826 



SCIENCE 



[N. S. Vol. XXXIII. No. 856 



universal principle in this family, although it 

 is by no means universal among mammals nor 

 even among the Herbivora. (2) Loss of parts. 

 This plays a very small part in the series of 

 titanotheres as compared, for example, with 

 the horses, since the chief parts lost are one 

 element in the carpus, the trapezium and cer- 

 tain upper and lower incisor teeth. (3) 

 Changes of proportion. This comprehends 

 one of the most important and significant 

 parts of titanothere evolution. Such change 

 it is proposed to designate as " allometric," 

 and new parts originating in this way may 

 he termed " allometrons." (4) Continuous 

 definite or adaptive origins of new characters, 

 which the writer has previously termed " recti- 

 gradations." 



Of the above phenomena (1) increase of 

 size and of (2) loss of parts may be left out 

 of consideration, and attention may be di- 

 rected upon the (3) allometrons and the (4) 

 rectigradations. It is found at once that their 

 mode of appearance or the laws governing 

 them are definite. 



First, as to rectigradations, as exemplified 

 by new cusps upon the teeth or by newly 

 arising horns upon the skull, we find them 

 subject to four important principles : (1) Rec- 

 tigradations appear under the law of ancestral 

 hereditary control, that is, the same rectigra- 

 dations arise independently at different times 

 in the descendants of remote common ances- 

 tors. This law has already been enunciated 

 at a previous meeting of the academy and 

 constitutes one of the most important gen- 

 eralizations brought out by the study of the 

 titanotheres. (2) Eectigradations are con- 

 tinuous, arising from infinitesimal and almost 

 invisible beginnings and passing into a stage 

 of usefulness; this principle was pointed out 

 many years ago by the author and described 

 as " definite variation." (3) Eectigradations 

 from the time of their first appearance are 

 subject to the allometric influence of sur- 

 rounding parts; thus a horn arising as a 

 rectigradation in a brachycephalic skull will 

 assume from the beginning a rounded form; 

 arising in a dolichocephalic skull it will as- 

 sume an elongate or oval form. (4) It is 



probable, but has not yet been demonstrated, 

 that rectigradations are subject to fluctua- 

 tions, that is, are more or less strongly devel- 

 oped around an average mean. 



Second, the allometrons or changes of pro- 

 portion follow partly the same and partly 

 different laws than those pursued by the recti- 

 gradations. The most fundamental diiierence 

 is the following: allometrons arise indepen- 

 dently of remote ancestral hereditary con- 

 trol, that is, from a mesaticephalic ancestor 

 there may arise, on the one hand, a dolicho- 

 cephalic, and, on the other hand, a brachy- 

 cephalic descendant; when, however, a trend 

 of development, a law which appears to be 

 coincident with the " Mutationsrichtung " of 

 Neumayr, is once established then a tendency 

 toward brachycephaly or dolichocephaly, re- 

 spectively, becomes increasingly manifest; in 

 this sense allometrons resemble rectigrada- 

 tions. The second law is that allometrons are 

 continuous. This is positively demonstrated 

 in certain phyla, and apparently will be dem- 

 onstrated in all the phyla as soon as a full 

 series or sequence is obtained. Any other 

 theory of change of proportion but continuity 

 is untenable in the face of the hundreds of 

 measurements which especially demonstrate 

 progressive brachycephaly. Measurements 

 demonstrating progressive dolichocephaly and 

 cytocephaly, or the bending down of the facial 

 upon the cranial region of the skull, are based 

 on less complete series. 



It has been found convenient to introduce 

 a series of cephalic indices of the ratios be- 

 tween breadth and length similar to the cra- 

 nial indices used in anthropology. Thus, 

 breadth -;- length gives the cranial index 

 of a titanothere, and the gradual trans- 

 formation from mesaticephaly into brachy- 

 cephaly or dolichocephaly may be expressed in 

 exact numerical terms. Every bone of the 

 skull enters into these remarkable transforma- 

 tions, and every single bone has its own in- 

 dividual percentage of increment. The evolu- 

 tion of every part is difl'erential. Thus there 

 is no general stretching of the skull in doli- 

 chocephaly, as if it were composed of india 

 rubber ; the elongation may be confined to cer- 



