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as will appear if we draw feveral incident rays with their 
refpeXive reflected ones, in fuch a manner that the angles 
of reflection may be equal to their refpeXive angles of in¬ 
cidence, as is done in the figure. The like is applicable 
to any of the other cafes of diverging and converging 
rays incident upon a fpherical furface. This is the reafon, 
that, when rays are confidered as refleXed from a fpherical 
furface, the diftance of the oblique rays from the perpen¬ 
dicular one is taken fo fmall, that it may be fuppofed to 
vanifh. 
From this it follows, that, if a number of diverging 
rays are incident upon the convex furface BD fit the fe¬ 
veral points B, D, D, &c. they will not proceed after re- 
fieXion as from any point in the line RB produced, but 
as from a curve line pafllng through the feveral points 
F,/,/, &c. 
Had the curve BD (fig. 7.) been an hyperbola, having 
its foci in R and F, then, R being the radiant (or the 
imaginary focus of incident rays), F would have been the 
focus of the refleXed ones, and vice verj'a, however diltant 
the points B and D might be taken from each other. In 
like manner, had the curve BD (fig. 8.) been an ellipfe 
having its foci in F and R, the one of thefe being made 
the radiant (or imaginary focus of incident rays), the 
other would have been the focus of the reflected ones; and 
vice verj'a. For, both in the hyperbola and ellipfe, lines 
drawn from each of their foci through any point make 
equal angles with the tangent to that point. Therefore, 
if the incident rays proceed to or from one of their foci, 
the reflected ones will all proceed as from or to the other 
focus. Therefore, in order that diverging or converging 
rays may be accurately reflected to or from a point, the 
refleXing furface mult be formed by the revolution of an 
hyperbola about its longer axis, when the incident rays 
are fuch, that their radiant or imaginary focus of incident 
rays fhall fall on one fide of the furface, and the focus of 
the reflected ones on the other; when they are both to 
fall on the fame fide, it muft be formed by the revolution 
of an ellipfe about its longer axis. However, as fpherical 
furfaces are more eafily formed than thofe which are ge¬ 
nerated by the revolution of any of the conic feXionsabout 
their axes, the latter are very rarely ufed. 
Whatever has been faid concerning the appearance of 
bodies feen through lenfes, by refraXed light, refpeXs 
alfo the appearance of bodies feen by refleXion. But, be- 
fides thefe, there is one thing peculiar to images by re¬ 
flexion, viz. that each point in the reprefentation of an 
objeX made by refleXion appears fituated fomewhere in a 
right line that paffes through its correfpondent point in 
the objeX, and is perpendicular to the refleXing furface. 
The truth of this appears fufliciently from the propofi- 
tions formerly laid down : in each of which, rays flowing 
from any radiant point are fliown to proceed after reflec¬ 
tion to or from fome point in a line that pafles through 
the radiant point, and is perpendicular to the refleXing 
furface. Forinftance, (fig. 1.) rays flowingfronr Y are col- 
leXedinX, a point in the perpendicular CD, which, being 
produced, pafles through Y; again, (fig. a.) rays flowing 
from G, proceed, after reflection, qs from N, a point 
in the perpendicular CD, which, being produced, pafles 
through G. 
This obfervation, however, except where an objeX is 
feen by refleXion from a plane furface, relates only to 
thofe cafes where the reprefentation is made by means of 
fuch rays as fall upon the refleXing furface with a very 
fmall degree of obliquity ; becaufe fuch as fall at a confi- 
derable diftance from the perpendicular, do not proceed 
after refleXion as from any point in that perpendicular, 
but as from other points fituated in a certain curve; on 
•which account thefe rays are negleXed, as making an in- 
diftinX and deformed reprefentation. And therefore it 
is to be remembered, that, however the fituation of the 
eye with refpeX to the objeX and refleXing furface may 
be reprefented in the following figures, it is to be fuppofed 
I c s. 
as fituated in fuch a manner with refpeX to the objeX, 
that rays flowing from thence, and entering it after reflec¬ 
tion, may be fuch only as fall with a very fmall degree of 
obliquity upon the furface; that is, the eye muft be fup¬ 
pofed to be placed almoft direXly behind the objeX, or 
between it and the refleXing furface. The reafon why it 
is not always fo placed, is only to avoid confufion in the 
figures. 
Prop. IV. When an objeft is feen by refleXion from a 
plane furface, the image of it appears at the fame dif¬ 
tance behind the furface that the objeX is before it, of 
the fame magnitude, and direXly oppofite to it. 
To explain this, let AB (fig. 10.) reprefent an objeX 
feen by refleXion from the plane furface SV ; and let the 
rays AF, AG, be fo inclined to the furface, that they 
fhall enter an eye at H after refleXion ; and let AE be per¬ 
pendicular to the furface : then, by the obfervation juft: 
mentioned, the point A will appear in fome part of the 
line AE produced, fuppofe I ; that is, the oblique rays 
AF and AG will proceed after refleXion as from that 
point; and further, becaufe the refleXed rays FII, GK, 
will have the fame degree of inclination to one another 
that their incident ones have, that point muft necefl'arily 
be at the fame diftance from the furface that the point A 
is ; the reprefentation therefore of the point A will be at 
the fame diftance from the furface that the point itfelf is 
before it, and direXly oppofite to it: confequently, fines 
the like may be fhown of any other point B, the whole 
image IM will appear at the fame diftance behind the fur¬ 
face that the objeX is before it, and direXly oppofite to it; 
and, becaufe the lines AI, BM, perpendicular to the 
plane furface, are parallel to each other, the image will 
alfo be of the fame magnitude with the objeX. 
Prop. V. When an objeX is feen by refleXion from a 
convex furface, its image appears nearer to the furface, 
and lefs than the objeX. 
Let AB (fig. 12.) reprefent the objeX, SV a refleXing 
furface whole centre of convexity is C : and let the rays 
AF, AG, be fo inclined to the furface, that after reflec¬ 
tion from it they fhall enter the eye at H: and let AE be 
perpendicular to the furface ; then will the oblique rays 
AF, AG, proceed after refleXion as from fome point in 
the line AE produced, fuppofe from I; which point, be¬ 
caufe the refleXed rays will diverge more than the inci¬ 
dent ones, muft be nearer to the furface than the point 
A. And, fince the fame is alfo true of the rays which 
flow from any other point B, the reprefentation IM will 
be nearer to the furface than the objeX; and, becaufe it is 
terminated by the perpendiculars AE and BF, which in¬ 
cline to each other, as concurring at the centre, it will 
alfo appear lefs. 
Prop. VI. When an objeX is feen by refleXion from a 
concave furface, the reprefentation of it is various, 
both with regard to its magnitude and fituation, ac¬ 
cording as the diftance of the objeX from the refleXing 
furface is greater or lefs. 
Cafe 1. When the objeX is nearer to the furface than 
its principal focus, the image falls on the oppofite fide of 
the furface, is more diftant from it, and larger than the 
objeX. Thus let AB (fig. 13.) be the object, SV the re¬ 
fleXing furface, F the principal focus, and C its centre. 
Through A and B, the extremities of the objeX, draw 
the lines CE, CR, which will be perpendicular to the fur¬ 
face; and let the rays AR, AG, be incident upon fuch 
points of it, that they fhall be refleXed into an eye at H. 
Now, becaufe the radiant points A and B are nearer the 
furface than the principal focus F, the refleXed rays will 
diverge, and therefore proceed as from fome points on 
the oppofite fide of the furface ; which points, by the ob¬ 
fervation laid down at the beginning of this feXion, will 
be in the perpendiculars AE, BR, produced, fuppofe in I 
and; 
