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Sur les Courbes Tautochrones. (and ) Nouvelles Réflexions sur les Tautochrones. - [THE CALCULUS OF VARIATIONS]
(Berlin, Haude et Spener, 1767 and Berlin, Ch. Fr. Voss, 1774). 4to. Without wrappers as issued in ""Mémoires de l'Academie Royale des Sciences et Belles Lettres"" Tome XXI, pp. 364-380 and "" Nouveau Mémoires..."", pp. 97-122.
Both first edition in the journal form. Huygens proved Geometrically in 1659 that the tautochrone was a cycloid curve. This solution was later used to attack the problem of the Brachistochrone curve. Jacob Bernoulli solved the problem by using calculus in a paper from 1690, which for the first time used the term 'integral'. Both Lagrange and Euler loked for an analytical solution to the problem. Lagrange, in the papers offered here, developed a formal calculus based on the analogy between Newton's theorem and the successive differentiations of the product of two functions. He also communicated this to Eule in a letter written in Latin slightly before the Italian publication. In a letter to D'Alembert in 1769 Lagrange confirmed that this method of maxima and minima was the first fruit of his studies - he was only 19 when he divised it - and that he regarded it as his best work.A paper by Leonhard Euler:Éclaircissement plus détailles sur La generation et Propagation du Son et sur la Formation de L'Echo"" Berlin Academy Royale 1767"" in first edition withbound.
Sur L'Integration des Equations à differences partielle du premier ordre. - [A MASTERPIECE OF ANALYSIS]
(Berlin, C.F. Voss, 1774). 4to. Uncut with wide margins, without wrappers as issued in ""Nouveaux Memoires de L'Academie Royales des Sciences et Belles- Lettres"", Année MDCCLXXII, pp. 353-372.
First edition of a work which is a breakthrough in the theory of ""First Order Partial Differential Equations"", generalizing the method of variation of parameters for solving differential equations. "" The oldest theory of integration of partial differential equations of the first order are due to Lagrange"" it is based on the fundamental fact that the most general solution of such differential equations can be calculated with the help of differentiations and eliminations if a complete integral of the differential equationn is known"" - ""This problem (of partial differential equations) had only been lightly touched on by Clairaut, Euler, d'Alembert, and Condorcet. Lagrange wrote: ""Finally I have just read a memoir that Mr de Laplace presented recently.....This reading aweakened old ideas that I had on the same subject and resulted in the following investigations...(which constitute) a new and complete theory."" Laplace wrote on 3 February 1778 that he considered Lagrange's essay ""a masterpiece of analysis, by the importence of the subject, by the beauty of method, and by the elegant manner in which it is represented."" (DSB). - Parkinson, Breakthroughs 1774 M.
Two Papers. 1. Sur une Nouvelle Espece de Calcul rélatifs à différentiation & à l'intégration des quantités variables. - 2. Sur la Forme des Racines Imaginaires des Equations. - [A NEW FOUNDATION OF THE CALCULUS]
(Berlin, Ch. Fr. Voss, 1774). 4to. Large uncut copy, broadmargins, without wrappers as issued in ""Nouvaux Memoires d l'Academie Royale des Sciences et Belles-Lettres. Année 1772."" pp. 185-221 and pp. 222-258.
First edition of these two important papers. In the first paper ""Sur une Nouvelle Espece de Calcul..."" he made the most ambitious attempt to rebuild the foundation of the calculus. The paper is in fact an outline of his ""Theorie des functions"", issued later in 1797.""This work greatly impressed Lacroix, Condorcet and Laplace. Based on the analogy between powers of biniminals and differentials, it is one of the sources of the symbolic calculuses of the nineteenth century. A typical example of Lagrange's thinking as an analyst in this sentence taken form the memoir: ""Although the principle of the analogy (between powers and diffrentials) is nor self-evident, nevertheless, since the conclusions drawn from it are not thereby less exact, I shall make use of it to discover various theorems...""""(Jean Itard in DSB). The second paper ""Sur la Forme...""When D'Alembert red the paper he gave the following respons:""Your demonstration on imaginary roots seem to me to leave nothing to be desired, and I am very much obliged to you for the justice you have rendered to mine, which, in fact, has the minor fault (pwerhaps more apparant than real) of not being direct, bu which is quite simple and easy. D'Alembert was alluding to his Cause de vents (1747).""(DSB).
Mémoire sur le Passage de Vénus du 3 Juin 1769. - [ON THE TRANSIT OF VENUS 1769]
(Berlin, Haude et Spener, 1768). 4to. Without wrappers as extracted from ""Memoires de l'Academie Royale des Sciences et Belles-Lettres"", tome XXII, pp. 265-333 and 2 foled engraved plates.
First edition. ""During his period in Berlin he produced an extraordinary series of papers on astronomy, on general dynamics, and on a variety of subjects in pure mathematics. Several of the most importent of the astronomical papers were sent to Paris and obtained prizes offred by the Academy.""(Arthur Berry). ""Lagrange's work in Berlin far surpassed the classical aspects of classical mechanics. Soon after his arrival he presented ""Memoire sur la Passage de Venus du 3 Juin 1769"", an occasional work that disconcerted the professional astronomers and contained the first somewhat extended example of an elementary astronomical problem solved by the method of three rectangular coordinates."" (DSB). The analyses of this Venus transit was meant to provided data for estimating the sun-earth distance as the Venus went across the disk of the sun.
Sur les Réfractions Astronomiques.
(Berlin, Ch. Fr. Voss, 1774). 4to. Uncut without wrappers as issued in ""Nouveau Mémoires de l'Academie Royale des Sciences et Belles-Lettres"", pp. 259-282.
First edition, the Journal issue.
"LAGRANGE, (LA GRANGE), JOSEPH LOUIS. - LAGRANGE'S CONTINUED FRACTIONS.
Reference : 45924
(1770)
Additions au Mémoire Sur la Résolution des Équations Numériques.
(Berlin, Haude et Spener, 1770). 4to. Clean and fine without wrappers as issued in ""Mémoires de l'Academie Royale des Sciences et Belles-Lettres"", Tome XXIV, pp. 111-180. With titlepage to ""Classe de Mathematique"".
First appearance of Lagrange's importent paper in which he developed continous fraction solutions of equations.
"LAGRANGE, (LA GRANGE), JOSEPH LOUIS. - A FUNDAMENTAL MEMOIR IN THE THEORY OF NUMBERS
Reference : 49805
(1769)
Sur La Solution des Problemes Indéterminés du Second Degré.
(Berlin, Haude et Spener, 1769). 4to. No wrappers as issued in ""Memoires de L'Academie Royale des Sciences et Belles Lettres"", tome XXIII, pp. 165-310. Clean and fine.
First edition of a fundamental paper in the Theory of Numbers in which Lagrange gives a solution in integers of indeterminate equations of the second degree - a remarkable turning point in Diophantine analysis. - Fermat had asserted that he could determine when the more general equation x2-Ay2=B was solvable in integers and that he could solve it when solvable, but Lagrange solved it in this paper and furthermore he gives the complete solution to the problem of giving all integral solutions of a general equation where the coefficients are integers. - Cajori calls Lagrange ""One of the greatest mathematicians of all times."" - Poggendorff I:1344.
Analytische Mechanik aus dem Französischen mit einigen Anmerkungen und erläuternden Zusätzen von Friedrich Wilhelm August Murhard. - [A LANDMARK IN MATHEMATICS AND MECHANICS]
Götingen, Vandenhoeck und Ruprecht, 1797. 4to. Contemp. hcalf. Gilt spine. Titlelabel with gilt lettering. A paperlabel pasted on top of spine. Stamps on title-page. XX,573,(1) pp. Faint browning to a few leaves, otherwise clean and fine.
First German edition of this milestone work in applied mathematics (Mecanique Analitique, 1788) in which Lagrange regarded mechanics as a geometry of four dimensions and here set down the principle of virtual velocities as applied to mechanics. The work can lay claim today to be ""one of the outstanding landmarks in the history of both mathematics and mechanics"" (Sarton).""This is the first textbook to treat theoretical mechanics in a purely analytic way. Its mathematical importence stems mainly from the application of Lagrange's new formalization of the calculus of variations, and its significance for rational mechanics from the fact that it summarizes, for the first time in a logically coherent way, the conformity the newtonian and continental mechanics of the 18th century on the basis of general variational principles.""(Helmut Pulte in ""Landmark Writings in Western mathematics 1640-1940).Horblit, 61 (""Discovery of the general equations of motions of any system of bodies""). - Dibner, 112 (French edition)
"LAGRANGE, (LA GRANGE), JOSEPH LOUIS. - A BREAKTHROUGH IN ""THE THEORY OF EQUATIONS"".
Reference : 46556
(1769)
Sur la Résolution des Équations Numériques. (On the Solution of numerical Equations).
(Berlin, Haude et Spener, 1769). 4to. Without wrappers as issued in ""Mémoires de l'Academie Royale des Sciences et Belles-Lettres"", Année 1767, Tome XXXIII, pp. 311-352.
First edition of a monumental paper in the theory of equations by ""one of the greatest mathematicians of all times"" (Cajori). In this memoir, which deals with the solutiuon of numerical equations, Lagrange examines the roots of algebraic equations and provides methods of separating the real and imaginary roots and of approximating the real roots with continued fractions.Parkinson ""Breakthroughs"" 1767 P.
