Ill] THE GOMPERTZ CURVE 157 



few and simple cases can a simple curve or single formula be found 

 to represent the growth-rate of an organism; and how our curves 

 mostly suggest cycles of growth, each spurt or cycle enduring for 

 a time, and one following another. Nothing can be more natural 

 from the physiological point of view than that energy should be 

 now added and now withheld, whether with the return of the 

 seasons or at other stages on the eventful journey from childhood 

 to manhood and old age. 



The symmetry, or lack of skewness, in the Verhulst-Pearl logistic 

 curve is a weak point rather than a strong; the Gompertz curve 

 is a skew curve, with its point of inflexion not half-way, but about 

 one- third of the way between the asymptotes. But whether in 

 this or in the logistic or any other equation of growth, the precise 

 point of inflexion has no biological significance whatsoever. What 

 we want, in the first instance, is an S-shaped curve with a variable, 

 or modifiable, degree of skewness. After all, the same difficulty 

 arises in all the use we make of the Gaussian curve : which has to 

 be eked out by a whole family of skew curves, more or less easily 

 derived from it. We are far from being confined to the Gaussian 

 curve {sensu stricto) in our studies of biological probabihty, or to the 

 logistic curve in the study of population. 



Yet another equation has been proposed to the S-shaped curve 

 of growth, by Gaston Backman, a very dihgent student of the 

 whole subject. The rate of growth is made up, he says, of three 

 components: a constant velocity, an acceleration varying with the 

 time, and a retardation which we may suppose to vary with the 

 square of the time. Acceleration would then tend to prevail in the 

 earlier part of the curve, and retardation in the latter, as in fact 

 they do; and the equation to the curve might be written: 



log H = ko + k^ log T-k^ log2 T. 



The formula is an elastic one, and can be made to fit many an 

 S-shaped curve; but again it is empirical. 



The logistic curve, as defined by Verhulst and by Pearl, has 

 doubtless an interest of its own for the mathematician, the statistician 

 and the actuary. But putting aside all its mathematical details and 

 all arbitrary assumptions, the generalised S-shaped curve is a very 

 symbol of childhood, maturity and age, of activity which rises to 



