Article 20. Definition of displacement
We have already compared the position of different points of a system at the same instant of time. We have next to compare the position of a point at a given instant with its position at a former instant, called the epoch.
The vector which indicates the final position of a point with respect to its position at the epoch is called the displacement of that point. Thus if A₁ is the initial and A₂ the final position of the point A, the line A₁A₂is the displacement of A, and any vector oa drawn from the origin o parallel and equal to A₁A₂ indicates this displacement.

Article 21. Diagram of displacement

Figure 3

If another point of the system is displaced from B₁ to B₂ the vector ob parallel and equal to B₁B₂ indicates the displacement of B.
In like manner the displacement of any number of points may be represented by vectors drawn from the same origin o. This system of vectors is called the diagram of displacement. It is not necessary to draw actual lines to represent these vectors; it is sufficient to indicate the points a, b, etc., at the extremities of the vectors. The diagram of displacement may therefore be regarded as consisting of a number of points, a, b, etc., corresponding with the material particles, A, B, etc., belonging to the system, together with a point o, the position of which is arbitrary, and which is the assumed origin of all the vectors.

Article 22. Relative displacement
The line ab in the diagram of displacement represents the displacement of the point B with respect to A.
For if in the diagram of displacement (fig. 3) we draw ak parallel and equal to B₁ A₁ and in the same direction, and join kb, it is easy to show that kb is equal and parallel to A₂B₂.
For the vector kb is the sum of the vectors ka, ao, and ob, and A₂B₂ is the sum of A₂ A₁, A₁ B₁, and B₁ B₂. But of these, ka is the same as A₁B₁, ao is the same as A₂A₁, and ob is the same as B₁ B₂, and by Article 10 the order of summation is indifferent, so that the vector kb is the same, in direction and magnitude, as A₂ B₂. Now ka, or A₁B₁ represents the original position of B with respect to A, and kb, or A₂B₂, represents the final position of B with respect to A. Hence ab represents the displacement of B with respect to A, which was to be proved.
In Article 20 we purposely omitted to say whether the origin to which the original configuration was referred, and that to which the final configuration is referred, are absolutely the same point, or whether, during the displacement of the system, the origin also is displaced.
We may now, for the sake of argument, suppose that the origin is absolutely fixed, and that the displacements represented by oa, ab, etc., are the absolute displacements. To pass from this case to that in which the origin is displaced we have only to take A, one of the movable points, as origin. The absolute displacement of A being represented by oa, the displacement of B with respect to A is represented, as we have seen, by ab, and so on for any other points of the system.
The arrangement of the points a, b, etc., in the diagram of displacement is therefore the same, whether we reckon the displacements with respect to a fixed point or a displaced point; the only difference is that we adopt a different origin of vectors in the diagram of displacements, the rule being that whatever point we take, whether fixed or moving, for the origin of the diagram of configuration, we take the corresponding point as origin in the diagram of displacement. If we wish to indicate the fact that we are entirely ignorant of the absolute displacement in space of any point of the system, we may do so by constructing the diagram of displacements as a mere system of points, without indicating in any way which of them we take as the origin.
This diagram of displacements (without an origin) will then represent neither more nor less than all we can ever know about the displacement of the system. It consists simply of a number of points, a, b, c, etc., corresponding to the points A, B, C, of the material system, and a vector, as ab represents the displacement of B with respect to A.

Article 23. Uniform* displacementWhen the displacements of all points of a material system with respect to an external point are the same in direction and magnitude, the diagram of displacement is reduced to two points—one corresponding to the external point, and the other to each and every point of the displaced system. In this case the points of the system are not displaced with respect to one another, but only with respect to the external point.
This is the kind of displacement which occurs when a body of invariable form moves parallel to itself. It may be called uniform displacement.

Article 24. On motion
When the change of configuration of a system is considered with respect only to its state at the beginning and the end of the process of change, and without reference to the time during which it takes place, it is called the displacement of the system.
When we turn our attention to the process of change itself, as taking place during a certain time and in a continuous manner, the change of configuration is ascribed to the motion of the system.

Article 25. On the continuity of motion
When a material particle is displaced so as to pass from one position to another, it can only do so by travelling along some course or path from the one position to the other.
At any instant during the motion the particle will be found at some one point of the path, and if we select any point of the path, the particle will pass that point once at least* during its motion. 
This is what is meant by saying that the particle describes a continuous path. The motion of a material particle which has continuous existence in time and space is the type and exemplar of every form of continuity.

Article 26. On constant* velocity
If the motion of a particle is such that in equal intervals of time, however short, the displacements of the particle are equal and in the same direction, the particle is said to move with constant velocity.
It is manifest that in this case the path of the body will be a straight line, and the length of any part of the path will be proportional to the time of describing it.
The rate or speed of the motion is called the velocity of the particle, and its magnitude is expressed by saying that it is such a distance in such a time, as, for instance, ten miles an hour, or one metre per second. In general we select a unit of time, such as a second, and measure velocity by the distance described in unit of time.
If one metre be described in a second and if the velocity be constant, a thousandth or a millionth of a metre will be described in a thousandth or a millionth of a second. Hence, if we can observe or calculate the displacement during any interval of time, however short, we may deduce the distance which would be described in a longer time with the same velocity. This result, which enables us to state the velocity during the short interval of time, does not depend on the body’s actually continuing to move at the same rate during the longer time. Thus we may know that a body is moving at the rate of ten miles an hour, though its motion at this rate may last for only the hundredth of a second.

Article 27. On the measurement of velocity when variable
When the velocity of a particle is not constant, its value at any given instant is measured by the distance which would be described in a unit of time by a body having the same velocity as that which the particle has at that instant.
Thus when we say that at a given instant, say one second after a body has begun to fall, its velocity is 980 centimetres per second, we mean that if the velocity of a particle were constant and equal to that of the falling body at the given instant, it would describe 980 centimetres in a second.
It is specially important to understand what is meant by the velocity or rate of motion of a body, because the ideas which are suggested to our minds by considering the motion of a particle are those which Newton made use of in his method of Fluxions,* and they lie at the foundation of the great extension of exact science which has taken place in modern times.

Article 28. Diagram of velocities
If the velocity of each of the bodies in the system is constant, and if we compare the configurations of the system at an interval of a unit of time, then the displacements, being those produced in unit of time in bodies moving with constant velocities, will represent those velocities according to the method of measurement described in Article 26.
If the velocities do not actually continue constant for a unit of time, then we must imagine another system consisting of the same number of bodies, and in which the velocities are the same as those of the corresponding bodies of the system at the given instant but remain constant for a unit of time. The displacements of this system represent the velocities of the actual system at the given instant.
Another mode of obtaining the diagram of velocities of a system at a given instant is to take a small interval of time, say the nth part of the unit of time, so that the middle of this interval corresponds to the given instant. Take the diagram of displacements corresponding to this interval and magnify all its dimensions n times. The result will be a diagram of the mean velocities of the system during the interval. If we now suppose the number n to increase without limit the interval will diminish without limit, and the mean velocities will approximate without limit to the actual velocities at the given instant. Finally, when n becomes infinite the diagram will represent accurately the velocities at the given instant.

Article 29. Properties of the diagram of velocities (fig. 5)
The diagram of velocities for a system consisting of a number of material particles consists of a number of points, each corresponding to one of the particles.

Figure 5

The velocity of any particle B with respect to any other, A, is represented in direction and magnitude by the line ab in the diagram of velocities, drawn from the point a, corresponding to A, to the point b, corresponding to B.
We may in this way find, by means of the diagram, the relative velocity of any two particles. The diagram tells us nothing about the absolute velocity of any point; it expresses exactly what we can know about the motion and no more. If we choose to imagine that oa represents the absolute velocity of A, then the absolute velocity of any other particle, B, will be represented by the vector ob, drawn from o as origin to the point b, which corresponds to B.
But as it is impossible to define the position of a body except with respect to the position of some point of reference, so it is impossible to define the velocity of a body, except with respect to the velocity of the point of reference. The phrase ‘‘absolute velocity’’ has as little meaning as ‘‘absolute position.’’ It is better, therefore, not to distinguish any point in the diagram of velocity as the origin, but to regard the diagram as expressing the relations of all the velocities without defining the absolute value of any one of them.

Article 30. Meaning of the phrase ‘‘at rest’’
It is true that when we say that a body is at rest we use a form of words which appears to assert something about that body considered in itself, and we might imagine that the velocity of another body, if reckoned with respect to a body at rest, would be its true and only absolute velocity. But the phrase ‘‘at rest’’ means in ordinary language ‘‘having no velocity with respect to that on which the body stands,’’ as, for instance, the surface of the earth or the deck of a ship. It cannot be made to mean more than this.
It is therefore unscientific to distinguish between rest and motion, as between two different states of a body in itself, since it is impossible to speak of a body being at rest or in motion except with reference, expressed or implied, to some other body.

Article 31. On change of velocity
As we have compared the velocities of different bodies at the same time, so we may compare the relative velocity of one body with respect to another at different times.

Figure 6

If a₁, b₁, c₁ be the diagram of velocities of the system of bodies A, B, C, in its original state, and if a₂, b₂, c₂ be the diagram of velocities in the final state of the system, then if we take any point w as origin and draw wa equal and parallel to a₁a₂, wb equal and parallel to b₁b₂, wg equal and parallel to c₁c₂, and so on, we shall form a diagram of points a, b, g, etc., such that any line ab in this diagram represents in direction and magnitude the change of the velocity of B with respect to A. This diagram may be called the diagram of total accelerations.

Article 32. On acceleration
The word acceleration is here used to denote any change in the velocity, whether that change be an increase, a diminution, or a change of direction. Hence, instead of distinguishing, as in ordinary language, between the acceleration, the retardation, and the deflection of the motion of a body, we say that the acceleration may be in the direction of motion, in the contrary direction, or transverse to that direction.
As the displacement of a system is defined to be the change of the configuration of the system, so the total acceleration of the system is defined to be the change of the velocities of the system. The process of constructing the diagram of total accelerations by a comparison of the initial and final diagrams of velocities is the same as that by which the diagram of displacements was constructed by a comparison of the initial and final diagrams of configuration.

Article 33. On the rate of acceleration
We have hitherto been considering the total acceleration which takes place during a certain interval of time. If the rate of acceleration is constant, it is measured by the total acceleration in a unit of time. If the rate of acceleration is variable, its value at a given instant is measured by the total acceleration in unit of time of a point whose acceleration is constant and equal to that of the particle at the given instant.
It appears from this definition that the method of deducing the rate of acceleration from a knowledge of the total acceleration in any given time is precisely analogous to that by which the velocity at any instant is deduced from a knowledge of the displacement in any given time.
The diagram of total accelerations constructed for an interval of the nth part of the unit of time, and then magnified n times, is a diagram of the mean rates of acceleration during that interval, and by taking the interval smaller and smaller, we ultimately arrive at the true rate of acceleration at the middle of that interval.
As rates of acceleration have to be considered in physical science much more frequently than total accelerations, the word acceleration has come to be employed in the sense in which we have hitherto used the phrase ‘‘rate of acceleration.’’
In future, therefore, when we use the word acceleration without qualification, we mean what we have here described as the rate of acceleration.

Article 34. Diagram of accelerations
The diagram of accelerations is a system of points, each of which corresponds to one of the bodies of the material system, and is such that any line ab in the diagram represents the rate of acceleration of the body B with respect to the body A.
It may be well to observe here that in the diagram of configuration we use the capital letters, A, B, C, etc., to indicate the relative position of the bodies of the system; in the diagram of velocities we use the small letters, a, b, c, to indicate the relative velocities of these bodies; and in the diagram of accelerations we use the Greek letters, a, b, g to indicate their relative accelerations.

Article 35. Acceleration a relative term
Acceleration, like position and velocity, is a relative term and cannot be interpreted absolutely.
If every particle of the material universe within the reach of our means of observation were at a given instant to have its velocity altered by compounding therewith a new velocity, the same in magnitude and direction for every such particle, all the relative motions of bodies within the system would go on in a perfectly continuous manner, and neither astronomers nor physicists, though using their instruments all the while, would be able to find out that anything had happened.
It is only if the change of motion occurs in a different manner in the different bodies of the system that any event capable of being observed takes place.

*If the path cuts itself so as to form a loop, as P, Q, R (fig. 4), the particle will pass the point of intersection, Q, twice, and if the particle returns on its own path, as in the path A, B, C, D, it may pass the same point, S, three or more times. BACK

*When the simultaneous values of a quantity for different bodies or places are equal, the quantity is said to be uniformly distributed in space.BACK

*When the successive values of a quantity for successive instants of time are equal, the quantity is said to be constant.BACK

*According to the method of Fluxions, when the value of one quantity depends on that of another, the rate of variation of the first quantity with respect to the second may be expressed as a velocity, by imagining the first quantity to represent the displacement of a particle, while the second flows uniformly with the time.BACK