706 SCIENCE AND HYPOTHESIS 



ized in planes at right angles, that they are then brought back again 

 to the same plane of polarization, and that we try to obtain inter- 

 ference. If the light were rigorously monochromatic, there would be 

 interference; but with our nearly monochromatic lights, there will be 

 no interference, and that, however narrow the ray may be. For it to be 

 otherwise, the ray would have to be several million times finer than 

 the finest known rays. 



Here then we should be led astray by proceeding to the limit. The 

 mind has to run ahead of the experiment, and if it has done so with 

 success, it is because it has allowed itself to be guided by the instinct of 

 simplicity. The knowledge of the elementary fact enables us to state 

 the problem in the form of an equation. It only remains to deduce 

 from it by combination the observable and verifiable complex fact. 

 That is what we call integration, and it is the province of the mathe- 

 matician. It might be asked, why in physical science generalization 

 so readily takes the mathematical form. The reason is now easy to 

 see. It is not only because we have to express numerical laws; 

 it is because the observable phenomenon is due to the superposition of 

 a large number of elementary phenomena which are all similar to each 

 other; and in this way differential equations are quite naturally intro- 

 duced. It is not enough that each elementary phenomenon should 

 obey simple laws: all those that we have to combine must obey the 

 same law; then only is the intervention of mathematics of any use. 

 Mathematics teaches us, in fact, to combine like with like. Its object 

 is to define the result of a combination without having to reconstruct 

 that combination element by element. If we have to repeat the same 

 operation several times, mathematics enables us to avoid this repetition 

 by telling the result beforehand by a kind of induction. This I have 

 explained before in the chapter on mathematical reasoning. But for 

 that purpose all these operations must be similar ; in the contrary case 

 we must evidently make up our minds to working them out in full 

 one after the other, and mathematics will be useless. It is therefore, 

 thanks to the approximate homogeneity of the matter studied by phy- 

 sicists, that mathematical physics came into existence. In the nat- 

 ural sciences the following conditions are no longer to be found: 

 homogeneity, relative independence of remote parts, simplicity of the 

 elementary fact; and that is why the student of natural science is 

 compelled to have recourse to other modes of generalization. 



The Theories of Modern Physics 



Significance of Physical Theories. The ephemeral nature of scien- 

 tific theories takes by surprise the man of the world. Their brief period 

 of prosperity ended, he sees them abandoned one after another; he 

 sees ruins piled upon ruins; he predicts that the theories in fashion 



