This book explores the background of a major intellectual revolution: the rigorous reinterpretation of the calculus undertaken by Augustin-Louis Cauchy and his contemporaries in the first part of the 19th century. Their generation changed the calculus from a method of solving problems to a collection of theorems, based on precise definitions, about limits, continuity, series, derivatives, and integrals. The book shows how Cauchy reshaped inherited 18th-century concepts to create an approach to rigor that we still accept today. In so doing, The Origins of Cauchy's Rigorous Calculus provides fresh insights and a new perspective on the foundations of analysis. After defining rigor and describing the characteristics of 19th-century thinking about analysis, the book examines 18th-century views of the calculus and the manifest lack of interest in the foundations of analysis. The greater part of the book concerns itself with tracing how specific achievements of 18th-century mathematics were transformed by Cauchy into the basis of his rigorous calculus (especially the development of the algebra of inequalities: ideas on limits, continuity, and convergence; and certain 18th-century treatments of the derivative and integral), with the work of Joseph-Louis Lagrange shown to be crucial in the transition to new ways of thinking.