502 ALGEBRA AND ANALYSIS 



foundations themselves of the doctrine are elucidated slowly, and 

 the mot attributed to d'Alembert, "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 problems 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 analytic 

 language was indispensable to the greatest extension of these prin- 

 ciples. 



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 

 themselves with the least effort. Such was particularly 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 into it. It is easy to answer, first, that the 

 new form under which one finds the things constitutes often of itself 

 an important discovery. 



But sometimes there is more; analysis, by the simple play of 

 its symbols, may suggest 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 mechanisms; the analytic 

 form which translates 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 

 calculus transformations have given the type of differential equations 

 to which one tends to carry back to-day the notion of mechanical 

 explanation. 



There are in science few examples comparable to this, of the 

 importance of the form 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 equations with a single independent vari- 



