724: THE THEORY OF TRANSFORMATIONS [ch. 



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 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 corresponding trans- 

 formation 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 xjy, or substituting 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 represents the old figure under strain, and 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 probability) to postulate the direction and 

 magnitude of the force capable of effecting the required trans- 

 formation. 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 



