Bibliography view/download PDF

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].