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