An earlier publication which contained part of his researches on elliptic functions was his Calcul intégral in three volumes (1811, 1816, 1817), in which he treats also at length of the two classes of definite integrals named by him Eulerian. He tabulated the values of for values of between 1 and 2.
One of the earliest subjects of research was the attraction of spheroids, which suggested to Legendre the function , named after him. His memoir was presented to the Academy of Sciences in 1783. The researches of Maclaurin and Lagrange suppose the point attracted by a spheroid to be at the surface or within the spheroid, but Legendre showed that in order to determine the attraction of a spheroid on any external point it suffices to cause the surface of another spheroid described upon the same foci to pass through that point. Other memoirs on ellipsoids appeared later.
The two household gods to which Legendre sacrificed with ever-renewed pleasure in the silence of his closet were the elliptic functions and the theory of numbers. His researches on the latter subject, together with the numerous scattered fragments on the theory of numbers due to his predecessors in this line, were arranged as far as possible into a systematic whole, and published in two large quarto volumes, entitled Théorie des nombres, 1830. Before the publication of this work Legendre had issued at divers times preliminary articles. Its crowning pinnacle is the theorem of quadratic reciprocity, previously indistinctly given by Euler without proof, but for the first time clearly enunciated and partly proved by Legendre.[48]
While acting as one of the commissioners to connect Greenwich and Paris geodetically, Legendre calculated all the triangles in France. This furnished the occasion of establishing formulæ and theorems on geodesics, on the treatment of the spherical triangle as if it were a plane triangle, by applying
