then stopping for an instant the ball again descends, to ascend on the other side, thus adding oscillation to oscillation. Were it not for the resistance of the atmosphere and certain mechanical imperfections these arcs would be the same, but, what is more important, the times of oscillating are the same.
The rapidity with which the pendulum descends depends upon its length and the amount of this impulse to drop vertically. This impulse is known as gravity. Therefore, with a pendulum of constant length the time of oscillation will be dependent upon gravity, and thus time and gravity are determinable one in terms of the other.
Newton had shown that gravity on the earth's surface depended upon distance from the center of the earth, and also the diminishing effect of the revolution of the earth on gravity. To this theory other mathematicians made valuable contributions, notably Clairaut, who demonstrated that the relative lengths of the equatorial and polar radii could be ascertained directly from the force of gravity at the equator and at one of the poles. Then, since the gravity is obtained directly from the time in which a pendulum makes an oscillation and its length, it was necessary to simply swing a pendulum at the equator and at one of the poles to have at once the coveted ellipticity—that is, the ratio of the difference between the equatorial and polar radii to the equatorial radius.
Unfortunately, it has not been possible to swing a pendulum at one of the poles. This inability, however, is made of no moment by a law which gives the value of the polar gravity whenever the gravity of a given place is known, together with the latitude of the place.
From this it appears that the earth's figure becomes known through a determination of the length of a pendulum and the time required for it to make an oscillation at the equator (or near it) and at the pole (or as near to it as possible). If the same pendulum is used and the constancy of its length assured, it becomes necessary to make sure of the length of time required for an oscillation at these two places. Inasmuch as the pendulum appears to stop for an instant when it reaches the highest point in its arc, it is a difficult matter to determine with exactness the time of an oscillation; but if one counts the number of oscillations in an hour, in two hours, or in any number of hours, a simple division will give the time of one oscillation.
The figure of the earth desired is an ideal figure, such a figure as it would have if one could remove all the land now standing higher than the surface of the sea—were a sea to occupy the place of the land. Hence it is the sea-level earth whose figure we want. Newton's law of gravity would require that a pendulum, if raised