1106:. 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. 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.
43:
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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
1449:
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
1064:
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.â Disappointed by the reception of his work in mathematical circles,
948:
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
499:
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
504:
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,
483:
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
840:
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
111:
941:
In the 1840s, mathematicians were generally unprepared to understand
Grassmann's ideas. 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.
872:, 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.
639:, who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition â the language was not available â but there is no doubt that he had the concept.
464:
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
709:
are real numbers, defines addition and multiplication by real numbers and formally proves the linear space properties for these operations. ... He then develops the theory of
2243:
1454:
in which each set of positive elements has a least member. was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.
1034:
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
327:, 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
945:
1837:. International Workshop on Automated Deduction in Geometry 1996. Lecture Notes in Computer Science. Vol. 1360. Toulouse, France: Springer. pp. 141â170.
541:
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
2218:
2228:
988:
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
1744:
1302:
836:
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
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608:
proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once
2158:â A compilation of English translations of three notes by Cesare Burali-Forti on the application of Grassmann's exterior algebra to projective geometry
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2238:
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138:
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2012:
1973:
1938:
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510:
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Grassmann lost his contacts with mathematicians as well as his interest in geometry. In the last years of his life he turned to historical
1873:
The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel
547:
64:
529:
Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the
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1911:
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414:
86:
968:. Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them.
663:, ..., he effectively defines the free linear space that they generate; that is to say, he considers formal linear combinations
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1323:
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in a way that is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of
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Starting during the political turmoil in Germany, 1848â49, Hermann and his brother Robert published a Stettin newspaper,
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Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of
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734:
377:
57:
51:
2138:
Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces.
1630:"A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann | The Online Books Page"
1606:"A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann | The Online Books Page"
2051:
1876:
857:
2101:, revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.
910:, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his
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1114:
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238:
526:, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.
68:
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777:
307:
1372:
Turner, R. Steven (1996). "The Origins of Colorimetry: What did Helmholtz and Maxwell Learn from Grassmann?".
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635:
and others published formal definitions. In fact, such a definition had been given thirty years previously by
612:
is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial
992:
according to Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was advancing his
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557:
519:
444:
Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to
431:
1629:
1605:
875:
In 1853, Grassmann published a theory of how colors mix; his theory's four color laws are still taught, as
1262:
781:
1833:
Fearnley-Sander, Desmond; Stokes, Timothy (1997). "Area in Grassmann Geometry". In Wang, Dongming (ed.).
1996:
1868:
1077:, collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the
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552:
530:
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2150:
Grassmann Bicentennial Conference (1809 â 1877), September 16 â 19, 2009 Potsdam / Szczecin (DE / PL)
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1653:
1035:
1008:
773:
449:
177:
1796:
Fearnley-Sander, Desmond (March 1982). "Hermann Grassmann and the Prehistory of Universal Algebra".
1376:. Boston Studies in the Philosophy of Science. Vol. 187. Springer Netherlands. pp. 71â86.
2119:
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1406:
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718:
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438:
259:
31:
1956:
Petsche, Hans-Joachim; Kannenberg, Lloyd C.; KeĂler, Gottfried; Liskowacka, Jolanta, eds. (2009).
864:: to devise a geometric calculus devoid of coordinates and metric properties (what Leibniz termed
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142:
2061:
Hermann GĂŒnther GraĂmann (1809â1877): Visionary Mathematician, Scientist and Neohumanist Scholar
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Hermann GĂŒnther GraĂmann (1809â1877): Visionary Mathematician, Scientist and Neohumanist Scholar
1103:
215:
1019:(1898), included the first systematic exposition in English of the theory of extension and the
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C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"
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2008:
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1991:
Petsche, Hans-Joachim; Liesen, Jörg; Lewis, Albert C.; Russ, Steve, eds. (September 2011).
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1109:
These philological accomplishments were honored during his lifetime. He was elected to the
2170:â An English translation of one Grassmann's papers on the applications of exterior algebra
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1683:
1580:(February 2005). "2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers".
1451:
1012:
973:
884:
846:
714:
1087:, Grassmann's work is often cited. In 1955 a third edition of his dictionary was issued.
456:, philosophy, and literature. He does not appear to have taken courses in mathematics or
17:
2045:
1396:
See p.74: "Helmholtz rejected almost as many of Grassmann's conclusions as he accepted."
952:
One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was
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few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
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1099:
1074:
981:
953:
911:
604:, which translates as "theory of extension" or "theory of extensive magnitudes". Since
570:
870:
Geometrische Analyse geknĂŒpft an die von Leibniz erfundene geometrische Charakteristik
853:, in the hope that these applications would lead others to take his theory seriously.
110:
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leading figures of Grassmann's day failed to recognize the value of his mathematics.
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166:
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Multiplanes and Multispheres: Notes on a Grassmann Algebra approach with Mathematica
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353:. In linguistics he helped free language history and structure from each other.
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1066:
1007:
of his extension theory. To establish the priority of Grassmann over Hamilton,
985:
811:
785:
722:
619:
Fearnley-Sander describes Grassmann's foundation of linear algebra as follows:
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methods he had been mulling over since 1832. This essay, first published in the
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316:
195:
2068:
2004:
1965:
1930:
1906:. Vita Mathematica (in German). Vol. 13. Basel, Switzerland: BirkhÀuser.
1547:
965:
860:
invited Grassmann to enter a competition to solve a problem first proposed by
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428:
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Hermann Grassmann was the third of 12 children of Justus GĂŒnter Grassmann, an
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1860:
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980:, which used Grassmann's approach to derive ancient and modern results in
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964:, he developed some of Hermann Grassmann's algebras and W.R. Hamilton's
609:
453:
312:
190:
146:
2063:. Boston Studies in the Philosophy of Science. Vol. 187. Springer.
906:. In 1862, Grassmann published a thoroughly rewritten second edition of
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of 1894â1911, contains the first known appearance of what is now called
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1842:
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256:
134:
1809:
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1342:
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urged Grassmann's heirs to have the 1840 essay on tides published.
933:
manner of exposition anticipates the textbooks of the 20th century.
2168:"Mechanics, according to the principles of the theory of extension"
1709:. Vol. I: Foundations. Eltham, Australia: Barnard Publishing.
1660:
Hermann Grassmanns gesammelte mathematische und physikalische Werke
1534:(2010). "Debating Grassmann's Mathematics: Schlegel Versus Klein".
522:. (This eventuated in 1871.) After writing a series of articles on
1983:
1948:
765:
636:
1582:
The Road to Reality: A Complete Guide to the Laws of the Universe
1207:
Die Ausdehnungslehre. VollstĂ€ndig und in strenger Form begrĂŒndet.
916:
Die Ausdehnungslehre: VollstÀndig und in strenger Form bearbeitet
879:. Grassmann's work on this subject was inconsistent with that of
587:
Die Ausdehnungslehre: VollstÀndig und in strenger Form bearbeitet
474:
Die Ausdehnungslehre: VollstÀndig und in strenger Form bearbeitet
1746:
La matematica da scienza delle grandezze a teoria delle forme: l
2164:(English translation of book by an early disciple of Grassmann)
1925:. Translated by Minnes, Mark. Basel, Switzerland: BirkhÀuser.
360:
36:
1271:
ed. Leipzig: B.G. Teubner. Reprinted 1972, New York: Johnson.
764:. (One should keep in mind that in Grassmann's day, the only
760:âouter productâ), the key operation of an algebra now called
579:
Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik
466:
Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik
448:
universities. Beginning in 1827, he studied theology at the
1758:
The Mathematics of Quantities to the Science of Forms: The
616:; the number of possible dimensions is in fact unbounded.
2024:
International Grassmann Conference: Potsdam and Szczecin
27:
German polymath, linguist and mathematician (1809â1877)
1902:
Petsche, Hans-Joachim (2006). Fellmann, Emil A. (ed.).
1113:
and in 1876 he received an honorary doctorate from the
511:
Deutsche Wochenschrift fĂŒr Staat, Kirche und Volksleben
1324:"Hermann Grassmann and the Creation of Linear Algebra"
2022:
Lenke, Peter C.; Petsche, Hans-Joachim, eds. (2010).
1567:, 1998 reprint, Woodbridge, CT: Ox Bow, pp. 113-116.
1337:(10). Mathematical Association of America: 809â817.
1804:(3). Mathematical Association of America: 161â166.
1477:. Translated by Leites, Dimitry A. Providence, RI:
1102:. In his honor, this phonological rule is known as
252:
229:
183:
173:
154:
120:
101:
2152:: From Past to Future: Grassmann's Work in Context
1565:Josiah Willard Gibbs: The History of a Great Mind
1083:(more than 1,000 pages). In modern studies of the
999:Comprehension of Grassmann awaited the concept of
752:, also called "combinatorial product" (in German:
311:; 15 April 1809 â 26 September 1877) was a German
1409:(June 1957). "The Axiomatization of Arithmetic".
1194:. Vol. 1: Arithmetik. Berlin: Adolph Enslin.
30:"Grassmann" redirects here. For the surname, see
1263:Gesammelte mathematische und physikalische Werke
1253:. Leipzig: Brockhaus. Translation in two vols.,
1213:English translation, 2000, by Lloyd Kannenberg,
1191:Lehrbuch der Mathematik fĂŒr höhere Lehranstalten
1993:From Past to Future: GraĂmann's Work in Context
1768:(PhD thesis) (in Italian). University of Genoa.
962:
621:
561:, but expositing this theory making use of the
596:In 1844, Grassmann published his masterpiece (
2156:"The Grassmann method in projective geometry"
2110:The MacTutor History of Mathematics archive:
2047:Hermann Grassmann: Sein Leben und seine Werke
1666:Bulletin of the American Mathematical Society
1060:the publisher wrote to Grassmann: âYour book
996:, which also expanded the scope of geometry.
577:. He went on to develop those methods in his
8:
1303:List of things named after Hermann Grassmann
898:In 1861, Grassmann laid the groundwork for
395:. Unsourced material may be challenged and
2244:Mathematicians from the Kingdom of Prussia
1468:
1466:
1464:
1462:
1322:Fearnley-Sander, Desmond (December 1979).
434:who taught mathematics and physics at the
109:
98:
1728:. Eltham, Australia: Barnard Publishing.
1677:
946:Adhémar Jean Claude Barré de Saint-Venant
744:Following an idea of Grassmann's father,
631:) became widely known around 1920, when
415:Learn how and when to remove this message
87:Learn how and when to remove this message
282:
270:
50:This article includes a list of general
2125:MacTutor History of Mathematics Archive
1475:Problems and Theorems in Linear Algebra
1314:
1090:Grassmann also noticed and presented a
642:Beginning with a collection of 'units'
1257:published 1876, vol. 2 published 1877.
2219:People from the Province of Pomerania
1152:. Translated by Kannenberg, Lloyd C.
306:
7:
2229:Humboldt University of Berlin alumni
1027:the exterior algebra was applied to
900:Peano's axiomatization of arithmetic
393:adding citations to reliable sources
335:, the space which parameterizes all
2026:(DVD). UniversitÀtsverlag Potsdam.
1958:Hermann GraĂmann â Roots and Traces
958:Theorie der complexen Zahlensysteme
2199:19th-century German mathematicians
1960:. Basel, Switzerland: BirkhÀuser.
1743:CantĂč, Paola (February 13, 2003).
1508:Dictionary of Scientific Biography
1139:(in German). Leipzig: Otto Wigand.
56:it lacks sufficient corresponding
25:
1798:The American Mathematical Monthly
1331:The American Mathematical Monthly
776:had yet to be defined.) In 1878,
1999:. Potsdam-Szczecin: BirkhÀuser.
1997:GraĂmann Bicentennial Conference
1835:Automated Deduction in Geomtetry
976:published the first part of his
780:joined this exterior algebra to
496:at all secondary school levels.
365:
308:[ËhÉÉmanËÉĄÊntÊ°ÉËÉĄÊasman]
41:
2239:19th-century German translators
2050:(in German). Leipzig, Germany:
1679:10.1090/S0002-9904-1907-01557-4
1003:, which then could express the
843:Neue Theorie der Elektrodynamik
772:, and the general notion of an
2214:19th-century German physicists
1921:Petsche, Hans-Joachim (2009).
1779:University of Notre Dame Press
1536:The Mathematical Intelligencer
1421:Association for Symbolic Logic
788:by replacing Grassmann's rule
600:) commonly referred to as the
441:, where Hermann was educated.
1:
2059:Schubring, Gert, ed. (1996).
1705:Browne, John (October 2012).
1634:onlinebooks.library.upenn.edu
1610:onlinebooks.library.upenn.edu
1584:. New York: Alfred A. Knopf.
1479:American Mathematical Society
1412:The Journal of Symbolic Logic
1289:(Grassmann was its precursor)
1219:American Mathematical Society
826:= â1.) For more details, see
1775:A History of Vector Analysis
1724:Browne, John (August 2020).
1473:Prasolov, Viktor V. (1994).
1136:Die Lineale Ausdehnungslehre
737:of elements onto subspaces.
289:Die lineale Ausdehnungslehre
277:Die lineale Ausdehnungslehre
1510:. Charles Scribner's Sons.
1382:10.1007/978-94-015-8753-2_8
1188:Grassmann, Hermann (1861).
1173:Grassmann, Hermann (1847).
1148:A New Branch of Mathematics
1144:Grassmann, Hermann (1994).
1133:Grassmann, Hermann (1844).
548:Traité de mécanique céleste
472:, later revised in 1862 as
2260:
2052:Friedrich Arnold Brockhaus
1877:Princeton University Press
1773:Crowe, Michael J. (1967).
883:. Grassmann also wrote on
275:1878 copy of Grassmann's "
29:
2234:Translators from Sanskrit
2069:10.1007/978-94-015-8753-2
2005:10.1007/978-3-0346-0405-5
1966:10.1007/978-3-0346-0155-9
1931:10.1007/978-3-7643-8860-7
1548:10.1007/s00283-009-9094-2
1181:Weidmannsche Buchhandlung
1111:American Oriental Society
585:) and its later revision
452:, also taking classes in
296:Hermann GĂŒnther Grassmann
265:
245:
115:Hermann GĂŒnther Grassmann
108:
103:Hermann GĂŒnther Grassmann
18:Hermann GĂŒnther Grassmann
2194:Scientists from Szczecin
2130:University of St Andrews
1015:'s first monograph, the
922:), fared no better than
778:William Kingdon Clifford
754:kombinatorisches Produkt
2136:Abstract Linear Spaces.
1243:Wörterbuch zum Rig-Veda
904:Lehrbuch der Arithmetik
520:constitutional monarchy
342:linear subspaces of an
71:more precise citations.
2209:Linguists from Germany
1869:Grattan-Guinness, Ivor
1542:(1). Springer: 41â48.
1179:(in German). Leipzig:
1115:University of TĂŒbingen
970:
845:and several papers on
782:William Rowan Hamilton
742:
476:, here referred to as
468:, here referred to as
315:known in his day as a
299:
292:
280:
239:University of TĂŒbingen
1246:. Leipzig: Brockhaus.
1025:differential geometry
531:University of Giessen
286:
274:
139:Province of Pomerania
2116:Robertson, Edmund F.
1447:. p. 145, 147:
1176:Geometrische Analyse
1094:that exists in both
1073:. He wrote books on
1062:Die Ausdehnungslehre
1036:mathematical physics
1009:Josiah Willard Gibbs
978:System der Raumlehre
623:The definition of a
573:and the notion of a
558:MĂ©canique analytique
450:University of Berlin
389:improve this section
331:. He introduced the
178:University of Berlin
2120:"Hermann Grassmann"
2114:O'Connor, John J.;
2099:online bibliography
1298:Multilinear algebra
1041:The Road to Reality
1023:. With the rise of
1005:multilinear algebra
814:, we have the rule
719:linear independence
711:linear independence
454:classical languages
32:Grassmann (surname)
2204:Linear algebraists
1843:10.1007/BFb0022724
1502:Crowe, Michael J.
1282:AmpĂšre's force law
1156:. pp. 9â297.
1029:differential forms
770:Euclidean geometry
733:of subspaces, and
524:constitutional law
516:German unification
319:and now also as a
293:
281:
143:Kingdom of Prussia
2078:978-94-015-8753-2
2033:978-3-86956-093-9
2014:978-3-0346-0404-8
1975:978-3-0346-0155-9
1940:978-3-7643-8859-1
1852:978-3-540-69717-6
1707:Grassmann Algebra
1504:"Hankel, Hermann"
1450:forms an ordered
1293:Geometric algebra
1210:. Berlin: Enslin.
1092:phonological rule
1069:and the study of
1017:Universal Algebra
748:also defined the
425:
424:
417:
269:
268:
247:Scientific career
206:Grassmann algebra
158:26 September 1877
97:
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16:(Redirected from
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2224:Color scientists
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2042:Schlegel, Victor
2037:
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1215:Extension Theory
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160:(1877-09-26)
83:
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2189:1877 deaths
2184:1809 births
1423:: 145â158.
1267:in 3 vols.
1260:1894â1911.
1249:1876â1877.
1067:linguistics
986:Felix Klein
966:quaternions
812:quaternions
786:quaternions
768:theory was
735:projections
196:Color space
69:introducing
2178:Categories
2097:Extensive
1984:2009930234
1949:2009929497
1895:j.ctt7rp8j
1699:References
1639:2023-10-18
1615:2023-10-18
1154:Open Court
810:= 1. (For
614:dimensions
494:mineralogy
304:pronounced
127:1809-04-15
52:references
2087:0068-0346
1861:0302-9743
1818:0002-9890
1672:: 33â36.
1407:Wang, Hao
1351:0002-9890
1309:Citations
893:mechanics
881:Helmholtz
856:In 1846,
766:axiomatic
727:dimension
551:and from
490:chemistry
439:Gymnasium
405:July 2024
376:does not
357:Biography
325:physicist
298:(German:
260:Gymnasium
234:PhD (Hon)
165:Stettin,
149:, Poland)
77:July 2024
2044:(1878).
1904:GraĂmann
1871:(2000).
1656:(1907).
1563:(1951),
1445:26896458
1276:See also
1251:Rig-Veda
1096:Sanskrit
1071:Sanskrit
1052:Linguist
972:In 1872
937:Response
851:surfaces
715:subspace
610:geometry
553:Lagrange
518:under a
446:Prussian
432:minister
429:ordained
317:linguist
313:polymath
300:GraĂmann
191:Bivector
147:Szczecin
1826:2320198
1688:1558534
1437:2964176
1359:2320145
1085:Rigveda
1080:Rigveda
902:in his
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543:Laplace
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436:Stettin
397:removed
382:sources
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891:, and
858:Möbius
563:vector
500:asked
492:, and
241:(1876)
230:Awards
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2095:Note:
1891:JSTOR
1822:JSTOR
1766:(PDF)
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1355:JSTOR
1327:(PDF)
1100:Greek
637:Peano
2083:ISSN
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