534 SCIENCE PROGRESS 



not (generally) obtain agreement between observed and calcu- 

 lated functions. If we suppose that the increase of weight is 

 proportional, not only to the cube of the length, but also to 

 the square of the length, and to the length itself, we shall 

 obtain a very much better agreement between observa- 

 tion and theory. It will then be possible to fit a curve, 

 W = 0o + aJ + a 2 l 2 + a z l 3 +...., to the empirical curve with 

 a very close agreement. It is open to us to regard 

 a + cixl + a 2 l 2 + a 3 / 3 + ... as, in a way, an expansion of al 3 , 

 and to say that not only must we consider the first differential 

 coefficient, but also the second and third derivatives ; but it 

 is likely, all the same, that increase in weight will depend on 

 length and surface as well as on volume. The intensity of 

 metabolism in an organism is, as we know, proportional not 

 so much to the volume and weight, as to the surface ; for 

 the surface (all, or part of it) is the boundary through which 

 metabolic exchange occurs. Therefore we find that the smaller 

 an organism is, the greater, proportionally to its weight, is 

 the rate of metabolism. 



But even when we fit a parabola a + aj + a 2 P + <z 3 / 3 + . . . 

 to a series of measurements of the weights of similar animals 

 of different ages, and obtain a very good correspondence 

 between empirical and theoretical curves, we shall find that 

 the parabola only describes this particular series well. An- 

 other series, as similar as possible to the first one, will be 

 described by an expression containing significantly different 

 coefficients <z > #i, ci 2 , etc. That means that the function 

 f(x) is really a much more complex one than we supposed. 

 The function y is not one of a single variable, nor even of two 

 or three variables, and we must write it y = f (a, b, c, . . . x). 

 That is to say, the causes of variability in an organism are 

 very numerous. And we must not even say that the variables, 

 a, b, c, ... x, are independent of each other, for any one of 

 them may be a function of some of the others. Further, it 

 is generally impossible to find what are these variables on which 

 the function depends, nor how they aree la ted to each other 

 and to the function, because of the complexity itself of the 

 relationships : the problem would be very similar to that of 

 attempting to express the behaviour of (say) a cubic centi- 

 metre of a perfect gas in terms of the number of molecules 

 in it, and of their velocities, directions of movements, and 



