Hermann Grassmann

“Grassmann” redirects here. For the surname, seeGrassmann (surname).

Hermann Günther Grassmann (German: Graßmann;April 15, 1809 – September 26, 1877) was a Germanpolymath, known in his day as a linguist and now also asa mathematician. He was also a physicist, neohumanist,general scholar, and publisher. His mathematical workwas little noted until he was in his sixties.

1 Biography

Grassmann was the third of 12 children of Justus Gün-ter Grassmann, an ordained minister who taught mathe-matics and physics at the Stettin Gymnasium, where Her-mann was educated.Grassmann was an undistinguished student until he ob-tained a high mark on the examinations for admissionto Prussian universities. Beginning in 1827, he studiedtheology at the University of Berlin, also taking classesin classical languages, philosophy, and literature. Hedoes not appear to have taken courses in mathematics orphysics.Although lacking university training in mathematics, itwas the field that most interested him when he returned toStettin in 1830 after completing his studies in Berlin. Af-ter a year of preparation, he sat the examinations neededto teach mathematics in a gymnasium, but achieved a re-sult good enough to allow him to teach only at the lowerlevels. Around this time, he made his first significantmathematical discoveries, ones that led him to the im-portant ideas he set out in his 1844 paper referred to asA1 (see references).In 1834 Grassmann began teaching mathematics at theGewerbeschule in Berlin. A year later, he returned toStettin to teachmathematics, physics, German, Latin, andreligious studies at a new school, the Otto Schule. Overthe next four years, Grassmann passed examinations en-abling him to teach mathematics, physics, chemistry, andmineralogy at all secondary school levels.In 1847, he was made an “Oberlehrer” or head teacher. In1852, he was appointed to his late father’s position at theStettin Gymnasium, thereby acquiring the title of Profes-sor. In 1847, he asked the PrussianMinistry of Educationto be considered for a university position, whereupon thatMinistry asked Kummer for his opinion of Grassmann.Kummer wrote back saying that Grassmann’s 1846 prize

essay (see below) contained "... commendably good ma-terial expressed in a deficient form.” Kummer’s reportended any chance that Grassmann might obtain a univer-sity post. This episode proved the norm; time and again,leading figures of Grassmann’s day failed to recognize thevalue of his mathematics.During the political turmoil in Germany, 1848–49, Her-mann and Robert Grassmann published a Stettin newspa-per calling for German unification under a constitutionalmonarchy. (This eventuated in 1871.) After writing aseries of articles on constitutional law, Hermann partedcompany with the newspaper, finding himself increas-ingly at odds with its political direction.Grassmann had eleven children, seven of whom reachedadulthood. A son, Hermann Ernst Grassmann, became aprofessor of mathematics at the University of Giessen.

2 Mathematician

One of the many examinations for which Grassmann satrequired that he submit an essay on the theory of the tides.In 1840, he did so, taking the basic theory from Laplace'sMécanique céleste and from Lagrange's Mécanique an-alytique, but expositing this theory making use of thevector methods he had been mulling over since 1832.This essay, first published in theCollectedWorks of 1894–1911, contains the first known appearance of what arenow called linear algebra and the notion of a vector space.He went on to develop those methods in his A1 and A2(see references).In 1844, Grassmann published his masterpiece, his DieLineale Ausdehnungslehre, ein neuer Zweig der Mathe-matik[1] [The Theory of Linear Extension, a New Branchof Mathematics], hereinafter denoted A1 and commonlyreferred to as the Ausdehnungslehre,[2] which translatesas “theory of extension” or “theory of extensive magni-tudes.” Since A1 proposed a new foundation for all ofmathematics, the work began with quite general defini-tions of a philosophical nature. Grassmann then showedthat once geometry is put into the algebraic form he ad-vocated, the number three has no privileged role as thenumber of spatial dimensions; the number of possible di-mensions is in fact unbounded.Fearnley-Sander (1979) describes Grassmann’s founda-tion of linear algebra as follows:Following an idea of Grassmann’s father, A1 also definedthe exterior product, also called “combinatorial prod-

1

2 4 LINGUIST

uct” (In German: äußeres Produkt[3] or kombinatorischesProdukt[4]), the key operation of an algebra now calledexterior algebra. (One should keep in mind that in Grass-mann’s day, the only axiomatic theory was Euclidean ge-ometry, and the general notion of an abstract algebra hadyet to be defined.) In 1878, William Kingdon Cliffordjoined this exterior algebra toWilliam Rowan Hamilton'squaternions by replacing Grassmann’s rule epep = 0 by therule epep = 1. (For quaternions, we have the rule i2 = j2= k2 = −1.) For more details, see exterior algebra.A1was a revolutionary text, too far ahead of its time to beappreciated. When Grassmann submitted it to apply fora professorship in 1847, the ministry asked Ernst Kum-mer for a report. Kummer assured that there were goodideas in it, but found the exposition deficient and advisedagainst giving Grassmann a university position. Over thenext 10-odd years, Grassmann wrote a variety of workapplying his theory of extension, including his 1845 NeueTheorie der Elektrodynamik[5] and several papers on alge-braic curves and surfaces, in the hope that these applica-tions would lead others to take his theory seriously.In 1846, Möbius invited Grassmann to enter a compe-tition to solve a problem first proposed by Leibniz: todevise a geometric calculus devoid of coordinates andmetric properties (what Leibniz termed analysis situs).Grassmann’s Geometrische Analyse geknüpft an die vonLeibniz erfundene geometrische Charakteristik,[6] was thewinning entry (also the only entry). Moreover, Möbius,as one of the judges, criticized the way Grassmann in-troduced abstract notions without giving the reader anyintuition as to why those notions were of value.In 1853, Grassmann published a theory of how col-ors mix; it and its three color laws are still taught, asGrassmann’s law. Grassmann’s work on this subjectwas inconsistent with that of Helmholtz. Grassmannalso wrote on crystallography, electromagnetism, andmechanics.Grassmann (1861) set out the first axiomatic presenta-tion of arithmetic, making free use of the principle ofinduction. Peano and his followers cited this work freelystarting around 1890. Lloyd C. Kannenberg published anEnglish translation of The Ausdehnungslehre and Otherworks in 1995 (ISBN 0-8126-9275-6. -- ISBN 0-8126-9276-4).In 1862, Grassmann published a thoroughly rewrittensecond edition of A1, hoping to earn belated recogni-tion for his theory of extension, and containing the defini-tive exposition of his linear algebra. The result, Die Aus-dehnungslehre: Vollständig und in strenger Form bear-beitet [The Theory of Extension, Thoroughly and Rigor-ously Treated], hereinafter denoted A2, fared no betterthan A1, even though A2's manner of exposition antici-pates the textbooks of the 20th century.

3 Response

In 1840s, mathematicians were generally unprepared tounderstand Grassmann’s ideas.[7] In the 1860s and 1870svarious mathematicians came to ideas similar to that ofGrassmann’s, but Grassmann himself was not interestedin mathematics anymore.[7]

One of the first mathematicians to appreciate Grass-mann’s ideas during his lifetime was Hermann Hankel,whose 1867 Theorie der complexen Zahlensysteme

... developed some of Hermann Grassmann’salgebras and Hamilton’s quaternions. Hankelwas the first to recognise the significance ofGrassmann’s long-neglected writings ...[8]

In 1872 Victor Schlegel published the first part of hisSystem der Raumlehre which used Grassmann’s approachto derive ancient and modern results in plane geometry.Felix Klein wrote a negative review of Schlegel’s book cit-ing its incompleteness and lack of perspective on Grass-mann. Schlegel followed in 1875 with a second part ofhis System according to Grassmann, this time develop-ing higher geometry. Meanwhile Klein was advancinghis Erlangen Program which also expanded the scope ofgeometry.[9]

Comprehension of Grassmann awaited the concept ofvector spaces which then could express the multilinearalgebra of his extension theory. To establish the prior-ity of Grassmann over Hamilton, Josiah Willard Gibbsurged Grassmann’s heirs to have the 1840 essay on tidespublished.[10] A. N. Whitehead's first monograph, theUniversal Algebra (1898), included the first systematicexposition in English of the theory of extension and theexterior algebra. With the rise of differential geometrythe exterior algebra was applied to differential forms.For an introduction to the role of Grassmann’s work incontemporary mathematical physics see The Road to Re-ality[11] by Roger Penrose.Adhémar Jean Claude Barré de Saint-Venant developeda vector calculus similar to that of Grassmann which hepublished in 1845. He then entered into a dispute withGrassmann about which of the two had thought of theideas first. Grassmann had published his results in 1844,but Saint-Venant claimed that he had first developed theseideas in 1832.

4 Linguist

Grassmann’s mathematical ideas began to spread only to-wards the end of his life. 30 years after the publication ofA1 the publisher wrote to Grassmann: “Your book DieAusdehnungslehre has been out of print for some time.Since your work hardly sold at all, roughly 600 copieswere used in 1864 as waste paper and the remaining few

3

odd copies have now been sold out, with the exception ofthe one copy in our library”.[12] Disappointed by the re-ception of his work in mathematical circles, Grassmannlost his contacts with mathematicians as well as his inter-est in geometry. The last years of his life he turned tohistorical linguistics and the study of Sanskrit. He wrotebooks on German grammar, collected folk songs, andlearned Sanskrit. He wrote a 2,000-page dictionary and atranslation of the Rigveda (more than 1,000 pages) whichearned him a membership of the American Orientalists’Society. In modern Rigvedic studies Grassmann’s workis often cited. In 1955 the third edition of his dictionaryto Rigveda was issued.[7]

Grassmann also discovered a sound law of Indo-European languages, which was named Grassmann’sLaw in his honor. These philological accomplishmentswere honored during his lifetime; he was elected to theAmerican Oriental Society and in 1876, he received anhonorary doctorate from the University of Tübingen.

5 See also

• bra–ket notation (as precursor)

• Exterior algebra

• Grassmann number

• Grassmannian

• Grassmann’s law (phonology)

• Grassmann’s law (optics)

6 References and citations

Primary sources

• A1: 1844. Die lineale Ausdehnungslehre.[13]Leipzig: Wiegand. English translation, 1995, byLloyd Kannenberg, A new branch of mathematics.Chicago: Open Court.

• 1847. Geometrische Analyse geknüpft an die vonLeibniz erfundene geometrische Charakteristik..[14]Available on quod.lib.umich.edu

• 1861. Lehrbuch der Mathematik für höhereLehranstalten, Band 1. Berlin: Enslin.

• A2: 1862. Die Ausdehnungslehre. Vollständig undin strenger Form begründet..[15] Berlin: Enslin. En-glish translation, 2000, by Lloyd Kannenberg, Ex-tension Theory. American Mathematical Society.

• 1873. Wörterbuch zum Rig-Veda.[16] Leipzig:Brockhaus.

• 1876–1877. Rig-Veda. Leipzig: Brockhaus. Trans-lation in two vols., vol. 1 published 1876, vol. 2published 1877.

• 1894–1911. Gesammelte mathematische undphysikalische Werke,[17] in 3 vols. Friedrich Engeled. Leipzig: B.G. Teubner. Reprinted 1972, NewYork: Johnson.

Secondary sources

• Crowe, Michael, 1967. A History of Vector Analy-sis, Notre Dame University Press.

• Fearnley-Sander, Desmond, 1979, "HermannGrassmann and the Creation of Linear Algebra,"American Mathematical Monthly 86: 809–17.

• Fearnley-Sander, Desmond, 1982, "HermannGrassmann and the Prehistory of UniversalAlgebra," Am. Math. Monthly 89: 161–66.

• Fearnley-Sander, Desmond, and Stokes, Timothy,1996, "Area in Grassmann Geometry ". AutomatedDeduction in Geometry: 141–70

• Ivor Grattan-Guinness (2000) The Search for Math-ematical Roots 1870–1940. Princeton Univ. Press.

• Roger Penrose, 2004. The Road to Reality. AlfredA. Knopf.

• Petsche, Hans-Joachim, 2006. Graßmann (Textin German). (Vita Mathematica, 13). Basel:Birkhäuser.

• Petsche, Hans-Joachim, 2009. Hermann Graß-mann – Biography. Transl. by M Minnes. Basel:Birkhäuser.

• Petsche, Hans-Joachim; Kannenberg, Lloyd;Keßler, Gottfried; Liskowacka, Jolanta (eds.),2009. Hermann Graßmann – Roots and Traces.Autographs and Unknown Documents. Text inGerman and English. Basel: Birkhäuser.

• Petsche, Hans-Joachim; Lewis, Albert C.; Liesen,Jörg; Russ, Steve (eds.), 2010. From Past to Fu-ture: Graßmann’s Work in Context. The GraßmannBicentennial Conference, September 2009. Basel:Springer Basel AG.

• Petsche, Hans-Joachim and Peter Lenke (eds.),2010. International Grassmann Conference. Her-mann Grassmann Bicentennial: Potsdam andSzczecin, 16–19 September 2009; Video Recordingof the Conference. 4 DVD’s, 16:59:25. Potsdam:Universitätsverlag Potsdam.

• Rowe, David E. (2010) “Debating Grass-mann’s Mathematics: Schlegel Versus Klein”,Mathematical Intelligencer 32(1):41–8.

4 7 EXTERNAL LINKS

• Victor Schlegel (1878) Hermann Grassmann: SeinLeben und seine Werke on the Internet Archive.

• Schubring, G., ed., 1996. Hermann Gunther Grass-mann (1809–1877): visionary mathematician, sci-entist and neohumanist scholar. Kluwer.

• Vasilʹevich Prasolov, Viktor (1994), Problems andTheorems in Linear Algebra, Translations of Math-ematical Monographs 134, American MathematicalSociety, ISBN 978-0-8218-0236-6

Extensive online bibliography, revealing substantial con-temporary interest in Grassmann’s life and work. Refer-ences each chapter in Schubring.

• Paola Cantù: La matematica da scienza dellegrandezze a teoria delle forme. L’Ausdehnungslehredi H. Grassmann [Mathematics from Science ofMagnitudes to Theory of Forms. The Aus-dehnungslehre of H. Grassmann]. Genoa: Univer-sity of Genoa. Dissertation, 2003, s. xx+465.

Citations

[1] Tr. The rulers extension theory, a new branch of mathe-matics

[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 character-istics invented by Leibniz

[7] Vasilʹevich Prasolov 1994, p. 46.

[8] Hankel entry in the Dictionary of Scientific Biography.New York: 1970–1990

[9] Rowe 2010

[10] LyndeWheeler (1951), JosiahWillard Gibbs: The Historyof a Great Mind, 1998 reprint, Woodbridge, CT: Ox Bow,pp. 113-116.

[11] Penrose The Road to Reality, chapters 11 & 2

[12] Vasilʹevich Prasolov 1994, p. 45.

[13] Tr. “The rulers extension theory”

[14] Tr. “Geometric analysis linked to the geometric charac-teristics 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”

7 External links• The MacTutor History of Mathematics archive:

• O'Connor, John J.; Robertson, Edmund F.,“Hermann Grassmann”, MacTutor History ofMathematics archive, University of St An-drews.

• Abstract Linear Spaces. Discusses the roleof Grassmann and other 19th century figuresin the invention of linear algebra and vectorspaces.

• Fearnley-Sander's home page.

• Grassmann Bicentennial Conference (1809 – 1877),September 16 – 19, 2009 Potsdam / Szczecin (DE/ PL): From Past to Future: Grassmann’s Work inContext

• “The Grassmann method in projective geometry” Acompilation of English translations of three notesby Cesare Burali-Forti on the application of Grass-mann’s exterior algebra to projective geometry

• C. Burali-Forti, “Introduction to Differential Ge-ometry, following the method of H. Grassmann”(English translation of book by an early disciple ofGrassmann)

• “Mechanics, according to the principles of the the-ory of extension” An English translation of oneGrassmann’s papers on the applications of exterioralgebra

5

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