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**Les méthodes nouvelles de la mécanique céleste. Tome I: Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotiques. Tome II: Méthodes de MM. Newcomb, Gyldén, Lindstedt et Bohlin. Tome III: Invariants intégraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques. Nouveau tirage augmenté d’un avertissement de Jean Kovalevsky.**
*(French)*
Zbl 0651.70002

Les Grands Classiques Gauthier-Villars. Paris: Librairie Scientifique et Technique Albert Blanchard. Tome I: XI, 385 p.; Tome II: VIII, 479 p.; Tome III: V, 414 p.; FF 544.00 (1987).

This is the second facsimile reprint of H. Poincaré’s “Les méthodes nouvelles de la mécanique céleste” [the first, published in 1957, was reviewed in Zbl 0079.23801]. The three volumes were first published in 1892, 1893 and 1899, see JFM 24.1130.01, JFM 25.1847.03, JFM 30.0834.08, respectively. In a foreword to the present facsimile reprint, J. Kovalevsky explains why modern specialists in the field of celestial mechanics should consider Poincaré’s work to be of more than historical interest. The reason is that the bases of the tracks along which modern researchers are most active were laid by Poincaré.

From the point of view of the historian, Poincaré represents a landmark in the development of modern celestial mechanics, in that he substituted a rigorous treatment of the mechanics of the solar system in place of the semi-empirical computations that had been prevalent before his time. In addition, he suggested many improvements and developments of the techniques of precursors and contemporaries, so that his work should be of interest to historians of earlier nineteenth century celestial mechanics who wish to place the classical problems and techniques in context. For example, one of the problems resolved by the mathematical astronomers in the second half of the nineteenth century was the elimination of secular terms from their expansions in series but the question of convergence remained. Although these expansions were adequate for the calculation of ephemerides, the question of convergence was crucial to considerations of stability. Poincaré proved that the expansions could not be uniformly convergent but that they could be used to provide asymptotic developments of the coordinates of a planet. Poincaré himself believed that his own results enabled him to combine in a sort of synthesis most of the new methods that had been proposed. The presentation of such a synthesis, he declared, was the principal aim of “Les méthodes nouvelles de la mécanique céleste”.

From the point of view of the historian, Poincaré represents a landmark in the development of modern celestial mechanics, in that he substituted a rigorous treatment of the mechanics of the solar system in place of the semi-empirical computations that had been prevalent before his time. In addition, he suggested many improvements and developments of the techniques of precursors and contemporaries, so that his work should be of interest to historians of earlier nineteenth century celestial mechanics who wish to place the classical problems and techniques in context. For example, one of the problems resolved by the mathematical astronomers in the second half of the nineteenth century was the elimination of secular terms from their expansions in series but the question of convergence remained. Although these expansions were adequate for the calculation of ephemerides, the question of convergence was crucial to considerations of stability. Poincaré proved that the expansions could not be uniformly convergent but that they could be used to provide asymptotic developments of the coordinates of a planet. Poincaré himself believed that his own results enabled him to combine in a sort of synthesis most of the new methods that had been proposed. The presentation of such a synthesis, he declared, was the principal aim of “Les méthodes nouvelles de la mécanique céleste”.

Reviewer: E.J.Aiton

### MSC:

70-03 | History of mechanics of particles and systems |

70F15 | Celestial mechanics |

01A75 | Collected or selected works; reprintings or translations of classics |

01A55 | History of mathematics in the 19th century |

37N05 | Dynamical systems in classical and celestial mechanics |