Euler, Lagrange, and Laplace

and Lagrange had left it, and for forty years was the only one to cultivate this new branch of analysis, until at last Jacobi and

Abel stepped in with admirable new discoveries.52 Legendre

imparted to the subject that connection and arrangement which belongs to an independent science. Starting with an integral depending upon the square root of a polynomial of the fourth degree in $x$ , he showed that such integrals can be brought back to three canonical forms, designated by $F (ϕ)$ , $E (ϕ)$ , and $Π (ϕ)$ , the radical being expressed in the form $Δ (ϕ) = \sqrt{1 - k^{2} \sin^{2} ϕ}$ . He also undertook the prodigious task of calculating tables of arcs of the ellipse for different degrees of amplitude and eccentricity, which supply the means of integrating a large number of differentials.

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 $\log Γ (p)$ for values of $p$ between $1$ and $2$ .

One of the earliest subjects of research was the attraction of spheroids, which suggested to Legendre the function $P_{n}$ , 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

certain corrections to the angles, and on the method of least squares, published for the first time by him without demonstration in 1806.

Legendre wrote an Éléments de Géométrie, 1794, which enjoyed great popularity, being generally adopted on the Continent and in the United States as a substitute for Euclid.