Mechanica sive motus scientia analytice exposita..

St. Petersburg: Academy of Sciences, 1736. First edition. 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 solving its problems; he was to include a summary of these in the Mechanica.

"In 1736 the Petersburg Academy, which had promised funding for the publication of most of Euler's future books, issued his 980-page Mechanica sive motus scientia analytice exposita (Mechanics, or the science of motion set forth analytically) in two volumes. These were each too large to appear in the Commentarii but were, as noted on the title page, supplements to it. Their title states Euler's central concept, though he briefly considered Dynamica as an alternative title. Johann I Bernoulli wrote in a letter that he thought Dynamica had merit, but Euler quickly settled on Mechanica and dedicated these volumes to his 'illustrious and most excellent' friend, academy president Korff.

"The preface to the Mechanica emphasizes its importance:

'[I]f analysis is needed anywhere, then it is certainly in mechanics. Although the reader can convince himself of the truth of the exhibited propositions, he does not acquire a sufficiently clear and accurate understanding of them, so that if those questions be ever so slightly changed he will not be able to answer them independently unless he turns to analysis and solves the same propositions using analytic methods. This in fact happened to me when I began to familiarize myself with Newton's Principia and Hermann's Phoronomia; although it seemed to me that I clearly understood the solutions of many of the problems, I was nevertheless unable to solve problems differing slightly from them. But then I tried, as far as I was able, to distinguish the analysis [hidden] in the synthetic method and to my own ends rework systematically those same propositions, as a result of which I understood the problem much better. . . . I expounded . . . [the propositions anew] using a systematic and unified method and reordered them more conveniently.'

"In putting forward simple and uniform analytic methods and logically arranging fundamental propositions and new definitions, thus making them intelligible, Euler had begun conceptually to organize infinitary analysis and was the first to apply it systematically to dynamics. This was a major feature of the Mechanica: Euler wanted to demonstrate that the proofs of the laws of mechanics are certain and necessary. He accepted Newton's dynamics, but his research program was not simply an extension of it; he was also to incorporate elements from the Cartesian and Leibnizian sciences and his own thought. In the preface to the Mechanica Euler declared that he had included 'both what I have found in the writings of others on the motion of bodies and what I myself have thought out in my ruminations.' Among his sources were Archimedes on statics; Galileo Galilei on kinematics, and Huygens, Newton, Johann I Bernoulli, and Jakob Hermann on dynamics. During his early Saint Petersburg years, he had become familiar with many of the writings of Galileo, Huygens, and Newton; he also referred to the second volume of Christian Wolff's Elementa matheseos universae of 1733, in part an exposition on statics and elementary mechanics. (Euler did not comment on its aerodynamics and hydraulics.) Clearly Newton's Principia mathematica and Hermann's Phoronomia most influenced him. For its systematic application of infinitary methods, the Mechanica's preface confidently concludes, '[Qui] in analysi tam finitorum quam in- finitorum satis fuerit exercitatus, is mira facilitate omnia intelligere atque sine ulla manuductione integrum hoc opus perlegere queat' (He who has enough practice in finite and infinite analysis will understand everything with wonderful ease and be able to read through this entire work without help). On the Continent, Euler was assuming the lead in applying infinitary analysis to mechanics. By 1750 analysis surpassed geometric synthesis.

"Among the many novelties in the Mechanica is the precision it gives to the concept of bodies, which Newton had employed only vaguely, giving it at least three disparate meanings. Seeing that in general Newton's laws correctly applied to masses reducible to points, Euler introduced the concept of mass points. The conclusions in his punctual mechanics apply only to punctiform - and not extended - bodies. Analogous to his treatment of motion in a plane, he introduced rectangular Cartesian coordinates in tracing the trajectories of mass points, explicitly decomposing the paths along the axes of forces acting on them. The binormal, principal normal, and tangent give the axes. It is mistakenly believed that Newton had shown this, but Newton's geometric method had only intrinsic coordinates. Euler asserted that motion is continuous, so we know each point in the space covered in a trajectory. This builds upon Leibniz's principle of continuity. From its initial to its final place, Euler demonstrated moreover that the straight-line path of a mass point to the center of force is the shortest. To Cartesian coordinates he added directed magnitudes or vectors at different points. Euler now also employed vectors, which had previously applied only to static forces, for changes in speed and direction; their addition lies at the heart of the Enlightenment concept of space-time.

"The concept of force was crucial to Euler's mechanics, and he treated it as an external entity to a body causing change in motion. His physical model was still incomplete and would generate criticisms; in the preface to his Traité de dynamique (1743), Jean-Baptiste le Rond d'Alembert described as vague and obscure purely mathematical explanations of motion without the inclusion of external observations. In 'Recherches sur l'origine des forces (1750) Euler would provide the absolute impenetrability of matter as the previously missing element in his explanation of force. Both Newton and d'Alembert had accepted this concept, but Euler went further in examining the relation between this property and impact phenomena.

"In the Mechanica's book 2, proposition 20, Euler proved Newton's second law: 'The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.' Newton had offered a dichotomy. His second law treated motion in two forms: in modern symbols F = Δ(mv) or F = ma, according to whether the force is acting continuously or discontinuously. Before Euler's time, that law was often auxiliary in studies of motion and mostly applied to one degree of freedom. Euler's proof reduced the ambiguity and argued that Newton's second law was an underlying principle of motion; among geometers and natural philosophers, Euler was prominent in developing it to apply to many degrees of freedom. He did not examine two bodies subject to the same forces, but his study of Kepler's planetary laws in book 1, chapter 5 of the Mechanica suggests that he could reduce these problems to determining the motion of a single body. Working only with mass points, he did not require Newton's third law on opposite and equal reaction ...

"Besides being significant in the development of mechanics, the Mechanica is notable in the history of mathematics. Through applying first-order ordinary differential equations and inventing many of the second order, Euler was participating especially with the Bernoulli family in founding differential equations as a core branch of calculus. Corollary 832 in the Mechanica's volume 2 places Euler among the first to introduce partial differential equations of functions with two independent variables; a few years earlier Nikolaus Bernoulli had worked on these, recasting the trigonometric lines - given as the traditional Ptolemaic chords and half chords - into numerical ratios and particularly working with the tangent. Euler was searching for deeper connections among the three elementary transcendental functions: the exponential, the Napierian logarithmic, and now the trigonometric. The Mechanica also touches upon cyclometric functions, which measure revolutions of a wheel, and elements of what would later become the calculus of variations, differential geometry, and the theory of geodesics. Attentive to a better symbolism, Euler adopted π in computing the area of a circle, extensively used e for the base of natural logarithms, and introduced his summation sign ∑. In section 184 of volume 1, the sign A·x represents the function of arctan x; in the second volume it stands for secant x, while At·x becomes the sign for arctan x. The continuing appearance of these symbols in Euler's many influential publications quickly made them conventional" (Calinger, pp. 124-130).

"[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" ().

"Throughout early Enlightenment Europe, most of the scientific community quickly recognized the Mechanica as a landmark in the history of physics and knew that a genius of the highest magnitude had arrived. The initial review, published in 1737 in the journal Bibliothéque germanique in Amsterdam, was by the Calvinist pastor and philosopher Jean-Henri-Samuel Formey, the leader of a scholarly circle in Berlin. Formey praised Euler for employing the analytical method both to prove established truths and to lead to generalizations and new theorems. Johann I Bernoulli, who wrote the next review, indicated the growing repute of Euler in 1737 by addressing him in a letter as the 'most famous and wisest man of science Leonhard Euler.' His review appeared in Leipzig's Nova acta eruditorum the next year. Bernoulli, whose enthusiasm recalled that of Leibniz for Hermann's Phoronomia, lauded the genius and acumen of Euler: 'Until now no book has appeared that so splendidly reached the inmost heart of mechanics. It has brought out in abundance sublime and hidden things ... The whole book is analytical, so that no synthetic ballast bores the reader.' Another commendation in 1738 came from Wolff, who cited Euler's application of differential calculus for the utility and ease in reading that it brought to the entire book. In his Remarks [Benjamin] Robins indicated a rising reputation for Euler, who for several recent articles 'has been celebrated in the various literary journals of different countries with high encomiums'" (ibid., p. 130).

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. Calinger, Leonhard Euler, 2016.

Two vols., 4to (270 x 200mm), pp. [xvi], 480; [viii], 500, with engraved vignette on dedication leaf and thirty-two folding engraved plates (foxing and spotting throughout (mostly marginal apart from the endpapers and title pages of both volumes), marginal wormholes throughout both volumes but particularly to the endpapers of volume one, pages dust-soiled at top margin). Contemporary full vellum with gilt lettering to spine, marbled edges, wormholes on upper and lower boards and spine of both volumes, some soiling to boards, corners slightly rubbed. A good, wide-margined set.