MOLECULAR THEORIES AND MATHEAiATICS.' 



By Il^MILE BOREL, 



Professevr d la Facultc des Sciences dc V Universile dc Paris, Sous-Direcleur de VEcole 



Normale Superieure. 



I. 



The relations between the niathematical sciences and the physical 

 sciences are as old as the sciences themselves. It is the study of 

 natural phenomena which leads man to set for himself the first 

 problems from which, through abstraction and generalization, has 

 gone forth the superb complexity of the science of numbers and of 

 space. Conversely, through a sort of preestablished harmony, it 

 has often happened that certain mathematical theories, after being 

 developed apparently far from the real, have been found to furnish 

 the key to phenomena concerning which the creators of these theories 

 had no thought at all. The most celebrated instance of this fact is 

 the theory of conic sections, an object of pure speculation among the 

 Greek geometers, but whose researches enabled Kepler, 20 centuries 

 later, to announce with precision the laws of tlie motions of the plan- 

 ets. In the same way, in the first half of the nineteenth century, it 

 was due to the theory of the imaginary exponentials that the study 

 of vibratory motions was rendered more profound, the importance of 

 which has been revealed on so large a scale in physics and even in 

 industrial art; it is to this study that we owe wireless telegraphy 

 and the transmission of energy by polyphase currents. More 

 recently stiU, we know what the utility of the abstract theory of 

 groups has been in the study of the ideas so profound and novel 

 whereby one lias tried to explain tlie results of the capital experi- 

 ments on relativity made by your illustrious compatriot Michelson. 



But these illustrations, whatever may be their importance, are 

 special and relate to particular theories. How mucli more striking 

 is the universal usage of the forms imposed on scientific thought by 



> Address delivered at Houston (Tex.) on the occasion of the inauguration of the Rice Institute (Oct. 10 

 to 12, 1912). 



Reprinted by permission of the author, the publishers, and the Rice Institute, from Revue gdndrale des 

 Sciences purcs ct appliqu6es, I'aris, Nov. 30, 1912. 



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