Editing Hermann Grassmann

{{Short description|German polymath, linguist and mathematician  (1809–1877)}}
{{redirect|Grassmann|the surname|Grassmann (surname)}}
{{More footnotes needed|date=July 2024}}
{{Infobox scientist
|name              = Hermann Günther Grassmann
|image             = Hermann Graßmann.jpg
|caption           = Hermann Günther Grassmann
|birth_date        = {{birth date|1809|04|15|df=y}}
|birth_place       = [[Stettin]], [[Province of Pomerania (1653–1815)|Province of Pomerania]], [[Kingdom of Prussia]] (present-day [[Szczecin]], Poland)
|death_date        = {{death date and age|1877|09|26|1809|04|15|df=y}}
|death_place       = Stettin, [[German Empire]]
|citizenship       =
|nationality       =
|field             =
|alma_mater        = [[University of Berlin]]
|work_institutions = [[Stettin]] [[Gymnasium (Germany)|Gymnasium]]
|doctoral_advisor  =
|doctoral_students =
|known_for         = {{plainlist|
*[[Bivector]]
*[[Color space]]
*[[Grassmannian]]
*[[Exterior algebra|Grassmann algebra]]
*[[Grassmann number]]
*[[Grassmann's law]] 
*[[Grassmann's laws (color science)|Grassmann's laws]]
}}
|influences        =
|influenced        =
|prizes            = [[PhD (Hon)]]:<br>[[University of Tübingen]] (1876)
}}
[[File:Grassmann-1.jpg|alt=1878 copy of Grassmann's "Die lineale Ausdehnungslehre"|thumb|298x298px|1878 copy of Grassmann's "''Die lineale Ausdehnungslehre''"]]
[[File:Grassmann-2.jpg|alt=First page of "Die lineale Ausdehnungslehre"|thumb|303x303px|First page of "''Die lineale Ausdehnungslehre''"]]
'''Hermann Günther Grassmann''' ({{langx|de|link=no|Graßmann}}, {{IPA|de|ˈhɛɐman ˈɡʏntʰɐ ˈɡʁasman|pron}}; 15 April 1809 – 26 September 1877) was a German [[polymath]] known in his day as a [[linguistics|linguist]] and now also as a [[mathematician]]. He was also a [[physicist]], general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a [[vector space]]. He introduced the [[Grassmannian]], the space which parameterizes all [[Dimension|''k''-dimensional]] linear subspaces of an ''n''-dimensional [[vector space]] ''V''.  In linguistics he helped free language history and structure from each other.

==Biography==
{{unreferenced section|date=July 2024}}
Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an [[Ordination|ordained]] [[Minister (Christianity)|minister]] who taught mathematics and physics at the [[Stettin]] [[Gymnasium (Germany)|Gymnasium]], where Hermann was educated.

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

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 ''Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik'', here referred to as '''A1''', later revised in 1862 as ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'', here referred to as '''A2'''.

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.

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 [[Ernst 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.

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]].

==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 ''[[Traité de mécanique céleste]]'' and from [[Joseph-Louis Lagrange|Lagrange]]'s ''[[Mécanique analytique]]'', but expositing this theory making use of the [[vector (geometry)|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 ''Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik'' ('''A1''') and its later revision ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'' ('''A2''').

In 1844, Grassmann published his masterpiece ('''A1''') 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 once [[geometry]] is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial [[dimension]]s; the number of possible dimensions is in fact unbounded.

Fearnley-Sander describes Grassmann's foundation of linear algebra  as follows:<ref>{{cite journal |last1=Fearnley-Sander |first1=Desmond |title=Hermann Grassmann and the Creation of Linear Algebra |journal=The American Mathematical Monthly |date=December 1979 |volume=86 |issue=10 |pages=809–817 |doi=10.2307/2320145 |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/DesmondFearnleySander.pdf |ref=Fearnley-Sander |publisher=Mathematical Association of America |issn=0002-9890 |jstor=2320145}}</ref>

{{blockquote|The definition of a [[linear space]] ([[vector space]]) [...] became widely known around 1920, when [[Hermann Weyl]] and others published formal definitions. In fact, such a definition had been given thirty years previously by [[Peano]], 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''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ..., he effectively defines the free linear space that they generate; that is to say, he considers formal linear combinations ''a''<sub>1</sub>''e''<sub>1</sub> + ''a''<sub>2</sub>''e''<sub>2</sub> + ''a''<sub>3</sub>''e''<sub>3</sub> + ... where the ''a<sub>j</sub>'' 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 of [[linear independence]] in a way that is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of [[linear subspace|subspace]], [[linear independence]], [[Linear span|span]], [[dimension]], [[join and meet]] of subspaces, and [[projection (linear algebra)|projection]]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 the [[exterior product]], also called "combinatorial product" (in German: ''kombinatorisches Produkt'' or ''äußeres Produkt'' “outer product”), the key operation of an algebra now called [[exterior algebra]]. (One should keep in mind that in Grassmann's day, the only [[axiom]]atic theory was [[Euclidean geometry]], and the general notion of an [[Algebra over a field|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 ''e<sub>p</sub>e<sub>p</sub>'' = 0 by the rule ''e<sub>p</sub>e<sub>p</sub>'' = 1. (For [[quaternions]], we have the rule ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = −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'' and several papers on [[algebraic curve]]s and [[algebraic surface|surface]]s, in the hope that these applications would lead others to take his theory seriously.

In 1846, [[August Ferdinand Möbius|Möbius]] invited Grassmann to enter a competition to solve a problem first proposed by [[Gottfried Wilhelm Leibniz|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). 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; his theory's four color laws are still taught, as [[Grassmann's laws (color science)|Grassmann's laws]]. Grassmann's work on this subject was inconsistent with that of [[Helmholtz]].<ref>{{cite book
 | last = Turner | first = R. Steven
 | contribution = The Origins of Colorimetry: What did Helmholtz and Maxwell Learn from Grassmann?
 | doi = 10.1007/978-94-015-8753-2_8
 | isbn = 9789401587532
 | pages = 71–86
 | publisher = Springer Netherlands
 | title = Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar
 | series = Boston Studies in the Philosophy of Science
 | year = 1996| volume = 187
 }} See p.74: "Helmholtz rejected almost as many of Grassmann's conclusions as he accepted."</ref> Grassmann also wrote on [[crystallography]], [[electromagnetism]], and [[mechanics]].

In 1861, Grassmann laid the groundwork for [[Peano axioms|Peano's axiomatization of arithmetic]] in his ''Lehrbuch der Arithmetik''.<ref>{{cite journal |last1=Wang |first1=Hao |author1-link=Hao Wang (academic) |title=The Axiomatization of Arithmetic |journal=[[The Journal of Symbolic Logic]] |date=June 1957 |volume=22 |issue=2 |pages=145–158 |doi=10.2307/2964176 |publisher=[[Association for Symbolic Logic]] |jstor=2964176 |s2cid=26896458 |quote=It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered [[integral domain]] in which each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis. |quote-page=145, 147}}</ref> 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'' ('''A2'''), fared no better than '''A1''', even though '''A2'''{{'s}} manner of exposition anticipates the textbooks of the 20th century.

==Response==
In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas.<ref name="Prasolov">{{cite book |last1=Prasolov |first1=Viktor V. |title=Problems and Theorems in Linear Algebra |date=1994 |publisher=[[American Mathematical Society]] |location=Providence, RI |isbn=0-8218-0236-4 |translator1-first=Dimitry A. |translator1-last=Leites}}</ref> 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.{{r|Prasolov|p=46}}

[[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.

One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was [[Hermann Hankel]], whose 1867 ''Theorie der complexen Zahlensysteme''.<ref>{{cite encyclopedia |last=Crowe |first=Michael J. |encyclopedia=Dictionary of Scientific Biography |isbn=0-684-10114-9 |publisher=Charles Scribner's Sons |entry=Hankel, Hermann |chapter-url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/hankel-hermann}}</ref>
{{blockquote|[…], he developed […] some of Hermann Grassmann's algebras and W.R. Hamilton's [[quaternion]]s. Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them.}}

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-dimensional geometry. Meanwhile, Klein was advancing his [[Erlangen program]], which also expanded the scope of geometry.<ref name="Rowe">{{cite journal |last1=Rowe |first1=David E. |author1-link=David E. Rowe |title=Debating Grassmann's Mathematics: Schlegel Versus Klein |journal=The Mathematical Intelligencer |year=2010 |volume=32 |issue=1 |pages=41–48 |doi=10.1007/s00283-009-9094-2 |publisher=Springer }}</ref>

Comprehension of Grassmann awaited the concept of [[vector space]]s, 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.<ref>[[Lynde Wheeler]] (1951), ''Josiah Willard Gibbs:  The History of a Great Mind'', 1998 reprint, Woodbridge, CT:  Ox Bow, pp. 113-116.</ref>  [[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 form]]s.

In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary [[mathematical physics]] see ''[[The Road to Reality]]'' by [[Roger Penrose]].<ref>{{cite book |last1=Penrose |first1=Roger |author1-link=Roger Penrose |title=The Road to Reality: A Complete Guide to the Laws of the Universe |date=February 2005 |publisher=Alfred A. Knopf |location=New York |isbn=0-679-45443-8 |chapter=2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers}}</ref>

==Linguist==
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."{{r|Prasolov|p=45}} Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. In 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). In modern studies of the ''Rigveda'', Grassmann's work is often cited. In 1955 a third edition of his dictionary was issued.{{r|Prasolov|p=46}}

Grassmann also noticed and presented a [[phonological rule]] that exists in both [[Sanskrit]] and [[Greek language|Greek]]. In his honor, this phonological rule is known as [[Grassmann's law]]. His discovery was revolutionary for historical linguistics at the time, as it challenged the widespread notion of Sanskrit as an older predecessor to other Indo-European languages.<ref>{{Cite web |title=A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann {{!}} The Online Books Page |url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp46747 |access-date=2023-10-18 |website=onlinebooks.library.upenn.edu}}</ref> This was a widespread assumption due to Sanskrit's more agglutinative structure, which languages like Latin and Greek were thought to have passed through to reach their more "modern" synthetic structure. However, Grassman's work proved that, in at least one phonological pattern, German was indeed "older" (i.e., less synthetic) than Sanskrit. This meant that genealogical and typological classifications of languages were at last correctly separated in linguistics, allowing significant progress for later linguists.<ref>{{Cite web |title=A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann {{!}} The Online Books Page |url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp46747 |access-date=2023-10-18 |website=onlinebooks.library.upenn.edu}}</ref>

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]].

==Publications==
* '''A1''': 
** {{cite book |last1=Grassmann |first1=Hermann |title=Die Lineale Ausdehnungslehre |date=1844 |publisher=Otto Wigand |location=Leipzig |language=de |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433017485941 }}
** {{cite book |last1=Grassmann |first1=Hermann |translator1-last=Kannenberg |translator1-first=Lloyd C. |title=A New Branch of Mathematics |date=1994 |publisher=[[Open Court Publishing Company|Open Court]] |isbn=9780812692761 |pages=9–297 |url=https://archive.org/details/newbranchofmathe0000gras |url-access=limited }}
* {{cite book |last=Grassmann |first=Hermann |title=Geometrische Analyse |url=https://books.google.com/books?id=cHGrfrQVq1oC |year=1847 |publisher=[[Weidmannsche Buchhandlung]] |location=Leipzig |language=de}}
* {{cite book |last=Grassmann |first=Hermann |title=Lehrbuch der Mathematik für höhere Lehranstalten |url=https://books.google.com/books?id=BeSD9fFZWDYC&pg=PR2 |volume=1: Arithmetik |year=1861 |publisher=Adolph Enslin |location=Berlin}}
* '''A2''':
**1862. ''[https://babel.hathitrust.org/cgi/pt?id=hvd.32044091872531 Die Ausdehnungslehre. Vollständig und in strenger Form begründet.]''. Berlin: Enslin.
** English translation, 2000, by Lloyd Kannenberg, ''Extension Theory'', [[American Mathematical Society]] {{ISBN|0-8126-9275-6}}, {{ISBN|0-8126-9276-4}}
* 1873. ''[http://gretil.sub.uni-goettingen.de/#GraSG Wörterbuch zum Rig-Veda]''. Leipzig: Brockhaus.
* 1876–1877. ''Rig-Veda''. Leipzig: Brockhaus. Translation in two vols., [https://archive.org/details/rigveda02grasgoog vol. 1] published 1876, vol. 2 published 1877.
* 1894–1911. ''[http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABW0785 Gesammelte mathematische und physikalische Werke],'' in 3 vols. [[Friedrich Engel (mathematician)|Friedrich Engel]] ed. Leipzig: B.G. Teubner.<ref>{{cite journal|doi=10.1090/S0002-9904-1907-01557-4|title=Book Review: ''Hermann Grassmanns gesammelte mathematische und physikalische Werke''|year=1907|last1=Wilson|first1=E. B.|author-link=Edwin Bidwell Wilson|journal=Bulletin of the American Mathematical Society|volume=14|pages=33–36|mr=1558534|doi-access=free}}</ref> Reprinted 1972, New York: Johnson.

==See also==
*[[Ampère's force law]]
*[[Bra–ket notation]] (Grassmann was its precursor)
*[[Geometric algebra]]
*[[Multilinear algebra]]
*[[List of things named after Hermann Grassmann]]

==Citations==
{{reflist}}

==References==
* {{cite book |last1=Browne |first1=John |title=Grassmann Algebra |date=October 2012 |publisher=Barnard Publishing |location=Eltham, Australia |isbn=978-1479197637 |volume=I: Foundations}}
* {{cite book |last1=Browne |first1=John |title=Multiplanes and Multispheres: Notes on a Grassmann Algebra approach with Mathematica |date=August 2020 |publisher=Barnard Publishing |location=Eltham, Australia |isbn=979-8657325379}}
* {{cite thesis |last=Cantù |first=Paola |date=February 13, 2003 |title=La matematica da scienza delle grandezze a teoria delle forme: l{{'}}''Ausdehnungslehre'' di H. Grassmann |trans-title=The Mathematics of Quantities to the Science of Forms: The ''Ausdehnungslehre'' of H. Grassmann |publisher=University of Genoa |degree=PhD |url=https://air.unimi.it/retrieve/handle/2434/62423/100549/La%20matematica%20da%20scienza%20delle%20grandezze%20a%20teoria.pdf |language=it}}
* {{cite book |last1=Crowe |first1=Michael J. |title=A History of Vector Analysis |date=1967 |publisher=[[University of Notre Dame Press]] |isbn=0-486-64955-5}}
* {{cite journal |last1=Fearnley-Sander |first1=Desmond |title=Hermann Grassmann and the Prehistory of Universal Algebra |journal=The American Mathematical Monthly |date=March 1982 |volume=89 |issue=3 |pages=161–166 |doi=10.2307/2320198 |publisher=Mathematical Association of America |issn=0002-9890 |jstor=2320198}}
* {{cite conference |title=Area in Grassmann Geometry |last1=Fearnley-Sander |first1=Desmond |last2=Stokes |first2=Timothy |date=1997 |conference=International Workshop on Automated Deduction in Geometry 1996 |editor-last=Wang |editor-first=Dongming |volume=1360 |book-title=Automated Deduction in Geomtetry |publisher=Springer |location=Toulouse, France |pages=141–170 |isbn=978-3-540-69717-6 |issn=0302-9743 |doi=10.1007/BFb0022724 |series=Lecture Notes in Computer Science}}
* {{cite book |last1=Grattan-Guinness |first1=Ivor |author1-link=Ivor Grattan-Guinness |title=The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel |date=2000 |publisher=[[Princeton University Press]] |isbn=9780691058580 |jstor=j.ctt7rp8j}}
* {{cite book |last1=Petsche |first1=Hans-Joachim |editor1-last=Fellmann |editor1-first=Emil A. |title=Graßmann |date=2006 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=3-7643-7257-5 |language=de |volume=13 |series=Vita Mathematica}}
* {{cite book |last1=Petsche |first1=Hans-Joachim |title=Hermann Graßmann |date=2009 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=978-3-7643-8859-1 |translator-last=Minnes |translator-first=Mark |doi=10.1007/978-3-7643-8860-7 |lccn=2009929497}}
* {{cite book |editor1-last=Petsche |editor1-first=Hans-Joachim |editor2-last=Kannenberg |editor2-first=Lloyd C. |editor3-last=Keßler |editor3-first=Gottfried |editor4-last=Liskowacka |editor4-first=Jolanta |title=Hermann Graßmann – Roots and Traces |date=2009 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=978-3-0346-0155-9 |lccn=2009930234 |doi=10.1007/978-3-0346-0155-9}}
* {{cite conference |date=September 2011 |conference=Graßmann Bicentennial Conference |conference-url=https://www.uni-potsdam.de/u/philosophie/grassmann/Grassmann-2009.htm |editor1-first=Hans-Joachim |editor1-last=Petsche  |editor2-first=Jörg |editor2-last=Liesen |editor3-first=Albert C. |editor3-last=Lewis |editor4-first=Steve |editor4-last=Russ |book-title=From Past to Future: Graßmann's Work in Context |publisher=Birkhäuser |location=Potsdam-Szczecin |isbn=978-3-0346-0404-8 |doi=10.1007/978-3-0346-0405-5}}
* {{cite AV media |editor1-first=Peter C. |editor1-last=Lenke |editor2-first=Hans-Joachim |editor2-last=Petsche |date=2010 |title=International Grassmann Conference: Potsdam and Szczecin |medium=DVD |publisher=Universitätsverlag Potsdam |isbn=978-3-86956-093-9}}
* {{cite book |last1=Schlegel |first1=Victor |author1-link=Victor Schlegel |title=Hermann Grassmann: Sein Leben und seine Werke |date=1878 |publisher=[[Friedrich Arnold Brockhaus]] |location=Leipzig, Germany |url=https://archive.org/details/hermanngrassman00schlgoog |language=de}}
* {{cite book |editor1-last=Schubring |editor1-first=Gert |title=Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar |date=1996 |publisher=Springer |isbn=978-94-015-8753-2 |doi=10.1007/978-94-015-8753-2 |series=Boston Studies in the Philosophy of Science |volume=187 |issn=0068-0346}}

'''Note:''' 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.

==External links==
{{Commons}}
{{Wikiquote}}
* 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.
* [https://web.archive.org/web/20080919130837/http://www.maths.utas.edu.au/old_web_stuff_from_2007/People/dfs/dfs.html Fearnley-Sander]'s home page.
* [https://web.archive.org/web/20090302075547/http://www.uni-potsdam.de/phi/Grassmann-2009.htm Grassmann Bicentennial Conference (1809 – 1877), September 16 – 19, 2009  Potsdam / Szczecin (DE / PL)]: From Past to Future: Grassmann's Work in Context
* [http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf "The Grassmann method in projective geometry"] – A compilation of English translations of three notes by Cesare Burali-Forti on the application of Grassmann's exterior algebra to projective geometry
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"] (English translation of book by an early disciple of Grassmann)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/grassmann_-_mechanics_and_extensions.pdf "Mechanics, according to the principles of the theory of extension"] – An English translation of one Grassmann's papers on the applications of exterior algebra

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{{DEFAULTSORT:Grassmann, Hermann}}
[[Category:1809 births]]
[[Category:1877 deaths]]
[[Category:Scientists from Szczecin]]
[[Category:19th-century German mathematicians]]
[[Category:Linear algebraists]]
[[Category:19th-century German linguists]]
[[Category:19th-century German physicists]]
[[Category:People from the Province of Pomerania]]
[[Category:Color scientists]]
[[Category:Humboldt University of Berlin alumni]]
[[Category:Translators from Sanskrit]]
[[Category:19th-century German translators]]
[[Category:Mathematicians from the Kingdom of Prussia]]