DEFINITIONS
Contents:
This is the measure of the matter of a body, arising from both the volume taken up by the body and the density of the body. Thus, doubling the volume and doubling the density together quadruple the quantity of matter. From this measure, we omit any medium present in the spaces between the parts of the body, e.g. air within a sponge. Furthermore, weight is proportional to matter, as can be seen in experiments on pendulums (shown later).
This is the measure of the motion (i.e. change in position) of a body, arising from both the velocity of the body and the quantity of the body’s matter. In other words, the motion of a body is the sum of the motion of the parts of the body. Thus, doubling the velocity and doubling the quantity of matter together quadruple to quantity of motion.
Also called:
Note that a body, given the inactivity of its matter (i.e. lack of motive force from the matter), is not put out of its state of rest of motion without some effort; hence the body’s matter exerts some force against any force impressed upon it. However, note that the body’s matter exerts its force only when another force is impressed upon it, i.e. the body’s matter as such is inactive when it is not acted upon. This innate force, i.e. inertia, has two aspects: (1) resistance, which refers to the aspect of inertia that withstands the force impressed so as to maintain the body’s present state, and (2) impulse, which refers to the aspect of inertia that, by withstanding the force impressed, acts to change the state of the body that is impressing the force.
A body that offers resistance also exerts an impulse, a reactive impulse amounts to resistance. Hence, these aspects of inertia are aspects of the same basic force, i.e. the innate force of a body’s matter. Also note that resistance and impulse both apply to a body whether it is at rest or in motion, because rest and motion (as usually understood) are relative, and not all bodies taken to be at rest from one point of reference are necessarily so with respect to another point of reference.
This is an action exerted upon a body so as to change its state (which may be a state of rest or uniform motion). This force is an aspect of action alone, and is no longer in the body once the action is over, because a body sustains its new states by inertia alone. Impressed forces can be from various sources: percussion, pressure, centripetal force.
This is a force by which bodies in any way tend toward a point, which is the centre of the force. Examples of such a force: earth’s gravity, magnetism, the force that draws back a stone whirled in a sling, the force that causes celestial bodies deviate from a rectilinear path and into a curvilinear path around a centre, etc.
Excerpt linking earth’s gravity to orbital movement:
A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and proceed in its motion in infinitum.
Hence, by identifying gravity as a centripetal force, and identifying that the curvature in a body’s line of motion due to any centripetal force decreases as the body’s velocity increases, Newton reveals the link between the forces making objects fall to the ground and forces making making celestial bodies go around in orbits.
Now, centripetal force can be quantified in three ways…
This is a measure of centripetal force proportional to the range and intensity of the force as propagated from the centre. In other words, it is a measure proportional to the efficacy of the cause of the centripetal force (that propagates it from the centre) through the spaces around the centre.
This is a measure of centripetal force proportional to the velocity it adds to body in a given time. For example, upto some distance from the earth’s centre, the earth adds the velocity of approximately 9.8 m/s to any body. Note that that velocity is not the same thing as motion; velocity is the measure of how much distance in a given direction is covered in a given time, whereas motion combines mass and velocity, i.e. motion refers to momentum. We see this distinction become clear in the distinction between the accelerative and motive quantity of centripetal force.
This is a measure of centripetal force proportional to the motion it adds to body in a given time. Hence, this is the measure of centripency, i.e. the tendency of the whole body toward the centre, which is in fact the body’s weight. Hence, the motive quantity of centripetal force is the weight of the body on which the force is being impressed. Note that a body’s weight is always know by the quantity of an equal and opposite force that is just enough to stop the descent of the body; for example, the innate force of the ground offers an equal and opposite force that is just enough to stop descent further toward the earth’s centre, and a weighing machine can measure this force.
For brevity, we can call these quantities of centripetal force as absolute, accelerative and motive forces. Their distinction is as follows: absolute force is with respect to the centre of force toward which bodies tend, accelerative force is with respect to the places of the bodies, and motive force is with respect to the bodies themselves, i.e. the momentum they gain as they tend toward the centre. In Newton’s own words, here is the distinction between motive and accelerative force:
I refer the motive force to the body as an endeavour and propensity of the whole towards a centre, arising from the propensities of the several parts taken together; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them; and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause be some central body (such as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats.
To continue:
Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter; and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several articles of the body, is the motive force of the whole.
NOTE: Celerity means velocity.
Change in motion is measured with respect to change within space and across time. However, change in motion is also measured in relation to a reference point. Given some reference point, it would not be evident whether the change in the motion of the body is due to a force impressed on the body or due to a force impressed on our reference point. Hence, it is valuable to distinguish between (1) motion observed without care taken about the nature of our own reference point, and (2) motion observed with respect to a fixed, i.e. unchanging reference point, i.e. with respect to a reference point to which no force is impressed, and with respect to which the motion of one or more bodies are observed. Such a distinction can be made using the terms “relative” and “absolute”.
However, in my understanding, Newton’s distinction of “relative” and “absolute” with respect to time, space and motion relies on the existence of absolute reference points, i.e. reference points that are inherently unchanging and universally applicable. While I agree with value of such a distinction, I do not agree with Newton’s view, because there is no empirical basis on which an absolute reference point be identified, and there is no theoretical basis for such a reference point in the first place. Nonetheless, as we shall see further, the distinction of “relative” and “absolute” is valuable, provided each term is validly defined.
NOTE: In this context, “absolute” = “true” = “mathematical”
Newton’s concepts:
Absolute time is a flow (i.e. constant self-contained change) without regard to anything external. Relative/apparent/common time is some sensible (i.e. self-perception-based) and external measure of duration by the means of motion (e.g. the movement of the sun, the swinging of a pendulum, etc.), which is commonly used instead of absolute/true time; such as an hour, a day, a month and a year.
Revised concepts:
Absolute time is a common unit of change with respect to which the change in other bodies is measured. Relative time is a change as measured by comparison with another change, without regard for a common unit. For Newton, any sensible measure of change is relative time, but there is no empirical or theoretical basis basis for some sort of “universal” time.
Newton’s concepts:
Absolute space is an unchanging and immovable region within which everything in existence resides. Relative space is some movable dimension or measure of the absolute spaces, and is empirically identified by its position to bodies. Newton illustrates the distinction between absolute and relative space in a helpful and valuable example:
Absolute and relative space, are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable.
Revised concepts:
Absolute space is a physical region which is defined by reference points that do not change or move with respect to each other, and which, as a whole, is not subject to any external forces. Relative space is the same, except it may be subject to forces that are impressed in the same direction and with the same magnitude on everything within this region.
Place is a part of space which a body takes up, and is, according to the space, either absolute or relative. Newton emphasis that “place” is a part of space and neither the position nor the surface of the body, because the places of equal solids are always equal, but their surfaces are often unequal due to their unequal shapes. Positions as such have no quantity nor are they places themselves; rather, they are properties of places. The concept of “place” is key to defining “motion”, and helps clarify discussions regarding bodies, their spaces, their movements and their interactions.
What exactly constitutes a body’s “place”?
The motion of the whole is the same as the sum of the motions of the parts, i.e. the translation of the whole, out of its place, is the same as the sum of the translations of the parts out of their places. Therefore, the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body.
Simply put, absolute motion is the translation of a body from one absolute place into another, and relative motion is the translation from one relative place into another. “Translation” refers to a kind of change wherein a body’s shape and matter are unchanged but the position of the body as a whole is changed. Tied to motion is the concept of rest. Absolute rest is the existence of a body within the same absolute place, and relative rest is the existence of a body within the same relative place. Newton gives a helpful illustration of these concepts:
Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of its cavity which the body fills, and which therefore moves together with the ship: and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space; partly from the relative motion of the ship on the earth; and if the body moves also relatively in the ship; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity expressed by 10 of those parts; but a sailor walks in the ship towards the east, with 1 part of the said velocity; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts.