238 GROWTH PRINCIPLES AND THEORY 2 



monotonic asymptotic approach to the final value or will be S-shaped, the point of 

 inflexion lying between o and 0.37 of final size, and the higher the greater the value of (7. 

 Length growth in time of an organ will show no inflexion if the relative growth of the organ 

 is negatively allometric or isometric, but will have an inflexion in the case of strong positive 

 allometry (a > i). Weight growth where the allometry constant a is multiplied by 3 will 

 be sigmoid with an inflexion, except in the case of strong negative allometry of the organ 

 (a < 1/3) where the curve of time growth will be qualitatively similar to a decaying 

 exponential. 



Figs. 37 and 38 illustrate the validity of this derivation, the brain as an organ 

 with strong negative allometry [cf. Table 16, p. 242) showing monotonic decrease of 

 growth rates, while other organs with higher allometry constants present S-shaped 

 curves. Numerical calculation would be easy but tiine-consuming except when 

 vising an electronic computer. 



Similar considerations apply to regressive changes. In first approximation, the weight 

 loss in a starving animal appears to follow the simple exponential: 



w = w^e-^' (5.17) 



Hence, organs should decrease according to : 



y = Ce-^^' (7.14) 



According to this, organs should decrease the more in starvation the higher their al- 

 lometry constants are in progressive development, and the ratio of organs in the starving 

 animal should be shifted in favor of those which grow slowly. If there are no irreversible 

 diflferentiations, as is the case in some lower organisms, the starving animal will seemingly 

 regress to an earlier developmental stage and appear "rejuvenated". This is well-known 

 for the body proportions of starving planarians and can be shown quantitatively for the 

 digestive tract (Table 15). Quantitative measurements of progressive and regressive 

 changes, applying equations (7.13) and (7.14) would allow another experimental verifi- 

 cation of the constants x and a and their relation. 



TABLE 15 



INTESTINAL SURFACE IN NORMAL AND STARVING PLANARIANS 



After BertalanflTy, 1940a 



Initial weight ca. 20 mg, 

 starved 6 weeks at ca. 19° C 



Unstarved animals 



Initial weight ca. 20 mg, 

 starved 9 weeks at ca. 1 1" C 



Unstarved animals 



VIII. GROWTH AND EVOLUTION 



(a) Transformation 



Changes in proportions represent a considerable part of evolutionary changes 

 within a given type or groundplan. Two approaches permit analysis of such 



