142 WHAT IS SCIENCE? 



inquire whether there is an actual experimental law 

 stating one of the invented numerical relations between 

 measured properties. The process is, in fact, part of 

 mathematics, not of experimental science ; and one of 

 the reasons why mathematics is useful to science is that 

 it suggests possible new forms for numerical laws. Of 

 course the examples that have been given are extremely 

 elementary, and the actual mathematics of to-day has 

 diverged very widely from such simple considerations ; 

 but the invention of such rules leads, logically if not 

 historically, to one of the great branches of modern 

 mathematics, the Theory of Functions. (When two 

 numbers are related as in our tables, they are technically 

 said to be " functions " of each other.) It has been 

 developed by mathematicians to satisfy their own 

 intellectual needs, their sense of logical neatness and of 

 form ; but though great tracts of it have no bearing 

 whatever upon experimental science, it still remains 

 remarkable how often relations developed by the mathe- 

 matician for his own purposes prove in the end to have 

 direct and immediate application to the experimental 

 facts of science. 



NUMERICAL LAWS AND DERIVED MEASUREMENT 



In this discussion there has been overlooked temporarily 

 the feature of numerical laws which, in the previous 

 chapter, we decided gave rise to their importance, 

 namely, that they made possible systems of derived 

 measurement. In the first law, taken as an example 

 (Table I), the rule by which the numerals in the second 

 column* were derived from those in the first involved a 

 numeral 7, which was not a member of those columns, 

 but an additional number applicable equally to all 

 members of the columns. This constant numeral, 

 characteristic of the rule asserted by the numerical law, 



