( 857 ) 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 denning it in the precise language of mathematics ; and the one method tends to follow the other in strict scientific order and historical continuity. Thus, fer 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 Natural 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). 123 858 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 LINNAEUS 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, I 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. I 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 has 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 equal 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 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 trabeculse, 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 ivords. 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.