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the planets. Their orbits gradually change their form back and forth through tens of thousands of years, like "great clocks of eternity which count off ages as ours count off seconds." The ordinary reader would be justified in some incredulity as to the correctness of these predictions for thousands of years to come, were it not for the striking precision with which the motions of the planets are actually predicted by the mathematical astronomer. This precision is reached by solving the very difficult problem of determining the effect of each planet on the motions of all the other planets. We might predict the motions of these bodies by assuming that each of them moves round the sun in a fixed ellipse, which, as I have just said, would be the case if it were not attracted by any other body. Our predictions would then, from time to time, be in error by amounts which might amount to large fractions of a degree; perhaps, in the course of a long time, to even more. To form an idea of this error we may say that one degree is the angle at which we see the breadth of an ordinary window frame at the distance of a hundred yards. The planet might then be predicted as in a line with one side of such a frame when in reality it would be at the other side or in the middle of the window.

But, taking account of the attraction of all the other planets, the prediction is so exact that the refined observations of astronomy hardly show any appreciable deviation. If we should mark on the side of a distant house a row of a hundred points, each apparently as far from the other as the average error of these predictions, the