Euler : Mechanica Vol. 2.

translated and annotated by
Ian Bruce

A Note on Differentials: Euler's method of presenting derivitaves such as dx and dy on his diagrams in the Mechanica is different from that used in modern elementary calculus texts; however, in both approaches, the variable x increases uniformly so that dx is constant while ddx is zero, while the variable y is a function of x and is allowed to have higher derivatives ddy, etc. Now in the modern approach, we at least pretend to draw an approximation to the value of the ratio dy to dx before the limit is taken, and refer to some kind of difference from which the limiting value of the ratio is extracted in general. Euler considers the limiting value of the ratio already taken, as dy and dx, which he places in a diagram as an evanescent triangle; usually along with this triangle there is presented a finite triangle similar to the evanescent triangle for the particular application; in this way the ratio of the sides in the evanescent case is equal to a ratio of the sides in the other finite triangle, and in this way a finite quantity is found for the ratio in the evanesent triangle, which can then be considered to shrink indefinitely. An alternate view is to consider two finite triangles, which differ by an infinitesemal amount that goes to zero. Euler uses several methods of this kind : sometimes the infinititesimal triangle collapses to a line, at other times he considers two similar infinitesimal triangles, one being of a higher order than the other, (as in Fig. 46 of vol. II), etc. Thus, in modern texts, we have an approximate limit on a precise diagram, while Euler presents an accurate limit on an approximate diagram. One way to visualise this, then, is to imagine that the limit has been taken and all the lines have assumed their final relative lengths ; the lines contracted in the limit are then 'pulled apart' a little like a concertina so that we can view the final ratios on a slightly distorted diagram. There is hence no need to make dx and dy small, a point of confusion for students who are meeting calculus for the first time. This is the process of differentiation, while integration sums the differences of ratios in the limit with a common differential : a concertina formed from infinitely many numerators of collapsed ratios. This is a personal heuristic view that you may or may not agree with. One of Euler's delightful ideas is to consider two such similar triangle, one of which is evanescent, while the other remains on taking the limit:and from such a combination he derives a formula for the radius of curvature of a curve at a point, etc. See e.g. p.564 of vol. 1

Click here : for Euler's Preface :

Translator's Preface : Most of what was said in the first preface to Vol. 1 applies here also. Euler sets out the kinds of motion he will discuss in the various chapters. He is particularly interested in the shortest curves joining points on a surface, both for the cases where a point mass is free to move on the surface and also where external forces act on the mass.

Click Here : For a list of the statements of the 99 Propositions in Vol. II with their section numbers, from which the appropriate file(s) can be opened below,

Contents :
Chapter One : Concerning Motion Not Free In General. (Sect.1 - Sect.82)
The first sections are concerned with defining motion along a line or channel, and later motion on surfaces is considered. Some reference is made to Huygens' cycloidal pendulum as an example of constrained motion on a surface. Motion in the absence of external forces, and then with such forces included, is treated in a general manner. This is an interesting chapter, as Euler has invented the analytical tools needed for the rest of the book. The radius of curvature of such restricted curves are elaborated on both in two and three dimensions, without and with external forces applied. The equation for a surface is given in the form dz = Pdx + Qdy. Section 82 sets out the chapter contents briefly. There is a great deal of mathematics in this chapter, regarding setting up normal planes to a surface, etc., most of which but not all has been annotated by me at present. There is a tendency to reach for a book on advanced calculus that one must resist as much as possible, to keep the due historical perspective in sight.

Chapter Two (part a) : Concerning The Motion Of A Point On A Given Line In A Vacuum. (Sect.83 - Sect.160)
The motion of a point mass sliding down concave and convex curves is considered in detail, incl. inclined planes. The cycloidal pendulum of Huygens is given as a means of containing a particle on a surface or curve. A treatment of tautochronous members of families of curves and associated examples is given; this relies on a theorem on homogeneous functions from E044 that is now translated. An introduction to oscillatory motion is then made.

Chapter Two (part b) (Sect.161 - Sect.223)
The motion of a particle on a given curve and the force of compression between the particle and the curve are calculated in general and for a number of cases, including the circle and the cycloid. S.H.M. is analysed on a curve in general where applicable and for certain cases. The case of vertical and horizontal forces acting on a body on a curve is discussed.

Chapter Two (part c) (Sect.224 - Sect.281)
The discussion continues : the curve on which a constant force acts at any point as a body descends along it due to its weight, is solved. Other curves satisfying other conditions such as an arbitrary ratio between this total reaction force(that includes a centrifugal term), and the normal reaction due to the weight of the body, are solved. The curve of constant descent is solved, and Neil's semi-cubic parabola is found; and curves for which the rate of descent along some straight line is constant are solved. Most of these curves had been investigated for special cases by Leibniz and Joh. Bernoulli; Euler provides general results for these.

Chapter Two (part d) (Sect.282 - Sect.327)
The discussion continues : Curves with constant angular velocity are considered. Orbital motion about a centre of force under a general force law is considered. Spiral and circular motions arise. Finally we reach curves which are compared so that the time to complete the arc on an unknown curve is equal the the square root of the y coordinate of a known curve. This leads to the occurrance of an infinitude of possible curves to satisfy the given time requirement. Soon we will investigate the curve that takes the least time to get from A to B.

Chapter Two (part e) (Sect.328 - Sect.366)
The discussion continues : an infinitude of curves (x, y) are found AMC that correspond to a given time t vs vertical displacement x curve AND for a body to descend from a fixed point A to either a line, a vertical line, a curve, a point on a given curve, etc. in a given time. These become more and more complex as Euler passes from Proposition to Proposition, and will test your ability to integrate ! Finally he tires of the game, and in Prop. 40 moves on to his presentation of the brachistochrone curve, which is more general than that produced previously by others, including forces along the curve, as well as vertical and horizontal ones.

Chapter Two (part f) (Sect.367 - Sect.429)
The discussion of the brachistochrone under varies force laws continues. This is followed by a discussion of general tautochrone curves, which are derived from a dimensionless quantity, with examples under various hypotheses of the force law. Some results are quoted from other papers by Euler that I have not yet got round to translating, such as E27 on isoperimetric curves.

Chapter Two (part g) (Sect.430 - Sect.464)
This is the final section in ch. 2. It contains Prop. 50, which even Euler admits caused him some trouble : it arose from a problem posed by Daniel Bernoulli. Some results are quoted from other papers by Euler that I have not yet got round to translating, such as E31 on an application of Ricatti's method for solving differential equations; when this has been done, perhaps a little extra light will be shed on some of the derivations. The origins of '-1 = e to the i pi ' are probably present here in Euler's logarithmic integration. The final propositions deal with isochronous oscillations of two connected curves.

Chapter Three (part a) : Concerning The Motion Of A Point On A Given Line In A Resisting Medium. (Sect.465 - Sect.504)
The motion of a point mass sliding up or down concave and convex curves is considered in detail, incl. inclined planes, where resistance proportional to some power of the speed is included. I have taken the opportunity in the first proposition to add a note that details the origins of Euler's dynamical equations, where the height proportional to the speed squared at a point is considered not as an energy equation, but rather a convenient transformation.

Chapter Three (part b) (Sect.505 - Sect.564)
A continuation of the previous section 3a ; motion up and down inclined planes and on connected planes is considered, leading to what is essentially the first ever analysis of damped simple harmonic motion treated for convenience on an inverted cycloid, with resistance varying as the square of the speed. Euler notes that the property of isochronism is lost by a dampled Huygens type pendulum. He cheerfully tells us that he is going to look in the following sections, using analysis, at Newton's study of such motion in a resistive medium, with the resistance having either a constant term, or proportional to the speeds, or to the squares of the speeds.

Chapter Three (part c) (Sect.565 - Sect.601)
Euler has in mind an analytical account of Newton's experiments on the oscillatory motion of damped pendulums. He considers in considerable detail the oscillatins of a particle on an inverted cycloid in the case where the resistance is proportional to the speed, which is specially suited to his theoretical scheme of analysis. The situations of overdamped, critically damped, and underdamped oscillations are presented here for the first time. The case of resistance being partially constant and partially proportional to the square of the speed on the cycloid is considered, along with some others.

Chapter Three (part d) (Sect.602 - Sect.648)
Euler completes his analysis of damped motion on the cycloid by considering a resistive force proportional to the fourth power of the speed; this is of theoretical interest as other starting points of the body on the curve can be related to the first starting point assumed. The analysis of the initial problem makes use of a series expansion. Euler next shifts his attention to the curve descended in order that the speed is a given function of the height. This leads him to the tractrix amongst other things.

Chapter Three (part e) (Sect.649 - Sect.690)
Euler considers the shape of the curve required to generate a constant speed either along the horizontal or vertical axes, with some form of resistance present, and with a uniform downwards force. He then moves on to one of his most wonderful propositions so far, no. 75, in which he demonstates the method by which the curve can be found that the body must follow to arrive at the lower point in the shortest possible time, by treating the corresponding height or the speed in each element as having a maximum (or minimum) value; as he observes, this sort of calculation only has meaning in a resistive medium. He then moves on to further considerations of the brachistochrone curve, this time in a resistive medium.

Chapter Three (part f) (Sect.691 - Sect.737)
The equations governing the shape of a damped brachistochrone are established. The curves of constant normal reaction, that Euler calls pressure, are examined. The tautochrones of descent and ascent are examined in detail. The number e is given its numerical value for the first time by Euler.

Chapter Three (part g) (Sect.638 - Sect.766)
The closing sections of this chapter are more challenging. Euler summarises his work on curves that are almost cycloids for very small resistance and is content to state results without too much proof. This process is extended to a wider range of curves. He then moves on to consider the conditions for curves continuous at the lowest point that are tautochrones for the descent and for the following ascent.

Chapter Three (part h) (Sect.767 - Sect.820)
The problem of fitting two curves together is a satisfactory manner is completed, and the use of a continuation of the same curve as the ascending tautochrone is examined. Such curves look rather like catenaries. However, the problem of completing the oscillation with the same returning curve cannot of course be solved. The chapter ends with means of comparing tautochrones in vacuo with those resisting in the usual manner. This has been an exceedingly long chapter, running to over 200 pages; on completing ch.4, I hope to present a list of the propositions, and to delve a little into the rights and wrongs of this mammoth work. There is of course no sense it 'correcting' a historical work, though there is a lot of sense in seeing how Euler handled the incomplete physical theory of the times regarding mechanics. Perhaps meanwhile it is a good maxim to regard Euler as Johan Bernoulli's student still, although vis viva is never mentioned in the text, it is certainly used with what can be termed a velocity potential, to which it is equated. If you have some thoughts on this, then I would like to hear from you!

Chapter Four (part a) (Sect.821 - Sect.855)
Euler returns in this brief last chapter to finish the work he started around section 68 of Chapter one of this book. It is a good idea to have the first chapter around if you decide to tackle this work. This part of the work can be classified as belonging to the differential geometry of simple curved surfaces. Initially he uses his previous resultts to find a differential equation for the radius of curvature of a path traced out by a body moving on a surface under the action of some force, and the angle between the radius of curvature of the surface itself and this direction. He then considers the shape of the path with no force present, and the new shape of the path traced out with added forces, initially normal to the surface, and then normal to the path but on the surface.

Chapter Four (part b) (Sect.856 - Sect.908)
Euler always starts from the most general situation, and proceeds to analyse individual cases later. In the present case he now sets out to show how threee forces present along the individual coordinate axes can each be resolved into a component normal to the curve described on the surface, a component normal to the curve but in the tangential plane, and a third component tangential to the curve. This is done with the help of elemental diagrams in stages, and is one of the great achievements of the Mechanica (in my opinion). Such diagrams have been thought out with great care, and do not fall into the trap of being useful for some cases only. Euler then applies his new found equations to describe the curves traced out on cylinders, surfaces of cones, surfaces of revolution, under various forces such as no external force - geodesics, uniform gravity, etc. Finally, the spherical pendulum is investigated, especially the properties of the apses. Finally on p. 500 of the original Book II, this wonderful exposition suddenly grinds to a halt, though one feels that it could go on for ever ............ And so the subject of analytical mechanics came into being, which was to keep mathematicians employed for at least 200 years.

Ian Bruce. Fri. Oct. 3, 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.