A Clock Model of Heterochroy

It is often impossible to decide whether we deal with speeding up or slowing down in evolution unless we have a standard criterion for measuring time [ein Orthochronisches Vergleichungsobject]. To lay hold of such a criterion is the most important thing we can do.

E. Mehnert, 1897

I find it remarkable that so little attention has been directed toward a synthesis of the two great literatures on size and shape: the quantitative measurement of allometry, long treated as bivariate in Huxley's (1932) formulation, but now attaining a multivariate generalization (Teissier, 1955; Jolicoeur, 1963; Gould, 1966; Hopkins, 1966; Mosimann, 1970; Sprent, 1972), and the study of heterochrony, a subject that has doggedly maintained a purely qualitative and descriptive approach.

The standard techniques of allometry do not provide an optimal metric for heterochrony because they subtly reinforce a prejudice directed against the dissociability upon which heterochrony depends. Mosimann (1970, p. 943) argues persuasively that the "functional relationships mold" of bivariate plotting places undue emphasis upon the functional association of size and shape. The form of a regression comes to be viewed as a primary feature. The abstracted straight line becomes a key character. Mosimann writes: "I do feel strongly that in many cases the use of functional relations in allometry has been a rebirth of the 'type' concepts of taxonomy" (personal communication, March 3, 1970). Any phyletic change is regarded as a "break" or "disruption" of this primary correlation. Association is primary, disassociation exceptional. The plotting of size and shape as a functional relation is inherently uncongenial to the notion of dissociability.

But the functional relation is but one method among many. Mosimann (1970) prefers a nonfunctional approach that considers the vectors of size and shape separately. In this model, isometry is no longer a rigid correlation of two variables with a slope of a = 1, but an expression of the "stochastic independence of some shape vector from some size variable" (1970, p. 931). The nonfunctional approach is rooted in the concept of dissociability. Heterochrony is no longer the disruption of a primary correlation, but rather the simple expression of differential changes in the independent vectors of size and shape.

The bivariate regression is not "truth"; it is a valid picture that directs thought in ways that are rarely appreciated because alternatives are not presented. A proper attention to dissociability requires a new picture in which the ordinate and abscissa of bivariate plots ac-