The broken and irregular character of the surface is most evident near the boundary between the dark and illuminated portions, about the time of first quarter. The most remarkable 3 to M, about N N' feature of the surface comprises the craters, which are scattered everywhere, and generally surrounded by an approximately circular elevated ring. Yet another remarkable feature comprises bright streaks, branching out in various directions and through long distances from a few central points, especially that known as Tycho.
The height of the lunar mountains is a subject of interest. It cannot be stated with the same definiteness that we can assign heights to our terrestrial mountains, because there is no fixed sea-level on the moon to which elevations can be referred. The only determination that can be made on the moon is that of the height above some neighbouring hollow, crater or plain. The most detailed measures of this sort were made by Beer and Madler, who give a great number of such heights. These generally range between 500 and 3000 toises, or 3000 and 20,000 English feet. The highest which they measured was Newton, 3727 toises, or 24,000 ft.
The general trend of lunar investigation has been against the view that there is any resemblance between the surfaces of the moon and of the earth, except in the general features already mentioned. No evidence has yet been found that the moon has either water or air. The former, if it existed at all, could be found only in the more depressed portions; and even here it would evaporate under the influence of the sun's rays, forming a vapour which, if it existed in considerable quantity, would in some way make itself known to our scrutiny. The most delicate indication of an atmosphere would be through the refraction of the light of a star when seen coincident with the limb of the moon. Not the slightest change in the direction of such a star when in this position has ever been detected, and it is certain that if any occurs it can be but a minute fraction of a second of arc. As an atmosphere equal to ours in density would produce a deviation of an important fraction of a degree, it may be said that the moon can have no atmosphere exceeding in density the b b l o o that of the earth.
Devoid of air and atmosphere, the causes of meteorological phenomena on the earth are non-existent on the moon. The only active cause of such changes is the varying temperature produced by the presence or absence of the sun's rays. The range of temperature must be vastly wider than on the earth, owing to the absence of an atmosphere to make it equable. Elaborate observations of the heat coming from the moon at its various phases were made and discussed in1871-1872by Lord Rosse. Among his results was that during the progressive phases from before the first quarter till the full moon the heat received increases in a much greater proportion than the light, from which it followed that the former was composed mainly of heat radiated from the moon itself in consequence of the temperature which it assumed under the sun's rays. So far as could be determined, 86% of the heat radiated was by the moon itself, and 14% reflected solar heat. But it seems probable that this disproportion may be somewhat too great. Rosse's determinations, like those of his predecessors, were made with the thermopile. After S. P. Langley devised his bolometer, which was a much more sensitive instrument than the thermopile, he, in conjunction with F. W. Very, applied it to determine the moon's radiation at the Allegheny observatory. His results for the ratio of the total radiation of the full moon to that of the sun ranged from 1: 70,000 to 1: 110,000, which were in substantial agreement with those of Rosse, who found 1: 82,000. When Langley published his work the law of radiation as a function of the temperature was not yet established. He therefore wrongly concluded that the highest temperature reached by the moon approximated to the freezingpoint of water. Stefan's law of radiation, on the other hand, shows that the temperature must have been about the boilingpoint in order that the observed amount of heat might be radiated. This is in fair agreement with the computed temperature due to the sun's radiation upon a perpendicular absorbing surface when no temperature is lost through conduction to the interior. The agreement thus brought about between the results deduced from the law of radiation and the most delicate observa tions of the quantity of heat radiated is of great interest, as showing that the theory of cosmicai temperature now rests upon a sound basis. There is, however, still room for improved determinations of the moon's heat by the use of the bolometer in its latest form. Possibility of Changes on the Moon. - No evidence of life on the moon has ever been brought out by the minutest telescopic scrutiny, nor does life seem possible in the absence of air and water. Some bright spots are visible by the earth-light when the moon is a thin crescent, which were supposed by Herschel to be volcanoes in eruption. But these are now known to be nothing more than spots of unusual whiteness, and if any active volcano exists it is yet to be discovered. Still, the question whether everything on the moon's surfaceis absolutely unchangeable is as yet an open one, with the general trend of opinion toward the affirmative, so far as any actual proof from observation is concerned. The spot which has most frequently exhibited changes in appearance is near the centre of the visible disk, marked on Beer and Madler's map as Linne. This has been found to present an aspect quite different from that depicted on the map, and one which varies at different times. But the question still remains open whether these variations may not be due wholly to the different phases of illumination by the sunlight as the latter strikes the region from various directions. Intensity of Moonlight. - An interesting and important quantity is the ratio of moonlight to sunlight. This has been measured for the full moon by various investigators, but the results are not as accordant as could be desired. The most reliable determinations were made by G. P. Bond at Harvard and F. Zollner at Leipzig, in 1860 and 1864. The mean result of these two determinations is the ratio 1: 570,000. We may therefore say that the intensity of sunlight is somewhat more than half a million times that of full moonlight. A remarkable feature of the reflecting power of the moon, which was made known by Miner's observations, is that the proportion of light reflected by a region on the moon is much greater when the light falls perpendicularly, which is the case near the time of full moon, and rapidly becomes less as the light is more oblique. This result was traced by Z011ner to the general irregularity of the lunar surface, and the inference was drawn that the average slope of the lunar elevation amounts to 47°.
Motion of the Moon
The orbit of the moon around the earth, though not a fixed curve of any class, is elliptical in form, and may be represented by an ellipse which is constantly changing its form and position, and has the earth in one of its foci. The eccentricity of the ellipse is in the general average about 0.055, whence the moon is commonly more than i' further from the earth at apogee than at perigee. The line of apsides is in continual motion, generally direct, and performs a revolution in about 12 years. The inclination to the ecliptic is a little more than 5°, and the line of nodes performs a revolution in the retrograde direction in 18.6 years. The parallax of the moon is determined by observation from two widely separated points; the most accurate measures are those made at Greenwich and at the Cape of Good Hope. The distance of the moon can also be computed from the law of gravity, the problem being to determine the distance at which a body having the moon's mass would revolve around the earth in the observed period. The measures of parallax agree perfectly with the computed distance in showing a mean parallax of 57' 2.8", and a mean distance of 238,800 miles. The period of revolution, or the lunar month, depends upon the point to which the revolution is referred. Any one of five such directions may be chosen, that of the sun, the fixed stars, the equinox,. the perigee, or the node. The terms synodical, sidereal, tropical, anomalistic, nodical, are applied respectively to these months, of which the lengths are as follow: - N This diagram is a key to some of the features reproduced in the photograph.
I. Tycho, Theophilus, 1900, October 12.
2. Mare Nubilm, &c., 1901, November 21.
?
Other numerical particulars relating to the moon are: Mean distance from the earth (earth's radius as I) .. 60.2634 Mean apparent diameter. .. 31 ' 515 Diameter in miles 2159.6 Moon's surface in square miles .... 14,600,000 Diameter (earth's equatorial diameter as I). 0.2725 Surface (earth's as I).. 0.0742 Volume (earth's as I). .. .. 0.0202 Ratio of mass to earth's mass 1.. I :81.53 t 047 Density (earth's as I) 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. .. I° 30' 113" 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 :- I. 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 1 A. R. Hinks, " Mass of the Moon, from Observations of Eros, 1900-1901," M. N. Roy. Ast. Soc., 1909, Nov., p. 73.
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 - I. 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+I 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 I°, 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 o017: y, 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 Pontecoulant 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 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 Theorie 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', y 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 in1877-1888Hill 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 Pontecoulant, 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 1 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 Pontecoulant stoutly maintained the correctness of the older result. But the demonstration of Adam's result was soon made 1 Philosophical Transactions, 1853.
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: The theoretical value of the acceleration, assuming the day to be constant, is ..... 5.91 " Hansen's value in his Tables de la lune is.. 12.19 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 111 The result from purely astronomical observation is 8.3 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 rpoon 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.
Bibliography. - Works on selenography: Hevelius, Selenographia sive lun g e descriptio (Danzig, 1647); Riccioli, Almagestum novum (Bologna, 1651); J. H. Schroeter, Selenotopographische Fragmente zur genauern Kenntniss der Mondflache (Lilienthal, 1791); W. Beer and J. H. Madler, Der Mond nach seinen kosmischen and individuellen Verhdltnissen, 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, Me'moires, tome xxii.
The following are among the works relating to the motion of the moon, which are of historic importance or present interest to the student: Clairaut, Theorie de la lune (2nd ed., Paris, 1765); L. Euler, Theoria motuum lunae nova methodo pertractata (Petropolis, 1772); G. Plana, Theorie du mouvement de la lune (3 vols., Turin, 1832); P. A. Hansen, Fundamenta nova investigationis orbitae verae quam luna perlustrat (Gotha, 1838); Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Storungen (Leipzig, 1862); C. Delaunay, Theorie du mouvement de la lune (2 vols., Paris, 1860-1867); F. F. Tisserand, Traite de mecanique celeste, tome iii., Exposé de l'ensemble des theories relatives au mouvement de la lune (Paris, 1894); E. W. Brown, " Theory of the Motion of the Moon," Memoirs of the Royal Astronomical Society, various vols.; also Transactions of the American Mathematical Society, vols. iv. and vi.; E. W. Brown, Introductory Treatise on the Lunar Theory (Cambridge University Press, 1896); Hansen, Tables de la lune (London, 1857) (Admiralty publication); W. Ferrel, " On the Effect of the Sun and Moon on the Rotary Motion of the Earth," Astron. Jour., vol. iii. (1854); S. Newcomb, " Researches on the Motion of the Moon " (Appendix to Washington Observations for 1875, discussion of the moon's mean motion); S. Newcomb, " Transformation of Hansen's Lunar Theory," Ast. Papers of the Amer. Ephemeris, vol. i.; R. Radau, " Inegalites planetaires du mouvement de la lune " (Annales, Paris Observatory, vol. xxi.); S. Newcomb, " Action of the Planets on the Moon," Ast. Papers of the Amer. Ephemeris, vol. v., pt. 3 (1896). Also, Publication 72 of the Carnegie Institution of Washington (1907); E. W. Brown, Inequalities in the Moon's Motion produced by the Action of the Planets (the Adams prize essay for 1907). (S. N.)
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Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Moon'. 1911 Encyclopedia Britanica. https://www.studylight.org/encyclopedias/eng/bri/m/moon.html. 1910.
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