2 PROBLEMS OF RELATIVE GROWTH 



to demonstrate the existence of certain broad empirical laws 

 which appear to govern most cases of differential growth so 

 far studied, to discuss their bearing on other branches of 

 biology, and to point the way to further attack on the subject 

 by those trained in other methods. 



The first step, it appeared to me, was to study a number 

 of clear-cut cases of differential growth and to see whether 

 they were capable of quantitative expression. My own mathe- 

 matics are regrettably deficient, but I was able (see Section 2) 

 to obtain a simple formula which appears to be at any rate 

 a first approximation to a general law for differential growth. 

 Among many morphologists and systematists there appears 

 still to linger a distrust of the application of even such element- 

 ary mathematics to biological problems. The usual criticism 

 is that the formulae arrived at may have a certain convenience, 

 but can tell us nothing new, and nothing worth knowing of 

 the biology of the phenomenon. This appears to me to be 

 very ill-founded. In the first place, to have a quantitative 

 expression in place of a vague idea of a general tendency is 

 not merely a mild convenience. It may even be a very great 

 convenience, and it may even be indispensable in making 

 certain systematic and biological deductions. But further, it 

 may suggest important ideas as to the underlying processes 

 involved ; and this is precisely what the quantitative analysis 

 of relative growth is doing. As will be seen in this and the 

 subsequent chapters, there are certain hypotheses which square 

 with the formula, others which do not : without the quanti- 

 tative expression, we should be largely theorizing in the air. 

 I would not trouble to spend my time on this point if it had 

 not been urged on several occasions in my hearing ; other- 

 wise, one would expect that the interaction of quantitative 

 theory with observation and experiment devoted to testing 

 the theory, so fruitful not only in other sciences but in 

 genetics within the field of biology, would automatically be 

 welcomed. 



Furthermore, the establishment of one quantitative rule 

 leads on to the discovery of others. Chapters I and II will 

 be devoted to showing that, when we consider the growth of 

 whole organs relative to the rest of the body, the results can 

 be understood if we postulate that the ratio between the 

 intensity (or relative rate) of growth of the organ and that 

 of the body remains constant over long periods of the animal's 

 life. To borrow a term from another branch of science, there 



