Hermann Grassmann

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Hermann Günther Grassmann
Hermann Grassmann.jpg
Hermann Günther Grassmann
Born(1809-04-15)15 April 1809
Died26 September 1877(1877-09-26) (aged 68)
Stettin, German Empire
Residence Prussia, Germany
Alma mater University of Berlin
Known for Multilinear algebra, Grassmannian, Exterior algebra
Awards PhD (Hon):
University of Tübingen (1876)

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.

Polymath person whose expertise spans a significant number of different subject areas

A polymath is a person whose expertise spans a significant number of subject areas, known to draw on complex bodies of knowledge to solve specific problems.

Linguistics is the scientific study of language. It involves analysing language form, language meaning, and language in context. The earliest activities in the documentation and description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of the Sanskrit language in his Aṣṭādhyāyī.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.



Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated.

Gymnasium (school) type of school providing advanced secondary education in Europe

A gymnasium is a type of school with a strong emphasis on academic learning, and providing advanced secondary education in some parts of Europe comparable to British grammar schools, sixth form colleges and US preparatory high schools. In its current meaning, it usually refers to secondary schools focused on preparing students to enter a university for advanced academic study. Before the 20th century, the system of gymnasiums was a widespread feature of educational system throughout many countries of central, north, eastern, and south Europe.

Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics.

Prussia state in Central Europe between 1525–1947

Prussia was a historically prominent German state that originated in 1525 with a duchy centred on the region of Prussia on the southeast coast of the Baltic Sea. It was de facto dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and de jure by an Allied decree in 1947. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organised and effective army. Prussia, with its capital in Königsberg and from 1701 in Berlin, decisively shaped the history of Germany.

Classics Study of the culture of (mainly) Ancient Greece and Ancient Rome

Classics or classical studies is the study of classical antiquity. It encompasses the study of the Greco-Roman world, particularly of its languages and literature but also of Greco-Roman philosophy, history, and archaeology. Traditionally in the West, the study of the Greek and Roman classics was considered one of the cornerstones of the humanities and a fundamental element of a rounded education. The study of classics has therefore traditionally been a cornerstone of a typical elite education.

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter, its motion, and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper referred to as A1 (see references).

In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all secondary school levels.

Chemistry is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.

Mineralogy Scientific study of minerals and mineralised artifacts

Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical properties of minerals and mineralized artifacts. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.

In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked Kummer for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "... commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics.

Ernst Kummer German mathematician

Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.

Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben , calling for German unification under a constitutional monarchy. (This eventuated in 1871.) After writing a series of articles on constitutional law, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.

Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen.


One of the many examinations for which Grassmann sat required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from Laplace's Mécanique céleste and from Lagrange's Mécanique analytique, but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what is now called linear algebra and the notion of a vector space. He went on to develop those methods in his A1 and A2 (see bibliography).

In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [1] [The Theory of Linear Extension, a New Branch of Mathematics], hereinafter denoted A1 and commonly referred to as the Ausdehnungslehre, [2] which translates as "theory of extension" or "theory of extensive magnitudes." Since A1 proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded.

Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:

Following an idea of Grassmann's father, A1 also defined the exterior product, also called "combinatorial product" (in German: äußeres Produkt [3] or kombinatorisches Produkt [4] ), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule epep = 0 by the rule epep = 1. (For quaternions, we have the rule i2 = j2 = k2 = −1.) For more details, see exterior algebra.

A1 was a revolutionary text, too far ahead of its time to be appreciated. When Grassmann submitted it to apply for a professorship in 1847, the ministry asked Ernst Kummer for a report. Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann a university position. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 Neue Theorie der Elektrodynamik [5] and several papers on algebraic curves and surfaces, in the hope that these applications would lead others to take his theory seriously.

In 1846, Möbius invited Grassmann to enter a competition to solve a problem first proposed by Leibniz: to devise a geometric calculus devoid of coordinates and metric properties (what Leibniz termed analysis situs). Grassmann's Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik, [6] was the winning entry (also the only entry). Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value.

In 1853, Grassmann published a theory of how colors mix; it and its three color laws are still taught, as Grassmann's law. Grassmann's work on this subject was inconsistent with that of Helmholtz. Grassmann also wrote on crystallography, electromagnetism, and mechanics.

Grassmann (1861) set out the first axiomatic presentation of arithmetic, making free use of the principle of induction. Peano and his followers cited this work freely starting around 1890. Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works in 1995 ( ISBN   0-8126-9275-6. -- ISBN   0-8126-9276-4).

In 1862, Grassmann published a thoroughly rewritten second edition of A1, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his linear algebra. The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet [The Theory of Extension, Thoroughly and Rigorously Treated], hereinafter denoted A2, fared no better than A1, even though A2's manner of exposition anticipates the textbooks of the 20th century.


In 1840s, mathematicians were generally unprepared to understand Grassmann's ideas. [7] In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore. [7]

One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme

... developed some of Hermann Grassmann's algebras and Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ... [8]

In 1872 Victor Schlegel published the first part of his System der Raumlehre which used Grassmann's approach to derive ancient and modern results in plane geometry. Felix Klein wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his System according to Grassmann, this time developing higher geometry. Meanwhile, Klein was advancing his Erlangen Program which also expanded the scope of geometry. [9]

Comprehension of Grassmann awaited the concept of vector spaces which then could express the multilinear algebra of his extension theory. To establish the priority of Grassmann over Hamilton, Josiah Willard Gibbs urged Grassmann's heirs to have the 1840 essay on tides published. [10] A. N. Whitehead's first monograph, the Universal Algebra (1898), included the first systematic exposition in English of the theory of extension and the exterior algebra. With the rise of differential geometry the exterior algebra was applied to differential forms.

For an introduction to the role of Grassmann's work in contemporary mathematical physics see The Road to Reality [11] by Roger Penrose.

Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed that he had first developed these ideas in 1832.


Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library”. [12] Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. The last years of his life he turned to historical linguistics and the study of Sanskrit. He wrote books on German grammar, collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the Rigveda (more than 1,000 pages) which earned him a membership of the American Orientalists' Society. In modern Rigvedic studies Grassmann's work is often cited. In 1955 the third edition of his dictionary to Rigveda was issued. [7]

Grassmann also discovered a sound law of Indo-European languages, which was named Grassmann's Law in his honor.

These philological accomplishments were honored during his lifetime; he was elected to the American Oriental Society and in 1876, he received an honorary doctorate from the University of Tübingen.


See also

Citations and references

  1. Tr. The rulers extension theory, a new branch of mathematics
  2. Tr. Expansion plan teachings
  3. Tr. outer product
  4. Tr. combinatorial product
  5. Tr. New theory of electrodynamics
  6. Tr. Geometric analysis linked to the geometric characteristics invented by Leibniz
  7. 1 2 3 Prasolov 1994, p. 46.
  8. Hankel entry in the Dictionary of Scientific Biography. New York: 1970–1990
  9. Rowe 2010
  10. Lynde Wheeler (1951), Josiah Willard Gibbs: The History of a Great Mind, 1998 reprint, Woodbridge, CT: Ox Bow, pp. 113-116.
  11. Penrose The Road to Reality, chapters 11 & 2
  12. Prasolov 1994, p. 45.
  13. Tr. "The rulers extension theory"
  14. Tr. "Geometric analysis linked to the geometric characteristics invented by Leibniz"
  15. Tr. "Higher mathematics for schools, Volume 1"
  16. Tr. "Dictionary of the Rig-Veda"
  17. Tr. "Collected mathematical and physical works"

Extensive online bibliography, revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.

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Exterior algebra algebraic construction used in Euclidean geometry

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Hermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg, Imperial Germany.

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