A TREATISE ON ASTRO-NOMY, DESCRIPTIVE, PHYSICAL, AND PRACTICAL. DESIGNED FOR SCHOOLS, COLLEGES, AND PRIVATE STUDENTS BY H. N. ROBINSON, A. M., ORXKmRLY PROFE8SOR OF MATHEMATICS IN THE U. S. NAVY; AUTHOR OF A TREATLIS ON ARITHMETIC; ALGEBRA; NATURAL PHILOSOPHY o ETC. ALBANY: ERASTUS H. PEASE & CO., 82 STATE STREET' CINCINNATI: JACOB ERNST, NO. 183 MAIN STREET. 1850. Entered, according to act of Congress, m the year 1849, BY HORATIO N. ROBINSON, Ia the Clerk's Office- of the District Court of the United States, for the District of Ohio. J. MUNSELL, STEREOTYPER AND PRINTER, ALBANY. PREFACE. To give at once a clear explanation of the design and in- PRErACE. 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 be 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 satisfaction to the real aspirant after astronomical knowledge. Of the less mathematical and more elaborate works on astronomy 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 immensity or grandeur of the heavens, but they give us no additional 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 motions 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 expressly to meet that demand. In the first part of the book we suppose the reader entirely unacquainted with the subject; but we suppose him competent 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 ) PREFACE. PasxcE. we have here designated, we believe, can take this book and leart 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 former 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 elementary 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 contracted form. The epochs for the sun and moon may be extended forward or backward, to any extent, by any one who understands the theory. The chapters on comets, variable stars, &c., are compilations, and are printed in smaller type; and the works to which we are most indebted, are Herschel'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, clearness 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 illustrations, particularly that of showing the rationale of tides rising on the opposite sides of the earth from the moon; and' in the general treatment of eclipses; but it is for others to determine how much merit should be awarded for such originalities; we have, however, used greater conciseness and perspicuity in general computations than is to be found in most of the books oft this subject; and this last remark will apply to the whole work. CONTENTS. SECTION I. Page. INTRODUcTION.-Defin ition of terms, &c.,..................... 1 CoNTlr1s, CHAPTER I. Preliminary Observations,................................... 6 A fixed point in the heavens —the pole and polar star,.......... 7 Index to the length of one year,............................. 11 Fixed stars-why so called,.............................. -12 CHAPTER II. Appearances in the heavQns,................................. 13 Important instruments for an Observatory,.............. 14 Standard measure for time................................... 16 An astronomical clock,....................................... 17 Movable and wandering bodies...................1............ 19 To find the right ascension of the sun, moon, and planets,...... 21 CHAPTER III. Refraction-'position of the equinox, &c.,..................... 21 Altitude and azimuth instrument............................. 23 Astronomical refraction-its effect, &c..................... 25 The declination of a star-how found,....................... 29 Observations to find the equinox,........................... 33 Length of the year-how observed,....................... 36 CHAPTER IV. Geography of the heavens,................................. 38 Method of tracing the stars,................................. 41 What constitutes a definite description,....................... 43 How to find the right ascension of any star,.................. 44 The southern cross and Magellan clouds,............... 48 SECTION II. DESCRIPTIVE ASTRONOMY. CHAPTER I. First consideration of the distances to the heavenly bodies-size and figure of the earth,................................... 49 How to find the diameter of the earth,.................. 53 (v) Vi CONTENTS. CONTENTS., Page. Dip of the horizon........................................ 54 The exact dimensions of the earth......................... 56 Gravity of the earth diminished on its surface by its rotation,... 58 A degree between two meridians-the law of decrease.........61 CHAPTER II. Parallax, general and horizontaly........................... 62 Relation between parallax and distance...................... 63 Lunar parallax- ow found,..................... 65 Variable distance to the moon............................ 67 Apogee and perigee..........................................67 Mean parallax -and parallax at mean distance,........... 68 Mean distance to the moon,............................. 69 Connection between semidiameter and the horizontal parallax of any celestial body....................................... 70 The earth a moon to the moon................... 71 CHAPTER III. The earth's orbit eccentric, &c........................ 72 Methods of measuring apparent diameters,...................... 73 Eccentricity of the earth's orbit- how found............ 76 Variations in solar motion................................... 76 Eccentricity of orbit and greatest equation of center connected,.. 85 CHAPTER IV. Causes of the change of seasons............................ 87 Temperature of the earth,..................... 89 Times of extreme temperature,............................. 90 CHAPTER V. Equation of time........................................... 90 Mean and apparent noon................................. 91 What is meant by sun slow and sun fast............... 94 Use of the equation of time.................-.............. 96 CHAPTER VI. Apparent motions of the planets,........................... 97 The morning and evening star.............................. 98 Motion of Venus in respect to the fixed stars................ 101 Retrograde motion of planets accounted for................. 102 CHAPTER VII. First approximnations of the relative distances of the planets from the sun................................................. 104 Whatto understand by stationary,......................... 106 Method of approximating to the orbits of planets,............ 108 CONTENTS vii CHAPTER VIII. Page..Methods of observing the periodical revolutions of the planets,.. 111 CII wrj., Diurnal motion of the planets,............................... 114 Times of revolution and distances compared,...............,. 117 Kepler's Laws,.......................... 118 CHAPTER IX. Transits of Venus and Mercury,..................... 119 Periods of the transits of Venus,.......120, 121, 122 Deductions from a transit made plain,........................ 124 CHAPTER -X. The horizontal parallaxes of the planets computed,.127 Real distance between the earth and sun determined,.. 128 How to find the magnitudes of the planets,............... 130 CHAPTER X-I. A general description of the planets,..................... 131 Professor Bode's law of planetary distances,......134 A bold hypothesis..........1......................... 135 Progressive nature of light —how determined,...'............. 139 CHAPTER XII. Comets................................................... 144 Inclinations of their orbits.......................... 147 Fears anciently entertained concerning comets,........... 150 CHAPTER XIII, On the peculiarities of the fixed stars,........................ 150 SECTION III. PHYSICAL ASTRONOMY CHAPTER I. General laws of motion-theory of gravity,................. 157 Attraction of a sphere —of a spherical shell, &c.,....... 161 A general expression for the mutual attraction of two bodies,.... 162 CHAPTER II. Demonstration of Kepler's Laws,..................... 163 A common error,...................167 How a planet finds its own orbit,.......................... 168 Kepler's third law rigorously true in circles and ellipses,........ 171 VUIIC ONT~ENTS. CHAPTER III. Page. CoTmirrs. Masses of the planets, &c,...................................... 174 i The diameter of the earth accurately determined from equations in physical astronomy, ( Art. 171 ), 180 The mass of the moon determined- densities of bodies,......... 183 CHAPTER IV. Lunar Perturbations,...................................... 185 Cause of nutation.......................................... 187 Mean radial force,...................................... 191 Acceleration of the moon's mean motion,..................... 197 A summary statement of the cause,......................... 198 The true mean value of the radial force,...................... 199 A summary statement of the lunar irregularities,.............. 200 CHAPTER V. The tides................................................. 201 A summary illustration of the physical cause of tides,.......... 202 Mass of the moon computed from the tides,.................... 204 CHA.PTER VI. Planetary perturbations,.................................... 206 Action and reaction equal and contrary,....................... 206 The effects of commensurate revolutions of the planets,........ 208 -The great inequalities of Jupiter and Saturn,.................. 209 These inequalities explained,................... 209, 210, 211, 212 The physical effects on Uranus that led to the discovery of Neptune, 213 CHAPTER VII. Aberration —nutation, and precession of the equinoxes,......... 213 The' velocity of light computed from the effect of aberration,.... 215 Cause of nutation explained,................................. 218 The physical cause of the precession of the equinoxes,......... 222 _Poper motion of the stars-how found,...................... 224 The latitude of the sun explained,........................... 226 SECTION IV. PRACTICAL ASTRONOMY. Preparatory remarks and trigonometrical formula,............. 229 CHAPTER I. Problems in relation to the sphere,...23................. 232 To find the time, from the latitude of the place —altitude, and declination of the sun,............................... 239 CONTENTS ix Page. An artificial horizon,..................................... 242 coNraEh,. Absolute and local time,................................... 244 Lunar observations,.........................2 246 Proportional logarithms,..................................... 247 CHAPTER II. Explanation of the tables.................................. 250 To compute the suh's longitude,.'.......................... 253 To find the equation of time to great exactness,................ 254 To compute the time of new and full moons,................. 255 Eclipses —when they occur, &c.,......................... 257 Limits of eclipses.................................... 259 Periods of eclipses,...................................261, 263 Elements of lunar eclipses,................................. 264 Semidiameter of the earth's shadow,..................... 265 CHAPTER III. Preparation for the computation of eclipses,................... 267 Directions for computing the moon's longitude, latitude, &c.,... 268 To construct a lunar eclipse,.............................. 275 To make exact computations in respect to lunar eclipses,....... 277 CHAPTER IV. Solar eclipses-general and local,....................... 279 Elements for the computation of a solar eclipse,............... 280 To construct a general eclipse,.............................. 281 How to determine the duration of a solar eclipse on the earth,... 282 To find where the sun will be centrally eclipsed at noon,........ 286 Results taken from the projection,........................... 287 Results from trigonometrical computations,.................... 288 CHAPTER V. Local eclipses, &c.,.,..................................... 291 How to construct a local solar eclipse,................ 291 How to find the time of greatest obscuration,............. 295 To find the time of the beginning, end, &c., of a local eclipse by the application of analytical geometry,..................... 296 Catalogue of eclipses which will take place between the years 1850 and 1900,................................... 300 AS TRONO MY. INTRODUCTION, AsTRONOMY is the science which treats of the heavenly.. Astamownl bodies, describes their appearances, determines their magni- defined. tudes, and discovers the laws which govern their motions. When we merely state facts and describe appearances as The divi. they exist in the heavens, we call it Descriptive Astronomy. sions or' ao tronomy, 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 astronomy, their corresponding computations, are called Nautical Astronomy. By nautical astronomy we determine positions on the Geographs earth, and subsequently, the magnitude of the earth; and mand astrono my united. 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 anti, the earliest age, the people could not have avoided observing qitomy.of a8. the successive returns of day and night,-and summer and winter. They could'not fail to perceive that short days corresponded 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. (1) ASTRONOMY. oNTRODUc. In this work, we shall not take facts unless they are within Facts alone the sphere of our own observations. We shall not peremptonot science. rily state that the 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 magnitude, and is thus far from the other heavenly bodies; and that will be knowledge of the most essential kind. The foun- All astronomical knowledge has its foundation in observaatronomical tion; and the first 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 supposition 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. Conven- Every science has its technicalities and conventional terms; tienal terms tia defini. and astronomy is by no means an exception to the general tions. rule; and as it will prepare the way for a clearer understanding 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 may be, 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 hxoizn. INTRODUCTION. Altitude.-The perpendicular hight from the horizon, INTrrODc. measured by degrees of a circle.:eridian. -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 Poles of overhead; and the Nadir is directly opposite to the zenith, or the horizon. under our'feet. The zenith and nadir are the poles to the horizon. Verticals. All lines passing from the zenith, perpendicu- rrime vet. lar to the horizon, are called Verticals, or Vertical Circles. tical. 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 meridian,t measured on the circle of the horizon, is called its Azimuth. The angular position, measured from its prime vertical, is Amplitude called its Amplitude. The sum of the azimuth and amplitude must always make 9O, degrees. Equator.- The Earth's Eputator 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 Celest,,' conceived to extend into the heavens. equator. When the sun, or any other heavenly body, meets the Equinoc. celestial equator, it is said to be in the Equinox, and the tial. 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 so-uth. If the measure is toward the north, it is north latitude; if toward the couth, south latitude. 4 ASTRONOMY. IrRODUC. The distance from the equator to the poles is 90 degreesone-fourth of a circle; and we shall know the circumference of the whole earth, whenever we can find the absolute length of onme 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. earth, 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. The first Other nations generally take the meridian of their princimeridian ar- pal observatories, or that of the capital of their country, as -bitrary the first meridian; but this is national vanity, and creates only trouble and confusion: it is important that the whole 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. We adopt In this work, we shall adopt the meridian of Greenwich as the meridian of Green. the zero line of longitude, because most of the globes and with; and maps, and all the important astronomical tables, are adapted why! to that meridian, and we see nothing to be gained by changing them. Declination.- Declination refers only to the celestial equator, 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 2700. As latitude is reckoned north and south, so longitude is INTRODUCTION. 5 reeckoned:east and west; but it would add greatly to syste- INTRODUC. matic regularity, and tend much to avoid confusion and am- Improve. biguity in computations, were this mode of expression aban- ment Sugdoned, and longitude invariably reckoned westward, from 0 to gested. 360 degrees. Latitude and longitude, on the earth, does not corre- Latitude, spond to latitude and longitude in the heavens. Latitude, on longitude,s the earth, corresponds with declination in the heavens; and cension. longitude, on the earth, has a striking analogy to right ascension in the heavens, thoughnot 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 fare forced to use these terms, that the lature and spirit of their import can be really understood. There are other technicalities, and terms of frequent use, Otherterms in astronomy, such as Conjunction, Opposition, Retrograde, not explain. 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 minds of our readers, that we shall put far more stress on the substance and spirit of a thing, than on its name. ASTRONOMY. SECTION I. CHAPTER I. PRELIMINARY OBSERVATIONS. CRaP. 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 directions of north, south, east, and west. The appa. At first, let us suppose our observer to be somewhere in rent motion of the stars,. the 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 instrument; but we would have him mark with attention the positions 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 horiz n; those at the south, and near his zenith, will apparently move westward; and those at thenorth 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 revoltion of of the night, and for several nights, and the observer cannot the heavenly bodies. 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 a fixedpoint; and that fixed point, as appears to us (in the middle and northern part of the United States), is about midway between the northern horizon and the zenith. Large and It should always be borne in mind, that the sun, moon, and small circles. stars, have an apparent diurnal motion round a fixed point, PRELIMINARY OBSERVATIONS. 7 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 apparient motion of this star, we must resort to nice observations, aided by mathematical 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' Positionof (i" from the pole, and on 1st of January, 1847, its distance the North Star. 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 pole as ii motion; but this is not so. The fixed stars are abso- in motion. hutely fixed; it is the pole itself that has a slow motion among the stars, but the cause of this motion cannot now be explained; 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 observer, it has always been taken as the infallible guide to direction; and every sailor of the ocean, and every wanderer t-)f the African and Arabian deserts, has held familiar acquaintance with it. (4.) If our observer now goes more to the southward, and Changes of makes the same observations on the apparent motions of the appearance on going stars, he will find the same general results; each individual southward. 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 mfficieatly far, the fixed point, the pole, will no longer be up 2 8 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 ance from equator, and from that line all the stars at or near the equathe equator. 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 this perpendicular equatorial circle, South of If the observer goes south of the equator, the apparent the equator. 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.) If the observer should go north, from the first appearance on going station, in 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 the earth, or at the latitude of 90 degrees: App"ear' (7.) Any celestial body, which is north of the equator, is ance from the north always visible from the north pole of the earth; hence the pole. sun, which is north of the equator from the 20th of March to the 23d of September, must be constantly visible during that period, in a clear sky. Just as the sun comes north of the equator, its diurnal progress, or rather, the progress of 24 hours, is around the horizon. When the sun's declination is 10 degrees north of the equator, the progress of 24 hours is around the horizon at the altitude of 10 degrees; and so for any other degree. From the north pole, all directions, on the surface of the earth, are south. North would be in a vertical direction toward the zenith. How to TWe have observed that the pole of the heavens rises as we find the cir. eumference go north, and sinks toward the horizon as we go south; and andeiameter when we observe that the pole has changed its position one of the earth. degree, in relation to the horizon, we know that we must have changed place one degree on the surface of the earth. PRELIMINARY OBSERVATIONS. 9 (8. ) Now we know by observation, that if we go north C1AsP, I about 691 English miles on the earth, the north pole will be one degree higher above the horizon, Therefore 69~ miles corresponds to one degree, on the earth; and hence the whole circumference of the earth must be 69{ multiplied by 360: for there are 360 degrees to every circle. This gives 24,930 miles for the circumference of the earth, and 7,930 miles for its diameter, which is not far from the truth. (9.) Here, in the United States, or anywhere either in Circumpo. Europe, Asia, or America, north of the equator, say in lati- lar stars. tude 400, the north pole of the heavens must appear at an altitude of 400 above the horizon; and as all the stars and heavenly bodies apparently circulate round this point as a center, it follows that all those stars which are within 400 of the pole can never go below the horizon, but circulate round and round the pole. All those stars which never go below the horizon, are called circum-polar stars. At the north, and very near the north pole, the sun is a The sun a sircumpolar body while it is north of the equator, and it is a circumpolareen eircumpolar body as seen from the south pole, while it is south from the of the equator; this gives six months day and six months north of lat. tude 66 denight, at the poles. grees ( 10. ) North of latitude 66~, and when the- sun's declinanation is more than 230 north (as it is on and about the 20th of June in each year), then the sun comes at, or very near, the northern horizon, at midnight; it is nearly east, at 6 o'clock in the morning; it is south, at noon, and about 23~ in altitude; and is nearly west at 6 in the afternoon, ( 11. ) In the southern hemisphere, there is no prominent star near the south pole; that is, no southern polar star; but, of course, there are circumpolar stars, and more and more as one goes south; and if it were possible to: go to the south pole, the whole southern hemnisphere would consist. of circumpolar stars, and the pole, or fixed point of the heavens, would be directly overhead; and the sun himself, when south of the equator, would be a circumpolar body, going round and round every-24 hours, nearly parallel with the horizon. (12. ) In all latitudes, and fromii allT plaC: the sun is 10' ASTRONOMY. CHAP. I. observed to circulate round the nearest pole, as a center; and The near- when the sun is on the same side of the equator as the ohest pole is server, more than half of the sun's diurnal circle is above the the center of the sun's di- horizon, and the observer will have more than 12 hours sunlrnal mo- light. tion. When the sun is on the equator, the horizon, of every latitude, cuts the sun's diurnal circle into two equal parts, and gives 12 hours day, and 12 hours night, the world over. When the sun is on the opposite side of the equator from the observer, the smaller segment of the sun's- diurnal circle is above the horizon, and, of course, gives shorter days than nights. We have, thus far, made but rude and very imperfect observations on the apparent motion of the heavenly bodies, and have satisfied'ourselves only of two facts: Facts set. 1. That all the stars, sun, moon, and planets included, apparently circulate round the pole, and round the earth, in a day, or in about 24 hours. 2. That the sun comes to the meridian, at different altitudes above the horizon, at different seasons of the year, giving long days in June, and short days in. December. (13.) Let us now pay attention to some other particulars. Let us look at- the different groups of stars, and individual stars, so that we can recognize them night after night. Necessity We should now have some means of measuring time; but, f hare vf g in early days, when astronomy was no further advanced than time, it is supposed to be in this work, a clock could hardly have had existence; and the advancement of timepieces has been nearly as gradual as the advancement of astronomy itself. But we will not dwell on the history, and difficulties, of clockmaking: whatever these difficulties may have been, or whatever niceties modern science and art may have attained, there never was a period when people had not a good general idea of time, and some means to measure it. For instance, sunrise and sunset could be always noted as distinct points of time; and the interval of a day and a night, or an astronomical day, which we now call 24 hours, was soon observed to be a constant quantity. PRELIMINARY OBSERVATIONS. 11 At first, only rude timepieces could be made, designed to CHAP. I. marks off equal intervals of time; but we will suppose, at once, that the reader of this work, or our imaginary observer, can have the use of a common clock, which measures mean solar time of 24 hours in a natural day, which is marked by the sun. (14.) Now, having power to recognize certain stars, or The parti. groups of stars, such as the Seven Stars,,the Belt of Orion, tilnfr posiAldebaran, Sirius, and the like, and having likewise the use in relation to of a clock, he can observe when any particular star comes to time. any definite position. Let a person place himself at any particular point, to the n'orth of any perpendicular line, as the edge of a wall or Ibuilding, and let him observe the stars as they pass behind the building, in their diurnal motions from the east to the west. For example, let us suppose that the observer is watching the star Aldebaran, and that, when the eye is placed in a particular'definite position, the star passes behind the building at exactly 8 o'clock. The next evening, the same star will come to the same point about 4 minutes before 8 o'clock; and it' will not come to the same point again, at 8 o'clock in the evening, until after the expiration of one year. (15.) But in any year, on the same day of the month, and at the same hour of the day, the same star will be at, or very near, the same position, as seen from the same point. For instance, if certain stars come on the meridian at a On stsr particular time in the evening, on the first day of December, coming to the same stars will not come on the meridian again, at the dian. same time of the night, until the first day of the next December. On the first of January, certain stars come to the meridian Index to at midnight; and ( speaking loosely) every first of January the length of the same stars come to the meridian at the same time; and yea there will be no other day during the whole year, when the same stars will come to the meridian at midnight. Thus, the same day of every year is observed to have the same position of the stars at the same hour of the night; and this is the most definite indexfor the expiration of a year. .1.2 ASTRONOMY. CaHP.. I ( 16.) The year is also indicated by the change of the sun's Another declination, which the most careless observer cannot fail to index of the notice. On the 21st of June. the sun declines about 23- delength 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 following 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 correspondence, in relation to time. Fixed In all our observations on the stars, we notice that their stars; why apparent relative situations are not changed by their diurnal this term is appa applied. motions. In whatever parts of their circles they are observed, or at whatever hour of the night they are seen, the same configuration 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 resembling 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 ing stars. respect to each other, they are called fixed 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 astronomy. Planets. In the earliest ages, those stars that changed their places, were called wandering stars; and they were subsequently found to be the planetary bodies of the solar system, like the earth on which we live. APPEARANCES JN THE HEAVENS. 13 CHAPTER II. APPEARANCES IN THE HEAVENS. IN the preceding chapter we have only called to mind the CHAP. iI. most obvious and preliminary observations, which force themselves 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 was on the equator, the poles, to him, would be in his horizon. find the latiIf he were at one of the poles, for instance, the north pole, the place of obequator would then bound the horizon. If he were half way seration. 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 horizon, 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 investigation of astronomy, and not making clocks or mathematical 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 Sourceo. must be taken into consideration, in making astronomical error in ma.-'e rvintg observe. observations. 1. Uncertainty as to the exact time. 2. Inex- tionB 14 ASTRONOMY. CHAP. II. pertness and want of tact in the observer; and 3. Imperfection in the instruments. Everything done by man is necessarily imperfect. Practical "It may be thought an easy thing," says Sir John Herdiffictie8s selle "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 considered that, owing to the application of telescopes to the purposes of angular measurement, every imperfection of structure 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 imperfection 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 temperature, and their unavoidable flexure or bending by their own weight, become perceptible and measurable." Necessary ('20.) The most important instruments, in an-observatory, stuments. aside from the clock, area 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 adjusted 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 fassit instrumetnt. tened on a horizontal axis, east and west, so that the telescope itself moves up and down in the plane of the meridian, but can never be turned aside from the meridian to the east or west. APPEARANCES IN THE HEAVENS. 15 T9 o:place the instrument in this' posi- CHAP. 1,. tion, is a very difficult matter; but it is a difficulty which, at present, should not come under consideration: we simply i conceive it so placed, ready for observations. "In the focus of the eyepiece, and at A line in right angles to the length of the tele- Transit Instument the transit instrument a scope, is placed a system of one horizontal and five equidis- visible meritant vertical threads or wires, as represented in the annexed dian. 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, introduced 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 Meridian Wires, position, that the middle one of the vertical wires shall intersect tAe 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 cplestial meridian. The instant of this event is noted by the clock or chronometer, which forms an indispensable accompaniment of the transit instrument. For greater precision, Practical the moment of its crossing each of the vertical threads is attain aes, noted, and a mean taken, which (since the threads are equi- racy. distant ) would give exactly the same result, were all the hbservations 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 between tle clock, we fix on some bright star, and mark when it comes to fixed stars the meridian, or appears to pass behind the central wire of the passing the meridian al. instrument. By noting the same event the next evening, theways connext, and the next, we find the interval to be very sensi- stant, 16 ASTRONOMY. CHAP. I. 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 Lacroix, in the small island of Formentera, in the Mediterranean, in December, 1807. Time of transit of the Intervals between Star ar Arietis. successive Transits. h. m. s. h. m. s. 1807, Dec. 24, 9 42 32.36'.... 25, 9 41 29.70 23 58 57.34!.. 26, 9 40 26.72 23 58 57.02 n.... 27, 9 39 23.90 23 58 57.18 e" "; 28, 9 38 21.38 23 58 57.48 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 astronoof measure mers in all ages, it completely establishes the fact, that all for time he fixed stars come to the meridian in exactly equal intervals 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 employs in turning on its axis; for if the star isfixed, it is a revolution on its axis. mark for the time that the meridian is in exactly the same position in relation to. absolute space. M. Arago's (23.) That the reader may not imbibe erroneous impresMock. sions, we remark, that the clock used for the preceding observations, made by M. Arago and Lacroix, ran too fast, if it was a common clock, and too slow, if it was an astronorical APPEARANCES IN THE HEAVENS. 17 clock. It was not mentioned which clock was used, nor was CHAP. Ii. 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 astroastronomical clock. Soon after the fact was established that ocal clock. 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,ils 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 an: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 unll pass the To deter. meridian at exactly'the same time, from one year's end to an- mno the ate of an astroother.* If the time is not the same, the clock does not run nomical clock. * 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 two successive transits of the same star over the meridian, to 24 hours, and to call this a sidereal day; which, infact, 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 sidereal day (not from the passage of any particular star across the meridian, but from the passage of the imaginary point in the heavens, where 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 successive 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 ideitical; the first point of Aries has a very slow motion westward among the stars, which is called the precession of the equinox, and 2 B* 18 ASTRONOMY. CHAP. 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 not equal meridian, and compare the elapsed intervalswith 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. 3 m. 56.5554 s. 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 sun From this fact, we are compelled to regard the sun, not as must have a fixed point; it must have motions, real or apparent, indereal or appatent 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 (which may be called a lunar day ), is observed to be about 24 h. 42 m.; the greatest, 25 h. 2 m.; and the mean, or average, 24 h. 54 m., 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, is (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.0 m. 0 s., the time between the transits of the same star, is - 24 h. 0 m. 0.00916 as. Every astronomer understands Art. (24) with this modification. APPEARANCES IN THE HEAVENS. 19 fixed body, like a fixed star, for then the interval would be CHAP. mII 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 10~ to 16 degrees,* traveling, or appearing to travel, through the whole circle 9f 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 ject of astro nomy. 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 Notions. (27.) Besides the sun and moon, several other bodies Other ~.movable cad were noticed as coming to the meridian at very unequal in- movable ain 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 sometimes equal to 24 sidereal hours. These facts show that these bodies have a real, or apparent 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 heavens from west to east. Only four such bodies ( ~tars ) were known to the ancients, Wanderin, namely, Venus, Miars, Jupiter, and Saturn. stars hnow I~-~~~, ~~~to the amn These stars are a portion of the planets belonging to our cients. solar system, and, by subsequent research, it was found that Modler discoveries the Earth was also one of the number. As we come down toimore modern times, several other planets have been discovered, namely, Mercury, U'ranus, Vesta, Juno, Ceres, Pallas, and, very recently (1846), the planet iVNptune.t * 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 discovery. As yet, we have really no idea of a planet, or a planetary aystem. 32Q.AST R ON TO MY. cAPv, UI. We shalblagain examine the meridian passages ofa the stt, moon, and planets, and deduce other important facts concerning them, besides that of their apparent, or real motions among the fixed stars. Observta. (28.) But let us return to the fixed stars. We have tions which tdetermine several times mentioned th9 fact, that the same star returns the meridiah to the same meridian again and again, after every interval of distances of 24 sidereal hours. So two different stars come to the merithe stars. 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 observed 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. Correosron- As 24 hours corresponds to the whole circle, 360 degrees, tlelce be- therefore one hour corresponds to 15 degrees; and 4 minutes, tween hEattsr and degrees. in time, to one degree of are, Hence, whatever be the observed interval of time between the passing of two stars over the meridian, that interval will determine the actual differencee of the meridians running through the stars; and when we know the position of any one, in relation to any celestial meridian, we know the positions of-all whose meridian observations have been thus compared. Right as, The position of a star, in relation to a particular celestial tension. meridian, is called Right Ascension, and may be expressed either in time- or degrees. Astrionomers have chosen that It is truAe, We might mention every fact, and every particular respecting' each planet; such as its period of revolution, size, distance from the sun, &e.; 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 deduced 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 presented. REFRACTION, 21 nmeiadian, for the first meridian, which passes through the CHAP. If. sun's center at the instant the sun crosses the celestial equa- First,ieri. tor in the spring, on the 20th of March. dian. Right ascension is measured from the first meridian, eastward, on the equator, all the way round the circle, from 0 to 360 degrees, or from Oh. 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 To find tho time between the coming of a star to the meridian, and the right oafscn coming of any other celestial body, as the sun, moon, planet, sun, moon, or comet, such difference, applied to the right ascension of the and planet,. 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 0 h. 0 m. 0 s. when the equinox is on the meridian; and, if it does so, and runs regularly, then the time that anybody passes the meridian by the clock, will give the right ascension of the body in time, without 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. WiFlREACTION. --- POSITION OF THE ]EQUINOX, AND OBILQUITY OF THE ECLIPTIC - HOW FOUND BY OBSERYATION. (30.) To determine the angular distance of the stars from CrAP. Iti. 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 22 ASTRON MY. cI.P. Mft. ( which is the latitude of the place of observation), he, of course, knows the distance between the zenith and the pole. Prepara. As the north pole is but an imaginary point, no star beingu tions fbr delefmining there, we cannot directly observe its altitude. But there is a the latitude very bright star near, the pole, called the, Polar Star, which, observa- as all other stars in the same region, apparently revolves ion.s. 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 gn a circle; but before we examine this fact, we will show how altitudes can be taken by the mural circle. The mural (31.) The mural, or tircle. Fig. 2. twall circle, is a large nmetallic circle, firmly fastened to a wall, so that its plane shall coincide with the plane of the meridian. h A perpendicular line through the center, Z2 V (Fig. 2), represents the zenith and nadir points; and at right angles to this, through the center, N is the horizontal line, Hh. fow toob. A telescope, Tt, and an index bar, Ii, at right angles to erve meri. the telescope, are firmly fixed together, and made to revolve.tfian altitudes. on the center of the mural circle. The circle is graduated from the zenith and nadir points, each way, to the horizon, from 0 to 90 degrees. When the telescope is directed to the horizon, the index points, I and i, will be at Z and V, and, of course, show 00 of altitude. When the telescope is turned perpendicular to Z, the inAex bar will be horizontal, and indicate 90 degrees of altitude. When the telescope is pointed toward any star, as in the REFRACTION. 28 figure, the index points, I and i, will show the position of the CHAP. I. 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 oir. freely round the whole circle, we can measure altitudes with clealso in transit init equally well from the north or the south; but as it turns strument. only in the plane of the meridian, we can observe only meridian altitudes with it. This instrument has been called a tranqsit 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 verified, as a transit may, by the reversal of the two ends of its axis, east for west.. Nothing, however, 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 simultaneous determination of the right ascensions and -polar distances of objects observed with it; the time of transit being noted by the clock, and the circle being read off by the lateral microscope." (32.) To measure altitudes in all directions, we must have Altituda and] azimuth another instrument, or a modification of this. i meut 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 azimuth instrument, because it can measure altitudes and azimuths at the same time. (33.) After astronomy is a little advanced, and the angular distance of each particular star, sun, moon, and planet, 3 X24 ASTRONOMY. CHaAr.. from the pole is known, then we can determine the latitude by The lati. observing the meridian altitude of any known celestial body.; tude takenl but before their positions are established ( as is now supposed by the altitude of the to be the case with the reader of this work ), the only way to pole. observe the latitude is by the altitudes of some circumpolar star, as mentioned in Art. 30. To settle this very important element, the observer turns the telescope of his mural circle to the pole star, and observes its greatest and least altitudes, and takes the half sum for his latitude. But is this really his latitude? Does it require any correction, and if so, what, and for what reason? A difficulty, At first, it was very natural to suppose that this gave the exact latitude; but astronomers, ever suspicious, chose to verify it, by taking the same observations on other circumpolar stars; and if the theory was correct, and the observations correctly taken, all circumpolar stars would give the same, or very nearly the same, result. Such observations were made, and stars at the same distance from the pole gave the same latitude, and stars at different distances from the pole:gave dilTerent latitudes; and the greater the distance of any star from the pole, the greater the latitude deduced from it. A star 30 or 35 degrees from the pole, observed from about the latitude of 40 degrees, will give the latitude 12 or 15 minutes of a degree greater than the pole star. New and Astronomers were now troubled and perplexed. These tmptstant great and manifest discrepancies could not be accounted for by imperfection of instruments, or errors of observations, and some unconsidered natural cause was sought for as a solution. Curves de. To bring more evidence to bear on the case, astronomers scribed by circumpolar examined the apparent paths of the stars round the pole, by stars. means of the altitude and azimutA instrument, and they were found to be not exact circles; but departed more and more from a circle, as the star was a greater and greater distance from the pole. These curves were found to be somewhat like ovals -the longer diameter passing horizontally through the pole - the ASTRONOMICAL REFRACTION. 25 upper segments very nearly semicircles, and the lower segments CHaP. m. flattened on their under sides. With such evidences before the mind, men were not long in deciding that these discrepancies were owing to ASTRONOMICAL REFRACTION. ( 34. ) It is shown, in every treatise on natural philosophy, General that light, passing obliquely from a rayer medium into a fect of re. denser, is bent toward a perpendicular to the new medium. Now, when rays of light pass, or are conceived to pass, from any celestial object, through the earth's atmosphere to an observer, the rays must be bent downward, unless they pass perpendicularly through the atmosphere; that is, come from the zenith. Let A B, CD Fig. 3. EF, &c. (Fig. 8 ), represent s different strata -of the earth's atmosphere. L et s be'a star, and conceive a line of light to pass from the star E through the va- c rious' strata of _ _a.i, to the observer, at 0.'When it meets the first strata, as EF, it is slightly bent Refraction downward; and: as the air becomes more and more dense, its increases alrefracting 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 determine the position of the star as seen by him. Hence the observer at 0 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 02 ARTTRONOMY.. MHkP. II. ward, therefore, as we may see by inspecting the figure, tMe 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 altitudes-and the resulting latitudes, deduced from them, all too great. The object is now to obtain the amount of the refraction eorresponding 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 amount of re. a star passes through, or very near his zenith, it has no refraction cor. fraction. But, at the equator, the diurnal circles are pereo every dde pendicular to the horizon; and those starswhich are very gree of alti. near the equator, really change their altitudes in proportion to Dude. the time. Now a star may be observed to pass the zenith, at the equator, at a particular moment: four hours afterward ( sidereal time ), the zenith distance of this star must be 4 times 15, or 60 degrees, and its altitude just 30 degrees. But, by observation, the altitude will be found to be 300 1' 38". From this, we perceive, that 1' 38" is the amount of refraction corresponding to 30 degrees of aItitude. In six sidereal hours from the time the star passed the zenith, the true p6sition 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 reof horizontal. ef hoationta fraction at the horizon. refraction, Thus, by taking observations at all intervals of time, between the zenith and the horizon, we can determine the refraction corresponding 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 .ASTRONOMICAL REFRACTION. 27 he 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 The mathewould be perfect circles, and the mathematician, by means of matician's method of the altitude and azimuth,'which can be taken at any and finding.the every point of a curve, tan determine how much it deviates amountofrefraction. 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 further approximate to the refraction. In this way, by a multitude of observations and computations, the table of refraction (which appears among the tables of every astronomical work ) was established and drawn out. (36.) The effect of refraction, as we have already seen, is Refraction to increase the altitude of all the heavenly bodies. There- increases the time of sunfore, 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?nore than half of -the earth's surface 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 thlie horizon is about 33' of'a degree, the length of a day, at the equator, is more thanfour 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 ia very materially increased, in length, by the effects of refrac- high lati. tudes. 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 28 ASTRONOMY. carP. m. zon; and this causes a visible distortion of both sun and moon, just as they rise or set. Distortion For instance, when the lower limb of the sun is just in the of the sun and moon in horizon, it is elevated, by refraction, 33'. tho 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 linib 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 illusion. near the horizon, has nothing to do with refraction: it is a mere illusion, and has no reality, as may be known by applying 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 student 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 invisible; 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. Application (37.) Having shown the possibility of making a table of ofrefraction. refraction corrtsponding to all apparent altitudes, we can now, by applying its effeets to the observed altitudes of the cireumpolar stars, obtain the true latitude of the plaee of obserl vation. Let it be borne in mind, that the latitude of any place 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 denionstrate this as. follows. Let E (Fig. 4) repre. ASTRONOMICAL REFRACTION. 29 sent the earth. Fig. 4. CHAP. m. Now, as an ob- A demonserver always con- stration. ceives himself to be on the topmost part of the earth, the vertical point, Z, truly and natu- H E O rally represents his zenith. Through E, draw iE 0, at right angles to E Z; then BE 0 will represent the horizon (for the horizon is always at right angles to the zenith). Let E Q represent the plane of the equator, and at right angles to it, from the center of the earth, must be the earth's axis; therefore, E P, at right angles to E Q, is the direction of the pole. Now the arcs, - ZP+P 0=900, Also, - - - ZP+Z Q=900, By subtraction, - P O —ZQ=O; Or, by transposition, the arc PO = ZQ; that is, the altitude of the pole is equal to the latitude of the place; which was to be demonstrated. In the same manner, we may demonstrate that the arc H Q is equal to the are Z P; that is, the polar distance of the zenith'is equal to the meridian altitude of the celestial equator. Now, we perceive, that by knowing the latitude, we know the several divisions of the celestial meridian, from the northern to the southern horizon, namely, 0 P, P Z, Z Q, and QH.L (38.) We are now prepared to observe and determine the declinations of the stars. The declination of a'star, or any celestial oyject, is its mert- Declinadian distancefrom the celestial equator. tion defind. This corresponds with latitude on the earth, and declination might have been called latitude. The term latitude, as applied in astronomy, is to be defined hereafter. 80 ASTRONOMY. CHAP. III. To determine the declination of a' star, we must observe How to its meridian altitude (by some instrument, say the mural lnatihed a te circle, Fig. 2 ), and correct the altitude for refraction (see clination ofa a star. table of refraction); the difference will be the star's true altitude. If the true meridian altitude of the star is less than the meridian altitude of the celestial equator, then the declination of the star is south. If the meridian altitude of the star is greater than the meridian altitude of the equator, then the declination of the star is north. These truths will be apparent by merelyinspecting Fig. 4. EX3:AMPLI;ES. Examples 1. Suppose an observer in the latitude of 400 12' 18",, thod pursued north, observes the meridian altitude of a star, from the to find any southern horizon, to be 310 36' 37"; what is the declination star's decli- of that star? nation. From - 900 - 0' 00" Take the latitude, - - 40 12 18 Diff. is the meridian alt. of the equator, 490 47' 42" Alt. of star, 31~ 36' 37" Refraction, 1 32 True altitude, 31~ 35' 5' -- 31~ 35' 5" Declination of the star, south, - 184 12' 37" 2. The same observer finds the meridian altitude of another star, from the southern horizon, to be 790 31' 42"; what is the declination of that star?.. Observed altitude, - - - 790 31' 42" Refraction, - True altitude, - - - -79 31 31 Altitude of equator, - - - 49 47 42 Star's declination, north, - 290 43' 49" 3. The same observer, and from the same place, finds the meridian altitude of a star, from the northern horizon, to be 510 29' 53"; what is the declination of that star? ASTRONOMICAL REFRACTION. 31 Observed altitude, - 51~ 29' 53" CHAp. im. Refraction, - - - - 46 True altitude of star, - - - 51 29 7 Altitude of pole ( or latitude), - 40 12 18 Star from the pole (or polar dist.), 11 16 49 Polar dist., from 90~, gives decl., north, 780 43' 11" In this way the declination of every star in the visible heavens can be determined. (39.) In Art. 28 we have explained how to obtain the Elements for a chart difference of the right ascensions of the stars; and in the last of the stars. article we have shown how to obtain their declinations. With the declinations and diferences of right ascensions, we may mark down the positions of all the stars on a globe or spherethe true representation of the appearance of the heavens. Quite a region of stars exists around the south pole, which are never seen from these northern latitudes; and to observe them, and define their positions, Dr. Halley, Sir John Herschel, and several other English and French astronomers, have, at different periods, visited the southern hemisphere. Thus, by the accumulated labors of the many astronomers, we at length have correct catalogues of all the stars in both hemispheres, even down to many that are never seen by the naked eye. (40.) In Art. 28, we have explained how to find the df- The Zerf meridian of ferences of the right ascensions of the stars; butwe have not right ascen. yet found the absolute right ascension of any star, for the want sion. 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 to find exactly where ( near what stars ) this meridian line is. Before we can define this line, we must take observations on the sun, and determine where it crosses the equator, and from the time we can determine the place. But before we can place much reliance on solar observations, we must ask ourselves this question. Has the sun any parallax? $82 ASTRONOMY. CoP. M. 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? Parallax. To all 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 parallax'; 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 rallax insen- thrown to the southward, as seen by a northern observer, it sible, in comrn. monobserva. Will cross the equator in the spring sooner than it appears tion0. to cross. But let an observer be in the southern hemisphere, and, to him, the sun would be apparently thrown over to the north, and it would appear to cross the equator before it really did cross. Hence, if the sun is thrown out of place by parallax, an observer in the southern hemisphere would decide that the sun crossed the equator quicker, in absolute time, than that which would correspond to northern observations. Northern But, in bringing observations to the test, it was found that and southern observations both northern and southern observers fixed on the same, or compared. v)ery nearly the same, absolute time- for the sun crossing the equator. This proves that the position of the sun was not sensibly affected by parallax. We will now suppose (for the sake of simplicity) that a sidereal clock has been so regulated as to run to the rate of sidereal time; that is, measure 24 hours between any two successive transits of the same star, over the same meridian, but the sidereal time not known. Also, suppose that, at the Observatory of Greenwich, in the year 1846, the following observations were made: * * In early times, such observations were often made. We took these results from the Nautical Almanac, and called them observations; but, for the purpose of showing principles, it is immaterial whether obserVations are real or imaginary. EQUINOCTIAL POINT. 33 CHAP. III. Face of the Side- Declination by Observa. real Clock. (Art. 38. ) Observa. tions to find h. m. s. O,,I the equinox, March 18, 1 3 20.00 0 58 53.4 south, and the side94 19, 1 6 58.62 0 35 11.3 real time. cc 20, 1 10 37.10 0 11 29.4 " 21, 1 14 15.47 0 12 12.0 north, " 22, 1 17 54.07 0 35 52.0 " From these observations, it is required to determine the sidereal time, or the error of the clock; the time that the sun crossed the equator; the sun's right ascension; its longitude, and the obliquity of the ecliptic. It is understood that the observations for declinations must have been meridian observations, and, of course, must have been made at the instant of apparent noon, local solar time. By merely inspecting these observations, it will be perceived that the sun must have crossed the equator between the 20th and 21st; for at the apparent noon of the 20th, the declination was 11' 29".4 south; and on the 21st, at apparent noon, it was 12' 12" north. Between these two observations, the clock measured out 24 h. 3 m. 38.37 s., of sidereal time. If the sun had not changed its meridian among the stars, the time would have been just 24 hours. The excess (3 m. 38.37 s.) must be changed into arc, at the rate of four minutes to one degree. Hence, to find the arc, we have this proportion: As 4m: 3m. 38.37s': 10: to the required result. The result is 54' 35".4; the extent of are which the sun changed right ascension during the interval between noon and noon of the 20th and 21st of March. To examine this matter understandingly, draw a line E-Q (Fig. 5), and make it equal to 54' 35".4. From E, draw ES at right angles to E Q, and make it Computa. equal to 11' 29".4. From Q, draw Q N at right angles to tion3 to find the equinox. EQ, and make it equal to 12' 12". Then S will represent bhe sun at apparent noon, March 20th, and N the position of bhe sun at apparent noon on the 21st, and SNis the line of 3 34' ASTRONOMY. CHAP. III. Fig. 5. the sunamong 0 N the stars, and the point cp (called the first point of &Q....T~ E Aries ), and it is where the sun crosses S the equator. Now we " " wish to find, P where the line E Q is crossed by the line S N; or, the object is, to find E ca, expressed in time. To facilitate the computation, continue E S to P, making SP- Q1V and draw the dotted line P Q. Then S P Q N is a parallelogram. EP=11' 29".4+12' 12"=23' 41".4; and the two triangles, P E Q and SE cm, are similar; therefore we have PE: EQ:: SE: Emq. To have the value of E m, in time, E Q must be taken in time; which is 3 m. 38.37 s. Hence, (23'41".4): (3m. 38.37S.): 11' 29".4: E@c. The result gives, Ecr=lm. 45.91s. But the clock time that the point E passed the meridian, was - Ih. 10m. 37.10s. Add, - - - - - 1 45.91 Error of The equi. passedmerid. (by clock) at 1 h. 12m. 23.01 the clock. tBut, at the instant that the equinox is on the meridian, the sidereal clock ought to show 0 h. 0 m. 0 s. The error of the clock was, therefore, h. 12m. 23.01 s. ( subtractive ). Sun's right As the whole line, E Q (in time), is - 3 m. 38.37 s. ascension. And the part E om is 1 45.91 Therefore, mc Q is - - - - 1 m. 52.46 But qc Q is the right ascension of the sun at apparent noon, EQUINOCTIAL POINT. g~ at Greenwich, on the 21st of March, 1846; a very importawt CHAP. IIt. element. The right ascension of any heavenly body, whether it be How to sun, moon, star, or planet, is the true sidereal time that it find the absolute right passes the meridian; and now, as we have the error of the ascension of clock, we can determine the true sidereal time that any star the star4, passes the meridian, and, of course, its right ascension; thus, aun,' laoet. for example, If a star passed the meridian at - 10 h. 15 m. 47 s. Error of the clock is (subtractive) 1 12 23 Right ascension of the star is 9 h. 3 m. 24s. (42.) To find the Greenwich apparent time, when the sun crossed the equinox, we refer to Fig. 5; and as the point E1 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, we have the following proportion: EQ: Ec p:: 24h.: x. Giving to EQ and Emc their numeral values in seconds of sidereal time, the proportion becomes: 218".37: 105".91:: 24h.: x. The result of this proportion gives 11 h. 38 m. 24s. for the Time of interval, after the noon of the 20th of March, when the sun the equinozs crossed the equator. This result is in apparent time. The difference between apparent time, and mean clock time, will be explained hereafter. At this period, the difference between the sun and the common clock was 7 m. 36 s., to be added to apparent time, Equinox of 1846, March - - 20 d. 11 h. 38m. 24s. Equation of time (add), - 7 36 Equinox, clock time (Greenwich), 20 d. 11 h. 46 m. 0 (43.) The two triangles, ES c and pc Q are really Obliquie spherical triangles; but triangles on a sphere whose sides are of the eclip less than a degree may be regarded as plane triangles, with- tic, how found. out any appreciable error. In the triangle ESm, Ecp~-1588".65, ES=689".4; 86 ASTrRO NOMYb cHAP. II. and, if we regard these seconds of are as mere numerals, and calculate the angle E mr S, we find it 230 27' 43"; which is the obliquity of the ecliptic. Sun's Ion- By computing the length of the line SN, we find it 59' 30"; gitude. which was tle variation the sun's longitude, between the noon of the 20th and 21st. Both longitude and right ascension are reckoned from the equinoctial point o: longitude along the line cp N (which line is called the ecliptic), and right ascension along the celestial equator cp Q. Computing the length of the line r NV; we find it equal to 30' 36".6; which was the sun's longitude at the instant of apparent noon, at Greenwich, March 21st, 1846. Latitude, Meridians of right ascension are at right angles to the celestial In astrono- equator (at right angles to cq Q). The first meridian runs my, from what line through the point m. Meridians of latitude are at right teekoned. angles to the ecliptic (at right angles to the line SN). Latitude, in astronomy, is reckoned north and south of the ecliptic. Thus a star at m (Fig. 5), mo n would be its longitude, n m its north latitude, cp o its right ascension, and o m its north declination. Path ofthe (44.) Thus, it may be perceived, that these observations sun. are very fruitful in giving important results; but, as yet, we have used only two of them — those made on the 20th and 21st. By bringing the other observations into computation, and extending Fig. 5, we can find the points where the sun was on the other days mentioned; and then, by taking observations every day in the year, the sun's right ascension and longitude can be determined for every day, and its exact pathLength of way through the apparent celestial sphere. The same kind a year, how of observations taken on the 20th, 21st, 22d, 23d,, and 24th observed. observed. days of September, will show when the sun crosses the equator from north to south;, and how long it remains north of the equator, and how long south of it. In March, 1847, the same observations might have been made, and the exact length of an equinoctial year determined: and in this way that important interval has been decided, even to seconds. The true length of an equinoctial year was early a very SOLAR YEA, 387 interesting problem to astronomers; and, before they had cHsAP. n1, good clocks and refined instruments, it was one of some difficulty to settle. But the more the difficulty, the greater the zeal and perseverance;- and we are often astonished at the accuracy which the ancients attained. The length of the equinoctial year, as stated in the tables of Days. hours, min. sees Ptolom6e, is - - 365 5 55 12 Tycho Brahe, made it - - 365 5 48 45 Kepler, in his tables, - - - 365 5 48 57 M. Cassini, in his tables, - 365 5 48 52 M. De Lalande, - - - 365 5 48 45 Sir John Herschel, - - - 365 5 48 49.7 The last cannot differ from the truth more than one or two Solar and seconds. Let the reader notice that this is the equinoctial sidereal year, year-the one that must ever regulate the change of seasons. There is another year — the sidereal year - which is about 20 minutes longer than the equinoctial year. The sidereal year is the time elapsed from the departure of the sun from the meridian of ANY STAR, until it arrives at the same omeridian again, and consists of 365 d. 6 h. 9 m. 9 s. As the stars are really the fixed points in space, this latter Cause of period is the apparent revolution of the sun; and the shorter difference, period, for the equinoctial year, is caused by the motion of the equinoctial points to the westward, called the precession of the equinoxes. Since astronomers first beganf to record observations, the fixed stars have increased, in right ascension, about 2 hours in time, or 30 degrees of arc. The mean annual precession of the equinoxes is 50".1 of are; which will make a revolution, among the stars, in 25868 years.* * The computation is thus: As 50".1 is to the number of seconds in 360 degrees; so is one year to the number of years. Which gives 25868 years, nearly. We say, the stars increase in right ascension; and this is true; but the stars do not move —they are fixed; the meridian moves from the stars, D 88 ASTRONOMY. CHAPTER IV. GEOGRAPHY OF THE HEAVENS. CHAP. IV. (45. ) THE fixed stars are the only landmar1ks in astronoGroups of my, in respect to both time and space. They seem to have stars. been thrown about in irregular and ill-defined groups and clusters, called constellations. The individuals of these groups and clusters differ greatly as to brightness, hue, and color; but they all agree in one attribute — a high degree of permanence, as to their relative positions in the group; and the groups are as permanent in respect to each other. This has procured them the title of fixed stars; an expression which must be understood in a comparative, and not in an absolute, sense; for, after long investigation, it is ascertained that some of them, if not all, are in motion; although too slow to be perceptible, except by very delicate observations, continued through a long series of years. Magni. The stars are also divided into different classes, according tades of the to their degree of brilliancy, called magnitudes. There are six magnitudes, visible to the naked eye; and ten telescopic magnitudes- in all, sixteen. The brightest are said to be of the first magnitude; those less bright, of the second magnitzde, etc.; the sixth magnitude is just'visible to the naked eye. One star The stars are very unequally distributed among these of the first classes; nor do all astronomers agree as to the number bemagnitude. longing to each; for it is impossible to tell where one class ends, and another begins; nor is it important, for all this is but a matter of fancy, involving no principle. In the first magnitude there is really but one star ( Sirius); for this is manifestly brighter than any other; but most astronomers put 15 or 20 into this class. The second magnitude includes from 50 fo 60; the third, about 200, the numbers increasing very rapidly, as we descend in the scale of brightness. From some experiments on the intensity of light, it has GEOG'RAPHY OF THE HEAVENS. 39 been d-etermined, that if we put the light of a& star, of the CHAP. nv average 1st magnitude, 100, we shall have: 1st magnitude 100 4th magnitude = 6 2d " =I 25 5th "; 2 3d " 12 6th " 1 On this seale, Sir William Herschel placed the brightness of SiiTus at 320. Ancient astronomy has come down to us much tarnished with superstition, -and heathen mythology. Every constella-'tion bears the name of some pagan deity, and is associated with some absurd and ridiculous fable,; yet, strange as it may appear, these masses of rubbish and ignorance -these clouds and fogs, intercepting the true light of knowledge, are still not only retained, but cherished, in many modern works, and lignified with the name ofastronomy.:iMerely as names, either to constellations or to individual Ancient stars, we shall make no objections; and it would be useless, bmes CmOs;if we did; for names long known, will be retained, however nued. improper or objectionable; hence, when we speak of Orion, the Little Dog, or the Great Bear, it must not be understood that we have any.:reat respect for mythology. It is not our purpose now to -describe the starry heavens — to point out the variable, doube, and ml4diple stars - the AMilky Way anid nebule; these will receive special attention in some future chapter: at present, our only aim is to point out the method of obtaining a knowledge of the mere appearance of the sky, to the common observer, which may be ealled the'gograpy qf the heavens. To give a person an idea of locality, on the earth, we refer to points and places supposed to be known. Thus, when we say that- a certain town is 15 miles north-west of Boston, a ship is 100 miles east of the Cape of Good Hope, or a certain mountain 10 miles north of Calcutta, we have a pretty definite idea of the localities of the town, the ship, and the mountain, on the face of the earth, provided we have a clear idea of: the face of the earth, and know thesposition of Boston, the Cape of Good Hope, and Calcutta. So it is with the geography of the heavens; the apparent 4 40 ASTRONO MY. CHAP. IV. surface of the whole heavens must be in the mind, and theLt the localities of certain bright stars must be known, as land-, marks, like Boston, the Cape of Good Hope, and Calcutta. Stars about We shall now make some effort to point out these landthepole, marks. The North Star is the first, and most important to be recognized; and it can always be' known to an observer, in any northern latitude, from its stationary appearance and iAltitude, equal to the latitude of the observer. At the distance of about 32 degrees from the pole, are seven bright stars, between the 1st and 2d magnitudes, forming a figure resembling a dipper, four of them forming the cup, and three the handle. The two forming the sides of the cup, opposite to the handle, are always in a line with the North Star; and are therefore callerd pointers: they always point to thle Xorth Star, The line juicning the equinoxes, or the first meridian of right aseension, runs from the pole, between the other two stars forming the cup. The first star inr the handle, nearest the cdup, is called Alioth, the' next iizar, near which is a small star, of the 4th magnitude; the last one is Benetnasch. The stars in the handle are said to be in the tail of the Great Bear. About four degrees from the pole star, is a star of the 3d magnitude, e. Ursee Minoris. A line drawn through the pole (not pole star) and this. star,will pass through, or very near1 the poles of the ecliptic and the tropics. A small constellation, near the pole; is, clled Ursa JMinor, or the Little Bear. An irregular semicircle of bright stars, between the dipper and the pole, is called the Serpent. imaginary If a line be drawn from e Ursce l~finoris, through the pole lines from starto star. star, and continued about 45 degrees, it will strike a very beautiful star, of the 1st magnitude, called Capella. Within five degrees of Capella are three stars, of about the 4th magnitude, forming a very exaet isosceles triangle, the vertical! angle about 28 degrees. A line drawn from Alioth, through the pole star, and continued about the same distance on the other side, passes through a cluster of stars called Cassiopia in her chair. The principal star in Cassiopea, with the pole star and Capella, form an isosceles triangle, Capella at the vertex. *EOGRAPHY OF THE HEAVENS. 41 (46.') More attention has been paid to the constellations CHAP. IV. along the equator and ecliptic,-than to others in remoter Ecliptic regions of the heavens, because the sun, moon, and planes, defined. traverse through them. The ecliptic is the sun's apparent annual path among the stars (so called because all eclipses, of both sun and moon, can take place only when the moon is either in or near this line). Eight degrees on each side of the ecliptic is called the Signs of zodiac; and this space the ancients divided into 12 equal the zodiab. parts (to correspond with the 12 months of the year ), and each part (800) is called a sign- and the whole, the signs of the zodiac. These divisions are useless; and, of late years, astronomers have laid them aside; yet custom and superstition will long demand a place for them in the common almanacs. The signs of the zodiac, with their symbolic characters, are as follows: Aries m, Taurus i, Gemini rr, Cancer, Leo Sl, lVirgo Ar, Libra -, Scorpio Aq, Sagittarius A, Capricornus Y, Aquarzius X Pisces B. 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, Method of no prominent star is near it; and, going along the ecliptic to' tracing th the eastward, there is nothing to arrest special attention, stars.,unt llwe come to the Pieiades, or Seven &Sars, though only six are visible to the naked eye. This little cluster is so well known, and so remarkable, that it needs no description. Southeast of the Seven Stars, 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 Ist magnitude, of a red color, called Aldebaran. It is one of the eight stars selected 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 ]D 42 AST RONOCMY OaH,. IV., a V when west of it. The Seven Stars, Aldebaran, and C7a 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 distinguished 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 Sirigs, the most brilliant star in the heavens, The ecliptic passes about midway between the Seven Stars ind Aldebaran, in nearly an eastern direction, Nearly due east from the northernmost and bright; est star in Orion, and at the distanee of about 25 degrees, is the star Procyon, a bright, lone star. The northernmost star in Orion, with Sirius and Procyonj form an equilateral triangle, The con, Directly north of Procyon, at the distances of 25 and 30 are above the degrees, are two bright strs, Castor and Pollu stor is horizon, and the most northern. Pollux is one of the eight lunar stars& visible every Thus we might run over that portion of the heavens which is evening duo ring the win- ever visible to us, and by this method every student of astroter season. noty can render himself faniliar with the aspect of the sky but it is not sufficiently definite and scientific to satisfy a.maca thematical mind, (47.) The only scientifi.c method of defining the position of a place on the earth, is to mention its latitude and lonpitude,/ and this method fully defines any and every place, howevet unimportant and unfrequented it may be: so in astronomy, the only scientific methods of definng the position of a star, is to mention its latitude and longitude, ol, more eonveniently, its General right ascension and declination; aad indefi- It is not sufficient to tell the navigator that a coast makes nite descrip- off in such a direction from a certain point, and that it is so tions not salisfatm. far to a certain cape; ana, from one cape to another, it is GEOGRAPHY OF THE HEAVENS. 13 about 40 miles south-west- he would place very little reli- CHAP. IV. ance on any such directions. To secure his respect, and Whatconcommand his, confidence, the latitude and longitude of every stitutes a definite deO point, promontory, river, and harbor, along the coast, must be scription. given; and then he can shape his course to any point, or strike in upon it from the indefinite expanse of a pathless sea. So with an astronomer; while he understands and appreciates the rough and general descriptions, such as we have just given, he requires the certain description, comprised in right ascension and declination. Accordingly, astronomers have given the right ascensions and declinations of every visible star in the heavens (and of very many that are invisible ), and arranged them in tables, in the order of right ascension. There are far too many stars, for each to have a proper John Bay. name; and, for the sake of reference, Mr. John Bayer, of ofre method of reference. Augsburg, in Suabia, about the year 1603, proposed to denote the stars by the letters of the Greek and Roman alphabets; by placing the first Greek letter cc to the principal star in the constellation, 9 to the second in magnitude, 7' to the third, and so on; and if the Greek alphabet shall become exhausted, then begin with the Roman, a, b, e, etc, " atalogues of particular stars, in sections of the heavens, Particular have been published by different astronomers, each author catalogues. numbering the individual stars embraced in his list, according to -the places they respectively occupy in the catalogue." These references to particular cat.agues are sometimes marked on celestial globes, thus: 79 HI, meaning that the star is the 79th in Herschel's catalogue; 37 M, signifies the 37th number in the catalogue of Mayer, etc. Among our tables will be found a catalogue of a hundred of the principal stars, inserted for the purpose.of teaching a definite and scientific method of making a learner acquainted with the geography of the heavens. To have a clear understanding of the method we are about to, explain, we again consider that right ascension is reckoned from the equinox, eastward along the equator, from 0 h. to 24 hours. When the sun comes to the equator, in March, its 44 ASTRONOMY. CHAP. IV. right ascension is 0; and from that time its right ascension increases about four minutes in a day, throughout the year, to 24 hours; and then it is again at the equinox, and the 24 hours are dropped. When it is But whatever be the right ascension of the sun, it is appaapparent rent noon when it comes to the meridian; and the more eastf~oo~n ward a body is, the later it is in coming to the meridian. Thus, if a star comes to the meridian at two o'clock in the afternoon (apparent time ), it is because its right ascension Is Two HOURS G-REATER than the right ascension of the sun. Therefore, if from the right ascension of a star we subtract the right ascension of the sun, the remainder will be the time for that star to come to the meridian. Connection If we put ( R *) to represent the star's right ascension, between R, and (R A) to represent that of the sun, and Tto represent A. and me- the apparent time that the star passes the meridian, then we ridirin' passage shall have the following equation: R *-R( =7T; By transposition.. R =R O +T: That is, the right ascension of a star ( or any celestial body ), is equal to the right ascension of the sun, increased yj the time that the star (or body ) comes to the meridian. The right ascension of the sun is given, in the Nautical Almanac (and in many other almanacs ), for every day in the year, when the sun is on the meridian of Greenwich; but many of the readers of this work may not have such an almanac at hand, and, for their benefit, we give the right ascension for every fifth day of the year 1846 ( Table III): the local time is the apparent noon at Greenwich. We take the year 1846, because it is the second year after leap year; and the sun's, right ascension for any day in that year, will not differ more than two minutes from its right ascension, on the same day, of any other year; and will correspond with the right ascension of the same day in 1850, by adding 7-3a seconds; and so on for each succeeding period of four years. To apply the preceding equation, the observer should adjust his watch to apparent time; that is, apply the equation GEOGRAPHY OF THE HEAVENS. 45 of time, and know the direction of his meridian,,at least CHAP. IV. approximately. In short, by the range of definite objects, he must be able to decide, within two or three minutes, when a celestial body is on his meridian. Thus, all prepared, we will give a few EIXAMP LES. 1. Onz the 20th of May ( no matter what year, if not many Examples, years from 1850 ), in the latitude of 400 NV, and longitude of to find stars. 800 W., at 9 h. 24 m. in the evening, clock time, I observed a lone, bright star, of about the 2d magnitude, on the meridian. It had a bland, white light; and, as I had no instrument to measure its altitude, I siimply judged it to be 420. What star was it? We decide the question thus: Time per watch, - - - 9 h. 24 m. 00 s. Equation of time ( see Table ), add 3 46 Apparent time, - - - 9 27 46 Lon. 80~ W., equal, in time, to 5 20 00 Apparent time, at Greenwich, - 14 47 46 The right ascension of the sun, on the 20th of May (noon, Correction Greenwich time ), is 3 h. 47 m. 15 sa.. ( see Table III). The of the sun's R. A. increase, estimated at the rate of 4 minutes in 24 hours, will give 1 minute in 6 hours, or 10 seconds to 1 hour; this, for 14 h. 47 m., gives 2 m. 27 s. Hence, the right ascension of the sun, at the time of observation, was 3 h. 49 m. 42s. Apparent time of observation, - 9 27 46 Right ascension of the star, - 13 h. 17 m. 28 s. By inspecting the catalogue of the stars ( Table II), we find the right ascension of Spica to be 13 h. 17 m. 08 s., and its declination, 10~ 21' 35". But, in the latitude of 400 N., the meridian altitude of the celestial equator must be 50~; and any stars south of that must be of a less altitude. Therefore, the meridian altitude of Spica must be 500, less 100 21', or 390 39'; but the star [ observed, I simply judged to have had an altitude of 42~. ASTRONOMY. CHaP. IV. It is very possible that I should. err, in altitude, two or tbree; degrees; * but, it is not possible that the star I observed should be any other star than Spica; for there is no other bright star near it. This is one of the lunar stars. Personal Being now certain that this- star is Spica, I can observe it servations in relation to its appearance — the small stars that are near recommendda. it, and the clustera of stars that are about it — or the fact, that no remarkable constellation is near it. In short, I can so make its acquaintance as to know it ever after; but I am unable to convey such acquaintance to others, by language:. true knowledge, in this particular, deman&d personal observation. Continsua 2. On the 3d day of July,. 1846, at 9 h. 34m., P. M., mean tion ofexam. ples to find time per watch, a star of the Ist magnitude came to the meridian.,tars. I was in latitude 390 N., and about 750 W. - The star was of a deep red color, and, as near as my judgment could decide,. its altitude was between 25~ and 300. Two small stars were near it, and a remarkable cluster of smaller stars were west and northwest of it, at the distances of 5~ 60, or 70. Whlat star was this, Time per watch, - - - - 9 h. 34 m. 00 s9 Equa. of time ( subtr. from mean time ) 3 48 Apparent time, 9 30 12' Longitude, 750, equal to - - 5 Apparent time, at Greenwich, - - 14 h. 30 mn 00s. By examining the table for the sun's R. A., I find that, On the Ist of July, it is - - 6h. 40m. 00 s. On the 5th, - - - 6 56 30 Variation, for 4 days, - - - 16 m. 30 s. At this rate, the variation for 2 days, 141 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 angle 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 object to be greater than it really is, especially if the altitude is less than 45 degrees. GEOGRAPHY OF THE HEAVENS. 47 far from 10 m. 10 s.; and the right ascension of the sun, at CHAP IV the time of observation, must have been A.n examNearly - - - - - h. 50 m. 10 s. pe of finding 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 260 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, hence (sub.) 26 4 Meridian altitude of Antares, in lat. 50~, 230 56 As the observation corresponds to the right ascension of Antares ( 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 tfouble of direct observation. 3. On the' night of the 20th of June, 1846, latitude 400 1V, and To fina longitude 750 W., at 1 h. 48 m.: past midnight, clock time, _ ob- Altair. served a star of the 1st magnitude nearly on the meridian; two other stars, of about the 3d magnitude, within 30 of it; the three stars forming nearly a right line, north and south; the altitude of the principal star about 60~. iWhat star was it? In these examples, the time must be reckoned on from noon to noon again; therefore h. 48 m. after midnight nmust be written, - 13h. 48m. 00s. 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 Ih 57 m. 40s. Time, - - - 13 46 48 Star's right ascension, - - - 19h. 44 m. 28s. 48 ASTRONOMY. CHAP. IV. By inspecting the catalogue of stars, we find the right ascension of Altair 19 h. 43 m. 15 s., and its declination 80 27' N. In latitude 40~ N., the declination of 8~ 27' N. will give a meridian altitude of 580 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 acquainted 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 corresponding star is found in the catalogue, such a star would probably be a planet; and if a planet, its right ascension will change. The South. (48.) The whole region of stars south of declination 500, erm Cross, and Magel- iS never seen in latitude 40~ north, nor from any place north Ian Clouds. of 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., andt-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 nebulce. The other two are black - extremely so - and are supposed to be places entirely devoid of all stars; yet they are in a very bright part of the Milky Way: right ascension 10 h. 40 m., declination 620 south. DESCRIPTIVE ASTRONOMY. 49 SECTION II. DESCRIPTIVE ASTRONOMY. CHAPTER I. FIRST CONSIDERATIONS AS TO THE DISTANCES OF THE HEAVENLY BODIES. — SIZE AND EXACT FIGURE OF THE EARTH. (49.) Hitherto we have con- Fig. 6. CAP. sidered only appearances, and Distance have not made the least inquiry is but rela. as to the nature, magnitude, or tive. 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 measure of one hundred miles, would be too trifling to mention. Now if we consider the dis- Are the heavenly botance to the stars, it must be dies remote? relative to some measure taken as a standard, or our inquiries will not be definite, or even intelligible. We now make this C: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 4 E 50 ASTRONOMY. CHAP. I. should call it near; if 100, 200, or 2000 times the diameter 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 the earth, these stars may be said to be near; but if a b c represents the circumference of the earth, the stars are many times the diameter of the earth, in distance, and therefore may Themeans be said to be remote. If A B C is the circumference of of deciding the.earth, in relation to these stars, the apparent distance of this questiona 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 6b are relatively the same as A and B. And, it isdan ocular demonstration that the angle under which the two stars would appear at a is the same, or nearly the same, as that Sunder 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, he -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 ), manifest not the slightest perceptible variation. Not only this, but COMPARATIVE DISTANCES. 51 at whatever point of the earth's surface the measurement is CHAP. I. performed, the results are absolutdy 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 Another maore intelligible to the general reader. illustration of the great Let Z Hfi distance to lgT. represent Fg..the stars& II represent z the celestial equator, as seen- from the aequator - on the earth; and if the earth be large, in relalion to the distance to the stars, the observer will be at o'; and the part oftho N celestial are above his horiton would be represented by A Z B, and the part below his horizon by A NVB, and these arcs are obviously unequaZ; and their relation would be measured by the time a star or heavenly body remains above the horizon; compared wjth the time below it; but by observation ( refraction being allowed for), We kno'w that the stars are as long above the horizon as they are below; Which shows that the ob-h server is not at', t at at 2, and even more near the center; so that the are A Z B, is imperceptibly unequal to the are i: VlH; that is, they are equal to each other; and the earth is comparatively but a point, in relation to the distance to the stars. This fact is well established, as applied to the fixed stars, The moot uin, -and planets; but witih the moon it s different: that body tion excp 62 A 1 1AST RONO MY. cHaP. I. is longer below the horizon than above it; which shows that its distance from the earth is at least measurable. (51.) It is improperj 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, inaccessible 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. Earth's In addition to this, any observer may convince himself, that surface con- the surface of the sea, or a lake, is not a plane, but everywhere vex. 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 strface of'the earth is convex on every part -hence it must be a globe, or nearlya:globe,:: These facts, last mentioned, are sufficiently illustrated by (2.) On the supposition that the earth is a sphere, there are several methods of measuring it, without the labor of applying the measitre 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. fow to Thus, if we can know the number of degrees, and parts of find the eircnferen cel a degree, in the arc A B (Fig. 9), and then measure the disofthe earth tance in miles, we in fact virtually know the whole circumfe. DIAMMETER'OF THE EARTH. 9 rene.; for Whatever part the are 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 be measured simply in distance, as miles, yards, or feet; but this is a laborious operation, requiring great care and perseverance. 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 dez viations, and reducing the meas'ure to a meridian line. Let m be the miles or feet in A B; then the whole circumference will be expressed by (arc AB) (53.) When we know the Fig. t. how ta hight of a mountain, as re M find the di.-' presented in Fig. 9, and at tdeter. 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 theorem. in geometry; thus, U;tx2 MB=())2 / Or, gJf ='(AA) o MB Now as the right hand e member of this equation is known, C.M is known, and apart of it (MMB) is already known, the other part, P C, the diameter of the earth, thus becomes known. This method would be a very practical one, if it were not Otjectijd for the uncertaintv and variable nature of refraction near the to this mehorizon; and for this reason, this method is never relied upon, thud. although it often well agrees with other methods. As an ex& ample under this method, we give the following; AST RONOMY. CHAP. t. A mountain, two. miles in perpendicular Light, was,eetn from sea at a distance of 126 miles. If these data are correot, what then is thl diameter of the earth? Solution: iMC (126 ) 63X126=7938. BC=7936. Dip of the (54. ) This same geometrical theorem serves to compute;horizon. the rdip 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, makling allowance for its dip. If the navigator's eye were on a level with the sea, ami the sea perfectly stable, the true and apparent horizon would be the same, Butitthe 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 A MC (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 BiM. For the benefrt of navigators, -a table has been formed, showing the dip for all common elevations,* * The dip is computed thus: The angle at the center Put B a (Fig. 9) =D, BM-hf=;'is equas to the dip. Then -EA -Th+); and(MA)2= CtfXYXB==(D+h)h. By trigonometry, (EA)2. ( 2A)2; R: tan.2AEiI; That is, -' -- (.D-h):.: 2: tan.?AE. for very moderate elevations, A's extremely small, in rela-.ion to D; and the second'term- of the ~p-oportion may be Dh. (R represents the radius of the tables.) Making this Donsideration, we have D2 -- ).` D W tan.2AEM; Or, - - sD: A::4R2: tan.2AEM; Or, - JID: h;-: 2R; tanAEM. DIP OF THIE HORIZON. 65 i (55.i All such computations are made on the supposition CHAP. I. 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 exactly concluded that its equatorial diameter might be greater than pheri its polar diameter; and investigations have been made to decide the fact. If the earth were exactly spherical, it is plain that the curvature 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 discordances 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 meridia: (astronomically determined as above described), exBy inspecting this last proportion, it will be perceived that the tangent of the dip varies as the square root of the elevation. 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 Jeet. What is the dip 27.. log. - - -: - 10.301030,/fh,.. log. -.5 - - 500000 Product of the means (log.), - - - 10.801030 J) miles, 7912, - - log. - 3.898286 Feet, - 5280, - - log. - 3.722684 2 ) 7.620920 JV infeet, ( log.) 3.810460.. 8810460 tan.3'~ 22'" - -.. 6.990570 5 66 ASTREONOMY. CHAP. I. 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 instruments, and furnished with every facility which could tend to insure a successful result of their important labors. o Latitude ength of [Country. of Mgriddlee Observers. the Arc'. concluded I Sweden......, 66 20 10 1 1~37' 19" 365782 Svanberg. IR ussia... 58 1737 3 35 5 365368 Struve. England....... 52 35 45 3 57 13 364971 Roy, Kater., France........ 46 52 2 8 20 0 364872 Lacaille, Cassini. France........44 4'51 2 12 22 13 364535 Delambre. Mechain. Roome..... 42 59 () 2 9 47 364262 Boscovtch. America, S... 39 12 0 1 28 45 363786 Mason, Dixon. Cape of G. Hope 33 18 30 1 13 171 364713 Lacaille. India......... 16 8 22 15 57 40 363044 Lambton, Everest. India........ 12 32 21 1 34 56 363013 Lambton. Peru......... 1 31 0 3 7 3 1 362808 Condamine, etc. The earth It is evident, from a mere inspection of the second and less curved at tse poles fourth columns of this table, that the measured lengt&h of a dethan'at the gree increases with the latitude, being greatest near the poles, equator 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, correa sponding to any assigned. length of the degree in a given latitude. - Without troubling the reader with the investigation (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:IFeet. Miles. Greater,obr equatorial diam., =41,847,426= —7925.648 Lesser, or polar diam., - =41,707,620=7899.170 Difference of diameters, or 139,806= 26.478 polar compression, - - - The: proportion.of the diameters is very nearly that of FORM OF THE EARTH. 57 298: 299, and their difference - of the greater, or a very CHAP. I.,little greater, than -0." (56. ) 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 hours; 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 Explanaits surface would tend directly toward the center of gravity tion of radius of curvature. of the body; but the earth being an ellipsoid, or an oblate spheroid, and the plumb lines, being perpendicular to the surface 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 Fig. 10.,A, yet the mathematical center, tr o and center of gravity of the figure, is at E. So at X the plumb line tends to the point G; A A and as the length of a degree at B A, is to the length of a degree at B so is IG to HE If, however, a passage were made through the earth, and a body let drop through it, the body would not pass from I to G: its first tendency at Iwould 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 of the force of gravity, on its surface, would be everywhere the gravity different on diffesame; but, it being neither-stationary, nor a perfect sphere, rent parts of the force of gravity, on the different parts of its surface, must the earth; be different. The points on its surface nearest its center of and why? 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 wsill be a diminution of gravity on that account. Let AB (Fig. 10) represent the equatorial diameter of 5:s8 ASTRONOMY. CHAP. L the earth, and C-0 the polar diameter; and it- is obvious that E will be the center of gravity, of the whole figure, and Gravity di- that the force of gravity at C and D will be greater than at minished by rotation. any other 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 L, also, 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 its axis, amounts, to its -9 part,* at the equator. By this fraeComputa. * Let D be the equatorial diameter tmount of ig. of the earth, P the versed sine of an are diminution. corresponding to the motion in a second of time, and c the chord or are (for the I B chord and are of so small a portion of the circumference will coincide, practically speaking). A portion of the earth's gravity, equal to F, is destroyed by the rotation of the earth, and we are now to compute its value. By proportional triangles, F: c:: c: 2; 62 Or () 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 revoluation is 23 h. 56 m. 4 s., - 86164 seconds. The whole circumference is (3.1416)D-; Therefore, -. --- (31416 )D (2) By this value of c, we bave IF — (.16)21 (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 P. Now the paTt of gravity des EFFECT OF FORM ON-GRAVITY. b9 tion, then, is the weight of the sea about the equator lightened, cHAP. IL and thereby rendered susceptible of being supported at a higher level than at the poles, where no such counteracting force exists. troyed by rotation, as we have just seen, is; therefore the whole force of gravity is (g+- ) Our next inquiry is: what part of the whole is Me part de- Ratioofthe diminution stroyed? Or what part of (gf- is 2? omputed. Which, by common arithmetic, is, C2. cc - C2 gD + c2 gqD. From (2) - D= (.86164)2 o, D (86164)2 (3.1416)2 c2 (3.1416)21 Hence, gD_ (86164)2g (86t164)2 (16.07) 2 (3.1416)2 D- (3.1416)2 (7925)(5280)~ By the application of logarithms, we soon find the value of this expression to be 288.4. Therefore, gD 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, g —, or c=-gD). But as often as c is contained in the whole circumference, is the corresponding number of seconds in a revolution; that is, the time in seconds must correspond to the expression, (3.1416)Dl., or (3.1416).! D 4 60 ASTRONOMY. CasP. I. (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, exercising at once a direct and an indirect influence. The amount Rotation of the former (as we may see by the note ) is easily calcuhas a direct and indirect lated; that of the latter is far more difficult, and requires a effect on gra. knowledge of the integral calculus, "But it has been clearly ityo. treated by Newton, Maclaurin, Clairaut, and many other eminent 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 th part; which, together with the 2 ~-th part, due from centrifugal force, make the whole quantity T th part; which corresponds with observations as deduced from the vibrations of pendulums."- See Natural Philosophy. (59.) The form of the earth Fig. 12. is so nearly a sphere, that it is considered such, in geography, Qff0 navigation, and in the general problems of astronomy. English The average length of a deand geogra. phical miles, gree is 69~ English miles; and, as this number is fractional, and inconvenient, navigators have tac i P citly agreed to retain the ancient, rough estimate of sixty miles to 4 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 application of logarithms, we find this expression to be 5069 seconds, or 1 h. 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. 61 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 CP. Let D F/Ibe any plane parallel to E QC; then we shall have the following proportion: EC: DI:: EQ: DE. In trigonometry, E C is known as the radius of the sphere; D I as the cosine of the latitude of the point D (the numerical values of sines and cosines, of all arcs, are given in trigonometrical 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 thte length of a degree on the equator, to the length of a parallel degree in that latitude. Calling a degree, on the equator, 60 miles, what is the Example length of a degree of longitude, in latitude 420~? SOLUTION BY LOGARITHMS. As radius (see tables), - - - 10.000000 Is to cosine 420 (see tables), - - - 9.871073 So is 60 miles (log.), - 1.778151 To 445U8-P miles, - - - 1.649224 At the latitude of 600, the degree of longitude is 30 miles; the diminution is very slow near the equator, and very rapid -near the poles. In navigation, the D's are the known quantities ob- To reduce tained by the. estimations from the log line, etc.; and the departure to navigator wishes to convert them into longitude, or, what longitude. is the same thing, he wishes to find their values projected on the equator, and he states the proportion thus: DI: -E:: DF:'EQ; That is, as cosine of latitude is to radius, so is departure to dference of longitude. GS2- A S T R ASTRONOMY. CHAPTER II. PARALLAX, GENERAL AND HORIZONTAL. -- RELATION BETWEEN PARALLAX AND DISTANCE. -- REAL DIAMETER AND MAGNITUDE OF THE MOON. CHAP. II. ( 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 ParalJax is the difference in position, of any body, as seen general-. from the center of the earth, and from its surface. When a body is in the zenith of any observer, to him it has 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 we 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 Fig. 13. Let C be parallax., the center of the earth, Z the observer, and P, or P, the position of a body. Fromn the center of the earth, the body is seen in the direction of the line CP, or Gp; from the observer at Z, it is seen in the PARALLAX. 63 direction of ZP, or Zp; and the dflerence in direction, of CHAP. II. these two lines, is parallax. When P is in the zenith, there is no parallax; when P is in the horizon, the angle Z P C is then greatest, and is the horizontal parallax. We now perceive that the horizontal parallax of any body Relation is equal to the apparent sernidiameter of the earth, as seen from between parallax and the body. The greater the distance to the body, the less the distance. horizontal parallax; and when the distance is so great that the semidiameter of the earth would appear only as a point, then the body has no parallax. Conversely, if we can detect no sensible parallax, we know that the body must be at a vast distance from the earth, and the earth itself appear as a point from such a body, if, in fact, it were even visible. Trigonometry gives the relation between the angles and sides of every conceivable triangle; therefore we know all about the horizontal triangle Z CP, when we know CZ and the angles. Calling the horizontal parallax of any body p, and the radius of the earth r, and the distance of the body from the center of the earth x ( the radius of the table always R, or unity), then, by trigonometry, we have, R x:: sin. p: r: Therefore, - - - _ ( r. From this equation we have the following, general rule, to find the distance to any celestial body: RULE. --- Divide the radius of the tables by the sine of the Rule to horizontal parallax. Multiply that quotient by the semidiameter find the dis. of the earth, and.the product will be the result. tances to tue This result will, of course, be in the same terms of linear bodiesavenly measure as the semidiameter of the earth: that is, if r is in feet, the result will be in feet; if r is in miles,, the result will be in miles, etc.: but, for astronomy, our terrestrial measures are too diminutive, to be convenient' (not to say inappropriate); and, for this reason, it is customary to call the semidiameter of the earth unity, and then the distance of any body from the earth is simply the quotient arising from dividing the radius by the sine of the horizontal parallax pertaining to 64: ~ ASTRONOMY. CHAP. II. the body; and it is obvious, that the less the parallax, the greater this quotient; that is, the greater the distance to the body; and the difficulty, and the only d7ijculty, is to obtain the horizontal parallax. Horizontal (61.) The horizontal parallax cannot be directly observed, parallax can, by reason of the great amount and irregularity of horizontal served. refraction; but if we can obtain a parallax at any considerable altitude, we can compute the horizontal parallax therefrom.* The fixed stars have no sensible horizontal parallax, as we have frequently mentioned; and the parallax of the sun is so small, that it cannot be directly observed (see 40 ); the moon is the only celestial body that comes forward and presents its parallax; and from thence we know that the moon is the only body that is within a moderate distance of the earth. That the moon had a sensible parallax, was known to the earliest observers, even before mathematical instruments were at all refined; but, to decide upon its exact amount, and detect its variations, required the combined knowledge and observations of modern astronomers. Deductibn * In the two triangles Zp C and ZP C (Fig. 13), call the pari ontal angle p the parallax in altitude, and the angle ZP C = x and Cp and CP each equal D. Then, by trigonometry, we have sin. pZC: sin.p:: D: r; And - - R: sin. x:: D: r. Therefore, by equality of ratios (see algebra), sin. pZC: sin. p R: sin. x. But the sine pZC is the sine of the apparent zenith distance. Therefore, R sin. p sin. zenith distance That is, the sine of the horizontcl parallax is equal to the sine of the parallax'in altitude, into the radius, and divided by the sine of the apparent zenith distance. LUNAR PARALLAX. 65 The lunar parallax was first recognized in European and CHAP. II. hrthern countries, by its appearing to describe more than a By what semicircle south of the equator, and less than a semicircle north observations - -.. the lunar paw of that line; and, on an average, it was observed to be a longer rallax was time south, than north of the equator; but no such inequality first indica. ted. could be otservedfrom the region of the equator. Observers at the south of the equator, observing the position of the moon, see it for a longer time north of the equator than south of it; and, to them, it appears to describe more than a semicircle nowth 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; and when we reflect that parallax must always depress the object (see Fig. 13), and throw it farther from the observer, it is therefore perfectly clear that a northern observer should see the moon farther to the south than it really is, and a southern observer see 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) Observaand the Cape of Good Hope. They would be more favorable tions to obtain the aif they were on the same meridian; but the small change in mount of padeclination, while the moon is passing from one meridian to rallax 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 observatory, and B the station at the Cape of Good Hope. (7 is the center of the earth, and Z and Z' are the zenith 5 C, B* 66 ASTRONOMY. CHAP. II. Fig. 14. points of the observers. Let Mi be the position of the moon, and the observer at A will see it projected on the sky at in', and the observer at B will see it projected on the sky at m. Illustration lNow the figure A CB M is a of primary quadrilateral; the angle A CB observations. - observations, is known by the latitudes of the two observers; the angles MA C and ilB C are the respective zenith distances, taken from 1800. But the sum of all the angles of any quadrilateral is equal to four right angles; and hence the angles at A, C, and B, being known, the parallactic angle at 2 is known. In this quadrilateral, then, we have two sides, A C and CB, 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 Ai;, MA!B, and finally MIC.* Now XC being known, the horizontal A mathe- * The direct and analytical method of obtaining 1.C, will be mlactical de- very acceptable to the young mathematician; and, for that reason, we give it. Put AC=CB-=r, C —=x, and the two parts of the observed parallactic angle, M. represented by P and Q, as in the figure. Also, let a represent the natural sine of the angle ilA C, and b the natural sine of the angle MB C: Then, by trigonometry, - x: a:: r: sin. Q; Also,- x: b:: r: sin. P; Hence, - - - -sin. P-sin. Q=(a-.. (1) X LUNAR PARALLAX. parallax can be computed, for it is but a function of the dis- CHAP. iX. tance (see 60). By the equation (Art. 60), xz=( s-n r By changing, - - sin. p= ( )r; and when, 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 55A%5, and 63-8?4A; distance 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 farther 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 Apogev which is most remote from the earth, it is said to be in apogee; and perigee. 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, ___Q P-Q sin.P+sin. Q-2sin. cos. 2 (2) Now by equating (1) and (2), and observing that P+Q-Q M, and that (cos. P-Q) must be extremely near unity; and, therefore, as a factor, may disappear; we then have, 2 sin. 2 (a+-b)r or (a+b)r or X 2sin. 1 A more ancient method is to compute the value of the little triangle B C -G, and then of the whole triangle A.AG, and then of a part AMC or M GC. 68 ASTRONOMY. OHAP. n. (64.) Many times, when the moon comes round to its per[gee, we find its parallax less than 61' 29", and, at the opposite 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 variable, therefore there must be a mean distance: we shall lshow, hereafter, that her motion is variable; therefore there is a mean motion; and, as the eccentricity is variable, there is a mean eccentricity. MEAN pa The extreme parallaxes, at mean eccentricity, are 60' 20" parallax and and 54' 05", and the corresponding distances from the earth MEAN dis. are 56.93 and 63.64, the radius of the earth being unity. tance. The mean parallax, 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 are different quantities; but the following investigation will set the matter at rest. Let d and D be extreme distances, and 2l the mean distance. Then, - ~ - d+D=-2M. (1) Also, let p and P be the parallaxes corresponding to the distances 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 1 1 1 d —- D and M= -. sin.p sin. P' sin. x Substituting these values of d, D, and M. in equation (1) we hv1 _ 1 _ 2 sinaP + sin. P = sin. x 2 sin. p sin. _P ~Or,- - - Esl~ si - -in.P+sn. n. (2) sin. x VARIATION OF PARALLAX. 69 55.92+63-84 CHAP. Im. The mean between extreme distances is 55 -6384or 59.88; but the true mean distance is 60.26, corresponding to the Mean disparallax 57' 3". The mean, between extremes, is a variable te to th quantity; but the true mean distance is ever the same, a little more than 604 times the semidiameter of the earth. (66.) The variations in the moon's real distance must correspond 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 pa. rallax. ae have, sin. P+sin.p= 2 sin.( +) cos. (2 )() By equating (3) and (2), and observing that the cosines of very small arcs may be practically taken as unity, or radius; therefore, sin. (+p) sin.P sin.p Ulna t:2 --- = Bsin. sin. P sin. p Or, - - sin. X sin.~(P-_p)' On applying this equation, we find x=57' 3". * Let A be the Fig. 15. point of vision, and d the diameter of any body at diffe- d rent distances,AB, A AC'. B c Now, by trigonometry, we have the following proportions: AC: d':: tan. CAD AB:: d::: tan.-BAE. 170 A STRONOMY. CHAP. U. Now if the moon has a real change in distance, as obserrations show, such change must be accompanied with apparent changes in the moon's diameter; and, by directing observations 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 horizontio, between tal parallax is the semidiameter of the earth, as seen from the semidiameter and hori. moon; and now we further say, that what we call the moon.sq zontal paral. semidiameter, an observer at the moon would call the earth's lax. horizontal parallax; and the variation of these two angles depends 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, - - - 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. CA/D. 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:: angleBAE: 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. 1 As spheres are to each other in proportion to the cubes of CHAP. nI. their diameters, therefore the bulk (not mass) of the earth. is to that of the moon, as 1 to js, nearly. A.s the moon's distance is 60~ times the radius of the earth, Augmen. it follows that it is about — th nearer to us, when at the tation of the ~~~~6 0~' ~tmoon's semi. zenith, than when in the horizon. Making allowance for this diameter: its (in proportion to the cosine of the altitude), is called the cause. augmentation of the semidiarmeter. (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 roll 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 revolves 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 Fig. 16. the earth (Frig. 16),m that of the D moon, D the distance between the two bodies; and let the radius\ of the tables be unity. Put P to represent the moon's horizontal parallax, and s its apparent semidiameter. Then, by trigonometry, -D::: 1: tan. P; and D: m:: 1: tan. s. From the first, D=)-; from the 2d, D-tPn an.P' tas E' mn tan. s. m Therefore, - tanp- tn- or taB Q. D. 6 72 ASTRONOMY. CHAP. In two degrees in diameter, invariably fixed in their shyj and the stars passing slowly behind it. The moon "BUt," says Sir John Herschel, " the moon's rotation on volves on her axis is uniform; and since her motion in her orbit is not an axis. 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 surface, 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 librations. In consequence of these two distinct kinds 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 ANGULAA MOTION OF THE SUN. NOT UNIFORM. — LAWS BETWEEN DISTANCE, REAL, AND ANGULAR MOTION. -E —-CCENTRICITY Of THE ORBIT. HalAP. III (69. ) THE sun's parallax is too small to be detected by The sun any common mieans of observation; hence it remained unlahgeear than known, for a long 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 are; 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 semidiamneter of the earth, as seen from the sun, must be less than 20"; but the semidia APPARENT DIAMETERS. 73 meter of the sun is 15' 56", or 956"; therefore the sun must CHAP..fl. be vastly larger than the earth —by at least 48 times its diameter; 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 suffer 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 the section of this work, clearly show, or give the appearance of tSn go round the earth, or the sun going round the earth once in a year; but the appear- the earth ance would be the same, whether the earth revolves round the round the 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: that 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 semidiameter and horizontal parallax of a body have a constant relation to each other; and, while we cannot discover the one, we will examine all the variations of the other ( f it have variations ), and thereby determine whether the earth and sun always remain at the same distance from each other. ilere it is very important that the reader should clearly Methods understand, how the apparent diameter of a heavenly body of measuring apparentdiacan 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 deductions are to be made, whatever body, moon, or planet, may be under observation. An instrument to measure the apparent diameter of a planet The micro. is called a mnicrometer. It is an eyepiece to a telescope, with meter. 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 an3; 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 G 74 ASTRONOMY. CsIAP. I. the mean result is regarded as a single observation. Generally speaking,. he 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 passthod by timae in bpassi ing in the perpendicular wires of the transit instrument. the meridian. All bodies (by the revolution of the earth ) come to the meridian af right angles, and 15 degrees pass by the meridian in one hour of sidereal time; and, in four minutes1 one degree 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 stationary. Hence we must have a correction, for the sun's,motion, applied to the time of its passing the meridian. Correctionsa We have also supposed the sun on the equator, and for a obemade, moment continue the supposition, and also conceive its diameter 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 ifafromr the equator, the longer it will be in passing the meridian on that account; the increase of time depending on the cosine of the 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 are: one for the progressive motion of the sun in right ascension; and one for the existing declination. We give an example. Method of On the first day of June, 1846, the sidereal time (time dciding the measured by the sidereal clock ) of the sun passing the me APPARENT DIAMETERS. 70 ridian wire, was observed to be 2 nm 16.6-4 s.; the declination CHAP. 1I1. was 22~ 2' 45", and the hourly increase of right ascension was exact appa10.235 s. What was the sun's semidiameter? rent diamneter of the 3600.: 10.235 s.:: 136.64 ~ 039 s. sun, moon, or planets. Observed dura. of tran., in sees., 136.64 Reduction for solar motion, -.39 136.'2.. 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 are of 1894".5, and its semidiameter 947".2, or 15' 47".2; which is the result given an 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 2 m. 22.2 s., or 142.2 s. The sun's hourly motion in right ascension, at that time, was 11.06 s., and the declination was 220 11'. What was the sun's semisdiameter?* Ans. 16' 17".3. These observations may be made every clear day through- Extreme values of the out the year; and they have been made at many places, and sun's appafor many years; and the combined results show that the rent semidiameter. * The following is the formula for these reductions: 15(t-c)cos. D R Here t is the observed interval in seconds, c is the correction for the increase in right ascension, D is the declination, R the radius of the tables, and s 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 latter 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 be disregarded. 781, ASTRONOMY. CBsP. D apparent 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. 66. Variation Therefore the distance to the sun on the 30th of Decemof the dis ber, must be to its distance on the first day of July, as the tance from the earth to number 15' 45".1 is to the number 16' 17".3, or as the numthe sun. ber 945.1 to 977.3; and all other days in the year, the proportional distance-must be represented by intermediate numbers. 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 longitude is 1800~. Eccentri- if we would consider the mean distance between the earth eithy' obf the and sun as unity (as is customary with astronomers), and then how known. put x to represent the least distance, and y the greatest distance, we shall have x+y=2. And,- - z: y:: 9451 9773. A solution gives x=0.98326, nearly, and y=1.01674, nearly; showing that the least, 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 meanz is unity ), is called the eccentricity of the orbit. SUN'S MOTION IN LONGITUDE. 77 The eccenricity, as just mentioned, must not be regarded as CHAP. m. accurate. It is only a first approximation, deduced from the first and most simple view of the subject; but we shall, hereafter, give other expositions that will lead to far more accurate results. In theory, the apparent diameters are sufficient to determine Ecentrieithe eccentricity, could we really observe them to rigorous ty from apparent diaexactness; but all luminous bodies are more or less affected meters only by irradiation, which dilates a little their apparent diameters; approximate. 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 230 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 Variationslongitude; and this is its most regular motion. in the distance of the When we compare the sun'A motion, in longitude, with its sun, comsemidiameter, we find a correspondence - at least, an apparent pared with its variations connection. in longitude. in longitude. 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, 10 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 ( A 78 ASTRONOMY. C-HAP. Li. about the earth in a circular orbit, and its real velocity uniform; but 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 if this is a true view of the subject, the variation in shows 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 the inverse should be to 977".3, as 57' 11".48 to 61' 9".95, if the supproportion to the square position of the first observerswere true. But these numbers of the dis- have not the same ratio; therefore this supposition is not tance. satisfactory; and it was probably abandoned for the want of this mathematical support.. The ratio between 945".1, and 9773 977"3 is -9451-1.0341, nearly 3669".95 between 57' 11".48, and 61'9".95, 3 - =-1.0694, nearly. 3431".48 If we square (1.0341) thefirst ratio, we shall have 1.06936, a number so near in value to the second ratio, that we conclude it ought to be the same, and would be the same, provided we had perfect accuracy in the observations. Law be. Thus we compare the angular motion of the sun in diffetween mou tion and dis rent parts of its orbit; and we always find, that the inverse tance. square of its distance is proportional to its angular motion; and this incontestiblefact is so exact and so regular, that we lay it down as a law; and if solitary observations do not correspond with it, we must condemn the observations, and not the law. (74.) To investigate this subject 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 O. The distance from O 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 enlarged view of the science- for to depart from it would lead to oftrepeated and troublesome technicalities, if not to confusion of ideas. Clearness does not always correspond with exactness of expression. VARIATIONS IN SOLAR MOTION. 79 any point in the or- Fig. 17. c,,p. m. bit is called the radius vector; and it is a varying quantity, conceived to sweep round the point O. D.Let D) be the value. of the radius vector at any point, and r D its value at some other point, as represented in the figure. Let y represent the real motion of the Variations sun, for a very short interval of time, at the extremity of the in real and angular moradius vector.D; and x represent the real motion, at the tion. extremity of the radius vector r ), in the 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, reA is the angle under which y appears from the same point. Now, considering the sectors as triangles, we have the following proportions: 1: A:: rD: x;: r2A:: D: y. From the first, - - ==rAD, From the second, - y==r2AD. Multiply the first of these equations by r, and we perceive that - y=rx. This last equation shows that the real velocity of the earth The real vt or a...velocity of in its orbit varies in the inverse ratio as the radius vector; or the earth in it varies directly as the apparent diameter of the sun. its orbit va(75.) If we multiply r D by x, the product will express the uies as the sun's appa. double- of an area passed over by the radius vector in a certain rent diame. 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 r)Dx, and the second is the same, as we shall see by taking the value of y (r fx); that is r D x=rDx; hence we announce this general law: 80 ASTRONOMY. CHAP. IhI. That the solar radius vector describes equal areas in equal The radius times. vector describes equal When expressed in more general terms, this is one of the areas in e- three laws of Kepler, which will be fully brought into notice qual times. in a subsequent part of 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 connect 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 following summary: Laws of 1. The eccentricity of the solar ellipse, as determinedfrom the motion in an apparent diameter of the suen, is.01674.* ellipse, 2. The sun's 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 consider the sun moving round the earth, or the earth round the sun; for, if the earth is in one position 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 same error of observation on longitude would not be as proportionally great as on apparent diameter. Let E'be the eccentricity of the orbit; then ( —E) is the least distance to the sun, and (1+E) the greatest disi;ance. Then, by observation, we have (1-E)2: (1+E)2:: 57' 11".48: 61' 9".95; Or, (1-E)2: (1+E)2:: 343148 366995; Or, 1 —E: 1E:: 343148: 366995. Whence E=.016788+. We shall give a still more accurate method of computing this important element. SUN'S ELLIPTICAL MOTION. Sl sun appears exactly in Fig. 18. CHAP. IE the opposite position, D and every motion made bv the earth' must correspond to an A apparent motion made B by the sun. b'6 But, for the purpose of getting nearer to fact, we will now suppose 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 distance 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; and 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 termine the to perigee, and from perigee to apogee; provided that these positions of points are stationary in the heavens, and they are so, very the solr 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 observations upon it until the earth comes to C, then the longitude of the sun. has changed 180~; and if the time is less than 1. The longer axis of the orbit, or apogee point, changes position by a 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 minute deviations, to be brought into view when we are better prepared to understand them. These minute variations, for short periods of time, do not sensibly affect general results. 6 -82 ASTRONOMY. CHAP. II, 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, We cannot determine' the exact point of the apogee or how avoided perigee, by direct observations on the sun's apparent diameter; 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 D a was described in little less than half a year, and the area a C b, 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 6, and we can proportion to it to great exactness. Longitude ( 77. ) The longitude of the apogee, for the year 1801, was of apogee 990 31' 9", and, of course, the 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 12hours on the north side; being longer in the northern hemisphere 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 unequally di. vieqd. 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 depends on SUN'S ELLIPTICAL MOTIONr. 83 the intensity multiplied by the time it is applied; and the CHAm,. m. product of the time and distance to 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 80 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 ihethod time of describing the arc D)A C is 178 days 17' w hours; and of obtaining -~rruu~ the eccentrithe time of describing the are CB ) is 186 days 121 hours. city of an orBut, as areas are in proportion to the times of their descrip- bit. tion; therefore, d. h. d. *h. area CSDA: area CBD)S:: 178 174 186 124. 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 diameter of the ellipse, as represented by Fig. 19. For the sake of simplicity we will suppose the observer at Prepararest -at the point o (one focus of the ellipse ), and the sun tion for finding the true really to move round on the ellipse, describing equal areas variation in in equal times round the point o. an ellipse. Conceive, also, an imaginary sun to pass round the circle, describing equal angles, in equal times, round the center m. Now suppose the two suns to be together at the point B: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 84 ASTRONOMY. CHAP IT. Fig. 19. at the point A, and again at the point B, /c -C< and so continue in each subsequent revolution. The imaginary sun A 3 iB on the circle everywhere 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 saine time, from the point B, and, after a certain interval of time, one is at s, the other at I'. Then we must have area oBs: area mBs':: area elli2se: area circle. Mean and The angle B m s' is the angle the sun would make, or its tlte ano- increase in longitude from the apogee; provided the angular maly. motion of the sun was uniform. The angle B o s is its true increase of longitude; the difference between these two angles is the angle mn o. The angle B m ns' is always known by the time; and if to every degree of the angle B m s' we knew the corresponding angle m no, the two would give us the angle Bo s; for, Bms' —mno-=mon, or Bos. The angle mins' is called the mean anomaly, and the angle Bo s is called the true anomaly. The eqa.- The angle B m s' is greater than the angle Bos, all the tlon of the way from the apogee to the perigee; but from the perigee to center. the apogec,the true sun, on the ellipse, is in advance of the imaginary sun on the circle. The angle n 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 mnno depends on the eccentricity of tthe ellipse; and its amount is put in a table corresponding to every ECCENTRICITY OF ORBIT. 85 degree of the mean anomaly; subtractive, from the apogee to CUAP. nII, the perigee, and additive from the perigee to the apogee.* (81.) Again: conceive the two suns to set out from the same The greatpoint, B; and as the angle B ms' increases uniformly, it will est equation of the center increase and become greater and greater than the angle B o s, gives the ec. until the true sun attains its mean angular motion, and no centricity of the orbit, longer. Then the angle m n o attains its greatest value, and, at that time the side an=no, and the point n is perpendicular over o m, and o s is a mean proportional between o B and o A. That is, when the sun, or any planet, attains the greatest equa. tion, of the center, the true sun is very near the extremity 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 com0. puting the eccentricity of the ellipse, as we may see by the note below.t t Let C (Fig. 20) be the place of the true sun, and Fig 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 sectorA mF G to the triangle om C, o m as the circle is to the ellipse. Now let mB=a, inC=b, om=ea, r —3.1416; Then, the area of the circle is ran2; the area of the ellipse is a cab rab; that of the sector is ( GtP),. and of the triangle e2. eab lta\ Hbence,- 2:: 7rab: a2; * By a mnere mechanical contrivance, the modern astronomical tables are so arranged, that all corrections are rendered additive; so that the mechanical operator 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. 86 ASTRONOMY. CHAP. II. IWVhen once the eccentricity of any planetary ellipse becomes 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, - -ea: (GF)a: b: a; deducieg- the eccentricity Or, - ea GF: 1 1 from the 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 eccenthe center. tric ellipse. Now we must compute the number of degrees in the arc F G. The whole circumference is 2zra. Therefore, 2 a: ea.:: 360 are FG: 180 e'Hence, - - are FGG= =angle n m C. But the angle o nm=nm C+-n Cm-=2nmn C, nearly; 360 e Therefore, 2 n m C_2 on m = 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 360 e 360e10 55' 24". From this equation, it is true, we have only the approximate value of e; but it is a very approximate value, and sufficiently accurate. Reducing both members to seconds, and we have, 3600'360e-=6924 7r, and e=0.0167842. The greatest equation of the center is at present diminishing at the rate of 17".17 in one hundred years: this corresponds to a diminution of eccentricity by 0.00004166 which is determined by a solution of the following equation: 360e 3600- =17"T17, CHANGE OF SEASONS. 8 CHAPTER IV.'THE CAUSES OF THE CHANGE OF SEASONS. (82.) THE annual revolution of the earth in its orbit, CHAP. A1. 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 pause there would be no change of seasons, and the sun would then ofthechange of seasons. 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, NS, to be indined to the paper at an angle of 660 32'. Fig. 21. A In-all representations of NS, one half of it is supposed to be above the paper, the other half below it.,NS is always parallel to itself; that is, it is always in the same position* always at the same inclination to the plane * Except minulte variations, which it would be improper to notice in this part of the work. 7 9SS a~TROASTRONOMY. CIAP. IV. of its orbit —always directed to the same point in the head vens, in whatever part of the orbit it may be. The plane of the equator, represented by Eg, is inclined to the plane of the orbit by an angle of 23~ 28'. Importance By inspecting the figure, the reader will gather a clearer of inspecting the figure. view of the 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. Position df Now conceive the line VNS to stand perpendicular to the the axis to plane of the paper, and continue so; then Eq would lie on cause no Pi change of the paper, and the sun would at all times be in the plane of seasons. the equator, and there would be no change of seasons. If NS 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 eqii The longitiude of the sun, at the vernal equinox, is 00; and hoctial and at the autumnal equinox, its longitude is 180~. solstitial points. The time of the greatest north declination is the 20th of June; the sun's longitude is then 900, and is said to be at the summner solstice. The time of the greatest south declination is the 22d of December; the sun's longitude, at that time, is 2700, and is 3aid to be at the winter solstice. CHANGES OF SEASONS. 89 By inspecting the figure, we perceive, that when the earth CHAP. IV. is at the summer solstice, the north pole, _P, and a conside- Long sea. rable portion of the earth's surface around. is within the en- sons of sunlightened half of the earth; and as the earth revolves on its darkness at axis NS, this portion constantly remains enlightened, giving and about a constant' day —or a day of weeks and months duration, the poles. 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 230 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 part of the earth's surface Temrperah depends mainly, if not entirely, on its exposure to the sun's earth. 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 received 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 average; 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 summer, 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,, the mean state is once more attained. From thence to D) and, finally, round again to A, all the same phenomena, it is obvious, must again occur, but reversed;it being now winter in the northern, and summer in the southern hemisphere." H* ~90 ~ ASTRONOMY. CHAP. IV. The inquiry is sometimes made why we do not have thef 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 extremetem. give extra warmth, on arriving at the summer solstice; but peratuge.e 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 intemperature, 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 placesl 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 essentially abate on or about the 16th. of February, in all northern, latitudes. CHAPTER V. EQUATIO'N 0'2 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 eonfusion in. the minds of the pupils; and such is the nature of the case, that it is difficult to understand even the facts, without investigating their causes. Sidereal Sidereal time has no equation; it is uniform, and, of itselfl time perfect. perfect and complete. EQUATION OF TIME. TA 7he time, by a;prfect clock, is theoretically perfect and CHAP. IV. ~complete, and is called mean time. The time, by the sun, is not uniform; and, to make it Solar tiJme agree with the peefedt cloGk, requires a correction — a quan- fit niho;f tity to make equality; and tlis quantity is called the equation of time.* If the sun were sta'tionary i'n tie heavens, like a star, it would come to the meridian after exact an'd equal intervals of time; and, in that case, there would be no equation of time. If the sun's motion, in right ascension, were aniform, 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 stationury in the heavens,'nor does it move nriformly; therefore it tannot 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 me idian, it is then apparent n'oon, Mean anr for that day: it is the real solar noon, or, as near as may be, apparsnt half way between sunrise and sunset; but it may not be noon. 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 livery 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 in'terval. But the Sun increases 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 over the meridian is just 24 hoars; 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 sun 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, then, the sun and clock must be constantly separating. On * In astronomy, the term equation is applied to all corrections to,onVert a mean to its true quantity. 92 ASTRONOMY. CHAP. V. and about the 20th of December, the time from one meridian of the sun to another is 24 h. 0 m. 24.3 s., 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, result of two we must separate the cause into its two natural divisions. 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 uniform, still there would be an equation of time; for that motion, 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 the year in agree, that is, show noon at the same instant, are April 15th, which the sun and June 16th, September 1st, and December 24th. clock agree. The greatest amount, arising from the first cause, is 7m. 42 s., and the greatest amount, from the second cause, is 9 m. 53 s.; but as these maximum results never happen exactly at the same time, therefore the equation of time can never amount to 17 m. 35 s. In fact, the greatest amount is 16 m. 17 s., and takes place on the 3d of November; antd, 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. 93 clock agree, in consequence of the slow and gradual change cHAP. V; in the position of the solar apogee. (See Art. 77.) (84. ) The elliptical form of the earth's orbit gives rise to The eqna. the unequal motion of the earth in its orbit, and thence to the tion of the apparent unequal motion of the sun in the ecliptic; and this and the first same unequal motion is what we have denominated the first part of the equation of cause of the equation of time. Indeed, this part of the equa- time, u lave 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. cause. The greatest equation for the sun's longitude (81, note ), is by observation 1~ 55' 30"; and this, proportioned into time, gives 7 m. 42 s., 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 otherwise 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 ini the other half of its orbit. For a more particular explanation of the second cause, we must call attention to Fig. 22. Let (Fig. Fig. 22. 22) represent the P ecliptic, and mc Cthe equator. By the first correction, the apparent motion along the ecliptic is rendered s uniform; and the sun. is then supposed to pass over equal spaces in equal intervals of time along the are c Sga. 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 cr S on 94 ASTRONOMY. CHAP, V. the ecliptic, is greater than to the same meridian along the equator. The difference between mr Sand c S', turned into time, is the equation of time arising from the obliquity of the ecliptic corresponding to the point S. At the points mr, a, and -, 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 to To compute the equation of time arising from this cause, compute the we must solve the spherical triangle cr S S'; cp S is the sun's of the epua. longitude, and the angle at mc is the obliquity of the ecliptic, tion of time. and at S' is a right angle. Assume any longitude, as 32~, 350, or 400, or any other number of degrees, and compute the base. The difference between this base and the sun's longitude, converted into time, is the quantity sought corresponding 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 meantbysun equator; and as uniform spaces along the equator, near the fast and slow of clock. point cp, will pass over more meridians than the same number of equal spaces on the ecliptic; therefore the sun, at S, will befast of clock, or come to the meridian before it is 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 quadrant 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 give the following table. EQUATION OF TIME. 95 Table showing the separate results of the two causes for the equa- CHAP. V. tion of time, corresponding to every flfth day of the second years after leap year; but is nearly correctfor any year. st cause. cause. st cause. id cause. Sun slow Sun slowSun fasto Su slow. of Clock. of Clock. m. S. m. s. m. s. m. s. januarv 5 0 41 5 8 July 1 0 0 3 32 10 1 22 6 35 7 0 40 5 815 2 2 7 48 12 1 19 6 35 20 2 41 8 45 17 1 57 7 48 25 3 19 9 26 22 235 8 45 29 3 56 9 49 28 3 12 9 26 Feb. 3 4 30 9 53 Aug. 2 347 9 49 8 5 2 9 40 7 421 9 53 13 5 32 9 9 12 452 9 40 18 5 39 8 23 17 522 9 9 23 6 24 7 22 22 550 8 23 28 6 45 6 9 28 614 7 22 March 5 7 3 4 46 Sept. 2 636 6 9 10 7 18 3 15 7 656 446 15 7 29 1 39 12 712 3 15 20 7 37 sun fast 17 724 1 39 25 7 42 1 39 23 7 34 sun fast 30 7 42- 3 15 28 740 1 39 April 4 7 40 4 46 Oct. 3 7 42 3 15 9 7 34 6 9 8 740 4 46 14 7 24 7 22 13 734 6 9 19 7 12 8 23 18 724 7 22 24 6 56 9 9 23 7 12 8 23 30 6 36 9 40 28 656 9 9 May 5 6 14 9 53 Nov. 2 636 9 40 10 5 50 9 49 7 614 9 53 15 5 22 9 26 12 550 9 49 20 4 52 8 45 17 5 22 9 26 26 4 21 7 48 22 452 8 45 31 3 47 6 35 27 422 7 48 JuIne 5 3 12 5 8 Dec. 2 3 47 6 35 10 235 332 7 312 5 8 16 1 57 1 48 12 235 3 32 21 1 19 sun slow 17 1 57 1 48 26 0 40 1 48 21 1 19 sun slow. 26 040 1 48 By this table, the regular and symmetrical result of each Use of ths cause is visible to the eye; but the actual value of the equa- preceding tlon of time, for any particular day, is the combined results table. of these two causes. Thus, to find the equation of time for fthe 5th day of March, we look at the table and find that 96 ASTRONOMY. CHAP. V. The first cause gives sun slow, - - - 7m. 3 s. 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 11 m. 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. 49 s. Use of the By inspecting the table, we perceive, that on the 14th of equatidn of April the two results nearly counteract each other; and consequently 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 frequently 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 to every particular point on the ecliptic, and, at the same time depend- pi.e 1 ing on the time, the particular point on the ecliptic has a definite relasnr's longi- tion to the equator, therefore any point, as S (Fig. 22), on tude can be the ecliptic, has the two corrections for the equation of time; formed. 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 as that change is very slow, a table of that kind will serve for PLANETARY MOTIONS. 97 many 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' clock regulator, should thoroughly understand the equation of the eutiomr time; and persons thus occupied pay great attention to it; but most people in common life are hardly aware of its existence. CHAPTER VI. TEE APPARENT MOTIONS OF THE PLANETS. (86.) Wr have often reminded the reader of the great CHAP. VI. regularity of the fixed stars, and of their uniform positions in relation to each other;'and by this very regularity and constancy 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- Recapitu. cerning the sun and moon, which we briefly recapitulate, as lation. 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 variable; 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 celes. But there are several other bodies that are not fixed stars; tial bodies. For I A ST RO NO M Y. CHIAD. Vi. and although not as conspicuous as the sun and moon, hat been known from time immemorial. They appear to belong to one family; but, before the true system of the world was discovered, 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. The plan. In general terms, these bodies are called planets — and ets. -Venus. 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 observedfacts, as extracted from the Cambridge astronomy The morn- (87.)'" There are few who have not observed a beautiful ing and even- star in the west, a little after sunset, and called, for this reasng star. 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 450, or 4th of the celestial hemisphere, after which it begins to return; and as we can ordinarily discern it with the naked eye only when the sun is below the horizon, it is visible only for a certain time immediately 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, anri 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 450, 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 mocre PLANETARY MOTION. g9 -than 2000 years, evidently indicate that the evening and CHAP. VL, morning star are one and the same body.: They indicate, also, that this star has a proper motion, in virtue of which it oscillates about the sun, sometimes preceding and sometimes following 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 T'e... is, in a great measure, taken away, and we perceive that it of VwU:. 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 Trhe nhrse:; now gradually increases to a semicircle, and finally to a full.f v..... 1,,,!, circle, as the planet again approaches the sun; but, as t/he its arpan'erlT iiamele, crescent increases, the apparent diameter cf the planet dciminzisaes; have i.are r i and at every alternate approach of the planet to the sun, the spolndinphase of the planet is full, and the apparent diameter small;. ehang 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 dim 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 1'. 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 100 ASTRONOMY. CaV. 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 al- as the sun —never departing more than 47~ on each side of ways near it- 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 meridian withinr three hours 20 minutes of the sun, and, of elongation. course, in daylight. But this does not prevent meridian ob.servations being taken upon it, through a good telescope;* a micrometer, which is two spider lines, always parallel, near the focus of a telescope, and so attached, by the mechanism of screws, as to open and close at pleasure. To understand its grade of adjustment, bring the two lines together, so as to form one line. Then take any object, whose angular diameter is known at that time, as the diameter of the sun, and open the lines so as just to take in its disc, counting the turns, and parts of a turn given to the index screw to open to this object. From this we can compute the angle corresponding to one turn, or to any part of a turn, of the index screw. Now if we wish to measure the apparent diameter of any planet, bring the lines together, and then open them, just to inclose the planet; and the number of.turns, or the part of a turn, given to the screw, will determine the result. This may not be the exact mechanism of every micrometer, but this is the principle of their construction. * Perhaps we ought to have informed the reader before, "that the stars continue visible through telescopes, during the day, as well as the night; and that, in proportion to the power of.the instrument, not only the largest and brightest of them, but even those of inferior luster, such as scarcely strike the eye, at night, as at all conspicuous, are readily found and followed even at noonday,- unless in that part of the sky which is very near the sun,-by those who possess the means of pointing a telescope accurately to the proper places. Indeed, from the bottoms of deep narrow pits, such as a well, or the shaft of a mine, such bright stars as pass the zenith may even be discerned by the naked eye; and we have ourselves heard it stated by a celebrated optician, that the PLANETARY 4MOTION, 101 and, as to this particular planet, it is sometimes so bright as CHAP. VI. to be seen by the unassisted eye in the daytime. (90.) Even without instruments and meridian observations, Motion of the attentive observer can determine that the motion of Venus, Venus in respect to thie in relation to the stars, is very irregular - sometimes its stars. motion is rapid - sometimnes slow - sometimes direct - sometimes 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 traverse round and round among the stars. (91. ) But Venus is not the only planet that exhibits the Mercury appearances we have just described. There is one other, and similar in all appearances only one — Mercury; a very small planet, rarely visible to the to Venos. naked eye, and not known to the very ancient astronomers. Whatever description we have given of Venus applies to Mercury, 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 appvearances clearly indicate that the sun must be A conclo the center, or near the center, of these motions, and not tle earth; sion, and that Mercury must revolve in an orbit within that of Venus. So clear and so unavoidable were these inferences, that ever the ancients (who were the most determined advocates for the immobiitq 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. clent astronomers found them variable in their apparent diam- rent diam&e 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."-I-Rerschel's Astro. nomy. * In astronomy, direct motion is eastward among the stars; stationary is no apparent motion, in respect to the stars; and retrograde is a westward motion. 1 * 102 AS T R 0 N 0 M i. CH.i. V-I elrs, and angular motions; so mnuch so, that it was -.ipossitle ters or' the to reconcile ajppecarances with the idea of a stationary point of I!anets are observation; unless the appearances were taken for realities, variable. and that was against all true notions of philosophy. The planet _,::1,s is most remarkable for its variations; and tl!e 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. The earth The greatest apparent diameter of Mars takes place when nof its on. the planet is in opposition to the sun, and it is then 17".1; and tion. its least apparent diameter takes place when in the neighborhood 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 rregularities stars, is 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 motionand those planets which show the greatest change in appaL 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 stationary, we cannot account for this retrograde motion of'the planets, unless that motion is real; and if real, why, anld how can it change from direct to stationary, and from stationary to retrograde, and the reverse? Retrograde But if we conceive t/he earth in motion, and going the sancew motion ofthe Way with the planet, a(nd moving snore rapidly tian the planet, planets acoounteza for. theen the planet will cppear to'un back; that is, retrolcgrade. 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 eircumstances take place in all positions from the sun, it is a sufficient 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. 10'3 Ao, 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 rianets nevindicated by the change in apparent diameter; and observa- erstationary. tions confirm this, and show that no planet is really stationary, although it may appear to be so. (95.) If we suppose the earth to be but one of a familyof The earth a bodies, called planets - all circulating round the sun at dif- planet. Trrent times —in the order of Mercury, Venus, Earth, Mars (omitting the small telescopic planets), Jupiter, Saturn, Hterschel, or Uranus, we can then give a rational and simple account 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- Copernieus Jedged, is called the Copernican system, from its discoverer, and the Copernican sysCopernicus, 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- vived. 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 must fall 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 his dialogue 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 three fictitious persons, and the points left undecided. But the caution of Galileo was not suflicient, or his dialogue 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. -8 104 $ ~ CASTRONOSMYg cmra. v1. But this is a digression. With the history of astronomy, ax interesting as it may be, we design to have little to do, and to proceed only with the science itself. CHAPTElR VII. PIRIST APPROXIMATIONS TOr THE HRLATIVE DISTANCES OF THl PLANETS FROM THE SUN.- HOW TEr 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. Distinction Mercurw and Venue, never coming in opposition to the sun, between" in- but revolving around that body in orbits that are within that ferior and superior pIan- of the earth, are called inferior planets. at$. 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 position 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 ali the superior planets; and after that, it will be sufficient to state results. For materials to operate with, we give the following table' of the planetary irregularities ( so called) drawn from obser.vation: Greatest- Least Angular Dist.i Apparent Apparent from Sun at the Mean arc of Planets. Diameters. Diameters. instant of being Retrogradationavl stationary. [ Mercury. 11.3 5.0 18 00 13 30 Venus. 59.6 9.6 28 48 16 12 Earth. Mars. 17.1 3.6' 136 48 16 12 Jupiter. 44.5 304 115 12 9 54 Saturn. 20.1 16.3 o08 54 6 18 Uranus. 4.1 3.7 103 30 3 36... ~. _.,............... PLANETARY MOTiON) 105 Er~~~-D_ —~~~ —?~ ~CHAP. Vnb Mean Duration of th Mean Duration of the Synodic Planets, grade otion. Revolution, or interval between grade Mtwo successive oppositions, Mercury. 23 days. 118 days. Venus. 42 " 584 " Earth. Mars. 73' 780 " Jupiter. 121 " 399 " Saturn. 139' 378'" Uranus. 151' a 370' In the preceding table, the word mean is used at the head Why the of several coluins, because these elements are variable- word MEAm should be sometimes more and sometimes less, than the numbers here used. given - which indicates that the planets do not revolve in ciricles 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 elongaz table, we must not fail to note the important element of the tions of Mer~ envy and Ve~ elongations of M!ecury and Venus. This term can be applied ansY to no other planets. It is very variable in regard to Metcury- showing that Thiselement the orbit of that planet is quite elliptical. The Variation is variable and what it 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 s 17~ 37', The greatest 6 " is - 280 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 - 440 58'. The greatest " " is - 470 30'. The mean (or at mean distances), is - 460 - The least extremes must happen when the planet is in its perigee and the earth in its apogee, and the greatest when the earth is in perigee and the planet in apogee; but it is 106 ASTRONOMY. CHAP. VnI. very seldom that these two circumstances take place at. the same time. How to Relying on these facts as established by observations, we find the comparative can easily deduce the relative orbits of Mercury and Venus. magnitudes Let S (Fig. 23) reof the orbits Fig. 23. of Mercury, present the sun, E the Venus, and earth, VVenus. the earth -the earth Conceive the planet to pass round the sun in the direction of A V B. The earth moves also in the same direction, but not so rapidly as Venus. Now it is clearly evident, from inspection, that when the planet is passing by the earth, as at B, it will appear to pass along in the heavens in the direction of m to n. But when the planet is passing along in its orbit, at A, and the earth about the popsition of E, the planet will appear to pass in the direction of n to m. When the planet is at, as represented in the figure, its absolute motion is nearly toward the earth, and, of course, its appearance is nearly stationary. What to It is absolutely stationary only at one point, and even then understand by station. but for a moment; and that point is where its apparent moary. 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 of the line SE. When the line E V touches the circumference A VB, the angle S.E V, or angle of elongation, is then greatest; and the triangle SE V is right angled at V; and if SE is made radius, S V will be the sine of the angle. SE 17: But the line SE is assumed equal to unity, and then S V PLANETARY MOTION. 107 will be the natural sine of 46~0 20', and can be taken out of CHAP, II. any table of natural sines; or it can be computed by logarithms, and the result is.72336. For the planet Mercury, the mean of the same angle is 224 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 The orbits of' Mercury SE V, of greatest elongation, would always be the same; and Venw u 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 SE 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 Computation of orbits find the approximate relations of their orbits. For instance, from aopab when the planet Venus is at B, and appears on the sun's rent diame. disc, its apparent diameter is 59'".6; and when it is at A, or ters as near A as can be seen by a telescope, its apparent diameter is 9".6. Now put SB=x; then EB=I — x, and AE=1+-x By Art. 66, 1 —x: 1+x:: 96: 596; Hence, - - - x-0.72254. By a like computation, the mean distance of Mercury from the 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 M one of the superior planets, say Mars. It is easy to decide, from observation, when the planet is in opposition to the sun. 108 ASTRONOMY, CHAP. vii. Fig. 24. This gives the position Method of of S, E, and 2, in one approximat- right line, in respect ing to the or to longitude. Now by superior pla- knowing the true angunets. _ lar motion of the earth about the sun (73), and the mean angular motion of the planet, * we can determine the angle mSe, corresponding to any definite future time; for, by the motion of the earth round the sun, we can determine the angle ESe; and by the motion of the planet in the same time, we can determine the angle MS nm; and the difThe relative By means of apparent diameters, we can determine the plianet from 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, termined by after a definite time, when the earth is at e, measure the apthe variation in its parent diameter again, and observe the angle S e m. Proapparen; diae duce Se to n. Then, by the apparent diameters, we have meter the proportion of e m and e n (e n is the same as EXi brought to this position); and in the triangle em n we have the proportion between the two sides and the included angle m e n. These are sufficient data to determine the angles e nm and emn; and their difference is the angle Sme. Now we can determine the side S m, of the triangle Sm e, and the triangle Sem is completely known. Subtract the angle e Sm from the whole angle e SAC; and the angle _M'Sm is left. That is, while the earth is describing the angle E Se, the planet describes the angle JlfSm. Put P for the periodical revoe Here we anticipate a little; for we have not shown how to determine the periodical time of revolution from observation: but this is shown in a future chapter, and in the above text note. PLANETARY MOTION. 109 ference of these two angles is the angle m S e. By direct CHAP. VIT. observation at e, we determine the angle Se m; and two angles, and the side S e, of the triangle S m e, are sufficient to determine the side Sm, the value sought. The triangle gives the following proportion: sin. Sem sin. Sine: 1:: sin. Sem: Sm_s sin. Sm e This is a general solution, for any superior planet; but the Why the result is only approximate; for, until we know the eccentri- result is apcity of the orbit in question, and the part of the orbit in proximate. which the planet then is, we cannot accurately know the angle MS2m. lution of the planet; then, on the supposition of uniform motion, we have arc MSm: arcESe: 365: P In this proportion the two ares are known, and from thence P becomes known; and thus, we perceive, that by the variations of the apparent diameter of a planet, we can determine its relative distancefrom the sun, and its periodical revolution.. We give the following hypothetical example, for the purpose of further illustration. The apparent diameter of.Mars, when in opposition to the sun, A problem was observed to be 17".1. One hundred and eleven days afterward, when the earth had passed over 1100 of its orbit, the apparent diameter of Mars was again observed, and found to be 7".4, and its angcular position, in lorgitude, was 87 fr'om the sun. From these data, it is required tojfind the relative approximate distance of the planet from the sung and the approximate time of its revolution round the sun. From these data we have the angle S1n-=1100, Se m=- Its sol 87~; therefore ne m-=930. tion. — Fig, By the observed apparent diameter, we have EXM to em as 7".4 to 17".1; but E Me n, therefore en em:: 74: 171. In the triangle n e m we can take en= —74, and E'm=171, for the purpose merely of finding the angles. Then, by trigononmetry, we have 110 ASTRONOMY. clAsP. Vr. ( 102.) By a perusal of the last text note, it will ]be Seent Results by those even who are not expert mathematicians, that it is from varia- not difficult to decide upon the relative distances of the tions in apparent dia- planets from the sun, by observing their changes in apparent meters diameter, as seen from the earth. Such observations have been often made, and the following table shows the results; which are compared with the results deduced from Kepler's Third Law.* Planets. Deduced from appa- From Kepler's Difference or rent Diameters. Law. Error. Mercury,.. 0.386400 0.387098 -.000698 Venus..... 1 0.722540 0.723331 —.000791 Earth...... 1.000000 1.000000 Mars... 1.533333 1.523692 +.009641 Jupiter..... 5.180777 5.202776 -.021999 Saturn.. 9.579000 9.538786 -.040214 Uranus. 19.500000 19.182390 + -.317610 870 Text note 171+74: 171-74 tan. -: tan. difference becontinued. tween the angle n and n m e. That is, - 245: 97.: tan. 430 30': tan Sm-ne Whence, Sme=410~ 11'. Now-in the triangle Sme, sin. 410 11' 1:: sin. 870: Sm —1.517. Secondly, as the angle Sime-410 11' and Sem 870, therefore, - - mSe=510 49', and.MSm 58c 11'. But the times of revolution, between any two planets, must be inversely as the angles they describe in the same time; the greater the angle, the shorter the periodic time; and therefore if we put P to represent the periodical revolution of Mars, we shall have 58-, 110 ~: 3654: P. Hence P=6902 days. The true time is 686.97964; showing an error of a little more than three days; but this is not a great error, considering 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. 111 The distances drawn from Kepler's law, are considered CHAP. vii. more accurate than conclusions drawn from most other con- Wh1y the siderations; and it is rather remarkable that these deduc- results from tions from the apparent diameters agree as well as they do, aaterstcadiowing to the difficulty of settling the exact apparent diam- not be relied eter, by observation. Take the apparent diameter of Ura- upon for aecuracy, nus, for example, 3".7 and 4".1, and change either of them -1 of a second, and it will make a great difference in the deduced result. CHAPTER -VIII. PCeTHtODS OF OBSERVING THE PERIODICAL REVOLUTIONS OF THE PLANETS, AND THEIR RELATIVE DISTANCES FROM THE SUN. (103.) THE subject of this chapter will be td explain the CHAP-. Vit. principles of finding the periodical revolutions of the planets Why direct around the sun. If observers on the earth were at the observations are not to the 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 mathematical corrections, to obtain true results; and this was an impossible 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 impor, the sun, he would see them in their true places among the tant posistars -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 Mi the planet is in opposition to the sun; and it is seen projected 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 Conjunctions conjunction with the sun; but a conjunction cannot be o6- served. 112 As:bi RONO MY. CHAP. vII. served on 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 called trfansits. ( 104.) All the planets move around the sun in the same Revolution direction, and not far from the same plane, and the rudest of inferior and most careless observations show that those planets nearplanets less, and of supe. est the sun, perform their revolutions in shorter periods than rior planets those more remote. From this, we decide at once that the greater 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 Its longitude is then 180 degrees from the longitude of the can be observed sun, and comes to the meridian nearly or exactly at midnight. If it is a little short of opposition at the time of one observation, and a little past at another, the observer can proportion 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 Fig. 25. opposition of the same kind and s of the same planet, can be observed and recorded. The elapsed time between two Synodical such oppositions is called the syrevolution. nodical revolution of the planet. We note the time that a \Mean ang. planet is in opposition to the ]ar motion of sun. Then S, E and lA are in the planets one planeas represented in Fig. determined p defrom their 25. If the planet X should synodical remain at rest while the earth,evolutions. ME made its revolution, then I V~ the synodical revolution would be the same as the length of our year. But all the planets move in the same direction ae PLANETARY MOTION. 113 the earth; and therefore the earth, after making a revolu- CHAP. VIII. tion, must pass onward and employ additional time to overtake 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 conriod must be proportionately long. The planet Jupiter siderations. moves about 310 in its orbit in a year; and therefore, after one opposition, the earth is round to the same point in 365k days, and to gain the 31~ requires about 32 days more; hence the synodical revolution of Jupiter must be about 397 days, by this very rough and imperfect computation. By inspecting the table on page 105, we perceive that the mean synodical revolution of Jupiter is 399 days, and this observed fact shows us that Jupiter passes over about 310 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 Computation in its orbit is very accurately known. Represent its daily to determine the mean an motion by a. The angular motion of the planet (any supe- gulal motion rior planet that may be under consideration) is unknown; oftheearth. therefore, represent its daily.motion by x. Let the angle E SC represent a, and the angle!f S 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 angle m SC is contained in 3600, will be the number of days in 360 a synodical revolution. Therefore, _ = the observed time of a synodical revolution; and by taking the times from the table (page 105), we have the following equations: Mars. Jupiter. Saturn. Uranus. 360 360 360 360 ='780, =399, - 378, __ 370.* a -- a —. a - a - a x d * These equations correspond to the general equation t —- in Robinson's Algebra, page 105, University edition. 8 J. 114 ASTRONOMY CHAP. VIII. The value of a is 59' 8", and then a solution of these sevw eral equations gives the mean angulqr motion, per day, of the several planets, as follows: Mars. Jupiter. Saturn. Uranus. 31' 27" 4' 59".4 I' 59".5 45".3 Times of Dividing the whole circle 360~ by the mean daily motion revolution of each planet, will give their respective times of revolution, derived from the angular and the following are the results: nlotion. Mars, Jupiter. Saturn. Uranus. 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 Yenus. 360 360 -....118; -= 584. x —a x-a x=40 2' 11"; x=10 36' 7". Mean an- These diurnal angular motions correspond to 89 days for gular motion the revolution of Mercury, and 224.8 days for the revolution ofthe inferior planets, and of Venus. All these results are, of course, understood as their revolu-first approximations, and accuracy here is not attempted. tion round the sun. 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 attempted, 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 following 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 Mercury 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 440 41' 35". Observa. Comparing this occultation with that which took place in tions carried 1723, the true time of conjunction was November 9th, at 5 h. through a long course 29m., P. M., and Mercury's longitude was 460 47' 20". PLANXETARY MOTrION. 115 The elapsed time was 92 years, 2 days, 9 h. 39 m. Twenty- CHAP. VIII. two of these years were bissextile; therefore the elapsed time of years, to was (92X365) days, plus 24 d. 9 h. 39 m. secure accuIn this interval, Mercury made 382 revolutions, and 20 5 racy. 45" over. That is, in 33604.402 days, Mercury described 137522.095826 degrees; and therefore, by 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 360 0923 or, 87.9701 days, or 87 days 23 h. 15 m. 57 s. 4.0923 ( 108. ) The following is another method of observing the Anothet periodical times of the planets, to which we call the student's observingthod of special attention. periodical re. The orbits of all the planets are a little inclined to the volutions cf 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 straight line, (See Geometry.) The reader must also recognize and acknowledge the following 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 in 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 i3 the two planes do not coincide. Passing the plane of the meant by 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 1fl16G - A AS o NT 0OMY. CHAP. VIIT. 1800, as seen from the sun (unless the nodes themselVe ati'e Two nodes in motion, which in fact they are; but such motion is not, 180 degrees sensible for one or two revolutions of Venus or Mars). from each theras seen 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 elapeed, is the time of a revolution of that planet round the sun. From such observations 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 motion3 they were enabled to caleulate the times and circumstances of the transits which happened in 1761, and in 1769; save those resulting from parallax alone. Pirst idea of (109.) On comparing the time that a planet remains on the perieo of the plan. each side of the ecliptic, we can form some idea of the position Cts. 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. Ypinalresults. (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 long course of years, and from age to age, the times of revolution, the mean relative distances of the planets from the sun, were approximated to, step by step, until a great degree of exactness Was attained, and the following were the results Sidereal Revolutions Mean distanlce from 0 Mercury, - - 87,969258 0.387098 Venus, - - 224.700787 0.723332 Earth,'- - 365.256383 1.000000 Mars, - - 686.979646 T.523692 Jupiter, ~ - 4332.584821 5i202776 Saturn, - - 10759.219817 9,538786'Uranus, - ~ 30686.820830 19.182390 PLANETARY MOTION. 17 (1) by inspecting the preceding table, we find that the CHAP. VIIL treater the distance from the sun, the greater the time of Times ofrev, revolution; but the ratio for the time is greater than the ratio olaution and distances corresponding to distance; yet we cannot doubt that some compared connection exists between these ratios. For instance, let us compare the Earth with Jupiter, The'atio between their times of revolution, is near 12. The ratio between their relative distances from the sun, as We perceive, is nearly 5.2. The Square of 12 is 144; the cube of 5.2 is near 1414 But 12 is a little greater than the real ratio between the times of revolutionj and 5.2 is not quite large enough for the ratio of distance; and by taking the correct ratios, they seem to bear the relation of siquare to cube. Without a very rigid or close examinations we perceive that five revolutions of Jupiter are nearly equal to two revolutions of Saturn; that is, J is nearly the ratio between their times of revolution. By inspecting the column of distances, we perceive that the ratio of the distances of these two planets, is nearly 25 and if we square the first ratio, and cube the second, we shall have nearly the same ratio. Now let us compare two other planets, say Venus and Result di -aars, more exactly.,er Their ratio of eVtolutlon 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 tiiie, 0.9070694 Their ratio of distance isj 15,23692 log. - 1.182883 7.23'32 log. - 859323 Log. of the ratios, 0.823560 Multiply by- - - Log. of the cube of the ratio of distance, 0.970680 Thus:we perceive that the squares of the times of revolution. are to each other as the cubes of the mean distances of 118 ASTRONOMY CHAP. vlII. the planets from the sun,* and this is called Kepler's third Kepler's law; and it was by such numerical comparisons that Kepler taws. discovered the law.t We may now recapitulate the three laws of the solar system, called Kepler's laws, as they were discovered by that philosopher, Ist. The orbits of the planets are ellipses, of which the sun occupies one of the foci. 2d. The radius vector in each case describes areas about the focus, which are proportionaZ to the times. 3d. 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 -n, -m; n and m taken to represent the ratios. t n d-m; n and e taken to represent the ratiosd Square the Ist equation and cube the 2d. Then T2 3 -- *n2, and d=m3 But by inspection we know that n2=m3; therefore, -- or, t2 T2 d:: D3. t 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 compare 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 impatience, he made such errors in computation, that he rejected the hypothesis 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 immortal SOLAR PARALLAX. 119 CHAPTER IX. TR.ANSITS 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 Attempts to orbits, and their relative magnitudes; but, as yet, we know find the sun's nothing of the distance to the sun; save the indefinite fact, parallax. 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 suficiently Difficulties refined to determine this parallax by direct observation, in the of ancient manner of finding that of the moon (Art. 60), and hence the astronomers. ingenuity of men was called into exercise to find some artifice to obtain the desired result. After Kepler's laws were established, and the relative distances 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 Parallax of the sun (as determined by Kepler's law) are as 1 to 1.5237; Mars. 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 Maraldi opposition to the sun, and very near a star of the 5th mag- obtains an to approxima. nitude, its parallax became sensible; and Mr. Maraldi, an tion to the Italian astronomer, pronounced it to be 27". The relative parallax of Mars. 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 ear'th appears; and, at a greater distance, it will appear under a 9 120 ASTRONOMYi CHsAP.IX. less angle. The distance of Mars from the earth, at Z:hat 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-l tal parallax. Observa- On the 6th- of October, 1751, Mars was attentively obj tions byWar. t gentin and served by Wargentin and Lacaille (it being near its opposiLacaille. 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, ey's sugges- an English astronomer, very happily conceived the idea of' finding the sun's' parallax by the comparisons of observations made from different parts of the earth, on a transit of Venus over the sun's'disc. If the plane of the orbft 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. Rut the orbit of Venus is inclined to the orbit of the ea;rth by an angle of 30 23' 28"''''; and, in the year 1800, the planet crossed the ecliptic from south to north, in longitude 740 54' 12'', and from north to south, i'd longitude 2540 54' 12": the first mentfoned point is calledThe nodes the ascending node; the last, the descending node. The nodes; f venus. retrograde 31' 10" in a century. What times (115.) The mean synodical revolution of 584 days correin the vear sponds with no ali'quot part of a year; and therefore, in tlhe transits may t take place. course of time, these conjunctions will happen at different points along the ecliptic. The sllnis in that part of the ecliptic near the nodes ofYVenus, June 5th and December 6bt or 7th; and the two Iast transits happened' in 1761 and in 1769; and from these periods we date our knowledge of the solar parallax. Revolu- ( 116.)' The periodical revolution of the earth is 365.256383 tions com- ~ys; and that of Venus is 224.700787; and as numbers they pared. are neqarly in proportion of 13 to 8. From this it follows, that eight' revolutons- of the earth SOLAR PARALLAX. 121 requite nearly the same time as 13 revolutions of Venus; CRAP. IX. 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 Comparah tive motions Compared, as terms of a fraction, thus, 365.25638 and it is of venus and 8365.256381' 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 Robinsot's Arithmetic ) The approximating fractions are 1 1 2 3 8 235 Vg 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 revolutions 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 hate 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 122 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 conjunctions point; but, as it is, the points of conjunction in the heavens at the same time of, the are east and west of a given point, and approximate nearer year. and nearer to that point as the periods are greater and greater. Only two To be more practical, however, the intervals between contransits cian junctions are such, combined with a slight motion of t/he nodes, happen at in. tervals of' 8 that the geocentric latitude of Venus, at inferior conjunctions years. near the 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 trsits tf over the center of the sun, as seen from the center of the Venus. earth - that is, pass conjunction and node at the same timeat 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 (118.) Knowing the relative distances of Venus, and the can be computed. earth, from the sun — the positions and eccentricities of both Dr. Halley orbits-also their angular motions and periodical revolutionsshows how to find the every circumstance attending a transit, as seen from the sun's paral. earth's center, can be calculated; and Dr. Halley, in 1677, lax. read a paper before the London Astronomical Society, in Text note in 1761, June 5th; eight years after, 1769, June 3d, there continued, was another; and the next that will occur, at that node, will be in 2004, June 7th, 235 years after 1769. SOLAR PARALLAX. 123 which he explained the manner of deducing the parallax of CHAP. IX. 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 Venus ate 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 give the solar paworld, to have correct observations on the last transit of rallax than Venus, in 1769, at remote stations, that the British, French, those of Met. and Russian governments were induced to send out expedi- cury tions to various parts of the globe, to observe it. "The famous 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 paralla~x, 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 impor. from observations on a transit of Venus, we regard as the tance of this problem. most important, for a student to understand, of any in astronomy; 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 explanation. 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 and the earth. As these two bodies move the same way, and nearly in the same plane, we may suppose the earth stationary, and Venus 124 ASTRONOMY. Cam. IX. Fig. 26. to move with an angular velocity, The case equal to the difference of the two. simplified. When the planet arrives at v, an v' J 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 are v q, with a velocity equal to the diference 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-.m,~ ~ sures the angle A v' G; an angle v q under which a definite portion of the earth appears as seen from the sun. An abstract (121.) To have a more definite proposition idea of the practicability of this mefor the pur. thod, let us suppose the parallactic ptraion. o jiA G angle, A v' G, equal to 10", and inquire how long Venus would be in passing the relative are v q. Venus, at its mean rate, passes - 10 36' 8" in a day. The earth, c" " 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 2220": 1440m.:: 10": 6m. 29s. Now if observation gave more than 6 minutes and 29 seconds, we shall conclude that the parallactic angle was more than 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 position of each in relation to a line joining the center of the SOLAR.PARA LLAX. 125 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 whichwe deduce the parallactic angle A v' G, and from thence the horizontal parallax. The same observations can be made when the planet passes A combinaoff the sun, and a great many stations can be compared with tion of many - observations 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 mesubject, which is in some respects more simple and preferable thodofdednucing the proto the one just explained. It is for the observers at every blem. 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 beginning 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 different observers. 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 observed 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 A B v and a b v ( Fig. 27) will be proportional, and we have A v: av: A B: a b. The resalt. But the first three of these terms are known; therefore the fourth, a b, is known also; and if any definite angular space _ A\C 126 ASTRONO 0MY. CHAP. IX. Fig. 27. on the sun becomes: known, the whole semidiameter becomes known, and from thence the horizontal parallax is immediately deduced.* Under what (123.) The accuracy of this imethod should be circurmstan- questioned when Venus passes near the sun's ces this me- thod should center, for the two chords are never more than not be used. | - 30" asunder, and hence they will not perceptibly differ in length when passing near the sun's 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 diameter of Venus was observed to be 59". Transits of (125.) The same general principles apply Mercury not to the transits of Mercury and Venus; but those important. 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 Mercury, its transits occur more frequently. The frequent appearance of this planet on the face of the sun, gives to astronomers fine opportunities to determine the position of its node and the inclination of its orbit. Revolutions In 1779, M. Delambre, from observations on the transit of of thMecahry May 7, placed the ascending node, as seen from the sun, in and the earth placed compared. longitude 45~0 57' 3". From the transit of the 8th of May, 1845, a~ observed at Cincinnati, it must have been in longitude 460 31' 10"; this gives ita progressive motion of about 10 10' in a century. The inclination of the orbit ism70 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 parallax. (See 66). PLANETARY PARALLAX. 127 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- Intervals be. tween tranlow each other by some combination of these numbers. sits. The following is a list of all the transits of Mercury that have occurred, *or will occur, between the years 1800 and 1900: At the ascending node. At the descending node. 1802, - - - Nov. 8. 1799, - - - May 7. 1822, - - - Nov. 4. 1832, - - -May 5. 1835, - - - Nov. 7. 1845, - - - May 8. 1848, - - -Nov. 9. 1878, - - -May 6. 1861, - - - Nov. 11. 1891, - - - May 9. 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 MAGNITUDES. (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 mag. the real distance, diameter, and magnitude, of every planet nitudes and distances can in the solar system. now be deIn Art. 60 we have explained, or rather defined, the hori- termined zontal parallax of any body to be the angle under which -the sermidiameter of the earth appears, as seen from that body; and if the earth were as large as the body, the apparent diameter of the body, and its horizontal 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 semidiameter. The mean horizontal parallax of the sun, as determined in 128 ASTRONOM Y. CHAP. x. the last chapter, is 8".6; the semidiameter of the sun, at the Real dia- corresponding mean distance, is 16' 1", or 961". Now let d meter of the represent the real diameter of the earth, and D that of the ind,d deter- sun, then we shall have the following proportion: d D:: 8".6: 961".0. But d is 7912 miles; and the ratio of the last two terms is 111.66; therefore D=(111.66)(7912)-=883454 miles. Real dis. (127'.) The sun's horizontal parallax is the angle at the tance be. base of a right angled triangle; and the side opposite to it is tween the t t t earthandsun the radius of the earth (which, for the sake of convenience, determined. we now call unity). Let x represent the radius of the earthis orbit; then, by trigonometry, sin. 8".6: 1:: sin. 90~: x; sin. 900 Therefore, x i -log. 10.00000-log. 5.620073.* sin.8".6 That is, the log. of x=4.379927, or x=23984; which is the distance between the earth and. sun, when the semidiameter 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 ch]ange the unit from the semidiameter of the earth to an English mile; and then the distance between the earth and sun is Distance in (3956)(23984)-94880706; round numbers. and, in round numbers, we say 95 millions of miles. By Kepler's third law, we know the 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 themselves, 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 sineof 1", therefore, is -4. 685575 Multiply by the number 8.6; that is, add log. - 0. 934498 The Sine of 8".6, therefore, must be, - - - - 5. 620073 In the same manner, find the sine of any other small arc. PLANETARY PARALLAX. 129 all the planets from the sun; and now, having found the real CHAP. 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 th distance of any ing to any other planet. Thus, for the distance of Venus, planet from we multiply 94880706 by.72333; and the result isthe sun 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 Tofind the 1769, it was concluded that the horizontal parallax of that Venus.te f planet was 30".4; and its semidiameter, at the same time, was 29".2. Hence (Art. 127), 304: 292:: 7912: to a fourth term; which gives 7599 miles for the diameter of Venus. (130.) We cannot observe the horizontal parallax of Ju- Parallax of the planets piter, Saturn, or any other very remote planet: - if known at cannotbe oball, it becomes known by computation; but the parallax can served. 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 horizontal parallax. For the horibzontal parallax of Jupiter, when ai 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 h equal to the horizontal parallax of Jupiter; then we have, - 5.202776: 1:: 8".6: A; or hA= 8".6 5.202776' From this, we perceive, that if we divide the sun's horizontal How to parallax by the ratio qf a planet's distance from the sun, the compute the parallax of quotiet will be the horizontal parallax of the planet, when at a the planet. distance from the earth equal to its mean distancefrom the sun. (131.) To find the diameter of a planet, in relation to the SHow to diameter of the earth, we have a similar proportion as in Art. findd the real 126: and to find the diameter of Jupiter, we proceed as the planets. follows: The greatest apparent diameter of Jupiter, as seen from 9 130 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 semidiameter 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 8".6 unknown diameter of Jupiter; 5202776 is its horizontal parallax, and 18".35 its corresponding semidiameter; then, as 8.6 in Art. 126, - 7912. D 277::6 18.35; 7912X18.35X5.202776 7912 Therefore D 8 7912 X =1.11 8.6 87900 miles. In the same manner, we may find the diameter of any other planet. Jupiternot 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. Howtofind (132.) But the magnitudes of similar bodies are to one the magni- another as the cubes of their like dimensions; therefore the tude of the 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 IV, 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 acquainted with the theory of universal gravity. SOLAR. SYSTEM. CHtAPTER 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 IV, 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, Factsreveal. must be determined by observing the motion of spots on the ed by spots on the sun or visible disc; and if no spots are visible, the fact of revolution planets. 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 repository of 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. Sometimes forty or fifty spots may be seen at the same time, and sometimes only'one. They are often so large 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 n$.-.nle. L0~~2 ASTRONOMY. CHArp. xIk.'" Ill resplect to the nature and design of these spets, ahtinbst eVFty astronomer has formed a different theory. Some have supposed them to be solid opaque masses of scoria, floating in the liquid fire of the sun; others as satellites, revolviig round him, and hiding his light from us; others as immense masses, which have fallen on his disc, and which are dark coldred, because they have not yet bebome sufficiently heated. H Dr. Herschel, from many observations with his great telescope, concludes, that the shining matter of the sun consists of a mass of phosphoric clouds3 and that the spots on his surface are owing to disc turbances in the equilibrium of this luminous matter, by which openings are made through it. There are3 however, objections to this theory, as indeed there are to all the others, and at present it can only be said, that no satisfactory explanation of the cause of these spots hag been given.'. Singular (135.) AiMrcUry.. Thls planet is the nearest to the sun) le~ansofdis. and has been the subject of considerable remark in the prep covering rotation. ceding pages It is rarely visible, owing to its small size and proximity to the sun, and it never appears larger to the naled eye than a Star of the fifth magnitude. Mercury is too near the sun to admit of atiy 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 elongabe seen, tion Ist of the sin.'It will then be vsilble 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 morning, most advantageously in August and September. The symbol for the greatest elongation of Mercury, as written in the common almanacs, is ~ Gr. Elon. High moun- ( 136) Tenus. This planet is second in order from the sun, tains on ye. and in relation to'ts position and motion, has been sufficiently described. The period of its rotation on its akis Is 23h. 21m. The position of the axis is alilays the same, And is not at right angles o the plane of its orbit, which gives it a change of seasons,:he tangent position of the mun's light across this SOLAR SYSTEM. planet:shows a very lough sur- CRAP. XIL face; indeed, high mountains. Telescopic By the radiating and glimmer- views ot'Ye~ ihg nature of the light of this planet, we infer that it must have a deep and dense atmosphere. i 137,) The Earth is the next planet in the system; but it The 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 buing furnished with an attendant, The earth' the moon; and insignificant as this latter body it, compared attendanti to the whole solar system, it iS the most important and interesting to the inhabitants of otlr earth. The two bodies, the earth and the moon, as seen friom the sitn, 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.) iMars — the frst superior planet-is of a red color Mars; hid and very variable in its apparent magnitude. About every phyesranace,' 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 Fig. 98, The moon I. /911 Mnotion con. L n.P be a portion of the leave toward earth's orbit equal to 14& 30',6 XL, the sun. L the position of the earth at the First Quarter of the moon, and P its position at the L-ast Quarter. Draw the, chord L F, and compute m n the versed sine of the are 70 15'. The mean radius of the earth's orbit is 397 times the ragdius of the lunar orbit. A radius of 397 awid'an angle 70 15' gives a versed sine of 3.49; but on this scale thei distance from the earth to the moon is unity, or less than one third of n m; hence, the moon's path must be between the:,h ord L F and the are L n P'- that is, always concave toward tV sun. 134 ASTRONOMY. CHAP. xl. other year, when it comes to the meridian near midnight, it is then most conspicuous; and the next year it is scarcely noticed by the common observer. "The physical appearance of 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 winters 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 a sense 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 extinguished as their rays pass through that medium." Apparentim- (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. Mars, that the void space between the two had often been considered as an imrpe:fection, and it was a general impression among astronomers that a planet ought to occupy that vacant space. Bode's law. Professor Bode, of Berlin, on comparing the relative distances of the planets from the sun, discovered the following remarkable 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, &c., The reason and this last series of numbers very nearly,'though not exwhy it should actly, corresponds to the relative distances of the planets from not be called pae a law. 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 better 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. 135 coincidence." But, mere accident as it may be, it suggested CHAP. XI. the possibility of some small, undiscovered planet revolving A bold hy. in this region, and we can easily imagine the astonishment of pothesis. astronomers, on finding four in place of one, revolving in orbits tolerably well corresponding to this law, or rather coincidence. Had they even found but one, it would seem to indicate something more than mere 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 circulated 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. Names of DisPlanets. Ncoverserofis. Residence of Discoverers. Date of Discovery. -_ -_____ - _ _ _ - ____ ___ ______ Ceres... M. Piazi, Palermo, Sicily, 1st Jan., 1801. Pallas... Dr. Olbers, Bremen, Germany, 28th Mar., 1802. Juno.. M. Harding, Lilienthal, near Bremen, 1st Sept. 1804.' Vesta.. Dr. Olbers, Bremenl, 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 fasvorable to this hypotbe. been made, but thje elements of the orbits are not regarded as sis. 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 Ilebe. His success induced others to a like exa- ets discovert ed in 1845 mination, and a Mr. Hind, of London, within the past year, and 1846. 10 136. ASTRO N GOMY. CGHP. 3x. 1848, claims a seventh and eighth asteroid, name1d Irs and-'Flora. Thus we have eight miniature worlds, supposed to have once composed a planet; and if the four last named are. veritable discoveries, we shall soon have the elements of their orbits in an unquestionable shape, The elements of the orbits of the four known asteroids, as given for the epoch 1820, are not as accurate as the following, 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. Sidere I1 Mean Distance from Eccentricity of Revolutions. the Sun. Orbits. Days. Vesta..3..2.... E34. 289 2. 36120 0. 08913 Junlo @ 1594. 721 2. 66514 0. 25385 Ceres.... 1683. 064 2. 76910 0. 07844 Pallas....... 1685. 162 2. 77125 0. 24050 Pla etc, Longitude of Inclination of Longitude of anescending Node. Orbits. Perihelion-.-'.... 0'' " 0' Vesta..... 103 20 47 F 8 29 I 251 4 34 Juno....... 70 53 0 13 2 53 54 18 32 Ceres.... 80 47' 5g 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:,,ig. 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 ifs to show that there are two' points in the- heavens, nearly opposite to each other, near toc which all these planets; pass. One of these points is about the longitude of 185 degrees, and' the Iatitu&e- 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 ids internal fires, from thatf SOLAR SYSTEM. 138 ionment they must have HAP. XI. started from the same Where the point, and the orbits of all original pla. hgave one c mnon distance have epod have explod. from the sun; and for ed, if tlh ages after such a catas- hypothesis of an origin at trophe, these fragments planet is trui must have had nearly a eommon node; and the fact that they do not, at 2Iresent, pass through a common point, nor have a common node, does not prove that they were not b'riginally in one body; for, owing to mutual dis1turbances, and the disturbantes of other pla- Figi 29. nets, the nodes must change positions; andthe longe'r axis of the orbits, especially the very ec-'centric ones, must change ~positionis; andnow (after we know not how "many'ages), i is not inconsistent with the theory bof an el:plosion, that we find the orbits as they -are. The hypothesis that these Fplanets were originally one, and must, thereforb, have two ciommon points in the heavens nea. which they must all pass, led to the discovery of Juno and T,* 138 ASTRONOMY. CHAP. XI. Vesta, by carefully observing these two portions of the heavens. The apparent diameters of these planets are too small to be accurately measured; and therefore we have only a very rough or conjectural knowledge of their real diameters. All of these planets are invisible to the naked eye, except Vesta, which sometimes can be seen as a star of the 5th or Cth magnitude. (141.) Jupiter. We now come to the most magnificent planet in the system- the well-known Jupiter —which is nearly 1300 times the magnitude of the earth. Jupiter's The disc of Jupiter is always observed to be crossed, in an belts. eastern and western direction, by dark bands, as represented in Fig. 30. Fig. 30.- Telescopic View of Jupiter. "These belts are, however, fby no means alike at all times; they vary in breadth and in situation on the disc (though never in their general'direction). They havei even-been seen' broken up, and distributed over-the'whole face'of Athe planet: but this phenomenon is extremely rare. Branches running out from them, and sabdivisi'ons, as represented in the'figure, as well as evident' dark spots, like strings of clouds, are by no means uncommon; and from these, attentively watched, it is concluded that this planet revolves in the surprisingly Diurnalre- short period of 9 h. 55 m. 50 s. (sid. time), on an axis perpendicular to volution. the direction of the belts. Now, it is very remarkable, and forms a most satisfactory comment on the reasoning by which the spheroidal figure of the earth has' been deduced from its diurnal rotation, that the outline of- Jupiter's disc is evidently not circular, but elliptic, being considerably flattened in the direction of its axis of rotation. SOLAR SYSTEM. 139 " The parallelism of the belts to the equator of Jupiter, their occa- CHAP. XI. 8ional variations, andi the appearances of spots seen upon them, render Atmosphere it extremely probable that they subsist in the atmosphere of the planet, of Jupiter. forming tracts of comparatively clear sky, determined by currents analogous to our tradewinds, but of a much more steady and decided chapacter, as might indeed be expected from the immense velocity of its rotation. That it is the comparatively darker body of the planet which appears in the belts, is evident from this,-that they do not come up in all their strength to the edge of the disc, but fade away gradually before they reach it. (142.) "When Jupiter is viewed with a telescope, even of moderate Jupiter's power, it is seen accompanied by four small stars, nearly in a straight satellites. line parallel to the ecliptic. These always accompany the planet, and are called its Satellites. They are continually changing their positions with respect to one another, and to the planet, being sometimes all to the right, and sometimes all to the left; but more frequently some on each side. The greatest distances to which they recede from the planet, on each side, are different for the different satellites, and they are thus distinguished: that being called the First satellite, which recedes to the least distance; that the Second, which recedes to the next greater distance, and so on. The satellites of Jupiter were discovered by Galileo, im 1610.," Sometimes a satellite is observed to pass between the sun and Jupiter, and to cast a shadow-which describes a chord across the disc. This produces an eclipse of the sun, to Jupiter, analogous to those which the moon produces on the earth. It follows that Jupiter and its satellites are opake bodies, which shine by reflecting the sun's light. " Careful and repeated observations show that the motions of the satellites are from west to east, in orbits nearly circular, and making small angles with the plane of Jupiter's orbit. Observations on the eclipses of the satellites make known their synodic revolutions, from which their sidereal revolutions are easily deduced. From measurements of the greatest apparent distances of the satellites from the planet, their real distances are determined. "A comparison of the mean distances of the satellites, with their sidereal revolutions, proves that Kepler's third law, with respect to the planets, applies also to the satellites of Jupiter. The squares of their sidereal revolutions are as the cubes of their mean distances from the planet. "The planets Saturn and Uranus are also attbnded by satellites, and the same law has place with them." (143.) By the eclipses of Jupiter's satellites, the progres- Progressive sive nature of light was discovered; which we illustrate in light. the following manner: 140 ASTRONOMY. OAr. Xl. Fig. 31. Let S (Fig. 31) represent the sun, JJupiter, Eearths and m Jupiter's first satellite. By careful and accurate observations astronomers have decided that the mean revolution of this satellite round its primary, is performed in 42 h. 28 m. and 35s.; that is, the mean time from one eclipse to another. Velocity of VBut when the earth is at E, and moving in a direction toward, or light, hoe nearly toward, the planet as represented in the figure, the mean time determined, between two consecutive eclipses is shortened about 15 seconds; and we can explain this on no other hypothesis than that the earth has advanced and met the successive progression of light. When the earth is in position as respects the sun and Jupiter, as represented in our figure at E", and moving from Jupiter, then the interval between two consecutive eclipses of Jupiter's first satellite is prolonged or increased about 15 seconds. But during the interval of one revolution of Jupiter's first satellite, the earth moves in its orbit about 2880000 miles; this, divided by 15, gives 19200(0 miles for the motion of light in one second of time; and this velocity will carry light-from the sun to the earth in about eight and one-fourth minutes. Longitude (144. ) As an eclipse of one of Jupiter's satellites may be found by the seen from all places where the planet is there visible, two eclipses of observers viewing it will have a signal for the same moment, Jupiter's satellites. at their respective places; and their difference in local time will give their difference in longitude. For example, if one observer saw one of these eclipses at 10 h. in the evening, and another at 8 h. 30 m., the difference of longitude between the observers would be 1 h. 30 m. in time, or 220 30' of arc. The absolute time that the eclipse takes place, is the same to all observers; and he who has the latest local time is the most eastward. These eclipses cannot be observed at sea, by reason of the motion of the vessel. SOLAR SYSTEM. 141 (145.) Saturn. The next planet in order of remoteness CAIp. xL from the sun, is Saturn, the most wonderful object in the Saturnsolar system. Though less than Jupiter, it is about 79000 his,ings. miles in diameter, and 1000 times greater than our earth. " This stupendous globe, besides being attended by no less than seven satellites, or moons, is surrounded with two broad, flat, extremely thin rings, concentric with the planet and with each other; both lying in one plane, and separated by a very narrow interval from each other throughout their whole circumference, as they are from the planet by a much wider. The dimensions of this extraordinary appendage are as follows: Exterior diameter of exterior ring,................. 176418. Interior ditto,................................ 155272. Exterior diameter of interior ring,....... 151690. Interior ditto,................ - 117339. Equatorial diameteA of the body,................ 79160. Interval between the planet and interior ring,......- 19090. Interval of the rings........................... 1791. Thickness of the rings not exceeding.............. t 00. Dimensions Fig. 32.- Telescopic View of Saturn. of the rings. "The figure represents Saturn surrounded by its rings, and having its The rings body striped with dark belts, somewhat similar, but broader and less are opake. strongly marked than those of Jupiter, and owing, doubtless, to a similar cause. That the ring is a solid opake substance, is shown by its throwing its shadow on the body of the planet, on the side nearest the sun, and on the other side receiving that of the body, as shown in the figure. From the parallelism of the belts with the plane of the ring, it may be conjectured' that the axis of rotation of the planet is perpendicular to that plane; and this conjecture is confirmed by the occasional appearance of extensive dusky spots on its surface, which when watched, like the spots on Mars or Jupiter, indicate a rotation in 10 h. 29 m. 17 s. about an axis so situated. "It will naturally be asked how so stupendous an arch, if composed of solid and ponderous materials, can be sustained without collapsing 142 A STR ONOMY. cHAP. xi. and falling in upon the planet? The answer to this is to be foulnd fin The stabi- a swift rotation of the ring in its own plane, which observation has lity of the detected, owing to some portions of the ring being a little less bright rings. than others, and assigned its period at 10h. 29m. 17 s., which, from what we know of its dimensions, and of the force of gravity in the Saturnian system, is very nearly the periodic time of a satellite revolving at the same distance as the middle of its breadth. It is the centrifugal 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 independently of the other in a similar state of equilibrium. The rings " Although the rings are, as we have saidy very nearly concentric revolve a- with the body of Saturn, yet recent micrometrical measurements, of round the extreme delicacy, have demonstrated that the coincidence is not matheplanet like satellites. matically exact, but that the center of gravity of the rings oscillates round that of the body, describing a very minute orbit, 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. Hience, 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 eeeentricityyand to bring the nearest parts of them together." Uranusalias (146.) UYranzus. The next planet, beyond Saturn, was ilerschel. discovered by Sir W. F. Herschel, in 1781, and, for a time, was called Herschel, in honor of its discoverer; but, according to custom, the name of a heathen deity has been substituted, and the planet is now called Uranus — the father of Saturn. This planet This planet is rarely to be seen, without a telescope. In a rarely visible clear night, and in the absence of the moon, when in a favorto the naked able position 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. 148 The existence of this planet was suggested by some CHAP. Xi. 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 mere 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 disco. very of Nepverrier; 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, Leverrier and Adams, fixed on the same region of the heavens, for the then position of their ]hypothetical planet; and by diligent search, the planet was actually discovered about the same 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 longitude, 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 physicallaws. " We shall close this chapter with an illustration calculated to convey fIow to. to the minds of our readers a general impression of the relative magni- obtain a cortudes 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 systemr,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 144 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 diameter. 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 theplanetary seconds; Venus, in 4 m. 14 s.; the earth, in 7 minutes; Mars, in 4 m. motions. 48 s.; Jupiter, in 2 h. 56 m.; Saturn, in 3 h. 13 m.; and Uranus, in 2h. 16 m."-Herschel's Astronomy. CHAPTER XII. ON COMETS. CHAP. XII. (147.) BESIDES the planets, and their satellites, there are Comets great numbers of other bodies, which gradually come into formerly in- view, increasing in brightness and velocity, until they attain spired ter. por. 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 appearbanishes 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. s" 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. 145 of a more compact substance capable of retarding for a longer time CRAP. XI. 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 comets are as permanent as the planets, but subject to a different kind of movement. "When 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 apparently less distinctly terminated. Some, however, have appeared to consist mere masses of merely a light vapor, without a sensible nucleus, since the stars are visible through it. During their revolution, they experience progressive variations 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 brightness 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, in 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 of system, it is natural to suppose that they move about the sun like comets. planets, but in orbits extremely elongated. These orbits must, therefore, 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 on the same comet at near the same time. At the times the observations 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. 10 a 146 ASTRONOMY. CHAP. XII. relate to the elliptical motion, we obtain those of the parabolic motion of comets. Comets des- " Hence it 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 roots qual times. of 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 by a comet in a given time after passing the perihelion, and fix its position in the parabola. It only remains then to bring this theory to the 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 vations 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 shall have the direction of three visual rays drawn at these times from comet. 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 intervals of which correspond to the number of days between the observations. rig. 33. "Or if we suppose the earth in motion 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 X s \) question is to find a parabola CC'C", T. I having its focus in S at the center of the sun, and cutting the three visual rays conformably to the conditions reTr" quired. The ofbit of a "These conditions are more than sufficient to determine completely comet found the elements of the parabolic motion, that is, the perihelion distance by three ob- of the comet, the position of the perihelion, the instant of passing this 5arvations. point, the inclination of the orbit to the ecliptic, and the position of its nodes. These five elements being known, we can assign the posi. tion of the comet for any time whatever, and compare it with the results of observation. But the calculation of the elements is very difficult, and can be performed only by a very delicate analysis, which Cannot here be made known. COMETS. 147 I" Abdut 1i0 comets have been calculated upon the theory of the CHAP. XI. parabolic motion, and the Observed places are found to answer to such a supposition. We can have no doubt, therefore, that this is conform- Inclinations able to the law of nature. We have thus obtained precise knowledge of their orof the motions of these bodies, and are enabled to follow them in space. bits. This discovery has given additional confirmation to the laws of Kepler, and led to several other important results. "' Comets do not all move from west to east like the planets. Some have a direct, and some a retrograde motion. "Their orbits are not iomprehended within a narrow zone of the heavens, like those of the principal planets. They vary through all degrees of inclination. There are some Whose plane is nearly coincident with that of the ecliptic, and others have their planes perpendicular to it. "It is farther to be observed that the tails of comets begin to appear, as the bodies approach near the sun; their length increases with this proximity, and they do not acquire their greatest extent, until after passing the perihelion, The direction is generally opposite to the stin, forming a curve slightly concave, the sun on the concave side. "The portion of the comet nearest to the sun must move more rapidly than its remoter parts, and this Will account for the lengthening of the tail; "The tail is, h6wever) by no means an invariable appendage of some coni. comets; Many of the brightest have been observed to have short and ets have no feeble tails, and not a few have been entirely Without them. Those tails. of 1585 and 1763 offered no vestige of a tail; and Cassini describes the comet of 1682 as being as round and as bright as Jupiter. On the other hand, instances are not wanting of comets furnished with many tails, or streams of diverging light. That of 1744 had no less than six, spread oUt like an immense fan, extending to a distance of nearly 30 degrees in length, " The smaller comets, such as aire Visible only in telescopes, or With difficulty by the naked eye, and Which are by far the most numerous, offer very frequently no appearance of a tail, and appear only as round or somewhat oval vaporous masses, more dense toward the center; Where, however, they appear to have no distinct nucidleus, or anything Which seems entitled to be considered as a solid body. "The tail of the comet of 1456 Was 60 degrees long, That of 1618, Othershave 100 degrees, so that its tail had not all risen when its head reached the several tails. middle of the heavens, The comet of 1680 was so great, that though its head set soon after the sun, its tail, 70 degrees long, continued visible all night. The comet of 1689 had a tail 68 degrees long. That of 1769 had a tail more than 90 degrees in length. That of 1811 had a tail 23 degrees long. The recent comet of 1843 had a tail 60 degrees in length." The following figure gives a telescopic view of the comet of 1811. 1148 ASTRONOM Y. CHAP. XII. "When we have determined the elements of a codmet's orbit, *e Guiisl pare them with those of comets beftre observed, and see whether there of comets is an agreement with respect to any of them. If there is a perfect how deter- identity as to the elements, we should have no hesitation in concluding:hinladd: -that they belonged to different appearances of the same comet. But this condition is not rigorously necessary; foir the elements of the orbit may, like those of other heavenly bodies, have Uindergone changes from the perturbations of the planets or their mutual attractions, Consequently, we have only to see whether the actual elements ate nearly the same with those of any cornet before observed, aiid then, by the dock trine of chances, we can judge what reliance is to be placed upon thiS resemblance'" Cormet of 1811. br. tIalley's -"Dr. Halley remarked that the comets obseriVed in 1531, 1607, 16829 prediction had nearly the same elements; and he hence concluded that they be~ verified. longed to the same comet, which, in 15i years, made two revolutions: its period being about 76 years. It actually appeared in 1759, agre~eably to the prediction of this great astronomei and again in 1832, by the computation of several eminent astionoineis; According to Kepler's third law, if we take for unity half the major axis of the earth's Particulars orbit, the mean distance of this comet rhust be equal to the cube root efcomets. of the square of 76j that is, tb 17.95. The majoi- axis of its orbit mustt therefore, be 35.9; and as its observed perihelion distance is found to be 0.58, it follows that its aphelion distance is equal to 35.32. It COM ETS. 149 ddPifts, fidrefoie, from the sun to thirty-five times the distance of the CHAP. XI!. earth, and afterward approaches neatly 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 romets are subject to perturbations, like those' of the planets, and to a much more sensible degree.' Elements of the Orbits of the three Comets, whicdh have appeared ace cording to predidtion, taken from the work of Professor Littrow. Halley. Encke. Biela,:Longitude of the a cending nbde~ - 54e 3350 2490 Inclination' of the'brbit-to-the ecliptic, 1620 130 130 Longitude of the perihelion. - - 3030 1570 1080 Greatest semidiameter' that of the earthI 158 2.2 3.6 being called 1,.-. Leapt semidiameter, - - 4.6 1. 2 Q.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 the signs of the zodiac, or have their motions diSect; the motion of that of Halley is retrograde. "Comets, in passing among and near the planets, are materially Juplite% drawn aside from their cotirses, and in some cases have their orbits en- and his satel., tirely changed. This is remarkably the case With Jupiter, which seems, lites, a great stumblingby somie strange fatality, to be constantly in their way, and to serve as block to th. a perpetual stumbling-block to them. In the case of the remarkable comets 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 predicted by him accordingly~ the prediction Was disappointed by the comet actually getting entangled among the satellites of Jupiterj. and being completely thrown out of its orbit by the attraction of that planet, and forced into a much larger ellipse. By this extraordinary relconter, the motions of the satellites suffered not the least perceptible derangementa sufficient proof of the smallness of the comet's mass."' The comet of 1456, represented as having a tail of 600 in length, is now found to be Halley's domet, which has made several returnsin 1531, 1607, 1682, 1759land recently, in 1835. In 1607 the tail wag said to have been over 300 in length; but in 1835 the tail did not exceed 120 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 awes always put forth? We ask these questionsi but cannot answer them, a TIC 150 a s T nz. ASTR NOMY. CHAP. XII. The following cut represents the appearance of the comet of 1819. Fears en- "Professor Kendall, in his Uranography, speaking of the fears occatertained, by sioned by comets, says: "Another source of apprehension, with regard Some, that to comets, arlses from the possibility of their striking our earth. It is comets may quite probable that even in the historical period the earth has been ultimately,ome into enveloped in the tail of a Comet. It is not likely that the effect would bollisionwith be sensible at the time. The actual shock of the head of a comet against aur 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 well known that the size of mountains on the earth is illustrated by comparing them to particles of dust on a common globe." CHAPTEit XIII. ON THE PECULIARITIES OF THE FIXED STAS., CHAP. XIII. FOR the facts as contained 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. 151 give some index by which to judge of their magnitudes, nature, CHAP. XII. and peculiarities. "It would be idle to inquire whether the fixed stars have a sensible Base from parallax, when observed from different parts of the earth. We have which to already had abundanit evidence that their distance is almost infinite. It measure to the stars. is only by taking the longest base accessible to ius, that we can hope to arrive at any satisfactory result. "Accordingly,we employ the major axis o'f the earth's orbit, which is nlearly 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 same or not at these two epochs. " It is obvious, indeed, that the star must appear more elevated above Axial the plane of the ectiptic 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 two 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'his straight line, is 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 of a sensible star at the intermediate epochs, we ought, if the annual parallax is sen- parallax. 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 thislast with the celestial sphere, will constitute a small ellipse, in wh'ich the star will always appear diametrically opposite 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- parallaxmust fectly exact, we have been able to discover nothing which indicates, be less than with certainty, even the existence of an annual parallax, to say nothing one second. 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 idf Dr. Bradley, who made many observations to discover it, and who, in this undertaking, fell unexpectedly upon the phenomena of aberration * 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. 152 ASTRONOMY. CHAP. XIII. place among observations, that it is hardly to be supposed nhat thd 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 ant angle of 1':; and to an observer' placed at this distance, our sun, wi;th the whole planetary system, would occupy a space scarcely exceeding the' thickness of a spider's thread. Conclusion "' If these restilts' do not make lknown the distance of ihe stars front to be drawn the earth, they at least teach us the' limit beyond which the stars must from these necessarily be situated. If we conceive a right-angled triangle, having facts. for its' base half the major axis of tie earth's orbitf and for its vertex an angle of 1", the-distance of this vertex from the earth, or the lengthof the visual ray,. will be expressed by 212907, the radius of the earth'y orbit being unity;; and as this radius contains 23987 times the semidiav meter of the earth, it follows that if the annual parallax of a star were only I", its distance from the earth would be equal to 5090209309 radii of the earth, or 20086868036404 miles - that is, more than 20 billionsB'ut if the annual parallax is less than F1"r the stars are beyond the limit which we have assigned. Clhanges 6' It is' evident that the stars undergo considerable changes, since these' in individual changes are sensible even at the distance at which we ate placed. There }tars. are' some which gradually lose their light, as the star 4d of Ursa Major; Others, as R of Cetus:, become more brilliant. Finally,e there are some which have been observed to assumne suddenly a new splendor, and thenr gradually fade away. Such was the new' s'tr which appeared in 1572,9 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 todiminish, and the star disappeared ifn 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 Saturii. Arnt Another other star which. appeared suddenly in 1604, in the constellation Serd newv star. pentarius, presented similar variations, and disappeared after several; months. These phenomena, seemn te 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' oiT globe, anat are yet to take place. Periodial " Some stars, without entirely disappearing, exhibit variations not less' changeso remarkable. Their light increased aind decreases alternately in regular periods. They are-called for this' eason varia6le stars. Such is the FIXED STARS. 15S 9;t -Al oi, in the head of Medusa, which has a period of about three CHAP. XIru days;. of Cepheus, which has one of five days; R2 of Lyra, six; ce of Antinous, seven; o of Cetus, 334; and many others. " Several attempts have been made to Explain these periodical varia- Attempts i6Ons. It is supposed that the stars whith are subjeCt to themr are, like to explain all the other stars, self-ldminous bodies, br true suns; turning on their periodical axkes, and having their surfaces partly covered with dark spots, Which changes. may be supposed to prevent themselves to us at certain times only, in c'nsequence of their r6tationm 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 diffefence 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 opako bodies, revolving about these stars, and occasionally intercepting a part b'f their lights. Time anid the multiplication of observations may piehaps decide w:vhich of these hypotheses is the true one. "One of the best methods of observing these phenomena is to compare Order id the stars toagether, designating them by letters or numbers, and dispos- these obsern ing them hii the order of their brilliancy. If'We finds by observation, vations. that this order changes, it is a proof that ohe of the stars thus compared, 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 manhner, we can only Compare each star With those Which are in the ifeighborhood, and visible at the same time. But by afterward comparing these with Others, We -danm by a series of intermediate terms, connect together the most distant extremes. This method, which is niow pracm t'iced, is far preferable to that of the ancient astronomers, who classed the stars after a very viague comparison, accofdilg to what they called the orde& of their mgnitudes, biMt which was) in reality, nothing but that of their brightnes~, estimated in a very imperfect manner. "By comparing the places of some of the fixed stars, as determined Suggestio from anCient and modern observations, Dr. Halley discovered that they ofDr.Hallte. had a proper motion), which could Hot atise from parallax, precesslonb br aberration. This romarkable circumstance was afterward noticed by Cassini and Le lbinniler, 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 treater part of them had a pioper motion. He suggested that the change of place might arise from a progressive motion of the si!n toward Onhe quarter of the heavens, but as the result of his observation did not accord with his theory, hO remarks that many centuries murst elapse before the true bause of this motion could be explained. " The probability of a progressive motion of the stn was suggested tpon theoretical principles by the late Dbr. Wilson of GlasgoW; and Lalande deduced a similar opinion from thi rotatory motion of the sun, by supp'6sing) that the samne mechaniical force which gives it a motioi 154 ASTRONOMY. CHAP. XIII. round its axis, would also displace its center, afid give it a moticin of translation in absolute space Conse- s" If the sun has a motion in absolute space, directed toward any quences of such t the. quarter of the heavens, it is obvious that the stars in that quarter must ory. appear to recede from each other, while those in the opposite region would seem gradually to approach, in the same manner as when walking through a forestf the trees toward which we advance are constantly separating, while the distance of those which we leave behind is gradud ally contracting. The proper motion of the stars, thetefore, in oppositd regions, as ascertained by a comparison of ancient with modern observations, ought to correspond with this hypothesis; and Sir W. Herschel 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 it 2500 52' 30", and whose north polar distance is 40~ 22'. Klugel found the right ascension of this point to be 2600, and Prevost made it 2300, with 650 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 centerd The attractive force capable of producing such an effect, he does not 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 starsd Double "There are stars which, when viewed by the naked eye, and even ad. multiple by the help of a telescope of moderate power, have the appearance of stares only a single star; but, being seen through a good telescope, they are found to be double, and in some cases a very marked difference is perceptible, both as to their brilliancy and the color of their light. These Sir W. Herschel supposed to be so near each other, as to obey reciprocally the power of each other's attractions revolving about their common center of gravity, in certain determinate periods. Castor,? Leonis, Rigel, Pole Star, 9rMonoc, g Cancri. Revolutions "The two stars, for example, which form the double star Castor, of the multi. have varied in their angular situation more than 45~0 since they were ple stars, observed by Dr. Bradley, in 1759, 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 positions1 at internmediate times, but never could perceive any change in their distance. The retrograde revolution of? 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 i of Bootes) FIXED'rTAR6. 155 pterorm a direct revolution in 1681 years, in a plane oblique to the sun. CHAP. XII, I'he stars Z of Serpens, perform a retrograde revolution in about 375 years; and those of?' in Virgo in 708 years, without any change of their distance. In 1802, the large star 3 of HIercules, eclipsed the smaller one, though they were separate in 1782. Other stars are supposed to be united in triple, quadruple, and still more complicated systems. "With respect to the determination of the real magnitude of the starts, Descriptiota and their respective distances, We have as yet made but little progress. of nebulwa 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 Nebulue, which appeal in the heavens as represented in the accompanying 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 ordinarily blended by irradiation, and thus present to the eye only a faint uniform With theet of light. such a bul That large, white, luminous track, which traverses the heavens from. one pole to the other, is at least a hudernes as great the namd of the Milky Way, is probably nothing but a nebula of thes The Milke kind, whsech appeas larger than the others, becausnce it is neare to us. Way a ehWith the aid of the telescope we discover in this zone of light such a buil, prodigious number that thise olimaginalection may appear as bewildered in Attempting to represent them. Yet from the angular distances of these stars, it is certain that the space wthich separates those which Beem 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 extent of the group. To what distance then must we withdraw, in order that this whole Collection may appear as small as the other nebulve which we perceive, some of which cannot, by the assistance 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 we attempt, in imagination, to fathom this abyss, it is in vain to think of prescribing any limits to 156 ASTRONOMY. CHAP. XM1I. 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 attentions on ta- tion of the reader to table II, in which will be seen, at a glance (in ble.II. the columns marked annual variation), the general effect of the precession 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 18th hour of right ascension, have a progressive motion to the southward (-), and all the stars from the 18th to the 6th hour of right ascension have a progressive motion to the northward (-+-), and the greatest variations are at 0 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, lax of 61 it was selected by Bessel as a fit subject for the investigation of stellar Cygni disco. parallax; and it is now contended, and in a measure granted, that the vered. 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 coqld not traverse in less than nine and one-fourth years. PHYSICAL ASTRONOMY. 157 SECTION III. PHYSICAL ASTRONOMY. CHAPTER I. GENERAL LAWS OF MOTION — THE THEORY OF GRAVITY. CHAP. I. (148.) IN a work like this, designed for elementary in-Whatshould struction, it cannot be expected that a full investigation of be expected physical astronomy shall be entered into; for that subject in 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- Elementary rai 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 measure of freely in void space, or in an unresisting medium a triple forceaure 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 elements, and then we have this important and general equation, tv=d (1) N 158 ASTRONOMY. CHAP. 1. d d F- rom this we derive v=- (2) and I= — (3) Expression t V for force. ( 149.) As forces are in proportion to velocities (when momentum is not in question), theref6re, if we put f 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 d D f: X::: T. (4)* The law of ( 150. ) A body at rest, has no power to put itself in moinertia. tion; and having no self power, no internal force or will, in 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. Some central The earth, moon, and planets move in curves —not in force must Fight lines. The directions of their motions are chazged. act on the motions of Something external from them must, therefore, change them; the earth, for the law of inertia would continue a motion once obtained moon, and. planets., in a straight line. Now this force must exist within the orbit of every curve; we therefore naturally refer it to the body round whieh 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 preservation of the system. The moon's The moon goes round the earth, constantly deviating from motion con- the tangent of its orbit, and the law of inertia is constantly sidered. * We number the proportions the same as equations, for a proportion is but an equation in another form. THE EARTH'S ATTRACTION. 159 urging it to rise from the center; the two on an average balan- CHsp. I. cing each other, retains the moon 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 Contempla. to have led the mind of Newton to the contemplation of this tions of Sir Isaac New. force which compels and causes bodies to fall, and he at once 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 surface 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 the theory of minished by the distance; and corresponding thereto he de- gravity. cided that the moon must then make a revolution in its orbit in 10h. 55m. But the actual time is 27d. 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 computations. fall from the tangent of its orbit in one second of time, it was found to be about'6-loo part of 16 A — 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 w'ir', 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- inphilosophy ing at a point, must diminish 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. 160 ASTRONOMY. CHIP.. L 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 directions from its central point must be enfeebled as the square of the distance. The theory From observations and considerations like these, Newton of universal gravityal 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 well estab- the truth of this theory in question, and substitute in its lished. place the influence of light, caloric, and electricity; but any thing like a close application shows how feebly all such substitutes stand the test. The theory of gravity so exactly accounts for all the physical 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 of an irregu. composing a body has its influence, and a very irregular body lar 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 the whole body. STANDARD OF FORCE. 161 In asphere composed of homogeneous particles, the aggre- CHAP. I. 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 inside of a of course would show no indication of being attracted at all. spherical 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 Attraectieon at 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 Expression and homogeneous sphere may be expressed by the mass of the for the atsphere divided by the square of the radius. traction of 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 2 - — g-16-'- feet. This attractive force, algebraically expressed by;i we call g, and it is sufficient to cause bodies to fall 16-'2 feet during the first second of time. If the earth had contained more matter, bodies would have fallen more than 16A-1 feet the first second; if less, a less distance. With the same matter, but more compact, so that r2 would The definite attraction of' be less with E the same, 2 would be greater, and the attrac- the earth. tive power at the surface greater, and bodies would then fall more than 16-' feet the first second of their fall. Now we say this 16-L' 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. 11 N ]62 ASTRONOMY. CHAP.. For this reason, we call the undivided attention to this force, the known - the noted - the all-imrportant 16-' feet. To find the ( 155. ) IBy the theory of gravity, we can readily obtain an attraction of analytical expression for the attraction of a sphere at any disa sphere at ~ anydistance. 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, g' x' 2-~ ~ D-q; D Or, x=g ~-5 (5) As g and r are constant quantities, the variations to x will correspond entirely to the variations of D2. We shall often refer to this equation. An expres- ( 156. ) As every particle of matter in the universe atsion for the tracts every other particle, therefore the moon attracts the traction earth as well as the earth attracts the moon; and the extent two bodies. by which they will draw together, depends on their mutual attraction. If m represents the mass of the moon, and R the radius of the lunar orbit; then, The earth will attract the moon by the force R_ The moon will attract the earth by the force C The two bodies will draw together by the force 1R2 If we substitute the value of g, as found in ( 154 ), in equation (5 ), and making R = D), then we have the expression 2The 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. 163 CHAPTER II. REPLER'S LAWS - DEMONSTRATION OF THE SECOND AND THIRDHOW A PLANETARY BODY WILL FIND ITS ORBIT. (157.) IN this chapter we design to make some examina- CHAP. It. tion of Kepler's laws, recapitulating them in order. ExaminaThe orbits of the planets are ellipses, having the sun at tionsofKeplers laws. one of their foci. 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 motion ) sweeps over equal areas in equal times is susceptible of a rigid mathematical demonstration, under the following general theorem. Any body, being in motion, and constantly, urged toward any A general 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, Its denmor. stration, having a velocity which would carry it to AB, say in one second of time. By the law of inerttia, it would move from B to C, an equal dis- A si 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 philosophy, describe the diagonal B E, of the parallelogram B'DEC. 164 tASTRONOMY. cHAP. oI. Now in the first interval of time, we Supposed the imoving body described the triangle S A B, The second interval, it would have described the triangle- S B C, if undisturbed by any force at S, but by such a force it describes the triangle S B E; but the triangle 8 B E it equal to the triangle S B C, 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. Therefore the triangle S A B is equal to the triangle S B', because they are both equal to the triangle S B C; 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, of as the square of the distance. rhe con- The converse of this theorem if, that when a body descfibes terse of the equal areas in equal times round any point, the body is cons 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 Ant toward any other point Within the orbits. Kepler's (58. ) The third law of Kepler is most important of alli thira law namely — The squares of the tihies of revolution are to each'proves that the sun's at, Othe as the cutbes of ~he distances from the sun. By this law traction is it is proved, that it is the same force which urges all the Inversely as planets to the same point, and that its intensity is inversely as the square of tihe distance. the Square of the distance from that point ( the center of thd ian), confirming the Newtonian thdory of gravity. Fig. 30. To show thin, let us suppose that the P planets revolve round the sun in circulai v Hi~ orbits (which is not far from the truth), and let P (Fig. 30 ) represent the position of a planet; F the distance which the planet is drawn froam a tangent during unity of time; in the same time that it describes the indefinite small are 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 revolutions SOLAR SYSTEM. 160 Utf IP s the diameter of the orbit and t the time of revolu- CHAp. I. tion, then will t — —,. d ~ (1) So for any other planet. If is the force urging it toward AA impotr the sun, a its corresponding arc, T its time of revolution, and tanttruthdnb R the radius of its orbit; then) reasoning as befores 2 R —- By comparing (1) and (2) we have D 21R C a D2 4R2 Bysquaring~ t2 72: 2 - By Kepler's laws, tP' T2 r3.R. By comparing the two last proportions, and observtin that 2r may be put for D, and reducing, we have 1 I 2 Iut by the wdll-known pioperty of the circle, we have F;c.: c: 2e; or, c =2c-2rF. in like manneP,.. a2 - 2 Rf. Substituting these vtalues in the iats proportion, and redua 6ing, We have i.1 ~; Rf: F:: r t 2F R. lencee, Z2ft2FS; or, F f R3 f r21 1 1 r2 R2' That is, the attractive fnrce of the sun is ieciprocally proportional to the square of the distance. (159.) If we commence with the hypothesis, that bodies T-i theoq tend toward a central point with a force inversely propor- of gtavit 166 ASTRONOMY, CHAP- II. tional to the squares of their distances, and then compute and laws of the corresponding times of revolution, we shall find that the motion result in Kepler's squares of thie times must be as the cubes of the distances. Hence thirld 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 considerable. Hlow answer- On one of these points of inquiry, the reader must take our ied. assertion; for its demonstration requires the use of the inte/ral calculus, a method that we designed not to employ in this work. Kepler's third law supposes all the force to be in the central body, and the planets only moving points. But we have seen in Art. (120) 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 very small compared 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 thsrd lawma- planets ( considered as bodies without masses) revolving in thematieallp 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 greatcr 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. 167 end the distance the pendulum was first drawn aside. The CHAP. I. motion of a planet depends on the force of gravity, its mean distance from the sun, and the original impulse first given to A common it. Most persons, who have not investigated this subject, errorofopinimagine 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 diferent, either as to amount or direction, or as to both, then by the action of gravity and inertia, the planet would have found a corresponding orbit. (162.) The force of gravity, from the action of any attract- Examinaing body, is always as the mass of the body divided by the square tion of the of its distance. Algebraically, if Mis the mass of the body, motianetar in r its distance, and F the force at that distance, then (see 118) elliptic orbits 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- 2rF= f —2 and that velocity at right angles to r (Fig. 28), then the planet's orbit would be a circle, with the radius r. If the velocity had been less in nmount than this expression, and still at right angles to r, then the planet would fall within the circle, and the action of gravity 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 sufficient further from to maintain the planet in a circle, at its then distance; but the the sun. BEdirection of the motion will not permit the planet to run into OW is neardirection of the motion will not permit the planet to run into er to it. 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 tkan 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 16$ ASTROXO'MY. CHAP. It If the velocity, on starting from the point P, wetfe tfy Gravity and much less than sufficient to maintain a circle, at that distanced original ve- then the orbit it would take would be very eccentric, and locity determine the ec- its mean distance much less than I. If the original velocity centricityand at, P were greater than to maintain it in a circle, it would mean distan-, mean distanhe or- pass outside of this circle, and the point P would be the perices of the or-. bits, helion point of the orbit, Thus, we perceive, that the eccentricity of orbits and mean distances fr'omn 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 defino'g impulse, either in amount or' direction, ta move, in an orbitif the direction is not directly to or from the sung A hypothe. (163.) For a more definitne explanation of this subject, let alase. us conceive a planet launched out into space with a velocity sufficient to maintain it in a cirele at the distance it then happened 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 depen4 on the direction of the motion; but the longe a&ais 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 How a / Let P (Fig, 31) be the posiIpanet finds- tion of a plhnet, S the sun; an, its orbit. /i let the velocity, a, be just suffit cient to maiftiain the planet ih. a circle, if it were at right angleas P-o.s............' o,. a l Now to find the orbit that this planet wolad' describe, draw the' I F-ne P C' at right angles' to a,. a and from S let fall a perpendicular on Po; ASC' will be theeccentricity of the orbit, and PC' will be the hlalf of its conjugate' axis; and with these lines the' whole ovbit is known. PLANETARY MOTION. 169 ( 164.) Now let us suppose that a planet is rather carelessly CsAP. L. launched into space, with a velocity neither at right angles to Planets -the sun, nor of sufficient amount to maintain it in a circle, at willfindthei orbits, whatV that distance front the sun. ever be the Let P (Fig. 32) represent the Fig. 32 direction and position of the planet, a the, original mte amount and direction of its ha- tibn. hlazard velocity during the first / unit of time. The direction of I the motion being within a right angle to SP, the action of gra / vity increases: the velocity O B 0 of the planet, a on the same principle that a falling body in- \ creases in velocity; and the planet \ goes on in a curvejdescribing equal areas in equal times round the point " S; and it will find a point, p, where * its increased velocity will be just equal to the velocity in a citcle whose radius is the diminished distance Sp. from the point p, and at right angles to a, draw p C, &c., fo-ming the right angled triangle p C S. S C is the eccentricity, S a the mean distance, and p C half the Conjugate axis of the orbits If the planet is launched into space in the other direction, The brbiti the action of gravity will diminish its motion, and will bring will be sym~ 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 cireie Bt that apogee and distance; and it will, of course, approach neater and neaiter perigee. to th~ sun by the same laws of motion and foree that it receded from the sun; hence the curve on each side of the apogee W 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 examil An impon. nati con of the laws of, otn, and, that whatever. iaatioa of the laws of ymtgy, motionr, and ine~ia9 that whatever sion. 170 ASTRONOMY.. CasP. IL. may be the primary force and direction given to a planetary body ( if not directly to or from the sun), the planet will find a correspondinzg 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 the 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 * Let S be the sun, and P the position of a planet as repreDr. Olbers concerning sented in the annexed figure, and we may now suppose it to the asteroids 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, &o. Fig. 33. As action is just equal to reaction, under all circumstanceds 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 the ain the opposite 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 orbits cone- in the same general direction, and with the same aggregate sponding to velocity as the original planet. The bursting of a rocket is their velocities andposi. a very minute, but a very faithful representation of such an tions. explosion. KEPLER'S LAWS. 171 (165.) To see whether Kepler's third law applies to ellipses, CHAP. Ir. we represent half the greater axis of any ellipse by A, and Kepler's half the shorter axis by B, and then (3.1416)AB is the area third law ri-,of the ellipse. Also, let a represent the velocity or distanee gorously true in relation to ellipses, as If the velocities of the several fragments were equal, the well as to times of their revolutions would be equal; but the eccentri- circles. cities of the several orbits would depend on the angles of a, b, c, &c., with SP. If a is at right angles to S P, and just sufficient to maintain the planet in a circle at that distance, then its orbit would have no eccentricity. If still at right angles, but not sufficient to maintain a circle at that distance, then SP would be the greatest radius of the orbit. Hence, we perceive, there is an abundance of room to have a multitude of orbits passing through the same point, during the first one or two revolutions; and the times of such revolutions may be equal, or very unequal. In short, there is no physical impossibility to be urged against the theory of Dr. Olbers, that the asteroids are but fragments of a planet. The objection is (if an objection it can be called) Ihat these planets have not, in fact, a common node, nor have an approximation to one; nor have they an approximation to a common radius vector, as S P. But the objection vanishes when we consider that the elements of the different orbits must be variable; and time, a suficient length of time, would separate the nodes and change the positions of the orbits so as to hide the common origin, as is now the case. But if it be true that these planets once had a common origin in one large planet, it is possible to find the variable nature of the elements of their orbits to such a degree of exactness as to trace them back to that origin — define the place where, and the time when, the separation must have occurred. If, however, a planet should burst at one time, and afterward one or more of the fragments burst, there could be no tracing to a common origin; hence it is possible that the asteroids in question may have a common origin, and it be wholly beyond the power of man to show it. 172 ASTRONOMY. CHaP. 1I. that the planet will move in a unit of time, when at the exs tremity of its shorter axis; then - a B will express the area described in that unit of time. But as equal areas are described in equal times, as often as this area is contained in the whole ellipse will be the number of such units in a revolution. Put t= that number, or the time of revolution; then (3.1416)AB 2(3.1416)A 2aB a Let A' and B' be the semiaxes of any other ellipse; a' the velocity at the extremity of B', and t' the time of revolution; then will - t,2(3.141)A' By comparing these equations, and rejecting common facA A' tors, we have t':: a a/ But by Art. 162, a= j- and a'= 2 M mass of sun); and putting the values of a and a', in the above proportion, we have t: t':: AJA: A'.; J2SM JM Or, - t: t':: AJA.: A'JA'. By squaring t2 t'2:: A A'3; which is Kepler's third law. Eccentrici- (166.) We have seen, in articles 163 and 164, that the ties of the eccentricity of an orbit depends on the direction of the motion planetary or. bits change to the radius vector, when the planet is at mean distance. If by their mu. that direction is at right angles to the radius vector at that tual attrac. ations. time, then the eccentricity is nothing. If its direction is very acute, then the eccentricity is very great, &c. Now suppose another planet to be situated at B (Fig. 32); its attraction on the planet, passing along in the orbit p a, is to give the velocity, a, a direction more at right angles to KEPLER'S LAWS. 173'SP, and thus to diminish the eccentricity of the orbit. If cHAP.. the disturbing body, B, were anywhere near the line C S, its The mean tendency would be to increase the eccentricity; and thus, in distancese. general, A disturbing body near a line of the shorter axis of ver vary. an orbit, has a tendency to diminish the eccentricity of the orbit of the disturbed body; and, anywhere near a line of the greater axis, has a kndency to increase the eccentricity. Hence the eccentricities of the planets change in consequence of their mutual attractions; but their mean distances ever change. (167.) As the time of revolution is always the same for the same mean distance, whatever be the eccentricity of the orbit, therefore if we conceive a planet to turn into an infinitely eccentric orbit, and fall directly to the sun, the time of such fall would be half a revolution, in an orbit of half its present mean distance, as we perceive, by inspecting Fig. 34. Hence, by Kepler's third law, we can compute the Fig. 34 The printime that would be required for any planet to fall to ciples and the computathe sun. Let x represent the time a planet would tion of the revolve in this new and infinitely eccentric orbit; then, time required for the planby Kepler's law, ets to fall to 2t2, o, the sun. tz' Xa "2~'13 O3, X2~. Therefore half of the revolution, or simply the time of the fall, must be expressed by -, or, -; 248 4J2 s that is, to find the time in which any planet would fall to the sun, if simply abandoned to its gravity, or the time in which any secondary planet would fall to its primary, divide its time of revolution by/four times the square root of two. Bfy applying this rule, we find that Days. h. m. Mercury would fall to the sun in............... 15 13 13 Venus,.............. 39 17 19 Earth,............................. 64 13 39 Mars....................1.................. 121 10 36 Jupiter................................... 765 21 36 Saturn,.................................... 1901 23 24 Uranus,...................................... 5424 16 52 The moon would fall to the earth in 4d. 19 h. 54m.,36 s. _.i 174 ASTRONOMY. CHAPTER III. MASSES OF THE PLANETS — DENSITIES - PRESSURE ON THEIS SURFACES. CHP. WII (168.) IF the earth contained more matter, it would Masses mea- attract with greater force; and if the sun has a greater sured by at- power of attraction than the earth, it is because it contains traction. more matter than the earth; and therefore, if we can find the relative degree of attraction between two bodies, we have their relative masses of matter. If the earth and sun have the same amount of matter, they will attract equally at equal distances. Let Mbe 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 the attraction of the earth, as i to E. But attraction is inversely as the square of the distance. M Hence the attraction of the sun at D distance, is —; and the attraction of the earth at R distance is ER2 Gravity of The earth is made to deviate from a tangent of its orbit the sun is by the attraction of the sun; and the moon is made to deviate measured by 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 earth from a tangentof its amounts of solar and terrestrial gravity. orbit. If we take any small period of time, as a minute or a second, and compute the versed sine of the are 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 are which the moon describes in the same time, that quantity will express the attraction of the earth. How to com- In Figure 30, Art. 158, F represents the versed sine of an parative com- arc; and if we take D to represent the mean distance bemassesofthe tween the earth and sun, and consider the orbit a circle sun and earth (as we may without error, 164 ), the whole circumference is MASSES OF THE PLANETS. 175 D (r=- 6.2832). Divide the whole circumference'by the CHAP. LU. number of minutes in a revolution; say T, and the quotient will represent the are a (Fig. 30). When T is very small, and of course a very small, the chord and arc practically coincide; and by the well known property of the circle, we have a2 2D'a: a:: a F; Or, F — 2D (1) g5 D 3 5r_22 a2 Do But a - -; hence, a2_ 2 and -- 7', T2 ) 2D272' r-2D That is, F = T-; which is an expression for the sun's attraction at the distance of the earth. But is also an expression for the sun's attraction at the same distance; therefore, M _ Or, M-XTi. D -- 2 T2' 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 2.R3 E 2t2 Therefore, B: MX::. 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, t2 D3 M t2D.. a(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, and T will be the corresponding time of revolution. tio.s equa Also R may represent the radius of the lunar orbit, or the 176 ASTRONOMY. CHAP. IS. orbit 6f 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 equa. the distance a planet falls from the tangent of its orbit, in a tion will not be perfectly M-+E accurate,and definite moment of time, is not, accurately 2-, but D2 why? (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. But 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 general be very small. The error will be greatest in obtaining Corrections for equation the mass of 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 inconsiderable part of the entire mass of both earth and moon; and also the attraction of the sun on the combined mass of the 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: A more ac- From Art. 155, we learn that the attraction of the earth curate equa. r \ tion. at the distance to the sun, is g. By Art. 168, we have just seen that the attraction of the sun on the earth, is therefore, N. 2,r2.D E:.2:: g ~ 2T Taking the mass of the earth as unity, we have /.2.D3 2 2gT2' (B) Equation (B) is more accurate than equation (A), MASSES OF THE PLANETS. 177 because ( B) does not involve the periodical revolution of the CHAP. nI. 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 taiictale nu right hand member of equation ( B ) must be rendered homo- suit. geneous; and as g, the force of gravity of the earth, is expressed 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 g (23984)3r But to ex2gT72 press the whole in numbers, we must give each symbol its value; that is, - = 6.2832; r- (3956) ( 5280 ); g = 16.1; 7= 31558150, the number of seconds in a sidereal year. Therefore,.zr:(6.2832)2 (23984) s (3956)(5280) (32.2)(31558150)2 It would be too tedious to carry this out, arithmetically, An example without the aid of logarithms, and accordingly we give the she wing the great utility logarithmetical solution, thus, of logarithms 6.2832 log. 0.798178X2... 1.596356 23.984 log. 4.380000X3 13.140000 3956 log... 3.597256 5280 log... 3.722632 Logarithm of the numerator,.. 22.056244 32.2 log..... 1.507856 The mass of 31558150 log. 7.499114X2. 14.998228 the sun determined. Logarithm of the denominator,.. 16.506084 Therefore M-= 354945, whose log. is 5.550160 Tlhat is, the mass or force of attraction in the sun is 354945 times the mass or attraction of the earth. La Place 12 178 ASTRONOMY. CHr. III. says it is 354936 times; but the difference is of no consequence. Equation (A) gives 350750; but equation (B), as we have before remarked, is far more accurate, and the result here given, agrees, within a few units, with the best authorities. 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 ofJu- same thing, the mass of the sun when Jupiter is taken as 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 (A), let the radius of the earth equal unity, then the radius of Jupiter must be 11.11 (Art. 131 ); and as observation shows the radius of Jupiter's 4th satellite is 26.9983 times its equatorial 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. t2 _D3 Therefore, by applying equation (A), (A= T R); we (16.6888)2(23984)3 have (365.256)2(11.11)3(26.9983)3' 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. 179 365.256, log. 2.562600X2. 5.125200 CHAP. 1II, 11.11,log. 1.045714X3. 3.137142 - 26.9983,1og. 1.431320X3. 4.293960 Logarithm of the denominator,.. 12.556302 Therefore M2= 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 Themass of find the mass of Jupiter to be 332 times that of the earth; Jupiared to that 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 Saturn times the mass of the earth. and Uranus. 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 great accuracy; but the results attached to sultsmay be considered others in table IV, must be regarded as near approximations. accurate. The slight variations which the earth's motion experiences The masses by the attractions of Venus and Mars, are sufficiently sensi- Mof Vens, ble to make known the masses of these planets; and M. Mercury. Burckhardt gives Tvvl8Tl for Venus, and' 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, give results a little different, as expressed in table IV. I 8 O ASTRONOMY, CHAP. II= 2-8; but this result is little more than hypothetlca1l 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. By means of (171. ) It may be astonishing, but it is nevertheless true, gravity and the lunar par- that by means of equations (A) and (B) we can find the allax, we diameter Qf the eath to a greater degree of exactness than by may find the liameter of any one actual measurement. te earth. We have several times observed that equation (A) is not accurate when used to find the masses of the earth and sui, because it contained the time of the revolution of the moon; which revolution is aceleiated by the gravity of the moon, and'retarded by the action of the sun. Therefore, to make equation (A) accurately express the thass of the sun, the element t2 requires two corrections, which will be determined by subsequent investigation. The first is an increase of,g'th part; the second is a diminution of 58,th part7and both corrections Will be made if we take 76.358 75359 2 in place of t2. A common Then hating two correet expressions for the mass of the t un, those two expressions must equal each other; that is, 76;358 2D3 D r2 D3 75-859 2R3f- 2g T2 By suppressing common factors, we have 76.358 t2 *2 75.359- B 2g2a In this equation P represents the mean radiu of the earth, aind we will suppose it unknown; the equation will then mAke it known. The relation between R, the mean Radius of the lunar or. bit, and ir, the mean radius of the earth, is given by means of the moon's horizontal parallax. h uatorial The moon's eqtado'rt horieontal parallax, as we have seeno nhorizontal Oarallax and (65) is 57' 3"; but the horizontal paratlax for the mean raa MASSES OF THE PLANETS. 181 dius, iS 66' 57"; this makes R = (60.36 ) r, whatever the CHAP. nLo numerical value of r may be. Put this value of R in the mean hori, preceding equations and suppress the common factor rib zental paral76'358 it 9l wve then have fi~ - 75 359(60,36 )3r 2g' Therefore, 2g'76358 t2 75.359( 60.36 )3r2 As g is expressed in feet and corresponds to t in seconds, Cdfide'nce the numerical value of ei Will be in feet, which divided by itS th result. 5280, the number of feet in a miles will give the number of miles in the mean radius or mean semidiimeter of the earth; and by applying the pteceding equation, giving g, t, and x, their proper Values; and by the help of logarithms, we readily find t- 3953 miles; only thred miles from the most approved result; and we do not hesitate to says that this result is more to be relied upon than any other. MASS do t HE MOoN. (172.) Approximations to the mass of the moon have The uMasso been determined, fionom time to time, by careful observations the moo~ on the tides; but it is in vain to look for Diathematical re~ determined bults from this source; for it is impossible to decide Whether trbm obserd any particular tide has been accelerated or retarded, aug- vations nd mented or diminished, by the Winds and weather; and if not affected at the place of observation, it might have been at temote distances; but notwithstanding this objection, the lilass of the moon can be pretty accurately determined by means of the tided, owing to the great number and variety of observationS that d be be bought into the account; and We Shall give an exposition of this deduction hereafter; but at present we shall confine oddr 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 matez tial -point, and was not disttlfbed by the attraction of the sun, then the distance that the moon Would fall from a tangent of its orbit, in one second of time, Wold be just equal r 182 ASTRONOMY CHAP. II. gr2 - to R2 (Art. 155.) In this expression.g, r, and/, represent the same quantities as in the last article, The distance that the moon actually falls from a tangent of its orbit, in one second of time, is equal to the versed sine of the are it describes in that time, and the analytical expression for it is found thus: Let Yr R represent the circumference of the lunar orbit, and if t is put for the number of seconds in a mean revolution, then rR represents the are corresponding to the moon's motion in one second (Fig. 30), and as this so nearly- coincides with a chord, we have r22R R An expres. Hence, we perceive, that is the distance that the sion for the 2 t distance the moon would fall from the tangent of its orbit in one second inoon falls in one second of time, if it were undisturbed by the action of the sun; but of time. 359 we can free it from such action by multiplying it by 359, as we shall show in a subsequent chapter. That is, the attraction of both the earth and moon, at the distance of the 859gz2R lunar orbit, is 358.2t2 But the attraction of the earth alone, at the same distance, is; and comparing these quantities with the more general expressions in Art. 156, we have ~E ~E+m g r2 359 tr 2 R 2 R2 R2 358.2 t-2 By suppressing the common denominator, in the first couplet, and calling I, the mass of the earth, unity, the proportion reduces to 359 7r2R3 1: l-+m:: gr2 -.358.2 MASSES OF THE PLANETS. 183 As in the last article, R=(60.36)r, and this value put for CHAP. III..R3, and reduced, gives 359r2 (60.36)3 r I: 1+m: g: 882; Therefore, 359 r2-(60.36)3 r Therefore, - += 3582 t2g The result. 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 1+-m=1.01301; hence m=.01301, which is a little greater than v,th. Laplace says -yth of the earth given by La. is the true mass of the moon; and this value we shall use. place. 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 for density. the standard of measure. The earth is generally taken for that standard. It 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 proportionalto the volume; and if the earth is taken for unity in mass, 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 its 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 ofthe 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 r3:: 1: d; hence d= r3 From this equation we readily find the density of the sun, for we have its mass (354945), and its. semidiameter 111.6 for the sdin times the semidiameter of the earth (Art. 156); therefore its sities of 13 184 ASTRONOMY, P * density must be l,-0.254, or a little more than cty spheres com- (111.6 4 pared to the the density of the earths density of the earth. The miass of Jupiter is 332 tiihes that of the earth, and its volume is 1260 times the volume of the earth; therefore the 332 density of Jupiter is 1=- 0.264; which is a little more than the density of the sun: Densities The mass of the moon is a, and its volume 4-1 thereforeits mooJ&c.tr density is -I- divided by -I or 4 =0.6533; about 2 the denmoon, &c. 593 sity of the earth. From these examples the reader will understand how the densities were found, as expressed in table IV. GRAVITY ON THE SURFACE OF SPHERES. Gravity on (174.) The gravity on the surface of a sphere depends on tie surfaces of the other fathe mass and volume. The attraction on the surface of a planets, how sphere is the same as if its whole mass were collected at its four.d center; and the greater the distance from the center to the surface, the less the attraction, in proportion to the square of the distance: but here, as in the last article, some one sphere 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 earth; therefore a pound, on the surface of the earth, is to the pressure of the same mass, if it were on the surface of 1 354945' the suin, as 1 to (1116)' 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 332 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 IV. tLUNAR PERTURBATIONS. 185 CHAPT ER IV. PROBLEM OF THE THREE BODIES. --- LUNAR PERTURBATIONS (175.) By the theory of universal gravitation, every body in CHAP. Iv. the universe attracts every other body, in proportion to its The theory mass; and inversely as the square of its distance; but simple ofg"ravty. 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 motions The cornm would be comparatively simple, and easily traced, for they plexity of results. would either fall together or circulate around each other in some one undeviating curve; but as it is, when two bodies circulate areand 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 cannot 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 probthird body; and to find all its results, in general terms, is letm of the three bodies, the famous problem of " the three bodies;" but its complete solution surpasses thepower of analysis, and the most skillful mathematician is obliged to content himself with approximations 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 Abstract another body A, moving in the direction PmAB, and dis-attraction. tarbed by the attraction of some distant body D. 186 ASTRO N ON MY. CHkP TV. Fig. 35. We now propose to show some of ID the most general effects of the action of D, without paying the least regard to quantity. If A and m were equally atTwo bodies- tracted by D, and the attraction equally at- exerted in parallel lines, then D would tracted inpa- not disturb the mutual relations of ralel lines are not af- A and m. But while m is nearer to fected in / 2L than A is to D, it must be- more their mutual B relations strongly atsractedl and let the line mp represent this excess of attraction. PIl \LFL A Deonmpose this force (see Nat. Phil.) into two others, m n and n p, the first,9 along the line Am, the other at right angles to it. Vie,, The first is a lifting. force (called by astronomers the radial force), the other- is a fanreftlal force,. and affects the motion of m. It will accelerate the motion of m, while acting with it, from P to B; and retard its motion, while acting against it, from B to Q. We must now examine the effect, when the revolving body is at m', a greater distance from D than A is from DNow A is more' strongly attracted than mn', and the result of this unequal attraction is the same as though A were not attracted at all, and m' attraeted the other way by a force equal to the difference of the attractions of D) on the two bodies A and m'. Let this difference be represented by the line m'/p',and d'ecompose i-t into two other forces, m' n' and %'/', the' first a lifting force, the other the tangental force. The rationale of this last position may not be perceived by every readfer-and to such we suggest, that they conceive A and m' joined together by an inflexible line A m', and both A and m' drawn toward A, but A drawn a greater distance than in'. Then it is plain that the position of the line Am' will be' changed: the angle D Am' will become greater, and the angl CAAm' less; that is, the motion of m' will be Vt NAR PERTURLBATVONS. s18'accelerated from Q to IC, but from C to P it will be re-'CHAP.IV.:tarded. In short, the motion of m will be accelerated when moving to- The dis. bvard the line D B C, and retarded while moving from that iine.'turbing body constantly That is, retarded from B to Q, accelerated from Q to, re- urges arevoltarded from C to P,:and again accelerated from P to B. ving body to If we conceive A to be the earth,.m the moon, anri D the he line of syz.ligies.'sun; then D B C is called the line of the syzigies, a term which means the plane in which conurrctions and oppositions take place. At tEh point B the moon falls in conjunction with the sun, and is new moon; at the point C it is in opposition,:,r full moon. Fig. 36. (176. ) Conceive a ring of mat~t& &round a sphere, as represented in Fig. 36,'and let it be either aftached or detached from the,sphere, and let D be not in the plate of the Ting. From what Was explained in the last article, the partides of matter at m are conitantly urged $oward the line D B C, and the particles at m' are constantly urged toward the sake line; that is, the at- 7L B Action ~e an attracting traction of D, on the ring, has a tendency to body on a tdiminish its inelination to the line b B C; ring. and its position would be changed by such -attraction from what it would otherwlse be;'c A' iand if the ring is attached to the sphere, the sphere itself will have a slight motion in consequence of the action on the ring. Now there is, in fact, a broad ring attached to the equatorlal part 0f the earth, giving the whole a spheroidal formn; and the lyarie of the equator is in the plane of the ring. When the sun or moon is without the plane of this ring, Caus o that is, ithout the plate of the equator, their attraction has nutation a tendency to draw the planeo'of the equlator toward the attracting body, and acatally does so draw it; which motion is called 7rtation. How this motion was {discovered, And its amount ascertained, will.be explained in a subsequent chapter. (177.) We may conceeive the line. B C to be in the 188 ASTRONO MY. CHAr. IV. plane of the ecliptic, D the sun, and the ring around the earth Applica. the moon's orbit, inclined to the plane of the ecliptic with an tion of the angle of about five derees; then when the sun is out of the ring to the re th t t lunar orbit. 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 moo,'s The point where the moon falls into the ecliptic is called nodes retro- the moon's node; and by this external action of the sun the e moon falls into the ecliptic'Fig. *7 from its greatest inclination S before it describes 900, and goes from node to node before it describes 180~- and hence we say that the moon's nodes fall backward on the ecliptic. The rate of retrogradation is 19~ 19' in a year, Z / making a whole circle in about 18.6 years. Lunar per C (178.) We are now pretuirbations \/pared to be a little more defiEB nite, and inquire as to the amount of some of the lunar irregularities. inesti g a. /Let S be the mass of the Investigae D i sun, E that of the earth, and ing a general / A. m the moon, situated at D. analytical expealyical \ it Let a be the mean distance expression for the lunar E between the earth and sun, z perturba- C the distance between the sun and moon, and r the mean radius of the lunar orbit. Let the moon have any indefinite position in its orbit. (It is represented in the figure at D.) The attraction of the sun on the earth is - the attract 0 the attrac-5" LUNAR PERTURBATIONS 189 S CHAP. IV. Lion of the sun on the moon is 2; and the attraction of the earth and moon, on the moon, is E (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 D A and D C; the first along the lunar radius vector, the other parallel to SE.i The two triangles CD B and D S E are similar, and give the proportion a: z:: CD DB. But DB= —; aS Therefore CD —. By a similar proportion we find rS D A = —. Let the angle SED be represented by x, then D G will be expressed by r cos. x, and S D G will be a right line nearly, for the angle D SE is never greater than 7'. Now if the force D C, which is parallel to SE, 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. The force of the sun's attraction on the earth is -; and as this must be less than the force of attraction on the moon when the moon is at D, conceive it represented by the line Cn, and subtracted from CD, 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= CD-Cfn.aS S. An expres, Z3 a2 nwsion for the whole distur Or Dn = aS ( -), the disturbing force. Decom- bing force. pose this force ( Dn) into two others, Dp and pn, by means of the right angled triangle Dp n; the angle p 1) n being equal to D E S, which we represent by x. 190 AsiTRo NOMY. Whence Dp =Sa1 -3 a CO3)S. And n Sa (z a )sin. x. The force DA, i. e. (-3 ) is called the additious force The radial the force Dp the ablatitious force. The difference of these force. two forces is called the radial force; that is Sa (3- S coS. x - 3rS the radial force; pn is the z3 a o tangental force. Expressiati When the angle x is equal to 90~, cos. x = o, SD - SbE, of the radial qruacatt the or = a; which values, substituted, give- for the value quadratus. gv a3 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. t = 1, and the radial force becomes Sa Sa rS S(a-r) Sa a or - g3 - a3 X3 X3 a3 Xitt at that point z = (a- r), which ialue substituted, and rejecting the comparatively very small quantities in both ihumerator and denominator, we have, for the radial force at 2r S conjunction, 2rS When the angle x -=1800 (the moon is in opposition to the sun), cos. x = - 1, and the. force becomes Sa Sa rS S AS ( a+-r ) - -; or d3 Z3 Z3 2 0a3 But at this point z a + r, which, substituting as before, 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 expression 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 PERttJRBATIONS, 191 (179.) As the radial force increases the gravity of the CHAP. IV. moon to the earth at the quadratures, and diminishes it at Points the syzigies, there must be points in the orbit symmetrically wherethe ra. situated, in respect to the syzigies, where the radial force dial force is neither increases nor diminishes the gravity, and of course zero. its expression for those points must be zero; and to find How to these points we must have the equation find them. X(J-4) COS.X — _O.. (1) By inspecting the figure we perceive that the line SD G is in value nearly equal to the line SE, and for all points in the orbit we have z=a+ ros.. x..... (2) Reducing equation ( 1 ), we have (a Z3 ) )cos. -ra2.,, (3) Cubing (2 ), X3-a 03 +C 3a2 r COS. $ X 3: r2 COS. 2X + r3 COS. 3$. a3= a3 -4-3a' r cos. x4 3ar' c r cos. ox. As ~ is very small in relation to a, the terms containing the powers of r, after the first, may be rejected; we then have (a3 -z3) — 3a r os. x. (4) This value substituted in (3), and reduced, gives = Result of J 3 cos. 2X = 1. the radial Hence cos. x 4/1 and x = -540 44', or the points fore at the quadratures are 350 16' from the quadratures. snd syzigies. This shows that at the quadratures, and about 350 on each side of them, the gravity of the moon is increased by the action of the sun, and at the syzigies, and about 540 on each side of them, the gravity is diminished; and the diminution in the one case is double the amount of increase in the Mean ra other, and by the application of the differential calculus we dial three. learn that the mean result, for the entire revolution, is a dimirS nution whose analytical expression is 2-3 an expression which holds a very prominent place in the lunar theory; the .192 ASTRONOMY. CHAP. IV. result of which we have used in Art. 171, and there stated it to be W-I 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 the mean ra. dial forcen very serious inquiry of the critical student? and 1Low The force that retains the moon in its orbit is E+mn found. r2 ( Art. 156); and if the radial force can be rendered homogeneous with this, some numerical ratio must exist between them. Let x represent that ratio, and we must find some numerical value for x to satisfy the following equation: rS E+m 2% x=-2 A.. Therefore x 2 (E+m)a3; r3 S calling E= 1, m=,5 (Art. 172), or E+ m is 1.013. S = 354945 (Art. 169), and the relation between the mean distance to the sun, and the mean radius of the lunar orbit, is 397.3,* therefore (2.026)(397.3)3 358 354945 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 circuFct of the radial force. late at 358th part greater distance from the earth than it otherwise would have, and its periodical revolution is increased 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. rS The radial But we perceive the expression 2-a contains two variable force varia. ble. 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. LUNAR PERTURBATIONS. 193 riable; and it will be least when the earth is at the greatest CHAr. LV. 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 anna. hence we perceive that the same cause which affects the ap- al equation ofthe moon's 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 varies 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, S a -- cos. x --- and banish the letter zfortheradial z3 as X3 force at any from it by means of the equation moon's orbit. z a - r cos. X Or, Z= 3 as3 3a2 r cos. x, (neglecting the powers of r) and we shall have, rS (3 cos. 2 x- 1) a3 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 tangental force, and banish z, as before, we have, 3rs cos. x sin. x tangental force - a3 By doubling numerator and denominator, this fraction can Expression take the following form: for the tan gental force, 3rs (2 cos. x sin. x)..2a3 But, by trigonometry, 2 cos. x sin. x - sin. 2x, 3rs sin. 2x. Therefore the tangental force 3 — in3 2.a This expression vanishes when x = o and x = 900; for then Its vanish. sin. 2x = sin. 180 = 0. Hence the tangental force van- ing points. ishes at the syzigies and quadratures, attains its maximum * This is equation I, in the Lunar Tables. 13 Q AS94 TO N AO S N MY. CIAP. IV. value at the octants, and varies as the sine of the dou8be angtdiar distance of the moon from the sun. The tan- The mean maximum for this force must be determined by geneatest observation. It is known by the name of variation, and by when the mere inspection we can see that its amount must correspond earth is in perigee. to the variations of r and of 3. HLence, to obtain the moon'S place, we must have'correction on correction. The variation amounts to about 35'. It increases the velocity 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. Application (1820) Let us now examine the effect of the radial force of the radial force to an on the lunar orbit, considered as elliptical. ellipt, al or- Let S E ( B ig. 38) be at right bit. angles to A B, the greater axis of the lunar orbit, and conteive 3A CB to represent the orbit that the moon would take if it'were undisturbed by the sun. But when the moon comes found 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 3. But here r is less than its mean value, a and the expression is less than its mean, and therefore the moon is A I B not crowded so near the earth as V E it otherwise would be, and, of course, at this point the moon D will run farther from the earth. At the point C, the radial force tends to increase the distance between the earth and moon, and to widen the orbith Whien the When the moon passes round to B, the radial force again radial force increases the gravity of the moon, and r, in the expression LUNAR PERTURBATION& 19~ CHAP. IVo, is greater than its mean value; and, of course, crowds the -- a3 a ~~~~~~~~~~~~~~~~decreases moon nearer to the earth than it otherwise would go; and the ecceftrs city of the l~a thus we perceive that the action of the radial force on an el- nar ellipse. liptical orbit has a tendency to decrease the eccentricity of the ellipse, when the sun is at right angles to its greater axis. (183.) Now conceive the sun to be in a line, or nearly in a line, with the longer axis of the lunar orbit, as represented in Fig. 39. The radial force at the quadratures, SO When the C and D, has a tendency to press in radial force increases the the orbit, or narrow it. At the point Fig.'39. eccentricity A, the tendency, it is true, is to in- of the lunaorbit~ crease the distance between the earth and moon; but that tendency is not so strong as it would be if the moon were at its mean distance from the earth. The tendency at B is to increase the distance, and it is a tendency greater than the medium. That is, the tendency at A is less than the medium; at B, greater than the me- A dium; and at C and D, the compressed parts of the orbit, the tendency is to a still greater compres- C D sion; therefore, the entire action of the radial force is to increase the eccentricity of the lunar orbit, when the sun is in line, or nearly in line, with the longer axis. B Thus, we perceive, that under the disturbing action of the sun, the eccentricity of the moon's orbit must be in a state of perpetual change, now more, now less, than its mean state. Corresponding with this change of eccentricity there must be changes in the lunar motion; and to keep account of it, and allow for it, astronomers have formed a table called EVEOCTION. 196 ASTRONOMY. CHAP. IV, (184.) Now let us examine the effect of the radial force Effect of on the position of the lunar apogee. the radial Let E (Fig. 40), be the earth, and, notion ofthe Fig. 40. for the sake of simplicity, we conceive lunar peri 0 the earth to be stationary, and the,$ Sd sun and moon both to revolve about it with their apparent angular velocities; the moon in the orbit A CB, and in the direction A C B; the saun in a distant orbit, part of which is represented by S St. Let A B be the greater axis of the moon's orbit, in its natural position, or as it would be if undisturbed by the sun; and being undisturbed, the ~,B, Sperigee and apogee would remain constant at the points A and B; and the time from A to B, or from B to A, would be just equal to the mean time of half a revolution, as explained in a former part of this work. k'ettograde Now let us conceive the sun to be:notibn ofthe porion'eb and in its orbit at S, then the moon will aipogee, A be in the syzigy when it comes round to s, and as the radial force at that point tends to increase the distance between the earth and the moon, the apogee will take place at s, or between s and B; and it is evident that the apogee in that case would recede or run back. But at The ma3or axis of the the next revolution of the moon, in a little more than twentylunar orbit is seven days, the sun at that time will, apparently, have moved ncllow th to S' about twenty-seven degrees. Now the syzigy will take bun. place at s', and the greatest distance between the earth and. moon will now be between B and s', that is, the apogee will advance, in one revolution, from near s to near s'; and thus, in general, the longer axis of the moon's orbit is strongly inclined to follow the sun; and this is the source of its progressive motion. It makes a revolution in 3232-. days; but its motion is very irregular, for, as we have just seen, %UNAR PERTURBATIONS 197 Wheri the line which joins the earth and sun makes a very c.HP. IYv acute angle with the longer axis of the lunar orbit, and is approaehing that axis, the motion of the apogee and perigee is retrograde; but, all of a sudden, when the sun passes the longer axis of the lunar orbit, the motion of the apogee becomes direct, and moves with considerable rapidity. When the sun is at right angles to the major axis of the Under what moon's orbit, the tendency of the radial force is to diminish position of the sun_. the the eccentricity of the orbit, but it has no tendency to change lunarperiged the position of the axis. relnains staFrom this investigation it follows, that when the sun has tionary. just passed the greater axis of the lunar orbit, the interval from apogee to apogee, or from perigee to perigee, will be greater than a revolution. Just before the sun arrives at the position of the longer axia, the time from one apogee to an-.other is less than a revolution; and when the sun is at right angles to the longer axis, the time is just equal to a revolution in longitude. (18;5,) By comparing eclipses of the moon, observed by Ancient the ancient Egyptians and Chaldeans, with those of more eclipses cornpared with modern times, Dr. Halley, and other astronomers, concluded modern ohb that the periodic time of the moon is now a little shorter sertationt, than at those remote periods; and to make these extreme observations agree with modern ones, it became necessary to conceive the moon's mean motion to be accelerated about 11 seconds per century. For a long time this fact seriously perplexed astronomers: the ted some were for condemning the theory of gravity as insufli- slt. cdent to explain the cause of the lunar perturbations, while others were for rejecting the facts, although as well established as any mere historical facts could be. In this dilemma, says Herschel, "Laplace stepped in to rescue physical astronomy from reproach by pointing out the real cause of the phenomenon in question." Although this subject troubled the greatest philosophers of the past age -the greatest mathematical philosophers the world ever saw the problem is quite simple, now the solution is pointed out, and we are sure that every reader of or 198 ASTRONOMY. CHAP. IV. dinary capacity can understand it, provided he gives his serious attention to the subject. A summary The secular acceleration of the moon's mean motion is statement of the cause. caused by a small change in the mean value of the radial force, occasioned by a change in the eccentricity of the earth's orbit. rS The expression 2a is the mean radial force of the sun acting on the moon's orbit, dilating it and increasing the time of the lunar revolution.. When the If the earth's orbit had no eccentricity, 2a3, the denominamoon's moe tion is intor of the fraction, would always have the same value, and creased. then regarding the numerator as constant, there would be no variation of the moon's motion arising from this cause. But in consequence of the earth and moon moving toward the apogee of the earth's orbit, a, of course, a3 becomes greater, and the value of the radial force becomes less than its mean value, and in consequence of this, the moon's motion is increased. And when the earth and moon move toWhen di. ward the earth's perigee, a and a3 become less, and the ninished. value of the radial force becomes greater than its mean; the moon's orbit is dilated to excess, and its motion is diminished; The ex. and the orbit is more dilated when the earth is in perigee than it pression for is contracted when the earth is in apogee. In other words, the the mean radial force is mean dilatation of the lunar orbit is greater, and the mean not the true motion of the moon less, in proportion as the earth's orbit is mean. mean. more eccentric. rS The less the value of 2 the greater is the moon's mean motion, and that value is least when a is greatest. But a would have no variation of value if the earth's orbit were circular. The earth's orbit, however, is eccentric, and in the course of a year the value of the radial force is exactly expressed by 23 only at two instants of time, when the earth passes the extremities of the shorter axis of its orbit. At all other times a is either greater or less than its mean value, and the variations are equal on each side, of it; that LUNAR PE.RTURBATIONS. 199 is, a becomes (a — d) or (a + d), and the radial force is cAP. IV. really rS rS or- or 2(a —d)3 2(a+d)3' which expressions correspond to equal distances on each side The true mean value of the mean distance, and d may have all values from 0 to of the radial ae, the eccentricity. The mean value of the radial force force. corresponding to the *whole year, is equal to 2a (a-d) + (a+d) Or, 4 (a-d)3 (a+d) 3) But this expression is always greater than - except The meat'2a value of the when d= 0; then it is the same, as any algebraist can verify. radial force will be least Hence the mean radial force for the whole year is greater of all when as the earth's orbit is more eccentric, and it will be least of the earth's orbit is a all when that orbit becomes a circle; and then, and then circle. r S only, it will be accurately represented by 23 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 eause mains constant. It must be so from the nature of motion, ofthe change -of eccentriforce, action, and reaction; but by the attraction of the city of the planets the eccentricity of the earth's orbit is in a state of per- earth's orbit, petual change; the change, however, is excessively slow. From the earliest ages the eccentricity of the orbit has been diminishing; 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 eccentric,. and increase in eccentricity to a certain moderate amount, and then again decrease. 14 200 ASTRONOMY, CHAP. Ivt. The period for these vibrations, " though calculable, has never The im- been calculated further than to satisfy us that it is not to be mense period reckoned by hundreds or even by thousands of years." It is a corresponding to these period so long that the history of astronomy, and of the whole changes. 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. ihe f tli (187. ) For the sake of simplicity, we have thus far conlunar orbit sidered the moon's orbit to be in the same plane as the taken into earth's orbit; but this is not true; the mean inclination of the account. lunar orbit to the ecliptic is 50 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 correction, 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 constant force urging the moon to the ecliptic, causes the moon's nodes to retrograde. We conclude this chapter by a brief summary of the principal causes which affect the moon's motion. A summary 1. The eccentricity of the earth's orbit; which gives rise to statement of the lnar iof the annual equation of the moon in longitude. regularities, 2. The eccentricity of the lunar orbit; producing the equation 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 axis of the lunar orbit; giving rise to the inequality called evection. 5. The inclination of the moon's orbit. THE TIDES. 201 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 3) THE alternate rise and fall of the surface of the CxAP. Vo 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 term tide. definitely known, it was recognized as having some hidden and mnysterious connection with the moon, for it rose and fell twice Connection in every lunar day. High water and low water had no con- with the 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 moonwere in conjunction, or in opposi- High tides. 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- Tides afence on the tides. When the declination is high in the north, fected by the declination the tide in the northern hemisphere, which is next to the moon, of 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 a superficial by its attraction should be able to raise a tide on the opposite reasoner. side of the earth. 202 ASTRONOMY. CHAP. v. That the. moon should attract the water- on the side qf the earth next to her, and thereby raise a tide,. seems rational and natural, but that the same simple action also raises the opposite 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. The true But the true cause is the sun and moon's attraction; and cause,. until this is clearly and decidedly Fig. 41. understood - not merely assented m. to, but fully comprehended- it is impossible to understand the common results of the theory of gravity, which are constantly exemplified in the solar system. We now give a rude, but striking, and, we hope, a satisfactory explanation. A summary Conceive the frame-work of the lthetratiodn earth -to be an inflexible solid, as it of the tides. really is, composed of rock, and incapable of changing its form under any degree of attraction; conceive A at also that this solid protuberates out of the sea, at opposite points of the earth, at A and B, as represented in Fig. 41,. being on the side of the earth next to the moon, m, and B opposite to it. Now in connection with this solid conceive a great portion of the earth to be composed of water, whose particles are inert, but readily B move among themselves. The solid A B cannot expand under the moon's attraction, 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. 203 more poqweful attraction than the solid, rise toward A, pro- O-AP. V. ducing a tide. The particles of water at b being less attracted toward mn than the solid, will not move toward in 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 lbetween the the expressions for the perturbations of the lunar motion. bations and What we have called the radial force, in treating of the the perturba-.tions of the lunar irregularities, is the same in its nature as the force that ocean. raises the tides; the tide force is a radial force, which diminishes 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 radial 2rS force as apthe syzigies, is expressed by -; in which expression S is plied to the amoon. 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 Converted into an ex. m, the mass of the moon, and conceive a to represent the dis- pression for tance to the moon, and r the radius of the earth. For the the tides. tides, then, we have 2rm and as the numerator is always cona, stant, the variation of the tides must correspond to the cube of the inverse distance to the moon. ( 190.) The sun's attraction on the earth is vastly greater Sen's attraction conthan that of the moon; but by reason of the great distance sideredlo to the sun, that body attracts every part of the earth nearly alike, and, therefore, it has much less influence in raising a tide than the moon. 204 ASTRONOMY. CHAP. V. From a long. course of observations made at Brest, in Observations France, it has been decided that the medium high tides, atBrest. when the sun and moon act together in the syzigies, is 19.317 feet; and when they act against each other (the moon in quadrature ), the tides are only 9.151 feet. Hence Compara- the efficacy of the moon, in producing the tides, is to that tive influencesofthesnn of the sun, as the number 14.23 to 5.08. and moon. Among the islands in the Pacific ocean, observations give the proportion of 5 to 2.2, for the relative influences of these two bodies; and, as this locality is more favorable to accuracy than that of Brest, it is the proportion generally taken. Having the relative influences of two bodies in raising the tides, we have the relative masses of those two bodies, provided they are at the same distance. But by the expression for the tides, as we have just seen, the variation for distance corresponds with the inverse cube of the distance, and the distance to the sun is 397.2 times the mean distance to the moon. Hence, to have the influence of the moon on the tides, when that body is removed to the distance of the sun, we must divide its observed influence by the cube of 397.2. Mass of the That is, the mass of the moon is, to the mass of the sun, as: moon computed. 5 the number 397 2) to the number 2.2. In all preceding computations we have called the mnass of the earth unity, and in relation thereto, the mass of the sun is 354945 (Art. 169). Let us represent the mass of the moon by m, then we have the following proportion: The result. m: 354945 2.2. (397.2)3 This proportion makes the mass of the moon a little less than 5-X-; but I have little confidence in the accuracy of the result, as the data, from their very nature, must be vague and indefinite. The times (191.) The time of high water at any given point is not of high wa- commonly at the time the moon is on the meridian, but two ter different in different or three hours after, owing to the inertia of the water; and! localities. places, not far from each other, have high water at very dif 'THE TIDES. 205 Serent times on the same day, according to the distance land CHAP. V. direction that the tide wave has to undulate from the main ocean. The interval between ihe meridian passage of the,moon and the time of high water, is nearly constant at the same place. It is about fifteen minutes less at the syzigies than at the quadratures; but whatever the mean interval is at any place, it is called the establisIhment of the port. It is high water at Hudson, on the Hudson river, before The tides it is high water at New York, on the same day; but the tide stantly ease wave that makes high water one day at Hudson, made high on the remowater at New York the day before; and the tide waves that aul of their make high water now, were, probably, raised in the ocean several days ago; and the tides would not instantly cease on the annihilation of the sun;and moon. The actual rise of the tide is very different in different Tides very places, being greatly influenced by local circumstances, such dchy lffocta as the distance and direction to the main ocean, the shape circumof the bay or river, &c., &c. stances. In the Bay of Fundy the tide is sometimes fifty and sixty feet; in the Pacific ocean it is about two feet; and in some places in the West Indies, it is scarcely fifteen inches. In inland seas and lakes there are no tides, because the moon's attraction is equal over their whole extent of surface. The following table shows the hight of the tides at the most important points along the coast of the United States, as,ascertained by recent observation. Feet. Annapolis (Bay of Fund),.................................60 Apple River,............................................50 Chicneito Bay (north part of the Bay of Fundy)..........60 Passamaquoddy River,.................................... 25 Penobscot River,.........................................10 Boston,................................................. 11 Providence, R. I.,................... 5 New Bedford......................................... 5 New Haven.......................................... 8 New York,....................................... 5 Cape May,.......................................... 6 Cape Henry.......................................... 4/p IL 206 ASTR'RONOMY. CHAPTER VI. PLANETARY PERTURBATIONS.CHP. VI. ( 192.) The perturbations of a planet, produced by the atPlanetary tractions of another planet, are precisely analogous to the perand lunar perturba- turbations of the mbon, produced by the action of the sun. tions analo. The disturbing forces are of the same kind, and they are g'ons. subject to similar variationrs from precisely the same causes. But the amount of the disturbances is, in most cases, very trifling, on account of the small mass of the disturbing planet compared with the mass of the sun, or its great distance from the body disturbed. Aetion and As action and reaction are everywhere equal, the planets reantion am. mutually disturb each other, and if one is accelerated in its ong the plan. ets recipro- motion, the other must be retarded; if the tendency of one tocal. ward the sun is diminished, that of the other must be increased. Examine Fig. 23, and conceive V, Venus, to be disturbed by the- attraction of the earth at EF, and if the motion of the planets is in the direction of VB, it is perfectly clear that Venus will be accelerated by the earth, and the earth will be retarded by Venus. One planet But Venus will be more accelerated in its motion than the Is accelerated while an- earth will be retarded, for the disturbance at this point is in other is re- a line with the motion of Venus, and not in a line with the tarded. motion of the earth. When the After Venus passes conjunction, that is, passes the varying action line SE, her motion becomes retarded; and the earth's is acchanges. 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, Juwhant piter, the moon, &c., are simply the effect that these bodies far perturba. have on the motion of the earth. tions. 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. 207 greater mass will move but half as far as the smaller, under C HAP. VI. their mutual action. But when the amount of disturbance is Angular irreferred to angular motion for its measure, regard must be regularities had to the distances of each planet from the sun; for the indicae the amount of same distance on a larger orbit corresponds to a less angle.* planetary Also, the whole amount of the disturbing force of a superior disturbance planet on an inferior will, at times, be a tangental force after certain' reductions. ( Fig. 23); but the reaction of the inferior planet on the superior 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 consideration, 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 al results in respect to the the inferior, and to maintain it at a greater distance from the times of revsun than it would otherwise have. The action of the inferior olution. 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 revolution of all its superior planets, would be a little increased, and the revolution of Mercury, its inferior planet, would be a littie diminished. If Jupiter were annihilated, the times of revolution 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 in circular oracceleration in one part of an orbit would be exactly compen- bits. X 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 intricate and curious calculation is fhlly confirmed by observation.HERSCHrEL. 208 ASTRONOMY. CHAP. vt. sated by the retardation in another; and in the course of a whole revolution, the mean motions of both planets (the disturber 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 oils of in- one part cannot be exactly counterbalanced by the retardaqualities de- tion of the other; and, therefore, there is commonly left a cerpending on rtain outstanding error, which increases during every synodiconjunctions in the same cal revolution of the two planets, until the conjunctions take parts of the place in opposite parts of the orbits, then it attains its maximum, which is as gradually frittered away as the line of conjunctions works round to the same point as at first. Some of Hence, between every two disturbing planets there is a common tsese inequalties too inequality depending on th/eir mutual conjunctions, in the same, minute to be or nearly in the same, parts of thieir orbits. But it would be noticed. folly to compute the inequalities for every two planets, by reason 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 considerable mutual perturbations, on account of their orbits being near each other, and both bodies far away from the sun. The effect (195.) Again, if the revolutions of two planets are exofratmreo- actly commensurate with each other, or, what is the same lutions ofthe thing, the mean motion of both exactly commensurate with planets. the circle, then the conjunctions of those two planets will always occur at the same points of the orbits ( just as the conjunctions of the two hands of a clock always occur at the same points on the dial plate), and, in that case, the conjunctions 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. A supposed For instance, if three times the mean angular motion of case for illus. tration. one planet were exactly equal to twice the mean angular mo PLANETARY PERTURBATIONS, 20% tion of another, then three revolutions of the one would ex- CHAP. VI. actly correspond to two of the other, and every second conjunction 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 symmetrical, unless the longer axes of the two orbits are in the same 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 the solar sys tern. 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 incommensurable; but, nevertheless, there are instances that approach 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- The great paring together ancient with modern observations of Jupiter inequalities of Jupiter and Saturn, their mean motions could not be uniform." The and Saturn. 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 calculated place, and Jupiter was as constantly in advance of his. On the other hand, in the eighteenth century, a process precisely the reverse was going on. The amount of retardations and accelerations, corresponding The per. to one, two, or three revolutions were not very great; but, as to the phie. they went on accumulating, material differences, at length, sophers. existed between the observed and calculated places of both 14 a* 210 ASTRONOMY. CHAP. 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 enAolved the deavor to account for 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 commensurability 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 elliptical, 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 synodieal 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 uaions of Ju. of Saturn, 21518 days. So that, in a period of two revolupiteli and Saturn compar- tions of Saturn (about sixty of our years), after any conjunced. 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 dical revolu- will take place, we compute the synodical revolution of these mined. two planets by dividing the circumference of the circle in seconds (1296000) by the difference of the mean daily motion of the planets in seconds (178".6),1 and the quotient is 7253.4 days; three times this period is 21760 days. In this period Jupiter performs five revolutions and 80 6' over; Saturn, smakes 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. - See problem of the two couriers, Robinson's Algebra. PLANETARY PERTURBATIONS. 211 of conjunction advances 80 6' in longitude during the period CHAP. VLf of 21760 days. In the year 1800, the longitude of Jupiter's perihelion point, was 110 8', and that of Saturn 8909'; the inclination of the greater axis of the orbits, therefore, was 78~ 1'. Fig. 42 Q Let AB ( 42) represent th major axis of Stus Th Let AB (Fig. 42) represent the major axis of Saturn's The seriei orbit, and a b that of Jupiter; the two are placed at an angle of conjunct tions exof 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 conjunc- Lineofcon. tion); in 7253.4 days afterward another conjunction will take junction ex. place. In this interval, however, Saturn will describe about plained. 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 R T the next conjunction will be at s i, 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 ellipsest for the purpose of magnifying the principle under consideration. 212 ASTRONOMY. CHAP. VL tertain portion of the orbits, between S and T, where the two Certain planets would come nearer together in their conjunction, than,conjunctions they do at conjunctions generally, and, of course, while any,bring the pla. nets nearer one of the three conjunctions is passing through that portion together tha of the orbits - Jupiter disturbs Saturn, and Saturn reacts on most others. Jupiter more powerfully than at other conjunctions; and this is the cause of " the great inequality of Jupiter and Saturn." Theperiodof (198.) To obtain the period of this inequality, we cornthisle remark- pute -the time requisite for one of these lines of conjunction quality com- to make a third of a revolution, that is, divide 120~ by 8~ 6', 9lated, and and we shall find a quotient of 1422, showing the period to be the computa- 2 tion confirm- 142 — times 21760 days, or nearly 883 years; which would be ed by obser- the actual period, provided the elements of the orbits revation. ation mined unchanged during that time. But in so long a period the relative position of the perigee points will undergo considerable variation; which causes the period to lengthen to about 918 years. Fig. 43. The maximum amount S 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 great An expla.- c —Y achievement of the powtation of the ers of mind in the solar principle that led to the system, was the discovery discovery of of the new planet NRepNeptune. tune, by Leverrier and Adams analyzing the inP Q equalities of the motion. of Uranus. To give 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. 213 S P, although no such body was ever seen or known to exist. CHAP. vI. The next time the planet comes round into the same portions Df its orbit,* suppose the center of the protuberance to have changed to the line S Q. This would indicate that the un- How computations known and unseen body was now in the line S Q, and that could be since the former observations it had changed positions by the made for the angle P S Q; and, by this angle, and the time of its descrip- revolution of 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 seen, and is called NYeptune. CHAPTER VII. ABERRATION, NUTATION, AND PRECESSION OB THE EQUINOXES. CHAP. VI1L (200.) ABOUT the year 1725 Dr. Bradley, of the Green~wich observatory, commenced a very rigid course of observa- leysDr. Brado.wich observatory, commenced a very rigid course of observa- ley's obsertions 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, finding their near the meridian of the sun, have a decrease of longitude of paralUnexpecte 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 revolution of Uranus, 214 ASTRONOMY. CHAP vn. perplexing, the results were directly opposite from such as would arise from parallax. These facts were thrown to the world as a problem demanding solution, and, for some time, it baffled all attempts at explanation; but it finally occurred to the mind of the Doctor, that it might be an effect 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 standAberration illustrated. i S ing still in a rain shower,.-c + when the rain falls perpenxi dicularly, the drops will strike directly on the top \\ Iof 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. I, of the drops, as if the person stood still and the rain came inclined from the direction 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 following explanation is more appropriate. Aslother and Conceive the rays of more appro- light' to be of a material priate illus. tration. _ __ substance, and its particles B A 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 D1B is the motion of light, and A B the motion of the earth, then the tele ABERRATION. 215 scope must be inclined in the direction of A D, to receive the cHAP. vu. light of the star, and the apparent place of the star would be at S', and its true place at S and the angle ADB is20".36, at its maximum, called the angle of aberration. By the known motion of the earth in its orbit, we have the value of A B corresponding to one second of time: we have the angle A DB by observation: the angle at B is a right angle, and (from these data) computing the side B D we have the velocity of light, corresponding to one second of time. To make the computation, we have D B: BA:: Rad.: tan. 20".36.* But B A, the distance which the earth moves in its orbitl The velo. Fig. 45. city of light 90 computed by means of ab* *' erration. 180 * *0 CD 270'To obtain the logarithmetic tangent of 20".36 see note on page 128. 15 <2'6 ASTRONOMY. Cissi. 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 B D 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 remarkable 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 ohf"'effect heavens, we give figure 45. Conceive the earth to be of aberra- moving in its orbit from A to B. The stars in the line ABE tion. whether at 0 or 180, are not affected by aberration. The stars, at right angles to the line A B, are most affected by' aberration, and it is obvious that the general effect of aberration 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 opposite to it, at 270, has its longitude decreased, by the effect of aberration; both being thrown more toward 180. The effect on each star is 20".36. But when the earth is in the opposite part of its orbit, and moving the other way, from G to 1, then the star at 90 is apparently thrown nearer to 0; so also is the star at 270, and the whole annual variation of each star, in respect to longitude, is 40".72. Proonfoftm (202. ) The supposition of the earth's annual motion fully annual motion of the explains aberration; conversely, then, the observed variations earth. of the stars, called aberration, are decided 2roofs 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 ecliptic, the ellipse be ABERRATION. 217 eomtes a circle of 40".72 in diameter, in respect to a great CHAP. VI_ circle; but a circle, however small, around the pole, will in-:elude 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 revolution round the pole by aberration; and the same is true inrelation to the pole of the celestial equator. All these ellipses have their longer axis 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, Aberratioi because the distance to the sun varies but little, and without of the sun. material error, 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 mooi noticed, as light passes that distance in about one second of not affected by aberratime. tion. (204. ) While Dr. Bradley was continuing his observa- Other inetions to verify his theory of aberration, be observed other qualities ob. served by Dr. small variations, in the latitudes and declinations of the stars, Bradl.y. that could not be accounted for on the principle of aberration. The period of these variations was observed to be about ber in 20".36 cos. (S-s) *Aber. in Lon. - cos. I Aber. in Lat. -20".36 sin. (S-.s) sin. 1. In these expressions S represents the longitude of the sun, * the longitude of the star, and I its latitude. 218 ASTRONOMY. CHAP. VII. the same as the revolution of the moon's node, and the amount of the variation corresponded with particular situations 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 equator, which gives the earth its spheroidal form, and the effect itself is called NUTATION. Fig. 46. *? utation 25 eh hwnnAt7thtteatato* utaltipn ( 205 ) We have shown, in Art. 176, that the attraction fully explain. edbyt*e the. of a body, m, on a ring of matter around a sphere, has the or of gravi effect of making the plane of the ring incline toward the attY. tracting body. Let B C, Fig. 46, represent the plane of the equator; and conceive the protuberant mass of matter, around the equator, to be represented by a ring, as in the figure. Let m be the NUTATION. 219 moon at its greatest declination, and, of course, without the CHAP. VII. 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, corresponding to the declinations of the sun and moon, was carefully 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 satisfactorily 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 gene. ral effect of when the sun and moon have their greatest northern declina- nutation il. 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 Fig.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- Where the feet must be when the declination is greatest. node must be The moon's declination is greatest when the longitude of tocorrespond to the moon's the moon's ascending node is 0, or at the first point of Aries. greatest deo The greatest declination is then 280 on each side of the clination, 220 ASTRONOMY. CHAP. vII. 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 opposite position. Hence, the amount of lunar nutation depends on the position of the moon's nodes. Monthly nu. It is very natural to suppose that the period of lunar nutaation, effect tion would be simply the time of the revolution of the moon; and so, 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 13~ 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 its variation from that line is only about five degrees on each moon on the,nass of mat- side; hence, the medium effect of the moon on the protuberant ter around mass of matter at the equator is the same as though the the equator. 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 equilibrium of position, 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 protuberant 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 nu- We have, thus far, made particular mention of the moon, tation. but there is also a solar nutation: its period is, of course, a year; and it is very trifling in amount, because the sun attracts 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 in THE EQUINOXES. 221 one direction), is too short a time to have any great effect on CHAP. Vn. 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 nutation-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, resulting from the same physical cause, we mean the PRECESSION OF THE EQUINOXES. We again return to first principles, and consider the mu- First printual attraction between a ring of matter and a body situated ciples agan examined. 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 attraction of the moon is in the plane of the The mean ecliptic. The sun is all the while in the ecliptic. Hence, the attrach ons of 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- the ecliptic. 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 preeession of the are, as is shown by the sun coming into the equinox, or equinoxes. crossing the equator at a point 50"..1 before it makes a revolution in respect to the stars. Perhaps it is clearer to the mind to say, that the sun is Natural mode of exdrawn to the equator by the protuberant mass of matter pression. 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, 222 ASTRONO0MY. CHAP. YII. 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. The true If the moon were all the while in the ecliptic, the precesphysical cause of the sion of the equinoxes would then be a constantlyJlowing quanprecession of tity, equal to 50".1 for each year; but, for a succession of the equinoxes. 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 this inequality, 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 of the equi- of the tropical year, and slightly, very slightly, affects sidenoxes. real time. Mean and There is a true equinox and a mean equinox; and, as sidetrue sidereal real time is measured from the meridian transit of the equitime. 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, generally, it is much less. Explanation ( 209.) In the hope of being more clear than some authors of Fig. 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, 230 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; cr and o_ are the two opposite points where the ecliptic and equator intersect; mcE is the first meridian of longitude; W P is the first meridian of right ascension. The angle EcP is 230 27', and E P, produced, is the meridian passing through the solstitial points. To obtain a clear conception of the precession of the equinoxes, the stars THE EQUINOXES. 223 the ecliptic, and its pole E, must be considered as FIXED, rc,,k. vuy and the line m9 = as having a slow motion of 50".1 per arlFig. 47. From the num, on the ecliptic, in a retrograde direction; and this must fixed posicarry the pole P, around the point F, as a center, carrying tion of the also the solstitial points backward on the ecliptic. Some ealso of the of the stars have proper motions; but, putting that circum- stars, the stance out of the question, the stars are fixed, and the eclip- stars never change lati., tic is fixed; therefore, the stars never change latitude, but tade. 224 ASTRONOMY. CAP. V&I. 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, 9c E, moves backward; they change right ascension, because the meridian, cm P, and all the meridians of right ascension, revolve backward. One hemi- By inspecting the figure, we readily perceive that all the ars ap-p f stars near cr must, apparently, approach the north pole, beproaches the cause the pole, in its revolution round X, is approaching tonorth pole, ward that part of the ecliptic; for the same reason, all the the other recedes from stars near. — are, apparently, moving southward, because the it. equator is being drawn over them. In short, all the stars, from the eighteenth hour of right ascension through cp, to the sixth hour of right ascension, must diminish in north polar distance, and all the stars, from the six hours through -i, 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 Table I. I, 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 corresponding to six hours, and eighteen hours of right ascension; and the rapidity of this variation is greater as the star is nearer to 0 hours, or twelve hours of right ascension. Annual va- When the right ascension of a star is 0 hours, or twelve riation in declination, hours. it is easy to compute its annual variation in declinahow comput- tion, corresponding to its precession along the ecliptic of ed. 50".1. Conceive a small plane triangle whose hypothenuse is 50".1, the angle at the base 230 27' 40" (i. 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- It is thus, by the motion of these imaginary lines over the tions, how discovered. whole concave of the heavens, that the annual variation of both right ascension and declination of each individual star THE EQUINOXES. 225 ic the catalogue is computed and put down; and if any par- C1hAP. vIt. titular star does not correspond with this, it is said to have 2Iroper 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 of preces. in a year, it will require 25868 years to perform a revolution; sion. p 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- Comparacession of the equinoxes is, to the mean effect of the sun, as tive effectof sun and five to two. The sun's action is nearly constant, because moon. the sun is always in the ecliptic; a small annual variation, however, is observed. The great inequality of 17".3, corresponding 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 Undulatory move round the pole of the ecliptic, E, in an even circumfe- motionofthe earth's axis rence of a circle, but it has a waving or undulating motion, as around the 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 contained in 25868. From this, we perceive, that the undulations in the figure are much exaggerated, and vastly too few in number; an exact linear representation of them would be impossible. (211.) From the foregoing, we learn that the positions of Mean and all the stars are affected by aberration, precession, and nuta- apparent 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 sa.nnll quantity as for a large one, and often greater; and to reduce the apparent place of a fixed star from its mean place, 15 2246 ASTRONOMY. CHAP. VII. and its mean place from its apparent place, is one of the most troublesome problems in practical astronomy. Generalfor- The mean place of a fixed star, reduced to the time of obpaulre, where found, servation, is sufficiently near its apparent place to be considered 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 NVautical 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 of physicalmy 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. 14 m.; 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 apogee, must be very nearly a constant quantity; but, on consulting and comparing the predictions to be found in the English nautical almanacs, he will find these periods to be (in comparison to his anticipations) very fluctuating. They differ from the stated 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 misprint; but, on examining concurrent facts, such as the logarithms 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 tudefexihne the first page of our Tables. There it will be found, that the sun explain. ed. sun has latitude; that it deviates to the north and south of the ecliptic, by a quantity too small ever to be observed: it is, therefore, a quantity wholly determined by theory, and, as THE EQUINOXES. 227 the sun's latitude changes with the latitude of the moon, we CHAP. VII. must seek for its cause in the lunar motions. Fig. 48. To understand the fact of the sun having S 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 ecliptic, Sc, the earth, E would be on the other m' 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 chanye his latitude, when the moon changes her latitude. E If the moon were all the while in the plane of the ecliptic, Longitude the sun would have no latitude (save some extremely minute ofcthe suyn f. 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 5~0 20 from the ecliptic, and, of course, the earth makes but a proportional deviation on the other side; but, in longitude, the moon deviates to a right angle on both sides, in respect 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 greatest 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 Longitlhde of the moon earth around the common center of gravity of the earth and affects the moon, that chiefly affects the time when the earth comes to time that the its apogee and perigee. When the moon is in conjunction to its apogee with the sun, the center of the earth is farther from the sun and perigee. than it otherwise would be; and when the moon is in opposition to the sun, the earth is about 3200 miles nearer the sun than it would be in its mean orbit; and thus, we periteive, that the longitude of the moon has a great influence in 228 A1S Tr 00SRONOMY, CHAP. VII. bringing the earth into, or preventing it from coming into, its perigee or apogee; but the perigee and apogee points,for the center of gravity, are quite uniform, agreeably to the views expressed in the first part of this article. These explanations will give a general insight into some of the apparent intricacies of physical astronomy. Stuall equa- The small equations of the sun's center are computed on ions of ththe principle explained by Fig. 48, the sun having a movun's center anxplained; tion round the center of gravity between itself and each of the planets. For example, the perturbation produced by Jupiter is greatest when Jupiter is in longitude 900 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 account, 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 separately. PRACTICAL ASTRONOMY.'29 S E CTION IV. PRACTICAL ASTRONOMY. PREPARATORY REMAR]So'WE have now done with general demonstrations, and with TRIx" Ininute 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 ]Lave a passable 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 comprehension of all we design to embrace in this section. To prompt the student in his knowledge of trigonometry, we give the following formulte: I. Relative to a single are or angle. 1. - sin. a = tan. a cos. a.* ltan. a 2. - - sna tan.a - vlftan.2a. 3. c - - coS. a -- jlV~tan.2a. 4. - - - cos. a=2 cos. 2- a — L sin. a 5.- tan. a= -sin. -1+cos. a 1- cos. a tan. cos. a~ 7. - - - sin. 2a=2 sin. a cos. a. * Radius is unity in all these equations 230 ASTRONOMY. Two. 8. 8 - - os. 2a=2 cos. 2a-1=1-2 sin. 2a. II. Relative to two arcs, a and b, of which a, is supposed to be the greater. 9. - sin. (a+b)=sin. a cos. b —-sin. b cos. a. 10. - cos. ('a+b)=cos. a cos. b-sin. a sin. b. 11. - sin. (a —b)=sin. a cos. b-sin. b cos. a. 12. - cos. (a-6)=cos. a cos. b-+sin. a sin. b. Sum of (9) and ( 11 ) gives 13, diff. gives 14. 13. - sin. (a+-b)sin. (a —b)=2 sin. a cos. b. 14. - sin. (a+b) —sin. (a-b)=2 cos. a sin. b. 15. - tan. (a+tb) tan. a- tan. 1-tan.a tan. b. tan. a tan. b 16. - tan. (a-b)tan. atan. -4-ltan. a tan. b sin. a-sin. b tan. - (a+b) 17. - sin. a-sin. b tan. I (a-b)' 18. - tan. a+tan. 6 sin. (a —b) tan. a-tan. b sin. (a-b)' 19 I tan. b - =tan. (45~0-4-b). 1-tan. b 1 —tan. 6tan. (b50) - =tan. (450 —b).! 1-{-tan. b We shall, probably, make an application of the following theorem; it applies to finding the unknown angles of a triangle, when the logarithms of two sides (not the sides themselves) and the angle included between the sides are given. The greater of two sides of a plane triangle is, to the less, as radius to the tangent of a certain angle. Take this angle from 450, and call the diference a. Lastly, radius is to the tangent, a, as the tangent of the half sum of the angles at the base is to the tangent of half their differenoe. III. Resolution of right-angled spherical triangles. In the following equations, h is the hypothenuse, s a given EQUATIONS. 231 isde, a. a given angle, and x the quantity sought. (The right TRIG. angle is unity, and always given.) Given, Required, Solution. h (side op. a 20. sin. x=sin. h sin. a. and side adj. a 21. tan. x= —tan. h cos. a.,a the other angle 22. cot. x=cos. h tan. a. h the other side 23. cos. x-=O. COS. s and ang. adj. s 24. cos. x=tan. s8 cot. a s 1sin. s ang. op. s 25. sin. x- -n, sin. h A 26. sin.x s. —-- and sin. a a the other side,. 27. sin. x=tan. s cot. a Qopposite, the other ang. 28. sin. x —S f R 29. cot. x=cos. a cot. s and I the other side, 30. tan. x=tan. a sin. s d the other ang. 31. cos. =-sin. a cos. s. adjacent,the other a. The h 32. cos. x-cos. s cos. other side two sides. 1 the angles, 33. cot. x=sin. adj. side Xcot. [opp. side. IV. Resolution of oblique angled spherical triangles. Let A B and C be the three angles of any spherical triangle, and ab 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 proporeional to the sines of the opposite sides. Therefore sin. sin. sin. sn. Therefore 3. = -_.= sin. a sin. -- sin.c Given the three sides a bc; Required one of the angles, A. 35. - - Sin. s A - in. (s-b) sin. (s —c) sin. b sin. c 16 232 ASTRONO MY. TsRIG. - - S 2 1 4= sin. S sin. (s-a) sin. b sin. c In 35 and 36, 2S=a — b+c CHAPTER I. AST RONOMICAL PROBLEMS o PROBLEM I. HAP. e, G6iven the qrght ascension and declination of any heavenly body,to find its latitude and longitude; or conversely, given the latitude and longitude of a body to find its corresponding right ascension and declination. A general Fig. 49, From any point as a center cpronnection for E (Fig. 49) describe a circle Q right ascen- EPx, &c. Let this circle sion, decli- represent the meridian, whicl nation, ]on- ~, gitude, and / passes through the pole of the latitude. ecliptic E, the pole of the c /P \1 j Q earth's axis P, and through the solstitial points St and W. W \ A / Then the point Aries (v ) will be at the center of the circle, P-p and k2 qp ~a and Q co q will be lines crossing each other by an angle equal to the obliquity of the ecliptic. Pp is the celestial meridian which passes through the equinoctial points, and is the first meridian of right ascension. E cT e is the first meridian of longitude, and, of course, the angle F mp P is equal to the obliquity of the ecliptic. The figure Let s be the position of any celestial body, and draw the trai consideparent, meridian of right ascension Psp; also draw the meridian of,and both longitude Es e draw also cT s. We have now two right-angled sides of it are represented. spherical triangles s D cr and qp Bs, having a common hypothenuse cm s; the first is the right ascension triangle, the PRACTICAL PROBLEMS. 233 Second is the longitude triangle. Let the student observe CHAP.I. that the line Q q represents a circle, the whole equator; and the point m represents, in fact, two points, the 0 degree of right ascension and the 180th degree. So the point s represents two points, and cr D is the right ascension from 0 degree, or from 180 degrees. In our figure, the point s is north of both ecliptic and equator; but it might have been between the two, or south of both; hence, to meet every case, the judgment of the operator must be called into exercise to perceive a general solution. Now, having the right ascension and declination of s, we find its latitude and longitude thus: In the triangle i Ds, mf D and Ds are given, and equation 32 gives cr s ( h); 33 gives the angle s mo D. From s p D subtract B cy D, the obliquity of the ecliptic, and there remains the angle s qc B,* With the angle s c B3 and the side cp s, equation 20 gives sB the latitude, and 21 gives rfnB the longitude. EXAMPLES. 1. The right ascension of a certain point in the heavens is 5 h. 7 m. 50 s., or in arc 760 57' 30"; and its declination is 260 11? 36" N.: Wrhat is the latitude and longitude of the same point? Four equa~ (32.) (33.) tions con tained in one PD 760 57' 30" cos. 9,353454 - - - sin. 9.988651 operation. sD) 26~ 11' 36" cos. 9.952952 - cot. 10.308104 mr s 78t 19' 3" cos. 9.306406 26047' 27"cot. 10.296755 B D - - 23 27 32 scpB - - - 3 19 55 - a * In general, take the difference between the angle s T)D and the obliquity of the ecliptic; and if the angle s T D) is the greater quantity, the body is north of the ecliptic, otherwise it is south of it. When the declination is south, the angle s T D must be added to the obliquity of the ecliptic in the first and second quadrants, and subtracted in the third and fourth. Hence the judgment of the operator must be called in to decide the particulars of the case; or he must bave a general formula that will give no exercise to the mind. T* 324 ASTRONOMY, CHAP. I. (20,) (21.) (h) 780 19' 3" sin. 9.990911 tan. 10.684611 (a).8 19 55 sin. 8.763965 cos. 9.999265 30 15' 36" sin. 8.754876 78 18 6 tan. 10.683876 Thus we determine that the longitude must be 78~ 18' 6'", and -the latitude 30 15' 36" N. 2. The longitude of the moon, at a certain time, according to computation, was 102~ 7'; and latitude 50 14' 15" S.: What was the corresponding right ascension and declination 9*:rom these (32.) (33.) examples we cpB 770 53' cos. 9.322019 sin, 9.990215 might foruml a sB 50 14' 15" cos. 9.998183 cot. 11.037780 general rule; but rulesthus Cp S 770 56' 12" cos. 9.320202 5~21' 27" cot. 11.027995 formed sel- B cr)D - - 23 27 42 dom reflect principles; 18 6 15 therefore for educational (20.) (21.) purposes, we (h) 776 56' 12" sin. 9.990302 tan. 10.670170 fall rark (a) 18 6 15 sin. 9.492400 cos. 9.977948 equations. 17 41 22 sin. 9.482702 770 19'41" tan. 10.648118 Thus we find that the right ascension distance on the equator, from the 180th degree, was 770 19' 41"; or its right ascension 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 170 51' 36" S.: what was its longitude and latitude? Ans. Lon. 2540 9' 14", Lat. 4~ 41' 12" N. Any nnm. In the following examples either right ascension and decliher of the nation may be taken may be taken for the data, and the longitude and latilike exam. loit ples can be tude the sought terms, or conversely; the longitude and found. latitude may be the given data, and the right ascension and * As the longitude is more than 90~ and less than 1800, the moon is in the second quadrant of right ascension, and 770 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. 235 declination the required terms. A Nautical Almanac will CaPr. I. furnish any number of similar examples. R. A. Dec. Lon. Lat. h. m. s. ~' t 0 i, O /, 4 15 47 36 15 58 15 south, 238 14 48 4 30 17 north, 5 613 22 18 23 2 north, 93 10 55 5 423 south, 6 11 24 44 1 45 28 north, 171 12 40 1 52 51 south, 7 20 23 33 14 11 9 south, 30447 15 5 2 23north. PROBLEM II. Given the latitude of the place, and the declination of the sun Tables for the semidia or star.; to find the semidiurnal arc, or the time the sun or star urnal arc and 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, ary computed where it will rise and set. lem. To illustrate this problem we draw Figure 50. Let P Zf These ex. amples do &c., represent the celes- Fig. 50. not take re tial meridian passing d fraction into through theplace. Make Q account. the are Q Z equal to the / latitude, then Z P will equal the co-latitude. The line HA is every-f /l h where 900 from Z. and represents the horizon. \ / /d Pp represents the earth's _ axis, and the meridian 900 distant from the me- p ridian of the place; Q N 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 o d', where this circle crosses the circle of the horizon H-h 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 represented by the same point in the projection ). Through P 0 p describe the meridian as in the figure, and the right-angled spherical triangle R O C appears; right angled at B. 236 ASTRONOMY. CAP, I:t In the triangle R Q C, there is given the side 1R Q, the declination, and the angle opposite R C (, which is equal to the co-latitude. R C, expressed in time, at the rate of 150 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 are C O 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 = — os; 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 latitude is the same as the sine of R C o, corresponding to a in the equation. EXAMPLE. The time In the latitude of 400 N., when the sun's declination is 200 in all these examples is, N., what time before and after six will it rise and set, and what of course, ap.- will be its amplitude? parent, because it re- (27.) (26.) fers directly 20~ tan. 9.561066 sin. 9.534052 to the sun, and not to a 40 tan. 9.923813 cos. 9.884254 clock. 170 47' sin. 9.484879 260~ 31' sin. 9.649798 Thus we find that the arc called the ascensional deiference, is 170 47', or, in time, lh. 11m. 8s., showing that the sun or heavenly body, whatever it may be (when not affected by parallax or refraction ), will be found in the horizon 7h. lim. 8s. before and after it comes to the meridian. Its amplitude for that latitude and declination is 260 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 Lat. 510 32' N., the sun's amplitude was observed to be 390 48' toward the north; what was its declination, and what was the apparent time of its rising and setting? Ans. Sun's declination, 230 27' 59" N. Sun's rising, 3h. 47m. 32s.; sun's setting, 8h. 12m. 28s. PRACTICAL PROBLEMS. 237 The amplitude of the sun is frequently observed, at sea, to CHAP. I discover the variation of the compass; but, by reason of re- Refraction fraction, the results are not perfectly accurate. not taken inFrom the right-angled spherical triangle (Fig. 50) P Z A(, to account, if it were, the 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 would remain The angle Z is a right angle, P Z is the co-latitude, and above the PO is the co-declination. horizon Equation (23) gives the cosine of Z (, or the sine of the would be inaltitude of the sun when it is east or west- the latitude and while it rose declination being given — and equation (24) will give the in altitude 33' of are. angle or time from 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 IV: C S, the given declination, will be its hypothenuse and P Ch, the latitude, will be the are of its angles. By means of right-angled spherical trigonometry, as comprised in the equations from 20 to 33, we can resolve all possible problems that can occur in astronomy, pertaining to the sphere; but, for the sake of brevity, mathematicians, in some cases, use oblique-angled spherical trigonometry, which is nothing more than right-angled trigonometry combined and condensed. PROBLEM 1II. Given, the latitude of the place of observation, the sun's de- The sun's cuination, and its altitude above the horizon, to find its meridian distance from the me.distance, or the time from apparent noon. ridian, as There is no problem more important in astronomy than measured from the pole that of time. No astronomer puts implicit faith in any chro- asr center, nometer or clock, however good and faithful it may have and on the been; and even to suppose that a chronometer runs true, it equator 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 time from apmethod, directly or indirectly, of finding the sun's distance parent noon. from the meridian. When the center of the sun is on any meridian, it is then and there apparent noon; and the equation of time will be the 238 ASTRONOMY CHAP. I. interval to orfrom mean noon; but none, save an astronomer Great im- in an observatory, can define the instant when the sun is on portance of the meridian; no one else has a meridian line sufficiently defithis problem. 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 is required ), we must resort to observations on the sun's altitude. Direct me- It is evident that the altitude of the sun is greater and vations nobser. greater from sunrise to noon, and from noon to sunset the algenerally ac- titude is continually becoming less. If we could determine, curate. by observation, exactly when -the sun had the greatest altitude, 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 observations are not the most proper to determine the time. Proper times From two to four hours before and after noon (depending of observation. 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 Fig. 51. time and altitude have a scientific connection, ad which can only be disenaQ, \ / I Hap tangled by spherical triDescription gonometry. But we of the figure. proceed to the problem. Draw a circle, P Z Q H, &c., (Fig. 51), representing the merid dian; Z is the zenith, and Z V is the prime vertical; Hh is the ho$ \ I Aq rizon; Z Q is an are equal to the given latiN tude; Q q is the equator, and, at right angles to it, we have the earth's axis, P S. PRACTICAL PROBLEMS. 239 Take Ha, ih a, equal to the observed altitude of the sun, CHAP. L and draw the small circle, a a, parallel to the horizon, HAh. From the equator take Qd, q d, equal to the declination of the sun, and draw the small circle, d d, parallel to Q q. Where these two small circles, aa, d d, intersect, is the position of the sun at the time. From Z draw the vertical, Z ( V, and from P draw the meridian through the sun, P 0 S. The triangle P Z 0 has all its sides given, from which the angle Z P O can be computed; which angle, changed into time at the rate of 150 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 O is the complement of the altitude, P ( The obser. is the complement of the declination, and P Z is the comple- nesion nde ment of the latitude, and equation (35) or (36) will solve points out a the problem; that is, find the angle P which can be made triangle. to correspond to A, in the equation. But, in place of using the complement of the latitude, we may use the latitude itself; 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 circumstance, 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 ticiansmake 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 practial opebe content with the following practical rule, to find the time rations. of day, from the observed altitude of the sun, together with. its polar distance, and the latitude of the observer. RULE 1.-Add together the altitude, latitude, and polar dis- Practical tance, and divide the sum by two. From this half sum subtract rule used at sea. the altitude, reserving the 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 the hal f sum, and the sine of the remainder. Add 240 A STRO NO MY. CHAP. I. these four loyariltzms together, and divide the sum by two; the result is the loyarithmetic 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 altirtant andsex-% 9tudes at sea, or wherever the observer tant and re- may happen to be, is a quadrant or fleeting cirt- sextant, according to the number of allythe same degrees of the arc. It is made on the instrument. 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 aniother mirror, one half silvered, the other half transparent, called the horizon glass. Directly opposite 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 instrument. To be in adjustment, the two mirrors, namely the index 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 the index mirror per.pendicular to the plane of the instrument? Put the index in about the middle of the arch, and look into the index mirror, and you will see part of the arch reflected, and the same part direct; and if the arch appears perfect, the mirror is in adjustment; but if the arch appears broken, the mirror is not in adjustment, and must be put so by a screw behind it, adapted to this purpose. 2. Are the mirrors parallel when the index is at O? 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. 241 EXAMPLE. CHAP. I. In latitude 390 46' north, when the sun's declination was 30 27' north, the altitude of the sun's center, corrected for refraction, index error, &c., was 320 20', rising; what was the apparent time? Altitude, 32 20 Latitude, 39 46 - cos. comple. -.114268 Polar dis., 86 33 - sine comple. -.000788 2)158 39 79 19 30 cosine - 9.267652 32 20 46 59 30 sine - 9.864090 2)19.246798' ZP O 24 50 30 sine 9.623399 2 The hourly angle is 49 41 0, which, converted into time, gives 3 h. 18 m. 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 horizon glass perpendicular to the plane of the instrumenlt? 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, unbroken 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 16 U 242 ASTRONOMY. CHAP. I. the sun was rising, it was before noon, and the apparent time was 8 h. 41m. 16s. An arc may A good observer, with a good instrument, in favorable cirbe measured by the quad- cumstances, can define the time, from the sun's altitude, to rant within within three or four seconds. one minute. An artificial At sea, the observer brings the reflected image of the sun horizon. 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 horizon; and, therefore, if we measure, by the instrument, the angle between the object and its image in the artificial horizon, 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: example. 1" On the 18th of May, 1848, at sea, in latitude 360 21' north, longitude 540 10' west, by account, at 7 h. 43 m. per watch; the altitude of the sun's lower limb was 320 51' rising; the bight of the eye was eighteen feet, and the index directions, it may be considered as in an approximate adjustment-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 index error, additive, if the greatest measure is off the arch, subtractive, if on the arch. PRACTICAL PROBLEMS. 243 error of the sextant was 2' 12" additive. What was the er- CHAP. l, ror of the watch?" PREPARATION. Time, per watch, 7 h. 43 m., morning. Preparations Longitude, 540~ 10', in time, - 3 38 to be made according to Estimated mean time at Greenwich, 11 h. 21 m. circum. The declination of the sun at mean noon, Greenwich time, stances. 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 190 38' 8" north, and the polar distance 70~ 21' 52". Observed altitude, - - - 320 51' 00" Index error, - - - - ~ + 2 12 Semidiameter, -.- - + 15 49 Refraction, -.- - - 1 28 Dip of the horizon, - - - 4 13 True altitude of sun's center, b - 330 3' 20" Altitude, 330 3' 20" Latitude, 36 21 cos. complement,.093982 Polar dis., 70 21 52 sin. complement,.026013 2)139 46 12 69 53 6 cosine, - - 9.536470 33 3 20 36 49 46 sine, - - 9.777770 2)19.434235 l hourly angle, 31 25 30 sine, - - 9.717117 This angle corresponds to 4 h. 11 m. 24 s., or, in reference to the forenoon, 7 h. 48 m. 36 s. apparent time. On the 18th of May, noon, Greenwich time, the equation PY ohser, vations thus of time was 3 m. 54 s. subtractive; therefore, the true mean taken at dif. time, at ship, was - - 7 h. 44 m. 42 s. ferent times Time, per watch, 7 43 at the same place, the Watch slow, 1 42 rate of the watch can be A short time before this observation was taken, the watch determined. 244 ASTRONOMY, CHAP. T. Was 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. s, 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 45s=530 56' west, how to de- The longitude is west, because it is later in the day, at isefrom thie Greenwich, than at the ship. This example explains all the observations whether the details of finding the longitude by a chronometer. longitude is By taking advantage of the observations for time on shores east or west. Howto de. we may draw a meridian line with considerable exactness; termine and for instance, in the last observation (if it had been on land), meridian in 4 h. 11 m. 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 accuracy, the direction from any point toward the sun's center, at the end of that interval, would be a meridian line. Several such meridians, drawn from the same point, from time to time, and the mean of them taken, will give as true a meridian 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 findiade oa. ing the difference of longitude between two places always resolves 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. 245 rent hours and minutes of time; but such difference would be cnP, L occasioned entirely by difference of longitude: the event is the same for all places - it is a 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 means of rence in local time will give the difference in longitude. But events. a perfect clock is a noter of events, it marks the event noter ois 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 it runs true, but not otnermarker of events9 nothing more would be necessary to give us wise, 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 cannot 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 are an event, but it occurs so seldom as to amount to no practical events, value. The eclipses of Jupiter's satellites are such events, ahsolnte but they cannot be observed without a telescope of consider- time, but for common purA able power, and a large telescope cannot be used at sea. poses they Hence these events are serviceable to the local astronomer are of little value. only; the sailor and the practical traveler can be little benefited by them. The moon has comparatively a rapid motion among the stars ( about 130 in a day), and when it comes to any' definite distance to or from any particular star, that circumstance 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 300 from a particular The motion star, that event must correspond to some instant of absolute mong the time; and if we are sufficiently acquainted with the moon, stars, maybe and its motion, so as to know exactly how far it will be from considered as an index certain definite points (stars) at the times, when it is noon, moving 3, 6, 9, &c., hours at Greenwich, then, if we observe these roundacircle marking abe events from any other meridian, we thereby know the Green- solute time, wich time, and, of course, our longitude, Finding the Greenwich time by means of the moon's angular distance from the sun or stars, is called taking a lunar; U* 246 ASTRONO MY. CHAP. 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. Lunar ob- In a sea journal of 1823, it is stated that the distance of servations il- Castrated by the moon from the star Antares was found to be 66~0 37' 8", an example. when the observation was properly reduced to the center of the earih, and the time of observation at ship was September 16th, at 7h. 24m. 44s. r. M. apparent time. By comparing this with the Nautical Almanac, it was found that at 9 P. mi., apparent time at Greenwich, the distance between the moon and Antares was 660 5' 2", and at midnight it was 670 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 660 37' 8" falls between the distances in the Nautical Almanac; and thus an observer, with a good instrument, can at any moment determine 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 computations may be adapted. Observed The angular distances from the moon to the sun, stars, itancesis and planets, as put down in the Nautical Almanac, corre-.ances as sponding to every third hour, are distances as seen from the rFen from center of the earth, and when observations are taken on the the center of,e eartl. surface the distance is a little different; the position of the moon is affected by parallax and refraction, the sun or star Clearing the by refraction alone; and therefore a reduction is necessary, dis'ance. which is called dearing the distance. This is done by spheri PROPORTIONAL LOGARITHMS. 247 tal trigonometry. The distance between the moon and star CHAP, 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 coaltitudes, 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 mathematicians, 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- Proportional portional problems to be resolved, and as the terms are logarithms — an explanamostly incommensurable, it would be very tedious, in most tion of the cases, to proceed arithmetically, we therefore resort to loga- construction of the table rithms, and to a prepared scale of logarithms, very appropri- given. *ately called preoorional 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 construction 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:; Or 3600: 13 23:: 33 17:. Divide the first and second terms of this proportion by the 17 2458 ASTRONOMY. CHAP.. 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 3600 3600 13.23' 33.17 Multiplying extremes and means, using logarithms, and remembering that the addition of logarithms performs multiplication, 3600 /3600 \ / 3600 Then we have log. log. (log. (36,) By the construction of the table, the proportional logarithm 3600 of 1" is the common logarithm of 1; of 2" is the com3600 3600 3600 mon logarithm of 2 of 3" is,&c., to hence the proportional logarithm of 3600 is 0. We now work the problem: I g! 13 23 - - - P. L. 6516 33 17 - - - P. L. 2559 Result, - 7 252 - - - P. L. 9075 Examples EXAMPLES FOR PRACTICE. given to illustrate the 1. When the sun's hourly motion in longitude is 2' 29', practical uti- what is its change of longitude in 37 m. 12 s.? Iity ofpropor-. tional logar- Ans. 1 32"'5. ithms.. 2. When the moon's declination 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 mi. 19 s.? Ans. 3' 21". 4. When the moon changes her right ascension I m. 58 sb in one hour, how much will it change in 13 m. 7 s.? Ans. 25".8. PROPORTIONAL LOGARITHMS 249 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. E XAMPLiS. 1. If the sun's declination changes 16' 33" in twenty-four Examples hours, what will be the change in 14 h 18 m.? gvlustrate til Statement, 24: 14.18: 16' 33" practical uti 24 lity ofpropor. Or, 12: 7.09 tional logar Or, 60: 35.45:: 16' 33t ithms. 16' 33" P. L. 5594 35' 45" P. L. 2249 Ans. 9' 51".5 P. L. 7843 2. If the moon changes her declination 10 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 terim, 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": Other et amplee. 30" P. L. 2.0792 60 m. P. L. 0.0000 Product of extremes, 2.0792 26' 30" P. L. sub. 3549 Result, 1 m. 7. P. L. 1.7243 38: 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. x., in longitude 740 west, on the same day? Ans. 6 m. 29- s 250 ASTRONOMY. CHAPTER II. GENERAL PROBLEM. CHAP. 11. Given, the motions of sun and moon, to determine their appaA general rent positions at any given time; from which results their appa. problem pre- rent relative situations, and the eclipses of the sun and moon. thea computa This problem covers two-thirds of the science of astronomy, tionsofeclip- and includes many minor problems; therefore a brief or hasty BeS. 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 particulars 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 instances; but some of them require special mention. All the tables are adapted to mean time at Greenwich. EXPLANATION OF TABLES. A very ge. Table IV contains the sun's mean longitude, the longineral and tude of its perigee (each diminished by 2~), and the Argua comprehensive explana- ments* for some of the small inequalities of the sun's appation of the rent motion. tables. Explanation * The term, ARGUMENT, in astronomy, means nothing more than a of the term correspondence in quantities. Thus, each and every degree of the argument. sun's longitude corresponds with a particular amount of declination; and therefore, a table could be formed for the declination, and the argument 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 semidiameter; hence, a table can be made for finding the semidiametere and the arguments would be the horizontal parallax. But the hori EXPLANATION OF TABLES, 251 Argument I, corresponds to the action of the moon; Ar- CHAP,. IE gument II, to the action of Jupiter; Argument III, to Venus; 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 solar or lunar tables. The object of diminishing the mean longitude and perigee Explanation of the sun by 20, is to render the equation of the center al- of the solar tables. ways additive; for if 20 are taken from the longitude, and 20 added to the equation of the center, the combination of the two quantities will be the same as before; and, as the equation of the center is always less than 20, therefore, 20 added to its greatest minus value, will give a positive result. By the same artifice all equations may be rendered always positive. The 20, taken from the mean longitude, are restored by adding 10 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.10 59' 3" from the quantity found in the table. Table XII, shows the time of the mean new moon, &c., in January, diminished by fifteen hours, to render the corrections always additive. The fifteen hours are restored by adding 4 h. 20 m. to the first equation, 10 h. 10 m. 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 difference 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 correspond to half a circle, and so on in proportion. This mode of dividing the circle has been adopted, with certain limitations, to avoid the greater labor of computing by denominate numbers. 252 ASTRONOMY. CHAP. II. orbit; Argument II, corrects for the mean eccentricity of the lunar orbit; Argument III, corrects for the different combinations of the solar and lunar perigee; and Argument IV, corrects 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 desynodicalthe pended upon within four minutes, and commonly much nearer; motionofthe but when great accuracy is required, the more circuitous and moon, by elaborate method of computing the longitudes of both sun which new and full and moon must be employed. moonscanbe Tables XIII, XIV, and XV, are used in connection with computed. Table XII. Explaftation Table XVI, shows the reduction of the latitude, and also of of the lunar the moon's horizontal parallax, corresponding to the latitude, table. occasioned by the peculiar shape of the earth, and the diminution 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, evection, &c., necessary in computing the moon's true place in the heavens. The method The argument for evection is diminished by 29'; the anoof cosmputing maly by 10 59', the variation by 80 59', and the longitude gitude of the by 90 44', and the balances are restored by adding the same sun, amounts to the various equations, which, at the same time, renders the equation affirmative, as explained in the solar tables. The arguments in Table xxxII,are also arguments for polar distance, or latitude, in Table xxvIII. 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 intermediate minute or second, by means of the hourly motions. PRACTICAL PROBLEMS. 253 CBAP, II. PROBLEM I. From the solar tables find the sun's longitude, hourly motion in longitude, declination, semidiameter and equation of time; rand for a specific example, find -these elements corresponding to mean time, at Greenwzch, 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 E's Ion. (650 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 0 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 corresponding 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 sun's bers, rejecting entire circles in every column, and the sums or distances from its perisurplus, as the case may be, will give the mean values of all gee point is the quantities for the given instant. called its mean anit Subtract the longitude of the perigee from the mean Ion- maly. 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 equations is beo cause it is sometimes negative. 254 ASTRONOMY. cHAP. b. noxes from Table X, and apply it according to its sign, and the result will be the true longitude from the true equinox. M. Lon. Lon. Perig. I. TI. III. N 1854 9 84848 9 82529 073 998 902 809 May 3281640 20 59 301 206 18 26 d 24 38 28 4 844 63 43 4 8h 19'43 11 o O 0 40m 1 39 987 362 151 831 2 2 518 9 82553 Eq. of center 3 642 2 2 518 I 10 4 23 39 25 = Mean anomaly. [I 13 iII' 8 2 5 12 31 Eq. of the equinox —16 Sun's hourly motion in Ion. 2' 24" True ion. 2 5 12 15 " semidiameter, 15' 49' These prin- To find the equation of time to great accuracy. ciples were explained on By equation 21, page 231, we find o pages 94 the sun's R. A., - - - 63 16 10 and 95. Subtract this from the sun's ion., - 65 12 15 Equatorial point is west of mean eastward motion by - - - 10 56' 5" (at From the equation of the center, as just found, - - 3 6 42 Subtract the constant of the table, - 1 59 30 The sun east of its mean place, - 1 7 12 (b) Subtract (6) from (a) because one is east, the other west, and we have the are - - 48' 53" This are, converted into time, gives 3 m. 15.5s. for the equation of time at this instant, and the sun will not come to the meridian at mean noon, but 3 m. 15- s. afterward, Hence, to convert mean into apparent time, in the month of May, add the equation of time. PRACTICAL PROBLEMS. 255 Thus, in general, we can determine the exact amount of CHAP. I[. 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 ascension, hourly motion in longitude, semidiameter of the sun, and equation of time corresponding to 20 minutes past 9, mean time at Albany, N. Y., on the 17th of July, 1860? N. B. At this time the sun will be eclipsed. Ans. Lon. 2140 38' 21"; Dec. 210 12' 48". R. A., in time, 7h. 46m. 15s.; Eq. of time to add to apparent time, 5m. 46.2s.; hourly motion in lon., 2' 23"; S.-D., 15' 45.6". PROBLEM II. From Tables X], XI,; and XIlI; 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 many 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 corresponding 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 the tion. Sometimes one more lunation than the number of the number of I. nations ne. month after January, will be required to bring the time to cessary to the required month, as it occasionally happens that two luna- bring the result to the retions occur in the same month. quired timeApply 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. 256 ASTRONOMY. CHAP. If. For the time at any other meridian apply the time corresponding 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 Cincinnati, Ohio. January. Mean N. Moon. I. II. Ill. IV.] N. 1854, 27d. 18h. 14m. 0761 1168 19 04 668 Four Luna. 118 2 56 3234 2869 61 961 341 145. 21 10 3995 4037 80 00 1009 Table XIII.120. May, 25 21 10 I.May, 25 21 146 N shows an eclipse of the III. 4 64 sun-visible in the United III. 41 States. III. 1 17 IV. 20 May, 26 8 47 New M mean time at Greenwich, - 8 h. 47 m., P. M. Boston, Longitude, - - - 4 44 New ~ Boston time, - - - 4 3 Cincinnati, Longitude from Boston, 53 New C Cincinnati time, - 3 10 2. Required the approximate time of full moon, in July, 1852, for,the meridian of Greenwich, and for Albany time, New York. January. Mean N. Moon. I. II. III.'IV. N.1 1852, 20d. 11h. 53m. 0549 3239 38 27 538 Five Luna. 147 15 40 4042 3586 76 95 426 Half Luna. 14 18 22 404 5359 58 50 43 1182 21 55 4995 2184 72 72 007 Tab. 13. Bis. 182 Tb.1.3. Bis. 182 The column N shows that July, 0 21 55 the moon is very near her I. 4 21 node. There will be a total II. 42 eclipse of the moon-invisiIII. 17 ble in the United States. IV. 10 July, 1 3 25 Mean time at Greenwich. ECLIPSES. 257 Full O Greenwich time, - 3 h. 25 m. r. M. CHAP. II. Albany, Longitude, - - 4 55 Full ( Albany time, - - - 10 30 A. M. Thus we can compute the 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 eclip. the sun at new moon, and an eclipse of the moon at full sce take place. moon; but eclipses do not happen at every new and full moon; and the reason of this must be most clearly comprehended 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 eclip. ses do not the moon's motion was along the ecliptic, there would be an take place 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 celestial arch does not coincide with the sun's path, the ecliptic; but is inclined to it by an angle whose average value is 50 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 be essential for more and 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 eclipses. 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 saime months of the year. For instance, if the longitude of one node was 300, the other would be in longi- Why an tude 30+180, or 2100; and, as the sun is at the first of eclipse should take these points about the 20th of April, and at the second about place in any the 20th of October, the moon could not pass the sun in particular montl. these months without coming very nearly in range with it, of course, producing eclipses in April and October. 17 v* 258 ASTRONO MY. aCHMP. i. 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 G l _, crossing the equator at two opposite points, represents the ecliptic; and the curved line F O) g2 represents the path of the moon crossing the ecliptic at the points 6 and 2; the first of these points is the descending, the other, the ascending node. As here represented, the ascending node is in longitude about 2100, 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 figure The sun and moon at conrepresents junction are represented in the at path s oe figure a little after the sun the sun and has passed the northern tropic, moonthrough which must be about the first of August; and it is perfectly evident that no eclipse can then take place, the moon running past the sun, at a distance of aboutfive degrees south; and at the opposite longitude, the moon must pass about five degrees north. The moon's nodes move backward at the mean rate of 190 19' per year; but the sun moves ECLIPSES. 259 over 100 in about twenty days; therefore, the eclipses, on CHAP. AL 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 longitude 216~; in eight years, at the rate of 190 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; therefore, 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 means ing of the coby the preceding examples, the columns marked N, not hith- lumns N, in erto used, indicate the distance of the sun and moon from the tables 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 certain space be at 500; and when the moon changes within 37 of 0, or along the 500, that is, 37 of either node, there must be an eclipse of ecliptic. 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 distance is greater than 25, and less than 35, the case is doubtful; but, like the limits to the new moon, the 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, eclipses of the sun ansd the lmits 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 260 A STR O NAMY. CHAP, I. sun 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, ini May, 1854, know that an eclipse of the it shows that the moon will then have passed the ascending sun will hap, node, and be north of the ecliptic, and the eclipse must be pen on the t 26th of May, visible on the northern portions of the earth, and not on the 1854, and southern. from what When the moon changes in south latitude, which will be circunistance we learn that shown by N being a little more than 500, or a little less than it will be an 1000, the corresponding eclipse, if of the sun, will' be visible eclipse to some north. on some southern portion of the earth, and not visible in the emr portionofnorthern portion; and if of the moon, the moon will run the earth through the southern portion of the earth's shadow. TableBp.31,shows the moon's latitude, approximately corWhat indti responding to the column N; or N is the argument for the Oates that a latitude, and the heading of the argument columns will solar eclipse will be visi. Show whether the moon is ascending to the northward, or deble on some scending to the southward. southern por- The tables from XVI to XVIII, together with the solar tion *of the earth. 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 principles, and give primary exercises to the student in calculating eclipses; but he who aspires to be an astronomer, must continue the subject, and compute from the lunar tables, farther on. The times, and the intervals of time, between eclipses, depend 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 s. Neglecting the seconds, and conceiving the sun, moon, and node to be together at any point of time, and after an unknown interval of time5 which we represent by P, suppose them together again. Then 2_- 444 represents the 29 12' 44 utth number of returns of the lunation to the node m the time cHAP. I. P The motione P, and'the expression 346 14 52 represents the number of of the sun and moon in returns of the sun to the node in the same time. Each re- relation to turn of either body to the node is unity; therefore, these ex- moon's node pressions are to each other as two whole numbers; say as m investigated P P to n; that is, 29 12 44 346 142 n; Tol m Orn m (29 12 44) (346 14 52)' Or, (346 14 52)n —— (29 12 44)m - - - (a) n 29 12 44 Or, Om 346 14 52' Reducing to minutes, and dividing numerator and denomin 10631 nator by 4, we have - 12478' As this last fraction is irm 124788 reducible, and as m and n must be whole numbers to answer' the, assumed condition, therefore, the smallest whole number for m is 1247833 and for n is 10631; that is, as we see by equation (a), the sun, moon, and node will not be exactly together 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 degree, through a period of so long a duration. This period, however, contemplates an exact return to the This period contemplate same positions of the sun, moon, and earth, so that a line practicmlaite. 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 Eactcoi to former position should be attained; a greater or less cidencesnev. approximation to former circumstances will produce a greater er happen, or less approximation to a former eclipse: but exact coincidences, in all particulars, can never take place, however long the period. To determine the time when a return of eclipses may hap ASTRONO MY. CHAP,.. pen (particularly if we reckon from the most favorable posie How to tions - that is, commence with the supposition that the sun, find the sue- moon, and node are together), it is sufficient to find the first cessiie return of 10631 eclipses, approximate values of the fraction -124783. If we find the'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 1 3 4 19 156 11 12 35 47 223 1831 A series of These fractions show that 11 lunations from the time an fractions the eclipse occurs, we may look for another; but if not at 11, it periods at will be at 12, and it may be at both 11 and 12 lunations; wheclipsesch and at five or six lunations, we shall find eclipses at the other eclipses occur. 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 32' 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 100 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. 31 m., taminationof and 4 revolutions of the sun to the node include 1386 d. the periodicalreturn of 11 h. 29 m.; the difference is 1 day 11 h. 29 m.; but in this eclipses. 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 Arithmnetic. ECLIPSES, 263 or.3 years and nearly 11 months afterward, would take place CHAP. II. 1o 32' short of the same node; and if it were the ascending node, the moon's latitude would be about 5' 40" south, and, if the descending node, about 5' 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, The Chal-" showing a difference of only a fraction of a day; and if the dean astronsun and moon were at the node, in the first place, they would omers called be only about 20' from the node, at the expiration of this this period Saros. 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 peand - enabled them to give general and indefinite predictions aiod we can make a sumof the edlipses that were to happen; and by it any learner, mary predichowever crude his imath6matical knowledge, can designate the tion of day on which an eclipse will occur from simply knowing the eclipses 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 A summary lunar eclipses, we wish to find the time when some future putiodeng othe 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 an eclipse must occur. 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 18 264 ASTRONOMY. COHP. I. June-therefore, on the first of July 1852, or at the tfme 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 gradually wear the eclipse away, and another eclipse will as gradually 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' for the comn must have the following data, commonly called elements of pnutation of lunar the ec-lipse: eclipses. 1.- The 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 la;titude. 5. Mooan's, hourly motiont 6. Moon's semidiameter. 7. The semidiameter of the moon and earth's shadow. seneraa di To find these elements, the approximate time of full moon rentions to and obtainthe el- is found from Table XI, and the tables immediately con-w ements of nected. For the time thus found, compute the longitude of eclipses the sun from Table IV, and the tables immediately connected, 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 following out the computation by a strict inspection of the examples we have given (rules, aside from the examples, would be of no avail); and, if the longitude of the moon is exactly 1800 in advance of the sun,. k is then just the time of full; moon; if not 1800, it is- not full mo-on; if more than 1800, it, is past full moon. ECLIPSES. 265 It will rarely, if ever, happen that the longitude of the cHAP. I. -noon 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 approximate and true time of full moon. To find the semidiameter of the earth's shadow, where the Rule to find the semidia, moon runs through it, we have the following rule meter of the To the moon's horizontal parallax, add the sun's, and, from earth's shal the sum, subtract the sun's semidiameter. This rule requires demonstration. Let S (Fig. 53) be Fig. 53. p the center of the sun, E 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 triangle p E P. By inspecting the figure, we perceive that the three Demonstra angles: tion of the $ Ep+pEP+m EP 180. rule. Also, the three angles of the triangle, PEp, are, together, equal to 1800; Therefore, SEp+p E P+m EP== P+p+p EP; Drop the angle, p EP, from both members of the equation, and transpose the angle SEp, we then have m EP' P+p —SEp. R* obinson's Algebra-problem of the couriers, 266 ASTRONOMY. CHAP. II. But the angle, mnEp, is the semidiameter of the earthlshadow at the distance of the moon; SEp is the semidiameter of the sun; P, that is, the angle E7Pp, is the moon's horizontal parallax; andp is the horizontal parallax of the sun; therefore, the equation is the rule just given.* What is The angle of the moon's visible path with the ecliptic is almeant by the angle of the ways greater than its real path with the ecliptic, and depends, moon's visi- in some measure, on the relative motions of the sun and ble path with the ecliptic, moon. 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, mAgB, Fig. 54. m is the angle of the moon's real path with the ecliptic. Conceive 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 6 in the same time; join b and m, and the angle m 6 B is the angle of the moon's visible path with the ecliptic, which is greater than the angle mA B; 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 one-sixtieth parts but we emphatically deny the propriety of the direction. ECLIPSES. 267 CHAPTER III. PREPARATION FOR THE COMPUTATION OF EICLIPSES. WE shall now go through the computation in full, that it CHAP. III. may serve for an example to guide the student in computing Computaother eclipses. tion of a lunar eclipse. Mean N. Moon. I. II. III. IV. N. The approx1851, id. 14h. 21m. 0038 3916 40 39 431 imfalltimeof Six Luna. 177 4 24 4851 4303 92 95 511 computed. Half Luna. 14 18 22 404 5359 58 50 43 193 13 7 5293 3578 90 84 985 181 As N is within 25 of 1000, July, 12 13 7 or O, there must be an eclipse. 1. 3 35 The sun is 15 short of the asII. 2 9 cending node, and the moon at full, being opposite, must be 15 III* 14 short of the descending node, IV. 11 and therefore, in north latitude, -Full 12 19 16 sending. We now compute the sun's longitude, hourly motion, and Sun's ion semidiameter for 1851, July 12, 19 h. 15 m. mean Greenwich gitude corn puted, corretime, as follows: sponding to the approxiO M1 Lon. Lon. Peri. I. II. III. N. mate time of ___8.__________ O ______________________, - -. a/ _ |full moon. st' S. 0 / /. 1851 9 83239 9 8 2224 958 250 025 648 July 52824 8 31 129 454 310 27 12 d 10 50 32 2 371 28 19 2 19 h 46 49 - 27 0 01 0 I- I 15m 0 37 1485 732 151 677 3183445 9 82257 Eq. of center 13938 3183445 I. 10. 1 6 10 11 48 = Mean anomaly. II. 18 III. 20 3 20 15 11 O's hourly motion, 2' 23" Eq. of equinox -16's semidiameter, 15' 46" Eqon. of equinox - 1655 0 Ion. 3 20 14 55 268 ASTRONOMY. CH&AP. IIT. We now compute the moon's longitude, latitude, semidiDirection ameter, horizontal parallax, and hourly motions for the same, for comput- mean Greenwich time, as follows: ing the moon's true longitude. FOtR 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 correction 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 tho center from Table XXV, and write it under the sum of the preceding equations, and find the sum of all, thusfar. 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 equation for variation from Table XXVI. 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 her orbit, reckoned from the mean equinox. Equation 6. Add the orbit longitude to the supplement of the node, of the equime and the sum is the argument of reduction to the ecliptic; it times called is also the first argument for polar distance. nutation in With the argument of reduction take out the reduction longitude. from Table XXVII, and add it to the longitude. ECLIPSES. 269 With argument 19, which is the same as N in the solar ta- CHAP nl. Nles, take- out the equation of the equinox, and apply it ac- - cording to its sign; the result will be the moon's true longitude reckoned on the ecliptic from the true equinox. FOR THE LATITUDE. Add the same correction ( to its nearest minute) to column General diII, as was added to the column of longitude, and add its rections for value, expressed in the 1000th part of a circle, to all the fol- fimndioongs latihe lowing columns, except column X. Add up these columns, tude. 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 longitude 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 ediptic, and, of course, will be in north latitude if the sun 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 0 sign, than 6 signs, it is tending north. For the equatorial horizontal parallax.- The arguments for Equatorial:Eveetion, Anomaly, and Variation are also arguments for parallax and semidiamehorizontal parallax, and with these arguments take out the ter depend corresponding equations from the tables adapted to this upon each other. purpose. For the semidiameter.- The equatorial parallax is the argument for semidiameter, Table XXXIV. For the hourly motion in longitude. —Arguments 2, 3, 4, and General di5 of longitude sensibly affect the moon's motion; they are, finding tfor therefore, arguments for hourly motion, Table 36 (the units hourly moand tens in the arguments are rejected). Take out these tion of the equations from table, also take out the equation corresponding to the argument of evection, Table XXXVII. With the we 270 ASTRONOMY. CHAP IrI. sum of the preceding equations, at the top, and 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 preceding 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- For the hourly motion in latitude. —With the 1st and 2d portion the arguments of latitude, take out the corresponding quantities first term is from Tables XLIII, and XLIV, and find their algebraic sum, the mean mo. tion of the noting the sign; call the result 1. moon. Then make the following proportion: L I 32' 56" ~:.. ~ 32' 56"; 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: 3600: 13~ 46':: 1000: x. The proportion resolved gives x = the number added to the several columns. But to avoid the formality of resolving a proportion for every example, we give the following skeleton of a table that ECLIPSES. 271 may be filled out to any extent to suit the convenience and CHAP. III, taste of the operator. Degrees decimal parts Degrees = parts. 0 0 1 5.003 5 24.015 1 26 =.004 7 12 -.020 1 48 =.205 9 0 =.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 correction for longitude, in a particular example is, 110 31' 25"; what is the corresponding decimal or numeral part? Thus 9~ =.030 2 31 -- 7 11 31 =.037 We now continue the examples, hoping to follow these precepts. ..a _ 1 2 1 3 1 4 5 1 6 1 7 18 1 9 101111 12 11314 15 116117 18I 19120 1851. 000519167 264415695 3467 4239 8539 3477 3885 306 958 918 570 487 493 960 201 569 648 038 July, 4955 76291827311942 0732 7341 0444 064314396 698 634 754 185 948 525 625 613 699 27 83 D. 12, 301 7149 1442 3157 3691 4093 635 4293 267 772 342 775 376 91 336 403 463 286 2 5 h. 19, 22 515 823 227 266 295 46 309 19 *56 25 56 27 78 24 29 33 21 0 0 m. 15, 0 7 11 3 3 4 1 4 0 1 0 1 0 1 0 0 0 0 0 0 65283 4467 3193|1024 81.597966- 8726I8567 833 959 504 158 605 378 017 io310 575 677 126 Evection. Anomaly; Variation. Longitude. Sup. of N, II. V. VI. VII.IVIII IX. X. S.. S S. 0 S. ( o 1851. 6 24 41 3514 3 1 38 11 6 6 36 8- 15 54 25 7 23 15 51 6 1 45 6358 359 504 506 g282 816 July, 8 8 17 16J6 24 45 48 1 16 31 32 7 14 55 40 9 35 5 7 8 32 156 147 112 103 4861 962 h. 19, 8 57 32 10 20 35 9 39 3 10 25 53 2 31 8 50 27 31 22 27 28 4 m. 15, 7 4 8 10 7 37 8 14 2 7 0 0 0 0 0 0 Sum of eq. J 31 341 1 2 57 13 8 19 13 45 54 13 46 1 23 23 23 23 23 17 17 23 55i4 3 2 21 5 29 39 1 920 6 31 8 2 27 6 6 4 40 938 994 972 030 213 840 Reduction, 8 54 9 20 6 31 9 20 15 25 5 22 34 27 Argument I of Latitude. Equation of Equinox, -16 Moon's true Longitude, 9 20 15 9 *From 10 to 20, the numbers are found by proportioning from Table xvIIi; thus, 19 hours is 9 of one day. Arg._ Moon's Longitude. Arg. Eq. Moon's Polar Dis. Moon's hourly motion in Longitude. I II 0! If 1 14 34 I. 89 10 40 Arguments. Equations. 2 3 21 II. 9 41 2 3 2l12. 9 41 2 of Longitude, 1" 4 5 14 2Ion. 1 3 of Longitude, 3 4 14 20 Ion. 44 of Longitude, 3 6 218 V. 120 Evection, 13 7 112 VII. 29 20 8 2 11 VIII. 17 Anom. and Sum of Eq. 12 9 5 IX. 4 Anomaly, 28 58 10 16 _X. 41 11 6 2930 12 f21 ( Polar Dis.89 22 48 Variation and Sum, 3 13 4 90 Variation, 1 19 14 27 Reduction, f C 14 27 ~ La t. N. 37 12 tending SR t 15 |3Lat.N. 7 12 tending S Hourly motion in Lon. 30 54 r 16 20 - 17 4 18 7 19 16 Moon's hourly motion in Latitude. 20 17 For the Equatorial Parallax. ~~~20 17 ___________________Argument I. — 2 56'' - EveiSumo, 31 34 Arg. Moon's Equation. II. - 4 Evection, 31 23 3 0 Sum, 1 2 57 Evection, 0' 251" Anom. 12 5 22 Anomaly, 53 54 Sum, 13 8 19 Variation, 3256"30' 54"::-3':-2' 4". Varia. 37 35 Parallax, 55 16 The result of this proportion gives -2' 49" Sum, 13 <45 54_ S. D. 15 4 for the hourly motion in Latitude. 274 ASTRONOMY. CHAP. lIt. S 0 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 0 0 14. Therefore, at the time for which these longitudes were computed, the moon will be past her full by 14" of are: 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 COr- The unit for t is one hour, and the denominator of the fracrection is subtractive tion is the difference of the hourly motions of the sun and because tile moon, as determined by the tables; the result is 29 seconds moon is past of time to be subtracted. conjunction, otherwise it The Greenwich time will be, 1851, July 12d. 19h. 15m. Os. would be ad. Subtract - - 29 ditive. 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 approximate and true time, however, do not commonly coincide as near as this: if they did, none but the most rigid astronomer 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 4".5. The moon's latitude, at the time of full moon, is, therefore, 37' 17".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. s. 1. True time of full moon, July, - - 12 19 14 31 2. Semidiameter of earth's shadow (page 265), 0 39' 39" 3. Angle of the moon's visible path with the ecliptic,* - - - 5 38 26 * This is the angle of the base of a right-angled triangle, whose base ECLIPSES. 275 I' CHAP. lad 4. Moon's latitude N. descending, - 37 17.4 - 6. Moon's.hourly motion from the sun, - 28 31 6. Moon's semidiameter, - - - 15 4 7. Semidiameter of I) 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 elements, 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 the ment in your dividers (54 43) = 543, and from C, as a center moon has 4very little ia. with that distance, describe the semicircle B D HE (Fig. 55). titude deTake CA = the 2d element, and describe the semidiameter scribe a full of the earth's shadow. From C the center of the shadow, Wh larcle. When large draw Cn at right angles to B X the ecliptic, above B E when south latithe latitude is north, as in the present example, but below, tude, deseribe only if south. the lower Fig. 55. semicircle. i:s —. T A ( E Take the moon's latitude from the scale of equal parts, and set it off from C to n. Through n draw D n, 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 perpendicular, the moon's hourly motion in latitude (2' 49"). See page 266, figure 54. 276 AS'tR 0 N 0OMY. CHAP. III center to run along the line from D to I, and from C draw Cm perpendicular to D H. When the moon is ascending in her orbit, D H must incline the other way, and Cm must lie on the other side of Cn. The eclipse commences when the moon arrives at D. It is the time of full moon when it arrives at n; the greatest obscuration occurs when it arrives at m, and the eclipse ends at H; The duration is the time employed in passing from D to't; and to find the duration apply D Hto the scale, and thus The 5thele. find its measure. Divide this measure by the 5th element, mient is the and we shall have the hours and decimal parts of an hour in moon's angular motion the duration. Also apply D)n to the scale and find its meatom the sun. sure. Divide this measure by the 5th element, for the time of describing Dn, also divide the measure nHifor the time of describing nith The time of describing D n, subtracted from the time of full moon, will give the time of the beginning of the eclipses and the time of describing n g, 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 motion 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 D n = 36' and niH= 44', These distances, divided by 28' 31", give 1 h.14 m. 16 s. for the time of describing D n, and lbh. 32 m. 40 s. for n H: whole time, or duration, 2 h. 27 m. 20 s. Astronomi- h. m tal time con- Therefore from lihe time of full O 19 14 31 Verted into Subtract - - 1 14 16 eivil time. Eclipse begins - 18 0 15 Add the duration - 2 47 20 This eclipse Eclipse ends - 20 47 35 This eclipse uot visible an That is, in 1851, July 12 d. 18 h. 0 m, 15 s. mean astrono; why. mical time, the eclipse begins; but this time corresponds with July 13, at 6 h. 0 m. in the morning; and at this time, the sun will be above the horizon of Greenwich, and, of course, the full moon, which is always opposite to the sun, will be below CHAP. mI, the horizon, and the eclipse will be invisible to all Europe. Visible iav In the United States, however, the eclipse will be visible; the U.. 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, I h. 5 m. morning, Middle -. 2 30 End of the eclipse 3 8 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 quafn the moon's diameter, which is immersed in the shadow, tity of the eclipse hov* at the time of greatest obscuration, and compare it with found. the whole diameter of the moon; and in the present example, 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 COH and Can, and the relative motion of the moon from the sun. In the right-angled triangle C nm, right-angled at m, Cn The Yeaae is the latitude of the moon = 3'7 17".4 = 2237".4, and the computati6i of the duma angle n Cm -- 50 38' 26"; with these data we find n =-tion of ~th 220"', and Cmn = 2212" eclipse. In the right-angled triangle C D m, or its equal Cm H, we have - - Cm2 +-m ll2 A CH2; Or, - m H CH -- CCm2,; Or, - - mH (C H+S Cnm) ( CH- Cm). CHis the 7th element = 3283", and Cm = 2212".6. Therefore, m H1= / ( 5495 ) (1071 ) 2426" This ~ T78 ASTRONOMY. CHAP. LII. divided by 1711", the 5th element, gives the time of half the duration of the eclipse 1 h. 25 m.; therefore the whole dau ration is 2 h. 5Om., which is 2 m. 40 s.morethan the time we obtained by the rough construction. The distance nrm, as just determined, is 220"', and the time of describing this space, at the rate of 1711" per hour, requires 7 m. 52 s.. which taken from and added to the semiduration, gives 1 h.17m. 8 s. from the beginning of the eclipse to full moon, and I h. 32 m. 52 s. from the full moon to the end of the eclipse. the trigoe For the magnitude of the eclipse,we add the moon's seminometrical computation diameter in seconds (904") to Cm ( 2212" ), and from the )ofthemagni- sum subtract the semidiameter of the shadow in seconds tucdipse (2379 ), and the remainder is the portion of the moon's diameter 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 — In all these computations we were guided by the- construestruction a sufficient tion; which will always prove a sufficient index, and all that guide to car- should be required. tyigonmetie We may determine, in any case, whether the eclipse will or cal computa. Will not be total, by the following operation: tions. Subtract the E's semidiameter from the semidiameter of the shadow, and if the moon's latitude, at th6 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 semidiameter 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 bv the hourly motion, will give the time of total darkness. When the moon's latitude is north, as in the present example, the southern limb of the moon is eclipsed — and conversely. EULIPSES. 279 CHAPTER IV.'SOLAR ECLIPSES - GENERAL AND LOCAL THm 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 difound -by the solar and lunar tables. reftions to The approximate time of new moon is first computed, and ments. for this time, compute the sun's longitude, declination, parallax, 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 proportion to the true time, as described in the last chapter. The elements for a general solar eclipse are: 1. The time of d * at some known meridian. 2. Longi- What eletude of 0 and 4). 3. 0's dedlination. 4. C)'s latitude. ments are necessary. 5. 0's hourly motion. 6. g)'s hourly motion in longitude. 7. C)'s hourly motion in latitude. 8. The angle of the )'s -visible path with the ecliptic. 9. I's horizontal parallax. 10. fI's semidiameter. 11. O's semidiameter. 12. 0'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 to - example proparticular emmple, we now propose to show the yener4a course posed. of an eclipse of the sun, which will occur in May 1854; where it will first commence on the earth; in what latitude and longitude the sun will be centrally eclipsed at noon, and uhere; 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 genethis we mean the, moon's shadow on the earth. If an observer ral preliminary explais in the moon's shadow, of course, the sun would be in an nations. eclipse to him; and, if a tangent line be drawn between the * Sign of conjunction. 19 280 AsT RO N O M Y. CHAP. IV. sun and moon, and that line strike the eye of an observer 0o the earth, to that observer the limbs of the sun and moon 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 earthd When no such lines can strike the earth there can be no eclipse. For the sake of simplicity in explaining a projection Point o of a solar eclipse, whether it be general or local, an observer view. 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 comn puted on page 254, and, if very close results are not required, we may compute the sun's longitude, declination, hourly motion, and semidiameter for this titne, and take out the moon's horizontal parallax, hourly motion, and semidiameter from Table IX; but we have computed the elements more accurately by the lunar tables, and finfd them as follows: d. h. m. s. 1. Greenwich mean time of - 1854, May 26 8 45 39 Accurate 2. Lon. of O and G) - -. 650 14' 6" elements for a, Declination of the O' 21 11 43 Nd the solar eclipse, 4. Latitude of the' i f - 21 19 Nio which will 5. I's hourly motion in lon., 2 24 take place May 26, 6, q's hourly motion in Ion., - 30 3 1854. 7. t'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 V's horizontal equatorial parallax, 54 30 10. The'"s semidiameter, - d 14 51 11. The Q's semidiameter, - - 15 48 12. The (D's horizontal paravllar, always taken at 9 Add together the (O's horizontal' parallax, the 1's hori'f zontal parallax, and the semidiameters' of 0- and', and if the moon's latitude is less than this sum, there will be aiv PI 10 I P 8 ES. -281 eclipse, othet*iWse 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 UB, equal to the difference of the parallaxes of ( and D (in the present example 54' 21", the minute is the unit), describe the semicircle C B'P H, or the whole circle, when the case requires it. When the moon has small latitude (less than 20') describe the whole circle; when the moon has large north latitude,describe the northern senihcircle; when south, describe the southern semicircle. Through C draw V CD P L perpendicular to HB. This perpendicular will represent the plane of the earth's axis, as seen from the moon. From P take PA, P F, each equal to the obliquity of the ecliptic 230 27' 30", and draw the chord A F. On A F, as a diamieter, describe the semicircle AL P. ind theaxis 2. Find the distance of the sun from the tropic, nearest to of the eclip6 it, by taking the difference between the sun's longitude and tie, 90~ or 2700, as the case miay be. In the present example we subtract 65~ 14' from 90~, the remainder is 240 46'. Take t T, equal to 240 4i6, and draw TLE parallel to L C. Draw CE the axis of the ecliptic. By the revolution of the earth round the sun, the axis of The axis the ecliptic appears to coincide with the axis of the equator, of thie edip'when the sun is at either tropic, and it appears to depart in position. from that line by the whole amount of the obliquity of the ecliptic; and the time of this greatest departure is when the sun is on the equator. That is, CE runs out to CA at the vernal equinox, and runs out to CF at the autumnal equlnox. As a general rule, CE, the axis of the ecliptic, is to the left of U'P, the axis of the equator, from the 20th of December to the 20th of June, and to the fight of that line the rest of the year. Draw C Gt the axis of the moon's orbit, so Hlow to Mid that the angle 6 Ct shall be equal to the angle of th e laxis of toon's visible path with the ecliptic, and C a is to the left qf bit. :2:82 ASTRONOMY. IHMP. IV. CE when the eclipse is about the ascending node, as in this example, but at the right when the eclipse is about the decending node. For this projection to appear natural, the reader should face the north, so that H will appear to the west, and B on the east of the figure. The shadow of the moon across the earth is from a western to an eastern direction, therefore, the moon is conceived to come in on the earth from the west side. The equa- The point C is perpendicular to the sun's declination, and tor. C V is the sine of the declination, and the curved line H VB is a representation of the equator as seen from the moon, When the sun has no declination, the equator draws up into a straight line. How to 3. Take C n from the scale of equal parts, making it equal draw the to the moon's latitude, and through the point n, and at right angles to C G, draw the line k i m n rp q, which represents the center of the shadow, or the moon's path across the disc. From C as a center, at the distance C O, describe the outer semicircle, equal to the sum of the moon's horizontal parallax, the sun's horizontal parallax, and the semidiameter of both sun and moon; then OH is the semidiameter of the sun and moon. When the eclipse first commences, the center of the moon is at k, and the center of the sun is on the circumference of the other circle, in a direct line to C, not represented in the figure, therefore, the two limbs must then just touch. As C is the center of the earth, and H on the equator, therefore CH 0 is a line in the plane of the equator, and the point k is a little below the equator; which shows that the eclipse first commences on the earth a little south of t'he equator. How to de. The time that the eclipse is on the earth is measured by termine the h.. ri s durtion of a the time required for the moon to pass from k to q with Its general true angular motion from the sun. eclipse. The length of this line, k q, can be found from the elements, and trigonometry, as in, an eclipse of the moon, and the time of describing it is found in the same way. !S,m VAZ tSXUJIDX 2S4 ASTRONOMY. CHAP. IV. When the moon's center comes to i, the central eclipse Howtode. commences, and the arc Hi shows that it must be about in termine in the latitude of 70 north. When the moon's center comes what latitides the to r, 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 enter, pass over, and clination, which, in this case, is about 23~; hence to the pass off the sun's declination, 210 12', add 23~, making 440 12'; showing, earth. as near as a mere projection can show, that the sun will be centrally eclipsed at noon on some meridian, in latitude 440 12' north. The central eclipse will end, or pass off the earth, when the moon's center arrives at p and the arc Bp 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, and for its limb to touch the sun, the sun's center must be at h, and the arc B h shows that the latitude must be about 300 north. The lines, c d and a b, parallel to the moon's path, and distant 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; c d is the southern line of simple contact, 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 central line. We may But we need not stop at coarse approximations: we have make accurate compr. all the data for correct mathematical results, on the same tations by principles as we determined those in relation to a lunar eclipse. nometry. In the triangle Cnr, we have the side Cn, the moon's latitude in seconds, which may be used as linear measure, as yards or feet and in proportion thereto, we may compute Cr and n r, when we know the angle n Cr. An equa. But the following equation always gives the tangent of the tion for the. position of angle E CD or n Cr, calling the sun's distance from the solthe axis of stice D, the obliquity of the ecliptic E, and the radius unity. the ecliptic. tan. E C D=-tan. E sin. D.* * The student who has acquired a little skill in analytical trigonometry can discover the preliminary steps to this equation; the principles are all visible in the construction of the figure. ECLIPSES. 285 To the angle E CD, add the angle G CE, the angle of the CHAP IV moon's visible path with the ecliptic, and we have the whole angle G CD, or m Cr. Cmn is a right angle; and in the two triangles Cmn and Canr, we have all the data, and can compute n r and r C. When the moon arrives at m, it is in the line of conjunction in her orbit;. when it arrives at n, it is in ecliptic conjunction; and when it arrives at r, it attains conjunction in right ascension. For the last six or eight years, the English Nautical Al- Recent manac has given the conjunctions and oppositions in right as- the English cension, in place of conjunctions and oppositions in longitude, Nautical Aland has given the difference of declinations between the sun manac 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 corn- The method mrences with the rising sun, and finally leaves the earth with offindingthe longitude the setting sun. In this example, we have decided that the where the eclipse must commence very near the equator, not more than eclipse first strikes the 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 motion 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 n. This shows that three hours, and a little more (we say 3 h. 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, 286 ASTRONOMYI. CHP. IV. the moon changes at 9 h. 3 m. in the morning,- apparent time 3; but the apparent time of new moon at Greenwich is 8 h. 49 m. P. M., making a difference of 11 h. 46 m. for mere locality: the absolute instant is the same; the difference is only in meridians which correspond to a difference of longitude of 1750 30'; and it is west, because it is later in the day at Greenwich. The method The central eclipse also first comes on the earth at a place of finding where the where the sun is rising. In this example it first strikes the central earth at the point 1, in latitude about 70 N.; but, in latitude eclipse first strikes the 70 N., and declination 210 N., the sun rises at 5 h. 48m., earth. 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 I 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 h. 44 m. in the morning. From this to 8 h. 49 m. in the evening, the time at Greenwich, gives a difference of 13 h. 5 m., reckoned eastward from the locality, or 10h. 55 m. reckoned westward; which corresponds to 1960 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, we had better call the longitude west'. To find the Where the sun is centrally eclipsed on the meridian, it is longitude just 12, apparent time; the moon's center is then at r, and, where the sun will be by the construction, it must be about seven minutes after centrall at conjunction in that locality; hence, the conjunction is seven eclipsed at 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 1340 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 210 11', as it now is, the sun sets at 7h. 15 in. apparent time; but this is h. 40m. after conjunction, therefore the conjunction in that locality must be at 5 h. 35 m.; but, at Greenwich, it is ECLIPSES. 287 8 h. 49 m., giving, for difference of longitude, 3 h. 14 mi, or CHAP. IV. 480 30' west. The eclipse finally leaves the earth in latitude 46~ north; To find the but, in this latitude, the sun sets at 6 h. 51 m., and the con- longitude where the junction will be 3 h. 0 m. sooner (the time required for the eclipse will moon to pass from n to q), therefore the conjunction in this leave the earth. locality must be at 3 h. 51 m.; but, at Greenwich, it will be h. 49m., giving 4h. 58 m. for difference of longitude, or 740 30' west. Thus, by the mere geometrical construction, we have roughly determined the following important particulars: App. time Gr. Lat. Longitude, h. m. ~ o I Eclipse commences, May 26, 5 46 1 S. 175 30 W. Results me chanically Cen. eclipse commences, 6 53 7 N. 196 15 W. taken from Cen. eclipse at local noon1 8 56 46 134 00 W. the projecCen. eclipse ends, 10 34 40 48 30 W. ion End of eclipse, 11 46 30 73 30 W. To find the latitude of the first commencement of simple The locali. ties of the contact on the southern line, all we have to do is to find the southern and are -c; and for the latitude on the northern line, we find the northern are Ha; the point c is in latitude about 270 south, and a in nes of sire. pie contact, about 540 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 obtain 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 trigonometrical computation, 0 In the triangle CGnm, we have Cn-21' 19"=1279, the angle m Cn=5~0 42' 50", and the angle m a right angle. Whence Cm=1273", and mn-127".3. 288 ASTRONOMY. CHAP. IV. tan. E C-D=n Cr=tan. (23~ 27' 32") sin. (240~ 45' 54") In these (page 284). computa- Whence BE CD-= 10~ 18' 8", tions the moon's lati- Add GCE= 50 42' 50", tude and the distances Sum is G CD= —m Cr=160 0' 58". from the cen- In the triangle mn Cr, we have Cm (1273), the perpendicuter, C, to the circumfer- lar, and the angle m Cr as just determined; whence, ences are giv- mr=365".3; Cr=1324".3. en lines. In the triangle C mp, 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, Cm2 +-mp2 = Cp2 Or mp2= Cp2-m2.=(C (p+Cm) (Cp —Cm), That is, mp=y-(4533)(19S7)- 3001l.7. Therefore, ip, the whole chord, is 6003".4, which, divided by 1659' (the moon's motion from the sun), gives 3.616 h. 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= ('q+ Cm) ( Cq —Cm), But, Cq=54' 21"+14' 5.1"+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: Accurate From k to I in 1 h. 9 m. 54 s. results on the condition of From 1 to m in 1 49 00 to d in orbit. invariable el- From m to n in 4 36 to d in ecliptic. ements. From n to r in 8 37 to d in right ascension. From rtop in 1 35 40 From p to q in 1 9 54 ECLIPSES. 289 The apparent time of ecliptic conjunction, at Greenwich, C,,. IV. as determined by the tables (and applying the equation of time), is at 8 h. 49 m. 0 s. Subtract from k to ecliptic 6, 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 2 236), 8 58 00 Central eclipse ends at (add 1 33 58), 10 33 48 End of eclipse at (add 1 9 54), 11 43 41 By comparing these times with those obtained simply by A careful the projection, we perceive that the projection is not far out projection more acca. of the way, notwithstanding the terms rough and roughly that rate, than is we have been compelled to use concerning it. Indeed, a good generally draftsman, with a delicate scale and good dividers, can decide supposed. 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 accurate with very little trouble: he perceives, at a glance, what elements 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 longitude, in the present example, the astronomer would notice in particular 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 k m is passed over by the moon in about 7 seconds less time than our computation made it, and the last parftrequires 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. 19 Y 290 ASTRONOMY. CHAP. IV. tor, we conceive k and C joined and if tLhe angle in Ck is greater than the angle m CH, the point k is south, otherwise north. By trigonometry, C k: k m: ~ sine 900: sine mC k; Or, 5138: 4900".3: sin. 90: sin.m Ck=75 31 20 To this add G CD, - - - - 16 0 58 Sum is the angle r C k - - 91 32 18 This angle shows that the eclipse will first touch the earth in latitude 10 32' 18" south. To find the arc HI, conceive the points Cl joined, and the two triangles Cim, m Cp are equal. And CI: m:: sin. 900: m CZ; Or, 3261: 3003.7:: sin. 90 sin. m CI=67 7 50 To this add G CD, - - - 16 0 58 The sum is, - - - - 83 8 48 Where the This angle shows the latitude of the point I to be 60 51' eclipse first 12" north. That is, the central eclipse first touches the strikes the earth earth in 60 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 CZ = 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 380 53' 8" north, and the general eclipse entirely leaves the earth in latitude 300 25' 38". To find the latitude of the point r, we consider Cr to be a sine of an are, 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 of the place time is 8 h. 57 m. 37 s., p. M.; this difference, changed into lonwhere the' sunis central- gitude, gives 1340 25' west, within a degree of the result delyeclipsedon termined from the projection; and it is not important to go the meridian, over a trigonometrical computation for the longitudes, since ECLIPSES. 291 we are sure of knowing h0ow to do it; and we are also sure CHAP. iv. that the results will not differ much from those already determined. In short, from the elements, the figure, and a knowledge Snfficient of trigonometry, we can determine all the important points in tfata in tilhe each of the three lines c d, k q, and a b, for between them we t 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 CHAI,. v. project and accurately compute every circumstance in relation 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 place, 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 equal parts, and The scale 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 Cm, the universal meridian, at right angles to CD, and from _ take e cm and a_, each equal to the obliquity of the ecliptic ( 23 27') and draw the straight line Ay,, on the right. Subtract the sun's longitude from 90~ or 2700 to find its distance from the nearest solstitial point, and note the difference (in this example 24~ 46' ). How to find From the point a, with a cr as radius, make a a equal-to ecliptic. 292 ASTRONOIMY. CHAP. V. the sine of 24~ 46',* and join C G, and produce It teo V; CA is the axis of the ecliptic; this line is variable, and is on the other side of the line Coi between June 20 and Deeember 21. How to find From E take the are EL equal to the moon's visible path the axis of the moonis with the ecliptic, to the right of E when the moon is descend. orbit. ing, but to the left when ascending as in the present examn ple. Join CL, 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, - 420 6' 39" N. Sun's declination, - 21 11 43 N. Sum is 630 19' 22"1, and difference is 200 54' 56". howto find From C, make C12 equal to the sine of the difference of tepoit n thepointsi the two arcs (200 54i 56" )j and Cd the sine of the sum thB ellipse marking the (630 19' 22"). visible path Divide (12) d into two equal parts at the point g; and on Cf the place bver the y (12), as radius, mark the sine of 150, 30~, 450, 600, 750, dartil's disc. 900; the line 7, 5, runs through the first point; 8, 4, through the second, &c. Subtract the latitude (42~ 6' 39") from 906, thus finding the co-latitude (47~ 53' 21"). On the semidiameter of the earth's disc, as radius, take the sine of the co-latitude (470 53'), and set off that distance from g, both ways to 6; thus Imaking a line, 6, 6, at right angles to the universal meridian, Cg. On g (6) as radius, and from the point g as a center, find the sine of 150, 300, 450, &c., and set off those distances each way from g and through the points thus found, draw lines parallel to y C; these lines, meeting the lines drawn parallel to 6 y 6, will define the points 5, 6, 7, 8, &c. to 12, and 1, 2, 3, &c. to 7, the hours of the day on the elliptic curves That is, our supposed obserVer at the mroon would see Boston of the hours (or any other place in the sanie latitude as Boston), at the round the el. point 9 when it is 9 o'clock at the place, and at 12 when it lipse. is noon at the place, &e. * The reader is supposed to understand how to draw a sine to any afcm corresponding to any radiug, either With or without a sector ma~~~~ I,*, \~~~~~~~~ 69 6~~~~~~~~~~~~~~a ~ ~ ~ ~ ~ G 868'~aS4TaD' 294 A,S TRO NOMY. CHAP. V. As this curve touches the disc before 5 and after 7, it shows that, in that latitude, on the day in question, the sun will rise before 5 in the morning, and set after 7 in the evening. If the declination of the sun had been as much south as now north, the point d would have been 12 at noon, and all the hours would have been on the upper part of the ellipse, which is not now represented. From C, as in the general eclipse, set off the distance C n equal to the moon's latitude, and, through the point n, draw the moon's path at right angles to CL. As the ellipse represents the sun's path on the disc, and as the point (12) refers, of course, to apparent noon, and not to mnean noon, therefore, we will mark off the time on the moon's path corresponding to apparent time. flow to mark When the moon's center passes the point n, it is at ecliptic time on the conjunction, apparent time, at Boston, or it must be considered moon's path. the apparent time corresponding to any other meridian for which the projection may be intended. The ecliptic d, apparent time, Greenwich, is 8 h. 49 m. 0 s. 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, part corresponds with a minute of the moon's motion from the the ellipse sun, and the distance a will correspond with one hour of the mente com- moon's 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 5 o'lock. time, will be just beyond the point 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 oh the moon's path, and the center 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 E C L t IPES 295 ing that the eclipse must have commenced prior to that time. CsAP. V. To find the time of commencement more exactly, let the hour To find the on the moon's path be subdivided into 10 or 6-minute spaces, more exact 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 6 h. 44m. To find the time of greatest obscuration, we must look Howto find the time of along the moon's path, and discover, as near as possible, from greatest ob; what point a line drawn at right angles from that path will scuration. strike the sun's path at the same hour and minute; the time, thus marked on both paths, will be the time of greatest obscuration. In this case it appears to be 5 h. 40 m., and the two centers 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 Howto find its continuance, we must take the semidiameter of the sun in the magni; tude of the the dividers from the scale; and, from the point of time on eclipse. 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 results of this projection are as follows: App. time. Mean time. Beginning of the eclipse, P. M., 4 h. 28 m. 4 h, 24 m. 39 s. aecuraoy of Greatest obscuration, 5 40 5 36 39 the result. End of the eclipse, 6 44 6 40 39 From the projection the two centers are- nearer together than' the -difference-of the, semdiameter of the sun and moon, 20 29S0B ASTRONOMY. CHAP. v. and the moon's diameter being least, the eclipse will be annular, 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 particular 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 anand impor- gles to the moon's path, and that point must also be 2 h. on Uant problem... the moon's path; running back to conjunction, we find it HIow solved, must take place at 1 h. 50 m.; but the conjunction for Greenwich time is 8 h. 49 m., the difference is 6 h. 59 m., corresponding to 1040 45' west longitude; we further perceive that the sun would there be about 9 digits eclipsed on the sun's southern limb. iHowtofind Now, admitting this construction to be on mathematical raoe aesu-l principles (as it really is, except the variability of the elements), we can determine the beginning and end of a local eclipse to great accuracy, by the application of ANALYTICAL GEOMETRY. General Let CD and C z be two rectangular co-ordinates, then equations to aid in cornm- the distance of any point in the projection from the center puting all the can be determined by means of equations. circumstances of an Let x and y be the co-ordinates of any point on the sun's eclipse as path or elliptic curve, and Xand Y the co-ordinates of any one place.t any point on the moon's path, then we have the following equations: (1) y=p sin, L cos. D+p cos, L sin. D cos. t solar (2) x=p cos. L sin. t co-ordin. (3) Y=d+h i sin. B lunar coordinates. (4) X=hicos.B In these remarkable equations, p is the semidiameter of projection, L the latitude, D the sun's declination, t the time from apparent noon, d the difference in declination between E CLIPS ES. 297:Sun'and moon at the instant of conjunction in right ascen- CVaP. V. Sion, h the moon's hourly motion from the sun, i the interval of time from conjunction in right ascension-mrninus, if before conjunction-plus, if after; and B is the angle L C A, or the angle which the moon's path makes with C D. In the equations, x and X are 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 Explanation the moon is northward, and the minus -sign when southward. "f the symbols. The sin. t, or cos. t, means the sin. or cos. of an are, corresponding to the time at the rate of 150 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- d,expresses the tions; the minus sign applies to forenoon, the p2us sign to time of conafternoon: junction in right ascension. To apply these equations, and, of course, the former ones, i, the interval of time from conjunction must be assumed, and, as the time of coinjunction is known, t thus becomes known; d, h, and B, are known by the elements; therefore, x, y, and XA Y; are all known. But the distance between any two points referred to co-ordinates, is always expressed by ( C:rX) 2 +( y ) Y)2. When an eclipse first commences, or just as it ends, this expression 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 llmb is over the sun, or how far asunder the limbs are, and will, of course, indicate what change in -the time mun:st be made to correspond with a contact, or a particular phase of the eclipse. For an eclipse absolutely central, and at the time of being ventral, the last expression must equal zero; and, in that 29'8 AST'RONO M Y. cuAP. V.: case, =-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, efficiently, and briefly cover the whole subject; and we now close by showing their application to the case before us. Application By the projection we decided that the beginning of the of the preceding expres- eclipse would be at 4h. 28 m., apparent time at Boston. Call sions. 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 4h. 28 m. is, in fact, the time, otherwise it is not, &c. h. m. S. An acrate Conjunc. in R. A., app. time, Boston, 4 13 21 computation foTthebegin- Assume i equal to 15 ning of the Therefore, t is equal to 4 28 21=67o 5' 15". eclipse as seen from p=54' 21"=3261. Reduced! lat., L-420 6' 38". Boston. D-=21~ 11' 43"; d==-C r=1324".3; h=1659 ii= B —160 0' 58". p 3261 - log. 3.513511 - log. 3.5136511 L 420 6' 38" sin. 9.826437 - cos. 9.870315' D 21 11 43 cos. 9.969583 - sin. 9.558149 t 67 5 15 cos. 9.590288 2039.1 log. 3.309531 346.3 log. 2.532263 346.3........ y=1692.8 p 3.5151311 cos. L 9.870315 sin. t 9.964303, x=2228.5 log. 3.348129 For Yand X: B 160 0' 58" sin. 9.440775 - cos. 9.982804 Ai 41411.75 - log. 2.617800 - log. 2.617800 114.5 2.058575 398.6 2.600604 add 1324.3 Y= 1438.8 X=398.6 (Ycn y)=264 (xa X)=.1829.9t ECLIPSES-. 299 fere are two sides of a right-angled triangle, and the bhy-:cHA. 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 semidiameter of the sun and moon is only 1853"; there- The eclipse fore the eclipse has not yet commenced, and will not until the must be annular. moon moves over 4".8; which will require about 9 s., as we determined by proportion, because the apparent motion of the mnopn 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 triangle, 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 computation must be made accordingly. Hence, to the assumed time of beginning, 4 h. 28 m. 21 s. Add 19 Beginning, apparent time, - 4 28 40 Mean time, - - 4 25 19 By the application of the same expressions, we learn that The moon's center appa. the greatest obscuration will take place at 4h. 41m. mean rently 18"N. time at Boston; and the apparent distance of the moon's een- of the sun's ter will be 18" north of the sun's center; and, as the moon's at appareint 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 i=1 h. 33 m., and as t is more than 6 h., the second part of y changes sign, as we see by the figure; the sun after 6, must be above the line 6g 6. Occultations of stars are computed on the same principles as an eclipse of the sun, the star having neither diameter nor parallax. As problems, to give practice to the learner, we take the elements of two solar eclipses for 1846, from the Nautical Almanac, with their results as answers to the problems: goo, AS TRO NOM 0 Y. CHAP. V. ELEMENTS OF THE ECLIPSES OF THIE SUN,. 1846. April 25. October A19 Examplers h. m, S. h. m. s. given for Greenwich M. T. of din R.A., 4 55 54'5 19 50 12.2 practice. o and )('s Right Ascension, 2 11 8'31 13 38 31-54 )'s declination, N. 13 25 19'8 S. 10 23 43'0 O's declination, N. 13 13 21 -2 S. 10 15 3'9 )'s hourly motion in R. A., 33 55'1 30 42'2 O('s hourly motion in R. A., 2 21'3 2 21'5 )'s hourly motion in dec. N. 8 23 -6 S. 8 37 (0 O's hourly motion in dee. N. 0 48'8 S. 0 5441 ('s equatorial hor. parallax, 57 53 -8 55 33'4 0's equatorial hor. parallax, 8 -5 8'-,]'s true semidiameter, 15 46'5 15 8'4 O's true semidiameter, 15 54'5 16 5' THE APRIL ECLIPSE. General re- Begins on the earth generally April 25 d. 2 h. 2 m. 4 s., mean ults., time at Greenwich, in longitude 119~ 40' W. of Greenwich, and latitude 6~ 15' S. Central Eclipse begins generally April 25 d. 3 h. 3 m. 3 s. in longitude 1350 51' W. of Greenwich, and lat. 2~ 11' S. Central eclipse at noon, April 25 d. 4 h. 55 m. 9 s. in longitude 740 31' WV. of Greenwich,. and lat. 25~ 21' N. Central eclipse ends generally April 25 d. 6 h. 37 m. 6 s. in longitude 30 43' W. of Greenwich, and lat. 24~ 56' N. Ends on the earth generally April 25 d. 7 h. 38 m. 5 s. in longitude 200 4' W. of Greenwich, and lat. 20~ 52' N. THE OCTOBER ECLIPSE. Begins on the earth generally October 19 d. 16 h. 46 m. 7 s. mean time at Greenwich, in longitude 16~ 21' E. of Greenwich, and latitude 90 50' N. Central eclipse begins generally October 19 d. 17 h. 52 m. 0 s. in longitude 00 32' W. of Greenwich, and lat. 6~ 44'N. Central eclipse at noon, October 19 d. 19 h. 50 m. 2 s in longitude 580 41' E. of Greenwich, and lat. 190 22' S. ECLIPSES. 301 Central Eclipse ends generally October 19 d. 21 h. 38 m. 9 s. CHAP. V. in longitude 1260 5' E. of Greenwich, and lat. 23~ 51' S. Ends on the earth generally October 19 d. 22 h. 44 m. 1 s. in longitude 1090 6' E. of Greenwich, and lat. 20~ 47' S. The following is a catalogue of the solar eclipses that will be visible in New England and New York, between the years 1850 and 1900; the dates are given in civil, not astronomical, time. 1851, July 28th. Digits eclipsed, 33, on sun's northern limb. Statistics 4 54, May 28th.of eclipses 1854, May 26th. As computed in the work. from 1850 to 1858, March 15th.. Sun rises eclipsed. Greatest obscura- 1900. tion, 5- digits on sun's southern limb. 1859, July 29th. Digits eclipsed, 2~, on sun's northern limb. 1860, July 18th. Digits eclipsed, 6, on sun's northern limb. 1861, December 31st. Sun rises eclipsed. Digits eclipsed at greatest obscuration, 43, on sun's southern limb. 1865, October 19th. Digits eclipsed, 83, on sun's southern limb. 1866, October 8th. ~ digit eclipsed. South of New York no eclipse. 1869, August 7th. Digits eclipsed, 10, on sun's southern limb. This eclipse will be total in North Carolina. 1873, May 25th. Sun and moon in contact at sunrise, Boston. 1875, September 29th. Sun rises eclipsed. This eclipse will be annular in Boston, Maine, New Hampshire, and Vermont. 1876, March 25th. Digits eclipsed, 3-, on sun's northern limb. 1878, July 29th. Digits eclipsed, 7~, on sun's southern limb. This is the fourth return of the total eclipse of 1806. 1880, December 31st. Sun rises eclipsed. Digits eclipsed at greatest obscuration, 5~, on sun's northern limb. 1885, March 16th. Digits eclipsed, 6~, on sun's northern limb. g 302 ASTRONOMY. CHAP. V. 1886, August 28th. North of Massachusetts no eclipse; Statistics south, sun eclipsed. off eclipses 1892, October 20th. Digits eclipsed, 8, on sun's northern from 1850 to 1900. limb. 1897, July 29th. Digits eclipsed, 4~, on sun's southern limb. 1900, May 28th. Digits eclipsed, 11, on sun's southern limb. The sun will be totally eclipsed in the State of Virginia. TABLES. EXTRACTS FROM THE NAUTICAL ALMANAC FOR JANUARY, 1846. THE SUN'S of the RTIHE MOON'S i0 -Apparent Radius D [Vector Longitude. Latitudeof the Longitude. Latitude. Semi- Hor. Earth. diam. Paral. Noon. Noon. Noon. Noon. Noon. Noon. o 0 I 11 0 1 11 / I 11 1 280 46 15.3 N.0.49 9.992661 330 42 13.9 N.4 54 8.5 16 21.6 60 2.3 281 47 26.1 0.45 9.99266 345 7 12.0 4 24 8.7 16 8.3159 13.5 3282 48 36.5 0.37 9.99267 359 4 55.4 3 39 5.9 15 53.9 58 20.5 4283 49 46.5 0.27 9.99267 12 35 34.7 2 43 1.9 15 39.8 57 28.7 5 84 50 56.1 0.16 9.99268 25 41 31.5 1 39 55.7 15 26.7 56 40.8 285 52 5.3 N.0.03 9.9926 38 26 25.0 N.O 33 28.3115 15.2 55 58.7 7 286 53 13.9 S. 0.11 9.99270 50 54 23.21S. 0 33 3.6 15 5.655 23.3 8 87 54 22.0 0.25 9.99271 63 9 30.11 1 36 46.8 14 57.6 54 54.1 9288 55 29.7 0.38 9.99272 75 15 21.81 2 35 8.6 14 51.5 54 31.6 10289 56 36.8 0.49 9.9927 87 14 56.3 3 25 55.414 46.9 54 14.6 11 290 57 43.4 0.58 9.99277 99 10 31.3 4 7 13.7 14 43.8 54 3.3 1 91 58 49.5 0.65.9927 111 3 50.8 4 37 30.7 14 42.1 53 57.0 13 292 59 55.3 0.70 9.99282 122 56 17.6 4 55 38.9 14 41.7 53 55.7 14 294 1 0.5 0.71 9.99285 134 49 7.9 5 0 56.414 42.8 53 59.8 15295 2 5.4 0.69 9.99288 146 43 48.4 4 53 7.6 14 45.5 54 9.7 16 296 3 9.9 0.64 9.99292 158 42 11.3 4 32 23.1 14 50.0 54 26.0 17 297 4 14.0 0.57 9.99295 170 46 44.8 3 59 17.1 14 56.3 54 49.0 [S 298 5 17.8 0.47 9.99299 183 0 38.7 3 14 47.1 15 4.6 55 19.7 1 99 6 21.2 0.35 9.9930 195 27 41.8 2 20 14.2115 15.2 55 58.4 20300 7 24.2 0.23 9.99308 208 12 10.4 1 17 27.8115 27.7 56 44.4 21 301 8 26.71 S. 0.09 9.9931 21 18 27 5 S. 0 8 53.1,15 42.0 57 37.0 22 02 9 28.9 N.0.4 9.99318 234 50 26.7lN.1 2 20.5j15 57.3 58 32.9 23 303 10 30.4 0.15 9.99323 248 50 42.5 2 12 11.716 12.5 59 28.8 24 304 11 31.31 0.25 9.99328 63 19 30.41 3 15 50.9116 26.2 60 19.0 125 305 12 31.51 0.33 9.9933 278 13 48.8 4 8 2.8116 36.860 57.9 306 13 30.91 0.38 9.99339 293 26 49.2 4 43 49.4116 42.9161 20.2 27 307 14 29.31 0.40 9.99345 308 48 22.8 459 32.4P16 43.5 61 22.6 128 308 15 26.8 0.40 9.99351 324 6 34.0 4 53 45.416 38.7161 4.9 29 309 16 23.3 0.37 9.99357 339- 9 55.3 4 27 32.9 16 28.9160 29.1 30310 17 18.5 0.30 9.99363 353 49 32.0 3 44 8.216 15.6 59 40.2 31 311 18 12.6 0.21 9.99369 8 0 13.1 2 47 58.716 0.2158 43.7 32 312 19 5.31 N.O.10 9.99375 21 40 34.3 N.1 43 50.6115 44.2157 45.1 TABLES. TABLE I. MEAN ASTRONOMICAL REFRACTIONS. Barometer 30 in. Thermometer, Fah. 50,. Ap. Alt. Refr. Ap. Alt. Refr. Ap. Alt. Refr. Alt. Refr.1 i 5' 33' 51" 4 0' 11' 52 12Q O' 428.1" I 4.6 5 32 53 10 11 30 10 4 24.4 43 1 2.4 10 31 58 20 1 110 20 4 20.8 44 1 0.3 15 31 5 30 10 50 30 4 17.3 45 0 58.1 20 30 13 40 10 32 40 4 13.9 46 56.1 25 29 24 50 15 50 4 10.7 47 54.2 30 28 37 5 0 9 58 13 0 4 7.5 48 52.3 35 27 51 10 9 42 10 4 4.4 49 50.5 40 27 6 0 9 27 20 4 1.4 50 48.8 45 26 24 30 911 30 358.4 51 47.1i 50 25 43 40 8 58 40 3 55.5 52 45.4 55 25 3 50 8 45 50 3 52.6 53 43.8 1 0 24 25 6 0 8 32 14 0 3 49.9 54 42.2 5 23 48 10 820 10 3 47.1 55 40.8 10 23 13 20 8 9 20 3 44.4. 56 39.3 i5 22 40 30 7 58 30 3 41.8 57 37.8 20 22 8 40 7 47 40 3 39.2 58 36.4 25 2137 50 737 50 3 36.7 59 35.0 30 21 7 70 727 15 0 3 34.3 60 33.6 35 20 38 10 7 17 15 30 3 27.3 61 32.3 40 2010 20 7 8 16 0 3 20.6 62 31.0 45 19 43 30 6 59 16 30 3 14.4 63 29.7 50 19 17' 40 6 51 17 0 3 8.5 64 28.4 55 18 52 50 6 43 17 30 3 2.9 65 27.2 2 0 18 29 8 0 6 35 18 0 2 57.6 66 25.9 5 18 5 10 6 28 19 2 47.7 67 24.7 10 17 43 20 6 21 20 2 38.7 68 23.5 15 17 21 30 6 14 21 2 30.5 69 22.4 201 17 0 40 6 7 22 2 23.2 70 21.2 25 16 40 50 6 0 23 2 16.5 71 19.9 30 16 21 9 0 5 54 24 2 10.1 72 18.8 35 16 2 10 5 47 25 2 4.2 73 17.7 40 15 43 20 5 41 26 1 58.8 74 16.6 45 15 25 30 5 36 27 1 53.8 75 15.5 50 15 8 40 5 30 28 1 49.1 76 14.4 55 14 51 50 5 25 29 1 44.7 77 13.4 3 0 14 35 10 0 5 20 30 1 40.5 78 12.3 1 5 14 19 10 5 15 31 1 36.6 79 11.2 10 14 4 20 5 10 3.2 1 33.0 80 10.2 15 13 50 30 5 5 33 1 29.5 81 9.2 20 1 35 40 5 0 34 1 26.1 82 8.2 25 1 3 21 50 456 35 1 23.0 83 7.1 30 1 3 7 11 0 451 36 1 20.0 84 6.1 35 12 53 10 4 47 37 1 17.1 85 5.1 140 12 41 20 4 43 38 1 14.4 86 4.1 45 12 28 30 4 39 39 1 11.8 87 3.1 50 1q 1 6 40 435 40 1 9.3 881 2.0 55 12 3 50 431 41 1 6.9 89 1.0 TABLE C. 3 CORRECTION OF MEAN REFRACTION. Hlght of the Thermometer. App. 24o 28~8 320 1 360 400 440 520 560 600 640 680 72C 76o 800 0. 2.18 +1.5,,, 133,,, 51 31 10' 48.07 1.25 43 2.011219 o0o.2.181.55 1.33 1.11 51 31 10 29 48 1.07 1.251432.0112.19 0.102.12 1.49i 1.28 1.08 48 29 9 27 45 1.04 1.21 1.381.542.12 0.,0 2.0511.44 1.24 1.04 46 28 9 26 44 1.01 1.1711.33 1.4912.05 0.30 1.5911.39 1.20 1.01 44 26 8 25 41 58 1.1311,281.4311.59 0.40 ],53 1.34{ 1.16 58 42 25 8 24 39 55 1.10 1,24 1.381.53i 0.50 1.48 1.291 1.12 55 40 24 8 23 37 52 1.06 1.20 1.3411.48 1.0011,4311.25 1.09 53 38 23 7 21 36 50 1.0311.1711.3011.43 1.101.38 1.21 1.06 50 36 22 7 20 34 48 1.001.13 1.261.38 1.20 1.33 1.17 1.03 48 34 21 6 19 32 45 57 1.09 1.21 1.33 1.301.29 1.14 1.00 46 32 20 6 18 31 43 54 1.06 1.18 1.29 1,40 1.25 1.11 57 44 31 18 6 18 30 41 521{.04{1.151.25 1.50 1.21 1.08 55 42 30 17 6 17 28 39 5011.01 1.11 1.21 2.00 1.18 1.05 53 39 29 17 5 16 27 37 48 58 1.0811.18 2.20 1.11 1.00 48 37 26 16 5 15 25 35 44 54 1.03 1.11 2.401.06 55 44 34 24 14 5 14 23 32 41 50 58 1.06 3.00 1.01 51 41 32 22 13 4 13 21 30 38 46 54/1.01 3.20 57 47 38 29 21 13 4 12 20 28 35 43 50.57 3.40 53 44 36 28 20 12 4 11 18 26 33 40 47 53 4.00 49 41 33 26 18 11 4 10 17 24 31 37 44 50 4.30 45 38 31 24 17 10 3 9 16 22 28 34 40 45 5.00 41 35 28 22 16 9 3 9 14 20 26 31 36 40 5.30 38 32 26 20 14 9 3 8 13 19 24 29 34 38 6.00 35 30 24 19 13 8 2 7' 12 17 22 26 31 35 6.30 331 28 22 17 1.2 7 2 7 11 15 20{ 24 29 33 7.00 31 26 21 161 12 7 2 6' 10 14 19 23 27 31 8 27 23 19 15 10 6 2 5 9 13 16 20 24 27 9 241 20 16 131 9, 5 2 5 8 11 14 18 21 24 10 22 18 15 121 8 5 1 4 7 10 13 16 19 22 11 201 17 141 11 8 5 1 4 7 9 12 I5 18 s 20 12 181 15 13 l10 7 4 1 4 6 9 11 -13 16 18 1:3 17 14 12 9 7 4 1 3 6 8 10 12 15 17 14 16 13 11 8l 6 4 1 3 5 7 9 11 14 16 15 15 12 10 8 6 3 1 3 5 7 9 11 13 15 16 14 12 91 7 5 3 1 3 5 6 8 10 129 14 17 13 11 9l 7 5 3 1 3 4 6 8 9 11 13 18 12 10l 8l 6 5 3 1 2 4 6 7 9 10{ 12 19' 11 9 81 6; 4 3 1 2 4 s 7 8 10 11 20 11 9 71 6 4 2 1 2 4 51 8 9 1l 1 10 9 7 5 4 2 1 2 3 51 1 7 9 10 22 10 8 7 5 4 2 1 2 3 51 7 8 10 23 9 8{ 6 5 4 2 1 2 3 41 6 7 8 9 241 91 71 6 51 3 2 1 2 3 4{ 5 6 8 9 25 l 7 6 5 3 2 1 2 3 4 5 6 7 8 2fi 13 7 6 4 3 2 1 2 3 4 5 6 7 8 27 8 6 5 4 3 2 1 2 3 4 5 6 7 8 2B 71 5 4 31 2 0 1 2 3 5 5 6 7 30 1 71 6 5 4 3 2 0 1!2 3 4 5 6 7 I_ - I_ + + + - - 828.26128.56128.8529.1 7530530.35 30.6430.93 Hight of the Barometer. 20 4 TABL$S. TABLE II. MEAN PLACES FOR 100 PRINCIPAL FIXED STARS, FOR JAN. 1, 18460 Star's Name. n RightAscen. Annual Var. Declination..Ann. Tar. h. m. s. s. deg. min. sec. sec. c ANDROMEDA,......... 1 0 0 26.257 + 3.0720 N.28 14 25.40 +20.055' PEGASI (Algenib)...... 2.3 0 5 18.691 3.0784 N.14 19 37.80 2~0.050 1/ Hydri................. 0 17 34.168 3.3054* S. 78 7 24.40 19.997 cc CASSIOPEi............3 0 31 48.294 3.3418 N.55 41 31.08, 19.862 R/ Ceti.................2.3 0 35 51.339 + 2.9995 S.18 49 59,01 19.810 ct URS. MIN. (Polaris),.. 2.3 1 3 52.226 17.1346 N.88 c9 17.88 19.279 01Ceti............... 3 1 16 19.692 3.0015 S. 8 58 45.93 18.952 c Eridani (Achernar),... 1 1 31 58.291 2.2339 S.58 1 14 8.461 a ARIETIS,......3 1 58 30.193 + 3.3475 N.22 43 53.86 +17.432 2- Ceti. 3 2 35 19.633 3.1085 N. 2 35 1.17 15.621 a CETI,............ 2.3 2 54 14.072 3.1266 N. 3 28 55.70 14.532 cc PERSEI.............. 23 3 13 21.403 4.2324 N,49 18 28.20 13.329 Tauri...............3 3 38 20.382 + 3.5473 N.23 37 27.73 +11.620?~ Eridani.............2 3 3 50 50.760 2.7898 S. 13 57 1.50 10.711 c TAuRI(Aldebaran)),..... 1 4 27 5.404 3.4274 N.16 11 41.39 7.907 c AURIGmA (Capella),...... 1 5 5 19.317 4,4082 N.45 50 6.56 4.737,B ORIONIS (Rigel)....... 1 5 7 8.383 + 2.8787 8, 8 23 3.33 + 4.583 TAR................ 2 5 16 33.662 3.7827 N.28 28 17.49 3.776 J' ORIONIS.............. 2 5 24 8.428 3.0609 S 0 25 4.86 3.123 cc Lepris,..... 3. 5 25 56.406 2.6425 S. 17 56 12.77 2.968 S ORIONIS............. 2.3 5 28 24.062+ 3.0404 S. 1 18 17.53 + 2.754 c Columboe....... 2 5 34 4.531 2.1691 S. 34 9 36,95 2.262 a, Orionis............. 1 5 46 50.189 3.2433 N. 7 22 22.32-+ 1.149,c Geminorum,......... 3 6 13 38.621 3.6257 N.22 35 13.16- 1.196 a Argus (Canopus)...... 1 6 20 32.145 + 1.3279 S.52 36 49.17 - 1.796 51 (Hev.) Cephei........ 6 6 26 30.287 30.7946 N.87 15 31.20 2.337 c CANIS MAJ. (Sirius),... 1 6 38 21.883 2.6459[ S. 16 30 32.83 4.484* t Canis Majoris......... 2.3 6 52 34.440 2.3558 S.28 45 59.38 4.562 J' Geminorum.......... 3.4 7 10 55.298 + 3.5918 N.22 15 37.47 - 6.110 cc2GEMINOR. (Castor),.... 3 7 24 46.065 3.8561 N.32 13 12.93 7.2'53 a CAN. MIN. (Procyon),.. 1.2 7 31 14.237 3.1445* N. 5 36 54.95 8.758* /3 GEMINOI. (Pollux),... 2 27 35 53.153 3.6829* N.28 23 34.06 8.152 15 Argus................3 8 0 59.232 + 2.5596 S. 23 51 50.94 —10.104 g Hydr............... 4 8 38 37.154 3.1966 N. 6 58 48.51 12.800 t Urse Majoris...3. 8 48 38.088 4.1261* N.48 38 32,35 13.464 Argus,.........2.... 2 9 12 58.192 1.6100 S. 58 37 49.78 14.961 d HYDRA............... 2 9 20 1.170 + 2.9499 S. 7 59 39.051 —15.366 0 Ursae Majoris,....... 3 9 22 31.453 4.0504* N.52 22 31.09 16.108* j Leonis,.............. 3 9 37 6.098 3.4258 N.24 28 49.46 16.283 c LEonIS (Regulus),.... 1 10 0 10.062 + 3.2211 N.12 43 2.96 — 17.377 TABLE IL Star's Name. c Right Ascen. Annual Var. Declination. Ann. Var h. m. S. S. deg. min. sec. s., Argus.............. 2 0 39 6.223 + 1a.3051 S. 58 52 34.26 -18.82 a URSs MAJOrIS,....... 1.2 10 54 10.737 3.8001 N.62 34 51.81 19.24 J LEorNs..............s 3 11 5 54.583 3.1928 N.21 21 59.86 19.50 J' Hydra et Crateris,... 3.4 11 11 38.718 3.0010 S. 13 56 46.85 19.61, LEONIS............ 2.3 11 41 12.066 + 3.0654* N.15 25 58.12-19.99 URSE MAJORI,........ 2 11 45 42.219 3.1874 N.54 33 3.18 20.02 R Chameleontis......... 5 12 9 26.893 3.3409 S. 78 27 26.15 20.04 l1Crucis,............ 112 18 4.916 3.2710 S. 62 14 39.74 19.99 1B Corvi..............2. 3112 26 18.465 + 3.1342 S. 22 32 39.93 -19.92 12 Canum Venaticorum,.. 2. 3 12 48 49.007 2.8403 N.39 9 4.18 19.60 C VIRGINIS (Spica)...... 1 13 17 5.233 3.1512 S. 10 21 20.80 18.94 s URSPE MAJORs,....... 2. 13 41 27.894 2.3525* N.50 5 1.45 18.12 m Bootis............... 3 13 47 21.140 + 2.8606 N.19 10 21.03 —17.89,/ Centauri........... 1 13 53 0.800 4.1508 S. 59 37 33.93 17.67 a BOOTIS, (Arcturus),... 14 8 38.366 2.7336* N.19 59 12.07 18.94* X2Centauri............. 14 29 11.925 4.0165* S. 60 11 37.00 15.12 BooTIS.............. 3 14 38 15.706 + 2.6229 N.27 43 35.23 -15.46 aSLIBRAE,.............. 3 14 42 22.132 + 3.3102 S. 15 23 53.52 15.23 R/ URSAc MINORIS,.... 3 14 51 13.199 - 0.2692 N.74 47 5.58 14.71 Libr,.............. 1. 15 8 43.595 + 3.2226 S. 8 48 38.53 13.63 C CORONA BOREALIS.... 2 15 28 10.083 + 2.5279 N.27 14 11.071-12.33 ac SERPENTIS......... 2.3 15 36 41.077 + 2.9391 N. 6 54 49.88 11.74 C Ursae Minoris.. 4 15 49 41.194- 2.3520 N.78 15 55.43 10.80 RI1Scorpii.....2 15 56 29.397 + 3.4742 S.19 22 44.18 10.29 S OPHIUCHI........O..... 3 16 6 16.830 + 3.1382 S. 3 17 35.67 - 9.55 c SCORPII, (Antares),.... 1 16 19 58.461 3.6638 S. 26 5 4.58 8.48 Draconis............. 3 16 21 55.119 0.7960 N.61 51 50.58 8.32 Trianguli Australis,... 2 16 32 25.090 + 6.2587 S. 68 44 4.75 7.48 Ursue Minoris,......... 4 17 1 55.988 — 6.5328* N.82 16 52.30 - 5.03 Ha HERCULIS........3... 17 7 37.617 + 2.7320 N.14 34 12.67 4.54. Octantis,........... O6 17 22 55.004 106.8627 S. 89 16 10.25 3.14 DRACONIS........... 2 17 26 57.473 1.3513 N.52 25 3.28 2.88 t OPrIUC,............ 2 17 27 47.219 + 2.7727 N.12 40 37.11 - 2.81 7, DRACONIS,........... 2 17 53 1.955 1.3900 N.51 30 33.50- 0.61 -i Sagittarii............3.4 18 4 33.276+ 3.5861 S.21 5 36.14 0.40 P URSAL MINOPS,........ 3 18 22 0.703 -19.2683* N.86 35 42.58 ~ 1.91 t LYRns (Vega),........ 1 18 31 43.386 + 2.0118 [N.38 38 35.33 + 2.77,B LYRXA........... 3 18 44 23.696 2.2124 N.33 11 14.80 3.86 AQUILME,......... 3 18 58 19.965 2.7566 N.13 38 20.49 5.05 J AQUILZc...4......, 3.4 19 17 43.889+ 3.0086 N. 2 48 43.64-+ 6.67 AQUILA,......... 3 19 38 56.278 + 2.8511 N.10 14 31.50+ 8.39 ac AQUILIE, (Altair)j.... 1.2 19 43 16.128 2.9254*N. 8 27 54.321 8.74 AUIL........ 3.4 19 47 44.866 2.9446 -N. 6 1 33.90{ 8.55' a 2CAPRICORN,'..:..:1:. 3 20 9 30.316 3.3315 S. 13 1 4.191 10.74 TABLES. Star's Name, | Right Ascen Annual Var. Declination. Ann. Vat I h. m s. s. deg. min. sec. sec. t Pavonis,............. 2 20 13 25.814 + 4.8046 S. 57 13 19.50 +11.03 Ursne Minoris......... 5 20 16 31309 -— 52.1273 N.88 50 53.54 11.22 a CYNI................ 1 20 36 11.005~ +.0418 N.44 43 57.43 12.64 11 CYGNI... X 55. 20 59 59.947 2.6908 N.37 59 42.08 17.48 Cygni....,. 3 21 6 23,073+ 2.5486 N.29 35 53.03 414.57 CEPE.......... 3 21 14 53.940 1.4163 N.61 56 4.55 15.07 fi AQUARI.............. 3 21 23'26.875 3.1628 S. 6 14 44.46 15.56,B CEPHEr,.............. 3 21 26 39.120 0.8059 N.69 53 7.21 15.73 ~ Pegasi,............ 2.3 21 36 37.346~ 2.9441 N. 9 10 17.35 +16.26 a- AtwAi............. 3 21 57 52.326 3.0831 S. 1 3 56.72 17.28 Gris................ 2 1 58 29.837 3.8134 S.47 42 12.42 17.30 Pegasi,.... 3 22 33 46.976 2.9837 N.10 1 44.67 18.65 aC Pis. AiUs. (Fomalhaut),. 1 22 49 7.531 + 3.3095 8.30 26 12.28 +19.11 a, PEGASI (MAarkab). 2 22 57 5.584 2.9776.14 22 40.12 19.31 Piscium, 4.5 23 32 1.736 3.0569 N. 4 47 30.74 19.36,_ Cephe,...,. 3 23 33 4.581+ 2.4042 N.76 46 22.01 +19.92 Those Annual Variations which include proper motion are distinguished by an Asterisk. SUN' IS IGR ASCENSION FOR 1846. Day - r - of January. February. March. April. May. June. Mo.,ry ay. Jue h. m. s. h. m. s. h, s.. h. m. s. h. m. s. h. m. s. 1 18 46 52 20 59 11 22 48 17 0 41 52 2 23 6 4 35 48 5 19 4 30 21 15 22 23 3 12 0 56 26 2 48 25 4 52 12 10 19 26 21 21 35 18 231 40 1 14 43 3 7 47 5 12 50 15 19 4 7 57 21 54 54 240 0 13 640 0 3 2724 5 3334 20 20 9 17 22 1412 23 58 14 1 51 38 3 47 15 5 54 22 25 20 30 19 22 3314 0 16 25 2 10 22 4 7 20 6 15 10 30 20 510 0 34 36 2 29 17 4 27 38,6 35 55 Day of July. August. I September. October. November. December. Mo. _ ___ 1'640 4 8 44 55 1 1 0 29 4 0 12 216 1629 4 1 5 656 34 9 023 1055 29 12 43 36 1441 2 1646 23 10 717 5 91929 11 1330 13 154 15 1 5 17 817 15 73725 93821 113128 132024 152128 173022 20 757 33 9 56 60 11 49 25 113 39 8 15 42 14 17 52 33 2s 81728 101527 12 724 1358 9 16 319 181446 30 837 7 10 3344112 25 27 141727 16243 183657 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 seven Peconds for each period. TABLE Il. 7 TA3ULJLAR VIEW OF THE SOLAER SYSTEM. Mean diameter Mean distance Mean dist.; Log. of Time of revoii- Log. of Names. f rom the Sun the Earth's mean tions round times of in mies. in miles. dist. unity. distance. the Sun.:evolution. iSun 883000 DAYS. Mercuty 3224 37 million 0.3870989.587818 87.9692581.944324 Venus 7687 68 " 0,7233329.859306 224.700787 2.351610 The Earth 7912 95 " 1.0000000.000000 365.256383 2.562598 Mars 4189 144 " 1.5236920.182810 686.979646 2.836942 Vesta 238 224,340,000 2.36120 0.373100.1324.289 3.12199i Iris 226 million 2.37880 0.376384 1327.973 3.123190 Hebe Unknown 230 6, 2.42190 0.384004 1375. nearly 3.13303 Flora U nown 240 " 2.52630 0.402487 1469.76 3.167300 Astrea 246 " 2.5895 0.413211 1512. nearly 3.179547 Juno 1420 253,600,000 2.66514 0.425710 1594.721 3.202700 Ceres Not well 160 263,236,000 2.76910 0.442334 1683.064 3.226086 Pallas known. 120 265 million 2.77125 0.442725 1685.162 3.226610 Jupiter 89170 490 " 5.20277610.716212 4332.584821 3.636738 Saturn 79040 900 " 9.5387860.979476 10759.219817.031718 Uranus 35000 1800 " 19.1823901.282853 30686.8208 4.486953 Neptune 35000 2850 129.59 11,477121 60128.14 4.7790761 TABLE III. ULEMENTS OF ORBITS FOR THE EPOCH 01 1850, JANUARY 19 MEAN NOON AT GREENWICH. Inclination Variation Long. of the Variation Longitude Variation |Mean longiPlanets. of orbits to in 100 ascehding in 100 of in 100 tude at ecliptic. years, nodes. years. Perihelion. years. epoch. 0 " ", 0, I 0 "o Mercury 7 0 18 +18,2 46 34 40 75 9 47 +93 327 17 9 Venus 3 23 26 -4. 6 75 17 40 +51 129 22 53 +78 243 58 4 Earth 100 22 10 103 100 47 1 Mars 1 51 6 - 0.2 48 20 24 42 333 17 57 +110 182 9 30 Vesta 7 8 29 -12. 103 20 47 -26 254 4 34- 157 113 28 12 Juno 13 2 53..... 170 53,,... 54 18 32........ 165 17 38 Ceres 10 37 17...... | 80-47 56...... 147 25 41..... 1 3 10 Pallas 34 37 44. 1~2 2 38... 121....... 2 42 38........ 121 30 13.......327 31 24 Jupiter 1 18 42 -22. 98 55 19 -57 11 56 0 - 95 160 21 50 Saturn 2 29 29 -15. 112 22 54 -51 90 7 0 4-116 13 58 13 Uranus 0 46 27 3. 73 12 0 +-24 168 14 47 + 87 28 20 22 * It is with -reluctance that we give these planets a place in the tables. The fact of their existence is as yet questionable, and their elements, at present, cannot be well known. We give the logarithms in the tables, that the data may be at hand to exercise the student on Kepler's third law, TABLE III. TABULAR YIEW OF THE SOLAR S~STE1M Nzmes. Mass. Density. ravity. Sidereal Light and Rotation Heat. h. m. s. Mercury.. 2 25 8 1 3.244 1.22 24 5 28 6.680 Venus.....; 1 0.994 L 0.96 23 21 7 1.911 Earth..... - 1.000 1.00 24 0 0 1.000 Mars.....' 8 0 0.973 0.50 24 39 21.431 Jupiter.... 4 8.7 0232 2.70 9 55 50.037 Saturn.... 50.2 0.132 1.25 10 29 17.011 Uranus... 9 71 a 0.246 1.06 Unknown..003 Sun....... 1 0.256 28.19 25 12 0 Moon...... v6 1 0 2 0.665'0.18 27 7 43 TABLE IIL. Rcce;triiti ~Variation in 100 otion in mean Mean Daily oPlanets.its long. in 1 year Motion in oots. ye as. P t. of orbits. ~ years. of 365 days. longitude. 0o " 0o Mercury... 0.20551494 +-.000003868 53 43 3.6 4 5 32.6 Venus...... 0.00686074 -—,.000062711 224 47 29.7 1 36 7.8 Earth...... 0.01678357 -,000041630 -0 14 19.5 0 59 8.3 Mars 0.09330700!+.000090176 191 17 9.1 0 31 26.7 Vesta....... 0.08856000 +.000004009.......... 0 16 17.9 Juno....... 0.25556000.................. 013 33.7 Ceres....... 0.07673780 -,.000005830.......... 0 12 49.4 Pallas..... 0.24199800............ 0 12 48.7 Jupiter...... 0.04816210 +.000159350 30 20 31.9 0 4 59.3 Saturn...... 0.05615050 -.000312402 12 13 36.1 0 0 0.6 Uranus..... 0.04661080 —.000025072 4 17 45.1 0 0 42.4 TABLE III. LUNAR PERIODS. Mean sidereal revolution,.. 27.321661418 Mean synodical revolution,. 29.530588715 Mean revolution of nodes (retrogradey,. 6793.391080 Mean revolution of perigee (direct,.... 3232.575343 Mean inclination of orbit,. 50 8' 48" Mean distance in measure, of the equatorial radius of the earth........ 29.98217 Mean distance in measure of the mean radius,.. 0.20000 TAB LE IVr.9 SUNrS EPOCHS. |Years.| M. Long. Long. Perigee. |I. |It. iIII. N. s. ~ # hi..0, t.t,1846 9 8 45- 8 9 8 1L7 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 S 1849 9 9 1 17 9 8 20'22 -238 420 775 540 1850 9 8 46 58 -9,821 23 598 336 400 594 1 8519 8 32 39 9 8 22 24 958 250 025 648 1852]B. 9 9 17 27 9 8S23 26 353 168 653 701 1853 9 9 3 8 9 ~824 27 713 083 277 755.1854 9 8 48 48 9 8 25 29 073 998 902 809 1855 9 8 34929 9 8 26-30 433 913 527 863 18561B. 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 1i 860 B. 9 9 21 8 9 8 3,1 38 30,1494 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 Bo 9 8 38 10 9 8 34 42 381 239 530 292 } 1864 9 9 ~2258 9 8 35 44 775 157 157 346, 1865 9 9 8 39 ~9 8 36 45 135 072 783 400:(t866 -9 8 54 20 9 8 37 47 495 985 408 453,1867 9 8 40 0 9 8 38 49 855 902 033 507 i1868 1B 9 9 24 49 9 8 39 50 249 820 659 561, 869 9 9 10 30 9 8 40 52 609 734 285 615 t 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 i 1871 9 8-41 51 9 8 42 54 329 564 534 721 l~ 1872 B. 9 9 26 39 9 8 43 -M 72.3 481 161 774 -i 1873 9 9 ~12 20 9 8 45 38 083 396 785 828 1874 9 8 58 1 9 8 47 0 443 311 -410 881! 1875 9 8 43 41 9 8 48 a 803 226 034 935 l.1876 B. 9 9 28 30 9 8 49 4 297 143 6-61 989 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 I 1880 B. 9 9 30 20 9 8 53:9 671 807 164 204 i1881 9 9 3]6 1 9 8 54 10 031 722 790 257 1882 9 9 1 41 9 8 55 12 391 637 415 311 11883 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 17561'9 8 58 16 505 385 292 471 1886 9 9 3 32'9 8 59 17 865 300O918 525 1887,9 8 49 12 9 8 0 19 225 216 544 579 1888 B. 9 9 34 1 9 8 1 20 619 133 169 632........... ~., a 21 TABLE V. SUN'S MOTIONS FOR MONTHS. Months. Longitude.'er I IL III. N S. 6 Jan Co... 0 0 0 0 0 0 0 0 0 a Bis.. Bis 11 29 0 52 0 966 997 998 0 Feb. Com.... 1 0 33 18 5 47 78 53 4 Bis... 0 29 34 10 5 13 75 51 4 March......... 1 28 9 11 I0 993 148 01 9 April.......... 2 28 42 30' 15 42 226 154 13 May........... 3 28 16 40 20 59 301 206 18 Yune........... 4 28 49 58 2 110 379 259 22 July........... 5 28 24 8 31 129 454 310 27 August....... 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 November...... 9 29 38 12 52 300 762 521 45 December...... 0 29 12 22 57 313 837 572 49 TABLE VI. SUN'S HOURLY MOTION. ARGUMENWr.-Sun's Mean Anomaly, Os Is I s sIIs IVs V __ rF I II_.... 0'! l q t! r! II l! ~~~~~~~~~~~0 0 2 33 2 32 2 30 2 8 225 2 24 30 10 233 2 32 2 29 2 7 2 25 2 23 20 20 233 2 31 2 29 2 2'6 2 24 2 23 I0O 30 2 32 2 30 2 28 2 25 2 24 2 23 0,~~~~~~~~~II,[1 XIS s AS"I Xs "-IXs VIII s Vis VIs SUN'S SEMIDIAMETERt. ARGUMENT. —Sun's Mean Anomaly. Os Is S IHs IVs V 0 s 1 I! I I ~1! I 1 0 16 1 615 16 9 16 15 53 15 48 30 10 1619 16 14 16 7 15 58 15 51 15 46 20 20 16 17 16 12 164 15 56 1549 15 46 110 30 16 15'i16 91 16 1 15 53 j 15 48 15 45 0 XIs Xs IXs V....s vi- Vf s TABLE VI, SUN'S MOTIONS FOI DAYS AND HOURS. Days. I Logitude. Per. I. II. III. Nb Hours. Long. I. 000 0 0 0 0 1 2 28 1 2 059 8 0 34 3 2 0 2 4 56 3 3 1 58 17 0 68 5 3 0 3 7 23 4 4 257 25 0 101 8 5 0 4 9 51 6 5 356 33 1 135 10 7 1 5 12 19 7 6 455 42 1 169 13 9 1 6 14 47 8 7 554 50 1 203 15 10 1 7 17 15 10 8 653 58 1 236 18 12 1 8 19 43 11 9 753 7 1 270 20141 9 22 11 13 tO 852 15 1 304 23 15 1 10 24 38 14 11 951 23 2 338 25 17 1 11 27 6 16 12 10 50 32 371 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 2'3 17 15'46 13 3 540 40 272 17 41 53 24 18 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 28 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 SUN S MOTIONS FOR MINUTES. Min. Longitude. Min. Longitude. 1-o I's'13o1'*' A- "'' 11 0 2 1 30 1 16 5 0 12 35 1 26 10 0 25 40 1 39 15 0 37 I 45 1 51 2 0 49 50 2 3 25 I 1 2 55 2 16 30 I 1 14 1 60 1 2 28 2A 12 TABLE VIII, EQUATIONS OF THE SUN S CENTER. ARGUMENT.-Sun's Mean Anomaly. Os Is IIs IIIS IVs Vs 0 0 I 0 r It 0 I t t I t It 0 0 1 59 30 2 58 15 3 40 27 3 54 50 3 38 21 2 56 9 1 2 1 33 3 0 0 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 43 153 54 33 3 35 8 2 50 54 4 2 7 43 3 5 9 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 0 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 52 228 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 0 19 2 37 49 3 27 46 3 53 10 3 47 49 3 14 0 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 23 245 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 28 30 2 58 15 3 40 27 3 54 50 3 38 21 2 56 9 1 59 30 TABLE VIII. 13 EQUATIONS OF THE SUN'S CYNTER. ARGUMENT.-Sun's Mean Anomaly. VIs VIIS VIIIs IXs Xs XIs 0 0 0 I0 I I o, 0 159 30 1 2 51 0 20 39 0 410 01833 1 045 1 57 32 1 1 8 0 19 38 0 48 01933 1 232 2 5533 0 5927 01839 0 4 9 02035 1 419 3 5335 0 5746 01742 0 412 02139 1 6 8 4 5137 0 56 6 0 1647 0 417 02244 1 758 5149 39 0 54 27 0 15 53 0 424 02352 1 948 6 47 41 0 52 47 0 15 2 0 433 025 1 1 1140 7 45 44 0 51 14 0 1412 0 444 0 2612 11332 8 4346 0 4938 0 1324 0 457 02725 1 1526 9 41 49 O 48 5 01238 0 513 0 28 40 1 1720 10 139 52 0 46 32 0 11 53 0 530 02956 1 1915 11 137 56 0 45 0 011 11 0 550 03114 12111 12136 0 0 4330 0 10 31 0 6 11 0 3234 1 23 8 13 1 34 4 0 42 1 0 9 53 0 6 35 0 33 55 1 25 5 14 132 9 040 34 09 16 071 0 3518 127 3 15 13014 0398 0 842 07 29 03642 129 2 16 12-820 037 43 0 89 0759 0389 1311I 17 12626 036 2o00739 0 831 03936 133 1 18 12433 034 58 0710 09 50 419 135 1 19 1 2241 033 38 06 44 0942 0 4236 137 1 20 120 49 032 190 6 20 010 20 0 44 9 1 39 3 21 1 1857 03120 557 011 0 04542 141 4 22 1 17 7 0 2)9 46 0 5 37 0 1i 43 0 47 17 1 43 6 23 -1 15 17 0 28 32 0 5 19 0 12 27 0 48 54 1 45 9'24 113 28 027 190 53 013 13 05032 147 11 25111 40 026 9 0449 014 2 05211 149 14 26 19 52024 59 043.7 014 52 053 51 15117 27 1 86 02352 04 27 01545 055 33 15320 28 16 20 0 224604 19 016 39 05716 155 23 29 1 435 021 4104 13 017 35 059 0 15727 30 1 2 51 0 20 39 0 4 10 0 18 33 1 0 45 1 59 30 14 TABLE IX. SMALL EQUATIONS OF THE SUN'S LONGITUDE. Arg. I II. III. I Argo. II. III. 0 10 10 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 40 12 14 7 550 8 10 6 60 12 14 7 560 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 3 100 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 0 150 15 18 9 650 5 7 C 160 15 18 9 660 5 6 0 170 15 18 10 670 5 6 1 180 15 18 10 680 5 6 1 190 16 18 11 690 4 5 2 200 16 18 11 700 4 5 2 210 16 18 12 710 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 7 480 16 16 17 780 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 19 870 6 2 12 380 14 12 19 880 6 3 13 390 14 12 19 890 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 8 6 13 450 12 10 14 950 8 6 13 460 11 10 13 960 9 7 12 470 11 10 13 970 9 8 12 480 1 10 12 980 9 9 11 490 10 10 11 990 10 9 11 500 10 10 10 1000 10 10 10 TABLE X. 15 NUTATIONS. ARGtcMENT.-Supplement of the Node, or N. N. Long. R. Asc. Obliq. N. Long. R. Asc. Obliq. 0 + 0 + 0 +10 530 - 0 - 0 -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 + 11 + 10 + 8 600 -11 10 -8 120 12 11 7 620 12 11 7 140 14 13 6 640 14 13 6 ]60 15 14 5 660 15 14 5 180 16 15 4 680 16 15 4 200 + 17 + 16 3 700 17 -16 -3 220 18 16 2 720 18 16 2 240 18 16 1 740 18 16 1 260 18 16 _ 1 760 18 16 + 1 280 18 16 2 780 18 16 2 300 + 17 + 16._3 800 -17 -16 + 3 320 16 15 4 820 16 15 4 340 15 14 5 840 15 14 5 360 14 13 6 860 14 13 6 380 12 11 7 880 12 11 7 400 + 11 + 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 o 980 2 2 10 500 + 0 + 0 10 1000 1 0 + 10 TABLE XI. EARTH'S RADIUS VEOTOR.-ARGUMIENT. Sun's Mean Anomaly. - | O09 | IS Is IIs Ills IVs Vs 0.98313 0.98545313 0.99173 1.00018 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 1.00947 1.01503 26 6 0.98322 0.98643 0.99331 1.10193 1.00994 1.01527 24 8 0.98330 0.98679 0.99386 1.002511 1.01040 1.01549 22 10 0.98339 0.98717 0.99441 1.00308 1.01084 1.01569 20 12 0.98350 0.98756 0.99497 1.00366 1.01128 1.01588 18 14 0.98364 0.98797 0.99554 1.00422 1.01170 1.01604 16 16 0.98380 0.98840 0.99611 1.00478 1.01210 1.01619 14 18 0.98397 0.98883 0.99668 1.00534 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 I 1.01666 2 1 30 0.98545 0.99173 1.00018 1.00850 1.01450 1.01667 1 0 XI I Xs lxs VIIIs VIiS VIS 16 TABLE XI. MEAN NEW MOONS AND ARGUMENTS IN JANUARY, - Mean New Moon in I. II. III. IV. N. I January.. - _ A. D. D, H. M. 1836 B. 17 10 32 0469 1246 17 08 669 1837 5 19 20 0171 9852 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 0125 7382 75 73 278 1849 22 20 43 0635 6705 73 61 386 1850 12 5 32 0337 5311 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 t845 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 249 24 0675 7702 84 70 820 1858 13 18 13 0376 6307 1 67 59 843 1859 3 3 1 0078 4913 50 48 865 1860 B. 22 0 34 0588 4236 48 36 972 1861 10 9 22 0290 2840 31 25 995 1862 29 655 0800 2163 30 14 102 1863 18 15 44 0504 0769 13 03 125 1864 B. 8 0 32 0204 9374 96 92 147 1865 25 22 5 0714 8698 94 80 256 1866 15 653 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 20 424 0542 1765 23 12 559 1872 B.I — 813 13 0244 0371 05 01 581 1873 X 27 10 46 0754 9694 03 89 689 1874 19 35 0456 8300 86 78 711 1875 i 4 24 0158 6906 69 67 733 1876 B?, 1 57 0668 6229 67 55 841 1877 3:4 10 41 0370 4835 50 863 j-~ 1 5 31878 -8-48-38- 0072 3441 33 23 885 1879 22 16 11 0582 2764 31 21 993 1 l7t1 1880 B. 11 15 0 0284 1370 14 10 015 -z// 23 o, o5foaS3 t t 3~ ~ r1/z: TABLE XII. 17 MEAN LUNATIONS AND CHANGES OF THE ARGUMENTS. Num. Lunations. I. II. III. IV. N. d. h. m. 2 14 18 22 404 5359 58 50 43 1 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 537 22 93 682 9 265 18 36 7276 454 37 92 767 10 295 7 20 8085 7117 53 91 852 11 324 20 5 8893 7889 68 90 937 12 354 8 49 9702 8606 83 89 22 13 38.3 21 33 510 9323 9J ad88., TABLE XIII. TABLE XIV. NUMIBER OF DAYS FROM THE I _ COMMENCEMENT OF THE YEAR Arg. d A d Arg. TO THE FIRST OF EACH MONTH. I. H. Par. S. D. H. Mo. II. Months. v —, I, Months. Com. Bis. 0 60 29 16 29 36 48 10000 -________ __ _- /250 60 26 16 26 36 44 9750 Days. Days. 500 60 17 16 25 36 19 9500 January... 0( 0 750 60 0 16 21 36 8 9250 1000 59 47 16 17 35 48 9000 February.. 31 31 1250 59 24 16 11 35 28 8750 March.... 59 60 1500 58 56 16 3 34 57 8500 1750 58 30 15 56 34 34 8250 April;..... 90 91 2000 58 7 15 50 33 58 8000 May.... 120 121 2250 57 37 15 42 33 32 7750.2500 57 1 15 31 32 42 7500 June 8.... 151 152 2750 56 32 15 23 32 9 7250 July..... 181 182 3000 56 2 15 16 31 36 7000 3250 55 40 15 10 31 13 6750 ~Augurst~.... 212 1213 3500 55 22 15 7 30 52 6500 September: 243 244 3750 55 12 15 3 30 29 6250 Oct r-". [* 4000 54 51 14 56 30 7 6000 October:... 273 274 4250 54 39 14 54 29 55 5750 November. 304 305 4500 54 26 14 50 29 43 5500 LDecember 3. 334 3.4750 54 18 14 48 29 37 5250 ___ 15000 54 13 14 45 29 35 5000 18 TABLE XV. EQUATIONS FOR NEW AND FULL MOON. Arg. I. II Arg. I. Arg. III. IV. Arg. m. h. m. h. m. h. m. m. m. 0 4 20 10 10 5000 4 20 10 10 25 3 31 25 oo i 436 9 36 - 5100 4 5 10 50 126 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 0 18 800 24 5 46 5800 2 21 15 13 33 4 9 17 900 6 38- 5 15 5900 2 8 15 46 34 4 29 16 1000 6 51 4 46 6000 1 55 16 18 35 4 29 15 1100 7 4 4 17 6100 1 42 16 48 36 5. 28 14 1200 7 15 3 50 6200 1 31 17 16- 37 5 28 13 1300 7 27 3 24 6300 1 19 17 42 38 5 27 12 1400 2? 37 2 59 6400 1 9 18 6 39 5 27 11 1500 7 47 2 35 6500 0 59 18 28 40 6 26 10 1600 7 55 2 14 6600 0 50 18 48 41 6 26 9 1700 8 3 — 1 53 6700 0 42 19 6 42 7 25 8 1800 8 0 1 35 6800 0 34 19 21 43 7 25 7 1900 816 1 18- 6900 0 28 19 33 44 7 24 6 2000 8 21 1 3 7000 0 22 19 44 45 8 23 5 2100 8 25 0 51 7100 0 17 19 52 46 8 23 4 2200 8 29 040 7200 014 1957 47 9 22 3 2300 8 31 0 32 7300 0 11 20 0 48 9 21 2 2400 8/32 0 25 7400 0 9 20 1 49 10 21 1 2500 8 32 F0 21 7500 0 8 19 59 50 10 20 0 2600 8 31 ) 19- 7600 0 8 19 55 51 10 19 99 2700 829 020 7700 0 9 19 48 52 11 19 98 2800 8 26 0 23 7800 0 11 19 40 53 11 18 97 2900 8 23 0 28 7900 0 15 i9 29 54 12 17 96 3000 8 18 0 36 8ooo0 019 1917 55 12 17 95 3100 8 12 0 47 81'00 0 24 19 2 56 13 16 94 3200 8 6 0 59 8200 0 30I 18 45 1 57 1,3 15 93 3300 7 58 1 14- 8300 0 37 18 27 58 13 15 92 3400 7 50 — 1 32 8400 0 45 18 6 59 14 14 91 3500 7 41 1 52 8500 0 53- 17 45 60 14 14 90 3600 7 31 2 14 8600 1 3 17 21 61 15 13 89 3700 7 21 2 38 8700 1 13 16 56 62 15 13 88 3800 7 9 3 4 8800 1 25 16 30 63 15 12 87 3900 6 58 3 32: 8900 1 36 16 3 64 15 12. 86 4000 6 45 4 2 9000 1 49- 15 34, 65 16 11 85 4100 6 32 4 34 9100 2 2 15 5. 66 1,6 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 5 361 6 54 9500 3 0 2. 58 70 17 10 80 4600 1 5 21 7 32 9600 3 164 12 25 71 17 10 79 4700 5 6 8 1- 9700 3 3 11 52- 72 17 10 78 4800 1 4 51 8 50- 9800 3 11 18 73 17 10 77 4900 1 4 35 9 30 9900 4 4 10 44 74 17 9 76 J5000 4 20 10 10 10000 4 20 10 10 75 17 9 75 TABLE E, SHOWING THE EQUATION OF TIME * ARGUMENT. —Sun's true Longitude. Os IIIs Is I IVs Vs VIs VIIS VIIIs] I IXs Xs iXs M. s. m. s. m.. m.. m. s. I m. s. m. s. m. s. m. s. m. s. m. s. m. s. 0 +7 31.2 -1 7.3 -3 43.5 +1 15.0 +6 8.2+-2 27.71- 7 33.4-15 33.3-13 39.41- 1 23.6+11 19.0 +14 9.8 2 6 56.0 1 32.4 3 35.1 1 50.7 6 9.7k 1 57.8 8 16.8 15 48.7 13 6.1'- 0 21.1 11 51.7 13 58.2 4 6 19.8 1 53.6 3 25.0 2 16.4 6 10.01 1 19.9 8 58.31 16 0.4 12 29.3 + 0 35.9 12 22.3 13 42.5 6 5 43.8 2 16.8 3 12.6! 2 42.2 6 8.0i 0 44.0 9 38.1 16 9.0 11 51.0 1 34.3 12 49.0 13 25.1 8 5 04.4 2 35.0 2 57.11 3 10.0 6 3.5i+0 6.6 10 18.0 16 14.3 11 9.1 2 31.2 13 12.3 13 4.8 10 4 25.6 2 55.5 2 38.01 3 23.7 5 58.1 —0 30.7 10 55.71 16 16.51 10 24.4 3 30.0 13 34.0 12 45.0 qm 12 3 49.4 3 9.5 2 20.4 3 56.8 5 46.21 1 12.4 11 32'4 16 15.3 9 37.0 4 25.8 13 50.3 12 19.1 1 14 3 13.2 3 22.0 2 0.0 4 18.3 5 33.2 1 42.3! 12 7.0! 16 11.4 8 48.0 5 20.8 14 6.6 11 52.8 16 2 39.4 3 32.8 1 37.7 4 38.1 5 13.3 2 32.01 12 40.61 16 3.8 7 57.0 6 11.9 14 17.5 11 22.8 18 2 2.4 3 44.1 1 14.8 4 57.9 5 0.21 3 15.9 13 12.11 15 52.9 7 2.9 7 2.6 14 25.0 10 52.4 20 1 25.6 3 50.6 0 49.0 5 15.7 4 41.2 4 04.0 13 42.21 15 40.7 6 9.4! 7 50.3 14 30.6 10 23.7 22 0 53.0 3 54.2 0 24.1 5 30.4 4 17.11 4 42.91 14 8.9' 15 11.0 5 11.1 8 37.5] 14 32.2 9 52.2 24 4-0 21.4 3 54.9 -0 22.0 5 42.7 3 53.0 5 25.8 14 34.11 15 0.6 4 15.3 9 21.5 14 30.9 9 18.1 26 — 0 8.6 3 53.3 +0 28.6 5 54.1 3 26.6 6 9.8 14 56.4! 14 36.2 3 17.9 10 3.2. 14 27.4 8 42.4 28 0 38.7 3 49.4 0 55.9 6 1.1 2 58.3 6 50.5 15 20.7 14 9.1 2 20.2 10 43.1 14 20.5i 8 7.7 30 1 7.3 3 43.5 1 15.0 6 8.2 2 27.7 7 33.41 15 33.3 13 39.4( 1 23.6 11 19.01 14 9.81 7 31.2 * The Sun gives apparent time; to convert this into mean time apply the minutes and seconds, as found in this Table; the MINUS sign indicates subtraction, the plus sign, addition. c. 20 TABLE XVI. MOON S EPOCHS. Years. 1 2 3 4 5 6 7 -8 9 - 1846 0013 2475 3275 1688 0773 4880 3179 0800 9542 1847 0006 9683 2941 6432 3245 0678 4239 3257 8406 1848 B. 0026 7542 3646 1463 6052 6847 5358 6106 7295 184' 0019 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 1852 B. 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 1858 0010 0925 2385, 9476 1439 5565 6078 1460 5981 1859 0003 8134 2051 4220 3911 1362 7139 3917 4845 1860 B. 0023 5992 2756 9551 6718 7531 8257 6765 3734 1861- 0016 3200 2423 3995 9190 3329 9317 9222 2597 1862 0009 0409 2088 8739 1661 9126 0378 1679 1461 186.3 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 2687 4678 1899 6940 1867 0001 7101 1457 2745 4355 8485 5738 4357 5804 1868 B. 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 11872 B. 0020 4432 1867 7039 7383 8215 1158 7423 0172 11873 0013 1640 1533 1783 9854 4012 2239 9880 9036 1874 0006 8848 1199 6527 2325 9809 3300 2337 7900 1675 9999 6056 0865 1271 4796 5606 4361 4794 6764 1876B. 0019 3914 1571 6292 7603 177a 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 1 1285 5545 7823 5335 9782 7859 1132 1881 0011 0604 0951 0289 0294 1 132 0843 0316 9996 1882 0004 7812 1 0617 5033 2765 6929 1904 2873 8860 1883 9997 5020 0283 1 9777 5236 2726 2965 5330 7724 11884 B. 0017 2878 0989 i 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 1888 B. 0016 2360 0693 4051 8263 2455 8386 8395 2092 1889 0009 9568 0359 8795 0734 8252 9447 0852 0956 189)0 0002 6776 0025 3539 3205 1 4049 0508 3309 9820 TABLE XVI. 21 MOON S EPOCHS. 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 810 484 970 644 613 901 486 749 643 433 371 1848 B. 486 876 759 151 905 072 881 143 144 487 539 1849 093 237 479 624 099 212 241 496 619 540 705 1850 700 597 199 097 29'3 352 600 848 094 594 871 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 1.96 072 147 023 168 944 074 300 020 809 539 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 692 546 095 949 042 537 548 1 752 947 024 206 1859 299 907 814 422 236 677 908 105 42? 078 372 1860 B. 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 1868 B, 967 247 500 781 277 032 X51 404 775 561' 874 1869 573 608 219 254 471 172 610 756 251 615 040 1870 180 968 939 737 665 313 969 109 726 668 207 1871 787 328 659 200 8.59 554 328 562 201 721 374 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 1876 B. 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 281 744 809 978 481 234 747 824 969 390 368 1884 B. 958 136 699 485 773 405 143 219 470 454 535 1885 565 496 419 958 967 646 502 672 945 507 702 1886 172 856 139 431 161 887 86'1 125 420 560 869 1887 779 216 859 904 355 128 320 578 895 613 036 1888 B. 456 608 749 411 647 299 7161 973 396 677 203 1889 063 968 469 884 841 540 075 426 871 730 370 1890 670 1328 189 357 035 781 434 879 346 783 537, ~...............,........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 22 ~Ti~A 1LE XVII, MOON S MOTIONS FOR MONTH8. Months. |1 j2 3 4 J 5| 6 7 8 9 Jan. Com. 0000 0000 0000 0000 0000 0000 0000 0000 0000 ) Bis, 9973 9350 8960 9713 9664 9628 9942 9610 9976 Feb. Con. 849 146 2246 8896 402 1533 1789 2099, 753 Bis. 821 9497 1205 8609 66 1161 1731 1709 729 March.... 1615 8343 1371 6931 9797 1951 3404 3027 1433 April...... 2464 8490 3616 5827 199 3484 I 5193 5126 2186 May 3285 7986 4822 4436 265 4646 6924 6835 2914 June..4134 8133 7067 3332 666 6179 8713 8934 3667 July....... 4955 7629 8273 1942 732 7341 444 643 4396 August..i. 5804 7776 518 838 1134 8874 2233 2742 5148 Septernbet. 6653 7922 2764 9734 1536 408 1 4021 4842 5901 October...., 7474 7419 3969 8343 1602 1569 5752 6550 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 8 9 I1 -0000 0000 0000 0000 0000 i 0000 0000 0000 0000 2 27 650 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 5201 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 5L99 8321 2296 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 1445 3157 3691 4093 635 4293 267 13 328 7799 248 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 9155 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 TA3BLE XVI. MOON S MOTIONS FOR MONTHS. Months. 10 11 12 13 14 15 16 17 18 19. 20 Jan. Ca. 000 000 000 000 000 000 000 000 000 000 000 J Bris. 930 969 930 966 901 969 963 958 974 000 000 F Com. 175 965 184 59 74 946 135 304 805 5 14 ebBis. 105 934 114 25 975 916 98 262 779 5 14 March.... 139 836 157 16 851 801 159 482, 5'32 9 27 April..... 314 801 342 76 925 747 294 786 336 13 41 May...... 419 735 556 101 899 663 392 47 115 18 1 55 June..... 593 700 640 160 973 609 527 1 351 920 1 22 6i9 July....... 698 634 754 185 948 525 625 613 699 27 83 August,.. 873 599' 938 245 22 471 759 917 503 31 97 September. 48 563 123 304 96 417 894 221 308 36 111 October.... 152 497 237 329 71 333 992 483 87 40 125 November.. 327 462 1 421 388 145 279 127 787 892 45 139 December.. 432 396 535 414 120 194 225 49 1 670 49 153 TABLE XVIII. MOON S MOTIONS FOR DAYS. Days. 10 11 12 13 14 15 16 117 18 19 20 1 000 000 000 000 000 000 000 000 000 000 000 2 70 31 70 34 99 31 37 42 26 0 0 3 140 62 141 68 198 61 73 84 52 0 1 4 210 93 211 103 297 92 110 126 78 0 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 7 421 187 423 205 595 183 220 252 156 1 3 8 491 218 493 239 694 214 256 1 294 182 1 3 9 561 249 564 273 793 944 293 336 2/08 1 4 10 6f31 280 634 308 892 275 329 379 234 1 4 11 702 311 705 342 1 992 305 366 421 260 1 5 12S 772 342 775 376 91 336 403, 463 2S6 2 5 13 842 374 845 410 190 366 439 505 312 2 5 14 912 405 916 i 444 289 397 476 547 337 2 6 15 982 436 986 478 388 427 512 589 368 2 6 16 52 467 57 513 487 458 549 631 389 2 7 17 122 498 127 547 587 488 586 673 1415 2 7 18 193 529 198 581 686 519 622 715 4411 2 8 19 263 560 268 615 785 549 659 757 467 3 8 20 333 591 339 649 884 580 695 799 493 3 9 21 1403 623 409 1 683 983 611 722 841 517 3 9 22 473 654 480 718 82 641 769 883 545 3 1 23 543 685 550 752 182 672 805 925 571 3 10 24 614 716 621 / 786 281 1702 842 967 1 597 3 11 25 684 747 691 820 389 733 878 9 623 4 11 26 754 778' 762 854 479 763 915 5 649 4- 11 27 / 824 809 832 888 578 794 952 94 675 4 12 I 28 894 840 903 923 677 824 988 136 701 4 12 29 1 964 872 973 957 777 855 25 178 727] 4 13 30 34 903 43 991 876 885 6i 220'753 4 13 31 105 1 934 114 25 975 916 98 262 L779 4 14 -- ~ _ TABLE XIX. MOON S MIOTIONS FOR HOURS. Iours. I 2 l 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 ]08 126 140 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 33 26' 623 997 275 322 357 55 374 23 24 27 650 1040 287 336 372 58 390 24 TABLE XIX. MOON S MOTIONS FOR MINUTES. Min. 2 3 4 5 7 8 9 10 11 121314 I1 0 0o1 0 o 0 0 0 o 5 0 2 4 1 1 1 10 0 0 0 0 0 10 0 5 7 2 2 3 0 3 0 0 0 0 0 1 15 0 7 11 3 3 4 1 4 0 1 0 1 0 1 20 0 9 14 4 5 51 0 1 0 1 0 1 125 0 11 18 5 6] 6 1 7 0 1 1 1 2 30 1 114 22 6 7 8 1 8 0 1 1 1 1 2 35 1 1 1625 7 8 9i 1 10 1 2 1 1 2 1 2 40 11 1829 8 910 211 1 2 1 2 1 3 45 1 20 32 9 10 12 2 i 12 1 2 1 I 1 3 50 1 1 23 36 1011 13 2 13 1 2 1 2 1 3 55 1 25 40 11 13 14 2 15 1 3 1 31 1 1 4 6011 2743 12 1 14 15 2 16 1 3 1 3 1 4 TAB3LES. 26 IrELIOCENTRIC LONGITUDES, ETC. OF THE PLANET VENUS, AT THE TIMES OF THE NEXT TWO TRANSITS OVER THE SUN'S DISC. The subject matter of this table is connected with Chapter IX, page 119. Times, Hel. Long. from Hel. Lat. Rad. Vec. __________Times__.true Equinox. 1874, Dec. 8th, at 12h. 760 41' 36.6" 4' 6.3" N.' 0.7203632 16h. 76 57 44.1 5 3.5 0.7203449 20h. 77 13 51.5 6 1.0 0,7203268 1882, Dec. 6th, at noon. 74 12 55.7 4 58.1 S. 0.7205502 4h. 74 29 2.5 4 0.8 0.7205315 8h. 74 45 9.7 3 3.5 0.7205127 DIP OF THE HORIZON. iFor the principle of computing the dip of the horizon see text-note, page 54. lHight Hight in Dip. in Dip. feet. feet. 1 1' 1" 16 4' 3" 2 1 26 17 4 11 3 1 45 18 4 18 4 i2 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 BUN3'S SEMIDIAMETER FOR EVERY TENTH DAY OF THE YEA3R. Days. Jan. July. Days. April. Oct. 1 16 18 15 46 1 16 1 16 1 11 16 17 15 46 11 15 58 16 3 21 16 17 15 46 21 15 55 16 7 Feb. August. May. Nov. 1 16 15 15 47 1 15 53 16 9 11 16 13 15 49 11 15 51 16 12 21 16 11 15 51 21 15 49 16 14 March. Sept. June. Dec. 1 16 10 15 53 1 15 48 16 16 11 16 17 15 56 11 15 46 16 17 -21 16 4 15 58 21 15 46 16 18 22 26 TABLE XX. MOON S EPOCHS. Years. Evection. Anomaly. Variation. Longitude. S O S 0 S O' " S O' " 1846 2 0 45 6 0 26 21 2 1 5 48 4 10 15 48 23 1847 7 21 16 35 3 25 4 23 5 15 25 29 2 25 11 28 1848B. 1 23 7 5 7 6 51 37 10 7 14 21 7 17 45 8 1849 7 13 38 35 10 5 34 57 2 16 51 46 11 27 8 14 1850 1 4 10 4 1 4 18 18 6 26 29 11 4 6 31 20 1851 6 24 41 35 4 3 1 38 11 6 6 36 8 15 54 25 1852B. 0 26 32 5 7 14 48 53 3 27 55 29 1 8 28 6 1853 6 17 3 34 10 13 32 13 8 7 32 53 5 17 51 11 1854 0 7 35 4 1 12 15 34 0 17 10 19 9 27 14 17 1855 5 28 6 33 4 10 58 54 4 26 47 43 2 6 37 22 1856B. 11 29 57 3 7 22 46 9 9 18 36 36 6 29 11 3 1857 5 20 28 33 10 21 29 29 1 28 14 1 11 8 34 9 1858 11 11 0 2 1 20 12 50 6 7 51 26 3 17 57 14 1859 5 1 31 33 4 18 56 10 10 17 28 52 7 27 20 20 18601B. 11 3 22 3 8 0 43 25 3 9 17 44 0 19 54 0 1861 4 23 53 33 10 29 26 45 7 18 55 9 4 29 17 6 1862 10 14 25 3 1 28 10 6 11 28 32 34 9 8 40 12 1863 4 4 56 33 4 26 53 27 4 8 10 0 1 18 3 18 1864 B. 10 6 47 2 8 8 40 41 8 29 58 51 6 10 36 58 1865 3 27 18 32 11 7 24 2 1 9 36 17 10 20 0 4 1866 917 50 2 2 6 7 23 5 19 13 42 2 29 23 10 1867 3 8 21 32 5 4 50 43 9 28 51 8 7 8 46 15 18681B. 9 10 12 2 8 16 37 58 2 20 40 0 0 1 19 56 1869 3 0 43 33 11 15 21 19 7 0 17 25 4 10 43 2 1870 8 21 15 2 2 14 4 40 11 9 54 50 8 20 6 8 1871 2 11 45 31 5 12 47 1 3 19 31 16 0 29 28 13.7 1872 B. 8 2 17 0 8 11 30 21.7' 7 29 8 41 5 8 51 19.4 1873 2 4 7 31 11 23 17 36.6 0 20 57 36 10 1 25 0.3 1874 7 24 39 0 2 22 0 57.3 5 0 35 0 2 10 48 6 1875 1 15 10 29 5 20 44 18 9 10 12 24 6 20 11 11.7 1876B. 7 5 41 59 8 19 27 38.7 1 19 49 50 10 29 34 17.4 1877 1 7 32 30 0 1 14 53.6 6 11 38 40 3 22 7 58.3 1878 6 28 3 59 2 29 58 14.3 0' 21 16 5 8 1 31 4 1879 0 18 35 28 5 28 41 35 3 0 53 30 0 10 54 9.7 18801B. 6 9 6 58 8 27 24 55.7 7 10 30 55 4 20 17 15.4 1881 0 10 57 29 0 9 12 10.6 0 2 19 47 9 12 50 56.3 1882 6 1 28 58 3 7 55 31.3 4 11 57 12 1 22 14 2.0 1883 1-1 22 0 27 6 6 38 52.0 8 21 34 37 6 1 37 7.7 1884 B. 512 31 56 9 5 22 12.7 1 1 12 2 10 11 0 13.4 1885 1 14 22 28 0 17 9 27.: 5,23 0 54 3 3 33 54.3 1886 5 4 53 57 3 15 52 48.3 10 2 38 19 7 12 57 0.0 1887 10 25 25 26 6 14 36 9.0 2 12:15 44 11 22 20 5.7 18881B. 4 I,5' 56 57 9 13 19 29.7 6 21 53 9 4 1 43 11.0 1.889 16 17 47 28 0 25 6 44.6 11 13 42 1, 8 24 16 51.9,'1890 48 18 57 3 23 50 5.3 3 23 19 26 1 3 39 57.6 TABLE X 27 MOON'S EPOCHS. Years. Supp. of Node. II. V. VI. VII. VIII. IX. X. S 0 t s 0 1846 4 16 35 9 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 7 0 9 116 122 582 587 042 997 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 f200 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 31 634 636 456 459 557 639 1855 10 10 38 51 9 5 3 047 048 763 765 637 579 1856 B. 11 0 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 0 8 41 8 8 17 48 323 325 715 718 912 402 1859 0 28 0 51 0 8 20 736 737 023 024 992 342 1860 B. 1 17 23 43 4 10 1 184 189 359 364 108 286 1861 2 6 43 27 8 0 33 598 601 666 670 187 226 1862 2 263 9 11 21 5 012 014 974 977 267 165 1863 3 15 23 11 3 11 37 426 426 282 283 347 105 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 1868 B. 6 22 7 43 10 16 36 563 567 877 882 818 812 1869 7 11 27 46 2 7 8 977 980 185 188 897 752 1870 8 0 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. 99 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 90 57 077 080 752 758 332 453 1875 11 7 28 48 0 21 29 490 492 054 064 412 392 1876 B. 11 26 48 31 4 12 1 904 905 364 370 492 331 1877 0 16 11 24 8 13 42 352 357 700 710 607 276 1878 1 5 31 6 0 4 14 765 769 008 018 687: 215 1879 ] 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 3 867 869 576 586 122 916 1884 B. 5 1 32 29 10 18 35 280 281 884 894 202 855 1885 5 20 55 22 e 20 16 728 733 220 234 317 800 1886 6 10 15 4 6 10 48 141 145 528 542 397 739 1887 6 29 $4 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 i8 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 501w. W.... - -..., -. And TABLE XX, MOON' S MOTIONS FOR MONTHS. Months. Evection. Anomaly. Variation. M. Longitude. 0! I! s 0 I I S 0 - Jan. Com.. 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Jb.j Bis..11 18411111656611174833111649 25 Feb. Com~..ll2048 42 1 15 t0O53 0 17 5448 118 28 6 eBis.11 92943 1 15659 0 54321 1 5 17 31 March.....10 7 4026 1 2050 411291515 1 27 24 27 April... 92829 8 3 55057 01710 3 3 15 52 32 May........ 97 5851 4 74756 0225324 4 21 10 3 June....... 8284733 5224849 1104811 6 9 38 9 July........ 8 8 1716 6244548 1163132 7 14 55 40 August..... 729 5 59 8 94642 2 42620 9 3 23 46 September.., 719 54 4i 9244735 22221 7 10 21 51 52 October..... 629 24 24 10264434 228 42811 27 9 22 November...620 13 6 0114527 3155916 1 15 37 28 December. 529 42 49, 1 134226 3214237 2 20 54:59 TABLE XX. MOON'S MOTIONS FOR DAYS. DaysJ Evection. Anomaly. Variation. Mean Longitude, 1 Os 0'0" Os 0o o' 0" 0 s 00' 0" Os oo O' 0" 2 0 11 18 59 0 13 3 54 0 12 11 27 0 13 10 35 3 0 22 37 59 0 26 7 48' 0 24 22 53 0 26 21 10 4 1 3 56 58 1 9 11 42 1 6 34 20 1 9 31 45 5 115 1558 1 22 15 36 1 18 45 47 1 22 42 20 6 1 26 34:57 2 5 19 30 2 0 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 1 27 18 25 20 7 3 2 14 5 9 3 0 31 55 3 14 31 12 3 7 31 34 3 15 24 40 10 3115055327356 319430 3235 15 11 3 23 954 4 10 39 0 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 5 8 7 0 14 4 27 653 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 B 19 44 51 6 15 58 29 6 -2 51 40 6 17 38 45 17 6 1 351 6 o2 23 6 15 3 7 7 0 49 20 18 6 1222 50 712 6 17 627 1434 7 1359 55 19 6 23 4150 7 25 1011 7926 1 7 27 1030 20 7 5049 88 14 57 2137207 81021 5 21 7 16. 19 49 8 21 17 59 8 3 48 54 8 23 31 40 22 727 33489 421 53 816 021 96 4216 23 8 85747 91725 47 8 28 1147 9 1952 51 24.8 20 164710 029 41 9 1023 14 103 3 26 259 135 46 101333 35 922 3441 1016 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 2910 16 51 44 0 5 49 11 11 11 20 28 0 8 56 21 30 10 28 1043 0 18 53 5 11 23 31 54 0 22 6 56 11 9 29 43 1 1 56 59 0 5 43 21 1 5 1731 TABLE XX, 29 MOON'S MOTIONS FOR MONTHS. Months. Supp. of Node. II. V. VI. VII. VIII. IX. X S O 0 Jan Comr.. 0 0 0 0 0 0 000 000 000 000 000 0' Jan Bis.. 11 29 56 49 11 18 51 966 961 972 966 964 995 Feb. Com.. 0 1 38 30 11 15 43 54 224 875 45 111 165 Bis. 0 1 35 19 11 4 34 20 185 847 11 75 159 March..... 0 3 7 27 9 27 59 7 330 66 989 114 313 April...... 0 4 45 57 9 13 42 61 554 542 34 225 478 May........ 0 6 21 16 8 18 15 81 738 389 46 300 638 June..... 0 7 59 46 8 3 58 136 962 264 91 411 802 July........ 0 9 35 5 7 8 32 156 147 112 103 486 962 August... 0 11 13 35 6 24 15 210 371 987 147 497 126 September... 0 12 52 5 6 9 58 265 595 862 193 708 291 October..... 0 14 27 24 5 14 32 285 780 710 204 783 451 November... 0 16 5 53 5 0 15 339 4 585 250 894 615 December... 0 17 41 13 4 4 49 359 188 432 261 969 775 TABLE XX. MOON S MOTIONS FOR DAYS, iv,., xxo Days. Supp. of Node. II. V. VI. VII. VIII. IX. X. 1 0o 0' 0" Os 00 0' 000 000 000 000 000 000 2 0 3 11 11 9 34 39 28 34 36 5 3 0 6 21 22 18 68 79 56 67 72 11 4 0 9 32 1 3 27 102 118 85 101 108 16 5 0 12 52 1 14 37 136 158 113 135 143 21 6 0 15 53 1 25 46 170 197 141 169 179 27 8 0 19 4 2 6 55 204 237 169 202 215 32 8 0 22 14 2 18 4 238 276 198 236 251 37 9 0 25 25 2 29 13 272 316 226 270 287 43 10 0 28 36 3 10 22 306 355 254 303 323 48 11 0 31 46 3 21 31 340 395 282 337 i358 53 12 0 34 57 4 2 40,,374 434 311 371 394 58 13 0 38 7 4 13 50 408 474 339 405 430 64 14 0 41 18 4 24 59 442 513 367 438 466 69 15 0 44 29 5 6 8 476 553 395 472 502 74 16 0 47 39 5 17 17 510 592 424 506 538 80 17 0 50 50 5 28 26 544 632 452 539 573 85 18 0 54 1 6 9 35 578 671 480 573 609 90 19 0 57 11 6 20 44 612 711 508 607 645 96 20 1 22 7 1 53 646 750 537 641 681 101 21 1 3 33 7 13 3 680 790 565 674 717 196 22 1 6 43 7 24 12 714 829 593 708 753 112 23 1 9 54 8 5 21 748 869 621 742 788 117 24 1 13 5 8 16 30 782 908 650 775 824 122 25 1 16 15 8 27 39 816 948 678 809 860 128 26 1 19 26 9 8 48 850 987 706 843 896 133 27 1 22 36 9 19 57 884 027 734 877 932 138 28 1 25 47 10 1 6 918 066 762 910 968 143 29 1 28 58 10 12 16 952 106 791 944 003 149 30 1 32 8 10 23 25 986 145 819 978 039 154 31 1 35 19 11 4 34 020 185 847 011 075 15) 30 TABLE XX. MOON S MOTIONS FOR HOURS. Hours. Evection. Anomaly. Variation. Longitude. 0 I. o 1 O' I 1 0 2817 03240 030 29 03256 2 05635 1 5 19 1 057 1 553 3 1 2452 1 3759 1 3126 1 3849 4 15310 21039 2 1 54 21146 5 2 21 27 2 43 19 2 3223 2 4442 6 2 4945 3 1558 3 252 31739 7 3 18 2 34838 33320 35035 8 34620 42118 4 349 42332 9 41437 45358 43417 45628 10 44255 52637 5 446 52925 11 51112 55917 53515 6 221 12 5 3930 63157 6 543 63517 13 6 747 7 437 63612 7 814 14 636 5 73716 7 6 40 74110 15 7 422 8 956 7 37 9 814 7 16 73240 84236 8 738 847 3 17 8 057 81516 838 6 920 0 18 82915 94755 9 835 95256 19 85732 10 2035 939 3 102553 20 925 50 105315 10 932 105849 21 954 7 112555 1040 1 113146 22 102224 11 58 34 111029 12 442 23 105042 123114.11 4058 123739 24 111859 13 354 121127 131035 TABLE XXI. MOON S MOTIONS FOR MINUTES. Sup. Min. Evec. Anomaly. Variations. Longitude. Node. II.,,, r 1t /I It 1 0 28 0 33 0 30 0 33 0 0 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 2?1 50 23 34 27 13 25 24 27 27 7 23 55 25 56 29 56 27 56 30 12 7 26 60 28 17 32 40 30 29 32 56 8 28 TABLE XX. 31 MOON'S MOTIONS FOR HOURS. H' |ours. Suppof II. V. VI. VII. VIII. IX. X. Node. 1J ) 0 8 0 28 1 2 1 1 1 O 2 0 16 0 56 3 3 2 3 3 0 3 0 24 1 24 4 5 4 4 4 1 4 0 32 1 52 6 5 6 6 1 5 0 40 2 19 7 8 6 7 7 1 6 0 48 2 47 9 1i 7 9 9 1 7 0 56 3 15 10 13 8 10 10 2 8 1 4 3 43 11 13 9 11 12 2 9 1 11 4 11 13 15 11 13 13 2 10 1 19 4 39 14 16 12 14 15 2 I1 1 27 5 7 16 18 13 15 16 2 12 1 35 5 35 17 20 14 17 18 3 13 1 43 6 2 18 21 15 18 19 3 14 1 51 6 30 20 23 16 19 21 3 15 1 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 8 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.* TABLE B. PERTURBATIONS OF EARTH S _ )'S APPROX. LAT.-ARG. N. RADIUS VECTOR. N. N. S. S.. s A. D. D. A. Lat. Arg. I. II. III. Arg. I. II. III. - __ ___ 0 500 500 1000 0 0 0 8 4 3 500 2 50 4 495 505 995 9 41 50 8 4 3 550 2 1 4 10 490 510 990 19 22 100 7 4 2 o600 3 1 3 15 485 515 985 29 3 150 7 4 11 650 3 2 2 2 20 480 520 980 38 -40 200 6 4 0 700 4 3 1 25'475 525 975 48 18 250 5.4 0 750 5 4 0 30 470 530 970 58 40 300 4 3 1 800 6 4 0 35 465 535 965 67 28 30 13 2 2 850 7 41 410 460 540 960 76 45 400 3 1 3 9007 4 2 45 455 545 955 86 21 450 2 1 1 4 9508 4 3 50 450 550 950 95 26 500 2 0.1 4 11000 8 4 3 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. —ARGUIUIENT 1. Arg. 1 Diff. Arg 1 Diff. 0 12 40 42 5000 12 40 40 100 11 58 42 5100 13 20 41 200 11 16 42 5200 14 1 40 300 10 34 41 5300 14 41 400 9 53 41 5400 15 20 40 500 9 12 40 5500 16 0 38 600 8 32 38 5600 16 38 700 7 54 38 5700 17 15 800 7 16 36 5800 17 52 900 6 40 5900 18 27 1000 6 6 6000 19 1 1100 5 33 31 6100 19 33 31 1200 5 2 30 6200 20 4 1300 4 32 27- 6300 20 33 1400 4 5 25 6400 21 1 26 1500 3 40 23 6500 21 27 23 1600 3 17 21 6r100 21 50 22 1700 2 56 21 6700 22 12 1800 2 32 16 6800 22 31 17 1900 2 22 13 6900 22 48 15 2000 2 9 1 7000 23 3 12 2100 1 58 7100 23 15 2200 1 50 8 7200 23 25 2300 1 44 7300 23 32 2400 1 41 7400 23 37 2 2500 1 41 0 7500 23 39 2600 1 43 7600 23 39 2700 1 48 7700 23 36 2800 1 55 7800 23 30 2900 2 5 7900 23 22 3000 2 17 12 8000 23 11 1 3100 2 32 8100 22 58 13 3200 2 49 17 8200 22 42 16 3300 3 8 1 8300 22 24 3400 3 30 8400 22 3 21 3500 3 53 23 8500 21 40 23 3600 4 19 26 8600 21 15 25 3700 4 46 27 8700 20 48 27 3800 5 16 30 8800 20 18 30 3900 5 5 47 3 8900 19 47 4000 6 19 32 9000 19 14 33 4100 6 53 9100 18 40 4200 7 28 35 9200 18 4 36 37 38 4300 8 5 9300 17 26 3 4400 8 42 3 9400 16 48 40 4500 9 20 38 9500 16 8 4 4i00 9 59 4 9600 15 27 41 470,0 10 39 40 9700 14 46 42 4800 11 19 40 9800 14 4 42 49q00 11 59 41 9900 13 22 42 5000 12 40 0000 12 40 ___ __ _ _ __- _ _ TABLE XXII. 33 EQUATIONS 2 TO 7 OF MOON'S LONGITUDE.-ARGUMENTS 2 TO 7. Arg. 2 3 4 5 6 7 Arg. 2500- 4 57 0 2 6 30 3 39 0 6 0 1 2500 2600 4 57 0 2 6 30 3 39 0 6 0 1 2400 2700 4 56 0 3 6 29 3 38 0 7 0 1 2300 2800 4 55 0 3 6 27 3 37 0 8 0 2 2200 2900 4 53 0 4 6 24 3 36 0 9 0 3 2100 3000 4 50 0 5 6 21 3 34 0 1( 0 4 2000 3100 4 47 0 6 6 17 3 32 0 12 0 5 1900 3200 4 43 0 8 6 12 3 29 0 14 0 6 1800 3300 4 39 0 9 6 7 3 26 0 17 0 8 1700 3400 4 34,0 11 6 1 3 22 0 19 0 10 1600 3500 4 29 0 13 5 54 3 18 0 22 0 12 1500 3600 4 23 0 15 5 47 3 14. 0 25 0 14 1400 3700 4 17 0 18 5 39 3 10 0 29 0 17 1300 3800 4 11 0 20 5 30 3 5 0 33 0 19 1200 3900 4 4 0 23 5 21 3 0 0 37 0 22 1100 4000 3 57 0 26 5 12 2 54 0 41 0 25 1000 4100 3 49 0 29 5 2 2 49 0 45 0 28 900 4200 3 41 0 32 4 52 2 43 0 50 0 31 800 4300 3 33 0 35 4 41 2 37 0 54 0 35 700 4400 3 24 0 39 4 30 2 30 0 59 0 38 600 4500 3 15 0 42 4 19 2 24 1 4 0 42 500 4600 3 7 0 46 4 7 2 17 1 9 0 45 400 4700 2 58 0 49 3 56 2 10 1 14 0 49 300 4800 2 48 0 53 3~ 44 2 4 1 19 0 53 200 4900 2 39 0 56 3 32 1 57 1 25 0 56 100 5000 2 30 1 0 3 20 1 50 1 30 1 0 0000 5100 2 21 1 4 38 1 43 1 35 1 4 9900 5200 2 11 1 7 2.56 1 36 1 40 1 7 9800 5300 2 2 1 11 2 44 1 29 1 46 1 11 9700 5400 1 53 1 14 2 33 1 23 1 51 1 15 9600 5500 1 44 1 18 2 21 1 16 156 1 18 9500 5600 1 36 1 21 2 10 1 10 2 1 1 22 9400 5700 1 27 1 25 1 59 1 3 2 6 1 25 9300 5800 1 19 1 28 1 48 0 57 2 10 1 28 9200 5900 1 11 1 31 1 38 0 51 2 15 1 32 9100 6000 1 3 1 34 1 28 0 46 2 19 1 35 9000 6100 0 56 1 37 1 19 0 40 2 23 1 38 8900 6200 0 49 1 39 1 10 0 35 2 27 1 40 8800 6300 0 33 1 42 1 1 0 30 2 31 143 8700 6400 0 36 1 44 0 53 0 26 2 35 1 46 8600 6500 0 31 1 47 0 46 0 21 2 38 1 48 8500 6600 026 1 49 0 39 0 18 2 41 1 50 8400 6700 0 21 1 51 0 33 0 14 2 43 1 52 8300 6800 0 17 1 52 0 28 0 11 2 46 1 54 8200 6900 0 13 1 54 0 23 0 8 2 48 1 55 8100 7000 0 10 1 55 0 19 0 6 2 50 1 56 8000 7100 0 7 1 56 0 16 0 4 2 51 1 57 7900 7200 0 5 1 57 0 -13 0 2 2 52 158 7800 7300 0 4 1 57 0 11 0 1 2 53 1 59 7700 74900 0 3 1 58 0 10 01 2 54 1 59 7600 7500 0 3 1 58 0418 0 1 2 54 1 59 7500 34 TABLE XXIII. EQUATIONS 8 TO 9 OF MOON S LONGITUDE.-ARGUMENTS 8 TO 9. Arg. 8 9 Arg. 8 9 0 1 20 1 I20 5000 1 20 I 0 1 20 - 1 20 5000 1 20 1 20 100 1 15 1 29 5100 1 24 1 26 200 1 11 1 37 5200 1 29 1 31 300 1 7 1 46 5300 1 33 1i 37 400 1 2 1 54 5400 1 37 1 42 500 0 58 2 1 5500 1 42 1 47 600 0 54 2 8 5609 1 46 1 51 700 0 50 2 15 5700 1 50 1 55 800 0 46 2 20 5800 1 54 1 58 900 0 42 2 25 5900 1 58 2 0 1000 0 38 2 29 6000 2 1 2 1 1100 0 35 2 32 6100 2 5 2 2 12!00 0 31 2 34 6200 2 8 2 2 1:300 0 28 2 35 6300 2 11 2 1 1400 0 25 2 35 6400 2 14 1 59 1500 0 33 2 34 6500 2 17 1 56 1600 0 20 2 32 6600 2 19 1 52 1700 0 18 2 29 6700 2 22 1 48 1800 0 16 2 26 6800 2 24 1 43 1900 0 14 2 21 6900 2 25 1 38 2000 0 13 2 iG6 7000 2 27 1 32 2100 0 11 2 11 7100 2 28 1 25 2200 0 10 2 4 7200 2 29 1 18 2300 0 10 1 58 7300 2 30 1 11 2400 0 9 1 51 7400 2 30 1 4 2500 0 9 1 43 7500 2 31 0 56 2600 0 10 1 36 7600 2 3e 0 49 2700 0 10 1 29 7700 2 30 0 42 2800 0 11 1 22 7800 2 29 0 36 2900 0 12 1 15 7900 2 28 0 29 3000 0 13 1 8 8000 2 27 0 24 3100 0 15 1 2 8100 2 26 0 18 3200 0 16 0 57 8200 2 24 0 14 3300 0 18 0 52 8300 2 22 0 10 3400 0 21 0 47 8400 2 20 0 8 3500 0 23 0 44 8500 2 17 0 6 3600 0 26 0 41 8600 2 15 0 5 3700 0 29 0 39 8700 2 12 0 5 3800 0 32 0 38 8800 2 9 0 6 3900 0 35 0 38 8900 2 5 0 8 4000 0 39 0 39 9000 2 2 0 11 4100 0 42 0 40 9100 1 58 0 15 4200 0 46 0 42 9200 1 54 0 20 4300 0 50 0 45 9300 1 50 0 25 4400 0 54 0 49 9400 1 46 0 32 4500 0 58 0 53 9500 1 42 0 39 4600 1 3 0 58 9600 1 38 0 46 4700 1 7 1 3 9700 1 33 0 54 4800 1 11 1 9 9800 1 29 1 3 4900 1 16 1 14 9900 1 24 1 11 5000 1 20 1 20 10000 1 20 1 20 L __ _ _ _ _ _ _ _ _ _ _ _ _ _ - __ _ _ _ __ _ _ TABLE XXIII. 35 EQUATIONS 10 AND 11. EQUATIONS 12 TO 19. QIUATION 20. Arg. 10 11.10 11 Arg. 121 131415161718 19 Arg.Arg. 20 Arg. It tl I I i 11 1 i ii I r 1 I I II 0 10 10 500 10 10 250 2 28 034 3 17 3 250 0 10 500 10 9 11 510110111 260 2 28 0 314 17 3 240 10 11 5310 20 9 12520 9 11 27012 28 0 34 3 17 3 230 20 1220 30 8113 530 9112 280 3 2 81 0133 317 3 220 30 13 530 40 7 14 540 8 13 290 3 28 0 33 4 16 3 210 40 13 540 50 7 15 1550 8 14 300 3 28 0133 4116 32 00 50 14 550 60 611611560 8 14 310 3 39 1331 31416 3 190 60 15 560 70 6 17 570 8 15 320 4 3 9 1 32 4 16 4 180 70 16 570 80 5117 580 7115 330 4 4 9 1 321 416 4 170 80 16 580 90 5 18 590 7 15 340 5 410 2 32 4 16 4 160 90 17 590 100 5118 600 7 16 350 6 5 10 2 31 5 15 4150 100 17 600 110 4 19 610 7116 360 6 611 2 31 5 15 5 140 110 17 610 120 4 191 620 7116 13707 711 3 301 5115 5 130 120 17 620 130 4119 630 7116 380 8 71121 3129 5115 5 120 130 17 030 140 4 19 640 7 15 390 9 8012 4 29 6 14 6 110 140 17 640 150 4 19 650 8 15 400 10 9131 4281 614 6 100 150 17 650 160 4 19 660 8 15 410 10110 131 51271 614 6 00 160 i7 660 170 4118 670 8141 142011111141 5 271 7 13 7 80 170 16 670 180 5118 680 9113 430 12 12 15 61261 713 7 70 180 16 680.190 517 6 190 913 440 1313151 6125 8112 7 60 190 1 690 200 5116 70010 12 450114 14|16 7 24 8 12 8 50 200 14 700 210 6 16 710110111 460 116115117 7123 812 8 40 2,10 13 710 220 6/15 720O11110 1470117 16181 8 231 9111 9 30 220 13 7.20 230 7 14 73011 1 1480181181891'2 911 9 20 230 127 240 7I13 740112 9 1490 19 19119 9 21110 10 10 10 240 11 740 250 8I12 750112 8 500 120 20120 10 20 1010 10 000 250 10 750 260 811 760113 7 510 121 212111 191110 10 10 990 260 9 760 270 1 10'7701 13 6 520 122 221 21111181 11 9111 980 270 8 770 2801 9110 780114 5 1530 23 2312212 17 11 9 11 70 280 7 780 290 10 9 790014 4 54024 25423 12 17112 8112 960 i 290 6 79( 300 10 8800115 3 1 550 12512624131612 8112 950 300 61 800 310 11 7. 810115 3 1 560 126 27124 14 15112 7 13 940 310 5 810 320 11 6 820115 2 570 27 28125 14114I13 7 13 930 320 4 820 330 12 6 830116 2 580 28 29126 15113113 7113 920 330 4 830 340 12 5 84016 1 590 129 30126 15 13 13 6 14 910 340 3 840 1350 12 51 850116 1 600 30 31,27 16112 14 61 14 900 350 81 850 360 12 5 860116 1 610 31 32128 16111 14 6 14 890 360 3 860 370 13 4 870116 1 1620 32 3328 17111i14 5 15 880 370 3 870 380131 41 8801 1 630 3333 171015 515870 380 3 880 390 13 4 890116 1 640 34 34 2918 915 15 860 390 3 890 400113 4 900115 21 650 34 3530 1 9115 5116 850 400 31 900 410 13 5 910115 2 660 35 36;30118 816 4116 840 410 3 910 420 12 51 920115 31 1 670 351 36I31 1'' 816 4116 830 420 4 920 430 12 5 93014 3 680 363731191 81169 4116 820 430 4 930 440 12 6 940114 4 690 36137 31 191 7 16 4117 810 440 5 940 450 12 6 950113 5 700 371371 32 1917116 4117 800 450 6 950 46011 11796013 6 17101373832 207163117 790 460 6 960 470 11 81 97012 7 720 371 383220 6116 3 17 780 470 7 970 480 11 8 980o11 8 730 38138 32120 61161 3117 770 480 8 980 490 10 99011 9 740 38 38 32 201 6117 3 17 760 490 9 990 500 10 1010 10 10 750.80 383832120o 61 317 7501 00 10 1000 22 2c 36 TABLE XXIV, EVEOTION. Argument. —Evection Corrected. Os Is I IS IIIs Vs Vs 00 1 30' 0" 210' 43"' 2040 0" 20~ 50' 25' 20 39' 8" 20 9' 42' 1 1 31 25 2 11 57 2 40 51 2 50 23 2 38 25 2 8 29 2 1 32 51 2 13 9 2 41 30 2 50 20 2 37 40 2 7 16 3 1 34 16 2 14 21 2 42 8 2 50 15 2 36 55 2 6 2 4 1 35 42 2 15 31 2 42 45 2 50 9 2 36 8 2 4 47 5 1 37 7 2 16 41 2 43 21 2 50 1 2 35 19 2 3 32 6 1 38 32 2 17 50 2 43 55 2 49 52 2 34 30 2 2 16 7 1 39 57 2 18 58 2 44 27 2 49 41 2 33 40 2 1 0 8 1 41 21 2 20 5 2 44 59 2 49 29 2 32 48 1 59 43 9 1 42 46 2 21 11 2 45 29 2 49 15 2 31 55 1 58 26 10 1 44 10 2 22 17 2 45 57 2 49 0 2 31 2 1 57 8 11 1 45 34 2 23 21 2 46 24 2 48 43 2 30 7 1 55 49 12 1 46 58 2 24 24 2 46 50 2 48 26 2 29 11 1 54 30 13 1 48 21 2 25 26 2 47 14 2 48 6 2 28 14 1 53 11 14 1 49 44 2 26 28 2 47 37 2 47 45 2 27 16 1 51 51 15 1 51 7 2 27 28 2 47 59 2 47 23 2 26 17 1 50 31 16 1 52 29 2 28 27 2 48 19 2 47 0 2 25 17 1 49 11 17 1 53 51 2 29 25 i2 48 37 2 46 35 2 24 16 1 47 50 18 1 55 12 2 30 21 2 48 54 2 46 8 2 23 14 1 46 29 19 1 56 33 2 31 17 2 49 10 2 45 41' 2 22 11 1 45 7 20 1 57 53 2 32 11 2 49 24 2 45 12 2 21 7 1 43 46 21 1 59 13 2 33 5 2 49 37 2 44 41 2 20 2 1 42 24 22 2 0 32 2 33 57 2 49 48 2 44 9 2 18 56 1 41 2 23 2 1 51 2 34 48 2 49 58 2 43 36 2 17 50 1 39 39 24 2 3 9 2 35 38 2 50 6 2 43 2 2 16 43 1 38 17 25 2 4 26 2 36 26 2 50 13 2 42 26 2' 15 34 1 36 54 26 2 5 43 2 37 13 2 50 19 2 41 49 2 14 25 1 35 32 27 2 6 59 2 59 25023 2 41 11 2 13 16 1 34 9 28 2 8 15 2 38 44 2 50 25 2 40 31 2 12 5 1 32 46 29 2 9 30 2 39 28 2 50 26 2 39 50 2 10 54 1 31 23 30 2 10 43 2 40 10 1 2 50 25 2 39 8 2 9 42 1 30 0 TABLE XXV. MooN's EQUATORIAL PARALLAX. Argument.-Arg. of the Evection. rOs Is IIs Ills IVs Vs oo 1' 28" 1' 23" 1' 9" 0' 50" 0' 32" 0' 18" 300 2 1 28 1 22 1 8 0 49 0 30 0 18 28 4 1 28 1 22 1 7 0 47 0 29 0 17 26 6 1 28 1 21 1 5 0 46 0 28 0 17 24 8 1 28 1 20 1 4 0 45 0',7 0 16 22 10 1 28 1 19 1 3 0 44 0 26 0 16 20 12 1 27 1 18 1 2 0 42 i 0 25 0 15 18 14 1 27 1 17 1 0 0 41 0 24 0 15 16 16 1 27 1 16 0 59 0 40 0 24. 0 15 14 18 1 26 1 15 0 58 0 39 0 23 0 14 12 20 1 26 1 14 0 57 0 37' 0 22 0 14 10 22 1 25 1 13 0 55 0 36 0 21 0 14 8 24 1 25 1 12 0 54 0 35 0 20 0 14 6 26 1 24 1 11 0 53, 0 34 0 20 0 14 4 28 1 24 1 10 0 51 0 33 0 19 0 13 2 30 1 23 1 9 0 50 0 32 018 0 13 0 I-_ Xs IXs VIlls I.VIIs VIS TAB LE XXIV. 37 EVRCTION. Argument.-Evection Corrected. VIs VIIs VIIs IXs Xs XIs 00 10 30' 0" 0o 50' 18" 00 20' 52" 00 9' 34" 0 19' 50" 0 49' 16" 1 1 28 37 0 49 6 0 20 10 0 9 34 0 20 32 0 50 30 2 1 27 14 0 47 55 0 19 29 0 935 0 21 16 0 51 45 3 1 25 51 0 46 44 0 18 49 0 9 37 0 22 1 0 53 1 4 1 24 28 0 45 34 0 18 11 0 9 41 0 22 47 0 54 17 5 1 23 6 0 44 26 0 17 34 0 9 47 0 23 34 0 55 33 6 1 21 43 d 43 17 0 16 58 0 9 54 0 24 22 0 56 51 7 1 20 20 0 42 10 0 16 24 0 10 2 0 25 12 0 58 9 8 1 18 58 0 41 4 0 15 50 0 10 12 0 26 3 0 59 28 9 1 17 36 0 39 58 0 15 19 0 10 23 0 26 55 1 0 47 10 1 16 14 0 38 53 0 14 48 0 10 36 0 27 48 1 2 7 11 1 14 52 0 37 49 0 14 19 0 10 50 0 28 43 1 3 27 12 1 13 31 0 36 46 0 13 51 0 11 5 0 29 39 1 4 48 13 1 12 10 0 35 44 0 13 25 0 11 23 0 30 35 1 6 9 14 1 10 49 0 34 43 0 13 0 0 11 41 0 31 33 1 7 31 15 1 9 29 0 33 43 0 12 37 0 12 1 0 32 32 1 8 53 16 1 8 09 0 32 44 0 12 14 0 12 23 0 33 32 1 10 16 17 1 6'49 0 31 46 0 11 54 0 12 45 0 34 34 1 11 39 18 1 5 30 0 30 49 0 11 34 0 13 10 0 35 36 1 13 2 19 1 4 11 0 29 53 0 11 16 0 13 35 0 36 39 1 14 26 20 - 1 2 52 0 28 58 0 11 0 0 14 3 0 37 43 1 15 50 21 1 1 34 0 28 5 0 10 45 0 14 31 0 38 48 1 17 14 22 1 0 17 0 27 12 0 10 31 -0 15 1 0 39 55 1 18 3 23 0 590 0 26 20 0 10 19 0 15 33 0 41 2 1 20 3 24 0 57 44 0 25 30 0 10 8 0 16 5 0 42 10 1 21 28 25 0 56 28 0 24 40 0 9 59 0 16 39 0 43 19 1 22 53 26 0 55 13 0 23 52 0 9 51 0 17 15 0 44 29 1 24 18 27 0 53 58 0 23 5 0 9 45 0 17 52 0 45 39 1 25 44 28 0 52 44 0 22 20 0 9 40 0 18 30 0 46 51 1 27 9 29 0 51 31 0 21 35 0 9 36 0 19 9 0 48 3 1 28 34 30 0 50 18 0 20 52 0 9 34 0 19 50 0 49 16 1 30 0 TABLE P. MOON'S EQUATORIAL PARALLAX. Argument. —Arg. of the Variation. Os IS s I Is IVs Vs' I 0 56" 42" 16' 411 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 0 XIs Xs IXs VIls VII s I I I TABLE XXV, EqUATxON OF MOON'S Cr~s~ra. Argument. —Anomaly corrected, I o~ i I~! m I n~ I IV~ v~ i 0o 7o 0' 0" ~oo~o, 58,,]1~o3s'4~,, ~3o~7,35,, ~2o~6,~1,, bo58 ~9';'! I [ 7 7 5 I10 2652]124143 113 17 5 1121248 95258 } 2 [ 7 1410 110 32421124435 /131628 ]12 911 94724 31721151103827{1247201131544112 529 4]72819/1044 81124959 }131453 ]12 I 94148 41 3610 5[73523/104943 [125230/1313561115749 93029 6]74226[lO55141125455[131252Ill5352 92446 7 174928/11 039 1125712{131141114950 919 1 8175628J ll 6 01125923/131024114544 91313 918328/111115}13 I26113 9 i 114133 9 724 1- ]81026]111624113 o23 /13 731113717 I32 }81722/112i29113 512 ]13 5541132,9 57 5539 1 ]82417]112627 [13 65~ [13 412112833 44 3110 [11 3120[13 ~30 [13 223 1124 5 49 ~' 43 47 1 /8381 [1136 8113 959 /13 027111932 [8 37 49 4450 [114049 1131120 ]125826 111455 31 49 i /85~36 t ll 45~5/~3 ~34/~2 56~a 1110~4 1' ~ 2548 I 58 20 Ill 49 54113 13 41 /12 54 5 11 5 30 19 46 1 ] 5 I II1' 5,i 18113 1441/12 5145 11 041 1342 11 / 11 39 Ill 5835 ]13 1534[12 4919 105549 38 88 ~ 18 15 32 12 2471131620/124647105053 8 55 26 ~ 24 47 112 6 52 /13 1659[12 4410 104553 2: [ 31 16 112 1050 ~13 1~31/12 4127 104050 18'2: 37 42 ]12 14 ~ 42113 1756]123838103543 ~ 49 10 43 9, [ 44 4 [12 1828[13 1814|123543103033 37'~i [ 5023 ]12 22 7 ]13 1824]12 3243 102520 ~ 30 52'~i [ 5638 /12 2540{13 1828/12 E937 1020 4 2442'~' ]1 249 }12 29 6}13 1825}12 2626 101445 1832 ~'Z I1 856 /12 3225/131816[12231010 922 21 2! 126 I1 1459 [123538~131759[12194810 357; 1~ 31 ]1 2055 [123844[131735|121621 95829 7 0 TABLE XXVI. MooN's E(~UATO~XAL P.~RALLAX. Argument? —Corrected Anomaly, Os Is IIs IIIs [Vs Vs..... 0o 58,' 57' s 55, 3~', ~4, ~" 5a' 3" 30o.8" 58t 27" " 2 5818 5823 57 2 5523 5357 53 0 28 5817 5819 5655 551[ 5352 5258 26 58~6 5814 5649 55 I t 5347 5256 24 58~5 5810 5642 55 t 5343 5254 22 1 58f4 158 5 5636 54: 5] 5338 5252 20 12 58~3 58 0 5629 545,) 5334 5250 18 14 58~1 ] 57 55 5622 544; 5330 5249 16 16 58 z. 9 5749 5616 544) 5326 5247 14 18 58 ~ 57 ~6 44 56 9 543L 5322 5246 12 20 58 ~.4 5738 56 3 542) 5319 5245 10 22 58 ~.1 5732 5556 542~ 5315 5244 8 24 58 5726 5549 541] 5312 5243' 6:'~ 26 58 5720 5543 541)~ 53 9 5243 4 28 58 5714 5536 54. ~ 53 6 5243 2 30 58'I 57 8 5530 54,):53 3 5243 0 ~. iI, IXs ~II[~ "His VIs TABMLE XXV QtQVA. ION OF MOON'S CENTER. Argument.-Anomaly corrected. VIs VIIs VIIs IXs Xs XX I 00 70 0' 0" 40 1' 31" 10 43' 39' 00 42' 25" 10 21' 16" 3039' 2" 1 6 53 49 3 56 3 1 40 12 0 42 1 1 24 22 3 45 1 2 6 47 39 3508 1 36 50 0 4'1 44 1 27 35 3 51 4 3 6 41 28 3 45 15 1 33 34 0 41 35 1 30 54 3 57 11 4 6 35 18 3 39 56 1 30 23 0 41 32 1 34 20 4 3 22 5 6 29 8 3 34 40 1 27 17 0 41 36 1 37 53 4 9 37 6 6 22 59 3 29 26 1 24 17 0 41 46 1 41 32 4 15 55 7 6 16 50 3 24 17 1 21 22 0 42 4 1 45 18 4 22 18 8 6 10 42 3 19 10 1 18 33 0 42 29 1 49 10 4 28 44 9 6 4 34 3 14 7 1 15 50 0 43 1 1 53 8 4 35 13 10 5 58 28 3 9 7 1 13 12 0 43 40 1 57 13 4 41 45 11 5 5222 3 411 1 10 41 0 44 26 2 1 24 4 48 21 12 5 46 17 2 59 19 1 8 15 0 45 19 2 5 42 4 54 59 13 5 40 14 2 54 30 1 5 55 0 46 19 2 10 5 5 1 40 14 53412 2 4946 1 3 42 04726 21435 5 8 24 15 5 28 11 2 45 5 1 1 34 0 48 40 2 19 11 5 15 10 16 5 22 11 2 40 28 0 59 33 0 50 1 2 23 52 5 21 59 17 5 16 13 2 35 55 0 57 37 0 51 30 2 23 39 5 28 50 18 5 10 16 2 31 27 0 55 48 0 53 5 2 33 32 5 35 43 19 5 421 2 27 3 0 54 6 0 54 47 2 38 31 5 42 37 20 4 58 28 2 22 43 0 52 29 0 56 37 2 43 35 5 49 34 21 4 52 36 2 18 27 0 50 59 0 58 33 2 48 45 5 56 32 22 4 46 47 2 14 16 0 49.36 1 0 37 2 54 0 6 3 31 23 4 40 59 2 10 10 0 48 19 1 2 48 q 59 21 6 10 32 24 4 36 14 2 6 8 0 47 8 1 55 3 4 46 6 17 34 25 4 29 31 2 2 11 0 46 4 1 7 30 3 10 17 6 24 37 26 4 23 50 1 58 19 0 45 7 1 1 1 3 15 52 6 31 41 27 4 18 11 1 54 31 0 44 16 1 12 40 3 21 33 6 38 45 28 4 12 35 1 50 49 0 43 32 1 15 25 3 27 18 6 45 50 29 4 7 2 1 47 11 0 42 55 1 1817 3 33 8 6 52 55 30 4 131 1 43 39 0 42 25 1 21 16 3 39 2 7 0 0 40 TABLE XXVII. VARIATION. ARGUMENT.-Variation, corrected. Os Is IIs II IVs Vs 0 0 fIt I t t0 )t it II to a! I ID 0 038 0 1 8 1 1 6 58 0 35 54 0 5 29 0 6 2 2 0 40 26 1 9 7 1 5 36 0 33 27 0 4 21 0 7 24 4 0 42 52 1 10 3 1 4 5 0 31'o 0 3 22 0 8 55 6 045 16 1 10 50 1 2 27 0 28 34 0 2 33 0 10 34 8 047 38 1 11 26 1 0 42 0 26 11 0 1 54 0 12 22 10 049 57 1 11 53 0 58 49 0 23 51 0 1 24 0 14 17 12 052 13 1 12 9 0 56 50 0 21 34 0 1 5 0 16 19 14 054 24 1 12 15 0 54 45 0 19 22 0 0 57 0 18 27 16 1 056 30 1 12 10 0 52 35 0 17 15 0 0 59 0 20 41 18 0 58 30 1 11 5500 50 21 0 15 13 0 1 11 0 23 0 -20 1 0 24 11 30 0 48 2 0 13 17 0 1 34 0 25 23 22 1 211 10 55 0 45 40 0 11 28 0 2 8 0 27 50 24 51 1 10 10 0 43 16 0 9 47 0 2 51 0 30 20 26 1 5 23 1 9 15 0 40 50 0 8 13 0 3 45 0 32 52 28 1 647 -1 8 11 038 22 0 6 47 0 4 48 0 35 26 30 1 8 1 1 6 58 0 35 54 0 5 26 0 6 2 0 38 0 VIs VIs VIIIs IXs Xs XIs u'O 11 1' 1 1 - - - - -- I -- 0 o I o Iot 0 I 0 I t I o t 0 o t 0 I o t 0 10 38 0 1 9 58 10 300 40 6 0 9 20 7 58 P2 l0 40 34 1 11 11 1 9 130 37 38 0 7 49 0 9 13 4 10 43 8 1 12 151 7 470 35 10 0 6 45 0 10 37 6 0 45 40 1 13 9 1 6- 130 32 44 0 5 50 0 12 9 8 0 48 10 1 13 52 1 4 310 30 19 0 5 5 0 13 49 10 0 50 37 1 14 26 1 2 42 0 27 58 0 4 29 0 15 36 12 0 53 0 114 48 1 0 470 25 39 0 4 410 17 30 14 0 55 19 1 15 1 0 58 450 23 25 0 3 50 0 19 30 16 0 57 33 1 15 3 0 56 380 21 15 0 3 45 0 21 36 18 0 58 41 1 14 54 0 54 25 0 19 10 0 3 51 0 23 47 2) 1 143 1 14 35 0 52 9 0 17 11 0 4 7 0 26 3 22 1 3 38 1 14 60 49 490 15 18 0 4 34 0 28 22 24 1 525 1 13 27 0 47 26 13 33 0 5 10 0 30 44 26 1 7 5 1 12 3810 45 0 011 54 0 5 57 0 33 8 28 1 8 3611 11 39 0 42 330 10 24 0 6 53 0 35 33 30 1 9 581 10 3010 40 60 9 2 0 7 58 0 38 0............... TABLE XXVIII. 41 MOON'S DISTANCE FROM THE NORTH POLE OF THE ECLIPTIC. ARGUMENT. Supplement of Node+Moon's Orbit Longitude. IIIs I Vs Vs VIs VIHs VIHs O6' 840 29' 16" 850 20' 43" 870 13' 47" 890 48' 0" 92b 22' 1311" 940 15' 1711 30w 2 84 39 27 85 26 16 87 23 12 89 58 46 92 31 27 94 20 31 28 4 84 40 1 85 329 87 32 48 90 9 31 92 40 30 94 25 25 26 6 8440 58 85 38 20 87 42 33 90 20 14 92 49 19 94 29 59 24 8 84 42 17 85 44 50 88 52 28 90 30 55 92 57 56 94 34 12 22. 10 84 43 58 85 51 37 88 2 31 90 41 33 93 6 18 94 38 4 20 12 84 46 2 85 58 4 88 12 42 90 52 7 93 14 27 94 41 85 18 14 84 48 27 86 6 3 88 23 0. 91 86 93 22 20 94 44 45 16 16 84.51 15 86 13 40 88 33 24 91 13 0 93 29 57 94 47 32 14 18 84 84 2 8 21 33 88 43 53 91 23 18 93 37 18 94 49 58 12 20 84 57 56 8 29 42 88 54 27 91 33 29 93 44 23 94 52 2 10 22 85 1 48 86 384 89 55 91 43 82 93 51 10 94 53 43 8 24 88 6 1 86 46 41 89 15 46 91 53 27 93 57 40 94 55 2 6 26 8510 35 86 55 30 89 26 29 92 3 12 94 3 51 94 55 59 4 28 85 15 29 87 4 32 89 37 14 92 12 48 94 9 44 94 56 03 2 30 85 20 48 87 13 47 89 48 0 92 2 13 94 15 17 94 5644 0. Hs _ Is O s XIs Xs IXs TABLE XXIX. EQUATION II OF THE MOON S POLAR DISTANCE. ARGUMENT Ii, corrected. Is s IVs Vs VIs VIIs VIIIs -—..-....,. __ _: _................. 00 0' 14"1 1' 24" 4 37"/ 9' O" 13' 23" 16t' 3" 300 2 0 14 1 34 4' 53 9 18 13 39 16 45 28 4 0 15 1 44 5 9 9 37 13 54 16 53 26 6 0 17 1 54. 5 26- 9 55 14 9 17 1 24 8 0 19 2 5 5 43 10 13 14 24 1 n 22 10 0 22 2 17 6 0 10 31 14 38 17 14 20 12 0 25 2 29 6- 17 10 49 14 52 17 20 18 14 0 29 2 41 6 35 11 7 15 5 17 26 16 16 0 34 2 54 6 53 11 25 15 18 17 31 14 18 0 40 3 8 7 11 11 43:15 31 17 35 12 20 0 45 3 22 7 29 12 0 15 43 17 38 10 22 0'52 3 36 7 47 12 17 15 55 17 41 8 24 0 59 3 51 8 5 12 34 16 6 17 43 6 26 1 7 4 6 8 23 12 51 16 16 17 45 4 28 1 15 4 21 842 13 7 16 26 17 46 2 30 1 24 4 37 9 0 13 23 16 36 17 46 0 HIs Is Os XIs Xs IXs TABLE XXX. EQUATION III OF THE POLAR DISTANCE. ARGUMENT. Moon's True Longitude. IIs Irs Vs VIs VIs VIUs 00 16" 15" 12" 8" 4" 4 " 30 6 16 14 11 7 3 1 24 12 16 14 10 6 3 0 18 18 16 183 10 5 2 0 12 la 30 15 12 8 4 1 0 0 -is Is Os XIs X. IXs .42 TA3LE XXXI. EQUATIONS OF POLAR DISTANCE. ARGUMENTS. —-20 of Longitude; V to IX, corrected; and X, not corrected, Arg. 20 V. VI. VII. VIII. IX. X Arg. 260 0"l 56" 6" 3" 25" 31" 11" 240 280 1 55 6 3 25 3 11'220 300 1 55 7 4 25 4 11 200 320 2 53 8 5 24 6 1]2 180 340 3 52 10 6 23 7 13 160 360 4 50 12 8 23 9. 14 140 380 5 48 14 10 S9 11 16 120 400 6 45 16 12 21 14 17 100 420 8 42 18 14 20 17 19 80 440 10 39 21 17 19 20 21 60 460 11 36 24 ] 9 17 23 23 40 480 13 33 27 22 16 27 25 20 500 15 30 30 25 15 30 27 000 52i 17 27 33 28 14 33 29 980 540 19 24 36 31 12 37 31 960 560 20 20 39 33 11 40 33 940 580 22 17 41 36 10 43 35 920 600 24 15 44 38 9 46 37 900 620 25 12 46 40 8 48 38 880 640 26 10 48 42 7 51 40 860 660 27 8 50 44 6 53 41 840 680 28 7 52 45 6 54 42 820 700 29 5 53 46 5 56 42 800 720 29 5 53 47 5 56 - 43 780 740 30 4' 54 47 5 57 43 760 TABLE XXXII. IEDUCTION ArtJmENwt.-Supplement of Node + Moon's Orbit Longitude. Os VIs Is VIIs i Ils VHIts IIIs IXs IVt Xs Vs XIs 00 7' o" 1, 3", 1' 3"1 7' 0"o 13' 57' 12' 57" 2 6 31 0 49 1 18 7 29 13 10 12 42 4 6 3 0 38 1 35 7 57 13 22 12 25 6 5 34 0 28 1 54 8 26 13 32 12 6 8 5 6 0 20 - 2 14 8 54 13 40 11 46 10 4 39 0 14 2 35 9 21 13 46 11 25 12 4 12: 10 2 58 9 48 13 50 11 2 14 3 46 0 8 3 22 10 13 13 52 10 38 16 3 22 0 8 3 46 10 38 13 52 10 13 18 2 58.0 10 4 1.2 11 2 13 50 9 48 20 2 35 0 14 4 39 11 25 13 46 9 21 22 2 14 0 20 5 6 11 46 13 40 8 54 24 1 54 0 28 5 34 12 6 13 32 8 26 26 1 35 0- 38 6 3 12 25 13 22 7 57 28 1 18 049 6 31 12 42 1 13 10 7 29 30 1 3 1 3 7 0 12 57 1 57 7 0 TABLE XXXIVb 43 MOON' S SEMIDIAMETER, ARGUMENT. Equatorial Parallax. Eq. Parallax. Semidiam. Eq. Parallax. Semidiam. IEq. Parallax.. Semidiasn. 53' 01' - 14' 27" 56' 0" 15' 16" 59' 0"' 16' 51".53 20 14 32 56 20 15 21 59 20 16 10 53 40 14 37 56 40 15 26 59 40 16 16 54 0 14 43 57 15 32 60 0 16 21 54 20 14 48 57 20 15 37 60 20 16 26 54 40 14 54 57 40 15 43 60 40 16 32 55 0 14 59 58 0 15 48 61 0 16 37 55 20 o 5, -58 20 15 54 61 20 16 43 55 40 15,1 58 40 15 59 61 40 16 48 56 0 15:6., 59 5 16 5 62 0 16 54 TABLE XXXV. TABLE XXXVI. AUGMENTATION OF MOON'S SEM[- MOON'S HOURLY MOTION IN LONDIAMETER,- GITUDE. ARGUMENT. Apparent Altitude. ARGUMENTS. 2, 3, 4, and 5 of Longitude. p. Alt. Augm,. Arg, 22 3 4 5 Arg. 60 2"12 3 0 6"Y' 3' 3 100 18, 5 5 5.2 3 3 95 24 6 10 s 2 3 3 90 30 8 15 4 2 3 3 85 20 4 3 2 2 80 16 9 42 11 25 -3 2 2'5 48 12 30 2 3 2 2 70 54 13 os5 2 4 1 1 65 60 14 40 1 4 1 1 60 45 1 4 1 1 55 66 15 50 _0 5 1 1 5) 72 15 78 16 84 16 90 6 TABLE XXXVII. -OON'S 1HOURLY MOTION IN LONGITUDE. ARGUMENT. Argu ment of the Evection.. Os Is s I llss IVs i Vs t0 1' 20" 1' 15"' 1t,0" 0' 39" 0' 20" 0' 6" 300 2 1 20 1 14 0 58 0 38 0 19 0 5 28 4 1 20 1 13 0 57 0 37 0 18 0 5 26 6 1 20 1 12 0 56 0 35 0 16 0 4 24 8 1 20 1 11 0 54 0 34 0 15 0 4 22 10 120 I 11 0 53 0 33 0 14 0 3 20 12 1 19. I 0 52 0 31 O 13 0 3 18 14 1 l9 i 0 9 1 0 0 12 0 2 16 16. 119 i 8 0 49 0 29 0 11 0 2 14 18 118 I 7 0 48 0 27 0 11 0 2 12 20. 1 -18 1 5 0 46 0 246 10 26 10 0 1 10 22 1 17 1 4 0 45 0 25 0 9 0 1 8 24. 1 17.1 3 0 44 0 23 0 8 0 1 6 26 1 16 12 0 4 0 22 0 7 0 1 4 28 1 15 1 1 0 41 0 21 0 7 0 1 2 30 1 15 1 0 0 39 0 20 0 6 0 1 0 XIs Xs IXs VIIIs VIis VIs 44 TABLE XXXVIII. MOON S HOURLY MOTION IN LONGITUDE. ARGUMENTS. Sum of preceding equations, and Anomaly, correectd 0" 20" 40" 60" 80" 1 100" Os 00 4" 6" 9" 11" 14" 16" XIIs 0~ 10 4 7 9 11 13 16 20 20 5 7 9 11 13 15 10 Is 0 5 7 9 11 13 15 XIs 0 10 6 7 9 11 13 14 20 20 7 8 9 11 12 13 10 IIs 0 7 8 9 11 12 13 Xs 0 10 8 9 10 10 11 12 1 20 20 9 10 10 10 10 11 I 10 Ils 0 10 10 10 10 10 10 10 IXs 0 10 11 11 10 10 9 9 20 20 12 11 10 10 9 8 10 IVs 0 - 13 12: 11 9 8''7 VIIIs 0 10 14 12 11 9 8 -6. 20 20 14 12 15 9 8 6 10 Vs 0 15 13 11 9 7 5 VIIs 0 10 15 13 11 9 7 5 20 20 15. 13 11 9 7 5 - 1 VIs 0 15 13 11 9 7 5 VIs 0 0".! 20"1 40"o 60-" 80";, 100" TABLE XXXIX. MOON'S HOURLY MOTION IN LONGITUD]$. ARGUMENT. Anomaly, corrected. Os Is s Is IIIs Vs Vs 00 34' 51," 34' 141" 32' 39" 30' 45" 29t 6" 28' 1"1 30 2 34 51 34 9 332 32 30 38: 29 0 27 58 28 4 34 51 34 4 323 24 30 31 28 55 27 55 26 6 34 50 33 59 323 17 30 23 28 50 27 53' 24 8 34 49 33 55 32 9 30 16 28 45 27 50 22 10 34 47 33 47 32 2 30 9 28 40 27 48 20 12 34 45 33 41 31 54 30 2 28 35 27 46 18 14 34- 43 33 35 31 46 29 56 28 30 27 45 1616 34 41 33 28 31 38 29 49 28 26 27 43 14 18 34 38 33 22 31 31 29 42 28 22 27 42 12 20- 34 34 33 15 31 23 29 36 28 18 27 41 10 22 34 31 33 8 31 15 29 30 28 14 27 40 8 24 34 27 33 1 31 8 29 23 28 10 27 39 6: 26 34 23 32 54 31 0 29 17 28 7 27 39 4 28 34 19 32 47 30 5, 29 12 28 4 27 38 2 30 34 14 32 39 30 45 29 6 28 1 27 38 0 XIs Xs IXs VIIIs VHs VIs TABLE XL. 45 X3OON'S HOURLY MOTION IN LONGITUDE. ARGUMENTS. Sum of preceding equations, and Argument of Variation. 271 29' 31' 33' 35' 37' Os 00 0"f 2"l 51t 7" 180" 2r XIIs 0" 10 0 3 5 7 9 12 20 20 1 3 5 7 9 11 10 Is 0 3 4 5 7 8 9 XIs 0 10 5 5 6 6 7 7 20 20 7 7 6 6 5 5 10 IIs 0 9 8 7 5 4 8 Xs 0 10 11 9 7 5 3 1 20 20 12 10 7 5 2 0 10 Hlls 0 12 10 7 5 2 0 IXs 0 10 12 10 7 5 2 0 20 20 11 9 7 5 3 1 10 Is 0 9 8 7 5 4 3 vmIIs 0 10 7 7 6 6 5 5 20 20 5 5 6 6 7 7 10 Vs 0 3 4 5 7 8 9 VIIs 0 10 1 3 5 7 9 11 20 20 0 2 5 7 10 12 10 VIs 0 0 2 5 7 10 12 VIs 0 27t 29-' 31 33'1 35' 37' TABLE XLI. MOON'S HOURLY MOTION IN LONGITUDE. ARGUMENT. Argument of the Variation. Os Is II s III s -s Vs 03 1' 17"1 0t 58" 0' 20"1 0' 2" 0' 22".1t' 0" 300 2 117 055 0 18 0 3 0 24 1 2 28 4 117 053 0 16 0 3 0 26 1 4 26 6 116 0 51 3O 14 0 3 0 29 1 6 24 8 116 0 48 0 12 0 4 0 31 1 8 22 10 115 045' 0 11 0 5 0 34 1 10 20 12 1 14 0 43 0 9 0 6 0 37 1 12 18 14 1 13 0 40 0 8 q 7 0 39 1 13. 16 16 1 11 0 38 0 6 0 8 0 42 1 15 14 18 \1 10 0 35 0 5 0 10 0 44 1 16 12 20 1 8 0 32 0 4 0 11 0 47 1 17 10 22 1 6 0 30.- 0 4 0 13 0 50 1 18 8 24 1 4 0 27 0 3 0 15 0 52 1 18 6 26 1 2 0 25 0 3 0 17 0 55 1 19 4 28 1 0 0 23 0 2 0 19 0 57 1 19 2 30 0 58 0 20 0 2 0 22 1 0 1 19 0 X Xl s IX s vIIIs IXs VIIVIs TABLE XLII. MOON'S HOURLY MOTION IN LONGITIUDEJ. ARGUMENT. Argument of the Reduction. Os Is Is IIs I Vs Vs 00 2"l 6"/i 14"1 18/" 14" 6'" 30~ 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 8 16 8 1 18 12 4 22 10- 3 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 9 3 12 20 4 17 16 9 3 10 22 4 12 I8 16 8 2 8 24 5 12 18 15 8 2 6 26 5 13 18 15 7 2 4 28 6 13 18 14 7 2 2 30 6 14 18 14 6 2 0 XIs Xs IXJs VIls VIIS: VIs TABLE XLIII. MOON S HOURLY MOTION IN LATITUDE. ARGUMENT. Argument I, of Latitude. Os+ Is Is+ II s — IVs- Vs00 2' 588" 2' 34' 1' 29" 0' 0" 1' 29" 2' 34" 300 a 2 58 2 31 1 24 0 6 1 35 2 37 28 4. 2 58 228 1 r8 0 12 1 40 2 40 26 6 a 57 2 24 1 13 0 19' 1 45 2 43 24 8 2 56 a 22 1 7 0 25 1 50 2 45 22 10 2 55 2 17 1 1 0 31 1 55 2 47 20: 12 2 54 2 12 0 55 0 37 1 59 2 49 18 14 2a 53 2 8 0 49 0 43 2 4 2 51 16, 16 2 51: 2a 4 0 43 0 49 2 8 2 53 14, 18 2 49 1 59 0 37 0 55 2 12 2 54 12 20 2 47 1 55 0 31 1 1 2 17 2 55 10 22 2 46 1 50 0 25 1 7 2 20 2 56 8 24 2 43 1 45 0 19 1 13 2 24 2 57 6 26 2 40 40 012 1 18 2 28 2 58 4 28 2 37 1 35 0 6 1 24 2 31 2a 58 2 i 30 a 34 1 29 1 29 2 34 2 58 0; XIs+ Xs+ IXs+ VIIs — VIIs — VIsTABLE XLIV. MOON'S HOURLY MOTION IN LATITUDE. ARGUMENT. Argument II, of Latitude. OS+ Is+ 1+II IIIs- I-s- Vs00 411 4" 2" 0"o 2" 4"1 300 6 4 3 2 0 3 4 24 12 4 3 1 1 3 4 18 18 4 3 1 1 3 4 12 24 4 3 0 2 3 4 6 30 4 2 0 2 4 4 0 XIs+ Xs+ IXs+ VI,/s-VII V- s — TABLE XLV.-PROPORTIONAL LOGARITHMS. 47 0' 1 2' 3' 4' 5' 6' 7 0"f 0 0000 1 7782 1 4771 1 3010 1 1761 10792 1 0000 9331 1 3 5563 1 7710 1 4735 1 2986 1 1743 10777 9988 932) 2 82553 1 7639 1 4699 1 2962 1 1725 1 0763 9976 9310 3 30792 1 7570 1 4664 12939 1 1707 1 0749 9964 9300 4 2 9542 1 7501 1 4629 1 2915 1 1689 1 0734 9952 9289 5 2 857.3 1 7434 14594 1 2891 1 1671 1 0720 9940 9279 6 2 7782 17368 1 4559 1 2868 1 1654 1 0706 9928 9269 7 2 7112 17302 1 4525 1 2845 1 1636 1 0692 9916 9259 8 2 6532 17238 1 4491 1 2821 11619 1 0678 9905 9249 9 26021 17175 1 4457 1 2798 1 1601 1 0663 9893 9238 10 2 5563. 7112 14424 1 2775 1 1584 10649 9881 9228 11 25149 1 7050 14390 12753 1 1566 10635 9S69 9218 12 24771 16990 14357 12730 1 1549 10621 9858 9208 13 24424 16930 14325 1 2707 1 1532 10608 9846 9198 14 24102 16871 14292 1 2685 1 1515 10594 9834 9188 15 23802 16812 1-4260 12663 1 1498 10580 9820 9178 16 23522 16755 1 4228 1 2640 1 1481 10566 9811 9168 17 23259 16698 14196 12618 11464 10552 9800 9158 18 2 3010 16642 1 4165 12596 1 1457 1 0539 9788 9148 19 2 2775 16587 14133 12574 1 1430 10525 9777 9138 20 2a 2553 16532 1 4102 12553 1 1413 1 0512 9765 9128 21 22941 16478 14071 12531 11397 1 0498 9754 9119 22 22139 16425 14040 12510 11580 10484 9742 9109 23 21946 16372 14010 12488 1 1363 10471 9731 9099 24 21761 16320 13979 12467 11347 10458 9720 9089 25 21584 1 6269 13949 12445 11331 10444 9708 9079 26 21413 16218 13919 12424 11314 10431 9697 9070 27 21249 16168 13890 12403 11298 10418 9686 9060 28 21091 16118, 13860 12382 1 1282 1 0404 9675 9050 29 20939 16069 13831 12362 1 1266 10391 9664 9041 30 2 0792 16021 13802 12341 11249 10378 9652 9031 31 20649 15973 1 3773 12320 11233 10365 9641 9021 32 20512 15925 13745 1 2300 11217 1 0352 9630 9012 33 20378 1 5878 13716 12279 1 1201 10339 9619 9002 34 20248 1 5832 13688 12259 1 1186 10326 9608 8992 35 20122 15786 13660 12239 1 1170 1 0313 9597 8983 36 20000 15740 13632 1 2218 11154 i 0300 9586 8973 37 19881 15695 13604 1 2198 11138 1 0287 9575 8964 38 19765 15651 1 3576 12178 11123 10274 9564 8954 ~39 19652 15607 13549 1 2159 11107 10261 9553 8945 40 19542 15563 13523 12139 11091 10248 9542 8935 41 19435 15520 13495 12119 11076 10235 9532 8926 42 19331 15477 13468 1 2099 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 1 2041 1 1015 1 0185 9488 8888 46 1 8935 15310 13362 12022 10999 10172 9478 8879 47 1 8842 15269 1 3336 12003 10984 1 0160 9467 8870 48 1 8751 15229 13310 11984 10969 1 0147 9456 8861 49 18661 15189 13284 1 1965 10954 1 0135 9446 8851 50 18573 15149 13259 11946 10939 10122 9435 8842 51 1 8487 15110 13233 11927 10924 1 0110 9425 8833 52 18403 15071 13208 11908 10909 10098 9414 8824 53 18.310 15032 13183 11889 10894 10085 9404 8814 54 18239 14994 13158 11871 10880 1 0073 9393 8805 55 11 1,9 14956 13133 11852 1 0865 1 0061 9383 8796 56 18081 14918 13108 1 1834 10850 10049 9372 8787 57 18004 14881 13083 1 1816 10835 1 0036 9362 8778 58 1 7929 14844 13059 1 1797 10821 1 0024 9351 8769 59 17855 14808 13034 11779 1 0806 1 0012 9341 8760 60 17782 1 4771 13010 11761 1 0792 1 0000 9331 8751 2D 48 TABLE XLV.-PROPORTIONAL LOGARITHMS. 8' 9' 10' 11' 12' 13' 14' 15' 16' 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 3 8724 8215 7760 7348 6972 6625 6305 6006 5727 4 8715 8207 7753 7341 6966 6620 6300 6001 5722 5 8706 8199 7745 7335 6960 6614 6394 5997 5718 6 8697 8191 7738 7328 6954 6609 6289 5992 5713 7 8688 8183 7731 7322 6948 6603 6284 5987 5709 8 8679 8175 7724 7315 6943 6598 6279 5982 5704 9 8670 8167 7717 7309 6936 6592 6274 5977, 5700 10 8661 8159 7710 7302 6930 6587 6269 5973 5695 11 8652 8152 7703 7296 6924 6581 6264 5968 5601 12 8643 8144 7696 7289 6,918 6576 6259 5963 5686 13 8635 8136 7688 7283 6912 6570 6254 5958 5682 14 8626 8128 7681 7276 6906 6565 6248 5954 5677 15 8617 8120 7674 7270 6900 6559 6243 5949 5673 16 8608 8112 7667 7264 6894 6554 6238 5944 5669 17 8599 8104 7660 7257 6888 6548 6233 5939 5664 18 8591 8097 7653 7251 6882 6543 6228 5935 5660 19 8582 8089 7646 7244 6877 6538 6223 5930 5655 20 8573 8081 7639 7238 6871 6532 6218 5925 5651 21 8565 8073 7632 7232 6865 6527 6213 5920 5646 23 8556 8066 7625 7225 6859 6521 6208 5916 5642 23 8547 8058 7618 7219 6853 6516 6203 5911 5637 24 8539 8050 7611 7212 6847 6510 6198 5906 5633 25 8530 8043 7604 7206 6841 6505 6193 5902 5629 26 8522 8035 7597 7200 6836 6500 6188 5897 5624 27 8613 8027 7590 7193 6830 6494 6183 5892 5620 28 8604 8020 7583 7187 6824 6489 6178 5888 5615 29 84.9 8012 7577 7181 6818 6484 6173 5883 5611 30 8487 8004 7570 7175 6812 6478 6168 5878 5607 31 8479 7997 7563 7168 6807 6473 6163 5874 5602 32 8470 7989 7556 7162 6801 6467 6158 5869 5598 33 8462 7981 7549 7156 6795 6462 6153 5864 5594 34 8453 7974 7542 7149 6789 6457 6148 5860 5589 35 8445 7966 7535 7143 6784 645i 6143 5855 5585 36 8437 7959 7528 7137 6778 6446 6138 5850 5580 37 8428 7951 7522 7131 6772 6441 6133 5846 5576 38 8420 7944 7515 7124 6766 6435 6128 5841 5572 39 8411 7936 7508 7118 6761 6430 6123 5836 5567 40 8403 7929 7501 7112 6755 6425 6118 5832 5563 41 8395 7921 7494 7106 6749 6420 6113 5827 5559 4 8386 7914' 7488 7100 6743 6414 6108 5823 5554 43 8378 7906 7481 7093 6738 6409 6103 5818 5550 44 8370 7899 7474 7087 6732 6404 6099 5813 5546 45 8361 7891 7467 7081 6726 6398 6094 5809 5541 46 8353 7884 7461 7075 6721 6393 6089 5804 5537 47 8345 7877 7454 7069 6715 6388 6084 5800 5533 48 8337 7869 7447 7063 6709 6383 6079 5795 5528 49 8328 7862 7441 7057 6704 6377 6074 5790 5524 50 8320 7855 7434 7050 6698 6372 6069 5786 5520 51 8312 7847 7427 7044 6692 6367 6064 5781 5516 52 8304 7840 7421 7038 6687 6362 6059 5777 5511 51 8296 7832 7414 7032 6681 6357 6055 5772 5507 54 8288 7825 7407 7026 6676 6351 6050 5768 5503 55 8279 7818 7401. 7020 6670 6346 6045 5763 5498 56 8271 7811 7394 7014 6664 6341 6040 5758 5494 57 8263 7803 7387 7008 6659 6336 6035 5754 5490 58 8255 7796 7381 7002 6653 6331 6030 5749 5486 59 8247 7789 7374 6996 6648 6325 6025 5745 5481 6t 8239 7782 7368 6990 6642 6320 6021 5740 5477 TABLE XLV.-PROPORTIONAL LOGARITHMS. 49 17' 18 19' 0o' 21' 22t 23' 24' 25'. — I...,I,....... 0" 5477 5229 4994 4771 4559 4357 4164 3979 3802 1 5473 5225 4990 4768 4556 4354 4161 3976 3799 2 5469 5221 4986 4764 4552 4351 4158 3973 3796 3 5464 5217 4983 4760 4549 4347 4155 3970 3793 4 5460 5213 4979 4757 4546 4344 4152 3967- 3791 5 5456 5209 4975 4753 4542 4341 4149 3964 3788 6 5452 5205 4971 4750 4539 4338 4145 3961 3785 7 5447 5201 4967 4746 4535 4334 4142 3958 3782 8 5443 5197 4964 4742 4532 4331 4139 3955 3779 9 5439 5193 4960 4739 4528 4328 4136 3952 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 5422 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 8934 3759 16 5409 5165 4933 4714 4505 4305 4114 3931 3756 17 5405 51el 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 4292 4102 3919 3745 21 5389 5145 4915 4696 4488 4289 4099 3017 s742 22 5384 5141 4911 4692 4484 4285 4096 3914 3739 23 5380 5137 4907 4689 4481 4282 4092 3911 3736 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 387,27 21 5364 5122 4892 4675 4467 4269 4080 3899 3725 28 5359 5118 4889 4671 4464 4266 4077 3896 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 3702 36 5326 5086 4859 4643 4437 4240 4052 3872 3699 37 5322 5082 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 5310 5071 4844 4629 4424 4228 4040 3860 3688 41 5306 5067 4841 4625 4420 4224 4037 3857 3685 42 5302 5063 4837 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 8674 46 5285 5048 4822 4608 4404 4209 4022 3843 3671 47 5281 5044 4819 4604 4400 4205 4019 3840 3668 48 5277 5040 4815 4601 4397 4202 4016 3837 3665 49 5273 5036 4811 4597 4394 4199 4013 3834 3663.50 5269 ) 5032 4808 4594 4390 4196 4010 3831 3660 51 5265 5028 4804 4590 4387 4193 4007 3828 3657 52 5261 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 3646 56 5245 a009 4786 4573 4370 4177 3991 3814 3643 57 5241 5005 4782 4570 4367 4174 3988 3811 3640 58 5237 5002 4778 4566 4364 4171 3985 3808 3637 59 5233 4998 4775 4563 4361 4167 3982 3805 3635 60 5229 4994 4771 4559 4357 4164 3979 4802 3632 50 TABLE XLV.-PROPORTIONAL LOGARITHMS. -26' 27' 28' 29' 30' 31' 32' 33' 34' 0" 3632 3468 3310 3158 3010 2868 2730 2596 2467 1 3629 3465 3307 3155 3008 2866 2728 2594 2465 2 3626 3463 3305 3153 3005 2863 2725 2593 2462 3 3623 3460 3302 3150 3003 2861 2723 2590 2460 4 3621 3457 3300 3148'38001 2859 2721 2588 2458 5 3618 3454 3297 3145 2998 2856 2719 2585 2456 6 3615 3452 3294.3143 2996 2854 2716 2583 2454 7 3612 3449 3292 3140 2993 2852 2714 2581 2452 8 3610 3446 3289 3138 2991 2849 2712 2579 2450 9 3607 3444 3287 3135 2989 2847 2710 2577 2448 10 3604 3441 3284 3133 2986 2845 2707 a574 2445 11 3601 3438 3282 3130 2984 2842 2705 2572 2443 12 3598 3436 3279 3128 2981 2840 2703 2570 2441 13 3596 3433 3276 3125 2979 2838 2701 2568 2439 14 3593 3431 3274 3123 2977 2835 2698 2566 2437 15 3590 3428 3271 3120 2974 2833 2696 2564 2435 16 3587 3425 3269 3118 2972 2831 2694 2561 2433 17 3585 3423 3266 3115 2969 2828 2692 2559 2431 18 3582 3420 3264 3113 2967 2826 2689 2557 2429 19 3579 3417 3261 3110 2965 2824 2687 2555 2426 20 3576 3415 3259 3108 2962 2821 2685 2553 2424 21 3574 3412 3256 3105 2960 2819 2683 2451 2422 22 3571 3409 3253 3103 2958 2817 2681 2548 2420 23 3568 3407 3251 3101 2955 2815 2678 2546 2418 24 3565 3404 3248 3098 2953 2812 2676 2544 2416 25 3563 3401 3246 3096 2950 2810 2674 2542 2414 B6 3560 3399 3243 3093 2948 2808 2672 2540 2412 27 3557 3396 3241 3091 2946 2805 2669 2538 2410 28 3555 3393 3238 3088 2943 2803 2667 2535 2408 29 3552 3391 3236 3086 2941 2801 2665 2533 2405 30 3549 3388 3233 3083 2939 2798 2663 2531 2403 31 3546 3386 3231, 3081 2936 2796 2060 2529 2401 32 3544 3383 3228 3078 2934 2794 2658 2527 2399 33 3541 3380 3225 3076 2931 2792 2656 2525 2397 34 3538 3378 3223 3073 2929 2789 2654 2522 2395 35 3535 3375 3220 3071 2927 2787 2652 2520 2393 36 3533 3372 3218 3069 2924 2785 2649 2518 2391 37 3530 3370 3215 3066 2922 2782 2647 2516 2389 38 3527 3367 3213 3064 2920 2780 2645 2514 2387 39 3525 3365 3210 3061 2917 2778 2643 2512 2384 40 3522 3362 3208 3059 2915 2775 2640 2510 2382 41 3519 3359 3205 3056 2912 2773 2638 2507 - 2380 42 35 16 3357 3203 3054 2910 2771 2636 2505 2378 43 3514 3354 3200 3052 2908 2769 2634 2503 2376 44 3811 3351 3198 3049 2905 2766 2632 2501 2374 45 3508 3349 3195 3047 2903 2764 2629 2499 2372 46 3506 3346 3193 3044 2901 2762 2627 2497 2370 47 3503 3344 3190 3042 2898 2760 2625 2494 2368 48 3500 3341 3188 3039 2896 2757 2623 2492 2366 49 3497 3338 3185 3037 2894 2755 2631 2490 2364 50 3495 3336 3183 3034 2891 2753 2618 2488 2362 51 3492 3333 3180 3032 2889 2750 2616 2486 2359 52 3489 3331 3178 3030 2887 2748 2614 2484 2357 53 3487 3328 3175 3027 2884 2746 2812 2482 2355 54 3484 3325 3173 3025 2882 2744 2610 2480 2353 55 3481 3323 3170 3022 2880 2741 2607 2477 2351 56 3479 3320 3168 3020 2877 2739 2605 2475 2349 57 3476 3318 3165 3018 2875 2737 2603 2473 21317 58 3473 3315 3163 3015 2873 2735 2601 2471 2341 59 3471 3313 3160 3013 2870 2732 2599 2469 2343 60 3468 3310 3138 3010 2868 2730 2596 2467 2341 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE XLV.-PROPORTIONAL LOGARITHMS. 51 35 36' 37'1 38' 39' 40' 41' 42' 43' 0" 2341 2218 2099 1984 1871 1761 1654 1549 1447 1 2339 2216f 2098 1982 1869 1759 1652 1547 1445 2 2337 2234 2096 1980 1867 1757 1650 1546 1443 3 2335 2212 2094 1978 1865 1755 1648 1544 1442 4 2233 2210 2092 1976 1863 1754 1647 1542 1440 5 2331 2208 2090 1974 1862 1752 1645 1540 1438 6 2328 2206 2088 1972 1860 1750 1643 1539 1437 7 2326 2204 2086 1970 1858 1748 1641 1537 1435 8 2324 2202 2084 1968 1856 1746 1640 1535 1433 9 2322 2200 2082 1967 1854 1745 1638 1534 1432 10 2320 2198 2080 1965 1852 1743 1636 1532 1430 11 2318 2196 2078 1963 1850 1741 1634 1530 1428 12 2316 2194 2076 1961 1849 1739 1633 1528 1427 13 2314 2192 2074 1959 1847 1737 1631 1527 1425 14 2312 2190 2072 1957 1845 1736 1629 1525 1423 15 2310 2188 2070 1955 1843 1734 1627 1523 1422 16 2308 2186 2068 1953 1841 1732 1626 1522 1420 17 2306 2184 2066 1951 1839 1730 1624 1520 1418 18 2304 2182 2064 1950 1838 1728 1622 1518 1417 19 2302 2180 2062 1948 1836 1727 1620 1516 1415 20 2300 2178 2061 1946 1834 1725 1619 1515 1413 21 2298 2176 2059 1944 1832 1723 1617 1513 1412 22 2296 2174 2057 1942 1830 1721 1615 1511 1410 23 2294 2172 2055 1940 182 1719 1613 1510 1408 24 2291 2170 2053 1938 1827 1718 1612 1508 1407 25 2389 2169 2051 1936 1825 1716 1610 1506 1405 26 2287 2167 2049 1934 1823 1714 1608 1504 1403 27 2285 2165 2047 1933 1821 1712 1606 1503 1402 28 2283 2163 2045 1931 1819 1711 1605 1501 1400 29 2381 2161 2043 1929 1817 1709 1603 1499 1398 30 2279 2159 2041 1927 1816 1707 1601 1498 1397 31 2277 2157 2039 1925 1814 1705 1599 1496 1395 32 2275 2155 2037 1923 1812 1703 1598 1494 1393 33 2273 2153 2035 1921 1810 1702 1596 1493 1392 34 2271 2151 2033 1919 1808 1700 1594 1491 1390 35 2269 2149 2032 1918 1806 1698 1592 1489 1388 36 2267 2147 03- 1916 1805 1696 1591 1487 1387 37 2265 2145 2028 1914 1803 1694 1589 1486 1385 38 2263 2143 2026 1912 1801 1893 1587 1484 1383 39 2261 2141 2024 1910 1799 1691 1585 1482 1382 40 2259 2139 2022 1908 1797 1689 1584 1481 1380 41 2257 2137 2020 1906 1795 1687 1582 1479 1378 42 2255 2135 2018 1904 1794 1686 1580 1477 1377 43 2253 2133 2016 1903 1792 1684 1578' 1476 1375 44 2251 2131 2014 1901 1790 1682 1577 1474 1373 45 2249 2129 2012 1899 1788 1680 1575 1472 1372 46 2247 2127 2010 1897 1786 1678 1573 1470 1370 47 2245 2125 2039 1895 1785 1677 1571 1469 1368 48 2243 2123 2007 1893 1783 1675 1570 1467 1367 49 2241 2121 2005 1891 1781 1673 1568 1465 1365 50 2239 2119 2003 1889 1779 1671 1566 1464 1363 51 2237 2117 2001,888 1777 1670 1565 1462 1362 52 2235 2115 1999 1886 1775 1668 1563 i 1460 1360 53 2233 2113 1997 1884 1774 1666 1561 1459 1359 54 2231 2111 1995 1882 1772 1664 1559 1457 1357 55 2329 2109 1993 1880 1770 1663 1558 1455 1355 56 2227 2107 1991 1878 1768 1661 1556 1454 1354 57 2225 212 5 1989 1876 1766 1659 1554 1452 1352 58 2223 2103 1987 1875 1765 1657 1552 1450 1350 59 2220 2191 1986 1873 1763 1655 1551 1449 1349 60 - 2218 2599 1984 1871 1761 1654 1549 1 1447 1347 23 2D* 52 TABLE XLV.-PROPORTIONAL LOGARITHMS. 44' 45' 46' 47' 48' 49' 50' 51 1 52 0" 1347 1249 1154 1061 969 880 792 706 621 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 1243 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 18 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 930 841 754 669 585 27 1303 1206 1112 1019 929 840 753 668 584 28 1301 1205 1110 1018 927 838 751 666 583 1300 1203 1109 1016 926 837 750 665 581 30 1298 1201 1107 1015 924 835 749 663 580 31 1286 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 917 828 741 656 573 36 1288 1192 1098 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 999 909 821 734 649 566 41 1280 1184 1090 998 908 819 733 648 565 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 989 899 811 724 640 557 48 1269 1173 1079 987 897 809 723 638 555 49 1267 1171 1078 986 896 808 721 637 554 50 1266 1170 1076 984 894 806 720 635 552 51 1264 1168 1074 983 893 805 719 634 551 52 1262 1167 1073 981 891 803 717 633 650 53 1261 1165 1071 980 890 802 716 631 548 54 1259 1163 1070 978 888 801 714 60 547 55 1257 1162 1068 977 877 799 713 628 546 56 1256 1160 1067 975 885 798 711 627 544 57 1254 1159 1065 974 884 796 710 626 553 58 1253 1157 1064 972 883 795 709 624 541 59 1251 1156 1062 971 881 793 707 623 540 00 1249 1154 1061 960 880 792 706 621 539 TABLE XLV. —PROPORTIONAL LOGARITHMS. 6 53', 54 t 55 | 56' 57' 58 0" 539 458 378 300 22 147 73 1 537 456 377 i 221 146. 72 2 536 455 375 297 220 146 71 3 535 454 374 296 219 143 69 4 533 452 373 294 218 142 68 5 532 451 371 293 216 141 67 6 531 450 370 2012 215 140 66 7 529 448 i 369 291 14 139 64 8. 528 447 367 289'113 137 - 63 9 526 416 366 288 211 136 62 10 525 444 365 287 ]10 135 61 11 524 443 363 285 209 134 60 12 522 442 362 284 208 132 58 13 521 440 361 283 206 131 57 14 520 439 359 2 205 130 56 15 518 438 358 280 204 129 55 A6 517 436 835 279 B02 127 53 17 516 435 356 278 201 126 52 18 514 434 354 276 200 125 5/ 19 513 432 353 275 199.124 50 20 512 431 352 274 137 122 [ 49 21 510 430 350 273 106 121 - 47 22 509 428 349 271 195 120 46 23 507 427 348 270 1~4 119.45 24 506 426 346 269 19.1 1171 44 25 505 424 346 267 191 110 42 26 503 423 344 26E 190 115 41 27 502 422 340 265 189 114 40 28 501 420 341 264 187 112 39 29 499 419 840 262 186 111 28 30 498 418 339 261 15 1101 36 31 497 416 337 260 184 109 35 32 495 415 336 258 182 107 34 33 494 414 335 257 181 106 33 34 - 493 412 33 256 180 105 31 35 491 411 332 255 179 104 30 36 490 410 331 253 177 103 29 37 489 408 329 252 176 101 28 38 487 407 328 251 175 100 27 39 486 406 337 250 174 99 25 40 484 404 326 248 172 96 24 41 483 403 324 241 171 96 23 43 482 402 323 246 170 95 22 43 480 400 322 244 169 94 21 44 479 399 320 243 167 93 19 45 478 398 319 242 166 91 18 46 476 396 918 241 165 90 17 47 475 395 316 239 163 89 16 48 474 394 315 238 162 88 15 49 472 393 314 237 161 87 13 50 471 391 313 235 160 85 12 51 470 190 311 234 158 84 11 52 468 388 310 233 157 S 10 53 467 387 309 232 156 82 8 54 466 386 307 280 155 80 7 55 464 384 306 229 153 7 6 56 463 383 305 228 152 78 5 57 462 882 304 227 151 77 4 58 460 3810 302 226 150 75 2 59 459 879 801 224 148 74 1 60 458 378 300 223 147 73 0 SATELL1TES OF JUPITER. |Sideral Itlclination of Mass; that of Sat. Mean Distance. Sidereal Orbit to that of Jupiter being Revolutio Jupiter. 1000000000 d. h. m. / 1 6 6.04853 1 18 28 3 5 30 17328 2 9.62347 3 13 14 Variable. 23235 3 15.35024 7 3 43 Variable. 88497 4 26.99835 16 16 32 2 58 48 42659 SATELLITES OF SATURN. Mean Sidereal Eccentricitles and Inclinations. t.Distance. Revolution. d. h, m. 1 3.351 0 22 38 The orbits of the six interior 2 I 4.300 1 8 53 satellites are nearly circular,and 3 5.284 1 21 18 very nearly in the plane of the 4 6.819 2 17 45 ring. That of the seventh is 5 9.524 4 12 25 considerably inclined to the rest, 6 22.081 15 22 41 and approaches nearer to coin7 64.359 79 7 55'Icidence with the ecliptic. SATELLITES OF URANUS. Sat. Mean Sidereal Period. Inclination to Ecliptic. d. h. m. s. 17 13.120 5 21 25 0 Their orbits are inclined about 2 17.022 8 16 56 5 78~ 58' to the ecliptic, and their 3? 19.845 10 23 4 0 motion is retrograde. The pe4 22.752 13 11 8 59 riods of the 2d and 4th require a 5? 45.507 38 1 48 0 trifling correction. The orbits' 6? 91.008 107 16 40 0 appear to be nearly circles.