"It is Legendre’s eternal glory to have discovered the kernel of an important branch of analysis and by the work of half a lifetime to have erected on these foundations an independent theory. [...] Only with the continued determination that enabled the great mathematician to return again and again to the subject was he able to overcome difficulties that, with the means he had at his command, must have seemed scarcely subduable." (P G Lejeune Dirichlet, 1852)
"Legendre's Law of Quadratic Reciprocity […] is […] the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic', and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. […] we find in the 'Opuscula Analytica' of Euler […] a memoir […] which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler […] expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined […] that he had obtained a satisfactory demonstration." (Henry J S Smith, "Report on the Theory of Numbers", 1859)
"The fear that the mathematician most chiefly concerned with the determination of numerical values had of the imaginary, was the reason that Legendre was prevented from taking the most important step in modern analysis, the introduction of doubly periodic functions." (Jacobi, 1832)
"Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; [...]" (James J Sylvester,"A Plea for the Mathematician", 1869)
"It is well known that the reluctance of Gauss to publish his discoveries was due to the rejection of his Disquisitiones arithmeticae by the French Academy, the rejection being accompanied by a sneer which, as Rouse Ball has said, would have been unjustifiable even if the work had been as worthless as the referees believed. It is the irony of fate that, but for this sneer, the Traite des fonctions elliptiques, the work of a Frenchman, might have assumed a different and vastly more valuable form, and Legendre might have been spared the pain of realizing that many years of his life had been practically wasted, had the method of inversion come to be published when Legendre’s age was fifty instead of seventy-six." (George N Watson, "The Marquis and the Land-Agent; A Tale of the Eighteenth Century", Math. Gazette 17, 1933)
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