580 OPT 
and M; but they will diverge in a lefs degree than their 
incident ones; and, therefore, the faid points will be far¬ 
ther from the furface than the points A and B. The 
image therefore will be on the oppofite fide of the furface 
with refpeCt to the object; it will be more diftant than it; 
and confequently, being terminated by the perpendiculars 
Cl and CM, it will alfo be larger. 
Cafe 2. When the object is placed in the principal focus* 
the reflected rays enter the eye parallel; in which cafe the 
image ought to appear at an infinite diftance behind the 
reflecting furface; but the reprefentation of it, for the 
realons given in the foregoing cafe, being large and dif- 
tinCt, we do not reckon it much farther from the furface 
than the image. 
Cafe 3. When the objeCt is placed between the principal 
focus and the centre, the image falls on the oppofite fide 
of the centre, is larger than the object, and in an in¬ 
verted pofition. Thus let AB (fig. 14.) be the object, 
SV the reflecting furface, F its principal focus, and C its 
centre. Through A and B draw the lines CE and CN, 
which will be perpendicular to the furface; and let AR, 
AG, be a pencil of rays flowing from A. Thefe rays, 
proceeding from a point beyond the principal focus, will 
after reflection converge towards fome point on the oppo¬ 
se fide of the centre, which will fall upon the perpendicu¬ 
lar EC produced, but at a greater diftance from C than 
the radiant A from which they diverged. For the fame 
reafon, rays flowing from B will converge to a point in 
the perpendicular NC produced, which lhall be farther 
from C than the point B ; whence it is evident, that the 
image IM is larger than the objeCt AB, that it falls on the 
contrary fide of the centre, and that their pofitions are in¬ 
verted with refpeft to each other. 
Cafe 4. If the objeCt be placed beyond the centre of 
convexity, the image is then formed between the centre 
and the focus of parallel rays, is lefs than the objeCt, and 
its pofition is inverted. This Propolition is the converfe 
of the preceding ; for as in that cafe rays proceeding from 
A were reflected to I, and from B to M ; lo rays flowing 
from I and M will be reflected to A and B : if therefore 
an objeCt be fuppofed to be fituated beyond the centre in 
IM, the image of it will be formed in AB between that 
and the focus of parallel rays, will be lefs than the objeCt, 
and inverted. 
Cafe 5. If the middle of the objeCt be placed in the cen¬ 
tre of convexity of the reflecting furface, the object and 
its image will be coincident; but the image will be in¬ 
verted with refpeCt to the objeCt. 
That the place of the image and the objeCt Ihould be 
the fame in this cafe requires little explication ; for, the 
middle of the objeCt being in the centre, rays flowing 
from it will fall perpendicularly upon the furface, and 
therefore neceflarily return thither again ; fo that the mid¬ 
dle of the image will be coincident with the middle of the 
objeCt. But that the image fhould be inverted is perhaps 
not fo clear. To explain this, let AB (fig. 15.) be the 
objeCt, having its middle point C in the centre of the re¬ 
flecting furface from SV ; through the centre and the point 
R draw the line CR, which will be perpendicular to the 
reflecting furface ; join the points AR and BR, and let 
AR reprefent a ray flowing from A ; this will be reflected 
into RB, for, C being the middle point between A and B, 
the angle ARCc=CRB ; and a ray from B will likewife be 
reflected to A; and therefore the pofition of the image 
will be inverted with refpeCt to that of the objeCt. In this 
Propolition it is to be fuppofed, that the object AB is fo 
fituated with refpeCt to the reflecting furface, that the 
angle ACR may be right; for otherwile the angles ARC 
and BRC will not be equal, and part of the image only 
will therefore fall upon the objeCt. 
Cafe 6. If in any of the three lalt cafes, in each of which 
the image is formed on the fame fide of the reflecting fur¬ 
face with the objeCt, the eye be fituated farther from the 
furface than the place where the image falls, the rays of 
each pencil, crofting each other in the feveral points of 
I c s. 
the image, will enter the eye as from a real objeCt fituated 
there ; fo that the image will appear pendulous in the air 
between the eye and the reflecting furface, and in the po¬ 
fition wherein it is formed, viz. inverted with refpeCt to 
the objeCt, in the fame manner that an image formed by 
refraCted light appears to an eye placed beyond it; which 
was fully explained under Prop. IV. 
But, as what relates to the appearance of the objeCt when, 
the eye is placed nearer to the furface than the image was 
not there fully inquired into, that point lhall now be more 
ftriCtly examined under the following cafe, which equally 
relates to refraCted and reflected light. 
Cafe 7. It the eye be fituated between the reflecting fur¬ 
face and the place of the image, the objeCt is then feen be¬ 
yond the lurface; and, the farther the eye recedes from 
the furface towards the place of the image, the more con¬ 
futed, larger, and nearer, the objeCt appears. To explain 
this, let AB (fig. 16.) reprefent the objeCt; IM its image, 
one of whofe points M is formed by the concurrence of 
the reflected rays DM, EM, &c. which before reflection 
came from B ; the other, I, by the concurrence of DI, El, 
&c. which came from A ; and let ab be the pupil of an 
eye, fituated between the furface DP and the image. This 
pupil will admit the rays Ha, Kb ; which, becaufe they 
are tending towards I, arefuch as came from A, and there¬ 
fore the point A will appear diffufed over the fpace RS. 
In like manner the pupil will alfo receive into it the re¬ 
flected rays K« and Lb, which, becaufe they are tending 
towards M, by fuppofition came from B ; and therefore 
the point B will be feen fpread as it were over the fpace 
TV, and the objeCt will feem to fill the fpace RV; but 
the reprefentation of it will be confuted, becaufe, the in¬ 
termediate points of the objeCt being equally enlarged in 
appearance, there will not be room for them between the 
points S and T, but they will coincide in part one with 
another : for inftance, the appearance of that point in the 
objeCt, whofe reprefentation falls upon c in the image, 
will fill the fpace mn ; and fo of the relt. Now, if the fame 
pupil be removed into the lituation ef, the reflected rays 
Ee and Gf will then enter the eye, and therefore one ex¬ 
tremity of the objeCt will appear to cover the fpace XY; 
and becaufe the rays O/'and Le will alfo enter it in their 
progrefs towards M, the point B, from which they came, 
will appear to cover ZV ; the object therefore will appear 
larger and more confufed than before. When the eye re¬ 
cedes quite to the image, it fees but one Angle point of 
the objeCt, and that appears diffufed all over the reflecting 
furface : for inftance, if the eye recedes to the point M, 
then rays flowing from the point B enter it upon whatever 
part of the furface they fall. The objeCt alfo appears 
nearer to the furface the farther the eye recedes from it 
towards the place of the image; probably becaufe, as the 
appearance of the objeCt becomes more _and more con¬ 
fufed, its place is not fo eafily diftinguifhed from that of 
the reflecting furface itfelf, till at lalt, when it is quite 
confuted, (as it is when the eye is arrived at M,) they 
both appear as one, the furface all'uming the Golour o’f 
the objeCt. 
As to the precife apparent magnitude of an objeCt feen 
after this manner, it is fuch, that the angle it appears un¬ 
der lhall be equal to that which the image of the fame 
objeCt would appear under were we to fuppofe it feen from 
the fame place: that is, the apparent objeCt (for fucti 
we mult call it, to diftinguilli it from the image of the 
fame objeCt) and the image lubtend equal angles at 
the eye. 
Here we mu ft fuppofe the pupil of the eye to be a point 
only, becaufe the magnitude of it caufes afinall alteration 
in the apparent magnitude of the objeCt. Let the point 
a reprefent the pupil, then will the extreme rays that can 
enter it be H« and K a\ the objeCt therefore will appear 
under the angle H«K=MaI, the angle under which the 
image IM would appear were it to be feen from a. Again, 
If the eye be placed in f, the objeCt appears under the an¬ 
gle Gf O—lfbA, which the image fubtends at the lame 
place. 
