dec. 1899] METHOD OF TREATING VARIATIONS 411 



iu order to contrast it with the position shown later. If it is sought 

 to connect directly the rate of A r ariation in any one particular organ 

 or part of an organ with a certain change in environmental conditions, 

 it is not difficult to show that the conclusions reached will depend 

 much more upon the pre-existing assumptions with regard to the 

 relations of the organism and environment than upon the actually 

 observed facts. 



This can be shown mathematically. If $(x) represent a group of organisms, 

 i.e. their characters, let x 1 be one single character ; 4>(-x\) will then represent 

 this character in the group and ^'(^1) its rate °f variation. Further, let f(x) 

 represent the group of environmental conditions, then f'(x) is the rate of 

 variation of this group. That <j>(x) varies with f(r) is the accepted position in 

 biology. 



That <f>'(x 1 ) = kf'(x) is the above position with regard to the rate of variation 

 in a single organ, where k is a constant depending on /"(•*')• Brit x is composed 

 of many variables, say x v x.,, x 3 , etc., hence the true relation between the rate of 

 variation in <f>(x) and f{x) must be <£'( J "i J '-> x z ■ • •) = %f '(#)• These two equa- 

 tions cannot both be true except on two extreme probabilities : that the other 

 organs do not vary, or that the rate of variation is the same for all. This 

 means that ^(j^), ^'('''n) • • • which include the variations due to growth and 

 correlation, are all equal and each = <f>(x 1 ). This assumption is obviously a 

 very great one, but even then we have only come to the observed fact that 

 4>'( x i) = kt'( x )- We come now to the conclusion that k measures the rate of 

 change of 4>( J -'i) "with regard to f(x), but what then 1 The meaning we give to 

 h must obviously depend upon the assumption we make as to the relations 

 between <}>(x) and f(x). In other words, I- cannot be taken to prove our 

 original assumption. There seem to be but two ways of regarding the relation 

 between the rate of variation of an organ and a change in the environment, the 

 one that the relation is direct, the change in the environment causing the 

 alteration of the organ during growth ; the other that the relation is indirect— 

 the change in the environment bringing about the alteration in the organ by 

 destroying the individuals which did not possess the actually observed altered 

 organ. In the former case, k is like any observed constant in the science of 

 physics ; in the latter, it is a measure of natural selection. 



The question then conies to be, which of these assumptions will best 

 explain the facts ? Hitherto the theory of natural selection has 

 nourished under the belief that it could explain the facts rather than 

 that the facts were rightly explained. In the conclusions of this 

 paper an endeavour will be made to show how this theory rests on an 

 assumption which, however probable in appearance, must always 

 remain unproven, and it will be suggested that the counter-theory 

 explains the facts better. 



If it is difficult to make a just comparison of the changes in a 

 single organ with the changes in the environment, it is equally 

 difficult, on the other hand, to make such a comparison for the species. 

 It is the " species " that has formed the starting-point of the theory of 

 natural selection, and by the light of the " species " the structures of 

 the individual, its birth, every portion of its life, and even its death, 

 have been interpreted. But the " species " is a quantity not easy to 

 measure, and it thus seems very wide of the mark to talk of a 



