DEVELOPMENT OF MATHEMATICAL ANALYSIS 515 



and where it needed some effort on the part of the chemists to 

 recognize, under their algebraic mantle, laws of high importance. 



It seems that chemistry has to-day gotten out of the premathe- 

 matic period, by which every science begins, and that a day must 

 come when will be systematized grand theories, analogous to those 

 of our present mathematical physics, but far more vast, and com- 

 prising the ensemble of physicochemic phenomena. 



It would be premature to ask if analysis will find in their develop- 

 ments the source of new progress; we do not even know before- 

 hand what analytic types one might find. 



I have constantly spoken of differential equations ruling phe- 

 nomena; will this always be the final form which condenses a theory? 

 Of this I know nothing certain, but we should, however, remember 

 that many hypotheses have been made of more or less experimental 

 nature ; among them, one is what has been called the principle of 

 non-heredity, which postulates that the future of a system depends 

 only on its present state and its state at an instant infinitely near, 

 or, more briefly, that accelerations depend only on positions and 

 velocities. 



We know that in certain cases this hypothesis is not admissible, 

 at least with the magnitudes directly envisaged; one has sometimes 

 misemployed on this subject the memory of matter, which recalls 

 its past, and has spoken in affected terms of the life of a morsel of 

 steel. Different attempts have been made to give a theory of these 

 phenomena, where a distant past seems to interfere; of them I need 

 not speak here. An analyst may think that in cases so complex it 

 is necessary to abandon the form of differential equations, and resign 

 one's self to envisage functional equations, where figure definite 

 integrals which will be the witness of a sort of heredity. 



To see the interest which is attached at this moment to functional 

 equations, one might believe in a presentiment of the future needs 

 of science. 



VIII 



After having spoken of non-heredity, I scarcely dare touch the 

 question of the applications of analysis to biology. 



It will be some time, no doubt, before one forms the functional 

 equations of biologic phenomena; the attempts so far made are 

 in a very modest order of ideas; yet efforts are being made to get 

 out of the purely qualitative field, to introduce quantitative meas- 

 ures. In the question of the variation of certain characteristics, 

 mensuration has been engaged in, and statistic measures which are 

 translated by curves of frequency. The modifications of these curves 

 with successive generations, their decompositions into distinct curves, 

 may give the measure of the stability of species or of the rapidity 



