This state of things arises from the inherent difficulties and complexities of the subject, and from the fact that no one method or system has yet been discovered by which all the difficulties can be surmounted and all the complexities 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 treatment 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 advantage. 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 determination 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 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 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 longitude 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 unknown. 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 unknown 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 0, 29 sin g (Equation of centre)

+ l o- 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 - l o- 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 0, 29 - l o ∙27 = δ o 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 inequalities already written with that found by Hipparchus, we see that the latter required the correction—

l o, 27 {si11<7 + sin (2Z>-y)} =

Γ ’ 27 {(1 - cos 2D) sin g + sin 2D cos g) . At quadratures we have D=±90°, 2D = 180°, and hence cos 2Z> = - 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 motions of the planets, and withal so intricate, that little interest attaches to it.

The student of Arabian science may find much to interest him in the astronomical speculations of the Arabs, but this people do not seem to have furnished anything in the way of suggestive theory. In the fourth book of De Reυolu-tionibus, where we find the lunar theory of Copernicus, no writer later than Ptolemy is referred to. Moreover, as already intimated, the work of Copernicus in this particular direction forms little more than an episode in the history of the subject. The working hypothesis of the great founder of modern astronomy was borrowed from the ancients, and was that the celestial motions were all either circular or compounded of circular motions. The hypothesis of equal circular motions, though accepted by Ptolemy in name, was so strained by him in its applications that little was left of it in the Almagest (the Arabic translation of his Syntaxis'). But, by taking the privilege of compounding circular motions indefinitely—in other words, of adding one epicycle to another—Copernicus was enabled to represent the planetary and lunar inequalities on a uniform system, though his heavens were perhaps worse “scribbled o’er” than those of Ptolemy. To one epicycle representing the equation of the centre he added another for the evection, and thus represented the longitude of the moon both at quadratures and oppositions. But the third inequality, “variation,” which attains its maxima at the octants and vanishes at all four quarters, was unknown to him. To Tycho Brahe is commonly and justly ascribed the discovery of the variation. Joseph Bertrand of Paris has indeed claimed the discovery for Abù Ί-Wefâ, an Arabian astronomer, and has made it appear probable that Abù Ί-Wefä, 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 phenomena as is necessary to make clear his claim to the discovery. As regards Tycho, although he discovered the fact, he could add nothing in the way of suggestive theory. To the double epicycle of Copernicus he was obliged to add a motion of the centre of the whole lunar orbit round a circle whose circumference passed through the centre of the earth, two revolutions round this circle being made in each lunation. Kepler, by introducing a moving ellipse having the earth as its focus, was enabled to make a nearer approach to the truth than any of his predecessors. But the geometrical hypotheses by which he represented the inequalities due to the action of the sun form no greater epoch in the progress of science than do the geometrical constructions of his predecessors. We may therefore dispose of 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 new inequalities, the representation of which required geometrical constructions increasing in complexity with every step.

The modern lunar theory commenced with Newton, and consists in determining the motion of the moon deductively from the theory of gravitation. But the great founder of modern mechanics did not employ the method best 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. He succeeded perfectly in explaining the elliptic motion of two mutually attracting bodies round their common centre of gravity by geometrical constructions. But when the prob lem was one of determining the variations from the elliptic motion which would be produced by a third body, such constructions could lead only to approximate results. The path to modern methods was opened up by the Continental mathematicians, whose great work consisted in reducing the problem to one of pure algebra. The chasm between the laws of motion laid down by Newton and a problem of algebra seems so difficult to bridge over that it is worth while to show in what the real spirit of the modern method consists. We call to mind the statement of Newton’s first two laws of motion: that a body uninfluenced by any force moves in a straight line and with uniform velocity for ever, and that the change of motion is proportional to the force impressed upon the body and in the direction of such force. These two laws admit of being expressed in algebraic language thus :—let us put m the mass of a material point; x its distance from any fixed plane whatever; t the time ; X the sum of the components of all the forces acting upon the point in the direction perpendicular to the fixed plane, it being supposed that each force is resolved into three mutually perpendicular components, one of which is perpendicular to the fixed plane; then the differential equation cPx m dt* X expresses Newton’s first two laws of motion with a completeness and precision which is entirely wanting in all statements in ordinary language. The latter can be nothing more than lame attempts to express the equation in language which may be understood by the non-mathematical reader, but which bear the same relation to the algebraic equation that a statement of the operations of the Bank of England in the symbolic language of a tribe of savages would bear to the bank statement in pounds, shillings, and pence. By taking two other planes, perpendicular to each other and to the first plane, we have three equations like the one last written. The law of gravitation and Newton’s third law of motion enable us to substitute for X and the other forces the masses and coordinates of the various attracting bodies. Thus the data of the problem are expressed by a triplet of three equations for each attracting body. The integration of these equations is a problem of pure algebra, which, when solved, leads to expressions that give the position of each body in terms of the time, which is what is wanted. The special form which it is necessary to give the equations has not been radically changed during the century and a half since this method of research was opened out. The end aimed at is the algebraic expression of all the quantities involved in the form of an infinite series of terms, each consisting of a constant coefficient multiplied by the sine or cosine of an angle increasing uniformly with the time. It is indeed a remarkable fact that, notwithstanding the great advances which modern mathematics has made in the discovery of functions more general than the old-fashioned sines and cosines of elementary trigonometry, especially of elliptic functions, yet the form of development adopted by the mathematicians of the last century has remained without essential change.

It will be instructive to notice the general and simple property of the trigonometric functions to which is due their great advantage in the problems of celestial mechanics. It may be expressed thus :— If we have any number of quantities, each of which is expressed in the form of a trigonometric series in which the angles increase uniformly with the time, then all the powers and products of these qτιantities, and all their differentials and integrals with respect to the time, may be expressed in series of the same form. This theorem needs only an illustration by an example. Let our quantities be X and Y, and let us suppose them expressed in the form X—a cos A + b cos B + c cos C ’ + , &c.

Y-a ’ sin A ’ + b ’ sin B ’ + c' sin C + , &c., in which λvθ may suppose that the quantities a, b, c, &c., converge towards zero. In forming their product, the first term will be aa' cos A sin A'. But we have cos A sin A' = ⅛ sin {A' + A) + ⅜ sin {A' - A). Hence the product X Y will be of the form X Y= ⅛ aa' sin {A' + A ) + ⅛ αα'sin (A' - A) + ⅜ ab' sin (Λ + B' + ),&c., which is another series of the same general form. Moreover, if we suppose the angles A, B, &c., to increase uniformly with the time—that is, to admit of expression in the form A = a + mt, A'=a' + m't, &c— we shall have, by integrating, 2 ∫ xrM = ∞ s < j '^' l -' kc ∙ which, again, is a trigonometric series of the same general form, which admits of being manipulated at pleasure in the same way as the original expressions X and Y. This property does not belong to the elliptic functions, and in consequence, notwithstanding the great length of the trigonometric series, no attempt to supersede them has been successful.

The efforts to express the moon’s motion by integrating the differential equations of the dynamical theory may be divided into three classes. (1) Laplace and his immediate successors found the problem so complex that they sought to simplify it by reversing its form; instead of trying from the beginning to express the moon’s coordinates in terms of the time, they effected the integration by expressing the time in terms of the moon’s true longitude. Then, by a reversal of the series, the longitude was expressed in terms of the time. Although it would be hazardous to say that this method is unworthy of further consideration, we must admit that its essential inelegance is such as to repel rather than attract study, and that it holds out no promise of further development. (2) By the second general method the moon’s coordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining the algebraic symbols which express 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 0Ό17 ; 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; and 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. It should be remarked that the solution by the first method appears in the same form as by this one after the true longitude is expressed in terms of the mean longitude. (3) By the method just mentioned 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 insure 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. Hence, when numerical precision alone is aimed at, it has been found best to avoid this difficulty by using the numerical values of the elements instead of their algebraic symbols. This method has the advantage of leading to the more rapid and certain determination of the numerical values of the several coefficients of sines and cosines. 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. And among these we must assign the first rank to the method of Delaunay, developed...