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whence motions so accurate and beneficial are produced ; it seems proper to mentiou previously, some of the principal laws of the planetary motions, discovered by that eminent astronomer, John Kepler, from actual observations, according to the Copernican hypothesis, among which are the following:

1st. The areas, which the planets, which revolve round the sun, describe by right lines drawn to it, are proportional to the times.

2nd. The orbits, which they describe, are not circles, as was before generally supposed, but ellipses, the sun being in one of the focuses.

3rd. The cubes of their mean distances from the sun are to each other, as the squares of their periodick times.

The two first laws being applicable to the moon's motion round the earth, and all three to the motion of Jupiter and Saturn's satellites round their primaries. It remained for the great Newton to deduce these and other laws of the system of the world, from the laws of motion, by mathematical reasoning. Some of his principal discoveries on this subject are delivered in the following elements,

DEFINITIONS.

The quantity of matter, is a measure thereof, arising from its density and magnitude jointly.

The air, for instance, its density being doubled, in a double space is four-fold, in a triple, six-fold. This quantity may be ascertained by its weight, especially in an exhausted receiver.

2. The quantity of motion, is a measure thereof, arising from the velocity and quantity of matter jointly.

The motion of the whole, is the sum of the motions of all the parts, and therefore in a body of double the quantity of matter, with an equal velocity, is double, and with a double velocity, four-fold. And ever so small a power may be made to move ever so great a weight; namely, by making the velocity of the power compared with that of the weight such, that the product of the quantity of matter of the power multiplied by its velocity, may be greater than the product of the quantity of matter of the weight by its velocity, and so much greater as to overcome such resistance as may arise from friction, &c.

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3. The force of inertness, or tis inertiæ, or vis insita of matter, is the power of resisting, by which every body, as much as is in it, perseveres in its state of rest, or of uniform motion in a right line. This force is proportional to the quantity of matter.

4. An impressed force, is an action exercised on a body, to change its state of rest, or uniform motion in a right line.

This force consists in the action alone, nor does it remain in the body after the action. For the body perseveres in every new state by its force of inertness alone. But the impressed force is of different origins, as from a stroke, a pressure, a centripetal or centrifugal force.

5. A centripetal force, is that, by which bodies are drawn, impelled, or any how tend towards any point as a centre.

Of this kind is gravity, by which bodies tend to the centre of the earth ; magnetism, by which iron is attracted towards a magnet ; and that force, whatever it be, by which the planets are perpetually drawn from rectilineal motions, and caused to be revolved in curve lines. A stone, whirled about in a sling, endeavours to recede from the hand which turns it ; and by that endeavour, distends the sling, and with so much the greater force, as it is revolved with the greater velocity and as soon as 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 towards the hand as the centre of the orbit, may be called 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, by which they are retained in their orbits, and which may therefore be called centripetal, would go off in right lines with a uniform motion. A projectile, if it were not for the force of gravity, would not deviate towards the earth, but would go off in a right line, and with a uniform motion, if the resistance of the air were taken away. gravity it is perpetually drawn aside from its rectilineal course, and made to deviate towards the earth more or less, according to the force of its gravity, and the velocity of its motion.-. By how much the less the force of gravity is, and the greater the velocity, with which it is projected, by so much the less it will deviate from a rectilineal course, and the farther it will go. If a leaden ball, projected from the top of a mountain, by the force of gun-powder, with a given velocity, in a horizontal direction, be carried to the distance of two miles before it falls to the

By its 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

cted, 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 eartlı, 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 conimon 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, wliether 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 inove, 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 motiom 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 common Deasures.

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.

Law 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.

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