  I provide for the first time in English a line-by-line translation of the
entire text of the monumental 1840 memoir of Olinde Rodrigues, "On the
geometrical laws governing the motions of a solid system..." published in
French in the Journal de Math\'{e}matiques Pures et Appliqu\'{e}es. I accompany
the translation with copious notes in italics, in which I explicate some
passages whose meaning is obscure in the direct translation, supply detailed
proofs, where lacking, of assertions in the original, and clarify the overall
organization of the memoir and the relation of its sections to one another.
