Ill] THE LOGISTIC CURVE 147 



The mathematics of the Verhulst-Pearl curve need hardly concern 

 us; they are fully dealt with in Raymond Pearl's, Lotka's and other 

 books. Verhulst starts, as Malthus does, with a population growing 

 in geometrical progression, and so giving a logarithmic curve: 



dp 



He then assumes, as his "loi d'affaibUssement," a coefficient of 

 retardation {n) which increases as the population increases: 



dp 2 



Integrating, p = ^^_^_^_. 



If the point of inflection be taken as the origin, k = 0; and again 

 for < = 00, jt? = — = L. We may write accordingly: 



1 + e-^ 



Malthus had reckoned on a population doubling itself, if unchecked 

 by want or "accident," every twenty-five years*; but fifty years 

 after, Verhulst shewed that this "grande vitesse d'accroissement" 

 was no longer to be found in France or Belgium or other of the 

 older countries!, but wa^ still being reaUsed in the United States 

 (Fig. 28). All over Europe, "le rapport de I'exces annuel des nais- 

 sances sur les deces, a la population qui I'a fourni, va sans cesse en 

 s'affaiblissant ; de maniere que Faccroissement annuel, dont la valeur 

 absolue augmente continuellement lorsqu'il y a progression geo- 

 metrique, parait suivre une progression tout au plus arithmetique." 



nourrie de bles etrangers, jamais un gouvernement sage ne consentira a faire 

 dependre I'existence de milliers de citoyens du bon vouloir des souverains etrangers." 

 On this and other problems in the growth of a human population, see L. Hogben's 

 Genetic Problems, etc., 1937, chap. vii. See^also [int. al.) Warren S. Thompson and 

 P. K. Whelpton, Population Trends in the United States, 1933; F. Lorimer and 

 F. Osborn, Dynamics of Population, 1934, etc. 



* An estimate based, like the rest of Malthus's arithmetic, on very slender 

 evidence. 



t In Quetelet's time the European countries, far from doubling in twenty-five 

 years, were estimated to do so in from sixty years (Norway) to four hundred years 

 (France); see M. Haushofer, Lehrbuch der Statiatik, 1882. 



