TENDENCIES IN MATHEMATICAL SCIENCE 329 



to the connections, only the components W, W, W" ... are effec- 

 tive, therefore the forces F, V, V" ... must be equilibrated by the 

 connections. We will call the forces P, P', P" ... the impressed 

 forces, the forces W, W, W" ..., which produce the actual motions, 

 the effective forces, and the forces F, F', V" ... the forces gained, and 

 lost, or the equilibrated forces. We perceive, thus, that if we resolve 

 the impressed forces into the effective forces and the equilibrated 

 forces, the latter form a system balanced by the connections. Mach. 



To d'Alembert is attributed the celebrated epigram concerning 

 Benjamin Franklin, "He snatched the thunderbolt from heaven, 

 the sceptre from tyrants" (Eripuit coelo fulmen sceptrumque 

 tyrannis] . 



J. L. Lagrange (1736-1813), a native of Turin, also spent many 

 years in Berlin and his later life in Paris, where he became pro- 

 fessor at the newly established Ecole poly technique. At the 

 age of 25 he was pronounced the greatest mathematician living. 

 His chief work, the Mecanique analytique, is a masterly discussion 

 of the whole subject, showing by the aid of the new mathematical 

 methods its dependence on a few fundamental principles. On the 

 death of his royal patron, Frederick the Great, in 1787, he was 

 invited from Berlin not only to Paris, but to Spain and to Naples, 

 accepting the first-named opportunity. Lagrange's works include 

 also very important contributions to differential equations and 

 the calculus of variations^ of which any detailed account would 

 be too technical for our purpose. The significance and impor- 

 tance of Lagrange's Mecanique analytique are within its field com- 

 parable with those of Newton's Principia. 



Lagrange like Newton has possessed in the highest degree the fortu- 

 nate art of discovering the universal principles which constitute the 

 essence of science. This art he understands how to unite with a rare 

 elegance in the development of the most abstruse theories. Laplace. 



In contrast with the predominantly geometrical and synthetic 

 methods of Newton, Lagrange's methods are mainly analytical. 



Generality of points of view and of methods, precision and ele- 

 gance in presentation, have become, since Lagrange, the common 



