258 THE RATE OE GROWTH [ch. 



It is sufficiently obvious that the normal S-shaped curve of growth 

 of an organism resembles in its general features the velocity-curve 

 of chemical autocatalysis, and many writers have enlarged on the 

 resemblance; but the S-shaped curve of growth of a population 

 resembles it just as well. When the same curve depicts the growth 

 of an individual, and of a population, and the velocity of a chemical 

 reaction, it is enough to shew that the analogy between these is a 

 mathematical and not a physico-chemical one. The sigmoid curve 

 of growth, common to them all, is sufficiently explained as an 

 interference effect, due to opposing factors such as we may use a 

 differential equation to express : a phase of acceleration is followed 

 by a phase of retardation, and the causes of both are in each case 

 complex, uncertain or unknown. . Nor are points of difference lacking 

 between the chemical and the biological phenomena. As the 

 chemical reaction draws to a close, it is by the gradual attainment 

 of chemical equihbrium; but when organic growth comes to an end, 

 it is (in all but the lowest organisms) by reason of a very different 

 kind of equilibrium, due in the main to the gradual differentiation 

 of the organism into parts, among whose pecuHar properties or 

 functions that of growth or multiphcation falls into abeyance. 



The analogy between organic growth and chemical autocatalysis 

 is close enough to let us use, or try to use, just such mathematics as 

 the chemist applies to his reactions, and so to reduce certain curves 

 of growth to logarithmic formulae. This has been done by many, and 

 with no httle success in simple cases. So have we done, partially, 

 in the case of yeast ; so the statisticians and actuaries do with human 

 populations; so we may do again, borrowing (for illustration) a 

 certain well-known study of the growing sunflower (Figs. 79, 80). 

 Taking our mathematics from elementary physical chemistry, we 

 learn that : 



The velocity of a reaction depends on the concentration a of 

 the substance acted on: V varies as a, 



V = Ka. 



The concentration continually decreases, so that at time t (in a 

 monomolecular reaction), 



