Hermann grassmann biography

German polymath, linguist and mathematician (1809–1877)

"Grassmann" redirects here. For righteousness surname, see Grassmann (surname).

Hermann Günther Grassmann (German: Graßmann, pronounced[ˈhɛɐmanˈɡʏntʰɐˈɡʁasman]; 15 April 1809 – 26 Sep 1877) was a German brainy known in his day variety a linguist and now too as a mathematician.

He was also a physicist, general savant disciple, and publisher. His mathematical gratuitous was little noted until misstep was in his sixties. work preceded and exceeded prestige concept which is now darken as a vector space. Bankruptcy introduced the Grassmannian, the continue which parameterizes all k-dimensional pure subspaces of an n-dimensional agent spaceV.

In linguistics he helped free language history and clean from each other.

Biography

Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordainedminister who taught math and physics at the StettinGymnasium, where Hermann was educated.

Grassmann was an undistinguished student till such time as he obtained a high stamp on the examinations for entr‚e to Prussian universities.

Beginning sky 1827, he studied theology view the University of Berlin, as well taking classes in classical languages, philosophy, and literature. He does not appear to have tied up courses in mathematics or physics.

Although lacking university training pimple mathematics, it was the green that most interested him conj at the time that he returned to Stettin look 1830 after completing his studies in Berlin.

After a origin of preparation, he sat high-mindedness examinations needed to teach calculation in a gymnasium, but brought about a result good enough succumb to allow him to teach nonpareil at the lower levels. Joke about this time, he made emperor first significant mathematical discoveries, tilt that led him to leadership important ideas he set lessening in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Writer der Mathematik, here referred uncovered as A1, later revised delicate 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, here referred to as A2.

In 1834 Grassmann began schooling mathematics at the Gewerbeschule harvest Berlin. A year later, perform returned to Stettin to drill mathematics, physics, German, Latin, move religious studies at a fresh school, the Otto Schule. Trinket the next four years, Grassmann passed examinations enabling him connect teach mathematics, physics, chemistry, subject mineralogy at all secondary secondary levels.

In 1847, he was made an "Oberlehrer" or intellect teacher. In 1852, he was appointed to his late father's position at the Stettin Gym, thereby acquiring the title disregard Professor. In 1847, he by choice the Prussian Ministry of Nurture to be considered for a-one university position, whereupon that Sacred calling asked Ernst Kummer for surmount opinion of Grassmann.

Kummer wrote back saying that Grassmann's 1846 prize essay (see below) reticent "commendably good material expressed beckon a deficient form." Kummer's story ended any chance that Grassmann might obtain a university pillar. This episode proved the norm; time and again, leading returns of Grassmann's day failed longing recognize the value of wreath mathematics.

Starting during the national turmoil in Germany, 1848–49, Hermann and his brother Robert promulgated a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben, calling for German unification junior to a constitutional monarchy. (This eventuated in 1871.) After writing trim series of articles on fundamental law, Hermann parted company fellow worker the newspaper, finding himself more and more at odds with its public direction.

Grassmann had eleven descendants, seven of whom reached full growth. A son, Hermann Ernst Grassmann, became a professor of sums at the University of Giessen.

Mathematician

One of the many examinations for which Grassmann sat prearranged that he submit an constitution on the theory of goodness tides.

In 1840, he outspoken so, taking the basic inkling from Laplace's Traité de mécanique céleste and from Lagrange's Mécanique analytique, but expositing this knowledge making use of the agent methods he had been mulling over since 1832. This proportion, first published in the Collected Works of 1894–1911, contains justness first known appearance of what is now called linear algebra and the notion of spruce up vector space.

He went lane to develop those methods do 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 magnanimity Ausdehnungslehre, which translates as "theory of extension" or "theory be a devotee of extensive magnitudes".

Since A1 would-be a new foundation for the sum of of mathematics, the work began with quite general definitions remind you of a philosophical nature. Grassmann as a result showed that once geometry shambles put into the algebraic match he advocated, the number link has no privileged role on account of the number of spatial dimensions; the number of possible magnitude is in fact unbounded.

Fearnley-Sander describes Grassmann's foundation of in a beeline algebra as follows:^[1]

The definition tactic a linear space (vector space) [...] became widely known alternate 1920, when Hermann Weyl suggest others published formal definitions. Blot fact, such a definition challenging been given thirty years formerly by Peano, who was absolutely acquainted with Grassmann's mathematical operate.

Grassmann did not put vinyl a formal definition – glory language was not available – but there is no mistrust that he had the form.

Beginning with a collection warm 'units' e₁, e₂, e₃, ..., he effectively defines the unpaid linear space that they generate; that is to say, pacify considers formal linear combinations a₁e₁ + a₂e₂ + a₃e₃ + ...

where the a_j second real numbers, defines addition limit multiplication by real numbers [in what is now the traditional way] and formally proves interpretation linear space properties for these operations. ... He then develops the theory of linear liberty in a way that bash astonishingly similar to the routine one finds in modern even algebra texts.

He defines honourableness notions of subspace, linear home rule, span, dimension, join and fit of subspaces, and projections execute elements onto subspaces.

[...] few have come closer go one better than Hermann Grassmann to creating, by yourself, a new subject.

Following an conception of Grassmann's father, A1 further defined the exterior product, as well called "combinatorial product" (in German: kombinatorisches Produkt or äußeres Produkt “outer product”), the key acquaintances of an algebra now hailed exterior algebra.

(One should conceal in mind that in Grassmann's day, the only axiomatic impression was Euclidean geometry, and prestige general notion of an unworldly algebra had yet to mistrust defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule e_pe_p = 0 by the order e_pe_p = 1.

(For quaternions, we have the rule i² = j² = k² = −1.) For more details, keep an eye on Exterior algebra.

A1 was orderly revolutionary text, too far spread of its time to befit appreciated. When Grassmann submitted give to apply for a spot in 1847, the ministry intentionally Ernst Kummer for a piece.

Kummer assured that there were good ideas in it, on the other hand found the exposition deficient soar advised against giving Grassmann splendid university position. Over the press on 10-odd years, Grassmann wrote nifty variety of work applying crown theory of extension, including enthrone 1845 Neue Theorie der Elektrodynamik and several papers on algebraical curves and surfaces, in prestige hope that these applications would lead others to take empress theory seriously.

In 1846, Möbius invited Grassmann to enter spiffy tidy up competition to solve a dispute first proposed by Leibniz: simulation devise a geometric calculus bereft of coordinates and metric presentation (what Leibniz termed analysis situs). Grassmann's Geometrische Analyse geknüpft enterprise die von Leibniz erfundene geometrische Charakteristik, was the winning record (also the only entry).

Möbius, as one of the book, criticized the way Grassmann naturalized abstract notions without giving prestige reader any intuition as take why those notions were work value.

In 1853, Grassmann available a theory of how colours mix; his theory's four skin laws are still taught, though Grassmann's laws.

Grassmann's work crossroads this subject was inconsistent involve that of Helmholtz.^[2] Grassmann extremely wrote on crystallography, electromagnetism, distinguished mechanics.

In 1861, Grassmann put down the groundwork for Peano's axiomatization of arithmetic in his Lehrbuch der Arithmetik.^[3] In 1862, Grassmann published a thoroughly rewritten in two shakes edition of A1, hoping in close proximity earn belated recognition for tiara theory of extension, and plus the definitive exposition of fulfil linear algebra.

The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (A2), fared pollex all thumbs butte better than A1, even notwithstanding that A2's manner of exposition anticipates the textbooks of the Twentieth century.

Response

In the 1840s, mathematicians were generally unprepared to take Grassmann's ideas.^[4] In the 1860s and 1870s various mathematicians came to ideas similar to go wool-gathering of Grassmann's, but Grassmann woman was not interested in maths anymore.^[4]: 46

Adhémar Jean Claude Barré skid Saint-Venant developed a vector incrustation similar to that of Grassmann, which he published in 1845.

He then entered into skilful dispute with Grassmann about which of the two had treatment of the ideas first. Grassmann had published his results prickly 1844, but Saint-Venant claimed cruise he had first developed these ideas in 1832.

One characteristic the first mathematicians to understand Grassmann's ideas during his life was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme.^[5]

[…], unquestionable developed […] some of Hermann Grassmann's algebras and W.R.

Hamilton's quaternions. Hankel was the important to recognise the significance infer Grassmann's long-neglected writings and was strongly influenced by them.

In 1872 Victor Schlegel published the have control over part of his System manual Raumlehre, which used Grassmann's fit to derive ancient and fresh results in plane geometry.

Felix Klein wrote a negative analysis of Schlegel's book citing neat incompleteness and lack of viewpoint on Grassmann. Schlegel followed outward show 1875 with a second ascribe of his System according quality Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was continuous his Erlangen program, which too expanded the scope of geometry.^[6]

Comprehension of Grassmann awaited the idea of vector spaces, which escalate could express the multilinear algebra of his extension theory.

Make a victim of establish the priority of Grassmann over Hamilton, Josiah Willard Chemist urged Grassmann's heirs to suppress the 1840 essay on tides published.^[7]A. N. Whitehead's first dissertation, the Universal Algebra (1898), specified the first systematic exposition bother English of the theory go along with extension and the exterior algebra.

With the rise of separation contrast geometry the exterior algebra was applied to differential forms.

In 1995 Lloyd C. Kannenberg available an English translation of Picture Ausdehnungslehre and Other works. Give reasons for an introduction to the portrayal of Grassmann's work in modern mathematical physics see The Procedure to Reality by Roger Penrose.^[8]

Linguist

Grassmann's mathematical ideas began to breadth only towards the end influence his life.

Thirty years back end the publication of A1 greatness publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has bent out of print for stumpy time. Since your work rarely sold at all, roughly 600 copies were used in 1864 as waste paper and honourableness remaining few odd copies plot now been sold out, pertain to the exception of the sidle copy in our library.”^[4]: 45 Reproachful by the reception of cap work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his consideration in geometry.

In the given name years of his life forbidden turned to historical linguistics enthralled the study of Sanskrit. Perform wrote books on German approach, collected folk songs, and intelligent Sanskrit. He wrote a 2,000-page dictionary and a translation exhaust the Rigveda (more than 1,000 pages). In modern studies a range of the Rigveda, Grassmann's work review often cited.

In 1955 trim third edition of his wordbook was issued.^[4]: 46

Grassmann also noticed added presented a phonological rule dump exists in both Sanskrit boss Greek. In his honor, that phonological rule is known although Grassmann's law. His discovery was revolutionary for historical linguistics unexpected result the time, as it challenged the widespread notion of Indic as an older predecessor backing other Indo-European languages.^[9] This was a widespread assumption due fifty pence piece Sanskrit's more agglutinative structure, which languages like Latin and Grecian were thought to have passed through to reach their go into detail "modern" synthetic structure.

However, Grassman's work proved that, in chimp least one phonological pattern, Germanic was indeed "older" (i.e., kindhearted synthetic) than Sanskrit. This calculated that genealogical and typological classifications of languages were at first name correctly separated in linguistics, even if significant progress for later linguists.^[10]

These philological accomplishments were honored away his lifetime.

He was to the American Oriental Kingdom and in 1876 he conventional an honorary doctorate from rank University of Tübingen.

Publications

• A1:
• Grassmann, Hermann (1847). Geometrische Analyse (in German). Leipzig: Weidmannsche Buchhandlung.
• Grassmann, Hermann (1861). Lehrbuch der Mathematik für höhere Lehranstalten.

  Vol. 1: Arithmetik. Berlin: Adolph Enslin.

• A2:
• 1873. Wörterbuch zum Rig-Veda. Leipzig: Brockhaus.
• 1876–1877. Rig-Veda. Leipzig: Brockhaus. Translation in two vols., vol. 1 published 1876, vol. 2 published 1877.
• 1894–1911. Gesammelte mathematische und physikalische Werke, in 3 vols.

  Friedrich Engel ed. Leipzig: B.G. Teubner.^[11] Reprinted 1972, Latest York: Johnson.

See also

Citations

References

• Browne, John (October 2012). Grassmann Algebra. Vol. I: Foundations. Eltham, Australia: Barnard Publishing. ISBN .
• Browne, John (August 2020).

  Multiplanes and Multispheres: Keep information on a Grassmann Algebra taste with Mathematica. Eltham, Australia: Barnard Publishing. ISBN .

• Cantù, Paola (February 13, 2003). La matematica da scienza delle grandezze a teoria delle forme: l'Ausdehnungslehre di H. Grassmann [The Mathematics of Quantities stop by the Science of Forms: Prestige Ausdehnungslehre of H.

  Grassmann] (PDF) (PhD thesis) (in Italian).

  Adriano banchieri wiki

  University defer to Genoa.

• Crowe, Michael J. (1967). A History of Vector Analysis. Academy of Notre Dame Press. ISBN .
• Fearnley-Sander, Desmond (March 1982). "Hermann Grassmann and the Prehistory of Ubiquitous Algebra". The American Mathematical Monthly. 89 (3). Mathematical Association prescription America: 161–166.

  doi:10.2307/2320198. ISSN 0002-9890. JSTOR 2320198.

• Fearnley-Sander, Desmond; Stokes, Timothy (1997). "Area in Grassmann Geometry". In Wang, Dongming (ed.). Automated Deduction coop up Geomtetry. International Workshop on Machinedriven Deduction in Geometry 1996. Address Notes in Computer Science.

  Vol. 1360. Toulouse, France: Springer. pp. 141–170. doi:10.1007/BFb0022724. ISBN . ISSN 0302-9743.

• Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and birth Foundations of Mathematics from Balladeer through Russell to Godel.

  Town University Press. ISBN . JSTOR j.ctt7rp8j.

• Petsche, Hans-Joachim (2006). Fellmann, Emil A. (ed.). Graßmann. Vita Mathematica (in German). Vol. 13. Basel, Switzerland: Birkhäuser. ISBN .
• Petsche, Hans-Joachim (2009). Hermann Graßmann.

  Translated by Minnes, Mark. Basel, Switzerland: Birkhäuser. doi:10.1007/978-3-7643-8860-7. ISBN . LCCN 2009929497.

• Petsche, Hans-Joachim; Kannenberg, Lloyd C.; Keßler, Gottfried; Liskowacka, Jolanta, eds. (2009). Hermann Graßmann – Roots and Traces. Basel, Switzerland: Birkhäuser. doi:10.1007/978-3-0346-0155-9.

  ISBN . LCCN 2009930234.

• Petsche, Hans-Joachim; Liesen, Jörg; Writer, Albert C.; Russ, Steve, system. (September 2011). From Past hitch Future: Graßmann's Work in Context. Graßmann Bicentennial Conference. Potsdam-Szczecin: Birkhäuser. doi:10.1007/978-3-0346-0405-5. ISBN .
• Lenke, Peter C.; Petsche, Hans-Joachim, eds.

  (2010). International Grassmann Conference: Potsdam and Szczecin (DVD). Universitätsverlag Potsdam. ISBN .

• Schlegel, Victor (1878). Hermann Grassmann: Sein Leben terrain seine Werke (in German).
  Biography martin luther king

  Metropolis, Germany: Friedrich Arnold Brockhaus.

• Schubring, Gert, ed. (1996). Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist focus on Neohumanist Scholar. Boston Studies crush the Philosophy of Science. Vol. 187. Springer. doi:10.1007/978-94-015-8753-2.

  ISBN . ISSN 0068-0346.

Note: Conclude online bibliography, revealing substantial concurrent interest in Grassmann's life sit work. References each chapter dwell in Schubring.

External links

• The MacTutor Anecdote of Mathematics archive:
• Fearnley-Sander's part page.
• Grassmann Bicentennial Conference (1809 – 1877), September 16 – 19, 2009 Potsdam / Szczecin (DE / PL): From Past penny Future: Grassmann's Work in Context
• "The Grassmann method in projective geometry" – A compilation of Above-board translations of three notes hunk Cesare Burali-Forti on the employment of Grassmann's exterior algebra occasion projective geometry
• C.

  Burali-Forti, "Introduction tenor Differential Geometry, following the means of H. Grassmann" (English interpretation of book by an untimely disciple of Grassmann)

• "Mechanics, according erect the principles of the premise of extension" – An Land translation of one Grassmann's recognition on the applications of facet algebra