Z04 THE POPULAR SCIENCE MONTHLY. 



positions are points of difference, as we find in the atomic grouping 

 of compound molecules, where the phenomena of isomerism appear ; 

 in the order of successive sounds, whether in language or music ; aud 

 as in the various series in which muscular and nervous forces coordi- 

 nate in animal movements. In all such cases the multiplication of 

 effects tends to follow a law of even greater increase than that of geo- 

 metrical progression namely, the law of permutations. 



If A B C be elements given, their permutations in groups of 3 are 

 G (3x2x1), in groups of 2, 6 more, and adding 3 for the elements 

 taken singly, 15 is obtained as the number of permutations of all 

 kinds. The addition of a new element increases them to 64 (15 x 4 + 4), 

 and so on in a ratio increasing with every additional element, until 

 we find that 10 produce 9,856,900 permutations, and but 1,024 combi- 

 nations. 



These abstract laws are paralleled by the multiplied results which 

 follow in the wake of any important invention or discovery. Forty 

 years ago the main arts of representation were five in number sculp- 

 ture, painting, printing, engraving, and lithography. The art of photog- 

 raphy, introduced by Daguerre in 1839, and since so beautifully de- 

 veloped, is continually increasing derivative arts. It is applicable to 

 every other main art, and may become an element in new permu- 

 tative groups of them. It has already given aid to the sculptor, the 

 painter, and the engraver, and in the heliotype and woodburytype 

 exhibits relations with lithography and printing; besides, it has added 

 to human power in many other ways, has made the stereoscope avail- 

 able, brino-inff the natural beauties and artistic treasures of distant 

 lands vividly near ; it has aided astronomy in fixing views of tran- 

 sits and eclipses of brief duration, and in mapping the sun and moon; 

 the physiologist has used it to preserve the evanescent exhibitions of 

 dissection ; and in observatories it accurately marks the minute move- 

 ments of delicate apparatus. It limns the interiors of pyramids, 

 catacombs, caves, and mines, giving incidental help to archaeology 

 and geology ; and, in regions inaccessible to man, pictures the depths 

 of the sea. It serves in war and might in peace to aid the topog- 

 rapher in mapping plans of city and country; in times of siege it 

 has reduced correspondence to microscopic limits for carriage in the 

 only possible way by birds ; and from year to year this wonderful 

 art continues to be applied in new and valuable uses. 



The illustration it affords of the manner in which human resources 

 are multiplied by the accession of a new discovery might be repeated, 

 were all the applications and results of the steam-engine, locomotive, 

 or telegraph, traced in their numerous ramifications. So far from 

 these mighty achievements exhausting the conquests possible to man, 

 they are merely centres of new circles of power from which he may 

 successively penetrate into the ever-boundless regions of the unknown. 



The late Mr. Mill, at a period of great depression in his early life, 



