■57 G 
OPT 
3. Let them diverge from X, a point between the focus , 
of parallel rays and the centre ; they then make lefs angles 
of incidence than the rays MS and MB, which became 
parallel by refleflion ; they will confequently have lefs 
angles of reflection, and therefore proceed converging 
towards fome 'point, as Y; which point will always fall 
on the contrary fide of the centre, becaufe a reflected ray 
always falls on the contrary fide of the perpendicular 
with refpeft to that on which its incident one falls; and, 
of confequence, it will be farther diftant from the centre 
than X. 
4. If the incident rays diverge from Y, they w ill, after 
reflection, converge to X ; thole which were the incident - 
rays in the former cafe being the reflected ones in this. 
5. If the incident rays proceed from the centre, they 
fall in with their refpedive perpendiculars ; and, for that 
reafon, are reflected thither again. 
V. Parallel rays rcjlcRcd from convex Jhrfaccs are ren¬ 
dered diverging. —For, let AB, GD, EF, (fig. 2.) be three 
parallel rays falling upon the convex furface BF, whofe 
centre is C, and let one of them, viz. GD, be perpendi¬ 
cular to the furface. Through B, D, and F, the points 
of reflection, draw the lines CV, CG, and Cl'; which 
•will be perpendicular to the furface at tbefe points. The 
incident ray GD, being perpendicular to the furface, 
will return after reflection in the fame line; but the ob¬ 
lique ones AB and EF will return in the lines BK and 
FL, fituated on the contrary fide of their refpeftive per¬ 
pendiculars BY and FT. They will therefore diverge, 
after reflection, as from fome point M in the line GD 
produced ; and this point will be in the middle between 
D and C. 
VI. Diverging rays reflected from convex fur faces are 
rendered more diverging .—For, things remaining as above, 
let GB, GF, be the incident rays. Thefe having 
greater angles of incidence than the parallel ones AB and 
EF in the preceding cafe, their angles of reflection will 
alfo be greater; they will therefore diverge after reflec¬ 
tion, fuppofe in the lines BP and FQ, as from fome 
point N, farther from C than the point M ; and the de¬ 
gree of their divergency will exceed their divergency be¬ 
fore reflection. 
VII. Converging rays rcf.cRcd from convex firfaces are 
parallel, converging, or diverging. —If they tend towards 
the focus of parallel rays, they then become parallel; if to 
a point nearer the furface, they converge, but in a lefs de¬ 
gree than Before reflection ; if to a point between that 
and the centre, they will diverge after reflection, as from 
fome point on the contrary fide of the centre, but fituated 
farther from it than the point to which they converged : 
if the incident rays converge to a point beyond the cen¬ 
tre, the reflected ones will diverge as from one on the 
contrary fide of it, but nearerto it than the point to which 
the incident ones converged; and, if the incident rays 
converge towards the centre, the reflected ones will feem 
to proceed from it. 
1. Let them converge in the lines KB and LF, tend¬ 
ing towards M, the focus of parallel rays; then, as the 
parallel rays AB, EF, were reflected into the lines BK and 
FL (by Prop. V.) thofe rays will now, on the contrary, be 
reflected into them. 
2. Let them converge in the lines PB, QF, tending to¬ 
wards N, a point nearer the furface than the focus of pa¬ 
rallel rays ; they will then be reflected into the converging 
lines BG and FG, in which the rays GB and GF pro¬ 
ceeded that were fiiown to be reflected into them by the 
laft Propofition : but the degree of their convergency will 
exceed their convergency before refleflion. 
3. Let them converge in me lines RB and SF, proceed¬ 
ing towards X, a point between the focus of parallel rays 
and the centre; their angles of incidence will then be lefs 
than thofe of the rays KB and LF, which became parallel 
after reflection : their angles of reflection will therefore 
be lefs; on which account they mull: necefl'arily diverge, 
Fuppofe in the lines Bli and FI, from fome point as Y; 
I c s. 
which point (by Prop. IV.) will fall on the contrary fide 
of the centre with refpeft to X, and will be farther from it 
than that. 
4. If the incident rays tend towards Y, the reflected 
ones will diverge as from X ; thofe which were the in¬ 
cident ones in one cafe being the refleCled ones in the 
other. 
5. If the incident rays converge towards the centre, 
they coincide with their refpeftive perpendiculars ; and 
will therefore proceed, after refleflion, as from that centre. 
We have already obferved, that in fome cafes there is a* 
very great reflection from the fecond furface of a tranfpa- 
rent body. The degree of inclination neceflary to caufe 
a total reflection of a ray at this furface, is that which re¬ 
quires that the refraCted angle (fuppofing the ray to pafs 
out there) fliould be equal to, or greater than, aright one ; 
and, confequently, it depends on the refraftive power of 
the medium through which the ray paffes, and is there¬ 
fore different in different media. When a ray paffes 
through glafs furrounded with air, and is inclined to its 
fecond furface under an angle of 42 0 or more, it will be 
wholly refleCled there. For, as 11 is to. 17 (the ratio of 
refraClion out of glafs into air), fo is the fine of an angle 
ot 42 0 to a fourth number that will exceed the fine of a 
right angle. Hence it follows, that, when a ray of light 
arrives at the fecond furface of a tranfparent fubftance 
with as great, or a greater, degree of obliquity than that 
which is neceffary to make a total reflection, it will there 
be all returned back to the-flrft : and, if it proceeds to¬ 
wards that with as great an obliquity as it did towards 
the other, (which it will do if the furfaces of the medium 
be parallel to each other,) it will there be all reflected 
again, &c. and will therefore never get out, but pafs from 
fide to fide, til! it be wholly extinguiflied within the body. 
From this may arife an obvious enquiry, how it comes to 
pafs, that light, falling very obliquely upon a glafs win¬ 
dow from without, fliould be tranfnutted into the room. 
In anfwer to this it mull be confidered, that, however ob¬ 
liquely a ray fails upon the furface of any medium whofe 
fides are parallel, as thofe of the glafs in a window, it will 
fuller luch a degree of refraClion in entering there, that 
it ihall fall upon the fecond with a lefs obliquity than that 
which is neceflary to caufe a total reflection. For, fince 
the medium is glafs, then, as 17 is to 11, fo is the line of 
the greateft angle of incidence with which a ray can fall 
upon any furface to the fine of a lefs angle than that of to¬ 
tal reflection. Therefore, if the Tides of the glafs be pa¬ 
rallel, the obliquity with which a ray falls upon the flrft 
furface cannot be fo great, that it Ihall pafs the fecond 
without fullering a total refleflion there. 
When light paffes out of a denier into a rarer medium, 
the nearer the fecond medium approaches the flrft in its 
refraflive power, the lefs of it will be refraCted in paffing 
from one to the other; and, when their refraCting powers 
are equal, all of it will pafs into the fecond medium. 
The above Propofitions may be all mathematically tie- 
monltrated in the following manner. 
Prop I. Of the reflection of rays from a plane furface. 
When rays fall upon a plane furface, if they diverge, the 
focus of the reflected rays will be at the fame diftance be¬ 
hind the furface, that the radiant point is before it: if 
they converge, it will be at the fame diftance before the 
furface that the imaginary focus of the incident rays is 
behind it. This Propofition admits of two cafes. 
Cafe 1. Of diverging rays. —Let AB, AC, (fig. 3.) be 
two diverging rays incident on the plane furface DE; the 
one perpendicularly, the other obliquely. The perpendi¬ 
cular one AB will be refleCled to A, proceeding as from 
fome point in the line AB produced: the obltque one 
AC will be reflected into feme line, as CF, fe that the 
point G, where the line FG produced interfefls the line 
AB produced alfe, Ihall be at an equal diftance from the 
furface DE with the radiant point A. For, the perpen¬ 
dicular CPI being drawn, ACH and KCF will be the an¬ 
gles 
