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Moon




THE MOON. The subject of the moon divides itself into two separate branches, the one concerned with the con-stitution of the lunar globe, the other with its motions. For the first subject the reader is referred to the article ASTRONOMY (vol. ii. p. 801 sq.); the present article is con-fined to the second, which is commonly called the Lunar Theory.

The lunar theory does not yet form a well-defined body of reasoning and doctrine, like other branches of mathematical science, but consists only of a series of researches, extending through twenty centuries or more, and incapable of being welded into a consistent whole.

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 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 byan 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 Apiogee.— 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°p29 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 2D = 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 - lo,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 g + sin (2D-g)} = l°-27 {(1 - cos 2D) sing + sin 2Z> cosg) . At quadratures we have D= ±90°, 22) = 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 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 Revolutionibus, 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 particu-lar direction forms little more than an episode in the his-tory 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 hypo-thesis 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 compound-ing circular motions indefinitely—in other words, of adding one epicycle to another—Copernicus was enabled to repre-sent 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 Abu 'l-Wefa, an Arabian astro-nomer, and has made it appear probable that Abu '1-Wefa 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 de-scription 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 intro-ducing 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 deter-mination 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 geometri-cal 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 con-struct 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 hy geometrical constructions. But when the problem 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 alge-braic 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 expresses Newton's first two laws of motion with a com-pleteness and precision which is entirely wanting in all statements in ordinary language. The latter can be no-thing more than lame attempts to express the equation in language which may be understood by the non-mathe-matical 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, perpen-dicular 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 attract-ing 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 ele-mentary 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 advan-tage in the problems of celestial mechanics. It may be expressed thus :—If we have any number of quantities, each of which is ex-piressed in the form of a trigonometric series in which the angles increase uniformly with the time, then all the powers and products of these quantities, and all their differentials and integrals with respect to the time, may be expressed in scries of the same form. This theorem needs only an illustration by an example. Let our quan-tities 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' + V sin B + c' sin C +, &c., in which we may suppose that the quantities a, b, c, &c., converge towards zero. In forming their product, the first term will be

act! cos A sin A'. But we have cos A sin A' = ^ sin (A' + A) + \ sin (A! -A). Hence the product X Y will be of the form
X Y=i aa' sin (A' + A) + J aa'sin (A! - A) + i 016' sin (^1 + _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,
f aa . aa'
2 XYdt= -—;—,cos (A +A) - —; cos (A - A), &c,
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, notwith-standing 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 0-017; y, the sine of half the inclination of the moon's orbit, about 0'(M6; 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 accord-ing to the powers of the above small numbers. This method was applied with success by Pontecoulant 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 in his Theorie du Mouvement de la Lune, because it contains a germ which may yet develop into the great desideratum of a general method in celestial mechanics. To explain it, we must call to mind the general method of " variation of elements," due to Lagrange. This method is applicable to cases in which a problem of dynamics can be completely solved when any small forces which come into play are left out, but which does not admit of direct solution when these forces are included. Omitting the small forces, commonly called " disturbing forces," let us suppose the problem of the motion of a body under the influence of the "principal forces" completely solved. This will mean that we have found algebraic expressions for the coordi-nates which determine the position of the body in terms of the time, and (in the case of a material point) of six constant quantities, to which we may assign values at pleasure. Then Lagrange showed how, by supposing these constant quantities to become variable, the same expressions could be used for the case in which the effect of the disturbing forces was included. In other words, the effect of the disturbing forces could be determined by assuming them to change the constants of the first approxi-mate solution into very slowly varying elements.

In the researches on the lunar theory before Delaunay the principal force was taken to be the attraction of the earth upon the moon, and the disturbing force was that due to the sun's attraction. When the action of the earth alone was included the moon would move in an ellipse, in accord-ance with Kepler's laws. The effect of the sun's action could be allowed for by supposing this ellipse to be mov-able and variable. But when it was required to express this variation the problem became excessively complicated, owing to the great number of terms required to express the sun's disturbing force. Now, instead of passing from the elliptic to the disturbed motion by one single difficult step, Delaunay effected the passage by a great number of easy steps. Out of several hundred periodic terms, the sum of which expressed the disturbing force of the sun, he first took one only, and determined the variations of the Keplerian ellipse on the supposition that this term was the only one. In the solution the variable elements of the ellipse would be expressed in terms of six new con-stants. He then showed how these new constants could be taken as variables instead of the elements of the original ellipse. Taking a second term of the disturbing force, he expressed the new constants in terms of a third set of con-stants, and so repeated the process until all the terms of the disturbing force were disposed of.

Among applications of the third or numerical method, the most successful yet completed is that of Hansen. His first work appeared in 1838, under the title Funda-menta nova investigationis orbitx verm quam luna perlustrat, and contained an exposition of his ingenious and peculiar methods of computation. During the twenty years follow-ing he devoted a large part of his energies to the numerical computation of the lunar inequalities, the re-determination 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 Saxon Boyal Society of Sciences.

It is found on comparing the results of Hansen and Delaunay that there are some outstanding discrepancies, which, though too small to be of great practical importance, are of sufficient magnitude to demand the attention of those interested in the mathematical theory of the subject. It is therefore desirable that the numerical inequalities should be again determined by an entirely different method. This is the object of Sir G. B. Airy's Numerical Lunar Theory, which is not yet completely published, but is sufficiently far advanced to give hopes of an early comple-tion. The essence of Sir George's method consists in starting with a provisional approximate solution (that of Delaunay being accepted for the purpose), and substituting the expressions for the moon's coordinates in the funda-mental differential equations of the moon's motion as dis-turbed by the sun. If the theory were perfect, the two sides of each equation would come out equal. As they do not come out exactly equal, Sir George puts the problem in the form : What corrections must be applied to the expressions for the coordinates that the two sides may be made equal 1 He then shows how these corrections may be found by solving a system of equations.

The several methods which we have described have for their immediate object the determination of the motion of the moon round the earth under the influence of the combined attractions of the earth and sun. In other words, the question is that of solving the celebrated "problem of three bodies" in the special case when one of the bodies, the sun, has a much greater mass than the other two, and is at a much greater distance from them than they are from each other. All methods lead to a solution of the same general form which we shall now describe. Lot us put g the moon's mean anomaly ; g' the mean anomaly of the sun (or earth) ; w the angular distance of the lunar perigee from the moon's node on the ecliptic ; w' the angular distance of the sun's perigee from the moon's node on the ecliptic. When no account is taken of the action of the sun the angles g and g' increase uniformly with the time, representing in fact the uniform motion of the moon round the earth and of the earth round the sun, while w and a' remain constant. When account is taken of the action of the sun all four of the angles change with a uniform progressive motion. In conse-quence, the mean orbit of the moon round the earth becomes a moving ellipse whose major axis makes a revolution round the earth in about nine years, and the line of whose nodes makes a revolution in about eighteen and a half years. All the other ele-ments of this ellipse—namely, its major axis, its eccentricity, and its inclination to the ecliptic—remain absolutely constant however long the motion may continue, unless some other disturbing force than that of the sun comes into play. But in the actual motion of the moon there are periodic deviations from this ellipse, which may be represented by an infinite trigonometric series, each term of which is of the form
c (sin or cos) (ig + i'g' + ju +/«'), in which the quantities c are absolutely constant coefficients, and i, i', j, and / are integers which may take all combinations of values —positive, negative, or zero. The circular function is, a sine in the expression for longitude or latitude, a cosine in the expression for the parallax. Also, j and / must be both even or both odd in the expressions for longitude and parallax, but the one even and the other odd in the ease of the latitude. For example, if we sup-pose j, f, and i all zero, we shall have terms of the form c1 sin g' + c2 sin Ig' + c3 sin 3g' +, &c. To write other terms, suppose 1 = 1, then we have terms of the form
«1 sm (g-d) + H sin (g+g) + % sin (g + 2gf) +, &c. Taking the case when j = 2 and / = - 2, we shall have terms of the form
»! sin (g - g' + 2« - loi') + m3 sin (g - 2g' + 2« - 2u') +, &c.

As the indices i, i!, j, and / become larger, the coefficients c, e, in, &c., become smaller; but the number of terms included in the theories of Hansen and Delaunay amount to several hundreds. In the analytical theories, like that of Delaunay, each of the coeffi-cients c, e, m, &c, is a complicated infinite series, but in the numerical theories it is a constant number. And the principal problem of the modern theory of three bodies is to find the appropriate co-efficient for each of these hundreds of terms.





Action of the Planets on the Moon.— For nearly two centuries it has been known from observations that the mean motion of the moon round the earth is not absolutely constant, as it ought to be were there no disturbing body but the sun. The general fact that the motion has been accelerated since the time of Ptolemy was first pointed out by Halley, and the amount of the acceleration was found by Dunthorne. After vain efforts by the greatest mathe-maticians of the last century to find a physical cause for the acceleration, Laplace was successful in tracing it to the secular diminution of the eccentricity of the earth's orbit, produced by the action of the planets. He computed its amount to be 10" per century—that is, if the place of the moon were calculated forward on its mean motion at the beginning of any century, it would at the end of the century be 10' in advance of its computed place. This theoretical result of Laplace agreed so closely with the acceleration found by Lalande from the records of ancient and mediaeval eclipses that it was not questioned for nearly a century. In 1852 Mr John C. Adams showed that Laplace had failed to take account of a series of terms, the effect of which was to reduce the acceleration to 6" or less. The result was inconsistent with the accounts of ancient eclipses of the sun, and a cause for the discrepancy had to be sought for. A probable cause was pointed out, first by Ferrel, and afterwards by Delaunay. The former, in papers published in Gould's Astronomical Journal, and in the Proceedings of the American Academy of Arts and Sciences, showed that the action of the moon on the tidal waves of the ocean would have the effect of increasing the time of the earth's axial rotation or the length of the day, which is necessarily taken as the unit of time. Since, as the days became longer, the moon would move farther in one day, though its absolute motion should remain unchanged, and hence an apparent acceleration would be the result. That this cause really acts there can be no doubt. But the data for determining its exact amount are discrepant. If we take only such data as are purely astronomical—namely, the eclipses recorded by Ptolemy between 720 B.C. and 150 A.D., and those observed by the Arabians between 800 and 1000 A.D.—the apparent excess of the observed acceleration to be accounted for by the tidal retarda-tion amounts to only 2" per century, and may be even less. But this small acceleration is entirely incompatible with conclusions drawn from certain supposed accounts of total eclipses of the sun, notably the eclipse associated with the name of Thales. This is the famous eclipse supposed to be alluded to by Herodotus wdien he describes a battle as stopped by a sudden advent of darkness, which had been predicted by Thales. If the true value of the co-efficient resulting from the combined effect of tidal retardation of the earth and secular acceleration of the moon is less than 10", then not only could the path of totality not have passed over the field of battle but the greatest eclipse could not have occurred till after sunset. In fact, to represent this and other supposed eclipses of the sun, the acceleration must be increased to about 12", wdrich is near the value found by Hansen from theory, and adopted in his tables of the moon. But his theoretical computation is un-doubtedly incorrect, because in computing in what manner the eccentricity of the earth's orbit enters into the moon's motion he took account only of the first approximation, as Laplace had done. The following is a summary of the present state of the question: — The theoretical value of the acceleration, assuming the day to be constant, is, according to Delaunay 6"'176
Hansen's value, in his Tables etc la Lunc, is 12"18
Hansen's revised but still theoretically erroneous result is 12 '56 The value which best represents the supposed eclipses
(1) of Thales, (2) at Larissa, (3) at Stikkelstad, is about 11 "7
The result from purely astronomical observations is 8'3
The result from Arabian and modern observations alone
is about 7'0

Inequalities of Long Period. — Combined with the question of the secular acceleration is another which is still entirely unsettled— namely, 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. All efforts to find a satisfactory explanation were, however, so unavailing that Poisson, in 1835, disputed the reality of the inequality. But Airy, from his discussion of the Greenwich observations between 1750 and 1830, conclusively proved its existence. About the same time Hansen announced that he had found from theory two terms of long period arising from the action of Venus which fxilly corre-sponded to the inequalities indicated by the observations. These terms, as employed in his Tables de la Lunc, are
15"-34 sin (~g- 163' +18/' + 33° 36') + 21"'47 sin (8/' - 13/ + 4° 44'),
in which g, g', and g" represent the mean anomalies of the moon, the earth, and Venus respectively. During the first few years after the publication of Hansen's tables they represented observa-tions so well that their entire correctness was generally taken for granted. But doubt soon began to be thrown upon the inequalities of long period just mentioned. Indeed, Hansen himself admitted that the second and larger term was partly empiri-cal, being taken so as to satisfy observations between 1750 and 1850. Delaunay re-computed both terms, and found for the first term a result substantially identical with that of Hansen. But he found for the second or empirical one a coefficient of only 0"'27, which would be quite insensible. With this smaller coefficient the obser-vations from 1750 could not be satisfied, so that, so far as observa-tions could go in deciding a purely mathematical question, the evidence was in favour of Hansen's result. But on comparing Hansen's tables with observations between 1650 and 1750 it was found that the supposed agreement with observation was entirely illusory. Moreover, since 1865 the moon has been steadily falling behind the tabular place. These inequalities of long period have not yet been satisfactorily explained. The most plausible supposi-tion is that they are due to the action of one or more of the larger planets. But the problem of the action of the planets on the moon is the most difficult and intricate of celestial mechanics, and no satisfactory general method of attacking it has yet been found. The sources of difficulty are two in number. First, the disturbing action of the planets is modified by that of the sun in such a way that the ordinary equations of disturbed elliptic motion are no longer rigorous, and hence new and more complicated ones must be constructed. And, secondly, the combination of the four bodies —moon, earth, sun, and planet—leads to terms so numerous and intricate that it has hardly been found possible to isolate them. The question has, indeed, been raised whether the rotation of the earth on its axis, and hence the unit of time, may not be subject to slow and irregular changes of a nature to produce apparent corre-sponding changes in the motion of the moon. But it has recently been found, from a discussion of the observed transits of Mercury since 1677, that, although such inequalities may exist, they cannot have the magnitude necessary to account for the observed changes of long period in the moon's motion.

The following is a summary of the present state of the various branches of the lunar theory. (1) The numerical solution of the problem of the sun's action on the moon may be regarded as quite satisfactory, at least when Hansen's results shall have been verified by an independent method. (2) The analytic theory needs to be perfected by finding some remedy for the slow convergence of the series by which it is expressed, but its general form may be regarded as quite satisfactory. (3) Except in one or two special cases, the action of the planets on the moon, when treated with the necessary rigour, is so intricate that no approach to a satisfactory solution has yet been attained. When this desideratum is reached, the mathematical theory will be complete. (4) The general discussion of ancient and modern observations with a view to finding what real or apparent inequalities of long period in the mean motion may exist is still to be finished. With it the astronomical theory will be complete. (S. N.)


Footnotes

The fall title, De Revolutionibus Orbium Coelestium Libri VI. (small folio, Nuremberg, 1543).




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