In recent research on the reception of Poincaré's methods in celestial mechanics, I have formulated the conjecture that the “new methods” referred to in the title of the famous book of Poincaré was part of a strategy to connect himself with the field of celestial mechanics. One of the main pieces of evidence was that there were other names known as practioners of “new methods.” Thus, I suggest that, although Poincaré made new contributions, his intention was not merely to label his own methods as “new”; he was seeking rather to promote the so-called new methods and to gain acceptance in the field.

When one investigates the writings of important actors of this context, it is noticeable that a great number of them speak of new methods associated with the names of Hill and Gyldén. It is known that Poincaré substantially used Hill’s approach in formulating his own methods and that Gyldén reacted negatively to the prize Poincaré won. The relationship between these individuals, all of whom were recognized as innovators, is however more complicated. Starting with their published papers, I propose to analyze the differences and similarities in their methods and how they were received by astronomers until the very beginning of the 20th century.

One of Poincaré’s propositions most efficiently used by contemporary astronomers involves the determination of periodic solutions in the restricted three-body problem, which starts with a known closed orbit and investigates the effects of perturbations. This practice is related to a method Gyldén disseminated some years before, that of Intermediate Orbits. On the other side, Poincaré’s developments are based on another way of treating these orbits, proposed by Hill in 1877 but only known in Europe after 1886. The renewal of interest in Hill’s methods probably has to do with the popularity of Gyldén. Yet, it was Hill’s approach that led Poincaré to consider Gyldén's methods as outdated: Gyldén's methods have only historical interest, stated Poincaré and other commentators after him.

This brief account shows how much this episode deserves to be reexamined, and it is an important part of understanding the network into which Poincaré desired to insert his works in celestial mechanics.