202 THE POPULAR SCIENCE MONTHLY. 



MATHEMATICS IX EVOLUTION. 



By GEOKGE ILES. 



WHILE we know that only Infinite Intelligence could reduce the 

 entire phenomena of the universe to mathematical expression, 

 it affords an observer constant surprise to find primitive laws of 

 order and number recur again and again amid the infinite variety of 

 Nature. 



The spectroscope would seem to indicate that the elements of our 

 present chemistry are really very complex structures, yet we find 

 them, when grouped in all sorts of proportions as molecules, capable 

 of crystallizing in forms of perfect geometrical symmetry, often of 

 much simplicity. In botany, where the factors both chemically and 

 mechanically are extremely various, we find simple laws obeyed in 

 the disposition of leaves, flowers, and parts of flowers ; a remarkable 

 instance of which occurs in the growth of leaves on spirally-leaved 

 plants. In the first order of them, a leaf is found in -| the circumfer- 

 ence of the stem, and throughout the series the arcs occupied by a 

 leaf are respectively -|, , f , T 5 , -g^, and -if, of a circle, the numerator 

 and denominator of each fraction being those of the two next pre- 

 ceding added together. 



In the highest plane of Nature, that of animal forms, the condi- 

 tions fulfilled are too complex to permit any formulation of lines and 

 angles, but natural history in its first chapters gives us the habita- 

 tions of the nautilus and other organisms low in the scale of life, 

 which in their beautiful volutes and spirals embody simple geometry. 

 So also does the architecture of our common insects, the bee, wasp, 

 and spider, which, wonderful as it is, must remain less so than the 

 work of the microscopic coral zoophytes, which, while severally living 

 and building where it is easiest, yet unconsciously cooperate through 

 successive generations to complete a structure of comparatively vast 

 proportions and much symmetrical unity. 



These few examples, which might be multiplied indefinitely, may 

 serve as bases for the opinion that complex wholes, acting in many 

 cases like simple ones, may be more easily reducible to mathematical 

 treatment than might at first view be supposed, from the number and 

 variety of ultimate factors concerned in any given problem. Nature 

 would seem to act by but few first principles, which she constantly 

 repeats in her various fields, and which, combined in different ways, 

 yield all her infinite manifestations. The scientific progress of our 

 times is marked by the continual absorption of diverse laws into 

 higher and more general ones ; thus the forms of force that used to be 

 thought distinct entities are now proved to be interchangeable, and 

 therefore essentially the same. A minor instance of a like kind occurs 



