Euler : Mechanica Vol. 1.

translated and annotated by
Ian Bruce

Click here for Euler's Preface :

Translator's Preface : The preface throws some light on Euler's thinking at the time, in which he is rather critical of previous works, including that of Newton and Hermann (perhaps one of his previous teachers at Basel, and certainly a colleague at St. Petersburg). The insights of Newton into the working of the physical universe were, of course, a necessary preliminary step to his own analytical refinements and extensions. Newton, like Huygens and others, were stuck in a sort of no-man's-land between the old classical methods and the emerging analytical methods. Euler's earlier papers show some evidence of this reluctance to fully grasp the new ideas. However, it was now a case of out with the old and in with the new. Thus, while Newton's Principia was fundamental in giving us our understanding of at least a part of mechanics, it yet lacked in analytical sophistication, so that the mathematics required to explain the physics lagged behind and was hidden or obscure, while with the emergence of Euler's Mechanica a huge leap forwards was made to the extend that the physics that could now be understood lagged behind the mathematical apparatus available. A short description is set out by Euler of his plans for the future, which proved to be too optimistic. However, Euler was the person with the key into the magic garden of modern mathematics, and one can savor a little of his enthusiasm for the tasks that lay ahead : no one had ever been so well equipped for such an undertaking. Although the subject is mechanics, the methods employed are highly mathematical and full of new ideas.

In addition, a summary of each chapter in Vol. I is presented. Euler has organised his work very carefully into sections of different kinds. There are Definitions of the basic concepts to be used, corresponding to the axioms of ancient Greek geometry; Propositions that are either Theorems introducing new ideas or Problems that are particular useful instances of the associated Theorem; Corollaries that add refinements to proofs and look at special cases, etc; and Scholia which are notes, these are occasionally of a historical nature, or relate his work to that of others, give an indication of what lies ahead, etc. Occasionally examples of a numerical nature are given. Thus, in this manner, the task of constructing an elaborate theory is split up into well-reasoned steps.

The sections presented in each chapter or part thereof, are given here in brackets (these are the result of splitting the chapters up into manageable lengths of around 50 pages per file for ease in transmitting, editing, etc.) : this is a useful aid in navigating your way around the work. It is possibly a good idea to make a copy of this page as a contents reference if you decide to make the Mechanica part of your mathematical education. Note that occasionally I make changes in earlier chapters, but I leave a message for a week or so to that effect in the introduction to the relevant part. The Latin original is available in the public domain at the Gallica website of the French National Library, and is part of Euler's Opera Omnia edited by Paul Stackel; this is the copy I have used in the translation ; the page numbers in square brackets refer to the original St. Petersburg edition of 1736, which I have referred to occasionally also. The chapter contents given below should only be taken as a rough guide of what is present in each section; below in the Contents, I have enumerated the statements of the propositions and definitions, so that you can navigate your way around this immense work. I have been content to list only what springs to mind on completing each part for each file. Finally, if you see some obvious faults, please let me know so that they can be corrected. This has been an immense labour for me running to more than 1000 hours of typing, translating, and understanding the mathematics, etc, and yet it seems to only scratch the surface of what Euler knew at the time, as he picked and chose what to put in his book. Perhaps I should explain a little about Euler's potential force function and its relation to the vis viva; there was confusion at the time as to the conservation laws, and certainly the idea of work and energy were not understood at all well. Euler sets up the differential of what came to be called a force potential function, though he includes non-conservative forces in his function. This he equates to mv^2 in the vis viva approach. In fact, people had to wait until the age of the machine, well into the 19th Century, before Rankin came up with the idea of energy and work in problems involving changes in height. Thus, writers such as Routh in his Rigid Body Dynamics hedged his bets by presenting the old vis viva explanations as well as the then modern energy related explanations, in the solutions of problems. Euler had to contend with Jonan Bernoulli in his understanding of physics, which is perhaps why he seems to slide around the issues, and refrains from putting a half into his mv^2 formula, as J.B. was very keen on the vis viva approach. All was well as long as ratios were taken of speeds; clearly the answers were out by a factor of 2 or root 2 for absolute cases. You can find a good explanation of the vis viva idea in Wikipedia. A late addition is the welcome translation of Stackel's Forward by Dr. Ernest Hirsch, which sheds some light on the reception at the time of the Mechanica.

Click Here : For a list of the statements of the 132 Propositions in Vol. I with their section numbers, from which the appropriate file(s) can be opened below,

Click here for a translated version of the O O preface.

Contents :
Chapter One : Concerning Motion In General. (Sect.1 - Sect.98)
This was the first major work carried out by Euler on being ensconced at the St. Petersburg Academy. It was and still is, a monumental undertaking. The first chapter follows Newton's Principia closely, though he is influenced by the work of Daniel Bernoulli on the foundations of mechanics and of course Galilio; the reader is introduced to some examples of relative and absolute forces and motions. S.H.M. is introduced as an example. This chapter and the following provide the basis for the mathematical developments to follow. The method adopted by Euler for linear motion does not rely on solving differential equations relating to time, but rather uses an early form of potential energy that is equated to the work done in a displacement under a given force. Later in Ch. 5 he treats curvilinear motion along the tangent in this way, and introduces a centripetal component along the radius of the circle of curvature. Time development then follows in each case from an integral. The final scholium indicates the contents of the first book.

Chapter Two : Concerning the effect of forces acting on a free point. (Sect. 99 - Sect.188)
The second chapter is also preliminary, some interesting analytical work on the forces acting on single point masses is presented, which is elaborated on in the following chapters. The first use of e as the base of natural logarithms appears here. Some (minor) corrections have been made to this chapter (Jan. '09).

Chapter Three : Concerning the rectilinear motion of a free point acted on by absolute forces.

Ch. Three (part a) (Sect.189 - Sect.285)
This is a larger chapter and includes the establishment of equations governing the descent of bodies under different forces of gravity. The case of centripetal forces in one dimension under various powers of the distance is examined, and the differential equation for the centripetal force is integrated in a time-independent manner to obtain the first integral (though Euler of course cannot talk about energy conservation at this time.)

Ch. Three (part b) (Sect.286 - Sect.366)
This concludes chapter 3, in which many different kinds of forces acting at a central point are considered; the velocity at any point is found, as is the time to fall to that point. Both centripetal and centrifugal [repulsive] forces are considered. In this chapter Euler demonstrates his great powers, and essentially establishes modern analytical methods.

Chapter Four : Concerning the motion of free points in a medium with resistance.

Ch. Four (part a) (Sect.367 - Sect.449)
Chapter 4a introduces resistance of various sorts into the differential equation for the motion. It is here that Euler establishes the superiority of his analytical approach, and one meets the exponential function.

Ch. Four (part b)(Sect.450 - Sect.542)
Chapter 4b continues with more complex examples, and concludes with an analysis of Simple Harmonic Motion with damping. A truly wonderful chapter in which so much that we take for granted is set out for the first time, although not in the manner in which it is now presented, of course.

Chapter Five : Concerning the curvilinear motion of free points acted on by absolute forces of any kind.

Ch. Five (part a) (Sect.543 - Sect.640)
Chapter 5a introduces motion in two dimensions; the equation for the tangential component of the motion along a curve is carried over from the previous chapters, but the normal component of a general force is introduced to account for the change in direction. Problems involving projectile motion and motion in bound orbits are considered in depth initially : the general orbit problem for any kind of central force is analyzed and shown how it may be separated into an angular and a radial component, ending in sect. 601. This is an amazing achievement, and necessitates an entry into the world of complex numbers; it is clear that Euler is familiar with his famous formula exp(iA) = cosA + isinA for some angle A at this time. We can now see the conservation of angular momentum in the equations, though at the time this was just an experimental fact from Kepler's Laws governing planetary motion, and the subsequent work of Newton. As previously, all squares of speeds are reduced to equivalent heights under uniform gravity, which works admirably in Euler's hands, doing away with all time dependence in the equations, where every quantity is handled by a work or potential energy function, from our point of view.

Ch. Five (part b) (Sect.641 - Sect.706)
Chapter 5b continues the work set up by the general theorem, and here are explained the three main situations where a body moving about a centre of force in a plane can be resolved : these are the cases where the force on the body is directly proportional to the distance from the centre of force C, established around sect. 641, where it varies inversely as the square of the distance, around sect. 644, and where it varies inversely as the cube of the distance, around sect. 671. The first two motions are ellipses with C at the centre or at a focal point respectively, and the third motion is that of a logarithmic or hyperbolic spiral. Other algebraic curves arising from higher order forces are considered. A start is made on dealing with situations where more than one centre of force acts, and the precession of the ellipse is considered.

Ch. Five (part c) (Sect.707 - Sect.762)
Chapter 5c is initially concerned with various instances of the inverse orbit problem : where the curve is given and the corresponding central force has to be found. The rest of this section is mainly concerned with establishing a mathematical framework for dealing with orbits that precess. Euler shows that in order for an orbit to vary in time, a hypothetical force proportional to the inverse cube of the distance must be added to the inverse square law force (sect. 729). There is some remarkably fine mathematics presented here. The work follows similar propositions of Newton, but set out in the analytical manner. This work is introductory to a discussion of the motion of the moon.

Ch. Five (part d) (Sect.763 - Sect.801)
Chapter 5d continues the preparatory work for Euler's lunar theory. The main theorem is concerned with establishing the motion of a body acted on by a central force, which is produced by a body itself in motion along another curve. At this stage a number of results are drawn together and presented in an attempt at describing the moon's motion, which draws on Newton's theory, but presented in the analytical manner. It is apparent, however, that planar motion will not do to express the moon's orbit, and the next propositions will examine motion in three dimensions, in the final section of this long chapter.

Ch. Five (part e) (Sect.802 - Sect.859)
Chapter 5e finally completes the propositions extended to three dimensions, and we have the curvilinear components of acceleration under the action of forces acting along the tangent and along the two principal normals to the element. Forces acting along the coordinate axis are then transformed into their equivalent components along the above directions. Several examples are produced, where the motion can be reduced to the planar motion already discussed. The motion of the moon is considered as due to the action of a force about the earth proportional to the distance, together with a motion due to a force in proportion to the distance of the moon from the plane of the ecliptic [clearly the sun has such an effect for the second SHM force, although it is hard to understand why the first force is not inversely proportional to the distance]. One has to wonder at the energy of this man Euler! It is probably a good idea to read the section 8.15 on p. 252 : The motion of the Lunar Apses, by George E. Smith, in Cohen's translation of the Principia, if you need more information. [Also, this is quite a large file, ~1MB.]

Chapter Six : Concerning the curvilinear motion of free points in a medium with resistance.

Ch. Six (part a) (Sect.860 - Sect.924)
Chapter 6 completes the motion of a point, with the curvilinear motion of a point traveling with resistance either in a plane or in space. The resistance is assumed to affect only the tangential force, and part 6a is concerned with planar motion, mainly with the resistance proportional to the speed. The applied force is vertically downwards in sections a and b.

Ch. Six (part b) (Sect.925 - Sect.1004)
Part 6b is still concerned with planar motion, and extends the treatment to cases where the resistance is proportional to the square of the speed and other powers. The treatment of a projectile with this square law resistance is treated for the first time. There are fewer notes added as most of the work is self-explanatory. Euler's acceptance of negative resistance should not be taken too seriously, as he does know that such cases are mathematical fictions.

Ch. Six (part c) (Sect.1005 - Sect.1062)
Part 6c is still concerned with planar motion, but the analysis is concerned with centripetal forces acting under various power laws, and where the resistance is put in different ways that are integrable. The logarithmic and hyperbolic spirals arises from such developments, and comparisons are made between the vacuum and resistive cases for given curves. The exponential decay function forms a part of some of the analysis, arising in a natural way as a damping term, as it still does in the solutions of differential equations.

Ch. Six (part d) (Sect.1063 - Sect.1116)
Part 6d continues with planar motion, and introduces the notion of angular motion. Subsequently, the extension to three dimensions is finally made, where the previous propositions of Ch. 5 are generalised to include resistance.

Ian Bruce. April 28, 2008 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.