64 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



(c) There is evidently a definite functional relation (in the mathe- 

 matical sense) between the number of leaves in a whorl and its position. 

 Biologically this means that the form of a particular whorl is in some 

 manner related to the number of whorls which the growing bud has 

 previously formed. Successively formed whorls show an increase in 

 the number of leaves. 



The values in table 34 and the diagrams indicate that this increase 

 follows a definite law. We can in general formulate this by mere in- 

 spection of the data. There is an increase in the mean number of 

 leaves in successively formed whorls, but the increment in leaf -number 

 diminishes with each successive whorl. So much is clear, but we are 

 only thrown back to the discovery of the law according to which the 

 increment diminishes. So far as I can see, any kind of biological rea- 

 soning is powerless to help us further. Observation shows that there 

 is some sort of a functional relation between number of leaves in the 

 whorl and position, just as observation indicates to the physicist that 

 there is a functional relation between two things. But to determine 

 the nature of the functional relation, or in other words, to find out the 

 law which the phenomena follow, we are compelled, so far as I can see, 

 to resort to mathematical treatment of the data. The physicist has 

 done this, of course, for a very long time. If this is a logical, scientific 

 method in physics (and I presume no biologist has any doubt that it is) , 

 why is it not equally logical and scientific when a precisely similar 

 problem is presented by biological phenomena? 



To formulate the law of growth according to which the change in 

 mean leaf -number in the successively formed whorls of a Ceratophyllum 

 plant occurs we must turn to mathematics, as biological reasoning will 

 not help us further. I have dwelt at some length on this point in order 

 to show that in one particular class of biological problems at least we are 

 compelled to call mathematics to our aid, or else to be content to stop 

 considerably short of a goal within easy reach. The idea seems to 

 prevail among many biologists that the application of higher math- 

 ematical methods to biology is altogether idle and futile. It seems 

 possible that something may be done towards removing this unfortunate 

 prejudice if clear and definite statements telling just why it is neces- 

 sary to resort to mathematics if we are to advance on particular prob- 

 lems, are more frequently made in biometrical writings. 



The mathematical problem before us is this: If we let y indicate the 

 mean number of leaves per whorl and x the position of the whorl on a 

 primary branch, then direct observation shows us that 



y=<l>{x). 



