## Mechanica sive motus scientia analytice exposita.

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 acknowledges 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).

“This is Euler’s outline of a program of studies embracing every branch of science, involving a systematic application of analysis. [It] thus laid the foundations of analytical mechanics and was the first published work in which e appeared. In addition, these two volumes were the result of Euler's consideration of the motion produced by forces acting on both free and constrained points.

“[The first] volume focuses on the kinematics and dynamics of a point-mass, introducing infinitely small bodies that can be considered to be points under certain assumptions. Euler focuses on single mass-points except for a few pages at the end of Chapter I, where he looks at the motion of one point relative to another moving point. He then looks at the nature of rest and uniform motion. In Chapter II, Euler states Newton's second law of motion. Throughout this volume, he considers the free motion of a point-mass in a vacuum and in a resisting medium so that all forces under consideration are known. Mathematically, acceleration is given to within an arbitrary multiplicand, and in each example he considers, the arguments of the force function are limited to position and speed. Thus, Euler devotes this volume to integrating particular second-order differential equations and to interpreting his results.

“For about half of this volume, Euler analyzes motion along straight lines. The remainder is mainly concerned with motion in a plane, with a few pages looking at motion along a skew curve. He introduces fixed rectangular Cartesian coordinates for the position of the mass-point but uses arc length as the independent variable to set up his differential equations of motion. He also resolves the enforced acceleration into components along the tangent and normal to the path. In three dimensions, he uses two orthogonal normals, one of which he forces to be parallel to a fixed plane.

“In [the second] volume, Euler considers the motion of a point-mass lying on a given curve or surface. He derives some differential equations of the geodesics governing the problem of free motion on a surface. In this way, he shows that the path of a mass-point that is free to move on a fixed surface is locally the shortest possible path between its initial and final points” (eulerarchive.org).

DSB IV: 479-80; Parkinson, Breakthroughs, p. 154; Poggendorff I: 689; Roberts and Trent, pp. 103-4; Timoshenko, History of Strength of Materials, p. 29.

Two vols., 4to (253 x 190 mm), pp. [16] [1] 2- 232 225-480 [i.e., 488]; [8], [1] 2-500, with engraved vignette on dedication leaf and thirty-two folding engraved plates. Contemporary calf, richly gilt spine with red and green spine labels, boards with some superficial wear, upper margin of title to volume 1 with a triangular piece (4 x 1 cm) torn and lost. Otherwise a very fine and clean set in unretsored condition.

Item #4767

Price: \$12,500.00

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