V GROWTH IN TIME OF THE TOTAL ORGANISM I77 



apply to many curves of growth in linear dimensions. A physiological meaning 

 can hardly be attached to the Gompertz equation which originally was introduced 

 to describe death rates within a population. 



As this survey shows, none of these expressions is apt to reproduce the essential 

 and basic characteristics of the usually observed curves of growth. It is important 

 to note that this criticism is not based upon the consideration that some particular 

 sets of growth data are not well fitted by a certain formula. Rather it has been 

 shown that none of these formulas is concordant with the trend and charac- 

 teristics generally found in empirical growth curves. 



//. The way to arrive at a rational theory of growth is outlined in the preceding 

 paragraph. One has to start with a model, that is, plausible and testable hypo- 

 theses about this phenomenon. These have to be mathematically formulated. 

 The supposed law must then be put to empirical test, that is, it has to be shown 

 that it allows calculation of the process in question. In doing so, one has to start 

 with cases that are as simple and lucid as possible. Subsequently, analysis w^ill 

 proceed to more complicated cases which may entail consideration of interfering 

 factors and complication, revision, generalization, or abandonment of the original 

 theory. A theory of growth should not only give satisfactory approximation of 

 observed growth data, but should also allow deduction, qualitatively but prefer- 

 ably quantitatively, of consequences which can be tested and verified in inde- 

 pendent experiments. Of particular importance is verification of the constants 

 appearing in the growth equations, that is, comparing the constants calculated 

 frow a growth curve with values independently established in physiological 

 determinations. If such verification is successful, it may be inferred that the model 

 under consideration is basically correct, although it may need further improve- 

 ment and correction with the inclusion of additional observations and the 

 progress of research. Eventually the theory should lead to a unification of pre- 

 viously separate fields, e.g. metabolism and growth. 



{d) Quantitative theory of animal growth 



An attempt toward a quantitative theory of growth must first emphasize an 

 essential fact. There is no general growth curve which could be expressed in a 

 universal formula. Rather the course of growth is different in different organisms, 

 in different dimensions, in the organism as a whole and its parts, etc. This is 

 self-evident because the physiological basis of growth is different in each case. 

 As growth ultimately depends on metabolism, it will be the scope of a theory 

 of growth to derive these differences from the comparative physiology of 

 metabolism. 



So far as the author knows, there is at present only one quantitative theory 

 which, based upon a rational hypothesis on the mechanism of growth, allows 

 deduction of quantitative laws, covers a considerable amount of empirical data, 

 and allows for theoretical predictions which were confirmed, often in a surprising 

 way, by experiment. This is the theory and equations of animal growth which, in 

 the basic ideas, was stated by Putter (1920) and further was developed by the 

 present author and his coworkers (Bertalanffy, 1934, 1948, 1949, 1951b, 1957). 



Literature p. 253 



