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XXVII. — Morphology and Mathematics. By D'Arcy Wentworth Thompson. 



(Read December 7, 1914. MS. received February 1, 1915. Issued separately June 22, 1915.) 



The study of Organic Form, which we call by Goethe's name of Morphology, is 

 but a portion of that wider Science of Form which deals with the forms assumed by 

 matter under all aspects and conditions, and, in a still wider sense, with Forms which 

 are theoretically imaginable. 



The study of Form may be descriptive merely, or it may become analytical. We 

 begin by describing the shape of an object in the simple words of common speech : 

 we end by defining it in the precise language of mathematics ; and the one method 

 tends to follow the other in strict scientific order and historical continuity. Thus, 

 for instance, the form of the earth, of a raindrop or a rainbow, the shape of the 

 hanging chain, or the path of a stone thrown up into the air, may all be described, 

 however inadequately, in common words ; but when we have learned to comprehend 

 and to define the sphere, the catenary, or the parabola, we have made a wonderful 

 and perhaps a manifold advance. The mathematical definition of a "form" has a 

 quality of precision which was quite lacking in our earlier stage of mere description ; 

 it is expressed in few words, or in still briefer symbols, and these 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, that " the Book of Nature is written 

 in characters of Geometry." 



Next, we soon reach through mathematical analysis to mathematical 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, 

 however 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 the support of Galileo's second aphorism, " Ignorato 

 motu, ignoratur Natura." 



In the morphology of living things the use of mathematical methods and symbols 



has made little progress ; and there are various reasons for this failure to employ a 

 TRANS. ROY. SOC. EDIN., VOL. L, PART IV (NO. 27). 123 



