ISI4.] M. Lagrange. 327 



continually to act upon each other and alter their positions in their 

 revolutions. The problem was that of six bodies. Lagrange 

 attacked the difficulty and overcame it, demonstrated the cause of 

 the inequalities observed by astronomers, and pointed out some 

 others too feeble to be ascertained by observations. The shortness 

 of the time allowed, and the immensity of the calculations, both 

 analytical and numerical, did not permit him to exhaust the subject 

 entirely in a first memoir. He was sensible of this himself, and 

 promised further results, which his other labours always prevented 

 him from giving. Twenty-four years after M. Laplace took up 

 that difficult theory, and made important discoveries in it, which 

 completed it and put it in the power of astronomers to banish em- 

 piricism from their tables. 



About the same time a problem of quite a different kind drew 

 the attention of M. Lagrange. Fermat, one of the greatest ma- 

 thematicians of his time, had left very remarkable theorems 

 resj>ecting the properties of numbers, which he probably discovered 

 by induction. He had promised the demonstrations of them ; but 

 at his death no trace of them could be found. Whether he had 

 suppressed them as insufficient, or from some other cause, cannot 

 now be ascertained. These theorems perhaps may appear more 

 curious than useful. But it is well known that difficulty constitutes 

 a strong attraction for all men, especially for mathematicians. 

 Without such a motive would they have attached so much import- 

 ance to the problems of the brachytochrone, of the isoperimetres, 

 and of the orthogonal trajectories? Certainly not. They wished to 

 create the science of calculation, and to perfect methods which 

 could not fail some day of finding useful applications. With this 

 view they attached the;nselves to the fiist question which required 

 new resources. The system of the world discovered by Newton 

 was a most fortunate event for them. N«ver could the transcend- 

 ental calculu'i find a subject more worthy or more rich. Whatever 

 progress is made in it, the first discoverer will always retain his rank. 

 Accordingly, M. Lagrange, who cites him often as the greatest 

 genius that ever existed, adds also, " and the most fortunate. We 

 do not find every day a system of the world to establish." It has 

 required 100 years of labours and discoveries to raise the edifice of 

 which Newton laid the foundation. But every thing is ascribed to 

 him, and we suppose him to have traversed the whole country upon 

 which he menly entered. 



Many mathematicians doubtless employed themselves on the 

 theorems of Fermat ; but none had been successful. Euler alone 

 had pnetrated into that difficult road in which M. Legendre and 

 M. (jjauss afterwards signalized themselves. M. Lagrange in de- 

 monstrating or rectifying some opinions of Euler, resolved a 

 proiilein wliich appears to be the key of all the others; and from 

 which he deduced a useful result; namely, the complete resolution 

 of equations of the second degree, with two indeternunates which 

 must be whole numbers. 



