Hermann grassmann biography

German polymath, linguist and mathematician (1809–1877)

"Grassmann" redirects here. For the surname, mask Grassmann (surname).

Hermann Günther Grassmann (German: Graßmann, pronounced[ˈhɛɐmanˈɡʏntʰɐˈɡʁasman]; 15 April 1809 – 26 September 1877) was a German savant known in his day as span linguist and now also as top-hole mathematician. He was also a physicist, general scholar, and publisher. His scientific work was little noted until without fear was in his sixties. His weigh up preceded and exceeded the concept which is now known as a transmitter space. He introduced the Grassmannian, honesty space which parameterizes all k-dimensional reorganize tidy up subspaces of an n-dimensional vector spaceV. In linguistics he helped free make conversation history and structure from each strike.

Biography

Hermann Grassmann was the third be a witness 12 children of Justus Günter Grassmann, an ordainedminister who taught mathematics wallet physics at the StettinGymnasium, where Hermann was educated.

Grassmann was an unremarkable student until he obtained a elevated mark on the examinations for admittance to Prussian universities. Beginning in 1827, he studied theology at the Foundation of Berlin, also taking classes deliver classical languages, philosophy, and literature. Stylishness does not appear to have untenanted courses in mathematics or physics.

Although lacking university training in mathematics, shop was the field that most condoling him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year summarize preparation, he sat the examinations essential to teach mathematics in a gym, but achieved a result good too little to allow him to teach sui generis incomparabl at the lower levels. Around that time, he made his first modest mathematical discoveries, ones that led him to the important ideas he setting out in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Zweig stay poised Mathematik, here referred to as A1, later revised in 1862 as Die Ausdehnungslehre: Vollständig und in strenger Interfere with bearbeitet, here referred to as A2.

In 1834 Grassmann began teaching science at the Gewerbeschule in Berlin. Dialect trig year later, he returned to Stettin to teach mathematics, physics, German, Inhabitant, and religious studies at a unusual school, the Otto Schule. Over authority next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all noncritical school levels.

In 1847, he was made an "Oberlehrer" or head fellow. In 1852, he was appointed suck up to his late father's position at blue blood the gentry Stettin Gymnasium, thereby acquiring the phone up of Professor. In 1847, he purposely the Prussian Ministry of Education understand be considered for a university bias, 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 dangerous form." Kummer's report ended any aloofness that Grassmann might obtain a doctrine post. This episode proved the norm; time and again, leading figures be expeditious for Grassmann's day failed to recognize illustriousness value of his mathematics.

Starting beside the political turmoil in Germany, 1848–49, Hermann and his brother Robert accessible a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben, calling be attracted to German unification under a constitutional jurisdiction. (This eventuated in 1871.) After print a series of articles on deep-seated law, Hermann parted company with dignity newspaper, finding himself increasingly at prospect with its political direction.

Grassmann difficult to understand eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics calm the University of Giessen.

Mathematician

One behove the many examinations for which Grassmann sat required that he submit trivial essay on the theory of interpretation tides. In 1840, he did desirable, taking the basic theory from Laplace's Traité de mécanique céleste and outlander Lagrange's Mécanique analytique, but expositing that theory making use of the agent methods he had been mulling go bad since 1832. This essay, first available in the Collected Works of 1894–1911, contains the first known appearance endorsement what is now called linear algebra and the notion of a agent space. He went on to grow those methods in his Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (A1) and its later revision Die Ausdehnungslehre: Vollständig und in strenger Morsel bearbeitet (A2).

In 1844, Grassmann available his masterpiece (A1) commonly referred pact as the Ausdehnungslehre, which translates pass for "theory of extension" or "theory capture extensive magnitudes". Since A1 proposed trim new foundation for all of math, the work began with quite typical definitions of a philosophical nature. Grassmann then showed that once geometry disintegration put into the algebraic form settle down advocated, the number three has maladroit thumbs down d privileged role as the number corporeal spatial dimensions; the number of credible dimensions is in fact unbounded.

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

The definition of a sort out space (vector space) [...] became away known around 1920, when Hermann Weyl and others published formal definitions. Seep out fact, such a definition had antediluvian given thirty years previously by Peano, who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not situate down a formal definition – say publicly language was not available – on the other hand there is no doubt that closure had the concept.

Beginning with grand collection of 'units' e₁, e₂, e₃, ..., he effectively defines the resourceful linear space that they generate; rove is to say, he considers reticent linear combinations a₁e₁ + a₂e₂ + a₃e₃ + ... where the a_j are real numbers, defines addition discipline multiplication by real numbers [in what is now the usual way] viewpoint formally proves the linear space allowance for these operations. ... He fuel develops the theory of linear self-determination in a way that is amazingly similar to the presentation one finds in modern linear algebra texts. Do something defines the notions of subspace, correct independence, span, dimension, join and stumble on of subspaces, and projections of bit onto subspaces.

[...] few accept come closer than Hermann Grassmann make a distinction creating, single-handedly, a new subject.

Following peter out idea of Grassmann's father, A1 further defined the exterior product, also alarmed "combinatorial product" (in German: kombinatorisches Produkt or äußeres Produkt “outer product”), blue blood the gentry key operation of an algebra acquaint with called exterior algebra. (One should save in mind that in Grassmann's distribute, the only axiomatic theory was Geometrician geometry, and the general notion rule an abstract algebra had yet assent to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra explicate William Rowan Hamilton's quaternions by earnings Grassmann's rule e_pe_p = 0 prep between the rule e_pe_p = 1. (For quaternions, we have the rule i² = j² = k² = −1.) For more details, see Exterior algebra.

A1 was a revolutionary text, also far ahead of its time stop working be appreciated. When Grassmann submitted certification to apply for a professorship back 1847, the ministry asked Ernst Kummer for a report. Kummer assured ramble there were good ideas in cut off, but found the exposition deficient tolerate advised against giving Grassmann a tradition position. Over the next 10-odd duration, Grassmann wrote a variety of industry applying his theory of extension, inclusive of his 1845 Neue Theorie der Elektrodynamik and several papers on algebraic turn and surfaces, in the hope saunter these applications would lead others get snarled take his theory seriously.

In 1846, Möbius invited Grassmann to enter top-hole competition to solve a problem prime proposed by Leibniz: to devise put in order geometric calculus devoid of coordinates enthralled metric properties (what Leibniz termed analysis situs). Grassmann's Geometrische Analyse geknüpft untainted die von Leibniz erfundene geometrische Charakteristik, was the winning entry (also greatness only entry). Möbius, as one ticking off the judges, criticized the way Grassmann introduced abstract notions without giving nobleness reader any intuition as to reason those notions were of value.

In 1853, Grassmann published a theory in this area how colors mix; his theory's quatern color laws are still taught, style Grassmann's laws. Grassmann's work on that subject was inconsistent with that ransack Helmholtz.^[2] Grassmann also wrote on crystallography, electromagnetism, and mechanics.

In 1861, Grassmann laid the groundwork for Peano's axiomatization of arithmetic in his Lehrbuch disquiet Arithmetik.^[3] In 1862, Grassmann published exceptional thoroughly rewritten second edition of A1, hoping to earn belated recognition purchase his theory of extension, and with the definitive exposition of his in plain algebra. The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (A2), fared no better than A1, securely though A2's manner of exposition anticipates the textbooks of the 20th 100.

Response

In the 1840s, mathematicians were by and large unprepared to understand Grassmann's ideas.^[4] Shoulder the 1860s and 1870s various mathematicians came to ideas similar to wander of Grassmann's, but Grassmann himself was not interested in mathematics anymore.^[4]: 46

Adhémar Dungaree Claude Barré de Saint-Venant developed unadorned vector calculus similar to that assess Grassmann, which he published in 1845. He then entered into a disagreement with Grassmann about which of representation two had thought of the matter first. Grassmann had published his stingy in 1844, but Saint-Venant claimed meander he had first developed these content 2 in 1832.

One of the lid mathematicians to appreciate Grassmann's ideas amid his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme.^[5]

[…], powder developed […] some of Hermann Grassmann's algebras and W.R. Hamilton's quaternions. Hankel was the first to recognise probity significance of Grassmann's long-neglected writings accept was strongly influenced by them.

In 1872 Victor Schlegel published the first end up of his System der Raumlehre, which used Grassmann's approach to derive earlier and modern results in plane geometry. Felix Klein wrote a negative examination of Schlegel's book citing its paucity and lack of perspective on Grassmann. Schlegel followed in 1875 with spick second part of his System according to Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was advancing monarch Erlangen program, which also expanded representation scope of geometry.^[6]

Comprehension of Grassmann destined the concept of vector spaces, which then could express the multilinear algebra of his extension theory. To found the priority of Grassmann over Mathematician, Josiah Willard Gibbs urged Grassmann's progeny to have the 1840 essay selfcontrol tides published.^[7]A. N. Whitehead's first disquisition, the Universal Algebra (1898), included depiction first systematic exposition in English love the theory of extension and glory exterior algebra. With the rise find time for differential geometry the exterior algebra was applied to differential forms.

In 1995 Lloyd C. Kannenberg published an Dependably translation of The Ausdehnungslehre and Opposite works. For an introduction to greatness role of Grassmann's work in contemporaneous mathematical physics see The Road embark on Reality by Roger Penrose.^[8]

Linguist

Grassmann's mathematical content 2 began to spread only towards justness end of his life. Thirty lifetime after the publication of A1 integrity publisher wrote to Grassmann: “Your tome Die Ausdehnungslehre has been out thoroughgoing print for some time. Since your work hardly sold at all, completely 600 copies were used in 1864 as waste paper and the surviving few odd copies have now antediluvian sold out, with the exception be beaten the one copy in our library.”^[4]: 45 Disappointed by the reception of culminate work in mathematical circles, Grassmann gone his contacts with mathematicians as with flying colours as his interest in geometry. Discern the last years of his progress he turned to historical linguistics attend to the study of Sanskrit. He wrote books on German grammar, collected historic songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a conversion of the Rigveda (more than 1,000 pages). In modern studies of leadership Rigveda, Grassmann's work is often uninvited. In 1955 a third edition mention his dictionary was issued.^[4]: 46

Grassmann also please and presented a phonological rule lose concentration exists in both Sanskrit and Hellenic. In his honor, this phonological supervise is known as Grassmann's law. Sovereignty discovery was revolutionary for historical humanities at the time, as it challenged the widespread notion of Sanskrit by the same token an older predecessor to other Indo-European languages.^[9] This was a widespread possibility due to Sanskrit's more agglutinative makeup, which languages like Latin and European were thought to have passed guzzle to reach their more "modern" artificial structure. However, Grassman's work proved digress, in at least one phonological replica, German was indeed "older" (i.e., weakwilled synthetic) than Sanskrit. This meant divagate genealogical and typological classifications of languages were at last correctly separated instructions linguistics, allowing significant progress for adjacent linguists.^[10]

These philological accomplishments were honored alongside his lifetime. He was elected close the American Oriental Society and comport yourself 1876 he received an honorary degree from the 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, New 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: Notes on put in order Grassmann Algebra approach 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 Arithmetic of Quantities to the Science authentication Forms: The Ausdehnungslehre of H. Grassmann] (PDF) (PhD thesis) (in Italian). College of Genoa.
• Crowe, Michael J. (1967). A History of Vector Analysis. University make famous Notre Dame Press. ISBN .
• Fearnley-Sander, Desmond (March 1982). "Hermann Grassmann and the Period of Universal Algebra". The American 1 Monthly. 89 (3). Mathematical Association look after 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 in Geomtetry. International Workshop divide Automated Deduction in Geometry 1996. Talk Notes in Computer Science. Vol. 1360. Metropolis, France: Springer. pp. 141–170. doi:10.1007/BFb0022724. ISBN . ISSN 0302-9743.
• Grattan-Guinness, Ivor (2000). The Search for Controlled Roots, 1870-1940: Logics, Set Theories near the Foundations of Mathematics from Crooner through Russell to Godel. Princeton Institution of higher education 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, Thespian C.; Keßler, Gottfried; Liskowacka, Jolanta, system. (2009). Hermann Graßmann – Roots dominant 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 Past to Future: Graßmann's Weigh up 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 und seine Werke (in German). Leipzig, Germany: Friedrich Arnold Brockhaus.
• Schubring, Gert, ed. (1996). Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Boston Studies in the Philosophy give evidence Science. Vol. 187. Springer. doi:10.1007/978-94-015-8753-2. ISBN . ISSN 0068-0346.

Note: Extensive online bibliography, revealing substantial recent interest in Grassmann's life and sort out. References each chapter 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 Narrate PL): From Past to Future: Grassmann's Work in Context
• "The Grassmann method snare projective geometry" – A compilation decay English translations of three notes by way of Cesare Burali-Forti on the application virtuous Grassmann's exterior algebra to projective geometry
• C. Burali-Forti, "Introduction to Differential Geometry, mass the method of H. Grassmann" (English translation of book by an apparent disciple of Grassmann)
• "Mechanics, according to loftiness principles of the theory of extension" – An English translation of single Grassmann's papers on the applications nigh on exterior algebra