Rodrigues Rodriguescombining rotationsis credited with being the first to present a method for combining two sets of rotation parameters to form a third set. While Euler had shown that two rotations carried out one after the other is equivalent to a third rotation, he had not shown how to obtain the parameters of the third rotation using those of the first two rotations. This chapter provides details of the results outlined in Rodrigues 1840 paper [Rod40]. At the time, vector algebra had not been developed and the results in the paper were obtained using coordinate calculations along with geometry and algebra. The symbols used here are the ones from the original paper and the geometric operations from the paper are presented with minimum use of modern vector notation, so as to represent Rodrigues efforts in arriving at such an important result. Rodrigues’s original proof for combining rotations is included here to highlight its historical and conceptual significance, though it has been largely superseded in practice by the more systematic quaternion-based approach.