Grassmann Biography

Hermann Günther Grassmann was born in 1809 in Stettin, a town in Pomerania a short distance
inland from the Baltic. His father Justus Günther Grassmann taught mathematics and physical
science at the Stettin Gymnasium. Hermann was no child prodigy. His father used to say that he
would be happy if Hermann became a craftsman or a gardener.
In 1827 Grassmann entered the University of Berlin with the intention of studying theology. As
his studies progressed he became more and more interested in studying philosophy. At no time
whilst a student in Berlin was he known to attend a mathematics lecture.
Grassmann was however only 23 when he made his first important geometric discovery: a
method of adding and multiplying lines. This method was to become the foundation of his
Ausdehnungslehre (extension theory). His own account of this discovery is given below.
Grassmann was interested ultimately in a university post. In order to improve his academic
standing in science and mathematics he composed in 1839 a work (over 200 pages) on the study
of tides entitled Theorie der Ebbe und Flut. This work contained the first presentation of a
system of spacial analysis based on vectors including vector addition and subtraction, vector
differentiation, and the elements of the linear vector function, all developed for the first time.
His examiners failed to see its importance.
Around Easter of 1842 Grassmann began to turn his full energies to the composition of his first
'Ausdehnungslehre', and by the autumn of 1843 he had finished writing it. The following is an
excerpt from the foreword in which he describes how he made his seminal discovery. The
translation is by Lloyd Kannenberg (Grassmann 1844).
The initial incentive was provided by the consideration of negatives in geometry; I was
used to regarding the displacements AB and BA as opposite magnitudes. From this it
follows that if A, B, C are points of a straight line, then AB + BC = AC is always true,
whether AB and BC are directed similarly or oppositely, that is even if C lies between A
and B. In the latter case AB and BC are not interpreted merely as lengths, but rather their
directions are simultaneously retained as well, according to which they are precisely
oppositely oriented. Thus the distinction was drawn between the sum of lengths and the
sum of such displacements in which the directions were taken into account. From this
there followed the demand to establish this latter concept of a sum, not only for the case
that the displacements were similarly or oppositely directed, but also for all other cases.
This can most easily be accomplished if the law AB + BC = AC is imposed even when A,
B, C do not lie on a single straight line.
Thus the first step was taken toward an analysis that subsequently led to the new branch of
mathematics presented here. However, I did not then recognize the rich and fruitful
domain I had reached; rather, that result seemed scarcely worthy of note until it was
combined with a related idea.
While I was pursuing the concept of product in geometry as it had been established by my
father, I concluded that not only rectangles but also parallelograms in general may be
regarded as products of an adjacent pair of their sides, provided one again interprets the
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product, not as the product of their lengths, but as that of the two displacements with their
directions taken into account. When I combined this concept of the product with that
previously established for the sum, the most striking harmony resulted; thus whether I
multiplied the sum (in the sense just given) of two displacements by a third displacement
lying in the same plane, or the individual terms by the same displacement and added the
products with due regard for their positive and negative values, the same result obtained,
and must always obtain.
This harmony did indeed enable me to perceive that a completely new domain had thus
been disclosed, one that could lead to important results. Yet this idea remained dormant
for some time since the demands of my job led me to other tasks; also, I was initially
perplexed by the remarkable result that, although the laws of ordinary multiplication,
including the relation of multiplication to addition, remained valid for this new type of
product, one could only interchange factors if one simultaneously changed the sign (i.e.
changed + into – and vice versa).
As with his earlier work on tides, the importance of this work was ignored. Since few copies
were sold, most ended by being used as waste paper by the publisher. The failure to find
acceptance for Grassmann's ideas was probably due to two main reasons. The first was that
Grassmann was just a simple schoolteacher, and had none of the academic charisma that other
contemporaries, like Hamilton for example, had. History seems to suggest that the acceptance of
radical discoveries often depends more on the discoverer than the discovery.
The second reason is that Grassmann adopted the format and the approach of the modern
mathematician. He introduced and developed his mathematical structure axiomatically and
abstractly. The abstract nature of the work, initially devoid of geometric or physical
significance, was just too new and formal for the mathematicians of the day and they all seemed
to find it too difficult. More fully than any earlier mathematician, Grassmann seems to have
understood the associative, commutative and distributive laws; yet still, great mathematicians
like Möbius found it unreadable, and Hamilton was led to write to De Morgan that to be able to
read Grassmann he 'would have to learn to smoke'.
In the year of publication of the Ausdehnungslehre (1844) the Jablonowski Society of Leipzig
offered a prize for the creation of a mathematical system fulfilling the idea that Leibniz had
sketched in 1679. Grassmann entered with 'Die Geometrische Analyse geknüpft und die von
Leibniz Characteristik', and was awarded the prize. Yet as with the Ausdehnungslehre it was
subsequently received with almost total silence.
However, in the few years following, three of Grassmann's contemporaries were forced to take
notice of his work because of priority questions. In 1845 Saint-Venant published a paper in
which he developed vector sums and products essentially identical to those already occurring in
Grassmann's earlier works (Barré 1845). In 1853 Cauchy published his method of 'algebraic
keys' for solving sets of linear equations (Cauchy 1853). Algebraic keys behaved identically to
Grassmann's units under the exterior product. In the same year Saint-Venant published an
interpretation of the algebraic keys geometrically and in terms of determinants (Barré 1853).
Since such were fundamental to Grassmann's already published work he wrote a reply for
Crelle's Journal in 1855 entitled 'Sur les différentes genres de multiplication' in which he
claimed priority over Cauchy and Saint-Venant and published some new results (Grassmann
1855).
It was not until 1853 that Hamilton heard of the Ausdehnungslehre. He set to reading it and soon
after wrote to De Morgan.
ABriefBiographyOfGrassmann.nb 2
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I have recently been reading … more than a hundred pages of Grassmann's
Ausdehnungslehre, with great admiration and interest …. If I could hope to be put in
rivalship with Des Cartes on the one hand and with Grassmann on the other, my scientific
ambition would be fulfilled.
During the period 1844 to 1862 Grassmann published seventeen scientific papers, including
important papers in physics, and a number of mathematics and language textbooks. He edited a
political paper for a time and published materials on the evangelization of China. This, on top of
a heavy teaching load and the raising of a large family. However, this same period saw only few
mathematicians — Hamilton, Cauchy, Möbius, Saint-Venant, Bellavitis and Cremona — having
any acquaintance with, or appreciation of, his work.
In 1862 Grassmann published a completely rewritten Ausdehnungslehre: Die
Ausdehnungslehre: Vollständing und in strenger Form. In the foreword Grassmann discussed
the poor reception accorded his earlier work and stated that the content of the new book was
presented in 'the strongest mathematical form that is actually known to us; that is Euclidean …'.
It was a book of theorems and proofs largely unsupported by physical example.
This apparently was a mistake, for the reception accorded this new work was as quiet as that
accorded the first, although it contained many new results including a solution to Pfaff's
problem. Friedrich Engel (the editor of Grassmann's collected works) comments: 'As in the first
Ausdehnungslehre so in the second: matters which Grassmann had published in it were later
independently rediscovered by others, and only much later was it realized that Grassmann had
published them earlier' (Engel 1896).
Thus Grassmann's works were almost totally neglected for forty-five years after his first
discovery. In the second half of the 1860s recognition slowly started to dawn on his
contemporaries, among them Hankel, Clebsch, Schlegel, Klein, Noth, Sylvester, Clifford and
Gibbs. Gibbs discovered Grassmann's works in 1877 (the year of Grassmann's death), and
Clifford discovered them in depth about the same time. Both became quite enthusiastic about
Grassmann's new mathematics.
Grassmann's activities after 1862 continued to be many and diverse. His contribution to
philology rivals his contribution to mathematics. In 1849 he had begun a study of Sanskrit and
in 1870 published his Wörtebuch zum Rig-Veda, a work of 1784 pages, and his translation of the
Rig-Veda, a work of 1123 pages, both still in use today. In addition he published on
mathematics, languages, botany, music and religion. In 1876 he was made a member of the
American Oriental Society, and received an honorary doctorate from the University of Tübingen.
On 26 September 1877 Hermann Grassmann died, departing from a world only just beginning
to recognize the brilliance of the mathematical creations of one of its most outstanding eclectics.