The Origins of Cauchy's Theory of the Derivative

Document Type



Mathematics (Pitzer)

Publication Date



Cauchy's theorem, Derivative, Limit


It is well known that Cauchy was the first to define the derivative of a function in terms of a rigorous definition of limit. Even more important, he used his definitions to prove theorems about the derivative. We trace the historical background of the property of the derivative which Cauchy used as his definition and of the proof techniques Cauchy used. We focus on Cauchy's theorem. (Cauchy's statement and delta-epsilon proof of this theorem are reproduced as an Appendix to this article). We show how J.-L. Lagrange used what later became Cauchy's defining property of the derivative, and the associated proof techniques—though differently conceived and inadequately justified—to prove facts about derivatives, including Cauchy's theorem (1). We show, looking at the work of Euler and Ampère, where Lagrange got these ideas, how he developed and used them, and by what means they reached Cauchy. Finally, we see how Cauchy, recognizing what was essential in earlier work, clarified and improved what had been done, and for the first time placed the theory of derivatives on a firm mathematical foundation.

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© 1978 Academic Press, Inc.