THE MATHEMATICAL THEORY OF 

 ORGANIC VARIABILITY 



By JAMES JOHNSTONE, D.Sc 



University, Liverpool 



Every character of an organism, or of a part or organ of an 

 organism, is variable. If, as is almost always the case, the 

 characters studied are measurable, this variability can be 

 treated mathematically ; and even if the character is one 

 which cannot be measured, its variability can still be investi- 

 gated in a similar manner. In considering organic variation 

 two series of variable values are formed : ( i ) the arbitrarily 

 chosen values of the dimensions, position, colour, etc., of the 

 variable character, and (2) the series of values representing 

 the frequency with which each value of the variate occurs. 

 The former series (following the usual terminology) is that of 

 the argument, or independent variable, and the latter is that 

 of the function, or dependent variable. Calling the values 

 of the argument ^-values, and those of the function jy-values, 

 we say that y=f(x), that is to say, as x varies uniformly 

 throughout a certain range of values, y also varies but not 

 (in general) uniformly. There is a certain mathematical 

 relationship between x and y which is expressed in saying that 

 y is a function of x. 



In very many cases the form of the function f(x) can be 

 found. Thus if we have a series of bodies of similar shape 

 but of different diameters, and if we weigh all these bodies 

 we shall find that the variation in weight can be expressed by 

 the equation W — ad z , W being the weight of any one of them, 

 d its diameter, and a a. constant. As a very general rule, 

 however, no series of organic variates exhibits this physical 

 simplicity. If we weigh a number of animals belonging to 

 the same species, but all of which are of different lengths, we 

 shall find that the weight is not proportional to the cube of 

 any one diameter (say the length). If we make an empirically 

 drawn curve representing the observed variation of weight with 

 uniformly increasing length, and then try to fit to this curve 

 a calculated one expressing the physical law W = a/ 3 , we shall 



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