176 GROWTH PRINCIPLES AND THEORY 2 



phase"', to which the decaying exponential was apphed. This sphtting into pieces 

 of a very characteristic and universal curve is gratuitous, and no physiological 

 credentials can be given. 



J. The logistic has been most favored in quantitative analysis of growth. It is 

 likely to give a correct interpretation of growth in higher green plants {cf. footnote, 

 p. 200). In the case of animal growth, however, the logistic is misplaced. The 

 logistic has a point of inflexion at 1/2 the final weight. This contradicts observed 

 growth curves which usually show inflexion at about 1/3 the final weight. The 

 logistic is ruled out by consideration of growth in length. If the logistic is to apply 

 for weight growth, the curve of growth in length should have an inflexion at 

 0.63 of final length. Such curve is never found in growth of linear dimensions. 



To adjust the logistic to actual growth curves, more complicated expressions 

 have been used. Thus, Robertson (1926) used an asymmetric logistic: 



^og^'^^Kit-T) (5.14) 



with an accessory constant B as compared to the solution of the logistic : 



log ~^=K{t-T) (5.15) 



(T = time for w = w*J2). 



However, to fit empirical data, Robertson had to pile up superimposed curves. 

 For the calculation of the growth of white mice he used no less than four curves 

 (two symmetric, one asymmetric logistic, and a linear equation) (Fig. 27, p. 221) 

 with 1 1 arbitrary constants. Another way to overcome the rigidity of the logistic 

 was used by Pearl and Reed (1925) by introducing, in place of the constant k, 

 a power series of time. For example. Pearl's formula for the growth in rats reads: 



243 



7 + 



I + e 



4.3204 — 7.2196f + 30.878/2 + 0.5291/3 



That complications of this sort yield better approximation is, of course, no merit of 

 a special equation but due to the profusion of arbitrary constants to which no 

 physiological meaning can be attached. Robertson's data, fitted by one single 

 formula according to (5.28) are shown in Fig. 27 (p. 221). Examples calculated by 

 Pearl are indicated in Figs. 11 (p. 191) and 26 (p. 221), and we see that these data 

 are well represented by equations (5.28, 5.29). 



4. The parabola has no steady-state solution or inflexion. Since empirical growth 

 curves, in general, have, the parabola is unfit to describe growth in time. Approxi- 

 mation of empirical data by means of parabolic equations is possible only by 

 siibdividing the curve into approximable segments which is arbitrary since no 

 physiological reason can be given. Furthermore, any function for specific growth 

 rates which contains t as the only independent variable, is a priori unfit to describe 

 growth. For growth rates are in general a function not of time, but of the size 

 attained (p. 213). In other words, formulas containing t as independent variable 

 do not allow for regulation as it does occur in many phenomena of growth. 



5. The Gompertz equation has a point of inflexion at 0.37 of final weight, which 

 corresponds tolerably well to empirical curves of weight growth, but does not 



