## Émile Mathieu's Preface to *Dynamique analytique*

We give below an English translation of Émile Mathieu's Preface to

*Dynamique analytique*(1878):**Dynamique analytique**

**M Émile Mathieu**

Professor at the Faculty of Sciences at Nancy

1878

When a problem of mechanics is solved, it can be achieved by employing geometrical considerations, or it is possible to use exclusively analytical reasoning. The geometer must be able to demonstrate all the elementary properties of mechanics in both ways; but when it comes to difficult questions, and those which require long calculations, it is almost always advantageous to employ only the method of analysis. This method also makes it possible, when an exact question cannot be solved, to more easily judge what one neglects and thus to obtain all the precision that one desires.

Geometrical and mechanical reasonings often have the advantage of being easier for those who are not very familiar with analysis; these are also sometimes more intuitive to everyone. But for those who are versed in analysis, there is a certain interest in treating all questions in a more uniform manner and in being built on a much smaller number of principles.

Moreover, the very complete Treatise on Mechanics by M Resal is conceived in the first style; it seemed to me useful to make a Treatise on Dynamics with the second approach.

When the second edition of Lagrange's

*Mécanique analytique*appeared at the beginning of this century, it was an accomplished work; but Poisson, Hamilton, Jacobi, and other mathematicians have since brought out important works on this subject. In truth, M Bertrand has added excellent Notes to Lagrange's Treatise to bring it to the present level of Science; but these discoveries were sufficiently important for us to wish to blend the new results with the old ones, and that is what determined me to compose the present work. We may judge the results by which science has been enriched on this matter since Lagrange, by only Section II, devoted to works which date back several years.

The Book which I publish, as its title indicates, does not contain Statics, nor does it contain Hydrodynamics; this limitation of the subject has the effect of adding to the uniformity of the work. These parts of mechanics were found in Lagrange's Treaty; but, on the other hand, Sections II, V, VI, VIII, IX contain questions which are almost entirely outside of

*Mécanique analytique*: Section VII also contains many results which are not found in the same work. As for what is due to me, it matters little to the reader, and, moreover, I have indicated it in the body of this Book for the principal things.