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Introduction

Hermann Grassmann's 1862 Ausdehnungslehre (literally, ``Theory of Extension'') is one of the great mathematical works of the nineteenth century. In it the foundations of linear and multilinear algebra are laid and much of the superstructure too is constructed. It is regrettable that such a book on such a subject should, from the moment of publication, have been not much read. Indeed, Grassmann's reputation for impenetrability has persisted to this day. Yet one may suspect that a writer who is, in many respects, a century ahead of his time will be somewhat more readable when that century has elapsed than he was to his contemporaries. It is my hope that this translation and commentary will make it easy for today's mathematically educated reader to appreciate Grassmann's presentation of the theory of basis and dimension --- it does not differ much from the initial chapter of a modern linear algebra text.

The work called simply Die Ausdehnungslehre, though its title page bears the date 1862, actually appeared in the latter half of 1861. It was Grassmann's second attempt to present his theory and was totally different in conception from Die Lineale Ausdehnungslehre, which had appeared in 1844 and had failed to make an impact in the mathematical community. One must grant that this first version was indeed difficult to read and remains so. It was an ambitious attempt to develop, motivate and apply what in the subtitle is called ``a new branch of mathematics'', and to do this from scratch without assuming any existing mathematics. The result was a tantalizing, highly original work, philosophically prolix and mathematically sketchy, replete with ideas that would have to wait decades for their full development.

When the first Ausdehnungslehre failed, Grassmann realised that the subject would have to be presented differently if it was to appeal to mathematicians rather than to philosophers, and so, though clearly it went against his personal inclination, he wrote a new work in which the theory was developed, as he said, backwards --- instead of trying to characterize linear spaces and the other algebraic objects he deals with by means of their properties, as he had originally, he now developed them constructively by starting with a system of generating elements forming linear combinations of them, defining addition in coordinate-wise fashion, and so on. Moreover, he now assumed familiarity with other parts of contemporary mathematics, and he adopted what he called ``the most rigorous mathematical form we know, the Euclidean'' and replaced a discursive format by a modern-looking definition-theorem-remark type of presentation. Despite the contrary opinion of Friedrich Engel, editor of the Collected Works ([4], III.2, p. 231), the result was a much more accessible exposition.

Grassmann's work is fascinating, not only for the extraordinary originality of his vision, but also from an historiographical point of view in that his ideas were repeatedly discovered by others over many decades. Sometimes his work was referred to, but often it was not. See [1], [2] and [3].

The Collected Works include both the first Ausdehnungslehre and the second (as I.1 and I.2 respectively). The material which is translated here comprises sections 1-26 of that version of the 1862 Ausdehnungslehre; although Engel did alter the original in some places (while indicating clearly what changes were made and why), there are no changes of consequence in this part of Chapter 1.

Grassmann's terminology is a problem, especially because he intentionally avoids geometrical language in order to emphasize the distinction between his ``pure'' mathematics and geometry, which he regarded as an application of it. The distinction between an aspect of the physical world and a mathematical model of it is now a commonplace, and it is no longer necessary to draw attention to it by a strict dichotomy of terminologies. Moreover, Grassmann's concepts, as distinct from his terms, have become the bread and butter of mathematics. It seems sensible, therefore, to attempt not a literal translation but a readable one, replacing his terms by their modern equivalents where they exist and where this can be done without losing the flavour of the original. For the historian, of course, no translation can replace the original.

The annotated translation presented here is intended to be a useful resource for the teaching of history of mathematics. All the footnotes to the translation are comments by the translator for the benefit of the modern reader. Students with a basic knowledge of linear algebra will be able to grasp Grassmann's theory of dimension if, without further ado, they read § 2 of what is translated here, while

1. assuming that the domain under the discussion is a fixed linear space whose elements are called ``quantities''.

2. assuming that (linear) dependence is defined as in § 1,2 and that a ``system of units'' is simply an linearly independent set, and

3. ignoring all references to ``primitive units'' and to ``degree''.

In the remainder of this discussion I shall concentrate on the subtleties which will be missed in such a reading. The opening section of the book, § 1, presents considerable difficulties of interpretation, which have nothing to do with basis and dimension but rather reflect what the author intends to do in subsequent chapters.

Grassmann's interests are by no means limited to the properties of linear spaces; later in the book he introduces various types of product and, in particular, he develops in some detail the theory of exterior algebras. One will understand essentially correctly the unfamiliar concepts of Chapter 1 if one thinks in terms of an exterior algebra. The primitive (or original) units are the elements of a fixed linearly independent set which generates the (linear); on the other hand, considered as a linear space, the algebra has as a basis the set of all non-zero products of the primitive units --- an element of this basis which is a product of k (necessarily distinct) primitive units is called a unit of degree k. A linear combination of units of degree k is called a quantity of degree k.

Unfortunately Grassmann uses the same word ``Stufe'' both for the degree of an element in this sense and for what we call the dimension of a space, though its use in the latter sense is confined to spaces whose elements are all of degree 1. For ease of reading, the German word has here been translated by whichever of these English terms is appropriate in the context. It should be noted that in an exterior algebra, although a nonzero product does not determine the 1st degree factors , it does determine the space they span (and hence its dimension).

One must also realise that, presumably for typographical convenience, Grassmann sometimes uses `` to denote the system of primitive units while at other times he means this to represent an arbitrary independent set.

It remains to comment on the most conspicuous difference between § 1 and § 2: whereas § 2 deals explicitly with finite-dimensional spaces, the definitions and statements (though not all the proofs) of § 1 seem to leave open the possibility that the number of units is infinite. I believe that the ambiguity here is intentional. Certainly Grassmann does deal implicitly with algebras which are infinite-dimensional (see Footnote 9, below), and assumes that the linear space properties given in Sections 8 and 12 are valid in such algebras (as indeed they are). It seems that he even wishes to leave open the possibility that his constructions can be carried out for infinite linear combinations of units.

In these remarks I have tried to ease the path for the reader by using clear modern terms; it should be remembered that the corresponding conceptions in the text are sometimes hazy. In particular, though it may seem paradoxical, while Grassmann clearly has the idea of a linear subspace he has trouble pinning down the ambient set in which he operates as a well-defined object. To do this may now appear a trivial matter, but its difficulty at the time is suggested by the fact that although Peano having read and appreciated Grassmann was able to define a linear space in modern fashion in 1888 ([5], p. 141), it was a further thirty years before this key idea of modern algebra became common currency.

References

1

M. Crowe, A History of Vector Analysis, Notre Dame, 1967.

2

D. Fearnley-Sander, Hermann Grassmann and the Creation of Linear Algebra, Am. Math. Monthly 86 (1979), 809-817.

3

D. Fearnley-Sander, Hermann Grassmann and the Prehistory of Universal Algebra, Am. Math. Monthly (1982), 161-166.

4

H. G. Grassmann, Gesammelte mathematische und physikalische Werke. Ed. F. Engel. 3 vols. in 6 parts, Leipzig,1894-1911.

5

G. Peano, Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann. Torino, 1888.