OPT 
formation of'the laft or fixth fedtor, cOd. Mr. Harris, in 
the XVIIth Prop, of his Optics, has evaded this difficulty, 
and given a falfe demonftratioii of the propofition. He 
remarks, that the lath fed!or cOd is produced “ by the re¬ 
fledtion of the rays forming either of the two laft images,” 
fnam-ely, bOd and aQc ;) but this is clearly ahfurd, for the 
fedtor cQd would thus be formed of two images lying 
above each other, which is impoffible. In order to un- 
derftand the true caufe of the formation of the fedtor cOd, 
we muft recolledt that rhe line OS is the line of jundtion 
of the mirrors, and that the eye is placed any where in 
jhe plane pairing through OE and Infecting AOB. Now, 
if the mirror BOE had extended as far as Or/, the fedtor 
cOd would have been the image of the factor bOd re- 
fledted from BO ; and in like manner, if the mirror AOB 
had extended as far as Or, the fedtor cOd would have been 
the image of the fedtor aOc refledted from AO ; but, as 
this overlapping or extension of the mirrors is impoffible, 
and as they muft neceflarily join at the line OE, it follows, 
that an image cGc, of-only half the fector bOrf, viz. LOr, 
can be feen by reflection from the mirror BO ; and that 
an image dOe, of only half the fedtor a Or, viz. aOs, can. 
be feen by reflection from the mirror AO. Hence it is 
manifeft,. that the lalt fedtor, cOd, is not, as Harris fup- 
pofes, a reflation from either of the two lafi images bOd, 
aOc; but is compofed of the images of two half-lectors, 
one of which is formed by the mirror AO, and the other 
by the mirror BO. 
Mr. Harris repeats the fame miftake in a more ferious 
fojwn, in his fecond Scholium, § 24.0. where he fnows that 
the 1 images are arranged in the circumference of a circle. 
The two images D, d, fays he, coincide, and make but one 
image. Mr. Wood has committed the very lame miftake 
in his Optics, at Cor. 2. Prop. XIV. and his demonftra- 
tion of that corollary is decidedly erroneous. This Co¬ 
rollary is flatted in the following manner : “ When a (the 
angle of the mirrors) is a meafure of 180 0 , two images 
coincide and it is demonftrated, that “ fmce two images 
of any objedt X (fig. a.) muft be formed, viz. one by each 
mirror ; and fmce thefe two images muft be formed at 180 0 
from the objedt X, placed between the mirrors, that is, at 
the fame point x; it follows that the two images muft co¬ 
incide.” Now it will appear from the fimpleft confidera- 
tions, that the aflumpti-on, as well as the conclulion, is 
here erroneo.us. The image a-is feen-by the laft refledtion 
from the mifror BOE, and another image would be feen 
at x, if the mirror AOE had extended as far as x; but as 
this is impoffible, without covering the part of the mirror 
BOE which gives the firfl: image ,r, there can be only one 
image feen cat x. When the objedt X is equi-diftant from 
A and B, then one half of the kail-refledted image x will 
be formed by the laft refledtion from the mirror BO, and 
the other half by the laft refledtion from the mirror AO ; 
and thefe two half-images will join each other, and form a 
whole image at e, as perfedt as any of the reft. In this 
laft cafe, when the angle AOB is a little different from an 
even aliquot part of 360°, the eye at E will perceive at e 
an appearance of two incoincident images ; but this arifes 
from the pupil of the eye being partly on one fide of E) 
and partly on the other; and therefore the apparent 
duplication of the image is removed by looking through 
a very fmali aperture at E. 
As the preceding remarks are equally true, whatever 
be the inclination of the mirrors, provided it is an even 
aliquot part.of a circle, it follows, 1. That, when AOB 
is -§> iV tV’ °f a circle, the number of refledted 
images of any objedt X, is 4—t, 6—1, 8—1, jo—i, 12—1. 
2. That, when X is nearer one mirror than another, the 
number of images feen by refledtion from the mirror to 
which it is nearelt will be , while the number 
of images formed by the mirror from which X is molt diftant 
will be — 1, J-—i, -|— 1, -—1; that is, an image more 
always reaches the eye from the mirror nearelt X than 
from the mirror iartheit from it. 3. That, when X is equi- 
diffant from AO and BO, the number of images which 
Vol. XVII. No. 1204, 
ICS. Oil 
reaches the eye from each mirror is equal, and is always 
■ 4 V~ - 1 -, w hich are fradtiona! va¬ 
lues, (bowing that the laft image is coihpofed of two half- 
images. 
When the inclination of the mirrors,‘or the angle AOB, 
fig. 3. is an odd aliquot part of a circle, fuch as -J, A, -jj-, 
&c. the different fedtors which compote the circular image 
are formed in the very fanTe manner as has been already 
defcribed ; but, as the number o f refit Bed fedtors. m u ft in 
this cafe always be even, the line OE, where the mirrors 
join, will feparate the two lafl-refledted fedlors, uOc, «Or. 
Hence it follows, 1. That, when AOB is -j, a A, a, &c. 
of a circle, the number of refledted images of any object 
is 3 — 1, 5—1, 7—-r, &c. 2. That the number of images 
which reach the eye from each mirror is 
which are always even numbers. 
Hitherto we have fuppofed the inclination of the mir¬ 
rors to be exactly either an even or an odd aliquot part of 
a circle. We lhali now proceed to conilder the eftedts 
which will be produced when this is not the cale. It the 
angle AOB, fig.’ 2. is made to increafe from being an even 
aliquot part of a circle, fuch as £th, till it becomes an odd 
aliquot part, fuch as yth, the laft-refledted image dOc, 
compofed of the two halves dOe, cOe, will gradually in¬ 
creafe, in confequence of each of the halves increafing; 
and, when AOB becomes J.th of the circle, the fedtor dOe 
will become double of AOB, and cOe, dOe, will become 
each complete fedtors, or equal to AOB. If the angle 
AOB is made to vary from |th to -Jt-h of a circle, the lalt 
fedtor dOc will gradually diminifh, in confequence of each 
of its halves, dOe, cOe, diminlihing; and, juft when the 
angle becomes -Jth of a circle, the fedtor dOo will have 
become infinitely fmali, and the two fedtors bad, aOc, 
will join each other exadtly at the line Oe, as in fig. 3. 
Principles of the.KaleidpJcope .—The principles juft laid 
down muft not be conlidered as in any refpedt the prin¬ 
ciples of the kaleidofcope. They are merely a feries of 
preliminary dedudtions, by means of which the principles 
of the inftrument may be illuftrated, and they go no far¬ 
ther than to explain the formation of an apparent circular 
aperture by means of fucceffive refledtions. 
Allfthe various forms which nature and art prefent to 
us may be divided into two claffes ; namely, Jhnple or 
irregular forms, and compound or regular forms. To the 
firfl: clafs belong all thofe forms which are called pic- 
turefque, and which cannot be reduced to two forms fimi- 
lar, and flmilarly fituated with regard to a given point ; 
and to the fecond clafs belong the forms of jnimals, the 
forms of regular architedtural buildings, the forms of all 
articles of furniture and ornament, the forms of many 
natural produdlions, and all forms, in fhort, which are 
compofed of two forms, fimilar, and flmilarly .fituated 
with regard to a given line. Now’, it is obvious that all 
compound forms of this kind are compofed of a diredfc 
and an inverted image of a Ample or an irregular form ; 
and therefore, every Ample form can be converted into 
a compound or beautiful form, by fkilfully combining it 
with an inverted image of itfeif, formed by reflection. 
The image, however, muft be formed by refledtion from 
the firfl. furface of the mirror, in order that the diredt and 
the refledted image may join, and conftitute one united 
whole ; for, if the image is refledted from the polterior 
furface, as in the cafe of a looking-glafs, the diredt and 
the inverted image can never coalelce into one form, but 
muft always be feparated by a fpace equal to the thicknefs 
of the mirror-glafs. If we arrange Ample forms in the. 
molt perfedt manner round a centre, it is impoffible by 
any art to combine them into a fymmetrical and beautiful 
pidture. The regularity of their arrangement may give 
fome fatisfadtion to the eye ; but the adjacent forms can 
never join, and muft therefore form a pidture compofed 
of difunited parts. The cafe, however, is quite different 
with compound forms. If we arrange a fucceifion of 
fimilar forms of this clafs round a centre, it neceflarily 
8 A ’ follows 
