NUMERICAL LAWS AND MATHEMATICS 153 



in the different positions, and thus to arrive at the desired 

 relation between the time and the positions successively 

 occupied by the pendulum. The whole of the calculation 

 which plays so large a part in modern science is nothing 

 but an elaboration of that simple example. 



MATHEMATICAL THEORIES 



We have now examined two of the applications of 

 mathematics to science. Both of them depend on the 

 fact that relations which appeal to the sense of the 

 mathematician by their neatness and simplicity are 

 found to be important in the external world of experiment. 

 The relations between numerals which he suggests are 

 found to occur in numerical laws, and the assumptions 

 which are suggested by his arguments are found to be 

 true. We have finally to notice a yet more striking 

 example of the same fact, and one which is much more 

 difficult to explain to the layman. 



This last application is in formulating theories. In 

 Chapter V we concluded that a theory, to be valuable, 

 must have two features. It must be such that laws 

 can be predicted from it and such that it explains these 

 laws by introducing some analogy based on laws more 

 familiar than those to be explained. In recent develop- 

 ments of physics, theories have been developed which 

 conform to the first of these conditions but not to the 

 second. In place of the analogy with familiar laws, 

 there appears the new principle of mathematical 

 simplicity. These theories explain the laws, as do the 

 older theories, by replacing less acceptable by more 

 acceptable ideas ; but the greater acceptability of the 

 ideas introduced by the theories is not derived from an 

 analogy with familiar laws, but simply from the strong 

 appeal they make to the mathematician's sense of 

 form, 



