OPT 
continuation of the firft femi-diameter to the firft focus, 
to EG orEF, the focal diftance of the lens; which, accord¬ 
ing as the lens is thicker or thinner in the middle than at 
its edges, rnuft lie on the fame iide as the emergent rays, 
or on the oppofite iide. 
Cor. 3. Hence, when rays fall parallel on both fides of 
any lens, the focal diftances EF, Ef, are equal. For, let 
rt be the continuation of the femi-diameter Er to the firlt 
focus t of rays falling parallel upon the furface A ; and 
the fame rule that gaverR : j-E=RT : EF, gives alfo rR : 
RE— rt : E f. Whence E/i=EF, becaufe the reCtangles 
cExBT=REX)’t. For rE is to rt, and alfo RE to RT, 
in the fame given ratio. 
Cor. 4. Hence, in particular in a double-convex or dou¬ 
ble-concave lens made of glafs, it is, As the fum of their 
femi-diameters (or in a menifcus as their difference) to 
either of them, fo is double the other to the focal dif¬ 
tance of the glafs. For the continuations RT, rt, are 
feverally double their femi-diameters ; becaufe in glafs, 
ET: TR and alfo E*: tr= 3 : 2. 
Cur. 5. Hence, if the femi-diameters of the furfaces of 
the glafs be equal, its focal diftance is equal to one of 
them j arid is equal to the focal diftance of a plano-convex 
or plano-concave glafs whofe femi-diameter is as fliort 
again. For, confidering the plane furface as having an 
infinite femi-diameter, the firft ratio of the laft-mentioned 
proportion may be reckoned a ratio of equality. 
Prof. II. The focus of incident rays upon a fingle fur¬ 
face, fphere, or lens, being given, it is required to find 
the focus of the emergent rays. 
Let any point Q, (fig. 13 to 18 ; Plate IV.) be the fo¬ 
cus of incident rays upon a fpherical furface, lens, or 
fphere, whofe centre is E ; and let other rays come paral¬ 
lel to the line QEg the contrary way to the given rays, 
and after refraftion let them belong to a focus F ; then, 
taking E/'equal to EF the lens or fphere, but equal to 
FC in the fingle furface, fay, As QF is to FE, fo is Ef to 
fq-, and, placing fq the contrary w ay from/ to that of FQ 
from F, the point q will be the focus of the refracted rays, 
without fenlible error; provided the point Q be not fo 
remote from the axis, nor the furfaces fo broad, as to caufe 
any of the rays to fall too obliquely upon them. 
For, with the centre E and femi-diameters EF and F f 
defcribe two arches FG,fg, cutting any ray QA itq in G 
and g, and draw EG and Eg-. Then, fuppoftng G to be 
a focus of incident rays (as GA), the emergent rays (as 
agq) will be parallel to GE ; and, on the other hand, fup- 
poiing g another focus of incident rays (as ga), the emer¬ 
gent rays (as AGQ) will be parallel to g-E. Therefore 
the triangles QGE, F '.gq, are equi-angular; and, confe- 
quently, QG : GE=Eg-: gq; that is, when the ray QA 
aq is the neareft to QE q, QF : FE=E/’: fg. Now, when 
Q accedes to F and coincides with it, the emergent rays 
become parallel, that is, q recedes to an infinite diftance; 
and conl'equently, when Q pafles to the other fide of F, the 
focus q will alfo pafs through an infinite fpace from one 
Iide of f to the other fide of it. Q. E. D. 
Cor. 1. In a fphere or lens, the focus q may be found by 
this rule : QF : QE, —QE: Q q, to be placed the fame 
way from Q as QF lies from Q. For, let the incident and 
emergent rays QA, qa, be produced till they meet in e; 
and the triangles QGE, Qoq, being equi-angular, we have 
QG : QE=Qe : Qq; and, when the angles of thefe tri¬ 
angles are vanilhing, the point e will coincide with E; 
becaufe in the fphere the triangle A ca is equi-angular at 
the bafe A a, and confequently Ae and ne will at laft be¬ 
come femi-diamsters of the fphere. In a lens, the thick- 
nefs A a is confiderable. 
The focus may alfo be found by this rule : QF: FE=c 
QE 1 E<7 ; for QG : GE^xQA : A//. And then the rule 
formerly demonftrated for fingle furfaces holds good for 
the lenfes. 
Cor. 2. In all cafes the diftance fq varies reciprocally 
as FQ does; and they lie contrarivvife from / and F ; be- 
Vol. XVII. No. 1198. 
ICS. 573 
caufe the reCtangle or the fquare under EF and E/ the 
middle terms in the foregoing proportions, is invariable. 
The principal focal diftance of a lens may not only be 
found by collecting the rays coming from the fun, con- 
fidered as parallel, but alfo (by means of this propofition) 
it may be found by the light of a candle or window. E°r, 
becaufe Q q-. qA-=.QE\ EG, we have (when A coincides 
with E) Q q -. ^E=QE s EF ; that is, The diftance obferved 
between the radiant object and its picture in the focus is 
to the diftance of the lens from the focus as the diftance 
of the lens from the radiant is to its principal focal dif¬ 
tance. Multiply therefore the diftances of the lens from 
the radiant and focus, and divide the product by their fum. 
Cor. 3. Convex lenfes of different ftiapes, that have equal 
focal diftance when put into each other’s places, have 
equal powers upon any pencil of rays to refraCt them to 
the fame focus. Becaufe the rules above mentioned de¬ 
pend only upon the focal diftance of the lens, and not 
upon the proportion of the femi-diameters of its furfaces. 
Cur. 4. The rule that was given for a fphere of an uni¬ 
form derifity, will ferve alfo for finding the focus of a pes- 
cil of rays refraCted through any number of concentric 
furfaces, which feparate uniform mediums of any different 
denfities. For, when rays come parallel to any line drawn 
through the common centre of thefe mediums, and are 
refraCted through them all, the diftance of their focus 
from that centre is invariable, as in an uniform fphere. 
Cor. 5. When the focufes Q, q, lie on the fame fide of 
the refraCting furfaces, if the incident rays flow from Q, 
the refracted rays will alfo flow from q ; and,.if the inci¬ 
dent rays flow towards Q, the refraCted will alfo flow to¬ 
wards q : and the contrary will happen when Q and q 
are on contrary lides of the refraCting furfaces; becaufe 
the rays are continually going forwards.- 
From this propofition we alfo derive an eafy method of 
drawing the progrefs of rays through any number of lenfes 
ranged on a common axis. Let A, B, C,'(fig. 19.) be the 
lenfes, and RA a ray incident on the firft of them. Let 
a, £, *, be their foci for parallel rays coming in the oppo¬ 
fite direction ; draw the perpendicular d a, cutting the in¬ 
cident ray in d, and draw da through the centre of the 
lens : then AB, parallel to da, will be the ray refracted 
by the firft lens. Through the focus of the fecond lens 
draw the perpendicular £ e, cutting AB in e; and draw 
eh through the centre of the fecond lens. BD parallel to 
he will be the next refraCted ray. Through the focus k 
of the third lens draw the perpendicular v.f, cutting ED 
in / and draw fc through the centre of the third iens. 
CE, parallel to fc, will be the refracted ray ; and fo on. 
Prop. III. To find the focal length of a compound lens. 
Let the two lenfes A and C, fig. 20. be placed clofe to¬ 
gether, in luch a manner that their axes may coincide; 
and let QA and DE be two rays of a parallel pencil inci¬ 
dent upon them, of which DE is coincident with their 
common axis. Take f, the principal focus of rays inci¬ 
dent upon the lens A, in the direction DE ;,a«d F, the 
principal focus of rays incident the contrary way. upon 
the lens C. Then, after refraction at the lens A, the rays 
converge to/, and are thus incident upon the lens C; if, 
therefore, we take/F : FC :: Cf : Cq, and meafure C<t 
and Cf in the fame or oppofite directions from C, accord¬ 
ing as F/'and FC are meafured in the fame, or oppofite 
directions from F, q is the focus of emergent rays, and 
Cq the focal length of the compound, lens. By proceed¬ 
ing in the fame manner, we may determine the focal 
length, when any number of lenfes are combined together. 
Cur. When F and /are coincident, the emergent rays 
are parallel. 
Prop. IV. If a prolate fpheroid be generated by an el- 
lipfe whofe major axis is to the diftance between its 
foci as the fine of incidence to the fine of refraction 
out of the ambient medium into the folid, a pencil of 
arallel rays, incident in the direction of its axis, will 
e refracted, convergingaccurately, to the farther focus. 
7 G Let 
