A New Branch of Mathematics: The Ausdehnungslehre of 1844

by Hermann Grassmann

This is a translation of the work of Grassmann, a 19th-century mathematician, who is acknowledged as one of the founders of vector analysis. His ideas have recently been applied in elementary particle physics and computer sciences, as well as in analytic geometry.

555 pages, Paperback

First published January 1, 1

About the author

Hermann Günther Grassmann (German: Graßmann; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguist and now also as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.

Reviews

[Manny]

I must reproduce the wonderful description in Crowe's History of Vector Analysis:

1844. Grassmann publishes the first full exposition of his system, his Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik dargestellt and durch Anwendungen auf die ubrigen Zweige der Mathematick, wie auch auf der Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert. Whereas on the title page of Hamilton’s Lectures on Quaternions, Hamilton was identified by a large array of titles and memberships, Grassmann’s title page identified him only as “Lehrer an der Friedrich Wilhelms Schule zu Stettin.” The book attracts almost no attention and about 600 copies of it were in 1864 used for waste paper.

Grassmann’s Die lineale Ausdehnungslehre (Linear Extension Theory) demonstrated deep mathematical insights. It also in one sense contained much of the modern system of vector analysis. This, however, was embedded within a far broader system, which included n-dimensional spaces and as many as sixteen different products of his base entities (including his inner and outer products, which are respectively somewhat close to the our modern dot and cross products). Moreover, Grassmann justifies his system by philosophical discussions that may have put off many of his readers. The abstractness of his presentation and the originality of his insights also contributed to the difficulties readers had in comprehending Grassmann’s message, as is evident from comments made by various mathematicians who had struggled with the book. Möbius, for example, labeled it unreadable, Baltzer reported that reading the book made him feel “dizzy and to see sky blue before my eyes,” and Hamilton wrote De Morgan that to read the Ausdehnungslehre he would have to learn to smoke.

1845. After Grassmann’s unsuccessful efforts to persuade Möbius to write a review of his book, Grassmann, at Möbius’s urging, writes a review of his own book—the only review his book received! He also publishes a paper containing a new discovery in electrical theory that he had derived using his new methods. The result: more neglect, until the 1870s when Clausius made the same electrical discovery and published it, only then realizing that Grassmann had preceded him.