with universal experience. They were assumed by Sir Isaac Newton as the fundamental principles of mechanicks; and the theory of all motions deduced from them, as principles, being found to agree, in all cases, with experiments and observations, the laws themselves are considered as mathematically true.

CHAP. II.

Of the Comparison of uniform Motions.

PROP. IV. The quantities of matter in bodies are in the compound ratio of their magnitudes and densities.

If the magnitudes of two bodies be given, the quantities of matter will be as the densities: If their densities be given, the matter will be as the magnitudes: therefore the matter is universally in the compound ratio of the magnitudes and densities. For example; If A and B be two balls equal in magnitude, the quantity of matter in A will be to that in B, as the density of A is to that of B: if both be of the same density, their quantities of matter will be as their magnitudes.

PROP. V. The velocities with which bodies move, are directly as the spaces they describe, and inversely as the times in which they describe these

spaces.

verse.

It is manifest, that the degree of velocity increases as the space a body passes over in a given time increases, and as the time in which it passes over a given space decreases; and the reFor example; If one ball A move over six feet, and another ball B over three feet in the same time, A has double the velocity of B: but if the ball A passes over six feet in two seconds of time, and the ball B passes over six feet in one second, the velocity of B is double of that of A.

PROP. VI. The spaces which bodies describe are in the compound ratio of their times and velocities.

It is evident, that the longer time any body continues to move, and the greater velocity it moves with, the greater space it will pass through; and the reverse. If, for example, the body A moves for one second, and the body B moves for two seconds, both moving with the same velocity, A will move through half as much space as B: If A moves with two degrees of velocity, and B with one degree of velocity, A will, in the same time, pass over twice as much space as B.

PROP.

PROP. VII. The times in which bodies move are directly as the and inversely as the velocities.

spaces,

The greater space any body passes through, and the less degree of velocity it moves with, the greater must be the portion of time taken up in the motion; and the reverse. For example; If the ball A moves with the same velocity with the ball B, but passes over double the space, A will move twice as long as B; If A moves over the same space with B, and with half the velocity, it must, in this case also, move twice as long as B.

PROP. A. If bodies be acted upon by different constant forces, the velocities communicated will vary in a ratio compounded of the forces and times.

Let F, V, T, represent force, velocity and time, and be supposed variable; it is evident that the velocity will be increased and diminished in the same ratio with both force and time, and these being independent of each other, V will be as FXT.

COR. If, therefore, F be compared with any other known force ƒ capable of generating a velocity equal to v in the time t, then V: v:: FXT: ƒxt.

PROP. VIII. The power required to move a body at rest is as the quantity of matter to be moved.

Each particle of matter in any body resisting motion, a force must be exerted upon each particle to overcome this resistance; if, therefore, two bodies containing different quantities of matter are to be moved, the greater body will require the greater force.

DEF. I. The momentum of any body is its quantity of motion.

PROP. IX. In moving bodies, if the quantities of matter be equal, the momenta will be as the velocities.

It is manifest, that if the body A be equal to the body B, but A has twice the velocity of B, A has twice as much motion as B.

PROP. X. The velocities of two bodies being equal, their momenta will be as their quantities of matter.

The bodies A and B moving with equal velocities, since every portion of matter in A has as much motion as an equal portion of B, it is evident, that if A has twice the quantity of matter in B, it must have twice as much motion.

PROP. XI. The momenta of moving bodies are in the compound ratio of their quantities of matter and velocities.

The greater quantity of matter there is in any body, and the greater velocity it moves with, the greater will evidently be its quantity of motion; and the reverse.

If, for

example,

example, the body A be double of the body B, and moves with twice its velocity, the momentum of A will be quadruple of that of B: For it will have twice the momentum of B from its double velocity, and also twice the momentum of B from its double quantity of

matter.

COR. Hence, if in two bodies the product of the quantities of matter and velocities are equal, their momenta are equal.

PROP. XII. The velocities of moving bodies are as their momenta directly, and their quantities of matter inversely.

The greater momentum any body has, and the less quantity of matter it contains, the greater must be its velocity. For example; If the body A is half of B, and their momenta are equal, A will move with twice the velocity of B; and if A and B are equal, and the momentum of A is double of that of B, its velocity will also be double.

PROP. XIII. The force, or power of overcoming resistance, in any moving body, is as its momentum.

Since a body having a certain degree of motion is able to overcome a certain degree of resistance, it is manifest, that with an increased momentum, it will be able to overcome a greater resistance.

COR. Hence the momentum of any body is measured by its power of overcoming resistance.

SCHOL. Let 2, q, denote the quantities of matter in any two bodies, D, d, their densities, and B, b, their bulk or magnitude, V, v, their velocities, T, t, the times of their motion, S, s, the spaces over which they pass, P, p, the moving powers, M, m, their momenta, and F, f, their force: The preceding propositions may be thus expressed :

Plate 1.
Fig. 4.

CHAP. III.

Of the Composition and Resolution of Forces.

DEF. A. Simple motion is that which is produced by the action, or impressed force of one cause. Compound motion is that which is produced by two or more conspiring powers, i. e. by powers whose directions are neither opposite nor coincident.

Equable motion is either simple or compound.

PROP. XIV. A body acted upon by two forces united, will describe the diagonal of a parallelogram, in the same time in which it would have described its sides by the separate action of these forces.

If in a given time, a body, by the single force M impressed upon it at the point A, would be carried from A to B; and by another single force N impressed upon it at the same point, would be carried from A to C; complete the parallelogram ABDC; and with both forces united, the body will be carried in the same time through the diagonal of the parallelogram from A to D. For since the force N acts in the direction of the right line AC parallel to BD, this force (by Prop. II.) has no effect upon the velocity with which the body approaches towards the line BD by the action of the force M. The body will therefore arrive at the line BD in the same time, whether the force N is impressed upon it or not; and at the end of that time will be found somewhere in the line BD. For the same reason

at the end of the same time it will be found somewhere in the line CD; therefore it must be found at the point D, the intersection of these two lines. And (by Prop. I.) it will move in a right line from A to D.

EXP. Two equal leaden weights, suspended at the end of a triangular frame of wood to give them a steady motion, and let fall at the same instant from equal heights, striking a ball suspended by a cord at the point in which their lines of direction meet, will carry it forwards in the diagonal of the parallelogram of those lines produced.

COR. I. Hence, the velocity produced by the joint action of two forces is to that with which the body moves by the action of each force singly, as the diagonal of the parallelogram to either side; for the diagonal is described in the same time with either side.

COR. 2. If two sides of a triangle represent the directions and quantities of two forces, the third side will represent the direction and quantity of a force equivalent to both acting jointly For the third side may be considered as the diagonal of a parallelogram.

COR.

COR. 3. A body may be moved through the same line by different pairs of forces. In plate 1. fig. 4. AD is the diagonal both to the parallelogram ABCD, and to the parallelogram AEDF; and consequently expresses a force equal to AB, AC, and to AF, AE.

COR. 4. Hence we learn why any heavy body let fall perpendicularly from the top of a mast, when a ship is under full sail, will fall to the bottom, in the same manner as if it had been

at rest.

SCHOLIUM. This proposition may be farther illustrated. If two men sit upon the opposite sides of a boat in full sail, and toss a ball to one another, they will catch the ball in their turn, just as they would have done if the boat had been at rest. The ball is here acted upon by two forces: (1.) it partakes of the motion of the boat, which is common to the ball, the boat and the men: (2.) the other force is that with which the man throws it across the boat. By these two forces together, the ball will describe the diagonal of a parallelogram, one of whose sides is the line that the boat has described whilst the ball is flying across; and the other side is a line drawn from one man to the other.

PROP. XV. The velocity produced by two joint forces, when they act in the same direction, will be as the sum of the forces, and when they act in opposite directions, will be as their difference; and the velocity will be the greater the nearer they approach to the same direction, and the reverse.

In the parallelograms ABCD, in which AB, AC, express the direction and quantity of two Plate 1. joint forces, the side AB being placed at different angles with AC, it is manifest, that as AB Fig. 5. approaches towards AC, the diagonal increases, till at length it becomes equal to AC, CD, that is, to AC, AB, and the velocity is as the sum of the forces, since they act in the same direction.

In the parallelograms ABCD, as AB recedes from CD, the diagonal decreases, till at length Plate 1. it vanishes with the angle, and the two sides AB, AC, constitute one right line, the parts of Fig. 6. which, AB, AC, representing forces acting in opposite directions, if the forces be equal, they will destroy each other; if unequal, the velocity will be as their difference.

PROP. XVI. Any single force or motion may be resolved into two forces or motions; and the directions of these may be infinitely varied: also any two forces may be compounded into single forces.

A body moving in the line AD, may be considered as receiving its direction and velocity Plate 1. from two forces acting jointly in the directions AB, AC, or from two other forces expressed Fig. 4. by AF, AE: For (Prop. XIV. Cor. 3) each pair would produce the same effect. In like

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manner