Joseph Louis Lagrange - Lectures on Elementary Mathematics

LECTURES ON
ELEMENTARY
MATHEMATICS

JOSEPH LOUIS LAGRANGE

Translated from the French by

Thomas J. McCormack

DOVER PUBLICATIONS, INC.
Mineola, New York

Bibliographical Note

This Dover edition, first published in 2008, is a republication of the work originally published by The Open Court Publishing Company, Chicago, in 1898. The biographical sketch from the Open Court edition has been replaced by the more detailed biography of Lagrange from A Short Account of the History of Mathematics by W. W. Rouse Ball, also available from Dover Publications, Inc. (0-486-20630-0).

Library of Congress Cataloging-in-Publication Data

Lagrange, J. L. (Joseph Louis), 17361813.

Lectures on elementary mathematics / Joseph Louis Lagrange ; translated from the French by Thomas J. McCormack. Dover ed.

p. cm.

Originally published: Chicago : Open Court, 1898.

Includes bibliographical references and index.

eISBN-13: 978-0-486-15502-9

1. Mathematics. I. Title.

QA7.L17 2008

510dc22

2007053034

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

PREFACE.

T HE present work, which is a translation of the Leons lmentaires sur les mathematiques of Joseph Louis Lagrange, the greatest of modern analysts, and which is to be found in Volume VII. of the new edition of his collected works, consists of a series of lectures delivered in the year 1795 at the Ecole Normale,an institution which was the direct outcome of the French Revolution and which gave the first impulse to modern practical ideals of education. With Lagrange, at this institution, were associated, as professors of mathematics. Monge and Laplace, and we owe to the same historical event the final form of the famous Gomhde descriptive, as well as a second course of lectures on arithmetic and algebra, introductory to these of Lagrange, by Laplace.

With the exception of a German translation by Niedermller (Leipsic, 1880), the lectures of Lagrange have never been published in separate form ; originally they appeared in a fragmentary shape in the Sances des Ecoles Normales, as they had been reported by the stenographers, and were subsequently reprinted in the journal of the Polytechnic School. From references in them to subjects afterwards to be treated it is to be inferred that a fuller development of higher algebra was intended,an intention which the brief existence of the Ecole Normale defeated. With very few exceptions, we have left the expositions in their historical form, having only referred in an Appendix to a point in the early history of algebra.

The originality, elegance, and symmetrical character of these lectures have been pointed out by DeMorgan, and notably by Dhring, who places them in the front rank of elementary expositions, as an exemplar of their kind. Coming, as they do, from one of the greatest mathematicians of modern times, and with all the excellencies which such a source implies, unique in their character as a reading-book in mathematics, and interwoven with historical and philosophical remarks of great helpfulness, they cannot fail to have a beneficent and stimulating influence,

The thanks of the translator of the present volume are due to Professor Henry B. Fine, of Princeton, N. J., for having read the proofs.

THOMAS J. MCCORMACK.

LA SALLE, ILLINOIS, August 1, 1898.

Joseph Louis Lagrange (17361813)

From A Short Account of the History of Mathematics
(4th edition, 1908) by W. W. Rouse Ball.

Joseph Louis Lagrange, the greatest mathematician of the eighteenth century, was born at Turin on January 25, 1736, and died at Paris on April 10, 1813. His father, who had charge of the Sardinian military chest, was of good social position and wealthy, but before his son grew up he had lost most of his property in speculations, and young Lagrange had to rely for his position on his own abilities. He was educated at the college of Turin, but it was not until he was seventeen that he shewed any taste for mathematicshis interest in the subject being first excited by a memoir by Halley, across which he came by accident. Alone and unaided he threw himself into mathematical studies; at the end of a years incessant toil he was already an accomplished mathematician, and was made a lecturer in the artillery school.

The first fruit of Lagranges labours here was his letter, written when he was still only nineteen, to Euler, in which he solved the isoperimetrical problem which for more than half a century had been a subject of discussion. To effect the solution (in which he sought to determine the form of a function so that a formula in which it entered should satisfy a certain condition) he enunciated the principles of the calculus of variations. Euler recognized the generality of the method adopted, and its superiority to that used by himself; and with rare courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus. The name of this branch of analysis was suggested by Euler. This memoir at once placed Lagrange in the front rank of mathematicians then living.

In 1758 Lagrange established with the aid of his pupils a society, which was subsequently incorporated as the Turin Academy, and in the five volumes of its transactions, usually known as the Miscellanea Taurinensia, most of his early writings are to be found. Many of these are elaborate memoirs. The first volume contains a memoir on the theory of the propagation of sound; in his he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely; in this paper he points out a lack of generality in the solutions previously given by Taylor, DAlembert, and Euler, and arrives at the conclusion that the form of the curve at any time t is given by the equation y = a sin mx sin nt. The article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in this volume are on recurring series, probabilities, and the calculus of variations.

The second volume contains a long paper embodying the results of several memoirs in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics.

The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the integral calculus; a solution of Fermats problem mentioned above, to find a number x which will make (x2n + 1) a square where n is a given integer which is not a square; and the general differential equations of motion for three bodies moving under their mutual attractions. In 1761 Lagrange stood without a rival as the foremost mathematician living; but the unceasing labour of the preceding nine years had seriously affected his health, and the doctors refused to be responsible for his reason or life unless he would take rest and exercise. Although his health was temporarily restored his nervous system never quite recovered its tone, and henceforth he constantly suffered from attacks of profound melancholy.

The next work he produced was in 1764 on the libration of the moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalized equations of motion, equations which he first formally proved in 1780.