1 7 4 Ss oe ore ow ns hs; ” 243 ga? T 9141 2434 might have inverted the Order of the Terms of the Binomial, ſo as to make it ſtand thus x3 ＋ ax P: and in that Caſe it would have been In which Example you * PD NM, and Q = =: and therefore Fr A, AQ == B, BQ — = = C, &c. and conſequently = Han? =x + 382 ANACYSIS Y EanaTions, &. ＋ wy — = + rr — = Sc. The former is to be choſen when x is ſmall; the latter, when it 48:great in Compariſon of a. 49. Ex. 3. Let it be y=/05 + cx — „In: and ſuppoſe we take the fecond Form of the Theorem, vzz. PPC= _ N == P* X 1+—Q+ TX 2 Sc, Here we have Pc, -.. 05 —_— 5 c -x IL K- W ** . ä cXðx -x, UP ITT. c c5 Here it may be obferved, when you make uſe of this Form of the +4 POL: 5 . . — * 4 wid.od Q = , M l, 5; Pc: and therefore it is y=c xy General Theorem, the Series may be eaſily produced, by conſidering that the Factors both in the Numerators and Denominators:of the Nu- meral Coethcients go on in Arithmetical Progreſſions. This Series by involving the Quantity to the ſeveral Powers, and multiplying 1 4x — x5 2c e —=4actx% 42x00 1 EOfx3 mr 8x7 4+ 18c42"16x'5 by c, 18 y=c+ .g4 "+ 3, on * | — . &c. | N In this Example you might have inverted the Order of the Terms, making it ſtand thus V Tc Pc,: and then it would have been x5 P=— x5, Q = Px, Sc. and bo you would have _ had y.= * * ＋ A c. The former Series con- verges when x is ſmall; this when x is great. Fo. Ex. 4. Suppoſe u NN. Here you have P 4 * Q =. Ne P A, Ge. Wherefore, by the ſe cond Form of the Theorem it is y = (aa + xwd* =) aix 14 eee | 4xlx=240 , 4x1 X=2X=5 elELIEEIEITELF 36 zx0x9 25 3x2 a 3 XG N Iz x 15 4ʃ0 4 8 moot Son da hn its huts Oe, EXPLAINED. — —EX-»ͤ——Uiͤu 4. — ab cc pry 51. Ex. 5. Let y = be propoſed : and it is required to ind the Value of y in an infinite Series, and then divide the one by the other 3 or you might firſt multiply both by the Denominator, by way of Preparation, as was ſhewn above, Art. 43: or laſtly you may bring the Quantity 4+ — ccx* from the Denominator to the Numera- tor, by changing the Sign of the Index +, into it's oppoſite, fo that the Expreſſion ſtand thus eK X #—cax®\*, and then reduce che two Factors into infinite Series by the Binomial Theorem; and afterwards multiply theſe two Series into one another. Suppoſe we take this laſt Method: then for Brevity fake, ſubſtitute a for cc and 4 for — cc + rr, ſo that it be y = & + bx*|* xa* — a . By com- paring the firſt of theſe with the general Theorem you have P = aa, Q= = n il, 2: by comparing the ſecond, P=aa, Q — - x — m=—1, 2. And according to the firſt Form of that Theorem, you'll have for the firſt Series PA, AQ 2a 32 m2 bix® mz =B, B= — . C. c = =D, D 3 4 — LA Sc. and according to the ſecond Form of the Theorem, . pay 8 « you'll have for the ſecond Series P aa — and therefore -a 1 i * 7 A331 —1*—3*—5 x7 en 8.1 2 + 2X4 a* 2x4x6 a3 | 2x4x0x8 a4 Ge. So | - ba* * , bix* Ars 3 that a ＋ 4 + 2 — Bas 76 — 857 Sc. and aa — ax*\ * 128a7 = — * 755 + Lond + 5 —— 2 &c. Multiply them as follows, b b* x4 þ3,6 564x5 6 _ 2a TE ＋ 1627 1287 Sc. I * 3x+ 5 35 TT TETIET I bax* b*x+ b3.x® TI. + === * * S* b2x6 þb3 x3 + 2a + 4a * ＋ 32a? Sc. 3 35 36*x8 * 8a rat (a Sc. 6 bs + & +555" $6 35x L20a% 324 and * | 383 | 384 ANALYSIS by Equations, &c. and you obtain this Product 1+ e + i ib E e Leeder Tell Cc. Which therefore is the Value «« 128 V a*—ax* Al | | 52. Hence it follows, by our Author's ſecond Rule, that a Cute having ſuch an Ordinate, has it's Area equal to x + * LID, * n ,. Which by 1124 115245 * % replacing c for 4, and rr — ce for b, makes & + x34 L en 86472 — 40 = 644864. 175, + — 1 x7 + — 15 IE x? Sc. Which expreſſes the Length of an Elliptical Arch, beginning at the Extremity of either of the Axes whoſe Halves are and c; and having x for it's right Sine (as appears from Art. 42.) And this is the Series which Mr. Fame; Gregory ſent to Mr. Collins. See Com. Epiſt. No 20. And the fame Series will ſerve for the Length of an Arch of the Hyperbola, if all the particular Parts of each Term be made poſitive ; and the whole of the 3d, 5th, 7th, &c. Terms be made negative: that is, if it ſtand thus x + op Eat = + 4% + 7 04% +48c4rt+-24 54-578 112044 1152078 x? &c. And if you ſuppoſe p to be inſerted for =, that is half the Parame- ter of the Axis 27; or, which is the ſame, inſert pr for ci every where: the Series for the Arch of the Ellipſe will appear in this | Form: | x + 5:33 + ore + 175 * o+ di Sc. ** _ — . a + . T Which is the Series given by Sir Jaac Newton for the Elliptical Arch, in his firſt Letter to Mr. Oldenburg, Com. Epiſt. No 48. And this Series may be continued by obſerving that the Numeral Coefficients of the uppermoſt Line, viz. . >. 5+ &c. are Fractions whoſe Deno- minators go on by the common Difference 4, having Unity their com- mon Numerator : and as to the Numeral Coefficients of all the inferior Terms 92 * — 7 - ” FR 5 . * * SL * ' 7 | | x | N REM off 94 * = * | 1 * 3 . A - | Y * 22 f 28 | . * 4 * . | - 2 00, 3 IJ) -.al PP LATWVNE D \ { > * ww 4 — 4 * x 1 Xx” 8 4 . 2 ** & * — Aa 4 . Terms in every Column, that they ariſe by multiplying the atop l Coefficient of the uppermoſt Term continually, be the erm of b Progreſſion = . =, Ls . 177 Oc. the Leiter n denoting the Index of the Power of þ, in the uppermoſt Term of the Column. 53. If in the preceding Gregorian Series, you ſuppoſe r = c; or in the Newfoman r p; in which Caſe the Ellipſe becomes a Circles 1 1 x5 x | x : __ the Series becomes x + 5- + ,- As + nr Ge. Which therefore denotes the Arch of a Circle whoſe Right Sine is x, the Radius being r. Which Series may be produced to any Length by multiply- FR; ROOM ee L 1* 1 3&3 5 * 7 * » » ing continually by he Nuibers 2 45 +5 . — Gc. in forming the numeral Coefficients of the Terms. Which is the Series given by Sir Jaac Newton for this Purpoſe at Art. 38 of this Analyſis, ſuppoſing El. ö 8 | 4. Tis evident from what has been ſaid, that this Binomial Theo- rem will . equally ſerve for reducing complicate Fractions to infinite Series; and alſo for involving any compound Quantity to any perfect Power: as it doth for reducing complicate Radicals. | Ex. 6. Let it be required to find the Value of y in ſimple Terms, ſup- poling y = @ Kl. Here Pa, Q==>, m=5, =I: conle- quently P = a5 = A, A ga = B, ——BQ = 10a3Ix* —= C, ——CQ = io =D, 2 DQ = Fa. = E, 7 EQ = x5 =P, EQ =o =G: and ſo the Series terminates, ſince G (So) enters into the Compoſition of every ſubſequent Term. So that you have y =a + x0 == a5 + gan + Loa + 109%3 + gax+ ＋ x3, And it's evident that in this Caſe of perfect Powers, the Series muſt always terminate; and that when the Number of Terms is one more than the Exponent of the Power. | 55. And in order to reduce complicate Fractions, you have no more to do but bring the Denominator into the Numerator, with the Sign of the Index changed into it's oppoſite : and then apply the Theorem, Ex. 7. Suppoſe y = — 2 J expreſs it thus y — 2 — * — x By comparing which with the ſecond Form of the general Theorem, Xx you have P a, QS -, urn, 1 =. Therefore it is y = | | | D dd E plied, in @ variety of Examples. But it may be proper in this Place to Tranſmutation of Curves, founded upon Prop. 9. of the Treatiſe of the Abſciſs z and Ordinate y of a Curve: and ſuppoſe that the Index 1 is ſuch, that the Quantity e T ANALYSIS by Equations, &c. 1 I I x* Tx—2x—3 x3 IX—2X=—Jx—4 a3. 1 a ix2 2 ixax3 "ai txax3x4 8. Ex, 87 Let it be y= . that is y=2=*x@ Tan. By comparing a* N with the ſecond Form of the Theorem, you x2> have P =a*, Q= >, m=—3, n=1. Therefore by a due Subſtitution FEAT (art x DE EIS EF SIP a* 1x2 1 1x2x3 @a* Ix2x3x4 @ =) ;— +2 — 22 c. Which being multiplic by a — x, produces — — — 2 + 4 7 = Ce. . And ſo the Area's of the Curves, having their Ordinates thus reduced into infinite Series, are eaſily found by the Author's ſecond Rule. 56. And thus far I have ſhewn how the Binomial Theorem is ap- obſerve that, notwithſtanding the Series for the Area; or Fluent, found by the Help of this Theorem, always becomes infinite, when the In- dex of the Power, Root, or Dimenſion of the Quantity expreſſing the Ordinate of the Curve; or Fluxion, is not a poſitive Integer : and con- ſequently exhibits the Area or Fluent, only by an Approximation: yet it often happens that the Curve is geometrically quadrable ; and ſo the Area or Fluent may be fully aſſigned. Which is done by Means of Propoſitions 5 and 6 of the Treatiſe of Quadratures. Which ee, with our Notes upon them. 57. The ſame Thing may be found likewiſe by Means of the Quadratures. Thus let y = d#* xe define the Relation betwixt [ 4 , „ being reduced, runs out into an infinite Series: and yon would know if the Curve be geometrical I * quadrable Put e =: hence a = 4 =, and * = Therefore, by taking the Fluxions of e + fs" =x, you have . &, thatisx*:2$::nſ2 *: 1:: h: z. Now, ſup- paſe v to be the Ordinate of a Curve, whoſe Abſciſs is x, * 4 TP EAT Area equal to the Area of the Curve propoſed, having ꝝ for it's Abſciſs, and y = dz* K N for it's Ordinate: then, by that Prop. (Sd * De v:: (&: à ::) : z. Whence v= 5 X — V . « Wd! e+f2” l == (by ſubſtituting the preceding Values) 0 — ES . . Therefore, if a — 1 be a poſitive Integer, the Quantity 7 +=] 4 is a perfect Power, which may be expreſt by a finite Number of ſimple Terms (See Art. 54.) Let it be done, and each Term multiplied by 4 and then ſquared by Rule 1ſt: So you ſhall have the Area re- quired : unleſs 3 the Values of E, / and n be ſuch, as to give ſome Term with the Root x in it's Denominator : which would make the Area infinite (See Art. 28.) Therefore, when, in the Curve pro- poſed y = dz* x =, the Expreſſion 2, is an Integer, the Curve is quadrable, with the Limitation mentioned. 58. Moreover, ſince the Ordinate or Expreſſion d“ x Y may be converted into this other Form dz*+* x fo ez (by Article 28. Treatiſe of Quad.) if you put x = f+ ez"; and proceed as in the former Article, you'll find that the Curve whoſe Abſciſs is 2, and Ordinate y = d# x', is equal to another Curve, having the ] Un 1... =. 1 Abſciſs x, and Ordinate — = X — 7 +=] 2 Therefore if — be a poſitive Integer, the Curve propoſed will be quadrable, . with the like Limitation as to an hyperbolical Term, as in the preceding Article : and the Area found the ſame Way. But theſe two Areas, found by this and the preceding Article, lye upon different Sides of the Ordinate (See Sect. 5 of the preceding Treatiſe.) And if they can both be exactly found, the Curve is of that Sort, which we call doubly Quadrable. | | el | 59, The gth and 1oth Corollaries of Prop. 9. of the Quadratures ſerve for the ſame Purpoſe. Thus, if in Cor. 9. you put y=o =O, F=I=T=q, —— H N-; and ſuppoſe it be a= — 75 b= 75 g So Gc. then in the Curve whoſe Abſciſs is 2 and D d d 2 Ordinate Ü .m— 52 pF ** - - P _ Yy 6s Fd ' "7 7 pe * * 2 1 1 a - 9 LO FP 4 = * 497 * * * —— „ 4a* LS WY i 1 = . * * 4 — . p 0 * A * = 4 a * . - , . : pe - - > o #4 * 4 +% hy 2 * * — 0 = / \ ** 4 — we * - 4. 5 - — - * . 0 1 4 „ T3 - — 5 ” l ar - _ 3. A * „ * 5 I 9 * A f * 1 * „ „ 20 i 22 4 . $;,* * 1 2 * * : een 8 f ay” 6 n * — — - —— * — — — — N Mii ad Ft 0 * 4 a# 8 5 - = b F * 3 — 5 . * bo” . F * FLY w 8 * 1 » "Ny <<” CY _ ER ©» - — * 1 1 * n = . - * « (7 | . 4 * 0 4 1 oy 4 * "a 4 2 © . D » a N 7 * id w N * 1 vx ” ; by 6 . ? 1 p WD: * k * ** th 7 . N A ors r ho Ke, 2 My a inate wet 12 K U . into another which i is equal 4 to it, 7 e Ortlinate } is & * — 22 = + „ Or (by multiplying bak by the given Quantity S) the . Sie Ordinate is dE -N THe” „ paſſes i into another equal Curve, whoſe Ordinate is - * * 7. Compate the Ordinate dz* K =), in Art. 555 ki former of tk two, _ you have q n; dre Subſti itution, 2 given Cue, al Ondinate | is dal xe 72 j , Pail. ad . es into another of an equal Area, whoſe Ordinate is - * == +; Whence, if —— is be a politive Integer, the Curve is indi quadrable, as baſire, And the ſame way, it may be thewn that the Curve propoſedis likewiſe quadrable, if — be a poſitive Integer. But I treated more fully of this Affair; and | in a different Manner, in the OP AER, of OP: 5. Quad. Which the Reader may conſult, L118 SN FU * _ E " - * W s E Cr. II. Explication of theNumeral Reſolution of affected Equations, 5055 contain d in Art. a 21 29. - Wo af L 66h 2 FN ker to prey EY our Anthor's Method of extraRing the Roots of affected Numeral Equations, with the Abbreviations made uſe of therein, it will be proper to explain ſome Things which reſpect 2 Fractions, and the Operations about them. 261, 1“. In Multiplication of Decimals, the Number of Places of 8 Fraftions in the Product, muſt be made equal to the Sum of the Places: of Decimal Fractions in both Factors. And in Diviſion, = Number of ſuch Places in the Quotient, muſt be made equal to ' 94 & 7 7 K « 4 2 . 81 . ; 4 = Exceſs of thoſe in the Dividend above thaſe in the Diviſor. * * - 0 as Ho _S p - * as ; # p a * 8 N. 1 12 — N 2 * " LL _ . 2 : 7 3 9 R | 8 >. a0 24% ; „ N n Nee —_— 1, 22 "4 = - ; Pe 7 997572 r Iz * 2 A ** $. * * * *. * * 2 1 o * A p - 9 . W oy . k Z ws 4 Fn \ * pe *. \# 3 * * 8 8 © * A p l F * 1 * 4 - y p * D 4. L 4A TT Mead, = * FI e E-» : : K. 10 4 * q - *7 {FL * 3 _ , - - * * SEES | 4 "© „ 1 7 ky . "» ;% 1 P : * : 7 F.Y a $4 * . w IE „ 2 ; 1 42 at ot 7 1 7 # 4 A » © \ * © - © F : # IJ - } *. 5 * „ , « — a - « 7. a E * 1 . ws i . * = w 4 * * y 7 . F . « * 1 i. l * N * * * ! | : : : | : 1 * * 4 - a «# © IT * 1 7 24 - | & I wo 7 8 0 a = * 4 * N. -* - 4 AF = 4 a i. g % PW.” 34K — « | 0 2 | Po __— 22 oon y od * * 5 - * 997 „ N p "4 -A$., * $ 5 0 l * * 7 * Ling = 1 _ 3 72 0 k | "1 « * © > \ : 5 0 20 «I = wed. * 2 8 1 5 "le 7 5 "37 Was . OTIS TT ITS COT STEIN r <4 * 4 , p. . ; * * : - * . / = = N 5 : ” * | 93239. 12 | WS: | 18 a f ZI tk © = ö 5 - | j >, "67 * nne - l . : ws = 4 g 4 * = * — 2 ry : * + = U PL p \ v . 2 i p A * 1 1 * * w » = * » þ iy 4 — , * N X * 14 — = | "4 . nen a | rf. TY 9 4 8 e a v3 MOI TS, . 2 : - - +| — ' * 2 a 4 e n ACS +4 F234 „ \ 45 . - - * a >. C > i | e of > Las LESLIE. - i \ -& ; — * "= = » * . * 0 4,4 » Yu 1 0 * .. 1 . p< * 5 % *- : 9 *& 14 g W 2.4 * * um, you m re ecima CC SD ok ſai 5 | n. the left J.. A ER 4 . E "yy 1 2 & Fay "A 25 #7 1+ IE * 5 - oY * 1 > i the Number of Figures in the Product be riot ſo e the Product, until the Number of Places altogether be g Reet 88 7 5 8 5 . 4 that Sum. And in Diviſion if there be a Dete& in the Number of Places in the Quotient, it muſt be ſupplied the fame way.” But if the Places of Decimal Fractions in the Dividend, be fewer than thoſe in the Diviſor, you muſt add Decimal Cyphers upon the right hand of the Dividend, untill the Places of Decimal Fractions be as many as in the Diviſor. Which Operations are commonly known; and the Reaſons of them underſtood. 4 51 1 other on MONwd 2% When any two Decimal Fractions are multiplied; : whereof one, or both, have any Number of Cyphers prefixt, the Number of Cyphers prefixt in the Product, muſt at leaſt be as great as the Sum of thoſe prefixt in both Factors. For, by the laſt, the Number of Decimal Places in the Product, is equal to the Sum of thoſe in the two Factors: but the Number of ſignificant Figures in the Product, towards the right hand, can never be greater than. the Number of ſignificant Figures in both Factors (which is evident from the Nature of Multiplication) : therefore the Number of Cyphers prefixt in the Product is at leaſt equal to thoſe prefixt in both Factors. And often times they are more. Thus 0.0024 x 0.03 = 0.000072. 125 ze. Hence it follows that if » denote any Decimal Fraction wi n Number of Cyphers prefixt, then the Number of Cyphers prefixt in its ſeveral Powers, vis. , 73, 74, 75, Sc. muſt be at leaſt an, zu, 4n, zu, &c. vis. ſo many times u as the Exponent of the Power denotes, ie . 4. If two Numbers are propoſed to be multiplied, whereof the one is either a Decimal Fraction or mixt Number; and the other a Decimal . Fraction, with Cyphers prefixt ; and the Product is uy requited to a certain Number of Decimal Places, you may cut off from the right hand of the firſt as many Decimal Places, as will make thoſe which remain, together with the prefixed Cyphers in the other, equal to the Number of Decimal Places required in the Product: and neglect thoſt which are cut off, in the Multiplication. Thus if it were required td find the Product of 123,4.567 and 0.000089, only. to 6 Places of De- cimals, you may neglect 67 in the Multiplication : if the Product were required only to 5 Places, you may neglect 567: ſo as to multiply 0.000089 by 123.4 only: and if the Product be required only to 4 Places of Decimals, you need only multiply 0.060089 by the tear Part 123. But if the Number of. Decimal Places, to which the Pros duct is required, be leſs than the Number of Cyphers * in * | ccimal * TY * 2 . I > $* - | a 4% * . 8 ** = FEY » : x” N - - 9 » 1 ö * + % N = * 2 * » £ 390 ANALYSIS by Eavartions, &c. Decimal Fraction, which is one of the Factors, e.g. 0.000089, then you may neglect, not only all the Decimal Places of the other F 2001 123, 4567; but as many of the integral Places as the Decimal Place, requited in the Product, want of the Cyphers prefixt in the firſt Factor. Thus if the Product be required only to 3, 2, 1, Places of Decimals, you need only multiply 0.000089 by 12, by 1, by o _ reſpectively. | PO wha For fince 0.000089 is leſs than 0.0001; and all after 123,45 le than 0.0 i: which two multiplied by one another produce only o 00000; by Obſ. 1. Therefore it is evident that 0.000089 multiplied by what follows 123,45; muſt produce ſomething leſs than 0.000001 : end therefore cannot affect the Product within the 6th Place of Decimals, And the reaſoning is the ſame in every other Caſe, by conſidering the Decimal Progreſſion of the Places. $, 62. Theſe things with reſpect to Decimals being premiſed, our Au- thor's Method of extracting the Roots of affected Numeral Equations, E with the Abbreviations he makes uſe of therein, will be the better underſtood. | 13 Let ar + br + cr3 + dri &c. = A, repreſent any Numeral Equz- tion, in which a, 6b, c, d, &c. and A are given Numbers, poſitive or negative, and 7 the Root whoſe Value is fought. Then by dividing by a, and tranſpoſing, you have r= ogy i yt Sc. Now when one of the Values of r is ſmall, ſo as it may be denoted by a Decimal Fraction with Cyphers prefixed, you may eaſily find the Limits within which the ſupplementary Parts - i 7 ; =, Sc. are con- tained: and conſequently the Limits within which the Difference betwixt the true Value of 7, and the Approximate Value of it, viz. r=, muſt be contained. For fince S is an Expreſſion, in which 7* is the Square of r a Decimal Fraction having Cyphers prefixt; and therefore twice as many at leaſt (by Obſ. 3.) you may always find how many Cyphers will be prefixt in the Expreſſion XZ from the Nature of Decimal Arithmetic. And after the ſame Manner, how many ſuch Cyphers will be prefixt in the Values of . =, Sc. For if 2 be leſs than 1 (in which Caſe it's Value in Decimal Arithmetick is entirely below the Place of Unites) there muſt be as many Cyphers, 77 g at EXPLAIMNE D. at leaſt, prefixt i ; ” as in * (by Obſ. 2.) But if - + contin any Number of Integral Places, then the Cyphers prefixt in — will, at leaſt, be equal to the Exceſs of the Cyphers prefixt in *, above the Integral Places in 5 Thus, if 7 had 4 Decimal Cyphers prefixt in it's m_— and ſuppoſe a=11, 16196; þ=6,3; then, fince . - (= 771 15 96706) is leſs than an Integer, and 7* has 8 decimal Cyphers pelt, muſt have as many prefixt at leaſt. After the ſame Manner - i would have 12 at leaſt; and univerſally , as many, at leaſt, as is — by 41. But if, the Value of r remaining, it were 4=0.243» þ=6.3 ; then fince — «(= a6) is more than 1, and contains two integral Places, therefore = may have only 6 (=8 — 2) Decimal Cyphers prefixt : although in ſome Caſes, when the Figures are ſmall, it may have more. And univerſally *- has 4. —2 at leaſt. And fo in other Caſes. Which Things eaſily follow from what was ſhewn in the preceding Obſervations. And therefore you may proceed thus: I, Conſider how many integral Places would be contained in 75 (which may be eaſily found by Inſpection) cut off as many of the Cyphers prefixt to the ſignificant Figures in *, the remaining Cyphers ue the feweſt that _ be prefixt to the ſignificant Figures in =; | 2%, If the Value of — ? have no integral Places, but only decimal Places, then conſider how many 1 Decimal Cyphers would be prefixt to the lgnificant Figures i in the Value of - — (which you may eaſily diſcover by Inſpection) and then the e Cyphers prefixt to the ſignifi- cant Figures in the Value of A will be equal to the Sum of theſe prefixt in < and 7*, both together. 63. From what has been ſaid it appears, that, if the Number of Decimal Cyphers Nein! in the Value o r, be known, in the preceding 391 Equation r = > — — — = Ge. in which 2 is an approximate Value 5 [2 SE tay WR e 3 5 | b 1 | 5 1 1 . | I 7 on r r oe $5” 39 ANA ENS IS by Bavarions, &c, Value of r, you. may eaſily know to what Number of Decimal Places ; t Se 2. Et BLISS YG the approximate Value of 7, VIZ, , may be continued, before any 4 | wary % „ a EY — { * 1 fas 5 2 1 Ly, | of the ſupplementary. Parts , , &c. can affect the, Root. Thu, p | WW you had this Equation 0.000541550536 ＋ 11. 1620474874 | —_ 2 N 9 14 W is. of, 4 1 ” | L Ins 2 387 +73 =0: Which is the third ſupplementary Equation in : * our Author 8 Example, ſuppoſing it to be fully drawn out: that Is t | — ©-c00541550536 _ + & — I 2c of! adi ts it. 1 | er dont 11.16204748 yy 11.16204748 11.16204748 ? and it is known that the Value of + is a decimal Fraction having 4 Cyphers prefixt, 1 Then becauſe the Term TT is equal to 77.185 Tc X, of | which the firſt Part is a Decimal Fraction with one Cypher prefut; | and the other a Decimal Fraction with 12 Cyphers prefixt, therefore | the firſt ſignificant Figure of it's Value will fall upon the 14th Place 8 of Decimals (by Art. 62.) ſo that the Value of the Root 7 cannot i be affected by it within the Limits of 13 Places. After the fame Manner the Term 229. will not affect the Root within the Limit of 8 Places of Decimalts. Whence you conclude that the Value of /, 1 deduced from ſuppoſing r =— 2225452532 will be true to 8 Deciml : 1116204748 Places, that is the Divifion of o. ooo 54 1550 536 by 11. 16204748 may de carried forward to ſo many Decimal Places: all which will be true unleſs perhaps the laſt Figute, Which may ſometimes be increaſed or | diminiſhed by 1, by the Addition or Subtractiom of the ſupplemen- | tary Part. And ſince 7* will, at, leaſt, have twice as many Decimal Cyphers prefixed in it's Value as 7 has, therefore, generally ſpeaking, the Number of Decimal Places or Figures, to which tne Quotient de- noted- by =, may be carried, will be twice as many as thoſe to which the Root hath been found already, by the former Apptoximation: agteeably to what Sir Jaac remarks Art. 22. So that in this Method of reſolving affected Numeral Equations, every new Operation, as it were, doubles the Number of Decimal Places formerly found in the Root: even ſuppoſing all the Terms of the ſupplementary Equations to be rejected, wherein the Supplements p, g, 7, &c. riſe above the iſt Power; unleſs perhaps when 4 exceeds 4: in which Caſe the ſupplementary Part ar. may affect the laſt Figures. 64. By * P LA - 7 1 % 0 . T E D. * * * — . E Xx 3 Ty «> 1 N; 4 « 4. f * q 64. By the Conſideration of theſe. Things you may abridge the at conſiderably towards the latter End, +5 neglecling ſuch Nn or Parts of Terms, as you foteſee will be of no Uſe within the Limits to which the Root is propoſed to be carried: as you ſee done by che Author in the Proſecution of the Root y, in the Example propoſed by him. However it may be obſerved that the Quotient 0.000048 53 &c. ens in the laſt figure 3, by bis omitting the Term 9? altogether, in transforming the Equation of q: for it ought to be 0.000048 51 &c. only: as you may find by Art. 63. : 65. Schol. Dr. Halley's Rational Formula for extracting the Roots of Equations is eaſily deduced from what hath been ſhewn above. Let fs + g2* + 23 + ia Sc. K be any Equation; z the unknown Quantity, the other Letters ſtanding for known Quantities : and let | be a Number as near a Value of à as may be; found by Trial, or any other Way: ſubſtitute / 4-7 inſtead of z in the given Equation fe + gz* + bs &c. = K: and let the transformed Equation be or + br* + cr3 + dr* Sc. = A: where r is the Supplement to /; or the Difference betwixt / the approximate Value of z, and it's true Value: which therefore is ſuppoſed to be ſmall. Wherefore ar = A or r= 2 nearly; or ar ＋ b = A more nearly, that is ==> fere, Subſtitute for 7 in the ſecond Member of this laſt Equation, it's approximate Value deduced from the preceding r —= = and you'll have r= (a=) aaa quem proxime Wherefore you havez =1 + 222 aA 393 "FTA nearly. Which is Dr. Halley's Rational Formula: publiſhed in Phil. Tranſactions for May 1694, Ne 210. 66. And if any one will be at Pains to conſider the Buſineſs of the Reſolution of Equations, there may be many Methods of approximating to their Roots diſcovered. Thus, retaining the ſame ſupplementary Equation ar + 6br* + cr3 + dr+ &c. = A, you have, as before r — a br* — nearly: or 7 = 2 — — more nearly: inſert in the latter Member of this laſt Equation, inſtead of 7 it's approximate Value 27, deduced: from the preceding, and you'll obtain r = 2. — - * = for a nearer Approximation to the ry Root r. Which approximate Value of r being found, it e: and in the Equation of , inſect Eee e+s | | | | ANAEYSIS h Bavarions, Oc. ＋ for v: from which: you may find an approximate Value of ,. and {6 on for the other ſabſequent Supplements. Re! 657. Our Author tells us that when there is any Doubt whether the Value of 5 in the firſt ſupplementaty Equation, approaches near enough the Truth, by ſuppofing 1056 1 = o, you may aſſume the three nal Terms; titus, 6p* . 10p — r= 0: and bythe Sofation of the Ou. dtatic, find the firſt Figure of the Value of p. Now by the $otation of the Quadratic 6p* + 105 — 1 =, yau have, p— r of which two-fold Value you take L = + oog quamproxime , which: is the leaſt Root, and that which approaches neareſtito the Va. lac of p, by putting 10p — 1== ©. 'Fhe Reaſon of which, ſhall. he ſhewn Art. 71. And thus you find! 0.09-anearer Approach to the Va. lus of p than . 1 is: and it gives you one Decimal! Place more in the Quotient. Moreover, as he tells us, it is convenient to aſcertain the ſecond; or third Figure, of the Quotient, the fame way, when in the laft reſulting Equation, the Square of the Coefficient of the Penult. Tern is not ten times. greater than ths Product of the laft Term multiplied: into the Coefficient of the Antepenult Tar ms 30 vin 9 68. To underſtand the Reaſon. of this: ſuppoſe ar + br A, to repreſent an Equation reſulting, when all the Terms except the three laſt of any ſupplementary Equation, are rqeGed:: and ar = A, that which ariſes by rejecting all ſave the two laſt Terms: then we ma eaſily find: the Diffetence betwixt the two correſponding Values of x ariſing from theſe two Equations. From the laſt; we have r— =; fromthe fitſt, r= DEVLIN, That we may compare bel two Values of 1, ſuppoſe-a#*== E * (where 2 repreſents the Num- ber which multiplying A, makes tho Product equal to 47) that is b— — : infert this Value of 5 inſtcad of it, in the Equation Pk ZE, and youll have r= — 1 T U TAN 5: of * — „ * which two-fold Value, be 25 + N/ 125-2 * 2 is to be taken, which s the leaſt of the two Roots, and that which approaches neateſt to the Root of the Equation ar = A, or r, betwixt which and it the Difference is ſcught. This Differencs! then is the ſame with the Difference betwixt — and 2 X < I'S +2 — 22: which will vary ac- cording + £XPLAIWED: | cording co the Value of 2.) H-itbe'z = 10, then Vial I = 22 2 0.91 6 Sc. and therefore the Difference bet wint che — 1 = this Caſe js T_TJ16.x = = 0.08386. x ©, Which is ieſs than n of 75 but if a 9, chon 7 ÞF'2— 0 Ge. do that the Difference af he Roots -wonld. be nearly ,0.0g1.89c. * 2. which is nigh to 28 2. And if 2 be 8 or any thing 'lefs, the Difference willhe more than -*-af & (the Diffprence ſtill inereaſing as diminiſhes) Whence-tt ap why, if z be Je than 10, that is, if the Square of the Coefficient of the Penult Term be not ten times greater than the product of the, laſt Term and Coefficient of the Antepenult Term, it is proper to aſcertain the firſt Figute of the Value of r, by aſſuming the three laſt Terms af the ſupplementary Equation ; and taking the lat Root of the Quadratic: heeauſe in this Caſe the approximate Root of r found oy long only the two laſt Terms of the Equatioo, may differ from the true Root, by. a Difference which, may alter the firſt Figure of that approximate Value af rx. | 59. Nay, as Sir ac obſerves, if in the.whole Courſe of the Ope- mation, vc aſſume therhree. laſt Terms of each ſupplementary Equation, inſtead of the two laſt only: and extract the leaſt of the two Roats out af the Quadratic, we ſhall obtain about twice as many Figures in the Quotient the one way, as the other, at each Operation: as may be. eaſily callected from hat. hath been ſaid in Art. 63, 64: by which it appears that each new Operation; or ſupplementary Equation, will, as it were, triple the Number of , Decimal Places already found in the Quotient. And. it may be obſerved that by aſſuming the three laſt Terms of the. ſupplementary Equation, as in the preceding Ar- tele, you have Dr. Haley s inratianal Formula, publiſhed an the Phi- Mr. .Raphſon's Mfethod of Approximation in the extracting the Roats of Equations, publiſhed in his Analysis Equationum uni venſalis, it is, in Heel, the very fame with our Author's Method here laid down ; which proceeds by aſſuming only the two laſt Terms of the ſupple- mentary Equations, at each new Operation. K 70. This Method of extracting che Roots of affected Numetal Equations, may be ſucceſsfully appicd to all kinds of them, how ſoeyer: Whether the Terms be al | wanting: and therefore alſo ſerves for extracting the Roots of , Ee e 2 Equations, ee compleat ; or any Number of them pure 395 kÞphical Tranſactions in the Place mentioned already . And as to * Art.65. | | | | 396 ANALYSIS by Eavartions, &c. Equations, that is the Roots of Numbers, eſpecially the higher Ones Moreover, after you have found the Quotient to any Degree of Ex. actneſs you pleaſe, the Quotient ſo found together with the Root of the laſt reſulting Equation, muſt be equal to the Root of the Equation at firſt. propoſed : as appears from the Proceſs itſelf. - And therefore, having continued the Work to = deſired Length, in ap roximating o the Root, you may prove or ere, as well as in other arithmetical Operation, which proceeds by Reſolution. For if you ſubſtitute for y, in the Original Equation, the Quotient already found -+ the Supplement g, r, or s, Sc. the tranſ- formed Equation ought to be the fame with the Cy Equation of q, r, or s, &c. Thus e. g. having found, in the Author's Example, y=(2+p=)2.1+9g: transform the original Equation y3 — 2y — 5 So, by ſubſtituting 2.1 2 for y, and you have 0.061 ＋ 11.234 4 6.37 + q* = ©, the ſame with the correſponding ſupplementary Equa- tion : which aſcertains the Truth of the Operation. 71. There ſeems to be no other Difficulty that can occur to the young Geometrician in this Matter: unleſs it be this. Every fupple- mentary Equation is of the fame Order or Rank with the original Equation propoſed ; and therefore the Roots or Values of the Supple- ment may be as many as the Dimenſions of that Equation : how doth it appear which Root of the ſupplementary Equation ought to be taken: and that the Value of the Supplement which arifes by aſſum- ing the two laſt Terms of that Equation; or the leaſt of the two Values of the Supplement, which ariſe by aſſuming the three laſt Terms, is an approximate Value of that which you want, to make up the true Root ? 5 In order to clear up this Difficulty, let ay + 6y* cy Gr. A, reprefent any Numeral Equation, y being the unknown Quantity, and the other Letters ſtanding for known Numbers; poſitive, or negative: and let the Roots of the Equation be called 7, g, b, &c: poſitive, or negative. Then let be a Number which approaches near to one of the Roots as F: which is found by Trial; or otherwiſe : and let it be n+p =y, where p is the Supplement. Subſtitate 1 ＋ p for y in the Equation ay by*-cy3 Sc. A: and thence ariſes a new Equation, v1z: ze A * . 5 an ap | bn T3. bps Daa, which you may call the Equation of p; ee he n | and the other the Equation of y. In this Equation. of p; the Roots or Values of p are juſt as many as the Roots or Values of y in the e | ues certain the Truth of the Operation | „ 6&0 P i 00 - > © * * — —— asg EXPLAINED. Values of p, one, at leaſt, is very ſmall in Compariſon of u, which is denoted by f— u, ſince n was ſuppoſed to differ but little from 7 ind this therefore is the Value of p, which is the Supplement ſought Wherefore in the Equation of p, the Terms which include the Powers of P, that is all before the two laft Terms, will be ſmall in Compa- riſon of theſe two laſt Terms, if p be ſmall enough in Compariſon of n: which is the Caſe when the Root f-— is ſought: therefore thefe Terms may be neglected, when you are ſeeking a near Value 3 | of the Root - n: ſo that you'll have 5 12 So nearly, Cn) + zen a & Ce. or e LL 7 a near Value of the Supplement p, that is of f—n. And you may eaſily diſcover how near this Approach will come to the true Value of the Supplement p or f-: which (ceteris paribus) will be different according to the different Propor- tion that 1 bears to p orf -: ſuppoſe p(=f—n)==, then in 10? the preceding Equation of p, each Term, proceeding from the left to the right, abſtracting from the numeral Coefficients, may be confidered as ten times greater than the Term immediately ſucceeding it : in which Caſe the two laſt Terms of the Equation upon the left hand, are, ab- ſtracting from the Coefficients, more than £2; of the whole; the three laſt Terms more than , of the whole, Cc. And therefore the Value of the Supplement p deduced from the two laſt Terms of the | Equation put equal to nothing, will not differ, generally ſpeaking, from the true Value, but by a few hundred Parts. This will further appear if you compare the Value of p deduced from the abridg'd Equa- tion, vix. þ = ee with ah 509 derived from the 188 . . . — an — bn* 4 Oc. '4+ c. entire Equation, which will be e — Senn 2 c. : . . JT An- cn! Ee. . ; that is (by inſerting — for p) = ET + 3cn® Nc. bx 0.01n* þ ze x 0.014) Ho. ot Wc. _ pw 3 Tree Ti from whence it ap- pears that the approximate Value differs from the full. Value, gene- rally ſpeaking, but by a few hundred Parts; finee the Value of the: Places in the Numerater of the ſecond Term of the Value of p, are but hundred Parts of the Value of the Places in the Numerator of the krſt Term, the Denominators being the ſame. And after * 1 of y, ſince p=y —n, that is / — n, g , b — n, Gc. but of theſe * a AN AU VS 18 by. Rogations, &c. Manner wh acre that. if þ ===, » the approximate Vee wit differ Koch th e true . N thouſand Pa Ge. but whatever, 1 8 gs. 85 to n, e G = ients: Fe the exact Limits, in 58 A . Art. 1 tp TA 2 4 % cog a u yo u maß . diſcover to whet diucaber of Phat th all true; when you be he Ratio — which the Ratio of » to þ —_ not deſaend. But if the Square of the Ctefficient of the penult Term be not ten times greater than the Product of the laſt Term multiplied into the NN of the 2 Term, when u is not much more than 100 (of which Cate I ſpoke Art. 68) then by aſſuming the three la Terms of the Equation "fp, „ and putting them equal to nothing, the — 'of 8 adratic, res à two-fold Value of p: now of theſe du take the ue fer ptement, becauſe till proceed _ dhe n of p which yau kg is a ſmall 778 N . which ma eee to be the leaſt Value of p, dere ngs concerning our Authors Mabot of relo olving num fr ge Tard ah; it 20. be proper here to remat 1. He "PILE that PETER TIE Value of y, which is aſſumed in the beginning, differs not a tenth part of itſelf from the true Value. Such an agproximate- Value may be found by the Methods for determining the Limits of the Roots of Equations, found in Al- | gebraical. Writers ; or perhaps by repeated Trials. 2. This. Method of reſolving numeral:Eguations, equally extends to the diſcoyering, of all the Daſfible Roots of any Equation, whether they be poſitive'or negative, commenſurable or incommenſurable: for whether the approximate Value of y in the prapoſed Equation be taken poſitive. or negative; if it be ach an approximation to any real Root pod * been mentioned, the Operation goes on àfter the fame Man- However, if any one chuſe, he may transform any propoſed Equation Having dne or more of it's Roots negative, into another, in Which all the Roots ſnall be poſitive, by increaſing the Roots by a Number greater than any of the negative Roots, by Tranſformation: and fo he»fhall only —— 3pofitive Roots to extract. Moreover this Transformation: is of Uſe for determining the Limits of the Roots: for when you liave foundithe two immediate Limits betwixt which EXP EAT TE D.. ny Root tek PP one is greater and the other lefs than the Roat fought, in the Manner taught by our Author in his Arith. Univ. Capi. 4 Limitibus kene, vou may cafily diſcover whether any 6f theſe Limits doth or doth not differ by g of itſelf from the true : for if you ſubſtitute the Limits in Place of the unknown tity'in' the Equation: and then ' ſubſtitute the leaſt Limit in- creaſed by Pe of itſelf; or the greateſt Limit diminiſhed by g of -it- ſelf, in Place of the unknown. Quantity; and obferve the Signs with which the Aggregates of all the Terms of the Equation ia every Caſe are affected, you'll thereby determine what yon want: for if the Ag- te ariſing from the Subſtitution of the leaſt Limit, be affected with a different Sign from that with which the A te is affected dy ſubſtituting chat Limit increaſed” by Ez; of iiſelf, the leaſt Limit doth not differ , of itſelf from the true. Root; otherwiſe if they be affected with the fame Sign, it differs more than : the ſame Rule is to be obſerved. with reſpedt to the greateſt Limit: but here I f uppoſe, that the Difference of the two Limits is greater than g of — leaſt Limit in the firſt Caſe; and greater than g of the ſt Limit in the other Caſe: for if the Limits don't differ that much from one another; they will not differ from * true e by de 7s of n ro- - 4 fin 4 This Method of of reſolving Equativas i — lagi applied for finding the incommenſurable Roots: becauſe the commenfatable Roots are found more conveniently and ee wy the Mcthods ee a in Algebra. 4. If the laſt Term of any ſupplementary Equation of #,'9; r, &c. auch, then the preceding Quotient already found is a true Root; or full Value of the unknown Quantity y: for in this Caſe the leaſt Root of the Supplementary Equation is equal to nothing, bn! is . Supplement is Ry to N — r ” Þ% 1 TY — s E C T. — 4 Explication of the Reſolution of affected . 2 l Equations, contain d in Art. Je 37. | 2 "_ HE ſpecious Equations, whoſe Reſolution our Author treats of in this Placa, contain two unknown Quantities, as æ and ): which. are to he conceived as the Abſciſs, and: redtangular an, ) ANALYSTS h Eqvatioxs, &c, of a Curve; wWhoſe Relation to one another is expreſt by any (@ Equation: the other letters ſtanding for definite known Quantitie, And it is propoſed. to expreſs the Root y in a Series of Terms in. cluding the Quantity x together with the other known Quantities Thus in the Equation y3 + a% — 243 + axy — x3 =o, which he propoſes, x is confi as the Abſcib, and y as the Ordinate, in , Curve, 'whoſe Nature is defined by that Equation. Now that the Value of y may be found in ſuch a converging Series, it is nece to conſider æ either, 1. as very ſmall ; or, 29. as very great; or then, 3*. as differing-very little from ſome given Quantity, That ſo by ſup. poſing x to be very ſmall, the Series may converge wherein the Root y is expreſt by a Progreſſion of Terms, in which the Dimenſions of x increaſe in the Numerators: , 25. By ſuppoſing x to be very great, that Series may converge, in which the Dimenſions of x continually increaſe in the Denominators of the Terms: Or that, 30. By ſuppoſing x to differ but very little from ſome given Quantity, ſome Species or Letter, as 2, being ſubſtituted for that Difference, it may come in Place of the Species x conſidered as very ſmall. And all theſe three different Suppoſitions, may have each it's Conveniency, according to the Circumſtances of the Caſe. ot it 74. The firſt is the Caſe Sir Iſaac chiefly inſiſts upon: becauſe moſt uſeful in the Affair of Quadratures: and to which the other Caſes may be reduced. MTs In which he after this Manner. He ſuppoſes x entirely to vaniſh : and from the reſulting Equation ſeeks a Value of y. This muſt be an approximate Value of y, becauſe when x is very ſmal, the Value of y cannot differ much from it's Value when x quite wa- niſhes. But as this is but an 8 Value of y (ſuppoſe you call it A) he puts y =A + p : where p is the Supplement to the Root, as in the Reſolution of Numeral Equations. Wherefore he ſubſtitutes A) inſtead of y, in the given Equation : whence ariſes a new Equa- tion {which we ſhall call the Equation of p) from whence the Value af 5 is to be derived. Now although p may have ſeveral different Values, in this new Equation, -yet it is the leaſt Value that makes the Supple- ment (for which See Art. 71) Wherefore, ſince both x and p ate very ſmall, he rejects all the other Terms unleſs thoſe where x and P are, ſeparately, of the feweſt Dimenſions : and ſuppoſes them, 2 it were, equal to nothing from whence he finds an approximate Va- luo of p: Which call Be: and put p BY . Then he ſubſti tutes + 7 inſtead of p in the Equation of p; 9 being the Sup plement to the approximate Value of p, viz. Bx; as p was * 1 | the —w , 5 a OJ a am aids = EXPLAINE D. the Supplement to A, the approximate Value of y. And from hence you obtain a third Equation for finding an approximate Value of 7 viz. by the like Means, that is by rejecting all the Terms of the Equation of 9, fave thoſe wherein and x are ſeparately of the feweſt Dimenſions, the other Terms being comparatively of very little Sig- nificancy. Which near Value call Cx”, org = Cx* + 7: » being the Supplement. And ſo he proceeds. Whence he finds y=A+p U ion A + B CM, &c. By which you find a Series of Terms continually approaching to the Root y, that is to one of it's Values. The Proceſs may be ſeen in the Treatiſe itſelf, 75. In which you may obſerve the following Things *. 16. That in the Equations of p, 9, r, &c. upon the right Hand, thoſe Parts of the laſt Term, in each Equation, will mutually deſtroy each other; in which x is of no higher Dimenſion than in the laſt, or immediately preceding Term of the Quotient. Thus in the Equa- tion of p, the Parts + a3 + a3 — 243 of the ultimate Term; in the Equation of 9, the Parts — a*x + &; in the Equation of r, the Parts + a — 7 a (and ſo for the reſt) do deſtroy one another. The Reaſon of which is evident, by conſidering that thoſe Parts which de- {troy one another in theſe laſt Terms, ſtand oppoſite to ſo many other Terms in the collateral Equation upon the left Hand, which were put equal to nothing. For ſince a, 5 x, + 55 Sc. are the Values of y, , q, Sc. in the Equations y3 ＋ 4 — 243 o, 4a + a*x = © Qs — a D o, &c. Hence it follows that the Subſtitution of By — Ax, + E, Sc. in place of y, p, 9. &c. muſt make the Whole equal to nothing : So that thoſe Parts of the laſt Terms of the Equa- tions of p, q, 7, &c upon the right Hand of the Diagram, which have been mentioned, muſt deſtroy each other. Whence it appears 2. That the Powers of x in the laſt Term of the transformed Equations of p, 9, r, &c. upon the right Hand, increaſe more and more continually ; the loweſt of which, after rejecting thoſe Parts To prevent being miſunderſtood, let it be obſerved that the Series expreſſing the Value of y, is called the Quotient: and with reſpect to the Equations of p, g, r, &c. which ſtand upon the right-hand Column of the Diagram, I call that the fr ft Term, in which p, g, r, Cc. 15 of the higheſt Dimenſion : although it be placed laſt in the Diagram; and that the /zf Term of the Equation, which ſtands firſt in that Part of the Diagram : becauſe the Order of the Terms ought to be computed by the Dimenſions of p, g, r, Cc. Moreover when any Term of an Equation is complex, I call the ſeveral ſimple 82 which make it up, Parts of that Term: thus 2, . b Parts of the penult Term of the Equation Der * * Fff which 402 Art. 71. 73. through all the ſubſequent Transformations in inf. So that in all _ tities p, 9, 7, Sc. continually diminiſhes, viz. the leaſt Value, { ANALYSIS by Equations, &c, which deſtroy one another (according to the laſt Remark) will be n higher Mi the higheſt Dimenſion of x in the preceding part of the p Quotient. | A 8 3. Wherefore, when x is very ſmall in Compariſon of any oe given Quantity, as a, in the given Equation of y, the laſt Term cf the ſupplementary Equation of p, 9, r, &c. is continually diminiſh. ing, ſo as at length to vaniſh. And therefore one Value of the Quay. that is the Supplement, as was obſerved already * : Conſequently ſince that Value of p, 9, r, &c. together with the Quotient as far as fox. merly found, makes up the Value of , it follows, that, by this Ope- ration, you approach continually to the Root y, ſo as to differ from it at length by leſs than any given Difference: Becauſe the laſt Term of the ſupplementary Equation, becoming at length leſs than any given Quantity, and vaniſhing, ſome Root of that Equation muſt become leſs than any given Quantity, and vaniſh ; ſince the laſt Term of any Equation is the Product of all the Roots with contrary Signs: which diminiſhing and evaneſcent Root is the Supplement, as hath been juſt now ſaid. | 4* Whereas in every new Step, by which a new Term of the Value of y is found, our Author deſires you to aſſume the Terms in which x and p; & and g; are ſeparately of the feweſt Dimenſions, it may be obſerved that the Species p, 9, 7, &c. will always be found ſepa- rately of one Dimenſion, without being affected with x : becauſe in the firſt Suppoſition, you aſſume an Equation in which the Letter x is not found, and therefore the approximate Value of y thence deduced doth not include x ; whence it 1s evident that the Parts of the Penult Term of the firſt transformed Equation, upon the right Hand Side of the Diagram, which ſtand collaterally to thoſe Terms of the propoſed Equation, wherein y is not affected with x, muſt contain p ſeparately of one Dimenſion only, in the Equation of p: which in his Example are 34% and a*p : therefore in the next ſubſequent Transformation, vi⁊. in the Equation of 9, upon the right Hand Side of the Diagram, that part of the Penult Term which ſtands collateral to 44*p upon the left Hand, where þ is ſeparately but of one Dimenſion, muſt contain q ſeparately of one Dimenſion only : and fo the Reaſoning is the ſame Equations of this Kind, where you can deduce the firſt approximate Value of y, by ſuppoſing x to be nothing, the Parts of the transformed Equations to be ſelected, for deriving the Terms of the Quotient, in the Progrefs of the Operation, are manifeſt ; and the Derivation of theſe EXPLAIMNE B. 103 theſe Terms of the Quotient, requires no more but the dividing the Part of the laſt Term wherein x is of the leaſt Dimenſion, with the Sign changed, by the Coefficient of ſuch Part of the Penult Term, as includes but one Dimenſion of p, 9, , &c. ſeparately, _. 76. And thus I have explained and demonſtrated the general Proceſs for deriving the ſeveral Terms of the converging Series for expreſſing the Value of y: but whereas in Art. 31, Sir 1/aac mentions an Abridg- ment, which may ſerve to contract the Work, I muſt ſhew upon what Foundation it is built. In order to which, let it be obſerved that, when the Quotient is propoſed to be continued only to a certain number of Terms; or (which is equivalent) to a given Dimenſion of x, all thoſe Terms of any Equation upon the right Hand fide of the Diagram, may be entirely omitted in the ſubſequent Transformation, in which, if, for the Supplement p, 9, 7, &c. that Power of x were ſubſtituted, which is found to be it's approximate Value, x would be of a higher Dimenſion than the propoſed Limit . The Reaſon of which is this: ſince the Species x is ſuppoſed indefinitely ſmall, any Term, that, by the Subſtitution juſt now mentioned, would ariſe to a higher Dimenſion of x, than the Dimenſion to which the Quotient is propoſed to be continued, is vaſtly lefs than any of the Terms in which the Dimenſion of x riſes not above that Limit: and as ſuch Term in the fubſequent Transformation, muſt make that Part of the laft Term of the Equation upon the right Hand, which ſtands colla- teral to it, of a higher Dimenſion of x than the propoſed Dimenſion ; and every ſubſequent Quantity upon that ſame Line of the Transfor- mation, of ſtill a higher Dimenſion of x, after a proper Subſtitution, as is evident, hence it appears that no ſuch Term can any way affect the Quotient within the propoſed Limits: ſince every new Quotient Term muſt have that Dimenſion of x, which is the loweſt in the laſt Term of the ſupplemental Equation. Thus in the Operation in the Author, after you have arrived at the Equation. of ; and found from it q= = nearly, the firſt Term 93 is entirely neglected in the * 2 2 - ſubſequent. Transformation : becauſe if 75 + 7 were ſubſtituted for i The Reader muſt obſerve that this Rule which is given by Sir /aac ; and the Reaſon given for it here, can only be extended to thoſe Caſes in which the Supplements p, 9, r, Cc. are found in ſome Part of the penult Term of their reſpective Equations, without any Mixture of x: in ocher Caſes the Rule will not hold. And therefore an univerſal Rule for all Caſes ſhall be delivered Fff 2 7 | | 404 ANALYSIS by Equations, &c. x*x% . x6 Sic A . 9, it would give g = gegn + Zi: + = ＋ , for it's call. teral Value ; in which the laſt Term —=— riſes above the fourth 26214443 Power, which is the propoſed Limit of the Dimenſion of x : and the other Terms would ſtill riſe to higher Dimenſions of x, fince the approximate Value of r muſt contain more Dimenſions of x than doth, which contains the Dimenſion x* : whence it appears, that none of the Terms of the Value of 95, in any ſubſequent Transformation, can give or produce any Quantity, but what ſhall contain either the 6th Dimenſion -of x; or one {89m in the ultimate Term : which therefore can have no Influence upon the Quotient within the pro. poſed Limit. And the reaſoning is exactly the ſame with reſpect to the Part — g of the ſecond Term of the Equation of 9 upon the right hand: which therefore is likewiſe entirely neglected in the ſub. ſequent Transformation. 77. But ſometimes all the Terms of the transformed Value of ; Quantity are not to be omitted, but only ſome of them, in order to carry the Quotient to a propoſed Limit, or Dimenſion of x: for which you are to obſerve this Rule, That after the firſt reſulting Tern of the transformed Value of any Quantity, ſtanding collateral to it per the Right-hand, ſo many more Terms are to be added, as the Index of the higheſt Power of x, in the Quotient, exceeds the Index of x in fuch - firſt reſulting Term of the transformed Value: provided the Dimenſuns ef x aſcend only by the Difference of one, as in the Example of the Author : where the Dimenſions of x in the Terms of the Quote are *, x", **, x3, Cc. Thus if the Quotient is to be continued only to the 4th Dimenſion of x, you omit all the Terms after &., and put one after x3 ; and ſo in other Caſes. Accordingly in the preſent Example, after you have arrived at the Equation of q, the transformed Value of the Quantity 3ag* is = + 2= + 3ar* : of which 32 the two Terms — + 34r* might be omitted, when the Quote is deſigned to be carried only to the 4th Power of x : for the ſame Reaſon, oppoſite to ;x*%q, the Term er might be omitted. The Reaſon of which may be eaſily deduced from what hath been faid in the preceding Article. Which Rule includes the particular Rulc given by Sir Iſaac, Art. 31: as will be evident to one that confiders it. But if the Dimenſions of x in the Quote proceed by greater Differences, as , xX, &, x*, Cc. or , x3, *, „, ©c. ber 1 W + Eo . ²— hg 0 ww” XX YT O29 EXPLATMNE D. when the Differences are 2, after the firſt reſulting Term of the transformed Value of any Quantity, you need only ſubjoin half the Number of Terms, which denotes the Exceſs of the Index of the higheſt Power of x, to which the Quotient is to be continued, above the Index of x in the firſt reſulting Term of the Value of the Quan · tity, to which that Term ſtands collateral: And when the Differences of the Dimenſions of x in the Terms of the Quote are 3, you need only adjoin a third of that Number of Terms, Cc. becauſe now the Differences of the Dimenſions of x in the Terms of the transformed Value of any Quantity taken from the laſt reſulting Equation, will be 2, 3, Cc. after ſubſtituting for 45 q, r, Sc. their approximate Values. And after the Operation, in this abridged way, is carried ſo far, as that the Supplement , r, &c. riſes no higher than the Root, in the transformed Equation upon the Right-hand Side of the Diagram, you obtain the reſt of the Terms by Diviſion: which is to be con- tinued untill the Quote be brought to the propoſed Limit *, 78. The Author's Example, with the Operation abridged, accord- ing to the preceding Rules, may ſtand thus : where the Mark * denotes 405 the Parts of the Terms which are omitted: the Quotient or Value of y being propoſed to be carried the Length of the 4th Power of x. * Theſe Things help to explain Art. 44. of the Author? N. B. ANALYSIS by — &c. "2 ö +. — N * St x 27 * — 2 * —_ + ber- 2 + itt + | AXDT FN — | = DX 0% nm = 4 44 + - + =b,ovþ + jeg OST. Soi : 2 By 8 £X — | = —_ 2 . | b 10 . e — b,vh + bob K Sz ——_ — 6.4 o0= — — . * 5 r e =. b+ Z==g om - bex3 + wit —=| = if + 25 fro + 3,9 + d + | XD = 2 — 2 10 ale + Af 133 —_— d.v P.-4 = 4.5 = | 3 24 Won if + dof + dot + w+|= A+ J+o += | 92 . E + 2 „ o tx — (x0 tr — ſoo + if | | i" ad — — aHõe ouo 4ozyop 82 yory am 4 sf pot ep 228 m nurn od. 0 N —_ — — — — — — — ́[ — 8 5 : 2 | = 49 ru = p—_ 2X47 = 2497 9 baxg3Þ + r — = d.xgaÞ + | * boy + 3997 — | deb + e = _ ER DD * 4 + | = ee — 242] — —=gÞ — * * E + — e9:X,29 + [o gue ＋ Bextor — bet * 77558 b ＋ Z—=0 | aa SENS _TE mT +. | „ EET ——-F — — 43,9 — = fon — | 2 © = 3,92 + Jv | 492 + £994 + % + deb — FF +| = + J+9+=4i } | wx e ere 2 + | = dx = wi 9 — zz | o = e TAY — | f noa Se pa2501d us pur O 10 G4 o = - 8323, 293 2wnye nox : 139 213 $1 ponunuoo q 03 pajodoad sf zuanor ay3 yo1yas 03 x 30 uomuamreg yauFry ay pur ⁊ ujj,m)ufd aq Aq po. 4uanony Nc uf x Jo Suomuaung 23 212Um O= ,7— K + Ax — A + 46x 2jdwexg 12qzoue 235ÞT, OL Suh | pohoſoi 21294 & Jo s Jlayfry pur 135 ay} Surp2ye su, L o parBay ut * x 30 H y3Þ q puoſeq poi q cu yaw yoga 42 58 + * 175 | 210g) 211 $2413 yoga D — wh A — 2522 + x5 $urprap Iq 21 αμjõꝗn iq punoy ax payjod 181 od yur] Sn St YIIUM fx jo 12M0q ur q sv ig SE ©uo73oney) 203 Jo νje L 23 Jo yo 2753 10 213 ury3 aa ou s zuduuoiddug qa = yo uonenbꝗ q $2413 yo UONREWIAOJSUEA,T, Uri ou u; Ynro2q ON 408 ® Art. 72. the given Quantity a, then he deſires you to ſelect thoſe Terms in the Root y out of this laſt Equation ; which will be found y = 1, | 7 ANALYSIS by Equations, &c. Here in this Example, becauſe the Differences of the Dimenſions of x in the Value of y are 2, and that Value or Quotient is to be ca. ried forward only to the 6th Dimenſion of x, therefore in transform. ing the ſeveral Quantities of the Equation of p, you will obſerye thy thoſe are entirely omitted in which if &, which is the Dimenſion f x in the approximate Value of þ were ſubſtituted for p, the Dime. fion would bo higher than the 6th : no Term placed after the firſt te. ſulting Term, when the Index of x in it is 6 : and only one Term x. ter it, when the Exponent of x is 4. By which the Equation of q riſes not above the firſt Power of q : therefore you find other two Terms of the Quotient by Diviſion, as appears by the Operation itſel And fo you proceed in other like Caſes, by which you ſhall obtain ; Value of y in an infinite Series of ſimple Terms: and therefore if ; and y denote the Abſciſs and perpendicular ordinate of a Curve, the correſpondent Curvilinear Area is found by our Author's ſecond Rule: which in this Caſe, where the Powers of x aſcend in the Numeraton, approaches the nearer to the Truth the leſs that x is. 80. But if x be ſuppoſed to be very great in Compariſon of any other known Quantity in the given Equation (which is the ſecond Cate we mentioned above * ; and of which the Author treats in Art, zz, 34,) in this Caſe, the Quotient, or Value of y, muſt be ſuch, thatthe Dimenſions of x diminiſh in the Numerators; or increaſe in the De- nominators of the Terms. Now in order to find ſuch a Value of y, our Author directs to proceed thus: let y3 + axy + x*y — a3 — 2 So be propoſed, in which x is ſuppoſed very great in Compariſon of which x and y either ſeparately or multiplied together are of the moſt and at the ſame Time equal Dimenſions, and put them as it were equal to nothing, and the Value of the Root y thence deduced, ſhall gie the firſt Term of the Quotient : therefore in this Example you put 93 + x*y — 2x3 = a, whence you find y x, an approximate Value of y, that is y & + p, p being the Supplement. It will be the ſame Thing as Sir //aac obſerves if you turn the aſſumed Equation 93 + x*y — 2x3 =0o, into this other 53 + y — 2 =o, and extrad and then multiply this Value of y by x : the Reaſon of which is thi, that if the given Equation y 4 x*y — 2x3 = o, be ttansformed by putting æ = > or 2x = y, it will give x323 þ x32 — 2x3 = 0; 0, by dividing by x3, 23 + z — 2 = o, which is the ſame with y* +) — 2 o, whoſe Root 1 being multiplied by x, is the Root of the given Equation y3 + x*y — 2x3 =o, ſince y x. 81, Now EXPLAINED. 81. Now the fi | covered in this 1 of the Quotient or VA eee ol y being dif- in the given Equation 2 are found by fabRti be ſhewn after forming it thereby — + axy + x*y 290 N * + þ for y mate Value of þ is to be 5 new Equation; from * o, and tranſ- Equation, which are va __ by aſſuming thoſe T ich an approxi- them, as it were, equal y greater than all th erms of this new 2 8 falcon: _=_ to nothing: and ſo car de reſt, and putting ready, in the other Caſe, © after the like ae is hes 29 The Proceſs of which e, in which x was ſuppoſe 3 as been ſhewn al- is here repreſented to 1 very ſmall, 4 „rr. „* = | + = = . 4x*þ + ax* So fore 3 — == I 2= (58 Aa? 43 nds r = -le. = — a * Buy. thu $6 5 7 = — — — e e 9 41 + os a Kok 25 * 64x 1 + 7 =+—— + za+r 3227 * 8 4096x* ＋ "Gaz + 73 0 — Ja =— 3as 3217 | —— 13143 __ e „ + * =+ 3 ye r= -— hw ＋ 3 1 + * g= © We by 1024x air — ax — az + 128 raxr 4 1 =+ © 6 — — + ir DOSE wn 65@3 64 64 A.- fax &c. 1314 1544 )+ 128 — e. NI gogat 4096 12 . r ＋ 131 512K * 163843 128 10245 &c | oga⸗ 1 512 16384 Ggg 0 Ms mi- a T 4+ —522—— ae or KEH + 441 | + 42 + 1329-04 Ec. In 410 — —— — — * Art. 78. 79. ANALYSIS by EquaTions, Ge. 82. In which Operation, having obtained the Equation of 5,1 throw by the firit and ſecond Terms, as containing the Cube and Squate of p: which being of ſmall Value in Compariſon of the other Terms, are to be neglected: likewiſe of the Penult Term, you only retain the Part 4x*%, in which x riſes to the higheſt poſitive Dimenſion , or if there were no poſitive Dimenſion of x in it, you muſt take that Part only where x is of the loweſt Negative Dimenſion *, the reſt yon neglect, becauſe by the Hypotheſis x is ſuppoſed to be vaſtly great in Compariſon of 4; likewiſe of the ultimate Term I retain (after throy. ing by repugnant Parts) only the Part ax*, viz. that in which x is of the higheſt poſitive Dimenſion ; otherwiſe, if there be no poſitive Di. menſion, where it is of the Joweſt Negative Dimenſion : and ſo I put 4x*þ + ax —= o almoſt, whence I deduce p = — 4a for an approxi. mate Value of p; and conſequently, for the ſecond Term of the con. verging Series for the Value of y : and then putting p = — {a +q, you transform the Equation of þ into a new Equation; as you fee; which you manage after the ſame Manner as the preceding, by put- ting 4x% — a = © almoſt, whence I deduce 9 = + = nearly, which is the third Term of the converging Series : and fo you proceed to find the other Terms by the like Means. But here it may be obſerved, that when the Series for the Value of y is propoſed to be continued only to a certain Limit, or Dimenſion of x, many unne- ceſſary Terms, or Parts of Terms, in the Transformations of the Equations, may be omitted, viz. thoſe that cannot affect the Quo- tient within the propoſed Limit, agreeably to what was ſhewn for- merly *, allowing for the Difference of the Caſes: according to which thoſe Quantities marked with an Aſteriſk above them, might have been omitted, if the Value of y was propoſed to be carried no further than the Dimenfion , or x=3. Moreover it is to be obſerved, that, after you have obtained a certain Number of Terms of the Series, by a continued Transformation, you may obtain ſo many more by Divi- fion, as you fee in the Operation. 'The Diviſion is always performed by dividing the laſt Term with the Signs changed, by the Complex 3 I call a negative Dimenſion of x, a Power having a negative Index in the Numerator ; or a poſitive Index in the Denominator as * = . accordingly I call - 22 _ a lower x negative Dimenſion of x than — = .: likewiſe a®*= 2 is to be accounted the loweſt Dimenſion of x, whether poſitive or negative. | EXPLAIMNE D. Coefficient of the Supplement 9, r, &c. in the Penult Term of the laſt Equation. And in order to know to how many Terms the uotient of the Diviſion may be carried, obſerve this Rule. In the laſt transformed Equation, ſuppoſe it be that of r, divide, or ſuppoſe to be divided, all the Terms of the Equation preceding the Penult, by r, ſome Power of which above the firſt, enters into them all : Then find the Power of x which enters into the firſt Term of the Value of r: which in the preceding Example J. find to be — or x=*, by putting 4x*r — Goat 1 5 ery "EY - — ig == 0: ſubſtitute that Dimenſion of x, viz. — in Place of r, or ſuppoſe it to be ſubſtituted, in the Terms preceding the Penult, after they have been ſuppoſed to be divided by r as above; and let the loweſt poſitive, or yet negative Dimenſion of x, thence æriſing, be x" or x reſpetively, according as x 1s ſuppoſed to be very ſmall, or very great: then you may continue the Diviſion untill you obtain that Term of the Quotient whoſe poſitive, or negative Dimenſion, is the next below x" or x=" reſpettively : only it muſt be obſerved, that, if there be any poftive Dimenſion of x in any of the Terms 2 ng the Penult Term of the Equation, after the Diviſion and Subſtitution mentioned above, and the poſiti ve Dimenſions of x decreaſe in the Value of y, then the Di- uon is to be continued only until you obtain that Term of the Quotient, whoſe poſitive Dimenſion is the very next above the higheſt in theſe Terms of the Equation. And you muſt take Care, in ſetting down the Parts of the Diviſor and Dividend, that you make the lower Dimenſiuns to precede the higher, when they are pofitive, and x very ſmall; but the bigher poſitive Dimenſions to precede the lower, and the lower negative Dimenſions to 44 the higher, when x 1s very great. In this Rule it is ſuppoſed that all the Terms of the transformed Equations are compleat, although this be not always neceſſary * : and the Rule will ſerve in all Caſes, whether the Dimenſions of x be all poſitive and increaſe from the beginning of the Series; or 29. the Dimenſions of x be firſt negative and decreaſe, and afterwards become poſitive, and then increaſe: in which Caſes x muſt be ſuppoſed to be very ſmall; or, 30. the Dimenſions of x be negative and increaſe from There is not any Neceſſity for _—— any of the Terms, of the laſt transformed Equati- on, which precede the Antepenult : becauſe the Dimenſions of x contained in the Quantities, which lye upon the ſame horizontal Line, proceed always in an arithmetical Progreſſion, after the Subſtitution mentioned in the Rule: nay and of the Antepenult Term, there is no Ne- ceſſit / for annexing any other Part or Quantity, but that one where x is of the leaſt poſitive or negative Dimenſion, according as x is very ſmall or very great ; fince the Continuation of the Terms of the Quotient, ariſing by the Diviſion, is regulated by it only. Ggg 2 the 411 412 are to proceed in extracting the Robts of literal Equations includin .ttanding that it has been ſhewn in the laſt Example, how to find the ing found in the preceding Example y = x — 7 + 55 + 221 ANALYSIS y Equvartions, &c. the beginning of the Series; or 4*. the Dimenſions of x be firſt poſitie and decreaſe, and then become negative and afterwards increaſe; in which two laſt Caſes, x muſt be ſuppoſed to be very great. Ang I] need not take up Time in ſhewing the Foundation of this Rule: ſince the Reader will eaſily underſtand it from what was demonſtrated formerly with reſpect to numeral Equations *, applied to this Caſe gf literal Equations, which is very eaſy to do. 83. From theſe Examples, I ſuppoſe, it will be manifeſt how you two unknown Quantities x and y; fo as to expreſs the Value of by a Series of Terms including different Powers of x, and that whe. ther x be ſuppoſed to be very ſmall, or very great, But, notwi Terms' of a converging Series expreſſing the Value of y, when x i; very great in compariſon of any other Quantity, as a, in the Equz- tion, by a Method differing in ſome reſpe& from that whereby the Value of y was found, when x was ſuppoſed to be very ſmall; yet the ſame thing might be obtained, by the firſt Method, if you only ſuppoſe that the given Quantity a, in Compariſon of which x is ſup- poſed to be very great, be conſidered as the indefinitely ſmall Quan- tity coming in Place of x : and that you 1 to yourſelf to find a converging Series expreſſing the Value of y in Terms proceeding ac- cording to the poſitive Dimenſions of a, inſtead of thoſe of x : which is taken Notice of by our Author at Art. 33: The Reaſon of which is, that the Smalneſs or Greatneſs of x is comparative, viz. with ref to the Magnitude of the known Quantity @ ſo that the ſuppoſing of x to be very great is equivalent to the ſuppoſing of à to be very little. 84. And thus by either of theſe Methods the Value of y being ex- tracted, and expreſſed by an infinite Series of ſimple Terms, includ- ing the Quantity x involved with known Quantities, the Area of the Curve, whoſe Abſciſs and Ordinate are x and y is found by the Au- thor's ſecond Rule. And the Area will approach the nearer to the Truth (ceteris paribus) the gruer that x is, becauſe this will evi- dently make both the Series for the Value of y; and that for the Va- lue of the Area, thence deduced, to converge the faſter : thus hay- 5123* a? 64x — — _$ogat _ 2 2 EA 3 — -6384x7 Gc. the Area belonging thereto is — + 131@3 oa lz — 3276 Cc. both of which converge the more quickly * : greater EXPLAINED. greater the Abſciſs x is. And as to the Curve, it approaches continual- ly towards a reCtilinear Aſſymptote, at which the Part of the Ordinate denoted- by the two firſt Terms of it's Value, vis. x — , always terminates z ſo that you may eaſily find the Poſition of that Aſſymptote, and any Portion of it, by aſſuming any two Points in the Abſciſs, and drawing two Perpendiculars at theſe two Points, equal to the Length of the Abſciſs leſs za; the Line drawn betwixt the Extremiities of the Perpendiculars, is the Portion of the rectilinear Aſſymptote, correſpond- ing to the Part of the Abſciſs lying betwixt the two Points aflumed therein. But that this, and likewiſe the way of computing the Area may appear the more clearly, I ſhall repreſent the thing to the Reade by an actual Deſcription, ſince the Author himſelf has juſt onl mentioned it, Art. 34; and it may be of ſome Uſe to the young Geometer. Draw the Right Line AB produced indefinitely towards B, upon which take AG = a, and calling the Abſciſs AB = x, let the Curve Da be deſcribed by the ndicular Ordinate 9 22 y, ſo that it be ; f 917 Sc. upon BD take draw GE, and it is an Aſſymptote to the Curve Da, to which the Curve infinitely produced to- : wards a, continually ap- _ . proaches. The Curvili- 5 4 near Area was found to A 4 as rede. — u 655 W . 2 4 64x — i — — 28 Ter Sc. of which that Part which is expreſt by the two 512x ” al # * * * firſt Terms La — 18 denotes a rectilinear Area adjacent to the Abſciſs AB: for if through A you draw AH perpendicular to AB, and meet- ing the Aſſymptote G E below the Abſciſs, in the point H, then the Triangle GBE == A* — aN + a, and the Triangle AGH = 7:7, therefore GBE AGH = — =. Again the Part — g- - 2 512 * ANALYSIS by Equarions, &c. A 4275807 Sc. denotes a curvilinear Area lying upon the other (4, of the Ordinate, terminated at the Curve infinitely produced.upon that Side: becauſe the poſitive Powers of x are found in the Denominaton of all the Terms: the other Terminations of this Curvilinear Are, may be determined by comparing the Ordinate 4- 71275 + 765105 Sc. with which it is deſcribed, with the preceding Parts of the Va, of y, viz. *— . and + —_ of which the firſt Part, viz. x is the right Line BE, which always touches the Aſſymptote GE; the other Part, vis. er will belong to an Hyperbola: for if in the Line E D you take EF = -— the third Term of the Value of y, de Point F will touch an Hyperbola having HE and HA for it; Aſſymptotes; which let be deſcribed, and ſet it be Fo: from whence it appears that the Curvilinear Area denoted by the Terms — == — 7 68 Sc. is terminated by the Part of the Ordinate FD and the two Legs of the Curves 9D and F infinitely extended beyond the Ordinate BD towards « : but whereas the third part of the Cur. vilenear Area denoted by — is an infinite hyperbolical Space, lying betwixt the Hyperbola and it's Aſſymptote, infinitely produced, there- fore we can only find the Curvilinear Area adjacent to any given Part of the Abſciſs as BB, viz. the Area BHD: which is done thus: ſubtract the Area belonging to the ſhorter Abſciſs A from the Area belonging to the longer Abſciſs AB, without taking in the Hyperbolical Area = and the Difference ſhall give the Sum of the redlilineat Area BBE and the curvilinear Area Fed D: (for which ſee our Author's ſecond Rule, Examp. 3. and Sect. 5. of the preceding Treatiſe) wherefore if to theſe you add the byperbolical Area E. F, you ſhall have the entire Area B89D. To find that hyperbolical Area Ee SF, you may proceed thus: Upon BD take BL—EF, and with the Center A and Aſſymptotes AB, AH deſcribe the rectangular Hyperbola La through the Point L, cutting G in the Point a, and the Area BBAL is equal to the Area Es, as is evident; wherefore let the Area Ba L belonging to the equilateral Hyperbola ; and adjacent to the given part of the Aſſymptote BB be found by what gy enn —̃ ꝶ'l1li.. , . ̃˙— ße * 1 " EXPLAINE D. ſhewn formerly“: and this added to the Areas BGE and Fd D- Art. 25. found already, will make up the whole Area BD required. 85. And thus I have endeavoured to illuſtrate the two different Caſes mentioned by Sir Jſaac in this Treatiſe, for diſcovering the Roots of affected literal Equations, and expreſſing them by infinite Series s, either when x is ſuppoſed to be very ſmall; or when it is ſuppoſed to be very great in Compariſon of any given Quantity in the propoſed Equation : but I mentioned a third Caſe, which is when x is ſuppoſed to differ very little from any given Quantity: in this Caſe you ſubſtitute ſome letter for that ſmall Difference in the propoſed Equation, abs z ; and ſo having the two unknown and variable Quan- tities y and z, of which 2 is conſidered as indefinitely ſmall in Com- pariſon of any given Quantity in the Equation, you may find a con- verging Series for the Value of y, in Terms made up of poſitive Pow- ers of x, and known Quantities, as formerly: from which you ma deduce a Series including x by replacing it in the Series for it's formerly ſubſtituted Value. Thus in the Equation y3 + axy + a — x3 — 24 = ©, it being known or ſuppoſed that x is nearly of the ſame Quantity with a, you ſubſtitute z for their Difference, 1. e. put z= - x or x - a, that is ſubſtitute — 2 or 4 ＋ 2 for x in the pro- poſed Equation: by ſubſtituting the firſt of which, there ariſes this new Equation y3 + 24*% — azy + 342 — 348 + 23 — 343 =o, from whence the Value of y may be found by the firſt Method, z being ſup- poſed very ſmall in Compariſon of a : which Value being found, you may replace x for a — 2, that is a —x for 2; and fo the Area ma be found in Terms made up of à and x. And this third Method is proper to be taken when the Series for the Value of the Ordinate y, and conſequently that for the Value of the Area, will converge the more quickly the nearer that the Length of the Abſciſs x is to the Length of the given Line a. 86. Moreover, hence it appears that all the three different Methods for extracting the Roots of ſuch affected literal Equations, may be re- duced to the firſt Method only: for this third Caſe mentioned in the preceding Art. falls in with it; and it was ſhewn in what Manner the in the propoſed Equation, which muſt make z very ſmall when x is very great. Thus in the preceding Example y3 + axy + x*%y — a3 = 2x3 =0, in which x is ſuppoſed very great in Moyes » a, | | tute 415 kcond Caſe, which ſuppoſes x to be very great, falls in with it alſo *.. Art. 83. Which Caſe likewiſe you may reduce to the firſt, by ſubſtituting— for x 416 i - : * : 1 l * | - 6%. * * a " 5 = F * ANALYSIS by. Equations, Wc. ſubſtitute — for x, and you will have y + 2＋2 — 41 — 3 2 , that is, by multiplying by 23, 23y3 + az*y y — 432 — 220 in which E is very ſmall : and from which the Root may be extracted upon that Suppoſition, by what will be ſaid afterwards upon the ſecond Remark under Art. 35 of the Author. Which Article I ſhould now enter upon the Explication of directly; were it not that it will fall in to be explained afterwards to more Advantage in the Progreſs of the following Section. 3 S EC T. V. Of the various Methods of extracting and expreſſing the | Roots of affected literal Equations, including two un- known Quantities, by Means of infinite Series's: in which the Author's 35th 42—46 Art. are explained, 9 87. NE and the ſame Equation may afford different Series's for expreſſing the Root or Value of y, by ſelecting two or more Terms differently, out of the propoſed Equation, and putting them equal to nothing, in order to diſcover from thence the firſt Term of a Series expreſſing the Value of y. In what goes before, it hath been ſhewn that one and the ſame Equation may afford two different Series's; one upon Suppoſition that x is very ſmall, and another upon Suppo- fition that x is very great, in Compariſon of any known Quantity in the Equation : but it frequently happens that from the ſame Equation, there may be more Series's than one expreſſing the Value of y, when x is very ſmall ; and likewiſe more than one agreeing to the Suppoſition of x being very great : but then there is ſome Difficulty in finding out the firſt Terms of all thoſe Series's : for underſtanding of which I ſhall demonſtrate the following Lemmas. / \ LEMM A I. 88. In any Equation including two unknown Quantities x and y, from whence a converging Series is to be derived for exprefling a Value of y in ſimple Terms involving different Powers of x ; let Ax" denote the firſt Term of ſuch a Series, where u denotes any indefinite Expo- nent, 2 p —— 19 < X EXPLAITIMNE D. nent, poſitive or negative; integral or fractional ; and A any indeter- minate Coefficient : then Ax” being ſubſtituted for y throughout the ſeveral Terms of the propoſed Equation, let the Dimenſions of x in two or more of the Terms be the ſame, and either leſs than the Di- menſions of x in all the other Terms; or greater than the Dimenſions of x in all the other Terms *, I ſay theſe two or more Terms are vaſtly greater than all the reſt of the Terms of the Equation, according as x is ſuppoſed to be indefinitely leſs, or indefinitely greater, than any known Quantity in the Equation, i. e. if x be ſuppoſed to be indefi- nitely ſmall, theſe two or more Terms in which the Dimenſions of * are equal, and leſs than in all the reſt, are vaſtly greater than all the reſt : but if x be indefinitely great, the two or more Terms in which the Dimenſions of x are equal and greater than in all the reſt, are vaſtly greater than all the reſt of the Terms of the Equation. Dem. For when x is ſuppoſed indefinitely little, the Dimenſions of x in the Terms of a converging Series, mult be ſuppoſed to increaſe as the Series proceeds, ſo as to be leaſt of all in the firſt Term, vis. Ax” : wherefore Ax” will be vaſtly greater than all the remaining Terms, which will vaniſh in Compariſon of it, if you make x ſmall enough, ſo that Ax” is, in that Caſe, nearly the Value of y: conſequently it Ax” being ſubſtituted for y in the Terms of the Equation, raiſe two or more of them to equal and lower Dimenſions of x than in all the reſt, theſe two or more Terms muſt be vaſtly greater than all the reſt taken together. This is when x is ſuppoſed to be indefinitely ſmall. And by the ſame Way of Reaſoning, if x be vaſtly great, it may be ſhewn, that, if there be two or more Terms in which the Dimen- fions of x are equal and higher than the Dimenſions of x in all the other Terms of the Equation, after Ax” is ſubſtituted for y, ſuch two or more Terms are vaſtly greater than all the reſt of the Terms taken together : by conſidering that in a converging Series anſwering to this Caſe, the Dimenſions of x muſt continually diminiſh ; and that any higher Dimenſion of x is vaſtly greater in Value than all the lower Dimenſions of x taken together. Q. E. D. Here it muſt be obſerved that any negative Dimenſion of x is conſidered as lower than any poſitive Dimenſion ; and of negative Dimenſions of x, the greater the negative Exponent is, the lower is the Dimenſion eſteemed to be: thus x—+ or = is reckoned a lower Dimenſion I , of x than & or —: ſo that . 1. . „ „ x3 ec. reprelents - * a Scale of Dimenſions of x aſcending. dis Hhh 89. If o 9 . o . 1 * = oo ! ́M . —— rn — — — roo -. — — ůD2j2vc — ——.ß—§ßÄ5tæcg˙nä᷑łv - —— — — — Ar, 3 - — U = — —— ro . ha - FE ⁵˙üJꝛ—ꝛ—̃ T — — => i \ -— - r — — — — — 2 — — — — — — - — — ij 4 ' } l =y „5 | | iN [1 418 ANALYSIS by Equations, LEMM A II. 89. If K*. x.y. exhibit the Dimenſions of x and yin any two Terms of ſuch an Equation as has been mentioned, in which Terms Ax" (fig nifying the ſame Thing as before) being ſubſtituted for y, makes the Ge. Dimenſions of x equal, and leſs in them; or greater in them, than the Dimenſions of x in all the reſt of the Terms of the Equation, after Subſtitution of Ax" for y in them likewiſe, theſe two Terms being put equal to nothing, will give an approximate Value of y, in which the Dimenſion of x ſhall be x", the ſame as in Ax” the firſt Term of the converging Series *. Dem. For the two Terms including xy and x"y* being put equal to nothing, will give a Value of y, which contains this Dimenſion of . . Vit. , as is evident: now I ſay = n. For, by inſerting for y in the two Terms of the Equation in which the Dimenſions of x and y are xy? and xy, their Dimenſions of x will be += ang *, which by the Hypotheſis are equal, that is p + gn = r + ©, whence you'll have — —=#=. Q. B. D. go. Cor. 1. From the laſt Lemma it follows, that if there be any two Terms of ſuch an Equation as has been mentioned, which be- ing put equal to nothing, give a poſſible Value of y, in which Va- lue the Dimenfion of x is ſuch, that, being ſubſtituted for y in all the Terms of the Equation, it makes them all of higher, or all of lower Dimenſions of x than in the two Terms mentioned, thefe two Terms, ſo aflumed and put equal to nothing, will give the firſt Term of a converging Series for the Value of y, according as x is ſuppoſed to be indefinitely ſmall ; or indefinitely great. | Example I. Let the Equation yi — axy — x3 o be propoſed, (which is that propoſed by the Author in Art. 35. Rem. 2.) put the Terms — axy — x3 = o, thence ariſes y = — - in which the Di- menſion of x is x*: which being ſubſtituted for y in the firſt Term of the Equation, viz, y3 would give x* for the Dimenſion of x; and that being higher than x3 ; which is the Dimenſion of x in the two Terms aſſumed, hence — = is the firſt Term of a converging Series for the Value of y, whence x is very little. The propoſed Equation including x and y, is ſuppoſed here and all along to have all it's Terms brought to one Side, ſo that the whole Aggregate of the Terms may be equal to ung, After EXPLATMNE ͤ D. After the ſame Manner, if you 1 5 — x3 D, you have y= x, which will give the firſt Term of another converging Series for the Value of y, when æ is very great: becauſe if x be ſubſtituted for y in the middle Term of the Equation, viz. — axy, it gives the Dimen- ſion **, lower than x3, which is it's Dimenſion in the two Terms aſſumed. | | Example II. Let the Equation y3 — 4% + axy — x3 = © be pro- poſed : if you put y' — 4*y = ©, you have thence y = -- @ or — a, where the Dimenſion of x is x* = 1, which being ſubſtituted for y through all the Terms, the Dimenſions of x will be „e, &, x*, x*:. which ſhews that + @ or — 4 may, either of them, be the firſt Term of a converging Series, when x is very little. 91. From which Example it is evident, that, in every ſuch Equa- tion, if there be two or more Terms including no Dimenſion of x ; from which put equal to nothing a Value of y may be deduced, ſuch Value will always give the firſt Term of a converging Series which ſhall be a Root of the Equation, when x is ſmall. . And this coincides with our Author's Rule, Art. 30. 92. In the fame Equation, aſſuming the two Terms — 4˙ — x# x3 So, you have y = — : by inſerting of which Dimenſion of x for y through all the Terms, the Dimenſions of x ſtand thus &, x3, *, x3: whence it appears that — 25 will be the firſt Term of another converging Series, when x is very little, | * Again if you aſſume ) — * = o, you have y = x: by inſerting of which for y, the Dimenſions of x are x3, x, x*, x* : whence it appears that -x will be the firſt Term of a converging Series for the Va- lue of y, when x is very great : becauſe the Dimenſions of x in the two aſſumed Terms are higher than in any of the reſt of the Terms of the Equation. | 93. Cor. 2. But if any two Terms of ſuch an Equation as has been mentioned, be aſſumed, and put equal to nothing, and from thence no poſſible Value of y can be deduced ; or if a real Value can be deduced, but ſuch a one that the Dimenſion of x it contains, being ſubſtituted in place of y through all the Terms, make the Dimenſions of x in ſome higher, and in ſome lower than in the two aſſumed Terms, in either Caſe the two aſſumed Terms are improper from whence to derive the firſt Term of a converging Series, either when x is ſmall or great. For in the firſt Caſe there is no real Value : and in the other Caſe where there is a real Value, it cannot be the firſt mOu_ Hhh2 0 419 - 4 „ . — — : — —___ ___ i | 1318 1 1 11 [ | 15 "= — — — + 4 2 * * A * —— — — = — — Uä—äñ — — —— ' Ls 420 ANALYSIS by EqvarTlions, &c. of a converging Series, ſince the firſt Term of a converging Series muſt be deduced from thoſe Terms only, which are much greater than all the reſt : which the two Terms aſſumed would not be in this Caſe, whether you ſuppoſe x to be ſmall or great: as eaſily appears from what hath been ſaid above, eſpecially in Lem. I. The ſame Thing is likewiſe evident of any two or more Terms in which the Dimenſion of y is the ſame: becauſe the Value of y thence deduced. is o. Thus in the laſt Equation y* — a*y + axy — & =o, if you put the two Terms y* ＋ axy = o, you have y = — ax, an ima- ginary Expreſſion : therefore no firſt Term of a Series conſiſting of real Terms, can be thence deduced. Suppoſe you put axy — x* S o, thence you have y = al by ſub. ſtituting of which Dimenſion of x through the Terms of the Equa- tion, the Dimenſions of x are theſe, x*, **, x*, x* : where it ap- pears that there is one Dimenſion of x higher, and another lower, than in the two aſſumed Terms: and therefore theſe two Terms are improper for deriving the firſt Term of a converging Series, either when x is very ſmall ; or very great. Finally, the two Terms — n + axy are improper : becauſe the Value of y deduced from putting them equal to nothing is y == 0. L EM M A III. 94. In any Equation including one variable or flowing Quantity only, as x, the reſt being conſtant or invariable Quantities, all the Terms muſt include the fame Dimenſion of K Dem, For, if poſſible, let 'ax* + bx* + cx* + cx* = © be an Equation whoſe Terms include different Dimenſions of x, viz. x“ and &, and x at the ſame Time a variable or flowing Quantity: then di- vide all the Terms by the loweſt Dimenſion of x, viz. x*, and it is ax + bx* e == o: whence * == a x22 V 55 : there- fore x is a conſtant Quantity, contrary to Hyp. Univerſally let ax? + bx7 + cx” —= © be an Equation including one variable Quantity x, which hath different Dimenſions, if poſſible, and let x” be the loweſt Dimenſion, whether the others be equal Di- menſions of x or not: divide by x”, and you have ax*”” + bx” e So: which being an Equation of a definite or certain degree, x can In this Lemma it is ſuppoſed that the Coefficients a, b, e, &c. are nat nothing. only only have a certain determinate Number of Values, which is contrary to the Nature of a variable Quantity; which admits of infinite dif- ferent Values. x % LEMMA IV. 95. Let ax*y?, bx y, cx y-, be three Terms of an Equation, which includes two unknown and variable Quantities x and y: and let theſe Terms be ſuch, as that, when any Dimenſion of x, as x”, is inſerted for y in them, the Dimenſions of x are equal: Moreover, when the three Terms axty? 4- bx'y' + cx'y” are put So, let a Va- lue of y thence deduced be ax, I ſay n =vy. Dem. For if you inſert ax” for y in the Equation ax*y! + bx'y' + oxy =0, you ſhall have a&x?t® + ba'x+® + canxt” = o: where x being a variable Quantity, it's Dimenſions through all the Terms muſt be equal ; therefore p + qy = r + 9; whence » ==: but . Art. 94. by inſerting x” for y in the ſame three Terms, the Dimenſions of x in them are K“. +=, x: which by the Hyp. are equal, that is p +qgn=r + on or n= = : therefore » =». Q. E. D. 96. Cor. 1. The reaſoning is the ſame if there be more than three Terms. And hence it follows, that, if there be three or more Terms. of any propoſed Equation, including two unknown Quantities x and , which being put equal to nothing, give a poſſible Value of y, con- | taining a certain Dimenſion of x * ſuch, as that the Dimenſion of x, being ſubſtituted for y through all the Terms of the given Equation, makes the Dimenſions of x greater in all the other Terms, or leſs in them all, than in the three or more aſſumed Terms, the Value of y thence deduced will give the firſt Term of a converging Series, for a Value of y in the propoſed Equation: according as x is ſuppoſed to be very little, or very great. But if the Dimenſions of x rife higher in ſome Terms, and lower in other Terms, of the propoſed Equa- tion, than they do in the three or more aſſumed Terms, theſe aſſumed Terms are improper for deriving the firſt Term of a converging Series, either when x is very ſmall, or very great. 97. Cor. 2. Let axfy! . bx'y' . cx'y* be three Terms ſelected out of the propoſed Equation, which being put equal to nothing, give y = Ax", whether Ax” be the firſt Term of a converging Series or not; then it ſhall be p—r : 5—9 :: r—# : v—s : or by Alternation, and Here x* = 1 is conſidered. as a Dimenſion of xæxxx H Compoſition 423 Art. 94 ANALYS 18 by 'EquaTioNs, &c. Compoſition p—r : p—t : : 5—q : v—q : that is the Differences of the Indexes of x and y are proportional, For by inſerting Ax" for , the Dimenſions of x in the three Terms ſelected, are K. yr+n *r which being equal “, you have 15. p--gn = ran, 29, 1 In: from the firſt 2 = = from the ſecond n== ; Whence s? p—=r:iS—q::ir—t:;0v—, And the reaſoning would be the fame if the Terms were more than three. The Converſe of which is likewiſe true, viz. If there be three or more Terms having the In- dexes of x and y proportional, and if & be inſerted for y in all of them, make the Dimenſion of x the ſame in any two of them, it will make it to be the fame in all of them. For let xft — += then I fay K r is equal to any of theſe : becauſe ſince it is þ mn Sr, hence 2 but whence 7 = — or —9 —7 7 in -n rt, and by Tranſpoſition, fu r. 98. Cor. 3. If there be any Number of Terms more than two, of any propoſed Equation, as the Terms ax%y* . bx'y* . cx'y*, from which put equal to nothing, the firſt Term of a converging Series is derived, viz. Ax* then any two of theſe Terms being put equal to nothing, will give a Value of y in which the Dimenſion of x is the fame as in Ax", For if you put any two of the Terms aſſumed, as fy axty! ix Do, you thence deduce y , where the Di- menſion of-x is the ſame as in Ax" by Lem. IV. 99. Cor. 4: From what has been ſaid it appears, that, if any two Terms of a propoſed Equation, being aſſumed, and put equal to nothing, a Value of y be thence deduced; and the Dimenſion of x contained in that Value, being ſubſtituted for y in all the Terms of the propoſed Equation, make the Dimenſion of x the fame in one or more Terms, as it is in the two aſſumed Terms; but higher in all the other Terms; or lower in all the other Terms; then ſuch one or more Terms being joined to the two aſſumed Terms, and the Aggregate put equal to nothing, any real Value of y thence deduced, will make the firſt Term of a converging Series. Moreover any two Terms of the propoſed Equation being taken, which belong to the Equation to be aſſumed for determining the firſt Term of a converging Series, you may always know if there are any more Terms to be joined with theſe two, from the Property of ſuch Terms mentioned Art. g7. fince the Differences of the Indexes of x and y muſt be proportional. Examp. JJ T OW Examp. Let the Equation y* — gxys - pa- + 6a*s' ＋ U Do be propoſed: aſſume the two Terms 6 — 74*x*y* = o, thence you have y + V X Vax, where the Dimenſion of x is &. which being ſubſtituted for y through the ſeveral Terms of the pro- poſed Equation, makes the Dimenſions of x theſe, viz. x. xt. x5, x3. X. *: wherefore you put the three Terms y* — 97a*x%* + baz So, and thence ſeek the firſt Term of a converging Series, which you will find to be fourfold, viz. + Vax, — Vax, + V2ax, — +/2ax, for all theſe are Roots of the Equation y* — 7 643x?, as will appear by ſubſtituting them for y, one after another: therefore + ax, or — Vax, or + Vaax, or — Vaax, may be the firſt Term of a converging Series, each of which Series's is a Root of the propoſed Equation, when x is very little. Moreover you ſee that the Differences of the Dimenſions of x and y are proportional in the three Terms y* — 74 + Ga or & — 7a*%x%y* + 643x3y*®, for 0—2:2—6::2—3J:0—20 — 2: —4:;—1: —2. From the Symptoms or Properties of the Terms of any propoſed ſpecious Equation, including two unknown Quantities x and y, which are fit for deriving the firſt Term of a converging Series, which have been obſerved and demonſtrated in the preceding Lemmas and Co- rollaries, there may be various Methods deduced for diſcovering ſuch Terms. METHOD L To be applied when there are two or more Terms of the propoſed Equation, which involve not the Quantity x ; and one or more of which involve the Quantity y. | 100. Put the Terms of the propoſed Equation which include not the Quantity x, equal to nothing, giving each Term it's proper 2 * 422 the Value of y deduced from this fictitious Equation, will be the firſt Term of a converging Series for the Value of y; and if this fictitious Equation have more real Roots than one, each of theſe will give the firſt Term of a converging Series: when x is very ſmall, This is the Rule propoſed by Sir 1/aac himſelf Art. 30. See the De- monſtration of it above, Art. 74. 91. and Examples in Art. 79. 80. and under Examp. 2. of Art go n But if y be of the ſame Dimenſion in all the Terms, which include not x, the firſt Term of a converging Series cannot be derived from 4 them “. 4% AN ATEYS IS by Bavarions, Wc. * Art. 93. them *. And what may be done when all the Roots of the fQiti. ous Equation are impoſſible ſhall be ſhewn afterwards. ö METHOD H. By which the firſt Terms of all the converging Series's may be diſcovered which are the Roots of the propoſed Equation, either when & is very [mall or very great. [ 997 101. This Method is by Trial, founded upon the Symptoms or Properties of the Terms to be ſelected for this Purpoſe, which have been explained and demonſtrated above. | The Method of proceeding is this: Take the firſt Term of the pro- poſed Equation, and join to it every one of the other Terms, each with it's proper Sign, one after another, forming ſo many fictitious Equations; from each of theſe ſeek the Value of y; and obſerve the Dimenſions of x which each of theſe Values contains : ſubſtitute the Dimenſion of x contained in the Value of y derived from the firſt of thoſe fictitious Equations, inſtead of y, through all the Terms of the | | 8 Equation, and obſerve the Dimenſions of x through all the —__— — — —_— —— erms; and if you find them either higher in all the other Terms, or lower in all the other Terms, than the Dimenſion of x in the two af. . ſumed Terms, of which the firſt fictitious Equation was formed, then | the Value of y deduced from that fictitious Equation, is the firſt Term | „Art. 9. of a converging Series *; or if there be more Values of y than one deducible from that fictitious Equation, each of them will be the firſt 3 Term of a converging Series. But if, beſides the two Terms forming | the firſt fictitious Equation, there be any other, in which the Dimen- | fion of x is the ſame as in theſe two Terms, and it's Dimenſions in all | the reſt of the Terms, either all higher or all lower, in that Caſe, the | other Term or Terms, in which the Dimenſion. of x is the ſame as | in the two firſt aſſumed Terms, muſt be joined with them, all with | their proper Signs, from which you are to form a new Equation, by | putting the Aggregate equal to nothing; and the Root or Roots of it will give the firſt Term or Terms, of one or more converging Series's, * Art. 96. which will be a Root, or Roots, of the propoſed Equation *, ee But if the Dimenſions of x, after Subſtitution through all the Terms of the propoſed Equation as above, be higher in ſome Terms, and lower in other Terms, than what the Dimenſion of x is in the two Terms of the firſt fictitious Equation, the aſſumed Terms are impro- f per to derive the firſt Term of a converging Series, whether x be very 95. little, or very great *. i Terms EXPLAIMNE D. 425 T bis done, you muſt next join the ſecond Term of the propoſed , Equation with all the other Terms except the firſt ; or perhaps ſuch other as it was joined with formerly, in deriving the firſt Term of a converging Series ; and thereby form ſo many new fictitious Equations, as there are diſtinct Terms to join it with: which are to be managed after the ſame manner as the former fictitious Equations. Then pro- ceed to the third Term of the propoſed Equation, and join it with the other Terms that follow it, one after another, with the exception juſt now mentioned ; and uſe theſe fiftitious Equations as the for- mer ones. And thus proceed to do with all the remaining Terms; and ſo you ſhall have diſcovered the firſt Terms of all the Converg- ing Series's, which expreſs the Roots of the propoſed Equation, f. e. the Values of y, either when x is very little; or very great. The Demonſtration of this Method is contained in the preceding Lemmas and Corollaries, particularly in the Articles referred to: And the following Example will illuſtrate it. Ex. Let the Equation y — by* + gbx* — x3 =o be propoſed: I form an Equation by joining the firſt two Terms together, which gives y5 — by* So, from whence you have y=Vb or Vb x xe: where the Dimenſion of x being * = 1, I ſubſtitute it for y in the Terms of the given Equation ; and they will contain theſe Dimenſi- ons of x, viz. &. X. Xx. x3: wherefore I conclude that Vs is the firſt Term of a converging Series for y, when x is very ſmall : ſince the Dimenſions of x in the other Terms are greater than in the two firſt Terms, which were aſſumed. | AS But this I might have concluded immediately by the firſt Method, Next I take the firſt and third Terms of the propoſed Equation, and form this Equation y5 ＋ 9 D o, from whence I have y = 3 96x, in which the Dimenſion of x is x*, which I ſubſtitute for y through all the Terms of the propoſed Equation, from whence they have theſe Di- menſions of x, viz. *. x* . x*. x3. where I obſerve that the Dimen- ſion of x in the ſecond Term is leſs, and in the fourth Term greater than the Dimenſion of x in the firſt and third Terms, which were thoſe aſſumed : wherefore I know thoſe two Terms are improper for r 9 — ch) 2 . 1* 2 OO CI = 1s deducing the firſt Term of a converging Series, either when x is very d ſmall, or very great. | | 70 3*. I form an Equation of the firſt and laſt Terms, viz. y5 — x3 - =0: whence I have y = ; by ſubſtituting which Dimenſion * of x for y, you have x3, *, „, x3: from which I con- 1 clude Ns | 426 ANALYSIS by Equations, &c. clude that x? is the firſt Term of a converging Series, when x is very Art. 95. great: becauſe the Dimenſion of x in the two aſſumed Terms, is greater than in the other two. | And thus having joined the firſt Term of the propoſed tion with all the reſt, I proceed to join the ſecond Term with thoſe that follow it; it having been joined with the firſt already. Wherefore by joining the ſecond and third you form this Equation — by* + 94a So, whence I find y= + 3x or — 3x; ſo that ſubſtituting this Dimenſion of x for y in the * of the propoſed Equation, the Dimenſions ſtand thus x5, x*, x*, x3 : whence, as before, I conclude that + 3x and — zx are the firſt Terms of two converging Series's, each of which will be a Value of y when x is very ſmall, in Com- pariſon of 5. | And now I join the ſecond Term of the propoſed Equation with the laſt, and thereby form this Equation, viz. — % —x3—0: in which I eaſily perceive that y = /— - =* xi, which being the Square Root of a negative Quantity, ſuppoſing 5 to be poſitive, is impoſſible: from whence I conclude that theſe two Terms — by* — 3 are unfit for finding the firſt Term of a converging Series, by them- ſelves : and I know they cannot be joined with any other Term, ſo as to form with it an Equation fit for that Purpoſe, becauſe that would have appeared in the foregoing Part of the Operation, either when the firſt and fecond Terms were aflumed ; or when the ſecond and third Terms were aſſumed, for forming theſe fictitious Equations “. 102, But if this had not appeared from the foregoing Part of the Operation, then I would have proceeded to inſert x* for y through the Terms of the propoſed Equation, in order to find whether there might not be fome other Term, which joined with theſe two, might have been proper to form an Equation, admitting of ſome real Root, for the firſt Term of a converging Series; for although two Terms don't yield a real Root when they are alone, yet I cannot from that con- clude that they may not be joined with ſame other Term, which with them may form an Equation with one or more real Roots, which wil be the firſt Term or Terms of one or more converging Series'ss Thus if the propoſed Equation had been — by* þ 2 i + y + gbx* — x3— ©, then by joining the two extreme Terms, which are the ſame _ with thoſe juſt now mentioned in the foregoing Equation, you have — by* — x3 =0, whence = 4 X * an impoſſible Value, when 1 wes > ©” A 9 * TP „ 03 r Ws (DP % e n to EXPLAIN KD. 427 when is poſitive; yet I ſubſtitute x* for y through the Terms of the Equation, and I find the Dimenſions theſe x3, x3. K . xl: whence I am directed to frame this Equation — by* 4 252 — == , of by changing all the Signs ) — 2 e, where bas a real Value, vis. + He; which will therefore make the firſt Term of a converging Series expreſſing one of the Roots of the Equation — by + 26*x*y +y o - * SD o, viz. when is ſuppoſed much greater than 6. 103. And now I have ſhewn what will be the Reſult of forming Equations by joining together every two Terms of the propoſed Equa- tion y5 — by* + gbx* — #3 — ©, except the laſt two, from which it is plain no Value of y can be derived, ſince y doth not enter into any of them, And it appears that there may be three convergin Series's when x is very little in Compariſon of 5, whoſe firſt 1 . + 3x. — zx. ; and one converging Series when & is very great in Compatiſon of 4, whoſe firſt Term is + * | 104. Schol. But here I ſuppoſe the Reader will eafily underſtand that the whole of this Trial, to know what Terms of any propoſed Equation are fit to derive converging Series's from, in the moſt Part of Caſes, may be made in his own Mind, without the Neceſſity of always writing down all the Steps. Thus if the Equation by3 — 2b*x*y — y — 9bx* + x3 = © be pro- poſed, I eaſily perceive, by running over the Terms in my own Mind, ſo as to join them two by two, that the only fictitious Equations that can ſerve for finding the firſt Terms of converging Series's, are 1“. by - y %“. 2%. XK o. 3% —y—9g6x* =0; the firſt · Art. ico. and third when x is ſuppoſed to be very little; and the ſecgnd, when it is ſuppoſed to be very great in Compariſon' of 5; which firſt x Terms therefore are 4 IF — Ip — Ip and — ba-. ; 428 ANATLYSIS by EqvaTtions, &c. METHOD III. By which, for moſt part, the firſt Terms of two converging Series's ma "- diſcovered, one of them for the Caſe of x being very ſmall ; the a for the Caſe of x being very great. | 105. - Let M repreſent the Dimenſion of x in that Term of the propoſed Equation in which æ is ſeparately (i. e. without any Mixture of y) of the loweſt Dimenſion *, and let xy repreſent the Dimenſions of x and y in every other Term of the Equation indifferently, one after another, (where 7 and s are ſuppoſed to put on as many different Values as the different Terms require) run over the ſeveral Terms, and obſerve where the Value of ＋ is greateſt ; then, that Term, or thoſe Terms, if there be more of them, where this happens, being joined to the Term where you have the Dimenſion x”, with their proper Signs, and put equal to nothing, the Root or Roots of this fictitious Equation will give the firſt Term or Terms of one or more converging Series's for the Value of y when x is very ſmall. Again if x” repreſent the Dimenſion of x in that Term of the pro- poſed Equation in which x is ſeparately of the higheſt Dimenſion; and x.y. repreſent the ſame thing as before, obſerve where the Expreſſion — is leaſt of all, and ſuch Term or Terms, being managed as before, will ſerve, in the fame way, to diſcover the firſt Term or Terms of one or more converging Series's, when x is very great *. Dem. Let x? denote the ſame thing as in this Art. formerly; and let xy xy. * V5 Sc. repreſent the Dimenſions of x and y in the other Terms of the propoſed Equation, where I have given the Letters r and s Marks of DiſtinCtion, according as they are ſuppoſed to denote the Indexes of x and y in the different Terms: Now I fay if ＋ be 4 the greateſt of all the correſponding Expreſſions, that is A, Kc. lere it is ſuppoſed that the Exponents of the Powers of x and y are poſitya: and that x* therefore is the loweſt Dimenſion of x in the propoſed Equation. | It muſt be obſerved that in computing the Value of the Expreſſions 2 ad} + & $ # 2 Value is leſs than any poſitive Value; and a greater negative is a leſs Value than a teſs Negative. then ET PEATVYEZ D. 40 then the two Terms having * and xh for their Dimenſions, being put equal to nothing, are proper to derive the firſt Term of a con- verging Series from, when x is very ſmall. For if theſe two Terms be put equal to nothing, the Value of y thence deduced, will contain pmr | this Dimenſion of x, viz. x * : put —— Sn; inſert & for y in the | other Terms of the Equation, and the Dimenſions of x in them will I be KT x +" &c. Now by Hyp. 1 (= Z =) > Land or 22 Kc. 2 3 wherefore, if you multiply by or“ Cc. you'll have us p-; . p- &c. (and this is true whether 2 = — be poſitive or ne- gative) and by Tranſpoſition . — ni g p, .. — 10 g p, &c. whence the thing is evident from Art. 90. And the Reader will eaſily under- ſtand, that, if there be any other Term, whoſe Dimenſions of x and y are denoted by &), ſuch, that — =p —, ſuch Term muſt be joined with the other two, in forming the fictitious Equation for de- riving the firſt Term of the converging Series *. * Art. 97. Again if x” denote the higheſt Dimenſion of x ſeparately, and 97 the other Symbols be uſed in the ſame Senſe as before; by putting the Terms having the Dimenſions x". x, equal to nothing, you have a Value of y whoſe Dimenſion of x is ; put Qn, then by the Hyp. 12 7 ; multiply both by 7, and it is 27 mn —r (and that whether » = — be poſitive or negative) wherefore, by Tranſpoſition, n Sm: and the reſt of the Reaſoning is as before, by Help of Art. 3o. and gg. Ex, 1. Let the Equation y3 — axy + x3 = © be propoſed, to find the firſt Term of a converging Series, when x is very ſmall in Com- pariſon of a, then p=3, and z likewiſe, r—1. and s=1, when applied to the middle Term — axy, and o, s— 3, when applied to the firſt Term y3, fo that the Values of © or —. are 5 —— = 2, and _ = 1; Wherefore I put — axy 4 x3 =© for finding the firſt Term of a converging Series, when x is very ſmall; and y3 + xi — ©, when x is very great: which give y = + <; and x reſpectively. 18 e Ex. 430 ANALYSIS by Equations, &c. Ex. 2. Let 51 — u + axy —x3 = o be propoſed: here you find, by the ſame Method, the Terms — @*y — x3 proper for deriying the firſt Term of a converging Series, when x is very ſmall; and yi <1 when x is very great: which therefore are — 5, and & reſpectively. Ex. 3. Let the Equation 35 — 26*x*y — —gb* N be propoſed: here you have — gb the Term wherein x is of the loweſt Dimenſion ſeparately, and therefore p = 2, and the Term ... y joined to it, is that which gives the Expreſſion _ the greateſt, which is —> = 2, therefore the Equation — y — ghx* = © gives the firſt Term of a converging Series, when x is very little, viz. y == — gh, Again + x* is the Term where x is ſeparately of the higheſt Dimenſion, therefore m — 3, and the Term ) is that which compared with + x, gives the Expreſſion —— the leaſt, wherefore I put by* + & o. and thence deduce y = — 57 for the firſt Term of a converging Series, when x is very great. | 136. Schol. 1. This Method ſuppoſes that the Exponents of the Powers or Dimenſions of x and y, through all the Terms of the pro- poſed Equation, are poſitive ; otherwiſe the Method will not anſwer, without other Limitations; as will appear, if you ſuppoſe that, in the firſt Part of the Demonſtration, x = = is poſitive ; and 2 7 and / both negative. 107. Scol. 2. The Expreſſion , is always the greater the leſs that 7 is, if s be the fame in different Terms of the propoſed Equation: and if r be the fame in different Terms of the Equation, Led is always the greater the leſs that s be, when p — 7 is poſitive; but the greater that s be, if p—r be negative: which is of Uſe in the Practice of ſelecting the Terms, to be uſed for diſcovering the firſt Term of a converging Series, when x is very ſmall; the like Obſervations may be made with Reſpect to the Expreſſion =, to direct you in the 4 ſelecting of the Terms proper for finding the firſt Term of ſuch a Series, when x is very great. So that what has been ſaid under the Head of this third Method, ſerves to illuſtrate our Author's ſecond Remark under Art, „ * EXPLAINED. #4 431 Art. 3 5. and, in general, his whole Method of proceeding in diſcover- i — ſeveral Terms of the convergi ws for the Value of y: . in 3 which he calls the Quotient. And this Eads us to METHOD WV. By Means of which the fir Terms of all the converging Series's may be found which are Roots of the given Equation, whether x be very ſmall, or very great. 108. This Method is by the help of a rectangular Parallelogram, ingeniouſly contrived by our celebrated Author. hk Let AB, AC be two right Lines at right Angles to one another, divided each into ſome Number of equal Paris: and the Parallelogram AD being completed, throagh all the Points of Section of AB and AC draw Perpendiculars to theſe Lines, as you ſee, dividing the whole into equal Squares: which let be denominated from 2 | 432 of x and y; ſo that thoſe lying along AB may contain the Dimenſions * ANALYSIS by Equations, Ofc. of x ſeparately, viz. 1 =x*.x. *. x3, * &c, thoſe along AC the ſeparate Dimenſions of y, -viz. 1 . y. . 53. „ Fc. ang every other Cell or Square, that Dimenſion of x that ſtands at the beginning of the ſame horizontal Row, and that Dimenſion of y tht ſtands at the foot of the ſame perpendicular Row with itſelf, in the Manner the Figure repreſents. Then any Equation being propoſe, as the Equation x#y7 — 4y5 -- 2axy3 — 2bx*%y3 ＋ 3x39* — abx%y 4. * +- 2ax5 =o, mark ſuch of the ſmall Squares or Cells, as correſpond to the ſeveral Terms of the propoſed Equation, according to the Di- menſions of x and y they contain: which you ſee here done, by plac- ing this Mark * in them: then let a Ruler as EF be applied to the loweſt of the cells that, being marked, lye along the line AB, (for in every Equation there muſt be at leaſt one ſuch Term) which in this Example is xi, touching it in the Point E, as you ſee, then make the Ruler EF to turn from EA towards the Right Hand, until it touch, beſides the Cell x“, one or perhaps more of the Cells that are marked, ſo as that all the reſt lye upon one Side of it: ſo here you find x*y to be the only one it meets with at firſt, Wherefore you ſele the two Terms of the propoſed Equation, which have the ſame Dimenſions of x and y, viz. — abx*y A K, and form an Equation of them, vis. — abx*y + x* = o, from whence the Value of y being found, vis. + 5 gives the firſt Term of a converging Series, which converges the faſter the leſs that x is. Then I make the Ruler revolve the ſame Way about the Cell x*y, until I find it touch, beſides it, one or more of the marked Cells, ſo as all the others lye upon one Side of the Ruler: here I find it to touch the three marked Cells xy. xy3 . ys : therefore I form another Equation of the three correſponding Terms of the given Equation, viz. — 4y5 + 2axy3 — abx*%y =© : from which you'll find a fourfold Root -|- Wd ia | > —＋ + 2 x & 3 e. Vie, = Fe — — — — „„ 44 — 4 — = e X x*, any of which will be the firſt Term of a converging Series, when x is very ſmall in Compariſon of a and þ, Next I make the Ruler, according to the ſame Method, to touch the two marked Cells ys and x*y”, ſo as to leave all the reſt of the * c g 7 * 0 . * a7 -— . N EXPLAIMNE D. Cells upon the ſame Side of the Ruler, as you ſee by the Figure; and ſo I form an Equation of the two correſponding Terms, viz. xy7—495=0 : whence you have y= + = or — =, for the firſt Terms of other two Series's, which will alſo converge when x is very ſmall. | Hitherto the Ruler has been made to touch the outward marked Cells below, ſo as to have all the reſt till lying as it were above it: and as No more of the extreme marked Cells can be touched by it in this Manner, you have got all the fictitious Equations from which the firſt Terms of Series's can be formed, which converge when x is very ſmall. And therefore you now begin to lay the Ruler to that marked Cell, which is the higheſt of ſuch that lye along the Line AB, which here is x5, and m:king the Ruler to touch it above, let it revolve the contrary Way, and obſerve what extreme marked Cells can be touched by it at once, ſo as to leave all the others below it; and theſe will afford fictitious Equations for finding the firſt Terms of Se- ries's, which will converge the faſter the greater that x is : and of ſuch I find only one, formed of the two Terms, which correſpond to the two Cells x5 and xt) ?: therefore I put xh + 2ax5 So, from whence you have y—= V2a xX, which will be the firſt Term of a converg- ing Series, wherein the Dimenſions of x ſhall continually diminiſh, correſponding to the Caſe of x being very great. Here you may conceive the Ruler beginning to revolve from the Poſition E A towards E F, until it has made a half of an entire revo- lution, making it to touch the outward marked Cells below, as it re- yolves ; and ſo you ſhall find the Terms of the propoſed Equation, which are proper for deriving the firſt Terms of all thoſe Series's, which converge the faſter the leſs that x is. (And again, if you con- ceive the Ruler, beginning at the point B, and lying in the direction of the Side AB of the Parallelogram, to turn about the contrary . Way to what it did before, until it ſhall have turned about, 180 De- grees, ſo as ta make it touch the extreme marked Cells above, always leaving. all the reſt entirely upon one Side of it, and as it were below it, you ſhall thereby obtain the Terms of the propoſed Equation, which are proper for forming fictitious Equations from which the firſt Terms of all ſuch Series's are to be derived, which converge when x is very great; that is in which, the Dimenſions of x continually de- creaſe. The fame Thing may be conceived to be done by drawing a right lined Figure, every one of whoſe Sides ſhall touch two or more of, the outward marked Cells, ſo as to leave the reſt entirely within the Figure, in the Manner repreſented in the Diagram itſelf, Kkk log. The 438 rt. 90, ſhall be the fame *. ANALYSIS by Equations, Ee. 109. The Demonſtration of theſe Things may be eaſily made out from a due Conſideration of the Parallelogram, and what has been for- merly ſaid. In order to which, all that is neceſſary to be done, is to ſhew that, if the Cells which are touched by the lower Sides of the right-lined Figure, that is the Sides lying betwixt B A and CD upan the lower Sides of the Figure, be put equal to nothing ; or (which, in this Caſe, is the ſame) the Terms of the propoſed Equation, which correſpond to theſe Cells, and the Dimenſion of x in the Value of -thence deduced, be incerted for y in all the reſt of the marked Cells, they ſhall contain higher Dimenſions of x than thoſe Cells which the Line touches, after the Dimenſion of x. is inſerted in place of y in them alſo: and that theſe laſt mentioned ſhall contain the ſame Di- menſion af x. * And after the like Manner, that, if the Cells which are touched by the upper Sides of the right-lined Figure above, i. e. the Sides which lye betwixt BA and D C upon the upper Side of the Figure, be put equal to nothing, the Dimenſion of x contained in the Value of thence deduced, being inſerted inſtead of y in all the marked Cells, will produce lower Dimenſions of x in them all, than in thoſe which the Side of the Polygon touches: in which laſt the Dimenſions of x Now in order to ſhew this, let it be conſidered that if any Num- ber of Cells are touched by any Line, as the Cells x“, xy*, and y* are by the Line EF; and as the Cells xy, xy? and y5 are touched by the next Side of the Polygon, and ſo for any others, whatever way the Line touch them, whether above or below, and you take any Num- ber of ſuch Cells more than two, and that in any Order or at any Di- ſtances from each other, the Differences of the Indexes of x and y will be proportional, in the Senſe of Art. 97. For let the right Line 1 touch the Cells x5, x x3y+, s. upon the ſame Side: compare the Indexes of x and y in the three Cells x5, x3y*, x*y* together, and you find 5—J : 4—0:: 5—2 : 6—o. and ſo it is univerſally: for let the Cells x5 K, xi; be touched by the Line g iin the Points g, i, J. and let i b, IE, be the horizontal Lines drawn through i and /, meet- ing AB in the Points b and 4, then the Triangles g h i, g EI being 9 — yon have gh: hi: : g4: +4}: but gh and g meaſure the ifferences of the Indexes of x, and hi and & | the Differences of the ladexes of y, as is evident by canſiderjng the Diagram : thereſore the Pifferences of the Indexes of x and y are proportional, in the Terms x5, X N, x%y*. And the Reaſoning is the ame, in every other Fo | 110. Fram 7 C__ TT EXPLAINED. 110, From this it follows, that, if any Dimenſion of x, being in- ſerted in place of y in all the Cells which are fimilatiy evuched by any right Line, tnake the Dimenſions of & in two of theſs Cells equal, then the Dimenſions of x in them all ſhall be equal *. | 111. Again, if a right Line touch any Number of Cells ſimilarly below them (as the Line E F touches the Cells x*, My, y*) then the Dimenſion of x, mentioned in the laſt Art. viz. that which being inferted for y in the touched Cells, makes them, or the Dimenſions of x in them, equal; when it is inſerted for y in the Cells that lye en- tirely above the Line E F, ſhall give higher Dimenſions of x. For take any of the Cells which the Line EF touches below, as xy, which it touches in the Point n: let # #1 und o be the dicular and horizontal Lines upwatds and towards the right Hand, then it will be evident, that all the Cells contained within the right Anyle mM no have either a greater Dimenſion of x, or a greater Dimenſion of 3, than x*y has; or both a greater Dimenſion of x and y: and the like muſt hold with reſpect to every other Cell that is fimilatly touched by E F, or by any other right Line which flopes downward from left to right: and theſe Cells which are contained within ſuch angular Spaces, are all that lye entirely above the Line: but in this Caſe the alue of y that atifes by putting any two of the Quantities contained in two of the touched Cells, equal to nothing, is a poſitive Dimenſion the Parallelogram : therefore, when this poſitive Dimenſion of x is 435 Art. of x, as will be evident to any one that conſiders the Conſtruction of inſerted for y in the other Cells which lye entirely above EF, the Di- menſion of x muſt be higher in every one of them. But if the Line or Ruler, ſtill touching below, flope downward from right to left, the Value of y which ariſes by putting any two of the Quantities, contained in two of the Cells which it touches ſimilarly, equal to nothing, muſt contain a negative Dimenſion of x, as will be evident to one that conſiders it ; and all the Cells contained within the right Angle which proceeds from any of the touched Cells towards the left Hand and upwards, contain either a higher Dimenſion of x with the ſame Dimenſion of y, or a lower Dimenſion of y with the ſame Dimenſion of x ; or then, both a higher Dimenſion of x and a lower Dimenſion of y; in all which Caſes, the negative Dimenſion of x in- ſerted for y in ſuch Cells, muſt make the Dimenſion of x higher in them than in the touched Cell: and this being true with refpett to every one of the Cells, which are touched by a Line floping downwards from right to left, hence the ſame Thing appears in this Caſe likewiſe, viz, that the negative Dimenſion of x, which inſerted for y in the Cells K kk 2 which 97+ 436 ANALYSIS by E warloxs, &c. which the Ruler touches ſimilarly below, makes the Dimenſions of x the ſame in them, will make the Dimenſions of x higher in all the Cells, which lye entirely above the Line, when it is inſerted for y in them. | | 112. And therefore, this being univerſally true, it plainly appears that the Cells which correſpond to the Terms of any given Equation, which are touched by the lower Sides of the Polygon, muſt point out the | Terms of the Equation, which are proper for deriving the firſt Terms Art. 100. Art. 80. of thoſe Series's, which converge when x is very ſmall. And by a like Way of Reaſoning it may be demonſtrated that the upper Sides of the Polygon placed as has been explained, muſt point out thoſe Terms of the propoſed Equation, which ſerve to diſcover the firſt Terms of thoſe Scries's which converge, when x is very great: which were the Things we propoſed to ſhew. | | 113. Schol. 1. In the foregoing Demonſtration I have not parti. cularly mentioned the Caſes in which the Ruler, touching any of the extreme marked Cells, either lies parallel to AB; or to AC: becauſe in the firſt Caſe, it's plain no Value of y can be deduced from the fictitious Equation *; and the Reaſoning in the other Caſe is moſt apparent. I 14. Schol. 2. It may be obſerved likewiſe that Sir I. Newton makes uſe of the rectangular Parallelogram, only for diſcovering the Terms of any Equation, which are proper for finding the firſt Term of a converging Series, that will converge when x is very ſmall ; becauſe he reduces all other Caſes to this one *. And in particular, he uſes it only for finding the firſt Term of ſuch a converging Series, as may be deduced from the Suppoſition of the Ruler touching that Cell which contains the loweſt Dimenſion of x ſeparately, repreſented in the Dia- gram by the Line EF touching the Cells x+ and x*y * : which, I ſup- poſe, he doth, partly that he might not perplex his Readers with too great a Variety of Caſes; and likewiſe becauſe he uſes the fame uni- form Method in finding the firſt, and all the other ſubſequent Terms of the Quotient or Value of y, as we ſhall ſee juſt now. | Of the different Methods of finding the ſubſequent Terms of the Quotient Ne. 63. or converging Series, for the Value of y. . 115. The firſt Term of the Quotient being found by any of the preceding Methods, the ſubſequent Terms may be diſcovered in diffe- See his Method of Fluxions and infinite Series, Introd. Art. 29, 30, and Com. Epiſt, rent PLANE 437 rent Ways: one of which is by Means of the Parallelogram already deſcribed. Thus, in any propoſed Equation, having found the firſt Term of any Root or Value of y, when is very ſmall, to be Ax*, you put y Ax +-p; and then transform the propoſed Equation of y, by ſubſtituting Ax" + p for y, whence ariſes a new Equation, vi. that of p : which you now ſuppoſe to- come in place of y : then mark thoſe Cells of the Parallelogram, which correſpond to the Terms of this ſecondary Equation : and laying a Ruler to the loweſt of theſe Cells which lye adjacent to the Side A B, make it to revolve towards the right Hand and upwards, until it touch one or more of the marked Cells beſides the firſt, in the Manner already explained; the Terms of the Equation of p, which correſpond to the Cells which the Ruler touches, are to be feigned equal to nothing, and an approximate Value of þ thence deduced, which will give the ſecond Term of the Quotient: which let be denoted by Bx“ t then you ſuppoſe p B A g, and transforming the former Equation of p into a new one, by ſubſti- tuting Bx"+* + g, in place of p, you muſt find an approximate Va- lue of 9, by Means of the Parallelogram, after the fame Manner as you find an approximate Value of p already, i. e. by applying the Ruler to the loweſt marked Cell adjacent to AB. And ſo you proceed to find the ſubſequent Terms of the Quotient as far as you pleaſe to roduce it. By proceeding thus, the Dimenſions of x which are in the marked Cells upon the left Hand, adjacent to A B, will continually increaſe, as will be evident to one that conſiders it, and therefore you will continually approach to the true Root, by diminiſhing conſtantly the Supplement, p, 9, &c *. * Art, 75. 116, Schol. In this Method of continuing the Terms of the Quo- tient, although you might find out the firſt Term variouſly, by diffe- rent Applications of the Ruler ; yet you find all the ſubſequent Terms in the ſame uniform Manner, viz. by applying the Ruler always to the loweſt marked Cell adjacent to the Line AB : the Supplement p. 9, r, &c. being ſuppoſed to be very ſmall ; as well as x. I have only conſidered the Caſe of a Root which converges when x is very ſmall ; becauſe all other Caſes may be reduced to this: and this Me- thod of finding the ſubſequent Terms of the Quote cannot be fo properly applied to the Caſe of x- being very great: upon the account of the increaſing negative Indexes of x. 117. But whereas the Parallelogram contains no negative nor fracted Powers of x and y, therefore it is proper to obſerve that, when any ſuch occur, either in the Equation propoſed at firſt, or in any of the ſecondary or ſupplementary Equations ariſing in the Progreſs of the Operation, 438 ANALYSIS by Eqvations, &c- Operation, they ought to be removed, before you proceed, by ſuch Methods as are practiſed in Algebra: thus, if 2 had the — x3 + 3x — 2 — 165-3 = ©, you free it from the negative Dimenſions of x and y by multiplying by the pofitive Dimenſions, which correſpond to the higheſt negative Dimenfion, viz. x and y3, by which Means it becomes x3 . 3x — 293 — 16x =o. Aſter and thence ariſes v —-a0* -þ xv* - 4K! ＋ 2 — the ſame Manner the Equation x —= = — = + 57 muſt be con- verted into xy9 = a*y* — 27% + 34 or xy3 — 4 + 24% — 33a So. Which change being made, either upon the primary, or - of the ſecondary Equations, when neceſſary, the Operation will oceed. 118. Likewiſe in the Caſe of fractional Indexes, remove them by making a proper Subſtitution : thus if the Equation y* — xy: + xt Doo, were propoſed, put y: =v and x3 , and by Subſtitution, you ſhall have v — ziv E-. ©, out of which having extracted the Root v, you replace ys for v and x3 for z : By extracting the Root v you would find v=2 +2* + 625 ++ 5127 &c. that is yz = * + x + 6x + 51xi &c. and by ſquaring both Sides, y = xi - 2x3 + 13x* + 114x7 &c. | 119. Moreover when any complicate Fractions, or ſurd Quantities, happen to occur, either in the propoſed Equation ; or to ariſe afterwards, they ought to be removed by the Methods afed in common Algebra. Thus if the Equation y* ++ — * — x? =©, was propoſed, you muſt free it of the complicate Fraction, by multiplying by a —'x, whence ariſes ay — X + a*y* — ax* + K* , out of which you extract the Root y: or you might ſubſtitute — for y, and thence would ariſe — + = —X' So, or v + a'o* -& 3a K — 3ax* +x® =0: from whence having found the Value of ©, divide it by a — &, and the Quotient gives the Root bp Likewiſe the Equation y3 — J — x Xy* + x3 = © being given, to extra the Root y, you may ſubſtitute z for /a — x, whence you will have ) — 25? + a? — 3a*2* + 3a + 25 o, out of which having extracted the Root y, replace /a — x for 2. Thus alſo the Equation y* — Va — x * — Va — xXx* = © being given, put V va x x5 x þ 4 0 P þ 0 0 EXPLAINE D. 439 x5 = ©, the Root of which laſt Equation being extracted, divide it by Ja mr at- La "ix — c. and fo you ſhall have 2 Value of y. Underſtand alſo that in the other Example which pre- cedes, it the Area of the Curve be required, you ſhould find the Ex- preffion of that Area in Terms of 2 before you replace /a — x for a. The Reduction and Preparation of Equations, mentioned in this Article, is always neceſſary in all the Methods of extracting the Roots of affected Equations, by Series ; but thoſe mentioned in the two pre- ceding Articles are not always neceſſary in the other Methods; but are ſo in that which proceeds by Means of the Parallelogram. 120. Another Method for diſcovering the ſubſequent Terms of the Root, is that of our Author's, which he delivers in this Treatiſe, and which was formerly explained &. But becauſe there are ſome Things“ Sed. 4-. in it about which the Reader may ſtill require ſome further Satisfac- tion; and which may be of uſe in Practice, I ſhall illuſtrate it a little further. Wherefore let ay” + bxty + c &c. + bx*= o be a 1757 neral Equation with indefinite Exponents, the Root y being * ed to be extracted, when x is ſuppoſed to be very ſmall ; to which all the other Caſes may be reduced “: in which there will be one Term, * Art. 86. at leaſt, which do not include x, repreſented here by ay", and another ons at leaſt, that includes not y, which is here repreſented by + h: let the Terms cx y + h be thoſe from which, put equal to nothing, the firſt Term of the Root may be deduced ; found by any of the 1 2 ' preceding Methods; which therefore is — 7 put = , then y =» Y + let the Equation be transformed, as follows, by the Help of the Binomial Theorem; omitting the numeral Coeffi- cients, | * * * „ A. Dh a gms e mm po Sc. + by =+8 „= „ e „N „enge. + af =o ear=+e; x} w==p+ex —_ Wc. &c. | Ge. Bs ＋ bx” = T. ; - By 440 ANALVYS IS % Eqvartions, &c. By conſidering of which the following Things will appear: 15. That in the transformed Equation, viz. the Equation of 5, thoſe Parts of the ultimate Term * which are collateral to the Terms * Art. 75. h + c in the given Equation, muſt deſtroy one another. * Rem. 1. 20. Theſe ſame Parts of the ultimate Term contain a lower Dimen- fion of x, than any of the other Parts of that ultimate Term do : be- cauſe they are the Quantities which ariſe by inſerting the Dimenſion of x, which the Value of y contains, deduced from the fictitious Equa- tion hx” + cx. == o, inſtead of y, in the ſeveral Terms of the pro- * Art, 88. poſed Equation “. Art. 73. 3. The Dimenſions of x in the ſeveral Quantities lying in the ſame horizontal Line of the Equation, conſtantly diminiſh by the ſame Dif- ference ; and thoſe of p increaſe by the ſame Difference; ſo that, abſtracting from the Coefficients, theſe Quantities go on in a continued Proportion, by the common N 16 xp: which is alſo the ſame in all the Lines. This appears from the Nature of Involution of Powers. | 4. From the laſt two Obſervations it follows, that the Part of the 0 . Penult Term, viz. c x =?) X x*—"þ == cep, contains a lower Dimenſion of x, than any of the other Parts of that Penult Term upon that ſame Line contains a lower Dimenſion of x than an other Part or Quantity belonging to. the Antepenult Term, &c. And moreover that, ſince the approximate Value of p; or the firſt Term of that Value, muſt contain a higher Dimenſion of x than x”, which is it's Dimenſion in the firſt Term of the Quotient or Root, expreſſing the Value of y *, therefore, if the Dimenſion of x contained in that approximate Value of p, were ſubſtituted for p through all the Quan- tities, belonging to the Equation of , the Dimenſions of x will in- creaſe, and that by the ſame common Difference in all the horizontal Lines, reckoning the Terms from the left Hand to the right : con- ſequently ze. If + bx a be that Part of the Ultimate Term, which, after throwing by thoſe Parts which deſtroy each other, contains See Note 1. Art. 75. the — Term : that the Part c x ==] XN p of the antepenult — ot. wed oa ads. oe eig M£© xz BXPLAIWED. 1 441 the loweſt Dimenſion of x, the approximate Value of the Supplement 7 muſt be found from the * Equation c ο +6 x EM ; and the ſame is to be anderfinad of all the other ſecondary or i pplementary Equations, Therefore you have this general Rule for finding $4 — ſubſ "ger" 3 of the Value of y after the firſt, viz. Let that Part of the ultimate Term 9 pplementary Equa- tion, which contains the loweſt . 5 x 2 it's Sign changed, be divided by the Coefficient of p in that of the Penult Term, wherex 1s of the loweſt Dimenſion, and that ſhall & next Term of the Quote to be annexed to what hath — formerly found. The Obſervation of which, together with a Rule — given *, will mightily facilitate the * Art. 82. Extraction of the Roots of ſuch ſpecious Equations. 6*. The Coefficient of p, 9. , &c. in the Part of the Penult Term of any ſupplementary Equation, by which you divide, as in the laſt Remark, contains the fame invariable Dimenſion of x throughout all theſe Equations: as will be evident if you conſider, that, in the next ſupplementary Equation, vi. that of q, it will be found in the Tranf- formation of the Equation of p, oppoſite to, or upon the fame Line with, the Quantity cx*—*p : ſo that if that Coefficient contain the Di- menſion x** in the firſt ſupplementary Equation, it will contain the fame in all the ſubſequent Ones. 7. The Difference betwixt the Exponents of x in the two firſt Terms of the Series expreſſing the Root, is the ſame with the Diffe- rence of the Exponents of x, in the Parts of the ultimate Term of the Equation of p, in which x is of the loweſt Dimenfion, and of the next higher, which are here bx” and 5 x =P „;: For the dif- ference of the Dimenſions of x in theſe Parts is . rn - u. the Value of the firſt Term of p, deduced from the fictitious *. FVV ALS emo, 2A.. tion + c X + 6 * art = ©, contains this Power of x, viz, „t, whoſe Exponent exceeds the Exponent of 7 which is the Power of x contained in the firſt Term of t e Series, by q + rn — ©, the ſame as the former: and the Reaſoning is the ſame, ed the Terms of the propoſed Equation from which, the firſt Term of Series is derived, be any other than cx'y* ＋ .. L II 121. \ Y | 442 ANALYSIS % EqvaTtions, &c. RN 121, Although we have hitherto +4 of the converging Series, or Quotient, expreſſing the Value of y, as if the Exponents of the Powers of x in the Terms of the Series proceeded in an Arithmetical Progreſſion ; yet the Reader muſt know that, albeit this moſt fre- quently happens, yet it is not always ſo: for which Reaſon I ſhall reſolve the following PROBLEM, To find whether. the Exponents of the Powers of x in the Terms of the converging Series go on in an Arithmetical Progreſſion, or. not: and if ſo, what the common Difference of that Progreſſion is. 122. Having found the Dimenſion or Power of x contained in the firſt Term of the Series, (viz. when x is very ſmall) by any of the preceding Methods, which let be x", ſubſtitute x* inſtead of y through | | all the Terms of the propoſed Equation ; and let the Exponents of the | Powers of x after Subſtitution be A, A, B, C, E, &c. of which A is the | leaſt (and therefore found at leaſt in two Terms) and the others B, C, | | E, &c. of which B is the leaſt; C the next; E the next, &c. in an aſ- | cending Scale: then if B A meaſure C B, E - C, &c. through- out all the reſt, the Exponents of x in the Terms of the Series pro- ceed in an Arithmetical Progreſſion, whoſe common Difference is B— A: but if B — A do not exactly meaſure all the Differences C—B, E — C, Sc. the Dimenſions of x in the Terms of the con- verging Series, wilp not conſtitute an Arithmetical Progreſſion. Which will be plain by Examples. Fs | Ex, 1, Take the Equation y* + ay — 24 braxy — x* =o, (which is the firſt of our Author's) by ſuppoſing y* + a*y — 24 „At. 100. = © ®, we have à or ax® for the firſt Term of the Root; whoſe Di- mention of x, viz. & = 1, being put for y, in the Terms of the propoſed Equation, the Exponents of x in the ſeveral Terms ſhall be o, o, o, 1, 3, Where the Difference betwixt o and 1, viz. 1. mea- ſures the Difference betwixt 1 and 3, wis. 2, therefore I conclude that the Exponents of the Powers of x in the Terms of the Quotient or Series, proceed in an Arithmetical Progreſſion, whoſe Difference Is 1. | Ex. SP . SP * propoſed : by feigning cy — c =o ®; you find A- c for the firſt, t. 15 Term of a converging Series; wherefore ſubſtitue x for y, you have EX-P:LATNED '.;. a4 Ex. 2. Let the Equation x%+ +. c*y+ — x43* ＋ c — C =0© be theſe Exponents of the Powers of x, viz. 2, o, 4, 2, o; which te- duced to Order, and the coincident ones thrown into one, gives this Progreſſion of diſtin Terms o, 2, 4, where 2 — 0 ( = 2) mea- ſures 4 — 2 (= 2) therefore the Exponents of the Powers of æ in the Terms of the Root, go on in an Arithmetical Progreſſion, having 2 for the common Difference. | ane > Ex. 3. Let the Equation y5 — by* + gbx* — x3 = o be propoſed: then you may form the firſt Term of the Root either from the fictitious Equation y5 — ) So; or — by* 4-gbx*=0o *: the firſt gives y= Art. 1oĩ0. VV or Vbxx*; wherefore x* being inſerted every where for y, the 2822 different Exponents & x in the ſeveral Terms, will be o. 2. 3: here 2 being the Difference betwixt the leaſt Exponents, meaſures not the Difference 3 — 2 = 1, therefore the Exponents of x in the Terms of the Series will not proceed in an Arithmetical Progreſſion. But if the firſt Term of the Root be found from the other fictitious = = N * , 4 8 "ZY » * 4 8 5 p = ” 4 * 1 A - 1 * = CY N * 8 4+ —_= = FA 4 1 i. 1 Tc * * * - F4 N Equation — 4 ＋ g6x*—=0; it is 128 zx: where the Di- or menſion of x, viz. x*, being inſerted for y gives theſe Exponents of x in the Terms of the propoſed Equation, viz. 5. 2. 3; which ſhews that 3—2=1, meaſuring 5 — 3 , the Exponents of x in the Terms of this Root, will conſtitute an Arithmetical Progreſſion, whoſe common Difference is 1. | | Ex. 4. If f + * —ax3 — a = © be propoſed, the firſt Term of the Root may be derived from putting * — 4 & , which will Art. 105. be alxł; inſert x+ for y, and the Dimenfions of x are 1. . 3. where the Difference betwixt the two leaſt 1 and 3, doth not meaſure the Difference betwixt 3 and Þ = 32, therefore the Exponents of & in the Terms of the Quotient will not proceed in an Arithmetical K * 3x7 | 7 Progreſſion: for the Root is a*x+ - TT — 5 7s 320 Sc. The Proceſs of which, according to this Method, I ſhall here ſet down oe, the Reader's Satisfaction. Putting then y = aix+ +, we have as follows : | 1 Ll1lz2 + + ANALYSIS by Equations, &c, + Ale = + auf + 4a = 0 2 TI = A botxip A- 4al xh . datxip— ar o ® fire | | x} Art. 120. 7 + Jai + 7 Rem. ;. In this Proceſs the Reader will obſerve, that I have omitted all thoſe Quantities which cannot affe& the Quotient within the Limit of that Power of x which is denoted by x7 incluſive, which Quantities may be „Art. 82. eaſily known and diſtinguiſhed by what was formerly ſhewn *, eſpecially when joined with the Remarks in Art. 120, 123, Having illuſtrated this Problem by Examples, I come now to give the Demonſtration of it. Wherefore let ay” + bxty + cx Sc. + Do repreſent any ſpecious Equation as formerly: from which: the firſt Term of the Series being found by any of the preced- ing Methods, let the Equation be transformed into the firſt ſupple- mental Equation, called that of ; in which Equation of p, let the Exponents of the Powers of x in the ultimate Term be A, A, B, C, E, Ult. Penult. N ; * &. ASD B BAD 8 4 2. Antepen. A2 B ＋ 2D E ＋ 2D Sc. as in the Margin: of which two at leaſt muſt be equal, which are here repreſented by A and A: the Parts of the ultimate Term correſponding to which, do always deſtroy one another . Then ſup- poſe . LW WP UV . w NT . SV 1 to the penult Term; EXPLAINED. - 448 poſe the Exponent of the Power of x, which is contained in the firſt Term of the Value of p, found as above, to be D: fo that after Sub- ſtitution of ſuch Power of x inſtead of þ in the ſeveral Terms of the transformed Equation, viz. that of p, the Exponents of the Powers of x in the Penult Term may be A D, B D, EAD Ge. in the Antepenult A 2D, B 2D, Sc. as in the Margin: for where there are Parts of theſe Terms of the Equation. of p upon any horizontal Line, the Exponents of x in them, muſt be ſuch as here repreſented, as appears by Remark 3 and 4, Art. 120. Wherefore it is evident, that, when the Equation of p is transformed into the ſecond ſupple- mentary Equation, viz. that of g, the ultimate Term of this new uation can contain no other Exponents of x but A D, A + 2D, Az, Ge. B. BD, B-+2D &c. C, CD, Ge. E, E-+D &c. But in the Equation of p, after throwing by A, the Exponent B is ſuppoſed to be the leaſt of all the remaining ones B, C, E, &c. in“ the ultimate Term, therefore A DB“; conſequently A + 2D *Art.120. BTD, A+3D= B+2D Fe. ſo that in the Equation of * there ſhall be no other Exponents of & in the ultimate Term but what may be deſigned thus, B, A D, A 2D, A+3D, &c. C, CD, C 2D, Sc. E, ED, ET 2D, Gc. as in the Margin; of which B and A D Being equal, the Parts of the ultimate Term correſponding to theſe muſt deſtroy one ano- Ult. . Penult. Antepen. ther, as before. Now after N 8 throwing by theſe, let A+2D A#D ATT 2D be the loweſt Quantity in the A 2D A 3D A+4D ultimate Term, then it is evi- A ＋ 3D A 4D A 5D Ge. dent that oppoſite to A + D, Ge. &c. Sc. in this Equation of 9, there C «+ Mi] | muſt be a Part or Quantity C+D C+z2D | +2D C+3D C+4D Oc. Sc. Sc. +D EEÆ 2D + 2D Sc. C but no Part belonging to any Term preceding the penult, E becauſe it is the Part of the E Equation of Þ which ariſes E by the Transformation of the Part of the penult Term of the | Equation of p, which correſponds to AD, in which the Supplement E+3D E+4D+ &c. Sc. — p is in the firſt Power. And if you ſuppoſe that Power of æ to be inſerted for 9, which is contained in the approximate Value of 4 found as above, the Exponent of x in this Part of the penult Term, muſt 440 ANALYSIS by Eavartions, &c. muſt be A + 2D, iz. the ſame as it is in that Part of the ultimate * Ibid. Art. 120. Rem, 3, 4 Term where it is loweſt “: conſequently the Exponents of x in the other Parts of the Penult Term, where there are ſuch Parts, muſt exceed the Exponents of x in the correſponding Parts of the ultimate Term, by the ſame Difference D, after Subſtitution for q : and the Exponents of x in the Antepenult Term, muſt exceed thoſe in the Penult by the ſame Difference ſtill, vig. D; where there are ſuch Parts belonging to the Antepenult Term: and ſo on for the reſt of the Terms “, as yon ſee repreſented in the Margin. And thus, by ſup- poſing the Equation of q to be transformed into the new ſupplemen- tal Equation, viz. that of 7; and fo the Transformation to go on continually, it appears that the ultimate Term of every ſuch Equation in inf. can contain no other Exponents of x but what fall in with ſome one or other of the Arithmetical Progreſſions A, A - D, A * 2D, A 4 zb, Ge. C, C D, C I 2b, &. E, E + D, &. and that therefore if the firſt Progreffion A, A + D=B, Sc. fall any ſupplemental Equations wil upon C, E, &c. the loweſt * — of x in the ultimate Term of exceed the loweſt Exponent of x in the ultimate Term of the immediately preceding ſupplemental Equa- Art. 105. tion, by the conſtant Difference D B — A : therefore the Ex nents of the Powers of x in the converging Series for the Root y of the propoſed Equation, ought likewiſe 'to proceed in an Arithmetical Progreflion (by Rem. 6. above) arid that' having the ſame Difference D or B — A (by Rem. 7.) 5 1 TO | But if the Atithmetical Ptogreffign A, A 4D=B, A4+2D, Sc. do not fall upon the Terms C, E, Gc. contained in the ultimate Term of the Equation of p, it appears from what has been ſaid, that the Exponents of x, in the Terms of the Series which expreſſes the Root y, will not proceed in an Arithmetical Progteſſion; but the Dif- ference of the Exponents, after certain Intervals, become leſs. And therefore, fince the ultimate Term of the Equation of þ, is the very fame with what ariſes by ſubſtitating the firſt Term of the converging Series in place of y, throughout the 'Terms of the propoſed Equation, the Thing we propoſed is demonſtrated, fl. 123. Schol. It makes no Difference in the Demonſtration, from whatever Terms ſelected out of the propoſed Equation, the firſt Term of the Series be deduced, provided they be proper Terms for this Pur- poſe : although the preceding Demonſtration repreſents the firſt Term as deduced by the third Method preceding *. wks. 124. Moreover it may be proper to advertiſe the Reader, that, al- though the Difference D = B — A do meaſure the other Differences C—B, 1 EXPEALNE Div | 'C—B, E- C, &c. yet it may happen, that the known Coefficient of one or more Terms of the Series for the Root of the propoſed Equation, may become equal to nothing: which may bappen from certain Parts of the ultimate Term of any ſecondary. or ſupplemental Equation deſtroying one another, ſooner than they ought to do: which may make a Chaſm in the Arithmetical Progreſſion: of the Exponents of x in the Series for the Root of the propoſed Equation, when that happens; but the Exponents of the Powers of * through all the Terms of the Series, even in ſuch a Caſe, muſt belong to the Arithmetical Progreffion mentioned in this Problem. And the Pro- blem muſt therefore be underſtood to be taken with this Limitation. 125. And it may likewiſe happen that all the Parts, or Quantities, conſtituting the- ultimate Term of any fupplemental Equation, ma 447 | Y totally deſtroy one another: in which Caſe, the Series for the Root of the propoſed Equation terminates ; and that Root is fully expreſſed by the Terms of the Quotient already found: fince the remaining Sup- plement vaniſhes “. 126. Beſides the two Methods I have“ already mentioned, and ex- plained, for extracting the Roots of affected ſpecious Equations, con- taining two variable and unknown Quantities, by Means of infinite Se- ries; there is a third. which T ſhall explain a little, leſt the Reader ſhould find Fault with the omitting of it entirely. It conſiſts in the Aſſumption of a Series of Terms with indeterminate Coefficients, in which the Exponents of the Powers of x praceed in an Arithmetical Progteſſion, either aſcending or deſcending, according as x is ſuppoſed to be very ſmall, or very great, in Compariſon of the other known Quantities in the given Equation.” Such a Series may be repreſented thus Ax" + BY + CX + Dx"+3, &c. which Series is ſuppoſed to expreſs the Root of the propoſed Equation ; and therefore to be equal to y. And in order to determine the Values of the Coefficients A, B, C, D, Sc. the propoſed Equation is transformed into a new Equation, by ſubſtituting Ax + Bat? - Cxv+#, &c. inſtead of y; and the Powers of it inſtead of the like Powers of , in the Terms of the propoſed Equation ; whence there ariſes a new Equation, includ- ing only one variable Quantity, viz. x : of which Equation all the Terms together being equal to nothing; each Term by itſelf muſt be equal to nothing; that is the complex Coefficient of each Term muſt each Term equal to nothing, there will ariſe ſo many Equations, by the Reſolution of which, the Values of the indeterminate Coefficients A, B, C, &c. may be found : and conſequently the Root of the | propoſed * Art; 75. Rem. 3. be equal to nothing: ſo that by putting the complex Coefficient of ANALYSIS by Bavations, Ge. propoſed Equation; provided the affumed Seris be taken of « right Mw. ver 127. To illuſtrate the Thing by an Example, let the Equation 93 + a*y + axy — 243 — x3 =0© be propoſed ; to find y: then aſ- ſume y AT B x + Cx* + Dx3, &c. and let the given Equation be transformed as you ſee | : = + A3 + 3ABx + 3A*Ce* 3A. Dx. + 3A*Ex+ &c.” 7: „ J 3AB) + 6ABC +6aBDr | | ＋ BBB + 3ACC | ＋ 3BBC 448 . + a*y = + @A + &Bx + @&Cx*. DIW Ex“ &c. | | + axy= * '+ ax + aBx* + aCx3 f aD! &c. — 24 = — 24 a — X — * * * — X3 4 In which transformed Equation, the complex Coefficient of each Term muſt be equal to nothing, otherwiſe the Whole could not be equal to nothing, as we ſhall ſee by and by: therefore, by putting theſe Coefficients equal to nothing, there ariſes ſo many Equations, viz. 1. A3 + @A — 243 = o. 2%. 3A*B +@&B + aA =o. 3% 3A*C + , + 3AB* + aB = o. 49. 3A*D D + 6ABC + BB + aC— io, &c. From the firſt of which find the Value of A : which Value of A being inſerted for it in all the reſt of the Equa- tions; you may, from the ſecond, find the Value of B: which Value of B being inſerted for it in all the ſubſequent Equations, you may, from the third, find the Value of C; and ſo on in inf. From firſt Equation A3 + a*A — 243 = o, you find A=-+ a: from the ſecond you will find B=— 2: from the third, C — .: from the fourth, D = + 7225. Sc. therefore, by ſubſtituting theſe Values of A, B, C, D, Sc. inſtead of them in the aſſumed Series y = A + Bx + Cx* + Dx3, &c. We have y = @ —=+E. 4 . 512 * Art, 78. the ſame as formerly “. | 3 128. The chief Difficulty in this Method is to find the right Form of a Series: for although the Series A + Bx + Cx* ＋ Dx3, Sc. was that which we aſſumed in the preceding Example; and which in- deed is the right Form for finding the Root of the Equation there propoſed ; yet in other Examples, the Series to be affumed, my be of a different a diffecent Form. And it's evident, that the Difference of * | F . " 4 _ - 1 N p ** p T e . . % * N | \ p _ . * . 72 * hp * , OG . : . Y e +, 1 N „ EAP L411 © Bw, > T — A. 9 — 5 N * 4 g — * * 4 "4 . 7 - As depends upon theſe two Things, viz. 1%. The Dimenſion or Power of x in the firſt Term of the Series. 2. The Difference betwixt the Exponents of the Powers of x in any two Terms- of the Series that are next to another. The firſt of theſe Things may be found by any of the Methods that were formerly explained for diſcovering the firſt Term of a converging Series, which (hall be the Root of the propoſed Equation, _* "© | 129. The ſecond Thing, viz. the common Difference of the Ex- ponents, may be found thus: inſert the Power or Dimenſion of & contained in the firſt Term of the Series, in place of y, in the ſeveral Terms of the propoſed ation; then ſet down all the different Ex- nents of x (among which o is included) in order, which let be a, * e, &c. find the greateſt Number, which will exactly meaſure all the Differences þ — 4, b, e— c, &c. call it p; then is p the Difference of the Indexes of x in the Terms of the converging Series: ſo that if be the Exponent of x in the firſt Term of the Series, the right Form of the Series to be aſſumed ſhall be Ax ++ BY Cx-t# ＋ Det, &c. The Thing will appear plainer by Examples. Ex. 1. Let the Equation y + ar + axy — 24* —x* — o, men- tioned juſt now, be propoſed: then by feigning y* + 4 — 24 =o, we have y = + 2 or ++ a; which is the firſt Term of the Root, the propoſed Equation, and the Indexes of the Powers of x are o, o, 1, o, 3; ſo that the different ones, reduced into order, are o, 1, 3, whoſe Differences 1, 2, have 1 ſor their greateſt common Meaſure: therefore A + Bx ＋ Cx* + Dx3, Sc. is the proper Form of the Se- ries to be aſſumed. 3 Ex. 2. Let the Equation ð — 495 ＋ 2axy* — 26ð ) A* 3x y* Dab] + x* + 2ax5 S o, be given: by putting the Terms — abx*y + x+ = ©, you will have + 3 for the firſt Term of a converging Series, when x is very ſmall * : wherefore, inſert x* for y, through Art. 10g. the ſeveral Terms of the propoſed Equation; and the Exponents of x 108. will be theſe, 18, 10, 7, 8, 7, 4, 4, 5, that is the different ones, reduced to order, are 4, 5, 7, 8, 10, 18, whereof the Differences are, 1, 2, 1, 2, 8, whoſe greateſt common Meaſure is 1 : therefore the Form of the Series is Ax + Bx* + Cxt, &c. But if the propoſed Equation remaining the ſame, you deduce the firſt Term of the Series from the fictitious Equation — 4y5 + 24% Mmm . - =—abx*y | 4 ds * the Forms 8 when x is very ſmall * : next inſert x* inſtead of y in the Terms of Art. 100. 450 ANALYSIS % Bavanions, Ce. — he =0 you'll find it to be + V 14 4 7 + 7 Xx or - MB ied” Sabi 1 . ; Art. 10o8.— e T + 16 * ** eh which Eves the firſt 'Term of the Series, fourfold ; but »=+ in all of them: and by inſerting x* for y, the Exponents of the Powers of x will be 72. 21. 2+, 3+. 4. 22 4.5 ; that is the different ones when reduced to Order, are 2+. 27. 4. 5. 71; whoſe Differences are 1. , 1.2% or +, I. 2+, whoſe reateſt common Meaſure is +: Therefore the proper Form of a Series Yor any of theſe Roots will be Axt + Bx + Cx ＋ Dx?, Ge. Ex. 3. Let the Equation yi ＋ pay — 34 + x*y — 8043 =0 be propoſed : by aſſuming the fictitious Equation y3 + gay? — 4a¹² — $043 = ©, and reſolving it, you'll find it to have theſe three = + 5a, — 2a, — Ba; any of which will be the firſt Term of a con- Art. 100. verging Series, when x is very ſmall “: therefore inſert & for y, and the Indexes of the Powers of x in the propoſed Equation, which are different, are none other but o and 2, i oh Difference 2 has 2 for it's greateſt meaſure: whence it follows that A + B 4+ C + Dx& Sec. is the true Form of the Series. Which Example, as to the Me- thod of Operation, is here repreſented to the Reader. + 933 =+þ+ A3 + 34Bx*) + 33A*Cxt+ 3A*Dx* +;3A*Ex* c. + 3AB* ABC —+6ABD | ＋ BBB —+3zACC ＋35˙C + 5ay* =+ 54A? +104ABx*+104ACx*þ+104ADxf-+1o00AEx*&c, 8 ＋ $5aBB -+10aBC -+104BD 2 | ＋ FCC —3 4 — 34A — 34% Bx- —344*Cxt 34a — 34 Ex? &c. II * + Ar Bu C5 + . Dx" & ol Where, by putting the complex Coefficients of each Term of the transformed Equation, equal to nothing, you'll have 1 A3 + 5aA® —— 344*A — 843 == 0. 7, 3A*B =_ 104AB — 344*B + Ao. 3*. 3ZA*C+ 3AB* ＋ 104AC + gaB* — 34 C＋ BS o: and ſo on, for the Coefficients of the ſubſequent Terms. The firſt of theſe Equations, wiz. A* + 5aA* — 344 A — 803 So, is the — $043 = — 8043 ſame with * ſuch, whoſe Dimenſion of x muſt fall in wi EXPLAINED. with the fictitious Equation at firſt aſſumed for wes, oe firſt Term of the Seties, vis. y* - gay* — 34a*y'— 80 (which will always that is A== + 5@ or — 2 or — 84: any of which may be taken, according to the Root that is propoſed to be extracted. Let the po- fitive Value 4- 5@ be taken: which therefore muſt be ſubſtituted in Place of A, in the ſubſequent Equations for diſcovering the Values of B, C, D, Sc. by ſubſtituting it for A in the ſecond Equation, viz. 3A*B + 10 AB — 34 + A = o, it becomes 7 5@*B + 5083 — 345 ＋ 5 o, whence B I Again, by ſubſtituting theſe Values of A and B, in Place of them, in the third Equation for determining C, viz. 3A*C + 3AB* + 104AC + _ — 34 C + B=o, it becomes 75 C + 2 + 50a*C + - 83 — 34 C — 82812 g 0 ; hene you have S ALA 1 After the ame Man- ner, by ſubſtituting theſe known Values of A, B and C, in the fourth Equation, you may diſcover the Value of D, vis. D = + EM and by Means of the fifth Equation, in like Manner, you would find ** * — nd ſo on in 7 Therefore, by rue theſe Values Ex* Ge. =0, we Do —— LY 5 e Se. which is an Approximation to one of 45 Roots of the ret Equation, when æ is very ſmall. 130, The Demonſttation of this practice may be caßly derived from 451 be the Caſe) therefore A will have the ſame Values here, as y had there; what was demonſtrated with reſpect to the preceding Method * : For Art. 122. when any ſpecious Equation, including two unknown Quantities x and '23: y, is propofed, to find the Value of y in a converging Series; of which Series AX” is the fitſt Term; and Ax” is inſerted for y through the Terms of the given Equation ; if the aſſumed Series Az" + BY +. Cy &c. be of ſuch a Form, as that the Dimenſion of x in every one of the'Terms of the propoſed Equation (after ſubſtituting Ax for y in them) fall in with the Dimenſion of x in ſome Term of the aſ- ſumed Series: I fay if this be the Caſe, the young Geometrician will eaſily underſtand, "from what hath been ſhewn in the foreſaid Place, that there can be no Term of the Root of the 17 * poſed Equation, but t the Dimenſion of x Mmma2 in 452 ANALVYSIS by Equations, &c. in ſome Term or other of the aſſumed Series; ſince p, which is the common Difference of the Exponents of x in the aſſumed Series, is a common Meaſure of the Dift-rences of the Dimenſions of x in the Terms of the propoſed Equation, after Subſtitution of Axꝰ for y in them: and ſince p is the greateſt common Meaſure of theſe Differences, there cannot be any other Series aſſumed, having the Exponents of x in an arithmetical Progreſſion, whoſe common Difference is greater, which ſhall have theſe * Properties. Conſequently, if the Values of the indeterminate Coefficients A, B, C, D, Cc. in the aſſumed Series can be found, which we have ſhewn how to do, by the Examples we have adduced, then the ſeveral Terms of the Root of the propoſed Equation, will thereby be found. Which is all that is 7 131. All that the young Geometrician can require to know further, for his Satisfaction with reſpect to the Grounds of this Method of extracting the Roots of Equations, by the Aſſumption of a Series with indetermi- pate Coefficients is, 1. Why the Exponents of x in the Terms of the transformed Equation, muſt conſtitute an arithmetical Progreſſion hav- ing the ſame common Difference as the Exponents of x in the Terms of the aſſumed Series: which is abſolutely neceſſary for finding the Values of A, B, C, D, &c. ? This Queſtion will be eaſily ſatisfied, by conſidering the Nature of the Involution of Powers: by which it appears, that, if there be any Quantity conſiſting of any Number of Terms, finite or infinite, in which the Exponents of the Powers of any Letter as x, are in arithmetical Progreſſion, any perfect Power of that Quantity, as the Square, Cube, Cc. will conſiſt of Terms in which the Exponents of x will conſtitute an arithmetical Progreſſion having that ſame Difference: ſo that any Power of Ax + BY + Cxn+# O Ec. or even any Power of it multiplied by any Power of x, will conſtitute a Series of Terms, in which the Exponents of x will make an arithmetical Progreſſion, having the ſame Difference as in Ax + Bxv+ Sc. viz. p. From which it follows, that the Expo- nents of the Powers of x, in the ſeveral horizontal Lines of the transformed Equation, as in the preceding Examples, will proceed by the fame common Difference in all, where there are more Terms than one upon any ſuch Line: and where there is but one, the Di- menſion of x in it, muſt fall in with the Dimenſion of x in ſome one, or more Quantities, of the other horizontal Lines ; fince the Progreſſion of the Exponents of x, in the Series Ax“ + B + C Se. falls upon the Exponent of x in ſuch Terms by the Hyp. Which Con- fiderations, by themſelves, may furniſh a Proof of this Method. 132. The 7 0 * N — EXPLAINED. © 132. The other Thing he may want to know - (which 1 likewiſe promiſed to ſhew) is this: how it appears that, in any Equation ſuch as X -N + RJ Cc. =o, in which x is a variable or flowing Quantity, and a, 6, c, d, e, , &c. conſtant Quantities, if the Terms contain different Powers of x, the Coefficients of every one of the Terms, that is a—5, c—4d, e—f, &c; muſt be equal to no- thing? The Thing appears thus: divide all the Terms of the Equation, by the loweſt Power of x, which let be , and you'll have a= + == N | FX" —=0: which being true whatever be the Value of x, it is true when x vaniſhes: wherefore, let x vaniſh, and then it will become 4 —b = ©: therefore - X M + i= x x &c. =0: conſequently, by the ſame way -of reaſoning, we prove that c - A o: and ſo on for all the reſt: as was to be ſhewn. 133. The Reader will eaſily perceive that ſometimes the Values of ſame of the indeterminate Coefficients B, C, D, &c. may be equal to nothing : which will always happen when the Exponents of in the Terms of the Series for the Root, do not conſtitute. an arithmetical Progreſſion : which we ſhew'd how to diſcover formerly. In which Caſes, there may perhaps Methods be fallen upon to abridge the Ope- ration, by the Aſſumption of Series's with different Properties, when the Root is propoſed to be carried only to a certain Limit: but upon this, and ſome other uſefu] Things relating to this Method, I will not now inſiſt, leſt I ſhould be thought to be too particular. Only the Reader may obſerve, that this Method of reſolving ſpecious Equations, ſerves for extracting the Roots, as well when the Species x is ſuppoſed very great, as when it is ſuppoſed very ſmall: and the fame Rules will ſerve for determining the Form of the Series to be aſſumed ; only the Exponents of x in the Numerators of it's Terms muſt - conſtitute: a deſcending arithmetical Progreſſion : thus Az” + BxT + CT Se. where n may be poſitive or negative, and p always poſitive. The Operation, in one Example, I ſhall here ſet down. Let the Equation | * + a'y + axy — 243 — x3 D o, be propoſed, to find the Value of y in a converging Series, when x-is very great: the ſame Equation, which hath been propoſed formerly, particularly under this third Me- thod. I take the fictitious Equation y — x3 = o, from whence I find + x for the firſt Term of the Root, and likewiſe of the Series to be aſſumed : and by ſubſtituting x* for y in the ' propoſed Equation, I find 1 for the greateſt common Meaſure of the Exponents of x : whence I know it muſt be y = As + BX + Cx c. Wherefore * © © 454 43—46. ANALYSIS H EqvatioNs, &c. $ =4- of Bx* 4- 3Cx + 3D& + 3ED &L +5 af WH ＋ 3B» 6BC ＋ 6BD . | + BBB 4+3CC, - |. + z3B*C | | = & + ay = * * —Þ+ aax + @Bx? TAC &c. | + axy = * 4 ax* + aBx + aCx*? + ADY Ge. — 243 = * * | * — 24x . . — 3 = — x} h : 2 In which transformed Equation, 1 is inſerted inſtead of A, becauſe it was found already that Ax = 1x: then for deriving the Values of B, C, D, E Ce. we have theſe Equations, 1% 3B+a=o, 2%. 3C+3B* ＋ 2 ＋ aB =o. 3% 3D +6BC+ B + aaB + 420 — 24 =0, 4% ZE + 6BD + 3C* + 3B*C +4 ND. &c. by reſolving of which, you'll find 1“. B=—=— 7 *. .C== — 925 3”. DT 4. E=+, Sc. whence you have y = (Ax + BY + C + Dx + EY &c =) * — 2 + 552 4. *®_ $56, the Root ſought. $1x3 2434+ 134. It may likewiſe be obſerved, that, in the foregoing Methods of reſolving Equations into infinite Series, the Reduction of complex Fractions and Radicals into infinite Series, is virtually included, Thus if it were required to reduce the Fraction Es into an infinite Series of fimple Terms, you ſuppoſe y — EET or (by multiplying by S -x) + y Da: from which you extract the Value of y. After the e Manner if it were propoſed to reduce Va — xx into an infinite Series, you might put y V , that is * = a* — K*: and if 3 1 6 a—x1|+ were propoſed, you may put y A - , whence y+= 2 or y* = — 3a*x + 3ax* — &: and ſo the Values of y being found by the Reſolution of Equations into Series, you'll obtain the Values of the Radicals in ſuch a Series as was propoſed. 135. The preceding Doctrine of the Extraction of the Roots of ſpe- cious Equations, may be conveniently applied to that Operation which goes under the Name of the Reverfron of Series: which is nothing elſe but this: the Value of one of the indefinite Quantities, e. g. y, be- ing expreſt by a Series of ſimple Terms, including different Powers of the EXPLAIMNE D. the other indefinite Quantity x, thence, by a kind of rever/e Opera- tion, the value of x is found, expreſt in 4 Series of 9 2 including different Powers of the Quantity y. By Means of this Operation, the Areas or Lengths of Curves being given in a converg- ing Series, the Baſes or Abſciſſes, and conſequently the Ordinates, of the Curves may be found: as our Author ſhews in Art, 43—46: likewiſe from the Logarithm given, we return to the Ratio of which it is the Logarithm. Let CD (fee the Fig. in the Author Art. 43) be an equilateral Hy- perbola, AB x, a Part of one of the Aſſymptotes, ſo that the Di- ſtance of the Point A from the Centre be called 1; AC and BD being two -Ordinates : then the Area ABDC is the hyperbolical Logarithm of the Ratio of AC to BD, that is of 1 +x to 1 : which Area, as has been ſhewn formerly, is x — 1 ＋ 4x3 — * + 4x5 &c. to that, if ou put / for the known Logarithm of the Ratio of 1 + x to æ, that is, if you put x — * + 4x3 — ix+ Cc. —[=0o, then by the pre- ceding Methods, you may extract the Root x, after the Manner Sir Jſaac teaches in this Art. which is nothing elſe but the Application of his Method of reſolving ſpecious Equations, applied to this particular Caſe : by which you find « =/ + 4/* + 4 + en + 55:15 &c. ſo that the Ratio, of which / was the given Logarithm is that of 1 +/ + 2]* + Tin Sc. to 1, that is according to the common way of ſpeaking, the Logarithm of the Number 1 +/ + i* + 41 Ge. But if the Logarithm that is given be the common Tabular Logarithm, you muſt firſt reduce it to the hyperbolical Logarithm by multiplying dy 2.30258 Sc. which Doctrine we explained at fuller Length in our Notes upon the Quadrature of Curves. Sir Jaac having ſet before the Reader the Reverſion of this Series, by the Method he lays down for extracting the Roots of Equations in this Treatiſe; I ſhall ſhew how it is done by the Aſſumption of a Series with indeterminate Coefficients. Since x is ſuppoſed to be ve ſmall, you'll find by Art. 128, 129, that the Form of the Series m be x AI + BA TC + DA Sc. wherefore transform thus: — — + TAT B, + c + D/ + E* Ge. — 2 = | * — 42A 1 — A ˖· D Sc. 11 T „ +APSABEACH G. . + AB* | % *.::. eb þ T= * 6 6 * SA Ge. 450 ANALYSIS by Equations, &c. Whence you obtain ſo many Equations, for determining the Values of the Coefficients A, B, C, &c. viz. 1%. A—1=0.' 2% B — A= =0. 3%. C—AB NAS o, Sc. by reducing of which you'll find A=1, B=+4, C=, D=#, E= &c. and therefore a= +p3# + +7 pri Se ini. .\ 5"! 136. There is, moreover, another. ingenious Method, which our Author makes uſe of for the Reverſion of Series: which is this ; let 3 $ 7 9. the Equation z r — za Ha- +45 &c. be propoſed, which was given by Mr. James Gregory * : by which the Length of the Arch of a Circle is found, from the Radius » and Tangent # bein given, the Arch being called z : and it is required, converſely, to nd the Tangent ? from the Arch 2 given, in a converging Series extend- ing to the gth Power of z. In order to this, let the given Series Powers, as follow : | | | 175 1 x 2 = 112 r YT f — 3 | t5 t7 #9 Was cu: Bow or tif 7 IF oe We 7 | t * . „ HT Ne. | | | 19 2) — % +#® | * + 1 1 2 Ge. 29 —= ** * * * + £9 Here Tobſerve, that, if the ſecond Line were multiplied by gr, and then added to the firſt, it would make the ſecond Term of the Series, vix.— 4 - to diſappear : by doing which you will obtain this Equa- a 23 a Se a ca 8 t tion 2 += *®—= + 345 3 — #77 x &c then J obſerve, with reſpect to this laſt Equation, that the next Term — £- 2 would I'S 4 be deſtroyed, by multiplying both Sides of the Equation that comes 1 See Com, Epiſt. Ne. 64. 2 gee Com. Epiſt. Ne. 20. . next 44. Li A4ATINED. next in dhe Example, wie, a5 = t% f L g Wt, by gr and then adding the two Equations together ; whence ariſes 2+ 72 4 = t * * — Tur + vr 4 G. Next 1 perceive that the Term — #7-5 in this laſt Equation may be deſtroyed by multiplying, the fourth Equation in the Example, viz. 27 =# 2 12 by r and then adding as before: for hence will ariſe 2 + = fs = Th —— 21 n: finally, by multiplying 2% =?" Gr. by = and adding as before, you'll deſtroy the Term w_ 7175 for chere will ariſe 3 + * ＋ r* 1 +- = Soe. = t, the Equation fought; whereby the Tangent is found from. the Arch; whieh is the fame with Mr. 8 And after the ſame Manner may any other Series be reverted. | OSS, 137. By this Method of extracting the Roots of affected Equations, we are enabled to reſolve Equations though infinitely affected: thus in the Equation : | | — 8 +2*— 44 + 92 — 1621 Ce.) I y X.2* — 22+ +-32® Tine 42 Ge. —* Xx * — 2+ + 2* — 2 &c. Irony + * N= rf = Let the Value of y be required as far as the 4th Dimenſion of z : then becauſe 2 is ſuppoſed to be ſmall, the firſt Term of the Root _ be derived from this fictitious Equation — 8 + yiz* = o*® ; which *Art,1os, therefare-muſt-be + 22: therefore, I conſider to what, Lengt I propoſe to have the Root y extracted, that is, to what Dimenſion o 2; that ſo the unneceſſary Terms may be thrown. by: and it being propoſed to carry the Quote only to the 4th Dimenſion of 2, all tha Terms may be neglected from the beginning, in which the Dimenſion ky See Com. Epiſt, Ns, 20. | : 2 22431 Nun SIC | of 458 Art. 82. 120. ANALYSIS by Equations, &c. of 2, after Subſtitution of 2 for y, exceeds 2 , becauſe they can- not affect the Quote within the propoſed Limit, as appears from what was ſhewn formerly * * : ſo that the Equation to be reſolved, comes to this 2289 — g) + 2%3 — 2%* + 20 — 2 — 2 + 2˙̊⁰ + 2*— 80: out of which the Root y may be extracted by the preceding Methods. 138. Hence alſo the Reader may ſee, how the Root y may be extracted out of this infinite Equation ay + by* + cy3 + a+ &c. = gx + b + ix + kx* &c. the Reſolution of which will give the Theorem for this Purpoſe, publiſhed by that famous Analyſt Mr. Abraham de Motvre, in the Phil. Tranſ. No. 249: and likewiſe, how ſtill more general Theorems may be inveſtigated, for the Solution of Equations in which the indefinite Species x and y are infinitely affected. rity m_ 139. One of the chief Difficulties in the Practice of reſolving affected Equations, by the Aſſumption of an infinite Series with inde- terminate Coefficients, is, the raifing of ſuch an infinite Series to any Power required: and therefore, to ſerve this, as well as other Pur- poſes, I ſhall here ſet down at length the diſtinct Terms, until the 7th, incluſive, of any indefinite Power of the infinite univerſal Series Ax" + Bet + Cort? + DH + Ex + F, + GY Sc. having m for it's Exponent, by Help of our Author's famous Binomial Theorem : by which duly applied we have Ax +Brt}Þ Cot Dent Ente Faure COT Nc.“ 1 A” K d- x Am x Bats +-xA*txC 3 K Fee 2 When the ropoſed Equation, viz. that of y, is transformed (by puttin = + 2z 4 + J into the fupplementary Equation of p, that Part of the penult Term of = Equation of 7, which ſtands oppoſite to the Term + y3z* in the Equation of Y, is + 1223p; in which the Dimenſion of K, viz. Ro is lower. than in any other Part of een Term of the Equation of þ hence it appears, that, in the Preſecution of the Root of the profolcg Equation, the ſeveral Parts of the ultimate Terms of the ſupplementary Equations, muſt always be divided by that Power of x, that is, the Dimenſion of x in them muſt be leſſened by 4 : whence it follows, that, when the Root of the propoſed Equation is to be carried no further than 24, all thoſe Terms of that Equation, may be neglected from the beginning, where the Dimenſion of x riſes above 21 after ſubſtituting 2 for y, in thoſe Terms, See Art. 120. * * ; 5 65 E ＋ P CL AT IN E D. a OO +ExXamzD. EN NA x 2BC = N AM + XA x E _ Ae wk, | 4 CC xt + XXX ASS x 3BC + -xA—xÞ ® =! $ A= , 4 2BE NA XA Lac EN x 43BD X + = X Am—" „ G J m m—l | 2BF Z + X— XA* x4 2CE D + XN XA x cb „ : C N 1 m Ml MWomnZ „—3 3D + EXE XX XAT x ER, + XR * = x x x Ax 5B*C 2 3 | MMI, M2 Mm; Hm * 1 * 3 * 4 * 4 460 Art. 35. ANALYSES h Eayartions, &. By Help of which any Power of the univerſal infinite Series A 4- Ba”+? + Cx"+# Sc. may be raiſed the Length of ſeven Terms, whateyer be the particular Values of the Numbers », p, m, which belong to the Exponents: e. g. if it was required to raiſe the infinite Series A + Bx* + Cx* + D. Gr. to the third Power, you'll have „ o, p 2, m=3: by inſerting of which Values into the general Theorem, it becomes | | A3 . \ A*Bx* 4 2Cxt + 3A Dxꝰ &c. eee Þ eanc e BW. The ſame Theotem will ſerve for extracting any Root of an infinite Series; and univerſally for exhibiting any Power, Root, Dimenſion, or Dignity whatſoever, of ſuch a Series, whatever it's Exponent be, poſitive or negative, integral or fractional : viz. by inſerting ſuch it's Exponent in Place of mm: and although the Exponents of x in the Terms of the Series, decreaſe inſtead of increaſing, it will, notwith- ſtanding, give you any defired Power, by only ſuppoſing p to be ne- gative; and u alſo negative if neceſſary. Which therefore, becauſe of it's great Excellency and Uſefulneſs, I have laid before the Reader, to apply according to Occaſion, The Demonſtration of the Binomial Theorem ſhall conclude this Work, after we have explained a little the firſt and third Obfervatiom under our Author's 4 5th Art. and ſhewn by one Example, how this Doctrine of the Reſolution of affected Equations is to be applied for finding the Areas of Curves. 140. Sit 1/aac takes Notice in the firſt of theſe Obſervations, that, although the Root of the fictitious Equation, by means of which the firſt Term of the converging Series for the Value of y in the propoſed Equation is to be found, were a ſurd Quantity; or wholly unknown, as being perhaps the inextricable Root of an high affected Equation, you may notwithſtanding, even in this Caſe, reſolve the propoſed Equation, provided the Root of the fictitious Equation be not im- poſſible (of which immediately) by ſubſtituting ſome Letter for that unknown Root, and then proceeding with it as before, after the Manner he ſhews. - | . Thus if the Equation 53 — 3a*y + axy ＋ a3 — x3 =© were propoſed, to find the Valde of y in a converging Series of ſimple Terms, when x is very ſmall: I find, by the Methods formerly taught, that there can be no other Equation” aſſumed for finding the firſt Term of the Series, but y3 — 3a*y + a3 o, whoſe Root is a Surd; d e 4 Surd; therefofe, inſtead of the Letter à, which denotes u known inoariable Quantity, 1 ſubſtitute any Number 1 think proper; e. F. +, by which it beeomes'97 = v5 T Ii S, Which 18 t be ect by our Author's Method of veſolving affected fameral Equations: b 1.53208 Ge. ſo chat, Robt of thut fictitious which ene of its Reots will be found y = having ſubſtitated aby Letter, ſuch 4s , fort Equation y3 = J 43 o, and extracted che Root of the pro- poltd Equation 5 3 ＋ Bhy A α e o, after the Manner our Author ſhews, to any deſired Length, you ſubſtitute + 1. 3205 r. 6vety Where fot 5; and 1 for 4, Wilk Wis the Value aflumed for it; and ſo you (hall have the required Number of Terms of a ct. verging Series, for one of the Roots of the propoſed Equation y* — fy u A- ==." Bat, If you put #= r, the propoſ 'fnay be firſt converted inte this 'y* — 4y T wy IL o: which being reſlved- by putting 'þ for the firſt Term of Series, as repteſenting the Root of the fictitious Equatioſ y — 35 1 0, you'll obtain y = 6 + fs + 2 x* + kts wh an Ce. where c ſtands for 3 36: and therefore, having found one of the Roots of 9% — y + 1 e t6 be 1.54208 &c. inſert it Inftead of the fictitious 6: and thus you Fj have fo many initial Terms of the converging Series for the Value of y: in which Value, Unity is, equal to the given Quantity, ot Line, à. * 141. Yet perhaps in fach Oaſes, ſome of the known Species ot Letters belonging to the propoſed Equation, may be retained in the Root : thus in the preceding Equation, having found that the firſt Term of the converging Series onght to be derived from y* = 3a*%y + a o, you may divide the Root of this fictitious Equation by a, atcording to the known Rules of Algebra, by which it will become 3y' = 3y ++ 1 =0, and having found any of the Values of y in this numeral Equation, multiply that Value by a, and you ſhall have an approximate Root of the Vader y' = 34% +a o, for thefirſt Term of the converging Series. A Therefore, all literal Equations are to be freed as much as poſſible frotti the Letters they contain, by the Rules of Algebra: but if, after all, there remaln ohe or mote kfown Letters in the fictitious Equation, frotn Which the firſt Term of the Series is to be derived, and the Root of that Equativh be fufd or w__ unknowyn, you may ſabſtitate Nutnbers for ſuch Lettets: which Numbers muſt bear the ſanie Pro- portion to one another, that the Quantities repreſented by ſuch Letters do bear, eithet exactly, if poſſible, or, if not, as near as may be; . 42 ANALYSTS: by Equations, &c. viz. when there are more than one: but if there be but one, ſuch Letter remaining, you may put any Number fox it, you think Proper: n_ _ you proceed. with the Solution, as you have been directed. I come now to ſpeak a little of that Caſe, Where che Equation to be 1 for deriving the firſt Term of the Series has no real! or poſſible Root, mentioned by the Author in Obſ. 3. Art. 35. In this Caſe he directs to increaſe, or diminiſh, the Quantity x, until you find ſuch a fictitious Pn a vo of; one or mote _ Roots. own AY Arte i Habit wo, ſ) Thus, if the Eq uation . * . * + 25 e o, were propoſed; $7 Equation from which the firſt Term of a conv ing Series ought to be derived, when x is very ſmall, is 51 ＋ 2y* 2y + o; whoſe Roots are all imaginary, therefore I. ſu * x > be increaſed, or . diminiſhed, by any given Quantity, whic you may call a, and then ſubſtitute the Quantity x fo increaſed, or dimi- niſhed, in place of the former x: thus if you put S & - 4, or 2 Ta =x, in the preceding Equation, it becomes y* — 2 —2a2y* Þ wp — &y* a +2 —2y + 1=0 : where, it you ſuppoſe 2 to vaniſh, this Equation reſults 11 — 4 +.ay* * 2y* — 2 +1 So: in which Equation, if there be any Number, which being put in for a, will give an Equation with one or more real and com- menſurable Roots, I inſert ſuch Number, and then take the Root of that Equation for the firſt Term of the converging Series: thus I find that if @ was ſuppoſed. to be 2, the laſt fictitious Equation would be- come 1 — 2y + 1 = ©, in which one of the Values of y is-þ 1: therefore I fuppol 2,+ 2= x, and ſo the transformed Equation y+ — 2*y* — 242)* — ef + * + af + 2y* — 2y + 1=0, is con- verted into y* — 2*y* — 32y* — 2) + 1 =o, where the firſt Term of a converging Series for the Value of y, is + 1; and if you proſecute the Extraction, you'll find y = 1 + - = 4 IP Sc. which therefore is one of the Values of y, 2 inline equal to : —2; and & being ſuppoſed to differ very little from 2. 143. The Reader may conceive the Matter thus : the propoſed Equation defines the Relation betwixt the Abſciſs x of ſome Curve, and it's Ordinate y: now if the Ordinate drawn through the beginning of the Abſciſs, cannot meet the Curve ; it is evident y can have no poſſible Value, when x the Abſciſs is ſuppoſed to vaniſh: therefore the Author's Rule for finding the farſt Term of the Value of y cannot be applied (although perhaps ſome of the other Methods de- monſtrated above may) immediately and directly: yet if you ſuppoſe the che beginning of the Abſciſs to be changed, by removing it to a pro- per and ſufficient Diſtance, the Ordinate paſfing through this new be- ginning of the Abſciſs, will meet the Curve in one or more Points; and then his Method for finding the firſt Term of the Root of the propoſed Equation, may be applied. Which may be done by inſert- ing a + 2, or a — , inſtead of x in the propoſed Equation; by Means of which, a new Equation ariſes, wherein y ſhall have one or more poſſible. Values when z:yaniſhes, if the Diſtance à betwixt the two beginnings of the Abſciſs, be taken of a proper Length. | And thus the Root of the, propoſed Equation may be found in a converg- ing Series, which ſhall converge the faſter the leſs that z is; or (which is the ſame Thing) the leſs that the Difference betwixt x and @ is. 144. It is with a Deſign to explain this Affair that Sir 1/aac intro» duces the Figure belonging to this Article, where it appears, that, when the Abſciſs AC is ſuppoſed to begin at the given Point A, there is no poſſible Value of the Ordinate; yet if that Abſciſs AC be leſſened by the given Line AB; ſo that the Point B be conceived as the Com- mencement of the Abſciſs BC, the Ordinate CD ſhall have a four fold Value, when BC vaniſhes: each of which Values may be the firſt Term of the Root or Quotient. | | a Suppoſe the Equation: to the Curye to be 50). + | RAY 9 — 2V51 X bay? + xy? 25 sr xy * by * = ©: here if you put x o, you have oy + 4+ o; where y has no poſſible Value: but if ybu ſuppoſe * to be diminiſhed! by the given Quantity Vg; or ſuppoſe x = 2 + 45 1x65: by ſubſtituting this for that, the given Equation will become go 50% + —— 295 51 + 2 — 9113. 1 10 15 reſponding to the four Values of the Ordinate, viz. BE, BF, BG, BH. Wherefore any of theſe may be made the firſt Term of a Series for expreſſing the Value of y, that is the Ordinate CD; and the other Terms muſt be found by the preceding Methods: And here it is The that, the nearer. the Ordinate CD is to EBH, ſo much the 3 e 1 1% 0 6. 2% J=T 5: 35. . 4% 5 = 6, cor- faſter muſt any of theſe Series's converge: for when the Points B an S the firſt Terms of the Series's, already found, are the full Values of the Ordinate; and therefore, the leſs the Diſtance BC Z-18, 463 464 ANALY'S 18% Kubarions, &c. Z is, the nearer. will the Lengths of the Ordinate CD approach to theſe : and accordingly the approximate Values of the Area's, thence deduced, wz.;BCDE, BCDF, BCDG, BCDH, will approach the nearer to the Truth, in any given Number of Terms, the leſs that wy ==2 is; or the leſs the Difference betwixt h and x is. 145. Schol. The Reader may. obſerve. from what hath been ſaid, how firſt to laſt, upon this Subje&, that the Roots of ſpecious Equa- tions, including two variable Quantities, may be . and ex- preſied after a variety of different Ways. And if one would deſire to find after how many Ways this may be done, you muſt try what are the Quantities, which being ſubſtituted for the indefinite Species x, in the propoſed Equation, will make it diviſible by y 4 or — any known Quantity; or perhaps by y alone: thus, in the Equation y3 + axy + a?y — x3 — 240 o, if you ſubſtitute + a, 4, — 24, or — V2 x a for x, there will reſult theſe Equations y* ＋ 24% — 3a” o, „ za. , y' —&*y . ö. = o, andy" — Vaxaty d- * = 0, reſpeively ; ; the firſt of which Equations is. diviſible by y— 4=0; the: ſecond by y — Vs * n the third by y. þ 20 ſtored. ET, F = 0; the fourth by y =o, and beſides by y—= Vegi V2 X A a o, andy + RE I ＋ V Xa—0: whence it follows that the known Quantities joined to y, in the. ſeveral, lateral Equations, having their Signs changed, may be the firſt Terms ef ſa. many converging Series's for the Value of y in the propoſed Wy a e —= x3 — 24) == 0; aceording as you ſuppoſe & to differ very little from ＋ or — 4 or = 24, or — V2 Na, which. were the, Quantities ſubſtituted for x © the Reaſon of which is h fince theſe will be Values of y when x is of the, ſuppoſed, Values: and therefore the ſeve- ral Roots may be proſecuted by inſerting 2 ＋T 4, or 2 — 4, of * 24, or M xa for x reſpectively, in the propoſed Equation; ; as has been ſhewn already : after which, x, if you Pleaſo may be re- And theſe you obtain by uppeflag the Differences botwixt K and + 4, x and — 4, x and — 24, » and - — Vi xa, that is 2, to be very ſmall; but you may obtain other Series 's, which ſnall be Roots of the 1 5 Equatian, by ſuppoſing theſe ſame Differences, that 18 2, to be indefinitely great, after the Magner thewn. above, 0 „ EXPLAINED \ all theſe different Series's you will obtain, beſides thoſe that are ob- tained by the Methods formerly taught. Nay moreover, you may ſubſtitate 2 ＋ or — any known Quantity you pleaſe, - for x, from whence an Equation will reſult, in which z being ſuppoſed to be very ſmall, or very great, a converging Series may be derived by any of the Ways which have been explained. Finally, you may obtain the ſame Thing by ſubſtituting any fictitious Values at Pleaſure for the indehite Quantity x, as.@z + 6b2?, Kfz, FS, Sc. and-proceeding with the reſulting Equations as before: fo that there may be infinite —.— Ways of extracting and expreſſing the Roots of ſuch ſpecious uations. | | 1 0 146. I come now to ſhew the Application of this Doctrine of the Extraction of the Roots of affected ſpecious Equations, to the Qua- drature of Curves, in one Example; by which the Reader may ſee how it is to be applied in other Examples of the Quadrature of Curves; for the ſake of which indeed it was, that I was firſt induced to explain this Treatiſe of Analyſis. | | Let DEFGH be a Curve deſcribed with the Ordinate CBcx, mov- ing perpendicularly along the Abſciſs or Baſe AB: call AB =, BC, Be, or Bx =y; and @ denoting a given Line, let the Nature of the Curve be defined by this Equation y3 + | gay — 344*y + x*y — 8043 = o, the D Area 1s required, | i . Drew ruh A the beginning of N 8 ME the Abſciſs, the Ordinate DA): tbenn „ Ing if the Area ABCD, ABcd, or AB«9, 5 x a. be ſought, whoſe initial Limit is DAad; G Dr in the given Equation y3 ＋ 5a — LL 344*y + x*y — 8043 = ©, ſuppoſe x 4 . to .vaniſh, and you have 55 ++ gay* — 344% — 8043 = o, whoſe Roots are AD, Ad, and AJ. In order to find theſe Roots, you may ſuppoſe them divided by the Quantity a, by which the Equation will ſtand thus y3 + 5y* — 34y — 80 =0; in which the Values of y 455 are y ＋ 5. y=—2, y=—8; ſo that the Values of y in the Equation y3 + 5ay* — 344 — 80a? are y = + 5a =AD, y=—20 = Ad and y=— 8a = A9: accordingly, any of theſe three will make the firſt Term of a Series which ſhall be a Root of the propoſed Equation, according as BC, or Bc, or Bx, is the Root required: and in order to diſcoyer the ſubſequent Terms of the Series, we muſt pro- ceed by ſome of the Methods formerly explained. Thus by proſecut- Ooo. N 466 ANALYSIS by Equations, &c. ing the Extraction of the Root amal Term is +s a, you would 2 x+ 6 9718 76184 o ind y=+ 5 - mp fan i a = BC. Wherefore by our Author's ſecond Rule the Curvilinear Area x3 $ 647037 ABCD = 1 Y 7 = — Sc. hb converges the faſter the leſs that x (= AB) is. After the ſame Manner, by proſecuting the Extraction of the Roots whoſe firſt Terms are — 24 = Ag, and — Ba — AJ, go would find * 114 4 = - 24 — 1 — = Ge. = Be, and y = — 84 + = + &c. Be: and therefore the Area ABcd = = — 24x — Je Gas Sc. and the Area AB. = — Bax — 4. = 3X 39@ 5* 39a Sc. which Areas lying below the Baſe AB, are repreſented negative; and therefore their true Values are obtained by changing the Signs of all the Terms: which Expreſſions are ſo much the nearer the Truth, the leſs that x (= AB) is. 147. But beſides the foregoing Value of y, expreſt by Series's which IP when x is very ſmall ; there is another Value of y, the firſt Term of which is found LY afluming the fictitious Equation + x*y — 8043 So, v2, + , which agrees to the Suppoſition of x being very great: and. which Value muſt denote the Ordinate BC lying above the Baſe, fince the firſt Term is poſitive : this Root may be further proſecuted, and it will give y == + Boa? 4. 272045 * ** a — &c. = BC: which converges only when, x Is great in Compariſon of a; and, by the Dimenſions of x increaſing « contin in the Denominatars of the Terms, ſhews that the Line AB is an Aſſymptote to that Branch of the Curve, viz. DCE. And the Area derived from this Exprefſion of the Ordinate (by the Author's fecond Rule) is — r — f + er Sc. which denotes the Area BCa lying upon the other Side of 1 = Ordinate, and adjacent to the Abſciſs infinitely extended with the Curve upon that Side. But upon the account of that Poſition, the Expreſſion gives it negative; and therefore the Signs of the Terms 1 be changed: ; but it wil only be of Uſe, £44 x (AB) is very great. 148. But if the curvilinear Area i Im A, adjacent to any given Part of = Baſe z /, be required, then you may find the Areas An d and 6048047 +. 22 631 —_— xs EEX LANE and Ad ſeparately, according to Art. 140, provided the Scries's by which theſe Areas are expreſſed, converge ſufficientiy; and ſubtract the latter from the former: or if theſe Serics's do not converge ſuſſicient- ly, then find the Arcas i A a and Im according to Art. 147, if the Series's for theſe Areas converge ſufficiently, and ſubtract the one from the other. But if none of theſe two Methods will ſerve your Purpoſe, you muſt ſuppoſe the beginning of the Abſciſs to be changed, and brought to ſome other Point, ſuch as you find to be moſt con- venient for the Purpoſe, In chafing this new beginning of the Abſciſs, it is proper to enquire 'whether there be any known Length, which ſubſtituted for x in the given Equation: y3 + gay* — 344% 4- */ — 8043 —o©, will produce an Equation having a commenſurable Root, at leaſt a Root which is not involved in a complex Radical Expreſſion, according to Art. 145, by which given —＋ 1 if the Abſciſs x be ſhortned, by ſubſtituting 2 + that Length the Root of the new / transformed Equation extracted, by ſuppoſing z to be very little, the Root may converge ſufficiently for your Purpoſe: in which Caſe, you ſhould proceed by ſuppoſing the Abſciſs x to be ſhortened by that known Length. But if this cannot be obtained, you muſt ſubſtitute z + any known Length for x, that will produce an Equation ſuch as is mentioned, and reſolved Art. 140, 141: which known Length ſhould be taken as convenient for the Purpoſe as poſſible. | | | 149. That this may be explained by the Example before us; let it be Az= ba, and A — 8a the Area iim might be found b finding the Areas Azkd and Almd ſeparately, it the Series whi ariſes by ſubtracting the two Series's denoting theſe two Areas, did 467 ſufficiently converge: but you would find it by Trial to converge ſo flowly as to make this Method improper. Wherefore, you may next try to find the Area ;: /m by Means of the Areas i E A and Ima: for finding the Area i & a, ſubſtitute 64 = Az for x in the general Series belonging to this Caſe, Article 147, vis. — — — 7” 60480a7 ' 63168049 . #7 | Boa® 2720. rr which it becomes — 6 — Fs + 504802* — ©3195%q" gc. that is, whenreduced, = 7 e. $x7776 7x279936 © | 749 xa. Then you find the Series for the Value of the Area Ima, by the like Means, by ſubſtituting 84—A/ for x in the fame general Expreſſion; which, after due Reduction, becomes — 10 — $5 +432 — ;724- &c. x“: both which Expreſſions are Oooz negative, 468 ANALYSIS by E vATIORSs, &c. negative, and therefore if you ſubtra& the firſt, which is greateſt, from the laſt, you'll obtain F. . 1757 = 1 T e Ke. X 4. = 4.8 &c. X a* for an approximate Value of the Area ilmk re- quired. "Oy 228 | * 3: ö But if none of the preceding Methods would anſwer (or even allow- ing that any of them did anſwer in ſome Meaſure, but ſo as the Convergency of the Series were very ſlow) you muſt ſuppoſe the begin- ning of the Abſciſs to be 1 ſo as it may fall in with the Point i or , or not be far diſtant from them, by the Method pointed out in Art. 145: but ſince I do not find any known Quantity ſuch as za, 6a, 7a, 8a, or /35Xa VA Xa &c. which ſubſtituted for x in the given Equation y3 + gay* — 344% + x*y — 8043 = o, would make it diviſible by y — any known Quantity, according to that Art. (which if there were and Icould find that Quantity, and it were ſuch as the Abſciſs being ſhortened by, would bring the beginning of it to be near to, or in the Line i I would ſhorten the Abſciſs by that known Length) therefore you may alter the Beginning of the Abſciſs variouſly ; but moſt conveniently by bringing it to the Point u, which biſſects i J: draw the Ordinate ; then is An = 74a: therefore, calling the new Abſciſs z, to be reckoned from the given Point u, ſubſtitute 7a ＋ 2 for x in the propoſed Equation, and manage it as directed in Art. 142. By ſuch Subſtitution, the propoſed Equation becomes y3 ＋ gay* 4- I5a*y + 14azy + 25 — 8043 =o; from which aſſuming the fictitious Equation y3 + gay* + 154*%y — 8043 =o, I put any letter as þ for it's Root; and ſo extract the Root of the Equation 93 + gay* + 15@y + 14azy + 2 — 8043 =o, in the Manner our Author ſhews Art. 35. Rem. 1. by which you find y — Cy 4 — ——— » — 14abc3 _ 13 6 ſtands for 154 ＋ 10a + 34*, and d for 29400 ＋ g8043#* — 2, And therefore the Area n/mo= bz + 122 + — Tn — + — — ——— _ 2+ &c. Wherefore ſince 21 . but ne- gative, as lying upon the other Side of the Point u, where the Abſciſs S begins, the Area 2 0&7 will be denoted by the ſame Expreſſion, only by making 2 negative, which is done by changing the Signs of the firſt, third, fifth, &c, Terms into their Oppoſites; that is the Area 10 E = — bs T= N ng 2:3 bo . + &c. Therefore, if you ſubtract this negative Value of 20; or add it's 23 E XPLAINE D. it's poſitive Value, which is the oppoſite of the former, you'll have the Area iI mk = ab + SEL ge, And in order to find the Value of ö, you muſt extract the Root of the Equation 5 + | OA + 15y — 80 = ©, by our Author's Method of extracting the oots of numeral Equations, and multiply it by a; by which you'll find þ— 2.42432070 &c. Xx a. So that if you ſubſtitute it for 5, and à for x —= 1, in the preceding Value, you ſhall have i Im — Area required. | 1 Fo. Schol. 1. By thus making the new Beginning of the Abſciſs to fall exactly in the middle of the given Baſe i i, the Series will be made to converge faſter than if you made it to fall in the Point i or /; and it is likewiſe more convenient for Calculation, than if it were brought to any other Point; ſince the ſame Series ſerves for the two different Parts IMO and niko, with the Change only of the Signs of ſome Terms: by the Addition of which Series to itſelf with ſuch Change of Signs, the reſulting Series, which expreſſes the Area i Im will con- verge the more quickly. | 152. Schol. 2. The Reader may obſerve that this Curve belongs to 469 the 4oth Species of our Author's Enumeration of Lines of the third | Order : and that the Areas. belonging to the different Branches of the Curve, might much more eaſily be found. by Prop. 5. of the Quad. of Curves, by taking the Abſciſs upon AD perpendicular to AB. It may be obſerved likewiſe that Cor. 7 and 8. Prop. 9. ibid. may be of conſiderable Uſe ; and when Occaſion requires, ought to be ap- lied, for reducing affected Equations, either to ſuch as are not af- ed ; or ſuch as are leſs affected, before we attempt to find the Areas of thoſe Curves, the Relations of whoſe Ordinates and Abſciſſes are defined by ſuch Equations. | 153. Schol. 3. With reſpe& to the Doctrine of the Extraction of the Roots of affected ſpecious Equations, contained in this Section, it: is to be obſerved, that although, in order to find an infinite converging; Series, which approaches continually to the true Value of the Root y; the Quantity x muſt, be ſuppoſed to be indefinitely ſmall; or inde-- finitely great, in Compariſon of any other known Quantity. in the: Equation, as appears from the Methods of extracting theſe. Roots ;. yet after ſuch Series for the Root is thus inveſtigated, it will ſtill de- note the Value of y, whatever be the Value of x : only the Series will not converge ſo faſt; or perhaps net at all. This appears from the: Analogy 470 ANALYSIS by FEqvaTtions, &c. Analogy of the Thing, eſpecially when x and y are conceived as de- noting the Abſciſs and Ordinate of a Curve. Only, in theſe Equations, the Value of x may be ſo great, or ſo ſmall, as to render the Root impoſſi- ble: which will happen when the Ordinate y cannot meet that Branch of the Curve, to which the infinite Equation belongs: and in this Caſe the Aggregate of all the Terms of the Series becomes infinitely great. —_ —_— —_— r Containing the Inveſtigation and Demonſtration of the Binomial Theorem. e HE Demonſtration of this famous Theorem I promiſed to give Art. 45, having there illuſtrated it by ſeveral Examples. 154. I fay then that PP =P" x my m—7 va tees WY - * err — 0-2 7.37 | 5 X Xx — Ec. 3 25 zu —— — 3 or PF +*AQ+*=BQ D +2 DQ &c. where P+ PQ denotes the Quantity whoſe Power, Root, or Dimenſion whatſo- ever, is to be inveſtigated; P the firſt Term, Q the Quotient of all the other Terms divided by the firſt ; = the Index or Exponent of the Power, Root or Dimenſion whatſoever, which denotes any Num- ber whatſoever, integral or fractional, poſitive or negative. Moreover, in the ſecond Form the Capitals A, B, C, Cc. denote the firſt, ſecond, third, &c. Terms of the Series. It likewiſe appears evident that both the Forms amount to the fame Thing: only when it is to be applied to particular Examples, the firſt Form may be more con- venient for continuing the Terms of the Series, by ſhewing more evidently the Law of the Progreſſion; the other is more convenient for ſumming up the Terms of the Series, when the Letters have the Numbers which denote their particular Values, put in Place of them. 155. Caſe 1. When » =1 and m7 poſitive: then it ſhall appear in this Shape P-+ PQ —=P" x1 Tua U e. | | * * Sun EXPLAINE D. In order to demonſtrate this firſt Caſe of the Prop. let 1 + de- note a fimple Binomial having Unity for it's firſt Term and x for the other; and let j=m+1: then if IT =1 ＋ mx + N — M—1 * + mx K Sc. I fay I+x} =1-+ x + þX H For it is evident from the Involution of Powers by Muttplication, that the ſeveral Terms of the Power of 14-x whoſe Exponent is pn A1, will be had by ſetting down in one Line the ſeveral Terms of the Power of 1-+-x, whoſe Exponent is in, in Ordet, and then theſe ſeveral Terms, each multiplied by x, in the ſame Order in another Line, advancing each Term one Step, and ſo adding the two Lines together as follows. | Oo 1 l I+mx + mX x TAX * Ge. Ix + uae? = + mx * Sc. od _ —— r —— — Sum. INA ＋ m x —x N + mx * x. Ge. Here the Conſtitution of the Coefficient of any Term of the Sum is manifeſt; which may be generally expreſſed thus: NMX — IX — 2 X — 23 XxX — Se. m X M—I X M—2 X t#-—7 Sc. hi h —— * „ which I F 5 ME 1x 2 | — — 8 * o e ay is = LEE , e where the Number of Factors in the Numerators and Denominators of the Terms muſt be the ſame; the Number of Factors in the firſt Term of the firſt Side of the Equation, the fame with thoſe in the Term ſtanding upon the other Side of the Equation ; but the Factors in the Numerator and Deno- minator of the fecond Term of the firſt Side of the Equation, one fewer. T 4, Bg: . For by inſerting #—1 for r in the firſt Side of the Equation, that Side ſtands thus LD. oo pix EN. — ——_— 1 * 2 X 3 * 4 x 5 Se. 1. X 2 K 3. * 4. Sc, — * a — 3 0 — — — — _ — - — — — ——— r „ wil BF nn Finn Uf nn Pe nf SET ONS 77 7 1 4 5 phony NV N 3 % 5 1 —— And it is evident that this will be troe, whatever 471 — — — 2 — n 472 ANALYSIS by Equations, &c. whatever be the Number of Factors by whoſe Multiplication the Terms of the Equation are made up, according to the forementioned Condi- tions. Now let us ſuppoſe m , then it is true that 14+ x "=1+ mx m X — x*+m X _ NO Sc. = 1+2x+x?, all the Terms after the third vaniſhing, becauſe they involve the Factor —= o: therefore 5 n being 3, it follows that IP = 1-þpx ＋ þ X — + þ NEE, And by the fame way of reaſon- ing if 9=p+1 then TT Le + gr, X Y + * * x = 2 and ſo on continually, Whence it appears that, if n be made to ſtand for any poſitive Integer what- ſoever, it ſhall be true that 1 PM = 1 + mx + mx D + m x * h N R Rr Ge. Now the Letter x may ſtand for any Quantity whatſoever, where- fore 1 + D —=14+mQ+ mx*=Q* + m * — * —Q3 + mn 2 8N—2 * — a wh. —Q &c. conſequently P + PQ! = (P' X INL —=)P”" x O bm 2 x * x 20 &c. Which is the firſt Caſe of the Prop. | 3PM, 1d 156. Schol. 1. Q may denote a Quantity conſiſting of any Number of Terms as a +6 + c +d &c. yet if we know how to raiſe the ſimple Binomial x1 + 4 to any poſitive integral Power, in which Caſe Q is only one Term; the ſame Rule will ſerve to raiſe the Quantity IAT &c. to any ſuch Power, where Q conſiſts of any Number of Terms whatſoever a + b + c + 4 &c. For 15%. ſuppoſe Q ＋ a Binomial, then Q = a* + 246 + >; Q3 —= a3 + 34% + 3a +83; QM = at + 44% + 64%* + 4463 —+ “: and fo on, by what hath been demonſtrated. Again 29, ſup- pole Q b +c, a Trinomial, you may conceive it as a Binomial, Whoſe two Members are @ and Ie: in which Caſe Q = @ + aa Xb+c + 22 5 1 a3 + 34 XU ＋ + 34 x Ve. 4 b+ : and ſo on for other higher Powers in this Caſe: and thus, by juppoting the Number of Terms of which Q conſiſts, to be in- creaſed continually, you may perceive by what Means the NN | * r une a Poy * as 255 . gude in Arg d 3g. e # t £06. the Series h 2 FER ole Sanger che Nane F Terms be one Bes dh the, ONES: Kar Uy ft / $==> x 158, Cafe 2.5 When the ha poſitive Fraftion Roper or improper, the Theorem will appear in M 0 Föf nn NN n += NL oF & e For take the fi * 2 + *'a before? TE HN =1 ＋ n Ge. in ie _To main iv aue hit e to be a variable or flowing Quan- ab. ahn = eit Wan Sik bt ug 11d iq the Power w Exponent n, and it is =I e Caſe 1.) 1 fr „„ — = ae — 3 , erer e AE G = o by- ſuppaſing x. to be very ſmall.-- ſhall Have 1 for the flat Term of a Series for that Root, by aſſuming the fictitious Equation. 3 — 1=0*; the Form of the Series for that Root ſhall bean. y=1 + Bx + Cx* + Dx? E &c. * Wherefore, by Help of Joh Caſe of this Prop. lat the 8 . — 2 —— ů — — (Ü ꝓ— — — [LOI * . 4 ge FE. I * % ˖ —*· * L* * * * Of: ==. @,-be. trans farmed thus: 18 66 EE I PN ee 7 B: 2 de tete 2263 Loo99) 2117 ν, N 42 n AX NK 2H &c. 02 las 550 nab h 0 a Horn dringen vnn ieee: Av e rides e 445 id: N 18 fop l DO at hoi 10 larger? au hs 8—1 ml. .. S 373 D „reer 912 2 N * 2 * XX 1 ANA LVS r Fc. which. Frome hf theft e 3 ag Tis | The rear. are-1% yy xD ='0. ., nD + nx = x 2BC + n x x = O0 r 97115 i Eb 4 deli n5: 72 1.8. 871 1 12 — EIS Ge. 185 Mn It „ 171990 Ir 241 15 ae iti 20 3 by, From the fiſt you find B=+' 4 by inferring which Yalue of B into the ſecond, ee i C KC D LE D kad e And ee 5 dhe Valoce, of he e D, Ac. H chaß f the fit 8 b 29191; Io St * A nous) 5 p. 2% "RE Þ 9% B- 1 27 778 2 52 FX EE FIrYl IX Po " 1-25 A1 25 N. an? by inſerting the Values of B as Cin "he id Equation, 1 U becomes e þ- id 22 Gu? . A ; $7 * 2 MO! oO 6. 1 944 7 of my = 9, that is, by reducing, be 3 * 22. | 1 27 "In : and thus you tay proceed to nate vine TP other i —_— of, —— „4 $\ $9 no + 1 2 indeterminate Coefficient in 5 4 80 that it appears that. Tak "OM. i 7” —.—.— _ 7 — | «+ F — =P 1420+ 28 to ESD = I 59; Schol. It is e that the Series in this Send Cafe muſt be infinite; fince no Repetition of u can be equal to m. 160. Caſe the third, When the Exponent M denotes any negative Number integral or fractional, the Theorem will appear In this Form: ao. AM. m r r e er 7 . 1 * | ns * of 14014 Salt, 50 end 600 II ie eb ee | th. e I bade 167 © N 2 1 | 4 1 4 wrt — 0 -A ' . | ewf ag 2 bp 7 5 10 2 + F Wc * if. ene ww 15 FIC} ca | voi 4 g 50 ee e ror which, e. FF 1 BITS =(by Cala rand 2) EN 0 Lb Sam 0 aten N Bald 48 . e 10 As 104.1 142. 0 1 12 divide the Nune 1 += DIST * e . 7 rator of this FraQtion, vis. 4, by the Denominator 144 © —X — "Is * 'Sc: and the . vil come out 9 —— 8 — * „ — „ „% —I— — „ 2 two Bre multiplied into one "ina of Spa 1, as they ddl to do, "he . in 2 — one another. Wbence Ac: in Pol * it * that the n . From all _ it appears, that, whatever be the Value of the nu- meral Exponent = = whether integral Te anc poſitive or negative, wg Ft it will ſtill be true that P TPU =P" . e — Se, of P* + = AQ +*ZBQ_ + . QF. D. We 61. This 4 00 Theorem may be he expeditiouſly inveſtigated and demonſtrated, by the Help of the Doctrine of Fluxions; 1 it is neceſſary that the Rule for finding the Fluxion of any Power, ſect or imperfect, of any flowing Quantity, be demonſtrated other Principles, without making uſe of this Theorem itſelf for that Pu ſe : which our Author does *. * Quad, ow that Rule may be demonſtrated independently of the Binomial _ Theorem thus: let n denote any poſitive whole Number, and x a flow- Wr nnn. P P 2 n ANA LY SIS 25 Eguxttons, Se. In order to demonſtrate which I proceed thus: let the Fluxion of x* be ſuppoſed to be nN, then. if che reren of is px . For * = x* x. x.: _ wherefore, itz denote that Term in the etrical Progreſſion 1, x, x, x, &c. whoſe Diſtante from the firſt is denoted by u, ſo that 2 , then it is „ 2x; therefore their Fluxions are equal: now the Fluxion of xx is zx + a (See is, by r ſt ing 12 5. j — nx is ſuppoſed to be the Fluxion of *) nx"x + x*x or n 4-1 x x"x that is px?*x which therefore is the Fluxion of „, if fo be nx*—'x be the Fluxion of x", Now; by the fame way of reaſoning, if g=p—+ 1, it appears, that, if the Fluxion of = px!—ix, the Fluxion of * ſhall be g9x%”'x and ſo on continually. ., But if 22g, it is demonſtrated that the Fluxion of x* (S*) is n= = 2xx, in the forecited Place: therefore it follows, that, if z be 3, 4, 5, &c. that is any poſitive * whatſoever, the fluxion of & ſhall be nx—x. And from what is ſhewn. in the ſame Place, it appears, that, if-nx"="x be the Fluxion of x, when. u is a poſitive Integer, it ſhall be ſo likewiſe when u denotes any other Number, whether it be a negatiye Integer; r poſitive or negative Frachon of any Kind; ſo chat the Rule is de- monſtrated to hold univerſally. n 1562. Cor. Hence it appears that the Fluxion of 1+xl- is » x ITI *; n denoting any numeral Index as before. 5H 163. Upon the preceding Foundation the Binomial Theorem is thus demonſtrated : ſuppoſe IU = I++ Bx + Cx + Dx3 + Ex. Sc. where 7 ſtands for any numeral Index, poſitive or negative, inte- gral or fractional; and the Letters B, C, D, E, &c. ſtand for inde- terminate Coefficients to be found: it being evidefit from the Nature of Powers and Roots, that the aſſumed Series ought to be of the Form here expreſt, and have 1 for it's firſt Term. Now by taking the Fluxions of both Sides of the preceding Equation (by Art. 161, 162.) we have 1X 1 | = Bx + 2Cxx + 3Da*x + 4Ex3zx, &c. and dividing by & ; A“ =B + 2Cx-+ 3D. 4Ex* &c. which being true whatever be the Value of the flowing Quantity x, it muſt be true likewiſe when x vaniſhes: wherefore, let x vaniſh, a you have n x 11 = B, that is 2 =B. Again, aſſuming the laſt Equa- tion, viz. n IF B + 2Cx + 3Dx* + 4Ex3 &c. and put- ting it into Fluxion, you have x x IX TFH # =2Cx +2 x 3Dxx * BE X 2 AT y E D A 5 4 45 3Dxx + 3 x 4Ex*x &c. and if you divide both Sides by x, and then ſuppaly' xjtq! pri: jay before, you fd NN C. Again by aſſuming the Equation n x z—1 x IFS =2C+2x RT | 3 £ 4Ex*. &. (which! ariſes by-dividirig the preceding by the Plug of x) _ putting i into Fluxions, you have a 1 I's = 2x 30% + 2X3 x bes, Se. which, being K 33; ! redgced as before, gises e = Diz'and by-procedd- ing after this Manner, it will be pl to any one that conſiders the 9 ration, that all the Coefficients of the ſubſequent Terms of the umed Series 1 + Bx + Cx* + Dx3 + Ex &c. will be found by the continual Multiplication ſa the Terms of this Series IX AN a7, KE 0D nt Ge. in inf. the Coefticicat” of each 2 3 4 Term of tle Series 3 +Bx + CO + Dr + Bot K. being the Product of as many Terms of the other Series, as it's Place +. at So that by fubſtituting/the Values of B. C, D, Ge. we find 1777 I 07.341 \ 31 2222 ar hann * 5 2 Mes Ge. 464. Whence it appears, from \ what hath been vid x formerly, _ FN = * 2. ä * n = * Qkee. 155 —— or p 4 N =D d WT T5 q. e any numeral Index whatſoever, — A, B, C, &c. ſtanding for the firſt, ſecond, third, &c. Terms in the laſt Form. | 165. What further remains of this Treatiſe of our celebrated Author with ſpect to the Application of the Doctrine of infinite Series and Quadratures to other Purpoſes in Geometry, being either ſufficiently plain in itſelf, to one that underſtands what hath been ſaid-to explain the Foundations upon which thoſe Things are built ; or having, been moſtly explained in the Commentary upon the Quadrature of Curves,, I thought it improper to trouble the Reader, now En this ng: vi Repetitions 0 T to no Purpoſe. G EN ER AL. aa A* LFSIS h Babs ene G.. 4 f wo; : Wit 15420 * Fo i, a 15 di? RP. * * Az. 15 BY =; a % 0 E N E R AL” 80 HO: L FU M13 ncht * 32 = . „iM adiſzipA ad: Sinns IF it Mankind be capable of ſock Beyer of Knowledge, ds the In- 6 — and Diſcoveries of a Sit Jſaac Newton prove they ate; hom much greater Degrees of Knowledge m In we ſuppoſe ſuper jor Intelli ences capable of arriving at, whioſe better Faculties and more extend- ed Views, furniſh them with a larger Stock of Ideas, the only Aa- terials of Knowledge ; and enable them to perceive and diſcern more accurately and quic ly their Agreements and various Relations. And bow mes higher Degrees of Knowledge, ſhall we outſelves arrive to, when we have got beyond the Confines of Mortality, and ſhaken off theſe Tabernacles of Clay, where our Souls bein confined as in a Priſon, are kept from extending and enlarging their Views; and be- ing bore down as by a heavy Weight, are greatt retarded i in their Progreſs in the Ways of Knowledge ?—Then we perceive more and better Objects, and judge and reafon more accurately, quickly, and diſtinctly about them, — Happy Time, when the Diſcovery of Truth ſhall be attended with no ſuch Toi! and Labour, as it often is here ! In this imperfe# State of Things, we are obliged to purſue long Trains of Reaſoning, in order to attain to the Knowledge of thoſe Truths; which, when our Faculties are enlarged and per fected, we ſhall perceive and diſcern as it were at one View z for as then our Know- ledge will be more extenſive, ſo it will be more diſtin and more intuitive: as a facred Writer expreſſes it, New we ſee through a Glaſs darkly, or in a Riddle ; but * 700 to Face . Happy Day! when 1 1 Coy. xiii. 12. BAinopor ya , iar bÞ Zulypalh 3 PET Wobownov. Here the Apoſtle Pau plainly Ane a remerkeble Difference in Sat: 7. , and the future State, 250 che Diſtinanehs of it, and the Manner of attaining it 3 as well as to the Extenſiveneſs of it. He deſcribes our Knowledge in this State, as bein being BEER and dark, by ſaying we know only in an Enigma or Riddle, which is a dark and obſcure Repreſentation of ſome Truth, cloathed in Figure and Metaphor. And further he deſcribes it as — & ioonr7ge, by the intervention of a Speculam or Mirror ; by which we behold the Images or Species's of things ; and not the very themſelves. Whereas 76s, in the future State, we ſhall ſee and know wecawro woe; r that is immediately, by a dire& Intuition of 3 — as they really are in — So that when this Manner of {ering and knowing takes Place, the former ſhall vaniſh or be done away : as the ſame Authar aw in the ſame Place. And by conſidering the whole Deſcription St. Paul gives in this Place, of the Knowledge of the future State, compared with that of the preſent, we ſhall find the Antitheſis to conſiſt in theſe Things, 72. It will be vaſtly greater, more thorough and extenſive. 29. More clear, diſtin& and ſatisfactory. be More immediate and intuitive. 4. Acquired with leſs Pain and Difficulty, Which Things Wuſtrats by a beautiful Variety of Expreſſions. 1 ä c not EXPLAINED. not only our intellectual, but our moral Faculties, ſhall- be extended and enlarged far beyond their preſent Pitch: and that u State begin, where our whole 2 ata in all it's Powers and Fa- ties, ſhall make further and further Progreſs and Advancement, in a Likeneſs to God himſelf, the i and moſt perfect Being. When we ſhall be admitted into the immediate Preſence and beatifical Vi- fion of the Divine Majeſty ; and for ever employed in ſearching, con- templating and —_— is Perfections, and his Works: which will afford an inexhauſtible ure of Knowledge and Delight, encreaſing to endleſs Ages. The E N D. gdf AANM TINA. 4 4 babmoizo 5d Hach n 8 110 z Jedi 289 ino 20 87872 V K NL Ach Bus: Had 303 91q n: dul: bond 11 bagiglas bas e bar 1870 4 Its nt gut} denotes viodw 1yo % giga ti c tοαν,,ꝭ A bas 811014 e bits 191101 24g lac ines ei e NN Roch ban! Te 8 2613 Aslan boa c C4 28124410 8 AY Sole 2 x 4, ts Wend — 5 a" £2145: be of ite» =" : , | \ * 1 TOY > — e 2 ett i 6 Þ 4: — * * r „ten ? 7 7 FD on | — 2 9 | * de al MA 10 11 5 E. TW itt * 2 Coll 7 I 1 CHIC 145 bt * 1 \ l! 44 ati 1221 10 Feri, 9 70tto . ti _ 5 Abos 98.4) 1 0 u IT He. a bien. 0 if a) © 4% 22A £ AA . 2 Law 4 aA 7 T7 x E. R R A T. A. la the Nuachage-r of Curves and Commentaly. © AG. 7 eee. p. 87. —— _ — © = P- 10. . J. 5. . p. * I. 27. _ 222 Po Thid. I. 21. Curves r. Caſes 35: 39 annex the N** Eck e. id. 2 1 orm 1, throughout all the cies's in the Denom* of Vas of yr. throu hout Prat of the the Cori te Tat v + — — ral Tbid. AN Ro in the Ares, r. aDGa Ibid. Species 3. r. zaDGa 27. Form 6. Species n. Ord. of the ſecond Conic Section r. ae Ibid Form 7. Species 1. in the Denopminator Wart” 48. L8. 8 ſider, and this cannot be altered or influ- enced by any former or future Fluxion of the Abſciſs 2. I. 1. . CE 55. I. 14. 7. n 56. 1.6, * Er 110 r. 2th 2 r. one uniformly flowing Quantity Fluxion of another flowing Quan- 67 ese 47 x 46, and 46 r. 51 68.1. 32. r. 15-—45 | — I. 5. r. ax 0? 71. J. 3. 7. 77. I. 35. dele &c. 78. I. 8. r. 2cxx==0 p. 79. I. 35. Times 1. Terms 81. 1. 38. or r. and 82. 1. =7 of the Nn 7. Mr. David Gre. gory 83.1. 28. BF. V. BC 87. in the ſecond Term of both Forms of + the Ordinate 2A r. An 168. 1. 8. R 88. 1. 22, r. ofTRAt Pete 9. I. 2. r. 26-44 ;) 91. F 95. 1. 8. nge c 99. 1. 11. Abs fi 101. I. 15. but . although. | Ibid. in 32. . may be finite 103.1. 28. 1. 109. 1. 12. x. ot jos Ibid. I. 30. Fraction 2. Fluxion | 110, I. 28. . above thrice 112. I. 1.1, — x3 117. L 19. r. 21 * 119. 1. 17. r. be not 121. I. 29. r. 2 "bs. 122. I. 26. r. may deen! 124. 1. 29. r. demonſtrated 132. I. 29. Forms . Terms, Terms r. Form Ibid, 1. 9. . 145. J. 3. 5. extending 146. J. 13. r. — 147. L 13. r. 4 * W- 148. 14 ABCP 150. I. 27. r, Now although the Series's above would converge very ſlowly, or not 2 e or x & A | 151. 1. 1 153. in 1 der 165. 1. 30. r. — 0 Ibid. 4 r. g Ibid. I. penult & L 5. r. NH Ibid. 1. 10.7 8 170. I. 14. 15 173. I. 32. r. ＋ card 1 55 I 78. l.5.r. "I 1 180, 14. r. D x TF TN 181. I. 7. r. TDA me Fact * 182, I. 2. . firſt, . and fourth 254 I. 29. r. 1 d. . KR 15.30 * 1 46.uf- r. for 196. 1. ult. after negative, r. ( W cepted) 199. 1. 8. - HE Form 201. on the Margin oppoſite to Art. ar. Form 1 P · 208. 1. 13. r. aDB 5 209. 1. 10. r. + 2 f 216. 1. 11. fe. 220. 1.4. r. = x 90 = 3s 222. 1. penult. r. z*” ra 230. I. 24. r. will be an Aſſymptote Ibid, 1. 25. are r. and | 231. 1.30. AO r. IO 233. 1. 14, 15, 17. JOHN r. JONH l L av. . T ; E. Io NH. Ibid, 1. 22. r 5 % 1 4 {4 235. I. 4. 4 r. 4 236. 1. 7. r. the loweſt Order 240. I. 1. . Tay —5 =. BX BC 2 242. L 1. 7. 243.1. 11, 13. BC. AC 244. 1. 34, 36. r. Au EGA M 247. I. 12. 7. the equilateral K 4 248. I. 47. r. the Fluxion of the 2 I. 20. DG, Deg, AG, 9, = De, „ 3 {eee ec Ibid. 1. 31. DG r. DE, and 1. 32 Dg.r. De 1 So. bt» 2 250, „ 14 253. I. 12, 14. Aac r. ABE Ibid. 1. 22. AO? : . AO? Ibid, I. 27. r. A ABE. 254- A 26. 1. 8 25 17. r. by a logarithmical Ibid. 1. — r. is to "=> 258. 1. 25. DB r. B 261. in e Fig. er. o 2 Ibid. I. 33. . MA — | k Ibid. 1. 14, 15. r. . Ibid. fx" r. fx" 267. I. 48. . DG —DH Ibid. I. 20. . VE = AG. _ Ibid. I. 21, 7. DH == f Ibid. 1. 27, — 72 269. I. 1. in in Denominator r. 4=2 | 270.1. 11. in the Denominator r. 4 1 1.5..— 274. 1. 5. v. 4 Ibid. I. 13. r. Difference of x Ibid. I. 18. r. In which Caſe a Difference of any Order is the ſame with the firſt Term of that Difference 277. I. 9.4. when the Fluxions * 1. 9 . 0 3 BR RB A TN. | pan oY — 283. I. 16. 37. 2 Ibid 1.18. 21.3 1886+ 1.86. 75 0p page: I. S. AT Ar . Ibid 1. — n 288. 1. 16. r. le the Arch GH meaſuring the Diſtance of. the Vertex of — 9 the Segment from the Circumference of its Baſe 289. |. 6. . 33=0 wia. l 27 Cr.G 290. . 6. 7. Ordinate BC | 291. 1. 15. r. 0* 292. I. 25. Point T. r. Line BT. 293. 1. 4. BC r. BC r. BT 294. L 32. SE r. S8 —1. Lg. „% Ibid. in the Fig. Dr. y 298. I. 20. K r. * Kia. 1. 26. AB = E 301. I. 1. after you r. have 304. I. penult . — Ex | 305. I. 23. r. A 308. 12 and r. about 7 . 10, 7, reyolving Parabola. id, I. 16. GB. Gg © bid. I. 30 A8r. Gg 314. I. 17. thence r. then 315. 1. 20. r. than an 316. J. 24 92 317. I. 13. in this and the three next Pages the Numbers of the Articles of Quad. of FF N wid. 1. 31. for r. from. In the Analyſis, Sc. and Commentary. $23. L 3. . I(= x=2) 1 at the End of the Line AB gBDF Gi, co ied P 224. EK 5 = 15 K in the Note . 406 — 408 329. I. 5. ＋ 27 —2. 331. I. 1. 4e r. 339. I. 17. r. BD bx Ibid, I. 33. . Quadratrix 341. 1. 10. r. 3x*%0 + 3x0* 346. I. 18. . this Principle 357. I. 26. r. effected 377. I. 16. CD r. BD 378. L 4. . AC the half Tranſverſe and AF the half Conjugate Ibid. I 5. AF. AC, and1.6. AC r. AF. Ibid. I. 16. . Treatiſe Art. 406——41 5 379. 1. 3. after the Extraction of the Root, zax* r. Zax* m m 380, 1. 8. +P" „ = P* p- 381. — —ͤ—ũ——2—ä—ä— 404 AGES: 8 ted A 0. &4 a . ** N — « 2 . * : 2 ® = 1 - p % . , . 1 \ « a. „ A * 13 | 3 , . E R R AT A. | p. 381. Genn Itid 1 DI] tofo tee; aghes ol | wank t ppt . P- l. penult and p. 407. I. 3. r. = I 22 — 4 407. I. 12. 7. ＋ 405 n 5 382. J. 7. at, WI ns 7 — 418. 1. ult. whence 4 when 2 #D ; 12867 420. I. 25. dele cx* 5 83. I. 2. 7. Series. In order to which h ; TR, : 7 may extract the Square Root of the Nu- n= 1a IEA We merator and Denominator, and then Cc. | . 9 - being * Ibid. 1. 78. r. — 423. J. 3. r. „ TV "= } 0 aa x . 384. 4 8. —5r 2 5 . 427. 1. 15. r. + b3 87. I. 12. 7. 2 tive Integer Wy WY; - - ” ” A. I. ult. LY = | 434. 3 6. sli . Ade 41795 wat. » be = - 1 | 438. 32. F. — 0 — 388. I. 4 * 7 Ibid. 1. 5. Eau r. fa 439. L 2 el which do . Ibid. 1. 7. r. — . bid. 1. 8, r. * — 11 here by ay” ; and another one 396. at the Bottom dele Values | 441. I. ult. ». the Series ' 401. 1.22. a, 3x, r. +a, — A 445. 1.2. r. to be ſuch, that, after Subſtitu- 402. |. 22. after , r. Cc. tion of that Power of x, &c. = _ 449. I. 5. . next to one another 404. I. 30. „ 80a * 32 466. I. 27. er 406. I. antepen. 7. + who © Fr — ADVERTISEMENT to the REA DER. In revifing the Sheets after they were printed off, for diſcovering the Errors of the Preſi, I found EF YE. ity of ſome Miftakes, which I B 4 HD to pr out to 1 2 in this Place, with the Correction; and a few other 22 or Aluſtrations. | Ee The Reader may be pleaſed to obſerve that by the Phraſe I»fant of Time made uſe of in the \ following Work, is not meant any Portion of Time, or Part of Duration, how ſmall ſoever, which muſt be diviſible into Parts, like all other kinds of Quantity; but an Inſtant of Time muſt be conceived as having no Parts, and to be with reſpe& to any Portion of Time or Du- ration, what a Point is with Reſpect to a Line in Geometry. F Becauſe Lemma 4. Page 41. may be thought not ſo clearly demonſtrated, the following Demonſtration may be ſubſtituted for the other. | Let V denote any given intermediate Velocity, betwixt M and N any two variable Velocities aſſumed, through which two Velocities, the variable Velocity, mentioned in this Lemma, paſſes ; ſo that M be upon the one Side of V, and N upon the other. Imagine M and N to approach continually towards one another, and towards V, which always lies betwixt them: | | | the Time which the variable Velocity takes in g from M to N, may be ſo ſmall, by the | continual Approach of M and N, as to be leſs than any given Time: by which Means M and N ſhall approach ſo near to one another, as to differ from one another (and conſequently from V) by iel than any given Difference (as appears from the Nature of the variable Velocity mentioned in this Lemma.) Therefore ſince M and N g_ continually to V, during a finite Time, fo as before. the End of that Time, to differ from it by leſs than any given Dif. ference, they muſt at laſt be equal to it (by Art, 7.) But V was taken any intermediate Velo- City whatſoever. Therefore, &fc. as in the Lemma. Q. E. D. | Page 103. Art. 134. The Reader may obſerve that the firſt Paragragh of this Article reſpecta F the Caſe in which = is lefs than 1. In which Caſe there are two Sppoſitions made; either that 3 is greater than 1; or leſs. Now becauſe upon this ſecond Suppoſition, the Demon - ſtration for proving that the Series there mentioned, muſt converge, may not appear ſo plain; it may be more clearly demonſtrated thus. | Again if it be 22 +1, the Quantity —p— muſt, at a finite Diſtance from the Beginning of the Series, become a poſitive Quantity greater than 1 ; and after that Time continually di- miniſh, ſince the Difference betwixt -g and 7-4-:-+1, is a given Number: ſo that the _ | | : | 7 . "BE * * l — . ADVERTISEMENT. Ds BALD Rigs tu I e #0 leſs, that is than the Ratio of + to fe" : therefbre 2 „eee eee 2 Rnd ws the Term pcs, cs 24 continually diminiſhes : and therefore the Series muſt converge. Page 104. laſt Paragraph Art 134. begins thus: ““ There is a 36 Caſe, &ec, In which Caſe the «« Convergency depends entirely upon the Coefficients : '” whereas it ſhould be ſaid that it de- pends upon the Coefficients and the Signs of the Terms, Again where it is ſaid, If A, < both the Series's for the Area will converge, read, One, or both the Series s for the Area will converge, viz. when the Signs of the erms become alternate ; otherwiſe not : Page 2 Art. 410. An Overight having been committed in the Operatian in this Article, the Length of the Curve Line is not rightly e e . 8 let on Article run thus. If we would have the Length of the Guere Line a 8 to what hath been delivered in 5. 10. then (ſee Fig. Art. 355) which A Coles in Plate Fig. 3. Tab 2. Quad. of Curves, ſuppoſing the Hyperbola deter bed with the ſame Latus Tranſverſum and Rectum as before: then it will be 2 22. — — — for Example, ſuppoſe in the given Parabola, the Length of the Ablciſs AB (x) dons, then in the Hyperbola, .by taking aB = (AB'=) I, 8 AB= 43 ; and from the Property of the Hyperbola, and (Se =+#, BC = . BD=4, DC =: the Triangle ABD =) 33% 3 and the Triangle AH (which is the Modylus) or === Fux: Whence you'll have A ABD - — _ 5 =- v l. that is to fay the Meaſure of the Ratio of 3 to 1 to the Modulus. 27; ſubtracted from eres But whereas the Modulus of the Tabular Logarithms is o. 434294 &. (ee Art. 349.) therefore the ſame Quantity expreſſed by Means of the I 22 Tab. Log. will be (ſee Art. 3 50) 154 — 72858 ESE * Log. 118 512x 06434294 &c. x 0.477122 &c, == (by reducing all to Decimals) 1. O nearly, Wherefore if you multiply this laſt Number by 2, you ſhall have 1.622444 the Length of the Curve Line AC . required : which, by Means of a Table of Logarichna of more Placovot Figures, may be carried to a proportionably greater Number of Fi igures. — 439. Art. 120. near the Begi : it is aid that in tho general Equation there men- tioned, there will be one Term at leaſt w doth not include x ; and anather one at leaſt that includes not y. Which is an Ove : for it is true that there will be one Term at leaſt that doth not include x : ſince if all the Terms of the ſed Equation do involve x, you may always free one Term of x by dividing by the loweſt Dimenſion of x : and for the like Reaſon there will be one Term at leaſt that doth not ladude y. But it will not always happen that theſe Terms are diſtin& : for the ſame Term that doth not include x, may be the only one likewiſe that doth not include . However this Overſight makes no Alteration in the Reaſoning there uſed. But it may be corrected by making it run thus: in which there will be one Term at leaſt which doth not include y, which is here repreſented by hi, omitting what is in the middle. Page 445. Art. 123. not far from the Begi of the Article, there is a Miſtake in ſup- poſing the Letter D to repreſent the — — of the Power of x contained in the firſt Term of the Value of p; and then, as a Conſequence of that and what was ſuppoſed: before that, making AD, B+D, ED, Cc. to repreſent the Dimenſions of x in the penult Term. Be- cauſe the Dimenſions of x in the. penult Term will not exceed theſe in the ultimate by ſo great a Difference as is the Dimenſion of x in the firſt Term of the Value of p, as appears by Rem. 3. Art. 120. preceding. Yet this Miſtake doth not affect the Reaſoning + and it may be rectified thus. Inſtead of ſaying, © Then ſuppoſe the Exponent of the Power of x, which is contained in « the firſt Term of the Value of p, found as above, to be D: ſo that after Subſtitution of ſuch « Power of x, &c.“ read thus, Then ſuppoſe the Exponent of the Power of x, which il contained in the firſt Term of the Value of p, found as above, to be ſuch, that, after Subſtitution of that e of x Cc. as it was formerly. And che Reaſoning proceeds and the Symbols wer. PAL — *