1086 THE THEORY OF TRANSFORMATIONS [ch. 



eye and nostril, and all the other great landmarks of cranial anatomy, 

 not only continue to exist but retain their relative order and position 

 throughout all our transformations. 



We can discover a certain invariance, somewhat more restricted 

 than before, between the mammahan skull and that of fowl, frog 

 or even herring. We have still something common to them all; 

 and using another mathematical term (somewhat loosely perhaps) 

 we may speak of the discriminant characters which persist unchanged, 

 and continue to form the subject of our transformation. But the 

 method, far as it goes, has its limitations. We cannot fit both 

 beetle and cuttlefish into the same framework, however we distort 

 it; nor by any coordinate transformation can we turn either of 

 them into one another or into the vertebrate type. They are 



Fig. 552. Skull of dog, compared with the human .skull of Fig. 548. 



essentially different; there is nothing about them which can be 

 legitimately compared. Eyes they all have, and mouth and jaws; 

 but what we call by these names are no longer in the same order 

 or relative position; they are no longer the same thing, there is no 

 invariant basis for transformation. The cuttlefish eye seems as 

 perfect, optically, as our own ; but the lack of an invariant relation 

 of position between them, or lack of true homology between them 

 (as we naturalists say), is enough to shew that they are unrelated 

 things, and have come into existence independently of one another. 

 As a final illustration I have drawn the outline of a dog's skull 

 (Fig. 552), and inscribed it in a network comparable with the Car- 

 tesian network of the human skull in Fig. 548. Here we attempt to 

 bridge over a wider gulf than we have crossed in any of our former 

 comparisons. But, nevertheless, it is obvious that our method still 

 holds good, in spite of the fact that there are various specific 

 differences, s.uch as the open or closed orbit, etc., which have to be 



