722 THE THEORY OF TRANSFORMATIONS [ch. 



The organic forms which we can define, more or less precisely, 

 in mathematical terms, and afterwards proceed to explain and 

 to account for in terms of force, are of many kinds, as we have 

 seen; but nevertheless they are few in number compared with 

 Nature's all but infinite variety. The reason for this is not far 

 to seek. The living organism represents, or occupies, a field of 

 force which is never simple, and which as a rule is of immense 

 complexity. And just as in the very simplest of actual cases we 

 . meet with a departure from such symmetry as could only exist 

 under conditions of ideal simplicity, so do we pass quickly to 

 cases where the interference of numerous, though still perhaps very 

 simple, causes leads to a resultant which lies far beyond our powers 

 of analysis. Nor must we forget that the biologist is much more 

 exacting in his requirements, as regards form, than the physicist ; 

 for the latter is usually content with either an ideal or a general 

 description of form, while the student of living things must needs 

 be specific. The physicist or mathematician can give us perfectly 

 satisfying expressions for the form of a wave, or even of a heap of 

 sand; but we never ask him to define the form of any particular 

 wave of the sea, nor the actual form of any mountain-peak or 

 hill*. 



* In this there lies a certain justification for a saying of Minot's, of the greater 

 part of which, nevertheless, I am heartily incUned to disapprove. "We biologists," 

 he says, "cannot deplore too frequently or too emphatically the great mathematical 

 delusion by which men often of great if hmited abihty have been misled into 

 becoming advocates of an erroneous conception of accuracy. The delusion is that 

 no science is accurate until its results can be expressed mathematically. The 

 error comes from the assumption that mathematics can express complex relations. 

 Unfortunately mathematics have a very limited scope, and are based upon a few 

 extremely rudimentary experiences, which we make as very little children and of 

 which no adult has any recollection. The fact that from this basis men of genius 

 have evolved wonderful methods of dealing with numerical relations should not 

 blind us to another fact, namely, that the observational basis of mathematics is, 

 psychologically speaking, very minute compared with the observational basis of 

 even a single minor branch of biology.... While therefore here and there the 

 mathematical methods may aid us, we need a kind and degree of accuracy of which 



mathematics is absolutely incapable With human minds constituted ap they 



actually are, we cannot anticipate that there will ever be a mathematical expression 

 for any organ ox even a single cell, although formulae wiU continue to be useful 

 for dealing now and then with isolated details..." (op. cit., p. 19, 1911). It were 

 easy to discuss and criticise these sweeping assertions, which perhaps had their 

 origin and parentage in an obiter dictum of Huxley's, to the effect that "Mathe- 

 matics is that study which knows nothing of observation, nothing of exjjeriment. 



