Cauchy's Cours d'analyse: An Annotated Translation

Description

He proves the Fundamental Theorem of Algebra in ch. 10 (Theorem 2), pp. 224ff [PDF pp. 236ff.].

Theorem 3 [PDF pp. 228ff.] is that n -th order polynomials have at most n real or immaginary roots.

A.-L. Cauchy's [Cours d'analyse de l'École royale polytechnique. I. Analyse algébrique, Debure frères, Paris, 1821], recently reissued in the Cambridge Library Collection [Cambridge Univ. Press, Cambridge, 2009], was republished [Cauchy 1897] as part of his [Oeuvres complètes, Ser. 2, Vol. 3, Imprimerie Royale, 1897]. It has appeared in German, Russian and Spanish translations. This first English translation is based on [Cauchy 1897] and includes an index and a page concordance between the 1821 and 1897 editions as well as discussion of the minor differences between them.
[Cauchy 1821] was intended as the first part (Analyse algébrique) of a multipart work. The other parts were never written due to curricular changes at the École Polytechnique. Nevertheless, [Cauchy 1821] became very influential as an early attempt at rigor in the foundations of analysis. It covers real functions, continuity, simultaneous linear equations, interpolation by polynomials, special functional equations, convergent and divergent series, complex numbers, functions and series, the fundamental theorem of algebra, the numerical solution of equations and infinite products, among other things.
The translators provide a preface, about 200 (mostly brief) footnotes and a bibliography of 64 items. Their policy is to "let Cauchy speak for himself'' even if this occasionally means awkward formulations or the use of terms which could be misleading to modern readers. In the footnotes, cross-references are sometimes given to pages in [Cauchy 1821] and [Cauchy 1897] but not to those in the current translation. The footnotes and bibliography give, or lead to, useful information on historical questions.

Reviewed by M. E. MuldoonA.-L. Cauchy's [Cours d'analyse de l'École royale polytechnique. I. Analyse algébrique, Debure frères, Paris, 1821], recently reissued in the Cambridge Library Collection [Cambridge Univ. Press, Cambridge, 2009], was republished [Cauchy 1897] as part of his [Oeuvres complètes, Ser. 2, Vol. 3, Imprimerie Royale, 1897]. It has appeared in German, Russian and Spanish translations. This first English translation is based on [Cauchy 1897] and includes an index and a page concordance between the 1821 and 1897 editions as well as discussion of the minor differences between them.

[Cauchy 1821] was intended as the first part (Analyse algébrique) of a multipart work. The other parts were never written due to curricular changes at the École Polytechnique. Nevertheless, [Cauchy 1821] became very influential as an early attempt at rigor in the foundations of analysis. It covers real functions, continuity, simultaneous linear equations, interpolation by polynomials, special functional equations, convergent and divergent series, complex numbers, functions and series, the fundamental theorem of algebra, the numerical solution of equations and infinite products, among other things.
The translators provide a preface, about 200 (mostly brief) footnotes and a bibliography of 64 items. Their policy is to "let Cauchy speak for himself'' even if this occasionally means awkward formulations or the use of terms which could be misleading to modern readers. In the footnotes, cross-references are sometimes given to pages in [Cauchy 1821] and [Cauchy 1897] but not to those in the current translation. The footnotes and bibliography give, or lead to, useful information on historical questions.

Reviewed by M. E. Muldoon