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SCIENCE. 



[N. S. Vol. I. No. 6. 



whom John the first and his three sons 

 were then active, worked in all fields of ma- 

 thematical research, and rendered especiallj^ 

 good service in extending the theorj^ of elas- 

 ticity founded by Galileo. The industrious 

 Euler, a pupil of John Bernoulli, and a com- 

 panion of his sons, em-iched analysis in 

 every direction, gave for the first time the 

 correct theory of rotating bodies, and -WTote 

 on almost every question in the mathema- 

 tics, physics, and astronomy of his day. It 

 is estimated that his memoirs if fully print- 

 ed would fill sixty to eighty quarto volumes. 

 Not the least noteworthy of his works are 

 his Letters to a German Princess, giving a 

 popular account of the principles of me- 

 chanics, optics, acoustics, and astronomy. 



Notwithstanding the broad foundation 

 for mechanics laid by Newton in his Prin- 

 cipia, and notwithstanding the indefatigable 

 labors of Clau-aut, d'Alembert, the Ber- 

 noullis, and Euler, there was near the end 

 of the eighteenth century no comprehensive 

 treatise on the science. Its leading prin- 

 ciples and methods were fairly well known, 

 but scattered through many works, and pre- 

 sented from divers points of view. It re- 

 mained for Lagrange to unite them into 

 one harmonious system. Mechanics had 

 not yet h-eed itself from the restrictions of 

 geometry, though progress since Newton's 

 time had been constantly toward analytical 

 as distinguished from geometrical methods. 

 The emancipation came with Lagrange's 

 Mecanique Analytique, published one hundred 

 and one years after the Prineijna. How 

 completely the geometrical method was sup- 

 planted by the analytical, at the hands of 

 LagTange, may be inferred from a para- 

 graph in the advertisement to his Mecanique 

 Analytique. " One will fiind " he says, " no 

 diagrams in this work. The methods I ex- 

 pose requu-e neither geometrical construc- 

 tion nor geometi-ical reasoning, but only al- 

 gebraical operations subjected to a regidar 

 and uniform procedure." 



From a philosophical and historical point 

 of view this characteristic feature of the 

 Mecanique Analytique is of the gi'eatest im-^ 

 portance. The mere statement of the fact, 

 however, convej'S no adequate idea of the 

 immense value of Lagrange's treatise. The 

 value of his work consists in the exposition 

 of a general method bj^ which every me- 

 chanical question may be stated in a single 

 algebraic equation. The entire history of 

 any mechanical sj'stem, as for example, the 

 solar system, may thus be condensed into a 

 single sentence ; and its detailed interpreta- 

 tion becomes simjjlj^ a question of algebra. 

 No one who has not tried to cope with the 

 difficulties presented by almost any mechan- 

 ical problem can form a just appreciation of 

 the great utility of such a labor-saving and 

 thought-sa-sdug device. It has been well 

 called ' a stupendous contribution to the 

 economjr of tliought.' But Lagrange did 

 more than this for the science of mechanics. 

 He not only perfected a unique and com- 

 prehensive method, and showed how to ap- 

 ply it to manj^ of the most important and rec- 

 ondite problems of his day, but he was the 

 first to di-aw sharpljr the line of demarcation 

 between physics and metaphysics. The me- 

 chanical ideas of Descartes, Leibnitz, Mau- 

 pertius, and even of Euler, had proved to 

 be moi-e or less hazy and unfruitful from a 

 failure to separate those two distinct re- 

 gions of thought. Lagrange piit an end to 

 this confusion, for no seiious attempt has 

 since been made to derive the laws of me- 

 chanics from a metaphysical basis. 



The age which witnessed the culmination 

 of the splendid generalization of LagTange 

 in his Mecanique Analytique was also the age 

 in which Newton's law of gi-avitation re- 

 ceived its verification, and the age in which 

 the foundations of the modern science of 

 mathematical phj-sics were laid. Lagrange 

 himself is closely identified with these two 

 important events in the history of me- 

 chanics ; but the names which outsliine all 



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