MORPHOLOGY AND MATHEMATICS. 859 



the hexagonal outline and the rhombic dodecahedral base of the cells of the honey- 

 comb ; while a more extended application of the theory of surface-tension, on the 

 lines laid down by Plateau, gives us a complete understanding of the frothlike con- 

 glomeration of cells which we observe in vegetable parenchyma and in many other 

 cellular tissues. A study of mechanical pressures acting on an elastic fluid-contain- 

 ing envelope enables us to define and to explain the conformation of a bird's egg. 

 Moseley and his followers, Neumann, Blake, and others have brought within the 

 field of strict mathematical analysis the logarithmic spiral of the Ammonite, the 

 snail shell, and all other spiral shells in general. And, to take a more complex case, 

 Meyer showed us how the form of a bone, and the minute configuration of its internal 

 trabeeulse, gives us a mathematical diagram of the stresses and strains of which in 

 life it is the subject. 



But the vast majority of organic forms we are quite unable to account for, or to 

 define, in mathematical terms ; and this is 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 deformation 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 trans- 

 lating 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 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 it into a table of numbers, 

 from which again we may at pleasure reconstruct the curve. 



But it is the next step in the employment of co-ordinates which is of special 



