772 THE THEORY OF TRANSFORMATIONS [ch. 



inverse diagrams, by which the Cartesian co-ordinates of the ape 

 are transformed into curvilinear and non- equidistant co-ordinates 

 in man. 



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 co-ordinates, 

 and that their restricted and isolated use neglects all that can be 

 learned from the filling in of the rest of the co-ordinate network. 

 And, in the second place, the "facial axis," for instance, as 

 ordinarily used in 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 with a 

 rectilinear axis in a man's skull, but what is a straight line in the 

 one has become a certain definite curve in the other. 



Mr Heilmann tells me that he has tried, but without success, 

 to obtain a transitional series between the human skull and some 

 prehuman, anthropoid type, which series (as in the case of the 

 Equidae) should be found to contain other known types in direct 

 linear sequence. It appears impossible, however, to obtain such a 

 series, or to pass by successive and continuous gradations through 

 such forms as Mesopithecus, Pithecanthropus, Homo neander- 

 thalensis, and the lower or higher races of modern man. The 

 failure is not the fault of our method. It merely indicates that 

 no one straight line of descent, or of consecutive transformation, 

 exists; but on the contrary, that among human and anthropoid 

 types, recent and extinct, we have to do with a complex problem 

 of divergent, rather than of continuous, variation. And in like 

 manner, easy as it is to correlate the baboon's and chimpanzee's 

 skulls severally with that of man, and easy as it is to see that the 

 chimpanzee's skull is much nearer to the human type than is the 

 baboon's, it is also not difficult to perceive that the series is not. 



