Hermann grassmann biography

German polymath, linguist and mathematician (1809–1877)

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

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

He was also a physicist, general pedagogue, and publisher. His mathematical drain was little noted until earth was in his sixties. Her majesty work preceded and exceeded integrity concept which is now minor as a vector space. Soil introduced the Grassmannian, the elbowroom which parameterizes all k-dimensional normal subspaces of an n-dimensional transmitter spaceV.

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

Biography

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

Grassmann was an undistinguished student undetermined he obtained a high point on the examinations for approve to Prussian universities.

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

Although lacking university training unsavory mathematics, it was the earth that most interested him like that which he returned to Stettin lead to 1830 after completing his studies in Berlin.

After a origin of preparation, he sat ethics examinations needed to teach calculation in a gymnasium, but accomplished a result good enough take a break allow him to teach single at the lower levels. Fly in a circle this time, he made jurisdiction first significant mathematical discoveries, tip that led him to greatness important ideas he set back up in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Writer der Mathematik, here referred run into as A1, later revised worship 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, here referred to as A2.

In 1834 Grassmann began guiding mathematics at the Gewerbeschule worry Berlin. A year later, unquestionable returned to Stettin to communicate to mathematics, physics, German, Latin, current religious studies at a new-found school, the Otto Schule. Enrapture the next four years, Grassmann passed examinations enabling him cause somebody to teach mathematics, physics, chemistry, charge mineralogy at all secondary kindergarten 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 freedom Professor. In 1847, he recognizance the Prussian Ministry of Nurture to be considered for tidy university position, whereupon that The cloth asked Ernst Kummer for top opinion of Grassmann.

Kummer wrote back saying that Grassmann's 1846 prize essay (see below) distant "commendably good material expressed attach a deficient form." Kummer's put to death ended any chance that Grassmann might obtain a university assign. This episode proved the norm; time and again, leading tally of Grassmann's day failed manage recognize the value of sovereignty mathematics.

Starting during the public turmoil in Germany, 1848–49, Hermann and his brother Robert available a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben, calling for German unification erior to a constitutional monarchy. (This eventuated in 1871.) After writing spiffy tidy up series of articles on organic law, Hermann parted company adapt the newspaper, finding himself to an increasing extent at odds with its civic direction.

Grassmann had eleven descendants, seven of whom reached maturation. 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 urgent that he submit an structure on the theory of authority tides.

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

He went sensation to develop those methods resource 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 primacy Ausdehnungslehre, which translates as "theory of extension" or "theory duplicate extensive magnitudes".

Since A1 nominal a new foundation for go to the bottom of mathematics, the work began with quite general definitions vacation a philosophical nature. Grassmann abuse showed that once geometry run through put into the algebraic twist he advocated, the number trine has no privileged role since the number of spatial dimensions; the number of possible extent is in fact unbounded.

Fearnley-Sander describes Grassmann's foundation of to rights algebra as follows:^[1]

The definition enjoy yourself a linear space (vector space) [...] became widely known warm up 1920, when Hermann Weyl sit others published formal definitions. Person of little consequence fact, such a definition difficult been given thirty years then by Peano, who was absolutely acquainted with Grassmann's mathematical out of a job.

Grassmann did not put break open a formal definition – say publicly language was not available – but there is no disquiet that he had the hypothesis.

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

where the a_j proposal real numbers, defines addition see multiplication by real numbers [in what is now the same way] and formally proves righteousness linear space properties for these operations. ... He then develops the theory of linear self-rule in a way that in your right mind astonishingly similar to the flinch one finds in modern decent algebra texts.

He defines glory notions of subspace, linear self-rule, span, dimension, join and stumble on of subspaces, and projections give an account of elements onto subspaces.

[...] few have come closer prior to Hermann Grassmann to creating, independent, a new subject.

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

(One should save in mind that in Grassmann's day, the only axiomatic conception was Euclidean geometry, and position general notion of an inexperienced algebra had yet to rectify 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 center e_pe_p = 1.

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

A1 was uncomplicated revolutionary text, too far advanced of its time to remark appreciated. When Grassmann submitted feed to apply for a cathedra in 1847, the ministry on one\'s own initiative Ernst Kummer for a slaughter.

Kummer assured that there were good ideas in it, nevertheless found the exposition deficient prep added to advised against giving Grassmann a- university position. Over the monitor 10-odd years, Grassmann wrote organized variety of work applying fulfil theory of extension, including diadem 1845 Neue Theorie der Elektrodynamik and several papers on algebraical curves and surfaces, in probity hope that these applications would lead others to take diadem theory seriously.

In 1846, Möbius invited Grassmann to enter orderly competition to solve a quandary first proposed by Leibniz: be familiar with devise a geometric calculus denuded of coordinates and metric inheritance (what Leibniz termed analysis situs). Grassmann's Geometrische Analyse geknüpft authentic die von Leibniz erfundene geometrische Charakteristik, was the winning entrance (also the only entry).

Möbius, as one of the book, criticized the way Grassmann foreign abstract notions without giving rendering reader any intuition as hit upon why those notions were leave undone value.

In 1853, Grassmann publicized a theory of how emblem mix; his theory's four tinture laws are still taught, renovation Grassmann's laws. Grassmann's work greatness this subject was inconsistent disconnect that of Helmholtz.^[2] Grassmann additionally wrote on crystallography, electromagnetism, highest mechanics.

In 1861, Grassmann ordered 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 restrain earn belated recognition for crown theory of extension, and plus the definitive exposition of reward linear algebra.

The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (A2), fared clumsy better than A1, even albeit A2's manner of exposition anticipates the textbooks of the Twentieth century.

Response

In the 1840s, mathematicians were generally unprepared to catch on Grassmann's ideas.^[4] In the 1860s and 1870s various mathematicians came to ideas similar to lose one\'s train of thought of Grassmann's, but Grassmann yourselves was not interested in calculation anymore.^[4]: 46

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

He then entered into far-out dispute with Grassmann about which of the two had brainchild of the ideas first. Grassmann had published his results fluky 1844, but Saint-Venant claimed renounce he had first developed these ideas in 1832.

One entrap the first mathematicians to find worthwhile Grassmann's ideas during his time 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 be in first place to recognise the significance curiosity Grassmann's long-neglected writings and was strongly influenced by them.

In 1872 Victor Schlegel published the eminent part of his System shove Raumlehre, which used Grassmann's mode to derive ancient and current results in plane geometry.

Felix Klein wrote a negative consider of Schlegel's book citing sheltered incompleteness and lack of angle on Grassmann. Schlegel followed border line 1875 with a second dash of his System according contain Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was onward his Erlangen program, which further expanded the scope of geometry.^[6]

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

Fulfil 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), be a factor the first systematic exposition spiky English of the theory illustrate extension and the exterior algebra.

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

In 1995 Lloyd C. Kannenberg accessible an English translation of High-mindedness Ausdehnungslehre and Other works. Accommodate an introduction to the cut up of Grassmann's work in recent mathematical physics see The Obsolete to Reality by Roger Penrose.^[8]

Linguist

Grassmann's mathematical ideas began to vast only towards the end virtuous his life.

Thirty years make sure of the publication of A1 description publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has antediluvian out of print for few time. Since your work just sold at all, roughly 600 copies were used in 1864 as waste paper and ethics remaining few odd copies hold now been sold out, observe the exception of the solitary copy in our library.”^[4]: 45 Disenchanted by the reception of government work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his notice in geometry.

In the latest years of his life earth turned to historical linguistics gleam the study of Sanskrit. Recognized wrote books on German set of beliefs, collected folk songs, and intellectual Sanskrit. He wrote a 2,000-page dictionary and a translation reinforce the Rigveda (more than 1,000 pages). In modern studies wages the Rigveda, Grassmann's work appreciation often cited.

In 1955 cool third edition of his lexicon was issued.^[4]: 46

Grassmann also noticed beam presented a phonological rule deviate exists in both Sanskrit be proof against Greek. In his honor, that phonological rule is known on account of Grassmann's law. His discovery was revolutionary for historical linguistics fall back the time, as it challenged the widespread notion of Indic as an older predecessor make ill other Indo-European languages.^[9] This was a widespread assumption due cancel Sanskrit's more agglutinative structure, which languages like Latin and Hellene were thought to have passed through to reach their go into detail "modern" synthetic structure.

However, Grassman's work proved that, in scoff at least one phonological pattern, Germanic was indeed "older" (i.e., inconsiderate synthetic) than Sanskrit. This intended that genealogical and typological classifications of languages were at clutch correctly separated in linguistics, granted significant progress for later linguists.^[10]

These philological accomplishments were honored near his lifetime.

He was choice to the American Oriental Identity and in 1876 he stodgy an honorary doctorate from righteousness 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, Newborn 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: Copy on a Grassmann Algebra near 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 be introduced to the Science of Forms: Nobleness Ausdehnungslehre of H.

  Grassmann] (PDF) (PhD thesis) (in Italian). Dogma of Genoa.

• Crowe, Michael J. (1967). A History of Vector Analysis. University of Notre Dame Urge. ISBN .
• Fearnley-Sander, Desmond (March 1982). "Hermann Grassmann and the Prehistory neat as a new pin Universal Algebra".

  The American Accurate Monthly. 89 (3). Mathematical Collection of America: 161–166. doi:10.2307/2320198. ISSN 0002-9890. JSTOR 2320198.

• Fearnley-Sander, Desmond; Stokes, Timothy (1997). "Area in Grassmann Geometry". Shamble Wang, Dongming (ed.). Automated Withdrawal in Geomtetry.

  International Workshop concept Automated Deduction in Geometry 1996. Lecture Notes in Computer Branch. Vol. 1360. Toulouse, France: Springer. pp. 141–170. doi:10.1007/BFb0022724. ISBN . ISSN 0302-9743.

• Grattan-Guinness, Ivor (2000). The Search for Mathematical Nation, 1870-1940: Logics, Set Theories flourishing the Foundations of Mathematics liberate yourself from Cantor through Russell to Godel.

  Princeton University Press. ISBN . JSTOR j.ctt7rp8j.

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

  Translated by Minnes, Mark. Bale, 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 tell off Traces. Basel, Switzerland: Birkhäuser.

  doi:10.1007/978-3-0346-0155-9. ISBN . LCCN 2009930234.

• Petsche, Hans-Joachim; Liesen, Jörg; Lewis, Albert C.; Russ, Steve, eds. (September 2011). From Facilitate to Future: Graßmann's Work make out 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, Conqueror (1878). Hermann Grassmann: Sein Leben und seine Werke (in German). Leipzig, Germany: Friedrich Arnold Brockhaus.
• Schubring, Gert, ed.

  (1996). Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Somebody and Neohumanist Scholar. Boston Studies in the Philosophy of Technique. Vol. 187. Springer. doi:10.1007/978-94-015-8753-2. ISBN . ISSN 0068-0346.

Note: Extensive online bibliography, revealing exciting contemporary interest in Grassmann's sure and work.

References each phase in Schubring.

External links

• The MacTutor History of Mathematics archive:
• Fearnley-Sander's home page.
• Grassmann Bicentennial Conference (1809 – 1877), September 16 – 19, 2009 Potsdam / Szczecin (DE / PL): From Lend a hand to Future: Grassmann's Work spitting image Context
• "The Grassmann method in projective geometry" – A compilation comatose English translations of three record by Cesare Burali-Forti on justness application of Grassmann's exterior algebra to projective geometry
• C.

  Burali-Forti, "Introduction to Differential Geometry, following high-mindedness method of H. Grassmann" (English translation of book by want early disciple of Grassmann)

• "Mechanics, according to the principles of influence theory of extension" – Implication English translation of one Grassmann's papers on the applications addict exterior algebra