MOON 799 This state of things arises from the inherent difficulties and complexities of the subject, and from the fact that no one method or system lias yet been discovered by which all the difficulties can be surmounted and all the com plexities disentangled. Hence each investigator, when he has desired to make any substantial advance beyond his predecessors, has been obliged to take up the subject from a new point of view, and to devise such method as might seem to him most suitable to the special object in hand. The historical treatment is therefore that best adapted to give a clear idea of the results of these investigations. The ancient and modern histories of the subject are quite distinct, the modern epoch commencing with Newton. The great epoch made by Copernicus did not extend to the case of the moon at all, because in every investigation of the moon s motion, modern as well as ancient, the motion is referred to the earth as a centre. Hence the heliocentric -system introduced no new conception of this motion, except that of taking place round a moving earth instead of round a fixed one. This change did not affect the consideration of the relative motion of the earth and moon, with which alone the lunar theory is concerned. The two stages of the lunar theory are therefore (1) that in which the treat ment was purely empirical, (2) that in which it was founded rationally on the law of gravitation. It is in the investigation of the moon s motion that the merits of ancient astronomy are seen to the best advan tage. In the hands of Hipparchus (see ASTRONOMY, vol. ii. p. 749) the theory was brought to a degree of precision which is really marvellous when we compare it, either with other branches of physical science in that age, or with the remarks and speculations of contemporary non- scientific writers. Whether this was wholly the work of Hipparchus, or whether he simply perfected a system already devised by his predecessors, it is now impossible to say ; but, so far as certain knowledge extends, the works of his predecessors did not embrace more than the deter mination of the mean motion of the moon and its nodes. Although the general fact of a varying motion may have been ascertained, the circumstances of the variation had probably never been thoroughly investigated. The dis coveries of Hipparchus were : 1. The Eccentricity of the Moon s Orbit. He found that the moon moved most rapidly near a certain point of its orbit, and most slowly near the opposite point. The law of this motion was such that the phenomena could be re presented by supposing the motion to be actually circular and uniform, the apparent variations being explained by the hypothesis that the earth was not situated in the centre of the orbit, but was displaced by an amount about equal to one- twentieth of the radius of the orbit. Then, by a well-known law of kinematics, the angular motion round the earth would be most rapid at the point nearest the earth that is, at perigee and slowest at the point most distant from the earth that is, at apogee. Thus the apogee and perigee became two definite points of the orbit, indicated by the variations in the angular motion of the moon. 2. The Motion of the Perigee and Apogee. As already defined, the perigee and apogee are at the ends of that diameter of the orbit which passes through the eccentrically situated earth, or, in other words, they are on that line which passes through the centre of the earth and the centre of the orbit. This line was called the line of apsides. On comparing observations made at different times, it was found that the line of apsides was not fixed, but made a complete revolution in the heavens, in the order of the signs of the zodiac, in about nine years. 3. The Numerical Determination of the Elements of the Moon s Motion. In order that the two capital discoveries just mentioned should have the highest scientific value it was essential that the numerical values of the elements involved in these complicated motions should be fixed with precision. This Hipparchus was enabled to do by lunar eclipses. Each eclipse gave a moment at which the longi tude of the moon was 180 different from that of the sun, and the latter admitted of ready calculation. Assuming the mean motion of the moon to be known and the perigee to be fixed, three eclipses observed in different points of the orbit would give as many true longitudes of the moon, which longitudes could be employed to determine three unknown quantities the mean longitude at a given epoch, the eccentricity, and the position of the perigee. By taking three eclipses separated at short intervals, both the mean motion and the motion of the perigee would be known beforehand, from other data, with sufficient accuracy to reduce all the observations to the same epoch, and thus to leave only the three elements already mentioned un known. In the hands of a modern calculator the problem would be a very simple one, requiring little more than the solution of a system of three equations with as many un known quantities. But without algebra the solution was long and troublesome, and not entirely satisfactory. Still, it was probably correct within the necessary limits of the errors of the observations. The same three elements being again determined from a second triplet of eclipses at as remote an epoch as possible, the difference in the longitude of the perigee at the two epochs gave the annual motion of that element, and the difference of mean longitudes gave the mean motion. Such was the method of determining the elements of the moon s motion down to the time of Copernicus. The determination of the eccentricity from eclipses, as above described, leads to an important error in the resulting value of the eccentricity, owing to the effect of the neglected evection. We know from our modern theory that the two principal inequalities in the moon s true longitude are 6 29 sin g (Equation of centre) + l-27 sin (-2D-g) (Evection), where g = mean anomaly, and D = mean angular distance of the moon from the sun. Now during a lunar eclipse we always have D 180 very nearly, and 2Z> 360. Hence the evection is then 1 27 sin g, and so has the same argument, g, as the equation of centre, and so is confounded with it. The value of the equation of centre derived from eclipses is thus (6 29 - 1 27 = 5 - 02) sin g. Therefore the eccentricity found by Hipparchus and Ptolemy was only 5, and was more than a degree less than its true value. The next important step in advance was the discovery of the "evection," which is described by Ptolemy (see ASTRONOMY, vol. ii. p. 750) as if made by himself. In view of the bad habit which Ptolemy had of making his own observations verify results previously arrived at, which were sometimes in error, we must view such a discovery by him as quite exceptional, and as best explainable by the large magnitude of the outstanding error. Although, as just shown, the erroneous eccentricity found by Hipparchus would always represent eclipses, so that the error could never be detected by eclipses, the case was entirely different when the moon was in quadratures. Comparing the in equalities already written with that found by Hipparchus, we see that the latter required the correction l-27 {sin^ + sin (-2D -g}} = l-27 {(1 - cos 2D) sin $r + sin 2Z> cos <?} . At quadratures we have D= 90, 2Z> = 180, and hence cos 2D = - 1 and sin 2D = 0. The omitted inequalities at these points of the orbit have therefore the value 2 -54 sin g, a quantity so large that it could not fail to be detected by careful observations with the astrolabe. Such an inequality as this, superposed upon the eccentric motion of the moon, was very troublesome to astronomers who had no way of representing the celestial motions except by geometrical construction. The construction proposed by
Ptolemy was so different from those employed for the