228 GROWTH PRINCIPLES AND THEORY 2 



where S' is the vahie for the biological similarity concerned. Equation (7.1 1) is isomorphic 

 with the allometry formula: 



y = bx'^ (7.1) 



with X = body weight, and a corresponding to the exponent in equation (7.1 1). 



It can further be argued that the exponent of biological similarity will be some- 

 where between the exponents of mechanic ("/) and electrodynamic (■/') similarity, 

 i.e. between (^ + 3Y + S/2) and ([^ + 3Y + §)• It then appears that, considering 

 the dimensionality of the biological function in question, the allometry constants 

 of physiological functions such as those of the cardiovascular and renal system, 

 heart and respiratory rate, etc. are within, and in close agreement to the theoret- 

 ical values (Giinther and Guerra, 1955). The general principle is that, in order 

 to remain functional, an organism cannot remain geometrically similar, i.e. 

 increase isometrically, but must increase by way of physiological similarity, 

 i.e. with various allometries, positive and negative, in respect to its functions 

 and parts. 



Further illustrations of the same principle are presented in Fig. 35 (heart rate 

 in relation to body weight) and in the considerations of p. 242 (allometry constants 

 of various organs). A functional analysis of the allometric growth of birds' wings 

 was given by Meunier (1951). Although it is, in general, not possible numerically 

 to deduce the allometry constants of organs and biological functions, these can 

 be imderstood qualitatively as guaranteeing "biological similarity", i.e. proper 

 functioning in spite of variations of absolute body size. Furthermore, they are 

 within the limits drawn by physical (mechanical and electrodynamic) similarity. 



J. In the last resort, relative growth, that is, the specific capacity of organs to 

 appropriate a certain share of available resources, is determined genetically. It is a 

 quantitative expression of the principle of harmonized reaction rates (Goldschmidt, 

 1927, 1938), known from genetics and physiology of development. This principle 

 states that genes determine chains of reactions which go on simultaneously and 

 whose quantitative ratio determines which of them will eventually gain the lead 

 (for a discussion of Goldschmidt's principle cf. Bertalanffy, 1952). On the other 

 hand, changes in the genome can lead to a displacement of this balance, that is, 

 to changes of relative growth rates and consequently to changes in proportion 

 manifest as evolutionary changes (p. 240!!.). 



The principle of allometry can be applied either intraspecifically, i.e. in com- 

 parison of individuals of the same species but of different sizes, or interspecijically, 

 i.e. in comparison of animals (usually adult) of different races, species, etc. 

 (allomorphosis after Huxley, Needham and Lerner, 1941). Intraspecific allometry 

 is mostly applied to different stages of development : ontogenetic allometry (heter- 

 auxesis after Huxley et al.). When species compared in interspecific allometry 

 belong to a phylogenetic series we can speak of evolutionary allometry. 



The principle of allometry was found to apply to numerous phenomena in 

 morphology, cytology, biochemistry, pharmacology, evolution, etc. (for a review 

 of earlier publications cf. Huxley, 1932; for more recent work Bertalanffy, 1951a). 

 In the following only a few examples are given. 



