We search the concepts and methods1) of the theory of deformable solids from Galileo to Lagrange. Neither of them achieved much in our subject, but their works serve as termini: With Galileo's Discorsi in 1638 our matter begins2) (for this is the history of mathematical theory), while Lagrange's Mechanique Analitique closed the mechanics of the Age of Reason with a formal treatise, since regarded universally, though most wrongly, as the definitive repository of the best from all that went before. As will appear, the giants of our subject are James Bernoulli and Euler. Here, for the first time, may be read the story of what these men really did for the theories of flexible or elastic bodies. Modern theories of materials are set chiefly upon the foundation laid down by Catjchy from 1822 to 1845. Thus our account serves as preface to his researches.