"Moon of harvest, herald mildOf plenty, rustic labor's child,Hail, O hail! I greet thy beam,As soft it trembles o'er the stream,And gilds the straw-thatch'd hamlet wide,Where innocence and peace reside;Tis thou that glad'st with joy the rustic throng,Promptest the tripping dance, th' exhilarating song."
"Moon of harvest, herald mildOf plenty, rustic labor's child,Hail, O hail! I greet thy beam,As soft it trembles o'er the stream,And gilds the straw-thatch'd hamlet wide,Where innocence and peace reside;Tis thou that glad'st with joy the rustic throng,Promptest the tripping dance, th' exhilarating song."
To understand the reason of the harvest moon, we will, as before, consider the moon's orbit as coinciding with the ecliptic, because we may then take the ecliptic, as it is drawn on the artificial globe, to represent that orbit. We will also bear in mind, (what has been fully illustrated under the last head,) that, since the ecliptic cuts the meridian obliquely, while all the circles of diurnal revolution cut it perpendicularly, different portions of the ecliptic will cut the horizon at different angles. Thus, when the equinoxes are in the horizon, the ecliptic makes a very small angle with the horizon; whereas, when the solstitial points are in the horizon, the same angle is far greater. In the former case, a body moving eastward in the ecliptic, and being at the eastern horizon at sunset, would descend but a little way below the horizon in moving over many degrees of the ecliptic. Now, this is just the case of the moon at the time of the harvest home, about the time of the Autumnal equinox. The sun being then in Libra, and the moon, when full, being of course opposite to the sun, or in Aries; and moving eastward, in or near the ecliptic, at the rate of about thirteen degrees per day, would descend but a small distance below the horizon for five or six days in succession; that is for two or three days before, and the same number of days after, the full; and would consequently rise during all these evenings nearly at the same time, namely, a little before, or a little after, sunset, so as to afford a remarkable succession of fine moonlight evenings.
The moonturns on her axisin the same time in which she revolves around the earth. This is known by the moon's always keeping nearly the same face towards us, as is indicated by the telescope, which could not happen unless her revolution on her axis kept pace with her motion in her orbit. Take an apple, to represent the moon; stick a knittingneedle through it, in the direction of the stem, to represent the axis, in which case the two eyes of the apple will aptly represent the poles. Through the poles cut a line around the apple, dividing it into two hemispheres, and mark them, so as to be readily distinguished from each other. Now place a candle on the table, to represent the earth, and holding the apple by the knittingneedle, carry it round the candle, and you will see that, unless you make the apple turn round on the axis as you carry it about the candle, it will present different sides towards the candle; and that, in order to make it always present the same side, it will be necessary to make it revolve exactly once on its axis, while it is going round the circle,—the revolution on its axis always keeping exact pace with the motion in its orbit. The same thing will be observed, if you walk around a tree, always keeping your face towards the tree. If you have your face towards the tree when you set out, and walk round without turning, when you have reached the opposite side of the tree, your back will be towards it, and you will find that, in order to keep your face constantly towards the tree, it will be necessary to turn yourself round on your heel at the same rate as you go forward.
Since, however, the motion of the moon on its axis is uniform, while the motion in its orbit is unequal, the moon does in fact reveal to us a little sometimes of one side and sometimes of the other. Thus if, while carrying the apple round the candle, you carry it forward a little faster than the rate at which it turns on its axis, a portion of the hemisphere usually out of sight is brought into view on one side; or if the apple is moved forward slower than it is turned on its axis, a portion of the same hemisphere comes into view on the other side. These appearances are called the moon'slibrations in longitude. The moon has also alibration in latitude;—so called, because in one part of her revolution more of the region around one of the poles comes into view, and, in another part of the revolution, more of the region around the other pole, which gives the appearance of a tilting motion to the moon's axis. This is owing to the fact, that the moon's axis is inclined to the plane of her orbit. If, in the experiment with the apple, you hold the knittingneedle parallel to the candle, (in which case the axis will be perpendicular to the plane of revolution,) the candle will shine upon both poles during the whole circuit, and an eye situated where the candle is would constantly see both poles; but now incline the needle towards the plane of revolution, and carry it round, always keeping it parallel to itself, and you will observe that the two poles will be alternately in and out of sight.
The moon exhibits another appearance of this kind, called herdiurnal libration, depending on the daily rotation of the spectator. She turns the same face towards thecentreof the earth only, whereas we view her from the surface. When she is on the meridian, we view her disk nearly as though we viewed it from the centre of the earth, and hence, in this situation, it is subject to little change; but when she is near the horizon, our circle of vision takes in more of the upper limb than would be presented to a spectator at the centre of the earth. Hence, from this cause, we see a portion of one limb while the moon is rising, which is gradually lost sight of, and we see a portion of the opposite limb, as the moon declines to the west. You will remark that neither of the foregoing changes implies any actual motion in the moon, but that each arises from a change of position in the spectator. Since the succession of day and night depends on the revolution of a planet on its own axis, and it takes the moon twenty-nine and a half days to perform this revolution, so that the sun shall go from the meridian of any place and return to the same meridian again, of course the lunar day occupies this long period. So protracted an exposure to the sun's rays, especially in the equatorial regions of the moon, must occasion an excessive accumulation of heat; and so long an absence of the sun must occasion a corresponding degree of cold. A spectator on the side of the moon which is opposite to us would never see the earth, but one on the side next to us would see the earth constantly in his firmament, undergoing a gradual succession of changes, corresponding to those which the moon exhibits to the earth, but in the reverse order. Thus, when it is full moon to us, the earth, as seen from the moon, is then in conjunction with the sun, and of course presents her dark side to the moon.
Soon after this, an inhabitant of the moon would see a crescent, resembling our new moon, which would in like manner increase and go through all the changes,from new to full, and from full to new, as we see them in the moon. There are, however, in the two cases, several striking points of difference. In the first place, instead of twenty-nine and a half days, all these changes occur in one lunar day and night. During the first and last quarters, the changes would occur in the day-time; but during the second and third quarters, during the night. By this arrangement, the lunarians would enjoy the greatest possible benefit from the light afforded by the earth, since in the half of her revolution where she appears to them as full, she would be present while the sun was absent, and would afford her least light while the sun was present. In the second place, the earth would appear thirteen times as large to a spectator on the moon as the moon appears to us, and would afford nearly the same proportion of light, so that their long nights must be continually cheered by an extraordinary degree of light derived from this source; and if the full moon is hailed by our poets as "refulgent lamp of night,"[10]with how much more reason might a lunarian exult thus, in view of the splendid orb that adorns his nocturnal sky! In the third place, the earth, as viewed from any particular place on the moon, would occupy invariably the same part of the heavens. For while the rotation of the moon on her axis from west to east would appear to make the earth (as the moon does to us) revolve from east to west, the corresponding progress of the moon in herorbit would make the earth appear to revolve from west to east; and as these two motions are equal, their united effect would be to keep the moon apparently stationary in the sky. Thus, a spectator at E, Fig. 38, page 175, in the middle of the disk that is turned towards the earth, would have the earth constantly on his meridian, and at E, the conjunction of the earth and sun would occur at mid-day; but when the moon arrived at G, the same place would be on the margin of the circle of illumination, and will have the sun in the horizon; but the earth would still be on his meridian and in quadrature. In like manner, a place situated on the margin of the circle of illumination, when the moon is at E, would have the earth in the horizon; and the same place would always see the earth in the horizon, except the slight variations that would occur from the librations of the moon. In the fourth place, the earth would present to a spectator on the moon none of that uniformity of aspect which the moon presents to us, but would exhibit an appearance exceedingly diversified. The comparatively rapid rotation of the earth, repeated fifteen times during a lunar night, would present, in rapid succession, a view of our seas, oceans, continents, and mountains, all diversified by our clouds, storms, and volcanoes.
"Some say the zodiac constellationsHave long since left their antique stations,Above a sign, and prove the sameIn Taurus now, once in the Ram;That in twelve hundred years and odd,The sun has left his ancient road,And nearer to the earth is come,'Bove fifty thousand miles from home."—Hudibras.
"Some say the zodiac constellationsHave long since left their antique stations,Above a sign, and prove the sameIn Taurus now, once in the Ram;That in twelve hundred years and odd,The sun has left his ancient road,And nearer to the earth is come,'Bove fifty thousand miles from home."—Hudibras.
Wehave thus far contemplated the revolution of the moon around the earth as though the earth were atrest. But in order to have just ideas respecting the moon's motions, we must recollect that the moon likewise revolves along with the earth around the sun. It is sometimes said that the earthcarriesthe moon along with her, in her annual revolution. This language may convey an erroneous idea; for the moon, as well as the earth, revolves around the sun under the influence of two forces, which are independent of the earth, and would continue her motion around the sun, were the earth removed out of the way. Indeed, the moon is attracted towards the sun two and one fifth times more than towards the earth, and would abandon the earth, were not the latter also carried along with her by the same forces. So far as the sun acts equally on both bodies, the motion with respect to each other would not be disturbed. Because the gravity of the moon towards the sun is found to be greater, at the conjunction, than her gravity towards the earth, some have apprehended that, if the doctrine of universal gravitation is true, the moon ought necessarily to abandon the earth. In order to understand the reason why it does not do thus, we must reflect, that, when a body is revolving in its orbit under the influence of the projectile force and gravity, whatever diminishes the force of gravity, while that of projection remains the same, causes the body to approach nearer to the tangent of her orbit, and of course to recede from the centre; and whatever increases the amount of gravity, carries the body towards the centre. Thus, in Fig. 33, 152, if, with a certain force of projection acting in the direction A B, and of attraction, in the direction A C, the attraction which caused a body to move in the line A D were diminished, it would move nearer to the tangent, as in A E, or A F. Now, when the moon is in conjunction, her gravity towards the earth acts in opposition to that towards the sun, (see Fig. 38, page 175,) while her velocity remains too great to carry her with what force remains, in a circle about the sun, and she therefore recedes from the sun, and commences herrevolution around the earth. On arriving at the opposition, the gravity of the earth conspires with that of the sun, and the moon's projectile force being less than that required to make her revolve in a circular orbit, when attracted towards the sun by the sum of these forces, she accordingly begins to approach the sun, and descends again to the conjunction.
The attraction of the sun, however, being every where greater than that of the earth, the actual path of the moon around the sun is every where concave towards the latter. Still, the elliptical path of the moon around the earth is to be conceived of, in the same way as though both bodies were at rest with respect to the sun. Thus, while a steam-boat is passingswiftlyaround an island, and a man is walkingslowlyaround a post in the cabin, the line which he describes in space between the forward motion of the boat and his circular motion around the post, may be every where concave towards the island, while his path around the post will still be the same as though both were at rest. A nail in the rim of a coach-wheel will turn around the axis of the wheel, when the coach has a forward motion, in the same manner as when the coach is at rest, although the line actually described by the nail will be the resultant of both motions, and very different from either.
We have hitherto regarded the moon as describing a great circle on the face of the sky, such being the visible orbit, as seen by projection. But, on a more exact investigation, it is found that her orbit is not a circle, and that her motions are subject to very numerous irregularities. These will be best understood in connexion with the causes on which they depend. The law of universal gravitation has been applied with wonderful success to their developement, and its results have conspired with those of long-continued observation, to furnish the means of ascertaining with great exactness the place of the moon in the heavens, at any given instant of time, past or future, and thus to enable astronomers to determine longitudes, to calculate eclipses,and to solve other problems of the highest interest. The whole number of irregularities to which the moon is subject is not less than sixty, but the greater part are so small as to be hardly deserving of attention; but as many as thirty require to be estimated and allowed for, before we can ascertain the exact place of the moon at any given time. You will be able to understand something of the cause of these irregularities, if you first gain a distinct idea of the mutual actions of the sun, the moon, and the earth. The irregularities in the moon's motions are due chiefly to the disturbing influence of the sun, which operates in two ways; first, by acting unequally on the earth and moon; and secondly, by acting obliquely on the moon, on account of the inclination of her orbit to the ecliptic. If the sun acted equally on the earth and moon, and always in parallel lines, this action would serve only to restrain them in their annual motions around the sun, and would not affect their actions on each other, or their motions about their common centre of gravity. In that case, if they were allowed to fall towards the sun, they would fall equally, and their respective situations would not be affected by their descending equally towards it. But, because the moon is nearer the sun in one half of her orbit than the earth is, and in the other half of her orbit is at a greater distance than the earth from the sun, while the power of gravity is always greater at a less distance; it follows, that in one half of her orbit the moon is more attracted than the earth towards the sun, and, in the other half, less attracted than the earth.
To see the effects of this process, let us suppose that the projectile motions of the earth and moon were destroyed, and that they were allowed to fall freely towards the sun. (See Fig. 38, page 175.) If the moon was in conjunction with the sun, or in that part of her orbit which is nearest to him, the moon would be more attracted than the earth, and fall with greater velocity towards the sun; so that the distance of the moon from the earth would be increased by the fall. If the moon wasin opposition, or in the part of her orbit which is furthest from the sun, she would be less attracted than the earth by the sun, and would fall with a less velocity, and be left behind; so that the distance of the moon from the earth would be increased in this case, also. If the moon was in one of the quarters, then the earth and the moon being both attracted towards the centre of the sun, they would both descend directly towards that centre, and, by approaching it, they would necessarily at the same time approach each other, and in this case their distance from each other would be diminished. Now, whenever the action of the sun would increase their distance, if they were allowed to fall towards the sun, then the sun's action, by endeavoring to separate them, diminishes their gravity to each other; whenever the sun's action would diminish the distance, then it increases their mutual gravitation. Hence, in the conjunction and opposition, their gravity towards each other is diminished by the action of the sun, while in the quadratures it is increased. But it must be remembered, that it is not the total action of the sun on them that disturbs their motions, but only that part of it which tends at one time to separate them, and at another time to bring them nearer together. The other and far greater part has no other effect than to retain them in their annual course around the sun.
The cause of the lunar irregularities was first investigated by Sir Isaac Newton, in conformity with his doctrine of universal gravitation, and the explanation was first published in the 'Principia;' but, as it was given in a mathematical dress, there were at that age very few persons capable of reading or understanding it. Several eminent individuals, therefore, undertook to give a popular explanation of these difficult points. Among Newton's contemporaries, the best commentator was M'Laurin, a Scottish astronomer, who published a large work entitled 'M'Laurin's Account of Sir Isaac Newton's Discoveries.' No writer of his own day, and, in my opinion, no later commentator, has equalled M'Laurin,in reducing to common apprehension the leading principles of the doctrine of gravitation, and the explanation it affords of the motions of the heavenly bodies. To this writer I am indebted for the preceding easy explanation of the irregularities of the moon's motions, as well as for several other illustrations of the same sublime doctrine.
The figure of the moon's orbit is an ellipse. We have before seen, that the earth's orbit around the sun is of the same figure; and we shall hereafter see this to be true of all the planetary orbits. The path of the earth, however, departs very little from a circle; that of the moon differs materially from a circle, being considerably longer one way than the other. Were the orbit a circle having the earth in the centre, then the radius vector, or line drawn from the centre of the moon to the centre of the earth, would always be of the same length; but it is found that the length of the radius vector is only fifty-six times the radius of the earth when the moon is nearest to us, while it is sixty-four times that radius when the moon is furthest from us. The point in the moon's orbit nearest the earth is called herperigee; the point furthest from the earth, herapogee. We always know when the moon is at one of these points, by her apparent diameter or apparent velocity; for, when at the perigee, her diameter is greater than at any time, and her motion most rapid; and, on the other hand, her diameter is least, and her motion slowest, when she is at her apogee.
The moon's nodes constantly shift their positions in the ecliptic, from east to west, at the rate of about nineteen and a half degrees every year, returning to the same points once in eighteen and a half years. In order to understand what is meant by this backward motion of the nodes, you must have very distinctly in mind the meaning of the terms themselves; and if, at any time, you should be at a loss about the signification of any word that is used in expressing an astronomical proposition, I would advise you to turn back to the previous definition of that term, and revive its meaning clearly in the mind, before you proceed any further. In the present case, you will recollect that the moon's nodes are the two points where her orbit cuts the plane of the ecliptic. Suppose the great circle of the ecliptic marked out on the face of the sky in a distinct line, and let us observe, at any given time, the exact moment when the moon crosses this line, which we will suppose to be close to a certain star; then, on its next return to that part of the heavens, we shall find that it crosses the ecliptic sensibly to the westward of that star, and so on, further and further to the westward, every time it crosses the ecliptic at either node. This fact is expressed by saying thatthe nodes retrograde on the ecliptic; since any motion from east to west, being contrary to the order of the signs, is called retrograde. The line which joins these two points, or the line of the nodes, is also said to have a retrograde motion, or to revolve from east to west once in eighteen and a half years.
Theline of the apsidesof the moon's orbit revolves from west to east, through her whole course, in about nine years. You will recollect that the apsides of an elliptical orbit are the two extremities of the longer axis of the ellipse; corresponding to the perihelion and aphelion of bodies revolving about the sun, or to the perigee and apogee of a body revolving about the earth. If, in any revolution of the moon, we should accurately mark the place in the heavens where the moon is nearest the earth, (which may be known by the moon's apparent diameter being then greatest,) we should find that, at the next revolution, it would come to its perigee a little further eastward than before, and so on, at every revolution, until, after nine years, it would come to its perigee nearly at the same point as at first. This fact is expressed by saying, that the perigee, and of course the apogee, revolves, and that the line which joins these two points, or the line of the apsides, also revolves.
These are only a few of the irregularities that attend the motions of the moon. These and a few others were first discovered by actual observation and have been long known; but a far greater number of lunar irregularities have been made known by following out all the consequences of the law of universal gravitation.
The moon may be regarded as a body endeavoring to make its way around the earth, but as subject to be continually impeded, or diverted from its main course, by the action of the sun and of the earth; sometimes acting in concert and sometimes in opposition to each other. Now, by exactly estimating the amount of these respective forces, and ascertaining their resultant or combined effect, in any given case, the direction and velocity of the moon's motion may be accurately determined. But to do this has required the highest powers of the human mind, aided by all the wonderful resources of mathematics. Yet, so consistent is truth with itself, that, where some minute inequality in the moon's motions is developed at the end of a long and intricate mathematical process, it invariably happens, that, on pointing the telescope to the moon, and watching its progress through the skies, we may actually see her commit the same irregularities, unless (as is the case with many of them) they are too minute to be matters of observation, being beyond the powers of our vision, even when aided by the best telescopes. But the truth of the law of gravitation, and of the results it gives, when followed out by a chain of mathematical reasoning, is fully confirmed, even in these minutest matters, by the fact that the moon's place in the heavens, when thus determined, always corresponds, with wonderful exactness, to the place which she is actually observed to occupy at that time.
The mind, that was first able to elicit from the operations of Nature the law of universal gravitation, and afterwards to apply it to the complete explanation of all the irregular wanderings of the moon, must have given evidence of intellectual powers far elevated abovethose of the majority of the human race. We need not wonder, therefore, that such homage is now paid to the genius of Newton,—an admiration which has been continually increasing, as new discoveries have been made by tracing out new consequences of the law of universal gravitation.
The chief object of astronomicaltablesis to give the amount of all the irregularities that attend the motions of the heavenly bodies, by estimating the separate value of each, under all the different circumstances in which a body can be placed. Thus, with respect to the moon, before we can determine accurately the distance of the moon from the vernal equinox, that is, her longitude at any given moment, we must be able to make exact allowances for all her irregularities which would affect her longitude. These are in all no less than sixty, though most of them are so exceedingly minute, that it is not common to take into the account more than twenty-eight or thirty. The values of these are all given in the lunar tables; and in finding the moon's place, at any given time, we proceed as follows: We first find what her place would be on the supposition that she moves uniformly in a circle. This gives hermeanplace. We next apply the various corrections for her irregular motions; that is, we apply theequations, subtracting some and adding others, and thus we find hertrueplace.
The astronomical tables have been carried to such an astonishing degree of accuracy, that it is said, by the highest authority, that an astronomer could now predict, for a thousand years to come, the precise moment of the passage of any one of the stars over the meridian wire of the telescope of his transit-instrument, with such a degree of accuracy, that the error would not be so great as to remove the object through an angular space corresponding to the semidiameter of the finest wire that could be made; and a body which, by the tables, ought to appear in the transit-instrument in the middle of that wire, would in no case be removed toits outer edge. The astronomer, the mathematician, and the artist, have united their powers to produce this great result. The astronomer has collected the data, by long-continued and most accurate observations on the actual motions of the heavenly bodies, from night to night, and from year to year; the mathematician has taken these data, and applied to them the boundless resources of geometry and the calculus; and, finally, the instrument-maker has furnished the means, not only of verifying these conclusions, but of discovering new truths, as the foundation of future reasonings.
Since the points where the moon crosses the ecliptic, or the moon's nodes, constantly shift their positions about nineteen and a half degrees to the westward, every year, the sun, in his annual progress in the ecliptic, will go from the node round to the same node again in less time than a year, since the node goes to meet him nineteen and a half degrees to the west of the point where they met before. It would have taken the sun about nineteen days to have passed over this arc; and consequently, the interval between two successive conjunctions between the sun and the moon's node is about nineteen days shorter than the solar year of three hundred and sixty-five days; that is, it is about three hundred and forty-six days; or, more exactly, it is 346.619851 days. The time from one new moon to another is 29.5305887 days. Now, nineteen of the former periods are almost exactly equal to two hundred and twenty-three of the latter:
For 346.619851 × 19=6585.78 days=18 y. 10 d.
And 29.5305887 × 223=6585.32 " = " " " "
Hence, if the sun and moon were to leave the moon's node together, after the sun had been round to the same node nineteen times, the moon would have made very nearly two hundred and twenty-three conjunctions with the sun. If, therefore, she was in conjunction with the sun at the beginning of this period, she would be in conjunction again at the end of it; and all things relating to the sun, the moon, and thenode, would be restored to the same relative situation as before, and the sun and moon would start again, to repeat the same phenomena, arising out of these relations, as occurred in the preceding period, and in the same order. Now, when the sun and moon meet at the moon's node, an eclipse of the sun happens; and during the entire period of eighteen and a half years eclipses will happen, nearly in the same manner as they did at corresponding times in the preceding period. Thus, if there was a great eclipse of the sun on the fifth year of one of these periods, a similar eclipse (usually differing somewhat in magnitude) might be expected on the fifth year of the next period. Hence this period, consisting of about eighteen years and ten days, under the name of theSaros, was used by the Chaldeans, and other ancient nations, in predicting eclipses. It was probably by this means that Thales, a Grecian astronomer who flourished six hundred years before the Christian era, predicted an eclipse of the sun. Herodotus, the old historian of Greece, relates that the day was suddenly changed into night, and that Thales of Miletus had foretold that a great eclipse was to happenthis year. It was therefore, at that age, considered as a distinguished feat to predict even the year in which an eclipse was to happen. This eclipse is memorable in ancient history, from its having terminated the war between the Lydians and the Medes, both parties being smitten with such indications of the wrath of the gods.
TheMetonic Cyclehas sometimes been confounded with the Saros, but it is not the same with it, nor was the period used, like the Saros, for foretelling eclipses, but for ascertaining theageof the moon at any given period. It consisted of nineteen tropical years, during which time there are exactly two hundred and thirty-five new moons; so that, at the end of this period, the new moons will recur at seasons of the year corresponding exactly to those of the preceding cycle. If, for example, a new moon fell at the time of the vernal equinox, in one cycle, nineteen years afterwards it wouldoccur again at the same equinox; or, if it had happened ten days after the equinox, in one cycle, it would also happen ten days after the equinox, nineteen years afterwards. By registering, therefore, the exact days of any cycle at which the new or full moons occurred, such a calendar would show on what days these events would occur in any other cycle; and, since the regulation of games, feasts, and fasts, has been made very extensively, both in ancient and modern times, according to new or full moons, such a calendar becomes very convenient for finding the day on which the new or full moon required takes place. Suppose, for example, it were decreed that a festival should be held on the day of the first full moon after the Vernal equinox. Then, to find on what day that would happen, in any given year, we have only to see what year it is of the lunar cycle; for the day will be the same as it was in the corresponding year of the calendar which records all the full moons of the cycle for each year, and the respective days on which they happen.
The Athenians adopted the metonic cycle four hundred and thirty-three years before the Christian era, for the regulation of their calendars, and had it inscribed in letters of gold on the walls of the temple of Minerva. Hence the termgolden number, still found in our almanacs, which denotes the year of the lunar cycle. Thus, fourteen was the golden number for 1837, being the fourteenth year of the lunar cycle.
The inequalities of the moon's motions are divided into periodical and secular.Periodicalinequalities are those which are completed in comparatively short periods.Secularinequalities are those which are completed only in very long periods, such as centuries or ages. Hence the corresponding termsperiodical equationsandsecular equations. As an example of a secular inequality, we may mention the acceleration of themoon's mean motion. It is discovered that the moon actually revolves around the earth in a less period now than she did in ancient times. The difference, however, is exceedingly small, being only about ten seconds in a century. In a lunar eclipse, the moon's longitude differs from that of the sun, at the middle of the eclipse, by exactly one hundred and eighty degrees; and since the sun's longitude at any given time of the year is known, if we can learn the day and hour when an eclipse occurred at any period of the world, we of course know the longitude of the sun and moon at that period. Now, in the year 721, before the Christian era, Ptolemy records a lunar eclipse to have happened, and to have been observed by the Chaldeans. The moon's longitude, therefore, for that time, is known; and as we know the mean motions of the moon, at present, starting from that epoch, and computing, as may easily be done, the place which the moon ought to occupy at present, at any given time, she is found to be actually nearly a degree and a half in advance of that place. Moreover, the same conclusion is derived from a comparison of the Chaldean observations with those made by an Arabian astronomer of the tenth century.
This phenomenon at first led astronomers to apprehend that the moon encountered a resisting medium, which, by destroying at every revolution a small portion of her projectile force, would have the effect to bring her nearer and nearer to the earth, and thus to augment her velocity. But, in 1786, La Place demonstrated that this acceleration is one of the legitimate effects of the sun's disturbing force, and is so connected with changes in the eccentricity of the earth's orbit, that the moon will continue to be accelerated while that eccentricity diminishes; but when the eccentricity has reached its minimum, or lowest point, (as it will do, after many ages,) and begins to increase, then the moon's motions will begin to be retarded, and thus her mean motions will oscillate for ever about a mean value.
——"As when the sun, new risen,Looks through the horizontal misty air,Shorn of his beams, or from behind the moon,In dim eclipse, disastrous twilight shedsOn half the nations, and with fear of changePerplexes monarchs: darkened so, yet shone,Above them all, the Archangel."—Milton.
——"As when the sun, new risen,Looks through the horizontal misty air,Shorn of his beams, or from behind the moon,In dim eclipse, disastrous twilight shedsOn half the nations, and with fear of changePerplexes monarchs: darkened so, yet shone,Above them all, the Archangel."—Milton.
Havingnow learned various particulars respecting the earth, the sun, and the moon, you are prepared to understand the explanation of solar and lunar eclipses, which have in all ages excited a high degree of interest. Indeed, what is more admirable, than that astronomers should be able to tell us, years beforehand, the exact instant of the commencement and termination of an eclipse, and describe all the attendant circumstances with the greatest fidelity. You have doubtless, my dear friend, participated in this admiration, and felt a strong desire to learn how it is that astronomers are able to look so far into futurity. I will endeavor, in this Letter, to explain to you the leading principles of the calculation of eclipses, with as much plainness as possible.
Aneclipse of the moonhappens when the moon, in its revolution around the earth, falls into the earth's shadow. Aneclipse of the sunhappens when the moon, coming between the earth and the sun, covers either a part or the whole of the solar disk.
The earth and the moon being both opaque, globular bodies, exposed to the sun's light, they cast shadows opposite to the sun, like any other bodies on which the sun shines. Were the sun of the same size with the earth and the moon, then the lines drawn touching the surface of the sun and the surface of the earth or moon (which lines form the boundaries of the shadow) would be parallel to each other, and the shadow would be a cylinder infinite in length; and were the sun less thanthe earth or the moon, the shadow would be an increasing cone, its narrower end resting on the earth; but as the sun is vastly greater than either of these bodies, the shadow of each is a cone whose base rests on the body itself, and which comes to a point, or vertex, at a certain distance behind the body. These several cases are represented in the following diagrams, Figs. 39, 40, 41.
Figs. 39, 40, 41.Figs. 39, 40, 41.
It is found, by calculation, that the length of the moon's shadow, on an average, is just about sufficient to reach to the earth; but the moon is sometimes further from the earth than at others, and when she is nearer than usual, the shadow reaches considerably beyond the surface of the earth. Also, the moon, as well as the earth, is at different distances from the sun at different times, and its shadow is longest when it is furthest from the sun. Now, when both these circumstances conspire, that is, when the moon is in her perigee and along with the earth in her aphelion, her shadow extends nearly fifteen thousand miles beyond the centre of the earth, and covers a space on the surface one hundred and seventy miles broad. The earth's shadow is nearly a million of miles in length, and consequently more than three and a half times as long as the distance of the earth from the moon; and it is also, at the distance of the moon, three times as broad as the moon itself.
An eclipse of the sun can take place only at new moon, when the sun and moon meet in the same part of the heavens, for then only can the moon come between us and the sun; and an eclipse of the moon can occur only when the sun and moon are in opposite parts of the heavens, or at full moon; for then only can the moon fall into the shadow of the earth.
Fig. 42.Fig. 42.
The nature of eclipses will be clearly understood from the following representation. The diagram, Fig. 42, exhibits the relative position of the sun, the earth, and the moon, both in a solar and in a lunar eclipse. Here, the moon is first represented, while revolving round the earth, as passing between the earth and the sun, and casting its shadow on the earth. As the moon is here supposed to be at her average distance from the earth, the shadow but just reaches the earth's surface. Were the moon (as is sometimes the case) nearer the earth her shadow would not terminate in a point, as is represented in the figure, but at a greater or less distance nearer the base of the cone, so as to cover a considerable space, which, as I have already mentioned, sometimes extends to one hundred and seventy miles in breadth, but is commonly much less than this. On the other side of the earth, the moon is represented as traversing the earth's shadow, as is the case in a lunareclipse. As the moon is sometimes nearer the earth and sometimes further off, it is evident that it will traverse the shadow at a broader or a narrower part, accordingly. The figure, however, represents the moon as passing the shadow further from the earth than is ever actually the case, since the distance from the earth is never so much as one third of the whole length of the shadow.
It is evident from the figure, that if a spectator were situated where the moon's shadow strikes the earth, the moon would cut off from him the view of the sun, or the sun would be totally eclipsed. Or, if he were within a certain distance of the shadow on either side, the moon would be partly between him and the sun, and would intercept from him more or less of the sun's light, according as he was nearer to the shadow or further from it. If he were atcord, he would just see the moon entering upon the sun's disk; if he were nearer the shadow than either of these points, he would have a portion of this light cut off from his view, and more, in proportion as he drew nearer the shadow; and the moment he entered the shadow, he would lose sight of the sun. To all places betweenaorband the shadow, the sun would cast a partial shadow of the moon, growing deeper and deeper, as it approached the true shadow. This partial shadow is called the moon'spenumbra. In like manner, as the moon approaches the earth's shadow, in a lunar eclipse, as soon as she arrives ata, the earth begins to intercept from her a portion of the sun's light, or she falls in the earth's penumbra. She continues to lose more and more of the sun's light, as she draws near to the shadow, and hence her disk becomes gradually obscured, until it enters the shadow, when the sun's light is entirely lost.
As the sun and earth are both situated in the plane of the ecliptic, if the moon also revolved around the earth in this plane, we should have a solar eclipse at every new moon, and a lunar eclipse at every full moon; for, in the former case, the moon would comedirectly between us and the sun, and in the latter case, the earth would come directly between the sun and the moon. But the moon is inclined to the ecliptic about five degrees, and the centre of the moon may be all this distance from the centre of the sun at new moon, and the same distance from the centre of the earth's shadow at full moon. It is true, the moon extends across her path, one half her breadth lying on each side of it, and the sun likewise reaches from the ecliptic a distance equal to half his breadth. But these luminaries together make but little more than a degree, and consequently, their two semidiameters would occupy only about half a degree of the five degrees from one orbit to the other where they are furthest apart. Also, the earth's shadow, where the moon crosses it, extends from the ecliptic less than three fourths of a degree, so that the semidiameter of the moon and of the earth's shadow would together reach but little way across the space that may, in certain cases, separate the two luminaries from each other when they are in opposition. Thus, suppose we could take hold of the circle in the figure that represents the moon's orbit, (Fig. 42, page 197,) and lift the moon up five degrees above the plane of the paper, it is evident that the moon, as seen from the earth, would appear in the heavens five degrees above the sun, and of course would cut off none of his light; and it is also plain that the moon, at the full, would pass the shadow of the earth five degrees below it, and would suffer no eclipse. But in the course of the sun's apparent revolution round the earth once a year he is successively in every part of the ecliptic; consequently, the conjunctions and oppositions of the sun and moon may occur at any part of the ecliptic, and of course at the two points where the moon's orbit crosses the ecliptic,—that is, at the nodes; for the sun must necessarily come to each of these nodes once a year. If, then, the moon overtakes the sun just as she is crossing his path, she will hide more or less of his disk from us. Since, also, the earth's shadow is always directly opposite to thesun, if the sun is at one of the nodes, the shadow must extend in the direction of the other node, so as to lie directly across the moon's path; and if the moon overtakes it there, she will pass through it, and be eclipsed. Thus, in Fig. 43, let BN represent the sun's path, and AN, the moon's,—N being the place of the node; then it is evident, that if the two luminaries at new moon be so far from the node, that the distances between their centres is greater than their semidiameters, no eclipse can happen; but if that distance is less than this sum, as at E, F, then an eclipse will take place; but if the position be as at C, D, the two bodies will just touch one another. If A denotes the earth's shadow, instead of the sun, the same illustration will apply to an eclipse of the moon.
Fig. 43.Fig. 43.
Since bodies are defined to be in conjunction when they are in thesamepart of the heavens, and to be in opposition when they are inoppositeparts of the heavens, it may not appear how the sun and moon can be in conjunction, as at A and B, when they are still at some distance from each other. But it must be recollected that bodies are in conjunction when they have the same longitude, in which case they are situated in the same great circle perpendicular to the ecliptic,—that is, in the same secondary to the ecliptic. One of these bodies may be much further from the ecliptic than the other; still, if the same secondary to the ecliptic passesthrough them both, they will be in conjunction or opposition.
In a total eclipse of the moon, its disk is still visible, shining with a dull, red light. This light cannot be derived directly from the sun, since the view of the sun is completely hidden from the moon; nor by reflection from the earth, since the illuminated side of the earth is wholly turned from the moon; but it is owing to refraction from the earth's atmosphere, by which a few scattered rays of the sun are bent round into the earth's shadow and conveyed to the moon, sufficient in number to afford the feeble light in question.
It is impossible fully to understand themethod of calculating eclipses, without a knowledge of trigonometry; still it is not difficult to form some general notion of the process. It may be readily conceived that, by long-continued observations on the sun and moon, the laws of their revolution may be so well understood, that the exact places which they will occupy in the heavens at any future times may be foreseen and laid down in tables of the sun and moon's motions; that we may thus ascertain, by inspecting the tables, the instant when these two bodies will be together in the heavens, or be in conjunction, and when they will be one hundred and eighty degrees apart, or in opposition. Moreover, since the exact place of the moon's node among the stars at any particular time is known to astronomers, it cannot be difficult to determine when the new or full moon occurs in the same part of the heavens as that where the node is projected, as seen from the earth. In short, as astronomers can easily determine what will be the relative position of the sun, the moon, and the moon's nodes, for any given time, they can tell when these luminaries will meet so near the node as to produce an eclipse of the sun, or when they will be in opposition so near the node as to produce an eclipse of the moon.
A little reflection will enable you to form a clear idea of the situation of the sun, the moon, and the earth, atthe time of a solar eclipse. First, suppose the conjunction to take place at the node; that is, imagine the moon to comedirectlybetween the earth and the sun, as she will of course do, if she comes between the earth and the sun the moment she is crossing the ecliptic; for then the three bodies will all lie in one and the same straight line. But when the moon is in the ecliptic, her shadow, or at least the axis, or central line, of the shadow, must coincide with the line that joins the centres of the sun and earth, and reach along the plane of the ecliptic towards the earth. The moon's shadow, at her average distance from the earth, is just about long enough to reach the surface of the earth; but when the moon, at the new, is in her apogee, or at her greatest distance from the earth, the shadow is not long enough to reach the earth. On the contrary, when the moon is nearer to us than her average distance, her shadow is long enough to reach beyond the earth, extending, when the moon is in her perigee, more than fourteen thousand miles beyond the centre of the earth. Now, as during the eclipse the moon moves nearly in the plane of the ecliptic, her shadow which accompanies her must also move nearly in the same plane, and must therefore traverse the earth across its central regions, along the terrestrial ecliptic, since this is nothing more than the intersection of the plane of the celestial ecliptic with the earth's surface. The motion of the earth, too, on its axis, in the same direction, will carry a place along with the shadow, though with a less velocity by more than one half; so that the actual velocity of the shadow, in respect to places over which it passes on the earth, will only equal the difference between its own rate and that of the places, as they are carried forward in the diurnal revolution.
We have thus far supposed that the moon comes to her conjunction precisely at the node, or at the moment when she is crossing the ecliptic. But, secondly, suppose she is on the north side of the ecliptic at the time of conjunction, and moving towards her descendingnode, and that the conjunction takes place as far from the node as an eclipse can happen. The shadow will not fall in the plane of the ecliptic, but a little northward of it, so as just to graze the earth near the pole of the ecliptic. The nearer the conjunction comes to the node, the further the shadow will fall from the polar towards the equatorial regions.
In a solar eclipse, the shadow of the moon travels over a portion of the earth, as the shadow of a small cloud, seen from an eminence in a clear day, rides along over hills and plains. Let us imagine ourselves standing on the moon; then we shall see the earth partially eclipsed by the moon's shadow, in the same manner as we now see the moon eclipsed by the shadow of the earth; and we might calculate the various circumstances of the eclipse,—its commencement, duration, and quantity,—in the same manner as we calculate these elements in an eclipse of the moon, as seen from the earth. But although the general characters of a solar eclipse might be investigated on these principles, so far as respects the earth at large, yet, as the appearances of the same eclipse of the sun are very different at different places on the earth's surface, it is necessary to calculate its peculiar aspects for each place separately, a circumstance which makes the calculation of a solar eclipse much more complicated and tedious than that of an eclipse of the moon. The moon, when she enters the shadow of the earth, is deprived of the light of the part immersed, and the effect upon its appearance is the same as though that part were painted black, in which case it would be black alike to all places where the moon was above the horizon. But it not so with a solar eclipse. We do not see this by the shadow cast on the earth, as we should do, if we stood on the moon, but by the interposition of the moon between us and the sun; and the sun may be hidden from one observer, while he is in full view of another only a few miles distant. Thus, a small insulated cloud sailing in a clear sky will, for a few moments, hide the sun from us,and from a certain space near us, while all the region around is illuminated. But although the analogy between the motions of the shadow of a small cloud and of the moon in a solar eclipse holds good in many particulars, yet the velocity of the lunar shadow is far greater than that of the cloud, being no less than two thousand two hundred and eighty miles per hour.
The moon's shadow can never cover a space on the earth more than one hundred and seventy miles broad, and the space actually covered commonly falls much short of that. The portion of the earth's surface ever covered by the moon's penumbra is about four thousand three hundred and ninety-three miles.
The apparent diameter of the moon varies materially at different times, being greatest when the moon is nearest to us, and least when she is furthest off; while the sun's apparent dimensions remain nearly the same. When the moon is at her average distance from the earth, she is just about large enough to cover the sun's disk; consequently, if, in a central eclipse of the sun, the moon is at her mean distance, she covers the sun but for an instant, producing only a momentary eclipse. If she is nearer than her average distance, then the eclipse may continue total some time, though never more than eight minutes, and seldom so long as that; but if she is further off than usual, or towards her apogee, then she is not large enough to cover the whole solar disk, but we see a ring of the sun encircling the moon, constituting anannular eclipse, as seen in Fig. 44. Even the elevation of the moon above the horizon will sometimes sensibly affect the dimensions of the eclipse. You will recollect that the moon is nearer to us when on the meridian than when in the horizon by nearly four thousand miles, or by nearly the radius of the earth; and consequently, her apparent diameter is largest when on the meridian. The difference is so considerable, that the same eclipse will appear total to a spectator who views it near his meridian, while, at the same moment, it appears annular to one who has the moonnear his horizon. An annular eclipse may last, at most, twelve minutes and twenty-four seconds.