5? 8 OPT 
having their imaginary focus in the point R, which was 
the radiant point in the foregoing cafe. Then, as RD 
was in that cafe reflefted into DE, KD will in this be re¬ 
flected into DF ; for, iince the angles of incidence in 
both cafes are equal, the angles of reflection will be equal 
alfo ; fo that F will be the focus of the reflected rays: but 
it was there demonftrated, that FB : RB=CF : CR ; that 
is, the diltance of the focus from the furface is to the 
diftance (in this cafe) of the imaginary focus of the in¬ 
cident rays, as the diltance of the focus from the centre 
is to the diltance of the imaginary focus of the incident 
rays from the fame. 
Cafe 3. Of converging rays falling vpon a convex furface, 
and tending to a point between the focus of parallel rays and 
the centre. —Let B (fig. 8.) reprefent a convex furface 
whofe furface is C, and whofe focus of parallel rays is P; 
and let AB, KD, be two converging rays incident upon 
it, and having their imaginary focus at R, a point be¬ 
tween P and C. Now, becaufe KD tends to a point be¬ 
tween the focus of parallel rays and the centre, the re¬ 
flected ray DE will diverge from feme point on the other 
lide the centre, fuppofe F ; as explained above. Through 
D draw the perpendicular CD, and produce it to H; then 
will KDH=HDE, being the angles of incidence and re¬ 
flection, and" conlequently RDC=CDF too. Therefore 
the triangle RDF is bifefted by the line DC : wherefore 
FD and DR, orBD, vanilhing, FB : BR=FC : CR; that 
is, the diltance of the focus of the reflected rays is to that 
of the imaginary focus of the incident ones, as the diftance 
of the former from the centre is to the diftance of the lat¬ 
ter from the centre. 
Cafe 4. Of diverging rays falling upo?i a concave furface, 
and proceeding from a point between the focus of parallel 
rays and the centre. —Let RB, RD, be the diverging rays 
incident upon the concave furface BD, having their ra¬ 
diant point in R, the imaginary focus of the incident rays 
in the preceding cafe. Then, as KD was in that cafe re¬ 
flefted intoDE, RD will now be reflected into DF. But 
we had FB : RB — CF : CR ; that is, the diftance of 
the focus is to that of the radiant, as the diftance of the 
former from the centre is to the diftance of the latter from 
the centre. 
The anglesof incidence and reflection being equal, it is 
evident, that if, in any cafe, the reflected ray be made the 
incident one, the incident will become the reflected one ; 
and therefore the four following cafes may be confidered 
refpeCtively £s the converfe of the four preceding; for in 
each of them the incident rays are fuppofed to coincide 
with the reflected ones in the other. Or they may be thus 
demonftrated independently of them. 
Cafe 5. Of converging rays falling upon a convex furface, 
and tending to a point nearer the Jurface than the focus of 
parallel rays. —Let ED, RB, fig. 7. be the converging rays 
incident upon the convex furface BD, whole centre is C, 
and principal focus P ; let the imaginary focus of the in¬ 
cident rays be at F, a point between P and B; and let 
DR be the reflected ray. From C and R draw the lines 
CH, RH ; the one palling through D, the other paral¬ 
lel to FE. Then RHD=HDE, the angle of incidence. 
But RIiE=:HDR, the angle of reflection : wherefore the 
triangle HDR is ilolceles, and DR=RH. Now, the lines 
FD and RH being parallel, the triangles FDC and RHC 
are limilar; and therefore RH or RD: FD:=CR : CF ; 
but, BD vanishing, RD and FD coincide with RB and FB, 
wherefore RB : FB=CR : CF; that is, the diftance of 
the focus from the furface is to the diftance of the imagi¬ 
nary focus of the incident rays, as the diftance of the fo¬ 
cus from the centre is to thediftance of theiinaginary focus 
of the incident rays from the centre. 
Cafe 6. Of diverging rays falling upon a concave furface, 
and proceeding from a point between the focus of parallel rays 
and the Jurface.— Let FD and FB be two rays diverging 
front the point F, which was the imaginary focus of the 
incident rays in the preceding cale. Then, as ED was 
in that cafe reflected into DR, FD will be reflected into 
ICS. 
DK, (for the reafon mentioned in Cafe a.) fo that the re¬ 
flected ray will proceed as from the point R : but it was 
demonftrated in Cafe 5. that RB : FB=CR : CF; that 
is, the diftance of the focus from the furface is to that of 
the radiant from the furface, as the diftance of the former 
from the centre is to that of the latter from the centre. 
Cafe 7. Of converging rays falling upon a convex furface, 
and tending towards a point beyond the centre. —Let AB, FD, 
fig. 8. be the incident rays tending to F, a point beyond 
the centre C, and let DK be the reflected ray of the inci¬ 
dent one ED. Then, becaufe the incident ray ED tends 
to a point beyond the centre, the reflected ray DK will 
proceed as from one on the contrary fide, fuppofe R ; fee 
Prop. VII. Through D draw the perpendicular CD, and 
produce it to FI. Then will EDH=HDK, being the 
angles of incidence and reflection ; but CDF=CDR, be¬ 
ing their verticals : confequently the angle FDR is bi¬ 
feCted by the line CD : wherefore, RD : DF; or, BD va- 
nifhing, RB : BF=RC : CF ; that is, the diftance of the 
focus of the reflected rays is to that of the imaginary focus 
of the incident rays, as the diftance of the former from 
the centre is to the diftance of the latter from the centre. 
Cafe 8. Of diverging rays falling vpon a concave furface, 
and proceeding from a point beyond the centre. —Let FB, FD, 
be the incident rays radiating from F, the imaginary focus 
of the incident rays in the preceding cafe. Then, as ED 
was in that cafe reflected into DK, FD will now be re¬ 
flected into DR ; fo that R will be the focus of the reflefted 
rays. But it was demonftrated in Cafe 7. that RB : FB 
—RC : CF; that is, the diftance of the focus of the re¬ 
flected rays from the furface is to the diftance of the radi¬ 
ant from the furface, as the diftance of the focus of the 
reflected rays from the centre is to the diftance of the ra¬ 
diant from the centre. 
The two remaining cafes may be confidered as the con¬ 
verfe of thofe under Prop. II. becaufe the incident rays in 
thefe are the reflected ones in them ; or they may be 
demonftrated in the fame manner with the preceding, as 
follows. 
Cafe 9. Converging rays falling upon a convex furface, and 
tending to the focus of parallel rays, become parallel after 
refection. —Let ED, RB, fig. 7. reprefent two converging 
rays incident on the convex furface BD, and tending to¬ 
wards F, which we (hall now fuppofe to be the focus of 
parallel rays; and let DR be the reflected ray, and C the 
centre of convexity of the reflecting furface. Through C 
draw CD, and produce it to H, drawing RH parallel to 
ED produced to F. K ow it has been demonftrated, (Cafe 
5. where the incident rays are fuppofed to tend to the point 
F,) that RB: FB^RC : CF; but, F in this cafe being 
fuppofed to be the focus of parallel rays, it is the middle 
point between C and B, (by Prop. II.) and therefore FB 
=FC, confequently RB=RC; which can only be upon 
the fuppofition that R is at an infinite diftance from B ; 
that is, that the reflefted rays BR and DR be parallel. 
Cafe 10. Diverging rays falling upon a concave furface, 
and proceeding from the focus of parallel rays, become paral¬ 
lel after reflection.- —Let RD, RB, fig. 8. be two diverging 
rays incident upon the concave furface BD, as fuppofed 
in Cafe 4. where it was demonftrated that FB : RB=CF: 
CR. But in the prefent cafe RB=CR, becaufe R is fup¬ 
pofed to be the focus of parallel rays ; therefore FB=FC ; 
which cannot be unlefs F be taken at an infinite diftance 
from B ; that is, unlefs the reflefted rays BF and DF be 
parallel. 
It may here be obferved, that, in the cafe of diverging 
rays falling upon a convex furface, the farther the point 
D is taken from B, fig. 9. the nearer the point F, the fo¬ 
cus of the reflefted rays, approaches to B, while the radi¬ 
ant point R remains the fame. For it is evident from the 
curvature of a circle, that the point D may be taken fo 
far from B, that the reflefted ray DE (hall proceed as 
from F, G, FI, or even from B, or from any point between 
B and R ; and, the farther it is taken from B, the fafter 
the point from which it proceeds approaches towards R : 
1 as 
