. < \mtm\i No Division Range Shelf. Received... PRESENTED TO THE If I li mm of tie University of California UNIVERSITY EDITION. REVIStl) AND ENLARGED. TREATISE ASTRONOMY. DESCRIPTIVE. THEORETICAL AND PHYSICAL, DESIGNED FO* SCHOOLS, ACADEMIES, AND PRIVATE STUDENTS. BY H. N. ROBINSON. A. M.. ffMtM2LY PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY ; AUTHOR Ol A TREATISE ON ARITHMETIC, iLGEBRA, GEOMETRY, TRIGONOMBTRT, SURVEYING. CALCULUS. NATURAL PHILOSOPHY. <&C. <&C. NEW YORK: IVISON, PHINNEY, BLAKEMAN & CO. CHICAGO: S. C. GRIGGS & 00. 1866. Entered, according to act of Congress in the year 1849, BY HORATIO N ROBINSON, In the Clerk's Offier or the District Court of the United States, for the District of Ohio. Entered according to act of Congress, in the year 1857, BY H. N. ROBINSON, U the Clerk's Office -If the District Court of the United States for the Northern District of Mew York. PREFACE. To give at once a clear explanation of the design and in- tended character of this work, it is important to state that its author, in early life, imbibed quite a passion for astronomy, and, of course, he naturally sought the aid of books ; but, in this field of research, he was really astonished to find how little substantial aid he could procure from that source, and not even to this day have his desires been gratified. Then, as now, books of great worth and high merit were to oe found, but they did not meet the wants of a learner ; the substantially good were too voluminous and mathematically abstruse to be much used by the humble pupil, and the less mathematical were too superficial and trifling to give satis- faction to the real aspirant after astronomical knowledge. Of the less mathematical and more elaborate works on as- tronomy there are two classes the pure and valuable, like the writings of Biot and Herschel ; but, excellent as these are, they are not adapted to the purposes of instruction ; and every effort to make class books of them has substantially failed. From the other class, which consists of essays and popular lectures, little substantial knowledge can be gathered, for they do not teach astronomy ; as a general thing, they only glorify it; they may excite our wonder concerning the im- mensity or grandeur of the heavens, but they give us no ad- ditional power to investigate the science. Another class of more brief and valuable productions were, and are always to be found, in which most of the important facts are recorded ; such as the distances, magnitudes, and mo- tions of the heavenly bodies; but how these facts became known is rarely explained : this is what the true searcher after science will always demand, and this book is designed ex- pressly to meet that demand. In the first part of the book we suppose the reader entirely unacquainted with the subject ; but we suppose him compe- tent to the task to be, at least, sixteen years of age to have a good knowledge of proportion, some knowledge of algebra, geometry, and trigonometry and then, and not until then, can the study be pursued with any degree of success worth mentioning. Such a person, and with such acquirements as iii iv PREFACE. PRKJMCK. we have here designated, we believe, can take this book and learn astronomy in comparatively a short time; for the chief design of the work is, to teach whoever desires to learn : and it matters not where the learner may be, in a college, academy, school, or a solitary student at home, and alone in the pursuit. The book is designed for two classes of students the well prepared in the mathematics, and the less prepared ; the for- mer are expected to read the text notes, the latter should omit them. With the text notes, we conceive it, or rather designed it to be, a very suitable book to give sound elemen- tary instruction in astronomy ; but we do not offer the work as complete on practical astronomy ; for whoever becomes a practical astronomer will, of course, seek the aid of complete and elaborate sets of tables, such as would be improper to insert in a school book. We have inserted tables only for the purpose of carrying out a sound theoretical plan of instruction, and, therefore, we have given as few as possible, and those few in a very con- tracted form. The epochs for the sun and moon may be ex- tended forward or backward, to any extent, by any one who understands the theory. The chapters on comets, variable stars, &c., are compila- tions, and are printed in smaller type ; and the works to which we are most indebted, are Hcrschel's Astronomy and the Cambridge Astronomy, originally the work of M. Biot. Other parts of the work, we believe, will be admitted as mainly original, by all who take pains to examine it. The chief merits claimed for this book are, brevity, clear- ness of illustration, anticipating the difficulties of the pupil, and removing them, and bringing out all the essential points of the science. Some originality is claimed, also, in several of our illustra- tions, particularly that of showing the rationale of tides rising on the opposite sides of the earth from the moon ; and in tho general treatment of eclipses ; but it is for others to deter- mine how much merit should be awarded for such originali- ties; we have, however, used greater conciseness and per- spicuity in general computations than is to be found in most of the books on this subject ; and this last remark will apply to the whole work. PREFACE TO THE REVISED EDITION. THE author and publisher of this volume have tlio satisfaction of p RKFACE knowing, that all teachers, who are really qualified to teach Astron-~ omy, and who have examined this hook, hold it in the highest estim- ation ; and they only regret that so few pupils are prepared to profit by it. This being more complimentary than discouraging, we resolved to make the book as useful as possible, in the hope that it will aid in imparting essential knowledge of Astronomy to many of the youths of the United States. With this object in view we have increased the present edition by fifty pages of very practical and important matter. In the former editions we were careful not to give more than could be received, and the subject of solar eclipses was not exhausted. In the Nautical Almanacs, a rude map generally represents the parts of the earth over which tho solar eclipses will be visible, and curved lines and loops mark tlu places where the eclipse will commence at sunrise, and end with sunset, noy todies, describes their appearances, determines their magni- defined - tudes, and discovers the laws which govern their motions. When we merely state facts and describe appearances a"s The dm. they exist in the heavens, we call it Descriptive Astronomy. 8I< When we compute magnitudes, determine distances, record observations, and make any computations whatever, we call it Practical Astronomy. The investigation of the laws which govern the celestial motions, and the explanation of the causes which bring about the known results, is called Physical Astronomy. When the mariner makes use of the index of the heavens, Nautical to determine his position on the earth, such observations, and M their corresponding computations, are called Nautical Astro- nomy. By nautical astronomy we determine positions on the Geography earth, and subsequently, the magnitude of the earth ; and thus, we perceive, that Geography and Astronomy must be linked together ; and no one can fully understand the former science, without the aid of the latter. Astronomy is the most ancient of all the sciences, for, in The ami- the earliest age, the people could not have avoided observing the successive returns of day and night, and summer and winter. They could not fail to perceive that short days cor- respon ded to winter, and long days to summer ; and it was thus, probably, that the attentions of men were first drawn to the study of astronomy. , 11 1* ASTRONOMY. iKxaoDtrc, i n tins work, we shall not take facts unless they are within Facu aion the sphere of our own observations. We shall not perempto- aot science. ^ gtate ^^ ^ Q earth, is 7912 miles in diameter; that the moon is about 240,000 miles from the earth, and the sun 95,000,000 of miles ; for such facts, alone, and of themselves, do not constitute knowledge, though often mistaken for knowledge. We shall direct the mind of the reader, step by step, through the observations and through the investigations, so that he can decide for himself that the earth must be of such a mag- nitude, and is thus far from the other heavenly bodies ; and that will be knowledge of the most essential kind. Th fowl. Al} astronomical knowledge has its foundation in observa- Mtamomfcai ** on ' an< * ^ e ^* st object of this book shall be to point out knowledge, what observations must be taken, and what deductions must be made therefrom ; but the great book which the pupil must study, if he would meet with success, is the one which spreads out its pages on the blue arch above ; and he must place but secondary dependence on any book that is merely the work of human art. As we disapprove of the practice of throwing to the reader astounding astronomical facts, whether he can digest them or not, and as we are to take the inductive method, and to lead the student by the hand, we must commence on the supposi- tion that the reader is entirely unacquainted even with the common astronomical facts, and now for the first time seriously brings his mind to the study of the subject ; but we shall suppose some maturity of mind, and some preparation, by the acquisition of at least respectable mathematical knowledge, c-onren- Every science has its technicalities and conventional terms ; and definu an< ^ astronomy is by no means an exception to the general rule ; and as it will prepare the way for a clearer understand- ing of our subject, we now give a short list of some of the technical terms, which must be used in our composition. Horizon. Every person, wherever he maybe, conceives himself to be in the center of a circle; and the circumference of that circle is where the earth and sky apparently meet. That circle is called the horizon. INTRODUCTION. 13 Altitude. The perpendicular bight from the horizon, measured by degrees of a circle. Meridian. An imaginary line, north and south from any point or place, whether it is conceived to run along the earth or through the heavens. If the meridian is conceived to divide both the earth and the heavens, it is then considered as a plane, and is spoken of as the plane of the meridian. Poles. The points where all meridians come together : poles of the earth the extremities of the earth's axis. Zenith. The zenith of any place, is the point directly Poiei f overhead ; and the Nadir is directly opposite to the zenith, or tie homOB * under our feet. The zenith and nadir are the poles to the horizon. Verticals. All lines passing from the zenith, perpendicu- ( Pn vw lar to the horizon, are called ^ 7 erticals, or Vertical Circles. tlcal> The one passing at right angles to the meridian, and striking the horizon at the east and west points, is called the Prime Vertical. Azimuth. The angular position of a body from the meri- dian, measured on the circle of the horizon, is called its Azi- muth. The angular position, measured from its prime vertical, is called its Amplitude. The sum of the azimulh and amplitude must always make 90 degrees. Equator. The Earth's Equator is a great circle, east and west, and equidistant from the poles, dividing the earth into two hemispheres, a northern, and a southern. The Celestial Equator is the plane of the earth's equator conceived to extend into the heavens. equator. When the sun, or any other heavenly body, meets the Equ celestial equator, it is said to be in the Equinox, and the tw * equatorial line in the heavens is called the Equinoctial. Latitude. The latitude of any place on the earth, is its distance from the equator, measured in degrees on the meridian, either north or south. If the measure is toward the north, it is north latitude; if toward the couth, south latitude. 4 ASTRONOMY. **roprc. The distance from the equator to the poles is 90 degrees one- fourth of a circle; and we shall know the circumference of the whole earth, whenever we can find the absolute length of one degree on its surface. Co-Latitude. Co-latitude is the distance, in degrees, of any place from the nearest pole. The latitude and co-latitude ( complement of the latitude) must, of course, always make 90 degrees. Parallels Parallels of latitude are small circles on the surface of the of latitude ear th, parallel to the equator. Every point, in such a circle, has the same latitude. Longitude. The longitude of a place, on the surface of the earth, is the inclination of its meridian to some other meridian which may be chosen to reckon from. English astronomers and geographers take the meridian which runs through Greenwich Observatory, as the zero meridian. TL firt Other nations generally take the meridian of their princi- ridiaii ar- p a j observatories, or that of the capital of their country, ag the first meridian; but this is national vanity, and creates only trouble and confusion : it is important that the wholt world should agree on some one meridian, from which to reckon longitude ; but as nature has designated no particular one, it is not wonderful that different nations have chosen different lines. w adopt j n this work, we shall adopt the meridian of Greenwich a f Green- tne zero ^ ne f longitude, because most of the globes and wich ; and maps, and all the important astronomical tables, are adapted wh7 1 to that meridian, and we see nothing to be gained by chang- ing them. Declination. Declination refers only to the celestial equa- tor, and is a leaning or declining, north or south of that line, and is similar to latitude on the earth. Solstitial Points. The points, in the heavens, north and south, where the sun has its greatest declination. The northern point we call the Summer Solstice, and the southern point the Winter Solstice; the first is in longitude 90, the other in longitude 270. As latitude is reckoned north and south, so longitude is INTRODUCTION. | 6 reckoned east and west ; but it would add greatly to syste- INTRUDUO. matic regularity, and tend much to avoid confusion and am- i mp rov. biguity in computations, were this mode of expression aban- ment rag. doned, and longitude invariably reckoned westward, from to s e * ted * 360 degrees. Latitude and longitude, on the earth, doeer not corre- Latitude, spond to latitude and longitude in the heavens. Latitude, on ^'^ tTt a- the earth, corresponds with declination in the heavens ; and cension. longitude, on the earth, has a striking analogy to right ascen- sion in the heavens, though not an exact correspondence. We shall more particularly explain latitude, longitude, and right ascension in the heavens, as we advance in this work ; for it is only when we are forced to use these terms, that the nature and spirit of their import can be really understood. There are other technicalities, and terms of frequent use, other term* in astronomy, such as Conjunction, Opposition, Retrograde, "* explaiap Direct, Apogee, Perigee, &c., &c., all of which, for the sake of simplicity, had better not be explained until they fall into use ; and, once for all, let us impress this fact on the Hinds of our readers, that we shall put far more stress on the substance ind spirit of a thing, than on its name. 6 ASTRONOMY. SECTION I. CHAPTER I. PRELIMINARY OBSERVATIONS. CHAT, i. To commence the study of astronomy, we must observe and call to mind the real appearances of the heavens. Take such a station, any clear night, as will command an extensive view of that apparent, concave hemisphere above us, which we call the sky, and fix well in the mind the direc- tions of north, south, east, and west. The appa- At first, let us suppose our observer to be somewhere in f the Stan. tn United States, or somewhere in the northern hemisphere, about 40 degrees from the equator. As yet, this imaginary person is not an astronomer, and neither has, nor knows how to use, any astronomical instru- ment ; but we would have him mark with attention the po- sitions of the heavenly bodies. ( 1. ) Soon he will perceive a variation in the position of the stars : those at the east of him will apparently rise ; those at the west will appear to sink lower, or fall below the hori- zon ; those at the south, and near his zenith, will apparently move westward; and those at the north of him, which he may see about half way between the horizon and zenith, will appear stationary. Apparent Let such observations be continued during all the hours ^ ^ n ig n *> an ^ for several nights, and the observer cannot fail to be convinced that not only all the stars, but the sun, moon, and planets, appear to perform revolutions, in about twenty-four hours, round & fixed point; and that fixed point, as appears to us (in the middle and northern part of the United States ), is about midway between the northern hori- zon and the zenith. ft gnou i J always be borne in mind, that the sun, moon, and clro! "' gtars, have an apparent diurnal motion round a fixed point, PRELIMINARY OBSERVATIONS. i7 and all those stars which are 90 degrees from that point, CHAP. i. apparently describe a great circle. Those stars that are nearer to the fixed point than 90 degrees, describe smaller circles; and the circles are smaller and smaller as the objects are nearer and nearer the fixed points. ( 2. ) There is one star so near this fixed point, that the small circle it describes, in about 24 hours, is not apparent from mere inspection. To detect the apparent motion of this star, we must resort to nice observations, aided by ma- thematical instruments. This fixed point, that we have several times mentioned, is The North the North Pole of the heavens, and this one star that we have just Star * mentioned, is commonly called the North Star, or the Pole Star. (3.) This star, on the 1st of January, 1820, was 1 39' PoUoof 6" from the pole, and on 1st of January, 1847, its distance * he Nort * from the pole was 1 30' 8"; and it will gradually and more slowly approach within about half a degree of the pole, and afterward it will as gradually recede from the pole, and finally cease to be the polar star. We here, and must generally, speak of the star, or the stars, The poll as in motion ; but this is not so. The fixed stars are abso- m motl u ' iuiety fixed ; it is the pole itself that has a slow motion among the stars, but the cause of this motion cannot now be ex- plained; it is one of the most abstruse points in astronomy, and we only mention it as a fact. As the North Star appears stationary, to the common ob- server, it has always been taken as the infallible guide to direction ; and every sailor of the ocean, and every wanderer of the African and Arabian deserts, has held familiar ac- quaintance with it. ( 4. ) If our observer now goes more to the southward, and changes oi makes the same observations on the apparent motions of the a PP earanr stars, he will find the same general results ; each individual , othwaid. star will describe the same circle ; but the pole, the fixed point, will be lower down, and nearer the northern horizon ; and it will be lower and lower in proportion to the distance the observer goes to the south. After the observer has gone sufficiently far, the fixed point, the pole, will no longer be up 2) 18 ASTRONOMY. CHAP. I. in the heavens, but down in the northern horizon ; and when Appear- the pole does appear in the horizon, the observer is at the aace from equator, and from that line all the stars at or near the equa- the eqn*tor. ^ appear to rise up directly from the east, and go down directly to the west; and all other stars, situated out of the equator, describe their small circles parallel to mis perpendi- cular equatorial circle. South of jf ^ e O k server g 0es south of the equator, the apparent north pole of the heavens sinks below the northern horizon, and the south pole rises up into the heavens at the south. Changes in / 5 \ jf fa e observer should go north, from the first appearance . . , -i i on going station, m place of going south, the north pole would rise north nearer to the zenith ; and, should he continue to go north, he would finally find the pole in his zenith, and all the stars would apparently make circles round the zenith, as a center, and parallel to the horizon ; and the horizon itself would be the celestial equator. ( 6. ) When the north pole of the heavens appears at the zenith, the observer must then be at the north pole, on Q n the o] s t of June, the sun declines about 23 1 de- length of the year. grees from the equator toward the north ; and, of course, to us in the northern hemisphere, its meridian altitude is so much greater, and the horizontal shadows it casts from the same fixed objects will be shorter; and the same meridian altitude and short shadow will not occur again until the fol- lowing June, or after the expiration of one year. Thus, we see, that the time of the stars coming on to the meridian, and the declination of the sun, have a close corre- spondence, in relation to time. Fixed In all our observations on the stars, we notice that their Is tennis a PP aren * relative situations are not changed by their diurnal applied. motions. In whatever parts of their circles they are observed, or at whatever hour of the night they are seen, the same con- figuration is recognized, although the same group, in the different parts of its course, will stand differently, in respect to the horizon. For instance, a configuration of stars resem- bling the letter A, when east of the meridian, will resemble the letter V, when west of the meridian. Wander- As the stars, in general, do not change their positions in respect to each other, they are called fated stars; but there are a few important stars that do change, in respect to other stars; and for that reason they become especial objects of attention, and form the most interesting portion of astro- nomy. j n t^ ear ii es t ages, those stars that changed their places, were called wandering stars; and the.? were subsequently found to be the planetary bodies of the lolar system, like the earth on which we live. stari * APPEARANCES IN THE HEAVENS. CHAPTER II. APPEARANCES IN THE HEAVENS. IN the preceding chapter we have only called to mind the CHAP. u. most obvious and preliminary observations, which force them- selves on every one who pays the least attention to the subject. We shall now consider the observer at one place, making more minute and scientific observations. (17.) We have already remarked, that if the observer HOW to were on the equator, the poles, to him, would be in his horizon. tu " de ^ e f * h ' e If he were at one of the poles, for instance, the north pole, the place of oi> equator would then bound the horizon. If he were half Way serv&tion - between the equator and one of the poles, that pole would appear half way between the horizon and the zenith. Therefore, by observing the altitude of the pole above the hori- zon, we determine the number of degrees we are from the equator, which is called the latitude of the place. (18.) To carry the mind of the reader progressively along, in astronomy, we must now suppose that he not only has the use of a good clock, but has also some instrument to measure angles. Clocks and astronomical instruments progressed toward perfection in about the same ratio as astronomy itself; but, as we are investigating or leading the young mind to the in- vestigation of astronomy, and not making clocks or mathe- matical instruments, we therefore suppose that the observer has all the necessary instruments at his command, and we may now require him to make a correct map of the visible heavens ; but to accomplish it, we must allow him at least one year's time, and even then he cannot arrive at anything like accuracy, as several incidental difficulties, instrumental errors, and practical inaccuracies, must be met and overcome. (19.) There are three principal sources of error, which Source. f must be taken into consideration, in making astronomical observations. 1. Uncertainty as to the exact time. 2. Inex- tion. 24 ASTRONOMY. CM^P. fl. pertness and want of tact in the observer ; and 3. Imperfec- tion in the instruments. Everything done by man is neces- sarily imperfect. Practical " It may be thought an easy thing," says Sir John Her- scne ^ " by one unacquainted with the niceties required, to of error. turn a circle in metal, to divide its circumference into 360 equal parts, and these again into smaller subdivisions, to place it accurately on its center, and to adjust it in a given position ; but practically it is found to be one of the most difficult. Nor will this appear extraordinary, when it is con- sidered that, owing to the application of telescopes to the purposes of angular measurement, every imperfection of struc- ture or division becomes magnified by the whole optical power of that instrument; and that thus, not only direct errors of workmanship, arising from unsteadiness of hand or imperfec- tion of tools, but those inaccuracies which originate in far more uncontrollable causes, such as the unequal expansion and contraction of metallic masses, by a change of tempera- ture, and their unavoidable flexure or bending by their own weight, become perceptible and measurable." Ncessary ( 20.) The most important instruments, in an observatory, nt8 ' aside from the clock, are a circle, or sector, for altitudes ; and a transit instrument. The former consists of a circle, or a portion of a circle, of firm and durable material, divided into degrees, at the rate of 360 to the whole circle. Each degree is divided into equal parts; and, by a very ingenious mechanical adjustment of an index, called a Vernier scale, the division of the degree is practically (though not really) subdivided into seconds, or 3600 equal parts. The whole instrument must now be firmly placed and ad- justed to the true horizontal ( which is exactly at right angles to a plumb line ), and so made as to turn in any direction. With this instrument we can measure angles of altitude. The tran. ( 21.) The transit instrument is but a telescope, firmly fas- tened on a horizontal axis, east and west, so that the telescope itself moves up and down in the plane of the meridian, but an never be turned aside from the meridian to the east or APPEARANCES IN THE HEAVENS. 25 Transit Instrument. Meridian Wires. To place the instrument in this posi- tion, is a very difficult matter ; but it is a difficulty which, at present, should not come under consideration: we simply conceive it so placed, ready for observa- tions. " In the focus of the eyepiece, and at right angles to the length of the tele- scope, is placed a system of one horizontal and five equidis- tant vertical threads or wires, as represented in the annexed figure, which always appear in the field of view when properly illuminated, by day by the light of the sky, by night by that of a lamp, intro- duced by a contrivance not necessary here to explain. The place of this system of wires may be altered by adjusting screws, giving it a lateral (horizontal) motion; and it is by this means brought to such a position, that the middle one of the vertical wires shall inter- sect the line of collimation of the telescope, where it is arrested and permanently fastened. In this situation it is evident that the middle thread will be a visible representation of that portion of the celestial meridian to which the telescope is pointed ; and when a star is seen to cross this wire in the telescope, it is in the act of culminating, or passing the celes- tial meridian. The instant of this event is noted by the clock or chronometer, which forms an indispensable accom- paniment of the transit instrument. For greater precision, the moment of its crossing each of the vertical threads is noted, and a mean taken, which ( since the threads are equi- distant ) would give exactly the same result, were all the observations perfect, and will, of course, tend to subdivide and destroy their errors in an average of the whole." ( 22. ) Thus, all prepared with a transit instrument and a clock, we fix on some bright star, and mark when it comes to the meridian, or appears to pass behind the central wire of the instrument. By noting the same event the next evening, the next, and the nexc, we find the interval to be very sensi- OHAP. A line in the transi'. instrument visible neri- dian. Practical artifices, to attain accu- racy. Intervals between the fixed stars passing the meridian al ways con slant. 26 ASTRONOMY. CHAP. II. bly less than 24 hours ; but the intervals are equal to each other ; and all the fixed stars are unanimous in giving equal intervals of time between, two successive transits of the same star, if measured by the same clock. The following observations were actually taken by M. Arago and Laeroix, in the small island of Formentera, in the Mediterranean, in December, 1807. Date of Observations. Time of transit of the Star * Arietis. Intervals between successive Transits. 1807, Dec. 24, " 25, " 26, " 27, " 28, h. m. s. 9 42 32.36 9 41 29.70 9 40 26.72 9 39 23.90 9 38 21.38 h. m. s. 23 58 57.34 23 58 57.02 23 58 57.18 23 58 57.48 f measure for time. These intervals between the transits agree so nearly, that it is very natural to suppose them exactly equal, and the small difference of the fraction of a second to arise from some slight irregularities of the clock, or imperfection in making the observations. The equality of these intervals is not only the same for all the fixed stars, in passing the meridian, but they are the same in passing all other planes. standard Now as this has been the universal experience of astrono- mers in all ages, it completely establishes the fact, that all the fixed stars come to the meridian in exactly equal inter- vals of time ; and this gives us a standard measure for time, and the only standard measure, for all other motions are variable and unequal. Time of Again, this interval must be the time that the earth the earth's em pi ovg j n turning on its axis; for if the star is fixed, it is a revolution on J & .... ts axis. mark for the time that the meridian is in exactly the same position in relation to absolute space. M. Arago't ^ 23.) That the reader may not imbibe erroneous impres- sions, we remark, that the clock used for the preceding ob- servations, made by M. Arago and Lacroix, ran too fast, if it was a common clock, and too slow, if it was an astronomical APPEARANCES IN THE HEAVENS. $7 aock. It was not mentioned which clock was used, nor was CHAP. 11. it material simply to establish the fact of equal intervals ; nor was it essential that the clock should run 24 hours, in a mean solar day : it was only essential that it ran uniformly, and marked off equal hours in equal times. If it had been a common clock, and ran at a perfect rate, the interval would have been 23 h. 56 m. 4.09 s. ( 24.) In the preceding section we have spoken of an An astro. astronomical clock. Soon after the fact was established that "J^. cm the fixed stars came to the meridian in equal times, and that interval less than 24 hours, astronomers conceived the idea of graduating a clock to that interval, and dividing it into 24 hours. Thus graduating a clock to the stars, and not to the sun, is called a sidereal, and not a solar, or common clock ; and as it was suggested by astronomers, and used only for the purposes of astronomy, it is also very appropriately called mi astronomical clock; but save its graduation, and the nicety of its construction, it does not differ from a common clock. With a perfect astronomical clock, the same star will ^>ass the To deter. meridian at exactly the same time. from one year's end to an- of an astro- clock minetherat other* If the time is not the same, the clock does not run _ __ Sidereal time-has been slightly modified since the discovery of the precession of the equinoxes, though such modification has never been distinctly noticed in any astronomical work. At first, it was designed to graduate the interval between iwo suc- cessive transits of the same star over the meridian, to 24 hours, and to call this a sidereal day ; which, in fact, it is. But it was necessary, in some way, to connect sidereal with solar time ; and, to secure this end, it was determined to commence the side- real day (not. from the passage of any particular star across the meri- dian, but from the passage of the imaginary point in the heavens, wbero the sun's path crosses the vernal equinox, called the first point of Aries), thus making the sidereal day and the equinoctial year commence at the same moment of absolute time. For some time, it was supposed that the interval between two suc- cessive transits of the first point of Aries, over the meridian, was the same as two successive transits of a star ; but the two intervals are not identical; the first point of Aries has a very slow motion westward among the stars, which is called the precession of the equinox, and 28 ASTRONOMY. ii. to sidereal time; and the variation of time, or the difference between the time when the star passes the meridian, and the time which ought to be shown by the clock, will determine the rate of the clock. And with the rate of the clock, and its error, we can readily deduce the true time from the time shown by the face of the clock. Solar days ( 25. ) When we examine the sun's passage across the meridian, and compare the elapsed intervals with the sidereal clock, we find regular and progressive variations, above and below a mean period, that cannot be accounted for by errors of observation. The mean interval, from one transit of the sun to another, or from noon to noon, when we take the average of the whole year, is 24 hours of solar time, or 24 h. 3m. 56.5554s. of sidereal time ; but, as we have just observed, these intervals are not uniform; for instance, about the 20th of December, they are about half a minute longer, and about the 20th of September, they are as much shorter, than the mean period. The un From this fact, we are compelled to regard the sun, not as must have a fj xe( j po int ; it must have motions, real or apparent, inde- real or appa- r . J x rent motion, pendent of the rotation of the earth on its axis. ( 26. ) When we compare the times of the moon passing the meridian, with the astronomical clock, we are very forcibly struck with the irregularity of the interval. General The least interval between two successive transits of the Motion of moon ^ w hich m ay be called a lunar day ), is observed to be about 24 h. 42 m. ; the greatest, 25 h. 2m. ; and the mean, or average, 24 h. 54m., of mean solar time. These facts show, conclusively, that the moon is not a which makes its transits across the meridian a fraction of a second shorter than the transits of a star. The time required for 366 transits of a star across the meridian, in ( 3".34), three seconds and thirty-four hundredths of a second of sidereal time, greater than for 366 transits of the equinox. This difference would make a day in about 25000 years. The time elapsed between two successive transits of the equinox being now called a sidereal day of ..... 24h. Om. Os., the time between the transits of the same star, is - 24 h. m. 0.00916 r Every astronomer understands Art. (24) with this modification. APPEARANCES IN THE HEAVENS. 29 fixed body, like a fixed star, for then the interval would be CHAP, it 24 hours of sidereal time. But as the interval is always more than 24 hours, it shows that the general motion of the moon is eastward among the stars, with a daily motion varying from IQi to 16 degrees,* traveling, or appearing to travel, through the whole circle of the heavens ( 360 ) in a little more than 27 days. Thus, these observations, however imperfectly and rudely Chief ob- taken, at once disclose the important fact, that the sun and ' ect moon are in constant change of position, in relation to the stars, and to each other ; and, we may add, that the chief object and study of astronomy, is, to discover the reality, the causes, the nature, and extent of such motions. ( 27. ) Besides the sun and moon, several other bodies Othel were noticed as coming to the meridian at very unequal in- w" tervals of time intervals not differing so much from 24 bodies, sidereal hours as the moon, but, unlike the sun and moon, the intervals were sometimes more, sometimes less, and some- times equal to 24 sidereal hours. These facts show that these bodies have a real, or appa- rent motion, among the stars, which is sometimes westward, sometimes eastward, and sometimes stationary ; but, on the whole, the eastward motion preponderates ; and, like the sun and moon, they finally perform revolutions through the hea- vens from west to east. Only four such bodies ( stars ) were known to the ancients, Wandering namely, Venus, Mars, Jupiter, and Saturn. 8tars ^ These stars are a portion of the planets belonging to our dents, solar system, and, by subsequent research, it was found that Mode the Earth was also one of the number. As we come down to more modern times, several other planets have been disco- vered, namely, Mercury, Uranus, Vesta, Juno, Ceres, Pallas, and, very recently ( 1846), the planet Neptune.^ * Four minutes above 24 hours corresponds to one degree of arc. t We have not mentioned the names of these planets in the order in which they stand in the system, but rather in the order of their dis- covery. As yet, we have really no idea of a planet, or a planetary system. 30 ASTRONOMY. CHAP. n. "We shall again examine the meridian passages of the sun, moon, and planets, and deduce other important facts con- cerning them, besides that of their apparent, or real motions among the fixed stars. observa. (28.) But let us return to the fixed stars. We have determine severa l times mentioned the fact, that the same star returns the meridian to the same meridian again and again, after every interval of tii" itar" f ^ sidereal hours. So two different stars come to the meri- dian at constant and invariable intervals of time from each other ; and by such intervals we decide how far, or how many degrees, one star is east or west of another. For instance, if a certain fixed star was observed to pass the meridian when the sidereal clock marked 8 hours, and another star was ob- served to pass at 9, just one sidereal hour after, then we know that the latter star is on a celestial meridian, just 15 degrees eastward of the meridian of the first mentioned star. Coireipon- As 24 hours corresponds to the whole circle, 360 degrees, >n h *** t ^ iere ^ ore one hour corresponds to 15 degrees ; and 4 minutes, nd degrew. i a time, to one degree of arc. Hence, whatever be the ob- served interval of time between the passing of two stars over the meridian, that interval will determine the actual difference of the meridians running through the stars ; and when we know the position of any one, in relation to any celestial meri- dian, we know the positions of all whose meridian observations have been thus compared. night as- The position of a star, in relation to a particular celestial ** n " on ' meridian, is called Right Ascenvw, and may be expressed either in time or degrees. Astronomers have chosen that It is true, we might mention every fact, and every particular re- specting each planet ; such as its period of revolution, size, distance from the sun, &c. ; but such facts, arbitrarily stated, would not convey the science of astronomy to the reader, for they can be told alike to the man and to the child to the intellectual and to the dull to the learned and to the unlearned. To constitute true knowledge to acquire true science the pupil must not only know the fact, but how that fact was discovered, or de- duced from other facts. Hence we shall mainly construct our theories from observations, as we pass along, and teach the pupil to decide the case from the facts, evidences, and circumstances pr< sented REFRACTION. 31 meridian, for the first meridian, which passes through the CHAP. n. sun's center at the instant the sun crosses the celestial equa- First mert tor in the spring, on the 20th of March. dian - Right ascension is measured from the first meridian, east- ward, on the equator, all the way round the circle, from to 300 degrees, or from h. to 24 h. The reason why right ascension is not called longitude will be explained hereafter. ( 29. ) If we observe and note the difference of sidereal T* find the time between the coming of a star to the meridian, and the ^ n l g J^ ^ coming of any other celestial body, as the sun, moon, planet, sun, moon, or comet, such difference, applied to the right ascension of the an p a " star, will give the right ascension of the body. But every astronomer regulates, or aims to regulate, his sidereal clock, so that it shall show Oh. Om. Os. when the equinox is on the meridian ; and, if it does so, and runs regu- larly, then the time that any body passes the meridian by the clock, will give the right ascension of the body in time, with- out any correction or calculation; but, practically, this is never the case : a clock is never exact, nor can it ever run exactly to any given rate or graduation. We have thus shown how to determine the right ascensions of the heavenly bodies. We shall explain how to find their positions in declination, in the next chapter. CHAPTER III. REFRACTION. POSITION OP THE EQUINOX, AND OBLIQUITY OP THE ECLIPTIC HOW FOUND BY OBSERVATION. ( 30. ) To determine the angular distance of the stars from CHAP. lu the pole, the observer must first know the distance of his zenith from the same point. As any zenith is 90 degrees from the true horizon, if the observer can find the altitude of the pole above the horizon ASTRONOMY CHAP.^III. ( \vhich is the latitude of the place of observation ), he, of course, knows the distance between the zenith and the pole. Prepara. ^ g fa e nor t n p O i e j s b u t an imaginary point, no star being tions for de- ' * lexinining there, we cannot directly observe its altitude. But there is a the latitude verv bright star near the pole, called the Polar Star, which, as all other stars in the same region, apparently revolves round the pole, and comes to the meridian twice in 24 sidereal hours; once above the pole, and once below it; and it is evident that the altitude of the pole itself must be midway between the greatest and least altitudes of the same star. provided the apparent motion of the star round the pole is really in a circle ; but before we examine this fact, we will show how altitudes can be taken by th\i mural circle. (31.) The mural, or wall circle, is a large me- tallic circle, firmly fas- tened to a wall, so that its plane shall coincide with the plane of the me- ridian. A perpendicular line through the center, ZN, (Fig. 2), represents the zenith and nadir points ; and at right angles to this, through the center, is the horizontal line, Hli. A telescope, Tt, and an index bar, li, at right angles to rrve men- ^ & telescope, are firmly fixed together, and made to revolve dian alti- . on the center of the mural circle. The circle is graduated from the zenith and nadir point? each way, to the horizon, from to 90 degrees. When the telescope is directed to the horizon, the index points, / and i, will be at Z and N, and, of course, show of altitude. When the telescope is turned perpendicular to Z, the index bar will be horizontal, and indicate 90 degree* of altitude. When the telescope is pointed toward any star, as in tho REFRACTION. 33 figure, the index points, /and i, will show the position of the CH\P. in. telescope, or its angle from the horizon, which is the altitude of the star. As the telescope, and index of this instrument, can revolve Mural cu. freely round the whole circle, we can measure altitudes with ^ nsi a t ls in * it equally well from the north or the south ; but as it turns stmment. only in the plane of the meridian, we can observe only meri- dian altitudes with it. This instrument has been called a transit circle, and, says Sir John Herschel, " The mural circle is, in fact, at the same time, a transit instrument ; and, if furnished with a proper system of vertical wires in the focus of its telescope, may be used as such. As the axis, however, is only supported at one end, it has not the strength and permanence necessary for the more delicate purposes of a transit ; nor can it be veri- fied, as a transit may, by the reversal of the two ends of its axis, east for west. .Nothing, howover, prevents a divided circle being permanently fastened on the axis of a transit instrument, near to one of its extremities, so as to revolve with it, the reading off being performed by a microscope fixed on one of its piers. Such an instrument is called a transit circle, or a meridian circle, and serves for the simulta- neous determination of the right ascensions and polar dis- tances of objects observed with it ; the time of transit being noted by the clock, and the circle being read off by the late- ral microscope." ( 32.) To measure altitudes in all directions, we must have A!tii* \i . ,./, .. /. ,7 . and azimwUi another instrument, or a modification of this. instrument Conceive this instrument to turn on a perpendicular axis, parallel to Z N, in place of being fixed against a wall ; and conceive, also, that the perpendicular axis rests on the center of a horizontal circle, and on that circle carries a horizontal index, to measure azimuth angles. This instrument, so modified, is called an altitude and azi- muth instrument, because it can measure altitudes and azi- muths at the same time. ( 33.) After astronomy is a little advanced, and the angu- lar distance of each particular star, sun, moon, and planet, 3 34 ASTRONOMY. CHAP. HI. from the pole is known, then we can determine the latitude by The lati- observing the meridian altitude of any known celestial body ; n Fig 3. EF, c (Fig. 3 ), represent different strata of the earth's at- mosphere. Let 8 be a star, and conceive a line of light to pass from the star through the va- rious strata of air, to the ob- server, at 0. When it meets the first strata, as E F, it is slightly bent downward; and as the air becomes more and more dense, its increases t! n ,. . , titudes. retracting power becomes greater and greater, which more and more bends the ray. But the direction of the ray, at the point where it meets the eye of the observer, will deter- mine the position of the star as seen by him. Hence the observer at will see the star at s', when its real position is at s. As a ray of light, from any celestial object, is bent down- 36 ASTRCNOMY. CHAP. ill. ward, therefore, as we may see by inspecting the figure, the altitude of all the heavenly bodies is increased by refraction. This shows that all the altitudes, taken as described in Art. 33, must, be apparent altitudes greater than true alti- tudes and the resulting latitudes, deduced from them, al 1 too great. The object is now to obtain the amount of the refraction corresponding to the different altitudes, in order to correct or allow for it. To determine the amount of refraction, we must resort to observations of some kind. But what sort of observations will meet the case ? How to Conceive an observer at the equator, and when the sun or find the a- . mount of re- a s * ar passes through, or very near his zenith, it has no re- fraction cor- fraction. But, at the equator, the diurnal circles are per- to'eve '"de P en dicular * * ne horizon ; and those stars which are very gree of niti- near the equator, really change their altitudes in proportion to tnde the time. Now a star may be observed to pass the zenith, at the equator, at a particular moment : four hours afterward ( side- real time ), the zenith distance of this star must be 4 times 15, or 60 degrees, and its altitude just 30 degrees. But, by ob- servation, the altitude will be found to be 30 1' 38". From this, we perceive, that 1' 38" is the amount of refraction corresponding to 30 degrees of altitude. In six sidereal hours from the time the star passed the zenith, the true position of the star would be in the horizon ; but, by observation, the altitude would be 33' 0", or a little more than the angular diameter of the sun. Amount From this, we perceive, that 33' 0" is the amount of re- of horizontal /, ,. , ,- -, redaction fraction at the horizon. Thus, by talcing observations at all intervals of time, between the zenith and the horizon, we can determine the refraction corre- sponding to every degree of altitude. ( 35. ) In the last article, we carried the observer to the equator, to make the case clear ; but the mathematician need not go to the equator, for he can manage the case wherever matlcian>8 ASTRONOMICAL REFRACTION. 37 be may be he takes into consideration the curves, as men- CHAP. m. tioned in Art. 33. If it were not for refraction, the curves round the pole would be perfect circles, and the mathematician, by means of z + method of the altitude and azimuth, which can be taken at any and finding the every point of a curve, can determine how much it deviates amount of re- from a circle, and from thence the amount of refraction, or nearly the amount of refraction, at the several points. By using the refraction thus imperfectly obtained, he can correct his altitudes, and obtain his latitude, to considerable accuracy. Then, by repeating his observations, he can fur- ther approximate to the refraction. In this way, by a multitude of observations and computa- ti :>ns, the table of refraction ( which appears among the tables of every astronomical work ) was established and draVn out. ( 36. ) The effect of refraction, as we have already seen, is Refraction to increase the altitude of all the heavenly bodies. There- ^^J*^ fore, by the aid of refraction, the sun rises before it otherwise light would, and does not set as soon as it would if it were not for refraction ; and thus the apparent length of every day is increased by refraction, and more than half of the earth's sur- face is constantly illuminated. The extra illumination is equal to a zone, entirely round the earth, of about 40 miles in breadth. As the refraction in the "horizon is about 33' of a degree, the length of a day, at the equator, is more than four minutes longer than it otherwise would be, and the nights four minutes less. At all other places, where the diurnal circles are oblique to the horizon, the difference is still greater, especially if we take the average of the whole year. In high northern latitudes, the long days of summer are Effects in very materially increased, in length, by the effects of refrac- b 'g h lati< tion ; and near the pole, the sun rises, and is kept above the horizon, even for days, longer than it otherwise would be, owing to the same cause. Refraction varies very rapidly, in its amount, near the hori- 38 ASTRONOMY. f CHAP, ui. zon; and this causes a visible distortion of both sun and moon, just as they rise or set. D ^ rt ^ For instance, when the lower limb of the sun is just in the and moon in horizon, it is elevated, by refraction, 33'. the horizon. But the altitude of the upper limb is then 32', and the refraction, at this altitude, is 27' 50", elevating the upper limb by this quantity. Hence, we perceive, that the lower limb is elevated more than the upper ; and the perpendicular diameter of the sun is apparently shortened by 5' 10", and the sun is distinctly seen of an oval form, which deviates more from a circle below than above. An optical The apparently dilated size of the sun and moon, when near the horizon, has nothing to do with refraction : it is a mere illusion, and has no reality, as may be known by apply- ing the following means of measurement. Roll up a tube of paper, of such a size and dimensions as just to take in the rising moon, at one end of the tube, when the eye is at the other. After the moon rises some distance in the sky, observe again with this tube, and it will be found that the apparent size of the moon will even more than fill it. The reason of this illusion is well understood by the stu- dent of philosophy; but we are now too much engaged with realities to be drawn aside to explain illusions, phantoms, or any Will-o'-the-wisp. When small stars are near the horizon, they become invi- sible ; either the refraction enfeebles and dissipates their light, or the vapors, which are always floating in the atmosphere, serve as a cloud to obscure them. Appiicatiom (37.) Having shown the possibility of making a table of refraction corresponding to all apparent altitudes, we can now, by applying its effects to the observed altitudes of the cir- cumpolar stars, obtain the true latitude of the place of obber- vation. Let it be borne in mind, that the latitude of any plao.e on the earth, is the inclination of its zenith to the plane of the equator ; which inclination is equal to the altitude of the pole above the horizon. We demonstrate this as follows. Let E ( Fig. 4 ) rtpre- ASTRONOMICAL REFRACTION. 3 of the first meridian, or zero line, from which to reckon. But astronomers have agreed to take that meridian for the zero meridian, which passes through the sun's center the instant the sun comes to the celestial equator, in the spring ( which point on the equator is called the equinoctial point ) ; but the difficulty is tojind exactly where ( near what stars ) this meridian line is. Before we can define this line, we must take obser- vations on the sun, and determine where it crosses the equa- tor, and from the time we can determine the place. But be- fore we can place much reliance on solar observations, we must ask ourselves this question. Has the sun any parallax? 42 ASTRONOMY. CHAP, ill, that is, is the position of the sun just where it appears to be? Is it really in the plane of the equator, when it appears to be there ? Paralla*. T;O a n northern observers, is not the sun thrown back on the face of the sky, to a more southern position than the one it really occupies? Undoubtedly it is; and this change of position, caused by the locality of the observer, is called paral- lax; but, in respect to the sun, it is too small to be considered in these primary observations. The early astronomers asked themselves these questions, and based their conclusions on the following consideration : Sun's pa- If the sun is materially projected out of its true place ; if it is raiiax msen- thrown to the southward, as seen bv a northern observer, it eible, in com- . . monobserva- will cross the equator in the spring sooner than it appears U, expressed in time. To facilitate the computation, continue E S to P, making SP=QN, and draw the dotted line P Q. Then SP Q tf is a parallelogram. EP=IV 29".4+12' 12"=23' 41".4; and the two triangles, P E Q and SE T, are similar; there- fore we have PE : EQ : : SE : E*r. To have the value of E T, in time, E Q must he taken in time ; which is 3 m. 38.37 s. Hence, (23'41".4) : (3 m - 38.37 s -) : 11'29".4 : Ev. The result gives, E is - - 1 45.91 Therefore, V Q is - - -1m. 52.46 But qp Q is the right ascension of the sun at apparent noon, EQUINOCTIAL POINT. 45 at Greenwich, on the 21st of March, 1846; a very important CHAP. in. elen? int. The right ascension of any heavenly body, whether it be HOW to sun. moon, star, or planet, is the true sidereal time that it fiml tlie ab .. solute rig lit passes the meridian ; and now, as we have the error of the ascension ot clock, we can determine the true sidereal time that any star the star *. passes the meridian, and, of course, its right ascension / thus, ,' for example, If a star passed the meridian at - 10 h. 15 m. 47 s. Error of the clock is (subtractive) 1 12 23 Eight ascension of the star is - 9 h. 3 m. 24 s. ( 42.) To find the Greenwich apparent time, when the sun crossed the equinox, we refer to Fig. 5 ; and as the point E corresponds to apparent noon of March 20th, and the Q to apparent noon of March 21st, and supposing the motion of the sun uniform ( as it is nearly ) for that short interval, w< have the following proportion : EQ : Ep : : 24h. : x. Giving to EQ and E and ^QJV^ are really spherical triangles ; but triangles on a sphere whose sides are of the less than a degree may be regarded as plane triangles, with- out any appreciable error. In the triangle 46 ASTRONOMY. CHAP, in. and, if we regard these seconds of arc as mere numerals, and calculate the angle E T S, we find it 23 27' 43" ; which in the obliquity of tJie ecliptic. Sun'i ion- By computing the length of the line S N, we find it 59' 30"-, which was the variation in the sun's longitude, between tlie noon of the 2Qth and 21st. Both longitude and right ascension are reckoned from the equinoctial point , Aquarius CO-, Pisces X. Owing to the precession of the equinoxes, these signs do not correspond with the constellations, as originally placed : the variation is now about 30 degrees; the stars remain in their places ; and the first meridian, or first point of Aries, has drawn back, which has given to the stars the appearance of moving forward. Beginning with the first point of Aries as it now stands, no prominent star is near it ; and, going along the ecliptic to facing the eastward, there is nothing to arrest special attention, 8 until we come to the Pleiades, or Seven Stars, though only six are visible to the naked eye. This little cluster is so well known, and so remarkable, that it needs no description. South- east of the Seven Sfars, at the distance of about 18 degrees, is a remarkable cluster of stars, said to be in the Bull's Head; the largest star in this cluster is of the 1st magnitude, of a red color, called Aldebaran. It is one of the eight stars se- lected as points from which to compute the moon's distance, for the assistance of navigators. This cluster resembles an A when east of the meridian, and 52 ASTRONOMY. IV - a V when west of it. The Seven Stars, Aldebaran, and Ca- pella, form a triangle very nearly isosceles Capella at the vertex. A line drawn from the Seven Stars, a little to the west of Aldebaran, will strike the most remarkable constella- tion in the heavens, Orion ( it is out of the zodiac, however ) ; some call it the Ell and Yard. The figure is mainly distin- guished by three stars, in one direction, within two degrees of each other ; and two other stars, forming, with one of the three first mentioned, another line, at right angles with the first line. The five stars, thus in lines, are of the 1st or 2d magnitude. A line from the Seven- Stars, passing near Aldebaran and through Orion, will pass very near to Sirius, the most bril- liant star in the heavens. The ecliptic passes about midway between the Seven Stars mean pie to find tinw per watch, a star of the \st magnitude came to the meridian. * I was in latitude 39 N., and about 75 W. The star was of a deep red color, and, as near as my judgment could decide, its altitude was between 25 and 30. Two small stars were near it, and a remarkable cluster of smaller stars were west and north- west of it, at the distances of 5, 6, or 7. What star was this ? Time per watch, - - - - 9 h. 34 m. 00 s, Equa. of time ( subtr. from mean time ) ,.3 48 Apparent time, - - - 9 30 12 Longitude, 75, equal to - - 5 Apparent time, at Greenwich, - - 14 h. 30 m. 00 & By examining the table for the sun's R. A., I find that, On the 1st of July, it is 6 h. 40 m. 00 s. On the 5th, - 6 56 30_ Variation, for 4 days, - 16 m. 30 s. At this rate, the variation for 2 days, 14 hours, cannot be Ten or twenty degrees, near the horizon, is apparently a much larger space than the same number of degrees near the zenith. Two stars, when near the horizon, appear to be at a greater distance asunder than when their altitudes are greater. The variation is a mere optical illusion; for, by applying instruments, to measure the angie in the different situations, we find it the same. Unless this fact is taken into consideration, an observer will always conceive the altitude of any ob- ject to be greater than it really is, especially if the altitude is less than 45 degrees. GEOGRAPHY OF THE HEAVENS. 5* fai from 10m. 10s.; and the right ascension of the sun, at CHAP rv the time of observation, must have been An exam Nearly h. 50m. 10s. ^ e fi g ndin To which add, apparent time, - - 9 30 12 Right ascension of the star, - - 16 h. 20 m. 22 s. By inspecting the catalogue of stars, I find Antares to have a right ascension of 16h. 20m. 2s. and a declination of 26 4', south. In the latitude mentioned, the meridian altitude of the celestial equator must be 50 0' Objects south of that plane must be less, Jtence (sub.) 26 4 Meridian altitude of Antares, in lat. 50, 23 56 As the observation corresponds to the right ascension of An- tares ( as near as possible, considering errors,*in observation, and probably in the watch), and as the altitudes do not differ many degrees ( within the limits of guess work ), it is certain that the star observed was ANTARES. By its peculiar red color, and the remarkable clusters of stars surrounding it, I shall be able to recognize this star again, without the trouble of direct observation. 3. On the night of the 2 other stars, of about the 3d magnitude, within 3 of it ; the three stars forming nearly a right line, north and south ; the altitude of the principal star about 60. What star was it? In these examples, the time must be reckoned on from noon to noon again ; therefore 1 h. 48 m. after midnight must be written, 13 h. 48m. OOs. Equation of time, to subtract, - 1 12 Apparent time, - - - 13 46 48 Longitude, .... 5 Greenwich apparent time, June 20, 18 h. 46m. 48s. Sun's right ascension, at this time, - 5 h. 57 m. 40 s. Time, - 13 46 48 Star's right ascension, - - 19 h. 44m. 28s. 53 ASTRONOMY. CHAP. IY. By inspecting the catalogue of stars, we find the right ascension of Altair 19 h. 43 m. 15 s.., and its declination 8 27' N. In latitude 40 N., the decimation of 8 27' N. will give a meridian altitude of 58 27'; and, in short, I know the star observed must be Altair, and the two other stars, near it, I recognize in the catalogue. By taking these observations, any person may become ac- quainted with all the principal stars, and the general aspect of the heavens; but no efforts, confined merely to the study of books, will accomplish this end. The equation in Art. 47 is not confined to a star ; it may be any heavenly body, moon, comet, or planet. The time of passing the meridian is but another term for right ascension. If observations are made on any bright star, and no corre- sponding star is found in the catalogue, such a star would probably be a planet; and if a planet, its right ascension will change. Tk 6 ^ Tlie ^jg re gj on O f stars ^th O f declination 50, ud Mage 1 & never seen in latitude 40 north, nor from any place north .an Ciouds 9 f that parallel ; and, to register these stars in a catalogue, it has been necessary for astronomers to visit the southern hemisphere, as we have before mentioned ; but these stars are mostly excluded from our catalogues. There are several constellations, in the southern region, worthy of notice the Southern Cross and the Magellan Clouds. The Southern Cross very much resembles a cross ; so much so, that any person would give the constellation that appellation. Its principal star is, in right ascension, 12 h. 20 m., and south declination 33. The Magellan Clouds were at first supposed to be clouds by the navigator Magellan, who first observed them. They are four in number ; two are white, like the Milky Way, and have just the appearance of little white clouds. They are nebula. The other two are black extremely so and arc supposed to be places entirely devoid of all stars ; yet they are in a very bright part of the Milky Way: right ascen- sion 10 h. 40 m., declination 62 south. DESCRIPTIVE ASTRONOMY. SECTION II. DESCRIPTIVE ASTRONOMY. CHAPTER I. FIRST CONSIDERATIONS AS TO THE DISTANCES OF THE HEAVENLF BODIES. SIZE AND EXACT FIGURE OF THE EARTH. ( 49.) Hitherto we have con- Fi S' 6 ' sidered only appearances, and have not made the least inquiry as to the nature, magnitude, or distances of the celestial objects. Abstractly, there is no such thing as great and small, near and remote ; relatively speaking, however, we may apply the terms great, and very great, as regards both magnitude and distance. Thus an error of ten feet, in the measure of the length of a building, is very great when an error of ten rods, in the mea- sure of one hundred miles, would be too trifling to mention. Now if we consider the dis- tance to the stars, it must be relative to some measure taken as a standard, or our inquiries will not be definite, or even in- telligible. We now make this general inquiry : Are the heavenly bodies near to, or remote from, the earth ? Here, the earth itself seems to be the natural standard for measure ; and if any body were but two, three, or even ten times the diameter of the earth, in distance, we CHAP. t. Are the heavenly bo- dies remote t 6tt ASTRONOMY. CHAP.J. should call it near; if 100, 200, or 2000 times the diametei of the earth, we should call it remote. To answer the inquiry, Are the heavenly bodies near or remote ? we must put them to all possible mathematical tests ; a mere opinion is of no value, without the foundation of some positive knowledge. Let 1, 2 ( Fig. 6 ), represent the absolute position of two stars ; and then, if A B C represents the circumference of tho earth, these stars may be said to be near ; but if a b c repre- sents the circumference of the earth, the stars are many times the diameter of the earth, in distance, and therefore may The means be said to be remote. If ABC is the circumference of this question tne earfc h, in relation to these stars, the apparent distance of pointed out. the two stars asunder, as seen from A, is measured by the angle 1 A 2 ; and their apparent distance asunder, as seen from the point B, is measured by the angle 1 B 2 ; and when the circumference A B C is very large, as represented in our figure, the angle A, between the two stars, is manifestly greater than B. But if a b c is the circumference of the earth, the points a and b are relatively the same as A and B. And, it is an ocular demonstration that the angle under which the two stars would appear at a is the same, or nearly the same, as that under which they would appear at b; or, at least, we can conceive the earth so small, in relation to the distance to the stars, that the angle under which two stars would appear, would be the same seen from any point on the earth. The con- Conversely, then, if the angle under which two stars appear is the same as seen from all parts of the earth's surface, it is certain that the diameter of the earth is very small, compared with the distance to the stars ; or, which is the same thing, the distance to the stars is many times the diameter of the earth. Therefore observation has long since decided this important point. Sir John Herschel says : " The nicest measurements of the apparent angular distance of any two stars, inter se, taken in any parts of their diurnal course ( after allowing for the unequal effects of refraction, or when taken at such times that this cause of distortion shall act equally on both ), mani- fest not the slightest perceptible variation. Not only this, but COMPARATIVE DISTANCES. 61 at whatever point of the earth's surface the measurement is CHAP. L performed, the results are absolutely identical. No instruments ever yet invented by man are delicate enough to indicate, by an increase or diminution of the angle subtended, that one point of the earth is nearer to or farther from the stars than another." ( 50.) Perhaps the following view of this subject will be more intelligible to the general reader Let Z Hjft p. 7 distance // represent ** 8tari ' the celestial equator, as seen from the equator on the earth; and if the earth be large, in rela- tion to the distance to the stars, the observer will be at z' ; and the part of the celestial arc above his horizon would be represented by A Z B, and the part below his horizon by A NB, and these arcs are ob- viously unequal ; and their relation would be measured by the time a star or heavenly body remains above the horizon, com- pared with the time below it ; but by observation ( refraction being allowed for ), we know that the stars are as long above the horizon as they are below; which shows that the ob- server is not at z', but at z, and even more near the center ; so that the arc A Z B, is imperceptibly unequal to the arc H NH\ that is, they are equal to each other; and the eartib is comparatively but a point, in relation to the distance to the stars. This fact is well established, as applied to the fixed stars, Th sun, and planets ; but with the moon it is different : that body Son. ASTRONOMY. is longer below the horizon than above it ; which shows thai its distance from the earth is at least measurable. ( 51.) It is improper, at present, or rather, it is too advanced an age, to pay any respect to the ancient notion, that the earth is an extended plane, bounded by an unknown space, inacces- sible to men. Common intelligence must convince even the child, that the earth must be a large ball, of a regular, or an irregular shape; for every one knows the fact, that the earth has been many times circumnavigated; which settles the question. Eanh' In addition to this, any observer may convince himself, that enrface con- the surface of the sea, or a lake, is not a plane, but everywhere convex ; for, in coming in from sea, the high land, back in the country, is seen before the shore, which is nearer the observer ; the tops of trees, and the tops of towers, are seen before their bases. If the observer is on shore, viewing an approaching vessel, he sees the topmast first ; and from the top, downward, the vessel gradually comes in view. This being the case on every sea, and on every portion of the earth, proves that the surface of the earth is convex on every part hence it must be a globe, or nearly a globe. These facts, last mentioned, are sufficiently illustrated by Fig. 8. ( 52.) On the supposition that the earth is a sphere, there are several methods of measuring it, without the labor of applying the measure to every part of it. The first, and most natural method (which we have already mentioned), is that of measuring any definite portion of the meridian, and from thence computing the value of the whole circumference. HOW to Thus, if we can know the number of degrees, and parts of end the cir- a Degree, in the arc A B (Fig. 9), and then measure the dis- fiii earth tance in miles, we in fact virtually know the whole circumfe DIAMETER OF THE EARTH. $ rence ; for whatever part the arc A B is of 360 degrees, the CHAP. I. same part, the number of miles in A B, is of the miles in the whole circumference. To find the arc A B, the latitudes of the two points, A and B, must be very accurately taken, and their difference will give the arc in degrees, minutes, and seconds. Now A B must bo measured simply in distance, as miles, yards, or feet; but this is a laborious operation, requiring great care and perse- verance. To measure directly any considerable portion of a meridian, is indeed impossible, for local obstructions would soon compel a deviation from any definite line ; but still the measure can be continued, by keeping an account of the de- viations, and reducing the measure to a meridian line. Let m be the miles or feet in A B\ then the whole circum- / 360 m \ , terence will be expressed by ( - 1~/. ' ( 53. ) When we know the Fig. 9. hight of a mountain, as re- presented in Fig. 9, and at the same time know the dis- tance of its visibility from the surface of the earth; that is, know the line MA ; then we can compute the line M C, by a simple theo- rem in geometry; thus, Now as thi3 right hand member of this equation is known, CM is known ; and as part of it ( MB ) is already known, the other part, B C, the diameter of the earth, thus becomes known. This method would be a very practical one, if it were not objection for the uncertainty and variable nature of refraction near the to thu horizon ; and for this reason, this method is never relied upon, thod ' although it often well agrees with other methods. As an ex- ample under this method, we give the following : 64 ASTRONOMY. A mountain, two miles in perpendicular hight, was seen from sea at a distance of 126 miles. If these data are cor- rect, what then is the diameter of the earth ? Solution: MC=-== 63x126=7938. C= Dip of u* ( 54. ) This same geometrical theorem serves to compute horizon. the dip of the horizon. The true horizon is a right angle from the zenith ; but the navigator, in consequence of the motion of his vessel, can never use the true horizon ; he must use the sea offing, making allowance for its dip. If the naviga- tor's eye were on a level with the sea, and the sea perfectly stable, the true and apparent horizon would be the same. But the observer's eye must always be above the sea ; and the higher it is, the greater the dip ; and the amount of dip will depend on the hight of the eye, and the diameter of the earth. The difference between the angle AMC (Fig. 9 ), and a right angle ( which is the same as the angle A EM), is the measure of the dip corresponding to the hight EM. For the benefit of navigators, a table has been formed, showing the dip for all common elevations.* * The dip is computed thus : atlhe 6 cent! Put EC (Fig. 9) =D, BM=h ; is equal to / 7} u dip. Then EM= +^) ; and ( MA) 2 = By trigonometry, (EA)* : (MA) 2 : : R 3 : t&n. 2 AEM; That is, - - - ~ i(D+h)h:i R 2 : iim*AEM. For very moderate elevations, k is extremely small, in rela- tion to D ; and the second term of the proportion may be Dh. (R represents the radius of the tables.) Making this consideration, we have ^- : Dh : : R* : iw*AEM; Or, - - D : h : : 4R a : tan.M^Jf; Or, - - jDi Jh. : : 2R : iim.AEM. DIP OF THE HORIZON. 65 ( 55. ) All such computations are made on the supposition CHAP. L that the earth is exactly spherical ; and it is, in fact, so nearly spherical, that no corrections are required in consequence of its deviation from that figure. After correct views began to be entertained, as to the mag- The earth nitude of the earth, and its revolution on an axis, philosophers not e * actl y concluded that its equatorial diameter might be greater than " P its polar diameter; and investigations have been made to decide the fact. If the earth were exactly spherical, it is plain that the cui- vature over its surface would be the same in every latitude ; but if not of that figure, a degree would be longer on one part of the earth than on another. " But," says Herschel, "when we come to compare the measures of meridional arcs made in various parts of the globe, the results obtained, although they agree sufficiently to show that the supposition of a spherical figure is not very remote from the truth, yet exhibit discord- ances far greater than what we have shown to be attributable to error of observation; and which render it evident that the hypothesis, in strictness of its wording, is untenable. The following table exhibits the lengths of a degree of the meri- dian ( astronomically determined as above described), ex- By inspecting this last proportion, it will be perceived that the tangent of the dip varies as the square root of the eleva- tion. To apply this proportion, we adduce the following problem : The diameter of the earth is 7912 miles ; the elevation of the eye, above the surface, is ten feet. What is the dip? 2J2 . . log. 10.301030 N /F. . log. _ : 500000 Product of the means (log.), .... 10.801030 I) mile*, 7912, - - log. - 3.898286 Feet, - 5280, - - log. - 3.722634 2)^620920 in feet, - - (log.) 3.810460 . . 8_810460_ tan. 3' 22" - - - 6.990570 66 ASTRONOMY. CHAP. r. pressed in British standard feet, as resulting from actual measurement, made with all possible care and precision, by commissioners of various nations, men of the first eminence, supplied by their respective governments with the best instru- ments, and furnished with every facility which could tend to insure a successful result of their important labors. Country. Latitude of Middle of the Arc. . iLength of *&u | co D n :f; e d e ed Observers. Sweden .... 66 20 10 58 17 37 52 35 45 4652 2 4451 2 4259 39 12 33 18 30 16 822 12 32 21 1 31 137'19" 3 35 5 3 57 13 8 20 12 22 13 2 9 47 1 28 45 1 13 174 15 57 40 1 34 56 373 365782 365368 364971 364872 364535 364262 363786 364713 363044 363013 362808 Svanberg. Struve. Roy, Kater. Lacaille, Cassini. Delambre. Mechain. Boscovich. Mason, Dixon. Lacaille. Lambton, Everest. Lambton. Condamine, etc. Knp'land France Rome America, U. S-. . Cap of G. Hope India India . ... Peru The earth < ; It is evident, from a mere inspection of the second and ! fourt h columns of this table, that the measured length of a de- A&n at. the gree increases with the latitude, being greatest near the poles, ** nator and least near the equator." "Assuming," continues Herschel, "that the earth is an ellipse, the geometrical properties of that figure enable us to assign the proportion between the lengths of its axes which shall correspond to any proposed rate of variation in its cur- vature, as well as to fix upon their absolute lengths, corre- sponding to any assigned length of the degree in a given latitude. Without troubling the reader with the investiga- tion (which may be found in any work on the conic sections), it will be sufficient to state that the lengths, which agree on the whole best with the entire series of meridional arcs, which have been satisfactorily measured, are as follow : Feet. Miles. Greater, or equatorial diam., =41,847,426=7925.643 Lesser, or polar diam., - - =41,707,620=7899.170 Difference of diameters, or _ jgg 806= 26 478 polar compression, - - - The proportion of the diameters is very nearly that of FORM OF THE EARTH. 6-r 298 : 299, and their difference ^g- of the greater, or a very CHA f. little greater than 3-^-." ( 5f). ) The shape of the earth, thus ascertained by actual measurement, is just what theory would give to a body of water equal to our globe, and revolving on an axis in 24 Lours; and this has caused many philosophers to suppose that the earth was formerly in a fluid state. If the earth were a sphere, a plumb line at any point on Expiana. its surface would tend directly toward the center of gravity ^ ion - fradiul of the body ; but the earth being an ellipsoid, or an oblate splieroid, and the plumb lines, being perpendicular to the sur- face at any point, do not tend to the center of gravity of the figure, but to points as represented in Fig. 10. The plumb line at H tends to F, yet the mathematical center, and center of gravity of the figure, is at E. So at I, the plumb line tends to the point (?; and as the length of a degree at A, is to the length of a degree at H, so is 16? to II F. If, however, a passage were made through the earth, and a body let drop through it, the body would not pass from /to G: its first tendency at /would be toward the point G\ but after it passed below the surface at I, its tendency would be more and more toward the point E, the center of gravity ; but it would not pass exactly through that point, unless dropped from the point A, or the point C. ( 57. ) If the earth were a perfect and stationary sphere, Force ot the force of gravity, on its surface, would be everywhere the gravity difie- ,,.,.,..., ,. -, rent on difle- same ; but, it being neither stationary, nor a perfect sphere, renl parts of the force of gravity, on the different parts of its surface, must the eartb be different. The points on its surface nearest its center of gravity, must have more attraction than other points more remote from the center of gravity ; and if those points which are more remote from the center of gravity have also a rotary motion, there will be a diminution of gravity on that account. Let .4 2? (Fig. 10) represent the equatorial diameter of 68 ASTRONOMY. CHAP - ! - the earth, and CD the polar diameter; and it is obvious that E will be the center of gravity, of the whole figure, and Gravity di- t ^ at t ] ie f orce O f g rav ity at C and D will be greater than at rotation. ^ an J ot ^ er points on the surface, because E C, or ED, are less than any other lines from the point E to the surface. The force of gravity will be greatest on the points C and D, aiso, because they are stationary : all other points are in a circular motion ; and circular motion has a tendency to depart from the center of motion, and, of course, to diminish gravity. The diminution of the earth's gravity by the rotation on it? axis, amounts to its | part,* at the equator. By this frao- Computa. lion of the amount of diminution. Fiff. 11 * Let D be the equatorial diametei of the earth, F the versed sine of an arc corresponding to the motion in a second of time, and c the chord or arc ( for the chord and arc of so small a portion of tho circumference will coincide, practically speaking). A portion of the earth's gravity, equal to F t is destroyed by the rotation of the earth, and we aro now to compute its value. By proportional triangles, F : c : : c : D; c 3 The value of c is found by dividing the whole circumference into as many equal parts as there are seconds in the time of revolution. But the time of revolution is 23 h. 56 m. 4 s., =? 86164 seconds. The whole circumference is (3.1416)2>; (3.1416)1? "(86164)" (3.1416)21) Therefore, (2) By this value of c, we have F- (86164) 2 The visible force of gravity, at the equator, is the distance a body will fall the first second of time, expressed in feet. Let us call this distance g. Now the part of gravity des- EFFECT OF FORM ON GRAVITY. 69 tion, then, is the weight of the sea about the equator lightened, and thereby rendered susceptible of being supported at a higher level than at the poles, where no such counteracting force exists. fcroyed by rotation, as we have just seen, is -= ; therefore tho whole force of gravity is (g-\--jr ) Our next inquiry is: what part of the whole is the part de- Ratio of * diminutiot ttnyedf Or what part of O+ is ? 3 mputea ' Which, by common arithmetic, is, (86164)* ~ (3.1416) 2 /> : Hence, ffl> = (86164)^ (86164)^(16.07) c 2 = " (3.1416) 2 D~(3.1416)2(7925)(5280)' By the application of logarithms, we soon find the value of 1 this expression to be 288.4. Therefore, gD =- . +1 289.4 We may now inquire, how rapidly the earth must revolve on its axis, so that the whole of gravity would be destroyed on the equator. That is, so that F shall equal g. Equation (1) then becomes, <7-fi, or c= But as often as c is contained in the whole circumference, is the corresponding number of seconds in a revolution; thai is, the time in seconds must correspond to the expression, 70 ASTRONOMY. CHAf - * ( 58. ) It is this centrifugal force itself that changed the shape of the earth, and made the equatorial diameter greater than the polar. Here, then, we have the same cause, exer- cising at once a direct and an indirect influence. The amount Rotation O f the former ( as we may see by the note ) is easily calcu- md indirect ^ a ted ; that of the latter is far more difficult, and requires a effect on gra- knowledge of the integral calculus, " But it has been clearly treated by Newton, Maclaurin, Clairaut, and many other emi- nent geometers ; and the result of their investigations is to show, that owing to the elliptic form of the earth alone, and independently of the centrifugal force, its attraction ought to increase the weight of a body, in going from the equator to the pole, by nearly its j^ th part; which, together with the ^| th part, due from centrifugal force, make the whole quan- tity 7ir 4 - th part; which corresponds with observations as deduced from the vibrations of pendulums." See Natural Philosophy. ( 59. ) The form of the earth is so nearly a sphere, that it is considered such, in geography, navigation, and in the general problems of astronomy. The average length of a de- S rcc * s ^'U ^ n g^ sn miles ; and, as this number is fractional, ar.d inconvenient, navigators have ta- citly agreed to retain the ancient, rough estimate of sixty miles to a degree ; calling the mile a geographical mile. Therefore, the geographical mile is longer than the English mile. D, in feet, = (7925)(5280) ; g = 16.076. By the applica- tion of logarithms, we find this expression to be 5069 seconds, or Ih. 24m. 29 s.; which is about 17 times the rapidity of its present rotation. In a subsequent portion of this work, we shall show how to arrive at this result by another principle, and through another operation. CONVERGENCY OF MERIDIANS. 71 As all meridians come together at the pole, it follows that CHAP. I. a degree, between the meridians, will become less and less as we approach the pole ; and it is an interesting problem to trace the law of decrease.* * This law of decrease will become apparent, by inspecting Fig. 12. Let EQ represent a degree, on the equator, and EQC a sector on the plane of the equator, and of course EC is at right angles to the axis C P. Let D ^/be any plane parallel to EQC\ then we shall have the following proportion : EG : DI : : EQ : DF. In trigonometry, E C is known as the radius of the sphere; D /as the cosine of the latitude of the point D (the nume- rical values of sines and cosines, of all arcs, are given in trigo- nometrical tables) : therefore we have the , following rule, to compute the length of a degree between two meridians, on any parallel of latitude. RULE. As radius is to the cosine of the latitude / so is the length of a degree on the equator, to the length of a parallel de- gree in that latitude. Calling a degree, on the equator, 60 miles, what is the length of a degree of longitude, in latitude 42 ? SOLUTION BY LOGARITHMS. As radius (see tables), 10.000000 Is to cosine 42 (see tables), - - 9.871073 So is 60 miles (log.), - 1.778151 To 44 r \V_ m fl eSj . . . 1.649224 Ai the latitude of 60, the degree of longitude is 30 miles; the diminution is very slow near the equator, and very rapid near the poles. In navigation, the DJ<"s are the known quantities ob- TO tained by the estimations from the log line, etc. ; and the d e n arture to navigator wishes to convert them into longitude, or, what is the same thing, he wishes to find their values projected on the equator, and he states the proportion thus : DI : EC : : DF : EQ ; That is, as cosine of latitude is to radius, so is departure to difference of longitude. ASTRONOMY. CHAPTER II. PARALLAX, GENERAL AND HORIZONTAL. RELATION BETWEEN PARALLAX AND DISTANCE. REAL DIAMETER AND MAGNI- TUDE OF TILE MOON. CHAP U. ( 60. ) PARALLAX is a subject of very great importance in astronomy : it is the key to the measure of the planets to their distances from the earth and to the magnitude of the whole solar system. Parallax in Parallax is tJie difference in position, of any body, as seen from the center of the earth, and from its surface. When a body is in the zenith of any observer, to him it hag no parallax; for he sees it in the same place in the heavens, as though he viewed it from the center of the earth. The greatest possible parallax that a body can have, takes place when the body is in the horizon of the observer ; and this parallax is called horizontal parallax. Hereafter, when wo speak of the parallax of a body, horizontal parallax is to be understood, unless otherwise expressed. A clear and summary illustration of parallax in general, is given by Fig. 13. Horizontal FUr. 13. Let (7 be parallax. ^ra^H>BMiKi*wwp*s^Hm^l^^^*inlcl parallax is equal to the sine of the parallax in altitude ', into the radius, and divided by the tine of tfo apparent zenith distance. LUNAR PARALLAX 7 The lunar parallax was first recognized in northern Europe C "*P- - by the moon appearing to describe more than a semicircle south By of the equator, and less than a semicircle north of that line; and, on an average, it was observed to be a longer time south, than north of the equator ; but no such inequality could fi j* 1 lndlc be observed from the region of the equator, Observers at the south of the equator, observing the posi- tion of the moon, see it for a longer time north of the equator than south of it ; on- ', to than, it appears to describe more t/ian a semicircle no th of the equator. Here, then, we have observation against observation, unless we can reconcile them. But the only reconciliation that can be made, is to conclude that the moon is really as long in one hemisphere as the other, and the observed discrepancy must arise from the positions of the observers ; an,d when we reflect that parallax must always depress the object ( see Fig. 13 ), and throw it farther from the observer, it is therefore per- fectly clear that a northern observer should see the moon farther to the south than it really is , and a southern observer sec the same body farther north than its true position. ( 62.) To find the amount of the lunar parallax, requires the concurrence of two observers. They should be near the same meridian, and as far apart, in respect to latitude, as possible ; and every circumstance, that could affect the result, must be known. The two most favorable stations are Greenwich (England) and the Cape of Good Hope. They would be more favorable if they were on the same meridian ; but the small change in mount ot > declination, while the moon is passing from one meridian to '* the other, can be allowed for ; and thus the two observations are reduced to the same meridian, and equivalent to being made at the same time. The most favorable times for such observations, are when the moon is near her greatest declinations, for then the change of declination is extremely slow. Let A ( Fig. 14 ) represent the place of the Greenwich ob- servatory, and B the station at the Cape of Good Hope. C is the center of the earth, and Z and Z' are the zenith jjjj" ASTRONOMY. OBAP.JI. Fig. 14. points of the observers. Let M be the position of the moon, and the observer at A will see it pro- jected on the sky at m', and tho observer at B will see it pro- jected on the sky at m. Now the figure A CBM is a quadrilateral; the angle A C B is known by the latitudes of the two observers; the angles MA C and MB C are the respective zenith distances, taken from 180. But the sum of all the angles of any quadrilateral is equal to four right angles ; and hence the angles at A, C, and , being known, the parallactic angle at M is known. In this quadrilateral, then, we have two sides, A C and C B % land all the angles; and this is sufficient for the most ordinary mathematician to decide every particular in connection with it ; that is, we can find AM, M j?, and finally MC.* Now MO being known, the horizontal A mam.. * The direct and analytical method of obtaining M C, will be 1 de " very acceptable to the young mathematician ; and, for that reason, we give it. Put AC=C=r, CM=x, and the two parts of the ob- served parallactic angle, M, represented by P and Q, as in the figure. Also, let a represent the natural sine of the angle MAC, and b the natural sine of the angle MB C: Then, by trigonometry, - x : a : : r : sin. Q ; Also, - - - - - - x : b : : doc t ion. sn. Hence, sin. P+sin. Q= . (I) J.UNAR PARALLAX. 77 parallax can be computed, for it is but a. function of the dls- CHA. n tance (see 60). By the equation (Art. GO), #=( -. jr (72 \ Jr; and whenar, the distance, is known, sin. p, or sine of the horizontal parallax, is known. ( 63. ) The result of such observations, taken at different Variable times, show all values to MC, between 55^, and 63 T Y - ; n to taking the value of r as unity. These variations are regular and systematic, both as to time and place, in the heavens ; and they show, without fur- ther investigation, that the moon does not go round the earth in a circle, or, if it does, the earth is not in the center of that circle. The parallaxes corresponding to these extreme distances, are 61' 29" and 53' 50". When the moon moves round to that part of her orbit which is most remote from the earth, it is said to be in apogee; and and, when nearest to the earth, it is said to be in perigee. The points apogee and perigee, mainly opposite to each other, do not keep the same places in the heavens, but gradually move forward in the same direction as the motion of the moon, and perform a revolution in a little less than nine years. But, by a general theorem in trigonometry, P.L.Q P Q sin. P+ sin. Q=^2 sin. ^ cos. -A . (2) Now by equating (1) and (2), and observing that P-^-Q=* (p Q\ cos. ^ ) must be extremely near unity; and, therefore, as a factor, may disappear ; we then have, . M (a+ft> 2 sm.- 5 - = v ' ' , or x=- 2 a? 2 sin. A more ancient method is to compute the value of the little eriangle B C G, and then of the whole triangle A MO, and then of a part A MC or M Q C. 78 ASTRONOMY. OHAP. n. (64.) Many times, when the moon comes round to its peri- gee, we find its parallax less than 61' 29", and, at the oppo- site apogee, more than 53' 50". It is only when the sun is in, or near a line with the lunar perigee and apogee, that these greatest extremes are observed to happen ; and when the sun is near a right angle to the perigee and apogee, then the moon moves round the earth in an orbit nearer a circle ; and thus, by observing with care the variation of the moon's parallax, we find that its orbit is a revolving ellipse, of variable eccentricity. (65.) Because the moon's distance from the earth is va- riable, therefore there must be a mean distance: we shall show, hereafter, that her motion is variable ; therefore there is a mean motion ; and, as the eccentricity is variable, there is a mean eccentricity. MBA* P a. The extreme parallaxes, at mean eccentricity, are 60' 20" parallax at an< ^ ^' ^" > an( ^ * ne corresponding distances from the earth MEAN dis- are 56.93 and 63. 04, the radius of the earth being unity. mean paral ] aXj or mean between 60' 20" and 54' 05", is 57' 12".5; but the parallax, at mean distance, is 57' 03"*. * It may seem paradoxical that the mean parallax, and the parallax at mean distance arc different quantities ; but the following investigation will set the matter at rest. Let d and D be extreme distances, and M the mean distance. Then, - --- rf+D=2Jf. . . . (1) Also, let p and P be the parallaxes corresponding to the dis- tances d and D ; and put x to represent the parallax at mean distance. Then, by Art. 60 ( if we call the radius of the tables unity), we have i i and M=- sm. x Substituting these values of d, D, and M, in equation (1) w 1 1 2 have, - - -. \- -. - = . ; sm.jp ' sin. P sin.* _ 2 sin. p sin. P X0x Or, ... sin. P + sin. p = (2) VARIATION OF PARALLAX. 79 . 55.92+63-84 C H*P. n. The mean between extreme distances is ^ or o9.oo; but the true mean distance is 60.26, corresponding to the Mean du- parallax 57' 3". The mean, between extremes, is a variable m>on quantity; but the true mean distance is ever the same, a little more than 60i times the semidiameter of the earth. ( 66.) The variations in the moon's real distance must cor- respond to apparent variations in the moon's diameter ; and if the moon, or any other body, should have no variation in apparent diameter, we should then conclude that the body was always at the same distance from us. The change, in apparent diameter, of any heavenly body, is numerically proportioned to its real change in distance; as appears from the demonstration in the note below.* But by a well known, and general theorem in trigonometry, Mean p* rallax. (P-L-p\ /P n\ 3r4-; J cos. ( ) (3) A / \ 'A ' By equating (3) and (2), and observing that the cosines of very small arcs may be practically taken as unity, or ra- dius ; therefore, 'P-\-p\ _ sin. P sin. p sin. sm. x _ . sin. P sin. p Or, ..... sm. x = On applying this equation, we find #=57' 3 . * Let A be the _ Fi g- 15 - point of vision, and d the diameter of | any body at diffe- 40. Now, by trigonometry, we nave the following proportions : AC : d :: H, : tan. CAD AB : d : : R : tan. BAE. 80 A 3 T R N O M Y. CHAP ii. Now if the moon has a real change in distance, as observa- tions show, such change must be accompanied with apparent changes in the moon's diameter; and, by directing observa- tions to this particular, we find a perfect correspondence; showing the harmony of truth, and the beauties of real science. Connec- We have several times mentioned that the moon's horizon- tion between ^j p ara }] ax [ s ^Q gemidiameter of the earth, as seen from the ieruiliame- ter and hon- moon ; and now we further say, that what we call the moon's rental parai- gemidiameter, an observer at the moon would call the earth's horizontal parallax ; and the variation of these two angles de- pends on the same circumstance the variation of the distance between the earth and moon; and, depending on one and the same cause, they must vary in just the same proportion. When the moon's horizontal parallax is greatest, the moon's semidiameter is greatest; and, when least, the semidiameter is the least ; and if we divide the tangent of the semidiameter by the tangent of its horizontal parallax, we shall always find the same quotient (the decimal 0.27293) ; and that quotient is the ratio between the real diameter of the earth and the diameter of the moon.* Having this ratio, and the diameter of the earth, 7912 miles, we can compute the diameter of the moon thus : 7912x0.27293=2169.4 miles. From the first proportion, - - - A C tan. CAD=dR ; From the second, AB tan. BAE=dR : By equality, - - - - A Ctan. CAD=AB tan. BAE. This last equation, put into an equivalent proportion, gives : AC : AB : tan. BAE :: tan. CAD. But tangents of very small arcs ( such as those under which the heavenly bodies appear) are to each other as the arcs themselves. Therefore, AC : AB : : angle BAE : angle CAD; That is ; the angular measures of the same body are inversely proportional to the corresponding distances. * This requires demonstration. Let E be the real semi- APPEARANCE FROM THE MOON 8l As spheres are to each other in proportion to the cults of CHAP. IT, their diameters, therefore the bulk (not mass) of the earth. is to that of the moon, as 1 to $, nearly. A.c< the moon's distance is 60 j- times the radius of the earth, Angmen- it follows that it is about r V tn nearer to us, when at the talion , of the u ^ moon's semi. zenith, than when in the horizon. Making allowance for this diameter : iu ( in proportion to the sine of the altitude ), is called the cause> augmentation of the semidiameter. ( 68. ) It may be remarked, by every one, that we always The earth see the same face of the moon ; which shows that she must a moon to -i *^ e m * roil on an axis in the same time as her mean revolution about the earth ; for, if she kept her surface toward the same part of the heavens, it could not be constantly presented to the earth, because, to her view, the earth revolves round the moon, the same as to us the moon revolv9S round the earth ; and the earth presents phases to the moon, as the moon does to us. except opposite in time, because the two bodies are opposite in position. When we have new moon, the lunarians have full earth ; and when -we have first quarter, they have last quarter, etc. The moon appears, to .us, about half a degree in diameter ; the earth appears, to them, a moon, about diameter of the earth (Fig.l6),iw that of the moon, D the distance be- tween the two bodies ; and let the radius of the tables be unity. Put P to represent the moon's horizontal parallax, and * its appa- rent semidiameter. Then, by trigonometry, D : E : : 1 : tan. P; and D : m : : 1 : tan. * From the first, 2)= ^; from the 2d, J9= vn tan. s nn f Em tan.* m Therefore,- - = - , or b = --, . Q. E< D tan. P tan. * tan. P E 82 ASTRONOMY CHAP. ii. two degrees in diameter, invariably fixed in their sty, arid the stars passing slowly behind it. The nu>on " But," says Sir John Herschel, "the moon's rotation on revolves .>n j^ ^^ j g un jf orm . an j since her motion in her orbit is not so, we are enabled to look a few degrees round the equatorial parts of her visible border, on the eastern or western side, according to circumstances ; or, in other words, the line join- ing the centers of the earth and moon fluctuates a little in its position, from its mean or average intersection with her sur- face, to the east, or westward. And, moreover, since the axis about which she revolves is not exactly perpendicular to her orbit, her poles come alternately into view for a small space at the edges of her disc. These phenomena are known by the name of libratiom. In consequence of these two dis- tinct kmds of libration, the same identical point of the moon's surface is not always the center of her disc ; and we therefore get sight of a zone of a few degrees in breadth on all sides of the border, beyond an exact hemisphere." CHAPTER III. THE EARTH'S ORBIT ECCENTRIC. THE APPARENT ANGULAR MOTION OP THE SUN NOT UNIFORM. LAWS BETWEEN DIS- TANCE, REAL, AND ANGULAR MOTION. ECCENTRICITY OP THE ORBIT. CHAP, in ( 69. ) THE gun's parallax is too small to be detected by The sun any common means of observation ; hence it remained un- tiufearth *" ^ nowri ' ^ or a ' on g series of years, although many ingenious methods were proposed to discover it. The only decision that ancient astronomers could make concerning it was, that it must be less than 20" or 15" of arc ; for, were it as much as that quantity, it could not escape observation. Now let us suppose that the sun's horizontal parallax is less than 20"; that is, the apparent semidiameter of the earth, as scpn from the sun, must be less than 20"; but the semidia- APPARENT DIAMETERS. 83 meter of the sun is 15' 56", or 956" ; therefore the sun must CHAP. m. be vastly larger than the earth by at least 48 times its diame*ter ; and the bulk of the earth must be, to that of the sun, in as high a ratio as 1 to the cube of 48. But as we do not allow ourselves to know the true horizontal parallax of the sun, all the decision we can make on this subject is, that the sun is vastly larger than the earth. ( 70. ) Previous observations, as we explained in the first Does t.h, section of this work, clearly show, or give the appearance of * elrtT^ the sun going round the earth once in a year ; but the appear- the eanh ance would be the same, whether the earth revolves round the round thc sun, or the sun round the earth, or both bodies revolve round a point between them. We are now to consider which is the most probable : (hat a large body should circulate round a much smaller one; or, the smaller one round a large one. The last suggestion corresponds with our knowledge and experience in mechanical philosophy ; the first is opposed to it. (71.) We have seen, in the last chapter, that the semidia- meter and horizontal parallax of a body have a constant rela- tion to each other ; and, while we cannot discover the one, we will examine all the variations of the other ( if it have va- riations ), and thereby determine whether the earth and sun always remain at the same distance from each other. Here it is very important that the reader should clearly Methods understand, how the apparent diameter of a heavenly body of measuring . . . apparent dia- can be determined to great precision. meters. As an example, we shall take the diameter of the sun ; but the same principles are to be followed, and the same deduc- tions are to be made, whatever body, moon, or planet, may be under observation. An instrument to measure the apparent diameter of a planet The mtcio is called a micrometer. It is an eyepiece to a telescope, with opening and closing parallel wires ; the amount of the opening is measured by a mathematical contrivance. For the measure of all small objects, the micrometer is exclusively used; and since it is impossible that any one observation can be relied upon as accurate ( on account of the angular space eclipsed by the wires), a great number of observations are taken, and meter. 84 ASTRONOMY. Clil? - "_' the mean result is regarded as a single observation. Gene- rally speaking, the following method is more to be relied upon, when large angles are measured, and to it we commend special attention. The me- The method depends on the time employed by the body in pass- thod by time ^ perpendicular wires of the transit instrument. IB passing v r r the meridian. All bodies (by the revolution of the earth) come to the meridian at right angles, and 15 degrees pass by the meridian in one hour of sidereal time ; and, in four minutes, one de- gree will pass; and, in two minutes of time, 30 minutes of arc will pass the meridian wire. Now if the sun is on the equator, and stationary there, and employs two minutes of sidereal time in passing the meridian, then it is evident that its apparent diameter is just 30' of arc; if the time is more than two minutes, the diameter is more; if less, less. But we have just made a supposition that is not true ; we have supposed the sun stationary, in respect to the stars ; but it is not so : it apparently moves eastward ; therefore it will not get past the meridian wire as soon as it would if station- ary. Hence we must have a correction, for the sun's motion, applied to the time of its passing the meridian. Corrections We have also supposed the sun on the equator, and for a to be made. moment con tinue the supposition, and also conceive its dia- meter to be just 30' of arc. Now suppose it brought up to the 20th degree of declination, on that parallel, it will extend over more than 30' of arc, because meridians converge toward the pole ; therefore the farther the sun, or any other body is from tfie equator, the longer it will be in passing the meridian on thai 1 account; the increase of time depending on the cosine of tlie declination. (See 59.) Hence two corrections must be made to the actual time that the sun occupies in crossing the meridian wire, before we can proportion it into an arc : one for the progressive motion of the sun in right ascension ; and one for the existing decli- nation. We give an example. Method of ^ n tlie firSt da ^ f June ' 1846 ' ^ 6 Sidereal time ( time the measured by the sidereal clock ) of the sun passing the me- APPARENT DIAMETERS. rtft ridian wire, wa,s observed to be 2 m. 16.64 s. ; the declination CHAP. HL was 22 2' 45", and the hourly increase of right ascension was exact a( , pa 10.235s. What was the sun's semidiameter ? rent diame- ter of the 3600s. : 10.235s. :: 136.64 : 0.39 a. sun, mo a or planets. Observed dura, of tran., in sees., 136.64 Reduction for solar motion, - .39 136.25 . . log. 2.134337 Dec. 22 2' 45"; cosine, 9.967021 Duration, if stationary on equa., 126.3 s. . . log. 2.101358 Minutes or seconds of time can be changed into minutes or seconds of arc, by multiplying by 15 ; therefore the diameter of the sun, at this time, subtended an arc of 1894". 5, and its gemidiameter 947".2, or 15' 47".2 ; which is the result given in the Nautical Almanac, from which any number of examples of this kind can be taken. We give one more example, for the benefit of those who may not have a Nautical Almanac. On the 30th day of December ( not material what year ), the sidereal time of the sun's diameter passing the meridian was observed to be 2m. 22.2s., or 142.2s. The sun's hourly motion in right ascension, at that time, was 11.06 s., and the declination was 23 11'. What was the sun's semi- diameter?* Ans. 16' 17".3. These observations may be made every clear day through- Extreme , iiii i , i raluesof th out the year ; and they have been made at many places, and gun , s apl>a . for many years; and the combined results show that the meter. * The following is the formula for these reductions : 15(i c)cos. D_ ~R~ Here t is the observed interval in seconds, c is the correction for the in- crease in right ascension, D is the declination, R the radius of the tables, and * is the result in seconds of arc. c is always very small ; for one hour, or 3600 s., the variation is never less than 8.976 s., nor more than 11.11 s. The former happens about the middle of September ; the lat- ter about the 20th of December. For the meridian passage of the moon, the correction c is considerable ; because the moon's increase of right ascension is comparatively very rapid. For the planets, c may t* dii- ngarded. 86 ASTRONOMY. CHAP, in. a pp aren t diameter of the sun is the same, on the same day of the year, from whatever station observed. The least semidiameter is 15' 45". 1 ; which corresponds, in time, to the first or second day of July ; and the greatest is 16' 17 ".3, which takes place on the 1st or 2d of January. Now as we cannot suppose that there is any real change in the diameter of the sun, we must impute this apparent change to real change in the distance of the body, as explained in Art. G6. Variation Therefore the distance to the sun on the 30th of Decem- )e dls " ber, must be to its distance on the first day of July, as the trie earth to number 15' 45". 1 is to the number 16' 17".3, or as the num- thesun. b er 945 i to 977.3 ; an d a ll other days in the year, the pro- portional distance must be represented by intermediate num- bers. From this, we perceive that the sun must go round the earth, or the earth round the sun, in very nearly a circle ; for were a representation of the curve drawn, corresponding to the apparent semidiameter in different parts of the orbit, and placed before us, the eye could scarcely detect its departure from a circle. ( 72.) It should be observed that the time elapsed between the greatest and least apparent diameter of the sun, or the reverse, is just half a year ; and the change in the sun's lon- gitude is 180. Eccentri- If we would consider the mean distance between the earth earths f rwt e &n *^ sun as ww % ( as * s customary with astronomers), and then now known, put x to represent the least distance, and y the greatest dis- tance, we shall have And, - - x : y : : 9451 : 9773. A solution gives *=0.98326, nearly, and y=1.01674, nearly; showing that the least j mean, and greatest distance to the sun, must be very nearly as the numbers .98326, 1., and 1.01674. The fractional part, .01674, or the difference between the extremes and mean ( when the ntean is unity ), is called th eccentricity of the orbit. SUN'S MOTION IN LONGITUDE. 8 r . The tccentricity, as just mentioned, must not be regarded as CHAP. H accurate. It is only a first approximation, deduced from the first and most simple view of the subject; but we shall, here- after, give other expositions that will lead to far more accu- rate results. In theory, the apparent diameters are sufficient to determine Eccentric* the eccentricity, could we really observe them to rigorous t ^ re ^ m d "^ exactness ; but all luminous bodies are more or less afl'ected meters only by irradiation, which dilates a little their apparent diameters ; a PP roximi and the exact quantity of this dilatation is not yet well ascertained. ( 73. ) The sun's right ascension and declination can be observed from any observatory, any clear day; and from thence we can trace its path along the celestial concave sphere above us, and determine its change from day to day ; and we find it runs along a great circle called the ecliptic, which crosses the equator at opposite points in the heavens ; and the ecliptic inclines to the equator with an angle of about 23 27' 40". The plane .of the ecliptic passes through the center of the earth, showing it to be a great circle, or, what is the same thing, showing that the apparent motion of the sun has its center in the line which joins the earth and sun. The apparent motion of the sun along the ecliptic is called Variations longitude ; and this is its most regular motion. m the dls " tance of the When we compare the sun s motion, in longitude, with its 8un> com . semidiameter, we find a correspondence at least, an apparent P ared with its variations connection. in i ongitDde< When the semidiameter is greatest, the motion in longitude . is greatest; and, when the semidiameter is least the motion in longitude is least ; but the two variations have not the same ratio. When the sun is nearest to the earth, on or about the 30th of December, it changes its longitude, in a mean solar day, 1 1' 9".95. When farthest from the earth, on the 1st of . July, its change of longitude, in 24 hours, is only 57' 11 ".48. A uniform motion, for the whole year, is found to be 59' 8".33. The ancient philosophers contended that the sun moved 88 ASTRONOMY. CHAP, m. about the earth in a circular orbit, and its real velocity uni- form ; Lut the earth not being in the center of the circle, the same portions of the circle would appear under different angles ; and hence the variation in its apparent angular motion. The result NOW jf ^ n j s j g a true v j ew O f tne su bi ec f the variation in hows that the angular angular motion must be in exact proportion to the variation in motion is in distance, as explained in the note to Art. 66; that is, 945". 1 ^portion" 6 should be to 977".3, as 57' 11".48 to 61' 9".95, if the sup- to the square position of the first observers were true. But these numbers e dls " have not the same ratio ; therefore this supposition is not satisfactory ; and it was probably abandoned for the want of this mathematical support. The ratio between 945". 1, and 0770 977".3is ..... - - - ^=1.0341, nearly: between 57' 11".48, and6l'9".95, f" 1.0694, nearly. If we square (1.0341) the first ratio, we shall have 1.06936, a number so near in value to the second ratio, that we con- clude it ought to be the same, and would be the same, pro- vided we had perfect accuracy in the observations. Law be. Thus we compare the angular motion of the sun in difie- m] d\a- ren ^ parts of its orbit ; and we always find, that the inverse square of its distance is proportional to its angular motion; and this incontestable /ad is so exact and so regular, that we lay it down as a law ; and if solitary observations do not corre- spond with it, we must condemn the observations, and not the law. ( 74.) To investigate this su eject thoroughly, we cannot avoid making use of a little geometry. Let Fig. 17 represent the solar orbit,* the sun apparently revolving about the observer at 0. The distance from to * We say solar orbit, when it is really the earth's orbit; so we speak of the sun's motion, when it is really the motion of the earth ; and it is customary, with astronomers, to speak of apparent motions as real and none object to this manner of speaking, who have a clear or en- larged view of the science for to depart from it would lead to oft- repeated and troublesome technicalities, if not to confusion of ideas Clearness does not always correspond with exactness of expression. VARIATIONS IN SOLAR MOTION. 8S any point in the or- Fig. 17 CHAP. in. bit is called the ra- dius vector; audit is| a varying quantity, conceived to sweep I round the point 0. Let D be the va- lue of tl^ radius vec- tor at any point, and rD its value at some other point, as repre- sented in the figure. Let y represent the real motion of the Variation! sun, for a very short interval of time, at the extremity of the "*' radius vector D\ and x represent the real motion, at the tion. extremity of the radius vector r D, in jthe same time. From 0, as a center, at the distance of unity, describe a circle. Put A to represent the angle under which x appears from 0; then, by observation, r 2 A is the angle under which y appears from the same point. Now, considering the sectors as triangles, we have the fol- lowing proportions : 1 : A :: rD : 0; : r*A : : D : y. From the first, - - x=rAD, From the second, - y=r 3 AD. Multiply the first of these equations by r, and we perceive that ------ yrx. This last equation shows that the real velocity of the earth The real in its orbit varies in the inverse ratio as the radius vector ; or J^ *^, jn it varies directly as the apparent diameter of the sun. its orbit . ( 75.) If we multiply rl) by x, the product will express the j^, g as a ** double of an area passed over by the radius vector in a certain rent diam* interval of time ; and if we multiply D by y, we shall have ter * the double of another area passed over by the radius vector in the same time. But the first product is rDx, and the second is the same, as we shall see by taking the value of y (r a?) ; that is rDx=rDx\ hence we announce this general law: 50 ASTRONOMY. CHAP, in. That the solar radius vector describes equal areas in equal The radius times. cribes e uli When expressed in more general terms, this is one of the reas in e- three laws of Kepler, which will be fully brought into notice qnal times. J Q ft su |j se q uen t p ar fc o f this WOrk. If we draw lines from any point in a plane, reciprocally proportional to the sun's apparent diameter, and at angles differing as the change of the sun's longitude, and then con- nect the extremities of such lines made all round the point, the connecting lines will form a curve, corresponding with an ellipse (see Fig. 18), which represents the apparent solar orbit ; and, from a review of the whole subject, we give the follow- ing summary: Laws of 1. The eccentricity of the solar ellipse, as determined from the ^7 ^ an a PP arent dwnwter of the sur, is .01674.* 2. The surfs angular velocity varies inversely as the square of its distance from the earth. 3. The real velocity is inversely as the distance. 4. The areas described by the radius vector are proportional to the times of description. (76.) We have several times mentioned, that, as far as appearances are concerned, it is immaterial whether we con- sider the sun moving round the earth, or the earth round the sun; for, if the earth is in one positipn of the heavens, the *,By making use of the 2d principle, above cited, we can compute the eccentricity of the orbit to greater precision than by the apparent diameters, because the samfr error of obser- vation on longitude would not be as proportionally great as on apparent diameter. Let E be the eccentricity of the orbit; then (1 E) is the least distance to the sun, and (\-\-E) the greatest dis- tance. Then, by observation, we have (\Ey : (1+^)2 :: 57' 11".48 : 61' 9".95; Or, (1 .ff) 2 : (1+.E) 2 :: 343148 : 366995; Or, lE : 1+E : : V 843148 : -^366996; Whence .#=.016788+. We shall give a still more accu- rate method of computing this important element. SUN'S ELLIPTICAL MOTION. <)j Bun appears exactly in Fig- 18- CHAP, in. the oppi site position, and ever y motion made by the earth must correspond to an apparent motion made by the sun. But, for the purpose of getting nearer to fact, we will now sup- pose the earth revolves round the sun in an elliptical orbit, as represented by Fig. 18. We have very much exaggerated the eccentricity of the orbit, for the purpose of bringing principles clearer to view. The greatest and least distances, from the sun to the earth, make a straight line through the sun, and cut the orbit into two equal parts. When the earth is at B, the greatest dis- tance from the sun, it is said to be in apogee, and when at A, the least distance, it is in perigee ; and the line joining the apogee and perigee is the major, or greater diameter of the orbit ; arid it is the only diameter passing through the sun, that cuts the orbit into two equal parts. Now, as equal areas are described in equal times, it follows Observa- that the earth must be just half a year in passing from apogee JJJ * ^" to perigee, and from perigee to apogee ; provided that these positions of points are stationary in the heavens, and they are so, very the 8olar a> * pogee and nearly.* perigee. If we suppose the earth moves along the orbit from D to A, and we observe the sun from D, and continue observa- tions upon it until the earth comes to C, then the longitude of the sun has changed 180; and if the time is less than * The longer axis of the orbit, or apogee point, changes position by b. very slow motion of about 12" per annum, to the eastward : but this motion must be disregarded, for the present, as well as many other mi- nute deviations, to be brought into view when we are better prepared to understand them. Those minute variations/ for short periods of time, do not sensibly affect general results. 92 ASTRONOMY. CHA. HI. half a year, we are sure the perigee is in this part of the orbit. If we continue observations round and round, and find where 180 degrees of longitude correspond with half a year, there will be the position of the longer axis ; which is sometimes called the line of the apsides. Difficulties, \y e cannot determine the exact point of the apogee or perigee, by direct observations on the sun's apparent diame- ter; for about these points the variations are extremely slow and imperceptible. If we take observations in respect to the sun's longitude, when the earth is at b, and watch for the opposite longitude, when the earth is about a, and find that the area b Da was described in little less thafti half a year, and the area aCb, in a little more than half a year, then we know that b is very near the apogee, and a very near the perigee. If we take another point, b', and its opposite, a', and find converse results, then we know that the apogee is between the points b' and b, and we can proportion to it to great exact- ness. Longitude ( 77. ) The longitude of the apogee, for the year 1801, was Md pe'rfcfT " 31/ 9 "' and > of course tne perigee was in longitude 279 31' 9". These points move forward, in respect to the stars, about 12" annually, and, in respect to the equinox, about 62" ; more exactly 61". 905, and, of course, this is their annual increase of longitude. In the year 1250, the perigee of the sun coincided with the winter solstice, and the apogee with the summer solstice ; and at that time the sun was 178 days and about 17^ hours on the south side of the equator, and 186 days and about 12^ hours on the north side ; being longer in the northern hemi- sphere than in the southern, by seven days and 19 hours: at present, the excess is seven days and near 17 hours. The year (78.) As the sun is a longer time in the northern than in the southern hemisphere, the first impression might be, that more solar heat is received in one hemisphere than in the other ; but the amount is the same ; for whatever is gained in time, is lost in distance ; and what is lost in time, is gained by a decrease of distance. The amount of heat depend 1 } OB SUN'S ELLIPTICAL MOTION. y^ the intensity multiplied by the time it is applied; and the CHAP. ra. product of the time .and distanced the sun, is the same in either hemisphere ; but the amount of heat received, for a single day, is different in the two hemispheres. ( 79.) Conceive a line drawn through the sun, at right angles to the greater diameter of the orbit D S C ( see Fig. 18 ), the point C is 8 21' from the first point of Aries; and if we observe the time occupied by the sun in describing 180 degrees of longitude, from this point (or from any point very near this point), that time, taken from the whole year, will give the time of describing the other 180 degrees. Without being very minute, we venture to state, that the A method time of describing the arc DA C is 178 days 17^ hours; and of obtaimn the time of describing the arc CBD is 186 days 12-1 hours, city of an or- But, as areas are in proportion to the times of their descrip- bit * tion ; therefore, d. h. d. h. area CSDA : area CBDS : : 178 17 : 186 12i. By taking half of the greater axis of the ellipse equal unity, and the eccentricity an unknown quantity, e, the mathematician can soon obtain analytical expressions for the two areas in question; and then, from the proportion, he can find the value of the eccentricity e : but there is a better method we only give an outside view of this, for the light it throws on the general principle. ( 80.) Now let us conceive the orbit of the earth inclosed by a circle whose diameter is the greatest dkmeter of the ellipse, as represented by Fig. 19. For the sake of simplicity we will suppose the observer at rest at the point o ( one focus of the ellipse ), and the sun * ion for fimi> really to move round on the ellipse, describing equal areas Ration "i! in equal times round the point o. an e ">p- Conceive, also, an imaginary sun to pass round the circle, describing equal angles, in equal times, round the center n/i. Now suppose the two suns to be together at the point E : they depart, one on the ellipse, the other on the circle; and, on account of both describing equal areas, in equal times, round their respective centers of motion, they will be together 94 ASTRONOMY. CH Fig. 19. at the point A, ami again at the point B } and so continue in each subsequent re- volution. The imaginary sun on the circle every- where describes equal angles in equal times ; and the true sun, on the ellipse, describes only equal areas in equal times ; but the angles will be unequal. Conceive the two suns to depart, at the same time, from the point B, and, after a certain interval of time, one is at s, the other at s'. Then we must have area oBs : area mBs' : : area ellipse : area circle. Mean and The angle Bms' is the angle the sun would make, or its increase in longitude from the apogee ; provided the angular motion of the sun was uniform. The angle Bos is its .true increase 01 longitude ; the difference between these two angles is the angle in n o. The angle Bms' is always known by the time; and if to every degree of the angle Bms' we knew the corresponding angle mno, the two would give us the angle Bos; for, Bms' mno=mon, or Bos. The angle Bms' is called the mean anomaly, and the angle B o s is called the true anomaly. The equa- The angle Bms' is greater than the angle Bos, all the n of the way f rom fo Q apogee to the perigee; but from the perigee to the apogee, the true sun, on the ellipse, is in advance of the imaginary sun on the circle. The angle m n o is called the equation of the center ; that is, it is the angle to be applied to the angle about the center m, to make it equal to the true anomaly. The angle mno depends on the eccentricity of the ellipse; and its amount is put in a table corresponding to every true air ECCENTRICITY OF ORBIT. 95 degree of the mean anomaly ; suhtractive, from the apogee to CHJU-. ni the perigee, and additive from the perigee to the apogee.* (81.) Again: conceive the two suns to set out from the same The great, point. B: and as the angle Bms' increases uniformly, it will est e i uatlon J \ ofthecentei increase and become greater and greater than the angle h o s, s i ves the ec- until the true sun attains its mean angular motion, and no centncit y of mi i i i longer. I hen the angle m n o attains its greatest value, and, at that time the side mn=no, and the point n is over the center of o m, and o s' is a mean proportional between o B ind o A. That is, when the sun, or any planet, attains the greatest equation of the center, the true sun is very near the txtremity of the shorter axis of the ellipse : o, the greatest equation of the center, can be determined by observation ; and, from the greatest equation, we have the most accurate method of computing the eccentricity of the ellipse, as we may see by the note below. f t Let C (Fig. 20) be the place of the true sun, and G\ the place of the imaginary! sun ; the line m F cuts off equal portions of the circle and the ellipse. Then we have to make the sector] m F G to the triangle om C ; ^^^^^^^ as the circle is to the ellipse. Now let mBa, mC=b, vm=ea, 5 r=3il416; Then, the area of the circle is *a? ; the area of the ellipse is *ab ; that of the sector is ( OF)^. and of the triangle ^-. r Hence,- 5? : O * By a mere mechanical contrivance, the modern astronomical tables are so arranged, that all corrections are rendered additive ; so that the mechanical oporator cannot make a mistake, as to signs, and he may continue to work without stopping to think. These arrangements have their advantages, but they cover up and obscure principles. 96 A S T R O N M Y. CHAP. ni. When once the eccentricity of any planetary ellipse be- comes known, the equation of the center, corresponding to all degrees of the mean anomaly, can be computed and put into a table for future use ; but this labor of constructing tables belongs exclusively to the mathematician. Method of Or,- - eab : (GF)a :: b : a; deducing the eccentricity Or, CO. I GF II 1 I 1 . greatest e- Consequently, GF=ea, and FG=om ; which shows that the quation of angle o Cm is nearly equal to Fm G, unless it is a very eccen- tric ellipse. Now we must compute the number of degrees in the arc FG. tfhe whole circumference is %7ra. Therefore, 2a : ea : : 360 : arcFG: Hence, - - - arc FG= -- =angle nm C. rr But the angle onm=nm C-\-n Cm=2nm C, nearly; Therefore, =2 nmC=onm= greatest equation of center, nearly. But the greatest equation of the center, for the solar orbit, is, by observation, 1 55' 30" ; and as the sun has not quite its greatest equation of the center, when at the point C, it will be more accurate to put =l 55' 24". From this equation, it is true, we have only the approxi- mate value of e ; but it is a very approximate value, and suffi- ciently accurate. Reducing both members to seconds, f and we have, 3600-360 *=6924?r, and e =0.0167842. The greatest equation of the center is at present diminish ing at the rate of 17". 17 in one hundred years: this corre- sponds to a diminution of eccentricity by 0.00004166. whir). is determined by a solution of the following equation : CHANGE OF SEASONS. 97 CHAPTER IV. THE CAUSES OF THE CHANGE OF SEASONS. ( 82. ) THE annual revolution of the earth in its orbit, CHA*. iv combined with the position of the earth's axis to the plane of its orbit, produces the change of the seasons. If the axis were perpendicular to the plane of its orbit, The cause there would be no change of seasons, and the sun would then ^ thechan t of season*. be all the while in the celestial equator. This will be understood by Fig. 21. Conceive the plane of the paper to be the plane of the earth's orbit, and conceive the several representations of the earth's axis, JV/S', to' be in- clined to the paper at an angle of 66 32'. Fig. 21. In all representations of NS, one half of it is supposed to bo above the paper, the other half below it. yS is always parallel to itself; that is, it is always in the same position* always at the same inclination to the plane Except minute variations, which it would be improper to notice in this part of the work. 7 08 ASTRONOMY. CHAP, iv. O f it s orbit always directed to the same point in the hea- vens, in whatever part of the orbit it may be. The plane of the equator, represented by Eq, is inclined to the plane of the orbit by an angle of 23 28'. inspecting the figure, the reader will gather a clearer f ^e subject than by whole pages of description : he will perceive the reason why the sun must shine over the north pole, in one part of its orbit, and fall as far short of that point when in the opposite part of its orbit ; and the number of degrees of this variation depends, of course, on the position of the axis to the plane of the orbit. Positioner Now conceive the line N S to stand perpendicular to the to plane of the paper, and continue so ; then Eq would lie on change of the paper, and the sun would at all times be in the plane of aions. t h e e q ua tor, and there would be no change of seasons. If N S were more inclined from the perpendicular than it now is, then we should have a greater change of seasons. By inspecting the figure, we perceive, also, that when it is summer in the northern hemisphere, it is winter in the southern ; and conversely, when it is winter in the northern, it is summer in the southern. When a line from the sun makes a right angle with the earth's axis, as it must do in two opposite points of its orbit, the sun will shine equally on both poles , and it is then in the plane of the equator ; which gives equal day and night the world over. Equal days and nights, for all places, happen on the 20th of March of each year, and on the 22d or 23d of September. At these times the sun crosses the celestial equator, and is said to be in the equinox. The equi. The longitude of the sun, at the vernal equinox, is ; and octiai and at t ] ie autumnal equinox, its longitude is 180. point" The time of the greatest north declination is the 20th of June ; the sun's longitude is then 90, and is said to be at the summer solstice. The time of the greatest south declination is the 22d of December ; the sun's longitude, at that time, is 270, and is said to be at the winter solstice. CHANGES OF SEASONS. 99 By inspecting the figure, we perceive, that when the earth CHAP. iv. is at the summer solstice, the north pole, N, and a conside- LCD* se rable portion of the earth's surface around, is within the en- i0ns of snn * lightened half of the earth ; and as the earth revolves on its aLkness "It axis JV/S', this portion constantly remains enlightened, giving and abou a constant day or a day of weeks and months duration, l e po e8 ' according as any particular point is nearer or more remote from the pole: the pole itself is enlightened full six months in the year, and the circle of more than 24 hours constant sunlight extends to 23 28' from the pole (not estimating the effects of refraction). On the other hand, the opposite, or south pole, S, is in a long season of darkness, from which it can be relieved only by the earth changing position in its orbit. " Now, the temperature of any par$ of the earth's surface depends mainly, if not entirely, on its exposure to the sun's ea rth. rays. Whenever the sun is above the horizon of any place, that place is receiving heat ; when below, parting with it, by. the process called radiation; and the whole quantities re- ceived and parted with in the year must balance each other at every station, or the equilibrium of temperature would not be supported. Whenever, then, the sun remains more than 12 hours above the horizon of any place, and less beneath, the general temperature of that place will be above the ave- rage ; when the reverse, below. As the earth, then, moves from A to B, the days growing longer, and the nights shorter in the northern hemisphere, the temperature of every part of that hemisphere increases, and we pass from spring to sum- mer, while at the same time the reverse obtains in the southern hemisphere. As the earth passes from B to C, the days and nights again approach to equality the excess of temperature in the northern hemisphere, above the mean state, grows less, as well as its defect in the southern ; and at the autumnal equinox, C, the mean state is once more attained. From thence to D, and, finally, round again to A, all the same phe- nomena, it is obvious, must again occur, but reversed; it being now winter in the northern, and summer in the southern hemisphere." 100 ASTKONOMY. CHAP. rv. The inquiry is sometimes made why we do not have the warmest weather about the summer solstice, and the coldest weather about the time of the winter solstice. Times of This would be the case if the sun immediately ceased to tem give extra warmth, on arriving at the summer solstice ; but if it could radiate extra heat to warm the earth three weeks, before it came to the solstice, it would give the same extra heat three weeks after ; and the northern portion of the earth must continue to increase in temperature as long as the sun continues to radiate more than its medium degree of heat over the surface, at any particular place. Conversely, the whole region of country continues to grow cold as long as the sun radiates less than its mean annual degree of heat over that region. The medium degree of heat, for the whole year, and for all places, of course, takes place when the sun is on the equator; the average temperature, at the time of the two equinoxes. The medium degree of heat, for our .northern summer, considering only two seasons in the year, takes place when the sun's declination is about 12 degrees north ; and the medium degree of heat, for winter, takes place when the sun's declination is about 12 degrees south ; and if this be true, the heat of summer will begin to decrease about the 20th of August, and the cold of winter must essen- tially abate on or about the 16th of February, in all northern latitudes. CHAPTER V. EQUATION OF TIME. ( 83.) WE now come to one of the most important subjects in astronomy the equation of time. Without a good knowledge of this subject, there will be constant confusion in the minds of the pupils ; and such is the nature of the case, that it is difficult to understand even the/wfe, without investigating their causes. Bidrai Sidereal time has no equation ; it is uniform, and, of itself time iwrftct perfect and complete. EQUATION OF TIME. 101 The time, by a perfect clock, is theoretically perfect and CHAI. iv. complete, and is called mean time. The time, by the sun, is not uniform; and, to make it Solar time agree with the perfect clock, requires a correction a quan- notumform tity to make equality; and this quantity is called the equa- tion of time.* If the sun were stationary in the heavens, like a star, it would come to the meridian after exact and equal intervals of time; and, in that case, there would be no equation of time. If the sun's motion, in right ascension, were uniform, then it would also come to the meridian after equal intervals of time, and there would still be no equation of time. But ( speaking in relation to appearances ) the sun is not station- ary in the heavens, nor does it move uniformly ; therefore it cannot come to the meridian at equal intervals of time, and, of course, the solar days must be slightly unequal. When the sun is on the meridian, it is then apparent noon, Mean sod for that day : it is the real solar noon, or, as near as may be, *PP ar5nt half way between sunrise and sunset ; but it may not be noon by the perfect clock, which runs hypothetically true and uniform throughout the whole year. A fixed star comes to the meridian at the expiration of uvery 23 h. 56 m. 04.09 s. of mean solar time ; and if the sun were stationary in the heavens, it would come to the meridian after every expiration of just that same interval. But the sun increaws its right ascension every day, by its apparent eastward motion ; and this increases the time of its coming to the meridian ; and the mean interval between its successive transits o^er the meridian is just 24 hours ; but the actual intervals' are variable some less, and some more than 24 hours. On and about the 1st of April, the time from one meridian of the win to another, as measured by a perfect clock, is 23 h. 59 m 52.4 s. ; less than 24 hours by about 8 seconds. Here, theo, the sun and clock must be constantly separating. On In astronomy, the term equation is applied to all corrections to convert a mean to its true quantity. 102 ASTRONOMY. CHAP. v. and about the 20th of December, the time from one meridian of the sun to another is 24 h. Om. 24.3s., more than 24 seconds over 24 hours ; and this, in a few days, increases to minutes and thus we perceive the fact of equation of time. Equation To detect the law of this variation, and find its amount, of time the we must/ se p ara t e the cause into its two natural divisions. result of two causes. 1. The unequal apparent motion of the sun along the ecliptic. 2. The variable inclination of this motion to the equator. If the sun's apparent motion along the ecliptic were uni- form, still there would be an equation of time ; for that mo- tion, in some parts of the orbit, is oblique to the equator, and, in other parts, parallel with it; and its eastward motion, in right ascension, would be greatest when moving parallel with the equator. From the first cause, separately considered, the sun and clock would agree two days in a year the 1st of July and the 30th of December. From the second cause, separately considered, the sun and clock agree four days in a year the days when the sun crosses the equator, and the days he reaches the solstitial points. When the results of these two causes are combined, the sun and clock will agree four days in the year ; but it is on neither of those days marked out by the separate causes ; and the intervals between the several periods, and the amount of the equation, appear to want regularity and symmetry. Days in The four days in the year on which the sun and clock 1116 h^thl* a <= ree ' * na * * s ' snow noon at the same instant, are April 15th, sun and June 16th, September 1st, and December 24th. The greatest amount, arising from the first cause, is 7m. 42s., and the greatest amount, from the second cause, is 9m. 53 s. ; but as these maximum results never happen exactly at the same time, therefore the equation of time can never amount to 17m. 35s. In fact, the greatest amount is 16m. 17 s., and takes place on the 3d of November ; and, for a long time to come, the maximum value will take place on the same day of each year ; but, in the course of ages, it will vary in its amount and in the time of the year in which the sun and EQUATION OF TIME. 10 clock agree, in consequence of the slow and gradual change CHAP. ?. in the position of the solar apogee. (See Art. 77.) ( 84. ) The elliptical form of the earth's orbit gives rise to The cause. the unequal motion of the earth in its orbit, and thence to the tlon of th * un' center, apparent unequal motion of the sun in the ecliptic; and this and the firm same unequal motion is what we have denominated the first P art of tho cause of the equation of time. Indeed, this part of the equa- J^ 1 " have tion of time is nothing more than the equation of the center a common ( 80), changed into time at the rate of four minutes to a degree. The greatest equation for the sun's longitude (81, note ), is by observation 1 55' 30"; and this, proportioned into time, gives 7 m. 42s., for the maximum effect in the equation of time arising from the sun's unequal motion. When the sun departs from its perigee, its motion is greater than the mean rate, and, of course, comes to the meridian later than it other- wise would. In such cases, the sun is said to be slow and it is slow all the way from its perigee to its apogee ; and fast in the other half of its orbit For a more particular explanation of the second cause, we must call attention to Fig. 22 Let V < ^ (Fig. Fig 22. 22 ) represent the ^ ecliptic, and the equator. By the first cor- rection, the apparent j motion along the ecliptic is rendered uniform ; and the sun is then supposed to pass over equal spaces in equal intervals of time along the arc qp 55. But equal spaces of arc, on the ecliptic, do not correspond with equal spaces on the equator. In short, the points on the ecliptic must be reduced to corresponding points on the equator. For instance, the number of degrees represented by T on 104 ASTRONOMY. , v. the ecliptic, is greater than to the same meridian along the equator. The difference between $>', turned into time, is the equation of time arising from the obliquity of the ecliptic corresponding to the point S. At the points qp, 2S, and =o=, and also at the southern tropic, the ecliptic and the equator correspond to the same meridian ; but all other equal distances, on the ecliptic and equator, are included by different meridians. HOW t To compute the equation of time arising from this cause, compute the we mu st solve the spherical triangle <>S S' ; 9P /Sis the sun 'a of the eqna- longitude, and the angle at T is the obliquity of thfc elliptic, Uonoftim. and at S' is a right angle. Assume any longitude, a? 32, 35, or 40, or any other number of degrees, and compute the base. The difference between this base and the pun's longitude, converted into time, is the quantity sought corre- sponding to the assumed longitude ; and by assuming every degree in the first quadrant, and putting the result in a table, we have the amount for every degree of the entire circle, for all the quadrants are symmetrical, and the same distance from either equinox will be the same amount. What is The perfect clock, or mean time, corresponds with the meant by sun equator; and as uniform spaces along the equator, near the f clock. point T, will pass over more meridians than the same num- ber of equal spaces on the ecliptic ; therefore the sun, at S, will be fast of clock, or come to the meridian before it ia noon by the clock and this will be true all the way to the tropic, or to the 90th degree of longitude, where the sun and. clock will agree. In the second quadrant, the sun will come to the meridian after the clock has marked noon. In the third qua- drant the sun will again be fast; and, in the fourth quadrant, again slow of clock. It will be observed, by inspecting the figure, that what the sun loses in eastward motion, by oblique direction near the equator, is made up, when near the tropics, by the diminished distances between the meridians. For a more definite understanding of this matter, we givt the following table. EQUATION OF TIME. 105 Table showing the separate results of the two causes for the equa- CHAP. v. (ion of time, corresponding to every f.fth day of the second years after leap year ; bat is nearly correct for any year. 1st cause. Sun slow of Clock. 2tl cause. Sun slow ofClock. 1st cause. Sun fast. 2d cause. Sun slow. m. s. m. s. m. 8. m. s. January 5 10 41 1 22 5 8 6 35 July 1 40 3 32 5 8 15 2 2 7 48 12 1 19 6 35 20 2 41 8 45 17 1 57 7 48 25 3 19 9 26 22 2 35 8 45 29 3 56 9 49 28 3 12 9 26 Feb. 3 4 30 9 53 Aug. 2 3 47 9 49 8 5 2 9 40 7 4 21 9 53 13 5 32 9 9 12 4 52 9 40 18 5 39 8 23 17 5 22 9 9 23 6 24 7 22 22 5 50 8 23 28 6 45 6 9 .' 28 6 14 7 22 March 5 7 3 4 46 Sept. 2 6 36 6 9 10 7 18 3 15 7 6 56 4 46 15 7 29 1 39 12 7 12 3 15 20 7 37 sun fast 17 7 24 1 39 25 7 42 1 39 23 7 34 sun fast 30 7 42 3 15 28 7 40 1 39 April 4 7 40 4 46 Oct. 3 7 42 3 15 9 7 34 6 9 8 7 40 4 46 14 7 24 7 22 13 7 34 6 9 19 7 12 8 23 18 7 24 7 22 24 6 56 9 9 23 7 12 8 23 30 6 36 9 40 28 6 56 9 9 May 5 6 14 9 53 Nov. 2 6 36 9 40 10 5 50 9 49 7 6 14 9 53 15 5 22 9 26 12 5 50 9 49 20 4 52 8 45 17 5 22 9 26 26 4 21 7 48 ' 22 4 52 8 45 31 3 47 6 35 27 4 22 7 48 Ji'Me 5 3 12 5 8 Dec. 2 3 47 6 35 10 2 35 3 32 7 3 12 5 8 16 1 57 1 48 12 2 35 3 32 21 1 19 sun slow 17 1 57 1 48 26 40 1 48 21 1 19 sun slow. 26 40 1 48 By this table, the regular and symmetrical result of each , cause is visible to the eye ; but the actual value of the equa- preceding tion of time, for any particular day, is the combined results tabl * of these two causes. Thus, to find the equation of time for the 5th day of March, we look at the table and find that 106 ASTRONOMY. CHAP, v The first cause gives sun slow, - - - 7m. 3 a. The second, " sun slow, - - - 4 46 Their combined result (or algebraic sum) is 11 49 slow. That is , the sun being slow, it does not come to the meridian until llm. 49 s. after the noon shown by a perfect clock ; but whenever the sun is on the meridian, it is then noon, apparent time ; and, to convert this into mean time, or to set the clock, we must add 11 m. 49s. Use of the By inspecting the table, we perceive, that on the 14th of J^* ' 9 April the two results nearly counteract each other ; and con- sequently the sun and clock nearly agree, and indicate noon at the same instant. On the 2d of November the two results unite in making the sun fast ; and the equation of time is then the sum of 6 36 and 9 40, or 16 m. 16 s. ; the maximum result. The sun at this time being fast, shows that it comes to the meridian 16 m. 16 s. before twelve o'clock, true mean time ; or, when the sun is on the meridian, the clock ought to show 11 h. 43 m. 44 s. ; and thus, generally, when the sun is fast, we must subtract the equation of time from apparent time, to obtain mean time ; and conversely, when the sun is slow. As no clock can be relied upon, to run to true mean time, or to any exact definite rate, therefore clocks must be fre- quently rectified by the sun. We can observe the apparent time, and then, by the application of the equation of time, we determine the true mean time. A table for (85.) As the sun has a particular motion, corresponding equation of_t every particular point on the ecliptic, and, at the same ing* on^the tmie * ne particular point on the ecliptic has a definite rela- m's long! tion to the equator, therefore any point, as S (Fig. 22), on be the ecliptic, has the two corrections for the equation of time ; consequently a table can be formed for the equation of time, depending on the longitude of the sun; and such a table would be perpetual, if the longer axis of the solar orbit did not change its position in relation to the equinoxes. But aa that change is very slow, a table of that kind will serve for PLANETARY MOTIONS. 107 n-any years, with a very trifling correction, and such a table CHAP, v is to be found in many astronomical works. It is very important that the navigator, astronomer, and Utility of dock regulator, should thoroughly understand the equation of time ; and persons thus occupied pay great attention to it ; but most people in common life are hardly aware of its ex- istence. CHAPTER VI. THE APPARENT MOTIONS OF THE PLANETS. (86.) WE have often reminded the reader of the great CHAP. VL regularity of the fixed stars, and of their uniform positions in relation to each other ; and by this very regularity and con- stancy of relative positions, we denominate them fixed; but there are certain other celestial bodies, that manifestly change their positions in space, and, among them, the sun and moon are most prominent. In previous chapters, we have examined some facts con- Recapit* cerning the sun and moon, which we briefly recapitulate, as at!0n * follows : 1. That the sun's distance from the earth is very great; but at present we cannot determine how great, for the want of one element its horizontal parallax. 2. Its magnitude is much greater than that of the earth. 3. The distance between the sun and earth is slightly va- riable ; but it is regular in its variations, both in distance and in apparent angular motion. 4. The moon is comparatively very near the earth; its distance is variable, and its mean distance and amount of variations are known. It is smaller than the earth, although, to the mere vision, it appears as large as the sun. The apparent motions of both sun and moon are always in one direction; and the variations of their motions are never far above or below the mean. other cefc* But there are several other bodies that are not fixed stars ; tw bodies. 108 ASTRONOMY. C ^__Z!' an< ^ a l tnou gh no * as conspicuous as the sun and moon, hava been known from time immemorial. They appear to belong to one family; but, before the true system of the world was discovei ed, it was impossible to give any rational theory concerning their motions, so irregular and erratic did they appear; and this very irregularity of their apparent motions induced us to delay our investigations concerning them to the present chapter. Th plan. j n general terms, these bodies are called planets and *"*" there are several of recent discovery and some of very recent discovery; but as these are not conspicuous, nor well known, all our investigations of principles will refer to the larger planets, Venus, Mars, Jupiter, and Saturn. We now commence giving some observed fads, as extracted from the Cambridge astronomy The morn. ( 87.) " There are few who have not observed a beautiful mg and even- g ar j n ^ wes ^ a little after sunset, and called, for this rea- son, the evening star. This star is Venus. If we observe it for several days, we find that it does not remain constantly at the same distance from the sun. It departs to a certain distance, which is about 45, or {th of the celestial hemi- sphere, after which it begins to return ; and as we can ordi- narily discern it with the naked eye only when the sun is below the horizon, it is visible only for a certain time imme- diately after sunset. By and by it sets with the sun, and then we are entirely prevented from seeing it by the sun's light. But after a few days, we perceive, in the morning, near the eastern horizon, a bright star which was not visible before. It is seen at first only a few minutes before sunrise, and is hence called the morning star. It departs from the sun from day to day, and precedes its rising more and more ; but after departing to about 45, it begins to return, and rises later each day ; at length it rises with the sun, and we cease to distinguish it. In a few days the evening star again appears in the west, very near the sun ; from which it departs in the same manner as before ; again returns ; disappears for a short time ; and then the morning star presents itself. These alternations, observed without interruption for more PLANETARY MOTION 109 than 2000 years, evidently indicate that the evening and CHAP TL morning star are one and the same body. They indicate, also, that this star has a proper motion, in virtue of which it oscil- lates about the sun, vsometimes preceding and sometimes fol- lowing it. These are the phenomena exhibited to the naked eye ; but the admirable invention of the telescope enables us to carry our observations much farther." ( 88. ) On observing Venus with a telescope, the irradiation Th pha* is, in a great measure, taken away, and we perceive that it has phases, like the moon. At evening, when approaching the sun, it presents a luminous crescent, the points of which are from the sun. The crescent diminishes as the planet draws nearer the sun ; but after it has passed the sun, and appears on the other side, the crescent is turned in the other direction ; the enlightened part always toward the sun, showing that it receives its light from that great luminary. The crescent now gradually increases to a semicircle, and finally to a full fVeni and circle, ag the planet again approaches the sun ; but, as the it8 ppnt crescent increases, the apparent diameter of the planet diminishes / have corrtc and at every alternate approach of the planet to the sun, the spending phase of the planet is full, and the apparent diameter small; c ange8 " and at the other approaches to the sun, the crescent diminishes down to zero, and the apparent diameter increases to its maximum. When very near the sun, however, the planet is lost in the sunlight ; but at some of these intervals, between disappearing in the evening, and reappearing in the morning, it appears to run over the sun's disc as a round, black spot ; giving a fine opportunity to measure its greatest apparent diameter.* When Venus appears full, its apparent diameter is not more than 10", and when a black spot on the sun, it is 59". 8, or very nearly V. Hence its greatest distance must be, to its least distance, as 59". 8 to 10, or nearly as 6 to 1. * Astronomers do not measure the apparent diameters of the planets by the process described for the sun and moon, because they pass the meridian too quickly. Most of them will pass the meridian in a small fraction of a second. They use 110 ASTRONOMY. vi. ( 89. ) When we come to form a theory concerning the real motion of this planet, we must pay particular attention to the fact, that it is always in the same part of the heavens Venus ai- as the sun never departing more than 47 on each side of ivnys near ft called its greatest elongation. In consequence of being always in the neighborhood of the sun, it can never come to the meridian near midnight. Indeed, it always comes to the Greatest mer i(ji ail w ithin three hours 20 minutes of the sun, and, of spect *o the m relation to the stars, is very irregular sometimes its , t ars. motion is rapid sometimes slow sometimes direct some- times stationary, and sometimes retrograde;* but the direct motion prevails, and, as an attendant to the sun, and in its own irregular manner, as just described, it appears to tra- verse round and round among the stars. ( 91. ) But Venus is not the only planet that exhibits the Mercwy appearances we have just described. There is one other, and 8imilar in a11 8i)pca.ra.nccs only one Mercury ; a very small planet, rarely visible to the to v en as. naked eye, and not known to the very ancient astronomers. Whatever description we have given of Venus applies to Mer- cury, except in degree. Its variations of apparent diameter are not so great, and it never departs so far from the sun ; and the interval of time, between its vibrations from one side to the other of the sun, is much less than that of Venus. (92.) These appearances clearly indicate that the sun must be A concim- the center, or near the center, of these motions, and not the earth ; ilon * and that Mercury must revolve in an orbit within that of Venus. So clear and so unavoidable were these inferences, that even the ancients (who were the most determined advocates for the immobility of the earth, and for considering it as the principal object in creation the center of all motion, etc.) were compelled to admit them ; but with this admission, they contended, that the sun moved round the earth, carrying these planets as attendants. (93.) By taking observations on the other planets, the an- The appa cient astronomers found them variable in their apparent diam- rent diam* earliest circumstance which drew his attention to astronomy, was the regular appearance, at a certain hour, for several successive days, of a considerable star, through the shaft of a chimney." Herschd'a Astro- nomy. * In astronomy, direct motion is eastward among the stars ; station" ary is no apparent motion, in respect to the stars ; and retrograde is a westward motion. 112 ASTRONOMY. CHAP. vi. eters, and angular motions; so much so, that it was impossible ter of the to reconcile appearances with the idea of a stationary point of planets are observation ; unless the appearances were taken for realities, and that was against all true notions of philosophy. The planet Mars is most remarkable for its variations ; and the great distinction between this planet and Venus, is, that it does not always accompany the sun; but it sometimes, yea, at regular periods, is in the opposite part of the heavens from the sun called Opposition at which time it rises about sunset, and comes to the meridian about midnight. Th earth The greatest apparent diameter of Mars takes place when uTr'ofiu mo' *ke pl anet * 8 i opposition to the sun, and it is then 17".l; and tion. its least apparent diameter takes place when in the neighbor- hood of the sun, and it is then but about 4"; showing that the sun, and not the earth, is the center of its motion. Systematic The general motion of all the planets, in respect to the ineguianties g |- arg j s direct ; that is, eastward; but all the planets that attain opposition to the sun, while in opposition, and for some time before and after opposition, have a retrograde motion and those planets which show the greatest change in appa- rent diameter, show also the greatest amount of retrograde motion and all the observed irregularities are systematic in their irregularities, showing that they are governed, at least, by constant and invariable laws. If the earth is really sta- tionary, we cannot account for this retrograde motion of tho planets, unless that motion is real; and if real, why, and how can it change from direct to stationary, and from station- ary to retrograde, and the reverse? Retrograde But if we conceive the earth in motion, and going the same motion of the wg y y^fa the planet, and moving more rapidly than the planet, counted for. then the planet will appear to run back ; thai is, retrograde. And as this retrogradation takes place with every planet, when the earth and planet are both on the same side of the sun, and the planet in opposition to the sun ; and as these cir- cumstances take place in all positions from the sun. it is a suf- ficient explanation of these appearances ; and conversely, then, these appearances show the motion of the earth. (94.) When a planet appears stationary, it must be really PLANETARY MOTION. 113 so, or be moving directly to or from the observer. And if it CHAP vi. be moving to or from the observer, that circumstance will be rianeu ner- indicated by the change in apparent diameter; and observa- ' tationaj y lions confirm this, and show that no planet is really station- ary, although it may appear to be so. (95.) If we suppose the eartb. to be but one of a family of The earth bodies, called planets all circulating round the sun at dif- P 1&Mt ' ferent times in the order of Mercury, Venus, Earth, Mars (omitting the small telescopic planets), Jupiter, Saturn, Her- schel, or Uranus, ?e can then give a rational and simple ac- count for every appearance observed, and without discussing the ancient objections to the true theory of the solar system, we shall adopt it at once, and thereby save time and labor, and introduce the reader into simplicity and truth. (96.) The true solar system, as now known and acknow- ledged, is called the Copernican system, from its discoverer, Copernicus, a native of Prussia, who lived some time in the tem. fifteenth century. But this theory, simple and rational as it now appears, and Lost and re. capable of solving every difficulty, was not immediately adop- lv * d * ted ; for men had always regarded the earth as the chief object in God's creation ; and consequently man, the lord of crea tion, a most important being. But when the earth was hurled from its imaginary, dignified position, to a more humble place, it was feared that the dignity and vain pride of man roust tall with it ; and it is probable that this was the root of the opposition to the theory. So violent was the opposition to this theory, and so odious Galileo and would any one have been who had dared to adopt it, that it appears to have been abandoned for more than one hundred years, and was revived by Galileo about the year 1620, who, to avoid persecution, presented his views under the garb of a dialogue between throe fictitious persons, and the points left undecided. But the caution of Galileo was not sufficient, or his dia- logue was too convincing, for it woke up the sacred guardians of truth, and he was forced to sign a paper denouncing the theory as heresy, on the pain of perpetual imprisonment. 8 114 ASTRONOMY, CHAP. VL But this is a digression. With the history of astronomy, an interesting as it may be, we design to have little to do, and to proceed only with the science itself. Distinction between in- terior and su- perior plan- ets. CHAPTER VII. FIRST APPROXIMATIONS TO THE RELATIVE DISTANCES OF THB PLANETS FROM THE SUN. HOW THE RESULTS ARE OBTAINED. (97.) BEING convinced of the truth of the Copernican system, the next step seems to be, to find the periodical times of the revolutions of the planets, and at least their relative distances from the sun. Mercury and Venus, never coming in opposition to the sun, b u t revolving around that body in orbits that are within that f tne earfcn ' are called inferior planets. Those that come in opposition, and thereby show that their orbits are outside of the earth, are called superior planets. We shall show how to investigate and determine the posi- tion of one inferior planet ; and the same principles will be sufficient to determine the position of any inferior planet. It will be sufficient, also, to investigate and determine the orbit of one superior planet ; and if that is understood, it may be considered as substantially determining the orbits of all the superior planets ; and after that, it will be sufficient tv state results. For materials to operate with, we give the following table of the planetary irregularities ( so called ) drawn from obser- vation : Planet*. Greatest Apparent Diameters. Least Apparent Diameters. Angular Dist. from Sun at the instant of being stationary. Mean arc of Retrogradation. Mercury. Venus. Earth. Mars. Jupiter. Saturn. Uranus. 11.3 59.6 17*1 44.5 20.1 4.1 5.0 9.6 3.6 30.1 16.3 3.7 18 00 28 48 136 48 115 12 108 54 103 30 u f 13 30 16 12 16 12 9 54 6 18 3 36 PLANETARY MOTION. 115 Planets. Mean Duration of the Retro- grade Motion. Mean Duration of the Synodic Revolution, or interval between two successive oppositions. Mercury. 23 days. 118 days. Venus. 42 " 584 " Earth. Mars. 73 " 780 " Jupiter. 121 " 399 Saturn. 139 " 378 Uranus. 151 " 370 CHAP. VIL In the preceding table, the word mean is used at the head why the of several columns, because these elements are variable word at:AK , . . should be sometimes more and sometimes less, than the numbers here use d. given which indicates that the planets do not revolve in cir- cles round the sun, but most probably in ellipses, like the orbit of the earth. On the supposition, however/ that the planets revolve in circles ( which is not far from the truth ), the greatest and least apparent diameters furnish us with sufficient data to compute the distances of the planets from the sun in relation to the distance of the earth, taken as unity* (98.) In addition to the facts presented in the preceding The eionga. table, we must not fail to note the important element of the tion * of Mer - dongaiions of Mercury and Venus. This term can be applied nus> to no other planets. It is very variable in regard to Mercury showing that This element the orbit of that planet is quite elliptical. The variation is much less in regard to Venus, showing that Venus moves shows. round the sun more nearly in a circle. The least extreme elongation of Mercury is - 17 37'. The greatest " " " is - 28 4'. The mean (or the greatest elongation when both the earth and planet are at their mean distances from the sun ) is - - - 22 46'. The least extreme elongation of Venus is - 44 58'. The greatest " " " is - 47 30'. The mean (or at mean distances), is - 46 30'. The least extremes must happen when the planet is in ita perigee and the earth in its apogee, and the greatest when the earth is in perigee and the planet in apogee; but it is and 116 A3TRONOMY. CHAH. vii. very seldom that these two circumstances take place at the same time. HOW to Relying on these facts as established by observations, wo can easily deduce the relative orbits of Mercury and Venus. Let .V (Fig. 23) re- present the sun, E the earth, V Venus. Conceive the planet to pass round the sun in the direction of A ^ 7 B. The earth moves also in the same direction, but not so rapidly as Venus. Now it is clearly evi- dent, from inspection, that when the planet is passing by the earth, as at B, it will appear to pass along in the hea- vens in the direction of m to n. But when the planet is passing along in its orbit, at A, and the earth about the position of E, the planet will appear to pass in the direction of n to w. When the planet is at V, as represented in the figure, its absolute motion is nearly toward the earth, and, of course, its appearance is nearly stationary. What to jj. ' ls l( j )SO lui e iy stationan/ only at one point, and even then understand . . , by station- but for a moment ; and that point is where its apparent nio- ry. tion changes from direct to retrograde, and from retrograde to direct; which takes place when the angle SE V is about 29 degrees on each side o*' the line /'. When the line E V touches the circumference A VB, the angle E \\ or angle of elongation, is then greatest ; and the triangle SE Vis right angled at V; and if SE is made ra- dius, S V will be the sine of the angle SE V. Bu* the line S E is assumed equal to unity, and then S V PLANETARY MOTION. jj 7 will be thfs natural sine of 46 20', and can be taken out of CHAP vu any table of natural sines ; or it can be computed by loga- rithms, and the result is .72336. For the planet Mercury, the mean of the same angle is *2"2 46'; and the natural sine of that angle, or the mean radius of the planet's orbit, is .38698. Thus we have found the relative mean distances of three planets from the sun, to stand as follows: Mercury, - 0.38698 Venus, - - - 0.72336 Earth, - - 1.00000 ( 99. ) If the orbits were perfect circles, then the angle Thfl ' bit SE V, of greatest elongation, would always be the same; nd Venn* but it is an observed fact that it is not always the same ; not circles, therefore the orbits are not circles ; and when S V is least, and S E greatest, then the angle of elongation is least ; and conversely, when S V is greatest and S E least, then the angle of elongation is the greatest possible ; and by observing in what parts of the heavens the greatest and least elongations take place, we can approximate to the positions of the longer axis of the orbits. ( 100. ) By means of the apparent diameters, we can also Comimta- find the approximate relations of their orbits. For instance, J^ r '" when the planet Venus is at B, and appears on the sun's rent diamo- disc, its apparent diameter is 59".6 ; and when it is at A, or ter$ as near A as can be seen by a telescope, its apparent diame- ter is 9".6. Now put SB=xj then EB=lx, and AE=\-\-x. By Art. 66, lx : 1+x : : 96 : 696; Hence, - - - - *=0.72254. By a like computation, the mean distance of Mercury from tne sun is 0.3864. (101.) To determine the mean relative distances of the superior planets from the sun, we proceed as follows : Let S (Fig. 24) represent the sun, E the earth, and J/"one of the superior planets, say Mars. It is easy to decide, from observation, when the planet is in opposition to the sun. 118 ASTRONOMY. CHAP. V!i. Fig. 24 This gives the position of S, E, and M, in one right line, in respect to longitude. Now by knowing the true angu- lar motion of the earth about the sun (73), and the mean angular mo- tion of the planet, * v:o can determine the anglt 1 mSe, corresponding to any definite future time ; for, by the motion of the earth round the sun, we can determine the angle E Se; and by the mo- tion of the planet in the same time, we can determine the angle M S m ; and the dif- Jjy means of apparent diameters, we can determine the values of the orbit. When the planet is in opposition to the the sun de- sun, at E( Fig. 24), measure its apparent diameter; and, tera:ned by a f ter a defimte time, when the earth is at e, measure the ap- tion limits parent diameter again, and observe the angle S cm. Pro- apparent =690f days. The true time is 686.97964; showing an error of a little more than three days ; but this is not a great error, consider- ing the remoteness of the data, and the want of minuteness .and unity in the supposed observations. Our object is only to teach principles ; not, as yet, to establish minute results. A principle to be explained in Physical Astronomy. PLANETARY MOTION. 12' The distances drawn from Kepler's law, are considered CHAP. vn. more accurate than conclusions drawn from most other con- Why tha siderations ; and it is rather remarkable that these deduc- result* from / ,1 T 11 A i apparent di- tions iroin the apparent diameters agree as well as they ao, ameter , can . owing to the difficulty of settling the exact apparent diain- not be relied eter, by observation. Take the apparent diameter of lira- nus, for example, 3".7 and 4".l,and change either of them y 1 ^ of a second, and it will make a great difference in the deduced result. CHAPTER VIII. H ETHODS OF OBSERVING THE PERIODICAL REVOLUTIONS OF THE PLANETS, AND THEIR RELATIVE ' DISTANCES FROM THE SUN. ( 103.) THE subject of this chapter will be to explain the CHAP, vm. principles of finding the periodical revolutions of the planets why direct around the sun. If observers on the earth were at the r e nt to th center of motion, they could determine the times of revo- point lution by simple observation. But as the earth is one of the planets, and all observers on its surface are carried with it, the observations here made must be subjected to mathemati- cal corrections, to obtain true results ; and this was an impos- sible problem to the ancients, as long as they contended for a stationary earth. If the observer could view the planets from the center of TWO i the sun, he would see them in their true places among the ^ P stars and there are only two positions in which an observer on the earth will see a planet in the same place as though he viewed it from the center of the sun, and these positions are conjunction and opposition. Thus, in Fig. 24, when the earth is at E, and a planet at M, the planet is in opposition to the sun ; and it is seen pro- jected among the stars at the same point, whether viewed from S or from E. In Fig. 23, if the planet is at B, or A, it is said to be in conjunction with the sun; but a conjunction cannot be ob- rvd. A&lRONOMY. viii. served OK account of the brilliancy of the sun, unless it be the two planets, Mercury and Venus, and then only when they pass directly before the face of the sun, and are projected on its surface as a black spot. Such conjunctions are culled transits. ( 104.) All the planets move around the sun in the same Revolution direction, and not far from the yarne plane, and the rudest of inferior ^ mogt care j ess observations show that those planets near- planets less, and of supe- est the sun, perform their revolutions in shorter periods than tint planets tnoge more remo t e . From this, we decide at once that the g renter than a year. mean angular motion of all the superior planets is less than the mean angular motion of the earth in its orbit; and the mean angular motion of the inferior planets, as seen from the sun, is greater than the mean motion of the earth. ( 105.) The time that any planet comes in opposition to Times of the sun, can be very distinctly determined by observation, opposition j tg longitude is then 180 degrees from the longitude of tho served sun, and comes to the meridian nearly or exactly at midnight. If it is a little short of opposition at the time of one obser- vation, and a little past at another, the observer can propor- tion to the exact time of opposition, and such time can be definitely recorded and by such observation, we have the true position of the planet, as seen from the sun. Another pi 25 opposition of the same kind and of the same planet, can be ob- served and recorded. The elapsed time between two Synodic*] ^^^^^^^HNHRBH^H| such oppositions is called the sy- nodical revolution of the planet. We note the time that a lt . , ........ . ^- ;; , ..,, . r .,._ planet is in opposition to tho hr motion oi K&O^S^-i J.l ^1 '.'., W'SJ SUn ' ^ n611 ^' ^ an( * -^" are m planet? ttS^^^^S P^S^^^*|H one P^ ane as represented in Fig. 5 If the P la " Ct M Sh u ^ synodical ^y?:^^^ J^ iWfcl^I^S ' remain at rest while the earth revolution.. |y |^ v ig|||^| E made its revolution, then the synodical revolution would be the same as the length of our year. But all the planets move in the same directior vi PLANETARY MOTION. 123 the earth; and therefore the earth, after making a revolu- CHip.vni tion, must pass onward and employ additional time to over- take the planet ; and the more rapidly the planet moves, the longer time it will require. Hence, in case two planets have but a small difference in angular motion, their synodical pe- General coo- riod must be proportionately long. The planet Jupiter ^derations moves about 31 in its orbit in a year; and therefore, after one opposition, the earth is round to the same point in 365} days, and to gain the 31 requires about 32 days more ; hence the synodical revolution of Jupiter must be about 397 days, by this vei-y rough and imperfect computation. By inspect- ing the table on page 105, we perceive that the mean synodi- cal revolution of Jupiter is 399 days, and this observed fact shows us that Jupiter passes over about 31 in a year, and of course its revolution must be a little less than 12 years; and by the same considerations, we can form a rough estimate of the periodical revolutions of all the planets. ( 106.) The general principle being understood, we may now be more scientific. The mean motion of the earth Computatio in its orbit is very accurately known. Represent its daily *? determil) * motion by a. The angular motion of the planet ( any supe- xniai motioa rior planet that maybe under consideration) is unknown ;' Ttbee * rth> therefore, represent its daily motion by x. Let the angle E SC represent a, and the angle MS m represent x\ then the angle m SCor (a x) will represent the daily angular advance of the earth over the planet ; and as many times as the an- gle m SC is contained in 360, will be the number of days in a. synodical revolution. Therefore, = the observed a x time of a synodical revolution ; and by taking the times from the table (page 105), we have the following equations: Mart. Jupiter. Saturn. Uranus. _. i __ 360 a x d * These equations correspond to the general equation f=ss_ _ In ltobinson*s Algebra, page 105, University edition. 124 ASTRONOMY . vin. The value of a is 59' 8", and then a solution of these sev- era: equations gives the mean angular motion, per day, of the several planets, as follows : Mars. Jupiter. Saturn. Uranus. 31' 27" 4'59".4 r 59".5 45".3 Times of Dividing the whole circle 360 by the mean daily motion dTrivecTfrom ^ eac ^ pl anet/ > w ^ g^ ve their respective times of revolution, the angular and the following are the results : Mars, Jupiter. Saturn. lLr>us. 687 days. 4331 days. 10840 days. 28610 days. ( 106.) For the inferior planets, Mercury and Venus, we have the same principle, only making x greater than a, and For Mercury. For Venus. 80_ *=42' 11"; s=l36'7". Mean an- These diurnal angular motions correspond to 89 days for guiar motion ^ revolution of Mercury, and 224.8 days for the revolution ofthe inferior * ' J planets, and of Venus. All these results are, of course, understood as their revoiu- g rgt approximations, and accuracy here is not attempted. e un. We are only showing principles ; and it will be noticed, that the times here taken in these considerations, are only to the nearest days , and not fractions of a day, as would be necessary for accurate results. By this method accuracy is never at- tempted, on account of the eccentricity of the orbits. No two synodical revolutions are exactly alike ; and therefore it is very difficult to decide what the real mean values are. (107.) To obtain accuracy, in astronomy, observations must be carried through a long series of years. The follow- ing is an example ; and it will explain how accuracy can be attained in relation to any other planet. On the 7th of November, 1631, M. Cassini observed Mer- cury passing over the sun ; and from his observations then taken, deduced the time of conjunction to be at 7 h. 50 m., mean time, at Paris, and the true longitude of Mercury 44 41' 35". Comparing this occultation with that which took place in 1723> the true time of con J unction was November 9th, at 5 h. 29m., P. M., and Mercury's longitude was 46 47 20". PLANETARY MOTION. 125 The elapsed time was 92 years, 2 days, 9 h. 39 m. Twenty- CHAP. VHI. two of these years were bissextile ; therefore the elapsed time O f years, to was (92 X 365) days, plus 24 d. 9 h. 39 ra. ecw In this interval, Mercury made 382 revolutions, and 2 5' n 45" over. That is, in 33604.402 days, Mercury described 137522.095826 degrees; and therefore, Toy division, we find that in one day it would describe 4.0923, at a mean rate. Thus, knowing the mean daily rate to great accuracy, the mean revolution, in time, must be expressed by the fraction or, 87.9701 days, or 87 days 23 h. 15m. 57 s. ( 108. ) The following is another method of observing the Another periodical times of the planets, to which we call the student's b e ser in t h, Special attention. periodical re- The orbits of all the planets are a little inclined to the volutions of 1 , the planets. plane of the ecliptic. The planes of all the planetary orbits pass through the center of the sun ; the plane of the ecliptic is one of them, and therefore the plane of the ecliptic and the plane of any other planet must intersect each other by some line passing through the center of the sun. The intersection of two planes is always a straigJit line. (See Geometry.) The reader must also recognize and acknowledge the fol- lowing principle : That a body cannot appear to be in the plane of an observer, unless it really is in that plane. For example : an observer is always in the plane of his meridian, and no body can appear to be in that plane unless it really is in that plane ; it cannot be projected into or out of that plane, by parallax or refraction. Hence, when any one of the planets appears to be in the plane of the ecliptic, it actually is in that plane ; and let the time be recorded when such a thing takes place. The planet will immediately pass out of the plane, because what the two planes do not coincide. Passing the plane of the ln( ' ant *" ecliptic is called passing the node. Keep track of the planet until it comes into the same plane ; that is, crosses the other node : in this interval of time the planet has described just 126 ASTRONOMY. CHAP. vm. 180, as seen front, the sun (unless the nodes themselves are Two nod., in motion, which in fact they are ; but such motion is not 180 degree* sensible for one or two revolutions of Venus or Mars), other as een Continue observations on the same planet, until it comes from the sun. into the ecliptic the second time after the first observation, or to the same node again; and the time elapsed, is the time of a revolution of that planet round the sun. From such observa- tions the periodical time of Venus became well known to astronomers, long before they had opportunities to decide it by comparing its transits across the sun's disc ; and by thus knowing its periodical time and motion, they were enabled to calculate the times and circumstances of the transits which happened in 1761, and in 1769; save those resulting from parallax alone. First idea of (109.) On comparing the time that a planet remains on of'the^piTn* eac ^ s ^ e f tne ecliptic* we can form some idea of the position ts. of its apogee and perigee. If it is observed to be on each side of the ecliptic the same length of time, then it is evident that the orbit of the planet is circular, or that its longer axis coin- cides with its nodes. If it is observed to be a shorter time north of the plane of the ecliptic than south of it, then it is evident that its perigee is north of the ecliptic; but nothing more definite can be drawn from this circumstance. Final remits. (110.) Finally. By the combination of the different methods, explained in articles (98 ), ( 100 ), ( 101 ), ( 105 ), (107 ), and (108), and extending the observations through a lo i Mars, more exactly. cowed. Their ratio of revolution is 686.979 log. - 2.836948 224.701 log- - 2.351601 Log. of the ratio, ^ - 0.485347 Multiply by - 2 Log. of the square of the ratio of time, 0.870694 Their ratio of distance is, 15.23692 log. - 1.182883 "7^23332 log. - 859323 Log. of the ratio, - - 0.323560 Multiply by - . . 3 Log. of the cube of the ratio of distance, 0.970680 Thus we perceive that the squares of the times of revolu- tion. are to each other as the cubes of the mean distances of 128 ASTRONOMY CHAP, vin. the planets from the sun,* and this is called Kepler's third Kepler's law : and it was by such numerical comparisons that Kepler discovered the law.f We may now recapitulate the three laws of the solar sys- tem, called Kepler's laws, as they were discovered by that philosopher. 1st. The orbits of the planets are ellipses, of which the sun occupies one of the foci. ^d. The radius vector in each case describes areas about the focus, which are proportional to the times. 3c?. The squares of the times of revolution are to each other as the cubes of the mean distances from the sun. * For a concise mathematical view of this subject, we give the following: Let d and D represent mean distances from the sun, and t and T the times of revolution. Then T D ~= n, ~j m > n an( J m taken to represent the ratios. Square the 1st equation and cube the 2d. Then T 2 D* ~=n 2 , and ~^=m\ But by inspection we know that n 2 =m 3 ; therefore, = --1, or, t 2 : T 2 ::d* : D 3 . t 2 a 3 f It appears that Kepler did not compare ratios, as we have done ; but took the more ponderous method of comparing the elements of the ratios (the numbers themselves ) ; for, says the historian : It was on the 8th of March, 1618, that it first came into Kepler's mind to com- pare the powers of the numbers which express their revolutions and distances ; and by chance he compared the squares of the times with the cubes of the distances ; but from too great anxiety and impa- tience, he made such errors in computation, that he rejected the hy- pothesis as false and useless ; but on examining almost every other relation in vain, he returned to the same hypothesis, and on the 15th of May, of the same year, he renewed his calculation with complete success, and established this law, which has rendered his name im- mortal SOLAR PARALLAX. CHAPTER IX. TRANSITS OF VENUS AND MERCURY. HOW SUN's HORIZONTAL PARALLAX DEDUCED ( 112. ) WE have thus far been very patient in our inves- CHAP, ix tigations groping along finding the form of the planetary Attem^ l orbits, and their relative magnitudes; but, as yet, we know find the iun' nothing of the distance to the sun ; save the indefinite fact, P arallax - that it must be very great, and its magnitude great; but how great we can never know, without the sun's parallax. Hence, to obtain this element, has always been an interesting problem to astronomers. The ancient astronomers had no instruments sufficiently Dim^iti., refined to determine this parallax by direct observation, in the of ancient manner of finding that of the moon (Art. 60), and hence the Mtronoineiri - ingenuity of men was called into exercise to find some artifice to obtain the desired result. After Kepler's laws were established, arid the relative dis- tances of the planets made known, it was apparent that their real distance could be deduced, provided the distance between the earth and any planet could be made known. (113.) The relative distances of the earth and Mars, from p aranajKflf the sun (as determined by Kepler's law) are as 1 to 1.5237 ; Mar. and hence it follows that Mars, in its oppositions to the sun, is but about one half as far from the earth as the sun is; and therefore its parallax (Art. 60) must be about double that of the sun ; and several partially successful attempts were made to obtain it by observation. On the 15th of August, 1719, Mars being very near its opposition to the sun, and very near a star of the 5th mag- nitude, its parallax became sensible ; and Mr. Maraldi, an tion u> u* Italian astronomer, pronounced it to be 27". The relative distance of Mars, at that time, was 1.37, as determined from its position and the eccentricity of its orbit. But horizontal parallax is the angle under which the earth appears ; and, at a greater distance, it will appear under a 9 130 ASTRONOMY. CHAP. ix. less angle. The distance of Mars from the earth, at that time, was .37, and the distance of the sun was 1 ; therefore, 1 : .37 :: 27" : 9".99, or 10", nearly, for the sun's horizon- tal parallax. On the 6th of October, 1751, Mars was attentively ob- serve< ^ ^J Wargentin and Lacaille (it being near its opposi- Lacaiiie tion to the sun), and they found its parallax to be 24" .6, from which they deduced the mean parallax of the sun, 10".7. But at that time, if not at present, the parallax of Mars could not be observed directly, with sufficient accuracy to satisfy astronomers ; for no observer could rely on an angu- lar measure within 2" ; for full that space was eclipsed by the micrometer wire. Dr. Hal- (114.) Not being satisfied with these results, Dr. Halley, '* ^ss 68 ' an English astronomer, very happily conceived the idea of finding the sun's parallax by the comparisons of observa- tions made from different parts of the earth, on a transit of Venus over the sun's disc. If the plane of the orbit of Venus coincided with the orbit of the earth, then Venus would come between the earth and sun, at every inferior conjunction, at intervals of 584.04 days. But the orbit of Venus is inclined to the orbit of the earth by an angle of 3 23' 28" ; and, in the year 1800, the planet crossed the ecliptic from south to north, in longitude 74 54' 12", and from north to south, in longitude 254 54' 12": the first mentioned point is called The nodes the ascending node ; the last, the descending node. The nodes of Venus. retrograde 3r 10 in a century. Whattimei (115.) The mean synodical revolution of 584 days corre- i* the year g p 0nds w j fc } 1 no a ]iq uo t part of a year ; and therefore, in the transits may * . * / take place, course of time, these conjunctions will happen at different points along the ecliptic. The sun is in that part of the ecliptic near the nodes of Vnnus, June 5th and December 6th or 7th ; and the two last transits happened in 1761 and in 1769; and from these periods we date our knowledge of the solar parallax. Revoin- ( 116.) The periodical revolution of the earth is 365.256383 tio, com. dayg and that of y ermg ig 224.700787 ; and as numbers they are nearly in proportion of 13 to 8. From this it follows, that eight revolutions of the earth SOLAR PARALLAX. 131 require nearly the same time as 13 revolutions of Venus; CHAP. ex and, of course, whenever a conjunction takes place, eight years afterward another conjunction will take place very near the same point in the ecliptic.* * The ratio of the times of these revolutions is directly Compara. 224.700787 tive motions compared, as terms of a fraction, thus, TTTTF-TFTTTTTTT; and it is ofVenns anu o05.25uo81 the earth. manifest that 365.256383 days, multiplied by the number 224700787, will give the same product as 224 700787 days multiplied by the number 365256383 ; that is, after an elapse of 224700787 years, the conjunction will take place at the same point in the heavens; and all intermediate conjunctions will be but approximations to the same point : and to obtain these approximate intervals,, we reduce the above fraction to its approximating fractions, by the principle of continued fractions. ( See Robinson's Arithmetic. ) The approximating fractions are 1 1 2 3 8 235 I' 2' 3' 5' 13' 382* To say nothing of the first two terms, these fractions show that two revolutions of the earth are near, in length of time, to three revolutions of Venus ; three revolutions of the earth a nearer value to five revolutions of Venus ; and eight revo- lutions of the earth a still nearer value to 13 revolutions of Venus ; and 235 revolutions of the earth a very near value to 382 revolutions of Venus. The period of eight years, under favorable circumstances, will bring a second transit at the same node : but if not in eight years, it will be 235 years, or 235+8=243 years. For a transit at the other node, we must take a period of 235 8 years, divided by 2, or 113 years; and sometimes the period will be eight years less than this, or 105 years The first transit known to have been observed was in 1639, December 4th ; to this add 235 years, and we have the time of the next transit, at the same node, 1874, December 8th; and eight years after that will be another, 1882, December 6th. The first transit observed at the ascending node, was 132 ASTRONOMY. CHAP. ix. If the proportion had been exactly as 13 to 8, then the Periods of conjunctions would always take place exactly at the same conjunction. -^ . k ut as j t j g fa Q points of conjunction in the heavens at the same r . J time of the are east and west of a given point, and approximate nearer ye&r - and nearer to that point as the periods are greater and greater. only two To be more practical, however, the intervals between con- *" " junctions are such, combined with a slight motion of the nodes, tervals of 8 that the geocentric latitude of Venus, at inferior conjunctions years. near ^] ie ascending node, changes about 19' 30" to the north, in the period of about eight years. At the descending node, it changes about the same quantity to the southward, in the same period ; and as the disc of the sun is but little over 32', it is impossible that a third transit should happen 16 years after the first; hence only two transits can happen, at the same node, separated by the short interval of eight years. Periods be. (117.) If at any transit we suppose Venus to pass directly ^e center of the sun, as seen from the center of the earth that is, pass conjunction and node at the same time at the end of another period of about eight years, Venus would be 19' 30" north or south of the sun's center; but as the semidiameter of the sun is but about 16', no transit could happen in such a case ; and there would be but one transit at that node until after the expiration of a long period of 235 or 243 years. After passing the period of eight years, we take a lapse of 105 or 113 years, or thereabouts, to look for a transit at the other node. Transits ^ ^g ^ Knowing the relative distances of Venus, and the pnted. earth, from the sun the positions and eccentricities of both Dr. Haiiey orbits also their angular motions and periodical revolutions iofind the everv circumstance attending a transit, as seen from the sun's parai- earth's center, can be calculated; and Dr. Halley, in 1677, lax * read a paper before the London Astronomical Society, in jrext note i n ly^l, j une 5 tn ; eight years after, 1769, June 3d, there was another ; and the next that will occur, at that node, will be in 2004, June 7th, 235 years after 1769. SOLAR PARALLAX. 133 which he explained the manner of deducing the parallax of CHAP, 1X the sun, from observations taken on a transit of Venus or Mercury across the sun's disc, compared with computations made for the earth's center, or by comparing observations made on the earth at great distances from each other. The transits of Venus are much better, for this purpose, why the than those of Mercury ; as Venus is larger, and nearer the transits of J ' m Venus are earth, and its parallax at such times much greater than that better adapt- of Mercury ; and so important did it appear, to the learned ed to iv9 world, to have correct observations on the last transit of ra j^ "thwi Venus, in 1769, at remote stations, that the British, French, those of Mer- and Russian governments were induced to send out expedi- cnry * tions to various parts of the globe, to observe it. " The fa- mous expedition of Captain Cook, to Otaheite, was one of them." (119.) The mean result of all the observations made on The result that memorable occasion, gave the sun's parallax, on the day of the transit (3d of June), 8".5776. The horizontal paral- lax, at mean distance, may be taken at 8". 6 ; which places the sun, at its mean distance, no less than 23984 times the length of the earth's semidiameter, or about 95 millions of miles. This problem of the sun's horizontal parallax, as deduced The hnpor. from observations on a transit of Venus, we regard as the tance of tbu most important, for a student to understand, of any in astro- Pr nomy ; for without it, the dimensions of the solar system, and the magnitudes of the heavenly bodies, must be taken wholly on trust; and we have often protested against mere facts being taken for knowledge. ( 120.) We shall now attempt to explain this whole matter A general on general principles, avoiding all the little minutiae which ex P Ianatioa render the subject intricate and tedious ; for our only object is to give a clear idea of the nature and philosophy of the problem. Let S (Fig. 26) represent the sun, and m n and P Q small portions of the orbits of Venus ftnd the earth. As these two bodies move the same way, and nearly in the same plane, we may suppose the earth stationary, and Venus 134 ASTRONOMY At abstract proposition for the pur- pose of illus- tration. to move with an angular velocity equal to the difference of the two When the planet arrives at v, an observer at A would see the planet projected on the sun, making a dent at v'. But an observer at G would not see the same thing until after the planet had passed over the small aie v g, with a velocity equal to the dit'- erence between the angular motion of the two bodies; and as this will require quite an interval of absolute time, it can be detected ; and it mea- sures the angle A v' G; an angle under which a definite portion of the earth appears as seen from the sun. (121.) To have a more definite idea of the practicability of this me- thod, let us suppose the parallactic angle, A v' G, equal to 10", and in- quire how long Venus would be in passing the relative arc v q. Venus, at its mean rate, passes - 1 36' 8" in a day. The earth, " 59' 8" " The relative, or excess motion of Venus for a mean solar day is then 37'. Now, as 37' is to 24h. so is 10" to a fourth term; or, as 2-220" : 1440m. :: 10" : 6 m. 29 s. Now if observation gave more than 6 minutes and 29 sec- onds, we shall conclude that the parallactic angle was more fchan 10"; if less, less. But this is an abstract proposition. When treating of an actual case in place of the mean motion, We must take the actual angular motions of the earth and Venus at that time, and we'must know the actual position of the observers A and G in respect to each other, and the po- sition of each in relation to a line joining the center of the SOLAR PARALLAX. 135 earth and the center of the sun ; and then by comparing the CHAP. IX local time of observation made at A, with the time at G, and referring both to one and the same meridian, we shall have the interval of time occupied by the planet in passing from v to q, from which we deduce the parallactic angle A v' G, and from thence the horizontal parallax. The same observations can be made when the planet passes off the sun, and a great many stations can be compared with A, as well as the station G. In this way, the mean result of a great many stations was found in 1761, and in 1769, and the mean of all cannot materially differ from the truth. ( 122.) There is another method of considering this whole Another me- subject, which is in some respects more simple and preferable thodo [ deda - J cing the pro- to the one just explained. It is for the observers at every b i em station to keep the track of the transit all the way across the sun's disc, and take every precaution to measure the length of chord upon the disc, which can be done by carefully noting the times of external and internal contacts, and the begin- ning and end of the transit, and at short intervals carefully measuring the distance of the planet to the nearest edge of the sun by a micrometer. If the parallax is sensible, it is evident that two observers, Situation of situated in different hemispheres, will not obtain the same chord. For example, an observer in the northern hemisphere, as in Sweden or Norway, will see Venus traversing a more southern chord than an observer in the southern hemisphere. Now if each observer gives us the length of the chord as ob- served by himself, and, knowing the angular diameter of the sun, we can compute the distance of each chord from the sun's center, and of course we then have the angular breadth of the zone on the sun's disc between them. But as this zone is formed by straight lines passing through the same point, the center of Venus, its absolute breadth will depend on its distance from the point v; that is, the two triangles ABv and a b v ( Fig. 27) will be proportional, and we have Av:av::A:ab. But the first three* of these terms are known ; therefore the fourth, a b, is known also ; and if any definite angular space 13G ASTRONOMY. CHAP. IX. Under what circumstan- ces this me- thod should not be used. Transits Oil Mercury not important. Revolutions of Mercury and the earth compared. Fig. 27. on the sun becomes known, the whole sem- idiameter becomes known, and from thence the horizontal parallax is immediately dedu- ced* (123.) The accuracy of this method should bd questioned when Venus passes near the sun' center, for the two chords are never more than 30" asunder, and hence they will not percepti- bly differ in length when passing near the sun'? center, and Venus will be upon the sun nearly the same length of time to all observers. ( 124.) The apparent diameter of Mercury and Venus can be very accurately measured when passing the sun's disc. In 1769 the di- ameter of Venus was observed to be 59". ( 125.) The same general principles applj- to the transits of Mercury and Venus ; but those of Mercury are not important, on account of the smaller parallax and smaller size of that planet : but owing to the more rapid revolution of Mer- cury, its transits occur more frequently. The frequent appearance of this planet on the face of the sun, gives to astronomers fine opportu- nities to determine the position of its node and the inclination of its orbit. In 1779, M. Delambre, from observations on the transit of May 7, placed the ascending node, as seen from the sun, in longitude 45 57' 3". From the transit of the 8th of May, 1845, as observed at Cincinnati, it must have been in longi- tude 46 31' 10"; this gives it a progressive motion of about 1 10' in a century. The inclination of the orbit is 7 0' 13". The periodical time of revolution is 87.96925 days; that of the earth is 365.25638 days, and by making a fraction of these numbers, and reducing as in the last text note, we find That is, as the real diameter of the sun, is to the real diameter of the earth, so is the sun's angular semidiameter to its horizontal par- allax. ( See 66). PLANETARY PARALLAX. 137 that 6, 7, 13, 33, 46, 79, and 520 years, or revolutions of the CHAP, ix. earth nearly correspond to complete revolutions of Mercury. Hence we may look for a transit in 6, 7, 13, 33, 46, &c., years, or at the expiration of any combination of these years after any transit has been observed to take place ; and by examining the following table, the years will be found to fol- [j,* 1 *** low each other by some combination of these numbers. 8 j, The following is a list of all the transits of Mercury that have occurred, or will occur, between the years 1800 an<* 1900: At the ascending node. At the descending node. May 7. - May 5. May 8. - May 6. May 9. 1802, - - - Nov. 8. 1799, - - 1822, - - - Nov. 4. 1832, - - 1835, - - - Nov. 7. 1845, - - 1848, - - - Nov. 9. , 1878, - - 1861, - - Nov. 11. ' 1891, - - 1868, - - - Nov. 4. 1881, - - - Nov. 7. 1894, - - - Nov. 10. CHAPTER X. THE HORIZONTAL PARALLAXES OF THE PLANETS COMPUTED, AND FROM THENCE THEIR REAL DIAMETERS AND MAGNITDDES. ( 126.) HAVING found the real distance to the sun, and the CHAP. X. sun's horizontal parallax, we have now sufficient data to find Real ma fr the real distance, diameter, and magnitude, of every planet nitudes and . A . . distances cat in the solar system. now be de . In Art. 60 we have explained, or rather defined, the hori- zontal parallax of any body to be the angle under which the semidiameter of the earth appears, as seen from that body ; and if the earth were as large as the body, the semi-diame- ter of the body, and its liorizontal parallax, would have the same value. And, in general, the diameter of the earth is to the diameter of any other planetary body, as the horizontal parallax of that body is to its apparent semidiametev. The mean horizontal parallax of the sun, as determined in J38 ASTRONOMY. CHAP x. the last chapter, is 8".6; the semidiaraeter of the sun, at th Re7i"=(111.74)(7912)=884087 miles. Real dis- ( 127.) The sun's horizontal parallax is the angle at the tance be- b ase O f a r ight, angled triangle; and the side opposite to it is tween the ~, . earth and sun ** radius ot the earth (which, tor the sake ot convenience, determined, we now call unity). Let x represent the radius of the earth's orbit; then, by trigonometry, sin. 8".6 : 1 : : sin. 90 : x\ cin Q0 Therefore, * = ^pL==l og . 10.00000 log. 5.620073 * That is, the log. of 3=4.379927, or a?=23984 ; which is the distance between the earth and sun, when the semidia- meter of the earth is taken for the unit of measure ; but, for general reference, and to aid the memory, we may say the distance is 24000 times the earth's semidiameter. (128.) Now let us change the unit from the semidiameter of the earth to an English mile ; and then the distance be- tween the earth and sun is Distance i (3956)(23984)=94880706 ; f*und num- l^ eif and, in round numbers, we say 95 millions of miles. By Kepler's third law, we know th,e relative distances of * Students generally would be unable to find the sine of 8". 6, or the sine of any other very small arc ; for the directions given in common works of trigonometry are too gross, and, indeed, inaccurate, to meet the demands of astronomy. On the principle that the sines of small arcs vary as the arcs them- selves, we can find the sine of any small arc as follows : Sine of 1', taken from the tables, is - - - - 6.463726 Divide by 60, that is, subtract the log. of 60, - - 1.778151 The sine-of 1", therefore, is 4. 685575 Multiply by the number 8.6 ; that is, add log. - 0. 934498 Tim nine of 8".6, therefore, must be, - - - - 5. 620073 In the same manner, find the sine of any other small are. PLANETARY PARALLAX. 139 all tiie planets from the sun ; and now, having found the real Cm?, x. distance of the earth, we may have the distance in miles, by HOW to multiplying the distance of the earth by the ratio correspond- find the dl * ing to any other planet. Thus, for the distance of Venus, planet from we multiply 94880706 by .72333 ; and the result is the su in 68629960 miles, for the distance of Venus: and proceed, in the same manner, for the distance of any other planet. (129.) By observations taken on the transit of Venus, in TO find the 1769, it was concluded that the horizontal parallax of that y* n e , ter f planet was 30".4; and its semidiameter, at the same time, was 29".2. Hence (Art. 126), 304 : 292 : : 7912 : to a fourth term; which gives 7599 miles for the diameter of Venus. (130.) ^Ye cannot observe the horizontal parallax of Ju- piter, Saturn, or any other very remote planet: if known at all, it becomes known by computation ; but the parallax can erved. be known, when the real distance is known; and, by Kepler's third law, and the solar parallax, we do know all the planetary distances ; and can, of course, compute any particular hori- zontal parallax. For the horizontal parallax of Jupiter, when at a distance from the earth equal to its mean distance from the sun, we proceed as follows : The parallax, or the semidiameter of the earth, when seen at the distance of the sun, is 8".6. When seen from a greater distance, the angle would be proportionally less. Put k equal to the horizontal parallax of Jupiter ; then we have, - 5.202776 : 1 :: 8".6 : k; or A =T |.'_. From this, we perceive, that if we divide the sun's horizontal HOW u> parallax by the ratio of a planet's distance from the sun, the com P ute fh quotient will be the horizontal parallax of the planet, when at a the planet. distance from the earth equal to its mean distance from the sun. (131.) To find the diameter of a planet, in relation to the HOW to diameter of the earth, we have a similar proportion as in Art. find the real i > i /.-ITT diameters of l'2b : and to find the diameter of Jupiter, we proceed as the p i aMU . follows : The greatest apparent diameter of Jupiter, as seen from 140 ASTRONOMY. CHAP. x. the earth, is 44".5; the least is 30". 1; therefore the mean, as seen from the sun. cannot be far from 37 ".3, and the semi- diameter 18". 65; La Place says it is 18". 35; and this value we shall use. Now, as in Art. 126, let d=7912, D= the O// /" unknown diameter of Jupiter ; OAOT-T/? is its horizontal parallax, and 18".35 its corresponding semidiameter ; then, as in Art 126. 7912. : D : : - : 18.85; Therefore p= 87900 miles. In the same manner, we may find the diameter of any other planet. Jupiter not We have just seen that the diameter of Jupiter is 11.11 spherical. times the diameter of the earth ; but this is the equatorial diameter of the planet. Its polar diameter is less, in the proportion of 167 to 177, as determined by the mean of many micrometrical measurements ; which proportion gives 82930 miles, for the polar diameter of Jupiter. These extremes give the mean diameter of Jupiter, to the mean diameter of the earth, as 10.8 to 1. HOW to find (132.) But the magnitudes of similar bodies are to one the magi- anot | )er as fa e Cll ]j es O f their like dimensions ; therefore the tude of i >e planets. magnitude of Jupiter is to that of the earth, as (10.8) 3 to 1, and from thence we learn that Jupiter is 1260 times greater than the earth. In the same manner we may find the magnitude of any other planet, and it is thus that their magnitudes have often been determined, and the results may be seen in a concise form in Table III, which gives a summary view of the solar system. The masses and attractions of the different planets will be investigated in physical astronomy, after we become acquain- ted with the theory of universal gravity. SOLA R SYSTEM. CHAPTER XI. A GENERAL DESCRIPTION OF THE PLANETS. ( 133.) WE conclude this section of astronomy by a brief CHAP. XI. description of the solar system, which we have purposely delayed lest we might interrupt the course of reasoning respecting the planetary motions. The reader is referred to Table III, for a concise and comparative view of all the facts that can be numerically expressed ; and aside from these facts, little can be said by way of explanation or description. The fact, that the sun or a planet revolves on an axis, Facts reveal- must be determined by observing the motion of spots on the onthenn 0tl visible disc ; and if no spots, -are visible, the fact of revolution pianeu. cannot be ascertained.* But when spots are visible, their motion and apparent paths will not only point out the time of revolution, but the position of the axis. THE SUN. ( 134.) The sun is the central body in the system, of im- The sun the mense magnitude, comparatively stationary, the dispenser of re P sitor y f light and heat, and apparently the repository of that force which governs the motion of all other bodies in the system. "Spots on the sun seem first to have been observed in the year 1611, since which time they have constantly attracted attention, and have been the subject of investigation among astronomers. These spots change their appearance as the sun revolves on its axis, and become greater or less, to an observer on the earth, as they are turned to, or from him ; they also change in respect to real magnitude and number, one spot, seen by Dr. Herschel, was estimated to be more than six times the size of our earth, being 50000 miles in diameter. Some- times forty or fifty spots may be seen at the same time, and sometimes only one. They are often so largo as to be seen with the naked eye ; this was the case in 1816. " In two instances, these spots have been seen to burst into several parts, and the parts to fly in several directions, like a piece of ice thrown upon the ground. * Mercury is an exception to this principle. 112 ASTRONOMY. CHAI-. XI. " In respect to the nature and design of these spots, almost every astronomer has formed a different theory. Some have supposed them to be solid opaque masses of scoritu, floating in the liquid fire of the sun ; others as satellites, revolving round him, and hiding his light from us; others as immense masses, which have fallen on his disc, and which are dark colored, because they have not yet become sufficiently heated. " Dr. Herschel, from many observations with his great telescope, concludes, that the shining matter of the sun consists of a mass of phosphoric clouds, and that the spots on his surface are owing to dis- turbances in the equilibrium of this luminous matter, by which open- ings are made through it. There are, however, objections to this theory, as indeed there are to all the others, and at present it can only ba said, that no satisfactory explanation of the cause of these spots has been given." Singular ( 135.) Mercury. This planet is the nearest to the sun, rdlfr an( j j iag k een ^ Q gu kj ec t O f considerable remark in the pre- covenng ro- m .... . ceding pages It is rarely visible, owing to its small size and proximity to the sun, and it never appears larger to the na- ked eye than a star of the fifth magnitude. Mercury is too near the sun to admit of any observations on the spots on its surface ; but its period of rotation has been determined by the variations in its horns the same ragged corner comes round at regular intervals of time 24h. 5m. Times when The best time to see Mercury, in the evening, is in the Mercury may spring of the year, when the planet is at its greatest elonga- tion east of the sun. It will then be visible to the naked eye about fifteen minutes, and will set about an hour and fifty minutes after the sun. When the planet is west of the sun, and at its greatest distance, it may be seen in the morn- ing, most advantageously in August and September. The symbol for the greatest elongation of Mercury, as written in the common almanacs, is $ Gr. Elon. High moan ( 136.) Venus. This planet is second in order from the sun, tains on v an( j j n re l a tion to its position and motion, has been sufficiently described. The period of its rotation on its axis is 23h. 21m. The position of the axis is always the same, and is not at right angles to the plane of its orbit, which gives it a change of seasons. The tangent position of the sun's light across thif SOLAR SYSTEM. 143 planet shows a very rough sur- ffl|HH8^BHBHBB^B CH4P - ** face; indeed, high mountains. W3H, i^r^m : -'" Ticopw By the radiating and glimmer- iewofv- ing nature of the light of this planet, we infer that it must have a deep and dense atmos- phere. ( 137.) TJie Earth is the next planet in the system; but it Th earth would be only formality to give any description of it in this a planet place. As a' planet, it seems to be highly favored above its neighboring planets, by being furnished with an attendant, The earth'* the moon; and insignificant as this latter body is, compared at to the whole solar system, it is the most important and in- teresting to the inhabitants of our earth. The two bodies, the earth and the moon, as seen from the sun, are very small : the former subtending an' angle of about 17" in diameter, the latter about 4", and their distance asunder never greater than between seven and eight minutes of a degree. Contrary to the general impression, the moon's motion in absolute space is always concave toward the sun.* ( 138.) Mars the first superior planet is of a red color, M * rt ; ...... ., , . , physical an. and very variable m its apparent magnitude. About every pearancefto * This may be shown thus the moon is inside the earth's orbit from the last quarter to the first quarter, on an average 14 days and 18 hours. During this time the earth moves in its orbit 14 30'. Let L n F be a portion of the earth's orbit equal to 14 30', L the position of the earth at the First Quarter of the moon, and F its position at the Last Quarter. Draw the chord L F, and compute mn the versed sine of the arc 7 15'. The mean radius of the earth's orbit is 397 times the ra- dius of the lunar orbit. A radius of 397 and an angle 7 15' givqs a versed sine of 3.17; but on this scale the distance from the earth to the moon is unity, or less than one third of nm: hence, the moon's path must lie between the chord LF and the arc L n F that is, always concave toward ike turn. 144 ASTRONOMY. CHAP^XI other year, when it comes to ;the meridian near midnight, ifc is then most conspicuous ; and the next year it is scarcely noticed by the common observer. . Tr . "The physical appearance of Telescopic View of Mars. ,, . , . Mars is somewhat remarkable His polar regions, when seen through a telescope, have a brilliancy so much greater than the rest of his disc, that there can be little doubt that, as with the earth so with this planet, accumulations of ice or snow take place during the wm- ters of those regions. In 1781 the south polar spot was extremely bright ; for a year it had not been exposed to the solar rays, The color of the planet most probably arises from adense atmosphere which surrounds him, of the existence of which there is other proof depending on the appearance of stars as they approach him ; they grow dim and are sometimes wholly extin- guished as their rays pass through that medium." Apparent im. C 139 ) The next planet, as known to ancient astronomers* perfection in is Jupiter ; but its distance is so great beyond the orbit of the system. jyf ars ^ fa^ ^ vo j ( j S p ace between the two had often been considered as an imperfection, and it was a general impression among astronomers that a planet ought to occupy that vacant space. Bode'siaw. Professor Bode, of Berlin, on comparing the relative dis- tances of the planets from the sun, discovered the following re- markable fact that if we take the following series of numbers : 0, 3, 6, 12, 24, 48, 96, 192, &c , and then add the number 4 to each, and we have, 4, 7, 10, 16, 28, 52, 100, 196, &o. f rhe reason and this last series of numbers very nearly, though not ex- Mb! cabled ac %> corresponds to the relative distances of the planets from (aw. the sun, with the exception of the number 28. This is sometimes called Bode's law ; but remarkable as it certainly is, it should not be dignified by the term law, until some bet- ter account of it can be given than its mere existence ; for, at present, all that can be said of it is, " here is an astonishing SOLAR SYSTEM. 145 coincidence." But, mere accident as it may be, it suggested CHAP. XL the possibility of some small, undiscovered planet revolving A~bo!d~hj. in this region, and we can easily imagine the astonishment of pothesu astronomers, on finding four in place of one, revolving in orbits tolerably well corresponding to this law, or rather co- incidence. Had they even found but one, it would seem to indicate something more than nere coincidence ; but finding four, proves the series to be simply accidental unless the four or more planets there discovered were originally one planet ; and then came the inquiry, is not this the case ? Thus originated the idea that these new and newly discovered small planets are but fragments of a larger one, which formerly cir- culated in that interval, and was blown to pieces by some internal explosion and we shall examine this hypothesis in a text note, under physical astronomy. The names of these planets, in the order of the times of their discovery, are, Ceres, Pallas, Juno, Vesta. The order of their distances from the sun, is Vesta, Juno, Ceres, Pallas Planets. Names of Dis- coverers. Residence of Discoverers. Date of Discovery. 1 Ceres... Pallas... j Juno . . . | Vesta... M. Piazzi, Dr. Olbers, M. Harding', Dr. Olbers, Palermo, Sicily, Bremen, Germany, Lilienthal, near Bremen, Bremen, 1st Jan., 1801. 28th Mar., 1802. 1st Sept. 1804. 29th Mar., 1807. If a planet has really burst, it is but reasonable to suppose that it separated into many fragments ; and, agreeably to this view of the subject, astronomers have been constantly on the alert for new planets, in the same regions of space ; and every Recent discovery of the kind greatly increases the probability of the discoveries theory. The following very recent discoveries are said to have been made, but the elements of the orbits are not regarded as sufficiently accurate to demand a place in the table. On the 8th of December, 1845, Mr. Hencke, of Dreisen, claims to have discovered a planet which he calls Astrea; and the same observer also claims another, discovered in New plan 1847, called Hele. His success induced others to a like exa- et <>vef mination, and a Mr. Hind, of London, within the past year, 10 146 ASTRONOMY. CHAP, xi 1848, claims a seventh and eighth asteroid, iiai.aed Iris and Flora. Thus we have eight miniature worlds, supposed to have once composed a planet ; and if the four last named are veri- table discoveries, we shall soon have the elements of their orbits in an unquestionable shape. The elements of the orbits of the four known asteroids, la given for the epoch 1820, are not as accurate as the follow- ing, which were deduced from the Nautical Almanac for 1846 and 1847 ; which have been corrected from more modern, extended, and accurate observations. (Epoch Jan., 1847.) On account of the small magnitude of these new planets, and their recent discovery, nothing is known of them save the following tabular facts, and these are only approximation to the truth. Planets. Sidereal Revolutions. Mean Distance from the Sun. Eccentricity of Orbits. Vesta Days. 1324. 289 2 36120 0. 08913 1594. 721 2 66514 0. 25385 Ceres 1683 064 2 76910 07844 Pallas . 1685 162 2 77125 24050 Planets. Longitude of Ascending Node. Inclination of Orbits. Longitude of Perihelion. Vesta O ' " 103 ^o 47 O ' " 7 8 29 " 251 4 34 Juno ...... 170 53 13 2 53 54 18 32 Ceres 80 47 56 10 37 17 147 25 41 Pallas 172 42 38 34 37 42 121 20 13 object of ( 140.) With the two elements, the longitude of the ascend- Fif . 29. ing nodes, and the inclination of the orbits to the ecliptic, we are enabled to give a general projection of these orbits around the celestial sphere, in relation to the ecliptic, as represented on page 37 ; and our object is to show that there are two points in the heavens, nearly opposite to each other, near to which all these planets pass. One of these points is about the longitude of 185 degrees, and the latitude of 15 degrees north ; and the other is the opposite point on the celestial sphere. If these planets are but fragments of one original planet, which burst or exploded by its internal fires, from that SOLAR SYSTEM moment they must have started from the same point, andi/ o11 f the r ' n g ' n lia own plane, which observation haa lity of the detected, owing to some portions of the ring being a little less bright riafcs. than others, and assigned its period at 10 h. 29m. 17 8., which, from what we know cf its dimensions, and of the force of gravity in the Saturnian system, is very nearly the periodic time of a satellite revolv- ing at the same distance as the middle of its breadth. It is the centri- fugal force, then, arising from this rotation, which sustains it ; and, although no observation nice enough to exhibit a difference of periods between the outer and inner rings have hitherto been made, it is more than probable that such a difference does subsist as to place each inde- pendently of the other in a similar state of equilibrium. The rings " Although the rings are, as we have said, very nearly concentric revolve a- w jth the body of Saturn, yet recent micrometrical measurements, of extreme delicacy, have demonstrated that the coincidence is not inathe- * mat ' ca "y exact, but that the center of gravity of the rings oscillates round that of the body, describing a very minute orbil, probably under laws of much complexity. Trifling as this remark may appear, it is of the utmost importance to the stability of the system of the rings. Supposing them mathematically perfect in their circular form, and exactly concentric with the planet, it is demonstrable that they would form ( in spite of their centrifugal force ) a system in a state of unstable equilibrium, which the slightest external power would subvert not by causing a rupture in the substance of the rings but by precipitating them, unbroken, on the surface of the planet. For the attraction of such a ring or rings on a point or sphere eccentrically situate within them, is not the same in all directions, but tends to draw the point or sphere toward the nearest part of the ring, or away from the center. Hence, supposing the body to become, from any cause, ever so little eccentric to the ring, the tendency of their mutual gravity is, not to correct, but to increase this eccentricity, and to bring the nearest parts of them together." (146.) Uranus. The next planet, beyond Saturn, was Henchei. discovered by Sir W. F. Herschel, in 1781, and, for a time, was called Herschel, in honor of its discoverer; but, accord- ing to custom, the name of a heathen deity has been substi- tuted, and the planet is now called Uranus the father of Saturn. This lanet ^ n * s P^ ane * i ' ls rarely to be seen, without a telescope. In a rarely visible clear night, and in the absence of the moon, when in a favor- o the naked ^^ p OS it,ion above the horizon, it may be seen as a star of about the 6th magnitude. Its real diameter is about 35000 miles, and about 80 times the magnitude of the earth. SOLAR SYSTEM. 153 The existence of this planet was suggested by some CHAP. XL of the perturbations of Saturn ; which could not be accounted for by the action of the then known planets ; but it does not appear that any computations were made, as a guide to the place where the unknown disturbing body ought to exist ; and, as far as we know, the discovery by Herschel was ?nere accident. But not so with the planet Neptune, discovered in the Facts led latter part of September, 1846, by a French astronomer, Le- to the ' Hs terrier ; and also a Mr. Adams, of Cambridge, England, who has tune put in his claim as the discoverer. The truth is, that the attention of the astronomers of Europe had been called to some extraordinary perturbations of Uranus ; which could not be accounted for without supposing an attracting body to be situated in space, beyond the orbit of Uranus ; and so distinct and clear were these irregularities, that both geometers, Le- verrier and Adams, fixed on the same region of the heavens, for the then position of their hypothetical planet ; and by dili- gent search, the planet was actually discovered about the game time, in both France and England. At present, we can know very little of this planet ; and according to the best authority I can gather, its longi- tude, January 1, 1847, was 327 24'. Mean distance from the sun, 30.2 ( the earth's distance being unity) ; period of revolution 166 years. Eccentricity of orbit 0.0084; mass, 1 23000"' According to Bode's law, the distance of the next planet from the sun, beyond Uranus, must be 38.8 ; and if Neptune really is at 30.2, it shows Bode's law to be only a remarkable coincidence ; for there can be no exceptions to positive physi- cal laws. " We shall close this chapter with an illustration calculated to convey How u to the minds of our readers a general impression of the relative magni- O bt a in a t r- tudes and distances of the parts of our system. Choose any well- rect concep- leveled field or bowling green. On it place a globe, two feet in diame- tion of the ter ; this will represent the sun ; Mercury will be represented by a grain solar system of mustard seed, on the circumference of a circle 164 feet in diameter, for its orbit; Venus a pea, on a circle 284 feet in diameter ; the earth 154 ASTRONOMY. CHAP. XI. also a pea, on a circle of 430 feet; Mars a rather large pin's head, on a circle of 654 feet; Juno, Ceres, Vesta, and Pallas, grains of sand, in orbits of from 1000 to 1200 feet ; Jupiter a moderate-sized orange, in a circle nearly half a mile across; Saturn a small orange, on a circle of four-fifths of a mile ; and Uranus a full-sized cherry, or small plum, upon the circumference of a circle more than a mile and a half in dia- meter. As to getting correct notions on this subject by drawing circles on paper, or, still worse, from those very childish toys called orreries, it is out of the question. To imitate the motions of the planets in the View of above-mentioned orbits, Mercury must describe its own diameter in 41 the planetary seconds; Venus, in 4 m. 14s. ; the earth, in 7 minutes ; Mars, in 4m. motions. 48 S . . Jupiter, in2h 56m. ; Saturn, in 3 h. 13m. ; and Uranus, in 2h. 16 m." HerscheVs Astronomy. CHAPTER XII. ON COMETS. CHAP. xn. (147.) BESIDES the planets, and their satellites, there are Comets great numbers of other bodies, which gradually come into formerly in- v i e w, increasing in brightness and velocity, until they attain ror " a maximum, and then as gradually diminish, pass off, and are lost in the distance. Knowledge "These bodies are comets. From their singular and unusual appear- b-irtishes ance, they were for a long time objects of terror to mankind, and were dread. regarded as harbingers of some great calamity. " The luminous train which accompanied them was particularly alarming, and the more so in proportion to its length. It is but little more than half a century since these superstitious fears were dissipated by a sound philosophy ; and comets, being now better understood, excite only the curiosity of astronomers and of mankind in general. These discoveries which give fortitude to the human mind are not among the least useful. " It was formerly doubted whether comets belonged to the class of heavenly bodies, or were only meteors engendered fortuitously in the air by the inflammation of certain vapors. Before the invention of the telescope, there were no means of observing the progressive increase and diminution of their light. They were seen but for a short time, and their appearance and disappearance took place suddenly. Their light and vapory tails, through which the stars were visible, and their whiteness often intense, seemed to give them a strong resemblance to those transient fires, which we call shooting stars. Apparently, they differed from these only in duration. They might be only composed COMETS 155 of a more compact substance capable of retarding for a longer time CHAP. XIL their dissolution. But these opinions are no longer maintained ; more accurate observations have led to a different theory. "All the comets hitherto observed have a small parallax,* which places Parallax of them far beyond the orbit of the moon ; they are not, therefore, formed comets, in our atmosphere. Moreover, their apparent motion among the stars is subject to regular laws, which enable us to predict their whole course from a small number of observations. This regularity and constancy evidently indicate durable bodies ; and it is natural to conclude that corm-ts are as permanent as the planets, but subject to a different kind of movement. " W hen we observe these bodies with a telescope, they resemble a mass Comets are of vapor, at the center of which is commonly seen a nucleus more or a PP arentl 7 less distinctly terminated. Some, however, have appeared to consist m * r * of merely a light vapor, without a sensible nucleus, since the stars are visible through it. During their revolution, they experience progres- sive variatious in their brightness, which appear to depend upon their distance from the sun, either because the sun inflames them by its heat, or simply on account of a stronger illumination. When their bright- ness is greatest, we may conclude from this very circumstance that they are near their perihelion. Their light is at first very feeble, but becomes gradually more vivid, until it sometimes surpasses that of the brightest planets ; after which it declines by the same degrees until it becomes imperceptible. We are hence led to the conclusion that comets, coming from the remote regions of the heavens, approach, ill many instances, much nearer the sun than the planets, and then recede to much greater distances. "Since comets are bodies which seem to belong to our planetary Orbits *f system, it is natural to suppose that they move about the sun like comets planets, but in orbits extremely elongated. These orbits must, there- fore, still be ellipses, having their foci at the center of the sun, but having their major axes almost infinite, especially with respect to us, who observe only a small portion of the orbit, namely, that in which the comet becomes visible as it approaches the sun. Accordingly the orbits of comets must take the form of a parabola, for we thus designate the curve into which the ellipse passes, when indefinitely elongated. " If we introduce this modification into the laws of Kepler, which . * The parallaxes of comets are known to be small, by two observers, at distant stations on the earth, comparing their observations taken or. the same como . at near the same time. At the times the observa- tions are made., neither observer can know how great the parallax is. It is only afterward, when comparisons are made, that judgment, in this particular, can be formed ; and it is not common that any more definite conclusion can be drawn, than that the parallax is small, and, of course, the body distant. 156 ASTRONOMY. . XII. relate to the elliptical motion, we obtain those of the parabolic motion of comets. Comets de- " ^ eace '* follows that the areas described by the same comet, in its cribe equal parabolic orbit, are proportional to the times. The areas described by areas in e- different comets in the same time, are proportional to the square root* qual times. o f their perihelion distances. " Lastly, if we suppose a planet moving in a circular orbit, whose radius is equal to the perihelion distance of a comet, the areas described by these two bodies in the same time, will be to each other as 1 to /2. Thus are the motions of comets and planets connected. " By means of these laws we can determine the area described bv a comet in a given time after passing the perihelion, and fix its posi- tion in the parabola. It only remains then to bring this theory to thi test of observation. Now we have a rigorous method of verifying it, by causing a parabola to pass through several observed places of a comet, and then ascertaining whether all the others are contained in it Three obser- " For this purpose three observations are requisite. If we observe rations suffi- the right ascension and declination of a comet at three different cient to find times, and thence deduce its geocentric longitude and latitude, we the orbit of a 8 \ m \\ nave the direction of three visual rays drawn at these times from the earth to the comet, and in the prolongation of which it must necessarily be found. The corresponding places of the sun are also known ; it remains then to construct a parabola, having its focus at the center of the sun, and cutting the visual rays in points, the inter- vals of which correspond to the number of days between the obser- vations. " Or if we suppose the earth in mo- tion and the sun at rest, let T, T, T", represent three successive positions of the earth, and TC, T'C, T"C", three visual rays drawn to the comet. The question is to find a parabola CC'C", having its focus in ba 0.58. it follows that itn aphelion distance is equal to 35.32. It COMETS. 159 departs, therefore, from the sun to thirty-five times the distance of the CHAP. XH earth, and afterward approaches nearly twice as near the sun as the earth is, thus describing an ellipse extremely elongated. "The intervals of its return to its perihelion are not constantly the same. That between 1531 and 1607 was three months longer than that between 1607 and 1682 ; and this last was 18 months shorter than the one between 1682 and 1759. It appears, therefore, that the motions of comets are subject to perturbations, like those of the planets, and to a much more sensible degree. " Elements of the Orbits of the three, Comets, which have appeared ac- cording to prediction, taken from the work of Professor Littrow. Halley. Encke. Biela. Longitude of the ascending node, - 54 C 335 249 Inclination of the orbit to the ecliptic, 162 13 13 Longitude of the perihelion, - - 303 157 108 Greatest semidiameter, that of the earth) jg QQ Qg being called 1, - - - - ) Least semidiametqf. - - - 4.6 1.2 2.4 Time of revolution in years, - 76 3.29 6.74 Nov. 16. May 4. Nov. 27 Time of the perihelion passage, - 1835 1832 1832 " The comets of Encke and Biela move according to the order of thr> signs of the zodiac, or have their motions direct; the motion of that of Halley is retrograde. "Comets, in passing among and near the planets, are materially Jupitei, drawn aside from their courses, and in some cases have their orbits en- andhissatel- tirely changed. This is remarkably the case with Jupiter, which seems, lltes> a gieat by some strange fatality, to be constantly in their way, and to serve as * ck h a perpetual stumbling-block to them. In the case of the remarkable come t s comet of 1770, which was found by Lexell to revolve in a moderate ellipse in the period of about five years, and whose return was pre- dicted by him accordingly, the prediction was disappointed by the comet actually getting entangled among the satellites of Jupiter, and being completely thrown out of its orbit by the attraction of that planet, and forced into a much larger ellipse. By this extraordinary renconter, the motions of the satellites suffered not the least perceptible derangement a sufficient proof of the smallness of the comet's mass." The comet of 1456, represented as having a tail of 60 in length, is now found to be Halley's comet, which has made several returns in 1531, 1607, 1682, 1759, and recently, in 1835. In 1607 the tail was said to have been over 30 in length ; but in 1835 the tail did not ex- ceed 12 Does it lose substance, or does the matter composing the tail condense ? or, have we received only exaggerated and distorted accounts from the earlier times, such as fear, superstition, and uwe, always put forth ? We ask these questions, but cannot answer them 1 eO A S T R N O M Y. CHAT. XII. The following cut represents the appearance of the comet of 1819. Fears en- "Professor Kendall, in his Uranography, speaking of the fears occa- tertained, by gjoned by comets, says: "Another source of apprehension, with regard ome, that ^ Q come j Sj ar } ses from the possibility of their striking our earth. It is may quite probable that even in the historical period the earth has been come into enveloped in the tail of a comet. It is not likely that the effect would collision with be sensible at the time. The actual shock of the head of a comet against our earth. the earth is extremely improbable. It is not likely to happen once in a million of years. " If such a shock should occur, the consequences might perhaps be very trivial. It is quite possible that many of the comets are not heavier than a single mountain on the surface of the earth. It is w^ll known that the size of mountains on the earth is illustrated by com- paring them to particles of dust on a common globe." CHAPTER XIII. ON THE PECULIARITIES OF THE FIXED STARS. CHAP, xin. p OR t h e f actg ag con tained in the subject matter of this chapter, we must depend wholly on authority ; for that reason we give only a compilation, made in as brief a manner as the nature of the subject will admit. In the first part of this work it was soon discovered that the fixed stars were more remote than the sun or planets ; and now, having determined their distances, we may make further inquiries as to the distances to the stars, which will FIXED STARS. 161 give some index by which to judge of their magnitudes, nature, CHAP. XIIL and peculiarities. " It would be idle to inquire whether the fixed stars have a sensible Bae from parallax, when observed from different parts of the earth. We have wnich *o already had abundant evidence that their distance is almost infinite. It mcaur " to is only by taking the longest base accessible to us, that we can hope to arrive at any satisfactory result. "Accordingly, we employ the major axis of the earth's orbit, which is nearly 200 millions of miles in extent. By observing a star from the two extremities of this axis, at intervals of six months, and applying a correction for all the small inequalities, the effect of which we have calculated, we shall know whether the longitude and latitude are the came or not at these two epochs. " It is obvious, indeed, that the star must appear more elevated above Annual the plane of the ecliptic when the earth is in the part of its orbit which parallax, is nearest to the star, and more depressed when the contrary takes place. The visual rays drawn from the earth to the star, in these ivvG positions, differ from the straight line drawn from the star to the center of the earth's orbit ; and the angle which either of them forms with this straight line, i* called the annual parallax. " As the earth does not pass suddenly from one point of its orbit to The effect the opposite, but proceeds gradually, if we observe the positions of a f a niuJ star at the intermediate epochs, we ought, if the annual parallax is sen- para ax sible, to see its effects developed in the same gradual manner. For example, if the star is placed at the pole of the ecliptic, the visual rays drawn irom it to the earth, will form a conical surface, having its apex at the star, and for its base, the earth's orbit. This conical surface being produced beyond the star, will form another opposite to the first, and the intersection of this last with the celestial sphere, will constitute a small ellipse, in which the star will always appear diametrically oppo- site to the earth, and in the prolongation of the visual rays drawn to the apex of the cones. "But notwithstanding all the pains that have been taken to multiply The annual observations, and all the care that has been used to render them per- parallaxmnst fectly exact, we have been able to discover nothing which indicates, ^ leBS lhan with certainty, even the existence of an annual parallax, to say nothing on of its magnitude. Yet the precision of modern observations is such, that if this parallax were only 1", it is altogether probable that it would not have escaped the multiplied efforts of observers, and especially those of Dr. Bradley, who made many observations to discover it, and who, in this undertaking, fell unexpectedly upon the phenomena of aberra- tion* and nutation. These admirable discoveries have themselves served to show, by the perfect agreement which is thus found to take Subject to be explained hereafter. 162 ASTRONOMY. CHAP. XIII. place among observations, that it is hardly to be supposed that the annual parallax can amount to 1". The numerous observations of the pole star, recently employed in measuring an arc of the meridian through France, have been attended with a similar result, as to the amount of the annual parallax. From all this we may conclude, that as yet there are strong reasons for believing that the annual parallax is less than 1", at least with respect to the stars hitherto observed. " Thus the semidiameter of the earth's orbit, seen from the nearest star, would not appear to subtend an angle of 1' ; and to an observer placed at this distance, our sun, with the whole planetary system, would occupy a space scarcely exceeding the thickness of a spider's thread. Conclusion " If these results do not make known the distance of the stars from to be drawn tne ear th, the^ at least teach us the limit beyond which the stars must 86 necessarily be situated. If we conceive a right-angled triangle, having for its base half the major axis of the earth's orbit, and for its vertex an angle of 1 , the distance of this vertex from the earth, or the length of the visual ray, will be expressed by 212207, the radius of the earth's orbit being unity ; and as this radius contains 23987 times the semidia- meter of the earth, it follows that if the annual parallax of a star were only 1", its distance from the earth would be equal to 5090209309 radii of the earth, or 20086868036404 miles ; that is, more than 20 billions. But if the annual parallax is less than 1", the stars are beyond the limit which we have assigned. Changes " It is evident that the stars undergo considerable changes, since these (n individual changes are sensible even at the distance at which we are placed. There ****** are some which gradually lose their light, as the star (T of Ursa Major. Others, as /fi of Cetus, become more brilliant. Finally, there are some which have been observed to assume suddenly a new splendor, and then gradually fade away. Such was the new star which appeared in 1572, A new star, in the constellation Cassiopeia. It became all at once so brilliant that it surpassed the brightest stars, and even Venus and Jupiter when nearest the earth. It could be seen at midday. Gradually this great brilliancy began to diminish, and the star disappeared in sixteen months from the time it was first seen, without having changed its place in the heavens. Its color, during this time, suffered great variations. At first it was of a dazzling white, like Venus ; then of a reddish yellow, like Mars and Aldebaran ; and lastly, of a leaden white, like Saturn. An- Another otner s t ar which appeared suddenly in 1604, in the constellation Ser- nw star. pentarias, presented similar variations, and disappeared after several months. These phenomena seem to indicate vast flames which burst forth suddenly in these great bodies. Who knows that our sun may not be subject to similar changes, by which great revolutions have perhaps taken place in the state of our globe, and are yet to take place. Periodical Some stars, without entirely disappearing, exhibit variations not less remarkable. Their light increases and decreases alternately in regular periods. They are called for this reason variable stars. Such is the FIXED STARS. 163 star Algol, in the head of Medusa, which has a period of about three CHAP. XIIL days ; 3 of Cepheus, which has one of five days ; @ of Lyra, six ; ft of Antinous, seven ; o of Cetus, 334 ; and many others. "Several attempts have been made to explain these periodical varia- Attempt* tions. It is supposed that the stars which are subject to them, are, like to explain all the other stars, self-luminous bodies, or true suns, turning on their periodical ax'is, and having their surfaces partly covered with dark spots, which chan S es - may be supposed to present themselves to us at certain times only, in consequence of their rotation. Other astronomers have attempted to account for the facts under consideration, by supposing these stars to have a form extremely oblate, by which a great difference would take place in the light emitted by them under different aspects. Lastly, it has been supposed that the effect in question is owing to large opake bodies, revolving about these stars, and occasionally intercepting a part of their light. Time and the multiplication of observations may per- haps decide which of these hypotheses is the true one. " One of the best methods of observing these phenomena is to compare Order in the stars together, designating them by letters or numbers, and dispos- these obsw- ing them in the order of their brilliancy. If we find, by observation, varans, that this order changes, it is a proof that one of the stars thus com- pared, has likewise changed ; and a few trials of this kind will enable us to ascertain which it is that has undergone a variation. In this man- ner, we can only compare each star with those which are in the neigh- borhood, and visible at the same time. But by afterward comparing these with others, we can, by a series of intermediate terms, connect together the most distant extremes. This method, which is now prac- ticed, is far preferable to that of the ancient astronomers, who classed the stars after a very vague comparison, according to what they called the order of their magnitudes, but which was, in reality, nothing but that of their brightness, estimated in a very imperfect manner. " By comparing the places of some of the fixed stars, as determined Suggestion from ancient and modern observations, Dr. Halley discovered that they ofDr.Halley. had a proper motion, which could not arise from parallax, precession, or aberration. This remarkable circumstance was afterward noticed by Cassini and Le Monnier, and was completely confirmed by Tobias Mayer, who compared the places of 80 stars, as determined by Roemer, with his own observations, and found that the greater part of them had a proper motion. He suggested that the change of place might arise from a progressive motion of the sun toward one quarter of the heavens ; but as the result of his observation did not accord with his theory, he remarks that many centuries mirst elapse before the true cause of this motion could be explained. " The probability of a progressive motion of the sun was suggested upon theoretical principles by the late Dr. Wilson of Glasgow ; and Lalande deduced a similar opinion from the rotatory motion of the sun, by supposing, that the same mechanical force which gives it a motion 164 ASTRONOMY. CHAP. XIII. round its axis, would also displace its center, and give it a motion of translation in absolute space ns "" " If the sun has a motion in absolute space, directed toward any such a the- q uarter f tne heavens, it is obvious that the stars in that quarter must OTV- appear to recede from each other, while those in the opposite region would seem gradually to approach, in the same manner as when walk- ing through a forest, the trees toward which we advance are constantly separating, while the distance of those which we leave behind is gradu- ally contracting. The proper motion of the stars, therefore, in opposite regions, as ascertained by a comparison of ancient with modern obser- vations, ought to correspond with this hypothesis ; and Sir W. Her- schel found, that the greater part of them are nearly in the direction which would result from a motion of the sun toward the constellation Hercules, or rather to a part of the heavens whose right ascension is 250 52' 30", and whose north polar distance is 40 22'. Klugel found the right ascension of this point to be 260, and Prevost made it 230, with 65 of north polar distance. Sir W. Herschel supposes that the motion of the sun, and the solar system, is not slower than that of the earth in its orbit, and that it is performed round some distant center. The attractive force capable of producing such an effect, he does no< suppose to be lodged in one large body, but in the center of gravity of a cluster of stars, or the common center of gravity of several clusters." The following figures, taken from Norton's Astronomy, represent the telescopic appearance of some of the double stars. Doable " There are stars which, when viewed by the naked eye, and even and multiple by the help of a telescope of moderate power, have the appearance of tan. on ] v a s i n g| e s t ar ; but, being seen through a good telescope, they are found to be double, and in some cases a very marked difference is per- ceptible, both as to their brilliancy and the color of their light. These Sir W. Herschel suppos/d to be so near each other, as to obey recipro* cally the power of each other's attraction, revolving about their com- mon center of gravity, in certain determinate periods. Castor, y Leonis, Rigel, Pole Star, jrMonoc, gCancri. Revolutions " The two stars, for example, which form the double star Castoi of the multi- have varied in their angular situation more than 45 since they were pie stars. observed by Dr. Bradley, in 1759 5 and appear to perform a retrograde revolution in 342 years, in a plane perpendicular to the direction of the sun. Sir W. Herschel found them in intermediate angular positions, at intermediate times, but never could perceive any change in their distance. The retrograde revolution of y in Leo, another double star, is supposed to be in a plane considerably inclined to the line in which we view it, and to be completed in 1200 years. The stars < of Bootes, FIXED STARS. 165 perform a direct revolution in 1681 years, in a plane oblique to the sun. CHAP. Yrq, The stars of Serpens, perform a retrograde revolution in about 375 years; and those of y in Virgo in 708 years, without any change of their distance. In 1802, the large star of Hercules, eclipsed the smaller one, though they were separate in 1782. Other stars are sup- posed to be united in triple, quadruple, and still more complicated systems. "With respect to the determination of the real magnitude of the stars, Descriptioa and their respective distances, we have as yet made but little progress, of nebula. Researches of this kind must be left to future astronomers. It appears, however, that the stars are not uniformly distributed through the heavens, but collected into groups, each containing many millions of stars. We can form some idea of them from those small whitish spots called Nebulae, which appear in the heavens as represented in the ac- companying illustration. By means of the telescope, we distinguish in these collections an almost infinite number of small stars, so near each Other, that their rays are ordina- rily blended by irradiation, and thus present to the eye only a faint uniform sheet of light. That large, white, lumi- nous track, which traverses the heavens from one pole to the other, under the name of the Milky Way, is probably nothing but a nebula of th*s The Milk} kind, which appears larger than the others, because it is nearer to u*. Way a ne With the aid of the telescope we discover in this zone of light such a bula ' prodigious number of stars that the imagination is bewildered in attempting to represent them. Yet from the angular distances of these stars, it is certain that the space which separates those which seem nearest to each other, is at least a hundred thousand times as great as the radius of the earth's orbit. This will give us some idea of the immense t-xtent of the group. To what distance then must we with- draw, in order that this whole collection may appear as small as the other nebulae which we perceive, some of which cannot, by the assist- ance of the best telescopes, be made to present anything but a bright speck, or a simple mass of light, of the nature of which we are able to form some idea only by analogy ? When w pttetnpt, in imagination, to fathom this abyss, it is in vain to think of prescribing any limits to 166 ASTRONOMY. CHAP. XIiI. the universe, and the mind reverts involuntarily to the insignificant portion of it which we are destined to occupy. " Observa- Before we close this chapter, we think it important to call the atten- tions on ta- t | on O f tne reader to table II , in which will be seen, at a glance (in the columns marked annual variation), the general effect of the preces- sion of the equinoxes ; and although we have called particular attention to the fact elsewhere, we here notice that all the stars, from the 6th to the 3th hour of right ascension, have a progressive motion to the southward ( ), and all the stars from the 18th to the 6th honr of right ascension have a progressive motion to the northward (-)-), and the greatest variations are at h. and 12 h. But these motions are not, in reality, the motions of the stars ; they result from motions of the earth. Whenever the annual motion of any star does not correspond with this common displacement of the equinox, we say the star has a proper motion ; and by such discrepancy it has been decided, that those stars marked with an asterisk, in the catalogue, have proper motions ; and the star 61 Cygni, near the close of the table, has the greatest proper motion. The paral- From this circumstance, and from the fact of its being a double star, i&x of 61 it was selected by Bessel as a fit subject for the investigation of stellar Cygni disco- p ara n ax ; an( j it is now contended, and in a measure granted, that the annual parallax of this star is 0".35, which makes its distance more than 592.000 times the radius of the earth's orbit ; a distance that light could not traverse in less than nine and one-fourth years. PHYSICAL ASTRONOMY. 16? SECTION III. PHYSICAL ASTRONOMY. CHAPTER I. GENERAL LAWS OF MOTION THE THEORY OF GRAVITY. CHAP. 1. (148.) IN a work like this, designed for elementary in- mat &ho ~ ld struction, it cannot be expected that a full investigation of be e x P ecte* physical astronomy shall be entered into ; for that subject n this work, alone would require volumes ; and to fully appreciate and comprehend it, requires the matured philosopher combined with the accomplished mathematician. We shall give, however, a sufficient amount to impart a good general idea of the subject if one or two points are taken on trust. For elementary principles we must turn a moment to natu- Elemental ral philosophy, and consider the laws of inertia, motion, and principles. force. Motion is a change of place in relation to other bodies which we conceive to be at rest ; and the extent of change in the time taken for unity is called velocity, and the essential cause of motion we denominate force. A double force will give a double velocity to bodies moving Velocity the / i i i * * ? measure of freely in void space, or in an unresisting medium a tnple force force, a triple velocity, &c. This is taken as an axiom and hence, when we consider mere material points in motion, the relative velocities measure the relative amounts of force. There are three elements to motion, which the philosopher never loses sight of; or we may say that he never thinks of motion without the three distinct elements of time, velocity, and distance, coming into his mind. Algebraically, we put t, v, and d, to represent the three ele- ments, and then we have this important and general equation, tv = d (1) 168 ASTRONOMY. CHAP. 1. d d From this we derive v= (2) and *=- C3) Expression t V ^ ' tv force ( 149.) As forces are in proportion to velocities (when mo- mentum is not in question ), therefore, if we put / and F to represent two forces corresponding to the distances d and D, which are described in the times t and T, then by making use of equation ( 2 ), in place of the velocities, we have f F - i ( 4 ^ * w * t T v/ The law of ( 150. ) A body at rest, has no power to put itself in mo- tion ; and having no self power, no internal force or will, m any shape, it cannot increase or diminish the motion it may have, or change the direction it may be moving. This is the law of inertia. It cannot of itself change its state ; and if it is changed it must be acted upon by some external force ; and this accords with universal experience ; and this law is the most natural and simple of any we can imagine, but it is only in the motion of the heavenly bodies that it is fully exemplified. Bwne central The earth, moon, and planets move in curves not in force mnst j-jg]^ l mes . The directions of their motions are changed. motions of Something external from them must, therefore, change them ; the earth, f or fa e l aw O f inertia would continue a motion once obtained planets. i Q a straight line. Now this force must exist within the or- bit of every curve; we therefore naturally refer it to the body round which others circulate. The earth and planets go round the sun, and if we could suppose a force residing in the sun to extend throughout the system sufficient to draw bodies to it, this would at once account not only for the planets deviating from a right line, but would account for a constant deviation of all bodies to that point, and the preser- vation of the system. Th moon's The moon goes round the earth, constantly deviating from Motion con. ^ tangent of its orbit, and the law of inertia is constantly sidered. We number the proportions the same aa equations, for a propor- tion is but an equation in another form. THE EARTH'S ATTRACTION. 169 urging it to rise from the center ; the two on an average balan- CHAP. I. cing each other, retains the inoou in an orbit about the earth. Now what and where is this force ? Is it around the earth, or within the earth ? Is it electrical or magnetic ? or is it that same force ( call it what we may ) that makes a body fall toward the earth's center when unsupported on a resting base ? A trifling incident, the fall of an apple from a tree, seems contempia. to have led the mind of Newton to the contemplation of this Hons of SH force which compels and causes bodies to fall, and he at once ^ ac Ne> conceived this force to extend to the moon and to cause it to deviate from the tangent of its orbit. The next consideration was, whether if this were the force, it was the same at the distance of the moon, as on the sur- face of the earth ; or if it extended with a diminished amount, what was the law of diminution ? Newton now resorted to computation, and for a test he incipient conceived the force in question to extend to the moon, undi- steps to ^ minished by the distance ; and corresponding thereto he de- gravity, cided that the moon must then make a revolution in its orbit in 10 h. 55m. But the actual time is 27 d. 7h. 43m., which shows that if the force is the same which pervades a falling body on the surface of the earth, it must be greatly diminished. Now by making a reverse computation, taking the actual important time of revolution, and finding how far the moon did really com P uta - fall from the tangent of its orbit in one second of time, it was found to be about ^^ part of 16 T ^ feet the distance a body falls the first second of time. But the distance to the moon is about 60 times the radius of the earth, and the inverse square of this is ^g-V^* which corresponds to the actual fall of the moon in one second. (151.) It is a well-established fact in philosophy, and A principle geometrically demonstrated, that any force or influence exist- ' B P hlloso P* ing at a point, must dimmish as it spreads over a larger space, and in proportion to the increase of space. But space increases as the square of linear distance, as we see by Fig. 28, 170 ASTRONOMY. CHAP, i. A double distance spreads the influence over four times the space, whatever that influence may be; a triple distance, nine times the space, etc., the space increasing as the square of Fig. 28. the distance. Therefore, any influence spreading in all di- rections from its central point must be enfeebled as the square of the distance. The theory From observations and considerations like these, Newton gravity. established the all-important and now universally admitted theory of gravity. This theory may be summarily stated in the following words : Every body of matter in the universe attracts every other body, in direct proportion to its mass, and in the inverse proportion to the square of the distance. This theory Some attempts have been made, from time to time, to call wen estab- fa e ^ ru th of this theory in question, and substitute in its place the influence of light, caloric, and electricity; but any thing like a close application shows how feebly all such sub- stitutes stand the test. The theory of gravity so exactly accounts for all the phy- sical phenomena of the solar system, that it is impossible it should be false ; and although we cannot determine its nature or its essence, it is as unreasonable to doubt its existence, as to doubt the existence of animate beings, because we know nothing of the principle of life. Attraction (152.) According to the theory of gravity, every particle * f *^!" esu * composing a body has its influence, and a very irregular body may be divided in imagination into many smaller bodies, and the center of gravity of each taken as the point of attraction, and all the forces resolved into one will be the attraction of thf whole body. STANDARD OF FORCE. 171 In a sphere composed of homogeneous particles, the aggre- CHAP. L gate attraction of all of them will be the same as if all were Attraction of compressed at the center; but this will be true of no other a sphere. body. The earth is not a perfect sphere, and two lines of attraction from distant points on its surface may not, yea, will not, cross each other at the earth's center of gravity. ( See Fig. 10.) (153.) A particle anywhere inside of a spherical shell of Attraction equal thickness and density, is attracted every way alike, and nide of of course would show no indication of being attracted at all. Hence a body below the surface of the earth, as in a deep pit or well, will be less attracted than on the surface, as it will be attracted only by the diminished sphere below it. At the center of the earth a body would be attracted by the earth every way alike, ,and there would be no unbalanced force, a sphere. and of course no perceptible or sensible attraction.* ( 154 .) The attractive power on the surface of any perfect Ex regsio and homogeneous sphere may be expressed by the mass of the for the at- *phere divided by the square of the radius. tbe^TiLeo" Consider the earth a sphere (as it is very nearly), and a sphere. put E to represent its mass, and r its mean radius, then E This attractive force, algebraically expressed by we call ^, and it is sufficient to cause bodies to fall 16 T ^ feet during the first second of time. If the earth had contained more matter, bodies would have fallen more than 16 T ^ feet the first second ; if less, a less distance. With the same matter, but more compact, so that r 2 would The definit- ... , .., T-, ,, E attraction of be less with E the same, would be greater, and the attrac- ,h e earth. tive power at the surface greater, and bodies would then fall more than 16-jL feet the first second of their fall. Now we say this 16 T ^ feet is the measure of the earth's attraction at its surface, and it is made the unit and standard measure, directly or indirectly, for all astronomical forces. * See Robinson's Natural Philosophy, page 16 172 ASTRONOMY. For this reason, we call the undivided attention to this force, the known the noted the all-important TO find the ( 155. ) By the theory of gravity, we can readily obtain an * t analytical expression for the attraction of a sphere at any dis- uy distance, tance from the center, after knowing the attraction at the surface. For example. Find the value of the attraction of the earth, at the distance of D from its center ; r being the radius of the earth, and g the gravity at the surface ; put x to represent the attraction sought. Then by the theory, 9 * -' 0r '* As g and r are constant quantities, the variations to x will correspond entirely to the variations of D 2 . We shall often refer to this equation. (156.) As every particle of matter in the universe at- skm for the tracts every other particle, therefore the moon attracts the "action & of eart h as we ^ as *be eartn attracts the moon ; and the extent wo bodies, by which they will draw together, depends on their mutual at- traction. If m represents the mass of the moon, and fi the radius of the lunar orbit ; then, E The earth will attract the moon by the force -^. The moon will attract the earth by the force -^-. jfi \ Im The two bodies will draw together by the force ~ . If we substitute the value of g, as found in ( 154 ), in equa- E tion (5 ), and making r = D, then we have the expression The spirit of these expressions will be more apparent when we make some practical applications of them, as we intend soon to do. KEPLER'S LAWS. 173 CHAPTER II. SEPLISB'S LAWS DEMONSTRATION OF THE SECOND AND THIRD HOW A PLANETARY BODY WILL FIND ITS ORBIT. ( 157. ) IN tbis chapter we design to make some examina- CHAP. n. tion of Kepler's laws, recapitulating them in order. The orbits of the planets are ellipses, having the sun at , J7 . - . ler'a laws wie oj weir joci. This law is but a concise statement of an observed fact, which never could have been drawn from any other source than observation ; but the second law, namely, That the radius vector of any planet ( conceived to be in mo- tion ) sweeps over equal areas in equal times is susceptible of a rigid mathematical demonstration, under the following gen- eral theorem. Any body, being in motion, and constantly urged toward any A g ener l fixed point, not in a line with its motion, must describe equal areas in equal times round that point. Let a moving body be at A, having a veloci- ty which would | carry it to JB, say in one sec- ond of time. By\ the law of iner- ti.a, it would move from B to C, an equal dis- 1 tance, in the next second of time. But during this second interval of time, let us suppose it must obey an impulse or force from the point S, sufficient to carry it to D. It must then, by the composition of forces explained in natural phi- losophy, describe the diagonal B E, of the parallelogram BDEG. 174 ASTRONOMY. CHAP ii. Now in the first interval of time, we supposed the moving body described the triangle SAB. The second interval, it would have described the triangle S C, if undisturbed by any force at S, but by such a force it describes the triangle S B E\ but the triangle S B E is equal to the triangle SBC, because they have the same base S B, and lie between the parallels S B and E C. Also the triangle S B C is equal to the triangle S A B, because they terminate in the same point S, and have equal bases A B and B C. There- fore the triangle S A B is equal to the triangle S B E, be- cause they are both equal to the triangle S B (7; that is, the moving body describes equal areas in equal times about the point S, and this is entirely independent of the nature of the force at S\ it may be directly or inversely as the distance, or as the square of the distance. The con- The converse of this theorem is, that when a body describes r the equal areas in equal times round any point, the body is con- stantly urged toward that point; and therefore as the planets are observed to describe equal areas in equal times round the sun, their tendency is toward the sun, and not toward any other point within the orbits. Kepler's (158.) The third law of Kepler is most important of all, third law name ly The squares of the times of revolution are to each prove* that the sun's at- ^ er as ^ e cuoe & f the distances from the sun. By this law traction is it is proved, that it is the same force which urges all the theTuare of P^ anets to tne same point, and that its intensity is inversely as the distance, the square of the distance from that point ( the center of the sun ), confirming the Newtonian theory of gravity. Fig. 30. To show this, let us suppose that the planets revolve round the sun in circular orbits ( which is not far from the truth), and let P ( Fig. 30 ) represent the posi- tion of a planet ; F the distance which the planet is drawn from a tangent during unity of time; in the same time that it describes the indefinite small arc c; and the number of times that c is contained in the whole circum- ference, so many units of time, then, must be in one revolution. SOLAR SYSTEM. 175 If D is the diameter of the orbit and fthe time of revolu- CHAf - " tion, then will "T ' ' : m So for any other planet. If f is the force urging it toward AH impor the sun, a its corresponding: arc, Tits time of revolution, and R the radius of its orbit ; then, reasoning as before, moustraUd. By comparing ( 1 ) and ( 2 ) we have I) 2 By squaring, f* : T 2 : : : By Kepler's law, t 2 : T 2 : : r* j ^ 3 . By comparing the two last proportions, and observing that 2r may be put for />, and reducing, we have But by the well-known property of the circle, we have F . : c :: c : 2r; or, c 2 =2rF. In like manner, . . . a 2 = 2 Rf. Substituting these values in the last proportion, and redu cing, we have Or, . . Rf : rF :: r : R. Hence, R 2 f=r 2 F', or, F if n R 2 : r 2 . 1 i Or, . . F'-f JL A T" J.C That is , the attractive force of the sun is reciprocally pro- portional to the square of the distance. ( 159.) If we commence with the hypothesis, that bodies tend toward a central point with a force inversely propor- 176 ASTRONOMY. CHIP - n - tional to the squares of their distances, and then compute and laws of the corresponding times of revolution, we shall find that the in Kepler's S 9 uares f &* times must be as the cubes of the distances. Hence third law. Kepler's third law is but the natural mathematical relation which must exist between times and distances among bodies moving freely, in circular orbits, animated by one central force which varies as the inverse square of the distance. An inquiry. ( 160. ) Having shown that Kepler's third law is but a mathematical theorem when the planets move in circles and their masses inappreciable in comparison to that of the sun's, we now inquire whether the law is true, or only approximately true, when the orbits are ellipses, and their masses consid- erable. HOW answer- On one of these points of inquiry, the reader must take our ed - assertion ; for its demonstration requires the use of the inte- gral calculus, a method that we designed not to employ in thia work. Kepler's third law supposes all the force to be in the central body, and the planets only moving points. But wo have seen in Art. ( 156 ) that the attracting force on any planet is the mass of both sun and planet divided by the square of their mutual distance; and therefore when the mass of the planet is appreciable, the force is increased, and Masses of the time of revolution a little shortened. But the fact that the planets Kepler's law corresponds so well with other observations compareT to proves that the masses of all the planets are inappreciable the sun. compared to the mass of the sun. Kepler's ( 161. ) As to the other point, we state distinctly that the third h.w ma- planets (considered as bodies without masses) revolving in ^*[! y ellipses of ever so great eccentricity, the squares of the times tic orbits. of revolution are to each other as the cubes of half the greater axes of the orbits. We shall not attempt a demonstration of this truth ; but hope, the following explanation will give the reader a clear view of the subject. Bodies revolving in ellipses round one of the foci, may be considered to have a rising and a falling motion; something like the motion of a pendulum. The motion of a pendulum depends on the force of gravity, the length of the pendulum, PLANETARY MOTION 177 and the distance the pendulum was first drawn aside. The CHAP, n, motion of a planet depends on the force of gravity, its mean distance from the sun, and the original impulse firct given to A common it. Most persons, who have not investigated this subject, wnwofopm- imagine that each planet must originally have had precisely the impulse it did have to maintain itself in its orbit ; and so it must, to maintain itself in just that definite orbit in which it moves. But had the original impulse been different, either as to amount or direction, or as to both, then lytlie action of gravity and inertia, the planet would have found a corresponding orbit. (162.) The force of gravity, from the action of any attract- Examina. ing body, is always as the mass of the body divided by the square tion of th * of its distance. Algebraically, if Mia the masa of the body, ^tumi^ r its distance, and F the force at that distance, then (see 156) "'ptte o* J we have - , - - ^-=F. (See Fig. 28.) Now if the planet has such a velocity, c, as to correspond with the proportion F : c : : c : 2r, Or,- - - - c=J2rF=^ , and that velocity at right angles to r (Fig. 31), then the planet's orbit would be a circle, with the radius r. If the velocity had been less in amount than this expression, and still at right angles to r, then the planet would fall within the circle, and the action of gra- vity would increase the motion of the planet ; and the motion would increase faster than the increased action of gravity : there would be a point, then, where the motion would be suflicient to maintain the planet in a circle, at its then distance ; but the **> n. B. direction of the motion will not permit the planet to run into * ****" the circle, and it must fall within it. The motion continues to increase until its position becomes at right angles to the radius vector ; the motion is then as much more than sufficient to maintain the planet in a circle, as it was insufficient in the first instance; it therefore rises, by the law of inertia, and returns to the original point P, where it will have the same velocity as before ; and thus the planet vibrates between two extreme distances. 12 7S ASTRONOMY. mean distan- ces of the or \its. A hypothe deal case. CHAP, ii If the velocity, on starting from the point P, were very Gravity and much less than sufficient to maintain a circle, at that distance, original ve ^hen ^he orbit it would take would be very eccentric, and locity deter- . mine the ec- its mean distance much less than r. If the original velocity at P were greater than to maintain it in a circle, it would " pass outside of this circle, and the point P would be the peri- helion point of the orbit. Thus, we perceive, that the eccentricity of orbits and mean distances from the sun depend on the amount and direction of the original impulse, or velocity which the planet has in some way obtained; and it is not necessary that the planet should have any definite impulse, either in amount or direction, to move in an orbit, if the direction is not directly to or from the sun (163.) For a more definite explanation of this subject, let us conceive a planet launched out into space with a velocity sufficient to maintain it in a circle at the distance it then hap- pened to be, but the direction of such velocity not at right angles to the sun, then the orbit will be elliptical, and the degree of eccentricity will depend on the direction of the motion ; but the longer axis of the orbit will be equal to the diameter of the circle, to which its velocity corresponds ; and the time of its revolution will be the same, whether the orbit is circular or more or less elliptical. Let P (Fig. 31) be the posi- tion of a planet, S the sun; and let the velocity, a, be just suffi- cient to maintain the planet in a circle, if it were at right angles toSP. Now to find the orbit that this planet would describe, draw the line P at right angles to a, and from S let fall a perpendi- cular on PC; SC will be the eccentricity of the orbit, and PC will be the half of its conjugate axis; and with these lines the whole orbit is known. PLANETARY MOTION. 170 (164.) Now let us suppose that a planet is rather carelessly CHAP, n launched into space, with a velocity neither at right angles to pianeti the sun, nor of sufficient amount to maintain it in a circle, at w llfin d theu orbits, what- that distance from the sun. ever be the Let P (Fig. 32) represent the Fig. 32. direction and \- .1, 1 A i ^^MBBI^ forceo' theii position of the planet, a thei amount and direction of its hap- hazard velocity during the first I unit of time. The direction of the motion being within a right I angle to S P, the action of gra- vity increases the velocity TJ/--N of the planet, >^ on the same principle that a falling body in- creases in velocity ; and the planet I goes on in a curve, describing equal areas in equal times round the point S; and it will find a point, p, where its increased velocity will be justj equal to the velocity in a circle whose radius is the diminished distance Sp. From the point p, and at right angles to , draw p C, c., forming the right angled triangle p C S. SO is the eccentricity, Sa the mean distance, and p C half the conjugate axis of the orbit. If the planet is launched into space in the other direction, The the action of gravity will diminish its motion, and will bring wilt be sym " r . i i metrical on it at right angles to the line joining the sun ; it is then at its each side of apogee, with a motion too feeble to maintain a circle at that a Ps ee anfl distance ; and it will, of course, approach nearer and nearer pe to the sun by the same laws of motion and force that it receded from the sun ; hence the curve on each side of the apogee will be symmetrical ; and the same reasoning will apply to the curve on each side of the perigee ; and, in short, we shall have an ellipse. To sum up the whole matter, it is found by a strict exanii- AB nation of the laws <& gravity, motion, and inertia, that whatever ,*on. 180 ASTRONOMY. CHAP. II. may be the primary force and direction given to a planetary body ( if not directly to or from the sun ), the planet mil find a corresponding orbit, of a greater or less eccentricity, and of a greater or less mean distance ; and whatever be the eccentricity of the. orbit, the real velocity, at tJie extremity of the shorter axis, will be just sufficient to maintain the planet in a circular orbit, at that mean distance from the sun* Theory of * L e t S be the sun, and P the position of a planet as repre- l)r. Gibers . . * concerning sentecl m e annexed ngure, and we may now suppose it to the asteroid* burst into fragments, the figure representing three fragments only; the velocity and direction of one represented by a; of another by b, and of a third by c, &c. Flff. 33. As action is just equal to reaction, under all circumstances, therefore the bursting of a planet can give the whole mass no additional velocity ; a small mass may be blown off at a great velocity, but there will be an equal reaction on other masses, On th direction. bursting of a planet, the The whole might simply burst into about equal parts, and fragments then they would but separate, and all the parts move along would take , J i.ii orbits corre- m *" e same general direction, and with the same aggregate spending to velocity as the original planet. The bursting of a rocket is a ry minute, but a very faithful representation of such an explosion KEPLER'S LAWS. 181 ( 165.) To see whether Kepler's third law applies to ellipses, CHAP. n. we represent half the greater axis of any ellipse by -4, and Kepler^ half the shorter axis by B, and then (3.1416)^4^ is the area third law ri- of the ellipse. Also, let a represent the velocity or distance f ^J^ ~~ ellipses, as If the velocities of the several fragments were equal, the well a to times of their revolutions would be equal ; but the eccentri- circle - cities of the several orbits would depend on the angles of a, 6, c, &c., with S P. If a is at right angles to e the mass of the sun, and E the mass of the earth, then ( at the same unit of distance), the attraction of the sun is, to tlie attraction of the earth, as M to E. But attraction is inversely as the square of the distance. M Hence the attraction of the sun at D distance, is -= ; and E the attraction of the earth at R distance is ~Ra* Gravity of The earth is made to deviate from a tangent of its orbit the snn is ^j the attraction of the sun; and the moon is made to deviate the devia. from a tangent of its orbit by the attraction of the earth, and tion of the the amount of these deviations will give the respective tangent of iu a m ounts of solar and terrestrial gravity, orbit. If we take any small period of time, as a minute or a sec- ond, and compute the versed sine of the arc which the earth describes in its orbit during that time, such a quantity will express the sun's attraction ; and if we compute the versed sine of the arc which the moon describes in the same time, that quantity will express the attraction of the earth. HOW to com- i n Figure 30, Art. 158, ^represents the versed sine of an parative 0in arc ; and if we take D to represent the mean distance be- masses of the twcen the earth and sun, and consider the orbit a circle *tm and earth ^ ag we ma y w jthout error, 164), the whole circumference is MASSES OF THE PLANETS. 185 it]} (* = 6.2832). Divide the whole circumference by the CHAP. m. number of minutes in a revolution ; say T, and the quotient will represent the arc a (Fig. 30). When T is very large, and of course a very small, the chord and arc practically coin- cide ; and by the well known property of the circle, we have 2D : a:: a: F; Or, F= ~. . (1) _ *D **D 2 a 3 VK* * But a = -^-; hence, a* = -^-, and ^= ; That is, F = _ a ; which is an expression for the sun's attraction at the distance of the earth. But -=r is also an expression for the sun's attraction at the same distance ; therefore, _ = -^- ; Or, #**- In the same manner, if R represents the radius of the lunar orbit; t the number of minutes in the revolution of the moon ; the mass of the central attracting body ( in this case the earth ) must be expressed by Therefore, E : M: : : -^-. This proportion gives a relation between the masses of the earth and sun expressed in known quantities. If we assume unity for the mass of the earth, we shall have for the mass of the sun, . . . (A) (169.) This is a very general equation, for D may repre- The general sent the radius of the earth's orbit, or the orbit of Jupiter or application Saturn, arid 3Twill be the corresponding time of revolution. ioB /* Also R may represent the radius of the lunar orbit, or the 186 ASTRONOMY. CHAP. in. orbit of one of Jupiter's or Saturn's moons, and then t will be its corresponding time of revolution. The results This equation, however, is not one of strict accuracy, as of the eqna- f^g distance a planet falls from the tangent of its orbit, in a tion will not accutend definite moment of time, is not, accurately j. -, but ~ why ? - a ( see 156 ), E being the mass of the planet. The force which retains a moon in its orbit is not only the attracting mass of the central body, but that of the moon also. Bu? the planets being very small in relation to the sun, and in general the masses of satellites being very small in respect to their primaries, the errors in using this equation will in gen- eral be very small. The error will be greatest in obtaining Corrections for equation the mass ot the earth, as in that case the equation involves < A) the periodic time of the moon; which period is different from what it would be were the moon governed by the attraction of the earth alone ; but the mass of the moon is no inconsid- erable part of the entire mass of both earth and moon ; and also the attraction of the sun on the combined mass of tho earth and moon, prolongs the moon's periodical time by about its 179th part. With these corrections the equation will give the mass of the sun to a great degree of accuracy ; but we can determine the mass of the sun by the following method : From Art. 155, we learn that the attraction of the earth curate (a- / 3 \ Uom * at the distance to the sun, is g \j^j- By Art. 168, we have just seen that the attraction of the fr 2 D sun on the earth, is = ; therefore, Taking the mass of the earth as unity, we have W 2 7)3 Equation (J?) is more accurate than equation (.4), MASSES OF THE PLANETS. 187 because ( B ) does not involve the periodical revolution of the CM IP. in. moon, which requires correction to free it from the effects of the sun's attraction. To obtain a numerical expression for HOW to ob. the mass of the sun, M, the numerator and denominator of the " right hand member of equation ( B ) must be rendered homo- nit. geneous ; and as g, the force of gravity of the earth, is ex- pressed in feet ( corresponding to T in seconds ), therefore r the mean radius of the earth, and D the distance to the sun, must be expressed in feet. But from the sun's horizontal parallax, we have the ratio between r and D ( see 127 ), which gives D = 23984 r. This reduces the fraction to - fT~rjia - But * ex ~ press the whole in numbers, we must give each symbol its value ; that is, * = 6.2832 ; r = ( 3956 ) ( 5280 ) ; g = 16.1 ; T= 31558150, the number of seconds in a sidereal year. (6.2832) 2 (23984W3956)(5280) It would be too tedious to carry this out, arithmetically, An without the aid of logarithms, and accordingly we give the showin g i -^ L- i i *- *v s reat ut loganthmetical solution, thus, of logarithms 6 .2832 log. 0.798178X2 . . . 1.596356 23984 log. 4.380000X3 . . 13.140000 3956 log. . . . . . 3 .597256 5280 log. ..... 3 .722632 Logarithm of the numerator, . . .22 .056244 32.2 log ...... 1 .507856 Themasaol 31558150 log. 7.499114X2 . . .14 .998228 the ' d " - terminal. Logarithm of the denominator, . 16 .506084 Therefore M= 354945, whose log. is 5 .550160 That is, the mass or force of attraction in the sun is 354945 times the mass or attraction of the earth. La Place 188 ASTRONOMY. CHAP, m. says it is 354936 times ; but the difference is of no conse- quence. Equation (A) gives 350750; but equation (-5), as we have before remarked, is far more accurate, and the result here given, agrees, within a few units, with the best author- ities. Equation ( B ) is not general ; it will only apply to the relative masses of sun and moon, because we do not know the element g, the attraction, on the surface of any other planet, except the earth. That is, we do not know it as a primary fact ; we can deduce it after we shall have determined the mass of a planet. Equation ( A) is general, and although not accurate, when applied to the earth and sun, is sufficiently so when applied to finding the masses of Jupiter, Saturn, or Uranus ; because these planets are so remote from the sun, that the revolutions of their satellites are not troubled by the sun's attraction. TO find the (170.) To find the mass of Jupiter (or which is the masses of Ju- game ^j n g fa e mass O f fa e gun w ^ en J U piter is taken as piter, Saturn, and Uranus, unity), we conceive the earth to be a moon revolving about the sun, and compare it with one of Jupiter's satellites revolving round that body. To apply equation (/I), let the radius of the earth equal unify, then the radius of Jupiter must be 11.11 (Art. 131 ) ; and by observation the orbit radius of Jupiter's 4th satellite is 26.9983 times Jupiter's radius, therefore the distance from the center of Jupiter to the orbit of its 4th satellite, must be the following product (11.11) (26.9983), which corresponds to R in the equation. D = 23984; T= 365.256; t= 16.6888. Therefore, by applying equation (A), (M= (16.6888) 2 (23984)3 have Jf= (365.256) 2 (1L11)3(26 .9983)'' By logarithms 16.6888 log. 1 .222410x2 . 2 .444820 23984 log. 4 .380000x3 . 13 .140000 Logarithm of the numerator, . 15. 584820 MASSES OF THE PLANETS. 18* 865.256, log. 2 .562600x2 . 5 .125200 CHAP. ID 11.11, log. 1.045714x3 . 3.137142 26.9983, log. 1.431320x3 . 4 .293960 Logarithm of the denominator, . . 12 .556302 Therefore M = 1068*, log. ... 3 .028518 This result shows that the mass of the sun is 1068 times the mass of Jupiter ; but we previously found the mass of the sun to be 354945 times the mass of the earth, and if unity is taken for the mass of the earth, and J for the mass of Jupiter, we shall have 1068 J= 354945; because each member of this equation is equal to the mass of the sun. By dividing both members of this equation by 1068, we The mass of find J,he mass of Jupiter to be 332 times that of the earth ; "red",, ^ but'in Art. 132, we found the bulk of Jupiter to be 1260 of the earth, times the bulk of the earth ; therefore the density of Jupiter is much less than the density of the earth. In the same manner we may find the masses of Saturn and The masses Uranus the former is 105.6 times, and the latter 18.2 of Satnra times the mass of the earth. and Uranu *' The principles embraced in equation ( A ) apply only to those planets that have satellites ; for it is by the rapid or slow motion of such satellites that we determine the amount of the attractive force of the planet. In short, the masses of those planets which have satellites, what re- are known to groat accuracy; but the results attached to sults ma y *** others in table IIT, must be regarded as near approximations. acc urate. The slight variations which the earth's motion experiences The m by the attractions of Venus and Mars, are sufficiently sensi- ble to make known the masses of these planets; and M. Mercnrj Burckhardt gives ^^VTT for Venus, and ??T _ ?T for Mars ( the mass of the sun being unity ) . Mercury he put down at This is a correct result according to these data; but more modern observations, in relation to the micrometic measure of Jupiter, and the distance of his satellites, ^ive results a little different, as expressed in table III. 190 ASTRONOMY. CHAP, ui _- r2 i- T _; but this result is little more than hypothetical, as it is drawn from its volume, on the supposition that the densities of the planets are reciprocal to their mean distances from the sun; which is nearly true for Venus, the earth, and Mars. of (171.) It may be astonishing, but it is nevertheless true, the lunar par. that by means of equations (-4) and (B~) we can find the alia*, we diameter of the earth to a greater degree of exactness than by may find the . diameter of an J one ac ^ual measurement. the earth. "We have several times observed that equation ( A ) is not accurate when used to find the masses of the earth and sun, because it contained the time of the revolution of the moon; which revolution is accelerated by the gravity of the moon, and retarded by the action of the sun. Therefore, to make equation ( A ) accurately express the mass of the sun, the element t 2 requires two corrections, which will be determined by subsequent investigation. The first is an increase of T l jth part ; the second is a diminution of ^|jth part, and both corrections will be made if we take 76-357 ^ K ogQ * 2 m place of t 2 . 75-358 A common Then having two correct expressions for the mass of the sun, those two expressions must equal each other ; that is, 76-357 75-358 By suppressing common factors, we have 76-357 1 2 ** 75-358 R* In this equation r represents the mean radius of the earth, and we will suppose it unknown ; the equation will then make it known. The relation between JB, the mean radius of the lunar or- bit, and r, the mean radius of the earth, is given by means of the moon's horizontal parallax. "^ e moon ' s equatorial horizontal parallax, as we nave seen, and (65) is 57' 3"; but the horizontal parallax for the mean ra- MASSES OF THE PLANETS. 191 dius, is 56' 57"; this makes R = ( 60.36 ) r, whatever the CHAP. m. numerical value of r may be. Put this value of R in the preceding equation, and suppress the common factor r 2 , zontai 76-357 * 2 * 2 we then have ___=- Therefore, ' 75-358(60.36)3** * As g is expressed in feet, and corresponds to t in seconds, confidence the numerical value of T will be in feet, which divided by in the result< 5280, the number of feet in a mile, will give the number of miles in the mean radius or mean semidiameter of the earth ; and by applying the preceding equation, giving g, t, and , their proper values ; and by the help of logarithms, we readily find r = 3955.8 miles ; less than a mile from the most approved result ; and we do not hesitate to say, that this result is more to be relied upon than any other. MASS OF THE MOON. ( 172. ) Approximations to the mass of the moon hav<* The ma3- ^ been determined, from time to time, by careful observations the moon on the tides ; but it is in vain to look for mathematical re- determined suits from this source ; for it is impossible to decide whether from obser. vations i the tides. any particular tide has been accelerated or retarded, aug- va mented or diminished, by the winds and weather; and if not affected at the place of observation, it might have been at remote distances ; but notwithstanding this objection, the mass of the moon can be pretty accurately determined by means of the tides, owing to the great number and variety of observations that can be brought into the account; and we shall give an exposition of this deduction hereafter ; but at present we shall confine our attention to the following simple and elegant method of obtaining the same result. If the moon had no mass ; that is, if it were a mere mate- rial point, and was not disturbed by the attraction of the sun, then the distance that the moon would fall from a tan- gent of its orbit, in one second of time, would be just equal 192 ASTRONOMY . III. OT~ to ~. (Art. 155. ) In this expression g, r, and R, repre- sent the same quantities as in the last article. The dis- tance that the moon actually falls from a tangent of its orbit, in one second of time, is equal to the versed sine of the arc it describes in that time, and the analytical expression for it is found thus : Let n- R represent the circumference of the lunar orbit, and if t is put for the number of seconds in a mean revolution, then *R represents the arc corresponding to the moon's motion in one second (Fig. 30), and as this so nearly coincides with a chord, we have ZR ^ .. 1? . t t Hence, we perceive, that -^- is the distance that the ion for the ^ t 2 distance the moon wou ](j f a ]j f r0 m the tangent of its orbit in one second moon falls m _ on* second of time, if it were undisturbed by the action of the sun ; but of iim. 359 we can free it from such action by multiplying it by 5^ as we shall show in a subsequent chapter. That is, the attraction of both the earth and moon, at the distance of the 858-** lunar orbit, is gg^. But the attraction of the earth alone, at the same distance, or 2 is ^- ; and comparing these quantities with the more gene- raJ expressions in Art. 156, we have gr^ 358 *R ^ ~wr$w By suppressing the common denominator, in the first couplet, and calling E, the mass of the earth, unity, the pro- portion reduces to 1 : H* 9+ : -357^1 MASSES OF THE PLANETS. 193 As in the last article, .#=(60. 36) r, and this value put for <**> ni> 9 , and reduced, gives 358 T* (60.36) 3 r 1+m :: g 357-2*3 . Therefore, - - l+ro= - 357 . 2 , 2 This fraction, as well as the one in the last article, can be reduced arithmetically; but the operation would be too tedious; they are both readily reduced by logarithms, by which we found 14-w=1.01333 ; hence m=.01333, which is very nearly ^th. Laplace says ^jth of the earth gi ve n by La- is the true mass of the moon ; and this value we shall use. P lace - THE DENSITIES OF BODIES. * * . (173.) The density of a body is only a comparative term, standard and to find the comparison, some one body must be taken as the standard of measure. The earth is generally taken for that standard. T t is an axiom, in philosophy, that the same mass, in a smaller volume, must be greater in density; and larger in volume, must be less in density ; and, in short, the density must be directly proportional to the mass, and inversely pro- portional to the volume ; and if the earth is taken for unity in ma,ss, and unity in volume, then it will be unity in density also ; and the density of any other planetary body will be its mass divided by Us volume; and if its volume is not given, the density may be found by the following proportion, in which d represents the density sought, and r the radius of the body ; the radius of the earth being unity. The proportion is drawn from the consideration that spheres are to one another as the cubes of their radii. 1 mass , , mass r : : : 1 : a: hence o= - . 1 r 3 r 3 9 From this equation we readily find the density of the sun, for we have its mass (354945), and its semidiameter 111.6 times the semidiameter of the earth (Art. 156) ; therefore its 13 194 ASTRONOMY. CH*P. m. , . 354945 -- density must be =0.'ZD t l, or a bttle more than ta pheies com. (lll.Oj 3 paied to the tne density of the earth. the earth. The mass of Jupiter is 332 times that of the earth, and its volume is 1260 times the volume of the earth ; therefore the oon density of Jupiter is =0.264 ; which is a little more than the density of the sun. Densities The mass of the moon is -^j, and its volume j 1 ^, therefore its moon s^c 1 "' density is T 'j divided by ? ^, or A =0.6533; about f the den- sity of the earth. From these examples the reader will understand how the densities were found, as expressed in table III. GRAVITY ON THE SURFACE OP SPHERES. Gravity on ( 174.) The gravity on the surface of a sphere depends on h<5 ne'v&el *^ e mass an< * volume. The attraction on the surface of a , aet, how sphere is the same as if its whole mass were collected at its foucd center; and the greater the distance from the center to the surface, the less the attraction, in proportion to the square o f the distance : but here, as in the last article, some one sphcr must be taken for the unit, and we take the earth, as before. The mass of the sun is 354945, and the distance from its center to its surface is 111.6 times the semidiameter of the arth ; therefore a pound, on the surface of the earth, is to the pressure of the same mass, if it were on the surface of the sun, as - to -, or as 1 to 28 nearly. That is, one pound on the surface of the earth would be nearly 28 pounds on the surface of the sun, if transported thither. The mass of Jupiter is 332, and its radius, compared to that of the earth, is 11.1 (Art. 131); therefore one pound, on ooo the surface of the earth^ would be , or 2. 48 pounds on the surface of Jupiter; and by the same principle, we can compute the pressure on the surface of any other planet. Results will be found in table III. LUNAR PERTURBATIONS. 195 CHAPTER IV. PROBLEM OF THE THREE BODIES. LUNAR PERTURBATIONS. (175.) By the theory of universal gravitation, every body CHAP ^ in the universe attractsevery other body, in proportion to its Thelhe <>T J . of gravity. mass ; and inversely as the square ot its distance ; but sim- ple and unexceptionable as the law really is, it produces very complicated results in the motions of the heavenly bodies. If there were but two bodies in the universe, their mo- The com " tions would be comparatively simple, and easily t.raoed, for ^"(^ they would either fall together or circulate around each other in some one undeviating curve ; but as it is, when two bodies circulate around each other, every other body causes a deviation or vibration from that primary curve that they would otherwise have. The final result of a multitude of conflicting motions can- not be ascertained by considering the whole in mass : we must take the disturbance of one body at a time, and settle upon its results ; then another and another, and so on ; and the sum of the results will be the final result sought. We, then, consider two bodies in motion disturbed by a The prob- third body : and to find all its results, in general terms, i? *? m of *** thre* \>o*ies, the famous problem of "the three bodies ;" but its complete solution surpasses the power of analysis, and the most skillful mathematician is obliged to content himself with approxi- mations and special cases. Happily, however, the masses of most of the planets are so small in comparison with the mass of the sun, and their distances so great, that their influences are insensible. We shall make no attempt to give minute results ; but we hope to show general principles in such a manner, that the reader may comprehend the common inequalities of planetary motions. Let m, Fig. 35, be the position of a body circulating around Abstram another body A, moving in the direction PmB, and dig- attnxcti the sun, and the ring around the earth the moon's orbit, inclined to the plane of the ecliptic with an angle of about five degrees ; then when the sun is out of the plane of the ring, or moon's orbit, the action of the sun has a constant tendency to bring the moon into the ecliptic, and by this tendency the moon does fall into the ecliptic from either side sooner than it otherwise would. The point where the moon falls into the ecliptic is called the moons node; and by this external action of the sun the moon falls into the ecliptic from its greatest inclination before it describes 90, and goes from node to node be- fore it describes 180 and hence we say that the moon's nodes fall backward on the ecliptic. The rate of retro- gradation is 19 19' in a year, making a whole circle in about 18.6 years. (178.) We are now pre- pared to be a little more defi- nite, and inquire as to the amount of some of the lunar irregularities. Let S be the mass of the sun, E that of the earth, and m the moon, situated at D. Let a be the mean distance between the earth and sun, z the distance between the sun and moon, and r the mean ra- dius of the lunar orbit. Let the moon have any indefinite position in its orbit. ( It is represented in the figure at c The attraction of the sun on the earth is , the attrae a 3 LUNAR PERTURBATIONS 199 tion of the sun on the moon is ; and the attraction of the ' inri earth and moon, on the moon, is ^ , ( Art. 156, ) Let the line D B, the diagonal of the parallelogram A C, be the attraction of the sun on the moon, and decompose it into the two forces DA and D (7; the first along the lunar radius vector, the other parallel to SJS. The two triangles C D B and D S E are similar, and give the proportion a : z : : CD : D B. But D = ; Therefore CD = ^-. By a similar proportion we find Let the angle SED be represented by x, then D G will be expressed by r cos. #, and SD G will be a right line nearly, for the angle D S E is, never greater than 1'. Now if the force D C, which is parallel to S E, is only equal to the force of the sun's attraction on the earth, it will not disturb the mutual relations of the earth and moon. cr The force of the sun's attraction on the earth is ; and as this must be less than the force of attraction on the moc-n when the moon is at D, conceive it represented by the line Cn, and subtracted from (77), will leave Dn the excess of the sun's attraction on the two bodies, the earth and the moon ; and this alone constitutes the disturbing force of the moon's motion ; That is, Dn = CDCn = ^f 4 5 An Mpw "~ Z 3 a 2 sion for the whole distnr. Or Dn = aS { -- J t the distorting force. Decoin- bing fol pose this force ( Dn ) into two others, Dp and jn, by means of the right angled triangle Dpn; the angle pDn being equal to DE S, which we represent by x. ASTRONOMY. /I 1 \ Whence Dp = : Sa ^ -J cos. a?; And The force D-4, i.e. ( ) is called the additions force; The radial the force Dp the aUaiitwus force. The difference of these force. two f orces i s called the radial force ; that is Sa ( ) cos. x = the radial force ; pn is the \ 33 a s / Z 3 tangental force. Expression \Vhen the ande x is equal to 90, cos. x = o, SD SK of the radial force at the __ w hich values, substituted, give for the value quadratures. #3 of the radial force at the quadratures, and its tendency there is to increase the gravity of the moon to the earth. When the angle x is zero ( the moon is in conjunction with the sun ) the cos. x = 1, and the radial force becomes &*__Sa___rS f S(a r) Sa ^F"~^3~~^' or Z 3 ~~a~3' But at that point z = ( a r ), "which value substituted, and rejecting the comparatively very small quantities in both numerator and denominator, we have, for the radial force at 2rS conjunction, . When the angle x = 180 ( the moon is in opposition to the sun ), cos. x = 1, and the force becomes Sa___ Sa __ rS^ S S (a+r) a3 3 z 3 ' >r a 2 " z 3 But at this point z = a -(- r, which, substituting as before, O Cf and we have for the radial force in opposition --, the same expression as at conjunction. If we compare the radial force at the syzigies with the ex- pression for it at the quadratures, we shall find it the same in form, but double in amount and opposite in sign, showing that it is opposite in effect. LUNAR PERTURBATIONS. 201 (179.) As the radial force increases the gravity of the CHAP. rv. moon to the earth at the quadratures, and diminishes it at Poinlfl thesyzigies, there must be points in the orbit symmetrically where the ra . situated, in respect to the syzigies, where the radial force dial force * neither increases nor diminishes the gravity, and of course its expression for those points must be zero; and to find HOW u> these points we must have the equation find thenu os.*- = . . (1) a/ z 3 By inspecting the figure we perceive that the line SD Q is in value nearly equal to the line SE, and for all points in the orbit we have z = a + r cos. x ...... ( 2 ) Reducing equation ( 1 ), we have (a 3 z 3 ) cos. # = ra 2 . . . . (3) Cubing (2), As r is very small in relation to a, the terms containing the powers of r, after the first, may be rejected; we then have ( 3_2 3 ) = zp3 a a rcos. a?. . . (4) This value substituted in ( 3 ), and reduced, gives Result of 4- 3 cos. 2 a? = 1. the radial Hence cos. x = Jl and x = 54 44 7 , or the points for< * at ** quadratures are 35 16' from the quadratures. *nd syzigieta This shows that at the quadratures, and about 35 on each side of them, the gravity of the moon is increased by the action of the sun, and at the syzigies, and about 54 on each side of them, the gravity is diminished ; and the diminu- tion in the one case is double the amount of increase in the Mean ra other, and by the application of the differential calculus we dl learn that the mean result, for the entire revolution, is a dimi- nution whose analytical expression is ^ - , an expression which holds a very prominent place in the lunar theory; the 202 ASTRONOMY. . result of which we have used in Art. 171, and there stated it to be ^1 g th part of the force that retained the moon in its orbit. Value of But how do we know this to be its numerical value, is a dili mea for^r verv 8er i us inquiry of the critical student? fold b W The force that retains tlie moon in its or bit is . E + m . ( Art. 156 ) ; and if the radial force can be rendered homoge- neous with this, some numerical ratio must exist between them. Let x represent that ratio, and we must find dome numerical value for x to satisfy the following equation : rS_ E+m Therefore r = calling JS= 1, m = ^ (Art. 172), or E -\- m is 1.013. S = 354945 ( Art. 169 ), and the relation between th* mean distance to the sun, and the mean radius of the lunai orbit, is 397.3,* therefore or the coefficient to x, in equation ( A ), is one three hundredth and fifty-eighth part of the force which retains the moon in its orbit. General ef. (180.) The mean radial force causes the moon to circu- diai force. ^ a ^ e a ^ 3^8^ P art grater distance from the earth than it otherwise would have, and its periodical revolution is in- creased by its 179th part ; but this would cause no variation or irregularity in its distance or angular motion, provided its orbit were circular, and the earth and moon always at the name mean distance from the sun. The radial But we perceive the expression ^-^ contains two variable force varia. w. quantities, r and a, which are not always the same in value ; and, therefore, the value of the expression itself must be va- * This relation is found by dividing the horizontal parallax of the moon, 56' 57", by the horizontal parallax of the sun, 8".6. iJNAR PERTURBATIONS. 203 riable ; and it will be least when the earth is at the greatest CHAP. tv. distance from the sun, and, of course, the moon's motion will then be increased. But the earth's variable distance from the sun depends on the eccentricity of the earth's orbit; and The annu. hence we perceive that the same cause which affects the ap- al e i uatlon r r of the moon'* parent solar motion, affects also the motion of the moon, and motion. gives rise to an equation called the annual equation* of the moon's motion. It amounts to 11' in its maximum, and va- ries by the same law as the equation of the sun's center. ( 181.) If we take the general expressions for the radial A general ^ expression force, Sa( -L) cos. x -, and banish the letter z J" theradlal \2 3 a 3 ' 2 3 force at any /. .. i /.i . point of tha from it by means or the equation moon's orbit. 2 = a 4- r cos. x Or, z 3 = a 3 + 3a 2 r cos. x, ( neglecting the powers of r ) and we shall have, rS (3 cos. 2 x 1) 3 for an expression of the radial force corresponding to any angle x from the syzigy. If we take the general expression for the line pn, the tan- gental force, and banish 2, as before, we have, 3rs cos. x sin. x tangental force = - . By doubling numerator and denominator, this fraction can Expression (2 cos. x sin. x\ take the following form : for the tan ' gental force. But, by trigonometry, 2 cos. x sin. x = sin. 2#, mi f .1 . i /. %rs sin. 2x Ineretore the tangental force = - ^ -- . This expression vanishes when x = o and x = 90 ; for then its vanish gin. 2x sin. 180 = 0. Hence the tangental force van- in *P inM xshes at the syzigies and quadratures, attains its maximum This is equation I, in the Lunar Tables. 204 ASTRONOMY. The tan- rCe when the earth is in perigee. Application force* to an on elliptical o: CHAP. iv. value at the octants, and varies as the sine of the double angular distance of the moon from the sun. The mean maximum for this force must be determined by observation. It is known by the name of variation, and by mere inspection we can see that its amount must correspond to the variations of r and of a 3 . Hence, to obtain the moon's place, we must have correction on correction. The variation amounts to about 35'. It increases the ve- locity of the moon from the quadratures to the syzigies, and diminishes it from the syzigies to the quadratures ; hence, in consequence of the variation, the velocity of the moon is greatest at the syzigies, and least at the quadratures. (182.) Let us now examine the effect of the radial force lunar orbit, considered as elliptical. Let SE(Y\g. 38) be at right angles to A JB, the greater axis of the lunar orbit, and conceive A C B to represent the orbit that the moon would take if it were undisturbed by the sun. But when the moon comes round to its perigee at A, it is in one of its quadratures, and the radial force then increases the gravity of the moon toward the earth by the expression . But here r is less than its mean value, and the expression is less than its mean, and therefore the moon is not crowded so near the earth ag it otherwise would be, and, of course, at this point the moon will run farther from the earth. At the point C, the radial force tends to increase the dis- tance between the earth and moon, and (o widen the orbit. when the Wh corresponding to the whole year t is equal to -(-* 2\(a i' s | * rS V i d)*^(a+d)*/' Or r8 ( l 4- l } 4V(a rf)^(5+5j>/' But this expression is always greater than - , except The meaB 2a 3 value of the when d = ; then it is the same, as any algebraist can verify. radial ' Hence the mean radial force for the whole year is greater O f ail when as the earth's orbit is more eccentric, and it will be least of the . earlh>i all when that orbit becomes a circle ; and then, and then circ i e . only, it will be accurately represented by ^--. But when the radial force is least, the mean motion must be greatest, and that force is less and less as the eccentricity of the earth's orbit becomes less and less; and corresponding thereto the moon's motion becomes greater and greater, as has been the case for more than 4000 years. ( 186. ) The mean distance between the earth and sun re- The can mains constant. It must be so from the nature of motion, * f '^JJJUJ! force, action, and reaction ; but by the attraction of the city of th planets the eccentricity of the earth's orbit is in a state of per- earth '* orblt petual change : the change, however, is excessively slow. From the earliest ages the eccentricity of the orbit has been dimin- ishing ; and this diminution will probably continue until it is annihilated altogether, and the orbit becomes a circle ; after which it will open out in another direction, again become ec- centric, and increase in eccentricity to a certain moderate amount, and then again decrease. 14 210 ASTRONOMY. CHAP. IV. The period for these vibrations, " though calculable, has never The im. fe en calculated further than to satisfy us that it is not to be nespo d reckoned by hundreds or even by thousands of years." It is a ding to these period so long that the history of astronomy, and of the whole human race, is but a point in comparison. The moon's mean motion will continue to increase until the earth's orbit becomes a circle; after which it will again decrease, corresponding with the increase of a new eccentricity. ( 187 ' ) For the sake of simplicity, we have thus far con- lunar orbit sidered the moon's orbit to be in the same plane as the account int eartn ' s or ki* kut this is not true ; the mean inclination of the lunar orbit to the ecliptic is 5 8', varying about 9' each way, according to the position of the sun. Owing to this inclination of the lunar orbit, the expressions which we have obtained for the tangental force need cor- rection, by multiplying them by the cosine of the inclination ; and for the effect of the same forces in a perpendicular direction to the moon's longitude, multiply them by the sine of the inclination of the orbit. The position of the moon's orbit, in relation to the sun, is strictly analogous to the ring in relation to the disturbing body D (Art. 176) ; the sun is constantly urging the moon into the plane of the ecliptic, which has a constant tendency to diminish the inclination of the lunar orbit ( except when the sun is in the positions of the moon's nodes) ; and this con- stant force urging the moon to the ecliptic, causes the moon's nodes to retrograde. We conclude this chapter by a brief summary of the prin- cipal causes which affect the moon's motion. A summary j > The eccentricity of the earth's orbit ; which gives rise to itatementof ,, \ f ,, . , ., , the lunar ir- *" e annua l equation of the moon in longitude. 2. The eccentricity of the lunar orbit ; producing the equa- tion of the center. 3. The tangental force; giving rise to the equation called variation. 4. The position of the sun in respect to the greater axil of the lunar orbit ; giving rise to the inequality called evettwn. 5. The inclination of the moon's orbit. THE TIDES. 211 6. The combination of the first cause, when differing from CHAP. IV. its mean state, augments or diminishes the result of every other thus making many additional small equations 7. The ellipsoidal form of the earth. CHAPTER V. THE TIDES. ( 188. ) THE alternate rise and fall of the surface of the CHAT. v. sea, as observed at all places directly connected with the Definition waters of the ocean, is called tide ; and before its cause was of the ternt tide. definitely known, it was recognized as having some hidden and mysterious connection with the moon, for it rose and fell twice Connection in every lunar day. High water and low water had no con- WIth th * J * moon. nection with the hour of the day, but it always occurred in about such an interval of time after the moon had passed the meridian. When the sun and moon were in conjunction, or in opposi- HigUtidei. tion, the tides were observed to be higher than usual. When the moon was nearest the earth, in her perigee, other circumstances being equal, the tides were observed to be higher than when, under the same circumstances, the moon was in her apogee. The space of time from one tide to another, or from high water to high water ( when undisturbed by wind ), is 12 hours and about 24 minutes, thus making two tides in one lunar day ; showing high water on opposite sides of the earth at the same time. The declination of the moon, also, has a very sensible influ- Tide at- ence on the tides. When the declination is high in the north, [ ect , ed b ? the 1 decimation the tide in the northern hemisphere, which is next to the moon, O f the moon. is greater than the opposite tide ; and when the declination of the moon is south, the tide opposite to the moon is greatest. A difficulty It is considered mysterious, by most persons, that the moon * by its attraction should be able to raise a tide on the opposite wagoner side of the earth. 212 ASTRONOMY. CHAP. V. The true can** Fig. 41. A summary illustration of the tides. That the moon should attract the water on the side of the earth next to her, and thereby raise a tide, seems rational and natural, but that the same simple action also raises the oppo- site tide, is not readily admitted ; and, in the absence of clear illustration, it has often excited mental rebellion and not a few popular lecturers have attempted explanations from false and inadequate causes. But the true cause is the sun and moon's attraction ; and until this is clearly and decidedly understood not merely assented to, but fully comprehended' it ia impossible to understand the com- mon results of the theory of gra- vity, which are constantly exem- plified in the solar system. We now give a rude, but strik- ing, and, we hope, a satisfactory explanation. Conceive the frame-work of the earth to be an inflexible solid, as it really is, composed of rock, and in- capable of changing its form under any degree of attraction ; conceive also that this solid protuberates out of the sea, at opposite points of the earth, at A and B, as repre- sented in Fig. 41, A being on the side of the earth next to the moon, m, and B opposite to it. Now in connection with this solid con- ceive a great portion of the earth to be composed of water, whoso particles are inert, but readily move among themselves. The solid A B cannot expand under the moon's attrac- tion, and if it move, the whole mass moves together, in virtue of the moon's attraction on its center of gravity. But the particles of water at a, being free to move, and being under a THE TIDES. 213 more powerful attraction than the solid, rise toward A, pro- CHAP. ?. ducing a tide. The particles of water at b being less attracted toward m than the solid, will not move toward m as fast as the solid, and being inert, they will be, as it were, left behind. The solid is drawn toward the moon more powerfully than the particles of water at b, and sinks in part into the water, but the observer at B, of course, conceives it the water rising up on the shore (which in effect it is), thereby producing a tide. ( 189. ) The mathematical astronomer perceives a strict Analogy analogy between the analytical expressions for the tides and i* n ^ e ^ rtnr . the expressions for the perturbations of the lunar motion. bations and What we have called the radial force, in treating of the the P 61 *"**. * tions of th lunar irregularities, is the same in its nature as the force that ocean raises the tides ; the tide force is a radial force, which dimi- nishes the pressure of the water toward the center of the earth under and opposite to the moon, in the same manner as the radial force diminishes the gravity of the moon toward the earth in her syzigies. In Art. 179 we found that the radial force for the moon, at The radiaj 2 r force as ap. the syzigies, is expressed by ; in which expression S is P lied to ^ Q moon. the mass of the sun, a its distance from the earth, and r the radius of the lunar orbit. The same expression is true for the tides, if we change S to Convertn w iW ^ e retarded, for the disturbance at this point is in other ii re- a line with the motion of Venus, and not in a line with the tard.d. mot i on O f t he earth. When the After Venus passes conjunction, that is, passes the varying line SJZ, her motion becomes retarded, and the earth's is ac- celerated; but every motion of the earth we ascribe to the sun; and in all modern solar tables, the corrections of the sun's longitude corresponding to the action of Venus, Mars, Ju- " e moon, &c., are simply the effect that these bodies t by lar pertu.-ba. have on the motion of the earth. The direct effect of any of these bodies on the position of the sun is absolutely insensible. The relative disturbances of two planets are reciprocal to their masses ; for if one is double in mass of another, the PLANETARY PERTURBATIONS. 217 greater mass will move but half as far as the smaller, under CHAP. vi. their mutual action. But when the amount of disturbance is .i . -i i Angular i- referred to angular motion for its measure, regard must be regularities had to the distances of each planet from the sun ; for the indicate tb same distance on a larger orbit corresponds to a less angle.* pI^Ly Also, the whole amount of the disturbing force of a superior disturbance planet on an inferior will, at times, be a tangental force ^ductum*" ( Fig. 23 ) ; but the reaction of the inferior planet on the su- perior can never be in a tangent directly with, or opposed to, the motion of the superior. If observations can give the mutual disturbance of any two planets, then these circumstances being taken into considera- tion, an easy computation will give the relative masses of the planets. ( 193.) As a general result, the attraction of a superior The gene- planet on an inferior, is to increase the time of revolution of ral resulu ' respect to the the inferior, and to maintain it at a greater distance from the t j meg O f rev . sun than it would otherwise have. The action of the inferior ointiou. is to diminish the time of revolution of the superior; and the general effect is greater than it would be, if the inferior planet were constantly situated at the distance of the sun. (Art. 185.) As an illustration of this truth, we say, that if Venus were annihilated, the length of our year, and the times of revolu- tion of all its superior planets, would be a little increased, and the revolution of Mercury, its inferior planet, would be a lit- tle diminished. If Jupiter were annihilated, the times of re- volution of all its inferior planets would be a little diminished ; for it acts as a radial force to keep them all a little farther from the sun. ( 194.) If the orbits of all the planets were circular, the inequalities acceleration in one part of an orbit would be exactly compen- m orbits. * Geometry demonstrates, that, on the average of each revolution, the proportion in which this reaction will affect the longitudes of the two planets, is that of their masses multiplied by the square roots of the major axes of their orbits, inversely; and this result of a very in- tricate and curious calculation is fully confirmed by observation.-- HERSOHEL. 518 ASTRONOMY. CAP. vj. sated by the retardation in another ; and in the course of a whole revolution, the mean motions of both planets (the dis- turber and the disturbed) would be restored, and the errors in longitude would destroy each other. But the orbits are not circles, and it is only in certain very rare occurrences that symmetry on each side of the line of conjunctions takes place ; and hence, in a single revolution the acceleration of ds of P ine- one P ar ^ cannot be exactly counterbalanced by the retarda- quaiities de- tion of the other; and, therefore, there is commonly left a cer- unctions ta * a outstanding error, which increases during every synodi- in the same cal revolution of the two planets, until the conjunctions take orbits be P^ ace i Q PP os ite parts of the orbits, then it attains its maxi- mum, which is as gradually frittered away as the line of con- junctions works round to the same point as at first. Some of Hence, between every two disturbing planets there is a common uaiities too wwquality depending on their mutual conjunctions, in the same, minute to be or nearly in the same, parts of their orbits. But it would be lce * folly to compute the inequalities for every two planets, by rea- son of the extreme minuteness of the amounts ; for instance, Mercury is not sensibly disturbed by Saturn or Uranus; and Mars, and Mercury, and Uranus, practically speaking, do not disturb each other ; but Jupiter and Saturn have very con- siderable mutual perturbations, on account of their orbits be- ing near each other, and both bodies far away from the sun. The effect (195) Again, if the revolutions of two planets are ex- surate^revoi acfc ly commensurate with each other, or, what is the same lotions of the thing, the mean motion of both exactly commensurate with pianeu. ^ c j rc j e> then th e conjunctions of those two planets will al- ways occur at the same points of the orbits ( just as the con- junctions of the two hands of a clock always occur at the same points on the dial plate), and, in that case, the conjunc- tions will not revolve and distribute themselves around the orbits, so that in time, the radial and tangental forces will have an opportunity to accelerate on one side of the line of conjunctions as much as they retard on the other; and, therefore, a permanent derangement would then take place. Aropposed -p or i n8 t ance jf three times the mean angular motion of ease for illos- , -, i uation. one planet were exactly equal to twice the mean angular TOO- PLANETARY PERTURBATIONS. 219 tion of another, then three revolutions of the one would ex- CHAP, v actly correspond to two of the other, and every second con- junction of the two would take place in the same points of the orbits ; and the orbits, not being circular, the portions of them on each side of the line of conjunctions cannot be sym- metrical, unless the longer axes of the two orbits are in the game line, and the conjunctions also taking place on that line. Here, then, is a case showing that the disturbing force may constantly differ in amount on each side of the line of conjunctions, and, of course, could never compensate each other, and a permanent derangement of these two planets would be the result. Hence, we perceive, that, to preserve the solar system, it stability of is necessary that the orbits should be circles, or their times thesolali y t< tKU of revolution incommensurable; but we do not pretend to say that the converse of this is true : we do say, however, that no natural cause of destruction has thus far been found. ( 196.) The times of the planetary revolutions are incom- mensurable; but, nevertheless, there are instances that ap- proach commensurability, and, in consequence, approach a derangement in motion, which, when followed out, produce Very long periods of inequality, called secular variation. The most remarkable of these, and one which very much perplexed the astronomers of the last century, is known by the term of " the great inequality " of Jupiter and Saturn. " It had long been remarked by astronomers that, on com- Th* gnat paring together ancient with modern observations of Jupiter 1 " e i n * llt1 ^ and Saturn, their mean motions could not be uniform." The and Satnm. period of Saturn appeared to have been increased throughout the whole of the seventeenth century, and that of Jupiter shortened. Saturn was constantly lagging behind its calcu- lated place, and Jupiter was as constantly in advance of his. On the other hand, in the eighteenth century, a process pre- cisely the reverse was going on. The amount of retardations and accelerations, corresponding ** to one, two, or three revolutions were not very great ; but, as *| they went on accumulating, material differences, at length, existed between the observed and calculated places of both 220 ASTRONOMY. . vi. these planets; and, as such differences could not then be ac- counted for, they excited a high degree of attention, and formed the subject of prize problems of several philosophical societies. Laplace For a long time these astonishing facts baffled every en- th * deavor to accounfc f r them, and some were on the point of declaring the doctrine of universal gravity overthrown ; but, at length, the immortal Laplace came forward, and showed the cause of these discrepancies to be in the near commensu- rability of the mean motions of Jupiter and Saturn ; which cause we now endeavor to bring to the mind of the reader in a clear and emphatic manner. ( 197.) The orbits of both Jupiter and Saturn are ellipti- cal, and their perihelion points have different longitudes, and, therefore, their different points of conjunction are at different distances from each other, and no line * of conjunction cuts the two orbits into two equal or symmetrical parts ; hence, the inequalities of a single synodical revolution will not destroy each other ; and, to bring about an equality of perturbations, requires a certain period or succession of conjunctions, as we are about to explain. The revo- Five revolutions of Jupiter require 21663 days, and two 'rterTd"^ of Saturn ' 21518 davs - So thafc > in a P eriod of two revolu- urn zompar- tions of Saturn (about sixty of our years), after any conjunc- tion of these two planets, they will be in conjunction again not many degrees from where the former took place. Their syno- To determine definitely where the third mean conjunction dicai revoin- w jjj ta ^ e p i ace we compute the svnodical revolution of these tion deter- tinea. two planets by dividing the circumference of the circle in sec- onds (1296000) by the difference of the mean daily motion of the planets in seconds (178".6),t and the quotient is 7253.4 days; three times this period is 21760 days. In this period Jupiter performs five revolutions and 8 6' over; Saturn makes two revolutions and 8 6' over ; showing that the line * Line of conjunction, an imaginary line drawn from the sun through the two planets when in conjunction. t See problem of the two couriers, Robinson's Algebra. PLANETARY PERTURBATIONS. 221 of conjunction advances 8 6' in longitude during the period CHA?. TI of 21760 days. In the year 1800, the longitude of Jupiter's perihelion point was 11 8', and that of Saturn 89 9'; the inclination of the greater axis of the orbits, therefore, was 78 1'. Fig. 42 Let AB (Fig. 42) represent the major axis of Saturn's The se orbit, and ab that of Jupiter; the two are placed at an angle ^J^^"" 8 Of 78.* plained. Suppose any conjunction to take place in any part of the orbits, as at JS (the line JS we call the line of conjune- Lineofcon- tion) ; in 7253.4 days afterward another conjunction will take J unctlon place. In this interval, however, Saturn will describe about 243 in its orbit, at a mean rate, and Jupiter will describe one revolution and about 243 over, and it will take place as re- presented in the figure, at P Q ( STB being the direction of the motion). The next conjunction will be 243 from PQ, or at R T. From RT the next conjunction will be at si, 8 6' in advance of JS, and thus the conjunction JS ( so to speak) will gradually advance along on the orbit from S to T. But, as we perceive, by inspecting the figure, there is a * We have very much exaggerated the eccentricities of these ellip- ses, for the purpose of magnifying the principle under consideration. 122 ASTRONOMY. CHAP. VI. The period of this remark- able ine- quality com- puted, and the computa- tion confirm- ed by obser- vation. certain portion of the orbits, between S and T, where the two planets would come nearer together in their conjunction, than they do at conjunctions generally, and, of course, while any one of the three conjunctions is passing through that portion of the orbits Jupiter disturbs Saturn, and Saturn reacts on Jupiter more powerfully than at other conjunctions ; and this is the cause of " the great inequality of Jupiter and Saturn" ( 198. ) To obtain the period of this inequality, we com- pute the time requisite for one of these lines of conjunction to make a third of a revolution, that is, divide 120 by 8 6', and we shall find a quotient of 14f , showing the period to be 14f times 21760 days, or nearly 883 years: which would be the actual period, provided the elements of the orbits re- mained unchanged during that time. But in so long a period the relative position of the perigee points will undergo con- siderable variation ; which causes the period to lengthen to about 918 years. The maximum amount of this inequality, for the longitude of Saturn, is 49', and for Jupiter 21', always opposite in effect, on the principle of action and reaction. (199.) The last gre^t achievement of the pow- ers of mind in the solar system, was the discovery of the new planet Nep- tune, by Leverrier and Adams analyzing the in- equalities of the motion of Uranus. To give 1 a rude explanation of the possibility of this problem, we present Figure 43. Let S be the sun, and the regular curve the orbit of Uranus, as corresponding to all known perturbations; but at a it departs from its computed track and runs out in the protuberance a c b. This indicated that some attracting body must be somewhere in the direction ABERRATION 223 S P, although no such body was ever seen or known to exist. CHAP> ** The next time the planet comes round into the same portions D its orbit,* suppose the center of the protuberance to have changed to the line S 0. This would indicate that the un- flow com ' nutations known arid unseen body was now in the line S Q, and that cou id be since the former observations it had changed positions by the made for tho angle P S Q\ and, by this angle, and the time of its descrip- an unsaen tion, something like a guess could be made of the time of its planet, revolution. With the approximate time of revolution, and the help of Kepler's third law, its corresponding distance from the sun can be known. With the distance of the unseen body, and the amount that Uranus is drawn from its orbit by it, we can approximate to its mass. Thus, we perceive, that it is possible to know much about an existing planet, although so distant as never to be seen. But the body that disturbed the motion of Uranus has been teen, and is called Neptune. CHAPTER VII. ABERRATION, NUTATION, AND PRECESSION OF THE EQUINOXES. (200.) ABOUT the year 1725 Dr. Bradley, of the Green- c l! wich observatory, commenced a very rigid course of observa- le , s r ' tions on the fixed stars, with the hope of detecting their vations on parallax. These observations disclosed the fact, that all the the fixed stars for the stars which come to the upper meridian near midnight, have purpose of an increase of longitude of about 20"; while those opposite, fin(iin s near the meridian of the sun, have a decrease of longitude of P 20" ; thus making an annual displacement of 40". These results. observations were continued for several years, and found to be the same at the same time each year ; and, what was most Leverrier and Adams had not the advantage of a complete revolu- tion of Unm UP tneu 224 ASTRONOMY. Aberration HliitraUd. CHAP vn. perplexing, the results were directly opposite from such ai would arise from parallax. These facts were thrown to the world as a problem demand- ing solution, and, for some time, it baffled all attempts at ex- planation; but it finally occurred to the mind of the Doctor, that it might be an eifect produced by the progressive motion of light combined with the motion of the earth ; and, on strict examination, this was found to be a satisfactory solution. Fig. 44. (201.) A person stand- ing still in a rain shower, when the rain falls perpen- dicularly, the drops will strike directly on the top of his head; but if he starts and runs in any di- rection, the drops will strike him in the face ; and the effect would be the same, in relation to the direction of the drops, as if the per- son stood still and the rain came inclined from the di- rection he ran. This is a full illustration of the principle of these changes in the positions of the stars, which is called aberration; but the follow- ing explanation is more appropriate. Conceive the rays of light to be of a material substance, and its particles progressive, passing from the star S (Fig. 44) to the earth at B-, passing directly through the telescope, while the telescope itself moves from A to B by the motion of the earth. And if D B is the mo- tion of light, and A B the motion of the earth, then the tele- A tother and T>r appro* pi; ate illu- 3-ation. ABERRATION. icope must be inclined in the direction of A D, to receive the CH * P - y U- light of the star, and the apparent place of the star would bo at &', and its true place at S and the angled/)/? is 120". 30, at its maximum, called the angle of aberration. By the known motion of the earth in its vrbit, we have tl.e value of AB corresponding to one second of time: we have the angle A D B by observation : the angle at H is a right angle, arid ( from these data ) computing the side B 1) we have the velocity of light, corresponding to one second of time. To make the computation, we have D B : BA : : Rad. : tan. '20". 30.* But B A, the distance which the earth moves in its orbit The veto Fig. 45. c ' l y ' ''s' lt .computed bj *To obtnin the lojrnrit.hmpf.ic tanjrpnt of 20''.?K sw note on pngo!28. 10 226 ASTRONOMY. PHAP, vii. in one second of time, is within a very small fraction of 19 miles; the logarithm of the distance is 1.278802, and, from this, we find that ED must be 192600 miles, the velocity of light in a second ; a result very nearly the same as before deduced from observations on the eclipses of Jupiter's moons. (Art. 143.) The agreement of these two methods, so disconnected and so widely different, in disclosing such a far-hidden and re- markable truth, is a striking illustration of the power of science, and the order, harmony, and sublimity that pervades the universe. A compre. To show the effects of aberration on the whole starry h r^ Ve ^ lCW heavens, we give figure 45. Conceive the earth to be ot tnt? enecti of aberra- moving in its orbit from A to B. The stars in the line AB, tion whether at or 180, are not affected by aberration. The stars, at right angles to the line A JS, are most affected by aberration, and it is obvious that the general effect of aberra- tion is to give the stars an apparent inclination to that part of the heavens, toward which the earth is moving. Thus the star at 90 has its longitude increased, and the star op- posite to it, at 270, has its longitude decreased, by the effect of aberration; both being thrown more toward 180 The ef- fect on each star is 20".36. But when the earth is in the opposite part of its orbit, and moving the other way, from C to D, then the star at 90 is apparently thrown nearer to ; so also is the star at 270, and the whole annual variation of each star, in respect to longitude, is 40".72. Proofed / 202, ) The supposition of the earth's annual motion fully annual no- \ . , t . , , . . ' .IOH of ,-e explains aberration; conversely, then, the observed variations ,..h. of the stars, called aberration, are decided proof s of the earth's annual motion. In consequence of aberration, each star appears to describe a small ellipse in the heavens, whose semi-major axis is 20".36, and semi-minor axis is 20".36 multiplied by the sine of the latitude of the star. The true place of the star is the center of the ellipse. If the star is on the ecliptic, the ellipse, just mentioned, becomes a straight line of 40".72 in length If the star is at either pole of the ecliptio, the ellipse be- ABERRATION. 227 comes a circle of 40".72 in diameter, in respect to a great CHAP, VH circle ; but a circle, however small, around the pole, will in- clude all degrees of longitude ; hence it is possible for stars very near either pole of the ecliptic, to change longitude very considerably, each year, by the effect of aberration ; but no star is sufficiently near the pole to cause an apparent revo- lution round the pole by aberration ; and the same is true in relation to the pole of the celestial equator. All tftese ellipses have their longer axes parallel to the ecliptic, and for this reason it is easy to compute the aberration of a star in latitude and longitude,* but it is a far more complex problem to compute the effects in respect to right ascension and declination. ( 203. ) The aberration of the sun varies but a very little, Aberration because the distance to the sun varies but little, and without material enror, it may be always taken at 20".*2, subtractive. The apparent place of the sun is always behind its true place by the whole amount of aberration ; but the solar tables give its apparent place, which is the position generally wanted. In computing the effect of aberration on a planet, regard must be had to the apparent motion of the planet while light is passing from it to the earth. The effects of aberration on the moon are too small to be The moon noticed, as light passes that distance in about one second of not affected . . by aberra- timC - lion. ( 204. ) While Dr. Bradley was continuing his observa- other ine- tions to verify his theory of aberration, he observed other qnalltl " b - serveil by Dr. small variations, in the latitudes and declinations of the stars, that could not be accounted for on the principle of ab- erration. The period of these variations was observed to be about *Aber. m Lon. cos./ Aber. in Lat. = 20".36sin. (Ss) sin. /. In these expressions S represents the longitude of the sun, 9 the longitude of the star, and / its latitude. ASTRONOMY. CHAP. m. the same as the revolution of the moon's node, and the amount of the variation corresponded with particular situa- tions of the node ; and, in short, it was soon discovered that the cause of these variations was a slight vibration in the earth's axis, caused by the action and reaction of the sun and moon on the protuberant mass of matter about the equa- tor, which gives the earth its spheroidal form, and the effect itself is called NUTATION. Fig. 46. / 205.) We have shown, in Art. 176. that the attraction folly explain* j ed by the the. * a body, m, on a ring of matter around a sphere, has the ory of gravi- effect of making the plane of the ring incline toward the at- tracting body. Let B C, Fig. 46, represent the plane of the- ecliptic ; and conceive the protuberant mass of matter, around the equator, tc be represented by a ring, as in the figure. Let m be thi NUTATION. 229 moon at its greatest declination, and, of course, without the CHAP. VH. plane of the ring. Let P be the polar star. The attraction of m on the ring inclines it to the moon, and causes it to have a slight motion on its center ; but the motion of this ring is the motion of the whole earth, which must cause the earth's axis to change its position in relation to the star P, and in relation to all the stars. When the moon is on the other side of the ring, that is, opposite in declination, the effect is to incline the equator to the opposite direction, which must be, and is, indicated by an apparent motion of all the stars. A slight alternate motion of all the stars in declination, cor- responding to the declinations of the sun and moon, was care- fully noted by Dr. Bradley, and since his time has been fully verified and definitely settled : this vibratory motion is known by the name of nutation, and it is fully and satisfac- torily explained on the principles of universal gravity ; and conversely, these minute and delicate facts, so accurately and completely conforming to the theory of gravity, served as one of the many strong points of evidence to establish the truth of that theory. ( 206.) By inspecting Fig. 46, it will be perceived that The %*** when the sun and moon have their greatest northern declina- t a t ion a. tions, all the stars north of the equator and in the same hemi- lustrated by sphere as these bodies, will incline toward the equator; or all Fl s- 46 - the stars in that hemisphere will incline southward, and those in the opposite hemisphere will incline northward ; the amount of vibration of the axis of the earth is only 9".6 ( as is shown by the motion of the stars ), and its period is 18.6, or about nineteen years , the time corresponding to the revolution of the moon's node. When the moon is in the plane of the equator, its attraction can have no influence in changing the position of that plane; and it is evident that the greatest ef- feet must be when the declination is greatest. node most b T he moon's declination is greatest when the longitude of tocorrespond . / A tothemoon'* the moon s ascending node is 0, or at the first point of Aries. The greatest declination is then 28 on each side of the 2 3o ASTRONOMY. CHAP. vn. equator; but when the descending node is in the same point, the moon's greatest declination is only 18. Hence there will be times, a succession of years, when the moon's action on the protuberant matter about the equator must be greater than in an opposite succession of years, when the node is in an oppo- site position. Hence, the amount of lunar nutation depends on the position of the moon's nodes. Monthly nn- j^ j s ver y na t u ral to suppose that the period of lunar nuta- gmaii/ tion would be simply the time of the revolution of the moon ; and to, in fact, it is ; but the corresponding amount is very small only about one-tenth of a second. This is because half a lunar revolution, about 13i days, while the moon is on one side of the equator, is not a sufficient length of time for the moon to effect much more than to overcome the inertia of the earth ; but, in the space of nine years, effecting a little more than a mean result at every revolution, the amount can rise to 9". 6, a perceptible and measurable quantity. The mean ( 207.) The mean course of the moon is along the ecliptic : effect of the j ts var i a ti on f rom that line is only about five degrees on each orioon on the mass of mat. side \ hence, the medium effect of the moon on the protuberant ter around m ass of matter at the equator is the same as though the moon was all the while in the ecliptic. But, in that case, its effect would be the same at every revolution of the moon ; and the earth's equator and axis would then have an equili- brium ofpos : tion, and there would be no nutation, save the slight monthly nutation just mentioned, which is too small to be sensible to observation ; and the nutation that we observe, is only an inequality of the moon's attraction on the protube- rant equatorial ring ; and, however great that attraction might be, it would cause no vibration in the position of the earth, if it were constantly the same. Solar nn. We have, thus far, made particular mention of the moon, tolion but there is also a solar nutation : its period is, of course, a year ; and it is very trifling in amount, because the sun at tracts all parts of the earth nearly alike; and the short period of one year, or half a year (which is the time that the unequal attraction tends to change the plane of the ring iu THE EQUINOXES. 231 one direction), is too short a time to have any great effect on CHAP. vil. the inertia of the earth. The solar nutation, in respect to declination, is only one second. (208.) Hitherto we have considered only one effect of nu- tation that which changes the position of the plane of the equator or, what is the same thing, that which changes the position of the earth's axis ; but there is another effect, of greater magnitude, earlier discovered, and better known, re- sulting from the same physical cause, we mean the PRECESSION OF THE EQUINOXES. We again return to first principles, and consider the mu- FiMt pna- tual attraction between a ring of matter and a body situated CIJ out of the plane of the ring; the effect, as we have several times shown, is to incline the ring to the body, or (which is the same in respect to relative positions), the body inclines to run to the plane of the ring. The mean uttraction of the moon is m the plane of the The mean ecliptic. The sun is all the while in the ecliptic. Hence, the mean attraction of both sun and moon is in one plane, the moon are in ecliptic; but the equator, considered as a ring of matter sur- rounding a sphere, is inclined to the plane of the ecliptic by an angle of 23 degrees, and hence the sun and moon have a constant tendency to draw the equator to the ecliptic, and actually do draw it to that plane ; and the visible effect is, to make both sun and moon, in revolutions, cross the equator sooner than they otherwise would, and thus the equinox falls back on the ecliptic, called the precession of the equinoxes. The annual mean precession of the equinoxes is 50". 1 of The P recc - , . A i , .1 ion of * arc, as is shown by the sun coming into the equinox, or eq uinoxet crossing the equator at a point 50". 1 before it makes a revo- lution in respect to the stars. Perhaps it is clearer to the mind to say, that the sun is Natural drawn to the equator by the protuberant mass of matter pregticil> around the earth, and, in consequence, arrives at the equator, in its apparent revolutions, sooner than it otherwise would. But the truth is, that the ecliptic is stationary in position, 232 A S T R N O M Y CHAP, vii. and the equator, by a slight motion, meets the ecliptic; which motion is caused by the attractions of the sun and moon, as has been several times explained, rhe tine j^ t ^ e 1110 on were all the while in the ecliptic, the preees- physical cause of the sion of the equinoxes would then be a constantly flowing quan- ,*ece,ion of ^ ec a j to 5Q//^ for eac h year : but, for a succession of the equinox- es about nine years, the moon runs out to a greater declination than the ecliptic, and, during that time, its action on the equatorial matter is greater than the mean action, and then comes a succession of about nine years, when its action is less than its mean ; hence, for nine years, the precession of the equinoxes will be more than 50". 1 per year, and, for the nine years following, the precession will be less than 50". 1 for each year ; and the whole amount of variation, for tJiis in- equality, in respect to longitude, is 17". 3, and its period is half a revolution of the moon's nodes. This inequality is called the equation of the equinoxes, and varies as the sine of the longitude of the moon's nodes. Equation The equation of the equinoxes, of course, affects the length equl " of the tropical year, and slightly, very slightly, affects side- real time. Mean and There is a true equinox ajid a mean equinox; and, as side- 8 real time is measured from the meridian transit of the equi- nox, there must be a true sidereal and a mean sidereal time; but the difference is never more than 1.1 s. in time, and, gene- rally, it is much less. Explanation ( 209.) In the hope of being more clear than some authors rFlg ' 47 have been, in explaining the results of precession, we present Fig. 47. E represents the pole of the ecliptic, and the great circle around it is the ecliptic itself. P is the pole of the earth, 23 27' from the pole E, and around P, as a center, we have attempted to represent the .equator, but this, of course, is a little distorted; qp and ^ are the two opposite points where the ecliptic and equator intersect; . vii and the line qp == as having a slow motion of 50". 1 per an- Fig. 47. From the num, on the ecliptic, in a retrograde direction ; and this must fixcd pogi . carry the pole P, around the point E, as a center, carrying tion of th also the solstitial points backward on the ecliptic. Some ^ S ' P '^' f ^ e of the stars have proper motions; but, putting that circum- stars, the stance out of the question, the stars are fixed, and the eclip- st tic is fixed; therefore, the stars never change latitude, but t nde 234 ASTRONOMY. .^YII. the whole frame-work of meridians from the pole P, the pole itself, and the equator, revolve over the stars ; and, in respect to that motion of the meridian and the equator, the stars change right ascension, declination, and longitude, but do not change latitude. The stars change longitude, simply because the first meridian of longitude, T E, moves backward ; they change right ascension, because the meridian, cp P, and all the meridians of right ascension, revolve backward. One hemi. By inspecting the figure, we readily perceive that all tbe un 1 " ap- stars near V must, apparently, approach the north pole, be- proaches the cause the pole, in its revolution round E, is approaching to- th e rt otherre' war( ^ ^at P ar ^ ^ *h e ecliptic ; for the same reason, all the cede* from stars near ^ are, apparently, moving southward, because the equator is being drawn over them. In short, all the stars, from the eighteenth hour of right ascension through , to the sixth hour of right ascension, must diminish in north po- lar distance, and all the stars, from the six hours through * , to the eighteenth hour of right ascension, must increase in north polar distance, in consequence of the precession of the equinoxes. inspection These observations may be confirmed by inspecting Table II, in which is registered the positions of the principal fixed stars, with their annual variations. The column of annual variation of declination changes sign at the point correspond- ing to six hours, and eighteen hours of right ascension ; and the rapidity of this variation is greater as the star is nearer to hours, or twelve hours of right ascension. Annual va- When the right ascension of a star is hours, or twelve hours, it is easy to compute its annual variation in declina- how comput- tion, corresponding to its precession along the ecliptic of 50".l. Conceive a small plane triangle whose hypothenuse is 50".l, the angle at the base 23 27' 40" (. e. the obliquity of the ecliptic ), the side opposite to this angle will be found to be a little over 20", corresponding to the figures in the table. Proper mo. jfc j g thus, by the motion of these imaginary lines over the wn l e concave of the heavens, that the annual variation of both right ascension and declination of each individual star THE EQUINOXES. 235 in the catalogue is computed and put down ; and if any par- CHAP, vn ticular star does not correspond with this, it is said to have proper motion ; and it is thus that proper motions are detected. As P must circulate round E by the slow motion of 50". 1 Final effect in a year, it will require 25868 years to perform a revolution; and the reader can perceive, by inspecting the figure, why the pole star is in apparent motion in respect to the pole, and why that star will cease to be the polar star, and why, at the expiration of about 12000 years, the bright star, Lyra, will be the polar star. ( 210.) The mean effect of the moon in producing the pre- son . cession of the equinoxes is, to the mean effect of the sun, as tive effect of five to two. The sun's action is nearly constant, because the sun is always in the ecliptic ; a small annual variation, however, is observed. The great inequality of 17".3, corre- sponding to about nineteen years, is caused entirely by the unequal action of the moon, depending on the longitude of the moon's ascending node. In consequence of this inequality, the pole, P, does not move round the pole of the ecliptic, E, in an even circumfe- motion of th rence of a circle, but it has a waving or undulating motion, as *" nn j "'* represented in this figure ; each wave pole of the corresponding to nineteen years ; and, ^^ -^^^ ecliptic, therefore, there must be as many of them in the whole circle as 19 is con- tained in 25868. From this, we per- ceive, that the undulations in the fig- ure are much exaggerated, and vastly too few in number; an exact linear representation of them would be im- possible. (211.) From the foregoing, we learn that the positions of Mean an all the stars are affected by aberration, precession, and nuta- ppnt tion : the amount for each cause is very trifling in itself, yet, star> in most cases, too great to be neglected, when accuracy is required ; and it is as difficult to make computations for a small quantity as for a large one, and often greater; and to reduce the apparent place of a fixed star from its mean place, 236 ASTRONOMY. CHAP, vn and its me?*n place from its apparent place, is one of the most troublesome problems in practical astronomy. Genera, for- ^he mean place of a fixed star, reduced to the time of ob- enulae, where ..-.., . . found servation, is sufficiently near its apparent place to be con- sidered the same. The practical astronomer, however, who requires the star as a point of reference, or uses it for the adjustment of his instruments, must not omit any cause of variation; but such persons will always have the aid of a Nautical Almanac, where general formulae and tables will be found, to direct and facilitate all the requisite reductions. Importance ( 212.) Physical astronomy brings many things to light that would otherwise escape observation, and some of these developments, at first, strike the learner with surprise, and he is not always ready to yield his assent. For instance, as a general student, he learns that the anomalistic year, the time that the earth moves from its perigee to its perigee again, is 365 d. 6 h. 14m. ; that the perigee is very slow in its motion, moves only about 12" in a year, and is subject to but few fluctuations. He has also learned that the earth, in its orbit, describes equal areas in equal times; hence, he concludes, that the time from perigee to perigee, or from apogee to apo- gee, must be very nearly a constant quantity; but, on con- sulting and comparing the predictions to be found in the En- glish nautical almanacs, he will find these periods to be (in comparison to his anticipations) very fluctuating. They differ from the state* mean times, not only by minutes and seconds, but by hours, and even days. The investigator is, at first, surprised, and fancies a mistake; at least, a mis- print; but, on examining concurrent facts, such as the lo- garithms of the distance from the sun, and the sun's true motion at the time, he finds that, if a mistake has been made, it is a very harmonious one, and every other circumstance has been adapted to it. The lati. But let us turn a moment from these facts, and examine explain. tne firsfc P a g e of our Tables - Tnere Jt wil1 De found, that the . sun has latitude ; that it deviates to the north and south of the ecliptic, by a quantity too small ever to le observed : it is, therefore, a quantity wholly determined by theory, and, as THE EQUINOXES. 237 the sun's latitude changes with the latitude of the moon, we CHAP, vn must seek for its cause in the lunar motions. Fig. 48. To understand the fact of the sun having latitude, we must admit that it is the center of gravity between the earth and moon, that moves in an elliptical orbit round the sun; and that center is always in the ecliptic ; and the sun, viewed from that point, would have no latitude. But when the moon, m, ( Fig. 48 ), is on one side of the plane of the eclip- tic, Sc, the earth, E would be on the other I side, and the sun, seen from the center of the] earth, would appear to lie on the same side of the ecliptic as the moon. Hence, the sun will change his latitude, when tlie moon changes her latitude. If the moon were all the while in the plane of the ecliptic, un ? itnd the sun would have no latitude (save some extremely minute fthesun * f v fecled by th quantities, from the action of the planets, when not in the position of plane of the ecliptic ) ; but the moon does not deviate more the moon ' than 5 20 from the ecliptic, and, of course, the earth makes but a proportional deviation on the other side ; but. in longi- tude, the moon deviates to a right angle on both sides, in re- spect to the sun, and when the moon is in advance in respect to longitude, the sun appears to be in advance also ; and when the moon is at her third quarter, the longitude of the sun is apparently thrown back by her influence : the great- est variation in the sun's longitude, arising from the motion of the earth and moon about their center of gravity, is about 6" each side of the mean. Now it is this motion of the Lon e itlul<> of the moon earth around the common center of gravity of the earth and a ff ec ts the moon, that chiefly affects the time when the earth comes to timo that the -, . TTI i earth comei its apogee and perigee. >\ hen the moon is in conjunction to JM apogM with the sun, the center of the earth is farther from the sun and than it otherwise would be ; and when the moon is in oppo- sition to the sun, the earth is about 3200 miles nearer the sun than it would be in its mean orbit ; and thus, we per- ceive, that the longitude of the moon has a great influence in 238 ASTRONOMY. CHAP. vn. bringing the earth into, or preventing it from coming into, it perigee or apogee ; but the perigee and apogee points,/or the center of gravity, are quite uniform, agreeably to the views ex- pressed in the first part of this article. These explanations will give a general insight into some of the apparent intrica- cies of physical astronomy. Small equa- The small equations of the sun's center are computed on nnns of lie ^ e principle explained by Fig. 48, the sun having a mo- tion round the center of gravity between itself and each of the planets. For example, the perturbation produced by Ju- piter is greatest when Jupiter is in longitude 90 from the sun, as seen from the earth ; the greatest effect is then about 8", and varies very nearly as the sine of Jupiter's elongation from the sun. When Jupiter is in conjunction with the sun, the sun is nearer the earth than it otherwise would be; and, on this ac- count, we have a small table to correct the sun's distance from the earth, called the perturbations of the sun's distance. The same remarks apply to other planets but, to avoid confusion, the effects of each one must be computed sepa- rately. PRACTICAL ASTRONOMY 39 SECTION IV. PRACTICAL ASTRONOMY. PREPARATORY REMARKS. WE have now done with general demonstrations, and with TKM> minute and consecutive explanations ; but we shall give all necessary elucidation in relation to the particular problems under consideration. To go through this part of astronomy with success and satisfaction, the reader must have a passa- ble understanding of plane and spherical trigonometry : and if to these he adds a general knowledge of the solar system, as taught in the foregoing pages, he will have a full compre- hension of all we design to embrace in this section. To prompt the student in his knowledge of trigonometry. w a "r^ ' bile utncr ciijg. COS. 9 /d ! h 29. cot. #=cos. a cot. * ana the other side, 30. tan. 2= tan. a sin. rl i a r*fi n f 1 the other ang. 31. cos. #=sin. a cos. *. The ( h two sides. ( the angles, 32. cos. #=cos. s cos. other side 33. cot. ar=sin. adj. sideXcot. [opp. side. IV. Resolution of oblique angled spherical triangles. Let A B and C be the three angles of any spherical triangle, and a b and C the sides opposite to them respectively, that is, the side a is opposite to A, &c. In spherical trigonometry, the sines of the angles are propor- tional to the sines of the opposite sides. sin. A sin. B sin. Bin. a sin. b sin.c _.. .. . sin. -a sin. j. Therefore 34. = - - Given the three sides abc; Required one of the angles, A. 35. - - Sin. a i A = - 1U sm.b sin. c 242 ASTRONOMY. Tuo. . 36. - - sin. b sin. c In 35 and 36, CHAPTER I. ASTRONOMICAL PROBLEMS. PROBLEM L CHAP. I. Given the right ascension and declination of any heavenly body, to find its latitude and longitude ; or conversely, given tht latitude and longitude of a body to find its corresponding righ, ascension and declination. Fig. 49. From any point as a center (Fig. 49) describe a circle Q EPo$, &c. Let this circle represent the meridian, which passes through the pole of the ecliptic E, the pole of the earth's axis P, and through the solstitial points 05 and y?. Then the point Aries ( ) will be at the center of the circle, and V? TP 25 and Q

D and Ds are given, and equa- tion 32 gives D subtract B <1P D, the obliquity of the ecliptic, and there remains the angle s T B.* With the angle s *(> B and the side

D and the obliquity of the ecliptic ; and if the angle S T D is the greater quan- tity, the body is north of the ecliptic, otherwise it is south of it When the declination it south, the angle S^ D must be added to the obliquity of the ecliptic in the first and second quadrants, and sub- tracted in the third and fourth. Hence the judgment of the operator uinst be called in to decide the particulars of the case ; or he must have a general formula that will give no exercise to the mind. 44 ASTRONOMY. CHAP t (20.) (21.) (A) 78 19' 3" sin. 9.990911 tan. 10.684611 (a) 3 19 55 sin. 8.763965 cos. 9.999265 3lW' sin. 8.754876 78 18 6 tan. 10.683876 Thus we determine that the longitude must be 78 18' 6", and the latitude 3 15' 36" N. 2. The longitude of the moon, at a certain time, according to computation, VMS 102 7'; and latitude 5 14' 15" S. : What was the corresponding right ascension and declination ?* From the,. (33.) examples we V B 77 53' cos .9.322019 sin. 9.990215 might form a general rale * sB 5 14' 15" COS .9.998183 cot. 11.037780 T77 56' 12" COS, , 9.320202 5 21' 27" cot. 11.027995 form dom ed sel- reflect BwD - - 23 27 42 principles ; 18 6 15 then " edoc sfore for sational (20.) (21.) purposes, we (A) 77 56' 12" sin. 9.990302 tan. 10.670170 the back on nrimarv (a) 18 6 15 sin. 9.492400 cos. 9.977948 equations. 17 41 22 sin. 9.482702 7719'41" tan. 10.648118 Thus we find that the right ascension distance on the equa* tor, from the 180th degree, was 77 19' 41"; or its right as- cension in arc was 102 40' 19", or in time, 6h. 50m. 41s. 3. By meridian observations on the moon, at a certain time, its right ascension was found to be 16h. 53m. 33s., and its decli- nation 17 51' 36". S. : what was its longitude and latitude? Ans. Lon. 254 9' 14", Lat. 4 41' 12" N. Any nnm- In the following examples either right ascension and decli ^ e nation may be taken for the data, and the longitude and lati pies c&n be tude the sought terms, or conversely ; the longitude and tound> latitude may be the given data, and the right ascension and As the longitude is more than 90 and less than 180, the moon it in the second quadrant of right ascension, and 77 53' in longitude from the equator; and as her latitude is south, it does not correspond to B S in the figure, and we give the example to exercise the judg- ment of the learner. PRACTICAL PROBLEMS 54$ declination the required terms. A Nautical Almanac will ** * furnish any number of similar examples. R. A. Dec. Lon. Lat. h. m. s. ' " ' ' " ' " 4 15 47 36 15 58 15 south, 238 14 48 4 30 17 north. 5 6 13 22 18 23 2 north, 93 10 55 5 4 23 south. 6 112444 145 28 north, 1711240 152 51 south, 7 20 23 33 14 11 9 south, 304 47 15 5 2 23 north! PROBLEM II. Given the latitude of the place, and the declination of the sun Tables fol or star ; to find the semidiurnal arc, or the time the sun or star urnal aro an ' d would remain above the horizon; and to find its amplitude, or the amplitudes number of degrees from the east and west points of the horizon, J**^^ 1 ** where it will rise and set. lem. To illustrate this problem we draw Figure 50. Let P Z ff, These ex. &c., represent the celes- Fig. 50. noTtlke rt tial meridian passing '^:^^ c Vy^^/^^E^v^;>^.:';^;;,;4'v^^;B fraction into through the place. Ma the arc Q Z equal to the latitude, then ZP will] equal the co-latitude. The line Hh is every- 1 where 90 from Z, and] represents the horizon. Pp represents the earth' F axis, and the meridian 90 distant from the me- 1 ridian of the place; Q} is the equator. From the points Q and q set off d and d', equal to the declination (north or south, as the case may be) and describe the small circle of declination, d Q d', where this circle crosses the circle of the horizon Hh is the point where the body ( sun, moon, or star ) will rise or set ( rise on one side of the meridian and set on the other, both are repre- sented by the same point in the projection ). Through P Q p describe the meridian as in the figure, and the right-angled spherical triangle R O appears ; right angled at R. 246 ASTRONOMY. j n the triangle R Q C, there is given the side 72 Q the declination, and the angle opposite ft C Q, which is equal to the co-latitude. R C, expressed in time, at the rate of 15 to one hour, will be the time before and after 6 hours, from the time the body is on the meridian to the time it is in the horizon ; and the arc C Q is the amplitude. The triangle is immediately resolved by equations 26 and 27. (27.) Sin. R C = tan. declin. X tan. lat. sin. declin. (26., Sin. CO cos. lat. ' Observing that the tangent of the latitude is the same as the cotangent of the angle R C Q, and the cosine of the lati- tude is the same as the sine of R C Q, corresponding to a in the equation. EXAMPLE. In the latitude of 40 N., when the sun's declination is 20 s is, N., what time before and after six will it rise and set, and what of course, ap- win b e its amplitude? parent, be- cause it re- (27.) (26.) !o\f '.It 20 tan ' 9 - 561066 sin - 9.534052 nd not "* 40 tan. 9.923813 cos. 9.884254 8lock - 17 47' sin. 9.484879 26 31' sin. 9.649798" Thus we find that the arc called the ascensional difference^ is 17 47', or, in time, Ih. llm. 8s., showing that the sun 01 heavenly body, whatever it may be (when not affected by parallax or refraction ), will be found in the horizon 7h. llm. 8s. before and after it comes to the meridian. Its amplitude for that latitude and declination is 26 31' north of east, or north of west, and, if observed by a compass, the apparent deviation would be the variation of the compass. 2. At London, in Lot. 51 32' N., the sun's amplitude wa* observed to be 39 48' toward the north ; what was its declina- tion, and what was the apparent time of its rising and setting / Ans. Sun's declination, 23 27' 59 ' N. Sun's rising, 3h. 47ra. 32s. ; sun's setting, 8h. 12m. 28s. PRACTICAL PROBLEMS. 247 The amplitude of the sun is frequently observed, at sea, to CHAP l discover the variation of the compass ; but, by reason of re- r~~ J Refractioi fraction, the results are not perfectly accurate. not taken in. From the right-angled spherical triangle (Fig. 50) PZQ, * we can Compute the time when the sun is east or west in po- time that ' the sition, and the altitude it must have, when in that position. sun w uld The angle Z is a right angle, P Z is the co-latitude, and JJ* the is the co-declination. horizon Equation (23) gives the cosine of Z Q, or the sine of the would ** in ' altitude of the sun when it is east or west the latitude and while u rose declination being given and equation ( 24 ) will give the itt altitude i ,- f 33' of are. angle or time trom noon. We may also find the altitude and azimuth of the sun, at 6 o'clock, by making use of a triangle formed by drawing a vertical through Z s N'. C S, the given declination, will be its hypothenuse and P Ch, the latitude, will be the arc of its By means of right-angled spherical trigonometry, as com- prised in the equations from 20 to 33, we can resolve all pos- sible problems that can occur in astronomy, pertaining to the sphere; but, for the sake of brevity, mathematicians, in some eases, use oblique-angled spherical trigonometry, which is nothing more than right-angled trigonometry combined and condensed. PROBLEM III. Given, the latitude of t/te place of observation, the sun's de- The sun'* clination, and Us attitude above the horizon, to find its meridian dlstance . - from * ne- distance, or the time from apparent noon. ridian, a. There is no problem more important in astronomy than m aured that of time. No astronomer puts implicit faith in any chro- a r * nometer or clock, however good and faithful it may have and on the been ; and even to suppose that a chronometer runs true, it e ? uat< " as a circumfer- can only show time corresponding to some particular me- ence, is the ridian; and hence, to obtain local time, we must have some measure of method, directly or indirectly, of finding the sun's distance paren t n0 on. from the meridian. When the center of the sun is on any meridian, it is than tnd there apparent noon ; and the equation of time will be the 248 ASTRONOMY CHAP. I. Grvat im- portance of this problem. Direct me- ridian obser- vations not generally ac- cnrate. Proper times of observa- iica Description of the figure. interval to or from mean noon; but none, save an astronomci in an observatory, can define the instant when the sun is OB the meridian ; no one else has a meridian line sufficiently defi- nite and accurate, and with him it is the result of great care, combined with a multitude of nice observations. To define the time, then (when anything like accuracy ii required ), we must resort to observations on the sun's al- titude. It is evident that the altitude of the sun is greater and greater from sunrise to noon, and from noon to sunset the al- titude is continually becoming less. If we could determine, by observation, exactly when the sun had the greatest alti- tude, that moment would be apparent noon ; but there is a considerable interval, some minutes, before and after noon, that it is difficult to determine, without the nicest observations, whether the sun is rising or falling ; therefore, meridian ob- servations are not the most proper to determine the time. From two to four hours before and after noon ( depending in some respects on the latitude ), the sun rises and falls most rapidly ; and, of course, that must be the best time to fix upon some definite instant ; for every minute and second of altitude has its corresponding time from noon ; and thus the time and altitude have a scientific connection, which can only be disen- tangled by spherical tri- gonometry. But we proceed to the problem. Draw a circle, P Z Q //, &c., ( Fig. 51), representing the meri- dian ; Z is the zenith, jand Z N is the prime vertical ; Hh is the ho- rizon; Z Q is an arc equal to the given lati- tude ; Q q is the equa- tor, and, at right angles to it, we have the earth's axis, P Fig. 51. PRACTICAL PROBLEMS. 249 Take Ha, ha, equal to the observed altitude of the sun, CHAP. i. and draw the small circle, a a, parallel to the horizon, H h. From the equator take Q d, qd, equal to the declination of the sun, and draw the small circle, d d, parallel to Q q. Where these two small circles, aa } dd, intersect, is the posi- tion of the sun at the time. From Z draw the vertical, Z Q &, and from P draw the meridian through the sun, P Q S. The triangle P Z Q has all its sides given, from which the angle Z P Q can be computed; which angle, changed into time at the rate of 15 to one hour, will give the time from noon, when the altitude was taken. If the time, per watch, should agree with the time thus computed, the watch is right, and as much as it differs is the error of the watch. The side Z Q is the complement of the altitude, P O The 1>seT is the complement of the declination, and P Z is the comple- fines and ment of the latitude, and equation ( 35 ) or ( 36 ) will solve points out a the problem ; that is, findi the angle P which can be made tnang e * to correspond to A, in the equation. But, in place of using the complement of the latitude, we may use the latitude it- self; and, in place of using the complement of the altitude, we may use the altitude itself; provided we take the cosine, when the side of the triangle calls for the sine ; for it would be the same thing. By thus taking advantage of every cir- cumstance, ingenious mathematicians have found a less troublesome practical formula than either (35) or (36) would Mathema. be : but we cannot stop to explain the modifications and tlc ' ans ma e great exer changes in a work like this; we contemplate doing so in tions to ab- a work more appropriate to such a purpose : the student must breviate be content with the following practical rule, to find the time rations. of day, from the observed altitude of the sun, toe/ether uith its polar distance, and the latitude of the observer. RULE 1. Add together the altitude, latitude, and polar dis- Prectic*. tance, and divide the sum by two. From this half sum subtract * the altitude, reserving tJie remainder. 2. Take the arithmetical complement of the cosine of the lati- tude, the arithmetical complement of the sine of the polar distance, the cosine of tfie half sum, and the sine of the remainder. Add 260 ASTRONOMY. CHAP - ' these four logarithms together, and divide the sum by two; the result is the logarithmetic sine of half the hourly angle. 3. This angle, taken from the Tables, and converted into time at the rate of four minutes to one degree, will be the time from apparent noon ; the equation of time applied, will give the mean time when the observation was made.* * The instrument for taking alti- tudes at sea, or wherever the observer may happen to be, is a quadrant or sextant, according to the number of degrees of the arc. It is made on the principle of reflecting the image of one body to another, by means of a small mirror revolving on a center of motion, carrying an index with it over the arch. Nearly opposite to the index mirror is another mirror, one half silvered, the other half transparent, called the horizon glass. Directly op- posite to the horizon glass is the line of sight, in which line there is sometimes placed a small telescope. The line of sight must be parallel to the plane of the instrument. The two mirrors must be perpendicular to the plane of the instru- ment. To be in adjustment, the two mirrors, namely the in- dex glass and horizon glass, must be parallel, when the index stands at 0. To examine whether a sextant is in adjustment or not, proceed as follows : 1. Is tJte index mirror perpendicular to the plane of the in* strument ? Put the index in about the middle of the arch, and look into the index mirror, and you will see part of the arch re- flected, and the same part direct; and if the arch appears perfect, the mirror is in adjustment ; but if the arch appeari broken, the mirror is not in adjustment, and must be put BO by a screw behind it, adapted to this purpose. 2. Are the mirrors parallel when the index is at ? Place the index at 0, and clamp it fast; then look at some well-defined, distant object, like an even portion of the dis- PRACTICAL PROBLEMS 251 EXAMPLE. In latitude 39 46' north, when the sun's declination was 3 27' north, the altitude of the sun's center, corrected for refraction, index error, &c., was 32 20', nsing ; what was the apparent time? 20 - cos. comple. - .114268 - sine comple. - .000788 10 - 9 .267652 CHAP. I Altitude, Latitude, Polar dis., 32 39 86 2)158" 20 46 33 ~39 30 79 32 19 20 46 59 30 Z P 24 50 30 sine 2 9 .864090 .246798 9 .623399 The hourly angle is 49 41 0, which, converted into time, gives 3h. 18m. 44s., the time from apparent noon, and, as tant horizon, and see part of it in the mirror of the horizon glass, and the other part through the transparent part of the glass ; and, if the whole has a natural appearance, the same as without the instrument, the mirrors are parallel; but, if the object appears broken and distorted, the mirrors are not parallel, and must be made so, by means of the lever and screws attached to the horizon glass. 3. Is the Jiorizon glass perpendicular to the plane of the in- strument ? The former adjustments being made, place the index at 0, and clamp it ; look at some smooth line of the distant horizon, while holding the instrument perpendicular ; a continued, un- broken line will be seen in both parts of the horizon glass ; and if, on turning the instrument from the perpendicular, the horizontal line continues unbroken, the horizon glass is in full adjustment; but, if a break in the line is observed, the glass is not perpendicular to the plane of the instrument, and must be made so, by the screw adapted to that purpose After an instrument has been examined according to these 52 ASTRONOMY. CHAP. i. the sun was rising, it was before noon, and the apparent time was 8 h. 41 m. 16 s. An are may ^ good observer, with a eood instrument, in favorable cir- tx measured by the quad- cumstances, can define the time, from the sun s altitude, to rant within within three or four seconds. An "artificial ^i sea > *^e observer brings the reflected image of the sun to the horizon, and allows for the dip ( Tables p.25). On shore, where no natural horizon can be depended upon, resort is had to an artificial horizon, which is commonly a little mercury turned out into a shallow vessel, and protected from the wind by a glass roof. The sun, or any other object, may be seen reflected from the surface of the mercury ( which, of course, is horizontal ),, and the image, thus reflected, appears as much below the natural horizon as the real object is above the hori- zon; and, therefore, if we measure, by the instrument, the angle between the object and its image in the artificial hori- zon, that angle will be double the altitude. When mercury is not at hand, a plate of molasses will do very well; and in still, calm weather, any little standing pool of water may be used for an artificial horizon. Observations taken in an artificial horizon are not affected by dip, but they must be corrected for refraction and index error, and, if the two limbs of the sun are brought together, its semidiameter must be added after dividing by two. A practical The following example is from a sailor's note book : On the lgth of May> 1848> at gea> in latitude 36 o 21 north, longitude 54 10' west, by account, at 7 h. 43 m. pei watch ; the altitude of the sun's lower limb was 32 51' ris- ing; the hight of the eye was eighteen feet, and the index directions, it may be considered as in an approximate adjust- ment a re-examination will render it more perfect and, finally, we may find its index error as follows : measure the sun's diameter both on and off the arch that is, both ways from 0, and if it measures the same, there is no index error ; but if there is a difference, half that difference will be the in- dex error, additive, if the greatest measure is off the ardi, sub tractive, if on the arch. PRACTICAL PROBLEMS. error of the sextant was 2' 12" ror of the watch?" additive. What was the er- 253 .L 7 h. 43 m., morning. 3 38 PREPARATION. Time, per watch, Longitude, 54 10', in time, Estimated mean time at Greenwich, 11 h. 21 m. The declination of the sun at mean noon, Greenwich time, was 19 38' 29" increasing, the daily variation being 13' ; the variation, therefore, for 39', the time before noon, was 21" subtractive. Hence, the declination of the sun, at the time of observation, was 19 38' 8" north, and the polar dis- tance 70 21' 52". Observed altitude, Index error, ... Semidiameter, ... Refraction, - ... Dip of the horizon, True altitude of sun's center, Altitude, 33 3' 20" Preparation* to be mad* according to circum- stance*. 32 51' 00" + 2 12 + 15 49 1 28 4 13 33 3' 20" Latitude, Polar dis., 36 70 21 21 52 2)139 46 12 cos. complement, sin. complement, 69 33 53 3 6 20 36 49 46 cosine, - sine, .093982 .026013 9.536470 9.777770 2)19.434235 9.717117 hourly angle, 31 25 30 sine, This angle corresponds to 4h. llm. 24s., or, in reference to the forenoon, 7 h. 48 m. 36 s. apparent time. On the 18th of May, noon, Greenwich time, the equation of time was 3 m. 54 s. subtractive ; therefore, the true mean time, at ship, was - - 7 h. 44 m. 42 s. Time, per watch, - - - 7 43 Watch slow, - - 1 42 A short time before this observation was taken, the watch Bjr obser- rations thni taken at dif- ferent time* at the same place, the rate of the watch can be determined. 254 ASTRONOMY. CHAP.I. wa s compared with a chronometer in the cabin, which was too fast for mean Greenwich time, 19 m. 12.5 s., according to estimation from its rate of motion. The chronometer was fast of watch by 3 h. 56 m. 39 s. What was the longitude of the ship? h. m. . Time of observation, per watch, 7 43 00 Diff. between watch and chron., 3 56 39 Time, per ch., at observation, 11 39 39 Chron. fast of Greenwich time. 19 12 Greenwich mean time, - 11 20 27 Mean time at ship, - - 7 44 42 Longitude in time, - 3 35 45=53 56' west. Howtode. The longitude is west, because it is later in the day, at Greenwich, tnan at tne ship- This example explains all the whether the details of finding the longitude by a chronometer. lonjitade is -g taking advantage of the observations for time on shore, east or west. e Howtode- we may draw a meridian line with considerable exactness; termine and f or instance, in the last observation ( if it had been on land ), - 24 s. after the observation was taken, the sun line. would be exactly on the meridian ; and if the watch could be depended upon to measure that interval with tolerable accu- racy, the direction from any point toward the sun's center, at the end of that interval, would be a meridian line. Sev- eral such meridians, drawn from the same point, from time to time, and the mean of them taken, will give as true a me- ridian as it is practical to find ; although, for such a purpose, a prominent fixed star would be better than the sun. Absolute The problem of time includes that of longitude, and find- ing the difference of longitude between two places always re- solves itself into the comparison of the local times, at the same instant of absolute time. When any definite thing occurs, wherever it may be, that is absolute time. For instance, the explosion of a cannon is at a certain instant of absolute time, wherever the cannon may be, or whoever may note the event ; but if the instant of its occurrence could be known at far distant places, the local clocks would mark very diffe- PRACTICAL PROBLEMS. 265 rent hours and minutes of time ; but such difference would be cmr I occasioned entirely by difference of longitude : the event is the same for all places it is & point in absolute time. Thus any single event marks a point in absolute time. If Absolute the same event is observed from different localities, the diffe- by meang O f rence in local time will give the difference in longitude. But events, a perfect clock is a noter of events, it marks the event & ^ O c te r c " of noon, the event of sunrise, the event of one hour after events, when noon, &c. ; and if we could have perfect confidence in this u rons trae| butnototner marker of events, nothing more would be necessary to give us w j ge . the local time at a distant place. The time, at the place where we are, can be determined by the altitude of the sun, or a star, as we have just seen. But, unfortunately, we can- not have perfect confidence in any chronometer or clock ; and therefore we must look for some event that distant observers can see at the same time. The passage of the moon into the earth's shadow is such Eclipses an an event, but it occurs so seldom as to amount to no practical J^*^*' maik value. The eclipses of Jupiter's satellites are such events, absolute but they cannot be observed without a telescope of consider- time ' but fo1 L common pur- able power, and a large telescope cannot be used at sea. pos es they Hence these events are serviceable to the local astronomer are of little only ; the sailor and the practical traveler can be little bene- fited by them. The moon has comparatively a rapid motion among the stars ( about 13 in a day ), and when it comes to any definite distance to or from any particular star, that cir- cumstance may be called an event, and it is an event that can be observed from half the globe at once. Thus, if we observe that the moon is 30 from a particular The motion star, that event must correspond to some instant of ^absolute among the time ; and if we are sufficiently acquainted with the moon, stars, may be and its motion, so as to know exactly how far it will be from ^"a^dex certain definite points ( stars ) at the times, when it is noon, moving 3, 6, 9, &c., hours at Greenwich, then, if we observe these events from any other meridian, we thereby know the Green- so iate tim. wich time, and, of course, our longitude. Finding the Greenwich time by means of the moon's angu- lar distance from the sun or stars, is called taking a, lunar; 256 ASTRONOMY. CHAF. i. and it is probably the only reliable method for long voyages at sea. If the motion of our moon had been very slow, or if the earth had not been blessed with a moon, then the only methods, for sea purposes, would have been chronometers and dead reckoning. For a practical illustration of the theory of lunars, we mention the following facts. e^atioUn". In & 86a J ournal of 1823 > ' li is stated that the distance of Uutrated by the moon from the star Antares was found to be 66 37' 8", an exampi*. V) } ien f/ te observation was properly reduced to the center of tJie earth, and the time of observation at ship was September 16th, at 7h. 24m. 44s. p. M. apparent time. By comparing this with the Nautical Almanac, it was found that at 9 P. M., apparent time at Greenwich, the dis- tance between the moon and Antares was 66 5' 2", and at midnight it was 67 35' 31"; but the observed distance was between these two distances, therefore the Greenwich time was between 9 and 12 p. M., and the time must fall between 9 and 12 hours in the same proportion as 66 37' 8" falls between the distances in the Nautical Almanac; and thus an observer, with a good instrument, can at any moment deter- mine the Greenwich time, whenever the moon and stars are in full view before him. The moon, in connection with the stars in the heavens, may be considered a public clock ( quite an enlargement of the town-clock ), by which certain persons, who understand the dial plate and the motion of the index, and who have the necessary instrument, can read the Greenwich time, or the time corresponding to any other meridian to which the com- putations may be adapted. observed The angular distances from the moon to the sun, stars, irtances^ and planets, as put down in the Nautical Almanac, corre- tancei ai spending to every third hour, are distances as seen from the soen from cen ^ er O f ^ ne earth, and when observations are taken on th Die earth, surface the distance is a little different ; the position of flu moon is affected by parallax and refraction, the sun or stai ^J refraction alone ; and therefore a reduction is necessary, which is called clearing the distance. This is done by spheri- PROPORTIONAL LOGARITHMS. 257 cal trigonometry. The distance between the moon and star CBAP. I. is observed, the altitudes of the two bodies are also observed. The co-altitudes come to the zenith, and the co-altitudes, with the distance, form three sides of a spherical triangle, from which the angle at the zenith can be computed. Then correct the altitude of the moon for parallax and refraction, and the star for refraction, and find the true altitudes and co- altitudes, and the true co-altitudes and angle at the zenith give two sides and the included angle of a spherical triangle, and the third side, computed, is the true distance. An immense amount of labor has been expended by mathe- maticians, to bring in artifices to abbreviate the computation of clearing lunar distances ; and they have been in a measure successful, and many special rules have been given, but they would be out of place in a work of this kind. PROPORTIONAL LOGARITHMS. In every part of practical astronomy there are many pro- portional problems to be resolved, and as the terms are 'og mostly incommensurable, it would be very tedious, in most t j on of y,. cases, to proceed arithmetically, we therefore resort to loea- construction *1 J 1 t 1 -^ Of * ** bl nthms, and to a prepared scale of logarithms, very appropri- giren> atcly called proportional logarithms. The tables of proportional logarithms commonly correspond to one hour of time, or 60' of arc, or to three hours of time. The table in this book corresponds to one hour of time, or 3600 seconds of either time or arc. To explain the construc- tion and use of a table of proportional logarithms, we propose the following problem : At a certain time, the moon's hourly motion in longitude was 33' 17" ; how much would it change its longitude in 13m. 23s. ? Put x to represent the required result, then we have the following proportion : m. m. s. ' " 60 : 13 23 : : 33 17 : x\ Or 3600 : 13 23 : : 33 17 : *. Divide the first and second terras of this proportion by the 17 258 ASTRONOMY. L second, and the third and fourth by the third, then we have 3600 x 13.23 : : 33.17 Divide the third and fourth terms by x, and multiply the same terms by 3600, and the proportion becomes 3600 a 3600 ( 3600 13.23 : x ! 33.17' Multiplying extremes and means, using logarithms, and re membering that the addition of logarithms performs multipli- cation, 3600 . /3600\ , . /3600\ Then we have log. = log. (-^ +log. (^--). By the construction of the table, the proportional logarithm of 1" is the common logarithm of S600 ; of 2" is the com- mon logarithm of 3600 . 3600 ; of 3 is 3600 , &c., to ; 1 hence the proportional logarithm of 3600 is 0. ,1 11 We now work the problem : 13 23 33 17 - - - P. L. - - - P. L. 6516 2559 Result, - -25i - - - P. L. 9075 Examples EXAMPLES FOR PRACTICE. lustrate the 1. When the sun's hourly motion in longitude is 2' 29", fa c h an g e O f longitude in 37 m. 12 s.? - AnS. 1 62 .5. practical nti- lity of proper. tional logar- 2. When the moon's decimation changes 57".2 in one hour, what will it change in 17 m. 31 s. ? Ans. 16".8. 3. When the moon changes longitude 27' 31" in an hour, how much will it change in 7 m. 19 s. ? Ans. 3' 21". 4. When the moon changes her right ascension 1 m. 58 s. in one hour, how much will it change in 13 m. 7 s. ? Ans. 25".8. PROPORTIONAL LOGARITHMS. 259 N. B. This table of proportional logarithms will solve any CHAP. i. proportion, provided the first term is 60, or 3600 ; therefore, when the first term is not 60, reduce it to 60, and then use the table. EXAMPLES. 1. If the sun's declination changes 16' 83" in twenty-four Exampiet hours, what will be the change in 14 h. 18m.? f iven to "' Instrate the Statement, 24 : 14.18 :: 16' 33" practical a*. hty of propor- Or, 12 : 7.09 tional Io g9 r Or, 60 : 35.45 : : 16' 33" ithn "- 16' 33" P. L. 5594 35' 45" P. L. 2249 Ans. 9' 51".5 P. L. 7843 2. If the moon changes her declination 1 31' in twelve hours, what will be the change in 7 h. 42 m. ? Ans. 58'. Conceive degrees and minutes to be minutes and seconds, and hours and minutes to be minutes and seconds. When 60 minutes or 3600 seconds are not the first term of a proportion, the result is found by taking the difference of the proportional logarithms of the other term for the P. L. of the sought term, as in the following example : The moon's hourly motion from the sun is 26' 30", what time will it require to gain 30" ? Statement, 26' 30" : 60m. : 30" : x other 30" P. L. 2.0792 60 m. P. L. 0.0000 Product of extremes, 2.0792 26' 30" P. L. sub. 3549 Besult, 1m. 7 s. P. L. 1.7243 3. The equation of time for noon, Greenwich, on a certain day, was 6 m. 21 s. ; the next day, at noon, it was 6 m. 43 s. : what was it corresponding to 3 h. 42 m. p. M., in longitude 74 west, on the same day ? Ang. 6 m. 29 B. ?60 ASTRONOMY. CHAPTER II. GENERAL PROBLEM. CHAP. ii. Given, the motions of sun and moon, to determine their appa- A general reni positions at any given time ; from which results their appa- problem pre- rent relative situations, and the eclipses of the sun and moon. thecmnpnta^ ^^ s problem covers two- thirds of the science of astronomy, tioEofeciip. and includes many minor problems ; therefore a brief or hasty "* solution must not be expected. From the foregoing portions of this work, the reader is supposed to have acquired a good general knowledge of the solar and lunar motions, and the tables give all the particu- lars of such motions; and all the artifices and ingenuity that astronomers could devise, have been employed in forming and arranging these tables, for the double purpose of facilitating the computations and giving accuracy to the results. The tables, in general, must be left to explain themselves, and the mere heading, combined with the good judgment of the reader, will furnish sufficient explanation, in most in- stances ; but some of them require special mention. All (he tables are adapted to mean time at Greenwich. EXPLANATION OF TABLES. , A very ge- Table IV contains the sun's mean longitude, the longi- tude of its perigee (each diminished by 2), and the Argu- ana. ments * for some of the small inequalities of the sun's appa- rent motion. tion of the tables. Explanation * The term, ARGUMENT, in astronomy, means nothing more than a f the term correspondence in quantities. Thus, each and every degree of the gun's longitude corresponds with a particular amount of declination { and therefore, a table could be formed for the declination, and the ar- gument would be the sun's longitude. The moon's horizontal parallax and semidiameter vary together, and each minute of parallax corresponds to a particular amount of se- midiameter; hence, a table can be made for finding the semidiameter, and the arguments would be the horizontal parallax. But the hori- EXPLANATION OF TABLES. 261 Argument I, corresponds to the action of the moon; Ar- CHAP. li. gument II, to the action of Jupiter; Argument III, to Ve- nus ; and Argument N, is for the equation of the equinoxes, and corresponds with the position of the moon's node ; and, by inspecting the column in the table, it will be perceived that the argument runs round the circle in a little more than eighteen years, as it should; and thus, by inspection, we can obtain an insight as to the period of any argument in the eolar or lunar tables. The object of diminishing the mean longitude and perigee Explanation of the sun by 2, is to render the equation of the center al- of the sola * ways additive ; for if 2 are taken from the longitude, and 2 added to the equation of the center, the combination of the two quantities will be the same as before ; and, as the equa- tion of the center is always less than 2, therefore, 2 added to its greatest minus value, will give a positive result. By the same artifice all equations may be rendered always posi- tive. The 2, taken from the mean longitude, are restored by adding 1 59' 30" to the equation of the center, and 10" to each of the other equations ; hence, to find the real equation of the center corresponding to any degree of the anomaly, subtract 1 59' 3" from the quantity found in the table. Table XI, shows the time of the mean new moon, &c., in January, diminished by fifteen hours, to render the correc- tions always additive. The fifteen hours are restored by add- ing 4h. 20m. to the first equation, 10 h. 10m. to the second, 10 m. to the third, and 20 m. to the fourth. Argument I, corrects for the action of the sun on the lunar zontal parallax and semidiameter of the moon depend (not solely) on the moon's distance from its perigee; hence, a table can be formed giving both horizontal parallax and semidiameter; which ARGUMENTS are the anomaly. In other words, an argument may be called an INDEX, and when the arguments correspond to points in a circle, or to the differ- ence of points in a circle, the circle may be considered as divided into 1000 or 100 parts, then 500, or 50, as the case may be, would corre- spond to half a circle, and so on in proportion. This mode of dividing the circle had been adopted, with certain limitations, to avoid the greater labor of computing by denominate numbers. ASTRONOMY. CHAI-. ii. orbit ; Argument II, corrects for the mean eccentricity of the lunar orbit ; Argument III, corrects for the different combina- tions of the solar and lunar perigee ; and Argument IV, cor- 1 rects for the variation occasioned by the inclination of the lunar orbit to the ecliptic ; N. shows the distance from or to the nodes. Tables ad- New and full moons, calculated by these tables, can be de- tno'dicV * P en d e( l upon within four minutes, and commonly much nearer; motion of the but when great accuracy is required, the more circuitous and moon, by e i a ij 0ra t e method of computing the longitudes of both sun which new and full and moon must be employed. moons can be Tables XIII, XIV, and XV, are used in connection with mpnteA Table XL Explanation Table XVI, shows the reduction of the latitude, and also of table * ne moon ' s horizontal parallax, corresponding to the latitude, occasioned by the peculiar shape of the earth, and the dimi- nution of its diameter as we approach the poles. The table is put in this place because of the convenient space in the page. Table XVII, and the following tables to No. XXX, contain the arguments and epochs of the moon's mean longitude, erec- tion, &c., necessary in computing the moon's true place in the heavens. rhe method The argument for evection is diminished by 29'; the ano- * mal J b J IQ 59 ' the variation by 8 59', and the longitude e by 9 44', and the balances are restored by adding the same amounts to the various equations, which, at the same time, renders the equation affirmative, as explained in the solar tables. The arguments in Table xxxn, are also arguments for polar distance, or latitude, in Table XXTIII. Anything like a minute explanation of these tables would lead us too far, and not comport with the design of this work. The use of the tables will be shown by the examples. We have carried the mean motions of the sun and moon only to five minutes of time and this is sufficient for all practical purposes for we can proportion to any interne diate minute or second, by means of the hourly motions. PRACTICAL PROBLEMS. 263 PROBLEM I. From the solar tables find the sun's longitude, hourly motion in longitude, declination, semidiameter and equation of time; and for a specific example, find these elements corresponding to mean time, at Greenwich, 1854, May 26 d.. 8 h. 40 m. To find the sun's declination, spherical trigonometry gives us the following proportion : (Eq. 20, page 231.) As radius - - 10.000000 Is to sin. of Q's Ion. (65 12' 15") - - 9.957994 So is sin. of obliq. of the eclip. ( 23 27' 32") 9.599900 To sin. declination N., 21 10' 54" - - 9^557894 In nearly all astronomical problems, time is reckoned from noon to noon from hour to 24 hours. When the given time is apparent, reduce it to mean time, and when not at Greenwich, reduce it to Greenwich time, by applying the longitude in time. ( This is necessary because the tables are adapted to Greenwich mean time. ^ From Table IV, and opposite the given year, take out the whole horizontal line of numbers ( headed as in the table ) and from Tables V, VII, VIII, take out the numbers corre- sponding to the month day of the month hour and minute of the day, as in the following example. Add up the perpendicular columns, as in compound num- The snn's bers, rejecting entire circles in every column, and the sums or duta ^ ces surplus, as the case may be, will give the mean values of all ge e point u the quantities for the given instant. called itf Subtract the longitude of the perigee from the mean Ion- m9 ^ t gitude, and the remainder will be the mean anomaly ; which is the argument for the equation of the center. With the respective arguments take out the corresponding equations, all of which add to the mean longitude, and the true longitude of the sun from the mean equinox will be found. With the argument N* take out the equation of the equi- The reason why N is not applied with the other equation* te be- cause it ia sometimes negative. 264 - 3 6 42 - 1 59 30 1 7 12 PRACTICAL PROBLEMS. 265 Thus, in general, we can determine the exact amount of CHAF. IL the equation of time, by means of the two arcs ( a ) and ( b ) ( which are roughly tabulated on page 95 ), and, without strictly scrutinizing each particular case, we can determine whether we are to take the sum or difference of the arcs by inspecting the table on page 95, or by referring our results to some respectable calendar. EXAM PLE. 2. What will be the sun's longitude, declination, right as- cension, hourly motion in longitude, semidiameter of the su: , and equation of time corresponding to 20 minutes past 9, mean time at Albany, N. Y., on the 17th of July, 1860 V N. B. At this time the sun will be eclipsed. Ans. Lon. 114 38' 21 ; Dec. 21 12' 48". R. A., in time, 7h. 46m. 15s. ; Eq. of time to add to apparent time, 5m. 46.2s.; hourly motion in Ion., 2' 23"; S. D., 15' 45.6' . PROBLEM II. From Tables XI, XII, and XIII, to find the approximate time of new and full moons. Take the time of new moon, and its arguments, from Table XI, corresponding to January of the given year, and take as ma iy lunations, from the following table, as correspond to the number of the months -after January, for which the new moon is required; add the sums, rejecting the sums corre- sponding to whole circles, in the arguments, and in the column of days, rejecting the number corresponding to the expired months, as indicated by Table XIII; the sums will be the mean new moon and arguments for the required month. When a full moon is required, add or subtract half a luna- Add th tion. Sometimes one more lunation than the number of the numberofln - month after January, will be required to bring the time to cessary t the required month, as it occasionally happens that two luna- brin s the re - tions occur in the same month. quired ^im*. Apply the equations corresponding to the different argu- of year, ments taken from Table XIV, and their sum, added to the mean time of new or full moon, will give the true mean time of new or full moon for the meridian of Greenwich, within four minutes, and generally within two minutes. 266 ASTRONOMY. CHAP. IL For the time at any other meridian apply the time corre- sponding to the longitude. EXAMPLES. 1. Required the approximate time of new moon, in May. 1854, corresponding to the day of the month, and the time of the day, at Greenwich, England, Boston, Mass., and Cincin- nati, Ohio. January. Mean N. Moon. I. II. I III. IV.) N. 1854, Four Luna. Table XIII. 27d. 18h. 14m. 118 2 56 0761 3234 1168 2869 19 61 04 1 668 96 ] 341 145 21 10 120 3995 4037 80 00 | 009 N shows an eclipse of the sun visible in the United States. May, J.. II. III. IV. 25 21 10 6 46 4 14 17 - 20 May, 26 8 47 - 4 New C> mean time at Greenwich, - 8 h. 47 m., p. M. Boston, Lorigitude, - 4 44 New f) Boston time, Cincinnati, Longitude from Boston, New ) Cincinnati time, 2. Required the approximate time of full moon, in Jufy, 1852, for the meridian of Greenwich, and for Albany time, New York. January. Mean N. Moon. I. II. III. IV. N. 1852, Five Luna. Half Luna. 20d. llh. 53m. 147 15 40 14 18 22 0549 4042 404 3239 3586 5359 38 76 58 27 95 50 538 426 43 Tab. 13. Bis. 182 21 55 182 4995 2184 72 72 007 The column N shows that the moon is very near her node. There will be a total eclipse of the moon invisi- ble in the United States. Mean time at Greenwich. July, IL III. IV. 21 55 4 21 42 17 10 1 8 25 July, ECLIPSES. 267 Full Greenwich time, - - 3 h. 25 m. p. M. CHA. a Albany, Longitude, - 4 55 Full m Albany time, - - 10 30 A. M. Thus we can compute tbe time of new or full moon for any month in any year ; but, as the numbers for the arguments correspond to mean or average motions, and cannot, without immense care and labor, be corrected for the true, variable motions, the results are but approximate, as before observed. ECLIPSES. Eclipses take place at new and full moons ; an eclipse of When cHp- the sun at new moon, and an eclipse of the moon at full *** c<> moon; but eclipses do not happen at every new and full moon; and the reason of this must be most clearly compre- hended by the student before it will be of any avail for him to prosecute the further investigation of eclipses. If the moon's orbit coincided with the ecliptic, that is, if wh - T "^ sea do not the moon's motion was along the ecliptic, there would be an take p ] ace eclipse of the sun at every new moon, and an eclipse of the every month moon at every full moon ; but the moon's path along the ce- lestial arch does not coincide with the sun's path, the ecliptic ; but is inclined to it by an angle whose average value is 5 8', crossing the ecliptic at two opposite points on the apparent celestial sphere, which are called the moon's nodes. If the moon's path were less inclined to the ecliptic, there What would would be more eclipses in any given number of years than j **""^*[ now take place. If the moon's path were more inclined to whatforfew. the ecliptic than it now is, there would be fewer eclipses. er ecli P 9e> The time of the year in which eclipses happen, depends on the position of the moon's nodes on the ecliptic; and if that position were always the same, the eclipses would always happen in the same months of the year. For instance, if the longitude of one node was 30, the other would be in longi- why a. tude 30+180, or 210; and, as the sun is at the first of eclips * should tak these points about the 20th of April, and at the second about p ] a c* in any the 20th of October, the moon could not pass the sun in P rtlcnl month. these months without coming very nearly in range with it, of oouro^, producing eclipses in April and October ASTRONOMY. Fig. 52. For a clearer illustration, we present Fig. 52: the right line through the center of the figure, represents the equator,the curved line qposrcz, crossing the equa- tor at two opposite points, re- presents the ecliptic; and the curved line Q O Q represents the path of the moon crossing the ecliptic at the points Q and Q; the first of these points is the descending, the other, the as- cending node. As here represented, the as- cending node is in longitude about 210, and the descending node in about 30; which was about the situation of the nodes in the year 1846, and, of course, the eclipses of that year must have been, and really were, in April and October. The sun and moon at con- junction are represented in the nas passed the northern tropic, wh j ch mugt be ab(mt the firgt of . .. August; and it is perfectly evi- dent that no eclipse can then take place, the moon running past the sun, at a distance of about Jive degrees south ; and at the opposite longitude, the moon must pass about Jive degrees north. The moon's nodes move back- ward at the mean rate of 19 19' per year; but the sun moves ECLIPSES. over 19 in about twenty days ; therefore, the eclipses, on CHA * n. an average, must take place about twenty days earlier each year, or at intervals of about 346 days. In May, 1846, the moon's ascending node was in longi- tude 216 ; in eight years, at the rate of 19 19' per year, it would bring the same node to longitude 61 28'. The sun attains this longitude each year on the 23d of May; there- fore, the eclipses for 1854 must happen in May, and in the opposite month, November. In computing the time of new and full moons, as illustrated The mean ' by the preceding examples, the columns marked N, not hith- Iu m s N.Tn erto used, indicate the distance of the sun and moon from the table the moon's node at the time of conjunction or opposition. The circle is conceived to be divided into 1000 parts, com- Eclipses are mencing at the ascending node ; the other node then must limited to a be at 500 ; and when the moon changes within 37 of 0, or along th 500, that is, 37 of either node, there must be an eclipse of ecli P tir the sun, seen from some portion of the earth. When the distance to the node is greater than 37, and less than 53, there may be an eclipse, but it is doubtful : we shall explain how to remove the doubt in the next chapter. When the moon fulls within 25 divisions of either node, there must be an eclipse of the moon : when the dis- tance is greater than 25, and less than 35, the case is doubtful ; but, like the limits to the new moon, the Trr . . . Comparative doubts are easily removed. We repeat, the ecliptic limits number of for eclipses of the sun are 53 and 37 ; for eclipses of the moon, eiipet f the limits are 35 and 25. Hence, in any long period of time, moon> the number of eclipses of the sun is, to the number of eclipses of the moon, as 53 to 35. In the same period of time, say in one hundred years, there will be more visible eclipses of the moon than of the sun ; for every eclipse of the moon is visible over half the world at once, while an eclipse of the sun is visible only over a very small portion of the earth ; therefore, as seen from any one place, there are more eclipses of the moon than of the sun. , In the preceding examples the columns N are far within the limits, and, of course, there must be an eclipse of the 270 ASTRONOMY. CHAP, ii. gun on the 26th of May, 1854, and an eclipse of the moon in July, 1852. HOW we As N is in value 9, at the time of new moon, in May, 1854, eclipse of the ^ shows that the moon will then have passed the ascending nn will hap. node, and be north of the ecliptic, and the eclipse must be aeth o^Maij, v^ble on the northern portions of the earth, and not on the 1854, and southern. ^ mgt ^* When the moon changes in south latitude, which will be we learn that shown by N being a little more than 500, or a little less than HiTse 1 * to 1^0, ^ e corresponding eclipse, if of the sun, will be visible some north. o n some southern portion of the earth, and not visible in the em portion of nor thern portion; and if of the moon, the moon will run through the southern portion of the earth's shadow. Table B,p.31, shows the moon's latitude, approximately cor- What indi- responding to the column N ; or N is the argument for the catet that a latitude, and the heading of the argument columns will will *> vh. show whether the moon is ascending to the northward, or de- We *a tme wending to the southward. le The tables from XVI to XVIII, together with the solar tables, will give approximate values of the elements necessary for the calculation of eclipses ; and if accurate results are not expected, these tables are sufficient to present general princi- ples, and give primary exercises to the student in calculating eclipses ; but he who aspires to be an astronomer, must con- tinue the subject, and compute from the lunar tables, far- ther on. The times, and the intervals of time, between eclipses, de- pend on the relative motion of the sun and moon, and the motion of the moon's nodes. The relative motion of the sun and moon is such as to bring the two bodies in conjunction, or in opposition, at the average interval of 29 d. 12 h. 44 m. 3 s., and the retrograde motion of the node is such as to bring the sun to the same node at intervals of 346 d. 14 h. 52 m. 16 e. Neglecting the seconds, and conceiving the sun, moon, and node to be together at any point of time, and after an un- known interval of time, which we represent by P, sup- p pose them together again. Then ^ ^ represents the ECLIPSES. 2~j number of returns of the lunation to the node m the time CHAP. 11. P, and the expression 14. RO re P resen * s ^ ne number of of the snn and moon in returns of the sun to the node in the same time. Eacn re- relation to turn of either body to the node is unity ; therefore, these ex- mooi ' >d pressions are to each other as two whole numbers ; say as m m to *; that is, ^-g : r : : m : ; Or, (29 12 44) (346 14 52)' Or, - (346 14 52)rc=(29 12 44)m - - - (a) n __ 29 12 44 ' f~346 14 52* Reducing to minutes, and dividing numerator and denomi- n 10631 nator by 4, we have = . As this last fraction is ir- reducible, and as m and n must be whole numbers to answer the assumed condition, therefore, the smallest whole number for m is 124783, and for n is 10631; that is, as we see by equation ( a ), the sun, moon, and node will not be exactly to- gether a second time, until a lapse of 124783 lunations, or 10631 returns of the sun to the same node ; which require a period of no less than 10088 years and about 197 days. We say about, because we neglected seconds in the computation, and because the mean motions will change, in some slight de- gree, through a period of so long a duration. This period, however, contemplates an exact return to the THU period same positions of the sun, moon, and earth, so that a line p^S'Tm! drawn from the center of the sun to the center of the moon possibilities, would strike the earth's axis in exactly the same point ; but to produce an eclipse, it is not necessary that an exact return Exactcoin . to former position should be attained; a greater or less cide nee. ne. approximation to former circumstances will produce a greater er ha PP n - or less approximation to a former eclipse : but exact coinci- dences, in all particulars, can never take place, however long the period. To determine the time when a return of eclipses may hap- *72 ASTRONOMY. CHAP ' fl - pen ( particularly if we reckon from the most favorable posi- HOW to tions that is, commence with the supposition that the sun, io4 the $uc- moon an( } no d e are together ), it is sufficient to find the first eessive ie- lorn of ] 0631 approximate values of the fraction VoTob* ^ we nn( ^ tne successive approximate fractions, by the rule of continued fractions,* we shall have the successive periods of eclipses, which happen about the same node of the moon The approximating fractions are _1 J. _3 4_ 19_ JL56 Tl 12 36 47 223" 1831* These fractions show that 11 lunations from the time an bowing the ec ^P se occurs, we may look for another; but if not at 11, it period* at will be at 12, and it may be at both 11 and 12 lunations; **l Kh and at five or six lunations, we shall find eclipses at the other eclipse* oc- cur. node, and the same succession of periods occurs at both nodes. ^ To be more certain of the time when an eclipse will occur, we must take 35 lunations from a preceding eclipse, which period is 1033 days 13 h. 40 m., and the sun at that time is about 6 40' farther from, or nearer to, the node, than before and, if the count is from the ascending node, the moon's latitude is about 38' farther south than before; and if from the descending node, the moon is about the same distance farther north. The double of 11, 12, and 35 lunations, from any eclipse, may also bring an eclipse. If an eclipse occurs within 10 of either node, it is certain that eclipses will again happen after the lapse of 47 lunations. A brief ex- The period of 47 lunations includes 1387 d. 22 h. 31m., and 4 rcvolutions of the sun to the node include 1386 d. of H h. 29m.; the difference is 1 day 11 h. 29m.; but in this eoiip*e. time the sun will move, in respect to the node, 1 32 and some seconds ; therefore, if the first eclipse were exactly at the node, the one which follows at the expiration of 47 lunations, See Robinson's Arithmetic. ECLIPSES. 273 or 3 years and nearly 11 months afterward, would take place CH*. *i. 1 32' short of the same node ; and if it were the ascending node, the moon's latitude would be about 8' 40" south, and if the descending node, about 8' 40" more to the north. The period, however, which is most known, and the most remarkable, appears in the next term of the series, which shows that 223 lunations have a very close approximate value to 19 revolutions of the sun to the node. The period of 223 lunations includes 6585.32 days, and 19 returns of the sun to the same node require 6585.78 days, showing a difference of only a fraction of a day ; and if the dis e ^ l ~ sun and moon were at the node, in the first place, they would omen called be only about 20' from the node, at the expiration of this ^ j>eriod period, and the difference in the moon's latitude would be less than 2', and therefore the eclipse, at the close of this period, must be nearly the same in magnitude as the eclipse at the beginning; and hence the expression "a return of the eclipse" as though the same eclipse could occur twice. This period was discovered by the Chaldaean astronomers, By this pe. and enabled them to give general and indefinite predictions ^akeTtnm" of the eclipses that were to happen ; and by it any learner, mary predic- however crude his mathematical knowledge, can designate the tic day on which an eclipse will occur from simply knowing the date of some former eclipse. The period of 6585 days is 18 years, including 4 leap years, and 11 days over; therefore from any eclipse, if we add 18 years and 11 days, we shall come within one day of the time of an eclipse, and it will be an eclipse of about the same magnitude as the one we reckon from. For the purpose of illustrating the method of computing Asnmmarj lunar eclipses, we wish to find the time when some future in e c t J eclipse of the moon will take place ; and from the American time when Almanac of 1833, we find that an eclipse of the moon took *" njt JjjJjJ"" place on the 1st day of July of that year, therefore "a re- turn of this eclipse" must take place on the 12th of July 1851. By a simple glance into the American Almanac for the year 1834, we find a total eclipse of the moon on the 21st of 18 574 ASTRONOMY. CHIP ii. June therefore, on the first of July 1852, or at the time that the moon fulls on or about the first of July, there must be a large eclipse of the moon, visible to all places from where the moon will then be above the horizon; and furthermore, 18 years and 11 days after this, that is, in the year 1870, on the 12th day of July, the moon will be again eclipsed; and, in this way, we might go on for several hundred years, but in time the small variations, which occur at each period, will gradu- ally wear the eclipse away, and another eclipse will as gradu- ally come on and take its place. In the same manner we may look at the calendar for any year, take any eclipse, that is anywhere near either node, and run it on, forward or backward. Let us now return to the eclipse of July 12th, 1851. Elements To decide all the particulars concerning a lunar eclipse we co " must have the following data, commonly called elements of lunar the eclipse : ecHpses. j ^he time of full moon. 2. The semidiameter of the earth's shadow. 3. The angle of the moon's visible path with the ecliptic. 4. Moon's latitude. 5. Moon's hourly motion. 6. Moon's semidiameter. 7. The semidiameter of the moon and earth's shadow General di- To find these elements, the approximate time of full noon ^taiTtheeT- is found from Tal)le XI ' and tbe tables immediately con- ement.8 of nected. For the time thus found, compute the longitude of t ^ e gun rom rj^bie jy ? an j the tables immediately con- nected, as illustrated by examples on page 254. Compute, also, the latitude, longitude, horizontal parallax semidiameter, and hourly motion in latitude and longitude, from the lunar tables, commencing with Table XVI, and fol- lowing out the computation by a strict inspection of the ex- amples we have given ( rules, aside from the examples, would be of no avail ) ; and, if the longitude of the moon is exactly 180 in advance of the sun, it is then just the time of full moon; if not 180, it is not full moon; if more than 180, it is past full moon. ECLIPSES. 275 It will rarely, if ever, happen that the longitude of the CHAP.II. moon will be exactly 180 in advance of the longitude of the sun ; but the difference will always be very small, and, by means of the hourly motions of the sun and moon, the time of full moon can be determined by the problem of the couriers* The moon's latitude must be corrected for its variation, corresponding to the variation in time between the approxi- mate and true time of full moon. To find the semidiameter of the earth's shadow, where the Rule to f a ike #tim, subtract the sun's semidiameter. This rule requires demonstration. Let S (Fig. 53) be Fig. 53. the center of the sun, h the center of the earth, and Pm a small portion of the moon's orbit. Draw p P, a tangent to both the earth and sun ; from p and P, draw P E and p E, forming the triande p P. By inspecting the figure, we perceive that the three Demonstra, angles: '7 of tbe Also, the three angles of the triangle, P Ep, are, together, equal to 180; Therefore, SEp+p E P+m EP=P+p+p EP ; Drop the angle, p E P, from both members of the equation, and transpose the angle SEp 9 we then have RoU MOB'S Algebra problem of the couriers. 276 ASTRONOMY. CHAP - n - But the angle, mP,ia the semidiameter of the earth's shadow at the distance of the moon; S Ep is the semidiame- ter of the sun ; P, that is, the angle Pp, is the moon's horizontal parallax ; and p is the horizontal parallax of the sun ; therefore, the equation is the rule just given.* The angle of the moon's visible path with the ecliptic is al- angie of the ways greater than its real path with the ecliptic, and depends, moon's visi- j n 8ome measure, on the relative motions of the sun and ble path with & ecliptic. m 0n - To explain why the real and visible paths of the moon are different, let A B ( Fig. 54 ) be a portion of the ecliptic, and A m a portion of the moon's orbit; then the angle, Fig. 54. **" "" is the angle of the moon's real path with the ecliptic. Con ceive the sun and moon to depart from the node, A, at the same time, the moon to move from A to m in one hour, and the sun to move from A to I in the same time ; join b and m, and the angle mbB is the angle of the moon's visible path with the ecliptic, which is greater than the angle mAB', which is the angle of the moon's real path with the ecliptic. On this principle we determine the angle in question. All the other elements are given directly from the tables. * Some writers have directed us to increase this value of the shadow by its cne-sixtlelli part, but we emphatically deny the propriety of the direction. ECLIPSES. 277 CHAPTER III. PREPABATION FOE THE COMPUTATION OP ECLIPSES. WE shall now go through the computation in full, that it CHAP, m. may serve for an example to guide the student in computing other eclipses. Mean N. Moon. I. II. 111. IV. N. 1851, Six Luna. Half Luna. Id. 14h. 21m. 177 4 24 14 18 22 0038 4851 404 3916 4303 5359 40 92 58 39 95 50 431 511 43 193 13 7 181 5293 3578 90 84 985 As N is within 25 of 1000, or 0, there must be an eclipse. The sun is 15 short of the as- cending node, and the moon at full, being opposite, must be 15 short of the descending node, and therefore, in north latitude, descending. July, II. III. IV. 12 13 7 3 35 2 9 14 11 Full 9 12 19 16 tion of a lu- nar eclipse. The approx- imate time of fall moon computed We now compute the sun's longitude, hourly motion, and Sun ion eemidiameter for 1851, July 12, 19 h. 15m. mean Greenwich gitud cora puted, corre- time, as follows: spending to the approxi- mate time of ful noon. 1851 Eq. ol July 12 d 19 h 15m f cente I. II II O M. Lon. Loh. Peri. I. II. III. N. 8 ' " 9 83239 52824 8 10 50 32 4649 037 3183445 r 1 39 38 10 18 L 20 S. O ' " 9 8 22 24 31 2 9 82257 3 18 34 45 958 129 371 27 250 454 28 025 310 19 648 27 2 677" 485 732 151 6 10 11 48 = Mean anomaly. O 's hourly motion, 2' 23" O's semidiameter, 15' 46" 3201511 Eq. of equinox 16 O Ion. 3 20 14 55 278 ASTRONOMY. CHAP. in. We now compute the moon's longitude, latitude, semidi- Direction anieter, horizontal parallax, and hourly motions for the same for comput- mean Greenwich time, as follows : ing the moon's true longitude. FOR THE LONGITUDE. 1. Write out the arguments for the first twenty equations, and find their separate sums. With these arguments enter the proper tables ( as shown by the numbers ), and take out the corresponding equations, and find their sum. 2. Write out the evection, anomaly, variation, longitude, supplement to node, and the several arguments for latitude, in separate columns, corresponding to the given time, and write the sum of the twenty preceding equations in the column of evection. 3. Add up the column of evection first ; its sum will be the corrected argument of evection, with which, take out the equation of evection ( Table XXIV ), and write it under the sum of the first twenty equations ; their sum will be the cor- rection to put in the column of anomaly. 4. Add up the column of anomaly, and the sum will be the moon's corrected anomaly, which is the argument for the equation of the center. With this argument take out the equation of the center from Table XXV, and write it under the sum of the preceding equations, and find the sum of all, thus far. Write this last sum in the column of variation, and then add up the Column of variation ; which sum is the correct argument of variation, and with it take out the equa- tion for variation from Table XXVII. 5. Add the equation for variation to the sum of all the preceding equations, and the sum will be the correction for longitude, which, put in the column of longitude, and the whole added up, will give the moon's longitude in lier orbit, reckoned from the mean equinox. Equation 6. Add the orbit longitude to the supplement of the node, noils' so^e" an< ^ tne suin * s *^ e ar g ument of reduction to the ecliptic ; it times called is also the first argument for polar distance, notation in ^j^ tlie a^^ O f reduction take out the reduction 'on^itude from Table XXXII, and add it to the longitude. ECLIPSES. 279 With argument 19, which is the same as N in the solar to- CHAP ra. lies, take out the equation of the equinox, and apply it ac- cording to its sign ; the result will be the moon's true longi- tude reckoned on the ecliptic from the true equinox. FOR THE LATITUDE. Add the same correction ( to its nearest minute ) to column General s i ati . lowing columns, except column X. Add up these columns, td rejecting thousands ( or full circles ), and the sums will be the 5th, 6th, 7th, 8th, 9th, and 10th arguments of latitude. The sum of the moon's orbit longitude, and supplement to node, is the first argument of latitude. The sum of column II is the second argument of latitude ; the moon's true longi- tude is the third argument, and the twentieth of longitude is the fourth argument. Then follow 5, 6, &c., up to 10. With these arguments enter the proper Tables, and take out the corresponding equations, and their sum will be the moon's true distance from the north pole of the ecliptic, and, of course, will be in north latitude if the sum is less than 90, otherwise in south latitude. N. B. When the first argument of latitude is nearer 6 signs than 12 signs, the moon is tending south; when nearer 12 signs, or sign, than 6 signs, it is tending north. For tlie equatorial horizontal parallax. The arguments for Equator... Eveetion, Anomaly, and Variation are also arguments for P arallax and horizontal parallax, and with these arguments take out the ter depend corresponding equations from the tables adapted to this P n eacb other. purpose. For the semidiameter. The equatorial parallax is the ar- gument for semidiameter, Table XXXIV. For the hourly motion in longitude. Arguments 2, 3, 4, and General di- 5 of longitude sensibly affect the moon's motion ; they are, Jj^j 8 ^ therefore, arguments for hourly motion, Table 36 ( the units hourly mo- and tens in the arguments are rejected ). Take out these tion of lh * equations from table, also take out the equation correspond- ing to the argument of evection, Table XXXVII With the t30 ASTRONOMY. CHAF. ill. sum of the preceding equations, at the top, aad the corrected anomaly at the side, take out the equations from Table XXXVIII. Also, with the correct anomaly, take out the equation from Table XXXIX. With the sum of all the pre- ceding equations at top, and the argument of variation at the side, take out the equation from Table XL. Also with the variation, take the equation from Table XLI. With the argument of reduction take out the equation from Table XLII. These equations, all added together, will give the true hourly motion in longitude. in this pro- F r the hourly motion in latitude. With the 1st and 2d portion the arguments of latitude, take out the corresponding quantities thTnaanmo* ^ rom Cables XLIII, and XLIV, and find their algebraic sum, tion of the noting the sign; call the result /. Then make the following proportion : 32' 56" : L : : I : ^ the true hourly motion in latitude, tending north, if the sign is plus, and south, if minus. In this proportion L is the true motion of the moon in longitude, and the first term is the moon's mean motion ; and the proportion is founded on the principle that the true motion in latitude must vary by the same ratio as the motion in longitude. N. B. In computing the moon's latitude we caution the pupil against omitting to add to the arguments II, V, VI, VII, VIII, and IX, the same correction as to the column of longitude ; its value must be changed into the decimal division of the circle for all the columns except column II. In the following example the correction for longitude is added to column II, and its value to all the following columns except column X. We find the value in question thus : 360 : 13 46' : : 1000 : x. The proportion resolved gives ar = the number added to the several columns. But to avoid the formality of resolving a proportion for every example, we g\ve the following skeleton of a table that ECLIPSES. 281 may be filled out to any extent to suit the convenience and CHAT. m. taste of the operator. Degrees = decimal parti Degrees = park. ' o ' 15 = .003 5 24 = .015 1 26 = .004 7 12 = .020 1 48 = .205 90 = .025 2 10 = .006 10 48 = .030 2 31 = .007 12 36 = .035 2 53 = .008 14 24 = .040 3 14 = .009 16 12 = .045 3 36 = .010 To make use of this table, we will suppose that the cor- rection for longitude, in a particular example is, 11 31' 25 '; what is the corresponding decimal or numeral part ? Thus 9 = .025 2 31 = 7 11 31 = .032 We now continue the examples, hoping to follow these precepts. 82 ASTRONOMY. CHAT. Ill 00 t- o co -O CO I- O t^ 00 r^ CM 3 -i CO fy. 00 "t O CO ~* J -i O t- O cs t r- I o 00 <* OJ O O IO CO TT CM OS CO CO CO 00 CM CO i O OS t- O CO CO I- # CO t^ OS O o; co ^ CO CM t- CO CO OS ^t i CO r- T os o CM ^ CO CM CO I l- co o * ! oo o i co co CO OS CO t^ CO OJ O CO CO OS < o . co rTco CM~CO -. ^ t- -^ CM -H CO CM Tj< 00 CM_00 - t^ Cs Os C t> CO < O Os r- r- ! S CO Oi CO <* O ' O O 00 TJ CO "^ 00 O 00 iQO -JO 00 OS CM CO l 0} -^ CO | CM eo !^r S lo - o t-J I' >_ > C5 1-- ^t ' O OO I O2 o -^ co co co |o CO -^ Tt 1 iCb 00 --D rf t^ O O O 1 O CO 1 CO O C^ CO O * -^ 00 CJ 00 CO co 3 go o J w CO CO ^t O CO So w . O CO "* CM -^ Tj" CO -H CO CO i t" 00 t- t~ Tj r-, CM O CO ^ ^ ^ cr CM OS O *"* ~* *~* < . ^ CO o I S .2 "5 -a S o 3 JS x _0 X3 I tfl c '5 i S S3 o o ECLIPSES. 283 CHAP. nL I ^ -^j orsoj cc e a _ 2 -2 - j> t'C ;-2 s s b 5 o - o o a * H .2 SS5 ^H "g 1 ftiO (2 C^ w 10 I0*~i 3 ji o i s Jq er a . ' B" tAs? . 1 .2^.2 g HI |Q wo euoj 00 C< " -"t co ift O r^ mm .84 ASTRONOMY CHAP. III. e ' ' - The moon's longitude, as just computed, will be 9 20 15 9 The sun's longitude, at the same time, will be 3 20 14 55 The difference will be - 6 14. Therefore, at the time for which these longitudes were computed, the moon will be past her full by 14" of arc : to correct the time, then, we must find how much time will be required for the moon to gain 14" ; which, by the problem of the couriers, is 14 _14^_ 14^ - (30.54) (2.23) 28' 31" ~~ 1711* The unit for t is one hour, and the denominator of the frac- wbtractive tion is the difference of the hourly motions of the sun and the rnoon, as determined by the tables ; the result is 29 seconds The Greenwich time will be, 1851, July 12d. 19h. 15m. Os. would be.i- Subtract - _ 29_ True time of full moon - 12 19 14 31 But the time given by the lunation table was 19 h. 14 m., differing only 31 seconds from the true time ; the approxi- mate and true time, however, do not commonly coincide as near as this : if they did, none but the most rigid astrono- mer would use the lunar tables for the time of conjunction or opposition. To be very exact we must correct the moon's latitude for what it will vary in 31 seconds ; that is, in this case, increase it 1".5. The moon's latitude, at the time of full moon, is, therefore, 37' 13".4. We have now all the elements necessary for computing the eclipse, or, at least, we have all the materials for finding them, and, for convenience, we collect the elements together : d. h. m. i. 1. True time of full moon, July, - - 12 19 14 31 2. Semidiameter of earth's shadow (page 265), - 39' 39" 3. Angle of the moon's visible path with the ecliptic,* - - 5 88 26 Thif 's the angle of the base of a right-angled triangle, whoM baw ECLIPSES. 285 4. Moon's latitude N. descending, 37 13.4 5. Moon's hourly motion from the sun, - 28 31 6. Moon's semidiameter, - 15 4 7. Semidiameter of f) and earth's shadow, 54 43 Whenever the moon's latitude, at the time of full moon, is less than this last element, the moon must be more or less eclipsed ; and it is by computing and comparing these two ele- ments, viz., 4 and 7, that all doubtful cases are decided. TO CONSTRUCT A LUNAR ECLIPSE. From any convenient scale of equal parts, take the 7th ele- When th ment in your dividers (54 43) = 54f , and from C, as a center with that distance, describe the semicircle B D HE (Fig. 55). t jt u de Take CA = the 2d element, and describe the semidiameter scribe a ful1 of the earth's shadow. From C the center of the shadow, \Vncn lsrg$ draw Cn at right angles to B E the ecliptic, above BE when g 0nt h lati- the latitude is north, as in the present example, but below, tude> de ' scribe only if south. th . lower Fig. 55. micircl*. Take the moon's latitude from the scale of equal parti, and set it off from C to n. Through n draw Dnll, the moon's path, so that the line shall incline to B E, the ecliptic, by an angle equal to the 3d element. Conceive the moon's is the hourly motion of the moon from the sun (28' 31"), and the per- pendicular, the moon's hourly motion in latitude (2' 49"). See page 266, figure 54 286 ASTRONOMY. JHAF III center to run along the line from D to ff, and from C draw Cm perpendicular to Dff. When the moon is ascending in her orbit, D ZTmust incline the other way, and Cm must lie on the other side of On. The eclipse commences when the moon arrives at D. It is the time of full moon when it arrives at n ; the greatest ob- scuration occurs when it arrives at m, and the eclipse ends at ff. The duration is the time employed in passing from D to H\ and to find the duration apply Dff to the scale, and thus The sth eU- find its measure. Divide this measure by the 5th element, ment 1S and we shall have the hours and decimal parts of an hour in moon's angu- lar motion the duration. Also apply Dn to the scale and find its mea- from the sun gure Divide this measure by the 5th element, for the time of describing Dn, also divide the measure nff for the time of describing nff. The time of describing Dn, subtracted from the time of full moon, will give the time of the beginning of the eclipse; and the time of describing nff, added to the time of full moon, will give the time when the eclipse ends. With lunar eclipses the time of greatest obscuration is the instant of the middle of the eclipse, provided the moon's mo- tion from the sun, for this short period of time, is taken as uniform, as it may be without sensible error. In reference to this example Dw = 36' and nff =44:'. These distances, divided by 28' 31", give 1 h.l4m.!6s. for the time of describing Dn, and 1 h. 32 m. 40 s. for nff: whole time, or duration, 2 h. 27 m. 20 s. h. m. s. Astronomi- cal time con Therefore from the time of full C 19 14 31 into Subtract - - - 1 14 16 civil time. Eclipse begins - - - 18 15 Add the duration - - 2 47 20 Eclipse ends - '- - 20 47 35 This eclipse That Is > in 1851 ' Jul y 12d * 18 L m ' 15 8 ' mean astr n - wby. mical time, the eclipse begins ; but this time corresponds with July 13, at 6 h. m. in the morning; and at this time, the sun will be above the horizon of Greenwich, and, of course, the ECLIPSES. 287 full moon, which is always opposite to the sun, will be below CHAP. ra. the horizon, and the eclipse will be invisible to all Europe. vi*iti fa In the United States, however, the eclipse will be visible; the u.s for, at these points of absolute time, the sun will not have risen nor the moon have gone down ; but, to be more definite, we demand the times of the beginning, middle, and end of the eclipse, as seen from Albany, N. Y. To answer this demand, all we have to do, is to subtract from the Greenwich time the difference of meridians between the two places, which, in this case, is 4 h. 55 m. ; and the result is, Beginning of the eclipse 13 d. 1 h. 5m. morning, Middle 2 30 End of the eclipse 3 52 In the same manner we would compute the time for any other place. For the quantity of the eclipse we take the portion of The qnan- the moon's diameter, which is immersed in the shadow, tlty of the eclipse how at the time of greatest obscuration, and compare it with found. the whole diameter of the moon; and in the present ex- ample, we perceive, that more than half of the diameter is eclipsed about 7 digits when the whole is called 12, or 0.6 when the diameter is 1. All these results, however, except the time of full moon, are approximate, because we cannot, nor do we pretend to construct to accuracy ; but any mathematician can obtain accurate results by means of the triangles D C H and C nm, and the relative motion of the moon from the sun. In the right-angled triangle Cnm, right-angled at m, Cn The exact is the latitude of the moon = 37' 17".4 = 2237".4, and the computation angle n Cm = 5 38' 26" ; with these data we find m n = tion Ol ^ , and Cm = 2212" clip". In the right-angled triangle C Dm, or its equal CmH, we Uve - Or, - - Or, - - mH 2 =(Cff+Cm) (Off Cm). Cffis the 7th element = 3283", and Cm = 2212 '.6. Therefore, m ff= 7(5495) (1071 ) = 2426" This 288 ASTRONOMY. "' divided by 1711", the 5th element, gives the time of half the duration of the eclipse Ih. 25m.; therefore the whole du- ration is 2 h. 50m., which is 2 m. 40 s.morethan the time we obtained by the rough construction. The distance nm, as just determined, is 220", and the time of describing this space, at the rate of 1711" per hour, re- quires 7 ra. 52 s., which taken from and added to the semi- duration, gives 1 h.lTm. 8 s. from the beginning of the eclipse to full moon, and 1 h. 32 m. 52 s. from the full moon to the end of the eclipse. The uigo- p or the magnitude of the eclipse, we add the moon's semi- diameter in seconds (904" ) to Cm ( 2212" ), and from the ofthemagni- sura subtract the semidiameter of the shadow in seconds f he ( 2379 ), and the remainder is the portion of the moon's di- ameter not eclipsed. Subtract this quantity from the moon's diameter, and we shall have the part eclipsed. Divide this by the whole diameter, and the quotient is the magnitude of the eclipse, the moon's diameter being unity. Following these directions, we find the magnitude of this eclipse must be 0.587. The con. j n a jj th ese computations we were guided by the construc- sufficient tion ; which uill always prove a sufficient index, and all that guide to car. should he required. uigonometrit ^ e mav determine, in any case, whether the eclipse will or cai computa- will not be total, by the following operation : Subtract the $)'s semidiameter from the semidiameter of the shadow, and if the moon's latitude, at the time of full moon, is less than the remainder, the eclipse will be total, otherwise not. To find the duration of total darkness. Diminish the semi- diameter of the shadow by the semidiameter of the moon, and from the center of the shadow describe a circle, with a radius equal to the remainder; a portion of the moon's path must come within this circle ; that portion, measured or divided by the hourly motion, will give the time of total darkness. When the moon's latitude is north, as in the present ex- ample, the southern limb of the moon is eclipsed and con- verielv ECLIPSES CHAPTER IV. SOLAR ECLIPSES GENERAL AND LOCAL. THE elements for a solar eclipse are computed in the same CHAP. iv. manner as the elements of a lunar eclipse ; all of which are General di found by the solar and lunar tables. rections tc _. J find the ele- The approximate time of new moon is first computed, and mentg for this time, compute the sun's longitude, declination, paral- lax, semidiameter, and hourly motion; and for the same time compute the moon's longitude, latitude, hourly motion in longitude and latitude, horizontal parallax, and semidiameter. If the longitudes of both sun and moon are found to be the same, then the approximate time of conjunction; found by the lunation tables, is the same as the true time ; if not, we pro- portion to the true time, as described in the last chapter. The elements for a general solar eclipse are : 1. The time of ^ * ai some known meridian. 2. Longi- what ele. tude of O and f). 3. Q's declination. 4. f)'s latitude. ments ara w v - / necessary. 5. O' s hourly motion. 6. O's hourly motion in longitude. 7. O's hourly motion in latitude. 8. The angle of the O's visible path with the ecliptic. 9. O's horizontal parallax. 10. f)'s semidiaraeter. 11. Q' s semidiameter. 12. Q's horizontal parallax. For a local eclipse, the latitude of the particular locality must also be given, or considered as one of the elements. As we can best illustrate general principles by taking a A definite particular example, we now propose to show the general course exal " p e yf an eclipse of the sun, which will occur in May 1854; where it will first commence on the earth ; in what latitude and longi- tude the sun will be centrally eclipsed at noon, and where ; in what latitude and longitude the eclipse will finally leave the earth. We speak of an eclipse of the sun being on the earth ; by Some ? ene- this we mean the moon's shadow on the earth. If an observer is in the moon's shadow, of course, the sun would be in an natiow eclipse to him ; and, if a tangent line be drawn ltween the Sign of conjunction 19 S90 ASTRONOMY CHAP. nr. BUn and moon, and that line strike the eye of an ohserver on the earth, to that observer the limbs of the sun and iLoon would apparently meet, and all projections of eclipses are on the principle of lines drawn from some part of the sun to some part of the moon, and those lines striking the earth. When no such lines can strike the earth there can be no eclipse. For the sake of simplicity in explaining a projection Point of f a s l ar eclipse, whether it be general or local, an observer i* is supposed to be at the moon, looking down on the earth, viewing the moon's shadow as it passes over the earth's disc; and, of course, the earth to him appears as a plane, equal to the moon's horizontal parallax. The approximate time of new moon will be found com- puted on page 254, and, if very close results are not required, we may compute the sun's longitude, declination, hourly mo- tion, and semidiameter for this time, and take out the moon's - horizontal parallax, hourly motion, and semidiameter from Table XIV; but we have computed the elements more accu- rately by the lunar tables, and find them as follows : d. h. m. . 1. Greenwich mean time of ^ 1854, May 26 8 45 39 2. Lon. of O and f) - - - 65 14' 6" 3. Declination of the Q - - 21 11 43 K 4. Latitude of the O - 21 19 N. 5. O's hourly motion in Ion., - - - 2 24 6. C>'s hourly motion in Ion., - - 30 3 7. f)'s hourly motion in lat., tending north, 2 46 From 5, 6, and 7 we obtain 8, as explained in the last chapter. 8. Angle of the moon's visible path o ' " with the eclip., - - 5 42 50 9. The f)'s horizontal equatorial parallax, 54 30 10. The f)'s semidiameter, - - 14 51 11. The O's semidiameter, - - 15 48 12. The O's horizontal parallax, always taken at 9 Subtract the O 's horizontal parallax from the ) 's ; and to the remainder the semidiameters of O aQ d ) , and if the moon's latitude is less than this sum, there will be an Accurate element* for the solar eclipse, which ' will take place Maj 28, 18S4. ECLIPSES. 291 eclipse, otherwise not; and it is by comparing this sum with CHAP.IV. the moon's latitude that all doubtful cases are decided TO CONSTRUCT A GENERAL ECLIPSE. 1. Make, or procure, a convenient scale of equal parts, and from any point as C ( Fig. 56 ) with the radius CJB, equal to the difference of the parallaxes of Q an d O (in the pre- sent example 54' 21", the minute is the unit), describe tho semicircle C B P H, or the whole circle, when the case re- quires it. When the moon has small latitude (less than 20') describe the whole circle ; when the moon has large north lati- tude, describe the northern semicircle; when south, describe the southern semicircle. Through C draw VCD PL perpendicular to HE. This perpendicular will represent the plane of the earth's axis, as seen from the moon. From P take P A, PF, each equal to the obliquity of the ecliptic 23 27' 30", and draw the chord A F. On A F, as a diameter, describe the semicircle ALF. ^ the M 2. Find the distance of the sun from the tropic, nearest to of ihe it, by taking the difference between the sun's longitude and Uo * 90 or 270, as the case may be. In the present example we subtract 65 14' from 90, the remainder is 24 46'. Take L T, equal to 24 46', and draw TE parallel to L C. Draw C E the axis of the ecliptic. By the revolution of the earth round the sun, the axis of T 1 * the ecliptic appears to coincide with the axis of the equator, ic when the sun is at either tropic, and it appears to depart iu p- The time that the eclipse is on the earth is measured by ie t " ne rcc l u i re d f r ^e moon to P asB fr m to 7 fwurfti true angular motion from the sun. clime. rpk e i en gth o f this line, k q, can be found from the ele- ments, and trigonometry, as in an eclipse of the moon, and the tirae of describing it is found in the same way. ECLIPSES. 298 294 ASTRONOMY. OH A*, iv. When the moon's center comes to I, the central eclipse HOW u> de- commences, and the arc HI shows that it must be about in ^ mine m the latitude of 7 north. When the moon's center comes t-i.ies the to **> the sun will be centrally eclipsed at apparent noon ; and eclipse will Cr is the sine of the number of degrees north of the sun's over,' and declination, which, in this case, is about 23 ; hence to the pass off the sun's declination, 21 12', add 23, making 44 12'; showing, as near as a mere projection can show, that the sun will be centrally eclipsed at noon on some meridian, in latitude 44 12 north. The central eclipse will end, or pass off the earth, when the moon's center arrives at p and the arc Ep from the equator, shows that the latitude must be about 41 north. The eclipse will entirely leave the earth when the moon's center arrives at q, arid for its limb to touch the sun, the sun's cen- ter must be at h, and the arc E h shows that the latitude must be about 30 north. The lines, cd and ab, parallel to the moon's path, and dis- tant from it equal to the sum of the semidiameters of sun and moon, represent the lines of simple contacts across the earth, or limits of the eclipse ; cd is the southern line of simple con- tact, and a b is the northern line of simple contact, and the latitudes at which these lines make their transits over the earth, are determined precisely as the latitudes on the cen- tral line. We may J$\it we need not stop at coarse approximations: we have rate compn- a ll the data for correct mathematical results, on the same * tations by principles as we determined those in relation to a lunar eclipse. ,"* In the triangle Cnr, we have the side Cn, the moon's latitude in secunds, which may be used as linear measure, as yards or feet and in proportion thereto, we may compute Cr and nr, when we know tJie anyle n Cr. An eqoa- jj ut ^ f l} ow j,,g. equation always gives the tangent of the position of angle E CD or n Cr, calling the sun's distance from the sol- the axu of gtice D, the obliquity of the ecliptic E, and the radius unity. tlm ecliptic. tan. JC2)=tan. E sin. D.* * The student who has acquired a little skill in analytical tri|?o- metry can discover the preliminary steps to this equation; the princi- ples are all visible in the construction of the figure. ECLIPSES 296 To the angle E CD, add the angle Q CJS, the angle of the CHAF nr. moon's visible path with the ecliptic, and we have the whole angle G C D, or m Or. Cmn is a right angle; and in the two triangles Cm n and Cmr, we have all the data, and can compute n r and r C. When the moon arrives at m t it is in the line of conjunction in her orbit ; when it arrives at , it is in ecliptic conjunction ; and when it arrives at r, it attains conjunction in right as- cension. For the last six or eight years, the English Nautical Al- Recent i . .1 ... -, . . i , changes in manac has given the conjunctions and oppositions in right as- the English eension, in place of conjunctions and oppositions in longitude, Nautical Al- and has given the difference of declinations between the sun m< and moon, in place of giving the moon's latitude ; that is, it has given the time that the moon arrives at r, in place of n, and given the line Cr in place of Cn. All lunar tables give the ecliptic conjunction at n, and from this we can compute the time at r by means of the triangle Cnr. Having explained the principle of finding the latitude on the earth, when a solar eclipse first commences, we are now ready to show another important principle how to find the longitude ; and with the latitude and longitude, we have the exact point on the earth. Where an eclipse first commences on the earth, it com- The method mences with the rising sun, and finally leaves the earth with "Jjf "f^JI U the setting sun. In this example, we have decided that the where the eclipse must commence very near the equator, not more than *** fi ^ one degree south; but in that latitude the sun rises at 6 h. earth. A. M. apparent time ; therefore, at the place where the eclipse commences, it is six in the morning, apparent time. From the scale of equal parts, take the moon's hourly mo- tion from the sun in the dividers (27' 39"), and apply it on the line k q: it will extend three times, and a little over, to the point . This shows that three hours, and a little more ( we say 3h. 3m.) must elapse from the first commencement of the eclipse to the change of the moon at n. Hence, by the local time at the place of the commencement of the eclipse, 295 ASTRONOMY. CHAP, iv. the moon changes at 9 h. 3 m. in the morning, apparent time ; but the apparent time of new moon at Greenwich is 8 h. 49 m. p. M., making a difference of 11 h. 46m. for mere locality: the absolute instant is the same; the difference is only in meridians which correspond to a difference of longitude of 175 30'; and it is west, because it is later in the day at Greenwich. of 'fiadiM ^ e centra l eclipse also first comes on the earth at a place where tbe where the sun is rising. In this example it first strikes the ce " tra e 1 ^ earth at the point /, in latitude about 7 N. ; but, in latitude strikes the ^ N., and declination 21 N., the sun rises at 5h. 48m., earlh A. M. apparent time ( Prob. II ); and from that time to the change of the moon, namely, the time required for the moon to move from / to n, is ( as near as we can estimate it by the construction ), 1 h. 56 m.; therefore, the time of new moon, in the locality where the central eclipse first commences, is 7 b. 44 m. in the morning. From this to 8 h. 49 m. in the even- ing, the time at Greenwich, gives a difference of 13 h. 5m, reckoned eastward from the locality, or 10 h. 55m. reckoned westward ; which corresponds to 196 15' west longitude from Greenwich, or 163 45' east longitude; the meridian is the same. If the longitude is called east, the day of the month must be one later; but, to avoid this, wo had better call the lungitude west. To find the Where the sun is centrally eclipsed on the meridian, it is whfre" 8 the J us ^ ^~' a PP arenfc ^ me > * ne moon's center is then at r, and, snn will be by the construction, it must be about seven minutes after ed 7 at Gon J unct * on * n ^at locality ; hence, the conjunction is seven noon minutes before 12, and at Greenwich it is 8 h 49 m. after 12, giving 8 h. 56 m. for difference of longitude, or 134 west longitude The central eclipse will leave the earth with the setting sun, when the center of the moon and sun are both at p; but the latitude of p we decided to be 40 north, and in this latitude, when the sun's declination is 21 11', as it now is, the sun sets at 7h. 15m. apparent time; but this is 1 h. 40 m. after conjunction, therefore the conjunction in that locality must be at 5 h. 35 m. ; but, at Greenwich, it if ECLIPSES 297 8h. 49m., giving, for difference of longitude, 3h. 14m., or 48 30' west. The eclipse finally leaves the earth in latitude 46 north ; To find th but, in this latitude, the sun sets at 6 h. 51 m.. and the con- lo ? ltnde where tn junction will be 3 h. m. sooner ( the time required for the eciipe will moon to pass from n to ^), therefore the conjunction in this locality must be at 3h. 51m.; but, at Greenwich, it will be 8h. 49m., giving 4h. 58m. for difference of longitude, or 74 30' west. Thus, by the mere geometrical construction, we have roughly determined the following important particulars : App. time Gr. Lat. Longitude. h. m. o / Eclipse commences, May 26, 5 46 IS. 175 30 W. Cen. eclipse commences, 6 53 7 N. 196 15 W. Cen. eclipse at local noon, 8 56 46 134 00 W. the P>jec. Cen. eclipse ends, 10 34 40 48 30 W. tlon< End of eclipse, 11 46 30 73 30 W. To find the latitude of the first commencement of simple The locali - contact on the southern line, all we have to do is to find the sout h er n and arc #c;and for the latitude on the northern line, we find the northern arc Ha ; the point c is in latitude about 27 south, and a in p]" e * ta *!*" about 54 north. The southern line of simple contact leaves the earth at d, between the seventh and eighth degrees of north latitude, and the northern line passes off beyond the pole. We have, thus far, taken the results but approximately from the projection, and the projection is sufficient to teach us principles ; and it must be our guide, if we attempt to ob- tain more minute results ; and with the elements and the figure we have the whole subject before us as minutely accurate as it is magnificent, and as simple as it is sublime. To complete our illustration, we now go through the trigo- nometrical computation. In the triangle Cnm, we have C7w=21' 19"=1279, the angle mCn=5 42' 50", and the angle m a right angle. Whence Cro=1273", and m=127".3. 298 ASTRONOMY. CHAP, iv. tan. E CD=n CV=tan. (23 27' 32") sin. (24 45' 54") In these ( page 284). tions2" Whence E CD= 10 18' 8", moon> 8 lati- Add G C E= 5 42' 50", tude and the - -- - distances Sum is Q C D=m CV=16 0' 58". In the trian S le m Cr > we have Om (1273), the perpendieu- lar, and the angle m Cr as just determined ; whence, mr=365".3 ; CV=1324".3. In the triangle Cmp, Cp is the horizontal parallax of moon and sun (54' 30") 9", or 54' 21"=3260". By the well-known property of the right-angled triangle, Or mp 2 = Cp 2 Cm 2 =(Cp+Cm) (CpCm), That is, w j p= > /(4533)(1987)=3001 / '.7. Therefore, Ip, the whole chord, is 6003".4, which, divided by 1659' (the moon's motion from the sun), gives 3.616h. or 3 h. 37m. 40s. for the time that the central eclipse will be on the earth. In the same manner the line m q is found. That is, mq=J(Cq-\-Cm)(Cq (7m), But, C?=54' 21"-fl4' 5.1'H-15' 48"=5100". Or m q= > /(6373)(3827)=4938".3. Therefore, the whole chord, kq, is 9876.6, which, divided by 1659", gives 5 h. 57 m. 20 s. for the entire duration of the general eclipse on the earth. On the supposition that the moon's motion from the sun is uniform for the six hours that the eclipse will be on the earth, the several parts of the moon's path will be passed over by the moon, as follows : From * to * in111 - 9m. 54s. condition of From / to m in 1 49 00 to , 3 3 30 Eclipse commences, Greenwich app. time, 5 45 30 Central eclipse commences (add 1 9 54), 6 55 24 Sun centrally eclipsed on some meridian, or d in right ascension, Greenwich time, at (add 2 2 36), 8 58 00 Central eclipse ends at (add 1 35 48), 10 33 48 End of eclipse at (add 1 9 54), 11 43 41 By comparing these times with those obtained simply by the projection, we perceive that the projection is not far out of the way, notwithstanding the terms rough and rougJdy that rate> than we have been compelled to use concerning it. Indeed, a good generally draftsman, with a delicate scale and good dividers, can decide inpl>08< the times within two minutes, and the latitudes and longitudes within half a degree ; but all mathematical minds, of course, prefer more accurate results; yet, however great the care, absolute accuracy cannot be attained ; the nature of the case does not admit of it.* To find whether the point k is north or south of the equa- *The astronomer, by making use of his judgment, can be very ac- curate with very little trouble: he perceives, at a glance, what ele- ments vary, and what the effects of such variation will be; but a learner, who is supposed not to be able to take a comprehensive view of the whole subject, must go through the tedious process of computing the elements for the times of the beginning and end of the eclipse, as well as the time of conjunction, if he aims at accuracy, but an astronomer can be at once brief and accurate. In computing the moon's longi- tude, in the present example, the astronomer would notice in particu- lar the moon's anomaly, and, by it, he perceives whether the moon's hourly motion is on the increase or decrease, and at what rate. It is on the decrease, and the first part of the chord km is passed over by the moon in about 7 seconds less time than our computation made it, and the last part requires about 7 seconds longer time ; but the times of passing m and n should be considered accurate, and the times of beginning and end should be modified for the variation of the moon's motion, making the beginning and end 7 seconds later, and the beginning and end of the central eclipse about 4 seconds later. 300 ASTRONOMY. .?jy. tor, we conceive k and C joined, and if the angle m Ck is greater than the angle m Off, the point k is south, otherwise north. By trigonometry, Ck : km : : sine 90 : sine mCk; o / // Or, 5138 : 4900".3 : sin. 90 : sin.w Cfe75 31 20 To this add G CD, - - 16 58 Sum is the angle r Ck - 91 32 18 This angle shows that the eclipse will first touch the earth in latitude 1 32' 18" south. To find the arc HI, conceive the points Cl joined, and the two triangles Clm, m C p are equal. Cl : Im : : sin. 90 : m Cl' Or, 3261 : 3003.7 : : sin. 90 : sin. m C7=67 7 50 To this add G C , v 160 58 The sum is, 83 8 48 Where the This angle shows the latitude of the point / to be 6 51' triEa iht"* * 2 " norfcn - That i s *be centr al eclipse first touches the earth earth in 6 51' 12" of north latitude; differing very little from the point determined by construction. To find the latitude of the point p, we have m Cl = m Cp = 67 7' 50"; and subtracting 16 0' 58", we have the polar distance, or co -latitude; the result is, that the central eclipse passes off at latitude 38 53' 8" north, and the gene- ral eclipse entirely leaves the earth in latitude 30 25' 38". To find the latitude of the point r, we consider Cr to be a sine of an arc, and C P the radius. Therefore, 3261" : 1324".3 : : R : sin. x = 23 58 00 To this add the sun's declination, 21 11 43 Sum is latitude where the sun will be centrally eclipsed on the meridian, - 45 9 43 N. HOW to find Wherever the sun is centrally eclipsed on the meridian, it the longitude is apparent noon at that place, but at Greenwich the apparent pla e time is 8 h. 57 m. 37 a., p. M. ; this difference, changed into lon- >i> central- gituole, gives 134 25' west, within a degree of the result de- ly eclipsed on termined from the projection; and it is not important to go the meridian. . , . ,, II-T . over a trigonometrical computation for tho longitudes, since ECLIPSES. 301 i we arc sure of knowing how to do it; and wo are also sure CHAP. IV. that the results will not differ much from those already de- termined. In short, from the elements, the figure, and a knowledge s n fficiem of trigonometry, we can determine all the important points in * in th each of the three lines c d, kq, and a b, for between them we have, or may have, a complete net-work of plane triangles. CHAPTER V. LOCAL ECLIPSES, ETC. WE now close the subject of eclipses by showing how to CHAP. v. project and accurately compute every circumstance in rela- tion to a local eclipse. For an example, we take the eclipse of May, 1854, and for the locality, we take Boston, Mass., because we anticipated a central eclipse at that pjace, but the result of computations shows that it will not be quite central even there. We use the same elements as for the general eclipse. THE CONSTRUCTION. Draw a line CD, and divide it into 65 *qual parts, and The scat* consider each part or unit as corresponding to one minute of the moon's horizontal parallax. From C, as a center, at a distance equal to the diff. of parallax of the sun and moon (54 21 ), describe a semicircle north or south according to the latitude, or describe a whole circle if the latitude is near the equator. From C draw C, the universal meridian, at right angles to CD, and from 05 take 25 qp and 05 ==, each equal to the obliquity of the ecliptic ( 23 27' ) and draw the straight line f^, =^ on the right. Subtract the sun's longitude from 90 or 270 to find its distance from the nearest solstitial point, and note the difference ( in this example 24 46' ). th^Tx"/ From the point a, with a T as radius, make a G equal to ecliptic. 302 ASTRONOMY CHA * v - the sine of 24 46',* and join C G, and produce it to E\ CE is the axis of the ecliptic : this line is variable, and is on the other side of the line C between June 20 and Decem- ber 21. How to find From E take the arc EL equal to the moon's visible path the moon'g with the ecliptic, to the right of E when the moon is descend. orbit i n g, but to the left when ascending as in the present exam- ple. Join C L, a line representing the axis of the moon's orbit. To and from the reduced latitude of the place add and sub- tract the sun's declination: Thus, Boston, reduced latitude, - 42 6' 39" N. Sun's declination, - 21 11 43 N. Sum is 63 19- 22", and difference is 20 54' 56". HOW to find From (7, make (712 equal to the sine of the difference of uHriiH I" the two arcs ( 20 54 ' 56 ")> and Cd the sine of the sum miking the (63 19' 22"). If the fcce Divide (12) apparent time, at Boston, or it must be considered the apparent time corresponding to any other meridian for which the projection may be intended. The ecliptic , apparent time, Greenwich, is 8h. 49m. Os. For the longitude of Boston, subtract 4 44 16 Conjunction, apparent time, at Boston, 4 4 44 The moon's hourly motion from the sun is 27 39": take this distance from the scale, in the dividers, and make the small scale ab, which divide into 60 equal parts; then each in this case, P ai> * corresponds with a minute of the moon's motion from the the ellipse SUI1) an( j fc De distance ab will correspond with one hour of the men!* C moon ' 8 motion along its path. At 4 h. 4 m. 44 s. the moon's tween 4 and center will be at the point n; the sun's center, at the same $ o'clock. time ^ w yj be j ugt jj e y 0n ^ t h e p i n t 4 on the ellipse ; and, as the distance between these two points is greater than the sum of the semidiameters of sun and moon, therefore the eclipse will not then have commenced ; but the moon moves rapidly along its path, and, at 5 o'clock, the center of the moon will be at the point marked 5 on the moon's path, and the cente* of the sun will be at the point marked 5 on the ellipse; and these two points are manifestly so near each other, that the limb of the moon must cover a part of that of the sun, show- ECLIPSES. 30& ing that the eclipse must have commenced prior to that time. CHAP. v. To find the time of commencement more exactly, let the hour TO find * on the moon's path be subdivided into 10 or 5-minute spaces, aaore MM| and take the sum of the semidiameter of the sun and moon in your dividers from the scale CD, and, with the dividers thus open, apply one foot on the moon's path and the other on the sun's path, and so adjust them that each foot will stand at the same hour and minute on each path as near as the eye can decide. The result in this case is 4 h. 28 m. The end of the eclipse is decided by the dividers in the same manner, and, as near as we can determine, must take place at 6h. 44m. To find the time of greatest obscuration, we must look How tofin * , , , ,. MI* Ul * *T along the moon s path, and discover, as near as possible, from grealegt ^. what point a line drawn at right angles from that path will curatio strike the sun's path at the same hour and minute; the time, thus marked on both paths, will be the time of great- est obscuration. In this case it appears to be 5 h. 40 m., and the two cen- ters are very nearly together ; so near, that we cannot decide on which side of the sun's center the moon's center will be, . without a trigonometrical calculation. To show a representation of an eclipse at any time during ROW to fiM its continuance, we must take the semidiameter of the sun in lhe ***&&' tude of the the dividers from the scale ; and, from the point of time on ec ii p e. the sun's path, describe the sun ; and, from the same point of time on the moon's path, describe a circle with the radius of the moon's semidiameter ; the portion of the sun's diameter eclipsed, measured by the dividers, and compared with the whole diameter, will give the magnitude of the eclipse as near as it can be determined by projection. The result* of this projection are as follows: A pp. time. Mean time. Beginning of the eclipse, P. M., 4h. 28m. 4h. 24m. 39s. Greatest obscuration, 5 40 5 36 39 End of the eclipse, 6 44 6 40 39 ' From the projection the two centers are nearer together than the difference of the semidiameter of the sun and moon 20 ASTRONOMY. v. and the moon's diameter being least, the eclipse will be an- nular, as represented in the projection. The above results are, probably, to be relied upon to within three minutes. We have now done with the projection, as far as the particu- lar locality, Boston, is concerned ; but, in consequence of the facility of solution, we cannot forbear to solve the following problem : In the same parallel of latitude as Boston, find the longitude where the greatest obscuration will be exactly at 2 p. M. apparent time. A very easy From the point 2, in the ellipse, draw a line at right an- ant obtem ^ es * ^ e moon ' s P atn an( ^ tna * point must also be 2h. on the moon's path; running back to conjunction, we find it How solved, must take place at 1 h. 10 m. ; but the conjunction for Green- wich time is 8 h. 49 m., the difference is 7 h. 39 m., correspond- ing to 114 45' west longitude ; we further perceive that the sun would there be about 9 digits eclipsed on the sun's north- ern limb. HOW to find Now, admitting this construction to be on mathematical ore accu- p r i nc jpi eg ( ag it really is, except the variability of the ele- ments ), we can determine the beginning and end of a local .eclipse to great accuracy, by the application of ANALYTICAL GEOMETRY. enerai -^ fc , p ^ (7 05 be two rectangular co-ordinates, then equations to aid in com- the distance of any point m the projection from the center putingaiithe can ^ e determined by means of equations. oes of an Let x and y be the co-ordinates of any point on the sun's eclipse as : p a th or elliptic curve, and Jfand Y the co-ordinates of any n n piac any P* nt on *^ e m on's path, then we have the following equa- tions : ( 1 ) y=p sin. L cos. D+p cos. L sin. D cos. t ( solar ( 2 ) x=p cos. L sin. t I co-ordin. (3) Y=dhis\n.B | lunar co . ordinates . (4) X=hieos.B ( In these remarkable equations, p is the semidiameter of pro- jection, L the latitude, D the sun's declination, t the time from apparent noon, d the difference in decimation between ECLIPSES. $07 iun and moon at the instant of conjunction in right ascen- Cm*. T. sion, h the moon's hourly motion from the sun, i the interval of time from conjunction in right ascension minus, if before conjunction plus, if after; and B is the angle L C s, or the angle which the moon's path makes with C D. In the equations, x and X arc horizontal distances. In equation ( 1 ), the plus sign is taken when the hours are on the upper side of the ellipse, as in winter ; when on the lower side, take the minus sign. In equation ( 3 ), the plus sign is taken when the motion of Ex P uatio the moon is northward, and the minus sign when southward. of the *y m - The sin. t, or cos. t, means the sin. or cos. of an arc, corre- sponding to the time at the rate of 15 to one hour. The solar and lunar co-ordinates, or equations ( 1 ), ( 2 ), The symbol ( 3 ), and ( 4 ), are connected together by the following equa- <4 , tions ; the minus sign applies to forenoon, the plus sign to time of eon- afternoon : junction i. <4 e= /; ri * ht M0 ton. To apply these equations, and, of course, the former ones, e, the interval of time from conjunction must be assumed, and, as the time of conjunction is known, t thus becomes known; (/, h, and B, are known by the elements; therefore, x t y, and X, Y, are all known. But the distance between any two referred to co-ordinates, is always expressed by When an eclipse first commences, or just as it ends, this ex- pression must be just equal to the semidiameter of the sun and moon ; and if, on computing the value of this expression, it is found to be less than that quantity, the sun is eclipsed ; if greater, the sun is not eclipsed ; and the result will show how much of the moon's limb is over the sun, or how far asunder the limbs are, and will, of course, indicate what change in the time mut be made to corresp ind with a con- tact, or a particular phase of the eclipse. For an eclipse absolutely central, and at the time of being central, the last expression must equal zero; and, in that 308 ASTRONOMY. CBAP. V. Application f the preced- ing eiprei* computation for the begin- ning of the eclipse as een from Beiton. case, x=X, and y= Y. In cases of annular eclipses, to find the time of formation or rupture of the ring, the expression must be put equal to the difference of the semidiameters of sun and moon. In short, these expressions accurately, ef- ficiently, and briefly cover the whole subject; and we now close by showing their application to the case before us By the projection we decided that the beginning of the eclipse would be at 4h. 28m., apparent time at Boston. Call this the assumed or approximate time, and for this instant we will compute the exact distance between the center of the sun and the center of the moon, and if that distance is equal to the sum of their semidiameter, then 4 h. 28 m. is, in fact, the time, otherwise it is not, &c. h. m. . Conjunc. in R. A., app. time, Boston, 4 13 21 Assume i equal to 15 Therefore, t is equal to 4 28 21=67 5' 15". jo=54' 21"=3261. Reduced lat., Z=42 6' 38". />=21 11' 43"; d=Cr=1324".3; J B=160'58". p 3261 - log. 3.513511 L 42 6' 38" sin. 9.826437 D 21 11 43 cos. 9.969583 < 67 5 15 log. 3.513511 cos. 9.870315 sin. 9.558149 cos. 9.590288 2039.1 346.3, y=1692j8 log. 3.309531 . 346.3 log. 2.532263 p cos. L 3.513511 9.870315 sin^J _ 9.961303 *=222&5 log. 3.348129 For Fand X: Ai414".75 114.5 add 1324.3 1438.8 :264 sin. 9.440775 log. 2.617800 2.058575 - cos. 9.982804 - log.^.617800 398.6 2.600604 ECLIPSES. 309 Here are two sides of a right-angled triangle, and the hy- CHAP. V. pothenuse of that triangle is 1857*'. 8, which is the distance between the center of the sun and moon at that instant ; but the semidiaraeter of the sun and moon is only 1853"; there- Theeoiip* fore the eclipse has not yet commenced, and will not until the m ' moon moves over 4".8; which will require about 9s., as we determined by proportion, because the apparent motion of the moon will be almost directly toward the sun. When the apparent motion of the moon is not so nearly in a line with the sun, as it is in this case, we cannot proportion directly to the result of the correction. In fact, the apparent motion of the moon is on one side of a plane right-angled tri- angle, and the distance between the center of sun and moon is the hypothenuse to that triangle, and the variation of the moon on its base varies the hypothenuse, and the computa- tion must be made accordingly. Hence, to the assumed time of beginning, 4 h. 28 m. 21 s. Add ... 9 Beginning, apparent time, - - 4 28 30 Mean time, - - - - 4 25 19 By the application of the same expressions, we learn that the greatest obscuration will take place at 4 h. 41 m. mean time at Boston ; and the apparent distance of the moon's cen- f the "' ter will be 18" north of the sun's center ; and, as the moon's " on j*n. t * semidiameter is 57" less than that of the sun, a ring will be formed of between 10" and 11" wide at the narrowest point. End of the eclipse, 6 h. 46 m. 58 s. mean time. In computing for the end of the eclipse, we assumed t=2h. 33m., and as t is more than 6h., the second part of y changes sign, as we see by the figure ; the sun after 6, must be above the line 6^6. Occultations of stars are computed on the same principle* as an eclipse of the sun, the star having neither diameter nor parallax. From forty to fifty occultations of the fixed stars by the moon, occur each month, but not more than three or four ire visible, as seen from any one place. Very few, if any, cave Aldebaran, are visible to the naked eye. 310 APPENDIX. APPENDIX TO ECLIPSES, AND OTHER MATTER. introductoiy WE have thus far treated solar eclipses in a genera] Remarki. 'manner, for the benefit of those who are not well versed in the principles of spherical sections, but now we propose to show the strict geometrical principles, which cover the whole subject, and define the latitudes and longitudes which bound the -visibility of solar eclipses on the earth. We shall take the. same eclipse, as before, for an example, and take the elements as we find them in the English Nau- . tical Almanac, for 1854, which are as follows : 1854, MAY 26. h. m. s. Greenwich, mean time of tf R. A. - 8 55 43.2 Q and Q)'s Right Ascension, - - 413 7.41 Q)'s Declination N. - 21 33' 31"8 9's Declination N. - 21 11' 16"8 Q)'s Horary motion in R. A. 31' 18"9 ^'s Horary motion in R. A. 2' 31 "8 Q)'s Horary motion in Decl. N. 8' 7"3 Q's Horary motion in Decl. N. 25"9 Q)'s Equatorial Hor. Parallax, 54' 32"6 Equatorial Hor. Parallax, 8"5 Q)'s true semidiameter, 14' 53"6 Q's true semidiameter, - 15' 48"9 From these elements, the following results have beeii comDuted, which we extract from the Nautical Almanac. APPENDIX. 31 LINE OF CENTRAL AND ANNULAR ECLIPSE. fORTl Longi ude. Latitude. Longitude. Latitude. * TAC>1 ^nr 51 53'W 36 J 18'N 73 53' 44 14' 92" 40' 48 3J' 11 6 24' 49 23' 134 45' W 4533'N 143 J 41'W 156 56' 169 28'W 179 24' E 162 51' E SOUTHERN LIN 41 37' N 32 39' 22 33' 14 52' 643'N E OF SIMPLE CO. IEBN LlNE OF SIMPLE CONTACT. Longitude. Latitude. I Longitude. Latitude. <. 40 16' E 48 4' 72 4' 90 54' 120 28' 130 17' 125 43' 11645'E 68 57' N 76 27' 79 57' 80 4' 76 0' 65 37' 57 42' 52 20' N 68 2'W 87 44' 101 53' 108 24' 126 24' 145 54' 163 10' W 1 77 22' E 5 40' N 12 49' 16 12' 16 49' 13 21' N 1 7' S 14 15' 24 16' S ECLIPSE BEGINS AT SUN-SET. ECLIPSE ENDS AT SUN-RISE. Longitude. Latitude. Longitude. Latitude >li-o/s rfj niJ edi 1o K* J - 38 21' E 10 15' 2 44' E 6 C 54' W 40 48' 54 23' 65 I8'W 68 55' X 65 22 63 17' 59 29' 29 37' 13 29' 6 9'N 17436'E 157 21' 150 39' 143 48' 124 19' 116 50' 117 7'E 23 48' S 9 43' S 14'S 10 56' N 41 16'N 50 37' 51 56' We now propose to show most clearly to the geometrical student, how such like results can be obtained. We shall make the same general construction, as before, but enlarge it, to show the sections of the sphere, and the application of spherical trigonometry. To make the following projection appear natural, the learner must conceive his eye to be in a line between the center of the earth and the center of the sun, and at a dis- tance from the earth equal to that of the moon. The diameter of the earth will then appear to cover & space in the heavens equal to the moon's horizontal par- allax, (4' 52".6, ) and as the sun's declination is north, in B*eofu* APPENDIX. this case, the north pole of the earth will be visible, as represented in the projection at the point P. In eclipses, we suppose the sun to be stationary, and the moon to move with the excess of motion between the SUQ and moon, therefore we subtract the parallax of the sun, 8". 5 from (54' 32".6), and we have 54' 24". 1, or 3264".!, for the value of CB or CA. As the earth is not a perfect sphere, those who desire to thewrfaceofbe extremely accurate, would subtract 11" from 3255"!, th earth as making 3244". 1 for the value of CH. Cd would be about * 3259", but we shall attempt no such accuracy. Any at- tempt to correct the projection from a true circle, would make it more inaccurate than it now is.* The perpendicular plane passing through CH is the plane of the meridian, that meridian on which the sun and moon are at the instant of conjunction in right ascension. Ob- serve that CG is a line perpendicular to the plane of the moon's orbit. The line K, m, r, d, h, is the plane of the moon's orbit, and the value of Cr is the difference in declination between the sun and moon. The inclination of the moon's path, Kr k, to the meridian Howtoob- ^^ * 8 determined by the elements, and very much will tain the mo. depend on the accuracy of that angle. tion of Uw -ITII tion to tb Q)> s motion in R. A. (per hour,) - 31' 18"9 """* @'s motion in R. A. - - 2' 3I"8 Q)'s motion from the sun, - - 28' 47"! = 1727* But the moon is now above the 21st degree of north de- clination, where the length of a degree is less than it is at the equator, in the proportion of cosine of the declination to the radius. * Here we would caution the learner not to assume, at the outset, that he cannot comprehend the figure, because it appears complex. It is a diagram which includes a great number of problems, and it was drawn with a design to solve them all : hence, the necessity of many lines and arcs. APPENDIX. 313 Hence, to reduce 1727"! to its equatorial value, or to Small cir- reduce it to a great circle, we must multiply by cosine of ^"J" 1 ^ the moon's declination, and divide by the radius. large ones Thus log. 1727"! 3.237292 cos. 21 33' 32" - - 9.968503 1606"2 3.205795 By the given elements, we have, for hourly increase of the moon's declination, - - - 8' 7"3 And of the sun's - 25"9 The excess is - - 7' 41"4=461"4 Conceive a plane triangle, whose base is 1606.2, and Howtoind perpendicular 461.4, and find the acute angles, and the [^ h JJJj greater of which (73 58' 20") is the angle at r, in the axis of the figure, and the less (16 1' 40") is the angle m Cr, which | a r " l * measures the arc 6fff. the earth. The hypotenuse of this triangle (1671") is the apparent motion of the center of the moon's umbra over the earth's disc, per hour. From Q)'s declination N". - 21 33' 31"8 Take Q's declination - - 21 11' 16"8 And we have Cr - - - 22' 15" = 1335" We must now compute the values of (7*71=1282", mr= $ev erai ime. J68"5. We have before remarked, that Kcmd represents com P uted * the line of central eclipse over the earth, and it is obvious that the extent of the eclipse, north and south of this line, must depend on the apparent semidiameters of the sun and moon. Their sum is (14' 53"5) added to (15 48"9) = 1842"4. Hence, we take mO and mt, each equal to 1842"4, and through the points and t, draw lines a b and ef, each parallel to the central line. The line a b is the diameter of a circular section of the p og i t j OBO f sphere, which defines the northern line of simple contact. the P' C f The line ef is the southern line of simple contact. The three lines a b, cd, and ef, are diameters of small circles APPENDIX, APPENDIX. 315 of the sphere one of these small circles is represented in the figure by the dotted line c R Q d. The point C is at the center of the plane of projection The eart>l *. J , th bate of or we can conceive it to be on the surface or the earth, p r0 j eo tion directly under the sun at noon. Then we should conceive the line C If to be the meridian arc of a great circle, C H being 90, whence the arc HP is equal to the sun's decli- nation. Whence G HP is a right angled triangle, on the surface of a sphere ; G R P is another spherical triangle, and PR is the co-latitude of the point R, on the surface of the earth. We will now continue the computation of all the lines and arcs, that we shall have occasion to use. We have already (7*71=1282" Add m<9=1842"4 Sum (70=3124"4 Diff. Ct=560"4. (7 3 17 6 19 6 20 6 21 6 22 6 23 6 24 6 25 16 6 3 6 7 6 10 6 14 6 16 6 19 8 20 6 21 6 23 6 24 6 25 6 27 6 28 6 29 18 6 4 6 8 6 12 6 16 6 19 6 21 o 23 6 24 6 26 6 27 6 29 6 30 6 32 6 33 20 6 4 6 9 6 13 6 18 8 21 6 24 3 26 6 27 6 29 6 30 6 32 6 34 6 36 6 37 22 6 5 6 10 6 15 6 20 6 23 6 27 8 28 6 30 6 32 6 34 6 36 6 38 6 39 6 41 24 6 5 6 11 6 16 6 22 6 26 6 29 o 31 6 33 6 35 6 37 6 39 6 41 6 44 6 4G 26 6 6 6 12 6 18 6 24 6 28 8 32 3 34 6 36 6 39 6 41 6 43 6 45 6 48 6 50 28 6 6 6 13 6 19 6 26 6 30 6 35 3 37 6 40 6 42 6 45 6 47 6 50 6 62 6 65 30 6 7 6 14 6 21 6 28 6 33 6 38 > 41 6 43 6 46 6 49 6 51 6 54 6 67 7 32 6 8 6 15 6 23 6 31 6 38 6 41 3 44 8 47 6 50 6 63 6 56 6 58 7 1 7 6 34 6 8 6 16 P 25 6 33 6 39 3 45 6 48 6 51 6 54 6 67 7 7 3 7 7 7 10 36 6 9 6 18 6 26 6 36 6 4-2 6 48 3 51 8 55 6 58 7 1 7 6 7 8 7 12 7 16 38 <> 9 6 19 6 28 6 38 6 45 6 52 3 55 6 69 7 2 7 6 7 10 7 14 7 17 7 81 40 6 10 6 20 6 31 6 41 6 48 6 56 3 59 7 3 7 7 7 11 7 15 7 19 7 23 7 28 42 6 1) 6 22 6 33 6 44 8 5-2 7 7 4 7 8 7 12 7 17 7 21 7 25 7 30 7 36 44 6 12 6 23 6 35 6 47 d 66 7 4 7 9 7 13 7 18 7 22 7 27 7 32 7 37 7 42 46 6 12 6 25 6 38 6 51 7 7 9 1 14 7 19 7 24 7 29 7 34 7 39 7 44 7 60 48 6 13 6 27 6 41 6 66 7 4 7 14 7 19 7 25 7 30 7 35 7 41 7 47 7 63 7 69 50 6 14 6 29 6 44 6 59 7 9 7 20 7 25 7 31 7 37 7 43 7 49 7 55 ' 2 8 8 62 6 15 6 31 6 47 7 3 7 14 7 26 7 3-2 7 38 7 45 7 61 7 58 8 5 8 12 8 19 54 6 17 6 33 o 50 7 8 7 20 7 33 7 40 7 46 7 63 8 8 8 8 15 8 23 8 31 56 6 18 6 36 6 54 7 13 7 27 7 41 7 48 7 65 8 3 8 11 8 19 8 27 8 36 8 46 68 6 19 6 39 6 59 7 20 7 34 7 49 7 57 8 5 8 14 8 2 : 2 8 32 8 41 8 51 9 2 60 6 21 6 42 7 4 7 26 7 42 7 59 3 8 8 17 8 26 8 36 8 4< 8 58 9 9 9 22 61 6 22 6 44 7 6 7 30 7 47 8 5 3 14 8 24 8 34 8 44 8 55 9 7 9 20 9 34 62 6 23 6 46 7 9 7 34 7 62 8 11 S 20 3 31 8 42 8 53 9 5 9 18 9 32 9 47 63 6 24 6 48 7 13 7 39 7 57 8 17 3 27 8 38 8 50 9 2 9 16 9 30 9 46 10 4 64 6 25 6 50 7 16 7 43 8 3 8 24 3 35 8 47 9 9 13 9 28 9 44 10 2 10 24 65 6 26 6 52 7 19 7 48 8 9 8 32 8 44 8 67 9 10 9 26 9 42)10 1 10 22 .0 61 The sun is supposed to be stationary and vertical over the point C. The earth revolves from west to east, hence observers on the curve LA eg see the sun in the eastern horizon as it appears to them. When the moon arrives at K, an observer at A would see its limb just meet the limb of the sun, and the sun would he riLlng to that observer. But that observer is in latitude 1 24' south, and the 322 APPENDIX. declination of the sun is 21 11' north, whence by the table, the sun must rise about two minutes after six.* . ' * h. m. s. 6 1 44 3 10 36 Sun rises, apparent time, Time from .AT to r, Time of tf at .this locality, - 9 12 20 morning. Time of tf at Greenwich, - 8 58 58 evening. Difference of meridians, 11 46 38=Lon. 176 39' W. When the moon arrives at c, the sun is centrally eclipsed that point, and it is the first point on the earth at which * L .'! the sun is centrally eclipsed, and its latitude is 6 37' K". But in this latitude, when the sun is 21 11' north, h. m. s. The sun rises at 5 50 30 Time the Q) passes from c to r is 2 57 ; Various ba ftudg App. time of tf at this locality, 7 51 27 morning. Time of cf at Greenwich, 8 58 58 evening. Difference of meridians 13 7 31=Lon. 163 7' E. When the moon arrives at d, the central eclipse passeg off of the earth ; but this point is in latitude 36 3' north, and the declination is 21 11' north. With this latitude and declination, we enter the table, and find that h. m. The sun sets at - - 7 5 The Q) passed from r to d m 1 35 Time of of at this place, Time of at Greenwich, Difference of meridians, 5 30 P. M. 8 59 P. M. 3 29=Lon. 5215'W. * When we wish to be very accurate, we must not use the table, but aolve the problem as taught on page 246, thus : tan. 1 P 24' - 8.388092 tan. 21 11' - - 9.588316 sin. 26' This arc, 26', corresponds to 1m. 44s. 7.976408 APPENDIX. 323 When the moon arrives at h, the eclipse leaves the earth, and the last observer must be at q but the latitude 6f q we have determine) 1 to be 28 16' N., and in this latitude h. m. s. The sun sets at - 6 49 The Q) moves from r to h in 2 44 42 Time of tf in this locality, 4 418 evening. Time of tf at Greenwich, 8 58 58 evening. Difference of meridians, 4 54 40=Lon. 73 39' W. In the same way we can determine the longitude of the extreme points on the northern line a b, and on the south- ern line ef. An observer at (a) will not see the moon touch the sun until the moon arrives at a distance from m equal to a 0. But an observer at a sees the sun in his horizon, and to him it is rising. Now we have determined the latitude of a to be 51 33' 24", and in this latitude, when the sun's declination is 21 11' north, h. m. s. The sun rises at 43 appa. time morn. The Q) moves from the perpendic - ular let fall from a to r* ( 1314") in 50 49 The time of tf in this locality, 4 53 49 morning. The time of tf at Greenwich, 8 58 58 evening. Difference of meridians, (if W.) 16 511 (ifE.) 7 52 49=Lon.ll810'E The point 5, is in latitude 68 48', and from this latitude, when the sun's declination is 21 11' north, we find that h. m. s. The sun must set at 1 1 50 28 apparent time. This is past We will now consider any other point on the circumfe- ^^"'""^rence of projection within the limits of the eclipse, and points taken determine the latitude and longitude of that point, which w ^ a ^ so ^ e a Pi nt where the eclipse will begin at sun-rise, if the point is on the western side of the projection, or a f projection, point where the eclipse will end at sun-set, if it be on the eastern side of the projection. Let g be a point taken at hap-hazard on the circle Ac O t and suppose the arc Qg be taken equal 42. Then OP will be the co-latitude of the point g, and the sun being vertical over C, is 90 distant from the observer/and of course, in his horizon. APPENDIX. We take the distance (gu), (gr), equal to mO, the sum of the semidiameters of the sun and moon (1842"4). When the moon comes to F, the eclipse commences, and when it passes u, the eclipse ends. To - 9.969604 Add cos. 59 1' 40" - 9.723738 cos. co-latitude or sin. lat. 29 34' 30" 9.693342 In like manner we may find the latitude of any point on A * eneral . investigation the arc within the limits of the eclipse, from e to a, and O f the pro. from b to/. Observe that the points,(^') and (g), are at blem for find ; equal distances from F; hence, the eclipse will be seen to ["aVand loo* commence at the same moment of absolute time, from (^)gitud* of p u- and (g'). The difference between the latitudes of these "j W te eiret be * two points is the difference between the two arcs Pg' andgimandeiuu Pg, and their difference in longitude is equal to the arc *^ d I * un ^"* The distances ru and r Fare obtained thus : Observe that m S is the sine of Og to the radius cm. LetiF=y. Then VS=y cm sin. Og. Sff=cmco8. Gg Cm. (ycm gin. Gg) 2 -{-(cm cos. GgCm) 2 =( \ 842"!)*. Whence y= "^/(184r fj' (cm cos. /e \ cos.m= . (5; cos.^4 cos.D The value of (cos.m COB. A) obtained from (5) and placed IQ (4), will give cos./) 33t APPENDIX. This equation contains (sm.A), which we wish to ex punge, and the spherical triangle Zmp will give us .Z sin./) cos.L cos. D. Or sin.^=cos.* cos.L cos.D-^-sm.L sin./). Or sin.^4 sin./)=cofi.*cos./i cos./) sin./)-|-sin.Zsin. 3 2). Whence sin.Z sin. A sin./)=sin.Z( 1 sin. 2 /)) cos.* cos L cos./) sin./). But 1 sin. a /)=cos. 3 /). This value put in the second member, and both members divided by cos./), we shall have sin.Z sin. A sin./) . r ^ -- = - =sm..L cos./) cos.* cos. //sin./). COS.// Comparing this with (6), we obtain am=p cos. L cos. D p cos. L sin./) cos.*. (7) Thus we have found Parallax in right ascension =p cos.L sin./. Par. in declination p sin.Z cos./) p cos.L sin./) cos.t. If we compare these expressions with the solar co-ordi- nates on page 306, we shall see that they are the same. Hence, the theory of the projection agrees with spherical trigonometry. If we place A=p cos.Z, /?=jt> sin. L cos./), and C=p cos.L sin./), we shall have Parallax in Right Ascension =A sin.*. And Parallax in Declination =/? Coos.*. The parallax in R. A. varies as the sine of the moon's meridian distance, and the parallax in Declination varies as the cosine of the meridian distance united to a constant. The high. We will now take the latitude of 49 20' north, and lon- utimde gi^ U( j e io5 west, and compute the eclipse as seen 11 from that place. The result ought to be very nearly a central eclipse. h. m. s. Conj. at Greenwich, app. time, 8 58 58 P. M. Longitude in time, - -7 Sun and moon west of merid. 1 58 58 ==29 58' 30**r-f. APPENDIX. 3S5 ^'s equatorial horizontal parallax, 54' 32 V .6 j^'s horizontal parallax, 8".5 54' 24". 1 Reduction for latitude 49 - 7".l 54 ' 17 " =3257"=;). P - - 3 ,512818 P 3.512818 sin, L 49 20' 9, 879963 cos .L . 9 .814019 cos.D 21 33' 32" 9 .968503 sin .1) - 9 .565187 , cos .*29 58' 30" 9 .937640 B 2297 6 3 .361284 675 6 . 2 .829664 1622"=parallax in declination. A= 3.326837 sin.*= 9.698641 (7=2.892024 Parallax in R. A. 1060".4 3.025478 We will now compute the parallax in right ascension and declination at the expiration of half hour intervals after this time, as follows : At time of conjunction in this locality, /=29 58' 30" 30 minutes interval, -f- 7 30' Q)'s motion from ^ during this interval, 14' 23".6 First half hour after conjunction, - /=37 14' 6" Motion from the meridian in 30m. 7 15' 36" One hour after conjunction, /=44 29' 42" Motion from the meridian in 30m. 7 15' 36" One hour and thirty minutes after conj. t=5l 45' 18 Two hours after conjunction, *=59 0' 54" Parallax in Right found as follows : 1st half hour. A 3.326837 A sin.* 9.781814 Ascension, 3.326837 9.845632 in half hour 3d. A 3.326837 9.895063 intervals, is Practical computatioa 4th. of parallHX. A 3.326837 9.933140 At^ 3.108651 1060."4 1284".3 3.172469 1487".4 3.221900 1666".7 3.259977 1819". APPENDIX. The right ascension of the moon is east of the sun at (f h. af. 863"6 Ih. 1727"2 lh. 2590"8 2h.3454"4 )par. 1060"4 1284"3 1487"4 1666"7 1819" Q) N. 1060"4 cos.Q)D..93 W. 420"7 .93 E. 239"8 .93 924"! .93 1635"4 .93 3181 1262 7194 2782.3 4906.2 95436 3786 21582 83469 147186 986.17 391.22 223.014 862.513 1529.922 <3)W.of Q. Q) W. of Q. Q) E. of Q. Q)E. of Q. Q)E. ^. Thus are the apparent distances in Right Ascension re- duced to the arc of a great circle. For the parallax in Declination, at these several inter- vals, we operate thus : (72.892024 (72.892024 (72.892024 (72.892024 cos.* 9.901001 9.853252 9.791717 9.711625 X 2.793025 2.745276 2.683741 2.603649 620"9 556"2 482"8 401"5 B 2297"6 2297"6 2297.6 2297"6 Q) par. in Q)N.of( D. 1676"7 P 1565"7 1741 "4 1796"4 1814"8 2027"! 1896"! . 2257"8 Q)app. S.of^lll" N. 65" N. 212"3 361"7 At conjunction the Q) was north of ^, 1335", but the parallax in declination was then 1622" southward. Hence, the apparent declination must have been 287" south. Now to determine the beginning and end of the eclipse, and other circumstances, we must resort to a partial pro- jection, as follows : Let S represent the center of the sun at the time of con- junction in right ascension, apparent time. The true right ascension of the moon is then the same as that of the sun. But the parallax in R. A. we have found to be C86"2, westward of course, because the moop is west of the meridian. APPENDIX. * Draw the horizontal a=986"2, and from a make am= 287", and ra is the apparent place of the moon at the time of , ca i e conjunction. Projection : One half hour afterwards the apparent place of the moon was 391" west of the ^, and 83" south. We therefore take a'=391"2, and 7'm'=M 1", andm' is the apparent place of the Q) 30 minutes after ^, and mm' is the apparent motion of the Q) during the half hour. At the expiration of the next half hour, the apparent place of the Q) was 223" east of the ^ at b, and 55" north of that point at m". In the same manner we determine the points m a and w 4 . The position of the center of the sun is stationary at 8 9 but the apparent places of the moon, are at m, m', m", m a , m 4 , during the interval of two hours. By merely inspecting this figure we perceive that about 46 minutes after ^ the two centers will be nearest to each other, and they will not be 3" asunder. That is, at (Ih, 69m. -{-46m. ), or 2h. 45m. apparent time, the sun will be centrally eclipsed. By the figure, we can determine the distance be- tween S and m", S and m', &c., and their rate of approach and departure, and thence we can determine the time of beginning and ending, the time of forming of the ring, l P"* apparent time, the right ascension of the meridian is one*' f hour greater than the right ascension of the sun. When a star or planet is on the meridian, the right ascension of the meridian is the same as the right ascension oj that star. If we subtract the right ascension of the sun from the right ascension of a star, the remainder is the apparent time for that star to pass the meridian. That is, Apparent time -^ on merid.=R. A. -^ R. A. ^ The time from the meridian to the horizon, or the time from the horizon to the meridian, is called the semi-diurnal nalare * arc, as we have before explained. Corresponding to cer- tain latitudes of the observer, and degrees of declination of the heavenly body, it is to be found in a table on page 331, or it can be computed in each case, independently of the table, as taught on page 236. EXAMPLES. 1. What time will the fixed star Sirius rise, pass the me- ridian, and set, on the 20th of January, 1858, as seen from Latitude 40 N. and Longitude 75 W? In Table II, we find the right ascension and declination of Sirius, with its annual variations. From 1846 to 1858, is twelve years. Hence 22 338 APPENDIX. R. A. h. m. s. Declination. 6 38 21.884 16 30' 32"83 8. Variation 12y +31.722 53.80 ^-'s position, 1858, 6 38 53.605 16 29' 39"03 Tb reason f^Q r jght ascension of the sun is zero, on or about the fa Uw'oplra 1 ! 20th of March in each year, and it increases about 2h. each tfc. month, therefore, on the 20th of January, it cannot be far from 20 hours. This subtracted mentally from the right ascension of the star, shows us that the star must p'ass all meridians on that day when the local time at each place, cannot be far from lOh. 30m. When it passes the meridian of Lon. 75 west, the local time in that longitude must be near lOh. 30m. and the time at Greenwich 15h. 30m. Howtoob- Therefore, to obtain a result nearly accurate, we mus* tain the snnj ]ave ^ Q j-jg^ ascension of the sun corresponding to Jan- ion. u(jry 20th, 15h. 30m. of Greenwich time, which is 8h, previous to noon of the 21st of January. The right ascension of the sun is given in the Nautical Almanacs for the noon of each day at Greenwich, and the hourly variation. h. m. s. 1858, Jan. 21, Q's R. A. 20 13 53.9 Variation Ih. 10s.54X8 1 29.6 20 12 24.3 h. ID 8. Right Ascension of Sirius, - - 6 38 53.6 U Right Ascension, - - 201224.3 Sirius passes meridian, (apparent time,) 10 26 29.3 Equation of time, (Nautical Almanac,) + 11 34.6 Star passes (Ion. 75) mean time, - 10 38 3.9 P. M. Now to find the time this star will rise and set, we must apply the semi-diurnal arc, so called, which can be found nearly, in the table on page 321, corresponding to Lat. 40 N. and Dec. 16 29'. The table will give us 6h. 57m. 30s., and this would be the interval sought, provided the decii- APPENDIX. 339 nation was north. It being south, we must subtract this sum from 12h. Hence the arc sought is 5h. 2m. 30s. h. m. Approximate Now the star passes the mer. mean time, 10 38 4 Subtract the semi-diurnal arc, - 5 2 30 Star rises (mean time) P. V 6 35 34 (Add semi-diurnal arc.) Star sets, A. &L. o 4U next morning. The time the star passes the meridian is tolerably accu- Corrections I to b - rate, but the times of rising and setting require correction and for whau for refraction. That cause would increase the semi-diurnal arc about 2 minutes; and in addition to this, we must con- sider that the sun changes its right ascension during the 6h. 2m. that the star requires to pass from the meridian to the horizon. The change in the sun's right ascension for one hour is 10s. 54; in 5h. 2m. the change will be 53s. That is, when the star actually rises, the sun's right as- cension is 53s. less than when it passes the meridian, and it is 53s. greater when the star sets. Hence a slight cor- rection is necessary. Or we may view this problem from another stand point. When the star is rising it is in the eastern horizon, and Anotht* h. m. 8. mode of com. Its Right Ascension is - - - 6 38 53.6 JT|f of * Semi-diurnal arc -f- refraction - 5 4 30 sing nd set. _ ting of tb* Diff. = Right Ascension of meridian, 1 34 23.6 * tar * Sun's Right Ascension at this time, 20 11 31.3 Star rises, apparent time, - 5 22 52.3 Equation of time, add ... +11 34.6 Star rises (mean time,) - - 5 34 27 When the star sets, it is in the western horizon, and the Right Ascension of the meridian is greater (or eastward,) than that of the star. 940 APPENDIX. h. m. s. :'s Right Ascension, - - 6 38 63.6 ;mi -diurnal arc -J- Refraction, - -}-5 4 30 Right Ascension of the meridian, - 1 1 43 23.6 Right Ascension of sun, 20 1317.3 Star sets (apparent time,) - - 15 30 6.3 Equation of time, ... -j- 11 34.6 Star sets (mean time,) - - - 15 41 41 Or, 3h. 41m. lls. A. M. on the 21st of January. Practical ^ n solving problems of this kind, practical men make no not attempt at accuracy, as the element of refraction is very unable. uncerta j n j n j^g re sults, and VQVJ often the stars cannot be seen at all, when near the horizon. 2. Giving the operator the use of a Nautical Almanac for 1858, we require him to determine what time of day tie planet Jl/ars will pass the meridian of Boston, (Lat. 42 22' N., and Lon. 4h. 44m. W.,) on the %d day of July. Also required the time it will rise and set. On the 2d of July, 1858, the Right Ascension of Mara is 14h. 53m. 20s., and Declination 19 1' south, and these elements may be taken as invariable during that day. The times The Right Ascension of the sun, on the 2d of July of "^'any year, is not far from 6h. 44m., and this mentally sub- rising attracted from 14h. 53m., leaves 8h. 9m., to which add the setting of the longitude of Boston in time, 4h. 44m., and we have Ifch. 53m - f Greenwich time, to which the sun's Right Ascen- sion must correspond. h. m. s. R. A. of sun, July 2d, '58, at noon, Gr. 6 44 33.9 Variation per hour, 10s.3X(12.53) -)-2 12.1 R. A. of sun when moon is on merid. 6 46 46 From Right Ascension of Planet, 14 53 20 Subtract Right Ascension of sun, 6 46 46 Mars passes meridian, (app. time,) 8 6 34 P. M. Equation of time, (add) - - +343 Mars on meridian, (mean time,) 8 10 17 APPENDIX. 341 For latitude 42 22', and Declination 19 1', the semi- diurnal arc is 7h. 13m., or 4h. 47m. In this case it is 4h. 47m., the Latitude being north and Declination south. h. m. s. Hence, from and to - - 8 10 17 Subtract and add - - - 4 47 Mars rises, (mean time,) - - 3 23 17 P. M. Mars sets, " - - 12 57 17 P. M. Or Oh. 57m. 17s. on the morning of the 3d of July. The planet rises when the sun is up, and r f course it will be invisible. Here neither refraction nor the change ir t the sun's B. A. are taken into account, and it is not important that they should be, however, for the improvement of the student, we will compute the time the planet sets, increasing the semi-diurnal arc 2m. for refraction, and 50s. for the in- crease in the sun's Right Ascension. h. m. B. Bight Ascension of Mars, - 14 53 20 Semi-diurnal arc -|-2m. 4 49 Bight Ascension of meridian^ 19 42 20 B. A. of sun at this time, 6 47 26 Mars sets, (app. time,) - 12 54 54 Equation of time, -(- - - 3 44 Mars sets, - - : ^- fl 57 38 A. M. July 3d. We are now prepared to apply these principles to the moon. Several years ago, when only the longitudes and latitudes O f this of the moon were given in the Nautical Almanac, the rising blem and setting of the moon was a problem of some complexity, but recently the right ascensions and declinations of the moon are computed and written down, corresponding to every hour, in both the English and American Nautical APPENDIX. Almanacs, and every student of Astronomy should hare a copy.* Whe *i the moon changes, as it is called, it sets about the time, or a little before the sun. When the moon fulls, it rises about the time of sun-set. We compute the successive times the moon sets, com- mencing with new moon and closing with full moon. Then commence computing for moon-rise, from full moon to ne^w moon. FOR EXAMPLE. Having a Nautical Almanac for 1857 before us, we find that the moon changes or passes the sun, January 25th, 5h. 49m. P. M., mean time at Cincinnati. It will go down invisible in the blaze of sun-light that day, a little before or a little after the sun, according to the relative declina- tion^ of the two bodies. The exact time, for the day of change is never computed, unless it be in connection with an eclipse. A definite What time will the moon set, January 26th, 1 857, as seen npie. f rom Cincinnati, Latitude 39 6' N., Longitude, in time, 5h. 37m. west of Greenwich? 4* ' SOLUTION. ^Preparation. On the 26th of January, in Lat. 39 N, the sun sets at 5h. 10m. mean time, and the moon I judge will set Ih, af- terwards, at 6h. 10m. To this add the longitude, 5h. 37m. and the Greenwich time is thus determined to be 1 Ih. 47m. In the Nautical Almanac we find that the right ascen- s ' on ^ *^ e moon at ^is time is 21h. 35m. 2s., and decli- nation 18 21' 15" S. The semi-diurnal arc corresponding to Lat. 39 6' N., and declination 18 21' S., is 7h. 3m. from 12h., or 4h. 57m Hence, the computation is as follows : * Nautical Almanacs are made at public expense, and sold very cheap for the promotion of science. The price of a single copy is from 50 cents to $1 , barely enough to cover the cost of printing and paper. APPENDIX. S43 Q)'s R. A. Jan. 26 (at 6 10 P. M.) 21 as 2 Semi-diurnal arc, add 4 57 Right A. of meridian, 26 32 2 R. A. of the , sub. 20 37 44 Q) sets (apparent time), Cincinnati, 5 54 18 P. M. Equation of time, Nautical Almanac, -|- 13 1 Q) sets, mean time, 6 7 19 This computation makes no allowance for refraction, or Allowances for parallax. Within the latitudes of 40 on each side of * * made * for refraction the equator, the moon is generally kept above the horizon an a parallax. about two minutes longer by refraction, and will set about four minutes earlier in consequence of parallax. The effect of the two causes combined make the moon set two minutes, or more, sooner than is given by the preceding result. Therefore, if we were making an Almanac for 1857, for the locality of Cincinnati, we would record the setting of the moon January 26th, at 6h. 5m. Having the Nautical Almanac before us, and knowing the moon to be near her perigee, and her south declination tinned decreasing, we know that the moon must set on the eve- ning of the 27th, more than one hour later we judge about 7 15. This will make the Greenwich time near 13h. Corresponding to which time, we find the moon's right ascension and declination, as before, and compute the time it sets. h. m. s. Thus, Q)'s R. A. N. A., - 22 31 (Q)'s Dec. 12 19' S.) semi-diurnal arc, 5190 R. A. of Meridian, - - 27 50 R. A. of Sun, N. A. (Sub.) - 20 42 Q) sets (apparent time), - - 780 Equation of time, N. A. - - - 13 12 Q) sets, mean time, Cincinnati, - 7 21 12 And thus we go on from day to day. 344 APPENDIX. TO COMPUTE THE TIME THE MOON RISKS. Anexanpi* On the 8th of February, between Band 7 P. M., Cincin- ** T * n - nati time, the moon fulls. What time will it rise on the 9th? We judge that it will rise about one hour after sunset, or about 6h. 13m. Adding 5 37 we obtain llh. 60m. for the Greenwich time. h. m. 8. Th com- Q)'s R. A. at that time, Nautical Almanac, 10 24 25 Semi-diurnal arc, (sub.) 6 43 20 Right ascension of meridian, 341 5 R. A. at that time, Nautical Almanac, 21 30 39 ) rises, apparent time, 6 10 26 Equation of time, Nautical Almanac, -|- 14 31 mean time, (unconnected), Correction for R. and P. ) rises, mean time, corrected, 6 27 The critical student will be desirous to learn how to compute the precise effect of parallax and refraction on a heavenly body, just at the point of rising or setting. To show this we take the following: example : In latitude 60 North, when the moon's declination is 20 North, horizontal parallax 58', and refraction 34', what is the semi-diurnal arc, refraction and parallax being duly allowed for ? CONSIDERATION. The moon is depressed 58' by parallax, and de- rated 34' by refraction ; therefore, the moon, in the horizon, is depressed 24' and will set when a star, or the sun, at the same point, as seen from the center of the moon, would be 24' above the horizon. Therefore, to find the true semi-diurnal arc for the moon, we con- ceive a star at the altitude of 24', latitude 60, and polar distance, and compute the polar angle, as explained on page 251. This gives the semi-diurnal arc for the moon, in this example, 8h. 31m. 58s. But the semi-diurnal arc, without refraction or parallax, is 8h. 36m. 20s. Hence, the effect of parallax and refraction in this example is 4m. 22s. That is, the moon will set 4m. 22s. earlier, or rise 4ci 22*. later than it would, unaffected by refraction and parallax. And thus we could compute exactly in any other example. APPENDIX. 34* On the 10th, the moon will rise about one hour later, or at 7h. 30m., to which add the longitude in time, 5h. 37m., and the sum is )3h., Greenwich time. At that time the moon's right ascension, by the Nau- tical Almanac, was llh. 11m. 34s. and its declination 7 11'30"N. h. in. s. Q)'s R. A. (at 13h., Greenwich time), 11 11 36 Semi-diurnal arc, (sub.) - 6 29 Right A. of Meridian. - 4 42 36 ^'s 11. A. (Nautical Almanac), - 21 38 44 Q) rises, apparent time, - 7 3 52 Equation of time, Nautical Almanac, 14 31 Correction for Ref. and P. - 2 Q) rises, mean time, - - - 7 1 9 23 From latitude 39 North, and Q)'s declination, 13 6' North, we find the semi-diurnal arc, to be 6h. 43m. 20s. as above. Thus we may go on from day to day. The labor of making these calculations is not so great The labor t as it appears to be in these pages. b^oiTe^fc! Here we are teaching the pupil, and were compelled to miliai. write out all our thoughts. In actual calculation we write out only the figures, and do not carry it to seconds for common almanacs. ARGUMENTS FOR EQUATING THE MOON*S LON- GITU DE. In the lunar table, we find 20 Arguments for the moon's Genei. longitude, and we have been requested to explain them, or P lan tioni - show to what 1, 2, 3, 4, An. The variations of ^An. and )An., combined with all the possible positions of the sun, moon, and moon's node, Till produce variations in the moon's motion. For the first ten Arguments the circle of 360 is sup- - posed to be divided into 10,000 equal parts; and from 10 to Arg. 20 it is divided into 1,000 equal parts. The first Argument corresponds to the annual equation, caused by the sun's variable distance. The sun moves from his perigee to perigee again, in 365 days 13h. Dividing 10,000 by 365d. 13h., gives us 27.36 for one day the mean motion of the sun's anomaly. The symbol (Q) )> indicates that the sun's mean motion must be subtracted from that of the moon. "With these explanations, we suppose that all the follow- ing indications will be understood: ARGUMENTS FOB LONGITUDE. Motion in 24 hours Arg. \=^An. = 27.36 2=2(Q) Q) ^An. = 650.08 3=Arg.2+(j)An.-\-QAn. =1040.35 4=Ar ff .2(3)An. = 287.17 = 335.55 = 372.03 = 57.68 = 390.27 9=^An. (^Perigee = 24.24 10=2(3 Q 3^. )+QAn.= 70.17 1 1 Evection Per. Node = 31.28 12=2(3 &)+9An. = 70.552 APPENDIX. 347 * = 34.19 14=3.4r0r.l3 gfrom QjAbcfe = 99.15 \b=Evecti&nZPer.% t Node 30.54 16=Q)'s motion from Node = 36.74 17=Z Ceti * Eridam (Achernar),. t AlUETIS y Ceti, a Cirrr N.23 37 27.73 S. 13 57 1.50 N.16 11 41.39 N.45 50 6.56 S. 8 23 3.33 N.28 28 17.49 S. 25 4.86 S. 17 56 12.77 S. 1 18 17 53 i TSridaui a TAURI, (Aldebaran),.. a. AUUJU/E, (Capella),.. . ft ORIOXIS, (#i?eJ) /3 TAURI, 1 1 1 2 2 3.4 2 .'- 2 1 3 1 6 1 3.4 3 1 .2 2 3.4 4 3.4 2 2 3 3 1 S.34 9 .S6.95 N. 7 22 22.32 N.22 35 13.16 S.52 36 49.17 N.87 15 31.20 S. 16 30 32.83 S.28 45 59.38 N.22 15 37.47 N.32 13 12.93 N. 5 36 5495 N.28 23 3406 S.23 51 50.94 N. 6 58 48.51 N.48 38 32 35 S.58 37 49.78 S. 7 59 39.05 N.52 22 31.09 N.24 28 49.46 N.12 43 2.96 * Argus, (Ca.nopus),. . . 51 (Hev.) Cephei CANIS MAJ., (Sirius), t Canis Majoris, f Gerninorurn, o 2 GEMINOR. (Castor),... A CAN. Mix., (Procyon), /8 GEMINOR, (Pollux),.. 15 Argus, i Hydrse * Ursje Majoris, * HYDR^E . . fl Ursae Majoris, Leonis, * LKONIS, (Rcgulus},. . . TABLE II. Star's Name. ti i ?. Right Ascen. Annual Var. Declination. Ann. Var. 2 f\ 3 .4 i 2 5 1 I 3 1 1 1 3 3 3 2 ' 2 2 .i 4 2 3 J 3 2 4 3. 6 2 2 2 3. 3 1 3 3 3. 3 3.< 3 39 6.223 54 10.737 1 5 54 583 1 11 38.718 1 41 12.066 1 45 42.219 12 9 26.b9.s 12 18 4916 12 26 18465 12 48 49007 13 17 52ltt 13 41 27.894 13 47 21.140 13 53 0.8i '0 14 8 38.366 14 29 11.925 14 38 15706 14 42 22.132 14 51 13.199 15 8 43.595 15 28 1008: 15 36 41.07" 15 49 41.194 15 56 29 397 -1- 2.3051 3.8001 3.1928 30010 + 3.0654* 3.1874 3.3409 3.2710 -i- 3.1342 2.8403 3.1512 2.3525* H- 2.86H6 4.1508 2.7336* 40165* + 2.6229 4- 3.3J02 0.2692 H- 3.2226 + 2.5279 4- 2.9.S91 2.3520 + 3.4742 + 3 1382 3.6638 0.7960 + 6.2587 6.5328* + 2.7320 106.8627 1.3513 -h 2.7727 1.S900 + 3.5861 19.2683 + 2.0118 22124 2.7566 -f- 3.U086 + 2.8511 2.9254 2.944H 3.3315 deg min. see. . 58 52 34.26 N.62 34 51.81 N.21 21 5986 S.13 56 46.85 NM5 25 58.12 N 54 33 3.18 b.78 27 26.15 b.62 14 39.74 S. 22 32 39.93 \.39 9 418 S. I'l 21 20.80 N.50 5 1.45 N.19 10 21.03 S.59 37 33.93 N.19 59 12.H7 S GO 11 37.00 N.27 43 35.23 S. 15 23 53.52 N.74 47 5.58 S. 8 48 38.53 N.27 14 11.07 N. 6 54 49 88 N.78 15 55.43 S.19 22 44.18 S. 3 17 35.67 S. 26 5 4 58 N.61 51 50.58 S. 68 44 4.75 N.82 16 52.30 N.14 U 12.67 S t9 16 10.25 N.52 25 3.28 N.12 40 37 11 N.5i 30 33.50 S. 1 5 36 14 N.86 35 42.5e N.38 38 35.33 N.33 11 14.80 N.I 3 38 20.49 N. 2 48 43.64 N.10 14 31.50 N. 8 27 54.32 N. 6 1 33.90 S. 13 1 4.19 18.33 1924 19.50 1961 1999 2:.02 20.04 19.99 19.92 19.60 18.94 18.12 17.89 17.67 18.94* 15.12 15.46 1523 14.71 13.63 12.33 11.74 10.80 10.29 9.55 8.48 8.o2 7.48 5.03 4.54 3.14 2.88 2.81 61 4-040 4- 1.91 4- 2.77 386 5.05 4- 6.67 4-8.39 8.74 8.55* 10.74 A URS./K MAJORIS . . . AQUIL^E, AQUIL.E, (Altair,).. . * 2 CAPBICORNI, TABLES. Star's Name. si S Right Ascen. Annual Var. Declination. Ann Var. 2 5 1 5.6 3 3 3 3 2.3 3 2 3 1 2 4.5 3 20 13 25.814 20 16 31.309 20 36 11.005 20 59 59 947 21 6 23.073 21 14 53.940 21 23 26.875 21 26 39.120 21 36 37.346 21 57 52.326 21 58 29.837 22 33 46.976 22 49 7.531 22 57 5.584 23 32 1.736 23 33 4.581 + 4.8046 52.1273 + 2.0418 2.6908* -h 2.5486 1.4163 3.1628 O.b059 + 2.9441 3.0831 3.8134 2.9837 + 3.3095 2.9776 3.0569 + 2.4042 S.57 13 19.50 N.88 50 53.54 N.44 43 57.43 N.37 59 42.08 N.29 35 53.03 N.61 56 4.55 S. 6 14 44.46 N.69 53 7.21 N. 9 10 17.35 S. 1 3 56.72 S. 47 42 12.42 N.10 1 44 67 S.30 26 12.28 N.14 22 40.12 N. 4 47 30.74 N.76 46 22.01 + H.03 11.22 12.64 17.48 + 14.57 15.07 15.56 15.73 -I- 16.26 17.28 17.30 18.65 4- 19.11 19.31 19.36* 4- 19.92 y Ursae Minoris, 61> CYGNI, $ CEPHEI, OL AQUARIA a Gruis * Pis. &.vs.(Fomalha.ut), a. PEGASI (Markab'), y Cephei, Those Annual Variations which includes proper motion are distinguished by an Asterisk. SUN'S RIGHT ASCENSION FOR 1846. By of Mo. January. February. March. April. May. June. 1 5 10 15 20 25 30 h. min. sec. 18 46 52 19 4 30 19 26 21 19 47 57 20 9 17 20 30 19 20 51 h. min. sec. 20 59 11 21 15 22 21 35 18 21 54 54 22 14 12 22 33 14 h. min. sec. 22 48 17 23 3 12 23 21 40 33 40 23 58 14 16 25 3*36 h. min. sec. 41 52 56 26 1 14 43 1 33 6 1 51 38 2 10 22 2 29 17 b. min. sec. 2 23 6 2 48 25 3 7 47 3 27 24 3 47 15 4 7 20 4 27 8 h. min. sec. 4 35 48 4 52 12 5 12 50 5 33 34 5 54 22 6 15 10 6 35 55 "3 Mo. July. August. September. October. November. December. 1 5 10 15 20 25 3C 6 40 4 6 56 34 7 17 5 7 37 25 7 57 33 8 17 28 8 37 7 h. min. sec. 8 44 55 9 23 9 19 29 9 38 21 9 56 60 10 15 27 10 33 44 h. min. sec. 10 41 10 55 29 11 13 30 11 31 28 11 49 25 12 7 24 12 25 27 h. mi >. stc. 12 29 4 12 43 36 13 1 54 13 20 24 13 39 8 13 58 9 14 17 27 h. min. sec. 14 25 16 14 41 2 15 1 5 15 21 28 15 42 14 16 3 19 16 24 43 h. rain. sec. 16 29 1 16 46 23 17 8 17 17 30 22 17 52 33 18 14 46 18 36 57 The R. A. in this table will answer for corresponding days, in other years* within four minutes ; and for periods of four years, the difference is only about ere a Mands for each period. TABLE III. TABULAR VIEW OF THE SOLAR SYSTEM. Names. Mean diameters in miles. Mean distance Mean dist.;| Log. of (Time of revolu- from the Sun the Earth's mean 1 tions round in mile?. dist. unity.] distance. Sun. Log. of ' times of revolution 1.944324 2.351610 2 562598 2.836942 3.121991 3.123190 3.138303 3.167300 3.179547 3.202700 3.226086 3.226610 3.636738 4.03171S 4.486953 4.779076 Sun Mercury . Venus .. . The Earth Mars Vesta . . . Iris 1 . Hebe 1 Flora f* AstreaJ . Juno 883000 3224 7687 7912 4189 238 > Unknown. 1420 Not well Q 60 known. Jl20 89170 79040 35000 35000 37 million 68 " 95 " 144 " 224,340,000 226 raiilion 230 " 240 246 " 253,600,001) 263,236,000 265 million 490 " 900 " 1800 2850 " 0.387098 0.723332 1.000000 1.52369-2 2.36120 2.37880 2.42190 252630 2.5895 2.66514 2.76910 2.77125 5.202776 9.538786 19182390 29.59 DAYS. 9.587818 1 87.969258 9.859306 224.700787 0.0(10000, 365.256383 0.182810 686979646 0.373100 1324.289 0.376384 1327.973 0384104 1375. nearly 0.402487 1469.76 0.413211 I5l2.nearly 0.425710 1594.721 " 0442334 1683.064 0.442725 1685162 0.716212 4332.584821 097947610759,219817 1 .282853 30686.8208 1.477121,60128 14 Ceres Pallas . . . Jupiter.. . Saturn . . . Uranus . . Neptune . TABLE III. ELEMENTS OF ORBITS FOR THE EPOCH OF 1850, JANUARY 1, MEAN NOON AT GREENWICH. Planets. Inclinati'u of orbits to ecliptic. Variation in 100 years. Long. of the ascending nodes. Variation in 100 years. Longitude of Perihelion. Variation in 100 years. Mean longi- tude at epoch. Mercury Venus. . Earth . . . Mars . . . Vesta... O ' " 7 18 3 23 26 1 51 6 7 8 29 13 2 53 H-18.2 4.6 0.2 12. O ' " 46 34 40 75 17 40 48 20 24 103 20 47 170 53 +51 +42 +26 O ' " 75 9 47 129 22 53 100 22 10 333 17 57 2.74 4 34 54 18 32 + 93 + 78 103 +110 157 C ' 327 17 9 243 58 4 100 47 1 182 9 30 113 28 12 165 17 38 10 37 17 80 47 56 147 25 41 1 3 10 Pallas . 34 37 44 172 42 38 121 30 13 327 31 24 Jupiter.. Saturn. . Uranus.. 1 18 42 2 29 29 46 27 22. 15. 3 98 55 19 112 22 54 73 12 +57 +-51 +24 11 56 90 7 168 14 47 + 95 +116 + 87 160 21 50 13 58 13 28 20 22 * Recently some thirty-two Astero Js have been discovered, by different iD- ervers. A table of twenty-two wi\ be found on page 55 of tables. We give the logarithms in the tables, that the data may be at hand to exercise the stu- dent cu Kepler's third law. TABLE III. TABULAR VIEW OF THE SOLAR SYSTEM. Names. Mass. Density. Gravity. Siderial. Rotation. Light and Heat. Mercury . . WlllTV 3.244 1.22 b. m. i. 24 5 28 6.680 Venus Earth 0.994 1 000 0.96 1.00 23 21 7 24 1.911 1 000 Mare ysstan 0.973 0.50 24 39 21 .431 Jupiter . . . TffiffT 0.232 2.70 9 55 50 .037 Saturn ssiff.y 0.132 1.25 10 29 17 .011 Uranus . . . T^iif 0.246 1.06 Unknown. .003 Sun Moon 1 0.256 0.665 28.19 0.18 25d. 12h. O m . 27*. 7h.43m. TABLE III. Planets. Eccentricities of orbits. Variation in 100 years. Motion in mean long, in 1 year of 365 days. Mean Daily Motion in longitude. Mercury... Venus Earth 0.20551494 0.00686074 01678357 + .000003868 .000062711 .000041630 ' " 53 43 3.6 224 47 29.7 14 19.5 O ' " 4 5 32.6 1 36 7.8 59 8.3 Mars 09330700 + .000090176 191 17 9.1 31 26.7 Vesta 08856000 4- 000004009 16 179 Juno . 25556000 13 33 7 Ceres . 07673780 000005830 12 494 Pallas 024199800 12 487 Jupiter Saturn Uranus 0.04816210 0.05615050 0.04661080 -+- .000159350 .000312402 .000025072 30 20 31.9 12 13 36.1 4 17 45.1 4 59.3 2 0.6 U 42.4 TABLE III. LUNAR PERIODS. d. Mean sidereal revolution, 27.321661418 Mean synodical revolution, 29.530588715 Mean revolution of nodes (retrograde), 6793.391080 Mean revolution of perigee (direct), 3232.575343 Mean inclination of orbit, 5 8' 48" Mean distance, in measure, of the equc*orial radius of the earth, .. 29.98217 Mean distance, in measure, of the mean rudius, 30.20000 TABLE IV SUN'S EPOCHS. Yeara. M. Long. Long. Perigee. I. II. III. N. 1. ' " 8. ' " 1846 9 8 45 8 9 8 17 17 124 673 897 379 1847 9 8 30 48 9 8 18 19 484 588 623 433 1848 B. 9 9 15 37 9 8 19 20 878 505 151 487 1849 99 1 17 9 8 20 22 2.S8 420 775 540 1850 9 8 46 58 9 8 21 23 598 336 400 594 1851 9 8 32 39 9 8 22 24 958 250 025 648 1852 B. 9 9 17 27 9 8 23 26 353 168 653 701 1853 9938 9 8 24 27 713 083 277 755 1854 9 8 48 48 9 8 25 29 073 998 902 809 1855 9 8 34 29 9 8 26 30 433 913 527 863 1856 B. 9 9 19 18 9 8 27 32 827 832 153 916 1857 9 9 4 58 9 8 28 34 187 746 779 970 1858 9 8 50 39 9 8 29 35 547 661 404 024 1859 9 8 36 19 9 8 30 37 907 576 029 078 1860 B 9 9 21 8 9 8 31 38 301 494 656 131 1861 9 9 6 49 9 8 32 39 661 409 281 185 1862 9 8 52 29 9 8 33 41 021 324 906 239 1863 9 8 38 10 9 8 34 42 381 239 530 292 1864 B. 9 9 22 58 9 8 35 44 775 157 157 346 1865 9 9 8 39 9 8 36 45 135 072 783 400 1866 9 8 54 20 9 8 37 47 495 985 408 453 1867 9 8 40 9 8 38 49 855 902 033 507 1868 B. 9 9 24 49 9 8 39 50 249 820 659 561 1869 9 9 10 30 9 8 40 52 609 734 285 615 1870 9 8 56 10 9 8 41 53 969 649 910 668 1882 9 9 1 41 9 8 54 10 391 638 416 313 1871 9 8 41 51 9 8 42 54 329 564 534 721 1872 B 9 9 26 39 9 8 43 56 723 481 161 774 1873 9 9 12 20 9 8 45 58 083 396 785 828 1874 9 8 58 1 9 8 47 443 311 410 881 1875 9 8 43 41 9 8 48 2 803 226 034 935 1876 B. 9 9 28 30 9 8 49 4 297 143 661 989 1877 9 9 14 10 9 8 50 5 657 058 286 042 1878 9 8 59 51 9 8 51 6 017 974 912 096 1879 9 8 45 32 9 8 52 7 377 889 537 150 IbSOB. 9 9 30 20 9 8 53 9 671 807 164 204 1881 9 9 16 1 9 8 54 10 031 722 790 257 1882 9 9 1 41 9 8 55 12 391 637 415 311 1883 9 8 47 22 9 8 56 13 751 552 040 364 1884 B. 9 9 32 10 9 8 57 15 145 469 666 418 1885 9 9 17 51 9 8 58 16 505 385 292 471 1886 9 9 3 32 9 8 59 17 865 300 918 525 1887 9 8 49 12 9 8 19 225 216 544 579 1888 B. 9 9 34 1 9 8 1 20 619 133 169 632 10 TABLE V. SUN'S MOTIONS FOR MONTHS. Months. Longitude. Per. I. II. III. N. T 1 Com. . . . Jan -jBis r* . I Com. . . . Feb 'jBis March s. o ' " 0000 11 29 52 1 33 18 29 34 10 1 28 9 11 5 5 10 966 47 13 993 997 78 7fi 148 998 53 51 01 4 4 9 April 2 28 42 30 15 42 226 154 13 May 3 28 16 40 20 59 301 206 18 4 28 49 58 26 110 379 259 22 July . 5 28 24 8 31 129 454 310 27 6 28 57 26 36 182 531 363 31 September. . . 7 29 30 44 41 233 609 416 36 October. . . 8 29 4 54 46 250 684 468 40 Novembf r . 9 29 38 12 52 300 762 521 45 10 29 12 22 57 313 837 572 49 TABLE VI. SUN'S HOURLY MOTION. ARGUMENT. Sun's Mean Anomaly. Os Is Us Ills IVs 2 25 2 25 2 24 2 24 V . 10 20 30 ' n 2 33 2 33 2 33 2 32 2 32 2 32 2 31 2 30 / n 2 30 2 29 2 29 2 28 2 28 2 27 2 26 2 25 2 24 2 23 2 23 2 23 30 20 10 XIs Xs IXs VIIIs VIIs Vis SUN'S SEMIDIAMETER. ARGUMENT. Sun's Mean Anomaly. Os Is Us Ills IVs VB o a / // / / / // / // i n o 16 18 16 15 16 9 16 1 15 53 15 48 30 10 16 18 16 14 16 7 15 58 15 51 15 46 20 20 16 17 16 12 16 4 15 56 15 49 15 46 10 30 16 15 16 9 16 1 15 53 15 48 15 45 XIj Xs IXs VIIIs vn* VI TABLE VII. SUN'S MOTIONS FOE DAYS AND IIOUBS. 11 j Days. Logitude. Per I. II. III N. Hours. Long. I. O i a , 1 000 1 2 28 1 2 59 8 34 3 2 2 4 56 3 3 1 58 17 68 5 3 3 7 23 4 4 2 57 25 101 8 5 4 9 51 ti 5 3 56 33 1 135 10 7 1 5 12 19 7 6 4 55 42 169 13 9 1 6 14 47 8 7 5 54 50 203 15 10 1 7 17 15 10 8 6 53 58 236 18 12 1 8 19 43 11 9 7 53 7 270 20 14 9 22 11 13 10 8 52 15 304 23 15 1 10 24 38 14 11 9 51 23 2 338 25 17 1 11 27 6 16 12 10 50 32 2 371 28 19 2 12 29 34 17 13 11 49 40 2 405 30 21 2 13 32 2 18 14 12 48 48 2 439 33 22 2 14 34 30 20 15 13 47 57 2 473 35 24 2 15 36 58 21 16 14 47 5 3 506 38 26 2 16 39 26 23 17 15 46 13 3 540 40 27 2 17 41 53 24 13 16 45 22 3 574 43 29 2 18 44 21 25 19 17 44 30 3 608 45 31 3 19 46 49 27 20 18 43 38 3 641 48 33 3 20 49 17 28 21 19 42 47 3 675 50 34 3 21 51 45 30 22 20 41 55 4 709 53 36 3 22 54 13 31 23 21 41 3 4 743 55 38 3 23 56 40 32 24 22 40 12 4 777 58 39 3 24 59 8 34 25 23 39 20 4 810 60 41 4 26 24 38 28 4 844 63 43 4 27 25 37 37 4 878 65 45 4 23 26 36 45 5 912 68 46 4 29 27 35 53 5 945 70 48 4 30 28 35 2 5 979 73 50 4 31 29 34 10 5 13 75 51 4 SONS MOTIONS FOR MINUTES. Min. Longitude. Min. Longitude. 1 5 10 15 20 25 30 2 12 25 37 49 1 2 1 14 30 35 40 45 50 55 6Q / a 1 16 1 26 1 39 1 51 2 3 2 16 2 28 12 TABLE VIII. EQUATIONS OP THE SUN^S CENTER. ARGUMENT. Sun's Mean Anomaly. Os Is Us Ills IV. Vs o O ' " Q t II O ' " Q t II o O ' n 1 59 30 2 58 15 3 40 27 3 54 50 3 38 21 2 56 9 1 2 1 33 300 3 41 25 3 54 47 3 37 18 2 54 25 2 2 3 37 3 1 44 3 42 21 3 54 41 3 36 14 2 52 40 3 2 5 40 3 3 27 3 43 15 3 54 33 3 35 8 2 50 54 4 2 7 43 359 3 44 8 3 54 23 3 34 1 2 49 8 5 2 9 46 3 6 49 3 44 58 3 54 11 3 32 51 . 2 47 20 6 2 11 49 3 8 28 3 45 47 3 53 57 3 31 41 2 45 32 7 2 13 51 3 10 6 3 46 33 3 53 41 3 30 28 2 43 43 8 2 15 54 3 11 43 3 47 17 3 53 23 3 29 14 2 41 53 9 2 17 56 3 13 18 3 48 3 53 3 3 27 58 2 40 3 10 2 19 57 3 14 51 3 48 40 3 52 40 3 26 41 2 38 11 11 2 21 58 3 16 24 3 49 18 3 52 16 3 25 22 2 36 19 12 2 23 59 3 17 54 3 49 55 3 51 50 3 24 2 2 34 27 13 2 25 59 2 19 24 3 50 29 3 51 21 3 22 40 2 32 34 14 2 27 59 3 20 51 3 51 1 3 50 51 3 21 17 2 30 40 15 2 29 58 3 22 18 3 51 31 3 50 18 3 19 52 2 28 46 16 2 31 57 3 23 42 3 51 59 3 49 44 3 18 26 2 26 52 17 2 33 55 3 25 5 3 52 25 3 49 7 3 16 58 2 24 56 18 2 35 52 3 26 26 3 52 49 3 48 29 3 15 30 2 23 19 2 37 49 3 27 46 3 53 10 3 47 49 3 14 2 21 4 30 2 39 45 3 29 4 3 53 30 3 47 7 3 12 28 2 19 8 21 2 41 40 3 30 24 3 53 47 3 46 22 3 10 55 2 17 11 22 2 43 34 3 31 35 3 54 3 3 45 36 3 9 22 2 15 14 23 2 45 28 3 32 48 3 54 16 3 44 48 3 7 46 2 13 16 24 2 47 20 3 33 59 3 54 27 3 43 58 3 6 10 2 11 19 25 2 49 12 3 35 8 3 54 36 3 43 7 3 4 33 2 9 21 26 2 51 2 3 36 16 3 54 43 3 42 13 3 2 54 2 7 23 27 2 52 52 3 37 21 3 54 48 3 41 18 3 1 14 2 5 25 28 2 54 41 3 38 25 3 54 51 3 40 21 2 59 33 2 3 27 29 2 56 28 3 39 27 3 54 52 3 39 22 2 57 52 2 1 25 30 2 58 15 3 40 37 3 54 50 3 38 91 2 56 9 1 59 30 i TABLE VIII. 13 EQUATIONS OF THE SUN 8 CENTER. ARGUMENT. Sun's Mean Anomaly. Vis VIIs VIIIs IXs Xs XIs o o ' " O ' " O ' " O ' " ' " Q 1 II 1 59 30 1 2 51 20 39 4 10 18 33 45 1 1 57 32 1 1 8 19 38 048 19 33 2 32 2 1 55 33 59 27 18 39 049 20 35 4 19 3 1 53 35 57 46 17 42 4 12 21 39 6 8 4 1 51 37 56 6 16 47 4 17 22 44 7 58 5 1 49 39 54 27 15 53 4 24 23 52 9 48 6 1 47 41 52 47 15 2 4 33 25 1 11 40 7 1 45 44 51 14 14 12 4 44 26 12 13 32 8 1 43 46 49 38 13 24 4 57 27 25 15 26 9 1 41 49 48 5 12 38 5 13 28 40 17 20 10 1 39 52 46 32 11 53 5 30 29 56 19 15 11 1 37 56 45 11 11 5 50 31 14 21 11 12 1 36 43 30 10 31 6 11 32 34 23 8 13 34 4 42 1 9 53 6 35 33 55 25 5 14 32 9 40 34 9 16 7 1 35 18 27 3 15 30 14 39 8 8 42 7 29 36 42 29 2 16 28 20 37 43 089 7 59 38 9 1 31 1 17 26 26 36 20 7 39 8 31 39 36 1 33 1 18 24 33 34 58 7 10 095 41 9 1 35 1 19 22 41 33 38 6 44 9 42 42 36 1 37 1 20 20 49 32 19 6 20 10 20 44 9 1 39 3 21 18 57 31 2 5 57 11 45 42 41 4 22 17 7 29 46 5 37 11 43 47 17 43 6 23 15 17 28 32 5 19 12 27 48 54 45 9 24 13 28 27 19 053 13 13 50 32 47 11 25 11 40 26 9 4 49 14 2 52 11 49 14 26 9 52 24 59 4 37 14 52 53 51 51 17 27 8 6 23 52 4 27 15 45 55 33 53 SO 28 6 20 22 46 4 19 16 39 57 16 1 55 23 39 4 35 21 41 4 13 17 35 59 1 57 27 30 1 2 51 20 39 4 10 18 33 1 45 1 59 30 TABLE IX. SMALL EQUATIONS OF THE SUN*S LONGITUDE Arg. I II. III. Arg. I. II. III 10 10 II 10 500 10 10 10 10 10 11 9 510 10 10 9 20 11 11 9 520 9 10 8 30 11 12 8 530 9 10 7 40 11 13 8 540 9 10 7 50 12 14 7 550 8 10 6 60 12 14 7 563 8 9 5 70 12 15 7 570 8 9 4 80 13 15 7 580 7 9 3 90 13 16 7 590 7 9 3 103 13 16 7 600 7 9 2 110 14 17 7 610 6 8 1 120 14 17 7 620 6 8 1 130 14 18 8 630 6 8 1 140 15 18 8 640 5 7 150 15 18 9 650 5 7 C 160 15 18 9 660 5 6 170 15 18 10 670 5 6 1 180 15 18 10 680 5 6 1 190 16 18 11 690 4 5 2 2)0 16 18 11 700 4 5 2 210 16 18 i2 710 4 4 3 220 16 18 12 720 4 4 3 230 16 18 13 730 4 4 4 240 16 17 14 740 4 3 5 250 16 17 14 750 4 3 6 260 16 17 15 760 4 3 6 270 16 16 16 770 4 2 7 280 16 16 17 7fO 4 2 8 290 16 16 17 790 4 2 8 300 16 15 18 800 4 2 9 310 16 15 18 810 4 2 9 320 15 14 19 820 5 2 10 330 15 14 19 830 5 2 10 340 15 14 20 840 5 2 11 350 15 13 20 850 5 2 11 360 15 13 20 860 5 2 12 370 14 12 19 H70 6 2 12 380 14 12 19 880 6 3 13 390 14 12 19 8!K) 6 3 13 400 13 11 18 900 7 4 13 410 13 11 17 910 7 4 13 420 13 11 17 920 7 5 13 430 12 11 16 930 8 5 13 440 12 11 15 940 P 6 13 450 12 10 14 950 8 6 13 463 11 10 13 96!) 9 7 12 470 11 10 13 970 9 8 12 480 11 10 12 983 9 9 11 490 10 10 11 990 10 11 500 10 10 10 1000 10 16 10 TABLE X NUTATIONS. ARGUMENT. Supplement of the Node, or N. 15 N. Long. R. Asc. Obliq. N. Long. R. Asc. Obliq. a pf n 4- o 4- 4- 10 500 10 20 2 2 10 520 2 2 9 40 4 4 9 540 4 4 9 60 7 6 9 560 7 6 9 80 9 8 8 580 9 8 8 100 I jo 600 11 10 8 120 12 11 7 620 12 11 7 140 14 13 6 640 14 13 6 160 15 14 5 660 15 14 5 180 16 15 4 680 16 15 4 200 4- 17 4- 16 t 3 700 17 16 3 220 r !8 16 2 720 18 16 2 240 18 16 1 740 18 16 1 260 18 16 1 760 18 16 + * 280 18 16 2 780 18 16 2 300 4- 17 4- 16 _ 3 800 17 16 4" 3 320 16 ^*5 4 820 16 15 4 340 15 14 5 840 15 14 5 366 14 13 6 860 14 13 6 380 12 11 7 880 12 11 7 400 4. 10 8 900 11 10 + 8 420 9 ~ 8 8 920 9 8 8 440 7 6 9 940 7 6 9 460 4 4 9 960 4 4 9 480 2 2 10 980 2 2 10 500 ~f" 10 1000 15 TABLE XI. EARTH'S RADIUS VECTOR. ARGUMENT. Sun's Mean Anomaly. Os Is Us Ills IVs V GO 0.98313 0.98545 0.99173 .00018 1.00850 1.01450 300 2 0.98314 0.98576 0.99225 1.00077 1.00899 1.01477 28 4 0.98317 0.98608 0.99278 1.00135 1/0947 1.01503 26 6 0.98322 0.98643 0.99331 .10193 1.00994 1.01527 24 8 0.98330 0.98679 0.99386 1.00251 1.01040 1.01549 22 10 0.98339 0.98717 0.99441 .00308 1.01084 1.01569 20 12 0.98350 0.98756 0.99497 .00366 1.01128 1.01588 18 14 0.98364 0.98797 0.99554 .00422 1.01170 1.01604 16 16 0.98380 0.98840 0.99611 .00478 1.01210 1.01619 14 18 0.98397 0.98883 0.99668 .00534 1.01249 1.01632 12 20 0.98417 0.98929 0.99726 1.00588 1.01286 1.01643 10 22 0.98439 0.98975 0.99784 1.00642 1.01322 1.01652 8 24 0.98462 0.99023 0.99843 1.00695 1.01357 1.01659 6 26 0.98486 0.99072 0.99901 1.00748 1.01389 1.01663 4 28 0.98515 0.99122 0.99960 1.00799 1.01420 1.01666 2 30 0.98545 0.99173 1.00018 1.00850 1.01450 1.01667 XIi Xs IXs vim vm VI. TABLE XI. MEAN NEW MOONS AND ARGUMENTS IN JANUARY. Mean New Moon in January. I. II. in. IV. N. A D D, H. M. 1836 B. 17 10 32 0469 1246 17 08 669 1837 5 19 20 0171 9S52 00 97 692 1838 24 16 53 0681 9175 99 85 799 1839 14 1 42 0383 7780 82 74 822 1840 B. 3 10 30 0085 6386 65 63 844 1841, 21 8 3 0595 5709 63 51 951 1842 10 16 51 0297 4314 46 40 974 1843 29 14 24 0807 3637 44 28 081 1844 B. 18 23 13 0509 2243 28 17 104 1845 7 8 1 0211 0848 11 06 126 1846 26 5 34 0721 0171 09 94 234 1847 15 14 22 0423 8777 92 84 256 1848 B. 4 23 11 01-25 7382 75 73 278 1849 22 20 43 0635 6705 73 61 386 1850 12 5 32 0337 53il 56 50 408 1851 1 14 21 0038 3916 40 39 431 1852 B. 20 11 53 0549 3239 38 27 538 1853 8 20 42 0251 1845 21 16 560 1854 27 18 14 0761 1168 19 04 668 1855 17 3 3 0463 9773 02 93 690 1856 B 6 11 51 0164 8379 85 82 713 1857 24 9 24 0675 7702 84 70 820 1858 13 18 13 0376 6307 67 59 843 1859 3 3 1 0078 4913 50 48 865 1860 B. 22 34 0588 4236 48 36 972 1861 10 9 22 0290 2840 31 25 995 1862 29 6 55 0800 2163 30 14 102 1863 18 15 44 0504 0769 13 03 125 1864 B. 8 32 0204 9374 96 92 147 1865 25 22 5 0714 8698 94 80 256 1866 15 6 53 0416 7303 77 69 277 1867 4 15 42 0118 5909 60 58 299 1868 B. 23 13 14 0628 5231 59 46 407 1869 11 22 3 0330 3837 42 35 429 1870 1 6 51 0032 2442 25 24 451 1871 20 4 24 0542 1765 23 12 559 1872 B. 8 13 13 0244 0371 05 01 581 1873 27 10 46 0754 9694 03 89 689, 1874 17 19 35 0456 8300 86 78 711 1875 7 4 24 0158 6906 69 67 733 1876 B. 26 1 57 0668 6229 67 55 841 1877 14 10 49 0370 4835 50 44 863 1678 3 18 38 0072 3441 33 23 885 1879 22 6 11 0582 2764 31 21 993 1880 B. il 15 0284 1370 14 10 015 TABLE XII. IT MEAN LUNATIONS AND CHANGES OF THE ARGUMENTS. Num. Lunations. I. II. III. IV. N. d. h. m. i/ 14 18 22 404 5359 58 50 43 j 29 12 44 808 717 15 99 85 2 59 1 28 1617 1434 31 98 170 3 88 14 12 2425 2151 46 97 256 4 118 2 56 3234 2869 61 96 341 5 147 15 40 4042 3586 76 95 425 6 177 4 24 4851 4303 92 95 511 7 206 17 8 5659 5020 7 94 596 8 236 5 52 6468 5737 22 93 682 9 265 18 36 7276 6454 37 92 767 10 295 7 20 8085 7171 53 91 852 11 324 20 5 8893 7889 68 90 937 IS 354 8 49 9702 8606 83 89 22 U 383 21 33 510 9323 93 88 108 TABLE XIII. KCMBER OP DAYS FROM THE COMMENCEMENT OF THE YEAR TO THE FIRST OF EACH MONTH. TABLE XIV. Months. Com. Bis. Days. Days. January, . . February. . 31 31 March .... 59 60 April 90 91 May. . , 120 121 151 152 July . . 181 182 August.. . . 212 213 September. 243 244 October... 273 274 November. 304 305 December . 334 335 A ff H. Par. m S.D. H.Mo. A iF i " / it 60 29 16 29 36 48 10000 250 60 26 16 26 36 44 9750 500 60 17 16 25 36 19 9500 750 60 16 21 36 8 9250 1000 59 47 16 17 35 48 9000 1250 59 24 16 11 35 28 750 1500 58 56 16 3 34 57 8500 1750 58 30 15 56 34 34 8250 2000 58 7 15 50 33 58 8000 2250 57 37 15 42 33 32 7750 2500 57 1 15 31 32 42 7500 2750 56 32 15 23 32 9 7250 3000 56 2 15 16 31 36 7000 3250 55 40 15 10 31 13 6750 3500 55 22 15 7 30 52 6500 3750 55 12 15 3 30 29 6250 4000 54 51 14 56 30 7 6000 4250 54 39 14 54 29 55 5750 4500 54 26 14 50 29 43 5500 4750 54 18 14 48 29 37 5250 5000 54 13 14 45 29 35 5000 18 TABLE XV. EQUATIONS FOR NEW AND FULL MOON. Arg. I. II. Arg. 1. II. Arg. III. IV. Arg. h. m. h. m. h. m. h. m. m. m. 4 20 10 10 5000 4 20 10 10 25 3 31 25 100 4 36 9 36 5100 4 5 10 50 26 3 31 24 200 4 52 9 2 5200 3 49 11 30 27 3 30 23 300 5 8 8 28 5300 3 34 12 9 28 3 30 22 400 5 24 7 55 5400 3 19 12 48 29 3 30 21 500 5 40 7 22 5500 3 4 13 26 30 3 30 20 600 5 55 6 49 5600 2 49 14 3 31 3 30 19 700 6 10 6 17 5700 2 35 14 39 32 4 20 18 800 6 24 5 46 5800 2 21 15 13 33 4 29 17 900 6 38 5 15 5900 2 8 15 46 34 4 29 16 1000 6 51 4 46 6000 55 16 18 35 4 29 15 1100 7 4 4 17 6100 42 16 48 36 5 28 14 1200 7 15 3 50 6200 31 17 16 37 5 28 13 1300 7 27 3 24 6300 19 17 42 38 5 27 12 uoa 7 37 2 59 6400 9 18 6 39 5 27 11 1500 7 47 2 35 6500 59 18 28 40 6 26 10 1600 7 55 2 14 6600 50 18 48 41 6 26 9 1700 8 3 1 53 6700 42 19 6 42 7 25 8 1800 8 10 1 35 6800 34 19 21 43 7 25 7 1900 8 16 1 18 6900 28 19 33 44 7 24 6 8000 8 21 1 3 7000 22 19 44 45 8 23 5 2100 8 25 51 7100 17 19 52 46 8 23 4 2200 8 29 40 7200 14 19 57 47 9 22 3 2300 8 31 32 7300 11 20 48 9 21 2 2400 8 32 25 7400 9 20 1 49 10 21 1 2500 8 32 21 7500 8 19 59 50 10 20 2600 8 31 19 7600 8 19 55 51 10 19 99 2700 8 29 20 7700 9 19 48 52 11 19 98 2800 8 26 23 7800 11 19 40 53 11 18 7 2900 8 23 28 7900 15 19 29 54 12 17 96 3000 8 18 36 8000 19 19 17 55 12 17 95 3100 8 12 47 8100 24 19 2 56 13 16 94 3200 8 6 0^59 8200 30 18 45 ! 57 13 15 93 3300 7 58 1 14 8300 37 18 27 58 13 15 92 3400 7 50 1 32 8400 45 18 6 59 14 14 91 3500 7 41 1 52 8500 53 17 45 60 14 14 90 3600 7 31 2 14 8600 3 17 21 61 15 13 89 3700 7 21 2 38 8700 13 16 56 62 15 13 88 3800 7 9 3 4 8800 25 16 30 63 15 12 87 3900 6 58 3 32 8900 36 16 3 64 15 12 86 4000 6 45 4 2 9000 49 15 34 65 16 11 85 4100 6 32 4 34 9100 2 2 15 5 66 16 11 84 4200 6 19 5 7 9200 2 16 14 34 67 16 11 83 4300 6 5 5 41 9300 2 30 14 3 68 16 10 82 4400 5 51 6 17 9400 2 45 13 31 69 17 10 81 4500 5 36 6 54 9500 3 12 58 70 17 10 80 4600 5 21 7 32 9600 3 16 12 25 71 17 10 79 4700 5 6 8 11 9700 3 32 11 52 72 17 10 78 4800 4 51 8 50 9800 3 48 11 18 73 17 10 77 4900 4 35 9 30 9900 4 4 10 44 74 17 9 76 , 5000 4 20 10 10 10000 4 20 I 10 10 75 17 9 75 TABLE E. " H K 4 I 20 TABLE XVI. MOON'S EPOCHS. Years 1 2 3 4 5 6 7 8 1846 0013 2475 3275 1688 0773 4880 3179 0800 9542 1847 0006 9683 2941 6432 3245 0678 4239 3257 8406 1848B. 0026 7542 3646 1463 6052 6847 5358 6106 7295 184) 001 Jl 4750 3312 6207 8524 2644 6418 8563 6158 1850 0012 1958 2978 0951 0995 8442 7479 1020 5022 1851 0005 9167 2644 5695 3467 4239 8539 3477 3885 1852B. 0025 7025 3350 0726 6274 0408 9658 6326 2774 1853 0018 4233 3016 5469 8746 6206 0718 8782 1637 1854 0011 1442 2681 0213 1217 2003 1778 1240 0501 1855 0004 8650 2347 4957 3689 7801 2839 3697 9365 1856 B. 0024 6509 3053 9988 6496 3970 3957 6446 8254 1857 0017 3717 2719 4732 8968 9767 5018 9002 7117 1 1858 0010 0925 2385 9476 1439 5565 6078 1460 5981 1859 0003 8134 2051 4220 3911 1362 7139 3917 4845 1860B. 0023 5992 2756 9551 6718 7531 8257 6765 3734 1861 1862 0016 0009 3200 0409 2423 2088 3995 8739 9190 1661 3329 9126 9317 0378 9222 1679 2597 1461 1863 0002 7617 1754 3483 4133 4923 1438 4137 0324 1864 B. 0022 5476 2460 8514 6941 1093 2557 6984 9212 1865 0015 2684 2126 3257 9412 6890 3617 9442 8076 1866 0008 9893 1792 8001 1883 26S7 4678 1899 6940 1867 0001 7101 1457 2745 4355 8485 5738 4357 5804 1868B. 0021 4959 2163 7776 7163 4654 6857 7204 4692 1869 OC14 2168 1829 2520 9634 0452 7917 9662 3556 1870 0007 9376 1495 7264 2105 6249 8978 2119 2420 1871 0000 6584 1161 2008 4576 2046 0039 4576 1284 1872 B. 0020 4432 1867 7039 7383 8215 1158 7423 0172 1873 0013 1640 1533 1783 9854 4012 2239 9880 9036 1874 0006 884S 1199 6527 2325 9809 3300 2337 7900 1875 9999 6056 0865 1271 4796 5606 4361 4794 6764 1876B. 0019 3914 1571 6292 7603 177o 5480 7641 5652 1877 0012 1122 1247 1036 0074 7572 6541 0098 4516 1878 0005 8330 0913 5780 2545 3369 7602 2555 3380 1879 9998 5538 0579 0524 5016 9166 8663 5012 2244 1880 B. 0018 3396 1285 5545 7823 5335 9782 7859 1132 1881 0011 0604 0951 0289 0294 1132 0843 0316 9996 1882 0004 7812 0617 5033 2765 6929 1904 2873 8860 1883 9997 5020 0283 9777 5236 2726 2965 5330 7724 1884 B. 0017 2878 0989 4798 8043 8895 4084 8177 6612 1885 0010 0086 0655 9542 0514 4692 5145 0634 5476 1886 0003 7294 0321 4286 2985 0489 6206 3091 4340 1887 9996 4502 9987 9030 5456 6286 7267 5548 3204 1888B. 16 2360 0693 4051 8263 2455 8386 8395 2092 1889 09 9568 0359 8795 0734 8252 9447 0852 0956 . 1890 0002 6776 0025 3539 3205 1 4049 0508 3309 9820 TABLE XVI MOON'S EPOCHS. 21 Years. 10 11 12 13 14 15 16 17 18 19 20 1846 203 123 250 171 419 760 126 396 167 379 204 1847 810 484 970 644 613 9iJl 486 749 643 433 371 1848 B. 486 876 759 151 905 072 881 143 144 4r<7 539 1849 093 237 479 624 099 212 241 496 619 540 705 1850 700 597 199 097 293 352 600 848 094 594 871 1851 306 958 918 570 487 493 960 201 569 648 038 1852 B. 983 350 707 077 780 664 355 595 070 701 206 1853 589 711 427 550 974 804 715 948 545 755 372 1854 196 072 147 023 168 944 074 300 020 809 539 1855 802 432 866 496 361 085 434 653 495 863 705 1856 B. 479 824 656 003 654 256 829 047 996 916 873 1857 086 185 375 476 848 396 189 400 471 970 039 1858 6'J2 546 095 949 042 537 548 752 947 024 206 1859 299 907 814 422 236 677 908 105 422 078 372 1860B, 975 298 604 929 529 848 303 499 923 131 540 1861 581 659 323 402 723 988 662 852 398 185 706 1862 187 020 042 875 916 129 021 204 873 239 873 1863 794 381 761 348 110 269 381 557 348 292 039 1864 B. 470 773 551 855 403 440 777 951 849 346 207 1865 077 134 271 328 597 580 136 304 324 400 373 1866 684 494 990 801 791 721 495 657 799 453 540 1867 290 855 710 274 985 861 855 009 274 507 707 1868B, 967 247 500 781 277 032 951 404 775 561 874 1869 573 6D8 219 254 471 172 610 756 251 615 040 1870 180 96t 938 737 665 313 969 109 726 668 207 1871 787 328 659 200 859 554 328 562 201 721 374 1872 B. 464 720 549 707 151 725 724 957 702 785 531 1873 071 080 269 180 345 966 083 410 177 838 698 1874 678 440 989 653 539 205 442 863 642 891 865 1875 285 800 709 126 733 446 801 316 117 944 032 1876B. 962 192 599 633 025 617 197 711 618 008 199 1877 569 552 319 106 219 858 556 164 093 061 366 1878 176 912 039 579 413 099 915 617 568 114 533 1879 783 272 759 052 607 340 274 070 043 167 700 1880 B. 460 664 649 559 899 511 670 465 544 231 867 1881 067 024 369 032 093 752 029 918 019 284 034 1882 674 384 089 505 287 993 388 371 494 337 201 1883 281 744 809 978 4S1 234 747 824 969 390 368 1884 B. 958 136 699 485 773 405 143 219 470 454 5H5 1865 565 496 419 958 967 646 502 672 945 507 702 1886 172 856 139 431 161 887 86i 125 420 560 869 1887 779 216 859 904 355 128 320 578 895 613 036 1888 B. 456 608 749 411 647 299 716 973 396 677 203 1889 063 968 469 884 841 540 075 426 871 730 370 1890 670 328 189, 357 035 781 434 879 346 783 537 TABLE XVII. MOON'S MOTIONS FOR MONTHS. Months. 1 2 3 4 5 6 7 8 9 i- 1 Com. 0000 0000 0000 0000 0000 0000 0000 0000 0000 Jan ' J Bis. 9973 9350 8960 9713 9664 9628 9942 9610 9976 rf , 1 Com. Feb 'jBis. 849 851 146 9497 2246 12;5 88H6 8609 402 66 1533 1161 1789 1731 2099 1709 753 729 March 1615 8343 1371 6931 9797 1951 3404 3027 1433 April.. 2464 8490 3616 5827 199 3484 5193 5126 2186 May 3285 7986 4J-22 4436 265 4646 6924 680 5 2914 4134 8133 7067 3332 666 6179 8713 8934 S667 July. . . 4955 7629 8273 1942 732 7341 444 643 4:i96 August... . 5804 7776 518 838 1134 8874 2233 2742 5148 September . 6653 7922 2764 9734 1536 408 4021 4842 5901 October 7474 7419 3969 8343 1602 1569 5752 655fr 6630 November.. 8323 7565 6215 7239 2004 3102 7541 8649 7382 December. . 9144 7062 7420 5848 2070 4264 9272 358 8111 TABLE XVIII. MOON'S MOTIONS FOR DAYS. Days. 1 2 3 4 5 6 7 8 9 1 0000 0000 0000 0000 0000 0000 0000 0000 0000 2 27 659 1040 287 336 372 58 390 24 3 55 1300 2080 574 671 744 115 781 49 4 82 1950 3121 861 1007 1116 173 1171 73 5 109 2600 4161 1148 1342 1488 231 1561 97 6 137 3249 5291 1435 1678 1860 289 1952 121 7 164 3899 6241 1722 2013 2232 346 2342 146 8 192 4549 7281 2009 2349 2604 404 2732 170 9 219 5199 8321 2236 2684 2976 462 3122 194 10 246 5849 9362 2583 3020 3348 519 3513 219 11 274 6499 402 2870 3355 3720 577 3903 243 12 301 7149 ~1442 3157 3691 4093 635 4293 267 13 328 7799 2482 3444 4026 4465 692 4684 291 14 356 8449 3522 3731 4362 4837 750 5074 316 15 383 9098 4563 4018 4698 5209 808 5464 340 16 411 9748 5603 4305 5033 5581 866 5854 364 17 438 398 6643 4592 5369 5953 923 6245 389 18 465 1048 7683 4878 5704 6325 981 6635 413 19 493 1698 8723 5165 6040 6697 1039 7025 437 20 520 2348 9763 5452 6375 7069 1096 7416 461 21 548 2998 804 5739 6711 7441 1154 7806 486 22 575 3648 1844 6026 7046 7813 1212 8196 510 23 602 4298 2884 6313 7382 8185 1269 8586 534 24 630 4947 3924 6600 7717 8557 1327 8977 559 25 657 5597 4964 6887 8053 8929 1385 9367 583 26 684 6247 6005 7174 8389 9301 1443 9757 607 27 712 6897 7045 7461 8724 9673 1500 148 631 28 739 7547 8085 7748 9060 45 1558 538 656 29 767 8197 9125 8035 9395 417 1616 928 680 30 794 8847 165 8322 9731 789 1673 1319 704 31 821 9497 1205 8609 66 1161 1731 1709 729 TABLE XVII. MOON'S MOTIONS FOB MONTHS. Months. 10 11 12 13 14 15 16 17 18 19 20 J-in ] Com - Jtlil - } Hrs. pi - \ Com. 1(>b -jBis. March . . . 000 930 175 105 IV) 000 969 9H5 934 836 000 930 184 114 157 000 966 59 25 16 000 901 74 975 F51 000 969 946 916 8U1 000 963 135 98 159 000 958 304 262 482 000 974 805 779 539 000 000 5 5 9 000 000 14 14 27 April Miy June July. .. H4 419 593 6M8 801 735 700 634 342 556 640 754 76 101 160 185 925 899 973 948 747 663 6)9 525 294 392 527 625 786 47 351 613 330 11 92) 699 13 18 22 97 41 55 *>9 83 August. .. . September . October November.. December. . 873 48 152 327 432 599 563 497 462 396 938 123 237 421 535 245 304 329 388 414 22 96 71 145 120 471 417 333 279 194 759 894 992 127 225 917 221 4t<3 787 49 503 30* 81 89* 670 31 36 4v 97 111 125 139 353 TABLE XVIII. MOON'S MOTIONS FOR DAYS. P,jys. 10 11 12 13 14 15 16 17 18 19 %> 1 090 000 000 090 000 000 000 090 000 000 000 2 70 31 70 34 99 31 37 42 26 3 149 62 141 68 198 61 73 84 52 1 4 210 93 211 103 297 92 110 126 78 1 5 281 125 282 137 397 122 146 168 104 1 2 6 351 156 352 171 496 153 183 210 130 1 2 7 421 187 423 205 595 183 229 252 156 1 3 8 491 218 493 239 694 214 256 294 Ib2 1 3 9 561 249 564 273 793 244 233 336 2<)8 1 4 10 6-31 28!) 634 398 892 275 329 379 24 1 4 11 702 311 705 342 992 305 366 421 260 1 5 12 772 342 775 376 91 336 493 463 2-6 2 5 13 842 374 845 410 190 366 4^9 505 312 2 5 14 912 495 916 444 289 397 476 547 337 2 G 15 92 436 986 478 388 427 512 589 363 2 6 16 52 467 57 513 487 458 549 631 389 2 7 17 122 498 127 547 5S7 488 586 673 415 2 7 18 193 529 198 581 6*6 519 622 715 441 2 8 19 263 569 263 615 785 549 659 757 467 3 8 21 333 591 339 649 884 580 695 799 493 3 9 21 4)3 623 409 633 983 611 722 841 517 3 9 22 473 634 480 718 82 641 769 883 545 3 10 23 543 685 550 752 182 672 805 925 571 3 10 24 614 716 621 786 281 702 842 967 597 3 11 gr 6S4 747 691 829 380 733 878 9 623 4 11 26 754 778 762 854 479 7H 915 52 649 4 11 27 824 8!)9 832 888 578 794 952 94 675 4 12 28 894 840 903 923 677 824 9h'8 136 701 4 12 29 964 872 973 957 777 855 25 178 707 4 13 30 34 903 43 991 876 885 61 220 753 4 13 31 105 934 114 25 975 916 98 262 779 4 14 _J 24 TABLE XIX. MOON'S MOTIONS FOB HOURS. Hours. 1 2 3 4 5 6 7 8 9 1 1 27 43 12 14 16 2 16 1 2 2 54 87 24 28 31 5 33 2 3 3 81 130 36 42 47 7 49 3 4 5 108 173 48 56 62 10 65 4 5 6 135 217 60 70 78 12 81 5 6 7 162 260 72 84 93 14 98 6 7 8 190 303 84 98 109 17 114 7 8 9 217 347 96 112 124 19 130 8 9 10 244 390 JOS 126 140 22 146 9 10 11 271 433 120 140 155 24 163 10 11 12 298 477 131 154 171 26 179 11 12 14 325 520 143 168 186 29 195 12 13 15 352 563 155 182 202 31 211 13 14 16 379 607 167 196 217 34 228 14 15 17 406 650 179 210 233 36 244 15 16 18 433 693 191 224 248 38 260 16 17 19 460 737 203 238 264 41 276 17 18 20 487 780 215 252 279 43 293 18 19 22 515 823 227 266 295 46 309 19 20 23 542 867 239 280 310 48 325 20 21 24 569 910 251 294 326 50 341 21 22 25 596 953 263 308 341 53 358 22 23 26 623 997 275 322 357 55 374 33 24 27 650 1040 287 336 372 58 390 24 TABLE XIX. MOON'S MOTIONS FOR MINUTES. Min. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 5 2 4 1 1 1 1 10 5 7 2 2 3 3 1 15 7 11 3 3 4 1 4 1 1 1 20 9 14 4 5 5 1 5 1 1 1 25 11 18 5 6 6 1 7 1 1 1 2 30 14 22 6 7 8 1 8 1 1 3 35 16 25 7 8 9 1 10 2 2 2 40 18 29 8 9 10 2 11 2 2 3 45 20 32 9 10 12 2 12 2 2 3 50 23 36 10 11 13 2 13 2 2 3 55 25 40 11 13 14 2 15 3 3 4 CO 27 43 12 14 15 2 16 3 3 4 TABLES. 26 HELIOCENTRIC LONGITUDES, ETC. OF THE PLANET TENUS, AT THE TIMES Of THE NEXT TWO TKANSITS OVER THE SUN'S DISC. The subject matter of this table is connected with Chapter IX, page 119. Times. Hel. Long, from true Equinox. Hel. Lat. Rad. Vec. 1874, Dec. 8th, at 12h. 16h. 20h. 76 41' 36.6" 76 57 44.1 77 13 51.5 4' 6.3" N. 5 3.5 6 1.0 0.7203632 0.7203449 0.7203268 1882, Dec. 6th, at noon. 4h. 8h. 74 12 55.7 74 29 2.5 74 45 9.7 4 58.1 S. 4 0.8 3 3.5 0.7205502 0.7205315 0.7205127 DIP OF THE HORIZON. For the principle of computing the dip of the horizon see text-note, page 54. Might fe'et. Dip. T feet. Dip. 1 1' 1" 16 4' 3" 2 1 26 17 4 11 3 1 45 18 4 18 4 2 2 19 4 25 5 2 16 20 4 32 6 2 29 21 4 39 7 2 41 22 4 45 8 2 52 23 4 52 9 3 2 24 4 58 10 3 12 25 5 4 11 3 22 26 5 10 12 3 31 28 5 22 13 3 39 30 5 33 14 3 48 35 6 1 15 3 56 40 6 25 TTN'S SEMIDIAMETER FOR EVERY TENTH DAT OF THE TEAK Days. 1 11 21 Jan. / n 16 18 16 17 16 17 July. 15 46 15 46 15 46 Days. 1 11 21 April. / // 16 1 15 58 15 55 Oct. / // 16 1 16 3 16 7 1 11 21 Feb. 16 15 16 13 16 11 August. 15 47 15 49 15 51 1 11 21 May. 15 53 15 51 15 49 Nor. 16 9 16 12 16 14 1 11 21 March. 16 10 16 7 16 4 Sept 15 53 15 56 15 58 1 11 21 June. 15 48 15 46 15 46 Dec. 16 16 16 17 16 18 26 TABLE XX. MOON'S EPOCHS. Yeure. Erection. Anomaly. Variation. Longitude. 1846 1847 1848 B. 1849 1850 s o ' 2 45 6 7 21 16 35 1 23 7 5 7 13 38 35 1 4 10 4 s ' " 26 21 2 3 25 4 23 7 6 51 37 10 5 34 57 1 4 18 18 s o " 1 5 48 4 5 15 25 29 10 7 14 21 2 16 51 46 6 26 29 11 8 ' " 10 15 48 23 2 25 11 28 7 17 45 8 11 27 8 14 4 6 31 20 1851 1852 B. 1853 1854 1855 6 24 41 35 26 32 5 6 17 3 34 7 35 4 5 23 6 33 4 3 1 38 7 14 48 53 19 13 32 13 1 12 15 34 4 10 58 54 11 6 6 36 3 27 55 29 8 7 32 53 17 10 19 4 26 47 43 8 15 54 25 1 8 23 6 5 17 51 11 9 27 14 17 2 6 37 22 1856 B. 1957 1858 1859 1860B. 11 29 57 3 5 20 28 33 11 11 02 5 1 31 33 11 3 22 3 7 22 46 9 10 21 29 29 1 20 12 50 4 18 56 10 8 43 25 9 18 36 36 1 28 14 1 6 7 51 26 10 17 28 52 3 9 17 44 6 29 11 3 11 8 34 9 3 17 57 14 7 27 20 20 19 54 1861 1862 1863 1864 B. 1865 4 23 53 33 10 14 25 3 4 4 56 33 10 6 47 2 3 27 18 32 10 29 26 45 1 28 10 6 4 26 53 27 8 8 43 41 11 7 24 2 7 18 55 9 11 28 32 34 4 8 10 8 29 58 51 1 9 36 17 4 29 17 6 9 8 40 12 1 18 3 18 6 10 36 58 10 20 4 1866 1867 1868 B. 1869 1870 9 17 59 2 3 8 21 32 9 10 J2 2 3 43 33 8 21 15 2 2 6 7 23 5 4 50 43 8 16 37 58 11 15 21 19 2 14 4 43 5 19 13 42 9 28 51 8 2 20 40 7 17 25 11 9 54 50 2 29 23 10 7 8 46 15 1 19 56 4 10 43 2 8 20 6 8 1871 1872 B. 1873 1874 1875 2 11 45 31 8 2 17 2 4 7,31 7 24 39 1 15 10 29 5 12 47 1 8 11 30 21.7 11 23 17 36.G 2 22 57.3 5 23 44 18 3 19 31 1C 7 29 8 41 23 57 36 5 35 9 10 12 24 29 28 13.7 5 8 51 19.4 10 1 25 0.3 2 10 48 6 6 20 11 11.7 1876 B. 1877 1878 1879 1880B. 7 5 41 59 1 7 32 30 6 28 3 59 18 35 28 6 9 6 58 8 19 27 38.7 1 14 53.6 2 29 58 14.3 5 28 41 35 8 27 24 55.7 1 19 49 50 6 11 38 40 10 21 16 5 3 53 30 7 10 30 55 10 29 34 17.4 3 22 7 58.3 8 1 31 4 10 54 9.7 4 20 17 15.4 1881 1882 1883 1884 B. 1885 10 57 29 6 1 28 58 11 22 27 5 12 31 56 11 14 22 28 9 12 10.6 3 7 55 31.3 6 6 38 52.0 9 5 22 12.7 17 9 27.6 2 19 47 4 11 57 12 8 21 34 37 1 1 12 2 5 23 54 9 12 50 56.3 1 22 14 2.0 6 1 37 7.7 10 11 13.4 3 3 33 54.3 1886 1887 1888B. J889 1890 5 4 53 57 10 25 25 26 4 15 56 57 10 17 47 28 4 8 18 57 3 15 52 48.3 6 14 36 9.0 9 13 19 29.7 25 6 44.6 3 23 50 5.3 10 2 38 19 2 12 15 44 6 21 53 9 11 13 42 1 3 23 19 26 7 12 57 0.0 11 22 20 5.7 4 1 43 11.0 8 24 15 51.9 1 3 39 57.6 TABLE X . MOON'S EPOCHS. Years. Supp. of Node. II. V. VI. VII. VIII. IX. X. 1846 4 16 :5 9 s ' 11 7 56 254 258 937 941 847 113 1847 5 5 54 52 2 28 38 668 670 245 247 927 053 1848 B 2 25 17 45 709 116 122 5b2 587 042 997 1849 6 14 37 27 10 20 41 531 535 889 893 122 937 1850 7 3 57 9 2 11 13 944 947 196 200 202 876 1851 7 23 16 51 6 1 45 358 359 504 506 282 816 1852 B. 8 12 39 44 10 3 27 806 811 841 846 398 760 1853 9 1 59 26 1 23 59 220 223 148 152 477 700 1854 9 21 19 9 5 14 31 6?4 6.J6 456 459 557 639 1855 10 10 38 51 953 047 048 76J 765 637 579 1856 B. 11 1 44 1 6 44 495 500 100 105 753 523 1857 11 19 21 26 4 27 16 909 912 407 411 832 463 1858 8 41 8 8 17 48 323 S25 715 718 912 402 1859 28 51 8 20 7S6 737 023 024 992 342 1860 B, 1 17 23 43 4 10 1 184 189 359 364 108 2-6 1861 2 6 43 27 8 33 598 601 666 670 187 226 1862 2 26 3 9 11 21 5 012 014 974 977 267 165 1863 3 15 23 11 3 11 37 426 426 2b2 283 347 105 1864 B. 4 4 45 44 7 13 18 873 878 618 623 463 049 1865 4 24 5 46 11 3 50 287 291 926 929 542 989 1866 5 13 25 28 2 24 22 701 703 233 236 622 928 1867 6 2 45 10 6 14 54 115 115 544 542 702 868 ; ll>68 B. 6 22 7 43 10 16 36 563 567 877 882 818 812 ! 1869 7 11 27 46 278 977 980 185 188 897 752 1870 8 47 28 5 27 40 390 392 493 495 977 691 1871 8 20 6 49 9 18 11 803 804 800 802 057 630 1872 B. 9 9 26 31 1 8 43 216 216 108 110 137 569 1873 9 28 49 24 5 10 25 664 668 444 450 252 514 1874 10 18 9 6 9 57 077 080 752 758 332 453 1875 11 7 28 48 21 29 490 492 054 064 412 392 1876 B. 11 26 43 31 4 12 1 904 905 364 370 492 331 1877 16 11 24 8 13 42 352 357 700 710 607 276 1878 1 5 31 6 4 14 765 769 008 018 687 215 1879 1 24 50 48 3 24 46 178 181 316 326 767 154 1880 B. 2 14 10 30 7 15 18 593 593 624 630 847 093 1881 3 3 33 23 11 16 59 041 045 960 970 962 038 1882 3 22 53 5 3 7 31 454 457 268 278 042 977 1883 4 12 12 47 6 28 3 867 869 576 5F6 122 916 1884 B. 5 1 32 29 10 18 35 280 281 884 894 202 855 1885 5 20 55 22 2 20 16 728 733 220 234 317 800 1886 6 10 15 4 6 10 48 141 145 52S 542 397 730 1887 6 29 34 46 10 1 20 554 557 836 850 477 678 1888 B. 7 18 54 28 1 21 52 967 969 144 158 557 617 1889 8 8 17 21 5 23 33 415 421 480 498 672 562 1890 8 27 36 3 9 14 5 828 833 788 806 752 501 TABLE XX. MOON'S MOTIONS FOR MONTHS. Months. Evection. Anomaly. Variation. M. Longitude. *-lr: HmT. March .... 6 ' " 0000 11 18 41 1 11 20 48 42 11 9 29 43 10 7 40 26 8 ' 0000 11 16 56 6 1 15 53 1 1 56 59 1 20 50 4 8 0000 11 17 48 33 17 54 48 5 43 21 11 29 15 15 8 0000 11 16 49 25 1 18 28 6 1 5 17 31 1 27 24 27 April 9 28 29 8 3 5 50 57 17 10 3 3 15 52 32 May 9 7 58 51 4 7 47 56 22 53 24 4 21 10 3 8 28 47 33 5 22 48 49 1 10 48 11 6 9 38 9 July.. 8 8 17 16 6 24 45 48 1 16 31 32 7 14 55 40 August September.. . October 7 29 5 59 7 19 54 41 6 29 24 24 8 9 46 42 9 24 47 35 10 26 44 34 2 4 26 20 2 22 21 7 2 28 4 28 9 3 23 46 10 21 51 52 11 27 9 22 November. . . De6ember . . . 6 20 13 6 5 29 42 49 11 45 27 1 13 42 26 3 15 59 16 3 21 42 37 1 15 37 28 2 20 54 59 T MOON ABLE X: 's MOTIONS F01 . El DAYS. Days. Evection. Anomaly. Variation. Mean Longitude. 1 Os 1 0' 0" Os GO 0' 0" Os GO 0' 0" Os GO 0' 0" 2 11 18 59 13 3 54 12 11 27 13 10 35 3 22 37 59 26 7 48 24 22 53 26 21 10 4 1 3 56 58 1 9 11 42 1 6 34 20 1 9 31 45 5 1 15 15 58 1 22 15 36 1 18 45 47 1 22 42 20 6 1 26 34 57 2 5 19 30 2 57 13 2 5 52 55 7 2 7 53 57 2 18 23 24 2 13 8 40 2 19 3 30 8 2 19 12 56 3 I 27 18 2 25 20 7 3 2 14 5 9 3 31 55 3 14 31 12 3 7 31 34 3 15 24 40 10- 3 11 50 55 3 27 35 6 3 19 43 3 28 35 15 11 3 23 9 54 4 10 39 4 1 54 27 4 11 45 50 12 4 4 28 54 4 23 42 54 4 14 5 54 4 24 56 25 13 4 15 47 53 5 6 46 48 4 26 17 20 5870 14 4 27 6 53 5 19 50 42 5 8 28 47 5 21 17 35 15 5 8 25 52 6 2 54 36 5 20 40 14 6 4 28 10 16 5 19 44 51 6 15 58 29 6 2 51 40 6 17 38 45 17 6 1 3 51 6 29 *> 23 6 15 3 7 7 49 20 18 6 12 22 50 7 12 6 17 6 27 14 34 7 13 59 55 19 6 23 41 50 7 25 10 11 7 9 26 1 7 27 10 30 20 7 5 49 8 8 14 5 7 21 37 27 8 10 21 5 21 7 16 19 49 8 21 17 59 8 3 48 54 8 23 31 40 22 7 27 38 48 9 4 21 53 8 16 21 9 6 42 16 23 8 8 57 47 9 17 25 47 8 28 11 47 9 19 52 51 24 8 20 16 47 10 29 41 9 10 23 14 10 3 3 26 25 9 1 35 46 10 13 33 35 9 22 34 41 10 16 14 1 26 9 12 54 46 10 26 37 29 10 4 46 7 10 29 24 36 27 9 24 13 45 11 9 41 23 10 16 57 34 11 12 35 11 28 10 5 32 45 11 22 45 17 10 29 9 1 11 25 45 46 29 10 16 51 44 5 49 11 11 11 20 28 8 56 21 30 10 28 10 43 18 53 5 11 23 31 54 22 6 56 31 11 9 29 43 1 1 56 59 5 43 21 1 5 17 31 / TABLE XX. MOON'S MOTIONS FOR MONTHS. Months. Supp. of Node. II. V. VI. VII. VIII. IX. X. T I Com.. Jan 'jBis. . ,-,,"1 Com.. Feb 'jBis. . March 8 ' " 0000 11 29 56 49 1 38 30 1 35 19 3 7 27 8 ' 000 11 18 51 11 15 43 11 4 34 9 27 59 000 966 54 20 7 000 961 224 185 330 000 972 875 847 666 000 966 45 11 989 000 964 111 75 114 000 995 165 159 313 April 4 45 57 9 13 42 61 554 542 34 995 478 May 6 21 16 8 18 15 81 738 389 46 300 638 J UI16 ...... 7 59 46 8 3 58 136 969 964 91 411 802 July 9 35 5 7 8 32 1^6 147 119 103 486 962 August September.. . October November. .. December . . . 11 13 35 12 52 5 14 27 24 16 5 53 17 41 13 6 24 15 6 9 58 5 14 32 5 15 4 4 49 210 265 285 339 359 371 595 780 4 188 987 862 710 585 432 147 193 204 250 261 497 708 783 894 969 126 291 451 615 775 TABLE XX. MOON'S MOTIONS FOR DAYS. Days Supp. of Node II. V. VI. VII. VIII. IX. X. 1 00 0' 0" Os GO 0' 000 000 000 000 000 000 2 3 11 11 9 34 39 28 34 36 5 3 6 21 22 18 68 79 56 67 72 11 4 9 32 1 3 27 102 118 85 101 108 16 5 12 52 1 14 37 136 158 113 135 143 21 6 15 53 1 25 46 170 197 141 169 179 27 8 19 4 2 6 55 204 237 169 202 215 32 8 22 14 2 18 4 238 276 198 236 251 37 9 25 25 2 29 13 272 316 226 270 287 43 10 28 36 3 10 22 306 355 254 303 323 48 11 31 46 3 21 31 340 395 282 337 358 53 12 34 57 4 2 40 374 434 311 371 394 58 13 38 7 4 13 50 408 474 339 405 430 64 14 41 18 4 24 59 442 513 367 438 466 69 15 44 29 568 476 553 395 472 502 74 16 47 39 5 17 17 510 592 424 506 538 80 17 50 50 5 28 26 544 632 452 539 573 85 18 54 1 6 9 35 578 671 480 573 609 90 19 57 11 6 20 44 612 711 508 607 645 96 20 22 7 1 53 646 750 537 641 681 101 21 3 33 7 13 3 680 790 565 674 717 1% 22 6 43 7 24 12 714 829 593 708 753 112 23 9 54 8 5 21 748 869 621 742 788 117 24 13 5 8 16 30 782 908 650 775 824 122 25 16 15 8 27 39 816 948 678 809 860 128 26 19 26 9 8 48 850 987 706 843 896 133 27 22 36 9 19 57 884 027 734 877 932 138 28 25 47 10 1 6 918 066 762 910 968 143 29 28 58 10 12 16 952 106 791 944 003 149 30 32 8 10 23 25 986 145 819 978 039 154 31 35 19 11 4 34 020 185 847 Oil 075 151 30 TABLE XX. MOON'S MOTIONS FOR HOURS. Hours. Evection. Anomaly. Variation Longitude. 1 2 3 4 5 ' " 28 17 56 35 1 24 52 1 53 10 2 21 27 O ' " 32 40 1 5 19 1 37 59 2 10 39 2 43 19 O ' a 30 29 1 57 1 31 26 2 1 54 2 32 23 O ' " 32 56 1 5 53 1 38 49 2 11 46 2 44 42 6 7 8 9 10 2 49 45 3 18 2 3 46 20 4 14 37 4 42 55 3 15 58 3 48 38 4 21 18 4 53 58 5 26 37 3 2 52 3 33 20 4 3 49 4 34 17 5 4 46 3 17 39 3 50 35 4 23 32 4 56 28 5 29 25 11 '12 13 14 15 5 11 12 5 39 30 6 7 47 6 36 5 7 4 22 5 59 17 6 31 57 7 4 37 7 37 16 8 9 56 5 35 15 6 5 43 6 36 12 7 6 40 7 37 9 6 2 21 6 35 17 7 8 14 7 41 10 8 14 7 16 17 18 19 20 7 32 40 8 57 8 29 15 8 57 32 9 25 50 9 42 36 8 15 16 9 47 55 10 20 35 10 53 15 8 7 38 8 38 6 9 8 35 9 39 3 10 9 32 8 47 3 9 20 9 52 56 10 25 53 10 58 49 21 22 23 24 9 54 7 10 22 24 10 50 42 11 18 59 11 25 55 11 58 34 12 31 14 13 3 54 10 40 1 11 10 29 11 40 58 12 11 27 11 31 46 12 4 42 12 37 39 13 10 35 TABLE XXI. MOON'S MOTIONS FOR MINUTES. Min. Evec. Anomaly. Variations. Longitude. Sup. Node. k II. 1 28 33 30 33 5 2 21 2 43 2 32 2 45 1 2 10 4 43 5 27 5 5 5 29 1 5 15 7 4 8 10 7 37 8 14 2 7 20 9 26 10 53 10 10 10 59 3 9 25 11 47 13 37 12 42 13 43 3 12 30 14 9 16 20 15 14 16 28 4 14 35 16 30 19 3 17 47 19 13 5 16 40 18 52 21 46 20 19 21 58 5 19 45 21 13 24 30 22 52 24 42 6 21 50 23 34 27 13 25 24 27 27 7 23 55 25 56 29 56 27 56 30 12 7 26 (VI 28 17 32 4!) 30 29 32 56 8 28 TABLE XX. MOON'S MOTIONS FOR HOURS. 31 Hours. j _: iSupn. of Node. II. V. VI. VII. VIII. IX. X. / / ' 1 8 28 1 2 1 1 1 2 1G Of\ A 56 If) A 3 3 2 3 3 4 mm 32 m I 52 6 7 5 6 6 1 5 40 2 19 7 6 7 7 1 6 48 2 47 9 n 7 9 9 1 7 56 3 15 10 13 8 10 10 2 8 4 3 43 11 13 9 11 12 2 9 11 4 11 13 15 11 13 13 2 10 19 4 39 14 16 12 14 15 2 31 27 5 7 16 18 13 15 16 2 12 35 5 35 17 20 14 17 18 3 13 43 6 2 18 21 15 18 19 3 14 51 6 30 20 23 16 19 21 3 15 59 6 58 21 25 18 21 22 3 16 2 7 7 26 23 26 19 22 24 4 17 2 15 7 54 24 28 20 24 25 4 18 2 23 ft 22 26 29 21 25 27 4 19 2 31 8 50 27 31 22 27 28 4 20 2 39 9 18 28 32 24 28 30 4 21 2 47 9 45 30 34 25 29 31 5 22 2 55 10 13 31 36 26 31 33 5 23 3 3 10 41 33 38 27 32 34 5 24 3 11 11 9 34 39 28 34 36 5 TABLE A.* PERTURBATIONS OF EARTH'S RADIUS VECTOR. TABLE B. APPROX. LAT. ARG. N. Arg. I. II. III. Arg. I. II. III. 8 4 3 500 2 4 50 8 4 3 550 2 1 4 100 7 4 2 600 3 1 3 150 7 4 1 650 3 2 2 200 6 4 700 4 3 1 250 5 4 750 5 4 300 4 3 1 800 6 4 350 3 2 2 850 7 4 1 400 3 1 3 900 7 4 2 450 2 1 4 950 8 4 3 500 2 4 1000 8 4 3 N. A. N. D. S. D. S. A. f>'s Lat. 500 500 1000 t a 5 495 505 995 9 41 10 490 510 990 19 22 15 485 515 985 29 3 20 480 520 980 38 40 25 475 525 975 48 18 30 470 530 970 58 40 35 465 535 965 67 28 40 460 540 960 76 45 45 455 545 955 86 21 50 450 550 950 95 26 55 445 555 945 04 56 Tables A. and B. are put in this place on account of the convenience in the page. 32 TABLE XXI fIRST EQUATION OF MOON's LONGITUDE. ARGUMENT 1. Arg. 1 Diff. Anr 1 Diff. 100 12 40 11 58 42 5000 5100 it 12 40 13 20 40 200 11 16 42 *c\ . 5200 14 1 41 300 10 34 I A 1 5300 14 41 40 400 9 53 41 5400 15 20 39 500 9 12 41 Ai\ 5500 16 40 600 8 32 40 qQ 5600 16 38 38 q7 700 7 54 oo qo 5700 17 15 O* Q7 800 7 16 OU Qfi 5800 17 52 o/ OK 900 6 40 oO 5900 18 27 OO 1000 6 6 no 6000 19 1 34 1100 5 33 33 6100 19 33 32 qi 1200 5 2 OA 6200 20 4 ol cin 1300 4 32 oU Cfc*7 6300 20 33 sv CfcO 1400 1500 4 5 3 40 mi 25 qq 6400 6500 21 1 21 27 28 26 1600 3 17 20 91 6^00 21 50 oo 1700 2 56 Zl IP. 6700 22 12 SESf -i(i 1800 2 38 lo 6800 22 31 IJ 1900 2 22 1 O 6900 22 48 2009 2 9 lo 7000 23 3 15 t Ck 2100 58 7100 23 15 IB in 2200 50 7200 23 25 ill 2300 44 ' - ^ 7300 23 32 2400 41 7400 23 37 2500 41 7500 23 39 2600 43 c 7600 23 39 lag 2700 48 O 7700 23 36 2800 55 -I f\ 7800 23 30 2900 3000 2 5 2 -17 10 12 7900 8000 23 22 23 11 11 1 Q 3100 2 32 17 8100 22 58 Jo 1 u 3200 3300 2 49 3 8 1 1 19 8200 8300 22 42 22 24 ID 18 Cll 3400 3 30 00 8400 22 3 21 99. 3500 3600 3 53 4 19 26 97 8500 8600 21 40 21 15 ZJ 25 97 3700 3800 4 46 5 16 ml 30 Ol 8700 8800 20 48 20 18 ml 30 Ol 3900 4000 4100 4200 5 47 6 19 6 53 7 28 ol 32 34 35 q7 8900 9000 9100 9200 19 47 19 14 18 40 18 4 31 33 34 36 qo 4300 8 5 Ol 07 9300 17 26 oO qo 4400 8 42 ol qo 9400 16 48 oO An 4500 9 20 OO Oil 9500 16 8 4U At 4600 9 59 39 A(\ 9600 15 27 41 At 4700 10 39 4U 9700 14 46 41 An 4600 11 19 JJ 9800 14 4 Vm AC) 4909 11 59 J.1 9900 13 22 VI 5000 12 40 41 10000 12 40 TABLE XXII. EQUATIONS 2 TO 7 OF MOON'S LONGITUDE. ARGUMENTS 2 TO 7. 33 Arg. 2 3 4 5 6 7 Arg. 2500 4 57 2 6 30 3 39 6 1 2500 2600 4 57 2 6 30 3 39 6 1 2400 2700 4 56 3 6 29 3 38 7 1 2300 2800 4 55 3 6 27 3 37 8 2 2200 2900 4 53 4 6 24 3 36 9 3 2100 3000 4 50 5 6 21 3 34 10 4 2000 3100 4 47 6 6 17 3 32 12 5 1900 3200 4 43 8 6 12 3 29 14 6 1800 3300 4 39 9 6 7 3 26 17 8 1700 3400 4 34 11 6 1 3 22 19 10 1600 3500 4 29 13 5 54 3 18 22 12 1500 3600 4 23 15 5 47 3 14 23 14 1400 3700 4 17 18 5 39 3 10 29 17 1300 3800 4 11 20 5 30 3 5 33 19 1200 3900 4 4 23 5 21 3 37 22 1100 400(1 3 57 26 5 12 2 54 41 25 1000 4100 3 49 29 5 2 2 49 45 28 900 4200 3 41 32 4 52 2 43 50 31 800 4300 3 33 35 4 41 2 37 54 35 700 4400 3 24 39 4 30 2 30 59 38 600 4500 'A 15 42 4 19 2 24 4 42 500 4600 3 7 46 4 7 2 17 9 45 400 4700 2 58 49 3 56 2 10 14 49 300 4-00 2 48 53 3 44 2 4 19 53 200 4930 2 39 56 3 32 1 57 25 56 100 5000 2 30 3 20 1 50 30 1 0000 510) 2 21 4 3 8 1 43 35 1 4 9900 5200 2 11 7 2 56 1 36 40 1 7 9800 5300 2 2 11 2 44 1 29 46 1 11 9700 5400 1 53 14 2 33 1 23 1 51 1 15 9600 S.iOO 1 44 18 2 21 1 16 1 56 1 18 9500 5600 1 36 21 2 10 1 10 2 1 22 3400 5700 1 27 25 1 59 1 3 2 6 25 9300 5800 1 19 28 1 48 57 2 10 '28 9200 5900 1 11 31 1 38 51 2 15 32 9100 6000 1 3 34 1 28 46 2 19 35 9000 6100 56 37 1 19 40 2 23 38 8900 6200 49 39 I 10 35 2 27 40 8806 6300 33 42 1 1 30 2 31 43 8700 6400 I 36 44 53 26 2 35 46 8600 6500 31 47 46 21 2 38 48 8500 6600 26 49 39 18 2 41 50 8400 6700 21 51 33 14 2 43 52 8300 6800 17 52 28 11 2 46 54 8200 6900 13 54 23 8 2 48 55 8100 7000 10 55 19 6 2 50 56 8000 7100 7 56 16 4 2 51 57 7900 720C 5 57 13 2 2 52 58 7800 7300 4 57 11 1 2 53 59 7700 740?) 3 58 10 1 2 54 59 7600 7500 3 1 58 18 1 2 54 59 7500 TABLE XXIII. EQUATIONS 8 TO 9 OF MOON's LONGITUDE. ARGUMENTS 8 TO 9. Arg. 3 9 Arg. 8 9 /; // it / // 20 1 20 5000 20 1 SO 100 15 1 29 5100 24 1 26 200 11 1 37 5200 29 1 31 300 7 1 46 5300 33 37 400 2 1 54 5400 37 42 500 58 2 1 5500 42 47 600 54 2 8 5600 46 51 700 50 2 15 5700 50 1 55 sao 46 2 20 5SOO 54 58 901) 42 2 25 5900 1 58 2 1000 38 2 29 6000 2 1 2 1 1?00 35 2 32 6100 2 5 2 2 ' 1^00 31 2 34 6200 2 8 2 2 1300 28 2 35 6300 2 11 2 1 1400 25 2 35 6400 2 14 59 1500 33 2 34 6500 2 17 56 1600 20 2 32 6600 2 19 52 1700 18 2 29 6700 2 22 48 1800 16 2 26 6800 2 24 43 1900 14 2 21 6900 2 25 38 2000 13 2 ,6 7000 2 27 32 2100 11 2 11 7100 2 28 25 2230 10 2 4 7200 2 29 18 2300 10 58 7300 2 30 11 2400 9 51 7400 2 30 4 2500 9 43 7500 2 31 56 2600 10 36 7600 2 30 49 2700 10 29 7700 2 30 42 2800 11 22 7800 2 29 36 2900 12 15 7900 2 28 29 3000 13' 8 8000 2 27 24 3100 15 2 8100 2 26 18 3200 16 57 8200 2 24 14 3300 18 52 8300 2 22 10 3400 21 47 8400 2 20 8 3500 23 44 8500 2 17 6 3690 26 41 8600 2 15 5 3700 29 39 8700 2 12 0- 5 3800 32 38 8800 2 9 6 3900 35 38 8900 2 5 8 4000 39 39 9000 2 2 11 4100 42 40 9100 58 15 4200 46 42 9200 54 20 4300 50 45 9300 50 25 4400 54 49 9400 46 32 4500 58 53 9500 42 39 4600 1 3 58 9600 38 46 4700 1 7 1 3 9700 33 54 4800 1 11 1 9 9800 29 1 3 4900 1 16 1 14 9900 24 1 11 5000 1 20 1 20 10000 20 1 20 EQUATIONS 10 AND 11. Arg. HI 11 Arg. 10 - 10 10 500 10 3 10 9 11 510 10 n 20 9 12 520 9 n 30 13 530 9 12 40 7 14 540 8 13 50 7 15 550 8 14 60 6 Iti 560 8 14 70 6117 570 8 15 80 5 17 580 7 15 90 5 18 590 7 15 100 5 18 600 7 16 110 4 IS) 610 7 16 120 4 19 620 7 16 i:-;o 4 IS) 6ro 7 16 140 4 1!) 640 7 15 150 4 IS) 650 8 15 160 4 19 660 8 15 170 4 18 670 8J14 180 5 18 680 9113 190 I 17 690 9 13 200 5 16 700 10 12 210 (5 16 710 10 11 220 6 15 720 11 10 230 7 14 730 11 9 240 13 740 12 9 250 8 12 750 12 8 260 8 11 760 13 7 270 9 10 770 13 6 280 10 780 14 5 290 10 9 790 14 4 300 10 8 800 15 3 310 11 7 810 15 3 320 11 6 820 15 2 330 12 6 8"0 16 2 340 12 5 840 16 1 350 12 5 850 16 360 12 5 8(50 16 370 13 4 870 16 380 13 4 880 16 390 13 4 890 16 400 13 4 BOO 15 2 410 13 5 910! 15 2 420 12 5 920J15 3 430 12 5 9301 14 3 440 12 6 940 14 4 450 12 6 950 13 5 460 11 7 960 13 6 470 11 8 970 12 7 480 11 8 980 11 8 490 10 9 990 11 9 500 10 10 1000 10 10 TABLE XXIII. EQUATIONS 12 TO 19. Arg. 12 13 14 15|16 17 18 19 Arg 600 'so 610! 31 620 32 630 33 640 34 650 34 660 35 670 35 680 36 690,36 700 '37 710137 31 32 28 33 28 33 34 29 29 17 1612 14 6!l 1611)14 Sjl 1711114 ft 1 10 15l 5 18! 15 6 35 3( 36 30 5 16 4 It 4 1 16 4 16 900 17 810 800 790 780 770 760 750 32 QUATION 20 Arg. 20 Arg. n 10 500 10 ,11 510 20 12 820 30 13 530 40 13 540 50 14 550 60 ! 15 560 70 1 16 570 SO 16 580 90 17 590 100 17 600 110 17 610 120 17 620 130 17 C30 140 17 640 150 17 650 160 17 660 170 16 670 180 1 16 680 190 15 690 200 14 700 240 13 710 220 13 720 230 12 730 240 11 740 250 10 750 260 9 760 270 8 7VO 7 780 [ 290 6 79 300 B 800 310 5 810 320 4 820 330 4 830 340 3 840 350 3 850 360 3 860 370 3 870 380 3 8SO 390 3 890 400 3 900 410 g 910 420 4 920 430 4 930 440 5 940 450 6 950 460 6 960 470 7 970 480 8 980 490 9 990 500 10 .OOOJ 36 TABLE XXIV. EVECTION. Argument Evection Corrected. Os Is Us Ills IVs Vs GO 130' 0" 2310' 43" 2 40 10" 20 50' 25" 2039' 8" 2 9' 42" 1 31 25 2 11 57 2 40 51 2 50 23 2 38 25 2 8 29 2 32 51 2 13 9 2 41 30 2 50 20 2 37 40 2 7 16 3 34 16 2 14 21 2 42 8 2 50 15 2 36 55 262 4 35 42 2 15 31 2 42 45 2 50 9 2 36 8 2 4 47 5 37 7 2 16 41 2 43 21 2 50 1 2 35 19 2 3 32 6 38 32 2 17 50 2 43 55 2 49 52 2 34 30 2 2 16 7 39 57 2 18 58 2 44 27 2 49 41 2 33 40 2 1 8 41 21 2 20 5 2 44 59 2 49 29 2 32 48 1 59 43 9 42 46 2 21 11 2 45 29 2 49 15 2 31 55 1 58 26 10 44 10 2 22 17 2 45 57 2 49 2 31 2 1 57 8 11 45 34 2 23 21 2 46 24 2 48 43 2 30 7 1 55 49 12 46 58 2 24 24 2 46 50 2 48 26 2 29 11 54 30 13 48 21 2 25 26 2 47 14 2 48 6 2 28 14 53 11 14 49 44 2 26 28 2 47 37 2 47 45 2 27 16 51 51 15* 51 7 2 27 28 2 47 59 2 47 23 2 26 17 50 31 16 52 29 2 28 27 2 48 19 2 47 n 2 25 17 49 11 17 53 51 2 29 25 2 48 37 5 4b 35 2 24 16 47 50 18 55 12 2 30 21 2 48 54 2 46 8 2 23 14 46 29 19 56 33 2 31 17 2 49 10 2 45 41 2 22 11 45 7 20 57 53 2 32 11 2 49 24 2 45 12 2 21 7 43 46 21 59 13 2 33 5 2 49 37 2 44 41 2 20 2 42 24 22 2 32 2 33 57 2 49 48 2 44 9 2 18 56 41 2 23 2 1 51 2 34 48 2 49 58 2 43 36 2 17 50 39 39 24 239 2 35 38 2 50 6 2 43 2 2 16 43 38 17 25 2 4 26 2 36 26 2 50 13 2 42 26 2 15 34 36 54 26 2 5 43 2 37 13 2 50 19 2 41 49 2 14 25 35 32 27 2 6 59 2 37 59 2 50 23 2 41 11 2 13 16 34 9 23 2 8 15 2 38 44 2 50 25 2 40 31 2 12 5 32 46 29 2 9 30 2 39 28 2 50 26 2 39 50 2 10 54 31 23 30 2 10 43 2 40 10 2 50 25 2 39 8 2 9 42 30 > TABLE XXV. MOON'S EQUATORIAL PARALLAX. Argument. Arg. of the Evection. Os U Hs Ills IVs Vs 1 28" 1' 23" ' 9" 0' 50" 0' 32" 0' 18" 30 2 1 28 1 22 8 49 30 18 28 4 1 28 22 7 47 29 17 26 6 1 28 21 5 46 J8 17 24 8 1 28 20 4 45 .!7 16 22 10 1 28 19 3 44 26 16 20 12 1 27 18 2 42 25 0- 15 18 14 1 27 17 41 24 15 16 16 1 27 16 59 40 24 15 14 18 26 15 58 39 23 14 12 20 26 14 57 37 22 14 10 22 25 13 55 36 21 14 8 24 25 12 54 35 20 14 6 26 24 11 53 34 20 14 4 28 24 10 51 33 19 13 2 30 23 9 50 32 18 13 XI* Xs IXs VIlIs VIIs VI* TABLE XXIV. EVECTION. Argument. Evection Corrected. 37 Vis VIIs VIIIs IXs Xs XIi QO 1030 0" 00 50' 18" GO 20' 52" 00 9' 34" GO 19' 50" Oo 49' 16" 1 1 28 37 49 6 20 10 9 34 23 32 50 30 2 1 27 14 47 55 19 29 9 35 21 16 51 45 3 25 51 46 44 18 49 9 37 22 I 53 1 4 24 28 45 34 18 11 9 41 22 47 54 17 5 23 6 44 26 17 34 9 47 23 34 55 33 6 .21 43 ) 43 17 16 58 9 54 24 22 56 51 7 20 20 42 10 16 24 10 2 25 12 58 9 8 1H 58 41 4 15 50 10 12 26 3 59 28 9 17 36 39 58 15 19 10 23 26 55 1 47 10 16 14 33 53 14 48 10 36 27 43 2 7 11 14 52 37 49 14 19 10 50 28 43 3 27 12 13 31 36 46 13 51 11 5 29 39 4 48 13 12 10 35 44 13 25 11 23 30 35 6 9 14 10 49 34 43 13 11 41 31 33 7 31 15 9 29 33 43 12 37 12 1 32 32 8 53 16 8 09 32 44 12 14 12 23 33 32 10 16 17 6 49 31 46 11 54 12 45 34 34 11 39 18 5 30 30 49 11 34 13 10 35 36 13 2 19 4 11 29 53 11 16 13 35 36 39 14 26 20 2 52 28 58 11 14 3 37 43 15 50 21 1 34 28 5 10 45 14 31 38 48 17 14 22 1 17 27 12 10 31 15 1 39 55 18 39 23 59 26 20 10 19 15 33 41 2 20 3 24 57 44 25 30 10 8 16 5 42 10 21 28 25 56 23 24 40 9 59 16 39 43 19 22 53 26 55 13 23 52 9 51 17 15 44 29 24 18 27 53 58 23 5 9 45 17 52 45 39 25 44 28 52 44 22 20 9 40 18 30 46 51 27 9 29 51 31 21 35 9 36 19 9 48 3 28 34 30 50 18 20 52 9 34 19 50 49 16 30 TABLE P. MOON'S EQUATORIAL PARALLAX. Argument. Arg. of tbe Variation. 0. I. Us Ills IVs Vs O 1 56" 42" 16" 4" 18" 44" 300 2 55 41 14 4 19 46 28 4 55 39 13 4 21 47 26 6 55 37 12 4 23 48 24 8 55 35 10 5 24 50 22 10 54 34 9 6 26 51 20 12 53 32 8 6 28 52 18 14 52 30 7 7 30 53 16 16 51 28 6 8 32 54 14 18 50 26 6 9 34 55 12 20 49 24 5 10 35 55 10 22 48 23 4 12 37 56 8 24 47 21 4 13 39 56 6 26 45 19 4 14 41 57 4 28 44 18 4 16 42 57 2 30 42 16 4 18 44 57 XIs x 1X8 VIIIs VIIi VI, | 38 TABLE XXV. EQUATION OF MOON'S CENTER. Argument. Anomaly corrected. Os Is III Ills IVa Vs 00 70 o' 0" 10 20' 58" 120 38' 44" 13 17' 35" 120 16' 21" 9058 29" 1 775 10 26 52 12 41 43 13 17 5 12 12 48 9 52 58 2 7 14 10 10 32 42 12 44 35 13 16 28 12 9 11 9 47 24 3 7 21 15 10 38 27 12 47 20 13 15 44 12 5 29 9 41 48 4 7 28 19 10 44 8 12 49 59 13 14 53 12 1 41 9 36 10 5 7 35 23 10 49 43 12 52 30 13 13 56 11 57 49 9 30 29 6 7 42 26 10 55 14 12 54 55 13 12 52 11 53 52 9 24 46 7 7 49 28 11 39 12 57 12 13 11 41 11 49 50 9 19 1 8 7 56 28 11 6 12 59 23 13 10 24 11 45 44 9 13 13 9 8 3 28 11 11 15 13 1 26 13 9 1 11 41 33 9 7 24 10 8 10 26 11 16 24 13 3 23 13 7 31 11 37 17 9 1 32 11 8 17 22 11 21 29 13 5 12 13 5 54 11 32 57 8 55 39 12 8 24 17 11 26 27 13 6 55 13 4 12 11 28 33 8 49 44 13 8 31 10 11 31 20 13 8 30 13 2 23 11 24 5 8 43 47 14 8 38 1 11 36 8 13 9 59 13 27 11 19 32 8 37 49 15 8 44 50 11 40 49 13 11 20 12 58 26 11 14 55 8 31 49 16 8 51 36 11 45 25 13 12 34 12 56 18 11 10 14 8 25 48 17 8 58 20 11 49 54 13 13 41 12 54 5 11 5 30 8 19 46 18 9 5 1 11 54 18 13 14 41 12 51 45 11 41 8 13 42 19 9 11 39 11 58 35 13 15 34 12 49 19 10 55 49 8 7 38 20 9 18 15 12 2 47 13 16 20 12 46 47 10 50 53 8 1 32 21 9 24 47 12 6 52 13 16 59 12 44 10 10 45 53 7 55 26 22 9 31 16 12 10 50 13 17 31 12 41 27 10 40 50 7 49 18 23 9 37 42 12 14 42 13 17 56 12 38 38 10 35 43 7 43 10 24 9 44 4 12 18 23 13 18 14 12 35 43 10 30 33 7 37 1 25 9 50 23 12 22 7 13 18 24 12 32 43 10 25 20 7 30 52 26 9 56 38 12 25 40 13 18 28 12 29 37 10 20 4 7 24 42 27 10 2 49 12 29 6 13 18 25 12 26 26 10 14 45 7 18 32 23 10 8 56 12 32 25 13 18 16 12 23 10 10 9 22 7 12 21 29 10 14 59 12 35 38 13 17 59 12 19 48 10 3 57 7 6 11 30 10 20 58 12 38 44 13 17 35 12 16 21 9 58 29 700 TABLE XXVI. MOON'S EQUATORIAL PARALLAX. Argument. Corrected Anomaly. Os Is Us Ills IVi Vs i Oo 58' 58" 58' 27" 57' 8 " 55' 30" 54' 2" 53' 3" 300 2 53 58 58 23 57 2 55 23 53 57 53 28 4 58 57 58 19 56 55 55 17 53 52 52 58 26 6 58 56 58 14 56 49 55 11 53 47 52 56 24 8 58 55 58 10 56 42 55 4 53 43 52 54 22 10 58 54 58 5 56 36 54 58 53 38 52 52 20 , 12 58 53 58 56 29 54 52 53 34 52 50 18 14 58 51 57 55 56 22 54 46 53 30 52 49 16 16 58 49 57 49 56 16 54 40 53 26 52 47 14 18 53 46 57 44 56 9 54 34 53 22 52 46 12 20 58 44 57 38 56 3 54 29 53 19 52 45 10 22 58 41 57 32 55 56 54 23 53 15 52 44 S 24 58 38 57 26 55 49 54 18 53 12 52 43 6 26 58 34 57 20 55 43 54 12 53 9 52 43 4 28 58 31 57 14 55 36 54 7 53 6 52 43 2 30 58 27 57 8 55 30 54 2 53 3 52 43 XIs X* IXs vm VIIi Vis TABLE XXV 39 EQUATION OP MOON'S CENTER. Argument. Anomaly corrected. Vis VIIs VIIIs IXs Xs Xls GO 70 0' 0" 40 i'3i 10 43' 39 ' 00 42' 25" 1021' 16" 3039' 2" 1 G 53 4 ( J 3 56 3 40 12 42 1 24 22 3 45 1 2 G 47 39 3 59 38 36 50 41 44 27 35 3 51 4 3 6 41 28 3 45 15 33 34 41 35 30 54 3 57 11 4 6 35 18 3 39 56 30 23 41 32 34 20 4 3 22 5 6 21) 8 3 34 40 27 17 41 36 37 53 4 9 37 6 6 22 59 3 29 26 24 17 41 46 41 32 4 15 55 4 G 16 5'J 3 24 17 21 22 42 4 45 18 4 22 18 8 G 10 42 3 19 10 18 33 42 29 49 10 4 28 44 9 G 4 34 3 14 7 15 50 43 1 53 8 4 35 13 1) 5 58 23 397 13 12 43 40 57 13 4 41 45 11 5 52 22 3 4 11 10 41 44 26 2 1 24 4 43 21 12 5 4G 17 2 59 19 8 15 45 19 2 5 42 4 54 59 13 5 4:) 14 2 54 30 5 55 46 19 2 10 5 5 1 40 14 5 34 12 2 49 46 3 42 47 26 2 14 35 5 8 24 15 5 2* 11 2 45 5 1 34 48 40 2 19 11 5 15 10 16 5 22 11 2 49 28 59 33 50 1 2 23 52 5 21 59 , 17 5 16 13 2 35 55 57 37 51 30 2 24 39 5 23 50 18 5 10 16 2 31 27 55 48 53 5 2 33 32 5 35 43 19 5 4 21 2 27 3 54 6 54 47 2 38 31 5 42 37 2) 4 58 23 2 22 43 52 29 56 37 2 43 35 5 49 34 21 4 52 36 2 18 27 50 5!) 58 33 2 48 45 5 56 32 22 4 46 47 2 14 16 49 36 1 37 2 54 6 3 31 2i 4 4 ) 59 2 10 10 48 19 1 2 48 2 59 21 6 10 32 21 4 36 14 268 47 8 1 5 5 3 4 46 6 17 34 25 4 29 31 2 2 11 46 4 7 30 3 10 17 6 24 37 26 4 23 50 1 58 19 45 7 10 I 3 15 52 6 31 41 27 4 18 11 1 54 31 44 16 12 40 3 21 33 6 38 45 Si 4 12 35 1 50 49 43 32 15 25 3 27 18 6 45 50 29 472 1 47 11 42 55 18 17 3 33 8 6 52 55 3!) 4 1 31 1 43 39 42 -25 21 16 3 39 2 700 TABLE XXVI. (Continued.) REDUCTION OF PARALLAX, AND ALSO OF THE LATITUDE. Argument. Latitude. Latitude. Uedu'Mion of Parallax. Reduction of LiitiUii'e. 3 0' 1' 12' (J 2 2- 9 3 :i2 12 4 39 15 1 5 4i 18 1 6 44 21 1 7 40 24 2 8 31 27 2 9 16 30 3 9 55 33 3 10 28 36 4 10 54 39 5 11 13 4-2 5 11 25 45 6 11 29 Latitude. Reduction of Parallax. Reduction of Latitude. 48 G 11' 15' 51 7 11 14 51 8 10 56 57 8 10 30 GO 9 9 57 G3 9 9 18 G6 10 8 33 69 10 7 42 72 10 6 46 75 11 5 45 78 11 4 41 81 11 3 33 84 11 2 24 87 11 1 12 90 11 40 TABLE XXVII. VARIATION. ARGUMENT. Variation, corrected. Us Is Us Ills IVs V o o ' " O ' " a O t n O ' " O 1 H 38 1 8 1 6 58 35 54 5 29 062 2 40 26 1 9 7 5 36 33 27 4 21 7 24 4 42 52 1 10 3 4 5 31 3 22 8 55 6 45 16 1 10 50 2 27 28 34 2 33 10 34 8 47 38 1 11 26 42 26 11 1 54 12 22 10 49 57 1 11 53 58 49 23 51 1 24 14 17 12 52 13 1 12 9 56 50 21 34 1 5 16 19 14 54 24 1 12 15 54 45 19 22 57 18 27 16 56 30 1 12 10 52 35 17 15 59 20 41 18 58 30 1 11 55 50 21 15 13 1 11 23 20 24 I 11 30 48 2 13 17 1 34 25 23 22 2 11 I 10 55 45 40 11 28 028 27 50 24 3 51 1 10 10 43 16 9 47 2 51 30 20 26 5 23 1 9 15 40 50 8 13 3 45 32 52 28 6 47 1 8 11 38 22 6 47 4 48 35 26 30 8 1 1 6 58 35 54 5 26 062 38 Vis VIIs VIIIs IXs Xs XIs o ' " ' n O ' " O ' " O ' " O ' n 38 9 58 10 30 40 6 092 7 58 2 40 34 11 11 9 13 37 38 7 49 9 13 4 43 8 12 15 7 47 35 10 6 45 10 37 6 45 40 13 9 6 13 32 44 5 50 12 9 8 48 10 13 52 4 31 30 19 055 13 49 10 50 37 14 26 2 42 27 58 4 29 15 36 12 53 14 48 47 25 39 044 17 30 14 55 19 15 1 58 45 23 25 3 50 19 30 16 57 33 15 3 56 38 21 15 3 45 21 36 18 58 41 14 54 54 25 19 10 3 51 23 47 20 1 1 43 14 35 52 9 17 11 047 26 3 22 1 3 38 14 6 49 49 15 18 4 34 28 22 24 1 5 25 13 27 47 26 13 33 5 10 30 44 26 1 7 5 12 38 45 11 54 5 57 33 8 28 1 8 36 11 39 42 33 10 24 6 53 35 33 30 1 9 58 10 30 40 6 092 7 58 38 TABLE XXVIII. MOON'S DISTANCE FROM THE NORTH POLE OF THE ECLIPTIC. ARGUMENT. Supplement of Node-f-Moon's Orbit Longitude. 41 Ills IVs Vs vis i viis VIIIs

Is Us in* IVi Vi OP 2" 6" 14" 18" 14" 6" SOP 2 2 7 14 18 13 6 28 4 2 7 15 18 13 5 26 6 2 8 15 18 12 5 24 8 2 8 16 18 13 4 22 10 2 9 16 17 11 4 20 12 3 9 16 17 11 4 18 14 3 10 17 17 10 3 16 16 3 10 17 17 10 3 14 18 4 11 17 16 3 12 20 4 11 17 16 3 10 22 4 12 18 16 2 8 34 5 12 18 15 2 6 as 5 13 18 15 2 4 28 6 13 18 14 2 2 30 6 14 18 14 2 XIs Xs IXs VIIIs VIIi VI TABLE XLIII. MOON'S HOURLY MOTION IN LATITUDE. ARGUMENT. Argument I, of Latitude. 08+ b+ Ils-f- Ills IVs Vs 2* 58" * 34" ' 29" (X 0" 1' 29" y 34" 39> 2 2 58 2 31 24 6 1 35 2 37 4 2 58 2 23 18 12 1 40 2 40 26 6 2 57 2 24 13 19 1 45 2 43 24 8 2 56 2 20 7 25 1 50 2 45 23 10 2 55 2 17 31 1 55 2 47 20 12 2 54 2 12 55 37 1 59 2 49 18 14 2 53 2 8 49 43 2 4 2 51 16 16 2 51 2 4 - 43 49 2 8 2 53 14 18 2 49 59 37 55 2 12 2 54 13 20 2 47 55 31 1 2 17 2 55 10 22 2 45 50 25 7 2 20 2 56 9 24 2 43 45 19 13 2 24 2 57 6 26 2 40 40 12 18 2 28 2 58 4 28 2 37 35 6 24 2 31 2 58 3 30 2 34 1 29 29 2 34 2 58 t XIs+ x+ 1X8+ VIIIs VIIs Vis TABLE XLIV. MOON'S HOURLY MOTION IN LATITUDE. ARGUMENT. Argument II, of Latitude. 0.+ I.+ II.+ IIIs ITi Vt (P 12 18 84 30 4" 4" 2" 0" 1 2 2 2" 3 3 8 3 4 4" 300 24 18 13 < t XI.+ X.+ IXt-f VIII* VII VI. TABLE XLV. PROPORTIONAL LOGARITHMS. 47 1' 2' 3' 4' 5' r 6' V 0" 00000 17782 14771 13010 11761 10792 10000 9331 1 35663 17710 14735 12986 11743 10777 9988 9320 2 82553 17639 14699 12962 11725 10763 9976 9310 3 80792 17570 14664 12939 11707 10749 9964 9300 4 29542 17501 14629 12915 11689 10734 9952 9289 5 28573 17434 14594 12891 11671 10720 9940 9279 6 87782 17368 14559 12868 11654 10706 9928 9269 7 27112 17302 14525 12845 11636 10692 9916 9259 8 26532 17238 14491 12821 11619 10678 9905 9249 9 26021 17175 14457 12798 11601 10663 9893 9238 10 25563 17112 14424 12775 11584 10649 9881 9228 11 25149 17050 14390 12753 11666 10635 9S69 9218 12 24771 16960 14357 12730 11549 10621 9858 9208 i 13 24424 16930 14325 12707 11532 10608 9846 9198 14 24102 16871 14292 12685 11515 10594 9834 9188 15 23802 16812 14260 12663 11498 10580 9826 9178 16 23522 16755 14228 12640 11481 10566 9811 9168 17 23259 16698 14196 1218 11464 10552 9800 9158 18 23010 16642 14165 12596 11457 10539 9788 9148 19 22775 16587 14133 12574 11430 10525 9777 9138 20 22553 16532 14102 12553 11413 10512 9765 9128 21 22341 16478 14071 12531 11397 10498 9754 9119 22 22139 16425 14040 12510 11380 10484 9742 9109 23 21946 16372 14010 12488 11363 10471 9731 9099 24 21761 16320 13979 12467 11347 10458 9720 9089 25 21584 162ti9 13949 12445 11331 10444 9708 9079 26 21413 16218 13919 12424 11314 10431 9697 9070 27 21249 13168 13890 12403 11298 10418 9686 9060 28 21091 16118 13860 12382 11282 10404 9tf75 9050 29 20939 16069 13831 12362 11266 10391 9664 9041 30 20792 16021 13802 12341 11249 10378 9652 9031 31 20649 15973 13773 12320 11233 10365 9641 9021 32 20512 15925 13745 12300 11217 10352 9630 9012 33 20378 15878 13716 12279 11201 10339 9619 9002 34 20248 15832 13688 12259 11186 10326 9608 QQQQ 35 20122 15786 13660 12239 11170 10313 9597 8983 36 20000 15740 13632 12218 11154 10300 9586 8973 J 37 19881 15695 13604 12198 11138 10287 9575 8964 38 19765 15651 13576 12178 11123 10274 9564 8954 39 19652 15607 13549 12159 11107 10261 9553 8945 40 19542 15563 13522 12139 11091 10248 9542 8935 41 19435 15520 13495 12119 11076 10235 9532 8926 42 19331 15477 13468 12099 11061 10223 9521 8917 43 19228 15435 13441 12080 11045 10210 9510 8907 44 19128 15393 13415 12061 11030 10197 9499 8898 45 19031 15351 13388 12041 11015 10185 9488 8888 46 18935 15310 13362 12022 10999 10172 9476 8879 47 18842 15269 13336 12003 10984 10160 9467 8870' 48 18751 15229 . 13310 11984 10969 10147 9456 8861 49 18661 15189 13284 11965 10954 10135 9446 8851 50 18573 15149 13259 11946 10939 10122 9435 8842 51 18487 15110 13233 11927 10924 10110 9425 8833 62 18403 15071 13208 11908 10909 10098 9414 8824 53 18320 15032 13183 11889 10894 10085 9404 8814 54 18239 149*1 13158 11871 10880 10073 9393 &305 55 1T>9 14956 13133 11852 10865 10061 9383 8796 56 18081 14918 13108 11834 10850 10049 9372 8787 57 18004 14881 13083 11816 10835 10036 9362 8778 i 58 17929 14844 13059 11797 10821 10024 9351 8769 - 59 17855 14808 13034 11779 10806 10012 9341 8760 W 17782 14771 13010 11761 10792 10000 9331 8751 | 48 TABLE XLV. PROPORTIONAL LOGARITHMS. * 8' 9' 10' 11' 12' 13' 14' 15' j 0" 8751 8239 7782 7368 6990 6642 6320 6021 5740 1 8742 8231 7774 7361 6984 6637 6315 6016 5736 2 8733 8223 7767 7354 6978 6631 6310 6011 5731 8724 8215 7760 7348 6972 6625 6305 6006 6727 8715 8207 7753 7341 6966 6620 6300 6001 5722 8706 8199 7745 7335 6960 6614 6294 5997 6718 8697 8191 7738 7328 6954 6609 6289 6992 6713 8688 8183 7731 7322 6948 6803 6284 6987 6709 8679 8175 7724 7315 6942 6598 6279 5982 5704 8670 8167 7717 7309 6936 6592 6274 1 5977 6700 10 8661 8159 7710 7302 6930 6587 6269 6973 6695 11 8652 8152 7703 7296 6924 6581 6364 6968 6691 12 8643 8144 7696 7289 6918 6576 6259 6963 6686 13 8635 8136 7688 7283 6912 6570 6254 6958 6682 14 8626 8128 7681 7276 ($06 6565 6243 6954 5677 15 8617 8120 7674 7270 6900 6559 6243 5949 6673 16 febus 8112 7667 7264 6894 6554 6238 5944 6669 17 855? 8104 7660 7257 6888 6548 6233 5939 6664 18 85H1 8097 7653 7251 6882 6543 6228 59;J5 6660 19' 8582 8089 7646 7244 6877 6538 6223 6930 56S5 20 8573 8081 7639 7238 6871 6532 6218 6925 6651 21 8565 8073 7632 7232 6865 6527 6213 5920 5646 22 8556 8066 7625 7225 6859 621 6208 5916 5642 23 8547 8058 7618 7219 685:} 6516 6203 5911 5637 24 8539 8050 7611 7212 6847 6510 6193 5fH)6 6633 25 8530 8043 7604 7206 6841 63)3 6193 6902 5629 26 8522 8035 7597 7200 6836 6500 6188 5897 5624 27 8513 8027 7590 7193 6830 6494 6183 5892 5H20 28 8504 8020 7583 7187 682-1 6489 6178 5888 615 29 8496 8012 7577 7181 6818 64S4 6173 5883 5611 30 8487 8004 7570 7175 6812 6478 6168 6878 5607 31 8479 7997 7563 7168 6807 6473 6163 5874 5602 32 8470 7989 7556 7162 6801 6467 6158 5ft>9 5598 33 8462 7981 7549 7156 6795 6462 6153 5864 5c94 34 8453 7974 7542 7149 6789 6457 6148 58HO 5589 ; 35 8445 7966 7535 7143 6784 6451 6143 5855 5585 | 36 8437 7959 7528 7137 6778 6-1-16 6138 5850 5580 37 8428 7951 7522 7131 6772 6141 6133 5846 5576 I 38 8420 7944 7515 7124 6766 6435 6128 5841 5572 00 39 8411 7936 7508 7118 6761 6430 6123 58136 5567 40 8403 7929 7501 7112 6755 6425 6118 5832 5563 8395 7921 7494 7106 6749 6420 6113 5827 5559 42 8386 7914 7488 7100 6743 6414 610S 5823 5554 43 8378 7906 7481 7093 6738 64U9 6103 5S18 5550 44 8370 7899 7474 7087 6732 6404 601)9 5813 5546 45 8361 7891 7467 7081 6726 6398 6094 5S09 5541 46 8353 7884 7461 7075 6721 6393 60S9 5F04 5537 47 8345 7877 7454 7069 6715 638S 60 H4 5800 6533 48 8337 7869 7447 7063 6709 6333 6079 5795 6528 49 8328 7862 7441 7057 6704 6377 6074 5790 5524 50 8320 7855 7434 7050 6372 6069 6786 6520 51 8312 , 7847 7427 7044 6692 6067 6064 6781 6516 - 52 8304 7840 7421 703S 6687 6362 6u:,9 6777 5511 53 8296 7832 7414 7032 6681 6357 6(155 5772 6507 54 8288 7825 7407 7036 6676 6351 6050 6768 6503 55 8279 7818 7401 70-JO 6670 6346 6045 6763 6498 56 8271 7811 7394 7014 6664 6341 6040 6758 6494 67 8263 7803 7387 7008 6659 633b 60:35 6754 5490 68 8255 7796 7381 7002 6653 63'<1 6(180 5749 5488 59 8247 7789 7374 6996 6648 6325 6025 6745 6481 M 8239 7782 7368 6990 6642 6320 6021 5740 6477 TABLR XLV. PROPORTIONAL LOGARITHMS 49 17' 18 19' 20' 21' 22' 23' 24' 25' 9 5477 5229 4994 4771 4559 4357 4164 3979 3802 1 5473 5225 4990 4768 4556 4354 4161 3976 379S 2 59 5221 4986 4764 4552 4351 4158 3973 3796 5-164 5217 4983 4760 4549 4347 4155 3970 3793 5460 5213 4979 4757 4546 4344 4152 3967 3791 6456 5209 4975 4753 4543 4341 4149 3964 3788 5452 52v)5 4971 4750 4539 4338 4145 3961 3785 5447 5201 4967 4746 4535 4334 4142 3958 3782 5443 5197 4964 4742 4532 4331 4139 3955 3779 5439 5193 4960 4739 4528 4328 4l: S952 3776 10 5435 5189 4956 4735 4525 4325 4133 3949 3773 11 5430 5185 4952 4732 4522 4321 4130 3946 3770 12 5426 5181 4949 4728 4518 4318 4127 3943 3768 13 54:32 5177 4945 4724 4515 4315 4124 3940 3765 14 5418 5173 4941 4721 4511 4311 4120 3937 3762 15 5414 5169 4937 4717 4508 4308 4117 3934 3759 16 5409 5165 4933 4714 4505 4305 4114 3931 3756 17 5405 51fl 4930 4710 4501 4302 4111 3928 3753 18 5401 5157 4926 4707 4498 4298 4108 3925 3750 19 5397 5153 4922 4703 4494 4295 4105 3922 3747 20 5393 5149 4918 4699 4491 422 4102 8919 3745 21 5389 5145 4915 4696 4488 4289 4099 3917 3742 22 5384 5111 4911 4692 4484 4285 40% 3914 3739 23 5380 5137 4907 4689 4481 4282 4092 3911 8736 24 5376 5133 4903 4685 4477 4279 4089 3908 3733 25 5372 5129 4900 4682 4474 4276 4086 3905 3730 26 5368 5125 4896 4678 4471 4273 4083 3902 3727 27 5364 5122 48H2 4675 4467 4269 40SO 3899 3725 28 5359 5118 4889 4671 4464 4266 4077 3a96 3722 29 5355 5114 4885 4668 4460 4263 4074 3893 3719 30 5351 5110 4881 4664 4457 4260 4071 3890 3716 31 5347 5106 4877 4660 4454 4256 4068 3887 3713 32 5343 5102 4874 4657 4450 4253 4065 3884 3710 33 5339 5098 4870 4653 4447 4250 4062 3881 3708 34 5335 5C94 4866 4650 4444 4247 4059 3878 3705 35 5331 5090 4863 4646 4440 4244 4055 3875 3703 36 5326 5086 4859 4643 4437 4240 4052 3872 3699 37 5322 5032 4855 4639 4434 4237 4049 3869 3696 38 5318 5079 4852 4636 4430 4234 4046 3866 3693 39 5314 5075 4848 4632 4427 4231 4043 3863 3691 40 531U 5071 4844 4629 4424 4228 4040 3860 3688 41 5306 5067 4841 4625 4420 4224 4037 3857 3685 42 5302 5063 48)7 4622 4417 4221 4034 3855 3682 43 5298 5059 4833 4618 4414 4218 4031 8852 3679 44 5294 5055 4830 4615 4410 4215 4028 3849 3677 45 5290 5051 4826 4611 4407 4212 4025 3846 3674 46 5285 5048 4822 4608 4404 4209 4022 3843 3671 47 5281 5044 4819 4604 4400 4205 4019 3S40 3668 48 5277 5040 4815 4601 4397 4202 4016 3837 3665 49 5273 5036 4811 4597 4304 4199 4043 3834 3663 50 5269 5032 4808 4594 4390 4196 4010 3831 3660 51 5265 5028 4804 4590 4387 4193 4007 3828 3657 52 6261 5025 4800 4587 4384 4189 4004 3825 3654 53 5257 5021 4797 4584 4380 4186 4001 3922 3651 54 5253 5017 4793 4580 4377 4183 3998 3820 3649 55 5249 5013 4789 4577 4374 4180 3995 3317 3616 H 5245 o009 4786 4573 4370 4177 3991 3814 3643 57 52-11 5005 4782 4570 4367 4174 3988 3811 3640 58 5237 5002 4778 4566 4364 4171 3985 8808 3637 59 5233 4998 4775 4563 4361 4167 3982 3805 3635 60 5229 4994 4771 4559 4357 4164 3979 4802 3633 50 TABLE XLV. PROPORTIONAL LOGARITHMS. 27' 29' 30' 31' 32' 33' 34' 3623 3621 3618 3615 3612 3610 3607 3604 3601 3590 3587 3576 3574 3571 3555 3544 3541 3535 3533 3530 3525 3519 3516 3514 3511 3503 3500 3497 3484 3479 3476 3473 3471 3468 3465 3463 3460 3457 3454 3452 3449 3446 3444 3441 3438 3436 3433 3431 3423 3420 3-117 3415 3412 3409 3407 3404 3401 3307 3305 3300 3274 3271 3264 3248 3241 3378 3372 3370 3367 3357 3346 3344 341 3313 3218 3213 3210 3203 3200 3198 3195 3190 3180 3178 3175 3173 3170 3160 3158 3155 3153 3150 3148 3145 3143 3140 3138 3135 3130 3123 3120 3118 3115 3113 3110 3108 3105 3103 3101 3098 3096 3091 3078 3076 3073 3071 3064 3061 3056 3054 3052 3049 3047 3044 3042 3039 3037 3010 3008 3005 3003 3001 2991 2977 2974 2972 2955 2953 2950 2943 2934 2915 2847 2730 2728 2725 2723 2721 2719 2716 2714 2712 2710 2707 2705 2703 2701 3018 3015 3013 3010 2877 2817 2815 2S10 2801 2796 2794 2787 2785 2782 2780 2778 2775 2773 2771 2769 2766 2762 2760 2757 2755 2753 2750 2748 2746 2744 2741 2737 2732 2730 2678 2672 2669 2667 2594 2581 2579 2577 2574 2572 2570 2564 2561 2544 2542 2540 2467 2465 2454 2452 2450 2448 2445 3443 2441 2439 2437 2431 2429 2426 2424 2418 2416 2414 2412 2410 2408 2405 2401 2647 2645 2643 2640 2634 2616 2614 2812 2610 2607 2518 2516 2514 2512 2510 2501 2490 2477 475 2473 2471 2469 2374 2372 2370 2351 2347 234? 2341 TABLE XLV PROPORTIONAL LOGARITHMS. 51 35' 2341 2339 2?37 2335 2233 2331 2324 2322 2320 2318 2316 2314 2312 2310 2294 2279 2277 2275 2273 2271 2261 2259 2257 2231 36' 2218 2216 2214 2212 2210 2J08 220(5 2204 2202 2200 2198 2196 2194 2192 2190 2188 2184 2182 2180 2178 2176 2174 2172 2170 2167 2165 2163 2161 2157 2155 2153 2151 2149 2147 2145 2143 2141 2137 2135 2131 2127 2125 2123 2121 2119 2117 2115 2113 2111 2109 2107 2J05 2103 2IUJ 37' 2086 2084 2082 2078 2076 2074 2072 2070 2066 2062 2061 2057 2055 9053 2051 2049 2047 2045 2043 2041 2037 2024 2018 2016 2014 2012 2010 2039 2007 2005 2001 1999 1997 1984 38' 1982 1980 1978 1976 1974 1972 1970 1968 1967 1965 1963 1961 1959 1957 1955 1951 1950 1948 1946 1944 1942 1940 1988 1936 1934 1933 1931 1919 1918 1916 1914 1912 1910 1908 1906 1904 1903 1901 1897 1895 1891 1884 1878 1876 1875 1873 1871 39' 1871 1869 1867 1854 1850 1849 1847 1845 1841 1839 1838 1836 1834 1832 1830 1821 1819 1817 1816 1814 1812 1810 1808 1806 1805 1803 1801 1799 1797 1795 1794 1792 1790 1788 1786 1785 1783 1781 1779 1777 1775 1774 1772 1770 1768 1766 1765 1763 1761 1761 1759 1757 1755 1754 1752 1750 1748 1746 1745 1743 1741 1739 1737 1736 1734 1732 1730 1728 1727 1725 1723 1721 1719 1718 1716 1714 1712 1711 1709 1707 1705 1703 1702 1700 1687 1686 1684 1678 1677 1675 1673 1671 1670 1663 1661 1(559 1657 1655 1654 41' 1654 1652 1650 1648 1647 1645 1643 1641 1640 1638 1636 1634 1633 1631 1629 1627 1617 1615 1613 1612 1610 1608 1606 1605 1603 1601 1596 1589 1587 1585 1584 1578 1577 1575 1573 1571 1570 1568 1566 1565 1563 1561 1559 1558 1554 1552 155* 1549 42' 1549 1547 1546 1544 1542 1540 1539 1537 1535 1534 1532 1530 1528 1527 1518 1516 1515 1513 1511 1510 1508 1506 1504 1503 1501 1499 1498 1496 1494 1493 1491 1487 1481 1479 1477 1476 1474 1472 1470 1469 1467 1465 1464 1460 1459 1457 1455 1454 1452 1450 1449 1447 43' 1447 1445 1443 1442 1440 1438 1437 1435 1433 1432 1430 1428 1427 1420 1418 1417 1416 1413 1412 1410 1408 1407 1405 T403 1402 1400 1398 1397 1395 1390 1388 1387 1382 1380 1378 1377 1375 1373 1372 1370 1363 1362 1360 1359 1357 1355 1354 1352 1350 1349 1347 52 TABLE XLV. PROPORTIONAL LOGARITHMS. 44' 45- 46' 47' 48' 49' 50' 51' 52* 0" 1347 1249 1154 1061 P69 880 792 706 621 1 1345 ! 1248 1152 1059 968 878 790 704 620 2 1344 1246 1151 1057 966 877 789 703 619 3 1342 1245 1149 1056 965 875 787 702 617 4 1340 1^3 1148 1054 963 874 786 700 616 5 1339 1241 1146 1053 962 872 785 699 615 6 1337 1240 1145 1051 960 871 783 697 613 7 1335 1238 1143 1050 959 769 782 696 612 8 1334 1237 1141 1048 957 868 780 694 610 9 1332 1235 1140 1047 956 866 779 693 609 10 1331 1233 1138 1045 954 865 777 692 608 11 1329 1232 1137 1044 953 863 776 690 606 12 1327 1230 1135 1042 951 862 774 689 605 13 1326 1229 1134 1041 950 860 773 687 603 14 1324 1227 1132 1039 948 859 772 686 602 15 1322 1225 1130 1037 947 857 770 685 601 16 1321 1224 1129 1036 945 856 769 683 599 17 1319 1222 1127 1034 944 855 767 682 598 1 1317 1221 1126 1033 942 853 766 680 596 19 1316 1219 1124 1031 941 852 764 679 595 20 1314 1217 1123 1030 939 850 763 678 594 21 1313 1216 1121 1028 938 849 762 676 592 22 1311 1214 1119 1027 936 847 760 675 591 23 1309 1213 1118 1025 935 846 759 673 590 24 1308 1211 1116 1024 933 844 757 672 588 25 1306 1209 1115 1022 932 843 756 670 587 26 1304 1208 1113 1021 930 841 754 669 685 27 1303 1206 1112 1019 929 840 753 668 584 28 1301 1205 1110 1018 927 838 751 666 583 29 1300 Ii03 1109 1016 926 837 750 665 581 30 1298 1201 1107 1015 924 835 749 663 680 31 1296 1200 1105 1013 923 834 747 662 579 32 1295 1198 1104 1012 921 833 746 661 577 33 1293 1197 1102 1010 920 831 744 659 576 34 1291 1195 1101 1008 918 830 743 658 574 35 1290 1193 1099 1007 917 828 741 656 673 36 1288 1192 k ]098 1005 915 827 740 655 572 37 1287 1190 1096 1004 914 825 739 654 570 38 1285 1189 1095 1002 912 824 737 652 569 39 1283 1187 1093 1001 911 822 736 651 568 40 1282 1186 1091 999 909 821 734 649 666 41 1280 1184 1090 998 908 819 733 648 665 42 1278 1182 1088 996 906 818 731 647 563 43 1277 1181 1087 995 905 816 730 645 562 44 1275 1179 1085 993 903 815 729 644 561 45 1274 1178 1084 992 902 814 727 642 559 46 1272 1176 1082 990 900 812 726 641 558 47 1270 1174 1081 989 899 811 724 640 657 48 1269 1173 1079 987 897 809 723 638 655 49 1267 1171 1078 986 896 808 721 637 664 50 1266 1170 1076 984 894 806 720 635 652 51 1264 1168 1074 983 893 805 719 634 651 62 1262 1167 1073 981 891 803 717 633 650 63 1261 1165 1071 980 890 802 716 631 648 64 1259 1163 1070 978 888 801 714 680 647 65 1257 1162 1068 977 877 799 713 628 646 86 1256 1160 1067 975 885 798 711 627 644 57 1254 1159 1065 974 884 796 710 626 553 68 1253 1157 1064 972 883 795 709 624 641 m 1251 fc ll$6 1063 971 881 793 707 623 640 60 1249 1154 1061 9ti8 880 792 706 621 639 TABLE XLV. PROPORTIONAL LOGARITHMS* 03 53' 54' 55' i 56' 57' 58' 59' 0" 539 458 378 300 220 147 73 1 537 456 377 298 221 146 72 2 536 455 375 297 2JO 145 71 3 5:35 454 374 296 219 143 69 4 533 452 373 294 218 142 68 5 532 451 371 2J3 216 141 67 6 531 450 370 292 815 140 66 7 529 448 369 291 U14 139 64 8 523 447 367 289 !)13 137 63 9 526 446 366 288 5111 136 62 10 525 444 365 287 felO 135 61 11 524 443 363 285 209 134 60 12 522 442 362 284 208 132 58 13 521 440 3H1 283 06 131 57 14 520 439 359 282 205 130 56 15 518 438 358 280 204 129 55 A 517 436 357 279 202 127 53 17 516 435 356 278 201 126 52 18 514 434 354 276 200 125 51 19 513 432 353 275 199 124 50 20 512 431 352 274 U7 122 49 21 510 430 350 273 t 121 47 22 509 428 349 271 195 120 46 23 507 427 348 270 K4 119 45 24 506 426 346 269 15-2 117 44 25 505 424 345 267 191 116 42 26 503 423 344 2.56 19o 115 41 27 502 422 342 2H5 189 114 40 28 501 420 341 2u4 187 112 39 29 499 419 340 2v2 186 111 38 30 498 418 339 2bl 185 110 36 31 497 416 337 280 184 109 35 32 495 415 336 258 182 107 34 33 494 414 335 257 181 106 33 34 493 412 333 256 180 105 31 35 491 411 332 255 179 104 30 36 490 410 331 253 177 103 29 37 489 408 329 252 176 101 23 38 487 407 328 251 175 100 27 39 486 406 327 250 174 99 25 40 484 404 326 248 172 98 24 41 483 403 324 247 171 96 23 42 482 402 323 246 170 95 22 43 4SO 400 322 244 169 94 21 44 479 399 320 243 167 93 19 45 478 398 319 242 166 91 18 46 476 396 318 241 165 90 17 47 475 395 316 239 163 89 16 48 474 394 315 238 162 88 15 49 472 392 314 237 Ifil 87 13 50 471 391 313 235 160 85 12 51 470 390 311 334 158 84 11 52 468 388 310 233 157 b3 10 53 467 387 309 232 156 82 8 54 466 3*6 307 230 155 80 7 55 464 384 306 2-jy 153 79 6 56 463 M 805 223 i58 78 5 57 462 3.S2 304 227 151 77 4 58 4tX) 381 i 302 225 150 75 a 59 459 379 301 224 148 74 1 60 458 378 300 223 147 73 TABLES. SATELLITES OF JUPITER. Sat. Mean Distance. Sidereal Revolu- tion. Inclination of Orbit to that of Jupiter. Mags ; that of Jupiter being 1000000000. 1 2 3 4 6 04853 9.62347 15.35024 26.99835 d. h. m. 1 18 28 3 13 14 7 3 43 16 16 32 O ' " 3 5 30 Variable. Variable. 2 58 48 17328 23235 88497 42659 SATELLITES OF SATURN. Sat. Mean Distance. Sidereal Revolu- tion. i Eccentricities and Inclinations. d. h. m. The orbits of the six inte- 1 33.51 22 38 rior satellites are nearly cir- 2 4.300 1 8 53 cular, and very nearly in the 3 5.284 1 21 18 plane of the ring. That of 4 6.819 2 17 45 the seventh is considerably 5 9.524 4 12 25 inclined to the rest, and ap- 6 7 22.081 64.359 15 22 41 79 55 proaches nearer to coincidence with the ecliptic. SATELLITES OF URANUS. Sat. Mean Distance. Sidereal Period. Inclination to Ecliptic. Their orbits are inclined 1? 13 120 5 21 25 about 78 58' to the ecliptic, 2 17.022 8 16 56 5 and their motion is retrograde. 3? 4 5? 19.845 22.752 45.507 10 23 4 13 11 8 59 38 1 48 The periods of the 2d and 4th require a trifling correction. The orbits appear to be nearly 6? 91.008 107 16 40 circles. TABLE OF ASTEROIDS. THE ASTEROIDS. The following tabular facts are from the most reliable sources the English Nautical Almanac and other Euro- pean publications. Planets. Mean dis- tance from the sun. Mean time of Revolu- tion. Eccentri- city of orbits. Lon. of the Ascending node. Inclination of orbit. Flora Earth's dis. 1. 22014 Days. 1193 16 15677 deg. m. 110 21 To Ecliptic. 5 53' Victoria 23348 1303 08 021854 2N5 40 8 23 * Vesta, 2.3627 132H.26 0.08945 103 24 7 08 23858 1345 6b 23232 259 44 5 28 Metis, 2.3868 1346.90 0.12274 68 28 5 36 Hebe 2.4256 1379.68 20200 138 32 14 47 Parthenope, . . . 2.4483 2.5H05 1399.06 1515.40 0.09800 0.16974 125 00 86 51 4 37 9 06 Astrea 26173 1547 58 18880 141 28 5 19 E-o-eria 25829 151582 08628 43 18 16 33 2.6679 1591.68 0.256.37 170 58 13 03 Ceres 27653 167986 07904 80 49 10 36 *Pallas 2.7715 1686.22 0.23894 172 37 34 42 2 6483 157408 18856 293 54 11 44 3.1512 2043.38 0.10090 287 38 3 47 Psyche 29771 1834 61 13082 150 36 3 04 Fortuna 25342 1 440 80 17023 211 17 1 32 Melpomene, Thetis 2.3292 2.4718 126981 1419.31 0.21644 0.12736 150 00 125 25 10 09 5 35 Lutetia 2 4353 1387 77 16104 80 34 3 05 'Calliope, Amphitrite,. . . 2.9054 2.5521 1809.00 1489.22 0.10308 0.06716 66 38 356 27 13 45 08 * We made an effort to arrange these planets in the order of their distances from the sun, and we have done so, as far as Hygeia The following ones were subsequent discoveries. Some future day, when their elements will be better known, by more varied and extended observations, a re -arrangement can be made. s *. BOOK IS D0E T ow