## Mémoire sur une propriété générale d’une classe trés-étendue de fonctions transcendantes, présenté à l'Académie le 30 Octobre 1826, pp. 176-264 in: Mémoires présentés par divers savants a l’Académie Royale des Scines de l’Institute de France … Tome Septième.

Paris: Imprimé par autorisation du roi a l’Imprimerie Royale, 1841.

Abel’s theorem is about evaluating integrals of the form

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 which depends algebraically on the original end-points (one might also have to add some rational and logarithmic functions).

The restriction to simple curves like y2 = h(x), where h is cubic or quartic, was severe but difficult to overcome. The vital breakthrough was made by Abel in the offered paper. “Here he states his main theorem, today known as ‘Abel’s theorem’, in the following form:

‘Si l’on a plusieurs fonctions don’t les derives peuvent être raciness d’une ‘même equation algébrique’, don’t tous les coefficients sont des fonctions ‘rationnelles’ d’une meme variable, on peut toujour exprimer la somme d’un nombre quelconque de semblables fonctions par une function ‘algébrique et logarithmique’, pourvu qu’on établisse entre les variables des fonctions en question un certain nombre de relations ‘algébriques’ … Le nombre de ces relations ne depend nullement du nombre des fonctions, mais seulement de la nature des fonctions particulières qu’on considère.’

“The proof of this general theorem proposed by Abel himself is very simple. In essence it follows from two elementary facts: first, any symmetric rational function of the roots of an algebraic equation is a rational function of its coefficients; second, the integral of a rational function is the sum of a rational function and a finite number of logarithms of rational functions. Thus, Émile Picard was induced to write: ’Sous cette forme, le theorem parait tout a fait èlémentaire, et il n’y a peut-être pas, dans l’histoire de la Science, de proposition aussi importante obtenue à l’aide de considérations aussi simple’” (Del Centina, pp. 1-2).

Abel “was born in 1802, into a family of very limited means. At the time Norway, as a dependency of Denmark, was suffering economically from the British efforts to eliminate or neutralize the Danish Navy to keep it out of the hands of Napoleon. In addition, his father was engaged in political activity, agitating for Norwegian independence, and seems to have been a heavy drinker as well, contributing to the family’s poverty. In 1815, the year after control of Norway shifted to Sweden, Abel and his older brother were sent to the Cathedral School in Christiania (Oslo). As most of the good teachers from the Cathedral School had gone to provide the teaching staff at the University of Christiania in 1813, Abel found few good teachers there and was not inspired. However, in 1817, a new mathematics teacher, Bernt Holmboe, arrived, and was to prove a constant friend and mentor to Abel. When Abel’s father died in 1820, Holmboe helped the young student to obtain a scholarship to continue his education.

“The following year Abel entered the University of Christiania. That same year, thinking he had succeeded in solving the quintic equation, he sent a paper to the Danish mathematician Ferdinand Degen, who asked him for a specific example of his method. Working out such an example revealed the mistake to Abel, and a few years later, he wrote the first draft of an impossibility proof, published privately in 1824 by Grøndahl. In this proof Abel recognized the importance of filling in the gap in Ruffini’s work. Unfortunately, his proof that the intermediate radicals in a supposed solution by formula can be expressed as rational functions of the roots suffers from some vagueness also, and the version that he finally published in the Journal für die reine und angewandte Mathematik was greatly expanded, with fuller explanations of the use of permutations.

“Degen had advised the talented young man to devote himself to elliptic functions, and this area, and its generalization to integrals of completely general algebraic functions formed the vast majority of Abel’s life work and perhaps the most profound theorem of the early 19th century, called Abel’s theorem at the suggestion of his rival in elliptic functions C .G. J. Jacobi (1804–1851). In 1825, after two years of intensive study of the German and French languages, Abel went on a tour of Denmark, Germany, and France, carrying some of his papers by way of introduction. The trip was only a partial success; he met Crelle, whose Journal für die reine und angewandte Mathematik provided the outlet for most of his work. He was less successful in his attempts to meet Gauss, A. M. Legendre, and others, and the referees from the French Academy (Cauchy and Legendre) took little interest in the brilliant paper containing Abel’s theorem. (This paper was published only in 1841, after considerable urging by Jacobi.) Abel returned home in 1827, in debt. He died in 1829 at the age of 26, his best work still unrecognized. Ironically, the Paris Academy awarded him a prize for his work on elliptic functions the following year” (Cooke, p. 399).

Cooke, ‘Niels Henrik Abel, paper on the irresolvability of the quantic equation (1826),’ Ch. 29 in: Landmark Writings in Western Mathematics 1640-1940 (Grattan-Guinness, ed.), 2005. Del Centina, ‘Abel’s manuscripts in the Libri collection: their history and their fate,’ pp. 87-103 in: Il manoscritto parigino di Abel preservato nella Biblioteca Moreniana di Firenze Abel’s Parisian manuscript preserved in the Moreniana Library of Florence, A. Del Centina ed., 2003. Bottazzini & Gray, Hidden Harmony – Geometric Fantasies, 2013.

4to (271 x 217 mm), pp. [vi], 647, with 42 plates, uncut (occasional light foxing). Original printed wrappers (a few closed tears and some minor loss of the edges, nowhere near printed area).

Item #5458

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