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The British development of the theory of invariants (1841–1895). (English) Zbl 1113.01013

The idea of an ‘invariant’ was given in an 1841 paper by George Boole. In the years especially 1850 to 1856 Arthur Cayley and James Joseph Sylvester laid the groundwork for the British theory of invariants; they then went on to develop it on and off throughout the rest of their lives. In 1845 Cayley showed that the invariants of binary forms satisfy an algebraic dependence relation (or syzygy as Sylvester would later dub them). Sylvester called also some invariants as discriminant and Hessian in 1850–51. He spent the summer of 1851 in regular communication with Cayley and single-mindedly pursuing the whole constellation of ideas and techniques that a theory of invariants would necessary involve. A letter from Cayley of 5 December will constitute the foundation of a new theory of Invariants’.
In the present paper this theory of Sylvester and Cayley is shortly described for binary \(2n\)-ic and \(n\)-ic forms by introducing the catalecticant. They drew into their newly emerging mathematical world the Irish mathematician, George Salmon, who gave its first systematic treatment in 1859 in his book. In Europe Paul Gordan proved that Cayley had been in error in his work on the binary quintic. He was able in 1868 to prove that given a binary form, a minimum generating set of invariants for that form is always finite. Cayley began the rehabilitation process in 1871. It is described how the young David Hilbert entered with his papers of 1889-96 the invariant-theoretic fray and developed the ‘naive’ and ‘formal’ phases to the ‘critical’ phase.

MSC:

01A55 History of mathematics in the 19th century
13-03 History of commutative algebra
13A50 Actions of groups on commutative rings; invariant theory
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[1] Philosophical Magazine 18 pp 67– (1859)
[2] Boole George, Cambridge Mathematical Journal 3 (1841)
[3] Cayley Arthur, Cambridge Mathematical Journal 4 pp 193– (1845)
[4] Cayley Arthur, Cambridge and Dublin Mathematical Journal 1 pp 104– (1846)
[5] DOI: 10.1098/rstl.1854.0010 · doi:10.1098/rstl.1854.0010
[6] DOI: 10.1098/rstl.1856.0008 · doi:10.1098/rstl.1856.0008
[7] DOI: 10.1098/rstl.1871.0003 · JFM 03.0040.01 · doi:10.1098/rstl.1871.0003
[8] DOI: 10.1007/BF01453440 · JFM 21.0104.01 · doi:10.1007/BF01453440
[9] Cayley Arthur, The collected mathematical papers of Arthur Cayley 14 (1889)
[10] Crilly Tony, unpublished doctoral dissertation (1981)
[11] DOI: 10.1016/0315-0860(86)90091-1 · Zbl 0631.01016 · doi:10.1016/0315-0860(86)90091-1
[12] DOI: 10.1016/0315-0860(88)90025-0 · Zbl 0662.01010 · doi:10.1016/0315-0860(88)90025-0
[13] DOI: 10.1016/j.hm.2004.03.001 · Zbl 1062.01011 · doi:10.1016/j.hm.2004.03.001
[14] Crilly Tony, Arthur Cayley: mathematician laureate of the Victorian age (2005)
[15] Despeaux Sloan Evans, unpublished doctoral dissertation (2002)
[16] Elliott Edwin Bailey, An introduction to the algebra of quantics (1895)
[17] DOI: 10.1515/crll.1868.69.323 · doi:10.1515/crll.1868.69.323
[18] DOI: 10.1515/crll.1844.28.68 · ERAM 028.0817cj · doi:10.1515/crll.1844.28.68
[19] Hilbert David, Mathematische Annalen 34 pp 223– (1889)
[20] DOI: 10.1007/BF01208503 · JFM 22.0133.01 · doi:10.1007/BF01208503
[21] DOI: 10.1007/BF01444162 · JFM 25.0173.01 · doi:10.1007/BF01444162
[22] Hilbert David, Mathematical papers read at the International Mathematical Congress held in connection with the World’s Columbian Exposition, Chicago 1893 pp 116– (1896)
[23] DOI: 10.1007/BF00327101 · Zbl 0232.01006 · doi:10.1007/BF00327101
[24] Meyer W Franz, Sur les progrés de la théorie des invariants projectifs (1897)
[25] Parshall Karen, Archive for History of Exact Sciences 38 pp 153– (1988)
[26] Parshall Karen, The History of Modern Mathematics 2 (1989)
[27] Parshall Karen, James Joseph Sylvester: life and work in letters (1998) · Zbl 0929.01018
[28] Parshall Karen, Nieuw Archief voor Wiskunde 17 pp 247– (1999)
[29] Parshall Karen Hunger, James Joseph Sylvester: Jewish mathematician in a Victorian world (2006) · Zbl 1100.01006
[30] Parshall Karen, The emergence of the American mathematical research community, 1876–1900: (1994)
[31] Salmon George, A treatise on conic sections, containing an account of some of the most important modern algebraic and geometric methods, 1. ed. (1848)
[32] Salmon George, Lessons introductory to the modern higher algebra (1859)
[33] Sylvester JJ, to Arthur Cayley , 3 February 1850, Sylvester Papers (1850)
[34] Sylvester JJ, Cambridge and Dublin Mathematical Journal 5 pp 262– (1850)
[35] Sylvester JJ, Cambridge and Dublin Mathematical Journal 6 pp 186– (1851)
[36] Sylvester JJ, Cambridge and Dublin Mathematical Journal 6 pp 289– (1851)
[37] Sylvester JJ, Cambridge and Dublin 6 pp 1– (1851)
[38] Sylvester JJ, Cambridge and Dublin Mathematical Journal 7 (1852)
[39] Sylvester JJ, Philosophical Magazine 4 pp 138– (1852)
[40] DOI: 10.1098/rstl.1853.0018 · doi:10.1098/rstl.1853.0018
[41] Sylvester JJ, The laws of verse or principles of versification exemplified in metrical translations: together with an annotated reprint of the inaugural presidential address to the Mathematical and Physical Section of the British Association at Exeter (1870)
[42] DOI: 10.2307/2369240 · JFM 11.0082.02 · doi:10.2307/2369240
[43] Sylvester JJ, The collected mathematical papers of James Joseph Sylvester 4 (1904)
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