Other numerical particulars relating to the moon are:—
| Mean distance from the earth (earth’s radius as 1) | 60·2634 |
| Mean apparent diameter | 31′ 51·5 |
| Diameter in miles | 2159·6 |
| Moon’s surface in square miles | 14,600,000 |
| Diameter (earth’s equatorial diameter as 1) | 0·2725 |
| Surface (earth’s as 1) | 0·0742 |
| Volume (earth’s as 1) | 0·0202 |
| Ratio of mass to earth’s mass[1] | 1:81·53 ± ·047 |
| Density (earth’s as 1) | 0·60736 |
| Density (water’s as 1, and earth’s assumed as 5) | 3·46 |
| Ratio of gravity to gravity at the earth’s surface | 1:6 |
| Inclination of axis of rotation to ecliptic | 1° 30′ 11·3″ |
The Lunar Theory.
The mathematical theory of the moon’s motion does not yet form a well-defined body of reasoning and doctrine, like other branches of mathematical science, but consists of a series of researches, extending through twenty centuries or more, and not easily welded into a unified whole. Before Newton the problem was that of devising empirical curves to formally represent the observed inequalities in the motion of the moon around the earth. After the establishment of universal gravitation as the primary law of the celestial motions, the problem was reduced to that of integrating the differential equations of the moon’s motion, and testing the completeness of the results by comparison with observation. Although the precision of the mathematical solution has been placed beyond serious doubt, the problem of completely reconciling this solution with the observed motions of the moon is not yet completely solved. Under these circumstances the historical treatment is that best adopted to give a clear idea of the progress and results of research in this field. Modern researches were developed so naturally from the results of the ancients that we shall begin with a brief mention of the work of the latter.
It is in the investigation of the moon’s motion that the merits of the ancient astronomy are seen to the best advantage. In the hands of Hipparchus 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 views of contemporary non-scientific writers. The discoveries 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 represented 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 an obvious 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.
These points 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.
2. 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 longitude of the moon was 180° different from that of the sun. 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 unknown. 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.
The eccentricity determined in this way is more than a degree in error, owing to the effect of the evection, which was unknown to Hipparchus. The result of the latter inequality is brought out when it is sought to determine the eccentricity of the orbit from the observations near the time of the first and last quarter. It was thus found by Ptolemy that an additional inequality existed in the motion, which is now known as the evection. The relations of the quantities involved may be shown by simple trigonometric formulae. If we put g for the moon’s anomaly or distance from the perigee, and D for its elongation from the sun, the inequalities in question as now known are—
| 6·29° sin g | (equation of centre) |
| +1·27° sin (2D−g) | (evection). |
During a lunar eclipse we always have D＝180°, very nearly, and 2D＝360°. Hence the evection is then −1·2° sin g, and consequently has the same argument g as the equation of centre, so that it is confounded with it. The value of the equation of centre derived from eclipses is thus—
6·29° sin g−1·27° sin g＝5·02° sin g.
Therefore the eccentricity found by Hipparchus was only 5°, and was more than a degree less than its true value. At first quarter we have D＝90° and 2D＝180°. Substituting this value of 2D in the last term of the above equation, we see that the combined equation of the centre and evection are, at quadrature—
6·29° sin g+1·27° sin g＝7·56° sin g.
Thus, in consequence of the evection, the equation of the centre comes out 2° 30′ larger from observations at the moon’s quarters than during eclipses.
The next forward step was due to Tycho Brahe. He found that, although the two inequalities found by Hipparchus and Ptolemy correctly represented the moon’s longitude near conjunction and opposition, and also at the quadratures, it left a large outstanding error at the octants, that is when the moon was 45° or 135° on either side of the sun. This inequality, which reaches the magnitude of nearly 1°, is known as the variation. Although Tycho Brahe was an original discoverer of this inequality, through whom it became known, Joseph Bertrand of Paris claimed the discovery for Abu ’l-Wefa, an Arabian astronomer, and made it appear that the latter really detected inequalities in the moon’s motion which we now know to have been the variation. But he has not shown, on the part of the Arabian, any such exact description of the inequality as is necessary to make clear his claim to the discovery. We may conclude the ancient history of the lunar theory by saying that the only real progress from Hipparchus to Newton consisted in the more exact determination of the mean motions of the moon, its perigee and its line of nodes, and in the discovery of three inequalities, the representation of which required geometrical constructions increasing in complexity with every step.
The modern lunar theory began with Newton, and consists in determining the motion of the moon deductively from the theory of gravitation. But the great founder of celestial mechanics employed a geometrical method, ill-adapted to lead to the desired result; and hence his efforts to construct a lunar theory are of more interest as illustrations of his wonderful power and correctness in mathematical reasoning than as germs of new methods of research. The analytic method sought to express the moon’s motion by integrating the differential equations of the dynamical theory. The methods may be divided into three classes:—
1. Laplace and his immediate successors, especially G. A. A. Plana (1781–1864), effected the integration by expressing the time in terms of the moon’s true longitude. Then, by inverting the series, the longitude was expressed in terms of the time.
2. By the second general method the moon’s co-ordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining as algebraic symbols the values of the various elements. Most of the elements are small numerical fractions: e, the eccentricity of the moon’s orbit, about 0·055; e′, the eccentricity of the earth’s orbit, about 0·017; γ the sine of half the inclination of the moon’s orbit, about 0·046; m, the ratio of the mean motions of the moon and earth, about 0·075. The expressions for the longitude, latitude and parallax appear as an infinite trigonometric series, in which the coefficients of the sines and cosines are themselves infinite series proceeding according to the powers of the above small numbers. This method was applied with success by Pontécoulant and Sir John W. Lubbock, and afterwards by Delaunay. By these methods the series converge so slowly, and the final expressions for the moon’s longitude are so long and complicated, that the series has never been carried far enough to ensure the accuracy of all the terms. This is especially the case with the development in powers of m, the convergence of which has often been questioned.
3. The third method seeks to avoid the difficulty by using the numerical values of the elements instead of their algebraic symbols. This method has the advantage of leading to a more rapid and certain
- ↑ A. R. Hinks, “Mass of the Moon, from Observations of Eros, 1900–1901,” M. N. Roy. Ast. Soc., 1909. Nov., p. 73.
