178 POPULAR SCIENCE MONTHLY 



progress of observations would only have served to create belief in 

 chaos. 



The second example is equally deserving of consideration. 



When Maxwell began his work, the laws of electro-dynamics ad- 

 mitted up to his time accounted for all the known facts. It was 

 not a new experiment which came to invalidate them. But in looking 

 at them under a new bias, Maxwell saw that the equations became 

 more symmetrical when a term was added, and besides, this term was 

 too small to produce effects appreciable with the old methods. 



You know that Maxwell's a priori views awaited for twenty years 

 an experimental confirmation; or if you prefer, Maxwell was twenty 

 years ahead of experiment. How was this triumph obtained? 



It was because Maxwell was profoundly steeped in the sense of 

 mathematical symmetry; would he have been so, if others before him 

 had not studied this symmetry for its own beauty ? 



It was because Maxwell was accustomed to ' think in vectors,' and 

 yet it was through the theory of imaginaries (neomonics) that vectors 

 were introduced into analysis. And those who invented imaginaries 

 hardly suspected the advantage which would be obtained from them 

 for the study of the real world; of this the name given them is proof 

 sufficient. 



In a word, Maxwell was perhaps not an able analyst, but this 

 ability would have been for him only a useless and bothersome baggage. 

 On the other hand, he had in the highest degree the intimate sense 

 of mathematical analogies. Therefore it is that he made good mathe- 

 matical physics. 



Maxwell's example teaches us still another thing. 



How should the equations of mathematical physics be treated? 

 Should we simply deduce all the consequences, and regard them as 

 intangible realities? Far from it; what they should teach us above 

 all is what can and what should be changed. It is thus that we get 

 from them something useful. 



The third example goes to show us how we may perceive mathe- 

 matical analogies between phenomena which have physically no rela- 

 tion either apparent or real, so that the laws of one of these phenomena 

 aid us to divine those of the otber. 



The very same equation, that of Laplace, is met in the theory of 

 Newtonian attraction, in that of the motion of liquids, in that of the 

 electric potential, in that of magnetism, in that of the propagation of 

 heat and in still many others. What is the result? These theories 

 seem images copied one from the other; they are mutually illuminating, 

 borrowing their language from each other; ask electricians if they do 

 not felicitate themselves on having invented the phrase flow of force, 

 suggested by hydrodynamics and the theory of heat. 



