OPT 
gles of incidence and refledfion ; which being equal, 
their complements ACB and FCE are alfo equal : but 
the angle BCG is equal to its vertical angle FCE : there¬ 
fore in the triangles ABC and GBC the angles at C are 
equal, the fide BC is common, and the right angles at B 
are equal ; therefore AB—BG : and, confequently, the 
point G, the focus of the incident rays AB, AC, is at the 
lame diftance behind the furface that the point A is be¬ 
fore it. 
Cafe 2. Of converging • rays. —This is the converfe of 
the former cafe. For, fuppofing FC and AB to be two 
converging incident rays, CA and BA will be the re¬ 
fledted ones (the angles of incidence in the former cafe 
being now the angles of refledfion and vice verfa), having 
the point A for their focus; but this is at an equal 
diitance from the refledting furface with the point G, 
which in this cafe is the imaginary focus of the incident 
rays FC and AB. 
It is not here as in the cafe of rays pafling through a 
plane furface, where fome of the refradted rays proceed as 
from one point, and fome as from another: but they all 
proceed after refledfion as from one and the fame point, 
however obliquely they may fall upon the furface ; for 
what is here demonllrated of the ray AC holds equally of 
any other, as AI, AK, See. 
The cafe of parallel rays incident on a plane furface is 
included in this Propofition : for in that cafe we are to 
fuppofe the radiantpointinfinitelydiftantfrom thefurface, 
and then by the propofition the focus of the refledted rays 
will be fo too : that is, the rays will be parallel after re- 
fledtion, as they were before it. 
Prop. II. Of the refledfion of parallel rays from a fpherical 
furface. 
When parallel rays are incident upon a fpherical furface, 
the focus of the refledfed rays will be the middle point be¬ 
tween the centre of convexity and the furface. 
Cafe i. Of parallel rays falling upon a convex furface .— 
Let AB, DH, (fig. 4.) reprefent two parallel rays incident 
on the convex furface BH, the one perpendicularly, the 
other obliquely j and let C be the centre of convexity. 
Suppofe HE to be the refledfed ray of the oblique one 
DH, proceeding as from F, a point in the line AB pro¬ 
duced. Through the point H draw' the line Cl, which 
will be perpendicular to the furface at that point; and 
the angles DHI and IHE, being the angles of incidence 
and refledfion, will be equal. But HCF— DHI, the lines 
AC and DH being parallel; and CHFrrIHE ; wherefore 
the triangle CFH is jfofceles, and confequently CFr=FH : 
but, fuppofing BH to vanifh, FH=FB ; and therefore, 
upon this fuppofition, FC=FB; that is, the focus of the 
refledfed rays is the middle point between the centre of 
convexity and the furface. 
Cafe a. Of parallel rays falling upon a concave furface. — 
Let AB, DH, (fig. 5.) be two parallel rays incident, the 
one perpendicularly, the other obliquely, on the concave 
furface BH, whofe centre of concavity is C. Let BE and 
HF be the refledted rays meeting each other in F ; this 
will be the middle point between B and C. For, drawing 
through C the perpendicular CH, the angles DHC —FHC, 
being the angles of incidence and refledfion ; but HCF= 
DHC its alternate angle, and therefore the triangle CFH 
is ifofceles. Wherefore CFizzFH: but, if we fuppofe BH 
to vanifh, FB=FH, and therefore CF=FB ; that is, the 
focal diftance of the refledted rays is the middle point be¬ 
tween the centre and the furface. 
It is here obfervable, that the farther the line DH, ei¬ 
ther in fig 4 or 5, is taken from AB, the nearer the point 
F falls to the furface. For, the farther the point H re¬ 
cedes from B, the greater the triangle CFH will become; 
and confequently, fince it is always ifofceles, and the bafe 
CH, being the radius, is every-where of the lame length, 
the equal legs CF and FH will lengthen ; but CF cannot 
grow longer unlefs the point F approach towards the l'ur- 
Vol. XVII. No. 1199. 
ICS. 577 
face. And the farther H is removed from B, the fafler F 
approaches to it. 
This is the reafon that, whenever parallel rays are 
confidered as refledfed from a fpherical furface, the dif¬ 
tance of the oblique ray from the perpendicular one is 
taken fo fmall with refpedt to the focal diitance of that 
furface, that without any phyfical error it may be fup- 
pofed to vanifh. Hence it follows, that, if a number of 
parallel rays, as AB, CD, EG, See. (fig. 6.) fall upon a 
convex furface, and if BA, DK, the refledfed rays of the 
incident ones AB, CD, proceed as from the point F, 
thofe of the incident ones CD, EG, viz. DK, GL, will 
proceed as from N, thofe of the incident ones EG, HI, as 
from O, &c. becaufe the farther the incident ones CD, 
EG, See. are from AB, the nearer to the furface are the 
points F, /, f in the line BF, from which they proceed 
after refledfion : fo that properly the foci of the reflected 
rays BA, DK, GL, &c. are not in the line AB produced, 
but in a curve line palling through the points F, N, O, 
See. The fame is applicable to the cafe of parallel rays 
refledted from a concave furface, as exprefled by the dotted 
lines on the other half of the figure, where PQ, RS, TV, 
are the incident rays ; QF, S f, V/, the refledted ones, in¬ 
terfiling each other in the points X, Y, and F; fo that 
the foci of thofe rays are not in the line FB, but in a 
curve paffing through thofe points. 
Had the furface BH, in fig. 4 or 5, been formed by the 
revolution of a parabola about its axis having its focus in 
the point F, all the rays refledted from the convex furface 
would have proceeded as from the pointT, and thofe re¬ 
fledfed from the concave furface would have fallen upon 
it, however diitant their incident ones AB, DH, might 
have been from each other. P'orin the parabola, all lines 
drawn parallel to the axis make angles with the tangents 
to the points where they cut the parabola (that is, with 
the furface of the parabola) equal to thofe which are made 
with the fame tangents by lines drawn from thence to the 
focus ; therefore, if the incident rays deferibe thofe paral¬ 
lel lines, the reflected ones will neceflarily deferibe thefe 
other, and fo will all proceed as from, or meet in, the 
fame point. 
Prop. III. Of the refledfion of diverging and converging 
raysfrom a fpherical furface. 
_ When rays fall upon any fpherical furface, if they 
diverge, the diftance of the focus of the refledfed ra'4 
from the furface is to the diftance of the radiant point 
from the fame (or, if they converge, to that of the 
imaginary focus of the incident rays), as the diftance 
of the focus of the refledfed rays from the centre is to 
the diftance of the radiant point (or imaginary focus of 
the incident rays) from the fame. This Propofition ad¬ 
mits often cafes. 
Cafe 1. Of diverging rays falling upon a convex furface. 
—Let RB, RD, (fig. 7.) reprefent two diverging rays 
flowing from the point R as from a radiant, and failin'.* 
the one perpendicularly, the other obliquely, on the 
convex furface BD, whofe centre is C. Let DE be the 
refledted ray of the incident one RD ; produce i D to F 
and through R draw the line RH parallel to FB. till it 
meets CD produced in H. Then RHD=EDH, the an¬ 
gle of refledfion ; and RHD=zRDH, the angle of inci¬ 
dence ; wherefore the triangle DRH is ilolceles, and DR 
=RH. Now, the lines FD and RH being parallel, the 
triangles FDC and RHC are fimilar, or the lides are cut 
proportionably; and therefore, FD : RH or RD=CF : 
CR; but, BD vanifhing, FD and RD differ not from 
FB and RB : wherefore, FB : RB=CF : CR ; that is, 
the diftance of the focus from the furface is to the diftance 
of the radiant point from the fame, as the diftance of the 
focus from the. centre is to the diftance of the radiant 
point from it. 
Cafe 2. Of converging rays falling upon a concave fur¬ 
face. —Let-KD and CB be the converging incident rays 
• 7 Id having 
