[56.] I shall now consider the case of pendulums. A pendulum is made by hanging a weight to a line, so that it may swing backwards and forwards. This motion the geometers have very carefully considered, because it is the most commodious instrument of any for the exact measurement of time.
[57.] I have observed already[61], that if a body hanging perpendicularly by a string, as the body A (in fig. 48.) hangs by the string A B, be put so into motion, as to be made to ascend up the circular arch A C; then as soon as it has arrived at the highest point, to which the motion, that the body has received, will carry it; it will immediately begin to descend, and at A will receive again as great a degree of motion, as it had at first. This motion therefore will carry the body up the arch A D, as high as it ascended before in the arch A C. Consequently in its return through the arch D A it will acquire again at A its original velocity, and advance a second time up the arch A C as high as at first; by this means continuing without end its reciprocal motion. It is true indeed, that in fact every pendulum, which we can put in motion, will gradually lessen its swing, and at length stop, unless there be some power constantly applied to it, whereby its motion shall be renewed; but this arises from the resistance, which the body meets with both from the air, and the string by which it is hung: for as the air will give some obstruction to the progress of the body moving through it; so also the string, whereon the body hangs, will be a farther impediment; for this string must either slide on the pin, whereon it hangs, or it must bend to the motion of the weight; in the first there must be some degree of friction, and in the latter the string will make some resistance to its inflection. However, if all resistance could be removed, the motion of a pendulum would be perpetual.
58. But to proceed, the first property, I shall take notice of in this motion, is, that the greater arch the pendulous body moves through, the greater time it takes up: though the length of time does not increase in so great a proportion as the arch. Thus if C D be a greater arch, and E F a lesser, where C A is equal to A D, and E A equal to A F; the body, when it swings through the greater arch C D, shall take up in its swing from C to D a longer time than in swinging from E to F, when it moves only in that lesser arch; or the time in which the body let fall from C will descend through the arch C A is greater than the time, in which it will descend through the arch E A, when let fall from E. But the first of these times will not hold the same proportion to the latter, as the first arch C A bears to the other arch E A; which will appear thus. Let C G and E H be two horizontal lines. It has been remarked above[62], that the body in falling through the arch C A will acquire as great a velocity at the point A, as it would have gained by falling directly down through G A; and in falling through the arch E A it will acquire in the point A only that velocity, which it would have got in falling through H A. Therefore, when the body descends through the greater arch C A, it shall gain a greater velocity, than when it passes only through the lesser; so that this greater velocity will in some degree compensate the greater length of the arch.
59. The increase of velocity, which the body acquires in falling from a greater height, has such an effect, that, if straight lines be drawn from A to C and E, the body would fall through the longer straight line C A just in the same time, as through the shorter straight line E A. This is demonstrated by the geometers, who prove, that if any circle, as A B C D (fig. 49.) be placed in a perpendicular situation; a body shall fall obliquely through every line, as A B drawn from the lowest point A in the circle to any other point in the circumference just in the same time, as would be imployed by the body in falling perpendicularly down through the diameter C A. But the time in which the body will descend through the arch, is different from the time, which it would take up in falling through the line A B.
60. It has been thought by some, that because in very small arches this correspondent straight line differs but little from the arch itself; therefore the descent through this straight line would be performed in such small arches nearly in the same time as through the arches themselves: so that if a pendulum were to swing in small arches, half the time of a single swing would be nearly equal to the time, in which a body would fall perpendicularly through twice the length of the pendulum. That is, the whole time of the swing, according to this opinion, will be four fold the time required for the body to fall through half the length of the pendulum; because the time of the body’s falling down twice the length of the pendulum is half the time required for the fall through one quarter of this space, that is through half the pendulum’s length. However there is here a mistake; for the whole time of the swing, when the pendulum moves through small arches, bears to the time required for a body to fall down through half the length of the pendulum very nearly the same proportion, as the circumference of a circle bears to its diameter; that is very nearly the proportion of 355 to 113, or little more than the proportion of 3 to 1. If the pendulum takes so great a swing, as to pass over an arch equal to one sixth part of the whole circumference of the circle, it will swing 115 times, while it ought according to this proportion to have swung 117 times; so that, when it swings in so large an arch, it loses something less than two swings in an hundred. If it swing through 1/10 only of the circle, it shall not lose above one vibration in 160. If it swing in 1/20 of the circle, it shall lose about one vibration in 690. If its swing be confined to 1/40 of the whole circle, it shall lose very little more than one swing in 2600. And if it take no greater a swing than through 1/60 of the whole circle, it shall not lose one swing in 5800.
61. Now it follows from hence, that, when pendulums swing in small arches, there is very nearly a constant proportion observed between the time of their swing, and the time, in which a body would fall perpendicularly down through half their length. And we have declared above, that the spaces, through which bodies fall, are in a two fold proportion of the times, which they take up in falling[63]. Therefore in pendulums of different lengths, swinging through small arches, the lengths of the pendulums are in a two fold or duplicate proportion of the times, they take in swinging; so that a pendulum of four times the length of another shall take up twice the time in each swing, one of nine times the length will make one swing only for three swings of the shorter, and so on.
62. This proportion in the swings of different pendulums not only holds in small arches; but in large ones also, provided they be such, as the geometers call similar; that is, if the arches bear the same proportion to the whole circumferences of their respective circles. Suppose (in fig. 48.) A B, C D to be two pendulums. Let the arch E F be described by the motion of the pendulum A B, and the arch G H be described by the pendulum C D; and let the arch E F bear the same proportion to the whole circumference, which would be formed by turning the pendulum A B quite round about the point A, as the arch G H bears to the whole circumference, that would be formed by turning the pendulum C D quite round the point C. Then I say, the proportion, which the length of the pendulum A B bears to the length of the pendulum C D, will be two fold of the proportion, which the time taken up in the description of the arch E F bears to the time employed in the description of the arch G H.
[63.] Thus pendulums, which swing in very small arches, are nearly an equal measure of time. But as they are not such an equal measure to geometrical exactness; the mathematicians have found out a method of causing a pendulum so to swing, that, if its motion were not obstructed by any resistance, it would always perform each swing in the same time, whether it moved through a greater, or a lesser space. This was first discovered by the great Huygens, and is as follows. Upon the straight line A B (in fig. 49.) let the circle C D E be so placed, as to touch the straight line in the point C. Then let this circle roll along upon the straight line A B, as a coach-wheel rolls along upon the ground. It is evident, that, as soon as ever the circle begins to move, the point C in the circle will be lifted off from the straight line A B; and in the motion of the circle will describe a crooked course, which is represented by the line C F G H. Here the part C H of the straight line included between the two extremities C and H of the line C F G H will be equal to the whole circumference of the circle C D E; and if C H be divided into two equal parts at the point I, and the straight line I K be drawn perpendicular to C H, this line I K will be equal to the diameter of the circle C D E. Now in this line if a body were to be let fall from the point H, and were to be carried by its weight down the line H G K, as far as the point K, which is the lowest point of the line C F G H; and if from any other point G a body were to be let fall in the same manner; this body, which falls from G, will take just the same time in coming to K, as the body takes up, which falls from H. Therefore if a pendulum can be so hung, that the ball shall move in the line A G F E, all its swings, whether long or short, will be performed in the same time; for the time, in which the ball will descend to the point K, is always half the time of the whole swing. But the ball of a pendulum will be made to swing in this line by the following means. Let K I (in fig. 52.) be prolonged upwards to L, till I L is equal to I K. Then let the line L M H equal and like to K H be applied, as in the figure between the points L and H, so that the point which in this line L M H answers to the point H in the line K H shall be applied to the point L, and the point answering to the point K shall be applied to the point H. Also let such another line L N C be applied between L and C in the same manner. This preparation being made; if a pendulum be hung at the point L of such a length, that the ball thereof shall reach to K; and if the string shall continually bend against the lines H M L and L N C, as the pendulum swings to and fro; by this means the ball shall constantly keep in the line C K H.
[64.] Now in this pendulum, as all the swings, whether long or short, will be performed in the same time; so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the circumference of a circle bears to its diameter.
65. It may from hence be understood in some measure, why, when pendulums swing in circular arches, the times of their swings are nearly equal, if the arches are small, though those arches be of very unequal lengths; for if with the semidiameter L K the circular arch O K P be described, this arch in the lower part of it will differ very little from the line C K H.