With these observations as a general guide to the cosmical factors involved in the planetary motions, the reader will be able to take a more intelligent interest in the foundations of his study than is the usual case from the pursuit of the subject by rule-of-thumb methods. When we come to the consideration of the Moon as a cosmical factor we are face to face with one of the most difficult and evasive problems. For many centuries astronomers grappled with this inconstant factor with small success, and at the present day the problems attaching to the vagaries of lunar motion are in anything but a satisfactory condition. Prior to the time of Ptolemy nothing was known of the Moon except that it had a certain mean motion and formed its syzygies at definite periods, the mean values of which were very closely known. But certainly nothing was known regarding the inequalities of motion which are found to exist. Ptolemy discovered the equation due to the action of the Sun upon the Moon in its orbit. This is called the Evection. Tycho later discovered that a further equation was due to the disturbance caused by the Sun along the vector. Both these equations were employed by Kepler. But of these, later astronomers have added one after another equation, going so far as to employ the action due to Venus and Jupiter, while ignoring that due to the action of the other planets upon the Moon. Buckhardt, whose formulae were used in the calculation of the Nautical Almanac for many years, employs no less than 37 equations of the Moon’s mean longitude. Indeed, the whole business has become farcical. The fact is that only three of these equations are necessary in order to obtain the Moon’s true place in the ecliptic at any time, and for the syzygy only one equation is necessary. The trouble has arisen from the fact that the eccentricity of the Moon’s orbit has been wrongly estimated, and most of these equations now employed by astronomers are effectual only in correcting this false estimate. Kepler gives the maximum equation to centre as 4° 59′ 59″, while modern astronomers have given it as 6° 18′ 28″. Neither of these is quite correct, though Kepler is much nearer the truth.