Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l'équation générale du cinquième degrée.
Christiana: Groendahl, 1824. 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 square roots is now taught to every student in high school. Similar formulas for solving cubic and quartic equations were not found until the 16th century, by Scipione del Ferro (1465-1525), Lodovico Ferrari (1522-60), and Niccolo Tartaglia (1506-59), and were first published by Girolamo Cardano (1501-76) in his Ars magna (1545). These formulas expressed the solutions in terms of ‘radicals,’ i.e., expressions involving rational functions (ratios of polynomials) of the coefficients of the equation and their square-, cube-, and higher roots. The search for a similar formula for quintic equations proved fruitless. “For two centuries thereafter, the resolution of the enigma was regarded as one of the most important problems of algebra and occupied the attention of the leading mathematicians of this epoch” (Ayoub, p. 257). In the Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin (1770-71), Lagrange published his remarkable memoir ‘Réflexions sur la résolution algébrique des equations.’ Katz & Parshall (p. 298) remark that “his introduction of the notion of permutations proved crucial to the ultimate proof that there was no algebraic solution of a fifth-degree polynomial equation,” but Lagrange himself still believed that a solution of the quintic would be found. “He concludes with this statement: ‘There, if I am not mistaken, are the true principles of the resolution of equations, and the most appropriate analysis which leads to solutions; all reduces, as we see, to a type of calculus of combinations by which we find results which we might expect a priori. It would be pertinent to make application to equations of the fifth and higher degrees whose solution is, up to the present, unknown: but this application requires a large number of combinations whose success is, however, very doubtful. We hope to return to this question at another time and we are content here in having given the fundamentals of a theory which appears to us new and general.’ So in spite of past failures in solving the quintic, Lagrange still harbors the hope that a careful analysis of his method will achieve the goal. “Did no one suspect that the solution of the quintic was impossible? Apparently not until 1799 when Ruffini published his book on the theory of equations: ‘General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than 4 is impossible.’ Parenthetically, we note that in the same year the young Carl Friedrich Gauss (1777-1855) wrote in his dissertation (in which he proved the fundamental theorem of algebra) as follows: ‘After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that the resolution is impossible and contradictory … Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place. Here it is enough to say that the general solution of equations understood in this sense [i.e., by radicals] is far from certain and this assumption [i.e., that any equation is solvable by radicals] has no validity at the present time.’ Gauss published nothing more on the subject” (Ayoub, pp. 262-3). “Although Gauss had proclaimed his belief that the unsolvability of the quintic might not be difficult to prove with all rigor, the Italian P. Ruffini, in 1799, was the first mathematician to state the unsolvability as a result and attempt to provide it with a proof. Ruffini’s style of presentation was long, cumbersome, and at times not free of errors; and his initial proof was met with immediate criticism for all these reasons. But convinced of the result and his proof, Ruffini kept elaborating and clarifying his theory in print for the next 20 years, producing a total of five different versions of the proof. The proofs were published in Italian as monographs in Bologna and in the mathematical memoirs of the Societa Italiana delle Scienze, Modena. Although published and distributed, the impact of Ruffini’s work was limited; among the few non-Italians to take a viewpoint on Ruffini’s work was Cauchy … Cauchy wrote Ruffini [a] letter in September 1821, in which he acknowledged Ruffini’s progress in the important field of solvability of algebraic equations: ‘I must admit that I am anxious to justify myself in your eyes on a point which can easily be clarified. Your memoir on the general solution of equations is a work which has always appeared to me to deserve to keep the attention of geometers. In my opinion, it completely demonstrates the algebraic unsolvability of the general equations of degrees above the fourth …’ “At least by 1821, the validity of Ruffini’s claim that the general quintic could not be solved by radicals was propounded, not only by a somewhat obscure Italian mathematician, but also one of the most promising and ambitious French mathematicians of the early 19th century. However, it would take further publications, notably by the young Abel, before this validity would be accepted by the broad international community of mathematicians … “In spite of the efforts of Ruffini and Gauss, the search for algebraic solution formulae for the quintic remained an attractive problem to a generation of young and aspiring mathematicians. In Norway, Abel thought he had solved it, but soon realized that he had been misled. In Germany, Carl Gustav Jacob Jacobi (1804–1851) worked on the problem, and in France [Évariste ] Galois (1811-32), too, thought he had found a solution, only to soon be disappointed. All of them attacked the problem while they still attended pre-university schools. The problem’s easy formulation and yet century-long history, and a general belief that its solution should be possible and not too difficult, made it appear as a good opening into doing creative mathematics. “Inspired by the stimulation of his new, and young, mathematics teacher B. M. Holmboe, Abel (1802-29) studied the masters and began to engage in creative mathematics of his own. In 1821, he thought he had produced a solution to the general fifth degree equation. In the incipient intellectual atmosphere of Christiania, few authorities capable of determining the validity of Abel’s reasoning could be found. But more importantly, the scientific milieu of Norway was still without a means of publication of technical mathematical results deserving international recognition. For these reasons, professor Chrisopher Hansteen (1784–1873) sent Abel’s manuscript to Professor Ferdinand Degen (1766–1825) in Copenhagen for evaluation and possibly publication in the transactions of the Royal Danish Academy of Sciences and Letters. The accompanying letter, which Hansteen must have written, and the paper are no longer preserved. Our only primary source of information is the letter which Degen wrote back to Hansteen, in which he asked for an elaborated version of the argument and an application to a specific numerical example. ‘As for the talented Mr. Abel, I will be happy to present his treatise to the Royal Academy of Science. It shows, even if the goal has not been reached, an extraordinary head and extraordinary insights, especially for someone his age. Nevertheless, I excuse myself to require the condition that Mr. A. sends an elaborated deduction of his result together with a numerical example, taken from, for instance, an equation such as x5 – 2x4 + 3x2 − 4x + 5 = 0 …’ “We have no indication that Abel ever produced an elaborated deduction; apparently the numerical examples worked their part — as the probes of truth — as Degen had suggested and led Abel to a radically new insight. In 1824, he published, at his own expense, a short work in French entitled Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l'équation générale du cinquième dégré. As Abel announced in the title, it demonstrated the impossibility of solving the general equation of the fifth degree. Abel intended the memoir to be his best self-introduction on his planned tour of the Continent. Since he had had to pay for the publication himself, it only covered six pages, and his style of presentation suffered accordingly. In numerous points he was unclear or left advanced arguments out. But when Abel came into contact with A. L. Crelle in Berlin, he found himself in a position to make his discovery available to a broader public. He rewrote the argument elaborating the ideas of the 1824 proof, and had Crelle translate it into German for publication in the very first issue of Journal für die reine und angewandte Mathematik. Through this treatise — and the French review which Abel wrote of it for Baron de Ferrusac’s (1776–1836) Bulletin des sciences mathématiques, astronomiques, physiques et chimiques — the world gradually came to know that a young Norwegian had settled the question of solvability of the general quintic in the negative” (Sørensen). “It is not certain when the printing was ready; Abel’s treatise was not included in the semi-annual list of Norwegian publications for the spring season of 1824, and when it was announced in Det Norske Rigstidende, the word equations, or équations, in the title: Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l'équation générale du cinquième dégré, came out as épurations; in other words, ‘purifications.’ It is likely that very few copies were sold, but Abel sent quite a number of copies to his friends in Copenhagen, and through Hansteen, some were also sent to Schumacher, and Schumacher saw to it that Gauss got to see Abel’s work at the end of July 1824 … [In] 1862, Hansteen stressed the fact that Gauss, upon first receiving Abel’s work on the fifth degree equations, had been negative, and he had said that he himself could probably prove the possibility of finding a solution to the quintic equation. But Gauss had little by little come to see that Abel was right. And the probable reason for Gauss’ initial discouraging response to Abel’s compact proof, apart from his constantly receiving prospective solutions to this popular mathematical challenge, was that Abel’s title was too short; it lacked the important precision in denoting that with regard to employment of the radical sign, a solution to the fifth degree equation was impossible” (Stubhaug, p. 306). After Gauss’s death, Abel’s memoir was found among his papers, unopened. Following the publication of his proof in Crelle’s journal, “Abel went to Paris, then the world centre for mathematics, where he called on the foremost mathematicians and completed a major paper on the theory of integrals of algebraic functions. His central result, known as Abel’s theorem, is the basis for the later theory of Abelian integrals and Abelian functions, a generalization of elliptic function theory to functions of several variables. However, Abel’s visit to Paris was unsuccessful in securing him an appointment, and the memoir he submitted to the French Academy of Sciences was lost. “Abel returned to Norway heavily in debt and suffering from tuberculosis. He subsisted by tutoring, supplemented by a small grant from the University of Christiania and, beginning in 1828, by a temporary teaching position. His poverty and ill health did not decrease his production; he wrote a great number of papers during this period, principally on equation theory and elliptic functions. Among them are the theory of polynomial equations with Abelian groups. He rapidly developed the theory of elliptic functions in competition with the German Carl Gustav Jacobi. By this time Abel’s fame had spread to all mathematical centres, and strong efforts were made to secure a suitable position for him by a group from the French Academy, who addressed King Bernadotte of Norway-Sweden; Crelle also worked to secure a professorship for him in Berlin. “In the fall of 1828 Abel became seriously ill, and his condition deteriorated on a sled trip at Christmastime to visit his fiancée at Froland, where he died. The French Academy published his memoir in 1841” (Britannica). The key concept in Abel’s proof – which is highly technical – is what is now known as a field (Abel expressed his ideas in a different, though equivalent way). A field is a collection of numbers that is closed under the operations of addition, subtraction, multiplication, and division (i.e., the sum of two number in a field is still in the field, etc.). The smallest field is the field Q of rational numbers, consisting of fractions m/n, where m and n are whole numbers (n ≠ 0); other fields are those of all real numbers, or all complex numbers. Suppose we are given a polynomial xn + a1xn-1 + a2xn-2 + … + an-1x + an, where n is the degree of the polynomial (n = 2 for a quadratic, n = 3 for a cubic, etc.) and a1, a2, … , an-1, an are its coefficients. A root of this polynomial is a value of x which makes the polynomial expression equal to zero. Gauss proved in 1799 that there are exactly n roots (which may not all be different) – this is called the ‘Fundamental theorem of algebra’. Let K be the smallest field containing Q and all the coefficients, and let L be the smallest field containing K and all the roots of the polynomial. If c is a number in K, and m is a whole number at least 2, we can form a new field K1 by adjoining the mth root of c, c1/m, to K – this is the smallest field containing K and c1/m. K1 will be larger than K if there is no number in K whose mth power is c; it is called a radical extension of K. More generally, we could repeat the procedure by adjoining to K1 a root of a number in K1, giving a field K2, and so on. Any field obtained by repeating this process a finite number of times is called a radical extension of K. This language allows us to state precisely what is meant by saying that the roots of the polynomial can be expressed in terms of square, cube, or higher roots: Lis contained in a radical extension of K. It also allows us to state clearly what Ruffini proved: for a ‘general’ polynomial of degree at least 5, L is not a radical extension of K – we have to say ‘general’ because there are, of course, ‘special’ polynomials that can be solved by radicals (e.g., x5 !). However, to prove the impossibility of solving such polynomials in radicals, it is necessary to prove the stronger statement that Lis not contained in a radical extension ofK. This is a subtle, but important difference from what Ruffini proved – it is possible that Ruffini did not appreciate the difference, and so believed his proof was complete. In his Mémoire, Abel first gave a simpler and clearer proof of Ruffini’s result, and then proved in addition that ifL is contained ina radical extension ofK, then Lisa radical extension ofK. By the Abel-Ruffini result, for a general polynomial of degree at least 5, L is not a radical extension of K, so it follows that L is not contained in a radical extension of K. The impossibility proof is thus complete. Ayoub, ‘Paolo Ruffini’s contributions to the quintic,’ Archive for History of Exact Sciences 23 (1980), pp. 253-277. Davitt, ‘William Marshall Bullitt and his amazing mathematical collection,’ The Mathematical Intelligencer 11 (1989), pp. 26-33. Katz & Parshall, Taming the Unknown, 2014. Ore, Niels Henrik Abel, Mathematician Extraordinary, 1974. Sørensen, Niels Henrik Abel and the theory of equations, 1999. Stubhaug, Niels Henrik Abel and his Times, 2000. For a general history of the Ruffini-Abel theorem, see Landmark Writings in Mathematics, Chapter 29.
4to, pp. [1-3], 4-8 (creased where folded horizontally with small holes along the crease in the title page, not affecting printing, some browning and soiling, particularly to the title page). Original marbled paper spine strip. A good, unrestored copy of a work that is virtually impossible to find.
Item #5834
Price: $165,000.00
