THE BEGINNING OF PEANO'S LOGIC AND THE "GEOMETRICAL CALCULUS"

PEANO, GUISEPPE.

Applicazioni geometriche del calcolo infinitesimale. + Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann. Preceduto dalle Operazioni della logica deduttiva.

Torino, Bocca, 1887 + 1888.

Royal 8vo. Bound uncut w. the original wrappers of both works in one very nice a bit later (ab. 1920) red hcalf w. five raied bands to back. Single gilt lines to raised bands and gilt title on spine. A bit of soiling to wrappers, which have minor lacks to the inner hinges, where they are mounted onto hinge-strips. Front-wrappers w. stamp from "Fratelli Bocca Editori". A bit of brownspotting, mainly to first work. A very fine and attractive copy of these two works, very finely bound together. XII, 334, (2) + X, (2), 170, (2) pp.

Two rare and important first editions by the famous Italian mathematician, logical philosopher, pioneer of symbolic logic, and a founder of mathematical logic and set theory, Giuseppe Peano, uniting his first publication in logic with his introduction of the basic elements of geometric calculus.

The present "Calcolo geometrico secondo l'Ausdehningslehre de H. Grassmann" contains a twenty-page long preliminary section on the operations of deductive logic, which constitutes Peano' s very first publication on the subject for which he is most famous, namely logic. This work appeared the year before his seminal "Arithmetices Principia...", in which he further improves his logical symbolism, which is introduced in the preliminary section of the present work. "This section, which has almost no connection with the rest of the text, is a synthesis of, and improvement on, some of the work of Boole, Schröder, Peirce, and McColl." (D.S.B. X:442).

In the other present work, "Applicazioni geometriche del calcolo infinitesimale", Peano introduces the basic elements of geometric calculus and gives new definitions for the length of an arc and for the area of a curved surface. This important work (in which not only his geometrical calculus is introduced, but in which he also presented several new geometrical discoveries) is based on his lectures on infinitesimal calculus and its application to geometry from 1885. "The treatise "Applicazioni geometriche del calcolo infinitesimal" (1887) was based on a course Peano began teaching at the University of Turin in 1885 and contains the beginnings of his "geometrical calculus" (here still influenced by Bellavitis' method of equipolences), new forms of remainders in quadrature formulas, new definitions of length of an arc of a curve and of area of a surface, the notion of a figure tangent to a curve, a determination of the error term in Simpson's formula, and the notion of the limit of a variable figure. There is also a discussion of the measure of a point set, of additive functions of sets, and of integration applied to sets. Peano here generalized the notion of measure that he had introduced in 1883." (D.S.B. X:443).

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 this 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. In 1890 Peano became extraordinary professor, and in 1895 ordinary professor, of infinitesimal calculus at the Unversity of Turin.

Cellerino (Guiseppe Peano e la sua scuola. Catalogo monografico): Nr. 2 + 3. 2: "Il più alto raggiunto dai matematici del XIX secolo nell'elaborazione della teoria delle funzioni di insiemi, è il V capitolo del libro di Peano..." F.A. Medvedev."

Order-nr.: 39042