Euler : Theoria Motus Corporum Solidorum seu Rigidorum

translated and annotated by
Ian Bruce


     The first 6 chapters is a sort of update of the Mechanica, which Euler acknowledges as being lacking in a number of ways, although a great improvement on what was available previously. The dynamics of particles is presented without reference to the originators of the theory, whom we might regard mainly as Galileo, Newton, and Huygens, mostly in an analytical manner, though of course dynamics has never escaped from the clutches of geometry completely. One sees here situations where vectors would now be used, and we have to remember that Euler's exposition set the stage as it were for future developments; his methods lend themselves to the use of different kinds of forces, knowledge of which was almost completely lacking at the time. The editor's preface of the Opera Omnia edition for the whole book has been translated and can be accessed here. It is a 'must read' in understanding Euler's treatise, and you should refer to it as needed. As indeed also you may wish to refer to the resume, where the main results used subsequently are set down again for your convenience. Note that all text and lines in red on the figures have been added by the translator to the original diagrams found in the St. Petersburg edition of this work in the Euler Archive; I am grateful for this resource supplied by Dartmouth College. I have of course read the O. O. edition of this work, and occasionally I note corrections made by the original editor of this work, Charles Blanc, for which I am also grateful.

Click here for the translated O O preface :

The Treatment of Motion in General.


Click here for the 1st Chapter : Introduction : In this chapter on kinematics, Euler takes a long hard look at the philosphical ideas then current regarding motion. Having convinced himself that he is right, he moves on to considering motion along a straight line, along a plane curve, and finally in three dimensions. Along the way he shows the great benefits to be made by resolving the speed of a point in either of these situations into components. One is struck by the apparent modern viewpoint adopted in this work, although it dates from 1765.

Click here for the 2nd Chapter : Concerning the Principles of Internal Motion : In this chapter basically on what we now call Newton's First Law, Euler wonders about 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 rel. to the fixed stars, and introduces the idea of inertia to account for the state of bodies not affected by external actions.

Click here for the 3rd Chapter : Concerning the External Causes of Motion or Forces : In this chapter basically on what we now call Newton's Second Law, Euler starts by considering the origin of force, from which he is led to the idea of impenetrability. He introduces the concept of inertia, which is identified with mass, and he finally produces what is Newton's Second Law analytically, as we know it. The need for unit quantities to be used in defining force is introduced. Euler considers the resolution of forces along orthogonal axis as a means of shortening calculations in general. We become aware of how far Euler has moved on in his thinking since he wrote the Mechanica, and these initial 6 chapters are a sort of rewriting of this great work, dealing with point masses or corpuscles. We have done Euler the honour of perpetuating his methods more or less, to the present day, which is why one has a certain sense of 'deja vu' when viewing this chapter : but this is where it all began, so to speak. A definite read for physics students, both young and old!

Click here for the 4th Chapter : Concerning Absolute Measures Sought from the Fall of Weights : In this chapter Euler replaces his former ratio method of comparing speeds and distances fallen by weights. There is a general move towards the use of absolute quantities rather than ratios, and we can see the beginnings of modern kinematics.

Click here for the 5th Chapter : Concerning the Absolute Motion of Bodies Acted on by Forces in General : Euler finally presents his new theory of dynamics, in which an analytical form of Newton II is put in place, forces are resolved along normal axes in 2 and 3 dimensions, and a number of problems are used to demonstrate dynamics more or less as we know it today. The absolute nature of physics is presented for the first time, and there is a move away from the relative ideas prevalent in the Mechanicae; Euler here tackles the theory of lunar motion from this new base with more success.

Click here for the 6th Chapter : Concerning the Relative Motion of Bodies Acted on by Forces in General : Euler gives a detailed discussion of relative motion in two and three dimensions.

Treatise on the Motion of Rigid Bodies .

Click here for a summary of the first 10 chapters :

Click here for the 1st Chapter : Concerning the Progressive or Rectilinear Motion of Rigid Bodies.

          The real work now begins, where the emphasis changes from point masses to extended bodies......At present chapters 1 - 10 are available in translation. Euler discusses the definitions of the terms rigid body, progressive motion, centre of mass, and introduces the idea of internal forces, along with various theorems and problems for extended bodies.

Click here for the 2nd Chapter : Concerning Rotational Motion about a Fixed Axis with no Disturbing Forces.

          Euler carefully discusses the definition of angular motion of a body about a fixed axis between two arbitrary points on a laminar body. He goes on to find the forces exerted on such an axis due to the uniform rotation of the body about this axis, and shows how such forces change on changing the reference points of the axis.

Click here for the 3rd Chapter : Concerning the Generation of Rotational Motion.

          Euler carefully discusses the starting conditions for the rotary motion of a body about a fixed axis between two arbitrary points on a body. The response of the individual particles of the body are here presented as if the particles were connected to the axis by threads, and the moment of inertia and related moments are defined for the first time. The forces and torques exerted on the supporting points on the axis are examined in depth : eventually these reactive forces are put to zero when the rotation of free bodies is considered, hence they are considered to be equal while still finite. This is a long and rather difficult chapter, containing much material. One notes of course, the initial idea that a body can have an acceleration, and yet be at rest initially, from which the dependence of the initial angular increment depends on the square of the time increment.

Click here for the 4th Chapter : Concerning the Disturbance of Rotational Motion Arising from Forces of any Kind.

          Euler carefully discusses the equation governing the accelerated motion of a body acted on by moments about a fixed axis between two arbitrary points on a body. This chapter is useful in explaining some of the lingering mystery of the preceding chapter, which Euler seems to acknowledge here as being rather vague at times. He has fixed on the need to develop theorems treating moments of inertia.

Click here for the 5th Chapter : Concerning the Moment of Inertia.

          Euler sets out the main theorems governing the calculation of moments of inertia, including the finding of the principal axes. This is a most useful chapter, and seems to surpass that available in modern texts on dynamics. One can see that the transition to vector and other notations follows in a simple manner from Euler's excellent presentation.

Click here for the 6th Chapter : The investigation of the Moment of Inertia for Homogeneous Bodies.

          Euler presents the moments of inertia of number of standard shapes, including a thin rod and a ring, various laminar shapes incl. a general triangle, and solid shapes incl. cylinder, sphere, solid of revolution, etc. Chapters 5 and 6 should be examined together.

Click here for the 7th Chapter : Concerning the Oscillatory Motion of Heavy Bodies.

          This is a most interesting chapter, in which the results concerning moments of inertia are applies to oscillating rigid bodies, incl. physical pendulums with one or two spherical bobs attached, and an analysis of the forces acting at the supports is presented for a body oscillating about one or two supports either horizontally or inclined at some angle. The use of pendulums in determining the acceleration of gravity is presented in detail.

Click here for the 8th Chapter : Concerning Free Axes, and the Motion of Rigid Bodies about such Axes.

          In this chapter Euler introduces free axes of rotation, which are soon established to act along the principal axes of the body, but which are not fixed in space. He goes on to investigate some simple kinds of motion about such axes, making use of the independence of linear motion and angular motion and shows that any force acting 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.

Click here for the 9th Chapter : Concerning the First Generation of Motion in Rigid Bodies.

          In this chapter Euler continues his treatment of the initial motion of a body, now free from fixed axes. There are a number of important results here, incl. the introduction of Euler Angles, the change from arbitrary axes to principal axes, and the minimal vis viva (read as kinetic energy doubled ) associated with principal axes. This is a challenging chapter, and may require considerable thought on the part of the reader to understand, as results are derived from first principles that nowadays are usually brushed off in text books using vector products, etc. In this chapter, 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 .

Click here for the 10th Chapter : Concerning the Momentary Change in the Axis of Rotation Produced by Forces.

          You need to be familiar with Ch. 9 to make progress here, esp. sect. 639. Someone should perhaps set out Euler's analysis in modern terms, although it is not the task of this translator, as it would take me too long. It is useful to have a printed copy of this chapter and the previous one on hand to make headway, as there is much to-ing and fro-ing amongst long formulas, although the basic ideas seem simple enough. A good book to have at hand is Rene Dugas's 'History of Mechanics', Ch. VI, which like all good books is out of print, if you are thinking of getting one; however, by trying to cover all of Mechanics, attention to individual authors is rather limited, and Euler has suffered considerably in this regard.

Click here for the 11th Chapter : Concerning the Free Motion of the said Bodies with Equal Principal Axes, Acted on by no Forces.

          This chapter is concerned with demonstrating the mixed or combined translational and rotational motion of a sphere or regular solid. The more complicated bodies to be considered in the next two chapters can of course be reduced to this simple case by making the moments of inertia about the principal axes equal.

Click here for the 12th Chapter : Concerning the Free Motion of the said Bodies with Two Equal Principal Axes, Acted on by no Forces.

          This chapter sets out the case for bodies with two equal axes. Here it is shown how general rotational motions can be compounded from motions about different axes. I will comment further on this chapter later, which it is a simpler version of the one to follow.

Click here for the 13th Chapter : Concerning the Free Motion of the said Bodies with Three Unequal Principal Axes, Acted on by no Forces.

          This chapter sets out the case for bodies with three unequal axes. Unfortunately the general equation of motion cannot be integrated using known functions and lapses into elliptic integrals, and this is presumably where Euler first came across these things. A compromise is found for some particular motions, which are presented, relating to the motion of the instantaneous axis of rotation on a sphere, using spherical triangles. The connection to absolute space via an extra transformation is also presented. The next chapter deals with external forces acting on spinning tops.

Click here for the 14th Chapter : Concerning the Motion of Spinning Tops on a horizontal plane with all Three Principal Axes equal to each other.

          The chapter 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 serious reader should read again perhaps, problems 62 and 68 of Ch. 10. Most of the mathematics is taken up with referring the axes to a reference frame at rest in an absolute space, as well as the local frame fixed in the top. Again, the motion of the centre of mass is abstracted out of the problem. The work is really one of absolute genius in my humble opinion : the derivation of results more fitting to vector analysis are produced by Euler referring to the motions of the axes on a unit sphere, and all from elementary geometrical considerations. A working knowledge of the sine and cosine rules for the arcs and angles associated with spherical triangles are needed. I have set out some notes dealing with these in the first problems, and revised problem 68, which in addition has some more notes written into the text. Euler tells us that due to the extremely cumbersome nature of his solution, he is to show another approach in the next chapter, more suited to such calculations, in which the moments of inertia are not all equal to each other.

Click here for the 15th Chapter : Concerning the Free Motion of Rigid Bodies acted on by any Forces.

          This chapter sets out Euler's new method for dealing with the problems handled to date, which he admits to being long and difficult; in a few pages 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.

Click here for the 16th Chapter : Concerning the Rotational or Turning Motions of Celestial Bodies .

          This chapter continues to set out Euler's new method for dealing with the problems handled to date, 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, which is not very convincing on physical grounds. He has much more success on revisiting his earlier work on the precession of the equinoxes treated earlier (E171). In the next chapter he returns to spinning tops.

Click here for the 17th Chapter : A fuller Explanation of the motion of Tops on a Horizontal Plane without Friction.

          In this 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.

Click here for the 18th Chapter : Concerning the motion of bodies with a spherical base on a horizontal plane.

          In this chapter Euler establishes his 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 s.h.m. 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.

Click here for the 19th Chapter : Concerning the motion of cylindrical bodies on a horizontal plane.

          In this chapter 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 , are established and solutions found for the vertical motion. A special kind of cylinder is considered and solved for s.h.m. as equivalent to a simple pendulum.  An Appendix to follow examines the effect of friction of motion on an inclined plane.

Click here for Ch's 1, 2, & 3 of the Appendix, Supplement concerning the motion of rigid bodies disturbed by friction.


            Euler has put a great deal of thought into the involvement of friction with dynamical processes. It is all very well to say that he was just looking for an excuse to find some interesting phenomena on which he could exercise his amazing calculating powers, but I think that is a very unfair comment. 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. The final Scholium gives details of the remaining chapters, which appear to be quite challenging, as they cover more complex instances of friction.

inclined plane.

Click here for Ch's 4 & 5 of the Appendix, Supplement concerning the motion of rigid bodies disturbed by friction.


            In Ch. 4 Euler considers the introduction of friction at the point of the 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.

            Ch. 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. The full treatment with torques of forces resolves along principal axes and forces applied the centre of inertia is then put in place. Euler did not use the laws of friction quite as we do now, but one can make allowances for this. This is a long and involved chapter. If someone could find the time to read it through very carefully and provide me with comments for improvement, I would be most grateful from them.



Ian Bruce. August 10, 2009 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish by clicking on my name, especially if you have any relevant comments or concerns.