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Original Articles

The British development of the theory of invariants (1841–1895)

Pages 186-199 | Published online: 23 Feb 2007
 

Abstract

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in Citation1859 and later Edwin Bailey Elliott in Citation1895 codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.

Notes

1 For his full-scale biography, see Parshall Citation2006.

2 For his full-scale biography, see Crilly Citation2005.

3 Both had been present at the annual meeting of the British Association for the Advancement of Science in Cambridge in 1845, but there is no evidence that they actually met there.

4 Despeaux (Citation2002, 371–382) gives a nice overview of nineteenth-century British algebraic research in her publication, from which this paragraph draws.

5 On the development of the calculus of operations in Britain and on the numerous contributors to it, see Koppelman Citation1971–1972; this paragraph has drawn from 155–157 and 187–189. Compare also Parshall Citation2006, 97–98. On the Cambridge Mathematical Journal, see Crilly Citation2004.

6In what follows, I have altered Boole's original notation so that it conforms with that later adopted by Sylvester and Cayley. The technical, invariant-theoretic discussions that follow draw from and parallel the expositions I have given previously in Parshall Citation1988, Citation1989, I, 157–206, Citation1998, Citation1999, Citation2006 and Parshall and Rowe Citation1994.

7Notice, here, the notation which literally indicates a linear change of the variables x and y to the variables x′ and y′. Boole only implicitly assumed that the determinant of the transformation was non-zero.

8See Cayley Citation1846. For the broader historical context of Cayley's work in invariant theory, see Crilly Citation1986, Citation1988. Compare the discussion of these matters in Parshall and Rowe Citation1994, 67–68.

9I have adapted Sylvester's notation here, using the variables [x,y,z instead of his ξ ,η ,ζ. In the paper, Sylvester also considered not merely □(UV) but also the somewhat more general □(λ U + μ V). See the commentary in Parshall Citation1998, 28–29.

10Sylvester's technique was, in fact, not the same as Hesse's. Compare the commentary in Parshall Citation1998, 30–31.

11 Compare Hesse Citation1844, especially 90–95. See also the commentary in Parshall Citation1998, 30–31.

12 This passage—in one of the footnotes that Sylvester added to this edition of his Exeter lecture—does not appear in the version reproduced in Sylvester Citation1904–1912.

13 Here, U is an invariant, and a, b, …, k are the coefficients of the underlying homogeneous polynomial of degree n in two unknowns.

14For the discussion in this and the next four paragraphs, compare Parshall Citation1999, 253–256.

15 Compare Elliott Citation1895, 267–268, for the analysis.

16Sylvester referred explicitly to one of Salmon's results in Sylvester Citation1851c.

17 Compare the discussion of this in Parshall Citation1988, 184–185.

18 See the anonymous review (Citation1859) as quoted in Despeaux Citation2002, 389.

19 See Cayley Citation1871. Tony Crilly discusses the “Ninth memoir” at some length in Crilly Citation1981, 135–139 and 2005, 301–302.

20 For this conclusion, compare Parshall Citation2006, 334.

21 See Hilbert 1896, 124. Meyer echoed Hilbert's assessment in Meyer Citation1897, 1–26.

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