642 OPT 
follows that they will all combine into one porfed whole ; 
in which all the parts either are, or may be, united, 
and which will delight the eye by its fymmetry and 
beauty. 
In order to illuftrate the preceding obfervations, we 
have reprefented, in fig. 4 and 5, the effects produced by 
the multiplication of iingle and compound forms. The 
line abccl, tor example, fig. 4. is a fimple form, and is ar¬ 
ranged round a centre in the fame way as it would be 
done by a perfect multiplying-glafs, if fuch a thing could 
be made. The confecutive forms are all difunited, and 
do notcompofea whole. Fige 5 reprefents the very lame 
fimple form, abed, converted into a compound form, and 
then, as it were, multiplied and arranged round a centre. 
In this cafe, every part of the figure is united, and forms 
a whole, in which there is nothing redundant and no¬ 
thing deficient ; and this is the precite effed which is 
produced by the application of the kaleidofcope to the 
fimple form abed. 
The fundamental principle, therefore, of the kaleido¬ 
fcope, is, that it produces fymmetrical and beautiful pic¬ 
tures, by converting fimple into compound or beautiful 
forms, and arranging them, by fucceffive refledions, 
into one perfect whole. This principle, it will be readily 
feen, cannot be difeovered by any examination of the 
luminous fedors which compote the circular field of the 
kaleidofcope, and is not even alluded to in any of the 
propofitions given by Harris and Mr. Wood. In looking 
at tlie circular field compofed of an even and an odd 
number of refledions, the arrangement of the fedors is 
perfed in both cafes ; but, when the number is odd, and 
the form of the objed fimple, and not fimilarly placed 
with regard to the two mirrors, a fymmetrical and united 
pidure 'cannot pofiibly be produced. Hence it is mani- 
feft, that neither the principles nor the eft'eds of the ka¬ 
leidofcope couid poffibly be deduced from any practical 
knowledge refpeding the luminous fedors. 
In order to explain the formation of the fj nunetrical 
picture fhown in fig. 5. we mult confider that the fimple 
form urn, fig. 2, is leen by dired vifion through the open 
fedor AOB ; and that the image no, of the objed mn, 
formed by one reflection in the (edor BOa, is necefiarily 
an inverted image. But, lince the image up, in the fedor 
(iQc, is a refleded (and confequently an inverted) image 
of the inverted image mt, in the fedor AO A ; it follows, 
that the whole nop is an inverted image of the whole nr,it; 
confequently the image no will unite with the image op, 
in the fame manner as mu unites with mt. But, as thefe 
two laft unite into a regular form, the two firft will all'o 
unite into a regular or compound form. Now, fmee the 
lialf dOe of the lalt fedor dOc was formerly fhown to be 
an image of the half-fedor aOs, the line qv will alio bean 
image of the line oz, and for the fame reafon the line vp 
will be an image of tij. But the image vp forms the fame 
angle with BO, or vq, that ty does, and is equal and fimi- 
lar to ti /; and qv forms the lame angle with AO that oz 
does, and is equal and fimilar to oz. Hence Ooz^Oq, and 
Gy. Or, and therefore qv and vp, will form one ftraight 
line, equal and fimilar to Iq, and fimilarly fituated with 
refped to BO. The figure mnopqt, therefore, compofed 
of one direct objed, and feveral refleded images of that 
objed, will be fymmetrical. 
As the fame reafoningis applicable to every objed ex¬ 
tending acrofs the aperture AOB, whether 1 fimple or com¬ 
pound, and to every angle AOB, which is 3n even ali¬ 
quot part of a circle, it follows, 1. That, when the in¬ 
clination of the mirror is an even aliquot part of a circle, 
tire objed feen by dired vifion acrofs the aperture, whe¬ 
ther it be fimple or compound, is fo united with the images' 
of it formed by repeated refledions, as to form a fymme¬ 
trical picture. 2. That the fymmetrical picture is com¬ 
pofed of a feries of parts, the number of which is equal 
to the number of times that the angle AOB is contained 
in 360°. And, 3. That thefe parts are alternately dired 
sndiiiverted pidures of the objed; a dired pidure of it 
I c s. 
being always placed between two inverted ones, and vice 
verfa, fo that the number of dired pidures is equal to the 
number of inverted ones. 
When the inclination of the biirrors is an odd aliquot 
part of 360°, fuch as £th, as fhown in fig. 3, the picture 
formed by the combination of the direct objed and its 
refleded images is fymmetrical only under particular cir- 
cumftances. 
If the objed, whether fimple or compound, is fimilarly 
fituated with refped to each of the mirrors, as the llraight 
line 1, 2, of fig.-6. the compound line 3, 4, the inclined 
lines 5, 6, the circular objed 7, the curved line 8, n, and 
the radial line 10, O, then the images of all thefe Objeds 
will alfo be fimilarly fituated with refped to the radial 
lines that feparate the fedors, and will therefore form a 
whole perfedly fymmetrical, whether the number of lec¬ 
tors is odd or even. But, when the objeds are not fimi¬ 
larly fituated with refped to each of the mirrors, as the 
compound line 1, z, fig. 7. the curved line 3, 4, and the 
flraight line 5, 6 ; and, in general, as all irregular objeds 
that are prefented by accident to the inflrument; then 
the image formed in the laft fedor aOe, by the mirror 
BO, will not join with the image formed in the laft fedor 
bOe, by the mirror AO. 
In order to explain this with fufficient perfpicuity, let 
us take the cafe where the angle is 72 0 , or-|fh part of the 
circle, as fhown in fig. 3. Let AO, BO, be the refleding 
planes, and mn a line, inclined to the radius which bifc.cts 
tlie angle AOB, fo that OmVOn; then mu', nm', will be 
the images formed by the firft refledion from AO and 
BO, and n'm", in' n" the images formed by the fecond 
refiedion ; but, by the principles of catoptrics, O m— 
Om'—Om", and On—On'zzzOn" ; confequently, fince O m 
is by hypothefis greater than On, we fhr.ll have O m" 
greater than On" ; that is, the images m'n", n'm", will 
notcoincide. As On approaches to an equality with Om, 
On'' approaches to an equality with Om"; and, w'hen Om 
—On, we have On"=zOm" ; and at this limit the imao-es 
are fymmetrically arranged, which is the cafe of the 
flraight line t, 2, in fig. 6. By tracing the images of the 
other lines, as is done in fig. 7, it will be feen, that in 
every cafe the pidure is deflitute of fymmetry when the 
objed has not tlie fame polition with refped to the two 
mirrors. 
This refult may be deduced in a more fimple manner, 
by confidering that the fymmetrical pidure formed by the 
kaleidofcope contains half as many pairs of forms as the 
number of times that the inclination of the mirrors is 
contained in 360° ; and that each pair confifls of a dired 
and an inverted form, fo joined as to form a compound 
form. Now. the compound form made up by each pair 
obvioufly conftitutes a fymmetrical pidure when multi¬ 
plied any number of times, whether even or odd ; but, if 
we combine fo many pair and half a pair, two dired 
images w ill come together, the half-pair cannot pofiibly 
join both with the dired and the inverted image on each 
fide of it, and therefore a fymmetrical whole cannot be 
obtained from fuch a combination. From thefe obferva- 
tions we may conclude, 
1. That, when the inclination of the mirrors is an odd 
aliquot part of a circle, the objed feen by dired vifion 
through the aperture unites with the images of if formed 
by repeated refledions, and forms a complete and fymme¬ 
trical pidure, only in the cafe w’hen the objed is fimilarly 
fituated with refped to both the mirrors ; the two laft 
fedors forming, in every other pofition of the objed, an 
impeded jundion, in confequence of thefe being either 
both dired or both inverted pidures of the objed. 
2. That the feries of parts which compofe the fymme¬ 
trical as well as the unlymmetrical pidure, confifts of di¬ 
red and inverted pidures of the objed, tlie number of 
dired pidures being always equal to half the number of 
fedors increafed by one, when the number of fedors is 
5, g, 13, 17, 21, Sic. and the number of inverted pidures 
being equal to half the number of' fedors diminiilied by 
one 
