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LECTURES 


Read in the 


PUBLICK SCHOOLS 


OF THE 


Univerſity of CAMBRIDGE, 
Anno Domini, 1669. 


By the late Sir ISAAC NEWTON, 


Then Lucaſian Profeſſor of the Mathematicks, 


r 


22 — — 


— 


Never before Printed. 
Tranſlated into Ergiiſh out of the Original Latin. 


__ 


— 


— — 


LO ND 0 N: 


Printed for Fxancis FayRraAn, at the 
South Entrance of the Royal Exchange. 


M. DCC. xxVIII. 


AIRS 


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1 


PREFACE. 


T was as long ago as the Tear 1566, 
when Sir Iſaac Newton fr /t pom 

out his Theory of Light and Colours. 
Upon Dr. Barrow's reſigning to him 
the Profeſſorſhip of the Mathematichs 
at Cambridge, he made A. 1669, this 
Diſcovery the Subject of his publick 
Lectures, in that Univerſity. In 1671 
he began to communicate it to the 
World, as alſo a Deſcription of his 
Reflecting Teleſcope, in the Philoſophi- 
cal Tranſactions. About the ſame time 
he intended to publiſh his Optical Lec- 
tures, wherein theſe Matters were hand- 
led more fully; together with a Trea- 
tiſe of Series and Fluxions. Hut the 
Diſputes, which were occaſioned, by 
what he had already ſuffered to come a- 
A 2 broad, 


o*s 


111 


iv The PREFACE. 


broad, deterred him from that Deſign. 
Aud hence he conceived ſo great au 
Horror for any thing, that looked like 
Controverſy, that the conſtant Importu- 
} nities of his Friends could not prevail 
upon him to print his Book of Opticks 
until the Tear 1704. As to his Lec- 
; tures, they were depoſited, at the time 
they were read, among ſi the Archives 
of the Univerſity. From whence many ' Þ 
Copies haue been taken, and handed a- 
bout by the Curious in theſe Matters. 


Theſe Lectures are divided into two 
Sets or Parts, What is treated of in 
the laſt Part, relates to the Doctrine 
of Colours, aud was left imper fest; 
but has been ſince publiſhed in the 
Opticks by Sir Iſaac himſelf with great 
Improvements. The firſt Part is com- 
pleat and preparatory to the other : 
And as it contains but little in common, 
with what has been already printed, we 

baue thought fit to make it now publick, 
The Reader will find in it Abundance of 
Particulars worthy their great Au- 
thor, and ſuch as will even at pre- 
ſent appear entirely new, It is divi- 


| 
| 2 ded 


The PR E FA CE. 


ded into four Sections. What is deli- 
vered in the two firſt Sections may be 
immediately ſeen in the Book itſelf; 
for there the Author has put in the 
Margin the Contents againſt their pro- 
per Places. But a ſhort View of the 
Whole tale, as follows. „ The firſt 
Section gives us avery full and plain 
Account of the different Refrangibility 
in the Rays of Light, with the Expe- 
riments from whence it was deduced ; 
and, amongſt many other curious things 
relating thereto, an elegant Demonſtra- 
tion of i.e Caſe, where the Image of 
the Sun made through a Priſm would 
be circular, provided the common Opi- 
nion of Refrattions was true. The 
3 Subjett of the ſecond Section 1s the 
3 Meaſure of Refrattions in tranſparent 
= Subſtances, as well Fluid as Solid, and 
XZ zhe comparing the Refrattions of hete- 
rogeneal Rays; and theſe are perform- 
ed not only in Mediums as contiguons to 
the Air, but when contiguous to one a- 
nother ; all which are illuſtrated by a 
Deſcription of the Inſtruments for ma- 
king the Experiments, and by Exam- 
A 3 ples, 


. — CC CORTIIITTRTSS 


The PR E F A CE. 


ples, together with ſuitable Demon. 
ſtrations. | 


The other two Seftions are in a man- 
ner purely Geometrical. In the firſt 
of them the Effects of the Refrattions 


of Rays arg conſidered, as they are 


incident upon one or two plane Surfa- 


ces. The firſt nineteen Propoſitions re- 
late to the Refratiions made by a ſingle 
Plane. Of theſe the firſt eight treat 
F homogeneal Rays; containing ſome 
of the Principles of Dioptricks. In 
the Scholium to the eighth Propoſi— 
tion our Author has a curious Specu- 
lation concerning the apparent F lace 
of the Image of an Object ſeen by Re- 
fraction. The reſt of theſe Propoli- 
tions are about the Divarications and 
Limits, of heterogeneal Rays; as they 
are refracted at a Surface ſeparating 
two Mediums, whoſe Denſities are con- 
ſidered either as permanent, or the 
Denſity of any one of the Mediums 
is ſuppoſed to be varied; amongſt 
which occurs, at the Concluſion of 
Prop. XII. what is very remark..ble, 
that, iu Rays of every Sort refratt- 
2 ed 


The PREFACE, 


ed at the ſame Point of a plane Sur- 
face, the Locus of the Centers of their 
Radiations, is the vulgar Cifloid. From 


thence to the End of this Seftion is 
concerning the Affectious of both homo- 
geneal and heterogeneal Rays refratt- 
ed by two Planes 5 which chiefly have 
Relation to the Experiments of the 
Priſm, from whence our Author de- 
duced his Theory of Light and Co- 


lours. And here is ſhewn, at Prop. XX. 


and XXI. that, if Rays diverge to 4 
Priſm, the homogeneal ones, after the 
double Refrattion, will ſtill continue 
to diverge, but ſome of the heteroge- 
neal ones will converge 3 therefore at 
Prop. XXII. that of Rays fo refrac- 
ted from an Object to the Eye, ſome 
' will fall gradually nearer to the Ver. 
tex of the Priſm than others, as they 
are more and more refrangible , whence 
from Prop. X. are defined the Orders 
of Colours in the Image made by Re- 
fraction; at Prop. XXIII. XXIV. that 
the bigger the vertical Angle of the 
Priſm, or the denſer its Matter is, the 
Difference of the Refraction will be (0 
much the greater, whence the Colours 


A 4 an 


The PREFACE 
in the Image will become the more ma- 
nifeſt ; at Prop. XXV. XXVI. that 
the Rays ſo falling on the Priſm, that 

the Nefraction on each Side may be 
equal, in homogeneal Rays, the Angle, 
which the incident and emerging Rays 
comprehend, will then become the great- 
eſt ; but in heterogeneal Rays the Dif- 
ference of thoſe Angles will become at 


that time the leaſt. And at the laſt 
Propoſition our Author ſets down a 


Mechanical Solution of the following 
Problem; Rays being refracted from 
one given Point to another given Point 
by a Priſm given in Poſition ; to find 
the Angles comprehended by the hetero- 
geneal Rays. He ſays, to perform this 
geometrically, would require ſuch a 
Conſtruction, as the Ancients called 
Linear, or what could not be effected 
by the Help of the Conicks. 


The laſt Section regards Rays, as 
they are refratted by curve Surfaces. 
Its chief Contents are, at Prop. XXIX. 
XXX. XXXII. XXXIII. the finding 
both the principal Focus, and al- 
ſo that of every particular Ray, not 


only 


The PREFACE. ix - 


only in Spheres, but in any curve Sur- 
face whatever 3 at Prop. XXXI. the 
computing the Errors ariſing from 
XZ the Figures of Optical Glaſſes; at 
Prop. XXXIV. the Invention of ſuch 
Curves, as will accurately refract the 
Rays of Light to any given Focus; 
and at Prop. XXXV. XXXVI. he 
determining the Rain-bow. In all this 
Section our Author makes no men- 
tion of heterogeneal Rays, until he 
comes to the laſt Propoſition, wherein 
he determines the Errors cauſed b the 
different Refrangibility of the Rays of 
Light. And from this Propoſition 
compared with the thirty firſt he dedu- 
ces this Concluſion, that the Imperfec- 
tion of Optical Inſtruments is not ow- 
ing, as has been all along thought, to 
the Unfitneſs in the Figure of the 
Glaſſes, but to the different Refrangi- 
bility in the Rays of Light. This 
Conſideration put our Author upon the 
noble Invention of the Reflecting Te- 
leicope; 4 very particular Account of 
which is given in his Opticks, This 
Inſtrument has been lately made to a 
greater Length, with a very curious 


C 0 


The PR E FA CE. 


Contrivance for managing it by that 
moſt ingenious Gentleman John Hadley, 
Eſa; A Deſeription whereof is pub- 
liſhed in the Philoſophical Tranſactions 
No 376. 


In the Preface of the learned Dr. 
Barrow's Lectiones Opticæ printed in 
1669, there is given a great Character 
of Sir Iſaac Newton, at a Time when 
he was altogether unknown to the World. 
And the Dr. :s ſo candid as farther 
to declare, that he there altered ſeue- 
ral Things upon his Advice, and inſert- 
ed ſome of his Inventions, as an addi- 
tional Ornament to his own. What Dr. 
Barrow publiſhed then of our Au- 
thors without their Proofs, the Reader 
will find demonſtrated in the follow- 
ing Diſcourſe. But it is needleſs to 
cite at this time of Day any Teſtimo- 
nies to our Author®s Praiſe, in order to 
recommend what ſhall come from One, 
who has acquired ſo univerſal a Repu- 
tation. It will be ſufficient, that the 
preſent Treatiſe is a faithful Tranſla- 
tion of a very correct Copy, taken from 
the Latin Original, as it was read in 


1669. 


The PREFACE. 

169. We have put it into the Lan- 
guage, Sir Iſaac Newton himſelf choſe 
to make Uſe of in his Opticks ; and at 
X the Bottom of the Pages we have ad- 
X ded here and there ſome very ſhort Re- 
marks, which, we preſume, will not be 
altogether uſeleſs to ſuch, as are not 
throughly informed of theſe Matters. 
By this our Labour we queſtion not, 
but we ſhall merit the Thanks of all, 
who are curious in theſe Speculations. 
And we ſhall moreover proceed in the 
ame manner to deſerve well at their 
Hands ; for we hope ſhortly to preſent 
the'Publick with ſeveral Mathematical 
Pieces, that were long ago written by 
our great Author, though never yet 
printed; whereby will be given ſtill 
farther Proofs, how early that Genius 
exerted it ſelf, which was at length able 
to produce thoſe divine Works, the 
PRINCIrIA and the OrTiCKs. 


London, June 29. 
1 7 2 7. , 


OPTICES 


* Ws enn 
a Wont 5 OWE 


— — an 


— 


* . r "7 
r n 
_ — —— 


1. 3. put Fig. 51 inthe Margin. p 166. l. 15. put Fig. $-in the 


ERRATA 


AG E 8. Line 20. put Fig. 3. in the Margio, p. 86. J. 23. 

read Fe and Pe. p. 88. I. 22. r. RefraQion, GX will be the 
Sine of Incidence and K L the Sive of Reſraction. p. 94. l. 14. . 
of. p. 112. J. 19. ſor ED G r. DG E. p. 117. l. 1. r. Radiation 
of the Riy RO. I. 7. r. diſſimilar, p- 130 1, 9. dele the Comma 
after D K. p. 146. 1, ult. r. and into. p 147. 1. 17. for Denſity r. 
Thickneſs, p. 153. 1. 7. r. Perpendicular I G. p. 155. I. 19. r. 
Refraction. p. 164. I. 4 r. FO R. l. 17. r. 286 E p. 165. 


Margin. 1, 22, r. and farther. p. 175. in the Citation l. I. r. C K;: 
p. 178. in Note 4. for convex Glaſſes r. Lens. p. 182. 1. 14. 
put a Point after aa. p. 183. I. J. r. IK — II. I. pen. r. N Gg. 
p- 184. |. 19. for g n. r. ng. p. 185. in the Citation 1.5. T. 1663 
and 1665. P. 136. I. 7. for 1 K 7. G K. l. 8. ſor R 1 _—_— 7. A. 
p. 195. I. 9. tor Incidence r. Re fraction. 1. 10. for Retraction r. 
Incidence. p. 202. l. 1. dele is. p. 203. l. antep. r. BC 4 · 
P. 204. 1. 3. dele the Point atter N. n 


NB. The Numbers in the Computation at Pages 209 and 210 
are erroneous, occalioned by C ſmall Miſtake made 1n Prop. XXXI. | 


For the Value of P Q is not 26 there ſet | down, dut 
RR y | l 


41 24 
+ This did not proceed from any Fault in our Author's 
41123. 2 2 | 
Method of — but only from his pucting in Coro. 6. at 
Page 196, N —1 the Value of C F inſtead of — the Value of - 
BF, from which it differs only by a Letter. This being re&ified 
the Value of PQ will come out, as I have here made it. For in thac 


2 . I a R R y L ; IN 4 4 

Corot. it will be V 2: — "= 
k<17 *'grTfRaBgia (IA KFT). 

R R y ? : RR * ? 9 


71177 which is equal to o Q, the double whereof FITC. 


will be the Value of PQ, I made no Alteration of the Num- 
bers inthe Nanuſcript ; becauſe the Reader may find a true Com- - 
putarton from more accurate Obſervations of the Laws of Nefrao- = 
tions in 0ur Author's Opticks Prop. VH. Book I. Part. I. 3 


o PTICERS 


Wor the Nefractiuns of the 
Rars of Licur. 


* 


SECTION I. 
17 he ann of Rays is dif- 


ferent. 
— HE late Invention of Telef- 
copes has ſo exerciſed moſt of 
the Geometers, that they ſeem 
1 to have left nothing unat- 
1 —.— in Opticks, no room for farther 
Wrnprovements. And beſides, * ſince the 
Diſeertationt, which you have not long 
1 ſince heard from this Place, were com- 
3 poſed with ſo great a Variety of optical 


* Viz. Dr.Barrow's Leflionts Optica, 
B Mat- 


- 


The Refrangibility Sxcr. I. 
Matters, ſuch Plenty of new Diſcove 
ries, and that confirmed by moſt accu- 
rate Demonſtrations; it may ſeem a vain 
Endeavour and an uſeleſs Labour, if I 
ſhall again undertake the handling this 
Science. But ſince I obſerve the Geo- 
meters hitherto miſtaking ina particular 
Property of Light, that belongs to its 
Refra&ions, tacitly founding their De- 
monſtrations on a certain Phyſical Hy- 
potheſis not well eſtabliſhed; I judge it 
will not be unacceptable, if I bring the 
Principles of this Science to a more ſtrict i 
Examination, and ſubjoin, what I have 
diſcovered in theſe Matters, and found | 
to be true by manifold Experience, to 
what my reverend Predeceſſor has laſt | 
delivered from this Place. 7 


Trey, that have been converſant in 
Dioptricks, imagine, that Optical Inſtru- 
ments may be brought to any Degree 
of Perfection, provided they were able 
to communicate to the Glaſſes in grind- 
ing, what Geometrical Figure they plea- 
ſed; and to that Purpoſe various me- 
chanical Contrivances have been thought 
of, whereby Glaſſes might be ground 

in⸗ 


SECT. I. of Rays V different. 


into hyperbolical or even parabolical 
Figures; but no Body has yet ſucceeded 
in the exact Deſcription of ſuch Figures: 
The Labour has been in vain; and 
that they may no longer take Pains 
in a fruitleſs Inquiry, I durſt promiſe 
them, that alrhough every thing ſhould 
ſucceed happily ; it would, nevertheleſs, 
not at all anſwer their ExpeQations. 
For Glaſſes, tho? they were formed into 
Figures, the beſt that could be deviſed 
for that End, yet they would not per- 
form twice as well as ſpherical ones, 
ground with the ſame Exactneſs. I do 
not ſay this, as if I contended, that the 
Writers in Opticks were to blame: For 
all they have advanced is accurate, and 
moſt true with regard to the Intention 
of their Demonſtrations ; but however 
they have left ſomething, and that of 
the greateſt Moment to be diſcovered 
by their Succeſſors; as in Refractions I 
find a certain Irregularity, that diſturbs 


all things; and not only cauſes, that the 
Figures of the conick Sections do not 


much ſurpaſs ſpherical ones, but alſo that 
ſpherical ones perform much leſs, than 
they would, if the ſaid Refraction was 
uniform. B 2 1 


|| 


K — — — 
— - — 4 — — 
r . eee ii Gl En — — — _ - * 
— — — — 1— 
— — — - 
, 


The Refrangibility Sect. I. 


I ſhall therefore treat of Dioptricks, 
not that IT intend to handle that Subject 
entirely, but only that I may firſt ſearch 
out this Property in the Nature of 
Light, then ſhew how much the Perfec- 
tion of Dioptricks is impeded by this 


Property, and by what Means that In- 


convenience, as far as the Nature of the 
thing will permit, may be avoided, 
Where I ſhall produce ſomething, that 
regards as well the * Theory, as the 
Praxis both of Teleſcopes and Microſ- 3 
copes: b Shewing that the greateſt Per- 
fection of Opticks is to be ſought, 
contrary to the common Opinion,/from 
Dioptricks and Catoptricks joined toge- 
ther. And in the mean while ] ſhall 
largely explain the Difference of Colours 


a This is not deliver'd in any of our Author's Ter- 
tures ; but for which ſee Huygen's Dioptricks, printed 
firſt in 4 at Leyden, A. 1703. amongſt his Opuſcula 
Poſt huma. 

b In our Author's refle&ing Teleſcope, firſt publiſhed 
in the Philoſophical Tranſactions No 81. and afterwards 
in his Opticks. 


© What is here promiſed is moſt wonderfully perform. 
ed in bis Opticks. 


= and 


SECT. I. of Ravs 2s different. 


and alſo from coloured Bodies. 


another. Of thoſe, that have all the 
ſame Angle of Incidence, ſome will 
have their Angle of Refraction ſome- 
what greater, others will have it ſome- 
what leſs. For a farther Illuſtration of 
this, let EFG (in Fig. I.) be any re- 
fracting Surface, ſuppoſe of Glaſs, and 
let any Line O F be drawn meeting it in 
E, and making with it the Angle OF 
E acute; conceive alſo the ſolar Rays 
to flow through that Line O E, conti- 
nually ſucceſſive to one another, ſo that 
they may one after another impinge on 
the Point E, and there be refracted into 
a denſer Medium; or, if you had rather, 
ſuppoſe parallel Rays to be indefinitely 
little diſtant from O F, and to fall on 
the Points the neareſt to F. Now from 
the commonly received Opinion, thoſe 
Rays having the ſame Incidence ought 
alſo to have all the ſame Refraction; as 
ſuppoſe into the Line FR. But I have 


and their Geneſis both from Priſms, 


B3 diſ- 


88 Light I have diſco- 1 All Ra 


/ 
vered, that its Rays in reſpect to the — Refrane 


Quantity of Refraction differ from one gibility. 


Fig. 1. 


6 


The Refrangibility Sxcr. I. 


diſcovered the contrary, vi. that after 
they are refracted, they diverge from one 
another, as if ſome were refracted into 
the Line F P, others into the Line F 
Q and others into the Lines F R, F 
S and F T, and innumerable others alſo 
through the intermediate Spaces; and 
ſome wandering on one Side and on the 
other, according as any Ray is ſuited 
to ſuffer a greater or leſs Refraction. I 
moreover find, that the Rays F P re- 


fracted the moſt produce purple Colours, : 
and thoſeF T the leaſt refracted produce 


red Colours; but theſe FQ, F R, FS, 


that proceed intermediate to thoſe, ge- 


4 
- 


nerate the intermediate Colours, viz. 
blue, green and yellow; and fo the Rays, 


more and more refracted than others, do 


generate theſe Colours in order, red, 


yellow, green, blue, and purple, toge- 


ther with all the intermediate ones, that 
may be ſeen in the Rain- bow; whence 
will eaſily appear the Production of Co- 


lours in a Priſm and the Rain- bow. But 
. theſe things being ſlightly obſerved, 1 
 Thall defer to hereafter, what J have to 4 


fay of Colours, 
Oux 


©E 
= * 
b 


as they are ſuited, that ſome may be 


Sker. I. Of Rays 2s different, 7 


9 Oux Opinion in this Matter being ener: 
thus briefly explained, that you may not gar Experi- 
think we have declared to you Fables in- — 


; gth of 
ſtead of Truth, we ſhall immediately ä 


produce the Reaſons and Experiments ſelar Image. 


on which theſe things are founded: And 
becauſe a certain very obvious Experi- 
ment of the Priſm gave me firſt an Oc- 
caſion of diſcovering the reſt, I ſhall firſt 
explain that. Let (in Fig. 2.) F be any Fs. 2. 
Hole in the Wall or Shut of a Chamber 
Window, through which the ſolar Rays 
O F may be tranſmitted, all other Ave- 
nues being every where carefully clo- 
FF fed, left the Light may enter elſewhere; 
this darkening of the Room is not neceſ- 
ſary, but only cauſes, that the Experi- 
ment may be ſomething more evident. 
XZ Then let be applied to that Hole the 
XZ triangular glaſs Prim A 2 B Cc, 
that may refract the Rays O F tranſ- 
8 mitted through it, towards PV T Z, 
where you will ſee formed a very ob- 
1 long Image, whoſe Length P T is four 
times its Breadth Y Z, and above. And 
hence it ſeems certainly to be evinced, 
that of the equally incident Rays, ſome 
B 4 ſuffer 


The Refrangibility Sect. I. 


ſuffer a greater RefraQtion than others ; 


for if the contrary was true, the ſaid I- 


mage of the Sun would appear nearly 


circular, and in a certain Poſition of the 
Priſm would be ſeen altogether as to 
Senſe circular, which is repugnant to all 
Experience; for in whatever Situation I 
diſpoſed the Priſm, I could never yet 
bring about, but that the Length of the 
Image was more than Quadruple its 
Breadth, the Angle of the Priſm A C B 
or 4c being about 60 Degrees, 


4. ACaſe Bur that there is a certain Poſition 


wherein the 
equally re- 


frangible 
Rays make 
A circular I. 
mage. 


of the Priſm, in which the Image of the 
Sun according to the received Opinion 
about Refractions would appear circu- 
lar, I thus ſhew. Near the Hole made in 


the Window-ſhut, let be placed abroad 


a Priſm, or what comes tothe fame thing, 
let E G be an opake Body placed/ on 
this Side the Priſm, in which let F be 


an indifferently ſmall and round Hole, 


a Plane cutting the refracting Planes 1 
11 | gC 


through which the refracted Rays may 


be tranſmitted to the directly oppoſite 


Wall to paint there the Image PIT 
Z, and let A B C be ſuppoſed to be 


YN 
*f 
"i 


SCT. I. f Ravs 1s different. 


2 C and 5 C perpendicularly, and al- 
ſo paſſing through the Hole E, as like- 


wiſe through the Center of the Sun D 


I H V, which let it biſect according 
to its Diameter DH, from whoſe Extre- 
mities let proceed the Rays D K and H 
N, lying in the ſame Plane, which after 
they are refracted, D K into K # and # 


T, and H N into N & and &, they 


both may go forward through the Cen- 


ter of the Hole F; and beſides, let there 


be ſuch an Inclination of the Priſm to 
thoſe Rays, that the Angles A K D and 
B * F may become equal, Then let 
I V be another Diameter of the Sun 
perpendicular to the faid Plane A B C, 
from whoſe Extremities let two other 
Rays V L and IM proceed, one IM on 
this Side the Plane A B C, which may 
be refracted into M and / Y, but the 
other V L, beyond that Plane, which 
may be refracted into L and Z, and 
let all the ſaid four Rays interſect one a- 
nother in the middle of the Hole F; 
laſtly let it be ſuppoſed, that the lucid 
Image PY T Z does directly reſpect 
the Hole, ſo that F P and F T, alſo 
FY and FZ may be equal. I now fay, 

that 


monſtration 
of this Caſe, 
its 1ſt Part. 


5. A De- 


The Refrangibility Sect. I. 


that in ſuch a Poſition of the Priſm the 
AnglesP F T andY F Z are equal, 


on a Suppoſition that all the Rays are 


equally refracted, that have the ſame 
Angle of Incidence, and therefore that 
this Image at leaſt to. Senſe ought to be 
circular, as whoſe Diameters P T and Y 
Z interſect one another perpendicularly, 
and ſubtend thoſe equal Angles. 


Bu r that thoſe Angles PF Tand Y Þ 
F are equal, I thus demonſtrate. Con-„t 
ceive any Ray to go backwards from P 
through & and N, whilſt another Ray 
proceeds from D through K and =; 
therefore ſince the Angles A K D and B 
k F are ſuppoſed equal, the Angles A K 
2 and B N made by the firſt Refracti- 
ons, will be alſo equal, whence the Tri- 
angles CK and C& N, will be ſimilar, 
and their external Angles ENA, K B 
equal, and conſequently the Angles made 
by the ſecond Refractions A NH and 


BA F are equal. Wherefore ſince the 


Angles A K D and B & E, alſo A NH 
and Bu F are equal, their Differences 
will be alſo equal, that is, the Angle 
F & or PF I equal to the Angle, _ 

eng 


Sect. I. of Rays is different. 


the Rays D K and HN comprehend, or 
to the Sun's apparent Diameter. 'The 
Angle PF T therefore is equal to the 
Sun's Diameter. Wherefore ſince it ſhall 
be moreover demonſtrated, that the An- 
ole Y F is equal to the aid Diameter, 
the Propoſition will be manifeſt. But 
in order to do that, a certain Theo- 
rem muſt be ſet down by way of 
Lemma, | 


LET there be (in Eig. 4.) two Planes 6. A Zem- 
ma to the 2d 
Part. Fig. 4+ 


AB C D and EF GH perpendicular 
to one another, whoſe common Inter- 
ſection let be KL, and let IP be any 


Ray, which falling on the Plane AB T 


D at the Point P is refracted by it into 
P R; I fay, that the Sine of the Angle, 
which the incident Ray IP makes with 
the perpendicular Plane F H, is to the 
Sine of the Angle, which the refracted 
Ray P R makes with the ſame Plane, as 
the Sine of Incidence to the Sine of Re- 


fraction, and conſequently in a given 
Ratio. For the Rays I P and PR being 


taken equal, and I Q and R V being let 
fall perpendicular to the Plane FH; and 
moreover at the Point of Incidence the 

Line 


Ix 


The Refrangibility sxcr. I. 


Line SP T being drawn perpendicular Þ 
to the refracting Plane B D, (which 
therefore coincides with the other Plane 

FH) and IS and R T being let fall 
perpendicular to 8 PT: The Angle IP 

Q will be that, which the incident Ray 

IP makes with the perpendicular Plane 

FH, and RP V the Angle, which the 
refracted Ray PR makes with the ſame 

Plane. Alſo IPS is the Angle of Inci- 

dence, and R PT the Angle of Refracti- 

on. Wherefore if IP or P R be ſuppoſed 

the Radius of a Circle, IQ, RV, IS 

and R T will be the Sines of the ſaid 

* 6,111 Angles. But I Q and RV are parallel, 
by reaſon they are perpendicular to the 
ſame Plane F H. Alſo IS and RT are 
1 * I. N b, becauſe lying in the Plane 
Wl S PT R they inſiſt perpendicularly on 
the ſame right Line 8 T. That is, the 

right Lines IQ, IS, which comprehend 

the Angle QIS, are parallel to the right 

Lines RV, RT, comprehending the 

Angle VRT. Wherefore thoſe Angles 

«10, 1 QISand VRT are equals, But Q8 
— and V T being drawn, the Angles I QS, 
«4 Def. 3. and RV T will become right ones; 4 
31. Elem becauſe the right Lines I Q and R V 
inſiſt 


Sgcr. I. Of Ravs it different. 


inſiſt perpendicularly on the Plane F H. 
Therefore the Triangles I Q S and 
RV T areſimilar,* and TQ.RV::TS. 
R T, that is, the Sines of the Angles, 
which the incident and refracted Rays 
make with any Plane F H perpendicular 
to the refracting Plane B D, are as the 
Sines of Incidence and Refraction, and 
conſequently in a given Ratio. For 
that the Ratio of theſe Sines is given, 
* Cartes has taught, and others have 
ſince experienced. 


Mok OVER the Truth of the Theo- 
rem now demonſtrated will remain 
good, though the Plane E G inſiſts 
perpendicular to the refracting Plane 
B D elſewhere, than at the refracting 
Point P. For from thence neither the 
Angles made with the Rays and Plane 
FH; nor therefore the Sines of thoſe 
Angles will be altered. 


* At this Time I ſuppoſe our Author thought Car- 
fes to be the Diſcoverer of this Proportion, which 
he deduced from a Manuſcript of Snellius, as was after. 
wards hinted by Sr. Iſaac himſelf in the Stholium | 
to Prop. 96. Lib. 1. of the Principia, and alſo by 
Huygens in his Dieptricks. | 65 


TazsE 


* 


4. 6. Elem. 


14 The Refrangibility Skcr. I. 


. end Pat Tuksx things being thus ſhewn, I 
P , now return to what was propoſed, v:z. 
to demonſtrate the Angle Y F Z (in 
Eg. 3. Fig. 3.) to be equal to the apparent 
Diameter of the Sun, and conſequently 
to the Angle PF T. From what was 
ſaid above it appears, that the Plane K 
D H N F x biſects the Angle contain ® 
ed by the Rays IM and V lying on 
each Side. Therefore ſeeing that Angle 
is equal to the Diameter of the Sun, 
the Angle, which one of the Rays, ſup⸗ 
poſe 1 M, makes with the ſaid Plane, 
will be equal to the Sun's Semidiameter, 
whoſe Sine let be A, and B the Sine 
of the Angle, which that Ray refracted 
M 7 makes with the ſame Plane. Now 
ſince that Plane is ſuppoſed perpendicu- 
lar to the refracting Plane A C of the 
Priſm, it will be from the foregoing 
Lemma, as the Sine A to the Sine B, 
ſo the Sine of Incidence to the Sine of 
Refraction out of a rarer Medium into 
a denſer Medium. Or on the contrary, 
as the Sine of Incidence to the Sine of 
Refraction from a denſer to a rarer Me- 


dium, ſo will be B to A. Wherefore 
ſince 


SkEcr. I. of Ray s is different. 
fince the ſaid Plane DH F is alſo per- 
pendicular to the other Plane B C of 
the Priſm, which refracts the Rays from 
a denſer into a rarer Medium, and more- 
over ſince B is ſuppoſed the Sine of the 
Angle, which the incident Ray MI 
makes with that perpendicular Plane D 
H F, A will be (by the preceding Lem- 
ma) the Sine of the Angle, which the 
refracted Ray / F makes with the ſame 
Plane D H E; but A is ſuppoſed the 
Sine of the Sun's Semidiameter ; there- 
fore that Angle, which the Ray #/ F 
makes with the Plane D H F, is equal 
to the Sun's Semidiameter, and its dou- 
ble F mor Y F Z equal to the Sun's 
Diameter: And ſince it was ſhewn a- 
bove, that the Angle P F T is equal 
to the ſame Diameter, thoſe two An- 
gles Y F Z and PF T will be equal. 
2. E. D. 


Now if the Plane Y F Z was per- 
pendicular to the Plane of the Image 
PYT Z as well as the Plane PF T, 
thoſe four Lines FP, FT, F Y and F 
Z, which comprehend equal Angles, 
would be all equal amongſt themſelves, 

and 


15 


| 


16 


* 


The Refrangibility Sxcœr. I. 


and conſequently the Subtenſes P T and 
1 Z would be alſo equal. But who 
carefully conſiders this Thing, will find 
the collateral Rays VLM F Z and IM 
ZF Y to be a little leſs refracted than 


the other two DK FT and HN + 


E P, and therefore the Plane Y F Z will 
decline a little more from the Ray FP 
than from F T, cutting the Line P T be- 
low its middle Point X, and ſo diva- 
ricating from the Perpendicular FX 
(which conceive to be drawn) will be 
ſomewhat oblique to the Plane of the 
Image PI I Z, and for that Cauſe 
the Lines F and F Z will be a little 
greater than FP and F T, and the Sub- 
tenſe Y Z a little greater than the Sub- 
tenſe PT. But I omit the Demonſtra- 
tion of this as being prolix, and not 
altogether neceſſary to my Purpoſe. 
For it matters not much, whether the 
Plane IF 2 is right to the Plane of 
the Image P Y T Z, or ſomething 
oblique, that is, whether Y Z is equal 
to, or greater than P T; it is ſuffici- 
ent, that it cannot be leſs. Lea, ſince 
by Reaſon of the Iſoceles Triangles 


P F T and I F 2 it is FP. F:: 
P T. 


Sect. I of RAV S 2s different. 17 


PT. Y Z, and F P and F X are 
very nearly equal, the Difference will 
be very ſmall between P T and Y Z, 
that as to Senſe they may be looked up- 
on as equal. 


Trex is therefore ſhewn a Caſe, . But in 
wherein the Length of the Solar Image, length of the 
tranſmitted through the Priſm, would be lune 8 


more than 
beheld equal to its Breadth; and confe- quadruple 


quently wherein that Image would ap- — 
pear as circular, provided the vulgar O- different Re- 
pinion was true: Moreover, although is evinced. 
the Poſition of the Priſm ſhould be o- 

therwiſe than I have deſcribed, ſuppo- 

ſing the Rays on each Side did not ſuft- 

fer a very unequal Refraction; yet the 

Figure of the Image would ſcarce on 

that Account be altered. Nor does it 

much ſignify, whether the opake Body 

E G, in which is bored the Hole F to 

tranſmit the Rays, be placed on this 

Side the Priſm or beyond it; nor is the 

Figure of the Hole to be much regarded, 

ſo it be but ſmall. For ſuch minute 
Variations will not alter the Image 

more, than perhaps a tenth or fifth 

Part of its Diameter, as will appear to 


one 


18 The Refrangibility Sxcr. I. 


one that conſiders. And ſo, that I may 
comprehend all in a few Words, it is 
manifeſt, that the refracted Image of 
the Sun ought for the moſt Part to be 
to Senſe, as if it were circular, provid- 
ed the Refraction of the ſame Incidence 
on the ſame Medium were to be al- 
ways the ſame. But the firſt is re- 
pugnant to Experience ; its Length 
exceeding, as it has been ſaid, its Breadth 

by more than four Times. Therefore 
the laſt is repugnant to Truth, and tlie 
Refraction of the ſame Incidence is 
different. 


Ty FROM the fame Experiment I could 
ſhorter ut;. here thus more briefly have ſhewa the 
on of the Propoſition, vi. ſince I ſo diſpoſed the 
fame Thing. Priſm, that the RefraQtion as well of the 

imerging as the emerging Rays, might 
be as it were equal, I meaſured (Fig. 2. 
or 3.) the Angles PFT, and YF Z, 
and found the Angle Y F Z, indeed, e- 
qual to halt a Degree or the Diameter 
of the Sun; but the Angle PF T ex- 
ceeded the fame Diameter four times, 
and above, to which however it ought 
to be equal, from the former 17 * 
2 O 


. 155 5 


Scr. I. of Rars ic different. 19 


of the foregoing Demonſtration; and 
thence the Propoſition moſt manifeſt- 
ly appears. But for the Sake of what 
will next follow, it ought to be de- 
monſtrated, that thoſe Rays, whoſe Re- 
frangibility is not unlike, will form an 
Image nearly circular, and on that Ac- 
count I have thought fit to deliver that 
Demonſtration, although it be ſome- 
what long, for the Illuſtration of this 
Experiment. 


Bur ſince in making the ſaid Experi- 10. By what 
eans the 


ments, I have ſuppoſed the Poſition of Prim may 
the Priſm to be ſuch, as that the Rays be eth Pla- 


may be equally refracted on both Sides * 


of the Priſm: I ſhall, for a Concluſion, — 
declare by what Means, that may be 
ſoon and eaſily done. If the Priſm be 
held in the Sun's Light, and by a gen- 
tle Motion turned about its Axis, you 
will ſee the Colours, which it makes, 
to be by a continual Motion tranſlated 
from Place to Place, in ſuch a manner, 
as ſometimes they will appear to aſcend, 
and then again deſcend, Obſerve there- 
fore the Middle between theſe contrary 
Motions, when the Colours now aſ- 


C 2 cending, 


* 


20 


11. The 


The Refrangibility Scr. I. 


cending, and preſently being about to 
deſcend, ſeem to be ſtationary, which 


when you ſee, ſtop the Priſm, and fix 
it in that Station. I fay it is done, 
for then the Sum of the Refractions 
made on both Sides, or the Inclination 
of the emerging Ray to the incident 
one, will become the leaſt of all. Which 
when it happens, the Refractions on 


both Sides are equal, as ſhall be demon- 
{trated hereafter.. 


IT is my Deſign now to proſecute the 


«#igureof the Various Circumſtances of this Experi- 


aid Im 
Heſcribe 
whic 


83 ment, not leſs pleaſant to the Experi- 


his com menter, than declarative of our Purpoſe. 
prehended 
partly by 
right Lines, 
partly by Se · 
micircles, 


And it is in the firſt Place to be remar- 
ked, that the Figure of this Image, ac- 
cording to its Length, was terminated 
by ſtraight Lines, and according to its 
Breadth by two (as well as I could 
judge by my Sight) Semicircles. In the 


5th Figure let P T be the Image of the 


Sun refracted by the Priſm; I obſerved 
this to be terminated on its Sides by 
two Lines AB and CD, ſtraight and 


parallel to one another, as to Senſe, 


but at the Extremities by two Semicir- 


cles 
2 In S1 r. III. Prop. xxv, 


Skcr. I. F Rays i different. 21 


cles APC, and B T D, the Cauſe 
whereof is thus determined, from what 
has been already declared. 


LET thoſe terminating Semicircles be 12. How 
| d i Circl 8 this ariſes 
comp eated into Circ CS, as you * In from the cir- 
the 6th Figure, and let another Circle on _ 
» . - - r 
Y Z be inſcribed intermediate to theſe. Sort of e- 
Now conceive certain Rays coming TG: 
| gible Rays 
from the Sun, that are ſo diſpoſed as produces) 


* 3 diſpoſed 

eing equally incident, they are alſo e- jength Ways. 

qually refracted. Theſe being tranſ- Fs © 

mitted through the Priſm, from what 

has been demonſtrated above will deli- 

neate (if they could be ſeen alone) as 

to Senſe a circular Image, as B D. 

Then conceive other Rays of the ſame 

Sun alſo uniform to one another, that 

are ſo diſpoſed as to be a little more 

refracted than the former. Theſe there- 

fore will paint another circular Image, 

as Y Z: And conceive other Rays 

ſtill more retrangible, that ſhall produce 

a third Image A C. Laſtly, imagine 

innumerable other Rays more and leſs 

refrangible, than the foregoing, and 

they will there form alſo other innu- 

merable circular Images both interme- 
ä diate 


- 


— = —ͤ —E— — — 


— 


, . ,, 
% 


* 
1 
p 
3 
bo 


22 The Refrangibility Sxkeœr. I, 


diate and extream to the former ones, 


illuminating the oblong Space P Y T 
Z, contained by two right Lines A B 
ani C D, and two Semicircles. But 
ſince thoſe Images are all nearly of the 
ſame Magnitude, and diſpoſed in a right 
Line between the Lines A B and C 
D, theſe Lines AB and C D, may be 
looked upon as right Lines parallel to 
one another, and as to Senſe they will 
appear ſuch; and ſo the whole Space P 

VI 2 illuminated by Rays variouſly 
refracted from the ſame Incidence will 
be terminated partly by parallel ſtraight 
Lines, and partly by oppoſite Semicir- 
cles, as is found by Experience. 


12. Hence is Bur that I might fully confirm this 
— libeg Conjecture, I bethought myſelf of the 
Wage Bore Image of the Sun tranſmitted without 
ders become any Refraction through a Hole to a 
very diſtinct. great Diſtance, as that it is ill defined, 
its Termination between the Light and 
Shade being very indiſtinct: But if thoſe 
Rays paſs through a convex Lens, whoſe 
Focus is at the Image, the Image will 
be terminated moſt diſtinaly. After the 
ſame Manner I perceived of equally re- 


g fran- 


- 
r* bY Fr 
_ 
WE. 


Stet. I. of Rays zs different. 23 


frangible Rays, that iſ they were tranſ- 
mitted through a Priſm to a great 
Diſtance, they would paint a circular 
Image ill defined, whoſe Termination 
notwithſtanding by the interpoſition of 
a convex Lens would become moſt di- 
ſtint. Therefore when I ſaw the Ter- 
minations of the refracted Image P Y 
T Z not very diſtinct ; of the Images 
B D, I Z, A C, and the reſt forming 


that oblong one I conjectured, that be- 


ing tranſmitted through a convex Lens 
they would be terminated much more 
diſtinctly than otherwiſe ; and upon 
Trial it proved ſo. For the right Lines 
A B and C D, in which all thoſe cir- 
cular Images on each Side were termi- 
nated, I ſaw very diſtinct, which before 
J had ſeen confuſed. 


BuT what ſeems very remarkable, 14. Why 
the circular Terminations A P C and | the circular 
B D T of that Image always appeared ors always 
very confuſed, the Light decreaſing by rad. 
Degrees, till at length it ended in the 
Shade; vig. the intermediate Circles as 
Y Z, are mixed with other Circles 
falling on each Side, with which they 


C 4 coin- 


\ 


hy” 


24 The Refrangibility Szcv. L 


coincide in ſome of their Parts ; but the 
circular Extremities A C and B D 
concur indeed with the others but in 
one Part, and their Meeting becomes 
continually rarer, and from thence the 
Light fainter, until it arrives at he 
Extremities P and T. But there is 
another Cauſe of this, which is, that the 
greateſt plenty of Rays -is ſuited to 
ſuffer a mean Refraction, and ſo to fall 
in the middle of the Image, and that 
their Number becomes continually leſs, 
to which belongs on each Side a more 
extream Degree of Refraction. 


16. An Ad. Bor in order to try theſe things I 
monition adviſe Lens's to be uſed, whoſe Foci 
concerning | . , 

the Figure are remote, being diſtant perhaps ſix 
and he Len or twelve Feet from the Lens's, pro- 
and Priſms. yided ſuch are at hand; at leaſt let 
them not be leſs diſtant than two Feet. 
And alſo the Sides of the Priſm ought 
to be exactly plain ; but if its Sides 
are ſomewhat convex, then it is more 
eligible to make uſe of a Lens, whoſe 
Focus is diſtant from it two or three 
Feet only. Theſe things being provi- 
ded, place the Lens near the Priſm on 
= Ie cither 


8 
1 


SEC. I. of Rays # different. 25 


either Side in ſuch a Manner, that it 
may directly reſpe& the Rays tranſmit- 
ted through it. Then let the Rays be 
received upon a Paper, which moye 
backwards and forwards till you ſee 
the coloured Image terminated on both 
Sides moſt diſtinctly by parallel ſtraight 
Lines. But it muſt be obſerved, that 
when the Priſm is placed beyond the 
Hole F, as in Fig. 3. or on this Side 
very near it, and the Lens is farther 
1 8 diſtant from that Hole than the Focus 
of the Lens, which would be made by 
the Rays falling parallel on it, is diſtant 
from the ſaid Lens; you will find js. oe - 
Caſes, in which the Image projected on cular Image. 
the Paper will become diſtin&; one, 4 
when all the ho mogeneal Rays, that fall 
parallel upon the Lens, are ſo refracted, 
as that they concur on that Paper in 
the ſame Point; which happens, when 
vyoau ſee the Image coloured, oblong, 
and terminated diſtinctly by parallel 
1 ſtraight Lines. The other Caſe is, when 
3 all the homogeneal Rays diverging from 
one Point of the Hole E, after they 
are refracted by the Lens, do converge 
again to one Point of the ſaid Paper. 
But 


26 


17. And 
of the Sha- 
dows of 
Clouds in- 
terpoſing be- 
tween the 
Sun. 


Fig. 7. 


\ 


The Refrangibility Secr. I. 


But this happens, when you ſee the 
Image white, circular, and well defined 
on all Sides, concerning which I ſhall 
ſpeak more fully elſewhere. It is ſuf- 
ficient to have given this Admonition 
in this Place, that any one experien- 
cing this with his own Eyes, may not 
be uncautiouſly deceived by the Am- 
biguity of the Effect, and thence call 
in Queſtion, what has been ſaid above. 


T ſhall further obſerve, that ſome 
very thin Clouds have intercepted the 
Sun's Diſc, not quite obſcuring the 
ſame, and have caſt Shadows on this 
Image P T, not like to themſelves, but 
extended in Length and parallel to the 
rectilineal Edges of the Image, which 
agrees accurately with the Reaſons juſt 
now alledged. For (Fig. 7.) conceive 
ſome Cloud on the Sun's Diſc to appear 
like the Sun itſelf, and let it, it the 
Rays moſt refrangible and bounded 
by the Circle A C are beheld, project 


a Shadow into the Place L; ſo that 


the Circle A C with the Shadow L 
may expreſs the Sun's Diſc obſcured 
by a Cloud. Which being ſuppoſed, if 

8 the 


- 
* 
1 = 
4 
— 
. 
84 FE 
24 
= 
_— ' 
* 4 
= . 
= v4 
® ” 
3”. 
we" 
- * 
7 
8 
"= 
EA 
7 
* - 
a8 
—_ 
E- 
- 
—- 
_ 
_— 
1 
1 
= 
_— 
& 


Sect. I. of Rars 2s different. 27 


the Rays the leaſt refrangible and cir- 
cumſcribed by the Circle B D, be be- 
held, the Shadow of the Cloud will be 
caſt by them into the Place N, whoſe 
Situation in the Circle BD will be ſuch, 

as is that of L in the Circle A C, 
vi g. this alſo repreſents the Sun's Diſc 
obſcured by a Cloud. And the ſame is 

to be likewiſe underſtood of any inter- 
mediate Circle with its ſmall Shadow 

M. So that on Account of the indefi- \ 
nite multitude of Circles poſſeſſing the 
whole Space A BDC, the Cloud may 
diſperſe its Shadows through the whole 
Length L N, and render it obſcure; 
and ſo, ſince many Clouds or Parts of 
Clouds may intervene between the Sun, 

its Image will be obſcured by many 
Clouds diffuſed in Length and Parallel. 


TraT I might ſearch out, as dili- 8. 5 


gently as poſſible, the ſaid Proprieties of the Figure of 


Light, I farther deviſed the following —_— 


Method, by which I might brin 75 them ange: 4 alſa 
to an Examination, Vig. (in Fig. 6 * — 10 


ſince the Magnitude of the Circles A C, becomes 
Y Z, B D depends on the Magnitude oblong. 
of the Sun, if the Sun's Diameter ** 6 


5 could 


— , — 


— — = 


— * 


bl 
! 
| 
! 
| 
{1 
| 
| 
| 
4 


* — 
* 2 , — — , . ao ore 


28 


Fig. 8. 


Fig · 9. 


The Refrangibility Sxcr. I. 


could be made ſomewhat leſs than it 
now really is, then thoſe Circles alſo 
would become leſs, without the Diſ- 


tance of the Centers H, I, K, being al- 
tered at all, as may be ſeen in Fig. 8; 


and ſo the Breadth of the Image com- 
pared with its Length, would become 
much leſs than before, as both being 
diminiſhed by the ſame Quantity. To 
prove this, I cauſed the Rays of the 
Sun to paſs through two ſmall Holes 
very diſtant from one another, before 
they fell upon the Priſm ; by which 
means the Rays coming from the ex- 
tream Parts of the Sun were excluded, 


and the Affair ſucceeded, as if the Sun's 


Diameter had been reall diminiſhed, 
For Illuſtration, let in Fig. 9. e 


be the Window-ſhut perforated with a 
ſmall Hole F, by which the ſolar Rays 
may enter the Chamber otherwiſe dark- 
ned; then let EFG bean opake Bo- 
dy perforated at E, and ſo placed in the 
middle of the Room, that the Rays 
may again pals through that Hole, be- 
fore they arrive at the Priſm AB C 
placed behind. Now the Diameter of 
theſe Holes being # of an Inch, and 

their 


e n N 
SOT _ * ons © WE 5 
RRR 


SECT. I. of RAV s ir differtnt. 


their Diſtance F F, being 12 Feet 
{vig. that the greateſt Inclination of 
the Rays paſſing through both the 
Holes might be an Angle of nearly 6 
Minutes, that is, about the fifth Part 
of the Sun's Diameter) and alſo the I- 
mage P T projected upon a Paper 10 
Feet diſtant from the Priſm, as the 
the Smallneſs of the Room would per- 
mit; I found the Length of the Image 
to be more than four Inches and an 
half, and the Breadth the third Part 
of an Inch ; that is, the Length more 
than 14 times greater than the Breadth, 
as it ought to be, from what has been 
faid. For ſince thoſe Rays only are let 
in, that are inclined to one another leſs 
than the fifth Part of the Sun's Dia- 
meter, the Diameters AC, Y Z and 


B D, diminiſhed by the Diameter of 


the Hole F, ought to be five times leſs, 
than otherwiſe they would be, as is 


— 


29 


to be ſeen in Fig. 6 and 7. As if Eg. 6, 7. 


they were produced by a Sun, whoſe 
Diameter was five times leſs than that 
of our Sun. But if the opake Body 


Fg (Fig. g. was removed, that the xj, 9. 


Rays might paſs only through one Hole 
F to 


The Refrangibility Sxcr. I. 

F to the Priſm, as was done in the 
former Experiments, the Breadth of 
the Image would become 14 of an 
Inch; and the Length more than 5 
Inches; the Angle of the Priſm be- 
ing 60 Degrees or a little more. There- 
fore the Diameter of the Circles A C, 
YZ and B D, which conſtitute the I. 
mage after che Manner declared above, 
would be 13 Inch; from whence let 
be taken the Diameter of the Hole, 
vis. ; Inch, and there will remain 
1:4 Inch, to whoſe fifth Part let there 
be again added the ſame Diameter of 
the Hole or 4 Inch, and there will be 
produced 3 Inch, the Diameter of the 
Circles A C, Y Z and BD in Figure 
8; which is leſs than the Diameter of 
thoſe Circles in #7g.6, by the Quantity 
z of an Inch; wherefore the 7 Figure 
is every where leſs than the 6, by the 
Quantity 4 Inch; and therefor its 
Length becomes more than 4 Inches; 
but its Breadth the third of an Inch, 
The {ame might alſo in ſome Meaſure 
happen, from the Image P T being far- 
ther removed from the Priſm. But it is 
to be obſerved, that I ſuppoſe the Holes 
Fand 


SECT. I. of Rays 7s different. 


Fand F reſpecting the Rays directly, 
although it does not much import, 
whether their Situation be a little ob- 
lique, as is done in the ninth Figure 
annexed. 


Moxeoves, if in this Experiment 19. This 
you ſhall make Uſe of, as before, PO 
convex Lens, whoſe Focus falls at the*? 
Image, the Hole E, if you pleaſe, being 
enlarged, or the opake Body E S en- 
tirely removed, that the Rays may only 
paſs through the diſtant Hole , and if 
you ſhall make the Hole F narrower 
than before, the reſt remaining the 
fame as at firſt, you will ſee a very 
long Image, and for its Length more 
lucid, than in the former Caſe. For ex- 
ample, if the Diameter of the Focus 
be the twentieth Part of an Inch, and 
if you ſhall place the Priſm with the 
Lens twelve Feet diſtant from thence, 
you will ſee the Length of the Image 
more than eighty or an hundred Times 
greater than the Breadth. But in ma- 
king theſe Experiments the Room 
ought to be every where cloſed, leſt 
the Light entering at any other Place 
be- 


The Refrangibilith src I. 
beſides the Hole F may diſturb the 


Image, and render it obſcure near its 
circular Extremities. And moreover 
if the Surfaces of the Priſm are ac- 
curately plane, a Lens ought to be 
made uſe of, that projects its Focus to a 
great Diſtance, ſuppoſe to 12 or 20 
Feet, provided the Bigneſs of the Room 
will permit, by which means you may 
make a more certain Judgment of the 
Proportions of the Image. But if the 
Sides of the Priſm are ſomewhat con- 
vex, as it ſometimes happens in thoſe, 
that are commonly ſold; ſuch a Priſm 


may be uſed alone without any Lens, 
and its Convexity will, inſtead of a 


Lens, collect the Rays at a great Diſ- 
tance. Beſides, if with the Priſm you 
make Uſe of a ſmall Lens, whoſe Fo- 
cus is not diſtant above two or three 
Feet, you will ſee an Image ſufficiently 
long indeed, but whoſe Breadtli will 
not be ſenſible. Which does not the 


leſs anſwer our Purpoſe, as well as if 


you could accurately judge of the Pro- 
portion of the Length to its Breadth, 
In making alſo theſe Experiments it 
may be farther remarked, that the 


1 Lens 


| O 
— 5 = 
OH 0 has 3. 
< ts 

U 2 N 

a, FP a 5 : _— 
— ee SY — 

M g = 
e FR 5% 4 
— 


Zaugg. 


n 


P— AS. 
. 
* - 3 


Sgcr. I of RAV Ss ts diferent. 33 


Lens ought not to be placed ſo far be- 
hind the Priſm, but that it may be ex- 
tended to tranſmit all the Rays toge- 
ther, leſt you be obliged to obſerve the 
Image ſucceſſively by its Parts only. 
And laſtly it may be obſerved, that if 
you place the Hole F beyond the Priſm, 
and the Lens beyond that Hole, at a 

greater Diſtance from 1t, than the Fo- 
cus of its Rays flowing from the more 
remote Hole f is diſtant from the Lens, 
there will be two Caſes in which the I- 
mage caſt upon the Paper will appear 
diſtin, according as the Rays co- 
ming from every Point of the Hole 
E, or from every Point of the Hole 7, 
are collected in as many Points of the 
Paper. In one Cale the Image will be 
white and circular, as I* obſerved be- N. 18. 
fore ; but in the other Caſe oblong and 
coloured, as the preſent Experiment re- 
qulres. 


Ir now appears from what has been 20. This ſtill 


ſaid, that the Breadth of the Image Pl Pro: 


moted by the 
T always becomes ſo much the leſs, as Image of the 


Pl : 
the remote Hole F is made the narrow- 9 


er, ſo that it is not to be doubted, but 
D the 


— 


—— — —— 


— 2 —xů —ͤ—Uéöö . — 


—— 


— — — — 
e 


> — * — Cs: . „„ —— 2 ¶ — — ù—— 
PPP ' 1 v 8 ES dh gd 


= — 


W „% oo, 


The Refrangibility Sxcr. I. 


the ſaid Breadth would altogether va- 
niſh, if inſtead of that lucid Hole, there 
was one moſt bright Point 1n its place; 
and that this would be ſo is confirmed 
from a not unlike Obſervation, which 
I formerly made of the Planet Venus. 
The room being every where cloſed, 


except a Hole a little more than two 


Inches broad, that it might be made 
very dark : In this Hole I placed the 


Object-Glaſs of a ſeven Foot Teleſcope, 


its Aperture being two Inches and more 
broad to tranſmit a ſufficient Quantity 
of Rays. Then at the Diſtance of ſe- 
ven Feet a Paper being placed tranſ- 
verſely, I ſaw projected upon it the I- 
mage of the Planet like a lucid Point; 
and a Priſm being interpoſed at the 
Diſtance of one or two Feet from 
that Paper, by which the tranſmitted 
Rays might be refracted again: Inſtead 
of that lucid Point, I faw at the Di- 
ſtance from thence of more than a Foot 
a very fine Line, tho" not very bright, 


| however very eaſy to be diſcerned, and 


whoſe Length exceeded half an Inch, 
but its Breadth as to Senſe, was none ; 
at leaſt not at than juſt to be per- 
Ex 2 ceiv'd. 


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Sect. I. of Rays zs different. 35 


ceiv'd. And I believe the ſame thing 

might be obſerved of Stars of the firſt 
Magnitude, as of Sirius, eſpecially if 

a Lens be uſed four or ſix Inches broad, 

that it may tranſmit many Rays. 

0 
How well this Experiment agrees 1. And it is 

with our Explanation, which we gave *PPlicd co 


= g the Deſerĩp- 
at the Beginning, of the different Re. tion of Re- 


fraction of Rays incident with the ſame — = 
Angle, it may be worth while to ob- iſt Figure. 
ſerve. In the firſt Figure I ſuppoſed di- 

vers Rays to be carried ſucceſſively a- 

long the ſame right Line to a refract- 

ing Surface, and there ſome to be gra- 
dually a little more refracted than o- 

thers. Which if it were conceived to 

be done, it would abundantly follow, 

that the Rays fo refracted, if they 

were afterwards intercepted by any o- 

pake Body, as Paper, they would there 

paint a ſmall bright Line. Now al- 

though the Rays coming from any Star 

do not all proceed in the ſame right 

Line, however what is equivalent, they 

may be looked upon as parallel, and by 

Reaſon they are by the convex Lens 

made to converge before they arrive at 

D 2 the 


36 The Refrangibility SECT; I. 


the Priſm, this does not deſtroy the A- 
nalogy, but greatly confirms it. Be- 
cauſe for every one proceeding in the 
ſame right Line, you ought only to 
conceive ſo many Pencils of Rays, 
which all have the ſame Axis, and the 
ſame Point of Concourſe; and thus 
of thoſe Pencils, ſome are more refrac- 
ted by the Priſm than others, ſo that 
their Points of Concourſe, or Foci, 
which before coincided, now every 
one fall ſeparate making a right Line, 
And therefore that the Axes of the 
Pencils, which coincided fo long, ſup- 
pole with the ſucceſſive Rays, until 
they arrived at the Priſm, and there 
by the various Refraction they are 
made to diverge, that they proceed to 
the Foci of the Pencils lying in a 
ſtraight Line. 


22 Ava. If you place the Priſm nearer to 
ried Circum-the Planet Venus than the Lens, that 
—— oo the Rays may be firſt tranſmitted thro 
the ſame De · jt, and then made converging by the 
ſcription Lens, you will ſee as before the ſame 
{mall Line, although leſs conſpicuous, 
and more difficult to be found. Now 


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S:cT. I. of Rays ii different. 37 


in this Specimen, ſince all the Rays 
come parallel, if they were equally re- 
fracted, as they thus paſſed through the 
Priſm, they would remain parallel at- 


terwards, until they fell upon the Lens, 


and in that they would therefore be ſo 


retracted, as all would thence forward 
proceed to the ſame Point, and ſo a 


lucid Point would be ſeen. Wherefore 
ſince inſtead of ſuch a Point a Line ap- 
pears, it muſt be concluded, that all 
the Rays are not equally refracted. 


Ir any one ſhould now object, that 23. That 
in Refractions there is indeed an Ine- d Ei- 
quality, but that it is contingent and ments che 

{ . . » Retra@ions 
not ariling from a previous Diſpoſition e not made 
of the Rays, or by any certain Laws, pad cr by 
I anſwer, that the above mentioned any ather 
Image of the Sun, if it became oblong ou mia 
by Rays refracted according to no Retrangibi- 
Law, it could not be terminated in its?” 
Length by ſtraight Lines, as has been 
ſhewn at the fifth Figure. Beſides it 
ought not at all to be oblong, but to 
be formed in its middle Part in the 
Shape of an Orb more ſplendid, and by 
a ſenſible Termination to be diſtinguiſh- 
= D 3 ed 


Hg · 10. 


The Refrangibility Sxcr. I. 


ed from the weaker, erratic Light diſ- 
perſed every where about. Juſt as the 
Sun appears, when he is almoſt obſcu- 
red by the Clouds, or as his Image is 
ſeen, when tranſmitted through a Glaſs 
Plate terminated by parallel Planes, and 
lightly o'erſpread with one's Breath or 
Smoke, that the Light in being re- 
fracted may be a little diſturbed. Add 
to this, if C Fig. 10.) two ſimilar 
Priſms ABC and a6 c be placed by 
one another parallel in reſpect to their 
Length, with their plane Sides A C, 
and ac, as allo B C and 6 c parallel, 
and if the Sun ſhines through both of 
them into the Place Z, where an o- 
pake Body is directly oppoſed to rhe 
Light; his Rays however being firſt 
tranſmitted through the circular Hole 
F. The Light incident on the ſaid Z 
will appear diſtinctly circular, not o- 
therwiſe than if it directly tended from 
F, the Priſm being not at all interpo- 
fed. It muſt therefore be acknow- 
ledged, that the Refractions of both 
Priſms conjointly are regular, and 
therefore alſo the Refractions of ei- 
ther of them. Vir. thoſe Rays alike 

inci- 


1 « 
+. = 
- x 


Sect. I. 7 Rars is different. 
incident are not all refracted equally 
in the firſt Priſm A B C, nor in the 


ſecond @ 6 c, yet ſince the Inequa- 


lity of the Refraction is not contin- 
gent, but ariſes from a previous Diſpo- 


lition of the Rays, therefore although 
various Rays are varioully retracted, 
yet of the ſame Ray there will be the 
ſame Quantity of Refraction in both 
Priſms, and as much as it is incurva- 
ted by the firſt A B C, ſo much will 
it be incurvated by the ſecond 4 6 c, 
whence any Ray, however refrangible 
it is, after it has emerged out of both 
Priſms, will become parallel to it ſelf, 
before it had been incident on them: 
And therefore ſince all tend towards the 
ſame Parts, to which they would free- 
ly have tended, if they had not been 
intercepted by the Priſms, it neceſſarily 
follows, that they exhibit the fame 
circular Image at Z, which they would 
exlybit, had they freely tended thither. 
But if the oblong Image made (as has 
been ſaid) by the Refraction of one 
Priſm only, did acquire its Figure by 
Raysdivaricating by ao certain Law, but 
refracted ſcateringly by chance here and 
D 4 there 


The Refrangibility Skcr. I. 


there, when the Refractions were dou- 
bled by two Priſms, the Errors alſo of 
the Rays would become twice as many 
and alſo twice as great, and from thence 
the Image at Z would become much 
more oblong, which yet by Experi- 


ence is found to be contracted i into an 
Orb. 


SOME may perhaps ſuſpect, that the 
Termination of the Light, or the con- 
fine of the quieſcent medium, may 
cauſe the Diverſity of Refraction; but 
to this Suſpicion a Remedy is at hand; j 
vix. by making (as in Fig. 3.) the 
Light to be terminated by the hinder- 
moſt Part of the Priſm only, that it 
may not become bounded by the Sha- 
dow before it ſhall be refracted. And 
therefore that no Suſpicion he raiſed 
about the various Thickneſs of the 
Glaſs, its Refraction may be tryed at 
different Thickneſſes, by moving the 
Priſm tranſverily with a parallel Motion 
near the Ingreſs of the Light, ſo that 
the Light at firſt may be tranſmitted at 
its Edge, then at thicker Parts of it; 
and in every Caſe the Appearance of 


the 


SECT. 1, Of RAVSs is different. 


the Colours will be alike. Nor does 
it much import, whether the Hole, 
through which the Light enters, 1s 
broader or narrower ; for from thence 
nothing elſe follows, beſides the Aug- 
mentation or Diminution of the Light 
exhibiting the Colours, and ſo great a 
Dilatation or Contraction of the Image, 
as there is of the Hole. 


FROM the Experiment of the two 
parallel Priſms already deſcribed, it is 
alſo manifeſt, that this dilating of the 
Image in Length does not ariſe from the 
ſpreading or ſplitting of any the ſame 
Ray into diverſe diverging Rays, for 
they by another ſpreading or ſplitting, 
in their Paſſage through the ſecond 
Priſm, ought then to be reſolved into 
a far greater Number and more diver- 
ging Rays, Moreoyer to all theſe Ob- 
jections is oppoſed the Experiment, 
where the latter Priſm is not placed pa- 
rallel to the former, but perpendicular- 
ly tranſverſe. For in that Caſe, if the 
former Priſm dilated the Image in 
Length from any other Cauſe than the 
different Refrangibility of different 


Rays, 


| 42 The Refrangibility Scr. I. 
Rays, then the latter Priſm by a tranſ. 
verſe Refraction ought to dilate that 
1 oblong Image in Breadth, and ſo would 
form a quadrilateral one. But u 
trying the Experiment, the Thing came 
out otherwiſe, viz, the Image not be- 
ing dilated in Breadth, but only ren- 
dered oblique by the leſs Refraction 
of the red Extremity than of the vio- 
Pg. 11. let, as is to be ſeen at Fig. 11. where 
the Image P IT by the Refraction of 
the ſecond Priſm is transferred to p t. 
From what has been ſaid, I believe, it 
more than ſufficiently appears what 1 
propoſed at firſt to demonſtrate : But be. 
caule the Agreement of many Things 
affords Delight to the Underſtanding, 
and often begets a firmer Affent, than 
the Teſtimony of a ſingle though 
the moſt ſcientific Argument, it will 
not be foreign to our Purpoſe if we 
briefly introduce another Sort of Expe- 
riments a-kin to the preceding ones. 


ry * * — 7 — ox > 4 
3 * ONE 
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— Other IN Fig. 12. let F be a very ſmall 
i to the Hole, through which the Light of the 


a· kin to the 
preceding Sun may be tranſmitted; and at a Diſ- 
upon. tance taken at Pleaſure, le be placed the 


Fig. 12. a 
— Priſm 


— 


=” . 


SECT. I. of RAV s 3s different. 


Priſm A B C, through which the Rays 
may paſs refracted, as I have explained 
in the former Experiments ; then with 
your Eye placed behind, you will ſee 
an oblong Image TP of the Hole F; 
whoſe Length compared to its Breadth, 
will be ſo much the greater, as the 
Hole F ſhall be made narrower, and 
thence it appears, that of the Rays 
ſome tending to the Eye by H, as if 
they had flowed from P, are more re- 
fracted, than others tending by TI, as 
if they had come from T, and the 
Rays not otherwiſe entering the Eye, 
than if they had flowed from the ob- 
long Space P T, it neceſlarily follows, 
that that long Space muſt appear il- 
luminated. 


Bor Care muſt be had, that the A- 
perture of the Hole F is not ſo large, as 
to hurt the Eye by the admiſſion of too 
much Light; nay it ought not to be 
greater, than that you may ſee through 
that Hole with your naked Eye a Par- 
ticle of the Sun like a lucid Point di- 
ſtint, and without any Circumradiati- 
on. But if the Light of the Sun be 

thought 


44 


The Refrangibility Sect. 
thought too much for this Experiment, 
the Light tranſmitted from the Clouds 
may be ſufficient, provided the Diſpo- 
ſition of your Eye be ſuch, as you 
may diſcern the Hole diſtin without 
any incompaſhng Rays, before you in- 
terpofe the Priſm, otherwiſe you will 

not ſee its Image diſtinct, nor dedu- 
ced into a due Length. Beſides you 
may alſo obſerve, if you look at a 
Thread through a Priſm, for the Thread 
Will appear much broader, when it is 
placed in a Situation parallel to the 
Length of the Priſm than when in a 
tranſverſe one. But that I may com- 
rehend all in one, if you behold a Star 
of the firſt Magnitude through the 
Priſm, its Image will alſo be ſeen long. 
But ſince the Rays of the Stars are e- 
ſteemed as parallel, if they all were re- 
fracted equally, they would alſo re- 
main parallel after they iſſued out of 
the Priſm, and ſo entering the Eye 
would make an Image altogether like 
the Star, or a lucid Point, not at all 
long; juſt as it happens, when the Star 
ſends parallel Rays directly into the 
Eye. You therefore ſee, that parallel 
Rays 


SrcT. I. of Ravs 1s different. 


Rays refracted by plane Surfaces be- 
come inclined, whence it neceſſarily fol- 
lows, that they ſuffer an unequal Re- 
fraction. But it may be remarked by 
the by, that a Teleſcope, if you think 
fit, being firſt applied, both that a ſuf- 


_ ficient Quantity of Light may be tranſ- 


mitted to the Eye, and that the Scintil- 
lation, wherewith the fixt Stars are 
wont as it were to be crowned, may 
be leſſened, and a Priſm then being in- 
terpoſed, you will ſee the whitiſh Line 
more diſtin than before, with a Breadth 
ſcarcely diſcernible. Theſe few things 
being declared concerning the different 
Refrangibility of Rays, the meaning 
whereof will appear more fully in what 
follows, when we come to treat of 
Colours : It remains, that the Quanti- 
ties and Meaſures of Refractions be 
now determined. 


. in the Philoſophical Tranſactions No; $0, C. 
and more fully in his Treatiſe of Opticks, 


; SECTION 


45. 


46 Of the Mrasun Seer, II. 


SECTION II. 


Of the MEASURE of 
REFRACTIONS. 
the meaſure 


s * E Ancierts determined Re- 
cf the Re- 122. by the Means of the 
fraction of : 


Rays of a Angles, which the incident and refrac- 
_ ted Rays made with the Perpendicular 
0 reffadting Plane, as if thoſe An- 
Fi. 13, les had a given Ratio. For Example, 
in Fig. 13. let I H be a refracting Plane, 
to which the Line DCE perpendicu- 
larly inſiſts at one of its Points C, and 
in this Point C let any Ray A C be in- 
cident, and let it be refracted to R: the 
refracted Ray C R being ſuppoſed to 
lie in the Plane A C I, that is perpendi- 
cular to the refrating Plane ; the An- 
tients fuppoſed, that the Angle of In- 
eidence A CD, the Angle of Refracti- 
on RCE, and the refracted Angle R 
CF are always in a certain given Ra- 
tio; or they rather believed it was a 
ſufficiently accurate Hypotheſis, when 
the Rays did not much divaricate from 


the 


25. Of 


A 3 
- 


— r RO . 
E N 2+ p 5 4 8 7 - 2 5 = * Bas: - 
7 a * * IT RE A WEEN r 
5 c wp 4 r - TEM "FF. a Cx; 0" ewe... 8 ” r e 
1 IF? 5 CP * "tt : 22 ** 2 2 — ä 3 e , WT , 4%, * 8 2 FI 1 hate 3 od; r 
n 8 es D r . . ES 


+ 


Szcr. II. of REFRACTI1ONS: 


the Perpendicular. So in Glaſs they 
made the Angle of Refraction to be a- 
bout tripple of the refracted Angle. But 
this eſtimating of the Refractions was 
found not to be ſufficiently accurate, 
to be made a Fundamental of Diop- 
tricks. And à Cartes was the firſt, that 
thought of another Rule, whereby it 
might be more exactly determined, by 
making the Sines of the faid Angles to 
be in a giving Ratio. In Fig. 13. if 
with the Center C, and any Diſtance 
A C a Circle be deſcribed cutting the 
aforeſaid Rays in A and R, and from 
thoſe Points to the Plane's perpendicu- 
lar DCE, the nomals AD and RE 
be drawn, the Proportion of A D and 
RE will be perpetually the ſame. The 
Truth whereof the Author had demon- 
ſtrated not inelegantly, provided he 


had left no room to doubt of the phyſi- 


cal Cauſes, which he aſſumed. And as 
alſo ſome have examined this with In- 
ſtruments accurately made for that Pur- 
poſe, and have found it (as to Senſe) 
exactly agreeing with the Truth; we do 


Of this ſuppoſed Diſcovery of Cartes, ſee the Note 
at Page 13. 


not 


47 


Fig. 13, 


48 


Fig. 14. 


4 

Of the MexsURE SECT. I. 
not ſcruple to receive it as a Funde 
mental, but with this Caution only, 
that ſince he affirmed it indifferently of 
any Sort of Rays, we only affirm it of 
each of their Kind conſidered apart, by 
laying down, that the Sine of Refrac- 
tion of equally refrangible Rays are 
as the Sines of Incidence. Let us con- 
ceive ſome Kinds of Rays to flow a- 
long the Line AC in Fig. 14. to the 
Point C, and there to be refracted by 
the Surface I H, ſuppoſe the mean re- 
frangible Rays into C R, the leaſt re- 


frangible into C T, and the moſt re- 


frangible into CP, and innumerable 
others of intermediate Degrees more 
or leſs refrangible to be diffuſed thro? 


the whole Space T C P. Now if 


D C G be drawn perpendicular to the 
refracting Plane I H, and with the Cen- 
ter C, and any Diſtance A C, let a 


Circle be deſcribed as before, cutting 


the ſaid Rays in A, P, R, IT, and from 
theſe Points let be fallen the perpendi- 


culars AD, PG, RE, T F, for the 


Sines of the Angles A 0 D, P CG, 


R CE, T CE, I ſuppoſe that howe- 
ver the Rays fall, yet it will be always 
AD 


gacr. II. of REKFRACTIORRHS. 49 


A D to P & in the ſame Ratio, which 
being once known, you have a Rule 
for meaſuring the Refraction of the 
moſt refrangible Rays, falling upon the 
ſame Surface at any Angle; and ſo it 
will always be A D to T E, in the fame 
Ratio, which being known, you have 
a Rule, whereby you may determine in 
any Incidence the Refraction of the leaſt 
refrangible Rays: And the ſame Thing 
may be conceived of the Ratio of AD 
toRE, and to the Sine of any inter- 
mediate Sort. 


BuT moreover, ſince the Sines PG, 2s. of 
RE, T E, and the reſt have a given pains 
Ratio to the Sine AD; they will have ons of Rays 
alſo a given Ratio to one another, and aq 
therefore it from one Obſervation you 
know the Proportion of the Sines PG, 

RE, T F, and the reſt belonging to the 
Rays refracted from the ſame Incidence, 
you will thence have a Rule, whereb 
having given the Sine of Refraction of 
any Sorr of Rays, and thoſe falling 
any how on that Surface, you may 
diſcover the Sines of all the reſt flow- 
ing from the ſame Incidence, although 


E what 


4 


50 Of the MEASURE SECT. IL 


what their Incidence is, be not known, 
Therefore that the Ratios of thoſe Sines 
may be inveſtigated, it 1s proper firſt to 
find, in ſome Sort of Rays, the Pro- 
portion of the Sine of Incidence, to the 
Sine of Refraction, then that there be 
determin'd the Proportions of the Sines 
of Refraction, for Rays of different 
Sorts, incident with the ſame Angle. 


27. To To compare the Sines of Inci- 
2 dence with the Sines of Refraction, it 


denceandRe- will be proper to uſe the mean Sort, 
fraction, 142. that Sort of Rays, which exhibit a 
be uſed the green Colour, or rather a Colour be- 
Rays. tween Green and Blue: For I believe 
thoſe, who hitherto have meaſur'd Re» 
fractions, (whether it was done in or- 
der to confirm the Hypotheſis of Car- 
zes, or for other Reaſons) I believe, Iſay, 
they accommodated their Meaſure to 
the Middle of the refracted Light; that 
is, if we regard the Space occupied by 
the Colours, to the Confine of Green 
and Blue. Or if we regard the Quan- 
tity, to the Middle of the Green: 
And beſides, that Point ſeems to have 


been looked upon as the principal Focus 


— — . — — —ů 
* "F< Be 
E 7 A 
. 8 1 * : 
2 2 0 *.” q 7 as * * * - "* ge” wt 9 "I WS 4 «A ta 
4 D 8 5 7 1 4 N * TO R 3s 222 2 89 * - - . 
Fry . 8 n F tl Fe l 4 i "On 4 q WWW 3 T3 . _ A" « 

4 N e %%ͤö? . ²˙·. ˙ ·¾—Sg̃⅛—ͤlĨ %Ü— .:, ĩ² a m . ²—ͤ _— 

3 * * * — 2 - 2 n PS "2, i: RIM 4 AW ba. __ * v_ ad TR >" „ 2 - « < < hat — 

0 * * ITS .. = . 1 9 5 1 * 5 FM 1 27 he" Wh, of 1 _— 2 


Sgr. It. Of REFRACTIONS. 
of convex Glaſſes, to which the in- 
termediate Sort of Rays converged. 


And alſo, ifat any Time we ſpeak in- 
diſtinctly of Rays, as has been hitherto 


the Cuſtom with the Writers in Op- 


ticks, the middle Sort may be more 


commodiouſly uſed for all than any 


of the extreme Sorts. 


Moxeovex, ſince perhaps there may 
be required a more accurate Examina- 
tion of the ſaid Rule of Cartes, than 
has hitherto been made, whilſt the va- 
rious Refrangibility of Rays was un- 
known, I ſhall firſt declare after what 
Manner, that may not be inconvenient- 
ly done. Becauſe the refracting Sur- 
faces of a tranſparent Fluid may be eaſily 
inclined to any given Angle, which is 
not permitted to a Solid, Fluids have 
been made Uſe of to this Purpoſe, 
but by an Inſtrument more operoſe 
than was neceſſary, and poſſibly more 
obnoxious to Errors, than if it wan- 
ted all that Apparatus, except a Beam 
to which a Veſſel full of Water is 
faltened; let therefore (in Fig. 15.) 
H K be a Beam of Wood Two or Three 

E 2 Yards, 


51 


28. The 


Manner of 
exploring 
the Propor- 
tions of thoſe 
Sines. 


Fig. 46. 


52 


Of the ME asuRE sSxcr. II. 
Vards, or more long, ſufficiently thick, 
that it may not in the leaſt band by its 
Length or Weight, quadrilateral, rect- 
angular, and ſtraight, with oppoſite 
Sides exactly parallel; then let two 
{mall Pieces, HI and K L be erected 
at right Angles, upon one of its Sides; 
K L near to one of its Ends, and H I 
about Four Inches diſtant from the 
other, the Length of which Two Pie- 
ces, let be Three or Four Inches, but 
their Breadth Two or Three: Then let 
be taken a cylindrical or priſmatic Veſ- 
ſel CF, Two or Three Inches broad, 
but Four or Five long; let its Baſe be 
faſtened upon the Piece HI, by ſome 
hard and tenacious Cement, and let it 
be fixed in that Situation by the Means 
of the Beam H K, produced beyond 
the faid Piece H 1 then let its Bot- 
tom, as alſo the piece be bored in the 
Middle with the ſmall Hole E, ſuppoſe 
the tenth Part of an Inch broad; and 


right againſt this Hole, let be marked 


in the other Piece the Point R, that 
may be at the ſame Diſtance from the 
Beam, with the Center of the faid 
Hole, in order that the Line FR, 


diawa 


SecT. Il, of REFRA cT1IONS, 


drawn through the Center of the Hole 
to R may be parallel to the Length of 
the Beam. Laſiiy, Let be taken a Glaſs 
Plate, plain, ſmooth, and uniformly 
thick, and let it be applied to the Side 
of the Piece H I, which is towards 
the Veſſel CE, over the Hole F; and 
let it be ſo faſtened with Cemeat, that 
the faid Veſſel may hold the Water it is 
filled with; and let Trial be made with 
a Square, whether the Glaſs Plate is 
perpendicular to the Beam; which if it 
happens not to be, let its Situation be 
corrected, till it is exactly perpendi- 
cular; for which Purpoſe it is fit, that 
the ſaid Glaſs Plate be Three or Four 
Inches broad and long, whereby a bet- 
ter Judgment may be made of its Situ- 
ation, This Inſtrument thus framed, 
and the Veſſel CF above half filled 
with Water, let it be ſo placed in the 
Sun's Rays, that they being refracted 
on the upper Surface of the Water, 
may emerge perpendicularly at the 
Hole E, and may proceed ſtraight for- 
wards, towards the Piece K L, the red 
Rays falling at T, the Purple at P, 
and the Green, or the Confine of Blue 

E 3 and 


54 Of the MEASURE Sxcr. II. 


and Green at R. It is convenient, that the 
ſaid Piece K L be painted white, or cove- 
red with a white Paper, whereby you 
may judge the better of the Colours. 
But in the mean while, with ſome large 
and exactly made Quadrant er, let 
be ſought the Inclination of the Beam 
H K to the Horizon, and you will have 


the Angle of Refraction er, and its 
Sine er; then let the Height of the 


Sun be immediately taken, and its 
Complement to go Degrees A &D will 
be the Angle of Incidence, and AD its 
Sine: Which Sines being compared to- 
gether, and the Experiment being re. 
peated at different Heights of the Sun, 
it will appear, whether the Proportion 
of the Sines is always the fame ; but if 
you would, that divers Experiments 
be made at the fame Time, or at a leſ- 
ſer Incidence than is the Compliment 
of the greateſt Height of the Sun, in- 
ſtead of the Rays flowing directly from 
the Sun, you may ule reflected ones. 
\ 
20. The SiNck it has been ſufficiently diſ- 


Manner of 
exploring Covered, that the fame Proportion of 
the retraQ- ; | | 
ing Force af any Solid encompaſſed with Air. 


the 


710 


— — 
„% 
— 

- * —- 

— 

+ 

- 

— 

- 

— 

- 


Sect. II. of REFR ACTIONS. 


the Sines of Incidence and Refraction 
does perpetually belong to any one 
Sort of Rays, however incident on any 
the ſame Surface: let it be propoſed to 
find that Proportion at a Surface ſepa- 


by one Experiment. If the Air be one 
of the given Mediums, and any Li- 
quor the other, the Inſtrument juſt now 
deſcribed may not incommodiouſly be 
uſed. But if the other Medium be a 
Solid, the Thing is expeditiouſly done 
at the 16th Diagram. For the ex- 
plaining of which, we ſhall premiſe the 
two following Lemmas. 


LEMMA I. 


IN Fig. 16. let ABC be a Priſm 
made of any tranſparent Subſtance , 
whoſe Axis let be parallel to the Ho- 
rigon and perpendicular to the Sun's 
Rays, and beſides let its Poſition be 
ſuch, as it may equally refract the 
ſaid Rays O X entering at X and go- 


done, was taught at No. 10. Now I 
ſay, that the Angle of Refraction, made 
at either of the refrafting Surfaces as 

E 4 0M C, 


rating any given Mediums, and that 


TY 


35 


Hg. 16, 


ing out at Y. But how that may be | 


% Of the MASH, Sr, II. 


AC, ic equal to half the vertical Au- 
ae of the Priſi A CB. 


Fon at the Point of TUFTS X. 
; 4 be erected the perpendicular H X, 
then H XY will be the Angle of Re- 
fraction at the Surface A C: Moreover 
let CI beſet full perpendicular on the 
Ray X X, and it will biſect the Angle 
CX, becauſe the Triangle Y CX 
(on account of the Equality of Re- 
fraction in X and V) is Ifoſceles; I 
therefore ſay, that the Angles HX Y 
and IC are equal. For the Angles 
AX MS ang. XI CICX (by 32. 
1. Elem.) but the Angles AX H and 
X I C are right; therefore the Remain- 
ders HX X and I S X are equal, 
* EW. 


915 LEMMA II. 
"Agar if the incident Ray O X and 
emerging one Y N be indefinitely produ- 
ted, meeting in G, aud beſides, if any 
right Line K L parallel to the Horizon 
be erofed by theſe Rays, making the Tri. 
angle G K L, and when the refracted 
. YN tende upwards, if. the . 
= 


1Secr;H. ARBFTIAerioxs. 


of the Angles LK X and KL I be 
taten, or their Difference, when that 
Ray Y N tends downwards : I ſay, 
that the half of that Sum or Difference 
together with the Angle of Refrattion 
HX Y will be equal to the Angle of 
Incidence H X G, 5 


Fon the ſaid Sum or Difference is 
. Equal to the Angle NG K (by 32. 1. 
Elem.) that is, to the Angles GXY 
+GYX, and ſince the Triangle G 
XX is Ifoſceles, the half of the id 
Sum or Difference will be equal to the 


* 


refracted Angle GX Y, which makes 
with the Angle of Refraction Y.X H 
the Angle of Incidence. Q. E. D .; 


Tus Things being premiſed, the 
Problem is thus done; in the firſt Place 
let be meaſured the vertical Angle of 
the Priſm A CB, and its half will be 
the Angle of RefraQtion, Then the 
Priſm being diſpoſed in the aforeſaid Po- 
ſition, through which the Rays enter- 
ing the Hole F may be tranſmitted, let 
þy the means of a. large and accurate 


Quadrant MN Q (rhe Diſtance gf 


whoſe 


* 


yo 


= An Ex- 
mple in the 
action oſ 


— — it to be 63 deg. 12 min. "whoſe 
of Glaſs. 


Of the MAU sker. If. 


whoſe Sig hts M 4nd N being one Foot 
at leaſt) be found the Angle Y L. K or 
P, which the refracted Rays Y 
MN make with the Horizon, by cau- 
ſing the mean refrangible Rays to paſs 
through the Sights M and N at the 
Diſtance of ten or twelve Feet from 
the Priſm, and at the ſame time let be 
obſerved the Heigth of the Sun X KL. 


Which two Angles let be added, if the 


refracted Rays Y MN tend upwards, 
as is deſcribed in the Scheme, other- 
wiſe let the leſs be taken from the 
greater; and half the Sum or Difference 
together with the Angle of Refraction 
found before will be the Angle of Inci- 
dence, as is manifeſt by the ſecond 
Lemma. Laſtly from the Angles of 
Incidence and Refraction being thus gi- 


ven, their Sines are given. Q. E. F. 


So in a certain Glaſs Priſm I meaſu- 
red its greateſt Angle A CB, and 


half H XY is 31 deg. 36 min. and 


its Sine 5240, the Sine of 90 deg. be- 
ing made 10000, Then ſince the Sun's 


1 O KL was obſerved to be 14 
deg. 


S8cT,IL of RE x KACTIORsSV. 32 


deg, 4 min. the other Angle M L K 
made by the Ray Y N tending to the 
middle of the green was 30 deg. 52: 
whole Sum 1s 44 deg. 56 . its 
half Y X K 22 deg. 28 min. which to- 
gether with the Angle of Refraction 
HX Y makes 54 deg. 4 min, the An- 
gle of Incidence, whoſe Sine is 8097, 
Laſtly by comparing the Sines now diſ- 
covered, that their Proportion may be 
had in the leaſt Terms, I found it to be 
as 11 to I7 nearly, Wherefore it may 
be laid down as a general Rule, that of 
Rays exhibiting a green Colour, the 
Sine of Incidence, out of Air into any 
Glaſs of equal Refraction with this 
Priſm, is to the Sine of Refraction, as 
ſeventeen to eleven, 


By meaſuring in like manner the 
Refraction of Rays exhibiting a Colour 
between green and blue, 45 deg. 8 min. 
was found for the double of the rer 
fracted Angle, whoſe half 22 deg. 34 
Min. together with the Angle of Re- 
fraction 31 deg. 35 min. gives the An- 
gle of Incidence 54 deg. 10 min, And 
its Sine 8107 is to the Sing of Refrac- 
tion 


10 


31. The 


of the afore. 


No 20. 


Fige 17. 


part infinitely ſmall. So for 


Of the MexsUrRE® Sect. II. 
tion 5240, as 82 to 53 nearly. But 


Convenieney the Conveniency of this Method in 


meaſuring Refractions may be gueſſed 
at from this, that there is here no Oc- 
caſion for any Inſtrument, beſides a 
Quadrant and the Priſm, whoſe Re- 
fraction is deſired ; that the Refraction, 
whilſt it is rendered double at X and Y, 
may be thence more certainly meaſured; 
and that it is very eaſy to diſpoſe the 
Priſm in the deſired Situation, as has 
been ſhewn above at No 10. And far- 
ther that a ſmall Error in the deſired Si- 
tuation is ſcarcely of any Account, 
whilſt as to Senſe the refracted Angle 
MGK will not thence be changed; as 
will appear upon Tryal. For that An- 
gle is here * the leaſt of all, and of 


Quantities generated by Motion, when 


they become the greateſt or the leaſt, 
that is in the Moment of their Re- 
greſs, the Motions are for the moſt 
ample, 
in Fig. 17. If with the Center, C be 
deſcribed a Circle  L 4 and without 


© This will appear from what our Author 
ſtrates hereafter at Sect. III. Prop. xxv, 


Sac. II. of REFRACTIONS: 

it be taken any Point G, and there be 
drawn G I C, and the 'perpenditukir 
G K erected. T hen if it be conceived, 

that the Point / be moved uniformly in 
the Circumference of that Circle, thro” 

which Point a certain ſtraight Line G / 
turning on the Center G, may conti- 
nually paſs, it is manifeſt, that the 


greater the Angle CG / is, or the leſs 


the Angle K G4, the leſs will be the 
angular Motion of the Line G 4, and 
when the Angle C G becomes the 
greateſt, or the Angle K G / the leaſt, 
that is, in the Moment of Regrefs (che 
right Line G then touching this Cir- 
cle in L) its Motion will be infinitely 
ſmall, and as to Senſe none, and a 
ſmall Error from the Point of Contact 
L will produce no ſenſible Variation 
in thoſe Angles K GL and CGL. 
And nearly after the ſame Manner a 
ſmall Convolution of the Priſm will 
not at all alter the Angle MG K (Fig. 
16) when it its the leaſt, or its Comple- 
ment the greareſt. But if the Priſm 
be diſpoſed in any other Situation than 
is hn deſcribed (as when the Rays 


entering perpendicularly are only refrac- 
ted 


Fig. 


65 


6 off the M As Sgr. II. 
ted at their going out) a very ſmall Er- 
ror from that deſired Situation will much 
alter the refracted Angle, and ſo the 


Experiment would be much more lia- 
ble to Uncertainty and Errors. 


32. A Rule Fox the greater Variety in this mat- 
to find the | : | 

Reſraction or ter, becauſe there are given ſome Caſes, 
Mediums 4 
— Ware the Refractions cannot be meaſu 


to one ano- red by the Ways hitherto deſcribed (as 
RefaRtions when the Refraction is made out of 
when conti- Glaſs into Cryſtal, out of Water into 
guous to the | | 5 
Air are Glaſs, or out of one Liquor into ano- 
known ther) and that there may be no refrac- 
| ting Surface, whoſe Refraction cannot 
be inveſtigated, it 1s thought proper to 


propoſe the following Problem. 


The Refrattions * given, which 
are made by two Mediums, conti- 
guous to a third, to find their Re- 


fraction, when contiguous to one 
another. 


Fig. 18. 2: Dig. 18, LeT the two Mediums 
propoſed be A and B, the RefraQtion of 
whoſe terminating Surface is ſought, 
_ let C be a third Medium, the Re. 


2 fractions 


fractions of whoſe Surface contiguous 
to A and B are given. And let the 
Sine of Incidence to the Sine of Re- 
fraction out of the Medium C into the 
Medium A be, as I to R, and the Sine 
of Incidence to the Sine of Refraction, 
out of the ſame Medium C into the o- 
ther Medium B, as ; to; I ſay, that 
it is I X 7. R X 7 : : Sine of Inci- 
dence to the Sine of Refraction out of 
the Medium B into the Medium A. 


Fox Example, let be propoſed the 
Inveſtigation of the Refraction out of 
Water into Glaſs, the Refraction out of 
Air into both being given; and let 
the Sine of Incidence out of Air into 
Glaſs be to the Sine of Refraction, as 
17 to 11, and the Sine of Incidence out 
of Air into Water to the Sine of Re- 
fraction, as 4 to 3. Wherefore by 
multiplying reciprocally theſe Sines, it 
will be as 17 X 3 to 11 X 4, or as 51 
to 44, fo the Sine of Incidence out of 


Water into Glaſs to the Sine of Re- 
fraction. And after the ſame manner 


the Refraction out of Air into any o- 


ther propoſed Mediums being known, 


you 


63 


Of the MEASURE SECT. II. 
you may find their Refractions amongſt 
themſelves, and on the contrary. 


Bur the Demonſtration of this muſt 
not be omitted, for which is premiſed 
the following Lemma. If the two 
propoſed Mediums A and B in Fig. 18. 
be conceived to be terminated by pa- 
rallel Planes, contiguous, and encom- 
paſſed by the ſaid third Medium, as 
ſuppoſe Air, and any Ray ON falling 
obliquely at N be refracted firſt to M, 
and next to L, and it emerging pro- 
ceeds to K, I ſay the incident Ray O N 
is parallel to itſelf, as emerging L K. 
The Truth whereof appears indeed by 
Experience: For let the Medium A. be 


ſuppoſed Glaſs, and the Medium B to 


be Water, and the third encompaſſing 
Medium to be Air; and let the Surface 
M R of the Glaſs Plate A be lightly 
ſmear'd with the Water B, and let it be 
placed parallel to the Horizon, that the 
Water may be of an equal Thickneſs ; 
which being done, you will ſee, that 
the Rays tranſmitted through both the 
Mediums A and B, will tend to the 


fame 
2 


Swe nv. II. of ReEeRACTIONS: 65 
ſame Parts, towards which they would 
tend directly from the Sun. 


Tuis being premiſed, let be erected 
Nr, HMG and R LI perpendicu- 
lars at the refracting Points N, M and 
L; it is therefore 7 to 7, as the Sine of 
the Angle O Ny, to the Sine of the 
Angle MNr, or NMH; and by 
multiplying the antecedent Ratio by I, 
it will become I X to I Xx x, as the 
Sine of ON/ to the Sine of NMH. 
Moreover, it is Ito R, that is, I x / 
is to R x y, as the Sine of the Angle 
K LI to the Sine of the Angle ML R, 
that is, as the Sine of the Angle ON 
to the Sine of LM G. Now, by per- 
muting the Terms of both the Pro- 
portions, it will become I X 7. Sine 
ONZ/'::Ixr7. Sine NMH, and IX. 

Sine ON/ :: R/. Sine LMG. here. 
fore, by Equality of Ratio it is I x 7. 
R xj :: Sine NMH. Sine LMG. Q. ED, 


From theſe thus ſhewn, a no un- 34. The 
anner f 


uſeful Problem ariſes, whereby the Re- meaſuring | 
fractions of Fluids may be meaſured by —— s 


the ſame Means, as is ſhewn of Solids accommoda- 


F at ted toF luids, 


66 


Of the METSsUxAE Sect. II. 


at Fig. 16, without uſing the Inſtru- 


ment HLK, that is deſcribed in Fig. 


15. For Example, let a Priſmatic Veſ- 
ſel be made of Glaſs Plates, connected 
in the Shape of a Wedge, whoſe Edge 
or vertical Angle let be about 86 or 
go Degrees. But you muſt have known 
the Quantity of that Angle by a very 


exact Meaſure, and always put the Sine 


of its half for the Sine of Refraction. 


Which being done, when the refractive 


Power of any Liquor is deſired, let 
the Veſſel be filled with that Liquor, 
and placed in ſuch a Situation, that the 
Edge made by the Concourſe of the 
refracting Planes may be parallel to the 
Horizon, and perpendicular to the ſo- 
lar Rays, and that thoſe Rays tranſ- 
mitted through the ſaid refracting Planes 
may ſuffer equal Refractions at their 
Ingreſs and Egreſs. And by the Help 
of a Quadrant, as was ſhewn at Fig. 
16, let be found the Angle of Inci- 
dence, whoſe Sine to the aforeſaid Sine 
of Refraction will be, as the Sine of 
Incidence to the Sine of Refraction 
out of Air into the propoſed Liquor. 


FOR 
2 


Sgcr. II. of REFRACTIONS: 


Fox Inſtance, that I might know 
the Refraction of Water, I procured a 


Refraction of 
Water, as 1 


wooden Priſm to be made, ſuch as is wyſelf have 


Ahe in Fig. 19, whoſe Angle A CB, 
which I deſigned for the vertical one, 


meaſured it, 
brought as 
an Example 
of this things 


was right, and the other two half a Rg. 19. 


right one, and I cauſed the refracting 
Planes Ac and Be to be bored through 
the middle by the Hole F parallel to 
the Baſe A, by which Hole the Light 
is to paſs, and the third Plane to be 
bored in G, till a Way is made tranſ- 
verſly to the Hole F. Then taking two 
Glaſs Plates, which a broken Looking- 
Glaſs furniſhed me with, I fixed one o- 
ver the very middle of the Plane Bc 
by Cement, and the other over the 
middle of the other Plane A c, that 
the Paſſage F might be cloſed on both 
Sides. Then I poured Rain Water thro? 
the Orifice G into the hollow Space, 
and ſhut it with a Stopple made of 
Cork. And fo the Water, included 
between two Glaſs Plates, which were 
inclined at right Angles, ſerved inſtead 
of a Priſm of Water having a right An- 
gle. But that theſe Plates contained 

F 2 exactly 


68 


Lemma 1. 


Ne 29. 


Of the MeasurRE sxcr. II. 


exactly a right Angle, I knew by the 
means of a Square; whole half 45 De- 
grees therefore is to be reckoned the 
Angle of Refraction. This Priſm I 
then ſo placed to the Ingreſs of the 
Light into a dark Room, that the 
Quantity of Refraction on both Sides 
might be the ſame, and from the Sun's 
Height, and the Inclination to the Ho- 
rizon of the refracted Rays exhibiting 
a green Colour, I found the refracted 
Angle to be 51 deg. 16 min. whoſe half 
25 deg. 33 min. together with the An- 
gle of Refraction 45 deg. will give the 
Angle of Incidence 70 deg. 38 min. 
But the Sines of theſe Angles 70 deg. 
38 min. and 45 deg. are 9434, and 
7071 in reſpect to the Sine of 90 deg, 
T0000. The Ratio of which Numbers 
is indeed a little leſs than Cartes's of 
250 to 187, anda little greater than 4 
to 3, Viz. 4,002 to 3, which however 
differs ſo little from the Ratio , that 
the Error would have been inſenſible, 
if Thad put it as 4 to 3, and that chief- 
ly ſince the Refraction of Water re- 
mains not always the ſame, but ſuffers 
ſomething from the Change of Colour, 

and 


Sect. II. of REFRACTIONS. 


and aſſumes various Degrees of Den- 
fity. Which fame Thing happens to 


the ambient Air, which alſo is not only 


variouſly incraſſated by Vapours, but 
likewiſe more cloſely (the Weight of 
the Atmoſphere being increaſed) or 
more laxly compreſſed. Add that 
there are different Denſities of Wa- 
ters ſpringing in different Regions of 
'the Earth, or by the Force of the Sun 
converted into Vapours, and thence 
into Rain; and their internal Diſpo- 
ſitions to refract are different, ariſing 
from various mineral TinCtures, which 
they extract from ſubterraneous Pla- 
ces, and from Exhalations variouſly 
denſe and copious, which are raiſed ay 
loft together with the Vapours. 


60 


TE Truth of this Problem of the _ 35. The 


emonſtra- 


Meaſure of the Refraction of Fluids tion of whae 
thus ſolved, will appear by ſhewing, that bas been ſaid. 


the Quantity of the Refraction in this 
Priſm compounded of Water and Glaſ- 
ſes is the ſame, as it would be, if the 
Glaſs was taken away, and the Water 
alone remained incompaſſed with the 


Air. Let therefore in Fig. 20, C Fig. 20, 


F 3 be 


S 
/ 


g. 18. 


Of the MEASURE SecT. II. 
be a Priſm made of Glaſs Plates A C 


Fd and B Ce (as has been faid) and 


filled with Water 4 / e,; and let it be 
conceived, that DE F is a Priſm of 
Water immediately incompaſſed by the 
Air, and altogether like to the aqueous 
one dee F incloſed in Glaſs, and ſimi- 
larly poſited ; and let parallel Rays 
ON, O K fall upon both, whereof 
one ON refracted in N, M, L and K 
tends to H, but the other O X refrac- 


ted in X and Y tends to Z. I now fay, 


that the emerging Rays K H and Y Z 
will be parallel, and therefore that in 
both Priſms the whole Quantity of Re- 
fractions will be the ſame. For in Fig. 
18, if the Ray o parallel to O N fall 
on the Glaſs Plate A, and emerges in 
4K, it is known, that the Ray / + will 
be parallel to o mn, that is to ON 
and LK, and ſince /4 and L K are pa- 
rallel, 21 and ML will be alſo parallel; 
whence appears the Propoſition, that 
the Quantity of Refraction out of 
Air into any propoſed Medium is the 
fame, whether the Rays enter imme- 
diately that Medium out of Air (as is 
done at om or they firſt pervade _—_ 

Fer 


SE Cr. II. Of REFRA CT1ON 8. 


ther Medium interpoſed, and termina- 
ted by parallel Planes, (as is done at 


71 


ON M L) and on the contrary. | 


And the ſame thing is to be underſtood, 
when inſtead of Air any other Medium 
is uſed. Wherefore in Fig. 20. ſince 
the parallel Rays O X and ON fall 


upon the Priſnis D FE and A CB ſi- 


milar and ſimilarly polited, the Quan» 
tity of the Refraction out of Air into 
Water will be the ſame, whether the 
Rays immediately enter, as is to be ſeen 
at D E FE, or firſt pervade the Glaſs 
Plate Ad f C, that is, the Ray X Y 
once refracted will be parallel to MI. 
twice refracted; and for the ſame Rea- 
ſon, ſince X Y and ML are parallel, 
the emerging Rays Y Z and KH will be 
alſo parallel. Wheretore ſince the inci. 
dent and emerging Rays are parallel, 
the whole Refraction of both Priſms 
will be the ſame. And theretore ſeeing 
an aqueous Priſm contiguous to the 
Air cannot be made by Reaſon of the 
Fluidity of the Water, in its Stead may 
be uſed a Glaſs Priſm filled with Water. 


Q. E. D. 


| a Ano 


Refractions 
of different 
Sorts of Rays 
are compa- 
red, and the 


Difference of the greateſt Reſrangibilities is inyeſtigated. 


Of the MEASURE Sxcr. II. 


Ap thus is ſhewn a general Me- 
thod, whereby the Refractions, out of 
Air into any propoſed Medium, may 
be determined, which is very eaſy, and 
little liable to Errors, eſpecially if the 
Angle of the Priſm is large, and ex- 
actly known, the Quadrant large and 
accurate, and the Obſervation made 
far behind the Priſm, where the Co- 
lours being much dilated, are the more 
eaſily to be diſtinguiſned. And beſides, 
ſince the Refraction between Air and 
the propoſed Mediums, are thus deter- 
mined by Experiments: A Rule is laid 
down, No. 32, whereby the Refracti- 
ons of the ſame Mediums, contiguous 
to one another, may be diſcovered. 
Which is ſufficient to have ſhewn in il- 
luſtrating the firſt Caſe, of meaſuring 
Refractions, when tlie Proportion of 
the Sines of Incidence and Refraction 
is ſought in the ſame Sort of Rays. 


THe other Caſe is now to be proſe- 
cuted, where the Refraction of Hetero- 
geneal Rays are to be compared. But 


that 


5 nd J)) rd doe! wo > 2. . 


* * <4 4 
n «= 
= _ 4 * 
c v * 
= 
0 "IP 
. 


'' 


N 
2 * 
* 


SECT. II. of REFRACTIONS. 
that the Sine of Refraction of any Sort 
of Rays is to the Sine of Incidence, in 
a certain given Ratio, you may try, by 
meaſuring the Refractions of every 
remarkable Sort Incident ſeparately, 
according to various Obliquities upon 
ſome retracting Medium, as upon Wa- 
ter (at Fig. 15.) ſtagnating in a Veſſel, 
or upon Glaſs Priſms, whoſe vertical 
Angles are of different Magnitudes. 
For by one Priſm you may find the 
Proportions of the Sines to every Sort of 
Rays, as was ſhewn at Fig. 16, then by 
other Priſms (or by the other Angles, 
whether greater or leſs of the ſame 
Priſm ) you may diſcover whether the 
ſame Proportions are obſerved in other 
Obliquities. And ſo (Obſervations being 
moſt accurately made) it will at the ſame 
Time appear, that the Refractions of any 
Sort of Rays are made, according to 
certain Ratios of the Sines, and the 
Ratios of thoſe Sines will be known. 
But at preſent, ſince I know the Re- 
fraction of any Ray to be the ſame, 
whether it is incident, mixed with he- 
terogeneal Rays, as in the Sun's Light 
not yet refracted, or is firſt ſeparated 
| 1 from 


© 


73 


Fg. 15. 


Fig . 16. 


—ͤ—ñ—ÿmũ—4 — — —  — — — 


74 


Fige 21, 


Of the MEASURE Sgr. II. 


from Heterogeneal Rays: I will ſhew 
how theſe Proportions may be obtained, 
by the Refraction of the Sun's immedi- 
ate Light, in the firſt Place by determi- 
ning the Proportions of the Sines of 
Nefraction amongſt themſelves, in Re- 
ſpect of the ſame Incidence, and then 
by Compariſon with the common Sine 
of Incidence. And becauſe it is eaſy to 
make a Judgment of the intermediate 
Sorts of Rays, provided the Refractions 
of the Extreams were known, it will 
be ſufficient, if I ſhall compare the moſt 
refrangible Rays of all with the leaſt 
Refrangible. Therefore, in Fig 21, 
let ABC be a Glaſs Priſm fo placed, 
that the Rays both entering and emer- 
ging, may ſuffer, as before, the ſame De- 
gree of Refraction: But let there be 
choſen a clear Day, and the Room be 
very dark, that the Colours, even to 
the Extremities of the Spaces they 
poſſeſs, may be ſeen ſufficiently diſtinct. 
Then, at the Diſtance of Twenty Feet 
or more from the Priſm, let the Rays 
be received on a Paper directly oppo- 
ſed to them, and let the Length and 


Breadth of the Space, (as P T) illumi- 


nated 


SEC T. II. of REFRACTIONS. 


nated by the Colours, be meaſured. So 
a Priſm being uſed, whoſe vertical 
Angle A C B was 63 Deg. 12 Min. 
and the Breadth of the Hole admitting 
the Rays being + of an Inch, at the 
Diſtance XP or XT Twenty two 
Feet, I found the greateſt Length of 
the Image PI to be 135 Inch. nearly, 
and the Breadth , Inch. Now, if the 
Breadth of this Image be taken from 
its Length, there will remain 10 + Inch. 
the Length it ought to have, if the 
Diſc of the Sun, and the Diameter of 
the Hole F was infinitely ſmall, that is, 
if all the Rays had flowed in the fame 
right Line OF. That Line therefore 
of 10 ＋ Inch. ſubtends an Angle, 
which two Rays incident alike, make 
by the Inequality of Refraction, where- 
of one is refracted the moſt of all, ha- 
ving the like Incidence, and the other 
the leaſt of all: Which Angle there- 
fore, by Calculation, is found to be 
2 Deg. 18 Min. But ſince that Angle is 
made by a double Refraction at X and 


Y; and beſides ſince both are ſuppoſed 


equal, a Calculation ſufficiently exact 
for this Buſineſs might have been = 
5 Or- 


73 


76 


Of the MEASURE SECT. II. 


formed from one Refraction only, as 
that made at the Side B C. For if the 
vertical Angle A CB be biſeQed by 
the Plane D C, and the other Half of 
the Prim D CB, or DC A be con- 
ceived to be taken away, the RefraCti- 
on made at the other Half by the Rays 
OF obliquely falling on AC, and 
emerging perpendicularly from the Side 
D C, or falling perpendicularly on the 
Side D C, according to one certain 
Line X I, and emerging obliquely 
from the Side B C; the Refraction, I 
fy, thus made at the other Half, would 
be Half the Refraction at the whole 
Priſm, provided one particular Sort of 
the mean refrangible Rays were on- 
ly regarded. Moreover, it all the other 
Sorts of Rays were at the ſame Time 
regarded, that Aſſertion, though it is 
not abſolutely true, yet it approaches 
the Truth ſo nearly, that as to Senſe 
and mechanical Calculation, it may be 
looked upon as true. Wherefore, ſince 
a Geometrical Calculation of both the 
Refractions made at X and Y is tedious 
to perform, I ſhall not fear to do it by 


a Way more accommodated to Pra-: 
 Qicez 


Sect. II. of REFPR ACTIONS. 

Qice, though a mechanical one; truſt» 
ing I ought not to be blamed, if whilſt 
I apply Computations to phyſical Mat- 
ters, I ſhall omit ſuch minute Circum- 
ſtances, as would occaſion a trouble- 
ſome Work to no Purpoſe: I ſhall 
therefore conſider the Refraction from 
one Side of the Priſm only, and becauſe 
all Rays, beſides the mean refrangible 
ones, ought to be twice refracted by 
the Half A C D, and once only from 
the other Half D CB, entering per- 
pendicularly the Plane Side D C, ac- 
cording to the Line XY: T herefore 
let the Calculation be made in the 
. Half DCB, that is, at the plane Side 
B C, it being ſuppoſed, that all the 
Rays flowing according to the fame 
Line X V, the Angle, which the moſt 
refrangible Rays would make with the 
leaſt refrangible, after they were re- 
fracted by the Side BC, would be 
Half the Angle PX T, that is, 1 Deg. 
9 Min. Now, ſince the Angle of Inci- 
' dence of the Ray X Y, from what has 
been ſhewn, is 31 Deg. 36 Min. and 
the Angle of the mean Refraction 


54 Deg, 10 Min. let all theſe be tranſ- 


I ferred 


- FE 


78 


Of the MexsuxeE Sxcr. II. 


Fig. 22. ferred to Scheme 22, by making C B 


the Surface terminating the Medium of 
Glaſs towards A, and that of Air to- 
wards F, and the Angle of Incidence 
XH to be 31 Deg. 36 Min. and 
the Angle of RefraQtion R Y F will be 
54 Deg. 10 Min. and the Angle P YT 
1 Deg. 9 Min. v:z. the Difference of 
Refraction between the moſt refran- 
gible Rays YP and the leaſt refran- 
gible JT: which Angle, let be bi- 
ſected by the mean refracted Ray YR, 
poſſeſſing the Confine of Blue and 
Green. And thence the Angle PR, 


or RY T will be 344 Min. Half the 


whole PY T. And therefore the Angle 
PXE will be 54 Deg. 4454 Min. and 
the Angle TYE 53 Deg. 35+ Min. 
and their Sines PG and F T will be 
81656 and 80481, whoſe Proportion 
being reduced to more ſimple Num- 
bers, will be TF to PG, as 69% to 
685 nearly. When I had often after 
this Manner made Experiments and 


Calculations, the Proportions of theſe 
Sines always came out between the Li- 
mits 67 to 66, and 72 to 71 ; but I oft- 
neſt fell upon the Proportions 69 to 

7 


Scr. II. of REFRACTIONS: 79 


68, 691 to 68; and 70 to 69, whoſe 
Difference is ſo ſmall, that it matters 
not, which is made Uſe of. 


Tar Ratio of the Sines of Refrac- 38. The 
tion for the extreme Sorts of Rays a- Nag of heſs 
like incident being thus Pound, their are compa- 
Computation to a Sine of Incidence is al- * 
ſo known: vigz. which has been before _ of In. 
found to be 52400; and by comparing * 
this 52400 to the Sines 81656 and 
80481, their Ratio in leſs Numbers will 
be found 44% to 69% and 68-, or 44+ to 
69 and 68 nearly; the Refractions being 
made out of Glaſs into Air. 


Bur if on the contrary the Rays are 5138. The 
alike incident out of Air into Glaſs, the Rays inci- 
Proportions of the Sines are without _—_ Pars 
any Trouble diſcovered from what has of 1 
been already found; as they are reci- Surface ale 
procal. Let I be the common Sine of *iprocally 
Incidence out of Glaſs into Air, P the 
Sine of Refraction of the moſt refran- 
gible Rays, R of the mean refrangible 
ones, and T of the leaſt refrangible. L 


ſay, that from the reciprocal Proportions 
of theſe, if I be put for the Sine of In- 
: cidence 


_ Of the MEASsSURKE Sxcœr. II. 


cidence out of Air into Glaſs, 5 will be 
the Sine of Refraction of the moſt re- 


frangible Rays, K the Sine of Refrac- 
tion of the mean refrangible Rays, and 
= of the leaſt refrangible. For ſince 
the Sine of ence of the moſt re- 
frangible Ray eut of Glaſs into Air is I, 
and the Sine of Refraction P, the Sine 
of Incidence of that Ray paſſing back- 
wards by the ſame Lines out of Air in- 
to Glaſs will be P, and the Sine of Re- 
fraction I; for now that is the Incident 
Ray, which before was the refracted. 
The Sine therefore of Incidence of a 
moſt refrangible Ray out of Air into 
Glaſs, however incident, is to the Sine 
of Refraction, as P to I, that is, (by 


applying the Ratios to P) as 1 to 5 
that is, (by applying laſtly to I) as 7 to 
F and by a like Argument it will ap- 


pear, that the like Sines of a mean re- 
- frangible Ray are, as 1 Tto x and the 


Sines of the leaſt refrangible as F to 


= It 1s manifeſt thereſore, that 7 1 7 be- 


ing 


Stor. II. of RETR ACTIONS: 81 
ing put for the common Sine of Inci- 

dence ” = and = will be the reſpec- 

tive Sines of each Sort. 


1 will illuſtrate it in N umbers. Since 145 The 
cti 
44% to 69 and 68 is the Ratio of the © of Ga. 1 
common Sine of Incidence to the Sines illuſtrated. 
of the greateſt diſagreeing Refractions 


out of Glaſs 15 Air, it will be as 


4 1 59 88 
444 to 53 and =; of = 447 (=106 nearly) 


to 68 and 69. That is, for the leaſt 
refrangible Rays as 106 to 69. Theſe 41. From 
being thus determined, the Proporti- — of che 
ons of the Sines for the intermediate — 
Rays are eaſily determined from know- ealy to make 
ing the Diſtances of the Colours, which fue 
they obſerve in the coloured Image. mediate ones. 
So the Rays which approach a little nea- 

rer to the Blue than the Yellow, ſince 

they fall on the middle of the Image, 

they will have the intermediate Propor- 

tion of the Sines 44+ to 684 or 106 to 


68+ nearly; and fo of others. 


G ArTER 


82 Of the Me xasurRE Sect. II. 


— Ar rx the ſame manner that the Re- 
Therrem bow fractions at Glaßß are determined, it 


from the Re- may be done for other Mediums, but i it 
fractions of 


heterogeneal will be worth while now to ſhew, how 
Rays at Glaſs the Meaſures of thoſe Refractions may 


or any other 


Medium be be determined for any other propoſed 
ned — Medium from their Sines being thus 


themſelves, determined for Glaſs; and that although 
tions at any it be contiguous to any other Medium 
other i: than Glaſs. In Fig. 23. Let AB bea 


ums contigu- | 
ous to the Surface terminating the Air on the Side 


— 1 and Glaſs on the Side G, at any 
ned among Point whereof & let be drawn the Line 
without the EX G perpendicular to it, and beſides 
ei Baer. let be conceived the right Line IX to 
ments be drawn in an Angle I X A infinitely 
8. % ſmall, according to which let all the 
Rays of all Forms be ſuppoſed to be in- 
cident, and in X to be refracted: As 
the moſt refrangible ones towards P, 
the mean refrangible towards R, ind 
the leaſt refrangible towards T, and the 
other intermediate ones n inter- 
mediate Parts: Moreover let be drawn 
any Line G H parallel to the Line of 
Incidence IX, that is, perpendicular to 
FG, but let it cut the Rays in the 


Toints 


Srzcr. II. of REFRACTI ONS. 
Points P, R and T, from which let 
fall PC, R D and TE perpendicular 


to the refracting Surface AB. Theſe 


Things being thus determined and de- 
ſcribed for Glaſs, if any Medium be 


now conceived to be ſubſtituted in the 


room of that of Glaſs, all things remain- 
ing the ſame, and a refracted Ray Xr, 
belonging to any mean refrangible Ray 
incident in the Line IX at X, be drawn 
cutting the right Line D R in r. 
Which I ſuppoſe to be done, becauſe I 
have before ſhewn, how the RefraQtions 
of the mean refrangible Rays may be 
inveſtigated for any Mediums. Then 
through the Point V let be drawn the 
tight Line t cutting the Lines CP 
and E I in p and ? perpendicularly, and 
let X and r X be joined. I fay, that 
the moſt refrangible Rays incident 3 in 
the ſaid Line IX will be refracted in- 
to the Line X 7, and the leaſt refrangi. 
ble into the Line Xr, and the Rays 
of any Sort, which Glaſs refracted to 
any Point of P T, will be refracted to 
a correſpondent Point of the right Ling 
et by the other Medium, that is ſup- 
poſed to be ſubſtituted for Glaſs ; thoſe 
G 2 Points 


83 


4 


Of the MrxsURE Sgr. II. 


Points of the Lines P T and 2 # being 


reckoned correſpondent, thro' which 
any right Line parallel to D R paſſes. 


The manner therefore appears, how 
the Refractions of any Rays incident 


with the greateſt Obliquity out of Air 


into any propoſed Medium may be de- 


termined, the Refraction of one Sort 
of Rays only into the Medium being 
known; and the Proportions of the 
Sines from that moſt oblique Refraction 
being determined, the Refractions of the 
ſame Rays will be given to any ener 


n Incidence. 


43. Of the THE Certainty indeed of this Theo- 


Certainty of 
2he Theorem. 


rem I have not yet derived from Ex- 
periments, but fince it ſeems ſcarcely 
to differ much from the Truth, I have 


not doubted for the preſent to aſſume 
it Gratis. Hereafter perhaps, I ſhall 

either confirm it by Experience, or cor- 

rekt it, if I ſhall find it to be falſe. 


44. Ofits As to Calculation, it may be eaſily 


Calculations made from this Proportionality, that 


the Sine of Incidence of the Ray IX 
(that is, the Sine of 90 deg.) is to the 
f. Sine 


Sect. II. of REFRACTIONS. 


Sine of Refraction (ſuppoſe that made 
in the Line XR) as X R to RG; fo 
in Glaſs it will be X R. R G:: us. 

635, and X P, PG :: 1c6. 68, and XT. 

T'G,.:: 206; Ga ad thence ir will be 

deduced, that. G P. G R. GT:: 39- 

39. 40. Which Proportions being once 

found they may be reſerved, to the End 

that by their means Refractions to o- 

ther Mediums than Glaſs may be deter- 

mined. For any Medium being propo- 
ſed, let be taken X E = 40, DE = & 
and CD = 2, and let the Perpendicu- 
lars CP, DR and E T be eretted. 

Then from the given Proportion of the 

Sines of Refraction of the mean refran- 

gible Rays, that is, from the given Pro- 

portion of Xr to X D, there will be 

given the Point 1 and the Length D 7, 

to which Cy and Ef are equal: And 

the Points p and # being thus given, 
there are given the Ratios of X p and 

X C, that is, of the Sines of Incidence 


and Refraction for the moſt refrangible 


Rays, as alſo the Ratios of X z and XE, 


that is, of the Sines of Incidence and 


Refraction for the leaſt refrangible Rays. 
So for a Surface bounding Water and 


» & ' 2 


85 


86 


45. Ano. 
ther Theo- 


Of the MEasurRE Sect. II, 


Air, thoſe Sines are as 68 to 90 for 
the leaſt refrangible, and as 68 to 91 
nearly for the moſt refrangible Rays, 
The Proportions of the Lines X C, 


XD and X E being thus found, the 


Meaſure of the Refractions out of Air 
into any propoſed Medium, and made 
a any Incidence, may be determined 
by another not inelegant Theorem. 


IN the Line FX (Lig. 24.) per- 


rem to per- pendicular to the refracting Plane A B 


ſorm the 


ſame Thing: 


Fig. 24. 


let be taken any Point F, which let be 
ſuppoſed lucid, and let any Line F d 
be drawn cutting A B in 4, and let it 


be conceived to be a mean refrangible 


Ray, whole refracted Ray out of Air 
into the propoſed Medium let be 4 M, 
which produced backwards let it cut 
FXin 4  Morcover let be made F d. 
RN KE (:: 39]. 0. and 
Ke: 
39.) and with the Center F and Hu- 
tervals Fc and F «© let be deſcribed 


Circles cutting A B in e and c, and let 


be joined Fe, Fc, Fe, F 6, and let Fe 
and fc be indefinitely produced to- 
wards N and L. Inow lay, if the leaſt 


refran- 


86 O the MEASURE SFr. II, 


Air, thoſe Sines are as 68 to 9o for 
the leaſt refrangible, and as 68 to 91 
nearly for the moſt refrangible Rays. 
The Proportions of the Lines X C, 
XD and X E being thus found, the 
Meaſure of the Refractions out of Air 
into any propoſed Medium, and made 
at any Incidence, may be determined 
by another not inelegant Theorem. 


, 45 _— IN the Line FX ig. 24.) per- 
ren to per · Pendicular to the refracting Plane A B 
fo king, let be taken any Point F, which let be 
Fg. 24. ſuppoſed lucid, and let any Line F d 
| be drawn cutting A B in d, and let it 
be conceived to be a mean refrangible 
Ray, whole reiracted Ray out of Air 
into the propoſed Medium let be 4 M, 
which produced backwards let it cut 
FX in 2 Morcover let be made F d. 
Fe $$S D. XE (:: 239-40.) and 
F i: KD XC : 20h 
39.) and with the Center F and In- 
tervals Fc and F c let be deſcribed 
Circles cutting A B in e and c, and let 
be joined Fe, Fc, fe, fc, and let fe 
and Fc be indefinitely produced to- 
wards N and L. Inow lay, it the leaſt 


refran- 


— 


— — 


0 


. 
4% a 


SECr. II. Of REFRR ACTIONS. 87 


refrangible Ray falls in the Line Fe, 
that it will be refracted into the Line 
e N; and if the moſt refrangible Ray 
falls in Fe, that it will be refracted 
into c L, and ſo the Rays of any Inter- 
mediate Sorts, flowing from the Point 
F, and falling upon their correſponding 
Points between c and e, will be ſo re- 
fracted by the propoſed Medium, as if 
they had all flowed from the Point F; 
thoſe Points between C and E, and 
c and e being looked upon as correſpon- 
ding, whoſe Diſtances from X and F 
reſpectively are in the ſame Ratio with 
DX to dF. For the demonſtrating 
which Theorem, the Two following 
 Lemmas are premiſed. 


1. Two Points c, 4 being taken (Fig. 46. For 
24.) in any Line A B, and other twoing this 
Points Fand F in its perpendicular F X,Th-orem. 
and fd, F d, fc and Ec being joined, 4g. 24. 
the Difference of the Squares of the 
two Lines d and Fd meeting in 4, 
will be equal to the Difference of the 
Squares of the other two fc and Fe 
meeting in c. For ſince fdq=fXq 
+X Aq, and Fd FX ＋ X Aq, 

G 4 the 


” 


Of the ME asone Stcr. II. 


the Difference fdq — F 4q will be 
—=fXq—FXq;. and for the ſame 


Reaſon the Difference eq — Feq will 
 be=fXq—FXq Wherefore the 


47. Ten- 
914 .2 
Fig. 25. 


ſaid Differences thus equal to the ſame 


Third, are equal amengſt themſelves. 
ol E. D. 


2. If any Ray F G (Fig. 25.) falls 
on the Surface A B, and is refracted to- 
wards H, the Line G H being drawn 
backwards, that it may cut the perpen- 
dicular FX in 7; I ſay, that fG. F G:: 
Sine of Incidence to the Sine of Re- 
fraction: And on the contrary, if FG. 
FG :: Sine of Incidence to the Sine of 
Reſraction, then F GH will be the Re- 


fracted Ray of FG. For let be taken 


FK = F G, and let fall K L perpendi- 
cular to FX, which being done, ſince 
the Angle G F X is equal to the Angle 
of Incidence, and the Angle GX to 
the Angle of Refraction, having Re- 
ſpect to aCircle, whoſe Semidiameter let 
be FG or FK; but. G. FK :: GX. KL. 
that 8 FG : GX. KL. Q. E. D. 


Tusk 


Stor. 11. of 'REFRACTIONS: 


N un SE Things being premiſed, 48. The 


8. 


the Theorem is thus demonſtrated. In Demonſtratl 


Fig. 24. Let be drawn IX the moſt "ig. 24. 
oblique Line, according to which 


Rays of all Forms are ſuppoſed to be 
incident, out; of Air at X, and to 
be refracted into the propoſed Me- 
dium, the moſt refrangible ones to. 
wards p, and the leaſt refrangible to- 
wards 1, and let theſe be cut by Lines 
erected perpendicular at the Points 


D, C and E in the Points 7, p and , 


as was explained at Fig. 23. Ne ow. 
ſince the Sines of Incidence and Re- 
fraction of theſe Rays are appointed to 
be as Xr to X D, XptoXC and X# 
to XE reſpeltively, if beſides it ſhall 
be demonſtrated, that 4 to F 4, fe 
to Fc and Fe to Fe are reſpectively 
in the fame Ratio; that is, that fd. 
Fd::Xr.XD:: Sine of Incidence to 
the Sine of Refraction of the mean re- 
frangible Rays, and fe. Fe :: Xp. X C:: 

Sine of Incidence to the Sine of Re- 
fraction of the moſt refrangible Rays, 
the Propoſition will be manifeſt by the 
ſecond Lemma. And as to the mean 


ICs. 


Of the Mason Sgr. II. 


refrangible Rays, ſince fd is ſuppoſed 

the refracted Ray of F a, it will be (by 
"the ſecond Lemma) fd to Fd as the 

Sine of Incidence to the Sine of Re- 


fraction, that is, as X to X D; But 


it is now propoſed to demonſtrate the 
ſame Proportionality in the other Sorts 


of Rays, as that it is Fc. Fe :: X. 


XC. For it is Fe FA :: XC. XD; 
as alſo Fd. fd :: XD. Xr by Hypo- 
theſis. Wherefore by Permutation and 
Converſion it is Fc. X C:: Fd. XD: 


Fa. Xr. and by ſquaring Fcq.X .Cq:: 


FA q. XDq.:: Faq. X q. and by 
diminiſhing by the Terms of the equal 


Ratio, Feq. X Cq :: eq. Ce q | 


RC q, (Xpq). Laſtly, By extracting 
the Roots of the Terms, and by Per- 
mutation it is fc. F.c:: Xp. X C. 
Wherefore fe or cL is the refracted 
Ray of Fc by the ſecond Lemma. 
QE. D. And by the fame Argument 
it will appear, that e N is the refraQted 
Ray of Fe. And the ſame is to be un- 
derſtood of other Rays variouſly poſ- 
ſeſſing the intermediate Spaces accor- 
ding to the various Degrees of Refran- 


TAG 
Or 


a _—_ A a * 3 1 - m K 
5 — * * . I a * * ow > K 9 N 
I A hs 29. hs e 8 = op S+ SME + 9 eta 2 * * 
D - Nee EE 1 . — FA 3 
rs 
7 * * 
at! — + . 


Z 


— II. CTIONS, 91 


* 6 5 . 15; 


0 F the Mealuting ihe Refraftions 'of 49. The 


: Refrationg 
Surfaces contiguous to the Air, wear heteroger 


have ſaid enough; bur if the ſame a noel Rayy 
Thing be required to be done for other 72 Saris: 


ces contigu- 
Sur faces contiguous to the Air on no $5 9 e 


Part, let (in Fig. 26.) AB H, and Air, — 
abu w be any two Mediums contigiu- 3 
ous in the Plane Surface H 6, and in- ren. 
compaſſed with Air. And let the Plane 
A B be parallel to Hb, and in it let be 
taken the Point X, to which let be 
drawn X V perpendicular, and IX the 
moſt oblique Line in which (as before) 


let the Rays of all Forms be incident; 


and according to the Degree of Reffan- 
gibility be refrafted to P, Rand T, and 


other intermediate Places. The Re- 
fractions of theſe Rays thus incident on 
the propofed Surface 4'b are now to be 
fought, and ſince the Refractions of the 
mean refrangible Rays to any Surfaces 
were before expoſed, let of the Ray XR 
the refracted one be R M, and let it be 


drawn backwards, till it cuts the Per- 
pendicular X V in , and let moreover 
be drawn fP, FT, and let them be 
produced to Land N. I fay, that = 
W 


92 


O the MeasuxE Stcr. II, 
will be the refracted Ray of XP, and | 


IN of XT, and all the Rays of the 


other Forms falling between P and T, 


will be ſo refracted, as afterwards to 
diverge from the Point f. For let it be 
conceived, that the Medium 4 6 u m 


be produced out further towards a , 
than the Medium A BH, fo that 
the Part of its Plane 4 H b between 
H and a, may be contiguous to the 


Air, and to ſome Point in it F let be 
_ drawn the perpendicular F g, alſo the 


moſt oblique Line / E, in which let the 


Rays of all Forms. be incident, and ac- 


cording to their Degree of Refrangibi- 


lity be refracted to p, r, t, and the 
1 Places, juſt as it was done 


at the Surface A B of the other Medi- 


um. Beſides, let be taken FD GR, 


and be drawn D parallel to F g, that 
it may cut the Ray Fr in r, whence 
let fall g perpendicular to Fg, and 


cutting the other Rays F and F f in 


P and t. Now, Since it is gr =G R, 
it will alſo be gp = GP, and g t = 
GT, from what was ſhewn at Fig. 23, 


And moreover, from what was ſhewn 


at Fig. 18. ſince the Refraction of the 
* 


SEC. II. of RETX ACTIONS. 
Rays incident in the parallel Lines IX 
and 7 F on the Medium 2b um is tlie 
ſame, whether they immediately enter 
out of the Air, as is done at F, or 
do firſt pervade another Medium, as 
AB 4H terminated by parallel Planes; 
it follows, that the Rays refracted after 
either of theſe Ways into the ſaid Me- 
dium 4 bun n, are parallel to the homo- 
geneal Rays refracted after the other 
Way into the ſame Medium, that is, 
that F is parallel to PL, Fr to 
RM, and Er to TN. Whereforeif 
the refracted Rays PL, RM and 
TN be drawn backwards, till each 
meet the perpendicular G X, they will 
conſtitute with it, and the Baſes G P, 
GR and G T. Triangles, ſimilar to the 
Triangles g E, grF and g E, andalſo 
equal to them; for their Baſes gy and GP, 
gr and GR, gt and G I are reſpeCtive- 
ly equal to one another. Wherefore, 
ſince the Vertex's of theſe Triangles 
meet at the ſame Point E, the Vertex's of 
thoſe Triangles ſhall meet at ſome other 
Point f. That is, the Rays PL, RM 
and T N, the refracted Rays of X P, 
| XX 


Is 


94 


50. The 


* is 
1 by 
Notes. 


'Of the MrAsURR Sect. II. 


XR and X T, ſhall all oo from the | 
fame Point f Q. E. D 


Tris being ſhewn, the following 
Things offer themſelves to our Obſerva- 
tion. 1. That the Proportions of the 
Sines of Incidence and Refraction, made 
at the Surface H 6, are eaſily determi- 
ned from theſe. For as to the moſt re- 
frangible Rays F T is to X T, Ge. 


2. HENex if tlie Proportions of the 
Sines of Refraction out of Air into any 
two propoſed Mediums, at like Inci- 
dences, be given ; the Proportions of 
the Sines of Refraction out of one © 
the Mediums into the other will be 
eaſily given, vis. by dividing the 
Sines of the latter Medium by the 
correſponding Sines of the anterior Me- 
dium. So when the Nefraction is 
made out of Air into Glaſs, the ſaid 
Sines are as 68, 682, 69; and when it 
is made out of Air into Water, they 


are as 90, 90g, 91. Therefore, when 
it is made out of Water into Glaſs, they 
wil 


will be as =} 55 52 52 that is, as 201, 
2015, 282 nearly. 20 


3. If any third Medium denſer than 


Air is placed behind the Medium & - x 
m, touching it in the Surface m u, that 


is conceived to be a Plane, and parallel 


to AB and 2 6, and if Rays diverging 
from the Point f (as it was juſt now 
ſhewn) fall upon it at the Points L, 
M and N; after they are refracted in 
the ſame, they will diverge again from 
any other Point x, that is ſituated in 
the Perpendicular X G, and fo on 12 
infnitum, however many the Mediums 
be, that are feparated from one ano- 
ther by parallel Planes, and follow one 


another in Order. But if Air immedi- 


ately ſucceeds the Medium 46 m, that 
Point x, from whence the emerging 


Rays tad will be ſituated at V in the 
very refraQting Surface, becauſe they 


will emerge parallel to the greateſt 


oblique Line I X, in which they 
were at firſt incident out of the Air; if 
they may be faid to emerge, which 


never 


95 


Of the MEASsURE Sgr. II. 


never d ivaricated : from the refracting 
Surface. 


Fig.24 4. Ir the Rays, diverging from any 
Point F ſituated in the Air, tend to the 
Points c, d, e, after the manner, that 
I explained at the 24 Scheme, and 
3 then paſs through various refracting 
Planes parallel to A B, they will e⸗ 
very one always diverge from ſome one 
Point, that is ſituated in the perpendi- 
cular of the Plane, that paſſes through 
E, not otherwiſe than if they had been 
incident on the Plane A B, proceeding 
in the moſt oblique Line I X, and the 
Lengths of the Rays intercepted be- 
tween the refracting Points and the 
ſaid Perpendicular, are as the Sines of 
Incidence and Refraction at every Plane, 
that they reſpect. The Demonitrations 
of which Aſſertions, ſince they are ea- 
fily derived from what has been faid 
| before, I omit, that I might not ſeem 
to dwell too much on this Affair. 


- SECTION 


FACE 
* « 


SECTION III. 
Of the REFRACTIONS of 
PLANES © 


HE Laws of RefraQtions having 
been laid down, other Affections 
of Rays tranſmitted through different 
Mediums are now to be delivered; and 
in the firſt Place I will deſcribe the Re- 
fractions of Planes for the Sake of the 
Doctrine of Colours hereafter to be ex- 
plained. Then I ſhall declare the Pro- 
perties of ſpherical and other Surfaces, 
both that the Phænomena of Colours 
thence ariſing may be detected, and alſo 
that the Conſtruction of the Inſtru- 
ments ſerving to optical Uſes may be 
the more truly known. But firſt 1 
ſhall conſider the Refractions of a ſin- 
gle Plane, then the repeated A 
ons of Planes. i 


. I - 4 n 
4 "> wb a 
. . : *4 N 
9 . at g* 
| * * LE 7 ns 
.\ \ ＋ ED 


8 


Of the Refraffions Sect. III. 


Of theRgrrACTIONS of a ſingle 


PLANE. 


As to what relates to Rays of any 
the fame Sort, their Affections (upon 
theſe Principles, that the Rays of Light 
in a ſimilar Medium are direct; that 
their Refraction is made in a Surface 
perpendicular to the Surface of the re- 
fracting Medium, and that the Sines of 
Incidence are conſtantly proportional to 
the Sines of the Refractions made in a. 
nother ſimilar Medium) are delivered 
in Dr. Barrow's Lectures; and there- 
fore it will be ſufficient to enumerate 
here ſome of them under the Form of 
Lemmatical Propoſit tions without their 
Demonſtrations. 


PROP. I. 


a The incident Ray of any refratted 
one becomes interchangeably the refrac- 
ted Ray of an incident one. 


C Barrow's Opti ca Lectures, Les, III. Art. III. 
PROP. 


PROP. II. 


b To an equal or a greater Angle of 
Incidence, there belong an equal and a 
greater both Angle of Refraction and 
refracted Angle, and the contrary. 


PROP. III. 


© Of Incident Rays to exhibit their 
refracted ones. 


TAxk an Inſtance in Rays diverging” 
out of a rarer Medium into a denſer one. 


IN Fig. 27, let F be a Point emit- Eg. 27: 
ting the Rays FR, Fr and innumera- 
ble others towards the refracting Sur- 
face A R, and let FA be the perpendi- 
cular Ray, which produce to K, that 
it may be A F to AK, as the Sine of 
Refraction to the Sine of Incidence, 
and at K ere& the perpendicular K L. 

Which being done, produce backwards 


b Barrow's Optical Tet. Le. III. Art. IV. & VI; 
* Ibid. Let. IV. Art. V. 


Ha any 


100 


Fig. 28. 


a 


Of the Refraftions Sxœr. III. 


any incident Rays FR, Fr, until they 
meet the aforeſaid K L in L and , and 
in the Angle FAR inſcribe RD — 
RLand rd Y I. Which being pro- 
duced towards M and m, you will have 


the refracted Rays RM andy m; and 


after the ſame manner you may ſpeedily 
draw many retracted Rays. 


PROP. IV. 


To deſign a Ray parallel to a given 
right Line, whoſe refracted Ray ſhall 
paſs through a given Point. 


IN Fig. 28. let A B be the refrac- 
ting Surface, M the given Point, and 


G the right Line, to which the in- 


cident Ray ought to be parallel. And 
firſt draw by Prop. III. HI the refrac- 
ted Ray of a Ray incident in GH, and 
draw MR parallel to it, and FR drawn 
parallel to the given Line G H will be 
the incident Ray. 


4 PROP. 


* 


Sect. IIL of a ſingle PLAN x. , as 


PROP. V. 


To\deſign a Ray proceeding from a 

given Point, whoſe refracted Ray ſhall 

become parallel to)a right Line given 
in Poſition. 


IT is performed after the manner of 
the IVin Propoſition; the Denomina- 
tion of the _ IF # ged accor- 

VI 


ding to Prop. I. 
PRO 


To deſign a Ray proceeding from 
@ given Point F, "whoſe refratted 
Ray ſhall paſs through another given 
Point M. 


TaHroven F and M Fig. 29.) let Fg. 29. 
be drawn Perpendiculars to the refrac- 
ting Surface, and (the Ray falling upon 
a denſer Medium) let it be made A E 
to A F as the Sine of Incidence to the 
Root of the Difference of the Squares 
of the Sines of Incidence and Refrac- 
tion ; ; alſo T to M as the Sine of 
| H 3 Refrac- 


Of the Refrattions Scr. III. 


Refraction to the ſame Root. And in 
the Angle AI M let be inſcribed the 
right Line R H paſſing through E and 
equal to T, and let be joined F R, 
RM; For F R, RM will be the 
Rays ſought. 


Wu the Ray is incident upon a 
rarer Medium, the Denomination be- 
ing (acording to Prop. I.) changed, it 
is ſolved, as before. 

Bur how in a right Angle is to be 
inſcribed a given right Line, which 
ſhall paſs through a given Point, is 
ſhewn in Dr. Barrow's Vth Lecture by 
the Interſe&tion of an Hyperbola and a a 
Circle. 


PROP. VII. 


b Of Rays, that diverge, are pa- 
rallel or converge at a plane Surface, 
their refratted Rays in like manner will 


* Viz. in Ted. Optic. Ted. V. Art. VII. But this 
was doneafterwards by the Dr. more generally, as well 
as more elegantly, at the Beginning of his VI. Geome* 
orical Lecture. 


© Barrow's Optical, Lectures, Le. IV. Art. II. &c 
diverge, 


| / 
Sect. III. of a ingle PLANE. 


diverge, be parallel or converge. And 
on the contrary. 


PROP. VIII. 


To find the Point from which theſe 
refracted Rays diverge 5 or to which 
they converge. 


CAS. 1. Wax the Inclination of 
the Rays 1s defined, draw the refracted 
ones by Prop. III, IV, V, or VI, and 
you will have the Interſection. 


_ CAS. 2. Bur when the Inclina- 
tion is indefinitely leſs than any given, 
the Problem comes to the fame Thing, 
as if you ſought a Point in an oblique 
retracted Ray, which diſtinguiſhes and 
falls between the Interſections of the 
Rays lying on both Sides, and which 
ought to be looked upon as the Cen- 
ter of Radiation, or the Place of 
the Image in reſpect of the Eye thro' 

the Center of whoſe Pupil that. Ray 1s 
to paſs. But its Invention is thus. In 


Fig. 30. let DR M be a refracted Fig. 30» 


Ray of ay incident ons FE RN; and 
H 4 let 


104 


Of the Refrattions Scr. III. 


let. F be the Center of Radiation 


of the incident Rays whether diver- 
ging or converging, and let F A inſiſt- 
ing perpendicular on the refracting 
Plane cut R M in D. Now from A 
let fall to thoſe Rays the Perpendiculars 
A G and A H, and make R F. R:; 
F G. DH, and F will be the Center 
of Radiation of R M, and of the o- 


ther refracted Rays near R M OG on 
each Side. 


SCHOL. Bur this F 1s the Con- 
courſe only of the Rays lying in the 


Plane FAR, the refracted Rays of the 


others lying out of the Plane FAR will 
cut theRay R F neither in the Point F, 
nor any where elſe at all, if you except 
thoſe only, whoſe incident Rays lie in a 
conical Surface, whoſe Axis is AE, Ver- 
tex F, and Semi-angle A FR, in as much 
as they will all cut the ſaid R in the 


Point D, which is placed in the Axis 
FA. And ſo the Centers of Radiation 


of this R Fare chiefly two, one f made 
by the refracted Rays of thoſe lying in 
the Plane FAR, and the other by the 


K. e Barrow's Optical Lefures, Lets V. Art. XV. Ce. 
Lrretra c. 


| Sgr. III. Of 4 Single- P L ＋ * * 


refracted Rays of thoſe lying in the co- 
nical Surfaces deſcribed with the Axis 
F A, and Angles AFR, A D R. But 
as to the reſt of the Rays placed other- 


wiſe every where about E R, their re- 


fracted Rays moſt nearly approach the 
Ray R 4 {ſomewhere between D and f. 
80 — in reſpect of the Eye, through 
the Center of whoſe Pupil the Ray 
RM paſles, the Place of the Image 
ought to be diffuſed through the whole 
Space F D; or rather, ſince the Space 
f Dis the Image of one Point only F, 
we ought to fix in it for the ſenſible L. 
mage ſome one Point, that may poſſeſs 
the very middle of all the Light pro- 
ceeding from it towards the Eye, lying 
between the Points D and F nearly in the 
middle Diſtance. But the accurate De- 
termination of that Point, when regard 
is to be had to all the Rays refracted 
from F towards the Pupil of the Eye, 
affords a Problem very difficult to be 
ſolved, unleſs the Aſſertion be founded 
on a certain Hypotheſis at leaſt proba- 
ble, if not accurately true. 5 ſince 
as equal a Number of Rays ſeems to 
flow towards the Eye from the Limit 

D and 


103 


105 


Of the Ref#aftihns Sxcr. III. 


D and other neighbouring Points, as 
from the Limit F and other Points a- 
like near to it; the Place of the Image 
ought ſo to be fixed in the middle of 
theſe Limits, that the Angle, which 
two Rays from D and F converging 
to any the ſame Point of the Pupil do 
include, may be always very nearly bi- 
ſected by a Ray proceeding from the 
Place of Viſion, to the ſame Point of 
the Pupil. Which Hypotheſis being 
admitted, there is nothing elſe to be 
done, than that it be made M f + 
MD.MD:: F D. D Z, and Z will 
be the ſought Place of Viſion of the 
Point F, it being ſuppoſed that at M 
is placed the Eye; for ſince it was put 
MFM D. MD :: F D. DZ; 

it will be by Diviſion, M 5 


F Z. D Z. And therefore three Lines 


being drawn from f, D and Z to M 
or rather to ſome Point indefinitely 
near to this M, the Angle, which the 
two external Lines contain, will (by 3, 
6. Elem.) be always very nearly bi- 
ſected by the interjacent Line. 


THESE 


Sgcr. III. Of a ſangle PLA Nx. 


Tusk few Things about homogene- 
al Rays being curſorily remarked for 
the Sake of what follows; in order to 
get a fuller Knowledge herein, I adviſe 
the Peruſal of the Lectures, which the 
Reverend Dr. Barrow has more largely 
compoſed on the ſame Subject; and I 
proceed immediately to diſcourſe of 


heterogeneal, or unequally refracted 
Rays. 


PROP. K 


Of Rays of different Sorts, flowing 
from a lucid Point, whoſe incident 
ones are neareſt to one another; thoſe 
alone can be refracted to a Focus or 
other common Point, which lie in a 
Plane paſſing through both Points, and 
perpendicular to the refracting Plane. 


For the Refraction of any Ray 
whatever is always made in a Plane 
perpendicular to the Surface of the re. 
fracting Medium, and two ſuch Planes 
cannot paſs through both Points. 


PROP, 


Io7 


108 


Eig. 31. 


Of the Refractions Sect. III. 


0 


Of Rays of aifferent Sorts flowing 
from a given Pot, whoſe refratted 
ones converge to another given Point, 
they, that are moſt refrangible, do moft 
divaricate from the right Line lying 
between the Points of their Concour ſe 
or of their INES: 


LET P/ FQF ( Fig. 31.) be 


diſſimilar Rays meeting on this and 


that Side in Fand 7, and it is manifeſt, 


that they will not entirely coincide; for 
ſo the Refraction would be equal con- 
trary to Hypotheſis. Nor can the 
greater refrangible Ray be nearer to the 
right Line Ff. For ſo on Account of 
the greater Obliquity from the Part 
of the denſer Medium, its Refraction 
would be greater by Prop. II. and 
Hypoth. that is, the Angle FF would 
be leſs than the Angle F QF contrary 
to 21. 1 Elem. E remains therefore, 
that EP is more refrangible, that di- 
yaricates more from the right Line F. 


L E M- 


SECT. HI. of a ſingle PLAN. 109 
LEMMA I. 


Four Lines GB, GC, GD, GE 
(Fig. 32.) being ſo drawn from a given Fig. 32. 
Point G to à given right Line EB, 
that it be GB. G C:: GD. GE. The 
4 Avgle BGC, which the leaſt GB con- 
| ſtitutes with either of the interme- 
diate ones as GC, 2s greater than the 
1 Angle DGE conſtituted by the other 
intermediate G D, and the great- 
% GE. 


Fox with the Center G, Radius GE 

let be deſcribed a Circle E K, and let 

the Radius G K be drawn, conſtituting 

the Angle D G K equal to the Angle 

B GC, and let the Points K, D be 

joined: And the Triangles G D K, 

1 GB C will be ſimilar on Account 

of the equal Angles at G and the Sides 

| about them proportional, * viz. G B. a66.Flem.et 

6 C:: G D. (GE) GK. Wherefore Hyp. 

| the Angle K DG = Angle CB G, but 

| Angle E DG* > angle C B G. b 16. 1. Elem. 

| Therefore the Line K DD E De and< 7. 3, Elem. 

Angle K GD > Angle DGE of that 421.1. Elem. 
is, 


110 


Of the | Refrattioms Sxcr. III. 
is, Angle C G B Angle EGD. | 
Q. E. D. N 


LEMMA II. 


Theſe Angles being ſuppoſed infinite. 
ly ſmall, and GA let fall perpendicu. 
lar to the Line E B, it will be Angle 
E GD. Angle CGB :: BA. DA. 


Fox from the Points B and D to the 
Lines G C, G E let fall the perpendicu- 
lars BR and Dr, and the aforeſaid 


Angles will be to one another as is 


D r B R . 5 


BR and Dy to be equipollent to the 
infinitely ſmall Arches, by which thoſe 
Angles are ſubtended. But it is B G. 
CG::DG. EG by Hypoth. and by 
Diviſion B G. CR :: DG. Er. Allo 
on Account of the ſimilar Triangles 
BAG, CRB, it is B A. AG:: CR. 
BR, and by a like Reaſon E A or DA. 
46 :: Er. Dr, or A G. DA :: D.. 


Er. Wheretore by adding equal Ra- 


tios it is BA A G A G. DA (:: 
B A. DA) :: CR. RB Dr. Er 
(and the Terms of the laſt Ratios be- 

|. ing 


ing permuted) :: C R. Er. + Dr. 
B R (and an equipollent Ratio being 


ſubſtituted for CR. Er) :: BG. DG 
＋ Dr. BR (and the Terms being 


applied to one another) :: 5 56. 58 it 
is therefore B A. D A.: + 8 a — that 


is, as the Angle E GD to the _— 
CGB. Q. E. D. 


7 


Heterogeneal Rays being incident , ac- 
cording to the ſame right Line, the more 
oblique their Incidence is, ceteris pari- 
bus, the greater will be the Difference 
of the Refraction. 


In Fig. 33. Let F G be the Line, ac- 
cording to which two Rays are inci- 


dent, whereof one the moſt refrangible 


proceeds towards P, and the leaſt re- 
frangible towards T, and the Angle 
PG T will be the Difference of Re- 


fraction. Alſo FH is an obliquer Line 


than F G, and according to this let 
other two like Rays be incident, 


whereof the moſt refrangible is refract- 


ed towards p, and the leaſt refrangible 


towards t, and in like Manner the 
Angle 


111 


Fig. 32. 


112 


Of the Nefractiout Sxcr. III. 
Angle ZH # will be their Difference of 
Refraction. I now ſay, that the Angle 
? Hf is > P GT. For let fall FA 
perpendicular to the refracting Plane, 
which may cut the refracted Rays con- 
tinued backwards in D, E, L and M, 
and to this from the Point G let be 
drawn two Lines GB, G C parallel to 
HL, HM. Now ſince the three 


No, 25, 26, LinesGF, GD, GE are (from the Na- 
Cc. 


Fg. 33. 


Baſes of equally high Triangles, where- 


ture of Refraction before deſcribed) in a 
given Ratio, and the other Three HF; 


HL, H M in the ſame Ratio, HL. HM:: 


GD. GE will be proportional, but it is 
HL. HM :: G B. GC, on Account of the 


ſimilar Triangles LMH and B CG. 


Wherefore G B. GC:: G D. GE. And 


conſequently the Angle BG C Angle 
E D G by Lem. I. that is Angle L H M 
> Angle D GE, or the AnglepH#* > 
Angle PG T. Q. E. D. 

But that a fuller Determination may 
be had of the mutual Proportions (in . | 
33.) of the Angles PGT and p Hz, I 
ſay moreover, that they are amongſt one 
another very nearly, as the Lines A B 
and A D; vig. the Segments of the 


of 


Skcr. III. of a fingle PLANE. 113 


of one EG D is conſtituted by the 
Rays GP and GT, meeting with the 
Perpendicular A F, and the other 
CG is ſimilar to the Triangle MH L 
conſtituted in like Manner by the Rays 
Hy and Hr. For the Angles E G D 
and C GB, if they were infinitely 
ſmall, would be to one another as AB 
to/ A D by Lem. II; but thoſe are by 
Hypotheſis equal to the Angles PG T 
and pp Hf. Wherefore alſo PGT 
and p Hz, provided they were infinite- 
ly ſmall; would likewiſe be as A B to 
AD; and by the like Reaſon it is 
plain, that they are very nearly, as 
AC to AE. vig. their Ratio does al- 
ways lie between theſe two Ratios, 
and therefore we ſhall ſtill approach the 
nearer to the Truth, by making Uſe of 
the intermediate Ratio. vg. which is 
PGT topHras AB ＋ A C to 
AD AE, or as AB Xx AC to 
/ AD X AE nearly. 


PROP. XII. 


To deſien Rays of different Sorts, 
owing from a given Point, whoſe re- 


fratted 


114 


Hg. 34. 


fracted Rays ſhall paſs through another 
given Point. | 


_ 


| Wax one of the Points is infinitely 
diſtant, that the Rays on that Side are 
parallel; the Buſineſs is done by the 


IVin and Ven Propoſi tons; and by 


the VI" Propoſition, when both are 
infinitely diſtant. 


 SCHOL. I T will be worth While 
to ſhew, how from the given Poſition 


of any Ray, all the reſt are more ex- 


peditiouſly determined. 


CAS. 1. Let in Fig. 34. F T, FR, 
FP be Rays proceeding from E, whoſe 
retracted ones TO, R M, PK are to 
be parallel. And of the Ray F J let 
the Sine of Incidence be to the Sine of 
Refraction, as I to T; as of the Rays 
F R and FP let thoſe Sines be, as J to 
R and to P. Now, that any of theſe 
being given in Poſition, the reſt may 
be readily deſigned, let fall F A perpen- 
dicular to the refracting Surface, and 
in the Angle F A T inſcribe T E, 
TD, on this Condition, that it may be 

2 T. R. 


— 


—— — 


3 
. 
"5p oh 
B31 
F 


Sect. III. CF a4 ſagle PI ANR. 


T. R, P:: T F. T E. TD, and draw 
FR, FP parallel to TE, TD. I fay 
it is done. vig. the refracted Rays 
TO, R M meeting with the Perpendi- 
cular DA in G and H, it will be I: 


T:: TG, TE, and beſides, ſince it 


is T.R:: TF. T E (Hyp.) it will be 
by Equality I. R:: T G. TE, but it 
is I. R:: RH. RF; therefore T G- 
TE :: RH. RF, and conſequently 
Since TE and R PF are parallel (by W- 
poth.) T G and R H will be alſo paral- 
lel. Q. E. D. And there is the ſame 
Reaſon in Relation to the Paralleliſm 
of the Ray P K. 


CAS. 2. Ir the incident Rays being 
parallel, the refracted Rays converge to 
a given Point, you may nevertheleſs 
perform, what was purpoſed, as ap- 
pears from the aforeſaid Prop. 


CAS. 3. Laſtly, If the incident 
Rays diverge, and the refracted ones 
converge, the Problem is ſolid, but is 
in ſome Sort reduced to a plane one, 
by feigning the Difference of Refrangi- 
bility to be infinitely ſmall, which ſince 
| 1 2 ir 


115 


116 


Ig. 35. 


Of the Refraftions Sgr. III. 


it is always very little, I ſhall exhibit 
not unwillingly a Solution on that Hy- 
potheſis. 


Surrosk F RX, Fe. 35. to be a 
Ray given in Poſition, and the Rays 
FPX, F T X (whoſe Ratio of Inci- 
dence and Refraction are given) are to 
be inſerted at the Points E and X. Now 
feign alſo other Rays equally refrangible, 
as the Rays FP, F T to be incident 
inthe Line F R, and deſcribe (by help of 
Prop. III.) their refracted ones R O, 
R K; and ſeek (by Prop. VIII.) the 
Centers Y and Z of their Radiations, 
and join Y X and ZX, meeting the 


refracting Surface in P and T. I ſay it 


is done; viz. F PX, F TX are the 
Rays, which were to be deſigned. For 
fince by Hypotheſis the Difference of 
Refrangibility, and therefore the Diſ- 
tance of the Points T, R and P, is 
infinitely little, it is manifeſt that the 
homogeneal Rays RO, P X are the 
neareſt to one another, and thence do 
diverge from the ſame Point of Radi- 
ation Y, ſo the Ray PX is rightly de- 
termined to paſs through the Center of 
T Radiation ; 


_— 


SECT. III. Of a ſmglte PLANE. 


Radiation ; and the like Reaſon holds 
in Relation to the Ray T X. 


Bur ſince the Determination of the 
Angle P XT is to the End, that it 
may be known, how great, on Ac- 
count of the unequal Refractions of 
diſſimular Rays, is the Confuſion of 
Objects ſeen by the Intervention of 

Refraction, and through how great a 
Space the Colours thence emerging are 
extended, as is maniteſt by conceiving 
F to be a lucid Point, which appears 
to the Eye placed in X to be dilated 
and diffuſed thf®ugh the whole angu- 
lar Space. PX T, that is comprehend- 
ed by the Rays PX and JX, the 
moſt and the leaſt refrangible of all: 
I ſhall adjoin a few Things concern- 
ing its Magnitude. Feign the curve 
Line Y Z to be deſcribed, in which 
lie the Centers of Radiations of all 
Sorts of Rays, incident in the Line 
E R, and fo refracted in the Point R, 
that they may divaricate through the 
whole Angle KR O, and that Curve 
will not be unaptly likened to a lucid 
Object, whoſe viſible Angle, or appa- 

1 rent 


5 : 


118 


Of the Refraction Sect. III. 
rent Magnitude, to the Eye placed in 
X, is YXZ, and its Diſtance from 


the ſame Eye, eſtimated from its mid- 
dle, FX, and hence it follows. 


7, THAT (ſince the apparent Magni- 
tude of any viſible Thing 1s nearly re- 
cipracally, as its Diſtance) the Point F 
remaining fixed, and the Point X taken 
any where in the Line R X, the Angle 
PXT or YXZ will be nearly reci- 
procally as the Length F X; and hence 
the Interval R X being diminiſhed, the 
Angle P X T will be augmented, and 
its Quantity in any Diſtance of the 
Point X will be given, provided it was 
ever given in any one Diſtance, 


2. Moxtover the Angle OR K be- 
ing known, any Angle PXT is known, 
by taking it in the Ratio to OR K, that 


RAT [K. ſeeing Y RZ (to 


which OR K is equal) is the apparent 


Magaitude of the Object IF Z in the 
ez. FR. 


SINCE 


. 


D 
þ# 
4 


SecT. III. H @ fugle PLANE. 


StiNck therefore the Angle O RK 
for any Obliquity of the Rays incident 
in R F has been determined above in 
the Schol. to Prop. XI. and the Point 
F isnot difficultly found, by making ac- 


cording to Prop, VIII. that it be RF. 


AFq, AD 
| ff #0 Nr Kö the Invention of the 


Angle PX T ſufficiently appears. 


Bur Tobſerveby the by, that the a- 
foreſaid Curve Y / Z, in which are pla- 
ced the Centers of Radiations of the 
Rays of all Sorts refracted in the Point 
R, is the vulgar or Dzocles's Ciſſoid 


accommodated to a Circle whoſe Dia- 


. AR.FR . 
meter is RE APA . That Circle 


RCE being deſcribed, let any right 
Line F B C be drawn perpendicular to 
RE, and terminated by the Circle in 
C and the Curvein f. And by reaſon 


of the analogous Sides of the fimilar 


Triangles RAD, RB it will be A 
D q, AR DR: B F q. BR x 
FR, and by applying the laſt Ratio to 


B R, it will become A Dq. AR Xx 
14 DR 


119 


120 


Of the Refractiuus Sxkcr. III. 


DR: : Bfs. f R, and again by draw- 


ing the Sete of the Ratios into 
R V and by applying to AR, there 


will ariſe A D q. DR R:: 3 


R fq F. 5: 
1 But it is FR : RF. R f 


as before, and the "os." AE being 
drawn into D R and the Antecedents in- 
to F R, there ariſes AFq. ADq :: 
F R q. DR x R, and alternately A 
E q. FR q :: A D q. DRxXx R 
Wherefore by connecting the Ratios a- 


greeing to the ſame third, there will be 


had 85 1 : AF q. FR q, and by 


drawing the Antecedents of the Ratios 
into B R, and the Conſequents into A R, 
there will come out B q. Rãĩq :: 
AFq X BR. FRq X AR; and 
moreover by applying the latter Ratio 
to A Eq, it will become B F q. R fq:: 


B R. = ol But when I had made 


RE AR: : FRq A E q, it will be 
La RE, and therefore Bf q. 


Rfq::B R. RE, and by Diviſion 
BE; 


SECT. III. Of 4 forge PL TNF. 


B E; and from the Nature of the Cir- 
cle BC is a mean Proportional between 
B R and B E, and therefore it is BR. 
B E:: BR q. BC q, and conſequently 
r B Rq :: BR q. BC q, or B . 
B R:: B R. BC, which manifeſts the 
Curve to be the C Mid, as I propoſed 
to ſhew. 


Taz Refractions at a Surface termi- 
nating two giving Mediums having 
been treated of, I now come to diſcover, 
what follows from the increafing the 
Rarity or Denſity of either Medium 


or to compare amongſt one another hd 
Effects of different Mediums. 


LEMMA II. 


If from two Points D, G (Fig. 36.) 
ſituated in any Line A D, there be 
drawn to two other Points L, N ſitua. 
ted in its Perpendicular, the four righs 
Lines D N, DL, GN, G L; the Ra- 


tio of the Lines drawn to the remoter 


Point N acceeds more to an Equality, 
than the Ratio 7 tho iſe drawn to the 
Beares 


121 


Fige 3% 


120 


Of the Re fractions StcT. III. 


DR: : 1 FR, and again by draw- 


ing the Conſroents of the Ratios into 
R / and by applying to AR, there 


will ariſe A D q. DR X RF:: 


R fq AF. 524, 
AR. But it is R BAR : R F. R/ 


as before, and the Daw being 
drawn into D R and the Antecedents in- 
to F R, there ariſes AFq. ADq: 

F R q. DR x R, and alternately A 
F q. FRq:: A D q. DRXR f. 
Wherefore by connecting the Ratios a- 


greeing to the ſame third, there will be 


had 85 : A F q. FRq, and by 


drawing the Antecedents of the Ratios 
into B R, and the Conſequents into A R, 
there will come out BF q. Ry q :: 
AFq XB R. FR qꝗ X AR; and 
moreover by applying the latter Ratio 


to A Eq, it will become B F q. Ryq:: 


B R. . a R. But when I had made 


RE AR: : ER. A F q, it will be 
AF TN RE, and therefore Bf q. 
R fq:: B R. RE, and by n 

BE ; 


SEC. III. Of 4 fengie PL T NE. 


B E; and from the Nature of the Cir- 
cle B C is a mean Proportional between 
B R and B E, and therefore it is B R. 
B E:: B R q. BC q, and conſequently 
B f q. B R q :: BR q. B C q, or B . 
B R:: B R. BC, which manifeſts the 
Curve to be the C Mid, as I propoſed 
to ſhew. 


Tre Refractions at a Surface termi- 
nating two giving Mediums having 
been treated of, I now come to diſcover, 
what follows from the increafing the 
Rarity or Denſity of either Medium 
or to compare amongſt one another th 
Effects of different Mediums. 


LEMMA UL 


If from two Points D, G (Fig. 36.) 
ſituated in any Line A D, there be 
drawn to two other Points L,, N ſitua. 
ted in its Perpendicular, the four righs 
Lines DN, DL, GN, G L; the Na- 
tio of the Lines drawn to the remoter 
Point N acceeds more to an Equality, 
than the Ratio 7 tho iſe drawn to the 

Bearex 


121 


Tig · 36. 


122 


7} No 46. 


Fg. 3 7. 


Of the Re fractions Sxcœr. III. 


nearer Point L, or it is G N. DN 
GL. DL. 


Fox let it be GN. DN:: GL. R, 
and it will be G N q. DNq :: GLq. 
R:: GN - GL q. DN - Rq. 
Wherefore ſince it is DN > GN, or 
DNq>GNq, it will be D N q — 
R q GN - GLq. But it is G 
Nq—GLq=DNq—DL q + 
and therefore D Nq— Rq>D N q— 
D L q. That is, DLqQ>Rq, or D 
L >R. And conſequently, ſince it was 
ſuppoſed GN. DN:: G L. R, it will 
be & N. DN GL. D L. Q. ED. 


PROP. XIII. 


There being ſuppoſed a common Sine 
of Incidence of Rays of a different Sort, 
the more different is the Denſity of 
the Mediums, the greater will be the 
Inequality of the Ratio of the Sines of 
Refrattion. 


IN Fig. 37.) let Fc be one of the 
leaſt refrangible Rays incident in any 


manner on the Surface Ac, and let its 
refract- 


= [EEE - 3 


F, 
« 
4 


| 19 $v- 
. 4 


Ss cr. III. of a ſingle PLANE. 123 


refracted Ray be c /, which continued 
backwards may cut the Perpendicular 

FA in f. Then let Ae be taken, that Fe 

be to F c in any given Ratio, ſuch as 

we have before deſcribed. * viz. with * Ne 44. 
this Condition, that F e being reckon-# * 
ed one of the moſt refrangible Rays, its 
refracted one may diverge from the 

ſome Point F. This being done, if for 

the latter Medium another however 

denſe or rare be ſubſtituted, two the 

like Rays incident in the ſame right 

Lines Fe, Fc ought to be always fo 
refracted, that they may diverge in like 

Sort from one and the ſame Point of 

that Perpendicular, b as from g to- „ Ne 45 8 
wards | and n, it being ſuppoſed, that 49. 

this latter Medium is of a Denſity more 
different from the former Medium, 

than the other latter Medium, that 

made the Rays diverge from f. It is 
therefore to be ſhewn, that the Inequa- 

lity of the Ratio of the Sines of Re- 
fraction is greater in the latter than in 

the former Cale. Vi. of the Ray Fc / 

the Sine of Incidence is to the Sine of 
RetraCtion as fc to Fc, that is, as 1 to 


Fc 
Je 


124 


Of the Refractiouus Sk Or. III. 


755 and fo of the Ray Fe n they are as 


I to fes wherefore the Sines of the Re- 
fractions of _ fame Rays are amongſt 
themſelves as = - to 7 = And by a like 


Diſquiſition it will appear, that of the 
refracted Rays diverging from g, the 
like Sines of their Nefractions are as 


to . It remains qr to be 


proved, that between = = and © I there 
is a er Diſproportion, _ between 


« Tem. r ll 


— and ; 5 — ; that is (lince 5 Te is 


77 
it remains to ſhew, that it is 27 — - > 


_ 75 Viz, it is g e. fe n 


Lem. III. and by taking the Recipros 


cals of the Ratios it will be . 7 : "75 > 


2X Tos and by drawing the former Ra- 


tio into F e, and the latter into F c, 


there will ariſe D # > _ 7 and by 


Alternation 3 777 f. 1 2 Q E D, 


ge rh 
SC HOL. 


SecT. III. of 4 ſingie PLAN R. 


after the ſame Manner in the greater 
Letters (by which I have deſigned the 
Refractions, when the latter Medium 
is rarer than the former One) provi- 
ded inſtead of the Sign > there be eve- 
ry where underſtood the Sign S, and 
> inſtead of . You will alſo obſerve, 
that in this T have made the Denſity of 
the latter Medium only to be varied, 
but it comes to the ſame Thing, if 
you conceive the former Mediums to 
be ſucceſſively varied, the latter not 


being changed, or which is all one, 


if the Refractions are contrarywiſe 
made out of the latter into the for- 
mer Medium. For indeed to Rays fal- 
ling on a Surface on either Side, there 
are the like Ratios of the Sines; but 
as to the exact finding the Ratio of 
theſe Sines, for any propoſed Mediums, 


I have treated of this before, and I had 


not touched on the preſent Propoſition, 
if the XV. Propoſition to be delivered 
by and by had not required it. 


LE M. 


SCHOL: The Demonſtration is 


125 


126 


Of the Refractioan Sect, III. 


LEMM A. IV. 
With the Center A, any Diſtance 


Fig. 38. A D (in Fig. 38.) let be deſcribed a 


Circle D g G: next with any Center C, 
diſtance A C let be deſcribed another 
Circle cutting the right Line A D in B 
and the former deſcribed Circle in G, then 
let the Arch B G be biſected in F and 
F K let fall perpendicular upon B D. 


Theſe Things being thus done, I ſay, 


that F K [0 let fall perpendicular will 
bi ſect the ſaid B D. 


Fox A E, A G, B E, FG and F D 
being joined, in the Triangles A F G, 
and AFD the Angles at A are equal 


on account of the equal Arches BF, F 


G, by which they are ſubtended. The 
Sides alſo about thoſe Angles A D and 
AG are equal, viz. the Radius's of 
the ſame Circle, and they have the 0- 
ther Side A F common, wherefore alſo 
the third Sides FG and F D are equal. 
But B F is equal to F G, by rea- 
ſon of the Equality of the Arches, 
which they ſubtend. And "Mm" 
B 


Sr CT. III. Of a ſangle Pi ins. 


B F == F D, and the Triangle FK B 


= Triangle F K D, and thence B K 


= KD. 


COROL. r. Hence the right Line 


K F, which biſeQs the baſe Line B D, 
and inſiſts on it perpendicularly, will alſo 
biſect the Arches B G of all the Circles 
paſſing through the two giyen Points 
AandB, and cutting ſomewhere in G 
the given Circle D G deſcribed with the 


Center A and Interval A G. And it will 


even biſect the Arches BG g in the o- 
ther Point of Interſection . 


C ORO L. 2. The ſame Thing 
will happen, when A and B coincide, 
that is, when the Circles A F G touch 
the right Line A D in the Point A. Al- 
ſo B may be taken on the other Side of 
A. It may alſo be obſerved by the by, 
that the Angles B F K, BG D, which 


the Circle A B F makes with the right 


Line F K and Arch G D, are equal. 


LE M- 


127 


. 
—— . — — -- — - a 


\ 


128 


Of the Refraftions Sf Cr. III. 


L E M M A. V. 


Four Lines Ab, AB, Ac and AG 


Fg. 39 (Fig. 39.) being in any Circle from the 


ſame Point of tht Circumference ſo in- 
feribed, that it is Ab. AB:: A c. A G, 
of all which let A 6 be the leaſt, I fay; 
that the Angle B A G is greater than 
the Angle b A c. 


Fox let another Circle A B be de- 
ſcribed, cutting the former in the Points 
A and B, whoſe Diameter let be to the 
Diameter of the Circle AB G, as AB to 
Ab, both their Centers lying on the 
ſame Side of A B. Then with the 
Center A, diftance A G deſcribe a 
third Circle & H, meeting the ſecond in 
g, and that Point g by Conſtruction 
will lie ſomewhere between G and H; 
and conſequently if A g be drawn, the 
Angle BAG will be greater than the 
Angle B Ag. But the Angle BAg 
is = Angle B Ac; by Reaſon that 
AB and A g are alike inſcribed 1n 
the Circle A Bg, as A and Ac are 
in A c, viz. having the ſame Ratios, 

both 


” " , 4 * 
8 5 * 
- 1 1 ry a ** 
- F it +. * * * 
: - 
* 


SrCr. III. of ®faitle PLANE. 
both amongſt one another, ( A 6. 
Ac::AB. AG or Ag) and to the 
Diameters of the Circles, in which 
they ate inſcribed. Since therefore it is 
BAG >BAg Ac, it will be 
B AGD bAc.QE: D. 


COROL. 1. Hence in any the ſame 
Sort of Rays, the greater the Refracti- 
on is, the = will be the refracted 
Angle. In Fig. 27. where it is F R. EE. 27. 
RD :: Fr. r ad, it will be the Angle 
Erd> Angle FE R D. 


CORO L. 2. Hence alſo if it is A G. 
AB>S Ac. As, it will be much 
more the Angle BAG Ac. That 
is in general, the greater the Subtenſes 
are, and at the ſame Time, the greater 
the Inequality is of their Ratio, the 
greater will be the Difference of the 
Angles, which they ſubtend ; and the 
fame is to be underſtood of Sines and 
their Angles, as the Halves of Subten - 
ks and their Angles: 


45 


Of the Refractious Sect. III. 


LEMM A. VL 


Moreover if the Arch cd be taken 
equal tobe, and AD be ſo mſcribed in 
the Circle ABD, as it ſhall be to A d, 
as AG to Ac; the reſt remaining, I 
ſay, that the Arch D G will be great. 
er than the Arch G B. 


Fox with the Centre A, Radius AD 
deſcribe a Circle D K, E cutting the 
Circle ABginK and the right Line 
A B in E, and let be drawn A K. Now 
ſince A K, Ag and AB are inſcribed 
alike in the Circle AB g K, as Ad, 
Ac and Ab are in the Circle Abc; 
the Arch gK will be = Arch Bg, 
wherefore g L being let fall perpendi- 
cular to B E, and produced till it 
cuts the Arch BDinF, that g L by 
Lem. TV. will biſe& both the right 
Line B E and the Arch D B. But be- 
cauſe g F by Conſtruction lies out of 
the Circle g G, the Point F will fall 
Fetween G and D. Wherefore D G 
> DF, or > FB and much more 
GB Q. E. D. 
1 COR OL. 


———— —  —— 


— 
— —— — — 


&& ct. III. of a ſingle PLANE. 


CORO L. 1. Hexcz if the Arch 
54 does not conſiſt only of two, but 
of any Number of equal Parts, the 
correſponding Parts of the Arch 4 D 
from the Termination & to the Termina- 
tion D, will gradually exceed one ano- 
ther in Length. So that if the Arch bc 
has any commenſurable Ratio to the 
Arch g 4, it will be Arch G D. Arch 
BG> Arch ed. Arch bc : for indeed 
the Number of equal Parts meaſuring 
the Arches & and e d, correſpond to 
the like Number of unequal Parts, con- 
ſtituting the Arches BG and G D, 


whereof thoſe in & D are all greater 


than the greateſt Part of BG. More- 
over, if 6e has any incommenſurable 
Ratio to ca, it will be in like Manner 
G D. BG > cd. bc. For the Simi- 
litudes of Ratios, which agree in- 
definitely to commenſurable Quanti- 
ties, on that Account do agree alſo to 
Incommenſurables alike affected, as 
may be ſhewn from Euclid's Definition 
of the like Ratios. But this is more eaſi- 
ly underſtood, by imagining that the 
Quantities, which they call incommen- 


K 2 ſurable, 


131 


Fige 40. 


Of the Refractious Sxòœr. III. 


ſurable, may be numbered by Parts in- 
definitely ſmall; and ſo in ſome Sort 
be reduced to the Nature of Commen- 
ſurables, eſpecially as to the Habitudes 
of Ratios. Conceive therefore the 
Arch 6 c to be divided into equal and 
indefinitely many Parts, and of theſe ſo 
many to be taken, as they ſhall differ 
leſs than by one Part (that is indefi- 
nitely little) from the Arch c 4, and 
therefore they ſhall be thought, accor- 
ding to the uſual Manner, equal to it. 
Conceive alſo B D to be divided into 
equal Parts, correſponding (as I before 
defined) to the Parts of 64, and on 
Account of as many unequal Parts 
greater indeed in G D and leſs in B G, 
as there are equal ones in d and be, 
it will be GD. BG. > cd. bc, 


COROL. 2. HENCRH beſides by 
compounding it follows, that B D. 
B GD d. be, and allo GD. BD 
> C d. b d. 


COROL. 3. It follows moreover, that 
the Subtenſes A 6b, Ac, Ad, Ae being 
any Way drawa in #7g. 40. and other 
| four 


szcr. III. Of a ſingle PLANE. 


four AB, AG, AD, AE, whereof 
each obſerves to each of the other the 
fame Ratio (__ AB A#:: a6 
Act: AD. Ad :: AE Ag); HAE 
Is the greateſt of them all, and A 
the leaſt, it will be Arch E D. Arch 
GB > Arche d. Arch c 6. For by Co- 
rol. 1. of this Prop. it is E D. DG 
>ed. de, and D G. GBS Ac. cb, 
and much more E D. G B> ed, 7 
Not otherwiſe it appears, that it is 
Arch E G. Arch DB > Arch ec. 
Arch 4, viz. by Corol. 2. of this, it 
is E G. DGS ec. dc, and D G. DB 
dc. db; and much more E G. DB 
Dec. db. Laſily, what has been ſaid 
of Subtenſes and their Arches, may be 
alſo underſtood of Sines and their 
Arches. 


PROP. XIV, 


Heterogeneal Rays being incident, out 
of a denſer Medium into a rarer, in the 
ſame given Line, on a Surface given in 
Poſition ; the rarer the Medium is, in- 
to which they are refracted, the greater 
will be the Difference of Refraction. 

K 3 LET 


133 


134 


as Fig. 41. 


2 No, 47 


Of the Refractious Scr, III. 


LET (Fig. 41.) F L be the Line, in 
which two Rays are incident on the 
Surface A L, whereof let the moit re- 
frangible Ray be refracted to P, and 
the leaſt refrangible to T. I fay, that 
if the rarer Medium was ſtill more rare, 
that it ſhould refrack the moſt refrans 
gible Ray to y and the leaſt refran- 
gible to t, then the Angle L. would 
be greater than the Angle PLT. For 
let fall FA perpendicular to the re. 
fracting Surface, which may cut the 
refracted Rays continued backwards 
in G, C, D and E. Then in the re- 
fracting Surface let be fought ſuch a 
Point as N, that it may be FN. 
DN:: FL. EL, and DN produced 
will bs the refracted Ray of the leaſt 
refrangible Ray, incident from E to 
Na. Now when the Poſition of 
FL and FN is ſuppoſed to be ſuch, 
that the refracted Rays D L and D N 
of the moſt refrangible Ray incident in 
FL, and of the leaſt refrangible Ray 
incident in FN, do diverge from the 
Point D, which is ſituated in the Per- 
pendicular F A, for that Cauſe, al- 
though 


SECT. III. Of a ſingle PLANE. 


though the Rarity of the Medium, in 
which the Refraction is performed, 
were different than is ſuppoſed, yet of 
the like Rays incident in the ſame 
Lines FN and F their refracted ones 
would always diverge from ſome Point, 
which is placed in the ſame F A, as is 
ſhewn in what went before d. So 
when the Rarity of the ſaid Medium is 
ſuppoſed to be ſuch, that the moſt re- 
frangible Ray incident in FL is re- 
fracted from any Point G; then the 
leaſt refrangible Ray incident in FN 
will be refracted from the ſame Point 
G. But when the moſt refrangible Ray 
was ſuppoſed to be refracted from the 
Point G, then alſo the leaſt refrangible 
Ray incident in the ſame Line FL 
was ſuppoſed to be refracted from 
Point C. Wherefore it is G N. FN: 


135 


b N® 49. 


CL. FL <, and beſides ſince I horn N. 25 
ſed it before to be FN. DN:: E L. 4d 47. 


E L, it will be by Equality G N. 
DN:: CL. EL. But by Lemma III. 
it is GN, DNS GL. DL, and con- 
ſequently C L. EL GL. DL. 
Wherefore if a Line B L be ſo drawn, 
that it may be CL. E L:: B L. DL, 


K 4 it 


136 


"Of the Refraftions SECT, III. 


it will be DL > GL, on Account of 
the greater Ratio, which it has to 


D L; and moreover C L will be great- 
er than B L, becauſe it is EL > D L, 


and therefore the Point B will fall be- 
tween Gand C, and the Angle G LC 
will be > Angle BLC; but ſince it 
SCE AL: BL DL, or alter nate- 
ly BL. CL:: DL. EL, the Angle 
B LC will be greater than the Angle 
DLE (Lem. I.) and much more the 


Angle G L C > Angle D L E. 
E. D. 


PROP. XV. 


Heterogeneal Rays being incident out 
of a denſer Medium into a rarer in the 


Jame given Line upon 4 Surface given 


in Poſition ; the denſer the Medium is, 
out of which the Rays are incident, the 


greater will be the Difference of the 
Refraction. 


Fox (on account of the greater R- 
fractions) the greater will be the Sines 
of the Refractions, in reſpect to a gi- 
ven Circle, to which they are referred, 
and at the ſame Time the greater will 

be 


SECT- III. Of a ſingle PLANE. 137 


be the Tnequality of the Ratio of thoſe 
Sines, by Prop. XIII. and conſequent- 
ly the greater will be the Difference of 


the Angles, which they ſubtend, by 
Corol. 2. to Lem. V. that is, the grea- 
ter will be the Difference of Refraction, 
Q. E. D. 


Heterogeneal Rays being incident 
out of a rarer Medium into a denſer 1 in 
the ſame given Line upon a Sur face groen 
in Poſition ; the rarer the Medium is, 
out of which the Rays are incident, 


the greater will be the Difference of 
the Nefraction. 


LET (in Fig. 42.) AD be a Surface, Fig. 42. 
on which two Rays are incident in the 
ſame given Line I X, whereof one the 
moſt refrangible let be refracted to P, 
and the other the leaſt refrangible to T. 
I fay, that if the Medium, out of which 
the Rays are incident, were {till rarer, 
that it might refract the ſaid Rays ill 
more, as the moſt refrangible one to- 
02 P. and the leaſt refrangible to- 

wards 


138 


Of the Refractious Sxcr. III. 


wards 2, then the Angle ꝓ X # would 
become greater than PX T. Which I 
thus gradually demonſtrate, 


CAS. 1. Ler us in the firſt Place 
ſuppoſe, that the right Line IX, in 
which the Rays are incident, 1s the 
moſt oblique to the refracting Surface, 
and let any right Line P D be drawn, 
perpendicular to the ſame Surface, and 
cutting the refracted Rays in the Points 
T, P, :, Pp, and let I X be produced, 


till i it cuts PD in F. Then in the Line 
AD let be ſought a certain Point B on 


this Condition, that B f, BP being 


drawn, it may be XF. X T:: By 
B P. It appears therefore, that if the 
leaſt refrangible Ray is incident in B, 
tending towards 7, it ought to be re- 
fracted towards P, for ſince by Hypo- 


theſis it is B P. B f.: X T. X, that 


is, the Sines of its Incidence and Re- 
fraction, as the Sines of Incidence and 


Refraction of another the leaſt refran- 


gible Ray I XF. Wherefore if we ſup- 
poſe thoſe Rays to recede backwards, 
viz. one of the leaſt Refrangibles from 
* to X. and the other from P to B, 
and 


Szcr. III. Of a ſingle PLANE. 


and the moſt refrangible Ray from P to 
X, the refracted ones of them all tend 
from the Point f, for it is a known 
Theorem, that of a Ray being incident 
backwards in its refracted Ray, the In- 
cident becomes the refratted. Now 
when the difform Rays PB, PX, flow- 
ing from the ſame Point P, are refrac- 
ted from the ſame Point f, which is ſi- 
tuated in the Perpendicular PD, the 
Proportion between PX and P B being 
once known, if from any other Point 
of the ſame Perpendicular there be 
drawn to the refracting Surface two 
Lines having the fame Ratio, that is, 
that one of them deſigning the moſt re- 
frangible Ray be to the other, that de- 
ſigns the leaſt refrangible Ray, as PX 
to BP: Then their refracted Rays 
(from what has been ſhewn before *) 
will diverge alſo from ſome Point, that 
is ſituated in the ſame Perpendicular 
P D; however rare the Medium is ſup- 
poſed, on the Side of the Ray IX, pro- 


vided the other Medium on the Side of 


the Ray P Xretains the ſame Denſity. As 
if the moſt refrangible Ray is incident 
in 2X and is refracted from J, vis. 

= 


139 


Ne. Jg. 


2 No 47. 


Of the Refract ion Scr. III. 


the Medium towards I X being now 
ſuppoſed rarer than before, then the 
right Line 2 & being fo drawn, that it 
be PX. BP: : PX. p6, the leaſt re- 
frangible Ray p would be refracted 
alſo from the ſame 7, whence it follows, 
that 2 bis to f 6, as the Sine of In- 
cidence of the leaſt refrangible Rays to 
the Sine of Refration ® But in the 
Ratio of thoſe Sines is alſo ? X to FX, 
becauſe the inflected Line I X # de- 
notes a Ray equally refrangible, whoſe 
Part I X produced paſles through 
the ſame Point 7; wherefore it is þ 6. 
Fb :: t X. F X. But ſince the Ray 


IX is ſuppoſed to be the moſt oblique 


to the retracting Surface, or inclined in 
an infinitely ſmall Angle, ſo that the 
right Line DF ought to be eſteemed as 
infinitely ſmall or nothing, it follows, 
that DY X DB BV and Db 
— 6 f: Which Values ſubſtituting 
for XV, Bf and 4 F in the above re- 
cited Proportions B P. BF:: TX. X. 
and p b. F:: t X. FX, there will 
come out B P. B D:: X T. X D, and 


5 b. Db: : t X. DX. From which it 


appears, that the right Lines B P and 
N XT; 


SreT. II. of a ſingle PLAN R. 


XT; 5 and Xr are parallel, and the 
Angles BP X and PX T; 6p XN 
and p X # are equal, but by Hypotheſis 
it is PX. B P:: X. 5 6, and therefore 


the Angle 6 X > Angle BPX by | 


Corol. 1. Lem. V. that is Angle p SF 
> Angle PX T. Q. E. D. 


C AS. 2. Bur the incident Rays 
making with the refracting Surface an 
Angle of a definite Magnitude, the 
Propoſition thus appears. Let H X be 
(in Fig. 43.) the right Line, in which 


the Rays are incident; and when they 
come out of a leſs rare Medium, let 


X M be the leaſt refracted, and X N 
the moſt refracted Ray. But when 
they come out of a more rare Medium 
let X n be the leaſt refracted, and X = 
the moſt refracted Ray. Let alſo be u- 
ſed the moſt oblique incident * 1 — 
with their refracted ones X T, 
Xr and X 2, ſuch as I have now aw 
ſcribed. vi. ſo, that when the Rarity 
of the anterior Medium is ſo great, 
that it makes the Rays H X to be bent 
towards M and N, then let it alſo bend 
the like Rays IX rowards T andP, but 
T when 


141 


Fg. 43s 


142 


Of the Refr ations SECT. III. 


when its Rarity is ſo much greater, as 


that it compels thoſe towards and », 
then at the ſame time let it compel 
theſe towards ? and . Let moreover 
APD be a Circle, deſcribed with the 
Center X and any Interval AX, that 
may cut theſe refracted Rays in T, P, 
M, N, , p, m, u, from which to the 
Perpendicular B X let fall the Sines of 
the Refractions TB, PC, MF, NG, 
tb, pc, mf, ng, and from the Law 
of Refractions it will appear, that it 


is TB. PC:: MF. NG, and #6- 


pc::mf.ng; and farther by Hy po- 
theſis and Conſtruction it will appear, 
that T B is the greateſt and # g the 
leaſt of thoſe Sines. And conſequent- 
ly by Corol. 3. Lem. VI. it is Angle 
T XP. Angle MXN Angle r Xp. 
Angle m Xu, and by Permutation it is 
Angle T XP. Angle r X p > Angle 
MXN. Angle 1 X A. But from what 
was ſhewn in the firſt Caſe, the Angle 
T XP is S Angle r X p, wherefore 
and much more it will be the Angle 
MXN S Angle 1 XA. Q. E. D. 


2 PROP. 


99 
” 
o ” 


SECT. III. of a ſingle PLANE. 


PROP. XVIL 


Heterogeneal Rays being incident out 
of a rarer Medium into a denſer, in 


the ſame Line, upon a Surface given 
in Poſition 3 the denſer the Medium is, 


on which the Rays are incident, the 


greater will be the Difference of the 
Refractions unto a certain Limit, and 
afterwards it will be leſs perpetually. 


For if the latter Medium exceeds 
very little the anterior in Denſity, ſo 
that it makes indefinitely ſmall Refrac- 
tions, the Difference of Refractions will 
be alſo indefinitely ſmall, and therefore 
leſs than it would be, it the latter Me- 
dium were ſuppoſed more denſe, that 
the Refractions might become greater. 
Wherefore the Denſity of the latter Me- 
dium being increaſed, the ſaid Diffe- 
rence of Refractions will be increa- 
ſed ; but if its Denſity be increaſed in 
infinitum, the Refractions alſo will be 
increaſed as much as poſſible. That 15, 
until all the refracted Rays emerge per- 


pendicularly, the Angles of mm 
and 


143 


144 


tis u 44. 


Of the Refraftions Sxer. III. 
and their Differences then altogether 
vaniſhing. Wherefore the Difference 
of Refractions is again diminiſhed, un- 
til it has vaniſhed into nothing. 


SCHOL. ArTHoven the Deter- 
mination of its Limit, where the Diffe- 
rence of Refraction becomes the great- 
eſt, may afford more Pains and Labour, 
than Utility; however, ſince it may 
perhaps be thought of ſome Moment 
to know the Denſity of a Medium, 
that may make by the Rays refracted 
into it the moſt conſpicuous Colours, 
T ſhall not think much to deſign this. 
And that in the firſt Place, when the 
Incidence is moſt oblique. 


CAS. 1. LeT (in Fig. 44.) I X be 
the common Way of Rays falling 
the moſt obliquely upon the Surface 
A X ſeparating any Mediums, and let 
their refracted Rays be, as before, Xp 
and Xz; and let a right Line # be 
drawn parallel to the aforeſaid Surface, 
that may meet thoſe Rays in P and z, 
from which to A X the Perpendiculars 
5 C, E being let fall, let C E be bi- 

ſected 


SECT. III. of a ſingle PLANE. 


ſected in D, and with the Center D, 
diſtance D X let a Circle be deſcribed 
cutting C in P, and Er in T, and let 
XP and X T be joined. I ſay, that 
when the Denſity of the latter Medi- 
um is ſuch, that of the Rays incident in 
the Line IX it refracts the moſt refran- 
gible ones to P, and the leaſt refran- 
gible to T, then the Angle P X T will 
become the greateſt. For however denſe 
the latter Medium is ſuppoſed, the re- 


fracted Rays will ſo cut the Lines CP 


and C T in the Points p and t, that the 
right Line p, may be parallel to AX. 
Wherefore if the Line Dy be drawn, 
that may biſect all the Lines pt, the 
Center of any Circle paſſing through 
p and t, will always lie in the ſame Dr. 
But the Angle pX# is the Angle in 
the Segment of a Circle paſſing through 
the Points p, , and X; which Angle 
therefore will be the greateſt, when 
the like Circle is the leaſt ; becauſe the 
Ratio of the Subtenſe pf to the Di- 
menſions of the Circle becomes then 
the greateſt. But that Circle becomes 
the leaſt of all, when its Center falls 
in D. For it then has for its Semidia- 


L meter 


145 


146 


Of the Refractions Scr. III. 


meter X D the leaſt of the refracted 
Lines, which can be drawn from & to 
R D. The Angle Þ Xz therefore is 
then the greateſt, when the Center of 
the Circle paſſing through the Points 
P, t and X falls on D. And conſe- 
quently, when the Circle X T and 
Angle PXT are of this Kind, the 
Propoſition is manifeſt. 


HENCE it appears by the by, that 
this Angle PX T then alſo becomes 
the greateſt, when the Denſity of the 
latter Medium is ſuch, that the Angle 
of Refraction of the mean refran- 
gible Rays, being incident moſt o- 
bliquely in I X, is Half a right Angle; 
and continually becomes leſs, the more 
the Angle of Refraction deviates (by 
Exceſs. or Defect) from Half a right 
Angle. As if the Refractions out of 
Air into Water, into Glaſs and into 
Cryſtal were compared, it will appear 
from Calculation, that, when the Angle 
of Incidence is Ninety Degrees nearly, 
then the Angle of Refraction into 
Water will be greater than Half a 
right Angle, into Glaſs it will be leſs. 

I Where- 


SECT. III. F a ſingle PLANE. 


Wherefore Water is leſs denſe, and 
Glaſs more denſe, than that they may 
make the Angle PX T the greateſt. 
And therefore ſince Cryſtal is ſtill 
denſer, it will make that PX T leſs, 
than Glaſs would make it. And ſo 
Glaſs, although it refracts leſs, yet in 
that Caſe it will more diſſipate from 
one another the heterogeneal Rays re- 
fracted into it, than Cryſtal; and by 
that Means it will project Colours 
more ſpread, on its oppoſite Surface. 
But theſe are moſt difficult to try, be- 
cauſe Glaſs and Cryſtal differ little in 
their Denſity, nor can they be had ſuffi- 
ciently thick; and if they could, then 
on Account of too great Denſity, they 
would not be per ſpicuous enough. 


CAS. 2. But if the Line, in which 
the Rays are incident, is not the moſt 
oblique, the Problem becomes ſolid. 
But I intend to ſhew a Way, whereby, 
its Conditions being ſomewhat chan- 
ged, it may be reduced to a Plane 
Problem. It muſt therefore be un- 
derſtood, that ſince between the ex- 
treme, or greateſt difform Rays, there 

L 2 are 


147 


148 


Of the Re fractions Scr. III. 


are innumerable intermediate ones, 
which by continually ſucceſſive, and 
infinitely ſmall Degrees, are ſome re- 
fracted more than others; the Diffe- 
rence of the extreme Rays will be 
made up of ſimilar Differences of the 
iatermediate ones, infinite in Number 
and ſmalneſs. Now, the Proporti- 
ons of thoſe infinitely ſmall Differences 
being known, we may from thence 
make a Judgment of all the Aggregates 
together, or of the finitely ſmall Dit- 
ferences, ſuch as intercede the Refracti- 


ons of the extreme Rays; eſpecially 


ſince thoſe Differences are very ſmall. 
So it being known, that the infinite 
{mall Differences are increaſed, dimi- 
niſhed, or at the ſame Time become 
greateſt, or leaſt; it will be conclu- 
ded, that the Sum of all is in the ſame 
Manner increaſed, diminiſhed, is the 
greateſt or the leaſt. But if they are not 
all at the ſame Time the greateſt or 
the leaſt, yet the Sum may be eſteemed 
the greateſt or the leaſt, when it hap- 
pens to an intermediate Part So the 
Breadth of all the Colours may be then 


thought the greateſt, when it happens 
to 


Szcr. III. Of 4 ſingle PLANE. 


to the green. Now although the pro- 
poſed Problem, when the Queſtion is 
about finite ſmall Parts, is ſolid; yet if 
it be made of infinitely ſmall Differen- 
ces, it may be reduced to a plane one. 
But in ſolving it I ſhall not take 
much Pains ; I ſhall only briefly ſhew, 
by what Means a Calculation in this 
and others the like is to be performed, 
that an Equation may be come at, from 
which the greateſt of the infinitely 
{mall Angles may be diſcovered. And 
farther from the ſame Fountain, I ſhall 
determine the Proportions of the Diffe- 
rences of Refraction in reſpect of dif- 
ferent Mediums, which in the four 
preceding Propoſitions I did but de- 
{cribe in general, 


IN the firſt Place therefore a Rule 
or Equation is to be inveſtigated, 
whereby from any given refracted 
Ray whatever, another refracted Ray 
making with it an infinitely ſmall Angle 
may be known. Rays being incident 
as before in the moſt oblique Line I X 
( Fig. 45.) out of a Medium of a gi- 
ven Denſity into a Medium of any Den- 

L 3 ſity 


149 


Fig. 45. 


150 


Of the Refrattions Scr. III. 


ſity whatever, let X R and Xx be two 
refracted Rays, whereof let one X R 
be a little more refrangible than the 
other X, yet by an infinitely ſmall 
Difference. And let be drawn any 
right Line R meeting theſe in R 
and 7, and parallel to the refraQting 
Surface. To which Surface alſo let fall 
the perpendiculars RD, 7 , which 
feign to have a given and finite 
Diſtance from X ; but from one ano- 
ther an infinitely ſmall one. But con- 
ceive the Lineola Rr with the Rays 
paſſing through Rr, to lie more or leſs 
from XD (as in the preceding) according 
to the various aſſumed Denſity of the 
latter Medium. Nov if the right Line 
DR cuts the Rays Xr in M, and IX 
in K, ſince tlie infinitely ſmall Triangle 
R Mr is ſimilar to the Triangle 
D MX, from which the Triangle 
K RX differs but by the infinitely 
{mall Differences RX M, and DX K, 
which do not make a Diſſimilitude, 
the Triangles allo RM and R DX 
ought to be looked upon as ſimilar; 
and therefore the Perpendiculars K L 
and R N being let fall, it will be DX. 

LR :: 


SECT. III. Of a ſongle PLANE. 
LR:: Rr. MN. And conſequently 
ſince it is LR =; for it is 


XR. KR (SANT: KN. 
RL, it will be alſo MN = N RN N 
into Rr. which is the Difference 
between X N, or XR, and XM. 


And then it will be X M = X 


R —: 1 into R. There 


is therefore found the Relation be- 
tween X K, X M and X R, when 
the Angle I X A is infinitely ſmall. 
Moreover however oblique it is ſup- 
poſed, thoſe Lines X K, X M and 
X R will obſerve the ſame Relation, 
for they are reciprocally, as the 
Sines of Incidence; and Refraction; 
and therefore there is alſo found 
the Relation between them for any 
Obliquity of the incident Ray I X. 
And fo X K and X R being known 
or any ways aſſumed at Pleaſure, 
thence at the fame Time X M is 
known. Which is the firſt Thing TI 
propoſed to determine. 


WIERETORE let I X be a Line ma- 
king with the refracting Surface any 
Lk. given 


151 


. Fig, 46. 


Of the Refrattions Seer, III. 
given Angle AX I, and the reſt remain- 
ing it will be MN = l into 
Rr. Fart her it is RD(=+/XRq-XDq)- 
X D:: M N. RN; and conſequently 
NES XR XK ista R . XD, 
a is NO = NRX XK AME X Dag. 
But it NR isdivided by X R, there will 
come out the Sine of the Angle RX 
N with regard to a Circle, whoſe Semi- 
diameter is Unity. Wherefore lince that 
Angle and its Sine are the greateſt, to 
determine the greateſt Angle, the greateſt 


Quantity N R muſt be ſought, that is) 
XRq- XK q into Rr XxX Ds» 
the greateſt R XK NRC N54 
or (a Diviſion being made by the given 
Quantity =} 2 there muſt be 
| | Rq-XK 
ſought the greateſt R — * 1K. 
Which may be done by the well known 
Method de Maximis & Minimis, and 
there will come out XR qq=3XKq 
Xx XRq—2XKq x XDq; the 
Conſtruction of which Equation is thus. 
From any Point (Fig. 46.) of the inci- 
dent Ray I X let tall the Perpendicular 
TA, and in it take AF = AX, and XI 
being 


SCT. III. Oy 4 fingle PI A NE. 


being produced to B that IB may be 
IX, upon B X deſcribe a Semicir- 
cle BE X, in which inſcribe X E = 
XF, then XB being produced to C, 
that B C may be B E, upon CX de- 
ſcribe a Semicircle C G X, which a 
Perpendicular erected to its Diameter at 
I may cut in G. Laſtly with the Cen- 
ter X and Interval G X let be deſcribed 
the Arch G H cutting A I produced in 
H. Let H X be drawn and produced 
towards R, and R X will be the refrac- 


ted Ray of IX, when the Denſity of 


the latter Medium is ſo great, that the 
Difference of Refraction R X M becomes 
the greateſt of all. Which being found, 
the Denſity of the latter Medium cau- 
ſing ſuch a Refraction will be eaſily gi- 
ven. Conceive therefore the Rays X 
Rand Xr to be mean refrangible ones, 
but in a different Degree, and the latter 

Medium ſo found will make not only 
between theſe, but alſo between the 
extreme, or the greateſt difform Rays, 
nearly the greateſt Difference, it poſſi» 
bly can. | 


Bur 


153 


154 


Fig. 45- 


Of the RefratFions Sxcr. III. 


Bur if there be required the Propor- 
tion of the like Sort of Differences to a 
various Rarity or Denſity of Mediums, 
this will be eaſily determined, from 
what has been now {hewn, provided 
they are ſuppoſed infinitely ſmall. So the 
Rarity or Denſity of the latter Medium 
only being varied, that the Rays 1nci- 
dent in I X are now refracted to M and 
R, then tomand 7 ; and any Line DK 
being drawn perpendicular to D X, 
that may cut them in K, M, R, mand r, 
the infinitely ſmall Angle M X R will 


be to the like Angle W Xr, as 
E N 2 XK Xrq-:XKq 15 
XR KD to NB. But if the 


Rarity or Denſity of the former Medi- 
um is varied, the latter Medium not be- 
ing changed: The Analyſt will eaſily 


diſcover, that (in Fig. 45.) it is MN 


1 XK 
—XKq Into Rx, and therefore 


Fig. 47. ik (in Fig. 47.) it is the Angle MXR. 


XRq=—: XKq, <3 PEP &« 4 
A mr.: XK, 
ngle "=. RB Xrir 


for it is not the ſame Thing, whether the 
Rarity or Denſity of the former Medium 
or of the latter Medium be varied, as 


ape 


SSC T. III. Of « ſingle PLANE. 


appears from What has been before 
ſhewn. 


THE preceding Propoſitions relate 
to the Diffuſion of Light flowing from 
a far. In the two following is treated 
of the Refraction of Light as proceed- 
ing from near at Hand. 


PROF aL 


Heterogeneal Rays being refracted 
from a given Point to a given Point, 
by a Surface given in Poſition 5 the 
more denſe the denſer Medium is, the 
greater will be their Inclination to one 
another on the Side of both the Mediums 


to à certain Limit, and afterwards 
it Will be the leſs. 


Fox when its Denſity 1s not greater 
than the Denſity of the other Medium, 
that the Refractions may be infinitely 
ſmall, then the Difference of Refraction 
will be alſo infinitely ſmall, and there- 
fore will be increaſed, by increaſing the 
Denſity. But if its Denſity be increa- 
ſed in infinitum, then of all the Rays 

Inci- 


— coy 


1 


156 


Of the Refraftims Sgor. III. 


incident upon it, the refracted ones do 


« No 42 & emerge perpendicularly *, and on the 


45. 


Eg. 48. 


contrary the Perpendiculars alone can 
enter the rarer Medium out of the 
denſer; whence all the Rays refracted 
from a Point to a Point will then pro- 
ceed in the ſame Lines, or coincide, 
and ſo the Difference of Refraction will 
again vaniſh into nothing. 


PROP. XIX. 


Heterogeneal Rays being refracted 
from a given Point to a given Point 
by a Surface given in Poſition 3 the more 
rare the rarer Medium is, the greater 
will be their Inclination to one ano- 


ther. 


O the Side of both Mediums, let 
AT be a Surface, ſo refracting the 
difform Rays FT X and FP X, that 


they flowing from the ſame Point F, 


they may again meet in the ſame Point X. 


I fay if the former Medium were rarer, 


that the aforeſaid Rays might be ſtill 


more refracted, as FT X into Ff X, and 


F PX into F y X, that the Angle P 


"MM 


src. III. of 4 fmgle PLANE. 
xt would be greater than the Angle 


PXT, as alſo the Angle 2 Ft greater 
than the AnglePF T. 


To abbreviate the Demonſtration of 
the firſt Caſe, let us ſuppoſe the Rays 
to be the leaſt difform, that on account 
of the infinitely ſmall Difference of Re- 
fraction, they may make the Angles 


157 


PX T and y Xr infinitely ſmall a; then . Conſult 
let be drawn T K the refracted Ray of — * 5 
a Ray conform to FP X, that the infi- p — XVII. 


nitely ſmall Angle K T * may be the 
Difference of Refraction of the Rays 
incident in the fame Line FT; and 


after the fame manner let be drawn r x 


the refracted Ray of a Ray conform to 
Fp X, that the infinitely ſmall Angle 
tt X may be the Difference of Refrac- 
tion of the Rays 1ncident in the ſame 
Line Fr. It appears therefore, that 
ſince the Ray F # is obliquer than F T, 
and is alſo incident on a denſer Medium, 
the Angle & X will be greater than 
the Angle K T X. Farther let be pro. 
duced K T and &, till they cut in the 
Points D and 4 the Line F A, which is 
perpendicular to the Plane A T, and 


let 


158 Of the Refractious Stcr. n.. 


let them be produced beyond to F and 


. F A 
g. ſo that it may be 4.3. :: T 


F. TV and = % . : f F. 7g, and 


the Points F and g fo found will be the 
Foci of the Rays FT X, and T: a 
Prop. VIII. Caf. 2. and aj. T:: 
Angle K T X. Angle PXT, as alſo 
Xg.tg:: Angle KX. Angle p X.. 
(Caf. 3. Sebol Prop. XII). Theſe Pro- 
portionalities indeed are not altogether 
true, when the aforeſaid Angles made 
by the Difference of Refraction are ſup- 
poſed to be of any definite Magnitude, 
But they approach the nearer to the 
Truth, the leſs theſe Angles are made, 
ſo that in infinitely ſmall Angles they 
ought to be looked upon as accurately 
true. Now ſince by Hypotheſis A # is 
AT, it will allo be X#>XT, as 
alſo 1g T 7, as appears from the 
Determination of the Points g and f 
given above. Wherefore it is f g. T f 
r KX. T X, or by Permutation 7g. 
t XS TF TX, and by compounding 
tg.Xg > TJ. XV, that is, by ſubſti- 
tuting Ratios equal to theſe, the Angle 
Xr. Angle Art X > Angle PX,. 
Angle 


SECT. III. of two PLANESs. 159 
Angle K TX, and by Permutation pX ;. 
PXTS S IX. K TX, as has been ſaid; 


and therefore much more is the Angle 
p XH > Angle PX T. Q. E. D. 


Bur from hence may be made a 
Conjecture of the latter Caſe, that the 
Angle p F t is always > Angle P F T; 
for it would require a far more difficult 
Demonſtration, and yet I am weary of 
having beſtowed many Words upon 
them already ; let theſe therefore ſuffice 
for the Refractions of a ſingle Surface. 


Of the Ar FECTIONS of RAYS 
twice refracted. 


Bur if the Refraction be double, as 
it happens in Priſms, whoſe Phznomena 
I chiefly intended to explain, the At- 
ſections of Rays ſo refracted, are fo 
manifeſt from the preceding, that it 
may ſeem a {mall Matter to treat about 
them. Of parallel Surtaces indeed no- 
thing farther occurs to be obſerved, 
than that the latter Surface as much re- 
fracts the Rays, as the former, but to- 

| wards 


160 


Of the Refractious Sxcr. III. 


wards a contrary Way. Of inclined 
ones what follows is to be obſerved. 


PR OP. XX. 


Homogeneal Rays diverging to 2 
Priſm, after both Refrattions proceed to 


_ diverge. 


Ix is manifeſt by Prop. VII. 


Axp the ſame is to be underſtood of 
parallel or converging Rays, viz. that 
after both Refractions they will conti- 
nue parallel or converging. 


SCHOL. Bor if the Point, from 
which any infinitely near Rays after 
both Refractions do diverge, or the 
Place of the Image ſeen thro? the Priſm 
ſhould be deſired, its Invention is ma- 
nifeſt from the Scholium of Prop. VIII, 
But that it may be readier done by 
Conjecture, this Mechanick Theorem 
may be uſed. That the Image will ap- 
pear at about that ſame Diſtance behind 
the Priſm, as the Object has, whoſe I. 

mage 
2 


EF TEE. 
— | 


SzcT. III. of two Prangs. 
mage it is, provided the Refractions on 
each Side be not very unequal. 


PROP. XXI. 


Of Heterogeneal Rays diverging to 
a Priſm, ſome after both Refrattions 
will converge. 


Tuts appears from Prop. X and XII. 
viz. Of thoſe, that lie in a Plane per- 
pendicular to both the refraQting Planes, 
the greater refrangible Rays from 4 
ſomewhat more oblique Incidence will 
meet with the leſs refrangible ones, and 
the ſame Thing happens of almoſt in- 
numerable others. 


PROP. XXII. 


Of Rays therefore ſo refracted from 
# Point to a Point, or from the Object to 
the Eye, ſome will paſs gradually near 
er to the Vertex of the Priſm than 0: 
thers, according as they are more and 
more refrangible. 


M | FRO 


162 


Of the Re fractions Sect. III. 


FROM Prop. X. whence the Or- 
ders of Colours are defined, of which 


hereafter. | 
\ PROP. XXIII. 


The greater the vertical Angle of 
the Priſm is, cæteris paribus , the 
greater will become the Difference of 
Refrattion, and thence the Appearance 
of the Colours the more diſtiuct. 


Ano this is manifeſt from Prop. II. 


P R O P. XXIV, 


The denſer the Matter of the Priſm 
is, or the rarer the incompaſſing Me- 
dium is, ceteris paribus, the greater 
will be the Difference of Refraction, 
and thence the Appearance of the Colours 
will be the more manifeſt. 


Tas latter Caſe is evident from 
Prop. XIV and XVI. But the former 
Caſe, that it may not be called in 
Queſtion from Prop. XVII. Ithus ſhew. 

2 Con- 


9e CT. III. of two Prints; 


163 


Conceive a more refrangible Ray PD Eg. 49: 


and a leaſt refrangible one T D, ſo to 
tall on a Priſm at any the fame Point 
D, that their refracted Rays may pro- 
ceed in the ſame Line D 4, and laftly 
being refracted at 4 may diverge to- 
wards and . Which being ſuppoſed, 
it appears by Prop: XV. that the An- 
gle det will be increaſed, from the 
Denſity of the Prifm being increaſed; 
and the Concluſion is the ſame of the 
Angle PDT, provided conſimilar Rays 
are conceived to recede in the ſame 
Lines. The Aſſertion therefore is ma- 
nifeſt of Rays coinciding i in the Priſm, 
and thence alſo of parallel Rays: 


LEMMA VII. 


Three homogeneal Rays b I. g 15 
refracted out of a denſer Medium into a 
rarer one by the Surface 1 K, if the 
Differences of the Incidences b I g, g Id 
are equal, the Sum of the refratted 
Angles made by the extreme Rays. will 
be more than double the refracted Angle 
made by the intermediate ones. That 
is, the refracted Rays being draw 

M 2 bath 


Of the RefrafFions StcT. III 


back to B, G and D, 1 fay, that the 
Angle B1l6+ DI 4i 2 Angle 
7 i ['> 


From any Circle A D G being de- 
ſcribed touching the refracting Surface 
in I, whoſe Diameter let be ATI, and 
which may cut the ſaid Rays in &, g, d. 
B, D, G. When indeed the Angles b 
Ig and g I d are equal, the Arches alſo 
bg and g will be equal. But A g, 
Ab Ge. being drawn: Ab, Ag, Ad 
will be the Sines of the Incidences, and 
conſequently amongſt one another, as 
are AB, AG, AD the Sines of the 
Refractions. Wherefore (by Lem. VI.) 
the Arch D G 1s greater than the Arch 
G B, and thence 29 6 2g G- GD 
+GB =gD 4 B= 9 D- 4 ++ 
g gB S DAB. That is : 
g GS DAB, or Angle BI4+ 
Angle dID> 2 Angle g IG. Q. E. D. 


PROP. XXV. 


 Homogeneal Rays being refracted by | 
@ Priſm, the Angle, which the inci. 
| N and erging Rye comprehend, 
. | then 


Scr. III. Of tuo PLAN E 5. 


then becomes the greateſt, when the Re- 
fraction 15 equal on both 8 _ 


LET A BC be a Priſm, GR 8 N a 
Ray refracted equally at Rand S8. Let IP 
QL be another Ray refracted unequally, 
more at P, leſs at Q, and let theſe Rays 
be produced, until they meet one ano- 
ther, I Pand QL in T, but GR and 
NS in V. I fay, that the Angle RVS 
is greater than the Angle PT Q. 
Which, that it may appear, conceive 
Rays proceeding each way in the Lines 
PQandRS to go out of the Priſm on 
each Side and ſo to be retracted out of a 
denſer Medium into a rarer one. For in 
the Triangles CPQ, CRS, ſince the 
Angle C is common, the Sums of the o- 
ther Angles will be equal, and therefore, 
ſince CR S is Iſoſceles, the double of 
the Angle CR S will be equal to the 


Angles CPQ + CQP. Wherefore 


the Incidence of the Ray Q Pat P, is 
as much greater than the Incidence of 
the Ray RS at 8, as the ſame Incidence 
is greater than the Incidence of P Q at 


Q. The Differences therefore of three 
Incidences are equal, and conſequen tly 
M 3 i. 


165 


166 


Of the Refraftions Sxcr. III. 
by the laſt Lemma, the Sum of the re- 
fracted Angles made by the greateſt 
and leaſt Incidence, will be greater than 
double the refracted Angle made by the 
middle Incidence. That is, Angle Q 
P T + Angle PQT > 2 Angle 


RSV or > Angle RSV+VRS. 


Therefore, ſince in the Triangles PT Q 


and RVS the Sum of the Angles at 
the Baſe PQ is greater than their Sum 


at the Baſe R S, the vertical Angle 
RVS will be greater than the ver- 


tical Angle P T Q. Q. E. D. 
L E MM A VIII. 


Fin three Lines b I, gl, 41 con- 
taining the equal Angles b 7 g. and gI d, 
three the leaſt refrangible Rays are 
incident at I upon the Surface 1 K, 
and are refracted from a rarer into a 
denſer Medium, whoſe refracted Rays 


' drawn backwards let be IB, IG, ID; 
farther if of thres the moſt re fras- 


gible Rays incident in the ſame ame Liner 
61, gl, 4 I, their refracted ones are 
drawn backwards Ib, I 72 Id, The 
Difference f Refyattion of the Rays, 

Whoſe 


SECT. III. Of two PLANE 8. 


whoſe I. ne is the leaſt, together 
with the Difference of Refratiion of 
thoſe, whoſe Incidence is the greateſt, 
will be greater than twice the Diffe- 
rence of thoſe, whoſe Incidence is in 
the Middle, that is Angle B Ib + Angle 
DId >2 Angle GIg. 


Fox any Circle A DG being de- 
deſcribed touching the refracting Sur- 
face in I, whoſe Diameter let be AI, 
and which may cut the aforeſaid Rays 
in the Points þ, g. d, B, b, G, g, 
D, d. Conceive Subtenſes to be drawn 
from A to theſe Points; and As, Ag, 
Ad will be amongſt one another, as 
are AB, AG, AD; and alſo. as are 
Ab, Ag, Ad: Whence it follows, 
that AB, AG, AD are amongſt one 
another, as Ab, Ag, Ad. And far- 
ther, by Lemma VI. that the Arch 
GD is > Arch B G, and Arch gd 
Arch bg. Now let the Arch G M 
be = B G, and it will be G D 
GM and AD SAM. Alſo in 
the Circumference A D take any Point 

N on this Condition, that if you 

conceive the Subtenſes A M, A N to 
M4 be 


168 


Fig: 53 


Of the Refrattions 8 Lect. Ur. 


be drawn, it may be AB. A b:: AM. 


AN, and AB, AG, AM will be a- 
mongft themſelves, as are A b, Ag, 
AN; and conſequently ſince the Arch- 
es B G and G Mare equal, the Sum of 
the arches Bb M N (by Lem- 
ma VIII.) will be greater than the 
Double of the Arch Gg. But 5 1 na 
is A M. AN (:: AB. A b):: 

Ad, or converſly A M. AD: 11 N. 
Dd; on account of A D>AM, it 
will be Arch Dd > Arch M N, and 
the Arch B b being added on both 
ſides, it will be Arch Bb + Arch 
Dd > Arch B b + Arch MN, and 
much more it will be Arch Bb + 
Arch Dd >2 Arch Gg, or Angle 
B Ib + Angle DI d > 2 Angle 
Glg QED. 


P R O P. XXVI. 


Heterogeneal Rays being refracted 
by a Priſm, the Difference of the 
Angles, which the incident Rays make 
with the emerging ones, will then be- 


come the leaſt, when the Refrattions on 
both Sides are equal. 


IN 


S:cT. III. of two PLANE S. 169 

I xn the Priſm ABC let CR be 
taken equal to C'S, and RS be drawn, 
and alſo any other Line PQ, that is 
not parallel to RS, and conceive within 
the Priſm Rays, proceeding each way in 
thoſe Lines PQ and RS, to go out at 
the Points P, 02.0 and S, and the 

greateſt refrangible Rays to be refracted 
towards K, M, H and O, and the leaſt 
refrangible towards I, L, G and N. 
I fay, that of the Refractions made 
unequally at P and Q, the Differences 
taken together IP K+LQ M are 
greater than HRG NS O the Dif- 
ferences taken together of the Refracti- 
ons made equally at Rand S. For the 
Differences of the Incidences at P, Q 
and S are equal, as was ſhewn in the 
preceeding Propoſition, and conſequent- 
ly by Lem. VIII. the Difference of the 
Refraction of the difform Rays at P, 
where is the greateſt Incidence, to- 
gether with the like Difference at Q, 
where is the leaſt Incidence, exceeds 
the Double of the like Difference at 8, 
where is the middle Incidence. That 
is, Angle I P K + Angle L QM 
>2 Angie NSO, or ſince N S O is 
equal 


350 


\ 


Of the Refractiuns Sect, UI. 


equal to Angle IP K + Angle LQM 
> Angle N 8 0 + Angle G RH. 
Q.: E. D. 


SCH OL. I have ſuppoſed indeed 


the Rays to go out of the Priſm on 
both Sides, but if they proceed from 
Land K through P and Q to L and 


M, and from G and, H, through R and 


8 towards N and O, the Poſitions of 


the Lines and Quantities of the Angles 
will not be thence changed, and there- 
fore the ſaid Demonſtration will then 
alſo hold; and for the ſame Reaſon it 
will alſo hold, when the Rays diver- 
ging to the Priſm, | become in the 
Priſm parallel, which is in like Manner 
to be underſtood of the Demonſtrations 
of the XXIV. and XXV. Propoſitions. 
Moreover, in any other Caſes what- 


ever; where they diverge before Re- 


fraction, and converge after, or are in- 
cident parallel on the Priſm, they will 


not ever ſo much recede from a Para- 
liſm within the Priſm, but that the 
| Angles or Differences of the Angles, 
which the incident, Rays make with 


the emerging ones, may be nearly 


eſteemed 


A 1 * 2 
l 
- 


SCT. III. Of two PLANES, 
eſteemed for the ſame, as if they 
were parallel within, ſo that the faid 
Propoſitions extend altogether to all 
Caſes. 


PROP, XXVII. 

Fe | 
1f Laſtly Rays being refracted from 
à given Point F to a given Point X, 
through a Priſm given in Poſition, 
there is required the Angles DFE, 


G XH, which the Heterogeneal Rays 


comprehend. 


THE Problem is of the Number, 
that the Ancients called Linear, but 
the following mechanical Solution ap- 
proaches the Truth, as much as practi- 
cal Things require. Feign the Sum of 
the Angles DFE ＋ G RX to be 
equal to the Angle NM O, made af- 
ter a double Refraction by two Rays 
ſimilar as to Refraction with the others 
FDand F E, and incident in the Line 
L M, nearly parallel to a right Line 
biſecting the Angle D FE. And of 
the Rays refracted to X produce ſome 
Qne, as G X meeting with the incident 


Ray 


fig. 5. 


172 Of the Refrattions &c. Stew. III. 


Ray F D in V. tof, that F may be the 
Place of the Image, which the Ob- 
ject F exhibits to the Eye at X. Then 
the Angle O M N and the Diſtan- 
ces F X and F V, being mechanically 
known, ſay as F X. FV: : Angle N 
MO. Angle GX H, and G XH will 
be very nearly what you ſeek. As is 
in ſome Part manifeſt, from what has 
been ſhewn at the Scholium of Prop. 
XII. When the Refractions on both 
Sides are not very unequal, the Buſi- 
neſs is more expeditiouſly done by the 
Schol. to the fame Propoſition, by ma- 
king as VX. FV :: Angle DFE, 

Angle GX H, or by Compoſition FV 
+ VX. EV:: Angle NM O. Angle 
GX H. 


SECTION 


| 173 


SECTION IV. 
| Of the REFRACTIONS of 
Curve Surfaces. 


HUS much of the Refractions of 
Planes; it 1s now time to trea 

of Curves and eſpecially ſpherical Surfa- 

ces; the Doctrine whereof in reſpect of 


homogeneal Rays I ſhall endeavour to * 


compriſe in the following Propoſitions, 


PROP. XXVIL 


A Ray bein incident on a curve Says 
face to draw its refracted Ray. 


Tux Refraction of a Ray by a Curve 
is the ſame, as by a Plane touching the 
Curve in the Point of Refraction. 
Seek therefore the Ray refracted by the 

touching Plane by Prop. III. 


PROP. XXIX. 


J N Rays, whether parallel or pro- 


ceeding 4 or to a Pont, fall upon a 
Sphere 


Fig. 55. 


Of the Refrattims Szcr. IV. 
Sphere to be refratted 3 to determine the 


Concourſe of the reſracted Rays near- 
eſt to the Axis or to find the Focus, + 


Lr A be a Point ſending out Rays 
towards a : ſpherical Surface B N P, de- 
ſcribed to the Center C: From the Ver- 
tex and Center erect to the Axis A C the 
Perpendiculars B H and CI; and thro? 
the Point A draw any Line H I meet- 
ing them in Hand I. Then from the 


Point C towards I, take C R, which 


may be to C I, as the Sine of Refrac- 
tion to the Sine of Incidence; and 
draw the right Line H R meeting A C 
in Z; and Z will be the Concourſe of 
the refrated Rays, which was to be 
determined. For let A N be a Ray 
falling the neareſt to the Axis at N, and 
meeting CI in K. Draw N Z meeting 


Cling, and as the manner is, conceive 


the infinitely ſmall Arch B N to be e-. 


qual to BM, a Segment of the right 
Line B H terminated at the Ray A K, 


+ This moſt elegant Way of finding the Focus in 
ſpherical Figures was firſt publiſhed by Dr. Barrow, at 
the End of his XI VH Optical Ze@urey as communicated 
to him from our Author, 


and 


ww 


Sgr. TV. of Coxve * ers. 


and it will be, CF. CR :: CR. Cg. 


* That is, CKtoC g, as is the 
Sine of Incidence to the Sine of Re- 
fraction. And therefore, ſince the An- 
gles CAK and C Z g by Hypotheſis 
are infinitely ſmall, and conſequently 


BN to K N, and C g to NZ Perpen- 


diculars, or at leaſt equipollent to Per. 


pendiculars, NZ will be the refracted 
Ray of AN. Q. E. D. 


COROL. 1. PouTTixG I to R, 


as the Sine of Incidence to the Sine of 
Refraction; it will be + A B. AC ::; 
B Z. C Z. For it is & AB. AB 


(T I. R) :: CI. C R, and A B. AC 
:B * CI, and by perturbated Equa- 


3 AB. A C (: B H. CR)::B 
6 


CO ROL. 2. Ir the Point A be 


infinitely diſtant, or ſends out parallel 


Rays, then by reaſon B H and CI are 
equal, it will be I. R:: B Z. C Z. And 


* For CI. CK: (BH. BM: !) CR.Cg, and by 
Permutation, CI, CR :: CR. C2. Pg 


fo 


7 


O the Refractions Ster. IV. 


ſo if the refracted Rays are parallel, 
then by Reaſon B H and CR are equal, 
it will be I. R:: AC. AB. 


COROL. 3. TIF of the four Points 
A, B, C and Z, any three are given, 
the fourth may be found, as will ap- 
pear by the following Examples. 


Ex aAMr. 1. LET A, 1 Nen 


and Z be ſought. It is K AB. Ac: 
B Z. C Z. And conſequently by Divi- 


ſion AB A C. AC:: BC. CZ. 


Ex AMr. 2. IF A, B and Z be gi- 
ven, and C ſought; ſince it is 1 A 


B. A C:: B Z. C Z, by Alternation it 


will be + A B. BZ: : AC. CZ, and 
by Compoſition = AB + B Z 
KA B.. AZ. Ac. 


EXAMP. 3. Is A, Cand Z — gi- 


ven, and B ſought; ſince i igis - X AB. 
1 A C 


SECT. IV. of Curve SURFACES. 


A C:: BZ. CZ, or AB. AC:: B 
EOSIN on be by 18 and 
Alternation f T AC. * AB. BZ, 


and by Compoſition © -AC+CE 
CZ: LY AF 4 


Tus ſame Things may be determi- 
ned by the drawing of Lines; as if A, 
B, and Q being given, C was ſought. 
Ere&BH of any Length perpendicular 
to A Z, and in it take BG, which may 
be to BH, as I to R, join AH and G 
Z meeting in I, and I C, being let fall 
perpendicular on A Z, will fall at the 
Point ſought C. 


NOTE 1. That Z is the Place of 
the Image of the Object A exhibited by 
Refraction, when the SpeQator's Eye 
is placed in the Axis beyond Z. 


2. Ie the refracted Rays diverge, or 
the incident Rays converge, or are pa- 
rallel, the Conſtruction of the Problem 
will be the ſame, thoſe Things being 
only changed, that ought to be changed, 

N 


3. IF 


177 


178 


Of the Refractious Scr. IV. 

3. Ir the Rays, emitted from the Point 
A, are tranſmitted ſucceſſively thro' ſeve- 
ral ſpherical Surfaces having the ſame 
Axis AC ; to determine the Concourſe 
after all the Refractions, ſeek firſt the 
Concourſe of the Rays after the firſt 
Refraction; then the Concourſe of the 
ſame after the ſecond Refraction, as if 


they had been primarily emitted from 
the Point of the preceding Concourſe; 


and ſo on till you are arrived at the 


laſt Concourſe. And by this means the 
Place of the Image of any Object ſeen 
through a Teleſcope or Microſcope may 
be determined. 


4. By Help of Corol. 3. convex 
Glaſſes may be made of ſpherical Sur- 
faces, which may ſerve for making Te- 
leſcopes after any deſigned manner. For 
it appears from that Corol. that not on- 
ly the Refractions of given convex 
Glaſſes may be inveſtigated, but alſo 
the Glaſſes be delineated, which ſhall 
produce given Refractions. 


L E Ms» 


. * * 
s W 


. = 7 * 
"WF. £ * 


* 8 


| SECT. IV. of Curve SURs ACES. 
LEMMA IX. 


In any given Curve, to determine the 
Concourſe of the Axis and the neareſt 
Perpendicular. 


In Fig. 56. Let BN be a Curve, 
and to any of its Points indifferently 
taken, ſeek a Perpendicular 2c by the 
known Method of drawing Perpendi- 
culars to Curves, and at the ſame Time 
you will find the Length B c. Then 
(t being let fall perpendicular to B) 
feign Br or t to be infinitely ſmall, 
or nothing, and there will come out the 
Length B C, whoſe Termination is at 
the Concourſe of the Axis with the 


neareſt Perpendicular. 


Ex AMr. I. Let B Nu be a Parabo- 
la, whoſe Latus rectum let be r, and 
Ber call x, it will be Bo Xxx. 
as is known. Now put x = o, and 
there will remain + » for the Length 
B C at the Vertex. 


N 2 ExAur. 


Fig. 56. 


Fig. 57. 


Of the ann SxcT. IV. 


Ex AMp. 2. Let B Nu be an Ellipſis 
whoſe Latus refum let be r and Latus 
ftranſuerſum q; it will be, as is known, 
BS ＋ + Zr. Now put x 
—0, and there will remain again I 
for the Length B C at the Vertex. 


Nor is the Proceſs different in more 
compounded Curves. 


PROP. XXX. 


Rays being incident very nearly Per- 
pendicularly on any curve Surface 5 to 
determine the Concourſe of the refract. 
ed Rays or their Focus. 


LET PB Q (in Fig. 37.) is any 
Curve. A the common Point or the 
Concourle of the incident Rays, A Ba 
perpendicular Ray or the Axis, and A 
N a Ray very nearly perpendicular, or 
the neareſt to the Axis. And let N C 


be perpendicular to the Curve, and 


meeting the Axis A B at C. And the 


Point C being found by Lem. IX. erect 


at Band C the Perpendiculars BH and 
| C1, 


SECT, IV, of Cuxvk SurrAcEs. 


CI, draw any Line ATI meeting them 
in H and I, towards I take CR, which 
may be to CI, as the Sine of Refraction 


to the Sine of Incidence; and the right 
Line H R will meet A B in Z the 
ſought Concourſe of the refracted Rays. 


Ir is proved after the manner of the 


preceding Propoſition, and to this be- 
long the like Corollaries and Notes. 


PRO P. XXXI. 


Parallel Rays being incident on 4 


Sphere ; to determine the Error from 


the principal Focus of Rays remote from 
the Axis. 


In Scheme 58. let N B A be a Sphere, 
C its Center, CB a Semidiameter paral- 
lel to the incident Rays, AN an inci- 


181 


Fig. 5% 


dent Ray, and N K its refracted Ray 


meeting the Axis, or the Semidiame- 
ter C B in K, and F being ſuppoſed the 
principal Focus, that is, into which the 
Rays lying near the Axis are collected, 


there is ta be ſought the Error EF K. Let 


fall therefore the Perpendiculars C E on 
N 3 NK 


182 


Of the Refractious Scr. IV. 


N K and NG on CK, and make C B 
= 4, GB = & and CK 2 and 
from the Nature of the Circle it will be 
NGq=24x—xx, to which add 
G K q, that is, 5 S ＋ 2X 2 — 2 42 
EAX - 24 * ＋Z 4 a, and there will 


come out NK 2 2 S ＋ 2X2 — 


2428 -+aa, Nou ſince N G is to C 


E, as the Sine of Incidence to the Sine 


of Refraction, or as I to R, and on ac- 
count of the fimilar Triangles C E K 
and N GK, N K and C K are in the 
ſame Ratio ʒ it will be II. RR (:: NK 
CKq):: S ＋ 2X 2 — 242 8 ＋ aa 
S Sand therefore I I X SS R 
R X Z E ＋TP— 2 RR Xx - 2 RR 
XxX az ＋ RR X 2 4. And by 
Reduction of the Equation, 2 & = 
2 R R. 4 * — 2 RR x2—RRaa 
SE RR —TII | 

and the Root being extracted & — 
RRa—RRx+RyIlas—2RRax;jRRxx, 
8 RR — 11 

i. e. (the radical Quantity being thrown 


into * an infinite Series) 5 Nl 


The Method of Series was a very early Invention of 
our Author's, viz. in 1665. A particular Account of 
which may be ſeen in a Book colled Commercium 
Epiſtolicum Jobanuis Collins & aliorum de Analyſt promota , 
firſt printed at London in 4%. A. 1712, and a ſecond 
Edition with Additions in 89, A. 1722. RR 


- SECT, IV. of Curvs Surxraces. 


RRE Ri- Ries? 
IR—II 21724 . c. Now 
ſince by Corol. 2 or 3. to Prop. XXIX. 
2 is =CF (which may be known 
from the Value of & now found, by 
teigning x =o) from this CF take the 


found Value of z and there will remain 
R R * * R 3 * x * 


[Regt 14 Ge. for the Value 


of the Error K F, which we ſought. 


CoOROL. I. * * be ſuppo- 
ſed very ſmall, [ — 1 will be very 


nearly equal to KF; for then the 


IE As RI » x 3 
Quantities . + 3TT7F 77 Sc. 


on Account of the aſcending Powers 
of the ſame x, will become very ſmall, 


and in Reſpe& of the Term IK . Ii may 
be looked on as nothing. 


COROL. 2. Moxeovex if you make 


N GS. it will be _TK — 


KF nearly, for it is NG = BG X 


BC+CG or = BG x 2 BC 
N 4 nearly, 


183 


184 


Of the Refrattions Sxcr. IV. 
nearly, that is, 5% = 2,4, x nearly, or 
— = x, and 2L being ae 


for x in the Value of KE, 5 7 1— 


IRomulla 


KF. 


COROL. 3. Hence the Errors 
K F areas the verſed Sines G B, or as 
the Squares of the half Chords N G. 


COROL. 4. Is the Ray ANK 
be given in Poſition, and the refracted 


Ray # & of any parallel Ray nearer 


the Axis, and falling on the other Side 


of the Axis, be drawn cutting the 


Axis in , and the refracted Ray N K 
in Q, and to the Axis be let fall the 
perpendicular Qo: the Line K o will 
become the greateſt of all, when the 
Ray 4 is nearly twice leſs diſtant 
from the Axis than the other Ray 


AN. For g being let fall perpen- 


dicular to the Axis, put #g—=v, Ko 


8, GK = F and K FS, and by 


Corol. 3. of this it will be 2 =: 


K k, farther it is GK, GN:: K 0. Qo, 
and 


SECT. IV. of Curve SukrAczks. 


and conſequently Qo = 22, alſo gn, 
GK(=gknearly):: Qo. 04, where- 
fore ok . To this add K o, 
and there will again come out K E 
= S. Wherefore it is 2225 


* 


5 v5 1 SEC 
= [, and Diviſion being made 


by v +, and the Equation reduced, 


there comes out $ 2 == — 
E, 


Now that the greateſt S may be 
found, multiply the Terms according 
to * Hudden's Method, by the Dimen- 
ſions of the indeterminate Quantity v, 


and there will come out o —= e 


or y =2v, that is N G = 2 »g. 


Here our Author refers to Hudden's Method de 
Maximis et Mininis printed Anno 1659 in Cartes's 
Geometry; becauſe his own Method of Fluxions was 
not yet made publick, though he had written ſe- 
veral ſmall Tra&s on this Subject in 1665, before he 
read theſe Lectures, and a larger one in 1671. Theſe 
have been never yet printed, though many Copies of 
them in Manuſcript are got abroad. The laſt is fre- 
quently mentioned in the Commerciurs Epiſtolicum. 


COROL, 


18 $ 


186 


Of the Refrattions Ste, Iv. 


COROL. 5. Axp hence K o, when 
it is greateſt, is equal nearly to a fourth 
Part of K F; for in the now found Va- 


lue of 8, F. you write 2 v for y, there 


will ariſe Sl 8. 


4 
COROL. 6. Ir is alſo o Q= FA 15 3 
For it is I K (= BE nearly). GN:: 


RRyy 


Ko. wh SIS ' $IRa—BIls 


YK F). 1 — 


CO ROL. 7. Is the Arch B M be 
taken equal to B N and B » = Ba, 
and the Rays refracted at the Points 
M and m be drawn meeting one ano- 
ther in P. It is manifeſt that the 

R y 3? 


SPACE P Qis = = 7155 viz. double 


of 0Q; and it is farther manifeſt, that 
the refracted Rays of all the Rays fal- 
— upon the ſpherical Surface between 

and M will converge into this Space 
> Q, and that the ſame Space PQ 


is the leaſt circular Space, into which 


al the Rays may be collected; and 
con- 


SECT. IV. of Cuxve SurxrACEs. 


conſequently is 'the Focus or Place of 
the Image of the Object ſending paral- 
lel Rays to the Lens, whoſe Aperture 
is the Limits M and N. v4z. no Rays 
can paſs beyond this Space, becauſe 
ſince 0Q is in a given Ratio to K o, 
and Q will be at the ſame Time the 


greateſt, and conſequently the Point Q 


is the remoteſt from the Axis of all the 
Points lying towards F, in which any 
Ray meets with the external Ray N K; 
nor can they be collected into a leſs 
Space, becauſe the Rays NK and M K 
cut the external Rays in the Points P 
and Q, by which the Space PQ is 
terminated. 


COROL. 8. Is NB M the Aper- 
ture of the Circle be enlarged, or dimi- 
niſhed, the lateral Error PQ will be as 
„or as the Cube of the Breadth of 
the Aperture N M. Alſo if the Aper- 
ture remaining unaltered, the Magni- 
tude of the Circle be changed, the Error 
P Q will be reciprocally as 4 4, or as 
CBq; and conſequently as B F q: 
For CB and B F are in a given Ratio. 


But if both the — of the 
Circle 


107 


188 


Fig. 59. 


07 the Re fractions SECT. IV. 
Circle and the Aperture be n 


that Error PQ will be as © = - Or as 


N M cub. R y 3 
">" as from _—_ 44 Value 


of PQ may be manifeſt. 


 SCHOL. NearLyY after the ſame 
Manner, as we have determined the 
Errors CF and PQ of parallel inci- 
dent Rays; may be ns. the like 
Errors of diverging or converging 
Rays, but with a more difficult Cal- 


culation. 


PROP. XXXII. 


1511 F Rays, whether parallel, or iucli- 


wed towards ſome common Point, are 
oppoſed to a Sphere to be refracted; to 
determine the Concourſe of the refracted 
Rays lying out of the Axis, the neareft 
to one another, and in the ſame Plane 


with the incident Rays. 


In Fig. 59. Let AN be an inci- 
dent Ray, N K its refracted one, and 
NY a right Line in the Plane of the 

Tri- 


N | 
SecT. IV. of Curve SuxrAces: 
Triangle AN K touching the Sphere 
at N. To AN draw NR perpendi- 
cular, and meeting the Axis A C in R; 
as alſo RV parallel, and meeting the 
Tangent NV in V. Alſo to NK 
draw NQ perpendicular and V Qpa- 
rallel meeting in Q, and draw QC 
meeting N K in Z; and Z will be the 
Concourſe of the Rays the neareſt to 
AN. For let A # be another of the 
incident Rays infinitely near to the for- 
mer A N, and meeting NR in G. Draw 
# Z meeting N Q in H; and to A N 


and N K from C the Center of the 


Sphere let fall the Perpendiculars C D 
and C E meeting Aa and Z in d and e. 
Now ſince A # is ſuppoſed infinitely 
near to A N, the infinitely ſmall Arch 
Nu may be looked on as a right Line 
coinciding with the Tangent N V, and 
the Triangles NG , NR V; and N 
H , N QV as ſimilar. Wherefore it 
AYR DA (:: NR. NG: : NV. NA 
NQ NH): EC. Ee. Whence by 
Coavertion and Alternation D C. E C:: 
4 C. e C. But it is DC to E C, as 
the Sine of Incidence to the Sine of Re- 
fraction, becauſe that N K is the re- 


fracted 


189 


150 


Of the Refractious Steer. IV. 
frated Ray of AN; and conſequently 
alſo 4 C to e C is, as the Sine of Inci- 
dence to the Sine of Refraction. And 
therefore ſince the Angles D A 4 and 
E Ze are infinitely ſmall, and for that 
Reaſon Cd to A and Ce to » 2 
Perpendiculars, or at leaſt equipollent to 
Perpendiculars, # Z will be the refracł- 
ed Ray of An. Q. E. D. 


'COROL. 1. 11 is NP. NE (or 
NP. NF) :: NR. N Q. For NC 
being 8 on account of the ſimilar 
Triangles NR V and NDC; NEC 
and NV, it is N D. NR: NC. 
NV):: NE. NQ, and by Permuta- 
tion N D. NE. :: NR. NQ. 


Hence is derived a more ready Re- 
ſolution of the Problem, viz. to the 
Rays AN, N K erect the Perpendicu- 
lars NR, N Q, whereofN R may meet 
the Axis AC; and NQ be to NR, as 
NF to NP. Then draw QC, which 
will meet with N K in the ſought 
Point 2 *. 

* This moſt 3 Conſtrusion was bliſhed as 
from our Author by Dr. Barrow in his 3 XIII Optical 


Lecture Art. XXVII. 
I COROL. 


— 


ssbr. IV. of Cunvs SuxrAcrs- 


/ 
/ 


COROL. 2. II is alſo AN x D 


CXxXNEADXECxXxXND::N 


Z. E Z. For it is AD. AN:: D C. 


N R, and thence N R — AN«DC: 


Alſo N D. N E:: NR. NQ, and 


NZD CNE 
thence N Q ANN; and con- 


ſequently AN x DC X NE. AD x 
N D XE C (:: NQ. EO) :: NZ. 
E Z. 


COROL. 3. Ir the radiating 
Point A is infinitely diſtant, or emits 
parallel Rays, putting I. R:: Sine Inci. 


dence. Sine Refraction, it will be I 


X NF.RXNP::NZ. EZ, For 
in this Caſe A N and A D ſeeing they 
are infinitely long, ought to be looked 
upon as equal; and conſequently by Co- 
rod, 2. of this, it will be DC X NE. 
ECXN D:: N Z. E Z. But by Hy- 
potheſis it is D C. E C:: I. R; and 
therefore I NE. R x ND(: : NZ. 
EZ) :: IX NF. R x NP. But of 
theſe ſee more in Dr. Barrow's Lec- 
tures, 


BuT 


97 


192 


Of the Refractious ITT IV. 


\ 


Bor there may be obſerved 1. That 
by making the proper Changes, the Re- 
ſolution of the Problem may be eaſily 
accommodated to any Caſe whatever ; 
whether the incident Rays diverge from 


any Point, or converge to the ſame, or 
fall haſt 


2. Since of the Rays neareſt to 
ANK, thoſe that lie in the Plane A 
N Rdo meet in Z, but thoſe, that lie in 
a conical Surface generated by the Re- 
volution of the Triangle A NK about 
its Side A K, do meet in K, the greateſt 
Conſtipation of the Rays neareſt on 
all Sides to A N K will be about the 
Middle of the Space K Z, as at Y; 
and therefore the Eye being placed in 
the Line N K beyond K, the Place of 
the viſible Image of the Object A, ſeen 
by the Refraction of the ſpherical Sur- 
face B N, will be at Y, or at leaſt with- 
in the Limits K and Z; for that Place 
is not preciſely determined. 


2 3: Wan 


ster. IV. of Curve SURFACES, 


3: Waen the Rays are ſucceſſively 
refracted by ſeveral Surfaces, that you 
may determine the Concourſe of the 
near Rays after all the Refractions, 


firſt ſeek the Concourſe after the firſt. 


Refraction, then the Concourſe of the 
ſame after the ſecond Refraction, as if 


they had flowed primarily from the 
Point of the preceding Concourſe ; and 
ſo on, as it was ſaid at Prop. XXIX. 


PROP: XXXIII. 


Rays being incident upon any curve 
Surface whatever, to determine the 
Concourſe of the refracted Rays neareſt 
fo one another, and lying in the ſame 
Plane with the incident ones. 


In Fig. 59. feign B NP now to re- 
preſent not a Sphere, but any other 
Curve, and let A be the common Point, 
or Concourſe of the incident Rays, 
AN one of the incident Rays, N K its 
refracted one, and NC a perpendicular 
to the Curve at the refracting Point, 
In this NC ſeek the Interſection of any 


Q the 


193 


Fig. 59« 


194 


Of the Refractions Scr. IV. 


the neareſt Perpendicular (ſuch as # C) 
inſiſting on another the neareſt refract- 
ing Point. Which thing ſhall be taught 
+ elſewhere. And let that Interſection 
beC; now AC being drawn, let fall 
to the Rays A N, N E, the Perpendicu- 
lars C D, CE, and erect N R, NQ, 
whereof let NR meet A C in R, and 
let it be NQto NR, as NE to ND, 
and Q C being drawn will meet with 
the refracted Ray N K in Z the deſired 
Concourſe of the neareſt refracted Rays. 
It is proved after the manner of the 
preceding Propoſition, and to this be- 
long alſo the like Corollaries and Notes. 


PROP. XXXIV. 


To determine the Figure which ſhall 
fo refrat4 homogeneal Rays, whether 
parallel, or terminated at ſome common 
Point, that all the refracted Rays may 


meet accurately in ſome other given 
Point, 


+ Viz. in his Treatiſes of Fluxions mentioned before, 
I 


Ix 


Sscr. IV, of Curve SurPACEs. 


"93 


In Fig. 60. let A be the Concourſe Eg. 6. 


of the incident Rays, and Z of the re- 
fracted Rays, and let any Point B be 
taken at Pleaſure in the right Line A Z 
for the Vertex of the Curve. From 
that Point B in the Line B Z towards 
the denſer Medium let be taken B G of 
any Length, and BR in the Proportion 
to B G, that the Sine of Incidence has 
to the Sine of Refraction. And with 
the Centers A and Z, and the Intervals 
A Gand ZR let be deſcribed Circles 
interſecting one another in N, and the 
Locus or Place of N will be the Curve, 
that will perform the deſired Refraction. 


Tur this may appear, let A N be 
produced to S, that it may be NS. N 
Z:: BG. B R; and to N S and NZ 
let be erected the Perpendiculars 8 T 
and Z T meeting in T, and NT being 
drawn will touch the Curve in N, as 
will be manifeſt from the Method of 
drawing Tangents elſewhere * explain. 
ed. Now ſince NS and NZ are as B 


* In our Author's Treatiſes of Hu xions. 


O 2 G and 


196 


Fig. 61. 


Of the Refrattions Sect. IV. 


GandBR, that is, as the Sine of In- 
cidence and Refraction; and in reſpe& 
of the whole Sine or Semidiameter N T, 
Nes is the Sine of the AngleN T'S, 
which is equal to the Angle of Incidence 
of the Ray A N, and NZ the Sine of 
the Angle NT Z, which is equal to the 
Angle of Refraction of the Ray NZ; 
it is manifeſt that N is the refracted 
Ray of AN. QE. D. 


NOTE 1. A Curve may alſo be de- 
ſcribed for this Purpoſe, which ſhall 
paſs through any given Point B placed 
out of the Axis A Z. V:z.in Fig. 61. 
let be drawn AB and Z B, and in them 
let be taken BG to BR as I to R, and 
let Circles be deſcribed meeting in N, 


and N will be in the Curve, that was to 
be deſcribed. 


2. TRE ſaid Reſolution of the Pro- 
blem by making neceſſary Changes, 
extends it ſelf to all Caſes, whether the 
incident or refracted Rays converge, 
di verge, or are parallel, or whether the 
the Refraction is made out of a rarer 
Medium into a denſer of out of a den- 


ſer 


SECT. IV. of Curve SurrFACEs. 


ſer into a rarer. And indeed if the 
Rays are on no Side parallel, that is, 
if neither of the Points A and Z is at 
an infinite Diſtance, the Curve will be 
one of the four Ovals, which Cartes 
has deſcribed for this End in his Geome- 
zry ; but if either of them is infinitely 
diſtant, ſo that the Rays reſpecting that 
Point become parallel, the Curve will 
be a conick Section, as is known. And 
in this Caſe the Circle R N or GN on 
account of the infinite Diſtance of the 
Center will become a right Line per- 
pendicular to A Z at R or G. * 


LEMMA X. 


Of parallel Rays refracted by a 
Circle, to determine that Ray, whoſe 


Part included within the Circle may 
have à given Ratio to the Part of its 


refracted Ray included within the ſame 
Circle. 


In Fig. 62. Let AN be the incident 
Ray, NK the refracted one, NP and 


* See his Principia Prop. 97, Lib. 1, 
O3 NF 


197 


Fig. 62 · 


198 


Of the Refractious Sor. IV. 


N F their Parts included within the 
Circle, C D and C E Perpendiculars let 
fall to thoſe Parts from the Center of 
the Circle, and B C a Semidiameter 
drawn parallel to AN. And let it be 
CD.CB::LR, and NP. NF:: . 
9. Theſe Things being ſuppoſed, that 
there may be known the Point N, 
which determines the Rays A N and 
N K, ere& at B C a Perpendicular BX, 


whoſe Square let be — the Square of B 


C, as T2 to Hf; and CX be- 


ing drawn will cut 1 Circle in the 
ſought Point N. For it is by Hypothe- 
ſis p. J (: NP. NF): : N D. NE, and 


L. R. CP, CE; wherefore ND 


=N E, and =C D SCE. Moreo- 
ver ſince it is NDq+ CDq (=N 
Cq) = N E q CE q, take from 
both Sides NDq+CE q, and there 
will remain CD CEA NEN 
— ND q, that is, by ſubſtituting the 
Values of — 1 and N E juſt now found, 
cDꝗ- H II CD ND — 
N D q, 8 Reduction being made, 
11 


SECT. IV. of Curvt Surracts. 


II — RR 77 — f 
Tr CD ND 4q. 


Which being reſolved into a Proportio- 


77 —5 P. II- RR 
nality becomes * I > (= 


Dq. NDq)::BXq BCq. Q. E. D. 


PROP. XXXV. 


The Sun ſhining upon a tranſparent 
Sphere, to determine the greateſt Incli- 
nation of the Sun's Rays to the Axis, 
as they emerge after one Reflexion. 


In Fig. 63. Let BN K be the pro- 
poſed Sphere ; B Q a Diameter or Axis 
parallel to the incident Rays, AN any 
one of the incident Rays, N F its re- 
fracted one, FG the reflected one, and 


G R the refracted one again; and 


there is required the greateſt Angle, 
which RG can make with the Axis B 
Q. For which Purpoſe it muſt be ob- 
ſerved, that in the Caſe only, when R 
G are inclined the moſt to BQ, the 
Rays the neareſt to AN can emerge pa- 


rallel to RG. For in other Caſes, of 


emerging Rays the neareſt to one ano- 
ther, ſome are continually more incli- 
O 4 ned 


199 


Fig. 63. 


200 


Of the Refrat7ions SECT, IV. 


ned, others leſs to BQ; and conſequent- 
ly are ſomewhat inclined to one ano- 
ther. It muſt be farther obſerved, that 
thoſe Rays emerge parallel, which meet 
at the Point of Reflexion. For draw 
the Ray an parallel and the neareſt to 
AN, and let its refracted Ray be 2, re- 
flected F , and refracted again g 7 5 
and the Points F and F coinciding , 
ſince the Angles N F and G F g are e- 
qual, and the Refractions at N # and 
G g alike, the emerging Rays GR and 
Ir will be equally parallel as the inci- 
dent Rays A N and 4. 


TukRE is therefore to be ſought a 
Ray A N, whoſe refracted Ray meets 
at F with the refracted one of the near- 
eſt Ray an; and indeed by Corol. 3. 


Prop. XXXII. (CD and CE being let 
fall Perpendiculars from the Sphere's 


Center to the Rays, and it being made 
I. R:: C D. CE) if the Rays themſelves 
concur at any Point Z, it will be I X 
NF. RX NP: N Z. E Z.: NF. 


E F (zz. the Paint Z falling by Hy- 


potheſis on F) ; : 2. 1. Wherefore I X 


NF=2RxNP, and I. 2 R:: NP 


NF. 


SECT. IV. of Curve SURFACES. 


N F. There is therefore given the 
Ratio of NP to NF; and thence, by 
Lemma X. there will be given the 
Point N. Thus to the Vertex of the 
Circle let be drawn the Tangent BX, 
whoſe Square may be to the Square of 
the Semidiameter BC, as 4 RR — II 
to II - RR, and let C be drawn; 

for this will meet the Circle in N ; and 


from N being found, the reſt are deter- 
mined with no Trouble. 


COROL. 1. Henxcst it becomes 
3 RR. IT - RR:: CNq. NDq. 
For ſince it is 4 RR - II. II - RR 
:: BX q. B C q; it will be by compoun- 
ding 3 RR. II — RR (:: BX ＋ 
B CS CX q. BCq) C Nq. N 
Dq. 


COQROL. 2. Ir is alſo I. 2 R:: N 
D. N E. For above it was I. 2 R:: NP, 
NF. And from theſe the Solution of 
the Problem becomes the more expe- 
ditious. 


SCH OL. TocgTRER with the great 
eſt Inclination of the Ray R G, there 


is given the greateſt of the Arches FQ 
is 


201 


202 


Of the Refrattions Secr. IV. 
is terminated at the refracted Rays N F. 
For the Angle FC Q, which F Q ſub- 


tends, is equal to the Angle, which 
CF and A N comprehend; that is, 


equal to Half the Angle, which R G 


and AN or BQ comprehend; and 
therefore of the Arches F Q, as well as 
of the Angles comprehended by RG 
and B Q, that is the greateſt, which is 
defined by the Ray A N incident at 


the Point now found. 


p R O P. XXXVI. 


The Sun ſhining upon a Tranſpa- 
rent Sphere, to determine the leaſt In- 
clination of the Rays to the Axis after 
two Reflections. 


LET AN and «a» be two inci- 
dent Rays the neareſt to one anorher, 
which after two Reflections in F, 
and Gg, let them emerge in HS and 
5, and it is manifeſt, that in that Caſe 
only, where the acute Angle, which 
BQ and 8 H comprehend, is the leaſt, 
thoſe Rays H S and hs can be parallel, 
as was ſaid above of the Rays G R and 


37 


Sect. IV. of Cuxvs SuxrAcks. 
gr; and where this happens, the Ray 
F G alſo will be parallel to fg. Whence 
2 arc FF ( arc Ff+Gg SD are 
EG — g S arc NF — »f) —arc 
Nu — Ff. And conſequently 3 arc 
FE f = are Na; and ſince N E is divi- 
ded in Z in the Ratio of theſe Arches, 
as is manifeſt; it will be NZ 3 Z E 
or 3 E Z. Since therefore by Corel, 3. 
Prop. XXXIL it is IXNF.R xXNP:: 
N Z. E Z. or: : 3. 1; it will be L XN 
FR XN P. or I. 3 R:: NP. NE. 
There is therefore given the Ratio of 
NP to NF; and thence by Lemma X. 
there will be given the Point N, vi. 
by drawing B X, that ſhall touch the 
Circle in the Vertex B, and whoſe 
Square may be to B C quad. as 9 R 
R —II to II- RR; and by draw- 
ing CX, that ſhall meet the Circum- 
ference in N; but N being found, 
the reſt are eaſily determined. 


CORO L. 1. HERNCx it is 8 RR. I 
IRR :: CN q. ND q. For 9 R 
R - II. II- RR:: B X q. B C, 
and by compounding 8 RR. II R 
R (: :CXq BCq):: CNq.NDq. 

COROL. 2. 


203 


204 


Of the Nefractions Sxœr. IV. 


_ COROL. a. I r is alſo I. 3 R:: ND. 
N Ez; ſeeing it was before I, 3 R:: 
N * . N F. | 


. 


SCHOL. Ar TER the ſame Manner 


will be found the greateſt Inclination 


to the Axis of the Ray K T emerging 
after three Reflections, as well as the 
greateſt of the Arches Q G . vig. in 
that Caſe FG and f'g will meet in G, 
and it will be the Arch Ff ( = Arch 
Fg—fg =ArchNF - 2 f) = N #— 
FE V, and thence 2 Arch Ff — Arch 
Nu, and NZ = 2 ZF. And con- 
ſequently 4. 1:: N Z. EZ:: (per 
Corol. 3. ad Prop, XXXII.) I XN 
F. RX NP, or I. 4 R:: NP. N F; 
and therefore by Lem. X. 16 RR 1 
I. II- RR:: BX q. B C q. Whence 
it follows, that 15 RR. II- RR:: C 
N q. ND q. And I. 4 R:: N D. NE. 


Anp ſo if there be required the leaſt 
Inclination of a Ray emerging after 
four Reflections, you will determine it 


by making 25 RR - II. II- RR:: 


BX. BC q, or 24 RR. II- R 
„ 


SecT. IV. of Cux vx SoxrAcEs. 
R:: CN q. ND q, and I. 5 R:: 
ND. NE. And ſo on in infinitum. * 


THE Refractions of homogeneal 
Rays having been handled, it now re- 
mains, that we come to heterogeneal 
ones. Of the Refractions of theſe in 
regard to Planes, we have treated the 
more diffuſely, that thereby the Affecti- 


ons of Priſms (whoſe Uſe in making 


Experiments will hereafter be very fre- 
quent) might be known. But the 
principal Thing about Curve Surfaces, 
that now occurs to be determined, is 
the Quantity of the Error of the Rays, 
from whence ariſes the Confuſion, or 
indiſtin& Viſion of Objects, which in 
Teleſcopes is wont to happen through 
the too large Aperture of the object 
Glaſs. And to this End, ſince there 
was premiſed Prop. XXXI. Whence 
the Errors are known, that are made 
in ſpherical Surfaces through the Un- 
fitneſs ofthe Figure: We ſhall now ſub- 


* The Uſe of the two laſt Propoſitions is to deter · 
mine the Rain-bow, as ſee our Author's Opticks, Book 1. 
Part 11. Prop. IX. 


2 join, 


205 


20⁰6 


Fig. 64. 


SECT. IV. 


Of the Refractiont 


join; how may be determined the 
Errors ariſing from the unequal Re- 
May of the Rays. 


PRO P. XXXVII. 


[ Heterogeneal Rays falling on a 


Sphere, to determine the Errors occa- 


ſioned by the unequal Refrattions of 
alike incident Rays. 


| FroM the Point A (Fig. 64.) upon 
the Sphere N B M, deſcribed to the 
Center C, let two the greateſt difform 
Rays be incident in any Line A N, 
whoſe refracted Rays are N F and Ny, 
meeting the Axis in F and 75, and on 
them let fall the perpendiculars C I, 
CP and CT. Now if an accurate 
Solution was required, the Refractions 
of the Rays N F and N muſt be ſe. 
parately computed, But ſince the Arch 
N M is ſuppoſed to be a very ſmall 
Portion of a Circle ; we ſhall attain to 
the Truth very nearly by aſſuming the 
Angles CNI, CNP and CNT 
to be almoſt as their Sines. Let there- 
fore I be the common Sine of Inci- 
dence, 


Sgcr. IV. of Curve SurrFAcEs. 


dence, P the Sine of Refraction of the 
greateſt refrangible Rays, and T the 
Sine of the leaſt refrangible ones; and 
it will be Angle C NI. Angle C N 
P:: I. P, and Angle C NP. Angle 
C NT:: P. T, and by Diviſion, 
Angle IN P. Angle CN P:: PI, 
P, and Angle CNP. Angle PN 
T:: P. P —T; and by Equality 
Angle IN P. Angle PN T:: P—L 
P — . 


TAkE now the Arch B M equal to 
the Arch B N, and of the Rays incident 
in AM draw ahi refracted ones M , 
M/, meeting the former in V and X: 


Draw VX, and produce it, till it meets 
the incident Rays at G and H; and 


it is manifeſt, that V is the Breadth 
of the leaſt Space, into which all the 
Rays may be collected. And it is 


G X. V X(:: Angle GN X. Angle 


VNX nearly :: Angle IN P. Angle 
PN T):: P— I. P— T; and 
G H- VX (2 GX). VH :: 2 
P — 2 1. P— T, and by Diviſion 
GH, VX: : PCT — 21. P— F. 
Whence P, IT and I being given; 

there 


207 


208 Of the Refrattions SECT. IV. 


there will be given the Ratio of G H 
to VX. 


Fox Example, ſincę I have above 
determined, that in Glaſs terminated 
by Air, it is I. P:: 44%. 692, and 
I. T:: 442. 682; if it be aſſumed 
LI = 442, it will be Pg 694, and 
T — 682, and P + T —2F= 49, 
and P—T = 1: and: conſequently 
„. 


SCH OL. By the Help of this 
and Prop. XXXI, the Errors of ho- 
mogeneal Rays, which happen in ſphe- 
rical Surfaces, through the Unfitneſs 
of the Figure, may be compared 
with the Errors of heterogeneal Rays 
and it will appear, that theſe are far 
greater in ſmall Portions of Spheres: 
and conſquently, that the Heterogenity 
of the Light, and not the the Unfitneſs 
of the ſpherical Figure is the Cauſe, 
that Teleſcopes are not yet arrived to a 
greater Degree of Perfection. 


bo | Lex us conceive for Example, that 
m. 58, 64. N M B. in Fig. 58. and 64, repre+ 
5 al ſents 


SecT. IV. of Curve SurrAcEs. 


ſents the Object Glaſs of a Teleſcope, 
whoſe anterior Surface N M let be plane, 
that it may only refract the Rays on 
its poſterior or ſpherical Surface N 
B M; and let us ſuppoſe C B the 
Semidiameter of this Sphere to be 10 
Feet, that it may make a Teleſcope of 
near 20 Feet (or 240 Inches) long. 
And let its Aperture be two Inches, as 
the greateſt that can be uſed, with ſuf- 
ficient diſtinctneſs of Viſion in theſe 
Sorts of Teleſcopes, that magnify the 
Object about 70 or 80 Times, and let 
the Sine of Incidence be to the Sine 
of Refraction, in the Confine of Glaſs 


Fand Air; as 11 to 17 nearly, as we 


have determined above. Theſe Things 
being ſuppoſed, there muſt be written 
120 for a, 1 for y, 11 for I, and 17 
for R in the Value of PQ, which we 


exhibited in Coro, 7. Prop. XXXI. 


2 


That is in the Term —— - and there 


17 Inches. 17 cakes LS 
421120120 633 600 © P O. 
And this is the lateral Error of the 
homogeneal Rays ariſing from the Un- 


| litnels of the Spherical Figure. Far- 
P ther 


emerges : 


209 


f 


210 


Of the Refrattions Sxcœr. IV. 


ther let us conceive the Rays A N and 
AM in Fig. 64. to be parallel, and the 


Aperture N M will be = 2 Inches = 


G H, which is to VX as49to 1, by 
the preceding. That is, V X = ++ 
Inches, or the Error, which ariſes 
from the Separation of the heteroge- 
neal Rays from one another, in the 
ſame Place of Concourſe, will be 


+3 


Inches. Now, confer theſe Errors, 


and it will appear, that VX is to PQ 


(or - —W7 45505 1267200 to 933, or 
as Ki to 1 nearly. 


ANp conſequently that VX is a- 
bove a Thouſand and five Hundred 
Times greater than PQ; ſo great 
a Diſproportion indeed, that P Q 
in Reſpect of V X may be looked 
upon as nothing. The Error VX 
ſince it is +; Inch is fo great, that I 
wonder, how Objects are ſeen fo di- 
ſtinctly through theſe Sorts of Tele- 
ſcopes. But = Error of the other 


ind P — 
K Qor 2 7 Inch that is 372577 


Inch nearly, is by far too little to be- 
come 


* 


„Ker. IV. of | Curve SURFACES: 1 


- come ſenſible, and therefore to be ne- 
3 glected; and the indiſtinct Viſion is 


only to be attributed to the Errors a- 


riſing from the Heterogeneity of Light. 
And hence it is manifeſt, that the 
Perfection of Teleſcopes i is not to be 
ſought from the Conick Sections, but 
that ſpherical Figures may be equally 
ſerviceable to this Purpoſe. In Micro- 
ſcopes indeed the Errors of the homo- 


geneal Rays from the ſpherical Surface 


of the Object Glaſs on account of the 
large Aperture do become enormous 
and very ſenſible. So that thoſe Glaſ- 
ſes, if they were duly formed into 
ſome Conick Section, would become 


a little more perfect. + But I am 


not ignorant of a Method of cor- 
recting thoſe Errors, without the Co- 
nick Sections, and by cauſing, that 
Glaſſes may be made of ſpherical Sur- 
faces, that ſhall refract ſufficiently 
exact the homogeneal Rays, not to 


/ 


+ I ſuppoſe our Author here means by compoſing 
the Obje&t-Glais of two Glaſſes with Water between 
them — See the Scholium at the End of the firſt Book 
of his Principia 3 and Prop. VII. Parti. Book 1. of his 
Optic ks, 


lay, 


21 


212 


Of the Refractiont, &c. SEC r. IV. 


» fay, that they ſhall refract far more 


accurately the oblique Pencils of Rays, 
than Glaſſes terminated by any other 
Figures whatſoever. So that I think, 
ſpherical Surfaces are better accom- 
modated to Dioptrical Uſes than all 


others,