In his fundamental "Essai sur le probleme des trois corps" (Oeuvres, vol. 6, pp. 229-324, 1772), Lagrange, well before Jacobi's "reduction of the node", carries out the first complete reduction of symmetries in this problem. Discovering the so-called homographic motions (Euler had treated only the collinear case), he shows that these motions necessarily take place in a fixed plane, a result which is simple only for the "relative equilibria". In order to understand the true nature of this reduction-and of Lagrange's equations-it is necessary to consider the n-body problem in an euclidean space of arbitrary dimension. The actual dimension of the ambient space then appears as a constraint, namely the angular momentum bivector's degeneracy. I describe in detail the results obtained in a joint paper with Alain Albouy published in French in 1998 (Albouy and Chenciner in Invent. Math. 131: 151-184, 1998): for a non-homothetic homographic motion to exist, it is necessary that the motion takes place in an even-dimensional space. Two cases are possible: either the configuration is "central" (that is, we have a critical point of the potential among configurations with a given moment of inertia) and the space where the motion takes place is endowed with a hermitian structure, or it is "balanced" (that is, we have a critical point of the potential among configurations with a given inertia spectrum) and the motion is a new type, quasi-periodic, of relative equilibrium. Only the first type is of Kepler type and hence corresponds to the absolute minimum in Sundman's inequality. When the space of motion is odd-dimensional, one can look for a substitute to the non-existing homographic motions: a candidate is the family of Hip-Hop solutions, which are "simple" periodic solutions naturally related to relative equilibria through Lyapunov families of quasi-periodic solutions (see Chenciner and Fejoz in Regul. Chaotic Dyn. 14(1): 64-115, 2009). Finally, some words are said on the bifurcation of periodic central relative equilibria to quasi-periodic balanced ones.
