Report on a memoir of Jacques Binet

 Jacque Binet submitted his memoir Sur l'expression analytique de l'élasticité et de la raideur des courbes à double courbure to be considered for publication. It was assigned to Lazare Carnot and Gaspard de Prony who were asked to report on it. Their report is contained in Thomas Thomson, Annals of Philosophy (Robert Baldwin, 1816). Here is an extract, but we have updated the language a bit:

After a purely geometric introduction, containing some new formulas, relating to polygons whose sides do not all lie in the same plane, and to curves of double curvature, the author considers problems of equilibrium which are the particular object of the paper. He successively introduces the effect of the action of forces on a polygon of the type just described and on curves of double curvature. He established in both systems a theory which is applicable both to the case of stiffness and to the case of elasticity. ...

M Binet has the merit of introducing explicitly and completely into his analysis all the elements of the question he has treated. We say 'explicitly' to distinguish the method which he has followed from that indicated by Lagrange's beautiful method of indeterminate quantities. ...

Lagrange arrives at three equations absolutely identical with those of M Binet. These equations contain three indeterminate quantities, which ought to be referred to as extensibility, flexibility and torsion. But this explanation is not found in the 'Mecanique Analytique', and it must have in general have escaped those who have studied that book. The reporter is even of the opinion that Lagrange has not paid attention to the part which these three indeterminate quantities play in his system, considered from a mechanical point of view; for he has not stated that the last two would furnish infinite forces, the one of first order and the other of second order; something which must have appeared singular to him if he had not overlooked it. M Binet deduces these values very simply from his analysis, and solves very clearly the kind of paradox that they present; leaving nothing to be desired respecting the value, the significance, and the functions of these quantities. If we add to these considerations the remarkable indetermination of the internal forces, which he first, as far as we know, pointed out in treating the equilibrium of the polygon, we conclude from them that he was right to announce that his researches might serve as an explanation and supplement of several chapters of the 'Mécanique Analytique'. In general the analysis is managed with much skill; and the geometric introduction, which would itself be an interesting memoir, ought to confirm, and even increase, the good opinion which has been formed of his scientific merit, from the different works which he has earlier submitted to the judgement of the Class.

In consequence of this report, the Class, in praising the memoir, ordered it to be printed in the collection of the Savans Etrangers.

JOC/EFR January 2015