MOLECULAR THEORIES AND MATHEMATICS — BOREL. 177 



only a small number of constants; on the other hand, the attempts 

 made for explaining the presence of this integer by hypotheses of 

 physical discontinuity have led to the placing of this discontinuity 

 in the interior of tlie atom itself. There is, then, no need of a very 

 great number of atoms; one alone is sufficient, whose structure 

 depends only on certain parameters, on magnetons in tlie theory of 

 Ritz, parameters whose number is far from being of the order of the 

 number of the atoms. 



This remark leads us to consider another category of phenomena, 

 to which we have already made allusion and in which the atoms or 

 corpuscles are observed individually. Does not the explanation of 

 these phenomena require atomic hypotheses? It seems difficult to 

 deny it without being paradoxical. We observe, however, that the 

 phenomena such as the emission of the ar-particles are susceptible 

 only of a globate explanation; it is not possible to foresee with pre- 

 cision a determinate emission, but only a mean number. It is, 

 then, only this mean number that exists, scientifically speaking. The 

 phenomenon which consists in the emission of one particle oc does not 

 present the characters which permit of rigorous experimentation. 

 We know not how to foresee it or how to reproduce it at will. It is 

 only the study of the trajectory after the emission that presents 

 these characters and in fact this study only requires equations suffi- 

 ciently restricted in number so that one can write them all. The 

 atomic hypotheses should permit one to foresee such mdividual emis- 

 sion, if one could actually calculate with reference to an extremely 

 great number of equations; but that is not possible; and in that 

 wliich concerns the glohate conjecture the atomic hypotheses is not, 

 at least a priori, necessary. 



We touch here upon the borders of science, since we attain from 

 phenomena accessible to our observation and which depend upon 

 causes so numerous that it would be impossible for us to know with 

 precision all their complexity. Science remains possible only for mean 

 values which one can calculate with precision by means of data accessi- 

 ble to observation. 



It is well understood, I think, that I do not contest the legitimacy 

 and the utUity of jnolecular theories. My remarks as a mathemati- 

 cian can not attain physical reality; at the bottom they reduce them- 

 selves to this: AH the calculations we shall ever really effect will 

 comprise only a sufficiently small number of equations actually writ- 

 ten. If we write one equation, and if we add that we consider some 

 billions of analogous equations, we do not calculate in fact these 

 equations which are not wiittcn, but only the written equation, in 

 taking count, perhaps, of the number of the equations which are not 

 written, a number which will also have been written. All mathe- 

 matical theory, then, reduces itself to a relatively small nujnber of 



