For those, like myself, who have struggled to comprehend the original works of Hermann Grassmann, the consoling factor has always been that most of the great 19^{th} century mathematicians were also baffled by his unique terminology and strange style of writing. But, in recent years, there have appeared several very good summaries of Grassmann’s mathematical ideas — including Martin J. Crowe’s history of vector analysis [1] and various articles by Fearnley-Sander [2]. And yet, prior to the publication of the two books under review, the most recent biography of Grassmann was that by Friedrich Engel, published in 1911.

Anyway, Grassmann’s importance arises from the fact that he is regarded as the founder of linear algebra, and the first mathematician to extend geometrical thinking beyond three dimensions. Yet these achievements emerged from the strange circumstances of his mathematically isolated existence in the Prussian town of Stettin — not to mention the tragic irony that his innovative work was largely unnoticed in his lifetime.

His major work was the *Ausdehnungslehre* (1844), which he hoped would earn him a professorship, and thereby provide opportunities for ongoing research. Unfortunately, this work attracted so few readers that these aspirations remained unfulfilled, and his professional life was mainly spent as a schoolteacher, burdened with a heavy teaching load. Cajori therefore says of Grassmann ‘At the age of fifty-three, this wonderful man, with a heavy heart, gave up mathematics, and directed his energies to the study of Sanskrit, achieving in philology results which were better appreciated, and which vie in splendour with those in mathematics’.

The mathematical goal that Grassmann had set himself had its origin in the thoughts of Leibniz, who sought an alternative to the algebra of Vieta and Descartes. He was hoping for a sort of universal algebra that would be distinctly geometrical or linear, and which would express *location *directly, as traditional algebra expresses *magnitude* directly. As a result, Grassmann developed a theory of linear forms Σx_{i}e_{i}, where x_{i} is a number and e_{i }is an undefined mathematical entity (an ‘extensive’). These forms satisfy all the conditions of linearity, but he also developed a ‘combinatorial method’, symbolically expressed as (Σ x_{i}e_{i} )(Σ y_{j}e_{j}) = Σ x_{i}y_{j}[e_{i},e_{j}], which led to the notion of the exterior product of a vector spaces.

Some of the ideas in this system were based upon concepts that were included in a much earlier dissertation, called ‘The Theory of Tides’, for which purpose he invented the idea of vector product and scalar triple product. Consequently, Grassmann is also considered to be one of the three founding fathers of vector analysis (the other two being Hamilton and Gibbs). Some time after his death, his ideas on exterior and linear algebra were utilized by a range of mathematicians that included Felix Klein, Peano and Henri Cartan.

Anyway, these two books, written in German by Hans-Joachim Petsche, are the first of three volumes celebrating the bicentenary of Grassmann’s birth, and they have been ably translated into English by Mark Minnes. The first volume, with which this review is principally concerned, provides a description of Grassmann’s life and an analysis of his mathematical ideas. The second, *Roots and Traces*, consists of previously unpublished letters and other documents of biographical interest, and the third volume will consist of an interdisciplinary anthology of Grassmann’s achievements in fields other than mathematics.

The biographical aspect of the first volume includes an absorbing account of Grassmann’s upbringing in Stettin and the mathematical influence his father and brother. It describes the political and social milieu in Prussia and its effect upon Grassmann’s outlook. But I couldn’t quite understand the author’s aim of trying to find out whether Grassmann’s work expressed ‘a social dimension in mathematics’. Nonetheless, Petsche’s account of Grassmann’s struggle for recognition as a mathematician, set in the backdrop of the German revolution of 1848, was most informative and enjoyable.

Another appealing aspect of the biography is its portrayal of Grassmann as an autodidactic polymath. He modified Newton’s theory of colours, he wrote treatises on electrodynamics and the theory of tides, and he was the first to provide a translation of the Rig Veda from Sanskrit into a European language (such is the nature of genius!).

In chapter 3, Petsche provides an introduction to Grassmann’s linear extension theory, which is the subject of the *Ausdehnungslehre*, and he partly describes its essential features in the vernacular of modern linear algebra. But the narrative is enriched because it is placed within the wider context of 19^{th} century mathematics. For example, Grassmann’s ideas are compared with those of Riemann, Hamilton, Moebius and many other mathematicians. Moreover, it is interesting to read that, rather like Cayley, who believed that all geometry lay within projective geometry, Grassmann hoped that his extension theory would also encapsulate all geometric ideas (but, because of his influence upon Felix Klein, that particular aspiration wasn’t far off the mark).

However, the aspects of this first volume that I find most difficult to comprehend, are those that pertain to the influence of the philosopher Friedrich Schleiermacher upon Grassmann the mathematician. The philosophical ideas in question relate to dialectics, but it isn’t clear how, and why, they led Grassmann to devise linear extension theory- as opposed to any other mathematical structure. A substantial part of the book is devoted to this theme, including a biographical vignette of Schleiermacher himself.

As I understand it, a central ingredient of the dialectical approach to mathematics consists of an examination, and resolution, of the tensions arising from the existence of conceptual opposites (continous/discrete, global/local, finite/infinite etc).

Whereas, Petsche says ‘We can hereby determine the nature of the true development of a mathematical theory … [as] the interlacing of two opposing methods, namely of a heuristic and predominantly philosophical method, with an architectonical method, characterized by the rigor of mathematical construction’. Now if this simply means that mathematics can be viewed as a way of knowing and as a body of knowledge, why can’t it more simply expressed? On the other hand, Schleiermacher is quoted as saying that ‘The particular existent, which mathematics posits as its object, is devoid of all content…’ and, if this is the case, then it explains why Grassmann gave his extensives no particular meaning. To that extent, I can see how he was influenced by Schleiermacher.

So, just as I have found Petsche’s discourse on dialectics to be the most mystifying aspect of his book, so did the first readers of Grassmann’s *Ausdehnungslehre*, and he therefore excluded philosophical preliminaries from the second edition.

In summary, I thoroughly enjoyed reading this biography, and it must surely be the seminal work on the life and times of Hermann Grassmann. Moreover, it provides an interesting perspective on 19^{th} century mathematics in general, and the birth of modern geometry in particular. The level of scholarship is consistently high, which is evidenced by the fact that almost one quarter of the book’s 306 pages is devoted to references and bibliographical information. It would, however, have greatly benefited from the addition of a mathematical index.

[1] *A History of Vector Analysis*, by Michael J. Crowe (Dover, 1985)

[2] ‘Hermann Grassmann and the Creation of Linear Algebra’, p. 291 of *Who Gave You the Epsilon* (MAA, 2009)

Peter Ruane** **is retired after many years spent in primary mathematics education. He is now 71 years old, and is reluctant to accept the fact of human mortality.