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Arthur Cayley, Robert Harley and the quintic equation: newly discovered letters 1859–1863. (English. French summary) Zbl 1370.01013

As its title indicates, this paper uses newly discovered letters, spanning the period 1859–1863, from Arthur Cayley to fellow-mathematician Robert Harley to shed new light on aspects of Cayley’s work on invariant theory – specifically, the calculation of resolvents of quintic equations. Further insight is provided by another recently discovered document: the manuscript of a memoir on the quintic equation that Cayley was working on at the time of his death in 1895.
The paper begins with a brief biographical overview of the two correspondents, together with an account of the circumstances (including the mathematical context) surrounding their exchange of letters, which are concerned largely with the efforts of Cayley, Harley and others to develop a theory for those quintic equations that may be solved in radicals (Abel having shown, of course, that the general quintic is not solvable in such terms). In particular, they sought to find methods for calculating the lower-degree resolvent equation of a given quintic, thereby to extend techniques that had been shown to be useful for polynomials of degrees four and lower. In the main parts of the paper, the mathematics of the efforts of Cayley and Harley, and the place of this within the correspondence, is outlined in considerable detail.
The authors view these letters as a means of seeing “a great mathematician at his workbench creating the mathematics that would be published in several of his papers” (p.167). They point in particular to the influence here of other nineteenth-century British mathematicians (such as James Cockle) who are now almost forgotten, and note how the long computations involved in the invariant-theoretic approach to this subject were at variance with the more abstract methods based on groups that eventually came to dominate. Nevertheless, the authors argue, the fact that Cayley returned to this topic with a draft memoir at the end of his life points to this work “as an important part of his legacy” (p.167).

MSC:

01A55 History of mathematics in the 19th century

Biographic References:

Cayley, Arthur; Harley, Robert
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References:

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