Paris: Chez de Bure Frères, 1829.
First edition, very rare. Following his earlier textbooks, the Cours d’Analyse (1821) and R sum des Leçons (1823), which provided the first rigorous formulation of the calculus of functions of a real variable, Cauchy in this work for the first time treats complex functions of a complex variable, defining their derivatives and extending to them many of the results of the differential calculus of a real variable obtained in the R sum , most notably Taylor’s theorem. “The discoveries with which Cauchy’s name is most firmly associated in the minds of both pure and applied mathematicians are without doubt his fundamental theorems on complex functions” (DSB). Cauchy also develops here a fully-fledged theory of infinitesimals, in many ways anticipating work of du Bois-Reymond in the late 19th century.
The twenty-three Leçons of the present work present the results on differential calculus from the first part of the R sum , but in much greater detail (and almost three times the length). Most significant, however, is that Cauchy here extends the results from the R sum to the case of complex functions of a complex variable. This begins with the 11th Leçon, in which complex functions of a complex variable are introduced, the next Leçon giving the definition of the derivatives (and differentials) of such functions, and continues with the 13th Leçon which gives the extension of Taylor’s theorem to the complex case. Further discussions of complex functions are found throughout the remaining Leçons. It is important to note that, although functions of a complex variable had been considered in earlier works by Cauchy, notably in his Mémoire sur les intégrales définies, prises entre des limites imaginaires (1825), those works had been concerned exclusively with the integration of such functions along paths in the complex plane (in modern terminology), and not on the properties of the functions themselves or their derivatives.
The present work is also important for containing Cauchy’s first and only fully-fledged theory of infinitesimals. The informal use of infinitesimals in the development of calculus began with Leibniz and Newton, but a more precise notion of infinitesimal had to wait until Cauchy’s Cours. Although Cauchy is widely recognized for having introduced rigour into calculus, in fact he rarely used (even implicitly) the ‘epsilon-delta’ type of argument favoured by modern textbooks. Most of his arguments are, in fact, couched in the language of infinitesimals. But even the R sum considers only the simplest kind of functions of a base infinitesimal, namely polynomials. Cauchy announces in the Pr liminaires of the present work his intention to develop a far more general theory of infinitesimals, and this is carried out in the 6th Leçon, ‘On the derivatives of functions which represent infinitely small quantities’. Cauchy defines the order of a function of an infinitesimal (which can by a real number or +∞), and gives examples of functions whose orders are 0 and +∞. Laugwitz has seen this as an anticipation of work of du Bois-Reymond and others in the late 19th century, and E. Borel, in his Leçons sur les S ries à Termes positifs (1902), explicitly refers to the 1829 Leçons in his discussion of du Bois-Reymond’s work. The use of infinitesimals was only fully accepted by mathematicians following the work of Abraham Robinson in 1961.
The present Leçons also contain some notable contributions to proving the existence of, and finding approximations for, the roots of algebraic and transcendental equations. The 14th Leçon proves a number of theorems on the existence of roots, which allow Cauchy to give, in particular, a new proof of the fundamental theorem of algebra. Finally, in a Note following the 23rd Leçon, Cauchy uses Taylor series to develop methods of finding approximate roots, recovering and extending results of Fourier.
Very rare, we have located only one copy at auction in the past 35 years.
D. Laugwitz, ‘Infinitely small quantities in Cauchy’s textbooks’, Historia Mathematica 14 (1987), 258–274.
4to (242 x 186 mm), pp. [iv] ii  289 , contemporary half calf, gilt spine lettering, upper part of spine slightly sunned, some light spotting throughout but in general very good, paper label with numbering to spine, top-edge gilt, ex-libris of Henri Viellard to free end-paper, book plate of the Institut Catholique de Paris to front pastedown, first and final text leaf with small rubber stamp of the Institut.