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 determination of the numerical quantities required. It has the disadvantage of giving the solution of the problem only for a particular case, and of being inapplicable in researches in which the general equations of dynamics have to be applied. It has been employed by Damoiseau, Hansen and Airy.

The methods of the second general class are those most worthy of study. Among these we must assign the first rank to the method of C. E. Delaunay, developed in his Théorie du mouvement de la lune (2 vols., 1860, 1867), because it contains a germ which may yet develop into the great desideratum of a general method in celestial mechanics.

Among applications of the third or numerical method, the most successful yet completed is that of P. A. Hansen. His first work, Fundamenta nova, appeared in 1838, and contained an exposition of his ingenious and peculiar methods of computation. During the twenty years following he devoted a large part of his energies to the numerical computation of the lunar inequalities, the redetermination of the elements of motion, and the preparation of new tables for computing the moon’s position. In the latter branch of the work he received material aid from the British government, which published his tables on their completion in 1857. The computations of Hansen were published some seven years later by the Royal Saxon Society of Sciences.

It was found on comparing the results of Hansen and Delaunay that there are some outstanding discrepancies which are of sufficient magnitude to demand the attention of those interested in the mathematical theory of the subject. It was therefore necessary that the numerical inequalities should be again determined by an entirely different method.

This has been done by Ernest W. Brown, whose work may be regarded not only as the last word on the subject, but as embodying a seemingly complete and satisfactory solution of a problem which has absorbed an important part of the energies of mathematical astronomers since the time of Hipparchus. We shall, try to convey an idea of this solution. We have just mentioned the four small quantities e, e′, and m, in terms of the powers and products of which the moon’s co-ordinates have to be expressed. Euler conceived the idea of starting with a preliminary solution of the problem in which the orbit of the moon should be supposed to lie in the ecliptic, and to have no eccentricity, while that of the sun was circular. This solution being reached, the additional terms were found, which were multiplied by the first power of the several eccentricities and of the inclination. Then the terms of the second order were found, and so on to any extent. In a series of remarkable papers published in 1877–1888 Hill improved Euler’s method, and worked it out with much more rigour and fullness than Euler had been able to do. His most important contribution to the subject consisted in working out by extremely elegant mathematical processes the method of determining the motion of the perigee. John Couch Adams afterwards determined the motion of the node in a similar way. The numerical computations were worked out by Hill only for the first approximation. The subject was then taken up by Brown, who in a series of researches published in the Memoirs of the Royal Astronomical Society and in the Transactions of the American Mathematical Society extended Hill’s method so as to form a practically complete solution of the entire problem. The principal feature of his work was that the quantity m, which is regarded as constant, appears only in a numerical form, so that the uncertainties arising from development in a series accruing to its powers is done away with.

The solution of the main mathematical problem thus reached is that of the motion of three bodies only—the sun, earth and moon. The mean motion of the moon round the earth is then invariable, the longitude containing no inequalities of longer period than that of the moon’s node, 18·6 y. But Edmund Halley found, by a comparison of ancient eclipses with modern observations, that the mean motion had been accelerated. This was confirmed by Richard Dunthorne (1711–1775). Corresponding to this observed fact was the inference that the action of the planets might in some way influence the moon’s motion. Thus a new branch of the lunar theory was suggested—the determination by theory of the effect of planetary action.

The first step in constructing this theory was taken by Laplace, who showed that the secular acceleration was produced by the secular diminution of the earth’s orbit. He computed the amount as about 10″ per century, which agreed with the results derived by Dunthorne from ancient eclipses. Laplace’s immediate successors, among whom were Hansen, Plana and Pontécoulant, found a larger value, Hansen increasing it to 12·5″, which he introduced into his tables. This value was found by himself and Airy to represent fairly well several ancient eclipses of the sun, notably the supposed one of Thales. But Adams in 1853 showed that the previous computations of the acceleration were only a rude first approximation, and that a more rigorous computation reduced the result to about one-half. This diminution was soon fully confirmed by others, especially Delaunay, although for some time Pontécoulant stoutly maintained the correctness of the older result. But the demonstration of Adam’s result was soon made conclusive, and a value which may be regarded as definitive has been derived by Brown. With the latest accepted diminution of the eccentricity, the coefficient is 5·91″.

The question now arose of the origin of the discrepancy between the smaller values by theory, and the supposed values of 12″ derived from ancient eclipses. In 1856 William Ferrel showed that the action of the moon on the ocean tidal waves would result in a retardation of the earth’s rotation, a result, at first unnoticed, which was independently reached a few years later by Delaunay. The amount of retardation does not admit of accurate computation, owing to the uncertainty both as to the amount of the oceanic friction from which it arises and of the exact height and form of the tidal wave, the action of the moon on which produces the effect. But any rough estimate that can be made shows that it might well be supposed much larger than is necessary to produce the observed differences of 6″ per century. It was therefore surprising when, in 1877, Simon Newcomb found, by a study of the lunar eclipses handed down by Ptolemy and those observed by the Arabians—data much more reliable than the vague accounts of ancient solar eclipses—that the actual apparent acceleration was only about 8·3″. This is only 2·4″ larger than the theoretical value, and it seems difficult to suppose that the effect of the tidal retardation can be as small as this. This suggests that the retardation may be in great part compensated by some accelerating cause, the existence of which is not yet well established. The following is a summary of the present state of the question:— Inequalities of Long Period.—Combined with the question of secular acceleration is another which is still not entirely settled that of inequalities of long period in the mean motion of the moon round the earth. Laplace first showed that modern observations of the moon indicated that its mean motion was really less during the second half of the 18th century than during the first half, and hence inferred the existence of an inequality having a period of more than a century.

The existence of one or more such inequalities has been fully confirmed by all the observations, both early and recent, that have become available since the time of Laplace. It is also found by computation from theory that the planets do produce several appreciable inequalities of long period, as well as a great number of short period, in the motion of the moon. But the former do not correspond to the observed inequalities, and the explanation of the outstanding differences may be regarded to-day as the most perplexing enigma in astronomy. The most plausible explanation is that, like the discrepancy in the secular acceleration, the observed deviation is only apparent, and arises from slow fluctuations in the earth’s rotation, and therefore in our measure of time produced by the motion of great masses of polar ice and the variability of the amount of snowfall on the great continents. Were this the case a similar inequality should be found in the observed times of the transits of Mercury. But the latter do not certainly show any deviation in the measure of time, and seem to preclude a deviation so large as that derived from observations of the moon. This suggests that inequalities in the action of the planets may have been still overlooked, the subject being the most intricate with which celestial mechanics has to deal. But this action has been recently worked up with such completeness of detail by Radau, Newcomb and Brown, that the possibility of any unknown term seems out of the question. (The enigma therefore still defies solution.

.—Works on selenography: Hevelius, Selenographia sive lunae descriptio (Danzig, 1647); Riccioli, Almagestum novum (Bologna, 1651); J. H. Schroeter, Selenotopographische Fragmente zur genauern Kenntniss der Mondfläche (Lilienthal, 1791); W. Beer and J. H. Mädler, Der Mond nach seinen kosmischen und individuellen Verhältnissen, oder Allgemeine 'vergleichende Selenographie (Berlin, 1837); Richard A. Proctor, The Moon (London, 1873; the first edition contains excellent geometrical demonstrations of the inequalities produced by the sun in the moon’s motion, which were partly omitted in the second edition); J. Nasmyth and J. Carpenter, The Moon, Considered as a Planet, a World and a Satellite (London, 1903; fine illustrations); E. Neison (now Neville), The Moon and the Conditions and Configurations of its Surface (London, 1876); M. Loewy and P. Puiseux, Atlas photographique de la lune (Imprimerie Royale, Paris, 1896–1908); W. H. Pickering, The Moon, from photographs (New York, 1904); G. P. Serviss, The Moon (London, 1908), a popular account illustrated by fine photographs.

On the subject of lunar geology, see N. S. Shaler in Smithsonian Contributions to Knowledge, vol. xxxiv. No. 1438, and P. Puiseux, “Recherches sur l’origine probable des formations lunaires,” in Annales de l’observatoire de Paris, Mémoires, tome xxii.