## E15 -- Mechanica, volume 1

(Mechanics, volume 1)

Originally published with the full title: Mechanica sive motus scientia analytice exposita. Auctore Leonhardo Eulero academiae imper. scientiarum membro et matheseos sublimioris professore. Tomus I. Instar supplementi ad commentar. acad. scient. imper. Petropoli. Ex typographia academiae scientarum. A. 1736.

Summary:
##### (based on C. Truesdell's An idiot's fugitive essays on science: methods, criticisms, training, circumstances)
This, along with E16, is Euler's outline of a program of studies embracing every branch of science, involving a systematic application of analysis. These two works 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.

This 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 arclength 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 addition to the dedication to J.A. Von Korff and the “Praefatio,” it contains 6 chapters:
1. De motu in genere.
2. De effectu potentiarum in punctum liberum agentium.
3. De motu rectilineo puncti liberi a potentiis absolutis sollicitati.
4. De motu rectilineo puncti liberi in medio resistente.
5. De motu puncti curvilineo libero a quibusdam potentiis absolutis sollicitati.
6. De motu puncti curvilineo libero in medio resistente.
Publication:
• Originally published as a book in 1736
• Opera Omnia: Series 2, Volume 1
B. Robins published a work in 1739: Remarks on Mr. Euler’s Treatise of motion, Dr. Smith’s Compleat system of opticks and Dr. Jurin’s Essay upon distinct and indistinct vision (London 1739), where “Remarks on Mr. Euler’s treatise entitled Mechanica” is found on p. 1-29.

Documents Available:
• Because the Mechanica is a large book (500+ pages), the Euler Archive has split the original into several files for easier downloading:
• Ian Bruce has made both a translation and transcription of E15 available at his page Mathematical Works of the 17th Century.
• The Euler Archive attempts to monitor current scholarship for articles an books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E15 include:
• Gaukroger, Stephen "The metaphysics of impenetrability: Euler's conception of Force", British Journal for the History of Science, 15 (1982), 132--154.
• Speiser D., “The Kepler problem from Newton to Johann Bernoulli.” Archive for History of Exact Sciences, 50 (2), pp. 103-116 (1996).

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