Read Ebook: Letters on Astronomy in which the Elements of the Science are Familiarly Explained in Connection with Biographical Sketches of the Most Eminent Astronomers by Olmsted Denison

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Observation has fully confirmed the prevalence of this law throughout the solar system; and recent discoveries among the fixed stars, to be more fully detailed hereafter, indicate that the same law prevails there. The law of universal gravitation is therefore held to be the grand principle which governs all the celestial motions. Not only is it consistent with all the observed motions of the heavenly bodies, even the most irregular of those motions, but, when followed out into all its consequences, it would be competent to assert that such irregularities must take place, even if they had never been observed.

Newton first published the doctrine of universal gravitation in the 'Principia,' in 1687. The name implies that the work contains the fundamental principles of natural philosophy and astronomy. Being founded upon the immutable basis of mathematics, its conclusions must of course be true and unalterable, and thenceforth we may regard the great laws of the universe as traced to their remotest principle. The greatest astronomers and mathematicians have since occupied themselves in following out the plan which Newton began, by applying the principles of universal gravitation to all the subordinate as well as to the grand movements of the spheres. This great labor has been especially achieved by La Place, a French mathematician of the highest eminence, in his profound work, the 'Mecanique Celeste.' Of this work, our distinguished countryman, Dr. Bowditch, has given a magnificent translation, and accompanied it with a commentary, which both illustrates the original, and adds a great amount of matter hardly less profound than that.

We have thus far taken the earth's orbit around the sun as a great circle, such being its projection on the sphere constituting the celestial ecliptic. The real path of the earth around the sun is learned, as I before explained to you, by the apparent path of the sun around the earth once a year. Now, when a body revolves about the earth at a great distance from us, as is the case with the sun and moon, we cannot certainly infer that it moves in a circle because it appears to describe a circle on the face of the sky, for such might be the appearance of its orbit, were it ever so irregular a curve. Thus, if E, Fig. 31, represents the earth, and ACB, the irregular path of a body revolving about it, since we should refer the body continually to some place on the celestial sphere, XYZ, determined by lines drawn from the eye to the concave sphere through the body, the body, while moving from A to B through C, would appear to move from X to Z, through Y. Hence, we must determine from other circumstances than the actual appearance, what is the true figure of the orbit.

As an example of a body revolving in an orbit under the influence of two forces, suppose a body placed at any point, P, Fig. 34, above the surface of the earth, and let P A be the direction of the earth's centre; that is, a line perpendicular to the horizon. If the body were allowed to move, without receiving any impulse, it would descend to the earth in the direction P A with an accelerated motion. But suppose that, at the moment of its departure from P, it receives a blow in the direction P B, which would carry it to B in the time the body would fall from P to A; then, under the influence of both forces, it would descend along the curve P D. If a stronger blow were given to it in the direction P B, it would describe a larger curve, P E; or, finally, if the impulse were sufficiently strong, it would circulate quite around the earth, and return again to P, describing the circle P F G. With a velocity of projection still greater, it would describe an ellipse, P I K; and if the velocity be increased to a certain degree, the figure becomes a parabola, L P M,--a curve which never returns into itself.

In Fig. 35, page 154, suppose the planet to have passed the point C, at the aphelion, with so small a velocity, that the attraction of the sun bends its path very much, and causes it immediately to begin to approach towards the sun. The sun's attraction will increase its velocity, as it moves through D, E, and F, for the sun's attractive force on the planet, when at D, is acting in the direction D S; and, on account of the small angle made between D E and D S, the force acting in the line D S helps the planet forward in the path D E, and thus increases its velocity. In like manner, the velocity of the planet will be continually increasing as it passes through D, E, and F; and though the attractive force, on account of the planet's nearness, is so much increased, and tends, therefore, to make the orbit more curved, yet the velocity is also so much increased, that the orbit is not more curved than before; for the same increase of velocity, occasioned by the planet's approach to the sun, produces a greater increase of centrifugal force, which carries it off again. We may see, also, the reason why, when the planet has reached the most distant parts of its orbit, it does not entirely fly off, and never return to the sun; for, when the planet passes along H, K, A, the sun's attraction retards the planet, just as gravity retards a ball rolled up hill; and when it has reached C, its velocity is very small, and the attraction to the centre of force causes a great deflection from the tangent, sufficient to give its orbit a great curvature, and the planet wheels about, returns to the sun, and goes over the same orbit again. As the planet recedes from the sun, its centrifugal force diminishes faster than the force of gravity, so that the latter finally preponderates.

I shall conclude what I have to say at present, respecting the motion of the earth around the sun, by adding a few words respecting the precession of the equinoxes.

The amount of precession annually is fifty seconds and one tenth; whence, since there are thirty-six hundred seconds in a degree, and three hundred and sixty degrees in the whole circumference of the ecliptic, and consequently one million two hundred and ninety-six thousand seconds, this sum, divided by fifty seconds and one tenth, gives twenty-five thousand eight hundred and sixty-eight years for the period of a complete revolution of the equinoxes.

Suppose we now fix to the centre of each of the two rings, before mentioned, a wire representing its axis, one corresponding to the axis of the ecliptic, the other to that of the equator, the extremity of each being the pole of its circle. As the ring denoting the equator turns round on the ecliptic, which, with its axis, remains fixed, it is easy to conceive that the axis of the equator revolves around that of the ecliptic, and the pole of the equator around the pole of the ecliptic, and constantly at a distance equal to the inclination of the two circles. To transfer our conceptions to the celestial sphere, we may easily see that the axis of the diurnal sphere would not have its pole constantly in the same place among the stars, but that this pole would perform a slow revolution around the pole of the ecliptic, from east to west, completing the circuit in about twenty-six thousand years. Hence the star which we now call the pole-star has not always enjoyed that distinction, nor will it always enjoy it, hereafter. When the earliest catalogues of the stars were made, this star was twelve degrees from the pole. It is now one degree twenty-four minutes, and will approach still nearer; or, to speak more accurately, the pole will come still nearer to this star, after which it will leave it, and successively pass by others. In about thirteen thousand years, the bright star Lyra will be within five degrees of the pole, and will constitute the pole-star. As Lyra now passes near our zenith, you might suppose that the change of position of the pole among the stars would be attended with a change of altitude of the north pole above the horizon. This mistaken idea is one of the many misapprehensions which result from the habit of considering the horizon as a fixed circle in space. However the pole might shift its position in space, we should still be at the same distance from it, and our horizon would always reach the same distance beyond it.

THE MOON.

HAVING now learned so much of astronomy as relates to the earth and the sun, and the mutual relations which exist between them, you are prepared to enter with advantage upon the survey of the other bodies that compose the solar system. This being done, we shall then have still before us the boundless range of the fixed stars.

The moon, which next claims our notice, has been studied by astronomers with greater attention than any other of the heavenly bodies, since her comparative nearness to the earth brings her peculiarly within the range of our telescopes, and her periodical changes and very irregular motions, afford curious subjects, both for observation and speculation. The mild light of the moon also invites our gaze, while her varying aspects serve barbarous tribes, especially, for a kind of dial-plate inscribed on the face of the sky, for weeks, and months, and times, and seasons.

The moon is distant from the earth about two hundred and forty thousand miles; or, more exactly, two hundred and thirty-eight thousand five hundred and forty-five miles. Her angular or apparent diameter is about half a degree, and her real diameter, two thousand one hundred and sixty miles. She is a companion, or satellite, to the earth, revolving around it every month, and accompanying us in our annual revolution around the sun. Although her nearness to us makes her appear as a large and conspicuous object in the heavens, yet, in comparison with most of the other celestial bodies, she is in fact very small, being only one forty-ninth part as large as the earth, and only about one seventy millionth part as large as the sun.

The moon shines by light borrowed from the sun, being itself an opaque body, like the earth. When the disk, or any portion of it, is illuminated, we can plainly discern, even with the naked eye, varieties of light and shade, indicating inequalities of surface which we imagine to be land and water. I believe it is the common impression, that the darker portions are land and the lighter portions water; but if either part is water, it must be the darker regions. A smooth polished surface, like water, would reflect the sun's light like a mirror. It would, like a convex mirror, form a diminished image of the sun, but would not itself appear luminous like an uneven surface, which multiplies the light by numerous reflections within itself. Thus, from this cause, high broken mountainous districts appear more luminous than extensive plains.

Some persons, when they look into a telescope for the first time, having heard that mountains and valleys are to be seen, and discovering nothing but these unmeaning figures, break off in disappointment, and have their faith in these things rather diminished than increased. I would advise you, therefore, before you take even your first view of the moon through a telescope, to form as clear an idea as you can, how mountains, and valleys, and caverns, situated at such a distance from the eye, ought to look, and by what marks they may be recognised. Seize, if possible, the most favorable period, and previously learn from drawings and explanations, how to interpret every thing you see.

Now, the moon is so distant that we could not easily distinguish places simply by their elevations, since they would be projected into the same imaginary plane which constitutes the apparent disk of the moon; but the foregoing considerations would enable us to infer their existence. Thus, when you view the moon at any time within her first quarter, but better near the end of that period, you will observe, on the side of the terminator within the dark part of the disk, the tops of mountains which the light of the sun is just striking, as the morning sun strikes the tops of mountains on the earth. These you will recognise by those white specks and little crooked lines, before mentioned, as is represented in Fig. 37. These bright points and lines you will see altering their figure, every hour, as they come more and more into the sun's light; and, mean-while, other bright points, very minute at first, will start into view, which also in turn grow larger as the terminator approaches them, until they fall into the enlightened part of the disk. As they fall further and further within this part, you will have additional proofs that they are mountains, from the shadows which they cast on the plain, always in a direction opposite to the sun. The mountain itself may entirely disappear, or become confounded with the other enlightened portions of the surface; but its position and its shape may still be recognised by the dark line which it projects on the plane. This line will correspond in shape to that of the mountain, presenting at one time a long serpentine stripe of black, denoting that the mountain is a continued range; at another time exhibiting a conical figure tapering to a point, or a series of such sharp points; or a serrated, uneven termination, indicating, in each case respectively, a conical mountain, or a group of peaks, or a range with lofty cliffs. All these appearances will indeed be seen in miniature; but a little familiarity with them will enable you to give them, in imagination, their proper dimensions, as you give to the pictures of known animals their due sizes, although drawn on a scale far below that of real life.

In the next place, let us see how valleys and deep craters in the moon might be expected to appear. We could not expect to see depressions any more than elevations, since both would alike be projected on the same imaginary disk. But we may recognise such depressions, from the manner in which the light of the sun shines into them. When we hold a china tea-cup at some distance from a candle, in the night, the candle being elevated but little above the level of the top of the cup, a luminous crescent will be formed on the side of the cup opposite to the candle, while the side next to the candle will be covered by a deep shadow. As we gradually elevate the candle, the crescent enlarges and travels down the side of the cup, until finally the whole interior becomes illuminated. We observe similar appearances in the moon, which we recognise as deep depressions. They are those circular spots near the terminator before spoken of, which look like bubbles of oil floating on water. They are nothing else than circular craters or deep valleys. When they are so situated that the light of the sun is just beginning to shine into them, you may see, as in the tea-cup, a luminous crescent around the side furthest from the sun, while a deep black shadow is cast on the side next to the sun. As the cavity is turned more and more towards the light, the crescent enlarges, until at length the whole interior is illuminated. If the tea-cup be placed on a table, and a candle be held at some distance from it, nearly on a level with the top, but a little above it, the cup itself will cast a shadow on the table, like any other elevated object. In like manner, many of these circular spots on the moon cast deep shadows behind them, indicating that the tops of the craters are elevated far above the general level of the moon. The regularity of some of these circular spots is very remarkable. The circle, in some instances, appears as well formed as could be described by a pair of compasses, while in the centre there not unfrequently is seen a conical mountain casting its pointed shadow on the bottom of the crater. I hope you will enjoy repeated opportunities to view the moon through a telescope. Allow me to recommend to you, not to rest satisfied with a hasty or even with a single view, but to verify the preceding remarks by repeated and careful inspection of the lunar disk, at different ages of the moon.

The heights of the lunar mountains, and the depths of the valleys, can be estimated with a considerable degree of accuracy. Some of the mountains are as high as five miles, and the valleys, in some instances, are four miles deep. Hence it is inferred, that the surface of the moon is more broken and irregular than that of the earth, its mountains being higher and its valleys deeper, in proportion to its magnitude, than those of the earth.

The varieties of surface in the moon, as seen by the aid of large telescopes, have been well described by Dr. Dick, in his 'Celestial Scenery,' and I cannot give you a better idea of them, than to add a few extracts from his work. The lunar mountains in general exhibit an arrangement and an aspect very different from the mountain scenery of our globe. They may be arranged under the four following varieties:

The mountains which form these circular ridges are of different elevations, from one fifth of a mile to three miles and a half, and their shadows cover one half of the plain at the base. These plains are sometimes on a level with the general surface of the moon, and in other cases they are sunk a mile or more below the level of the ground which surrounds the exterior circle of the mountains.

But those, who are anxious to furnish the moon and other planets with all the accommodations which they find in our own, have a subterfuge in readiness, to which they invariably resort in all cases like the foregoing. "There may be," say they, "some means, unknown to us, provided for retaining water on the surface of the moon, and for preventing its being wasted by evaporation: perhaps it remains unaltered in quantity, imparting to the lunar regions perpetual verdure and fertility." To this I reply, that the bare possibility of a thing is but slight evidence of its reality; nor is such a condition possible, except by miracle. If they grant that the laws of Nature are the same in the moon as in the earth, then, according to the foregoing reasoning, there cannot be water in the moon; but if they say that the laws of Nature are not the same there as here, then we cannot reason at all respecting them. One who resorts to a subterfuge of this kind ruins his own cause. He argues the existence of water in the moon, from the analogy of that planet to this. But if the laws of Nature are not the same there as here, what becomes of his analogy? A liquid substance which would not evaporate by such a degree of solar heat as falls on the moon, which would not evaporate the faster, in consequence of the diminished atmospheric pressure which prevails there, could not be water, for it would not have the properties of water, and things are known by their properties. Whenever we desert the cardinal principle of the Newtonian philosophy,--that the laws of Nature are uniform throughout all her realms,--we wander in a labyrinth; all analogies are made void; all physical reasonings cease; and imaginary possibilities or direct miracles take the place of legitimate natural causes.

On the supposition that the moon is inhabited, the question has often been raised, whether we may hope that our telescopes will ever be so much improved, and our other means of observation so much augmented, that we shall be able to discover either the lunar inhabitants or any of their works.

Some writers, however, suppose that possibly we may trace indications of lunar inhabitants in their works, and that they may in like manner recognise the existence of the inhabitants of our planet. An author, who has reflected much on subjects of this kind, reasons as follows: "A navigator who approaches within a certain distance of a small island, although he perceives no human being upon it, can judge with certainty that it is inhabited, if he perceives human habitations, villages, corn-fields, or other traces of cultivation. In like manner, if we could perceive changes or operations in the moon, which could be traced to the agency of intelligent beings, we should then obtain satisfactory evidence that such beings exist on that planet; and it is thought possible that such operations may be traced. A telescope which magnifies twelve hundred times will enable us to perceive, as a visible point on the surface of the moon, an object whose diameter is only about three hundred feet. Such an object is not larger than many of our public edifices; and therefore, were any such edifices rearing in the moon, or were a town or city extending its boundaries, or were operations of this description carrying on, in a district where no such edifices had previously been erected, such objects and operations might probably be detected by a minute inspection. Were a multitude of living creatures moving from place to place, in a body, or were they even encamping in an extensive plain, like a large army, or like a tribe of Arabs in the desert, and afterwards removing, it is possible such changes might be traced by the difference of shade or color, which such movements would produce. In order to detect such minute objects and operations, it would be requisite that the surface of the moon should be distributed among at least a hundred astronomers, each having a spot or two allotted to him, as the object of his more particular investigation, and that the observations be continued for a period of at least thirty or forty years, during which time certain changes would probably be perceived, arising either from physical causes, or from the operations of living agents."

FOOTNOTE:

THE MOON.--PHASES.--HARVEST MOON.--LIBRATIONS.

The moon does not pursue precisely the same track around the earth as the sun does, in his apparent annual motion, though she never deviates far from that track. The inclination of her orbit to the ecliptic is only about five degrees, and of course the moon is never seen further from the ecliptic than about that distance, and she is commonly much nearer to the ecliptic than five degrees. We may therefore see nearly what is the situation of the ecliptic in our evening sky at any particular time of year, just by watching the path which the moon pursues, from night to night, from new to full moon.

It is a natural consequence of this arrangement, to render the moon's light the most beneficial to us, by giving it to us in greatest abundance, when we have least of the sun's light, and giving it to us most sparingly, when the sun's light is greatest. Thus, during the long nights of Winter, the full moon runs high, and continues a very long time above the horizon; while in mid-summer, the full moon runs low, and is above the horizon for a much shorter period. This arrangement operates very favorably to the inhabitants of the polar regions. At the season when the sun is absent, and they have constant night, then the moon, during the second and third quarters, embracing the season of full moon, is continually above the horizon, compensating in no small degree for the absence of the sun; while, during the Summer months, when the sun is constantly above the horizon, and the light of the moon is not needed, then she is above the horizon during the first and last quarters, when her light is least, affording at that time her greatest light to the inhabitants of the other hemisphere, from whom the sun is withdrawn.

"Moon of harvest, herald mild Of 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, 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.

FOOTNOTES:

Dick's 'Celestial Scenery.'

"As when the moon, refulgent lamp of night, O'er heaven's clear azure sheds her sacred light, When not a breath disturbs the deep serene, And not a cloud o'ercasts the solemn scene, Around her throne the vivid planets roll, And stars unnumbered gild the glowing pole; O'er the dark trees a yellower verdure shed, And tip with silver every mountain's head; Then shine the vales, the rocks in prospect rise, A flood of glory bursts from all the skies; The conscious swains, rejoicing in the sight, Eye the blue vault, and bless the useful light."

MOON'S ORBIT.--HER IRREGULARITIES.

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. 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 was in 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.

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 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 above those 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 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 to its 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:

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, 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.

ECLIPSES.

HAVING now 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.

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 would be parallel to each other, and the shadow would be a cylinder infinite in length; and were the sun less than the 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.

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.

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 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 lunar eclipse. 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.

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 come directly 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, 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 the sun, 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.

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.

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 descending node, 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.

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