1032 THE THEORY OF TRANSFORMATIONS [ch. 



the shape of a snail-shell, the twist of a horn, the outline of a leaf, 

 the texture of a bone, the fabric of a skeleton, the stream-lines of 

 fish or bird, the fairy lace-work of an insect's wing. Even to do 

 this we must learn from the mathematician to ehminate and to 

 discard; to keep the 'type in mind and leave the single case, with 

 all its accidents, alone ; and to find in this sacrifice of what matters 

 little and conservation of what matters much one of the pecuUar 

 excellences of the method of mathematics*. 



In a very large part of morphology, our essential task lies in the 

 comparison of related forms rather than in the precise definition 

 of each; and the deformation of a complicated figure may be a 

 phenomenon easy of comprehension, though the figure itself have 

 to be left unanalysed and undefined. This process of comparison, 

 of recognising in one form a definite permutation or deformation of 

 another, apart altogether from a precise and adequate understanding 

 of the original "type" or standard of comparison, Hes within the 

 immediate province of mathematics, and finds its solution in the 

 elementary use of a certain method of the mathematician. This 

 method is the Method of Coordinates, on which is based the Theory 

 of Transformations f. 



I imagine that when Descartes conceived the method of co- 

 ordinates, as a generalisation from the proportional diagrams of the 

 artist and the architect, and long before the immense possibilities 

 of this analysis could be foreseen, he had in mind a very simple 

 purpose ; it was perhaps no more than to find a way of translating 

 the form of a curve (as well as the position of a point) into numbers 

 and into words. This is precisely what we do, by the method of 

 coordinates, every time we study a statistical curve ; and conversely, 

 we translate numbers into form whenever we "plot a curve," to 

 illustrate a table of mortality, a rate of growth, or the daily variation 

 of temperature or barometric pressure. In precisely the same way 



* Cf. W. H. Young, The mathematical method and its limitations, Congresso 

 dei Matematici, Bologna, 1928. 



f The mathematical Theory of Transformations is part of the Theory of Groups, 

 of great importance in modern mathematics. A distinction is drawn between 

 Substitution-groups and Transformation-groups, the former being discontinuous, 

 the latter continuous — in such a way that within one and the same group each 

 transformation is infinitely little different from another. The distinction among 

 biologists between a mutation and a variation is curiously analogous. 



