NEWTONIAN Philosophy, the doctrine of the universe, and particularly of the heavenly bodies, their laws, affections, &c. as delivered by Sir Isaac Newton.
The term Newtonian Philosophy is applied very differently; whence divers confused notions relating thereto. Some authors under this philosophy include all the corpuscular philosophy, considered as it now stands corrected and reformed by the discoveries and improvements made in several parts thereof by Sir Isaac Newton. In which sense it is that Gravefande calls his elements of physics, Introductio ad Philosophiam Newtonianam. And in this sense the Newtonian is the same with the new philosophy; and stands contradistinguished from the Cartesian, the Peripatetic, and the ancient Corpuscular.
Others, by Newtonian philosophy, mean the method or order which Sir Isaac Newton observes in philosophizing; viz. the reasoning and drawing of conclusions directly from phenomena, exclusive of all previous hypotheses; the beginning from simple principles; deducing the first powers and laws of nature from a few select phenomena, and then applying those laws, &c. to account for other things. And in this sense the Newtonian philosophy is the same with the experimental philosophy, and stands opposed to the ancient corpuscular.
Others, by Newtonian philosophy, mean that wherein physical bodies are considered mathematically, and where geometry and mechanics are applied to the solution of the appearances of nature. In which sense the Newtonian is the same with the mechanical and mathematical philosophy.
Others again, by Newtonian philosophy, understand that part of physical knowledge which Sir Isaac Newton has handled, improved, and demonstrated, in his Principia.
Others, lastly, by Newtonian philosophy, mean the new principles which Sir Isaac Newton has brought into philosophy; the new system founded thereon; and the new solutions of phenomena thence deduced; or that which characterizes and distinguishes his philosophy from all others.—Which is the sense wherein we shall chiefly consider it.
As to the history of this philosophy, we have nothing to add to what has been given in the preceding article. It was first made public in the year 1687, by the author, then a fellow of Trinity College, Cambridge, and in the year 1713, republished with considerable improvements.—Several authors have since attempted to make it plainer; by setting aside many of the more sublime mathematical researches, and substituting either more obvious reasonings or experiments in lieu thereof; particularly Whiston in his Prælect. Phys. Mathemat. Gravefande in Element. et Instit. and Dr Pemberton in his View.
The whole of the Newtonian philosophy, as delivered by the author, is contained in his Principia or Mathematical Principles of Natural Philosophy. He founds his system on the following definitions:
1. The quantity of matter is the measure of the same, arising from its density and bulk conjointly.—Thus air of a double density, in a double space, is
quadruple in quantity; in a triple space, sextuple in quantity, &c.
2. The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. This is evident, because the motion of the whole is the motion of all its parts; and therefore in a body double in quantity, with equal velocity, the motion is double, &c.
3. The vis inertia, or innate force of matter, is a vis inertia power of resisting, by which every body, as much as defined and in it lies, endeavours to persevere in its present state, whether it be of rest, or moving uniformly forward in a right line.—This definition is proved to be just, only by the difficulty we find in moving any thing out of its place; and this difficulty is by some reckoned to proceed only from gravity. They contend, that in those cases where we can prevent the force of gravity from acting upon bodies, this power of resistance becomes insensible, and the greatest quantities of matter may be put in motion by the very least force. Thus there have been balances formed so exact, that when loaded with 200 weight in each scale, they would turn by the addition of a single drachm. In this case 400 lb. of matter was put in motion by a single drachm, i. e. by part of its own quantity: and even this small weight, they say, is only necessary on account of the inaccuracy of the machine: so that we have no reason to suppose, that, if the friction could be entirely removed, it would take more force to move a tun weight than a grain of sand. This objection, however, is not taken notice of by Sir Isaac: and he bestows on the resisting power above mentioned the name of vis inertia; a phrase which is perhaps not well chosen, and with which inferior writers have endeavoured to make their readers merry at the expence of Newton. A force of inactivity, it has been said, is a forceless force; and analogous to a black white, a cold heat, and a tempestuous calm.
But objections of more importance have been made to the whole of this doctrine than those which merely respect the term vis inertia. "An endeavour to remain at rest (we are told *) is unnecessary, whilst nothing attempts to disturb the rest. It is likewise impossible to be conceived, as it implies a contradiction, as it implies a contradiction. A man, by opposing force to force, may endeavour not to be moved; but this opposition is an endeavour to move, not with a design to move, but by counteracting another force to prevent being moved. An endeavour not to move therefore cannot exist in bodies, because it is absurd; and if we appeal to fact, &c. we shall find every body in an actual and constant endeavour to move." It has been likewise observed, and we think justly, that "if bodies could continue to move by any innate force, they might also begin to move by that force. For the same cause which can move a body with a given velocity at one time, could do it, if present, at any other time; and therefore if the force by which bodies continue in motion were innate and essential to them, they would begin to move of themselves, which is not true." Newton indeed says that this innate force is the cause of motion under certain circumstances only, or when the body is acted upon by a force impressed ab extra. But if this impressed force do not continue as well
Newtonian as begin the motion, if it cease the instant that the impression is over, and the body continue to move by its vis inertiae, why is the body ever stopped?
"If in the beginning of the motion the body, by its innate force, overcomes a certain resistance of friction and air, in any following times, the force being undiminished, it will overcome the same resistance for ever. These resistances, therefore, could never change the state of a moving body, because they cannot change the quantity of its motive force. But this is contrary to universal experience." For these reasons we are inclined to think that bodies are wholly passive; that they endeavour nothing; and that they continue in motion not by any innate force or vis infinita, but by that force, whatever it be, which begins the motion, and which, whilst it remains with the moving body, is gradually diminished, and at last overcome by opposite forces, when the body of course ceases to move.
4. 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 inertiae only.
It is here implied, and indeed fully expressed, that motion is not continued by the same power that produced it. Now there are two grounds on which the truth of this doctrine may be supposed to rest.
"First, On a direct proof that the impressed force does not remain in the body, either by showing the nature of the force to be transitory and incapable of more than its first action; or that it acts only on the surface, and that the body escapes from it; or that the force is somewhere else, and not remaining in the body. But none of these direct proofs are offered.
"Secondly, It may rest on an indirect proof, that there is in the nature of body a sufficient cause for the continuance of every new state acquired; and that therefore any adventitious force to continue motion, though necessary for its production, is superfluous and inadmissible. As this is the very ground on which the supposition stands, it ought to have been indubitably certain that the innate force of the body is sufficient to perpetuate the motion it has once acquired, before the other agent, by which the motion was communicated, had been dismissed from the office. But the innate force of body has been shown not to be that which continues its motion; and therefore the proof, that the impressed force does not remain in the body, fails. Nor indeed is it in this case desirable to support the proof, because we should then be left without any reason for the continuance of motion *." When we mention an impressed force, we mean such a force as is communicated either at the surface of the body or by being diffused through the mass.
5. A centripetal force is that by which bodies are drawn, impelled, or any way tend towards a point, as to a centre.—The quantity of any centripetal force may be considered as of three kinds, absolute, accelerative, and motive.
6. The absolute quantity of a centrifugal force is the measure of the same, proportional to the efficacy of the
cause that propagates it from the centre, through the Newtonian spaces round it. Philosophy.
7. The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.
8. The motive quantity of a centripetal force is a measure of the same, proportional to the motion which it generates in a given time. This is always known by the quantity of a force equal and contrary to it, that is just sufficient to hinder the descent of the body.
I. Absolute, true, and mathematical time, of itself, Of Time. and from its own nature, flows equably, without regard to any thing external, and, by another name, is called duration. Relative, apparent, and common time, is some sensible and external measure of duration, whether accurate or not, which is commonly used instead of true time; such as an hour, a day, a month, a year, &c.
II. Absolute space, in its own nature, without re-Space. gard to any thing external, remains always similar and immovable. Relative space is some moveable dimension or measure of the absolute spaces; and which is vulgarly taken for immovable space. Such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position to bodies, and which is vulgarly taken for immovable space; as the distance of a subterraneous, an aerial, 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 earth passes; at another time it will be another part of the same; and so, absolutely understood, it will be perpetually mutable.
III. Place is a part of space which a body takes Place de-up; and is, according to the space, either absolute or fixed. relative. Our author says it is 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.
IV. Absolute motion is the translation of a body Of Motion. 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
Newtonian 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 towards the east, with a velocity of 10010 parts; while the ship itself with a fresh gale is carried towards the west, with a velocity expressed by 10 of these parts; but a sailor walks in the ship towards the east with one part of the said velocity: then the sailor will be moved truly and absolutely in immovable space towards the east with a velocity of 10011 parts: and relatively on the earth towards the west, with a velocity of 9 of those parts.
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 or 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 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.
8 Immutability of time and space. 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 we may be allowed the expression) out of themselves. For times and spaces are, as it were, the places 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.
But because the parts of space cannot be seen, or distinguished from one another by the 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
Newtonian motions, considering bodies as transferred from some Newtonian of those places into others. And so, instead of absolute places and motions, we use relative ones; and Philosophy. 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.
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 of each other. 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, though it be 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.
9 It is a property of motion, that the parts which retain given positions to their wholes do partake of the motion of their wholes. For all parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forwards arises from one and 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 only which 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 was not made, they would not really be 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.
A property near akin to the preceding is, 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 above-mentioned example of the sailor. Wherefore entire and absolute motions can be no otherwise determined than by immovable places. Now, no other places are immovable but those that from infinity to infinity do all retain the same given positions one to another; and upon this account must ever remain unmoved, and do thereby constitute what we call immovable space.
The causes by which true and relative motions are distinguished one from the other, are the forces impressed
Newtonian philosophy pressed 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 the 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 changes by such force. 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 motion suffers some change. Upon which account true motion does by no means consist in such relations.
10 Absolute and relative motion distinguished.
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 let go, it will be whirled about the contrary way; and while the cord is untwisting itself, 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; 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 in the water 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 plane 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 translations. 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, otherwise than they may perhaps participate of that only true motion. And therefore, in the system which supposes 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 the heavens, participate of their motions, and, as parts of revolving wholes, endeavour to recede from the axis of their motion.
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 then, if the meaning of words be determined by their use, 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.
It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from those that are only 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 we have some things to direct us in this intricate affair; and these arise partly from the apparent motions which are the difference 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 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 hindmost faces, or those which follow in the circular motion. 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 determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes might 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
Newtonian be at rest; and then, lastly, from the translation of the Philosophy globes among the bodies, we should find the determination of their motions.
Having thus explained himself, Sir Isaac proposes to show how we are to collect the true motions from their causes, effects, and apparent differences; and vice versa, how, from the motion, either true or apparent, we may come to the knowledge of their causes and effects. In order to this, he lays down the following axioms or laws of motion.
1. 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 UPON IT.—Sir Isaac's proof of this axiom is as follows: "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 motion, 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 their motions, both progressive and circular, for a much longer time."—Notwithstanding this demonstration, however, the axiom hath been violently disputed. It hath been argued, that bodies continue in their state of motion because they are subjected to the continual impulse of an invisible and subtle fluid, which always pours in from behind, and of which all places are full. It hath been affirmed, that motion is as natural to this fluid as rest is to all other matter. It is said, moreover, that it is impossible we can know in what manner a body would be influenced by moving forces if it was entirely destitute of gravity. According to what we can observe, the momentum of a body, or its tendency to move, depends very much on its gravity. A heavy cannon-ball will fly to a much greater distance than a light one, though both are actuated by an equal force. It is by no means clear, therefore, that a body totally destitute of gravity would have any proper momentum of its own; and if it had no momentum, it could not continue its motion for the smallest space of time after the moving power was withdrawn. Some have imagined that matter was capable of beginning motion of itself and consequently that the axiom was false; because we see plainly that matter in some cases hath a tendency to change from a state of motion to a state of rest, and from a state of rest to a state of motion. A paper appeared on this subject in the first volume of the Edinburgh Physical and Literary Essays; but the hypothesis never gained any ground.
2. 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.—Thus, if any force generates a certain quantity of motion, a double
force will generate a double quantity, whether that Newtonian force be impressed all at once, or in successive moments. Philosophy. To this law no objection of consequence has ever been made. It is founded on this self-evident truth, that every effect must be proportional to its cause. Mr Young, who seems to be very ambitious of detecting the errors of Newton, finds fault indeed with the expressions in which the law is stated; but he owns, that if thus expressed, The alteration of motion is proportional to the actions or resistance which produces it, and is in the direction in which the actions or resistances are made, it would be unexceptionable.
3. TO EVERY ACTION THERE ALWAYS IS OPPOSED AN EQUAL RE-ACTION: OR, THE MUTUAL ACTION OF TWO BODIES UPON EACH OTHER IS ALWAYS EQUAL, AND DIRECTED TO CONTRARY PARTS.—This axiom is also disputed by many. In the above-mentioned paper in the Physical Essays, the author endeavours to make a distinction between re-action and resistance; and the same attempt has been made by Mr Young. "When an action generates no motion (says he) it is certain that its effects have been destroyed by a contrary and equal action. When an action generates two contrary and equal motions, it is also evident that mutual actions were exerted, equal and contrary to each other. All cases where one of these conditions is not found, are exceptions to the truth of the law. If a finger presses against a stone, the stone, if it does not yield to the pressure, presses as much upon the finger; but if the stone yields, it re-acts less than the finger acts; and if it should yield with all the momentum that the force of the pressure ought to generate, which it would do if it were not impeded by friction, or a medium, it would not re-act at all. So if the stone drawn by a horse, follows after the horse, it does not re-act so much as the horse acts; but only so much as the velocity of the stone is diminished by friction, and it is the re-action of friction only, not of the stone. The stone does not re-act, because it does not act; it resists, but resistance is not action.
"In the loss of motion from a striking body, equal to the gain in the body struck, there is a plain solution without requiring any re-action. The motion lost is identically that which is found in the other body; this supposition accounts for the whole phenomenon in the most simple manner. If it be not admitted, but the solution by re-action is insisted upon, it will be incumbent on the party to account for the whole effect of communication of motion; otherwise he will lie under the imputation of rejecting a solution which is simple, obvious, and perfect; for one complex, unnatural, and incomplete. However this may be determined, it will be allowed, that the circumstances mentioned, afford no ground for the inference, that action and re-action are equal, since appearances may be explained in another way" (A.)
Others
(A) If there be a perfect reciprocity betwixt an impinging body and a body at rest sustaining its impulse, may we not at our pleasure consider either body as the agent, and the other as the resistant? Let a moving body, A, pass from north to south, an equal body B at rest, which receives the stroke of A, act upon A from south to north, and A resist in a contrary direction, both inelastic: let the motion reciprocally communicated be called fix. Then B at rest communicates to A fix degrees of motion towards the north, and receives fix degrees towards the south. B having no other motion than the fix degrees it communicated, will, by its equal
Others grant that Sir Isaac's axiom is very true in respect to terrestrial substances; but they affirm, that, in these, both action and re-action are the effects of gravity. Substances void of gravity would have no momentum; and without this they could not act; they should be moved by the least force, and therefore could not resist or re-act. If therefore there is any fluid which is the cause of gravity, though such fluid could act upon terrestrial substances, yet these could not re-act upon it; because they have no force of their own, but depend entirely upon it for their momentum. In this manner, say they, we may conceive that the planets circulate, and all the operations of nature are carried on by means of a subtle fluid; which being perfectly active, and the rest of matter altogether passive, there is neither resistance nor loss of motion. See MOTION.
From the preceding axiom Sir Isaac draws the following corollaries.
1. 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.
2. Hence we may explain the composition of any one direct force out of any two oblique ones, viz. by making the two oblique forces the sides of a parallelogram, and the direct one the diagonal.
3. 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: because the motion which one body loses is communicated to another; and if we suppose friction and the resistance of the air to be absent, the motion of a number of bodies which mutually impelled one another would be perpetual, and its quantity always equal.
4. The common centre of gravity of two or more bodies do 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.
5. The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion. The truth of this is evidently shown by the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or proceeds uniformly forward in a straight line.
6. If bodies, anyhow 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.
The whole of the mathematical part of the Newtonian philosophy depends on the following lemmas; of which the first is the principal.