572 OPT 
of glafs, and let the incident rays be parallel, or tending 
to R, infinitely diftant: they will be difperfed by refrac¬ 
tion from the principal focus O within the glafs. As they 
are made more converging, R comes nearer, and F retreats 
backward, till R comes to Q, the principal focus without 
the glafs 5 when F is now at an infinite difiance within 
the glafs, and the refracted rays are parallel. R (till com¬ 
ing nearer, F now appears before the glafs, overtakes R 
at the centre C, and is again overtaken by it at N. R 
now becoming a radiant point within the glafs, F follows 
it backwards, and arrives at O, when R has receded to an 
infinite diltance, and the feries is completed. 
4. Let the incident light, moving in glafs, fall on the 
concave furface of air, or come out of the convex furface 
of glafs. Let it tend to a point R at an infinite diltance 
■without the glafs. The refrafted rays will converge to 
O, the principal focus without the glafs. As the incident 
light is made more converging, R comes towards the glafs, 
■while F, fetting out from re, alfo approaches the glafs, and 
R overtakes it at the furface V. R now becomes a radi¬ 
ant point within the glafs, receding backwards from the 
furface. F recedes flower at firft, but overtakes R at the 
centre C, and pafles it with an accelerated motion to an 
infinite diftance; while R retreats to Q, the principal fo¬ 
cus within the glafs. R fiill retreating, F appears before 
the glafs; and, while R retreats to an infinite diltance, F 
conies to V, and the feries is completed. 
Of LENSES. Plate III. IV. 
The word lens properly fignifies a fmall roundilh piece 
of glafs of the figure of a lentil; but is extended to denote 
any piece of glafs, or other tranfparent fubftance, which 
has its two furfaces fo formed, that the rays of light, in 
palling through it, have their direftion changed, and 
made to converge and tend to a point beyond the lens, 
or to become parallel after converging or diverging, or 
laftly, to diverge as if they had proceeded from a point be¬ 
fore the lens. Some lenfes are convex, or thicker in the 
middle ; others concave, or thinner in the middle ; while 
others are plano-convex, or plano-concave; and fome 
again are convex on one fide and concave on the other, 
as the menifcus. When the particular figure is not con- 
fidered, a lens that is thickeft in the middle is called a 
convex lens, and that which is thinneft in the middle is 
called a concave lens, without farther diftinftioh. 
The nine different forms of lenfes are reprefented at 
Plate III. fig. x. where B, C, D, are convex lenfes, and 
E, F, G, H, are concave ones. Alfo, A is a plane glafs, 
flat on both fides, and of equal thicknefs in all its parts. 
B, a plane or fingle convex lens ; i. e. plane on one fide 
and convex on the other. C, a flat plano-convex, whofe 
convex fide is ground into feveral little flat furfaces. D, 
a convexo-convex, or double convex. E, a plano-con¬ 
cave ; that is, plane on one fide, and concave on the 
other. F, a concavo-concave, or double concave. G, FI, 
two different forms of the menifcus, both being concave 
on one fide and convex on the other, or concavo-con¬ 
vex : it feems belt to deferve its name of pienijcus, or little 
moon, when of the fliape reprefented at H : but the me¬ 
nifcus is feldom made ufe of in optical inftruments. Laftly, 
I is a prifm, which has three flat fides, and, when viewed 
endwife, appears like an equilateral triangle. 
In a lens, of whatever form, the right line perpendicu¬ 
lar to the two furfaces is called the axis, as K L ; the points 
where the axis cuts the furface are citlled the vertices of 
the lens ; alfo the middle point between them is called 
the centre; and the diltance between them, the diameter. 
We now proceed to explain the properties of lenfes, and 
the phenomena they prelent. 
A ray of light G/i, fig. z, falling perpendicularly on a 
plane glafs EF, will pafs through the glafs in the fame 
direction hi, and go out of it into the air in the fame 
ftraight line iH. Alfo, a ray of light AB, falling ob¬ 
liquely on a plane glafs, will go out of the glafs in the 
fame di reft ion, but not in the fame ftraight line ; for, in 
I c s. 
touching the glafs, it will be refrafted in the line BC, 
and, in leaving the glafs, it will be refrafted in the line CD. 
Lemma. There is a certain point E, (fig. 3, 4,) within 
every double convex or double concave lens, through 
which every ray that pafles will have its incident and 
emergent parts, QA, aq, parallel to each other: but in a 
plano-convex or plano-concave lens, that point E is re¬ 
moved to the vertex of the concave or convex furface; 
and, in a menifcus, and in that other concavo-convex 
lens, (fig. 5, 6,) it is removed a little way out of them, 
and lies next to the furface which has the greateft cur¬ 
vature. 
For, let REr be the axis of the lens joining the centres 
R, r, of its furfaces A, a. Draw any two of their femi- 
diameters RA, ra, parallel to each other, and join the 
points A, re, and the line Are will cut the-axis in the point 
E above defcribed. For the triangles REA, ?-Ere, being 
equiangular, RE will be to Erjn the given ratio of the 
femidiameters RA, ra ; and confequently the point E is 
invariable in the fame lens. Now, fuppofing a ray to pafs 
both ways along the line Are, it, being equally inclined 
to the perpendiculars to the furfaces, will be equally bent, 
and contrariwife in going out of the lens; fo that its emer¬ 
gent parts, AQ, aq , will be parallel. Nov/ any of thefe 
lenfes will become plano-convex or plano-concave, by 
conceiving one of the femidiameters RA, ra, to become 
infinite, and confequently to become parallel to the axis 
of the lens, and then the other femidiameter will coincide 
with the axis; and fo the points A, E, or re, E, will co¬ 
incide. Q.E.D. 
Cor. Hence when a pencil of rays falls almoft perpen¬ 
dicularly upon any lens, whofe thicknefs is inconfiderable, 
the courfe of the ray which pafles through E, above de¬ 
fcribed, may be taken for a ftraight line pafling through 
the centre of the lens without fenfible error in fenfible 
things. For it is manifeft from the length of Are, and 
from the quantity of the refraftions at its extremities, 
that the perpendicular diltance of AQ, aq, when pro¬ 
duced, will be diminiflied both as the thicknefs of the lens 
and the obliquity of the ray is diminiflied. 
Prop. I. To find the focus of parallel rays falling almoft 
perpendicularly upon any given lens. 
Let E, (fig. 7 to ia.) be the centre of the lens, and r 
the centres of its furfaces, Rr its axis, g’EG a line parallel 
to the incident rays upon the furface B, whofe centre is 
R. Parallel to g-E draw a femidiameter BR, in wdiich pro¬ 
duced let V be the focus of the rays after their firft refrac¬ 
tion at the furface B, and joining Vr let it cut gdS pro¬ 
duced in G, and G will be the focus of the rays that 
emerge from the lens. 
For, fince V is alfo the focus of the rays incident upon 
the fecond furface A, the emergent rays muft have their 
focus in fome point of that ray which pafles ftraight 
through this furface; that is, in the line Vr, drawn through 
its centre r : and, lince the whole courfe of another ray is 
reckoned a ftraight linegEG, its interfeftion G with Vr 
determines the focus of them all. Q. E. D. 
Cor. 1. When the incident rays are parallel to the axis 
rR, the focal diftance EF is equal to EG. For, let the 
■ incident rays that were parallel to gE be gradually more 
inclined to the axis till they become parallel to it; and 
their firft and fecond foci V and G will defcribe circular 
arches NT and GF, whofe centres are R and E. For the 
line RV is invariable; being in proportion to RB in a 
given ratio of the fmaller of the fines of incidence and 
refraftion to their difference (by a former propofition) ; 
confequently, the line EG is alfo invariable, being in pro¬ 
portion to the given line RV in the given ratio of rE to 
rR, becaufe the triangles EGr, RVr, are equiangular. 
Cor. z. The laft proportion gives the following rule for 
finding the focal diftance of any thin lens: As Rr, the in¬ 
terval between the centres of the furfaces, is to rE, the 
femi-diameter of the fecond furface, fo is RV 7 or RT, the 
continuation 
