174 GROWTH PRINCIPLES AND THEORY 2 



(c) Empirical growth formulas 



The above considerations apply to the topic of the present monograph, and 

 allow quickly to dispose of a considerable number of formulas proposed in the 

 study of animal growth. 



What is called a growth curve is in fact a set of usually scattered points rep- 

 resenting observed measurements. Curves of higher degree, even if they are 

 rigidly fixed, often can be fitted by quite different mathematical expressions. 

 A classical example is Kavanagh and Richard's demonstration (1934) that the 

 logistic curve can well be approximated by the probability integral. Such con- 

 sideration applies even more when a set of points is smoothed more or less 

 arbitrarily into a continuous curve. Hence, approximation of empirical growth 

 data is a necessary but by no means sufficient condition that an equation proposed 

 represents a "law" of growth. Good approximation may be obtained with 

 formulas which are meaningless, or with quite different formulas. 



/. The two basic approaches discussed above apply to the phenomenon of 

 growth. The first possibility is a purely empirical approach, that is, mathematical 

 expressions are sought which describe observations as closely and simply as 

 possible. The most general procedure is application of series which can represent 

 any set of observations with any degree of approximation desired, breaking the 

 series after the quadratic or cubic term being usually satisfactory. Such series can 

 be used either in the straightforward form of the Taylor series (equation 5.7), or 

 in some modified form. 



Some equations frequently used for describing growth are series either in j : 



dyjdt = k, + k,y + k^y^ + ... (5.8) 



or in t: 



dyjdt ^k^ +k^t -{- k^t^ + ... (5.9) 



The exponential is the Taylor series retaining one term: 



dyjdt - k^y (5.10) 



{k^ = o as otherwise dyjdt would not vanish with y = o); the logistic retaining 

 two terms: 



dyjdt = k^y + k^y^ (5. 11) 



Equation (5.9) was first proposed by Enriques (1909). The parabola (Schmal- 

 hausen, 1929) 



y=kt'' (5.12) 



is another simplification of this series. Similarly, Backman's (1943) formula is a 

 series in the logarithms : 



logji; = ^0 + '^i log t +k2\ogt^ + . . ., (5.13) 



Pearl's (p. 1 76) formula is a logistic with a power series in t, etc. 



Mathematical description of growth data by formulas of this kind can be useful 

 for purposes of intrapolation and, if necessary caution is used, also of extra- 



