A FOUNDATIONAL WORK OF THE SCIENCE OF INFINITESIMAL CALCULUS

GENOCCHI, ANGELO (recte GIUSEPPE PEANO).

Calcolo differenziale e principii di calcolo integrale, pubbicato con aggiunte dal Dr. Giuseppe Peano.

Torino, Fratelli Bocca, 1884.

8vo. Cont. full green cloth w. gilt title and ornamentation to spine. Minor occasional browning. A very nice and clean copy. XXXII, 333, (5, -index and errata) pp.

The scarce first edition of Peano's first major publication, his first book, the work that brought him international fame, and one of the most important calculus texts since the time of Euler and Cauchy.

The present book, which has a somewhat strange history, contributing to its scarcity, is considered a constitutional work of the science of infinitesimal calculus. In 1899 it was translated into German, and in 1903 into Russian.

The famous Italian mathematician, logical philosopher, pioneer of symbolic logic, and a founder of mathematical logic and set theory, Giuseppe Peano (1858 -1932), studied mathematics at the University of Turin, where he was employed just after graduating (1880), and where he stayed almost all of his life, devoting his life to mathematics. After having graduated with honours, he was employed to assist first Enrico D'Ovidio, and then the renowned Angelo Genocchi, who possessed the chair of Infinitesimal calculus. At this time, Genocchi's health was declining, and the teaching of the infinitesimal calculus course was handed over to Peano already in 1882. In 1880 Peano had published his first paper, and the following year he published another three; in 1884 he published his first book, the foundational "Calculus and Principles of Integral Calculus", which constitutes one of "the most important works on the development of the general theory of functions since the work of the French mathematician Augustin-Louis Cauchy (1789-1857)". (Encycl. Britt.)

As is evident from the title-page, the work was based on Genocchi's lectures on calculus; however, the book turned out to be much more than, and in fact something completely different from, that. Peano stands as the editor of the work, but in fact most of the book is written by Peano himself. Apparently, Genocchi had given his approval to the publication of an edited version of his lectures, but when he saw the final result, he regretted the fact that it had appeared under his name. Genocchi stated in a letter that "... the volume contains important additions, some modifications, and various annotations, which are placed first. So that nothing will be attributed to me which is not mine, I must declare that I have had no part in the compilation of the aforementioned book and that everything is due to that outstanding young man Dr Giuseppe Peano ...".
Peano assumed full responsibility for the work and also recognised it as his own. He later saw the importance that this book has had on the development of the science of infinitesimal calculus. "In 1915 he (Peano) printed a list of his writings, adding: "My works refer especially to infinitesimal calculus, and they have not been entirely useless, seeing that, in the judgment of competent persons, they contributed to the constitution of this science as we have it today." This "judgment of competent persons" refers in part to the "Encyclopädie der mathematischen Wissenschaften", in which Alfred Pringsheim lists two of Peano's books among nineteen important calculus texts since the time of Euler and Cauchy. The first of these books was Peano's first major publication and is something of an oddity in the history of mathematics, since the title page gives the author as Angelo Genocchi, not Peano: "Angelo Genocchi, Calcolo differenziale e principia de calcolo integrale, publicato con aggiunte dal Dr. Guiseppe Peano." The origin of the book is that Bocca Brothers wished to publish a calculus text based on Genocchi's lectures. Genocchi did not wish to write such a text but gave Peano permission to do so. After its publication Genocchi, thinking Peano lacked regard for him, publicly disclaimed all credit for the book, for which Peano then assumed full responsibility." (D.S.B. X:441).

Later the same year, after the publication of this his first major work, Peano became professor at the university of Turin.

His first work now stands, not only as one of the founding texts of modern infinitesimal calculus, but also as a prime example of Peano's excellent style, which perfectly mixes simplicity and rigour. "Beginning with a strict definition of real number, essentially that of Dedekind, he develops the calculus systematically, formulating every theorem with the greatest possible accuracy and precision, and strictly avoiding in the proofs any illegitimate appeal to intuitive properties of curves. When the customary enunciations of theorems are too loose, or conditions that need to be satisfied are not as a rule clearly stated, Peano often constructs counter-examples to show that assertions made in standard textbooks are incomplete or erroneous...." (Kneebone, Mathematical Logic and Foundations of Mathematics, p. 142).

Cellerino nr. 1. "Prima edizione del primo libro di Peano che venne tradotto nel 1899 in tedesco e nel 1903 in russo. Pubblicato sotto il nome di Genocchi di cui Peano era assistente, il volume è in realtà interamente opera sua tanto che Genocchi lo disconobbe publicamente dando origine ad una breve polemice. Questa è l'opera che diede a Peano notorietà internazionale." (Cellerino, Guiseppe Piano e la sua scuola. Catalogo monografico. Milano, 2004).

Order-nr.: 38779