ground; the same, with a double or ten-fold velocity, would go about double or ten times as far, provided the resistance of the air was taken away. 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 described by it, till at length it might describe ten, twenty, or ninety degrees, or even go round the whole earth, before it would fall; or finally might never fall, but might go off into the celestial spaces, and by the motion of going off, might proceed in infinitum. And in the same manner, as a projectile may, by the force of gravity, be made to revolve in an orbit, and go round the whole earth; the moon also may, either by the force of gravity, if it have gravity, or by any other force, by which it may be urged towards the earth, be perpetually drawn from a rectilineal course towards the earth, and made to revolve in its orbit: and without such a force the moon could not be retained in its orbit. If this force were too small, it would not sufficiently turn the moon from a rectilineal course; if too great, it would turn it too much, and draw it down from its orbit towards the earth. It is requisite, that the force be of a just quantity, and it belongs to mathematicians to find the force, by which a body may be accurately retained in any given orbit, with a given velocity; and again, to find the curvilineal path, into which a body going from a given place, with a given velocity, would be turned, by a given force.

Scholium.

When the word centripetal force, attraction, impulse or propension is used, the reader is to be aware, that, by these words, it is not meant, to determine the species or mode of action, or the physical cause or reason of it; or to ascribe these forces truly and physically to the centres, which are mathematical points, when the centres are said to attract, or forces are called centripetal; these forces being in this tract considered, not physically, but mathematically.

The terms, time, space, place and motion, are not explained in the above definitions, as being well known to all. But in order to avoid certain prejudices which may arise from the conmon conceptions of these things, it seems proper to distinguish them into absolute and relative, true and apparent, mathematical and common. In explaining the distinction between these, which appears extremely obvious, I shall be very brief.

1. Absolute, true, and mathematical time, in itself and its nature, flows equably, without relation to any thing external; and by another name is called, duration: Relative, apparent and common time, is some sensible and external measure of duration, by motion, whether accurate or inequable, commonly used for a true measure of time, as a day, a month, a year. Natural days are unequal, and astronomers correct the inequality, for the purpose of calculating the celestial motions. It is possible there may be no perfectly equable measure of time. All motions may be accelerated or retarded, but the flow of absolute time cannot be changed, and is the same, whether motions be swift or slow or none.

2. Absolute space, in its own nature, without relation to any thing external, remains always the same and immoveable. Relative space, is some moveable measure or dimension of this space, which is determined by our senses from its situation with respect to bodies, and is commonly reckoned immoveable: as the dimension of a subterraneous, aerial or celestial space determined by its situation with respect to the earth; and if the earth be moved, a space of our air, which, relatively and with respect to the earth, always remains the same, becomes at one time, one part of absolute space, and, at another time, another, and so its portion of absolute space is continually changed.

3. Place, is the part of space which a body occupies, and is, according to the nature of the space, either absolute or relative. Thus a body on this earth, which, apparently and with respect to the earth, remains in the same place, if the earth move, is continually changing its place or situation with respect to absolute or immoveable space; but situations, properly speaking, have not quantity, and are not so much places, as affections of places.

4. Absolute motion, is the translation of a body from an absolute place to an absolute place; Relative from a relative one to another. Thus if the earth move, a body on it may be relatively at rest, that is, with respect to the earth, and yet, with respect to absolute space, be in motion.

Hence it appears, that relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them, 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, by those names of time, space, place and motion, these measures are to be understood; and the language will be unusual, though purely mathematical, if the measured quantities themselves be meant.

Therefore they do violence to the sacred writings, who there interpret these terms for the measured quantities themselves. Nor do they less contaminate mathematicks and philosophy, who confound the true quantities, with their relations and com

mon measures.

To discover indeed the true motions of particular bodies, and actually to discriminate them, from those which are apparent, is a matter of no little difficulty; because the parts of that immoveable space. in which bodies are really moved do not strike our senses. Yet the cause is not entirely desperate. For arguments are within our reach, partly from the apparent motions, which are the differences of true ones, and partly from the forces, which are the causes of the true motions.

AXIOMS, OR LAWS OF MOTION.

Law 1. That every body perseveres in its state of rest, or of moving uniformly in a right line, unless so far as it is compelled, by forces impressed thereon, to change that state.

Projectiles persevere in their motions, unless so far as they are retarded by the resistance of the air, and impelled downward by the force of gravity. But the greater bodies of the planets and comets preserve their progressive and rotatory motions, in less resisting spaces, for a very long time.

Law 2.

That a change of motion is proportional to the force impressed, and according to the right line, in which that force is impressed.

If any force generate a motion, a double force will generate a double motion, a triple force, a triple one, whether the force be impressed together and at once, or gradually and successively. And if the impressed body were previously in motion, and the impressed force be in the same direction, as that previous motion, the impressed motion is added to, or taken from, the previous motion, according as the two motions conspire with, or are contrary to each other. But if the impressed force be in an oblique direction, with respect to the previous motion, a new motion will arise compounded of the determination of both.

Lar 3. That reaction is always equal and contrary to action: or, that the actions of two bodies on each other are always equal, and directed to contrary parts.

Whatever presses or draws another, is as much pressed or drawn by that other. If any one, with his finger press a stone, his finger is also pressed by the stone. If a horse draw a stone tied to a rope, the horse, if I may so speak, will be equally drawn back towards the stone for the stretched rope, by the endeavour of relaxing itself, will urge the horse towards the stone, and the stone towards the horse, and will as much impede the progress of one, as it advances that of the other. If a globular body, as an ivory ball, impinging on another similar one, by its force change in any way the motion of that other, the same will also by the force of that other, on account of the equality of the mutual pressure, undergo an equal change in its motion, to the contrary part. By these actions, if the bodies be unequal, equal changes are made, not of velocities, but of motions; namely, in bodies not otherwise obstructed: for the changes of velocities, made towards contrary parts, because the motions are equally changed, are reciprocally proportional to the magnitudes of the bodies.

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In attractions, which are the principal object of this tract, the truth of this law may be thus shewn. Between any two bodies A and B, mutually attracting each other, conceive any obstacle to be placed, by which their coming together may be hindered. If either body A be more attracted towards the other B, than that other B towards the former A, the obstacle would be more urged by the pressure of the body A, than by that of B, and therefore would not remain in an equilibrium. The stronger pressure would prevail, and cause, that the system of the two bodies and the obstacle would be moved directly towards the part, on which B lies, and in free spaces would go forward in infinitum, with a motion continually accelerated; which is absurd and contrary to the first law. For, by the first law, the system ought to persevere in its state of rest, or of moving uniformly forward in a right line; therefore the bodies must equally press the obstacle, and are equally attracted by each other. The truth of this may be shewn by experiment, in the attraction between a magnet and iron. If these, placed apart in proper vessels touching each other, float near each other

in still water, neither will propel the other, but, by the equality of attraction both ways, they will sustain each other's pressure, and at length rest in an equilibrium.

So also the gravity between the earth and its parts is mutual. Let the earth EH be cut by any plain DF into two unequal parts DEF and DHF; their weights towards each other are mutually equal. For if the greater part DHF be, by another plane GK parallel to the former DF, cut into two parts DFKG and GHK, of which the exterior part GHK is equal to the less part first cut off DEF: it is manifest, that the middle part DFKG will, by its own weight, tend to neither of the extreme parts, but will, if I may so speak, be suspended, and rest in an equilibrium between both. But the extreme part GHK would press with all its weight on the middle part, and urge it towards the other extreme part DEF; therefore the force with which the sum of the parts GHK and DFKG tends towards the third part DEF is equal to the weight of the part GHK, or, to the weight of the third part DEF. Therefore the weights of the two parts DEF and DHF towards each other are equal, as was proposed to be proved. And unless these weights were equal, the whole earth, floating in a free ether, would yield to the greater weight, and, in going from it, would be carried off in infinitum.