IRLF SEfi DM1 / OF Miss Sue Dunbar 1 s+ *> / * t THE AND ASTRONOMICAL CALCUljATOR, Containing the Distances) Diameters, Periodical and Diurnal Revolu- tions of all the Planets in the Solar System, with the Diameters of their Satellites) their distances from, and the periods of their Revolutions around their respective Primaries $ together with the method of calcula- ting those Distances^ Diameters and Revolutions* Also the method of calculating Solar and Lunar Eclipses ; being a compilation from various celebrated authors, with Notes, Examples and Interrogations ) prepared for the use of Schools and Academies* BY TOBIAS OSTRANI>ER, TJSMCHJEM OF MATHEMATICS, AND AUTHOR OF THE ELEMENTS OF NUMBERS, EASY INSTRUCTOR, MATHEMATICAL EXPOSITOR, &C. "Consult with reason, reason will reply, "Each lucid point, which glows in yonder sky, "Informs a system in the boundless space, And fills with glory its appointed place ; "With beams unborrow'd brightens other skies, "And worlds, to thee unknown, with heat and life supplies." PRINTED AT THE OFFICE OF THE WESTERN 1832, Nerthtrn Dittrlet of New- York, TO WIT: BE it remembered, that on the sixteenth day of July, Anno Domini, 1832, TOBIAS OSTRANDER of the said district, hath deposited in this office the title of a book, the title of which is in the words following, to wit : The Planetarium and Astronomical Calculator, containing the Distan- ces, Diameters, Periodical and Diurnal Revolutions of all the Planets in the Solar system, with the Diameters of their Satellites, their distances from, and the periods of their Revolutions around their respective Prima- ries ; together with the method of calculating those Distances, Diameters and Revolutions ; and the method of calculating Solar and Lunar Eclipses ; being a compilation from various celebrated authors, with Notes, Exam- ples and Interrogations ; prepared for the use of Schools and Academies. By TOAIAS OSTRANDER, Teacher of Mathematics, and author of the Elements of Numbers, Easy Instructor, Mathematical Expositor, &c. "Consult with reason, reason will reply, " Each lucid point which glows in yonder sky, ; "Informs a system in the boundless space, " And fills with glory its appointed place ; "With beams unborrow'd brightens other skies, "And worlds, to thee unknown, with heat and life supplies." The right whereof he claims as Compiler and Proprietor, in confer- ity with an act of Congress, entitled an Act to amend the several Acts respecting Copy Rights. RUTGER B. MILLER, Cltrk of the Northern District of New-York. I^ 3* PREFACE. IN presenting the following pages to the public, I will briefly remark, that the people generally are grossly ignorant in the important and engaging science of As- tronomy. Scarcely one in a county is found capable of calculating with exactness, and accuracy the precise time of an eclipse, or conjunction and opposition of the Sun and Moon. Is it for lack of abilities ? No. There are no people on the surface of this terraqueous globe, who possess better natural faculties of acquiring knowledge of any description, than those who inhabit the United States of America. In this land of liberty, much has been done, and much still remains to be done, for the benefit of the rising generation. Schools, Academies and Colleges have been erected, for the purpose of facilitating, and extending information and instruction among the youth of this delightful section. Gentlemen possessing the most profound abilities and acquirements, have engaged in the truly laudable em- ployment of disseminating a knowledge of all the scien- ces ; both of useful and ornamental description. Still this branch of the Mathematical science, called Astronomy, has been almost totally neglected, especially among the common people. From what source has this origina- ted 1 I answer, from a scarcity of books, well calcu- M288838 lated to give the necessary instruction. Though there are many productions possessing merit, and are of importance to the rising generation, yet they are defi- cient in the tables necessary for the calculation, and protraction of eclipses. The works of Ferguson, En- field and others, from which this is principally com- piled, contain all that is necessary ; but the expense renders them beyond the means of many, who perhaps posses the best abilities in our land. Extensive vol- umes are not well calculated for the use of Schools ; for a Student is under the necessity of reading so much unessential, and uninteresting matter, that the essence is lost, in the multiplicity of words ; and for these rea- sons, many of the teachers have neglected this useful, and important branch of the Mathematical science. I have long impatiently beheld the evil, without an op- portunity of providing a remedy, until the present period. I now present to this enlightened community, a volume within the means of almost every person ; con- taining all the essential parts of Astronomy, adapted to the use of Schools and Academies ; made so plain and easy to be understood, that a lad of twelve years of age, whose knowledge of Arithmetic extends to the single rule of proportion, can, in the short space of one or two weeks, be taught to calculate an eclipse ; and many possessing riper years, from the precepts and examples given in the work, will be found capable of accomplishing it, without the aid of any other teacher, The tables, (with the exception of two,) I have wholly calculated, and then duly compared them with those of Ferguson. Great care has also been taken, to present the work to the public, free from errors. Should the following pages meet the approbation of a generous and enlightened community, and be the means of extending the knowledge of this important branch of Education ; not only to the rising generation, but to those of maturer years, the Compiler, whose best abilities have hitherto been employed in endeav- oring to meliorate the condition of man, by improving the mind and enlightening the understanding, will have the sublime satisfaction, of removing some of the shackles of ignorance, and building up a fund of useful and interesting knowledge upon its ruins. THE COMPILER. SECTION FIRST. OF ASTRONOMY IN GENERAL. OF all the sciences cultivated by mankind, Astron- omy is acknowledged to be, and undoubtedly is, the most sublime, the most interesting, and th^most use- ful. By the knowledge derived from this science, not only the magnitude of the earth is discovered, the sit- uation and extent of the Countries and Kingdoms as- certained, trade and commerce carried on to the re- motest parts of the world, and the various products of several countries distributed, for the health, comfort, and conveniency of its inhabitants ; but our very facul- ties are enlarged, with the grandeur of the ideas it con- veys, our minds exalted above the low contracted pre- judices of the vulgar, and our understandings clearly convinced, and affected with the conviction, of the ex- istence, wisdom, power, goodness, immutability, and 8 Of Jlstronomy in General. Sec. 1. superintendency of the Supreme Being. So that without any hyperbole, every man acquainted with this science, must exclaim with the immortal Dr. Young : " An undevout Astronomef is mad." From this branch of Mathematical knowledge, we also learn by what means, or laws, the Almighty Power and Wisdom of the Supreme Architect of the Uuiverse, are administered in continuing the wonderful harmony, order and connexion, observable throughout the plan- etary system ; and are led by very powerful arguments, to form this pleasing and cheering sentiment, that minds capable of such deep researches, not only derive their origin from that Adorable Being, but are also incited to aspire after a more perfect knowledge of his nature, and a more strict conformity to his will. By Astronomy we discover, that the earth is at so great a distance from the sun, that if seen from thence, it would appear no larger than a point ; although its diameter is known to be nearly 8,000 miles : yet that distance is so small, compared with the earth's distance from the fixed stars, that if the orbit, in which the earth moves round the sun, were solid, and seen from the nearest star, it would likewise appear no larger than a point; although it is at least 190 millions of miles in diameter ; for the earth in going round the sun, is 190 millions of miles nearer to some of the stars, at one time of the year than at another ; and yet their apparent magnitudes, situations, and distances still remain the same ;and a telescope which magnifies above 200 times, does not sensibly magnify them; Sec. I Of Astronomy in General. 9 which proves them to be at least, one hundred thou- sand times further from us, than we are from the sun. It is not to be imagined, that all the stars are placed in one concave surface, so as to be equally distant from us; but that they are placed at immense distances from one another, through unlimited space, so that there may be as great a distance between any two neighboring stars, as between the sun from which we receive our light, and those which are nearest to him. Therefore, an observer who is nearest any fixed star, will look upon it alone as a real sun ; and consider the rest as so many shining points, placed at equal distances from him in the firmament, By the help of telescopes, we discover thousands of stars which are entirely invisible, without the aid of such instruments, and the better our glasses are, the more become visible. We therefore can set no limits to their numbers, or to their immeasurable distances. The celebrated Huygens carried his thoughts so far, as to believe it not impossible, that there may be stars at such inconceivable distances, that their light has not yet reached the earth since their creation ; although the velocity of light, be a million of times greater than the velocity of a cannon ball at its first discharge ; and as Mr. Addison justly observes, " This thought is far from being extravagant, when we consider that the Universe is the work of Infinite Power, prompted by Infinite Goodness, and having an Infinite space to exert itself in ; therefore our finite imaginations can set no bounds to it." A 10 Of Astronomy in General Sec. 1. The Sun appears very bright and large, in compar- ison of the fixed stars ; because we constantly keep near the Sun, in comparison to our immense distance from them. For a spectator placed as near to any star, as we are to the Sun, would see that star to be a body as large and bright as the Sun appears to us : and a spectator as far distant from the Sun, as we are from the stars, would see the Sun as small as we see a star, divested of all its circumvolving planets, and would reckon it one of the stars, in numbering them. The stars being at such immense distances from the Sun, cannot possibly receive from him so strong alight as they appear to have, nor any brightness sufficient to make them visible to ua ; for the Sun's rays must be so scattered before they reach such remote objects, that they can never be transmitted back to our eyes ; so as to render these objects visible by reflection. Therefore the stars, like the Sun, shine with ther own native and unborrowed lustre ; and since each particular one, as well as the Sun, is confined to a particular por- tion of space, it is evident that the stars are of the same nature with the Sun; formed of similar materials, and are placed near the centres of as many magnificent systems ; have a retinue of worlds inhabited by intelli- gent beings, revolving round them as their common centres ; receive the distribution of their rays, and are illuminated by their beams; all of which, are losttous, in immeasurable wilds of ether. It is not probable that the Almighty, who always acts with Infinite Wisdom, and does nothing in vain, Sec* 1 Of Astronomy in General. 1 1 should create so many glorious Suns, fit for so many important purposes, and place them at such distances from each other without proper objects near enough to be benefit-ted by their influences. Whoever ima- gines that they were created only to give a faint glim- mering light to the inhabitants of this globe, must have a very superficial knowledge of Astronomy, and a mean opinion of the Divine Wisdom ; since by an infinitely less exertion of creating power, the Deity could have given our earth much more light by one single addi- tional Moon. Instead of cur Sun, and our world only in the Uni- verse, (as the unskillful in Astronomy may imagine ;) that science discovers to us, such an inconceivable number of Suns, Systems and Worlds, dispersed through boundless space, that if our Sun, with all the planets, Moons and Comets, belonging to the whole So- lar System, were at once annihilated, they would no more be missed by an eye that could take in the whole compass of Creation, than a grain of sand from the Sea shore ; the space they possess, being comparatively so small, that their loss would scarcely make a sensible blank in the Universe. Although Herschel, the out- ermost of our planets, revolves about the Sun, in an orbit of three thousand, six hundred millions of miles in diameter, and some of our Comets, make excursions more than ten thousand millions of miles beyond his or- bit, and yet at that amazing distance, they are incom- parably nearer the Sun, than to any of the fixed stars, as is evident, from their keeping clear of the attractive 12 Of Astronomy in General. power of all the stars, and returning periodically by virtue of the Sun's attraction. From what we know of our own System, it may be reasonably concluded, that all the rest are with equal wisdom contrived, situated and provided with accom- modations for the existence of intelligent inhabitants. Let us therefore take a survey of the System to which we belong, the only one accessible to us, and from thence we shall be better able to judge of the nature and end of other systems of the Universe. Altho' there are almost an infinite variety in the parts of Creation, which we have opportunities of examining ; yet there is a general analogy running ihrough, and connecting all the parts into one scheme , one design of dissemmi- nating comfort and happiness to the whole Creation. To an attentive observer, it will appear highly proba- able, that the planets of our System, together with their attendants called Satellites or Moons, are much of the same nature with our earth, and destined for similar purposes ; for they are all solid opaque globes, capable of supporting animals and vegetables ; some are larg- er, some less, and one nearly the size oi our earth. They all circulate round the Sun, as the earth does, in a shorter, or longer time, according to their res- pective distances from him, and have, where it would not be inconvenient,) regular returns of Summer and Winter, Spring and Autumn. They have warmer and colder climates, as the various productions of our earth require, and of such as afford a possibility of dis- covering it, we observe a regular motion round their Sec. 1 Of Astronomy in General. 13 axes, like that of our earth, causing an alternate return of day and night, which is necessary for labour, rest, and vegetation, and that all parts of their surfaces may be exposed to the rays of the Sun. Such of the planets as are farthest from the Sun, and therefore enjoy least of his light, have that deficiency made up by several Moons, which constantly accom- pany and revolve about them, as our Moon revolves around the earth. The planet Saturn has over and above a broad ring, encompassing it, which no where touches his body; which like a broad zone in the Heavens, reflects the same light very copiously on that planet ; remote planets have the Sun's light fainter by day, than we, they have an addition to it, morning and evening, by one or more of their Moons, and a greater quantity of light in the night time. On the surface of the Moon, (because it is nearer to us than any other of the celestial bodies,) we discover a nearer resemblance of our earth, for by the assistance of telescopes we observe the Moon to be full of high mountains, large vallies, and deep cavities. These sim- ilarities leave us no room to doubt, but that all planets, Moons and Systems, are designed to be commodious habitations for creatures endowed with capacities of knowing, and adoring their beneficent Creator. Since the fixed stars are prodigious spheres shining by their own native light like our Sun, at inconceiv- able distances from each other, as well as from us, it is reasonable to conclude that they are made for simi- lar purposes, each to bestow light, heat and vegetation 14 Of Astronomy in General. Sec. 1 on a certain number of inhabited planets, kept by grav- itation within the sphere of its activity. When we therefore contemplate on those ample and amazing structures, erected in endless magnificence over all the ethereal plains, when \ve look up- on them as so many repositories of light, or fruitful abodes of life, when we consider that in all probability there are orbs vastly more remote than those which appear to our unaided sight, orbs whose effulgence, though travelling ever since the Creation, has not yet arrived upon our coast What an august, what an amazing conception does this give of the works of the Omnipotent Creator ; who made use of no preparatory measures, or long circuit of means. He spake, and ten thousand times ten thousand Suns, multiplied without end, hanging pendulous in the great vault of Heaven, at immense distances from each other, attended by ten thousand times ten thousand worlds, all in rapid motion, yet calm, regular, and harmonious, invariably keeping the paths prescribed, rolled from his creating hand. But when we contemplate on the power, wisdom, goodness, and magnificence of the Great Creator ; let us use the language of the immortal Dr. Young, in his appeal to the starry Heavens : -" Say proud arch, Built with Divine ambition, in disdain Oflimit built ; built in the taste of Heaven, Vast concave, ample dome. Wast thou designed A meet apartment for the Deity ? Not so ; that thought alone thy state impairs, Thy lofty sinks, and shallows thy profound, And straightens thy diffusive." Sec. 1 Interrogations for Section First. 15 INTERROGATIONS FOR SECTION FIRST. What is ASTRONOMY ? It is a mixed Mathematical Science, teaching the knowledge of the celestial bodies, their magnitudes, motions, distances, periods, eclipses and order. What are its uses 1 What conviction does a knowledge of this branch of science give to the understanding 1 What cheering sentiment is formed from a knowl- edge of this science 1 What is the diameter of the earth ? How many miles is the diameter of the earth's orbit ? How is it known that the stars are at immense dis- tances from us ? How is it known that they are at immense distances from each other 1 What instruments have been invented to aid the sight of man 1 Who supposed there were stars, whose light had not yet reached the earth since their first creation],? Who confirmed the idea ? Why cannot the same rays be reflected back from the stars to our eyes 1 With what light do the stars shine 1 How could the Deity have given us greater light in the night time, than by the w r hole starry host? How is it known that the Comets belong to the So- lar Svstem 7 1 6 Interrogations for Section First. Sec. I From what parity of reasoning, is it believed that the stars are so many suns, and have worlds revolving about them ? How are those planets supplied with light, which are farthest from the Sun ? SECTION SECOND. OF THE SOJL*& THE Solar System consists of the Sun, with all the Planets and Comets that move around him as their centre. Those which are near the Sun, not only fin- ish their circuits sooner, but likewise move with great- er rapidity in their respective orbits, than those which are more remote. Their motions are all performed from West to East, in Elliptical orbits. Their names, distances, magnitudes, and periodical revolutions, are as follows : The Sun is placed near the common cen- tre, or rather in the lower focus of the orbits of all the planets and comets, and turns round on his axis once in 25 days, 14 hours and 8 minutes; as has been pro- ved, from the motion of the spots, seen on his surface. His diameter is computed at 883,246 miles, and by the various attractions of the convolving planets, he is agi- tated by a small motion round the centre of gravity of the system. His mean apparent diameter as seen from the earth, is 32 minutes and one second. His *o- B 18 Of the Solar System. Sec. 2. lidity, and indeed that of every other planet, may be found by multiplying the cube of their diameters by ,5236. All the planets as seen by a spectator, placed on the sun, move the same way, and according to the order of the signs, Aries, Taurus, Gemini, &e. which represent the great ecliptic. But, to a spectator placed on any one of the planets, the others sometimes appear to go backward, sometimes forward, and at others sta- tionary ; not moving in proper circles, nor elliptical or- bits, but in looped curves, which never return into them- selves. The Comets, also appear to come from all parts, and appear to move in various directions. These proofs are sufficient to establish the fact, that the sun is placed near the centre, and that all the other planets revolve around him : are irradiated by his beams : receive the distribution of his rays, and are dependant for the enjoy- ment of every blessing on this grand dispensor of divine munificence. The orbits of the planets are not in the same plane with the ecliptic,* but crosses it in two points directly opposite to each other, called the planet's nodes, f That from which the planet ascends northward above the ecliptic, is called the ascending node ; and the other which is directly opposite, (and consequently 6 signs asunder,) is called the descending node. * The ecliptic is an imaginary great circle in the Heavens, in the plane of which the earth performs her annual revolutions round the sun. t The node is the intersection of the orbit of any planet with that of the rtfc. Sec. 2 Of the Solar System. 19 It was discovered on the first of January 1805, that the ascending node of the planet Herschel was in twelve degrees and fifty -three minutes of the sign Gemini, and advances 16 seconds in a year. Saturn in twenty-one degrees and 59 minutes of Cancer, and advances 32 seconds in a year. Jupiter in 8 degrees and 27 minutes of Cancer, and advances 36 seconds yearly. Mars in 18 degrees, and four minutes of Taurus, and advances 28 seconds yearly. Venus in 14 degrees and 55 min- utes of Gemini, and advances 36 seconds yearly. Mer- cury in 16 degrees of Taurus, and advances 43 seconds every year. In these observations, the earth's orbit is considered the standard, and the orbits of all the ether planets obliquely to it. The nearest planet to the sun is Mercury. The great brilliancy of light emitted by this planet: the shortness of the period during which observations can be made upon his disk ; and his posi- tion among the vapors of the horizon when he is obser- ved, have hitherto prevented Astronomers, from ma- king interesting discoveries to be relied on with cer- tainty respecting this planet. This planet, when view- ed at different times with a good telescope, appears in all the various shapes of the Moon, which is a plain proof that he receives, (like~ the Moon,) all his light from the Sun. That he moves round the Sun in an orbit, within the orbit of the earth, is also plain ; be- cause he is never seen opposite to the Sun, nor above 56 times the Sun's diameter from his centre. It has been said by Authors, that his light and heat from the Sun must be almost seven times as great as our's ; 20 Of the Solar System. Sec. 2 judging from his nearness to it. His light and heat however, depend more on the height and density of his atmosphere, than to his near approach to that lu- minary. His distance from the Sun is computed at 73,000,000 of miles, is 3,225 in diameter, and performs a revolu- tion round the Sun, in 87 days 23 hours 15 minutes and 28 seconds : his apparent diameter as seen from the earth, is ten seconds. His orbit is inclined,7 degrees to the ecliptic ; and that node from which he ascends northward above it, is in the 1 6th degree of Taurus, the opposite in the 16th degree of Scorpio. The earth is in these points on the 6th of November, and 4th of May; when he comes to either of his nodes at his infe- rior conjunction about these times, he will appear to pass over the face of the sun like a dark round spot. But in all other parts of his orbit, his conjunctions are invisible ; because he either goes above or below the Sun. On the 5th day of May, at 6 hours 43 minutes 22 seconds in the morning, in the year 1832, in the longitude of Washington, he was in conjunction with the Sun. His next visible conjunction will be on the 7th day of November 1835. Venus, the next planet in order is 68,000,000 of miles from the Sun by computation, and by moving at the rate of 69,000 miles every hour in her orbit, she performs her revolution round the Sun in 224 days, 16 hours and 49 minutes of our time ; in which, (though it be the full length of her year,) she has only 9 days and a quarter, according to observations made by Bi- Sec. 2 Of the Solar System. 2 1 anchini ; so that her every day and night together, is as long as 348^ days and nights with us. This odd quar- ter of a day in every year, makes in every fourth year a leap year to Venus, as the like does to the earth which we inhabit. Her diameter is computed at 7687 miles, and performs her diurnal revolutions in 23 hours 20 minutes, and 54 seconds ; with an inclination ol her orbit to the ecliptic, of 3 degrees 23 minutes, and 35 seconds. Her orbit includes the orbit of Mercury within it, for at her greatest elongation, or apparent distance from the Sun, she is about 96 times his diam- eter from his centre ; while that of Mercury is not above 56. Her orbit is included within the orbit of the earth, for if it were not, she would be as often seen in oppo- sition as in conjunction with the Sun. But she never departs from the Sun to exceed 47 degrees, and that of Mercury 28, it is therefore certain that the orbit of Mercury is within the orbit of Venus, and that of Ve- nus within the orbit of the earth. When this planet is west of the Sun, she rises in the morning before him, and hence she is called the morning star ; and when she sets after the Sun, she is called the evening star ; so that in one part of her orbit she rides foremost in the procession of night, and in the othe'r, anticipates the dawn ; being each in its turn 290 days. The axis of Venus is inclined 75 degrees to the axis of her orbit, which is 51 degrees and 32 minutes more than the axis of the earth is inclined to the axis of the ecliptic ; and therefore her seasons vary much more than our's. 22 Of the Solar System. Sec. 2 The north pole of her axis, inclines towards the 20th degree of Aquarius; the earth's to the beginning of Cancer. Consequently the northern parts of Venus have Summer, in the signs where those of the earth have Winter, and vice versa. The orbit of Venus is inclined three and one half degrees to the earth's, and crosses it in the'l 4th degree of Gemini, and Sagittarius, and therefore when the earth is near the points of the ecliptic, at the time when Venus is in her inferior con- junction,* she appears like a spot on the Sun, and it furnishes a true method of calculating the distances of all the planets from the Sun. It will not be uninteresting to those who peruse this treatise, to be put in the possession of all the elements of the transits, both of Mercury and Venus over the Sun's disk ; from this period to the end of the present century, I therefore insert the following tables : TRANSIT OF MERCURY OVER THE SUN'S DISK. Transit of Mercury May 4th 1832. D H. M. 8. Mean time of conjunction, May - 4 23 51 22 8 D. M. S. Geocentric longitude of the Sun and Mercury - 1 14 56 45 H M. 8. Middle apparent time, 18 1 Semi-duration of the transit, 3 28 2 Nearest approach to centres, - 8 16 North. * Inferior conjunction is, when the planet is between the earth and the Sun, in the nearest part of its orbit. Sec. 2 Of the Solar System. November 7th 1835. Mean time of conjunction, Geocentric longitude of the Sun and Mercury, Middle apparent time, Semi duration of the transit, Nearest approach of centres, May Sth 1845, Mean time of Conjunction. Geocentric longitude of the Sun and Mercury, Middle apparent time, Semi-duration of the transit, Nearest approach of centres, November 9th 1848. Mean time of conjunction, Geocentric longitude of the Sun and Mercury, Middle apparent time, Semi-duration of the transit, Nearest approach of centres, November llth 1861. Mean time of conjunction, Geocentric longitude of the Sun and Mercury- Middle apparent time, Semi-duration of the transit, Nwirett approach of cemtret, 23 H M. 8. 7 47 54 8 D. M. 8. 7 14 43 8 H M. 8. 8 12 22 2 33 53 - 5 37 South. H M. 8. 7 54 18 8 D. M. 8. 1 18 1 49 H M. 8. 7 32 58 - 3 22 33 -08 58 South. IX M. 8. 1 37 43 8 D. M. 8. 7 17 19 19 H M. 8. 1 49 43 2 41 33 2 36 North. H M. 8. 19 20 13 8 D. M. 8. 7 19 54 44 H M. 8. 19 20 14 2 23 10 02N0rtfc, 24 Of the Solar System. November 4th 1868. Stc. 2 Mean time of conjunction, Middle apparent time, Semi-duration of the transit, Geocentric longitude of (he Sun and Mercury, Nearest approach of centres, May 6th 1878, Mean time of conjunction, Middle apparent time, - Semi-duration of the transit, Geocentric longitude of the Sun and Mercury, Nearest approach of centres, November 7ih 1881. II M. B. 18 43 45 19 18 24 1 45 21 * D. M. 8. 7 13 9 42 12 20 South. H af. L 6 38 30' 6 55 14 3 53 31 8 D. M. S. 1 16 3 50 4 39 North. Mean time of conjunction, Middle apparent time, Semi-duration of the transit, H M. 8. 12 39 38 12 59 33 2 39 6 Geocentric longitude of the Sun and Mercury, Nearest approach of centres, May 9th 1891. S D. M. 8. 7 15 46 57 3 57 South, Mean time of conjunction, Middle apparent time, Semi-duration of the transit, II M. 8. 14 44 57 14 13 46 2 34 20 Geocentric longitude of the Sun and Mercury, Nearest approach of centres, November 10th 1894. 8 D. M. 8. 1 19 9 1 12 21 South. Mean time of conjunction, Middle apparent time, Semi-duration of the transit, H M. 8. 6 17 5 6 36 29 2 37 36 Geocentric longitude of the Sun and Mercury, Nearest approach of centre*, 6 D. M. S. 7 18 22 9 4 SONorth, Sec. 2 Of the Solar System. TRANSITS OF VENUS OVER THE SUN'S DISK, FROM THE YEAR 1769 TO THE YEAR 2004 INCLUSIVE. Jwi* 3d 1769. Mean time of conjunction, Middle apparent time, Duration of the transit, - * Geocentric longitude of the Sun and Venus, Nearest approach of centres, December 8tk 1874, H ac. s. 9 58 34 10 27 3 5 59 46 . * D. fir. s, 2 13 27 8 10 lONorth. Mean time of conjunction, Middle apparent time, Duration of the transit, Geocentric longitude of the Sun and Venus, Nearest approach of centres, December 6th 1882. II M. S. 16 8 24 15 43 28 4 9 22 B D. M. B. 8 16 67 49 13 51 North. Mean time of conjunction, Middle apparent time, Duration of the transit, Geoncentric longitude of the Sun and Venus, Nearest approach of centres, June 7th 2004. Mean time of conjunction, Middle apparent time, Duration of the transit, Geocentric longitude of the Sun and Venus, Nearest approach of centres, The earth is the next planet above Solar Sytem ; it is 95,000,000 of miles and performs a revolution around him, tice, or Equinox, to the same again, c H M. B. 4 16 24 4 49 42 6 8 26 S D. M. 8. 8 14 29 14 10 29 South, H M. B. - 20 51 24 20 26 59 5 29 40 8 D. M. S. 2 17 54 23 11 19 South. Venus, in the from the Sun, from any Sols- in 365 days, 5 25 Of the Solar System. Sec. 2 hours and 49 minutes : but from any fixed star to the same again, in 365 days, 6 hours and 9 minutes. The former being the length of the tropical year, and the latter the sidereal. It travels at the rate of 58,000 miles every hour, in performing its annual revolution. It revolves on its own axis from West to East, once in 24 hours. Its mean diameter as seen from the Sun, is 17 seconds and two tenths of a degree. Which, by calculation, will give about 7,970 miles for its diame- ter. The form of the earth is an oblate spheroid, whose equatorial axis exceeds its polar by 36 miles, and is surrounded by an atmosphere extending 45 miles above its surface. The Seas, and unknown parts of the earth, (by a measurement of the best Maps,) contain 160 millions, 522 thousand, and 26 square miles. The inhabited parts 38 millions, 990 thousand, 569. Europe four millions, 456 thousand, and 65. Asia 10 millions, 568 thousand, 823. Africa 9 millions, 654 thousand, 807. America 14 millions, 110 thousand, 874 : the whole amounting to 199 millions, 512 thousand, 595 ; which is the number of square miles, on the whole surface of the Globe we inhabit. The Moon is not a planet, but only a satellite, or an attendant of the earth, performing a revolution round it in 29 days, 12 hours and 44 minutes ; and with the earth, is carried round the Sun once in every year. The diameter of the Moon is 2,180 miles, and her mean distance from the earth's centre, is estimated at 240,000 miles. She goes around her orbit in 27 days, Sec. 2 Of the Solar System. 27 7 hours and 43 minutes ; moving about 2,290 miles ev- ery hour, and performs a revolution on her own axis exactly in the time that she goes round the earth ; con- sequently the same side of her is continually presented towards the earth, and the length of her day and night taken together, is equal to a lunar month. Her mean apparent diameter, as seen from the earth, is 31 minutes and 8 seconds of a degree. The orbit of the Moon, crosses the ecliptic in two opposite points, called the Moon's nodes, consequently one half of her orbit is above the ecliptic, and the other below ; the angle of its obliquity is 5 degrees, and 20 minutes. The Moon has scarcely any .difference of seasons, be- cause her axis is nearly perpendicular to the ecliptic, and consequently the Sun never removes sensibly from her equator. The earth which we inhabit, serves as a satellite to the Moon, waxing and waning regularly, but appearing thirteen times as large, and affording her, thirteen times as much light as the Moon does to us. When she is new to us, the earth appears full to her ; and, when she is in her first quarter as seen from the earth, the earth is in its third quarter as seen from the Moon. The Moon is an opaque globe, like the earth, and shines only by reflecting the light of the Sun ; there- fore whilst that half of her which is towards the Sun, is enlightened, the other half must be dark and in- visible. Hence she disappears when she comes between us and the Sun ; because her dark side is then to- wards us. 28 Of the Solar System. Sec. 2 The planet Mars is next in order, being the first a- bove the earth's orbit. His distance from the Sun is computed at 144,000,000 of miles, and by travelling at the rate of 54,000 miles every hour, he goes round the Sun in 686 of our days, 23 hours and 30 minutes, which is the length of his year, equal to 667 and 3-4th of his days ; and every day and night together, being nearly 40 minutes longer than with us. His diameter is computed at 4,189 miles, and by his diurnal rotation, the inhab- itants at his equator are carried 528 miles every hour. The Sun appears to the inhabitants of Mars, nearly two-thirds the size that it does to us. His mean apparent diameter, as seen from the earth is 27 seconds, and as seen from the Sun, ten seconds of a degree. His axis is inclined to his orbit 59 de- grees and 22 minutes. To the inhabitants of the planet Mars, our Earth and Moon appear like two Moons ; the one being 13 times as large as the other ; changing places with each other, and appearing sometimes horned, sometimes half or three-quarters illuminated but never full, nor at most above one quarter of a degree from each oth- er ; although they are in fact 240,000 miles asunder. This Earth appears almost as large from Mars as Venus does to us. It is never seen above 48 degrees from the Sun, at that planet. Sometimes it appears to pass the disk of the Sun, and likewise Mercury and Venus. But Mercury can never be seen from Mars by such eyes as our's (unless assisted by proper in- strument*,) and Venus will be as seldom seen as we Sec. 2 Of the Solar System. 29 see Mercury. Jupiter and Saturn are as visible to Mars as to us. His axis is perpendicular to the eclip- tic, and his inclination to it is one degree and 51 min- utes. The planet Mars is remarkable for the redness of its light, the brightness of its polar regions, and the variety. of spots which appear upon its surface. The atmosphere of this planet, which Astronomers have long considered to be of an extraordinary height and density, is the cause of the remarkable redness of its appearance. When a beam of white light passes through any medium, its color inclines to redness, in proportion to the density of the medium; and the space through which it has travelled. The momen- tum of the red, or least refrangible rays being greater than that of the violet, or most refrangible, the former will make their way through the resisting medium, while the latter are either reflected or absorbed. The color of the beam therefore when it reaches the eye, must partake of the color of the least refrangible rays, and must consequently increase with the number of those of the violet, which have been obstructed. Hence we discover, that the morning and evening clouds are beautifully tinged with red, that the Sun, Moon and Stars appear of the same color, when near the horizon, and that every luminous object seen through a mist, is of a ruddy hue. There is a great difference of color among the planets, we are there- fore, (if the preceding observations be correct,) under the necessity of concluding, that those in which the red color predominates, are surrounded with the most 30 Of the Solar System. Sec. 2 extensive and dense atmospheres. According to this idea, the atmosphere of Saturn, must be the next to that of Mars, in density and extent. The planet Mars is an oblate spheroid, whose equa- torial diameter is to the polar as 1,355 is to 1,272, or nearly as 16 to 15. This remarkable flattening at the poles of Mars, probably arises from the great variation in the density of his different parts. VESTA. Some Astronomers supposed that a planet existed between the orbits of Jupiter and Mars ; judging from the regularity observed in the distances of the former discovered planets from the Sun. The discovery of Ceres confirmed this conjecture, but the opinion which it seemed to establish respecting the harmony of the Solar System, appeared to be completely overturned, by the discovery of Pallas and Juno. Dr. Olders, however imagined that these small celestial bodies were merely the fragments of a larger planet which had burst asunder by some internal convulsion, and that several more might yet be discovered between the orbits of Mars and Jupiter. He therefore concluded that though the orbits of all these fragments might be inclined to the ecliptic, yet as they must have all diver- ged from the same point, they ought to have two com- mon points of reunion, or two nodes in opposite regions of the Heavens, through which all the planetary frag- ments must sooner or later pass. One of these nodes Sec. 2 Of the Solar System. 31 he found to be in Virgo, and the other in the Whale, and it was actually in the latter of these regions, that Mr. Harding discovered the planet Juno. With the intention therefore of detecting other fragments of the supposed planet, Dr. Olders examined thrice every year all the little stars in the opposite constellations of the Virgin, and the Whale, till his labors were crowned with success on the 29th of March, 1807, by the dis- covery of a new planet in the constellation Virgo, to which he gave the appropriate name of Vesta. The planet Vesta is the next above Mars, and is in appear- ance of the fifth or sixth magnitude, and may be seen in a clear morning by the naked eye. Its light is more intense, pure and white, than either of the three follow- ing Ceres, Juno, or Pallas. Its distance from the Sun is computed at 225 millions of miles, and its diameter at 238 : its revolutions have not hitherto been sufficient- ly ascertained. ON JUNO. The planet Juno, the next above Vesta, and between the orbits of Mars, & Jupiter was discovered by Dr. Har- ding, at the Observatory near Bremen, on the evening of the 5th of September, 1804. This planet is of a reddish color, and is free from that nebulosity which surrounds Pallas. It is distinguished from all the other planets by the great excentricity of its orbit, and the effecc of this is so extremely sensible, that it passes over that half of its orbit which is bisected by its perihelion in half the 32 Of the Solar System. Sec. 2 time that it employs in describing the other half, which is further from the Sun-, from the same cause its great- est distance from the Sun is douhle the least. Tne dif- ference between the two being about 127 millions of miles. Its mean distance from the Sun is computed at 252 millions of miles, and performs its tropical revolu- tion in 4 years and 128 days. Its diameter is estimated at 1,425 miles, and its apparent diameter as seen from the Earth, three seconds of a degree, and its inclination of orbit twenty-one degrees. ON CERES. The planet Ceres was discovered at Palermo, in Si- cily, on the first of January, 1801, by M. Riazzi, an ingenious observer, who has since distinguished him- self by his Astronomical labors. It was however again discovered by Dr. Ciders, on the first of January, 1807, nearly in the same place where it was expected from the calculations of Baron Zach. The planet Ceres is of a ruddy color, and appears about the size of a star of the 8th magnitude. It seems to be surrounded with a large dense atmosphere of 675 miles high, according to the calculations of Schroeter, and plainly exhibits a disk, when examined, with a magnifying power of 200. Ceres is situated between the orbits of Mars and Ju- piter. She performs her revolution round the Sun in four years, 7 months and ten days ; and her mean dis- tance is estimated at 263 millions of miles from that lu- minary. The observations which have been hitherto Sec. 2 Of the Solar System. S3 made upon this celestial body, do not appear sufficient- ly correct to determine its magnitude with any degree of accuracy. ON PALLAS. The planet Pallas was discovered at Bremen, in Lower Saxony, on the evening of the 28th of March, 1802, by Doctor Olders, the same active Astronomer, who re-discovered Ceres. It is situated between the or- bits of Mars and Jupiter, and is nearly of the same magnitude and distance with Ceres, but of a less ruddy color. It is seen surrounded with a nebulosity of al- most the same extent, and performs its annual revolution in nearly the same period. The planet Pallas however is distinguished in a very remarkable manner from Ce- res, and all the primary planets, by the immense inclin- ation of its orbit. While these bodies are revolving round the Sun in almost circular paths, rising only a few de- grees above the plane of the ecliptic ; Pallas ascends a- bove this plane, at an angle of about 35 degrees, which is nearly five times greater than the inclination of Mer- cury. From the eccentricity of Pallas being greater than that of Ceres, or from a difference of position in the line of their apsides, where their mean distances are nearly equal, the orbits of these two planets mutually intersect each other ; a phenomenon which is altogether anomalous in the Solar System. Pallas performs its tropical revolution in feur years 7 months and 11 days. The distance of this planet, from 34 Of the Solar System. Sec. 2 the Sun, is estimated at 265 millions of miles. It is surrounded with an atmosphere 468 miles high. OF JUPITER, Jupiter, the largest of all the planets, is still higher in the Solar System, being four hundred and ninety mil- lions of miles from the Sun, and by performing his an- nual revolution round the Sun in eleven years, 314 days, 20 hours and 27 minutes, he moves in his orbit at the rate of 29,000 miles in an hour. The diameter of this planet is estimated at 89,170 miles, and performs a revolution on its own axis in nine hours, 55 minutes and 37 seconds ; which is more than 28,000 miles every hour, at his equator, the velocity of motion on his axis being nearly equal to the velocity with which he moves in his annual orbit. This planet is surrounded by faint substances called belts, in which so many changes appear, that they have been regarded by some, as clouds or openings in the at- mosphere of the planet ; while others imagine that they are of a more permanent nature, and are the marks of great physical revolutions which are perpetually chang- ing the surface of the planet. The axis of Jupiter is so nearly perpendicular to his orbit, that he has no sensible change of seasons, which is a great advantage, and wisely ordered by the Author of nature; for if the axis of this planet were inclined any considerable number of degrees, just so many degrees round each pole would in their turn, be almost six of Sec. 2 Of the Solar System. 35 our years together in darkness, and, as each degree of a great circle on Jupiter contains 778 of our miles at a mean rate ; judge ye what vast tracts of lands would be rendered uninhabitable by any considerable inclination of his axis. The difference between the equatorial and polar di- ameters of this oblate spheroid is .computed at 6,230 miles ; for his equatorial diameter is to his polar, as!3 is to 12; consequently his poles are 3,115 miles nearer his centre than his equator. This results from his rapid motion round his axis, for the fluids together with the light particles which they can carry, or wash away with them, recede from the poles, which are at rest towards the equator, where the motion is more rapid, until there be a sufficient number of such particles accumulated to make up the deficiency of gravity occasioned by the cen- trifugal force, which arises from a quick motion round an axis j and when the deficiency of weight or gravity of the particles is made up by a sufficient accumulation, the equilibrium is restored, and the equatorial parts rise no higher. The orbit of Jupiter is inclined to the eclip- tic one degree and 20 minutes. His north node is in the 7th degree of Cancer, and his south node in the 7th degree of Capricorn. His mean apparent diameter as seen from the earth is 39 seconds, and as seen from the Sun, 37 seconds of a degree. This planet being situated at so great a distance from the Sun, does not enjoy that degree of light emanating from his rays, which is enjoyed by the earth. To sup- ply this deficiency, the great Author of our existence has 36 Of the Solar System. Sec. 8 provided 4 satellites, or Moons to be his constant atten- dants, which revolve around him, in such manner, that scarcely any part of this large planet but is enlightened during the whole night, by one or more of these Moons* except at his poles, where only the farthest Moons can be seen ; there, however this light is not wanted ; be- cause the Sun constantly circulates in or near the hori- zon, and is very probably kept in view of both poles by the refraction of his atmosphere. The first Moon, or that nearest to Jupiter performs a revolution around him in one day, 18 hours and 36 minutes of our time, and is 229 thousand miles distant from his centre : the second performs his revolution in 3 days 13 hours and 15 minutes at a distance of 364 thousand miles : the third in seven days, three hours and 69 minutes, at the dis- tance of 580 thousand miles, and the fourth, or farthest from his centre in 16 days, 18 hours, and 30 minutes, at the distance of one million of miles from his centre. The angles under which these satellites are seen from the earth, at its mean distance from Jupiter, are as fol- lows : The first three minutes and 65 seconds : the second six minutes and 15 seconds : the third 9 minutes and 58 seconds, and the fourth 17 minutes and 30 sec- onds. This planet when seen from its nearest Moon, must appear more than one thousand times as large as our Moon does to us. The threl nearest Moons to Jupiter, pass through his shadow and are eclipsed by him, in every revolution, but the omt oi the fourth is so much inclined, that it passes by itsopposition to Jupiter without entering his shadow, 2 Of th* Solar System. 37 two years in every six. By these eclipses, Astronomers have not only discovered that the Sun's light is about 8 minutes in coming to us ; but they have also determ:n:d the longitude of places on this earth with greater certain- ty, and facility than by any other method yet known. OF SATURN. Saturn is the most remarkable of all the planets, it is calculated at 9 hundred millions of miles from the Sun, and travelling at the rate of 21,900 miles every hour, and performs its annual circuit in 29 years, 167 days and 2 hours of our time ; which makes only one year to that planet. Its diameter is computed at 79,042 miles, and performs a revolution on its own axis once in ten hours, 16 minutes and two seconds. Its mean apparent diameter as seen from the earth, is 18 seconds, and as seen from the Sun, 16 seconds of a degree; its axis is supposed to be 60 degrees inclined to its orbit. This planet is surrounded by a thin broad ring, which no where touches its body, and when viewed by the aid of a good telescope appears double. It is inclined 30 degrees to the ecliptic, and is about 21 thousand miles in breadth ; which is equal to its distance from Saturn on all sides. This ring performs a revolution on its axis in the same space of time with the planet, namely, ten hours 16 miutes and two seconds. This ring seen from the planet Saturn, appears like a vast luminous circle in the Heavens, and, as if ft does not belong to the planet When we see the ring most 38 Of the Solar System. Sec 2. open, its shadow upon the planet is broadest, and from that time the shadow grows narrower, as the ring ap- pears to do to us, until by Saturn's annual motion, the Sun comes to the plane ojf the ring, or even with its edge ; which being then directed towards the earth, it becomes to us invisible on account of its thinness. The ring nearly disappears twice in every annual revo- lution of Saturn, when he is in the 19th degree, both of Pisces and Virgo. But, when Saturn is in the 19th degree either of Gemini or Sagitarius, his ring appears most open to us, and then its longest diameter is to its shortest as 9 to 4. This planet is surrounded with no less than seven satellites, which supply him with light during the ab- sence of the Sun. The fourth of these was first dis- covered by Huygens, on the 25th of March, 1655. Cassini discovered the fifth in October, 1671. The third on the 23d of december, 1672 : And the first and second in the month of March, 1684. The sixth and seventh were discovered by Dr. Herschel in the year 1789. These are nearer to Saturn than any of the others. These Moons perform their revolutions round this planet on the outside of his ring, and nearly in the same plane with it. The first, or nearest Moon to Saturn, performs its periodical revolution around him in 22 hours and 37 minutes, at the distance of 121,000 miles from his centre ; the second performs its period- ical revolution in one day, 8 hours and 53 minutes, at the distance of 156 thousand miles ; the third in one Sec. 2 Of the Solar System 39 day, 21 hours, 18 minutes and 26 seconds, at the dis- tance of 193 thousand ; the fourth in two days, 17 hours 44 minutes and 51 seconds, at the distance of 247 thou- sand ; the fifth in 4 days, 12 hours, 25 minutes and 1 1 seconds, at 346 thousand ; the 6th in 15 days, 22 hours, 41 minutes and 13 seconds, at the distance of 802 thou- sand ; and the 7th or outermost in 49 days, 7 hours, 53 minutes and 43 seconds, at the distance of two millions, 337 thousand miles from the centre of Saturn their primary. When we look with a good telescope, at the body of Saturn, he appears like most of the other planets, in the form of an oblate spheroid, arising from the rapid rota- tion about his axis. He however appears more flat- tened at the poles, than any of the others, and although his motion on his axis is not equal to that of Jupiter, yet he does not appear to be in form, so near that of a globe as that planet. When we consider that the ring by which Saturn is encompassed, lies in the same plane of his equator, and, that it is at least equal if not more dense than the planet, we shall find no difficulty in ac- counting for the great accumulation of matter, at the the equator of Saturn. The ring acts more powerful- ly upon the equatorial regions of Saturn; than upon any part of his disk ; and by diminishing the gravity of these parts, it aids the centrifugal force in flattening the poles of the planet. Had Saturn indeed never re- volved upon his axis, the action of the ring would of itself have been sufficient, to have given it the form of a spheroid. 40 Of the Solar System. Sec. S The following, are the dimensions of this luminous zone, as determined by Dr. Herschel. Inside diameter of the interior ring, - 146,345 Outside diameter of the interior ring, -;> 184,383 Inside diameter of the exterior ring, * 190,240 Outside diameter of the exterior ring, .* 204,833 Breadth of the interior ring, ... 20,000 Breadth of the exterior ring, - - '- 7,200 Breadth of the dark space between the two rings, 2,839 Angle which it subtends when seen at the mean M. s. distance of the planet. - - - - 7 25 ON HERSCHEL, OR URANUS. From inequalities in the motion of Jupiter and Saturn, for which no rational account could be given, and from the mutual action of these planets, it was inferred by some Astronomers, that another planet existed beyond the orbits of Jupiter and Saturn ; by whose action these irregularities were produced. This conjecture was confirmed on the 13th of March, 1781 ; when Dr. Her- schel discovered a new planet, which in compliment to his Royal Patron, he called Georgium Sidus, although it is more generally known by the name of Herschel, or Uranus. This new planet, (which had been former- ly observed as a small star by Flamstead, and likewise by Tobias Mayer, and introduced into their catalogue of fixed stars,) is situated, one thousand, eight hundred millions of miles from the centre of the System, and performs its revolution round the Sun in 83 years, 150 Sec. 2 Of the Solar System. 41 days, and 18 hours. Its diameter is computed at 35,112 miles. When seen from the earth, its mean apparent diameter is three and J seconds, and as seen from the Sun, 4 seconds of a degree. As the distance of this planet from the Sun is twice as great as that of Saturn, it can scarcely be distinguished without the aid of instruments. When the sky however is serene it appears like a fixed star of the sixth magnitude, with a bluish white light, and a brilliancy between that of Venus and the Moon ; but seen with a power of two or three hundred, its disk is visible and well defined. The want of light arising from the distance of this planet from the Sun, is supplied by six satellites, all of which were discovered by Dr. Herschel. The first of those satellites is twenty five and half seconds from its primary, and revolves round it in 5 days, 21 hours and 25 minutes ; the second is nearly 34 seconds distant from the planet, and performs its rev- olution in 8 days, 17 hours, 1 minute and 19 seconds. The distance of the third satellite is 38,57 seconds, and the time of its periodical revolution is ten days, 23 hours, and four minutes. The distance of-the fourth satellite is 44,22 seconds, and the time of its periodical revolution is 13 days, 11 hours, 5 minutes and 30 sec- onds. The distance of the fifth is one minute and 28 seconds, and its revolution is completed in 38 days, 1 hour, and 49 minutes. The sixth satellite, or the fur- thest from the centre of its primary, at the distance of two minutes, and nearly 57 seconds, and therefore re- quires 107 days, 16 hours, and 40 minutes to complete 42 Of the Solar System. ' Sec. 2 one revolution. The second and fourth of these were discovered on the llth of January, 1787, the other four were discovered in 1 790, and 1 794 ; but their distan- tances, and times of periodical revolution, have not been so accurately ascertained as the^ other two. It is however, a remarkable circumstance, that the whole of these satellites move in a retrogade direction, and in orbits lying in the same plane, and almost perpendicu- lar to the ecliptic. When the Earth is in its perihelion, and Herschel in its aphelion, the latter becomes stationary, as seen from the Earth, when his elongation, or distance from the Sun is 8 signs, 17 degrees, and 37 minutes, his retro- gradations continue 151 days, and 12 hours. When the Earth is in its aphelion, and Herschel in its perihelion, it becomes stationary, at an elongation of 8 signs, 16 degrees, and 27 minutes, and its retrogadations con- tinue 149 days and 18 hours. ON COMETS. Comets are a class of celestial bodies, which occa- sionly appear in the Heavens. They exhibit no visible or defined disk, but shine with a pale and cloudy light, accompanied with a tail or train, turned from the Sun. They traverse every part of the Heavens, and move in every possible direction. When examined through a good telescope, a Comet resembles a mass of aquious vapors, encircling an opaque nucleus, of different degrees of darknes in dif- Sec. 2 Of the Solar System 43 ferent Comets ; though sometimes, as in the case of sev- eral discovered by Dr. Herschel, no nucleus can be seen. As the Comet advances towards the Sun, its faint and nebulous light becomes more brilliant, and its lu- minous train gradually increases in length. When it reaches its perihelion, the intensity of its light, and the length of its tail reaches their maximum, and then it sometimes shines with all the splendor of Venus. During its retreat from the perihelion,it is shorn of its splendor, and it gradually resumes its nebulous appearance ; and its train decreases in magnitude, until it reaches such a distance from the Earth, that the attenuated light of the Sun, which it reflects, ceases to make an impression on the organ of sight. Traversing unseen the remote portion of its orbit, the Comet wheels its etherial course far beyond the limits of the Solar System. What region it there visits, or Upon what destination it is sent, the limited powers of man are un- able to discover. After the lapse of years, w r e per- ceive it again returning to our System, and tracing a portion of the same orbit round the Sun, which it had formerly described. Various opinions have been entertained by Astrono- mers respecting the tails of Comets. These tails or trains, sometimes occupy an immense space in the Heavens. The Comet of 1681, stretched its tail across an arch of 104 degrees ; and the tail of the Comet of 1769 subtended an angle of 60 degrees at Paris, 70 at Bologna, 97 at the Isle of Bourbon, and 90 degrees at 44 Of the Solar System. Sec 2. Sea, between Teneriffe and Cadiz. These long trains of light are maintained by Newton, to be a thin vapor, raised by the heat of the Sun from the Comet. If we knew their uses in our System, we could form more probable conjectures as to the chronology of their creation. They have been noticed from the earliest era of our Astronomical History, and if our modern Philosophers had not discovered, that some (at least,) leave us to return again into our System, and there- fore describe a vast elliptical orbit round our Sun, we might have fancied that the periods of their first recor- ded appearances in our field of science, were the eras of their individual formation. But their recurring presence proves, that their first existence ascends in- to unexplored and unrecorded antiquity. Yet, from whence they came to us, we as little know as for for what purpose. Tycha Brache proved that they were further from the Earth than the Moon, and were nearly as distant as the planets. The Comet of 1682, re-appeared in 1 759, in the interval described, an orbit in the form of an ellipsis, answering to a revolution of 27,937 days. It will therefore re-appear in Novem- ber, 1835. In its greatest distance, it is supposed not to go above twice as far as Uranus. This is indeed a prodigious sweep of space, and it has been justly obser- ved, that the vast distance to which some Comets roam, proves how* very far the attraction of the Sun extends ; for though they stretch themselves to such depths in the abyss of space, yet by virtue of the Solar power, they return into its effulgence. But it ha been Sec. 2 Of the Solar System. 45 recently discovered, that three Comets (at least,) never leave the planetary system. One whose period is three years and a quarter, is included within the orbit of Ju- piter ; another of six years and three quarters, ex- tends not so far as Saturn ; and a third of twenty years* is found not to pass beyond the circuit of Uranus. The transient effect of a Comet passing near the Earth, could scarcely amount to any great convulsion, but if the Earth were actually to receive a shock by collision, from one of those bodies, the consequences would be awful. A new direction would be given to its rotatory motion, and the Globe would revolve around a new axis. The Seas, forsaking their ancient beds, would be hurried by their centrifugal force to the new equatorial regions ; islands and continents, the abodes of men and animals, would be covered by the univer- sal rush of the waters to the new Equator, and every vestige of human industry and genius at once de- stroyed. Although the orbits of all the planets in the Solar System be crossed by five hundred different Comets, the chances against such an event however, are so ve- ry numerous, that there need be no dread of its occur- rence ; besides, that Almighty arm which first created them, and described for them their various orbits that Omnipotent Wisdom which directed the times of their periodical revolutions, still continues to guide and protect all the workmanship of his hands. 46 Interrogations for Section Second. Sec. 2 Interrogations for Section Second. Of what does the SOLAR SYSTEM consist 1 What planets finish their circuits soonest 1 Which moves with the greatest rapidity ? In what direction do they move in their orbits ? What is jthe form of the orbits described by the planets ? Where is the Sun placed 7 In what time does he turn round on his own axis ? How is that proved 1- What is his diameter 1 What is his mean apparent diameter as seen from the Earth 1 How is his solidity calculated ? What is the Ecliptic ? What is meant by the Nodes ? Which is the Ascending Node ? Which is the Descending Node 1 How many signs are they asunder ? What additional Astronomical discoveries were made in the year 1805 ? What planet is nearest the Sun 1 What reasons are given to suppose that^this planet receives its light from the Sun ? What is the computed distance of Mercury from the Sun? What is its diameter ? Sec. 2. Interrogations for Section Second. 47 In what time does it perform a revolution around the Sun ? What time on its own its axis 1 How many miles does it move in an hour in its mo- tion round the Sun ? How many degrees is his orbit inclined towards the Ecliptic ? What planet is next to Mercury ? What is the distance of this planet from the Sun ? , How many miles in an hour, does Venus move in performing her revolution round the Sun ? In what time does she perform her annual revolution ? In what time does she perform a revolution on her axis ? What is her diameter ? How is it known that the orbits of Mercury and Ve- nus are included within that of the Earth ? How many degrees at most does Mercury depart from the Sun ? How many Venus ? What is meant by inferior conjunction ? What is a transit ? % What is the name of the third planet from the Sun ? What is its distance from the Sun ? What is its diameter ? In what time does it perform a revolution around the Sun? What is its hourly progress 1 In what time does it perform a revolution on its axis ? What is its form ? 48 Interrogations for Section Second. Sec. 2 How many miles difference in the two diameters ? What is the Moon 1 In what time does she perform a revolution round the Sun ? Around the Earth ? Around her own axis ? What is her distance from the* Earth ? How many miles in diameter ? What is her mean apparent diameter as seen from the Earth ? Do the orbits of the Earth and Moon coincide ? How much more light does the Earth give to the Moon, than the Moon gives to us ? What is the name of the fourth planet from the Sun ? What is his distance from that luminary ? In what time does he perform his annual revolution ? What is his hourly progress ? What time his revolution on his axis ? What is his mean apparent diameter as seen from the Earth ? What as seen from the Sun ? For what is the planet Mars remarkable ? What have Astronomers concluded to be the cause of this remarkable appearance ? What is its form ? What is the name of the fifth planet from the Sun ? By whom was it discovered ? And when ? In what sigh of the Ecliptic can this planet be seen without the aid of a telescope ? What is its distance from the Sun ? Sec. 2 Interrogation* for Section Second 49 What is its diameter ? What is the name of the sixth ? By whom was it discovered 1 And when ? What is its color ? For what is it distinguished ? What is its distance from the Sun ? What is the time of its tropical revolution 7 What is its diameter ? What is its apparent diameter as seen from the Sun? What is the name of the seventh ? By whom was it first discovered ? In what year ? What is the height of its atmosphere 1 In what time does this planet perform its revolution round the Sun ? What is her distance from that luminary ? What is the name of the eighth planet from the Sun ? When was it discovered ? And by whom 1 What is its distance from the Sun 1 In what time does it perform its annual revolution around him ? What is the height of its atmosphere ? By what name are the last four collectively called ? They are called Asteroids. What is the name of the ninth planet from the Sun ? How many miles distant from the Sun ? What is his diameter 1 In what time does he perform his annual revolution ? In what time on his own axis ? What is his mean motion in his orbit 1 50 Interrogations for Section Second. Sec. 2 What is his mean motion on his axis 1 Have the inhabitants of Jupiter any sensible change of seasons ? How many miles constitute a degree on this planet 1 How many miles difference between his equatorial and polar diameters 1 Why is this great difference 1 How many degrees is the orbit of Jupiter inclined to the Ecliptic 1 In what sign of the Zodiac is his north node ? In what sign his south node 1 How many satellites attend this planet 7 What is his apparent diameter as seen from the Sun ? What as seen from the Earth 1 Of what benefit have those Moons been to the inhab- itants of this Earth ? Can either of them be seen by us without the aid of telescopes ? What name is given to the next, or tenth planet from the Sun ? What is its distance from the Sun ? How many miles does this planet move in an hour ? In what time does it perform its revolution around the Sun ? In what time on its own axis 1 What is its diameter ? What is his mean apparent diameter as seen from the Sun 1 What as seen from the Earth ? How many degrees is its axis inclined to its orbit ? What encircles his body 1 How does it appear when viewed with a telescope 1 What is its breadth 1 How does it appear to the inhabitants of Saturn 1 Sec. 2. Interrogations for Section Second. 5 1 Why is it sometimes invisible to us 1 How many times does it appear in one revolution of the planet ? In what signs and degrees of the Ecliptic does it disappear 1 In what signs and degrees does this ring appear most open ? How many Satellites has this planet 1 Where are they situated, inside or outside of the ring? What is the form of this planet 1 What is the name of the next, or outermost planet ? When was it discovered 1 How far is it situated from the Sun 1 In what time does it perform its annual revolution 1 What is its diameter ? What is its apparent diameter as seen from the Sun ? What as seen from the Earth ? Can it be seen without the aid of a telescope ? How many Satellites attend it ? In what direction do those Satellites move ? What are Cornets ? In what direction do they move 1 In what part of its orbit is its train most brilliant ? What was Newton's opinion concerning the Comet's tail, or train ? How many Comets are supposed to belong to the Solar System ? 52 Interrogations for Section Second. Ssc. 2 How many are known not to exceed the circuit o f Uranus ? What would be the result if a Comet should come in actual collision with this Earth ? Is it probable that there will ever be such an oc- currence ? Why is it not probable ? SECTION THIRD. THE power by which bodies fall towards the Earth, is Called GRAVITY, or Attraction. By this power in the Earth, it is that all the bodies on whatever side^ fall in lines perpendicular to its surface. On opposite parts of the Earth, bodies fall in opposite directions, all towards the centre, where the whole force of gravity appears to be accumulated. By this power constantly acting on bodies near the Earth, they are kept from leaving it, and those on its surface are kept by it, that they cannot fall from it. Bodies thrown with any ob- liquity, are drawn by this power from a straight line into a curve, until they fall to the ground. The great- er the force with which they are projected, the greater is the distance they are carried before they fall. If we suppose a body carried several miles above the surface of the Earth, jind there projected in an horizontal di- rection, with so great a velocity that it moves more than semidiameter of the Earth in the line, it would take to 54 On Gravity. Sec. 3 fall to the Earth by gravity, in that case, if there were no resisting medium, the body would not fall to the Earth at all ; but continue to circulate round the Earth, keep- ing always the same path, and returning to the point from whence it was projected with the same velocity with which it moved at first. We find that the Moon therefore must be acted upon by two powers, one of which would cause her to move in a right line, another bending her motion from that line into a curve. This attractive power must be seated in the Earth, for there is no other body within the Moon's orbit to draw her.* The attractive power of the Earth therefore extends to the Moon, and in combination with her projectile force, causes her to move round the Earth in the same manner, as the circulating body above supposed. The Moons of Jupiter, Saturn and Herschel, are ob- served to move around their primary planets ; there- fore there is an attractive power in these planets, op. erating on their Satellites in the same manner, as the attraction of the Earth operates on the Moon. All the planets and Comets move round the Sun, and respect it as their centre of motion, therefore the Sun must be endowed with an attracting power, as well as the * If the Moon revolves in her orbit in consequence of an attractive power residing in the Earth, she ought to be attracted as much from the tan- gent of her orbit in a minute, as heavy bodies fall at the Earth's surface in a second of time. It is accordingly found by calculation, that the Moon is deflected from the tangent 16,09 feet in a minute, which is the very space through which heavy bodies descend in a second of time at the Earth's surface. Sec. 3 On Gravity. 55 Earth and planets. Consequently all the bodies, or matter of the Solar System are possessed of this attrac- tive power, and also all matter whatsoever. As the Sun attracts the planets with their Satellites;, and the Earth the Moon, so the planets and Satellites re-attract the Sun, and the Moon the Earth. This is also confirmed by observation ; for the Moon raises tides in the Ocean ; the satellites and planets disturb each other's motions. Every particle of matter being possessed of an attracting power, the effect of the whole must be in proportion to the quantity of matter in the body. Gravity also, like all other virtues, or emanations, ei- ther drawing or impelling a body towards a centre, de- creases as the square of the distance increases ; that is, a body at twice the distance, attracts another with only a fourth part of the force ; at four times the dis- tance, with a sixteenth part of the force. By considering the law of gravitation which takes place throughout the Solar System, it will be evident that the Earth moves round the Sun in a year. It has been stated and shown, that the power of gravity de- creases as the square of the distance increases, and from this it follows with mathematical certainty, that when two or more bodies move round another as their centre of motion, the squares of the time of their peri- odical revolutions, will be in proportion to each other, as the cubes of their distances from the central body. This holds precisely with regard to the planets round 56 On Gravity. Sec. 3 the Sun, and the satellites round their primaries, the relative distances of which are well known. All Globes which turn on their own axis, will be ob- late spheroids, that is, their surfaces will be further from their centres in the equatorial, than in the polar regions ; for as the equatorial parts move with greater velocity, they will recede farthest from the axis of mo tion, and enlarge the equatorial diameter. That our Earth is really of this figure, is demonstrable from the unequal vibrations of a pendulum, and the unequal length of degrees in different latitudes. Since then the Earth is higher at the equator than at the poles, the Seas naturally would run towards the polar regions, and leave the equatorial parts dry, if the centrifugal force of these parts, by which the waters were carried thither, did not keep them from return- ing. Bodies near the poles are heavier than these near- er the Earth's centre, where the whole force of the Earth's attraction is accumulated. They are also heavier, because their centrifugal force is less on ac- count of their diurnal motions being slower. For both these reasons, bodies carried from the poles towards the Equator, gradually lose part of their weight Experiments prove that a pendulum which vibrates seconds near the poles, vibrates slower near the Equa- tor, which shows that it is lighter, or less attracted there. To make it oscillate in the same time, it is found necessary to diminish its length. By comparing the different lengths of pendulums vibrating seconds at the equator, and at London ; it is found that a pen- Sec. 3 Interrogations for Section Third. 57 dulum must be 2,542 lines * shorter at the Equator than at the poles. Interrogations for Section Third. What is GRAVITY ? Do falling bodies strike the surface of the Earth at right angles 1 Do falling bodies near the Earth, always direct their course to its centre ? Where is the centre of Gravity situated 1 When bodies are projected in a right line, what brings them to the Earth ? If there were no attractive power at the centre of the Earth, what would be the consequence were a bo- dy so projected, and not meeting any resistance from the air? We find that the Moon moves round the Earth in an orbit nearly circular. Why is it so 1 Where is that attractive power situated ? Have the other planets attractive powers also * How is it known 1 * A line is l-12th part of an inch. O 58 Interrogations for Section Third. Sec. 3 Where is the centre of attraction of the Solar Sys- tem placed 1 How is it known 1 Do the planets attract the Sun as well as the Sun the planets ? Has every particle of matter an attractive power ? In what proportion does Gravity increase 1 How far is the Moon deflected by Gravity from a tangent in one minute of time 1 How far does a failing body descend in one second 1 In what proportion are the squares of the times of the periodical revolutions of all the planets ? What will be the form all planets which revolve on their own axis 1 Why will they be of that form ? How is it ascertained to a certainty, that our Earth is of that form ? Why are bodies near the poles heavier than those at the Equator 1 Why is a pendulum vibrating seconds shorter at the Equator than at the poles ? What is the length of a pendulum vibrating seconds at the Equator 1 Jlns. 39,2 inches. SECTION FOURTH. PHENOMENA OF THE HEAVENS, AS SEEN FROM DIFFERENT PARTS OF THE EARTH. THE magnitude of the Earth is only a point when com- pared to the Heavens, and therefore every inhabitant upon it, let him be in any place on its surface, sees half of the Heavens. The inhabitant on the North Pole of the Earth, constantly sees the Northern Hemisphere, and having the North Pole of the Heavens directly over his head, his horizon coincides with the celestial Equa- tor. Therefore all the Stars in the Northern Hemi- sphere, between the Equator and the North Pole, ap- pear to turn round parallel to the horizon. The Equa- torial Stars keep in the horizon, and all those in the Southern Hemisphere are invisible. The like phe- nomena are seen by an observer at the South Pole. Hence, under either pole, only one half of the Heavens is seen ; for those parts which are once visible never set, and those which are once invisible never rise. 60 Phenomena of the Heavens, fyc /Sec. 4 But the ecliptic or orbit, which the Sun appears to de- scribe once a year by the annual motion of the Earth, has the half constantly above the horizon of the north pole ; and the other half always below it. Therefore whilst the Sun describes the northern half of the eclip- tic, he neither sets to the north pole, nor rises to the south ; and whilst he describes the southern half, he neither sets to the south pole, nor rises to the north. The same observations are true with respect to the Moon, with this difference only, that as the Sun de- scribes the ecliptic but once a year, he is, during half that time, visible to each pole in its turn, and as long invisible. But, as the Moon goes round the ecliptic in 27 days, 8 hours, she is only visible during 13 days and 16 hours, and as long invisible to each pole by turns. All the planets likewise rise &, set to the polar regions because their orbits are cut obliquely in halves by the horizon of the poles. When the Sun arrives at the sign Aries, which is on the twentieth of March, he is just rising to an observer on the north pole, and set- ting to another on the south pole.* From the Equator, he rises higher and higher in every apparent diurnal revolution, till he comes to the highest point of the ecliptic on the 21st of June, and then he is at his great- est altitude, which is 23 degrees and 28 minutes ; * It is therefore evident when the Sun is on the Equator, an observer placed at each pole, sees about one half of the Sun above the horizon, and likewise ] osi'e to the Sun, must be below the horizon during that half of the year. But when the Sun is in the southern half of the ecliptic, he never rises to the north pole ; during this half of the year, every full Moon happens in seme part of the northern half of the ecliptic which never sets. Conse- quently, as the polar inhabitants never see the full Moon in Summer, they have her always in the Win- ter ; before, at, and after the fuP, shining during 14 of our days and nights. And when the Sun is at his greatest distance bekw the horizon, being then in Capricorn, the Mcon is at ! er first quarter in Aries, full in Cancer, and at her third quarter in Libra. And Sec. 9 Of the Moon's Phases. *, J 1 5 as the beginning of Aries is the rising point of the eclip- tic, Canrcr the highest, ami Libra UK* setting point, the M-.ioii ris 'S at her first quarter in Aries, is most elevated .*! >w me horiz >n, an;i hi I in Cancer, and sets at the beginning of Libra in iu r third qsiartcr, having eontin- ui'tl visible for 14 iliurnnl rotations of the earth. Thus the poles are supplied one half of the Wintertime with constant M >o!i-light, in the absence of the bun, and only lose sight of the Moon from her third to her first quarter, while she gives but very liulc light, and could be but of little, and sometimes of no service to them. 110 Interrogations for Section J\finth. Sec. 9 Interrogations for Section Xinth. What is understood by the MOON'S PHASES 1 What is discovered by observing the moon with a telescope ? Of what use is the ruggedness to us ? If the surface of the moon is uneven, why does it not so appear when viewed by the eye only 1 What is the moon ? What part of the mo;m do we discover 1 When is she said to be in conjunction with the Sun ? When she is in her first octant, how much of her en- lightened side is visible 1 How much of her enlightened side does she show in her first quarter ? When she has gone half around her orbit, how does she appear ? How does she appear when viewed from the Sun ? Are the moon's morions faster or slower than the earth's, from her change to her first quarter ? How far does she fall behind the earth ? 'er first qu.utcT to her full, which moves with tvi rapidity ? Sec. 9 Interrogations for Section Ninth. 117 Which from the full to her third quarter ? Which from the third quarter to the change 1 Is the gravity of the moon at any time greater towards the Sun, than towards the earth, and at what time ? How much greater is the_quantity of matter in the Sun, than in the earth ? In what proportion does the attraction of each body diminish ? How far from the earth is the point of equal attrac- tion between the earth and the Sun ? Why does not the moon leave the earth and go to the Sun ? What is understood by the Harvest Moon ? How many minutes later at the equator does the moon rise every day, than on the preceding ? Is there any material difference in high northern, of southern latitudes 1 At what time in northern latitudes, does the full moon rise? How many days together does the moon in such cases rise at nearly the same time ? What is the cause of this small difference ? How far does the earth advance in her orbit, while the moon goes round the ecliptic ? How mnny conjunctions and oppositions of the Sun and Moon can take place in any particular part of the ecliptic, in the course of a year? How many full Moons in the course of a year, that rise with so little difference near the time of Sun- setting ? 1 18 Interrogations for Section Ninth. Sec. 9 Do s this singularity appear \i-. ."Oniht rs: >ntittlr,s <\* \vcll as in nott 1 ern ? Docs the M on's orbit lie ex. e ly in 1 1 e ec ipiic r Does the moon's orhii intersect ihr- ecliptic ? What is understood by the moon's no:i< s ? How many times from change to change, is the moon in her nodes ? Which is called the ascending node ? Wiiich is called the descending node? Ho\v many degrees are they asunder? How much does these nodes shift in ihc course of a year ? Which way do they shift ? In what length of time do they go around the eclip- tic ? How many degrees can the Sun go below the hori- zon of the poles ? How many degrees must the Sun be below the hori- zon before the twilight is wholly gone ? Is the full moon in the Summer season ever seen at the north pole ? Is it cominually seen in Winter, from the first to her third quarter ? Is it the same at the south pole ? SECTION TENTH. On the Ebbing and Flowing of the Sea. THE cause of the tides was first discovered by Kep- ler, whj thus explains it. The orb of the attracting power, (which is in the Moon,) is extended as far as the earth, and draws the waters under the torrid zone; acting upon places where it is vertical ; insensibly on confined Seas and Bays ; but sensibly on the Ocean, whose beds are larger, and the waters have the liberty of reciprocation,that is of rising and falling. And in the 70th page of his lunar Astroncmy he says : But the cause of the Tides of the Sea, appears to be the bo- dies of the Si:n and Moon, drawing the waters of the Sea. This hint being g^ven, the immortal Sir Isaac New- ton improved it, and wrote so amply on the subject as 120 On the Ebbing and Flowing of the Sea. Sec. 10 to make the theory of Tides in a manner quite his own, by discovering the cause of their rising on the side of the earth opposite to the Moon. For Kepler believed that the presence of the Moon occasioned an impulse which caused another in her absence. It has been already mentioned, that the power of gravity diminishes as the square of the distance in- creases, and therefore the waters on the side of the earth, next the Moon, are more attracted than the cen- tral parts of the earth by the Moon, and the central parts are more attracted by her, than the waters on the opposite side of the earth. Therefore the distance be- tween the earth's centre, and the water, or its surface, __>vill be increased. If the attraction be unequal, then that body which is most strongly attracted, will move with greater rapidity, and this will increase its distance from the other body. As this explanation of the ebbing and flowing of the Sea is deduced from the earths' constantly falling to- ward the Moon by the po\ver of gravity, some may find a difficulty in conceiving how this is possible when the Moon is full, or in opposition to the Sun ; since the earth revolves about the Sun, and must continually fall towards it ; and therefore cannot fall contrary ways at the same time : or if the earth is constantly falling towards the Moon, they must come together at last. To remove this difficulty, let it be considered that it is not the centre of the earth that describes the annual orbit round the Sun, but the common Sec. 10 On the Ebbing and Flowing of the Sea. 1 2 1 centre * of gravity of the earth and Moon together ; and that while the earth is moving round the Sun, it also describes a circle around that centre of gravity, going as many times around it in one revolution about the Sun, as there are lunations, or courses of the Moon around the earth; is constantly falling towards the Moon from a tangent to the circle it describes* around the said centre of gravity. The influence of the Sun in raising the Tides, is but small in comparison of the Moon's : though the earth's diameter bears a considerable proportion to its distance from the Moon, it is next to nothing when compared to its distance from the Sun. Therefore, the difference of the Sun's attraction on the sides of the earth, under, and opposite to him, is much less than the difference of the Moon's attraction on the sides of the earth un- der, and opposite to her : therefore the Moon must raise the Tides much higher than they can be raised by the Sun. On this theory, (so far as it has been ex- plained,) the Tides ought to be the highest directly under, and opposite to the Moon ; that is, when the Moon is due north or south. But we find in open Seas, where the water flows freely, the Moon is gen- * This centre is as much nearer the earth's centre, than the Moons, as the earth is heavier, or contains a greater quantity of matter than the Moon which is about 40 times. If both bodies were suspended from it, they would hang in equilibrio. Therefore divide the Moon's distance from the earth's centre, (240,000 miles,) by 40, and the quotient, will be the distance from the centre of the earth to the centre of gravity, which is 6,000 miles, or 2,000 from the earth's surface. O On Tides. Sec. 10 erally past the north and south meridian, when it is high water. The reason is obvious, were the Moon's attraction to cease wholly when she was past the me- ridian, yet the motion of ascent communicated to the water before that time, would make it continue to rise for some time afterward, much more must it continue to rise, when the attraction is only diminished. A lit- tle impulse given to a moving ball, will cause it to move further than it otherwise would have done. Or, as experience shows that the weather in Summer, is warmer at 2 o'clock in the afternoon, than when the Sun is on the meridian, because of the increase made to the heat already imparted. The Tides do not always answer to the same dis- tance of the Moon from the meridian at the same pla- ces, but are variously affected by the action of the Sun } which brings them on sooner when the Moon is in her first and third quarters, and keeps them back later when she is in her second and fourth ; because, in the one case, the Tide raised by the Sun alone, would be earlier than the Tide raised by the Moon, and in the other case later. The Moon goes round the earth in an elliptical or- bit, and therefore in every lunar month she approach- es nearer to the earth than her mean distance, and re- cedes further from it. When she is nearest, she at- tracts strongest,and so raises the Tides most ; the con- trary happens when she is farthest, because of her weaker attraction. Sec. 10 On Tides. 123 When both- luminaries are in the equator, and the Moon in perigee, (or at her least distance from the earth,) she raises the Tides highest of all ; especially at her conjunction and opposition, both because the equatorial parts have the greatest centrifugal force from their describing the largest circle, and from the concurring actions of the Sun and Moon. At the change, the attractive forces of the Sun and Moon be- ing united, they diminish the gravity of the waters un- der the Moon, and their gravity on the opposite side is diminished by means of a greater centrifugal force. At the full, while the .Moon raises the Tide under, and opposite to her ; the Sun acting in the same line, raises the Tide under, and opposite to him ; whence their conjoint effect is the same as at the change, and in both cases, occasion what is called, Spring Tides. But at the quarters, the Sun's action diminishes the action of the Moon on the waters, so that they rise a little under, and opposite to the Sun, and full as much under, and opposite to the Moon, making what we call Neap Tides; because the Sun and Moon then act crosswise to each other. But strictly speaking, these Tides happen not till some time after, because in this, as in other cases, the actions do not produce the greatest effect, when they are at the strongest, but sometime afterward. The Sun being nearer the earth in Winter than in Summer, is of course, nearer to it in February and October, than in March and September, and therefore the greatest Tides happen not till some time after the autumnal equinox : and return a little before the ver- 124 On Tides. Sec. 10 nal. The Sea being thus put in motion, would contin- ue to ebb and flow for several times, though the Sun and Moon should be annihilated, or their influence cease. When the Moon is in the equator, the Tides are equally high in both parts of the lunar day, or time of the Moons'* revolving from the meridian to the meri- dian again ; which is 24 hours and 50 minutes. But, as the Moon declines from the equator towards either pole, the Tides are alternately higher and lower at places having north or south latitude. One of the highest elevations, (which is that under the Moon,) follows her towards the pole to which she is nearest, and the other declines towards the opposite pole ; each elevation describing parallels as far distant from the equator on opposite sides, as the Moon declines from it to either side, and consequently, the parallels described by these elevations of the water, are twice as many degrees from each other, as the Moon is from the equator ; increasing their distance as the Moon increas- es her declination', till it be at the greatest ; when the said parallels are at a mean state 47 degrees asunder, and on that^'day, the Tides are most unequal in their heights. As the Moon, returns toward the equator,the parallels described by the opposite elevations approach towards each other, until the Moon comes to the equa- tor, and then they coincide. As the Moon declines toward the opposite pole at equal distances, each ele- vation describes the same parallel in the other part of the lunar day, which its opposite elevations described Sec. 10 On Tides. 125 before. While the Moon has north declination, the greatest Tides in the northern hemisphere, are, when she is above the horizon, and the reverse when her de- clination is south. Thus it appears, that as the Tides are governed by the Moon, they must turn on the axis of the Moon's orbit, which is inclined 23 degrees and 28 -minutes to the earth's axis at a mean state, and therefore the poles of the Tides, must be so many degrees from the poles of the earth, or in opposite points of the polar circles, go ing around them in every revolution of the Moon from any meridian to the same again. It is not, however, to be doubted, but that the quick rotation of the earth on his axis, brings the poles of the Tides nearer to the poles of the werld, than they would be, if the earth were at rest, and the Moon re- volved about it. only once a month, otherwise the Tides would be more unequal in their heights, and times of their returns, than we find they are. But how near the earth's rotation may bring the poles of its axis, and those of the Tides together, or how far the preceding Tides may effect those that follow, so as to make them keep up nearly to the same heights and times of ebbing and flowing, is a problem more fit to be solved by ob- servation than theory. In open Seas, the Tides rise but to very small heights in proportion to what they do in broad River?, whose waters empty in the direction of the stream of'Ii Ic : For, in channels growing narrower gradually, the wa- ter is accumulated by the opposition of the contracting 126 On Tides. Sec. 10 bank. The Tides are so retarded in their passage through different shoals and channels, and otherwise so variously affected by striking against Capes and Headlands, that to different places, they happen at all distances of the Moon from the meridian, and conse- quently at all hours of the lunar day. * Air being lighter than water, and the surface of the atmosphere nearer to the Moon, than the surface of the Sea j it cannot be doubted that the Moon raises much higher Tides in the air, than in the Sea. * In a register of the barometer kept for 30 years, the Professor Toal- do of Padua, added together all the heights of the mercury, when the Moon was in syzigy, when she was in quadrature, and when she was in the apo- geal and perigeal points of her orbit. The apogeal exceeded the perigeal heights by 14 inches, and the heights in syzigy exceeded those in quadra- ture by 11 inches. The difference in these heights, is sufficiently great to show that the air is accumulated and compressed by the attraction of the Moon. Sec. 10 Interrogations for Section Tenth. 127 Interrogations for Section Tenth. By whom was the cause of the TIDES first dis- covered ? How does he explain it ? Who improved the idea of Kepler ? By what does he consider the waters to be at- tracted ? Why are the waters on the side of the earth next to the Moon, more attracted than the central parts 1 Why are the central parts more attracted, than the wa- ters on the opposite side ? From what source is this explanation deduced ? By what power is the earth constantly falling towards the Moon, and the Moon towards the earth 1 If this be actually the case, why do they not come together ? Is it the centre of the earth that describes the annual orbit round the Sun ? Where is the centre of gravity between the earth and Moon ? How much more matter does the earth contain, than the Moon 1 What is the centre of gravity between the two bodies ? How is it found ? , Which has the greatest influence in raising Tides,the Sun or Moon ? Are the Tides at the highest when the Moon is due north, or south ? What is the reason ? 128 Interrogations for Section Tenth. Sec. 10 Do the Tides always answer to the same distance of the Moon from the meridian at the same places ? Does the Moon approach nearer, and recede farther iVom the earth in each of her revolutions'? At what time does she attract the earth most? At what time does she attract it the least ? In what position are the Sun and Moon when the highest Tides are raised 1 What are Spring-Tides ? What are Neap-Tides? In what manner do the attractions of the Sun and Moon act on each other, to produce Spring-Tides ? In what manner to produce Neap-Tides ? Where is the moon when the Tides are equally high in both parts of the lunar day ? What is understood by the lunar day ? What is the length of the lunar day ? At what time are the Tides most unequal ? In which hemisphere are the highest Tides, when the muon has north declination ? Which when in her south declination ? Do the Tides rise very high in open Seas ? Are the Tides ever retarded in their passage ? What retards them ? What are aerial Tides ? How were they discovered, and by whom ? SECTION ELEVENTH. PROBLEMS. PROBLEM I. To convert Time into degrees, minutes, fyc. RULE. As one hour is to 15 degrees, so is the time given to the answer. 1. How many degrees are equal to 8 hours, 20 min- utes, and 30 seconds 1 2d. The Sun passes the meridian of Detroit 1 hour, 19 minutes after 12 o'clock, noon at Boston, how far are those places asunder ? PROBLEM II. To convert degrees, minutes, Sfc. into Time. RULE. As 15 degrees are to an hour, so are the number of degrees given to the time. p 130 . Astronomical Problems. Sec. 11 EXAMPLES. 1. The apparent distance of Venus from the Sun, can never be above 50 degrees, and when at that dis- tance, how long does she rise before the Sun, or set after him ? 2. The greatest elongation of Mercury is said to be 28 degrees, 20 minutes and 19 seconds, how long can he set after the Sun, when an evening star ? PROBLEM III. The diurnal arc of the Sun, or of any planet being given, to find the time of the rising or setting of the Sun. RULE. Bring the diurnal arc into time by Problem 2d. Divide this time by two, and the quotient will be the time at which the Sun sets. Take this time from 12 hours, and the remainder \vill be the time at which the Sun rises. EXAMPLES. 1. Suppose the Sun's diurnal arc be 174 degrees and thirty minutes, at what time does he rise and set. Jlns. 5 hours 49 minutes, the time of the Sun's set- ting, and he rises at 6 hours and 1 1 minutes. 2. The diurnal arc of Venus is found to be 96 degrees and 44 minutes, at what hours does the Sun rise,and when does he set ? 3. The diurnal arc of Mars, is 198 degrees, 14 min- utes and 50 seconds. Sec. 1 1 Astronomical Problems. 131 The diurnal arc of Jupiter, is 201 degrees, 33 minutes and 16 seconds. The diurnal arc of Saturn, is 1 96 degrees, and 1 4 min- utes : and the diurnal arc of Herschel is 213 degrees, 41 minutes, and 58 seconds ; when, according to the above mentioned numbers, does the Sun rise and set? PROBLEM IV. The time which the Sun, or any planet remains above the horizon being given, to find the length of his diurnal, or nocturnal arc. RULE. Divide the given time by two, and the quotient will be the time of the Sun's setting. Take this time from 12 hours, and the remainder will be the time of his ri sing. Multiply the given time by 1 5 degrees, and the product will give the Sun's, or planet's diurnal arc ; this subtracted from 360 degrees, will leave the noc- turnal arc. EXAMPLES. 1. On the fourth of July, the Sun rose at 43 minutes past 5 o'clock ; at what time did he set on that day, and what was the length of his diurnal arc 1 2. September 7th, 1825, the Sun rose at 5 o'clock and 52 minutes, at what time did he set, and what are the dimensions of both arcs ? 132 Jlstr onomical Problems. Sec. 11 PROBLEM V. , To find the time which elapses between two conjunc- tions, or' two oppositions, or between one conjunction, and one opposition of any two planets. RULE. Find the difference between the given daily motions of the two given planets, as given in the following table of the daily motions, then say, as the difference ofttheir daily motions, is to one day, so is 360 degrees, to the difference in the times of the two conjunctions, or op- positions required. But for one conjunction, and lone opposition, or, for a superior and an inferior conjunction ; say as the difference of their daily mo- tions is to one day, so is 180 degrees to the time, which elapses between a conjunction, and an opposition of the two given planets. TABLE. D. Mercury's daily motion is .... 4,0928 degrees. Venus', do. do. ..... 1,6021 The earth's, do* do. . . . . . 0,9856 Mars', do. do 0,5240 Jupiter's, do. do 0,0831 Saturn's, do. do 0,0335 Herschel's, do. do 0,0118 Sec. 11 Astronomical Problems. 133 fvC. EXAMPLES. 1. How many days elapse between a conjunction, and an opposition of Mercury and \enus. Thus Mercury's daily motion, 4,0928 degrees Less the daily motion of Venus, 1,60212,4907 Degrees,then as 2,4907 d: 1 day : : 180d: 72,25 days, the time required. 2. How many days is Venus a morning and an evening star, alternately to the earth ? 3. How many days is Jupiter a morning and evening star, alternately to the earth ? 4. How many days is Mercury east, and how many west of the Sun to us ? PROBLEM VI. The heliocentric longitude of any two planets being given, to find when they will be in heliocentric con- junction. RULE. Subtract the given' longitude of the planet nearest the Sun, from that of the planet farthest from him, if practicable, but if not, add'to the latter 360 degrees, and then subtract, say, as the difference of the daily motions of the given planets is to one day, so is the difference of their longitudes, to the time when the given planets will be in conjunction. 134 Astronomical Problems. Sec. 11 EXAMPLES. 1. At what time were Mars and Venus in conjunction, after the first of January, 1823. Venus' daily motion is 1,60210,5240=1,0781 Mars' longitude for January 1st. 1823, was 311 de- grees and 41 minutes : less by 285 degrees, 16 minutes, the longitude of Venus at the same time=26 degrees and 25 minutes. Then as 1 ,0781 : 1 day : : 26 d. 25 m. =24,5, or January 25th, 1823. TABLE. The Sun's geocentric longitude for January 1st. 1823, was, Degrees. 28029 win. The heliocentric longitude of Mercu- ry, January 1st. 1823, was 27725 That of Venus, was 28516 The Earth's, 10020 That of Mars, 31141 Jupiter's, 64 51 Saturn's, 38 56 Herschel's, 27730 On what day of the year, 1823, was Venus in con- junction with the earth ? 3. When was Jupiter in conjunction with the earth in the year 1824 ? 4. When were Venus and Jupiter in conjunction, in 1825 ? 6. On what day in the year 1832, did Jupiter set ; at the moment the Sun arose ? Sec. 11 Astronomical Problems. 135 PROBLEM VII. When the heliocentric longitude of any planet, for any given day is known, to find it for any required day. RULE. Find the number of days between the given, and re- quired day : then as one day is to the given planet's daily motion, so are the days so found, to the distance which the planet has revolved during that time. Add this distance to the planet's known longitude, and the sum, if less than 360 degrees, will be the longitude for the required day, but if more than 360 degrees, then subtract 360 degrees from it, and the remainder will be the true longitude, & EXAMPLES, 1. On the first of January, 1823, the heliocentric longi- tude of Venus was 285 degrees, 16 min. ; what was it on the 4th of July, in the same year ? Jim. 220 degrees and three minutes. 2. On the first of January, 1823, the earth's longitude was 100 degrees and 20 minutes , what was its longi- tude on the 4th of July, 1825 ? PROBLEM VIII To determine whether Venus or Jupiter will be the morning or evening star on any given day. 136 Astronomical Problems Sec. II RULE. Find the longitude of Venus and the longitude of the earth for the given day. If the difference in longitude, counting from the earth's place eastward, be less than 180 degrees, Venus will be east of the Sun, and conse- quently evening star : but if that difference be greater than 180 degrees, she will be west of the Sun, and therefore morning star. EXAMPLES. On tiie 4th of July, 1823, was Venus a morning or an evening star ? The longitude of Venus on the given day, will be found by Problem 7th, to be 220 degrees and 3 min- utes, and the earth's longitude, for the same day by the same Problem, 281 degrees and 41 minutes ; the dif- ference=61 degrees and 38 minutes; this difference be- ing less than 180 degrees, shows that Venus is east of the Sun, and consequently an evening star. Did Jupiter rise before, or after the Sun, July 4th, 1832 ? How many days in succession, can Venus be a mor- ning, or an evening star ? ' ' / How many- days in succession, can Jupiter be a mor- ning or an evening star ? Sec. 11 Astronomical Problems. 137 PROBLEM IX. To determine the day on which any particular planet shall have a given longitude. RULE. Subtract the longitude of the given planet found in the preceding table,from the given longitude,if practic- able ; but if the longitude of the planet found in the Ta- ble, be greater than the given longitude, increase the latter by 360 degrees, and then subtract; divide the remainder by the planet's daily motion, as recorded in the Table, and the quotient will show the number of days from the first of January, when the planet will have the given longitude. EXAMPLES. 1. On what day of the year 1823, did Venus have 220 degrees of heliocentric longitude ? Answer fourth of July. 2. On what days in the year 1825, did each of the planets enter Virgo ? PROBLEM X. To find whether Venus or Mercury will cross the Sun's disk in any given year. RULE. Find by Problem 9th when Venus will pass her node. Find the earth's heliocentric longitude for that day, and if it equals the longitude of Venus' node, there will be a transit of Venus, and in no other case. The same may be said of the planet Mercury. Q 138 Astronomical Problems. Sec. 1 1 EXAMPLES. Were there a transit of Venus in the year 1824, or not ? The longitude of the ascending node of Venus, is 75 degrees and 8 minutes, which she passed on the 26th of June. The earth's longitude on that day, was 274 degrees and 44 minutes. The longitude of the descen- ding node of Venus, was 258 degrees and 8 minutes, which she passed on the 5th of March. The earth's longitude on that day, was 164 degrees and 55 minutes, consequently there was no transit of Venus in 1824. PEOBLEM XL To find when any two given planets shall have a given heliocentric aspect, taking their longitudes as sta- ted in the Table for 1823. RULE. Add the degrees in the aspect given to the heliocen- tric longitude of either given planet. Find the differ- ence between that sum and the heliocentric longitude of the other given planet : Then say, as jthe difference in the daily motions of the two given planets, is to one day,so is the difference in their longitude found as above to the answer required., EXAMPLES. At what time in the year 1824, did the earth and Ve- nus have a trine aspect ? Sec. 1 1 Astronomical Problems. 139 The longitude of the earth for January 1st. for that 3 ear, was 100 degrees and 6 minutes, to the earth's lon- gitude, add 120 degrees, (the given aspect,) and the sum is 220 degrees and 6 minutes. The longitude of Venus on the first day of January, 1824, was 150 degrees and two minutes ; the difference was 70 degrees and 4 minutes 5 Then 1,6021 degrees ,9S56=,6165 difference of daily motion. Then ,6165 : 1 day : : 70 d. 4 minutes : 113 days, or the 22d. of April. 2. On what day were the earth and Jupiter in conjunc- tion in the year 1826 ? 3. When in 1835, will the earth and Venus be in con- junction 1 NOTE. The preceding PROBLEMS would be correct, if the Planets moved in perfect circular orbits, which however is not the fact, yet they approach so near to circles, that deductions founded upon their figure as circles, are sufficiently accurate for ordinary calculations. SECTION TWELFTH. ON ECLIPSES. IN the Solar System, the Sun is the great fountain of Light, and every planet and satellite is illuminated by him, receive the distribution of his rays, and are ir- radiated by his beams. The rays of light are seen in direct lines, and consequently are frequently intercept- ed by the dark and opaque body of the Moon, passing directly between the earth and the Sun ; and hiding a portion, or the whole of his disk from the view of those parts of the earth where the penumbra, or the shadow of the Moon happens to fall. This is called an ECLIPSE OF THE SUN. It is only at the time of new Moon, that an Eclipse of this kind can possibly take place, and then only when the Sun is within seventeen degrees of either the as- cending or descending nodes ; for if his distance at the time of new Moon be greater than seventeen degrees from either node, no part of the Moon's shadow will touch the earth, and consequently there will be no Eclipse. The orbit in which the Moon really moves, is differ- ent from the ecliptic, one half being elevated five and Sec. 12 On Eclipses. 141 one -third degrees above it, and the other half as much depressed below. The Moon's orbit therefore inter- sects the ecliptic in two points diametrically opposite to each other, and these intersections are called the Moon's nodes. The Moon, therefore can never be in the ecliptic, but when she is in either of her nodes ; which is at least twice in every lunation, or course from change to change, and sometimes thrice. That node from which the Moon begins to ascend northward, or above the ecliptic in northern latitudes is called the ascending node ; and the other the descending node ; because the Moon when she passes by it descends below the ECLIPTIC southward. The ECLIPTIC is the great circle which the earth describes in its annual revolution around the Sun, and is divided into twelve equal parts, of thirty degrees each called signs. Six of these, namely, Aries, Taurus, Gemini, Cancer, Leo and Virgo are north ; and the other six, to wit, Libra, Scorpio, Sagitarius, Capricor- nus, Aquarius and Pisces, south of the equotor.* When the earth comes between the Sun and Moon, the Moon passes through the earth's shadow, and hav- ing no light of her own, she suffers a real Eclipse ; the rays of the Sun being intercepted by the earth. This can only happen at the time of full Moon ; and, when the Sun is within twelve degrees of the Moon's ascen- ding or descending nodes. Should the Sun's distance * The Equator is an imaginary circle passing round the earth from east to west, dividing it into two equal parts, called Hemispheres. 142 On Eclipses. Sec. 12 from the node exceed twelve degrees, the shadow of the earth would no where touch the surface of the Moon, and consequently she could not suffer an Eclipse. When the Sun is Eclipsed to us, the inhabitants of the Moon on the side next the earth, see her shadow like a dark spot travelling over the earth about twice as fast as its equatorial parts move, and the same way. When the earth passes between the Sun and Moon, the Sun appears in every part of the Moon where the earth's shadow falls totally Eclipsed ; and the duration is as long as she remains in the earth's shadow. If the earth and Sun were of equal sizes, the shadow of the earth would be infinitely extended, and wholly of the same breadth, and the planet Mars when in ei- ther of her nodes, and in opposition to the Sun, (al- though forty-two millions of miles from the earth,) would be Eclipsed by the shadow. If the earth were larger than the Sun, her shadow would be sufficient to Eclipse the larger planets, Jupiter and Saturn with all their satellites, when they were opposite to him ; but the shadow of the earth terminates in a point long be- fore it reaches any of the primary planets. It is there- fore evident, that the earth is much less than the Sun, or its shadow could not end in a point at so short a dis- tance. If the Sun and Moon were of equal sizes, she would cast a shadow on the earth's surface of more than two thousand miles in breadth, even if it fell directly against its centre. But the shadow of the Moon is seldom more than one hundred and fifty miles in breadth at . 12 On Eclipses. 143 the earth, unless in total Eclipses of the Sun, her shad- ow strikes on the earth in a very oblique direction. In annular Eclipses, the Moon's shadow terminates in a point at some distance before it reaches the earth ; and consequently the Moon is mnch less than the Sun. If the Moon were actually thrice its present size, it would still in many instances, be totally Eclipsed. A sufficient proof of this, is given by her long continuance in the earth's shadow, during any of her total Eclipses. Therefore the diameter of the earth is more than three times the diameter of the Moon. Though all opaque bodies, on which the Sun shines, have their shadows ; yet such is the magnitude of the Sun, and the distances of the planets, that the prima- ries can never Eclipse each other. A primary can only Eclipse its secondary, or be Eclipsed by it, and never by those except when they are in opposition or conjunction with the Sun, as before stated. The primary planets are very seldom in such positions, but the Sun and Moon are, in every month. If the Moon's orbit were coincident with the plane of the ecliptic, in which the earth wheels its stated courses, the Moon's shadow would fall on the earth at every change, and the Sun be eclipsed to every part of the earth where the penumbra happened to fall. In the same manner, the Moon would have to travel through the middle of the earth's shadow, and be to- tally Eclipsed at every full. The duration of total darkness in every instance, exceeding an hour and a half. %; 144 On Eclipses. Sec. 12 A question like the following naturally arises : Why is it that the Sun is not Eclipsed at every change, if the Moon actually passes between the Sun and the earth. And why is not the Moon Eclipsed at every full, if the earth passes between the Sun and Moon in every month 1 One half of the Moon's orbit, is elevated 5 degrees and twenty minutes above the ecliptic, and the other half is. as much depressed below it ; and, as before has been observed, the Moon's orbit intersects the ecliptic, in two opposite points, called the MOON'S NODES. When these points are in a right line with the cen- tre of the Sun at new or full Moon, the Sun, Moon,and earth are all in a right line ; and if the Moon be then new; her shadow falls upon the earth, but if she be full, the earth's shadow falls upon her. When the Sun and Moon are more than 1 7 deg's. from either of the nodes at the time of conjunction,the moon is generally too high or too low in her orbit to cast any part of her shadow on the surface of the earth. And when the Sun is more than 12 degrees from either of the nodes at the time of full Moon, the Moon is generally either two high or too low to pass through any part of the earth's shadow ; therefore in both these cases there can be no Eclipse. This howevever admits of some variation, for in apogeal Eclipses, the solar limit is only sixteen de- grees and thirty minutes, and in perigeal it is eighteen degrees and twenty minutes. When the full Moon is ee: 12 On Eclipses. 145 in her apogee, * she will be Eclipsed, if she be within 16 degrees and thirty minutes of the node ; and when in her perigee, if within twelve degrees and two minutes. The moon's orbit contains 360 deg's. of which the lim- its of seventeen degrees at a mean rate for Solar Eclip- ses, fy twelve for lunar are only small portions, and the Sun generally passes by the nodes only twice in a yean and consequently impossible that Eclipses should hap- pen in every month. If the line of the nodes, like the axis of the earth, were carried parallel to itself around the Sun, there would be exactly half a year between the conjunctions of the Sun and nodes. But the nodes shift backward, or contrary to the earth's annual mo- tion, nineteen degrees and twenty minutes every year ; and therefore the same node comes round to the Sun nineteen days sooner every year, than in the one pre- ceeding. 173 days, therefore, after the ascending node has passed by the Sun, the descending node also passes by him. In whatever season of the year the luminaries are Eclipsed, in 1 73 days after, we may expect Eclipses about the opposite node. The nodes shift through all the signs and degrees of the ecliptic in 18 years and 225 days, in which time there would always be a regular periodical return of Eclipses, if any number of lunations were completed without a * The fartherest point of each orbit from the earth's centre is called the apogee, and the nearest point is called the perige"e. These points are directly opposite each other, and consequently exactly six signs asunder. R 146 On Eclipses. Sec. 12 fraction. But this never happens, for if both the Sun and Moon should start from a line of conjunction with either of the nodes in any point of the ecliptic, the Sun would perform 18 annual revolutions and 222 degrees of the 19th, and the Moon 230 lunations, and 85 de- grees of another by the time the node came around to the same point of the ecliptic again. The Sun would then be 138 degrees from the node, and the Moon 85 degrees from the Sun. In 223 mean lunations after the Sun, Moon and node, have been in a line of conjunction, they return so nearly to the same state again, that the same node which was in conjunc- tion with the Sun and Moon at the commencement of these lunations, will be within 28 minutes, and 12 sec- onds df a degree of a line of conjunction with the Sun and Moon again, when the last of these lunations is completed. In that time, there will be a regular pe- riod of Eclipses, or rather a periodical return of the same eclipse for many ages. In this period, (which was first discovered by the Chaldeans,) there are 18 Julian years, 1 1 days, 7 hours, 43 minutes, and 21 seconds, when the 29th day of February in leap years, is four times included ; but one day less when included 5 times. Consequently, if to the mean time of any Eclipse, whether of the Sun or Moon, the above named time be added, you will have the mean time of its pe- riodical return. But the falling back of the line of con- junctions, or oppositions of the Sun and Moon, namely, 28 minutes, 12 seconds, with respect to the line of the nodes in every period, will wear it out in process of 12 On Eclipses. 147 time, so that the shadow will not again touch the earth or Moon, during the space of 12,492 years. Those Eclipses of the Sun which happen about the ascending node, and begin to come in at the north pole of the earth, will continue at each periodical return to ad- vance southwardly, until they leave the earth at the south pole; and the contrary with those that happen about the descending node, and come in at the south pole. From the time that an Eclipse of the Sun first touches the earth, until it completes its periodical re- turns, and leaves the same, there will be 77 periods equal to 1388 years. The same Eclipse cannot then again touch this earth, in a less space than 12492 years as above stated. If the motions of the Sun, Moon, and nodes were the same in every part of their orbits, we should need no- thing more than what has been said to find the exact time of all Eclipses ; but as this is not the case, we are under the necessity of forming Tables so constructed, that the mean time can be reduced to the true. By the following example, it will be found, that by the true motions of the Sun, Moon and nodes, the Eclipse cal- culated, leaves the earth five periods sooner than it would have done, by mean equable motions. To ex- emplify this matter more fully, I will take the Eclipse of the Sun, which happened in the year 1764, March 21st. Old Style, (or April 1st. in the new,) according to its mean revolutions, and also according to its true equated time. m 148 On Eclipses. Sec. 12 The shadow, or penumbra of the Moon, fell in open space at each return, without touching the earth ever since the creation, until the year of our Lord, 1 295 ; then on the 1 4th day of June, at 52 minutes, and 59 seconds in the morning, Old Style, the Moon's shadow touched the earth at the north pole. In each suc- ceeding period since that time, the Sun has come 28 minutes and 12 seconds nearer the same node, and the Moon's shadow has gone more southwardly. In the year 1962, on the 18th of July, Old Style, (or 31st. in the new,) at 10 hours. 36 minutes, 21 seconds in the afternoon, the same Eclipse will have returned 38 times. The Sun will then be only 24 minutes and 45 seconds from the ascending node, and the centre of the Moon's shadow will fall a little north of the equator. At the end of the next following period, in the year 1980, July 29th, Old Style, (or August llth in the new,) at 6 hours, 19 minutes and 41 seconds in the morning, the Sun will have receded back three min- utes and twenty-seven seconds from the ascending node ; the Moon will then have a small degree of south latitude, and consequently cast her shadow a lit- tle south of the equator. After this, at every follow- ing period, the Sun will be 28 minutes and 12 seconds further back from the ascending node than at the pre- ceding, and the Moon's shadow will continue at each succeeding period to approach nearer the south pole, until September 13, Old Style, (or October 1st. in the new,) at 1 1 hours, 46 minutes and 22 seconds in the morning, in the year 2665, when the Eclinse will have Sec. 12 On Eclipses. 149 completed its 77th periodical return, and the shadow of the Moon leaves the earth at the south pole to re- turn no more, until the lapse of 12492 years. But on account of the true, (or unequable) motions of the Sun, Moon, fy nodes, the first coming m of this Eclipse at the north pole of the earth, was on the 24th of June, 1313, at 3 hours, 57 minutes, and 3 seconds in the afternoon, and it will finally leave the earth at the south pjde on the 18th day of August, (according to New Style,) in the year 2593, at 10 hours, 25 minutes and 31 seconds afternoon, at the 72d. period. So that the true motions do not only alter the true times from the mean, but they also cut off, five periods from those of the mean returns of this Eclipse. In any year, the number of Eclipses of both lumina- ries cannot be less than two, nor more than seven ; the most usual number is four, and it is very rare to have more than six. The Eclipses of the Sun are more frequent than those of the Moon, because the Sun's ecliptic limits are greater than those of the Moon's. (The proportions being as 17 is to 12,) yet we have more visible Eclipses of the Moon, than of the Sun ; because Eclipses of the Moon are seen from all parts of that hemisphere of the earth which is next her ; and are equally great to each of those parts ; but Eclipses of the Sun are only visible to that small portion of the hemisphere next him, whereon the Moon's shadow happens to fall. The Moon's orbit being elliptical, and the earth in one of its focuses ; she is once at her least distance 150 On Eclipses. Sec. 12 from the earth, and once at her greatest in every lu- nation or revolution around the earth. When the Moon changes at her least distance from the earth, and so near the node that her dark shadow falls on the earth ; she appears sufficiently large to cover the whole disk of the Sun from that part on which her shadow falls, and the Sun appears totally eclipsed for the space of four minutes. But when she changes at her greatest distance from the earth, and so near the node that her dark shadow is directed towards the earth, her diameter subtends a less angle than the Sun's, and therefore cannot hide the whole disk from any part of the earth, nor does her shadow reach it at that time ; and to the place over which the point of her shadow hangs, the Eclipse is annular, and the edge of the Sun appears like a lumin- ous ring around the whole body of the Moon. When the change happens within 17 degrees of the node, and the Moon at her mean distance from the earth, the point of her shadow just touches the earth, and the Sun is totally eclipsed to that small spot on which the Moon's shadow falls ; but the duration of total dark- ness is not of a moment's continuance. The Moon's apparent diameter when largest, exceeds the Sun's when least, according to the calculations of modern Astronomers, two minutes and five seconds, the dura- tion of total darkness, therefore may at such time con- tinue four minutes and six seconds ; casting a shadow on the earth's surface of 180 miles broad. When the Moon changes exactly in the node, the penumbra is See. 12 On Eclipses. 151 circular on the earth at the middle of the general Eclipse, because at that time it falls perpendicularly on the earth's surface ; but in every other moment, it falls obliquely, and therefore will be elliptical, and the more so, as the time is longer after the middle of the gen- eral Eclipse ; and then much greater portions of the earth are involved in the penumbra. When the penumbra first touches the earth the gen- eral Eclipse begins, and it ends when it leaves the earth : from the beginning to the end, the sun appears. Eclipsed in some part of the earth or other., When the penumbra touches any place, the Eclipse begins at that place, and ends, when the penumbra leaves it. When the moon changes exactly in the node the pe- numbra goes over the centre of the earth as seen from the moon, and consequently by describing the longest line possible on the earth continues the longest upon it ; namely at a mean rate five hours and fifty minutes : more, if the moon be at her greatest distance from the earth, because she then moves slowest, and less if she be at her nearest approach, because of her ac- celerated motion. The moon changes at all hours, and as often in one node as in the other, and at all distances from them both, at different times as it happens ; the variety of phases of Eclipses are therefore almost innumerable, even at the same places, considering also how various- ly the same places are situated on the enlightened disk of the earth with respect to the motion of the penum- bra, at the different hours when Eclipses happen. 152 On Eclipses. Sec. 12 When the Moon changes 17 degrees short of her descending node, the penumbra just touches the north- ern part of the earth's disk near the north pole, and as seen from that place, the Moon appears to touch the Sun, but hides no part of him from sight. Had the change been as far short of the ascending node, the penumbra would have touched the southern part of the disk near the south pole. When the Moon changes 12 degrees short of the descending node, more than a third part of the penumbra, falls on the northern parts of the earth at the middle of the general Eclipse. Had she changed as far past the same node, as much of the other side of the penumbra would have fallen on the southern parts of the earth, all the rest in open space. When the Moon changes 6 degrees from the node, almost the whole penumbra falls on the earth at the the middle of the general Eclipse. The further the Moon changes from either node within 17 degrees of it, the shorter is the penumbra's continuance on the earth ; because it goes over a less portion of the disk. The nearer the penumbra's cen- tre is to the equator at the middle of the general Eclipse, the longer is its duration at places where it is central ; because the nearer that any place is to the equator, the greater is the circle it describes by the earth's motion on its axis, and the place moving quick keeps longer in the penumbra, whose motion is the same way with that of the place, though faster as has been mentioned. That Eclipses of the Moon can never happen only at the time of full, and the reason why she Sec. 12 On Eclipses. 153 is not eclipsed at every full, has already been men- tioned. The Moon when totally eclipsed, (though a dark opaque body, and shines only by reflection,) is not in- visible, if she be above the horizon, and the sky clear ; but generally appears of a dusky color which some have thought to be her native light. But the true cause of her being visible, is the scattered beams of the Sun, bent into the earth's shadow by going through the at- mosphere, which being more dense near the earth, than at considerable heights above it, refracts, or bends the rays of the Sun more inward the nearer they are pass- ing by the earth's surface, than those rays which go through higher parts of the atmosphere where it is less dense; according to its height, until it be so thin, or rare as to lose its refractive power. When the MocJn goes through the centre of the earth's shadow, she is directly opposite to the Sun, yet the Moon has been often seen totally eclipsed in the hori- zon, when the Sun was also visible in the opposite part of it ; for the horizontal refraction being almost 34 minutes of a degree, and the diameter of the Sun and Moon being each at a mean state but 32 minutes, the refraction causes both luminaries to appear above the horizon, when they are actually below it. When the Moon is full at 12 degrees from either node, she just touches the earth's shadow, but does not enter into it. When she is full at 6 degrees from either node, she is totally, but not centrally immersed in the earth's shad- ow, she takes the longest line possible, which is the 154 On Edipses. Sec. 12 diameter through it, and such an Eclipse, (being both total, and central,) is of the longest duration, namely, three hours, 57 minutes and 6 seconds from the begin- ning to the end, if the Moon be at her greatest distance from the earth ; and 3 hours, 37 minutes and 26 sec- onds, if she be at her least distance. The reason of this difference is, that when the Moon is farthest from the earth, her motions are retarded, but when nearest to the earth, her motions are accelerated. Sec. 12 Interrogations for Section Tkctlftk. 165 . X Interrogotions for Section Twelfth. Arc the rays of light proceeding from the Sun, fre- quently intercepted ? By what are they intercepted ? What is understood by the penumbra ? What is an Eclipse of the Sun ? At what stage of the Moon does an Eclipse of the Sun happen 1 How near to either of the nodes must the Sun be to suffer an Eclipse 1 Does the Moon's orbit differ from the ecliptic ? What is the ecliptic 7 What are the Moon's nodes 1 Why cannot the Sun be eclipsed unless he be within ; 17 degrees of the node ? How often is the Moon in the ecliptic ? Which is called the ascending node ? Which is called the descending node 1 What is an Eclipse of the Moon ? At what stage of the Moon does this happen ? How near must the Sun be to either of the node.?, so that the Moon can suffer an Eclipse ? What causes it ? 156 Interrogations for Section Twelfth. Sec. 12 Should the same distance from either node at the time of full Moon exceed twelve degrees, could the shadow of the earth touch the surface of the Moon ? As the Moon passes between the Sun and the earth at every new Moon, why is not the Sun eclipsed at eve- ry new Moon ? Why is not the Moon eclipsed at every full ? What is the farthest point of each orbit from the earth's centre called 1 What the nearest point ? How many times a year does the Sun generally pass by the nodes ? In what time do the nodes pass through all the signs of the ecliptic ? How many lunations after the Sun, Moon and nodes have been in conjunction, before they return nearly to the same state again 1 What is a periodical return of an Eclipse ? Are the motions of the Sun, Moon and nodes the same in every part of their orbits ? How can the mean time of these conjunctions be re- duced to the true ? How many are the greatest number of Eclipses that can possibly happen in one year ? How many the least ? What the most usual number ? Sec. 12 Interrogations for Section Twelfth. 157 Which is the most frequent, those of the Sun or Moon ? What is the reason 1 Are there more visible Eclipses of the Moon than of the Sun ? What is the reason 1 What is a total Eclipse of the Sun ? How long can the Moon hide the whole face of the Sun from our view ? In what part of her orbit must the Moon be to cause a total Eclipse ? What is an annular Eclipse ? How many miles in diameter would the shadow of the Moon be on the earth, in an Eclipse when total darkness continues four minutes ? When the Moon changes exactly in the node, what is the form of the shadow, and where does it strike the earth ? When does an Eclipse begin ? When does it end 1 When does it begin and end at any particular place 1 When the Moon changes 17 degrees short of her de- scending node, where will her shadow touch the earth ? If as far short of her ascending node, where on the earth will her shadow fall ? Why in total Eclipses of the Moon is she not invisi- ble, if she be a dark opaque body ? 158 Interrogations for Section Twelfth. Sec. 12 Is it possible for the Moon to be visibly eclipsed while the Sun is in sight? When the Moon is full, within six degrees of cither node, will she be totally eclipsed ? When she passes by the node in the earth's shadow, how much of the Moon will be eclipsed 1 What is the longest time that the Moon can suffer an Eclipse ? What the shortest if she be at her least distance 1 Why is this difference 1 What is the time of the longest duration of an Eclipse of the Sun ? What the shortest, if the Eclipse be central ? SECTION THIRTEENTH. SHOWING THE PRINCIPLES ON WHICH THE FOLLOWING ASTRONOMICAL TABLES ARE CONSTRUCTED, AND THE METHOD OF CALCULATING THE TIMES OF NEW & FULL MOONS & ECLIPSES BY THEM. THE nearer that any object is to the eye of an obser- ver, the greater is the angle under which it appears. The farther from the eye, the less it appears. The diameters of the Sun and Moon subtend differ- ent angles at different times. And at equal intervals of time, these angles are once at the greatest, and once at the least, in somewhat more than a complete revolution of the luminary through the ecliptic from any given fixed star, to the same star again. This proves that the Sun and Moon are constantly changing their distances from the earth and that they are once at their greatest distance, and once at their least, in a little more than a complete revolution. The gradual differences of these angles are not what they would be, if the luminaries moved in circular orbits, the earth beiug supposed to be placed at some distance from the centre. But they agree perfectly with elliptical orbits, suppo- sing the lunar focus of each orbit to be at the centre of the earth. 160 On the Construction of the follmving Tables. Sec. 13 The farthest point of each orbit from the earth's cen- tre, is called the apogee ; & the nearest point the perigee. These points are directly opposite each other. Astronomers divide each orbit into 12 equal parts, called signs; and each sign into 30 equal parts called degrees , each degree into sixty equal parts, called min- utes,and each minute into 60 equal parts, called seconds. The distance, therefore, of the Sun or Moon from any point of its orbit, is reckoned in Signs, Degrees, Minutes and Seconds. The distance here meant, is that through which the luminary has moved from any given point, (not the space it falls short thereof,) in coming round again,be it ever so little. The distance of the Sun or Moon from its apogee at any given time, is called its mean anomaly, so that in the apogee, the anomaly is nothing, in the perigee, it is six signs. The motions of the Sun and Moon are observed to be continually accelerated from the apogee to the perigee; and as gradually retarded from the perigee to the apo- gee , being slowest of all when the mean anomaly is nothing, and swiftest when it is six signs. When the luminary is in its apogee or perigee, its place is the same as it would be if its motions were equable in all parts of its orbit. The supposed equable motions are called mean, the unequable are justly called the true. The mean place of the Sun or Moon is always for- warder than the true, whilst the luminary is moving from its apogee to its perigee; and the true place is always Sec. 13 On the Construction of the follotmng Tables. 161 forwarder than the mean, whilst the luminary is mo- ving from its perigee to its apogee. In the former case, the anomaly is always less than six signs, in the latter more. It has been discovered by a long series of observa- tions, that the Sun goes through the ecliptic, from the vernal equinox to the same again, in 365 days, 5 hours, 43m, and 54s. And from the first star of Ariep, to the same star again, in 365 days, 6 hours, 9 minutes, and 24 seconds. And from his apogee to the same again in 365 days, 6 hours, and 14 minutes. The first of these, is called the Solar year ; the second the syde- real, and the third the anamolistic year. The solar year is 20 minutes and 29 seconds shorter than the sydereal; and the sydereal year is 4 minutes and 36 seconds shorter than the anamolistic. Hence it ap- pears, that the equinoxial point, or intersection of the ecliptic and equator at the beginning of Aries, goes backward, with respect to the fixed stars, and that the Sun's apogee goes forward. The yearly motion of the earth's or Sun's apogee, is found to be one minute and six seconds, which being subtracted from the Sun's yearly motion, in longitude, the remainder is the Sun's mean anomaly. It is also observed, that the Moon goes through her orbit from any given fixed star to the same again, in "21 days, 7 hours, 43 minutes, and 4 seconds, at a mean rate ; frcm her apcgee to her apogee again in 27 days, 13 hturs, 18 miiuites, ar.d <3 seccrds: si d fnni tie Suo to the Sun again in 9 days, 12 hours, 44 uai;utes. 162 On the Constniction of the, following Tables. Sec. 13 and 3 and ^ seconds. This confirms the idea that the Moon's apogee moves forward in the ecliptic, and that at a much greater rate than the Sun's apogee ; since the Moon is 5 hours, 55 minutes, and 39 seconds long- er in revolving from her apogee to her apogee again, than from any star to the same again. The Moon's orbit crosses the ecliptic in two oppo- site points, which are called her nodes, and it is obser- ved that she revolves sooner from any node to the same node again, than from any star to the same star again, by 2 hours, 38 minutes and 27 seconds ; which shows that her nodes move backward, or contrary to the or- der of signs in the ecliptic. To find the Moon's mean motion in a. common year of 3 Go days, the proportion is D H M s As the Moon's period, 27 7 43 5 Is to her whole orbit, or 360 degrees, So is a common year of 365 days, To 13 revolutions and 4s. 9d. 23 minutes, 5 seconds. The thirteen revolutions are rejected, and the remainder is taken for the Moon's motion in 365 days. To calculate the Moon's mean anomaly : The Moon's apogee moves once round her whole orbit in 8 years, 309 days, 8 hours, and 20 minutes, or, (adding two days for leap years,) in 3231 days, eight hours and 20 minutes. Then, Sec. 13 On the Construction of the following Tables. 163 As 323 Id. 8h. 20 Is to the whole circle, or 360 degrees, So is a common year of 365 days, To the motion of the Moon's apogee in one year= 40 degrees, 39 minutes, and 50 seconds. From the Moon's mean motion in longitude, during one year, s D M s 4 9 23 5 Subtract the motion of her apogee, 1 10 39 50 for the same time, and there remains, 2-28 43-15 the Moon's mean anomaly in one year. To find the mean motion of the Moon's node : The Moon's node moves backward round her whole orbit in 18 years, 224 days, 5 hours, therefore for its motion in 365 days, As 18 years, 224 days, 5 hours Is to the whole circle or 360 degrees, So is the year of 365 days To the motion of the Moon's node in 365 days=19 degrees, 19 minutes and 43 seconds. To find the mean motion of the Moon from the Sun. The Moon's mean motion in a common year of 365 days, is 4 signs, 9 degrees, 23 minutes and 5 seconds over and above 13 revolutions, and the Sun's apparent mean motion in the same time is 1 1 signs, 29 degrees, 45 minutes and 40 seconds. Then from the Moon's mean motion for one year, subtract the mean motion of the Sun for the same time, and the remainder will be the mean motion of the Moon from the Sun in one . year=4 signs, 9 degrees, 37 minutes and 25 seconds. 164 On the Construction of the following Tables. Sec. 13 The time, in which the Moon revolves from the Sun to the Sun again, (or from change to change,) is called a lunation, which would always consist of 29 days, 12 hours, 44 minutes, 3 seconds, 2 thirds and 53 fourths, if the motions of the Sun and Moon were always equa- hle. Hence 12 mean lunations contain 354 days, 8 hours,4S minutes,36 seconds, 35 thirds, and 40 fourths ; which is 10 days, 21 hours, 11 minutes, 23 seconds, 24 thirds, and 20 fourths less than the length ofaccmmon Julian year, consisting of 365 days and 6 hours ; and 13 mean lunations contains 383 days, 21 hours, 32 minutes, 39 seconds, 38 thirds, and 38 fourths, which exceeds the length of a common Julian year by 18 days, 15 hours, 32 minutes, 39 seconds, 38 thirds, and 38 fourths. The mean time of new Moon being found for any given year and month, as, suppose for March, 1 700, Old Style ; if this new Moon happens later than the llth of March, then 12 mean lunations added to the time of this mean new Moon, will give the time of the mean new Moon in March, 1701, after having thrown off 365 days. But, when the mean new 7 Moon hap- pens before the llth of March, we must add 13 mean lunations, to have the mean time of mean new Mcon .in March, following, always taking care to ?ubiracto65 days in common years, and in leap years, 366, from the sum of this addition. Thus in the year 1700, Old Style, the time of mean new Moon in March, was the 8th day, at 16 hours, 1 1 minutes, and 25 seconds past four, in the morning; of Sic. 13 O.i thz Construction of the following Tables. 105 the 9th (by, (according to common reckoning.) To this we must adJ 13 mean lunations, from which sub- tract S 65 days, because the year 1701 is a common year, and there will remain 27 days, 13 hours, 44 min- utes, 4 seconds, 38 thirds and 33 fourths, for the time of mean new Moon in March, in the year 1701. By carrying on this addition and subtraction, until the year 1703, we iind the time of new Moon in March that year, to be on tti3 6th day, at 7 hours, 21 minutes, 17 seconds, 49 thirds, and 46 fourths, past noon ; to which add 13 mean lunaiions, and subtract 66 days, (the year 1704 being leap year,) and there will re- main 24 days, 4 hours, 53 minutes, 57 seconds, 28 thirds and 20 fourths, for the time of mean new Moon in March, 1704. In this manner, was the first of the following Tables constructed to seconds, thirds, and fourths, and then written to the nearest seconds. The reason why we chose to begin the year with March, was to avoid the inconvenience of adding a day to the tabular time in leap years, after the month of Feb'y. or subtracting a day therefrom, in January, or February in those years; to which all tables of this kin 1 are sul j-'ct, (which begin the year with Jan- uary,) in calculating th? times of new or full Moon. Tho mean an;mu!ie,-< of the Sun and Mpon, and the Sun's mean motion from the ascending node of the Moon's orbit, are set down in Table 3d. from one to 13 lunations. The-e numbers for 13 lunations being added to the radical anomalies of the Sun and Moon ; and to the 166 On the Construction of the following Tables. Sec. 13 Sun's mean distance from the ascending node, at the time of mean new Moon in March, 1700, (Table first,) will give their mean anomalies, and the Sun's mean distance from the node, at the time of mean new Moon in March, 1701 ; and twelve mean lunations more with their mean anomalies, *c. added, will give them for the time of mean new Moon, in March, 1 702. And thus proceed to continue the Table as far as you please, always throwing off 12 signs, when their sum exceeds that number, and setting down the re- mainder as the proper quantity. If the numbers belonging to 1700, (in Table first,) be subtracted from those of 1800, we shall have their whole differences in 100 complete Julian years ; which accordingly we find to be 4 days, 8 hours, 10 minutes, 52 seconds, 15 thirds, and 40 fourths, with respect to the time of new Moon. These being added together 60 times, (taking care to throw off a ^0hole lunation, when the days exceed twenty -nine and a half,) make up 60 centuries, or 6,000 years, as in Table 6th,which was was carried on to seconds, thirds and fourths, and then written to the nearest seconds. In the same man- ner were the respective anomalies, and the Sun's dis- tance from the node found for these centurial years, and then (for want of room,) written to the nearest minutes, which is sufficiently exact for whole cen- turies. By means of these two Tables, we may readily find the time of any new Moon in Marchy together with the anomalies of the Sun and Moon, and the Sun's mean Sec. 13 On the Construction of the following Tables. 167 distance from the node at these times, within the limits of 6,000 years, either before or after the 18th century ; and the mean time of any new, or full Moon, in any given month after March, by means of the third and fourth Tables, within the same limits, as will be shown in the precepts for calculation. This Table is calculated in conformity to the Old Style, for the purpose of calculating Eclipses, which have made their appearances in former ages, and likewise for those, which will take place after the year 1900, which however, are easily made to conform to the New Style. It would be a very easy matter to calculate the time of new or full Moon, if the Sun and Moon moved equa- bly in all parts of their orbits. But, we have already shown that their places are never the same, as they would be by equable motions, except when they are in apogee, or perigee, which is when their mean anoma- lies are either nothing or 6 signs. And that their mean places are always forwarder than their true, whilst the anomaly is less than 6 signs ; and their true places more forward than their mean, when the anom- aly is more. Hence it is evident, that whilst the Sun's anomaly is less than 9 signs, the moon will overtake him,or be op- posite to him sooner, than she would,if his motion were equable ; and later whilst his anomaly is more than 6 signs. The greatest difference that can possibly hap- pen between the mean, and true time of new, or full Moon, on account of the Sun's motion, is three hours, 163 On the Construction of the following Tables. See. 13 43 minutes and 28 seconds ; and that is when the Sun's anomaly is three signs one degree, or eight signs and 23 degrees; sooner in the first ease, and later in the last. la all other signs and degrees of aaoimly, the difference is gradually less, and vanishes when the anomoly is nothing, or six signs. The Sun is in his apogee on the 30th of Jane, and in perigee on the 30th of December, in the present age. He is therefore, nearer the earth in our Winter than in Summer. The proportional difference of distance deduced frc m the Sun's apparent diameter, at these times, is as 983 to 1017. The Moon's orbit is dilated in Winter, and ccntrac- terl in Summer, therefore the lunations are longer in Winter than in Summer. The greatest difference is found to be 22 minutes and 29 seconds: the lunations are gradually increasing in length, whilst the Sun is moving from his apogee to his perigee, anJ decreasing in length while he is moving from his perigee to his apogee. On this account, the Moon will be later ev- ery time in coming to her conjunction with the Sun,or being in opposition to him, from December until June; and sooner from June until December, than if her or- bit ha 1 continued of the same size dur.'n ; all the year. These differences depend wholly on the Sun's anom- aly, they are therefore put together into one Table, & called the annual, or first equation of the mean to the true syzygy.* [See Table Seventh.] Tiiis equational * The word syzygy signifies both the conjunction and opposition of the Sua and Moon. Sfec. 13 On the Constntctionofthtfottotoing Tables. 169 difference is to b^ subtracted from the time of the mean syzygy, when the Sun's anomaly is less than six signs, and added when it is more. At the greatest, it is 4 hours, 10 minutes, and 57 seconds; viz: 3 hours, 48 minutes and 28 seconds, on the Sun's unequal mo- tion ; and 22 minutes and 29 seconds on the account of the dilation of theJMoon's orbit This compound equation would be sufficient, for re- ducing the mean time of new, or full Moon to the true, if the Moon's orbit were of a circular form, and her motion quite equable in it. But the Moon's orbit is more elliptical than the Sun's and her motion in it so much the more unequal. The difference is so great, that she is sometimes in conjunction with the Sun, or in opposition to him sooner, by 9 hours, 47 minutes and 54 seconds, than she would be, if her motion w r ere equable ; and at other times as much later. The for- mer happens when her mean anomaly is 9 signs, and 4 degrees ; and the latter, when it is 2 signs and 26 de- grees. [See Table 9th.] At different distances of the Sun from the Moon's apogee, the figure of the Moon's orbit becomes different. It is longest, or most eccen- tric, when the Sun is in the sign and degree, either with the Moon's apogee, or perigee. Shortest, or least ec- centric, when the Sun's distance from the Moon's apo- gee is either three signs, or nine signs; and at a mean state when the distance is either one sign and fifteen degrees; four signs and fifteen degrees; seven signs and fifteen degrees ; or ten signs and fifteen degrees. When the Moon's orbit is at its greatest eccentricity, 170 On the Construction of the f allotting Tables. Sec. 13 Ii2r apogeal distance from the earth's^ centre, is to her perigeal distance therefrom, as 1067 is to 933 : when least eccentric, as 1043 is to 957 ; and when at the mean state as 1055 is to 945. But the Sun's distance from the Moon's apogee is equal to the quantity of the -Moon's mean. anomaly, at the time of new Moon ; and by. the addition of 6 signs, it becomes equal in quantity to the Moon's mean anom- aly, at the time of full Moon. A Table therefore will be constructed to answer all the various inequalities, depending on the different eccentricities of the Moon's orbit in the syzygies, and called the second equation of the mean, to the true syzygy. [See Table Ninth.] The Moon's anomaly when equated by Table Eighth, becomes the proper argument for taking out the sec- ond equation of time, which must be added to the for- mer equated time, when the Moon's anomaly is less than six signs, and subtracted when the anomaly is more. There are several other inequalities in the Moon's motion, which sometimes bring on the true syzygy a little sooner, and at other times keep it back a little la- ter, than it vvo^ld otherwise be ; but they are so small that they may be all omitted except two; the former, of which [see Table 10th.] depends on the difference be- tween the anomalies of the Sun and Moon in the syzygies ; and the latter [see Table 1 1th.] depends on the Sun's distance from the Moon's nodes at these times. The greatest difference arising from the for- mer is four minutes and 58 seconds ; and from the lat- Sec. 13 On the Construction of the following Tables. 171 ter one minute and 34 seconds. Having described the phenomena arising from the inequalities of the solar and lunar motions, we shall now explain the reasons of these inequalities. In all calculations and observations relating to the Sun and Moon, we have considered the Sun as a mo- ving body, and the earth as being at rest ; since all the appearances are* the same, whether it be the Sun, or earth that moves. But the truth is that the Sua is at , rest, and the earth actually moves around him, once in every year, in the plane of the ecliptic. Therefore, whatever sign and degree of the ecliptic the earth is in at any given time, the Sun will then appear to be in the opposite sign and degree. The nearer any body is to the Sun, the more it is attracted by him, and this attraction increases, as the square of their distances diminishes, and vice versa. The earth's annual orbit is elliptical, and the Sun is placed in one of its foci. The remotest point of the earth's orbit is called the earth's aphelion, and thte nearest point of the earth's orbit to the Sun, is called the earth's perihelion. When the earth is in its aphe- lion, the Sun appears to be in its apogee ; and when the earth is in its perihelion, the Sun appears to be in its perigee. As the earth moves from its aphelion to its perihe- lion, it is constantly more and more attracted by the Sun ; and this attraction by conspiring in some degree with the motion of the earth, must necessarily acceler- ate its motion. 172 On the Construction of the following Tables. Sec. 13 But, as the earth moves from its perihelion to its ap- helion, it is continually less and less attracted by the Sun ; and as their attraction acts then just as much against the earth's motion, as it has acted for it in the other half of the orbit; it retards iLe n.otion in the like degree. The faster the earth moves, the faster will the Sun appear to move ; the slower the^arth moves, the slower is the Sun's apparent motion. The Moon's orbit is also elliptical, and the earth keeps constantly in one of its focuses. The earth's at- traction has the same kind of influence on the Moon's motion, that the Sun's attraction has on the motion of the earth. Therefore, the Moon's motion must be continually accelerated, whilst she is passing from her apogee to, her perigee ; and as gradually retarded in moving from her perigee to her apogee. At the time of new Moon, she is nearer to the Sun than the earth is at that time, by the whole semi diameter of the Moon's orbit ; which, at a mean state,-is 240,000 miles; and at the full she is as many miles farther from the Sun, than the earth then is. Consequently, the Sun attaacts the Moon more than it attracts the earth, in the former case, and less in the latter. The difference is greatest, when the earth is nearest the Sun ; and least when it is farthest from him. The obvious result of this is, that, as the earth is nearest to the Sun in Winter, and farthest from him in Summer ; the Moon's orbit must be dilated in Winter, and contracted in Summer, The^e are the principal causes of the differ- Sec. 13 On the Construction of iht following Tables. 173 ence of time, that generally happens between the mean and true times of conjunction or opposition, of the Sun and Moon. The other two differences, which depend on the difference between the anomalies of the Sun and Moon; and upon the Sun's distance from the lunar nodes in the syzygies, are occasioned by the different degrees of attraction of the Sun anil earth upon the Moon, at greater or less distances, according to their respect- ive anomalies, and to the position of the Moon's nodes, with respect to the same. If it should ever happen, that the anomalies of both the feun and Moon, were either nothing, or six signs at the mean time of new or full Moon ; and the bun should then be in conjunction with either of the Moon's nodes, all the above mentioned equations would then vanish ; and the mean, and true time of the syzygy, would coincide ; but if ever this circumstance did hap- pen, we cannot expect the like again in many ages af- terwards. Every 49th lunation returns very nearly to the same tima of the day as before ; for 49 mean luna- tions, wants only one minute, 30 seconds, 34 thirds of being equal to 1477 days. In 2,953,059,085,108 days, there are 100,000,000,000 lunations, exactly, and this is the smallest number of natural days, in which any exact number of mean lunations are completed. The following Tables are calculated for the meri- dian of WASHINGTON, excepting Table first, which is calculated for the meridian of LONDON, but they equal- ly serve for any other place by adding 4 minutes 174 On ths Construction of the following Tables. See. 13 to the tabular time, for every degree that the given place is eastward from WASHINGTON ; or subtracting 4 minutes for every degree that the given place is west ward from WASHINGTON. These Tables also begin the day at noon, and reckon forward to the noon following, for one day. Thus, March 31st. at 22 hours, 30 minutes, and 25 seconds of tabular time, (in common reckoning,) will be April 1st. at 30 minutes, 25 seconds after ten o'clock in the morning. Interrogations for Section Thirteenth. . . Does an object appear at a less angle when far off, than when near 1 Do the Sun and Moon subtend different angles at different times 1 Are the angles subtended by the Sun and Moon once at the greatest, and once at the least in one revolution ? Are these gradual differences the same as they would be, if those luminaries moved in circular orbits ? Do they agree perfectly with elliptical orbits ? Where must the lower focus of each orbit be placed to have them agree ? What is meant by the term apogee 1 What by perigee ? Sec. 13 Interrogations for Section- Thirteenth. 175 Into how many parts do Astronomers divide each orbit ? What is meant by the distance of the Sun or Moon from any point of its orbit ? What is the distance at any given point, of the Sun or Moon from its apogee called ? What is the anomaly of the Sun or Moon when in apogee 1 What in perigee ? In what part of their orbits, are the Sun and Moon continually accelerated ? In what part retarded ? 1 % What are the mean motions of the Sun and Moon called 1 What are the unequable called ? In what parts of their orbits are the mean motions forward of the true 1 In what part are the true forward of the mean ? How many signs is the anomaly in the former case? How many in the latter ? Does the Moon's apogee move forward in the ecliptic 1 Does it move faster or slower than the Sun's? Does the Moon revolve sooner from any node to the same again, than from any fixed star to the same again? If so, what is the difference ? What is meant by a lunation ? Why Do Astronomers begin the year with March ? What does Table third contain ? "what Table first ? Why was Table first calculated for Old Style ? 176 Interrogations for Section Thirteenth. Sec. 13 What is the greatest difference between the mean or true time of new or full Moon, on account of the Sun's motion ? Are the lunations longer in winter than in summer? What reason can you advance 1 What the greatest difference ? On what does these differences depend '? What are they called ? Why is this equation not sufficient to reduce the mean time to the true ? Is the Moon's orbit more elliptical than the Sun's ? What is meant by the word syzygy 1 Is the Moon sometimes sooner or later in counjunc- tion or opposition with the Sun, than she would be if her motions were equable in evefy part of her orbit ? If so, what is the greatest difference? On what account does the Mootf's orbit become different ? When is it the most eccentric ? When the least ? What is equal to the Sun's distance from the Moon's apogee ? On what does the first of these differences depend ? On what the second ? V, hat is the remotest point of the earth's orbit called ? What is the nearest point to the Sun called? Has the attraction of the earth any influence on the motion of the iVioon 1 In what case is the motion continually accelerated ? In what case retarded ? Why is the Moon's orbit dilated in winter ? Why contracted in summer ? For what place are the following Tables calculated ? By what means do they serve for any other place ? At what time do the Tables commence the day ? SECTION FOURTEENTH. Precepts Relative to the following Tables. To calculate the true time of new or full Moon, and Eclip- ses of the Sun or *\foon, by the following Tables. IF the required new or full Moon be between the years 1800 and 1900, takeout the mean time of new Moon in March, for the proposed year, from Table 16th together with the anomalies of the Sun and Moon, and the Sun's mean distance from the Moon's ascending noda. But if the time of full Moon be required in March, add the half lunation at the bottom of the page, from Table 3, with its anomalies,&c. to the former num- bers, if the new Moon falls before the 15th of March ; but if after the 15th of March, subtract the half lunation before mentioned, with the anomalies, &c. and write down the respective remainders. In these additions and subtractions, observe that 60 seconds make a minute, 60 minutes make a degree, 30 degrees make a sign, and 12 signs a circle. When the number of signs exceed 12 in addition, re- ject 12, and set down the remainder. IF 178 Precepts relating to Ike following Tables. Sec. 14 \Vh: n the number of signs to be subtracted is greater thin the number you subtract from, add 12 signs to the minued, yon will then have a renninder to set down. When the required new, or fail Moon is in any month after Mai eh, write out as mat y lunations, with their anomalies, ami the S MI'S rt of any in the column, under the given month ; add from Table 3d, one luna- tion, with its anomalies, &.c. to tlie aforesaid sum, and you will then hav,; a new s:un of days, wherewith to enter Ttible fourth, under the given month, where you are sure to find it the s? c >r.d time, if the first fails. With the signs and degrees of the Sun's anomaly, enter Table 7th, and therewith take out the annual, or first equation, for rc:h:ci:)g the m an to tie truj s\zyg> ; taking care t > make propoit ;>: s in il.e T blc for the odd minutes of anomaly, as the Table gives the equation only for whole degrees. Sec. 14 Precepts relating to the following Tables. 179 Observe in this arid every other case of finding equa- tions, that if the signs be \\\ the head of the Table, their degrees are at the L j ft hand, and arc reckoned down- wards. But if the signs be at the foot of the Table, their degrees arc at the right hand, and are counted up- wards; the equation being in the body of the Table, under or over the signs, in a collateral line with the de- grees. The terms add, or subtract, at the head or the foot of the Tables, where the signs are found, show whether the equation is to be added to the mean timo of new or full Moon, or subtracted from it. In Table 7th, the equation is to be subtracted, if the signs of the Sun's anomaly be found at the head of the Table; but ir is to be added, if the signs be at the foot. YViththo signs and degrees of the Sun's* anomaly, at the m:'un time of m-\v or full Moon, enter Table 8th, and take out the equation of the Moon's mean anoma- ly, subtract this equation from her mean anomaly, if the signs of the Sun's anomaly be at the head of the Table; but add it, if they be at the foot, the result will be the M -on's equated anomaly. With the signs and degrees of the Moon's equated anomaly, enter Table 9th, and take out the second equation, for reducing the mean to the trih time of new M.)on, adding this equation, if ihe signs of the Moon's equaled anomaly be at the head ol the Table , but sub- tracting it, if they be at the foot, and the result will be the mean time of the new or full M HUS,. twice equated. Subtract the Moon/s equated anomaly from the Sun's mean anomaly, and with the remainder, ia signs and 180 Precepts relating to the folloiving Tables. Sec. 14 degrees; enter Table tenth, and take out the third equa- tion, applying it to the former equated tim^, as the titles add, or subtract direct, and the result will be the mean time of new, or full Moon thrice equated. With the Sun's mean distance from the ascending node*, enter Table llth, and take out the equation answering to that argument ; adding it to, or subtracting it from, the thrice equated time, as the titles direct , to which apply the equation of natural days, from Table 17th, subtract- ing it, if the clock be faster than the Sun, and adding it, if the Sun be faster than the clock, the result will be the true time of new or full Moon, and consequently of an Eclipse; agreeing with solar time. The method of calculating an Eclipse, for an}' given year, will be shown further on, and a few examples com- pared with the precepts, will render the whole work plain, and easily understood. The Tables begin the day at noon, and reckon for- ward to the noon following. They are also calculated for the latitude and longitude of WASHINGTON, except- ing Table first, but serve for any place on the surface of the Globe, by subtracting four minutes for ever} 7 de- gree that the place lies west of WASHINGTON, from the true solar time of conjunction or opposition, &, adding four minutes to the true solar time for every degree that the place lies eastward of WASHINGTON, if Table 16th be used, and the same from LONDON, if Table first be used, the result will be the true solar time of the new or full moon, ^consequently of an eclipse corn sponding with the place for which the calculations are made* Sec. 14 Astronomical Tables. 181 TABLE I. OLD STYLE. The mean lime, of new Moon in March, Old Style; with the mean anom- alies of the Sun and Moon; and the Sun's mean distance from the moon's ascending node, from the year 1700 to 1800 inclcusive. Year of Christ Mean moon in D H. new March M. S. Sun's mean anomaly S D. M. s. Moon's mean anomaly S D. M. S. Sun smean tance from node S D. M. dis- the s. 1700 1701 1702 1703 1704 1705 1703 1707 8 27 16 6 24 13 2 21 16 13 22 7 4 13 22 20 11 25 44 5 32 41 21 18 53 57 42 34 31 11 3 50 8 19 9 8 8 27 8 16 9 5 8 24 8 13 9 2 58 20 36 52 14 30 46 8 48 59 51 43 54 47 89 50 1 22 30 28 7 11 7 55 9 17 43 8 23 20 739 5 12 57 4 18 34 37 42 47 52 57 2 7 13 6 7 8 8 9 9 10 11 14 31 23 14 1 16 9 19 18 2 26 5 4 8 12 51 7 8 55 42 43 30 17 18 1703 1709 1710 1711 1712 1713 1714 1715 10 29 18 7 25 15 4 23 4 2 11 20 17 2 11 8 52 27 25 7 13 43 2 20 34 59 23 36 12 13 44 52 8 21 9 9 8 29 8 18 9 6 8 25 8 15 9 3 24 46 2 13 40 5(5 12 34 43, 2 23 22 55 2 3 59 47 13 47 3910 23 35 51 9 29 12 43 8 9 35 6 18 48 47 5 24 25 18 24 SO 36 42 47 52 57 li 1 1 2 3 3 4 20 54 29 37 7 30 15 42 :4 2J 10 31 19 14 5 6 54 41 43 17 13 1716 11 17 33 39 8 22 50 39 4 4 14 2 4 27 17 05 1717 1 2 22 58 12 6 32 2 14 2 855 19 52 J718 19 23 54 45 9 28 44! 1 19 39 13 6 14 2 54 1719 9 8 43 22 8 19 44 37 11 29 27 18 6 22 5 41 1720 27 6 16 ' 1 9 8 6 4911 5 4 24 8 48 43 1721 16 15 4 38 8 27 22 411 9 14 52 29 8 8 51 29 1722 5 23 53 14 8 16 38 33 7 24 40 34 8 16 54 16 1723 24 21 25 54 9 5 45 7 17 40 9 25 37 18 1724 13 6 14 31 8 24 16 37 5 10 5 45 10 3 40 5 1725 2 15 3 7 8 13 32 29 3 19 53 50 10 11 42 52 1726 21 12 35 47 9 1 54 41 2 25 39 56 11 20 25 54 1727 10 21 24 23 8 21 10 34 1 5 19 1 11 28 28 41 1728 28 18 57 3 9 9 52 46 10 56 7 1 7 11 42 1729 13 3 45 40 8 23 48 39 10 20 44 12 1 15 14 29 1730 7 12 34 16 8 18 4 31 9 32 17 1 23 17 16 1731 26 10 6 56 9 6 26 42 8 6 9 23 3 2 17 1732 14 18 55 33 8 25 42 34 6 15 57 28 3 10 3 4 1733 4 3 44 9 8 14 58 26 4 25 45 33 3 18 5 51 1734 23 1 16 49 9 3 20 38 4 1 22 39 4 26 48 53 1735 12 10 3 25 8 22 36 30 2 11 10 44 5 4 51 40 1736 18 54 2 8 11 52 22 20 58 49 5 12 54 27 1737 19 16 26 42 9 14 34 11 26 35 55 6 21 37 29 1738 1 15 18 8 19 30 26 10 6 21 6 29 40 16 182 Astronomical Tabks. Sec. 1 4 TABLE I. OLD STYLE, CONTINUED. Year of Christ Mean new moon in March I> H. M. S. Sun's mean anomaly S D. M. S. Moon's mean anomaly S D. M.S. Sun's mean dis- tance fro ;n tho node. 1 S. P. M, S. 17d9 1710 1741 1742 1743 1744 21 22 16 7 5 16 24 13 13 22 2 7 47 36 25 57 45 35 58 34 11 52 127 4 9 7 8 27 8 16 9 4 8 24 8 13 52 8 24 46 2 13 3:. 30 27 o; 27 20 9 1-2 7 21 6. 1 5 7 3 17 1 28 1 6 49 11 37 16 14 2-2 2 27 50 32 8 8 8 16 8 24 10 3 10 18 10 19 23 26 18 11 14 17 18 5 52 54 44 28 1745 1746 1747 1748 1749 1750 1751 21 5 10 13 29 11 17 20 7 5 26 2 15 11 7 56 23 17 6 33 27 44 9 1 23, 8 23 099 36 8 28 13 8 17 53 9 6 29 8 25 40 56 18 34 53 12 23 bl: 24 30 2 20 3i 24 1 2 tl 12 U 17 8 27 7 7 6 13 4 22 27 38 15 4-J 52 4!'. 43 54 23 50 6 5 54 1 ) 11 23 6 1 14 1 22 ,2 3 9 3 17 3 46 49 51 34 37 30 17 13 5 52 5J 40 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1763 1769 1770 1771 1772 .1773 3 23 22 17 12 2 1 11 19 8 8 17 27 15 16 48 37 25 53 47 19 6 8 14 45 9 3 22 8 22 59 8 11 38 9 151 S 19 54 9 7 44 6 22 33 16 33 1: 2>- 23 1(5 2S 3 2 2 8 13 13 27 10 3 8 13 7 18 42 LJ 19 21 7 26 55 31 32 37 23 42 57 43 3 25 5 4 5 12 5 20 6 29 7 7 8 15 43 23 26 23 12 14 57 27 28 15 2 3 50 52 i>9 26 27 14 1 2 49 17 5 8 24 6 13 15 3 20 21 10 6 6 57 29 18 7 39 28 31 8 47 24 1 40 17 d 23 8 16 9 4 8 23 8 13 9 1 8 23 64 13 32 48 4 26 42 20 12 24 16 8 20 13 5 2j 4 3 3 14 1 23 3 11 9 9 19 45 o4 34 11 G 59 11 47 16 24 21 12 26 8 24 9 2 10 10 13 13 10 26 5 13 3 46 49 52 35 37 29 4 IS 12 6 21 25 19 15 3 4 12 22 10 11 19 49 38 10 59 43 23 9 66 9 9 33 8 28 10 8 17 40 9 5 2-3 8 25 2 8 14 43 9 2 19 8 22 4 23 36 58 14 33 52 8 23 17 9 21 13 5 17 9 8 24 7 4 5 14 4 23 2 29 1 9 15 10 25 4-J 32 37 3? 25 42 2 48 53 53 33 53 16 4 4 9 1 22 2 2 8 3 17 3 25 4 3 5 11 5 23 5 28 7 6 7 14 8 23 9 1 9 9 10 18 10 2S 20 23 26 9 12 15 58 51 33 25 27 14 1 3 53 1774 1775 1776 1777 1778 1779 1783 1781 1 3 23 1 8 10 27 7 16 16 6 1 23 23 13 7 o/ 30 1-9 51 40 29 1 50 63 2-3 12 51 28 4 44 21 8 11 8 29 8 19 9 7 8 26 8 15 8 4 8 23 24 46 2 24 40 56 18 34 1 l l 17 9 1 13 5 y 4 8 10 6 20 5 25 4 5 2 15 1 21 02 14 29 23 17 25 54 31 42 36 30 41 7 47 5 52 6 49 49 32 35 88 21 23 b7 88 25 26 13 1 4$ Sec. 14 Astronomical Tables. 183 TABLE I. OLD STYLE, CONCLUDED. Year of Christ Mean new moon in March r> ii. M. s. I Sun's mean anomaly Moon's mean anomaly s r. M. s. Sun's mean tance from node S. D. M dis- the . s i7J , 1783 1781 1785 1783 1787 1788 2 21 9 23 18 7 2-> id 14 23 20 5 14 11 3d J7 11 37 13 32 53 21 10 10 6 42 46 3 12 9 1 8 20 9 8 8 28 8 17 9 5 49 12 28 50 6 21 44 58 10 3 1,5 7 59 11 10 U 9 16 7 26 7 1 5 11 3 21 2 26 43 57 21 3 9 8 46 U 34 U 22 24 59 30 11 1 2 2 3 4 4 5 5 4 26 13 9 21 12 29 55 7 58 16 24 44 35 36 23 25 12 59 1 i7Bj 1790 1791 17P2 14 4 23 ; ! 20 5 2 11 31 3 19 :9 52 19 41 15 3 2o 8 14 9 2 8 21 15 38 53 3 56 SB i 6 11 16 10 22 9 2 47 c5 35 40 12 M S Moon's mean anomaly. SUMS Sun's mean dist. from the node. S D M S i 2 fr 4 i 29 12 44 3 59 1 28 6 S3 14 12 9 118 2 56 12 147 15 40 15 29 6 19 1 28 12 39 2 27 18 58 3 26 25 17 4 25 31 37 25 49 1 21 38 1 2 17 27 1 3 13 16 2 4952 1 40 1- 2 1 20 2S 3 2 04'. 4 2 40 56 5 3 21 1C 6 7 8 9 10 177 4 24 18 206 17 8 21 233 5 52 24 265 18 36 27 295 7 20 30 5 24 37 56 6 23 44 15 7 22 50 35 8 21 56 54 9 21 3 14 5 4 54 3 6 43 3 6 26 32 3 7 22 21 4 8 v 18 10 4 6 4 1 24 ^7 4 41 3t 8 5 21 5i 9626 10 .6 42 2^ r 12 13 324 20 4 33 354 8 48 36 383 21 32 40 10 20 9 33 11 19 15 52 18 22 12 9 13 59 5 10 9 43 5 11 5 37 6 11 7 22 34 8 24- 1 8 41 1 "i" _ 3 C ' 14 13 22 2 14 33 10 6 12 54 30 15 20 7 (JC^ The half lunation above, is used in finding the mean time of fall JJ/ocm, and likewise in calculating her Eclipses. 186 Astronomical Tables. TABLE IV. Sec. 14 The days of the Year, reckoned from the beginning of March. 1 s c 3 || CO "S. 2" O | o a C 9 EM to s c a f fft 5 *T E 5 ff f 1 * ?" c- n ~~[ j 32 62 93 123154 165215 246276 307 M* 2 33 63 91 186216 247.;:77 308:3:;9 3 g 34 64 95 125 1?6 187 217 248 278 309 34t> 4 4 35 65 ye 126157 188il>18 2491279 3!0 34! 5 5 J6 66 97 '27 ir ;8 189 219 250*280 31 ! 342 ~g ~6 ^ 67 ~98 128 [59 F90 '"^m 251281 ~7:2 543 7 7 i8 99 129 160 191 221 252 2f 2 313 344 8 8 i9 69 100 J3o 161 192 222 253 283 3!4 345 9 10 70 10! I3i 193 223 254:284 315 3^6 10 H n 71 102 132 163 194 224 255 285 3'6 347 n 1 12 72 108 13;\ 164 19f> 225 256 l;86 317 34? 12 1- [:^ 73 H>4 !34 165 196 226 257 287 318 349 13 |f '1 74 105 i 3"> 166 197 227 258 288 3!9 350 14' 1 75 106 136 l<>7 198 228 259 289 320 361 .'5 1 6 76 107 137 168 199 219 26(. 290 an aw Hi i 7 77 108 538 769 200 23T> 261 291 322363 1 7 I 8 78 109 139 170 201 231 262 292 323 354 18 ! r9 79 HO 140 171 202 232 263 293 324 ' 3 55 19 i ;> H' 111 141 172! 203 233 264 294 325 356 21) .' >i 81 112 ;42 173 204 234 265 295 326 357 21 '2. >2 -s- 113 143 174 205 235 26 240 27 1 30 1 1:s32 863 27;27<58 S8 119 49 180J211 241 272 802SS3364 28 28 59 89 120 150 181 212 242 27313031334 365J 29:29 60 :)) 121 151 i 82 2 i 3 248 274304^335 30 3061 9' 122 152 183 2141244 275 3 5;33f> 81 31 00 9 - 153 184 (245 306:337 Sec. 14 Astronomical Tables. TABLE V. THE SUN'S DECLINATION, 187 ARCrUMElVT.Tlie Sun's trne place. De- grees Signs. 0. N. 6 S. Signs. 1. N. 7 S. Signs. ,2. N. 8 S. De- grees U. M. U. M. D. M. 1 2 3 4 5 24 48 1 12 1 36 1 59 11 30 11 51 12 11 12 32 12 53 13 13 20 11 20 24 20 36 20 48 20 59 21 10 30 29 28 27 26 25 6 7 8 9 10 2 47 3 11 3 34 3 58 13 33 13 53 14 12 14 31 14 50 21 21 21 31 21 41 21 50 21 59 24 23 22 2t 20 1 1 12 13 14 15 4 22 4 45 5 9 5 32 5 55 t 1 5 9 15 28 15 46 16 4 16 22 22 8 22 16 22 24 22 31 22 38 19 18 17 16 15 16 17 18 19 20 6 18 6 41 7 4 7 27 7 50 k 16 39 16 57 17 14 17 30 17 -46 22 45 22 51 22 56 23 2 23 6 i4 IS 12 11 10 2 -' 23 2-1 25 8 13 8 35 9 57 9 20 9 4' 18 2 18 18 18 33 18 48 19 3 23 11 23 14 23 18 23 21 22 23 9 8 7 6 5 ' ' I ! 17 28 29 30 10 4 10 25 10 47 11 8 I \ SO 19 17 19 31 19 45 19 58 20 1 ! 23 25 23 27 23 28 23 29 23 29 4 3 2 1 . c- Signs. Signs. Si ;n -. De- gtees. 11 S. 5 N. 10 IS. 4 N. y s. S N. grees. 188 Astronomical Tables. Sec. 14 TABLE VI. Equation of the Sun's centre, or the difference between his mean and true place. ARGUMENT Sun's mean anomaly [f i ' Subtract. P | < rj 0) Signs. D M S 1 aJ.ga. D M S 2^. gas. - 3 sSi^us. DM S |D M s 4 Signs-. D M S o fSi.rio. n *M s 1 2 3 4 5 00 1 59 3 57 5 o 075 095 U 56 47 58 30 12 [ 1 55 [ 3 3 [ 5 11 I 3J 6 I 40 7 [ 41 t I 42 S [ 42 5' ( 4:{ o 1 55 37 55 3 55 38 55 36 55 31 55 1 41 U 1 40 i 1 39 1C 38 6 I 37 C : -'i .5 i . '., 03 JO 57. 7 55 19 t.J 30 51 40 49 40 iU 29 28 27 It 25 o 7 8 9 10 OH. 13 4t 15 46 17 4. 19 4' i ., 0\J I 8 ?. I 10 i I 11 3( [ 1 ; 9 L 44 4 [ 45 34 f 43 2 I 47 8 [ 47 L 55 15 55 0, 54 5C 54 33 54 17 i Ji 43 1 33 3i 1 32 1'.; 1 31 4 1 29 47 U 17 j'i i 46 05 44 11 42 16 40 21 H4 23 22 21 I 20 11 12 13 14 15 21 j t 23 33 25 29 27 25 0. 29 20 i 14 41 I 16 11 [ 17 40 I 19 8 f 20 34 I 49 15:1 53 3o I 49 541 53 12 I 50 301 52 46 T 51 fcjl 52 IS 27 9 25 4F 24 2 23 Jj -Ji ) 33 28 ) 34 30 ) 32 32 30 33 tJ 13 17 16 15 ! 16 17 18 19 20 31 15 33 9 35 3S 5- 38 47 i 21 59 I 23 2-2 I 21 44 I 26 5 I 27 2^ I 51 a, I 52 8 I 52 3^ [ 53 : I 53 27 Ji 43 51 15 I 50 41 [ 50 5 [ 4) 26 21 31 20 6 IS 3 17 5 33 J -28 33 26 33 3 24 3J ) -11 32 20 30 U 13 w ir 10 21 22 23 24 25 40 39 42 30 44 20 43 S 47 57 I 28 4 L 29 57 [ 31 1, [ 32 2 t 33 3 I i>J 6L [ 54 H I 54 28 L 54 4 f 54 5 [ *, -. 48 3 [ 47 IS [ 46 3-. I 45 4 U 60 12 2 : 10 47 9 i, 7 9 > 18 2d 16 28 14 24 J 12 21 1 10 i 8 7 6 5 -) ."' 27 23 29 90 49 451 51 32 i 53 18 55 3 56 47 I 34 45 I 35 53 ( 33 51 I 38 8 [ 39 6 10 Signs. I 53 1 [ 55 20 [ 55 28 I 55 3 [ b> 37 9 Sterna L 44 .3' r 44 1 r 4$ 7 [ 42 lOj I 41 12 8 Si>rns. L J 'i [ 4 7 [ 2 24 3. 58 53 i) 611* 047 024 TOO 4 3 2 J? ' ^ | o 12 48 .1 2 1 5f> . 30 57 .1 39 M : 48 3? "> 57 1 7 8 47 8 51 4f 8 56 H 9 2; 9 43! 9 fi 0f 9 46 44 9 45 3 9 45 12 3 44 11 ) 42 59 D 41 31 3 b 59 8 3 12 7 57 2S 7 51 35 7 45 4f 7 39 41 t 34 33 4 26 1 4 17 25 4 8 47 407 3 51 23 30 29 28 27 26 25 6 7 8 9 10 1 5 4: 1 18 41: 1 27 44 I 38 4<; I 49 3/3 '> 5 5 1 o 14 19 8 22 4 1 i 30 57 u 39 4 9 12 9 9 1 5 4H 9 19 5 9 22 M i) 23 1 % :) 40 8 9 38 1 9 9 36 24 .> 34 1! 9 32 1 ; 33 3; 7 27 2 7 21 7 14 3* 7 7 5' 3 4s 3- } 33 Sa ; 24 4 3 1 5 41 3 6 4: 24 23 22 21 20 11 in 13 14 15' 2 23 2 11 1<> 2 21 54 2 32 3 i 2 43 9 i 47 "> 54 4' 7 2 24 7 9 5JC 7 17 H :> 27 04 t 30 3 9 32 5; ) 35 i : ) 37 14 . 29 33 :J 26 54 .) 24 4 J 21 3 J 17 51 7 I i 54 1; 6 47 { ij 40 t H 32 5f 2 57 43 2 48 39 2 39 31 2 30 28 2 21 1 !) 19 18 17 16 15 ItJ 17 18 ' 19 20 2 53 Si -3 4 3 3 14 24 .-I 024 4.2 3 34 5 7 24 I 'J 7 31 IS 7 38 & 7 44 51 7 51 24 .) 39 t 9 40 51 9 42 21 9 43 4i 9 44 5? :) 14 2fc > 1 54 979 9 3 13 8 59 6 o 25 4(! ti 18 lit !i 10 41' 6 3 ir, 5 55 S 212 t 2 2 5.S 1 53 Si 1 44 1( 1 ' 34 54 14 13 12 11 10 21 22 23 24 25 3 45 11 3 55 21 4 5 26 4 25 26 4 25 20 7 57 45 8 3 56 S 9 57 3 15 46 S 21 24 9 45 52 9 46 38 9 47 13 9 47 36 9 47 49 8 54 50 3 50 24 8 45 48 S 41 2 8 36 6 5 47 54 5 40 4 5 32 9 5 24 9 5 16 5 1 25 31 1 16 7 1 6 41 57 13 10 47 44 9 8 7 6 5 26 27 28 29 30 ; 4 35 6 ;4 44 42 14 54 11 5 3 33 5 12 48 3- 26 53 3 32 11 3 37 19 13 42 18 8 47 8 |9 47 54 9 47 46 9 47 33 9 47 14 9 46 44 IS 31 8 25 44 8 20 18 8 .14 33 8 8 59 5 7 56 4 59 42 4 51 15 4 43 2 4 34 33 JO 38 13 ;0 28 41 19 8 9 34 003 4 3 2 s? tfq 11 Signs 10 Signs 9 Signs 8 Signs 7 Signs 6 Signs a < Subtract 192 Astronomical Tables. Sec. 14 TABLE X. The third equation of the mean, to the true syzygy. AStGUMEJBT r-Su.ti*s mean anomaly Moon's equated anomaly. B Si-ns. Si t jns. SLns. | 'K 3 ' >! ti , Niii,; ru 1 :! . 1 >i M, -...f.- , i S n , -a, , . 0^ 3 6 .Si .'M--, a id. 7 S5I..M1*, adj. 8 01 11--, il :'l. rt M 5? M S M S 2 3 4 5 5 10 15 20 25 2 22 2 26 2 30 2 34 2 33 2 42 4 12 4 13 4 18 4 21 4 24 4 27 30 21 28 27 26 25 ' 6 7 3 9 10 30 35 40 45 50 2 46 2 50 2 4 2 58 3 2 4 30 4 32 4 34 4 36 ' 4 33 -.4 23 22 21 20 11 12 13 14 15 o5 5 10 15 3 6 3 10 3 14 3 18 3 22 4 40 4 42 4 44 4 46 4 48 19 18 17 16 15 16 17 18 19 20 20 25 SO 35 40 3 26 3 30 3 34 3 38 3 42 4 50 4 51 4 52 4 53 . 4 54 14 13 12 11 10 21 22 23 24 25 45 49 52 56 2 3 45 3 43 3 b\ 3 54 S 57 4 55 4 56 4 57 4 57 4 57 y 8 7 6 5 26 27 ?8 29 SO 2 4 2 9 2 13 2 18 2 22 4 4 3 4 6 4 9 4 12 4 58 4 58 4 58 4 58 4 58 4 3 2 1 s? Sins. Signs. Si~ns. H Cfq 5 Si n<, subtrar-t. 4 ->i"ns, si.b rart.; 3 Siens, subtract. H S 1 1 Sign*, add. 10 Signs, add. y Signs, add. s Sec. 14 Astronomical Tables. 193 TABLE XL The fourth equation of the mean, to the true Syzygy. AR.G-UME1NT--T1IC sun's mean, distance from flue node* Add. V 1 (A IT i 2 3 4 5 ~6~ 7 8 9 10 ii" 12 13 14 15 Signs. 6 Signs. 1 Sign. 7 Signs 2 Signs 8 Signs 1 1JO 29 28 27 26 25 ~24 23 22 21 20 M. S. M. S. M. S. 4 7 10 13 16 1 22 1 23 1 24 1 25 1 26 1 27 1 22 1 21 1 20 1 18 1 16 1 14 20 23 26 29 32 1 28 1 29 1 30 1 31 1 32 1 12 1 10 1 8 1 6 1 3 35 38 41 44 47 1 33 1 33 1 34 I 34 1 34 1 57 54 51 49 19 18 17 16 15 ~14 13 12 11 10 ~~9 8 7 6 I ~~4 3 2 1 16 17 18 19 20 2T 22 23 24 25 50 52 54 57 1 1 34 1 34 1 34 1 33 1 38 45 41 37 34 31 1 2 1 5 1 8 I 10 1 12 1 32 1 31 1 30 1 29 1 28 28 25 22 19 16 26 27 28 29 30 1 14 1 16 1 18 I 20 1 22 1 27 1 26 1 25 I 24 1 22 13 10 6 3 5 Signs. 4 Signs. ! 3 Signs. 11 Signs. 10Signs.|9 Signs. Subtract. 194 Astronomical Tables. TABLE XII. Stc. 14 THE SUN'S MEAN LONGITUDE MOTION AND ANOMALY. Sun's' mean I*ongittitlcBSun'8 mean anomaly* Years beginning. s D M S , 6 D M 8 Old Style 1 9 7 53 10 6 23 48 1 1! 201 9 9 23 50 6 26 67 301 9 10 9 10 6 26 1 401 9 10 54 30 6 25 5 501 9 11 39 50 6 24 9 1001 9 15 26 30 6 19 32 1101 9 16 11 50 6 18 36 1201 9 16 57' 10 6 17 40 1301 9 17 42 30 6 16 44 1401 9 18 27 50 6 15 49 1501 9 19 13 10 6 14 53 1601 9 19 '58 30 6 13 57 1701 9 20 43 50 6 13 1 1801 9 21 29 10 6 12 6 N .S.1797 9 10 37 33 6 1 8 17 1798 9 10 23 13 6 52 51 1799 9 10 8 54 6 37 26 1800 9 9 54 35 6 22 1 1801 9 9 40 16 6 6 36 1802 9 9 25 56 5 29 51 10 1803 9 9 11 37 5 29 35 45 1804 9 9 56 26 6 19 28 1805 9 9 42 6 6 4 2 1806 9 9 27 48 5 29 48 38 1807 9 9 13 29 5 29 33 13 *. 1808 9 9 58 17 6 16 48 1809 9 9 43 57 6 1 31 1810 9 9 29 37 5 29 45 57 1811 9 9 15 17 5 29 30 32 1812 9 10 5 6 14 15 1813 9 9 45 45 5 29 58 49 1814 9 9 31 25 5 29 43 26 1815 9 9 17 5 5 29 27 58 1816 9 10 1 53 6 11 41 1817 9 9 47 33 5 29 56 15 1818 9 9 33 13 5 29 40 50 1819 9 9 18 53 5 29 25 24 1820 9 10 3 41 6 9 7 1821 9 9 49 22 * 3 ill 42 Sec. 14 Astronomical Tables. TABLE XII. CONTINUED. THE SUN'S MEAN LONGITUDE MOTION AND ANOMALY. 195 _ ? Sun's mean Mt>tioii*3nu.*s mean anomaly* Years complete. s D 31 s s D M 1 11 29 45 40 11 29 45 2 11 29 31 21 11 29 29 3 11 29 17 20 11 29 14 4 1 50 11 29 58 5 11 29 47 31 11 29 42 A 6 11 29 33 11 11 29 27 7 11 29 18 52 11 29 11 8 3 41 11 29 55 , 9 11 29 49 21 11 29 40 10 11 29 35 2 11 29 24 11 11 ,29 20 42 11 29 9 12 5 31 11 29 53 13 11 29 51 12 11 29 37 14 11 29 36 52 11 29 22 15 11 29 22 33 11 29 7 16 7 22 11 29 50 17 11 29 53 2 11 29 35 18 11 29 38 43 11 29 20 19 11 29 24 23 11 29 4 20 0' 9 12 11 29 48 40 18 24 11 29 37 60 27 36 11 29 26 80 36 48 11 29 15 100 46 11 29 4 200 1 32 11 28 8 300 2 18 11 27 12 400 3 4 11 26 16 500 3 50 11 25 21 600 4 32 11 24 25 700 5 17 20 11 23 29 800 6 2 40 11 22 331 900 6 48 11 21 37 1000 7 40 11 20 41 2000 15 20 11 11 22 3030 22 40 11 2 3 4000 1 13 20 10 22 44 5000 1 7 46 40 10 13 25 6000 1 15 30 10 4 I - 196 Astronomical Tables. Sec. 14 TABLE XIL CoKTimrED. THE SUN'S MEAN LONGITUDEMOTION AND ANOMALY. Sun's mean MotionSun's mean anomaly* MONTHS. s D M s s D M January, 8 February, 1 33 18 1 3'd March, 1 28 9 11 1 28 9 April, 2 28 42 SO 2 28 42 May, 3 28 16 40 3 28 17 June, 1 4 28 49 58 4 28 50 July, 5 28 24 8 5 28 24 August, 6 28 57 26 6 28 57 September, 7 29 30 44 7 29 30 " October, 8 29 4 54 8 29 4 November, 9 29 38 12 9 29 37 December. 10 29 12 22 10 29 11 Sun's mean motion &> anomalySun's mean motion & anomaly* DAYS S D M S IDAYS |S D M S 1 59 8 17 16 45 22 2 1 58 17 18 17 44 30 3 2 57 25 19 18 43 38 4 3 56 33 20 19 42 47 5 4 55 42 21 20 41 55 6 5 54 50 22 21 41 3 7 6 53 58 23 22 40 12 8 7 53 7 24 23 39 20 9 8 52 15 25 24 38 28 10 9 51 23 26 25 37 37 1110 10 50 32 27 26 36 45 12 11 49 40 28 27 35 53 13 12 48 48 29 28 35 2 14 13 47 57 30 29 34 10 15 14 47 5 31 30 33 18 16 15 46 13 See. 14 Astronomical Tables. 197 TABLE XII. CONCLUDED. THE SUN'S MEAN LONGITUDE MOTION AND ANOMALY. Sun's mean, motion and anomaly. H M S ~I 2 3 4 5 6 7 8 9 10 Sun's mean motion and anomaly. Sun's mean dist. from the Node. Sun's mean motion snd anomaly. Sun's mean dist. from the Node. DM S M 8 3dg s 3ds 4ths DM 8 MS T 8 T F H M S DM S MS T S T F DM S MS T S T F 2 280 2 36 4 560 5 12 7 240 7 48 9 510 23 12 190 12 59 31 32 33 34 35 36 37 38 39 40 1 16 23 1 18 51 1 21 19 1 23 47 1 26 15 1 20 30 1 23 6 1 25 42 1 2 8 18 1 30 54 14 470 15 35 17 150 18 11 19 430 20 47 22 110 23 23 24 280 25 58 1 28 42 1 31 10 1 33 38 1 36 6 1 38 34 1 33 29 1 36 5 1 38 40 1 41 16 1 43 52 11 12 13 14 15 16 17 18 19 20 2f 22 23 24 25 27 60 28 34 29 340 31 10 32 2;0 33 45 34 300 36 21 36 580 38 57 ,41 42 43 44 45 1 41 21 46 28 1 43 301 49 44 1 45 571 51 39 1 48 251 54 15 1 50 531 55 51 39 26 41 53 44 21 46 49 49 17 41 33 44 8 46 44 49 20 51 56 46 47 48 49 50 1 53 21 1 55 49 1 58 17 2 44 2 3 12 1 59 27 22" 2 4 39 2 7 13 2 9 50 51 45 54 13 56 40 59 8 1 1 36 54 32 57 8 59 43 1 2 19 1 4 55 1 52 53 54 55 56 57 58 59 60 2 5 40 288 2 10 36 2 13 4 2 15 32 2 12 25 2 15 2 2 17 38 2 20 14 2 22 50 26 27 28 29 30 144 1 6 32 1 9 1 11 23 1 13 55 1 7 31 1 10 7 1 12 43 1 15 19 1 17 55 2 17 59 2 20 27 2 22 55 2 25 23 2 27 51 2 25 26 2 28 2 2 30 38 2 33 14 2 35 50 In Leap- Year after February, add one day, and one day's motion. 198 Astronomical Tables. Sec. M TABLE XIII. Equation of the Sun's mean distance from the Node. ARGUMENT Sun's mean anomaly. > Subtract ! 0S 1*0 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Signs. D M 1 Sign D M 2 . Signs D M 3 Signs D M 4 Signs D M 5 Signs. D M 1 2 3 4 5 2 4 6 9 11 I 2 I 4 I 6 I 8 I 10 I 12 1 47 1 48 1 49 1 50 1 51 1 52 2 5 2 5 2 5 2 5 2 5 2 5 I 50 I 48 I 47 I 46 I 45 I 44 1 4 1 2 1 58 56 54 6 7 8 9 10 13 15 17 19 21 I 14 I 16 I 17 I 18 I 19 1 53 1 54 1 55 1 56 1 57 2 5 2 4 2 4 2 4 2 4 I 43 I 41 I 40 T 38 I 37 52 50 48 46 44 11 12 13 14 15 16 17 18 19 20 21" 22 23 24 25 23 25 28 30 32 I 21 I 22 I 24 I 26 I 27 1 58 1 58 1 59 2 2 2 3 2 3 2 3 2 2 2 2 I 36 I 34 I 33 I 31 I 30 42 40 37 35 33 34 36 D 38 40 42 I 28 I 30 I 31 I 34 I 35 2 1 2 1 2 2 2 2 2 3 2 1 2 1 2 2 1 59 1 28 I 27 I 25 I 24 I 23 31 21 27 24 22 44 46 48 50 52 1 36 I 37 I 39 40 41 2 3, 2 4 S i 2 4 1 59 1 58 1 57 1 56 1 55 1 21 19 17 15 13 20 18 16 13 11 26 27 28 29 30 54 56 58 1 1 2, 43 44 45 I 46 I 47 2 5 2 5 2 5 2 5 2 5 54 53 52 51 50 11 9 3 6 I 4 9 7 5 3 4 3 2 1 J? 11 10 Signs 1 Signs | 9 j 8 Signs] Signs 7 Signs 6 Signs S? Add. See. 14 Astronomical Tables, TABLE XIV. THE MOON'S LATITUDE IN ECLIPSES. 199 ARGUMENT Moon's Equated Distance from the Node. ) SIGNS NORTH ASCENDING. 6 SIGNS SOUTH ASCENDING. Degrees. DM s [Degrees. ~0 0~ 30 1 5 15 29 2 10 30 28 3 15 45 27 4 20 59 ' 26 5 26 13 25 6 31 26 24 . 7 36 39 23 8 41 51 22 9 47 22 2l - - 10 52 13 20 11 57 23 19 12 1 2 31 8 13 1 7 38 17 14 12 44 16 15 17 49 15 16 22 52 14 17 27 53 13 18 32 52 12 19 37 49 1 1 FIVE SIGNS. NORTH DESCENDING. ELEVEN SIGNS. SOUTH DESCENDING. This Table shows the Moon's true Latitude a little beyond the utmost limits of Eclipses. 200 Astronomical Tables. Sec. 14 TABLE XIV. The Moon's horizontal parallax, with the semi-diameters, and the true horary motions of the Sun and Moon, to every sixth degree of their mean anomalies ; the quantities for the intermediate degrees, being ea- sily proportioned by sight. . I c 1 f c c ^" t/T of = - Cft > ll B. 3 1. ffi o ffi o 11 go v M Cj s3 SE f> P T 1 | I o f PIT S D M S M S M S M S M S S D 54 29115 50 14 54 30 10 2 23 12 ) 6 ; 54 31 15 50 14 55 30 122 23 11 24 12 54 34 15 50 14 56 30 152 23 11 18 18 54 40 15 51 14 57 30 19 ! 2 23 11 12 24 54 47 15 51 14 58 30 26 2 23 11 6 54 56 15 52 14 59 30 342 24 11 y 6 55 6 15 53 15 1 30 44 2 24 10 24 12 55 17 15 54 15 4 30 55 2 24 10 18 18 55 29 15 55 15 8 31 09 2 24 10 12 24 55 42 15 56 15 12 31 23 2 25 10 6 , 2 55 56 15 58 15 17 31 40 2 25 10 2 6 56 12 15 59 15 22'31 56 2 26 9 24 2 12 56 29 16 1 15 2632 17 2 27 9 18 2 18 56 48 16 2 15 30 32 39 2 27 9 12 2 24 |57 8 16 4 15 3633 11 2 28 9 6j 3 o !57 30 16 615 41 33 23 2 28 9 0' 3 6 57 5216 815 47 33 47 2 29 8 24! 3 12 58 12'16 10 15 52 34 11 2 29 8 18 3 18 58 3lil6 11 15 5834 24 2 29 8 12 3 24 :58 49 16 13 16 334 58 2 30 8 6 4 59 616 14 16 9;35 22 2 eo 8 4 6 59 21 16 15 16 1435 45 2 1 7 24 4 12 59 35|16 17 16 1936 2 31 7 18 4 18 59 48 16 19 16 24 96 20 2 LZ 7 12 4 24 60 016 20 16 28,36 40 2 32 7 6 5 60 11 16 21 16 31 37 2 32 7 5 6 60 21 16 21 16 32 37 10 2 33 6 24 5 12 60 30 16 22 16 37 37 19 2 33 6 18 5 18 60 38 16 22 16 38 37 -28 2 33 6 12 5 24 60 45 16 23 16 39 37 36 2 33 6 6 6 60 45 16 2316 39 37 40 2 33 6 Sec. 14 Astronomical Tables. 201 TABLE XVI. MEAN NEW MOON, &c. IN MARCH, NEW STYLE, FROM 1800 TO 1900 INCLUSIVE : CALCULATED FOR THE MERIDIAN OF WASHINGTON. 76 DEGREES AND 56 MINUTES WEST LONGITUDE FROM LONDON. Year of chrisi Mean new Moon. Sun's mean anomaly. Moon's mean anomaly. Sun's mean dis- tance from the node- D H. M. S. b D. M. S. S D. M. S. S D. M. S. 1800 i-4- iU 14 3 i ^;i .., .;; 10 7 52 3( 11 3 5 24 1801 14 4 3 2 , 12 3-> 4/' 8 17 40 41 11 12 1 10 1802 3 H 51 4 1 51 39 6 27 28 4r 11 3 57 1803 22 10 24 2.' ; 20 13 51 6 3 5 5i^ 8 46 58 1804 10 19 13 i. ' 9 29 4:> 4 12 53 57 1 6 49 45 I80f> 4 1 3 ; 7 28 45 3 2 22 42 1 14 52 35 Unit; 19 1 34 1. J 17 7 43 i gti 19 1 2 s:3 3i> 3(3 1807 8 10 ^2 5f, J 6 23 S:, 8 7 Ifc 3 1 38 23 1808 2ii 7 55 S. c i 24 45 47 11 13 44 IS? 4 10 21 24 1809 15 16 44 1 5 14 1 39 9 23 32 2-1 4 18 24 11 181(1 5 1 32 45 3 3 17 31 8 3 20 2U 4 26 26 -58 181 i 23 aid 5 2: i 21 39 4-0 7 8 57 tf., 6 5 9 59 1812 12 7 54 < : 10 55 3f 5 18 45 4f 6 13 12 46 1813 1 16 42 4! ; 11 27 3 28 33 4. r > 6 21 15 23 1814 20 14 15 2 5 18 33 39 a 4 10 51 7 29 58 24 1 H 1 S 9 23 3 57 * 7 53 31 i 13 58 5' ft X 1 11 1 1 I, ai / iiO 3d 3/ 8 ao \.) 4.-- 19 30 i 9 tl. 44 12 1817 17 5 25 13 3 15 31 35 10 29 24 ? 9 24 46 59 18 to 6 14 13 5(i 8 4 47 27 9 9 12 ICT 10 2 49 46 1819 -25 11 46 29 8 23 9 39 8 14 49 18 11 11 32 47 182( 13 20 35 8 12 25 31 6 24 37 23 11 19 35 34 18-2! 3 :> *rj 4; 3 l 4! 23 5 4 25 sa 11 2/ rS 21 18 '2', -2-2 2 56 2i 8 20 3 35 4 10 2 29 1 6 22 22 1828 tl 11 44 59 8 9 19 27 2 19 50 34 1 14 25 9 1824 29 9L 17 39 3 27 41 39 1 25 27 40 2 23 9 10 1825 18 18 6 15 8 16 57 31 5 15 45 3 1 11 57 202 Astronomical Tables. Sec. 14 TABLE Year of Christ. Mean new Moon in March. Sun's mean Anomaly. Moon's mean Anomaly. Sun's mean dis- tance from the Node. I) II M S S D M S S D M S S D M S 1826 1327 1823 1829 8 2 54 52 27 27 31 15 9 16 8 4 18 4 45 23 15 37 24 8 6 13 23 8 24 35 35 8 13 51 27 8 3 7 19 8 21 29 31 10 15 3 50 9 20 40 56 8 29 1 6 10 17 6 5 15 54 12 3 9 14 44 4 17 58 45 4 26 1 32 5 4 4 19 6 12 48 20 1831 1832 1833 1834 1835 13 28 1 1 9 14 37 20 6 47 17 9 15 35 54 28 13 8 33 8 10 45 23 8 "0 1 15 8 18 23 27 8 7 39 19 8 26 1 31 3 25 42 17 2 5 30 22 1 11 7 28 11 20 55 33 10 26 32 39 6 20 51 7 6 28 53 54 8 7 37 55 8 15 40 42 9 24 24 43 1836 1837 1838 1839 1840 16 21 57 10 6 6 45 46 25 4 18 26 14 13 7 2 2 21 55 39 8 15 17 23 8 4 33 15 8 22 55 27 8 '12 11 19 3 1 27 11 9 6 20 44 7 16 8 49 6 21 45 55 5 1 34 3 11 22 5 10 2 27 30 10 10 30 17 11 19 14 18 11 27 17 5 5 19 52 1841 1842 1843 1844 1845 21 19 28 19 11 4 16 55 30 1 49 35 18 10 38 12 7 19 26 48 8 19 49 23 8 9 5 15 8 27 27 27 8 16 43 19 8 5 59 11 2 16 59 11 26 47 16 11 32 24 22 10 12 12 27 8 22 32 1 14 3 53 1 22 6 40 3 50 41 3 8 53 28 3 16 56 15 1846 1847 1848 1849 1850 26 16 59 28 16 1 48 5 4 10 36 41 23 8 9 21 12 16 57 57 8 24 21 23 8 13 37 15 8 2 53 7 8 21 15 19 8 10 31 11 7 27 37 38 6 7 25 43 4 17 13 48 3 22 50 54 2 2 38 59 4 25 40 16 5 3 43 3 5 11 45 50 6 20 29 51 6 28 32 38 iSji 1852 1853 1854 1855 2 1 46 33 19 23 19 13 9 8 7 49 28 5 40 29 17 14 29 5 7 29 47 3 8 18 9 15 ,8 7 25 7 8 25 47 19 8 15 3 11 12 27 4 11 18 4 10 9 27 52 15 9 3 29 21; 7 13 17 26; 7 6 35 25 8 15 19 28 8 23 22 13 10 2 6 14 lO 10 9 1 14 Tables. 203 TABLE XVI CONTINUED. Year of Christ. Mean new Moon in March. Sun's mean Anomaly. Moon's mean Anomaly. Snn's mean dis- tance from the Node. D II M S S D M S S D M S S D MS 1856 1857 1858 1859 1860 5 23 17 42 24 20 50 22 14 5 28 58 3 14 27 35 21 12 15 8 4 19 2 8 22 41 15 S 11 57 7 3 1 12 59 3 19 35 11 5 23 5 31 4 28 42 37 3 8 30 42 1 18 18 47 23 55 53 10 18 11 48 11 26 55 49 4 58 36 13 1 23 1 21 45 24 1861 1862 1863 1864 1865 10 20 48 51 29 18 21 31 19 3 10 7 7 11 58 44 26 9 31 24 8 8 51 3 8 27 13 15 8 16 29 7 3 5 45 8 24 7 12 11 3 43 58 10 9 21 4 8*19 9 9 6 28 57 14 6 4 34 20 1 29 48 11 3 8 32 12 3 16 34 59 3 24 37 46 5 3 21 47 1866 1867 1868 1869 1870 15 18 20 5 3 8 37 23 41 16 12 9 29 53 1 18 18 30 8 13 23 4 8 2 38 56 8 21 1 S 8 10 17 7 29 32 52 4 14 22 25 2 24 10 30 1 29 47 36 9 35 41 10 19 23 46 5 11 24 34 5 19 27 21 6 28 11 22 7 6 14 9 7 14 16 56 1871 1872 1873 1874 1875 20 15 51 9 9 39 46 27 22 12 26 17 7 1 2 6 15 49 39 8 17 55 4 8 7 10 56 8 25 33 8 8 14 49 8 4 4 52 9 25 5 8 4 48 10 7 10 25 16 5 20 13 21 4 1 26 8 23 57 9 1 3 44 10 9 47 45 10 17 50 32 10 25 53 19 1876 1877 1878 1879 1880 24 13 22 18 13 22 10 55 3 6 59 32 22 4 32 11 10 .13 20 48 8 22 27 4 8 11 42 56 8 58 48 8 19 21 8 8 36 52 3 5 38 32| 1 15 26 37 11 25 14 42 11 51 48 9 10 39 53 4 37 20 12 40 7 20 42 54 1 29 28 55 2 7 29 42 1881 1882 1883 1884 1885 29 10 53 28 18 19 42' 4 8 4 30 41 26 2 3 20 15 10 51 57 8 26 59 5 8 16 14 57 8 5 30 49 8 23 53 1 8 13 8 53 8 16 16 59 6' 26 5 4 5 5 53 9' 4 11 30 15! 2 21 18 20j 3 16 13 43! 3 9,-i 16 30 i 4 2f 19 17 5 11 3 18 i 5 19 6 5: 204 Astronomical Tables. Sec. 14 TABLE XVI CONCLUDED. yw of Ciirisi Mean new Moon in March. Sun's mean Anomaly. Moon's mean Anomaly. Sun's mean dis- tance from the Node. D H M -8 s n M s S D M S S D M S 1886 1837 1888 1839 1890 4 19 40 33 23 17 13 13 12 2 1 50 1 10 50 26 20 8 23 6 8 2 24 45 3 20 46 57 3 10 2 49 7 29 18 41 8 17 40 53 1 1 6 25 6 43 31 10 16 31 36 8 26 19 41 8 1 56 47 5 27 52 7 5 52 53 7 13 55 40 7 21 58 27 9 42 28 1891 1892 1893 1894 1895 9 17 11 43 27 14 44 22 16 23 32 59 6 8 21 35 25 5 54 15 3 6 56 45 8 25 18 57 8 14 34 49 8 3 50 41 3 22 12 5. 6 11 44 52 5 17 21 58 3 27 10 3 2 6 53 8 1 12 34 14 9 8 -io 15 10 17 2 24 10 25 32 3' 11 3 34 50, 11 18 51 1896 1897 1898 1899 1903 13 14 42 5'2 2 23 31 28 21 21 4 9 11 5 52 4' 30 3 25 24 8 11 28 43 3 44 37 3 19 6 4 8 8 22 41 8 26 44 53 11 22 22 lt> 10 2 10 24 9 7 46 33 7 17 34 35 | 6 23 13 41 19 21 33 27 24 25 2 6 8 26 2 14 11 13 | 3 22 53 14 The year 1900, will not be Leap-Year, the differ- ence then will be 13 days, between the Old and New Style. Sec. 14 Astronomical Tables. TABLE XVII. 205 A concise EQUATION TABLE, adapted to the second year after Leap- Year, within one minute of the truth, for every year, (excepting the second) showing to tha> nearest full minute, how much a Clock should be faster or slower than the Sun. ? t 3 rf* I S ff I ! if f 1 ll 5. S' s- *5 f S- M a f I S^ v. ft 3 i* * =' p 5 p: CO " r - January, 1 3 4 5' April, 1 4 || August, 1 i^ October,^'. November,!- i ib Id 5 6J 7 2( 3r- 2( I4C 7 10 1 -r ') - 24 132 8 F 15 Oa 28 II 2? 12* 15 15 9rt 10: IS _ 31 0? 30 December, g 11 SE. 10* If n? 24 2' September, ? 1 5 <} t 2: l'2 = 3 35 A ty **' *J Or 13 T May, 1 3 1 ^ 4 ^ f 3f r 3 1 1 Q x . H A 7" i t 6- February,!*" June, 5 V 2 5 16 05 1:? * 21 i If 6 J 4 'f 2* lo o' 21 7 ^ IF 3 = March, * 12 11 2: ;' 2* I 1 I to 2f 2: 22 S'' Octobe;, ^ 10 L 11 24 |J q i ^ 3' f. 12 r 2(J 1 - 2 7 Jil| y> o 4 ? re 13 = 2f 2 r-. i 8 ii 14 14 3' 3 t 2 5 ti 6^ I! 15 - $^>This Table is near enough to the truth, for regulating common Clocks and Watches, and was for that purpose calculated by Mr. Smeatou, SECTION FIFTEENTH. EXAMPLE I. Required the true time of new Mooon in July, 1832, and also whether there were an Eclipse of the Sun or not. Mc;in nov/ arch 1832 5 lunaiions, D 147 it =. 15 M "14" 40 15'; from Table 7 149 1 54 46 52 , ;ime once eq'td. Table 9th. 26 23 2 8 12 43 8 Table 10th. 26 20 56 3 35 38 Table llth. * 26 20 62 57 49 26 20 53 6 46 Eq'tn. of the | Sun's centre. 26 20 47 46 ' .:a!y. Moon's mean Anomaly. rSun's mean dist from the Node. j;I ' S 4 25 &1 3? S D M , V S 2 7 T"30"~22 49 5-2 S D . M S 6~28~53 54 5 3 21 10 25 38 5"2 6 14 35 24 2 14 04 6 13 55 6 4 11 a? 4J 40 IS Argument 4th. Argument 2. Argument 3d. Equal to the 27th day of July, 8 hours, 47 minutes, and 46 seconds in the morning, at WASHINGTON ; the true time of new Moon. The Sun being then only two degrees and 14 minutes from the Moon's ascending mode, w*i consequently eclipsed. Sec. 15 Examples. 207 EXAMPLE II. Required the true time of new Moon in May, 1836, and whether there will be an Eclipse of the Sun or not. Mean new Moon in March, 1836. Sun's mean Anomaly. Moon's mean Anomaly. Sun's m. dist. from the node. D 16 H M s S D M s S I> 31 S S D JVI s ' IQQfi 21 1 57 28 10 8 15 6 1 28 17 12 23 39 9 6 20 44 1 21 : 8 1 10 2 2 1 27 20 30 28 Table 3d Table 7th. 14 - 23 3 2~ 5 ~w 15 IF 21 57 1 16 ' 24 40 50 44 20 10 13 30 6 E r; 10 27 53 45 \ 8 2 3 47 JTable 9th. Table 10th. 11 14 23 i 10 29. 6 47 Table llth. 14 20 59 4 13 Table 17lh. 14 20 59 4 17 1 114 21 3 18 Equal to the loth day of May, 9 hours, 3 minutes, and 18 seconds ; true time of new Moon at WASHING- TON. The Sun being then only 13 degrees and 48 minutes from the Moon's Node, the Sun will conse- quently be visibly eclipsed. 208 Examples. Sec. 15 EXAMPLE III. Required the true time of New Moon, in Decem- ber, in the year 1850; and whether there will be an Eclipse at that time or not. Mean New Moon in March, 1350. Sun's mean Anomaly. Moon's mean Anomaly. Sun's mean distance from ' the Node. D 11 MS S D M S 8 D M S S D M S 12 16 57 57 265 18 36 27 8 10 31 11 8 21 56 54 2 2 38 59 7 22 21 4 6 f.8 32 38 9626 3 11 34 24 5 2 28 5 9 25 3 4 4 34 44 I 53 3 9 24 15 25 44 38 . ', 3 9 36 21 7 8 12 40 9 24 15 25 9 11 13 " 3 25 8 2 54 3 28 2 1 27 3 26 35 9 42 3 36 7 True time of new Moon in December, 1850, will be the 3 J. 36;;: 17s, afternoon. The Mm will then be more thin 56 degrees from the node, and conse- quently there can be no Eclipse at that time. Sec. 15. Examples. 209 EXAMPLE I. Required the true time of full moon in July 1833, and whether or not there will be an eclipse of the moon, at that time. Mean new Moon in March D H. M. S. Sun's mean anomaly S D. M. S. Moon's mean anomaly S D. M. S. Sun's mean distance from the node S D. M. S. 20 ! ) 6 14 18 47 17 12 9 22 2 8 IS 2 27 14 23 27 18 5S 33 1 (i i 11 2 17 6 12 i zo 27 1 54 30 8 7 3 2 15 37 20 5") 42 7 i 15 il 28 1 7 ) vj '0 11 1 o i5o 28 31 .0 11 28 oy 25 1 1 24 58 4-J L 10 7 20 21 42 8 1 18 47 1 10 11 in the afternoon. 28 34, i 7 33 13 3 38 1 7 34 35 16 1 7 34 19 3 22 I 7 37 41 =To the first day of July, 1833 the true time of full moon in the longitude of Washington, at 7 hours 37 minutes and 41" seconds in the afternoon, the sun, be- ing then within five degrees at a mean rate from the Moon's node, consequently the Moon will then be e- clipsed. 210 Examples. Sec. 15. Required the true time of full Moon in April, in the year 1836 at Rochester; and also, whether there will be an eclipse of the Moon, or not. The true time of full Moon, in April, in the year 1836, will be on the first day, 5 hours 19 minutes and 53 seconds in the afternoon, in the longitude of Ro- chester ; the sun will then be more than 40 degrees from the Moon's node, and consequently there will be no eclipse on that day. EXAMPLE VII. Required the true lime of full Moon, in September in the year 1848, in the longitude of Utica ; and whether there will be an eclipse at that time. Meantime of full Moon in March Sun's mean anomaly. Moon's mean anomaly. Sun's mean dis- distance from the node. 1) H. M. S. S D, M. 8. S D. M. S. S D. M. S. 4 10 177 4 14 18 36 24 22 4i 18 2 8 3 5 24 14 37 33 7 56 10 4 5 6 17 4 12 13 48 54 3 54 30 (3 11 45 4 1 15 20 53 24 7 12 9 3 23 57 1 O 2 12 4 3 4 32 13 30 4 5 1 2 21 29 61 p 1 7 21 12 5 .T 25 48 h9 40 10 8 31 True time at True time at 43J4 Lyons Utica. a 32 30 12 13 14 3 46 12 13 18 25 5 12 13 18 4 30 8 12 13 22 6 38 12 J3 28 38 The Moon will be full in the year 1848, on the 13th day of September, at 1 o'clock and 28 minutes in the morning, in the longitude of Utica. the Sun, then will be only one degree and seven minutes from the Moon's node; the Moon there f9re, will be eclipsed at that time. Sec. 15 Examples. 211 To calculate the true time of any new, or full Moon, and consequently Eclipses, in any given year and month, between the commencement of the Christian Era, and that of the 18th Century. Find a year of the same number in the 18th Centu- ry, with that of the year in the proposed Century from Table First; and take out the mean time of New Moon in March, Old Style, for that year ; with the mean anomalies of the Sun and Moon, and the Sun's mean distance from the node at that time, as before instructed. Take as many complete Centuries of years from Table Second as when subtracted from the year of the 18th Century, the remainder will answer to the given year, with the anomalies, and Sun's dis- tance from the node ; subtract these from those of the 18th Century, and the remainder will be the mean time of new Moon in March, with the anomalies, *c. for the proposed year ; then proceed, in all respects, for the true time of new or full Moon, as shown in the Precepts, or former Examples. If the day's annexed to these Centuries, exceed the number of daysfr om the beginning of March, taken out in the 18th Century, subtract a lunation, and its anom- alies, fyc. from Table 3d, to the time, and anoma- lies of new Moon in March, and then proceed as above stated this circumstance happens in Example Fifth. 212 Examples. EXAMPLE IV. Sec. 15 Required the true time of New Moon in June, in the year of Christ, 36, at the City of JERUSALEM. BY THE PRECEPTS Mean N. MOOI. in March. Sun's mean Anomaiy. Moon -; n ean Anomaly. Sun's mean dist. from the Node. D H M fe D D AI S S * D M S S D M S 8 D M S March, 1736. Add one lunation. 18 54 .9 12 44 '. 6 11 52 22 29 6 H 8 20 58 49 i5 49 5 12 24 27 1 9 40 14 'jast n. moon, March 173\ Subtract, 1700 vear?. ,0 7 38 5 9 19 11 25 3 10 58 41 3 19 ;8 48 1 1U 47 4o 1 22 10 37 b 13 34 41 6 14 31 7 .nean n.moon March in C Add three lunations. l\ 15 26 4( 38 14 12 J 20 59 53 2 27 . 18 58 11 24 17 K 2 17 27 1 21 29 3 34 3 2 4$ By Table Fourth= 110 June 18 5 38 4 3 18 IS 51 2 11 44 13 J 1 4 16 First equation 3 54 4i Arg for 1st eqt. 1 30 55 18 1 39 C< 9 24 1$ 3 ly 18 51 2 10 13 18 2 10 13 It Arg. 2:1. eqlion. 3 r 4 16 Ar?.4 h eqi'u Third equation.- 18 11 3 21 2 54 1 8 5 33 Ar. 3d. 'qt'n 18 11 26 3 True time of New Mooi. at LONDON 18 11 24 2 -20 True time al JERUSALEM 18 13 20 24 The true time of New Moon in June, in the year of our Lord, 36, on the 19th day, at one hour, 20 minutes and 24 seconds, in the morning. The mean distance of the Sun being 3 signs, 1 de- gree, 4 minutes, and 16 seconds from the Moon's as- cending node, consequently there was no Eclipse at that time. Sec. 15 Precepts and Examples. 213 To calculate the true time of new, or full Moon, and also to know whether there will be an Eclipse at the time, in any given year and month before the Christian Era. Find a year in (he 18th Century from Table 1st. which being added to the given number of years before Christ diminished b> one, shall make a number of com- plete Centuries. Find ihis number of Centuries in Ta- ble seeond, and subtract tfte time, anomalies and dis- tanccs from the node belonging to it, from those of the mean new Moon in March, the* above found year, in the 18th Century, and the remainder will c'enote the time, and anomalies, LC. of the mean new Moon in M .irch, the given year,, before Christ, ; then for the true time thereof in any month of that year, proceed us be- fore directed, 214 Precepts and Examples. EXAMPLE V. Sec. 15 Required the true time of new Moon in May, Old Style, the year before Christ, 585, at ALEXANDRIA, in EGYPT. The year 584, added to 1716, make 2300, or 23 Centuries. Mean N. Moon in March. Sun's mean Anomaly. Moon's mean Anomaly. Sun's mean dis- tance from the Node. D H M S S D M S S D M S S D M S March, 1716 11 17 33 29 8 22 50 39 4 4 14 2 4 27 17 5 March, 2000 Do. 300. . . 27 18 9 1. 13 32 37 8 50 10 3 15 42 Ilfi fi f) 6 27 45 1 9R 99 H Subtract 1 lunation . . 40 18 52 56 1 29 12 44 3 18 53 29 6 19 2 1 48 25 49 8 26 07 1 40 14 New Moon, 2300,.... 11 5 58 53 11 J9 46 41 1 5 59 7 25 26 46 Which sub't. m!716 March, B.C. 585... Add 3 lunations .... 11 34 36 88 14 12 9 9 3 03 58 2 27 18 58 2 28 15 2 2 17 27 1 9 1 50 19 3 2 42 New Moon March 58,. First equation 28 1 46 45 1 37 22 56 Argt. 1st. eqi'n. 5 15 42 3 46 3 51 01 28 1 45 08 22 56 5 15 41 17 5 15 41 17 Argt, 2d. eq'tn. 3 51 01 Second equation 28 1 45 08 2 15 1 6 14 41 3i) Arg't. 3d. eqt'n. 5 15 41 17 3 51 1 Third equation 28 4 00 9 1 9 6 14 41 39 5 15 41 17 3 51 1 Argt. 4th. eqt'n. Fourth equation... 28 4 01 18 12 Clock slower.... 28 4 1 30 3 Time at LONDON Difference of longitude. 28 4 4 30 2 2 May 28 6 6 30 The true time of new Moon at ALEXANDRIA in May, 585 years before Christ, was on the 28th day, 6 hours, 6 minutes and 30 seconds, afternoon. The Sun being then only three degrees and 51 minutes from the Moon's ascending node ; was consequently eclipsed. The above Eclipse was central, and total in NORTH AMERICA at eleven o'clock in the morning; it also pas- sed centrally over the south parts of FRANCE and ITALY. The duration of total darkness being about 3 minutes. Sec. 15 Precepts and Examples. EXAMPLE VIII. 215 Required the true time of full Moon, at ALEXANDBIA in EGYPT, in September, Old Style, in the year 201 be- fore the Christian Era. 200 years added to 1800, make 2000, or 20 Centuries. By the PRECEPTS Mean n. Moon in March. Sun's mean Anomaly. Moon's mea n Anomaly. Sun's dis.frm the Moon's ascnd'g node D H M S S D M S S D M S S D M S March, 1800. Add 1 lunation. 13 22 17 29 12 44 3 8 23 19 55 29 6 19 10 7 52 S3 25 49 11 8 58 24 1 40 14 From the same, Subtract 2003 yr's. 42 13 06 20 27 18 9 19 9 22 26 14 8 50 11 3 41 36 15 42 4 38 38 6 27 45 m. n. moon b.C.201 Add 6 lunations. 14 18 57 01 177 4 24 18 9 13 36 14 5 24 37 56 10 17 59 36 5 4 54 3 5 6 53 38 6 4 1 24 191 n.m.Sept.b.C.201 Add \ lunation, 7 23 21 19 14 18 22 2 3 8 14 10 14 33 10 3 22 53 39 6 12 54 30 11 10 55 2 15 20 7 full moon Sep. 201 First equation. 22 17 43 21 3 52 6 3 22 47 20 Arg. 1st. eqt. 10 5 48 9 1 28 14 11 26 15 9 Time once eqt'd. Second equation. 22 13 52 15 8 25 4 3 22 47 20 10 4 19 55 10 4 19 55 Arg. 2d. eqt. 11 26 15 9 Time twice eqt'd. Third equation. 22 5 26 11 58 5 18 27 25 Arg. 3d. eqt. 10 4 19 55 11 26 15 9 Arg. 4th eqt- 99 ^ 9^ 1 3 12 true T.at LONDON 22 5 25 1 add for diffoflong 2 2 Add S. Clock. 22 7 27 1 7 33 True Time at ALEXANDRIA. 22 7 34 34 The true time of full Moon, at ALEXANDRIA in EGYPT in the year before Christ, 201, in September, was on the 22d day, 7 hours, 34 minutes, and 34 seconds, the actu- al time of opposition. The Sun being within three de- 216 Examples and Precepts Sec. 15 grees and 45 minutes of the Moon's ascending node, consequently the Moon was visibly eclipsed at that time at ALEXANDRIA. To calculate the true time of new or full Moon, and Eclipses in any given year ; and month after the 18th Century. Find a year of the same number in the 18th centu- ry with that of the year proposed, and take out the mean time, and anomalies Sfc, of new moon in March, old style, from table first for that year. Take so many years from table second, as when added to the above mentioned year in the 18th century will answer to the given year in which the new, or full moon is required; and take out the first new Moon with its anomalies &,c for these complete centuries. Add all these together, and then proceed as before directed, to reduce the mean to the true syzygy. It is however necessary to remember, to subtract a lu- nation with its anomalies, when the above said .addi- tion carries the new Moon beyond the 31st day of March, as in the following example. Sec. 15 Examples and Precepts. EXAMPLE IX. 217 Required the true time of New Moon in July, old style, 2180 at Washington. Four centuries added'* to 178O make 2180. New Moon in March Sun's mean anomalies Moon's mean anomalies. Sun's dis- tance from the node. By the precepts. D. H. M. S S. D. M. S. S. D. M. S. S. D. M. S. March 1780 add 400 years 23 23 1 44 17 8 43 29 9 4 18 13 13 24 1 21 7 47 10 1 28 10 18 21 1 6 17 49 Subtract 1 lunation 41 7 45 13 29 12 44 3 9 17 42 13 29 6 19 11 22 35 47 25 49 5 6 10 1 1 49 14 Mean time new Moon March 2180 add 4 lunations 11 19 1 10 118 2 56 12 129 8 18 35 54 3 26 25 17 10 26 46 47 3 13 16 2 4 5 29 47 4 2 40 56 tf. M. July 2180 First equation 7 21 57 22 1 3 39 15 1 11 Arg. Isteqt. 2 10 2 49 24 12 8 8 10 43 Time once equated Second equation 7 20 53 43 9 24 8 15 1 11 2 9 38 37 2 9 38 37 Arg. 2nd eqt 8 8 10 43 Arg. 4th eqt i j.i t. Third equation, 8 6 17 51 3 56 10 5 22 34 Arg. 3d eqt. \ i t M Fourth equation 8 6 21 47 1 8 H f 11 F. Clock 8 6 22 55 4 30 T. time at London Difference of long. 8 6 IS 25 5 8 True time at Wash- ington the present Capitol of the TJ. S. 8 1 10 25 The true of time New Moon, old style, will then be on the 8th day of July, 1 hour 10 minutes and 25 sec- onds after noon ; or the 22d day, at the same hour minute and second new style. 218 Examples and Precepts EXAMPLE IX. Sec. 15 Required the Sun's true place, March 20th, 1764 O. Stvle, at 22 hours, 30 minutes, 25 seconds past noon. In common reckoning, March 21st, at 10 hours 30 min- utes and 25 seconds in the morning. Sun's Long. Sun's Anmly SDMSJSDMS To the Radical year after Christ, 1701 9 20 43 50 6 13 1 Add complete years, 60; 27 1211 29 26 3'11 29 17 11 29 14 March, Bissextile days, 20 Hours, 22 -,- Minutes, 30 Seconds, 25 Sun's mean place at the given time, Add equation of the Sun's centre, from table 6 ? Sun's true place, That is ARIES, 12deg's. 10 minutes, 12sec'ds. 1 28 9 111 1 28 9 20 41 55 20 41 5 54 13 1 14 10 14 36 54 13 1 14 1 9 1 27 23 12 10 12 with which enter table 6., Sec. 15 Precepts and Examples. EXAMPLE XI. 219 Required the Sun's true place,October 23, (X Style, at 16 hours, 57 minutes past noon, in the 4008th year before Christ L which was the 4007th year before the year of his^ birth, and the year of the Julian period, 706. This' is supposed by some to be the very instant of the Creation. BY THE PRECEPTS. Sun's Long Sun's anomy S D M S S D M S 9 7 53 10 1 7 46 40 6 28 48 10 13 25 Remains for a new Radix, To which add C 900 8 6 30 6 48 36 16 5 26 8 29 4 54 22 40 12 39 26 2 20 8 15 23 11 21 37 11 29 15 11 29 15 8 29 4 22 40 12 39 26 2 20 Complete years, < 80 C 12 Hours, Minutes, Sun's mean place at the given time, 6034 3 4 5 28 3? 58 Subtract equation of the Sun's centre, Argt. eqt'n. Sun's centre 6000 Which was just entering the Sign LI^RA. 220 Concerning Eclipses of tin Sun and Moon. Sec. 15 Concerning Eclipses of the um & To find the Sun's true distance from the Moon's as- cending node, at the time of any given new or full Moon, and consequent!} to know whether there be an Eclipse at that time or not. The Sun's mean'distance from the Moon's ascending node, is the Argument for finding the Moon's fourth equation in the syzygies, and therefore it is taken into all the foregoing Examples, in finding the true times thereof. Thus aj the time of mean new Moon in March> 1764, Old Style, or April in the new, the Sun's mean distance from the ascending node, is signs, 35 min- utes, 2 seconds. [See Table First.] The descending node is opposite to the ascending one, and consequent- ly are exactly 6 signs distant from each other. When the Sun is within 17 degrees of either of the nodes at the time of new Moon ; he will be eclipsed at that time, as before stated, and at the time of full Moon, if ihe Sun be within 12 degrees of either node, she will be eclipsed. Thus we find from Table First, that there was an Eclipse of the Sun, at the time of new Moon, April 1st. at 30 minutes, 25 seconds after 10 in the morning at LONDON, New Style, when the old is reduced to the new, and the mean time reduced to the true. It will be found by the Precepts, that the true time of that new Moon is 50 minutes, 46 seconds later, than the mean time, and therefore we must add the Sun's motion from the node during that interval to the abov,e mean distance Os. 6d. 35m, 2s. which motion is found Sec. 15 Elements for Solar Eclipses. 221 in Table Twelfth, for 50 minutes and 46 seconds to be 2 minutes, 12 seconds, and to this apply the equation of the Sun's mean distance from the node in Table 13th, which at the mean time of new Moon, April 1st. 1764, is 9 signs, 1 degree, 26 minutes, and 20 seconds, and we shall have the Sun's true distance from the node at the true time of new Moon, as follows : Sun from node, s D M s At the mean time of N.Moon in April, 1764, 5 35 2 Sun's motion from node for 50 minutes, 2 10 For 46 seconds, 2 Sun's mean dist from node at true N. Moon, 5 87 14 Equation from mean dist from node, add, 250 Sun's true dist. from the ascending node, 7 42 14 Which being far within the above named limits of 17 degrees, the Sun was at that time eclipsed. The man- ner of projecting this or any other Eclipse, either of the Sun or Moon, will now be shown. SECTION SIXTEENTH. To Project an Eclipse of the Sun. To project an Eclipse of the Sun, we must from the Tables find the ten following Elements : 1st. The true time of conjunction of the Sun and Moon, and 2d. The semi-diameter of the earth's disk, as seen from the Moon, at the true time of conjunction ; which is equal to the Moon's horizontal parallax. 3d. The Sun's distance from the solstitial colure ? to which he is then nearest. 4th. The Sun's declination. . 5th. The angle of the Moon's visible path with the ecliptic. 6th. The Moon's latitude. 7th. The Moon's true horary motion from the Sun. 8th. The Sun's semi-diameter, 9th. The Moon's semi-diameter. 10th. The semi-diameter of the penmb ra. Sec. 16 Elements for Protracting Solar Eclipses. 223 EXAMPLE XII. Required the true time of New Moon at LONDON, in April, 1764, New Style, and also whether there were an Eclipse of the Sun or not at that time ; arid likewise the elements necessary for its protraction, if there were at that time an Eclipse. By the Precepts. mean time of New Moon in March. Sun's mean anomaly. Moons mean anomaly. Sun's mean dist'.from the node. D H M S S D M S S D M S S D M S March 1764 Add 1 lunation 2 8 55 36 29 12 44 3 8 2 20 29 6 19 10 13 35 21 25 49 11 4 54 48 1 40 14 Mean New Moon First equation 31 21 39 39 4 10 40 9 1 26 19 Arg 1st eqt'n 11 9 24 21 1 34 57 5 35 2 Second equation. 32 1 50 19 3 24 49 9 1 26 19 11 10 59 18 11 10 59 18 Arg 2nd eqt' Third equation 31 22 25 30 4 37 9 20 27 1 'Arg 3d eqt'n 11 10 59 18 5 35 2 Arg 4th eqt' 31 22 30 7 18 Sun from node True New Moon Equation of days 31 22 30 25 3 48 5 35 2 31 22 26 35 The true time is April 1st, 10 hours, 26 minutes, 35 seconds in the morning, tabular time. The mean dis- tance of the Sun at that time, being only 5 degrees, 35 minutes and 2 seconds past the ascending node, the Sun was at that time eclipsed. Now proceed to find the elements, necessary for its protraction. The true time being found as above. To find the Moon's horizontal parallax, or semi-di- ameter of the Earth's disk as seen from the Moon. 224 Declination of Solar Eclipses. Sec. 16 Enter Table 15th with the signs and degrees of the Moon's anomaly, (making proportion because the anomaly is in the Table calculated only to every 6th degree,) and fyom it take out the Moon's horizontal parallax, which for the above time is 54 minutes, and 53 seconds, answering to the anomaly of 11s. 9d. 24m. 21 seconds. To find the Sun's distance from the nearest solstice, namely, the beginning of Cancer, which is 3 signs, or 90 degrees from the beginning of Aries. It appears from Example 1st. for calculating the Sun's true place, the calculation being made for the same time, that the Sun's longitude from the beginning of Aries, was then Os. 12d. 10m. 12 seconds, that is, the Sun's place was then in Aries, 12 degrees, 10m. 12 seconds, therefore from s D M s 3000 Subtract the Sun's longitude or place. 12 10 12 Remains Sun's distance from the solstice, 2 17 49 48 Which is equal to 77 degrees, 49 minutes, 48 seconds, each sign containing 30 degrees. To find the Sun's declination, enter Table 5th with the signs and degrees of the Sun's true place, anomaly 10s. 2d. and making proportions for the 10m. 12 sec- onds, take out the Sun's declination, answering to his true place, and it will be found to be 4 degrees^49 min- utes north. To find the Moon's latitude, this depends on her true distance from her ascending node, which is the same as the Sun's true distance from it at the time of new Moon, Sec. 16 Declination of Solar Eclipses. 225 and is thereby found in Table 14th. But we have al- ready found, [see Example.] by calculation the Sun's true place, thatat the true time of new Moon in April, 1764, the Sun's equated distance from the node was Os. 7d. 42m. 14s. therefore enter Table 14th with the above equated distance, (making proportions for the minutes and seconds,) her true latitude will be found to be 40 minutes and 18 seconds north ascending. To find the Moon's horary motion from the Sun, with the Moon's anomaly, namely, 11s. 9d. 24m. 21s. enter Table 15th, and take out the Moon's horary mo- tion, which, by making proportions in that Table, will be found 30 minutes, 22 seconds. Then with the Sun's anomaly, namely, 9s. Id. 26m. 19s. (in the present case,) take out his horary motion, 2 minutes and 28 seconds from the same Table ; sub- tract the latter from the former, and the remainder will be the Moon's horary motion from the Sun ; namely, 27 minutes and 54 seconds. To find the angle of the Moon's visible path with the ecliptic. This in the projection of Eclipses, may be al- ways rated at 5 degrees, and 35 minutes without any sensible error. To find the semi-diameters of the Sun and Moon. These are found iu the same Table, [15] and by the same Argument, as their horary motions. In the pres- ent case, the Sun's anomaly gives his semi-diameter, 16 miuutes, and 6 seconds, and the Moon's anomaly gives her diameter 14rn. and 27 seconds. 226 Declination of Solar Eclipses: Sec, 1 fr To find the semi-diameter of the penumbra. Add the , Sun's semi-diameter to the Moon's, and their sum will be the semi-diameter of the penumbra ; equal to 31 minutes and 3 seconds. Collect these elements together, that they may the more readily be found when they are wanted in the construction of this Eclipse. Thus- - D H M s 1st. The true time of new Moon in April, 1 10 30 25 D M s 2d. Semi-diameter of the Earth's disk, 54 53 3d. Sun's distance from the nearest solistic, 77 49 48 4th. Sun's declination north, 4 49 5th. Moon's latitude, north ascending, 40 18 6th. Moon's horary motion from the Sun, 27 54 7th. Angle of Moon's visible path with ecliptic. 5 35 8th. Sun's semi-diameter. 016 6 9th. Moon's semi- diameter. 14 57 10th. Semi-diameter of the penumbra. 51 3 To project an Eclipse of the Sun Geometrically : Make a scale of any convenient length, A. C. and di- vide it into as many equal parts, as the Earth's semi- disk contains minutes of a degree 5 which, at the time of the Eclipse in April, 1764, was 54 minutes and 53 seconds ; then with the whole length of the scale as a radius, describe the semicircle A. M. B. upon the cen- tre C. which semi-circle will represent the northern half of the Earth's enlightened disk, as seen from the Sun, &ec. 16 Declination of Solar Eclipses. 227 Upon the centre C. rises the straight line, Ch. per- pendicular to the diameter, A. C. B. then will A. C. B. be a part of the ecliptic, and C. H. its axis. Being provided with a good sector, open it to the ra dius C. A. in the line of Chords, and taking from thence the chord of 23 degrees and 28 minutes in your com- passes, set it off both ways from H. to g. fy g. to h. in the periphery of the semi-disk, and draw the straight line g. V. h. in which the north pole of the disk will be always found. When the Sun is in Aries, Taurus, Gemini, Cancer, Leo and Virgo, the north pole of the Earth is enlight- ened by the Sun, but while the Sun is in the other six signs, the south pole is in the dark. When the Sun is in Capricorn, Aquarius, Pisces, Aries, Taurus and Gemini, the northern half of the Earth's axis, C. XII. P. lieslo the right hand of the axis of the ecliptic, as seen from the Sun ; and to the left hand, whilst the Sun is in the other 6 signs. Open * the sector, till the radius, [or distance of the two 90s] of the signs be equal to the length of V. h. and take the sign of the Sun's distance from the solis- tice 77 degrees, 49 minutes, and 48 seconds in your * To persons acquainted with Trigonometry, the angle contained be- tween the Earth's axis, and that of the ecliptic, may be found more ac- curately by calculation. RULE. As Radius is to the sine of the Sun's distance from the sol- stice, so is the tangent of the distance of the poles, (23 degrees and 28 minutes,) to the tangent of the angle contained by the axis. Then set off the chord of the angle, from H. to h. and join C.'H. which will cut F. G. in P. the place of the north pole. S28 Declination of Solar Eclipses, Sec. 16 compasses from the line of sines, and set off that dis- tance from V. to P. in the line of g. V. h. because the Earth's axis lies to the right hand of the axis of the ecliptic in this case, [the Sun being in Aries,] and draw the straight line C. XII. P. for the Earth's axis, of which P. is the north pole. If the Earth's axis had lain to the left hand from the axis of the ecliptic, the distance V. P. would have been set off from V. towards g. To draw the parallel of latitude of any given place, as suppose for LONDON in this case, or the path of that place on the Earth's enlightened disk, as seen from the Sun, from Sim-rise to Sun-set take the following method. Subtract the latitude of LONDON in this case, 51 de- grees and 30 minutes,from 90 degrees, and the remain- der 38 degrees and 30 minutes will be the co-latitude, which take in your compasses from the line of chords, making C. A, or C. B. the radius, and set it from h. to the place where the Earth's axis meets the periphery of the disk to VI. and VI. and draw the occult or dot- ted line VI. K. VI. then from the points where this line meets the Earth's disk set off the chord of the Sun's declination, [4 degrees and 49 minutes,] to A. and F. and to E. and G. and connect these points by the two occult lines. F. XII. G. and A. L. E. Bisect L. K. XII. in K. and through the point K. draw the black line VI. K. VI. then making C. B. the radius of a line of sines on the sector, take the co -lati- tude of LONDON, (38 and J degrees,) from the sines in Sec. 16 Declination of Solar Eclipses. 229 your compasses, and set it both ways from K. to VI. and VI. These hours will be just in the edge of the disk at the equinoxes, but at no other time in the whole year. With the extent K. VI. taken into your com- passes, set one foot in K. in the black line below the occult one as a centre, and with the other foot describe the semi-circle VI. 7. 8. 9. 1(X &c. and divide it into 12 equal parts ; then from these points of division,draw the occult 7. p. 8. 0. 9. n. parallel to the Earth's axis, C. XII. P. With the small extent K. XII. as a radius, describe the quadrantal arc XII. f. and divide it into six equal parts, as XII. a. a. b. be. cd. de. ef. and through the division points a. b. c. d. e. draw the occult lines VII. e. V.VIII. d. IV. IXC. III. X. b. II. and XI. a. I. all parallel to VI. K. VI. and meeting the former occult lines 7. p. 8. 0. #c. in the points VII. VIII. IX. X. XL V. IV. III. II. and I. which points will mark the several situations of LONDON on the Earth's disk at these hours respectively, as seen from the Sun, and the elliptic curve VI. VII. VIII. fyc. being drawn through these points, will represent the parallel of latitude, or path of LONDON on the disk as seen from the Sun from its rising to its setting. If the Sun's declination had been south, the diurnal path of LONDON would have been on the upper side of the line VI. K. VI. and would have touched the line D. L. E. in L. It is necessary to divide the hourly spaces into quarters, and if possible into minutes also. 230 Declination of Solar Eclipses. Sec. 16 Make C. B. the radius of a line of chords on the sec- tor, and taking therefrom the chord of 5 degrees and 35 minutes, (the angle of the Moon's visible path with the ecliptic ;) set it off from H. to M. on the left hand of C. H. (the axis of the ecliptic,) because the Moon's latitude in this case is north ascending. Then draw C. M. for the axis of the Moon's orbit, and bisect the angle M. C. H. by the right line C. Z. If the Moon's latitude had been north descending, the axis of her or- bit would have been on the right hand from the axte of the ecliptic. The axis of the Moon's orbit lies the same way when her latitude is south ascending, as when it is north as- cending, and the same way when south descending, as when north descending. Take the Moon's latitude (40 minutes and 18 sec- onds,) from the scale C. A. in your compasses, and set it from i. to x. in the bissecting line C. Z. making i. x. parallel to C. y. and through x. at right angles, to the Moon's orbit, (C. M.) draw the straight line N, w. x. y. s. for the path of the penumbra's centre over the Earth's disk. The point w. in the axis of the Moon's orbit, is, that, where the penumbra's centre approaches nearest to the centre of the Earth's disk, and consequently is the middle of the general Eclipse, The point x. is where the conjunction of the Sun and Moon falls, according to equal time, as calculated by the Tables, and the point y. is the ecliptical conjunction of the Sun and Moon. Sec. 1 6 Declination of Solar Eclipses. 23 1 Take the Moon's true horary motion from the Sun, (27 minutes and 54 seconds,) in your compasses, from the scale C. A. (every division of which is a minute of a degree,) and with that extent, make marks along the path of the penumbra's centre, ^nd divide each space from mark to niark, into 60 equal parts, or horary minutes by dots, and set the hours to every 60th -min- ute in such manner, that the dot signifying the instant of new Moon by the Tables, may fall into the point x. halfway between the axis of the Moon's orbit, and the axis of the ecliptic ; and then the remaining dots will be the points on the Earth's disk, where the penum- bra's centre is at the instants denoted by them, in its transit over the Earth. Apply one side of a square to the line of the penum- bra's path, and move the square backwards and for- wards, until the other side of it cuts the same hour and minute, (as at m. and m.) both in the path of the pe- numbra's centre, and the particular minute, or instant which the square cuts at the same time in both paths, will be the instant of the visible conjunction of the Sun and Moon, or the greatest obscuration of the Sun at the place for which the construction is made, (namely, LONDON- in this Example,) and this instant is at 47 minutes and 29 seconds past 10 o'clock in the morning, which is 17 minutes, 5 seconds later than the tabular time of true conjunction. Take the Sun's semi-diameter, (16 minutes and six seconds,) in your compasses, from the scale C, A. and setting one foot in the path of LONDON, at m. viz. at 232 Declination of Solar Eclipses. Sec. 16 47 minutes and thirty seconds past 10, with the other foot describe the circle U. Y. which will represent the Sun's disk as seen from LONDON at the greatest ob- scuration. Then take the Moon's semi-diameter, fourteen min- utes and 57 seconds in your compasses from the same scale, and setting one foot in the path of the penumbra's centre at m. in the 47 and J minute after 10, with the other foot describe the circle T. Y. for the Moon's disk, as seen from LONDON, at the time when the Eclipse is at the greatest, and the portion of the Sun's disk, which is hidden, or cut off by the disk of the Moon, will show the quantity of the Eclipse at that time, which quantity may be measured on a line equal to the Sun's diameter, and divide it into 12 equal parts for digits, which, in this Example,is nearly eleven digits. This Eclipse was annular at PARIS. Lastly, take the semi-diameter of the penumbra, 3 1 minutes and 3 seconds from the scale A. C. in your compasses, and setting one foot in the line of the penum- bra's path, on the left hand, from the axis of the eclip- tic, direct the other foot towards the path of LONDON, and carry that extent backwards and forwards, until both the points of the compasses fall into the same in- stants in both the paths, and these instants will denote the time when the Eclipse begins at LONDON, Proceed in the same manner on the right hand of the axis of the ecliptic, and where the points of the compasses fall into the same instants in both the paths, they will show at what time the Eclipse ends at LONDON. Sec. 16 Defoliation of Solar Eclipses. 233 d i&mter of th&TenWftCbrti .< kor&ry matter* fre** <* Su * Sun'* semi dt&mtter semi diameter Sec. 16 Defoliation of Solar Eclipses. 235 According to this construction, this Eclipse began at 20 minutes after 9 in the morning, at LONDON, at the points N. and O. 47 minutes and 30 seconds after 10, at the points m. and m. for the time of the greatest ob- scuration, and 18 minutes after 12, atE. and S. for the time when the Eclipse ends. In this construction, it is supposed that the angles under which the Moon's disk is seen during the whole time of the Eclipse, continues invariably the same, and that the Moon's motion is uniform, and rectilinear du- ring that time. But these suppositions do not exactly agree with the truth and therefore supposing the ele- ments given by the Tables to be accurate, yet the times and phases of the Eclipse deduced from its con- struction, will not answer to exactly what passes in the Heavens, but may be at least two or three minutes wrong, though the work may be done with the great- est care and attention. The paths also, of all places of considerable latitudes are nearer the centre of the Earth's disk as seen from the Sun, than those constructions make them ; be- cause the disk is projected as if the Earth were a per- fect sphere, although it is known to be a spheroid. The Moon's shadow will consequently go farther north- ward in all places of northern latitude, and farther southward in all places of southern latitude, than can be shown by any projection. SECTION SEVENTEENTH, The Projection of JLunar JEcltpses. WHEN the Moon is within 12 degrees of either of her nodes, at the time when she is full, she will be eclipsed, otherwise not, as before stated, Required the true time of full Moon, at LONDON, in May, 1762, New Style, and also whether there were an Eclipse of the Moon at that time or not. It will be found by the Precepts, that at the true time of full Moon in May, 1762, the Sun's mean dis- tance from the ascending node was only 4 degrees, 49 minutes and 36 seconds, and the Moon being then op- posite to the Sun, must have been just as near her de- scending node, and was therefore eclipsed. The ele- ments for the construction of Lunar Eclipses are eight in number, as follows : Sec. 17 Delini ation of Lunar Eclipses. 237 1st, The true time of full Moon. 2d. The Moon's horizontal parallax. 3d. The Sun's semi-diameter. 4th. The Moon's semi-diameter. 5th. The semi-diameter of the Earth's shadow at the Moon. 6th. The Moon's latitude. 7th. The angle of the Moon's visible path with the ecliptic. 8th. The Moon's true horary motion from the Sun. To find the true time of full Moon, proceed as di- rected in the Precepts, and the true time of full Moon in May, 1762, will be found on the 8th day, at 50 min- utes, and 50 seconds past 3 o'clock in the morning. To find the Moon's horizontal parallax, enter Table 15th with the Moon's mean anomaly, ^at the time of the above full Moon,) namely, 9s. 2d. 42m. 42 seconds, and with it take out her horizontal parallax, which, by making the requisite proportions will be found to be 57 minutes and 23 seconds. To find the semi-diameters of the Sun and Moon, enter Table 15th, with their respective anomalies, the Sun's being 10s. 7d. 27m. 45 seconds, and the Moon's 9s. 2d. 42m. 42 seconds, (in this case,) and with these take out their respective semi-diameters, the Sun's 15 minutes and 56 seconds, and the Moon's 15 minutes and 38 seconds. To find the semi-diameter of the Earth's shadow at the Moon, add the Sun's horizontal parallax, (which is always 9 seconds,) to the Moon's which in the pres- 238 Deliniation of Lunar Eclipses. cc. 17 ent case is 57 minutes and 23 seconds, the Sun will be 57 minutes and 32 seconds ; from which subtract the Sun's semi-diameter, 15 minutes and 56 seconds, and there will remain 41 minutes and 36 seconds for the semi-diameter of that part of the Earth's shadow, which the Moon then passes through. To find the Moon's latitude. Find the Sun's true distance from the Moon's ascending node, (as already taught,) in the first Example for finding the Sun's true place, at the true time of full Moon, and this distance increased by 6 signs, will be the Moon's true distance from the same node, and consequently the Argument for finding her true latitude. The Sun's mean distance from the ascending node was at the true time of full Moon, Os. 4d. 49m. 35 sec- onds; but it appears by the Example that the true time thereof, was 6 hours, 33 minutes and 38 seconds sooner, than the mean time, and therefore we must subtract the Sun's motion from the node during this interval, from the above mean distance Os. 4<1. 49m. and 35 seconds, in order, to have his mean distance from the node, at the time of true full Moon. Then, to this apply the equation of his mean distance from the node, found in Table 13th, by his mean anomaly, 10s. 7d. 27m. 45 seconds, and lastly, add six signs, and the Moon's true distance from the ascending node, will be found as follows ; Stc. 17 Deliniation of Lunar Eclipses. 239 S D M S Sun from node at mean time of full Moon, 4 49 35 {6 hours, 15 35 33 minutes, 1 26 38 seconds. 2 Subtract the sum, - - - - "- - - _ Remains his mean dist. at true full Moon, 4 32 32 Equation of his mean distance, add, 1 38 Sun's true distance from the node, 6 10 32 To which add, 6000 Moon's true distance from the node, 6 6 10 32 And it is the argument used to find her true latitude at that time. Therefore, with this Argument, enter Ta- ble 14th, making proportions between the latitudes belonging to the 6th and 7th degree of the Argument for the 10 minutes, and 32 seconds, and it will give 32 minutes and 21 seconds for the Moon's true latitude, which appears by the Table to be south descending. To find the angle of the Moon's visible path with the ecliptic. This may be always stated at 5 degrees and 35 minutes without any error of consequence, in the projection of either Solar or Lunar Eclipses. To find the Moon's true horary motion from the Sun. With their respective anomalies, take out their horary motions from Table 15th, and the Sun's horary motion, subtracted from the Moon's, leaves remaining the Moon's true horary motion from the Sun, in the present case, 30 minutes and 52 seconds. The above elements are collected for use, 240 Deliniation of Lunar Eclipses. Sec. 11 D H M s 1st. Tr ue time of F. Moon in May, 17628 3 50 50 D 2d. Moon's horizontal parallax, 57 23 3d. Sun's semi-diameter, 15 56 4th. Moon's semi-diameter, 15 38 5th. S. diameter of Earth's shadow at Moon, 41 36 6th. Moon's true latitude south descending. 32 21 7th. Angle of Moon's visible path with eclp'tc 5 35 8th. Moon's true horary motion from Sun, 30 52 These elements being found for the construction of the Moon's Eclipse in May, 1 762, proceed as follows : Make a scale of any convenient length, as W. X. and divide it into 60 equal parts, each part standing for a minute of a degree. Draw the right line A. C. B, for part of the ecliptic, and C A perpendicular thereto for the southern part of its axis, (the Moon having south latitude,) Add the semi-diameters of the Moon and Earth's shadow together, which in this case, make 57 minutes and 14 seconds ; and take this from the scale in your compasses, and setting one foot in the point C. as a centre, with the other describe the semi-circle S. D. B. in one point of which the Moon's centre will be at the beginning of the Eclipse, and the other at the end. Take the semi-diameter of the Earth's shadow, (41 minutes and 36 seconds,) in your compasses from the scale, and setting one foot in the centre C. with the other describe the semi-circle K. L. M. for the south- Sec. 17 Deliniation of Lunar Eclipses. 241 ern half of the Earth's shadow, because the Moon's lat- itude is south in this Eclipse. Make C. D. equal to the radius of aline of chords en the sector, and set off the angle of the Moon's visible path with the ecliptic, (5 degrees and 35 minutes,) fromD. to E. and draw the right line C. F. E. for the southern half of the axis of the Moon's orbit, lying to the right hand from the axis of the ecliptic C. A. be- cause the Moon's latitude is south descending in this Eclipse. It would have been the same way on the other side of the ecliptic, if her latitude had been north descending, but contrary in both cases, if her latitude had been either north, or south ascending. Bisect the angle A. C. E. by the right line C. g. in in which the true equal time of opposition of the Sun aud Moon falls, as found from the Tables. v Take the Moon's latitude, 32 minutes and 21 sec- onds, from the scale in your compasses, and set it from C. to G. in the line C. G. g. and through the point G. at ridit angles to C. F. E. draw the rfeht line P. H. G. o o o F. N. for the path of the Moon's centre. Then F. shall be the point in the Earth's shadow, where the Moon's centre is, at the middle of the Eclipse ; G. the point where her centre is at the tabular time of her being full ; and H. the point where her centre is,at the instant of her ecliptical opposition. Take the Moon's horary motion from the Sun, (30 minutes and 52 seconds,) in your compasses from the scale \V. X. and with that extent, make marks along the line of the Moon's path, P. G. N. then divide each 242 Declination of Lunar Eclipses, Sec. 17 space from mark to mark, into 60 equal parts, or hora- ry minutes, and set the hours to the proper dots in such manner, that the dots signifying the instant of full Moon, namely,(50 minutes and 50 seconds after 3 in the mor- ning,) may be in the point G. where the line of the Moon's path enters the line that directs the angle D. a E. Take the Moon's semi-diameter, 15 minutes and 38 seconds in your compasses from the scale, and with that extent, as a radius upon the points N. F. and P. as centres, describe the circle Q. for the Moon at the be- ginning of the Eclipse, when she touches the Earth's shadow at V. ; the circle R. for the Moon at the mid- dle, and the circle S. for the Moon at the end of the Eclipse, just leaving the Earth's shadow at W. The point N. denotes the instant when the Eclipse begins, namely, at 15 minutes and 10 seconds after two in the morning. The point F. the middle of the Eclipse at 47 minutes and 45 seconds after three, and the point P. the 3nd of the Eclipse, at eighteen minutes after five^ at the greatest obscuration, the Moon was ten digits eclipsed. The Moon's diameter, (as well as the Sun's,) is sup- posed to be divided into 12 equal parts, (called digits,) and so many of these parts as are darkened by the Earth's shadow, so many digits is the Moon eclipsed. All that the Moon is eclipsed above 12 digits, show how far the shadow of the Earth is over the body cf the Moon, on that edge, to which she is nearest, at the middle of the Eclipse, Sec. 17 Declination of Lunar Eclipses. 243 It is difficult to observe exactly, either the beginning or ending of a Lunar Eclipse, even with a good Tele- scope ; because the Earth's shadow is so faint, and ill- defined about the edges, that when the Moon is either just touching or leaving it, the obscuration^of her limb is scarcely sensible, and therefore the closest observers can hardly be certain to four or five seconds of time. But both the beginning and ending of Solar Eclipses, are instantaneously visible, for the moment that the edge of the Moon's disk touches the Sun's, his round- ness appears to be broken on that part, and the moment she leaves it, he appears perfectly round again. In Astronomy, Eclipses of the MoorTare of great use ' in ascertaining the periods of her motions, especially such Eclipses as are observed to be alike in all circum- stances, and have long intervals of time between them. In Geography, the longitude of places are found by Eclipses. The Eclipses of the Moon are more useful for this purpose, than those of the Sun ; because they are more frequeutly visible, and the same Lunar Eclipse is equally large, at all places where it is seen. In Chronology, both Solar and Lunar Eclipses serve to determine exactly the time of any past event,* for there are so many particulars observable in every Eclipse with respect to its quantity the places where it is perceivable, (if of the Sun,) and the time of the day, or night, that it is impossible that there can be two So- lar Eclipses in the course of many ages, which are alike in all circumstances. From the preceding explanation of the doctrine of Eclipses, it is evident that the dark- 244 Ddiniation of Lunar Eclipses. Sec. 17 ness at the CRUCIFIXION of our SAVIOUR was not occa- sioned by an .Eclipse of the Sun. For he suffered on the day on which the PASSOVER was eaten by the JEWS, namely, the thlrJ day of April, A. D. 33 : on that day it was impossible that the Moon's shadow could fall on the Earth's. For the JEWS kept the PASSOVER at the time of full Moon ; nor does the darkness in total Eclip- ses of the Sun, last above four minutes and six seconds in any place ; whereas the darkness at the CRUCIFIXION Jested three hours, and overspread, at least, ail the Land of JUDEA. / Sec. 17 Deliniatlon of Lunar Eclipses. 245 SECTION EIGHTEENTH. THE FIXED THE Stars are said to be fixed, because 'they have been generally observed to keep at the same distances from each other ; their apparent diurnal revolutions being caused solely by the Earth's turning on its axis. They appear of a sensible magnitude to the eye, because the retina is affected, not onl} by the rays of Jight which are remitted directly from them, but by many thou- sands more, which, falling upon our eye-lids, and upon the aerial particles about us, are reflected into our eyes so strongly, as to excite vibrations, not only in those points of the retina, where the real images of the stars are formed, but also in other points of some distance round. This makes us imagine the Stars to be much larger than they would appear, if we saw them only by the few rays which come directly from them, so as to enter our eyes, without being intermixed with oth- ers. Any person may be sensible of this, by looking at a Star of the first magnitude, through a long, narrow 24,8 Of the Fixed Stars. Sec. 1 8 tube, which, though it takes in as much of the sky as would hold a thousand such stars, yet scarcely renders that one visible. The more a telescope magnifies, the less is the aper- ture through which the star is seen ; and consequently the less number of rays it admits into the eye. The stars appear less in a telescope which magnifies 200 times, than they do to the naked eye ; insomuch that they seem to be only indivisible points ; it proves at once that the stars are at immense distances from us, and that they shine by their own proper light. If they shone by reflection, they would be as invisible -without telescopes, as the satellites of Jupiter. These eatclliics appear larger when viewed with a go;>d telescope, than any of the fixed stars. The number of stars discoverable in cither hr nil- sphere by the unaided sight, is not above a thousand. This at first, may appear incr< ditable : because they seem to herd most innumerable, but the deception ari- ses from our looking confusedly upon them without re- ducing them to any order : look steadfastly upon a large portion of the sky, and count the number of stars in if, and you will be surprised to find thorn so few. Consid- er only how seldom the Moon's passes between us and any star, (although there are as many about her path, as in any other parts of the Heavens ,) and you will soon be convinced that the stars are much thinner sown,- than you expected. The British catalogue, which, be- sides the stars visible to the naked eye, includes a great number which cannot be seen, without the assistance Sec. 18 Of the Fixed Star*. 249 of a telescope, contains no more than three thousand in bath hemispheres. As we have incomparably more light from the Moon, than from all the stars together, it is the greatest absurd- ity to imagine, that the stars were made for no other purpose than to cast a faint light upon the earth ; espe- cially, since many more require the assistance of a good telescope to find them out, than are visible without that instrument. Our Sun is surrounded by a system of plan- ets, and comets, all of which would be invisible from the nearest fixed star : And, from what we already know of the immense distance of the star?, the nearest may be computed at 32 billions of miles from us, which is far- ther than a cannon ball can fly in 7 millions of years, though it proceeded \\ ith the same velocity as at its first discharge. Hence it is easy to prove, that the un, seen from such a distance would appear no larger than a star of the first magnitude. From the foregoing observations ir is highly probable, thai each star is the centre of a Bftgm&cenl system of worlds, moving round it, though unseen by us, and are irradiated by its beams: espe- cially, as the doctrine of plurality of worlds is rational, a *id ^re.itly manifests the power, wisdom and goodness of the great Creator. The stirs, on account of their apparently various mignitudes have been distributed iutj several classes, or orders. Those which appear largest are called stars of the first magnitude, the next to them in lustre, stars of the second magnitude, and so on to the sixth, which are the smallest that are visible to the unaided sight. E* 250 Of the Fixed Stars. Sec. 18 This distribution, having been undo long before the in- vention of telescopes, die stars which cannot he se* n without the assistance of these instruments, are distin- guished by the name of telescopic stars. The'ancients divided the starry spheres info particu- lar constellations, or systems of stars, 5i ste-i.itions : And, Third, that region en the south side of the ZoJiffc, containing 15 constellations. There is a remarkable track around the Heavens, called the Galaxy, or Milky Way from its peculiar whiteness. It was formerly thought to be owing to a vast number of very small stars, closely CGiinected,anil the observation of Dr. Her^chel have fully confirmed the opinion. lie therefore considers the Galaxy as a very extensive brandling congeries of many millions of stars, which prcbibly owes its origin to several re- markable large, as well as very closely scattered small stars, that may have drawn together the rest. OX GROUPS OF STARS. Groups of Stars, succeed to clustering Stars in Dr. I erschel's arrangement. A group is a collection of Stars, clcsely, and almost equally compressed, and of any iigure or outline. There is no particular conden- Kiii.ni oft!).' Star,; to in lijate tho exis:e:i.;c of a central f/rce, ami ll.e; r q-s i^re jiifikitnlly fcparatcd from neighboringptars, to show that they form peculiar sys- terns of tkeirown. 252 Of the Fixed Stars. Sec. 1 S ON CLUSTERS OF STARS, Dr. Herschel regards Clusters of Stars as the most magnificent objects in the Heavens. They differ from groups in their beautiful and artificial arrangement. Their form is generally round, and their condensation is such as to produce a mottled lustre, somewhat re- sembling a nucleus. The whole appearance of a clus- ter indicates the existence of a central force, residing either in a central body, or in the centre of gravity of the whole system. Sec. 18 Interrogations for Section Eighteenth. 253 Interrogations for Section Eighteenth. What is a fixed Star 1 Why do they appear of sensible magnitude to the eye? Do the Stars appear larger when viewed through a telescope, than viewed with the eye only 1 What does it prove ? Which appear the largest, the satellites of Jupiter, or the Stars, when viewed with a telescope ? About how many Stars in a clear night can be seen by the naked eye ? How many in the British catalogue ? Are some of that number telescopic 1 Would the planets and comets of the Solar System be invisible from the nearest Star ? At how many miles distant may we with propriety, suppose the nearest fixed Star ? 254 Interrogations far Section Eighteenth. Fee. 18 How long would a cannon ball be in flying that dis- tance, supposing it should continue to move with the same velocity, as at its first discharge ? Of what size would the Sun probably appear from the nearest fixed Star ? Is it not probable that every the centre of a ? Qn~3Eiat accoun.t have they beei. ;uted into classes-? Whatare those called which appear largest ? What are Constellations 1 What is the use of dividing them into Constella- tions ? How many Constellations on the celestial globes ? What is the Zodiac 1 How many Constellations in the Zodiac ? What is the breadth of the Zodiac ? What the Galaxy, or Milky \\ ay ? \\ hat is a Group of Stars 1 What are Clusters of Stars 1 SECTION NINETEENTH. AC "3" XT O7 THE GREGORIAN, OR NEW STYLF, TCGETZIEH WITH SDM5 CHRONOLOGICAL PilOELSMP, FOR FINDING THE EPACT, GOLDEN NUMBI?, DOMINICAL LETTER, &C. POPE GREGORY THIRTEENTH, made a reformation of the calendar. r i he Julian calender, (or Old Style,) had before that time, been in general use all over Europe. The year, according to the Julian calendar consists of 365 days and 6 hours, which 6 hours being 4 part of a day, the common years consisted of 365 days : and every fourth year, one day was added to the month of February, which made each of thope years consist of 336 days, commonly called leap years, This computation ear the truth,) is mre than the Solar year, by 1 1 minutes and 3 seconds, whi ;!>, in 13] ill E [uirn fen da tim^of the pener .:ii of N^re, liefd in (;. p 3.,.^ of the Christian Era, to the time of Pope Gregory, who therefore caused ten days to be taken cut of the 250 Of the Gregorian Calendar. Sec. 19 month of October, 1582, to make the Equinox full on the 2 1st of March, as it dU at the time of that council ; and to prevent the like variation for the future, he or- dered that three days should be abated in every four hundred years, by reducing the leap year at the close of each century, for three successive centuries, to common ysars, and retaining the leap year at the close of each fourth century only. This, at that time, was esteemed as exactly conformable to the true solar year. But since that time, the true solar year is found to consist of 365 days, 5 hours, 43 minutes arid 49 sec- onds, which in 50 centuries will make another day's variation. Though the Gregorian Calendar, (or New Sts !e,) ha:l long been in^ use throughout the greatest part of Europe, it did not take place in GREAT BRITAIN and AMERICA, till the first of January, 1752, and in Sep- tember following, the 1 1 days were adjusted by calling the third day of that month, the fourteenth, and contin- uing the rest in their order. CHRONOLOGICAL PROBLEMS. As there are three leap years to be abated in every four centuries, to find which century is to be leap year and which not. RULE. Cat off two cyphers from the given year, and divide the remaining figures by 4, if nothing re- main, the year will be leap year. The year 4 i^^r there being a remainder of 3, it will not be leap year. But the year *~p^ will. Sec. 19 Chronological Problems. 257 To find the Dominical or Sunday Shelter. RULE. To the given year, add its fourth part, re- jecting remainders, divide the sum by 7, and if there be no remainder, A is the Sunday letter ; but if any number remains, then the letter standing under that number, is the Dominical letter, and the day of the week on which the year commences. A leap year has two Dominical letters, the first of which commences the year, and continues to the 24th of February, and the other to the end of the year. EXAMPLE. 1234557 Required the Dominical letters for the years 1832. ^ 4)1832 Jt * | &% 458 "I H 1| 1 % Days in a week 7)2290 j -o jC - So* i c 327 1 _ 0123456 The year 1832, was leap year, A G F E D C B according to the work, the re- mainder being 1, the first Sunday letter was A, and G was the second,the year also commenced on Sunday. To find tlie Golden Number. R ULE . Add 1 to the given year, divide the sum by 19, and the remainder will be the Golden Number : if nothing remain, then 19 will be the number sought. Required the Golden Number for the year 1832, To the given year 1832 Add _ 1 19)1833(96 171 123 114 Golden Number, 9 258 Chronological Problems &*e. 19 Toitsid RULE. Subtract 1 from the Golden Number, divide the remainder by 3, if 1 remain, add 10 to the dividend, the sum will be the Epact ; if nothing remain, the divi- dend is the Epact. Required the Epact for the year 1832. The Golden Number, as found above, is 9, therefore, subtract l,and the remainder is 8, divide 8 by 3, and the quotient is two, and 2 remains, multiply this remainder by 10, and the product is 20, to which add the dividend, and the sum is 29, the Epact for 1832. To find the year of the Dionysian Period. RULE. Add to the given year 457, divide the sum by 532, and the remainder will be the number required. Required the year of the Dionysian Period, for the year 1832. To the given year, 1832. Add 457 532)2289(4 2128 161 =Dionysian Period. To find tiie Julian Period. RULE. Add 4713 to the given year, and the sum will be the Julian Period. Required the Julian Period for the year of the Christian Era, 1832. 1832 4713 6545 year of the Julian period. *To find tiie Cyelo cf tlio STZSI, Goldca Kwmljer, aad indiction.for any Current Year* Rule. To the current year add 4713, divide the sum by 28, 19, and 15, respectively, and the several remainders will be the numbers required. If nothing remains, the divisors are the required numbers. * A Cycle is a perpetual round, or circulation ci the mine parts ol lime of an} 7 sort. The Cycle of the Sun, is a revolution of 28 years, in which the days of the mouths return again to the same days of the Sec. 19 Chronological Problems. 259 Required the Cycle of the Sun, Golden Number, r Indiction for the year 1832. 1832 4713 56 94 84 1832 4713 1832 4713 19)6545(344 67 84 76 Io)6545(433 60 ~oT 45 80 76 95 SO 105 84 2i=ivycle of the, Sun 2=Golden Number 5=lndiction. T find on vrSiot day Easier vrSll Ir.appeu. It was ordered by the Nicene Council, that Easter Sunday, should be kept on the first Sunday after the 1st full Moon which happened upon,or after the twenty- first day of March, the day on which they thought the Vernal Equinox happened, though this was a mistake, for the vernal equinox that year fell on the 20th of March ; but yet, the full Moon which fell on, or next after the twenty-first of March, they called the Paschal full Moon ; and by the introduction of the Gregorian, or New Style, the equinox will new always happen on the twentieth, or twenty-first of March : and if the full Moon happen on a- Sunday, Easter Day is to be the next Sunday after. Therefore, find the time of the next full Moon after the 21st. of March, and the following Sunday is Easter. of the week, the Sun's place to the same signs and degrees of the ecliptic, on the same months and days, so as not to differ one degree in an hundred years, and the leap years begin the same course over again, with respect to the days of the week, on which the days oflho months fall. The Cycle of the Moon, (commonly called the Golden Number,) is a revolution of 19 years, in which the conjunctions, oppositions, and other aspects of the Moon, are within an hour and a half, of being the same as they were, on the same days of the months 19 years before. The indiction is a revolution of 15 years", used only by the ROMANS, for indicating the times of certain payments made by the subjects of the REPUBLIC : It was es- tablished by CONSTANTINE, A. D. 312, 260 Of the Fixed Stars. Sec. 19 Iill|l||ll|li|lf!l1ll|!l|lll COCCCO-OOCOCOCOCOCO'^^^^rrrhT^-rH^TfiO^^^lOO t O QDCOOOGOCX)OOOOQOOOOOOOOOOOQOOOOOOOOOaOOOGOGOOOGOOOCOD i ^ ^ I-H nn n r 1 04 CM i| . 1 o.'* CO CO Ol 1,024 6,4 5 Pallas 80 20,99 265,000,000 279,100 > 5 l 6,55 Jupiter 89,170 490,000,000 520,279 39 37 Saturn 79,042 900,000,000 954,072 18 16 Herschel 35,112 1,800,000,000 1,908,352 3 54 4 TABULAR VIEW OF THE SOLAR SYSTEM. If AMES OF THE Tropical Revo- lutions. Sydereal Revo- lutions. Place of Aphe- lion in Jan. 1800 Planets. Sun S I) M S D II 31 S D II M S Mercury 87 23 14 32 87 23 15 34 8 14 20 50 Venus 224 16 41 27 224 16 49 10 10 7 59 1 Earth 365 5 48 49 365 6 9 12 9 8 40 12 Moon Mars 686 22 18 27 686 23 30 35 5 2 24 4 Vesta 1155 4 2 9 42 53 Juno 1588 I 7 29 49 33 Ceres 1681 I 4 25 57 15 Pallas 1703 16 48 J10 1 7 Jupiter 4330 14 39 2 4332 14 27 10! 6 11 8 20 Saturn 10746 19 16 15 10759 1 51 11 8 29 4 11 Herschel 130637 400 i 30737 18 11 16 30 31 CONTENTS. PAGK. Astronomy in General, 7 Interrogations for Section First, 15 Description of the Solar System, 17 of Mercury, 19 of Venus, 20 Transits of Mercury, 22 of Venus, 25 Description of the Earth, 25 of the Moon, 26 of Mars, 28 of Vesta, 30 of Juno, 31 of Ceres, 3t of Pallas, S3 of Jupiter, of Saturn, 37 of Herschel, or Uranus, of Comets, 42 Interrogations for Section Second, On Gravity, 53 Interrogations for Section Third, 57 Phenomena of the Heavens, as seen from different parts of the Earth, 59 Interrogations for Section Fourth, 67 Physical Causes of the Motions of the Planets, Interrogations for Section Fifth, [ 76 Of Light, 78 Of Ai?, 81 Interrogations for Section Sixth, To find the distances of the Planets from the Sun, 87 Interrogations for Section Seventh, Of the Equation of Time, > Of the precession of the Equinoxes, Of the Moon's Phases, Of the Phenomena of the Harvest Moon, *y|> Interrogations for Section Ninth, On Tides, }J Interrogations for Section Tenth, *** Astronomical Problems, On Eclipses, } Interrogations for Section Twelfth, *?J On the Construction of theJAstronomical Tables, y>y Interrogations for Section Thirteenth, *,*4 Directions for the Calculation of Eclipses, Astronomical Tables, ** Examples for the Calculation of Eclipses, 2yt> To find tho Sun's true place, **" To find tho Sun's true distance from the Moon s ascending node, ~-J To project an Eclipse of the Sun, Projection of Lunar Eclipses, _^ On the Fixed Stars, i; ' On Groups of Stars, ^ Clusters of Stars, , . E Interrogations for Section Eighteenth, ~ An account of the Gregorian, or New Style, g g Chronological Problems. Ostrander, planetarium and O r\ 1 \ calculator QB43 08 1832 M288338 THE UNIVERSITY OF CALIFORNIA LIBRARY UK*