860 D'ARCY WENTWORTH THOMPSON ON 



interest and use to the morphologist ; and this step consists in the alteration, or 

 " transformation," of our system of co-ordinates and in the study of the correspond- 

 ing transformation of the curve or figure inscribed in the co-ordinate network. 



Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, 

 however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. 

 We may now treat this complicated figure, in general terms, as a function of x, y. 

 If we submit our rectangular system to " deformation," on simple and recognised 



lines, altering, for instance, the direction of the axes, the ratio of ' , or substituting 



y 



for x and y some more complicated expressions, then we shall obtain a new system 

 of co-ordinates, whose deformation from the original type the inscribed figure will 

 precisely follow. In other words, we obtain a new figure, which is a function of the 

 new co-ordinates in precisely the same way as the old figure was of the original 

 co-ordinates x and y. 



The problem is closely akin to that of the cartographer who transfers identical 

 data to one projection or another ; and whose object is to secure (if it be possible) a 

 complete correspondence, in each small unit of area, between the one representation 

 and the other. The morphologist will not seek to draw his organic forms in a new 

 and artificial projection ; but, in the converse aspect of the problem, he will inquire 

 whether two different but more or less obviously related forms can be so analysed 

 and interpreted that each may be shown to be a transformed representation of the 

 other. This once demonstrated, it will be a comparatively easy task (in all prob- 

 ability) to postulate the direction and magnitude of the force capable of effecting the 

 required transformation. Again, if such a simple alteration of the system of forces 

 can be proved adequate to meet the case, we may find ourselves able to dispense 

 with many widely current and more complicated hypotheses of biological causation. 

 For it is a maxim in physics that an effect ought not to be ascribed to the joint 

 operation of many causes if few are adequate to the production of it : " Frustra jit 

 /><-!■ plura, quod fieri potest per pauciora." 



It is evident that by the combined action of appropriate forces any material 

 form can be transformed into any other : just as out of a " shapeless" mass of clay 

 the potter or the sculptor models his artistic product; or just as we attribute to 

 Nature herself the power to effect the gradual and successive transformation of the 

 simplest into the most complex organism. In like manner it is possible, at least 

 theoretically, to cause the outline of any closed curve to appear as a projection of 

 any other whatsoever. But we need not let these theoretical considerations deter 

 us from our method of comparison of related forms. We shall strictly limit ourselves 

 to c;iscs where the transformation necessary to effect a comparison shall be of a simple 

 kind, and where the transformed, as well as the original, co-ordinates shall constitute 

 an harmonious and more or less symmetrical system. We should fall into deserved 

 and inevitable confusion if, whether by the mathematical or any other method, we 



