720 THE THEORY OF TRANSFORMATIONS [ch. 



words or symbols are so pregnant with meaning that thought 

 itself is economised ; we are brought by means of it in touch with 

 Galileo's aphorism (as old as Plato, as old as Pythagoras, as old 

 perhaps as the wisdom of the Egyptians), that "the Book of 

 Nature is written in characters of Geometry." 



Next, we soon reach through mathematical analysis to mathe- 

 matical synthesis; we discover homologies or identities which 

 were not obvious before, and which our descriptions obscured 

 rather than revealed: as for instance, when we learn that, how- 

 ever we hold our chain, or however we fire our bullet, the contour 

 of the one or the path of the other is always mathematically 

 homologous. Lastly, and this is the greatest gain of all, we pass 

 quickly and easily from the mathematical conception of form in 

 its statical aspect to form in its dynamical relations : we pass from 

 the conception of form to an understanding of the forces which 

 gave rise to it; and in the representation of form and in the 

 comparison of kindred forms, we see in the one case a diagram 

 of forces in equilibrium, and in the other case we discern the 

 magnitude and the direction of the forces which have sufficed to 

 convert the one form into the other. Here, since a change of 

 material form is only effected by the movement of matter, we have 

 once again the support of the schoolman's and the philosopher's 

 axiom, Ignorato motu, ignoratur Natura." 



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 w^ould seem to be a psychological 

 reason lying in the fact that the student of living things is by 

 nature and training ap 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 showed the application of mathematics to 



