MOLECULAR THEORIES AND MATHEMATICS BOREL. 175 



This geometrical study of figures having a very great number of 

 dimensions deserves, I believe, to be thoroughly investigated. It 

 puts ill evidence the abstract bases of the theories of mechanics and 

 statistical physics; that is to say, it enables us to distinguish among 

 the propositions to which physicists are led, those which are a conse- 

 quence of physical hr[3otheses from those wliich are derived only 

 from statistical hypotheses. But, independent of its physical 

 utility, this geometrical study of spaces having a very great number 

 of dimensions presents an interest of its own. It is to the molec- 

 ular theories that we are indebted for this new branch of mathe- 

 matics. 



V. 



We can, however, ask ourselves whether it is legitimate to regard 

 as connected with the molecular hypothesis a theory which ought, in 

 fine, to depend upon only a small number of constants. To say that 

 an ellipsoid with a very great number of dimensions is entirely defined 

 by five or six constants is to say that all the consequences that we 

 are to deduce from its study shall be expressible by means of five or 

 six constants. We can not suppose in such a case that it will be pos- 

 sible to imagine an analytical mechanism which would permit us to 

 obtain the same consequences, expressed by means of five or six con- 

 stants, without its being necessary to cause to intervene the equation 

 possessmg a very great number of terms; that is to say, without its 

 being necessary to utilize the molecular hypothesis. 



It is well that we pause on account of this objection, though it 

 may recall the controversy between the energetists and the atomists, 

 a controversy in which the atomists appear to have had tlie decided 

 advantage. In the first place, we can reply with an argument of 

 fact: It matters little that we might conceive the possibility, without 

 making use of molecular hyjiotheses, of combining among themselves 

 the consequences of these hypotheses ; the important thing is to know 

 whether this possibility is actually realized or if, on the contrary, 

 tliey are calculations based uj)on molecular hypotlieses which con- 

 stitute the most simple, if not the only, mode of deduction. If this 

 latter alternative be correct, and it seems difficult to deny it, molecu- 

 lar hypotheses are, then, actually very necessary, and that alone 

 ought to be of consequence to us. 



Under this modest form, which the future holds in reserve, this 

 reply appears preemptory; but I believe that many physicists would 

 not judge it too categorically. It is necessary to observe, however, 

 that the question is independent of experimental proofs of the reality 

 of the molecules. Should we succeed in seeing, by means of an instru- 

 ment more powerful than a microscope, the molecules of a solid body, 

 it would not follow, however valuable this knowledge might be, that 



