Bibliography to Grassmann’s work

At the end of the twentieth century there were still only a few
dozen important sources in English to the foundational
aspects of Grassmann’s algebra.

Bibliography as a PDF
Since Volume 1 is limited to discussing the foundational aspects
of the algebra, this bibliography is limited mostly to relevant
publications appearing before the year 2000.

### Grassmann’s collected works

The best source for Grassmann’s contributions to science is his Collected Works [Grassmann 1896]
which contain in volume 1 both Die Ausdehnungslehre von 1844 and Die Ausdehnungslehre von
1862, as well as Geometrische Analyse, his prizewinning essay fulfilling Leibniz’s search for an
algebra of geometry. Volume 2 contains papers on geometry, analysis, mechanics and physics, while
volume 3 contains Theorie der Ebbe und Flut.
Die Ausdehnungslehre von 1862, fully titled: Die Ausdehnungslehre. Vollständig und in strenger
Form is perhaps Grassmann’s most important mathematical work. It comprises two main parts: the
first devoted basically to the Ausdehnungslehre (212 pages) and the second to the theory of functions
(155 pages). The Collected Works edition contains 98 pages of notes and comments. The discussion
on the Ausdehnungslehre includes chapters on addition and subtraction, products in general,
progressive and regressive products, interior products, and applications to geometry. A Euclidean
metric is assumed.
### Translations into English

Both Grassmann’s Ausdehnungslehre have been translated into English by Lloyd C Kannenberg. The
1844 version is published as A New Branch of Mathematics: The Ausdehnungslehre of 1844 and
Other Works, Open Court 1995. The translation contains Die Ausdehnungslehre von 1844,
Geometrische Analyse, selected papers on mathematics and physics, a bibliography of Grassmann’s
principal works, and extensive editorial notes. The 1862 version is published as Extension Theory. It
contains work on both the theory of extension and the theory of functions. Particularly useful are the
editorial and supplementary notes.
Kannenberg has also translated Giuseppe Peano’s Calcolo geometrico secondo l’Ausdehnungslehre
di H. Grassmann [Peano, 1888] as Geometric Calculus According to the Ausdehnungslehre of H.
Grassmann.
### Whitehead, Forder and Hyde

Apart from these translations, probably the best and most complete exposition on the
Ausdehnungslehre in English is in Alfred North Whitehead’s A Treatise on Universal Algebra
[Whitehead 1898]. Whitehead saw Grassmann’s work as one of the foundation stones on which he
hoped to build an algebraic theory which united the several new mathematical systems which
emerged during the nineteenth century — the algebra of symbolic logic, Grassmann’s theory of
extension, quaternions, matrices and the general theory of linear algebras.
The second most complete exposition of the Ausdehnungslehre is Henry George Forder’s The
Theory of Extension [Forder 1941]. Forder’s interest is mainly in the geometric applications of the
theory of extension.
The only other books on Grassmann’s algebra written in English during the nineteenth and twentieth
centuries are those by Edward Wyllys Hyde, The Directional Calculus [Hyde 1890] and
Grassmann's Space Analysis [Hyde 1906]. They treat the theory of extension in two and three-
dimensional geometric contexts and include some applications to statics. Several topics such as
Hyde’s treatment of screws are original contributions.
### Other related algebras

Seminal papers on the Ausdehnungslehre (and Clifford algebra) can be found in William Kingdon
Clifford’s collected works Mathematical Papers [Clifford 1882], republished in a facsimile edition
by Chelsea.
Fortunately for those interested in the evolution of the emerging ‘geometric’ algebras, The
International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics
published a bibliography [Macfarlane 1913] which, together with supplements to 1913, contains
about 2500 articles. This therefore most likely contains all the works on the Ausdehnungslehre and
related subjects up to 1913.
The only other recent text devoted specifically to Grassmann algebra (to the author’s knowledge as
of 2001) is Arno Zaddach’s Grassmanns Algebra in der Geometrie, [Zaddach 1994].
### Grassmann Bicentennial Conference

Hermann Grassmann was born in 1809. In 2009 a conference on Grassmann was held in Potsdam,
Germany to commemorate his diverse and seminal contributions to science, philology and
mathematics. Three volumes emerged from the commemoration - see [Petsche 2009], [Petsche et al
2009] and [Petsche et al 2011]. This set is an invaluable resource for students of Grassmann’s work,
particularly the biography [Petsche 2009].

## Sources to Grassmann’s work

This page is an extract from “Grassmann Algebra Volume 1: Foundations - Exploring extended vector
algebra with Mathematica” by John Browne. First Edition 2012.

## Bibliography to Grassmann Algebra Volume 1: Foundations

© John Browne 2012
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