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The Physics Book.

### Principia 1687: Book 1, Section 1

##### Reference: A Logical Approach to Theoretical Physics

This paper presents BOOK 1, SECTION 1from the English translation of NEWTON’S PRINCIPIA, American edition, 1846.

The paragraphs of original material are accompanied by brief comments in color based on present understanding. The heading below links to the original materials.

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## Book 1, Section 1

BOOK I: Of the Motion of Bodies

SECTION 1: Of the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow.

LEMMA I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

There is continuity and convergence that leads to certain limits. The limits show equality as difference reduces to zero.

LEMMA II. If in any figure AacE, terminated by the right lines Aa, AE, and the curve acE, there be inscribed any number of parallelograms Ab, Bc, Cd, etc., comprehended under equal bases AB, BC, CD, etc., and the sides, Bb, Cc, Dd, etc., parallel to one side Aa of the figure; and the parallelograms aKbl, bLcm, cMdn, etc., are completed. Then if the breadth of those parallelograms be supposed to be diminished, and their number to be augmented in infinitum; I say, that the ultimate ratios which the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and curvilinear figure AabcdE, will have to one another, are ratios of equality.

At the limit, the areas of inscribed and circumscribed rectangles for a curve are equal to the area under the curve.

LEMMA III. The same ultimate ratios are also ratios of equality, when the breadths, AB, BC, DC, etc., of the parallelograms are unequal, and are all diminished in infinitum.

Both equal and unequal widths of rectangles reduce to the same significance when infinitesimal.

LEMMA IV. If in two figures AacE, PprT, you inscribe (as before) two ranks of parallelograms, an equal number in each rank, and, when their breadths are diminished in infinitum. the ultimate ratios of the parallelograms in one figure to those in the other, each to each respectively, are the same; I say, that those two figures …, are to one another in that same ratio.

Ratio of areas under the two curves may be approximated by the ratio of the number of rectangles drawn under those curves the same way.

LEMMA V. In similar figures, all sorts of homologous sides, whether curvilinear or rectilinear, are proportional; and the areas are in the duplicate ratio of the homologous sides.

Areas of similar figures are in duplicate ratio of the sides.

LEMMA VI. If any arc ACB, given in position, is subtended by its chord AB, and in any point A, in the middle of the continued curvature, is touched by a right line AD, produced both ways ; then if the points A and B approach one another and meet, I say, the angle BAD, contained between, the chord and the tangent, will be diminished in infinitum, and ultimately will vanish.

At the point of contact, the tangent has the same direction (angle) as the arc.

LEMMA VII. The same things being supposed, I say that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality.

At the limit the ratio of the arc, chord, and tangent is the ratio of equality.

LEMMA VIII. If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB. RACB, RAD, and the points A and B approach and meet: I say, that the ultimate form of these evanescent triangles is that of similitude, and their ultimate ratio that of equality.

At the limit the ratio of areas of triangles thus formed is the ratio of equality.

LEMMA IX. If a right line AE and a curve line ABC, both given by position, cut each other in a given angle, A; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C: and the points B and C together approach towards and meet in the point A: I say, that the areas of the triangles ABD, ACE, will ultimately be one to the other in the duplicate ratio of the sides.

Areas of triangles are in the duplicate ratio of the sides.

LEMMA X. The spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion one to the other in the duplicate ratio of the times.

Forces are in the duplicate ratio of the times.

LEMMA XI. The evanescent subtense of the angle of contact, in all curves which at the point of contact have a finite curvature, is ultimately in the duplicate ratio of the subtense of the conterminate arc.

The angles of contact are in the duplicate ratio of the arc.

SCHOLIUM

These lemmas introduce the mathematics of limits in geometry of curves, their tangents, and the area under them. This is calculus of integration and differentiation.

It is assumed that the curvature at the point of contact is neither infinitely small nor infinitely great. The demonstration uses limits that reduce a lot of tedious demonstration by older methods. The ultimate quantities and ratios are not of any determinate magnitude but such as are conceived to be always diminished without end toward a certain limit.

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### Principia 1687: Laws of Motion

##### Reference: A Logical Approach to Theoretical Physics

This paper presents the chapter on LAWS OF MOTION from the English translation of NEWTON’S PRINCIPIA, American edition, 1846.

The paragraphs of original material are accompanied by brief comments in color based on present understanding. The heading below links to the original materials.

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## Laws of Motion

LAW I:  Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then motions both progressive and circular for a much longer time.

The uniform motion of a body is characterized by constant velocity. This velocity changes only when the body is being pushed around by forces. The greater is the mass of a body, the lesser are the fluctuations in its velocity. This is seen as the uniform motion of the body being maintained by the inertia of matter (see Definition III).

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LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The greater is the push the bigger is the change in the resulting velocity of a body. The magnitude of the push is determined by its force and duration. It is noted that for a push to continue, the body pushing must attempt to move faster than the body being pushed. The overall alteration of motion is proportional to the push.

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LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

A push (impressed force) requires a contact. Here actions are considered to be reciprocal because they are balanced for the duration of the contact, for example, in the case of a person standing on the floor, or in an elastic collision. But, when the push is continuous over a long duration, it is balanced by the mass (inertia) coming into play through acceleration.

#### CRITICAL COMMENT

These three laws of Newton, when examined closely, lead to the conclusion that the natural uniform velocity of a body in space shall ultimately depend on its mass or inertia. The higher is the mass the lesser is the velocity and vice versa. You cannot keep increasing the velocity of a body without decreasing its mass (inertia). This is not the same consideration as given by relativity.

This explains why the velocity of light (that has no mass) is many degrees of magnitude higher than the velocity of matter (that has mass).

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COROLLARY I: A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart.

When a body is pushed in two different directions simultaneously, it is equivalent to a single push along the diagonal of the parallelogram formed as above by the two pushes M and N. This geometry is based on the uniform velocity due to inertia.

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COROLLARY II: And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the resolution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from mechanics.

Thus, two forces AC and CD may be shown as equivalent to a single force AD. Similarly, a single force AD may be resolved in any two directions AC and CD.

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COROLLARY III: The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

This is the principle of conservation of momentum of a system. In a collision, motion may transfer from one body to another as velocity. This change in mass is so small that it is ignored. If there is a change in the mass of a body it is in the inertia (quantization) of each particle, and not in the number of particles.

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COROLLARY IV: The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.

This geometry is based on the rigidity of matter, and on the fact that the effect of a force is inversely proportional to the distance from the point of effect.

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COROLLARY V: The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

Newton is assuming that the same relative velocities shall be maintained by the objects of a system if the same absolute velocity is added to all of them. This may be so if the velocity added is small. But when that velocity added is high the mass starts to reduce significantly (see the CRITICAL COMMENT under Law III).  Objects of different masses may respond differently, and the relative speeds may not be maintained.

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COROLLARY VI: If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.

Again, the objects of different masses may behave differently when the push is very large and/or applied for a long duration. The relative velocities among objects may not be maintained.

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SCHOLIUM

The experiments described by Newton in this scholium deal with low velocities only. Hence deviations from these conclusions at high velocities shall not be detectable.

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### Principia 1687: Definitions

##### Reference: A Logical Approach to Theoretical Physics

This paper presents the chapter on DEFINITIONS from the English translation of NEWTON’S PRINCIPIA, American edition, 1846.

The paragraphs of original material are accompanied by brief comments in color based on present understanding.  The heading below links to the original materials.

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## Definitions

DEFINITION I: The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

A body of matter exists in space. Matter may be defined as hard substance that is made up of atoms. Space may be defined as soft substance that is made up of non-atomic radiation. We perceive substance by its force and resistance. Substance has a variable density and amount.

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DEFINITION II: The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

A body of matter moves in space. This motion is perceived as the velocity of matter in space. A velocity may be assessed relative to a reference-body at “rest”. Newton uses the background of fixed stars as the reference-body at “rest”. He, however, forwards a concept of an “absolute, immovable space”.

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DEFINITION III: The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.

This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly are taken to be so.

Matter is seen as consisting of mass upon which gravity acts to produce weight. The greater is the mass of a body, the lesser is the change produced in its velocity by the same push or force. This property of mass is called inertia. It is the inertia of the body that maintains its state of motion.

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DEFINITION IV: An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertia only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force.

It takes an impressed force to change the motion of a body. By an “impressed force” is meant a finite action like a push. By “change in motion” is meant a change in the uniform velocity of a body of matter. The new uniform motion is then maintained by the inertia of the body.

It appears that a motion that is maintained is in some sort of a balance. The amount of inertia in the body may then adjust to maintain the new balance of motion. This is the key to what Newton may have missed. This key is provided by Faraday’s principle of Conservation of Force.

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DEFINITION V: A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. 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. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes; nor could the moon without some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course: if it was too great, it would turn it too much, arid draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.

The quantity of any centripetal force may be considered as of three kinds; absolute, accelerative, and motive.

A centripetal force originates from a center. Mathematically, a center would be a location represented by a dimensionless point. In reality, a center would be a location of a body of matter. It will draw lesser bodies of matter toward it by gravitational force.

A body of matter is fixed in space to the degree it is difficult to move it by a push. A body of great inertia shall be very fixed in space, but a body of lesser inertia shall be less so. A less fixed body shall revolve around a body that is more fixed. Space itself shall not be fixed at all. It is difficult to conceive of the “absolute immovable space” of Newton. Newton probably had in mind a space filled with fixed stars.

A rectilinear motion is the limiting condition of curvilinear motion around a center at a great distance. A balance exists between gravitational attraction toward the center and the pull away due to velocity of the rotating body. If the gravitational force becomes stronger, or the velocity decreases, then the rectilinear path shall increasingly become curvilinear. There has to be a balance between the mass and velocity of the rotating body for it to be in a stable orbit. For if a push is applied to slow the velocity of the rotating body, the mass will increase, and with that the gravitational force shall also increase and the two bodies will collapse.

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DEFINITION VI: The absolute quantity of a centripetal force is the measure of the same proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.

In case of gravity, the absolute quantity of the centripetal force seems to depend upon the quantity of “mass” at the center. It acts like is a center of force from which lines of force spread out in space in all directions.

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DEFINITION VII: The accelerative quantity of a centripetal force is the measure, of the same, proportional to the velocity which it generates in a given time.

Thus the force of the same load-stone is greater at a less distance, and less at a greater: also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth; but at equal distances, it is the same everywhere; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

The accelerative quantity of the centripetal force is the force spread out in space. The total force thins out as it moves away from the center. It thins out in the ratio of the areas of the spherical surfaces around the center.

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DEFINITION VIII: The motive quantity of a centripetal force, is the measure of the same, proportional to the motion which it generates in a given time.

Thus the weight is greater in a greater body, less in a less body; and, in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight; and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body.

The motive quantity of the centripetal force is force felt by the falling body. It is the product of the accelerative force and the mass of the falling body.

These quantities of forces, we may, for brevity s sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction s sake, consider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend; that is to say, 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.

Thus, motive force applies to the body that is falling toward the center. The accelerative force applies to the place where the body is due to the absolute force concentrated at the center.

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. Hence it is, that near the surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body: but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less.

Velocity is motion per unit mass of the falling body. Acceleration is force per unit mass of the falling body. Motive force is then force of the whole body, which is its weight.

I likewise call attractions and impulses, in the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

Newton’s treatment of these forces is primarily mathematical, and he does not differentiate them as to their actual nature.

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SCHOLIUM

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

Words used here must be defined very precisely, and the notions must be distinguished into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.

According to Newton “absolute time” is the duration of an event. Any duration, to be absolutely consistent, must be measured against the background of infinite duration. Any other background of finite duration shall only lead to a relative time. An object of infinite mass shall provide infinite duration. In our case, this would be the core of universe that keeps it persisting. This persistence of universe shall provide the background for absolute consistency of time. Time may be subdivided to any degree to get the exact duration.

II. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. 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.

Space, to be real, must define the extents of something substantial; otherwise it is nothing more than an abstract or mathematical idea. There can be no real space without substance. Empty space is void of matter, but it is not void of substance, such as, force and quantized radiation. Newton defines “absolute space” as that which remains always similar and immovable. Space is similar because it always represents the extents of substance.  Space is immovable only when it consists of substance of infinite inertia. The core of the universe provides that immovable space of infinite inertia. Space is absolutely consistent when it is measured against the immovable space of infinite inertia. Space is relative otherwise.

III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their superficies, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and 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.

For Newton, the “place” is the space taken up by a body’s volume. He has a complex idea of absolute and relative places. The primary importance of a measure being absolute is that it is then consistent. A relative measure is not consistent because it does not have a consistent basis.

IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. 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 ex pressed 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.

Newton’s immovable space seems to full of fixed stars. It would be a space of infinite inertia and it shall be at absolute rest. A space of infinitesimal inertia shall be all over the place. The absolute motion of space has to be tied to inertia filling that space. The consistency of motion between two spaces shall depend on the ratio of their inertia. Space empty of matter is not homogenously movable either as it is filled with forces of different densities.

Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all: and therefore it ought to be distinguished from what are only sensible measures thereof; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for determining the times of a phenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.

Newton is pointing out the inconsistency of relative time. Astronomers correct this inconsistency through equations based on celestial motions. Newton is laying out the necessity of establishing absolute consistency of time through experiments with material and celestial phenomena.

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute motions.

Time and space are more than abstract and mathematical entities. Real space and time are characteristics of a substance. Real time is duration of a substance. Real space is extent of a substance. The absolute basis of space and time is the universe itself. The core of the universe is substance of infinite inertia (no motion) that has infinite duration. Infinite inertia provides fixed location from which less fixed locations may be measured. These locations and their motion constitute space.

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

In our common affairs, we choose earth to be immovable and use it to define positions, distances, places and motions. But when it comes to astronomy, we find earth to be in motion relative to the fixed stars. The ultimate immovable reference-body is the universe itself as a body of infinite inertia. Wherever we find infinite inertia we may choose those locations as absolutely fixed and use them to define positions, distances, places and motions. It may be possible to relate natural uniform velocities of bodies generally to their inertia.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions.

According to Newton, absolute rest cannot be determined from the position of bodies in our regions. It is obvious that that Newton is not relating absolute velocities of bodies to their masses or inertia.

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

Newton’s argument is arising from the fact that we cannot view any body to be absolutely at rest. So we have to deal with motion in the heavens in a relative sense only.

A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than by immovable places: and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute immovable space.

Newton is assuming space to be rigid like matter and that space is rigidly attached to matter. But this applies only to a space filled with matter. Immovable space shall be filled with material that has infinite density of mass. Relative to that, space filled with lesser density of mass shall be movable. Such motion based on the density of mass (or inertia) shall be consistent on an absolute basis.

The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved: but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body; but relative motion does not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be preserved, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations.

True motion is generated and altered only when forces are impressed. Relative motion may be generated or altered without any force impressed upon the body. Just because there is a change in motion is not enough to tell us whether it is true or relative. True motion brings inertia into the picture.

The effects, which distinguish absolute from relative motion, are the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion; they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion; the surface of the water will at first be plain, as before the vessel began to move: but the vessel; by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps par take of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions.

In a true circular motion, material recedes from the axis of rotation because it involves centripetal force. There can be relative circular motion where such effect is not observed. We may, thus, observe that earth is rotating and not the heavens. Absolute motion brings the mass (inertia) into picture.

Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place and motion, their measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities themselves with their relations and vulgar measures.

Relative quantities are how they appear to us. They are not the real quantities because they are not absolutely consistent. To make them consistent we resort to mathematics. But there too we must watch the assumptions we make.

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate: for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions: and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindermost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to collect the true motions from their causes, effects, and apparent differences; and, vice versa, how from the motions, either true or apparent, we may come to the knowledge of their causes and effects, shall be explained more at large in the following tract. For to this end it was that I composed it.

Newton finds it difficult to distinguish true from apparent motions. The difficulty seems to coming from Newton’s definition of space. Empty space cannot be immovable. Only the space filled with mass of infinite density can be immovable. Compared to that, space filled with substance of lesser density shall be movable. No confusion arises between apparent and true motions if we can calculate the absolute motion of a body from its mass (inertia).

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### A Geometry for Outer Space

##### Reference: A Logical Approach to Theoretical Physics

From Newton’s Principia, page 77 (pdf 83)

II. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. 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.

Newton made the assumption that space is immovable. But space is absence of matter, and it cannot be assigned such a characteristic as immovability. So there are no positions in space that are automatically fixed. The only thing fixed in space is a theoretical object of infinite mass. The fixity of the location of any other object shall be proportional to its mass (see The Uniform Velocity in Space).

So earth, moon and sun have locations in space with uncertainties attached to them according to their fixity. We may project abstract positions using these locations, such as, a position half-way between the earth and the moon. A projected position shall have uncertainty depending on the actual reference locations.

The location of an abstract position far from any mass shall be totally uncertain. This gives us a new non-Euclidean geometry.

This subject may be called “space geometry”. The uniform absolute speed of a body shall determine the uncertainty associated with its location in space geometry. The distance between two locations shall be determined by the difference in the uncertainties of those locations. Two locations with similar uncertainties shall be at equal distance from the location to no uncertainty.

The gravitational field around a body shall be defined by increasing uncertainties of positions around it.

This geometry shall be based on a universal constant that establishes the absolute motion of a body and the uncertainty of its location. The absolute velocity and mass of a body, or a system, may change but the absolute motion of the body, or the system, shall remain constant.

### The Uniform Velocity in Space

##### Reference: A Logical Approach to Theoretical Physics

Terms, such as, force, mass, velocity and energy acquire their meaning from the mathematical formulations of Newtonian mechanics. The system created from this Newtonian “particles in void” framework is, however, unable to explain “action at a distance”. Therefore, we must evaluate its terms carefully beyond their narrow mathematical meanings.

Newton states in his Principia 1,

DEFINITION III: The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.

Force constitutes the very power of matter. We perceive matter based on its effect on our senses as force. Force is the primary concept. Other concepts, such as inertia, mass and energy are derived from force.

Faraday looked at force as the “power of substance” where substance includes not only the “mass” of matter, but also the “energy” of radiative phenomena. Faraday’s concept of force goes beyond the mathematical description of force as “mass x acceleration”. Faraday’s principle of conservation of force includes the conservation of mass, energy and motion (momentum). Faraday says 2,

There is no question which lies closer to the root of all physical knowledge, than that which inquires whether force can be destroyed or not. The progress of the strict science of modern times has tended more and more to produce the conviction that “force can neither be created nor destroyed,” and to render daily more manifest the value of the knowledge of that truth in experimental research. To admit, indeed, that force may be destructible or can altogether disappear, would be to admit that matter could be uncreated; for we know matter only by its forces: and though one of these is most commonly referred to, namely gravity, to prove its presence, it is not because gravity has any pretension, or any exemption amongst the forms of force, as regards the principle of conservation; but simply that being, as far as we perceive, inconvertible in its nature and unchangeable in its manifestation, it offers an unchanging test of the matter which we recognize by it.

Therefore, Faraday insisted that the principle of conservation of force be included in every hypothesis, theory or definition, because unless this principle is satisfied, no study can be considered complete.

Supposing that the truth of the principle of the conservation of force is assented to, I come to its uses. No hypothesis should be admitted nor any assertion of a fact credited, that denies the principle. No view should be inconsistent or incompatible with it. Many of our hypotheses in the present state of science may not comprehend it, and may be unable to suggest its consequences; but none should oppose or contradict it.

As an example, we may apply this conservation principle of Faraday to the following anomaly in Newtonian Mechanics 1:

When a force is applied to a body, it accelerates. When the force is removed there is no more acceleration, but the body continues to move with the increased velocity.

What keeps the body moving at the higher velocity?

This question arises specifically in the case when the body is drifting in space light minutes away from the bodies influencing it through their gravitational fields. Let us consider the following example:

When a constant force is applied to an object it starts to accelerate at a constant rate. Its velocity starts to increase. The force is constant and so is the acceleration, then what is converting into increased velocity and kinetic energy?

Per Newton’s definition,

DEFINITION II: The quantity of motion is the measure of the same arising from the velocity and quantity of matter conjointly.

To conserve the motion, as the velocity increase, the mass (inertia) must reduce. The mass reduces by converting into kinetic energy. The decrease in mass (inertia) would, however, be extremely small and imperceptible to cause a significant and perceptible increase in velocity.

Furthermore, when the force is removed the acceleration goes away. But if the object continues to move at a higher speed, it must mean that its inertia stays reduced. We may, therefore, conclude:

The velocity of an object drifting in space is a function of its mass (inertia). As its velocity increases, its mass decreases commensurately.

Theoretically, a body in space can be at absolute rest if its mass is infinite. Its inertia would be so large that the gravitational forces cannot push it around. But as its mass decreases it gets pushed around and it acquires a velocity inversely proportional to its mass. A body of near zero mass shall have a velocity of near infinity. We observe this phenomenon with the velocity of light, where the mass is zero and the velocity is extremely large. Light has some inertia due to quantization, which makes its velocity finite.

Newton did not discount absolute velocities as he says 1,

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions.

That remote body absolutely at rest could be at the center of our milky way. Therefore, we may determine the absolute velocity of a body in terms of its mass (inertia).

The exact relationship of absolute velocity to mass still remains to be determined, but it is likely to be inversely proportional.

It is possible that the constant of proportionality, if it exists, may turn out to be a universal constant. The application of the Faraday’s principle of Conservation of Force thus fixes the uncertainty associated with the uniform velocity of a body drifting in space.

Hopefully Faraday’s principle of Conservation of Force shall lead us to a better understanding of the nature of the particles and void.

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