December 23, 1904.] 



SCIENCE. 



871 



of new progress ; we do not even know be- 

 forehand what analytic types one might 

 find. 



I have constantly spoken of differential 

 equations ruling phenomena; will this al- 

 ways 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 nature 

 more or less experimental; among them, 

 one is what has been called the principle of 

 non-heredity, which postulates that the fu- 

 ture 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 hy- 

 pothesis 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 com- 

 plex it is necessary to abandon the form of 

 differential equations, and resign oneself 

 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 fu- 

 ture 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 measures. In the question of 

 the variation of certain characteristics, 

 mensuration has been engaged in, and sta- 

 tistic 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 cf 

 species or of the rapidity of mutations, and 

 we know what interest attaches itself to 

 these questions in recent botanic researches. 

 In all this so great is the number of para- 

 meters that one questions whether the 

 infinitesimal method itself could be of any 

 service. Some laws of a simple arithmetic 

 character like those of Mendel come occa- 

 sionally to give renewed confidence in the 

 old aphorism which I cited in the begin- 

 ning, that all things are explained by num- 

 bers ; but, in spite of legitimate hopes, it is 

 clear that, in its totality, biology is still 

 far from entering upon a period truly 

 mathematical. 



It is not so, according to certain econo- 

 mists, with potential economy. After Cour- 

 not, the Lausanne school made an effort 

 extremely interesting to introduce mathe- 

 matical analysis into political economy. 



Under certain hypotheses, which fit at 

 least limiting cases, we find in learned 

 treatises an equation between the quantities 

 of merchandise and their prices, which re- 

 calls the equation of virtual velocities in 

 mechanics : this is the equation of economic 

 equilibrium. A function of quantities 

 plays in this theory an essential role recall- 

 ing that of the potential function. More- 

 over, the best authorized representatives of 

 the school insist on the analogy of economic 

 phenomena with mechanical phenomena. 

 "As rational mechanics, says one of them, 

 considers material points, pure economy 

 considers the homo osconomicus. " 



Naturally, we find there also the ana- 



