EULOGY ON AMPERE. 135 



thougbt at first sight. The ordinary formuhis require to be changed 

 in order to be used in the solution of problems of mechanics. Ampere 

 gives these transformations and applies them to the ancient problem of 

 the catenary. 



This problem, which consisted in determining the curve formed by an 

 iuextensible chain of uniform weight when attached to two fixed points, 

 is famous under more than one name. Galileo tried, ineffectually, to 

 solve it. His conjecture that the curve sought might be a parabola, 

 was found false, in spite of all the paralogisms accumulated by Peres 

 Pardies, and de Lamis to prove its accuracy to the singular adver- 

 sarj^ who brought to oppose them proofs from mechanics. In 1G91 

 Jaques Bernoulli challenged the scientific world with the same problem. 

 Only three geometers had the courage to take up the gauntlet — Leib- 

 nitz, Huygens, and Jean Bernoulli, who, we may remark in passing, at 

 this time, evinced the first symptoms of his jealousy of his master, bene- 

 factor, and brother; thus demonstrating that the love of fame is capa- 

 ble of becoming the most ungovernable, most unjust, and blindest of 

 the x^assious. The four illustrious geometers were not content to give 

 the true differential equation of the curve ; tbey also pointed out the 

 consequences deduced from it. Everything now seemed to authorize 

 the belief that the subject was exhausted ; but this was a mistake. The 

 treatise of Ampere contains, in fact, new and very remarkable ijroperties 

 of the ^atenary and its development. There is no small merit, gentlemen, 

 in discovering hiatuses in subjects explored by such men as Leibnitz, 

 Huggens, and the two BernouUis. I must not forget to add that the 

 analysis of our associate unites elegance with simplicity. Ampere 

 gave, moreover, a new demonstration of the celebrated mathematical 

 relation known as Taylor's theorem, and calculated the finite expression, 

 neglected when the series are arrested at any term whatsoever. 



Called to the chair of mathematics at the polytechnic school, Ampere 

 could not fail to seek a demonstration of the principle of virtual veloci- 

 ties, disengaged from the consideration of infinitesimals. Such is the 

 object of a treatise published in 180(3, in the thirteenth number of the 

 journal of the school. 



Whilst candidate for the position left vacant by the death of La- 

 grange in 1813, Ampere presented to the academy, first: General con- 

 siderations on the integrals of equations of partial differences ; and 

 afterwards, an application of these considerations to the integration of dif- 

 ferential equations of the first and second order. These two treatises give 

 superabundant proof that analysis, in its most difficult form, was per- 

 fectly familiar to him. 



Ampere was not inactive after becoming a member of the academy ; 

 he busied himself with the application of analysis to the physical 

 sciences. xVmongst these productions we may cite : 



1. Benionstratioii of the laws of viariotre, read at the academy Jan- 

 uary 24, 1814. 



