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< The Mathematical Principles of Natural Philosophy (1729)
Philosophiae Naturalis Principia Mathematicaby Isaac Newton
Definitions

THE

Mathematical

PRINCIPLES

OF

Natural Philosophy.

Definitions

The Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly.

HUS 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.

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.

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 forwards 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 inertiæ or Force of Inactivity. But a body exerts this force only, when another force, impress'd 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 impress'd force of another, endeavours to change the state of that other. Resistance is usually ascrib'd 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.

An impress'd 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 Inertiæ only. Impress'd forces are of different origines; as from percussion, from pressure, from centripetal force.
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 continually 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 revolv'd 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 'tis 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, revolv'd 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. 'Tis 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 curved 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 forwards 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 impells it towards the Earth, may be continually drawn aside towards the Earth, out of the rectilinear way, which by its innate force it would pursue; and be made to revolve in the orbit which it now describes; nor could the Moon without some such force, be retain'd 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, and 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.

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.

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 shewn) 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 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; 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 ever known by the quantity of a force equal and contrary to it, that is just sufficient to hinder the descent of the body.

These quantities of Forces, we may for brevity's sake, call by the names of Motive, Accelerative, and Absolute forces; and for distinction 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 is some central body, (such as is the Load-stone, in the centre of the Magnetism, or the Earth in the centre of the gravitating force), or any thing 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.

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 arisses from the accelerative force multiplied by the same quantity of matter. For the sum of the actions of the accelerative force, upon the several particles of the body, is the motive force of the whole. Hence it is, that near the suffice 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.

Virtual dj 8 mac license key. 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.

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 common people 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.

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I. Absolute, True, and Mathematical Time, of it self, and from its own nature, flows equably without relation 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.

II Absolute Space, in its own nature, without relation 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 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 Air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetuallymutable.

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 translationof 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 thewhole body.

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 it self, 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 truemotion 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 mov'd toward the East, with a velocity of 10010 parts; while the Ship it selfwith a fresh gale, and full sails, is carry'd towards the West, with a velocity exprss'd 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 bemoved truly and absolutely 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.

Absolute time, in Astronomy, is distinguish'd from Relative, by the Equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly consider'd 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 distinguish'd 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 phænomenon, is evinc'd as well from the experiments of the pendulum clock, as by eclipses of the Satellites of Jupiter.

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 mov'd out of their places, and they will be moved (if the expression may be allowed) out ofthemselves. 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 translationsout of those places, are the only Absolute Motions.

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 consider'd as immovable, we define all places:and then with respect to such places, we estimate all motions, considering bodies as transfer'd 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 maybe that there is no body really at rest, to which the places and motions of others may be referr'd.

But we may distinguish Rest and Motion, absoluteand relative, one from the other by their Properties,Causes and Effects. It is a property of Rest, that bodiesreally at rest do rest in respect of one another. Andtherefore as it is possible, that in the remote regionsof the fixed Stars, or perhaps far beyond them, theremay be some body absolutely at rest; but impossibleto know from the position of bodies to one another inour regions, whether any of these do keep the sameposition to that remote body; it follows that absolute rest;cannot be determined from the position of bodies inour regions.

It is a property of motion, that the parts, whichretain given positions to their wholes, do partake of themotions of those wholes. For all the parts of revolvingbodies endeavour to recede from the axe of motion;and the impetus of bodies moving forwards, arisesfrom the joint impetus of all the parts. Therefore,if surrounding bodies are mov'd, those that are relativelyat rest within them, will partake of their motion.Upon which account, the true and absolute motion ofa body cannot be determin'd by the translation of itfrom those which only seem to rest: For the externalbodies ought not only to appear at rest, but to be reallyat rest. For otherwise, all included bodies, beside theirtranslation from near the surrounding ones, partake likewiseof their true motions; and tho' that translation wasnot made they would not be really at rest, but onlyseem to be so. For the surrounding bodies stand inthe like relation to the surrounded, as the exterior partof a whole does to the interior, or as the shell does tothe kernel; but, if the shell moves, the kernel will alsomove, as being part of the whole, without any removalfrom near the shell.

A property near a kin to the preceding, is this,that if a place is mov'd, whatever is placed thereinmoves along with it; and therefore a body, which ismov'd from a place in motion, partakes also of themotion of its place. Upon which account all motionsfrom places in motion, are no other than parts of entireand absolute motions; and every entire motion is composedout of the motion of the body out of its firstplace, and the motion of this place out of its place,and so on; until we come to some immovable place,as in the before mention'd example of the Sailor.Wherefore entire and absolute motions can be nootherwise determin'd than by immovable places; andfor that reason I did before refer those absolute motionsto immovable places, but relative ones to moveableplaces. Now no other places are immovable, butthose that, from infinity to infinity, do all retain thesame given positions one to another; and upon this account,must ever remain unmov'd; and do therebyconstitute, what I call, immovable space.

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The Causes by which true and relative motions aredistinguished, one from the other, are the forces impress'd upon bodies to generate motion. True motionis neither generated nor alter'd, but by some force impress'dupon the body moved: but relative motionmay be generated or alter'd without any force impress'dupon the body. For it is sufficient only to impresssome force on other bodies with which the former iscompar'd, that by their giving way, that relation maybe chang'd, in which the relative rest or motion ofthis other body did consist. Again, True motion suffersalways some change from any force impress'd uponthe moving body; but Relative motion does not necessarilyundergo any change, by such forces. For ifthe same forces are likewise impress'd on those other bodies, with which the comparison is made, that therelative position may be preserved, then that conditionwill be preserv'd, in which the relative motion consisits.And therefore, any relative motion may be changed,when the true motion remains unalter'd, and the relativemay be preserv'd, when the true suffers some change.Upon which accounts, true motion does by no meansconsist in such relations.

The Effects which distinguish absolute from relativemotion are, the forces of receding from the axe ofcircular motion. For there are no such forces in circularmotion purely relative, but in a true and absolutecircular motion, they are greater or less, according tothe quantity of the motion. If a vessel, hung by along cord, is so often turned about that the cord isstrongly twisted, then fill'd with water, and held atrest together with the water; after by the sudden actionof another force, it is whirl'd about the contrary way,and while the cord is untwisting it self the vesselcontinues for some time in this motion; the surface ofthe water will at first be plain, as before the vessel beganto move: but the vessel, by gradually communicatingits motion to the water, will make it begin sensiblyto revolve, and recede by little and little from themiddle, and ascend to the sides of the vessel, formingit self into a concave figure, (as I have experienced)and the swifter the motion becomes, the higher willthe water rise, till at last, performing its revolutions inthe same times with the vessel, it becomes relatively atrest in it. This ascent of the water shews its endeavourto recede from the axe of its motion; and thetrue and absolute circular motion of the water, whichis here directly contrary to the relative, discovers it selfand may be measured by this endeavour. At first,when the relative motion of the water in the vessel was greatest it produc'd no endeavour to recede from theaxe: the water shew'd no tendency to the circumference,nor any ascent towards the sides of the vessel,but remain'd of a plain surface, and therefore its Truecircular motion had not yet begun. But afterwards,when the relative motion of the water had decreas'dthe ascent thereof towards the sides of the vessel, prov'dits endeavour to recede from the axe; and this endeavourshew'd the real circular motion of the waterperpetually increasing, till it had acquir'd its greatestquantity, when the water rested relatively in the vessel.And therefore this endeavour does not depend upon anytranslation of the water in respect of the ambient bodies,nor can true circular motion be defin'd by suchtranslations. There is only one real circular motion ofany one revolving body, corresponding to only onepower of endeavouring to recede from its axe of motion,as its proper and adequate effect: but relative motionsin one and the same body are innumerable, accordingto the various relations it bears to external bodies,and like other relations, are altogether destitute of anyreal effect, any otherwise than they may perhaps participateof that one only true motion. And therefore intheir system who suppose that our heavens, revolvingbelow the sphere of the fixt Stars, carry the Planetsalong with them; the several parts of those heavens,and the Planets, which are indeed relatively at rest intheir heavens, do yet really move. For they changetheir position one to another (which never happensto bodies truly at rest) and being carried together withtheir heavens, participate of their motions, and as partsof revolving wholes, endeavour to recede from the axeof their motions.

Wherefore relative quantities, are not the quantitiesthemselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate) whichare commonly used instead of the measur'd quantitiesthem'elves. And if the meaning of words is to bedetermin'd by their use; then by the names Time,Space, Place and Motion, their measures are properly tobe understood; and the expression will be unusual, andpurely Mathematical, if the measured quantities themselvesare meant. Upon which account, they dostrain the Sacred Writings, who there interpret thosewords for the measur'd quantities. Nor do thoseless defile the purity of Mathematical and Philosophicaltruths, who confound real quantities themselves withtheir relations and vulgar measures.

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It is indeed a matter of great difficulty to discover,and effectually to distinguish, the True motion ofparticular bodies from the Apparent: because the partsof that immovable space in which those motions areperform'd, do by no means come under the observationof our senses. Yet the thing is not altogether desperate;for we have some arguments to guide us, partly fromthe apparent motions, which are the differences of thetrue motions; partly from the forces, which are thecauses and effects of the true motions. For instance,if two globes kept at a given distance one from theother, by means of a cord that connect them, were revolv'dabout their common centre of gravity; we might,from the tension of the cord, discover the endeavourof the globes to recede from the axe of their motion,and from thence we might compute the quantity oftheir circular motions. And then if any equal forcesshould be impress'd at once on the alternate faces of theglobes to augment or diminish their circular motions;from the encrease or decrease of the tension of the cord,we might infer the increment or decrement of theirmotions; and thence would be found, on what faces those forces ought to be impress'd, that the motions ofthe globes might be most augmented. that is, wemight discover their hindermost faces, or those which,in the circular motion, do follow. But the faces whichfollow being known, and consequently, the oppositeones that precede, we should likewise know the determinationof their motions. And thus we mightfind both the quantity and the determination of thiscircular motion, ev'n in an immense vacuum, wherethere was nothing external or sensible with which theglobes could be compar'd. But now if in that spacesome remote bodies were plac'd that kept always agiven position one to another, as the Fixt Stars do inour regions; we cou'd not indeed determine from the relativetranslation of the globes among those bodies,whether the motion did belong to the globes or to thebodies. But if we observ'd the cord, and found thatits tension was that very tension which the motions ofthe globes requir'd, we might conclude the motion tobe in the globes. and the bodies to be at rest; andthen, lastly, from the translation of the globes amongthe bodies, we should find the determination of theirmotions. But how we are to collect the true motionsfrom their causes, effects, and apparent differences; andvice versa, how from the motions, either true or apparent,we may come to the knowledge of theircauses and effects, shall be explain'd more at large inthe following Tract. For to this end it was that Icompos'd it.

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