Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degrée

ABEL, Niels Henrik

Edité par Groendahl, Christiana, 1824

A propos de cet article

"ONE OF THE GREATEST RARITIES IN MATHEMATICAL LITERATURE" . First edition of this "publication now considered one of the greatest rarities in mathematical literature" (Ore, p. 73), as well as one of the most famous. This is likely the only copy in private hands. It contains the first complete proof (albeit with some vagueness due to the extreme brevity of the work) that the general polynomial of degree at least 5 is not solvable by 'radicals', i.e., expressions involving only square, cube and higher roots. The formula for solving quadratic equations was discovered in antiquity, while those for cubic and quartic equations were found in the first half of the 16th century - these formulas only involve rational operations (addition, subtraction, multiplication, division) and square and cube roots. It was believed by virtually all mathematicians up to the end of the 18th century that similar formulas would exist for the solution of quintic and higher degree equations (possibly involving fifth and higher roots). However, in his Teoria Generale delle Equazioni (1799), Paolo Ruffini (1765-1822) proposed a proof that the quintic could not, in fact, be solved in terms of radicals. Ruffini's proof was long and difficult to follow, and was not accepted by most mathematicians (with the notable exception of Cauchy). Today it is accepted that Ruffini's work was incomplete, although it did introduce many of the ideas necessary for the proof and it changed the mind-set of many mathematicians toward accepting that the quintic may be unsolvable. Abel was unaware of Ruffini's work. Indeed, in 1821 Abel believed he had a proof that the quintic equation can be solved by radicals, but he soon found his error and proved the opposite (see below). In his Mémoire, Abel gave a much simpler proof of what Ruffini had achieved, and closed the gap in Ruffini's argument, thereby giving the first complete proof of the impossibility of solving the quintic equation by radicals. Abel published this pamphlet at his own expense, in a very small print run, and made his proof extremely concise to reduce printing costs. "The resulting brevity probably made it difficult to understand; at any rate, there was no reaction from any of the foreign mathematicians-including the great C.F. Gauss, to whom a copy was sent" (DSB). His achievement was only fully recognised when, two years later, he published a greatly expanded version of his proof in a leading European mathematical journal. We are not aware of any other copy of this pamphlet having appeared on the market, and there appear to be only a handful of copies in institutions. William Marshall Bullitt, the renowned collector of mathematical works, had virtually completed his collection by the end of World War II, but had been unable to find a copy of the present work. "Finally, in 1951, after a fourteen-year pursuit, through the unstinting efforts of [Oswald] Veblen, Paul Heegaard (a distinguished Norwegian mathematician), and Maggs Brothers, Bullitt was able to purchase a copy of the privately-printed 1824 pamphlet from the widow of a Professor of Actuarial Mathematics at the University of Oslo . In a letter to Bell [i.e., Eric Temple Bell, the historian of mathematics] dated 10 December 1951, Bullitt writes: 'I received today, in perfect order, the original Abel Mémoire. It is true that I had to pay an outrageous price [$500] for it, but at least I have added almost the last item that I needed to complete my collection'" (Davitt, pp. 30-31). After Bullitt's death, his widow presented his collection to the University of Kentucky, Louisville, where it remains. The first three works she presented were the Halifax copy of Newton's Principia (annotated by Newton), copies of the journal issues of Einstein's 1905 papers (inscribed to Bullitt by Einstein), and the present work. The method of solving quadratic equations was known to the Baghdad mathematician and astronomer Al-Khwarizmi (c. 780-850), and the formula involving squ.

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Détails bibliographiques

• Titre: Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degrée
• Éditeur: Groendahl, Christiana
• Date de parution: 1824
• Reliure: Couverture souple
• Édition: First edition.
• Catalogues du vendeur:
  • Mathematics, Analysis, Algebra