On Niels H Abel

"Other mathematicians confess that they have been unable to understand this proof and some have made the correct observation that Ruffini, perhaps by proving too much, had proved nothing in a satisfactory manner. Monsieur Abel has shown by a more penetrating analysis that there can be no algebraic [radical] roots, but he does not deny the possibility of transcendental roots. We recommend this problem to the attention of mathematicians specializing in this field." (Augustin-Louis Cauchy, [commenting on Abel's Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré], 1824)

"All of Abel's works carry the imprint of an ingenuity and force of thought which is unusual and sometimes amazing, even if the youth of the author is not taken into consideration. One may say that he was able to penetrate all obstacles down to the very foundations of the problems, with a force which appeared irresistible; he attacked the problems with extraordinary energy; he regarded them from above and was able to soar so high over their present state that all difficulties seemed to vanish under the victorious onslaught of his genius. [...] But it was not only his great talent which created the respect for Abel and made his loss infinitely regrettable. He distinguished himself equally by the purity and nobility of his character and by a rare modesty which made his person cherished to the same unusual degree as was his genius." (August L Crelle, "Crelle's Journal", 1841)

"He [Abel] appears to have fully developed in his own mind the subject of the separation of symbols of operation and quantity, not indeed to the extent of founding its results upon an algebraical theory, but to that of giving the theory a wider amount of application. He was a daring generalizer, and sometimes went too far: had he lived, he would have corrected some of his writings. And yet he appears to have been deeply impressed with the notion that a great part of mathematical analysis is rendered unsound by the employment of divergent series." ("A. de M. The Biographical Dictionary of the Society for the Diffusion of Useful Knowledge", 1842)

"Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged Legendre, who greatly admired Abel's genius, called it 'monumentum aere perennius'. During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century." (Florian Cajori, "A History of Mathematics", 1893)

"In Leibnitz's day [...] equations of the 2d, 3d, and 4th degrees were reduced to pure equations, but the reduction of equations of higher degrees than the 4th remained an unsolved problem, on which mathematicians spent much labor, until Niels Henrik Abel [...] a Norwegian mathematician of great ability and acuteness, demonstrated (1824) that the quintic equation and a fortiori the general equation of any order higher than five, is incapable of solution by radicals." (Alfred G Langley,[ footnote to Gottfried Wilhelm Leibniz, New Essays Concerning Human Understanding] 1916)

"Having found a method differing from that of Ferrari for reducing the solution of the general biquadratic equation to that of a cubic equation, Euler had the idea that he could reduce the problem of the quintic equation to that of solving a biquadratic, and Lagrange made the same attempt. The failures of such able mathematicians led to the belief that such a reduction might be impossible. The first noteworthy attempt to prove that an equation of the fifth degree could not be solved by algebraic methods is due to Ruffini (1803, 1805), although it had already been considered by Gauss. [...] The modern theory of equations is commonly said to date from Abel and Galois. [...] Abel showed that the roots of a general quintic equation cannot be expressed in terms of its coefficients by means of radicals." (David E Smith, "History of Mathematics", 1925)

"Abel criticised the use of infinite series, and discovered the well-known theorem which furnishes a test for the validity of the result obtained by multiplying one infinite series by another. He also proved the binomial theorem for the expansion of (1+x)^n when x and n are complex. As illustrating his fertility of ideas [...] notice his celebrated demonstration that it is impossible to express a root of the general quintic equation in terms of its coefficients by means of a finite number of radicals and rational functions; this theorem was the more important since it definitely limited a field of mathematics which had previously attracted numerous writers." (W W Rouse Ball, "A Short Account of the History of Mathematics", 1912)

"Abel's theorem [...] may be described as a theorem for evaluating the sum of a number of integrals which have the same integrand, but different limits - these limits being the roots of an algebraic equation. The theorem gives the sum of the integrals in terms of the constants occurring in this equation and in the integrand. We may regard the inverse of the integral of this integrand as a new transcendental function, and if so the theorem furnishes a property of this function." (W W Rouse Ball, "A Short Account of the History of Mathematics", 1912)

"Abel proposed himself the problem of finding all equations solvable by radicals, and succeeded in solving all equations with commutative groups, now called Abelian equations. Among Abel's other achievements are the discovery of the elliptic functions and their fundamental properties, his famous theorem on the integration of algebraic functions [and] theorems on power series." (Øystein Ore, "Abel On the Quintic Equation", [in "A Source Book in Mathematics"] 1929)

"It is most remarkable that two men as different in character and outlook as Abel and Galois should have been interested in the same problem and should have attacked it by similar methods. Both approached the problem of the quintic equation in the conviction that a solution by radicals was possible; Abel at eighteen, Galois at sixteen. In fact, both thought for a while that they had discovered such a solution; both soon realized their error and attacked the problem by new methods." (Tobias Dantzig, "Number: The Language of Science", 1930)

"He took up the problem of the division of the lemniscate (solving xn - 1 = 0 is the equivalent of the problem of the division of the circle into n equal arcs) and arrived at a class of algebraic equations [...] Abelian equations, that are solvable by radicals. The cyclotomic equation [xp - 1 = 0, where p is a prime] is an example [...] In this last work he introduced two notions (though not the terminology), field and polynomial irreducible in a given field. By a field of numbers he, like Galois later, meant a collection of numbers such that the sum, difference, product, and quotient of any two numbers in the collection (except division by 0) are also in the collection. [...] A polynomial is said to be reducible in a field (usually the field to which its coefficients belong) if it can be expressed as a product of two polynomials of lower degrees and with coefficients in the field. [...] Abel then tackled the problem of characterizing all equations which are solvable by radicals and had communicated some results [...] just before death overtook him in 1829." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

"Abel did not deny that we might solve quintics using techniques other than algebraic ones of adding, subtracting, multiplying, dividing, and extracting roots. ...the general quintic can be solved by introducing [...] 'elliptic functions', but these require operations considerably more complicated than those of elementary algebra. In addition, Abel's result did not preclude our approximating solutions [...] as accurately as we [...] wish. [...] What Abel did do was prove that there exists no algebraic formula... The analogue of the quadratic formula for second-degree equations and Cardano's formula for cubics simply does not exist [...] This situation is reminiscent of that encountered when trying to square the circle, for in both cases mathematicians are limited by the tools they can employ. [...] the restriction to 'solution by radicals' [...] hampers mathematicians [...] what Abel actually demonstrated was that algebra does have [...] limits, and for no obvious reason, these limits appear precisely as we move from the fourth to the fifth degree."  (William Dunham, "Journey Through Genius: The Great Theorems of Mathematics", 1990)