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

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 1859 and later Edwin Bailey Elliott in 1895 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 2006.

² For his full-scale biography, see Crilly 2005.

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 (2002, 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 1971–1972; this paragraph has drawn from 155–157 and 187–189. Compare also Parshall 2006, 97–98. On the Cambridge Mathematical Journal, see Crilly 2004.

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 1988, 1989, I, 157–206, 1998, 1999, 2006 and Parshall and Rowe 1994.

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 1846. For the broader historical context of Cayley's work in invariant theory, see Crilly 1986, 1988. Compare the discussion of these matters in Parshall and Rowe 1994, 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 □(U +λ V) but also the somewhat more general □(λ U + μ V). See the commentary in Parshall 1998, 28–29.

¹⁰Sylvester's technique was, in fact, not the same as Hesse's. Compare the commentary in Parshall 1998, 30–31.

¹¹ Compare Hesse 1844, especially 90–95. See also the commentary in Parshall 1998, 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 1904–1912.

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 1999, 253–256.

¹⁵ Compare Elliott 1895, 267–268, for the analysis.

¹⁶Sylvester referred explicitly to one of Salmon's results in Sylvester 1851c.

¹⁷ Compare the discussion of this in Parshall 1988, 184–185.

¹⁸ See the anonymous review (1859) as quoted in Despeaux 2002, 389.

¹⁹ See Cayley 1871. Tony Crilly discusses the “Ninth memoir” at some length in Crilly 1981, 135–139 and 2005, 301–302.

²⁰ For this conclusion, compare Parshall 2006, 334.

²¹ See Hilbert 1896, 124. Meyer echoed Hilbert's assessment in Meyer 1897, 1–26.