Mechanica sive motus scientia analytice exposita.

EULER, Leonhard

Published by Academy of Sciences, St. Petersburg, 1736

THE BIRTH OF ANALYTICAL MECHANICS. First edition of "Euler s famous work on mechanics in which he introduced the use of analytical methods instead of the geometrical methods of Newton and his followers" (Timoshenko, p. 29). Mechanica won the praise of many leading scientists of the time: Johann Bernoulli said of the work that "it does honour to Euler s genius and acumen," while Lagrange in his own Mécanique analytique acknowledged Euler s mechanics to be "the first great work where Analysis has been applied to the science of motion." "In an introduction to the Mechanica Euler outlined a large program of studies embracing every branch of science. The distinguishing feature of Euler s investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling the problems to be solved in a clear and direct way. Euler s concept is manifest in both the introduction and the very title of the book, Mechanica sive Motus Scientia analytice exposita. This first large work on mechanics was devoted to the kinematics and dynamics of a point-mass. The first volume deals with the free motion of a point-mass in a vacuum and in a resisting medium; the section on the motion of a point-mass under a force directed to a fixed center is a brilliant analytical reformulation of the corresponding section of Newton s Principia; it was sort of an introduction to Euler s further works on celestial mechanics. In the second volume, Euler studied the constrained motion of a point-mass; he obtained three equations of motion in space by projecting forces on the axis of a moving trihedral of a trajectory described by a moving point, i.e. on the tangent, binormal and principal normal. Motion in the plane is considered analogously. In the chapter on the motion of a point on a given surface, Euler solved a number of problems on the differential geometry of surfaces and of the theory of geodesics" (DSB). "On the path to rational mechanics, the principal question facing Euler was how to reorganize mechanics by applying the analytic method, along with how to formulate the fundamental, general principles at its base. The analytic method, which refers to his systematic employment of the terms and symbols of differential and integral calculus, was for Euler a heuristic mode of thought and a means to an end rather than simply an algorithmic approach. Behind the synthetic geometric format of Newton s Principia, including conic sections and quadratures, Euler saw a host of methods: infinitesimals, geometric limit procedures, interpolation techniques, and infinite series. Still, Newton had not devised the partial differential equations of motion. Euler was more indebted to the Continental geometers, led by the Bernoullis, who in an ad hoc fashion were applying differential calculus, then barely half a century old, to achieve precise solutions of problems of motion, its generations, and its alteration including instantaneous acceleration in physics that neither analytic geometry nor the ancient synthetic method of Euclidean geometry could attain. Breaking decisively with those two geometries, Euler sought to express Newtonian mechanics with Leibnizian differential equations and the partial differential equations that the Bernoullis had introduced, and he added many more of the latter. He differed from Leibniz, however, in evolving his concept of function by examining formulas and the relations between quantities rather than curves; he thus formalized the subject. By 1734 Euler had returned to writing the Mechanica with a singleness of purpose and soon completed the first volume. By then he had written for the Commentarii nine articles on mechanics and eleven on s. Seller Inventory # 5538

Bibliographic Details

Title: Mechanica sive motus scientia analytice ...
Publisher: Academy of Sciences, St. Petersburg
Publication Date: 1736
Edition: First edition.