Hermann Grassmann

Hermann Grassmann was born on April 15, 1809 in Stettin (by chance the birthday of Leonhard Euler) and died on September 26, 1877 in Stettin. His father was Justus Günther Grassmann[?] and his mother was Johanne Luise Friederike Grassmann (maiden name: Medenwald).

Hermann Grassmann was the son of the school teacher Justus Grassmann, Gymnasial-Professor, who wrote several influential books on physics and mathematics and various notes (Schulprogramme) which influenced his son Hermann. According to a biographical sketch by H. Grassmann himself, he was slow in school, and his father pointed him to an career as gardener. However, he finished the Gymnasium with a high grade and went on to Berlin together with his brother studying theology.

During that time his interest in mathematics arose and he wrote a treatise on the theory of the tides (Theorie der Ebbe und Flut, Prüfungsarbeit 1840, published by Justus Grassmann) to grade for a mathematics teacher position.

His Geometrische Analyse was submitted to the Fürstliche Jablonowski'schen Gesellschaft, as the only candidate, to reestablish or to newly invent a coordinate-free geometric calculus in the spirit of Gottfried Wilhelm Leibniz. The award was given on July 1, 1846.

The main mathematical works of Grassmann is found in his two books on the theory of extensive magnitudes (Die lineale Ausdehnunglehre, ein neuer Zweig der Mathematik, 1844 and Die Ausdehnunglehre: Vollständig und in strenger Form bearbeitet, Berlin 1862, cited and known widely as A1 and A2). Unfortunately these works did not receive the attention they deserved. The A1 was submitted as a Ph. D. thesis, but Ferdinand Möbius felt unable to evaluate the work and pushed it forward to Kummer[?] who rejected it without having studied it carefully. H. Grassmann was awarded the title of Gymnasial-Professor and had to stay as school treacher in Stettin.

Grassmann was not only a great mathematician, he was doing research in physics, (crystallography, electromagnetism, mechanics, etc), physiology (theory of colours, theory of vocals). His colour theory and the three Grassmann laws are still widely known and taught by practioners. His book on arithmetics could still be printed having an astonishing modern style.

After the complete failure of reception of his mathematical works, Grassmann turned to linguistics. He wrote books on German grammar, collected folk songs, and started to study Sanskrit. His dictionary to the Ajurveda[?] and the translation of this holy book (still in print in Germany!) were well recognized among philologists and he received a honorary doctorate from the University of Tübingen in 1876.

Mathematical Works

Theory of extensive magnitudes

Following an idea of his father, as Grassmann himself quotes in the A1, he invented a new type of product, the exterior product which he calls also combinatorial product (In German: äußeres Produkt or kombinatorisches Produkt). Since his aim in the A1 was to provide a new foundation of whole mathematics, he started with philosophical and quite general definitions. His method to construct algebraic structures uses generators and relations and is not manifestly basis independent. Grassmann used only real algebras, i.e. algebras whose scalars are real numbers (He made no distingtion between real number and real valued functions, which however changes algebra theory drastically). However, We follow this attitude here and give definitions for some of his products:

product (general definition)

A product is a linear map from the tensor product of a space with itself into a linear space. 'Note:' such a product is left and right distributive, but may not be unital or associative.

exterior product[?] (wedge product)

Let {e_i} be a set of generators[?] of a linear space 'V'. An exterior product of two such generators is defined by demanding the following computational rules (relations):

• e_i /\ e_j = e_ij = -e_ji if and only if i <> j.
• e_i /\ e_j = 0 if i=j.
• extend this process recursively to products of higher grade via e_i /\ e_j..k = e_ij..k = -e_ji..k etc.

Note, that the product takes values in a new space V/\V (doubly indexed) which is a factor space of the tensor product V (x) V. The product is associative by definition and alternating[?], i.e. it vanishes if two indices are equal. A short combinatorial calculation shows that one finds for n linear independent generators 2ⁿ linear independent products of generators. They build the linear space V^{^} a Grassmann algebra is build over. In this way, the exterior product is extended to the whole space V^{^}.

The Grassmann algebra is a graded algebra. We define the grade of scalars to be zero and the grade of generators to be 1. The grade of a non zero product of generators is the number of involved generators. The space of a Grassmann algebra can hence be decomposed into a direct sum of homogenous subspaces of definite grade, i.e. the linear spave of all products having exactly k generators:
V^{^} = V^{^0} (+) V^{^1} (+) ... (+) V^{^n}
where V^{^0} = R is identified with R, the real numbers.

interior product

not yet ...

regressive product

not yet ...

related to physics

not yet present ...

Linguistical Works

not yet present ...

To be continued...

External Links

• Hermann Grassmanns gesammelte mathematische und physikalische Werke, Friedrich Engel[?] ed., B.G. Teubner, Leipzig 1911, Band 3 II
• Hermann Grassmann, Sein Leben und seine Werke, Victor Schlege, F.A. Brockhaus, Leipzig, 1878
• See also biography in St. Andews History of Mathematics[?]:
  • http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grassmann