Plate 1.

Fig. 7.

Plate 1.

Fig. 8.

manner the direction and quantities of the forces will be diversified with every change of the sides of the parallelogram, the diagonal remaining the same.

It is also manifest, that any two joint forces may be compounded into one, being expressed by the sides of a parallelogram, or its diagonal.

PROP. XVII. If a body is acted upon by three forces, which are proportional to, and in the direction of, the three sides of a triangle, the body will be kept at rest.

Let a body placed at D be acted upon by three forces AD, GD, FD, proportional to, and in the direction of, the three sides of the triangle GED: complete the parallelogram GEFD; and make AD equal to, and in the direction of, the diagonal ED.

If the body at D be acted upon by the forces AD, ED, equal and in opposite directions, it will be kept at rest. But the force ED (Prop. XVI.) is equivalent to the two forces DG, DF, that is, DG, GE; therefore the body acted upon by the three forces AD, DG, DF, that is, by three forces proportional to, and in the direction of, the sides of the triangle GED, will be

at rest.

EXP. Let three weights in the proportions of 3, 4, 5, be suspended from cords, which pass over pullies and meet in a point; if the directions of the cords be parallel to the sides of a triangle (drawn in a plane parallel to the plane of the cords) whose sides are to each other as the weights, a ball at the point in which the cords meet will be kept at rest.

COR. The body will be at rest if the three forces are proportional to the three sides of a triangle drawn perpendicular to the direction of the forces; for such a triangle is similar to the former Draw Ag, Cd, and Be, perpendicular to the sides GE, GD, DE, forming a triangle ged, which is equiangular to GED; hence, the sides about their equal angles being proportional, the forces which are proportional to the lines GE, GD, and DE, are also proportional to ge, gd, and d e.

SCHOLIUM. A boy's kite, as it rests in the air, is an instance of a body resting whilst three forces act upon it. For the kite is acted upon by the wind; by its own weight; and by the string that holds it.

PROP. XVIII. The force of oblique percussion is to that of direct or perpendicular action, as the sine of the angle of incidence to radius.

Let a body strike upon the plane AD, at the point D, in the direction BD: the line BD expressing the force of direct impulse may be resolved into two others, BC, BA, the one parallel, the other perpendicular to the plane: Of these, the force BC, parallel to the plane, cannot af fect it: The whole force upon the plane may therefore be expressed by BA. But B is to BD as the sine of the angle of incidence BDA is to radius.

SCHOL.

SCHOL. If the surface to be struck be a curve, let AD be made tangent to the curve at D, and the proof will be the same.

PROP. XIX. The force of oblique action produced by percussion is to that of direct action, as the cosine of the angle, comprehended between the direction of the force and that in which the body is to be moved, to radius.

Let FD represent a force acting upon a body at D, and impelling it towards E; but let DM Plate I. be the only way in which it is possible for the body to move. The force FD may be resolved Fig. 9. into two forces FG, FH, or GD; of which only the force GD impels it towards M. And, FD being radius, GD is the cosine of the angle FDG, or MDE, comprehended between the direction of the force, and that in which the body is to be moved.

CHAP. IV.

Of Motion, as communicated by Percussion in Non-Elastick and Elastick Bodies.

DEF. II. Bodies are non-elastick, which, when one strikes another, do not rebound, but move together after the stroke.

COR. Hence their velocities after the stroke are equal.

DEF. III. Bodies are elastick, which have a certain spring, by which their parts, upon being pressed inwards by percussion, return to their former state, throwing off the striking body with some degree of force; when the elasticity is perfect, the body restores itself with a force equal to that with which it is compressed.

EXP. The existence of this property is visible in a ball of wool, cotton, or sponge com. pressed.

PROP. XX. When one non-elastick body in motion, strikes upon another at rest, or moving with less velocity in the same direction, the sum of their momenta remains the same after the stroke as before.

For (Prop III Cor 1.) as much motion as the striking body communicates, so much it loses; whence, if the motions of the bodies are in the same direction, whatever is added to the

Plate 1.

Fig. 7.

Plate 1.
Fig. 8.

manner the direction and quantities of the forces will be diversified with every change of the sides of the parallelogram, the diagonal remaining the same.

It is also manifest, that any two joint forces may be compounded into one, being expressed by the sides of a parallelogram, or its diagonal.

PROP. XVII. If a body is acted upon by three forces, which are proportional to, and in the direction of, the three sides of a triangle, the body will be kept at rest.

Let a body placed at D be acted upon by three forces AD, GD, FD, proportional to, and in the direction of, the three sides of the triangle GED: complete the parallelogram GEFD; and make AD equal to, and in the direction of, the diagonal ED.

If the body at D be acted upon by the forces AD, ED, equal and in opposite directions, it will be kept at rest. But the force ED (Prop. XVI.) is equivalent to the two forces DG, DF, that is, DG, GE; therefore the body acted upon by the three forces AD, DG, DF, that is, by three forces proportional to, and in the direction of, the sides of the triangle GED, will be

at rest.

EXP. Let three weights in the proportions of 3, 4, 5, be suspended from cords, which pass over pullies and meet in a point; if the directions of the cords be parallel to the sides of a triangle (drawn in a plane parallel to the plane of the cords) whose sides are to each other as the weights, a ball at the point in which the cords meet will be kept at rest.

COR. The body will be at rest if the three forces are proportional to the three sides of a triangle drawn perpendicular to the direction of the forces; for such a triangle is similar to the former Draw Ag, Cd, and Be, perpendicular to the sides GE, GD, DE, forming a triangle. ged, which is equiangular to GED; hence, the sides about their equal angles being proportional, the forces which are proportional to the lines GE, GD, and DE, are also proportional to ge, gd, and d e.

SCHOLIUM. A boy's kite, as it rests in the air, is an instance of a body resting whilst three forces act upon it. For the kite is acted upon by the wind; by its own weight; and by the string that holds it.

PROP. XVIII. The force of oblique percussion is to that of direct or perpendicular action, as the sine of the angle of incidence to radius.

Let a body strike upon the plane AD, at the point D, in the direction BD: the line BD expressing the force of direct impulse may be resolved into two others, BC, BA, the one parallel, the other perpendicular to the plane: Of these, the force BC, parallel to the plane, cannot affect it: The whole force upon the plane may therefore be expressed by BA. But B is to BD as the sine of the angle of incidence BDA is to radius.

SCHOL.

SCHOL. If the surface to be struck be a curve, let AD be made tangent to the curve at D, and the proof will be the same.

PROP. XIX. The force of oblique action produced by percussion is to that of direct action, as the cosine of the angle, comprehended between the direction of the force and that in which the body is to be moved, to radius.

Fig. 9.

Let FD represent a force acting upon a body at D, and impelling it towards E; but let DM Plate I. be the only way in which it is possible for the body to move. The force FD may be resolved into two forces FG, FH, or GD; of which only the force GD impels it towards M. And, FD being radius, GD is the cosine of the angle FDG, or MDE, comprehended between the direction of the force, and that in which the body is to be moved.

CHAP. IV.

Of Motion, as communicated by Percussion in Non-Elastick and Elastick Bodies.

DEF. II. Bodies are non-elastick, which, when one strikes another, do not rebound, but move together after the stroke.

COR. Hence their velocities after the stroke are equal.

DEF. III. Bodies are elastick, which have a certain spring, by which their parts, upon being pressed inwards by percussion, return to their former state, throwing off the striking body with some degree of force; when the elasticity is perfect, the body restores itself with a force equal to that with which it is compressed.

EXP. The existence of this property is visible in a ball of wool, cotton, or sponge com. pressed.

PROP. XX. When one non-elastick body in motion, strikes upon another at rest, or moving with less velocity in the same direction, the sum of their momenta remains the same after the stroke as before.

For (Prop III Cor 1.) as much motion as the striking body communicates, so much it loses; whence, if the motions of the bodies are in the same direction, whatever is added to the

the motion of the preceding body will be subducted from that which follows, and the sum will remain the same.

PROP. XXI. When two non-elastick bodies. moving in an opposite direction, strike upon each other, the sum of their momenta, after the stroke, will be equal to the difference of their momenta before the stroke.

For (from Prop III. Cor. 1.) that body which had the least motion will destroy a quantity equal to its own in the other; after which they will move together with the remainder, that is, the difference.

EXP. Let two cylinders, filled with clay, A, B, be of equal weight, and suspended by cords from equal heights; let two other cylinders of the same kind, C, D, but in weight as 2 to 1, be suspended from the same height The heights from which they are let fall, in the arc formed by the motion of the cylinder (from the nature of the pendulum, afterwards to be explained) will be the measure of their velocity; and (by Prop. XI.) their momenta will be as their velocities multiplied into their quantities of matter; whence the cases of the two preceding propositions may be established by the following experiments. N. B. Quantity of matter is expressed by q, velocity by v, and momentum by m.

No. 1. Prop. XX. Case 1. Let the cylinder A fall from the height of 18 inches, upon the cylinder B at rest. The momentum of A before the stroke (by Prop. XI.) is 18; for the quantity of matter is 1. and the velocity 18; whence q xv 18 = m 18. After the stroke, the quantity of matter being (Def. II.) 2, and the velocity of each cylinder 9, the momentum will be 18: q 2 × v9 = m 18.

No. 2. Case 2. Let A fall from 18 inches, and B from 9, in the same direction; their momenta before the stroke are 18+ 9 = 27; after the stroke, the quantity of matter will be 2, and the velocity 13; whence v 13 x 92 = m 27.

No. 3. Prop. XXI. Case 1. Let the equal cylinders A and B fall in opposite directions, from the height of 12 inches; the momenta being equal and opposite, the motion of both will be destroyed.

No. 4. Case 2. Let A fall from the height of 12 inches, and meet B falling in the opposite direction from 6 inches; their velocity after the stroke being 3, and quantity of matter 2, the momentum will be 6; q 2 x v 3 = m 6.

No. 5. Prop. XX. Case 3. Let the cylinder C, double of the cylinder D, fall from 12 inches on D at rest. Before the stroke, the quantity of matter in C is 2, and its velocity is 12; whence its momentum is 24; q 2 x v 12m 24. After the stroke, the velocity will be 8, and quantity of matter 3; whence q 3 x v8=m 24.

No.