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'A MONUMENT MORE LASTING THAN BRONZE' (LEGENDRE). First edition, journal issue in original printed wrappers, of "the most important single result in the theory of integrals of algebraic functions . Abel's theorem" (Bottazzini & Gray, p. 236), a result which Legendre called 'a monument more lasting than bronze'. It can be regarded as the birth of algebraic geometry. Abel had already begun to radically transform and generalize the theory of elliptic functions when, in July 1826, he visited Paris, hoping to make the acquaintance of the great mathematicians there. "The visit to Paris was to prove disappointing. The university vacations had just begun when Abel arrived, he found that they were aloof and difficult to approach; it was only in passing that he met Legendre, whose main interest in his old age was elliptic integrals, Abel's own specialty. For presentation to the French Academy of Sciences Abel had reserved a paper that he considered his masterpiece. It dealt with the sum of integrals of a given algebraic function. Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments. The minimal number p is the genus of the algebraic function, and this is the first occurrence of this fundamental quantity. Abel's theorem is a vast generalization of Euler's relation for elliptic integrals. Abel spent his first months in Paris completing his great memoir; it is one of his longest papers and includes a broad theory with applications. It was presented to the Academy of Sciences on 30 October 1826, under the title 'Mémoire sur une propriété générale d'une classe trés-étendue de fonctions transcendantes.' Cauchy and Legendre were appointed referees, Cauchy being chairman. A number of young men had gained quick distinction upon having their works accepted by the Academy, and Abel awaited the referees' report. No report was forthcoming, however; indeed, it was not issued until Abel's death forced its appearance. Cauchy seems to have been to blame; he claimed later that the manuscript was illegible" (DSB). In July 1829 Cauchy finally reported on the paper and recommended that it be published. But again nothing was done, until in 1840 a formal protest by the Norwegian government eventually forced the Académie to place Guglielmo Libri in charge of publishing the paper, which he duly completed in 1841. Abel's manuscript then once again went missing, until in 1952 all but eight pages of the original manuscript was found among a collection of Libri's papers in the Biblioteca Moreniana in Florence; the remaining eight pages were only discovered, also at the Moreniana, in 2002. ABPC/RBH list one copy of the complete journal volume (in 2014), and the Honeyman copy of the journal extract. We are not aware of any other copy in original printed wrappers having appeared on the market. Abel's theorem is about evaluating integrals of the form â « f(x, y)dx, where f is a rational function (i.e., a quotient of two polynomials) in x and y, and x and y are connected by an equation, say g(x, y) = 0. (One says that the integral is 'evaluated along' the curve whose equation is g(x, y) = 0.) The evaluation is to be given as a function of the upper end-point v of the integral. If g is of the form y = h(x), so that y can be eliminated from the integrand, the problem is easy and has a well-known solution - the integral is a sum of rational and logarithmic functions of v. If g is of the form y2 = h(x), the problem is much more complicated (unless h is quadratic). If h is of degree 3 or 4 (cubic or quartic), the integral is said to be elliptic; for h of higher degree it is hyperelliptic. The elliptic case was solved by Abel and Jacobi. An important consequence of their work was that the sum of any number of elliptic integrals, having the same integrand but different end-points, may be written as a single elliptic integral, the variable end-point of w. Seller Inventory # 5458
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