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Cauchy's Calcul Infinitésimal

An Annotated English Translation

  • Book
  • © 2019

Overview

Authors:
  1. Dennis M. Cates

    1. Sun City, USA

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  • The first English translation of Résumé des leçons sur le calcul infinitésimal

  • Complete with annotations throughout

  • Allows wider audience to experience Cauchy's original, complete text on calculus

  • Provides historical supplement to traditional calculus texts

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Table of contents (40 chapters)

  1. Front Matter

    Pages i-xxiv
    Download chapter PDF

  2. Differential Calculus

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. OF VARIABLES, THEIR LIMITS, AND INFINITELY SMALL QUANTITIES.

      • Dennis M. Cates
      Pages 3-6

    3. OF CONTINUOUS AND DISCONTINUOUS FUNCTIONS. GEOMETRIC REPRESENTATION OF CONTINUOUS FUNCTIONS.

      • Dennis M. Cates
      Pages 7-10

    4. DERIVATIVES OF FUNCTIONS OF A SINGLE VARIABLE.

      • Dennis M. Cates
      Pages 11-15

    5. DIFFERENTIALS OF FUNCTIONS OF A SINGLE VARIABLE.

      • Dennis M. Cates
      Pages 17-20

    6. THE DIFFERENTIAL OF THE SUM OF SEVERAL FUNCTIONS IS THE SUM OF THEIR DIFFERENTIALS. CONSEQUENCES OF THIS PRINCIPLE. DIFFERENTIALS OF IMAGINARY FUNCTIONS.

      • Dennis M. Cates
      Pages 21-25

    7. USE OF DIFFERENTIALS AND DERIVED FUNCTIONS IN THE SOLUTION OF SEVERAL PROBLEMS. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. VALUES OF FRACTIONS WHICH ARE PRESENTED UNDER THE FORM \(\frac{0}{0}\).

      • Dennis M. Cates
      Pages 27-30

    8. VALUES OF SOME EXPRESSIONS WHICH ARE PRESENTED UNDER THE INDETERMINATE FORM \(\frac{\infty }{\infty }, \infty ^{0}, \, \dots .\)  A RELATIONSHIP WHICH EXISTS BETWEEN THE RATIO OF FINITE DIFFERENCES AND THE DERIVED FUNCTION.

      • Dennis M. Cates
      Pages 31-35

    9. DIFFERENTIALS OF FUNCTIONS OF SEVERAL VARIABLES. PARTIAL DERIVATIVES AND PARTIAL DIFFERENTIALS.

      • Dennis M. Cates
      Pages 37-41

    10. USE OF PARTIAL DERIVATIVES IN THE DIFFERENTIATION OF COMPOSED FUNCTIONS. DIFFERENTIALS OF IMPLICIT FUNCTIONS.

      • Dennis M. Cates
      Pages 43-47

    11. THEOREM OF HOMOGENEOUS FUNCTIONS. MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

      • Dennis M. Cates
      Pages 49-52

    12. USE OF INDETERMINATE FACTORS IN THE STUDY OF MAXIMA AND MINIMA.

      • Dennis M. Cates
      Pages 53-57

    13. DIFFERENTIALS AND DERIVATIVES OF VARIOUS ORDERS FOR FUNCTIONS OF A SINGLE VARIABLE. CHANGE OF THE INDEPENDENT VARIABLE.

      • Dennis M. Cates
      Pages 59-64

    14. DIFFERENTIALS OF VARIOUS ORDERS FOR FUNCTIONS OF SEVERAL VARIABLES.

      • Dennis M. Cates
      Pages 65-69

    15. METHODS THAT WORK TO SIMPLIFY THE STUDY OF TOTAL DIFFERENTIALS FOR FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. SYMBOLIC VALUES OF THESE DIFFERENTIALS.

      • Dennis M. Cates
      Pages 71-75

    16. RELATIONSHIPS WHICH EXIST BETWEEN FUNCTIONS OF A SINGLE VARIABLE AND THEIR DERIVATIVES OR DIFFERENTIALS OF VARIOUS ORDERS. USE OF THESE DIFFERENTIALS IN THE STUDY OF MAXIMA AND MINIMA.

      • Dennis M. Cates
      Pages 77-80

    17. USE OF DIFFERENTIALS OF VARIOUS ORDERS IN THE STUDY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

      • Dennis M. Cates
      Pages 81-84

    18. CONDITIONS WHICH MUST BE FULFILLED FOR A TOTAL DIFFERENTIAL TO NOT CHANGE SIGN WHILE WE CHANGE THE VALUES ATTRIBUTED TO THE DIFFERENTIALS OF THE INDEPENDENT VARIABLES.

      • Dennis M. Cates
      Pages 85-89

    19. DIFFERENTIALS OF ANY FUNCTION OF SEVERAL VARIABLES EACH OF WHICH IS IN ITS TURN A LINEAR FUNCTION OF OTHER SUPPOSED INDEPENDENT VARIABLES. DECOMPOSITION OF ENTIRE FUNCTIONS INTO REAL FACTORS OF FIRST OR OF SECOND DEGREE.

      • Dennis M. Cates
      Pages 91-96
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Keywords

About this book

This book is a complete English translation of Augustin-Louis Cauchy's historic 1823 text (his first devoted to calculus), Résumé des leçons sur le calcul infinitésimal, "Summary of Lectures on the Infinitesimal Calculus," originally written to benefit his École Polytechnique students in Paris. Within this single text, Cauchy succinctly lays out and rigorously develops all of the topics one encounters in an introductory study of the calculus, from his classic definition of the limit to his detailed analysis of the convergence properties of infinite series. In between, the reader will find a full treatment of differential and integral calculus, including the main theorems of calculus and detailed methods of differentiating and integrating a wide variety of functions. Real, single variable calculus is the main focus of the text, but Cauchy spends ample time exploring the extension of his rigorous development to include functions of multiple variables as well as complex functions.


This translation maintains the same notation and terminology of Cauchy's original work in the hope of delivering as honest and true a Cauchy experience as possible so that the modern reader can experience his work as it may have been like 200 years ago. This book can be used with advantage today by anyone interested in the history of the calculus and analysis. In addition, it will serve as a particularly valuable supplement to a traditional calculus text for those readers who desire a way to create more texture in a conventional calculus class through the introduction of original historical sources. 

Reviews

“This book is a very welcome contribution. I will be using it as a source for student readings in my history of mathematics classes.” (Fernando Q. Gouvêa, MAA Reviews, 19 May, 2019)

Authors and Affiliations

  • Sun City, USA

    Dennis M. Cates

About the author

Dennis Cates holds a Ph.D. degree in Mathematics from Arizona State University, as well as Engineering Physics and Electrical Engineering degrees from the University of California at Berkeley.

Bibliographic Information

  • Book Title: Cauchy's Calcul Infinitésimal

  • Book Subtitle: An Annotated English Translation

  • Authors: Dennis M. Cates

  • DOI: https://doi.org/10.1007/978-3-030-11036-9

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Hardcover ISBN: 978-3-030-11035-2Published: 15 April 2019

  • eBook ISBN: 978-3-030-11036-9Published: 03 April 2019

  • Edition Number: 1

  • Number of Pages: XXIV, 267

  • Number of Illustrations: 1 b/w illustrations

  • Topics: History of Mathematical Sciences, Calculus

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