S2S 



TITANOTHERES OF ANCIENT WYOMING, DAKOTA, AND NEBRASKA 



received the enormous increment of 79 per cent, 

 which may be contrasted with the actual closure of 

 this same region in the brachvcephalic Palaeosyops. 



SLOW BUT DIFFERENTIAL EVOLUTION IN BREADTH OF 

 SKULL OF DOLICHORHINUS 



While the skull of DolichorMnus was rapidly 

 elongating (increment 33 per cent) it was much more 

 slowly evolving in breadth (increment 9 to 15 per 

 cent). Thus the disparity between the length and 

 breadth was constantly increasing, as shown in the 

 following comparisons: (1) The transverse measure- 

 ment across the zygomata increased 15 per cent, or 

 less than one-half the increment in length of the skull 

 (33 per cent). Similarly, the breadth across the post- 

 glenoid and post-tympanic processes increased only 

 13 per cent. (2) The space across the infraorbital 

 shelves increased only 9 per cent, partly because 

 these shelves were already relatively broad in M- 

 peter soni. (3) Consistent with the bracing of this 

 long and narrow skull we observe that the occipital 

 condyles widen much more rapidly than the remainder 

 of the skull, the increment being 28 per cent. (4) 

 Similarly, the occiput increased in breadth 68 per 

 cent in order to provide a large surface of attachment 

 for the cervical and cephalohumeral muscles. The 

 fact that the occiput increased in height only 20 per 

 cent indicates that the horns were used in an obliquely 

 lateral rather than in a tossing motion. (5) The 

 transverse measurement across the horn region shows 

 the relatively high increment of 35 per cent, while the 

 transverse measurement across the postorbital proc- 

 esses of the frontals increased 23 per cent. (6) A 

 noteworthy fact is that the palatal space between the 

 anterior premolars receives the large increment of 45 

 per cent. This broadening of the palate anteriorly 

 would allow greater space for the incoming food. 



DIFFERENTIAL EVOLUTION IN THE GRINDING TEETH 

 OF DOLICHORHINUS 



It is a striking fact that the teeth as a whole in- 

 creased in length at practically the same rate as the 

 skull as a whole, namely, 32 per cent. A second fact 

 of great interest is that the dental series lengthens 

 much more rapidly than the individual teeth increase 

 in width, the transverse increment of the teeth being 

 but 17 per cent while the total longitudinal increment 

 is 32 per cent. The teeth thus share the general 

 elongation of the skull. The grinding areas of the 

 cheek teeth as a whole become larger. Measuring the 

 grinding areas by the sum of the rectangles circum- 

 scribed by each tooth in the series we find that the 

 grinders of D. TiyognatTius have received an increment 

 of 68 per cent over those of M. petersoni. This 

 increase is differential, however, because the grinding 

 area of the premolars has increased only 41 per cent 

 as against 76 per cent increase in the true molars. 



This is the reverse of what occurs in the horse, in 

 which the premolar increment is greater than the 

 molar increment. 



The increment of the total grinding area of the 

 crowns among the individual teeth is distributed as 

 follows: P', a very slight increment in size; p^, an 

 -ncrement in grinding area of 48 per cent; p^~^, an 

 ncrement in grinding area of 36 per cent; true 

 molars, m'"^, an increment in grinding area of 80 per 

 cent. Thus the true molars received by far the 

 greatest increment. 



CONTRASTS OF DIFFERENTIAL EVOLUTION IN BRACHV- 

 CEPHALIC AND DOLICHOCEPHALIC SKULLS 



The net results of these increments in length and 

 width may be summarized as follows: 



Percentages of increase in size of skull and teeth of DolichorMnus 

 and Palaeosyops 



SUMMARY OF HARMONIC AND DIFFERENTIAL ALIOME- 

 TRONS IN THE SKULLS AND FEET AND AN INTERPRE- 

 TATION OF THE PHYLOGENY OF THE TITANOTHERES 



By W. K. Gregory 



HISTORY OF RESEARCH 



The first step toward the discovery of the allometric 

 principle was the analysis of dolichocephaly and 

 brachycephaly in the titanotheres by Osborn in 1902 

 (1902.208). At that time it seems to have been 

 thought that the tendencies toward the long-headed 

 and short-headed forms were mutually exclusive and 

 that their divergent mass effects indicated a very 

 ancient separation between the corresponding genera. 

 It was also apparently assumed (Osborn, 1902.207) 

 that the tendencies acted more or less uniformly on 

 all parts of the skull. 



