if a body be put in motion by a certain force, and another force operate upon it in a contrary direction, so as to tend to bring it to a state of equilibrium, such motion is called retarded motion.*

*The simplest example of accelerated motion, is exhibited in the action of the force of gravitation upon a falling body, where the force continues to act during its descent, and regularly increases in velocity; so that if a body A, fig. 3, be allowed to fall from that position towards the earth, it will pass through sixteen feet during the first second of time, forty-eight feet during the next, and eighty feet during the third. Had its motion been regular during these three seconds of time, it would have passed through only three times sixteen, or forty-eight feet, whereas it has passed through one hundred and forty-four feet, by reason of the force which first caused its motion continuing to act upon it. Now as its velocity increases regularly, we may conclude, that during the performance of the first half of the sixteen feet, it was not proceeding at the rate of sixteen feet per second; and if we suppose it was proceeding only at half that velocity, then it must have travelled through the second half at the rate of thirty-two feet per second; or, if the first eight feet took three quarters of the second, the second eight feet must have been performed in the remaining quarter, therefore, when the body arrived at B, it would be proceeding at the rate of thirty-two feet per second; to which, if we add the force that continues to urge it at the rate of sixteen feet per second, it will exhibit, for the second space, a velocity of fortyeight feet per second: and if for the third space we double its increasing velocity of thirty-two feet, and add that created by the continued force, we shall have twice thirty-two, and sixteen, are eighty, which is the result of experiment. The velocity of bodies under the continuous action of any given force, will, it appears, increase as the odd numbers 1, 3, 5, 7, 9, &c., that is, sixteen feet during the first second, thrice sixteen feet during the next second, five times sixteen during the third second, and so on; or, as the relative portions of the superficial space under equal parts of the perpendicular in a right angled triangle, as represented at fig. 3: where 0 to 1 represents the first second of time, 1 to 2 the second, and 2 to 3 the third. It will be perceived, that under each of these portions, the space contained in the triangle will be as 1, 3, 5; such is uniform accelerated motion. But if the continuous force, which has been shown to increase the velocity, vary in its action upon the body, it is plain the increase will be no longer uniform.

From a clear comprehension of the acceleration of motion in bodies the retardation of motion will be easily conceived: for example, if a body be cast perpendicularly from the earth, as in the firing of a shot from a cannon upwards, the force of the powder, overcoming the force of gravitation, will cause the ball to rise with a certain velocity, whilst that attraction continuing to operate in the opposite direction, checks by regular gradations the created force, and eventually destroys it. Thus the distance which the shot would have accomplished during the first second of time, is reduced by sixteen feet; that which it would have accomplished during the next second, by forty-eight; and so on until the created power is counterbalanced by the force of gravity, and the ball arrives at a state of rest; when the force of gravity acting upon it solely, will cause it to move in the opposite direction, till it descends to the earth

When a ball, attached to a centre by a flexible cord, is put in motion by any one force, which, in common with all other forces, acts in a right line, the motion will be circular. The tendency which such body has to fly from the centre, is called the centrifugal force; and that exerted by the cord to draw it towards the centre, the centripetal force.

When a body is set in motion by any force, it is enabled, to a certain extent, to act on other bodies, and create motion in them; and, as the velocity it obtained was as the power expended to create that motion, so is the power of transmitting that motion to its velocity. This power of communicating motion, or, in other words, this force possessed by matter in motion, is termed momentum, or the moving force; and the mode of transmitting it, impact: as this force is proportional to the velocity possessed by every particle of matter composing any body, the momentum must be represented by the quantity of matter multiplied by its velocity. For instance, suppose one hundred particles of matter were moving at the rate of one foot per second, the power requisite to overcome their force is exactly the same as that which would be necessary to arrest the motion of one particle moving at the rate of one hundred feet per second: for the velocity of the hundred particles being one foot per second each, their total force would be the force existing in one of them multiplied by one hundred: and again, as the force is in proportion to the velocity, one particle moving at the rate of one foot per second, multipled by one hundred in regard to velocity, will produce a similar result. Also, if a body of one pound weight be moving at the rate of one foot per second, it will possess a certain momentum, and if either its weight or its velocity be doubled, its momentum will be likewise doubled: if both be doubled, the momentum will be quadrupled.

Having now considered the action of one and two forces acting together in opposite and similar directions, we will proceed to examine the action of two forces upon a body, acting neither in the same, nor in contrary directions. Thus, if the line A B, fig. 4, represent a force sufficient to carry the body A to the point B, and AC represent another force sufficient to carry the body A to the point C, then AC and A B being equal to CD and BD, and those two forces act upon the body subsequently to each other, we may conceive that the body would, by passing over the lines A B and B D, or AC and CD, be carried to the point D. Now, if they act upon the body at the same instant,

the result will be the same, and the total expenditure of the forces will place the body, passing by the line AD, at the point D. Likewise, if the forces A B and A C be not at right angles, as in fig. 5, still as CD and BD are equal, and in similar directions to A B and A C, the motion received from them by A will be represented in amount and direction by the line A D. But supposing A B shall be twice or thrice the power of A C, then the effect will be the same as is shown in fig. 6, where the line AB represents thrice the power of A C. The separate actions of A B and AC will be represented as before by BD and CD, which would place the body A at the point D; therefore their combined force will cause it to pass by the diagonal line AD, as in the former instance. This proves that any number of forces acting upon a body in however many lines, not directly opposite to each other, will be compounded into one force: for suppose three forces, AB, AC, and A F, fig. 7, to operate in their several directions at the same instant, on the body A, they will be compounded into the force represented by AI; for if we describe a parallelogram as before by the lines A B and AC, those two forces will be compounded into a force represented by AD; and again, if we do the same with the two forces AC and A F, we shall have the force A H composed of them. We have therefore two forces AD and AH compounded of the three original forces. If we proceed with these two in the same manner, they will be compounded into the force represented by AI; DI and HI completing the parallelogram of which AI is the diagonal: so that any number of forces acting in any number of directions, excepting in opposite ones, may be compounded into one, which is termed their composant, and which is always represented by the diagonal of a parallelogram, like that already shown.

The resolution of forces is exhibited by reversing this problem; for as any number of forces may be combined into one force, so may one force be resolved into any number. If a single force be represented by a ball moving with a certain velocity in the direction of the line A B, fig. 8, when it shall come in contact with and act upon the balls C and D, these two balls will each of them move with one half of the velocity with which B was impelled, and in the direction of the lines C H and DI, drawn from the centre of B through each of their centres: so that if the force of B be divided into two equal portions, each of those portions may, by a similar process, be again divided, resolving the original force to infinity.

The next effect of forces upon bodies producing motion, is that in which a body receives motion from one force, whilst it is under the continuous action of another force, not acting upon it in an opposite direction. Suppose the ball A, fig. 9, to be ejected from the mouth of a cannon, the instant it has left it at A, it will be under the influence of the force of gravitation, which will cause it to descend towards the earth in the manner already shown when speaking of accelerated motion, and ultimately will bring it to a state of rest at the point B: for supposing that the ball, by the force of the powder, leaves A, and travels in the first second of time a given number of feet, expressed by the line A C, the gravitating force during such action will cause it to descend sixteen feet, expressed by the line C D; and during the next second, supposing the powder to have impelled it the distance expressed by the line D E, the gravitating force will cause it to fall forty-eight feet, as is shown by E F; and during the next portion of its horizontal motion, expressed by F G, its descent by gravitation will amount to eighty feet, represented by G B. The line, therefore, in which the body would move when acted upon by these two forces only, would be that of a parabolic curve; but as the resistance of the air is to be taken into account in all practical cases, the line of motion changes very considerably, and assumes one that involves a problem of exceeding complexity; which, together with many other results of the effects of combined forces, is of such intricacy as to demand much more room for their solution than the limits of this work will permit us to give.

OF FRICTION.

THE surfaces of bodies, however smooth they may appear to be, will be found, upon a minute inspection, to possess certain irregularities: so that if the body A B, fig. 10, have to move upon the surface of the body CD, and the lower surface of A B possesses prominences which enter into cavities in C D, it is manifest that A B cannot be moved along unless it either rises and falls the height of the several prominences, or breaks them off: in the first, it will have to overcome the attraction of gravitation; in the second, the attraction of cohesion. Again, if the body A B, fig. 11, be placed between CD and E F, which are pressed against its sides by any applied force, and their surfaces be similar to those in the former instance, to effect the movement of AB, the attraction of cohesion must be overcome, as before shown, or the applied force must be conquered. Such is the

almost universal nature of that resistance called friction; for although the irregularities upon the surfaces of bodies are by no means so manifest as those here represented, still, upon minute examination, we are enabled to discover that the smoothest surfaces contain them; and as the amount of resistance increases in direct proportion as the irregularities present themselves, we are warranted in concluding that all resistance arising from friction owes its origin solely to this cause.

OF THE MECHANICAL POWERS.

THE mechanical powers are six in number, the LEVER, the WHEEL and AXLE, the PULLEY, the INCLINED PLANE, the WEDGE, and the SCREW. A perfect knowledge and thorough appreciation of which should be clearly understood by those who purpose to examine into the effects of mechanical combinations; the whole of which, however intricate, originate from, and are reducible to, one or more of the laws which govern these simple machines.

In demonstrating the mechanical powers, that which is not strictly true must be admitted: the force of gravitation, the retardation of friction, the resistance of the atmosphere, and the irregularity arising from the partial elasticity of the substances of which they are formed, must be excluded, and supposed not to exist.

The first-mentioned power is the lever, which is divided into three classes. In fig. 12, A B is a lever, and C the fulcrum, or immovable point on which it rests: now, if a force be applied at B, and the resistance, or the force or weight to be overcome, is at A, then, with the fulcrum so situate between the forces, it is called a lever of the first class; and the operation of the force at B to overcome the resistance at A, will be in proportion as the distance A Cis to the distance BC; that is to say, if B C be four times the distance of A C, the force applied at B will be exactly equal to four times the same amount of force at A; or one pound weight at B will counterbalance four pounds weight at A; but to whatever height (suppose one foot) the weight at A be raised, B must descend four times that space, and consequently, to place B in its original position, the force applied must be equal to the raising of four single pounds one foot each, which is the same as the raising of four pounds one foot, as was effected at A.

An actual gain of power does not exist, but the gain in convenience is great; for, by the operation of one pound, four pounds is moved, which, but for the invention of the