- Hermann Grassmann
Infobox Scientist

name = Hermann Günther Grassmann

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caption = Hermann Günther Grassmann

birth_date =April 15 ,1809

birth_place =Stettin (Szczecin )

death_date =September 26 ,1877

death_place =Stettin

residence = German

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**Hermann Günther Grassmann**(April 15 ,1809 ,Stettin (Szczecin ) –September 26 ,1877 ,Stettin ) was a Germanpolymath , renowned in his day as a linguist and now admired as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher. His mathematical work was not recognized in his lifetime.**Biography**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. Hermann often collaborated with his brother Robert.Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to

Prussia n universities. Beginning in 1827, he studiedtheology at theUniversity of Berlin , also taking classes in classical languages,philosophy , and literature. He does not appear to have taken courses inmathematics orphysics .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. In the spring of 1832, he was made an assistant at the Stettin Gymnasium. 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 below).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. This wide range of topics reveals again that he was qualified to teach only at a low level. Over the next four years, Grassmann passed examinations enabling him to teach

mathematics ,physics ,chemistry , andmineralogy at all secondary school levels.Grassmann felt somewhat aggrieved that he was writing innovative mathematics, but taught only in secondary schools. Yet he did rise in rank, even while never leaving Stettin. 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.During the political turmoil in Germany, 1848-49, Hermann and Robert Grassmann published a Stettin newspaper calling for

German unification under aconstitutional monarchy . (This eventuated in 1866.) After writing a series of articles onconstitutional 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 .**Mathematician**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 are now calledlinear algebra and the notion of avector space . He went on to develop those methods in his**A1**and**A2**.In 1844, Grassmann published his masterpiece, his "Die Lineare Ausdehnungslehre, ein neuer Zweig der Mathematik" [The Theory of Linear Extension, a New Branch of Mathematics] , hereinafter denoted

**A1**and commonly referred to as the "Ausdehnungslehre," 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 oncegeometry is put into the algebraic form he advocated, then the number three has no privileged role as the number of spatialdimension s; the number of possible dimensions is in fact unbounded.[

*http://www.maths.utas.edu.au/People/dfs/Papers/GrassmannLinAlgpaper/GrassmannLinAlgpaper.html Fearnley-Sander (1979)*] describes Grassmann's foundation of linear algebra as follows:cquote|The definition of a

linear space (vector space )... became widely known around 1920, whenHermann Weyl and others published formal definitions. In fact, such a definition had been given thirty years previously byPeano , who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition --- the language was not available --- but there is no doubt that he had the concept.Beginning with a collection of 'units' "e

_{1}, e_{2}, e_{3}, ...", he effectively defines the free linear space which they generate; that is to say, he considers formal linear combinations "a_{1}e_{1}+ a_{2}e_{2}+ a_{3}e_{3}+ ..." where the "a_{j}" are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. ... He then develops the theory oflinear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions ofsubspace ,independence ,span ,dimension ,join andmeet of subspaces, andprojection s of elements onto subspaces....few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.

Following an idea of Grassmann's father,

**A1**also defined theexterior product , also called "combinatorial product" (In German: "äußeres Produkt" or "kombinatorisches Produkt"), the key operation of an algebra now calledexterior algebra . (One should keep in mind that in Grassmann's day, the onlyaxiom atic theory wasEuclidean geometry , and the general notion of anabstract algebra had yet to be defined.) In 1878,William Kingdon Clifford joined this exterior algebra toWilliam Rowan Hamilton 'squaternions by replacing Grassmann's rule "e_{p}e_{p}" = 0 by the rule "e_{p}e_{p}" = 1. "(Forquaternions , we have the rule "i^{2}" = "j^{2}" = "k^{2}" = -1.)" For more details, seeexterior algebra .**A1**was a revolutionary text, too far ahead of its time to be appreciated. Grassmann submitted it as a Ph. D. thesis, but Möbius said he was unable to evaluate it and forwarded it toErnst Kummer , who rejected it without giving it a careful reading. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 "Neue Theorie der Elektrodynamik" 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", was the winning entry (also the only entry). Moreover, 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. Grassman's work on this subject was inconsistent with that of

Helmholtz . Grassmann also wrote oncrystallography ,electromagnetism , andmechanics .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. Curiously, Grassmann (1861) has never been translated into English.In 1862, Grassman published a thoroughly rewritten second edition of

**A1**, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of hislinear 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.The only mathematician to appreciate Grassmann's ideas during his lifetime was

Hermann Hankel , whose 1867 "Theorie der complexen Zahlensysteme" helped make Grassmann's ideas better known. This workGrassmann's mathematical methods were slow to be adopted but they directly influenced

Felix Klein andÉlie Cartan .A. N. Whitehead 's first monograph, the "Universal Algebra" (1898), included the first systematic exposition in English of the theory of extension and theexterior algebra . The theory of extension led to the development ofdifferential form s and to the application of such forms to analysis andgeometry .Differential geometry makes use of theexterior algebra . For an introduction to the role of Grassmann's work in contemporarymathematical physics , see Penrose (2004: chpts. 11, 12).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 (and there is little reason to doubt him) that he had first developed these ideas in 1832.**Linguist**Disappointed at his inability to be recognized as a mathematician, Grassmann turned to historical

linguistics . He wrote books on German grammar, collected folk songs, and learnedSanskrit . His dictionary and his translation of theRgveda (still in print) were recognized among philologists. He devised a sound law ofIndo-European languages , named Grassmann's Law in his honor. These philological accomplishments were honored during his lifetime; he was elected to theAmerican Oriental Society and in 1876, he received a honorary doctorate from theUniversity of Tübingen .**ee also***

exterior algebra

*Grassmann number

*Grassmannian

*Grassmann's law (phonology)

*Grassmann's law (optics) **References****Primary**:

*1844. "Die lineare Ausdehnungslehre". Leipzig: Wiegand. English translation, 1995, by Lloyd Kannenberg, "A new branch of mathematics". Chicago: Open Court. This is**A1**.

*1861. "Lehrbuch der Mathematik für höhere Lehranstalten, Band 1". Berlin: Enslin.

*1862. "Die Ausdehnungslehre, vollständig und in strenger Form bearbeitet". Berlin: Enslin. English translation, 2000, by Lloyd Kannenberg, "Extension Theory". American Mathematical Society. This is**A2**. [*http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/Papers/GrassmannTranslation/Grass.html Excerpt*] translated by D. Fearnley-Sander.

*1894-1911. "Gesammelte mathematische und physikalische Werke," in 3 vols. Friedrich Engel ed. Leipzig: B.G. Teubner. Reprinted 1972, New York: Johnson.**Secondary**:

*Crowe, Michael, 1967. "A History of Vector Analysis". Notre Dame University Press.

*Fearnley-Sander, Desmond, 1979, " [*http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/Papers/GrassmannLinAlgpaper/GrassmannLinAlgpaper.html Hermann Grassmann and the Creation of Linear Algebra,*] " "American Mathematical Monthly 86": 809-17.

*--------, 1982, " [*http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/Papers/GrassmannUAlgpaper/GrassmannUAlgpaper.html Hermann Grassmann and the Prehistory of Universal Algebra,*] " "Am. Math. Monthly 89": 161-66.

* -------, and Stokes, Timothy, 1996, " [*http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/Papers/ADGPaper97.pdf Area in Grassmann Geometry*] ". "Automated Deduction in Geometry": 141-70

*Ivor Grattan-Guinness (2000) "The Search for Mathematical Roots 1870-1940". Princeton Univ. Press.

*Roger Penrose , 2004. "The Road to Reality". Alfred A. Knopf.

*Schlege, Victor, 1878. "Hermann Grassmann: Sein Leben und seine Werke". Leipzig: F.A. Brockhaus.

*Schubring, G., ed., 1996. "Hermann Gunther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar". Kluwer.

*Petsche, Hans-Joachim, 2006. "Graßmann". (Vita Mathematica, 13) Birkhäuser.Extensive [*http://www-history.mcs.st-andrews.ac.uk/history/References/Grassmann.html online bibliography,*] revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.

*Paola Cantu':" La matematica da scienza delle grandezze a teoria delle forme. L’Ausdehnungslehre di H. Grassmann" [Mathematics from Science of Magnitudes to Theory of Forms. The Ausdehnungslehre of H. Grassmann] . Genoa: University of Genoa. Dissertation, 2003, s. xx+465.**External links***The MacTutor History of Mathematics archive:

** MacTutor Biography|id=Grassmann

** [*http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_linear_spaces.html#24 Abstract Linear Spaces.*] Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces.

* [*http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/dfs.html Fearnley-Sander*] 's home page.

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