i82 POPULAR SCIENCE MONTHLY 



to recall how Felix Klein, in a question relative to Riemann surfaces, . 

 has had recourse to the properties of electric currents. 



It is true, the arguments of this species are not rigorous, in the 

 sense the analyst attaches to this word. x\nd here a question arises : 

 How can a demonstration not sufficiently rigorous for the analyst 

 suffice for the physicist ? It seems there can not be two rigors, that 

 rigor is or is not, and that, where it is not there can not be deduction. 



This apparent paradox will be better understood by recalling under 

 what conditions number is applied to natural phenomena. Whence 

 come in general the difficulties encountered in seeking rigor? We 

 strike them almost always in seeking to establish that some quantity 

 tends to some limit, or that some function is continuous, or that it 

 has a derivative. 



Now the numbers the physicist measures by experiment are never 

 known except approximately; and besides, any function always differs 

 as little as you choose from a discontinuous function, and at the same 

 time it differs as little as you choose from a continuous function. The 

 physicist may, therefore, at will suppose that the function studied is 

 continuous, or that it is discontinuous; that it has or has not a deriva- 

 tive; and may do so without fear of ever being contradicted, either by 

 present experience or by any future experiment. We see that with 

 such liberty he makes sport of difficulties which stop the analyst. He 

 may always reason as if all the functions which occur in his calculations 

 were entire polynomials. 



Thus the sketch which suffices for physics is not the deduction which 

 analysis requires. It does not follow thence that one can not aid in 

 finding the other. So many physical sketches have already been trans- 

 formed into rigorous demonstrations that to-day this transformation 

 is easy. There would be plenty of examples did I not fear in citing 

 them to tire the reader. 



I hope I have said enough to show that pure analysis and mathe- 

 matical physics may serve one another without making any sacrifice 

 one to the other, and that each of these two sciences should rejoice in 

 all which elevates its associate. 



