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 The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers; definite integrals were studied by Thomae, Du Bois-Reymond, and Darboux along the lines indicated by the definitions of such integrals given by Cauchy, Dirichlet, and Riemann. Dini wrote a text-book on functions of a real variable (1878), which was translated into German, with additions, by J. Lüroth and A. Schepp. Important works on the theory of functions are the Cours de M. Hermite, Tannery's Théorie des Fonctions d'une variable seule, A Treatise on the Theory of Functions by James Harkness and Frank Morley, and Theory of Functions of a Complex Variable by A. R. Forsyth.

"Mathematics, the queen of the sciences, and arithmetic, the queen of mathematics." Such was the dictum of Gauss, who was destined to revolutionise the theory of numbers. When asked who was the greatest mathematician in Germany, Laplace answered, Pfaff. When the questioner said he should have thought Gauss was, Laplace replied, "Pfaff is by far the greatest mathematician in Germany; but Gauss is the greatest in all Europe."[83] Gauss is one of the three greatest masters of modern analysis,—Lagrange, Laplace, Gauss. Of these three contemporaries he was the youngest. While the first two belong to the period in mathematical history preceding the one now under consideration, Gauss is the one whose writings may truly be said to mark the beginning