1028 THE THEORY OF TRANSFORMATIONS [ch. 



There is yet another way — we learn it of Henri Poincare — to regard 

 the function of mathematics, and to reahse why its laws and its 

 methods are hound to underlie all parts of physical science. Every 

 natural phenomenon, however simple, is really composite, and every 

 visible action and effect is a summation of countless subordinate 

 actions. Here mathematics shews her pecuUar power, to combine 

 and to generahse. The concept of an average, the equation to 

 a curve, the description of a froth or cellular tissue, all come within 

 the scope of mathematics for no other reason than that they are 

 summations of more elementary principles or phenomena. Growth 

 and Form are throughout of this composite nature; therefore the 

 laws of mathematics are bound to underlie them, and her methods 

 to be peculiarly fitted to interpret them. 



In the morphology of living things the use of mathematical 

 methods and symbols has made slow progress ; and there are various 

 reasons for this failure to employ a method whose advantages are 

 so obvious in the investigation of other physical forms. To begin 

 with, there would seem to be a psychological reason, lying in the 

 fact that the student of living things is by nature and training an 

 observer of concrete objects and phenomena and the habit of mind 

 which he possesses and cultivates is alien to that of the theoretical 

 mathematician. But this is by no means the only reason; for in 

 the kindred subject of mineralogy, for instance, crystals were still 

 treated in the days of Linnaeus as wholly within the province of 

 the naturalist, and were described by him after the simple methods 

 in use for animals and plants: but as soon as Haiiy shewed the 

 application of mathematics to the description and classification of 

 crystals, his methods were immediately adopted and a new science 

 came into being. 



A large part of the neglect and suspicion of mathematical methods 

 in organic morphology is due (as we have partly seen in our opening 

 chapter) to an ingrained and deep-seated belief that even when we 

 seem to discern a regular mathematical figure in an organism, the 

 sphere, the hexagon, or the spiral which we so recognise merely 

 resembles, but is never entirely explained by, its mathematical 

 analogue ; in short, that the details in which the figure differs from 

 its mathematical prototype are more important and more interesting 



