Editing Joseph-Louis Lagrange (section)

==== ''Miscellanea Taurinensia'' ====
In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the [[Turin Academy of Sciences]], and most of his early writings are to be found in the five volumes of its transactions, usually known as the ''Miscellanea Taurinensia''. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by [[Isaac Newton|Newton]], obtains the general [[differential equation]] for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a [[vibrating string|string vibrating transversely]]; in this paper, he points out a lack of generality in the solutions previously given by [[Brook Taylor]], [[Jean le Rond d'Alembert|D'Alembert]], and Euler, and arrives at the conclusion that the form of the curve at any time ''t'' is given by the equation <math>y = a \sin (mx) \sin (nt)\,</math>. The article concludes with a masterly discussion of [[echo (phenomenon)|echo]]es, [[beat (acoustics)|beat]]s, and compound sounds. Other articles in this volume are on [[recurrence relation|recurring]] [[series (mathematics)|series]], [[probability|probabilities]], and the [[calculus of variations]].

The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations, and he illustrates its use by deducing the [[principle of least action]], and by solutions of various problems in [[dynamics (mechanics)|dynamics]].

The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the [[integral calculus]]; a solution of a [[Pierre de Fermat|Fermat]]'s problem: given an integer {{math|''n''}} which is not a [[square number|perfect square]], to find a number {{math|''x''}} such that {{math|''nx''<sup>2</sup> + 1}}{{verify source|reason=Not sure that this is the correct formula|date=January 2022}} is a perfect square; and the general differential equations of [[N-body problem|motion for three bodies]] moving under their mutual attractions.

The next work he produced was in 1764 on the [[libration]] of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of [[virtual work]]. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780.