# Mathematical Treasure: Cauchy’s Cours d’Analyse

Author(s):
Frank J. Swetz (The Pennsylvania State University)

Augustin-Louis Cauchy (1789–1857) was a French mathematician. He studied at and graduated from the École Polytechnique and then spent his career as a professor there. In 1821, Cauchy wrote Cours d’Analyse for his students. During his lifetime he was a prolific author, producing five textbooks. However, his most influential textbook was his Cours d’Analyse because it established rigorous foundations for calculus. In this book, Cauchy used series and sequences extensively to prove his results, introduced the $$\delta-\varepsilon$$ (delta-epsilon) notation, and provided the definitions used today for function, limit, and derivative. The complete volume in French can be found online at Gallica-Math. Springer has published an annotated English language translation (2009) by Robert E. Bradley and C. Edward Sandifer that can be purchased as a hardcover book or consulted online.

These pages are from the fifth chapter of the book, “Research on a continuous function formed so that if two such functions are added or multiplied together, their sum or product is the same function of the sum or product of the same variables.”  Problem 1 requires:

To determine the function $$\phi(x)$$ in such a manner that it remains continuous between any two real limits of the variable $$x,$$ and so that for all real values $$x$$ and $$y$$ we have:  $$\phi(x+y)=\phi(x)+\phi(y)$$.

On these pages for Chapter 7, “On the imaginary expressions and their moduli,” Cauchy considered what is referred to today as “De Moivre’s Theorem.” In particular, he considered the polar expression for a complex number, $$\cos(\theta)+i\sin(\theta),$$ under various conditions, such as when the argument $$\theta$$ is negative and when the expression is raised to the power $$m$$ or $$-m.$$

The Special Collections staff at the Linderman Library of Lehigh University in Bethlehem, Pennsylvania, is pleased to cooperate with the Mathematical Association of America to exhibit this and other items from the Library’s holdings in Mathematical Treasures. In particular, Convergence would like to thank Lois Fischer Black, Curator, Special Collections, and Ilhan Citak, Archives and Special Collections Librarian, for their kind assistance in helping to make this display possible. You may use these images in your classroom; all other uses require permission from the Special Collections staff, Linderman Library, Lehigh University.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Cauchy’s Cours d’Analyse," Loci (June 2014)