858 D'ARCY WENTWORTH THOMPSON ON 



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 LlNNJSTJS 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 Hauy 

 showed 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, 1 think, due 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 ; and, in short, that the details in which the 

 figure differs from its mathematical prototype are more important and more interest- 

 ing than the features in which it agrees. This view seems to me to involve a mis- 

 apprehension. There is no essential difference between these phenomena of organic 

 form and those which are manifested in portions of inanimate matter. No chain 

 hangs in a perfect catenary and no raindrop is a perfect sphere : and this for the 

 simple reason that forces and resistances other than the main one are inevitably at 

 work. The same is true of organic form, but it is for the mathematician to unravel 

 the conflicting forces which are at work together. And this process of investigation 

 may lead us on step by step to new phenomena, as it has done in physics, where 

 sometimes a knowledge of form leads us to the interpretation of forces, and at other 

 times a knowledge of the forces at work guides us towards a better insight into form. 

 1 would illustrate this by the case of the earth itself. After the fundamental advance 

 had been made which taught us that the world was round, Newton showed that the 

 forces at work upon it must lead to its being imperfectly spherical, and in the course 

 of time its oblate spheroidal shape was actually verified. But now, in turn, it lias 

 been shown that its form is still more complicated, and the next step will be to seek 

 for the forces that have deformed the oblate spheroid. 



The organic forms which we can define, more or less precisely, and afterwards 

 proceed to explain and to account for in terms of Force, are few in number. They 

 are mostly, but not always, limited to comparatively simple cases. Thus we can at 

 once define and explain, from the point of view of surface-tension, the spherical form 

 of the cell in a simple unicellular organism. When e<jual cylindrical cells are set 

 together, we can understand how mechanical pressure, uniformly applied, converts 

 the circular outlines into a pattern of regular hexagons, as in the hexagonal facets of 

 the insect's eye or the hexagonal pigment-cells of our own retina. On similar lines. 

 Pappus, Maclaurin, and many other mathematicians have contributed to elucidate 



