146 THE RATE OF GROWTH [ch. 



of the population was estimated by the number of births, and the 

 births by the baptisms in the Church of England, "de maniere que 

 les enfants des dissidents ne sont point portes sur les registres 

 officiels." A law of population, or "loi d'affaibhssement" became 

 a mere matter of conjecture, and the simplest hypothesis seemed to 

 Verhulst to be, to regard "cet affaibhssement comme proportionnel 

 a I'accroissement de la population, depuis le moment ou la difficulty 

 de trouver de bonnes terres a commence a se faire sentir*." 



Verhulst was making two assumptions. The first, which is beyond 

 question, is that the rate of increase cannot be, and indeed is not, 

 a constant; and the second is that the rate must somehow depend 

 on (or be some function of) the population for the time being. 

 A third assumption, again beyond question, is that the simplest 

 possible function is a linear function. He suggested as the simplest 

 possible case that, once the rate begins to fall (or once the struggle 

 for existence sets in), it will fall the more as the population continues 

 to grow; we shall have a growth-factor and a retardation-factor in 

 proportion to one another. He was making early use of a simple 

 differential equation such as Vito Volterra and others now employ 

 freely in the general study of natural selec.tionf. 



The point wher^ a struggle for existence first sets in, and where 

 ipso facto the rate of increase begins to diminish, is called by Verhulst 

 the normal level of the population ; he chooses it for the origin of his 

 curve, which is so defined as to be symmetrical on either side of 

 this origin. Thus Verhulst's law, and his logistic curve, owe their 

 form and their precision and all their power to forecast the future 

 to certain hypothetical assumptions; and the tentative solution 

 arrived at is one "sous le point de vue mathematiquej." 



♦ Op. cit. p. 8. 



t Besides many well-known papers by Volterra, see V. A, Kostitzin, Biologie 

 mathematique, Paris, 1937. Cf. also, for the so-called "Malaria equations," Ronald 

 Ross, Prevention of Malaria, 2nd ed. 1911, p. 679; Martini, Zur Epidemiologie d. 

 Malaria, Hamburg, 1921; W. R. Thompson, C.R. clxxiv, p. 1443, 1922, C. N. 

 Watson, Nature, cxi, p. 88, 1923. 



J Verhulst goes on to say that " une longue serie d 'observations, non interrompues 

 par de grandes catastrophes sociales ou des revolutions du globe, fera probablement 

 decouvrir la fonction retardatrice dont il vient d'etre fait mention." Verhulst 

 simplified his problem to the utmost, but it is more complicated today than ever; 

 he thought it impossible that a country should draw its bread and meat from 

 overseas: "lors meme qu'une partie considerable de la population pourrait etre 



