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; and as in the various series in which muscular and nervous forces coordinate in animal movements. In all such cases the multiplication of effects tends to follow a law of even greater increase than that of geometrical progression—namely, the law of permutations.
If A B C be elements given, their permutations in groups of 3 are 6 (3 2 1), 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 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 combinations.
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—sculpture, painting, printing, engraving, and lithography. The art of photography, introduced by Daguerre in 1839, and since so beautifully developed, is continually increasing derivative arts. It is applicable to every other main art, and may become an element in new permutative groups of them. It has already given aid to the sculptor, the painter, and the engraver, and in the heliotype and woodbury type exhibits relations with lithography and printing; besides, it has added to human power in many other ways, has made the stereoscope available, bringing the natural beauties and artistic treasures of distant lands vividly near; it has aided astronomy in fixing views of transits 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 movements 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 topographer 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,
