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I provide for the first time in English a line-by-line translation of theentire text of the monumental 1840 memoir of Olinde Rodrigues, "On thegeometrical laws governing the motions of a solid system..." published inFrench in the Journal de Math\'{e}matiques Pures et Appliqu\'{e}es. I accompanythe translation with copious notes in italics, in which I explicate somepassages whose meaning is obscure in the direct translation, supply detailedproofs, where lacking, of assertions in the original, and clarify the overallorganization of the memoir and the relation of its sections to one another.

Rodrigues, Olinde: "Des lois g\'{e}om\'{e}triques qui r\'{e}gissent les d\'{e}placements d'un syst\`{e}me solide...", translation and commentary