125 years of Guiseppe Peano’s Geometric Calculus according to the Ausdehnungslehre of H. Grassmann

G. Peano’s Calcolo Geometrico was published 125 years ago in 1888!

Giuseppe Peano, Calcolo Geometrico secondo l’Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171.
  • Here is a brief excerpt of G. Peano’s own preface of February 1888:

“… I will be satisfied with my work in writing this book (which would be the only recompense I could expect), if it serves to disclose among mathematicians some of the ideas of Grassmann. It is however my opinion that, before long, this geometric calculus, or something analogous, will be substituted for the methods actually in use in higher education. It is indeed true that the study of this calculus, as with that of every science, requires time; but I do not believe that it exceeds that necessary for the study of, e.g., the fundamentals of analytic geometry; and then the student will find himself in possession of a method which comprehends that of analytic geometry as a particular case, but which is much more powerful, and which lends itself in a marvellous way to the study of geometric applications of infinitesimal calculus, of mechanics, and of graphic statics; indeed, some part of such sciences are already observed to have taken possessions of that calculus. …” (from page xi in L. Kannenberg’s translation of 2002)

  • Second, the description of the work in the Kannenberg’s 2002 edition:

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’s lucid and crisp translation, Geometric Calculus, will appeal to historians of mathematics, researchers, graduate students, and general readers interested in the foundations of mathematics and the development of a formal logical language.

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.

Despite its uniqueness, Calcolo Geometrico has been strangely neglected by historians of mathematics, and even by scholars of Peano. The book has never been reprinted in its entirety, and only two chapters have ever been translated into English. In part, this neglect has been due to Peano’s organization of the work. That is, the section on mathematical logic bears almost no relation to the rest of the book, and the material there was superseded only a year after its publication by Peano’s second book. Since all but this first section was generally thought to be expository rather than original work, it was regarded lightly, if noticed at all, and ultimately all but forgotten. Only in very recent years have the book’s unique merits begun to be recognized.

Among these merits are Peano’s presentation of the essential features of Grassmann’s notoriously obscure Ausdehnungslehre, a clarification and improvement upon Grassmann’s theory of extensive magnitudes, and a dissemination of other hard-to-understand material.

Readers of this valuable translation will gain insight into the work of a distinguished mathematician and founder of mathematical logic.


  • Third, what the Univ. of St. Andrews Biography says about the work:

“… In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic. This was his first work on the topic that would play a major role in his research over the next few years and it was based on the work of Schröder, Boole and Charles Peirce. A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself. This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano. …” (http://www-history.mcs.st-and.ac.uk/Biographies/Peano.html)

Source: http://www.springer.com/birkhauser/history+of+science/book/978-0-8176-4126-9, http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l%27Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf, http://www-history.mcs.st-and.ac.uk/Biographies/Peano.html

1 Comment

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One response to “125 years of Guiseppe Peano’s Geometric Calculus according to the Ausdehnungslehre of H. Grassmann

  1. Filippo Demonte-Barbera

    A new Italian edition of Peano’s “Calcolo Geometrico etc.” is available :


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