in the recent investigation of wave-motion. The old notion was that the particles in water-waves moved up and down in straight lines, but the fact has been demonstrated that they roll in circles having a diameter equal to the amplitude of the wave; this holds of all wave-motion, including light, so that the movements of the planets, as they turn on their axes and circle round the sun, are conveyed to our sight by an ethereal motion of precisely the same kind.
Although mathematical studies find ample illustration in Nature, an exaggerated love of symmetry may be induced by them, causing an enthusiast to pass legitimate bounds in an effort to over-simplify intricate problems; thus Kepler attempted to harmonize the orbits of the five planets with the boundaries of the five regular solids successively contained in each other. Such a vagary, however, could be pardoned in the author of the three immortal laws of astronomy.
In the present stage of knowledge so few of the sciences are exact, that any application of mathematics to the vast and complex processes of evolution is only allowable when the laws considered would be so powerful, did they work in an open field, that, though veiled by many weaker ones, they remain distinctly discernible in the salient features of Nature.
A valid application of this kind is made by Mr. Darwin in his theory of natural selection, where he states the tendency of organisms to multiply according to the law of geometrical progression—a tendency which he shows counts throughout the mazy conflict of forces affecting organic life. The purpose of this paper is to trace some effects of other such laws, in their theoretical simplicity so extremely potent, that their results persist through all practical qualifications, and so, when shown to account for observed facts, may serve as tenable ground for inference and deduction.
In evolution heterogeneity is a constant measure of progress, hence the laws stating the variety of effects producible from given elements have a direct interest and value. These are the laws of combination and permutation. Combinations, mathematically, are groups where the presence and not the position of an element counts for difference—thus B C A and B A C are the same combination but different permutations. As additions are made to the elements, combinations increase in geometrical progression with 2 as constant factor. Thus 2 elements yield 4; 3, 8; 4, 16; until, when we reach 63, the number of elements in chemistry, we find more than nine quintillions of combinations to be possible. This law tends to hold only in cases where the particular position of an element in a group is indifferent, as in the superimposition of colors in light; as in the simple molecules of chemistry, where, for instance, the result is the same, whether H2 unites with O, or O with H2; and as in all merely mechanical mixing of ingredients in manufacture, as pottery, gunpowder, and so on. Such cases are less common in Nature and art than those in which definite
