The Absolute and Relative Force

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. 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.

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.