60 THE STRUGGLE FOR EXISTENCE 



fluctuations about it. . . . He would like an expression, exact if 

 possible, or, failing that, approximate, of the law if there be one. 

 This means a mathematical expression of the functional relation 

 between the variables. . . . 



"It should be made clear at the start that there is, unfortunately, 

 no method known to mathematics which will tell anyone in advance 

 of the trial what is either the correct or even the best mathematical 

 function with which to graduate a particular set of data. The choice 

 of the proper mathematical function is essentially, at its very best, 

 only a combination of good judgment and good luck. In this realm, 

 as in every other, good judgment depends in the main only upon 

 extensive experience. What we call good luck in this sort of connec- 

 tion has also about the same basis. The experienced person in this 

 branch of applied mathematics knows at a glance what general class 

 of mathematical expression will take a course, when plotted, on the 

 whole like that followed by the observations. He furthermore knows 

 that by putting as many constants into his equation as there are 

 observations in the data he can make his curve hit all the observed 

 points exactly, but in so doing will have defeated the very purpose 

 with which he started, which was to emphasize the law (if any) and 

 minimize the fluctuations, because actually if he does what has been 

 described he emphasizes the fluctuations and probably loses com- 

 pletely any chance of discovering a law. 



"Of mathematical functions involving a small number of constants 

 there are but relatively few. ... In short, we live in a world which 

 appears to be organized in accordance with relatively few and rela- 

 tively simple mathematical functions. Which of these one will 

 choose in starting off to fit empirically a group of observations de- 

 pends fundamentally, as has been said, only on good judgment and 

 experience. There is no higher guide" (pp. 407-408). 



(2) We are now confronted by an entirely different problem which 

 has often arisen in other domains of exact science and which repre- 

 sents the next step after establishing the first empirical relations with- 

 out any mathematical theory. The problem is that from clearly 

 formulated hypotheses which appear probable on the ground of collected 

 experimental material certain mathematical consequences are deduced, 

 connecting the experimental values in equations accessible to experimental 

 verification. As a result a mathematical theory of the phenomena 

 observed in a given field of science is obtained. The equations of the 



