Cor. 2. The sum of the motions of all the bodies in the world, estimated in one and the same line of direction, and always the same way, is eternally and invariably the same; esteeming these motions affirmative which are progressive, or directed the same way; and the regressive motions negative. And, therefore, in this sense, motion can neither be increased nor diminished. But,

Cor. 3. If you reckon the motions in all directions to be affirmative, then the quantity of motion may be increased or decreased an infinite number of ways. As, suppose two equal non-elastic bodies to meet one another with equal velocities, they will both stop and lose all their motions.

For let M be the motion of each, then, before meeting the sum of their motions is M+M; and after their meeting, it is o. But in the sense of this Prop. M-M is the sum of the motions before they meet, because they move contrary ways, which is o; and it is the same after they meet. And thus a man may put several bodies into motion with his hands, which had no motion before; and that in as many several directions as he will.

PROP. XI.

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. For if a body be moving in any right line, and there be any force equally impressed, both upon the body and the right line in any direction; and, in consequence of this, they both move uniformly with the same velocity; now, as there is no force to carry the body out of that line, it must still continue in it as before; and as there is no force to alter the motion of the body in the right line, it will (by Ax. 1.) still continue to move in it as before. For the same reason, the motions of any number of bodies moving in several directions, will still continue the same; and their motions among themselves will be the same, whether that space be at rest, or moves uniformly forward.

Likewise, since the relative velocities of bodies, (that is, the difference of the real velocities the same way; or their sum, different ways) remain the same, whether that space be at rest, or it and the bodies move uniformly forward all together; therefore their mutual impulses, collisions, and actions upon one another, being (by Cor. 3. Prop. IX.) as the relative velocities, must (by Ax. 3.) remain the same in both cases.

SCHOLIUM. Before I end this section, it may not be amiss to mention a certain kind of force, called, by the foreigners, vis viva. This they term a faculty of acting; and distinguish it from the vis mortua, which, with them, signifies only a solicitation to motion,

such as pressure, gravity, &c.: concerning this vis viva they talk so obscurely, that it is hard to know what they mean by it. But they measure its quantity by the number of springs which a moving body can bend to the same degree of tension, or break; whether it be a longer or shorter time in bending them. So that the vis viva is the total effect of a body in motion, acting till its motion be all spent. And, according to this, they find that the force (or vis viva) to overcome any number of springs, will always be as the body multiplied by the square of the velocity.

Suppose any number of equal and similar springs placed at equal distances in a right line, and a body be moved in the same right line against these springs; then the number of springs which that body will break before it stop, will be as the square of its velocity; whatever be the law of the resistance of any spring in the several parts of its tension; for, from the foregoing Prop. it appears, that the swifter the body moves, so much the less time has any spring to act against it to destroy its motion; and, therefore, the motion destroyed by one spring will be as the time of its acting; and by several springs, as the whole time of their acting; and, consequently, the resistance is uniform. And since the resistance is uniform, the velocity lost will be as the time, that is as the space directly and velocity reciprocally; whence the space, and therefore, the number of springs, is as the square of the velocity. And upon this account they measure the force of a body in motion, by the square of the velocity. So at last the vis viva seems to be the total space passed over, by a body meeting with a given resistance, which space is always as the square of the velocity. And this comes to the same thing as the force and time together, in the common mechanics.

Now it seems to be a necessary property of the vis viva, that the resistance is uniform; but there are infinite cases where this does not happen; and, in such cases, this law of the vis viva must fail. And since it fails in so many cases, and is so obscure itself, it ought to be weeded out, and not to pass for a principle in mechanics.

Likewise, if bodies in motion impinge on one another, the conservation of the vis viva can only take place when the bodies are perfectly elastic. But as there are no bodies to be found in nature which are so; this law will never hold good in the motion of bodies after impulse, but, in this respect, it must eternally fail. This notion of the vis viva was first introduced by M. Leibnitz, who believed that every particle of matter was endued with a living soul.

SECTION SECOND.

THE LAWS OF GRAVITY, THE DESCENT

OF

HEAVY BODIES, AND THE MOTION OF PRO-
JECTILES, IN FREE SPACE.

PROP. XII.

THE SAME QUANTITY OF FORCE IS REQUISITE TO KEEP A BODY

IN ANY UNIFORM MOTION, DIRECTLY UPWARDS, AS IS REQUIRED TO KEEP IT SUSPENDED, OR AT REST.

AND IF A BODY DESCENDS UNIFORMLY, THE SAME FORCE THAT IS SUFFICIENT TO HINDER ITS ACCELERATION IN DESCENDING, IS EQUAL TO THE WEIGHT OF IT.

For the force of gravity will act equally on the body in any state whether of motion or rest; therefore, if a body is projected directly upwards or downwards, with any degree of velocity, it would for ever retain that velocity if it were not for the force of gravity that draws it down, (by Ax. 1.) If, therefore, a force equal to its gravity were applied directly upwards; then (by Ax. 9.) these two forces destroy each other's effects; and it is the same thing as if the body was acted on by no force at all; and, therefore, (by Ax. 1.) it would retain its uniform motion.

Cor. But if a body be moved upwards with an accelerated motion, the force to cause that motion will be greater than its weight; and that in proportion to its acceleration. (by Ax. 10.)

PROP. XIII.

THE VELOCITIES OF FALLING BODIES ARE AS THE TIMES OF

THEIR FALLING FROM REST.

For by Postulate 3. the body is uniformly acted on by gravity, which is its accelerating force downwards; therefore, by Cor. 3. Prop. V. the velocity is as the force and time directly, and the matter reciprocally. But by Ax. 5. the force of gravity is as the quantity of matter; and, consequently, the velocity will be as the

time.

Cor. 1. All bodies falling by their own weight, gain equal velocities in equal times.

Cor. 2. Whatever velocity a falling body gains in any time, if it be thrown directly upwards, it will lose as much in an equal time; by Ax. 12. And, therefore,

Cor. 3. If a body be projected upwards with the velocity it acquired by falling in any time; it will, in the same time, lose all its motion. Hence, also,

Cor. 4. Bodies thrown upwards lose equal velocities in equal times.

PROP. XIV.

THE SPACES DESCRIBED BY FALLING BODIES ARE AS THE SQUARES OF THE TIMES OF THEIR FALLING FROM REST.

For by Postul. 3. gravity is a uniformly accelerating force; therefore, (by Prop. VI.) the space described is as the time and velocity. But by the last Prop. the time is as the velocity; and, therefore, the space described is as the square of the time.

Cor. 1. The spaces described by falling bodies are also as the squares of the velocities; or the velocity is as the square root of the height fallen.

Cor. 2. Taking any equal parts of time; then the spaces described by a falling body, in each successive part of time, will be as the odd numbers, 1, 3, 5, 7, 9, 11, &c.

For, in the times 1, 2, 3, 4, &c., the spaces described will be as their squares 1, 4, 9, 16, &c. And, therefore, in the differences of the times, or in these equal parts of time, the spaces described will be as the differences of the squares, or as 1, 3, 5, 7, &c.

Cor. 3. A body moving with the velocity acquired by falling through any space, will describe twice that space in the time of its fall. By Cor. 1. Prop. VI.

Cor. 4. If a body be projected upwards with the velocity it acquired in falling, it will, in the same time, ascend to the same height it fell from; and describe equal spaces in equal times, both in rising and falling, but in an inverse order, and will have the same velocity at every point of the line described.

For by Cor. 2. of the last Prop. equal velocities will be gained or lost in equal times, (reckoning from the last moment of the descent.) Therefore, since, at the several correspondent points of time, the velocities will be equal, the spaces described in any given time will be equal, and the wholes equal.

Cor. 5. If bodies be projected upwards with any velocities, the

heights of their ascent will be as the squares of the velocities, or as the squares of the times of their ascending.

For, in descending bodies, the spaces descended are as the squares of the last velocities, by Cor. 1. And by Cor. 4. the spaces ascended will be equal to those descended.

Cor. 6. If a body is projected upwards with any velocity, with the same velocity undiminished, it would describe twice the space of its whole ascent, in the same time. By Cor. 3. and 4.

Cor. 7. Hence, also, all bodies from equal altitudes descend to the surface of the earth in equal times.

SCHOLIUM. It is known by experiments, that a heavy body falls 16 feet in a second of time, and acquires a velocity which will carry it over 32 feet in a second; which being known, the spaces described in any other times, and the velocities acquired, will be known by the foregoing propositions; and the contrary. These propositions are exactly true, where there is no resistance to hinder the motion; but because bodies are a little resisted by the air, descended bodies will be a little longer in falling; and a body projected upwards, will be something longer in descending than in ascending, and falls with a less velocity; and, consequently, a body projected upwards with the velocity it falls with, will not ascend quite to the same height; but these errors are so small, that, in most cases, they may safely be neglected.

If the force by which a body is accelerated in falling was directly as the height fallen from; it may be computed (by Cor. 2. Prop. V.) that the velocity acquired will also be as the height; or the space described directly as the velocity. And, therefore, if bodies were projected upwards, they would in this case ascend to heights, which are as the velocities with which they are projected. This being compared with Cor. 5. of the last Prop. it is easy to conclude that bodies projected upwards, and acted upon by a force which is neither of a given quantity, nor in proportion to the distance of the body from the top of the ascent, but between them; that these bodies will then ascend to heights which are between the simple and duplicate ratio of the velocities.

And as these propositions lead us to the knowledge of the relation between the velocities and spaces described, from the forces being given, so, vice versa, from that relation being given, the forces may be known. Whence, if bodies are projected with any velocities into a resisting medium, and the spaces described within that body, measured, the constitution of that body, and the law of its resistance, will be found.