## Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes.

Lausanne & Geneve: Marcum-Michaelem Bosquet & Socios, 1744.

First edition of “Euler’s most valuable contribution to mathematics in which he developed the concept of the calculus of variations” (Norman). “’This work displays an amount of mathematical genius seldom rivalled – Cajori. It may be said to have created the calculus of variations” (Evans). “The book brought him immediate fame and recognition as the greatest living mathematician” (Kline, p. 579). Du Pasquier (pp. 50-51) called the Methodus inveniendi “one of the finest monuments of the genius of Euler” who, he continued, “founded the calculus of variations which has become, in the twentieth century, one of the most efficient of the means of investigation employed by mathematicians and physicists. The recent theories of Einstein [i.e., general relativity] and the applications of the principle of relativity have greatly increased the importance of the calculus of variations which Euler created”. “The Methodus inveniendi is of two-fold interest for historians of mathematics. First, it was a highly successful synthesis of what was then known about problems of optimization in the calculus, and presented general equational forms that became standard in the calculus of variations. Euler’s method was taken up by Joseph Louis Lagrange (1736–1813) 20 years later and brilliantly adapted to produce a novel technique for solving variational problems. The two appendices to Euler’s book applied variational ideas to problems in statics and dynamics, and these too became the basis for Lagrange’s later researches. Second, in Euler’s book some of his distinctive contributions to analysis appear for the first time or very nearly the first time: the function concept, the definition of higher-order derivatives as differential coefficients; and the recognition that the calculus is fundamentally about abstract relations between variable quantities, and only secondarily about geometrical curves. The Methodus inveniendi is an important statement of Euler’s mathematical philosophy as it had matured in the formative years of the 1730s” (Fraser, p. 169). The Methodus inveniendi consist of six chapters, delivered to the publisher in July 1743, and two appendices, delivered in December 1743, the first on elastic curves, the second on the principle of least action.

“By the spring of 1741 at the latest, Euler had finished in Saint Petersburg a draft of Methodus inveniendi lineas curvas maximiminimive proprietate gaudentes (A method of finding curves that show some property of maximum or minimum), though without the two appendixes that would come later. In May 1743 the publisher Marcus-Michael Bousquet visited Berlin to present the king a copy of Johann I Bernoulli’s Opera omnia; Bousquet was impressed with the work of Euler, who handed him the main body of the Methodus inveniendi manuscript. That month Euler wrote to the Genevan mathematician Gabriel Cramer, asking him to proofread and correct for Bousquet that small book written on isoperimetric problems. Cramer was known to Euler in part for his commentaries and annotations to Bernoulli’s Opera. Finding the manuscript admirable, Cramer agreed. Daniel Bernoulli, attempting at the time to determine maxima and minima of elastic curves, recommended to Euler the addition to the Methodus inveniendi text of two appendixes on elasticity. Among all curves of a stated length that had tangents at the ends, Euler would minimize the integral of an element of the arc length divided by the radius of curvature squared. By the end of that summer Euler had completed the appendixes, but they were not sent to Bousquet in Lausanne until December” (Calinger, pp. 202-3).

“The 320-page Methodus inveniendi in quarto format was the first book Euler published in the 1740s. A landmark treatise, it appeared in print in September 1744. This was fast, for Euler had submitted its appendixes only the previous December. The study made him the principal creator of the first stage of a new branch of mathematics, the classical calculus of variations, which in the paths of motion sought to determine maximal or minimal lengths of plane curves, if any existed, and pursued extreme values for integrals (often named functionals). Its first section asserts that ‘since the fabric of the universe is most perfect, and is the work of a most Wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear.’ A letter of December 1745 to Maupertuis repeats Euler’s conclusion from the Methodus inveniendi that ‘in the natural course of movements there is a constant maximum or minimum, and I have determined … that all trajectory curves, and all bodies drawn toward a fixed center or mutually drawn together have been so described.’ The baroque title of the work derives from Euler’s perception of the new field as a Leibnizian ars inveniendi, or method of discovery. Its twentieth-century editor Constantin Caratheodory called it ‘one of the most beautiful mathematical works ever written.’

“Through skillful organization in arranging more than a hundred increasingly complex problems in eleven categories and providing new direction and ideas in the Methodus inveniendi, Euler replaced the previous ad hoc procedures for special case problems in the formative stage of the calculus of variations, instead offering standard differential equations for general solutions and giving techniques for reaching these equations. His work impressively extended and refined that of Brook Taylor, Jakob Hermann and Jakob and Johann I Bernoulli, and was the culmination of their efforts; Euler’s success where they had failed in creating the new field magnified his reputation. His methods were closest to Jakob Bernoulli’s in that they require two degrees of freedom to extremalize or optimize a curve. Euler’s attention to curves, including the tautochrone, isochrone, and brachistochrone, and his use of isoperimetry, a subject popular in the late seventeenth century, kept the new field largely geometric. This was one of the several occasions on which Euler searched for a different solution for the brachistochrone problem [that of determining the curve of quickest descent joining two given points]. Chapter 3 of the Methodus invenendi generalizes this early optimization problem, essentially finding sets of extremals, and section 45 provides the most elegant solution up to 1744, correcting Hermann's solution of the brachistochrone problem in a resisting medium. Chapter 2 contains the vital innovation, the Euler differential equation or the first necessary condition for extremals (now known as the Euler-Lagrange Equation, it is the basic equation in the modern calculus of variations), and chapter 4 takes up the problem of its invariance, but Euler did not recognize that the equation was insufficient to guarantee an extremum. As was typical for Euler, he gave some hundred examples to illustrate its results.

“In ‘De curvis elasticis,’ the first of two significant appendixes to the Methodus inveniendi that Daniel Bernoulli had proposed in a series of letters, Euler – over sixty-six pages with ninety-seven problems – presented the earliest study in print to employ the calculus of variations to solve problems in the theory of elastic curves and surfaces. It is thus the initial general tract on the mathematical theory of elasticity. To obtain his equations, Euler employed the methods of final causes and efficient causes; these came initially from Aristotle, and among others Leibniz had recently studied them. Final causes are teleological, giving the purpose or design of something and contrast with mechanical explanations, which Euler believed to draw upon existing variational principles. Closest to the modern definition of the term cause, efficient causes probe the properties of matter and mechanics explaining phenomena. Euler computed the shape of elastica from the forces of efficient causes, and checked to confirm that both approaches led to the same answer; without an appeal to both, the best explanations might not be reached. In his inventory of problems, Euler enumerated nine species of elastic curves and explained how elastic bands bend and oscillate.

“The appendix’s topics include the problem of the vibrating membrane, at the same time that Daniel Bernoulli was investigating the simpler vibrating string. Euler's buckling formula first appears here; it determines the maximum critical load, now called the first elastostatic eigenvalue, which an ideal, slender, long rod pinned at its ends can carry before it buckles. The critical axial load applied at its center of gravity needed to bend the rod depends upon the stiffness of its material and how the rod is supported at its ends, and it is proportional to the inverse square of the length of the rod. Euler also computed elastokinetic eigenvalues, eigenfrequencies of oscillations of the rod's transversal, and associated eigenfunctions, giving the shapes of a deformed rod.

“The other appendix, the ten-page ‘De motu projectorum in medio non resistente, per methodum maximorum ac minimorum determinando’ (On the motion of bodies in a non-resisting medium determined by the method of maxima and minima), introduces a general form of the principle of least action and experimentally determines absolute elasticity. His letters to Daniel Bernoulli show that by late 1738 Euler had mastered that principle” (ibid., pp. 223-7)

Dibner 111; Evans 9; Horblit 28; Nomran 731; Sparrow 60. Craig G. Fraser in: Landmarks Writings in Western Mathematics, chapter 12; Cajori, History of Mathematics; Calinger, Leonhard Euler. Mathematical Genius of the Enlightenment; Du Pasquier, Léonard Euler et ses amis; Kline, Mathematical Thought from Ancient to Modern Times; Roberts & Trent, Bibliotheca Mechanica, p. 104: “For the purposes of mechanics, the significance of this work lies in the appendix, which deals with geometrical forms of elastic curve … The present work illustrates the first solution to the problem of the buckling of a column.”

4to (240 x 193 mm), pp. [2], [1], 2-322, [1], with five engraved folding plates. Title in red and black with large engraved device and large woodcut headpiece and initial on first page of text. Contemporary calf rebacked, spine gilt in compartments red lettering-piece.

Item #5857

Price: \$9,500.00

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