thought at first sight. The ordinary formulas require to be changed in order to be used in the 
solution of problems of mechanics. Ampère gives these transformations and applies them to the 
ancient problem of the catenary. 
This problem, which consisted in determining the curve formed by an inextensible 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 Pères Pardies, and de Lamis to prove its 
accuracy to the singular adversary who brought to oppose them proof’s from mechanics. In 1691 
Jacques Bernoulli challenged the scientific world with the same problem. Only three geometers had 
the courage to take up the gauntlet - Leibnitz, Huggens, and Jean Bernoulli, who, we may remark in 
passing, at this time, evinced the first symptoms of his jealousy of his mater, benefactor, and 
brother; thus demonstrating that the love of fame is capable of becoming the most ungovernable, most 
unjust, and blindest of the passions. The four illustrious geometers were not content to give the 
true differential equation of the curve; they 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 Ampère contains, in fact, new and very remarkable properties of the 
catenary and ifs development. There is no small merit, gentlemen, in discovering hiatuses in 
subjects explored by such men as Leibnitz, Huggens, and the two Bernoullis. I must not forget to add 
that the analysis of our associate unites elegance with simplicity. Ampère 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, Ampère could not fall to seed: a 
demonstration of the principle of virtual velocities, disengaged from the consideration of 
infinitesimals. Such is the object of a treatise published in 1806, in the thirteenth number of the 
journal of the school. 
Whilst candidate for the position let vacant by the death of Lagrange in 1813, Ampère presented to 
the academy, first: General considerations on the integrals of equations of partial differences 
and afterwards, an application of these considerations to the integration of differential equations 
of the first and second order. These two treatises give superabundant proof that analysis, in 
its most difficult form, was, perfectly familiar to him. 
Ampère was not inactive after becoming a member of the academy; he busied himself with the 
application of analysis to the physical sciences. Amongst these productions we may cite 
I. Demonstration of the laws of Mariotte, read at the academy January 24, 1814.