Theoria motus corporum solidorum seu rigidorum. Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata.
Rostock & Greifswald: A. F. Röse, 1765. First edition of this landmark work, a continuation of Euler’s Mechanica (1736) in which he moves on from the treatment of the motion of point-masses in the earlier work to that of rigid bodies, studying rotational problems (some motivated by the problem of the precession of the equinoxes) and introducing many now familiar concepts such as the ‘moment of inertia,’ ‘principal axes’ and ‘Euler angles.’ This work also introduced some of the basic ideas of vector analysis, such as the dot and cross products (although vector notation is not used). “The Theoria motus corporum solidorum, published almost thirty years later (1765), is related to the Mechanica. In the introduction to this work, Euler gave a new exposition of punctual mechanics and followed Maclaurin`s example (1742) in projecting the forces onto the axes of a fixed orthogonal rectilinear system. Establishing that the instantaneous motion of a solid body might be regarded as composed of rectilinear translation and instant rotation, Euler devoted special attention to the study of rotatory motion. Euler thus laid the mathematical foundation of the numerous studies on variational principles of mechanics and physics which are still being carried out” (DSB). “In 1765 Euler’s landmark work Theoria motus corporum solidorum seu rigidorum (Theory of the motion of solid or rigid bodies) was published, and it reflected Euler’s recognition that his earlier Mechanica sive motus scientia analytice exposita on the motion of point masses was lacking in several ways. He not only brought it up to date but went beyond. Called his second Mechanica, Theoria motus corporum solidorum treated the kinematics and dynamics of particles without referring to his predecessors—mainly Galileo, Christiaan Huygens, and Newton. While not completely without geometry, most of it is analytical, providing and calculating differential equations. Euler based dynamics either on d’Alembert’s principle, which was a variation of Newton’s second law, or on the principle of action and reaction. Theoria motus corporum solidorum applied to a solid the system of equivalent forces … The text began with a six-chapter introduction containing illustrations and necessary additions to the motion of solid bodies, followed by a nineteen-chapter ‘Tractatus’ on the mechanics of rigid bodies. The book gave inertia as its elementary force and employed three permanent axes to describe it. Euler examined moments of inertia in wires, plane figures, and major solids, and he proceeded to the general case in which the moments were unequal. He also noted the importance of rotational motion and studied varieties of rotation under different forces; he had earlier thought that straight line motion was sufficient. In 1758 he made the important discovery of the equations of motion for the rotation of rigid bodies, and he subsequently solved that problem in 1759. The stress on rotational motion was something new, and Euler also investigated gyrations when a heavy pendulum is moved” (Calinger, pp. 443-4). The book concludes with a section of five chapters on the effects of friction. This is one of Euler’s rarer books, with no other copy at auction in the past decade. A second edition appeared in 1790 with additions by Euler’s son Johann Albrecht. Provenance: Vienna Observatory (ink stamp to title recto and verso, 18th-century bookplate on front pastedown). Numerous books from the Observatory library were sold in the 1930s. The Theory of motion of solid bodies begins with an introduction of six chapters, dedicated to the mechanics of a point mass. This introduction is not just a simple summary of the Mechanica but sets out the progress made since the publication of that work. The first chapter of the Introduction sets out the kinematics of a point mass – the idea of using the methods of analytical geometry for studying the motion of a single point mass. Chapter 2 is on what we now call Newton's First Law. Euler considers the situation of particles within a body and introduces the idea of external and internal actions. After considerable reflection he introduces the idea of absolute rest and motion, in practice relative to the fixed stars, and the idea of inertia to account for the state of bodies not affected by external actions. The third chapter introduces the principles of the dynamics of a point mass, which contains the essential ideas of mass, force, inertia, speed, acceleration, and Newton’s Second Law. The need for unit quantities to be used in defining force is emphasized. In Chapter 4 there is a general move towards the use of absolute quantities rather than ratios, and we can see the beginnings of modern kinematics. The final two chapters of the Introduction give the application of these principles and choice of units to the dynamics of a point. One finds in §205 and the following sections the intrinsic equations of motion along curves, from §218 the treatment involves planes, and from §228 the study of a point in a channel. The absolute nature of physics is presented for the first time, and there is a move away from the relative ideas prevalent in the Mechanica; Euler here tackles the theory of lunar motion from this new base with more success. Finally, in chapter 6, Euler writes down the equations of relative motion with respect to a reference point, such as a ship and a landmark, both in two and three dimensions. In the treatise proper, Euler notes that his treatment of dynamics rests on the application of d’Alembert's principle, or the principle of action and reaction, which can be stated in the following fashion: one defines in statics, what is called a system of equivalent forces (applied to a solid); Euler assumes that this idea is known to his reader (see §279); he introduces the idea of the elementary force (i.e., the force of inertia), tied to that concerning the cohesion of the solid (§295–§304); he calls the elementary force applied to a point of the solid, at some instant in its motion, the force that would be applied for which, were the point free, there would be the same motion. This force is then, by Newton's law, proportional to the acceleration. D'Alembert's principle postulates (§295) that these elementary forces form a system equivalent to these external forces providing the motion. Some of the different chapters of the treatise are dedicated to general problems of the dynamics of solids, others to particular problems that need to be resolved with the help of the general theory. In the first chapter, Euler treats the special problem of the motion of a body under the effect of forces acting on each element, parallel to each other and proportional to the mass of the elements (§275–§279). The movement is then a translation; or such forces are equivalent to a single force applied at a determined point, independent of their direction (§280–§288); this point he calls the centre of inertia. Euler shows much later (§690) that the whole motion can be decomposed into a translation and a rotation about an axis. Chapters 2 and 4 treat the dynamics of these motions. Chapter 2 treats the motion of a solid about a fixed axis in the absence of external forces besides those which hold the axis of rotation in place. After some definitions and kinematic considerations (§309–§320), Euler establishes that the angular speed remains constant (§321–§326). He then moves on to the problem of determining the forces which keep the axis in its position (we would now say, to determine the reaction from the axis). These forces are equivalent to elementary forces (§327–§331); if the solid can itself be reduced to a plane figure, perpendicular to the axis, one can reduce these forces to a unique resultant (§332–§337); in the case of a solid the reduction gives in general two forces. In passing, Euler considers the case where one of the points of application is at infinity, which is equivalent to our notion of a couple. In chapter 3, Euler studies the motion of a solid which is rotating about an axis. The systems of forces that are able to produce the motion sought (§352–§356) can be reduced in an infinite number of ways to two forces only (§357–§360). From this Euler deduces a law similar to Newton's law: the angular acceleration is proportional to the moment of the external forces and inversely proportional to a magnitude which depends on the solid and on the axis, and which is called the moment of inertia. The end of the chapter is then devoted to the calculation of the reactions of the axis in different cases; one looks in particular for that case in which the reaction is zero, where the idea of the centre of oscillation arises. It is then easy to study the rotation at some instant (chapter 4), Euler determines first of all the elementary forces corresponding to a given angular acceleration (§398–§407), then he calculates the acceleration produced by the given forces (§408–§412), and finally the reaction of the axis in such a motion (§413–§417). Chapter 5 is dedicated to a systematic study of the moment of inertia. Having recalled the definition and enunciated some evident properties (§422–§426), Euler studies the variation of the moment of inertia when the axis is varied: at first (§428–§431) in a parallel displacement, then (§432–§437) after a rotation about the centre of inertia. The principal axes of inertia are defined from their extreme properties; the existence of the principal axes follows from the existence of a maximum for a function defined over the surface of a sphere, which Euler accepts as obvious. The end of the chapter (§464–§470) considers the moment of inertia of a solid formed from the union of two given solids. In chapter 6, Euler calculates various moments of inertia: a wire (§471–§476); plane figures (§477–§500), and noteworthy solids (§501–§521) including a cylinder, sphere, solid of revolution, etc. The subject of Chapter 7 is the movement of a pendulum formed from a heavy body. The search for an analogy with translational motion leads to a certain point being considered, the centre of oscillation, or what amounts to the same thing, by considering the equivalent simple pendulum (§522–§546); then, and always with numerical examples, there is the study of pendulums formed from a number of solid parts (§547–§560), and finally the reaction of the axis (§561–§571). The three chapters that follow are dedicated to the rotational motion about a variable axis; they constitute one of the essential parts of the treatise. In chapter 8, Euler determines the permanent axes of a solid, calling these the free axes, that is to say the axis about which a rotation can be made without reaction on a support. A rotation about such an axis can be sustained without being changed if there are no external forces. It is easy, by means of the results of chapter 2 (§338), to find what these axes are: they are the principal axes of inertia (§572–§581). Euler shows that any force acting on a solid body can be replaced by a couple and a force through the centre of mass; the couple handles the rotational motion, while the force takes care of the linear motion. In Chapter 9 Euler considers the more general motions of a solid, now free from fixed axes. Here we find the introduction of Euler angles, the change from arbitrary axes to principal axes, and the minimal vis viva (kinetic energy) associated with principal axes. Use is made of spherical trigonometry to lessen the pain of the increasingly complex three-dimensional diagrams used to resolve forces and establish moments. The internal forces acting on an arbitrary axis are evaluated, and the initial change in motion found about an axis. In chapter 10, Euler treats the variation of rotation under the effect of given forces. He begins with a preliminary kinematics problem: a solid body rotating about an axis passing through the centre of inertia is subject to applied forces which, if the body were at rest, would cause a rotation about a different axis – to determine the variation in the rotation (§650–§660). Euler then solves successively the following problems: a solid body rotates about an axis passing through the centre of inertia – determine the elemental forces, i.e., the forces which act to modify the motion (§661–§664); these forces being known, to determine the instantaneous rotation which they impress on the body considered, as if it were at rest (§ 665–§ 668); finally, a solid body being subject to a rotation about an axis passing through the centre of inertia, to determine the instantaneous variation of this axis. Chapter 11 is concerned with demonstrating the mixed or combined translational and rotational motion of a sphere or regular solid for which the moments of inertia are equal. Chapter 12 treats the same problem for bodies with two moments of inertia; here it is shown how general rotational motions can be compounded from motions about different axes. Chapter 13 treats the case of bodies with three moments of inertia. Unfortunately, the general equation of motion cannot be integrated using known functions and involves elliptic integrals, and this is presumably where Euler first encountered them. A compromise is found for some particular motions relating to the motion of the instantaneous axis of rotation on a sphere, using spherical triangles. Chapter 14 deals with the external force of gravity acting on a spinning top on a smooth horizontal plane, with the three moments of inertia equal to each other for any choice of axes through the centre of mass. The next chapters set out Euler’s new method for dealing with the problems handled previously, which he admits to being long and difficult. In Chapter 15 he resets the stage by realizing that the angular velocity can be resolved along the principal axes and that the velocity of a point in a body rotating about any axis can be resolved by a new kind of product, which is now known to us as the vector cross product. This is derived via spherical triangles and can be contrasted with what we know as the dot product, which was introduced in Problem 68. The previous problems regarding rates of change of axes, etc are reworked and the same results obtained by much shorter expositions: Euler's equations for the motion of a rigid body are the confirmed result of this exercise, and so in this chapter we see the beginnings of vector analysis. Chapter 16 continues to set out Euler’s new method, but here he turns his interest to astronomy. He establishes the torques acting on a celestial body produced by a distant source of gravitational forces obeying the inverse square law. He then considers the moon’s librations as a candidate for such torques. He has much more success on revisiting his earlier work on the precession of the equinoxes. In the next chapter Euler establishes his differential equations arising from the moments of the reaction force for tops which are figures of revolution. He looks for integrable solutions to these equations and finds a minimum rotational speed for stability in such cases; in the next chapter he looks at hemispherical tops. In the next chapter Euler establishes the differential equations arising from the moments of the reaction force for tops which in general rotate on a base which is part of a sphere. He looks for integrable solutions to these equations but must be content with simple harmonic motion solutions for small oscillations. This was the introduction of what is now called the tippy top, as it is able under certain conditions to invert itself repeatedly. In the final Chapter 19 of the main work Euler establishes the differential equations arising from the moments of the reaction force for loaded cylinders, for which the centre of inertia does not coincide with the geometrical centre, and solutions are found for the vertical motion. A special kind of cylinder is considered and solved for simple harmonic motion as equivalent to a simple pendulum. Euler’s book concludes with an Appendix which examines the effect of friction on the motion of solid bodies. At the time, around 1760, most but not all the facts about friction were known. The main defect was the unreliable nature of experimental results, which masked the distinction between static and kinetic friction. The first chapter sets out the laws of friction as then understood and applies them to linear motion on a horizontal surface. The second chapter examines motion with friction down an inclined plane, including tipping or rolling of the object. The third chapter is much more extensive and deals with the rotational motion of a regular body between end cylinders constrained by some kind of bracket – not very sophisticated, but amenable to a detailed analysis; the case of a compound pendulum of some kind held by such a construction is considered in detail. In Chapter 4 Euler considers the introduction of friction at the point of a rotating top, proportional to the normal force and opposing motion. He is able to write down the general equations for such a top, but as yet cannot provide a solution to this more general problem. Chapter 5 is concerned with the motion of a ball or globe on a surface with friction. He considers the general situation where the ball or globe is proceeding in one direction on a horizontal plane with friction present, while it spins and abrades or grazes in another direction along the surface. The initial propositions deal with the kinematics of the situation, and the curve traced out by the centre of mass. DSB IV: 480; Parkinson, Breakthroughs, p. 154; Poggendorff I: 690; Roberts and Trent, pp. 105-6. Bruce (translator and annotator), ‘Euler: Theoria Motus Corporum seu Rigidorum’ (https://www.17centurymaths.com/contents/mechanica3.html). Calinger, Leonhard Euler, 2016.
4to (212 x 164 mm), pp. [xxxii], 520, with 15 engraved plates (occasional light browning and foxing). Contemporary calf, spine gilt in compartments with red lettering piece (extremities a bit bumped, a few scratches to boards, later gilt number at head of spine).
Item #6562
Price: $9,500.00
