,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,illustrations have been moved so they do not break up paragraphs4 ,old or antiquated spellings have been preserved4 ,typographical and punctuation errors have been silently corrected4 ,the list of works by ,sir ,david ,brewster has been moved form the first page to the end of the text4 ,illustration ,,the ,,kaleidoscope1 ,,its ,,history1 ,,theory1 ,,and ,,construction4 ,,with ,,its ,,application ,,to ,,the ,,fine ,,and ,,useful ,,arts4 ,,by ,,sir ,,david ,,brewster1 ,k4,h41 ,m4,a41 ,d4,c4,l41 ,f4,r4,s41 ,v4,p4,r4,s41 ,,edin41 ,m4,r4,i4,a41 ,f4,g4,s41 ,f4,r4,a4,s41 ,,associate ,,of ,,the ,,imperial ,,institute ,,of ,,france1 ,,honorary ,,or ,,corresponding ,,member ,,of ,,the ,,academies ,,of ,,petersburgh1 ,,vienna1 ,,berlin1 ,,copenhagen1 ,,stockholm1 ,,brussels1 ,,gottingen1 ,,modena1 ,,and ,,of ,,the ,,national ,,institute ,,of ,,washington1 ,,and ,,principal ,,of ,,the ,,united ,,colleges ,,of ,,st4 ,,salvator ,,and ,,st4 ,,leonard1 ,,st4 ,,andrews4 ,nihil tangit quod non ornat4 ,illustration ,third ,edition1 greatly enlarged4 ,,with ,,fifty-,,six ,,wood ,,engravings ,,and ,,one ,,plate4 ,,london3 ,,john ,,camden ,,hotten1 ,,piccadilly4 #ahgj4 ,,contents4 ,,page ,,introduction--,,history ,,of ,,the ,,kaleidoscope1 #a ,,chap4 ,i4--,,preliminary ,,principles ,,respecting ,,the ,,effects ,,of ,,combining ,,two ,,plain ,,mirrors1 #i ,,ii4--,,on ,,the ,,principles ,,of ,,the ,,kaleidoscope1 ,,and ,,the ,,formation ,,of ,,symmetrical ,,pictures ,,by ,,the ,,combination ,,of ,,direct ,,and ,,inverted ,,images1 #af ,,iii4--,,on ,,the ,,effects ,,produced ,,by ,,the ,,motion ,,of ,,the ,,object ,,and ,,the ,,mirrors1 #bf ,,iv4--,,on ,,the ,,effects ,,produced ,,upon ,,the ,,symmetry ,,of ,,the ,,picture ,,by ,,varying ,,the ,,position ,,of ,,the ,,eye1 #cg ,v4--,,on ,,the ,,effects ,,produced ,,upon ,,the ,,symmetry ,,of ,,the ,,picture ,,by ,,varying ,,the ,,position ,,of ,,the ,,object1 #df ,,vi4--,,on ,,the ,,intensity ,,of ,,the ,,light ,,in ,,different ,,parts ,,of ,,the ,,field1 ,,and ,,on ,,the ,,effects ,,produced ,,by ,,varying ,,the ,,length ,,and ,,breadth ,,of ,,the ,,reflectors1 #ea ,,vii4--,,on ,,the ,,construction ,,and ,,use ,,of ,,the ,,simple ,,kaleidoscope1 #ei ,,viii4--,,on ,,the ,,selection ,,of ,,objects ,,for ,,the ,,kaleidoscope1 ,,and ,,on ,,the ,,mode ,,of ,,constructing ,,the ,,object-,,box1 #fg ,,ix4--,,on ,,the ,,illumination ,,of ,,transparent ,,objects ,,in ,,the ,,kaleidoscope1 #gf ,x4--,,on ,,the ,,construction ,,and ,,use ,,of ,,the ,,telescopic ,,kaleidoscope1 ,,for ,,viewing ,,objects ,,at ,a ,,distance1 #ha ,,xi4--,,on ,,the ,,construction ,,and ,,use ,,of ,,polyangular ,,kaleidoscopes1 ,,in ,,which ,,the ,,reflectors ,,can ,,be ,,fixed ,,at ,,any ,,angle1 #hh #a4 ,bate's ,polyangular ,kaleidoscope with ,metallic ,reflectors1 #hi #b4 ,bate's ,polyangular ,kaleidoscope with ,glass ,reflectors1 #id ,,xii4--,,on ,,the ,,construction ,,and ,,use ,,of ,,annular ,,and ,,parallel ,,kaleidoscopes1 #ih #a4 ,mr4 ,dollond's ,universal ,kaleidoscope1 #ajj #b4 ,ruthven's ,universal ,kaleidoscope1 #ajb ,,xiii4--,,on ,,the ,,construction ,,and ,,use ,,of ,,polycentral ,,kaleidoscopes1 #aje #a4 ,on combinations of four mirrors forming a square1 #ajg #b4 ,on combinations of four mirrors forming a rectangle1 #aji #c4 ,on combinations of three reflectors at angles of #fj1 #aji #d4 ,on combinations of three reflectors at angles of #ij1 #de1 and #de1 #aaa #e4 ,on combinations of three reflectors at angles of #ij1 #fj1 and #cj1 #aab ,,xiv4--,,on ,,kaleidoscopes ,,in ,,which ,,the ,,effect ,,is ,,produced ,,by ,,total ,,reflexion ,,from ,,the ,,interior ,,surfaces ,,of ,,transparent ,,solids1 #aad ,,xv4--,,on ,,the ,,application ,,of ,,the ,,kaleidoscope ,,to ,,the ,,magic ,,lantern1 ,,solar ,,microscope1 ,,and ,,camera ,,obscura1 #aag ,,xvi4--,,on ,,the ,,construction ,,of ,,kaleidoscopes ,,which ,,combine ,,the ,,colours ,,and ,,forms ,,produced ,,by ,,polarized ,,light1 #abb ,,xvii4--,,on ,,the ,,construction ,,of ,,stereoscopic ,,kaleidoscopes1 #abf ,,xviii4--,,on ,,the ,,construction ,,of ,,microscopic ,,kaleidoscopes1 #abh ,,xix4--,,on ,,the ,,changes ,,produced ,,by ,,the ,,kaleidoscope1 #aca ,,xx4--,,on ,,the ,,application ,,of ,,the ,,kaleidoscope ,,to ,,the ,,fine ,,and ,,useful ,,arts1 #acd #a4 ,architectural ,ornaments1 #acg #b4 ,ornamental ,painting1 #ada #c4 ,designs for ,carpets1 #add ,,xxi4--,,on ,,the ,,photographic ,,delineation ,,of ,,the ,,pictures ,,created ,,by ,,the ,,kaleidoscope1 #adh ,,xxii4--,,on ,,the ,,advantages ,,of ,,the ,,kaleidoscope ,,as ,,an ,,instrument ,,of ,,amusement1 #aed ,,xxiii4--,,history ,,of ,,the ,,combinations ,,of ,,plane ,,mirrors ,,which ,,have ,,been ,,supposed ,,to ,,resemble ,,the ,,kaleidoscope1 #afb #a4 ,baptista ,porta's multiplying speculum1 #afd #b4 ,kircher's combination of plane mirrors1 #afh #c4 ,bradley's combination of plane mirrors1 #age ,,appendix1 #ahe ,,on ,,the ,,kaleidoscope4 ,,introduction4 ,,history ,,of ,,the ,,kaleidoscope4 ,the name ,kaleidoscope1 which ,i have given to a new ,optical ,instrument1 for creating and exhibiting beautiful forms1 is derived from the ,greek words 1 beautiful2 1 a form2 and 1 to see4 ,the first idea of this instrument presented itself to me in the year #ahad1 in the course of a series of experiments on the polarization of light by successive reflexions between plates of glass1 which were published in the ,philosophical ,transactions for #ahae1 and which the ,royal ,society did me the honour to distinguish by the adjudication of the ,copley ,medal4 ,in these experiments1 the reflecting plates were necessarily inclined to each other during the operation of placing their surfaces in parallel planes2 and ,i was therefore led to remark the circular arrangement of the images of a candle round a centre1 and the multiplication of the sectors formed by the extremities of the plates of glass4 ,in consequence1 however1 of the distance of the candles1 c41 from the ends of the reflectors1 their arrangement was so destitute of symmetry1 that ,i was not induced to give any farther attention to the subject4 ,on the #gth of ,february #ahae1 when ,i discovered the development of the complementary colours1 by the successive reflexions of polarized light between two plates of gold and silver1 the effects of the ,kaleidoscope1 though rudely exhibited1 were again forced upon my notice2 the multiplied images were1 however1 coloured with the most splendid tints2 and the whole effect1 though inconceivably inferior to the creations of the ,kaleidoscope1 was still far superior to anything that ,i had previously witnessed4 ,in giving an account of these experiments to ,m4 ,biot on the #fth of ,march #ahae1 ,i remarked to him1 8that when the angle of incidence 7on the plates of silver7 was about #he or #hf1 and the plates almost in contact1 and inclined at a very small angle1 the two series of reflected images appeared at once in the form of two curves2 and that the succession of splendid colours formed a phenomenon which ,i had no doubt would be considered1 by every person who saw it to advantage1 as one of the most beautiful in optics40 ,these experiments were afterwards repeated with more perfectly polished plates of different metals1 and the effects were proportionally more brilliant3 but notwithstanding the beauty arising from the multiplication of the images1 and the additional splendour which was communicated to the picture by the richness of the polarized tints1 it was wholly destitute of symmetry1 as ,i was then ignorant of those positions for the eye and the objects1 which are absolutely necessary to produce that magical union of parts1 and that mathematical symmetry throughout the whole picture1 which1 independently of all colouring1 give to the visions of the ,kaleidoscope the peculiar charm which distinguishes them from all artificial creations4#a #a ,an account of the experiments above alluded to was given in the ,analyse des ,travaux de la ,classe des ,sciences ,mathmatiques et ,physiques de l',institut ,royal de ,france1 pendant l'anne #ahae1 par ,m4 le ,chev4 ,delambre1 p4 #bi1 c4 ,the colours produced by repeated reflexions from plates of silver are those of ,elliptical ,polarization1 and are explained at great length in my paper on that subject1 published in the ,philosophical ,transactions for #ahcj4 ,although ,i had thus combined two plain mirrors1 so as to produce highly pleasing effects1 from the multiplication and circular arrangement of the images of objects placed at a distance from their extremities1 yet ,i had scarcely made a step towards the invention of the ,kaleidoscope4 ,the effects1 however1 which ,i had observed1 were sufficient to prepare me for taking advantage of any suggestion which experiment might afterwards throw in the way4 ,in repeating1 at a subsequent period1 the very beautiful experiments of ,m4 ,biot1 on the action of homogeneous fluids upon polarized light1 and in extending them to other fluids which he had not tried1 ,i found it most convenient to place them in a triangular trough1 formed by two plates of glass cemented together by two of their sides1 so as to form an acute angle4 ,the ends being closed up with pieces of plate glass cemented to the other plates1 the trough was fixed horizontally1 for the reception of the fluids4 ,the eye being necessarily placed without the trough1 and at one end1 some of the cement1 which had been pressed through between the plates at the object end of the trough1 appeared to be arranged in a manner far more regular and symmetrical than ,i had before observed when the objects1 in my early experiments1 were situated at a distance from the reflectors4 ,from the approximation to perfect symmetry which the figure now displayed1 compared with the great deviation from symmetry which ,i had formerly observed1 it was obvious that the progression from the one effect to the other must take place during the passage of the object from the one position to the other2 and it became highly probable1 that a position would be found where the symmetry was mathematically perfect4 ,by investigating this subject optically1 ,i discovered the leading principles of the ,kaleidoscope1 in so far as the inclination of the reflectors1 the position of the object1 and the position of the eye1 were concerned4 ,i found1 that in order to produce perfectly beautiful and symmetrical forms1 three conditions were necessary4 #a4 ,that the reflectors should be placed at an angle1 which was an even or an odd aliquot part of a circle1 when the object was regular1 and similarly situated with respect to both the mirrors2 or the even aliquot part of a circle when the object was irregular1 and had any position whatever4 #b4 ,that out of an infinite number of positions for the object1 both within and without the reflectors1 there was only one where perfect symmetry could be obtained1 namely1 when the object was placed in contact with the ends of the reflectors4 ,this was precisely the position of the cement in the preceding experiment with the triangular trough4 #c4 ,that out of an infinite number of positions for the eye1 there was only one where the symmetry was perfect1 namely1 as near as possible to the angular point1 so that the circular field could be distinctly seen2 and that this point was the only one out of an infinite number at which the uniformity of the light of the circular field was a maximum1 and from which the direct and the reflected images had the same form and the same magnitude1 in consequence of being placed at the same distance from the eye4 ,this1 also1 was the position in which the eye was necessarily placed when looking through the fluid with which the glass trough was partially filled4 ,upon these principles ,i constructed an instrument1 in which ,i fixed permanently1 across the ends of the reflectors1 pieces of coloured glass1 and other irregular objects2 and ,i showed the instrument in this state to some members of the ,royal ,society of ,edinburgh1 who were much struck with the beauty of its effects4 ,in this case1 however1 the forms were nearly permanent2 and slight1 though beautiful1 variations were produced by varying the position of the instrument with respect to the source of light4 ,the great step1 however1 towards the completion of the instrument remained yet to be made2 and it was not till some time afterwards that the idea occurred to me of giving motion to objects1 such as pieces of coloured glass1 c41 which were either fixed or placed loosely in a cell at the end of the instrument4 ,when this idea was carried into execution1 and the reflectors placed in a tube1 and fitted up on the preceding principles1 the ,kaleidoscope1 in its simple form1 was completed4 ,in this form1 however1 the ,kaleidoscope could not be considered as a general philosophical instrument of universal application4 ,the least deviation of the object from the position of symmetry at the end of the reflectors1 produced a deviation from beauty and symmetry in the figure1 and this deviation increased with the distance of the object4 ,the use of the instrument was therefore limited to objects in contact with the ends of the reflectors1 or held close to them1 and consequently to objects1 or groups of objects1 whose magnitudes were less than its triangular aperture4 ,the next1 and by far the most important step of the invention1 was to remove this limitation1 and to extend indefinitely the use and application of the instrument4 ,this effect was obtained by employing a draw tube1 containing a convex lens1 or1 what is better1 an achromatic object-glass of such a focal length1 that the images of objects1 of all magnitudes and at all distances1 might be distinctly formed at the end of the reflectors1 and introduced into the pictures created by the instrument in the same manner as if they had been reduced in size1 and placed in the true position in which alone perfect symmetry could be obtained4 ,when the ,kaleidoscope was brought to this degree of perfection1 it was impossible not to perceive that it would prove of the highest service in all the ornamental arts1 and would1 at the same time1 become a popular instrument for the purposes of rational amusement4 ,with these views ,i thought it advisable to secure the exclusive property of it by a ,patent2#b but in consequence of one of the patent instruments having been exhibited to some of the ,london opticians1 the remarkable properties of the ,kaleidoscope became known before any number of them could be prepared for sale4 ,the sensation excited by this premature exhibition of its effects is incapable of description1 and can be conceived only by those who witnessed it4 8,it very quickly became popular10 says ,dr4 ,roget1 in his excellent article on the ,,kaleidoscope in the ,encyclopdia ,britannica1 8and the sensation it excited in ,london throughout all ranks of people was astonishing4 ,it afforded delight to the poor as well as the rich2 to the old as well as the young4 ,large cargoes of them were sent abroad1 particularly to the ,east ,indies4 ,they very soon became known throughout ,europe1 and have been met with by travellers even in the most obscure and retired villages in ,switzerland40 ,according to the computation of those who were best able to form an opinion on the subject1 no fewer than two hundred thousand instruments were sold in ,london and ,paris during three months4 ,out of this immense number there were perhaps not one thousand constructed upon scientific principles1 and capable of giving anything like a correct idea of the power of the ,kaleidoscope2 and of the millions who have witnessed its effects1 there is perhaps not a hundred individuals who have any idea of the principles upon which it is constructed1 who are capable of distinguishing the spurious from the real instrument1 or who have sufficient knowledge of its principles to be able to apply it to the numerous branches of the useful and ornamental arts4 ,under these circumstances ,i have thought it necessary to draw up the following short treatise1 for the purpose of explaining1 in as popular a manner as ,i could1 the principles and construction of the ,kaleidoscope2 of describing the different forms in which it is fitted up2 of pointing out the various methods of using it as an instrument of recreation2 and of instructing the artist how to employ it in the numerous branches of the useful and ornamental arts to which it is applicable4 #b ,as this ,patent1 in so far as the simple ,kaleidoscope is concerned1 was to a great extent infringed1 it has been supposed that it was reduced in a ,court of ,law4 ,the validity of the ,patent was never questioned by any lawyer1 or any philosopher acquainted with its theory and construction1 as will appear from the opinion of four competent judges1 given in the ,appendix4 ,in a trial for the infringement of a ,patent several years ago1 a distinguished judge 7we believe it was ,judge ,alderson7 stated it as a fact1 that the ,patent for the ,kaleidoscope had been set aside in a ,court of ,law4 ,the party whose case was prejudiced by this erroneous assertion1 applied to me for an affidavit1 by which he was enabled to contradict it in ,court1 and remove any unfavourable impression it might have made upon the jury4 ,,chapter ,i4 ,,preliminary ,,principles ,,respecting ,,the ,,effects ,,of ,,combining ,,two ,,plain ,,mirrors4 ,the principal parts of the ,kaleidoscope are two reflecting planes1 made of glass1 or metal1 or any other reflecting substance ground perfectly flat and highly polished4 ,these reflectors1 which are generally made of plate glass1 either rough ground on their outer side1 or covered with black varnish1 may be of any size1 but in general they should be from four or five to ten or twelve inches long2 their greatest breadth being about an inch when the length is six inches1 and increasing in proportion as the length increases4 ,when these two plates are put together at an angle of #fj1 or the sixth part of a circle1 as shown in ,fig4 #a1 and the eye placed at the narrow end ,e1 it will observe the opening ,a ,o ,b multiplied six times1 and arranged round the centre ,o1 as shown in ,fig4 #b4 ,illustration3 ,,fig4 #a4 ,illustration3 ,,fig4 #b4 ,in order to understand how this effect is produced1 let us take a small sector of white paper of the shape ,a ,o ,b1 ,fig4 #b1 and having laid it on a black ground1 let the extremity ,a ,o of one of the reflectors be placed upon the edge ,a ,o of the sector4 ,it is then obvious that an image ,a ,o b of this white sector of paper will be formed behind the mirror ,a ,o1 and will have the same magnitude and the same situation behind the mirror as the sector ,a ,o ,b had before it4 ,in like manner1 if we place the edge ,b ,o of the other reflector upon the other side ,b ,o of the paper sector1 a similar image ,b ,o a will be formed behind it4 ,the origin of three of the sectors seen round ,o is therefore explained3 the first1 ,a ,o ,b1 is the white paper sector seen by direct vision2 the second1 ,a ,o b1 is an image of the first formed by one reflexion from the mirror ,a ,o2 and the third is another image of the first formed by one reflexion from the other mirror ,b ,o4 ,but it is well known1 that the reflected image of any object1 when placed before another mirror1 has an image of itself formed behind this mirror1 in the very same manner as if it were a new object4 ,hence it follows1 that the image ,a ,o b being1 as it were1 a new object placed before the mirror ,b ,o1 will have an image a ,o of itself formed behind ,b ,o2 and for the same reason the image ,b ,o a will have an image b ,o of itself1 formed behind the mirror ,a ,o1 and both these new images will occupy the same position behind the mirrors as the other images did before the mirrors4 ,a difficulty now presents itself in accounting for the formation of the last or sixth sector1 ,o 4 ,mr4 ,harris1 in the ,x,v,i,ith ,prop4 of his ,optics1 has evaded this difficulty1 and given a false demonstration of the proposition4 ,he remarks1 that the last sector1 ,o is produced 8by the reflexion of the rays forming either of the two last images0 7namely1 b ,o and a ,o 72 but this is clearly absurd1 for the sector ,o would thus be formed of two images lying above each other1 which is impossible4 ,in order to understand the true cause of the formation of the sector ,o 1 we must recollect that the line ,o ,e is the line of junction of the mirrors1 and that the eye is placed any where in the plane passing through ,o ,e and bisecting ,a ,o ,b4 ,now1 if the mirror1 ,b ,o1 had extended as far as ,o 1 the sector ,o would have been the image of the sector b ,o 1 reflected from ,b ,o2 and in like manner1 if the mirror ,a ,o had extended as far as ,o 1 the sector ,o would have been the image of the sector a ,o reflected from ,a ,o2 but as this overlapping or extension of the mirrors is impossible1 and as they must necessarily join at the line ,o ,e1 it follows1 that an image ,o e1 of only half the sector b ,o 1 viz41 b ,o r1 can be seen by reflexion from the mirror ,b ,o2 and that an image ,o e1 of only half the sector a ,o 1 viz41 a ,o s1 can be seen by reflexion from the mirror ,a ,o4 ,hence it is manifest1 that the last sector1 ,o 1 is not1 as ,mr4 ,harris supposes1 a reflexion from either of the two last images1 b o 1 a o 1 but is composed of the images of two half sectors1 one of which is formed by the mirror ,a ,o1 and the other by the mirror ,b ,o4 ,mr4 ,harris repeats the same mistake in a more serious form1 in his second ,scholium1 #bdj1 where he shows that the images are arranged in the circumference of a circle4 ,the two images ,d1 d1 says he1 coincide and make but one image4 ,mr4 ,wood has committed the very same mistake in his second ,corollary to ,prop4 ,,xiv41 and his demonstration of that ,corollary is decidedly erroneous4 ,this ,corollary is stated in the following manner3--8,when a 7the angle of the mirrors7 is a measure of #ahj two images coincide10 and it is demonstrated1 that since two images of any object ,x 7,fig4 #b7 must be formed1 viz41 one by each mirror1 and since these two images must be formed at #ahj from the object ,x1 placed between the mirrors1 that is1 at the same point x1 it follows that the two images must coincide4 ,now1 it will appear from the simplest considerations1 that the assumption1 as well as the conclusion1 is erroneous4 ,the image x is seen by the last reflexion from the mirror ,b ,o ,e1 and another image would be seen at x1 if the mirror ,a ,o ,e had extended as far as x2 but as this is impossible1 without covering the part of the mirror ,b ,o ,e1 which gives the first image x1 there can be only one image seen at x4 ,when the object ,x is equidistant from ,a and ,b1 then one-half of the last reflected image x will be formed by the last reflexion from the mirror ,b ,o1 and the other half by the last reflexion from the mirror ,a ,o1 and these two half images will join each other1 and form a whole image at e1 as perfect as any of the rest4 ,in this last case1 when the angle ,a ,o ,b is a little different from an even aliquot part of #cfj1 the eye at ,e will perceive at e an appearance of two incoincident images2 but this arises from the pupil of the eye being partly on one side of ,e and partly on the other2 and1 therefore1 the apparent duplication of the image is removed by looking through a very small aperture at ,e4 ,as the preceding remarks are equally true1 whatever be the inclination of the mirrors1 provided it is an even aliquot part of a circle1 it follows1-- #a4 ,that when ,a ,o ,b is 1 1 1 1 _/1 etc41 of a circle1 the number of reflected images of any object ,x1 is #d - #a1 #f - #a1 #h - #a1 #aj - #a1 #ab - #a4 #b4 ,that when ,x is nearer one mirror than another1 the number of images seen by reflexion from the mirror to which it is nearest will be _/1 _/1 _/1 _/1 _/1 while the number of images formed by the mirror from which ,x is most distant will be _/ - #a1 _/ - #a1 _/ - #a1 _/ - #a2 that is1 an image more always reaches the eye from the mirror nearest ,x1 than from the mirror farthest from it4 #c4 ,that when ,x is equidistant from ,a ,o and ,b ,o1 the number of images which reaches the eye from each mirror is equal1 and is always #d - #a1 #f - #a1 #h - #a1 #aj - #a1 #ab - #a ----- ----- ----- ------ ------ #b #b #b #b #b which are fractional values1 showing that the last image is composed of two half images4 ,when the inclination of the mirrors1 or the angle ,a ,o ,b1 ,fig4 #c1 is an odd aliquot part of a circle1 such as 1 1 1 1 etc41 the different sectors which compose the circular image are formed in the very same manner as has been already described2 but as the number of reflected sectors must in this case always be even1 the line ,o ,e1 where the mirrors join1 will separate the two last reflected sectors1 b ,o e1 a ,o e4 ,hence it follows1-- ,illustration3 ,,fig4 #c4 #a4 ,that when ,a ,o ,b is 1 1 1 1 etc41 of a circle1 the number of reflected images of any object is #c - #a1 #e - #a1 #g - #a1 etc41 and1-- #b4 ,that the number of images which reach the eye from each mirror is #c - #a #e - #a #g - #a1 -----1 -----1 -----1 #b #b #b which are always even numbers4 ,hitherto we have supposed the inclination of the mirrors to be exactly either an even or an odd aliquot part of a circle4 ,we shall now proceed to consider the effects which will be produced when this is not the case4 ,if the angle ,a ,o ,b1 ,fig4 #b1 is made to increase from being an even aliquot part of a circle1 such as th1 till it becomes an odd aliquot part1 such as th1 the last reflected image ,o 1 composed of the two halves ,o e1 ,o e1 will gradually increase1 in consequence of each of the halves increasing2 and when ,a ,o ,b becomes th of the circle1 the sector ,o will become double of ,a ,o ,b1 and ,o e1 ,o e will become each complete sectors1 or equal to ,a ,o ,b4 ,if the angle ,a ,o ,b is made to vary from th to th of a circle1 the last sector ,o will gradually diminish1 in consequence of each of its halves1 ,o e1 ,o e1 diminishing2 and just when the angle becomes th of a circle1 the sector ,o will have become infinitely small1 and the two sectors1 b ,o 1 a ,o 1 will join each other exactly at the line ,o e1 as in ,fig4 #c4 ,,chapter ,,ii4 ,,on ,,the ,,principles ,,of ,,the ,,kaleidoscope1 ,,and ,,the ,,formation ,,of ,,symmetrical ,,pictures ,,by ,,the ,,combination ,,of ,,direct ,,and ,,inverted ,,images4 ,the principles which we have laid down in the preceding chapter must not be considered as in any respect the principles of the ,kaleidoscope4 ,they are merely a series of preliminary deductions1 by means of which the principles of the ,instrument may be illustrated1 and they go no farther than to explain the formation of an apparent circular aperture by means of successive reflexions4 ,all the various forms which nature and art present to us1 may be divided into two classes1 namely1 simple or irregular forms1 and compound or regular forms4 ,to the first class belong all those forms which are called picturesque1 and which cannot be reduced to two forms similar1 and similarly situated with regard to a given point2 and to the second class belong the forms of animals1 the forms of regular architectural buildings1 the forms of most articles of furniture and ornament1 the forms of many natural productions1 and all forms1 in short1 which are composed of two forms1 similar and similarly situated with regard to a given line or plane4 ,now1 it is obvious that all compound forms of this kind are composed of a direct and an inverted image of a simple or an irregular form2 and1 therefore1 every simple form can be converted into a compound or beautiful form1 by skilfully combining it with an inverted image of itself1 formed by reflexion4 ,the image1 however1 must be formed by reflexion from the first surface of the mirror1 in order that the direct and the reflected image may join1 and constitute one united whole2 for if the image is reflected from the posterior surface1 as in the case of a looking-glass1 the direct and the inverted image can never coalesce into one form1 but must always be separated by a space equal to the thickness of the mirror-glass4 ,if we arrange simple forms in the most perfect manner round a centre1 it is impossible by any art to combine them into a symmetrical and beautiful picture4 ,the regularity of their arrangement may give some satisfaction to the eye1 but the adjacent forms can never join1 and must therefore form a picture composed of disunited parts4 ,the case1 however1 is quite different with compound forms4 ,if we arrange a succession of similar forms of this class round a centre1 it necessarily follows that they will all combine into one perfect whole1 in which all the parts either are or may be united1 and which will delight the eye by its symmetry and beauty4 ,in order to illustrate the preceding observations1 we have represented in ,figs4 #d and #e the effects produced by the multiplication of single and compound forms4 ,the line a b c d1 for example1 ,fig4 #d1 is a simple form1 and is arranged round a centre in the same way as it would be done by a perfect multiplying glass1 if such a thing could be made4 ,the consecutive forms are all disunited1 and do not compose a whole4 ,fig4 #e represents the very same simple form1 a b c d1 converted into a compound form1 and then1 as it were1 multiplied and arranged round a centre4 ,in this case every part of the figure is united1 and forms a whole1 in which there is nothing redundant and nothing deficient2 and this is the precise effect which is produced by the application of the ,kaleidoscope to the simple form a b c4 ,illustration3 ,,fig4 #d4 ,illustration3 ,,fig4 #e4 ,the fundamental principle1 therefore1 of the ,kaleidoscope is1 that it produces symmetrical and beautiful pictures1 by converting simple into compound or beautiful forms1 and arranging them1 by successive reflexions1 into one perfect whole4 ,this principle1 it will be readily seen1 cannot be discovered by any examination of the luminous sectors which compose the circular field of the ,kaleidoscope1 and is not even alluded to in any of the propositions given by ,mr4 ,harris and ,mr4 ,wood4 ,in looking at the circular field composed of an even and an odd number of reflexions1 the arrangement of the sectors is perfect in both cases2 but when the number is odd1 and the form of the object simple1 and when the object is not similarly placed with regard to the two mirrors1 a symmetrical and united picture cannot possibly be produced4 ,hence it is manifest1 that neither the principles nor the effects of the ,kaleidoscope could possibly be deduced from any practical knowledge respecting the luminous sectors4 ,in order to explain the formation of the symmetrical picture shown in ,fig4 #e1 we must consider that the simple form m n1 ,fig4 #b1 is seen by direct vision through the open sector ,a ,o ,b1 and that the image n o1 of the object m n1 formed by one reflexion in the sector ,b ,o a1 is necessarily an inverted image4 ,but since the image o p1 in the sector a ,o 1 is a reflected and consequently an inverted image of the inverted image1 m t1 in the sector ,a ,o b1 it follows1 that the whole n o p is an inverted image of the whole n m t4 ,hence the image n o will unite with the image o p1 in the same manner as m n unites with m t4 ,but as these two last unite into a regular form1 the two first will also unite into a regular or compound form4 ,now1 since the half ,o e of the last sector ,o was formerly shown to be an image of the half sector a ,o s1 the line q v will also be an image of the line o z1 and for the same reason the line v p will be an image of t y4 ,but the image v p forms the same angle with ,b ,o or n q that t y does1 and is equal and similar to t y2 and q v forms the same angle with ,a ,o that o z does1 and is equal and similar to o z4 ,hence1 ,o o o q1 and ,o y ,o v1 and therefore q v and v p will form one straight line1 equal and similar to t q1 and similarly situated with respect to ,b ,o4 ,the figure m n o p q t1 therefore1 composed of one direct object1 and several reflected images of that object1 will be symmetrical4 ,as the same reasoning is applicable to every object extending across the aperture ,a ,o ,b1 whether simple or compound1 and to every angle ,a ,o ,b1 which is an even aliquot part of a circle1 it follows1-- #a4 ,that when the inclination of the mirror is an even aliquot part of a circle1 the object seen by direct vision across the aperture1 whether it is simple or compound1 is so united with the images of it formed by repeated reflexions1 as to form a symmetrical picture4 #b4 ,that the symmetrical picture is composed of a series of parts1 the number of which is equal to the number of times that the angle ,a ,o ,b is contained in #cfj4 ,and-- #c4 ,that these parts are alternately direct and inverted pictures of the object2 a direct picture of it being always placed between two inverted ones1 and1 vice versa1 so that the number of direct pictures is equal to the number of inverted ones4 ,when the inclination of the mirrors is an odd aliquot part of #cfj1 such as th1 as shown in ,fig4 #c1 the picture formed by the combination of the direct object and its reflected images is symmetrical only under particular circumstances4 ,if the object1 whether simple or compound1 is similarly situated with respect to each of the mirrors1 as the straight line #a1 #b of ,fig4 #f1 the compound line #c1 #d1 the inclined lines #e1 #f1 the circular object #g1 the curved line #h1 #i1 and the radial line #aj1 ,o1 then the images of all these objects will also be similarly situated with respect to the radial lines that separate the sectors1 and will therefore form a whole perfectly symmetrical1 whether the number of sectors is odd or even4 ,illustration3 ,,fig4 #f4 ,but when the objects are not similarly situated with respect to each of the mirrors1 as the compound line #a1 #b1 ,fig4 #h1 the curved line #c1 #d1 and the straight line #e1 #f1 and1 in general1 as all irregular objects that are presented by accident to the instrument1 then the image formed in the last sector a ,o e1 ,fig4 #g1 by the mirror ,b ,o1 will not join with the image formed in the last sector b ,o e1 by the mirror ,a ,o4 ,in order to explain this with sufficient perspicuity1 let us take the case where the angle is #gb1 or th part of the circle1 as shown in ,fig4 #g4 ,let ,a ,o1 ,b ,o1 be the reflecting planes1 and m n a line1 inclined to the radius which bisects the angle ,a ,o ,b1 so that o m o n2 then m n1 n m1 will be the images formed by the first reflexion from ,a ,o and ,b ,o1 and n m1 m n1 the images formed by the second reflexion2 but by the principles of catoptrics1 o m o m ,o m1 and ,o n ,o n ,o n1 consequently since ,o m is by hypothesis greater than ,o n1 we shall have ,o m greater than ,o n2 that is1 the images m n1 n m1 will not coincide4 ,as ,o n approaches to an equality with ,o m1 ,o n approaches to an equality with ,o m1 and when ,o m ,o n1 we have ,o n ,o m1 and at this limit the images are symmetrically arranged1 which is the case of the straight line #a1 #b in ,fig4 #f4 ,by tracing the images of the other lines1 as is done in ,fig4 #h1 it will be seen1 that in every case the picture is destitute of symmetry when the object has not the same position with respect to the two mirrors4 ,illustration3 ,,fig4 #g4 ,illustration3 ,,fig4 #h4 ,this result may be deduced in a more simple manner1 by considering that the symmetrical picture formed by the ,kaleidoscope contains half as many pairs of forms as the number of times that the inclination of the mirrors is contained in #cfj2 and that each pair consists of a direct and an inverted form1 so joined as to form a compound form4 ,now the compound form made up by each pair obviously constitutes a symmetrical picture when multiplied any number of times1 whether even or odd2 but if we combine so many pair and half a pair1 two direct images will come together1 the half pair cannot possibly join both with the direct and the inverted image on each side of it1 and therefore a symmetrical whole cannot be obtained from such a combination4 ,from these observations we may conclude1-- #a4 ,that when the inclination of the mirrors is an odd aliquot part of a circle1 the object seen by direct vision through the aperture unites with the images of it formed by repeated reflexions1 and forms a complete and symmetrical picture1 only in the case when the object is similarly situated with respect to both the mirrors2 the two last sectors forming1 in every other position of the object1 an imperfect junction1 in consequence of these being either both direct or both inverted pictures of the object4 #b4 ,that the series of parts which compose the symmetrical as well as the unsymmetrical picture1 consists of direct and inverted pictures of the object1 the number of direct pictures being always equal to half the number of sectors increased by one1 when the number of sectors is #e1 #i1 #ac1 #ag1 #ba1 etc41 and the number of inverted pictures being equal to half the number of sectors diminished by one1 when the number of sectors is #c1 #g1 #aa1 #ae1 #ai1 etc41 and vice versa4 ,hence1 the number of direct pictures of the object must always be odd1 and the number of inverted pictures even1 as appears from the following table3-- ,inclination ,number ,number of ,number of of the of ,inverted ,direct ,mirrors4 ,sectors4 ,pictures4 ,pictures4 #abj #c #b #a #gb #e #b #c #ea_/ #g #d #c #dj #i #d #e #cb_/ #aa #f #e #bg_/ #ac #f #g #bd #ae #h #g #ba_/ #ag #h #i #ah_/ #ai #aj #i #ag #ba #aj #aa #c4 ,that when the number of sectors is #c1 #g1 #aa1 #ae1 #ai1 etc41 the two last sectors are inverted2 and when the number is #e1 #i1 #ac1 #ag1 #ba1 etc41 the two last sectors are direct4 ,when the inclination of the mirrors is not an aliquot part of #cfj1 the images formed by the last reflexions do not join like every other pair of images1 and therefore the picture which is created must be imperfect4 ,it has already been shown at the end of ,chap4 ,i4 that when the angle of the mirrors becomes greater than an even or less than an odd aliquot part of a circle1 each of the two incomplete sectors which form the last sector becomes greater or less than half a sector4 ,the image of the object comprehended in each of the incomplete sectors must therefore be greater or less than the images in half a sector2 that is1 when the last sector ,o 1 ,fig4 #b1 is greater than ,a ,o ,b1 the part q v in one half must be the image of more than o z1 and v p the image of more than t y1 and vice versa1 when ,o is less than ,a ,o ,b4 ,hence it follows that the symmetry is imperfect from the image in the last sector being greater or less than the other images4 ,but besides this cause of imperfection in the symmetry1 there is another1 namely1 the disunion of the two images q v and v p4 ,the angles ,o q v and ,o o p are obviously equal1 and also the angles ,o p v1 ,o p o2 but since the angle ,o 1 or q ,o p1 is by hypothesis greater or less than p ,o o1 it follows that the angles of the triangle q ,o p are either greater or less than two right angles1 because they are greater or less than the three angles of the triangle p ,o o4 ,but as this is absurd1 the lines q v1 v p1 cannot join so as to form one straight line1 and therefore the completion of a perfect figure by means of two mirrors1 whose inclination is not an aliquot part of a circle1 is impossible4 ,when the angle ,o is greater than p ,o o1 or ,a ,o ,b1 the lines q v1 v p1 will form a re-entering angle towards ,o1 and when it is less than ,a ,o ,b1 the same lines will form a salient angle towards ,o4 ,,chapter ,,iii4 ,,on ,,the ,,effects ,,produced ,,by ,,the ,,motion ,,of ,,the ,,object ,,and ,,the ,,mirrors4 ,hitherto we have considered both the object and the mirrors as stationary1 and we have contemplated only the effects produced by the union of the different parts of the picture4 ,the variations1 however1 which the picture exhibits1 have a very singular character1 when either the objects or the mirrors are put in motion4 ,let us1 first1 consider the effects produced by the motion of the object when the mirrors are at rest4 ,illustration3 ,,fig4 #i4 ,if the object moves from ,x to ,o1 ,fig4 #i1 in the direction of the radius1 all the images will likewise move towards ,o1 and the patterns will have the appearance of being absorbed or extinguished in the centre4 ,if the motion of the object is from ,o to ,x1 the images will also move outwards in the direction of the radii1 and the pattern will appear to develop itself from the centre ,o1 and to be lost or absorbed at the circumference of the luminous field4 ,the objects that move parallel to ,x ,o will have their centre of development1 or their centre of absorption1 at the point in the lines ,a ,o1 ,b ,o1 a ,o1 b ,o1 etc4 where the direction in which the images move cuts these lines4 ,when the object passes across the field in a circle concentric with ,a ,b1 and in the direction ,a ,b1 the images in all the sectors formed by an even number of reflexions will move in the same direction ,a ,b1 namely1 in the direction b1 a 2 while those that have been formed by an odd number of reflexions will move in an opposite direction1 namely1 in the directions a ,b1 ,a b4 ,hence1 if the object moves from ,a to ,b1 the points of absorption will be in the lines ,b ,o1 ,o1 and b ,o1 and the points of development in the lines ,a ,o1 a ,o1 and ,o1 and vice versa1 when the motion of the object is from ,b to ,a4 ,if the object moves in an oblique direction m n1 the images will move in the directions m t1 o n1 o p1 q t1 q p1 and m1 o1 q1 will be the centres of development1 and n1 p1 t1 the centres of absorption2 whereas1 if the object moves from n to m1 these centres will be interchanged4 ,these results are susceptible of the simplest demonstration1 by supposing the object in one or two successive points of its path m n1 and considering that the image must be formed at points similarly situated behind the mirrors2 the line passing through these points will be the path of the image1 and the order in which the images succeed each other will give the direction of their motion4 ,hence1 we may conclude in general1 #a4 ,that when the path of the object cuts both the mirrors ,a ,o and ,b ,o like m n1 the centre of absorption will be in the radius passing through the section of the mirror to which the object moves1 and in every alternate radius2 and that the centre of development will be in the radius passing through the section of the mirror from which the object moves1 and in all the alternate radii3 and1 #b4 ,that when the path of the object cuts any one of the mirrors and the circumference of the circular field1 the centre of absorption will be in all the radii which separate the sectors1 and the centre of development in the circumference of the field1 if the motion is towards the mirror1 but vice versa if the motion is towards the circumference4 ,when the objects are at rest1 and the ,kaleidoscope in motion1 a new series of appearances is presented4 ,whatever be the direction in which the ,kaleidoscope moves1 the object seen by direct vision must always be stationary1 and it is easy to determine the changes which take place when the ,kaleidoscope has a progressive motion over the object4 ,a very curious effect1 however1 is observed when the ,kaleidoscope has a rotatory motion round the angular point1 or rather round the common section of the two mirrors4 ,the picture created by the ,instrument seems to be composed of two pictures1 one in motion round the centre of the circular field1 and the other at rest4 ,the sectors formed by an odd number of reflexions are all in motion in the same direction as the ,kaleidoscope1 while the sector seen by direct vision1 and all the sectors formed by an even number of reflexions1 are at rest4 ,in order to understand this1 let ,m1 ,fig4 #aj1 be a plane mirror1 and ,a an object whose image is formed at a1 so that a ,m ,a ,m4 ,let the mirror ,m advance to ,n1 and the object ,a1 which remains fixed1 will have its image b formed at such a distance behind ,n1 that b ,n ,a ,n2 then it will be found that the space moved through by the image is double the space moved through by the mirror2 that is1 a b #b ,m ,n4 ,since ,m ,n ,a ,m - ,a ,n1 and since ,a ,m a ,m1 and ,a ,n b ,n1 we have ,m ,n a ,m - b ,n2 and adding ,m ,n or its equal b ,m b ,n to both sides of the equation1 we obtain #b,m ,n a ,m - b ,n b ,n b ,m2 but -b ,n b,n #j1 and a ,m b ,m a b2 hence #b,m ,n a b4 ,this result may be obtained otherwise1 by considering1 that if the mirror ,m advances one inch towards ,a1 one inch is added to the distance of the image a1 and one subtracted from the distance of the object2 that is1 the difference of these distances is now two inches1 or twice the space moved through by the mirror2 but since the new distance of the object is equal to the distance of the new image1 the difference of these distances1 which is the space moved through by the image1 must be two inches1 or twice the space described by the mirror4 ,illustration3 ,,fig4 #aj4 ,let us now suppose that the object ,a advances in the same direction as the mirror1 and with twice its velocity1 so as to describe a space ,a #b ,m ,n a b1 in the same time that the mirror moves through ,m ,n1 the object being at when the mirror is at ,n4 ,then1 since ,a a b and b ,n ,a ,n1 the whole ,n is equal to the whole a ,n1 that is1 a will still be the place of the image4 ,hence it follows1 that if the object advances in the same direction as the mirror1 but with twice its velocity1 the image will remain stationary4 ,illustration3 ,,fig4 #aa4 ,if the object ,a moves in a direction opposite to that of the mirror1 and with double its velocity1 as is shown in ,fig4 #aa2 then1 since b would be the image when ,a was stationary1 and when ,m had moved to ,n1 in which case a b #b ,m ,n1 and b the image when ,a had advanced to through a space ,a #b ,m ,n1 we have b ,n ,a ,n1 and b ,n ,n1 and1 therefore1 b b ,a ,n - ,n ,a #b ,m ,n1 and a b b b or its equal a b #d ,m ,n4 ,hence it follows1 that when the object advances towards the mirror with twice its velocity1 the image will move with four times the velocity of the mirror4 ,if the mirror ,m moves round a centre1 the very same results will be obtained from the very same reasoning1 only the angular motion of the mirror and the image will then be more conveniently measured by parts of a circle or degrees4 ,illustration3 ,,fig4 #ab4 ,now1 in ,fig4 #ab1 let ,x be a fixed object1 and ,a ,o1 ,b ,o1 two mirrors placed at an angle of #fj and moveable round ,o as a centre4 ,when the eye is applied to the end of the mirrors 7or at ,e1 ,fig4 #a71 the fixed object ,x1 ,fig4 #ab1 seen by direct vision will1 of course1 be stationary1 while the mirrors describe an arch ,x of #aj for example2 but since ,a ,o has approached ,x by #aj1 the image of ,x formed behind ,a ,o must have approached ,x by #bj1 and consequently moves with twice the velocity in the same direction as the mirrors4 ,in like manner1 since ,b ,o has receded #aj from ,x1 the image of ,x formed by ,b ,o must have receded #bj from ,x1 and consequently must have moved with twice the velocity in the same direction as the mirrors4 ,now1 the image of ,x in the sector b ,o is1 as it were1 an image of the image in ,b ,o a reflected from ,a ,o4 ,but the image in ,b ,o a advances in the same direction as the mirror ,a ,o and with twice its velocity1 hence the image of it in the sector b ,o will be stationary4 ,in like manner it may be shown1 that the image in the sector a ,o will be stationary4 ,since ,o e is an image of b ,o r reflected from the mirror ,b ,o1 and since all images in that sector are stationary1 the corresponding images in ,o e will move in the same direction as the mirrors2 and for the same reason the images in the other half-sector ,o e will move in the same direction2 hence1 the image of any object formed in the last sector ,o will move in the same direction1 and with the same velocity as the images in the sectors ,a ,o b1 ,b ,o a4 ,by a similar process of reasoning1 the same results will be obtained1 whatever be the number of the sectors1 and whether the angle ,a ,o ,b be the even or the odd aliquot part of a circle4 ,hence we may conclude1 #a4 ,that during the rotatory motion of the mirrors round ,o1 the objects in the sector seen by direct vision1 and all the images of these objects formed by an even number of reflexions are at rest4 #b4 ,that all the images of these objects1 formed by an odd number of reflexions1 move round ,o in the same direction as the mirrors1 and with an angular velocity double that of the mirrors4 #c4 ,that when the angle ,a ,o ,b is an even aliquot part of a circle1 the number of moving sectors is equal to the number of stationary sectors1 a moving sector being placed between two stationary sectors1 and vice versa4 #d4 ,that when the angle ,a ,o ,b is an odd aliquot part of a circle1 the two last sectors adjacent to each other are either both in motion or both stationary1 the number of moving sectors being greater by one when the number of sectors is #c1 #g1 #aa1 #ae1 etc41 and the number of stationary sectors being greater by one when the number of sectors is #e1 #i1 #ac1 #ag1 etc4 ,and1 #e4 ,that as the moving sectors correspond with those in which the images are inverted1 and the stationary ones with those in which the images are direct1 the number of each may be found from the table given in page #bd4 ,when one of the mirrors1 ,a ,o1 is stationary1 while the other1 ,b ,o1 is moved round1 and so as to enlarge the angle ,a ,o ,b1 the object ,x1 and the image of it seen in the stationary mirror ,a ,o1 remain at rest1 but all the other images are in motion receding from the object ,x1 and its stationary image2 and when ,b ,o moves towards ,a ,o1 so as to diminish the angle ,a ,o ,b1 the same effect takes place1 only the motion of the images is towards the object ,x1 on one side1 and towards its stationary image on the other4 ,these images will obviously move in pairs2 for1 since the fixed object and its stationary image are at an invariable distance1 the existence of a symmetrical arrangement1 which we have formerly proved1 requires that similar pairs be arranged at equal distances round ,o1 and each of the images of these pairs must be stationary with regard to the other4 ,now1 as the fixed object is placed in the sector ,a ,o ,b1 and its stationary image in the sector ,a ,o b1 it will be found that in the semicircle ,m b e1 containing the fixed mirror1 the #ast reflected image and direct object1 #b;d #c;d reflected image #dth #eth are stationary with #fth #gth respect to each other4 #hth #ith while in the same semicircle ,m b e1 the #ast reflected image and #b;d reflected image #c;d #dth #eth #fth are movable with #gth #hth respect to each other4 #ith #ajth ,on the other hand1 in the semicircle ,m a e1 containing the movable mirror1 the phenomena are reversed1 the images which were formerly stationary with respect to each other being now movable1 and vice versa4 ,in considering the velocity with which each pair of images revolves1 it will be readily seen that the pair on each side1 and nearest the fixed pair1 will have an angular velocity double that of the mirror ,b ,o2 the next pair on each side will have a velocity four times as great as that of the mirror2 the next pair will have a velocity eight times as great1 and the next pair a velocity sixteen times as great as that of the mirror1 the velocity of any pair being always double the velocity of the pair which is adjacent to it on the side of the fixed pair4 ,the reason of this will be manifest1 when we recollect what has already been demonstrated1 that the velocity of the image is always double that of the mirror1 when the mirror alone moves towards the object1 and quadruple that of the mirror when both are in motion1 and when the object approaches the mirror with twice the velocity4 ,when ,b ,o moves from ,a ,o1 the image in the sector ,b ,o a moves with twice the velocity of the mirror2 but since the image in b ,o is an image of the image in ,b ,o a reflected from the fixed mirror ,a ,o1 it also will move with the same velocity1 or twice that of the mirror ,b ,o4 ,again1 the image in the sector a ,o 1 being a reflexion of the stationary image in ,a ,o b from the moving mirror1 will itself move with double the velocity of the mirror4 ,but the image in the next sector ,o is a reflexion of the image in b ,o from the moving mirror ,b ,o2 and as this latter image has been shown to move in the direction b 1 with twice the velocity of the mirror ,b ,o1 while the mirror ,b ,o itself moves towards the image1 it follows that the image in ,o will move with a velocity four times that of the mirror4 ,the same reasoning may be extended to any number of sectors1 and it will be found that in the semicircle ,m b e1 containing the fixed mirror1 ,the #b and #c #b times the images #d and #e reflexions1 move with #d velocity of formed #f and #g #h the mirror2 by #h and #i #af whereas in the semicircle ,m a e1 containing the movable mirror1-- ,the #a and #b #b times the images #c and #d reflexions1 move with #d velocity of formed #e and #f #h the mirror2 by #g and #h #af a progression which may be continued to any length4 ,before concluding this chapter1 it may be proper to mention a very remarkable effect produced by moving the two plain mirrors along one of two lines placed at right angles to each other4 ,when the aperture of the mirrors is crossed by each of the two lines1 the figure created by reflexion consists of two polygons with salient and re-entering angles4 ,by moving the mirrors along one of the lines1 so that it may always cross the aperture at the same angle1 and at the same distance from the angular point1 the polygon formed by this line will remain stationary1 and of the same form and magnitude2 but the polygon formed by the other line1 at first emerging from the centre1 will gradually increase till its salient angles touch the re-entering angles of the stationary polygon2 the salient angles becoming more acute1 will enclose the apices of the re-entering angles of the stationary polygon1 and at last the polygon will be destroyed by truncations from its salient angles4 ,when the lines cross each other at a right angle1 the salient angles of the opening polygon can never touch the salient angles of the stationary polygon1 but always its re-entering angles4 ,if the lines1 however1 form a less angle than the complement of the angle formed by the mirrors1 then the salient angles of the opening polygon may touch the salient angles of the stationary polygon1 by placing the mirrors so as to form re-entering angles in the polygon4 ,when the lines form an angle between #ij and the complement of the angle formed by the mirrors1 the salient angle of the opening polygon may be made to touch the salient angles of the stationary one1 but in this case the stationary polygon can have no re-entering angles4 ,the preceding effects are finely exemplified by the use of a ,kaleidoscope with a draw-tube and lens described in ,chapter ,x41 and by employing the vertical and horizontal bars of a window1 which may be set at different angles1 by viewing them in perspective4 ,,chapter ,,iv4 ,,on ,,the ,,effects ,,produced ,,upon ,,the ,,symmetry ,,of ,,the ,,picture ,,by ,,varying ,,the ,,position ,,of ,,the ,,eye4 ,it has been taken for granted in the preceding chapters1 not only that the object seen by direct vision is in a state of perfect junction with the images of it formed by reflexion2 but that the object and its images have the same apparent magnitude1 and nearly the same intensity of light4 ,as these conditions are absolutely necessary to the production of symmetrical and beautiful forms1 and may be all effected by particular methods of construction1 we shall proceed to investigate the principles upon which these methods are founded1 in so far as the position of the eye is concerned4 ,when any object is made to touch a common looking-glass in one or more points1 the reflected image does not touch the object in these points1 but is always separated from it by a space equal to the thickness of the glass1 in consequence of the reflexion being performed by the posterior surface of the mirror4 ,the image and the object must therefore be always disunited2 and as the interval of separation must be interposed between all the reflected images1 there cannot possibly exist that union of forms which constitutes the very essence of symmetry4 ,in mirror-glass there is a series of images reflected from the first surface1 which unite perfectly with the object1 and with one another4 ,when the angles of incidence are not great1 this series of images is very faint1 and does not much interfere with the more brilliant images formed by the metallic surface4 ,as the angles of incidence increase1 the one series of images destroys the effect of the other1 from their overlapping or imperfect coincidence--an effect which increases with the thickness of the glass2 but when the reflexions are made at very oblique incidences1 the images formed by the metallic surface become almost invisible1 while those formed by the first surface are as brilliant and nearly as perfect as if the effect of the posterior metallic surface had been entirely removed4 ,in the following observations1 therefore1 it is understood that the images are reflected either from a polished metallic surface1 or from the first surface of glass4 ,illustration3 ,,fig4 #ac4 ,in order to explain the effects produced upon the symmetry of the picture by a variation in the position of the eye1 we must suppose the object to be placed at a small distance from the end of the mirror4 ,this position is represented in ,fig4 #ac1 where ,a ,e is a section of the mirror in the direction of its length2 ,m ,n ,o ,p an object placed at a distance from the extremity ,a of the mirror1 and m n o p1 its image seen by an eye to the right hand of ,e1 and which1 by the principles of catoptrics1 will be similar to the object and similarly situated with respect to the mirror ,a ,e4 ,now1 if the eye is placed at 1 it will see distinctly the whole object ,m ,n ,o ,p1 but it will only see the portion n r s o of the image cut off by drawing the line ,a r through the extremity of the mirror1 so that there cannot be a symmetrical form produced by observing at the same time the object ,m ,n ,o ,p and this portion of its image2 and the deviation from symmetry will be still greater1 if ,m ,n ,o ,p is brought nearer the line ,b ,a1 for the image m n o p will be entirely included between the lines ,a r and ,a ,b1 so that no part whatever of the image will be visible to an eye at 4 ,as the eye of the observer moves from to e1 the line ,a r will move into the position e ,a x1 and when it has reached the point e1 the whole of the image m n o p will be visible4 ,the symmetry1 therefore1 arising from the simultaneous contemplation of the object and its image will be improved2 but it will still be imperfect1 as the image will appear to be distant from the plane of the mirror1 only by the space m x1 while the distance of the object is ,m x4 ,as the eye moves from e to ,e1 the line e ,a x will move into ,e ,a ,b1 and the object and its image will seem to be placed at the equal distances ,m ,b1 m ,b from the plane of the mirror1 and will therefore form a symmetrical combination4 ,when the object is moved1 and arrives at ,b ,a the image will touch the object1 and they will form one perfect and united whole1 whatever be the shape of the line ,m ,p4 ,hence we conclude1 that when an object is placed at a little distance from the extremity of a plain mirror1 its image formed by reflexion from the mirror cannot unite with the object in forming a conjoined and symmetrical picture1 unless the eye is in the plane of the mirror4 ,illustration3 ,,fig4 #ad4 ,when two mirrors1 therefore1 are combined1 as in ,fig4 #ad1 the eye must be in the plane of both1 in order that the object and its image may have a symmetrical coincidence1 and therefore it must be at the point ,e where the two planes cut each other4 ,the necessity of this position1 and the effects of any considerable deviation from it1 will be understood from ,fig4 #ad1 where ,a ,o ,b is the angle formed by the mirrors1 and ,m ,n the place of the object4 ,then if the eye is placed at 1 the aperture ,a ,o ,b will be projected into a b upon a plane passing through ,m ,n and at right angles to ,e ,o2 but the orthographic projection of ,a ,b ,o upon the same plane is ,a ,b ,o1 or1 what is the same thing1 the reflecting surfaces of which ,a ,o1 ,b ,o are sections1 will1 when prolonged1 cut the plane passing through ,m ,n in the lines ,a ,o1 ,b ,o2 hence1 rays from the objects situated between ,a ,o ,b and a b cannot fall upon the mirrors ,a ,o ,e1 ,b ,o ,e1 or images of these objects cannot be formed by the mirrors4 ,the images1 therefore1 in the different sectors formed by reflexion round ,o as a centre1 cannot include any objects without ,a ,o ,b2 and since the eye at sees all the objects between ,a ,o ,b and a b1 there can be no symmetry and uniformity in the picture formed by the combination of such an object with the images in the sectors4 ,when the eye descends to e1 the aperture ,a ,o ,b is projected into a o b1 which approaches nearer to ,a ,o ,b2 but for the reasons already assigned1 the symmetry of the picture is still imperfect4 ,as the eye descends1 the lines a o1 b o approach to ,a ,o1 ,b ,o1 and when the eye arrives at ,e1 a point in the plane of both the reflecting surfaces1 the projection of the aperture ,a ,o ,b will be ,a ,o ,b1 and the images in all the sectors will be exactly similar to the object presented to the aperture4 ,hence we conclude in general1 that when an object is placed at any distance before two mirrors inclined at an angle1 which is an even aliquot part of #cfj1 the symmetry of the picture is perfect1 when the eye1 considered as a mathematical point1 is placed at ,e1 and that the deviation from symmetry increases as the eye recedes from ,e towards 4 ,if the object were a mathematical surface1 all the parts of which were in contact with the extremities ,a ,o1 ,b ,o of the mirrors1 then it is easy to see that the symmetry of the picture will not be affected by the deviation of the eye from the point ,e1 and1 in consequence of the enlargement of the sector1 seen by direct vision4 ,the symmetry of the picture1 is1 however1 affected in another way1 by the deviation of the eye from the point ,e4 ,we have already seen1 that1 in order to possess perfect symmetry1 an object must consist of two parts in complete contact1 one of which is an inverted image of the other4 ,but in order that an object possessing perfect symmetry may appear perfectly symmetrical1 four conditions are required4 ,the two halves of the object must be so placed with respect to the eye of the observer1 that no part of the one half shall conceal any part of the other2 that whatever parts of the one half are seen1 the corresponding parts of the other must also be seen2 and that the corresponding parts of both halves1 and both halves themselves1 must subtend the same angle at the eye4 ,when we stand before a looking-glass1 and hold out one hand so as to touch it1 the hand will be found to conceal various parts of its image2 and1 in some positions of the eye1 the whole image will be concealed1 so that a symmetrical picture cannot possibly be formed by the union of the two4 ,if the eye is placed so obliquely to the looking-glass1 that the hand no longer interferes with its image1 it will still be seen1 that parts of the hand which are not directly visible1 are visible in its reflected image1 and therefore that a symmetrical picture cannot be created by the union of two parts apparently dissimilar4 ,if the eye of the observer is placed near his hand1 so that he can see distinctly both the hand and its image1 the angular magnitude of his hand is much greater than that of its image2 and therefore1 when the two are united1 they cannot form a symmetrical object4 ,this will be better understood from ,fig4 #ae4 ,when the eye is placed at 1 the object ,m ,n ,o ,p is obviously nearer than its image m n o p1 and must therefore appear larger2 and this difference in their apparent magnitudes will increase as the eye rises above the plane of the mirror ,a ,e4 ,as the eye approaches to ,e1 the distances of the object and its image approach to an equality2 and when the eye is at ,e1 the object ,m ,n ,o ,p1 and its image m n o p1 are situated at exactly the same distance from the eye1 and therefore have the same angular magnitude4 ,hence it follows1 that when they are united1 they will form a perfectly symmetrical combination4 ,illustration3 ,,fig4 #ae4 ,when the eye is placed in the plane of both the mirrors1 the field of view arising from the multiplication of the sector ,a ,o ,b1 ,fig4 #ad1 will be perfectly circular2 but as the eye rises above the plane of both the mirrors1 this circle will become a sort of ellipse1 becoming more and more eccentric as the eye comes in front of the mirrors1 or rises in the direction ,e 4 ,if the observer were infinitely distant1 these figures would be correct ellipses2 but as the eye1 particularly when the mirrors are broad1 must be nearly twice as far from the last reflected sector as from the sector seen by direct vision1 the field of view1 and consequently every pattern which it contains1 must be distorted and destitute of beauty1 from this cause alone4 ,hitherto we have alluded only to symmetry of form1 but it is manifest that before the union of two similar forms can give pleasure to the eye1 there must be also a symmetry of light4 ,if the object ,m ,n ,o ,p1 ,fig4 #ae1 is white1 and its image m n o p black1 they cannot possibly form1 by their combination1 an agreeable picture4 ,as any considerable difference in the intensity of the light will destroy the beauty of the patterns1 it becomes a matter of indispensable importance to determine the position of the eye1 which will give the greatest possible uniformity to the different images of which the picture is composed4 ,it has been ascertained by the accurate experiments of ,bouguer1 that when light is reflected perpendicularly from good plate glass1 only #be rays are reflected out of #ajjj2 that is1 the intensity of the light of any object seen perpendicularly in plate glass1 is to the intensity of the light of its image as #ajjj to #be1 or as #dj is to #a4 ,when the angle of incidence is #fj1 the number of reflected rays is #aab1 and the intensities are nearly as #i to #a4--,when the angle of incidence is #hg1 the number of reflected rays is #ehd1 and the intensities are nearly as #ag to #aj1 so that the luminousness of the object and its image approach rapidly to an equality4 ,it is therefore clear1 that1 in order to have the greatest uniformity of light in the different images which compose the figure1 the eye must be placed as nearly as possible in the plane of both the mirrors1 that is1 as near as possible to the angular point4--,but as it is impracticable to have the eye exactly in the plane of both mirrors1 the images formed by one reflexion must always be less bright than the direct object1 even at the part nearly in contact with the object4 ,the second and third reflexions1 etc41 where the rays fall with less obliquity1 will be still darker than the first2 though this difference will not be very perceptible when the inclination of the mirrors is #cj or upwards1 and the eye placed in the position already described4 ,it is a curious circumstance1 that the positions of the eye which are necessary to effect a complete union of the images--to represent similar parts of the object and its images--to observe the object and its image under the same angular magnitude1 and to give a maximum intensity of light to the reflected images--should all unite in the same point4 ,had this not been the case1 the construction of the ,kaleidoscope would have been impracticable1 and hence it will be seen how vain is the attempt to produce beautiful and symmetrical forms from any combination of plain mirrors in which this position of the eye is not a radical and essential principle4 ,,chapter ,v4 ,,on ,,the ,,effects ,,produced ,,upon ,,the ,,symmetry ,,of ,,the ,,picture ,,by ,,varying ,,the ,,position ,,of ,,the ,,object4 ,having ascertained the proper position of the eye1 we shall now proceed to determine the position of the object4 ,if the object is placed within the reflectors at any point ,d1 ,fig4 #af1 between their object end ,o1 and their eye-end ,e1 a perfectly symmetrical picture will obviously be formed from it2 but the centre of this picture will not be at ,o1 the centre of the luminous sectors1 but at the point ,d1 where the object is placed1 or its projection d1 so that we shall have a circular luminous field enclosing an eccentric circular pattern4 ,such a position of the object is therefore entirely unfit for the production of a symmetrical picture1 unless the object should be such as wholly to exclude the view of the circular field1 formed by the reflected images of the aperture ,a ,o ,b4 ,as the point ,d approaches to ,o1 the centre d of the symmetrical picture will approach to ,o1 and when ,d coincides with ,o1 the centre of the picture will be at ,o1 and all the images of the object placed in the plane ,a ,o ,b will be similarly disposed in all the sectors which compose the circular field of view4 ,hence we may conclude1 that a perfectly symmetrical pattern cannot be exhibited in the circular field of view1 when the object is placed between ,o and ,e1 or anywhere within the reflectors4 ,if the eye could be placed exactly at the angular point ,e1 so that every point of the line ,e ,o should be projected upon ,o1 then the images would be symmetrically arranged round ,o2 but this is obviously impossible1 for the object would1 in such circumstances1 cease to become visible when this coincidence took place4 ,but independent of the eccentricity of the pattern1 the position of the object within the mirrors prevents that motion of the objects1 without which a variation of the pattern cannot be produced4 ,an object between the reflectors must always be exposed to view1 and we cannot restrict our view to one-half1 one-third1 or one-fourth of it1 as when we have it in our power to move the objects across the aperture1 or the aperture over the objects4 ,illustration3 ,,fig4 #af4 ,another evil arising from the placing of the objects within the mirrors1 is1 that we are prevented from giving them the proper degree of illumination which is so essential to the distinctness of the last reflexions4 ,the portions of the mirrors1 too1 beyond the objects1 or those between ,d and ,o1 are wholly unnecessary1 as they are not concerned in the formation of the picture4 ,hence it follows1 that the effects of the ,kaleidoscope cannot be produced by any combination of mirrors1 in which the objects are placed within them4 ,let us now consider what will happen1 by removing the object beyond the plane passing through ,a ,o ,b4 ,in this case the pattern will lose its symmetry from two causes4 ,in the first place1 it is manifest1 as already explained1 that as the eye is necessarily raised a little above the point ,e1 and also above the planes ,a ,o ,e1 ,b ,o ,e1 it must see through the aperture ,a ,o ,b a portion of the object situated below both of these planes4 ,this part of the object will therefore appear to project beyond the point1 or below the plane where the direct and reflected images meet4 ,if we suppose1 therefore1 that all the reflected images were symmetrical1 the whole picture would lose its symmetry in consequence of the irregularity of the sector ,a ,o ,b seen by direct vision4 ,but this supposition is not correct2 for since the image m n1 ,fig4 #c1 seen by direct vision does not coincide with the first reflected images mn1 nm1 it is clear that all the other images will likewise be incoincident1 and1 therefore1 that the figure formed by their combination must lose its symmetry1 and1 consequently1 its beauty4 ,as the eye must necessarily be placed above a line perpendicular to the plane ,a ,b ,o at the point ,o1 it will see a portion of the object situated below that perpendicular continued to the object4 ,thus1 in ,fig4 #af1 if the eye is placed at e above ,e1 and if ,m ,n is the object placed at the distance ,p ,o1 then the eye at e will observe the portion ,p ,o of the object situated below the axis ,p ,o ,e1 and this portion1 which may be called the aberration1 will vary with the height ,e e of the eye1 and with the distance ,o ,p of the object4 ,illustration3 ,,fig4 #ag4 ,let us now suppose ,e e and ,o ,p to be constant1 and that a polygonal figure is formed by some line placed at the point ,q of the object ,m ,n4 ,then if ,p ,q is very great compared with ,p ,o1 the polygonal figure will be tolerably regular1 though all its angles will exhibit an imperfect junction1 and its lower half will be actually1 though not very perceptibly1 less than its upper half4 ,but if ,q approaches to ,p1 ,p ,o remaining the same1 so that ,p ,o bears a considerable ratio to ,p ,q1 then the polygonal figure will lose all symmetry1 the upper sectors being decidedly the largest1 and the lowest sectors the smallest4 ,when ,q arrives near ,p1 the aberration becomes enormous1 and the figure is so distorted1 that it can no longer be recognised as a polygon4 ,the deviation from symmetry1 therefore1 arising from the removal of the object from the extremity of the reflectors1 increases as the object approaches to the centre of the luminous sectors or the circular field1 and this deviation becomes so perceptible1 that an eye accustomed to observe and admire the symmetry of the combined objects1 will instantly perceive it1 even when the distance of the object or ,p ,o is less than the twentieth part of an inch4 ,when the object is very distant1 the defect of symmetry is so enormous1 that though the object is seen by direct vision1 and in some of the sectors1 it is entirely invisible in the rest4 ,the principle which we have now explained is of primary importance in the construction of the ,kaleidoscope1 and it is only by a careful attention to it that the instrument can be constructed so as to give to an experienced and fastidious eye that high delight which it never fails to derive from the exhibition of forms perfectly symmetrical4 ,from these observations it follows1 that a picture possessed of mathematical symmetry1 cannot be produced unless the object is placed exactly at the extremity of the reflectors1 and that even when this condition is complied with1 the object itself must consist of lines all lying in the same plane1 and in contact with the reflectors4 ,hence it is obvious1 that objects whose thickness is perceptible1 cannot give mathematically symmetrical patterns1 for one side of them must always be at a certain distance from ,o4 ,the deviation in this case is1 however1 so small1 that it can scarcely be perceived in objects of moderate thickness4 ,in the simple form of the ,kaleidoscope1 the production of symmetrical patterns is limited to objects which can be placed close to the aperture ,a ,o ,b2 but it will be seen in the sequel of this treatise1 that this limitation may be removed by an optical contrivance1 which extends indefinitely the use and application of the instrument4 ,,chapter ,,vi4 ,,on ,,the ,,intensity ,,of ,,the ,,light ,,in ,,different ,,parts ,,of ,,the ,,field1 ,,and ,,on ,,the ,,effects ,,produced ,,by ,,varying ,,the ,,length ,,and ,,breadth ,,of ,,the ,,reflectors4 ,when we look through a ,kaleidoscope in which the mirrors are placed at an angle of #ah or #bb1 the eye will perceive a very obvious difference in the intensity of the light in different parts of the field4 ,if the inclination of the mirrors be about #cj1 and the eye properly placed near the angular point1 the intensity of the light is tolerably uniform2 and a person who is unaccustomed to the comparison of different lights1 will find it extremely difficult to distinguish the direct sector from the reflected ones4 ,this difficulty will be still greater if the mirrors are made of finely polished steel1 or of the best speculum metal1 and the observer will not hesitate in believing that he is looking through a tube whose diameter is equal to that of the circular field4 ,this approximation to uniformity in the intensity of the light in all the sectors1 which arises wholly from the determination of the proper position of the eye1 is one of the most curious and unexpected properties of the ,kaleidoscope1 and is one which could not have been anticipated from any theoretical views1 or from any experimental results obtained from the ancient mode of combining plain mirrors4 ,it is that property1 too1 which gives it all its value2 for1 if the eye observed the direct sector with its included objects distinguished from all the rest by superior brilliancy1 not only would the illusion vanish1 but the picture itself would cease to afford pleasure1 from the want of symmetry in the light of the field4 ,illustration3 ,,fig4 #ah4 ,before we proceed to investigate the effects produced by a variation in the length of the reflecting planes1 it will be necessary to consider the variation of the intensity of the light in different parts of the reflected sectors4 ,in the direct sector ,a ,o ,b1 ,fig4 #b1 the intensity of the light is uniform in every part of its surface2 but this is far from being the case in the images formed by reflexion4 ,in ,fig4 #ag1 take any two points m1 o1 and draw the lines m n1 o p1 perpendicular to ,a ,o1 and meeting ,o in n and p4 ,let ,o ,e1 ,fig4 #ah1 be a section of the reflector ,a ,o seen edgewise1 and let ,o p1 ,o n1 be taken equal to the lines m n1 o p1 or the height of the points n1 p1 above the plane of the reflector ,a ,o4 ,make ,o ,r to ,r ,e as ,o p is to ,e e the constant height of the eye above the reflecting plane1 and ,o r to r ,e as ,o n to ,e e1 and the points ,r1 r1 will be the points of incidence of the rays issuing from p and n2 for in this case ,o ,r p ,e ,r e1 and ,o r n ,e r e4 ,hence it is obvious1 that ,e ,r e is less than ,e r e1 and that the rays issuing from p1 by falling more obliquely upon the reflecting surface1 will be more copiously reflected4 ,it follows1 therefore1 that the intensity of the light in the reflected sector ,a ,o is not uniform1 the lines of equal brightness1 or the isophotal lines1 as they may be called1 being parallel to the reflecting surface ,a ,o1 and in every sector parallel to the radius1 between the given sector and the reflecting surface by which the sector is formed4 ,as it is easy from the preceding construction to determine the angles at which the light from any points m1 n1 is reflected1 when the length ,o ,e of the reflectors1 and the position of the eye at ,e is given1 we may calculate the intensity of the light in any point of the circular field by means of the following table1 which shows the number of rays reflected at various angles of incidence1 the number of incident rays being supposed to be #ajjj4 ,part of this table was computed by ,bouguer for plate glass not quicksilvered1 by means of a formula deduced from his experiments4 ,by the aid of the same formula ,i have extended the table considerably4 ,table showing the quantity of light reflected at various angles of incidence from plate glass4 ------------------------------ ,complement of ,rays ,reflected the ,angles out of ,incidence4 of #ajjj4 ------------------------------ #b #ehd #e #edc #g #dgd #aj #dab #ab #cef #ae #bii #bj #bbb #ba #baj #be #aeg #bf #adi #cj #aab #ca #aje #cd #he #ce #gi #cf #gd #cg #fi #ch #fe #ci #fa #dj #eg #df #dj #ej #cd #ee #bi #fj #bg #gj #be #hj #be #ij #be ------------------------------ ,in order to explain the method of using the table1 let us suppose that the angle of incidence1 or ,o ,r p1 ,fig4 #ai1 is #he3 then the number of rays in the corresponding point of the reflected sector ,a ,o b 7,fig4 #ag7 will be #edc4 ,by letting fall perpendiculars from the points 1 1 upon the mirror ,b ,o1 and taking ,o p1 ,o n1 ,fig4 #ai1 equal to these perpendiculars1 we may ascertain the angles at which the light from the points 1 1 suffer a second reflexion from the mirror ,b ,o4 ,let the angle for the point be #aj1 then the number of rays out of #ajjj reflected at this angle1 according to the table1 is #dab2 but as the number of rays emanating from 1 and incident upon ,b ,o1 is not #ajjj1 but only #edc1 we must say as #ajjj3 #dab #edc3 #bbd1 the number of rays reflected from ,b ,o1 or the intensity of the light in a point in the line ,o b corresponding to 4 ,illustration3 ,,fig4 #ai4 ,the preceding method of calculation is applicable only with strictness to the two sectors ,a ,o b1 ,b ,o a1 formed by one reflexion1 for the intensity of the light in the other sectors which are formed by more than one reflexion1 must be affected by the polarization which the light experiences after successive reflexions2 for light which has acquired this property is reflected according to laws different from those which regulate the reflexion of direct light4 ,when the mirrors are metallic1 the quantity of reflected light is also affected by its polarization1 but it is regulated by more complicated laws4 ,in ,kaleidoscopes made of plates of glass1 the last reflected image ,o 1 ,fig4 #b1 is more polarized than any of the rest1 and is polarized in a plane perpendicular to ,x ,e1 or in the same manner as if it had been reflected at the polarizing angle from a vertical plane parallel to ,x ,e4 ,illustration3 ,,fig4 #bj4 ,let us now consider what will take place by a variation in the length of the reflecting planes1 the angular extent of the field of view remaining always the same4 ,if ,a ,o ,e1 ,a ,o ,e1 ,fig4 #bj1 be two reflecting plates of the same breadth ,a ,o1 but of different lengths1 it is manifest that the light which forms the direct sector must be incident nearer the perpendicular1 or reflected at less obliquities in the short plate than in the long one1 and1 therefore1 that a similarly situated point in the circular field of the shorter instrument1 will have less intensity of light than a similarly situated point in a larger instrument4 ,but in this case1 the field of view in the short instrument is proportionally enlarged1 so that the comparison between the two is incorrect4 ,when the long and the short instrument have equal apparent apertures1 which will be the case when the plates are ,a ,o ,e1 ,a ,o ,e1 then similarly situated points of the two fields will have exactly the same intensity of light4 ,this will be better understood from ,fig4 #ai1 where ,o ,e may represent the long reflector and ,o ,e the short one4 ,then1 if these two have exactly the same aperture1 or a circular field of the same angular magnitude1 the rays of light which flow from two given points1 p1 n1 of the long instrument1 will be reflected at a certain angle from the points ,r1 r2 but as the points p1 n1 are the corresponding points in the field of the shorter instrument1 the rays which issue from them will be reflected at the same angles from the points ,r1 r1 the eye being in both cases placed at the same point e4 ,hence it is obvious1 that the quantity of reflected light will in both cases be the same1 and1 therefore1 that there is no peculiar advantage to be derived1 in so far as the light of the field is concerned1 by increasing the length of the reflectors1 unless we raise the eye above e1 till every part of the pupil receives the reflected rays4 ,there is1 however1 one advantage1 and a very important one1 to be derived from an increase of length in the mirrors1 namely a diminution of the deviation from symmetry which arises from the small height of the eye above the plane of the mirrors1 and of the small distance of the objects from the extremity of the mirrors4 ,as the height of the eye must always be a certain quantity1 ,e e1 ,fig4 #ag1 above the angular point ,e1 whatever be the length of the reflectors1 it is obvious1 that when the length of the reflectors is e ,o1 the deviation from symmetry will be only ,p o1 whereas when the length of the reflectors is reduced to e ,o1 the height of the eye e ,e being still equal to e ,e1 the aberration will be increased to ,p o4 ,this advantage is certainly of considerable consequence2 but in practice the difficulty of constructing a perfect instrument1 increases with the length of the reflectors4 ,when the plates are long1 it is more difficult to get the surface perfectly flat2 the risk of a bending in the plates is also increased1 which creates the additional difficulty of forming a good junction1 on which the excellence of the instrument so much depends4 ,by augmenting the length of the reflectors1 the quantity of dust which collects between them is also increased1 and it is then very difficult to remove this dust1 without taking the instrument to pieces4 ,from these causes it is advisable to limit the greatest length of the reflectors to seven or eight inches4 ,,chapter ,,vii4 ,,on ,,the ,,construction ,,and ,,use ,,of ,,the ,,simple ,,kaleidoscope4 ,in order to construct the ,kaleidoscope in its most simple form1 we must procure two reflectors1 about five1 six1 seven1 or eight inches long4 ,these reflectors may be either rectangular plates1 or plates shaped like those represented in ,fig4 #a1 having their broadest ends ,a ,o1 ,b ,o1 from one to two inches wide1 and their narrowest ends a ,e1 b ,e1 half an inch wide4 ,if the reflectors are of glass1 the newest plate glass should be used1 as a great deal of light is lost by employing old plate glass1 with scratches or imperfections upon its surface4 ,the plate glass may be either quicksilvered or not1 or its posterior surface may be ground1 or covered with black wax1 or black varnish1 or anything else that removes its reflective power4 ,this1 however1 is by no means absolutely necessary1 for if the eye is properly placed1 the reflexions from the posterior surface will scarcely affect the distinctness of the picture1 unless in very intense lights4 ,if it should be thought necessary to extinguish1 as completely as possible1 all extraneous light that may be thrown into the tube from the posterior surface of the glass plates1 that surface should be coated with a varnish of the same refractive and dispersive power as the glass4 ,if the plates of glass have been skilfully cut with the diamond1 so as to have their edges perfectly straight1 and free from chips1 two of the edges may be placed together1 as in ,fig4 #ag 7p4 #di71 or one edge of one plate may be placed against the surface of the other plate1 as shown in the section of ,mr4 ,bates's ,kaleidoscope4 ,but if the edges are rough and uneven1 one of them may be made quite straight1 and freed from all imperfections1 by grinding it upon a flat surface1 with very fine emery1 or with the powder scraped from a hone4 ,when the two plates are laid together1 so as to form a perfect junction1 they are then to be placed in a brass or any other tube1 so as to form an angle of #de1 #cf1 #cj1 or any even aliquot part of a circle4 ,in order to do this with perfect accuracy1 direct the tube containing the reflectors to any line1 such as m n1 ,fig4 #b1 placed very obliquely to one of the reflectors ,a ,o1 and open or shut the plates till the figure of a star is formed1 composed of #h1 #aj1 or #ab sectors1 or with #d1 #e1 or #f points1 corresponding to angles of #de1 #cf1 and #cj4 ,when all the points of the star are equally perfect1 and none of the lines which form the salient and re-entering angles disunited1 the reflectors must be fixed in that position by small arches of brass or wood ,a ,b1 a b1 ,fig4 #ba1 filed down till they exactly fit the space between the open ends of the plates4 ,the plates must then be kept in this position by pieces or wedges of cork or wood1 or any other substance pushed between them and the tube4 ,the greatest care1 however1 must be taken that these wedges press lightly upon the reflectors1 for a very slight force is capable of bending and altering the figure even of very thick plates of glass4 ,when the reflectors are thus placed in the tube1 as in ,fig4 #ba1 their extremities a ,e1 b ,e1 next the eye1 must reach to the very end of the tube1 as it is of the greatest importance that the eye get as near as possible to the reflectors4 ,the other ends of the reflectors ,a ,o1 ,b ,o1 must also extend to the other extremity of the tube1 in order that they may be brought into contact with the objects which are to be applied to the instrument4 ,in using transparent objects the cell or box which contains them may be screwed into the end of the tube1 so as to reach the ends of the reflectors1 if they happen to terminate within the tube2 but an instrument thus constructed is incapable of being applied to opaque objects1 or to transparent objects seen by reflected light4 ,illustration3 ,,fig4 #ba4 ,if the plates are narrower at the eye-end1 as in ,fig4 #ba1 the angular point ,e should be a little on one side of the axis of the tube1 in order that the aperture in the centre of the brass cap next the eye may be brought as near as possible to ,e4 ,when the plates have the same breadth at both ends1 the angular point ,e will be near the lower circumference of the tube1 as it is at ,o2 and in this case it is necessary to place the eye-hole out of the centre1 so as to be a little above the angular point ,e4 ,this construction is less elegant than the preceding2 but it has the advantage of giving more room for the introduction of a feather1 or a piece of thin wood covered with leather1 for the purpose of removing the dust which is constantly accumulating between the reflectors4 ,in some instances the plates have been put together in such a manner that they may be taken out of the tube1 for the purpose of being cleaned2 but though this construction has its advantages1 yet it requires some ingenuity to replace the reflectors with facility1 and to fix them at the exact inclination which is required4 ,one of the most convenient methods is to support the reflector in a groove cut out of a solid cylinder of dry wood of nearly the same diameter as the tube2 and after a slip of wood1 or any other substance1 is placed along the open edges of the plates1 to keep them at the proper angle given by the groove1 the whole is slipped into the tube1 where it remains firm and secure from all accident4 ,if the length of the reflectors is less than the shortest distance at which the eye is capable of seeing objects with perfect distinctness1 it will be necessary to place at the eye-end ,e a convex lens1 whose focal length is equal to1 or an inch or two greater than1 the length of the reflectors4 ,by this means the observer will see with perfect distinctness the objects placed at the object end of the ,kaleidoscope4 ,this lens1 however1 must be removed when the instrument is to be used by persons who are short-sighted4 ,the proper application of the objects at the end of the reflectors is now the only step which is required to complete the simple ,kaleidoscope4 ,the method of forming1 selecting1 and mixing the objects1 will be described in the next chapter4 ,at present1 we shall confine our attention to the various methods which may be employed in applying them to the end of the reflectors4 ,illustration3 ,,fig4 #bb4 ,the first and most simple method consists in bringing the tube about half an inch beyond the ends of the reflectors4 ,a circular piece of thin plane glass of the same diameter as the tube1 is then pushed into the tube1 so as to touch the reflectors4 ,the pieces of coloured glass being laid upon this piece of glass when the tube is held in a vertical position1 another similar disc of plane glass1 having its outer surface ground with fine emery1 is next placed above the glass fragments1 being prevented from pressing upon them1 or approaching too near the first plane glass by a ring of copper or brass2 and is kept in its place by burnishing down the end of the tube4 ,the eye being placed at the other end of the instrument1 the observer turns the whole round in his hand1 and perceives an infinite variety of beautiful figures and patterns1 in consequence of the succession of new fragments1 which are brought opposite the aperture by their own gravity1 and by the rotatory motion of the tube4 ,in this rude state1 however1 the instrument is by no means susceptible of affording very pleasing exhibitions4 ,a very disagreeable effect is produced by bringing the darkest sectors1 or those formed by the greatest number of reflexions1 to the upper part of the circular field1 and though the variety of patterns will be very great1 yet the instrument is limited to the same series of fragments1 and cannot be applied to the numerous objects which are perpetually presenting themselves to our notice4 ,these evils can be removed only by adopting the construction shown in ,fig4 #bb1 in which the reflectors reach the very end of the tube4 ,upon the end of the tube a b1 c d1 ,fig4 #bb1 is placed a ring of brass1 m n1 which moves easily upon the tube a b c d1 and is kept in its place by a shoulder of brass on each side of it4 ,a brass cell1 ,m ,n1 is then made to slip tightly upon the moveable ring m n1 so that when the cell is turned round by means of the milled end at ,m ,n1 the ring m n may move freely upon the tube4 ,the fragments of coloured glass1 etc41 are now placed in a small object-box1 as it may be called1 consisting of two glasses1 the innermost of which1 m n1 is transparent1 and the other ground on the outside ,p1 and kept at the distance of th or th of an inch by a brass rim3 this brass rim generally consists of two pieces1 which screw into one another1 so that the object-plate can be opened by unscrewing it1 and the fragments changed at pleasure4 ,this object-box is placed at the bottom of the cell ,m ,n1 as shown at ,o ,p1 and the depth of the cell is such as to allow the side ,o to touch the end of the reflectors1 when the cell is slipped upon the ring m n4 ,when this is done1 the instrument is held in one hand with the angular point ,e1 ,fig4 #ba1 downwards1 which is known by a mark on the upper side of the tube between a and b1 and the cell is turned round with the other hand1 so as to present different fragments of the included glass before the aperture ,a ,o ,b4 ,the tube may be directed to the brightest part of the sky in the day-time1 or in the evening to a candle1 or an ,argand ,lamp1 so as to transmit the light directly through the coloured fragments2 but it will always be found to give richer and more brilliant effects if the tube is directed to the window-shutter1 a little to one side of the light1 or is held to one side of the candle--or1 what is still better1 between two candles or lamps placed as near each other as possible4 ,in this way the picture created by the instrument is not composed of the harsh tints formed by transmitted light2 but of the various reflected and softened colours which are thrown into the tube from the sides and angles of the glass fragments4 ,when the pattern remains fixed in any position of the instrument1 a variety of beautiful changes may be effected by making the end of the tube revolve round a candle or a bright gas flame1 placed near the object-plate4 ,the general pattern remains the same1 but its colours vary both in their position and intensity1 as the light falls upon different sides of the fragments of glass4 ,in the preceding method of applying the objects to the reflectors1 the fragments of coloured glass are introduced before the aperture1 and pass across it in concentric circles2 and as the fragments always descend by their own gravity1 the changes in the picture1 though infinite in number1 constantly take place in a similar manner4 ,this defect may be remedied1 and a great degree of variety exhibited in the motion of the fragments1 by making the object-plates rectangular instead of circular1 and moving them through a groove cut in the cell at ,m ,n1 in the same manner as is done with the pictures or sliders for the magic lantern and solar microscope4 ,by this means the different fragments that present themselves to the aperture may be made to pass across it in every possible direction1 and very interesting effects may be produced by a combination of the rotatory and rectilineal motions of the object-plate4 ,when the object or objects are fixed1 and the tube with the reflectors moved round a centre1 as described in ,chapter ,,ii41#c we have the same succession of symmetrical pictures2 but in this case every alternate sector is stationary1 and the same number in motion1 the moving figures always changing their form1 and assuming that of the figures in the stationary sectors1 which of course change1 while the ends of the mirror pass over the fixed objects4 #c ,see ,chapter ,x4 ,when the simple ,kaleidoscope is applied to opaque objects1 such as a seal1 a watch-chain1 the seconds hands of a watch1 coins1 pictures1 gems1 shells1 flowers1 leaves1 and petals of plants1 impressions from seals1 etc41 the object1 instead of being held between the eye and the light1 must be viewed in the same manner as we view objects through a microscope1 being always placed as near the instrument as possible1 and so as to allow the light to fall freely upon the object4 ,the object-plates1 and all transparent objects1 may be viewed in this manner2 but the most splendid exhibition of this kind is to view minute fragments of coloured glass1 and objects with opaque colours1 etc41 placed in a flat box1 the bottom of which is made of mirror-glass4 ,the light reflected from the mirror-glass1 and transmitted through the transparent fragments1 is combined with the light reflected both from the transparent and opaque fragments1 and forms an effect of the finest kind4 ,as dust is apt to collect in the angle formed by the reflectors1 it may be removed when the reflectors are fixed1 either by the end of a strong feather1 or blown away with a pair of bellows4 ,when the dust is lodged upon the face of the reflectors1 it should be removed by a piece of soft leather4 ,,chapter ,,viii4 ,,on ,,the ,,selection ,,of ,,objects ,,for ,,the ,,kaleidoscope1 ,,and ,,on ,,the ,,mode ,,of ,,constructing ,,the ,,object-,,box4 ,although the ,kaleidoscope is capable of creating beautiful forms from the most ugly and shapeless objects1 yet the combinations which it presents1 when obtained from certain shapes and colours1 are so superior to those which it produces from others1 that no idea can be formed of the power and effects of the instrument1 unless the objects are judiciously selected4 ,when the inclination of the reflectors is great1 the objects1 or the fragments of coloured glass1 should be larger than when the inclination is small2 for when small fragments are presented before a large aperture1 the pattern which is created has a spotted appearance1 and derives no beauty from the inversion of the images1 in consequence of the outline of each separate fragment not joining with the inverted image of it4 ,the objects which give the finest outlines by inversion1 are those which have a curvilineal form1 such as circles1 ellipses1 looped curves like the figure #h1 curves like the figure #c and the letter ,s2 spirals1 and other forms1 such as squares1 rectangles1 and triangles1 may be applied with advantage4 ,glass1 both spun and twisted1 and of all colours1 and shades of colours1 should be formed into the preceding shapes2 and when these are mixed with pieces of flat coloured glass1 blue vitriol1 native sulphur1 yellow orpiment1 differently coloured fluids1 enclosed and moving in small vessels of glass1 etc41 they will make the finest transparent objects for the ,kaleidoscope4 ,when the objects are to be laid upon a mirror plate1 fragments of opaquely-coloured glass should be added to the transparent fragments1 along with pieces of brass wire1 of coloured foils1 and grains of spelter4 ,in selecting transparent objects1 the greatest care must be taken to reject fragments of opaque glass1 and dark colours that do not transmit much light2 and all the pieces of spun glass1 or coloured plates1 should be as thin as possible4 ,when the objects are thus prepared1 the next step is to place them in the object-box4 ,the distance between the interior surfaces of the two plane glasses1 of which the object-plates are generally composed1 should be as small as possible1 not exceeding th of an inch4 ,the outermost of these glasses has its external surface rough ground1 or is what is called a grey glass1 the principal use of which is to prevent the lines of external objects1 such as the bars of the window1 or the outlines of the illuminating flame from being introduced into the picture4 ,when a strong light is used1 a circular disk of fine thin paper placed outside of the object-box may be advantageously employed in place of the ground glass4 ,the thickness of the transparent glass plate next the reflectors should be just sufficient to keep the glass from breaking2 and the interior diameter of the brass rings into which the transparent and the grey or ground plates of glass are burnished1 should be so great that no part of the brass rim may be opposite the angular part of the reflectors during the rotatory motion of the cell4 ,if this precaution is not attended to1 the central part of the pattern1 where the development of new forms is generally the most beautiful1 will be entirely obliterated by the interposition of the brass rim4 ,instead of using transparent or grey glasses on the sides of every object-box1 some of the boxes should be made with disks of flint glass1 the interior surface of which have been stamped while in a state of fusion with a sort of pattern1 or with curved lines of a pleasing form4 ,in others1 the outer surface alone of the plate next the reflectors might be thus formed4 ,an object-box might also be formed of disks of glass1 one side of which is colourless1 and the other coloured1 some of the coloured portions being ground away irregularly1 as in certain ,bohemian articles of glass2 and the colour in one disk may be complementary to that in the other4 ,in object-boxes of this kind1 pieces of coloured glass may also be placed4 ,when the two parts of the object-boxes thus constructed are screwed or fixed together1 the box should be nearly two-thirds filled with the mixture of regular and irregular objects1 already mentioned4 ,if they fall with difficulty during the rotation of the cell1 two or three turns of the screw backward1 when there is a screw1 will relieve them2 and if they fall too easily1 and accumulate1 by slipping behind one another1 the space between the glasses may be diminished by placing within the box another glass in contact with the grey glass4 ,when the object-box1 now described1 is placed in the cell1 and examined by the ,kaleidoscope1 the pictures which it forms are in a state of perpetual change1 and can never be fixed1 and shown to another person4 ,to obviate this disadvantage1 an object-box with fixed objects generally accompanies the instrument2 the pieces of spun and coloured glass are fixed by a transparent cement to the inner side of the glass of the object-plate1 next the eye1 so that the patterns are all permanent1 and may be exhibited to others4 ,after the cell has performed a complete rotation1 the same patterns again recur1 and may therefore be at any time recalled at the pleasure of the observer4 ,the same patterns1 it is true1 will have a different appearance1 if the light falls in a different manner upon the objects1 but its general character and outline will1 in most cases1 remain the same4 ,the object-boxes which have now been described1 are made to fit the cell1 but at the same time to slip easily into it1 so that they themselves have no motion separate from that of the cell4 ,an object-plate1 however1 of a less diameter1 called the vibrating object-plate1 and containing loose objects1 is an interesting addition to the instrument4 ,when the ,kaleidoscope is held horizontally1 this small object-plate vibrates on its lower edge1 either by a gentle motion of the tube1 or by striking it slightly with the finger2 and the effect of this vibration is singularly fine1 particularly when it is combined with the motion of the coloured fragments4 ,another of the object-boxes1 in several of the instruments1 contains either fragments of colourless glass1 or an irregular surface of transparent varnish or indurated ,canada balsam4 ,this object-box gives very fine colourless figures when used alone2 but its principal use is to be placed in the cell between an object-box with bright colours and the end of the instrument4 ,when this is done1 the outline of the pieces of coloured glass are softened down by the refraction of the transparent fragments1 and the pattern displays the finest effects of soft and brilliant colouring4 ,the colourless object-box supplies the outline of the pattern1 and the mass of colour behind fills it up with the softest tints4 ,some of the object-boxes are filled with iron or brass wires1 twisted into various forms1 and rendered broader and flatter in some places by hammering4 ,these wires1 when intermixed with a few small fragments of coloured glass1 produce a very fine effect4 ,other object-plates have been made with pitch1 balsam of tolu1 gum lac1 and thick transparent paints2 and when these substances are laid on with judgment1 they form excellent objects for the ,kaleidoscope4 ,lace has been introduced with considerable effect1 and also festoons of beads strung upon wire or thread2 but pieces of glass1 with cut and polished faces1 are ill fitted for objects4 ,when the object-box is wide1 certain insects may be introduced temporarily1 without killing or injuring them1 and the crystallization of certain salts from their solution1 and other chemical changes1 may be curiously exhibited4 ,hitherto we have supposed all colours to be indiscriminately adopted in the selection of objects2 but it will be found from experience1 that though the eye is pleased with the combination of various objects1 yet it derives this pleasure from the beauty and symmetry of the outline1 and not from the union of many different tints4 ,those who are accustomed to this kind of observation1 and who are acquainted with the principles of the harmony of colours1 will soon perceive the harshness of the effect which is produced by the predominance of one colour1 by the juxtaposition of others1 and by the accidental union of a number2 and even those who are ignorant of these principles1 will acknowledge the superior effect which is obtained by the exclusion of all other colours except those which harmonize with each other4 ,in order to enable any person to find what colours harmonize with each other1 ,i have drawn up the following table1 which contains the harmonic colours4 ,deepest ,red1 ,blue and ,green equally mixed4 ,red1 ,blue and ,green1 with most ,blue1 mixed4 ,orange ,red1 ,blue mixed with much ,indigo4 ,orange1 ,blue and ,indigo1 the ,indigo predominating4 ,orange-,yellow1 ,indigo unmixed4 ,yellow1 ,violet and ,indigo nearly in equal portions4 ,greenish ,yellow1 ,pale ,violet4 ,green1 ,violet4 ,greenish ,blue1 ,violet and ,red in equal portions4 ,blue1 ,orange ,red1 ,red4 ,indigo1 ,orange-,yellow4 ,violet1 ,green4 ,it appears from the preceding table that ,bluish ,green harmonizes with ,red1 or1 in other words1 ,red is said to be the accidental colour of ,bluish ,green1 and vice versa4 ,these colours are also called complementary colours1 because the one is the complement of the other1 or what the other wants of white light2 that is1 when the two colours are mixed1 they will always form white by their combination4#d #d ,see the article ,accidental ,colours1 in the ,,edinburgh ,,encyclop,,dia1 vol4 i4 p4 #hh4 ,the following general method of finding the harmonic colours will enable the reader to determine them for tints not contained in the preceding table4 ,let ,a ,b1 ,fig4 #bc1 be the prismatic spectrum1 containing all the seven colours1 namely1 ,red1 ,orange1 ,yellow1 ,green1 ,blue1 ,indigo1 and ,violet1 in the proportion assigned to them by ,sir ,isaac ,newton1 and marked by the dotted lines4 ,bisect the ,a ,b at m1 so that ,a m is equal to ,b m1 and let it be required to ascertain the colour which harmonizes with the colour in the ,indigo space at the point p4 ,take ,a m1 and set it from p to o1 and the colour opposite o1 or an orange-yellow1 will be that which harmonizes with the indigo at p4 ,if p is between m and ,a1 then the distance ,a m must be set off from m towards n4 ,illustration3 ,,fig4 #bc ,in order to show the method of constructing object-boxes on the preceding principles1 we shall suppose that the harmonic colours of orange-yellow and indigo are to be employed4 ,four or five regular figures1 such as those already described1 must be made out of indigo-coloured glass1 some of them being plain1 and others twisted4 ,the same number of figures must also be made out of an orange-yellow glass2 and some of these may be drawn of less diameter than others1 in order that tints of various intensities1 but of the same colour1 may be obtained4 ,some of these pieces of spun glass1 of an indigo colour1 may be intertwisted with fibres of the orange-yellow glass4 ,a few pieces of white flint-glass1 or crystal spun in a similar manner1 and intertwisted1 some with fibres of orange-yellow1 and others with fibres of indigo glass1 should be added2 and when all these are joined to some flat fragments of orange-yellow glass1 and indigo-coloured glass1 and placed in the object-plate1 they will exhibit1 when applied to the ,kaleidoscope1 the most chaste combinations of forms and colours1 which will not only delight the eye by the beauty of their outline1 but also by the perfect harmony of their tints4 ,by using the thin and highly-coloured films or flakes of decomposed glass1 very brilliant and beautiful patterns are produced4 ,these films may be placed either upon a mirror plate or upon black wax1 and they may be placed among other objects1 or fixed in movable cells4 ,by applying the ,kaleidoscope to crystals in the act of formation1 shooting out in different directions1 symmetrical patterns are instantaneously created4 ,the effect produced by objects of only one colour is perhaps even superior to the combination of two harmonic colours4 ,in constructing object-plates of this kind1 various shades of the same colour may be adopted2 and when such objects are mixed with pieces of colourless glass1 twisted and spun1 the most chaste and delicate patterns are produced2 and those eyes which suffer pain from the contemplation of various colours1 are able to look without uneasiness upon a pattern in which there is only one4 ,in order to show the power of the instrument1 and the extent to which these combinations may be carried1 ,i have sometimes constructed a long object-plate1 like the slider of the magic lantern1 in which combinations of all the principal harmonic colours followed one another in succession1 and presented to the eye a series of brilliant visions no less gratifying to some persons1 and to some others even more gratifying1 than those successions of musical sounds from which the ear derives such intense delight4 ,we cannot conclude this chapter without noticing the fine effects which are produced by the introduction of carved gems1 and figures of all kinds1 whether they are drawn or engraved on opaque1 or transparent grounds4 ,the particular mode of combining these figures will be pointed out in a subsequent chapter4 ,,chapter ,,ix4 ,,on ,,the ,,illumination ,,of ,,transparent ,,objects ,,in ,,the ,,kaleidoscope4 ,when the ,kaleidoscope is directed to the sky1 or to a luminous object1 such as a gas flame1 or the flame of a candle1 a uniform tint is seen through the pieces of glass1 or other transparent fragments that have flat surfaces1 and there is a certain degree of hardness in their outlines4 ,when the instrument is not opposite the flame1 but directed to one side of it1 the light enters the transparent fragments obliquely1 and a much finer effect is produced4 ,the pattern1 indeed1 changes very considerably by making the ,kaleidoscope move round the flame4 ,an excellent effect is obtained1 as we have already stated1 by directing the tube between two bright lights2 and the richness of the symmetrical pattern increases with the number of lights which illuminate the objects4 ,as it would be inconvenient to adopt such a mode of illumination1 it becomes of importance to have some contrivance attached to the instrument1 by which we can illuminate the objects by light falling upon them in different directions4 ,illustration3 ,,fig4 #bd4 ,the simplest method of thus illuminating the objects1 is to fix on the end of the ,kaleidoscope the portion of a metallic or silvered-glass cone4 ,the light of a bright flame1 placed in front of the cone1 will be reflected from its interior surface1 and fall obliquely on the fragments of coloured glass4 ,in many cases the effect will be increased by placing in the base or mouth of the cone a circular stop1 or opaque disk1 in order to prevent any light from falling directly upon the objects1 their oblique illumination being produced solely by the rays reflected from the interior surface of the cone4 ,this will be understood from the annexed figure1 where ,m ,a ,n ,c is a portion of the cone1 fixed to the end ,e ,f of the ,kaleidoscope1 and m n o p the object-box4 ,if the angle formed by the sides of the cone is such that a ray of the sun's light falling upon ,m1 the upper margin of the reflecting surface of the cone1 is reflected to o1 the lower side of the object-box1 then all the rays of a beam of the sun's light incident upon the upper half of the conical surface1 will be reflected upon the object-box2 and1 for the same reason1 if a ray falling on ,n is reflected to m1 all the other rays falling on the lower half of the conical surface will be reflected upon the object-box1 and illuminate obliquely the objects which it contains4 ,in the ,kaleidoscopes of more recent construction1 the object-box is made transparent throughout1--the plates of glass m n1 o p being fixed in a cylindrical case of glass m n o p1 so that rays ,r ,r1 either parallel or diverging1 may be reflected from the cone ,a ,b ,c ,d1 and after passing through the transparent cylindrical rim m n o p1 illuminate the objects4 ,another cone ,a ,m ,n ,c1 with its angle less than a right angle1 may be joined to ,a ,b ,d ,c1 so as to throw the rays obliquely1 into m n and o p1 and also1 if desired1 upon the front glass m o of the object-box1 the direct rays being excluded by an opaque disk ,s ,s4 ,in this construction1 the outer face of the glass m o should not be ground1 as it would prevent the admission of the light to the objects1 the exclusion of external objects1 the purpose for which the grey glass is required1 being effected by the stop ,s ,s4 ,the illuminating cone may be of tin1 or1 what is much better1 of plated copper1 which reflects more light than any other metal1 and it must be so attached to the tube containing the reflectors1 as to have a rotatory motion4 ,the same kind of lateral illumination may be obtained from polyhedral cones or hemispheres of solid glass4 ,if ,a ,b ,c ,d1 ,fig4 #be1 is the section of a polyhedral cone of flint or plate glass1 a portion m n o p is cut out of its base ,a ,b1 to form an object-box for the reception of the pieces of coloured glass1 or other objects4 ,the sides m n1 m o1 o p1 being highly polished1 rays of light1 either parallel1 as emanating from the sun1 or diverging from artificial sources of light1 will be refracted and fall obliquely upon the faces of the object-box1 and illuminate its contents with the irregular prismatic spectra which are formed by refraction4 ,the apex ,d ,c ,e may1 in some cases1 be cut off1 and the polygonal section ,d ,e blackened in order to prevent the introduction of direct light1 and act as the stop ,s ,s in the preceding figure4 ,illustration3 ,,fig4 #be4 ,a similar effect will be obtained from a polyhedral solid1 of a hemispherical form1 as shown in ,fig4 #bf1 where ,a ,c1 ,c ,d1 ,d ,e1 ,b ,f1 ,f ,g1 ,g ,e1 are polished facets1 by which parallel or diverging rays immediately before it1 or incident in any lateral direction1 may be refracted so as to illuminate by the prismatic rays the objects in the box m n o p4 ,the front ,d ,e ,c may have an apex1 as in ,fig4 #be1 or may be made spherical to act as a condensing lens1 the surface of which may be blackened when necessary1 for the purpose of excluding the direct light4 ,these illuminators may be attached in various ways to the tube containing the reflectors1 so to have a rotatory motion in front of them4 ,illustration3 ,,fig4 #bf4 ,when the light is strong1 a circular disk of fine grained white paper may be advantageously placed upon the outer face m o of the object-box4 ,,chapter ,x4 ,,on ,,the ,,construction ,,and ,,use ,,of ,,the ,,telescopic ,,kaleidoscope1 ,,for ,,viewing ,,objects ,,at ,a ,,distance4 ,we have already seen1 in explaining the principles of the ,kaleidoscope1 that a symmetrical picture cannot be formed from objects placed at any distance from the instrument4 ,if we take the simple ,kaleidoscope1 and holding an object-box in contact with the reflectors1 gradually withdraw it to a distance1 the picture1 which is at first perfect in every part1 will1 at the distance of one-tenth of an inch1 begin to be distorted at the centre1 from the disunion of the reflected images2 the distortion will gradually extend itself to the circumference1 and at the distance of eighteen inches1 or less1 from the reflectors1 all the symmetry and beauty of the pattern will disappear4 ,an inexperienced eye may still admire the circular arrangement of the imperfect and dissimilar images2 but no person acquainted with the instrument could endure the defects of the picture1 even when the slightest distortion only is visible at the centre4 ,as the power of the ,kaleidoscope1 therefore1 in its simple form1 is limited to transparent objects1 or to the outline of opaque objects held close to the aperture of the reflectors1 it becomes a matter of consequence to extend its power by enabling it to produce perfectly symmetrical patterns from opaque objects1 from movable or immovable objects at a distance1 or from objects of such a magnitude that they cannot be introduced before the opening of the reflectors4 ,without such an extension of its power1 the ,kaleidoscope might only be regarded as an instrument of amusement2 but when it is made to embrace objects of all magnitudes1 and at all distances1 it takes its place as a general philosophical instrument1 and becomes of the greatest use in the fine1 as well as the useful arts4 ,illustration3 ,,fig4 #bg4 ,in considering how this change might be effected1 it occurred to me1 that if ,m ,n1 ,fig4 #bg1 were a distant object1 either opaque or transparent1 it might be introduced into the picture by placing a lens ,l ,l1 single or achromatic1 at such a distance before the aperture ,a ,o ,b1 that the image of the object may be distinctly formed in the air1 or upon a plate of glass1 the inner side of which was finely ground1 and in contact with the ends ,a ,o1 ,b ,o of the reflectors1 the plane passing through ,a ,o ,b4 ,by submitting this idea to experiment1 ,i found it to answer my most sanguine expectations4 ,the image formed by the lens at ,a ,o ,b became a new object1 as it were1 and was multiplied and arranged by successive reflexions in the very same manner as if the object ,m ,n had been reduced in the ratio of ,m ,l to ,l ,a1 and placed close to the aperture4 ,illustration3 ,,fig4 #bh4 ,the ,compound or ,telescopic ,kaleidoscope is therefore fitted up as shown in ,fig4 #bh1 with two tubes1 ,a ,b1 ,c ,d4 ,the inner tube1 ,a ,b1 contains the reflectors as in ,fig4 #bg1 and at the extremity ,c1 of the outer tube ,c ,d1 is placed a lens which1 along with the tube1 may be taken off or put on at pleasure4 ,the focal length of this lens should always be much less than the length of the outer tube ,c ,d1 and should in general be such that it is capable of forming an image at the end of the reflectors1 when ,a ,b is pulled out as much as possible1 and when the object is within three or four inches of the lens4 ,when it is required to introduce into the picture very large objects placed near the lens1 another lens of a less focal length should be used2 and when the objects are distant1 and not very large1 a lens1 whose principal focal length is nearly equal to the greatest distance of the lens from the reflectors1 should be employed4 ,when this compound ,kaleidoscope is used as a simple instrument for viewing objects held close to the aperture1 the tube ,a ,b is pushed in as far as it will go1 the cell with the object-plate is slipped upon the end ,c of the outer tube1 and the instrument is used in the same way as the simple ,kaleidoscope4 ,in applying the compound ,kaleidoscope to distant objects1 the cell is removed and the lens substituted in its place4 ,the instrument is then directed to the objects1 and the tube ,a ,b drawn out till the inverted images of the objects are seen perfectly distinct1 or in focus1 and the pattern consequently perfectly symmetrical4 ,when this is done1 the pattern is varied1 both by turning the instrument round its axis1 and by moving it in any direction over the object to which it is pointed4 ,when the object is about four inches from the lens1 the tube requires to be pulled out as far as possible1 and for greater distances it must be pushed in4 ,the points suited to different distances can easily be determined by experiment1 and marked on the inner tube1 if it should be found convenient4 ,in most of the instruments there is1 near the middle of the tube ,a ,b1 a mark which is nearly suited to all distances beyond three feet4 ,the object-plates held in the hand1 or the mirror-box placed upon a table1 at a distance greater than five or six inches1 may be also used when the lens ,l is in the tube4 ,the furniture of a room1 books and papers lying on a table1 pictures on the wall1 a blazing fire1 the moving branches and foliage of trees and shrubs1 bunches of flowers1 horses and cattle in a park1 carriages in motion1 the currents of a river1 waterfalls1 moving insects1 the sun shining through clouds or trees1 and1 in short1 every object in nature may be introduced by the aid of the lens into the figures created by the instrument4 ,the patterns which are thus presented to the eye are essentially different from those exhibited by the simple ,kaleidoscope4 ,here the objects are independent of the observer1 and all their movements are represented with the most singular effect in the symmetrical picture1 which is as much superior to what is given by the simple instrument1 as the sight of living or moving objects is superior to an imperfect portrait of them4 ,when the flame of a blazing fire is the object1 the ,kaleidoscope creates from it the most magical fireworks1 in which the currents of flame which compose the picture can be turned into every possible direction4 ,in order to mark with accuracy the points on the tube ,a ,b1 suited to different distances1 the instrument should be directed to a straight line1 inclined like m n1 ,fig4 #c1 to the line bisecting the angular aperture ,a ,o ,b1 and brought nearer to the centre ,o of the field4 ,the perfect junction of the reflected images of the line at the points m n1 c41 so as to form a star1 or a polygon with salient and re-entering angles1 will indicate with great nicety1 that the tube has been pulled out the proper length for the given distance4 ,in this way1 a scale for different distances1 and scales for different lenses1 may be marked on the tube4 ,in the construction of the ,tele-,kaleidoscope1 as it may be called1 the greatest care must be taken to have the lens of sufficient magnitude4 ,if it is too small1 the field of view will not coincide with the circular pattern1 that is1 the centre of the circular pattern will not coincide with the centre of the field2 and this eccentricity will increase as the distance of the lens from the reflectors is increased1 or as the object introduced into the picture approaches to the instrument4 ,the boundary of the luminous field is also an irregular outline1 consisting of disunited curves4 ,these irregularities are easily explained4 ,when the lens is too small1 the luminous field is bounded by the brass rim in which the glass is fixed2 and as this brass rim is at a distance from the reflectors1 the portion of it presented to the angular aperture cannot be formed by successive reflexions into a continuous curve2 and for the same reason1 the upper sectors of the luminous field are larger than the lower ones1 and consequently the centre of the pattern cannot coincide with that of the field4 ,in order to avoid these defects1 therefore1 the diameter of the lens should be such1 that when it is at its greatest distance from the reflectors1 the field of view may be bounded by the arch ,a ,b1 ,fig4 #ac1 and not by the brass rim which holds the lens4 ,this may be readily known by removing the eye-glass1 and applying the eye at ,e when the lens is at its greatest distance4 ,if the eye cannot see the brass rim1 then the lens is sufficiently large2 but if the brass rim is visible1 the lens is too small1 and must be enlarged till it ceases to become visible4 ,sometimes the lens has been made so small that the brass rim is seen not only at ,a ,b1 but appears also above the angular point ,o1 and produces a dark spot in the centre of the picture4 ,instead of using two tubes1 a lens is sometimes fitted into a tube about an inch longer than the focal length of the glass1 and this tube is slipped upon the object end ,a ,b ,o1 ,fig4 #ba4 ,this mode of applying the lens is1 however1 inferior to the first method1 as there is little room for adjusting it to different distances2 whereas with the long tube all objects at a greater distance than four inches from the lens may be introduced into the picture--a property which possesses very peculiar advantages4 ,the extension of the instrument to distant objects is not the only advantage which is derived from the use of the lens4 ,as the position for giving perfect symmetry is rather within the extremities of the reflectors than without them2 and as it is impossible to place movable objects within the reflectors1 we are compelled to admit a small error1 arising principally from the thickness of the objects1 and from the thickness of the plate of glass which is necessarily interposed between the objects and the reflectors4 ,the compound ,kaleidoscope1 however1 is entirely free from this defect4 ,the image of a distant1 or even of a near object1 can be formed within the reflectors1 and in the mathematical position of symmetry2 while1 at the same time1 the substitution of the image for the object itself1 enables us to produce all the changes in the picture which the motion of the object could have effected1 merely by turning the instrument round its axis1 or by moving it horizontally1 or in any other direction across the object4 ,this instrument may be advantageously placed upon a stand like a telescope1 and may either have a partial motion of rotation by means of a ball and socket1 as shown in the figure1 or what is better1 a complete motion of rotation round the axis of the tube ,c ,d1 within a brass ring1 occupying the place of the ball and socket4#e #e ,an instrument called ,the ,improved ,kaleidoscope has been recently brought out in ,paris4 ,it is merely the ,telescopic ,kaleidoscope deteriorated4 ,it consists of a lens fixed at the distance of about two inches in front of the reflectors1 and can therefore give symmetrical pictures of objects only at one distance1 while it cannot be used as an ordinary ,kaleidoscope4 ,the instrument described in the preceding page1 with a lens that can be slipped off1 is a much better ,kaleidoscope4 ,,chapter ,,xi4 ,,on ,,the ,,construction ,,and ,,use ,,of ,,polyangular ,,kaleidoscopes1 ,,in ,,which ,,the ,,reflectors ,,can ,,be ,,fixed ,,at ,,any ,,angle4 ,in all the preceding instruments1 the reflecting planes are fixed at an invariable angle1 which is some even aliquot part of #cfj2 and therefore1 though the forms or patterns which they create are literally infinite in number1 yet they have all the same character1 in so far as they are composed of as many pairs of direct and inverted images as half the number of times that the inclination of the reflectors is contained in #cfj4 ,it is therefore of the greatest importance1 in the application of the ,kaleidoscope to the arts1 to have it constructed in such a manner1 that patterns composed of any number of pairs of direct and inverted images may be created and drawn4 ,with this view1 the instrument may be fitted up in various ways1 with paper1 cloth1 and metallic joints1 by means of which the angle can be varied at pleasure2 but the most convenient methods are shown in the ,figures from ,fig4 #bi to #ce1 inclusive1 which represent two different kinds of ,polyangular ,kaleidoscopes1 as made by the late ,mr4 ,r4 ,b4 ,bate1 ,optician1 ,london1 who had devoted much time and attention to the perfection of this species of ,kaleidoscope4 ,bate's ,polyangular ,kaleidoscope with ,metallic ,reflectors4 ,illustration3 ,,fig4 #bi4 ,illustration3 ,,fig4 #cj4 ,the three ,figures1 viz41 #bi1 #cj1 and #ca1 represent the ,polyangular ,kaleidoscope with metallic reflectors1 as made by ,mr4 ,bate4 ,fig4 #bi shows the complete instrument1 when mounted upon a stand2 ,fig4 #cj is a section of it in the direction of its length2 ,fig4 #ca is a transverse section of it through the line ,s ,t1 ,fig4 #cj1 and ,fig4 #cb shows the lens of the eye-hole ,e4 ,the tube of this instrument is composed of two cones1 ,m ,m1 ,n ,n1 ,fig4 #cj1 connected together by a middle piece or ring1 ,r ,r1 into which they are both screwed4 ,these two cones enclose two highly polished metallic reflectors1 ,a ,o1 ,b ,o1 ,fig4 #ca1 only one of them1 viz41 ,b ,o ,e1 being seen in ,fig4 #cj4 ,one of these reflectors1 ,b ,o ,e1 is fixed to the ring ,r ,r1 by the intermediate piece ,k ,g ,l4 ,the reflector is screwed to this piece by the adjustable screws ,k1 ,l2 and the piece ,k ,g ,l is again fixed to the ring ,r ,r1 by two screws seen above and below ,g1 in ,fig4 #ca4 ,hence the tube1 consisting of the cones ,m ,m1 ,n ,n1 and the ring ,r ,r1 are immovably connected with the mirror ,b ,o ,e4 ,the surface of the reflector ,b ,o ,e is adjusted by the screws at ,k and ,l1 till it passes accurately through the axis of the cones and ring as seen in ,fig4 #ca4 ,the other reflector ,a ,o1 is fixed to an outer ring r r1 by means of an intermediate piece1 similar to ,k ,g ,l1 the arm ,f of which1 corresponding to ,g1 passes through an annular space or open arch1 of more than #ij1 cut out of the circumference of the inner ring ,r ,r4 ,the arm ,f is fixed to the outer ring r r by two screws1 seen above and below ,f2 and the reflector ,a ,o is fixed to the bar corresponding to ,k ,l1 ,fig4 #cj1 by similar screws1 for the purpose of adjusting it4 ,illustration3 ,,fig4 #ca4 ,illustration3 ,,fig4 #cb4 ,the lower edge ,o ,e of the reflector ,b ,o ,e extends about the #aeth of an inch below the axis of the cones1 as represented by the dotted line in ,fig4 #cj2 but the lower edge ,o ,e of the other reflector ,a ,o ,e1 which is finely ground to an acute angle1 forming a perfectly straight and smooth line1 is placed exactly in the axis of the cones1 so as just to touch a line in the reflector ,a ,o ,e1 which coincides with the axis of the cone1 and to form a junction with that line in every part of the two meeting planes4 ,the very nice adjustments which are necessary to produce so exact a motion are effected by the screws corresponding to ,k and ,l4 ,if we now fix the outer ring r r into the ring of a stand ,s ,t1 so as to be held fast1 and turn the cones with the hand1 we shall give motion to the reflector ,b ,o1 so as to place it at any angle we please1 from #j to #ij2 and during its motion through this arch1 the junction of the two reflectors must remain perfect1 if the touching lines are adjusted1 as we have described them1 to the axis of motion1 which must also be the axis of the cones and rings4 ,if1 on the contrary1 we take away the stand1 and1 holding the instrument in the hand by either of the cones ,m1 ,n1 turn the ring ,r ,r with the other1 we shall give motion to its reflector ,a ,o1 and produce a variation in the angle in the same manner as before4 ,the same effect may be produced by an endless screw working in teeth1 cut upon the circumference of the outer ring r r4 ,in order to enable the observer to set the reflectors at once to any even aliquot part of a circle1 or so as to give any number of pairs of direct and inverted images1 the most convenient of the even aliquot parts of the circle are engraven upon the ring r r2 so that we have only to set the index to any of these parts1 to the number #ab1 for example1 and the reflectors will then be placed at an angle of #cj 7#ab #cj #cfj71 and will form a circular field with twelve luminous sectors1 or a star with six points1 and consequently a pattern composed of six pairs of direct and inverted images4 ,as the length of the plates is only about five inches1 it is necessary1 excepting for persons very short-sighted1 to have a convex lens placed in front of the eye-hole ,e1 as shown in ,figs4 #cj and #cb4 ,a brass ring containing a plane glass screws into the outer ring ,c ,d1 when the instrument is not in use2 and there is an object-box containing fragments of differently coloured glass4 ,this object-box consists of two plates of glass1 one ground and the other transparent1 set in brass rims4 ,the transparent one goes nearest the reflector1 and the brass rim which contains it screws into the other1 so as to enclose between them the coloured fragments1 and regular figures of coloured and twisted glass4 ,a loose ring surrounds this object-box2 and when this ring is screwed into the circular rim ,c ,d1 the object-box can be turned round so as to produce a variety of patterns1 without any risk of its being detached from the outer cone4 ,in applying this instrument to opaque objects1 such as engravings1 coins1 gems1 or fragments of coloured glass laid upon a mirror1 the aperture of the mirrors is laid directly over them1 the large cone ,m ,m having been previously unscrewed1 for the purpose of allowing the light to fall freely upon the objects4 ,this property of the ,kaleidoscope is of great importance1 as in every other form of the instrument opaque objects must be held obliquely1 and therefore at such a distance from the reflectors as must affect the symmetry of the pattern4 ,as the perfection of the figures depends on the reflectors being kept completely free of dust1 particularly at their junction1 where it naturally accumulates1 the greatest facility is given by the preceding construction in keeping them clean4 ,for this purpose1 the large cone must be unscrewed2 the reflectors having been previously closed1 by turning the index to #fj on the ring4 ,they are next to be opened to the utmost1 and the dust may in general be removed by means of a fine point wrapped in clean and dry wash-leather4 ,if any dust1 however1 still adheres1 the small screw in the side of the ring opposite to the index should be removed1 and the smaller cone1 ,n ,n1 also unscrewed4 ,by easing the supporting screws of either of the reflectors1 their touching sides will separate1 so as to allow a piece of dry wash-leather to be drawn between them4 ,when every particle of dust has been thus removed1 the metals should be re-adjusted and closed before the cones are replaced2 both of which should be screwed firmly into the ring ,r ,r4 ,as the axis of motion in the preceding construction is necessarily the axis of the cones and rings1 the diameter of these cones and rings must everywhere be double the breadth of the reflectors4 ,from this cause1 the tube1 and consequently the object-box1 are wide1 and the instrument is1 to a certain degree1 not very portable4 ,this defect is completely avoided in another ,polyangular ,kaleidoscope constructed by ,mr4 ,bate1 upon entirely different principles1 which we shall now proceed to describe4 ,bate's ,polyangular ,kaleidoscope with ,glass ,reflectors4 ,illustration3 ,,fig4 #cc4 ,illustration3 ,,fig4 #cd4 ,illustration3 ,,fig4 #ce4 ,illustration3 ,,fig4 #cf4 ,a section of the whole of this instrument1 in the direction of its length1 is shown in ,fig4 #cc4 ,a section through ,m ,n or ,o ,p1 near the eye-end1 is shown in ,fig4 #ce1 ,fig4 #cd representing the mode of supporting the fixed reflector1 and ,fig4 #cf the mode of supporting the movable reflector4 ,the tube of the ,kaleidoscope1 in ,fig4 #cc1 is represented by b c d e f g h1 and consists of two parts1 b c g h1 and c d e f g4 ,the first of these parts unscrews from the second1 and the second contains all the apparatus for holding and moving the reflectors4 ,at the parts ,m ,n ,o ,p1 of the tube1 are inserted a short tube1 a section of which is shown in ,fig4 #cd4 ,the object of these tubes is to support the fixed mirror ,a ,o1 which rests with its lower end ,o upon the piece of brass t4 ,it is kept from falling forwards by the tongue r1 connected with the upper part s s1 and from falling backward by the piece of cork ,q1 which may be removed at any time1 for the purpose of taking out and cleaning the reflectors4 ,this little tube is fixed to the outer tube by the screws s1 s4 ,the contrivance for supporting and moving the second reflector ,b ,o1 is shown in ,fig4 #cf1 in section2 and a longitudinal view of it is given in ,fig4 #cc4 ,the mirror ,b ,o lies in an opening1 cut into two pieces of brass1 v ,b p1 one of which is placed at ,m ,n1 and the other at ,o ,p4 ,these two pieces of brass are connected by a rod m n1 ,fig4 #cc2 and in the middle of this rod there is inserted a screw k1 which passes through the main tube c d e f g1 into a broad milled ring w w1 which revolves upon the tube4 ,as the screw k1 therefore1 fastens the ring w w to the rod m n1 the reflector ,b ,o will be supported in the tube by the ring w w4 ,the lower part of the mirror ,b ,o1 or rather of the brass piece v ,b p1 rests at y1 upon the piece of watch-spring x y z1 fastened to the main tube at z4 ,this spring presses the face of the reflector ,b ,o against the ground and straight edge of the other reflector ,a ,o1 so as always to effect a perfect junction in every part of their length3--,the apparatus for both reflectors is shown in ,fig4 #ce4 ,an arch of about #de is cut out of the main tube1 so as to permit the screw k to move along it2 and hence1 by turning the broad ring w w1 the reflector ,b ,o may be brought nearly to touch the reflector ,a ,o1 and to be separated from it by an arch of #de1 so as to form every possible angle from nearly #j to #de1 which is a sufficient range for the ,kaleidoscope4 ,the main tube terminates in a small tube at ,e1 upon which may be screwed1 when it is required1 a brass cap e f1 containing a convex lens4 ,a short tube1 or cell1 a a a a1 for containing the object-boxes1 slips upon the end of the tube1 and should always be moved round from right to left1 in order that the motion may not unscrew the portion of the tube b c g h1 upon which it moves4 ,when the instrument is used for opaque objects1 the end piece1 b c g h of the tube1 screws off1 so as to admit the light freely upon the objects4 ,the advantages which the ,polyangular ,kaleidoscopes possess over all others are-- #ast1 ,that patterns of any number of sectors1 from the simplest to the most complicated1 can be easily obtained4 #b;d1 ,that the reflectors can be set1 with the most perfect accuracy1 to an even aliquot part of a circle4 #c;d1 ,that the reflectors can be at any time completely cleaned and freed from all the dust that accumulates between them1 and the instrument rendered as perfect as when it came from the hands of the maker4 ,in order to apply this ,kaleidoscope to distant objects1 or make it telescopic1#f a piece of tube with a lens at the end of it is put upon the end piece1 b c g h1 and may be suited to different distances within a certain range4 #f ,see ,chapter ,x4 ,,chapter ,,xii4 ,,on ,,the ,,construction ,,and ,,use ,,of ,,annular ,,and ,,parallel ,,kaleidoscopes4 ,in the instruments already described1 the pictures which they create1 though they may be made of various outlines1 have all a centre to which the reflected images are symmetrically related4 ,the same instruments give an annular pattern1 or a pattern returning into itself1 and included between two concentric circles1 by keeping the objects from the central part of the aperture2 but as such a pattern can never have its greatest radius more than the breadth of the mirror1 and as annular patterns of a very great radius1 where the eye can see only a portion of them at a time1 are often required1 it becomes of importance to adapt the ,kaleidoscope for this species of ornament4 ,illustration3 ,,fig4 #cg4 ,let ,a ,c ,b ,d1 ,fig4 #cg1 be two plane mirrors1 and let their inclination be measured by the angle ,a ,o ,b2 then1 if the eye is placed between ,c and ,d1 it will observe the reflected images of the objects which are placed before the aperture ,a ,c ,b ,d1 arranged1 in the annular segment ,m ,a ,b ,n1 round ,o1 as a centre4 ,the effect is exactly the same as if the reflectors had been continued to ,o1 with this difference only1 that the annular segment can never be complete4 ,this defect in the segment arises from two causes3 ,when the centre ,o is near ,c ,d1 the defect is occasioned by the want of a reflecting surface to complete the ring1 and not from any want of light in the reflected images2 but when the centre ,o is remote from ,c ,d1 the defect arises from the want of light in the last reflexions1 as well as from the want of a reflecting surface4 ,the theory of the ,annular ,kaleidoscope is exactly the same as that of the common instrument2 and therefore all the contrivances for producing symmetrical pictures1 from near and distant objects1 are applicable to this instrument4 ,as the picture1 however1 never can return into itself1 it is of no importance that the angle ,a ,o ,b be the aliquot part of a circle1 the picture being equally complete at all angles4 ,in order to have the most perfect symmetry with this ,kaleidoscope1 the eye should be placed at ,e1 between the nearest ends of the reflectors1 as it will there be nearer the plane of both reflectors than in any other position4 ,if the two mirrors are brought nearer each other1 so that their surfaces always pass through the point ,o1 the deviation from perfect symmetry will diminish as the eye becomes more and more in the plane of both2 and for the same reason the light of the field will be more brilliant4 ,when the point ,o is infinitely distant1 the two reflectors become parallel to each other1 as in ,fig4 #ch1 and the series of reflected images extends in a straight line1 forming beautiful rectilineal patterns for borders1 c4 ,in this position of the reflectors the eye should be placed in the centre at ,e1 and the symmetry of the picture and the light of the field will increase as the distance of the reflectors diminishes1 or as their length is increased4 ,illustration3 ,,fig4 #ch4 ,two different kinds of instruments have been constructed on the preceding principles1 the one by ,mr4 ,dollond1 and the other by ,mr4 ,john ,ruthven1 both of which possess very valuable properties4 ,mr4 ,dollond's ,universal ,kaleidoscope4 ,illustration3 ,,fig4 #ci4 ,illustration3 ,,fig4 #dj4 ,the instrument constructed by ,mr4 ,dollond is represented in ,figs4 #ci1 #dj1 and #da1 in section1 and is intended to unite the properties of a common ,kaleidoscope1 in which the reflectors are inclined at an angle of #cj1 and also those of an ,annular and a ,parallel ,kaleidoscope4 ,fig4 #ci represents the reflectors1 etc41 when they act as a common ,kaleidoscope2 and ,fig4 #dj shows them when they form a parallel ,kaleidoscope1 an annular ,kaleidoscope being formed when they have an intermediate position4 ,the tube of the instrument is shown1 in section1 by ,t ,t2 and to this tube is fixed1 by the screws s s1 a frame of metal1 a b1 to which the reflectors are fastened4 ,the reflectors1 which are made of the finest speculum metal1 are shown at ,a ,o1 ,b ,o1 and are attached to plates of brass1 c d1 c d1 whose breadth exceeds that of the reflectors so as to allow their extremities to descend below the point ,o1 ,fig4 #ci4 ,a double spring1 y x x y1 is placed in the tube1 so as to press upon the back of the reflectors1 and keep them in contact1 as shown in ,fig4 #ci1 and is sufficiently elastic as to allow them to open1 as in ,fig4 #dj4 ,the milled head ,m ,n1 which passes through the lower part of the tube1 carries1 at its lower end1 a very eccentric button or wheel1 the least diameter of which is seen at m1 ,fig4 #ci1 and the greatest at m1 ,fig4 #dj4 ,in the first position it has allowed the reflectors to come into contact at ,o4 ,in the other position it has forced them open into the position of parallelism4 ,by turning the milled head1 the lower ends 7,o ,o7 of the reflectors may be brought to any distance less than ,o ,o1 so as to form an annular ,kaleidoscope4 ,the eye-end of the instrument is shown in ,fig4 #da4 ,the lens ,e is placed in a slider1 ,c ,d1 which is to be moved according to the position of the reflectors1 being a little above ,o1 in ,fig4 #ci1 opposite the centre of the tube in ,fig4 #dj1 and at an intermediate position in the intermediate position of the reflectors4 ,illustration3 ,,fig4 #da4 ,this instrument is attached to a stand with a draw-tube1 which screws into the bottom of a mahogany box4 ,the object-plates1 and the lens for introducing distant objects1 are placed at the end of the instrument1 in the same manner as those of the usual construction4 ,particular kinds of objects are selected for giving rectilineal borders4 ,ruthven's ,universal ,kaleidoscope4 ,illustration3 ,,fig4 #db4 ,illustration3 ,,fig4 #dc4 ,the instrument constructed by ,mr4 ,ruthven is also a ,universal ,kaleidoscope1 which unites the properties of a ,polyangular one with those of ,annular and ,parallel ,kaleidoscopes4 ,its construction will be understood from ,figs4 #db1 #dc1 and #dd1 where ,a ,b ,e ,f ,g ,h represents a frame of iron or brass1 which slips into the tube4 ,the two sides ,a ,b1 ,f ,h1 of this frame1 are kept together by four cross pieces1 a b1 c d1 etc41 the other two corresponding to these being invisible in the figure4 ,the two reflectors1 the ends of which are seen at ,a ,o1 ,b ,o1 are each fixed to a plate of metal p p1 a section of which is seen in ,fig4 #dd4 ,each plate of metal has four cylindrical pins1 p1 p1 etc41 both on its upper and under edge4 ,the two pins nearest the ends of the reflectors pass through openings in the cross pieces a b1 c d4 ,on the top of the frame is placed a plate of brass ,m ,n ,q ,p1 in which are cut grooves e f1 g h1 k l1 m n2 e f and k l1 and also g h and m n1 being parallel to each other4 ,this plate can be moved forwards and backwards between the cross pieces a b1 c d1 by means of a small screw ,s ,s1 working in a female screw fixed upon the edge ,n ,o of the plate1 and as the middle pins p p1 attached to the plates which carry the reflectors1 pass through the grooves1 any change in the position of the plate ,m ,q1 produces a change in the distance1 p p1 of the pins1 and consequently in the distance of the upper edges of the reflectors4 ,by turning the screw ,s ,s1 therefore1 the upper edges of the reflectors may be either brought into contact1 or separated to a distance regulated by the inclination of the grooves e f g h4 ,a similar plate with a similar screw is placed upon the lower edges of the reflectors1 so that we are furnished with the means of giving the plates any inclination to each other1 or placing them at any distance within certain limits4 ,for example1 if the lower edges of the plates are in contact1 we can vary the angle of their inclination by separating or closing their upper edges by means of the upper screw4 ,by separating the lower edges1 we give them the position for annular patterns1 and by making the distance of the lower and upper edges the same1 we obtain from them rectilineal patterns2 and the figures of these annular and rectilineal patterns may be either contracted or expanded1 by altering the distance of the plates when in this parallel position4 ,illustration3 ,,fig4 #dd4 ,,chapter ,,xiii4 ,,on ,,the ,,construction ,,and ,,use ,,of ,,polycentral ,,kaleidoscopes4 ,hitherto we have considered the effects of combining two reflectors1 by means of which the reflected images are arranged around one centre1 either visible or invisible2 but it must be obvious1 from the principles already explained1 that very singular effects will be obtained from the combination of three or more reflectors4 ,as in instruments of this kind the reflected images are arranged round several centres1 we have distinguished them by the name of ,polycentral4 ,as #ij is the greatest angle which is an even aliquot part of #cfj1 and as all regular polygons1 with a greater number of sides than four1 must have their interior angles greater than #ij1 it follows1 that symmetrical pictures cannot be created by any number of reflectors greater than four1 arranged like the sides of a regular polygon4 ,if the polygon is irregular1 and consists of four sides1 or more1 then one of its angles must exceed #ij1 and consequently it cannot give symmetrical patterns4 ,in constructing ,polycentral ,kaleidoscopes1 we are limited to combinations of four or three reflectors4 ,the only modes in which we can combine four reflectors1 are so as to form a hollow square1 or a hollow rectangle2 but though these combinations afford regular patterns1 from their angles being even aliquot parts of #cfj1 yet these figures are composed merely of a great number of squares1 or rectangles1 the point where every four squares or rectangles meet being the centre of a pattern4 ,those1 however1 who may wish to construct such instruments1 must make the plates as narrow as possible at the eye-end1 so as to bring the eye1 as much as can be done1 into the plane of all the four reflectors4 ,in combining three reflectors1 the limitation is nearly as great2 but the effect of the combination is highly pleasing4 ,since the angles at which the reflectors must be placed are even aliquot parts of #cfj1 such as #ij1 #fj1 #de1 #cf1 #be-_/ #bb1 #bj1 #ah1 etc41 which are the quotients of #cfj1 divided by the even numbers1 #d1 #f1 #h1 #aj1 #ab1 #ad1 #af1 #ah1 #bj1 etc42 and since the reflectors are combined in the form of a prism1 the section of which is everywhere a triangle1 the sum of whose angles is #ahj1 we must select any three of the above even aliquot parts which amount to #ahj2 and when the reflectors are combined at these angles1 they will afford forms perfectly symmetrical4 ,now1 it is obvious1 that these conditions will be complied with when the angles are-- #ij #de #de #ahj #ij #fj #cj #ahj #fj #fj #fj #ahj ,the ,polycentral ,kaleidoscopes are therefore limited to five different combinations1 namely1-- #a4 ,four reflectors of equal breadth1 forming a square4 #b4 ,four reflectors1 two of which are broader than the other two1 and form1 a rectangle4 #c4 ,three reflectors at angles of #ij1 #de1 and #de4 #d4 ,three reflectors at angles of #ij1 #fj1 and #cj4 #e4 ,three reflectors at angles of #fj1 #fj1 and #fj4 #a4 ,on combinations of four mirrors forming a square4 ,illustration3 ,,fig4 #de4 ,the first of these ,kaleidoscopes is represented in ,fig4 #de1 where ,a ,b1 ,b ,c1 ,c ,d1 ,d ,a1 are the four equal and similar reflectors placed accurately at right angles to each other4 ,if we consider the effect only of the two reflectors ,a ,b1 ,b ,c1 and regard ,a ,d1 ,d ,c as only the limits of the aperture1 it is obvious1 from the principles explained in ,chap4 ,,ii41 that we shall have a regular figure ,d m k h ,d1 composed of four squares1 one of which ,d ,b is seen by direct vision2 other two ,a l1 ,c i formed by one reflexion from each mirror2 and the fourth ,b k composed of two half squares1 each half being formed by a second reflexion from each mirror4 ,in like manner1 if we suppose ,a ,d1 ,d ,c to act alone1 they will form a square pattern ,b b d f1 composed like the last2 and the same result will be obtained by supposing ,b ,a1 ,a ,d1 and ,b ,c1 ,c ,d to act alone4 ,the combination of these effects will produce a square a d g k1 composed of nine squares1 four of which1 formed by second reflexions1 are placed at the angles2 other four formed by first reflexions in the middle2 and one1 seen by direct vision1 in the centre4 ,hence1 it follows1 that the light of the different squares is symmetrical as well as the patterns1 a property which does not belong to all polycentral instruments4 ,the pattern1 however1 does not terminate with the square a d g k1 but extends indefinitely on all sides till the squares become invisible1 from the extinction of the light by repeated reflexions4 ,in order to discover the law according to which the squares succeed each other1 we shall examine in what manner a still larger square ,e ,f ,g ,h is completed round the central square1 seen by direct vision4 ,by considering every square in the large square a d g k as an object placed before the four reflectors1 and recollecting that the reflected images must be similarly situated behind the reflectors1 we shall find that the larger square ,e ,f ,g ,h is completed by images that have suffered two1 three1 and four reflexions1 as marked in the figure1 and that all these are symmetrically arranged with regard to the central square4 ,the squares which are crossed with a dotted diagonal line1 are those composed of two halves1 each half being formed by a different reflector4 ,when a ,kaleidoscope is formed out of the preceding combination1 the aperture1 or the breadth of the plates next the eye1 should not exceed one-sixth of an inch4 ,the effect is very pleasing when the reflectors are accurately joined and nicely adjusted1 and when distant objects are introduced by means of a lens4 #b4 ,on combinations of four mirrors forming a rectangle4 ,when the reflectors are of different breadths1 so as to form a rectangle1 the very same effects are produced as in the preceding combination1 with this difference only1 that the images are all rectangular1 in place of being square4 #c4 ,on combinations of three reflectors at angles of #fj4 ,illustration3 ,,fig4 #df4 ,when three reflectors are combined at angles of #fj1 as shown at ,a ,o ,b1 ,fig4 #df1 they form an equilateral triangle1 and therefore all the images will also be equilateral triangles4 ,the figure ,c ,d ,e ,f ,g ,h1 which is a truncated equilateral triangle1 is obviously composed of three hexagonal patterns1 of which the sectors1 or rather triangles1 are arranged round the three centres ,a1 ,o1 ,b2 the triangle ,a ,o ,b being common to all the three4 ,the three triangles1 adjacent to the sides of ,a ,o ,b1 are formed by one reflexion from each mirror4 ,the three which spring from the vertices ,a1 ,o1 ,b1 of the triangle1 consist of two halves1 each of which is formed by three reflexions1 the last reflexion of the one half being made from one of the nearest mirrors1 and that of the other half from the other nearest mirror4 ,if we consider the formation of a more extended figure1 ,i ,l ,m ,p ,n ,k1 which is also a truncated equilateral triangle1 with its truncations corresponding to the sides of the former figure1 we shall find that it has been completed by an addition1 to each side of the former1 of three equilateral triangles1 two of which are formed by three reflexions1 and the third1 consisting of two halves1 formed by four reflexions4 ,this figure consists of three entirely separate hexagons1 ,i ,c ,a ,o ,h ,k1 ,l ,d ,a ,b ,e ,m1 and ,b ,o ,g ,n ,p ,f1 all of which are formed of reflected images2--of one triangle ,a ,o ,b seen by direct vision1--and of three triangles ,a ,c ,d1 ,b ,e ,f1 ,o ,g ,h1 consisting of half sectors4 ,in constructing this ,kaleidoscope1 which1 like the two former1 has the equally luminous images symmetrically arranged round the aperture ,a ,o ,b1 it is unnecessary to shape all the reflectors with accuracy4 ,when two of them1 both of which have a greater breadth than is wanted1 are placed together1 with the edge of the one resting upon the face of the other1 the third reflector1 which must be ground with great accuracy to the desired shape of the tapering equilateral prism1 may then be placed so that each of its edges rests upon the faces of the other two4 ,when this instrument is nicely executed with metallic plates1 and when all the junctions are perfect1 the effects which it produces are uncommonly splendid4 #d4 ,on combinations of three reflectors at angles of #ij1 #de1 and #de4 ,illustration3 ,,fig4 #dg4 ,the effect produced by the combination of three reflectors at angles of #ij1 #de1 and #de1 is shown in ,fig4 #dg4 ,the two reflectors ,a ,o1 ,a ,b produce a pattern ,c ,d ,b ,i1 composed of eight triangles2 the reflectors ,b ,o1 ,b ,a1 likewise give a pattern ,a ,f ,g ,h1 composed of eight triangles2 and the reflectors ,a ,o1 ,o ,b1 give a pattern ,a ,b ,h ,i1 composed of four triangles4 ,the triangle ,i ,h ,k is an image formed by three reflexions1 one half of it being a reflexion of half of ,a ,i a1 from the mirror ,b ,o1 and the other half a reflexion of half of ,b ,h b1 from the mirror ,a ,o2 and the triangle ,d ,e ,f consists of two half images1 which are reflexions of the two half images in ,i ,o ,h4 ,the remaining triangle ,d ,l ,f is a reflexion of ,i ,k ,h1 from the mirror ,a ,b1 and is therefore formed by four reflexions4 ,as the three mirrors are not symmetrically placed1 with regard to each other1 the equally luminous images are not arranged symmetrically round the open triangle ,a ,o ,b1 as in the preceding combinations4 ,the effect is1 however1 very pleasing1 and all the reflected images included in the figure ,c ,l ,g ,k are sufficiently bright4 #e4 ,on combinations of three reflectors at angles of #ij1 #fj1 and #cj4 ,illustration3 ,,fig4 #dh4 ,the most complicated combinations of three reflectors is represented in ,fig4 #dh4 ,in the first combination1 all the angles were equal2 in the second1 two of the angles only were equal2 but in the present combination1 none of them are equal4 ,the field of view1 represented in the figure by ,d ,e ,h ,l ,m ,p1 is a truncated rhomb1 consisting of no fewer than thirty-one images of the aperture ,a ,o ,b4 ,the figure is composed of two hexagons ,d ,e ,f ,b ,r ,c1 ,r ,b ,k ,l ,m ,n1 every division of the hexagon consisting of two reflected images1 and of two rhombs ,c ,r ,n ,p1 ,f ,b ,k ,h1 each of which is composed of four reflected images4 ,in this combination1 as in the last1 the equally luminous sectors are not symmetrically arranged round the centre ,o of the figure4 ,in the rhomb ,c ,r ,n ,p1 for example1 the four images are formed by three1 four1 and five reflexions2 whereas in the corresponding rhomb ,f ,h ,k ,b1 they are formed by two1 three1 and four reflexions4 ,the effects produced by a ,kaleidoscope constructed in this manner are very beautiful1 particularly when the reflectors are metallic4 ,in the four figures which represent the different combinations of the reflectors1 the small figures indicate the number of reflexions by which each image is produced4 ,,chapter ,,xiv4 ,,on ,,kaleidoscopes ,,in ,,which ,,the ,,effect ,,is ,,produced ,,by ,,total ,,reflexion ,,from ,,the ,,interior ,,surfaces ,,of ,,transparent ,,solids4 ,when light is incident upon the most perfectly polished metals1 a very considerable quantity of it is absorbed1 and even when the reflexion is made at the greatest obliquities1 there is a very manifest difference in the intensity of the direct and the reflected pencil4 ,in the total reflexion of light from the second surfaces of transparent bodies1 the loss of light is very inconsiderable1 and the reflexion is made with a degree of brilliancy far surpassing that of the most resplendent metals4 ,illustration3 ,,fig4 #di4 ,in constructing a ,kaleidoscope upon this principle1 we must procure a piece of glass entirely free from veins1 and cut it into the form shown in ,fig4 #di4g ,the two surfaces ,b ,o ,e1 ,a ,o ,e1 must be inclined at an angle which is the even aliquot part of a circle4 ,they must be ground perfectly flat and highly polished1 and the junction ,o ,e must be made as fine as possible4 ,the upper surface ,a ,b ,e should be rough-ground1 and the side ,a ,b ,o1 and the side at ,e1 should be parallel and well polished4 ,if the glass is colourless and good1 the eye1 when placed at ,e1 will see the very same appearance as in the simple ,kaleidoscope2 and objects placed at ,a ,b ,o will be arranged into the same beautiful figures4 ,the only defects attending this form of the ,kaleidoscope1 are the loss of light occasioned by its passing through a mass of solid glass1 not perfectly transparent1 and the difficulty of obtaining a perfect junction of the two reflecting planes4 ,the first of these evils is1 however1 counterbalanced by the great intensity of the light which suffers total reflexion2 and the second does not exist when the ,kaleidoscope is intended to give rectilineal or annular patterns4 #g ,when this chapter was written 7#ahah71 it was very difficult to procure glass sufficiently homogeneous for this purpose3 but it can now be procured from the ,glass ,works of ,messrs4 ,chance ,co41 at ,smethwick1 near ,birmingham4 ,in the construction of instruments of this kind1 it is necessary to make the prism of glass longer than the distance at which the eye can see objects with perfect distinctness2 that is1 if the eye is capable of seeing objects distinctly at the distance of five inches1 it will not perceive the same objects distinctly when they are placed at the end of a prism of glass five inches long4 ,this singular effect arises from a property of plain lenses or pieces of plain glass1 in consequence of which1 they cause divergent rays to diverge from a point nearer the lens or plate1 than that from which they radiated4 ,it will therefore be more convenient1 for many reasons1 to make the glass prism only two or three inches long1 and obtain distinct vision by means of a lens placed at the eye-end of it2 but1 for the reason already mentioned1 the focal length of the lens must be less than the length of the glass prism4 ,the lens may even be joined to the prism1 by grinding the eye-end into a spherical form1 but the degree of convexity must be calculated upon the principles already stated4 ,the solid form of the ,kaleidoscope is peculiarly fitted for polycentral instruments1 as we have only to polish the side1 which would otherwise have been left rough1 the prism being supposed to be cut to the angles which are necessary to give symmetrical forms1 according to the principles stated in ,chapter ,,xiii4 ,,chapter ,,xv4 ,,on ,,the ,,application ,,of ,,the ,,kaleidoscope ,,to ,,the ,,magic ,,lantern1 ,,solar ,,microscope1 ,,and ,,camera ,,obscura4 ,illustration3 ,,fig4 #ej4 ,in the various forms of the ,kaleidoscope which have been described in the preceding chapters1 the pictures which it creates are visible only to one person at a time2 but it is by no means difficult to fit it up in such a manner as to exhibit them upon a wall to any number of spectators4 ,the necessary limitation of the aperture at the eye-end of the instrument1 however1 is hostile to this species of exhibition1 as it requires a very intense light for the purpose of illuminating the objects4 ,the general principle of the apparatus requisite for this purpose is shown in ,fig4 #ej1 where ,c ,d ,g ,f is the tube containing the reflectors ,a ,o ,e1 etc4 ,the objects from which the pictures are to be created are placed in the cell ,c ,d1 which may be made either to have a rotatory movement round the axis of the tube1 or to slide through a groove1 like the sliders of a magic lantern4 ,these objects are powerfully illuminated by a lens ,b1 which concentrates upon them the direct light of the lamp or candle ,h1 and also the part of the light which is reflected from the mirror ,m ,n4 ,at the eye-end ,e of the ,kaleidoscope1 is placed a lens ,l ,l1 close to the end of the reflectors1 and having its centre coincident with the centre of the aperture at ,e4 ,in order that this lens may form behind it an image ,p ,p of the objects placed in the object-plate ,c ,d1 its focal length must be less than the length ,a ,e of the plates4 ,if the focal length of ,l ,l is so small as one-half of ,a ,e1 then it follows1 from the principles of optics1 that the distance ,l ,p at which the image is formed behind the lens1 will be precisely equal to the distance ,a ,e of the object2 but this is obviously too small a distance1 for the diameter of ,p ,p would be equal only to the apparent diameter of the circular aperture of the ,kaleidoscope1 or to twice ,a ,o4 ,hence it is necessary1 that the focal length of the lens ,l ,l be less than ,a ,e1 and greater than half of ,a ,e4 ,two-thirds1 or three-fourths of ,a ,e will be found to be a suitable focal length2 for if it is larger than this1 the image will be formed upon the wall or screen at too great a distance from the instrument4 ,when the instrument is thus fitted up1 an enlarged image of the pattern will be thrown upon the wall1 which must be covered with white paper1 or some white ground1 in order to exhibit the colours to advantage4 ,by turning the object-plate round its centre1 or1 if it is a rectilineal one1 by pushing it through the groove1 and at the same time giving it a rotatory motion1 the pattern on the wall will undergo every possible transformation1 and exhibit to the spectators1 in a magnified form1 all those variations which have been observed by applying the eye to the ,kaleidoscope4 ,when the preceding apparatus is used in daylight1 so that the objects are illuminated by the rays of the sun1 the mirror ,m ,n is unnecessary4 ,the ,kaleidoscope1 etc41 must1 however1 be attached to the part of the frame of a ,solar ,microscope1 which goes into the aperture in the window-shutter4 ,as the most brilliant light is obtained from the burning of oxygen1 either by itself1 or along with coal gas1 as in the ,bude light1#h a lamp of this kind is peculiarly fitted for displaying the pictures of the ,kaleidoscope to a number of spectators4 ,one of ,mr4 ,bate's ,polycentral ,kaleidoscopes was1 many years ago1 fitted up with a lamp of this kind1 for exhibition1 at the lectures on natural philosophy1 delivered at ,guy's ,hospital1 by that eminent chemist1 the late ,william ,allen1 ,esq41 ,f4,r4,s4 ,the patterns which are created by the ,telescopic ,kaleidoscope from natural objects1 or from objects independent of the instrument1 may1 in like manner1 be exhibited to several spectators at once4 ,if the objects are in a room1 such as bunches of flowers1 statues1 human figures1 or large pictures1 they must be placed in one apartment1 and strongly illuminated4 ,the lens must then be placed upon the end ,a ,o of the ,kaleidoscope1 the object-plate ,c ,d having been removed1 and must be so adjusted that the images of the objects may fall exactly upon the end ,a ,o of the reflectors4 ,the objects may be placed at any distance from the lens1 from six inches to twelve feet1 according to their magnitude1 and the pictures will be exhibited with great distinctness and effect upon the wall of the other apartment4 ,if a blazing fire is employed1 the most brilliant display of fireworks may be exhibited4 ,when the objects are out of doors1 such as trees1 shrubs1 etc41 the ,kaleidoscope1 with its two lenses1 must be fixed in the circular opening of a window-shutter1 and the picture received upon white paper1 or any other suitable ground1 as in the ,camera ,obscura4 #h ,this light was first proposed by myself4 ,see ,edinburgh ,review1 ,april #ahcc1 vol4 lvii4 p4 #aib4 ,similar effects may be produced in a portable ,camera ,obscura1 by placing the apparatus ,c ,f ,l ,l ,g ,d in the moveable drawer of that instrument4 ,if the lens ,l ,l is of such a focal length as to admit the formation of the image within the instrument1 the picture will be finely displayed upon the ground glass1 and may be copied with considerable exactness4 ,in the preceding applications of the ,kaleidoscope1 the great difficulty to be overcome arises from the smallness of the aperture which can be obtained at the eye-end of the reflectors4 ,if we take a larger aperture1 for the purpose of gaining more light1 the light of the reflected images is diminished by this very circumstance1 and the picture loses its symmetry at the centre4 ,the only method by which we can remove this evil is to lengthen the reflectors1 and consequently increase their breadth in the same proportion4 ,let us suppose1 for example1 that when the reflectors are five inches long1 we can safely employ an aperture one-fourth of an inch in diameter2 then if the plates are made ten inches long1 we may use an aperture two-fourths in diameter1 the symmetry continuing as complete1 and the light of the reflected images as intense1 as when their length was only five inches4 ,by increasing the length of the reflectors1 therefore1 we increase also the quantity of light2 but1 unfortunately1 this increase of length is unfavourable to the other properties of the instrument2 for we must now use a lens ,l ,l of a great focal length1 which will render it necessary to receive the image at a great distance from the instrument4 ,in order to render the effect as brilliant as possible1 the inclination of the reflectors should1 in the present case1 never exceed #cj1 and might be even #cf or #de4 ,the objects should be selected as thin as possible1 and none with dark tints ought to be admitted into the object-boxes4 ,,chapter ,,xvi4 ,,on ,,the ,,construction ,,of ,,kaleidoscopes ,,which ,,combine ,,the ,,colours ,,and ,,forms ,,produced ,,by ,,polarized ,,light4 ,in the preceding chapters we have supposed that the objects are illuminated by common light1 and that the forms which compose the symmetrical figure are those of material bodies4 ,if we employ polarized light we may introduce into the ,kaleidoscopic figures1 the splendid colours produced by crystallized bodies1 and also the forms which these colours assume1 round the optical axes of crystals1 or through different thicknesses of the doubly-refracting substance4 ,the part of the polarizing apparatus which polarizes the light1 may be a large ,nicol's prism1 or a bundle of thin glass plates1 or a single plate of black glass1 fixed at the object end of the ,kaleidoscope4 ,the analysing part of the apparatus may be a ,nicol's prism1 or plates of the sulphate of iodo-quinine1 discovered by ,dr4 ,william ,herapath1 of ,bristol4 ,owing to the thickness of a ,nicol's prism1 it is not well fitted for the analyser1 as it prevents the eye of the observer from getting sufficiently near the small eye-hole of the ,kaleidoscope4 ,the plates of the sulphate of the iodo-quinine1 are therefore peculiarly adapted for analysers2 and when they can be obtained of the same size as the angular aperture of the ,kaleidoscope1 with fixed reflectors1 or of the whole circular aperture when the reflectors are movable1 they will also form the best polarizers4 ,the crystals which are to give the colours and forms produced by polarized light and its subsequent analysis1 may be either uniaxal crystals1 such as calcareous spar1 or quartz1 or beryl1 or biaxal crystals1 such as selenite1 topaz1 mica1 arragonite1 nitre1 etc4#i ,these crystals must be placed at the end of the reflectors1 and when they transmit polarized light1 their brilliant colours and forms will vary by turning the cell which contains them1 or by giving a motion of rotation to the analyser4 ,thin films1 or lamin of selenite of different thicknesses1 and generally of such a thickness as gives the bright rings of the second order of colours in ,newton's scale1 may be placed in a narrow cell or object-box1 and may have their outlines of various curvatures1 so as to combine both form and colour in the ,kaleidoscopic figure4 ,different forms may also be obtained by using pieces of colourless glass of different shapes1 or pieces of thin wire bent into a variety of curves4 ,if the outlines of the pattern are to be obtained from pieces of glass or wire1 the films of selenite might be cemented to one of the glass plates of the object-box1 so as to have their axes lying in different directions4 #i ,see my ,treatise on ,optics1 ,edit4 #ahec1 ,chaps4 ,,xxviii1 ,,xxix4 ,the coloured figures produced by glass quickly cooled1 might also be advantageously employed1 and1 likewise1 the remarkable forms of circular crystals and crystalline groups1 when they are sufficiently large to be seen by the naked eye4 ,very beautiful objects may be made by cementing a plate of sulphate of lime 7selenite71 #j4jahah of an inch thick1 to a plate of glass1 and cutting out1 upon a turning-lathe1 grooves or bands of such different depths1 as to give different colours by polarized light4 ,beautiful patterns may be thus executed in bands or lines of different colours2 but for the purposes of the ,kaleidoscope1 it will be sufficient to have curves or portions of curves of such different forms and curvatures1 as will produce agreeable figures by their combination4 ,these curves and irregular forms of any kind1 may be scratched or excavated in the plate of selenite by the point of a sharp knife1 and afterwards polished1 or even deepened1 by the action of water1 which has the property of slowly dissolving the selenite4 ,in order to observe the effects produced by polarized light1 it is not necessary to have the polarizing and analysing apparatus attached to the ,kaleidoscope4 ,the polarizer1 in the form of a plate of black glass1 or a bundle of plates of common window-glass1 may be laid on the table so as to reflect the light into the ,kaleidoscope at an angle of #ef1 and the analyser may be held in one hand1 and the ,kaleidoscope in the other4 ,in this case1 however1 it would be better to have the ,kaleidoscope fixed1 in order that the observer may have the use of his left hand1 to turn the object-box1 which contains the doubly-refracting crystal4 ,without the use of polarized light1 very fine forms1 and these splendidly coloured1 may be obtained by means of the coloured rings exhibited in the ,iriscope4#aj ,when two such systems of rings are formed by breathing through a tube upon black glass1 over which soap has been rubbed and subsequently wiped off1 by a piece of chamois leather1 they form a double system1 similar to the biaxial system in mica and topaz1 with curves of various shapes1 which exhibit beautiful combinations in the ,kaleidoscope4 #aj ,treatise on ,optics1 p4 #abj4 ,,chapter ,,xvii4 ,,on ,,the ,,construction ,,of ,,stereoscopic ,,kaleidoscopes4 ,if we apply the ,kaleidoscope to any statue or architectural ornament1 or any other solid object represented1 photographically1 on a transparent binocular slide1 the figures will be combined into a flat symmetrical pattern1 as shown in a future chapter4 ,but if1 in the lenticular stereoscope1 we place a ,kaleidoscope between each of the two semi-lenses and the statue1 or other object in the binocular slide1 we shall then see the statue or other object in full relief in the symmetrical figure4 ,this1 in a rude form1 is the ,stereoscopic ,kaleidoscope4 ,in order to construct the instrument independently of the stereoscope1 we have only to combine two equal ,kaleidoscopes1 with their reflecting mirrors equally inclined to each other1 and place at the eye-end of them two semi-lenses or quarter lenses1 at the distance of two and a half inches1 and having their focal lengths equal to the length of the stereoscope4 ,if the two right and left eye photographs1 to which we apply them1 are opaque1 upon paper or silver plate1 an opening must be left above the object end of the reflectors1 of sufficient size to allow light to be thrown upon the photographs4 ,when the figures are transparent1 this aperture must be closed4 ,if the objects in the binocular slides represent either bas-reliefs or alto-relievos1 or statues1 or vases1 or portions of any solid bodies1 such as animals1 plants1 or flowers1 which can be combined into symmetrical patterns1 they will appear in their true relief in the ,stereoscopic ,kaleidoscope4 ,in this way the artist may avail himself of the instrument in designing circular ,gothic windows1 the circular decorations of ceilings1 and rectilineal or curvilineal belts1 that are to be cut out of metals1 marble1 freestone1 or wood1 or formed of plaster of ,paris1 or metals susceptible of fusion4#aa #aa ,see my ,treatise on the ,stereoscope1 ,chap4 xi4 p4 #hf4 ,if we combine two ,telescopic ,kaleidoscopes1 in the manner already described1 so that the centres of their object-lenses are distant two and a half inches1 and receive the right and left eye pictures of real objects upon disks of ground glass1 with the ground side touching the ends of the reflectors1 these objects1 though themselves in true relief1 will be reduced to plane pictures on the ground glass1 and again brought into stereoscopic relief1 and combined into symmetrical patterns by the instrument4 ,the effect thus produced is different from what we should see were the ground glass removed1 and the image of the object formed in the air at the end of the reflectors1 for the same reason that the picture of an object in the stereoscope is different from what it appears if viewed directly by the eyes4 ,,chapter ,,xviii4 ,,on ,,the ,,construction ,,of ,,microscopic ,,kaleidoscopes4 ,the name of ,microscopic ,kaleidoscope may be given to the instrument1 under two forms1 namely1 when it is made to produce symmetrical patterns from microscopic objects1 or when it is made so short that a lens of a high power is necessary at one end of the reflectors1 to see distinctly1 and magnify the objects at the other end4 ,in both these forms ,i have often constructed them so small as one inch and one inch and a half in length4 ,as the ,kaleidoscope1 in this minute state1 has been applied both in this country and abroad1 as a female ornament1 we shall proceed to point out the best method of constructing it4 ,since the aberration from symmetry increases1 as the length of the reflectors is diminished1 and since the light of the field diminishes from the same cause1 it becomes extremely difficult to obtain correct figures1 and uniformity of light in small instruments4 ,in order to overcome these difficulties1 as far as possible1 the reflectors should be metallic1 and may be either made of polished steel or polished speculum metal4 ,the inclination at which they are fixed should not be less than #cf or #de2 and the eye-hole1 which should not exceed _/th of an inch in diameter1 must be placed as near as possible to the angular point4 ,since the aberration from symmetry increases with the distance of the object from the reflectors1 and is much augmented in small instruments1 the greatest care must be taken to have the objects at the least possible distance from the reflectors4 ,to accomplish this1 the objects themselves should be as thin and slender as they can be made2 the colours should be brilliant and not gloomy2 and they should be separated from the reflectors by a thin film of the most transparent mica1 which is superior to glass of equal thickness1 even if it could be got1 from its extreme toughness and elasticity4 ,the mica1 indeed1 is easily scratched1 but if this should take place to any extent1 it can easily be replaced by a new film4 ,it would even be of consequence to bend the mica into a slight concavity1 so as to permit the objects to lie rather within than without the extremity of the reflectors4 ,in order to see the pattern with perfect distinctness1 a lens must be placed at the end of the instrument2 the focal length of this lens1 however1 must not be exactly equal to the distance of the objects from the eye1 but as much greater as possible1 so that the eye1 by a little exertion1 may be able to obtain distinct vision4 ,the reason of this will be understood1 by considering that the images of the objects1 seen by reflexions1 are thrown to a greater distance1 as it were1 from the eye1 and could not therefore be seen distinctly by using a lens adjusted exactly to the nearest part of the picture4 ,consequently1 the focal length of the lens must be a mean between the distances of different parts of the picture1 that is1 a little greater than that which is suited to the sector seen by direct vision4 ,when small ,kaleidoscopes are made with four or with three mirrors1 the preceding directions are equally applicable1 the greatest care being taken that the reflectors taper nearly to a point at the eye-end1 so as not to leave an aperture greater than th of an inch in diameter4 ,when they are made of solid glass1 the focal length of the lens must be determined from the principles contained in the last chapter4 ,the preceding instruments may be fitted up with a draw-tube and lens1 and when it is required to introduce drawings of pictures and statues1 small microscopic photographs may be employed4 ,,chapter ,,xix4 ,,on ,,the ,,changes ,,produced ,,by ,,the ,,kaleidoscope4 ,the property of the ,kaleidoscope1 which has excited more wonder1 and therefore more controversy than any other1 is the number of combinations or changes which it is capable of producing from a small number of objects4 ,many persons1 entirely ignorant of the nature of the instrument1 have calculated the number of forms which may be created from a certain number of pieces of glass1 upon the ordinary principles of combination4 ,in this way it follows1 that twenty-four pieces of glass may be combined #a1cia1gbd1bhh1hhg1beb1iii1dbe1abh1dic1d jb1bjj times--an operation1 the performance of which would require hundreds of thousands of millions of years1 even upon the supposition that twenty of them were performed every minute4 ,this calculation1 surprising as it appears1 is quite false1 not from being exaggerated1 but from being far inferior to the reality4 ,it proceeds upon the supposition that one piece of glass can exhibit only one figure1 and that two pieces can exhibit only two figures1 whereas it is obvious that the two pieces1 though they can only be combined in two ways1 in the same straight line1 yet the one can be put above and below the other1 as well as upon its right side and its left side1 and may be joined1 so that the line connecting their centres may have an infinite number of positions with respect to a horizontal line4 ,it follows1 indeed1 from the principles of the ,kaleidoscope1 that if only one object is used1 and if that object is a mathematical line without breadth1 the instrument will form an infinite number of figures from this single line4 ,the line may be placed at an infinite number of distances from the centre of the aperture1 and equally inclined to the extremities of the reflectors4 ,it may be inclined at an infinite variety of angles to the radii of the circular field1 and it may be placed in an infinite variety of positions parallel to any radius4 ,in all these cases1 the ,kaleidoscope will form a figure differing in character and in magnitude4 ,in the first case1 all the figures are polygons of the same character1 but of different sizes4 ,in the second case1 they are stars1 differing from each other in the magnitude of their salient and re-entering angles2 and in the third case1 they form imperfect figures1 in which the lines unite at one extremity and are open at the other4 ,if1 instead of supposing a mathematical line to be the object1 we take a single piece of coloured glass1 with an irregular outline1 we shall have no difficulty in perceiving1 from experiment1 that an infinite variety of figures may be created from it alone4 ,this system of endless changes is one of the most extraordinary properties of the ,kaleidoscope4 ,with a number of loose objects1 it is impossible to reproduce any figure which we have admired4 ,when it is once lost1 centuries may elapse before the same combination returns4 ,if the objects1 however1 are placed in the cell1 so as to have very little motion1 the same figure1 or one very near it1 may1 without difficulty1 be recalled2 and if they are absolutely fixed1 the same pattern will recur in every revolution of the object-plate4 ,,chapter ,,xx4 ,,on ,,the ,,application ,,of ,,the ,,kaleidoscope ,,to ,,the ,,fine ,,and ,,useful ,,arts4 ,if we examine the various objects of art which have exercised the skill and ingenuity of man1 we shall find that they derive all their beauty from the symmetry of their form1 and that one work of art excels another in proportion as it exhibits a more perfect development of this principle of beauty4 ,even the forms of animal1 vegetable1 and mineral bodies1 derive their beauty from the same source4 ,the human figure consists of two halves1 one of which is the reflected image of the other2 and the same symmetry of form presents itself in the shapes of almost all the various tribes of animated beings4 ,in the structure of vegetables1 the principle of symmetry is less perfectly developed4 ,from the extreme delicacy and elasticity of its parts1 a plant1 regularly constructed1 would have lost all its symmetry from the influence of gravitation1 or from the slightest breath of wind2 and therefore a symmetrical combination of parts has been effected only in its leaves and flowers4 ,when the laws of crystallization are allowed to perform their functions uncontrolled1 the beautiful geometrical forms which they create are marked with the most perfect regularity4 ,even their physical properties are symmetrically related to some axis or fixed line2 and though all their functions are performed in utter silence and repose1 yet their physiology1 if we may apply that name to the actions of apparently dead matter1 is not less wonderful than that which embraces the busy agencies of animal and vegetable bodies4 ,the irregular forms which are the foundation of picturesque beauty constitute a single exception to the general law1 and therefore the landscape painter is the only artist who is not professionally led to the study of that species of beauty which arises from the inversion and multiplication of simple forms4 ,when we consider the immense variety of professions connected both with the fine and the useful arts1 in which the creation of symmetrical ornaments forms a necessary part1 we cannot fail to attach a high degree of utility to any instrument by which the operations of the artist may be facilitated and improved4 ,we are disposed to imagine that no machine is really useful1 unless it is directly employed in providing for our more urgent wants4 ,this1 however1 is a vulgar error4 ,an engine which forms the head of a pin1 has1 in reality1 as much importance as an engine for raising water1 or for manufacturing cloth2 for in these cases the three machines have the same object1 which is merely that of abridging manual labour4 ,the water would still be raised1 and the cloth and pins manufactured1 if the machines did not exist2 but the machinery insures us a more regular supply of these articles1 and enables us to receive them at a cheaper rate4 ,the operations of machinery have1 however1 a still higher character in comparison with those of individual exertion1 when they enable us to obtain any article1 either of necessity or of luxury1 in a more perfect state4 ,in this case1 the machine effects what is beyond the reach of manual labour2 and instead of being the mere representative of animal force1 it exhibits a concentration of talent and skill which could not have been obtained by uniting the separate exertions of living agents4 ,when we consider1 that in this busy island thousands of individuals are wholly occupied with the composition of symmetrical designs1 and that there is scarcely any profession into which these designs do not enter as a necessary part1 so as to employ a portion of the time of every artist1 we shall not hesitate in admitting1 that an instrument must have no small degree of utility which abridges the labour of so many individuals4 ,if we reflect further on the nature of the designs which are thus composed1 and on the methods which must be employed in their composition1 the ,kaleidoscope will assume the character of the highest class of machinery1 which improves at the same time that it abridges the exertions of individuals4 ,there are few machines1 indeed1 which rise higher above the operations of human skill4 ,it will create1 in a single hour1 what a thousand artists could not invent in the course of a year2 and while it works with such unexampled rapidity1 it works also with a corresponding beauty and precision4 ,the artist who forms a symmetrical design1 is entirely ignorant of the effect till it is completed--and if the design is to be embodied in coloured materials1 or in stone1 or any other solid substance1 he has no means of predicting the final effect which it is to produce4 ,every result1 in short1 is a matter of uncertainty3 and when the work is completed1 it must remain as it is4 ,the art of forming designs1 therefore1 is in a state of extreme imperfection2 and a more striking proof of this could not be obtained than from the servility with which we copy1 at the present moment1 the mouldings and ornaments of ,greek and ,gothic architecture1 and the decorations which embellish the furniture1 the dresses1 and the utensils of the ,romans4 ,if the ,kaleidoscope had been an instrument which merely enabled us to project upon a plane surface a variety of designs of the same character as those which the artist forms with his pencil1 it would still have been an instrument of great utility4 ,but it does much more than this4 ,when properly constructed1 and rightly applied1 it exhibits the final effect of the design1 when executed in the best manner2 and it does this1 not only by embodying the very materials out of which the reality is to be produced2 but by exhibiting1 instead of lights and shades1 the very eminences and depressions which necessarily exist in every design the parts of which lie in various planes4 ,in proceeding to point out the practical methods of obtaining these effects from the ,kaleidoscope1 we take it for granted that the artist has one or other of the correct instruments described in the preceding chapters1 and that they are mounted upon a stand1 and furnished with ,dr4 ,wollaston's ,camera ,lucida1 for enabling him to copy the designs which he wishes to perpetuate4 #a4 ,architectural ,ornaments4 ,almost every public and private edifice1 with the exception of picturesque cottages1 and buildings erected for the purposes of defence1 has a regular form1 consisting of two halves1 one of which is the inverted image of the other4 ,the inferior parts of the building1 such as the doors and windows1 have the same regular character2 and hence it necessarily follows1 that all the decorations1 whether in the form of rectilineal borders1 circular patterns1 or groups of figures1 should not only have the same symmetry1 but should also be symmetrically related to the bisecting line which separates the building into two halves4 ,if1 for example1 a rectilineal border1 surrounding a building like a belt1 consists of a pattern1 or of lines inclined in one direction1 such a border is not symmetrically related to the vertical and horizontal lines of which the building consists4 ,hence it will follow1 that sculptures1 representing an action of any kind1 or statues representing living objects1 when they are sufficiently large to be seen at the same time with the whole building1 can never connect themselves with its regular outline4 ,if these sculptures1 or statues1 are inverted so as to form a ,kaleidoscope pattern1 like the beautiful sculpture in the door of the temple of ,jun-wassa1#ab they may then be employed without the risk of destroying the general symmetry of the edifice4 ,these remarks are equally applicable to every object which derives its beauty from symmetry2 and it is curious to observe the numerous deviations from this principle1 and the bad effects they produce on some of the finest vases and ornaments of the ,romans4#ac ,if ,mr4 ,cockerell's ingenious theory#ad of the original composition of the statues of ,niobe and her children be correct1 the mode of grouping the figures will show how much the artist was disposed to sacrifice every other kind of effect1 to obtain something like a symmetrical group within the pediment4 ,the gradual increase in the height of the statues towards the middle of the tympanum1 and their inclination on both sides towards the same point1 form strong proofs of ,mr4 ,cockerell's hypothesis1 and afford a singular example1 the only one with which we are acquainted1 of an attempt to reconcile the apparently incongruous effect of a real picture and a symmetrical group4 ,had the statues been confined to one-half of the tympanum1 while the other half was a reflected image of the first1 we are persuaded1 that though the effect1 as a picture1 would have been diminished1 yet the effect1 as a part of the temple1 would have been greatly increased4 ,a very remarkable example of this species of symmetry is shown in the fine painting of the ,four ,sibyls1 by ,raphael1 which is now in the church of ,santa ,maria della ,pace1 at ,rome4 #ab ,see ,edinburgh ,encyclopdia1 ,art4 ,civil ,architecture1 plate ,,cli4 #ac ,see ,hope's ,costume of the ,ancients1 vol4 ii4 plates #bcj1 #bfd1 #bgh1 figs #a and #bih1 fig4 #b4 #ad ,see ,journal of the ,royal ,institution1 vol4 v4 p4 #ii4 ,after the architect has fixed upon the nature and character of his ornaments1 he must cut1 upon the surface of a large stone1 or place in relief upon it1 the elements of a variety of patterns4 ,these elements need not be exact representations of any object1 or any portion of it1 though1 in some cases1 an approximation to this may be desirable4 ,when this stone is set in a vertical position1 and so that the light may fall obliquely upon its surface1 for the purpose of giving light and shade to the pattern1 the ,kaleidoscope should be placed exactly opposite the stone1 and at a distance from it corresponding to the magnitude of the pattern which is wanted4 ,the tube containing the lens or lenses1 being put on1 or the inner tube being drawn out1 if the instrument consists of two complete tubes1 it must be adjusted to the distance of the stone1 or till an image of the stone is formed at the end of the reflectors4 ,when this adjustment is perfect1 the ,kaleidoscope must be directed to the carved part of the stone1 out of which it is proposed to form the pattern2 and by slight changes in its position1 by turning it round its axis1 and by varying the inclination of the reflectors1 an immense variety of the most beautiful designs will be exhibited1 in the finest relief1 and as perfect as if they had been carved out of the stone by the most skilful workman4 ,the architect has therefore only to select from the profusion of designs which are thus presented to him2 and when he has made his choice1 he may either copy it photographically1 or with his eye1 or by means of the ,camera ,lucida2 or he may trace upon the stone the projection of the angular aperture of the instrument1 in order that1 in the execution of his work1 he may have constantly before his eyes the real element out of which the picture is created4 ,if1 in the course of this selection1 the picture should become capable of improvement1 either by giving it depth in particular parts1 or by altering the outline1 this alteration can be easily made1 and its effect throughout the whole ornament will be instantly seen4 ,if the architect is desirous to introduce into his ornament a natural object1 such as a leaf1 he may first try the effect which it will produce when applied in its natural state to the instrument1 and he may then carve either the whole or the half of it in stone1 and then examine what will be its final effect4 ,if one half of the leaf is an inverted image of the other half1 it is necessary only to carve one half of it1 and place the reflectors at an angle contained as many times in a circle as twice the number of times that he wishes the whole leaf to be multiplied3 for example1 if the whole leaf is to be multiplied six times1 the angle of the mirrors must be _/th of #cfj1 or #cj4 ,the very same effect would be obtained by applying the whole leaf to the instrument when the inclination of the reflectors is #fj1 or th of #cfj4 ,but when the whole leaf is not symmetrical in itself1 or consists of two dissimilar halves1 it must be applied in its entire state to the instrument4 ,in the formation of circular ,gothic windows1 the architect will find the ,kaleidoscope a most important auxiliary4 ,by applying it to a mullion drawn upon paper1 with a portion of the curves which he wishes to introduce1 or by placing it upon various ornamental parts of the drawing of a ,gothic cathedral1 he will obtain combinations to which he has never observed the slightest approximation4 ,in designing the decorations for ceilings1 which are generally made of plaster of ,paris1 the same method should be followed as that which we have described for architectural ornaments4 #b4 ,ornamental ,painting4 ,in ornaments carved out of stone or marble1 or formed from plaster of ,paris1 the idea of colour does not enter into the consideration of the architect4 ,the forms1 however1 which are necessary in ornamental painting1 are always associated with colour2 and therefore1 in the invention and selection of these forms1 the ,kaleidoscope performs a double task4 ,while it creates the outline1 it at the same time fills it up with colour2 and by representing the effect of the two in a state of combination1 it enables the artist to judge of the harmony of his tints1 as well as of the proportion of his forms4 ,in the decoration of public halls and galleries1 there is no species of ornament more appropriate than those which consist in the combination of single figures1 or of groups of heads1 which are either directly or metaphorically associated with the history or object of the institution4 ,regular historical paintings on the ceiling of a room are quite incompatible with the symmetrical character of a public gallery4 ,if they are well executed1 they can never be seen to advantage1 and therefore their individual effect is lost1 while1 from their very nature1 they cannot possibly produce that general effect as an ornament which good taste imperiously requires4 ,in employing regularly combined groups of figures1 there is sufficient scope given to the powers of the artist1 while the systematic arrangement of his work prevents it from interfering with the general character of the place which it is to embellish4 ,the effects which the ,kaleidoscope develops1 when applied to the representations of living objects1 will1 we have no doubt1 give very great surprise to those who have not previously examined them4 ,in order to enable the reader to form some notion of them1 we have given1 in the annexed ,plate1 a series of reduced figures1 taken principally from the antique4 ,in order to separate an individual figure from the rest1 we have only to cut an opening of nearly the same size in a piece of paper1 and lay it upon the surface of the plate1 so as to conceal all the adjoining figures1 and permit the required figure to be seen through the aperture4 ,by applying a ,kaleidoscope1 in which the inclination of the mirrors does not exceed #cj1 the figure will be combined into a fine pattern1 exhibiting1 perhaps1 the head and a part of the body in every sector2 while the hands1 or the lower extremities1 are thrown into the stiff part of the design4 ,the singular ease and grace with which the figure necessarily rises out of the formal part of the pattern1 and with which it connects itself with the general picture1 produces a new effect1 which1 so far as we can learn1 no artist had ever attempted to produce4 ,in order to group these single figures with perfect accuracy1 the ,kaleidoscopes constructed by ,mr4 ,bate should be employed1 as both his instruments have a contrivance which allows the light to fall freely upon the surface of the picture4#ae #ae ,see p4 #ic4 ,illustration3 ,,fig4 #ea4 ,when the figures which we wish to introduce are larger than the aperture of the ,kaleidoscope1 we must use the lens1 and place them at such a distance as to reduce them to the proper magnitude4 ,the effect which is produced by these simple outlines will convey some idea of the beauty which must characterize the designs when the figures are finely shaded1 or chastely coloured4 ,by the application of the lens1 paintings or statues of any size may be reduced and admitted into the figure4 ,in order to convey to the reader some idea of the effect produced by the ,kaleidoscope in grouping figures1 we have given in ,fig4 #ea a design created by the instrument4 ,the inclination of the mirrors by which the figures were arranged is #cf1 or th of a circle2 and therefore the object is multiplied ten times1 so as to give five pair of direct and inverted images4 #c4 ,designs for ,carpets4 ,there is none of the useful arts to which the creations of the ,kaleidoscope are more directly applicable than the manufacture of carpets4 ,in this case1 the manufacturer requires not merely the outline of a design1 but a design filled up with the most brilliant colours2 and upon the nature of the figure which he selects1 and the tints with which he enriches it1 will depend the beauty of the effect which is produced4 ,a carpet1 indeed1 is in general covered with a number of ,kaleidoscope designs1 arranged in lines parallel to the sides of the apartment2 and while this instrument creates an individual pattern1 it may also be employed1 by the assistance of the lens1 in exhibiting the effect or arranging or grouping these individual patterns1 according to the form of the apartment1 and other circumstances which should invariably be attended to4 ,when a plasterer ornaments the ceiling of a room1 the figure which he chooses is always related to the shape of the ceiling1 and varies according as it is circular1 elliptical1 square1 or rectangular4 ,in like manner1 a carpet should always have a relation to the form of the apartment1 not only in the shape and character of the individual designs1 but in the mode in which they are combined into a whole4 ,although the designs given by the ,kaleidoscope are in general circular1 yet1 when they are once drawn1 their outline may be made either triangular1 square1 rectangular1 elliptical1 or of any other form that we please1 without destroying their beauty4 ,the outline of the pattern may be varied in the instrument1 by varying the shape of the part of the tube or aperture which bounds the field of view at the widest end of the angular aperture2 but it is only at certain inclinations of the reflectors that any of the regular figures can be produced in this way4 ,if the bounding line is circular1 the field of view will be a circle2 if the bounding line is rectilineal1 and equally inclined to the reflectors1 the field of view will be a regular polygon1 of as many sides as the number of times that the angle of the reflectors is contained in #cfj2 if the bounding line is rectilineal1 but placed at right angles to one of the reflectors1 the figure will still be a regular polygon1 but its number of sides will be equal to half the number of times that the angle of the reflectors is contained in #cfj4 ,hence it follows1 that a square field may be obtained in two ways1 either by placing the mirrors at an angle of #de1 and making the bounding line perpendicular to one of the reflectors2 or by inclining the mirrors #ij1 and making the bounding line equally inclined to both reflectors2--and that a triangular field may be obtained1 either by inclining the mirrors #fj1 and setting the bounding line at right angles to one of the reflectors1 or by making the inclination #abj1 and placing the bounding line at an angle of #fj and #cj to the reflectors4 ,an elliptical field may be obtained1 by giving the bounding line the shape of one quarter of an ellipse1 and placing it in such a manner that the vertex of the conjugate axis falls upon one of the reflectors1 and the vertex of the transverse axis upon the other4 ,the form of the pattern being determined1 the next step is to select an outline1 and the colours which are to enter into its formation4 ,in order to do this to the greatest advantage1 the differently coloured worsteds which the manufacturer proposes to employ should be placed upon a plane surface1 either in the state of thread1 or1 what is much better1 when they are wrought into cloth4 ,these differently coloured pieces of carpet1 which we may suppose to be blue1 green1 and yellow1 must then be placed at the distance of a few feet from the ,kaleidoscope1 so that their image may1 by means of the lens1 be formed at the end of the reflectors4 ,in this state a very perfect pattern will be created by the instrument1 and the blue1 green1 and yellow colour will predominate according as a greater or a lesser portion of these colours happens to be opposite to the angular aperture4 ,by shifting the position of the ,kaleidoscope1 any one of the colours may be made to predominate at pleasure2 and the artist has it thus in his power1 not only to produce any kind of outline that he chooses1 but regulate the masses of colour by which it is to be filled up2 and to try the effects which will be produced by the juxtaposition of two colours1 by the separation of others1 or by the transference of the separate or combined masses to different parts of the design4 ,it would be foreign to our object to describe the apparatus by which these changes in the quantities of colour1 and in their relative position1 may be most easily and conveniently effected2 the artist can have no difficulty in constructing such an apparatus for himself1 and by means of it he will be enabled to obtain results from the ,kaleidoscope which he would have sought for in vain from any other method4 ,as the methods we have described of using the ,kaleidoscope in ornamental architecture1 or ornamental painting1 and in the manufacture of carpets1 will apply to the various other professions in which the formation of symmetrical designs is a necessary part1 ,i shall merely state1 that it will be found of the greatest advantage to the jeweller in the arrangement of precious stones2 to the bookbinder1 the wire-worker1 the paper-stainer1 and the artist who forms windows of painted glass4 ,in this last profession1 in particular1 the application of the ,kaleidoscope cannot fail to indicate combinations far superior to anything that has yet been seen in this branch of art4 ,from the uniformity of tint in the separate pieces of glass which are to be combined1 the effect produced by the instrument from portions of the very same glass that is to be used for the windows1 may be considered as a perfect fac-simile of the window when well executed on a large scale4 ,,chapter ,,xxi4 ,,on ,,the ,,photographic ,,delineation ,,of ,,the ,,pictures ,,created ,,by ,,the ,,kaleidoscope4 ,in a preceding chapter we have referred to the delineation of the pictures created by the ,kaleidoscope1 when they are received on the ground glass of a camera obscura1 or when a camera lucida is placed at the eye-end of the instrument4 ,both of these methods are very imperfect1 and when the pictures have been copied for useful purposes1 we believe that they have generally been executed by a skilful draughtsman1 who delineated carefully one of the sectors of which the figure is composed1 and then repeated it so as to complete the picture4 ,illustration3 ,,fig4 #eb4 ,since the invention of the ,kaleidoscope1 the discovery of the photographic art--of the ,daguerreotype and ,talbotype processes1 has given a new value to the instrument4 ,by means of a ,kaleidoscope ,camera1 the most complex figures can be almost instantaneously transferred to paper1 or to plated copper1 and hundreds of designs offered to the choice of the artist who is to employ them4 ,the very same figure which is obtained by one mode of illumination1 may be altered indefinitely1 by merely changing the direction or the intensity of the light1 all the objects which give the figure remaining fixed in the object-box4 ,the optical arrangement by which these figures may be copied1 photographically1 is substantially the same as that which is shown in ,fig4 #eb1 where ,c ,g ,d ,f is the ,kaleidoscope1 ,a ,o the object-box1 and ,l ,l a small achromatic lens1 of rock crystal or glass1 a quarter of an inch in diameter1 placed in contact with the extremities of the glass or metallic reflectors1 and having its centre immediately behind the small opening at ,e4 ,when the rays of the sun1 or any other strong light1 containing the actinic rays1 are thrown obliquely upon the object-box ,a ,o1 an image of the ,kaleidoscopic figure1 produced by the objects in the object-box1 will be formed at ,p ,p1 in the focus of the lens ,l ,l4 ,if the focal length of the lens ,l ,l is equal to one-half ,c ,g1 the length of the ,kaleidoscope1 the image will be formed behind ,l ,l1 at a distance equal to ,c ,g1 and of the same size as the object-box2 but if the focal length of ,l ,l is greater than the half of ,c ,g1 the image will be formed at a greater distance than ,c ,g1 behind the lens1 and the size of it will be greater than that of the object-box1 the distance and the size of the picture increasing as the focal length of the lens increases2 and when it becomes equal to ,c ,g1 the size and the distance of the picture will be infinitely great4 ,when the focal length of the lens is less than half ,c ,g1 the figure will be smaller than the object-box1 and nearer the lens4 ,the ,kaleidoscope ,camera4 ,every camera employed for the purposes of photography may be readily adapted for taking the pictures formed by the ,kaleidoscope4 ,we have only to take out the lens or lenses which belong to it1 and place the ,kaleidoscope furnished with its lens ,l ,l1 ,fig4 #eb1 in the inner tube1 which is movable by means of the rack and pinion4 ,if the picture can be made distinct on the grey glass by the rack and pinion1 a negative or positive copy of it may be taken on collodion or paper1 in the same manner as other photographs4 ,in some cameras the end of the box which contains the grey glass is movable1 backwards and forwards1 so that the adjustment for rendering the picture distinct may be effected1 though the ,kaleidoscope is fixed in the front portion of the box4 ,when the camera is made for the express purpose of taking ,kaleidoscope pictures1 it becomes a very simple instrument and may be constructed easily and cheaply4 ,it requires no lens excepting the small one ,l ,l of rock crystal or of glass1 a quarter of an inch in diameter4 ,the aperture required for this lens is so small that the spherical and chromatic aberration cannot injure1 in any sensible manner1 the distinctness of the picture4 ,the difference between the chemical and luminous focus1 which cannot be made to coincide with a single lens1 may be easily determined by experiment1 or in order to avoid this1 the small lens may be made achromatic4#af #af ,a thin lens of rock crystal will transmit more of the actinic rays than one rendered achromatic4 ,the form of the camera as fitted up with one of ,mr4 ,bate's ,kaleidoscopes1 with metallic reflectors1 is shown in the annexed figure1 where ,m ,c ,d ,n is the body of the camera1 m n o p the tube which is moved out and in by a milled head attached to the pinion which drives the rack on the tube m n o p4 ,the ,kaleidoscope ,a ,b ,l ,l1 exhibiting the figure which is to be copied1 is temporarily fixed in this tube by pieces of cork or wood1 so that its axis1 or the line of junction of the reflectors1 may be perpendicular to the surface of the grey glass p q1 which can be taken out1 so as to allow the collodion or paper slide to be placed in the same groove4 ,illustration3 ,,fig4 #ec4 ,the pictures should always be taken1 when it is possible1 by sunlight incident obliquely upon the object-box1 or with the illuminators already described4 ,when this cannot be done1 artificial light1 containing the actinic or chemical rays1 may be thrown upon the object1 as shown in ,fig4 #eb4 ,the light of coal gas1 which may be condensed for any length of time upon the objects in the object-box1 will be found to answer the purpose2 but if a more rapid process is required1 we may1 as proposed by ,mr4 ,moule1 use the ,bengal lights1 which contain a large quantity of the actinic rays4#ag ,these lights are composed of six parts of nitre1 two of sulphur1 and one of the sulphuret of antimony4 ,the powder is made up in the shape of a cone1 which is ignited at the top2 but it would be better to spread it out widely1 in order that it may be condensed upon the object-box4 ,the lime-ball light may also be employed1 but the ,bude light1 which is a coal gas flame1 rendered more intensely luminous by a stream of oxygen1 is preferable to any other artificial flame4 ,in using it1 or any other of the lights already mentioned1 for obtaining ,kaleidoscope figures1 it must either be placed very near the object-box1 or condensed upon it by a large lens1 on account of the small size of the aperture at the eye-end of the ,kaleidoscope4 #ag ,see the ,journal of the ,photographic ,society1 vol4 iv4 p4 #acg4 ,in the formation of the ,kaleidoscope figures1 it is necessary to pay particular attention to the colour of the objects of which they are to be composed1 red1 orange1 and dark yellow objects1 which transmit few1 if any1 actinic rays1 should not be employed4 ,blue glasses1 which transmit no red rays1 are actinically more luminous than colourless glasses1 and therefore pale blue1 pale green1 and pale yellow glasses may be advantageously combined with colourless glasses as objects for the object-box4 ,when we wish to copy photographically the pictures obtained from external objects1 we must use the ,telescopic ,kaleidoscope of ,mr4 ,bates1 by which the images of these objects are formed at the extremity of the reflectors4 ,these objects may be flowers1 plants1 architectural ornaments1 or parts of them1 paintings1 photographs1 statues1 or loose and irregularly placed materials of any kind4 ,in the figures created by the common ,kaleidoscope1 we must trust to chance for the formation of the figures1 and can only choose those which we like out of a large number accidentally formed2 but when we make use of external objects1 we can group them and illuminate them in any manner we choose1 selecting those forms and colours which1 when combined1 produce the effect that we desire4 ,this power1 of arranging the objects1 when in contact with the ends of the reflectors1 may be obtained with the ,kaleidoscope of ,mr4 ,bates1 shown in ,fig4 #bi1 by removing the anterior half of the cone when the objects are opaque1 and placing the camera in a vertical position4 ,the objects being placed upon a horizontal plate1 may1 if opaque1 be illuminated1 by reflected light1 and the picture of them1 when combined by the ,kaleidoscope1 taken in the same manner as when the instrument was in the position shown in ,fig4 #eb4 ,if the figure is not what we like1 we can shift any individual object1 or remove it1 or replace it by another1 till we are satisfied with the combination4 ,the very same power of altering the combination1 may be obtained for transparent objects4 ,we have only to place these objects on a horizontal plate of glass1 not connected with the ,kaleidoscope1 but capable of being removed from it1 and again brought close to the ends of the reflectors4 ,when the objects on the glass plate have been put into their proper place1 so as to give the desired picture1 the plate is brought as near as possible to the ends of the reflectors1 and the objects are illuminated either by solar or other light reflected upwards upon the object-box4 ,,chapter ,,xxii4 ,,on ,,the ,,advantages ,,of ,,the ,,kaleidoscope ,,as ,,an ,,instrument ,,of ,,amusement4 ,the splendid discoveries which have been made with the telescope and microscope have invested them with a philosophical character which can never be attached to any other instrument4 ,it is only1 however1 in the hands of the astronomer and naturalist that they are consecrated to the great objects of science2 their ordinary possessors employ them solely as instruments of amusement1 and it is singular to remark how soon they lose their novelty and interest when devoted to this inferior purpose4 ,the solar microscope1 the camera obscura1 and the magic lantern1 are equally shortlived in their powers of entertainment2 and even the wonders of the electrical and galvanic apparatus are called forth1 at long intervals1 for the occasional purposes of instructing the young1 or astonishing the ignorant4 ,a serenity of sky very uncommon in our northern climate1 is absolutely necessary for displaying the powers of some of the preceding instruments2 and the effects of the rest can only be exhibited after much previous preparation4 ,from these causes1 but principally from a want of variety in their exhibitions1 they have constantly failed to excite1 in ordinary minds1 that intense and continued interest which might have been expected from the ingenuity of their construction and the splendour of their effects4 ,the pleasure which is derived from the use of musical instruments is of a different kind1 and far more intense in its effects1 and more general in its influence than that which is obtained from any of the preceding instruments4 ,there are1 indeed1 few minds that are not alive to the soothing and exhilarating influence of musical sounds1 or that do not associate them with the dearest and most tender sympathies of our nature4 ,but the ear is not the only avenue to the heart2 and though sorrow and distress are represented by notes of a deep and solemn character1 and happiness and gaiety by more light and playful tones1 the same kind of feelings may also be excited by the exhibition of dark and gloomy colours1 and by the display of bright and arial tints4 ,the association1 indeed1 is not so powerful in the one case as in the other1 for we have been taught from our infancy1 in consequence of the connexion of music and poetry1 to associate particular sentiments with particular sounds2 but there can be no doubt that the association of colour is naturally as powerful as that of sound1 and that a person who has never listened to any other music but that of nature1 nor seen any other colours but those of the material world1 might have his feelings as powerfully excited through the medium of the eye as through that of the ear4 ,the first person who attempted to supply the organ of vision with the luxuries of light and colour1 was ,father ,castel1 a learned ,jesuit1 who had distinguished himself chiefly by his opposition to the splendid optical discoveries of ,newton4 ,about the year #agbe or #agbf1 he published in the ,mercure de ,paris1 his first ideas of an organ1 or ocular harpsichord4 ,a full account of this curious instrument was afterwards published at ,hamburgh1 in #agci1 by ,m4 ,tellemann1 a ,german musician1 who had seen one of the harpsichords in the possession of the inventor1 when he was on a visit to ,paris4 ,this account was afterwards translated into ,french1 and printed at the end of ,castel's ,l',optique des ,couleurs1#ah which appeared at ,paris in #agdj4 ,the ocular harpsichord is a common harpsichord1 fitted up in such a manner1 that when a certain sound is produced by striking the keys1 a colour related to that sound is at the same instant exhibited to the eye in a box or frame connected with the harpsichord2 so that when a piece of music is played for the gratification of the ear1 the eye is simultaneously delighted by the display of corresponding colours4 #ah ,this work is entitled1 ,l',optique des ,couleurs1 fonde sur les simples observations et tourn surtout la pratique de la peinture1 de la teinture1 et des autres arts coloristes4 ,par le ,r4 ,p4 ,castel1 ,jesuite4 ,paris1 #agdj4 ,in adjusting the colours and the sounds1 ,castel lays down the following six propositions3-- #ast1--,there is a fundamental and primitive sound in nature1 which may be called ut1 and there is also an original and primitive colour1 which is the foundation of all other colours1 namely1 blue4 #b;d1--,there are three chords1 or essential sounds1 which depend upon the primitive sound ut1 and which compose with it a primitive and original accord1 and these are ut1 mi1 sol4 ,there are also three original colours depending on the blue1 which1 while they are not composed of any other colours1 produce them all1 namely1 blue1 yellow1 and red4 ,the blue is here the note of the tone1 the red is the fifth1 and the yellow is the third4 #c;d1--,there are five tonic chords1 ut1 re1 mi1 sol1 la1 and two semitonic chords1 fa and si4 ,there are also five tonic colours1 to which all the rest are ordinarily related1 namely1 blue1 green1 yellow1 red1 and violet1 and two semitonic or equivocal colours1 namely1 aurora and violant 7related to the orange and indigo of ,newton74 #dth1--,out of these five entire tones1 and two semitones1 is formed the diatonic scale1 ut1 re1 mi1 fa1 sol1 la1 and si2 and1 in like manner1 out of the five entire or tonic colours1 and the two demi-colours1 are formed the gradation of colours1 blue1 green1 yellow1 aurora1 red1 violet1 and violant2 for the blue leads to green1 which is demi-blue2 the yellow to aurora1 which is gilded yellow2 the aurora leads to red1 the red to violet1 which is two-thirds of red1 and one-third of blue2 and the violet to violant1 which has more blue than red4 #eth1-- ,the entire tones divide themselves into semitones2 and the five entire tones of the scale or gamut1 comprehending in this the two natural semitones1 make twelve semitones1 viz41 ut natural1 ut dieze1 re1 re dieze1 mi1 fa1 fa dieze1 sol1 sol dieze1 la1 la dieze1 and si4 ,in like manner there are twelve demi-colours1 or demi-tints1 and there can be neither more nor less1 according to the opinion of painters themselves1 and as may be demonstrated by other means4 ,these colours are blue1 sea-green1 green1 olive1 yellow1 aurora1 orange1 red1 crimson1 violet1 agathe1 and violant4 ,blue leads to sea-green1 which is a greenish blue2 sea-green leads to green2 green to olive1 which is a yellowish green2 olive to yellow2 yellow to aurora2 aurora to orange2 orange to red1 the colour of fire2 red to crimson1 which is red mixed with a little blue2 crimson to violet1 which is still more blue2 violet to agathe1 or bluish violet2 and agathe to violant4 #fth1--,the progression of sounds is in a circle1 setting out from ut1 and returning back3 thus1 ut1 mi1 sol1 ut1 or ut1 re1 mi1 fa1 sol1 la1 si1 ut4 ,this is called an octave1 when the last ut is one-half more acute than the first4 ,the colours also have their progression in a circle4 #gth1--,after an octave1 ut1 re1 mi1 fa1 sol1 la1 si1 there recommences a new one1 which is one-half more acute1 and the whole circle of music produces several octaves4 ,such are the principles upon which the ocular harpsichord was founded2 but though the instrument1 from its singularity1 excited great attention when it was first constructed1 we have not been able to learn that it was ever supposed to possess the power of affording pleasure to the eye4 ,it must be obvious1 indeed1 to any person who considers the subject1 that colour1 independent of form1 is incapable of yielding a continued pleasure4 ,masses of rich and harmonious tints1 following one another in succession1 or combined according to certain laws1 would no doubt give satisfaction to a person who had not been familiar with the contemplation of colours2 but this satisfaction would not be permanent1 and he would cease to admire them as soon as they ceased to be new4 ,colour is a mere accident of light1 which communicates richness and variety to objects that are otherwise beautiful2 but perfection of form is a source of beauty1 independent of all colours2 and it is therefore only from a combination of these two sources of beauty that a sensation of pleasure can be excited4 ,those who have been in the habit of using a correct ,kaleidoscope1 furnished with proper objects1 will have no hesitation in admitting that this instrument realizes1 in the fullest manner1 the formerly chimerical idea of an ocular harpsichord4 ,the combination of fine forms1 and ever-varying tints1 which it presents in succession to the eye1 have already been found1 by experience1 to communicate to those who have a taste for this kind of beauty1 a pleasure as intense and as permanent as that which the finest ear derives from musical sounds4 ,an eye for admiring and appreciating the effect of fine forms1 seems1 indeed1 to be much more general than an ear for music2 and we have heard of many cases where the tedium of severe and continued indisposition has been removed1 and where many a dull and solitary hour has been rendered cheerful1 by the unceasing variety of entertainment which the ,kaleidoscope afforded4 ,in one respect1 indeed1 this instrument is superior to all others4 ,when it is once properly constructed1 its effects are exhibited without either skill or labour2 and so numerous are its applications1 and so inexhaustible its stores1 that the observer is constantly flattered with the belief that he has obtained results which were never seen before1 and that he has either improved the instrument1 or extended its power1 by new applications4 ,such are the advantages1 as an instrument of amusement1 which the ,kaleidoscope possesses1 even in its present imperfect state4 ,to what degree of perfection it may yet arrive1 it is not easy to anticipate2 but we may venture to predict1 because we see the steps by which the prediction is to be fulfilled1 that combinations of forms and colours may be made to succeed each other in such a manner as to excite sentiments and ideas with as much vivacity as those which are excited by musical composition4 ,if it be true that there are harmonic colours which inspire more pleasure by their combination than others2 that dull and gloomy masses1 moving slowly before the eye1 excite feelings of sadness and distress2 and that the arial tracery of light and evanescent forms1 enriched with lively colours1 are capable of inspiring us with cheerfulness and gaiety2 then it is unquestionable1 that1 by a skilful combination of these passing visions1 the mind may derive a degree of pleasure far superior to that which arises from the immediate impression which they make upon the organ of vision4 ,a very simple piece of machinery is alone necessary for introducing objects of different forms and colours1 for varying the direction of the motion across the angular aperture1 and for accommodating the velocity of their motion to the effect which it is intended to produce4 ,these combinations of colours and forms may be adapted to a piece of music1 and their succession exhibited on a screen by means of the electric1 or lime-ball1 or other lights to which we have already had occasion to refer4 ,the coloured objects might be fixed between long stripes of glass1 moved horizontally or obliquely across the ends of the reflectors2 and the effects thus obtained might be varied by the occasional introduction of revolving object-boxes1 containing objects of various colours and forms1 partly fixed and partly movable4 ,similar forms in different colours1 and in tints of varying intensity1 losing and resuming their peculiar character with different velocities1 and in different times1 might exhibit a distinct relation between the optical and the acoustic phenomena simultaneously presented to the senses4 ,flashes of light1 coloured and colourless1 and clouds of different depths of shadow1 advancing into1 or emerging from the centre of symmetry1 or passing across the radial lines of the figure at different obliquities1 would assist in marking more emphatically the gay or the gloomy sounds with which they are accompanied4 ,a slight idea of the effects which might be expected from an ingenious piece of mechanism for creating and combining the various optical phenomena1 and exhibiting them in connexion with musical sounds1 may be obtained by a single observer1 who looks into a fine ,kaleidoscope1 firmly fixed upon a stand1 and produces with his two hands all the variations in form and colour which he can effect by such inadequate means1 and which he considers appropriate to the musical piece that accompanies them4 ,,chapter ,,xxiii4 ,,history ,,of ,,the ,,combinations ,,of ,,plane ,,mirrors ,,which ,,have ,,been ,,supposed ,,to ,,resemble ,,the ,,kaleidoscope4 ,it has always been the fate of new inventions to have their origin referred to some remote period2 and those who labour to enlarge the boundaries of science1 or to multiply the means of improvement1 are destined to learn1 at a very early period of their career1 that the desire of doing justice to the living is a much less powerful principle than that of being generous to the dead4 ,this mode of distributing fame1 injurious as it is to the progress of science1 by taking away one of the strongest excitements of early genius1 has yet the advantage of erring on the side of generosity2 and there are few persons who would reclaim against a decision invested with such a character1 were it pronounced by the grave historian of science1 who had understood and studied the subject to which it referred4 ,the apparent simplicity1 both of the theory and the construction of the ,kaleidoscope1 has deceived very well-meaning persons into the belief that they understood its mode of operation2 and it was only those that possessed more than a moderate share of optical knowledge1 who saw that it was not only more difficult to understand1 but also more difficult to execute1 than most of the philosophical instruments now in use4 ,the persons who considered the ,kaleidoscope as an instrument consisting of two reflectors1 which multiplied objects1 wherever these objects were placed1 and whatever was the position of the eye1 provided that it received only the reflected rays1 were at no loss to find numerous candidates for the invention4 ,all those1 indeed1 who had observed the multiplication and circular arrangement of a fire blazing between two polished plates of brass or steel2 who had dressed themselves by the aid of a pair of looking-glasses1 or who had observed the effects of two mirrors placed upon the rectangular sides of a drawing-room1 were entitled1 upon such a definition1 to be constituted inventors of the ,kaleidoscope4 ,the same claim might be urged for every jeweller who had erected in his window two perpendicular mirrors1 and placed his wares before them1 in order to be multiplied and exhibited to advantage2 and for every ,dutch toy-maker1 and dealer in optical wonders1 who had manufactured show-boxes1 for the purpose of heaping together1 in some sort of order1 a crowd of images of the same object1 of different intensities1 seen under different angles1 and presenting different sides to the eye4 ,this mode of grouping images1 dissimilar in their degree of light1 dissimilar in their magnitude1 and dissimilar in their very outlines1 produced such a poor effect1 that the reflecting show-boxes have for a long series of years disappeared from among the number of philosophical toys4 ,from these causes1 the candidates for the merit of inventing the ,kaleidoscope have been so numerous1 that they have started up in every part of the world2 and many individuals1 who are scarcely acquainted with the equality of the angles of incidence and reflexion1 have not scrupled to favour the world with an account of the improvements which they fancy they have made upon the instrument4#ai #ai ,it would be an easy matter to amuse the reader with an account of these improvements4 ,one of the most notable of them consists in covering the back of the reflectors with white paint1 for the purpose of increasing the light of the circular field4 ,this scheme is identically the same as if the author had proposed to improve the magnificent telescopes of ,herschel1 when rendered dark with a high magnifying power1 by white-washing the interior of the tube4 ,the earliest writer who appears to have described the use of two plane mirrors1 was ,baptista ,porta1 who has given an account of several experiments which he performed with them1 in the second chapter of the seventh book of his ,magia ,naturalis4 ,as the combination of plane mirrors which he there describes has been represented as the same as the ,kaleidoscope1 we shall give the passage at full length3-- ,speculum planis multividum construere4 8,speculum construitur1 quod polyphaton id est multorum visibilium dicitur1 illud enim aperiendo et claudendo solius digiti viginti et plura demonstrat simulacra4 ,sic igitur id parabis4 rea duo specula vel crystallina rectangula super basim eandem erigantur1 sintque in hemiolia proportione1 vel alia1 et secundum longitudinis latus unum simul colligentur1 ut libri instar apte claudi et aperiri possint1 et anguli diversentur1 qualia ,venetiis factitari solent3 faciem enim unam objiciens1 in utroque plura cernes ora1 et hoc quanto arctius clauseris1 minorique fuerint angulo3 aperiendo autem minuentur2 et obtusiori cernes angulo1 pauciora numero conspicientur4 ,sic digitum ostendens1 non nisi digitos cernes1 dextra insuper dextra1 et sinistra sinistra convisuntur2 quod speculis contrarium est3 mutuaque id evenit reflexione1 et pulsatione1 unde imaginum vicissitudo40 --,edit4 ,amstelod1 #affd4 ,the following is an exact and literal translation of this passage3-- ,how to construct a multiplying speculum out of plane ones4 ,a speculum is constructed1 called polyphaton1 that is1 which shows many objects1 for by opening and shutting it1 it exhibits twenty and more images of the finger alone4 ,you will therefore prepare it in the following manner3--,let two rectangular specula of brass or crystal be erected upon the same base1 and let their length be one and a half times their width1 or in any other proportion2 and let two of their sides be placed together1 so that they may be opened and shut like a book1 and the angles varied1 as they are generally made at ,venice4 ,for by presenting your face1 you will see in both more faces the more they are shut1 and the less that the angle is2 but they will be diminished by opening it1 and you will see fewer as you observe with a more obtuse angle4 ,if you exhibit your finger1 you will see only fingers1 the right fingers being seen on the right side1 and the left on the left side1 which is contrary to what happens in looking-glasses1 and this arises from the mutual reflexion and repulsion which produce a change of the images4#bj #bj ,we request that the reader will take the trouble of comparing with the original the following translation of ,baptista ,porta's description1 which was published in ,london1 and copied into all the foreign newspapers1 etc4 ,we hope the translator of it had no improper motive in altering the obvious meaning of the original2 yet it is singular1 that in the journal where this translation appeared1 the specification of the patent was published under the title of ,directions for making the ,kaleidoscope1 purporting to be an original communication to that journal1 the name of the patentee1 and the technical parts of the specification having been left out1 apparently to promote the belief that there was no patent1 and that every person might make them with impunity4 ,the following is the translation alluded to3-- 8,in the following manner we may construct a mirror for seeing a multitude of objects on a plain surface4 ,this kind of mirror1 when constructed1 is what is called polyphaton1 that is to say1 multiplying1 for by opening and shutting1 it shows twenty and more images of one single finger4 ,if1 therefore1 you wish to prepare it1 let two brazen or crystal rectangular mirrors be erected on the same base1 and let the proportion of length be one and a half of the width1 or any other proportion2 and let each side for the whole of its length be so connected together that they may easily be shut and opened like a book1 and that the angles may be varied1 as they are usually constructed at ,venice2 for if you place one object opposite to the face of each1 you will see several figures2 and this in proportion as you shut it closer1 and the angle shall be less4 ,but1 by opening1 the objects will be reduced in number1 and the more obtuse the angle under which you see it1 the fewer objects will be seen4 ,so if you exhibit your finger as the object1 you will see nothing but fingers4 ,the right fingers will be seen on the right side1 and the left on the left side1 which is contrary to the usual custom with looking-glasses2 but this happens from the mutual reflexion and repulsion which produce a change of the images40 ,it is quite obvious1 from the preceding passage1 that the multiplying speculum described by ,porta was not an invention of his own1 but had been long made at ,venice4 ,in the very next chapter1 indeed1 where he describes a speculum theatrale or amphitheatrale 7the show-box of ,harris and other modern authors71 he expressly states1 that a multiplying speculum was invented by the ancients4--8,speculum autem planis compactum1 cui si unum spectabile demonstrabitur1 plura illius rei simulachra demonstrabit1 prudens invenit vetustas2 ut ex quibusdam ,ptolemi scriptis qu circumferuntur percipitur20 that is1 8a speculum consisting of plane ones1 which1 when one object is presented to it1 will exhibit several images of it1 was invented by skilful antiquity1 as appears from some of the writings of ,ptolemy40 ,this speculum1 which consists of several mirrors arranged in a polygon1 with the object within it1 is characterized by ,porta as puerile1 and much less wonderful and agreeable than one of his own1 which he proceeds to describe4 ,this new speculum consists of ten mirrors1 placed within a box1 in a sort of polygonal form1 with one of the sides of the polygon open4 ,architectural columns1 pictures1 gems1 pearls1 coloured birds1 etc41 are all placed within the box1 and their images are seen heaped together in inextricable disorder1 as a whole1 but so as to astonish the spectator by their number1 and by the arrangement of individual groups4 ,porta speaks of this effect as so beautiful1 8ut nil jucundius nil certe admirabilius oculis occurset nostris10--that nothing more agreeable1 and certainly nothing more admirable1 was ever presented to our eyes4 ,it would be an insult to the capacities of the most ordinary readers to show that the instruments here described by ,baptista ,porta have no farther connexion with the ,kaleidoscope than that they are composed of plane mirrors4 ,the sole purpose of these instruments was to multiply objects by reflexion2 and so little did the idea of producing a symmetrical picture enter into ,baptista ,porta's contemplation1 that he directs the mirrors to be placed at any angle1 because the multiplying property of the mirrors is equally developed1 whatever be the angle of their inclination4 ,the show-box of which ,porta speaks with such admiration1 has so many mirrors1 and these are placed at such angles1 that not one of the effects of the ,kaleidoscope can be produced from them4 ,its beauty is entirely derived from the accumulation of individual images4 ,the next competitor for the invention of the ,kaleidoscope is the celebrated ,kircher1 who describes1 as an invention of his own1 the construction of two mirrors which can be opened and shut like the leaves of a book4 ,this instrument is represented in ,fig4 #ed1 and is described in the following passage3-- ,illustration3 ,,fig4 #ed4 ,,parastasis ,i4 ,specula plana multiplicativa sunt specierum unius rei4 ,vide ,fig4 #ed4 ,mira qudam et a nemine1 quod sciam observata proprietas elucescit in duobus speculis ita constructis1 ut ad instar libri claudi et aperiri possint2 ponantur illa in plano quopiam1 in quo semicirculum in gradus suos descriptum habeas4 ,si enim punctum1 in quo specula committuntur1 in centro semicirculi statuas1 ita ut utrumque speculi latus diametro insistat1 semel tantam videbitur rei imago1 apparebuntque du res1 una extra speculum vera1 altera intra1 phantastica3 ,si vero specula ita posueris1 ut divaricatio laterum #abj gradus intercipiat1 videbis rei intra latera est1 quia angulus reflexionis et incidenti tantus est1 quantus est angulus interceptus a lateribus1 videlicet #abj grad4 qui cum obtusius sit1 non nisi binam imaginem causare potest1 ut in ,propos4 v4 fol4 #hdh1 ostensum est4 ,si vero specula interceperint angulum #ij graduum1 videbis in plano circulum in quatuor partes divisum1 in quibus totidem simulacra rei posit comparabunt1 tria phantastica1 et unum verum2 cum enim reflectio fiat ad angulos rectos utrumque latus reflectens formam causabit intra se alias duas formas1 unde et consequentur pro multiplicatione laterum form multiplicabuntur1 qu et in reflectione laterum normam servabunt uti in ,propos4 v4 fol4 #hdh1 ostendimus4 ,porro si speculorum latera interceperint angulum #gb graduum1 videbis in plano horizontali efformari perfectum et regulare pentagonum1 in quo totidem form apparebunt1 item1 si sexaginta graduum interceperint angulum1 videbis hexagonum totidemque formas quinque nimirum phantasticas unam veram4 ,ita1 si speculorum angulus interceperit #ea gradus cum _/ comparebit perfectum heptagonum1 cum totidem rei intra specula collocat formis2 non secus angulus speculorum #de graduum dabit octogonum2 #dj graduum dabit enneagonum2 #cf graduum decagonum2 #cb graduum angulus cum _/ dabit endecagonum et denique angulus #cj graduum referet dodecagonum cum totidem formis1 et sic in infinitum2 ita ut semper tot laterum sit futurum polygonum anacampticum totidemque formarum1 quot polygonum cuivis latus speculorum intercipit divaricatio1 latera habuerit3 quorum omnium rationes dependent a ,propos4 v4 precedentis distinctionis4 --,kircheri1 ,ars ,magna ,lucis et ,umbr4 ,rom4 #afdf1 p4 #hij4 ,the following is a translation of the preceding passage3-- ,,parastasis ,i4 ,plane specula may be made to multiply the images of one object4 ,see ,fig4 #ed4 ,a wonderful1 and1 so far as ,i know1 a new property1 is exhibited by two specula1 so constructed that they may be opened and shut like a book4 ,if they are placed upon any plane1 in which there is a semicircle divided into degrees1 in such a manner that the point where the specula meet is in the centre of the circle1 and the edge of each speculum stands upon the diameter of the semicircle1 one image only will be visible1 and there will appear two things1 namely1 a real one1 without the specula1 and another formed by reflexion behind them1#ba and so on1 as in the following table1 where the first column shows the inclination of the specula1 and the second the figure which is produced3-- #ba ,i have thrown the rest of the passage into a tabular form1 that the reader may see1 more readily1 the effect produced by the variation of the angle4 ,angle of ,specula4 ,effect produced4 #ahj one image and one object1 99#abj two images and one object1 #ij four images1 99#gb a pentagon and five forms1 #fj a hexagon and six forms1 99#ea_/ a heptagon and seven forms1 #de an octagon and eight forms1 99#dj an enneagon and nine forms1 #cf a decagon and ten forms1 99#cb_/ an endecagon and eleven forms1 #cj a dodecagon and twelve forms1 and so on1 ad infinitum1 the polygon formed by reflexion having always as many sides as the number of times that the angle of the specula is contained in #cfj4 ,the combination of plane mirrors1 which ,kircher describes in the preceding extract1 is precisely the same as that which is given by ,baptista ,porta4 ,the latter1 indeed1 only mentions1 that the number of images increases by the diminution of the angle1 whereas ,kircher gives the number of images produced at different angles1 and enumerates the regular polygons which are thus formed4 ,it must be quite obvious to any person who attends to ,kircher's description1 that the idea never once occurred to him of producing beautiful and symmetrical forms by means of plane mirrors4 ,his sole object was to multiply a given regular form a certain number of times2 and he never imagined that1 when the mirrors were placed at the angles marked with an asterisk1 there could be no symmetry in the figure1 and no union of the two last reflected images1 unless in the case where a regular object was placed1 either by design or by accident1 in a position symmetrically related to both the reflectors4 ,in ,kircher's mirrors the eye was placed in front of them4 ,the object therefore was much nearer the eye than the images1 and the light of the different reflected images was not only extremely unequal1 but the difference in their angular magnitude was such that they could not possibly be united into a symmetrical whole4 ,from the accidental circumstance of his using lines upon paper as an object1 the distortion of the pictures arising from the erroneous position of the eye was prevented2 but if the same combination of mirrors were applied to the object-plates of the ,kaleidoscope1 it would be found utterly incapable of producing any of the fine forms which are peculiar to that instrument4 ,if it were necessary to prove that ,kircher and his pupils were entirely ignorant of the positions of the eye and the object which are necessary to the production of a picture1 symmetrical in all its parts1 and uniformly illuminated1 and that they went no farther than the mere multiplication of forms that were previously regular and symmetrical1 we would refer the reader to ,schottus' ,magia ,universalis ,natur et ,artis1 printed at ,wurtzbourg in #afeg1 where he repeats1 almost word for word1 the description of ,kircher1 and adds the following curious observation3--8,but it is not only the objects placed in the semicircle in the angle of the glasses that are seen and multiplied1 but also those which are more distant2 for example1 a wall1 with its windows1 and in this case the multiplication produced by the mirrors will create an immense public place1 adorned with edifices and palaces40 ,this passage shows1 in the clearest manner1 not only that the multiplication of an object1 independent of the union of the multiplied objects into a symmetrical whole1 was all that ,kircher and his followers proposed to accomplish2 but also that they were entirely unacquainted with the effects produced by varying the distance of the object from the mirrors4 ,if any person should doubt the accuracy of this observation1 we would request him to take ,kircher's two mirrors1 to direct them to a 8wall with its windows10 either by ,kircher's method1 or even by any other way that he chooses1 and to contemplate 8the public place adorned with edifices and with palaces40 ,he will see heaps of windows and of walls1 some of the heaps being much larger than others2 and some being farther from1 and others nearer to1 the centre2 and some being dark1 and others luminous2 while all of them are disunited4 ,let him now take a ,telescopic ,kaleidoscope1 and direct it to the same object2 he will instantly perceive the most perfect order arise out of confusion1 and he will not scruple to acknowledge1 that no two things in nature can be more different than the effects which are produced by these two combinations of mirrors4 ,illustration3 ,,fig4 #ee4 ,illustration3 ,,fig4 #ef4 ,we come now to consider the claims of ,mr4 ,bradley1 ,professor of ,botany in the ,university of ,cambridge4 ,in a work1 entitled1 ,new ,improvements in ,planting and ,gardening1 published in #agag1 this author has drawn and described ,kircher's apparatus as an invention of his own2 and1 instead of having in any respect improved it1 he has actually deteriorated it1 in so far as he has made the breadth of the mirrors greater than their height4 ,an exact copy of the mirrors used by ,bradley is shown in ,fig4 #ee1 from which it will be at once seen1 that it is precisely the same as ,kircher's1 shown in ,fig4 #ed4 ,we are far from saying that ,bradley stole the invention from ,kircher1 or that ,kircher stole it from ,baptista ,porta1 or that ,baptista ,porta stole it from the ancients4 ,there is reason1 on the contrary1 to think that the apparatus had been entirely forgotten1 in the long intervals which elapsed between these different authors1 and there can be no doubt that each of them added some little improvement to the instrument of their predecessors4 ,baptista ,porta saw the superiority of two mirrors1 as a multiplying machine1 to a greater number used by the ancients4 ,kircher showed the relation between the number of images and the inclination of the mirrors2 and ,bradley1 though he rather injured the apparatus1 yet he had the merit of noticing1 that figures upon paper1 which had a certain degree of irregularity1 like those in ,fig4 #ef1 could still form a regular figure4 ,in order that the reader may fully understand ,bradley's method of using the mirrors1 we shall give it in his own words3-- 8,we must choose two pieces of looking-glass 7says he71 of equal bigness1 of the figure of a long square1 five inches in length1 and four in breadth2 they must be covered on the back with paper or silk1 to prevent rubbing off the silver1 which would else be too apt to crack off by frequent use4 ,this covering for the back of the glasses must be so put on1 that nothing of it may appear about the edges on the bright side4 8,the glasses being thus prepared1 they must be laid face to face1 and hinged together1 so that they may be made to open and shut at pleasure1 like the leaves of a book2 and now the glasses being thus fitted for our purpose1 ,i shall proceed to explain the use of them4 8,draw a large circle upon paper2 divide it into three1 four1 five1 six1 seven1 or eight equal parts2 which being done1 we may draw in every one of the divisions a figure1 at our pleasure1 either for garden-plats or fortifications2 as1 for example1 in ,fig4 #ee1 we see a circle divided into six parts1 and upon the division marked ,f is drawn part of a design for a garden4 ,now1 to see that design entire1 which is yet confused1 we must place our glasses upon the paper1 and open them to the sixth part of the circle1 i4e41 one of them must stand upon the line b1 to the centre1 and the other must be opened exactly to the point c2 so shall we discover an entire garden-plot in a circular form 7if we look into the glasses71 divided into six parts1 with as many walks leading to the centre1 where we shall find a basin of an hexagonal figure4 8,the line ,a1 where the glasses join1 stands immediately over the centre of the circle1 the glass ,b stands upon the line drawn from the centre to the point ,c1 and the glass ,d stands upon the line leading from the centre to the point ,e3 the glasses being thus placed1 cannot fail to produce the complete figure we look for2 and so whatever equal part of a circle you mark out1 let the line ,a stand always upon the centre1 and open your glasses to the division you have made with your compasses4 ,if1 instead of a circle1 you would have the figure of a hexagon1 draw a straight line with a pen from the point c to the point b1 and1 by placing the glasses as before1 you will have the figure desired4 8,so likewise a pentagon may be perfectly represented1 by finding the fifth part of a circle1 and placing the glasses upon the outlines of it2 and the fourth part of a circle will likewise produce a square1 by means of the glasses1 or1 by the same rule1 will give us any figure of equal sides4 ,i easily suppose that a curious person1 by a little practice with these glasses1 may make many improvements with them1 which1 perhaps1 ,i may not have yet discovered1 or have1 for brevity sake1 omitted to describe4 8,it next follows that ,i explain how1 by these glasses1 we may1 from the figure of a circle1 drawn upon paper1 make an oval2 and also1 by the same rule1 represent a long square from a perfect square4 ,to do this1 open the glasses1 and fix them to an exact square2 place them over a circle1 and move them to and fro till you see the representation of the oval figure you like best2 and so1 having the glasses fixed1 in like manner move them over a square piece of work till you find the figure you desire of a long square4 ,in these trials you will meet with many varieties of designs4 ,as for instance1 ,fig4 #ef1 although it seems to contain but a confused representation1 may be varied into above two hundred different representations1 by moving the glasses over it1 which are opened and fixed to an exact square4 ,in a word1 from the most trifling designs1 we may1 by this means1 produce some thousands of good draughts4 8,but that ,fig4 #ef may yet be more intelligible and useful1 ,i have drawn on every side of it a scale1 divided into equal parts1 by which means we may ascertain the just proportion of any design we shall meet within it4 8,i have also marked every side of it with a letter1 as ,a1 ,b1 ,c1 ,d1 the better to inform my reader of the use of the invention1 and put him in the way to find out every design contained in that figure4 8,example ,i4--,turn the side ,a to any certain point1 either to the north1 or to the window of your room2 and when you have opened your glasses to an exact square1 set one of them on the line of the side ,d1 and the other on the line of the side ,c1 you will then have a square figure four times as big as the engraved design in the plate3 but if that representation should not be agreeable1 move the glasses 7still open to a square7 to the number #e of the side ,d1 so will one of them be parallel to ,d1 and the other stand upon the line of the side ,c1 your first design will then be varied2 and so by moving your glasses1 in like manner1 from point to point1 the draughts will differ by every variation of the glasses1 till you have discovered at least fifty plans1 differing from one another4 8,example ,,ii4--,turn the side marked ,b1 of ,fig4 #ef1 to the same point where ,a was before1 and by moving your glasses as you did in the former example1 you will discover as great a variety of designs as had been observed in the foregoing experiment2 then turn the side ,c to the place of ,b1 and1 managing the glasses in the manner ,i have directed in the first example1 you may have a great variety of different plans1 which were not in the former trials2 and the fourth side1 ,d1 must be managed in the same manner with the others2 so that from one plan alone1 not exceeding the bigness of a man's hand1 we may vary the figure at least two hundred times2 and so1 consequently1 from five figures of the like nature1 we might show about a thousand several sorts of garden-plats2 and if it should happen that the reader has any number of plans for parterres or wilderness-works by him1 he may1 by this method1 alter them at his pleasure1 and produce such innumerable varieties1 that it is not possible the most able designer could ever have contrived40 ,in reading the preceding description1 the following conclusions cannot fail to be drawn by every person who understands it4 #a4 ,dr4 ,bradley1 like ,kircher1 considers his mirrors as applicable to regular figures1 such as are represented in ,fig4 #ee1 and was entirely unacquainted with the fact1 that the inclination of the mirrors must be an even aliquot part of a circle4 ,this is obvious1 from his stating that the mirrors may be set at the third1 fourth1 fifth1 sixth1 seventh1 or eighth part of a circle2 for if he had tried to set an irregular object between the mirrors when placed at the third1 fifth1 or seventh part of #cfj1 he would have found that a complete figure could not possibly be produced4 #b4 ,from the erroneous position of the eye in front of the mirrors1 there is such an inequality of light in the reflected sectors1 that the last is scarcely visible1 and therefore cannot be united into a uniform picture with the real objects4 #c4 ,as the place of the eye in ,bradley's instrument is in front1 and therefore much nearer the object1 or sector1 seen by direct vision1 the angular magnitudes of all the different sectors are different1 and hence they cannot unite into a symmetrical figure4 ,this is so unavoidable a result of the erroneous position of the eye--a position too1 rendered necessary from the absurd form of the mirrors--that ,bradley actually employs his mirrors to convert a circle into an ellipse1 and a square into a rectangle6 #d4 ,in ,bradley's instrument1 the sectors thus unequally shaped1 and unequally illuminated1 are all separated from one another1 by a space equal to the thickness of the glass plates2 and from the same cause1 the images reflected from the first surface interfere with those reflected from the second1 and produce a confusion or overlapping of images entirely incompatible with a symmetrical picture4 ,these results are deduced upon the supposition that the object consists of lines drawn upon the surface of paper1 the only purpose to which ,bradley ever applied his mirrors2 but when we attempt to use these mirrors as a ,kaleidoscope1 and apply them to the objects which are usually adapted to that instrument1 we shall fail entirely1 as the mirrors are utterly incapable of producing the beautiful and symmetrical forms which belong to that instrument4 ,we come now to consider the method of applying the mirrors to the object shown in ,fig4 #ee1 which is an exact1 but reduced copy of ,bradley's figure4 ,because the angles of this figure are #ij1 he directs that the mirrors 8be opened and fixed to an exact square2 and that the edge of one of the mirrors must always be placed so as to coincide exactly with one of the sides of the rectangle1 and carefully kept in this line when the position of the mirrors is changed40 ,hence it is manifest1 that ,bradley was entirely ignorant of the fundamental principle of the ,kaleidoscope1 namely1 that if the inclination of the mirrors is an even aliquot part of a circle1 and if they are set in any position upon any object or set of objects1 however confused1 or distorted1 or irregular1 they will create the most perfect and symmetrical designs4 ,but even if ,bradley had been aware of this principle in theory1 neither he nor any other person was acquainted with the mode of constructing and fitting up the reflectors1 in order to render them capable of producing the effect4 ,the position of the eye1 for symmetry of light1 and for symmetry of form2--the position of the mirrors1 to produce a perfect junction of the last sectors2--the position of the object necessary to produce an equality and perfect junction between the object and the reflected images2--and the method of accomplishing this for objects at all distances1 were fundamental points1 in the combination of plane mirrors1 which1 so far as ,i know1 have never been investigated by any author4 ,but even though a knowledge of these theoretical points had been obtained1 numerous difficulties in the practical construction of the instrument remained to be surmounted2 and it required no slight degree of labour and attention to enable the instrument to develop1 in the most simple and efficacious manner1 the various effects which1 in theory1 it was found susceptible of producing4 ,as the combination of mirrors1 described by ,kircher and ,bradley1 had been long known to opticians1 and had excited so little attention that they had even ceased to be noticed in works on optical instruments1 it became necessary to discover some other origin for the ,kaleidoscope4 ,the thirteenth and fourteenth ,propositions of ,wood's ,optics1 and some analogous propositions in ,harris's ,optics1 were therefore presumed to be an anticipation of the invention4 ,professor ,wood gives a mathematical investigation of the number and arrangement of the images formed by two reflectors1 either inclined or parallel to each other4 ,these theorems assign no position to the eye1 or to the object1 and do not include the principle of inversion1 which is absolutely necessary to the production of symmetrical forms4 ,the theorems1 indeed1 which have no connexion whatever with any instrument1 are true1 whatever be the position of the eye or the object2 and ,mr4 ,wood has frankly acknowledged1 8that the effects produced by the ,kaleidoscope were never in his contemplation40#bb #bb ,see ,appendix1 p4 #ahe4 ,the propositions in ,harris's ,optics relate1 like ,professor ,wood's1 merely to the multiplication and circular arrangement of the apertures or sectors formed by the inclined mirrors1 and to the progress of a ray of light reflected between two inclined or parallel mirrors2 and no allusion whatever is made1 in the propositions themselves1 to any instrument4 ,in the propositions respecting the multiplication of the sectors1 the eye of the observer is never once mentioned2 and the proposition is true1 if the eye has an infinite number of positions2 whereas1 in the ,kaleidoscope1 the eye can only have one position4 ,in the other proposition 7,prop4 ,,xvii47 respecting the progress of the rays1 the eye and the object are actually stated to be placed between the reflectors2 and even if the eye had been placed without the reflectors1 as in the ,kaleidoscope1 the position assigned it1 at a great distance from the angular point1 is a demonstration that ,harris was entirely ignorant of the positions of symmetry1 either for the object or the eye1 and could not have combined two reflectors so as to form a ,kaleidoscope for producing beautiful or symmetrical forms4#bc ,it is important also to remark1 that all ,harris's propositions relate either to sectors or to small circular objects2 and that he supposes the very same effects to be produced when the inclination of the mirrors is an odd1 as when it is an even1 aliquot part of a circle4 ,it is clear1 therefore1 that he was neither acquainted with the fundamental point in the theory of the ,kaleidoscope1 nor with any of its practical effects4 ,the only practical part of ,harris's propositions is the fifth and sixth scholia to ,prop4 ,,xvii4 ,in the fifth scholium he proposes a sort of catoptric box1 or cistula1 known long before his time1 composed of four mirrors1 arranged in a most unscientific manner1 and containing opaque objects between the speculums4 8,whatever they are10 says he1 when speaking of the objects1 8the upright figures between the speculums should be slender1 and not too many in number1 otherwise they will too much obstruct the reflected rays from coming to the eye40 ,this shows1 in a most decisive manner1 that ,harris knew nothing of the ,kaleidoscope1 and that he has not even improved the common catoptric cistula1 which had been known long before4 ,the principle of inversion1 and the positions of symmetry1 were entirely unknown to him4 ,in the sixth scholium1 he speaks of rooms lined with looking-glasses1 and of luminous amphitheatres1 which were known even to the ancients1 and have been described and figured by all the old writers on optics4 #bc ,see ,chap4 ,i4 pp4 #aa1 #ab1 where we have shown that ,harris was not even acquainted with the way in which the last sector is formed by reflexion4 ,,appendix4 ,although ,i have no doubt that the observations contained in the preceding ,chapters are sufficient to satisfy every candid and intelligent person respecting the true nature of all the combinations of plane mirrors that preceded the invention of the ,kaleidoscope1 yet as there are many who are incapable1 from want of optical knowledge1 to understand the comparison which has been made between them1 ,i shall here present the opinions of four of the most eminent mathematicians and natural philosophers4 ,the first of these is contained in a note from the late ,professor ,wood1 ,master of ,st4 ,john's ,college1 ,cambridge1 written in reply to a letter1 in which ,i requested him to say1 if he had any idea of the effects of the ,kaleidoscope when he wrote the thirteenth and fourteenth ,propositions of his works on ,optics4 8,,st4 ,,john',s1 ,may #ai1 #ahah4 8,,sir1--,the propositions ,i have given relating to the number of images formed by plane reflectors1 inclined to each other1 contain merely the mathematical calculation of their number and arrangement4 ,the effects produced by the ,kaleidoscope were never in my contemplation4 ,my attention has for some years been turned to other subjects1 and ,i regret that ,i have not time to read your ,optical ,treatise1 which ,i am sure would give me great pleasure4--,i am1 ,sir1 your obedient humble servant1 8,j4 ,,wood40 ,the following is the opinion of the late celebrated ,mr4 ,james ,watt3-- 8,it has been said here10 says ,mr4 ,watt1 8that you took the idea of the ,kaleidoscope from an old book on gardening4 ,my friend1 the ,rev4 ,mr4 ,corrie1 has procured me a sight of the book4 ,it is ,bradley's ,improvements of ,planting and ,gardening4 ,london1 #agca1 ,part ,,ii4 ,chap4 ,i4 ,it consists of two pieces of looking-glass1 of equal bigness1 of the figure of a long square1 five inches long1 and four inches high1 hinged together upon one of the narrow sides1 so as to open and shut like the leaves of a book1 which1 being set upon their edges upon a drawing1 will show it multiplied by repeated reflexions4 ,this instrument ,i have seen in my father's possession seventy years ago1 and frequently since1 but what has become of it ,i know not4 ,in my opinion1 the application of the principle is very different from that of your ,kaleidoscope40 ,the following is the opinion of the late ,mr4 ,playfair1 ,professor of ,natural ,philosophy in the ,university of ,edinburgh3-- 8,,edinburgh1 ,may #aa1 #ahah4 8,i have examined the ,kaleidoscope invented by ,dr4 ,brewster1 and compared it with the description of an instrument which it has been said to resemble1 constructed by ,bradley in #agag4 ,i have also compared its effect with an experiment to which it may be thought to have some analogy1 described by ,mr4 ,wood in his ,optics1 ,prop4 ,,xiii4 and ,,xiv4 8,from both these contrivances1 and from every optical instrument with which ,i am acquainted1 the ,kaleidoscope appears to differ essentially1 both in its effect and in the principles of its construction4 8,as to the effect1 the thing produced by the ,kaleidoscope is a series of figures presented with the most perfect symmetry1 so as always to compose a whole1 in which nothing is wanting and nothing redundant4 ,it matters not what the object be to which the instrument is directed2 if it only be in its proper place1 the effect just described is sure to take place1 and with an endless variety4 ,in this respect the ,kaleidoscope appears to be quite singular among optical instruments4 ,neither the instrument of ,bradley1 nor the experiment or theorem in ,wood's book1 have any resemblance to this2 they go no further than the multiplication of the figure4 8,next1 as to the principle of construction1 ,dr4 ,brewster's instrument requires a particular position of the eye of the observer1 and of the object looked at1 in order to produce its effect4 ,if either of these is wanting1 the symmetry vanishes1 and the figures are irregular and disunited4 ,in the other two cases1 no particular position1 either for the eye or the object1 is required4 8,for these reasons1 ,dr4 ,brewster's invention seems to me quite unlike the other two4 ,indeed1 as far as ,i know1 it is quite singular among optical instruments2 and it will be matter of sincere regret1 if any imaginary or vague analogy1 between it and other optical instruments1 should be the means of depriving the ,doctor of any part of the reward to which his skill1 ingenuity1 and perseverance1 entitle him so well4 8,,john ,,playfair1 ,professor of ,natural ,philosophy in the ,university of ,edinburgh40 8,p4,s4--,granting that there were a resemblance between the ,kaleidoscope and ,bradley's instrument1 in any of the particulars mentioned above1 the introduction of coloured and movable objects1 at the end of the reflectors1 is quite peculiar to ,dr4 ,brewster's instrument4 ,besides this1 a circumstance highly deserving of attention1 is the use of two lenses and a draw-tube2 so that the action of the ,kaleidoscope is extended to objects of all sizes1 and at all distances from the observer1 and united1 by that means1 to the advantages of the telescope4 8,j4 ,p40 ,professor ,pictet's opinion is stated in the following letter3-- 8,,sir1--,among your friends1 ,i have not been one of the least painfully affected by the shameful invasion of your rights as an inventor1 which ,i have been a witness of lately in ,london4 ,not only none of the allegations of the invaders of your patent1 grounded on a pretended similarity between your ,kaleidoscope and ,bradley's instrument1 or such as ,wood's or ,harris's theories might have suggested1 appear to me to have any real foundation2 but ,i can affirm1 that1 neither in any of the ,french1 ,german1 or ,italian authors1 who1 to my knowledge1 have treated of optics1 nor in ,professor ,charles's justly-celebrated and most complete collection of optical instruments at ,paris1 have ,i read or seen anything resembling your ingenious apparatus1 which1 from its numberless applications1 and the pleasure it affords1 and will continue to afford to millions of beholders of its matchless effects1 may be ranked among the most happy inventions that science ever presented to the lovers of rational enjoyment4 8,m4 ,a4 ,,pictet1 ,professor of ,natural ,philosophy in the ,academy of ,geneva40 ,those who wish to examine farther the ancient combinations of plane mirrors1 and other subjects connected with the ,kaleidoscope1 are referred to the following works3-- ,baptista ,porta's ,magia ,naturalis4 ,kircher's ,ars ,magna ,lucis et ,umbr4 ,schottus's ,magia ,universalis ,natur et ,artis4 ,bradley's ,treatise on ,planting and ,gardening4 ,harris's ,treatise on ,optics4 ,wood's ,optics4 ,dr4 ,roget on the ,kaleidoscope1 in the ,annals of ,philosophy1 vol4 xi4 p4 #cge4 ,encyclopdia ,britannica1 ,art4 ,,kaleidoscope1 by ,dr4 ,roget2 and the ,compte ,rendu des ,travaux de ,l',acadmie de ,dijon1 pour #ahah1 pp4 #ajh-aag4 ,works by ,,sir ,,david ,,brewster4 ,,more ,,worlds ,,than ,,one3 the ,creed of the ,philosopher1 the ,hope of the ,christian4 ,cr4 #hvo41 #ds4 #f;d4 ,,the ,,martyrs ,,of ,,science3 ,galileo1 ,tycho ,brahe1 and ,kepler4 ,cr4 #hvo41 #ds4 #f;d4 ,,the ,,kaleidoscope ,practically ,described4 ,cr4 #hvo41 with ,illustrations1 #ds4 #f;d4 ,,the ,,stereoscope ,practically ,described4 ,cr4 #hvo41 with ,illustrations1 #ds4 #f;d4 ,,london3 ,,john ,,camden ,,hotten1 #gd1 ,,piccadilly4