 ## Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann. Preceduto dalle Operazioni della logica deduttiva.

Turin: Bocca Brothers, 1888.

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 contract 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 operation, 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.

“The (unnumbered) chapter on the ‘operations of deductive logic’ that opens the book is Peano’s first publication in mathematical logic. It is based on his study of E. Schröder, G. Boole, C. S. Peirce, and others. Although he says that “several of the notations introduced here are used in the geometrical calculus,” this part of the book actually has very little connection with what follows. His interest seems to be prompted by the analogy with the operations of algebra and geometrical calculus that is presented by the operations of mathematical logic. He says: “Deductive logic, which belongs to the mathematical sciences, has made little progress up to now … The few questions treated in this introduction constitute an organic whole, that can serve in many kinds of research”” (Kennedy, p. 21).

In this preliminary chapter on logic, Peano develops first a calculus of classes and introduces the modern symbols for union and intersection of sets, as well as symbols which have now been superseded for the empty set, the complement of a set, and set inclusion. He then develops a parallel calculus of propositions using some of the same symbols. For example, conjunction and alternation are represented by the symbols for intersection and union. Peano’s primary concern seems to be to show that the propositions of logic can be treated in an algebraic fashion.

“This publication gives only a hint of Peano’s future development of mathematical logic … But his own systematic development of logic will begin almost immediately, with the introduction of new symbols (several used here are abandoned) and new concepts. Noteworthy here, however, is his recognition of an equivalence between the calculus of classes and the calculus of propositions.

“The main part of the book, the geometrical calculus, is, by contrast, a polished work … Peano made no claim to originality for the ideas contained in it, but there can be no doubt that the extreme clarity of his presentation, in contrast with the notorious difficulty of reading Grassmann’s work, helped to spread Grassmann’s ideas and make them more popular … The preface (dated 1 February 1888) begins: “The geometrical calculus, in general, consists of a system of operations to be carried out on geometrical entities, analogous to those of algebra for numbers. It permits the expression by formulas of results of geometrical constructions, the representation by equations of geometrical propositions, and the substitution of the transformation of equations for proofs. The geometrical calculus presents an analogy with analytic geometry; it differs from it in this, that while in analytic geometry calculations are made on the numbers that determine the geometrical entities, in this new science the calculations are made on those very entities” …

“Although it seems to have aroused no attention at the time, one of the most remarkable features of this book historically is that here, for the first time, is presented [at the beginning of Chapter 9] axioms for a vector space … Peano then gives several examples of these ‘linear systems’ (vector spaces) of various finite dimensions, mentioning the possibility of an infinite dimensional vector space. In the ‘Applications’ at the end of the chapter he gives as an example of an infinite-dimensional vector space the set of algebraic entire functions of all degrees” (Kennedy, pp. 21-24).

Giuseppe Peano (1858-1932) “became a lecturer of infinitesimal calculus at the University of Turin in 1884 and a professor in 1890. He also held the post of professor at the Academia Militare in Turin from 1886 to 1901. Peano made several important discoveries, including a continuous mapping of a line onto every point of a square, that were highly counterintuitive and convinced him that mathematics should be developed formally if mistakes were to be avoided. His Formulaire de mathématiques (Italian Formulario mathematico, “Mathematical Formulary”), published from 1894 to 1908 with collaborators, was intended to develop mathematics in its entirety from its fundamental postulates, using Peano’s logical notation and his simplified international language... Peano is also known as the creator of Latino sine Flexione, an artificial language later called Interlingua. Based on a synthesis of Latin, French, German, and English vocabularies, with a greatly simplified grammar, Interlingua was intended for use as an international auxiliary language. Peano compiled a Vocabulario de Interlingua (1915) and was for a time president of the Academia pro Interlingua” (Britannica).

DSB X: 442; Kannenberg, Introduction to the English translation, Geometric Calculus – According to the Ausdehnungslehre of H. Grassmann, 2000; Kennedy, Peano: Life and Work of Giuseppe Peano, 1980; Van Heijenoort: From Frege to Gödel. A Source Book in Mathematical Logic, 1879-1931.

8vo (237 x 156 mm), pp. x, [ii], 170, . Contemporary half cloth, manuscript title label to spine, a very fine and clean copy.

Item #4029

Price: \$2,850.00