( This is a reproduction of a book published before 1923....)
This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Mecanique Analytique, Nouvelle Edition, Revue Et Augmentee Par L'auteur. - Paris, Veuve Courcier 1811-1815 Joseph Louis comte de Lagrange Veuve Courcier, 1811
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(This is a reproduction of a book published before 1923. T...)
This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Traite De La Resolution Des Equations Numeriques De Tous Les Degres. Nouvelle Ed. Revue Et Augmentee Par L'auteur Joseph Louis comte de Lagrange Courcier, 1808
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(This is a reproduction of a book published before 1923. T...)
This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Theorie Des Fonctions Analytiques. Nouvelle Edition Revue Et Augmentee Par L'auteur Joseph Louis comte de Lagrange Veuve Coucier, 1813
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(This is a reproduction of a book published before 1923. T...)
This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Leçons Sur Le Calcul Des Fonctions. With; Volume 14 Of Journ., École Polytechn., Cahier 12; Leçons Sur Le Calcul Des Fonctions. With; Joseph Louis Lagrange (comte.) Joseph Louis Lagrange (comte.) Mathematics; Calculus; Mathematics / Calculus; Mathematics / Differential Equations
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Born as Giuseppe Lodovico Lagrangia on 25 January 1736
in Turin, Piedmont-Sardinia, Lagrange was of Italian and French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, and married an Italian; so did his grandfather and his father. His mother was from the countryside of Turin.
He studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician.
By 1761 Lagrange was already recognized as one of the greatest living mathematicians. In 1764 he was awarded a prize offered by the French Academy of Sciences for an essay on the libration of the Moon (i. e. , the apparent oscillation that causes slight changes in position of lunar features on the face that the Moon presents to the Earth). In this essay he used the equations that now bear his name. His success encouraged the academy in 1766 to propose, as a problem, the theory of the motions of the satellites of Jupiter. The prize was again awarded to Lagrange, and he won the same distinction in 1772, 1774, and 1778. In 1766, on the recommendation of Euler and the French mathematician Jean d’Alembert, Lagrange went to Berlin to fill a post at the academy vacated by Euler, at the invitation of Frederick the Great, who expressed the wish of “the greatest king in Europe” to have “the greatest mathematician in Europe” at his court.
Lagrange stayed in Berlin until 1787. His productivity in those years was prodigious: he published papers on the three-body problem, which concerns the evolution of three particles mutually attracted according to Sir Isaac Newton’s law of gravity; differential equations; prime number theory; the fundamentally important number-theoretic equation that has been identified (incorrectly by Euler) with John Pell’s name; probability; mechanics; and the stability of the solar system. In his long paper “Réflexions sur la résolution algébrique des équations” (1770; “Reflections on the Algebraic Resolution of Equations”), he inaugurated a new period in algebra and inspired Évariste Galois to his group theory.
A kind and quiet man, living only for science, Lagrange had little to do with the factions and intrigues around the king. When Frederick died, Lagrange preferred to accept Louis XVI’s invitation to Paris. He was given apartments in the Louvre, was continually honoured, and was treated with respect throughout the French Revolution. From the Louvre he published his classic Mécanique analytique, a lucid synthesis of the hundred years of research in mechanics since Newton, based on his own calculus of variations, in which certain properties of a mechanistic system are inferred by considering the changes in a sum (or integral) that are due to conceptually possible (or virtual) displacements from the path that describes the actual history of the system. This led to independent coordinates that are necessary for the specifications of a system of a finite number of particles, or “generalized coordinates. ” It also led to the so-called Lagrangian equations for a classical mechanical system in which the kinetic energy of the system is related to the generalized coordinates, the corresponding generalized forces, and the time. The book was typically analytic; he stated in his preface that “one cannot find any figures in this work. ”
The Revolution, which began in 1789, pressed Lagrange into work on the committee to reform the metric system. When the great chemist Antoine-Laurent Lavoisier was guillotined, Lagrange commented, “It took them only an instant to cut off that head, and a hundred years may not produce another like it. ” When the École Centrale des Travaux Publics (later renamed the École Polytechnique) was opened in 1794, he became, with Gaspard Monge, its leading professor of mathematics. His lectures were published as Théorie des fonctions analytiques (1797; “Theory of Analytic Functions”) and Leçons sur le calcul des fonctions (1804; “Lessons on the Calculus of Functions”) and were the first textbooks on real analytic functions. In them Lagrange tried to substitute an algebraic foundation for the existing and problematic analytic foundation of calculus - although ultimately unsuccessful, his criticisms spurred others to develop the modern analytic foundation. Lagrange also continued to work on his Mécanique analytique, but the new edition appeared only after his death.
Generality was the characteristic goal of all his researches.
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.
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 [dgr ]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.
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.
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.
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.
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.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he has transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and presented the so-called mechanical "principles" as simple results of the variational calculus.
( This is a reproduction of a book published before 1923....)
(This is a reproduction of a book published before 1923. T...)
(This is a reproduction of a book published before 1923. T...)
(This is a reproduction of a book published before 1923. T...)
He was raised as a Roman Catholic, but later on became an agnostic.
Euler proposed Lagrange for election to the Berlin Academy and he was elected on 2 September 1756. He was elected a Fellow of the Royal Society of Edinburgh in 1790, a Fellow of the Royal Society and a foreign member of the Royal Swedish Academy of Sciences in 1806. In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Count of the Empire.
In 1767 he married his cousin Vittoria Conti.
