## ONE OF THE NINETEEN MOST IMPORTANT CALCULUS TEXTS SINCE EULER AND CAUCHY

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PEANO, Giuseppe [Angelo GENOCCHI].

# Calcolo differenziale e principii di calcolo integrale pubblicato con aggiunte dal D.r Giuseppe Peano.

### Rome, Turin and Florence, Fratelli Bocca, 1884.

8vo, pp. xxxii, 336, [2, corrections], [2, blank]; text a little toned, with occasional spotting, otherwise clean; ink ownership inscription of ‘U. Broggi’ to blank first leaf; marbled endpapers; very good in contemporary gilt panelled half morocco over marbled boards, a little rubbed.

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Calcolo differenziale e principii di calcolo integrale pubblicato con aggiunte dal D.r Giuseppe Peano.

**First edition of Peano’s first book. **The publication was credited to Angelo Genocchi (1817–1889), whose assistant Peano became after graduating at the Università degli Studi di Torino in 1880. Giuseppe Peano (1858–1932) is now more famous as one of the founders of modern mathematical logic and set theory, and for the Peano axioms, named after him, which provide the standard axiomatisation of the natural numbers.

After graduating with honours, Peano was immediately assigned as the assistant of Enrico D’Ovidio (1843–1933), who held the Chair of algebra and analytic geometry. The following academic year (1881-1882) he was transferred to serve as Genocchi’s assistant in infinitesimal calculus, later working as his substitute before the latter’s death in 1889. He then assumed full responsibility for the calculus course, while in 1886 he has also taken on a professorship ay the Military Academy, next door to the university. ‘Thus Peano’s life and career advanced without disturbance … One incident almost disturbed the advance of Peano’s career. A publisher had been trying to get Genocchi to write up his calculus course for publication. When he substituted for Genocchi, Peano was approached about this. Peano obtained permission from Genocchi to make up a text from his course, and this was published in 1884. Genocchi’s name was on the title page, and the title was *Differential Calculus and Fundamentals of Integral Calculus*, “published with additions by Dr. Giuseppe Peano.” The text was probably better than Genocchi’s lessons, and, of course, the additions were the best part of all. Naturally this irritated the quick-tempered Genocchi, and he had published in mathematical journals of Italy, France, and Belgium a declaration that he had had nothing to do with the book. Peano managed to weather the embarrassment caused by this, and the book made him an immediate reputation. Why? Pringsheim, in the *Encyklopädie der Mathematischen Wissenschaften*, lists this as one the nineteen most important calculus texts since the time of Euler and Cauchy (Peano’s calculus text of 1893 is also one of the nineteen) and cites the following results contained in it: theorems and remarks on limits of indeterminate expressions, pointing out errors in the better texts then in use; a generalization of the mean-value theorem for derivatives; a theorem on uniform continuity of functions of several variables; theorems on the existence and differentiability of implicit functions; an example of a function whose partial derivatives do not commute; conditions for expressing a function of several variables with a Taylor’s formula; a counterexample to the current theory of minima; and rules for the integrating rational functions when roots of the denominator are not known’ (Hubert Kennedy, *Twelve articles on Giuseppe Peano* 2002, pp. 17-18).

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