DEC. 1899] METHOD OF TREATING VARIATIONS 411 
in order to contrast it with the position shown later. If it is sought 
to connect directly the rate of variation 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 f(x) represent a group of organisms, 
i.e. their characters, let 7, be one single character; ¢(#,) will then represent 
this character in the group and ¢/(#,) its rate of variation. Further, let f(x) 
represent the group of environmental conditions, then f(x) is the rate of 
variation of this group. That (7) varies with /(z) is the accepted position in 
biology. 
That ¢'(x,) =k7'(«) is the above position with regard to the rate of variation 
in a single organ, where & is a OE depending on /f’(#). But x is composed 
of many variables, say x, %, #, etc., hence the true relation between the rate of 
variation in $(x) and (x) ie be $a, Lo L,...)=kf (x). These two equa- 
tions cannot both be true except on two extreme A steaaities : that the other 
organs do not oe or that the rate of variation is the same for all. This 
means that ¢$(x,), ¢'(#,) ... which include the variations due to growth and 
correlation, are all equal and each =¢(x,). This assumption is ‘obviously a 
very great one, but even then we have only come to the observed fact that 
'(x,)=kf' (x). We come now to the conclusion that / measures the rate of 
change of #(x,) with regard to f(x), but what then? The meaning we give to 
k must obviously depend upon the assumption we make as to the relations 
between (x) and f(x). In other words, & 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, & is like any observed constant in the science of 
physics ; in the latter, it is a measure of natural selection. 
The question then comes to be, which of these assumptions will best 
explain the facts? Hitherto the theory of natural selection has 
flourished 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 
