i-t V , 4k -- ' S f: & i / M ■'Tg. /if y /f /^>2" T ■ BURNDY LIBRARY Charter* d m I?-#! GIFT OF Bern Dibner The Dibner Library of the History of Science and Technology SMITHSONIAN INSTITUTION LIBRARIES 1*. ( / / / ' .Kiwis ‘ h- ' * . ■ '' . ■Ha,. . m ■ ■ v ' ■ . Ki -f.V> I ■■ BE ‘v V ■ . . I?* >‘J.' ■ - ‘ 'V ■' • . • v : 5 * •» ■ - ' *' +-■ ■ 1'- J ' I • ' • ' \ r ■ S » . . ■ • •‘SKI il' • V ‘v - - - -■ - •V ' ' - - A - >*- 4 '« V-.. - « ; - ■r >• . V * * ■: * ? ' H: ' - • -U V ■ > : • ♦ ■* •• ... '• • ..s ■» a - .^5 • , p > A. * . r /.#■ / V *' .f: ' ! r' r ' *1 / T- • » V — -T,. . - ! J j J 3L ■■■ ■ V • . - -■ < ; ; ' J ■ r . -f v-, t * i. ; . ■ r c Vw v: : i * . •/ .. 1 i - ■ 1 ' ' V MM t&k ADVERTISEMENT. PArt of the enfuing Difcourfe about Light was written at the defire of fome Gentlemen of the Royal Society, in the Tear 1675. and then fent to their Secretary, and read at their Meetings , and the reft was added about Twelve Tears after to complete the Theory ; except the Third Book , and the lajt Proportion of the Second , which were Jince put together out offcattered Papers. To avoid being engaged in Difputes about thefe Matters , I have hitherto delayed the Printing, and jhould fill have de¬ layed it, had not the importunity of Friends prevailed upon me. If any other Papers writ on . this Suh\eB are got out of my Hands they are imperfeff, and were perhaps written before / had tried all the Experiments here fet down , and fully fatisfied my felf about the Laws of Ref rations and Compofition of Colours. I have here Publifhed what I think proper to come abroad, wijhing that it may not be Tranflated into another Language without my Confent. The Crowns of Colours, which fometimes appear about the S un and Moon, I have endeavoured to give an Ac¬ count of ; but for want of fufficient Obfervations leave that Matter to be further examined. 'The Subject of the Third Book I have alfo left imperfeffi, not having tried all the Expe- Experiment# which I intended when J was about thefe Matters , nor repeated fome of thofe which I did try , until I had J'atisfied my felf about all their Circumftances. To communicate what I have tried , and leave the resit to others for further Enquiry, is all my Dejign in publifhing thefe Tapers. In a Letter written to Mr. Leibnitz in the T ear 1 67 6. and pubhfhed by Dr. Wallis, I mentioned a Method by which I had found fome general Theorems about fquaring Curvilinear Figures , or comparing them with the Conic Sections , or other the Jimple ft Figures with which they may he compared. And fome Tears ago I lent out a Manufcript containing fuch Theorems , and having Jince met with fome Things copied out of it, / have on this Occajim made it publick , prefixing to it an Introdudlion and fubjoyning a Scholium concerning that Method. And I have pined with it another f mall Trail concerning the Curvilinear Figures of the Second, R ind, which was alfo written, many Tears ago, and made known to fome Friends, who have foMdied the making it publick ; ~ - a - - 4 I N. DS! The FIRST BOOK O F O P T I C K S. PART I. MY Defign in this Book is not to explain the Pro¬ perties of Light by Hypothefes, but to propofe and prove them by Reafon and Experiments : In order to which , I fhall premife the following Defini¬ tions and Axioms. DEFINITIONS. D E F I N. I. BY the (Rays of Light I under ft and its leaf Tarts, and thofe as wellSucceffiVe in the fame Lines as Contemporary in fe¬ deral Lines . For it is manifeft that Light confifts of parts both Succeflive and Contemporary 5 becaufe in the fame place you may flop that which comes one moment, and let pals that which comes prefently after 5 and in the fame time you may flop it in any one place, and let it pafs in any other. For that part of Light which is ftopt cannot be the fame with that which is let pafs. The leaft Light or part of Light , which may be ftopt alone without the reft of the Light, or propagated alone, or do or fuffer any A thing 0 3 thing alone, which the reft of the Light doth not or fuft ers not, I call a Ray of Light. D E F I N. IT. ^ efrangihility of the Bays of Light, is their Difpofition to he refra&ed or turned out of their Way in pafjing out of one tranfi parent Body or Medium into another . And a greater or lefs Bp- frangihility of Bays, is their Difpofition to he turned more or lefs out of their Way in like Incidences on the fame Medium . Mathe¬ maticians ufually confider the Rays of Light to be Lines reaching from the luminous Body to the body illumina¬ ted, and the refraCtion of thofe Rays to be the bending or breaking of thofe Lines in their paffing out of one Me¬ dium into another. And thus may Rays and Refractions be confidered, if Light be propagated in an inftant. But by an Argument taken from the ./Equations of the times of the Eclipfes of Jupiter’s Satellites it feems that Light is propagated in time, (pending in its pafiage from the Sun to us about Seven Minutes of time : And therefore I have chofen to define Rays and Refractions in fuch general terms as may agree to Light in both cates. DEFIN. III. Bpfiexihility of Bays, ts their Difpofition to he turned hac^ into the fame Medium from any other Medium upon whofe Surface they fall. And Bays are more or left refiexihle , which are returned hack^ more or lefs eafily. As if Light pafs out of Glafs into Air , and by being inclined more and more to the conrr- mon Surface of the Glafs and Air, begins at length to be totally reflected by that Surface 3 thofe forts of Rays which at like Incidences are reflected moft copioufly , or by in¬ clining the Rays begin fooneft to be totally reflected, are moft reflexible, D E- E $ ] DEFIN. IV. The Angle of Incidence , is that Angle which the Line defcribed by the incident Pay contains with the (perpendicular to the refle¬ cting or refraMing Surface at the Point of Incidence . DEFIN. V. The Angle of Py flexion or Py fruition, is the Angle which the Line defcribed by the refleBed or ref railed Pay containeth with the Perpendicular to the reflecting or refralling Surface at the Point of Incidence . D EFI N. VI. The Sines of Incidence , Reflexion , and pefraltion, are the Sines of the Angles of Incidence , peflexion, and Pefraltion. DEFIN. VII. The Light whofe Pays are all alike Pyf Tangible, I call Sim¬ ple , Homogeneal and Similar $ and that whofe Pays are flome more Pyfrangible than others, l call Compound, Heterogene at and Dijfimilar. The former Light I call Homogeneal , not becaufe I would affirm it fo in all relpe&s $ but becaufe the Rays which agree in Refrangibility 5 agree at leaft in all thofe their other Properties. Which I confider in the following Difcourfe. DEFIN. VUE The Colours of Homogeneal Lights y I call Primary , Homo¬ geneal and Simple y and thofe of Heterogeneal Lights, Heteroge- neal and Compound . For thefe are always compounded of the colours of Homogeneal Lights 5 as will appear in the following Difcourfe. A z " A XI- C 4 1 AXIOMS. AX. I. THE Angles of Incidence, Inflexion, and RefraSion, lye in one and the fame Plane. AX. II. The Angle of Reflexion is equal to the Angle of Incidence „ A X. III. If the refraBed (Ray he returned direSly hack L to the Point of Incidence , it Jhall he ref railed into the Line before defer L led by the incidmt Ray. AX. IV. Re fr aBion out of the rarer Medium into the denfer , is made towards the Perpendicular 3 that is, fo that the Angle of Refra - Sion he left than the Angle of Incidence. A X. V. The Sine of Incidence, is either accurately 1 or Very ?iearly in a gtyen Ratio to the Sine of Refra&ion . Whence if that Proportion be known in~ any one Incli¬ nation of the incident Ray, *tis known in all the Inclina^ tions, and thereby the Refra&ion in all cafes of Incidence on the fame refra&ing Body may be determined. Thus if the Refra&ion be made out of Air into Water, the Sine ©f Incidence of the red Light is to the Sine of its Refra¬ ction as 4 to 3 . If out of Air into Glafs, the Sines are ’ * as m as 1 7 to it . In Light of other Colours the Sines have other Proportions : but the difference is fo little that it need feldom be confidered. Suppofe therefore, that R S reprefents the Surface of Fig. i ftagnating Water , and C is the point of Incidence in- which any Ray coming in the Air from A in the Line A C is reflected or refracted, and I would know whether this Ray {hall go after Reflexion or Refraction : I ereCt upon the Surface of the Water from the point of Inci¬ dence the Perpendicular C P and produce it downwards to Q., and conclude by the firfl: Axiom, that the Ray af¬ ter Reflexion and Refraction, fhall be found fomewhere in the Plane of the Angle of Incidence A C P produced. I let fall therefore upon the Perpendicular C P the Sine of Incidence A D, and if the reflected Ray be defired , I pro¬ duce A D to B fo that D B be equal to A D, and draw C B. For this Line C B fhall be the reflected Ray ; the Angle of Reflexion B C P and its Sine B D being7 equal to the Angle and Sine of Incidence, as they ought to be by the fecond Axiom. But if the refraCted Ray be de¬ fired, I produce A D to H, fo that D H may be to A B as the Sine of Refraction to the Sine of Incidence; that is as 3 to 4 ; and about the Center C and in the Plane AGP with the Radius C A defcribing a Circle ABE I draw Parallel to the Perpendicular GPQ, the Line H E cutting the circumference in E, and joyning CE, this Line CE • (hall be the Line of the refraCted Ray. For if E F be let fall perpendicularly on the Line P Q. , this Line E F fhall , be the Sine of Refraction of the Ray C E, the Angle of Refraction being ECQ; and this Sine EF is eoual to DH, and confequently in Proportion to the Sine of Inci- dence. AD as 3 to .4., I11 Fix . z . <_> -[6] In like manner, if there be a Prifm of Glafs (that is a Glafs bounded with two Equal and Parallel Triangular ends, .and three plane and well polifhed Sides, which meet in three Parallel Lines running from the three Angles of one end to the three Angles of the other end) and if the Refradion of the Light in paffing crofs this Prifm be defi- red : Let AC B reprefent a Plane cutting this Prifm tranf- verfly to its three Parallel lines or edges there where the Light pafleth through it, and let JE be the Ray inci¬ dent upon the firft fide of the Prifm A C where the Light goes into the Glafs 5 And by putting the Proportion of the Sine of Incidence to the Sine of Refradion as 1 7 to 1 1 find E F the firft refraded Ray. Then taking this Ray for the Incident Ray upon the fecond fide of the Glafs B C where the Light goes out, find the next refraded Ray F G by putting the Proportion of the Sine of Incidence to the Sine of Refradion as 1 1 to 17. For if the Sine of Inci¬ dence out of Air into Glafs be to the Sine of Refradion as 1 7 to 1 1 , the Sine of Incidence out of Glafs into Air muft on the contrary be to the Sine of Refradion as 1 1 to 17, by the third Axiom. Much after the fame manner , if A C B D reprefent a Glafs {pherically Convex on both fides (ufually called a Lem, fuch as is a Burning- glafs, or Spectacle-glafs, or an Objed-glafs of a Telefcope) and it be required to know how Light falling upon it from any lucid point Q {hall be refracted, let Q_M reprefent a Ray falling upon any point M of its firft fpherical Surface A CB, and by ereding a Perpendicular to the Glafs at the point M, find the firft refracted Ray M N by the Proportion of the Sines 17 to 1 1. Let that Ray in going out of the Glafs be inci¬ dent upon N, and then find the fecond .refracted Ray N q by the Proportion of the Sines 1 1 to 17. And after the C 7 J feme manner may the Refraction be found when the Lens is Convex on one fide and Plane or Concave on the other, or Concave on both Sides. AX. VI. Homogeneal (Rays which flow from federal (points of any Qk~ and fall almofi Perpendicularly on any reflecting or refra* Bing Plane or Spherical Surface , flhall afterwards diverge from fo many other Points , or he Parallel to fo many other Lines , or converge to fo many other Points , either accurately or without any fenjible Error. And the fame thing will happen , if the pays he refleBed or refraBed fuccejfiVely by two or three or more Plane or flherical Surfaces . The Point from which Rays diverge or to which they converge may be called their Focus. And the Focus of the incident Rays being given, that of the reflected or re¬ fracted ones may be found by finding the Refraction of any two Rays, as above y or more readily thus. Caf. i. Let ACBbe a reflecting or refracting Plane, Fg and Q the Focus of the incident Rays, and Qf C a per¬ pendicular to that Plane. And if this perpendicular be produced to q, fo that q C be equal to QC, the point q fhall be the Focus of the reflected Rays. Or if q C be taken on the fame fide of the Plane with QC and in Pro¬ portion to QC as the Sine of Incidence to the Sine of Refra&ion, the point q fhall be the Focus of the refrac¬ ted Rays. Caf. 2 Let A C B be the reflecting Surface of any Fk Sphere whofe Center is E. Bifect any Radius thereof (fup- pole EC) in T, and if in that Radius on the fame fide the point T you take the Points Q and q, fo that T Q, T E, and Tq be continual Proportionals, and the point Qbe the: [ 8 ] the Focus of the incident Rays , the point q (hall be the Focus of the refleded ones. Caj \ 4. Let ACB be the refracting Surface of any Sphere whofe Center is E, In any Radius thereof E C produced both ways take E T and C t feverally in fuch Proportion to that Radius as the leffer of the Sines of Incidence and Refradion hath to the difference of thofe Sines. And then if in the fame Line you find any two Points Q_ and q , fo that T Q be to E T as E t to t qy taking t q the contrary way from t which T Q. lieth from T, and if the Point Q.be the Focus of any incident Rays, the Point q fhall be the Focus of the refracted ones. And by the fame means the Focus of the Rays after two or more Reflexions or Refradions may be found. Caf. 4. Let A C B D be any refrading Lens , fpheri- cally Convex or Concave or Plane on either fide, and let CD be its Axis (that is the Line which cuts both its Sur¬ faces perpendicularly, and paffes through the Centers of the Spheres,) and in this Axis let F and /be the Foci of the refraded Rays found as above , when the incident Rays on both fides the Lens are Parallel to the fame Axis 5 and upon the Diameter F / bifected in E, defcribe a Circle. Suppofe now that any Point Q. be the Focus of any inci¬ dent Rays. Draw QE cutting the faid Circle in T and tJ and therein take t q in fuch Proportion to t E as t E or T E hath to T Q_. Let t q lye the contrary way from t which T Q. doth from T, and q fhall be the Focus of the refrac¬ ted Rays without any fenfible Error , provided the Point Q_ be not fo remote from the Axis, nor the Lens fo broad as to make any of the Rays fall too obliquely on the refracting Surfaces. And by the like Operations may the reflecting or re¬ fracting Surfaces be found when the two Foci are given, -- " : • and [ 9 ] and thereby a Lens be formed, which fhall make the Rays flow towards or from what place you pleafe. So then the meaning of this Axiom is , that if Rays fall upon any Plane or Spherical Surface or Lens, and before their Incidence flow from or towards any Point b a and AC ca meet in the edge of the refraCting Angle A 4. This edge A a being upward, is parallel both to the Horizon and to the parallel edges of the Paper D J and H E. And de re- p relents the Image of the Paper feen by RefraCtion up¬ wards in fuch manner that the blew half D G is carried hiaher to dg than the red half F E is to fey and therefore D fuffers [15] fuffers a greater Refradion. If the edge of the refrading; Angle be turned downward, the Image of the Paper will be refraded downward fuppofe to «N, and the blew half will be refraded lower to ^ y than the red half is to Exper. 2. About the aforefaid Paper, whofe two halfs were painted oyer with red and blew, and which was fluff like thin Paftboard, I lapped feveral times a flender thred of very black Silk, in fuch manner that the feveral parts ; of the thred might appear upon the Colours like fo many black Lines drawn over them, or like long and flender dark Shadows caft upon them. I might have drawn black Lines with a Pen, but the threds were fmaller and better defined. This Paper thus coloured and lined I fet againft a Wall perpendicularly to the Horizon, fo that one of the Colours might Hand to the right hand and the other to the left. Clofe before the Paper at the confine of the Co¬ lours below I placed a Candle to illuminate the Paper ftrongly : For the Experiment was tried in the Night* The flame of the Candle reached up to the lower edge of the Paper, or a very little higher. Then at the diftance of Six Feet and one or two Inches from the Paper upon the Floor I ereded a glafs Lens four Inches and a cjuarter broad, which might colled the Rays coming from the feveral Points of the Paper, and make them converge to¬ wards fo many other Points at the fame diftance of fix Feet and one or two Inches on the other fide of the Lens, and fo form tne Image of tne coloured Paper upon a white Paper placed there 5 after the fame manner that a Lens at a hole in a Window cafts the Images of Objeds abroad i upon a Sheet of white Paper in a dark Room. The afore¬ faid white Paper, ereded perpendicular to the Horizon and to the Rays which fell upon it from the Lens, I moved! fometimes towards- the Lens, fomerimes from it, to find1 thee tltf] the places where the Images of the blew and red parts of the coloured Paper appeared mod diftind. Thofe places I eafily knew by the Images of the black Lines which I had made by winding the Silk about the Paper. For the Images of thofe fine and (lender Lines (which by reafon of their blacknefs were like Shadows on the Colours) were ^onfufed and fcarce vifible, unlefs when the Colours on ei¬ ther fide of each Line were terminated mod diftindly. Noting therefore, as diligently as I could, the places where the Images of the red and blew halfs of the coloured Pa¬ per appeared mod diftind , I found that where the red half of the Paper appeared diftind, the blew half appeared eonfufed, fo that the black Lines drawn upon it could fcarce be feen 5 and on the contrary where the blew half appeared mod diftind the red half appeared eonfufed, fo that the black Lines upon it were fcarce vifible. And be¬ tween the two places where thefe Images appeared diftind there was the diftance of an Inch and a half : the diftance of the white Paper from the Lens, when the Image of the red half of the coloured Paper appeared mod diftind, be- ing greater by an Inch and an half than the diftance of the fame white Paper from the Lens when the Image of the blew half appeared moft diftind. In like Incidences there¬ fore of the blew and red upon the Lens, the blew was re- fraded more by the Lens than the red, fo as to converge fooner by an Inch and an half, and therefore is more refran¬ gible. . Illuftratio?i. In the Twelfth Figure, DE fignifies the co¬ loured Paper, D G the blew half, F E the red half, M N the Lens, H J the white Paper in that place where the red half with its black Lines appeared diftind, and hi the fame Paper in that place where the blew half appeared diftind. The place hi was nearer to the Lens MN than the place H J by an Inch and an half. Scholium . [ Scholium. The fame things fiicceed notwithstanding that fome of the Circumftances be varied : as in the firft Ex¬ periment when the Prifm and Paper are any ways inclined to the Horizon , and in both when coloured Lines are drawn upon very black Paper. But in the Defcription of thefe Experiments , I have fet down fuch Circumftances by which either the Phenomenon might be rendred more conlpicuous, or a Novice might more eafily try them, or by which I did cry them only. The fame thing I have often done in the following Experiments : Concerning all which this one Admonition may fuffice. Now from thefe Experiments it follows not that all the Light of the blew is more Refrangible than all the Light of the red 3 For both Lights are mixed of Rays differently Refrangible, So that in the red there are fome Rays not lefs Refrangible than thofe of the blew , and in the blew there are fome Rays not more Refrangible than thofe of the red 3 But thefe Rays in Proportion to the whole Light are but few, and ferve to diminifh the Event of the Experiment , but are not able to deftroy it. For if the red and blew Co¬ lours were more dilute and weak, the diftance of the Ima¬ ges would be lefs than an Inch and an half 3 and if they were more intenfe and full, that diftance would be greater, as will appear hereafter. Thefe Experiments may fuffice for the Colours of Natural Bodies. For in the Colours made by the Refradtion of Prifms this Propofition will appear by the Experiments which are now to follow in the next Propofition. C tp^O (P. \ ■ P R 0 P. II. Theor. II. The Light of the Sun confifls of Rays differently Refrangible, The Proof by Experiments. er.g. I N a very dark Chamber at a round hole about j[ one third part of an Inch broad made in the Shut of a Window I placed a Glafs Prifm, whereby the beam of the Suns Light which came in at that hole might be refracted upwards toward the oppofite Wall of the Chamber , and there form a coloured Image of the Sun.- The Axis of the Prifm (that is the Line paffing. through the middle of the Prifm from one end of it to the other end Parallel to the edge of the Refracting Angle) was in this and the following Experiments perpendicular to the incident Rays. About this Axis I turned the Prifm flowly , and faw the refraCted Light on the Wall or co¬ loured Image of the Sun firft to defcend and then to af- cend. Between the Defcent and Afcent when the Image feemed Stationary 5 I ftopt the Prifm, and fixt it in that Pofture, that it flhould be moved no more. For in that pofture the Refractions of the Light at the two fides of the RefraCting Angle, that is at the entrance of the Rays into the Prifm and at their going out of it, were equal to one another. So alfo in other Experiments as often as I would have the Refractions on both fides the Prifm to be equal to one another, I noted the place where the Image of the Sun formed by the refraCted Light flood ftill be¬ tween its two contrary Motions, in the common Period of its progrefs and egrefs 3 and when the Image fell upon that place, I made fail the Prifm. And in this pofture, as the [J9] the moft convenient, it is to be underftood that all the Prifms are placed in the following Experiments, unlefs where feme other pofture is defcribed. The Prifm therefore being pla¬ ced in this pofture, I let the refraded Light fall perpendi¬ cularly upon a Sheet of white Paper at the oppofite Wall of the Chamber, and obferyed the Figure and Dimenfions of the Solar Image formed on the Paper by that Light. This Image was Oblong and not Oval, but terminated with two Redilinear and Parallel Sides , and two Semi¬ circular Ends. On its Sides it was bounded pretty diftindly, but on its Ends very confufedly and indiftindly, the Light there decaying and vanifhing by degrees. The breadth qf this Image anfwered to the Suns Diameter, and was about two Inches and the eighth part of an Inch , including the Penumbra. For the Image was eighteen Feet and an half diftant from the Prifm, and at this diftance that breadth if diminilhed by the Diameter of the hole in the Window-lhut, that is by a quarter of an Inch, febtended an Angle at the Prifm of about half a Degree, which is the Sun’s apparent Diameter. But the length of the Image was about ten Inches and a quarter, and the length of the Rectilinear Sides about eight Inches 3 And the refrading Angle of the Prifm where¬ by fo great a length was made, was -6a degr. With a Ids Angle the length of the Image was lefs , the breadth re¬ maining the fame. If the Prifm was turned about its Axis that way which made the Rays emerge more obliquely out of the fecond refolding Surface of the Prifm, the Image foon became an Inch or two longer, or more 3 and if the Prifm was turned about the contrary way, fo as to make the Rays fall more obliquely on the firft refrading Surface, the Image foon became an Inch or two (hotter. And therefore in try¬ ing this Experiment, I was as curious as I could be in pla¬ cing the Prifm by the above-mentioned Rule exadiy in C i fuch [20] fuch a pofture that the Refractions of the Rays at their emer¬ gence out of the Prifm might be equal to that at their inci¬ dence on it. This Prifm had feme Veins running along within the Glafs from one end to the other , which feat- tered feme of the Sun’s Light irregularly, but had no fen- fible effeCt in encreafing the length of the coloured Spec¬ trum. For I tried the fame Experiment with other Prifms with the fame Succefs. And particularly with a Prifm which feemed free from fuch Veins, and whofe refracting Angle was 62- Degrees, I found the length of the Image 92 or 10 Inches at the diftance of i8J Feet from the Prifm, the breadth of the hole in the Window- flhut being i of an Inch as before. And becaufpit is eafie to commit a mi- ftake in placing the Prifm in its due pofture, I repeated the Experiment four or five times, and always found the length of the Image that which is fet down above. With another Prifm of clearer Glafs and better Polliflh, which feemed free from Veins and whofe refracting Angle was 63 Degrees, the length of this Image at the fame diftance of 18; Feet was alfo about 10 Inches, or 10k Beyond tkefe Meafures for about -1 or 1 of an Inch at either end of 4 3 the SpeCtrum the Light of the Clouds feemed to be a little tinged with red and violet, but fo very faintly that I fufpe- Cted that tinClure might either wholly or in great meafure arife from feme Rays of the SpeCtrum fcattered irre¬ gularly by feme inequalities in the Subftance and Polifh of the Glafs , and therefore I did not include it in thefe Meafures. Now the different Magnitude of the hole in theWindow-fliut, and different thicknefs of the Prifm where the Rays paffed through it, and different inclinations of the Prifm to the Florizon, made no fenfible changes in the length of the Image. Neither did the different matter of the [21 ] the Prifms make any : for in a Veffel made of polifihedr Plates of Glafs cemented together in the Ihape of a Prifmn and filled with Water, there is the like Succefs of the Ex¬ periment according to the quantity of the Refraction. It is further to be obferved, that the Rays went on in right Lines from the Prifm to the Image, and therefore at their very going out of the Prifm had all that Inclination to one another from which the length of the Image pro^ ceeded, that is the Inclination of more than two Degrees and an half. And yet according to the Laws of Opticks vulgarly received, they could not poflibly be fo much in¬ clined to one another. For let EG reprefent the Window- fliut, F the hole made therein through which a beam of the Sun’s Light was tranfmitted into the darkned Chamber, and ABC a Triangular Imaginary Plane whereby the Prifm is feigned to be cut tranfverfly through the middle of the Light. Or if you pleafe, let AB C reprefent the Prifm it felfj looking direCtly towards the Spectator s Eye with its nearer end : And let X Y be the Sun, M N the Paper upon which the Solar Image or Spectrum is call, and P T the Image it felf whofe fides towards V and W are Rectili¬ near and Parallel, and ends towards P and T Semicir¬ cular. YKHP and X L J T are the two Rays, the firft of which comes from the lower part of the Sun to the higher part of the Image, and is refraCted in the Prifm at K and H, and the latter comes from the higher part of the Sun to the lower part of the Image, and is refraCted at L and J. Since the Refractions on both fides the Prifm are equal to one another, that is the RefraCtion at K equal to the RefraCtion at J, and the RefraCtion at L equal to the RefraCtion at H, fo that the Refractions of the inci¬ dent Rays at K and L taken together are equal to the RefraCtions. of the. emergent Rays at Hand J taken toge¬ ther t: i>2] thef : it follows by adding equal things to equal things, that the RefraCtions at K and H taken together, are equal to the Refractions at J and L taken together , and there¬ fore the two Rays being equally refracted have the fame Inclination to one another after RefraCiion which they had before, that is the Inclination of half a Degree anfwering to the Sun's Diameter. For fo great was the Inclination of the Rays to one another before RefraCHon. So then, the length of the Image P T would by the Rules of Vul¬ gar O-p ticks fubtend an Angle of half a Degree at the Prifm, and by confequence be equal to the breadth Vw $ and therefore the Image would be round. Thus it would be were the two Rays X L J T and Y K H P and all the reft which form the Image P w T >, alike Refrangible. And therefore feeing by Experience it is found that the Image is not round but about five times longer than broad, the Rays which going to the upper end P of the Image fuffer the greateft Refraction, muft be more Refran¬ gible than thofe which go to the lower end T , unlefs the inequality of Refraction be cafual. This Image or SpeCtrum P T was coloured, being red at its lead: refracted end T, and violet at its mod: refraCted end P, and yellow green and blew in the intermediate fpaces. Which agrees with the firft Fropofition, that Lights which differ in Colour do alfo differ in Refrangibility. The length of the Image in the foregoing Experiments I meafured from the fainted; and outmoft red at one end, to the fainted: and outmoft blew at the other end. Exper. 4. In the Sun's beam which was propagated in¬ to the Room through the hole in the Window-fhut, at the diftance of feme Feet from the hole, I held the Prifin in fech a pofture that its Axis might be perpendicular to that beam. Then I looked through the Prifm upon the [ 23 ] hole, and turning the Prifm to and fro about its Axis to make the Image of the hole afcend and defcend, when be¬ tween its two contrary Motions it feemed ftationary, X ftopt the Prifm that the Refractions on both fides of the refracting Angle might be equal to each other as in the former Experiment* In this Situation of the Prifm view¬ ing through it the faid hole, I obferved the length of its refracted Image to be many times greater than its breadth, and that the mod refracted part thereof appeared violet, the leaf: refracted red, the middle parts blew green and yellow in order. The fame thing happened when I re¬ moved the Prifm out of the Suns Light , and looked through it upon the hole fhining by the Light of the Clouds beyond it. And yet if the Refraction were done regularly according to one certain Proportion of the Sines of Incidence and Refraction as is vulgarly fuppofed, the refracted Image ought to have appeared round. So then, by thefe two Experiments it appears that m equal Incidences there is a confiderable inequality of Re¬ fractions : But whence this inequality arifes, whether it be that feme of the incident Rays are refracted more and--, others lefs, conftantly or by chance, or that one and the fame Ray is by^ Refraction difturbed, flattered, dilated, and as it were Iplit and Ipread into many diverging Rays,,, as Grimaldo fuppofes, does not yet appear by thefe Experi¬ ments, but will appear by thofe that follow. Exper. 5 . Confidering therefore, that if in the third Ex¬ periment the Image of the Sun fheuld be drawn out into- an oolong form, either by a Dilatation of every Ray, or by any other cafual inequality of the Refractions, the fame, oblong Image would by a fecond Refraction made Side¬ ways be drawn out as much in breadth by the like Dila¬ tation. of the. Rays or other cafual inequality of the Re¬ fractions. I24] fractions Sideways, I tried what would be the Effects of fuch a fecond Refraction. For this end I ordered all things as in the third Experiment, and then placed a fecond Prifm immediately after the firft in a crofs Petition to it, that it might again refrad the beam of the Suns Light which came to it through the firft Prifm. In the firft Prifm this beam was refraded upwards, and in the fecond Sideways. And I found that by the Refradion of the fecond Prifm the breadth of the Image was not increafed, but its fupe- rior part which in the firft Prifm buffered the greater Re¬ fradion and appeared violet and blew, did again in the fecond Prifm buffer a greater Refradion than its inferior part, which appeared red and yellow , and this without any Dilation of the Image in breadth. 4- llluflratton. Let S reprefent the Sun, F the hole in the Window, ABC the firft Prifm, D H the fecond Prifm, Y the round Image of the Sun made by a dired beam of Light when the Prifms are taken away, P T the oblong Image of the Sun made by that beam palling through the •firft Prifm alone when the fecond Prifm is taken away, and ft the Image made by the crofs Refradions of both Prifms together. Now if the Rays which tend towards the feveral Points of the round Image Y were dilated and Ipread by the Refradion of the firft Prifm, fo that they ftiould not any longer go in fingle Lines to Angle Points, but that every Ray being fplit, fhattered, and changed from a Linear Ray to a Superficies of Rays diverging from the Point of Refradion, and lying in the Plane of the Angles of Incidence and Refradion, they Ihould go in thofe Planes to fo many Lines reaching almoft from one end of the Image P T to the other, and if that Image Ihould thence become oblong : thofe Rays and their feveral parts tending towards the feveral Points of [^5] the Image P T ought to be again dilated and fpread Side¬ ways by the tranfverfe RefraCtion of the fecond Prifm, fo as to compofe a fourfquare Image, fuch as is reprefented at «r7. For the better underftanding of which, let the Image PT be diftinguiftied into five equal Parts PQK, KQRL, LRSM, MSVN, NVT. And by the fame irregularity that the Orbicular Light Y is by the RefraCtion of the firft Prifm dilated and drawn out into a long Image P T, the the Light PQK which takes up a fpace of the fame length and breadth with the Light Y ought to be by the Refra¬ ction of the fecond Prifm dilated and drawn out into the long Image *qkp, and the Light K QR L into the long Image 4 q r h an^ ^ Lights LRSM, MSVN, NVT into fo many other long Images l r s w, m s v w, ny tl 3 and all thefe long Images would compofe the fourfquare Image *1- Thus it ought to be were every Ray dilated by Re¬ fraction, and fpread into a triangular Superficies of Rays diverging from the Point of RefraCtion. For the fecond RefraCtion would fpread the Rays one way as much as the firft doth another , and fo dilate the Image in breadth as much as the firft doth in length. And the fame thing ought to happen, were fome Rays cafually refraCted more than others. But the Event is otherwife. The Ima^e P T was not made broader by the RefraCtion of the Ifecond Prifm, but only became oblique, as ?tis reprefented at p its tipper end P being by the RefraCtion tranflated to a greater diftance than its low7er end T. So then the Light which went towards the upper end P of the Image, was (at equal Incidences) more refraCted in the fecond Prifm than the Light which tended towards the lower end T, that is the blew and violet, than the red and yellow 3 and therefore was more Refrangible. The fame Light was by the RefraCtion of the firft Prifm tranflated further from the [2(5] place Y to which it tended before Refraction 5 and there¬ fore fuffered as well in the firft Prifm as in the fecond a greater Refraction than the reft of the Light, and by con- lequence was more Refrangible than the reft, even before its incidence on the firft Prifm. Sometimes I placed a third Prifm after the fecond, and foitietimes alfo a fourth after the third , by all which the Image might be often refracted fide ways : but the Rays which were more refracted than the reft in the firft Prifin were alfo more refracted in all the reft, and that without any Dilatation of the Image fide ways : and therefore thofe Rays for their conftancy of a greater Refraction are de- fervedly reputed more Refrangible. But that the meaning of this Experiment may more clearly appear, it is to be confidered that the Rays which are equally Refrangible do fall upon a circle anfwering to the Suns Difque. For this was proved in the third Experi¬ ment. By a circle I underftand not here a perfect Geo¬ metrical Circle, but any Orbicular Figure whofe length is equal to its breadth, and which, as to fenfe, may feem circular. Let therefore A G reprefent the circle which all the moft Refrangible Rays propagated from the whole Difq ue of the Sun, would illuminate and paint upon the oppofite Wall if they were alone 5 E L the circle which all the leaft Refrangible Rays would in like manner illuminate and paint if they were alone 3 B H, C J, D K, the circles which fo many intermediate forts of Rays would fuccef- fively paint upon the Wall, if they were fingly propagated from the Sun in fucceffive Order, the reft being always in¬ tercepted 3 And conceive that there are other intermediate Circles without number which innumerable other inter¬ mediate forts of Rays would fucceffively paint upon the Wall if the Sun fhould fucceffively emit every fort apart. [27] And feeing the Sun emits all thefe forts at once, they muft all together illuminate and paint innumerable equal cir¬ cles, of all which, being according to their degrees of Re- frangibility placed in order in a continual feries, that ob¬ long Spedrum PT is compofed which I defcribed in the third Experiment. Now if the Sun's circular Image Y which is made by an unrefraded beam of Light was by any dilatation of the fingle Rays, or by any other irregu¬ larity in the Refraction of the firft Prifm, converted into the Oblong Spedrum, P T : then ought every circle A G, B H, C J, tsrc. in that Spedrum, by the crofs Refra¬ ction of the fecond Prifm again dilating or otherwife fcattering the Rays as before, to be in like manner drawn out and transformed into an Oblong Figure, and thereby the breadth of the Image P T would be now as much aug¬ mented as the length of the Image Y was before by the Re¬ traction of the firft Prifm 5 and thus by the Refradions of both Prifms together would be formed a fourfqnare Figure T*t 7 as I defcribed above. Wherefore fince the breadth of the SpeCtrum P T is not iiicreafed by the Refraction fi de¬ ways, it is certain that the Rays are not fplit or dilated, or otherways irregularly fcattered by that RefraCtion, but that every circle is by a regular and uniform RefraCtion tranflated entire into another place, as the circle A G by the greateft RefraCtion into the place agy the circle B H by a Ids RefraCtion into the place bh, the circle C J by a Re¬ fraCtion frill lefs into the place cz, and fo of the reft 5 by which means a new Spedrum p t inclined to the former P T is in like manner compofed of circles lying in a right Line 3 and thefe circles muft be of the fame bignefs with the former, becaufe the breadths of all the Spe- drums Y, PT and pt at equal diftances from the Prifms are equal. D 2 I con- [28] t confidered further that by the breadth of the hole F through which the Light enters into the Dark Chamber* there is a Penumbra made in the circuit of the SpeCtrum Y, and that Penumbra remains in the rectilinear Sides of the SpeCtrums P T and pt . I placed therefore at that hole a Lens or ObjeCt-glafs of a Telefcope which might call the Image of the Sun diftinCtly on Y without any Penum- bra at all, and found that the Penumbra of the Rectili¬ near Sides of the oblong SpeCtrums P T and pt was alfo thereby taken away, fo that thofe Sides appeared as di¬ ftinCtly defined as did the Circumference of the firft Image Y. Thus it happens if the Glafs of the Prifms be free from veins, atd their Sides be accurately plane and well polifihed without thofe numberlefs waves or curies which ufually arife from Sand-holes a little fmoothed in polifh- ing with Putty. If the Glafs be only well polifbed and free from veins and the Sides not accurately plane but a little Convex or Concave, as it frequently happens 3 yet may the three SpeCtrums Y, P T and pt want Penumbras, but not in equal diftances from the Prifms. Now from this want of Penumbras, I knew more certainly that every one of the circles was refraCted according to fome moft regular, uniform, and conftant law. For if there were any irregularity in the RefraCtion, the right Lines A E and G L which all the circles in the SpeCtrum P T do touch, could not by that RefraCtion be tranflated into the Lines a e and gl as diftinCt and ftraight as they were before, but there would arife in thofe tranflated Lines fome Penumbra or crookednefs or undulation, or other fenfible Perturba¬ tion contrary to what is found by Experience. Whatfo- ever Penumbra or Perturbation fbould be made in the circles by the crofs RefraCtion of the fecond Prifm , all that Penumbra or Perturbation would be conlpicuous in [29] the right Lines a e and g l which touch thofe circles. And therefore fince there is no fuch Penumbra or Perturbation in thofe right Lines there mu ft be none in the circles* Since the diftance between thofe Tangents or breadth of the Sped rum is not increafed by the Refradions, the Dia¬ meters of the circles are not increafed thereby. Since thofe Tangents continue to be right Lines, every circle which in the firft Prifm is more or lefs refraded , is exadly in the fame Proportion more or lefs refraded in the fecond. And feeing all thefe things continue to fucceed after the fame manner when the Rays are again in a third Prifm, and again in a fourth refraded Sideways, it is evident that the Rays of one and the fame circle as to their degree of Refrangibility continue always Uniform and Homogeneal to one another, and that thofe of feveral circles do differ in degree of Refrangibility , and that in fome certain and conftant Proportion. Which is the thing I was to prove. There is yet another Circumftance or two of this Ex- Fig. periment by which it becomes ftill more plain and con¬ vincing. Let the fecond Prifm D H be placed not imme- ately after after the firft, but at fome diftance from it 5 Suppofe in the mid-way between it and the Wall on which the oblong Spedrum P T is caft, fo that the Light from the firft Prifm may fall upon it in the form of an oblong Spedrum, *7 Parallel to this fecond Prifm, and be refraded Sideways to form the oblong Spedrum p t upon the Wall. And you will find as before, that this Spedrum p t is in¬ clined to that Spedrum P TL , which the firft Prifm forms alone without the fecond 3 the blew ends P and p beino^ fur¬ ther diftant from one another than the red ones T and t and by confluence that the Rays which go to the blew end *■ of the Image *1 and which therefore fuffer the greateft Refradion in the firft Prifm, are again in the fecond Prifm more refraded than the reft, 1 3°] The fame thing I try’d alfo by letting the Sun’s Light into a dark Room through two little round holes F and p made in the Window, and with two Parallel Prifins ABC and plaoed at thofe holes ( one at each ) refra&ing thofe two beams of Light to the oppofite Wall of the Chamber, in fuch manner that the two colour’d Images P T and m n which they there painted were joyned end to end and lay in one ftraight Line, the red end T of the one touching the blew end m of the other. For if thefe two refra&ed beams were again by a third Prifm D H pla¬ ced croft to the two firft, refrafted Sideways, and the Spe- Ctrums thereby tranflated to feme other part of the Wall of the Chamber , fuppofe the Spectrum P T to p t and the Spedxum M N to m w, thefe tranflated SpeCtrums p t and m n would not lie in one ftraight Line with their ends contiguous as before, but be broken off from one another and become Parallel, the blew end of the Image m n being by a greater Refraction tranflated farther from its former place M T, than the red end t of the other Image p t from the fame place MT which puts the Propofition paft di- fpute. And this happens whether the third Prifm D H be placed immediately after the two firft or at a great diftance from them , fo that the Light refraCfed in the two firft Prifms be either white and circular, or coloured and ob¬ long when it falls on the third. Exper. 6. In the middle of two thin Boards I made round holes a third part of an Inch in Diameter, and in the Window- fhut a much broader hole, being made to let into my darkned Chamber a large beam of the Sun’s Light 3 I placed a Prifm behind the Shut in that beam to refraCl it towards the oppofite Wall, and clofe behind the Prifm I fixed one of the Boards, in fuch manner that the middle of the refraCted Light might pafs through the hole made [30 made in it, and the reft be intercepted by the Board. Then at the diftance of about twelve Feet from the firft Board I fixed the other Board, in fuch manner that the middle of the refracted Light which came through the hole in the firft Board and fell upon the oppofite Wall might pafs through the hole in this other Board, and the reft be-r ing intercepted by the Board might paint upon it the co* loured SpeCtrum of the Sun. And clofe behind this Board I fixed another Prifm to refrad: the Light which came through the hole. Then I returned Ipeedily to the firft Prifm, and by turning it flowly to and fro about its Axis, I caufed the Image which fell upon the fecond Board to move up and down upon that Board, that all its parts might fucceflively pafs through the hole in that Board and fall upon the Prilm behind it. And in the mean time, I noted the places on the opppfite Wall to which that Light after its Refraction in the fecond Prifm did pafs $ and by the difference of the places I found that the Light which being moft refraCted in the firft Prifm did go to the blew end of the Image, was again more refrafted in the fecond Prifm than the Light which went to the red end of that Image, which proves as well the firft Propofition as the fecond. And this happened whether the Axis of the two Prifms were parallel, or inclined to one another and to the Horizon in any given Angles. Illuftration. Let F be the wide hole in the Window-ftiut, through which the Sun ftiines upon the firft Prifm ABC, ^ and let the refra&ed Light fall upon the middle of the Board D E, and the middle part of that Light upon the hole G made in the middle of that Board. Let this tra- jeCted part of the Light fall again upon the middle of the fecond Board d e and there paint fuch an oblong coloured Image of the Sun as was defcribed in the third Experiment. By [ 32] By turning the Prifm ABC flowly to and fro about its Axis this Image will be made to move up and down the Board d e, and by this means all its parts from one end to the other may be made to pafs fucceffively through the hole g which is made in the middle of that Board. In the mean while another Prifm ale is to be fixed next after that hole£ to refradt the trajedted Light a fecond time. And thefe things being thus ordered, I marked the places M and N of the oppofite Wail upon which the refradted Light fell, and found that whilft the two Boards and fecond Prifm remained unmoved, thofe places by turning the firft Prifm about its Axis were changed perpetually. For when the lower part of the Light which fell upon the fecond Board d e was call through the hole g it went to a lower place M on the Wall , and when the higher part of that Light was call through the fame hole^, it went to a higher place N on the Wall, and when any intermediate part of the Light was call through that hole it went to fome place on the Wall between M and N. The unchanged Pofition of the holes in the Boards, made the Incidence of the Rays upon the fecond Prifm to be the fame in all cafes. And yet in that common Incidence fome of the Rays were more refradted and others lefs. And thofe were more refracted in this Prifm which by a greater Refradtion in the firft Prifm were more turned out of the way, and therefore for their conftancy of being more refradted are defervedly cal¬ led more Refrangible. Exper . 7. At two holes made near one another in my Window- fhut I placed two Prifms , one at each, which might caft upon th§ oppofite Wall (after the manner of the third Experiment ) two oblong coloured Images of the Sun. And at a little diftance from the Wall I placed a long (lender Paper with ftraight and parallel edges, and ordered [ 33 1 ordered the Prifms and Paper fo, that the red Colour of one Image might fall direddy upon one half of the Paper, and the violet colour of the other Image upon the other half of the fame Papery fo that the Paper appeared of two Colours , red and violet , much after the manner of the painted Paper in the firft and fecond Experiments. Then with a black Cloth I covered the Wall behind the Paper, that no Light might be reflected from it to difturb the Experiment, and viewing the Paper through a third Prifm held parallel to it, I faw that half of it which was illumi¬ nated by the Violet-light to be divided from the other half by a greater Refradion, efpecially when I went a good way off from the Paper. For when I viewed it too near at hand, the two halfs of the Paper did not appear fully divided from one another, but Termed contiguous at one of their Angles like the painted Paper in the firft: Expe¬ riment. Which alfo happened when the Paper wras too broad. Sometimes inftead of the Paper I ufecl a white Thred, and this appeared through the Prifm divided into two Pa¬ rallel Threds as is reprelented in the i pth Figure , where Fig. D G denotes the Thred illuminated with violet Light from D to E and with red Light from E to G, and d e °fg are the parts of the Thred feen by Refraction. If one half or the 1 hred be conftantly illuminated with red, and the other half be illuminated with all the Colours lucceffively, (which may be done by caufihg one of the Prifms to be turned about its Axis whilft the other remains unmoved) tins other half in viewing the Thred through the Prifm, will appear in a continued right Line with the firft half when illuminated with red , and begin to be a little divi¬ ded from it when illuminated with Orange, and remove further from it when illuminated with Yellow, and ftill E further [34] further when with Green, and further when with Blew, and go yet further off when illuminated with Indigo, and fur- theft when with deep Violet. Which plainly flhews, that the Lights of feveral Colours are more and more Refran¬ gible one than another, in this order of their Colours, Red, Orange, Yellow, Green, Blew, Indigo, deep Violet 3 and fo proves as well the firft Propofition as the fecond. Fig-. 1 7. I caufed alfo the coloured SpeCtrums P T and M N made in a dark Chamber by the Refractions of two Prifrns to lye in a right Line end to end, as was defcribed above in the fifth Experiment, and viewing them through a third Priftn held Parallel to their length, they appeared no longer in a right Line, but became broken from one another, as they are reprefented at p t and m n\ the violet end m of the SpeCtrum m n being by a greater RefraCtion tranflated further from its former place M T than the red end £ of the other SpeCtrum pt. _f/V. 20. I further caufed thofe two SpeCtrums P T and MN to become co-incident in an inverted order of their Colours, the red end of each falling on the violet end of the other, as they are reprefented in the oblong Figure P T M N 3 and then viewing them through a Prifm D H held Paral¬ lel to their length, they appeared not co-incident as when viewed with the naked Eye , but in the form of two di- ftinCt SpeCtrums p t and mn crofting one another in the middle after the manner of the letter X. Which fhews that the red of the one SpeCtrum and violet of the other, which were co-incident at P N and M T , being parted from one another by a greater RefraCtion of the violet to p and m than of the red to n and t, do differ in degrees of Refrangibility. I illuminated alfo a little circular piece of white Paper all over with the Lights of both Prifms intermixed, and when when it was illuminated with the red of one Spedhrum and deep violet of the other , fo as by the mixture of thofe Colours to appear all over purple , I viewed the Paper, firft at a lefs diftance , and then at a greater , through a third Prifm 3 and as I went from the Paper, the refradted Image thereof became more and more divided by the un¬ equal Refradfion of the two mixed Colours, and at length parted into two diftindt Images, a red one and a violet one, whereof the violet was furtheft from the Paper, and there¬ fore buffered the greateft Refradtion. And when that Prifm at the Window which call: the violet on the Paper was ta¬ ken away, the violet Image difappeared} but when the other Prifm was taken away the red vanifhed : which (hews that thefe two Images were nothing elfe than the Lights of the two Prifms which had been intermixed on the purple Pa¬ per, but were parted again by their unequal Refradtions made in the third Prifm through which the Paper was viewed. This alfo was obfervable that if one of the Prifms at the Window, fuppofe that which caft the violet y on the Paper, was turned about its Axis to make all the Colours in this order, Violet, Indigo, Blew, Green, Yel¬ low, Orange, Red, fall fucceffively on the Paper from that Prifm, the violet Image changed Colour accordingly, and in changing Colour came nearer to the red one, until when it was alfo red they both became fully co-incident. I placed alfo two paper circles very near one another, the one in the red Light of one Prifm, and the other in the violet Light of the other. The circles were each of them an Inch in Diameter, and behind them the Wall was dark that the Experiment might not be difturbed by any Light coming from thence. Thefe circles thus illuminated, I viewed through a Prifm fo held that the Refradtion might be made towards the red circle , and as I went from them E 2 they they came nearer and nearer together, and at length be¬ came co-incident 3 and afterwards when I went ftill further off, they parted again in a contrary order, the violet by a greater Refraction being carried beyond the red. Exper. 8. In Summer when the Stiffs Light ufes to be ftrongeft, I placed a Prifrn at the hole of the Window- flint, as in the third Experiment, yet fo that its Axis might be Parallel to the Axis of the World, and at the oppofite Wall in the Stiffs refra&ed Light, I placed an open Book. Then going Six Feet and two Inches from the Book, I placed there the abovementioned Lens,by which the Light reftedled from the Book might be made to converge and meet again at the diftance of fix Feet and two Inches be¬ hind the Lens , and there paint the Species of the Book upon a fiheet of white Paper much after the manner of the iecond Experiment. The Book and Lens being made faft, 1 noted the place where the Paper was, when the Letters of the Book, illuminated by the fulled: red Light of the Solar Image falling upon it, did caff their Species on that Paper moft diftinctly • And then I flay'd till by the Mo¬ tion of the Sun and confequent Motion of his Image on the Book, all the Colours from that red to the middle of the blew pafs’d over thofe Letter and when thofe Letters were illuminated by that blew, I noted again the place of the Paper when they caff their Species moft diftindtly upon it : And I found that this laft place of the Paper was nearer to the Lens than its former place by about two Inches and an half, or two and three quarters. So much fooner there¬ fore did the Light in the violet end of the Image by a grea¬ ter Refraction converge and meet , than the Light in the red end. But in trying this the Chamber was as dark as I could make it. For if thefe Colours be diluted and weak- ned by the mixture of any adventitious Light, the diftance between. between the places of the Paper will not be fo great. This diftance in the fecond Experiment where the Colours of natural Bodies were made ufe of, was but an Inch and a half, by reafon of the imperfedtion of thofe Colours. Elere in the Colours of the Prifm , which are manifeftly more full, intenfe, and lively than thofe of natural Bodies, the diftance is two Inches and three quarters. And were the Colours ftill more full , I queftion not but that the di¬ ftance would be considerably greater. For the coloured Light of the Priim, by the interfering of the Circles de~ fcnbed in the i nh Figure of the fifth Experiment, and alfo by the Light of the very bright Clouds next the Sun's Body intermixing with thefe Colours, and by the Light foartered by the inequalities in the pobfti of the Prifm, was io very much compounded, that the Species which thofe faint and dark Colours, the Indigo and Violet, call upon the Paper were not diftindt enough to be well obferved. Exper. o. A Prifm, wliofe two Angles at its Bafe were equal to one another and half right ones, and the third a right one, I placed in a beam, of the Sun's Light let in¬ to a dark Chamber through a hole in the Window7- flint as in the thirA Experiment. And turning the Prifm flowly about its Axis until all the Light which went through one of its Angles and was refracted by it began to be reflected by its Bale , at which till then it went out of the Glafs, I obferved that thofe Rays which had fuffered the greateft Refradtion were fooner refledted than the reft. I conceived therefore that thofe Rays of the refledted Light, which were moft Refrangible, did firft of all by a total Reflexion- become more copious in that Light than the reft , and that afterwards the reft alfo, by "a total Reflexion, be¬ came as copious as thefe. To try -this , I made the. re¬ flected Eight pafs through another Prifm, and being refra- [38] £ktd by it to fall afterwards upon a fheet of white Paper placed at fome diftance behind it, and there by that Re¬ fraction to paint the ufual Colours of the Prifm. And then caufing the firft Prifm to be turned about its Axis as above, I obferved that when thofeRays which in this Prifm had fuffered the greateft Refraction and appeared of a blew and violet Colour began to be totally reflected , the blew and violet Light on the Paper which was moft refraCted in the fecond Prifm received a fenfible increafe above that of the red and yellow, which was leaf! refraCted 5 and afterwards when the reft of the Light which was green, yellow and red began to be totally reflected in the firft Prifm, the light of thofe Colours on the Paper received as great an increafe as the violet and blew had done before. Whence 'tis manifeft, that the beam of Light reflected by the Bafe of the Prifm, being augmented firft by the more Refrangible Rays and afterwards by the lefs Refrangible ones, is compounded of Rays differently Refrangible. And that all fuch reflected Light is of the fame Nature with the Sun's Light, before its Incidence on the Bafe of the Prifm, no Man ever doubted : it being generally al¬ lowed, that Light by fuch Reflexions fuffers no Alteration in its Modifications and Properties. I do not here take notice of any Refractions made in the Sides of the firft Prifm, becaufe the Light enters it perpendicularly at the firft Side, and goes out perpendicularly at the fecond Side, and therefore fuffers none. So then, the Stiffs incident Light being of the fame temper and conftitution with his emergent Light, and the laft being compounded of Rays differently Refrangible , the firft mull be in like manner compounded. Fig. 2 1 . lllujlration. In the 2 1 dr Figure, A B C is the firft Prifm, B C its Bafe, B and C its equal Angles at the Bafe, each of [39] of 45 degrees, A its Rectangular Vertex, F M a beam of the Sun’s Light let into a dark Room through a hole F one third part of an Inch broad, M its Incidence on theBafe of the Prifm, M G a lefs refra&ed Ray, M H a more refract¬ ed Ray, M N the beam of Light reflected from the Bafe , V X Y the fecond Frifm by which this beam in palling through it is refraCted, N t the lefs refracted Light of this beam, and N p the more refracted part thereof. When the firft Prifm A B C is turned about its Axis according to the order of the Letters ABC, the Rays M H emerge more and more obliquely out of that Prifm, and at length after their moft oblique Emergence are reflected towards N, and going on to p do increafe the number of the Rays N p* Afterwards by continuing the motion of the firft Prifm, the Rays MG are alfo reflected to N and increafe the number of the Rays N t. And therefore the Light M N admits into its Competition, firft the more Refrangible Rays, and then the lefs Refrangible Rays, and yet after this Compofition is of the lame; Nature with the Sun’s immediate Light F M, V the Reflexion of the fpecular Bafe B C caufing no Altera¬ tion therein. Exper. 10. Two Prifms, which were alike in Aiape, I tied fo together, that their Axes and oppofite Sides being Parallel, they compofed a Parallelepiped, And, the Sun fliining into my dark Chamber through a little hole in the Window- (hut, I placed that Parallelepiped in his beam at fome diftance from the hole, in fuch a pofture that the Axes of the Prifms might be perpendicular to the incident Rays, and that thofe Rays being incident upon the firft Side of one Priim, might go on through the two contiguous Sides of both Prifms, and emerge out of the laft Side of the fe¬ cond Prifm. This Side being Parallel to the firft Side of the firft Prifm , caufed the emerging Light to be Parallel to [ 4° ] to the Incident. Then, beyond thefe two Prifms I placed a third, which might refract that emergent Light, and by that Refraction call: the ufnal Colours of the Prifm upon the oppofite Wall, or upon a {heet of white Paper held at a convenient diftance behind the Prifm for that refraCled Light to fall upon it. After this I turned the Parallelepiped about its Axis, and found that when the contiguous Sides of the two Prifms beeame fo oblique to the incident Rays that thofe Rays began all of them to be reflected , thofe Rays which in the third Prifm had fuffered the greateft Re¬ fraction and painted the Paper with violet and blew, were iirft of all by a total Reflexion taken out of the tranfmitted Light, the reft remaining and on the Paper painting their Colours of Green, Yellow, Orange, and Red as before 5 and afterwards by continuing the motion of the two Prifms, the reft of the Rays alio by a total Reflexion vaniihed in order, according to their degrees of Refrangibility. The Light therefore which emerged out of the twTo Prifms is compounded of Rays differently Refrangible , feeing the more Refrangible Rays may be taken out of it while the lefs Refrangible remain. But this Light being trajeCted only through the Parallel Superficies of the twTo Prifms, if it fuffered any change by the RefraCtion of one Superficies it loft that impreffion by the contrary RefraCfcion of the other Superficies, and fo being reftored to its priftine con- ftitution became of the fame nature and condition as at firft before its Incidence on thofe Prifms 5 and therefore, before its Incidence, was as much compounded of Rays differently Refrangible as afterwards. lllujlration . In the 22th Figure ABC and B C D are the the two Prifms tied together in the form of a Parallele¬ piped, their Sides BC and CB being contiguous, and their Sides A B and C D Parallel. And H J K is the third Prifm, UO Priftn, by which the Suns Light propagated through the hole F into the dark Chamber, and there paffing through thofe fides of the Frifms AB, BC, CB and CD, is refra¬ cted at O to the white Paper PT, falling there partly upon P by a greater Refraction, partly upon T by a lefs Refra¬ ction, and partly upon R and other intermediate places by intermediate Refractions. By turning the Parallelopiped ACBD about its Axis, according to the order of the Let¬ ters A,C,D,B, at length when the contiguous Planes BC and CB become fufficiently oblique to the Rays F M, which are incident upon them at M, there will vanifli to¬ tally out of the refracted Light OPT, firft of all the moft refracted Rays O P, (the reft OR and O T remaining as before) then the Rays O R and other intermediate ones, and laftly, the leaft refracted Rays O T. For when the Plane B C becomes fufficiently oblique to the Rays inci¬ dent upon it, thofe Rays will begin to be totally reflect¬ ed by it towards N 5 and firft the moft Refrangible Rays \\ will be totally reflected (as was explained in the preceding experiment) and by confequence muft firft difappear at P, and afterwards the reft as they are in order totally reflect¬ ed to N, they muft difappear in the fame order at R and T. So then the Rays which at O fuffer the greateft Re¬ fraction, may be taken out of the Light M O whilft the reft of the Rays remain in it, and therefore that Light MO is Compounded of Rays differently Refrangible. And be- caufe the Planes A B and C D are parallel, and therefore by equal and contrary Refractions deftroy one anothers Effects, the incident Light F M muft be of the fame kind and nature with the emergent Light M O, and therefore doth alfo confift of Rays differently Refrangible. Thefe two Lights FM and MO, before the moft refrangible Rays are feparated out of the emergent Light M O agree in Co- F lour, [ 42 ] lour, and in all other properties fo far as my obfervation reaches, and therefore are defervedly reputed of the fame Nature and Conftitution, and by confequence the one is compounded as well as the other. But after the molt: Re¬ frangible Rays begin to be totally reflected, and thereby feparated out of the emergentLightMO,that Light changes its Colour from white to a dilute and faint yellow, a pretty good orange, a very full red fucceffively and then totally vanifhes. For after the moil Refrangible Rays which paint the Paper at P with a Purple Colour, are by a total re¬ flexion taken out of the Beam of light M O, the reft of the Colours which appear on the Paper at R and T being mixed in the light M O compound there a faint yellow, and after the blue and part of the green which appear on the Paper between P and R are taken away, the reft which appear between R and T (that is the Yellow, Orange, Red and a little Green) being mixed in the Beam M O com¬ pound there an Orange 5 and when all the Rays are by re¬ flexion taken out of the Beam MO, except the leaft Refran¬ gible, which at T appear of a full Red, their Colour is the fame in that Beam M O as afterwards at T, the Re¬ fraction of the Prifm HJK ferving only to feparate the differently Refrangible Rays, without making any alteration in their Colours, as (hall be more fully proved hereafter. All which confirms as well the firft Propofition as the fe- cond. Scholium . If this Experiment and the former be conjoyned 'fig. z2, and made one, by applying a fourth Prifm VXY to re¬ fract the reflected Beam MN towards tp, the conclufion will be clearer. For then the light Np which in the 4th Prifm is more refrafted, will become fuller and ftronger when the Light O P, which in the third Prifm H J K is more refracted, vanifhes at P 3 and afterwards when the lefs refracted refracted Light OT vaniflies at T, the lefs mfraded Light N t will become encreafed whilft the more refraded Light at p receives no further encreafe. And as the trajeded Beam M O in vanifliing is always of fuch a Colour as ought to refult from the mixture of the Colours which fall upon the Paper P T. fo is the refleded Beam MN al¬ ways of fuch a Colour as ought to refult from the mix¬ ture of the Colours which fall upon the Paper pt. For when the moil refrangible Rays are by a total Reflexion taken out oi the Beam MO, and leave that Beam of an Orange Colour, the excefs of thofe Rays in the refleded Light, does not only make the Violet, Indigo and Blue at p more full, but alfo makes the Beam M N change from the yellowifli Colour of the Sun's Light, to a pale white in¬ clining to blue, and afterward recover its yellowifli Co¬ lour again, fo foon as all the reft of the tranfmitted light MOT is refleded. Now feeing that in all this variety of Experiments, whether the trial be made inLight refleded, and that either from natural Bodies, as in the firfl: and fecond Experiment, or Specular, as in the Ninth 3 or in Light refraded, and that either before the unequally refraded Rays are by di¬ verging feparated from one another, and lofing their white- nefs which they have altogether, appear feverally of feve- ral Colours, as in the fifth Experiment 3 or after they are feparated from one another, and appear Coloured as in the fixth,^ feventh, and eighth Experiments 3 or in Light tra- jedea through Parallel fuperficies, deftroying each others Effeds as in the 1 oth Experiment 3 there are always found Rays, which at equal Incidences on the fame Medium fuf- fer unequal Refradions, and that without any Iplitting or dilating of fingle Rays, or contingence in the inequality or the Refradions, as is proved in the fifth and fixth Ex- F 2 periments 3 [44] periments $ and feeing the Rays which differ in Refrangibi- lity may be parted and forted from one another, and that either by Refraction as in the third Experiment, or by Re¬ flexion as in the tenth, and then the feveral forts apart at equal Incidences fuffer unequal Refractions, and thofe forts are more refraCted than others after feparation, which were more refraCted before it, as in the fixth and following Ex¬ periments, and if the Suns Light be trajeCted through three or more crofs Prifms fucceffively, thofe Rays which in the firft Prifm are refraCted more than others are in all the fol¬ lowing Prifms, refraCted more then others in the fame rate and proportion, as appears by the fifth Experiment 3 it’s manifeft that the Sun’s Light is an Heterogeneous mixture of Rays, fome of which are conftantly more Refrangible then others, as was to be propofed. PROP. III. Theor. III. 'The Suns Light conflfts of flays differing in fle flexibility, and thofe (flays are more fleflexible than others which are more frangible . THIS is manifeft by the ninth and tenth Experi¬ ments : For in the ninth Experiment, by turning the Prifm about its Axis, until the Rays within it which in going out into the Air were refraCted by its Bafe, became fo oblique to that Bafe, as to begin to be totally reflected thereby 5 thofe Rays became firft of all totally reflected, which before at equal Incidences with the reft had fuffered the greateft RefraCtion. And the fame thing happens in the Reflexion made by the commonBafe of the two Prifms in the tenth Experiment.

j about the tenth part of an Inch, and its height an Inch or more. For by this means , if the Axis of the Prifm be Parallel to the Perpendicular of the Triangle , the Image 5. pt will now be formed of Equicrural Triangles ag, bhy ci , e 1^ f m , &cc. and innumerable other intermediate ones anfwering to the Triangular hole in flhape and bignefs,and lying one after another in a continual Series between two Parallel Lines af an Agm. Thefe Triangles are a little intermingled at their Bafes but not at their Vertices, and therefore the Light on the brighter fide af of the Image where the Bafes of the Triangles are is a little compounded, but on the darker fide^ m is altogether uncompounded, and in all places between the fides the Compofition is Proportional to the diftances of the places from that ob- fcurer fide g m. And having a Spedtrum pt of fuch a Compofition, we may try Experiments either in its ftronger and lefs fimple Light near the fide af or in its weaker and Ampler Light near the other fide l m , as it fihall feem moft convenient. But in making Experiments of this kind the Chamber ought to be made as dark as can be, leaft any forreign Light mingle it felf with the Light of the Spectrum p and render it compound 3 efpecially if we would try Ex¬ periments in the more fimple Light next the fide g m of the Spedtrun^ which being fainter, will have a lefs Pro¬ portion to the forreign Light, and fo by the mixture of that Light be more troubled and made more compound. The Lens alfo ought to be good, fuch as may ferve for Optical Ufes, and the Prifm ought to have a large Angle, fuppofe of 70 degrees, and to be well wrought, being made of Glafs free from Bubbles and Veins, with its fides not a little Convex or Concave as ufually happens but truly Plane,and its pollifh elaborate, as in working Optick- c$o glaffes , and not fuch as is ufually wrought with Putty, whereby the edges of the Sand-holes being worn away, there are left all over the Glafs a number lefs company of very little Convex polite rifings like Waves. The edges alfo of the Prifrn and Lens fo far as they may make any irregular Refradion, muft be covered with a black Paper glewed on. And all the Light of the Suns beam let into the Chamber which is ufelefs and unprofitable to the Ex¬ periment, ought to be intercepted with black Paper or other black Obftacles. For otherwife the ufelefs Light being refleded every way in the Chamber , will mix with the oblong Spedrum and help to difturb it. In trying thefe things fo much Diligence is not altogether neceffary, but it will promote the fuccefs of the Experiments, and by a very fcrupulous Examiner of things deferves to be applied* Its difficult to get glafs Prifms fit for this purpofe, and and therefore I ufed fometimes Prifmatick Venels made with pieces of broken Looking-glades, and filled with rain Water. And to increafe the Refradion, I fometimes im¬ pregnated the Water ftrongly with Saccharum SaturuL PROP . V. Theor. IV. Homogeneal Light is refralied regularly without any Dilatation fplitting or /battering 0/ the [ays , and the confu/ed Vifion of Objects feen through <%ef ratting Ladies by Heterogeneal Light ari/es from the different Lgfrangibihty of federal forts of (ffays. TH E firft Part of this Propofition has been already diffidently proved in the fifth Experiment, and will further appear by the Experiments which follow. G 2 Exper . t2. Exper . i 2a In the middle of a black Paper I made a round hole about a fifth or fixth part of an Inch in Dia¬ meter. Upon this Paper I caufed the Spectrum of Homo- geneal Light defcribed in the former Propofition , fo to fall, that forne part of the Light might pafs through the hole of the Paper. This tranlmitted part of the Light I refraded with a Prifm placed behind the Paper, and let¬ ting this refra£fed Light fall perpendicularly upon a white Paper two or three Feet diftant from the Priftn, I found that the Spedtrum formed on the Paper by this Light was not oblong, as when 'tis made (in the third Experiment) by Refrafting the Suns compound Light, but was (fo far as I could judge by my Eye) perfectly circular, the length being no greater than the breadth. Which fhews that this Light is refradted regularly without any Dilatation of the Rays. Exper . 15. In the Homogeneal Light I placed a Circle of ^ of an Inch in Diameter, and in the Sun’s unrefradted Heterogeneal white Light I placed another Paper Circle of the fame bignefs. And going from the Papers to the diftance offomeFeet, I viewed both Circles through a Prifm. The Circle illuminated by the Sun's Heterogeneal Light appear¬ ed very oblong as in the fourth Experiment , the length being many times greater than the breadth : but the other Circle illuminated with Homogeneal Light appeared Cir¬ cular and diftindtly defined as when 'tis viewed with the naked Eye. Which proves the whole Propofition. Exper . 1 4. In the Homogeneal Light I placed Flies and fuch like Minute Objedls, and viewing them through a Prifm , I faw their Parts as diftindtly defined as if I had viewed them with the naked Eye. The fame Objects pla¬ ced in the Sun’s unrefradted Heterogeneal Light which was white I viewed alfo through a Prifm, and faw them moil confufedly eonfufedly defined, fo that I could not diftinguifii their fmal- ler Parts from one another. I placed alfo the Letters of a fmall Print one while in the Homogeneal Light and then in the Heterogeneal, and viewing them through a Prifm, they appeared in the latter cafe fo confufed and indiftinCt that I could not read them 3 but in the former they ap¬ peared fo diftinCt that I could read readily, and thought I faw them as diftinCt as when I viewed them with my naked Eye. In both cafes I viewed the fame Objects through the fame Prifm at the fame diftance from me and in the fame Situation. There was no difference but in the Light by which the Objects were illuminated , and which in one cafe was Simple and in the other Compound, and therefore the diftinCt Vifion in the former cafe and confu¬ fed in the latter could arife from nothing elfe than from that difference of the Lights. Which proves the whole Propofition. And in thefe three Experiments it is further very remar¬ kable, that the Colour of Homogeneal Light was never changed by the Refraction. PROP. VI. Theor. V. The Sine of Incidence of every [ay confidered apart , is to its Sim ’ of fRefraBwn in a given I^atio* THAT every Ray confidered apart is conftant to it felf in fame certain degree of Refrangibility, is fufficiently manifeft out of what has been faid. Thofe Rays which in the firft RefraCtion are at equal Incidences mo ft refraCted, are alfo in the following Refractions at equal Incidences mod refraCted 3 and fo of the leaft Re¬ frangible , and the reft which have any mean degree of Refran- Refrangibility, as is tnanifeft by the 5 th, 6th, 7th, 8th, and 9th Experiments. And thofe which the firft time at like Incidences are equally refracted, are again at like In¬ cidences equally and uniformly refra&ed , and that whe¬ ther they be refraCted before they be feparated from one another as in the 5 th Experiment, or whether they be re¬ fracted apart, as in the 1 2 th, 1 3 th and 14th Experiments. The RefraCtion therefore of every Ray apart is regular, and what Rule that RefraCtion obferves we are now to fliew. The late Writers in Opticks teach, that the Sines of In¬ cidence are in a given Proportion to the Sines of Refra¬ ction, as was explained in the 5 th Axiom 3 and fome by Inftruments fitted for meafuring Refractions, or otherwise experimentally examining this Proportion, do acquaint us that they have found it accurate. But whilft they, not underftanding the different Refrangibility of feveral Rays, conceived them all to be refraCted according to one and the fame Proportion, 5tis to be prefumed that they adapted their Meafures only to the middle of the refraCted Light 3 fo that from their Meafures we may conclude only that the Rays which have a mean degree of Refrangibility , that is thofe which when feparated from the reft appear green, are refraCted according to a given Proportion of their Sines. And therefore we are now to fliew that the like given Proportions obtain in all the reft. That it fhould be fo is very reafonable, Nature being ever confor¬ mable to her felf : but an experimental Proof is defired. And fuch a Proof will be had if we can fliew that the Sines of RefraCtion of Rays differently Refrangible are one to another in a given Proportion when their Sines of Incidence are equal. For if the Sines of RefraCtion of all the Rays are in given Proportions to the Sine of RefraCtion LS5] of a Ray which has a mean degree of Reffangibilky, and this Sine is in a given Proportion to the equal Sines of Incidence, thofe other Sines of Refraction will alfo be in given Proportions to the equal Sines of Incidence. Now when the Sines of Incidence are equal, it will appear by the following Experiment that the Sines of Refradion are in a given Proportion to one another. Exper. 15. The Sun Alining into a dark Chamber through a little round hole in the Window-fhut, let S re-Tg*. prefent his round white Image painted on the oppofite Wall by his dired Light, P T his oblong coloured Image made by refrading that Light with a Prifm placed at the Window 3 and pt, or ip it, or 3/? 3 ty his oblong coloured Image made by refrading again the fame Light tideways with a fecond Prifm placed immediately after the firft in a crofs Pofition to it, as was explained in the fifth Experi¬ ment : that is to fay, pt when the Refradion of the fecond Prifm is fmall, 2 p 2 1 when its Refradion is greater, and 3 pit when it is greateft. For fuch will be the diverfity of the Kefradions if the refrading Angle of the fecond Prifm be of various Magnitudes 3 fuppofe of fifteen or twenty degrees to make the Image p t , of thirty or forty to make the Image ip zt, and of fixty to make, the Image 3 p 3 1. But for want of folid Glafs Prifm s with Angles of convenient bigneffes, there may be Veffels made of poliflied Plates of Glafs cemented together in the form of Prifms and filled with Water. Thefe things being thus ordered, I observed that; all the folar images or co¬ loured Spedrums PT, pt, 2p 2t, %p did very nearly converge to the place S on which the dired Light of the Sun fell and painted his white round Image when the Prifms were .taken away. The Axis of the Spedrum PT, that is the Line, drawn through the middle of it Parallel to C5«] its Redilinear Sides, did when produced pafs exactly through the middle of that white round Image S. And when the Refraction of the fecond Prifm was equal to the Refraction of the firll, the refracting Angles of them both being about do degrees, the Axis of the Spectrum ip 3 £ made by that Refraction, did when produced p^fs alfo through the mid¬ dle of the fame white round Image S. But when the Re¬ fraction of the fecond Prifm was lefs than that of the firft, the produced Axes of the SpeCtrums t p or it zp made by that RefraCtion did cut the produced Axis of the Spe¬ ctrum TP in the Points m and ft, a little beyond the Cen¬ ter of that white round Image S. Whence the Proportion of the Line ^Tto the Line 3^ P was a little greater than the Proportion of z t T to 2 pP} and this Proportion a little greater than that of fT topP. Now when the Light of the SpeCtrum P T falls perpendicularly upon the Wall, thofe Lines 3 1 T, 3 p P, and 2 1 T, ip P and t T, p P,are the T an- gents of the Refractions 3 and therefore by this Experiment the Proportions of the Tangents of the Refractions are ob¬ tained, from whence the Proportions of the Sines being deriv¬ ed, they come out equal, fo far as by viewing the SpeCtrums and ufing fome Mathematical reafoning I could Eftimate. Eor I did not make an Accurate Computation. So then the Propofition holds true in every Ray apart, fo far as ap¬ pears by Experiment. And that it is accurately true may be demonftrated upon this Suppofition, That Bodies ref rah Light hy acting upon its (fays in Lines Perpendicular to their Surfaces . But in order to this Demonftration, I mull di- ftinguilh the Motion of every Ray into two Motions, the one Perpendicular to the refracting Surface, the other Pa¬ rallel to it, and concerning the Perpendicular Motion lay down the following Propofition. If If any Motion or moving thing whatfoever be incident with any velocity on any broad and thin Space termina¬ ted on both fides by two Parallel Planes, and in its paflage through that fpace be urged perpendicularly towards the further Plane by any force which at given diftances from the Plane is of given quantities 5 the perpendicular Velo¬ city of that Motion or Thing, at its emerging out of that fpace, fhall be always equal to the Square Root of the Summ of the Square of the perpendicular Velocity of that Motion or Thing at its Incidence on that fpace $ and of the Square of the perpendicular Velocity which that Motion or Thing would have at its Emergence, if at its Incidence its perpendicular Velocity was infinitely little. And the fame Propofition holds true of any* Motion or Thing perpendicularly retarded in its paflage through that (pace, if inftead of the Summ of the two Squares you take their difference. The Demonftration Mathematicians will eafily find out, and therefore I fhall not trouble the Rea¬ der with it, Suppofe now that a Ray coming moll obliquely in the Fig. Line MC be refra&ed at C by the Plane RS into the Line CN, and if it be required to find the Line CE into which any other Ray AC fhall be refracted 3 let MC, AD, be the Sines of incidence of the two Rays, and NG, EF, their Sines of Refraction, and let the equal Motions of the In¬ cident Rays be reprefented by the equal Lines M C and AC, and the Motion MC being confidered as parallel to the refracting Plane, let the other Motion AC be diftin- guifihed into two Motions. AD; and DC, one of winch AD is parallel, and she other DC perpendicular to the re¬ fracting Surface, hi like manner, let the Motions of the emering Rays be diftlngniflfd into two, whereof the per- H pendicular perpendicular ones are ^ CG and ^ CF. And if the force of the refracting Plane begins to aCt upon the Rays either in that Plane or at a certain diftance from it on the one fide, and ends at a certain diftance from it on the other fide, and in all places between thofe two Limits aCts upon the Rays in Lines perpendicular to that rafrafting Plane, and the Actions upon the Rays at equal diftances from the refracting Plane be equal, and at unequal ones ei¬ ther equal or unequal according to any rate whatever • that motion of the Ray which is Parallel to the refracting Plane will fuffer no alteration by that force 5 and that mo¬ tion which is perpendicular to it will be altered according to the rule of the foregoing Propofition. If therefore for the perpendicular Velocity of the emerging Ray CN you write ^ CG as above, then the perpendicular Velocity of any other emerging Ray CE which was ^ CF, will be equal to the fquare Root of CDq + CGq. And by fquaring thefe equals, and adding to them the Equals ADq and MCq — CD and dividing the Summs by the Equals CF$ + EF q and CG^-|-NGf, you will have equal to Whence AD, the Sine of Incidence, is to EF the Sine of RefraCtion, as MC to NG, that is, in a given ratio . And this Demonfiration being general, without determining what Light is, or by what kind of force it is refraCted, or affuming any thing further than that the refracting Body aCts upon the Rays in Lines per¬ pendicular to its Surface 3 I take it to be a very convincing Argument of the full Truth of this Propofition* So then* if the ratio of the Sines of Incidence and Re- fra&ion of any fort of Rays be found in any one Cafe, us given in all Cafes 3 and this may be readily found by the Method in the following Propofition. PROP . VII Theor. VI ‘The (Perfe&ion ofTelefcopes is impeded by the different %efran~ gibility of the Ofays of Light, rpH E imperfection of Telefcopes is vulgarly attri- I buted to the fpherical Figures of the Glaffes, and therefore Mathematicians have propounded to Figure them by the Conical Sections. To Chew that they are mifta- ken, I have inferred this Propofition^ the truth of which will appear by the meafures of the Refractions of the feve- ral forts of Rays 5 and thefe meafures I thus determine* In the third experiment of the firft Book, where the re¬ fracting Angle of the Prifrn was 6z] degrees, the half of that Angle 3 1 deg. 1 5 min. is the Angle of Incidence of the Rays at their going out of the Glafs into the Air 3 and the Sine of this Angle is 5188, the Radius being ioooo. When the Axis of this Prifrn was parallel to the Horizon, and the Refraction of the Rays at their Incidence on this Prifrn equal to that at their Emergence out of it, I obferved with a Qiiadrant the Angle which the mean refrangible Rays (that is, thofe which went to the middle ofthe Sun's colour¬ ed Image ) made with the Horizon and by this Angle and th^ Sun s altitude oblerved at the fame time, I found the Angle which the emergent Rays contained with the incident to be 44 deg. and 40 min. and the half of this Aiwle ^d- ded to the Angle of Incidence 3 1 deg. 1 5 min. males the H t Antrim [So]' Angle of RefraCtion, which is therefore 5 3 degi 3 j min. and its Sine 8047. Thefe are the Sines of Incidence and Re¬ fraCtion of the mean refrangible Rays, and their proportion in round numbers is 20 to 3 1 . This Glafs was of a colouring dining to green. The laft of the Prifms mentioned in the third Experiment was of clear white Glafs. Its reloading Angle 63 i degrees. The Angle which the emergent Rays contained, with the incident 45 deg. 50 min. The Sine of half the firft Angle 5 2 dr. The Sine of half the Summ of the Angles 815 7. And their proportion in round num¬ bers 20 to 3 1 as before. From the Length of the Image, which was about 9-! or 1 o Inches, fubduCt its Breadth, which was 2 ~ Inches, and the Remainder 7* Inches would be the length of the Image were the Sun but a point, and therefore fubtends the An¬ gle which the moft and lead: refrangible Rays, when inci¬ dent on the Prifm in the fame Lines, do contain with one another after their Emergence. Whence this Angle is 2 deg. 0/ 7." For the diftance between the Image and the Prifm where this Angle is made, was 1 8| Feet, and at that diftance the Chord /j Inches fubtends an Angle of 2 deg. 0/ 7." Now half this Angle is the Angle which thefe e- mergent Rays contain with the emergent mean refrangible Rays, and a quarter thereof, that is 30. 2," may be ac¬ counted the Angle which they would contain which the fame emergent mean refrangible Rays, were they co-inci¬ dent to them within the Glafs and differed no other Re- fraCtiomthen that at their Emergence. For if two equal Refractions, the one at the incidence of the Rays on the Prifm, the other at their Emergence, make half the Angle 2 deg. of 7." then one of thofe Refractions will make about a quarter of that Angle, and this quarter added to and ' [S'] and fubduded from the Angle of Refradion of the mean, refrangible Rays, which was 53 deg* 35", gives the An¬ gles of Refradion of the moft and leaft refrangible Rays 54 deg. 5* z", and 53 deg. 4 5 8 ^ whofe Sines are 8099 and 7995, the common Angle of Incidence being 3 1 deg. 1 5' and its Sine 51885 and thefe Sines in the leaft round numbers are in proportion to one another as 78 and 77 to 50. Now if you fubdud the common Sine of Incidence 50 1 from the Sines of Refradion 77 and 78, the remainders 27 and 28 fhew that in fmall Refradions the Refradion > of the leaft refrangible Rays is to the Refradion of the moft refrangible ones as 27 to 28 very nearly, and that the dif¬ ference of the Refractions of the leaft refrangible and moft . refrangible Rays is about the 27{th part of the whole Re-* fraction of the mean refrangible Rays. Whence they that are skilled in Opcicks will eafily un- derftand, that the breadth of the leaft circular fpace into which Object' Glades of Telefcopes can collect all forts of Parallel Rays, is about the 27-th part of half the aperture of the Glafs, or 5 5 th part of the whole aperture 5 and that the Focus of the moft refrangible Rays is nearer to the Object-Glafs than the Focus of the leaft refrangible ones, by about the 27jth part of the diftance between the Object- Glafs and the Focus of the mean refrangible ones. And if Rays of all forts, flowing from any one lucid point in the Axis of any convex Lens, be made by the Refraction ^ of the Lens to converge to points not too remote from the Lens , the Focus of the moft refrangible Rays flhall be nearer to the Lens than the Focus of the leaft refrangible ones, by a diftance which is to the 27^th part of the di¬ ftance of the Focus of the mean refrangible Rays from the Lens as the diftance between that Focus and the lucid point... . ■<; 1 point from whence the Rays flow is to die diftance be¬ tween that lucid point and the Lens very nearly. Now to examine whether the difference between the Re- fractions which the moft refrangible and the leaft refran¬ gible Rays flowing from the fame point fuffer in the Ob- jed-Glalies of Telefcopes and fuch like Glafles, be fo great , as is here defcribed, I contrived the following Experi- : ment. Exper. 1 6. The Lens which I ufed in the fecond and eighth Experiments, being placed fix Feet and an Inch dif- tant from any Objed, colleded the Species of that Object by the mean refrangible Rays at the diftance of fix Feet and an Inch from the Lens on the other fide. And there¬ fore by the foregoing Rule it ought to colled the Species of that Objed by the leaft refrangible Rays at the diftance of Fix Feet and 3 1 Inches from the Lens, and by the moft re¬ frangible ones at the diftance of five Feet and lof Inches from it : So that between the tw o Places where thefe leaft and moft refrangible Rays colled the Species, there may be the diftance of about 5* Inches. For by that Rule, as tfix Feet and an Inch ( the diftance of the Lens from the lucid Object ) is to twelve Feet and two Inches ( the di¬ ftance of the lucid Object from the Focus of the mean re¬ frangible Rays ) that is, as one is to two, fo is the 17 [th part of fix Feet and an Inch (the diftance between the Lens .and the fameFocus ) to the diftance between the Focus of the moft refrangible Rays and the Focus of the leaft re¬ frangible ones, which is therefore 5- inches, that is very nearly 5 ~ Inches. Now to know whether this meafure was true, I repeated the fecond and eighth Experiment of this Book with coloured Light, which was lefs compound¬ ed than that I there made me of : For I now feparated the hetero- fi eterogencous Rays from one another by the Method I de~ fc ribed in the i itn Experiment, fo as to make a coloured Spedrum about twelve or fifteen times longer than broach This Spedrum I call on a printed book, and placing the above-mentioned Lens at the diftance of fix Feet and an Inch from this Spedrum to colled the Species of the illu¬ minated Letters at the fame diftance on the other fide, I found that the Species of the Letters illuminated with Blue were nearer to the Lens than thofe illuminated with deep Red by about three Inches or three and a quarter : but the Species of the Letters illuminated with Indigo and Violet appeared fo confufed and indiftind, that I could not read them : Whereupon viewing the Prifm, I found it was full of Veins running from one end of the Glafs to the other 5 fo that the Refradion could not be regular. I took ano¬ ther Prifm therefore which was free from Veins, and in- ftead of the Letters I ufed two or three Parallel black Lines a little broader than the ftroakes of the Letters, and call¬ ing the Colours upon thefe Lines in fuch manner that the Lines ran along die Colours from one end of the Spedum to the other, I found that the Focus where the Indigo, or confine of this colour and Violet call the Species of the black Lines mort diftindly,tobe about 4 Inches or 4* near¬ er to the Lens than the Focus where the deepeft Red caft the Species of the fame black Lines moft diftindfy. The violet was fo faint and dark, that I could not difeern the Species of the Lines diPdnctly by that Co¬ lour 5 and therefore confidering that the Prifm was made ©f a dark coloured Glafs inclining to Green, I took another Pifm of clear white Glafs 3 but the Spedrum of Colours which this Prifm made had long white Streams of faint Light Shooting out from both ends of the Colours, which made me conclude that fomediing was axmifs 3 and view- t«4] ing the Prifm, 1 found two or three little Bubbles in the Glafs which refracted the Light irregularly. Wherefore I covered that part of the Glafs5 with black Paper, and let¬ ting the Light pais through another part of it which was free from fiich Bubles, the Spectrum of Colours became free from thofe irregular Streams of Light, and was now fuch as I defired. But Mill I found the Violet fo dark and faint, that I could fcarcefee the Species of the Lines by the Violet, and not at all by the deepeft part of it, which was next the end of the Spedlrum. I fulpedted therefore that this faint and dark Colour might be allayed by that Mat¬ tering Light which was refracted, and refledted irregularly partly by fome very final! Bubbles in the Glaffes and partly by the inequalities of their Poliffr. which Light, tho5 it was but little, yet it being of a White Colour, might fuffice to affedt the Senfe fo ftrongly as to difturb the Phenomena of that weak and dark Colour the Violet, and therefore I tried, as in the 12th, 13 th, 14th Experi¬ ments, whether the Light of this Colour did not confift of a fenfible mixture of heterogeneous Rays, but found it did not. Nor did the Refradtions caufe any other fenfible Colour than Violet to emerge out of this Light, as they would have done out of White Light, and by con- fequence out of this Violet Light had it been fenfi- bly compounded with White Light. And therefore I con¬ cluded, that the reafon why I could not fee the Species of the Lines diftindtly by this Colour, was only the darknefs of this Colour and Thinnefs of its Light, and its dif- tance from the Axis of the Lens 3 I divided therefore thofe Parallel Black Lines into equal Parts, by which I might readily know the diltances of the Colours in the Spedfrum from one another, and noted the diftances of the Lens from the Foci of fuch Colours as caft the Species of the Lines Lilies diflin&ly, and then confidered whether the diffe¬ rence of thofe diflances bear fuch proportion to 5 ^Inches, the greatefl difference of the diflances which the Foci of the deepefl Red and Violet ought to have from the Lens, as the diflance of the obferved Colours from one another in the Spedlrum bear to the like diflance of the deepefl Red and Violet meafured in the rectilinear fides of the Spect¬ rum, that is, to the length of thofe fides or excefs of the length of the Spectrum above its breadth. And my Ob- fervations were as follows. When I obferved and compared the deepefl fenfibleRed, and the Colour in the confine of Green and Blue, which at that rectilinear fides of the Spectrum was diflant from it half the length of thofe fides, the Focus where the confine of Green and Blue call the Species of the Lines diflinCtly on the Paper, was nearer to the Lens then the Focus where the Red cafl thofe Lines diCtinCtly on it by about or 2 - Inches. For fometimes the Meafures were a little grea¬ ter, fometimes a little lefs, but feldom varied from one another above ~ of an Inch. For it was very difficult to define the Places of the Foci, without fome little Errors. Now if the Colours diflant half the length of the Image, { meafured at its rectilinear fides ) give z~ or 2 ~ difference of the diflances of their Foci from the Lens, then the Co¬ lours diflant the whole length ought to give j or 5^ Inches difference of thofe diflances. But here its to be noted, that I could not fee the Red to the full End of the SpeCtrum, but only to the Center of the Semicircle which bounded that End, or a little far¬ ther 5 and therefore I compared this Red not with that Co¬ lour which was exadly in the middle of the Spectrum, or confine of Green and Blue, but with that which verged a little more to the Blue than to the Green : And as I reck- I oned [66] dttt& tile whole length of the Colours not to be the whole- length of the SpeCtrum, hot the length of its rectilinear fides, fo completing the Semicirlar Ends into Circles, when either of the obferved Colours fell within thofe Circles, I meafured the diftance of that Colour from the End of the SpeCtrum, and fubduCling half the diftance from the mea¬ fured diftance of the Colours, I took the remainder for their corrected diftance 3 and in thefe Obfervations fet down this corrected diftance for the difference of their di- fiances from the Lens. For as the length of the rectilinear fides of the SpeCtrum would be the whole length of all the Colours, were the Circles of which ( as we (hewed) that SpeCtrum confifts contracted and reduced to Fhyfical Points, fo in that Cafe this corrected diftance would be the real diftance of the obferved Colours. When therefore I further obferved the deepeft fenfible Red, and that Blue whofe corrected diftance from it w^as ^ parts of the length of the rectilinear fides of the SpeCtrum, the difference of the diftances of their Foci from the Lens was about 3 ^ Inches, and as 7- to 12 fo is 3^ to 5 j. When I obferved the deepeft fenfible Red, and that Indi¬ go whofe corrected diftance w^as ^ or ~ of the length of the rectilinear fides of the SpeCtrum, the difference of the di¬ ftances of their Foci from the Lens, was about 3 ^Inches,, and as 2 to 3 fo is 3 *-to 5b When I obferved the deepeft fenfible Red, and that deep- Indigo whofe corrected diftance from one another was ^ or 1 of the length of the rectilinear fides of the SpeCtum, the difference of the diftances of their Foci from the Lens was about 4 Inches 5 and as 3 to 4 fo is 4 to 5 |. When I obferved the deepeft fenfible Red, and that part of the Violet next the Indigo whofe corrected diftance from Red was or £ of the length of the rectilinear fides of the. r.«7] the Spe&rum, the difference of the diftances of their Foci from the Lens was about 4^ Inches 5 and as 5 to 6, fo is 4^ to 57. For fometimes when the Lens was advantage* oufly placed, fo that its Axis relpe&ed the Blue, and all things elfe were well ordered, and the Sun flione clear, and I held my Eye very near to the Paper on which the Lens call the Species of the Lines, I could fee pretty diftinctly the Species of thofe Lines by that part of the Violet which was next the Indigo 3 and fometimes I could fee them by above half the Violet. For in making thefe Experiments I had obferved, that the Species of thofe Colours only ap¬ peared diftinct which were in or near the Axis of the Lens : So that if the Blue or Indigo were in the Axis, I could fee their Species diftinctly 3 and then the Red appeared much lefs diftinct than before. Wherefore I contrived to make the Spectrum of Colours fhorter than before, fo that both its Ends might be nearer to the Axis of the Lens. And now its length was about z[ Inches and breadth about l- or \ of an Inch. Alfo inftead of the black Lines on which the Spectrum was caft, I made one black Line broader than thofe, that I might fee its Species more eafily 3 and this Line I divided by Abort crofs Lines into equal Parts, for meafuringthe diftancesof the obferved Colours. And now I could fometimes fee the Species of this Line with its divi- fions almoft as tar as the Centers of the Semicircular Violet End of the Spectrum, and made thefe further Observations. When I obferved the deepeft fallible Red, and that part of the Violet whofe corrected diftance from it was about | Parts of the rectilinear fidesof the Speftrum the difference of the diftances of the Foci of thofe Colours from the Lens, was one time 4^, another time 4J, another time 4I, Inches, and as 8 to 9, fo are 4*, 4*, 4$, to 5 * 5^5^ refpedively. I 2 When [68] When I obferved the deepeft fenfible Red, and deepeft fenfible Violet, (the Corrected diftance of which Colours when all things were ordered to the beft advantage, and the Sun fihone very clear, was about £ or ^ parts of the length of the rectilinear fides of the coloured Spectrum, ) I found the difference of the diftances of their Foci from the Lens fbmetimes 4^ fometimes 5^ and for the mod part 5 Inches or thereabouts : and as 11 to 1 2 or 15 to 26, fo is five Inches to 5 or 5 l Inches. And by this progreffion of Experiments I fatisfied my felf, that had the light at the very Ends of the Spectrum been ftrong enough to make the Species of the black Lines ap¬ pear plainly on the Paper, the Focus of the deepeft Vio¬ let would have been found nearer to the Lens, than the Fo¬ cus of the deepeft Red, by about 5 *- Inches at leaft. And this is a further Evidence, that the Sines of Incidence and RefraCtion of the feveral forts of Rays, hold the fame pro¬ portion to one another in the fmalleft Refractions whicE they do in the greateft. My progrefs in making this nice and troublefome Expe¬ riment I have fet down more at large, that they that fhali try it after me may be aware of the CircumfpeCtion re- quifite to make it fucceed well. And if they cannot make it fucceed fo well as I did, they may notwithftanding coi¬ led by the Proportion of the diftance of the Colours in the SpeCtrum, to the difference of the diftances of their Foci from the Lens, what would be the fuccefs in the more di- ftant Colours by a better Trial. And yet if they ufe a broader Lens than I did, and fix it to a long {freight Staff by means of which it may be readily and truly directed to the Colour whofe Focus is defired, I queftion not but the Experiment will fucceed better with them than it did with me,. For I directed the Axis as nearly as I could to the middle [ Glafles, and the Charges or magnifying Powers, ought to be as the Cubes of the fcjuare Roots of their lengths 5 which doth not anfwer to Experience. But the errors of the Rays ariflng from the different refrangibility, are as the apertures of the Ob¬ ject- Glafles,. and thence to make Telefcopes of various lengths, magnify with equal diftinCtnefs, their apertures and charges ought to be as the fquare Roots of their lengths 5 and this anfwers to experience as is well known. For in- ftance, a Telefcope of 64 Feet in length, with an aperture of 1- Inches, magnifies about 1 20 times, with as much dif¬ tinCtnefs as one of a Foot in length, with ~ of an Inch aper¬ ture, magnifies 1 5 times. Now were it not for this different refrangibility of Rays, Telefcopes might be brought to a greater Perfection than we have yet defcribed, by compofing the ObjeCt-Glafs of two Glafles with Water between them. Let ADFC repre¬ sent the ObjeCt-Glafs compofed of two Glafles ABED and and BEFC, alike convex on the outfides AGD and CFIF, and alike concave on the infides BME, BNE, with Water in the concavity BMEN. Let the Sine of Incidence out of Glafs into Air be as I to R and out of Water into Air as K to R, and by confequence out of Glafs into Water, as I to K : and let the Diameter of the Sphere to which the convex fides AGD and CHF are ground be D, and the Diameter of the Sphere to which the concave fides BME and BNE are ground be to D, as the Cube Root of KK — KI to the Cube Root of RK — RI: and the Refractions on the con¬ cave fides of the Glafles, will very much correct the Errors of the Refractions on the convex fides, fo far as they arife from the fphericalnefs of the Figure. And by this means might. _ [7$] might Telefcopes be brought to fufficient perfection, were it not for the different refrangibility of feveralforsof Rays. But by reafon of this different refrangibility, I do not yet fee any other means of improving Telefcopes by Refractions alone than that of increafing their lengths, for which end the late contrivance of Hugenius feems well accommodated. For very long Tubes are cumberfome, and fear ce to be readily managed, and by reafon of their length are very apt to bend, and fhake by bending fo as to caufe a continual trembling in the ObjeCts, whereby it becomes difficult to fee them diftin&ly : whereas by his contrivance the GlafTes are readily manageable, and the ObjeCtiGlafs being fixt up¬ on a ftrong upright Polebecomes more fteddy. Seeing therefore the improvement of Telefcopes of given lengths by Refractions is defperate 3 I contrived heretofore a PerfpeCtive by reflexion, ufing inftead of an ObjeCt Glafs a concave Metal. The diameter of the Sphere to which the Metal was ground concave was about 25 Englifli Inches, and by confequence the length of the Inftrument about fix Inches and a quarter. The Eye-Glafs was plano-convex, and the Diameter of the Sphere to which the convex fide was ground was about L of an Inch, or a little lefs, and by con¬ fequence it magnified between 50 and 40 times. By ano¬ ther way of meafuring I found that it magnified about 35 times. The Concave Metal bore an aperture of an Inch and a third part 3 but the aperture was limited not by an opake Circle, covering the Limb of the Metal round about, but by an opake circle placed between the Eye- Glafs and the * Eye, and perforated in the middle with a little round hole for the Rays to pafs through to the Eye. For this Circle by being placed here, ftopt much of the erroneous Light, which otherwife would have difturbed the Villon. By com¬ paring it with a pretty good PerfpeCtive of four Feet in K 2 length, [ 76 ] length, made with a concave Eye- Glafs, L could read at a greater diftance with my own Inftrument than with the. Glafs. Yet Objects appeared much darker in it than in the Glafs, and that partly becaufe more Light was loft by re¬ flexion in the Metal, then by refraction in the Glafs, and partly becaufe my Inftrument was overcharged. Had it. magnified but 50 or 2 5 times it would have made the Object appear more brisk and pleafant. Two of thefelmade about 16 Years ago, and have one of them ftill by me by which L ean prove the truth of what I write. Yet it is not fo good as at the firft. For the concave has been divers times tar- niftied and cleared again, by rubbing it with very foft Lea¬ ther. When I made thefe, an Artift in London undertook to imitate it 3 but ufing another way of polifhing them than I did, he fell much -Abort of what I had attained to, as I afterwards underftood by difeourfing the under-Work¬ man he had imployed. The Polifih I ufed was on this man¬ ner. I had two round Copper Plates each fix Inches in Diameter, the one convex the other concave, ground ve¬ ry true to one another. On the convex I ground the. Oh- jeCl-Metal or concave which was to hepoliflbd, till it had. taken the Figure of the convex and was ready for a Polifh. Then I pitched over the convex very thinly, by droppings melted pitch upon it and warming it to keep the pitch foft, whilft I ground it with the concave Copper wetted to make it fpread evenly all over the convex. Thus by work¬ ing it well I made it as thin as a Groat, and after the .con¬ vex was cold I ground it again to give it as true a Figure as, I could. Then I took.Putty which I had made very fine by wafhing it from all its grolfer Particles,, and laying a lit¬ tle of this upon the pitch, I ground it upon the Pitch with the concave Copper till it had done making a noife 3 and then upon the pitch I ground the Objeft-Metal with a brisk ■ • " *' " * .Motion, C 77' } Motion, for about two or three Minutes of time, leaning hard upon it,. Then I put frefh Putty upon the Pitch and ground it again till it had done making a noife, and after¬ wards ground the Gbjed Metal upon it as before. And this Work I repeated till the Metal was poliflied, grinding it the laft time with all my ftrength for a good while toge¬ ther, and frequently breathing upon the Pitch to keep it moift without laying on any more frefli Putty. The Ob- jed-Metal was two Inches broad and about one third part of an Inch thick, to keep it from bending. I had two of thefe Metals, and when I had polifihed them both I tried which was bell, and ground the other again to fee if I could make it better than that which I kept. And thus by many Trials I learnt the way of polifhing, till I made thofe two refleding Peipedives I fpake of above. For this Art of poliihing will he better learnt by repeated Pradice than by my defcription. Before I ground the Qbjed MetaTon the Pitch, I always ground the Putty on it with the concave Copper till it had done making a noife, becaufe if the Par¬ ticles of the Putty were not by this means made to ftick faft in the Pitch, they would by rolling up and down grate and fret the Objed Metal and fill it full of little holes. But becaufe Metal is more difficult to polifih than Glafs and is afterwards very apt to be fpoiled by tamifhing? and refleds not fo much Light as Glafs quick-filvered over does: I would propound to uleinftead of theMetal, a Glafs ground concave on the forefide, and as much convex on the back- fide, and quicksilvered over on the convex fide. The Glafs s muft be every where of the fame thicknefs exadly. Other- wife it will make Objeds look coloured and indiftind. By fuch a Glafs I tried about five or fix Years ago to make a refleding Telefcope of four Feet in length to magnify a- - bbut 1 50 times, and I fatisfied my felf that there wants no-- •vj? tiling but a good Artift to bring the defign to Perfection. For the Glafs being wrought by one of our London Artifts after fuch a manner as they grind Glaffes for Telefcopes., tho it feemed as well wrought as the ObjeCi Glaffes ufe to be, yet when it was quick-filvered, the reflexion difcovered innumerable Inequalities all over the Glafs. And by reafon of thefe Inequalities, ObjeCts appeared indiftinCt in this In- ftrument. For the Errors of reflected Rays caufed by any Inequality of the Glafs, are about fix times greater than the Errors of refraCled Rays caufed by the like Inequalities. Yet >by this Experiment I fatisfied my felf that the reflexion on the concave fide of the Glafs, which I feared would difturb the vifion,didno fenfible prejudice to it, and by confequence that nothing is wanting to perfect thefe Telefcopes, but good Workmen who can grind and polilh Glaffes truly fphe- rical. An ObjeCt-Glafs of a fourteen Foot Telefcope, made by one of our London Artificers, I once mended confidera- bly, by grinding it on Pitch with Putty, and leaning ve¬ ry eafily on it in the grinding, left the Putty flhould fcratch it. Whether this way may not do well enough for poliflh- ing thefe reflecting Glaffes, I have not yet tried. But he that flhall try either this or any other way of poliflhing which he may think better, may do well to make his Glaffes rea¬ dy for poliflhing by grinding them without that violence, wherevuth our London Workmen prefs their Glaffes in grind¬ ing. For by fuch violent preffure, Glaffes are apt to bend a little in the grinding, and fuch bending will certainly Ip oil their Figure. To recommend therefore the confideration of thefe reflecting Glaffes, to fuch Artifts as are curious in figuring Glaffes, I flhall defcribe this Optical Inftrument in , the following Propofition. PROP. [79] PROP. VII. Prob. II. To Jhorlen Tele/copes. TE T ABDC reprefent a Glafs fpherically concave on the forefide AB,1 and as much convex on the back- fide CD, fo that it be every where of an equal thicknefs. Let it not be thicker on one fide than on the other, left it make Objects appear coloured and indiftindt, and let it be very truly wrought and quick-filveredoveron the backfide 3 and fet in the Tube VXYZ which muft be very black within. Let EFG reprefent a Prifm of Glafs or Cryftal placed near the other end of the Tube, in the middle of it, by means of a handle of Brafs or Iron FGK, to the end of which made flat it is cemented. Let this Prifm be rectangular at E, and let the other two Angles at F and G be accurately equal to each other, and by confequence equal to half right ones, and let the plane tides FE and GE be fquare, and by confe¬ quence the third fideFG a rectangular parallelogram, whofe length is to its breath in a fubduplicate proportion of two to one. Let it be fo placed in the Tube, that the Axis of the Speculum may pafs through the middle of the fquare fide EF perpendicularly, and by confequence through the middle of the fide F G at an Angle of 45 degrees, and let the fide EF be turned towards the Speculum, and the diftance - of this Prifm from the Speculum be fuch that the Rays of the light PQ, RS, ScCi which are incident upon the Speculum in * Lines Parallel to the Axis thereof, may enter the Prifm at the fide EF, and be reflected by the fide F G, and thence go out of it through the fide GE, to the point T which muft be the common Focus of the Speculum ABDC, and of a Plano-convex Eye- Glafs FI, through which thofe Rays muft. pafs to the Eye. . And let the Rays at their corning out [ 8o ] Out of the Glafs pafs through a fmall round hole, or aper¬ ture made in a little Plate of Lead, Brafs, or Silver, where¬ with the Glafs is to be covered, which hole muff be no bigger than is neceiTary for light enough to pafs through. For fo it will render the Objedt diftindt, the Plate in which "tis made intercepting all the erroneous part of the Light which comes from the Verges of the Speculum AB. Such an Inftrument well made if it be 6 Foot long, ( reckoning the length from the Speculum to the Prifm, and thence to the Focus T) will bear an aperture of 6 Inches at the Spe¬ culum, and magnify between two and three hundred times. But the hole H here limits the aperture with more advan¬ tage, then if the aperture was placed at the Speculum. If the Inftrument be made longer or fihorter, the aperture mu ft be in proportion as the Cube of the fquare Root of the length, and the magnifying as the aperture. But its con¬ venient that the Speculum be an Inch or two broader than the aperture at the leaft, and that the Glafs of the Speculum be thick, that it bend not in the working. The Prifm EFG niuft be no bigger than is neceflary, and its back fide FG muft not be quick-filvered over. For without quick-filver it will refledt all the Light incident on it from the Speculum. In this Inftrument the Objedt will be inverted, but may be eredted by making the fquare fides EF and EG of the Prifm EFG not plane but fpherically convex, that the Rays may crofs as well before they come at it as afterwards be¬ tween it and the Eye-Glafs. If it be defired that the Inftru¬ ment bear a larger aperture, that may be alfo done by com- pofing the Speculum of two Glaffes with Water between them. THE Book I. Plate I. Part i. Fig. y. A / R \ Q. Book I. Plate I. Parti. [8i] THE FIRST BOOK O F O P T I C K s. PAR T II. PROP. I, THEOR. I. 1 he Phenomena of Colours in ref ratted or refietted Light are not caufed by new modifications of the Light varioufi \\ ly imfrejl , according to the variom terminations of the Light and Shadow . The Proof by Experiments. EX PER. I. FO R if the Sun fhine into a very dark Chamber pV % through an oblong Hole F, whofe breadth is the °* lixth or eighth part'of an Inch, or fomething lefs ; and his Beam F H do afterwards pafs firft through a very large Prifm ABC, diftant about oo Feet from the L • Hole, '[82] Hole, and parallel to it, and then (with its white part) through an oblong Hole H, whole breadth is about the fortieth or fixtieth part of an Inch, and which is made in; a black opake Body G I, and placed at the diftance of two or three Feet from the Prifm, in a pa¬ rallel lituation both to the Prifm and to the former Hole, and if this white Light thus tranfmitted through the Hole H, fall afterwards upon a white Paper PL placed after that Hole H, at the diftance of three or four Feet from it, and there paint the ufual Colours of the Prifm, fuppofe red at t, yellow at s, green at r, blue at q, and violet at p ; you may with an iron Wire, or any fuch like (lender opake Body, whole breadth is about the tenth part of an Inch, by intercepting the rays at k, 1, m, n or o, take away any one of the Colours* at t, s, r, q or p, whilft the other Colours remain up¬ on the Paper as before ; or with an obftacle fomething bigger you may take away any two, or three, or four Co¬ lours together, the reft remaining : So that any one of the Colours as well as violet may become outmoft in the confine of the fhadow towards p, and any one of them as well as red may become outmoft in the confine of the lhadow towards t, and any one of them may alfo border upon the fhadow made within the Colours by the obftacle R intercepting fome intermediate part of the Light ; and, laftly, any one of them by * being left alone may border upon the fhadow on either hand. All the Colours have themfelves indifferently to any confines of fhadow, and therefore the differences of thefe Colours from one another, do not arife from the diffe¬ rent confines of fhadow, whereby Light is varioufly modified as has hitherto been the Option of Philofo- phers. r «3 ] phers. In trying thefe things Lis to be obferved, that by how much the Holes F and H are narrower, and the intervals between them, and the Prifm greater, and the Chamber darker, by fo much the better doth the Ex¬ periment fucceed ; provided the Light be not fo far diminifhed, but that the Colours at p t be fufficiently viiible. To procure a Prifm of folid Glafs large enough for this Experiment will be difficult , and therefore a prifmatick Veffel muft be made of polifhed Glafs-plates cemented together, and filled with Water, EXPEL II. The Sun’s Light let into a dark Chamber through Fig. the round Hole F, half an Inch wide, paffed firft through the Prifm ABC placed at the Hole, and then through a Lens PT fomething more than four Inches broad, and about eight Feet diftant from the Prifm, and thence con¬ verged to O the Focus of the Lens diftant from it about three Feet, and there fell upon a white Paper DE„ If that Paper was perpendicular to that Light incident up- on it, as Lis reprefented in the pofture D E, all the Co¬ lours upon it at O appeared white* But if the Paper being turned about an Axis parallel to the Prifm, be¬ came very much inclined to the Light as Lis reprefeiv ted in the pofttions de and " - ; | I I . , - * , i. ■ B ook I . Plate HI. Par t 1 r\K *% \\ \ 1 Book I. Plate BE. Parti. a Book, I, fh te,W, Fig. 23. BOOK, I. Part I. Plate V, B C D E F ■1 I O'**'-"'' Fig. 23. A B C D E F BOOK, I, Part I. Plate V. Fig 25 . £ d e f h 1 h I 71L Fig. 2 7. A /'/gT 2 £ ABC // I g:i;;mh|p \\ DIF whS ■jm in Fig 2 (J. W [?l] gate this or that Motion into the Senforium, and in the Senforium they are fenfations of thole Mo¬ tions under the forms of Colours. PROP. III. PRO EL I. To define the refrcmgibihty of the fever d forts of homo* genecd Light anfwering to the fever d Colours . For determining this Problem I made the following Experiment. EXPER, V1L 1 When I had caufed the redilinear line fides AF, GM, Fig . 4, of the Spedrum of Colours made by the Prifm to be diftindly defined, as in the fifth Experiment of the firft Book is defcribed, there were found in it all the homogeneal Colours in the fame order and Situation one among another as in the Spedrum of fimple Light, defcribed in the fourth Experiment of that Book. For the Circles of which the Spedrum of compound Light FT is compofed, and which in the middle parts of the Spedrum interfere and are intermixt with one ano¬ ther, are not intermixt in their outmoft parts where they touch thofe redilinear fides A F and G M. And therefore in thofe redilinear fides when diftindly defi¬ ned, there is no new Colour generated by refradiom I obferved alio, that if any where between the two out¬ moft Circles TMF and P G A a right line, as 5 was crofs to the Spedrum, fo as at both ends to fall per¬ pendicularly upon its redilinear fides, there appeared M a one' i 0 3 one and the fame Colour and degree of Colour from one end of this line to the other. 1 delineated therefore in a Paper the perimeter of the SpeCtrum F APGMT, and in trying the third Experiment of the firft Book, I held the Paper fo that the Spectrum might fall upon this delineated Figure, and agree with it exaCtly, whilft an Affiftant whofe Eyes for diftinguifhing Colours were more critical than mine, did by right lines drawn crofs the SpeCtrum, note the confines of the Co¬ lours that is of theredM*£F of the orange * 7 * of the yellow 7 g i of the green ^ 0 1 , of the blue >1 * x.e ^ of the indico « and of the violet xGAm, And this operation being divers times repeated both in the fame and in federal Papers, I found that the Ob- fervations agreed well enough with one another, and that the rectilinear fides M G and F A were by the laid crofs lines divided after the manner of a mufical Chord. Let GM be produced to X, that MX may be equal to GM, and conceive G X, xX, JX, * X, *X, 7X, M X, to be in proportion to one another, as the num¬ bers 1, L L p b *6’ L and fo to reprefent the. Chords of the Key, and of a Tone, a third Minor, a fourth, a fifth, a fixth Major, a feventh, and an eighth above that Key : And the intervals «y, 7% ..** 9 xx ? alld xG, will be the ipaces which the leveral Co¬ lours (. red, orange, yellow, green, blue, indico, violet ) take up.^ Now thefe- intervals or fpacts fubtending the diffe¬ rences of the refraftions of the rays going to the limits, of thefe Colours, that is^ to the points M, <*, #,..h *,/, x, G, may without any fenfible.. Error be accounted propor¬ tional to the differences of the fines of net raCt ion of.th.ofe. rays [93]. . rays having one common fine of incidence, and there¬ fore fince the common fine of incidence of the inoft and leaft refrangible rays out of Glafs into Air was, ( by a method defcribed above ) found in proportion to their fines of refraction, as 50 to 77 and 78, divide the dif¬ ference between the fines of refraction 77 and 78, as the line G M is divided by thole intervals, you will have n, n\y 77} » 77j, 1% 77 a 77^,78,, the fines of refraCtion of thofe rays out of Glais into. Air , their common fine of incidence being 50. So then the fines of the incidences of all the red-making rays out of Glafs into Air, were to the fines- of their refraCtions, not greater than 50 to 77, nor lefs than 50 to 77-, but varied from one another according to all interme¬ diate Proportions. And the fines of the incidences of the green-making rays were to the fines of their refraCtions in all proportions from that of 50 to 77-1, unto that of 50 to 77-f And by the like limits above-mentioned were the refraCtions of the rays be¬ longing to the reft of the Colours defined, the fines of the red-making; rays extending from 77 to 77^ thofe of the orange-making from 77 \ to , thofe of the yel¬ low-making from 77~ to 77 thole of the green-making from 77^ to 7 7G thofe of the blue-making from jj~ to 77“, thofe of the indico-making from 77- to 77*, and thofe of the violet from 77 z9 to 78, . Thefe are the Laws of the refraCtions made out of Glafs into Air, and thence by the three Axioms of the firft Book the Laws of the refraCtions made out of Air. into Glafs are eafily derived. EXPER. [9+] EXPER. VIII. I found moreover that when Light goes out of Air through feveral contiguous refracting Mediums as through Water and Glafs, and thence goes out again into Air, whether the refracting fuperficies be parallel or inclined to one another, that Light as often as by contrary refraCtions "tis fo corrected, that it emergeth in lines parallel to thofe in which it was incident, continues ever after to be white. But if the emer¬ gent rays be inclined to the incident, the whitenefs of the emerging Light will by degrees in paffing on from the place of emergence, become tinged in its edges with Colours. This I tryed by refraCting Light with Prifms of Glafs within a prifmatick Veffel of Water. Now thofe Colours argue a diverging and reparation of the hetero¬ geneous rays from one another by means of their un¬ equal refraCtions, as in what follows will more fully appear. And, on the contrary, the permanent white¬ nefs argues, that in like incidences of the rays there is no fuch reparation of the emerging rays, and by confe- quence no inequality of their whole refraCtions. Whence 1 feem to gether the two following Theorems. i . The Exceffes of the fines of refraCtion of feveral forts of rays above their common fine of incidence when the refraCtions are made out of divers denfer mediums immediately into one and the fame rarer medium, are do one another in a given Proportion 2, The [95] a. The Proportion of the fine of incidence to the line of refraction of one and the fame fort of rays out of one medium into another, is compofed of the Proportion of the fine of incidence to the fine of refraction out of the firft medium into any third medium, and of the Pro¬ portion of the line of incidence to the fine of refraCtioiv out of that third medium into the fecond medium* By the firft Theorem the refractions of the rays of every fort made out of any medium into Air are known , by having the refraction of the rays of any one fort. As for inftance, if the refractions of the rays of every fort out of Rain-water into Air be defired, let the common line of incidence out of Glafs into Air be fubduCted from the fines of refraCtion, and the Exceffes will be ay, ay£,-.ayj, if-, ij[, if-r,if9yi%. Suppofe now that the fine of incidence of the leaft refrangible rays be~ to their fine of refraction out of Rain-water into Air as three to four, and fay as i the difference of thofe fines is to 3 the fine of incidence, fo is ay the leaft of the. Exceffes above-mentioned to a fourth number 8 1 j and 8.1- will be the common fign of incidence out of Rain¬ water into Air, to which fine if you add all the above- mentioned Exceffes you will have the defired fines of the refractions 108., io8i? io8f, io8i, io8t, io8f, I C89 ? IO9. By the latter Theorem the refraCtion out of., one me¬ dium into another is gathered as often as you. have the refractions out of them both into any third medium. As if the fine of incidence of any ray out of Glafs into Air be to its fine of refraCtion as ao to 3 1, and the fine of incidence of the fame ray out of Air into Water, be to [ 9^ ] to its fine of refra£tion as four to three ; the fine of incidence of that ray out of Glafs into Water will be to its fine of refraction as ao to 3 1 and 4 to 3 joyntly, that is, as the FaCtum of 20 and 4 to the Falfum of 3 1 and 3, or as 80 to 93. And thefe Theorems being admitted into Opticks, there would be fcope enough of handling that Science voluminoufly after a new manner ; not only by teaching thofe things which tend to the perfection of vifion, but alfo by determining mathematically all kinds of Pheno¬ mena of Colours which could be produced by refra¬ ctions. For to do this, there is nothing elfe requifite than to find out the reparations of heterogeneous rays, and their various mixtures and proportions in every mixture. By this way of arguing I invented almoft all the Phenomena defcribed in thefe Books, befide fome others lefs neceflary to the Argument ; and by the fuccefles I met with in die tryals, I dare promile, that to him who (hall argue truly, and then try all things with good Glaffes and fufficient circumfpeCtion, the expefted event will not be wanting. But he is firft to know what Colours will arife from any others mixt in any affigned Proportion. PROP. IV. THE OR. III. Colours may be produced by comfofition which jhatt be like to the Colours of homogene al Light as to the appearance of Colour , but not as to the immutability of Colour and conjlitutim of Light, Lind thofe Colours by ho‘w much they are more compounded by jo much are they lejs f ull .md inteufe , and by too much compofition they may be diluted [97] diluted and weakened till they ceaje . ‘ There may he aljo Colours produced by comnojition^ which are not fully like any oj the Colours oj homogeneal Light . For a mixture of homogeneal red and yellow com¬ pounds an orange, like in appearance of Colour to that orange which in the feries of unmixed prifmatick Co¬ lours lies between them ; but the Light of one orange is homogeneal as to refrangibility, that of the other is heterogeneal, and the Colour of the one , if viewed through a Prifm, remains unchanged, that of the other is changed and refolved into its component Colours red and yellow. And after the fame manner other neigh¬ bouring homogeneal Colours may compound new Co¬ lours, like the intermediate homogeneal ones, as yel¬ low and green, the Colour between them both, and af¬ terwards, if blue be added, there will be made a green the middle Colour of the three which enter the compo- lition. For the yellow and blue on either hand, if they are equal in quantity they draw the intermediate green equal¬ ly towards themfelves in competition, and fo keep it as it were in equilibrio, that it verge not more to the yellow on the one hand, than to the blue on the other, but by their mixt actions remain ftill a middle Colour. To this mixed green there may be further added feme red and violet, and yet the green will not prefent- ly ceafe but only grow lets full and vivid, and by in- creafing the red and violet it will grow more and more dilute, until by the prevalence of the added Colours it be overcome and turned into whitenefs, or feme other Colour, do if to the Colour of any homogeneal Light, the Sun's white Light compofed of all forts of rays be N added. added, that Colour will not vanifh or change its fpe* cies but be diluted., and by adding more and more white it will be diluted more and more perpetually. Lafi> ly, if red and violet be mingled, there will be generated according to their various Proportions various Purples, fuch as are not like in appearance to the Colour of any homogeneal Light, and of thefe Purples mixt with yel¬ low and blue may be made other new Colours. PROP. Y. THEOR. IV. Whitenefs and all grey Colours between white and blacks may be compounded of Colours , and the whitenefs of the Suns Light is compounded of all, the primary Colours mixt in a due proportion ... The Proof by Experiments ; ; ■ EX PER. IX. ' pig, ^ The Sun Aiming into a dark Chamber through a ° ' little round Hole in the Window fhut, and his Light being there refrafted by a Prifm to cafi his coloured- image PT upon the oppofite Wall : I held a white Pa¬ per V to that Image in fuch manner that it might be illuminated by the coloured Light refected from thence, and yet not intercept any part of that Light in its paL fage from the Prifm to the Spedtrum. And I found that when the Paper was held nearer to any Colour than to the reft, it appeared of that Colour to which it ap¬ proached neareft ; but when it. was equally or almoft equally [ 99 3 equally difrant from all the Colours, fo that it might be equally illuminated by them all it appeared white. And in this la ft fituation of the Paper, iffome Colours were intercepted, the Paper loft its white Colour, and appeared of the Colour of the reft of the Light which was not intercepted. So then the Paper was illuminated with Lights of various Colours, namely, red, yellow, green, blue and violet, and every part of the Light re¬ tained its proper Colour, until it was incident on the Paper, and became reflected thence to the Eye ; fo that if it had been either alone (the reft of the Light being intercepted) or if it had abounded moft and been pre¬ dominant in the Light reflected from the Paper, it would have tinged the Paper with its own Colour; and yet be¬ ing mixed with the reft of the Colours in a due propor¬ tion, it made the Paper look white, and therefore by a eompofition with the reft produced that Colour, The feveral parts of the coloured Light reflected from the -Sped rum, whilft they are propagated from thence thro" \ the Air, do perpetually retain their proper Colours, becaufe wherever they fall upon the Eyes of any Specta¬ tor, they make the feveral parts of the SpeCtrum to appear under their proper Colours. They retain there¬ fore their proper Colours when they fall upon the Pa¬ per V, and fo by the confufion and perfect mixture of thofe Colours compound the whitenefs of the Light reflected from thence, EX PER, X, Let that Spectrum or tolar Image P T fall now upon Fig, 6 the .Lens M N above four Inches broad, and about fix N 2 Feet [ i oo ] Feet diftant from the Prifm ABC, and fo figured that it may caufe the coloured Light which divergeth from the Prifm to converge and meet again at its Focus G, about fix or eight Feet diftant from the Lens, and thereto fall perpendicularly upon a white Paper DE. And if you move this Paper to and fro, you will per¬ ceive that near the Lens, as at dey the whole folar Image (fuppofe at pt) will appear upon it intenfly coloured after the manner above-explained, and that by receding from the Lens thofe Colours will perpetually come to¬ wards one another, and by mixing more and more di¬ lute one another continually, until at length the Paper come to the Focus G, where by a perfect mixture they will wholly vanifh and be converted into whitenefs, the whole Light appearing now upon the Paper like a little white Circle, And afterwards by receding further from the Lens, the rays which before converged will now crofs one another in the Focus G, and diverge from thence, and thereby make the Colours to appear again, but yet in a contrary order ; fuppofe at ^ , where the red t is now above which before was below, and the violet p is below which before was above. Let us now flop the Paper at the Focus G where the Light appears totally white and circular, and let us eonfider its whitenefs. I fay, that this is com poled of the converging Colours. For it any of thofe Colours be intercepted at the Lens, the whitenefs will ceafe and degenerate into that Colour which arifeth from the compofition of the other Colours which are not inter¬ cepted. And then if the intercepted Colours he let pais and fall upon that compound Colour, they mix with it, and by their mixture reftore the whitenefs. [ 101 ] So if the violet, blue and green be intercepted, the re¬ gaining yellow, orange and red will compound upon the Paper an orange, and then if the intercepted Co¬ lours be let pals they will fall upon this compounded orange, and together with it decompound a white. So alfo if the red and violet be intercepted, the remaining yellow, green and blue, will compound a green upon the Paper, and then the red and violet being let pafs will fall upon this green, and together with it decom¬ pound a white. And that in this compofition of white the feveral rays do not fuffer any change in their colori¬ fic qualities by ading upon one another, but are only mixed, and by a mixture of their Colours produce white, may further appear by thefe Arguments. If the Paper be placed beyond the Focus G, fuppofe at ^ , and then the red Colour at the Lens be alternate¬ ly intercepted, and let pafs again, the violet Colour on the Paper will not fuffer any change thereby, as it ought to do if the feveral forts of rays aded upon one another in the Focus G, where they crofs. Neither will the red upon the Paper be changed by any alternate flop¬ ping, and letting pafs the violet which crofleth it. And if the Paper be placed at the Focus G, and the white round Image at G be viewed through the Prilm HIK, and by the reflation of that Prilm be tranflated to the place rv, and there appear tinged with various Colours, namely, the violet at v and red at r, and others between, and then the red Colour at the Lens be often ftopt and let pafs by turns, the red at r will ac¬ cordingly difappear and return as often, but the violet at v will not thereby fuffer any change. And fo by flopping and letting pafs alternately the blue at the Lens, I *02 'Lens, the blue at r will accordingly difappear and re- turn, without any change made in the red at r. The red therefore depends on one fort of rays, and the blue on another fort, which in the Focus G where they are commixt do not aft on one another. And there is the fame reafon of the other Colours. I confidered further, that when the mod refrangible rays P p, and the leaft refrangible ones T t, are by con¬ verging inclined to one another, the Paper, if held very oblique to thofe rays in the Focus G, might refleft one fort of them more copioufly than the other fort, and by that means the reflected Light would be tinged in that Focus with the Colour of the predominant rays, pro¬ vided thofe rays feverally retained their Colours or co¬ lorific qualities in the compofition of white made by them in that Focus. But if they did not retain them in that white, but became all of them feverally endued there with a difpoiition to ftrike the fenfe with the pew ception of white, then they could never lofe their white- nefs by fuch reflexions. I inclined therefore the Paper to the rays very obliquely, as in the fecund Experiment of this Book, that the rnoft refrangible rays might be more copioufly reflefted than the reft, and the white- nefs at length changed fucceffively into blue, indico and violet. Then I inclined it the contrary way, that the moft refrangible rays might be more copious in the reflefted Light than the reft, and the whitenefs turned fucceffively to yellow, orange and red. . . Laftly, I made an Inftrument X Y in fafhion of a Comb, whofe Teeth being in number fixtecn were about an Inch and an half broad, and the intervals of the Teeth about two Inches wide. Then by interpofing fuc- C io3 ] iucceffively the Teeth of this Inftrument near the Lens, I intercepted part of the Colours by the interpofed Toothy whilft the reft of them went on through the in¬ terval of the Teeth to the Paper D E, and there pain¬ ted a round folar Image. But the Paper I had firft pla¬ ced fo, that the Image might appear white as often as the Comb was taken away; and then the Comb be¬ ing as was faid interpofed, that whitenefs by reafon of the intercepted part of the Colours at the Lens did al¬ ways change into the Colour compounded of thofe Colours which were not intercepted, and that Colour was by the motion of the Comb perpetually varied fo, that in the paffing of every Tooth over the Lens ail thefe Colours red, yellow, green, blue and purple, did always fucceed one another. I caufed therefore all the Teeth to pafs fucceffively over the Lens, and when the motion was How, there appeared a perpetual fuccefiion of the Colours upon the Paper : But if I fo much acce¬ lerated the motion, that the Colours by reafon of their quick fuccefiion could not be diftinguiftied from one another, the appearance of the fingle Colours ceafed. There was no red, no yellow, no green, no blue, nor purple to be feen any longer, but from a confufion of them all there arofe one uniform white Colour. Of the Light which now by the mixture of all the Colours ap¬ peared white, there was no part really white. One part was red, another yellow, a third green, a fourth blue, a fifth purple, and every part retains its proper. Colour till it ftnke the Senforium. If the imprefiions follow one another (lowly, fo that they may be feve- rally perceived, there is made a diftinft fenfation of all the Colours one after another in a continual fuccefiion. But..., [io4] But if the impreffions follow one another fo quickly that they cannot be feverallv perceived, there arifeth out of them all one common fenfation, which is nei¬ ther of this Colour alone nor of that alone, but hath it felf indifferently to ’em all, and this is a fenfation of whitenefs. By the quicknefs of the fucceffions the im¬ preffions of the feveral Colours are confounded in the Senforium, and out of that confufion arifeth a mixt fen¬ fation, If a burning Coal be nimbly moved round in a Circle with Gyrations continually repeated, the whole Circle will appear like fire ; the reafon of which is, that the fenfation of the Coal in the feveral places of that Circle remains impreft on the Senforium, until the Coal return again to the fame place. And fo in a quick confecution of the Colours the impreffion of every Colour remains in the Senforium, until a revolution of all the Colours be compleated, and that firft Colour re¬ turn again. The impreffions therefore of all the fucceffive Colours are at once in the Senforium, and joyntly ftir up a fenfation of them all ; and fo it is manifeft by this Ex¬ periment, that the commixt impreffions of all the Co¬ lours do ftir up and beget a fenfation of white, that is, that whitenefs is compounded of all the Colours, And if the Comb be now taken away, that all the Colours may at once pals from the Lens to the Paper, and be there intermixed, and together reflected thence to the Spectators Eyes ; their impreffions on the Senfo- rium being now more lubtily and perfectly commixed there, ought much more to ftir up a fenfation of white- nefs* You [10$] ' Y6u may inftead of the Lens ufe two Prifms HIK and L M N, which by refracting the coloured Light the contrary way to that of the firft refra&ion, may make the diverging rays converge and meet again in G, as you fee it repreiented in the feventh Figure. For Fig . y. where they meet and mix they will compote a white Light as when a Lens is ufed. EXPEL XL Let the Sun’s coloured Image PT fall upon the Wall Fig* 8* of a dark Chamber, as in the third Experiment of the firft Book, and let the fame be viewed through a Prifm a be, held parallel to the Prifm ABC, by whofe refra¬ ction that Image was made, and let it now appear lower than before, fuppofe in the place S over againft the red- colour T. And if you go near to the Image PT, the Speftrum S will appear oblong and coloured like the Image P T • but if you recede from it, the Colours of the Spectrum S will be contracted more and more, and at length vaniih, that Spectrum S becoming perfectly round and white ; and if you recede yet further, the Colours will emerge again, but in a contrary order. Now that SpeCtrum S appears white in that cafe when the rays of leveral forts which converge from the feve- ral parts of the Image PT, to the Prifm a be, are fo retraCted unequally by it, that in their paffage from the Prifm to the Eye they may diverge from one and the lame point of the SpeCtrum S, and fo fall afterwards upon one and the fame point in the bottom of the Eye, and there be mingled. O And Fig. 9. [lod] And further, if the Comb be here made ufe of, by whole Teeth the Colours at the Image PT may be fuo ceffively intercepted ; the Spedxum S when the Comb is moved flowly will be perpetually tinged with fuc- ceffive Colours : But when by accelerating the motion of the Comb, the fucceffion of the Colours is fo quick that they cannot be feverally feen, that Spedfrum S, by a confufed and mixt lenfation of them all, will appear white. EXPER. XII. The Sun Alining through a large Prifm ABC upon a Comb X Y, placed immediately behind the Prifm, his Light which paffed through the interftices of the Teeth fell upon a whitePaper BE. The breadths of the Teeth Were equal to their interftices, and feven Teeth together with their interftices took up an Inch in breadth. Now when the Paper was about two or three Inches diftant from the Comb, the Light which paffed through its feveral interftices painted fo many ranges of Colours kl, mn, op, qr, tor. which were, parallel to one another and contiguous, and without any mixture of white. And thefe ranges of Colours, if the Comb was moved continually up and down with a re¬ ciprocal motion, afcended and defcended in the Paper, and when the motion of the Comb was fo quick, that the Colours could not be diftinguilhed from one another, the whole Paper by their confufion and mixture in the Senforium appeared white* Let ' C I07 3 Let the Comb now reft, and let the Paper be remo¬ ved further from the Prifin, and the feveral ranges of Colours will be dilated and expanded into one another more and more, and by mixing their Colours will di¬ lute one another, and at length, when the diftance of the Paper from the Comb is about a Foot , or a little more ( fuppofe in the place 2 D 2 E ) they will fo far dilute one another as to become white,, With any Obftacle let all the Light be now ftopt wdiich paffes through any one interval of the Teeth, lb that the range of Colours which comes from thence may be taken away, and you will fee the Light of the reft of the ranges to be expanded into the place of the range taken away, and there to be coloured. Let the inter¬ cepted range pafs on as before, and its Colours falling upon the Colours of the other ranges, and mixing with them, will reftore the whitenefs. Let the Paper 2D 2 E be now very much inclined to the rays, fo that the mo ft refrangible rays may be more copioufly reflected than the reft, and the white Colour oi the Paper through the excefs of thofe rays will be changed into blue and violet. Let the Paper be as much inclined the contrary way, that the leaft refran¬ gible rays may be now more copioufly reflected than the reft, and by their excefs the whitenefs will be changed into yellow and red. The feveral rays there- fore in that white Light do retain their colorific qua¬ lities, by which thole of any fort, when-ever they be- come more copious than the reft, do by their excels and predominance caufe their proper Colour to ap¬ pear, - x ; And [ io8 ] And by the fame way of arguing, applied to the third Experiment of this Book, it may be concluded, that the white Colour of all refracted Light at its very firft emergence, where it appears as white as before its inci¬ dence, is compounded of various Colours. EX P ER. XIII. In the foregoing Experiment the feveral intervals of the Teeth of the Comb do the office of fo many Prifms., every interval producing the Phenomenon of one Prifm. Whence inftead of thofe intervals ufing feveral Prifms, I try'd to compound whitenefs by mixing their Colours, and did it by ufing only three Prifms, as alfo by ufing only fig. io. two as follows. Let two Prifms ABC and a be, whole refrafting Angles B and b are equal, be fo placed parallel to one another, that the refracting Angle B of the one may touch the Angle c at the bafe of the other, and their planes CB and cb, at which the rays emerge, may lye in direftum. Then let the Light trajeCted through them fall upon the Paper M M, diftant about 8 or 1 2 Inches from the Prifms. And the Colours generated by the interior limits B and c of the two Prifms, will be mingled at PT, and there compound white. For if either Prifm be taken away, the Colours made by the other will appear in that place PT, and when the Prifm is reflated to its place again, fo that its Colours may there fall upon the Colours of the other, the mixture <*f them, both will reftore the whitenefs. [ lop ] This Experiment fueceeds alfo, as 1 have try ed, when the Angle b of the lower Prifm, is a little greater than the Angle B of the upper , and between the interior Angles B an Ac, there intercedes fome fpace Be, as is reprefented in the Figure, and the refracting planes BC and be, are neither in dire&um, nor parallel to one another. For there is nothing more requifite to the fuccefs of this Experiment, than that the rays of all forts may be uniformly mixed upon the Paper in the place PT. If the mod refrangible rays coming from the fuperior Prifm take up all the fpace from M toP, the rays of the fame fort which come from the inferior Prifm ought to begin at P, and take up all the reft of the fpace from thence towards N. If the leaft refrangible rays coming from the fuperior Prifm take up the fpace M T, the rays of the fame kind which come from the other Prifm ought to begin atT, and take up the remain¬ ing fpace T N». If one fort of the rays which have in¬ termediate degrees of refrangibility, and come from the fuperior Prifm be extended through the fpace MQ, and another fort of thofe rays through the fpace MR, and a third fort of them through the fpace M S, the fame forts of rays coming from the lower Prifm, ought to il¬ luminate the remaining fpaces QM, RN, SN refpe- lively. And the fame is to be underftood of all the other forts of rays. For thus the rays of every fort will be fcattexed uniformly and evenly through the whole fpace MN, and fo being every where mixt in the fame, proportion, they muft every where produce the fame Colour. And therefore fince by this mixture they pro- duce white in the exterior fpaces M P and TN, fhey muft alio produce white in the interior fpace P T, This [no] is the reafon of the compofition by which whitenefs was produced in this Experiment, and by what other way foever I made the like compofition the refult was whitenefs. Laftly, If with the Teeth of a Comb of a due fize. the coloured Lights of the two Prifms which fall upon the fpace P T be alternately intercepted , that fpace P T, when the motion of the Comb is flow, will always appear coloured, but by accelerating the motion of the Comb fo much, that the fucceflive Colours can¬ not be diftingui filed from one another, it will appea r white. EXPEL XIV. i Hitherto I have produced whitenefs by mixing the Colours of Prifms. If now the Colours of natural Bo¬ dies are to be mingled, let Water a little thickned with Soap be agitated to raife a froth, and after that froth has flood a little, there will appear to one that fliall view it intently various Colours every where in the furfaces of the feveral Bubbles ; but to one that (hall go fo far off that he cannot diftinguifh the Colours from one another, the whole froth will grow white with a perfect whitenefs. EX PER. XV. Laftly, in attempting to compound a white by mixing the coloured Powders which Painters ufe, I confidered that all coloured Powders do fupprefs and flop in them a very confiderable part of the Light by which they [ill ]' they are illuminated. For they become coloured by reflecting the Light of their own Colours more copioufly, and that of all other Colours more fparinglv, and yet they do not refleCt the Light of their own Colours fo copioufly as white Bodies do. If red Lead, for inftance, and a white Paper, be placed in the red Light of the coloured SpeCtrum made in a dark Chamber by the re¬ fraction of a Prifm, as is defcribed in the third Eperi- ment of the firft Book ; the Paper will appear more lu¬ cid than the red Lead, and therefore reflects the red- making rays more copioufly than red Lead doth. And if they be held in the Light of any other Colour, the Light reflected by the Paper will exceed the Light re¬ flected by the red Lead in a much greater proportion. And the like happens in Powders of other Colours. And therefore by mixing Inch Powders we are not to expeCt a ftrong and full white, fuch as is that of Paper, but fome dusky obfcure one, luch as might arife from a mixture of light and darknefs, or from white and black, that is, a grey, or dun, or ruffet brown, fuch as are the Colours of a Man’s Nail, of a Moufe, of Allies, of or¬ dinary Stones, of Mortar, of Dull and Dirt in High¬ ways, and the like. And fuch a dark white I have often produced by mixing coloured Powders. For thus one part of red Lead, and five parts of Viride H£m,com- pofed a dun Colour like that of a Moufe. For thefe two Colours were feverally fo compounded of others, that in both together were a mixture of all Colours ; and there was lefs red Lead ufed than Viride ALrhj becaufe of the fuinefs of its Colour. Again, one part of red Lead, and four parts of blue Bife, compofed a dun Co¬ lour verging a little to purple, and by adding to this a sextain [ II21 certain mixture of Orpiment and ViricLi Arts in a due proportion, the mixture loft its purple tinCture, and be¬ came perfectly dun. But the Experiment fucceeded beft without Minium thus. To Orpiment I added by little and little a certain full bright purple, which Painters ule until the Orpiment ceafed to be yellow, and became of a pale red. Then I diluted that red by adding a little Vtride Arts, and a little more blue Bife than Vu ridi Arm, until it became of luch a grey or pale white, as verged to no one of the Colours more than to ano¬ ther. For thus it became of a Colour equal in white- nefs to that of Allies or of Wood newly cut, or of a Man's Skin. The Orpiment reflected more Light than did any other of the Powders, and therefore conduced more to the whitenefs of the compounded Colour than they. To aflign the proportions accurately may be difficult, by realon of the different goodnefs of Pow¬ ders of the fame kind. Accordingly as the Colour of any Powder is more or lefs full and luminous, it ought to be ufed in a lefs or greater proportion. Now confidering that thefe grey and dun Colours may be alfo produced by mixing whites and blacks, and by confequence differ from perfeCt whites not in Species of Colours but only in degree of luminoufnefs, it is ma- nifeft that there is nothing more requifite to make them perfeftly white than to increafe their Light fuffi- ciently ; and, on the contrary, if by increafing their Light they can be brought to perfeCt whitenefs, it will thence alfo follow, that they are of the lame Species of Colour with the beft whites, and differ from them only in the quantity of Light. And this I tryed as follows. I took the third of the above-mentioned grey mixtures [i 13 3 ( that Which was compounded of Qtpiment, Purple, Bife and Vinde Mr is ) and rubbed it thickly upon the floor of my Chamber, where the Sun (hone upon it through the opened Calement ; and by it, in the (ha- dow, I laid a piece of white Paper of the fame bignef^ Then going from them to the diftance of 1 2 or 1 8 Feet," fo that I could not difcern the unevennefs of the furface of the Powder, nor the little (hadows let fall from the gritty particles thereof ; the Powder appeared intently white, fo as to tranfcend even the Paper it felf in white- nets, efpecially if the Paper were a little (haded from the Light of the Clouds, and then the Paper compared with the Powder appeared of fuch a grey Colour as the Powder had done before. But by laying the Paper where the Sun dimes through the Glafs of the Window, or by (hutting the Window that the Sun might fhine through the Glafs upon the Powder, and by fuch other fit means of increafing or decreafing the Lights where¬ with the Powder and Paper were illuminated , the Light wherewith the Powder is illuminated may be made (Longer in fuch a due proportion than the Light wherewith the Paper is illuminated, that they (hall both appear exactly alike in whitenefs. For when I was trying this, a Friend coming to vifit me, I ftopt him at the door, and before I told him what the Colours were, or what I was doing ; I askt him, Which of the two whites were the beft, and wherein they differed ? And after he had at that diftance viewed them well, he anlwered, That they were both good whites, and that he could not fay which was beft, nor wherein their Co¬ lours differed. Now if you confider, that this white of the Ponder in the Sumfhine was compounded of the P Colours C r 14 1 Colours which the component Powders ( Orpimentr Purple, Bile, and Viride JEris) have in the fame Sun- fhine, you inuft acknowledge by this Experiment, as well as by the former, that perfect whitenefs may be compounded of Colours. From what has been faid it is alfo evident, that thee whitenefs of the Sun’s Light is compounded of all the Colours wherewith the feveral forts of rays whereof that Light confifts, when by their feveral refrangibili- ties they are feparated from one another, do tinge Paper? or any other white Body whereon .they fall. For thofer Colours by Prop. 1. are unchangeable, and whenever all thole rays with thofe their Colours are mixt again, they reproduce, the fame white Light as before. PROP, VI. PRGB. IL M a mixture of primary Colour r, the quantity and quaiitfr of, each -being given , t a know the Colour of the com- With the Center O and Radius O B defcribe a Circle A DF, and diftinguifti its circumference intodeven parts DE, EE, FG$ GA, AB, BG, G D, proportional to the feven mufical Tones or Intervals of the eight Sounds, Sof la , /^, folr la1 fa , fof contained in an Eight, that is, proportional to the numbers L, C C Lev the fir ft part DE reprefent a red Colour, the fecond E F orange, the third F G yellow,, the fourth G H green, the fifth A B blue, . the fixth BC indico, and the.feventh CD violet. And. conceive, that thefe are all the Colours of uncompounded Light gradually \ paffing [”$] paffing into one another, as they do when made by Prifms ; the circumference DE FG A BCD, reprefen- ting the whole l’eries of Colours from one end of the Sun’s coloured Image to the other, fo that from D to E be ail degrees of red, at E the mean Colour between red and orange, from E to F all degrees of orange, at F the mean between orange and yellow, from F to G all de¬ grees of yellow, and fo on. Let p be the center of gravity of the Arch DE, and q, r, s, t, v, x, the centers of gravity of the Arches EF, FG, GA, AB, BC and C D refpeftively, and about thofe centers of gra¬ vity let Circles proportional to the number of rays of each Colour in the given mixture be defcribed ; that is, the circle p proportional to the number of the red-ma¬ king rays in the mixture, the Circle q proportional to the number of the orange-making rays in the mixture, and fo of the reft. Find the common center of gravity of all thofe Circles p, q, r, s, t, v, x. Let that center be Z ; and from the center of the Circle A D F, through Z to the circumference, drawing the right line O Y, the place of the point Y in the circumference lhall fhew the Colour arifing from the compofition of all the Co¬ lours in the given mixture, and the line O Z fnall be proportional to the fulnefs or intenfenefs of the Colour, that is, to its diftance from whitenefs. As if Y fall in the middle between F and G, the compounded Colour lhall be the beft yellow 3 if Y verge from the middle to¬ wards F or G, the compounded Colour lhall according¬ ly be a yellow, verging towards orange or green. IfZ fall upon the circumference the Colour lhall be intenfe and florid in the higheft degree; if it fall in the mid way between the circumference and center, it lhall be P -2 but [lid] but half fo mtenfe, that is, it ihall be fucli a Colour as would he made by diluting the intenfeft yellow with an equal quantity of whitenefs ; and if it fall upon the center O, the Colour ihall have loft all its intenfenefiq and become a white. But it is to be noted, That if the point Z fall in or near the line O 33, the main ingredients being the red and violet, the Colour compounded ihall not be any of the prifmatic Colours, but a purple, in¬ clining to red or violet, accordingly as the point Z lieth on the fide of the line D O towards E or towards C, and in general the compounded violet is more bright and more fiery than the uncompounded. Alio if only two of the primary Colours which in the Circle are oppofite to one another be mixed in an equal proportion, the point Z ihall fall upon the center O, and yet the Co¬ lour compounded of thofe two ihall not be perfectly white, but fome faint anonymous Colour. For I could never yet by mixing only two primary Colours produce a perfect white. Whether' it may be compounded of a mixture of three taken at equal diftances in the circum¬ ference I do not know, but of four or five I do not much queftion but it may. But thefe are curiofities of little or no moment to the underftanding the Phenomena of nature. For in all whites produced by nature, there ufes to be a mixture of all forts of rays, and by confe- quence a competition of all Colours. To give an inftance of this Rule; fuppofe a Colour is compounded of thefe homogeneal Colours, of violet i part, of indico i part, of blue a parts, of green 3 parts, of yellow 5 parts, of orange 6 parts, and of red 1 o parts. Proportional to thefe parts I deferibe the Circles x, v, t, % ri T P refpedtively, that is, la that if the Circle x [ ri7 ] _ be i, the Circle v may be i, the Circle t r, the Circle S3, and the Circles r, q and p, 5, 6 and 10. Then I find Z the common center of gravity of thefe Circles, and through Z drawing the line OY, the point Y falls upon the circumference between E and F, fome thing nearer to E than to F, and thence I conclude, that the Colour compounded of thefe ingredients will be an orange, verging a little more to red than to yellow. Alfo I find that O Z is a little lefs than one half of OY, and thence 1 conclude, that this orange hath a little lels than half the fulnefs or intenfenefs of an un¬ compounded orange ; that is to fay, that it is fuch an orange as may be made by mixing an homogeneal orange- with a good white in the proportion of the line O Z to the line Z Y, this proportion being not of the quantities of mixed orange and white powders, but of the quan¬ tities of the lights refiedted from them. This Rule 1 conceive accurate enough for pradtife, though not mathematically accurate ; and the truth of it may be fufficiently proved to fenfe, by flopping any of the Colours at the Lens in the tenth Experiment of this Book. For the reft of the Colours which are not flopped, but pafs on to the Focus of the Lens, will there compound either accurately or very nearly fuch a Colour as by this Rule ought to refult from them mixture. PRO Vi .• » —I PROP. VII. THE OR* V. >,+411 the Colours in the Univerje -which are made by Light , and depend not on the power of imagination , are either the Colours of homogeneal Light r, or compounded of thefe and that either accurately or very nearly , ac¬ cording to the Rjule of the foregoing Tdr&blem. For it has been proved ( in Prop.i. Lib.o..) that the ^changes of dolours made by refradions do not arife from any new modifications of the rays impreft by thofe refradions, and by the various terminations of light andfhadow, as has been the conftant and general opi¬ nion of Philofophers. It has alfo been proved that the leveral Colours of the homogeneal rays do conftantly anfwer to their degrees of refrangibility, (Prop, i . Lik i . and Prop.i. Lib.'i.) and that their degrees of refrangi- bility cannot be changed by refradions and reflexions, (Prop.i. Lib A.) and by confequence that thofe their Colours are likewife immutable. It has alfo been pro¬ ved diredly by refrading and refieding homogeneal Tights apart, that their Colours cannot be changed, (Prop.i. Lib. o..) It has been proved alfo, that when the feveral forts of rays are mixed, and in croffing ;pafs through the fame fpace, they do not ad on one another So as to change each others colorifick qualities, (Exper. lo.Lik'i.) but by mixing their adions in the Senfio- rium beget a ienfation differing from what either would do apart, that is a fenfation of a mean Colour between fheir proper Colours ; and particularly when by the ^oncourfe and .mixtures of all forts of rays, a white Colour [H5>] Colour is produced, the white is a mixture of all the Colours which the rays would have apart, ( Prop. 5, Lik c. ) The rays in that mixture do not lofe or alter their feveral colorifick qualities, but by all their various kinds of actions mixt in the Senforiumr beget a fenfa- tion of a middling Colour between all their Colours- which is whitenefs. For whitenefs is a mean between ■ all Colours, having it felf indifferently to them a Fg fo- as with equal facility to be tinged with any of them. A red Powder mixed with a little blue, or a blue with a little red, doth not prelently lofe its Colour, but a white Powder mixed with any Colour is prefen tly tin¬ ged with that Colour, and is equally capable of being tinged with any Colour what-ever. It has been (hewed ’ aifo, that as the Sun’s Light is mixed of all forts of rays, lb its whitenefs is a mixture of the Colours of all forts. *- of rays ; thofe rays having from the beginning their fe- veral colorific qualities as well as their feveral refrangL bilities, and retaining them perpetually unchang’d not- withftanding any refractions or reliexions they may at any time fuffer, and that whemever any fort of the Sun’s- rays is by. any means (as by reflexion in Ex per. y , audio. Lik 1. or by refraCtion as happens in all re¬ fractions ) feparated from the : reft, they then manifeft their proper Colours, ... Thefe things have been proved/ and the fum of all this amounts to the Prop ofition here to be proved. . For if the Sun’s Light is mixed of 1L, ~ veral forts of rays,- each of which have originally their feveral refrangibilities and colorifick qualities, and not- - withftanding their refraCtions and reflections, and their . various reparations or . mixtures, keep .thofe their ori- - ginal properties perpetually the fame without, altera¬ tion ] ; [ 120 ] "tion ; then all the Colours in the World mu ft: be fuch as conftantly ought to arife from the original colorific qua¬ lities of the rays whereof the Lights coniift by which thofe Colours are feen. And therefore if the reafon of any Colour what-ever be required., we have nothing elfe to do then to confider how the rays in the Sun’s Light have by reflexions or refractions, or other caufes been par¬ ted from one another, or mixed together ; or otherwile to find out what forts of rays are in the Light by which that Colour is made, and in what proportion ; and then by the laft Problem to learn the Colour which ought to arife by mixing thofe rays (or their Colours) in that proportion. I fpeak here of Colours fo Aar as they arife from Light. For they appear fometimes by other caufes, as when by the power of phantafy we fee Colours in a Dream, or a mad Man fees things before him which are not there ; or when we fee Fire by ftriking the Eye, or fee Colours like the Eye of a Peacock’s Feather, by preffing our Eyes in either corner whilft we look the other way. Where thefe and fuch like caufes interpofe not, the Colour always anfwers to the fort or forts of the rays whereof the Light confiffs, as I have conftantly found in what-ever Phenomena of Colours I have hitherto been able to exarniir 1 flrall in the following Propoiitions give infiances of this in the Phenomena of ehiefeft note. PROP. PROP. VIII. P ROB. III. By the dij covered ‘Properties of Light to explain the Colours made by Prifms . Let ABC reprefent a Prifm refracting the Light of pia the Sun, which comes into a dark Chamber through a ° Hole F

face on the upper Glafs, and my Eye was about eight or nine Inches diftance from the Glafs, almoft perpen¬ dicularly over it, and the Glafs wTas \ of an Inch thick, and thence it is eafy to colled! that the true Diameter of the Ring between the Glaffes was greater than its meafured Diameter above the Glaffes in the proportion of 80 to 79 or thereabouts, and by confequence equal to ~ parts of an Inch, and its true Semi-diameter equal to ~ parts. Now as the Diameter of the Sphere ( 1 8a In¬ ches) is to the Semi-diameter of this fifth dark Ring ( — parts of an Inch) io is this Semi-diameter to the0 thicknefs of the Air at this fifth dark Ring ; which fe therefore or mm Parts an Inch, and the fifth part thereof; viz. the ^7th part of an Inch, is the thicknefs of the Air at the firft of thefe dark Rings; The fame Experiment { repeated with another dou¬ ble convex ObjedTgiafs ground on both fides to one and the fame Sphere. Its Focus was diftant from it 168- Inches, and therefore the Diameter of that Sphere was 1 84 Inches. This Glafs being laid upon the fame plain Glafs, the Diameter of the fifth of the dark Rings, when the black Spot in their center appeared plainly without preffing the Glaffes, was by the mea- fure of the Compafies upon the upper Glais im parts of an Inch, and by confequence between the Glaffes it was|~, For the upper Glafs was § of an Inch thick, B b and and my Eye was diftant from it 8 Inches. And a third proportional to half this from the Diameter of the Sphere is ^ parts of an Inch. This is therefore the thicknefs of the Air at this Ring, and a fifth part there¬ of, viz. the g^-th part of an Inch is the thicknefs there¬ of at the firft of the Rums as above. o , I tryed the fame thing by laying thefe Objeft-Glaffes upon Hat pieces of a broken Looking-glafs, and found the fame meafures of the Rings : Which makes me rely upon them till they can be determined more ac¬ curately by Glades ground to larger Spheres, though in fuch Glafies greater care muft be taken of a true nil. Thefe Dimen fior is were ta n when my Eye was placed almoft perpendicularly over the Glades, being about an Inch , or an Inch and a quarter, diftant from ° the incident rays, and eight Inches diftant from the Glafs ; fo that the rays were inclined to the Glafs in an Angle of about \ degrees. Whence by the following Observation you will underftand , that had the rays been perpendicular to the Glades, the thicknefs of the Air at thefe Rings would have been lefs in the propor¬ tion of the Radius to the lecant of 4 degrees, that is of loooo. Let the thicknedes found be therefore dimi- nilhed in this proportion, and they will become and or ( to ufe the neareft round number) the8~th part of an Inch. This is the thicknefs of the Air at the darkeft part of the firft dark Ring made by perpendi¬ cular rays, and half this thicknefs multiplied by the progredion,i,3,5,7,,9, 1 i,l 5>c. gives the thicknedes of the Air at the mod: luminous parts of all the brighteft Rings, vk. ^3 i7C)OO05- IJtiQCQ) IJijCCO? *70000?. i9V\ their arithmetical means. [II] means — 0c. being its thiekneffes at the darkeft parts of all the dark ones. obs, vir. The Rings were leaft when my Eye was placed per¬ pendicularly over the Glaffes in the Axis of the Rings : And when I viewed them obliquely they became big¬ ger, continually fwelling as I removed my Eye further trom the Axis. And partly by meafuring the Diameter ol the fame Circle at feveral obliquities of my Eye, partly by other means, as alio by making ule of the two Frilms for very great obliquities. I found its Dia¬ meter, and confequently the thicknefs of the Air at its perimeter in all thole obliquities to be very nearly in the proportions expreffed in this Table. Angle cidence Air, of In- on the Angle of Re¬ fraction into the Air . Diameter of the Ring, Thicknefs of the Air . deg. OO min. OO OO OO IO f. 1 > IO 06 26 IO OO *°iS O 1 2 45 20 OO Ici I0i . 1 8 49 3° OO IO; Hi 24 3° 4° OO I l| *3 151 I 29 37 5° OO 33 58 60 OO 14 20 35 47 65 00 *5? 37 l9 70 OO 1 6| 38 33 75 OO J94 37 39 27 SO OO 22f 5^ 40 OO 85 OO 2 9 84,-; 1 40 1 1 90 OO j 35 I 2 2l St [12] In th&tfco firft Columns are e&preffed the obliquities of the incident and emergent rays to the plate of the Air, that is, their angles of incidence and refraction. In the third Column the Diameter of any coloured Ring at thole obliquities is expreffed in parts, of which ten conftitute that Diameter when the rays are perpendicu¬ lar. And in the fourth Column the thicknefs of the Air at the circumference of that Ring is expreffed in parts of which alio ten conftitute that thicknefs when the rays ate perpendicular.. And from thefe meafures I feem to gather this Rule : That the thicknefs of the Air is proportional to the fe- cantwof an angle, whole Sine is a certain mean proper- tionai between the Sines of incidence and refraCtionr And that mean proportional, lb far as by thefe meafures I can determine it, is the firft of an hundred and fix arithmetical mean proportionals between thofe Sines counted from the Sine of refraction when the refra¬ ction is made out of the Glafs into the plate of Air, or from the Sine of incidence when the refraCtion is made out of the plate of Air into the Glafs. | • J I j - l | . O O C | (*-j- - i f -• 1 ‘ O B S. ¥ 1 1 L V ; - - \ 1 ! ^ i 1 O ay : ; * The dark Spot in the middle of the Rings increafed alfo by the obliquation of the Eye, although almoft in- fenlibly. But if inftead of the Object -Glades the Prifms were madeufeof, itsincreafe was more manifeft when ¥iewed fo obliquely that no Colours appeared about it. It was leaft when the rays were incident moft obliquely on the interjacent Air, and as the obliquity decreafed it increafed more and more until the coloured Rings ap- hi peared. peared, and then decreafed again,1 but not fo much as it increafed before. And hence it is evident, that the tranfparency was not only at the abfolute contact of the Glaffes, but alfo where they had feme little interval. I have fometimes obferved the Diameter of that Spot to be between half and two fifth parts of the Diameter of the exterior circumference of the red in the firft cir¬ cuit or revolution of Colours when viewed almoft per¬ pendicularly ; whereas when viewed obliquely it hath wholly vanifhed and become opake and white like the other parts of theGlafs; whence it may be collected that the Glaffes did then fcarcely, or not at all, touch one another, and that their interval at the perimeter of that Spot when viewed perpendicularly was about a fifth or fixth part of their interval at the circumference of the faid red. O B S. IX . ' ' f . { i , V f u - ’ ' . t • - - 'jj' ..! : ■' „ ; ' J • f\ : * : . i :■ - " • ; V; ; *; , ; -■ By ■ looking through the two contiguous Obje£L Glaffes, I found that the interjacent Air exhibited Rings of Colours, as well by tranfmitting Light as by reflect¬ ing it. The central Spot was now white, and from it the order of the Colours were yello wife red; black ; violet, blue, white, yellow, red ; violet, blue, green, yellow, red, bv. But thefe Colours were very faint and dilute unlefs when the Light was trajefted very obliquely through the Glaffes : For by that means they became pretty vivid. Only the fir ft yellowife red, like the blue in tire fourth Obfervation, was fo little and faint as fcarcely to be difcerned. Comparing the co¬ loured Rings made by reflexion , with thefe made by tranff [1+] tranfmiffion of the Light ; I found that white was op- polite to black, red to blue, yellow to violet, and green to a compound of red and violet. That is, thofe parts of the Glals were black when looked through, which when looked upon appeared white, and on the con¬ trary. And fo thofe which in one cafe exhibited blue, did in the other cafe exhibit red. And the like of the Fig. 5. other Colours. The manner you have reprefen ted in the third Figure, where A B, CD, are the furfaces of the daffes contiguous at E, and the black lines be¬ tween them are their diffances in arithmetical progrel- lion, and the Colours written above are feen by re¬ flected Light, and thofe below by Light tranfmitted. O B S. X. Wetting the ObjeCt-Glaffes a little at their edges, the water crept in (lowly between them, and the Cir¬ cles thereby became lels and the Colours more ^ faint : Infomuch that as the water crept along one half of them at which it firft arrived would appear broken off from the other half, and contracted into a lets room. By meafuring them I found the proportions of their Diameters to the Diameters of the like Circles made by Air to be about leven to eight, and conlequently the in¬ tervals of the Glaffes at like Circles, cauled by thofe two mediums Water and Airfare as about three to four. Perhaps it may be a general Rule, That if any other medium more or lefs denfe than water be compreffed between the Glaffes, their intervals at the Rings caufed thereby will be to their intervals caufed by interjacent [!$] Air, as the Sines are which meafure the refraction made out of that medium into Air. OBS. XI. When the water was between the Glalfes, if I pref- fed the upper Glafs varioufly at its edges to make the* Rings move nimbly from one place to another, a little white Spot would immediately follow the center of them, which upon creeping in of the ambient water into that place would prefently vanifh. Its appearance was fuch as interjacent Air would have caufed, and it exhibited the fame Colours. But it was not Air, for where any bubbles of Air were in the water they would not vanifh. The reflexion mult have rather been caufed by a fubtiler medium,, which could recede through the Glalfes at the creeping in of the water. O B S. XIL Thefe Obfervations were made in the open Air. But- further to examin the effects of coloured Light falling on the Glalfes, I darkened the Room, and viewed them by reflexion of the Colours of a Prifm call on a Sheet: of white Paper,. my .Eye being fo placed that I cou ld fee the coloured Paper by reflexion in the Glalfes, as in a Looking-glafs. And by this means the Rings be¬ came diftinfter and vilible to a far greater number than in the open Air. I have fometimes feen more than twenty of them, whereas in the open Air I could not difcern above eight or nine. O B S. XIII. Appointing an afliftant to move the Prifm to and fro about its Axis, that all the Colours might fuccef- fively fall on that part of the Paper which I faw by reflexion from that part of the Glades, where the Cir¬ cles appeared, fo that all the Colours might be fuccef- fively reflected front the Circles to my Eye whilft I held it immovable, I found the Circles which the red Light made to be manifeftly bigger than thofe which were made by the blue and violet. And it was very plea- fant to fee them gradually fwell or contra dt according as the Colour of the Light was changed. The inter¬ val of the Glades at any of the Rings when they were made by the utmoft red Light, was to their interval at the fame Ring when made by the utmoft violet, greater than as 3 to a, and lefs than as 1 3 to 8, by the mod of my Obfervations it was as 14 to 9. And this proportion feemed very nearly the lame in all obliquities of my Eye 3 unlefs when two Prifms were made ufe of inftead of the Obje6f-Glades.: For then at a' certaimgreat obliquity of my Eye, the Rings made by the feveral Colours feemed equal, and at a greater obliquity thofe made by the violet would be greater than the fame Rings made by the red. The refraction of the Prifm in this cafe caufing the mod: refrangible rays to fall more obliquely on that plate of the Air than the lead: refrangible ones. Thus the Experiment fucceeded in the coloured Light, which was fufficiently ftrong and copious to make the Rings fenflble. And thence it maybe gathered, that if the moft refrangible and leaft refran* [ J7 ] refrangible rays had been copious enough to make the Rings fenfible without the mixture of other rays, the .proportion which here was 1 4 to 9 would have been a little greater, fuppofe 14 ~ or 14 l to 9 . O B S. XIV. Whilft the Prifm was turn'd about its Axis with an uniform motion, to make all the feveral Colours fall fucceffively upon the ObjeCt rGlaffes, and thereby to make the Rings contract: and dilate : The contraction or dilation of each Ring thus made by the variation of its Colour was fwifteft in the red, and iloweft in the violet, and in the intermediate Colours it had inter¬ mediate degrees of celerity. Comparing the quantity of contraction and dilation made by all the degrees of each Colour, I found that it was greateft in the red ; lefs in the yellow, (till lefs in the blue, and leaft in the violet. And to make as juft an eftimation as 1 could of the proportions of their contractions or dilations, I obferved that the whole contraction or dilation of the Diameter of any Ring made by all the degrees of red, was to that of the Diameter of the fame Ring made by all the de¬ grees of violet, as about four to three, or five to four, and that when the Light was of the middle Colour between yellow and green, the Diameter of the Ring was very nearly an arithmetical mean between the greateft Dia¬ meter of the fame Ring made by the outmoft red, and the leaft Diameter thereof made by the outmoft violet : Contrary to what happens in the Colours of the oblong SpeCtrum made by the refraCt ion of a Prifm, where the red is moft contracted , the violet moft expanded, and D d m [i8] in the midft of all the Colours is the confine of green and blue. And hence I feem to colled that the thick- neffes of the Air between the Glaffes there., where the Ring is lucceffively made by the limits of the five prin¬ cipal Colours (red, yellow, green, blue, violet) in order ( that is, by the extreme red, by the limit of red and yellow in the middle of the orange, by the limit of yellow and green, by the limit of green and blue, by the limit of blue and violet in the middle of the in- digo, and by the extreme violet ) are to one another very nearly as the fix lengths of a Chord which found the notes in a fixth Major, fol, la , mi, fa, fol , la. But it agrees fomething better with the Obiervation to fay, that the thicknefies of the Air between the Glaffes there, where the Rings are fucceffively made by the limits of the feven Colours, red, orange, yellow, green, blue, in¬ digo, violet in order, are to one another as the Cube- roots of the Squares of the eight lengths of a Chord, which found the notes in an eighth, fol , la, fa, fol, la, mi, fa, fol ; that is, as the Cube-roots of the Squares of the Numbers, i , f, % \ \ \ ^ \. O B S. XV. Thefe Rings were not of various Colours like thole made in the open Air, but appeared all over of that prifmatique Colour only with which they were illu¬ minated. And by projecting the prifmatique Colours immediately upon the Glaffes, I found that the Light which fell on the dark Spaces which were between the coloured Rings , was tranfmitted through the Glaffes without any variation of Colour. For on a white [I 9 1 white Paper placed behind, it would paint Rings of the fame Colour with thofe which were reflected, and / of the bignefs of their immediate Spaces. And from thence the origin of thefe Rings is manifeft ; namely, That the Air between the Glaffes, according to its va¬ rious thicknefs, is difpofed in fome places to reflect, and in others to tranimit the Light of any one Co¬ lour (as you may fee reprefented in the fourth Figure ) fig, ^ and in the fame place to reflect that of one Colour where it tranfmits that of another. O B S. XVI.1 The Squares of the Diameters of thefe Rings made by any prilmatique Colour were in arithmetical pro- greflion as in the fifth Obfervation. And the Diameter of the fixth Circle, when made by the citrine yellow, and viewed almoft perpendicularly, was about s~ parts of an Inch, or a little lefs, agreeable to the fixth Ob¬ fervation. ■ The precedent Obfervations were made with a rarer thin medium, terminated by a denfer, fuch as was Air or Water compreffed between two Glaffes. In thofe that roliow are fet down the appearances of a denfer medium thin’d within a rarer, fuch as are plates of Muicovy-glafs, Bubbles of Water, and fome other thin fubftances terminated on all fides with Air. - 0 B 3. , Dd a O B S. XVII. If a Bubble be blown with Water firfl: made tenacious by diffolving a little Soap in it, ?tis a common Obfer¬ vation, that after a while it will appear tinged with a great variety of Colours, To defend thefe Bubbles from being agitated by the external Air (whereby their Colours are irregularly moved one among another, fe that no accurate Obfervation can be made of them,) as foon as I had blown any of them I covered it with a clear Glafs, and by that means its Colours emerged in a very regular order, like fo many concentrick Rings incompaffing the top of the Bubble. And as the Bubble grew thinner by the continual fubfiding of the Water, thefe Rings dilated llowly and over-fpread the whole Bubble, defending in order to the bottom of it, where they vanifhed fucceffively. In the mean while, after all the Colours were emerged at the top, there grew in the Center of the Rings a fmall round black Spot, like that in the firfl: Obfervation, which conti¬ nually dilated it felf till it became Fometimes more than - or f of an Inch in breadth before the Bubble broke. At firfl: I thought there had been no Light refledted from the Water in that place, but obferving it more cu- rioufly, I faw within it levend fmaller round Spots, which appeared much blacker and darker than the reft, whereby I knew that there was fome reflexion at the other places which were not fo dark as thofe Spots. And by further tryal I fouud that I could fee the Images ©f fome things ( as of a Candle or the Sun ) very faint¬ ly reflected, not only from the great black Spot, but alfo [21 ] aifo from the little darker Spots which were with¬ in it. Befides the aforefaid coloured Rings there would often appear fmall Spots of Colours, afcending and de¬ fending up and down the fides of the Bubble, by reafon of feme inequalities in the fubfiding of the Water. And fometimes fmall black Spots generated at the fides would afcend up to the larger black Spot at the top of the Bubble, and unite with it. OBS. XVIIL Becaufe the Colours of thefe Bubbles were more ex¬ tended and lively than thofe of the Air thin’d between two Glafles, and fo more eafy to to diftinguiftied , I fhall here give you a further defcription of their order,, as they were obferved in viewing them by reflexion of the Skies when of a white Colour, whilft a black Sub- ftance was placed behind the Bubble. And they were thefe, red, blue ; red, blue ; red, blue • red , green ; red, yellow, green, blue, purple ; red, yellow,, green, blue, violet ; red, yellow, white, blue, black. The three firft Succeffions of red and blue were very dilute and dirty, efpecialiy the firft, where the red feemed in a manner to be white^ Among thefe there was fcarce any other Colour fenfible befides red and blue, only the blues ( and principally the fecond blue inclined a little to green. The fourth red was alfo dilute and dirty, but not fo much as the former three ; after that fucceeded little, or no yellow, but a copious green, which at firft incli¬ ned a little to yellow, and then became a pretty brifque and . [ M ] and good willow green, and afterwards changed to a bluifh Colour; but there fucceeded neither blue nor violet. The fifth red at firft inclined very much to purple, and afterwards became more bright and brifque, but yet not very pure. This was fucceeded with a very bright and intenfe yellow, which wras but little in quantity, and foon changed to green : But that green was copious and fomething more pure, deep and lively, than the former green. After that followed an excel¬ lent blue of a bright sky-colour , and then a purple, which was lefs in quantity than the blue, and much inclined to red. / The fixth Red was at firft of a very fair and lively Scarlet, and foon after of a brighter Colour, being very pure and brifque , and the beft of all the reds. Then after a lively orange followed an intenfe bright and copious yellow , which was alfo the beft of all the yellows, and this changed firft to a greenifti yellow, and then to a greenifti blue ; but the green between the yellow and the blue, was very little and dilute, fceming rather a greenifti white than a green. The blue which fucceeded became very good, and of a very fair bright sky-colour, but yet fomething inferior to the former blue ; and the violet was intenfe and deep with little or no rednefs in it. And lefs in quan¬ tity than the blue. In the laft red appeared a tinfture of fcarlet next to violet, which foon changed to a brighter Colour, inclining to an orange ; and the yellow which followed was at firft pretty good and lively , but afterwards it grew more dilute, until by degrees it ended in perfedt white- 1 ( BooTc,n. Plate, H. 03] , whitenefs. And this whitenefs, if the Water was very tenacious and well-tempered, would ftowly fpread and dilate it felf over the greater part of the Bubble ; con¬ tinually growing paler at the top, where at length it would crack in many places, and thofe cracks, as they dilated, would appear of a pretty good, but yet obfcure and dark sky-colour j the white between the blue Spots diminiftiirig, until it refembled the threds of an irre¬ gular Net-work, and foon after vanifhed and left all the upper part of the Bubble of the laid dark blue Colour. And this Colour, after the aforefaid manner, dilated it felf downwards , until fometimes it hath overfpread the whole Bubble. In the mean while at the top, which was of a darker blue than the bottom, and appeared alfo full of many round blue Spots, feme- thing darker than the reft , there would emerge one or more very black Spots, and within thofe other Spots of an interifer blacknefs , which I mentioned in the former Obfervation j and thefe continually dilated themfelves until the Bubble broke. If the Water was not very tenacious the black Spots would break forth in the white, without any fenlible intervention of the blue. And fometimes they would break forth within the precedent yellow , or red, or perhaps within the blue of the fecond order, before the intermediate Colours had time to difplay them¬ felves. By this defcription you may perceive how great an affinity thefe Colours have with thofe of Air defcri- bed in the fourth Obfervation, although fet down in a contrary order, by reafon that they begin to appear when the Bubble is thickeft , and are molt conve* niently niently reckoned from the loweft and thickeft part of the Bubble upwards. OBS, XIX. Viewing in feveral oblique pofitions of my Eye the Rings of Colours emerging on the top of the Bubble, I found that they were fenfibly dilated by increafing the obliquity , but yet not fo much by far as thofe made by thin’d Air in the feventh Obfervation. For there they were dilated fo much as , when viewed moft obliquely, to arrive at a part of the plate more than twelve times thicker than that where they ap¬ peared when viewed perpendicularly ; whereas in this cafe the thicknefs of the Water, at which they arrived when viewed moft obliquely, was to that thicknefs which exhibited them by perpendicular rays, fome- thing lefs than as 8 to 5 . By the beft of my Observations it was between 15 and 15; to 10, an increafe about ap times lefs than in the other cafe. Sometimes the Bubble would become of an uniform thicknefs all over, except at the top of it near the black Spot, as I knew, becaufe it would exhibit the fame appearance of Colours in all pofitions of the Eye. And then the Colours which were feen at its apparent cir¬ cumference by the obliqueft rays, would be different from thofe that were feen in other places, by rays lefs oblique to it. And divers Spectators might fee the fame part of it of differing Colours, by viewing it at very differing obliquities. Now obferving how much the Colours at the fame places of the Bubble, or at di¬ vers places of equal thicknefs , were varied by the feveral [* 5] feveral obliquities of the rays ; by the affiftance of the 4th, 14th, 1 6 th and 18 th Obfervations, as they are hereafter explained, I colled the thicknefs of the Water requifite to exhibit any one and the fame Colour, at fe¬ veral obliquities , to be very nearly in the proportion expreffed in this Table. Incidence on the Water . Refraction in - to the Water , Thicknefs of the Water . deg. min. deg. min. OO OO OO OO IO 15 OO I I I I l°5 3° OO 22 I 10-; 45 OO 32 2 I 60 OO 4° 3° J3 • Hi 75 OO 46 25 90 OO 48 35 Hi _ In the two firft Columns are expreffed the obliqui¬ ties of the rays to the fuperficies of the Water, that is, their Angles of incidence and refradion. Where I fuppofe that the Sines which meafure them are in round numbers as 3 to 4, though probably the diffo- lution of Soap in the Water , may a little alter its reiradive Vertue. In the third Column the thicknefs of the Bubble, at which any one Colour is exhibited in thofe feveral obliquities, is expreft in parts, of which ten conftitute that thicknefs when the rays are perpen¬ dicular. I have fometimes obferved, that the Colours which arife on polilhed Steel by heating it, or on Bell-metal, and fome other metalline fubftances, when melted and E e poured [2d] poured on the ground , where they may cool in the open Air, have, like the Colours of Water-bubbles, been a little changed by viewing them: at divers ob¬ liquities, and particularly that a deep Hue, or violet, when viewed very obliquely, hath been changed to a deep red. But the changes of thefe Colours are not fo great and fenfible as of thofe made by Water. For the Scoria or vitrified part of the Metal, which rnoft Me¬ tals when heated or melted do continually protrude, and fend out to their furface, and which by covering the Metals in form of a thin glaffy skin, caules thefe Colours, is much denfer than Water ; and I find that the change made by the obliquation of the Eye is lead in Colours of the aenfeft thin fubftances. O B 3. XX. As in the ninth Obfervation, fo here, the Bubble, by tranfmitted Light, appeared of a contrary Colour to that which it exhibited by reflexion. Thus when the Bubble being looked on by the Light of the Clouds re- fltefited from it, feemed red at its apparent circumfe¬ rence, if the Clouds at the fame time, or immediately after, were viewed! through it, the Colour at its cir¬ cumference would be blue; And, on the contrary,, wftetr by reflected Light it appeared blue, it would ap¬ pear red by tranfmitted* Light. O B S. XXL By wetting vfery thin plates of Mufcovy-glafs, whofe thinnefs made the like Colours appear,, the Colours became c 27 ] became mote faint and languid ; especially by wetting the plates on that fide oppofite to the Eye : But I could not perceive any variation of their fpecies. So then the thieknefs of a plate requifite to produce any Co¬ lour, depends only on the denfity of the plate, and not on that of the ambient medium : And hence, by the 10th and 16th Oblervations, may be known the thick- nefs which Bubbles of Water, or Plates of Mufcovy- glafs, or other fubftances, have at any Colour pro¬ duced by them. O B S. X X I I. A thin tranfparent Body, which is denfer than Its ambient medium, exhibits more brifque and vivid Co¬ lours than that which is fo much rarer * as I have particularly obferved in the Air and Glafs. For blow¬ ing Glafs very thin at a Lamp-furnace, thofe plates incompafled with Air did exhibit Colours much more vivid than thofe of Air made thin between two Glaffes. O B S. XXIII. Comparing the quantity of Light reflected from the feveral Rings, I found that it was moft copious from the firft or inmoft, and in the exterior Rings be¬ came gradually lefs and left. Alfo the whitenefs of the firft Ring was ftronger than that reflected from thofe parts of the thinner medium which were with¬ out the Rings ; as I could manifeftly perceive by view¬ ing at a dirtance the Rings made by the two Object- E e 3 Glafles? [28] Glaffes | or by comparing two Bubbles of Water blown at diftant times, in the fir ft of which the whitenefs appeared, which fucceeded all the Colours, and in the other, the whitenefs which preceded them all OBS. XXIV. When the two ObjeCt-Glaffes wrere lay’d upon one another, fo as to make the Rings of the Colours ap¬ pear, though with my naked Eye I could not difcern above 8 or 9 of thofe Rings, yet by viewing them through a Prifm I have feen a far greater multitude, infomuch that I could number more than forty, befides many others, that were fo very final! and dole toge¬ ther, that I could not keep my Eye fteddy on them feverally fo as to number them, but by their extent I have fometimes eftimated them to be more than a hundred. And I believe the Experiment may be improved to the difcovery of far greater numbers. For they feem to be really unlimited, though vifible only fo far as they can be feparated by the refraction, as I (hall hereafter explain. But it was but one fide of thefe Rings, namely, that towards which the refraction was made, which by that refraCtion was rendered diftinCt, and the other fide be¬ came more confufed than when viewed by the naked Eye, infomuch that there I could not difcern above one or two, and fometimes none of thofe Rings, of which I could difcern eight or nine with my naked Eye, And their Segments or Arcs, which on the other fide appeared fo numerous, for the moll part [ 29 ] exceeded not the third part of a Circle. If the Re¬ fraction was very great, or the Prifm very diftant from the ObjeCt-Glaffes, the middle part of thofe Arcs be¬ came alfo confufed, fo as to difappear and conftitute an even whitenefs, whilft on either fide their ends, as alfo the whole Arcs furtheft from the center, became di- ftindter than before, appearing in the form as you fee them defigned in the fifth Figure. Fig The Arcs, where they feemed diftinCteft, were only white and black fucceffively, without any other Co¬ lours intermixed. But in other places there appeared Colours, whofe order was inverted by the refraCtion in fuch manner, that if I firft held the Prifm very near the Objedt-Glafles , and then gradually removed it further off towards my Eye, the Colours of the ad, 3d, 4th, and following Rings ihrunk towards the white that emerged between them , until they wholly va- nifhed into it at the middle of the Arcs, and after¬ wards emerged again in a contrary order. But at the ends of the Arcs they retained their order un¬ changed. I have fometimes fo lay’d one Objedt-Glafs upon the other, that to the naked Eye they have all over feemed uniformly white, without the lead appearance- of any of the coloured Rings -} and yet by viewing them through a Prifm, great multitudes of thofe Rings have difcovered themfelves. And in like manner plates, of Mufcovy-glafs , and Bubbles of Glafs blown at a Lamp- furnace, which were not fo thin as to exhibit any Colours to the naked Eye, have through the Prifm exhibited a great variety of them ranged irregu- larly up and, down in the form of waves. And fo Bubbles. Babbles of Water, before they began to exhibit their Colours to the naked Eye of a By-ftander, hare ap¬ peared through a Prifm, girded about with many pa¬ rallel and horizontal Rings ; to produce which effe®, it was neceflary to hold the Prifm parallel, or very nearly parallel to the Horizon, and to difpofe it fo that the rays might be refracted upwards, -.ft THE i C P ] THE SECOND BOOK PART II. Remarks ufon the foregoing Oifervations.. Aving given my Obfervations of thefe Colours, _ before I make ufe of them to unfold the Caufes of the Colours of natural Bodies, it is convenient that by the fimpleft of them, fuch as are the ad, 3d, 4th, 9th, rath, 18th, 20th, and a 4th , I firft explain the more expounded. And firft to fhew how the Colours in the fourth and eighteenth Obfervations are produ- eed, let there be taken in any right line from the point % the lengths ¥A, YB, Y C, YD, YE, YF, Y GyFig.$ YH, in proportion to one another, as the Cube-roots of the Squares of the numbers, ^ 3, f, ? 1, where¬ by the lengths of a mufical Chord to found all the Notes 1 ins an Eighth are reprefented ; that is, in the propor- > tion of the numbers 6 30a, 6814, 7114, 7631, 8255,, 8855, 2243,: ioooo.. And at the points A, B, C, D E,F. 51 CS2! E, F, G, H, let perpendiculars A B bv. be ere&ed, by whole intervals the extent of the feveral Colours fet underneath againft them, is to be reprefented. Then divide the line A « in fuch proportion as the numbers i, 2, 3, 5, 6, 7, 9, io, 1 1, 1 9V. fet at the points of divi¬ sion denote. And through thofe divilions from Y draw lines i I, 2 K, 5 L, 5 M, 6 N, 7 O, be. Now if A 2 be iuppofed to reprefent the thicknefs of any thin transparent Body , at which the outmoft violet is moft copioufly refie&ed in the firft Ring, or Series of Colours, then by the 1 3th Obfervation H K, will reprefent its thicknefs, at which the utmoft red is moft copioufly refle&ed in the fame Series. Alfo by the 5 th and 1 6th Obfervations, A 6 and HN will denote the thickneffes at which thofe extreme Colours are moft copioufly reflected in the fecond Series, and A 1 o and H Q the thickneffes , at which they are moft copioufly reflected in the third Series, and fo on. And the thicknefs at which any of the intermediate Colours are reflefted moft copioufly, will, according to the 1 4th Obfervation, be defined by the diftance of the line A H from the intermediate parts of the lines 2 K, 6N, 10Q, be. againft which the names of thofe Co¬ lours are written below. ; • But further, to define the latitude of thefe Colours in each Ring or Series, let A 1 defign the leaft thicknefs, and A 3 the greateft thicknefs, at which the extreme violet in the firft Series is reflected, and let H and H L, defign the like limits for the extreme red, and let the intermediate Colours be limited by the inter¬ mediate parts of the lines 1 1, and 3 L, againft which the names of thofe Colours are written, and fo on : But yet [33] yet with this caution, that the reflections be fuppofed ftrongeft at the intermediate Spaces, a K, 6 N, 10Q fee. and from thence to decreafe gradually towards thefe li¬ mits, 1 1, 3 L, 5 M, 7O, he. on either fide; where you mull not conceive them to be precifely limited, but to decay indefinitely. And whereas I have afiigned the fame latitude to every Series, I did it, becaufe al¬ though the Colours in the firft Series feem to be a little broader than the reft, by reafon of a ftronger reflexion there, yet that inequality is fo infenfible as fcarcely to be determined by Oblervation. Now according to this defeription, conceiving that the rays originally of feveral Colours are by turns re¬ flected at the Spaces 1 1 L 3, 5 M O 7, 9 P R 1 1, he. and tranfmitted at the Spaces AHIi,3LM5,70P9, he. it is eafy to know what Colour muft in the open Air be exhibited at any thicknefs of a tranfparent thin body. For if a Ruler be applied parallel to A H, at that di- ftance from it by which the thicknefs of the body is reprefented, the alternate Spaces 1 1 L 3, 5 M O 7,3 5V. wrhich it croffeth will denote the reflected original Co¬ lours, of which the Colour exhibited in the open Ay: is compounded. Thus if the conftitution of the green in the third Series of Colours be defired, apply the Ruler as you fee at * e * and by its palling through fome of the blue at * and yellow at0', as well as through the green at ?, you may conclude that the green exhi¬ bited at that thicknefs of the body is principally con- ftituted of original green, but not without a mixture of fome blue and yellow', Ff. By C34] $y this means you may know how the Colours front the center of the Rings outward ought to fucceed in order as they were defcribed in the 4th and 1 8th Ob- fervations. For if you move the Ruler gradually from AH through all diftances, having paft over the firft fpa.ee which denotes little or no reflexion to be made by thinneft fubftances, it will firft arrive at 1 the violet, and then very quickly at the blue and green, which together with that violet compound blue, and then at the yellow and red, by whofe further addition that blue is converted into whitenefs, which whitenefs con¬ tinues during the tranfit of the edge of the Ruler from 1 to 5, and after that by the fucceffive deficience of its component Colours, turns firft to compound yellow, and- then to red: and1 laft of all the red ceafeth at L. Then begin the Colours of the fecund Series, which fucceed in order during the tranfit of the edge of the Ruler from 5 to O, and are more lively than before, becaufe more expanded and fevered. And for the fame reafon, inftead of the former white there inter¬ cedes between the blue and yellow a mixture of orange, yellow, green, blue and indico, all which together ought to exhibit a dilute and imperfed green. So the Co¬ lours of the third Series all fucceed in order ; firft, the violet, which a little interferes with the red of the fe~ cond order, and is thereby inclined to a reddiih purple ; then the blue and green, which are lefs mixed with other Colours, and confequently more lively than be¬ fore, efpecially the green: Then follows the yellow, fomeof which towards the green is diftind and good, but that part of it towards the fucceeding red, as alfo that red is mixed with the violet and blue of the fourth Se- ries, whereby Various degrees of red very much incli¬ ning to purple are ’Compounded* This violet and blue, which fhould fucceed this red, being mixed with, and hidden in it, there fucceeds a green. And this at firft is much inclined to blue, but foon becomes a good green , the only unmixed and lively Colour in this fourth Series. For as it verges towards the yellow, it begins to interfere with the Colours of the fifth Series, by whofe mixture the fucceeding yellow and red are very much diluted and made dirty , efpecially the yel¬ low, which being the weaker Colour is fcarce able to (hew it felf. After this the feveral Series interfere more and more, and their Colours become more and more intermixed, till after three or four more revolutions ( in which the red and blue predominate by turns ) all forts of Colours are in all places pretty equally ben¬ ded, and compound an even whitenefs. And fince by the 1 5th Obfervation the rays indued with one Colour are tranfmitted, where thofe of ano¬ ther Colour are reflected, the reafon of the Colours made by the tranfmitted Light in the 9th and 29th Ob- fervations is from hence evident. ■ , If not only the order and fpecies of thefe Colours^ but alfo the precife thicknefs of the plate, or thin body at which they are exhibited, be defired in parts ;of an Inch, that may be alfo obtained by affiftance of the 6th or 1 6th Obfervations, For according to thofe Obferva- tions the thicknefs of the thinned Air, which between two Glades exhibited the molt luminous parts of the firft fix Rings were ^ ^ gfe parts of an Inch. Suppofe the Light reflected moft copioufiy at thefe thickneffes be the bright citrine yellow, or con- F f 2 fine [3*] fine of yellow and orange, and thefe thickneffes will be G Gv, G ?, G», G i. And this being known, it is to h • But farther, fincebythe 10th Obfervation the thick- nefs of Air was to the thicknefs of Water, which be¬ tween the fame Glaffes exhibited the fame Colour, as 4 to 3, and by the a 1 th Obfervation the Colours of thin bodies are not varied by varying the ambient me¬ dium ; the thicknefs of a Bubble of Water, exhibiting any Colour, will be ^ of the thicknefs of Air producing the fame Colour. And fo according to the fame 10th and aith Obfervations the thicknefs of a plate of Glafs, whofe refraftion of the mean refrangible ray, is meafured by the proportion of the Sines 31 to 20, may be of the thicknefs of Air producing the fame Colours y and the like of other mediums. I do not affirm, that this proportion of no to 31, holds in all the rays ; for the Sines of other forts of rays have other proportions. But the differences of thofe proportions are fo little that I do not here confider them. On thefe Grounds I have compofed the following Table^ wherein the thicknefs of Air, Water, and Glafs, at which each Colour is moft intenfe and fpecifick, is ex- preffed in parts of an Inch divided into Ten hundred thoufand equal parts. etermine what thicknels or Air is reprelented r by any other diftance of the ruler from The ‘The ibicknefs of coloured Plates and ‘Particles of i 'rAea *®9 hr ft Order, rVery Black Black Beginning of Their Colours of the j Bllfekck White Yellow- Orange : vRed r Violet Indica. Blue Of the fecond Order, , Qv*:?n 7 < Yellow c ■r Orange Bright Red Scarlet Air, Water . GUfs . Indico Blue Of the third Order, 4 Green Yellow Red LBluifh Red ( Bluilh Green Of the fourth Order, jGreen YYellowiih Green CRed Of the fifth Order, S§reenilh Blue Of the fixth Order, S G reenilh Blue • ^Red Of the feventh Order, 5 £ reeniilv Blue ’ (Ruddy Whitt I 2. t O % 8 3 1 J. 2 O I 4 3 1 2 % 17 it i x 1*0 n 3.5 3t 7? 51 4f 8 6 Si 9 <>\ 5f 1 1 6 Si yt 1 2g 9* Oi On 14 ioi 9 nf 9? i6f I2f iof 1-7 9 1J iif i Sf Hi i9f Hi I2f . 21 1 3 11 22ri 16* Hi 2-3 f J7ii 157! 2 51 i8* 1 0 I6f 2Df iyi 29 214 1 si 32 24 20f 34 25I 22 351 2 6f 22 f 36 2.7 23t 4°3 3°i 26 46 34l '29I 39i 34' ^n\ 00 4>|w 44 38 65 481 42. 71 53-1 45? 77 57l 49? . _ ■Now if this Table be compared with the 6th Scheme* you will there fee the conliitution of each Colour* as to its Ingredients* or the original Colours of which it is compounded* and thence be enabled to judge of its intenfenefs or imperfection; which may fuffice in ex¬ plication of the 4-th and 18th Obfervations* unlefs it be further deiired to delineate the manner how the Co¬ lours appear* when the two ObjeCt-Glaffes are lay’d upon one another. To do which* let there be de- fcribed a large Arc of a Circle* and a ftreight Line which may touch that Arc* and parallel to that Tan¬ gent feveral occult Lines* at fuch distances from it* as the numbers fet againft the feveral Colours in the Table denote. For the Arc* and its Tangent* will reprelent the fuperficies of the Glafles terminating the interjacent Air ; and the places where the occult Lines cut the Arc will fhow at what diftances from the Center* or Point of contaft* each Colour is reflected. There are alfo other ufes of this Table : For by its afliftanee the thicknefs of the Bubble in the 1 9th 01> fervation was determined by the Colours which it ex¬ hibited. And fo the bignefs of the parts of natural Bodies may be conjectured by their Colours* asfhall be hereafter fhewn. Alfo* if two or more very thin plates be lay’d one upon another* fo as to compofe one plate equalling them all in thicknefs* the refulting Colour may be hereby determined. For inftance* Mr. Hook in his Mifcrografhm obferves* that a faint yellow- platemf Mufcovy-glafs lay’d upon a blue one* conftituted a very deep purple. The yellow of the firft Order is a faint one* and the thicknefs of the plate exhibiting it* ac¬ cording to the Table is 4f, to which add 9* the thick- nefs exhibiting blue of the fecond Order, and the furn will be which is the thicknefs exhibiting the purple of the third Order. To explain, in the next place, the Circumftances of the ad and 3d Obfervations ; that is, how the Rings of the Colours may (by turning the Prifms about their common Axis the contrary way to that expreffed in thofe Obfervations) be converted into white and black Rings, and afterwards into Rings of Colours again, the Colours of each Ring lying now in an inverted order; it muft be remembred, that thofe Rings of Colours are di¬ lated by the obliquaticn of the rays to the Air which intercedes the Glaffes, and that according to the Table in the 7th Obfervation, their dilatation or increafe of their Diameter is moft manifeft and fpeedy when they are obliqueft, Now the rays of yellow being more re- frafted by the firft fiiperficies of thefaid Air than thofe of red, are thereby made more oblique to the fecond fu- perficies, at which they are reflected to produce the co¬ loured Rings, and confequently the yellow Circle in each Ring will be more dilated than the red ; and the excefs of its dilatation will be fo much the greater, by how much the greater is the obliquity of the rays, until at laft it be¬ come of equal extent with the red of the fame Ring. And for the fame reafon the green, blue and violet, will be alia fo much dilated by the ftill greater obliquity of their rays, as to become all very nearly of equal extent with the red, that is, equally diftant from the center of the Rings. And then all the Colours of the fame Ring mud be coincident, and by their mixture exhibit a white Ring. Arid thefe white Rings muft have black and dark Rings between them , beeaufe they do not [ 4° 3 fpread and interfere with one another as before. And for that reafon alio they muft become diftin&er and vi~ fible to far greater Numbers. But yet the violet being obliqueft will be fomething more dilated in proportion to its extent then the other Colours, and fo very apt to appear at the exterior verges of the white. Afterwards, by a greater obliquity of the rays, the violet and blue become more fenfibly dilated than the red and yellow, and fo being further removed from the center or the Rings, the Colours muft emerge out of the white in an order contrary to that which they had be¬ fore, the violet and blue at the exterior limbs of each Ring, and the red and yellow at the interior. And the vio¬ let, by reafon of the greateft obliquity of its rays, being in proportion moft of all expanded, will fooneft appear at the exterior limb of each white Ring, and become more confpicuous than the reft. And the feveral Series of Colours belonging to the feveral Rings , will, by their unfolding and fpreading, begin again to interfere, and thereby render the Rings lefs diftindf, and not vifL ble to fo great numbers. If inftead of the Prifms the Obje£Tglafies be made ufe of, the Rings which they exhibit become not white and diftinft by the obliquity of the Eye, by reafon that the rays in their paflage through that Air which inter¬ cedes the Glaffes are very nearly parallel to thofe Lines in which they were fir ft incident on the Glaffes, and con- fequently the rays indued with feveral Colours are not inclined one more than another to that Air, as it hap¬ pens in the Prifms. There is yet another circumftance of thefe Experiments tp be confidered, and that is why the black and white Rings [40 Rings which when viewed at adiftance appear diftind, fhould not only become confufed by viewing them near at hand , but alfo yield a violet Colour at both the edges of every white Ring. And the reafon is, that the rays which enter the Eye at feveral parts of the Pupil, have feveral obliquities to the Glaffes, and thofe which are moft oblique, if confidered apart, would reprefent the Rings bigger than thofe which are the leaft oblique. Whence the breadth of the perimeter of every white Ring is expanded outwards by the obliqued rays, and inwards by the leaft oblique. And this expan fion is fo much the greater by how much the greater is the difference of the obliquity ; that is, by how much the Pupil is wider, or the Eye nearer to the Glaffes. And the breadth of the violet mu ft be moft expanded, be- caufe the rays apt to excite a fenfation of that Colour are moft oblique to a fecond, or further fuperficies of the thin’d Air at which they are refle&ed, and have alfo the greateft variation of obliquity , which makes that Colour fooneft emerge out of the edges of the white. And as the breadth of every Ring is thus aug¬ mented, the dark intervals muft be diminifhed, until the neighbouring Rings become continuous, and are blended, the exterior firft, and then thofe nearer the Center , fo that they can no longer be diftinguilh’d apart, but feem to conftitute an even and uniform whitenel's. Among all the Obfervations there is none accompa¬ nied with fo odd circumftances as the 24th. Of thofe the principal are, that in thin plates, which to the naked Eye ieem of an even and uniform tranfparent G g white- r +* ;i wtiitfcnefs, without any terminations of fhadows, the refra&ion of a Prifm fhould make Rings of Colours ap¬ pear, whereas it ufually makes ObjeCts appear coloured only there where they are terminated with fhadows, or have parts unequally luminous; and that it fhould make thofe Rings exceedingly diftinCt and white, although it ufually renders ObjeCts confided and coloured. The caufe of thefe things you will underhand by considering, that all the Rings of Colours are really in the plate, when viewed with the naked Eye, although by reafon of the great breadth of their circumferences they fo much interfere and are blended together, that they feem to conftitute an even whitenefs. But when the rays pafs through the Prifm to the Eye, the orbits of the Several Colours in every Ring are refracted, fome more than others, according to their degrees of refrangibility : By which means the Colours on one fide of the Ring (that is on one fide of its Center) become more unfolded and dilated, and thofe on the other, fide more compli¬ cated and contracted. And where by a due refraCtion they are fo much contracted, that the fevral Rings be¬ come narrower than to interfere with one another, they mu ft appear diftinCt, and alfo white, if the conftituent Colours be fo much contracted as to be wholly coincident. But, on the other fide, where the orbit of every Ring is made broader by the further unfolding of its Co¬ lours, it mu ft interfere more with other Rings than before, and fo become lefs diftinCt. To explain this a little further, fuppofe the concern °pige 7. trick Circles A V, and BX, reprefent the red and violet of any order, which, together with the intermediate Colours, [ 43 1 Colours, conftitute any one of thefe Rings. Now thefe being viewed through a Prifm, the violet Circle B X, will by a greater refraction be further tranflated from its place than the red A V, and fo approach nearer to it on that fide, towards which the refractions are made. For inftance, if the red be tranflated to av, the viole;t may be tranflated to b x , fo as to approach nearer to it at x than before, and if the red be further tranflated to a v, the violet may be fo much further tranflated to b x as to convene with it at x, and if the red be yet further tranflated to «■ T, the violet may be ftill fo much further tranflated to 0 5 as to pals beyond it at 5, and convene with it at e and/. And this being underftood not only of the red and violet, but of all the other in¬ termediate Colours, and alfo of every revolution of thofe Colours, you will eafily perceive how thofe of the fame revolution or order, by their nearnefs at xv and T ?, and their coincidence at xv, e and/, ought to con¬ ftitute pretty diftinCt Arcs of Circles, efpecially at xv, or at e and /, and that they will appear feverally at x and at x v exhibit whitenefs by their coincidence, and again appear feveral at T 5, but yet in a contrary order to that which they had before, and ftill retain beyond and /. But, on the other fide, at ab, ab, or * 0, thefe Colours mu If become much more confu¬ ted by being dilated and fpread fo, as to interfere with thofe of other Orders. And the fame confufion will happen at T 5 between e and/, if the refraction be very great, or the Prifm very diftant from the ObjeCt-Glaffes : In which cafe no parts of the Rings will be feen, fave only two little Arcs at e and/, whofe diftance from one Gg 2 another, another will be augmented by removing the Prifhr ftill further from the ObjedbGlaffes : And thefe little Arcs muft be diftin cave Sphere of Gold filled with Water, and fodered up, has upon preffing the Sphere with great force, let the Water fqueeze through it, and ftand all over its out- fide in multitudes of fmall Drops, like dew, without burfting or cracking the Body of the Gold as I have been informed by an Eye-witnefs. From all which we may conclude, that Gold has more pores than folid parts, and by confequence that Water has above forty- times more pores than parts. And he that ftiall find out anHypothefis, by which Water may befo rare, and yet not be capable of compreflion by force, may doubtleis by the fame Hypothefis make Gold and Water, and all other [ 7° 3 other Bodies as much rarer as he pleafes, fo that Light may find a ready paflage through tranfparent fub- ftances. PROP. IX. Bodies reflect and refrafi Light by one and the fame fower varioujly exercifed in variom circumflances. This appears by feveral Confiderations. Firft, Be- caufe when Light goes out of Glafs into Air, as ob¬ liquely as it can poffibly do, if its incidence be made ftill more oblique, it becomes totally reflected. For the power of the Glafs after it has refracted the Light as obliquely as is poffible if the incidence be ftill made more oblique, becomes too ftrong to let any of its rays go through, and by confequence caufes total reflexions. Secondly , Becaufe Light is alternately reflected and tranfmitted by thin Plates of Glafs for many fucceflions accordingly, as the thicknefs of the Plate increafes in an arithmetical Progreffion. For here the thicknefs of the Glafs determines whether that power by which Glafs ads upon Light ftiall caufe it to be refleded, or luffer it to be tranfmitted. And, Thirdly, becaufe thofe furfaces of tranfparent Bodies which have the greateft refrading power, refled the greateft quantity of Light, as was Shewed in the firft Propofition. PROP. X. If Light be fwifter in Bodies than in Vacuo in the proportion of the Sines which measure the refraffion of the Bodies , the forces of the Bodies to refleiB and refrafl Light , [ 7 1 3 are very nearly proportional to the denjities of the fame Bodies , excepting that unduom and fulphureom Bodies re - frad more than others of this fame denfity . Let A B reprefent the refracting plane furface of any Body, and I C a ray incident very obliquely upon the Body in C, fo that the Angle A C I may be infinitely little, and let CR be the refraCted ray. From a given point B perpendicular to the refracting furface ereCt B R meeting with the refrafted ray CR in R, and if CR reprefent the motion of the refraCted ray, and this motion be diftinguilhed into two motions CB and BR, whereof CB is a parallel to the refraCting plane, and BR perpendicular to it : CB {hall reprefent the motion of the incident ray, and B R the motion generated by the refraCtion, as Opticians have of late explained. Now if any body or thing in moving through any fpace of a giving breadth terminated on both tides by two parallel plains, be urged forward in all parts ofi that {pace by forces tending direCtly forwards towards the laft plain , and before its incidence on the firft plane, had no motion towards it, or but an infinitly little one *. and if the forces in all parts of that fpace, # between the planes be at equal diftances from the planes equal to one another, but at feveral diftances be bigger or lefs in any given proportion, the motion generated- by the forces in the whole paflage of the body or thing- through. [70 through that fpace lha.ll be in a fubduplicate proportion of the forces, as Mathematicians will eafily underftand. And therefore if the fpace of activity of the refracting luperficies of the Body be confidered as fuch a fpace, the motion of the ray generated by the refraCting force of the Body , during its paflage through that fpace that is the motion BR muft be in a fubduplicate proportion of that refraCting force : I fay therefore that the lquare of the Line B R, and by confequence the refrading force of the Body is very nearly as the den- fity of the fame Body. For this will appear by the fol- lowingTable, wherein the proportion of the Sines which meafure the refraxions of feveral Bodies, the fquare of BR fuppofing CB an unite, the denfities of the Bodies eftimated by their fpecifick gravities, and their refraCtive power in refpeCt of their denfities are fet down in feveral Columns. The [73] The refrafting Bodies. The Proportion of the Sines of incidence and refraction of yellow Light , The Square of B R, to which the refracting force oftheBo - dy is propor¬ tionate. The denfity and fpeci - fc gravity of the Bo- dy. The refra¬ ctive power of the Body in refpeCt of its den- fity. A Pfeudo-Topazius, be¬ ing a natural, pellucid, brittle, hairy Stone, of 23 t0 14 ^699 4’27 3979 a yellow Colour 385° 9 4160 Air 2851 to o’ooo 5 2 o’oOI2 5 Glafs of Antimony 17 to 2*568 1*213 5*28 2*252 '4864 A Selenitis 61 to 41 5386 Glafs vulgar 31 to 20 1*4025 1 ’44 5 275§ 5436 Cryltal of the Rock 25 to 1 6 2^65 2?72 2?I43 5450 Hand Cryftal 5 3 1*778 6536 Sal Gemmae 17 to 1 1 1388 1T267 6477 Alume 3 5 to 24 *’7*4 *’7*4 i?9 6 57° Borax 22 tO *5 1*1511 6716 Niter 3 2 to 21 t’345 1*295 1*041 7°79 Dantzick Vitriol 3°3 to 200 I’7I5 7 5 5 * Oyl of Vitriol 10 to 7 x*7 6124V Rain Water 529 to 396 °’7845 I*179 I. 7S45 Gumm Arabic 31 to 21 i’375 8574 Spirit of Wine well recti¬ fied 100 to 73 °J8y6] And hence Light is in fits of eafy reflexion and eafy tranfmiflion, before its incidence on tranfparent Bodies. And probably it is put into fuch fits at its firft emiflion from luminous Bodies, and continues in them during all its progrefs, For thefe fits are of a lading Nature, as will appear by the next part of this Book* In this Propofition I fuppofe the tranfparent Bodies to be thick, becaufe if the thicknefs of the Body be much lefs than the interval of the fits of eafy reflexion and tranfmiflion of the rays, the Body lofethits reflecting power. For if the rays, which at their entering into the Body are put into fits of eafy tranfmiflion, arrive at the furtheft lurface of the Body before they be out of thofe fits they muft be tranfmitted. And this is the reafon why Bubbles of Water lofe their reflecting power when they grow very thin, and why all opake Bo¬ dies when reduced into very fmall parts become tranf¬ parent. PROP. XIV. Thofe fur faces of tranfparent Bodies , 'which if the ray he in a fit of refraction do refraCt it mofl ftronglyy if the fay be in a fit of reflexion do refleCt it mofl eafily. For we fhewed above in Prop. 8. that the caufe of reflexion is not the impinging of Light on the folid impervious parts of Bodies, but fome other powrer by which thofe folid parts a£t on Light at a diftance. We fhewed alfo in Prop. 9. that Bodies reflect and refra£t Light by one and the fame power varioufly exercifed in various circumftances, and in Prop. 1. that the mofl: ftrongly refracting furfaces reflect the mofl Light : All which which compared together evince and ratify both this and the laft Propolition, PROP, XV, In any one and the fame fort of rays emerging in any jingle out of any refracting fur face into one and the fame medium , the interval of the following fits of eafy reflexion and tranfmijflon are either accurately or very nearly , as the Re Cl angle of the fee ant of the Angle of refraSion , and of the fecant of mother Angle y whofe fine is the firfl of 106 arithmetical mean proportionals y between the fines of incidence and refraction counted from the fine of re~ fradion . This is manifeft by the 7th Oblervation. PROP, XVL In fever al forts of rays emerging in equal Angles out of any refraCting fur face into the fame medium, , the inter * vals of the following fits of eafy reflexion and eafy tranfl mijfion are either accurately , or very nearly , as the Cube* roots of the Squares of the lengths of a Chord, which found the notes in an Eighty fol, la, fa, folr la, mi, fa, fol, with all their intermediate degrees answering to the Colours of thofe raysy according to the Analogy deferibed in the fe* venth Experiment of the fecond Booh This is manifeft by the 13 th and 1 4th Obfervations* Mm 2 PROP, [8+] PROP. XVII. If rays of any one fort pafs perpendicularly into feveral mediums , the intervals of the fits of eafy reflexion and tranfmijfion m any one medium , is to thofe intervals in any other as the fine of incidence to the fine of refraHion , when the rays pafs out of the fir ft of thofe two mediums into the fecond. This is inanifeft by the loth Obfervatiom PROP. XVIII. If the rays which paint the Colour in the confine of yellow and orange pafs perpendicularly out of any medium into ylir^ the intervals of their fits of eafy reflexion are the zf~-ath part of an Inch . iAnd of the fame length are the intervals of their fits of eafy tranfmijfion. This is manifeft by the 6th Qbfervation. From thefe Propofitions it is eafy to colled the in¬ tervals of the fits of eafy reflexion and eafy tranfmif- fion of any fort of rays refraded in any Angle into any medium, and thence to know, whether the rays Ihall be refleded or tranfmitted at their fubfequent incidence upon any other pellucid medium. Which thing being ufeful for underflanding, the next part of this Book was here to be fet down. And for the lame -reafon I add the two following Propofitions. PROP. [85] PROP. XIX. If any fort of rays falling on the polite fur face of am pellucid medium be reflecled back^ the fits of eafy re¬ flexion which they have at the point of reflexion , J, hall fiill continue to return , and the returns Jhall be at di¬ fiances from the point of reflexion in the arithmetical progreflion of the numbers 2, 4, 6, 8, 10, I2,&c. and be¬ tween thefe fits the rays Jhall be in fits of eafy tranfl mijflon . For fince the fits of eafy reflexion and eafy tranf- miflion are of a returning nature, there is no reafbn why thefe fits, which continued till the ray arrived at the reflecting medium, and there inclined the ray to reflexion, fliould there ceafe, And if the ray at the point of reflexion was in a fit of eafy reflexion, the progreflion of the diftances of thefe fits from that point •rnuft begin from o, and fo be of the numbers o, a, 4, 6, 8, 1 Sc.. And therefore the progreflion of the di¬ ftances of the intermediate fits of eafy tranfmiflion rec¬ koned from the fame point, muft be in the progreflion of the odd numbers 1, 4, 5, 7, 9, If 1c, contrary to what happens when the fits are propagated from points of refraction, PRO P. XX The intervals of the fits of eafy reflexion and eajf tranfmiflion , propagated from points of reflexion into any medium , are equal to the intervals of the like fits which the fame rays would have , if refracted into the fame medium / [8<5] medium in Angles of refraction equal to their Angles of reflexion . For when Light is reflected by the fecond furface of thin plates, it goes out afterwards freely at the firft fur- face to make the Rings of Colours which appear by reflexion, and by the freedom of its egrefs, makes the Colours of thefe Rings more vivid and ftrong than thofe which appear on the other fide of the plates by the tranfmitted Light. The reflefted rays are therefore in fits of eafy tranfmiffion at their egrefs ; which would not always happen, if the intervals of the fits within the plate after reflexion were not equal both in length and number to their intervals before it. And this confirms alfo the proportions fet down in the former Propofition. For if the rays both in going in and out at the firft furface be in fits of eafy tranfmiffion, and the intervals and num¬ bers of thofe fits between the firft and fecond furface, before and after reflexion, be equal • the diftances or the fits of eafy tranfmiffion from either furface, muff be in the fame progrefiion after reflexion as before ; that is, from the firft furface which tranfmitted them, in the progrefiion of the even numbers o, 2, 4, 6, 8, )3c. and from the fecond which reflected them, in that of the odd numbers 3^ 5^ 7^ &V* But thefe two Pro- pofitions will become much more evident by the Gbfer- various in the following part of this Book* THE THE SECOND BOOK OPTICK S. PART IV. Observations concerning the Reflexions and Colours of thick tranfparent folijhed 'plates. THere is no Glafs or Speculum how well foever polifhed, but, befides the Light which it refracts or refle&s regularly , fcatters every way irregularly a faint Light, by means of which the polifhed furface, when illuminated in a dark Room by a beam of the Sun’s Light, may be eafily feen in all pofitions of the Eye. There are certain Phenomena of this fcattered Light, which when I firft obferved them, feemed very ft range and furprifing to me. My Operations were as follows. OBS. [ S8 ] O B S. I. The Sun firming into my darkened Chamber through a Hole ] of an Inch wide, I let the intromitted beam of Light fall perpendicularly upon a Glafs Speculum ground concave on, one fide and convex on the other, td a Sphere of five Feet and eleven Inches Radius, and quickTilvered over on the convex fide. And holding a white opake Chart, or a Quire of Paper at the Center of the Spheres to which the Speculum was ground, that is, at thediftance of about five Feet and eleven Inches from the Speculum, in fuch manner, that the beam of Light might pafs through a little Hole made in the middle of the Chart to the Speculum, and thence be reflected back to the fame Hole : I obferved upon the Chart four or five concentric Irifes or Rings of Colours, like Rain-bows, encompaffing the Hole much after the manner that thofe, which in the fourth and following Obfervations of the firftpart of this third Book appeared between theObje«] of Air, Water, or Glafs of an even thicknefs appeared of one Colour when the rays were perpendicular to it, of another when they were a little oblique, of another when more oblique, of another when ftill more oblique, and fo on ; fo here, in the lixth Obfervation, the Light which emerged out of the Glafs in feveral obliquities, made the Glafs appear of feveral Colours, and being propagated in thofe obliquities to the Chart, there pain¬ ted Rings of thofe Colours. And as the reafon why a thin plate appeared of feveral Colours in feveral obli¬ quities of the rays, was, that the rays of one and the fame fort are reflected by the thin plate at one obliquity and tranfmitted at another, and thofe of other forts tranf- mitted where thefe are reheated, and reflected where thefe are tranfmitted : So the reafon why the thick plate of Glafs whereof the Speculum was made did ap¬ pear of various Colours in various obliquities, and in thofe obliquities propagated thofe Colours to the Chart, was, that the rays of one and the fame fort did at one obliquity emerge out of the Glafs, at another did not emerge but were reflected back towards the Quick-fil- ver by the hither furface of the Glafs, and accordingly as the obliquity became greater and greater emerged and were relieved alternately for many fucceffions, and that in one and the fame obliquity the rays of one fort were reflected, and thofe of another tranfmitted. This is manifeft by the firft Obfervation of this Rook : For in that Obfervation, when the Speculum was illumi¬ nated by any one of the prifmatick Colours, that Light made many Rings of the fame Colour upon the Chart with dark intervals, and therefore at its emergence out of the Speculum was alternately tranfmitted, and not tranf- [ 9? 1 tranfmitted from the Speculum to the Chart for many fucceffions, according to the various obliquities of its emergence. And when the Colour caft on the Specu¬ lum by the Prifm was varied, the Rings became of the Colour caft on it, and varied their bignefs with their Colour, and therefore the Light was now alternately tranfmitted and not tranfmitted from the Speculum to the Lens at other obliquities than before. It feemed to me therefore that thefe Rings were of one and the fame original with thofe of thin plates, but yet with this difference that thofe of thin plates are made by the al¬ ternate reflexions and tranfmiffions of the rays at the fecond furface of the plate after one paffage through it : But here the rays go twice through the plate before they are alternately reflected and tranfmitted ; fir ft, they go through it from the firft furface to the Quick- filver, and then return through it from the Quick -filver to the firft furface, and there are either tranfmitted to the Chart or reflected back to the Quick-filver, ac¬ cordingly as they are in their fits of eafie reflexion or tranfmiffion when they arrive at that furface. For the intervals of the fits of the rays which fall perpendicu¬ larly on the Speculum, and are reflected back in the fame perpendicular Lines, by reafon of the equality of thefe Angles and Lines, are of the fame length and num- ber within the Glafs after reflexion as before by the 19th Propofition of the third Part of this Book. * And therefore ftnce all the rays that enter through the firft furface are in their fits of eafy tranfmiffion at their en¬ trance, and as many of thefe as are reflected by the fe¬ cond are in their fits of eafy reflexion there, all thefe mull be again in their fits of eafy tranfmiffion at their Q o return return to the firft, and by confequence there go out of the Glafs to the Chart., and form upon it the white Spot of Light in the center of the Rings. For the rea- fon holds good in all forts of rays , and therefore all forts mu ft go out promifcuoully to that Spot, and by their mixture caufe it to be white. But the intervals of the fits of thole rays which are reflected more ob¬ liquely than they enter, muft be greater after reflexion than before by the 15 th and 20th Prop. And thence it may happen that the rays at their return to the firft furface, may in certain obliquities be in fits of eafy re¬ flexion, and return back to the Guick-filver, and in other intermediate obliquities be again in fits of eafy tranfmiflion, and fo go out to the Chart, and paint on it the Rings of Colours about the white Spot. And becaufe the intervals of the fits at equal obliquities are greater and fewer in the lefs refrangible rays, and lefs and more numerous in the more refrangible, therefore the lefs refrangible at equal obliquities fhall make fewer Rings than the more refrangible, and the Rings made by thofe fhall be larger than the like number of Rings made by thefe ; that is, the red Rings fhall be larger than the yellow, the yellow than the green, the green than the blue, and the blue than the violet, as they were really found to be in the 5th Obfervation. And therefore the firft Ring of all Colours incompaffing the white Spot of Light fhall be red without and violet within, and yellow, and green, and blue in the middle, as it was found in the fecond Obfervation ; and thefe Colours in the fecond Ring, and thofe that follow fhall be more expanded till they fpread into one another, and blend one another by interfering* Thefe [99 3 i Thefe feem to be the reafons of thefe Rings in ge¬ neral, and this put me upon obferving the thicknefs of the Glafs, and confidering whether the dimenfions and proportions of the Rings may be truly derived from it by computation. O B S. VIII. I meafured therefore the thicknefs of this concavo- convex plate of Glafs, and found it every-where i of an Inch precifely. Now, by the 6th Obfervation of the firft Part of this Book, a thin plate of Air tranfmits the brighteft Light of the firft Ring, that is the bright yel¬ low, when its thicknefs is the 8^-0th part of an Inch, and by the i oth Obfervation of the fame part, a thin plate of Glafs tranfmits the fame Light of the fame Ring when its thicknefs is lefs in proportion of the fine of refraction to the fine of incidence, that is, when its thicknefs is the ~ath or ,-3;— th part of an Inch, fup- pofing the fines are as 1 1 to 17. And if this thicknefs be doubled it tranfmits the fame bright Light of the fecond Ring, if tripled it tranfmits that of the third, and fo on, the bright yellow Light in all thefe cafes be¬ ing in its fits of tranfmiffion. And therefore if its thick¬ nefs be multiplied 34.386 times fo as to become^ of an Inch it tranfmits the fame bright Light of the 34.386th Ring. Suppofe this be the bright yellow Light tranf- mitted perpendicularly from the reflecting convex fide of the Glais through the concave fide to the white Spot in the center of the Rings of Colours on the Chart : And by a rule in the feventh Obfervation in the firft Part of the firft Book, and by the 1 5 th and 20th Propofitions O o 2 of / [ioo] of the third Part of this Book, if the rays he made ob¬ lique to the Glafs, the thicknefs of the Glafs requh fite to tranfmit the fame bright Light of the fame Ring in any obliquity is to this thicknefs of of an Inch, , as the fecant of an Angle whole line is the firft of an hun¬ dred and fix arithmetical means between the fines of incidence and refraction, counted from the fine of inci¬ dence when the refraction is made out of any plated Bo¬ dy into any medium incompafliog it, that is, in this cafe, out of Glals into Air. Now if the thicknefs of the Glafs be increafed by degrees, fo as to bear to its firft thicknefs, ( viz . that of a quarter of an Inch ) the proportions which 343 86 (the number of fits of the perpendicular rays in going through the Glafs towards the white Spot in the center of the Rings,) hath to 34385, 34384, 34383 and 3438a (the numbers of thefits of the oblique rays in going through the Glafs towards the firft, fe- cond, third and fourth Rings of Colours,) and if the firft thicknefs be divided into 100000000 equal parts, the increafed thickneffes will be 100002908, 1000058165 100008725 and 10001163 3, and the Angles of which thefe thickneffes are fecants will be 26' 13", 357' 5', 45' 6' and 5, a' a 6", the Radius being ioooooooo ; and the fines of thefe Angles are 76a, 1079, 1321 and 1-5.25, and the proportional fines of refraction 117a, 1659, 2031 and 2345, the Radius being 100000. For fince the fines of incidence out of Glafs into Air are to the fines of refraCtion as ri to 17, and to theabove-mentioned fecants as. 1 1 to the firft of 106 arithmetical means between 11 and 17,, that is as 11 to 11^, thefe fe¬ cants will he to the fines of refraCtion as 11-5 to 1 7, and by this Analogy will give thefe fines. So then .[ 101 ] if the obliquities of the rays to the concave furface of the Glals be fuch that the lines of their relradtion in palling out of the Glafs through that furface into the Air be 117:2, 1659, 2031, 2345, the bright Light of the 34386th Ring lhall emerge at the thicknefles of the Glafs which are to I of an Inch as 34386 to 34385, 34384, 34383, 34382, refpedtively. And therefore if the thicknels in all thefe cafes be - of an Inch (as it is in the Glafs of which the Speculum was made) the bright Light of the 34385th Ring lhall emerge where the line of refraction is 1 172, and that of the 34384th, 384383th and 34382th Ring where the line is 1659, 2031, and 2345 refpedtively. And in thefe Angles of refraction the Light of thefe Rings lhall be propagated from the Speculum to the Chart, and there paint Rings about the white central round Spot of Light which we laid was the Light of the 34386th Ring. And the Semidiame¬ ters of thefe Rings lhall fubtend the Angles of refraCtion made at the concave, lurface of the Speculum, and by confequence their Diameters lhall be to the diftance of the Chart from the Speculum as thole lines of refratition doubled are to the Radius that is as 1 172, 1 659, 203 r, and 2.345, doubled are to 100000. And therefore if, the diftance of the Chart from the concave furface of the Speculum be fix Feet ( as it was in the third of thefe Obfervations) the Diameters of the Rings of this bright yellow Light upon the Chart lhall be x7688? 2’925j F375 Inches : For thefe Diameters are to A Feet as the above-mentioned fines doubled are to the Radius. No thefe Diameters of the bright yellow Rings, thus found by computation are the very fame with thole found in the third of thefe Obfervations by meafuring therrq [ 102 ] them, (viz. with ijj. a|> a|i, and 3’- Inches, and there¬ fore the Theory of deriving thefe Rings from the thick- nefs of the plate of Glafs of which the Speculum was made, and from the obliquity of the emerging rays agrees with the Obfervation. In this computation I have equalled the Diameters of the bright Rings made by Light of all Colours, to the Diameters of the Rings made by the bright yellow. For this yellow makes the brighteft part of the Rings of all Colours. If you defire the Diameters of the Rings made by the Light of any other unmixed Colour, you may find them readily by putting them to the Diameters of the bright yellow ones in a lubduplicate proportion of the intervals of the fits of the rays of thole Colours when equally inclined to the refracting or reflecting furface which caufed thofe fits, that is, by putting the Diameters of the Rings made by the rays in the extremities and limits of the feven Colours, red, orange, yellow, green, blue, indico, violet, proportional the Cube-roots of the numbers, 1 , f , | , * , > , * , % , { , which exprefs the lengths of a Monochard founding the notes in an Eight : For by this means the Diameter of the Rings of thefe Colours will be found pretty nearly in the fame proportion to one another, which they ought to have by the fifth of thefe Obfer- vations. And thus I fatisfied my felf that thefe Rings were of the fame kind and original with thofe of thin plates, and by conlequence that the fits or alternate difpofi- tions of the rays to be reflected and tranfmitted are pro¬ pagated to great diftances from every reflecting and re¬ fracting furface. But yet to put the matter out of doubt I added the following Obfervation. OBS. C io3 ] O B S. IX. If thefe Rings thus depend on the thicknefs of the plate ofGlafs their Diameters at equal di fiances from feveral Speculums made of luch concavo-convex plates of Glals as are ground on the fame Sphere, ought to be recipro¬ cally in a fubduplicate proportion of the thickneffes of the plates of Glals. And if this proportion be found true by experience it will amount to a demonftration- that thefe Rings ( like thofe formed in thin plates ) do- depend on the thicknefs of the Glafs. I procured there¬ fore another concavo-convex plate of Glafs ground on both fides to the fame Sphere with the former plate : Its thicknefs was f2 parts of an Inch ; and the Diameters of the three firft bright Rings meafured between the brighteft parts of their orbits at the diftance of 6 Feet from the Glafs were 3. 4.5. 5k Inches. Now the thick¬ nefs of the other Glafs being of an Inch was to thick¬ nefs of this Glafs a s j to f, > that is as 31 to 10, or 3 rooooooo to iooqoooo©5 and the roots of thefe numbers are 17607 and 10000, & in the proportion of the firft of thefe roots to the fecond are the Diameters of the bright Rings made in this Obfervation by the thinner Glafs, 3. 4k 5; to the Diameters of the fame Rings made in the third of thel'e Oblervations by the thicker Glafs i[k af that is, the Diameters of the Rings are reci¬ procally in a fubduplicate proportion of thickneffes of the plates of Glafs. So then in plates of Glafs which are alike concave on one fide, and alike convex on the other fide, and alike quick-filvered on the convex fides, and differ in nothing but [I°+] but their thicknefs, the Diameters of the Rings are re¬ ciprocally in a fubduplicate proportion of the thickneffes of the plates. And this fhews fufficiently that the Rings depend on both the furfaces of the Glafs. They de¬ pend on the convex furface becaufe they are more lu¬ minous when that furface is quick-filvered over than when it is without Quick-lilver. They depend alfo upon the concave furface, becaufe without that furface a Speculum makes them not. They depend on both furfaces and on the diftances between them., becaufe their bignefs is varied by varying only that diftance. And this dependance is of the fame kind with that which the Colours of thin plates have on the diftance of the furfaces of thofe plates, becaufe the bignefs of the Rings and their proportion to one another and the variation of their bignefs arifing from the varia¬ tion of the thicknefs of the Glafs, and the orders of their Colours, is fuch.as ought to refult from the Propo- fitions in the end of the third Part of this Book, derived from the the Phenomena of the Colours of thin plates fet down in the firft Part. There are yet other Phenomena of thefe Rings of Colours but fuch as follow from the fame Propofitions and therefore confirm both the truth of thofe Propofi¬ tions, and the Analogy between thefe Rings and the Rings of Colours made by very thin plates. I fhall fubjoyn fome of them. t 10$ ] O B S. X. When the beam of the Sun’s Light was reflected back from the Speculum not direftly to the Hole in the Win¬ dow, but to a place a little diftant from it, the common center of that Spot, and of all the Rings of Colours fell in the middle way between the beam of the incident Light, and the beam of the reflected Light, and by confequence in the center of the fpherical concavity of the Speculum, whenever the Chart on which the Rings of Colours fell was placed at that center. And as the beam of reflected Light by inclining the Speculum re¬ ceded more and more from the beam of incident Light and from the common center of the coloured Rings be¬ tween them, thofe Rings grew bigger and bigger, and fo alfo did the white round Spot, and new Rings of Co¬ lours emerged fucceffively out of their common center, and the white Spot became a white Ring encompaffing them ; and the incident and reflected beams of Light always fell upon the oppoiite parts of this Ring, illumi¬ nating its perimeter like two mock Suns in the oppoiite parts of an Iris. So then the Diameter of this Ring, meafured from the middle of its Light on one fide to the middle of its Light on the other fide, was always equal to the diftance between the middle of the Incident beam of Light, and the middle of the reflected beam meaiured at the Chart on which the Rings appeared : And the rays which formed this Ring were reflected by the Speculum in Angles equal to their Angles of inci¬ dence, and by confequence to their Angles of refrafif ion at their entrance into the Glafs, but yet their Angles of ' P p reflexion [io6] reflexion were not in the fame planes with their Angles of incidence. O B S. XL The Colours of the new Rings were in a contrary order to thofe of the former, and arofe after this man¬ ner. The white round Spot of Light in the middle of the Rings continued white to the center till the diftance of the incident ond reflected beams at the chart was about | parts of an Inch, and then it began to grow dark in the middle. And when that diftance was about if6of an Inch, the white Spot was become a Ring en- compafling a dark round Spot which in the middle in¬ clined to violet and indico. And the luminous Rings incompaffing it were grown equal to thofe dark ones which in the four firft Obfervations encompaffed them, that is to fay, the white Spot was grown a white Ring equal to the firft of thofe dark Rings, and the firft of thofe luminous Rings was now grown equal to the fe- cond of thofe dark ones, and the fecond of thofe lumi¬ nous ones to the third of thofe dark ones, and fo on. For the Diameters of the luminous Rings were now Inches. When the diftance between the incident and reflected beams of Light became a little bigger, there emerged out of the middle of the dark Spot after the indico a blue, and then out of that blue a pale green, and foon after a yellow and red. And when the Colour at the center was brighteft, being between yellow and red, the bright Rings were grown equal to thofe Rings which in the four firft Obfervations next encompaffed them ; that [I°7 3 that is to fay, the white Spot in the middle of thofe Rings was now become a white Ring equal to the firft of thofe bright Rings, and the firft of thofe bright ones was now become equal to the fecond of thofe, and fo on. For the Diameters of the white Rings, and of the other luminous Rings incompaffing it, were now lit ai? aU, ^8,2 or thereabouts. When the diftance of the two beams of Light at the Chart was a little more increafed, there emerged out of the middle in order after the red, a purple, a blue, a green, a yellow, and a red inclining much to purple, and when the Colour was brighteft being between yel¬ low and red, the former indico, blue, green, yellow and red, were become an Iris or Ring of Colours equal to the firft of thofe luminous Rings which appeared in the four firft Qbfervations, and the white Ring which was now become the fecond of the luminous Rings was grown equal to the fecond of thofe, and the firft of thofe which was now become the third Ring was be¬ come the third of thofe, and fo on. For their Diame¬ ters were i*« , at , ? gi Inches, the diftance of the two beams of Light, and the Diameter of the white Ring being a- Inches. When thefe two beams became more diftant there emerged out of the middle of the purplifti red, firft a darker round Spot, and then out of the middle of that Spot a brighter. And now the former Colours (purple, blue, green, yellow, and purplifti red ) were become a Ring equal to the firft of the bright Rings mentioned in the four firft Qbfervations , and the Ring about this Ring were grown equal to the Rings about that re*- fpe&ively ; the diftance between the two beams of P p ^ Light [i°8] Light and the Diameter of the white Ring ( which was now become the third Ring ) being about 3. In¬ ches. The Colours of the Rings in the middle began now to grow very dilute, and if the diftance between the two beams was increafed half an Inch, or an Inch more, they vanifhed whilft the white Ring, with one or two of the Rings next it on either fide, continued ftili vi- fible. But if the diftance of the two beams of Light was ftili more increafed thefe alfo vanifhed : For the Light which coming from feveral parts of the Hole in the Window fell upon the Speculum in feveral Angles of incidence made Rings of feveral bigneffes, which diluted and blotted out one another, as I knew by intercepting fome part of that Light. For if I intercepted that part which was neareft to the Axis of the Speculum the Rings would be lefs, if the other part which was re- moteft from it they would be bigger. OBS. XIL When the Colours of the Prifm were caft fucceflively on the Speculum, that Ring which in the two laft Ob> fervations was white, was of the fame bignefs in all the Colours, but the Rings without it were greater in the green than in the blue, and ftili greater in the yellow, and greateft in the red. And, on the contrary, the Rings within that white Circle were lefs in the green than in the blue, and ftili lefs in the yellow, and lea ft in the red. For the Angles of reflexion of thofe rays which made this Ring being equal to their Angles of incidence, the fits of every reflected ray within, the Glafs after [lop] after reflexion are equal in length and number to the fits of the fame ray within the Glafs before its incidence on the reflecting furface; and therefore fince all the rays of all forts at their entrance into the Glafs were in a fit of tranfmiffion, they were alfo in a fit of tranfmiflion at their returning to the fame furface after reflexion ; and by confequence were tranfmitted and went out to the white Ring on the Chart. This is the reafon why that Ring was of the fame bignefs in all the Colours, and why in a mixture of all it appears white. But in rays which are reflected in other Angles, the intervals pf the fits of the leaft refrangible being greateft, make the Rings of their Colour in their progress from this white Ring, either outwards or inwards, increafe or decrease by the greateft fteps • fo that the Rings of this Colour without are greateft, and within leaft. And this is the reafon why in the laft Obfervation, when the Specu¬ lum was illuminated with white Light, the exterior Rings made by all Colours appeared red without and blue within, and the interior blue without and red within. Thefe are the Phenomena of thick convexo-concave plates of Glafs, which are every where of the fame thicknefs. 1 here are yet other Phenomena when thefe plates are a little thicker on one fide than on the other, and others when the plates are more or lefs con- cave than convex, or plano-convex, or double-convex. For in all thefe cafes the plates make Rings of Colours, but after various manners ; all which, fo far as I have yet obferved, follow from the Propofitions in the end of the third part of this Book, and fo confpire to con¬ firm the truth of thofe Propofitions. But the Pheno¬ mena,, [no] mena are too various, and the Calculations whereby they follow from thofe Propofitions too intricate to be here profecuted. I content my felf with having profe- cuted this kind of Phenomena fo far as to difcover their caufe, and by difcovering it to ratify the Propofitions in the third Part of this Book, OBS. XII L As Light reflected by a Lens quick-filvered on the back- fide makes the Rings of Colours above de- Icribed, fo it ought to make the like Rings of Colours in palling through a drop of Water; At the firft re¬ flexion of the rays within the drop, fome Colours ought to be tranfmitted, as in the cafe of a Lens, and others to be reflected back to the Eye. For inftance, if the Diameter of a fmall drop or globule of Water be about the 500th part of an Inch, fo that a red-making ray in palling through the middle of this globule has 150 fits of eafy tranfmiflion within the globule, and that all the red-making rays which are at a certain diftance from this middle ray round about it have 249 fits within the globule, and all the like rays at a certain further di- fiance round about it have 24.8 fits, and all thofe at a certain further diftance 247 fits, and fo on ; thefe con- centrick Circles of rays after their tranfmiflion, falling on a white Paper, will make concentrick rings of red upon the Paper , fuppofing the Light which pafles through one Angle globule ftrong enough to be fenfible. And, in like manner, the rays of other Colours will make Rings of other Colours. Suppofe now that in a fair day the Sun Ihines through a thin Cloud of fuch globules [Ill] globules of Water or Hail, and that the globules are all of the fame bignefs,and the Sun feen through this Cloud lhall appear incompaffed with the like concent rick Rings of Colours, and the Diameter of the firft Ring of red fhall be 7-* degrees, that of the fecond 1 o'- degrees, that of the third 12 degrees 33 minutes. And accordingly as the globules of Water are bigger or Ids, the Rings fhall be lefs or bigger. This is the Theory, and expe¬ rience anfwers it. . For in June 169a. I law by reflexion in a Veflel of ftagnating Water three Halos Crowns or Rings of Colours about the Sun, like three little Rain¬ bows, concentrick to his Body. The Colours of the firft or innermoft Crown were blue next the Sun, red without, and white in the middle between the blue and red. Thofe of the fecond Crown were purple and blue within, and pale red without, and green in the middle. And thofe of the third were pale blue with¬ in, and pale red without; thefe Crowns inclofed one another immediately, fo that their Colours proceeded in this continual order from the Sun outward : blue, white, red ; purple, blue, green, pale yellow and red ; pale blue, pale red. The Diameter of the fecond Crown meafured from the middle of the yellow and red on one fide of the Sun, to the middle of the fame Colour on the other fide was degrees, or thereabouts. The Dia¬ meters of the firf^and third I had not time to meafure, but that of the firft feemed to be about five or fix de¬ grees, and that of the third about twelve. The like Crowns appear fometimes about the Moon ; for in the beginning of the year 1664, Febr. 1 9th at night, I faw two fuch Crowns about her. The Diameter of the firft or innermoft was about three degrees, and that of the fecond. [H2] feeond about five degrees and an half. Next about the Moon was a Circle of white, and next about that the inner Crown which was of a bluifli green within next the white, and of a yellow and red without, and next about thefe Colours were blue and green on the infide of the outward Crown, and red on the outfide of it. At the fame time there appeared a Halo about 22 degrees 35* diftant from the center of the Moon. It was Elliptical, and its long Diameter was perpendicular to the Horizon verging below fartheft from the Moon. I am told that the Moon has fometimes three or more concen trick Crowns of Colours incompafling one another next about her Body. The more equal the globules of Water or Ice are to one another, the more Crowns of Colours will appear, and the Colours will be the more lively. The Halo at the diftance of 22- degrees from the Moon Ys of another fort. By its being oval and remoter from the Moon below than above, I conclude, that it was made by refraCtion in fome fort of Hail or Snow floating in the Air in an horizontal Pofture, the refracting Angle being about 58 or 60 degrees, T H E [ri3] THE THIRD BOOK r O F O P T I C K s. Obfervations concerning the Inflexions of the rays of Light , and the Colours made thereby . GRimaldo has informed us, that if a beam of the Sun’s Light be let into a dark Room through a very fmall Hole, the fliadows of things in this Light will be larger than they ought to be if the rays went on by the Bodies in Freight Lines, and that thefe Ilia- dows have three parallel fringes, bands or ranks of co¬ loured Light adjacent to them. But if the Hole be enlarged the fringes grow broad and run into one ano¬ ther, lo that they cannot be diftinguifhed. Thefe broad fhadows and fringes have been reckoned by fome to pro¬ ceed from the ordinary refraction of the Air, but with¬ out due examination of the matter. For the circuit- fiances of the Phenomenon, fo far as I have obferved them-, are as follows, Q q OBS, [iH] O B S. I. I made in a piece of Lead a fmall Hole with a Pin, whofe breadth was the 41th part of an Inch. For 21 of thofe Pins laid together took up the breadth of half an Inch. Through this Hole I let into my darkened Chamber a beam of the Sun's Light, and found that the fhadows of Hairs, Thred, Pins, Straws, and fuch like lien- der fubftances place! in this beam of Light, were confider- ably broader than they ought to be, if the rays of Light palled on by thefe Bodies in right Lines. And particu¬ larly a Hair oi a Man’s Head, whofe breadth was but the 280th part of an Inch, being held in this Light, at the diftance of about twelve Feet from the Hole, did eaft a fhadow which at the diftance of four Inches from the Hair was the fixtieth part of an Inch broad, that is, above four times broader than the Hair, and at the di¬ ftance of two Feet from the Hair was about the eight and twentieth part of an Inch broad, that is, ten times broader than the Hair, and at the diftance of ten Feet was the eighth part of an Inch broad, that is 3 5 times broader. Nor is it material whether the Hair be incompaffed with Air, or with any other pellucid fubftance. For I wetted a poliftied plate of Glais, and laid the Hair in the Water upon the Glafs, and then laying another po- lifhed plate of Glafs upon it, fo that the Water might fill up the fpace between the GlaiTes, I held theni in the aforefaid beam of Light, lb that the Light might pais through them perpendicularly, and the fhadow of the Hair was at the lame diftances as big as before. The C»$] The fhadows of fcratches made in poliflied plates of Glafs were alfo much broader than they ought to be, and the Veins in poliffied plates of Glafs did alfo call the like broad fhadows. And therefore the great breadth of thefe fhadows proceeds from feme other caufe than the refraction of the Air. Let the Circle X reprefent the middle of the Hair ; pin\ A D G, B E H, C F I, three rays paffing by one fide of the Hair at feveral diftances; KNQ, LOR, M PS, three other rays paffing by the other fide of the Hair at the like diftances ; D, E, F and N, Q, P, the places where the rays are bent in their paffage by the Hair ; G, H, I and Q, R, S, the places where the rays fall on a Paper GQ; IS the breadth of the ffiadow of the Hair caft on the Paper, and T I, V S, two rays paffing to the points I and S without bending when the Hair is taken away. And it’s manifeft that all the Light between thefe two rays A I and V S is bent in paffing by the Hair, and turned afide from the ffiadow IS, becaufe if any part of this Light were not bent it would fall on the Paper within the ffiadow, and there illuminate the Paper contrary to experience. And becaufe when the Paper is at a great diftance from the Hair, the ffiadow is broad, and therefore the rays TI and VS are at a great diftance from one another, it follows that the Hair a£ts upon the rays of Light at a good diftance in their paffing by it. But the aftion is ftrongeft on the rays which pafs by at leaft diftances, and grows weaker and weaker accordingly as the rays pafs by at diftances greater and greater, as is reprefented in the Scheme : For thence it comes to pals, that the ffiadow of the Hair is much broader in proportion to the diftance of (2 q a the [nd] the Paper from the Hair, when the Paper is nearer the Hair than when it is at a great diftance from it. O B S. II. * H The ffiadows of all Bodies ( Metals, Stones, Glafs, Wood, Horn, Ice, toV. j in this Light were bordered with three parallel fringes or bands of coloured Light, whereof that which was contiguous to the lhadow was broadeft and mod luminous, and that which was re- moteft from it was narroweft, and fo faint, as not eafily to be vifible. It was difficult to diftinguiffi the Colours unlefs when the Light fell very obliquely upon a fmooth Paper, or fome other fmooth white Body, fo as to make them appear much broader than they would otherwife do. And then the Colours were plainly vifible in this order : The firft or innermoft fringe was violet and deep blue next the ffiadow, and then light blue, green and yellow in the middle, and red without. The fecond fringe was almoft contiguous to the firft, and the third to the fecond, and both were blue within and yellow and red without, but their Colours were very faint efpecially thofe of the third. The Colours therefore proceeded in this order from the ffiadow, violet, indico, pale blue, green, yellow, red ; blue, yellow, red ; pale blue, pale yellow and red. The fhadows made by fcratches and bubbles in poliflied plates of Glafs were bordered with the like fringes of coloured Light. And if plates of Looking-glafs floop’d off near the edges with a Diamond cut, be held in the fame beam of Light, the Light which paffes through the parallel planes of the Glafs will be be bordered with the like fringes of Co¬ lours [ 1 1 7 3 lours where thofe Planes meet with the Diamond cut, and by this means there will fometimes appear four or five fringes of Colours. Let A B, C D reprefent the Fig par allel planes of a Looking-glafs, and BD the plane of the Diamond-cut, making at B a very obtufe Angle with the plane A B. And let all the Light between the rays E N I and F B M pafs diredtly through the parallel planes of theGlafs, and fall upon the Paper between I and M, and all the Light between the rays GO and HD be refracted by the oblique plane of the Diamond cut B D,and fall upon the Paper between K and L ; and the Light which paffes direftly through the parallel planes of the Glafs, and falls upon the Paper between I and M, will be bordered with three or more fringes at M. O B S. III. When the Hair was twelve Feet diftant from the1 Hole, and its fiiadow fell obliquely upon a flat white fcale of Inches and parts of an Inch placed half a Foot beyond it, and alfo when the fliadow fell perpendicu¬ larly upon the fame fcale placed nine Feet beyond it; I meafured the breadth of the lhadow and fringes as accurately as I could, and found them in parts of an Inch as follows* [n8] At the diftance of ^oot^ *Feet The breadth of the Shadow S4 £ 9 The breadth between the middles of the brighteft Light of the innermoft fringes on either fide the fhadow T— or J- 3* 3* 7. S° The breadth between the middles of the brighteft Light of the middlemoft frin¬ ges on either fide the lhadow I ~ I 237 1 lT The breadth between the middles of the brighteft Light of the outmoft fringes on either fide the fhadow s °r 78] 10 The diftance between the middles of the brighteft Light of the fir ft and fecond fringes T _ 120 I 21 The diftance between the middles of the brighteft Light of the fecond and third fringes jr^ 17a r Si The breadth of the luminous part (green, white, yellow and red ) of the firft fringe r 173 1 Si The breadth of the darker fpace between the firft and fecond fringes. I 1 4? The breadth of the luminous part of the fecond fringe I 290 I__ A The breadth of the darker fpace between the fecond and third fringes. I 340 T_ 03 | Thcfe [ up] Thefe meafures I took by letting the Ihadow of the Hair at half a Foot diftance fall lb obliquely on the fcale as to appear twelve times broader than when it fell perpendicularly on it at the fame diftance, and bet¬ ting down in this Table the twelfth part of the mea¬ fures I then took. OBS. IV. When the fhadovv and fringes were caft obliquely upon a fmooth white Body, and that Body was remo¬ ved further and further from the Hair, the firft fringe began to appear and look brighter than the reft of the Light at the diftance of lefs than, a quarter of an Inch from the Hair, and the dark line or Ihadow between that and the fecond fringe began to appear at a lefs di¬ ftance from the Hair than that of the third part of an Inch* The fecond fringe began to appear at a diftance from the Hair of lefs than half an Inch, and the Ihadow between that and the third fringe at a diftance lefs than an Inch, and the third fringe at a diftance lefs than three Inches. At greater diftances they became much more fenfible, but kept very nearly the fame proportion of their breadths and intervals which they had at their firft appearing. For the diftance between the middle of the firft and middle of the fecond fringe, was to the diftance between the middle of the fecond and middle of the third fringe, as three to two, or ten to feven. And the laft of thefe two diftances was equal to the breadth of the bright Light or luminous part of the firft fringe. And this breadth was to the breadth of the bright Light of the fecond fringe as feven to four, and to the dark interval. [II2] interval of the firft and fecond fringe as three to two, and to the like dark interval between the fecond and third as two to one. For the breadths of the fringes feemed to be in the progreffion of the numbers i, i/^ , f/ \ and their intervals to be in the fame progreffion with them ; that is, the fringes and their intervals to¬ gether to be in the continual progreffion of the numbers i, / { ? f/~ , f/1-^ f/] > or thereabouts. And thefe pro¬ portions held the fame very nearly at all diftances from the Hair 5 the dark Intervals of the fringes being as broad in proportion to the fringes at their firft appea¬ rance as afterwards at great diftances from the Hair, though not ib dark and diftinft. OBS, V. The Sun fhining into my darkened Chamber through a Hole a quarter of an Inch broad ; I placed at the di- ftance of two or three Feet from the Hole a Sheet of Paft-board, which was black’d all over on both fides, and in the middle of it had a Hole about three quarters of an Inch fquare for the Light to pafs through. And behind the Hole I fattened to the Paft-board with Pitch the blade of a fliarp Knife, to intercept fome part of the Light which patted through the Hole. The planes of the PafTboard and blade of the Knife were parallel to one another, and perpendicular to the rays. And when they were fo placed that none of the Sun’s Light fell on the Paft-board, but all of it patted through the Hole to the Knife, and there part of it fell upon the blade of the Knife, and part of it palled by its edge : I let this part of the Light which patted by, fall on a white [121] white Paper two or three Feet beyond the Knife, and there faw two ftreams of faint Light fhoot out both ways from the beam of Light into the ftiadow like the tails of Comets. But becaufe the Sun’s direct Light by its brightnefs upon the Paper obfcured thefe faint ftreams, fo that I could fcarce fee them, 1 made a little Hole in the midft of the Paper for that Light to pafs through and fall on a black cloth behind it ; and then I law the two ftreams plainly. They were like one another, and pretty nearly equal in length and breadth and quantity of Light. Their Light at that end nex the Sun’s direft Light was pretty ftrong for the fpace o about a quarter of an Inch, or half an Inch, and in al its progrefs from that diredt Light decreafed gradually till it became infenfible. The whole length of either o thefe ftreams meafured upon the Paper at the diftano of three Feet from the Knife was about fix or eigh Inches ; fo that it fubtended an Angle at the edge of the Knife of about ioorn, dr at moft 14 degrees. Yet fometimes I thought I faw it (hoot three or four degrees further, but with a Light fo very faint that I could fcarce perceive it, and fulpefted it might ( in fome meafure at leaft) arile from fome other caufe than the two ftreams did. For placing my Eye in that Light beyond the end of that ftream which was behind the Knife, and looking towards the Knife, I could fee a line of Light upon its edge, and that not only when my Eye was in the line of the ftreams, but alfo when it was without that line either towards the point of the Knife, or towards the handle. This line of Light ap¬ peared contiguous to the edge of the Knife, and was narrower than the Light of the innermoft fringe, and R r narrowed [ 122 ] narroweft when my Eye was furtheft from the direct Light, and therefore feemed to pafs between the Light of that fringe and the edge of the Knife , and that which paffed neareft the edge to be mo ft bent, though not all of it. O B S. VI I placed another Knife by this fo that their edges might be parallel and look towards one another, and that the beam of Light might fall upon both the Knives, and fome part of it pafs between their edges. And when the diftance of their edges was about the 400th part of an Inch the ftream parted in the middle, and left a lhadow between the two parts. This fhadow was fo black and dark that all the Light which paffed between the Knives feemed to be bent, and turned afide to the one hand or to the other. And as the Knives ftill approached one another the fhadow grew broader, and the ftreams fhorter at their inward ends which were next the lhadow, until upon the con tad: of the Knives the whole Light vanilhed leaving its place to the lhadow. And hence I gather that the Light which is leaffc bent, and goes to the inward ends of the ftreams, paf- fes by the edges of the Knives at the greateft diftance, and this diftance when the fhadow begins to appear be¬ tween the ftreams is about the eight- hundredth part of an Inch. And the Light which paffes by the edges of the Knives at diftances ftill lefs and lets is more and more bent, and goes to thofe parts of the ftreams which are further and further from the dired Light, becaufe when r 1 2 3 i when the Knives approach one another till they touchy thole parts of the ftreams vanilh laft which are furtheft from the direct Light* O B S. VIL In the fifth Obfervation the fringes did not appear, but by reafon of the breadth of the Hole in the Win¬ dow became fo broad as to run into one another, and by joyning make one continued Light in the beginning of the ftreams. But in the fixth, as the Knives ap¬ proached one another, a little before the Shadow ap¬ peared between the two ftreams, the fringes began to appear on the inner ends of the ftreams on either fide of the direct Light, three on one fide made by the edge of one Knife, and three on the other fide made by the edge of the other Knife. They were diftinfteft when the Knives were placed at the greateft diftance from the Hole in the Window, and ftill became more diftinft by making the Hole lefs, infomuch that I could fometimes fee a faint lineament of a fourth fringe beyond the three above-mentioned. And as the Knives continually ap¬ proached one another, the fringes grew diftinTer and larger until they vanilhed. The outmoft fringe va¬ nished firft, and the middlemoft next, and the inner- moft laft. And after they were all vanilhed, and the line of Light which was in the middle between them was grown very broad, enlarging it fielf on both fides into the ftreams of Light defcribed in the fifth Obfer¬ vation, the above-mentioned fhadow began to appear in the middle of this line, and divide it along the middle into two lines of Light, and increafed until the whole Rr 2 Light . . . . t*H] Light vanifhed. This inlargement of the fringes was fo great that the rays which go to the innermoft fringe feemed to be bent above twenty times more when this fringe was ready to vanilh, than when one of the Knives was taken away. And from this and the former Obfervation compared, I gather, that the Light of the firft fringe paflfed by the edge of the Knife at a diftance greater than the eight- hundredth part of an Inch, and the Light of the fecond fringe palled by the edge of the Knife at a greater di¬ ftance than the Light of the firft fringe did, and that of the third at a greater diftance than that of the fe¬ cond, and that of the ftreams of Light defcribed in the fifth and fixth Obfervations palled by the edges of the Knives at lefs diftances than that of any of the fringes.. o b a vin. I caufed the edges of two Knives to be ground truly ftreight, and pricking their points into a board fo that their edges might look towards one another, and meet¬ ing near their points contain a rectilinear Angle, I fafl> ned their handles together with Pitch to make this Angle invariable. The diftance of the edges of the ATnives from one another at the diftance of four Inches from the angular point, where the edges of the Knives met, was the eighth part of an Inch, and therefore the Angle contained by the edges was about i degr. 54/. The Knives thus fixed together I placed in a beam of the Sun's Light, let into my darkened Chamber through a Hole the 42th part of an Inch wide, at the diftance of [ 12$ ] of ten or fifteen Feet from the Hole, and let the Light which pafled between their edges fall very obliquely upon a lmooth white Ruler at the diftance of half an Inch, or an Inch from the Knives, and there faw the fringes made by the two edges of the Knives run along the edges of the lhadows of the Knives in lines parallel to thole edges without growing fenfibly broader, till they met in Angles equal to the Angle contained by the edges of the Knives, and where they met and joyned they ended without eroding one another. But if the Ruler was held at a much greater diftance from the Paper, the fringes became fomething broader and. broader as they approached one another, and after they met they crofted one another, and then became much broader than before. Whence I gather that the diftances at which the fringes pafs by the Knives are not increafed nor altered by the approach of the Knives, but the Angles in which the rays are there bent are much increafed by that ap¬ proach ; and that the Knife which is neareft any ray determines which way the ray lhall be bent, and the other Knife increafes the bent. O B S.. IX,. When the rays fell very obliquely upon the Ruler at the diftance of the third part of an Inch from the Knives, the dark line between the firft and fecond fringe of the lhadow of one Knife, and the dark line between the firft and fecond fringe of the lhadow of the other Knife met with one another, at the diftance of the fifth part of an Inch from the end of the Light which pafled be¬ tween. tween the Knives at the concourfe of their edges. And therefore the diftance of the edges of the Knives at the meeting of thefe dark lines was the i6othpart of an Inch, For as four Inches to the eighth part of an Inch, fo is any length of the edges of the Knives rneafured from the point of their concourfe to the diftance of the edges of the Knives at the end of that length, and fo is the fifth part of an Inch to the 1 6oth part. So then the dark lines above-mentioned meet in the middle of the Light which pafles between the Knives where they are diftant the 1 6oth part of an Inch, and the one half of that Light pafles by the edge of one Knife at a diftance not greater than the 320th part of an Inch, and falling upon the Paper makes the fringes of the lhadow of that Knife, and the other half pafles by the edge of the other Knife, at a diftance not greater than the 320th part of an Inch, and falling upon the Paper makes the fringes of the lhadow of the other Knife. But if the Paper be held at a diftance from the Knives greater than the third part of an Inch, the dark lines above-men¬ tioned meet at a greater diftance than the fifth part of an Inch from the end of the Light which pafled be¬ tween the Knives at the concourfe of their edges ; and therefore the Light which falls upon the Paper where thofe dark lines meet pafles between the Knives where their edges are diftant above the 1 60th part of an Inch, For at another time when the two Knives were di¬ ftant eight Feet and five Inches from the little Hole in the Window, made with a fmall Pin as above, the Light which fell upon the Paper where the aforefaid dark lines met. pafled between the Knives, where the di~ ftance [127] fiance between their edges was as in the following Table, when the diftance of the Paper from the Knives was alio as follows* Difiances of the Paper from the Kjvives in Inches . Difiances between the edges of the }(jiives in mtlle- fimal parts of an Inch . ij. o’oi 2. 3? ooao. 8t- q’o’34. 32. O'C^y, 96, o’o8i. 0^087. And hence I gather that the Light which makes the fringes upon the Paper is not the fame Light at all di- fiances of the Paper from the Knives, but when the Pa¬ per is held near the Knives, the fringes are made by Light which paffes by the edges of the Knives at a lefs diftance, and is more bent than when the Paper is held at a greater diftance from the Knives, O B S* X. When the fringes of the lhadows. of the Knives fell perpendicularly upon a Paper at a great diftance from the Knives,- they were in the form of Hyperbolas, and their dimenfions were as follows* LetCA, CB repre- fent lines drawn upon the Paper parallel to the edges of the Knives, and between which all the Light would fall, if it paffed between the edges of the Knives with¬ out inflexion; DE a right line drawn through C making [ 128 ] the Angles A CD, BCE, equal to one another, and terminating all the Light whith falls upon the Paper from the point where the edges of the Knives meet ; eis, fkt, and glv, three hyperbolical lines reprefenting the ter¬ minus of the fhadow of one of the Knives, the dark line between the firft and fecond fringes of that lhadow, and the dark line between the fecond and third fringes of the fame fhadow ; x i p, y k q and z 1 r, three other Hy¬ perbolical lines reprefenting the terminus of the fhadow of the other Knife, the dark line between the firft and fecond fringes of that fhadow, and the dark line be¬ tween the fecond and third fringes of the fame fhadow. And conceive that thefe three Hyperbolas are like and equal to the former three, and crofs them in the points i, k and 1, and that the fhadows of the knives are termi¬ nated and diftinguifhed from the firft luminous fringes by the lines eis and xip, until the meeting and crof¬ ting of the fringes, and then thofe lines crofs the fringes in the form of dark lines, terminating the firft luminous fringes within fide, and diftinguifhing them from ano¬ ther Light which begins to appear at i, and illuminates all the triangular fpace ipDEs comprehended by thefe dark lines, and the right line DE. Of thefe Hy¬ perbolas one Afymptote is the line D E, and their other Afymptotes are parallel to the lines C A and C B. Let r v reprefent a line drawn any where upon the Paper parallel to the Afymptote D E, and let this line crofs the right lines A C in m and B C in n, and the fix dark hyperbolical lines in p, q, r ; s, t, v ; and by meafuring the diftances ps, qt, rv, and thence colleiting the the lengths of the ordinates n p, n q, n r or m s, m t, mv, and doing this at feveral diftances of the line rv, [ I2p] from the Afymptote DE you may find as many points of thefe Hyperbolas as you pleafe, and thereby know that thefe curve lines are Hyperbolas differing little from the conical Hyperbola. And by meafuring the lines C i , C k , Cl, you may find other points of thefe Curves. For inftance, when the Knives were diftant from the Hole in the Window ten Feet, and the Paper from the Knives 9 Feet, and the Angle contained by the edges of the Knives to which the Angle ACB is equal, was fut> tended by a chord which was to the Radius as 1 to 52, and the diftance of the line rv from the Afymptote DE was half an Inch : I meafured the lines ps, qt, rv, and found them 0^5, 0*65, 0*98 Inches refpedivefy, and by adding to their halfs the line [ mn (which here was the 1 28th part of an Inch, or o’ooyS Inches ) the fums np, nq, nr, were o?i8i8, 0^318, 0^978 In¬ ches. I meafured alfo the difiances of the brightefi parts of the fringes which run between pqarid st, qr and tv, and next beyond r and v, and found them o’j, o’8, and 1 ’ 1 7 Inches, OBS. XL JL The Sun fhihing into my darkened Room through a finall round Hole made in a plate of Lead with a llender Pin as above ; I placed at the Hole a Pilfer to refrad the Light, and form on the oppofite Wall theSpedrum of Colours, defcribed in the third Experiment of the firftBook. And then I found that the ihadows of all Bodies held in the coloured Light between the Pf life and the Wall, were bordered with fringes of the Cdlotif S s ‘ J of [ 13° 3 of that Light in which they were held. In the full red Light they were totally red without any fenfible blue or violet, and in the deep blue Light they were totally blue without any fenfible red or yellow ; and fo in the green Light they were totally green, excepting a little yellow and blue, which were mixed in the green Light of the Prifm. And comparing the fringes made in the feveral coloured Lights, I found that thofe made in the red Light were largeft, thofe made in the violet were leaf!:, and thofe made in the green were of a middle bignefs. For the fringes with which the fhadow of a * Man’s Hair were bordered, being meafured crofs the fhadow at the diftance of fix Inches from the Hair ; the difiance between the middle and moft luminous part of the firft or innermoft fringe on one fide of the fhadow, and that of the like fringe on the other fide of the lha- dow, was in the full red Light ^ of an Inch, and in the full violet tg. And the like diftance between the middle and moft luminous parts of the fecond fringes on either fide the fhadow was in the full red Light h , and in the violet • of an Inch. And thefe diftances of the fringes held the fame proportion at all diftances from the Hair without any fenfible variation. So then the rays which made thefe fringes in the red Light pafled by the Hair at a greater diftance than thofe did which made the like fringes in the violet; and there¬ fore the Hair in caufing thefe fringes adted alike upon the red Light or leafi refrangible rays at a greater di¬ ftance, and upon the violet or moft refrangible rays at a lefs diftance, and by thofe actions difpofed the red Light into larger fringes, and the violet into fmaller, and the Lights of intermediate Colours into fringes of inter- Cl 31 3 intermediate bigneffes without changing the Colour of of any fort of Light, When therefore the Hair in the fir ft and fecond of thefie Oblervations was held in the white beam of the Sun’s Light, and caft a fhadow which was bordered with three fringes of coloured Light, thofe Colours arofe not from any new modifications impreft upon the rays of Light by the Hair, but only from the various inflections whereby the feveral forts of rays were feparated from one another, which before feparation by the mixture of all their Colours, compofed the white beam of the Sun’s Light, but whenever fepaaated comp ole Lights of the feveral Colours which they are originally dilpo- fed to exhibit. In this 1 3 th Obfervation, where the Colours are feparated before the Light pafles by the Hair, the leaft refrangible rays, which when fepara¬ ted from the reft make red, were infleCted at a greater diftance from the Hair, fo as to make three red fringes at a greater diftance from the middle of the fhadow of the Hair 3 and the moft refrangible rays which when feparated make violet, were infleCted at a lefs diftance from the Hair, fo as to make three violet fringes at a lefs diftance from the middle of the fhadow of the Hair. And other rays of intermediate degrees of refrangible lity were infleCted at intermediate diftances from the Hair, fo as to make fringes of intermediate Colours at intermediate diftances from the middle of the fhadow of the Hair. And in the fecond Obfervation, where all the Colours are mixed in the white Light which pafles by the Hair, thefe Colours are feparated by the various inflexions of the rays, and the fringes which they make appear all together , and the innennolf S § a fringes . . - , t r32J fringes being contiguous make one broad fringe compo- fed of all the Colours in due order, the violet lying on the infide of the fringe next the fhadow, the red on the outfide furtheft from the fhadow, and the blue, green and yellow, in the middle. And, in like man¬ ner, the middlemoft fringes of all the Colours lying in order, and being contiguous, make another broad fringe compofed of all the Colours ; and the outmoft fringes of all the Colours lying in ordejr, and being contiguous, make a third broad fringe compofed of all the Colours. Thefe are the three fringes of coloured Light with which the lhadows of all Bodies are bordered in the fe- cond Obfervation. When I made the foregoing Obfervations, I defigned to repeat moft of them with more care and exaftnefs, and to make fome new ones for determining the man¬ ner how the rays of Light are bent in their paffage by Bodies for making the fringes of Colours with the dark lines between them. But I was then interrupt ted, and cannot how think of taking thefe things into further confederation. And iince I have not finifhed this part of my Defign, I fhall conclude, with propo- fing only lbme Queries in order to a further learclr to be made by others. .... A • ‘ "r * • f J^uery i . Do not Bodies ad upon Light at a diftance, and by their action bend its rays, and is not this addon (cateris fariAm ) ftrongeft at the leaft diftance ? G>u. a. Do not the rays which differ in refrangibility differ alfo in flexibility, and are they not by their dif¬ ferent inflexions feparated from one another, fo as after reparation to make the Colours in the three fringes above C 1 33 1 above defcribed ? And after what manner are they in- fleCted to make thofe fringes ? 3. Are not the rays of Light in paffing by the edges and fides of Bodies, bent feveral times backwards and forwards, with a motion like that of an Eel ? And do not the three fringes of coloured Light above-men¬ tioned, arife from three fuch bendings ? Qu. 4. Do not the rays of Light which fall upon Bo¬ dies, and are reflected or refracted, begin to bend be¬ fore they arrive at the Bodies ; and are they not re¬ flected, refracted and infleCted by one and the fame Principle, adding varioufly in various circmnftances ? 5* Do not Bodies and Light add mutually upon one another, that is to lay, Bodies upon Light in emit¬ ting, reflecting, refraCting and inflefting it, and Light upon Bodies for heating them, and putting their parts into a vibrating motion wherein heat con hits ? <%. 6. Do not black Bodies conceive heat more eafily from Light than thofe of other Colours do, by reafon that the Light falling on them is not reflected outwards, but enters the Bodies, and is often reflected and re- fradted within them, until it be {tilled and loft ? 7. Is not the ftrength and vigor of the aCtion between Light and fulphureous Bodies obferved above, one reafon why fulphureous Bodies take fire more readily, and burn more vehemently, then’ other Bor dies do ? 8, Do not all fixt Bodies when heated beyond a certain degree, emit Light and fhine, and is not this emiflion performed by the vibrating motions of their parts ? C 134] «%. 9. Is not fire a Body heated fo hot as to emit Light copioufly ? For what elfe is a red hot Iron than fire ? And what elfe is a burning Coal than red hot Wood ? 10. Is not flame a vapour, fume or exhalation heated red hot, that is, fo hot as tolhine? For Bodies do not flame without emitting a copious fume, and this fume burns in the flame. The Ignis Fatuus is a vapour j fhining without heat, and is there not the fame diffe¬ rence between this vapour and flame, as between rot¬ ten Wood fhining without heat and burning Coals of fire ? In diftilling hot Spirits, if the head of the ftill be taken off, the vapour which afcends out of the Still will take fire at the flame of a Candle, and turn into flame, and the flame will run along the vapour from the Candle to the Still. Some Bodies heated by motion or fermen¬ tation, if the heat grow intenfe fume copioufly, and if the heat be great enough the fumes will fhine and be¬ come flame. Metals in fufion do not flame for want of a copious fume, except Spelter which fumes copioufly, and thereby flames. All flaming Bodies, as Oyl, Tal- low, Wax, Wood, foffil Coals, Pitch, Sulphur, by ^flaming wafte and vanifh into burning fmoke, which fmoke, if the frame be put out, is very thick and vifible, and fometimes fmells ftrongly, but in the flame loles its fmell by burning, and according to the nature of the fmoke the flame is of feveral Colours, as that of Sul¬ phur blue, that of Copper opened with Sublimate green, that of Tallow yellow* Smoke paffing through flame cannot but grow red hot, and red hot Imoke can Vhave no other appearance than that of flame. 1 1 . - [!?;] <$u. n . Do not great Bodies conferve their heat the longeft, their parts heating one another, and may not great denfe and fix’d Bodies, when heated beyond a certain degree, emit Light fo copiouily, as by the emif- fion and reaction of its Light, and the reflexions and re¬ fractions of its .rays within its pores to grow ftill hot¬ ter, till it comes to a certain period of heat, luch as is that of the Sun ? And are not the Sun and fix’d Stars great Earths vehemently hot, whofe heat is conferved by the greatnefs of the Bodies, and the mutual aCtion and reaction between them, and the Light which they emit, and whofe parts are kept from fuming away, not only by their fixity, but alfo by the vaft weight and denfity of the Atmofpheres incumbent upon them, and very ftrongly comprefling them, and condenfing the va¬ pours and exhalations which arife from them ? £$u. i a. Do not the rays of Light in falling upon the bottom of the Eye excite vibrations in the "Tunica re - tina ? Which vibrations, being propagated along the folid fibres of the optick Nerves into the Brain, caufe the fenfe of feeing. For becaufe denfe Bodies conferve' their heat a long time, and the denfeft Bodies conferve their heat the longeft, the vibrations of their parts are of a lafting nature, and therefore may be propagated along folid fibres of uniform denfe matter to a great di- ftance, for conveying into the Brain the impreflions made upon all the Organs of fenfe. For that motion , which can continue long in one and the fame part of a Body, can be propagated a long way from one part to another, fuppofing the Body homogeneal, fo that the motion may not be reflected, refraCted, interrupted or; difordered by any unevennefs of the Body. §>u. 13. [i36] «%£. 1 3 . Do not feveral fort of rays make vibrations of feveral bigneffes, which according to their bigneffes excite fenfations of feveral Colours, much after the manner that the vibrations of the Air, according to their feveral bigneffes excite fenfations of feveral founds ? And particularly do not the mo ft refrangible rays ex¬ cite the fhorteft vibrations for making* a fenfation of deep violet, the leaft refrangible the largeft for making a fenfation of deep red, and the feveral intermediate forts of rays, vibrations of feveral intermediate bignef¬ fes to make fenfations of the feveral intermediate Co¬ lours ? 14. May not the harmony and difcord of Co¬ lours arife from the proportions of the vibrations propa¬ gated through the fibres of the optick Nerves into the Brain, as the harmony and difcord of founds arifes from the proportions of the vibrations of the Air ? For feme Colours are agreeable, as thofe of Gold and Indico, and others difagree* J : ; . .. • Y •: V - : ' • , ■ : - - . : — . • i ' :Y;n :Y : \o' i YTM i;JY-; ( i . , , f ' ; r 1 : ■it ■■ ■ . - . I ! . . • - - n~ rnam leptimam ubi tranfit per concurfum ilium, quo Fis- 3P- cai’u termini b ac d delunt, & concurfus afyrnpto- 32. ton ell centrum figura ab omnibus ejus partibus 33* oppofitis F/>. 34, Fig. 35. '%* 37* %* .38. C1^] , oppofitis aequaliter diftans. Et hx quatuor fpecles Diametnim non habent. Fertuntur etiam fpecies decima quarta ac decima fexta in vigefimam oftavam, decima quinta ac de¬ cima feptima in vigefimam nonam, decima octava & decima nona in tricefimam, & vigefima cum vige- lima prima in tricefimam primam. Et hx fpecies unicam habent diametrum. Ac deniq; fpecies vigefima fecunda & vigefima tertia vertuntur in fpeciem tricefimam fecundam cu- jus tres funt Diametri per concurfum afymptotdn tranfeuntes. Quae omnes! converfiones facillime in- telliguntur faciendo ut triangulum ab afymptotis comprehenfum diminuatur donee in pun&um eva- nefcat. Si in primo aequationum cafu terminus ax3 ne- gativus eft, Figura erit Hyberbola defediva unicam trum nm habrn- liabens afymptoton & duo tantum crura Hyperbo- lica juxta afymptoton illam in plagas contrarias in¬ finite progredientia. Et afymptotos ilia eft Ordi- nata prima & principalis AG. Si terminus ey non deeft figura nullam habebit Diametrum, fi deeft ha- bebit unicam. In priori cafu fpecies fic enume- rantur. Si sequationis hujus ax4 = bx5-j-cxx -[■ d x - e e, radices omnes At, AP, A A .w, funt reales & in- oequales, Figura erit Hyperbola anguinea afympto¬ ton flexu contrario amplexa, cum Ovali conjugata. Quse fpecies eft tricefima tertia. Si radices duae mediae AP & Af aequentur inter fe, Ovalis & Anguinea j ungun tur feie decuflantes in forma Nodi, Qua’ eft fpecies tricefima quarta. Si xix. tes„ ■ £' 3 9 * •:a. 40., [ ] Si tres radices funt sequaks, Nodus vertetur cuff idem acutiffimum in vertice anguinese. Et hxc eft fpecies tricesima quinta. Si e tribus radicibus ejufdem figni dux maxknae fi£' 43- A / & A -or fibi rnutuo sequantur, Ovalis in fundtum evanuit, Q.use fpecies eft tricefima fexta. Si radices dux quaevis imaginary funt7 fola ma- nebit Anguinea fur a, fine Ovalq decuflatione5 oil- pide vel pundo conjugate. Si Anguinea ilia non 42*., tranfit per pundum A fpecies eft tricefima feptima> fin tranfit per pundum illud A ( id quod contin^it Fl£* 43* ubi termini b ac d defunt,) pundum illud A erit centrum figure redas omnes per ipfum dudas & ad Curvam utrinq; terminatas bifecans. Et haec eft fpecies tricefima odava. In altero cafu ubi terminus ey deeft 8c propterea xx. figura Diametrum habet, fi aequationis hums ax3 Hyppp^fip- i j i J J * ■ a 't* \ a ' r> iZYft dsfeEiiV&.di- = bXX-rCX+.d radices OmneS AT, At, At, funt amnrum habex, reales, inasquales & ejufdem figni, figura erit Hyper- p- bola Conchoidalis cum Ovali ad convexitatem, ""Qua* 4’* ' eft fpecies tricefima nona. Si duse radices funt insequales & ejufdem figni & Bg, 44, tertia eft figni contrarii, Ovalis jacebit ad concavi- tatem Conchoidalis. Eftq; fpecies quadragefima. Si radices duse min ores AT, At, funt requales i%. 47' gefima tertia. Si aquationis illius radices duae funt impoflibiles, Fi£- 58. figure hyperbolo'parabolicae junguntur fefe de- cuffantes in morem crucis, & fpeciem quinquagefi¬ mam quartam conftituunt. Si radices illae funt inaequales & ejufdem figni, ha- Fi£- is- betur Hyperbola Conchoidalis cum Parabola ex eodem latere Afymptoti. Eftq; fpecies quinquage¬ fima quinta. Xx Si C 1 54 D *%•*<> : Si radices ilk funt figni contrariir habetur Con- choidalis cum Parabola ad alteras partes AfymptotL Ctuse fpecies eft quinquagefima fexta. xxiii. f^Siquando in primo ^quationum cafu terminus utei;qj ?x? & b.xx dee.ftv figura erit Hyperbolifmus b&u, ’ ‘ fedionis alicujus Conicce, Hyperbolifmum figure voco cujus Ordinata proditapplicandoconteptumfub Ordinata figure illius & red# data ad AbfciiTam com- munem. Hac ratione linea reda vertitur in hyper- bolam Gonicam, & fedio omnis Gonica vertitur in aliquam figurarum quas hie Hyperbolifmos fedio- num Conicarum voco. Nam sequatio ad figuras de quibus agimus , nempe x y y -j- e y = c x -j- d, feu e+z^eei- ^dx -ft 4 exx generatur appli- y 2 x cando contentum fub Ordinata fedionis Conicae e+//ee-b4dx-|-4cxx & reda data m ad curvarum 2 m ' “ AbfciiTam communem x. Unde liquet quod figura genita Hyperbolifmus erit Hyperbola?, Ellipfeos vel Parabolae perinde ut terminus cx affirmativus eft vel negativus vel nullus. Hyperbolifmus Hyperbola? tres habet afymptotos quarum una eft Ordinata prima & principalis A d alterae duse funt parallels Ablciffae A B & ab eadem bine inde sequaliter diftant. In Ordinata principali A d cape Ad, hinc inde aequales quantitati //c & per punUa d ac ^ age dg, Afymptotos Ab- feifife A B parallelas. Ubi terminus ey non deeft figura nullam ha¬ bet diametrum. In hoc cafu fi aequationis hujus cxx-ydx-j-Jee^o radices duae A P, Ap funt reales 8c [ !$$ ] Sc ina'quales (nam tequales efle nequeunt nifi figura fit Conica feftio ) figura conftabit ex tribus Hyper- bolis fibi oppofitis quarum una jacet inter afymp- totos parallelas & altera duae jacent extra. Et haec eft fpecies quinquagefima feptima. Si radices illae duae i’unt impoffibiles,habentur Hy¬ perbolae duae oppofitae extra afymptotos parallelas & Anguinea hyperbolica intra eafdem. Haec figura duarum eft lpecierum. Nam centrum non habet ubi terminus d non deeft ; fed fi terminus ille deeft pun£tum A eft ejus centrum. Prior fpecies eft quin¬ quagefima oftava, pofterior quinquagefima nona. TQuod fi terminus e y deeft, figura conftabit ex Kg. 64 tribus hyperbolis oppofitis quarum una jacet inter afymptotos parallelas & altera duae jacent extra ut in fpecie quinquagefima quarta, & praterea diame- trum habet quae eft abfciffa A B. Et haec eft fpecies fexagefima. Hyperbolifmus Ellipfeos per hanc aequationem de- XXIV. finitur x y y ey = cx -j- d, & unicam habet afymp- Tr‘t toton qua eft Ordinata principalis Ad. Si terminu s k£ 6^' ey non deeft, figura eft Hyperbola anguinea fine dia~ metro atq; etiam fine centre fi terminus d non deeft. Quae fpeeies eft fexagefima prima. ^ ^ At ft terminus d deeft, figura habet centrum fine%* 66* diametro & centrum ejus eft pun&um A, Species vero eft fexagefima fecunda, Et fi terminus ey deeft & terminus d non deeft, Fig* $7* figura eft Conchoidalis ad afymptoton A G, habetq; diametrum fine centro, & diameter ejus eft Abfciffa A EL Quae fpecies eft fexagefima tertia, X x 2 Hyper- xxv. Duo Hyperbo- lifmi Parabola. Fig. 68. Fig . 6p, <6- XXVI. Tridens . B>e ?6. XXVII* Parabola quin que divergentes . Fig. 70,71, 73* [I57$< data in vertice. Et haec eft Parabola Neiliana quo? vulgo femicubica dicitur. Si radices duae funt impoffibiles, habetur Parabola Fig. 73, 74, fura' campaniformis fpeciem feptuagefimain primam conftituens. I11 quarto cafu aequato erat y - ax -pbxx-f-cx xxvni. & haec aequatio Parabolam illam WaUiftanam Parabola cuhca. defignat quae crura habet contraria & cubic a du ci folet. Et ilc fpecies omnino funt feptuaginta duae. Si in planum infinitum a pundto lucido illumina- ^ xxix* turn umbrae ngurarum projiciantur, umbrae lectio- rum ^ vmfouv. num Conicarum femper erunt fedtiones Conicae, eae Curvarum fecundi generis femper erunt Curvae fe¬ cund! generis, eae curvarum tertii generis femper erunt Curvae tertii generis, & fic deinceps in infini¬ tum* Et quemadmodum Circulus umbram proji- ciendo generat fedtiones omnes conicas, tic Parabolae 4 quinq; divergentes umbris fuis generant & exhi- bent alias omnes fecundi generis curvas , & fic Curvae qusedam fimpliciores aliorum generum inve- niri poffunt quae alias omnes eorundem generum curvas umbris fuis a pundto lucido in planum pro** jedtis formabunt. Diximus Curvas fecundi generis a linea redta in xxx; pundtis tribus fecari poffe, Horum duo nonnun- quam coincidunt lit cum redta per Ovalem infi¬ nite parvam tranfit vel per concurfum duarum par- tium Curvae fe mutuo fecantium vel in cufpidem coeuntium ducitur* Et fiquando redtae omnes in plagana XXXI. ' Theoremata de Cur varum de- fcriytione orga~ nica. [158] plagam cruris alicujus infiniti tendentes Curvam in unico tantum pundto fecant ( ut fit in ordinatis Parabolae Cartefianae & Parabolae cubicse, nec non in redtis Abfciffae Hyperbolifmorum Hyberbolae & Para¬ bolae ySarallelis ) concipiendum eft quod redtae illae per alia duo Curvae pundta ad infinitam diftan- tiam lita ( ut ita dicam ) tranfeunt. Hujufmodi interfedtiones duas coincidentes five ad finitam fint diftantiam five ad infinitatn, vocabimus pun- dtum duplex. Curvae autem quae habent pun* dtum duplex defcribi poflimt per fequentia Theo« remata. 1 . Si anguliduo magnitudine dati PAD, PBD circa polos pofitione datos A, B rotentur, & eorum crura A P, B P concurfu fuo P percurrant lineam redtam ; crura duo reliqua A D, B D concurfu fuo D defcri- bent fedtionem Conicam per polos A, B tranfeun- tem : proeterquam ubi lineailla redta tranfit per po- lorum alterutrum A vel B, vel anguli BAD, A B D fimul evanefcunt, quibus in cafibus pundtum D de* fcribet lineam redtam. Si crura prima A P, B P concurfu fuo P percurrant fedtionem Conicam per polum alter- utrum A tranfeuntem, crura duo reliqua AD, BD concurfu fuo D defcribent Curvam fecundi gene¬ ris per polum alterum B tranfeuntem & pun¬ ctual duplex habentem in polo primo A per quern fedtio Conica tranfit : praeterquam ubi anguli BAD, ABD fimul evanefcunt , quo cafu pun¬ dtum [ r$9 1 £tum D defcribet aliam fefitionem Conicam per po- lum A tranfeuntem. g , At fi feftio Conica quam punfitum P percur- rit tranfeat per neutrum polorum A, B, punftum D defcribet curvam fecundi vel tertii generis pun- Sum duplex habentem. Et punStum illud duplex in concurfu crurum defcribentium, A D, BD in- venietur ubi anguli B AP, A BP fimul evanefcunt, Curva autem defcripta fecundi erit generis fi an¬ guli BAD, ABD limul evanefcunt, alias erit ter¬ tii generis & alia duo habebit punCta duplicia in polis A & B* Jam feftio Conica determinate' ex datis ejus xxxn. pundis quinq; & per eadem fie defcribi poteft. Dentur ejus punda quinq; A, B, C, D, E. Jun- tk pr dat*. quin- gantur eorum tria qusevis A, B, C & trianguli ABC rue(unna- rotentur anguli duo quivis C A B, CBA circa ver- V tices fuos A & B, & ubi crurum A C, B C interfedio C fucceffive applicatur ad punda duo reliqua D, E, incidat interfedio crurum reliquorum A B & BA in punda P & Q, Agatur & infinite producatur reda P Q, & anguli mobiles ita rotentur ut inter- fedio crurum AB, BA percurrat redam PQ, & crurum reliquorum interfedio C defcribet propofi- tam fedionem Conicam per Theorema primum. xxxin. Curvae omnes fecundi generis _ pundum duplex cuSuJZ™ habentes determinantur ex datis earum punctis &um duplex ha- feptem, quorum unum eft pundtum illud duplex, & tern pimcta. [ itfo ] & per eadem pundta fic defcribl poffunti Dentur Curvae defcribendse pundta quaelibet feptem A, B, C, D, E, F, G quorum A eft pundtum duplex, Jun- gantur pundtum A & alia duo quaevis e pundtis puta B & C ; & trianguli ABC rotetur turn angulus CAB circa, verticem fuum A, turn angulorum reli- quorum alteruter ABC circa verticem fuum B. Et ubi crurum AC? BC concurfus C lucceffive appli- catur ad pundta quatuor reliqua D, E, F, G incidat concurfus crurum reliquorum A B & B A in pundta quatuor P, Q, R, S. Per pundta ilia quatuor & quintum A defcribatur fedtio Conica, & anguli pro¬ fit! CAB, CB A ita rotentur ut crurum AB, B A concurfus percurrat fedtionem illam Conicam , & concurfus reliquorum crurum AC} B C defcribet Curvam propofitam per Theorema fecundum. Si vice pundti C datur pofitione redta B C quae Curvam defcribendam tangit in B, lineae AD, A P coincident, & vice anguli D AP habebitur linea redta circa polum A rotanda. Si pundtum duplex A infinite diftat debebit Redta ad plagam pundti illius perpetuo dirigi & motu pa- rallelo ferri interea dum angulus ABC circa polum B rotatur. Defcribi etiam poffunt hx curvae paulo aliter per Theorema tertium, fed defcriptionem fimpliciorem pofuilTe fufficit. Eadem methodo Curvas tertii, quarti & fuperio- rum generum defcribere licet, non omnes quidem fed quotquot ratione aliqua commoda per moturn localem defcribi poffunt* Nam curvam aliquam fecund i E ] ■fecundi vel fuperioris generis pundum duplex non habentem commode defcribere Problema eft inter difficiliora numerandum. Curvarum ufus in Geometria eft ut per earum xxxiv. interfediones Problemata folvantur. Proponatur sequatio conftruenda dimenfionum novem x^ft-bx7 fcnpiomm Cm- -j-cx6-i-dx5-|-ex44- f x3-j-gxx-|-hx-j-k = o, Ubi wum- -ft m b, c, d, )Sc. fignificant quantitates quafvis datas fignis fuis ft - & -< affefitas. Affumatur sequatio ad Parabolam cubicam x! = y, & sequatio prior, fcri- bendo y pro x?, evadet y5ft-bxy y ft- cyyft-dxxy -ft e x y ft- m y ft- f x3 ft-g x x ft- h x -ft k = o, xquatio ad Curvam aliam fecundi generis. Ubi m vel f deefife poteft vel pro lubitu affumi. Et per harum Curva¬ rum defcriptiones & interfediones dabuntur radices aequationis conftruendse. Parabolam cubicam feme! defcribere fufficit. Si xquatio conftruenda per defectum duorum ter- A minorum ultimorum h x & k reducatur ad feptem dimenfiones, Curva altera delendo m, habebit pun- fit um duplex in principio abfciffae, & inde facile de- lcribi poteft ut fupra. Si sequatio conftruenda per defefitum termino- rum trium ultimorum gxxft-hxft-k reducatur ad fex dimenfiones, Curva altera delendo f evadet fedio Conica. Et fti per defedum fex ultimorum terminorum xquatio conftruenda reducatur ad tres dimenfiones, incidetur in conftrudionem Wallifimam per Para¬ bolam cubicam & lineam redam. Conftrui etiam poffunt aequationes per Hyperbo- filmum Parabola cum dlametro. Ut fi conftruenda fit hsec sequatio dimenfionum novem tormina penul- tirno carens, a-pcxxfi-dx34'ex4'rfx ,-j-gx6-|-hx7 ^kx8-l'lx9=o j affumatur sequatio ad Hyperbolif- mum ilium xxy = i, & fcribendo y pro sequatio conftruenda vertetur in hanc ay 5 J|- c y y -j- d x yy -|- e y + f x y -j- m x x y •«-(- g h x -f k x x'-fi 1 x^ — o, qux cur- vam fecund! generis defignat cujus defcriptione Problema folvetun Et quantitatum m ac g alter- utra hie deeffe poteft, vel pro lubitu affumi. Per Parabolam cubicam & Curvas tertii generis conftruuntur etiam sequationes omnes dimenfionum non plufquam duodecinq & per eandem Parabolam & curvas quarti generis conftruuntur omnes dimen¬ fionum non plufquam quindecinq Et fic deinceps in infinitum. Et curvae illse tertii quarti & fuperiorum generum deferibi femper poffunt inveniendo eorum pundta per Geometriam planam. lit fi conftruenda fit sequatio xIS * d-axIOJpbx9'|-cx8Jpdx7+ex6-|-fx? ft- g x4 fi- hx3 -p ixx -y kx -j- l = o 5 & deferipta habeatur Parabola Cubica ; fit sequatio ad Pa¬ rabolam illarn cubicam x3 = & fcribendo y pro x2 sequatio conftruenda vertetur in hanc j4 -j-axy3 -J-cxxyy -^fxxy 4~ixx = o 5 quae eft rfb dx gx h -|^k x +1 sequatio ad Curvam tertii generis cujus defcriptione Problema folvetur. Deferibi autem poteft haec Curva inveniendo ejus punfta per Geometriam plananqprop- terea quod indeterminata quantitas x non nifi ad duas dimenfiones afeendit. fa ( -n /varum. Tab: I. / Cu/nm/rwrv Tab. H. ✓ \ / IP . Cjjrvwrwnv Tab. IE. / J ) >• * •/ t / I ( urvnrum Tab-. 14. TRACTATUS D E Quadratura Cur va nun , [Itf*] QUantitates Mathematicas non ut ex partibus quam minimis conftantes, fed ut motu conti- nuo defcriptas hie confidero. Lines deferi- buntur ac deferibendo generantur non per appofi- tionem partium fed per motum continuum pundto- rum, fuperficies per motum linearum, folida per motum fuperficierum, anguli per rotationem late- rum, tempora per fluxum continuum, & fic in teris. Hs Genefes in rerum natura locutn vere ha- bent & in motu corporum quotidie cernuntur. Et ad hunemodum Veteres ducendo reddas mobiles in longitudinem re&arum immobilium genefin docue- runt redtangulorum, Confiderando igitur quod quantitates squalibus temporibus crefcentes & crefcendo genitae, pro velo- citate majori vel minori qua crefcunt ac generantur, evadunt majores vel minores ; methodum qusrebam deter- ! 1x66-] : _ . determinandi quantitates ex velocitalibus rnotuum vel incrementorum quibus generantur ; & has mo tuum vel incrementorum velocitates nominando Flu¬ xiones & quantitates genitas nominando Fluent es , in- cidi paulatim Amis 1665 & 1666 in Methodum Flu- xionum qua hie ufus fum in Quadratura Curvarum. ■ Fluxiones funt quam proxime ut Fluentium aug- menta sequalibus temporis particulis quam minimis gen'ita, & ut accurate loquar, funt in prima ratione dugmentorum nafeentmm; exponi autem poffunt per lineas quafeunq; quae funt ipfis proportion? ;les. Ut Fig, t fi areae ABC., ABDG Ordinatis B C . B D fuper bah A B uniform! cum motu progredientibus deferi- bantur, harum arearum fli xiones erunt inter fe ut Ordinates defenbentes BC A BD5 & per Ordinatas- alias exponi poffunt, propterea quod Ordinatae ilia? funt ut arearum aug rienta nalcentia. Progre- diatur Ordinata BC de loco fuo BC in locum quemvis novum b c. Compleatur parallelogram- mum BC E b, ac ducatur reda VTH quse Cur- vam tangat in C ipfifq; be & BA produdis occur- /rat in T & V : & Abfciffs A Ordinatae B C5 & Lineoe Curvae A C c augmenta modo genita erunt Bb^ Ec & Cc; & in horurii augmentortim nafcen- tium ratione prima funt latera trianguli CE fpdeoq; fluxiones ipfarum AB5 BC & AC funt ut trianguli illius CET latera CE5 ET &CT & per eadem latera exponi poffunt, vel quod perinde eft per la¬ tera trianguli coiiftmilis V B C. Eodem recidit fi fuinantur fluxiones in ultima '.ratione par tium evanefeentium. Agatuf reda Cc & _producat.ur eadem ad K, Redeat Ordinata be in C 167 ] in locum fuum priorem B C, & coeuntibus ptm£tis C & c, reda C K coincidet cum tangente C H, & triangulum evanefcens CEc in ultima fua forma evadet fimile triangulo GET., & ejus latera evanef- centia CE, Ec & Cc erunt ultimo inter feut funt trianguli alterius GET latera CE, ET&CT, & propterea in hac ratione funt fluxiones linearum A B? BC & AC. Si punda C & c parvo quovis inter- vallo ab invicem diftant reda C K parvo interval!© a tangente CHdiftabit. lit reda CK cum tangente C H coincidat Sc ratiories ultimo linearum C E, E c & Cc inveniantuiq debent punda C Sc c coire & om- nino coincidere. Errores quam minimi in rebus-, mathematicis non funt contemnendL Simili argumento fi circulus centre B radio BC deferiptus in longitudinem Abfciffae AB ad angulos rectos uniform! cum motu ducatur, fluxio folidi ge~ niti ABC erit ut circulus ille generans? & fluxio fu- perficiei ejus erit ut perimeter Circuli illius & fluxio linege curvse A C conjundtim. Nam quo tem¬ pore folidum ABC generator ducendo circulum ilium in longitudinem AbfcifSe A B, eodem fuper- ficies ejus generator ducendo perimetrum circuli il¬ lius in longitudinem Curvse A Cs Rjeda TB circa plum datum T revolvensjecet aliam Fig , 2 0 pfitione datum re dam AB : quaritur froprtio fluxio- mm red arum illarum AB Id T B . Progrediatiir reda P B de loco fuo P B in locum novum P b* In P b capiatur P C ipfi P B sequalis1 & ad AB ducatur PD fic, ut angulus bPD sequalis fit angulo bBC ; & ob fimilitudinem triangulorum bBC, b P D erit augmentum B b ad augmentum C b ut Pb ad Db, Redeat 1*58 3 Redeat jaili P b in locum fuum prioreni P B ut aug- menta ilia evanefcant, & evanefcentium ratio ulti¬ ma, id eft ratio ultima Pb ad Db, ea erit quae eft PB ad DB, exiftente angulo PDB redo, & prop- tereain hac ratione eft fluxio ipfius A B ad fluxionem ipfius P B. Recta PB circa datum Polum P revolvens fecet ■ alias duos fo/itione datas rectos y4 B ly yl E in B E : qudcntur frofortio fiuxionum reel arum ittarum ElB .1? A E. Progrediatur reda revolvens PB de loco fuo P B in locum novum P b redas A B, A E in pundis b & e fecantem, & redse A E parallela B G ducatur ipfi Pb occurrens in C, & erit Bb ad BG ut A b ad A e, & B C ad E e ut P B ad P E, & conjundis rationibus Bb ad Ee ut AbxPB ad AexPE. Redeat jam linea P b in locum fuum priorem P B, & augmentum evanefeens Bb erit ad augmentum eva- nefeens Ee ut ABkPB ad AExPE, ideoq; in hac ratione ell fluxio reda: AB ad fluxionem recta' A E._ Hinc fi recta revolvens P B lineas quafvis Curvas pofitione datas fecet in pundis B & E, & recta’ jam mobiles AB,AE Curvas illas tangant in Sedionum pundis B & E : erit fluxio Curva quam reda, A B 'tangit ad fluxionem Curvae quam reda A E tangit ut A BxP B ad A E/P E. Id quod etiam eveniet fi reda P B Curvam aliquam pofitione datam perpe- tuo tangat in pundo mobili P. Fluat quantitas x unifermiter l? invenienda fit fluxio qumtitatis xn. Quo tempore quantitas x fluendo evadit xj-o, quantitas xn evadet x-j-o|n, id eft per anethodum ferierum infinitarum, xn-j noxlw [Itfp] '-j-«|?ooxn‘2-|-if5V. Et augmenta o & noxa'H "±£oox®'2 -\-lfjc. funt ad invicem ut i & nxn'I-|--2£foxn‘2-j- yc. Evanefcant jam augmenta ilia, & eorum ratio ultima erit i ad nx11'1 : ideoq; fluxio quantitatis x eft ad fluxionem quantitatis xn ut i adnx”'1. Similibus argumentis per methodum rationum primarum & ultimarum colligi poffunt fluxiones li- nearum feu redtarum feu curvarum in cafibus qui- bufcunque, ut & fluxiones fuperficierum, angulo- rum & aliarum quantitatum. In finitis autem quan- titatibus Analyfin lie inftituere, & finitarum nalcen- tium vel evanefeentium rationes primas vel ultimas inveftigare, confonum eft Geometric Veterum : & volui oftendere quod in Methodo Fluxionum non opus fit figuras infinite parvas in Geometriam intro- ducere. Peragi tamen poteft Analyfis in figuris qui- bufeunq; feu finitis feu infinite parvis qua figuris evanefeentibus finguntur fimiles, ut & in figuris qua' pro infinite parvis haberi folent, modo caute pro- cedas. Ex Fluxionibus invenire Fluentes Problema dif- ficilius eft, & folutionis primus gradus aequipollet Quadrature Curvarum : de qua fequentia olim feripfi. D E TRACTATUS D E Quadratura Curvarum. QUantitates indeterminatas ut motu perpetuo . creicentes vel decrefcentes, id eft ut fluen- tes vel defluentes in fequentibus confidero,deftgnoq; literis z5 y, x? v? & earum fiuxiones feu celeritates 9 * * « erefcendi noto iifdem literis pun&atis z, y, v. Sunt & harum fluxionum fiuxiones feu mutationes magis aut minus celeres quas. ipfarum z, y, x5 v fiuxiones fecundas nominare licet & fic dignare *-• I? •• <9%x z5 y, x? v, & harum fiuxiones primas feu ipfarum 9 • * * » ft «• » . ft * ♦ z5 y7 x, v fiuxiones tertias fic z, y5 x5 v, & quartas fic *••••••• • o,® !5 <1 ft SO z5 y, x} v. Et quemadmodum z5 y, x, v funt flu- xiones quantitatum z5 yr v5 & hoe funt fiuxiones quantitatum z, y5 x, v & hoe funt fiuxiones quantita¬ tum primarum z, y , x5 v : fic hoe quantitates confide- rari poffunt ut fiuxiones aliarum quas fic defignabo, [ I7I ] z, y, x, v, & hse utfluxiones aliarum z,y5 x, v, & hoc ut nuxiones aliarum z*> y5 x., v. Dengnant lgitur j « • « ® f * • «• * # •* ®«i; z5 z, z, z, z, z} z7 z rtor. feriem quantitatum quarum quoelibet pofterior eft fluxio prsecedentis & quaelibet prior eft fluens quantitas fluxionem habens fubfe- r 1 1 f.J - .-r-XX, CS quentem. Similis eft feries //az— zz? //az— zz? f ~ 1 <-• 1 -wu. f - ■ - r, -nn'#-w.i.*.-*a=J - - ,-iS 8 @ — ■ — ^»«=a : az — zz , f/ az — zz , V\ az — zz , ^ az — zz , ut & _ az-|-z2 az-f-z2 az-pz2 az-j-z2 az-j-z2 a— z Si Si if fj a- a- •z a- az-f-z2 . Et notandum eft quod quantitas quaelibet prior in his feriebus eft ut area figurae curviliniae cujus ordinatim applicata re&angula eft quantitas pofterior & abfcifla eft z : uti //az— zz area curvae cujus ordinata eft / az— zz & abfcifla z. Quo au«* tem fpedant haec omnia patebit in Propofitionibus quae fequuntur. Zz a PROP, [ r72 ] PROP. I. PROB. I. 2) at a ® temporis incrementis fuis oz, oy, ox au fite, evadent • « • z-|-oz, y-^oy, xq-ox, quae in aequatione prima pro z, y & x fcriptae dant aequationem x3 -j-^xxox • • • * 3X00XX o3x3 — xyy — oxyy — 2xoyy — ixooyy a 9 • ® * • — xooyy— xo3yy -j-aaz^-aaoz — b^ = o. Subducatur aequatio prior, & refiduum divifum per o erit 3XX2 9 V » 9'"' 9 * ® ® e • A •^3xxox~^x3oo — xyy— 3xyy — ^xoyy — xoyy — xooyy 4-aaz = b. Minuatur quantitas o in infinitum, & neg~ * » leftis terminis evanefcentibus reftabit gxx2 —xyy » • — 2xyy Taaz — °- Q- e. d, _ « Explicatio plenior . Ad eundem modum fi asquatio effet X3 — xyv ^E2S3ca=s=ssasK=s=3a=9 ^ v • % -J-aa f^ax— yy— b^ — o, producer etur :jx2x-— xyy - • * ' • . ~2xyy+aa^ ax— yy = o. Xlbi fi fiuxionern i/\ a?.— yy tollere velis, pone K ax— yy = z, & erit ax— yy = za & [i74] # ♦ ' ♦ & (.per hanc Fropofitionem ) ax— ! 2yy = ?/:i feu ax -^yy = z , hoc eft ax— ayy ^ax — - jry • E f* ax— yy . * • ,, a3x— iaayy inde g x2x — -xyy — 2xy y -p — . ~ = o 2 / ax — -yy Et per operationem repetitam pergitur ad fiuxio- mes fecundas, tertias & fequentes. Sit sequatio zy3— Z4-|-a4 = o, & fiet per operationem primam -* • t « * * -zy^Vim2 — A-zz?"°-> per fecundani zy3-j-6zyy2 ® « • « « * +3zyy2+6zy2y— 4ZZ3“I ° , per tertiam - c®# 9 *S •» *** **"*> zy1 v 9zyy2 + 9*yy* -h i8w -V 3zyy2-l- l8zyyy 8 9 9 *19 © 6zy3— qzz3— 3 6zzz2— aqz3z= o. * tlbi vero fic pergitur ad fluxiones fecundas, ter¬ tias & feqtientes, convenit quantitatem aliquam ut uniformiter fiuentem conliderar.eT& pro ejus iluxione prima unitatem teribere, pro fecunda vero & fe- quentibus nihil. Sit aequatio zy3 — z4 -p a4 = o, ut iupra ; Sc fluat Z uniformiter, fitq; ejus fluxio unitas, 8c fiet, per operationem, primam y^-^syy2 — 4Z3 = o, ■ * «« « ;.per fecundam 6yy2 -ji 3zyy2 ~^6zy2y — iaz2 = o, .per tertiam 9yf+ i Bpy+JzyyH’ iS^yy-j-^zyg -•*- — aqz = o. V ii7$ ] In hujus autem generis aequationibus concipien- dum eft quod ftuxiones in fingulis terminis fint ejul- ® * deni ordinis, id eft vel -omnes primi ordinis y, z, • • » • • 9 vel omnes fecundi y, y29 yz? z2, vel omnes tertii ••• »• • • • « •• • y, yy, yz, y3, y2z, yz2 Z3 &c. Et ubi res aliter fe habet complendus eft ordo per fubintelle&as ftuxio¬ nes quantitatis uniformiter fluentis, Sic aequatio « « » noviffima complendo ordinem tertium fit 2 Jf 1 8zy2y-|-3Zyy2 -'r 1 8zyyy-j-,6zy3— aqzz? = o. * PROP. II. PROB.IL Invmire Curvets qua qmdrari poffunt. Sit ABC figura invenienda, BC Ordinatim ap- Erg. 4. ' plicata reftangula , & AB abfeifla, Producatur CB ad E ut fit BE— 1 ? & compleatur parallelo- grammum ABED: & arearum ABC, ABE D ftuxiones erunt ut BC &BE, Aftumatur igitur aequatio quaevis qua relatio arearum definiaturr & inde dabitur relatio ordinatarum BC &BE per Prop. I. Q E, L Hujus rei . exempla habentur in Propofitionibus duabus fequentibus. PR OR \ C 176] PROP. III. THEOR. I. Si pro abfcifla A B ,8i area AE feu ABxi pro- milcuefcribaturz5& fipro e -Egz2fl-|-hz3«-j-’&cs fcribatur R : fit autem area Curvae z®Rh erit. ordinatim applicata BC — &c. in z 6-1 R K~l. Demonjlratio. Nam fi fit z8R,i=V, erit per Prop. 1, 9zz6'!Ra ® • ® ^-^z8RRVI = v. Pro R* in primo aequationis ter- mino & z9 in fecundo fcribe RRVI & zz9'1, & fiet « * • §zR-j-*zR in z8'*R\r = v. Erat autem R=e-|-fz» ~\-gz2«-\-hz^ &c. & inde per Prop. 1. fit R =j ® • ® ^fzz^1 Jp2„gzz2M'I-|-’3whzz3«‘i>'|- &c. quibus fubftitu- tis & fcripta BE feu 1 pro z7 fiet in *M R Q. E. D. Vl = v= BC. PROP. [ 3 "in PROP. IV, THEOR, II. Si Curvae abfciffa A B fit z, 8c fi pro e-j-> fz» -j-gz1* 8cc. fcribatur R, 8c pro k-}-lz»-}-mz2*4- &c. fcri- batur S ; fit autem area Curvse z® R* : erit or- dinatim applicata BC=, &e | S jgW “^0 ®^S9®89^*»e* 4" 0 0 1 7H -ta« c 1 z i f 1 z*" P» Slz3" * . I i^R^S*** ±l„ eraz2" il ftnz3,ZL §mz40-* i-2f« 1'2^S Demonftratur ad modum Propofitionis fuperioris. \\ PROP. V. THEOR. III. SiCurvae abfciffa AB fit z, & pro e-|-fz,'-j-gzI* -’p hz3" -j- 8cc. Icribatur R : fit autem ordinatim ap¬ plicata z9lRVI in a -f-bz" +cz2» -[-dz3"-|- 8cc. 8c po- natur ? = r. r-f-*=s. s4-* = t. t4-* = v.8cc. erit area a 2*R> i„;a AilMfA.. : •c-.fB-'SA . , >‘d r.fC^gB-vhA - T -TT_-Z" T— — — ■ -z2" + — - ■ Z3» r — f- 1 9e " ' * 5“ r *4— 2j e r —l. — s f r\ —t p —v L n H- ~3 ~2g z*” -I- 8cc. Ubi A, B, C, D, 8cc. 3?e r-H4,e Aaa denotant i. «eA l| a ■ * ‘ ' r +g+2»,eCz2''ji H'2"fCz3" &c. e c « e •+8-h»,eDz3"&c.j & earundem ares, Az9 R\ Bz9+” R . > Cz9+2» RA. Dz9'^ Ra. Et (i fitmma ordinatarum ponatur aequalis ordi¬ nate a^kzM'C^d'dz3"-^ &c. in z9'1 RA'\ fumma a rearum z9RA in A ^Bz» ^Cz2«4~Dz3», d- &c. equa- lis erit area? Curve cujus ifta eft ordinata. iEquen- tur igitur Ordinatarum termini correfpondentes, & fiet a=eeA, b = +9„fA e B , c= +2L§^ fB '4- FF*T,e C &c. & inde $; = A. ^4— — j • 1 5 _____ _ _ ^ 0-h«3e C s=== i§"l~ 1 rl A/jji B z’ j * » » - - — 7 f^2h e~~ - iy= Cu Et lie deinceps m infi¬ nitum Bo [i?9] nitum. Pone jam £ = r. r-|-A = s. s-j-A = t &c. & in area z9RAx A^-Bzw'jr-Cz2w-j-Dz3B &c» fcribe ip- forum A, B, G, &c. valores inventos & prodibit feries propoiita. Q. E. D. Et notandmn eft quod Ordinata omnis duobus modis iu feriem refolvitur. Nam index » vel affir- mativus eft poteft vel negativus. Proponatur Ordi¬ nata ' — 1 - . Hac vel fxc fcribi poteft z~» x 3 k — Izzxk — -lzz^-mz^Tq vel lie zx -l-j-^kz'* xm-lz_1'l-kz_ij -i. In cafu prioreeft a~ gk.b = o. c== — k e=k. f=o. g=-l. h— m. *=~i. »—i. s-i=-b 9=-i=r. s=-i. t=_i. v=o. In pofteriore eft a=— 1. b=o. c=3k. e— m. f=—l. g=o, h=i. A=-j. «=— i. e— i = i. »=2. r=— 2. s=— i;. t==— x. v— Tentandus eft cal’us uter- que. _ Et fi ferierum alterutra ob terminos tandem deficientes abrumpitur ac terminatur, habebitur area Curva? in terminis finitis. Sic in exempli hujus priore cafu feribendo in ferie valores ipforum a, b, e, e, f, g, h, a, sy r, s, t, v, termini omnes poll pri- mum evanefeunt in infinitum & area Curvae prodit ayk~~d-™’_ p;t }ia;C area ob lignum negativum adjacet abfeifise ultra ordinatam produdtse. Nam area omnis affirmativa adjacet tain abfcilTae quam ordinata?, negativa vero cadit ad contrarias par¬ tes ordinata & adjacet abfeilfe produtlse, manente fcilicet figno Ordinatae. Hoc modo . feries alter¬ utra & nonnunquam utraque femper terminatin' & finita evadit fi Curva geometrice quadrari po¬ teft. At fi Curva talem quadraturam non admit- tit, feries utraq; continuabitur in infinitum. & ea- A aa 2 rum [ i8o ] rum altera converget & aream dabit approximando, praeterquain ubi r ( propter aream inffoitam ) vel nihil eft vel numerus integer & negativus, vel ubi i- aequalis eft unitati. Si * minor eft unitate, conver- get feries in qua index „ affirmativus eft : fin l unita te major eft, converget feries altera. In uno cafu area adjacet abfciffe ad ufq; ordinatam duda?, in altero adjacet abfciffae ultra ordinatam produdae. Nota infuper quod fi Ordinata contentum eft fub fadore rationali Q & fadore furdo irreducibili R*, & fadoris furdi latus R non dividit fadorem ratio- nalem Q; erit a-i =tt & R*-1 = R» Sin fadoris fur- di latus R dividit fadorem rationalem femel, erit *— 1 = ‘jr-y i & R*-1 : fi dividit bis, erit 1=7 & RV1 ==R*'\~2: fi ter, erit i=rr-|p^ & R/-1— R^' 3 : 8c fic deinceps. Si Ordinata eft fradio rationalis irreducibilis cum Denominator ex duobus vel pluribus terminis com*- pofito : refolvendus eft denominator in divifores fuos omnes primos. Et fi divifor fit aliquis cui nullus alius eft aequalis , Curva quadrari nequit : Sin duo vel plures fint divifores aequales, rejicien- dus eft eorumunus, 8c fi adhuc alii duo vel plures fint fibi mutuo aequales 8c prioribus inaequales, re- jiciendus eft etiam eorum unus, 8c fie in aliis omnir- bus aequalibus fi adhuc plures fint : deinde divifor qui relinquitur vel contentum fub diviforibus omni¬ bus qui relinquuntur, fi plures funt, ponendum eft pro R, 8c ejus quadrati reciprocum R'2 pro R^,prae- terquam ubi contentum illud eft quadratum vel cu- bus vel quadrate quadratum, 8cc. quo eafu ejus latus ponem [i8i] ponendum eft pro R & poteftatis index 2 vel 5 vel 4 negative fumptus pro a. & Ordinata ad denomimu- torem Rz vel R3 vel R4 vel R? &c. reducenda. Ut ft ordinata fit —jfel ZJ~8z3 — ; quoniam hxc Z5-J-Z4 — 5Z3 — z2-|-8z — 4 7 1 fraftio irreducibilis eft & denominatoris divifores funt pares, nempe z — 1, z — 1, z — 1 & z^a, z-\-2 , rejicio magnitudinis utriufque diviforem unum & reliquor um z — 1, z — 1 , z\-2 conten- turn z3- — ’^z-j— 2 pono pro R & ejus quadrat! re- ciprocum qp feu R—2 pro R*"1. Dein Ordina¬ ta m ad denominatorem R2 feu R1-* rediico, & fit z6— 9z4'|-8z3 z3 —^zipoj quad. , id eft Z3 X 8 - 9 z 4-z3 x a — 3 z 4- z3f 2 1, &c. Et inde eft a — 8. b=— 9. c — o. d e=a. f=-3, g=o. h=i. k~i = ~7. k= _n 1. 9-1 = 5. s = 4 = r. s= 3. t = a. v=r. Et his in ferie feriptis prodit area nibus in tota ferie poft primum evanefeentibus terminis om- Si deniq; Ordinata eft fradio irreducibilis & ejus denominator contentum eft fub fadore rationali Q. & fadore furdo irreducibili R’, inveniendi funt la- teris R divifores omnes primi, & rejiciendus eft di- vifor unus magnitudinis cujufq; & per divifores qui reftant , fiqui fint , multiplicandus eft fador rationalis Q. : & ft fadum asquale eft lateri R vel later is illius poteftati alicui cujus indexed: numerus integer, efto index ille m, & erit *,~i = m, & RV. = R"T"m» m fi Ordinata fit poft tertium evanefcentibus. PROP. VI. THEO.R. IV. Si Curvse abfciffa A B fit z, & fcribantur R pro eJ[-fz» Jpgz^^hz311--]- &c. & S pro k -j- lz» \-njz2» -\-nz3B & c. fit autem ordinatim applicata z^'R^1 S'-1'1 in a-Wbz» -\~cz2» -\-dz3» &c. & fi terminorum, e, f, g, h, &c. & k, 1, m, n. &c. reftangula fint. ,ek fk gk hk &c el fl gl hi &c. em fm gm hm &c. en fn gn hn &c. Et [ *%] Et fi re&anguloriim illorum coefficients nuine- rales lint refpeftive »9 = r. = s. = t. t -]-a = v . &c, r--^ = s. s-f> = t. = v. v-|-^ — w. &c. S-|-|K = t. — v. v-\-^ = w. W-f-(*=X.&C. y/ y// JW JUU m a Hi t~ j-#*=v. V~rj-/* = wv = x. = y. &c. area Curvoe erit hxc *“ ~t g k - o AU — sfkA ln — -s -l-i, fk 73 *— t' f 1 \ ar, a ■ » a »b— s'elA »C-s'+i,el ^-t"emA z'fiObf4 in • — _j- — - — — — - - — a- rek * r-j-i, e k _ _ — V h k A _ *~Z -J-!, g kj^ _V' g 1 “-s-|-2, fk(^ *— £-|-i, f 1 -D — v"fm n Al — s%j-2, e 1 v<# C'-j-ije m —v'"e n i-|-2, e k H~3> vel — , B coeffieientem datam fecundi, C coeffieientem datam tertii, & fie deinceps. Terminorum vero, a, b, c, &c. k, 1, m, &c. unus vel plures deeffe poffunt. Demonftratur Propofitio ad modum prsecedentis, & quae ibi notantur hie ob- tinent. Pergit autem feries talium Propofitionum in infinitum, & Progreffio feriei manifefta eft. PROP. C 184] PROP. VII. THEOR. V. Si pro e4-fzw~[rgz2M-* &c. fcribatur R ut fupra? & in Curvae alicujus Ordinata RkA maneant quantitates datae », A, e5 £, g, See. & pro <7 ac T fcri- bantur fucceffive numeri quicunq; integri : Sc ft detur area unius ex Curvis quae per Ordinatas in- numeras ftc prodeuntes defignantur ft Ordinatae funt duorum nominum in vinculo radicis, vel fi dentur areae duarum ex Curvis fi Ordinatae funt trium no^ minum in vinculo radicis, vel areae trium ex Curvis fi Ordinatae funt quatuor nominum in vinculo radi- cis, Sc fic deinceps in infinitum : dico quod dabun- tur areae curvarum omnium. Pro nominibus hie habeo terminos omnes in vinculo radicis tarn de- ficientes quam plenos quorum indices dignitatum funt in progreffione arithmetica. Sic ordinata \/a4— ax3 4"X4 ob terminos duos inter a4 Sc —ax3 deficlentes pro quinquinomio haberi debet. At binomium eft Sc \/a4~Hx4 — 'It tanoniurm cumprogreffio jam per majores differentias proce- da£. Propofitio vero fic demonftratur. C A S. L Santo Curvarum duarum Ordinatae pz8'1 R*-1 & qz6frI RA“E5 Sc areae pA Sc qB5 exiftente R quanti¬ tate trium nominum e~j-fz» gz2». Et cum per Prop. [■85] Prop. III. fit z9R* area cmvx cuius Ordinata eft eel 0 fz" pz2* in z^R^.fubduc Ordinatas & areas priores de area & Ordinata pofteriori, & manebit 9 c gz2" in zVI RVI Ordinata nova Curvse, & — ft ZH z9p*_pA— qB ejufdem area* Pone ee-=p & =’q & Ordinata evadet gz2® in z9 RA & area zsRx — aeA — afB — »»fB. Divide utramq* per #g-t,A*g» & aream prodeuntem die C, & affumpta utcunq; r, erit r C area Curvse cujus Ordinata eft rjre-Hn-'gvi; Et qua ratione ex areis pA & qB aream rC Ordinata; rz9'^”'1 R*-'1 congruentem inve- nimus, licebit ex areis qB & rC aream quartam puta sD, ordinatae sz9 ' 3"'IR'l‘I congruentem invenire, & fic deinceps in infinitum. Et par eft ratio pro- greffionis ab areis B & A in partem contrariam pergentis. Si terminorum 9, e aliquis de¬ ficit & feriem abrumpit, affumatur area pA in prin- cipio progreffionis unius & area qB in principio al- terius, & ex his duabus areis dabuntur ares omnes in progreffione utraque. Et contra, ex aliis duabus areis aflumptis fit regreffus per analyfin ad areas A & B, adeo ut ex duabus datis cseterae omnes den- tur. Q. E. O. Hie eft cafus Curvarum ubi ipfius z index o augetur vel diminuitur perpetua additione vel fubdudtione quantitatis ». Cafus alter eft Curva¬ rum ubi index x augetur vel diminuitur unitatibus. Bbb CAS, [i86] C A S. II. Ordinate pz9'xRA & qz^^R*, quibus areae pA & qB jam refpondeant, ft in Rfteu ejj-fz^gz2” du« cantur ac deinde ad R viciffim applicentur, eva- dunt pe -j- "h Pgz2?l x z^R^1 & qez11 -J- qfz2« -Uqgz3” x z9-^^1. Et per Prop. III. eft az6R^ area Curvae cujus Ordinata eft sae ~j_9^ afz”qt9A)j agz2« in z9-iRa'1 j & bz8+MRA area Curvas cujus ordinata eft ^bez® Il9bfz2rt jlabgz3w in z9lRAI. Et harum qua- 1 — Ail — tuor arearum fumma eft pA-\-q@rj- az@RA-|- bz9 i *RA & fumma refpondentium ordinatarum aae +pe +0 afzM -rhn I"9 be ft* ft pf + qe -|-2AM jigZ2" -s bf ~n '+pg + qf 9 bgz3H in z^R*"1. -2A» + 02 Si terminus primus tertius & quartus ponantur fe- orfim sequales nihilo, per primum fiet eae-|-pe=o feu -8a = p, per quartum - 8b - »b - = q , & per tertium (eliminando p & q) “/= b. Unde fecundus fit adeoq; fumma quatuor Ordinatarum eft — 7®e fumma totidem refpondentium arearum eft a z9 RA -\- 2|fz8-bi R*_8aA — ■2^~2r^*u> agB. Divi- C 187 ] Dividantur hae fummse per >-"aff~'jPl"ah^ & }j Quotum pofterius dicatur D5 erit D area curvas cujus ordi- nata eft Quotum prius z®+1hIRA'1. Et eadem ratione ponendo omnes Ordinatae terminos praster primum asquales nihilo poteft area Gurvae inveniri cujus Or- dinata eft z8 lRVI. Dicatur area ifta C, & qua ra¬ tione ex areis A & B invents funt areas C ac D. ex his areis C ac D inveniri poffunt alias duas E & F ordinatis z^R*'2 & z°+w'IRv2 congruentes5 & fic de- inceps in infinitum. Et per analyfin contrariam regredi r licet ab areis E&Fad areas CacD, & inde ad areas A & B, aliafq; quas in progreffione fe- quuntur. Igitur fi index x perpetua unitatum ad¬ ditione vel lubdudione augeatur vel minuatur, & ex areis quae Ordinatis fic prodeuntibus refpondent duas fimpliciffimas habentur ; dantur alias omnes in infinitum. Q. E. O. C A $. III. Et per cafus hofce duos conjunctos^ fi tarn in¬ dex 8 perpetua additione vel fubduftione ipfius n quam index \ perpetua additione vel fubdufitione unitatis , utcunq; augeatur vel minuatur, dabuntur areas fingulis prodeuntibus Ordinatis refpondeiiteso Q.E.O. B b b 1 CAS. C *88 J t CAS . IV. Et fimili augmento fi ordinata conftat ex qua- tuor nominibus in vinculo radical! & dantur tres arearunq vel fi conftat ex quinq; nominibus & dantur quatuor arearum, & fic deinceps : dabun- tur areae omnes quae addendo vel fubducendo nume- rum * indici $ vel unitatem indici x generari poflunt. Et par eft ratio Curvaruin ubi ordinatae ex binomiis conflantur, & area una earum quae non funt geome- trice quadrabiles datur. Q. E. O. PROP. VIII. THEOR, VL Si pro e^-fzw4'gz2"~|~’&c. & k + \zn -j-mz2^-|-&:c, fcribantur R & S ut fupra5& in Curvae alicujus Or¬ dinata z9'^ R*±7 maneant quantitates datae ^ ^ ^ /q e, f, g, k, 1, m? &c. & pro & y) fcri¬ bantur fucceflive numeri quicunq; integri : & fi dentur areae duarum ex curvis quae per ordinatas fic prodeuntes defignantur fi quantitates R & S funt binomia, vel fi dentur areae trium ex curvis fi R & S conjunftim ex quinq; nominibus conftant, vel areae quatuor ex curvis fi R & S conjundtim ex fex nominibus conftant, & fic deinceps in infinitum : dico quod dabuntur areae curvarum omnium. Demonftratur ad modum Propofitionis fuperioris. PROP. [ iBp] PROP. IX. THEOR. VII. JEquantur Curvarum arete inter fe quarum Or- dinatae font reciproce ut fluxiones Abfciffarum. Nam contenta fob Ordinatis & fluxionibus Ab¬ fciffarum erunt aequalia, & fluxiones arearum font ut ha?c contenta. COROL. I. Si affumatur relatio quaevis inter Abfciffas dua- rum Curvarum, & inde per Prop. i. quteratur relatio fluxionum Abfciffarum, & ponantur Ordi¬ nate reciproce proportionales fluxionibus, inveniri poffunt innumerte Curvse quarum arete fibi mutua sequales erunt. COROL. II. Sic enim Curva omnis cujus htec eft Ordinata ze-i in e fz» g z2» &c.|* affumendo quantitatem quamvis pro » & ponendo j = s & zs = x, migrat in aliam fibi sequalem cujus ordinata eft £xfir» in e -\- f x" gx2" -j-r &c. jA- CO [ 19° 3 COROL. III. Et Curva omnis cujus Ordinata eft z9'1 in a bz» ~p cz2" -\- & c. x e-p-fz»-p gz2" &cj%affumen- do quan titatem quamvis pro " & ponendo n=s & V x, migrat in aliam fibi cequalem cujus ordinata eft in a-p bx'-ft- cx2t -p &c. xe -p fx'-pgx2' -p &c.f • COROL. IV. Et Curva omnis cujus Ordinata eft z®'1 in a + bzM cz2» -j- &c. x e -|- fz» gz2« &c~j* x k -j- lzM ~]-r mz2w Scc.J^ affiimendo quantitatem quamvis pro ? & ponendo 1 = s & zs~x? migrat in aliam fibi sequalem cujus ordinata eft jx-^ina-j-bx, -j- cx2v-p & c • x ejf fx" _p gx2'-p &c.px k-pix'-J-mx^-p COROL. V. Et Curva omnis cujus Ordinata eft z9'1 pfz" -p gz2» 4- &c* ponendo i= x migrat in in aliam fibi aequalem cujus ordinata eft x e -pfx'“ -Pgx 2» -p Sc? M eft xife * {l duofimt nomina in vinculo radicis vel x g-p fx*-p exiB| fi tria funt nomina ; & fic deinceps. CO- [ l9l ] t, CO ROL. VI, Et Curva omnis cujus Ordinata eft zu in e -j- f z" gz2” &c.j* x k lz8 -\- mz2“|- &c.[,“ ponendo i = x migrat in aliarn libi aequalem cu¬ jus ordinata eft — ppj x e -ft fx'fl gx"28 -|- dcc.j'1 X xM-lx-8-^mx-^-i-&c.l “ id eft Iiy X xl-j~kx"f* ft bina funt nomina in vinculis radicum, vel xH-i+aft-bt* x g fx» - j- ex2“ft x 1-j-kx"/'2 ft tria funt nomina in vinculo radicis prioris ac duo in vinculo pofterioris : & fie in aliis. Et nota quod area duae aequales in noviffimis hifce duobus Co- rollariis jacent ad contrarias partes ordinata rum. Si area in alterutra curva adjacet abfciffe , area huic aequalis in altera curva adjacet abfcilfse pra- ducta.', ' CO ROL. VII. Si relatio inter Curvse alicujus Ordinatam y & Abfciffam z definiatur per sequationem quamvis feftam hujus formas, v“ in e -I- fvz^-S- q v^z20'1-!- + &c. - # ta k? ir*VV£ + fay L figure affumendo 8=^, x=jz^x=^, migrat ia a ham nbi aequalem cujus Abfciffa x5 ex data Ordinata C 192 ] Ordinata v , determinatur per aequationem non § affeftam x e-j- fv*q- gv2M -\- hv3n &c.iA x k -\- lv# -p mv2w-|- &c.f* = x. COROL . VIII. Si relatio inter Cur vx alicujus Ordinatarn y & Abfciffam z definitur per aequationem quamvis affeftam hujus formas, ya in e^-’fywzcfS--j“gy2Mz2cP -f-Scc. . = z£ in k ly»zd' 4^my2«z2^--&c -$~z? in p qy V ry2rtz2^-|-&c. haec figura affumendo — gzs7 ^-7=^ & ^ migrat in aliam iibi aequalem cujus Abfciffa x ex data Ordinata v determinatur per aequationem minus affeftam va in e fvw-j-gv2» &c. = in k iv« mv2W dj- &c sV in p-pqv«^- rv2” &c. COROL . IX, Curva omnis cujus Ordinata eft t/1 jn fe ;j;p f I{4„ gz2" -H &c- x e -\- 1 z" -f- gz^&c.^'1 x |a -j- b |ezf -ft fzr+» -|- gz’"!'” ~j- &c.|Tj ? ft fits = & affumantur x = ez' -j- fzp+" -j- gz’'+2° •, migrat in aliam fibi squalem cujus ordi¬ nata eft x* x a-j-bx'l “• Et nota quod ordinata prior in ll9$l In hoc Corollario evadit fimplicior ponendo x — t , vel ponendo x = i & efficiendo ut radix dignitatis extrahi poffit cujus index eft », vel etiam ponendo = i = ut alios cafus praete- ream. CO ILO L. X. Pro ez' -J-r gzTP** &c. -j-2» g2^ 2"'1 "1“ &c- k -J- lz» ~j- mz2» -j- &c. & ilz*'1 ‘4" -nnz2"'I4'&c. fcribantur R, r, S & s refpe£tive, & Curva omnis cujus ordinata eft xSr 1 Rs in R*-1 S**-1 x aSv -\- bRx| , ft fite=”“ = " = • , z = a , *-* = . & RtS? = x, migrat in aliam fibi aequalem cujus or¬ dinata eft x3- x a-j-bx^j" Et nota quod Ordinata prior evadit fimplicior, ponendo unitates pro r, v% & » vel tfi & faciendo ut radix dignitatis extrahi poffit cujus index eft *>, vel ponendo * _= — i vel o . PROP. X. P ROB. III. Invenire figuras fimpliciffimas cum quibus Curva quoevis geometrice compari poteft, cujus ordinatim applicata y per aequationem non affe&am ex data ab- fcifla z determinatur* C c c CAS, ft [ m 1 C A S. I. Sit Ordinata az8'1, & area erit jaz8, ut ex Prop.V. ponendo b=o=c=d=f~g=h & e~ i , facile col- ligitur. CAS. II. Sit Ordinata az8'1 x e -j- fz" j- gz2jK"1 ■-]- &c- & ft curva cum figuris reftilineis geometrice comparari poteft, quadrabitur per Prop. V. ponendo b = o = c ■ = d. Sin minus convertetur in aliam eurvam fibi squalem cujus Ordinata eft ~x~ x e-|-fx-*j-gxl&c. gen Carol, a. Prop. IX. Deinde fi de dignitatum indicibus i per Prop. VII. rejiciantur uni¬ te tes donee dignitates ills fiant quam minima:, de- venietur ad figuras fimpliciffimas qua; hac rations colligi poffunt. Dein harurn unaquaeq; per Corol.5. Prop. IX. dat aliam quae nonnunquam llmplicior eft. Et ex his per Prop. III. & Corol. 9 & 10, Prop* IX. inter fe collatis, figurse adhuc fimpliciores quandoq- prodeunt, Deniq; ex figuris fimplicif- ftmis affumptis- fa£to regreffu computabitur area qusfita. CAS. [ IPS ] C A S. III. Sit Ordinata z9'1 x a bz* cz 2» -j-- &c. x e-jp fz« 4"' gz2M &c-|^ 1 •> & haec figura fi quadrari poteft, quadrabitur per Prop, V. Sin minus, di~ ftinguenda eft ordinata in partes z9-1 x axe^fz® ~!r gz2M -J-* Scc-p'1, z9'1 x bzw x e fz« \-g See. & per Caf. a. invenienda funt figura fimpli- ciffima cum quibus figura partibus illis refpon- dentes comparari poflunt. Nam area figurarum partibus illis refpondentium fub fignis fuis 4* & — conjun&a component aream totam quafitam. CAS. IV. C ) V i Sit Ordinata z9'1 x a -fi b z« -fi-cz2>| ~fi Sec, x e-J-’fz” gz2» -fi &c.H x k -fi lzw ^mz^-fi&c.J^1: & fi Curva quadrari poreft, quadrabitur per Prop. VI Sin minus, convertetur in fimpliciorem per Coral 4. Prop. IX. ac deinde comparabitur cum figuris fim- pliciffimis per Prop. VIII. & Coral 6, 9 & 10, Propa IX, ut fit in Cafu a & 3. • C A S. V, - Si Ordinata ex variis partibus conftat , partes lingula pro ordinatis curvarum totidem liabenda funt5& curva ilia quotquot quadrari poffunt,figilk- C c c a ‘ tim c*9 Y' [ 199 ] 1 6ee— <8 efz #-1- 6ffz2» — — c IR = t. -—96e3-|-48eefzM“-3i5effz2l)-]- 3of3 zjH io5»4 cl R. ~ t . TABULA Curvarum Jimpliciornm (pace cum Ettipft & Hyperbola compart pojfunt. Sit jam aGD vei PGD vei GDS Seftio Conica cujus area ad Quadraturam Curvae pro- Fig. 5 polite reqmiritur, fitq; ejus centrum A, Axis K a, Vertex a, Semiaxis conjugatus AP, datum Abfciffae principium A vei a vei ^ AbfciiTa AB vei aB vei * B - x , Ordinata reStangula BD = v5 & Area ABDP vei aBDG vei aBDG^s, exiftente «G Or¬ dinata ad pundtum Jungantur KDyAD, aD. Du- catur Tangens DT occurrens Abfciffe AB in T. & compleatur parallelogrammum A EDO, Et fiquando ad quadraturam Curvae propolit^e requri runtur areas duarum Sefitionein Conicaruuq dica- tur pofterioris AbfciiTa Ordinata T, 8c Area ^ Sit autem differentia duarum quantitatum ubi in- certum eft utrum pofterior de priori an prior de po» fteriori fubduci debeat. Curva- Curvarum Formse. Settionis Conicse. Curvarum Area;, Forma [201 ] c3 • i— » 4- ) 5— 4 -j-j txi O £ti Q H P3 Q PH d « ^ «*» H Q O c3 -G « »— i ii 4-> eo I <«|M «3 , <+5 O oo i S d >=sr-^ 1 C5* ■i l! i -M >18 SK 4-1 s e > 11 x vi u X M I 01 1 co 'O JC oo ! «5i Jo t o rs PQ 1 ^4- I P i Ph 1 w 1 H" xi in 02 > IS 41 > &4 n o d X i> lr ii 4~i 11 > X x I CO a 11 X X, aj ; *~i > •» [M I-*-* ■> XX. { vj w X X X 11 -f ® X l! 11 11 -is t-i i ® 1 N » Pd rs U il f*' u OG «— < 02 > -a i SI Ph Q PQ S“"H 02 > PQ P o id d ' ss 11 4-> 11 CO 1* / s » > II X X aj J— 1 X > X II 5S M -M I aj ■> > Xl C4 & Xi ro 4J s O > ii X X QJ I l £ > X I ST I M II <-M I _ l u w cs ■ Cl oo TSl *4- Ni -4* <5 Q O d a » *H g" < 02 > P < pH G * 1-K xi I'+j. 11 X X > X 1! ■*/ ?? u id 1,N! (-M Xf « i. Q O < c • $ *aJ > Q O p-i a d i j V i +-> 11 11 m > > Xt M 1 C< X s •l* ! 1 CO > CO X w 1 G ® n^t G • w-l G * d fe ¥i^ 00 1 * « & > X X 11 H I S? 1 jN II xi d o Q d G « *- ( %t4l 11 +-» n] co •I- > X M *0< ■3-f* > 11» ,x >. 11 ll| « X /x r v,X IX CO 1 hr r i G > k X ii t ( a 1 N IS G T" v > *-h N U 02 > SJ CO iec r\ c- •> 'SO SO ^0 W-( PQ Q OS <3 -I- o p cS m a © •II * ; I! y> i x Cl' * C/5 . on C e *HI *0 j S © > iM 8 4- .& > X ’1! wj* >v II *o ,N 4-i i! X X M I £ nS O > ir X X 3J> I nr >- X J N II t r> ^-f N crn ,N f > *x cd 4-» C » w-«i o* rd g s§ II II CO 6* 1 M > II X X bJD X II ns II S£ .N N bJ3 £ t U3 i . i I c r> CS b£ WD -d- > II X X bJ3 X J to i_ i 3£ . M 04 M 04 N to 1 t**1 u > [ 202 ] fXA/^ > 11 X X bJD a •*? I! •# 1 o -d/bo > X II (X N bJ3 ) L- i V Xxj» II 35 CS N bX3 1 _ sc . N "> c/nro x. s CS is b B CS sc fsl . N -d rV bJ3 a> nr I CK > o S-i a» a, 3 x> «o J3 P cd 4-i X cd s J-t O Px II « 4~> fc> nt- li i 1 £-• > AW \\ «1 } 05 6J CO f | i > X n? 1 cs 5S- 0*1 Akf is AW "d* I > x a I r* sc X X AW A>J> (X x II II 4“. -d > Acr a is 4> >4 I! C4 rs . W) a "■lN ss nS i X oo i £ nd i aj X <4L > ii x x bJD III t .X *+-« JAJ' Wf QJ ,AAf P-i r T"' bXD > wo> i 1) > X .11 ll i3 -In LXV^ >-» II X 1 N ll C/5 X pq Q O « a • »— < X; S SS ll V °° ll > C/5 _ . I toD — < i X = X J S ! Oft > > f P~< m an X 4 X VO bJ3 bQ s VO bJ3 se x, bS X vo 3.0 bJ3 £ *3- CH X * X X X Of) OB f'T“ -h £ f ~T~ X -V-* i T" QJ cu > I! x X OB 4~ 43 *> *> X X 11 ‘ll N N >-> >, ii \\ X II NS i=4 » «N N3 OD P-< br a> i > S21 tnj a *? » N J2 -d ^ [203 ] cd > nJ cO o cd 5 5-* o pp <1 PQ p X bfl X X ! tn bJO x 00 I QJ 3 > II X X bD *f j_ 6 ■> x 3& N ST N -d II ■ > is N fe w N W3 3? [ & cn N3 n3 I! > bOt^ fi aj o 4J 1' x x O VO HI M 4-1 X X ^Pcc D U X x 00 o IN M 14 > X X bQ^bX) IS SI X X X X OD m *h 4- £ 4~“ 1 1 QJ cu >• *> X X il II N > ll- X X OB J— 1 , <& [_ QJ X II N1 9 sr -d N W5 t t > es B Un O & m a m r-- fs SO .feCo SO .^0 Ddd 2 Forma nona. [204] i! C4 X I > X ii X X £P i _f« 4) * 4i-4 »t3 >> 9 X ss M »£5 fe .*> Q 11 . N 4-i 5? ? NJ *d 4) .bO £S3 43 I , OD rt a « *- 1 o c3 a u >% II ,N N V I W3 tt Cl N -d e Cl X hi * NJ 43 1 bD' *> U 5=1 ,N NS -r-r «,t»0 > > * cs ,te<) ^4 [*°$] In Tabulis hifce, feries Curvarum cujufq; Forms utrinq; in infinitum continuari poteft. Scilicet in Tabula prima, in numeratoribus arearurn for¬ ms tertise & quarts, numeri coefficientes initialium terminorum (2, — 4., 1 6, — 96, 868, &c.) generan- tur multiplicando numeros — 2, — 4, — 6, — 10, &c. in fe continuo, & fubfequentium terminorum coef¬ ficientes ex initialibus derivantur multiplicando ipfos gradatim, in Forma quidem tertia , per — J — — I, — f» —To &c. in quarta vero per -f’ ~h~~b — l’ &c- Et Denominator-urn coefficientes 3, 15, 105, &c. prodeunt multiplicando numeros 1, 3, 5, 7, 9, &c. infe continuo. In fecunda vero Tabula, Feries Curvarum forms prims, fecund®, quints, (ext®, non® & decims ope folius divifionis, & forms reliqus ope Propofitio- nis tertis & quarts, utrinq; producuntur in in¬ finitum. Quinetiam hs feries- mutando lignum numeri variari folent. Sic enim, e. g. Curva |\/"e_pfz»= y evadit ■ P R O P. IX. T H E O R. VIII, '‘Sit A D I C Curva qusvis Abfciffam habens v;a 0 . AB==z & Ordinatam BD=y, 8c fit A E K C Curva * 9 ‘ alia cujus Ordinata B E squalis eft prioris ares ABC [ 20 6 ] ADB ad unitatem applicate, &AFLC Curva tertia cujus Ordinata B F asqualis eft fecundas areas A E B ad unitatem applicatse, & AGMC Curva quarta cujus Ordinata B G squalls eft tertias areas AFB ad unitatem applicatas, &AHNC Curva quinta cujus Ordinata BH asqualis eft quartae areas A G B ad unitatem applicatae, & ftc deinceps in infinitum. Et funto A, B, C7 D, E, &c. Areae Cur¬ va rum Ordinatas habentium y, zy7 z2y, z3y? z4y? & Abrciflfam communem z. Detur Abfcifla quaevis AC=t, fitq; BC=t— z & funto P7 Q, R? S5 T areas Curvarum Ordi¬ natas habentium x, xy7 xxy, x*y, x4y & Abfciflam communem x. Terminenter autem has areas omnes ad Abfciflam totam datam A C7 nec non ad Ordinatam pofitione datam & infinite produdam C I : & erit arearum tub initio pofitarum prima ADIC=A=P5 fecunda AEKC=tA — B=Q.Tertia AFLC = «£=?£+£ = iR. Quarta AGMC — ^a— sttB-htc— p _ig . Qui'nta A H N C — iiA 4tD -|-E __ i -p 24 24 a • CO- [ 207 ] \ CORO L. llnde fi Curvs quarum Ordinata* funt y, zy, z'yr. z3y, &c. vel y, xy, xly, x!y , &c. quadrari poffunt, quadrabuntur etiam Curva? ADIC^ AEKC, AFLC, AGMC?&c. & habebuntur Ordinatf BE, BF-, BG5 BH areis Cur varum proportionates. SCHOLIUM. Quantitatum fiuentium fluxiones effe primas 9 fecundas, tertias, quartas 9 aliafq; diximus fiipra. Hce liuxiones funt ut termini ferierum infinita- rum convergentium. lie fi zM fit quantitas fluens & fluendo evadat z-|-o|«9 deinde refolvatur in feriem convergentem zM'j'HOzrI-^~ooz n3-^^±^o^zyr3 c. terminus primus hujus feriei z” erit quan¬ titas ilia fluens9 lecundus «ozrI erit ejus incremen- tum primum feu differentia prima cui nafeenti pro- portionalis eft ejus fluxio prima 9 tertius — ozr2 erit ejus inerementum fecundum feu differentia fe- cunda cui nafeenti proportionalis eft ejus fluxio fecunda, quartus ~~ 3 ^ o3zK'3 erit ejus increment turn tertium feu differentia tertia cui nafeenti fluxio tertia proportionalis eft. & fic deinceps in infinitum. Exponi r [ 208 ] Exponi autem poffunt hsefluxiones per Curvarum Ordinatas BD, BE, BE, BG, BH, &c. Ut fi Ordinata BE (— fit quantitas fluens , erit ejus fluxio prima ut ordinata B D, Si BF (=^~) fit quantitas fluens, erit ejus fluxio prima ut Or¬ dinata BE & fluxio iecunda ut Ordinata BD. Si B H (— fit quantitas fluens, erunt ejus fluxio- nes, prima, fecunda, tertia & quarta, ut Ordinate BG, BF, BE, BDrefpeaive. Et hinc in aequationibus quas quantitates tantum duas incognitas involvunt, quarum una eft quan¬ titas uniformiter fluens & altera eft fluxio quslibet quantitatis alterius fluentis , inveniri poteft fluens ilia altera per quadraturam Curvarum. Exponatur enina fluxio ejus per Ordinatam BD, & fi hcec fit fluxio prima, quaeratur area ADB=BExi5 fi fluxio iecunda, quaeratur area AEB = BFx i, fi fluxio tertia, quaeratur area AFB — BGxq&c & area inventa erit exponens fluentis quaeiitse. Sed & in aequationibus quae ftuentem & ejus fluxionem primam fine altera fluente , vel duas ejufdem fluentis fluxiones, primam & fecundam, vel fecundam & tertiam, vel tertiam & quartam, See. fine alter utra fluente involvunt : inveniri pofi font fluen tes per quadraturam Curvarum, Sit aequatio aav = av A- vv , exiftente v = B E , v =BD, z =AB & z = i, & aequatio ilia com- plendo dimenfiones fluxionum, evadet aav = avz feu jtxt? ==Z. Jam float v uniformiter & fit [ 209 ] fit ejus fluxio v=i & erit-q— = z, & quadrando Curvam cujus Ordinata eft & Abfcifla v, ha-= bebitur ftuens z. Adhsec fit aequatio aav=av-j-vv exiftente v=BF, v=BE, v=BD & z— AB & per relationem inter v & v feu B D & B E invenie- tur relatio inter A B & B E ut in exemplo fuperiore. Deinde per hanc rektionem invenietur relatio in¬ ter A B & BF quadrando Curvam AEB* iEquationes quae tres incognitas quantitates invol- vunt aliquando reduci poflunt ad aequationes quae duas tantum involvunt, & in his cafibus fluentes invenientur ex fluxionibus ut fupra. Sit aequatio a— -bxm=cxy«y +dy 2«yy. Ponatur y«y=v & erit a— bxracxv~}-dvv, Hxc aequatio quadrando Cur¬ vam cujus Abiciffa eft x & Ordinata v dat aream v5 & aequatio altera y«y=v regrediendo ad fluentes dat =v. Unde habetur fluens y. Quinetiam in aequationibus quse tres incognitas involvunt & ad aequationes quae duas tantum in- volvunt reduci non poflunt , fluentes quandoq; prodeunt per quadraturam Cur varum. Sit aequatio a xm -j- b xf — rexr’rys sexry j w — f y y\ exiftente x = i . Et pars pofterior r e x^1 ys -j- sex^y ys~l— f y y£, regrediendo ad fluentes, fit exrys — - JL qU^ proinde eft ut area Curvae cujus Abfcifla eft x & Ordinata axm -] i xlj^ & inde datur fluens y. E e e Sit C 2FO J _ _ , 9 Sit aequatio x * axraT bx"|p =— err:1 Et fluens cujus fluxio eft x xaxm ft- bx"|p erit ait area Curv^e cujus Abfcifla eft x & Ordinata eft a xm ft- bx«lp. Item fluens cuius fluxio eft dyYrI- erit ut area Curvs v'e+fyo cuius Abfcifla eft y & Ordinata -yyi’— id eft (per Cafum i- Formae quartae Tab. I.) ut area Ve-f fy”- Pone ergo^Ve-j-fy" aequalem areae Curvae cujus Abfcifla eft x & Ordinata ax® -\- b x"^ Sc habebitur fluens y, Et nota quod fluens omnis quae ex fluxione prima colligitur augeri poteft vel minui quantitate quavis non fluente. Qua: ex fluxione fecunda colligitur augeri poteft vel minui quantitate quavis cujus fluxio fecunda nulla eft. Quae ex fluxione tertia colligitur augeri poteft vel minui quantitate quavis cujus fluxio tertia nulla eft. Et fic deinceps in in¬ finitum, Poftquam vero- fluentes ex fluxionibus col left?: £unt?, fi de veritate Condufionis dubitatur, , fluxio- nes fluentium inventarum viciffim colligendae funt Sc cum fluxionibus fub initio propofitis comparand^ Mam fi' prodeunt aequales Conclufio rede fe ha- bet [ 21 1 1 Bet : fin minus , corrigenda? funt fiuentes fic , ut earum fluxiones fiuxionibus fub initio propofitis aequentur. Nam & Fluens pro lubitu aflfumi po- teft & affumptio corrigi ponendo fluxionem flu- entis affumpt® cequalem fluxioni propofitae, & ter- minos homologos inter fe comparand©* Et his principiis via ad majora fternitur- IT** i\j •■d ERRATA. BOOK I. Of Oftick*. PArt i. p.3. 1.20. Properties which, ib-p- 5- 1.$* and that C, p .6. I.9. DE,^ p.2i. I.23. are two Pays, p.27. 1.6. in the Margin put Fig. 14 ££ 15, p.30.1.7. Mat, I.9, M, p. 44. 1.15. aswttfpropofed, 11.52. 1. 17. a paper Circle, p.57. l.uit. emerging, p,6o. I.25. contain with the, p.64. 1.18. and 14th, p.65. i.13. at the, p.66. [^.Semicircular, p .67. I.25. Center, I.31. ^ Inches, p.68. 1.8. to 16, 1 .9. or p.71.1.1. bife&9 p.72.1.13. falls , I.20. being. Part II. p.86. 1.5. lelopipede, p.89. 1.9. made by, p.93. l.i 8. to 77^ 1.28,29, by the third Axiom of the firft Pan of this Book , the Laws, p.105. I.5. fee repre- femei, p. 144. 1. 244, b, r«, 5. P- for Lib.i. Ub.l. write Ttmi.Pam. p.122. 1 .^.iniico, jj.130.la9.ro the Jttgle, p.132. l.S. by tie bright- vcfs, p.1-3-5. 1.14. Ferlf in the, l.i6.firfl Fanyou, p.136. [.26. firfi Part, [.27. lights, p.137. 1.20. green, accordingly an, i^. 138. 1. 21. Prop.6. Part. 2. p.139. I.5. on which , ^.i^.l.ij.XTwhichhave been, p.143* 1*7 •purple, \.i6.feveral Lights, 1.2^. of white. BOOK. II. P.5. 1. 5 .nicely the, p. 7. 1.9.^, x denote, I.28. them divers ,p, 10. 1. 24. xooo to 1024, p.ii.l.n. olicpiities, I. p.17. 1*4* Hf P* 25. 1. 11. 10^ p»3i. !• 12. more com¬ pounded, p.55. 1 .3. foes re fie ft, I.24. and therefore their Colours arife, p.65. I.5. cqrpuf^ eles can , p.71. 1.17. given breadth, p.84* 1* 4* a.re tothofe, p. 96. 1. 24. Ohfervation of this Pan of this Book, p.103. 1.17. was to thelhickhefs, p. 105. 1. 19. of this white Ping, p.107. 1.20. become equal to the third of thofe. Enumeratio Line arum. p.143. 1.20. datas [ignis fm, p. 144. 1.27. refipiciunt, p. 146° 1 .^fmtJfympmo, p. 154. 1.13. cx-\-d dat Qrdinatamy =5 » 1.14* qux gener atur . Ouadratura Curvarum. p.x68.1.24.re^i ^Bjp.r/^.l.ult. J fz«, p.183. 1. 1 q.a,b, c,tic. e, fig, tic. k,l, m* ‘tic. p. 18$. I.4. in 2,9-i> p.188. 1.14. z9dr « ■ . ;* .. - ,v ■ - -- \ . v / ■' ' “ ' ' > li.-.vk:-' ‘ ■* . & ••*?. ’ 7 1 i . v- - - '