Cor. 1. (Fig. 1. Pl. I.) The forces, in the directions AB, AC, AD, are respectively proportional to the lines, AB, AC, AD.

For by Cor. Prop. IV. the time and the quantity of matter being given; the force is directly as the space described. And in accelerated motion, the same is true, by Prop. V. Cor. 3. and Prop. VI.

Cor. 2. The two oblique forces, AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal of the parallelogram AD.

Cor. 3. Any single direct force AD, may be resolved into the two oblique forces whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram whose diagonal is AD.

Cor. 4. A body being agitated by two forces at once, will pass through the same point, as it would do if the two forces were to act separately and successively. And if any new motion be impressed on a body already in motion, it does not alter its motion in lines parallel to its former direction.

Cor. 5. (Fig. 2. Pl. I.) If two forces, as AB, AC, act in the directions AB, AC, respectively, draw AR to the middle of the right line BC, and 2 AR is the force compounded out of these, and AR its direction.

PROP. VIII. (Fig. 3. Pl. I.)

LET THERE BE THREE FORCES, A, B, C, OF THE SAME KIND, ACTING AGAINST ONE ANOTHER, AT THE POINT D, AND WHOSE DIRECTIONS ARE ALL IN ONE PLANE; AND IF THEY KEEP ONE ANOTHER IN EQUILIBRIO, THESE FORCES WILL BE TO EACH OTHER RESPECTIVELY, AS THE THREE SIDES OF A TRIANGLE DRAWN PARALLEL TO THEIR LINES OF DIRECTION, DI, CI, CD.

Let DC represent the force C, and produce AD, BD, an complete the parallelogram DICH: And by the last Prop. the force DC is equivalent to the two forces DH, DI; put, therefore, the forces DH, DI instead of DC, and all the forces will still be in equilibrio. Therefore, by Ax. II., DI is equal to its opposite force A, and DH or CI equal to its opposite force B. Therefore, the three forces A, B, C are respectively as DI, CI, CD.

Cor. 1. Hence the forces, A, B, C are respectively as the three sides of a triangle, drawn perpendicular to their lines of direction, or in any given angle to them, on the same side. For such a triangle will be similar to the former triangle.

Cor. 2. The three forces ABC will be to each other as the sines of the angles through which their respective lines of direction do pass, when produced:

For DI: CI:: S.DCI or CDB : S.CDI or CDA. And CI: CD :: S.CDI or CDA : S.CHD or HDI or ADB. Cor. 3. If there be ever so many forces acting against any point in one plane, and keep one another in equilibrio, they may be all reduced to the action of three, or even of two equal and opposite ones.

For if HD, ID be two forces, they are equivalent to the single force DC. and, in like manner, A and B may be reduced to a single force.

Cor. 4. And if ever so many forces in different planes, acting against one point, keep one another in equilibrio, they may be all reduced to the actions of several forces in one plane, and, consequently, to two equal and opposite ones.

For if the four forces, A, B, H, I, act against the point D, and H, I, be out of the plane ABD, let DC be the common section of the planes ADB, HDI; then the forces H, I, are reduced to the force C, in the plane ADB.

SCHOLIUM.-This Prop. holds true of all forces whatever, whether of impulse or percussion, thrusting, pulling, pressing, or whether instantaneous or continual, provided they be all of the same kind.

Hence, if three forces act in one plane, their proportions are had; and if one force be given, the rest may be found. And if four forces act, and two be given, the other two may be found; but if only one be given, the rest cannot be found; for in the three forces, A, B, C, the force C may be divided into other two, an infinite number of ways, by drawing any parallelogram DICH about the diagonal DC; and, in general, if there be any number of forces acting at D, and all be given but two, these two may be found; otherwise not, though the positions of them all be given.

PROP. IX. (Fig. 4. Pl. I.)

IF ONE BODY ACTS AGAINST ANOTHER BODY, BY ANY KIND OF FORCE WHATEVER, IT EXERTS THAT FORCE IN THE DIRECTION OF A LINE PERPENDICULAR TO THE SURFACE WHEREON IT ACTS.

Let the body B be acted on by the force AB in the direction AB. Let the body C and the obstacle O, hinder the body B from moving; divide the force AB into the two forces AD, AE, or EB, DB, by Cor. 5, Prop. VII. the one perpendicular, the other parallel to the surface DB; then the surface DB receives the perpendicular force EB, and the obstacle O the parallel force DB; take away the obstacle O, and the force DB will move the body B in a direction parallel to the surface, with no other effect than what arises from the friction of B against that surface, occasioned

by the pressure of B against it by the force EB; which, if the surface be perfectly smooth and void of tenacity, will be nothing. The force DB, therefore, having no effect, the remaining force EB will be the only one whereby the body B acts against the surface DB, and that in direction EB, perpendicular to it.

Cor. 1. If a given body B strike another body C obliquely, at any angle ABD, the magnitude of the stroke will be directly as the velocity and the sine of the angle of incidence ABD; and the body C receives that stroke in the direction EB perpendicular to the surface

DB.

For if the angle ABD be given, the stroke will be greater in proportion to the velocity; and if the velocity AB be given, the stroke will be as AD, or S. Z ABD; or the magnitude of the stroke is as the velocity wherewith the body approaches the plane.

Cor. 2. (Fig. 5. Pl. I.) If a perfectly elastic body A impinges on a hard or elastic body CB at B, it will be reflected from it, so that the angle of reflection will be equal to the angle of incidence.

For the motion at B parallel to the surface is not at all changed by the stroke; and, because the bodies are elastic, they recover their figure in the same time they lose it by the stroke; therefore, the velocity in direction BE is the same after as before the stroke. Let AE, BE, represent the velocities before the stroke, and ED (AE) and BE the respective velocities after the stroke; then in the two similar and equal triangles, AEB, BED, ▲ ABE is equal to EBD.

But since no bodies in nature are perfectly elastic, they are something longer of regaining their figure; and, therefore, the an gle DBF will be something more acute than the ABG.

Cor. 3. If one given body impinges upon another given body, the magnitude of the stroke will be as the relative velocity between the

bodies.

For the magnitude of the stroke is as the line BE, (Fig. 4. Pl. I.) or the velocity wherewith the bodies approach each other; that is, as the relative velocity.

Cor. 4. And in any bodies whatever, if a body in motion strike against another, the magnitude of the stroke will be as the motion lost by the striking body.

For the motion impressed on the body that receives the stroke is equal to the magnitude of the stroke; and the same motion by Ax. 3, is equal to the motion lost in the striking body.

Cor. 5. A non-elastic body striking another non-elastic body, only loses half as much motion as if the bodies were perfectly elastic.

For the non-elastic bodies only stop; but elastic bodies recede with the same velocity they meet with.

Cor. 6. Hence, also, it follows, that if one body acts upon another, by striking, pressing, &c. the other re-acts upon this in the direction of a line perpendicular to the surface whereon they act. By Ax. 3.

SCHOLIUM. Though the momentum, or quantity of motion, in a moving body, is a quite distinct thing from the force that generates it, yet, when it strikes another body and puts it into motion, it may, with respect to that other body, be considered as a certain quantity of force proportional to the motion it generates in the other body.

Also, although the motion generated by the impulse of another body is considered as generated in an instant, upon account of the very small time it is performed in, yet, in mathematical strictness, it is absolutely impossible that any motion can be generated in an instant, by impulse or any sort of finite force whatever. For when we consider that the parts of the body which yield to the stroke, are forced into a new position, there will be required some time for the yielding parts to be moved through a certain space into this new position. Now, during this time, the two bodies are acting upon each other with a certain accelerative force, which, in that time, generates that motion, which is the effect of their natural impulse. So that it is plain that this is an effect produced in time; and the less the time, the greater the force; and if the time be infinitely small, the force ought to be infinitely great, which is impossible. But by reason that this effect is produced in so small a time as to be utterly imperceptible, so that it cannot be brought to any calculation, upon this account the time is entirely set aside, and the whole effect imputed to the force only, which is therefore supposed to act but for a moment.

The quantity of motion in bodies has been proved to be as the velocity and quantity of matter. But the momentum or quantity of motion may be the same in different bodies, and yet may have very different effects upon other bodies, on which they impinge. For if a small body with a great velocity impinge upon another body; and if, by reason of its great velocity, it act more strongly upon the small part of the body which it impinges, than the force of cohesion of the parts of that body; then the part acted on will, by this vigorous action, be separated from the rest; whilst, by reason of the very small time of acting, little or no motion is communicated to the rest of the body. But if a great body with a small velocity strike another body, and if, by reason of its slow motion, it does not act so vigorously as to exceed the force of cohesion, the part struck will communicate the motion to the rest of the body, and the whole body will be moved together. Thus, if a bullet be shot out of a gun, the momentum of the bullet and piece are equal; but the bullet will shoot through a board, and the gun will only jump a little against him that discharges it. Therefore small bodies, with great velocity, are more proper to tear in

pieces; and great bodies with small velocity, to shake or move the whole.

PROP. X.

THE SUM OF THE MOTIONS OF ANY TWO BODIES IN ANY ONE LINE OF DIRECTION, TOWARDS THE SAME PART, CANNOT BE CHANGED BY ANY ACTION OF THE BODIES UPON EACH OTHER, WHATEVER FORCES THESE ACTIONS ARE CAUSED BY, OR THE BODIES EXERT AMONG THEMSELVES.

Here I esteem progressive motions, or motions towards the same part, affirmative; and regressive ones, negative.

CASE I.

Let two bodies move the same way, and strike one another directly. Now, since action and re-action (by Ax. 3.) are equal and contrary, and this action and re-action is the very force by which the new motions are generated in the bodies, therefore, (by Ax. 2.) there will be produced equal changes towards contrary parts. And, therefore, whatever quantity of motion is gained by the preceding body will be lost by the following one; and, consequently, their sum is the same as before.

And if the bodies do not strike each other, but are supposed to act any other way, as by pressure, attraction, repulsion, &c., yet still, since action and re-action are equal and contrary, there will be induced an equal change in the motion of the bodies, and in contrary directions; so that the sum of the motions will still remain the same.

CASE II.

Suppose the bodies to strike each other obliquely; then, since (by the last Prop.) they act upon each other in a direction perpendicular to the surface in which they strike, the action and reaction in that direction being equal and contrary, the sum of the motions, the same way, in that line of direction, must remain the same as before. And since the bodies do not act upon each other in a direction parallel to the striking surface, therefore there is induced no change of motion in that direction. And therefore, universally, the sum of the motions will remain the same, considered in any one line of direction whatever. And if the bodies act upon one another by any other forces whatever, still (by Ax. 2, and 3,) the changes of motion will be equal and contrary, and their sum the same as before.

Cor. 1. The sum of the motions of any system or number of bodies, in any one line of direction, taken the same way, remains always the same, whatever forces these bodies exert upon each other; esteeming contrary motions to be negative, and, therefore,