THE OMGIN OF SPECIES 



strongly as they approach full size (Figure 99). In this series, k = 4.25, 

 a very high value. In deer, antlers show positive allometry. In the red 

 deer, Cervus elaphus, k is about 3 in young bucks, but it declines to about 

 1.6 as they grow fatter. Similarly, the "tails" of swallowtail butterflies show 

 positive allometry (Figure 100). 



Thompson and Huxley showed that phylogenetic changes in relative 

 size can be analyzed in the same way as are ontological changes. That is, 

 if selection favors increase in total body size, as long as the genes for 

 allometry remain unchanged, this will require further increase in the pro- 

 portionate size of specific parts. Thus the Irish Elk, Cermis antiquus, 

 shared with other deer the positive allometry of the antlers. The species 

 grew steadily larger throughout the late Pleistocene, and the antlers, con- 

 tinuing their allometric growth, became truly immense. It is often said 

 that the Irish Elk became extinct because its antlers grew so large that 

 it could not hold up its head. There is no evidence for this improbable 

 assertion. Had selection against increased antler size been so severe, it 

 would no doubt have favored genes for smaller body size. The cause of 

 the extinction of this giant deer is unknown. 



General body outline can be analyzed by Cartesian transformation, a 

 special aspect of allometry, particularly in highly variable groups. The 

 outline of a primitive member of the group is drawn upon a rectangular 

 grid, then the grid is distorted by stretching particular parts in one direc- 

 tion or another. The results simulate the outlines of related species with 

 different factors for allometric growth. Figure 101 shows this for a series 

 of crabs. This is a strong indication that such evolutionary changes are 

 based upon mutations of the genes influencing allometry. This conclusion 

 is also supported by Kurten's study on bear teeth. 



REFERENCES 



Huxley, J. S., 1932. "Problems of Relative Growth," Methuen & Co., Ltd., London. 

 A thorough study of problems of allometry. 



KuRTEN, B., 1955. "Contribution to the History of a Mutation During 1,000,000 Years," 

 Evolution, 9, 107-118. An interesting study, reported without difficult mathematics. 



Li, C. C, 1955. "Population Genetics," University of Chicago Press. An authoritative 

 treatment which, however, requires a good mathematical background. ( Hardy, 

 Weinberg. ) 



Simpson, George Gaylord, Anna Roe, and Richard C. Lewontin, 1960. "Quantita- 

 tive Zoology," Revised Ed., Harcourt, Brace, & Co., Inc., New York, N.Y. An un- 

 usually lucid introduction to biometry, based upon zoological examples. 



Thompson, D'Arcy, 1952. "On Growth and Form," 2nd Ed., Cambridge University 

 Press. This book is a beautifully written classic, and the foundation of the allometry 

 concept. 



Wright, S., 1940. "The Statistical Consequences of Mendelian Heredity in Relation to 

 Speciation," in Huxley, "The New Systematics," Oxford University Press. A succinct 

 statement by one of the major architects of evolutionary mathematics. 



262 



