Lectures on Elementary Mathematics, by Joseph Louis Lagrange docx

The Project Gutenberg EBook of Lectures on Elementary Mathematics, by Joseph Louis Lagrange This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Lectures on Elementary Mathematics Author: Joseph Louis Lagrange Translator: Thomas Joseph McCormack Release Date: July 6, 2011 [EBook #36640] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK LECTURES ON ELEMENTARY MATHEMATICS *** Produced by Andrew D. Hwang. transcriber’s note The camera-quality files for this public-domain ebook may be downloaded gratis at www.gutenberg.org/ebooks/36640. This ebook was produced using OCR text provided by the University of Toronto Gerstein Library through the Internet Archive. Minor typographical corrections and presentational changes have been made without comment. This PDF file is optimized for screen viewing, but may easily be recompiled for printing. Please consult the preamble of the L A T E X source file for instructions and other particulars. ON ELEMENTARY MATHEMATICS IN THE SAME SERIES. ON CONTINUITY AND IRRATIONAL NUMBERS, and ON THE NATURE AND MEANING OF NUMBERS. By R. Dedekind. From the German by W. W. Beman. Pages, 115. Cloth, 75 cents net (3s. 6d. net). GEOMETRIC EXERCISES IN PAPER-FOLDING. By T. Sundara Row. Edited and revised by W. W. Beman and D. E. Smith. With many half-tone engravings from photographs of actual exercises, and a package of papers for folding. Pages, circa 200. Cloth, $1.00 net (4s. 6d. net). (In Preparation.) ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. By Augustus De Morgan. Reprint edition with portrait and bibliographies. Pp., 288. Cloth, $1.25 net (4s. 6d. net). LECTURES ON ELEMENTARY MATHEMATICS. By Joseph Louis Lagrange. From the French by Thomas J. McCormack, With portrait and biography. Pages, 172. Cloth, $1.00 net (4s. 6d. net). ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. By Augustus De Morgan. Reprint edition. With a bibliography of text-books of the Calculus. Pp., 144. Price, $1.00 net (4s. 6d. net). MATHEMATICAL ESSAYS AND RECREATIONS. By Prof. Hermann Schubert, of Hamburg, Germany. From the German by T. J. McCormack. Essays on Number, The Magic Square, The Fourth Dimension, The Squaring of the Circle. Pages, 149. Price, Cloth, 75c. net (3s. net). A BRIEF HISTORY OF ELEMENTARY MATHEMATICS. By Dr. Karl Fink, of T¨ubingen. From the German by W. W. Beman and D. E. Smith. Pp. 333. Cloth, $1.50 net (5s. 6d. net). THE OPEN COURT PUBLISHING COMPANY 324 DEARBORN ST., CHICAGO. LONDON: Kegan Paul, Trench, Tr¨ubner & Co. [...]... 1 Systems of numeration — Fractions — Greatest common divisor — Continued fractions — Terminating continued fractions — Converging fractions — Convergents — A second method of expression — A third method of expression — Origin of continued fractions — Involution and evolution — Proportions — Arithmetical and geometrical proportions — Progressions — Compound interest — Present values and annuities... numerical equations — Position of the roots of numerical equations — Application of geometry to algebra — Representation of equations by curves — Graphic resolution of equations — The consequences of the graphic resolution — Intersections indicate the roots — Case of multiple roots — General conclusions as to the character of the roots — Limits of the real 87 contents xix roots of equations — Limits of... Biquadratic equations — The method of Descartes — The determined character of the roots — A third method — The reduced equation — Euler’s formulæ — Roots of a biquadratic equation Lecture IV On the Resolution of Numerical Equations Limits of the algebraical resolution of equations — Conditions of the resolution of numerical equations — Position of the roots... by 7 — Negative remainders — Test of divisibility by 11 — Theory of remainders — checks on multiplication and division — Evolution — Rule of three — Applicability of the rule of three — Theory and practice — Alligation — Mean values — Probability of life — Alternate alligation — Two ingredients — Rule of mixtures — Three ingredients — General solution — Development — Resolution by continued fractions... equations — The development and solution — A machine for solving equations Lecture V On the Employment of Curves in the Solution of Problems 115 Geometry applied to algebra — Method of resolution by curves — Problem of the two lights — Various solutions — General solution — Minimal values — Preceding analysis applied to biquadratic equations — Consideration of equations... curve of errors — Solution of a problem by the curve of errors — Problem of the circle and inscribed polygon — Solution of a second problem by the curve of errors — Problem of the observer and three objects — Employment of the curve of errors — Eight possible solutions contents xx of the preceding problem — Reduction of the possible solutions in practice — General conclusion on the method of curves... another and similar fraction, which is to be added to the last denominator, and which by our supposition is one divided by three And so with the rest You will then have the fraction 1 3+ 2+ 1 3 + as the expression of your ratio between the proposed length and the adopted measure Fractions of this form are called continued fractions, and can be reduced to ordinary fractions by the common rules Thus, if we... Laplace and Monge upon the activity which made these schools for generations to come exemplars of practical scientific education, systematising by his lectures there, and putting into definitive form, the science of mathematical analysis of which he had developed the extremest capacities Lagrange s activity at Paris was interrupted only once by a brief period of melancholy aversion for mathematics, a... fraction, i.e., if we consider only the first remainder and neglect the second, we shall have 3 + 1 , which is 2 equal to 7 Considering only the first and the second remainders, 2 4 on arithmetic we stop at the second fraction, and shall have 3 + 2+ 1 3 1 Now 2+ 1 3 = 7 We shall have therefore 3 + 3 , which is equal to 3 7 24 7 And so on with the rest If we arrive in the course of the operation at... 7, 3 The following is a rule by which we can write down at once the convergent fractions which result from these quotients, without developing the continued fraction 5 on arithmetic The first quotient, supposed divided by unity, will give the 3 first fraction, which will be too small, namely, 1 Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity . w2 ha" alt="" LECTURES ON ELEMENTARY MATHEMATICS BY JOSEPH LOUIS LAGRANGE FROM THE FRENCH BY THOMAS J. McCORMACK SECOND EDITION CHICAGO THE OPEN COURT PUBLISHING COMPANY LONDON AGENTS Kegan. numeration. — Fractions. — Greatest common divisor. — Continued fractions. — Termi- nating continued fractions. — Converging fractions. — Convergents. — A second method of expression. — A third method. second course of lectures on arithmetic and algebra, introductory to these of Lagrange, by Laplace. With the exception of a German translation by Niederm¨uller (Leipsic, 1880), the lectures of Lagrange