516 ALGEBRA AND ANALYSIS 



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 parameters that one questions whether the infinitesimal 

 method itself could be of any service. Some laws of a simple arith- 

 metic character like those of Mendel come occasionally to give 

 renewed confidence in the old aphorism which I cited in the begin- 

 ning, that all things are explained by numbers; 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 economists, with potential econ- 

 omy. After Cournot, the Lausanne school made an effort extremely 

 interesting to introduce mathematical analysis into political econ- 

 omy. 



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 recalls the equation of virtual 

 velocities in mechanics: this is the equation of economic equilib- 

 rium. A function of quantities plays in this theory an essential role 

 recalling that of the potential function. Moreover, the best author- 

 ized 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 con- 

 siders the hom.o oeconomicus." 



Naturally, we find there also the analogues of Lagrange's equa- 

 tions, indispensable matrix of all mechanics. 



While admiring these bold works, we fear lest the authors have 

 neglected certain hidden masses, as Helmholtz and Hertz would 

 have said. But although that may happen, there is in these doctrines 

 a curious application of mathematics, which, at least, in well-circum- 

 scribed cases, has already rendered great services. 



I have terminated, messieurs, this summary history of some of 

 the applications of analysis, with the reflections which it has at 

 moments suggested to me. It is far from being complete; thus I have 

 omitted to speak of the calculus of probabilities, which demands 

 so much subtlety of mind, and of which Pascal refused to explain the 

 niceties to the Chevalier de Mere because he was not a geometer. 



Its practical utility is of the first rank, its theoretic interest has 

 always been great; it is further augmented to-day, thanks to the 

 importance taken by the researches that Maxwell called statistical 

 and which tend to envisage mechanics under a wholly new light. 



I hope, however, to have shown, in this sketch, the origin and 

 the reason of the bonds so profound which unite analysis to geometry 

 and physics, more generally to every science bearing on quantities 

 numerically measurable. 



