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.