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Moreover, the value of c tended rather rapidly to zero, so that the convergence was quite rapid. Legendre illustrated this with an example. 2460561. 19) from which is followed that it was often enough to use just the first two terms. In the Exercises he gave a table of values of elliptic integrals to 14 decimal places. Legendre also sought to show how useful his new functions would be in various parts of mathematics. 20) 1 − c2 sin2 ψ where l is the length of the pendulum, c2 = 2lh where h is the height of the pendulum due to its speed at its lowest point (in units where the acceleration due to gravity =1) and the angle ψ is related to the angle of displacement from the vertical by the formula sin( φ2 ) = c sin ψ .

His success as an engineer, allied to his skills as an organiser, made him a very powerful supporter of pure mathematics. Even more valuably for Abel he was in the process of establishing a new journal for mathematics, the first of its kind in Germany. Crelle hoped thereby to raise what he saw as the poor standards of German mathematics, and he invited Abel to contribute to it. This journal, the Journal f¨ur die reine und angewandte Mathematik, rapidly became the leading journal devoted to mathematics in Germany, and after Crelle’s death ever more securely the journal of mathematics as Berlin saw it.

But when φ = 0, ψ = μ , so a comparison with Euler’s result shows that cos φ cos ψ − sin φ sin ψΔ (μ ) = cos μ . 4) for elliptic integrals. The bulk of the paper was given over to a sketch of how tables of the values of F(φ ) could be calculated for specified values of the modulus c. That Legendre had thoroughly adopted the function-theoretic point of view is even clearer in his Exercises de calcul int´egral (1811–1817) which begins with his study of “Des fonctions elliptiques”. He meant F(φ ), E(φ ), and related functions, but to avoid confusion with their inverses, which we nowadays call elliptic functions, we shall call Legendre’s functions elliptic integrals.

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