Concerning Motion Which In General Is Not Free

DEFINITION 1.

1. A body is said not to move freely, [i.e. to be constrained ] when external obstacles impede its progress, and in a like manner its motion in that direction is less than it should be moving, by reason of the absolute forces acting on it.

Scholium 1.

2. In the motion of free points that we set out in the first part, the space in which we assume the body moves is a vacuum free from all obstacles; now truly we put in place a space for comparison, so that it is not permitted for the body to progress in any direction because of solid walls across which it is not allowed to pass.

Corollary 1.

3. Therefore when a body finds an obstacle to its own motion it is not able to keep moving in that direction which it held, then either it comes to rest or the motion must continue in another direction.

Corollary 2.

4. Moreover, in what direction the body progresses after meeting the obstacle must be determined from the circumstances both of the motion and of the position of the obstacle. Scholium 2.
5. It seems that this is relevant to the theory of the collisions of bodies, in which the body is not yet free to move in this way or that. Truly in this book we assume obstacles of other kinds, which do not require that acquaintance. These are continuous obstacles that restrict the motion of points and neither do they allow any turning back; and a pipe or channel which is either straight or curved is an obstacle of this kind, along which the motion of a small body must continue. In this case the path inside is prescribed in which the body is to progress, and it is not able to escape because of the firmness of the pipe [or tube]. Whereby, since here in place of a body we consider a point, a point on a given line must be moving from this position, and it is unable to leave this line.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 2 Scholium 3.
6. Moreover in this book we deal with the two motions of the impeded or restricted kind, [p. 3] of which the first we have made mention includes the motion of points on a given line or curve. The other kind restricts the freedom of the motion less ; for it only prescribes a surface on which the body must always be moving. And we are to explain these two kinds of impediments to the motion in this book. Corollary 3.
7. Therefore these properties are sought for the first kind of motion which are : the speed of the body or rather of a point in the position of any prescribed line; the force on this line; and the time in which a given point traverses a portion of the path. Corollary 4.
8. Concerning the motion of the other kind, more than the motion on this line has to be found, as the body describes the motion upon a given surface. Concerning which we uncover the principles in this first chapter. Scholium 4.
9. Truly in this first chapter we investigate both kinds of motions, for which the body is acted on by no forces, where we show with what speed it should be progressing, and what force is must exert everywhere not only on a given line but also on a given surface. But if only a surface is given, then in addition we must determine the path along which the body moves when acted on by no forces. Then we set out the principles, by which it can be determined, what changes in the path arise with forces acting, both absolute and relative, [p. 4] and from which in the following chapters we can deduce particular individual cases. Scholium 5.
10. Moreover, for both motion on a given line as for motion on a given surface, we imagine that all friction has been removed and we put no retardation to the motion in place. On this account the lines and the surfaces upon which the points are placed to be moved, are considered to be the smoothest and free of all asperities, least the motion should be liable to be slowed down on that account. All rotational motion also we imagine to be removed everywhere, which is to be explain at length later. Because of this, a point is considered to be moving as if by creeping along, in order that any part of this, if in this manner a point can be considered as made up of parts of points [this is an idea introduced in Ch. I of Book I and not yet used], then they have the same motion. Scholium 6.
11. Therefore what has been treated in the preceding book, and what is to be treated concerning the motion of points in this book, can be adapted to bodies of finite size also, but only if the movement of these is always parallel to themselves, and all the parts of the body are provided with equal motion. This indeed will become clearer from the following books, for which there is no disagreements between the case of the motion of finite bodies from the motion of points. On which account therefore in these books we considerEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 3 only points since, as they do not have different parts, thus also they are unable to have parts with different motions. PROPOSITION 1. [p. 5] Theorem.
12. A body or a point, which is moving on a given line and is acted on by no forces, always keeps the same speed, only if any two adjoining elements of this line nowhere constitute a finite angle. Demonstration. Since the body, while it is moving on the line AM (Fig. 1), is acted on by no force and neither does the curve have any friction, and the motion of the body is unable otherwise to be changed, except in as much as it is impeded by the line AM , on which a small body is able to move freely; from which any change in the speed which should arise is to be investigated. Let the speed which the body has at M be equal to c; here therefore with this speed, the body progresses along the tangent Mv, if it could move freely ; which now, since the body is unable to leave the curve AM, is unable to happen, for the body is forced to progress along Mm. Hence for this reason the motion of the body along Mv can be considered to be resolved into the motion along Mm and the motion along Mn, with the right- angled parallelogram Mmvn arising. It is evident that this motion along Mn, the direction of which is normal to the element of the curve Mm, is to be absorbed by the curve unless there is no change in the motion. Therefore the body progresses with another motion along Mm with a speed, which is to the former speed as Mm to Mv; whereby the speed, .c . Since truly with which the body describes the element of the curve Mm, is equal to Mm Mv Mvm is the triangle for the rectangle mn, thus Mm < Mv, [p. 6] this speed is less than the previous speed c and the decrease in the speed is equal to ( Mv − Mm )c . For the value of Mv this can be found MO, the radius of osculation of the curve at M which is equal to r and the element Mm is equal to ds; and this is, on account of the angle O = angle mMv, MO : Mm = Mm : mv , from which there comes about mv = dsr and 2 From this the decrement in the speed can now be obtained, while the element of the curve ds is traversed, equal to cds2 , of which with the whole gives the decrement of the speed, 2 2rEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 4 2 while the body traverses the finite portion of the curve AM. But the expression cds2 is 2r equivalent to a differential of the second order ; therefore the integral of this is a differential of the first order. On account of which the decrease of the speed, after the body has traversed an arc of some given size, is infinitely small and the body is carried with a uniform motion along the whole curve AM, only if the radius of osculation r was nowhere infinitely small. Q.E.D. [We meet this kind of geometrical argument time after time in Euler’s work, where we would now refer to the principle of conservation of energy; in the present case, no work is done on the particle.] Corollary 1.
13. Therefore in any curve, in which the radius of osculation is nowhere infinitely small, the body moves uniformly, if indeed it is not allowed to be acted on by any forces or friction. Corollary 2.
14. If the radius osculation is infinitely small, then cds2 is either a finite quantity or is a 2 2r differential of the first order. [p. 7] In the first case the body parts with a finite change in the speed, in the other truly it is infinitely small. Corollary 3.
15. Moreover since points of this kind are rare in all curves and are widely scattered between each other, and the body still travels uniformly along the arc intercepted by two such points. Scholium 1.
16. The case, in which the body suffers a sudden finite decrease in speed, is only possible where the curve has cusps. For with these in place the body is forced to turn back directly and normally on the point of the cusp it strikes. Therefore the body then not only loses a finite step in its speed, but it must lose all of its motion entirely, except perhaps is put to be elastic, in which case it may be reflected with the speed with which it arrived, and thus the uniform motion is conserved. In a cusp, two elements of an infinitely acute angle are put in place. Scholium 2.
17. Truly as well as cusps, other points can be given on curves, in which the radius of curvature is infinitely small ; because any two touching elements are placed in almost the same direction and following this the angle is infinitely small, but it cannot happen that the body suffers a finite decrease in the speed, as the above demonstration shows. On account of which, since points of this kind are rare, the body nevertheless moves with a uniform motion. [p. 8]EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 5 Corollary 4.
18. Therefore if the motion of the body were elastic, then the motion is always carried on uniformly on any curve ; but if it is not elastic, then the cusps only upset the motion, while evidently they destroy it. Scholium 3.
19. In order that these may be made clearer, let two elements of the curve be AB and BC (Fig. 2) and following the angle ABC that they make, CBD is next put in place infinitely small, and the sine of this angle is dz, with the total sine put equal to 1. Because the body, after it describes the element AB, by the force of inertia, [vi insita] tries to progress along BD with the previous speed which was c, this motion is taken in two parts, the one in the direction BC, and the other in the direction normal to BC, which cannot be effected. Therefore by sending the perpendicular DC from D to BC, the body moves along BC with another motion, with a speed which is to the first speed as BC to BD, i. e. as ( 1 − dz 2 ) to 1. Therefore the speed along BC is equal to c ( 1 − dz 2 ) or c − cdz ; whereby the decrement of the speed is cdz , which is equivalent to a 2 2 2 2 differential of the second order. From which it is understood, as long as the angle CBD of the curve is infinitely small, the motion of the body progresses at a steady rate. But on the curve the angle CBD is either infinitely small or the angle ABC itself is infinitely small, when the point falls on cusps. Consequently only cusps disturb the uniformity of the motion, unless the body was elastic, in which case the motion nevertheless is conserved. [p. 9]EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 6 PROPOSITION 2. Theorem.
20. While the body is moving uniformly along the curve AM (Fig. 1), it pressed the curve normally with a force at the individual points M with a force which is to the force of gravity as the height corresponding to that speed is to half the radius of osculation. Demonstration. If the body on the curve AM must be moving freely with a uniform motion, then it is necessary for the force to be present everywhere only acts normally along MO, which has the ratio to itself to the force of gravity as the height corresponding to the speed of the body to half the radius of osculation MO, as appears from the demonstrations of the preceding book (165, 209, 552). For unless such a force is present then the body travels in a straight line. Moreover in this case the channel AM, in which the body is considered to be enclosed, impedes the motion, in which the body progresses less in a straight line. On account of which the body presses the channel normally with so great a force following the direction Mn. If indeed the body is pressed by such a normal force present, then it can move freely in the channel AM ; and neither might it press the channel; but truly with this force missing, as we put here, it is necessary in order that the body presses the channel itself with such a force. [Newton’s third law.]Q.E.D. Corollary 1.
21. Therefore if the height corresponding to the speed of the body is put as v and the radius of osculating MO is equal to r and the gravity on the body is equal to 1, as clearly it has if it has been put on the surface of the earth, the force will be by which the body pressing on the channel at M along Mn, is equal to 2rv .[p. 10] Corollary 2.
22. If the body moves with a greater or smaller speed along the curve AM, then the force pressing at M is greater or less in the square ratio of the speed, since the height v is proportional to the square of the speed.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 7 Corollary 3.
23. The direction of this force is normal to the curve and is in the opposite direction to the position of the radius of osculation MO. Whereby the radius of osculation in the other part of the curve produced gives the direction of this force. Corollary 4.
24. If the body is moving in a straight line, this force is zero on account of the infinite radius of osculation. This is also evident from the nature of the motion. For the body moves in a straight line uniformly spontaneously and on this account is not pressed by the channel. Corollary 5.
25. If the curve AM is a circle, the force is the same everywhere. Truly with that to be greater as the radius of the circle is made less. Indeed with the same speed present, the force varies inversely as the radius of the circle. Scholium 1.
26. Where the body is able to move freely along the curve AM uniformly, it is necessary that [p. 11] it is drawn along the normal MO by a force equal to 2rv . From which it is to be understood that the body struggles with such a force in the opposite region, otherwise the body cannot be kept on the curve by that force. Therefore while the body is forced to move along the channel AM it is being carried along by the struggle with this force, and this force is exercised by the channel itself. On account of which the channel must have such firmness, in order that it is able to sustain such force. Corollary 6.
27. Therefore the body is able to carry out the motion without any expenditure of the speed, which clearly is consistent with the definition of the force. Corollary 7.
28. Therefore the force arises from the motion alone. On account of which, just as motion is generated from forces, thus forces can arise from the motion. Scholium 2.
29. Hence it is understood, as now in the first book above we have agreed with the notion (102), that it is unclear whether motion is owing to forces or whether forces to motion. For we see each in the world, truly forces and motion to arise; therefore one is the cause of the other, the question is to be decided from reasoning as well as from observation. Indeed there seems to be hardly any agreement about forces that arise on bodies that remain at rest, with much less forces being decided to arise form these. Besides truly, that everything can be shown to arise from motion, is considered to be the natural cause to be given of all phenomena. [p. 12] For motion once in existence must always to be conserved we have clearly shown above (63); this we have truly elaboratedEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 8 upon, as forces arise from motion. As truly forces without motion that are able either to be present or to be conserved cannot be conceived. On account of which we can conclude that all forces, which are observed in the world, arise from motion; and it falls upon the diligent investigator, that for each and every motion of bodies from any kind of force in the world their origin has to be observed. Scholium 3.
30. Since it is difficult to understand how such an effect clearly of a continuous force can act on a body without any change arising in the speed, it is a worthwhile task to inquire about the cause of this effect. We have seen in the preceding proposition that the motion of the body was not exactly uniform, but the speed is actually allowed to decrease, while the body is moving through singular elements in the curve. Truly these decrements are equivalent to second order differentials, as also only infinitely small changes in the speed repeated infinitely often are able to diminish the speed. Therefore I declare that the force acting is ascribed to an infinitely small decrement in the speed; and I become more confirmed in this belief, because when the decrement in the speed becomes greater, so also does the force present increase. Since the force at M is equal to 2rv and while it is being traversed, the whole element Mm is acted upon by this force [p. 13], it is permitted to express the effect of this force on the element Mm = ds by 2vds . Truly the above r decrement in the speed, while the element Mm has been traversed, is found to be cds 2 (12). But because this is equal to c there, which here we have as 2r 2 2 equation arises : − dc = cds2 , and it becomes 2r v ; hence the Therefore we have − 4vdv = 4v ds2 equal to the square of the force that supports the 2 2 2r element Mm. Corollary 8.
31. Therefore the square of the force acting on Mm is equivalent to the decrement of 2v 2 . And if this decrement is equal to ds 2 , then the force is equal to the force of gravity, from which the comparison of these forces is known. Corollary 9.
32. Therefore it is conceded that an infinite number of infinitely small decrements in the speed suffices in the production of a finite force. For as long as the decrement v 2 itself is the homogeneous ds 2 , the force is finite; but truly if that infinite number of infinities becomes greater than ds 2 , then the force also becomes infinitely large.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 9 Definition 2.
33. This force, which the body exercises on the body in the line of the curve is called the centrifugal force, since the direction of this pulls from the centre of osculation O. [p. 14] Corollary 1.
34. The centrifugal force is therefore to the force of gravity as the height corresponding to the speed to half the radius of osculation. Corollary 2.
35. Therefore when the body is forced to move along the line of the curve the centrifugal force presses against the curve, even if no external force is acting. Scholium.
36. Therefore when the body is acted on by some external forces, a force also arises in the channel from these forces as well as from the channel itself, both pressing in a two- fold ratio, truly partially from the external forces and partially from the centrifugal force. Now therefore, what the force shall be that prevails on the constrained body is to be found. PROPOSITION 3. Theorem.
37. If the body, which is moving in the channel AM (Fig. 3), is acted on at M by the force MN, the direction of which is normal to the curve AM, then the speed is neither increased or decreased and the whole force is taken up in pressing against the channel. Demonstration. From the first book (164) it was shown that the force, the direction of which is normal to the direction of the motion, neither increases nor decreases the speed. Though indeed there for free motion it has a place for stiffness, [p. 15] since the normal force neither before or after pulls on the body. Truly in free motion the direction of the normal force does not change, as it is not able to have an effect in this situation. Therefore the body is pressed by this force on the channel and consequently only the force of the channel presses in the direction MN. Q.E.D.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 10 Corollary 1.
38. Therefore the direction of such a normal force is either incident in the direction of the centrifugal force or is in the direction contrary to that. In the first case the centrifugal force is increased, and in the other it is diminished. Corollary 2.
39. Because the direction of the centrifugal force falls on the convex part of the curve, the effect of this is to increase the force, if the normal force falls in the same place; but if the normal force is directed to a concave part of the curve, the effect is diminished the force. Corollary 3.
40. If the normal force is equal to N and the centrifugal force as before is equal to 2rv , the curve is pressed either by the force 2rv + N , if the forces act together, or by the force 2v − N , if they act in the opposite directions. r Corollary 4.
41. If the normal force is equal and opposite to the centrifugal force, then the curve sustains no force, or the body does not try to escape from the curve. Therefore in this case the body is free to describe the same curve; [p. 16] it is also evident that the normal force is equal to 2rv ; for it is brought about here, in order that the body is free to move on any curve.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 11 PROPOSITION 4. Theorem.
42. If the body, which is moving in the channel AM (Fig. 3), is acted on at M by a force, the direction of which is along the tangent MT, the effect of this is consistent with this, that the speed of the body is either increased or diminished in the same way as for free motion. Demonstration. Since the direction of this force is the tangent MT of the channel, the effect of the channel cannot impede the effect of this force ; nor can this force exercise any effect on the channel. On account of which the force either augments or diminishes the speed of the body, according as the direction of this either acts in the same direction as the body or in the opposite direction, and clearly if the body is moving freely. And with the height corresponding to the speed at M equal to v, the element Mm = ds and with the force MT = T, there is dv = Tds with the accelerating force T; but with retardation that becomes dv = −Tds . Q.E.D. Corollary 1.
43. Therefore in the motion of bodies on given lines, the normal force only generates a force on these lines, and the tangential force truly only affects the speed. Corollary 2.
44. Since the force of the retarding resistance may be greater than the tangential force, [p. 17] it acts in the same manner in the motion of bodies on given lines as in the case of free motion. If therefore as well as the accelerating tangential T there is the resistance R present, then with both joined together we have dv = Tds − Rds .EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 12 PROPOSITION 5. Problem.
45. If a body is moving on some given line AM (Fig. 4) in some medium with resistance and in addition it is acted on by some absolute force, the direction of which is MP, to determine the effect of the absolute force as well as of the resistance as well as the force supported by the curve AM. Solution. Let the height corresponding to the speed at M be equal to v, the force of the resistance is equal to R and the absolute force MP = P, the direction of which is such that, as with the element Mm taken equal to ds , the perpendicular mn from m sent to MP is equal to dx and Mn = dy = ( ds 2 − dx 2 ) . The force P is resolved into these two forces along the normal MN to the curve and the force pulling along the tangent MT ; on this account the triangles MPT and Mmn are similar and the normal force MN or PT = Pdx and the tangential force ds Pdy MT = ds increased the speed. Since truly the force of resistance decreases the speed, the Pdy speed is only increased by the excess ds − R ; on this account there is (42) dv = Pdy − Rds. The normal force truly is effected by Pdx , as the curve is pressed just as much at M along ds the direction MN to the convex part of the curve in place. [p. 18] Whereby, since the centrifugal force acting at the same place is equal to 2rv , with the radius of curvature designated by r the radius of osculation at M, the total force by which the curve is pressed on normally at M, along MN, is equal to Pdx

• 2rv . Hence the motion of the body on the ds given curve as well as the force acting on the curve can be found at individual points. Q.E.I. Corollary 1.

46. Therefore from these two formulas both the acceleration and the force on the wall can be expressed can all be deduced from the expressions, which pertain to the motion on the given lines.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 13 Scholium 1.
47. Here indeed we have put in place a single absolute force; yet nevertheless from that it is understood how the effect of many forces can be understood. Of course as we made in free motion, thus also here the individual forces are to be resolved into two parts, truly the normal and the tangential, from which by gathering these together a single normal force and a single tangential force arises; the effect of which can be determined by Propositions 3 and 4. Scholion 2.
48. Therefore up to the present we have set out the fundamentals, from which in the following it is permitted to determine the motion of bodies on given. But before we treat the similar principles of the motion on given surfaces, it is expedient that we consider a few cases in which the motion on a given line in effect can be deduced.[p. 19] In as much as with the help of channels, in which the body is contained, it is of minimal use to produce such motion on account of friction and other obstacles, which by no means are able to be removed. Moreover constrained motion of this kind is most conveniently brought into being with the aid of pendulums, as was first done by Huygens[Original reference presumably used by Euler: Chr. Huygens, Horologium oscillatorum sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris 1673; Opera varia, Vol. 1, Lugduni Batavorum 1724, p. 89. See the English translation in this series.]; why we arrange matters to make use of pendulums we explain in the following proposition. PROPOSITION 6. Problem.
49. How a body is able to move on a given line with the help of a pendulum. Construction. Let AMB (Fig. 5) be the proposed curve, in which the body must move; the evolute AOC of this curve is constructed and a plate is curved following this figure and set in place. Then a thread is led around this plate, which has one end fixed to the plate, and the body A to be moved is fastened to the other end. Therefore when the body begins, it is evident that it must move on the curve AMB, because the thread, as it then separates from the place, describes the evolute of this curve. Q.E.F.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 14 Corollary 1.
50. Therefore for this reason the body progresses along the given curve and is not liable to friction. Whereby in this manner such motion along curves as are found in theories can be conveniently put to the test. [p. 20] Corollary 2.
51. From the theory of evolutes it is understood that the separated part of the thread MO is normal to the curve AMB and is the radius of osculation of this curve. Corollary 3.
52. When the body is moving on the periphery of the circle AMB (Fig. 6), the curved plate is not needed, but the thread has only to be fastened at the end C to the centre of the C of the periphery. Corollary 4.
53. Since the thread MO (Fig. 5) is the radius of osculation, the total centrifugal force is devoted to stretching this thread. Whereby this thread has enough strength not to be liable to be extended. For unless the length is always kept the same, the desired curve is not described. Corollary 5.
54. By adding the absolute force besides the centrifugal force the normal force is obtained, which also pulls the thread, if the centrifugal force is to be added. But if it acts in the opposite direction, it diminishes the tension in the thread, indeed also, if this force is greater, then the thread is compressed, in which case it is of no use as an evolute. For since the thread must be flexible, it is not able to resist compression and neither does it offer any impediment, in which case the body recedes from the curve AMB towards the evolute. Scholium 1.
55. Besides this difficulty, the generation of curves by evolutes also labours under this weakness, because the straight line is unable to be produced [p. 21] ; indeed for that to be generated the thread is requires to be infinitely long. In a similar manner this evolute cannot be adapted for curves that have an infinitely great radius of curvature somewhere. Then also neither curves with cusps nor with contrary bends can be described in this manner. Thus on this account the practice only has a place with curves having a finite curvature everywhere, to which it must be added, so that the total force acting on the curve is anywhere directed to the concave part of the curve.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 15 Scholium 2.
56. Huygens, who first developed the principle of the evolute, at once put it to this use, that is apparent from the unusual need of the swinging pendulum clock. For since it can be shown that the swings on a cycloid are all isochronous, he wished to bring about cycloid motion in clocks, which is effected by the pendulum swinging between cycloidal plates. For since the evolute of a cycloid is a cycloid, for this reason it was obtained that a body tied to the end of a thread moves in a cycloid. Scholium 3.
57. Moreover in this motion of pendulums it is appropriate to note specially that besides the motion of the body, the thread too must be moving, but as it is the custom in this book in which only the motion of points is to be carried out, it is too small to be a concern. In addition, neither do we touch on the motion of the body attached as the pendulum which is not parallel to itself, but rather circular, and which is permitted around the centre of any circle of osculation on the curve. [p. 22] Therefore in this book we submit to be examined only the motion of a point in a given line or surface, and we do not consider either the motion of the thread or the case of circular motion. Moreover in the following motion of pendulums, where both the motion of the thread and circular motion are deduced from a computation, we reduce the motion to that of points only, thus in order that these which are treated in this book, are nevertheless found to have a practical use. On which account, as we have now advised, the point [acting as the pendulum] is considered to be always carried by moving parallel to itself either on a curve or surface without any friction. PROPOSITION 7. Theorem.
58. If a body under the action of no forces is moving in a vacuum or in a medium without resistance on some surface ABC (Fig. 7), then it is carried in a uniform motion, with all friction removed from the air. Demonstration. When a body moving on a given line is able to continue to be pressed, it is able to move much more on a given surface because there the freedom is less restricted. Therefore, let DMm be a line on which the body is progressing ; this is either a straight line or a curve. If this is a straight line, then there is no doubt that the body progresses with a uniform motion. But if it were a curve that could be expressed by an equation, and two adjoining elements of this are either situated nearly in the same direction, or they constitute an acute angle because cusps occur. [p. 23] In the above case it has been shown that the body suffers no decrease of the motion (12). Now with cusps indeed all the motion is lost, unless [the collision] is elastic. On accountEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 16 of which, if the motion is made on a curve, or on the part of a curve with the cusps missing, then the motion of the body is uniform. Q.E.D. Corollary 1.
59. For a decrease in the speed of the body is permitted, as often as the direction is forced to change, now this is therefore equivalent to a second order differential, and even if it is integrated then an infinitely small decrement is produced. Corollary 2.
60. Clearly if the speed of the body is c and the radius osculation MO = r, then the decrease in the speed while the body traverses the element ds is equal to cds2 (12). 2 2r Scholium.
61. The demonstration of this proposition clearly agrees with the above proposition except for this difference, in the former case the body is forced to move along a given line, while now in the above case it is free to have any path on the given surface. On account of which all the notes that have been made for the first proposition prevail here too. [p. 24] Therefore we will see what path on any given surface the body should traverse. PROPOSITION 8. Theorem.
62. The path DMm (Fig. 7), which the body moving on some given surface ABC describes, is the shortest line that can be drawn between the two terminal points D and M, clearly if the body is moving in a vacuum and is acted on by no forces. Demonstration. The body now describes the curve DM; it is evident that the body will be moving along the tangent Mn from M unless it is forced to persevere on the surface. Therefore, since the motion along Mn cannot be made, it is resolved into two components, of which the one is set out on the surface, and the other now is in a direction perpendicular to the surface, and thus removed from the surface it is not in effect possible to be deduced. On this account, the perpendicular nm is sent from n to the surface; Mm is the element of the line along which the body progresses from M . Hence the plane nMm is normal to the surface, in which are placed both the element mM and that previous element which has just been described. But the shortest line drawn on any surface has this property, that the plane on which any two contiguous elements are placed is normal to the surface. On account of which the line DMm, which is described by the body, is the shortest line on the surface ABC. Q.E.D. [p.EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 17 24] [Note that Dn is in effect the previous element transposed, according to Newton’s First Law, and the whole proposition is a generalization of the linear case presented above.] Corollary 1.
63. Hence if from the point A, at which the motion begins, the shortest line ABC is drawn following the direction of the motion, the path is obtained, along which the body is moving uniformly. Corollary 2.
64. Since the tense thread on the surface designates the shortest path, the tense thread also shows the path along which a body on that surface will move uniformly. Corollary 3.
65. Therefore if the proposed surface were a plane, then the line described by the body would be straight, since in the plane this is the shortest line. And on the surface of a sphere the body moves on a great circle. Corollary 4.
66. Since the plane in which the two adjoining elements of the curve DMm are placed is normal to the surface, then the normal to this plane lies on the surface, and the radius of osculation MO of the described curve is now put in the same plane, normal to the surface. Scholion.
67. As the shortest line to be found on any given surface has been demonstrated by me in Book III Comment. Acad. Imp. Petrop. [Concerning the shortest line joining any two given points on any surface: linea brevissima in superficie quacunque duo quaelibet puncta iungente. See E09 in this series of translations.] Moreover there I determined the shortest length from another principle, and the elements of that matter shall not yet be put in place: the shortest line or that which is described by a body that I have decided to determine in the following proposition. [p. 26] [We should note that the motion of a body on a surface is a dynamics problem, while the shortest distance between two points on a given surface is a purely geometrical problem.]EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 18 PROPOSITION 9. Problem.
68. On any given surface, to determine the line described by a body moving on the surface, acted on by no forces. Solution. In order that the nature of the proposed surface can be expressed the fixed plane APQ is taken, and in that plane the line (Fig. 8) AP it taken for the independent axis . Then from some point M on the surface the perpendicular MQ sent to this plane and from Q the perpendicular QP is sent to the axis AP. Now on putting AP = x, PQ = y and QM = z the nature of the surface is given an equation between these three variables x, y, z and constants. Let the differential equation of this surface be given by : dz = Pdx + Qdy , from which the shortest line on this surface or the line that the body describes must be determined. Now this line is determined from this consideration, that the radius of osculation is declared to be perpendicular to the surface. On this account, first the normal to the surface is drawn, and then we determine the radius of osculation of that curve drawn in this plane, in which later from the coincidence of these lines the nature of the line sought can be inferred. [p. 27] In order to find the normal to the surface, first the surface is cut by the plane MQB, with the line BQ proving to be in the plane APQ [the xy plane] parallel to the axis AP, and the curve BM is produced by this section; the nature of this curve is expressed by this equation dz = Pdx , which arises from the surface dz = Pdx + Qdy , with y constant or dy = 0 put in place. To this curve BM the normal ME [in the xz plane] is drawn crossing the line BQ produced in E; the subnormal QE = zdz = Pz . dx [since dz = ∂∂xz dx + ∂∂yz dy = Pdx + Qdy , in modern notation.] Now with the line EN drawn perpendicular to BE, any line MN drawn from M to NE is normal to the curve BM. [To understand this statement, consider the line MN, for any N along EH, to rotate about an element of MB centred on M as axis, keeping the same right angle to the element as the coplanar line ME, as the rotation is about an axis normal to the element. Similarly for the other case treated below.] In a like manner, the surface is cut by the plane PQM and the curve CM is produced by the section, the nature of which is expressed by the equation between z and y by keeping x constant, which is given by dz = Qdy . Let MF be the normal to this curve [in the yzEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 19 zdz − plane]; the subnormal is given by QF = dy = −Qz ; I use this with the negative sign, since the subnormal QF I put in place falls towards P. Now with the line FN drawn parallel to the axis AP, any line drawn from M to FN is normal to the curve CM [as above]. Therefore the line MN, which falls at the intersection N of the lines FN and EN, is perpendicular to each curve BM and CM and on account of this is perpendicular to the surface. Hence the locus of the normal is found [a basic necessity for Ch.4] by taking [The use of the subtangents is set out in the following sketches : ]EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 20 Now in the determination of the radius of osculation of any curve on the given surface let the position for two elements of the curve be Mm and mμ (Fig. 9), to which there corresponds the elements Qq and qρ in the plane APQ , and on the axis AP, I assume that the elements Pp and pπ are equal. Therefore we have : and Qq and Mm are produced in both directions, of which the first meets πρ in r, and the other crossed the normal rn to the plane APQ in n, [p. 28] and as Pp = pπ qr = Qq and mn = Mm , also πr = y + 2dy , and rn = z + 2dz . Now the normal mS is drawn to the element Mm in the plane Qm, crossing Qq produced at S; hence we have [Note : The triangles with hypotenuse Mm and mS are similar; the original equation below has QS rather than qS as the subject of the formula, which is incorrect; this mistake is perpetuated in the O.O. version as well. If QS is required, then it is given by : QS = dx 2 + dy 2 + dz 2 + zdz dx 2 + dy 2 ; all quantities to first order in the elements.]:EULER’S MECHANICA VOL. 2. Chapter one. page 21 Translated and annotated by Ian Bruce. Now with SR drawn in the plane APQ perpendicular to QS, then all the lines drawn from m to SR are normal to the element Mm. Therefore the radius of osculation of the curve Mmμ is one of these normals. Now that one of these normals agrees with the radius of osculation, which lies in that plane, in which the elements Mm and mμ have been placed. On account of which it is required to determine this plane. Now the elements mn and nμ are in this plane; thus each produced as far as the plane APQ gives the intersection of that plane with the plane APQ. But nm or mM crosses with the plane APQ at T, where it crosses the element Qq produced. Therefore we have : nμ itself is parallel to MV, situated in the plane mnμ ; now this line MV is incident at V in the plane APQ and from this ratio gives QV: and thus QV becomes rρ [since rn = z + 2dz ; rρ = dy + ddy ; ρμ = z + 2dz + ddz ; then QV = z . rn − ρμ = z( − ddy ) ][p. ddz 29] On account of this, the line TV produced is the intersection of the plane mnμ with the plane APQ, whereby the line MR, which is drawn from the crossing of the lines SR and TV is likewise normal to Mm and lies in the plane mnμ , and therefore the line MR is put as the radius of osculation of the curve at M. [Thus, above it is shown that all lines drawn from m intersecting SR are normal to the element Mm, and now we have the intersection of the plane containing the adjoining element mμ with this plane as the line TV extended to R. Euler asserts that MR is also normal to Mm , at the other end of the element, as RM lies in the plane of the curve mnμ , and is normal to one element; for in the limiting process, we can presume that m and M coalesce.] From these, the point R can be found in this manner : that is, with RX drawn perpendicular to AP produced, andEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 22 [Digression : It seems that Euler used an ab initio cross product method twice to get these results, as we now demonstrate. We have annotated Fig. 9 and put some of the coordinates on this diagram. Note that derivatives are always put positive by Euler in his diagrams, and we have followed this rule, even though they must be negative in the diagram; thus, the restrictions of the diagram does not interfere with the mathematics. Mn and nμ are elements on the surface; they are small enough compared to the local curvature that they can be considered as straight, and according to Euler’s habit, all the ‘bending’ occurs at the beginning of the second element nμ. The first element extended an equal length can be considered as an element of a tangent line at n in the direction Mn. The normal to the plane Mmμ is the vector N formed from the cross product of the vectors representing Mn and mμ : i j k = i( dyddz − dzddy ) − jdxddz + kdxddy , where i, j, and k are N = dx dy dz dx dy + ddy dz + ddz unit vectors along the x, y, and z axes. Now, the direction of the radius of curvature RM is perpendicular to both N and to the line TM, which acts in the direction (dx, dy, dz). Thus, the direction ratios of RM, or d.r.RM are found from the cross product of N with (dx, i j k dy, dz) : i. e. d .r .RM = dx dy dz . dyddz − dzddy − dxddz dxddyEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 23 MR hence has the direction ratios dx( dyddy + dzddz ), dydzddz − ( dx 2 + dz 2 )ddy , ( dx 2 + dy 2 )ddz − dydzddz . Any point on the line MR can hence be represented by : x + t .dx( dyddy + dzddz ); y + t [ dydzddz − ( dx 2 + dz 2 )ddy ], and z + t [( dx 2 + dy 2 )ddz − dydzddz ]. for different values of the parameter t. The point R is taken to lie in the z = 0 plane, and hence t = z /[( dx 2 + dy 2 )ddz − dydzddz ] , from which the corresponding x and y values can be found, as in Euler’s equations above. We now return to the text.] In which therefore the normal MN (Fig. 8) falling in the direction of the radius of curvature, must have AH = AX [= x + Pz ] and XR = HN [= − y − Qz ]; Hence, and Which equations indeed agree with each other ; for with these solved together, we obtain Pdx + Qdy = dz , which is the equation setting out the nature of the surface itself. Therefore either of these equations solved with this equation dz = Pdx + Qdy gives the curve traversed by the body on the proposed surface. Q.E.I. Corollary 1.
69. Therefore for the proposed line described on the surface, we have from the above equations : But since dz = Pdx + Qdy , this becomes or Corollary 2. [p. 30]
70. If the other equation is taken and from both sides is subtracted there is obtainedEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 24 Which is the actual equation given for the shortest line described in any surface in the Transactions of the St. Petersburg Academy of Sciences [Comm. Acad. Petr. E009], Vol. III. Scholium 1.
71. As in this case in which the body is not acted on by any forces, the direction of the radius of curvature must agree with the normal to the surface, thus in the other cases, when the body is acted on by forces, these lines must constitute a given angle : On account of which the angle is generally found between MN the normal to the surface and MR the direction of the radius of osculation ; that is, as we may now either put in place or as we have found (Fig. 10), and With NR drawn from N and the perpendicular NO is sent to MR ; there is produced andEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 25 NO with the whole sine put equal [p. 31] Now the tangent of the angle RMN is equal to MO to 1. Moreover with the assumed variables substituted above and with the equation called upon dz = Pdx + Qdy the tangent of the angle NMR becomes equal to Hence with the angle vanishing the equation becomes as above (69). Scholium 2.
72. Now the length of the radius of osculating MO (Fig. 9) is found from the angle nmμ with the aid of this ratio : as the sine of the angle nmμ is to the total sine, thus Mm is to MO. Now we have hence the perpendicular from n in mμ produced is equal to Whereby this perpendicular is to ( dx 2 + dy 2 + dz 2 ) as ( dx 2 + dy 2 + dz 2 ) is to MO, and thus the radius of osculation is produced : Moreover this formula for the radius of osculation is of help in the following proposition, in which we investigate the force that the body exercises on the surface. [p. 32] Scholium 3.
73. From this general expression for the radius of curvature there arises an expression for the shortest line, if it is solved with this equation :EULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 26 Moreover the radius of osculation is produced : And this expression gives the radius of osculation of the curve described on the proposed surface by a body acted on by no forces. PROPOSITION 10. Theorem.
74. The force that a body moving on a surface under the action of no external forces exercises on the surface is made normally to this surface towards the convex side and has the ratio to the force of gravity as the height corresponding to the speed of the body is to half the radius of osculation of the curve described by the body. Demonstration. Let DMm (Fig. 7) be the curve on the surface ABC described by the body, the height corresponding to the speed of the body is equal to v and the radius of osculation of the curve MO is equal to r. Because the body is able to move freely from M, it can progress along the element Mn, now the surface brings it about that the body advances along the element Mm, with the distance nm being perpendicular to the surface, and the surface is required to exert a force of such a size along mn that the body follows the direction of Mm along the surface, departing from the direction Mn. [p. 33] Now this is performed by the force 2rv acting normally to the surface along the direction of the radius of osculation MO. On this account the force of the body is normal to the surface , clearly acting along mn, and is equal to 2rv with the force of gravity acting on the body taken as equal to 1. Q.E.D.

Corollary 1.

75. This is therefore the centrifugal force that the body exerts on a surface in a similar way to that when it is forced to move on a given line.

Scholium 1.

76. The force acting on the surface must necessarily be normal. For unless it is normal, it is possible to resolve it into two components, of which one is normal and the other is placed along the surface. Now of these only the normal devotes itself to pressing on the surface, while the other changes the motion of the body.

Corollary 2.

77. We find that the length of the line of the radius of osculation r, that the body describes with no forces acting on a proposed surface (73). With this assumed, the centrifugal force is equal to :

Scholion 2. [p. 34]

78. This centrifugal force acting on the surface and the above centrifugal force acting on a given curve that has been discussed above, have the same place in the equation; see Prop. 2 (20) with the adjoining corollaries and scholium. For the shortest line that the body can describe on the surface can be considered to resemble the channels along which a body can move, and then all that has been said concerning the motion in these channels prevails, which have been produced above for the motion upon a given line with no external forces acting. PROPOSITION 11. Problem.
79. To determine the effect of any kind of force that acts upon a body on a given surface either in a vacuum or in a resisting medium. Solution. For any body, the direction of the external force acting can be resolved into three parts : the first of which that we call M, the normal direction to the surface; secondly, we designate by N that direction normal to the motion of the body as well as being normal to M, and the direction of this is in the tangent plane of the surface; and the direction of the third force called T agrees with the direction of the motion, which is therefore the tangential force ; surely the first two are the normal forces. Now since the directions of these three forces are mutually normal to each other, neither is able to disturb the others. [p. 35] Whereby, we investigate the effect that each can produce. The first external force M, the direction of which is normal to the surface, has no effect in changing the motion of the body, as the whole is expended in pressing upon the surface. Therefore M either diminishes or increases the force arising from the centrifugal force, as the direction of this falls on convex or concave parts of the curve. For that forceEULER’S MECHANICA VOL. 2. Chapter one. Translated and annotated by Ian Bruce. page 28 acting towards an inner part of the curve ; the total force acting on the surface towards the outside is equal to (77). For the force arising from the centrifugal force is diminished in this case by the force M. The second force N, since the direction of this is put both normal to the direction of the surface and to the direction in which the body moves, can only change the direction of the body and neither increase nor decrease the speed. Therefore this force makes the body move along the shortest line deduced, so that it no longer moves in a plane normal to the surface; therefore it is necessary to find the inclination of this plane of the shortest line in which the body moves, to the normal to the surface. Now this angle of inclination is equal to the angle that the radius of osculation of the line described makes with the normal to the curve, and which we have determined previously in general (71). After the body describes the element Mm with a speed corresponding to the height v [Italic v], [p. 36] is progressing, unless acted upon by the force N , along the element mv (Fig. 11) to v [Italic Greek ’nu’ : Microsoft seems to think these two letters are the same.], thus as Mm and mv are two elements of the shortest line and placed in a plane normal to the surface; the direction of the normal force N in the plane; the direction of the normal force in the plane of the paper is reduced to that above, if indeed we put this force N above to be put in this position of the elements, as represented in the figure. Therefore this force has the effect, that the body is moving along the mμ and by the angle vmμ deflected by the direction mv. For this angle corresponding to the radius of osculation is . Whereby when the force N generates this angle and the speed of the equal to = mv μν 2 curve corresponds to the height v, from the effect of the normal force it becomes Now where the inclination of the plane Nmμ , in which the body actually moves, to the plane Mmv, which with the normal in the surface is found, the perpendicular vn is sent from v to the element Mm produced ; μn is also in the perpendicular mn and thus the angle μnv is the angle of inclination of the plane μmM to the plane vmM; and since μv is μv normal to vn, the tangent of this angle is equal to nv = N2v.mv . But nv is determined from .nv 2 the inclination of the elements Mm and mv or the radius of osculation of the shortest line, = r and of which Mm and mv are elements. Here the radius of osculation is r, it is mv nv 2 thus the tangent of the angle μnv is equal to with the value found substituted in place of r (73). [p. 37] For now this angle is equal to the angle, which the radius of osculation of the elements Mm and mμ actually described by the body agrees with the radius of osculation of the elements Mm and mv or with the normals on the surface. Moreover we have found the tangent of the above angle (71). Whereby with the equation made we have from which equation the effect of the force N is determined. Or since it is the case that this equation is found The third force T, since it is placed in the direction of the body, either only increases or decreases the speed. We can put this force to be an acceleration, the effect of this is expressed by this equation : And if the motion is made in a medium with resistance and the resistance is equal to R, only the tangential force T is to be diminished by the resistance R. On which account we have : Q.E.I.

Corollary.

80. Therefore from the two equations, from one of which v is determined, and from the other dv, the one containing v solved with the position of the surface dz = Pdx + Qdy [p. 38] determines the curve that the body describes on the above proposed surface.

Scholium 1.

81. The force N needs to be attended to well, in which place it acts, that is whether it inclines either to the right or to the left hand region of the motion of the body. For this indeed the different tangent of the angle μnv either positive or negative has to be taken. Concerning which we will not now be concerned, but defer further inquiry of this to the last chapter of this book.

Scholium 2.

82. Therefore we progress to the following chapter, in which we examine the motion of the body upon a given line in a vacuum. In the third chapter we investigate the motion of a body on a given line with a resisting medium. Finally in the fourth chapter we carefully examine motion on a given surface both for the vacuum and resistive medium cases.