Decbmbek 23, 1904.] 



SCIENCE. 



861 



tion idea, on which analysis rests, is thus 

 developed little by little. 



The celebrated works of Euler hold then 

 a considerable place. However, the nu- 

 merous problems which present themselves 

 to the mathematicians leave no time for a 

 scrutiny of principles; the foundations 

 themselves of the doctrine are elucidated 

 slowly, and the mot attributed to d'Alem- 

 bert 'allez en avant et la foi vous viendra' 

 is very characteristic of this epoch. 



Of all the problems started at the end 

 of the seventeenth century or during the 

 first half of the eighteenth, it will suffice 

 for me to recall those isoperimetric prob- 

 lems which gave birth to the calculus of 

 variations. 



I prefer to insist on the interpenetration 

 still more intimate between analysis and 

 mechanics when, after the inductive period 

 of the first age of dynamics, the deductive 

 period was reached where one strove to 

 give a final form to the principles. The 

 mathematical and formal development 

 played then the essential role, and the an- 

 alytic language was indispensable to the 

 greatest extension of these principles. 



There are moments in the history of the 

 sciences and, perhaps, of society, when the 

 spirit is sustained and carried forward by 

 the words and the symbols it has created, 

 and when generalizations present them- 

 selves with the least effort. Such was par- 

 ticularly the role of analysis in the formal 

 development of mechanics. 



Allow me a remark just here. It is often 

 said an equation contains only what one 

 has put in it. It is easy to answer, first, 

 that the new form under which one finds 

 the things constitutes often of itself an im- 

 portant discovery. 



But sometimes there is more; analysis, 

 by the simple play of its symbols, may sug- 

 gest generalizations far surpassing the 

 primitive outline. Is it not so with the 

 principle of virtual velocities, of which the 



first idea comes from the simplest me- 

 chanisms; the analytic form which trans- 

 lates it will suggest extensions leading far 

 from the point of departure. 



In the same sense, it is not just to say 

 analysis has created nothing, since these 

 more general conceptions are its work. 

 Still another example is furnished us by 

 Lagrange's system of equations; here cal- 

 culus transformations have given the type 

 of differential equations to which one tends 

 to carry back to-day the notion of mechan- 

 ical explanation. 



There are in science few examples com- 

 parable to this, of the importance bf the 

 foi-m of an analytic relation and of the 

 power of generalization of which it may be 

 capable. 



It is very clear that, in each case, the 

 generalizations suggested should be made 

 precise by an appeal to observation and 

 experiment, then it is still the calculus 

 which searches out distant consequences for 

 checks, but this is an order of ideas which 

 I need not broach here. 



Under the impulse of the problems set 

 by geometry, mechanics and physics, we see 

 develop or take birth almost all the great 

 divisions of analysis. First were met equa- 

 tions with a single independent variable. 

 Soon appear partial differential equations, 

 with vibrating cords, the mechanics of 

 fluids and the infinitesimal geometry of 

 surfaces. 



This was a wholly new analytic world; 

 the origin itself of the problems treated 

 was an aid which from the first steps per- 

 mits no wandering, and in the hands of 

 Monge geometry rendered useful services 

 to the new-born theories. 



But of all the applications of analysis, 

 none had then more renown than the prob- 

 lems of celestial mechanics set by the 

 knowledge of the law of gravitation and 

 to which the greatest geometers gave their 

 names. 



