cult problems of mathematical astronomy. The trouble is that so many factors enter. If the Moon moved simply under the mutual attraction between it and the Earth, the problem would be the comparatively simple one known as the two-body problem. But the Sun's attraction is a powerful disturbing—or, in technical terms, perturbing—force; Venus exercises a strong attraction; the other planets, in smaller degree, enter, each with a force determined by its mass and distance; even the fact that the Earth is not a sphere, but bulges at the equator, is a factor by no means to be neglected. The Moon, therefore, does not move in a simple elliptic orbit, but in a very irregular curve, following the line of the ellipse only in a general way, and it is so near the Earth, relatively speaking, that every departure from simple elliptic motion is detected in our observations. To account for the observed motion under the law of gravitation, taking all the disturbing factors into consideration, is a problem that has exercised the highest powers of great mathematicians from Newton's time to the present day. We may well be proud of the fact that three American astronomers—the late Simon Newcomb, the late George William Hill, and Professor Ernest W. Brown of Yale University—have taken distinguished parts in the solution of this great problem. Professor Brown's lunar tables, now being printed, are the most accurate ever constructed.
Returning, after this digression, to the Moon's apparent motion, the diurnal motion due to the rotation of the Earth on its axis is the first to be noticed. We see the Moon rise above the eastern horizon, circle the sky towards the west, and set below the western horizon. The points of rising and setting are not always the same nor does the Moon cross the meridian always at the same altitude, and the times of rising change from day to day. The observer will quickly learn to associate the times of rising and setting with the Moon's age and its phases. For a day or two at new Moon time he will not see it rise or set at all. Then, if he is sharp-eyed and the air is very clear, he will see it rise shortly after sunrise, a slender crescent. As the crescent grows from day to day, the time of rising becomes later and later until, when the crescent has rounded thru the half Moon and gibbous phases to full Moon, it rises in the east about the time the Sun is setting in the west. As it wanes again, first to the half Moon phase, and then, in the last quarter of the month, to an ever nar-
