MORPHOLOGY AND MATHEMATICS. 



885 



thetical structures, on the assumption that they have varied from a known form in 

 some definite way. And this process may be especially useful, and will be most 

 obviously legitimate, when we apply it to the particular case of representing inter- 

 mediate stages between two forms which are actually known to exist, in other 

 words, of reconstructing the transitional stages through which the course of evolution 



















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Fig. 48. — Pelvis of Archmopteryx. 



Fig. 49. — Pelvis of Apatornis. 



must have successively travelled if it has brought about the change from some 

 ancestral type to its presumed descendant. While this paper was passing through 

 the press I sent to my friend, Mr Gerhard Heilmann of Copenhagen, some of my 

 own rough co-ordinate diagrams, including some in which the pelves of certain 

 ancient and primitive birds were compared one with another. Mr Heilmann, who 

 is both a skilled draughtsman and an able morphologist, has returned me a set of 

 diagrams which are a vast improvement on my own, and which are reproduced in 

 fio-s. 48-54. Here we have, as extreme cases, the pelvis of Archaeopteryx, the most 

 ancient of known birds, and that of Apatornis, one of the fossil "toothed" birds 



