At the same time, however, that I assert the inefficacy of any experiments with the pendulum as tending to show the earth's rotation, I admit that, provided a pendulum could be made to preserve its plane of oscillation for twenty-four hours, it would oscillate independently of the rotation of the earth, and actually describe a circle round a fixed table in that interval. The _mathematical proof_ of this proposition is of a most abstruse nature; so much so, indeed, that it is understood to have been relinquished by one of our ablest mathematicians. But that it is likely to be true, and one not difficult to comprehend, I think I can show to A. E. B.'s satisfaction in a few lines. If a pendulum be placed at one of the poles of the earth, it is obvious, that while it swings in one plane, the revolution of the earth beneath it will cause it to appear to describe a complete circle in twenty-four hours. This position is simple enough, but it is true also in any latitude, excepting near the equator. For there is no doubt, that, as gravity acts on the pendulum, _only in the line which joins the point of suspension and the centre of the earth_ (thereby merely drawing the "bobs" towards that line) it can have no effect on the _plane_ of oscillation; for the line of gravitation remains unchanged with respect to the pendulum, during a whole revolution of the earth on its axis. Take a map of a hemisphere, and on any parallel, say 60° of latitude, draw three pendulums, extended as in motion, with their centres of gravity directed toward the earth's centre, one on each extremity of the parallel of latitude, and one midway between the two; extend the "bobs" of the first two north and south, and those of the middle one east and west. Number them 1, 2, and 3, from the westward. It will then be observed that the _plane of oscillation_ of the three pendulums, thus placed, is one and the same--that of the _plane of the paper_; and moreover, that the lower "bob," which is south at No. 1., is west at No. 2., and north at No. 3. By this it will be evident, that the revolution of the pendulum will be through the whole circle, or 360° in twenty-four hours, at all points of the earth's surface, excepting near the equator; _the line joining the "bobs"_ remaining in a parallel plane.