# Hermann Grassmann

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**Hermann Günther Grassmann** (April 15, 1809 – September 26, 1877) was a German mathematician, physicist, linguist, scholar, and neohumanist.

Hermann Grassmann was born in Stettin (by chance, on the birthday of Leonhard Euler) and died there in 1877. 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 a 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 are 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 [1]. The A1 was submitted as a Ph. D. thesis, but Ferdinand Möbius felt unable to evaluate the work and forwarded it to Ernst Kummer; who rejected it without having studied it carefully. H. Grassmann was awarded the title of Gymnasial-Professor and had to stay as school teacher in Stettin.

Not only was Grassmann a great mathematician, he was doing research in physics (crystallography, electromagnetism, mechanics etc.) and physiology (theory of colours, theory of vocals). His colour theory and the three Grassmann laws are still widely known and taught by practitioners. 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 Ayurveda 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.

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## Works on mathematics

### 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 all of mathematics, he started with philosophical and quite general definitions. For more details, see the historical section at exterior power.

## Works on linguistics

Grassmann is best known to linguists for his work on an important sound law of Indo-European, today called Grassmann's Law in his honor.

## Children

His son Hermann Ernst Grassmann (1857-1922), also called Hermann Grassmann der jungere (the younger), was also a mathematician.

## References

*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