XVII] THE COMPARISON OF RELATED FORMS 723 



For various reasons, then, there are a vast multitude of organic 

 forms which we are unable to account for, or to define, in mathe- 

 matical terms ; and this is not seldom the case even in forms which 

 are apparently of great simplicity and regularity. The curved 

 outline of a leaf, for instance, is such a case ; its ovate, lanceolate, 

 or cordate shape is apparently very simple, but the difficulty of 

 finding for it a mathematical expression is very great indeed. 

 To define the complicated outline of a fish, for instance, or of a 

 vertebrate skull, we never even seek for a mathematical formula. 



But 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 deforwation of another, apart altogether from a precise and 

 adequate understanding of the original "type" or standard of 

 comparison, lies 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 Co-ordinates, 

 on which is based the Theory of Transformations. 



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 into numbers and into words. 



This is precisely what we do, by the method of co-ordinates, 



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 it is possible to inscribe in a net of rectangular 



co-ordinates the outline, for instance, of a fish, and so to translate 



nothing of induction, nothing of causation" {rit. Cajori, Hist of Eletn. Mathematics, 

 p. 283). But Gauss called mathematics "a science of the eye"; and Sylvester 

 assures us that "most, if not all, of the great ideas of modern mathematics have 

 had their origin in observation" {Brit. Ass. Address, 1869, and Laws of Verse, p. 120, 

 1870). 



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