The Effect Of Forces Acting On A Free Point

DEFINITION 10.

99. A force is an action on a free body that either: **- leads to the motion of the body at rest, or

• changes the motion of that body.**

The force of gravity is a force of this kind.

Through gravity, bodies fall freely downwards, and the descending motion is one of continuous acceleration.

Corollary

100. Any body left to itself will continue in a state of rest or of uniform motion in a fixed direction. Therefore to what extent it occurs that a free body at rest begins to move, or the uniform motion of a body becomes non-uniform, or the motion changes direction, the cause is to be ascribed to some external force, which we call the force acting on the body, according to whatever state of the body it produces.

Scholium 1.

101. Concerning the principles of external forces, in as far as many external forces are applied to a body in equilibrium and they keep the body at rest, that has now been explained in Statics.

There the external force has also been defined, as that force denotes all the force that prevails to make the body move.

The motion of the body is not to be considered in Statics, but only these cases are to be investigated in which several forces cancel each other out [p. 40], and the body upon which they act remains in a state of rest.

Mechanics explains how forces acting on a body, which do not cancel each other out, can produce motion in a body at rest, and indeed for a body in motion, change that motion.

Scholium 2.

102. Whether forces of this kind have their origin in the bodies themselves, or indeed through such forces that do exist in the world, these I will not consider here.

For here it is sufficient, in place of the forces that really arise in the world, only to consider the force of gravity, by which all bodies on the earth try to move downwards.

Besides truly forces of this nature are evident disturbing the motion of the planets, which unless they were influenced by a certain force, would be progressing in straight lines.

Similar forces arise from magnetic and electric bodies. These can attract bodies.

• The followers of Descartes think that all the forces arise from the motion of some subtle matter
• The followers of Newton think that the force is from the attraction or repulsion between the bodies themselves

Certainly, the forces of this kind can arise from vortices, and from elastic [energetic] bodies.

What is the effect of any forces on bodies?

[Thus, Euler intends to talk only about gravitational forces, for which experimental laws exist, until the other forces of nature have been more fully investigated. Daniel Bernoulli had a lot of influence on Euler’s thoughts about mechanics at this time, as Euler actually stayed with him, and both worked at the St. Petersburg Academy; both published papers on the topics considered in this chapter at the time in the early volumes of the St. Petersburg Commentarii.]

DEFINITION 11.

103. The direction of the force is the straight line along which the body is trying to move. Thus the direction of the force of gravity is that vertical line, for a heavy body tries to fall along that line.

Scholium 1.

104. In statics, everything is put in place to remain at rest.

All the forces have their directions set up, serving to keep the body at rest all the time.

But in mechanics [which we now call dynamics], when the body is always arriving at another place, the direction of the force acting will constantly be changing direction.

For different positions of the body the directions of the forces are either parallel to each other, or converging to a fixed point, or they act in response to some other law, from which so many different forms of the laws governing mechanics arise.

Scholium 2.

105. The comparison and measurement of different forces, likewise from statics, should be recalled.

In which some force has been treated a that has the ratio to another force b as m to n, when the force a is applied n times in turn on the point A (Fig. 11) along AB, and the force b is applied m times along the opposite direction AC, and the point A continues in equilibrium. Then indeed the force a, taken n times, is equivalent to the force b, taken m times, and will be related by na = mb or a : b = m : n.

Scholium 3.

106. Now in this regard, the measurement of forces differs in mechanics from statics, since in statics all the forces put in place are able to keep the same magnitude, while in mechanics, as with the body arriving in another place, the directions of these are made changeable, thus the magnitude of these is able to be variable following some law.

PROPOSITION 13. THEOREM

107. When a point is acted on by many forces, the same motion comes about from these, as if the point is acted on by a single force equivalent to all of these forces.

DEMONSTRATION

Let the point A (Fig. 12) be acted on by the forces AB, AC, AD, and AE, to which the force AM is equivalent. The equal and opposite position to this, AN, is taken, and as it is known from statics, it will cancel the action of the forces AB, AC, AD, and AE.

Therefore [harking back to (104), ] in the first place only the force AN is impressed on the point A and the motion is along AN, the magnitude of the forces AB, AC, AD, and AE acting together impress on the point A the force along the line AM in the central direction.

Truly the force AM alone, since it is equal to the force AN, is of such a size that it moves the point A along AM, to the same extent as the force AN moves the point towards AN. Whereby the force AM alone impresses on the point A a motion along AM, as much as the forces AB, AC, AD, AE acting together along the same direction [p. 43] AM. In each case therefore, the effect is the same. Q. E. D.

Corollary 1.

108. If a point is therefore influenced by many forces, that can be considered to be influenced as it were by a single force, which is equivalent to all these forces.

Corollary 2.

109. And in turn in place of a single force acting on a point, there can be considered to be many forces acting, to which that is equivalent ; that which, as has been shown from statics, can be made in a limitless number of ways. Scholium.

110. Because truly, as the body first has moved from its position, the forces acting on it change their directions and magnitudes or they are put in place to move with the body, there will be some other equivalent force for any moment. Hence on account of this circumstance, for any time, the equivalent force of all the forces acting on the point ought to be investigated, and thus not the long term effect of the same force is to be put in place [i. e. the time average force,] but rather that from an infinitely small element of time.

DEFINITION 12

111. The absolute force is the force that acts equally on a body either at rest or moving. An absolute force of this kind is the force of gravity. which acts equally downwards on a body which is either at rest or moving.

Corollary.

112. If therefore the absolute effect of a force acting on a body at rest is known, then the effect of the force on the body is also known for any kind of motion.

DEFINITION 13

113. A force is relative, which acts in one way on a body at rest, in another on a body in motion. A force of this kind is the force acting on a body dragged away by a river ; where indeed the faster the body moves in the river, the smaller the force shall be : and that therefore the force vanishes when the speed of the body is the same as that of the river. Corollary 1.

114. If therefore the speed of the body is given together with the law of the relative force, it is possible to find the strength, the magnitude of the force is exerted on the body. And hence from this as the absolute force has to be considered, as long as the body has the same speed, the effect of this from the action of the absolute action can be determined. For the strength of the relative force is to be determined from the given motion of the body.

Corollary 2

115. Therefore these relative and absolute forces in turn are different from each other; for the magnitude and direction of the absolute force acting on the body may only depend on the location of the body ; while truly the magnitude and direction of the relative force acting on the body depends in addition on the speed of the body. [p. 45]

Scholium 1

116. Relative forces are chiefly to be considered in the relative motion of bodies in fluids; for the action of these forces on bodies depends on their relative speed; for when that is greater, it is apparent that the force of the fluid acting on the body is also greater.

Moreover as well as the other causes of motion in fluids, which require a greater understanding of fluids, there are two which are easier to handle; the one when the fluid is at rest, and the other when the fluid is moving uniformly in a given direction. Truly it is always possible for the one to be substituted for the other, with the relative motion always reduced to the absolute ; likewise, clearly for the state of a fluid considered to be at rest the proper forces will remain [i. e. those of a relative nature have vanished]. Therefore in what follows, concerning the relative forces which may be proposed, those properties which also pertain to fluids at rest will be the chief concern. For truly the action of fluids on the motion of bodies consists wholly in the diminution of their speed and on this account it is called the resistance, which is also greater when bodies move faster, and which always disappears when the bodies are at rest. On this account we can put the true motions in place of the relative motion in what follows, and these are only affected by the absolute forces, and can be placed in a vacuum [in general].

Scholium 2.

117. The motion in mediums with resistance, if we wish the greatest order to be followed in the following chapters, must be referred to the last chapter [of vol. 1], in which the motion of the fluid is to be determined, since also it is not now agreed upon by which law the fluids resist the motions of bodies.

This matter is usually considered from many points of view, in order that the nature of fluids can be examined in a straight forward manner, these have been revoked and instead a purely mathematical hypothesis to be used : I have decided to retain this method as many elegant problems are passed over, which otherwise are not to be found in discussions on fluids. However I will apply this method only to the motion of points in fluids, as the calculations associated with bodies of finite magnitudes become insurmountable. Moreover when the shape of a body taken as a finite number of points is considered, there is a convenient outcome from this, which is that the direction of the resistive force is in agreement with the direction of the motion, since indeed that arises from a fluid at rest. Moreover, on account of this we agree that the motion of points in relative motion in fluids are always to be considered with relative forces in the same direction as the point itself, and that always as we will consider the motion to be decreasing.

PROPOSITION 14. PROBLEM.

118. For the given effect of an absolute force on a point at rest, to find the effect of the same absolute force on the same point in some kind of motion.

SOLUTION.

Let the point be placed at A (Fig. 13), from where it can be moved with speed c following the direction AB, and indeed the direction of the force acting on it is in the direction AC. Some element of time is taken dt, and in this short length of time the point A is pulled forwards, if it were at rest at A, through the small distance AC, [p. 47] that may be called dz, so that after the time dt the point is no longer at A, but at C.

This motion of the point along AC is the effect of the force acting on the point at rest. The effect of the same force, which is put as absolute, should be the same on a moving point as on one at rest (111). Now the direction of the point is taken so that it travels the distance AB along AB, since it travels the distance AB = cdt (30), with its speed c acting for the short time dt, if it is not influenced by any force. Truly with the force acting after the short element of time dt, the point will no longer be found at B, but to be elsewhere at D, thus so that the effect, which is to be measured by the deviation from the point B, which is the distance BD, which is equal to the effect of the same force on the resting point (111), i. e. AC. Hence BD = AC. Besides indeed BD is parallel to AC itself, since BD has been the effect of the force, and thus should be acting in the same direction, which does not change during the indefinitely short time dt. On account of this the point A having the speed c along the direction AB and influenced by the absolute force, for the elapse of the short time, is to be found not from B but from D, with BD equal and parallel to AC itself. Truly the distances traversed in the infinitely short time can be considered to be straight lines; on account of which the distance AD traversed in the very short time dt is agreed upon. Q. E. I.

Corollary 1

119. Since the motions in infinitely short distance traversed can be uniform, the speed with which the element AD (33) is traversed, is equal to AD (30)

dt

Corollary 2.

120. The speed along AD is put equal to c + dc, which preceding was c (35), will be given by c  dc  AD ; but before AB = cdt, from which c  AB . Hence there is dt

produced: dc  ADdt AB . Therefore on cutting off the portion Ab = AB from AD, there is left the equation dc  Db . dt

Scholium 1.

121. Moreover, it is to be understood that AC or BD is infinitely smaller than AB, since AB is the distance traveled with a finite speed in the time dt, but the absolute small distance AC is traversed in the same time element with an infinitely small speed; indeed it is not possible to infer a finite speed for a body at rest in an infinitely small increment of time.

[Euler’s way of saying that the distance AB is a first order increment, while AC or BD is a second or higher order increment.]

Corollary 3

122. Hence on account of this, the angle BAD is indefinitely small, and with the points B and b joined, the line increment Bb is perpendicular to AD. The sine of the angle BAC, which is surely to be given, is called k, with the total sine taken as 1, then the sine of the angle BDb is k also, and since BD = AC = dz, then Db  dz (1  kk ) and Bb = kdz.

Corollary 4.

123. Therefore, the increment of the speed dc, that we found before to be equal to Db , dt will be dz (1 kk )

It is understood that the distance dz is infinitely smaller than dt; dt for dz is infinitely small with respect to AB, i. e. cdt, and likewise with respect to dt, since c is put of finite magnitude.

Thus, we see that even at this time, 1733, the problems involved with an inadequate notation for orders of increments had not been resolved, and much was left to the intuitive powers of the practitioner. It does appear in general that Euler considers his infinitesimal elements set out in the diagrams as dx, dy, etc, as initially being small but finite linear quantities, from which relations are established, before undertaking a limiting process where dx, dy, etc are made infinitely small, while their ratio can remain finite. ]

Corollary 5.

124. With the increment of the speed dc found due to the force, the angle BAD of the change of direction of the point from the original direction represented by AB should be considered also, which likewise is found from the force. Truly the sine of the angle is Bb  kdz .

equal to AB cdt

Corollary 6.

125. Therefore there is a two-fold effect of the force affecting the motion of the point. One way is in agreement with the change in the speed, and the other with the change in the direction of this point. The first gives a change of the speed dc  second gives the declination of the sine of the angle kdz . cdt dz (1 kk ) , the dt

Corollary 7.

126. If the angle BAC is right then likewise k = 1, and dc = 0. Therefore in this case the speed remains unchanged by the force. Truly the sine of the angle of declination BAD dz . will be cdt

Corollary 8.

127. If the angle BAC is obtuse or greater than a right angle, then the cosine of this angle (1 kk ) is negative, and therefore the increment of the speed dc will be made negative, dz (1 kk ) . Which shows that the speed is diminished by the force. dt Declination kdz remains the same as it was before. cdt and equal to 

Corollary 9.

128. If the direction of the force AC agrees with the direction AB of the point A, this makes k = 0. Therefore in this case [p. 50] the direction of the motion is not changed by , if the the direction of the force. Truly the increment of the speed dc becomes equal to dz dt direction of the force agrees with the direction of the motion. But if moreover it should be in the opposite direction, then it becomes dc   dz . dt

Scholium 2.

129. Thus it is apparent from the solution to this proposition, how the absolute effect on a point in any kind of motion ought to be found, if the effect of the same force on the point at rest were known. Hence on account of this for the following propositions of this chapter, it will be sufficient for the force acting on a point to be placed either at rest or to be moving in the same direction as the force. For if the point A (Fig. 14) has the speed c and is moving along the direction AB; meanwhile truly it is acted upon by a force in the same direction AB, thus so that in the passing of a small interval of time dt it will not be at B that the body will be found, for only with the speed c will the width be traversed, but at b, and the effect of the force will be the small distance Bb. And by the same small distance ao the point A, if it were at rest at a, would have been treated in the same small increment of time dt. Therefore from the motion of the point A by the force the effect of the same force acting on the point at rest will be known, and again hence the effect of the force on any moton of the point.

PROPOSITION 15. PROBLEM.

130. For a given increment of the speed, that a certain force produces on the point A in the small increment of time dt, to find the increment of the speed,[p. 51] that the same force produces on the same point in the time increment dτ.

SOLUTION.

The point A (Fig. 14) has the speed c in the same direction AB as the force has acting on it, and ao is the small distance, through which the force pulls the point A, if it were at rest, in the small increment of time dt. If again the distance AB is taken, that the point A will traverse with speed c in the increment of time dt, then it will traverse the same distance beyond Ab with the force acting, by taking Bb = ao : and thus the distance, since it is infinitely small, is considered to be described by a uniform motion. [In modern terms, while we do not follow Euler’s development closely, the first distance gone without accelerating, is cdt; the acceleration a provides an extra distance gone ao, which is, if A starts from rest, given by 1 adt 2  vdt / 2  vav .dt , where vav is the average 2 speed over the increment, in which the force acting and the acceleration are considered to be constant.

Thus we have Ab = AB + ao.] Therefore in the following small time dt, the body travels a distance bC = Ab with this speed, unless acted upon by a force; and with the force acting again, which it can be put to remain unchanged even through the infinitely short time it will arrive beyond C at c, by taking Cc = ao. [Again, the second distance gone without further accelerating in the next equal time increment, is Ab = AB + ao; the acceleration a provides an extra distance gone ao, as above with the same conditions. Thus we have Ab = AB +2ao.]In the same manner in the third increment of the time dt it will traverse the distance de = dE + Ee, where again dE = Cd and Ee = ao. Indeed we have :

Ab  AB  ao; bc  AB  2ao; cd  AB  3ao; de  AB  4ao. ao is the is the increment of the speed produced by the force in the time dt ; 2dt Hence ao dt ao increment in the small increment of the speed acquired in the time 2dt ; similarly 3dt time 3dt; and generally in the short time ndt the speed c of the point will increase by the element nao . Put ndt  d , then n  ddt . Therefore the increment of the speed acquired dt in the time increment d will be ao.d2 . Since indeed the increment of the speed in the dt ao short time dt is dt , it will be produced in this ratio : (increment of the speed acquired in the increment of the time dt) is to (increment of the speed acquired in the time increment d ) as dt is to d . Consequently the increments of the speed are in proportion to the times in which they are produced. Q. E. I.

Corollary 1.

131. It is apparent that this increment of the speed does not depend on the speed c itself, but it will have the same value, however large or small a value is put for c. And from this the nature of the absolute force is better understood, since they act equally on moving bodies and on bodies at rest.[p. 52]

Corollary 2.

132. If c = 0, and the point A at rest is urged into moving by a force, the speeds acquired from the motion itself will be as the times : obviously twice the speed in twice the time, with the time tripled, so it will obtain three times the speed.

Corollary 3.

133. If therefore from the start, the speed of the motion acquired in a small time t is called c, and the distance traversed is s, then t = nc. But also t   ds (37). Hence the equation is c 2 2 produced nc   ds or ncdc = ds and hence s  nc2  2t n . Therefore the distances c described from the start of the motion are in the square ratio of the times or of the speeds acquired in that distance. [p. 53; we see that the constant n is the inverse of the acceleration.]

Scholium 1

134. The truth of this proposition, that the increments of the speed are in proportion with the increments of time in which they are generated, is also in agreement with finite quantities, as long as the force acting on the point stays the same, and always retains the same direction as the motion of the point itself. For infinitely small times it is not necessary to have this restriction; for no force of any variation greater than the smallest is to be considered. Moreover we are soon to show what the effect of the different forces may be, and also how points are influenced by forces of different kinds that we put in place, as another larger or smaller force can be put in the given ratio. Truly this does not adversely affect extremely small points, but indeed not only points that we understand to be mathematical but also the physical points that arise from the composition of bodies. For two or more points can be conceived to be merged into one, since the points still remain of infinitely small magnitude, and nevertheless it is better with single points.

Scholium 2.

135. GALILEO was the first person to use [this principle] in the investigation of a falling weight, for the solution of the problem found for this theorem. Indeed he did not give a demonstration of this, but yet on account of the conspicuous nature of this principle from many similar phenomena, he did not wish to be doubted. Indeed he had refuted other opinions regarding this matter, from which he had greatly confirmed his own views. [p. 54] Others indeed were of the opinion that increments not with time but with distance traversed were in proportion; truly the absurdity of this had been established by Galileo, and then by many philosophers. Moreover it appears, if the actions of the forces follow this law, that no body is able to lead in the motion at any time. Indeed it might be that dc , that one body might evade the other is = nds and c = ns, truly the time t, that is  ds c   1n l c  1n l ns + const., which constant ought to be   1n l 0n. Clearly equal to 1n ds c the times described by the logarithm of the distance divided by the zero distance should be in proportion, but is thus infinite. Therefore no body can ever lead in the motion from rest.

[In modern notation, t  1n ln c  1n ln ns  const. , and the constant should be

• 1n ln(n0) ;leading to t  1n ln c  1n ln(s / 0). Euler had no qualms about writing down the log of zero! We should note that he has already used the argument that such a motion takes an infinite time to come to the distance 1 from negative infinity; it would seem better perhaps to have used purely physical arguments rather than the mathematical argument produced here, to show that all bodies fall at the same rate.] Thus correctly Galileo responded to his opponents arguments, that instantaneous finite motion would have to arise from that [initial] position, otherwise forwards motion would not be possible. Although indeed in the beginning the speed at the point is made infinitely small, yet that speed from an imaginary force of this kind will never be able to be finite. Now from the given solution of the problem it is understood that it is necessary to find a law, so that neither can any other force with contradictions arise in the beginning. [Roughly speaking, to refute opponents : if the body is at zero, then it will always be at zero, unless a speed can be produced spontaneously somehow; according to the other view, a sum of infinitesimally increasing speeds over time leads to a finite speed : a variation on Zeno’s paradox, but using accelerated motion (see Book II of Gregorius' Geometry in this series of translations.)]

PROPOSITION 16. THEOREM.

136. The force q at the point b has the same effect that the force p has at the point a, if the ratio between the forces and distances is of the form q : p = b : a.

DEMONSTRATION.

[To establish this, if] q is put equal to np; i. e. q = np, then b = na. Now it is understood that the point na is divided into n equal parts, any of which is equal to a; [p. 55] of which each of the parts is acted on by an nth part of the force np, that is by the force p. With these put in place, any part is pulled in the same manner by its own force, as by which the point a itself is pulled by the force p. Neither are these points of na parts to be acted on by their own forces in turn on being separated; for they will always remain united, if they were indeed connected together initially. Moreover it is evident that these two cases revert to the same and do not disagree with each other, whether the point na is drawn by the force np, or if some part a of the point na is pulled by a similar part p of the force np, provided the parts are not separated from each other in turn. On account of which the proposition is agreed upon, that equally na parts are to be acted on by the force np, as a is acted on by the force p. Q. E. D.

Corollary 1.

137. Therefore the point na obtains the same acceleration from the force np as the point a from the force p.

Corollary 2.

138. In the same manner, it follows that for a point to have a greater speed induced than a smaller one, then it is necessary for a larger force, and with that force to be even so much greater, with that point so much greater than this one.

Scholium 1.

139. This Proposition embraces the foundation of measuring the inertial force, here indeed the ratio of all is advanced, whereby the matter or masses in Mechanics must be considered. For it is necessary that the number of points is attended to, from which the body to be moved has been agreed upon, and the mass of the body must be made proportional to this. Truly the points must be taken amongst themselves as equal to each other, not in the sense that they are equally small, but in that the force exerts an equal effect on each. If therefore we consider that the whole body has been divided up into a number of equal points or elements in this manner, then it is necessary to estimate the quantity of the matter of each body from the number of points, from which it is composed. Moreover the force of inertia is proportional to this number of points or the quantity of material in the body, as we will show in the following proposition.

Corollary 3.

140. Therefore two bodies which have been made from the same number of points are equal, because each contains the same amount of matter. And two bodies are in the ratio m if n, if the numbers of the points, upon which they agree, keep the ratio m to n. Scholium 2.

141. Truly it will be shown in the following propositions that this ratio of the quantity of the matter to be measured for the bodies themselves is to be put to use and to be undertaken in all work. For from the weight of each body it is usual to investigate the mass, and it is agreed that the weight and the quantity of matter are in proportion. Moreover it is agreed by experiments that all bodies in an empty space fall equally, and therefore all are accelerated equally by the force of gravity. Concerning which it is necessary that, in order that the force of gravity acting on individual bodies shall be proportional to their quantity of matter. [p.57] Truly the weight of the body indicates the force of gravity, by which that body is acted on. Whereby since that shall be proportional to the quantity of matter, with the weight of the quantity of matter known, from that itself wt .mass 22 considered, that we have divided here for the matter. [Thus, wt .1  mass 1. ]EULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce.

142. The force of inertia of any body is proportional to the quantity of matter, upon which it depends. DEMONSTRATION. The force of inertia is a force in place in any body in its own state of rest or of uniform motion in a direction to be kept the same.(74). That therefore is to be estimated from the strength or the force applied to the body, with the aid of which it is to be disturbed from its state. Truly different bodies equally in their state are disturbed by forces which are as the quantities of material contained in these. Therefore the forces of inertia of these are proportional to these forces. Consequently also the quantities of matter are in proportion. Q. E. D.

Corollary 1.

143. Likewise it is evident by demonstration that the same body, either in a state of rest or of motion, always has the same force of inertia. For, either at rest or moving, clearly it is affected by the same absolute motion. [p.58]

Corollary 2.

144. Nor indeed is the force of inertia homogeneous with any force : for it is not able to become so, as any body of any great size is not affected by a small force, as is shown in the following.

Scholium.

145. Hence it is apparent that the origin of the said force of inertia, comes from that which we have introduced above (76), since the force of inertia resists the action of any kind of force. Which Newton too had decided on, and who in Definition III of the Princ. Phil;. Nat. joined together the force of inertia with the same idea of a force being resisted, and each was set up to be in proportion to the quantity of matter. [Thus, to change the state of motion of a free body, an external force has to be exerted, that overcomes the ‘innate power of resisting’ or the force of inertia present in the body.]

PROPOSITION 18. PROBLEM.

146. With the effect of one force on some point given, to find the effect of any number of forces acting on the same point. [Or, how the distance moved in an element of time is related to the size of the force acting on a mass, and for which the parallelogram of forces is assumed. This development follows that of Daniel Bernoulli, who tried to give mechanics an axiomatic foundation, following along the line of Euclidian geometry.

Part of this development was to show how any force could be decomposed into the sides of a rhombus: see Die Werke von Daniel Bernoulli, Band 3; Examinen Principiorum Mechanicae …… A commentary on this Latin paper is presented in English (there are also I believe French and German versions) in this book by David Speiser; part of an on-going edition of the works of the Bernoulli family. Birkhäuser (1987). One should note that no translations are actual presented, merely discussions of what the writer considers has been said, and many papers have not even been discussed, and are listed at the end of the book.]

SOLUTION.

The point is at rest at A (Fig. 15) and the effect of a given force AB on this point is agreed upon, which is that in an element of time dt it is drawn through a small distance Ab. Now it is required to be found, by what distance in the element of time dt, the same point is drawn by another force AC. The lines AB and AC are drawn thus, in order that the joining line BC is normal to AC, since that can always be done if AC < AB. [i. e. there is a condition placed on AC, which represents that force arising from the parallelogram of forces as the resultant of the two ‘half’ forces AE and AF acting symmetrically as shown]

But if AC > AB, the solution can be easily deduced from the other condition. From the other side the line AD is drawn thus, in order that BAD is an isosceles triangle [p.59]. AB and AD are bisected in E and F, and half of the force AB may be represented by AE, and half the other force by AF. It is clear that the force AC is the greater [force] at the point A, because the two forces AE and AF act jointly (107), and since AC is equivalent, on account of the parallelogram AECF, to both AE and AF.

Therefore in place of the force AC we consider the point A to be acted on by the forces AE and AF. Truly we can understand the matter in this way : as if any forces AE and AF should each affect half [the mass] of the point A. Truly these half forces themselves act for the element of time dt as if [for the two half masses] freed from each other, and these in turn finally we suddenly combine again. Because now, as the force AB draws the point A in the element of time dt through the distance Ab, then half the force AE draws half the point in the same time element dt through the same distance Ab (136). Similarly in the time element dt, the other half of the point A will be drawn by AF through the distance Ad = Ab. Therefore at the end of the element of time dt the one half point A will be at b, the other at d. Now they may suddenly fit together again with each other, or be drawn together by an infinite force of cohesion, and they come together at the mid-point c of the little line bd : indeed there is no reason why they should meet nearer to b rather than d . Therefore with the forces AE and AF jointly acting in the element of time dt , the point A will be drawn through the small distance Ac. On account of which, the force AC, also being equivalent to the forces AE and AF, acting for the element of time dt, draws the point through the small distance Ac. Indeed bd is parallel to BD and therefore Ab : Ac = AB : AC. Therefore with the small distance Ab given, through which the point A is pulled by the force AB, [p.60] the small distance Ac is given, through which the same point A is pulled by the other force AC in the same element of time. And likewise it is apparent, if the effect Ac of the smaller force AC itself were given, so much greater will the effect Ab be of the greater force AB. Q. E. I.

Corollary 1.

147. Therefore the distances, through which equal points are to be pulled by any forces in equal time intervals, are as the forces themselves.

Corollary 2.

148. Since the distances moved from the beginning described by unequal time intervals are in the square ratio of the times (133), the distances will be, by which equal points by any forces for unequal time intervals will be dragged, in the ratio composed from simple forces and the square of the times.

Scholium.

149. From the first principles that we have used in the solution of this problem, it is well known in this respect that a body under the influence of many forces can be divided into an equal number of parts, any one part of which is pulled by the one force. Then when the individual parts are pulled forwards by their forces for an instant of time, it is understood that finally they are compelled to come together suddenly into a single point. The place where this happens, in which they gather together, is that same point to which the whole body would be pulled, acted upon likewise by all of the forces together for the same time for the whole body. The truth of this principle can be seen according to this, that the parts of bodies can be conceived to be connected together most strongly elastically [as by elastic threads] [p.61], and which, though the applied forces act incessantly during the interval, yet they fail at the end suddenly and the parts are able to contract as if by an infinite force put in place; thus in order that the time taken for the free parts to be reduced into one in turn shall be as nothing. Truly many other mechanics problems can now be solved by the use of this same principle. And many other problems have been adopted to non-separated bodies, in the case where the forces are not continuous, but are able to suddenly exercise their effect. Moreover with this principle admitted, it is clear that there are two equal lines upon which the points approach each other and they have to meet in the middle.

PROPOSITION 19. THEOREM.

150. A point can be moved along the direction AM (Fig. 16) and it is acted on, while it traverses the small distance Mm, by a force p pulling in the same direction ;the increase in the speed, that the point meanwhile acquires, is as the product of the force by the short time, in which the element of distance Mm is traversed.

DEMONSTRATION.

Let the element of time be dt, and in this time the point completes the distance Mμ, if it is not acted on by a force, but has the speed that it had at M, and it can go on moving uniformly. Truly the effect of the force is in accordance with this : as the point is drawn further forwards by μm, this extra small distance is equal to that, by which the same point initially at rest, will be drawn forwards by the same force acting for the same element of time dt, as the force is absolute, [p. 62] (111).

The increment of the speed is proportional to the time for this given distance. But if the force is the constant, then the increase of the speed is in proportion to the element of the time dt (130). [For if the force is not constant, then it will not have the same effect at different places.] Whereby when the small distance mμ, or increment of the speed shall be as the force p for a given short time interval, then the increment of the speed for any short time and for any force shall be as pdt, i. e. as the product of the force taken with the short time.

Q. E. D.

Corollary 1.

151. Let the speed of the point at M be c and the element of distance be Mm = ds, then dt  ds , since the time determined from the uniform motion in the element Mm can be c pds put in place. Moreover since dc shall be as pdt, also dc will be as c or cdc will be as pds. Therefore the increment of the square of the speed is proportional to the product of the force by the length of the element traversed. [Thus, the time to travel the distance Mm with speed c is simply dt = Mm/c; the extra speed dc generated in this time dt is equal to the acceleration a times by the time, or a × pds dt; which is proportional to pdt, or to c as shown. Hence, cdc  pds ; and on integrating for a constant force : c 2  p.s . This would now be thought of as the conversion of work done by the constant force into kinetic energy. The reader should bear in mind that the terms work, power , strength, etc, did not have the specific physics- related meanings then that they now have; this makes the business of translating more difficult, as these words with special meaning should not be used without qualification. In addition, results of this nature cannot be referenced to a general principle; the work-energy principle still lay some time in the future, though there were rumblings about it in the Bernoulli camp. Euler in sect. 153 then considers the extra distance as proportional to the force by the square of the time, from which the increase in the speed is again proportional to the force times the time increment.]

Corollary 2. 152. Therefore it is apparent that not only is this theorem true, but also it is true by necessity, as thus it would involve a contradiction to put dc  p 2 dt or p3dt or another function in place of p. All of which and equally commendable are considered by the most distinguished Daniel Bernoulli in Comment. Tom. I, and I have been greatly influenced with the rigor of the demonstration of these propositions.

[Daniel Bernoulli, Examen principiorum mechanicae. Comment. acad.sc.Petrop.I. (1726), 1728, p. 126 – 141; see in particular p. 127. See above.]

Scholium.

153. The demonstration of this proposition follows easily from (148), from which the element of distance mμ emerges as proportional to the [p.63] to the force p multiplied by the square of the time dt, thus so that mμ shall be as pdt 2 . But mμ divided by the time dt gives the increment of the speed, whereby the increment of the speed is as pdt, as was enunciated in the proposition. PROPOSITION 20. THEOREM.
154. The motion of the point in a direction in agreement with the direction of the force, the increment of the speed will be as the force taken with the element of time, and divided by the quantity of matter of the point is composed. DEMONSTRATION. Let there be two points or unequal bodies A and B (Fig. 17) in motion along the line AM and BN. These are influenced by the forces p and π respectively, while they traverse the distances Mm and Nn, and the times in which these are traversed are dt and dτ. It is clear that the point B is affected by the force π in the same manner as the point A by the force A (136). Whereby by putting in place of the point B equal to the point B A, for the force π there must be substituted the force AB , and in this way we obtain the case of the preceding proposition, for which the points are put equal. Hence on account of this, the increment of the speed in traveling through the distance Mm is as the increase of the speed through pdt the distance Nn as pdt is to AB d or as A to dB (150). From which the proposition is agreed upon, that the increase of the speed is as the product of the force and the time divided by the mass or quantity of matter of the point .Q. E. D. [p.64]EULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 64 Corollary 1. npdt
155. If the speed of the point A were c, then dc  A , where n in all cases denotes the same number; that depends neither on the size of the force, on the element of the time, or on the quantity of matter. [There is now a constant of proportionality, enabling equations to be used rather than proportionalities. Eventually Euler sets this to 1⁄2 for convenience, as he is not required to adhere to a set or units as we now are. Later, when the need arises, he absorbs this constant into his equations to produce the correct experimental acceleration in the units chosen. In his later works, Euler does adopt standards of mass and length and time, and moves away from ratios : see e.g. his introduction to the motion of rigid bodies presented in these translations in vol. 3] Corollary 2.
156. The quantity of matter comes under consideration here, in as much as , in as much as it resists the tendency to be disturbed by the force i. e. as much as it agrees with the force of inertia. On this account, the increment of the speed is directly as the force acting and the of the element of time acting, and inversely as the force of inertia of the body. Corollary 3.
157. With the distance put as Mm = ds the the time dt  ds . Hence the increment in the c npds npds speed becomes dc  Ac or cdc  A . Whereby the increment of the square of the speed is proportional to the product from the force by the distance traveled divided by the mass or the mass or the force of inertia of the body. Scholium.
158. This proposition embraces all the principles expounded upon this far concerning the nature of motion to be defined and the laws of motion. On account of which if this proposition is joined with fourteen, (118), by which the effect of forces acting at angles are determined, all the principles will be put in place, from which the motion for any points can be found from any forces acting. [p.65] Corollary 4. npdt
159. Since dc  A , the distance through which the point A is lead by the force p in the element of time dt will be equal to npdt 2 . This distance is indeed the product of dc by dt. A For by saying that this distance is dz, then dc  dz (128) and thus dz  dcdt  dt npdt 2 . A [Recall that in Ch.I, Euler said that he would consider steady motion in increments or intervals, with the step in the speed occurring at the start of the interval, rather than a continuously acting force; in this he was of course just following Newton, who adopted this procedure in evaluating centripetal force by a sequence of small forces acting inEULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 65 succession in steps. If a Newtonian posture is adopted, as it were, then it can be said, as even Euler said in his preface, that he was merely re-inventing the Principia using analysis, which one must admit to being true; but it is not the whole truth, for the analytical method finally freed people from the shackles of geometry, and the whole subject of dynamics was given a re-birth and was enabled to move on. Thus, a glance at a formula reveals in a second what may take hours to appreciate geometrically. Euler was quite cynical in the preface, of the state of play of this early calculus, which he has also used in his earlier papers, and from which he was now free to apply and change as he saw fit, which could not always be done with the geometrical reasoning used initially.] PROPOSITION 21. PROBLEM.
160. To determine the effect of any oblique forces acting on a moving point. SOLUTION. Let the point A (Fig. 13 repeated, [in which BAb is isosceles and the angle BAb is incremental.]) have the speed c in the direction AB. [A also refers to the mass of the body, while np/A is the acceleration in some set of units with constant of proportionality n.] Indeed it is acted on by a force p, the direction of which AC makes an angle with AB, the sine of which is k. It is evident that the point A left to itself unless acted on by a force progresses along the line AB and in an element of time dt travels through the distance AB = cdt (30). Truly with the force p acting the point A will be deflect from the line AB and meanwhile travel along the element of distance AD, as has been shown in Prop. 14 (118). Moreover we have put AC or BD = dz there in the diagram, which is the element of distance through which the point A, if it should be at rest would be drawn forwards by the force p in the time dt. Hence it follows that dz  npdt 2 (159) [Recall that the final speed in the increment is A taken as the speed throughout the increment]. Therefore the sine of the angle BAD which has been found, is equal to kdz (124) [as sinθ = k and dz / sin BAD  cdt / k ], which is cdt nkpdt . And the increment of the speed dc [= Db/dt] which was equal Ac npdt (1 kk ) dz (1 kk ) to (123), is now equal to . Q. E. I. [p. 66] dt A equal toEULER’S MECHANICA VOL. 1. Chapter two. page 66 Translated and annotated by Ian Bruce. Corollary 1.
161. The distance AD is called ds (Fig. 18), and the element of time by dt  ds , then with ds in place of dt above there is c c produced dc  npds (1 kk ) . The perpendicular DF is drawn Ac from D to the direction of the force AE; let AF = dy and DF = dx, then [the element of the distance squared] ds 2  dx 2  dy 2 and k  dx and ds dy (1  kk )  ds . Hence it npdy comes about that dc  Ac or Acdc = npdy. Corollary 2.
162. [A circle] is drawn to the curve described by the small body in this way, at the point A with radius of osculation AO, and Bb : AB = AD : AO. [Essentially the law of the . AD . Since Bb is the sine of the angle centripetal force at that point.] Whereby AO  ABBb AB BAD, which was found to equal nkpdt Bb  npdxdt , and on account of AD = ds . Hence AB Ac Acds 2 Acds . it comes about that : AO  npdxdt Corollary 3. 2 ds . The radius of the osculating circle
163. Since it is the case that dt  ds , then AO  Ac c npdx is taken as AO = r, and hence we have nprdx  Ac 2ds . Corollary 4.
164. If the direction AE of the force p is incident along the normal AO, then there arises AF = dy = 0 and DF = dx = AD = ds. On account of which it follows that cdc = 0, and therefore the force does not change the speed. Corollary 5. 2
165. Again in this case it follows that npr  Ac 2 on account of dx = ds, and r  Ac . np Therefore this force, the direction of which is normal to the direction of the body [p. 67] results in the arc of the [equivalent circular] curve, as the body is not able to complete its 2 rectilinear motion. [The centripetal force is np  Acr . This situation arises in projectile motion at the highest point, where dy is zero. We should perhaps recall that the force always acts downwards, while the initial speed is at any angle we choose, in Euler’s derivation of the equations governing motion in two dimensions that he has presentedEULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 67 here step by step, in a very careful manner, that I have tried to reproduce in this translation.] Corollary 6.
166. If the direction of the force p is incident along the tangent AB, then dx = 0 and dy = ds. In this case we have Acdc  npds . Therefore the force acting in this direction will give the body the greatest increase in speed. [This corresponds to motion vertical downwards under gravity.] Corollary 7.
167. If the direction of the force p is incident in the opposite direction to AB , thus in order that it is contrary to the direction of the motion of the body, the quantity p becomes negative, and we have Acdc  npds . Therefore in this case the speed is decreased by the same amount as it was increased before. [Motion vertically upwards under gravity.] Corollary 8.
168. Moreover in each case in which the direction of the force p is incident along the 2 ds , on account of dx = 0. Therefore in that case the direction of the tangent, and r  Ac np.0 body will not then change, and it will accelerate in a straight line. Corollary 9.
169. Therefore any case the value of the constant letter n determined by experiment will be put in place for all cases. Therefore than everything which could be wished for in the motion will be given absolute values. Corollary 10. npdy
170. From the first corollary there arises A  cdc . With this value substituted in the third corollary, we have nprdx  npcdyds or rdxdc  cdyds . [p. 68] dc In which equation neither n, A, nor p is present, and this prevails for any motion of the body whatsoever, and for any force to be acting. Corollary 11.
171. Nevertheless however, although the force p itself is not to be found in this equation, yet the direction of this, upon which the relation of the elements dx et dy depends, still remains. Therefore from the given direction of the force acting at any point on the curve and from the curve itself, along which the point may be moving, from these alone the speed at any point can be determined. For indeed it will be given by : dyds dc  dyds or c  e  rdx c rdx is 1. where e specifies the number the hyperbolic logarithm of whichEULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 68 [This was a great moment in mathematics: for it was the first time that e had been used as the inverse function of the hyperbolic logarithm, for which log e = 1, and for some reason it stayed attached. We should note however that other letters had been used for the same constant quantity previously in Euler’s early papers, and that he had already used the letters b, c, and d for other constant quantities here. There was thus no special meaning to be assigned to the choice ; as we have seen, he tended to use letters from the beginning of the alphabet to represent constant quantities, and letters from the end to represent variables. Euler also used π for the first time in his work in the next chapter : he was not the first to use π as the periphery to diameter ratio for the circle, but no doubt his adoption of the symbol made it popular. You find in Euler’s work a sort of ’loving care’ for symbols and numbers; he occasionally used verbs associated with human emotions to express happenings in the world of numbers.] Corollary 12.
172. Again since dt  ds , then t   e c dyds   rdx ds . Hence therefore it should be noted that likewise for the time in which any part of the motion is to be described, only the curve itself and the direction of the force needs to be given. Corollary 13.
173. If from O the perpendicular OE is sent to the direction of the force AE, then ds : dx  AO : AE. [See Fig. 18.] Hence with the position AE = q, which line is called the co-radius by certain people, will be rdx  q . Hence the speed becomes ds  dy  dy c  e q and t   e q ds . [Joh. Bernoulli, Concerning the motion of heavy bodies, pendula and projectiles. Acta erud. Lips. 1713; Opera Omnia, Book I, Lausanne et Geneve 1742, p. 531.] [p. 69] [Thus, if dx = ds as in (165) then r = q and dy = 0 and we have circular motion with c given by a constant. Again, for motion vertically downwards (166), dx = 0 and dy = ds, which is a degenerate case.] Scholium.
174. From the solution of this problem it appears to be possible to determine the motion of a small body, acted on by any kind of forces. For the motion is defined by two equations : both the speed of the body anywhere, and the curvature or the radius of osculation of the curve traversed. Indeed with these known, likewise the time can be found, in which the motion along some part of the curve is completed, which is sufficient for the motion to be determined. [As no examples are provided here, we are to imagine that Euler has just come across this result, and has not yet followed through with any consequences. He now moves on to another ‘pet theory’.]EULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 69 Definition 13.
175. The force of restitution is that imaginary infinite force, which restores the separate parts of the body again to their previous state. We considered a force of this kind to be present in the solution of Prop. 18 (146), by which the two parts of the small body, which momentarily we considered to be free, recombined again. Corollary 1.
176. If a point is considered to be divided into two points and these were separated by forces, the restoring force draws these into the middle line, as was shown (146) with sufficient explanation at the start. Corollary 2.
177. Since the effect of the restoring force must be produced instantaneously, the restoring force must be considered as provided by an infinite elastic force, by which the separated parts are again joined together. [p. 70] Scholium.
178. The use of this force of restitution now is clear in a certain way from proposition 18, yet the use of this will be most fully understood when we are to begin investigating the motion of bodies of finite magnitudes in what follows. Truly we will investigate the effect of this here with many separate parts of a point joined together, which will be of great use in what follows. Hence a certain principle of restitution is embraced, with the help of which many questions are easily resolved, that we call the principle of restitution.EULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 70 PROPOSITION 22. THEOREM.
179. Let there be two parts of a point separated at b and d (Fig. 19); I say that these are to be joined together by a force of restitution in the point c, at the centre of gravity of the particular b and d. DEMONSTRATION. In the first place these parts are joined together at A, and these are then pulled apart by the forces AB and AD to the points b and d in the same element of time dt. Truly the force AC is equivalent to these two forces, which in the same element of time act on the point and pull the element to c from A. Therefore it is clear that the parts b and d must be drawn together by the force c, since the force AC has the same effect acting on the whole point A as both AB and AD acting on the two parts of this (149). Hence it is therefore to be understood that c is the point of concurrence, into which the small parts b and d are driven together by the resisting force. By which moreover [p. 71] the small mass b is pulled forwards by the force AB for the element of time dt through the small distance Ab, ought to be Ab  n. ABb.dt 2 (159) or AB  Ab.b2 . Likewise on account of the ratio, AD  Ad .2d and AC  ndt ndt Ac.(b  d ) ndt 2 . Hence truly AC is the diagonal of the parallelogram, that is constituted from the forces AB and AD, which is equivalent to these. Moreover from these equations it can be deduced that AB  AD  AC ; truly AB, AD, and AC are between themselves as the sines of the Ab Ad Ac DAC  sin BAC  sin BAD , from angles DAC, BAC, and BAD. On account of which sin Ab Ad Ac which property it follows that the points b, c , and d are in the directions given. From which, bc : cd  sin BAC. Ab : sin DAC. Ad  AD. Ab : AB. Ad . But AD : AB = Ad.d : Ab.b. Consequently it follows that bc:cd = d:b or b.bc = d.dc. From which it is understood that the point c is the centre of gravity of the particular b and d. Q. E. D. Corollary 1.
180. Therefore the point A taken in any place, always falls upon the position of the same point of concurrence c, from which it is apparent that the constant restoring force does not depend either on the position of the point A nor on the particular particles of A b and d acted upon

Scholium.

181. This force of restitution agrees uncommonly well with the effect of an elastic force, that it is permitted to put in its place. For an elastic string bd can join together the small parts b and d, that by acting together make b and d meet in c. Moreover this force acts to draw the particles b and d together equally, since each is trying to contract equally. For truly the distance, by which b and d are drawn together in the same time, vary inversely as the sizes of the particles themselves (159), since they are affected by the same force. Whereby if the point of concurrence is c, bc and dc will vary inversely as b to d or b.bc = d.cd. From which also it is understood that the point c is the centre of gravity of the particular masses b and d.

Corollary 2.

182. Therefore although the force of restitution is imaginary and only exists in the form of thoughts, yet the effect of this follows the real laws of motion. And from this we can be more sure with the aid of this principle of restitution to always arrive at the truth. [Note that this is not to be confused with the coefficient of restitution that is a more modern concept involving elastic and inelastic collisions.]

PROPOSITION 23. THEOREM.

183. Let a, b, c, and d be parts of a point mutually separated, which are to be joined together again by the force of restitution, and these are in agreement with a common centre of gravity g.

DEMONSTRATION.

Initially we put the whole mass at some point O (Fig. 20), from which these individual parts a, b, c, and d in the element of time dt by the forces OA, OB, OC, OD are to be drawn out into a, b, c, d. OG is taken as the equivalent of these forces, which in the same element of time, pulls the whole point mass forwards from O to g; and g is the point at which all the parts a, b, c, and d will be drawn together by the force of restitution (149). [p. 73] Through the point O some line KN is drawn, and to that line perpendiculars are sent from the points A, a; B, b; C, c; D, d; G, g. Moreover, these are given by :EULER’S MECHANICA VOL. 1. Chapter two. Translated and annotated by Ian Bruce. page 72 Og .( a  b  c  d ) OA  Oa.2a , OB  Ob.b2 , OC  Oc.c2 , OD  Od .2d and OA  (159). But from ndt 2 ndt ndt ndt ndt the similar triangles OAK, Oak; OBL, Obl, etc, it follows that : .ak  ak .a , BL  OB.bl  bl .b , CM  OC .cm  cm.c , AK  OA 2 2 2 Oa Ob Oc ndt ndt ndt . .(    ) OG gs gs a b c d .dn  dn.d , and GS  DN  OD  , and Od Og ndt 2 ndt 2 .Ok  Ok .a , OL  OB.Ol  Ol .b , OM  OC .om  Om.c , OK  OAOa Ob Oc ndt 2 ndt 2 ndt 2 .On  On.d , and OS  OG.Os  Os.( a  b  c  d ) . ON  OD Od Og ndt 2 ndt 2

But since OG is the force equivalent to the forces OA, OB, OC, OD, it is agreed from statics that AK + BL + CM + DN = GS, et OK + OL – OM – ON = OS. Hence we find ak .a  bl.b  cm.c  dn.d  gs.(a  b  c  d ) and Ok .a  Ol.b  Om.c  On.d  Os.(a  b  c  d ). From which properties it is understood that the point g is the centre of gravity of the particles a, b, c, d. Hence the force restoring these particles agrees with the common centre of gravity g. Q. E. D.

Corollary 1.

184. Therefore the effect of the force of restitution is in agreement with this, that bodies separated into any number of parts can be brought together at the common centre of gravity. [p. 74]

Corollary 2.

185. Therefore in this manner, the motion of a point acted on by many forces can be determined without considering equivalent forces, provided that from the individual forces some number of parts are put in place to be affected for any short time interval, and to be brought together again by the restituting force. Scholium.

186. The demonstration of this theorem with the help of contracting elastic strings can be done in the same manner as we did before (181). For let the separate particles (Fig. 21) be at a, b, c, and d, and in the first place we only put a string to that draws the particles a and b together, these fit together with the centre of gravity e. Now we consider the particles a and b located at e to be joined with the particle at c, it will be the point of concurrence f, which is the centre of gravity of the three points a, b, and c. Now these three placed at f are joined to the fourth particle located at d, and it will be the point of concurrence g, the centre of gravity of the four points a, b, c and d. Whereby from the force of restitution all the particles come together at the common centre of gravity.

Corollary 3.

187. Therefore it is agreed once more correctly that the restoring force arising from the contraction of elastic fibers can be used to represent the join between each two particles.

GENERAL SCHOLIUM

188. Therefore from these principles put in place, from which the motion of a free point can be determined from the action of some number of forces, we will progress to the motion of free points that have to be investigated. Indeed it will be convenient to separate this tract into two parts, in the first of which only rectilinear motion will be examined, and in the other any kind of curvilinear motion. The first rectilinear motion, as it is understood from what has been said, arises when the motion agrees with the direction of the force ; and truly the other, when these directions disagree. Truly we will deal with each part in two ways for the two kinds of forces involved, and then indeed for both the absolute and the relative forces taken together. Indeed, in place of the relative force we will substitute a medium with resistance, which as now we have reminded, are to be considered; for the relative forces are to be determined for the motion of bodies in fluids (116), on account of which these will be mainly for the kinds found in nature, and neither shall we say much about these which are only to be found in the imagination, and over which we will not tarry. First therefore we will handle the rectilinear motion of a free point acted on by absolute forces. Then we will investigate the rectilinear motion of a free point in a resisting medium. Thirdly we set out the curvilinear motion of a free point acted on by any absolute forces. Fourthly we set out the curvilinear motion of a free point in a resisting medium.