204 GROWTH PRINCIPLES AND THEORY 2 



Similar considerations apply to the Third Growth Type (2/3 < m < i) for 

 m < n 7^ I . Also in this case, the point of inflexion would be somewhat shifted, 

 but the characteristic of this type, namely, a sigmoid curve of length growth, 

 remains unchanged. 



Variations of the point of inflexion within the limits calculated above {e.g. 

 between 0.24a'* and 0.29!^;*) are hardly detectable in the usually scattered growth 

 data of observation. Hence, barring further analysis based upon future experimen- 

 tation, the approximation of the general growth equation (5.18, p. 179) is satis- 

 factory and it would at present be inadvisable to complicate the formulas. 



The above derivations show that, assuming the principle that animal growth 

 is the result of a counteraction of synthetic and degradative processes, the family 

 of growth equations indicates the necessary as well as suflficient conditions for 

 the variety of growth curves found in observation. This, of course, is not to say 

 that the formulations given are final and do not allow for further analysis, 

 approximation, and refinement which would find their expression in a resolution 

 of the overall terms used into components, introduction of yet unconsidered 

 factors which would appear as new terms in the growth equations, etc. It does 

 say, however, that the general characteristics of empirical growth curves (as well 

 as a considerable number of related facts, p. 212) are consequences of the general 

 model of growth described. 



(/) Further discussion of the growth model 



There is a rather extensive literature on the Bertalanffy growth equations, the under- 

 lying experimental resuhs and theory, extending over the fields of physical chemistry 

 {e.g. Bray and White, 1954, 1957; Precht, Christophersen and Henzel, 1955) ; biochemistry 

 (Duspiva, 1955); theoretical biophysics (Rapoport, 1955a): cell physiology (Crandall and 

 Smith, 1952; Kunkel and Campbell, 1952; Schmidt-Nielsen, Bertalanffy and Pirozynski, 

 1951); cytology (Weiss and Kavanau, 1957); microbiology (Christophersen and Precht, 

 1954); comparative physiology (Duyff and Bruins, 1942; Fiisser and Kriiger, 1951; 

 Kienle and Ludwig, 1956; Krywienczyk, 1952a, b; Ludwig and Krywienczyk, 1950; 

 Roberts, 1957; Sattel, 1956; Will, 1952; Zeuthen, 1955); zoology (Harms, 1955; Klatt, 

 1949; Medawar, 1945); anatomy (Haardick, 1956; Linzbach, 1950, 1955); physiology 

 of nutrition (Mayer, 1949; L. and T. F. Zucker, 1942); anthropology (Keiter, 1951/52); 

 fisheries biology (Beverton, 1954; Beverton and Holt, 1957; Graham, 1956; Yoshida, 

 1956); theory of evolution (Rensch, 1954); etc. 



Hence, the discussion is fairly comprehensive as an evaluation of facts and theory from 

 different viewpoints. Objections on theoretical grounds have been raised {e.g. Duyff and 

 Bruins, 1942; Linzbach, 1955; Mayer, 1949; Medawar, 1945; Weiss and Kavanau, 

 1957; Zeuthen, 1955) which, so far as they deserve attention, are taken into account in 

 the present review. However, with the exception of the data reviewed below, no factual 

 evidence contradictory to the theory has been advanced. 



On the other hand, there is a large volume of confirmatory evidence. For example 

 Ludwig, whose laboratory with that of Bertalanffy, has done the most extensive research 

 on comparative physiology of body size-metabolism relations, states that "no contra- 

 dictions were found in the experiments to BertalanfTy's theory" (Ludwig and Krywienczyk, 

 1950). As already mentioned (p. igsflf.), the large material of the British Fisheries Laboratory 

 and the Food and Agricultural Organization on growth offish and other aquatic animals 

 has led to the acceptance of the equations discussed as a general standard in this branch 

 of applied biology. Careful examination of the various growth equations proposed led to 

 the conclusion that "von Bertalanffy's growth equation is the most satisfactory of any that 



