In addition to the various works of Brewster already noticed, the following may be mentioned: - Notes and Introduction to Carlyle's translation of Legendre's Elements of Geometry (1824); Treatise on Optics (1831); Letters on Natural Magic, addressed to Sir Walter Scott (1831); The Martyrs of Science, or the Lives of Galileo, Tycho Brake, and Kepler (1841); More Worlds than One (1854).
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers), which Riemann read, mastered and returned within a few days.
His first published writings consist of articles forming part of the Traite de mecanique (1774) of the Abbe Marie, who was his professor; Legendre's name, however, is not mentioned.
This is the subject with which Legendre's name will always be most closely connected, and his researches upon it extend over a period of more than forty years.
Legendre's researches connected with the " gamma function " are of importance, and are well known; the subject was also treated by K.
- Legendre's Theorie des nombres and Gauss's Disquisitiones arithmeticae (1801) are still standard works upon this subject.
The symbol (alp) which is known as Legendre's symbol, and denotes the positive or negative unit which is the remainder when au s (-1) is divided by a prime number p, does not appear in this memoir, but was first used in the Essai sur la theorie des nombres.
Legendre's formula x: (log x- I.
Legendre's second memoir was communicated to the Academie in 1784, and relates to the conditions of equilibrium of a mass of rotating fluid in the form of a figure of revolution which does not deviate much from a sphere.
(See Todhunter's History of the Mathematical Theories of Attraction and the Figure of the Earth (1873), the twentieth, twenty-second, twenty-fourth, and twenty-fifth chapters of which contain a full and complete account of Legendre's four memoirs.
The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.
Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.
In 1806 appeared Legendre's Nouvelles Methodes pour la determination des orbites des cometes, which is memorable as containing the first published suggestion of the method of least squares (see Probability).
- Legendre's name is most widely known on account of his Elements de geometrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry.
Legendre's proof is similar in principle to Lambert's, but much simpler.
It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics.
He received £50 for a translation of Legendre's Geometry; and an introduction, explaining the theory of proportion, is said by De Morgan to show that he could have gained distinction as an expounder of mathematical principles.