MORPHOLOGY AND MATHEMATICS. 



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skull, conspicuous as they are at first sight, will be found easy to bring under the con- 

 ception of a simple and homogeneous transformation, such as would result from the 

 application of some not very complicated stress. For instance, the corresponding 

 co-ordinates of Aceraiherium tridactylum, as shown in fig. 56, indicate that the 

 essential difference between this skull and the former one may be summed up by 

 saying that the long axis of the skull of Aceratherium has undergone a slight double 



Fig. 55. — Skull of Hyrachyus agrarius. (After Osborn. ) 



Fig. 56. —Skull of Aceratherium tridactylum. (After Osbop.n.) 



curvature, while the upper parts of the skull have at the same time been subject to a 

 vertical expansion, or to growth in somewhat greater proportion than the lower 

 parts. Precisely the same changes, on a somewhat greater scale, give us the skull of 

 an existing rhinoceros. 



Anion o- the species of Aceratherium, the posterior, or occipital, view of the skull 

 presents specific differences which are perhaps more conspicuous than those furnished 

 by the side view ; and these differences are very strikingly brought out by the series 

 of conformal transformations which I have represented in fig. 57. In this case it 



TRANS. ROY. SOC. EDIN., VOL. L, PART IV (NO. 27). 127 



