1084 THE THEORY OF TRANSFORMATIONS [ch. 



in an increased intensity or degree of deformation* . These anthropoid 

 skulls, then, which we can transform one into another by a "con- 

 tinuous transformation," are admirable examples of what Listing 

 called "topological similitude." 



In both dimensions, as we pass from above downwards and from 

 behind forwards, the corresponding areas of the network are seen 

 to increase in a gradual and approximately logarithmic order in the 

 lower as compared with the higher type of skull; and, in short, 

 it becomes at once manifest that the modifications of jaws, brain-case, 

 and the regions between are all portions of one continuous and 

 integral process. It is of course easy to draw the inverse diagrams, 

 by which the Cartesian coordinates of the ape are transformed into 

 curvilinear and non-equidistant coordinates in manf. 



From this comparison of the gorilla's or chimpanzee's with the 

 human skull we realise that an inherent weakness underlies the 

 anthropologist's method of comparing skulls by reference to a small 

 number of axes. The most important of these are the "facial '"and 

 " basicranial " axes, which include between them the "facial angle." 

 But it IS, in the first place, evident that these axes are merely the 

 principal axes of a system of coordinates, and that their restricted 

 and isolated use neglects all that can be learned from the filling in 

 of the rest of the coordinate network. And, in the second place, the 

 "facial axis," for instance, as ordinarily used m the anthropological 

 comparison of one human skull with another, or of the human skull 

 with the gorilla's, is in all cases treated as a straight line; but our 

 investigation has shewn that rectilinear axes only meet the case in 

 the simplest and most closely related transformations ; and that, for 

 instance, in the anthropoid skull no rectilinear axis is homologous 



* The empirical coordinates which I have sketched in for the chimpanzee as a 

 conformal transformation of the Cartesian coordinates of the human skull look as 

 if they might find their place in an equipotential elliptic field. They are indeed 

 closely analogous to some already figured by MM. Y. Ikada and M. Kuwaori, 

 Some conformal representations by means of the elliptic integrals, Sci. Papers 

 Inst. Phys. Research, Tokyo, xxvi, pp. 208-215, 1936: e.g. pi. xxxi&. 



t Speaking of "diagrams in pairs," and doubtless thinking of his own "reciprocal 

 diagrams," Clerk Maxwell says (in his article Diagrams in the Encyclopaedia Britan- 

 nica): "The method in which we simultaneously contemplate two figures, and 

 recognise a correspondence between certain points in the one figure and certain 

 points in the other, is one of the most powerful and fertile methods hitherto known 

 in science. . . .It is sometimes spoken of as the method or principle of duality." 



