Every branch of mathematics was enriched by the contributions of the Italian-born French mathematician Comte Joseph Louis Lagrange (1736-1813). He is best known for his analytical formulations of the calculus of variations and mechanics.

Joseph Louis Lagrange was born in Turin on Jan. 25, 1736; both his parents had French ancestors, and Lagrange wrote all his works in French. At the College of Turin he studied classics until, at the age of 17, his interest in mathematics was aroused by reading Edmund Halley's memoir on the utility of analytical methods in the solution of optical problems. Within 2 years Lagrange had made sufficient progress to be appointed professor of mathematics at the artillery school in Turin.

After reading Leonhard Euler's work on isoperimetric problems Lagrange developed an analytical method of solution in 1756. Two years later he helped to found a society which later became the Turin Academy of Sciences. He contributed many papers to its transactions, usually described as Miscellanea Taurinensia. The Paris Academy of Sciences awarded Lagrange prizes for his essays on the libration of the moon (1764), the satellites of Jupiter (1766), and the three-body problem (1772).

In 1766 Frederick the Great appointed Lagrange president of the Berlin Academy of Sciences. When Frederick died in 1786, Lagrange moved to Paris at the invitation of Louis XVI. Lagrange spent the remainder of his life in Paris. The successive Revolutionary governments honored him, and when the École Polytechnique was founded in 1797, Lagrange was appointed professor. He was president of the commission for the reform of weights and measures and a member of the Board of Longitude. Napoleon made him a senator and a count. Lagrange, a gentle and unassuming man, died on April 10, 1813.

Theory of Numbers and Algebra

Like Euler, Lagrange turned his attention to the many results that had been stated without proof by Pierre de Fermat. In particular, he completed Euler's work on the Diophantine equation x² - ay² = 1. Lagrange demonstrated that a general solution is always possible and that all the solutions can be found by developing a as a continued fraction. He also proved the theorem that an integer is either a square or the sum of two, three, or four squares, as well as Wilson's theorem that if n is a prime, (n - 1)! + 1 is a multiple of n.

Third-order determinants were used implicitly by Lagrange in a memoir of 1773; in particular he expressed the square of a determinant as another determinant. Work on the binary quadratic form ax² + 2bxy + cy² led him to the result that the discriminant was unchanged by a particular linear transformation. This was the first step in the development of the theory of algebraic invariance, which has found important applications in the general theory of relativity.

Lagrange also sowed the seed of another important branch of mathematics, namely, the theory of groups. Generality was the characteristic goal of all his researches. In seeking a general method of solving algebraic equations, he found that the common feature of the solutions of quadratics, cubics, and quartics was the reduction of these equations to equations of lower degree. Applied to a quintic equation, however, the method led to an equation of degree six. Attempts to explain this result led him to study rational functions of the roots of the equation. The properties of the symmetric group, that is, the group of permutations of the roots, provide the key to the problem. Lagrange did not explicitly recognize groups, but he obtained implicitly some of the simpler properties, including the theorem known after him, which states that the order of a subgroup is a divisor of the order of the group. Évariste Galois introduced the term "group" and proved that quintic equations were not in general solvable by radicals.

Differential Equations

An early memoir written by Lagrange in Turin is devoted to the problem of the propagation of sound. Considering the disturbance transmitted along a straight line, he reduced the problem to the same differential equation arising in the study of the transverse vibrations of a string. The form of the curve assumed by such a string, he deduced, can be expressed as y =a sin mx sin nt. Discussing previous solutions of the partial differential equation, he supported Euler in supposing that Jean d'Alembert's restriction to functions having Taylor expansions was not necessary. He failed, however, to recognize the generality of Daniel Bernoulli's solution in the form of a trigonometric series. Later he failed to recognize the importance of J. B. J. Fourier's ideas, first stated in 1807, which are fundamental for the solution of partial differential equations with given boundary conditions. Yet it was Lagrange who, in a series of memoirs written between 1772 and 1785, transformed the study of partial differential equations into a definite branch of mathematics; previously, mathematicians had treated only a few particular equations without a general method. Among Lagrange's important contributions to the subject was the explanation of the relationship between singular solutions and envelopes.

Calculus of Variations

Euler gave the name calculus of variations to the new branch of mathematics which he invented for the solution of isoperimetric problems. Lagrange thought that the method Euler employed lacked the simplicity desirable in a subject of pure analysis; in particular, he objected to the geometrical element in Euler's method. Lagrange developed the theory, notation, and applications of the calculus of variations in a number of memoirs published in the Miscellanea Taurinensia. If y = f(x), the value of y can be changed either by changing the variable x or by changing the form of the function. The first type of change is represented by the differential dy. Lagrange represented the second type of change, the variation, by y. In applications the problem is essentially that of maximizing or minimizing integrals by variation in the form of the function integrated.

The basic ideas of the calculus of variations are quite difficult and were not fully grasped by Lagrange's contemporaries. He did not attempt a rigorous justification of the principles, but the results amply vindicated the method.

Work in Mechanics

A century separated the publication of Lagrange's Mécanique analytique (1788) and Isaac Newton's Principia (1687). With Newton, as Lagrange recognized, mechanics became a new science, but his characterization of Newton's method as synthetic is a distortion which unfortunately is still widely believed. To the eye, Newton's Principia may have the appearance of Greek geometry; yet detailed study of the text leaves no doubt of the analytical foundation of the work. Certainly Lagrange himself brought analytical mechanics to perfection, though he recognized Euler as his precursor in the application of analysis to mechanics. In the preface of his work, Lagrange remarked that no diagrams would be found, but only algebraic equations.

The aim of Mécanique analytique, undoubtedly Lagrange's greatest work, was to present a mechanics of general applicability based on a minimum of principles. Moreover, Lagrange regarded the principles of mechanics as suppositions, not eternal truths, so that the purpose of mechanics was not to explain but simply to describe. To Lagrange we owe the first suggestion that this could be accomplished in terms of a geometry of four dimensions.

With the aid of the calculus of variations, Lagrange succeeded in deducing both solid and fluid mechanics from the principle of virtual work and D'Alembert's principle. The general formulation of the first of these he attributed to Johann Bernoulli. Lagrange did not regard the principle as an axiom but rather as a general expression of the law of equilibrium deduced from the laws of the lever and the composition of forces or, alternatively, from the properties of strings and pulleys. Statics then appeared as a consequence of the law of virtual velocities. In one of its formulations D'Alembert's principle states that the external forces acting on a set of particles and the effective forces reversed are in equilibrium; dynamical problems are thereby reduced to statics and consequently can be solved by the application of the principle of virtual velocities.

Instead of applying the principles to particular problems, Lagrange sought a general method; this led him to the idea of generalized coordinates. From dynamical equations he deduced the principle of conservation of vis viva and also the principle of least action, which Euler had formulated correctly for the special case of a single particle. Moreover, Lagrange removed the mystery that had surrounded the principle of least action by pointing out that it was based essentially on that of vis viva.

Work in Calculus

Lagrange's Théorie des fonctions analytiques (1797) was the most important of several attempts that were made about this time to provide a logical foundation for the calculus. While admitting that operations with differentials were expeditious in solving problems, he believed that compensating errors were involved in this method. To avoid these, he attempted to develop the calculus by purely algebraic processes.

First Lagrange derived by algebra the Taylor series, with remainder, for the function f(x + h), and then he defined the derived functions f(x), f(x), ... in terms of the coefficients of the powers of h. His view that this procedure avoided the concepts of limits and infinitesimals was in fact illusory, for these notions enter into the critical question of convergence, which Lagrange did not consider. Again, he was mistaken in supposing that all continuous functions could be expanded in Taylor series. Despite its defects, Lagrange's Théorie des fonctions analytiques was the first theory of functions of a real variable and focused attention on the derived function, as he termed it, the quantity which has become the central concept of the calculus.

Further Reading

• Extracts from Lagrange's work on the theory of equations and the calculus of variations are given in D. J. Struik, A Source Book in Mathematics, 1200-1800 (1969), and an extract on the principle of virtual velocities is given in William Francis Magie, A Source Book in Physics (1935). An introduction to the calculus of variations is in F. B. Hildebrand, Methods of Applied Mathematics (1954; 2d ed. 1965). For a readable account of Lagrange's dynamical equations see M. R. Spiegel, Theoretical Mechanics (1967). Background studies of mathematics which discuss Lagrange are Eric T. Bell, Men of Mathematics (1937); Alfred Hooper, Makers of Mathematics (1948); and Herbert Westren Turnbull, The Great Mathematicians (1961).