MATHEMATICS IN EVOLUTION. 203 



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 suc- 

 cessively contained in each other. Such a vagary, however, could be 

 pardoned in the author of the thi-ee immortal laws of astronomy. 



In the present stage of knowledge so few of the sciences are ex- 

 act, 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 organ- 

 isms 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 ex- 

 tremely potent, that their results persist through all practical quali- 

 fications, 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 BCA and B A C are the same combination but different per- 

 mutations. As additions are made to the elements, combinations in- 

 crease 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 com- 

 binations 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 H 3 

 unites with O, or O with H 2 ; 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 



