Olinde Rodrigues' paper of 1840 on transformation groups

It is this paper which will be discussed. In it, as the title indicates, he studied the motions (deplacements) of a rigid body (systeme solide) in three dimensional space independently of any dynamical considerations. He began by giving a complete description of motions in synthetic terms, establishing successively that a body is fixed in space once three non-collinear points have been determined; that if two points are fixed the motion is a rotation about an axis through those points; that translations when composed give a translation which is independent of the order of composition and can be found by the 'loi du polygone des translations' (p. 383) and that a translation is equal to an infinitesimal rotation about an axis perpendicular to the direction of the translation but situated at an infinite distance. RODRIGUES described the resultant as