Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann. Preceduto dalle Operazioni della logica deduttiva
PEANO, Guiseppe
Published by Turin: Bocca Brothers, 1888
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First edition, rare. "Calcolo Geometrico, G. Peano's first publication in mathematical logic, is a model of expository writing, with a significant impact on 20th century mathematics" (Kannenberg). It may be regarded (in part) as a preliminary version of his more famous work Arithmetices Principia, published in the following year. "Peano's first publication in logic was a twenty-page preliminary section on the operations of deductive logic in Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann (1888)" (DSB). "Peano had [in Calcolo Geometrico] used the logic of Boole and Schröder in mathematical investigations and introduced into it a number of innovations that marked a definite advance upon the work of his predecessors: for instance, the use of different signs for logical and mathematical operations, and a distinction between categorical and conditional propositions that was to lead him to quantification theory" (van Heijenoort, p. 83). The remainder of the work is a development of Hermann Grassmann's Ausdehnungslehre, first published in 1844, which had introduced a novel algebra of geometrical quantities which pre-figures modern vector algebra. Peano's exposition is not only a model of clarity, in contrast to Grassmann's which was notoriously obscure, but he also followed a more axiomatic style than had Grassmann. "In Chapter IX, with the innocent-sounding title 'Transformations of a linear system,' one finds the crown jewel of the book: Peano's axiom system for a vector space, the first-ever presentation of a set of such axioms. The very wording of the axioms (which Peano calls 'definitions') has a remarkably modern ring, almost like a modern introduction to linear algebra. Peano also presents the basic calculus of set operations, introducing the notation for 'intersection,' 'union,' and 'element of,' many years before it was accepted" (Kannenberg). COPAC lists copies at BL, Cambridge, Oxford and Mathematical Association only; no copies listed on ABPC/RBH. Kannenberg, Introduction to the English translation, Geometric Calculus - According to the Ausdehnungslehre of H. Grassmann, 2000; Van Heijenoort: From Frege to Gödel. A Source Book in Mathematical Logic, 1879-1931. 8vo, pp. x, [ii], 170, [2] (light browning, a few marginal notations in pencil). Contemporary half-cloth, paper label on spine (a little rubbed, label chipped). Seller Inventory # ABE-1633015770216
Bibliographic Details
Title: Calcolo Geometrico secondo ...
Publisher: Turin: Bocca Brothers
Publication Date: 1888
Binding: Hardcover
Condition: Very Good
Edition: 1st Edition
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Applicazioni geometriche del calcolo infinitesimale. + Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann. Preceduto dalle Operazioni della logica deduttiva. - [THE BEGINNING OF PEANO'S LOGIC AND THE "GEOMETRICAL CALCULUS"]
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Softcover
Seller: Herman H. J. Lynge & Søn ILAB-ABF, Copenhagen, Denmark
Seller rating 4 out of 5 stars
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.". Seller Inventory # 39042
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