TREATISE ON ASTROXOMYO BY ELIAS LOOMIS, LL. D. I i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I, ~ i i!,'! i', II 111,, I,,,~ 11111,,,,, R l ~ L,,,,~,~~,, ~~~,f~~~,,, l A TREATISE ON BY ELIAS LOOMIS, LL.D.,'ROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY IN YALE COLLEGE; AUTHOR OF "AN INTRODUCTION TO PRACTICAL ASTRONOMY," AND OF A SERIES OF MATHEMATICS FOR SCHOOLS AND COLLEGES. NEW YORK: HARPER & BROTHERS, PUBLISHERS, 327 AND 335 PEARL STREET. 1868. Entered, according to Act of Congress, in the year 1865, by HARPER & BROTIHERS, In the Clerk's Office of the Southern District of New York. THE design of the following treatise is to furnish a text-book for the instruction of college classes in the first principles of Astronomy. My aim has accordingly been to limit the book to such dimensions that it might be read entire without omissions, and to make such a selection of topics as should embrace every thing most important to the student. I have aimed to express every truth in concise and simple language; and when it was necessary to introduce mathematical discussions, I have limited myself to the elementary principles of the science. The entire book is divided into short articles, and each article is preceded by a caption, which is designed to suggest the subject of the article. Whenever it could be done to advantage, I have introduced simple mathematical problems, designed to test the student's familiarity with the preceding principles. At the close of the book will be found a collection of miscellaneous problems, many of them extremely simple, which are to be used according to the discretion of the teacher. I have dwelt more fully than is customary in astronomical text-books upon various physical phenomena, such as the constitution of the sun, the condition of the moon's surface, the phenomena of total eclipses of the sun, the laws of the tides, and the constitution of comets. I have also given a few of the results of recent researches respecting binary stars. It is hoped that the discussion of these topics will enhance the interest of the subject with a class of students who might be repelled by a treatise exclusively mathematical. My special acknowledgments are due to Professor H. A. Newton, who has read all the proofs of the work, and to whom I am indebted for numerous important suggestions. C ONTENTS. CHAPTER I. THE EARTH-ITS FIGURE AND DIMENSIONS.-DENSITY.-ROTATION. Page The Phenomena of the Diurnal Motion................................................... 9 The Figure of the Earth-how determined........................................... 15 Dimensions of the Earth-how determined.............................................. 16 The Celestial and Terrestrial Spheres...................................................... 18 Effects of Centrifugal Force upon the Form of the Earth............................ 22 Measurement of an Are of the Meridian.................................................. 25 The Density of the Earth-how determined............................................. 28 Direct Proof of the Earth's Rotation...................................................... 32 Artificial Globes-Problems on the Terrestrial Globe................................. 35 CHAPTER II. THE PRINCIPAL ASTRONOMICAL INSTRUMENTS. The Astronomical Clock-its Error and Rate......................................... 38 The Transit Instrument-its Adjustments................................................ 39 The Mural Circle-Reading Microscope.................................................. 43 The Altitude and Azimuth Instrument................................................... 47 The Sextant-its Adjustments and Use.......................................... 49 CHAPTER III. ATMOSPHERIC REFRACTION.-TWILIGHT. The Law of Atmospheric Refraction..................................................... 52 Refraction determined by Observation..................................................... 53 The Cause of Twilight-its Duration..................................................... 5 CHAPTER IV. EARTIH'S ANNUAL MIOTION.-EQUATION OF TIME.-CALENDAR. The Sun's apparent Motion-the Equinoxes, Solstices, etc......................... 5 The Change of Seasons-its Cause.................................................... 62 The Form of the Earth's Orbit.............................................................. 64 Sidereal and Solar Time-Mean Time and Apparent Time......................... 7 The Equation of Time explained.......................................................... 69 The Calendar-Julian and Gregorian.................................................... 72 Problems on the Celestial Globe...................................................... 74: vi CONTENTS. CHAPTER V. PARALLAX.-ASTRONOMICAL PROBLEMS. Page Diurnal Parallax-Horizontal Parallax................................................. 76 How to determine the Parallax of the Moon............................................. 77 Astronomical Problems-Latitude-Time, etc............................ 80 To find the Time of the Sun's Rising or Setting........................................ 83 CHAPTER VI. THE SUN-ITS PHYSICAL CONSTITUTION. Distance and Diameter of the Sun............................................. 88 The Physical Constitution of the Sun...................................................... 90 Theory of the Constitution of the Sun.................................................... 94 The Zodiacal Light described............................................................... 99 CHAPTER VII. PRECESSION OF THE EQUINOXES.-NUTATION.-ABERRATION. Precession of the Equinoxes-its Cause.................................................. 100 Nutation, Solar and Lunar................................................................... 104 Aberration of Light-its Cause..............................................10 105 Line of the Apsides of the Earth's orbit................................................... 107 CHAPTER VIII. THE MOON-ITS MOTION-PHASES-TELESCOPIC APPEARANCE. Distance, Diameter, etc., of the Moon..................................................... 109 Phases of the Moon-Harvest Moon................................................. 113 Has the Moon an Atmosphere?...................................................... 116 Telescopic appearance of the Moon............................................. 118 Librations of the Moon-Changes of the Moon's Orbit.............................. 124 CHAPTER IX. CENTRAL FORCES.-GRAVITATION.-LUNAR IRREGULARITIES. Curvilinear Motion-Kepler's Laws....................................................... 128 Theorems respecting Motion in an Orbit................................................. 129 Motion in an Elliptic Orbit................................................................... 132 The Law of Gravitation-Motions of Projectiles........................................ 137 The Problem of the Three Bodies.......................................................... 142 Sun's disturbing Force computed........................................................... 146 Evection, Variation, Annual Equation, etc.............................................. 148 Motion of the Moon's Nodes................................................................. 150 CHAPTER X. ECLIPSES OF THE MOON. Dimensions, etc., of the Earth's Shadow.................................................. 153 Lunar Ecliptic Limits-how determined............................................ 155 The Earth's Penumbra-its Dimensions................................................. 157 The Computation of Lunar Eclipses....................................................... 15I Computation illustrated by an Example.................................................. 161 CONTENTS. Vii CHAPTER XI. ECLIPSES OF THE SUN. Page Dimensions, etc., of the Moon's Shadow.................................................. 166 Different Kinds of Eclipses of the Sun............................................. 169 Phenomena attending Eclipses of the Sun.............................................. 173 Corona-Baily's Beads-Flame-like Protuberances.................................... 175 CHAPTER XII. METHODS OF FINDING THE LONGITUDE OF A PLACE. Method of Chronometers explained.................................................... 179 Method by Eclipses, Occultations, Lunar Distances, etc............................. 181 Method by the Electric Telegraph-Velocity of Electric Fluid..................... 182 CHAPTER XIII. THE TIDES. Definitions-Cause of the Tides....................................................... 185 Cotidal Lines-Velocity of Tidal Wave................................................ 188 The Tides modified by Conformation of the Coast..................................... 192 The Diurnal Inequality in the Height of the Tides.................................... 194 Tides of the Pacific Ocean, Gulf of Mexico, etc..................................... 196 CHAPTER XIV. THE PLANETS-ELEMENTS OF THEIR ORBITS. Number of the Planets-The Satellites...................................... 198 Apparent Motions of the Planets explained.............................................. 200 The Elements of the Orbit of a Planet................................................... 204 To determine the Distance of a Planet from the Sun................................. 207 To determine the Position of the Nodes-Inclination of Orbit, etc................. 209 CHAPTER XV. THE INFERIOR PLANETS. Greatest Elongations-Phases................................................ 213 Mercury and Venus-their Periods, Distances, etc..................................... 214 Transits of Mercury and Venus across the Sun's Disc................................. 217 Sun's Parallax-how determined........................................................... 219 CHAPTER XVI. THE SUPERIOR PLANETS. Mars-Distance-Phases-Form, etc..................................... 222 The Minor Planets-Discovery-Number, etc......................................... 224 Jupiter-Distance-Diameter-Belts, etc......................................... 227 Jupiter's Satellites-Distances-Eclipses-Occultations, etc......................... 228 Velocity of Light-how and by whom discovered..................................... 232 Saturn-Distance-Diameter-Rotation.............................................. 233 His Rings-their Disappearance explained........................................ 234 Saturn's Satellites-their Number, Distance, etc................................ 239 Uranus-Discovery-Distance-Diameter, etc......................................... 240 Neptune-History of its Discovery-its Satellite............................. a 243 V11i CONTENTS. CHAPTER XVII. QUANTITY OF MATTER IN THE SUN AND PLANETS.-PLANETARY PERTURBATIONS. Page How to determine the Mass of a Planet.............................................. 247 The Perturbations of the Planets-how computed..................................... 250 The Secular Inequalities of the Planets................................................... 253 The Stability of the Solar System....................................................... 255 CHAPTER XVIII. COMETS.-COMETARY ORBITS.-SHOOTING STARS. Number of Comets-the Coma-Nucleus-Tail, etc................................. 257 Comets' Tails —Dimensions-Position.................................................... 260 Cometary Orbits-how computed........................................................... 265 Halley's Comet-its History and Peculiarities.......................................... 269 Encke's Comet-Hypothesis of a resisting Medium......................... 271 Biela's Comet-Faye's Comet-Brorsen's Comet, etc.................................. 272 Comet of 1744-Comet of 1770-Comet of 1843, etc.................................. 276 Shooting Stars-Detonating Meteors —Erolites..................................... 2 279 CHAPTER XIX. THE FIXED STARS-THEIR LIGHT, DISTANCE, AND MOTIONS. Classification of the Fixed Stars-their Brightness.................................... 284 The Constellations-How Stars are Designated.................................. 287 Periodic Stars-Cause of this Periodicity-Temporary Stars........................ 290 Distance of the Fixed Stars-Parallaxes determined.......................... 292 Proper Motion of the Stars-Motion of the Solar System................... 295 CHAPTER XX. DOUBLE STARS.-CLUSTERS.-NEBULJE. Double Stars-Colored Stars-Stars optically Double............................... 297 Binary Stars-GammaVirginis-Alpha Centauri, etc................................ 299 Mass of a Binary Star computed-Triple Stars......................................... 303 Clusters of Stars-Nebule................................................................... 304 Planetary Nebulme-Variable Nebula.................................................... 308 The Milky Way-its Constitution and Extent........................................ 310 The Nebular Hypothesis-how tested...................................................... 314 MISCELLANEOUS PROBLEMS................................................................. 316 TABLES-ELEMENTS OF THE PLANETS, etc......................................... 321 EXPLANATION OF THE TABLES............................... C33 EXPLANATION OF TIIE PLATES.................................................. 337 AST RONOMY. CHAPTER I. GENERAL PHENOMENA OF THE HEAVENS.-FIGURE AND DIMENSIONS OF THE EARTH.-DENSITY OF THE EARTH.-PROOF OF THE EARTH'S ROTATION.-ARTIFICIAL GLOBES. 1. Astronomy is the science which treats of the heavenly bodies. The heavenly bodies consist of the sun, the planets with their satellites, the comets, and the fixed stars. Astronomy is divided into Spherical and Physical. Spherical Astronomy treats of the appearances, magnitudes, motions, and distances of the heavenly bodies. Physical Astronomy applies the principles of Mechanics to explain the motions of the heavenly bodies, and the laws by which they are governed. 2. Diurnal motion.-If we examine the heavens on a clear night, we shall soon perceive that the stars constantly maintain the same position relative to each other. A map showing the relative position of these bodies on any night, will represent them with equal exactness on any other night. They all seem to be at the same distance from us, and to be attached to the surface of a vast hemisphere, of which the place of the observer is the centre. But, although the stars are relatively fixed, the hemisphere, as a whole, is in constant motion. Stars rise obliquely from the horizon in the east, cross the meridian, and descend obliquely to the west. The whole celestial vault appears to be in motion round a certain axis, carrying with it all the objects visible upon it, without disturbing their relative positions. The point of the heavens which lies at the extremity of this axis of rotation is fixed, and is called the pole. There is a star called the pole star, distant about 1-~ from the pole, which moves in a small circle round the pole as a centre. All other stars appear 10 ASTRONOMY. also to be carried around the pole in circles, preserving always the same distance from it. 3. Axis of the celestial sphere.- This motion of rotation is perfectly uniform, as may be proved by observations with a telescope. Suppose the telescope of a theodolite, to be directed to the pole star; the star will appear to move in a small circle whose diameter is about three degrees; and the telescope may be so pointed that the star will move in a circle around the intersection of the spider-lines as a centre. This point of intersection is then the pole. The surface of the visible heavens to which all the heavenly bodies appear to be attached, is called the celestial sphere. 4. Use of a telescope mounted equatorially.-IHaving determined the axis of the celestial sphere, a telescope may be mounted capable of revolving upon a fixed axis which points toward the celestial pole, in such a manner that the telescope may be placed at any desired angle with the axis, and there may be attached to it a graduated circle by which the magnitude of this angle may be measured. A telescope thus' mounted is called an equatorial telescope, and it is frequently connected with clock-work, which gives it a motion round the axis corresponding with the rotation of the celestial sphere. 5. Diurnal paths of the heavenly bodies.-Let now the telescope be directed to any star so that it shall be seen in the centre of the field of view, and let the clock-work be connected with it so as to give it a perfectly uniform motion of rotation from east to west. The star will follow the telescope, and the velocity of motion may be so regulated, that the star shall remain in the centre of the field of view from rising to setting, the telescope all the time maintaining the same angle with the axis of the heavens. The same will be true of every star to which the telescope is directed; from which we conclude that all objects upon the firmament describe circles at right angles to its axis, each object always remaining at the same distance from the pole. 6. Time of one revolution of the celestial sphere.-If the telescope be detached from the clock-work, and, having been pointed upon GENERAL PHENOMENA OF THE HEAVENS. 11 a star, be left fixed in its position, and the exact time of the star's passing the central wire be noted, on the next night at about the same hour the star will again arrive upon the central wire. The time elapsed between these two observations will be found to be 23h. 56m. 4s., expressed in solar time. This, then, is the time in which the celestial sphere makes one revolution; and this time is always the same, whatever be the star to which the telescope is directed. 7. A sidereal day.-The time of one complete revolution of the firmament is called a sidereal day. This interval is divided into 24 sidereal hours, each hour into 60 minutes, and each minute into 60 seconds. Since the celestial sphere turns through 360~ in 24 sidereal hours, it turns through 15~ in one sidereal hour, and through 1~ in four sidereal minutes. 8. The diurnal motion is never suspended. —With a telescope of considerable power, all the brighter stars can be seen throughout the day, unless very near the sun; and by the method of observation already described, we find that the same rotation is preserved during the day as during the night. All the heavenly bodies, without exception, partake of this diurnal motion; but the sun, the moon, the planets, and the comets appear to have a motion of their own, by which they change their position among the stars from day to day. 9. The celestial equator is the great circle in which a plane passing through the earth's centre, and perpendicular to the axis of the heavens, intersects the celestial sphere. 10. If a plummet be freely suspended by a flexible line and allowed to come to a state of rest, this line is called a vertical line. The point where this line produced meets the visible half of the celestial sphere, is called the zenith; and the point where it meets the invisible hemisphere, which is under the plane of the horizon, is called the nadir. Every plane passing through a vertical line is called a vertical plane, or a vertical circle. That vertical circle which passes through the celestial pole is 12 ASTRONOMY. called the meridian. The vertical circle at right angles to the meridian is called the prime vertical. 11. A horizontal plane is a plane perpendicular to a vertical line. The sensible horizon of a place is the circle in which a plane passing through the place, and perpendicular to the vertical line at the place, cuts the celestial sphere. The rational horizon is the circle in which a plane passing through the earth's centre, and parallel to the sensible horizon, cuts the celestial sphere. On account of the distance of the stars, these two planes intersect the celestial sphere sensibly in the same great circle. The meridian and prime vertical meet the horizon in four points, called the cardinal points; or the north, south, east, and west points. 12. The altitude of a heavenly body is its elevation above the horizon measured on a vertical circle. The zenith distance of a body is its distance from the zenith measured on a vertical circle. The zenith distance is the complement of the altitude. The azimuth of a body is the arc of the horizon intercepted between the north or south point of the horizon, and a vertical circle passing through the body. Altitudes and azimuths are measured in degrees, minutes, and seconds. The amplitude of a star is its distance from the east or west point at the time of its rising or setting. 13. Consequences of the diurnal motion.-If an observer could Fig.. z. A watch the whole apparent J^<^^~^^^ ~ path of any star in the sky, / \\\X s\V he would see it describe a ~/ ^\\X. A\\ circle around the line PP'; F<^~MX^^r ^!^ ^^but as only half the celestial sphere is visible, it is evident' -- \\\\\ ~IfN that a part of the path of a star may lie below the hori~G X a zon and be invisible. Thus, in Fig. 1, let PP' be the axis V<^\ \^\\ / of rotation of the celestial sphere; NILSMK be the ho.' i rizon produced to intersect GENERAL PHENOMENA OF THE HEAVENS. 13 the sphere, and dividing it into two hemispheres, NS being the north and south line. If the parallel circles passing through A, C, E, and G be the apparent diurnal paths of four stars, then it is evident that 1st. The star which describes the circle AB will never descend below the horizon. 2d. The star which describes the circle Gi will never come above the horizon. 3d. The star which describes the circle ICKD will be above the horizon while it moves through ICK, and below the horizon through the portion KDI. 4th. The star which describes the circle LEMF will be above the horizon through the portion of the circle LEM, and below the horizon through the portion MFL. These stars are said to rise at I and L, and to set at K and M. They rise in the eastern part of the horizon, and set in the western. With the star C, the visible portion of its path ICK is greater than the invisible portion KDI; while with the star E, the visible portion of its path LEM is less than the invisible portion MFL. 14. Culminations of the heavenly bodies.-When stars cross the meridian above the pole they are said to culminate, or attain their greatest altitude. All stars cross the meridian twice every day; once above the pole, and once below the pole. The former is called their upper culmination, the latter is called their lower culmination. Thus the star which describes the circle AB has its upper culmination at A, and its lower culmination at B. It is evident from the figure that all stars which lie to the north of the equator, will remain above the horizon for a longer period than below it; all stars south of the equator will remain above the horizon for a shorter time than below it; and stars situated in the plane of the equator will remain above the horizon and below it for equal periods of time. 15. How the pole star may befound.-Among the most remarkable of the stars which never set in the latitude of New York, is the group of stars known as Ursa Major, shown in Fig. 2, which also represents the constellations Ursa Minor and Cassiopea. The constellation Ursa Major (represented on the left), is easily 14 ASTRONOMY. recognized by its resemblance to the figure of a dipper, and may be used to find the pole star by drawing a line through 3 and a Fig. 2. (called the Pointers), which will pass through the pole star a Ursae Minoris. A line drawn through 8 Ursve Majoris and the pole star, will pass nearly through 3 Cassiopeae (represented on the right). 16. What stars never set.-If a circle were drawn through N, the north point of the horizon, parallel to the equator, it would cut off a portion of the celestial sphere having P for its centre, all of which would be above the horizon; and a circle drawn through S, the south point of the horizon, parallel to the equator, would cut off a portion having P' for its centre, which would be wholly below the horizon. Stars which are nearer to the visible pole than the point N never set, and those which are nearer to the invisible pole than the point S never rise. 17. Why a knowledge of the dimensions of the earth is important. — The bodies of which astronomy treats are all (with the exception of the earth) inaccessible. Hence, for determining their distances, we are obliged to employ indirect methods. The eye can only judge of the direction of objects, and is unable to determine directly their distances; but by measuring the bearings of an inaccessible object from two points whose distance from each other is known, we may compute the distance of that object by the methods of trigonometry. In all our observations for determining the distance of the celestial bodies, the base line must be drawn upon the earth. It is therefore necessary to determine with the utmost precision the form and dimensions of the earth. FIGURE OF THE EARTH. 15 18. Proof that the earth is globular.-The figure of the earth is nearly globular. This is proved, 1st. By its having been many times sailed round in different directions. This fact can only be explained by supposing that the earth is rounded; but it does not alone furnish sufficiently precise information of its exact figure. 2d. By the phenomena of eclipses of the moon. These eclipses are caused by the earth coming between the sun and moon, so as to cast its shadow upon the latter. The form of this shadow is always such as one globe would project upon another. This argument is conclusive, but its force would not be admitted by those who deny the Copernican theory of the universe. 3d. By our seeing the top-mast of a ship, as it recedes from the observer, after the hull has disappeared. If the earth was a plane surface, the top-mast, having the smallest dimensions, should disappear first, while the hull and sails, having the greatest dimensions, should disappear last; but, in fact, the reverse takes place. Fig. 3. Land is visible from the top-mast when it can not be seen from the deck. The tops of mountains can be seen from a distance when their base is invisible. The sun illumines the summits of mountains long after it has set in the valleys. An aeronaut, ascending in his balloon after sunset, has seen the sun reappear with all the effects of sunrise; and on descending, he has witnessed a second sunset. 4th. If we travel northward, following a meridian, we shall find the altitude of the pole to increase continually at the rate of one degree for a distance of about 69 miles. This proves that a section of the earth made by a meridian plane is very nearly a circle, and also affords us the means of determining its dimensions, as shown.in Art. 20. 16 ASTRONOMY. 19. First method of determiining the earth's diameter.-The facts just stated not only demonstrate that the earth is globular, but afford us a rude method of computing its diameter. For this purpose we measure the height of some mountain, and also the distance at which it can be seen at sea. Let BD represent a mountain Fig. 4. (Chimborazo, for example), 4 miles in height; A 1B and suppose the distance, AB, at which it can be seen at sea, is 179 miles. Then, in the triangle ABC, representing the radius of the earth by R, we shall have V X (R+4)2=R2+ 1792 from which we find that R= 4000 miles nearly. Thus we learn that the radius of the earth is about 4000 miles. Similar observations made in all parts of the earth, give nearly the same value for the radius, which can only be explained by supposing that the earth is nearly a sphere. The earth is known to be globular by the most accurate measurements, as will be more fully explained hereafter. 20. Second method of determining the earth's diameter.-Having ascertained the general form of the earth, we wish to determine, as accurately as we can, its diameter. For this purpose we first ascertain the length of one degree upon its surface; that is, the distance between two points on the earth's surface so situated Fig 5. that the lines drawn from them to the centre of the earth st as may make with each other an angle of one degree. Let P and P' be two places on the earth's surface, distant from each other about 70 miles, and let C be the centre of the earth. Suppose two persons at the places P and P' observe two stars S and S', which are at the same instant vertically over the two places-that is, in the direction of plumb-lines suspended at those places. Let the directions of these plumb-lines be continued downward so as to intersect at C the centre of the earth. The angle which the directions of these stars make at 1P r' P is SPS', and the angle as seen from C is SCS'; but, on account of the distance of the stars, these angles are c ) sensibly equal to each other. If, then, the angle SPS' be measured, and the distance between the places P and FIGURE AND DIMENSIONS OF THE EARTH. 17 P' be also measured by the ordinary methods of surveying, the length of one degree can be computed. In this way it has been ascertained that the length of a degree of the earth's surface is about 69 statute miles, or 365,000 feet. Since a second is the 3600th part of a degree, it follows that the length of one second is one hundred feet very nearly. Since the plumb-line is perpendicular to the earth's surface, its change of direction in passing from one place to another may be found by allowing one second for every hundred feet, or more exactly by allowing 365,000 feet for each degree. 21. The circumference of the earth may be found approximately by the proportion 1 degree: 360 degrees:: 69 miles: 24,840 miles; and hence the diameter is found to be about 7900 miles; which results are a little too small, but may be employed as convenient numbers for illustration. The earth being globular, it is evident that the terms up and down can not every where denote the same absolute direction. The term up simply denotes from the earth's centre, while down denotes towards the earth's centre; but the absolute direction denoted by these terms at New York is very different from that denoted by the same terms at London or Canton. 22. Irregularities of the earth's surface.-The highest mountain peaks do not exceed five miles in height, which is about T,- of the earth's diameter. Accordingly, on a globe 16 inches in diameter, the highest mountain peak would be represented by a protuberance having an elevation of T-o inch, which is about twice the thickness of an ordinary sheet of writing-paper. The general elevation of the continents above the sea would be correctly represented by the thinnest film of varnish. In other words, the irregularities of the earth's surface are quite insignificant in comparison with its absolute dimensions. 23. Cause of the diurnal motion. The apparent diurnal rotation of the heavens may be caused either by a real motion of the celestial sphere, or by a real motion of the earth in a contrary direction. The former supposition is felt to be absurd as soon as we learn the distances and magnitudes of the celestial bodies. B 18 ASTRONOMY. The latter supposition is in itself not improbable, and perfectly explains all the phenomena. Moreover, we find direct proof of the rotation of the earth, in the descent of a body falling from a great height, which falls a little to the eastward of a vertical line. The figure of the earth, which is not that of a perfect sphere, affords independent proof of its rotation. Analogy also favors the same conclusion. All the planets which we have been able satisfactorily to observe, rotate on their axes, and their figures are such as correspond to the time of their rotation. The rotation of the earth gives to the celestial sphere the appearance of revolving in the contrary direction, as the forward motion of a boat on a river gives to the banks an appearance of backward motion; and since the apparent motion of the heavens is from east to west, the real rotation of the earth which produces that appearance must be from west to east. 24. The earth's axis is the diameter around which it revolves once a day. The extremities of this axis are the terrestrial poles; one is called the north pole, and the other the south pole. The terrestrial equator is a great circle of the earth perpendicular to the earth's axis. Meridians are great circles passing through the poles ofthe earth. 25. The latitude of a place is the arc of the meridian which is comprehended between that place and the equator. Latitude is reckoned north and south of the equator, from 0 to 90~. A parallel of latitude is any small circle on the earth's surface parallel to the terrestrial equator. These parallels continually diminish in size as we proceed from the equator to the pole. The polar distance of a place is its distance from the nearest pole, and is the complement of the latitude. The longitude of a place is the arc of the equator intercepted between the meridian of that place and some assumed meridian to which all others are referred. The English reckon longitude from the observatory of Greenwich, the French from the observatory of Paris, and the Germans from the observatory of Berlin, or from the island of Ferro, which is assumed to be 20~ west of FIGURE AND DIMENSIONS OF THE EARTH. 19 the observatory in Paris. In the United States we sometimes reckon longitude from Washington, and sometimes from Greenwich. Longitude is usually reckoned east and west of the first meridian, from 0 to 1800. The longitude and latitude of a place determine its position on the earth's surface. 26. The latitude of at place.-Let SIENQ represent the earth surrounded by the distant starry sphere IIZOK. The diameter of the earth being insignificant in comparison with the distance of the stars, the appearance of the heavens will be the same Fig. 6. Z whether they are viewed from the centre of the earth, or from any point on its surface. Suppose the observer to be at P, a point on the surface between the equator iE and the north pole N. The latitude of this place is zEP, or the angle 1ECP. If the line PC be continued to the firmament, it will pass through the point Z, which is the zenith of the observer. If the terrestrial axis NS be continued to the firmament, it will pass through the celestial poles N' and S'. If the terrestrial equator IEQ be continued to the heavens, it will constitute the celestial equator IE'Q'. The observer at P will see the entire hemisphere HZO, of which his zenith Z is the pole. The other hemisphere will be concealed by the earth. The arc N'O contains the same number of degrees as FE'Z, and 20 ASTRONOMY. the arc ZN' is the complement of ON'; that is, the altitude of the visible pole is equal to the latitude of the place, and the zenith distance of the visible pole is the complement of the latitude. 27. How the latitude of a place may be determined.-If there were a star situated precisely at the pole, its altitude would be the latitude of the place. The pole star describes a small circle around the pole, and crosses the meridian twice in each revolution, once above and once below the pole. The half sum of the altitudes in these two positions is equal to the altitude of the pole; that is, to the latitude of the place. The same result would be obtained by observing any circumpolar star on the meridian both above and below the pole. 28. Circles which pass through the two poles of the celestial sphere are called hour circles. If two such circles include an arc of 15~ of the celestial equator, the interval between the instants of their coincidence with the meridian will be one hour. 29. The right ascension of a star is the arc of the celestial equator comprehended between a certain point on the equator called the first point of Aries, and an hour circle passing through that star. Right ascension is sometimes expressed in degrees, minutes, and seconds of arc, but generally in hours, minutes, and seconds of time. It is reckoned eastward from zero up to 24 hours, or 360 degrees. If the hands of the sidereal clock be set to Oh. Om. Os. when the first point of Aries is' on the meridian, the clock (if it neither gains nor loses time) will afterward indicate at each instant the right ascension of any object which is then on the meridian, for the motion of the hands of the clock corresponds exactly with the apparent diurnal motion of the heavens. While 15~ of the equator pass the meridian, the hands of the clock move through one hour. The sidereal day therefore begins when the first point of Aries crosses the meridian, and the sidereal clock should always indicate Oh. Om. Os. when the first of Aries is on the meridian. 30. The distance of an object from the celestial equator, measured upon the hour circle which passes through it, is called its declination, and is north or south according as the object is on FIGURE AND DIMENSIONS OF THE EARTH. 21 the north or south side of the equator. North declination is indicated by the sign +, and south declination by the sign -. The position of an object on the firmament is indicated by its declination and right ascension. Its declination expresses its distance north or south of the celestial equator, and its right ascension expresses the distance of the hour circle upon which it is situated, from a fixed point upon the celestial equator. The north polar distance of a star is its distance from the north pole. 31. A right sphere.-The celestial sphere presents different appearances to observers in different latitudes. If the observer were situated at the terrestrial equator, the poles would lie in the horizon, the celestial equator would be perpendicular to the plane of the horizon, and hence the horizon would bisect the equator and all circles parallel to it. Therefore all celestial objects would be for equal periods above and below the horizon, and they would appear to rise perpendicularly on the eastern side of the horizon, and set perpendicularly on the western side. Such a sphere is called a right sphere, the diurnal motion being at right angles to the horizon. 32. A parallel sphere.-At one of the poles of the earth, the celestial pole being in the zenith, the celestial equator would coincide with the horizon, and by the diurnal motion all celestial objects would move in circles parallel to the horizon. This is called a parallel sphere. In a parallel sphere, an object upon the equator will be carried by the diurnal motion round the horizon, without either rising or setting. 33. An oblique sphere.-At all latitudes between the equator and the pole, the celestial equator is inclined to the horizon at an angle equal to the distance of the pole from the zenith; that is, equal to the complement of the latitude. The parallels DF, GK, Fig. 6, are unequally divided by the horizon; that is, all objects between the celestial equator and the visible pole are longer above than below the horizon, and all objects on the other side of the equator are longer below than above the horizon. A parallel, BO, whose distance from the visible pole is equal to the latitude, is entirely above the horizon; and the same is true 22 ASTRONOMY. of all parallels still nearer to that pole. Also the parallel HL, whose distance from the invisible pole is equal to the latitude, is entirely below the horizon; and the same is true of all parallels still nearer to that pole. Hence, in the United States, stars within a certain distance of the north pole never set, and stars at an equal distance from the south pole never rise. The circle BO is called the circle of perpetual apparition, because the stars which are included within it never set. The radius of this circle is equal to the latitude of the place. The circle HL is called the circle of perpetual occultation, because the stars which are included within it never rise. The radius of this circle is also equal to the latitude of the place. The celestial sphere here described is called an oblique sphere, the diurnal motion being oblique to the horizon. Whether the sphere be right or oblique, one half of the celestial equator will be below the horizon, and the other half above it. Every object on the equator will therefore be above the horizon during as long a time as it is below, and will rise and set at the east and west points. 34. Effects of centrifugal force.-We have discovered that the earth has a globular figure, and that it rotates upon its axis once in 24 sidereal hours. But, since the earth rotates upon an axis, its form can not be that of a perfect sphere; for every body revolving in a circle acquires a centrifugal force which tends to make it recede from the centre of the circle. Every particle, P, upon the earth's surface acquires, therefore, a force which acts in a direcrig. 7. N tion, EP, perpendicular to the axis of rotation. This centrifugal force, =E A~-h which we will represent by PA, may / be resolved into two other forces PB Q and PD, one acting in the direction of a radius of the earth, and the other at right angles to the radius. The former, being opposed to the earth's ~s ~ attraction, has the effect of diminishing the weight of the body; the latter, being directed toward the equator, tends to produce motion in the direction of the equator. The intensity of the centrifugal force increases with the radius of the circle described, and is therefore greatest at the equator. FIGURE AND DIMENSIONS OF THE EARTH. 23 Moreover, the nearer the point is to the equator, the more directly is the centrifugal force opposed to the weight of the body. The effects, therefore, produced by the rotation of the earth are, 1st. All bodies decrease in weight in going from the pole to the equator; and, 2d. All bodies which are free to move, tend from the higher latitudes toward the equator. 35. The effect of centrifugal force computed.-Let A be a ball attached to a string AS; let S be a fixed point, and ACE the circle in which the ball revolves, and AC the arc rig.8. which the ball describes in a given time. When B the ball was at A, it was moving in the direc- tion of the tangent AB, and it would continue in this direction if it were acted upon by no other force than the first impulse; but we find it deflected into the diagonal AC, and this diagonal is the resultant of two forces represented E by AB, AD. Now AB represents the path which the ball would describe under the first impulse, and therefore AD represents the motion impressed upon it by the tension of the string, and which deflects the ball from the tangent to the circle, and this is equal to the centrifugal force generated by the revolution of the ball. Now AD: AC:: AC: AE; whence AD =2 2AS'' But AC represents the velocity of the revolving body. If, then, C represents the centrifugal force of a revolving body, V its velocity in feet per second, and R the length of the string in feet, V2 we shall have C=2. 36. -Centrfugal force compared with the force of gravity. —We may compare the centrifugal force of a body with the force of gravity, by comparing the spaces through which the body would move in a given time, under the operation of these two forces. Let W=the weight of the revolving body, and g=16 feet, the space through which W would fall freely in one second. Then V2 we shall have W: C:: g:; 2iR W.V2 whence C=. 2^7 24 ASTRONOMY. We may also express the centrifugal force of a revolving body by reference to the number of revolutions made in a given time. Let N represent the number of revolutions, or the fraction of a revolution performed by the body in one second. The circumference of the circle which the body describes will be 27rR. The space through which the body moves in one second, that is, its velocity, is 27rR.N. Hence we have c = W.(2RN) _ 27x RN2.W 1.2275 x RN2.W. 2R.y g The amount of the loss of weight produced at the equator by centrifugal force, may be computed as follows: The radius of the equator is 20,923,600 feet; and since the time of one rotation is 23h. 56m. 4s., or 86164 seconds, N= I -—. Hence C= 1.2275 x 20,923,600 x xW, W or C- 2 289' Thus we find that at the equator the centrifugal force of a body arising from the earth's rotation, is - part of the weight; and since this force is directly opposed to gravity, the weight must sustain a loss of - part. 37. Centrifugal force at any latittude.-The centrifugal force at the equator is to the centrifugal force in any other latitude as radius to the cosine of the latitude. But the entire centrifugal force at any latitude is to that part of the centrifugal force which is opposed to the weight of the body, as radius to the cosine of the latitude; that is, the loss of weight of a body caused by the centrifugal force at any latitude, is o of the weight multiplied by the square of the cosine of the latitude. 38. Effect of centrifugal force upon the form of a body.-A portion, PD, of the centrifugal force causes a tendency to move toward the equator. If the surface of the globe were entirely solid, this tendency would be counteracted by the cohesion of the particles. But since a portion of the earth's surface is fluid, this portion must yield to the centrifugal force, and flow toward the equator. Thus the water must recede from the higher latitudes in either hemisphere, and accumulate around the equator. The earth, therefore, instead of being an exact sphere, must become FIGURE AND DIMENSIONS OF THE EARTH. 25 an oblate spheroid. A globe consisting of any plastic material would be reduced to such a figure by causing it to rotate rapidly upon an axis. The amount of the ellipticity of the earth must depend upon the centrifugal force, and the attraction exerted by the earth upon bodies placed on its surface. 39. Weightz of a body at the pole and the equator.-We have found that at the equator the loss of weight due to centrifugal force is -g. From a comparison of observations of the length of the seconds' pendulum made in different parts of the globe, it is found that the weight of a body at the pole actually exceeds its weight at the equator, by TyT. The difference between these fractions is T4-~-~ = - o; that is, the actual attraction exerted by the earth upon a body at the equator is less than at the pole, by the 590th part of the whole weight. This difference is due to the elliptic form of the meridians, by which the distance of the body at the equator from the centre of the earth is increased. 40. How an arc of a meridian is measured.-Numerous arcs of the meridian have been measured, for the purpose of accurately determining the figure and dimensions of the Fg. 9. earth. These arcs are measured in the fol- lowing manner: / A level spot of ground is selected, where a / base line, AB, from five to ten miles in length, c! is measured with the utmost precision. A third station, C, is selected, forming with the base line a triangle as nearly equilateral as is convenient. The angles of this triangle are measured with a theodolite, and the two remaining sides may then be computed. A fourth station, D, is now selected, forming with ~E two of the former stations a second triangle, in which all the angles are measured; and since one side is already known, the others may be computed. A fifth station, E, is then selected, forming a third triangle; and thus we proceed forming a series of triangles, fol- lowing nearly the direction of a meridian, by I 26 ASTRONOMY. means of which we can compute the distance of the extreme stations from each other. The latitude of the most northerly and also that of the most southerly station must be determined, whence we obtain the difference of latitude corresponding to the arc measured. This method is the most accurate known for determining the distance between two remote points on the earth's surface, because we may choose the most favorable site for measuring accurately the base line; and after this, nothing is required but the measurement of angles, which can be done with much less labor, and with much greater accuracy, than the measurement of distances. 41. Verification of the work.-In order to verify the entire work, a second base line is measured near the end of the series of triangles, and we compare its measured length with the length as computed from the first base, through the intervention of the series of triangles. In the great arc measured in France between the years 1792 and 1799, the base of verification was distant 400 or 500 miles from the first base, and was seven miles in length, yet the difference between its observed and computed length did not amount to twelve inches. In the Ordnance Survey of Great Britain and Ireland, six base lines have been measured, the longest being 7.88 miles in length, and the shortest 4.64 miles. In one instance the observed length of a base differs from its computed length by 19 inches. In each of the other cases, the discrepancy is less than three inches. 42. Results of measurements.-In this manner, arcs of the meridian have been measured in nearly every country of Europe. One has also been measured in India, one in South America, and one in South Africa. The operations for the survey of the coast of the United States will ultimately furnish several other arcs of a meridian. The following is a list of the principal arcs already completed: The Peruvian - - - arc is 214 miles in length 3~ 7'; " East Indian - - 1468 " " =21 21; " French - - - - 854 " " =12 22; English - - - " 756 " " 10 56; Russian- - - - 1753 " " =25 20; Cape of Good Hope 275 " " 4 0. FIGURE AND DIMENSIONS OF THE EARTH. 27 The sum of these arcs exceeds 60 degrees, not counting double measurements of the same part of the meridian; that is, we have measured nearly two thirds of the distance from the equator to the north pole. We can therefore compute the remaining distance with but small liability to error. The result is that a degree at the equator=68.702 miles. a degree at the pole =69.396 " the difference =.694 " 43. Conclusion from these results.-If the earth were perfectly spherical, a terrestrial meridian would be an exact circle, and every part of it would have the same curvature; that is, a degree of latitude would be every where the same. But we have found that the length of a degree increases as we proceed from the equator toward the poles, and the amount of this difference affords a measure of the departure of a meridian from the figure of a circle. The plumb-line must every where be perpendicular to the surface of tranquil water, and can not, therefore, every where point exactly toward the earth's centre. Let A, B be two plumb-lines suspended on the same meridian near Fig. 10. the equator, and at such a distance D _ C from each other as to be inclined at an angle of 1i. Let C and D be two other plumb-lines on a meridian near A one of the poles, also making with each other an angle of 1~. The distance from A to B is found to be less than from C to D, from which we conclude that the meridian curves more rapidly near A than near C. It is found that all the observations in every part of the world are very accurately represented by supposing the meridian to be an ellipse, of which the polar diameter is the minor axis. The equatorial diameter of this ellipse is 7926.708 miles. the polar diameter " " 7899.755 the difference is 26.953 " That is, the equatorial diameter exceeds the polar diameter by -~ th of its length. This difference is called the ellipticity of the earth. The meridional circumference of the earth is 24,857.5 miles. 28 ASTRONOMY. From measurements which have been made at right angles to the meridian, it appears that the equator and parallels of latitude are very nearly, if not exactly, circles. Hence it appears that the form of the earth is that of an oblate spheroid; which is a solid generated by the revolution of a semi-ellipse about its minor axis. 44. Loss of weight at the equator explained. —It has been mathematically proved that a spheroid whose ellipticity is I, and whose mean density is double the density at the surface, exerts an attraction upon a particle placed at its pole, greater by - oth part than the attraction upon a particle at its equator; and this we have seen is the fraction which must be added to the loss of weight by centrifugal force, to make up the total loss of weight at the equator, as shown by experiments with the seconds' pendulum. This coincidence may be regarded as demonstrating that the earth does rotate upon its axis once in 24 hours. 45. Equatorial protuberance.-If a sphere be conceived to be inFig. 1. N scribed within the terrestrial spheroid, having the polar axis NS for its diameter, a spheroidal shell will be included between its A( C D surface and that of the spheroid, having a thickness, AB, of 13 miles at the equator, and -^^_ ^ becoming gradually thinner toward the poles. S This shell of protuberant matter, by means of its attraction, gives rise to many important phenomena, as will be explained hereafter. The Density of the Eart]h. 46. Three methods have been practiced for determining the average density of the earth. These methods are all founded upon the principle of comparing the attraction which the earth exerts upon any object, with the attraction which some other body, whose mass is known, exerts upon the same object. Fi;rst method.-By comparing the attraction of the earth with that of a small mountain. In 1774, Dr. Maskelyne determined the ratio of the mean density of the earth to that of a mountain in Scotland, called Schehallien, by ascertaining how much the local attraction of the mountain deflected a plumb-line from a vertical position. This TIlE DENSITY OF THE EARTH. 29 mountain stands alone on an extensive plain, so that there are no neighboring eminences to affect the plumb-line. Two stations were selected, one on its northern and the other on its southern side, and both nearly in the same meridian. A plumb-line, at, tached to an instrument called a zenith Fi ig 12. C sector, designed for measuring small ze- A c nith distances, was set up at each of these stations, and the distance from the direc- tion of the plumb-line to a certain star was measured at each station, the instant \ that the star was on the meridian. The difference between these distances gave 3B IED the angle formed by the two directions of the plumb-lines AE, CG. Were it not for the mountain, the plumb-lines would take the positions AB, CD; and the angle which they would, in that case, form with each other, is found by measuring the distance between the two stations, and allowing about one second for every hundred feet. In Dr. Maskelyne's experiment, the distance between the two stations was 4000 feet; so that if the direction of gravity had not been influenced by the mountain, the inclination of the plumblines at the two places would have been 41 seconds. The inclination was actually found to be 53". The difference, or 12", is to be ascribed to the attraction of the mountain. It was computed that if the mountain had been as dense as the interior of the earth, the disturbance would have been about 21". Therefore, the ratio of the density of the mountain to that of the entire earth, was that of 12 to 21. The mean density of the mountain was ascertained by numerous borings to be 2.75 times that of water. Hence the mean density of the earth was concluded to be 4.95 times that of water. In the year 1855, observations were made for ascertaining the deviation of the plumb-line produced by the attraction of Arthur's Seat, a hill 822 feet high, near Edinburg, from which the mean density of the earth was computed to be 5.32. 47. Second method.-The mean density of the earth has been determined by experiments with the torsion balance. In the year 1798, Cavendish compared the attraction of the 30 ASTRONOMY. earth with the attraction of two lead balls, each of which was one foot in diameter. The bodies upon which their attraction was exerted were two leaden balls, each about two inches in diameter. They were attached to the ends of a slender wooden rod six feet in length, which was supported at the centre by a fine wire 40 inches long. The balls, if left to themselves, will come to rest when the supporting wire is entirely free from torsion, but a very slight force is sufficient to turn it out of this plane. The position of the supporting rod was accurately observed with a fixed telescope. The large balls were then brought near the small ones, but on opposite sides, so that the attraction of both balls might conspire to twist the wire in' the same direction, when it was found that the small balls were sensibly attracted by the larger ones, and the amount of this deflection was carefully measured. The large balls were then moved to the other side of the small ones, when the rod was found to be deflected in the contrary direction, and the amount of this deflection was recorded. This experiment was repeated seventeen times. These experiments furnish a measure of the attraction of the large balls for the small ones, and hence we can compute what would be their attraction if they were as large as the earth. But we know the attraction actually exerted by the earth upon the small balls, it being measured by the weight of the balls. Thus we know the attractive force of the earth compared with that of the lead balls; and since we know the density of the lead, we can compute the average density of the earth. From these experiments, Cavendish concluded that the mean density of the earth was 5.45. These experiments were repeated by Dr. Reich, at Freyberg, in Saxony, in the year 1836, and the mean of 57 trials gave a result of 5.44. In the years 1841-'2 a similar series of experiments was conducted with the greatest care by Sir Francis Baily in England, and from over 2000 trials he concluded the mean density of the earth to be 5.67. 48. Third method.-The mean density of the earth may be determined by means of pendulum experiments at the top and bottom of a deep mine. The rate of vibration of a pendulum depends upon the intensity of the earth's attraction, and thus be THE DENSITY OF THE EARTH. 31 comes a measure of this intensity. If we vibrate the same pendulum at the top and bottom of a mine whose depth is 1000 feet, we shall have a measure of the force of gravity at the bottom of the mine compared with the force at the top. Now at the top of the mine the pendulum is attracted by every particle of matter in the globe; but, since a spherical shell may be shown to exert no influence upon a point situated within it, the pendulum at the bottom of the mine will only be influenced by a sphere whose radius is 1000 feet less than that of the earth. We thus obtain the attraction of this external shell, whose thickness is 1000 feet, compared with the attraction of the entire globe; and since the volumes of both these bodies may be computed, we are able to deduce the average density of the globe, compared with that of the external shell. Now, by actual examination, we can determine the density of the strata penetrated by the mine, and hence we are able to compute the mean density of the globe. This method was applied in one of the mines of England, near Newcastle, in the year 1854. The depth of the mine was 1256 feet; and it was found that a pendulum which vibrated seconds at the top of the mine, when transferred to the bottom of the mine gained 2- seconds per day. From this it was computed that the force of gravity at the bottom of the mine was 19 29 greater than at the top of the mine; and hence it was computed that the average density of the globe was 2.62 times that of the external shell. By actual examination, it was found that the average density of the rocks penetrated by the mine was 2.5, whence it follows that the mean density of the earth is 6.56. 49. Fourth method.-In a somewhat similar manner, we may determine the density of the earth by comparing the length of the pendulum vibrating seconds on the summit of a mountain, with that at the base of the mountain. In 1824, the vibrations of a pendulum on Mount Cenis, in Italy, at an elevation of 673-1 English feet, were compared with the vibrations near the level of the sea, and the density of the earth was hence deduced to be 4.84. The average of these seven determinations is 5.46, which must be a tolerable approximation to the truth. These results verify, in a remarkable manner, the conjecture of Newton, who, in 1680, estimated that the average density of the earth was 5 or 6 times greater than that of water. 32 ASTRONOMY. 50. Volume and weight of the earth. —Having determined the dimensions of the earth, we cal easily compute its volume, and, knowing its density, we can also compute its weight. Its volume is found to contain 259,400 millions of cubic miles. Also the total weight of the earth is 6 sextillions of tons-a number expressed by the figure 6 with 21 ciphers annexed. 51. Direct proof of the earth's rotation. —A direct proof of the earth's rotation is derived from observations of a pendulum. If a heavy ball be suspended by a flexible wire from a fixed point, and the pendulum thus formed be made to vibrate, its vibrations will all be performed in the same plane. If, instead of being suspended from a fixed point, we give to the point of support a slow movement of rotation around a vertical axis, the plane of vibration will still remain unchanged. This may be proved by holding in the fingers a pendulum composed of a simple ball and string, and causing it to vibrate. Upon twirling the string between the fingers, the ball will be seen to rotate on its axis, without, however, changing its plane of vibration. Suppose, then, a heavy ball to be suspended by a wire from a fixed point directly over the pole of the earth, and made to vibrate; these vibrations will continue to be made in the same invariable plane. But the earth meanwhile turns round at the rate of 15~ per hour; and since the observer is unconscious of his own motion of rotation, it results that the plane of vibration of the pendulum appears to revolve at the same rate in the opposite direction. If the pendulum be removed to the equator, and set vibrating in the direction of a meridian, the plane of vibration will still remain unchanged; and since, notwithstanding the earth's rotation, this plane always coincides with a meridian, the plane of vibration appears to remain unchanged. 52. Phenomena in the middle latitudes.-At places intermediate between the pole and the equator, the apparent motion of the plane of vibration is less than 15~ per hour, and diminishes as we recede from the pole. This may be proved in the following manner: PROOF OF THE EARTH'S ROTATION. 33 Let NPSE represent a meridian of Fig. 13. the earth, AP a tangent to this circle A at P, meeting the earth's axis produced / in A. Suppose a pendulum to be set up at the point P,and vibrated in the plane of the meridian. When by the, rotation of the earth the point P is brought to P', the plane of vibration will tend to preserve its parallelism \ \ with the plane ACP; but the merid- ian of the place will have the position AP'C; that'is, the plane of vibration will now make an angle AP'B, s or PAP', with the plane of the meridian. The angle PAP', bePP' cos. Lat. ing taken very small, may be considered equal to —, or cot. Lat.' which equals sin. Lat.; that is, the apparent motion of the plane of vibration is every where proportional to the sine of the latitude. The hourly motion of the plane of vibration of a pendulum set up at New Haven, is therefore equal to 15~ x sin. 41" 18', which is a little less than 100 per hour. When this experiment is performed with the greatest care, the observed rate of motion coincides very accurately with the computed rate; and this coincidence may be regarded as a direct proof that the earth makes one rotation upon its axis in 24 sidereal hours. 53. Second proof of the earth's rotation.-A second proof of the earth's rotation is derived from the motion of falling bodies. If the earth had no rotation upon an axis, a heavy body let fall from any elevation would descend in the direction of a vertical line. But if the earth rotates on an axis, then, since the, top of a tower describes a larger circle than the base, its easterly motion must be more rapid than that of the base. And if a ball be dropped from the top of the tower, since it has already the easterly motion which belongs to the top of the tower, it will retain this easterly motion during its descent, and its deviation to the east of the vertical line will be nearly equal to the excess of the motion of the top of the tower above that of the base, during the time of fall. C 34 ASTRONOMY. Let AB represent a vertical tower, and AA' the space through ig. 14. which the point A would be carried by the earth's roA' zA tation in the time that a heavy body would descend ~ through AB. A body let fall from the top of the ~tower will retain the horizontal velocity which it had at starting, and, when it reaches the earth's surface, will have moved over a horizontal space, BD, nearly equal to AA'. But the foot of the tower will have moved only through BB', so that the body will be found to the east of the tower by a space equal to B'D nearly. This space B'D, for an elevation of 500 feet, c in the latitude of New Haven, is a little over one inch, so that it must be impossible to detect this deviation except from experiments conducted with the greatest care and from an elevation of several hundred feet. 54. Results of experiments.-In the year 1791 this method was first tried at Bologna, in Italy, Lat. 44~ 30', from a tower whose height was 256 English feet. According to a mean of 12 trials, the deviation amounted to 0.74 inch to the east of a vertical line, and 0.47 inch to the south of a vertical. According to theory, the easterly deviation should have been 0.43 inch, and the southerly deviation should have been zero. The result of these experiments was not therefore satisfactory. This discrepancy has been ascribed to a possible change in the position of the tower, due to the change of temperature, inasmuch as the experiments were made in summer, but the exact position of the plumb-line was not determined until the subsequent winter. In the year 1802 the experiment was repeated by Benzenberg, at Hamburg, Lat. 53~ 33', from a tower whose height was 250 English feet. The mean of 31 experiments gave a deviation of 0.35 inch to the east of a vertical, and 0.11 inch to the south. According to theory, the easterly deviation should have been 0.34 inch, which accords remarkably well with the observations. The southerly deviation of 0.11 inch is not in accordance with theory. This experiment was again repeated in 1804 by Benzenberg in Germany, Lat. 51~ 25', in a coal mine near Diisseldorf, whose depth was 280 English feet. According to 28 experiments, the average deviation to the east was 0.45 inch, while the computed ARTIFICIAL GLOBES. 35 deviation was 0.41 inch; showing a discrepancy of only 0.04 inch, which is quite satisfactory. The experiments also showed a deviation of 0.06 inch to the north, which is not explained by theory. In the year 1832 these experiments were repeated with great care by Prof. Reich, at Freyberg, Saxony, Lat. 50~ 53', in a mine whose depth was 520 English feet. The balls employed were of metal, 1.59 inch in diameter, and had a specific gravity of 7.88. According to the mean of 106 trials, the easterly deviation was 1.12 inch, while the deviation by theory should have been 1.08 inch. The experiments also showed a southerly deviation of 0.17 inch, which is not accounted for by theory. These experiments must be regarded as proving that the earth does rotate upon an axis, although the results exhibit discrepancies greater than might have been anticipated, and which, perhaps, are not fully explained. ARTIFICIAL GLOBES. 55. Artificial globes are either terrestrial or celestial. The former exhibits a miniature representation of the earth, the latter exhibits the relative position of the fixed stars. The mode of mounting is usually the same for both, and many of the circles are the same for both globes. An artificial globe is mounted on an axis which is supported by a brass ring, which represents a meridian, and is called the brass mneridian. This ring is supported in a vertical position by a frame in such a manner that the axis of the globe can be inclined at any angle to the horizon. The brass meridian is graduated into degrees, which are numbered from the equator toward either pole. The horizon is represented by a broad ring, whose plane passes through the centre of the globe. It is also graduated into degrees, which are numbered in both directions from the north and south points, to denote azimuths; and there is usually another set of numbers which begin from the east and west points, to denote amplitudes. It also usually contains the signs of the ecliptic, showing the sun's place for every day in the year. On the terrestrial globe, hour circles are represented by great circles drawn through the poles of the equator; and on the celestial globe corresponding circles are drawn through the poles of the ecliptic, and a series of small circles parallel to the ecliptic 36 ASTRONOMY. are drawn at intervals of ten degrees. These are for determining celestial latitude and longitude. The ecliptic, tropics, and polar circles are drawn upon the terrestrial globe, as well as upon the, celestial. About the north pole is a small circle, graduated so as to indicate hours and minutes, while a small index, attached to the brass meridian, points to one of the divisions upon this hour circle. This index can be moved so as to be set in any required position. There is usually a flexible strip of brass, equal in length to one quarter of the circumference of the globe, which is graduated into degrees, and may be applied to the surface of the globe so as to measure the distance between two places, or the altitude of any point above the wooden horizon. Hence it is usually called the quadrant of altitude. PROBLEMS ON THE TERRESTRIAL GLOBE. 56. To find the latitude and longitude of a given place. Turn the globe so as to bring the place to the graduated side of the brass meridian; then the degree of the meridian directly over the place will indicate the latitude, and the degree on the equator under the brass meridian will indicate the longitude. Example. What are the latitude and longitude of Cape Horn? 57. Given the latitude and longitude, to find the place. Bring the degree of longitude on the equator under the brass meridian, then under the given latitude on the brass meridian will be found the place required. Example. Find the place which is situated in Lat. 30~ N. and Long. 90~ W. 58. To find the bearing and distance from one place to another on the earth's surface. Elevate the north pole to the latitude of the first-mentioned place, and bring this place to the brass meridian. Screw the quadrant of altitude to this point of the brass meridian, and make it pass through the other place. Then the bearing of the second place from the first will be indicated on the wooden horizon, and the number of degrees on the quadrant of altitude will show the distance between the two places in degrees, which may be reduced ARTIFICIAL GLOBES. 37 to miles by multiplying them by 691, because 691 miles make nearly one degree. Example. What is the bearing and distance of Liverpool from New York? 59. To find the antipodes of a given place. Bring the given place to the wooden horizon, and the opposite point of the horizon will indicate the antipodes. The one place will be as far from the north point of the wooden horizon, as the other is from the south point. Example. Find the antipodes of London. 60. Given the hour of the day at any pace, to find the hour at any other place. Bring the first-mentioned place to the brass meridian, and set the hour index to the given time. Turn the globe till the other place comes to the meridian; the hour circle will show the required time. Example. What time is it at San Francisco when it is 10 A.M. in New York? 61. To find the time of the sun's rising and setting at a given place, on a given day. Elevate the pole to the latitude of the place. On the wooden horizon find the day of the month, and against it is given the sun's place in the ecliptic, expressed in signs and degrees. Bring the sun's place to the meridian, and set the hour index to 12. Turn the globe till the sun's place is brought down to the eastern horizon; the hour index will show the time of rising. Turn the globe till the sun's place comes to the western horizon; the hour index will tell the time of setting. Example. Required the time of rising and setting of the sun at Washington, August 18th. 38 ASTRONOMY. CHAPTER II. INSTRUMENTS FOR OBSERVATION. - THE CLOCK. -TRANSIT INSTRUMENT. - MURAL CIRCLE. - ALTITUDE AND AZIMUTH INSTRUMENT, AND THE SEXTANT. 62. Why observations are chiefly made in the meridian. —Whenever circumstances allow an astronomer to select his own time of observation, almost all his observations of the heavenly bodies are made when they are upon the meridian, because a large instrument can be more accurately and permanently adjusted to describe a vertical plane than any plane oblique to the horizon; and there is no other vertical plane which combines so many advantages as the meridian. The places of the heavenly bodies are most conveniently expressed by right ascension and declination, and the right ascension is simply the time of passing the meridian, as shown by a sidereal clock. Moreover, when a heavenly body is at its upper culmination, its refraction and parallax are the least possible; and in this position refraction and parallax do not affect the right ascension of the body, but simply its declination; while for every position out of the meridian, they affect both right ascension and declination. 63. The Clock.-The standard instruments of an astronomical observatory are the clock, the transit instrument, and the mural circle. In a stationary observatory, a pendulum clock is used for measuring time. The clock should be so regulated that if a star be observed upon the meridian at the instant when the hands point to Oh. Om. Os., they will point to Oh. Om. Os. when the same star is next seen on the meridian. This interval is called a sidereal day, and is divided into 24 sidereal hours. If the clock were perfect, the pendulum would make 86,400 vibrations in the interval between two successive returns of the same star to the meridian. But no clock is perfect, and it is therefore necessary to determine the error and rate of the clock daily, and in all our observations to make an allowance for the error of the clock. THE TRANSIT INSTRUMENT. 39 The error of a clock at anytime is its difference from true sidereal time. The atte of the clock is the change of its error in 24 hours. Thus, if, on the 8th of January, when Aldebaran passed the meridian, the clock was found to be 30.84s. slow, and on the 9th of January, when the same star passed the meridian, the clock was 31.66s. slow, the clock lost 0.82s. per day. In other words, the error of the clock January 9th was -31.66s, and its daily rate -0.82s. The Transit Instrument. 64. Most of the observations of the heavenly bodies are made when they are upon the celestial meridian; and, in many cases, the sole business of the observer is to determine the exact instant when the object is brought to the meridian, by the apparent diurnal motion of the firmament. This phenomenon of passing the meridian is called a transit, and an instrument, mounted in such a manner as to enable an observer, supplied with a clock, to ascertain the exact time of transit, is called a transit instrument. 65. Description of the Tran- rig. 15. sit Instrument.-Such an instrument consists of a telescope, TT, mounted upon an axis, AB, at right angles to the tube, which axis occupies a horizontal position, and points east and west. The. tube of the telescope, when horizontal, will therefore be directed north and south; and Iil if the telescope be revolved L I I on its axis through 180~, the central line of the tube will move in the plane of the me- iil ridian, and may be directed III to any point on the celestial meridian. For a large transit instrument, two stone piers, PP, are erected on a solid foundation, 40 ASTRONOMY. standing on an east and west line. On the top of each of the piers is secured a metallic support, in the form of the letter Y, to receive the extremities of the axis of the telescope. At the left end of the axis there is a screw, by which the Y of that extremity may be raised or lowered a little, in order that the axis may be made perfectly horizontal. At the right end of the axis is a screw, by which the y of that extremity may be moved backward or forward, in order to enable us to bring the telescope into the plane of the meridian. In order that the pivots of the axis may be relieved from a portion of the weight of the instrument, there is raised upon the top of each pier a brass pillar supporting a lever, from one end of which hangs a hook passing under one extremity of the axis, while a counterpoise sliding on the other end of the lever may be made to support as much of the weight of the instrument as is desired. 66. The Spirit Level.-When the instrument is properly adjusted, its axis will be horizontal, and directed due east and west. If the axis be not exactly horizontal, its deviation may be ascertained by placing upon it a spirit level. This consists of a glass Fig. 1. tube, AB, nearly filled with alcohol C D or ether. The tube forms a portion.An~~S d B~ gof a ring of a very large radius, and when it is placed horizontally, with its convexity upward, the bubble, CD, will occupy the highest position in the middle of its length. A graduated scale is attached to the tube, by which we may measure any deviation of the bubble from the middle of the tube. To ascertain whether the axis of the telescope is horizontal, apply the level to it, and see if the bubble occupies the middle of the tube. If it does not, one end must be elevated or depressed, In order to accomplish this, one of the supports of the axis is constructed so as to be moved vertically through a small space by means of a fine screw. The level must now be taken up and reversed end for end, and this operation must be repeated until the bubble rests in the middle of the tube in both positions of the level. 67. Method of observing transits.-In the focus of the eye-piece of the transit instrument, at F, is placed a system of 5 or 7 equi METHOD OF OBSERVING TRANSITS. 41 distant and vertical wires, intersected by 1 or rig. 17. M 2 horizontal wires. When the instrument has been properly adjusted, the middle wire, MN, will be in the plane of the meridian, and when an object is seen upon it, this object will be on the celestial meridian. The fixed stars appear in the telescope as bright points of light without sensible magnitude, and by the di- urnal motion of the heavens a star is carried successively over each of the wires of the transit instrument. The observer, just before the star enters the field of view, writes down the hour and minute indicated by the clock, and proceeds to count the seconds by listening to the beats of the clock, while his eye is looking through the telescope. He observes the instant at which the star crosses each of the wires, estimating the time to the nearest tenth of a second; and by taking a mean of all these observations, he obtains with great precision the instant at which the star passed the middle wire, and this is regarded as the true time of the transit. The mean of the observations over several wires, is considered more reliable than an observation over a single wire. In many observatories it is now customary to employ the electric circuit to record transit observations. By pressing the finger upon a key at the instant a star is seen to pass one of the wires of the transit, a mark is made upon a sheet of paper which is graduated into seconds by the pendulum of the observatory clock, according to the mode more fully explained in Art. 337. During the day, the wires are visible as fine black lines stretched across the field of view. At night they are rendered visible by a lamp, L, by which the field of view is faintly illumined. When we observe the sun or any object which has a sensible disc, the time of transit is the instant at which the centre of the disc crosses the middle wire. This time is obtained by observing the instants at which the eastern and western edges of the disc touch each of the wires in succession, and taking the mean of all the observations. When the visible disc is not circular, special methods of reduction are employed. 68. Rate of the diurnal motion.-Since the celestial sphere revolves at the rate of 150 per hour, or 15 seconds of arc in one second of time, the space passed over between two successive 42 ASTRONOMY. beats of the pendulum will be 15" of arc. When the sun is on the equator, and its apparent diameter is 32' of arc, the interval between the contacts of the east and west limbs with the middle wire will be 2m. 8s. 69. To adjust a transit instrument to the meridian.-A transit instrument may be adjusted to describe the plane of the meridian, by observations of the pole star. Direct the telescope to the pole star at the instant of its crossing the meridian, as near as the time can be ascertained. The transit will then be nearly in the plane of the meridian. Having leveled the axis, turn the telescope to a star about to cross the meridian, near the zenith. Since every vertical circle intersects the meridian at the zenith, a zenith star will cross the field of the telescope at the same time, whether the plane of the transit coincide with the meridian or not. At the moment the star crosses the central wire, set the clock to the star's right ascension which is given by the star catalogues, and the clock will henceforth indicate nearly sidereal time. The approximate times of the upper and lower culminations of the pole star are then known. Observe the pole star at one of its culminations, following its motion until the clock indicates its right ascension, or its right ascension plus 12 hours. Move the whole frame of the transit so that the central wire shall coincide nearly with the star, and complete the adjustment by means of the azimuth screw. The central wire will now coincide almost precisely with the meridian of the place. 70. Final verification.-The axis being supposed perfectly horizontal, if the middle wire of the telescope is exactly in the meridian, it will bisect the circle which the pole star describes in 24 sidereal hours round the polar point. If, then, the interval between the upper and lower culminations is exactly equal to the interval between the lower and upper, the adjustment is complete. But if the time elapsed while the star is traversing the eastern semicircle, is greater than that of traversing the western, the plane in which the telescope moves is westward of the true meridian on the north horizon; and vice versa if the western interval is greatest. This error of position must be corrected by turning the azimuth screw. The adjustment must then be verified by further observations, until, by continued approximations, the instrument is fixed correctly in the meridian. THE MURAL CIRCLE. 43 Other methods of adjusting a transit instrument to the plane of the meridian, will be found in works specially devoted to Practical Astronomy. The Mural Circle. 71. The mural circle is a graduated circle, aaaa, usually made of brass, and having an axis passing through its centre. This axis should be exactly horizontal; and it is supported by a stone pier or wall, so as to be directed due east and west. To the circle is attached a telescope, MM, so that the entire instrument, including the telescope, turns ini the plane of the meridian. Fig. 18.... l c s h b mde, I Mural circles have been made eight feet in diameter, but generally they have been made six feet; and at present astronomers are pretty well agreed that a circle of five feet is better than any larger size, being less liable to change of form from its great weight. At the great Russian observatory at Pulkova, the largest circle employed is only four feet in diameter. The circle is divided into degrees, and subdivided into spaces of five minutes, and sometimes of two minutes, the divisions being numbered from 0~ to 360~ round the entire circle. The smallest spaces on the 44 ASTRONOMY. limb are further subdivided to single seconds, sometimes by a verr nier, but generally by a reading microscope. 72. Use of the Vernier.-A vernier is a scale of small extent, graduated in such a manner that, being moved by the side of a fixed scale, we are enabled to measure minute portions of this scale. The length of this movable scale is equal to a certain number of parts of that to be subdivided; but it is divided into parts either one more, or one less, than those of the primary scale taken for the length of the vernier. Thus, if we wish to measure hundredths of an inch, as in the case of a barometer, we first divide an inch into ten equal parts. We then construct a vernier equal in length to 11 of these divisions, but divide it into 10 equal parts, by which means each division on the vernier is l-th longer than a division of the primary scale. Fig. 19. Thus, let AB be the upper end of a ba, A' rometer tube, the mercury standing at the point C; the scale is divided into inches and 30 tenths of an inch, and the middle piece, numbered from 1 to 9, is the vernier, that may c r,..i~ _ _ be slid up or down, and having 10 of its di2 _ visions equal to 11 divisions of the scale; that is, to ths of an inch. Therefore, each 5~-s 9 division of the vernier is -,ths of an inch; 7 -_ 9 or one division of the vernier exceeds one l ~8 _ division of the scale, by -ro-th of an inch. Now, as the sixth division of the vernier (in:^ ^1111 - the figure) coincides with a division of the scale, the fifth division of the vernier will stand a-oth of an inch above the nearest division of the scale; the fourth division -aoths of an inch; and the top of the vernier will be a-2-ths of an inch above the next lower division of the scale; i. e., the top of the vernier coincides with 29.66 inches upon the scale. In practice, therefore, we observe what division of the vernier coincides with a division of the scale; this will show the hundredths of an inch to be added to the tenths next below the vernier at the top. 73. Vernier applied to graduated circles.-A similar contrivance is often applied to graduated circles to obtain the value of an arc THE READING MICROSCOPE. 45 with greater accuracy. If a circle is graduated to half degrees, or 30', and we wish to -measure single minutes by the vernier, we take an arc equal to 31 divisions upon the limb, and divide it into 30 equal parts. Then each division of the vernier will be equal to 3lths of a degree, while each division of the scale is 3-ths of a degree; that is, each space on the vernier exceeds one on the limb by I'. In order, therefore, to read an angle for any position of the vernier, we pass along the vernier until a line is found coinciding with a line of the limb. The number of this line from the zero point, indicates the minutes which are to be added to the degrees and half degrees taken from the graduated scale. 74. The reading Microscope.-The large circles employed in astronomical observations are divided into spaces as small as 5', and sometimes as small as 2'. By a vernier these spaces are sometimes subdivided so as to give single seconds. The vernier is generally employed in instruments made by German artists, but upon large circles made by English artists the subdivisions are usually effected by the reading microscope. Fig. 20 represents the appearance of one of Fig. 20. these microscopes. It is a compound microscope, consisting of three lenses, one of which is the ob- = ject lens at L, and the other two are formed into a positive eye-piece, GH. In the common focus of the object lens and the eye-piece at K, is placed the spider-line micrometer. It consists of a small rectangular frame, across which are stretched two spider-lines forming an acute cross, and is moved laterally by means of a screw, M. The figure on the right shows the field of view, with the magnified divisions on the instrument, as seen through the microscope. When the microscope is properly adjusted, the image of the divided limb and the spider-lines are distinctly visible together; and also five revolutions of the screw must exactly measure one of the 5' spaces on the limb. One revolution of the head of the screw will therefore carry the spider-lines over a space of 1'. The circumference of the circle attached to the head, M, is divided into 46 ASTRONOMY. 60 equal parts, so that the motion of the head through one of these divisions, advances the spider-lines through a space of 1". There are six of these microscopes, A, B, C, D, E, F, placed at equal distances round the circle, and firmly attached to the pier. 75. To determine the horizontal point.-In order to ascertain the horizontal point upon the limb of the circle, we direct the telescope upon any star which is about crossing the meridian, and bring its image to coincide with the horizontal wire which passes through the centre of the field of the telescope. The graduation is then read off by the fixed microscopes. On the next night, we place a vessel containing mercury in a convenient position near the floor, so that, by directing the telescope of the mural circle toward it, the same star may be seen reflected from the surface of the mercury, and we bring the reflected image to coincide with the horizontal wire of the telescope. The graduation is then read off as before. Now, by a law of optics, the reflected image will appear as much below the horizon as the star is really above the horizon; therefore half the sum of the two readings at either of the microscopes, will be the reading at the same microscope when the telescope is horizontal. 76. To determine the altitude of any object. —Having determined the reading of each of the microscopes when the telescope is directed to the horizon, if we wish to determine the altitude of any object, we direct the telescope to it, so that it may be seen on the horizontal wire as the star passes the meridian, and then read off the microscopes. The difference between the last reading, and the reading when the telescope is horizontal, is the altitude required. The zenith distance of an object is found by subtracting its altitude from 90~. The pole star crosses the meridian, above and below the pole, at intervals of 12 hours sidereal time; and the true position of the pole is exactly midway between the two points where the star crosses the meridian; therefore half the sum of the readings of either microscope when the pole star makes its transit above and below the pole, will be the reading for the pole itself. The readings for the pole being determined, those which correspond to the point where the celestial equator crosses the meridian, are easily found, since the equator is 90~ from the pole. ALTITUDE AND AZIMUTH INSTRUMENT. 47 Having determined the position of the celestial equator, the declination of any star is easily determined, since its declination is simply its distance from the equator. 77. The Transit Circle.-Since the mural circle has a short axis, its position in the meridian is unstable, and therefore it can not be relied upon to give the right ascension of stars with great accuracy. It was formerly thought necessary at Greenwich to have two instruments for determining a star's place; viz., a transit instrument to determine its right ascension, and a mural circle to determine its declination. The German astronomers have, however, combined both instruments in one, under the name of meridian circle, which is essentially the transit instrument already described, with a large graduated circle attached to its axis; and a large transit circle is Fig. 21 now in use at the Greenwich Observatory. Altitude and Azimutlh Instruzment. 78. The altitude and azimuth instrument consists of one graduated circle confined to a horizontal plane; a second graduated circle perpendicular to the former, and capable of being / turned into any azimuth; and a telescope firmly fastened to the second \ circle, and turning with it in altitude. The ap- I pearance of this instrument will be learned from the annexed figure. EE are two legs of the'\ tripod upon which the instrument rests; and in close contact with the 48 ASTRONOMY. tripod is placed the azimuth circle, FF. Above the azimuth circle, and concentric with it, is placed a strong circular plate, which sustains the whole of the upper part of the instrument, and also a pointer, to show the degree and nearest five minutes to be read off on the azimuth circle; the remaining minutes and seconds being obtained by means of the two reading microscopes C and D. The pillars, IHH, support the transit axis I by means of the projecting pieces LL. The telescope, MM, is connected with the horizontal axis in a manner similar to that of the transit instrument. Upon the axis, as a centre, is fixed the double circle NN, each circle being placed close against the telescope. The circles are fastened together by small brass pillars, and the graduation is made on a narrow ring of silver, inlaid on one of the sides, which is usually termed the face of the instrument. The reading microscopes, AB, for the vertical circle, are carried by two arms, PP, attached near the top of one of the pillars. In the principal focus of the telescope, are stretched spider lines, as in the transit instrument, and the illumination is effected in a similar manner. 79. Adjustments of the instrument.-Before commencing observations with this instrument, the horizontal circle must be leveled, and also the axis of the telescope. The meridional point on the azimuth circle is its reading when the telescope is pointed north or south, and may be determined by observing a star at equal altitudes east and west of the meridian, and finding the point midway between the two observed azimuths; or the instrument may be adjusted to the meridian in the same manner as a transit. The horizontal point of the altitude circle is its reading when the axis of the telescope is horizontal, and may be found, as with the mural circle, by alternate observations of a star directly and reflected from the surface of mercury. This instrument has the advantage over the transit instrument and mural circle, in its being able to determine the place of a star in any part of the visible heavens; but we ordinarily require the place of a star to be given in right ascension and declination instead of altitude and azimuth, and to deduce the one from the other requires a laborious computation. Hence the altitude and azimuth instrument is but little used in astronomical observations, except for special purposes, as, for example, to investigate the laws of refraction. THE SEXTANT. 49 The Sextant. 80. The arc of a sextant, as its name implies, contains sixty degrees, but, on account of the double reflection, is divided into 120 degrees. The annexed figure represents a sextant, the frame being generally made of brass; Fig. 22. the handle, H, at its back, is c made of wood. When observing, the instrument is l to be held with one hand by the handle, while the other hand moves the index G. The arc, AB, is di- vided into 120 or more de-, grees, numbered from A to-\' A ward B, and each degree is divided into six equal parts of 10' each, while the vernier shows 10". The divisions are also continued a short distance on the other side of zero toward A, forming what is called the arc of excess. The microscope, M, is movable about a centre, and may be adjusted to read off the divisions on the graduated limb. A tangent screw, D, is fixed to the index, for the purpose of making the contacts more accurately than can be done by hand. When the index is to be moved a considerable distance, the screw I must be loosened; and when the index is brought nearly to the required division, the screw I must be tightened, and the index be moved gradually by the tangent screw. The upper end of the index G terminates in a circle, across which is fixed the silvered index glass C, over the centre of motion, and perpendicular to the plane of the instrument. To the frame at N is attached a second glass, called the horizon glass, the lower half of which only is silvered. This must also be perpendicular to the plane of the instrument, and in such a position that its plane shall be parallel to the plane of the index glass C, when the vernier is set to zero on the limb AB. The telescope, T, is carried by a ring, K; and in the focus of the object glass are placed two wires parallel to each other, and equidistant from the axis of the telescope. Four dark glasses, of different depths of shade and color, are placed at F, between the inD 50 ASTRONOMY. dex and horizon glasses; also three more at E, any one or more of which can be turned down, to moderate the intensity of the light before reaching the eye, when a bright object, as the sun, is observed. 81. To measure the altitude of the sun by refection from mercury. -Set the index near zero. Hold the instrument with the right hand in the vertical plane of the sun, with the telescope pointed toward the sun. Two images will be seen in the field of view, one of which, viz., that formed by reflection, will apparently move downward when the index is pushed forward. Follow the reflected image as it travels downward, until it appears to be as far below the horizon as it was at first above, and the image of the sun reflected from the mercury also appears in the field of view. Fasten the index, and, by means of the tangent screw, bring the upper or lower limb of the sun's image reflected from the index glass, into contact with the opposite limb of the image reflected from the artificial horizon. The angle shown on the instrument, when corrected for the index error, will be double the altitude of the sun's limb above the horizontal-plane; to the half of which, if the semi-diameter, refraction and parallax be applied, the result will be the true altitude of the centre. If the observer is at sea, the natural horizon must be employed. Direct the sight to that part of the horizon beneath the sun, and move the index till you bring the image of its lower limb to touch the horizon directly underneath it. 82. To measure the distance between two objects.-To find the distance between the moon and sun, hold the sextant so that its plane may pass through both objects. Look directly at the moon through the telescope, and move the index forward till the sun's image is brought nearly into contact with the moon's nearest limb. Fix the index by the screw under the sextant, and make the contact perfect by means of the tangent screw. The index will then show the distance of the nearest limbs of the sun and moon. In a similar manner may we measure the distance between the moon and a star. 83. Dip of the horizon.-In observing an altitude at sea with the sextant, the image of an object is made to coincide with the ATMOSPHERIC REFRACTION. 51 visible horizon; but since the eye is elevated above the surface of the sea, the visible horizon will be below the' true horizontal plane. Let AC be the radius of the earth, AD Fig. 23. the height of the eye above the level of the E D) s sea, EDH a horizontal plane passing through the place of the observer; then HDB will be the dclp or depression of the horizon, which may be found as follows: C The angle HDB is equal to the angle BCD; and in the right-angled triangle BCD, BD2 \ CD2- BC2 =(AC + AD)2 —AC2. Whence BD becomes known. Then, in the same triangle, CD: rad.: BD: sin. BCD(= HDB), the depression of the horizon. The depression thus obtained is the true depression; but this must be lessened by the amount of terrestrial refraction, which is very uncertain. About ~-th or y —th of the whole quantity is usually allowed. The following table shows the dip or apparent depression of the horizon for different elevations of the eye, allowing -lth for terrestrial refraction: Height. Depression. Height. Depression. 5 feet. 2' 9" 30 feet. 5' 15" 10 " 3 2 50 " 6 46 15 " 3 42 70 " 8 1 20" 4 17 100 " 9 35 CHAPTER III. ATMOSPHERIC REFRACTION.-TWILIGHT. 84. THE air which surrounds the earth decreases gradually in density as we ascend from the surface. At the height of 4 miles, the density is only about half as great as at the earth's surface; at the height of 8 miles about one fourth as great; at the height of 12 miles about one eighth as great, and so on. From this law it follows that at the height of 50 miles, its density must be extremely small, so as to be nearly or quite insensible. 52 ASTRONOMY. 85. Law of atmospheric refraction.-According to a law of optics, when a ray of light passes obliquely from a rarer to a denser medium, it is bent toward the perpendicular to the refracting surig. 24. face. Let SA be a ray of light coming Z I'S/ from any distant object, S, and falling /S on the surface of a series of layers of air, increasing in density downward. The ray SA, passing into the first layer, will be deflected in the direction BI~ ~, A AB, toward a perpendicular to the sur~___~ face, MN. Passing into the next layV___/. ____ er, it will be again deflected in the diD rection BC, more toward the perpendicular; and passing through the lowest layer, it will be still more deflected, and will enter the eye at D, in the direction of CD; and, since every object appears in the direction from which the visual ray enters the eye, the object S will be seen in the direction DS', instead of its true direction AS. Since the density of the earth's atmosphere increases gradually from its upper surface to the earth, when a ray of light from any of the heavenly bodies enters the atmosphere obliquely, its path is not a broken line, as we have here supposed, but a curve, concave toward the earth. The density of the upper parts of the atmosphere being very small, the curve at first deviates very little from a straight line, but the deviation increases as it approaches the earth. Both the straight and curved parts of the ray lie in the same vertical plane; that is, the refraction of the atmosphere makes an object appear to be nearer the zenith than it really is, but does not affect its azimuth. 86. How the refraction may be computed.-It is a difficult probN lem to compute the exact amount of the refraction of the atmosphere; but for altitudes exceeding 10 degrees, the entire refraction may be assumed to take place at a single surface, as MN, and may be computed approximately in the following manner: Let z denote the apparent zenith distance of a star, and r the effect of refraction; then, if there were no refraction, the zenith distance would be z+r. But we have found in Optics, Art. 691, that the sine of incidence=m x sine of refraction, where m represents the index of refraction. Hence ATMOSPHERIC REFRACTION. 53 sin. (z+r) =m sin. z. But by Trigonometry, Art. 72, sin. (z+r)=sin. z cos. r+ cos. z sin. r. For zenith distances less than 80~, r is less than 6', and therefore its sine may be considered equal to the arc, and its cosine equal to unity. Hence we find sin. z- r cos. z m sin. z. sin. Dividing by cos. z, and putting s- =tangent (Trigonometry, Art. 28), we obtain tang. tang. 28), we obtain tang. z+r-m tang. z, or r= (m —1) tang. z. r is here expressed in parts of radius. If we wish to have its value expressed in seconds, we must multiply it by 206265, which is the number of seconds in an arc equal to radius. If r" represents the refraction expressed in seconds, then r'"=206265 r. At the temperature of 50~, and pressure 29.96 inches, the refractive index of air is 1.0002836. Hence we have r" =0.0002836 x 206265 x tang. z. or r" =58".49, tang. z; that is, the refraction is equal to 58".49 x tangent of the zenith distance. For altitudes exceeding ten degrees, this formula will furnish the refraction pretty nearly; but near the horizon the law of refraction is exceedingly complicated. The following table shows the average amount of refraction for different altitudes: Altitude. Refraction. Altitude. Refraction. Altitude. Refraction. 0~ 34' 54" 6~ 8' 23" 20~ 2' 37" 1 24 25 7 7 20 30 1 40 2 18 9 8 6 30 40 1 9 3 14 15 9 5 49 50 48 4 11 39 10 5 16 70 21 5 9 46 15 3 32 90 0 We perceive from this table, that the refraction is nothing in the zenith, and is greatest in the horizon, where it amounts to about 35'. 87. How the refraction may be determined by observation.-The amount of refraction for different altitudes, may be determined by observation as follows: In latitudes greater than 45~, a star which 54 ASTRONOMY. passes through the zenith of the place, may also be observed when it passes the meridian below the pole. Let the polar distance of such a star be measured both at the upper and lower culminations. In the former case there will be no refraction; the difference between the two observed polar distances will therefore be the amount of refraction for the altitude at the lower culmination; because if there were no refraction, the apparent diurnal path of the star would be a circle with the celestial pole for its centre. This method is strictly applicable only in latitudes greater than 45~, and by observations at one station we can only determine the refraction corresponding to a single altitude. Since, however, for zenith distances less than 45~, the amount of refraction is quite small, and is given with great accuracy by the Tables, we may safely extend the application of this method. We may therefore select any star within the circle of perpetual apparition, and observe its polar distance at the upper and lower culminations, and correct the former for refraction. The difference between this corrected value and the observed polar distance at the lower culmination, will be the refraction corresponding to the latter altitude. 88. Second method of determininng refraction. —The following method is more general in its application, and requires no previous knowledge of the amount of refraction: Observe the altitude of a star whose declination is known, and note the time by the clock. Observe also when the star crosses the meridian, and the difference of time between the observations will give the hour angle of the star from the meridian. Fig. 25. z Let PZH be the meridian of the place of observation, P the pole, Z / \ g the zenith, and S the true place of the star. Let ZS be a vertical circle passing through the star, and IL ~ 0 PS an hour circle passing through the star. Then, in the triangle ZPS, PZ=the complement of the latitude, PS=the north polar distance of the star, and ZPS=the angular distance of the star from the meridian. In this triangle we know, therefore, two sides and the included angle, from which we can compute ZS, or the true zenith distance of the star. The difference between the computed value of ZS ATMOSPHERIC REFRACTION. 55 and its observed value, will be th,e refraction corresponding to this altitude. If we commence our observations when the, star is near the horizon, and continue them at short intervals until it reaches the meridian, we may, by a proper selection of stars, determine the amount of refraction for all altitudes from zero to 90~. 89. Corrections for temperature and pressure.-The amount of refraction at a given altitude is not constant, but depends upon the temperature, and weight of the air. Tables have been constructed, partly from observation and partly from theory, by which we may at once obtain the mean refraction for any altitude; and rules are given by which a correction may be made for the state of the barometer and thermometer. 90. Effect of refraction upon the time of sunrise.-Since refraction increases the altitudes of the heavenly bodies, it must accelerate their rising and retard their setting, and thus render them longer visible. The amount of refraction at the horizon is about 35', which being a little more than the apparent diameters of the sun and moon, it follows that these bodies, at the moment of rising and setting, are visible above the horizon, when in reality they are wholly below it. 91. Effect of refraction upon the figure of the sun's disc. —When the sun is near the horizon, the lower limb, being nearest the horizon, is most affected by refraction, and therefore more elevated than the upper limb, the effect of which is to bring the two limbs apparently closer together by the difference between the two refractions. The apparent diminution of the vertical diameter sometimes amounts at the horizon to one fifth of the whole diameter. The disc thus assumes the form of an ellipse, of which the major axis is horizontal. 92. Enlargement of the sun near the horizon.-The apparent enlargement of the sun and moon near the horizon is an optical illusion. If we measure the apparent diameters of these bodies with any suitable instrument, we shall find that they subtend a less angle near the horizon, than they do when near the zenith. It is, then, wholly owing to an error of judgment that they seem to us larger near the horizon. 56 ASTRONOMY. Our judgment of the absolute magnitude of a body is based upon our judgment of its distance. If two objects at unequal distances subtend the same angle, the more distant one must be the larger. Now the sun and moon, when near the horizon, appear to us more distant than when they are high in the heavens. They seem more distant in the former position, partly from the number of intervening objects, and partly from diminished brightness. When the moon is near the horizon, a variety of intervening objects shows us that the distance of the moon must be considerable; but when the moon is on the meridian no such objects intervene, and the moon appears quite near. For the same reason, the vault of heaven does not present the appearance of a hemisphere, but appears flattened at the zenith, and spread out at the horizon. Our estimate of the distance of objects is also affected by their brightness. Thus, a distant mountain, seen through a perfectly clear atmosphere, appears much nearer than when seen through a hazy atmosphere. 93. Cause of twilight.-The sun continues to illumine the clouds and the upper strata of the air, after it has set, in the same manner as it shines on the summits of mountains after it has set to the inhabitants of the adjacent plains. The air and clouds thus illumined reflect light to the earth below them, and produce twilight. As the sun continues to descend below the horizon, a less part of the visible atmosphere receives his direct light; less light is transmitted by reflection to the surface of the earth; until, at length, all reflection ceases, and night begins. This takes place when the sun is about 18~ below the horizon. Before sunrise in the morning, the same phenomena are exhibited in the reverse order. If there were no atmosphere, none of the sun's rays could reach us after his actual setting, or before his rising. Fig.26. Let ABCD represent a portion of the earth, A a point on its surface where the. "rD \sun, S, is in the act of setting, and let SAH i be a ray of light just grazing the earth at TWILIGHT. 57 A, and leaving the atmosphere at the point H. The point A is illuminated by the whole reflective atmosphere HGFE. The point B, to which the sun has set, receives no direct solar light, nor any reflected from that part of the atmosphere which is below ALH, but it receives a twilight from the portion HLF, which lies above the visible horizon BF. The point C receives a twilight only from the small portion of the atmosphere HMG, while at D the twilight has ceased altogether. 94. Duration of twilight at the equator.-The duration of twilight varies with the season of the year, and with our position upon the earth's surface. At the equator, where the circles of daily rotation are perpendicular to the horizon, when the sun is in the celestial equator, it descends through 18~ in an hour and twelve minutes (-1 =1 hours); that is, twilight lasts lh. and 12m. When the sun is not in the equator, the duration of twilight is somewhat increased. 95. Duration of twilight at the poles.-At the north pole there is night as long as the sun is south of the equator; but whenever it is not more than 18~ south, the sun is never more than 18~ below the horizon. About the close of September, the sun sinks below the horizon, and there is continual twilight until November 12th, when it attains a distance of 18~ from the equator. From this date there is no twilight until January 29th, from which time there is continual twilight until about the middle of March, when the sun rises above the horizon, and continues above the horizon uninterruptedly for six months. 96. Duration of twilight in middle latitudes.-At intermediate points of the earth, the duration of twilight may vary from lh. 12m. to several weeks. In latitude 40~, during the months of March and September, twilight lasts about an hour and a half, while in midsummer it lasts a little over two hours. In latitude 50~, where the north pole is elevated 50~ above the horizon, the point which is on the meridian 18~ below the north point of the horizon, is 68~ distant from the north pole, and therefore 22~ distant from the equator. Now, during the entire month of June, the distance of the sun from the equator exceeds 22~; that is, in latitude 50~ there is continual twilight from sunset to sunrise, during a period of more than a month. 58 ASTRONOMY. At places nearer to the pole, the period of the year during which twilight lasts through the entire night, is still longer. 97. Consequences if there were no atmosphere. — If there were no atmosphere, the darkness of midnight would instantly succeed the setting of the sun,and it would continue thus until the instant of the sun's rising. During the day the illumination would also be much less than it is at present, for the sun's light could only penetrate apartments which were directly accessible to his rays, or into which it was reflected from the surface of natural objects. On the summits of mountains, where the atmosphere is very rare, the sky assumes the color of the deepest blue, approaching to blackness, and stars become visible in the daytime. CHAPTER IV. THE EARTH'S ANNUAL MOTION.-SIDEREAL AND SOLAR TIME.THE EQUATION OF TIME.-THE CALENDAR.-THE CELESTIAL GLOBE. 98. Sun's apparent motion in right ascension.-If we observe the exact position of the sun with reference to the stars, from day to day through the year, we shall find that it has an apparent motion among them along a great circle of the celestial sphere, whose plane makes an angle of 23~ 27' with the plane of the celestial equator. This motion may be determined by observations with the transit instrument and mural circle. If the sun's transit be observed daily, andits right ascension be determined, it will be found that the right ascension increases each day about four minutes of time, or one degree, so that in a year the sun makes a complete circuit round the heavens, moving constantly among the stars from west to east. This daily motion in right ascension is not uniform, but varies from 215s. to 266s., the mean being about 236s., or 3m. 56s. 99. Sun's apparent motion in declination.-If the point at which the sun's centre crosses the meridian be observed daily with the mural circle, it will be found to change from day to day. Its declination is zero on the 20th of March, from which time its north THE EARTH'S ANNUAL MOTION. 59 declination increases until it becomes 230 27' on the 21st of June. It then decreases until the 22d of September, when the sun's centre is again upon the equator. Its south declination then increases until it becomes 23~ 27' on the 21st of December, after which it decreases until the sun's centre returns to the equator on the 20th of March. If we trace upon a celestial globe the course of the sun from day to day, we shall find its path to be a great circle of the heavens, inclined to the equator at an angle of 23~ 27'. This circle is called the ecliptic, because solar and lunar eclipses can only take place when the moon is very near this plane. 100. The equinoxes and solstices.-The ecliptic intersects the celestial equator at two points diametrically opposite to each other. These are called the equinoctial points; because, when the sun is at these points, it is for an equal time above and below the horizon, and the days and nights are therefore equal. The point at which the sun passes from the south to the north side of the celestial equator, is called the vernal equinoctial point, and the other is called the autumnal equinoctial point. The times at which the sun's centre is found at these points are called the vernal and autumnal equinoxes. The vernal equinox, therefore, takes place on the 20th of March, and the autumnal on the 22d of September. Those points of the ecliptic which are midway between the equinoctial points are the most distant from the celestial equator, and are called the solstitial points; and the times at which the sun's centre passes those points are called the solstices. The summer solstice takes place on the 21st of June, and the winter solstice on the 21st of December. 101. The equinoctial colure is the hour circle which passes through the equinoctial points. The solstitial colure is the hour circle which passes through the solstitial points. The solstitial colure is at right angles both to the ecliptic and to the equator, for it cuts both these circles 90 degrees from their common intersection; that is, from the equinoctial points. The distance of either.solstitial point from the celestial equator is 23~ 27'. The more distant the sun is from the celestial equator, the more unequal will be the days and nights; and, therefore, 60 ASTRONOMY. the longest day of the year will be the day of the summer solstice, and the shortest that of the winter solstice. In southern latitudes the seasons will be reversed. 102. The zodiac is a zone of the heavens extending eight degrees each side of the ecliptic. The sun, the moon, and all the principal planets, have their motions within the limits of the zodiac. The zodiac is divided into twelve equal parts, called signs, each of which contains 30 degrees. Beginning with the vernal equinox, they are as follows: Sign. Symbol. Sign. Symbol. I. Aries. T VII. Libra. II. Taurus. 6 VIII. Scorpio.' III. Gemini. IT IX. Sagittarius. p IV. Cancer. 5 X. Capricornus. V. Leo. St XI. Aquarius. VI. Virgo. X XII. Pisces. The vernal equinox is at the first point of Aries, and the autumnal equinox at the first of Libra. The summer solstice is at the first of Cancer, and the winter solstice at the first of Capricorn. 103. The tropics are two small circles parallel to the equator, and passing through the solstices. That on the north of the equator is called the Tropic of Cancer, and that on the south the Tropic of Capricorn. The Polar circles are two small circles parallel to the equator, and distant 23~ 27' from the poles. One is called the Arctic and the other the Antarctic circle. A great circle of the celestial sphere passing through the poles of the ecliptic, is called a circle of latitude. The latitude of a star is the distance of the star from the ecliptic, measured on a circle of latitude. It may be north or south, and is counted from zero to 90 degrees. The longitude of a star is the distance from the vernal equinox to the circle of latitude passing through the star, measured on the ecliptic in the order of the signs. Longitude is counted from zero to 360 degrees. 104. Appearances produced by the earth's annual motion.-The THE EARTH'S ANNUAL MOTION. 61 apparent annual motion of the sun may be explained either by supposing a real revolution of the sun around the earth, or a revolution of the earth around the sun. But it follows from the principles of Mechanics that the earth and sun must both revolve around their common centre of gravity, and this point is very near the centre of the sun. If the earth could be observed by a spectator upon the sun, it would appear among the fixed stars in the point of the sky opposite to that in which the sun appears as viewed from the earth. Thus, in Fig. 27, let S represent Fig. 2T. the sun, and ABPD the earth's Spr orbit: a spectator upon the earth ~ will see the sun projected among the fixed stars in the point of the f \h sky opposite to that occupied by the earth; and, as the earth moves A " from A to B and P, the sun will appear to move among the stars \ I from P to D and A, and in the course of the year will appear to trace out in the sky the plane of the ecliptic. When the earth is in Libra we see the sun in the opposite sign Aries; and as the earth moves from Libra to Scorpio, the sun appears to move from Aries to Taurus, and so on through the ecliptic. 105. Phenomena within the arctic circle.-At the summer solstice, on the arctic circle, the sun's distance from the north pole is just equal to the latitude of the place, and the sun's diurnal path just touches the horizon at the north point. Within the arctic circle, there will be several days during which the sun never sinks below the horizon. So, also, near the winter solstice, within the arctic circle, there will be several days during which the sun does not rise above the horizon. 106. Division of the earth into zones.-The earth is naturally divided into five zones, depending on the appearance of the diurnal path of the sun. These zones are, ist. The two frigid zones, included within the polar circles. Within these zones there are several days of the year during 62 ASTRONOMY. which the sun does not rise above the horizon, and other days during which the sun does not sink below the horizon. 2d. The torrid zone, extending from the Tropic of Cancer to the Tropic of Capricorn. Throughout this zone, the sun every year passes through the zenith of the observer, when the sun's declination is equal to the latitude of the place. 3d. The north and south temperate zones, extending from the tropics to the polar circles. Within these zones the sun is never seen in the zenith, and it rises and sets every day. 107. Cause of the change of seasons.-While the earth revolves annually round the sun, it has a motion of rotation upon an axis which is inclined 230 27' from a perpendicular to the ecliptic; and this axis continually points in the same direction. Hence result the alternations of day and night, and the succession of seasons. In June, when the north pole of the earth inclines toward the sun, the greater portion of the northern hemisphere is enlightened, and the greater portion of the southern hemisphere is dark. The days are, therefore, longer than the nights in the northern hemiFig. 28. MA 0 C~ A;rX ^^ /^ 8 T es CI 3;~~ ~~C~~~*j^^^^Ji iv i5Ja, ^^ THE EARTH'S ANNUAL MOTION. 63 sphere. The reverse is true in the southern hemisphere; but on the equator, the days and nights are equal. In December, when the south pole inclines toward the sun, the days are longer than the nights in the southern hemisphere. In March and September, when the earth's axis is perpendicular to the direction of the sun, the circle which separates the enlightened from the unenlightened hemisphere, passes through the poles, and the days and nights are equal all over the globe. These different cases are illustrated by Fig. 28. Let S represent the position of the sun, and ABCD different positions of the earth in its orbit, the axis ns always pointing toward the same fixed star. At A and C the sun illumines from n to s, and as the globe turns upon its axis, the sun will appear to describe the equator, and the days and nights will be equal in all parts of the globe. When the earth is at B, the sun illumines 23~~ beyond the north pole n, and falls the same distance short of the south pole s. When the earth is at D, the sun illumines 23-~ beyond the south pole s, and falls the same distance short of the north pole n. 108. Under what circumstances would there have been no change of seasons?-If the earth's axis had been perpendicular to the plane of its orbit, the equator would have coincided with the ecliptic; day and night would have been of equal duration throughout the year, and there would have been no diversity of seasons. 109. In what case vould the change of seasons have been greater than it now is?-If the inclination of the equator to the ecliptic had been greater than it is, the sun would have receded farther from the equator on the north side in summer, and on the south side in winter; and the heat of summer, as well as the cold of winter, would have been more intense; that is, the diversity of the seasons would have been greater than it is at present. If the equator had been at right angles to the ecliptic, the poles of the equator would have been situated in the ecliptic; and at the summer solstice the sun would have appeared at the north pole of the celestial sphere, and at the winter solstice it would have been at the south pole of the celestial sphere. To an observer in the middle latitudes, the sun would therefore, for a considerable part of summer, be within the circle of perpetual apparition, and for several weeks be constantly above the horizon. So, also, for a con 64 ASTRONOMY. siderable part of winter, he would be within the circle of perpetual occultation, and for several weeks be constantly below the horizon. The great vicissitudes of heat and cold resulting from such a movement of the sun, would be extremely unfavorable to both animal and vegetable life. 110. To determine the obliquity of the ecliptic.-The inclination of the equator to the ecliptic, or the obliquity of the ecliptic, is equal to the sun's greatest declination. It may therefore be ascertained by measuring, by means of the mural circle, the sun's declination at the summer, or at the winter solstice. The greatest declination of the sun is found to be 23~ 27' 25", both north and south of the equator. This arc is, however, diminishing at the rate of about half a second annually. 111. Form of the earth's orbit.-The path of the earth around the sun is nearly, but not exactly, a circle. The relative distances of the sun from the earth may be found by observing the changes in the sun's apparent diameter. The apparent diameter of the sun, at different distances from the spectator, varies inversely as A Fig. 29. the distance. Thus, in Fig. 29, R: sin. E::ES: AS. (St = ==-s. E AS 1 S\9E.. sin. E=-E-, or varies as Since the sines of small angles are nearly proportional to the angles, E varies as S, very nearly. By measuring, therefore, the sun's apparent diameter from day to day throughout the year, we have the means of determining the relative distances of the sun from the earth. Ex. 1. On the 1st of January, 1864, the sun's apparent diameter was 32' 36".4, and on the 1st of July, 1864, his diameter was 31' 31".8. Find the relative distances of the sun at these two periods. Ans. 0.96698. Ex. 2. On the 1st of April, 1864, the apparent diameter of the sun was 32' 3".4. Find the ratio of its distance to the distances in July and January. Ans. 0.98357 and 1.01716. 112. The earth's orbit is an elipse.-By observations of the sun's apparent diameter continued throughout the year, we find that THE EARTH'S ANNUAL MOTION. 65 the true form of the earth's orbit is an ellipse, having the sun in one of the foci. The sun's apparent diameter is least on the 1st of July, and greatest on the 1st of January. We may then construct a figure showing the form of the orbit, by setting off lines, SA, SB, SC, etc., corresponding to Fig. 30. the sun's distances, and making an- gles with each other equal to the sun's angular motion between the \ times of observation. The figure \ thus formed is found to be an ellipse, G A with the sun occupying one of the, foci, as S. 113. To find the eccentricity of the earth's orbit.-The point A of the orbit where the earth is nearest the sun, is called the perihelion, and this happens on the 1st of January. The point G most distant from the sun is called its aphelion, and this happens on the 1st of July; that is, the earth is more distant from the sun in summer than in winter. The distance from the centre of the ellipse to the focus, divided by the semi-major. axis, is called the eccentricity of the ellipse, and its value may be determined as follows: If a denote the semi-major axis, and e the eccentricity of the earth's orbit, then the earth's aphelion distance =a( +e); the earth's perihelion distance=a(1l —e). If we represent the aphelion distance by A, and the perihelion distance by P, we have A_1+e P 1-e Solving this equation, we obtain P A-P A e-A+P-1 P ~But ~ P 31' 31".8 IBut A =32 36",4=0.96698. A 32' 33 F-A Hence e=0.01678, which is about lOth. This eccentricity is subject to a diminution of 0.000042 in one E 66 ASTRONOMY. hundred years. If this change were to continue indefinitely, the earth's orbit must eventually become circular; but LeVerrier has proved that the diminution is not to continue beyond 24,000 years, when the eccentricity will be equal to.0033, and after that time the eccentricity will increase. 114. Law of the earth's motion in its orbit.-The radius vector of the earth's orbit describes equal areas in equal times. Let A and Fig. 31. B be the positions of the earth in its orbit on two successive days; let 0 rep-?^ ^ \^ ~ ~ resent the angle ASB, and R represent AS. Draw AC perpendicular to SB. Then AC AS sin. ASB = R sin. 0; and the area ASB=-R2 sin. 0. But since the earth's diurnal motion in the ecliptic is small, we may assume that the arc 0 is equal to its sine, and hence the area=-R20. If this area described by'the radius vector in one day, is a constant quantity, then R20 will evidently be a constant quantity. But R varies inversely as the apparent diameter of the sun. Hence, putting D for the sun's apparent diameter,'- must be a constant quantity; or 0: 0':: D2: D2 that is, the sun's diurnal motion in different parts of its orbit, must vary as the square of its apparent diameter. Now we find this supposition verified by observation. Thus: From noon of January 1st to noon of January 2d, 1864, the sun moved through 1~ 1' 9".9 of the ecliptic; and his apparent diameter at the same time was 32' 36".4. From noon of July 1st to noon of July 2d, 1864, the sun moved through 57' 12".9; and his apparent diameter at the same time was 31' 31".8. Reducing these values to seconds, we have 3669.9: 3432.9:: 1956.42: 1891.82. We find the same law to hold true in other parts of the orbit, and hence it is considered as established by observation, that the radius vector of the earth's orbit describes equal areas in equal times. SIDEREAL AND SOLAR TIME. 67 115. Why the greatest heat and cold do not occur at the solstices.The influence of the sun in heating a portion of the earth's surface depends upon its altitude above the horizon, and upon the length of time during which it continues above the horizon. The greater the altitude, the less obliquely will the rays strike the surface of the earth at noon, and the greater will be their heating power. Both these causes conspire to produce the increased heat of summer, and the diminished heat of winter. It might be inferred that the hottest day ought to occur on the 21st of June, when the sun rises highest, and the days are the longest. Such, however, is not the case, for the following reason: As midsummer approaches, the quantity of heat imparted by the sun during the day is greater than the quantity lost during the night, and hence each day there is an increase of heat. On the 21st of June this daily augmentation reaches its maximum; but there is still each day an accession of heat, until the heat lost during the night is just equal to that imparted during the day, which happens, at most places in the northern hemisphere, some time in July or August. For the same reason, the greatest cold does not occur on the 21st of December, but some time in January or February. Sidereal and Solar Time. 116. Sidereal Time. —The interval between two successive returns of the vernal equinox to the same meridian, is called a sidereal day. This interval represents the time of the rotation of the earth upon its axis, and is not only invariable from one month to another, but has not changed so much as the hundredth part of a second, in two thousand years. 117. Solar Time.-The interval between two successive returns of the sun to the same meridian, is called a solar day. The sun passes through 360 degrees of longitude in one year, or 365 days 5 hours 48 minutes and 47.8 seconds; so that the sun's mean daily motion in longitude is found by the proportion one year: one day:: 360~: daily motion-59' 8".33. This motion is not uniform, but is greatest when the sun is nearest the earth. Hence the solar days are unequal; and to avoid the inconvenience which would result from this fact, astronomers have recourse to a mean solar day, the length of which is equal to the mean or average of all the apparent solar days in a year. 68 ASTRONOMY. 118. Sidereal and solar time compared.-The length of the mean solar day is greater than that of the sidereal, because when the mean sun, in its diurnal motion, returns to a given meridian, it is 59' 8".3 eastward of its position on the preceding day. An arc of the equator, equal to 360~ 59' 8".3, passes the meridian in a mean solar day, while only 360~ pass in a sidereal day. To find the excess of the solar day above the sidereal day, expressed in sidereal time, we have the proportion 360~: 59' 8".3:: one day: 3m. 56.5s. Hence 24 hours of mean solar time are equivalent to 24h. 3m. 56.5s. of sidereal time. To find the excess of the solar day above the sidereal day, expressed in solar time, we have the proportion 360~ 59' 8".3: 59' 8".3:: one day: 3m. 55.9s. Hence 24 hours of sidereal time are equivalent to 23h. 56m. 4.1s. of mean solar time. 119. Civil day, and astronomical day.-The civil day begins at midnight, and consists of two periods of 12 hours each; but modern astronomers commence their day at noon, because this is a date which is marked by a phenomenon which can be accurately observed, viz., the passage of the sun over the meridian; and because observations being chiefly made at night, it is inconvenient to have a change of date at midnight. The astronomical day commences 12 hours later than the civil day, and the hours are numbered continuously up to 24. Thus July 4th, 9 A.M. civil time, corresponds to July 3d, 21 hours of astronomical time. 120. Apparent time, and mean time.-The interval between two successive returns of the sun to the same meridian, is an apparent solar day; and apparent time is time reckoned in apparent solar days, while mean time is time reckoned in mean solar days. The difference between apparent solar time and mean solar time, is called the equation of time. If a clock were required to keep apparent solar time, it would be necessary that its rate should change from day to day according to a complicated law. It has been found in practice impossible to accomplish this, and hence clocks are now regulated to indicate mean solar time. A clock, therefore, should not indicate 12h. when the sun is on the meridian, but should sometimes indi CAUSE OF INEQUALITY IN SOLAR DAYS. 69 cate more than 12h. and sometimes less than 12h., the difference being equal to the equation of time. 121. Cause of the inequality of the solar days.-The inequality of the solar days depends on two causes, the unequal motion of the earth in its orbit, and the inclination of the equator to the ecliptic. While the earth is revolving round the sun in an elliptical orbit, its motion is greatest when it is Fig. 32. nearest the sun, and slowest when it ~ C is most distant. Let ADGK repre- \ \ sent the elliptic orbit of the earth, \ i with the sun in one of its foci at S, and let the direction of motion be G A from A toward E. We have found that the sun's mean daily motion as seen from the / I earth, or the earth's mean daily mo- ~tion as seen from the sun, is 59' 8".3. But when the earth is nearest the sun its daily motion is 61' 10". In passing from A toward E its daily motion diminishes, and at G it is only 57' 12". While moving, therefore, from A through E to G, the earth will be in advance of its mean place, while at G, having completed a half revolution, the true and the mean places will coincide. For a like reason, in going from G to A, the earth will be behind its mean place; but at A the mean and true places will again coincide. This point A in the diagram, corresponds to about the 1st of January. Now the apparent direction of the sun from the earth, is exactly opposite to that of the earth from the sun. Hence, when the earth is nearest to the sun, the apparent solar day will be longer than the mean solar day. If, then, we conceive a fictitious sun to move uniformly through the heavens, describing 59' 8" per day, and that the true and fictitious suns are together on the 1st of January, it is evident that on the 2d of January the fictitious sun will come to the meridian a few seconds before the true sun; on the 3d of January the fictitious sun will be still more in advance of the true sun, and this difference will go on increasing for about three months, when it amounts to a little more than 8 minutes. From this time the difference will diminish until about the 1st of July, when the positions of the true and fictitious suns will coin 70 ASTRONOMY. cide. But on the 2d of July the fictitious sun will come to the meridian a few seconds later than the true sun; on the 3d of July it will have fallen still more behind the true sun, and this difference will go on increasing for about three months, when it amounts to a little more than 8 minutes. From this time the difference will diminish until the 1st of-January, when the positions of the true and fictitious suns will again coincide. So far, then, as it depends upon the unequal motion of the earth in its orbit, the equation of time is positive for six months' and then negative for six months, and its greatest value is 8m. 24s. 122. Second cause for the inequality of the solar days.-Even if the earth's motion in its orbit were perfectly uniform, the apparent solar days would be unequal, because the ecliptic is inclined to Fig. 33. p the equator. Let AgN represent the equator, and AGN the northern half of the ecliptic. Let the ecliptic be divided into equal portions, AB, BC, CD, etc., supposed /rT'?. }}IF Lto be described by the a o &^ e d y 2i Lf m 5 m& x sun in equal portions of time; and through the points B, C, D, etc., let hour circles be made to pass, cutting the equator in the points b, c, c, etc. The arc AGN is equal to the arc AgN, for all great circles bisect each other; also AG is equal to Ag, since the former is one half of AGN, and the latter of AgN. Now, since ABb is a right-angled triangle, AB is greater than Ab; for the same reason, AC is greater than Ac; AD is greater than Ad, and so on. But AG is equal to Ag; therefore Ag is divided into unequal portions at the points b, c, d, etc. Now B and b come to the meridian at the same instant; so also C and c, D and d, and so on. Suppose now that a fictitious sun moves in the equator at the rate of 59' 8" per day, while the real sun moves in the ecliptic at the same rate, and let them start together from A at noon on the 20th of March. On the 21st of March, at noon, the real sun will have advanced toward B 59' 8", which distance projected on the equator will be less than 59' 8", while the fictitious sun will have advanced toward b 59' 8"; that is, the fictitious sun will be east EQUATION OF TIME. 71 ward of the real sun, and the real sun will come to the meridian sooner than the fictitious one. The same will happen during the motion of the sun through the entire quadrant AG. The two suns will reach the points G and g on the 21st of June, and then they will both come to the meridian at the same instant. During the motion of the sun through the second quadrant, the real sun will come to the meridian later than the fictitious one, but both will reach the point N on the 22d of September at the same instant. During the motion through the third quadrant, the real sun will come to the meridian sooner than the fictitious one, until the 21st of December, when they will be found 180~ from the points G and g. During the motion through the last quadrant, the real sun will come to the meridian later than the fictitious one, but both will reach the point A at the same instant on the 21st of March. Thus we see that, so far as it depends upon the obliquity of the ecliptic, the equation of time is positive for three months; then negative for three months; then positive for three months; and then negative for another three months. The amount of the equation of time due to this cause, may be computed as follows: Suppose the sun to have advanced 45~ from A; then, in the right-angled triangle ADd, the angle at A is 230 27', and the hypothenuse is 45~. Ad is then computed from the equation, tang. Ad=cos. A tang. AD, whence Ad is found to be 42~ 31' 47". The difference between AD and Ad is 2~ 28' 13", or 9m. 52.8s. in time; and this is about the greatest amount of the equation of time, due to the obliquity of the ecliptic. 123. Resulting values of the equation of time.-The influence of each of these causes upon the equation of time, is artificially represented in the following figure, where AE is supposed to represent a year divided into twelve equal parts to represent the months; and the ordinates of the curve ABCDE, measured from the line AE as an axis, represent the values of the equation of time, so far as it depends upon the unequal motion of the earth in its orbit; and the ordinates of the curve FG-HI represent the values of the equation of time, so far as it depends upon the inclination of the equator to the ecliptic. The actual equation of time will be found by taking the algebraic sum of the effects due to these two separate causes. The result is the curve MNOPQR, 72 ASTRONOMY. Fig. 34. the ordinates being measured from the lower horizontal line in the figure. From this we see that the equation of time has two annual maxima and two annual minima, and there are four periods when the equation is zero. These dates and the corresponding values of the equation of time are as follows: February 11, +14m. 32sa. July 26, + 6mo. 12s. April 15, 0 0 September 1, 0 0 May 14, - 3 55 November 2, 16 18 June 14, 0 0 December 24, 0 0 These dates and the values of the equation of time changes slightly from one ear to anotherual m a, wan the rea accuracy is riods when the equation is zero. These dates and the correspondrequired, a table of the equation of time is required followsr each year. FebrSuch a table is annua14m. 32s. July ished in the Nautical Almanac. 12s. April 15, he September 1, ndr. May 14, - 3 55 lNovember 2, — 16 18 June 14, 0 0 December 24, 0 0 These dates and the values of the equation of time change slightly from one year to another, so that, where great accuracy is required, a table of the equation of time is required for each year. Such a table is annually published in the Nautical Almanac. The Calendar. 124. The Julian Calendar.-The interval between two successive returns of the sun to the vernal equinox, is called a tropical year. Its average length expressed in mean solar time is 365d. 5h. 48m. 47.8s. But in reckoning time for the common purposes of life, it is most convenient to have the year contain a certain number of whole days. In the calendar established by Julius Caesar, and hence called the Julian Calendar, three successive years were made to consist of 365 days each, and the fourth of 366 days. The year which contained 366 days was called a bissextile year, because the 6th of the Kalends of March was twice counted. It is also frequently called leap-year. The others are called common years. The odd day inserted in a bissextile year is called the intercalary day. The reckoning by the Julian calendar supposes the length of b~~~~~~~~~~~~~~~~~~~~~Z THE CALENDAR. 73 the year to be 365- days. A Julian year, therefore, exceeds the tropical year by 11m. 12s. This difference amounts to a little more than 3 days in the course of 400 years. 125. The Gregorian Calendar.-At the time of the Council of Nice, in the year 325, the Julian calendar was introduced into the Church, and at that time the vernal equinox fell on the 21st of March; but in the year 1582 the error of the Julian calendar had accumulated to nearly 10 days, and the vernal equinox fell on the 11th of March. If this erroneous reckoning had continued, in the course of time spring would have commenced in September, and summer in December. It was therefore resolved to reform the calendar, which was done by Pope Gregory XIII., and the first step was to correct the loss of the ten days, by counting the day after the 4th of October, 1582, not the 5th, but the 15th of the month. In order to keep the vernal equinox to the 21st of March in future, it was concluded that three intercalary days should be omitted every four hundred years. It was also agreed that the omission of the intercalary days should take place in those years which were not divisible by 400. Thus the years 1700,1800, and 1900, which, according to the Julian calendar, would be bissextile, would, according to the reformed calendar, be common years. The calendar thus reformed is called the Gregorian Calendar. The error of this calendar amounts to less than one day in 4000 years. 126. Adoption of the Gregorian. Calendar.-The Gregorian calendar was immediately adopted at Rome, and soon afterward in all Catholic countries. In Protestant countries the reform was not so readily adopted, and in England and her colonies it was not introduced till the year 1752. At this time there was a difference of 11 days between the Julian and Gregorian calendars, in consequence of the suppression in the latter, of the intercalary day in 1700. It was therefore enacted by Parliament that 11 days should be left out of the month of September in the year 1752, by calling the day following the 2d of the month, the 14th instead of the 3d. The Gregorian calendar is now used in all Christian countries except Russia. The Julian and Gregorian calendars are frequently designated by the terms old style and new style. In consequence of the intercalary days omitted in the years 1700 and 1800, there is now 12 days difference between the two calendars. 74 ASTRONOMY. 127. When does the year begin? —n the different countries of Europe, the year has not always been regarded as commencing at the same date. In certain countries, the year has been regarded as commencing at Christmas, on the 25th of December; in others, on the 1st of January; in others, on the 1st of March; in others, on the 25th of March; and in others at Easter, which may correspond to any date between March 22d and April 25th. In England, previous to the year 1752, the legal year commenced on the 25th of March; but the same act that introduced the Gregorian calendar established the 1st of January as the commencement of the year. In this manner the year 1751 lost its month of January, its month of February, and the first 24 days of March. This change in the calendar explains the double date which is frequently found in English books. For example, Feb. 15, 17 5- means the 15th of February, 1751, according to the old mode of counting the years from the 25th of March, and 1752 according to the new method prescribed by Parliament. In order to distinguish the one mode of reckoning from the other, it was for a long time customary to attach to each date the letters 0. S. for old style, or N. S. for new style. Thus the date of General Washington's birth was either written Feb. 11, 1731, 0. S., or Feb. 22,1732, N. S. 128. First and last days of the year.-Since a common year consists of 365 days, or 52 weeks and 1 day, the last day of each common year must fall on the same day of the week as the first; that is, if the year begins on Sunday it will end on Sunday. But if leap-year begins on Sunday it will end on Monday, and the following year will begin on Tuesday. PROBLEMS ON THE CELESTIAL GLOBE. 129. To find the right ascension and declination of a star. Bring the star to the brass meridian; the degree of the meridian over the star will be its declination, and the degree of the equinoctial under the meridian will be its right ascension. Right ascension is sometimes expressed in hours and minutes of time, and sometimes in degrees and minutes of arc. Ex. Required the right ascension and declination of Arcturus. 130. The right ascension and declination of a star being given, to find the star upon the globe. PROBLEMS. 75 Bring the degree of the equator which marks the right ascension to the brass meridian; then under the given declination marked on the meridian will be the star required. Ex. Required the star whose right ascension is 10h. im. 7s., and declination 12~ 37' N. 131. To set the celestial globe in a position similar to that of the heavens, at a given place, at a given day and hour. Set the brass meridian to coincide with the meridian of the place; elevate the pole to the latitude of the place; bring the sun's place in the ecliptic to the meridian, and set the hour index at 12; then turn the globe westward until the index points to the given hour. The constellations would then have the same appearance to an eye situated at the centre of the globe, as they have at that moment in the heavens. Ex. Required the appearance of the heavens at New Haven, Lat. 410 18', June 20th, at 10 o'clock P.M. 132. To determine the time of rising, setting, and culmination of a star for any given day and place. Elevate the pole to the latitude of the place; bring the sun's place in the ecliptic for the given day to the meridian, and set the hour index to 12. Turn the globe until the star comes to the eastern horizon, and the hour shown by the index will be the time of the star's rising. Bring the star to the brass meridian, and the index will show the time of the star's culmination. Turn the globe until the star comes to the western horizon, and the index will show the time of the star's setting. Ex. Required the time when Aldebaran rises, culminates, and sets at Cincinnati, October 10th. 133. To determine the position of the planets in the heavens at any given time and place. Find the right ascension and declination of the planets for the given day from the Nautical Almanac, and mark their places upon the globe; then adjust the globe as in Art. 131, and the position of the planets upon the globe will correspond to their position in the heavens. We may then determine the time of their rising and setting as in Art. 132. The time of rising and setting of a comet may be determined in the same manner. 76 ASTRONOMY. CHAPTER V. PARALLAX —ASTRONOMICAL PROBLEMS. 134. Diurnalparallax defined.-The direction in which a celestial body would be seen if viewed from the centre of the earth, is called its true place; and the direction in which it is seen from any point on the surface, is called its apparent place. The arc of the heavens intercepted between the true and apparent placesthat is, the apparent displacement which would be produced by the transfer of the observer from the centre to the surface, is called the diurnalparallax. X rig. 83. Let C denote the centre of the Flg. 36. ad' ~ earth; P the place of the observer on its surface; M an object seen in the zenith at P; M' the same object seen at the zenith distance MPM'; and M" the same object.i//.~,~, ~seen in the horizon. It is evident that M will appear /( ^c^T ^- ^in the same direction whether it be (c ~J ~ viewed from P or C. Hence, in the zenith, there is no diurnal parallax, and there the apparent place of an object is its true place. If the object be at M', its apparent direction is PM', while its true direction is CM', and the parallax corresponding to the zenith distance MPM' will be PM'C. As the object is more remote from the zenith, the parallax increases; and when the object is in the horizon, as at M", the diurnal parallax becomes greatest, and is called the horizontal parallax. It is the angle PM"C which the radius of the earth subtends at the object. It is evident that parallax increases the zenith distance, and consequently diminishes the altitude. Hence, to obtain the true zenith distance from the apparent, the parallax must be subtracted; and to obtain the true altitude from the apparent, the parallax PARALLAX. 77 must be added. The azimuth of a heavenly body is not affected by parallax. 135. To deduce the parallax at any altitude from the horizontal parallax.-In the triangle CPM' we have CM': CP:: -sin. CPM'(-sin. MP') sin. MPCM'P. (1) Also, in the triangle CPM", we have CM"-: CP:: 1: sin. CM"P. (2) Hence 1: sin. CM"P:: sin. MPM': sin. CM'P, or sin. CM'P =sin. CM"P x sin. MPM'; that is, the sine of the parallax at any altitude, is equal to the product of the sine of the horizontalparallax, by the sine of the apparent zenith distance. The parallax of the sun and planets is so small that we may, without sensible error, employ the parallax itself instead of its sine; that is, the parallax at any altitude is equal to the product of the horizontal parallax, by the sine of the apparent zenith distance. 136. Relation of the parallax of a heavenly body to its distance. Let us put z=the zenith distance MPM'; p= the parallax CM'P; r=CP, the radius of the earth; lR=CM' the distance of the heavenly body. Then, by equation (1), R: r:: sin. z: sin.p, r. or sin.p=R sin. z, ro or p=~ sill. z, very nearly. The parallax at any given altitude varies, therefore, inversely as the distance, very nearly. When the zenith distance becomes 90~, sin. z becomes unity; and if we denote the horizontal parallax by P, we shall have sin. P= r or P=-R, very nearly. 137. To determine the parallax of the moon by observation.-Let A, A' be two places on the earth situated under the same merid 78 ASTRONOMY. z Fig. T. ian, and at a great distance from each other; let C be the centre of the earth, and M the moon. Let AC be denoted by r, and CM by R, and let ZAM, Z'A'M, which / A _ Izf ^are the moon's zenith distances as ~~JR~' measured at the two observatories, be denoted by z, and z'. Then the moon's parallax, AMC, at the station A, will be r?. sin. z, and the parallax A'MC at the station A' will be' - sin. z'. Adding these equations together, we find p+P = —R (sin. z -sin. z'). But the angle p+~p', or AMA', is equal to the difference between ZCZ' and the sum of the angles z and z'; and since, if the places be situated one north and the other south of the equator, we have ZCZ' equal to the sum of the latitudes of the stations 1+', we obtain p+p'-=z+Z- I —'. Substituting this value in the preceding equation, we filnd Z+Z - =' -- (sin. z +sin. z'), r z+z'-l-I-' or -R sin. z-+ sin. z' But is the horizontal parallax of the moon, which was required to be found. 138. Stations of observation.-It is not essential that the two observers should be exactly on the same meridian; for if the meridian zenith distances of the moon be observed on several consecutive days, its change of meridian zenith distance in a given time will be known. Then, if the difference of longitude of the two places is known, the zenith distance of the moon as observed at one of the meridians, may be reduced to what it would have been found to be, if the observations had been made in the same latitude at the other meridian. PARALLAX. 79 139. Results obtained by this method.-There is an observatory at the Cape of Good Hope, in Lat. 33~ 56' S., where the moon's meridian altitude has been observed daily for many years, whenever the weather would permit; and similar observations are regularly made at Greenwich Observatory, in Lat. 51~ 28' N., as also at numerous other observatories in Europe. By combining these observations, the moon's parallax has been ascertained with great precision. It is found that the parallax varies considerably from one day to another. The equatorial parallax, when greatest, is about 61' 32", and when least, 53' 48". Its average value is 57' 2". By the preceding method the sun's parallax may be ascertained to be about 9". It can, however, be found more accurately by observations of the transits of Venus, as will be explained hereafter. The parallax of the planets can also be determined in the same manner as that of the moon; but in the case of the nearest planet the parallax never exceeds 32", and that of the remoter planets never amounts to 1"; and there are other methods by which these quantities can be more accurately determined. 140. To compute the distance of a heavenly body.-When we know the earth's radius and the horizontal parallax of a heavenly body, we can compute its distance. For (Fig. 36) sin. PM"C: PC:: radius: CM", or the distance of the object equals the radius of the earth, divided by the sine of the horizontal parallax. 141. Effect of the ellipticity of the earth upon parallax.-The horizontal parallax of the moon is the angle which the earth's radius would subtend to an observer at the moon. On account of the spheroidal figure of the earth, this horizontal parallax is not the.same for all places on the earth, but varies with the earth's radius, being greatest at the equator, and diminishing as we proceed toward either pole. It is necessary, therefore, always to compute the earth's radius for the place of the observer, and this may be done from the known properties of an ellipse. The moon's horizontal parallax for any given latitude is equal to the horizontal parallax at the equator multiplied by the radius of the earth at the given latitude, the radius of the equator being considered as unity. 80 ASTRONOMY. It is this corrected value of the equatorial parallax which should be employed in all computations which involve the parallax of a particular place. ASTRONOMICAL PROBLEMS. 142. To find the latitude of any place.-The latitude of a place may be determined by measuring the altitude of any circumpolar star, both at its upper and lower culminations, as explained in Art. 76. It may also be determined by measuring a single meridian altitude of any celestial body whose declination is known. Fig. 38. Let S or S' be a star on the meridian; Z SE or S'E its declination. Measure SH, p the altitude of the star S, and correct it /? \~ / ~\ for refraction. Then \ \^ - EH=SH-SE=S' + S'E. Ia -,Q 0 But EH is the complement of PO, which is the latitude sought. The declinations of all the brighter stars have been determined with great accuracy, and are recorded in catalogues of the stars. 143. To find the latitude at sea.-At sea the latitude is usually determined by observing with the sextant the greatest altitude of the sun's lower limb above the sea horizon at noon. The observations are commenced about half an hour before noon, and the altitude of the sun is repeatedly measured until the altitude ceases to increase. This greatest altitude is considered to be the altitude on the meridian. To this altitude we must add the sun's semi-diameter in order to obtain the altitude of the sun's centre, and this result must be corrected for refraction. To this result we must add the sun's declination if south of the equator, or subtract it if north, and we shall obtain the elevation of the equator, which is the complement of the latitude. The Nautical Almanac furnishes the sun's declination for every day of the year. 144. To find the time at any place.-The time of apparent noon is the time of the sun's meridian passage, and is most conveniently found by means of a transit instrument adjusted to the meridian. Mean time may be derived from apparent time by applying the equation of time with its proper sign. The time of apparent noon may also be found by noting the ASTRONOMICAL PROBLEMS. 81 times when the sun has equal altitudes before and after passing the meridian, and bisecting the interval between them. When great accuracy is required, the result obtained by this method requires a slight correction, since the sun's declination changes between morning and evening. 145. To find the time by a single altitude of the sun.-The time may also be computed from an altitude of the sun measured at any hour of the day, provided we know the sun's declination and the latitude of the place. Let PZH be the meridian of the place Fig. 3 Z of observation, P the pole, Z the zenith, and S the place of the sun. Measure the / zenith distance, ZS, and correct it for re- \ fraction. Then, in the spherical triangle ZPS, we know the three sides, viz., PZ, the complement of the latitude, PS, the distance of the sun from the north pole, and ZS, the sun's zenith distance. In this triangle we can compute (Trigonometry, Art. 223) the angle ZPS, which, if expressed in time, will be the interval between the moment of observation and noon. This observation can be made at sea with a sextant, and this is the method of determining time which is commonly practiced by navigators. 146. A meridian mark, and sun-dial.-If, upon a horizontal plane, we trace a meridian line, and at the south extremity of this line erect a vertical rod freely exposed to the sun, we may determine the time of apparent noon by the passage of the shadow of the rod over the meridian line. Or, if we set up a straight rod in a position parallel to the axis of the earth, its shadow, as cast upon a horizontal plane, will have the same direction at any given hour, at all seasons of the year. If, then, we graduate this horizontal plane in a suitable manner, and mark the lines with the hours of the day, we may determine the apparent time whenever the sun shines upon the rod. Such an instrument is called a sundial, and it may be constructed with sufficient precision to answer the ordinary purposes of society. This instrument will always indicate apparent time; but mean time may be deduced from it by applying the equation of time. F 82 ASTRONOMY. 147. To compute the longitude, right ascension, and declination of the sun, any one of these quantities, together with the obliquity of the ecliptic, being given. Fig. 40. Let EPQP' represent the equinocF~. 40. \ ftial colure, EMQ the equator, ESQ the ecliptic, E the first point of Aries, S the place of the sun, PSP' an hour f s\~~^ \ ~circle passing through the sun; then E^ ~- ~~~ — \ EM is the sun's right ascension, SM his declination, ES his longitude, and MES the obliquity of the ecliptic. Then, in the triangle ESM, we have, by Napier's rule, a' R cos. E =tang. ME cot. SE; that is, representing the obliquity by w, and the right ascension by R. A. tang. R. A. = tang. Long. cos. w, (1) and tang. Long. tang. R.A. (2) COS. o Also, IR sin. ME = tang. M3S cot. E; that is, sin. R. A.= tang. Dec. cot. w, (3) and tang. Dec. =sin. R. A. tang. w. (4) Also, R sin. MS= sin. E sin. ES; that is, sin. Dec.=sin. o sin. Long., (5) sin. Dec. and sin. Long. sin. De (6) sin. W' Also, R cos. ES= cos. ME cos. MS; that is, cos. Long. =cos. R. A. cos. Dec., (7) cos. Long. and cos. R. A. cos.Dec. (8) cos. Dec.' Ex. 1. On the 1st of June, 1864, at Greenwich mean noon, the sun's right ascension was 4h. 38m. 27.75s., and his declination 22~ 7' 55".2 N.; required his longitude. Ans. 710 10' 35".9. Ex. 2. On the 1st of January, 1864, the sun's longitude was 280~ 23' 52".3, and his declination 23~ 2' 52".2 S.; required his right ascension. Ans. 18h. 45m. 14.70s. Ex. 3. On the 20th of May, 1864, the sun's longitude was 590 40' 1".6, and the obliquity of the ecliptic 23~ 27' 18".5; re, quired his right ascension and declination. Ans. R. A. 3h. 49m. 52.62s. Dec. 20~ 5' 33".9 N. ASTRONOMICAL PROBLEMS. 83 Ex. 4. On the 27th of October, 1864, the sun's right ascension was 14h. 8m. 19.06s., and the obliquity of the ecliptic 23~ 27' 17".8; required his longitude and declination. Ans. Long. 214~ 20' 34".7. Dec. 12~ 58' 34".4 S. Ex. 5. On the 8th of August, 1864, the sun's declination was 16~ 0' 56".4 N., and the obliquity of the ecliptic 23~ 27' 18".2; required his right ascension and longitude. Ans. R. A. 9h. 14m. 19.20s. Long. 136~ 7' 6".5. 148. Given the latitude of a place and the sun's declination, to find the time of his rising or setting. Let PEP' represent the hour circle, Fig. 41. which is six hours from the meridian, and which intersects the horizon in the east point, E. Let S or S' be the posi- tion of the sun in the horizon, and 11 - i o through S draw the hour circle PSP'; also through S' draw the hour circle PS'P'. Then,in the right-angled spher- ical triangle EMS, or EM'S', EM or EM'-the distance of the sun from the six o'clock hour circle. MS or M'S'=the sun's declination, which we will represent by 8. MES= M'ES' -the complement of the latitude. Now, by Napier's rule, R sin. EM=tang. MS cot. MES. Representing the latitude by 9, sin. EM itang. 8 tang. p. The time from the sun's rising to his passing the meridian 6 hours -+ EM. Ex. 1. Required the time of sunrise at New York, Lat. 40~ 42', on the 10th of May, when the sun's declination is 17~ 49' N. Ans. 4h. 56m. Ex. 2. Required the time of sunset at Cincinnati, Lat. 39~ 6', on the 5th of November, when the sun's declination is 15~ 56' S. Ans. 5h. 6m. apparent time. 84 ASTRONOMY. 149. To find the time when the sun's upper limb rises, allowance being made for refraction.-The preceding method gives the time when the sun's centre would rise if there were no refraction. The effect of refraction is to cause the sun to be seen above the sensible horizon sooner in the morning, and later in the afternoon, than he actually is; and moreover, when the sun's upper limb coincides with the horizon, the centre is about 16' below. At the instant, therefore, of sunrise or sunset, his centre is 90~ 50' from the zenith; the semi-diameter being about 16', and the horizontal refraction 34'. In order, therefore, to compute the apparent time of rising of the sun's upper limb, we must compute when the sun's centre is 90~ 50' from the zenith. This may be done as follows: Fig. 42. Z Let PZH be the meridian of the place of x/-^/ ^\ ~observation, P the pole, Z the zenith, and S the place of the sun. In the spherical triangle ZPS, the three sides are known, a3T~" ~ - o~ 0 Tviz., PZ=the co-latitude=+; ZS =the zenith distance =z; PS= the north polar distance of the sun=d. In this triangle we can compute ZPS, which is the angular distance of the sun from the meridian. By Trigonometry, sin. A /sin. (S-b) sin. (S —c) sin. b sin. c Put 2S=z+d+~; then s in.ip /sin. (S-) sin. (S-d) then sin. v1P /-'.~ sin. sin. d Ex. 1. Required the time of sunset at New York, Lat. 400 42', on the 10th of May, when the sun's declination is 17~ 49' N. Here b= 49~ 18' sin. (S-~)=9.922892 d= 72 11 sin. (S-d)=9.747281 z= 90 50 cosec., =0.120254 S=106 9- cosec. cd=0.021345 S-4= 56 511 2)9.811772 S-d= 33 5819 -P= 530 37' sin. =9.905886 P=107~ 15'=7h. 9m. Hence the sun sets at 7h. 9m. apparent time; or, subtracting 4m. for equation of time, we have 7h. 5m. mean time. ASTRONOMICAL PROBLEMS. 85 Ex. 2. Required the mean time of sunrise at Boston, Lat. 42. 21', on the 15th of October, when the sun's declination is 8~ 47' S., mean time being 14 minutes slow of apparent time. Ans. 6h. 14m. 150. To find the time of beginning or end of twilight.-At the beginning or end of twilight, the sun is 18~ below the horizon; that is, his zenith distance is 108~. Hence this problem can be solved by the formula of the last article. Ex. 1. Required the time of the commencement of twilight at Washington, Lat. 38~ 53', on the 1st of June, when the sun's declination is 22~ 10' N., mean time being 2 minutes slow of apparent time. Ans. 2h. 41m. mean time. Ex. 2. Required the time of ending of twilight at New Orleans, Lat. 29~ 57', on the 19th of February, when the sun's declination is 11~ 19' S., mean time being 14 minutes fast of apparent time. Ans. 7h. 12m. mean time. 151. To compute the distance between two stars whose right ascensions and declinations are known. Let P be the pole, and S and S' two stars whose Fig. 43. places are known. Then PS and PS' will represent their polar distances, and SPS' will be the difference of their right ascensions. Draw SM / perpendicular to PS' produced. Then R cos. P=tang. PM cot. PS. Therefore, tang. PM=cos. P tang. PS. Also, S'M=PM-PS'. And cos. PM: cos. S'M:: cos. PS: cos. S'S. Ex. 1. Required the distance from Aldebaran, R. A. 4h. 27m. 25.9s., polar distance 73~-47' 33", to Sirius, R. A. 6h. 38m. 37.6s., polar distance 106~- 31' 2". P=h2h.llm.11.7s.= 32~ 47' 55" cos.=9.924579 PS=106 31 2 tang.=0.527916 PM=109 25 55 tang. =0.452495 PS'= 73 47 33 S'M= 35 38 22 cos. 9.909930 PS=106 31 2 cos.=9.453782 PM =109 25 55 sec. 0.477964 SS'= 46 0 44 cos. 9.841676 86 ASTRONO-MY, Ex. 2. Required the distance from Regulus, R. A h. Oh. m. 29.1s., polar distance 77~ 18' 41", to Antares, R. A. 16h. 20m. 20.3s., polar distance 116~ 5 55. Ans. 990 55' 45'. 152. Distance between two stars on the same parallel of declination. — If two stars have the same declination, their distance can be computed as follows: p Fig. 44. Let P be the pole, EQ a portion of the equator, and SS' a portion of any parallel of declination, and PCE, PCQ two meridians passing A -S through S and S'. Then, by Geometry, ~c~q arc EQ: arc SS':: CQ: AS:: 1: cos. Dec. E Therefore SS'=EQ cos. Dec. =EPQ cos. Dec. That is, the distance between the two stars is equal to their difference of right ascension, multiplied by the cosine of their declination. This distance is, however, not measured on an arc of a great circle, but on a parallel of declination. 153. To find the longitude and latitude of a star, when its right ascension and declination are known. Fig. 45. A Let P represent the pole of the equator, E the pole of the ecliptic, C the first point of Aries, PSP' an hour circle passing through the ls / \ star S, and ESE' a circle of lati~A {~ ~i'- / f\~B tude passing through the same star. Then AEBE' represents the solstitial colure, EP represents the ob\^ / ^/ / y/ liquity of the ecliptic, PS the polar ^^^C t/^ ~ distance of the star, ES its co-latiSE; ~ tude; SPB is the complement of its right ascension, and SEB is the complement of its longitude. Draw SM perpendicular to PB. Represent PM by a; also represent the longitude of the star S by L, its latitude by 1, and the obliquity of the ecliptic by o. Now, by Napier's rule, we have R cos. SPM=tang. PM cot. PS; that is, sin. R. A. =tang. a tang. Dec., or tang. a=sin. R. A. cot. Dec. (A) ASTRONOMICAL PROBLEMS. 87 Also, EM=EP+PMa+ o. Again, Trig., Art. 216, Cor. 3, sin. EM: sin. PM:: tang. SPM: tang. SEM; that is, sin. (a+w): sin. a:: cot. R. A.: cot. L:: tang. L: tang. R.A., or tang L _tang. R. A. x sin. (a+ w) sin. a Also, R cos. SEM=tang. EM cot. ES; that is, tang. I= cot. (a+ o) sin. L. (2) Ex. 1. On the 1st of January, 1864, the R. A. of Capella was 5h. in. 42.01s., and its Dec. 45~ 51' 20".1 N.; required its latitude and longitude, the obliquity of the ecliptic being 23~ 27' 19".45. By equation (A), BR.A. 760 40' 30".15 sin. 9.988148 Dec. 45 51 20.1 cot. 9.987028 a=43 21 48.2 tang. 9.975176 w=23 27 19.45 a+w-66 49 7.65 By equation (1), tang. R. A. =0.625527 sin. (a+ w)=9.963440 cosec. a=0.163282 L=79 58' 3".5 tang.=0.752249 By equation (2), cot. (a+ )=-9.631659 sin. L-9.993308 1=22~ 51' 48".3 tang.=9.624967 Ex. 2. On the 1st of January, 1864, the R. A. of Regulus was 10h. im. 9.34s., and its Dec. 120 37' 36".8 N.; required its latitude and longitude, the obliquity of the ecliptic being 23~ 27' 19".45. Ans. Latitude, Longitude, 88 ASTRONOMY. CHAPTER VI. THE SUN-ITS PHYSICAL CONSTITUTION. 154. Distance of the sun. -The distance of the sun from the earth can be computed when we know its horizontal parallax, and the radius of the earth. The mean value of the horizontal parallax of the sun has been found to be 8".58, and the equatorial radius of the earth is 3963 miles. Hence sin. 8".58: 3963:: 1: the sun's distance, which is found to be 95,300,000 miles; or, in round numbers, 95 millions of miles. 155. Velocity of the earth's motion in its orbit.-Since the earth makes the entire circuit around the sun in one year, its daily motion may be found by dividing the circumference of its orbit by 365k, and thence we may find the motion for one hour, minute, or second. The circumference of the earth's orbit is very nearly that of a circle whose radius is the sun's mean distance. We thus find the circumference of the orbit to be 598,800,000 miles; that the earth moves 1,639,000 miles per day; 68,300 miles per hour; 1138 miles per minute; and nearly 19 miles per second. By the diurnal rotation, a point on the earth's equator is carried round at the rate of 1037 miles per hour, or 17 miles per minute. The motion in the orbit is, therefore, 66 times as rapid as the diurnal motion at the equator. 156. The diameter of the sun.-The sun's absolute diameter can be computed, when we know his distance and apparent diameter. The apparent diameter, as well as the distance, is variable, but the mean value of his apparent diameter is 32' 3".64. Hence we have the proportion rad.: ES (95 millions):: sin. 16' 1".8: sun's radius, which is found to be 444,406 miles; or his diameter is 888,812 miles. The diameter of the sun is therefore 112 times that of the earth; THE SUN. 89 A Fig. 46. and, since spheres are as the cubes of their diameters, the volume of the sun is more than 1,400,000 times that of the earth. The density of the sun is about one quarter that of the earth; and, therefore, his mass, which is equal to the product of his volume by his density, is found to be 355,000 times that of the earth. 157. Figure of the sun's disc.-Since the sun rotates upon an axis, as shown Art. 171, his figure can not be that of a perfect sphere. The oblateness of a heavenly body depends chiefly upon the ratio of the centrifugal force to the force of gravity upon its surface. Now, on account of its slow rotation, the centrifugal force of a point upon the sun's equator, is only about one sixth what it is upon the earth, while the force of gravity is nearly thirty times as great; hence the oblateness of the sun should be only about -oth part of that of the earth. But the oblateness of the earth is about. —th. Hence the oblateness of the sun should be only about s, which corresponds to a difference of less than one twentieth part of a second between the equatorial and polar diameters. This quantity is too small to be detected by our observations; and although the sun's diameter has been measured many thousand times, still, with the exception of the effect due to refraction, explained in Art. 91, his disc is sensibly a perfect circle. 158. Force of gravity on the sun.-The attraction of a sphere being the same as if its whole mass were collected in its centre, will be proportional to the mass directly, and the square of the distance inversely; hence the force of gravity on the surface of the sun, will be to the force of gravity on the surface of the earth, as 355,000 3500 - to unity, which is 27.9 to 1; that is, a pound of terres1122 trial matter at the sun's surface, would exert a pressure equal to what 27.9 such pounds would do at the surface of the earth. A body weighing 200 pounds on the earth, would produce a pressure of 5580 pounds on the sun. At the surface of the earth, a body falls through 16-th feet in one second; but a body on the sun would fall through 16- x 27.9 =448.7 feet in one second. 90 ASTRONOMY. PHYSICAL CONSTITUTION OF THE SUN. 159. Solar spots.-When we examine the sun with a good telescope, we frequently perceive upon his surface, black spots of irregular shape, sometimes extremely minute, and at other times of vast extent. They usually make their first appearance at the eastern limb of the sun; advance gradually toward the centre; pass beyond it, and disappear at the western limb, after an interval of about 14 days. They remain invisible about 14 days, and then sometimes reappear at the eastern limb in nearly the same position as at first, and again cross the sun's disc as before, having taken 27d. 7h. in the entire revolution. The appearance of a solar spot is that of an intensely black, irregularly-shaped patch, called the nucleus, surrounded by a fringe which is less dark, and is called the penumbra. The form of this fringe is generally similar to that of the inclosed black spot; but this is not always the case, for several dark spots are occasionally included in a common penumbra. Black spots have occasionally been seen without any penumbra; and sometimes we see a large penumbra without any central nucleus; but generally both the nucleus and penumbra are combined. 160. Changes of the spots.-These spots change their form from day to day, and sometimes from hour to hour. They usually commence from a point of insensible magnitude, grow very rapidly at first, and usually attain their full size in less than a day. Then they remain stationary, with a well-defined penumbra, and continue for ten, twenty, and some even for fifty days. Then the nucleus usually becomes divided by a narrow line of light; this line sends out numerous branches, which extend until the entire nucleus is covered by the penumbra. Decided changes have been detected in the appearance of a spot within the interval of a single hour, indicating a motion upon the sun's surface of at least 1000 miles per hour. The duration of the spots is very variable. A spot has appeared and vanished in less than 24 hours, while others have lasted for weeks, and even months. In 1840, a spot was identified for nine revolutions, which corresponds to a period of about eight months. PHYSICAL CONSTITUTION OF THE SUN. 91 161. liMagnitude and number of the spots.-Solar spots are sometimes of immense magnitude, so that they have repeatedly been visible to the naked eye. In June, 1843, a solar spot remained for a whole week visible to the naked eye. Its breadth measured 167", which indicates an absolute diameter of 77,000 miles. The number of spots seen on the sun's disc is very variable. Sometimes the disc is entirely free from them, and continues thus for weeks, or even months together; at other times a large portion of the sun's disc is covered with spots. Sometimes the spots are small, but numerous; and sometimes they appear in groups of vast extent. In a large group of spots which appeared in 1846, upward of 200 single spots and points were counted. In 1837 a cluster of spots covered an area of nearly 5 square minutes, or nearly 4000 millions of square miles. 162. The black nucleus.-It is not certain that the black nucleus of a spot is entirely destitute of light; for the most intense artificial light, when seen projected on the sun's disc, appears as dark as the spots themselves. Sir W. Herschel estimated that the light of the penumbra was less than one half that of the brighter part of the sun's surface, and the light of the nucleus less than one hundredth of the brighter surface. 163. Upon what part of the sun do the spots appear?-The spots do not appear with equal frequency upon every part of the sun's disc. With few exceptions, they are confined to a zone included between 30~ of N. Latitude and 30~ of S. Latitude, measured from the sun's equator. According to a series of observations, extending over a period of ten years, and comprehending 1700 spots, the distribution in zones is as follows: Per Cent. Per Cent. Beyond 30~ N. Latitude 3 Beyond 30" S. Latitude 2 Between 20~ and 30~ N. ] 7 Between 20~ and 30~ S. 15 " 10 " 20 23 " 10 " 20 17 "_ 0 " 10 11 " 0 " 10 12 We thus perceive that 95 per cent. of all the spots are found within 30~ of the sun's equator. There are only three cases on record, in which spots have been seen as far as 45~ from the sun's equator. 92 ASTRONOMY. 164. Appearance of the bright part of the sun's disc.-Independently of the dark spots, the luminous part of the sun's disc is not uniformly bright. It exhibits a mottled appearance, like that which would be presented by a stratum of luminous clouds of irregular shape and variable depth. This mottled appearance is not confined, like the black spots, to a particular zone, but is seen on all parts of the surface, even near the poles of rotation. Sometimes we observe upon the sun's disc curved lines, or streaks of light, more luminous than the rest of the surface. These are called faculce, and they generally appear in the neighborhood of the black spots. 165. Proof that the sun's outer envelope is not solid.-The rapid changes which take place upon the surface of the sun, prove that his outer envelope is not solid. Admitting that the great mass of the sun is solid, that portion which we ordinarily see, must be either liquid or gaseous; and the rapid motion of 1000 miles per hour, which has been observed in solar spots, indicates that the luminous matter which envelops the sun must be gaseous, since liquid bodies could hardly be supposed to move with such velocity. 166. The solar spots are not planetary bodies.-It is evident that the solar spots are at the surface of the sun; for if they were bodies revolving around the sun at some distance from it, the time during which they would be seen on the sun's disc Fig.4T. would be less than that occupied in the remainder of their revolution. Thus, let S represent the sun, E the earth, and suppose ABC to represent the path of an opaque body revolving about the sun. Then AB represents that part of the orbit in which the body would appear projected upon the sun's disc, and this is less than half the entire circumference; whereas the spot reappears on the opposite limb of the sun after an interval nearly equal to that required to pass across the disc. 167. The dark spots are depressions in the luminous matter which envelops the sun. This was first proved by an observation made by Dr. PHYSICAL CONSTITUTION OF THE SUN. 93 Wilson, of Glasgow, in November, 1769. IHe first noticed a spot November 22d, when it was not far from the sun's western limb; and he observed that the penumbra was about equally broad on every side of the nucleus. The next day the eastern portion of the penumbra had contracted in breadth, while the other parts remained nearly of their former dimensions. On the 24th the peFig. 48. E/^~ ~~~NW \ A A numbra had entirely disappeared from the eastern side, while it was still visible on the western side. On the 11th of December the spot reappeared on the sun's eastern limb, and now there was no penumbra on the western side of the spot, although it was distinctly seen on the remaining sides. The next day the penumbra came into view on the western side, though narrower than on the other sides. On the 17th the spot had passed the centre of the sun's disc, and now the penumbra appeared of equal extent on every side of the nucleus. From these observations, it is inferred that the penumbra is lower than the general level of the sun's bright surface, and the nucleus lower than the penumbra. Dr. Wilson computed that the depth of the spot just described was nearly 4000 miles. 94 ASTRONOMY. Similar observations were repeatedly made by SirW. Herschel. In 1794 he observed that, as a spot approached near the western limb of the sun, the black nucleus gradually contracted in breadth, while its length remained unchanged. It became reduced to a narrow black line, and then disappeared, while the penumbra was still visible. Similar observations have repeatedly been made by other astronomers. In 1801 Sir W. Herschel observed that when a spot came near the western margin of the sun, he was able to distinguish the thickness of the stratum on the western border, but not on the eastern; and he hence computed that the depression of the penumbra below the bright surface of the sun was not less than 1800 miles. Similar observations have been made by M. Secchi at Rome. 168. The bright streaks orfaculce are elevated ridges rising above the general level of the sun's surface. This is proved by an observation made in 1859 by Mr. Dawes, of England. He had the good fortune to observe a bright streak of unusual size precisely at the edge of the sun's disc, and he perceived that it projected beyond the circular outline of the disc in the manner of a mountain ridge. In 1862, as an uncommonly large spot was passing off the sun's disc, Mr. Howlett perceived a small notch in the sun's margin, precisely over the place where the great nucleus had previously been seen, and on either side of it the photosplhere appeared to be heaped up above the general level of the sun's surface. 169. To determine the time of the sun's rotation.-It is found that Fig.49. D a spot generally employs 27{ days in passing from one limb of the sun around to the same limb again, and it is inferred ~/ \^ —^ ^that this apparent motion is caused by a A A Bs' \ rotation of the sun upon his axis. But A\ \ ) ] the period above mentioned is not the time in which the sun performs one rotation about his axis; for, let AA'B represent the sun, and EE'D the orbit of the E earth. When the earth is at E, the visible disc of the sun is AA'B; and if the earth was stationary at PHYSICAL CONSTITUTION OF THE SUN. 95 E, then the time required for a spot to move from the limb B round to the same point again would be the time of the sun's rotation. But while the spot has been performing its apparent revolution; the earth has advanced in her orbit from E to E', and now the visible disc of the sun is A'B', so that the spot has performed more than a complete revolution in the time it has taken to move from the western limb to the western limb again. Since an apparent rotation of the sun takes place in 271 days, the number of apparent rotations in a year will be 35 or 13.4. But, in consequence of the motion of the earth about the sun, if the sun had no real rotation, it would in one year make an apparent rotation in a direction contrary to the motion of the earth. Hence, in one year, there must be 14.4 real rotations of the sun, 365-k and the time of one real rotation is 14.4' or 25.3 days. Thus the time of a real rotation is found to be nearly two days less than that of an apparent rotation. 170. Temperature of different parts of the sun's disc.-By receiving the image of different portions of the sun upon a very sensitive thermometer, it has been discovered that the sun's disc has not throughout exactly the same temperature. The rays proceeding from the centre of the disc are hotter than those which proceed from the margin, and the black spots radiate less heat than the neighboring bright surface. The luminous intensity of different portions of the sun's disc exhibits corresponding variations, the borders of the disc being found less luminous than the centre. This difference is quite noticeable in a photographic picture of the sun. 171. Influence of solar spots upon terrestrial temperatures.-It has been supposed that the presence of an unusual number of large spots on the sun's disc must influence the temperature of the earth, and there are some facts which favor this supposition. At Paris, out of 26 years of observations, the mean temperature of those years in which the spots were most numerous was half a degree lower than that of those years in which the spots were least frequent. But during the same years a slight effect of the opposite kind was observed upon the temperature of places in the 96 ASTRONOT'MY~. United States, so that we seem obliged to ascribe the differences in question to other causes than the solar spots. 172. Position of the sun's equator.-Besides the time of rotation, observations of the solar spots enable us to ascertain the position of the equator with reference to the ecliptic. The angle between the solar equator and the ecliptic has been determined to be about 7~. About the first weeks of June and December, the spots, in traversing the sun's disc, appear to us to describe straight lines, but at other times the apparent paths of the spots are somewhat elliptical, and they present the greatest curvature about the first weeks of March and September. 173. Periodicity in the number of the solar spots.-The number of the solar spots varies greatly in different years. Some years the sun's disc is never seen entirely free from spots, while in other years, for weeks and even months together, no spots of any kind can be perceived. From a continued series of observations, embracing a period of 38 years, it appears that the spots are subject to a certain periodicity. The number of the spots increases during 5 or 6 years, and then diminishes during about an equal period of time, the interval between two consecutive maxima being from 10 to 12 years. As this period corresponds to the time of one revolution of Jupiter, it suggests the idea that possibly Jupiter may have the power of sensibly disturbing the sun's surface. 174. The sun not a solid body.-A comparison of the dark lines in the solar spectrum has led to the conclusion that the elements of which the sun is composed are to a great extent the same as those found upon the earth. The existence of iron, nickel, and several other well-known metals in the sun's atmosphere is considered as proved; and since the density of the sun is only one fourth that of the earth, while the force of gravity is 28 times its force upon the earth, we can not suppose that any large part of the sun's mass is in the condition of a solid or even a liquid body. The most refractory substances, iron and nickel, exist upon the sun in the state of elastic vapor. Hence the temperature of the sun's surface is extremely elevated, far beyond the heat of terrestrial volcanoes. It is possible that the centre of the sun consists PHYSICAL CONSTITUTION OF THE SUN. 97 of matter in the liquid or even the solid state; but it is probable that the principal part of the sun's volume consists of matter in the gaseous condition. 175. Nature of the sun's photosphere.-The bright envelope of the sun, which we call its photosphere, consists of matter in a state analogous to that of aqueous vapor in terrestrial clouds; that is, in the condition of a precipitate suspended in a transparent atmosphere. This photosphere is not only intensely luminous, but intensely hot, and the thermoscope indicates that it radiates more heat than the solar spots; but this does not prove that the photosphere is really hotter than the nucleus of a solar spot, for gases radiate heat more feebly than solids of the same temperature. The matter of the photosphere probably consists of particles precipitated in consequence of their being cooled by radiation. The sun's gaseous envelope extends far beyond the photosphere. During total eclipses we observe protuberances rising to a height of 80,000 miles above the surface of the sun, which requires us to admit the existence of bodies analogous to clouds floating at great elevations in an atmosphere; and if the extent of the solar atmosphere compared with the height of the visiFig. 50. ble clouds corresponds with what exists upon the earth, we must conclude that the:.....,,. — solar atmosphere extends to at least a million of miles beyond his surface.:176. Natures of the feum— ~~I ~bra. -The penumbra of a formed of filaments of photospheric light converging toward the centre of the nucleus, each of the filaments having the same light i, X. as the photosphere, and the L. -?: sombre tint results from the dark interstices between the luminous streaks, as in a steel engraving shades are produced by G 98 ASTRONOMY. dark lines separated by white interstices. The convergence of the luminous streaks of the penumbra toward the centre of the spot indicates the existence of currents flowing toward the centre. These converging currents probably meet an ascending current of the heated atmosphere, by contact with which the matter of the photosphere is dissolved, and becomes non-luminous. The faculae are ascribed to commotions in the photosphere, by which the thickness of the phosphorescent stratum is rendered greater in some places than in others, and the surface appears brightest at those points where the luminous envelope is thickest. 177. Motion of the solar spots.-The spots are not stationary on the sun's disc, for the apparent time of revolution of some of the spots is much greater than that of others. In one instance, the time of the sun's rotation, as deduced from observations of a solar spot, was only 24d. 7h., while in another case it amounted to 26d. 6h. This difference can only be explained by admitting that the spots have a motion of their own relative to the sun's surface, just as our clouds have a motion relative to the earth's surface. The motion of the solar spots in latitude is very small, and this motion is sometimes directed toward the equator, but generally from the equator. The motion of the spots in longitude is more decided. Spots near the equator have an apparent movement of rotation more rapid than those at a distance from the equator. While at the equator the daily angular velocity of rotation is 865', in lat. 20~ it is only 840', and in lat. 30~ it is 816'. Hence a point on the sun's equator makes a complete rotation in 25 days, but a point in lat. 30~ makes one rotation in 26- days. 178. Cause of the movements of the solar spots.-The heat of the sun must be continually dissipated by radiation. If this radiation is more obstructed in some regions than in others, heat must accumulate in such places. Now the phenomena observed during total eclipses indicate in the sun's atmosphere the existence of large masses analogous to terrestrial clouds. Wherever these clouds prevail, the free radiation of heat from the sun must be obstructed, and heat must rapidly accumulate. The solar atmosphere tends to move toward these heated centres, and this must be accompanied by an upward motion at the centre. The heated air thus ascending.partly dissolves and partly divides the matter PHYSICAL CONSTITUTION OF THE SUN. 99 of the photosphere, causing it to heap up in a ring around the opening, producing thus around the margin of the penumbra the appearance of a border of light more intense than the general photosphere. A general movement of the atmosphere toward one point must create a tendency to revolve around this centre, for the same reason that terrestrial storms sometimes rotate about a vertical axis. Such a motion of the solar spots has been repeatedly indicated Fig. 51. by observation. Moreover, solar spots have. l...., \ A sometimes exhibited a spiral structure such, A!., as might be supposed to result from rotation,II,''fil' about a vertical axis. Fig. 51 represents such a spot observed by M. Secchi at Rome in 1857., ^lv,^,''K"lSl The nucleus exhibited two centres perfectly';. black, while the penumbra showed numerous. l"'i"" dark lines extending spirally from these centres, and a large spiral filament, in the form of an eagle's beak, extended far within the nucleus. 179. Zodiacal light.-The zodiacal light is a faint light, somewhat resembling that of the Milky Way, or more nearly that of the tail of a comet, and is seen at certain seasons of the year in the west after the close of twilight in the evening, or in the east before its commencement in the morning. Its apparent form is nearly that of a cone with its base toward the sun, and its axis Fig. 53. iS situated nearly in the plane of the ecliptic. The season most favorable for observing this phenomenon, is when its direction, or the -direction of the ecliptic, is most nearly perpendicular to the horizon. For places near the latitude of New York, this occurs about the 1st of March for the evening, and about the 10th of' October for the morning. The distance to which the zodiacal light extends from the sun, varies from 20~ or 30~ to 80~ or 90~, and sometimes even more than 90~. Its breadth at its base perpendicularly to its length, varies from 8 to 100 ASTRONOMY, 30~. It is brightest in the parts nearest the sun, and in its upper part its light fades away by insensible gradations, so that different observers at the same time and place assign to it different limits. Under favorable circumstances, it has been seen to extend entirely across the heavens. It is probable that the zodiacal light is an envelope of very rare matter surrounding the sun, and extending beyond the orbits of Mercury and Venus, and at times even beyond the orbit of the earth. If the sun could be viewed from one of the other stars, it would probably appear to be surrounded by a nebulosity, similar to that in which some of the fixed stars appear to be enveloped, as seen from the earth. CHAPTER VII. PRECESSION OF THE EQUINOXES.-NUTATION. —ABERRATION.LINE OF THE APSIDES. 179. Fixed position of the ecliptic.-By comparing catalogues of stars formed in different centuries, we find that the latitudes of the stars continue always nearly the same. Hence the position of the ecliptic among the stars must be well-nigh invariable. 180. Precession of the equinoxes.-It is found, that the longitudes of the stars are continually increasing, at the rate of about 50" in a year. Since this increase of longitude is common to all the stars, and is nearly the same for each star, we can not ascribe it to motions in the stars themselves. We hence conclude that the vernal equinox, the point from which longitude is reckoned, has a backward or retrograde motion along the ecliptic, amounting to 50" in a year, while the inclination of the equator to the ecliptic remains nearly the same. This motion is called the precession of the equinoxes, because the place of the equinox among the stars each year precedes (with reference to the diurnal motion) that which it had the previous year. The amount of precession is 50".2 annually. In order to determine how many years will be required for a complete revolution of the equinoctial points, we divide 1,296,000, the number of seconds in the circumference of a circle, by 50".2, and obtain 25,800 years. PRECESSION OF THE EQUINOXES. 101 181. The pole of the equator revolves round the pole of the ecliptic.Since the position of the ecliptic is fixed, or nearly so, it is evident that the equator must change its position, otherwise there could be no motion in the equinoctial points; and a motion of the equator implies a motion of the poles of the equator. Since the obliquity of the ecliptic remains nearly constant, the distance from the pole of the equator to the pole of the ecliptic must remain nearly constant; and we may conceive the phenomena of precession to arise from the revolution of the pole of the celestial equator around the pole of the ecliptic, in the period of 25,800 years, at a constant distance of about 23] degrees. 182. The signs of the zodiac and the constellations of the zodiac.At the time of the formation of the first catalogue of stars, 140 years before Christ, the signs of the ecliptic corresponded very nearly to the constellations of the zodiac bearing the same names. But in the interval of 2000 years since that period, the vernal equinox has retrograded about 28~; so that the sign Taurus now corresponds nearly with the constellation Aries, the sign Gemini with the constellation Taurus, and so for the others. 183. The pole star varies from age to age.-The pole of the equator in its revolution about the pole of the ecliptic, must pass in succession by different stars. At the time the first catalogue of the stars was formed, the north pole was nearly 12~ distant from the present pole star, while its distance from it is now less than 11 degrees. The pole will continue to approach this star till the distance between them is about half a degree, and will then recede from it. After a lapse of about 12,000 years, the pole will have arrived within about 5~ of a Lyrse, the brightest star in the northern hemisphere. 184. Cause of the precession of the equinoxes.-The earth may be considered as a sphere surrounded by a spheroidal shell, thickest at the equator, Art. 45. The matter of this shell may be regarded as forming a ring round the earth, in the plane of the equator. Now the tendency of the sun's action on this ring, except at the time of the equinoxes, is always to make it turn round the intersection of the equator with the ecliptic, toward the plane of this latter circle. 102 ASTRONOMY. The solar force exerted on the part of this ring that is above the ecliptic, may be resolved into two forces, one of which is in the plane of the equator, and the other perpendicular to it. The latter force tends to impress on the ring a motion round its intersection with the ecliptic. So, also, the solar force exerted on the part of the ring that is below the ecliptic, may be resolved into two, one in the plane of the equator, and the other perpendicular to it. The sun's attraction upon the nearest half of the ring, tends to bring the plane of the ring nearer to the plane of the ecliptic; while its attraction upon the remoter half of the ring produces an opposite effect. But on account of the greater distance, the latter effect is less than the former; so that, besides the motion of translation produced by the various other forces, the ring would turn slowly around its intersection with the ecliptic, and the two planes would ultimately coincide. 185. How to find the resultant of two rotary motions.-While the equatorial ring has this tendency to turn about the line of the equinoxes, it also rotates on an axis perpendicular to its plane in twenty-four hours; that is, it has a tendency to rotate simultaneously about two different axes. The result is a tendency to rotate about an intermediate axis, whose position is determined by the following theorem: Fig. 54. If a body is revolving freely round the axis AB, D with the angular velocity V, and if a force be imc/ pressed upon it which would make it revolve about the axis AC with an an- Fig. 5. gular velocity V', then, PA the body will not revolve Vf about either of the axes AB, AC, but " about a third axis AD, situated in the \ plane BAC, and the angle BAC will be divided so that p\ sin. BAD: sin. CAD:: V' Y./ \ Let PP' represent the axis of diurnal rotation of the equato- S rial ring, and AB the line of the equinoxes, about which it also tends slowly to revolve. PRECESSION OF TIIE EQUINOXES. 103 The new axis of rotation, Ep', will be situated in the plane pEA, and the sine of its angular distance from each of the former axes will be in the inverse ratio of the angular velocity round that axis. Repeating the same construction for the following instant, we shall find the new position of the axis will be E3p", and so on; that is, the point p will be made to describe a curve around C, the pole of the ecliptic. 186. Illustration from the Gyroscope.-This motion of the earth's equatorial ring may be very closely imitated by a modified form Fig.. of the gyroscope. Let AB represent a brass ring, -F G —~- supported bywires AD, BD, which are connected \j" ~ with an axis, DC, whose extremity is a little above' the centre of gravity of the ring AB, and rests /\. A upon a support, CE. When the ring AB is at (, B rest, its axis DC will have a vertical position. If, \,0l — - however, the axis be inclined from the vertical, and be made to rotate by twirling it with the fingers, the plane of the ring will turn slowly round in azimuth, preserving, however, a nearly constant inclination to the horizon; that is, the axis of the ring will describe the surface of a cone, or the point F will describe the circumference of a circle about the point G. 187. IVlhy the precessiqn is so slow.-If the earth were a perfect sphere, the solar forces acting on the opposite hemispheres would exactly balance one another, and could produce no motion in the earth or its axis. If now we conceive the equatorial ring already described, to be attached to the spherical part of the earth, which is far heavier than the ring, it is evident that the ring, having to drag around with it this great inert mass, will have its velocity of retrogradation proportionally diminished. Thus, then, the entire globe must have a motion similar to that ascribed to the ring, but the motion will be extremely slow. The moon produces a similar retrogradation in the intersection of the equator with the plane of the lunar orbit, but, on account of its nearness to the earth, its effect is more than double that of the sun. The planets also, by their attraction, exert a small influence upon the position of the equatorial ring, but the result is slightly to diminish the amount of precession. The whole effect 104 ASTRONOM:Y. of the sun and moon is 50".37, and that of the planets 0".16, leav. ing the actual amount of precession 50".21 annually. Natation. 188. The effect of the action of the sun and moon upon the earth's equatorial ring, depends upon their position with regard to the equator. When either body is in the plane of the equator, its action can have no tendency to change the position of this plane, and consequently none to change the positions of the equinoctial points. Its effect in producing these changes, increases with the distance of the body from the equator, and is greatest when that distance is greatest. Twice a year, therefore, viz., at the equinoxes, the effect of the sun to produce precession is nothing, while at the solstices the effect of the sun is a maximum. On this account, the precession of the equinoxes, as well as the obliquity of the ecliptic, is subject to a semi-annual variation, which is called the solar nutation. There is also an inequality depending upon the position of the moon which is called lunar nutation. The maximum value of the lunar nutation in longitude is 17".2, and that of the solar nutation 1".2. In consequence of this oscillatory motion of the equator, its pole, in revolving about the pole of the ecliptic, does not move Fig. ST. strictly in a circle, but in a waving curve, rg...stwhich passes alternately within and without the circle, somewhat similar to that in ig. 57. t..' 189. Tropical and sidereal years. -The time occupied by the sun in moving from the vernal equinox to the vernal equinox again, is called a tropical year. The time occupied by the sun in moving from one fixed star to the same fixed star again, is called a sidereal year. On account of the precession of the equinoxes, the tropical year is less than the sidereal year, the vernal equinox having gone westward so as to meet the sun. The tropical year is less than the sidereal year, by the time that the sun takes to move over 50".2 of his orbit. This amounts to 20m. 22s. The mean length of a tropical year expressed in mean solar time is 365d. 5h. 48m. 48s. The length of the sidereal year is therefore 365d. 6h. 9m. 10s. ABERRATION. 105 Aberration. 190. The annual motion of the earth, combined with the motion of light, causes the stars to appear in a direction different from their true direction. This displacement is called aberration. The nature of this effect may be understood from the following illustration: If we suppose a shower of rain to fall during a dead calm in vertical lines, if the observer be at rest the rain will appear to fall vertically; and if the observer hold in his hand a tube in a vertical position, a drop of rain may descend through the tube without touching the sides; but if the observer move forward, the rain will strike against his face; and, in order that a drop of Fitg. rain may descend through the tube without,Sr touching the sides, the tube must be inclined forward. Suppose, while a rain-drop is falling from E to D with a uniform velocity, the spectator moves from C to D, and carries the tube inclined in the direction EC. A drop of rain entering the tube at E, when the tube has the position EC, would reach the ground at D when the tube has come into the posiC^ ^ CD B tion FD; that is, the drop of rain will appear to follow the direction EC. Now CD =ED x tang. CED; that is, the velocity of the observer=velocity of the rain x tangent of the apparent deflection of the rain-drop. 191. To determine the amount of aberration.-The aberration of light is explained in a similar manner. Let AB be a small portion of the earth's orbit, and S the position of a star. Let CD be the distance through which the observer is carried in Is., and ED the distance through which light moves in Is. If a straight tube be conceived to be directed from the eye at C to the light at E, so that the light shall be in the centre of its opening, and if the tube moves with the eye from C to D, remaining constantly parallel to itself, the light, in moving from E to D, will pass along the axis of the tube, and will arrive at D when the earth reaches the same point. It is evident that the star will appear in the direction of the axis of the tube; that is, the star appears in the direc 106 ASTRONOMY. tion S'D instead of SD. The velocity of the earth in its orbit is 19 miles per second; the velocity of light is 192,000 miles per second. In the triangle ECD, we have tang. CED =D 1 -ED 192,000' I-ence CED -20"; that is, the aberration of a star which is 90~ from the path in which the earth is moving, amounts to 20". 192. Effect of aberration upon a star situated at the pole of the ecliptic.-It is obvious that the aberration is always in the direction in which the earth is moving. Its effect, therefore, upon the apparent position of a star, will vary with the season of the year. rig. 59. Let ABCD represent the annual g.~) path of the earth around the sun; - let S be the place of the sun, and s the place of a star so situated that the line Ss is perpendicular to the D=~ ~T plane of the ecliptic. W /hen the earth is at the point A, moving toward B, the aberration ~~ 1 C t^ Swill be in the direction sa; that is, the star appears at the point a. When the earth has arrived at B, the aberration will be in the di~~B ~ rection sb; that is, the star appears at the point b. When the earth has arrived at C, the star appears at the point c; and when the earth has arrived at D, the star appears at the point d. But sa, sb, sc, sd are each 20", and therefore the star will appear annually to describe a small circle in the heavens, 40" in diameter. 193. Effect upon a star situated in the plane of the ecliptic.-If the star were situated in the plane of the ecliptic, in the direction of the line AC produced, then, when the earth is at C, the aberration will be 20", as before; but when the earth is at D, the aberration will be nothing, because the earth and the light of the star are moving in the same direction. When the earth is at A, the ab LINE OF THE APSIDES. 107 erration will again be 20", but in a direction opposite to what it was at C; and when the earth is at B, the aberration will again be nothing. Hence we see that if a star be in the plane of the ecliptic, it will appear to oscillate to and fro along a straight line, 20" on each side of the true position of the star, and this line will be situated in the plane of the ecliptic. A star situated between the ecliptic and its poles, will appear annually to describe an ellipse whose major axis is 40", but its minor axis will increase with its distance from the plane of the ecliptic. 194. The apsides of the earth's orbit.-The points of perihelion and aphelion of the earth's orbit, are called by the common name of apsides. The major axis of the earth's orbit is therefore called the line of the apsides. By comparing very distant observations, it is found that the line of the apsides has a progressive motion, or a motion eastward amounting to about 12" annually. Since the equinox from which longitude is reckoned moves in the opposite direction 50" annually, the longitude of the perihelion increases about 62" annually. At this rate, the line of the apsides would complete a sidereal revolution in 108,000 years, or a tropical revolution in 20,900 years. For the cause of this motion, see Arts. 279 478. 195. Changes in the position of the line of the apsides.-The line of the apsides, thus continually moving round, must at one period have coincided with the line of the equinoxes. The longitude of the perihelion in 1864 was 100~ 16', which point the earth passed on the 1st of January. The time required to move over an arc of 100-~ at the rate of 62" annually, is about 5818 years, which extends back nearly 4000 years before the Christian era-a period remarkable for being that to which chronologists refer the creation of the world. At this time the winter and spring were equal, and longer than the summer and autumn, which were also equal. 196. Mean place and true place; mean anomaly and true anomaly.-The mean place of a body revolving in an orbit, is the place where the body would have been if its angular velocity had been 108 ASTRONOMY. uniform; the true place of a body is the place where the body actually is at any time. Equations are corrections which are applied to the mean place of a body, in order to get its true place. The angular distance of a planet from its perihelion, as seen from the sun, is called its anomaly. If an imaginary planet be supposed to move from perihelion to aphelion with a uniform angular motion round the sun, in the same time that the real planet moves between the same points with a variable angular motion, the angular distance of this imaginary planet from perihelion is called its mean anomaly, while its actual distance at the same moment in its orbit is called its true anomaly. 197. Equation of the centre.-The difference between the mean and the true anomaly is called the equation of the centre. Let ABCD be the orbit of a plapet having the sun in one of the foci at S. With the centre S, and a radius,f //)^~ \^~ ^ equal to the square root of F ~ a X1 the product of the semi-axes of the ellipse, describe the circle EBFD; the area of this circle will be equal to that of the ellipse. At the same time that a planet departs from A, the perihelion, to describe the orbit ABCD, let an imaginary planet start from E, and describe the circle EBFD with a uniform motion, and perform a whole revolution in the same period that the planet describes the ellipse. The imaginary planet will describe around S, sectors of circles which are proportional to the times, and equal to the elliptic areas described in the same time by the planet. Suppose the imaginary planet to be at G; then take the sector ASHl=ESG, and H will be the place of the planet in the ellipse. The angle ESG is called the mean anomaly; ASH is the true anomaly; and GSH is the equation of the centre. If we consider the mean and the true anomaly as agreeing at A, the angles ESG and ASH must increase unequally, and the true anomaly must exceed the mean. The equation of the cen ANOMALISTIC YEAR.-THE MOON. 109 tre increases till the planet reaches the point B. From B to C the mean anomaly gains upon the true, until at C they coincide -that is, the equation of the centre is nothing. Proceeding from C, the mean anomaly must exceed the true, and the equation of the centre increases until the planet reaches the point D. From D to A the true anomaly gains upon the mean, until at A they coincide again. At the points B and D the equation of the centre is the greatest possible. The greatest value of the equation of the centre for the sun is 10 55' 27". 198. The anomalistic year.-The time occupied by the earth in moving from the perihelion to the perihelion again, is called the anoralistic year. This period must be a little longer than the sidereal year, since the earth must describe a further arc of 11".8 before reaching the perihelion; and the difference will be equal to the time necessary for the earth to describe 11".8 of its orbit, or 4m. 35s., which gives 365d. 6h. 13m. 45s. for the length of the anomalistic year. This period is occasionally used in astronomical investigations, but mankind are generally more concerned in the tropical year, on which the return of the seasons depends. CHAPTER. YVII. THE MOON-ITS MOTION PHASES-TELESCOPIC APPEARANCE. 199. Distance of the moon.-The distance of the moon can be computed when we know its horizontal parallax. This parallax varies considerably during a revolution of the moon round the earth. The equatorial parallax, when least, is 53' 48", and when greatest, 61' 32". The mean horizontal parallax of the moon at the equator is 57' 2".3. Hence the mean distance will be found by the proportion sin. 57' 2".3: 3963.35:: 1: the moon's distance, which is found to be 238,883 miles. In the same manner, the moon's greatest distance is found to be 253,263 miles, and its least distance 221,436 miles. 200. Diameter of the moon.-The absolute diameter of the moon can be computed when we know its apparent diameter, and its 110 ASTRONOMY. distance from the earth. The apparent diameter varies according to its distance from the earth. When nearest to us, it is 33' 31".1; but at its greatest distance it is only 29' 21".9. At its mean distance the apparent diameter is 31' 7".0. Hence the absolute diameter will be found by the proportion: 238,883:: sin. 15' 33".5: the moon's semi-diameter, which is found to be 1081.1 miles. Hence the moon's diameter is 2162 miles. Since spheres are as the cubes of their diameters, the volume of the moon is — th that of the earth. Its density is about 3ths (.615) the density of the earth, and its mass (- x.615) is about Ao-th of the mass of the earth. 201. Definitions.-A body is said to be in conjunction with the sun when its longitude is the same as that of the sun; it is said to be in opposition to the sun when their longitudes differ 180~; and to be in quadrature when their longitudes differ 90~ or 270~. The term syzygy is used to denote either conjunction or opposition. The octants are the four points midway between the syzygies and quadratures. The two points in which the orbit of the moon or a planet is cut by the plane of the ecliptic are called nodes. That node at which the body passes from the.south to the north side of the ecliptic is called the ascending node, and the other the descending node. 202. Revolution of the moon.-If the situations of the moon be observed on successive nights, it will be found that it changes its position among the stars, moving among them from west to east; that is, in a direction contrary to that of the diurnal motion. By this motion it makes a complete circuit of the heavens in about 27 (lays. Hence either the moon revolves round the earth, or the earth round the moon. Strictly speaking, the earth and moon both revolve about their common centre of gravity. This is a point in the line joining their centres, situated at an average distance of 2690 miles from the centre of the earth, or about 1270 miles beneath the surface of the earth. 203. Sidereal and synodic revolutions.-The interval of time oc THE MOON-' ITS MOTION. 111 cupied by the moon in pertorming one sidereal revolution round the earth, or the time which elapses between her leaving a fixed star until she again returns to it, is 27d. 7h. 43m. 11s. The moon's mean daily motion is found by dividing 360~ by the number of days in one revolution. The mean daily motion is thus found to be 13~.1764, or about 138 degrees. The synodical revolution of the moon is the interval betweer two consecutive conjunctions or oppositions. The synodical revolution of the moon is longer than the sidereal by 2d. 5h. Om. 51s., which is the time required by that body to describe with its mean angular velocity of 13- degrees per day the arc traversed by the sun since the previous conjunction. Hence we find the duration of the synodical period to be 29d. 12h. 44m. 2s. 204. How the synodical period is determined.-The mean synodical period may be determined with great accuracy by observations of eclipses of the moon. The middle of an eclipse is very near the instant of opposition, and from the observations of the eclipse the exact time of opposition may be easily computed. Now eclipses have been very long observed, and the time of the occurrence of some has been recorded even before the Christian era. By comparing an eclipse observed by the Chaldeans, 720 B.C., with recent observations, the duration of the mean synodic period has been ascertained with great accuracy. 205. How the sidereal period is derived from the synodical.-The sidereal period may be deduced from the synodical as follows: Let P=the length of the sidereal year, p=the sidereal revolution of the moon, T=the synodical period of the moon. Then the arc which the moon describes in order to come into conjunction with the sun, exceeds 360~ by the space which the sun has passed over since the preceding conjunction. This excess is found by the proportion P: T:: 360: 360T Then, as the whole distance the moon must move from the sun to reach it again, is to one circumference, so is the time of describing the former, to the time of describing the latter; that is, 112 ASTRONOMY. 360T 360+ P:360::T:p; or 1+: 1:: T:p PT 365.25 x 29.53 Whence p + T365.25+29.53 =27.32 days; and this is the most accurate mode of determining the sidereal period of the moon. 206. Moon's path.-The moon's observed right ascension and declination enable us to determine her latitude and longitude. By observing the moon from day to day when she passes the meridian, we find that her path does not coincide with the ecliptic, but is inclined to it at an angle of 5~ 8' 48", and intersects the ecliptic in two opposite points, which are called the moon's nodes. 207. Form of the moon's orbit.-It can be proved in a manner similar to that given for the sun, Arts. 111 and 114, that the moon in her orbit round the earth obeys the following laws: 1st. The moon's path is an ellipse, of which the earth occupies a focus. 2d. The radius vector of the moon describes equal areas in equal times. The point in the moon's orbit nearest the earth is called her perigee, and the point farthest from the earth her apogee. The line joining the apogee and perigee is called the line of the apsides. 208. Eccentricity of the moon's orbit.-The eccentricity of the lunar orbit may be found by observing the greatest and least apparent diameters of the moon, in the same manner as was done in the case of the sun, Art. 113. Example. In the month of October, 1862, the greatest apparent diameter of the moon was 33' 0".6, and the least was 29' 34".0. From these data determine the eccentricity of the lunar orbit during that month. The ratio of A to P is 0.89569. A-P Hence, by the formula e= A+P' we find e=0.0550, or about -. THE MOON-1ITS PHASES. 113 209. Interval of moon's transits.-The moon's mean daily motion in right ascension is 13~.17, or 12~.19 greater than that of the sun. Hence, if on any given day we suppose the moon to be on the meridian at the same instant with the sun, on the next day she will not arrive at the meridian till 51m. after the sun; that is, the interval between two successive meridiafi passages of the moon is, on the average, 24h. 51m. In consequeice of the inequalities in the moon's motion in right ascension, this interval varies from 24h. 38m. to 25h. 6m. 210. Mloon's meridian altitude.-The moon's altitude when it crosses the meridian is very variable. The meridian altitude of the sun at the summer solstice is 46~ 54' (twice the obliquity of the ecliptic) greater than it is at the winter solstice. Now, since the moon's orbit is inclined 5~ 9' to the plane of the ecliptic, the moon will sometimes be distant from the ecliptic by this quantity on the north side, and at other times by the same quantity on the south side; hence the greatest meridian altitude of the moon will exceed its least by 46~ 54'+10~ 18', or 57~ 12'. In latitude 41~ 18', the greatest meridian altitude of the moon is 77~ 18', and its least 20~ 6'. 211. The moon's phases.-The different forms which the moon's visible disc presents during a synodic revolution are called phases. The moon's phases are completely accounted for by assuming her to be an opaque globular body, rendered visible by reflecting light received from the sun. Let E be the earth, and ABCDH the orbit of the moon, the sun being supposed to be at a great distance in the direction AS. When the moon is in conjunction at A, the enlightened half is turned directly from the earth, and she must then be invisible. It is then said to be new moon. About 71 days after new moon, when she is in quadrature at C, one half of her illumined surface is turned toward the earth, and her enlightened disc appears as a semicircle. She is then said to be in herfirst quarter. About 15 days after new moon, when she is in opposition at F, the whole of her illumined surface is turned toward the earth, and she appears as a full circle of light. It is then said to be full moon. H 114 ASTRONOMY. Fig. 61. From new moon to first quarter, and from last quarter to new to be bbous. This phase is represented at D and. Tese 0D D 1^ About 71- days after full moon, when she is again in quadrature at H, one half of her illumined surface being turned toward the earth, she again appears as a semicircle. She is then said to be at her lst quarter. From new moon to first quarter, and from last quarter to eis monon, her enlightened disc is called a crescent. This phase is represented at B and.o The two extremities of the crescent are called cusps, or horns. From first quarter to full moon, and from full moon to last quarter, the form of her enlightened disc is said to be gibbous. This phase is represented at D and r. These phases prove conclusively that the moon shines by light borrowed from the sun. The interval from one new moon to the next new moon is called a lunation, or lunar month. It is evidently the same as a synodical revolution of the moon. 212. Obscure part of the moona's disc.-hen the moon is just visible after new moon, the whole of her disc is quite perceptible, the part not fully illumined appearing with a faint light. As the moon advances, the obscure part becomes more and more faint, and it entirely disappears before full moon. This phenomenon depends on light reflected from the earth to the moon, and from the moon back to the earth. WVhen the moon is near to A, she receives light from nearly THE HARVEST MOON. 115 the whole of the earth's illumined surface, and this light, being in part reflected back, renders visible that portion of the disc that is not directly illumined by the sun. As the moon advances toward opposition at F, the quantity of light she receives from the illumined surface of the earth decreases; and its effect in rendering the obscure part visible, is farther diminished by the increased light of the part which is directly illuminated by the sun's rays. It is obvious that, to an observer at the moon, the earth must appear as a splendid moon, presenting all the phases of the moon as seen from the earth, and having more than three times its apparent diameter. 213. Daily retardation of the moon's rising or setting.-The average daily retardation of the moon's rising or setting is the same as that of her passage over the meridian; but the actual retardation, being affected by the moon's changes in declination, as well as by the inequalities of her motion in right ascension, is subject to greater variation. In the latitude of New York, the least daily retardation is 23 minutes, and the greatest is lh. 17m. 214. Harvest Moon. —The less or greater retardation of the moon's rising attracts most attention when it occurs at the time of full moon. When the retardation has its least value near the time of full moon, the moon rises soon after sunset on several successive evenings; whereas, when the retardation is greatest, the moon ceases in two or three days to be seen in the early part of the evening. When the moon is in that part of her orbit which makes the least angle with the horizon, 13 degrees of her orbit (which is her average progress in a day) rises above the horizon at New York in less than 30 minutes. This happens for the full moon near the time of the autumnal equinox. As this is about the period of the English harvest, this moon is hence called the Harvest Moon. 215. Effect of altitude on the moon's apparent diameter.-The apparent diameter of the moon is not the same at the same instant for all points of the earth, on account of their different distances from the moon. As the moon rises above the horizon (if we sup 116 ASTRONOMY. pose its distance from the centre of the earth to remain constant), its distance from the place of observation must diminish, while its altitude increases, and, consequently, its apparent diameter must increase. This effect attains its maximum when the moon is in the zenith of the spectator. The distance AB is about equal to CB or CG, and exceeds AG Fig.62. 62 by AC, the radius of the earth, which is about one./^ \Em-~ ^sixtieth of the moon's distance. I-ence the angle GAI-, which the moon's radius subtends when in /xa/ \T Gthe zenith, exceeds the \ ______ 3 = = =~~ angle BAD, which the moon's radius subtends when in the horizon, by about one sixtieth of the whole quantity; that is, the augmentation of the moon's diameter on account of her apparent altitude may amount to more than half a minute. The apparent enlargement of the moon near the horizon is an optical illusion, as explained Art. 92. 216. Has the moon an atmosphere?-There is no considerable atmosphere surrounding the moon. This is proved by the absence of twilight. Upon the earth, twilight continues until the sun is 18~ below the horizon; that is, day and night are separated by a belt 1200 miles in breadth, in which the transition from light to darkness is not sudden, but gradual-the light fading away into the darkness by imperceptible gradations. This twilight results from the refraction and reflection of light by our atmosphere; and if the moon had an atmosphere, we should notice, in like manner, a gradual transition from the bright to the dark portions of the moon's surface. Such, however, is not the case. The boundary between the light and darkness, though irregular, is perfectly well defined and sudden. Close to this boundary, the unillumined portion of the moon appears just as dark as any portion of the moon's unillumined surface. 217. Argument from the absence of refraction.-The absence of an atmosphere is also proved by the absence of refraction when the moon passes between us and the distant stars. Let AB represent LIGHT. OF THE FULL MOON. 117 ig. 63. c the disc of the':3..........8 7'- moon, and CD an'........... S' atmosphere supposed to surround ---- ------ it. Let SAE represent a straight line touching the moon at A, and proceeding toward the earth, and let S be a star situated in the direction of this line. If the moon had no atmosphere, this star would appear to touch the edge of the moon at A; but if the moon had an atmosphere, this atmosphere would refract light; and a star behind the edge of the moon in the position S' would be visible at the earth, for the ray S'A would be bent by the atmosphere into the direction AE'. So, also, near the opposite limb of the moon, a star might be seen at the earth, although really behind the edge of the moon. Hence we see that if the moon had an atmosphere, the time during which a star would be concealed by the moon would be less than if it had no atmosphere; and the amount of this effect must be proportional to the density of the atmosphere. Many thousand occultations of stars by the moon have been observed, and no appreciable effect of refraction has ever been detected. This species of observation is susceptible of such accuracy, that if the refraction amounted to 4" of arc, it is believed that it could not fail to be detected in the mean of a large number of observations. Now the earth's atmosphere changes the direction of a ray of light more than half a degree when it enters the atmosphere, and the same when it leaves it, making a total deflection of over 4000". IHence we conclude that if the moon have an atmosphere, its density can not exceed one thousandth part of the density of our own. Such an atmosphere is more rare than that which remains under the receiver of the best air-pump when it has reached its limit of exhaustion. 218..Light of the fall mnoon. —The light received from the full moon was compared by Bouguer with the light received from the sun, by comparing each with the light of a candle. The light of the sun being admitted into a dark room through a small aperture, he placed in front of the operator a concave lens, to diminish the intensity of the sun's rays by causing them to diverge. He then placed a candle at such a distance that its light received 118 ASTRO'NO,. upon a screen was exactly equal to that of the sun received upon the same screen. Repeating this experiment at night with the full moon, he compared the light of the moon with that of the candle. By several experiments of this kind, he arrived at the conclusion that the sun illumines the earth 300,000 times more than the full moon. Professor G. P. Bond compared the light of the moon -with that of the sun by placing in the sun's light a glass globe with a silvered surface, and comparing the brightness of the reflected image of the sun with an artificial light, and afterward comparing the light of the full moon with the same standard. He hence inferred that the light of the sun was 470,000 times that of the full moon. 219. Heat of the moon.-Until recently, the most delicate experiments had failed to detect any heat in the light of the moon. The light of the full moon has been collected into the focus of a concave mirror of such a magnitude as, if exposed to the sun's light, would have been sufficient to evaporate platinum; yet no sensible effect was produced upon the bulb of a differential thermometer so delicate as to show a change of temperature amounting to the 500th part of a degree. This experiment, if reliable, would indicate that the moon reflects a less proportion of the heating rays than of the luminous rays of the sun. In 1846 Melloni repeated this experiment on the top of Mount Vesuvius with a lens of three feet diameter, and found feeble indications of heat when the light of the moon was concentrated upon a delicate thermo-multiplier. In the summer of 1856, Professor Smyth repeated this experiment on the summit of Teneriffe, over 10,000 feet above the sea, and found that the heat of the full moon was equal to one third that of an ordinary candle pTaced at a distance of 15 feet. Even this small amount of heat appears to be absorbed by the atmosphere before reaching the earth; and near the earth's surface, the moon's heat is inappreciable by the most delicate means of observation hitherto employed. 220. Telescopic appearance of the moon. If with a telescope we examine the bounding line between the illumined and dark portions of the moon's surface, especially about the time of the first THE MOON-PARTICULAR PHENOMENA DESCRIBED. 119 quarter, we shall find it to be very broken and irregular. At some distance from the generally illumined surface we may notice bright spots, often entirely surrounded by a dark ground; and we also find dark spots entirely surrounded by an illumined surface. These appearances change sensibly in a few hours. As the light of the sun advances upon the moon, the dark spots become bright; and at full moon they all disappear, and we only notice that certain regions appear more dusky than others. The moon's surface is therefore uneven; and, by observing the passage of the sun's light over these spots, we may form a judgment of their dimensions and figure. The most favorable time for observing these inequalities is near the first or third quarter, because then the shadows of the mountains appear of their greatest length, and are not shortened by being seen obliquely. See Plate II., Fig. 2, which gives a representation of a small portion of the moon's surface as seen through a powerful telescope. 221. Particular _phenomena described.-Near the bounding line of the moon's illumined surface we frequently observe the following phenomena: A bright ring nearly circular; within it, on the side next the sun, a black circular segment; and without it, on the side opposite to the sun, a black region with a boundary more or less jagged. Near the centre of the circle we sometimes notice a bright spot, and a black stripe extending from it opposite to the sun. After a few hours, the black portions are found to have contracted in extent, and in a day or two entirely disappear. After about two weeks these dark portions reappear, but on the side opposite to that on which they were before seen; and they increase in length until they pass entirely within the dark portion of the moon. These appearances can only be explained by admitting the existence of a circular wall, rising above the general level of the moon's surface, and inclosing a large basin, from the middle of which rises a conical peak. 222. Height of the lunar mountains.-If the distance of the illuminated summit of a mountain from the enlightened part of the disc be measured with a micrometer, and the positions of the sun and moon at the time be obtained by observation or computation, the height of the mountain may be computed. J0 120.. ASTRONOMY. F. 64. Let AFE be the illuminated hemisphere Fig. 64. 3A of the moon, SA a ray of the sun touch-~O ing the moon at A, and let BD be a mountain so elevated that its summit just reaches to the ray SAB, and is illumined while the intervening space AB is dark. 1 Vi /Suppose now the earth to be in the direc-' I / tion of the diameter AE produced. Let the angle which AB subtends at the earth be measured with a micrometer; then, since the distance of the moon from the earth is known, the absolute length of AB can be computed. Then, in the right-angled triangle ABC, AC, the radius of the moon, is known, whence BC can be computed; and subtracting AC from BC, gives BD, the height of the mountain. If the earth is so situated that the line AB is not seen perpendicularly, since we know the relative positions of the sun and moon, we can determine the inclination at which AB is seen, and hence the absolute length of AB. The height of a mountain may also be computed fiom the measured length of the shadow it casts. The greatest elevation of any lunar mountain which has been observed is 23,800 feet. The altitudes of the higher mountains in the moon are probably as accurately known as those of the highest mountains on the earth. 223. Circular craters.-Mountain ranges, approaching nearly to the form of circles, are very common on the moon's surface. They sometimes have a diameter of over 50 miles, and a height of 2 or 3 miles. Tycho, Kepler, and Copernicus are among the most remarkable of these mountain ranges. See Plate II., Fig. 1. Tycho, No. 1, is near the moon's southern limb; Kepler, No. 2, near the eastern limb; and Copernicus, No. 3, a little west of Kepler. These circular mountains bear an obvious analogy to the volcanic craters upon the earth. 224. The crater of Kilauea, on one of the Sandwich Islands, is a vast basin, more than three miles in its longer diameter, and nearly 1000 feet deep. From the bottom of the basin rise numerous little cones, from which smoke is almost constantly emitted, and sometimes melted lava. The craters of most volcanoes exhibit THE MOON-VOLCANOES. 121 an irregular circular wall of considerable height, sometimes 2 or Fig. 65. 3 miles, and within this wall rise one or two cones formed by the occasional overflowing of the lava. 225. Lunar volcanoes compared with terrestrial.-The lunar volcanoes differ from the terrestrial in their enormous dimensions and immense number. This may be due, in some degree, to the feeble attraction of the moon, since objects on the moon's surface weigh only one sixth what they would on the earth. 226. It is certain that most of the lunar volcanoes are entirely extinct; and it is doubted whether any signs of eruption have ever been noticed. The spot called Aristarchus, marked 4 on Fig. 1, Plate II., is so brilliant that some have concluded it to be an active volcano. Herschel observed on the dark portion of the moon three bright points, which he ascribed to volcanic fires; but the same lights may be seen every month, and they are probably to be ascribed to mountain peaks which have an unusual power of reflecting the feeble light which is emitted by the earth. It is believed that all the inequalities of brightness observed on the moon's surface (with the exception of the shadows described in Arts. 220-1) result from a difference in the nature of the reflecting materials. Two distinguished astronomers, Beer and Madler, who have studied the moon's surface with greater care than any one else, assert that they have never seen any thing that could au 122 ASTRONOM-WY. thorize the conclusion that there are in the moon volcanoes now in a state of ignition. 227. Streaks of light from Tycho.-Very remarkable streaks of light are seen diverging from several of the lunarcraters. These are quite conspicuous about Tycho, Kepler, Copernicus, and Aristarchus. One of these streaks of light diverging from Tycho can be traced 1700 miles. These streaks cross ridges and valleys without interruption; and some of them have been noticed to cast shadows. They are thought to have resulted from some violent volcanic eruption, by which enormous crevices were opened in the moon's surface. These crevices are supposed to have been filled with melted lava, which congealed into a glassy rock, having a more brilliant reflecting surface than the general disc of the moon. Similar phenomena,but upon a far less extensive scale, have taken place on the earth's surface. 228. There is no water on the moon's surface. The dusky regions, which were once supposed to be seas, are regions comparatively level; but upon which, with a good telescope, we can detect black shadows, indicating the existence of permanent inequalities, which could not exist on a fluid surface. Moreover, if there were any water on the moon's surface, a portion of it would rise in vapor, and form an atmosphere which would refract light to an extent far beyond what we actually observe. 229. Can volcanoes exist without air or water?-It may be objected that volcanoes could not exist without air or water. It is not certain that the presence of air is necessary to the activity of a volcano. Volcanoes may be ascribed to the primitive heat of the globe, or to galvanic action on a large scale. A commotion of the melted lava would be instantly produced by the introduction of water, which would suddenly generate large quantities of steam; and it might also be produced by the presence of various other bodies; as, for example, sulphur, which almost invariably accompanies volcanic eruptions. Some similar substance might cause an eruption of a lunar volcano without the agency of water. 230. Can animal life exist upon the moon? —Air and water are necessary to the support of both animal and vegetable life. It is THE MOON-DOES IT INFLUENCE THE WEATHER? 123 doubtful, therefore, whether even the humblest form of life with which we are acquainted could exist upon the moon. Nothing has ever been discovered upon the moon's surface to indicate the agency of human beings, or the presence of any form of animal or vegetable life. The extremes of temperature upon the moon's surface must be far more violent than they are upon the earth. For 14 successive days the sun shines uninterruptedly upon the same portion of the moon, and for the next 14 days his light is entirely withdrawn. During the first period, the moon's surface must become intensely heated; and during the next fortnight the cold must be equally severe, since there is no atmosphere or clouds to obstruct the radiation of heat. While, then, we are compelled to say that Infinite wisdom and power can create beings to live in such a world, we can safely assert that no varieties of animal or vegetable life with which we are acquainted can exist in the moon. 231. Does the moon influence the weather?-The impression is almost universal (especially among uneducated men) that the moon exerts a sensible influence upon the weather. Various and even opposite effects have been ascribed to the moon's action; but most people are confident that the moon exerts a powerful influence of some sort. One opinion, which has been defended even by some men of science, is that the full moon has a decided influence in dissipating the clouds. But, from a comparison of seven years' observations at Greenwich, I have found that at full moon the average cloudiness of the sky is precisely the same as at new moon. It has also been claimed that the amount of rain is affected by the moon's age; and some observations appear to indicate that from the first quarter to the full, the amount of rain exceeds the average; but the results obtained at different stations are diverse and often contradictory, indicating that these differences are due mainly to other causes than the moon. 232. Does the moon influence the pressure of the air?-Many have imagined that inasmuch as the moon elevates the water of the ocean, its disturbing influence ought to be much greater upon a fluid of such mobility as our atmosphere. The moon does indeed influence the pressure of the air, but its disturbing force is extremely small. At Singapore, under the equator, when the moon 124 ASTRONOMY. is on the meridian, the barometer is higher by -rTth of an inch than when the moon is six hours from the meridian; at St. Helena, in Lat. 15~ 55', this difference amounts to - ro-th of an inch; and in our latitude the difference should be still less. This effect is so minute that it can only be detected by the most accurate observations, continued for a period of several years. Indeed, it has never been shown that the moon exerts any influence upon the weather, except that which is of the feeblest kind, and which is only appreciable after a very long series of the best observations. 233. Moon's rotation upon an axis.-The various spots on the moon always occupy nearly the same positions upon the disc, from which it follows that nearly the same surface is always turned toward the earth. Hence we conclude that the moon rotates upon an axis in the same time that she makes a revolution in her orbit. If the moon had no motion of rotation, then in opposite parts of her orbit she would present opposite sides to the earth. In order that a globe which revolves in a circle around a centre should turn continually the same hemisphere toward that centre, it is necessary that it should make one rotation upon its axis in the time it takes to revolve about the centre. 234. Librations of the moon.-Although it is true that nearly the same hemisphere of the moon is always turned toward the earth, yet the moon has apparently a slight oscillatory motion, which allows us to see a portion of the opposite hemisphere. This oscillatory motion is called libration. Libration in longitude.-While the moon's angular velocity on its axis is rigorously uniform throughout the month, its angular velocity in its orbit is not uniform, being most rapid when nearest the earth. Hence we see at one time a little more of the eastern or western edge of the moon than we do at another time. This is called the libration in longitude. Libration in latitude.-The axis of the moon is not quite perpendicular to the plane of her orbit, but makes an angle with it of 831 degrees. On account of this inclination, the northern and southern poles of the moon incline alternately 6-~ to and from the earth. When the north pole leans toward the earth, we see a little more of that region, and a little less when it leans the contrary way. This variation is called the libration in latitude. THE MOON-LUKNAR DAY. 125 Diurnal libration.-By the diurnal motion of the earth, we are carried with it round its axis; and if the moon presented exactly the same hemisphere toward the earth's centre, the hemisphere visible to us when the moon rises, would be different from that which would be visible to us when the moon sets. This is another cause of a variation in the edges of the moon's disc, and is called the diurnal libration. In consequence of all these librations, we can see somewhat more than half of the surface of the moon; yet there remains about 3ths of its surface which is always hidden from our view. 235. Lunar day.-The rotation of the moon upon its axis, being equal to that of its revolution in its orbit, is 271 days. The intervals of light and darkness to the inhabitants of the moon, if there were any, would be altogether different from those upon the earth. There would be about 328 hours of continued light, alternating with 328 hours of continued darkness. The heavens would be perpetually serene and cloudless. The stars and planets would shine with extraordinary splendor as well in the day as in the night. The inclination of her axis being small, there Would be no sensible change of seasons. The inhabitants of one hemisphere could never see the earth; while the inhabitants of the other would have it constantly in their firmament by day and by night, and always nearly in the same position. To those who inhabit the central part of the hemisphere presented to us, the earth would appear stationary in the zenith, with the exception of the small effect due to libration. The earth illumined by the sun would appear as the moon does to us, but with a superficial magnitude about fourteen times as great. Its phases would also be similar to those which we see in the moon. 236. Equality of the periods of rotation and revolution.-That the moon should rotate on an axis in exactly the same time that is required for a revolution around the earth, can not be supposed to be accidental. We are forced, then, to seek for some physical cause to explain this coincidence. If we admit that originally these two motions were nearly equal, the exact equality may be explained as follows: The moon, like the earth, was probably once in a plastic condi 126 ASTRONOMY. tion. The earth would then act upon the moon as the moon acts upon the earth in raising the tides, only with much greater power; that is, it would give the moon an elongated figure, its major axis pointing toward the centre of the earth. If the moon has such an elongated figure, the earth must act upon it as upon a pendulum. When a pendulum is deflected from the vertical position, the earth's attraction brings it back again, causing it to oscillate to and fro. So, also, if the longer axis of the moon were deflected from pointing toward the earth, the earth's attraction would tend to bring it back to this position, thus tending to establish a rigorous equality between the times of rotation and revolution of the moon. 237. Position of the moon's centre of gravity.-From a careful study of the moon's motions, Hansen concludes that the centre of gravity of the moon does not coincide with its centre of figure, and that the centre of figure is nearer to us by 33 miles than the centre of gravity; in other words, the hemisphere which is turned toward the earth is lighter than the opposite hemisphere, and may be regarded as an enormous mountain, rising 33 miles above the mean level of the moon. This lightness may be the result of volcanic energy, upheaving the crust, and leaving large cavities beneath; and these cavities must be mainly on the side of the moon which is turned toward the earth. This cause may have contributed to produce that elongated figure of the moon which enables us to explain the exact equality between the time of rotation upon its axis and of revolution about the earth. This conclusion of Hansen is notaccepted by all astronomers. 238. Path of the moon in its motion about the sun.-While the moon revolves about the earth, it also accompanies the earth in its motion about the sun. The actual path described by the moon will then be an undulating line, alternately within and without the orbit of the earth. The undulations are, however, so small, in comparison with the dimensions of the earth's orbit, that the path of the moon is always concave toward the sun. The distance, AB, passed over by the earth in a fortnight, is about 24 millions of miles. If we draw a chord connecting these points, this chord, at its middle point, will fall about 700,000 miles within the orbit of the earth, while the greatest distance of the moon from THE MOON-CHANGES OF ITS ORBIT. 127 Fig. 66. - e \ the earth is only 253,000 miles. The moon's path, therefore, approaches so near to that of the earth as to be always concave toward the sun. 239. Changes of the moon's orbit.-The elliptic path described by the moon, changes gradually from month to month both in form and position. Its eccentricity varies within certain limits, being sometimes as great as 0.065, and sometimes as small as 0.049. Its mean value is 0.05484, or about y-th. The major axis of the moon's orbit is not fixed, but has a direct motion on the ecliptic at the rate of about 41~ in a year, accomplishing a complete revolution in a little less than nine years; so that in 41 years the perigee arrives where the apogee was before. This motion of the line of the apsides is not equable throughout the whole of a lunar month; for when the moon is in syzygies, the line of apsides advances in the order of the signs, but is retrograde in quadratures. The direct motion is, however, greater than the retrograde. 240. Motion of the line of the nodes.-The line in which the plane of the moon's orbit cuts the ecliptic, is called the line of the nodes. The position of the nodes is found by observing the longitude of the moon when she has no latitude; and it appears, by a comparison of such observations, that the line of the nodes is not fixed, but has a slow retrograde motion at the rate of about 19~ in a year. By this motion the nodes make a mean tropical revolution in 18 years and 224 days, nearly. It is not, however, an equable motion throughout the whole of the moon's revolution. The node is generally stationary when the moon is in quadrature, or in the ecliptic; in all other parts of the orbit it has a retrograde motion, which is greater the nearer the moon is to the syzygies, or the greater the distance from the ecliptic. Thus we see that the path of the moon does not return into it 128 ASTRONOMY. self, but is a curve of the most complicated kind, whose form and position are both in a state of continual change. 241. The lunar cycle.-The lunar cycle consists of 235 synodical revolutions of the moon, which differ from 19 years of 365I days only by about an hour and a half. For 29.5305887 x 235 = 6939.688 days. And 365- x19=6939.75 days. If, then, full moon should happen on the 1st of January in the first year of the cycle, it will happen on that day (or within a very short time of its beginning or ending) again after a lapse of 19 years; and all the full moons in the interval will occur on the same days of the month as in the preceding cycle. This period of 19 years is sometimes called the Metonic Cycle, and the year of the Metonic cycle is called the Golden Number. This cycle of 19 years is used for finding Easter. Easter day is the first Sunday after the fall moon which happens upon or next after the 21st day of March. The present lunar cycle began in 1862, when full moon occurred April 14th. Full moon also occurred on the same day of April in 1843, 1824, etc. The following are the dates of the full moons next following the vernal equinox for several lunar cycles: Year. Year. Year. Year. Full Moon. 1805 1824 1843 1862 April 14 1806 1825 1844 1863 April 3 1807 1826 1845 1864 March 22 etc. etc. etc. etc. CHAPTER IX. CENTRAL FORCES. -LAW OF GRAVITATION. -LUNAR IRREGULARITIES. 242. Curvilinear motion.-If a body at rest receive an impulse in any direction; it will, if entirely at liberty to obey that impulse, move in that direction, and with a uniform rate of motion. When a body moves in a curve line, there must then be some force which at every instant deflects it from the rectilinear course it tends to pursue in virtue of its inertia. We may then consider this motion in a curve line to arise from two forces: one a primitive im CENTRAL FORCES. 129 pulse given to the'body, which alone would have caused it to describe a straight line; the other a deflecting force, which continually urges the body toward some point out of the original line of motion. 243. Kepler's laws.-Before Newton's discovery of the law of universal gravitation, the paths in which the planets revolve about the sun had been ascertained by observation; and the following laws, discovered by Kepler, and afterward called Iepler's laws, were known to be true: 1st. The radius vector of every planet describes about the sun equal areas in equal times. 2d. The path of everyplanet is an ellipse, having the sun in one of its foci. 3d. The squares of the times of revolution are as the cubes of the mean distances from the sun, or as the cubes of the major axes of the orbits. From these facts, revealed by observation, we may deduce the law of attractive force upon which they depend. 244. Theorem.- When a body moves in a curve, acted on by a force tending to a fixed point, the areas which it describes by radii drawn to the centre of force are in a constant plane, and are proportional to the times. Let S be the centre of' -gf attraction; let the time be divided into short and // equal portions, and in the // first portion let the body describe AB. In the sec- ond portion of time, if no new force were to act upon the body, it would proceed S -- --— Cto c in the same straight line, describing Be equal / to AB. But when the body has arrived at B, let a force tending to the centre S act on it by a single instantaneous impulse, and I 130 ASTRONOMY. compel the body to continue its motion along the line BC. Draw Cc parallel to BS, and at the end of the second portion of time, the body will be found in C, in the same plane with the triangle ASB. Join SC; and because SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, and therefore also to the triangle SAB, because Be is equal to BA. In like manner, if a centripetal force toward S act impulsively at C, D, E, etc., at the end of equal successive portions of time, causing the body to describe. the straight lines CD, DE, EF, etc., these lines will all lie in the same plane, and the triangles SCD, SDE, SEF will each be equal to SAB and SBC. Therefore these triangles will be described in equal times, and will be in a con stant plane; and we shall have polygon SADS: polygon SAFS:: time in AD: time in AF. Let now the number of the portions of time in AD, AF be augmented, and their magnitude be diminished in infinitum, the perimeter ABCDEF ultimately becomes a curve line, and the force which acted impulsively at B, C, D, E, etc., becomes a force which acts continually at all points. Therefore, in this case also, we have curvilinear area SADS: curvilinear area SAFS:: time in AD: time in AF. 245. Theorem.-The velocity of a body moving in a curve and attracted to a fixed centre, is inversely as the perpendicular from the fixed centre upon the tangent to the curve. For the velocities in the polygon at two points, A, E, are as AB, EF, because these lines are described in equal portions of time. But if SY, SZ be drawn perpendicular to these lines, SY.AB = SZ.EF, because the triangles SAB, SEF are equal. Therefore velocity at A: velocity at E:: SZ: SY. And ultimately, the velocity in the polygon becomes the velocity in the curve, and the lines AY, EZ are the tangents to the curve at A and E. 246. Theorem.-If a body moves in a curve line in a constant plane, and by a radius drawn to a fixed point, describes areas about that point proportional to the times, it is urged by a central force tending to that point. Every body which moves in a curve line is deflected from a straight line by some force acting upon it. If the body were to CENTRAL FORCES. 131 describe the polygon ABCDEF, describing the equal triangles SAB, SBC, etc., in equal times, it must at B be acted on by a force directed toward S. For in AB produced, take Be equal to AB. Then the triangle ASB=BSc. But, by supposition, ASB =BSC. Therefore, BSC=BSc; and, consequently, Cc is parallel to SB. Now BC may be regarded as the resultant of two forces, one the impulse in the direction of AB produced, and the other a deflecting force Cc, which is parallel to SB; that is, the deflecting force at B is directed toward the sun. But ultimately the motion in the polygon will coincide with the motion in the curve, and the force in the polygon will be the same as the force in the curve. Therefore in the curvilinear motion the proposition is true. Now since the planets describe about the sun equal areas in equal times, it follows that the force which deflects them from a straight line is directed toward the centre of the sun. 247. Theorem. - When bodies describe different circles with uniform motions, the forces tend to the centres of the circles, and are as the squares of the velocities divided by the radii of the circles. By Art. 246 the forces tend to Fig. 6. the centres of the circles. Let AC, ac be arcs described in two different circles in equal times. Draw the tangents AB, ab; draw BC, be perpendicular to the tangents, and CD, cd parallel to them. Draw also the e chords AC, ac. Then BC, be, or AD, ad, are the spaces through which the bodies are deflected from the tangents by the action of the forces to S and s. Then AC2 AD: AC:: AC: AE; whence AD 2A'. ac2 Also ad= 2 Now when the are is taken indefinitely small, we shall have the centripetal force at A: centripetal force at a:: the square of the arc AC divided by the radius AS: the square of the arc ac divided by the radius as. But the arcs AC, ac, described in equal times, are as the velocities; hence in circles, if F represent the centripetal force, V the velocity, and R the radius of the circle, 132 ASTRONOMY. V2 v2 we shall have F:f:: -.: r' V2 or F varies as-e 248. Theorem.- When bodies describe different circles with uniform motions, the central force is as the radius of the circle divided by the square of the time of one revolution. Let R be the radius of the circle, V the velocity, and T the time of describing the whole circle. The circumference of the circle 27rR will be represented by 27rR, which equals VT; hence V = T. V2 4w2R2 r But by Art. 247, F varies as —, or T2,which varies as R- r that is, F::::t2 249. Theorem.-If a body describes an ellipse, being continually urged by a force directed toward the focus, that force must vary inversely as the square of the distance. Fig. 69. _~ t'Let APB represent u p]" Q9^ ^ the elliptic orbit of a /~~ ~ \,/^4 z\ planet, and S the fo~D,/; A // \ \^ Gcus occupied by the sun. Let PQ be an arc described by the Ls^ ~ ^~ ~ P^ E g planet in an indefi/ itely short time, t. / Draw the diameter /PG; also the ordinate ^^^c~~~~ _ ^^^~Qv parallel to the tangent at P; and let DK be the diameter which is conjugate to PG. Draw the radius vector SP, cutting the diameter DK in E, and the ordinate. Qv in x, and complete the parallelogram QxPR. Also draw QT perpendicular to SP, and PF perpendicular to D)K. If the arc PQ be taken indefinitely small, it may be considered as a straight line described by the joint action of the force which is directed toward S, and of the projectile force which acts in the direction of PR. That is, the force PQ may be resolved into the CENTRAL FORCES. 133 two forces Qx and Px. During the time, the deflecting force, if it acted alone, would cause the body to describe Px. Hence, denoting the intensity of this force by F, we have, by Mechanics, Px= Ft2; and taking t for the unit of time, we have F=2Px. By similar triangles, Px: Pv:: PE: PC. By Ellipse, Prop. XIX., Gv. Pv: Qv2:: PC2 CD2. Compounding these two proportions, we have Gv. Px: Qv2: PC. AC: CD2, since PE=AC, Geom., Ellipse, Prop. VII. But when the arc PQ is taken indefinitely small, Qv=Qx, and Gv=2PC. Gv. Px. CD2 2Px. CD" Hence QG2 PC.AC - AC (1) Again, by similar triangles, Qx: QT:: PE(=CA):PF. Also (Ellipse, Prop. XVI.), CD. PF=CG. CB, or CA: PF:: CD: CB. Hence Qx: QT:: CD: CB; QT. CD d QQT. CD CB,and = Q CB2 (2) From equations (1) and (2) we have 2Px. D2 QT2. CD2 AC - CB2 Represent half the major axis of the ellipse by a, and half the minor axis by b; then QT2. AC QT2. a 2P= CB2 - 2 (3) If now we denote the area of the elliptical sector SQP by k, we have k=~SP.QT; 2k 472 and hence QT=p, and QT2=Sp2. Substituting this value in equation (3), we have 2 4k2 a F=2Px=If we consider the action of the deflecting force at some other point of the ellipse, as P', and denote the intensity of the force by 4 we2 a F', we shall have F'-_ 2 SP'2'b2' 134 ASTRONOMY. But by Kepler's first law, k is a constant quantity; hence we have F:F'::SP2: SP2; or the deflecting force varies inversely as the square of the distance of the planet from the sun. 250. Theorem. —When several bodies revolve in ellipses about the same centre offorce, varying inversely as the square of the distance, the squares of the periodic times will vary as the cubes of the major axes. Let T denote the periodic time of a planet, expressed in seconds, and k the area described by the radius vector in one second; then the entire area of the ellipse will be represented by Tk. But this area is also represented by rab (Ellipse, Prop. XXI.). Hence o rab Tkc=rab, or k= TRepresent the distance of the planet from the sun by R; then, by the last article, 4w2a2b2 a 47r2a3 T2 b2R2 T2R2 If we represent by f the value of the deflecting force F at the distance of unity, then, by hypothesis, 1 1 ~ Cf f::F..2:; that is, F=2. f 4rr2a3 Hence T2TR2 3 47r2a3 27a7 that is T2=, or T= f vf' If T' denote the periodic time of a second planet, and a' half the major axis of its orbit, we shall have 3 - 2rra' 2 T/ V 3 3 whence T: T':: a: a', or T2: T'2:: a3 a3 Thus we perceive that Kepler's first law would hold true, whatever might be the law by which the deflecting force depended upon the distance; but the second and third laws prove that in the solar system this deflecting force varies inversely as the square of the distance. LAW OF GRAVITATION. 135 251. Modification of Kepler's third law.-Kepler's third law is strictly true only in the case of planets whose quantity of matter is inappreciable in comparison with that of the central body. In considering the motion of a planet, for instance Jupiter, round the sun, it is necessary to remember that while the sun attracts Jupiter, Jupiter also attracts the sun. The motion which the attraction of Jupiter produces in the sun, is less than the motion which the attraction of the sun produces in Jupiter, in the same ratio in which Jupiter is smaller than the sun. If the sun and Jupiter were allowed to approach one another, their rate of approach would be the sum of the motions of the sun and Jupiter, and would therefore be greater than their rate of approach if the sun were not movable, in the same ratio in which the sum of the masses of the sun and Jupiter is greater than the sun's mass. Consequently, in comparing the orbits described by different planets round the sun, we must suppose the central force to be the attraction of a mass equal to the sum of the sun and planet. If we regard the mass of the sun as unity, and represent the masses of two planets by m and in', then we shall have a3 a'3 T2: T'2 a a ~ ^ l+?'l+ m + and this proportion is rigorously true. 252. The force that retains the moon in her orbit is the same as that which causes bodies tofall near the earth's surface, theforce being diminished in proportion to the square of the distancefrom the earth's centre. Let E be the centre of the earth, A a point i. 70. on its surface, and BC a part of the moon's or- D ln B bit assumed to be circular. When the moon is G at any point, B, in her orbit, she would move on in the direction of the line BD, a tangent to the orbit at B, if she was not acted upon by some deflecting force. Let F be her place in her orbit one second of time after she was at B, and let FG be drawn parallel to BD, and FHl parallel to EB. The line FH, or its equal BG, is the distance the moon has been drawn, during one second, from the tangent toward the earth at E. If we divide the circumference of the moon's 136 ASTRONOMY. orbit by the number of seconds in the time of one revolution, we shall have the length of the arc BF. Now, by Geometry, 2BE: BF: BF: BG. But the chord BF does not differ sensibly from the arc BF, already obtained. BG is thus found, by computation, to be 0.0534 inch. At the equator, a body falls through 1921 inches in the first second. At the distance of the moon, the force of gravity (if it diminishes in proportion to the square of the distance from the earth's centre) will be found by the proportion 59.9642: 12:: 192-1: 0.0535 inch, which agrees very nearly with the distance above computed. The space through which the moon actually falls toward the earth in one second is a little less than that computed from the force of gravity at the earth's surface, because (as we shall see hereafter) the action of the sun diminishes by a small quantity the moon's gravity toward the earth. 253. Deductions from Kepler's laws.-We are thus led, by the laws of Kepler, to consider the centre of the sun as the focus of an attractive force, which extends infinitely in every direction, decreasing in the ratio of the square of the distance. The law of the proportionality of the areas described by the radius vector to the times.of description, shows that the principal force acting on the planets and comets is always directed toward the centre of the sun. The ellipticity of the planetary orbits, and the almost parabolic orbits of the comets, prove that, for each planet and comet, this force is inversely proportional to the square of the distance of the body from the sun; and from the law of the proportionality of the squares of the times of revolution to the cubes of the major axes of the orbits, it follows that this force is the same for all the planets and comets, placed at equal distances from the sun; so that, in this case, these bodies fall toward it with the same ve. locity. 254. Motion of the satellites.-If from the planets we pass to the satellites, we shall find that as the laws of Kepler are very nearly observed in the motions of the satellites about their primary planets, they ought to gravitate toward the centres of these planets in the inverse ratio of the square of their distances from those cen LAW OF GRAVITATION. 137 tres; the satellites ought likewise to gravitate toward the sun in nearly the same manner as their planets, in order that the relative motions about their primary planets may be very nearly the same as if these planets were at rest. For each system of satellites, the squares of the times of their revolutions are as the cubes of their mean distances from the centre of the planet. The satellites are, therefore, attracted toward the planets, and toward the sun, by forces inversely proportional to the squares of the distances. 255. The law of gravitation extends to all the bodies of the solar system.-Hence it follows that the sun, and the planets which have satellites, are endowed with an attractive force, extending indefinitely, decreasing inversely as the square of the distance, and including all bodies in the sphere of their activity. Moreover, since a body can not act on another without experiencing an equal and contrary reaction, and since the planets and comets are attracted toward the sun, they must, in like manner, attract that body. For the same reason, the satellites attract their planets; this attractive property is therefore common to the planets, comets, and satellites; and, consequently, we may consider the gravitation of the heavenly bodies toward each other as a general law of the universe. The law of gravitation in the inverse ratio of the square of the distance, represents with the greatest precision all the known inequalities of the motions of the heavenly bodies; and this accordance, taken in connection with the simplicity of the law, authorizes the belief that it is rigorously the law of nature. 256. Gravitation is also proportional to the masses; for if the planets and comets are supposed to be at equal distances from the sun, they would fall freely toward it through equal spaces in equal times; consequently, their gravities would be proportional to their masses. The motions of the satellites about their primary planets prove that the satellites gravitate, like the planets, toward the sun in the ratio of their masses. Hence we see that the comets, planets, and satellites, placed at the same distance from the sun, would gravitate toward it in the ratio of their masses; and since action and reaction are equal and contrary, it follows that they attract the sun in the same ratio; consequently, their actions on the sun are proportional to their masses, divided by the square of their distances from its centre. 138 ASTRONOMY. This universal gravitation is the cause of various perturbations of the motions of the heavenly bodies. For the planets and comets in obeying their mutual attractions must vary a little from the elliptical motion which they would ex'actly follow if they were attracted only by the sun. The satellites, disturbed in their motions about their planets by their mutual attractions, and by that of the sun, vary also from these laws. 257. The heavenly bodies all move in conic sections.-It was demonstrated by Newton that if a body (a planet, for instance) is impelled by a projectile force, and is continually attracted toward the sun's centre by a force varying inversely as the square of the distance, and no other forces act upon the body, the body will move in one of the following curves —a circle, an ellipse, a parabola, or an hyperbola; that is, it will move in one of the conic sections. The form of the orbit will depend upon the direction and intensity of the projectile force. ig. 71. r If we conceive F to be the centre of an attractive force, and a body at A to be / _~/; \^ projected in a direction at /UB-~ ^\ \^~ ~right angles to the line AF, then there is a certain veS ~A.~ ~~ lO~~locity of projection which \A^~^' / J ~ would cause the body to describe the circle ABC; a greater velocity would cause it to describe the ellipse ADE, or the more eccentric ellipse AGH; and if the velocity of projection be sufficient, the body will describe the semi-parabola AKL. If the velocity of projection be still greater, the body will describe an hyperbola. The curve can not be a circle unless the body be projected in a direction perpendicular to AF, and, moreover, unless the velocity with which the planet is projected is neither greater nor less than one particular velocity, determined by the length of FA and the mass of the central body. If it differs little from this particular velocity (either greater or less), the body will move in an ellipse; but if it is much greater, the body will move in a parabola or an hyperbola. MOTIONS OF PROJECTILES. 139 If the body be projected in a direction AB Fig. 72. \ oblique to SA, and the velocity of projection is small, the body will move in an ellipse; but if the velocity is great, it may move in a parabola or hyperbola, but not in a circle. If a body describe a circle, the sun is in the centre of the circle. If the body describe an ellipse, the sun is not in the centre of the ellipse, \ but in one focus. If the body describe a parabola or an hyperbola, the sun is in the focus. The planets describe ellipses which differ little from circles. A few of the comets describe very long ellipses; and nearly all the others that have been observed are found to move in curves which can not be distinguished from parabolas. There is reason to think that two or three comets which have been observed move in hyperbolas. 258. Motions oflprojectiles.-The motions of projectiles are governed by the same laws as the motions of the planets. If a body be projected in a horizontal direction from the top of a mountain, it is deflected by the attraction of the earth from the rectilinear path which it would otherwise have pursued, and made to describe a curve line which at length brings it to the earth's surface; and the greater the velocity of projection, the farther it will go before it reaches the earth's surface. We may therefore suppose the velocity to be so increased that it shall pass entirely round the earth without touching it. Let BCD represent the sur- Fig.. A face of the earth; AB, AC, AD the curve lines which a body would describe if projected horizontally from the top of //__ a high mountain, with success- ively greater and greater ve- I -r locities. Supposing there were no air to offer resistance, and the velocity were sufficiently great, the body would pass entirely round the earth, and re- ~ turn to the point from which it was projected. 140 ASTRONOMY. 259. Time of revolution near the earth's surface.-By means of Kepler's third law, we are able to compute the time required to complete a revolution in such an orbit near the earth's surface. We may regard such a body as a satellite revolving round the earth's centre in an orbit whose radius is equal to the radius of the earth, while the moon completes one revolution in 27.32 days in an orbit whose radius is 59.96 times the radius of the earth. If we put T to represent the periodic time of such a satellite, we shall have the proportion 59.963: 13:: 27.322: T2 from which we find T=0.0588 days, or lh. 24m. 35s. If the velocity of projection were too small to carry it entirely round the earth, and the impenetrability of the earth did not prevent, it would describe an ellipse, of which the earth's centre would occupy the lower focus, andit would return again to the point from which it started. This conclusion is easily reconciled with the doctrine of Mechanics, that the path of a projectile is a parabola, for it is there assumed that gravity acts in parallel directions, and that it is a constant accelerating force. These principles are sensibly true for small distances, but they are not true when great distances are considered. Problem. — ow much faster than at present must the earth rotate upon its axis, in order that bodies on its surface at the equator may lose all their gravity? Ans. 17 times. 260. Why a planet at perihelion does not fall to the sun.-Since the sun's force of attraction is greatest when the distance is least, it might seem that when a planet has reached its perihelion it must inevitably fall to the sun. The planet, however, recedes from the sun, partly on account of the increased velocity near perihelion, and partly on account of the gradual change in its direction. The curvature of any part of a planetary orbit depends not solely upon the force of the sun's attraction, but also on the velocity with which the planet is moving. The greater the velocity of the planet, the less will be the curvature of the orbit. Suppose a planet to have passed the aphelion A with so small a velocity that the sun's attraction bends the path very much, and causes it immediately to begin to approach toward the sun; the sun's attraction will increase its velocity as it moves through B, C, and D; for when the planet is at B, the sun's attractive MOTION IN AN ORBIT. 141 Fig. 7. A force acts in the direction BS; and, on account of the small inclination of BC to BS, the:B( \]: force acting in the direction of BS increases the planet's velocity. Thus the planet's velocity is continually increasing as the planet C G- moves through B, C, and D; and although, on account of the planet's nearness, the sun's D\ \s F attractive force is very much increased, and tends therefore to make the orbit more curved, yet the velocity is so much increased that the orbit is no more curved at E than it was at A; and at perihelion the velocity is so great that the planet begins immediately to recede from the sun. A similar course of reasoning will explain why, when the planet reaches its greatest distance from the sun, where the sun's attraction is least, it does not altogether fly off from the sun. As the planet passes through F, G, H, the sun's attraction, which is always directed toward S, retards the planet in its orbit, and when it has reached A its velocity is extremely small; and therefore, although the sun's attraction at A is small, yet the deflection which it produces in the planet's motion is such as to give its path the same curvature as at E. Then the planet again approaches the sun, and goes over the same orbit as before. 261. Could the rotary and orbital motions of the earth have been caused by a singleforce?-It is possible that the rotary motion of the earth, and its motion in its orbit about the sun, are both the result of a single primitive impulse. If a sphere were to receive an impulse in the direction of its centre of gravity, it would have a progressive motion without any rotation upon an axis. But if the impulse were given in any other direction, it would produce also a rotary motion. It is possible to compute at what distance from the centre of gravity an impulse must be given to produce the actual progressive and rotary motions observed in a body. In order to explain the motion of the earth in its orbit, and that of its rotation upon an axis in 24 hours, the impulse must have been given in a line passing 24 miles from the centre of the earth. 142 ASTRONOMY. 262. PROBLEMS. Prob. 1. The mean distance of the planet Hygeia from the sun is 3.14937 (the distance of the earth being taken as unity); required its periodic time? By Art. 250, a3: a3:: T2: T2; that is, 13: 3.149373:: 365.252: T2. Ans. 2041.4 days. Prob. 2. The periodic time of the planet Flora is 1193 days; required its mean distance from the sun? Ans. 2.2013. Prob. 3. What would be the periodic time of a planet revolving about the sun at a mean distance of ten million miles? Prob. 4. What would be the periodic time of a planet revolving about the sun at a mean distance of one million miles? Prob. 5. Suppose there exists a planet revolving about the sun at a mean distance of 5000 millions of miles, what must be its periodic time? Prob. 6. What would be the periodic time of a satellite revolving about the earth at a mean distance of 10,000 miles from the earth's centre? Prob. 7. Suppose the earth had a satellite making one revolution in a year, what would be its mean distance from the earth? 263. The problem of the three bodies.-When there are only two bodies that gravitate to one another with forces inversely as the squares of their distances, they move in conic sections, and describe about their common centre of gravity equal areas in equal times. But if there are three bodies, the action of any one on the other two, changes the form of their orbits, so that the determination of their motions becomes a problem of great difficulty, distinguished by the name of the problem of the three bodies. The solution of this problem, in its utmost generality, has never been effected. Under certain limitations, however, and such as are quite consistent with the condition of the heavenly bodies, it admits of being resolved. The most important of these limitations is that the force which one of the bodies exerts upon the other two is, either from the smallness of that body or its great distance, very inconsiderable, in respect of the forces which these two exert on one another. LUNAR IRREGULARITIES. 143 The force of this third body is called a disturbing force, and its effects in changing the places of the other two bodies are called the disturbances, or perturbations of the system. Though the small disturbing forces may be more than one, or though there be a great number of remote disturbing bodies, their combined effect may be computed, and therefore the problem of three bodies, under the conditions just stated, may be extended to any number. 264. How the moon's elliptic motion is disturbed.-The only body in the solar system which produces a sensible disturbing effect upon the moon is the sun; for although several of the planets sometimes come within less distances of the earth, their masses are too inconsiderable to produce any sensible disturbing effect upon the moon's motion. The mass of the sun, on the contrary, is so great, that, although the radius of the moon's orbit bears a small ratio to the sun's distance, and although lines drawn from the sun to any part of that orbit are nearly parallel, the difference between the forces exerted by the sun upon the moon and earth is quite sensible. 265. Relative attractions of the sun and earth upon the moon.-It was shown, Art. 252, that the earth draws the moon from a tangent 0.0534 inch in a second. If a similar calculation be made in relation to the orbit of the earth, it will be found that the sun draws the earth from a tangent 0.119 inch in a second. Also, the average force which the sun exerts upon the moon must be the same as that which it exerts upon the earth; that is, the sun exerts upon the moon a force 2-1 times as great as the earth does. The moon is therefore much more under the influence of the sun than of the earth. 266. Mass of the sun compared with that of the earth. —The force which the sun exerts on the earth is 21 times greater than that which the earth exerts on the moon. But the force of attraction varies inversely as the square of the distance, and the distance of the sun from the earth is about 400 times the distance of the moon. Hence, if the sun were at the same distance as the moon, his force of attraction would be the square of 400, or 160,000 times as great as it is now; that is, it would be 2- x 160,000, or 144 ASTRONOMY. 352,000 times as great as the earth's attraction, and, consequently, must have 352,000 times as much matter. The best determination of the sun's mass is considered to be 354,936. 267. How the sun's attraction acts as a disturbing force.-If the sun were at an infinite distance, the earth and moon would be attracted equally and in parallel straight lines, and, in that case, their relative motions would not be in the least disturbed. But although the distance of the sun compared with that of the moon is very great, it can not be considered infinite. The moon is alternately nearer to the sun and farther from him than the earth, and the straight line which joins her centre and that of the sun forms with the terrestrial radius vector an angle which is continually varying. Thus the sun acts unequally and in different directions on the earth and moon, and hence result inequalities in her motion, which depend on her position in respect of the sun. 268. General effect of the sun's disturbing action.-Let us suppose that the projectile motions of the earth and moon are destroyed, and that they are allowed to fall freely toward the sun. If the moon was in conjunction with the sun, it would be more attracted than the earth, and fall with greater velocity toward the sun, so that the distance of the moon from the earth would be increased in the fall. If the moon was in opposition, she would be less attracted than the earth by the sun, and would fall with a less velocity toward the sun than the earth, and the moon would be left behind by the earth, so that the distance of the moon from the earth would be increased in this case also. If the moon was in one of the quarters, then the earth and moon, being both attracted toward the centre of the sun, would approach the sun, and at the same time would necessarily approach each other, so that their distance from each other would in this case be diminished. Now whenever the action of the sun would increase their distance if they were allowed to fall toward the sun, it produces the same effect as if their gravity to each other was diminished; and whenever the action of the sun would diminish their distance, their gravity to each other is increased. Hence we conclude that the sun's action increases the gravity of the moon to the earth at the quadratures, and diminishes it at the syzygies. LUNAR IRREGULARITIES. 145 269. How to estimate the amount of the sun's disturbing force.We may estimate the amount of this disturbing force in the following manner: Fig. 5. c Let ABCD represent the orbit of the moon, with the earth at E, and let the sun be at S and the moon at M. Let the line SE be taken Dn - to represent the force with which the sun atj\ / tracts the earth; then we may determine the \ A \ magnitude of the force with which the sun acts on the moon at M by the proportion SE3 X S2.SM2: SE2:: SE: SM2 In the line MS, proSE3 duced if necessary, take MG-=SM, and it will represent the force with which the sun attracts the moon. We may suppose the force MG to result from the combined action of two forces, MF and MH (MG being the diagonal of the parallelogram MFGH), of which one, MF, is equal and parallel to ES. Now if the earth and moon were only acted I upon by the equal and parallel forces ES and I /3 MF, their relative motions would not be affected. Therefore it is only MH which disturbs this relative motion; that is, MII repG resents the quantity and direction of this disturbing force. This force, MH, may be resolved into two forces, MK, ML, the first being in the direction of the radius vector ME, and the other having the direction of a tangent to the orbit. The force MK augments or diminishes the moon's gravitation to the earth; while the force ML affects. the moon's angular motion round the earth, sometimes accelerating and sometimes retarding it. It is evident that the tangential force LM retards the moon's motion when going from A to B. If we construct a similar figure for each of the other quadrants, we shall find that the tangential force accelerates the moon's motion from D to A, and also from B to C, but retards the moon's motion when going from C to D. This force becomes zero at each of the points A, B, C, and D, and has its maximum value near the octants. K 146 ASTRONOMY. When the moon is in conjunction, the disturbing force of the sun is wholly employed in drawing the moon away from the earth; that is, in diminishing the moon's gravitation to the earth. When the moon is in opposition, the force with which the sun draws the earth is greater than that with which it draws the moon, so that the effect of the sun's attraction is to increase the distance of the moon from the earth; that is, it is the same as if the sun's force drew the moon away from the earth, or diminished the moon's gravitation to the earth. When the moon is in quadrature, the tangential force disappears, and the disturbing force is wholly employed in augmenting the moon's gravitation to the earth. The sun attracts the earth and moon equally, but not in parallel lines. If we suppose the projectile motions of the earth and moon to be destroyed, and that they are allowed to fall freely toward the sun, the earth and moon, both moving toward the centre of the sun, would approach each other, and in one second (their distance from the sun being 400 times the radius of the moon's orbit) their distance from each other would be diminished by l,-,th part of the space fallen through. Hence, if ES represents the force of the moon's gravitation to the sun, then BE will represent the augmentation of the moon's gravitation to the earth in quadratures. 270. Numerical estimate of the sun's disturbing force.-The ratio of the line ME to ES may be computed by Trigonometry when we know the distance of the sun and moon from the earth, and also the angular distance of the moon from the sun. Also the disturbing force of the sun upon the moon may be compared with the earth's attraction upon the moon by the following proportions: 1st. Disturbing force: sun's attraction on earth:: MH: ES; 2d. Sun's attraction on earth: earth's attr'n on sun:: 354,936: 1; 3d. Earth's attraction on sun: earth's attraction on moon:: EM2: ES2. Compounding these three proportions, we have Disturbing force: earth's attraction on moon:: 354,936 x MH x EM2: ES3. Since the values of MU, EM, and ES are known, we can compute the ratio of the disturbing force to the earth's attraction. Ex. 1. Compare the disturbing force of the sun upon the moon LUNAR IRREGULARITIES. 147 with the earth's attraction upon the moon at the time of conjunction, assuming the distance of the sun to be 399.32 times the distance of the moon, and the sun's mass 354,936 times that of the earth. Fig. 76. Sun's att. on moon: sun's att. on _E ___ _ _ - earth: SE2: SM2:: 1.00502: 1. Hence, Disturbing force: sun's attraction on earth:: 0.00502: 1. And, Disturbing force: earth's attraction on moon:: 354,936 x 0.00502: 399.322:: I: 89; that is, by the disturbing action of the sun at conjunction, the moon's gravity to the earth is diminished by -Ith part. Ex. 2. Compare the disturbing force of the sun upon the moon with the earth's attraction upon the moon at the time of opposition. Fig. T7. Sun's att. on moon: sun's att. on ~ b -~sS earth:: SE2 SM2::.99501: 1. Hence, Disturbing force: sun's attraction on earth:: 0.00499: 1. And, Disturbing force: earth's attraction on moon:: 354,936 x 0.00499: 399.322:: 1:90; that is, by the disturbing action of the sun at opposition, the moon's gravity to the earth is diminished by -Ith part. Ex. 3. Compare the disturbing force of the sun upon the moon with the earth's attraction upon the moon at the time of quadra. ture. Disturbing force: sun's attraction on earth:: 1: 399.32. Art. 269. Hence, Disturbing force: earth's attraction on moon:: 354,936: 399.323:: 1:179; that is, by the disturbing action of the sun at quadrature, the moon's gravity to the earth is increased by -Both part. Thus we see that at the quadratures, the gravity of the moon to the earth is increased by about the 179th part, while at the opposition and conjunction it is diminished by about twice this quantity; and, by a computation extending to every part of the orbit, it is found that the average effect is to diminish the moon's gravity by A-It/ part. In consequence of this diminution of her gravity, the moon describes her orbit at a greater distance from the earth, with a less angular velocity, and in a longer time, than if she were urged to the earth by her gravity alone. 148 ASTRONOMY. 271. The equation of the centre depends upon the eccentricity of the orbit. The eccentricity of the moon's orbit was stated in Art. 208 to be -I-th, and the greatest value of the equation of the centre is 6~ 18' 17", being more than three times that of the sun. 272. Evection.-After the equation of the centre, the most important inequality affecting the motion of the moon is that termed the Evection, the discovery of which we owe to the famous astronomer HIipparchus, in the second century before the Christian era. The evection is an inequality in the equation of the centre depending on the position of the major axis of the moon's orbit, in respect of the line drawn from the earth to the sun. 273. Cause of evection.-Any cause which at the perigee should have the effect to increase the moon's gravitation toward the earth beyond its mean, and at the apogee to diminish the moon's gravitation toward the earth, would augment the difference between the gravitation at the perigee and apogee, and, consequently, increase the eccentricity of the orbit. But any cause which at the perigee should have the effect to diminish the moon's gravitation toward the earth beyond its mean, and at the apogee to increase it, would diminish the difference between the two, and, consequently, diminish the eccentricity. Fig. 78. " Let E represent the earth, J ~~ ^ f~ ~ABCD the moon's orbit, of which A is the perigee and / o \ C the apogee, and let SS' S"S"' be the apparent orbit of the sun. If the sun be at iD|_ EB B S, so that the major axis of the moon's orbit is directed to the sun, the distance of the moon at A from the earth is less than if it moved in a circle, and the sun's disturbing force, computed as in Ex. 1, Art. 270, will be found to be less than -1th of the moon's gravity. So, also, the distance of the moon from the earth at C is greater than if it moved in a circle, and the disturbing force computed, as in Ex. 2, LUNAR IRREGULARITIES. 149 Art. 270, will be found to be greater than -clth part of the moon's gravity; that is, when the transverse axis of the moon's orbit is directed to the sun, the moon's gravity to the earth when at perigee is diminished less than the mean, and at apogee is diminished more than the mean. Hence the moon, when at perigee, is drawn away from the earth by less than the mean quantity, and when at apogee, is drawn away from the earth by more than the mean quantity. Thus the inequality between the two distances of the moon from the earth is increased; that is, the eccentricity of the moon's orbit is increased. But if the sun be at S' and the moon at A, the sun's disturbing force, computed as in Ex. 3, Art. 270, will be found to be less than T-9th part of the moon's gravity; but if the moon be at C, and the sun at S', the disturbing force of the sun will be found to be greater than T,-T-th part of the moon's gravity; that is, when the line of the apsides is in quadrature, the gravitation at the apogee is most augmented, and that at perigee is least augmented. Hence the effect of the sun's action is to diminish the inequality between the two distances of the moon from the earth at these two points; that is, to diminish the eccentricity of the orbit. Thus we find, in general, that the moon's orbit is most eccentric when the line of the apsides is in syzygy, and is least eccentric when the line of the apsides is in quadrature. The greatest value of evection is 1~ 16' 27". 274. Variation.-Another large inequality in the moon's motions is called the Variation. By comparing the moon's observed place with the place computed from the mean motion, the equation of the centre, and the evection, Tycho Brahe, in the sixteenth century, discovered that the two places did not generally agree. They agreed only at the syzygies and quadratures, and varied most in the octants, where the inequality amounted to 39' 30". 275. Cause of variation.-This inequality is occasioned by that part of the sun's disturbing force which acts in the direction of a tangent to the moon's orbit, Art. 269. This force is nothing at the syzygies and quadratures, and is greatest near the octants. It accelerates the moon's motion in going from quadrature to conjunction; and when the moon is past conjunction, the tangential force changes its direction and retards the moon's motion. 150 ASTRONOkMY 276. The annual equation is an inequality in the moon's motion arising from the variation of the sun's distance from the earth. When the earth is at perihelion, the sun's disturbing force is greater than its average value; the moon's gravity to the earth is diminished more than usual; and its velocity is therefore slower than the mean. For the same reason, at aphelion the moon's velocity is greater than the mean. The period of this inequality is one year, and its maximum effect upon the moon's longitude amounts to 11' 9' 277. Other inequalities in the moon's motion.-These three inequalities, evection, variation, and annual equation, are the largest of the inequalities in the moon's motion. The other inequalities are more minute; but, in order to represent the moon's place with the greatest possible accuracy, it is necessary to take into account a large number of corrections. The moon's place for every hour of the year is computed several years beforehand, and published in the Nautical Almanac. These places are now computed from Tables published by Professor Hansen in 1858. The average difference between the observed places of the moon and the places computed from these Tables does not exceed 3", and only once or twice in a year does the difference amount to so large a quantity as 10". 278. Cause of the retrograde motion of the moon's nodes. —The plane of the moon's orbit is inclined to the ecliptic about 5~; that is, in half of her revolution she is on the north side of the ecliptic, and in half is on the south side of the ecliptic. The sun is seldom in the plane of the moon's orbit, and his action generally has a tendency to draw the moon out of the plane in which she is moving. This oblique force may be resolved into two other forces-one lying in the plane of the ecliptic, and the other perpendicular to it. Let ENN" represent the ecliptic, and AN a Fig.9. 9 A portion of the moon's OJ^^ ^is orbit. Let the moon E C be atA, and approach~.=.-.,ing the descending KIic node N. The sun being situated in the plane EN, his attrac LUNAR IRREGULARITIES. 151 tion tends to draw the moon toward that plane. Let that part of the sun's disturbing force which is perpendicular to the plane EN be represented by AB, and suppose that in the time that the perpendicular force would cause it to describe AB, the moon, if undisturbed, would have advanced from A to D. By the combined action of these two forces, the moon will describe the diagonal AC, and cross the ecliptic in the point N'. Thus the node has shifted from N to N' in a direction contrary to that of the moon's motion, and the inclination of the orbit to the ecliptic has increased. After the moon has crossed the ecliptic, the sun's disturbing action tends to draw the moon northward toward the ecliptic. Suppose the moon to be at F, and let that part of the sun's disturbing force which is perpendicular to the ecliptic be represented by FK, while FG represents the moon's velocity in her orbit. The resultant of these two forces will be a motion in the diagonal FHI, as if the moon had come, not from N', but from N", a point still farther to the westward. Thus the node has traveled farther westward, but the inclination of the orbit to the plane of the ecliptic has diminished. Thus it appears that both in approaching the node, and in receding from it, the node shifts its place in a direction contrary to that of the moon's motion; but the inchnation of the moon's orbit increases while the moon approaches the node, and diminishes while the moon is receding from it. When the line of the nodes of the moon's orbit passes through the sun, there is no disturbing force tending to draw the moon out of the plane of its orbit; but in every other position the line of the nodes is constantly regressing, making a complete revolution in about 19 years. See Art. 240. The inclination of the plane of the orbit to the ecliptic increases and diminishes alternately. This variation is, however, confined within very narrow limits, so that there is no permanent change in the inclination of the orbit. 279. Cause of the progression of the line of the apsides.-The apsides of the moon's orbit are distant from each other more than 180~. This is caused by the disturbing action of the sun, which tends to diminish the moon's gravity to the earth. If the moon was only acted upon by the earth's attraction, she would describe an ellipse, and her angular motion from perigee to apogee would be just 1800; but when the effect of the sun's action is to dimin 152 ASTRONOMY. ish the moon's gravity, she will continually recede from the ellipse that would otherwise be described; her path will be less curved, and she must move through a greater distance before the radius vector intersects the path at right angles. She must therefore move through a greater angular distance than 180~ in going from perigee to apogee, and, consequently, the apsides must advance. On the contrary, when the moon's gravity is increased by the sun's action, her path will fall within the ellipse which she would otherwise describe; its curvature will be increased, and the distance through which she must move before the radius vector intersects her path at right angles will be less than 1800. The apsides will therefore move backward. Now it has been shown that the sun's action alternately increases and diminishes the moon's gravity to the earth. The motion of the apsides will therefore be alternately direct and.retrograde. But as the diminution of gravity has place during a much longer part of the moon's revolution, and is also greater than the increase, the direct motion will exceed the retrograde; and in one revolution of the moon, the apsides have a progressive motion of about 30, making a complete revolution in about nine years. See Art. 239. 280. Periodical and secular inequalities.-The perturbations in the elliptic movements of the planets and their satellites may be divided into two distinct classes. Those of the first class depend simply on the configurations of the planets, and complete the cycle of their values upon each successive return of the same configuration. These are called periodic inequalities. Their periods, generally speaking, are not long; and their general effect is slightly to accelerate or retard a planet in its orbit. The perturbations of the second class depend on the configuration of the nodes and perihelia. They vary with extreme slowness, requiring centuries to complete the cycle of their values, and they are hence denominated secular inequalities. Laplace has indeed demonstrated that the last-mentioned inequalities are also periodic, but the periods are much longer than those of the other inequalities, and are independent of the mutual configurations of the planets. 281. Secular acceleration of the moon's mean motion.-The mean motion of the moon exhibits a secular inequality which has become very celebrated. By comparing the results of recent ob ECLIPSES OF THE MOON. 153 servations with the Chaldean observations of eclipses at Babylon in the years 719 and 720 before the Christian era, Dr. Halley discovered that the periodic time of the moon is now sensibly shorter than at the time of the Chaldean eclipses. The mean motion of the moon increases at the rate of more than 10' in one hundred years. If this acceleration of her motion, and the consequent diminution of her distance, were perpetually to continue, it would follow that she would eventually be precipitated to the earth. But Laplace has shown that this acceleration of the moon is occasioned by the change in the eccentricity of the earth's orbit. It has been stated, Art. 113, that the eccentricity of the earth's orbit has been diminishing from the time of the earliest observations. The mean action of the sun upon the moon tends to diminish the moon's gravity to the earth, and thereby causes a diminution of her angular velocity. This diminution being once supposed to occur, the angular velocity would afterward remain constant, provided the mean solar action always retained the same value. This, however, is not the case, for it depends, to a certain extent, on the eccentricity of the earth's orbit. Now the eccentricity of the earth's orbit has been continually diminishing from the date of the earliest recorded observations down to the present time; hence the sun's mean action must also have been diminishing, and, consequently, the moon's mean motion must have been continually increasing. This acceleration will continue as long as the earth's orbit is approaching toward a circular form; but as soon as the eccentricity begins to increase, the sun's mean action will increase, and the acceleration of the moon's mean motion will be converted into a retardation. CHAPTER X. ECLIPSES OF THE MOON. 282. Cause of eclipses.-An eclipse of the sun is caused by the moon passing between the sun and the earth. It can therefore only occur when the moon is in conjunction with the sun; that is, at the time of new moon. An eclipse of the moon is caused by the earth passing between the sun and moon. It can therefore only occur when the moon is in opposition; that is, at the time of full moon. 154 ASTRONOMY. 283. Why eclipses do not occur every month.-If the moon's orbit coincided with the plane of the ecliptic, there would be an eclipse of the sun at every new moon, since the moon would pass directly between the sun and earth; and there would be an eclipse of the moon at every full moon, since the earth would be directly between the sun and moon. But since the moon's orbit is inclined to the ecliptic about 5~, an eclipse can only occur when the moon, at the time of new or full, is at or near one of its nodes. At other times, the moon is too far north or south of the ecliptic to cause an eclipse of the sun, or to be itself eclipsed. 284. Form of the earth's shadow.-Since the magnitude of the sun is far greater than that of the earth, and both bodies are of a globular form, the earth must cast a conical shadow in a direction opposite to that of the sun. Let AB represent the sun, and CD A Fig. 80. "B the earth, and let the tangent lines AC, BD be drawn, and produced to meet in F. -Then CFD will represent a section of the earth's shadow, and EF will be its axis. If the triangle AFS be supposed to revolve round the axis SF, the tangent CF will describe the convex surface of a cone, within the whole of which the light of the sun must be intercepted by the earth. 285. The semi-angle of the cone of the earth's shadow is equal to the sun's apparent semi-diameter, minus his horizontal parallax. In Fig. 80 the semi-angle of the cone of the earth's shadow is EFC or EFD. Now SEB-EFB+~EBF; that is, EFB=SEBEBD; or half the angle of the cone of the earth's shadow is equal to the sun's apparent semi-diameter, minus his horizontal parallax. Putting s for the sun's semi-diameter, and p for his horizontal parallax, we have the semi-angle of the earth's shadow, EFC ==s-po, ECLIPSES OF THE MOON. 155 286. The length of the earth's shadow varies according to the distance of the sun from the earth; its mean length being 856,200 miles, or more than three times the distance of the moonfrom the earth. In the right-angled triangle EFC, right-angled at C, CE CE sin. EFC: CE::R' EF — sin. EFC-sin. (s-p)' The mean value of the sun's apparent semi-diameter is 16' 1".8, and the sun's horizontal parallax is 8".6; hence s-p=- 15' 53".2. Also, the mean diameter of the earth =3956.6 miles. Hence the 3956.6 average length of the earth's shadow sin. 15' 53". 856,200 miles. Since the mean distance of the moon from the earth is only 238,880 miles, the shadow extends to a distance more than three times that of the moon. 287. The average breadth of the earth's shadow, at the distance of the moon, is almost three times the moon's diameter. Let M'M" represent a portion of the moon's orbit. The apparent semi-diameter of the earth's shadow at the distance of the moon is the angle MEH. But EHD-=FEIH+HFE. Hence MEH =EHD-HFE. But EIID=the moon's horizontal parallax; and HFE =the sun's semi-diameter minus his horizontal parallax (=s —p); therefore half the angle subtended by the section of the shadow is equal to the sum of the parallaxes of the sun and moon, minus the sun's semi-diameter. If we represent the moon's horizontal parallax by p', we shall have MEH =p +_p'-s. The mean value ofp' is 57' 2".3, and s-p=15' 53".2; hence pp' —s=41' 9".1. The mean value of the moon's apparent semi-diameter is 15'39".9. Hence the diameter of the shadow is almost three times the moon's diameter, and therefore the moon may be totally eclipsed for as long a time as she takes to describe about twice her own diameter. The eclipse will begin when the moon's disc at M' touches the earth's shadow, and the eclipse will end when the moon's disc touches the earth's shadow at M". 288. Lunar ecliptic limits.-There is a certain distance of the moon's node from the centre of the earth's shadow beyond which a lunar eclipse is impossible, and a certain less distance within: 156 ASTRONOMY. which an eclipse is inevitable. These distances are called the lunar ecliptic limits. The first is called the major limit, and the second the minor limit. Fig. 81. Let NE represent the ecliptic, NM the moon's orbit, and N the moon's ascending node. Let EA be the semi-diameter of the earth's shadow, and MA the semi-diameter of the moon. When the line ME, joining the centres of the moon and shadow, becomes equal to the sum of the semi-diameters, the moon will touch the earth's shadow; and if ME be less than this limit, the moon will enter the shadow, and be partially or totally eclipsed. The line NE represents that distance of the moon's node from the centre of the earth's shadow beyond which there can be no eclipse. 289. To compute the values of the ecliptic limits.-We may regard EMN as a spherical triangle, right-angled at M, in which EM represents the sum of the radii of the moon and of the earth's shadow, and N is the inclination of the moon's orbit to the ecliptic. Now, by Napier's rule, sin. EM R sin. EM=sin. EN sin. N; or sin. EN=' sin. N Since EM and N are both variable, the ecliptic limit is variable. To obtain the distance beyond which a lunar eclipse is impossible, we must employ the greatest possible value of EM, and the least possible value of N. To obtain the distance within which an eclipse is inevitable, we must employ the least possible value of EM, and the greatest possible value of N. The greatest possible value of EM is 62' 38", and the least inclination of the moon's orbit to the ecliptic is 50, from which we obtain the major limit of lunar eclipses, 12~ 4'. The least possible value of EM is 52' 20", and the greatest possible inclination of the moon's orbit to the ecliptic is 5~ 17', from which we obtain the minor limit of lunar eclipses, 9~ 30'. If, then, at the time of opposition, the moon's node is distant from the centre of the earth's shadow less than 9~ 30', or if the sun be distant from the opposite node of the moon less than 9~ 30', ECLIPSES OF THE MOON. 157 there will certainly be an eclipse of the moon; but if the sun be distant from the node of the moon's orbit more than 12~ 4', there can not be an eclipse. When the distance falls between these limits, it will be necessary to make a more minute calculation in order to determine whether there will or will not be an eclipse. 290. Different kinds of lunar eclipses. -When the moon just touches the earth's shadow, but passes by it without entering it, the circumstance is called an appulse. When a part, but not the whole of the moon enters the shadow, the eclipse is called a partial eclipse; when the moon enters entirely into the shadow, it is called a total eclipse; and if the moon's centre should pass through the centre of the shadow, it would be called a central eclipse. It is probable, however, that a strictly central eclipse of the moon has never occurred. 291. The earth's penumbra. —Long before the moon enters the cone of the earth's shadow, the earth begins to intercept from it a portion of the sun's light, so as to render the illumination of its surface sensibly more faint. This partial shadow is called the earth's penumbra. Its limits are determined by the tangent lines AD, BC produced. Throughout the space included between the lines CK and DL, the light of the sun is more or less obstructed by the earth. If a spectator were situated at L, he would see the entire disc of the sun; but between L and the line DF, he would see only a portion of the sun's surface, and the portion of the sun which was hidden would increase until he reached the line DF, beyond which the sun would be entirely hidden from view. 292. The semi-angle of the earth's penumbra is equal to the sun's apparent semi-diameter, plus his horizontal parallax. The angle KNF=BNS=BEN+NBE. But BEN is the sun's 158 ASTRONOMY. apparent semi-diameter, and NBE is the sun's horizontal parallax. Hence the semi-angle of the penumbra is represented by s -p. 293. The apparent semi-diameter of a section of the penumbra at the moon's distance is equal to the sum of the parallaxes of the sun and moon, plus the sun's semi-diameter. The angle MEr=ENm+EmN. But ENm=s+p, Art. 292. And EmN=the moon's horizonal parallax=p'. Hence MEm_=p -p' +s, which equals the apparent semi-diameter of the shadow, plus the sun's diameter. 294. Effect of the earth's atmosphere. —In obtaining the above expression for the dimensions of the earth's shadow, the shadow is assumed to be limited by those rays of the sun which are tangents to the sun and earth. It is, however, found that the observed duration of an eclipse always exceeds the duration computed on this hypothesis. This fact is accounted for in part by supposing that most of those rays which pass near the surface of the earth are absorbed by the lower strata of the atmosphere; but we must also admit that those rays of the sun which enter the atmosphere, and are so far from the surface as not to be absorbed, are refracted toward the axis of the shadow, and are thus spread over the entire extent of the geometrical shadow, thereby diminishing the darkness, but increasing the diameter of the shadow, and, consequently, the duration of the eclipse. In consequence of the gradual diminution of the moon's light as it enters the penumbra, it is difficult to determine with accuracy the instant when the moon enters the dark shadow; and astronomers have differed as to the amount of correction that should be made for the effect of the earth's atmosphere. It is generally found necessary, however, to increase the computed diameter of the shadow by about,-th part. 295. loon visible when entirely immersed in the earth's shadow.When the moon is totally immersed in the earth's shadow, she does not, except on some rare occasions, become invisible, but assumes a dull reddish hue, somewhat of the color of tarnished copper. This arises from the refraction of the sun's rays in passing through the earth's atmosphere, as explained in the preceding ECLIPSES OF THE MOON. 159 Article. Those rays from the sun which enter the atmosphere, and are so far from the surface as not to be absorbed, are bent toward the axis of the shadow, and fall upon the moon, causing sufficient illumination to render the disc distinctly visible. 296. Computation of lunar eclipses.-By the solar tables we may ascertain the apparent position of the centre of the sun from hour to hour, and hence we may learn the position of the centre of the earth's shadow. From the lunar tables we ascertain, in the same manner, the position of the moon's centre from hour to hour. The eclipse will begin when the distance between the centre of the moon and that of the shadow is equal to the sum of the apparent semi-diameters of the moon's disc and the shadow; the middle of the eclipse will occur when this distance is least; and the eclipse will end when the distance between the centres is again equal to the sum of the apparent semi-diameters. The Nautical Almanac for each year furnishes the places of the sun for every day of the year, as computed from the solar tables, and the places of the moon are given for every hour of the year. With this assistance, it is easy to compute the times of beginning and end of an eclipse. 297. Construction of the diagram.-First find the time of opposition, or the time of full moon. For this time compute the declination, horizontal parallax, and semi-diameter both of the sun and moon; also the hourly motion of the moon from the sun both in right ascension and declination. Let C represent the Fig. 83. centre of the earth's / shadow. Draw the line D ACB parallel to the / - equator, and DCM perpendicular to it. Select a convenient scale of __-_ equal parts, and from it A take CG, equal to the - i moon's declination, minus the declination of the centre of the shadow, and set it on CD 160 ASTRONOMY. from C to G, above the line AB, if the centre of the moon is north of the centre of the shadow, but below if south. Take CO, equal to the hourly motion of the moon from the sun in right ascension, reduced to the arc of a great circle, and set it on the line CB, to the right of C. Take CP, equal to the moon's hourly motion from the sun in declination, and set it on the line CD from C to P, above the line AB, if the moon is moving northward with respect to the shadow, but below if moving southward. Join the points O and P. The line OP will represent the hourly motion of the moon from the sun; and parallel to it, through G, draw NGL, which will represent the relative orbit of the moon, the earth's shadow being supposed stationary. On this line are to be marked the places of the moon before and after opposition, by means of the hourly motion OP, in such a manner that the moment of opposition may fall exactly upon the point G. 298. To determine the beginning and end of the eclipse.-The semidiameter of the earth's shadow is equal to the horizontal parallax of the moon, plus that of the sun, minus the sun's semi-diameter, which result must be increased by Ioth part, on account of the earth's atmosphere. With this radius, describe the circle ADB about the centre C. Add the moon's semi-diameter to the radius CB, and, with this sum for a radius, describe about the centre C a circle, which, if there be an eclipse, will cut NL in two points, E and H, representing, respectively the places of the moon's centre at the beginning and end of the eclipse. Draw the line CKR perpendicular to LN, and cutting it in K. The hours and minutes marked on the line LN, at the points E, K, and H, will represent respectively the times of the beginning of the eclipse, middle of the eclipse, and end of the eclipse. If the circle does not intersect NL, there will be no eclipse. With a radius equal to the moon's semi-diameter, describe a circle about each of the centres E, H, and K. If the eclipse is total, the whole of the circle about K will fall within ARB; but if part of the circle falls without ARB, the eclipse will be partial. In either case, the magnitude of the eclipse will be represented by the ratio of the obscured part Rl to the moon's diameter. When the eclipse is total, the beginning and end of total darkness may be found by taking a radius equal to CB, diminished by the moon's semi-diameter, and describing with it round the centre C a circle cutting LN in two points, ECLIPSES OF THE MOON. 161 representing respectively the places of the moon's centre at the beginning and end of total darkness. Example 1. 299. Required the times of beginning, end, etc., of the eclipse of the moon, March 30, 1866, at Greenwich. By the Nautical Almanac, the Greenwich mean time of opposition in right ascension is, March 30, 16h. 39m. 18.9s. Corresponding to this time, the Nautical Almanac furnishes the following elements: Declination of the moon -... S. 4~ 12' 55".5 Declination of the earth's shadow - - S. 4 3 42.3 Moon's equatorial horizontal parallax - - 54 28.1 Sun's horizontal parallax - -. 8.6 Moon's semi-diameter --- 14 52.0 Sun's semi-diameter -- 16 2.2 Moon's hourly motion in right ascension - 28 48.0 Sun's hourly motion in right ascension - 2 16.4 Hourly motion of moon in declination - S. 9 14.1 Hourly motion of shadow in declination - S. 58.1 The figure of the earth being spheroidal, that of the shadow will deviate a little from a circle, so that instead of the equatorial horizontal parallax, we should employ the horizontal parallax belonging to the mean latitude of 45~. The reduction for latitude, A4L L~~ 162 ASTRONOMY. by Table VIII., is 5".4, so that the moon's reduced parallax is 54' 22".7. Then, to obtain CB, the semi-diameter of the earth's shadow, we have 54' 22".7+8".6-16' 2".2, which is equal to 38' 29".1. Increasing this by -th part of itself, or 38".5, we have 39' 7".6=CB; to which adding the moon's semi-diameter, we obtain CE=53' 59".6. From the centre C, with a radius CB, taken from a convenient scale of equal parts, describe the circle ARB, representing the earth's shadow. Draw the line ACB to represent a parallel to the equator, and make CG perpendicular to it, equal to 9' 13".2, which is the moon's declination, minus the declination of the centre of the shadow; the point G being taken below C, because the centre of the moon is south of the centre of the shadow. The hourly motion of the moon from the sun in right ascension is 26' 31".6, which must be reduced to the arc of a great circle by multiplying it by the cosine of the moon's declination, 4~ 12' 55", Art. 152, thus: 26' 31".6=1591".6= 3.201834 cos. dec. 4~ 12' 55" =9.998824 Reduced hourly motion =1587".3 =3.200658 Make CO = 1587".3, and CP, perpendicular to it, equal to 8" 16".0, which is the hourly motion of the moon from the shadow in declination, the point P being placed below C, because the moon was moving southward with respect to the shadow. Join OP; and parallel to it, through G, draw the line NGL, which represents the path of the moon with respect to the shadow. On NL let fall the perpendicular CK. Now at 16h. 39m. 18.9s. the moon's centre was at G. To find X, the place of the moon's centre at 16h., we must institute the proportion 60m.: 39m. 18.9s.:: OP: GX; which distance, set on the line GN, to the right of G, reaches to the point X, where the hour, 16h. preceding the full moon, is to be marked. Take the line OP, and lay it from 16h., toward the right hand, to 15h., and successively toward the left to 17h., 18h., etc. Subdivide these lines into 60 equal parts, representing minutes, if the scale will permit; and the times corresponding to the points E, K, and HI will represent respectively the beginning of the eclipse, 14h. 38m.; the middle of the eclipse, 16h. 33m.; and the end of the eclipse, 18h. 28m. If the results obtained by this method are not thought to be sufficiently accurate, we may institute a rigorous computation. ECLIPSES OF THE MOON. 163 Computation of the Eclipse. 300. The phases of the eclipse may be accurately computed in the following manner: In the right-angled triangle OCP, we have given CO=1587".3, and CP=496".0, to find OP and the angle CPO, thus: CP: R:: CO: tang. CPO. CO 1587.3 = 3.200658 CP= 496.0=2.695482 CPO =72 38' 49" tang. =0.505176 Also, sin. CPO: R:: CO: OP. CO 3.200658 sin. CPO= 9.979769 OP= 1663".0 3.220889 301. Beginning, middle, and end of the eclipse.-The middle of the eclipse is found by means of the triangle CGK, which is similar to CPO, because EG and OP are parallel, and CK is perpendicular to PO. Hence the angle CGK= 72~ 38' 49"; and GG, the difference of declination between the moon and the centre of the shadow=9' 13".2=553".2. To find CK and KG, we have the proportions R: CG:: sin. CGK: K:: cos. CGK: GK. sin. CGK= 9.979769 cos. GK = 9.474593 CG =2.742882 CG 2.742882 CK =528".0=2.722651 GK= 165".0 =2.217475 Then, to find the time of describing GK, we say, As OP (1663".0) is to GK (165".0), so is 1 hour to the time (357.2s.), 5m. 57.2s., between the middle of the eclipse and the time of opposition in right ascension, 16h. 39m. 18.9s., which gives the time of middle of the eclipse 16h. 33m. 21.7s. Now, in the triangle CKE, we have the hypothenuse CE=53' 59".6=3239".6, and CGK528".0, to find KE, thus: IKE =CE2-CK2 = CE + CKx CE- CK= 3196".3. To find the time of describing KE, we form the proportion 1663".0: 3196".3:: 3600s.: 6919.3s. =lh. 55m. 19.3s.; which, subtracted from 16h. 33m. 21.7s., the time of middle, gives 14h. 38m. 2.4s. for the begiinnig of the eclipse; and, added to the time of middle, gives for the end of the eclipse 18h. 28m. 41.0s. 164 ASTRONOMY. 302. Magnitude of the eclipse.-Subtracting CK, 8' 48".0, from CR, 39' 7".6, we have KR, 30' 19".6; to which adding KI, 14' 52".0, we obtain RI, 45' 11".6. Dividing this by the moon's diameter, 29' 44".0, we obtain the magnitude of the eclipse, 1.520 (the moon's diameter being unity); and the eclipse takes place on the moon's north limb. The magnitude of an eclipse is sometimes expressed in digits, or twelfths of the moon's diameter. In the present instance, the eclipse amounts to 18 digits. 303. Beginning and end of total darkness.-The beginning and end of total darkness may be found in the same manner. With a radius equal to CB, diminished by the moon's semi-diameter (that is, 39' 7".6-14' 52".0, which equals 24' 15".6, or 1455".6), describe about the centre C a circle cutting LN in the points S and T, which will represent the places of the moon's centre at the beginning and end of total darkness. In the triangle CKS, CS=1455".6, and CK=528".0. Hence KS = 1455.62-528.02= 1356".5. Then, to find the time of describing KS, we say, 1663".0: 1356".5:: 3600s.: 2936.4s. =48m. 56.4s.; which, being subtracted from 16h. 33m. 21.7s., gives the beginning of total darkness 15h. 44m. 25.3s.; and, being added to the time of middle, gives for the end of total darkness 17h. 22m. 18.1s. 304. Contacts with the penumbra.-The contacts with the penumbra may be found in a similar manner. The semi-diameter of the penumbra is equal to the semi-diameter of the shadow, plus the sun's diameter, or 39' 7".6+32' 4".4=71' 12".0. If we take the circle ARB, Fig. 84, to represent the limits of the penumbra, CE will be equal to 71' 12".0+14' 52".0=86' 4".0. Then, in the triangle CKE, we have given CE=5164".0, and CK- 528".0. Hence KE- =51642- 5282 5136".9. To find the time of describing KE, we say, 1663".0: 5136".9:: 3600s.: 11120.3s. 3h. 5m. 20.3s.; which, being subtracted from 16h. 33m. 21.7s., gives the first contact with the penumbra at 13h. 28m. 1.4s.; and, being added to the time of middle, gives for the last contact with the penumbra 19h. 38m. 42.0s. ECLIPSES OF THE MOON. 165 305. Results. —The results thus obtained are as follows: First contact with the penumbra at 13h. 28m. 1.4s. First contact with the umbra - -14 38 2.5 Beginning of total eclipse - - 15 44 25.3 Meantime Middle of the eclipse - - - - -16 33 21.7 at End of total eclipse - - - 17 22 18.1 Greenwicho Last contact with the umbra - -18 28 40.9 Last contact with the penumbra -19 38 42.0 Magnitude of the eclipse, 1.520 on the northern limb. 306. Timesfor any other meridian.-To obtain the times for any other place, we have only to add or subtract the longitude. For New Haven, whose longitude is 4h. 51m. 41.6s. west of Greenwich, the times will accordingly be First contact with the penumbra at 8h. 36m. 20s. First contact with the umbra - 9 46 21 Beginning of total eclipse - - - 10 52 44 Mean time Middle of the eclipse - - 11 41 40 at End of total eclipse - -- 12 30 37 New Haven. Last contact with the umbra - - 13 36 59 Last contact with the penumbra - 14 47 0 J Ex. 2. Compute the phases of the eclipse of March 19,1867, for New York city, longitude 4h. 56m. 0.2s. west of Greenwich, from the following elements: Greenwich mean time of opposition in right ascension..- 20h. 28m. 30.9s. Declination of the moon - - - -N. 0 50' 56".0 Declination of the earth's shadow - - N. 0 17 4.5 Moon's equatorial horizontal parallax - - 57 2.7 Sun's horizontal parallax ----- 8.6 Moon's semi-diameter -i- 15 34.2 Sun's semi-diameter.- 16 5.3 Moon's hourly motion in right ascension - 31 25.4 Sun's hourly motion in right ascension -- 2 16.5 Hourly motion of moon in declination - - S. 10 19.7 Hourly motion of shadow in declination - S. 0 59.2 Ans. First contact with the penumbra at 13h. 9.3m.' First contact with the umbra - - 14 20.0 I Mean time Middle of the eclipse --- 15 52.8 \ at Last contact with the umbra -- - 17 25.6 New York. Lasl contact with the penumbra - - 18 36.3 166 ASTRONOMY. CHAPTER XI. ECLIPSES OF THE SUN. 307. Length of the moon's shadow.-The length of the moon's shadow is about equal to the distance of the moon from the earth, being alternately a little greater and a little less. Suppose the moon at conjunction to be at one of her nodes. Her centre will then be in the plane of the ecliptic, and in the straight line passing through the centres of the sun and earth. Ar Fig. 85. Let ASB be a section of the sun, KEL that of the earth, and CMD that of the moon interposed directly between them. Draw AC, BD, tangents to the sun and moon, and produce these lines to meet in V. Then V is the vertex of the moon's shadow; and these lines represent the outlines of a cone, whose base is AB, and whose vertex is V. The angle SMB = MVB + MBV; hence MVB = SMB -MBV. But SMB: SEB:: SE: SM (Art. 111):: 400: 399; 400 therefore SMB =39 SEB. Now SEB, the sun's mean semi-diameter as seen from the earth =16' 1".8; hence SMB=16' 4".2, which is the sun's semi-diameter as seen from the moon. Put p=the sun's horizontal parallax. p'=the moon's horizontal parallax. s'=the moon's semi-diameter. Since the parallaxes of bodies at different distances are inversely as the distances, Art. 136, we shall have': p:: SE: ME, or p'-p: p:: SM: ME. ECLIPSES OF THE SUN. 167 But since the apparent diameters of the same body at different distances are inversely as the distances, Art. 111, we shall have SM: ME:: s' MBC; hence MBC = -s P -P Now p, p', and s' are known quantities; hence MBC=2".3, which is the sun's horizontal parallax as observed from the moon. Hence MVB, the semi-angle of the cone of the moon's shadow, equals 16' 4".2-2".3=16' 1".9. Then sin. 16' 1".9: DM (1080 miles):: rad.: MV = 231,590 miles. But the mean distance of the moon from the earth's centre is 238,883 miles. Hence, when the moon is at the mean distance from the earth, her shadow will not quite reach to the earth's surface. When the earth is at its greatest distance from the sun, the sun's apparent semi-diameter is 15' 45".5; and the angle MVB= 15' 45".6. In this case MV-235,582 miles. Now when the moon is nearest the earth, her distance from the centre of the earth is only 221,436 miles. Hence, when the moon is nearest to us, and her shadow is the longest, the shadow extends 14,000 miles beyond the earth's centre, or about three and a half times the earth's radius; and there must be a total eclipse of the sun at all places within this shadow. 308. Breadth of the moon's shadow at the earth.-The greatest breadth of the moon's shadow, at the earth, when it falls perpendicularly on the surface, is about 166 miles. In the triangle FEV, FE: EV:: sin. FVE: sin. VFE. But when the moon is nearest, and the shadow is the longest, EV-14,146 miles; and the angle FVE=15' 45".6. Also, FE =3956.6 miles. In this case VFE 56' 20".9. But the angle FEG = VFE+FVE = 56' 20".9 + 15' 45".6 = 1 12' 6".5 arc FG. Hence the arc FH=2~ 24' 13"; and if we allow 69 miles to a degree, the breadth of the moon's shadow is 166 miles, nearly. When the moon is at some distance from the node, the shadow falls obliquely on the earth, and its greatest breadth will evident ly be increased. 168 ASTRONOMY. 309. Breadth of the moon's penumbra at the earth.-The greatest breadth of the moon's penumbra at the earth's surface, when it falls perpendicularly on the surface, is about 4800 miles. If we draw the tangent lines AD, BC, and produce them to meet the earth, the sun's rays will be partially excluded from the space included between DV and DL, as also between CV and CK. Any point on the line CK will receive light from all points of the sun's disc. As the point advances toward CV, it will receive less and less of the sun's light, since a larger portion of the moon, M, will be interposed between it and the sun. At the boundary CV, all the rays of the sun are intercepted. This space, KCV, from which the sun's light is partially intercepted, is called the moon's penumbra. The semi-angle of the penumbra CIM= SCB + CSM, of which SCB is the sun's apparent semi-diameter at the moon, and CSM is the sun's horizontal parallax at the moon. The breadth of the penumbra will be greatest when the moon's distance from the earth is greatest, and the sun's distance is least. The sun's greatest apparent semi-diameter at the moon is 16' 20".2. Hence CIM 16' 22".5. In the triangle IKM, the angle CKM, when least, is 14' 41".0; and KM, when greatest, is 249,307 miles. Then sin. CIM: sin. CKMI:: KM: TIM 223,552 miles. Hence IE=476,815 miles. Then, in the triangle IEK, EK: IE:: sin. EIK: sin. EKI=144~ 58' 10". Hence EKN=350 1' 50". The angle KEI=EKN-EIK=34~ 45' 28"=the arc KG. Hence the entire arc KL=69~ 30' 56"; and if we allow 691 miles to a degree, the breadth of the penumbra is 4808 miles, nearly. 310. Velocity of the moon's shadow over the earth. —The moon advances eastward among the stars about 30' per hour more than the sun; and 30' of the moon's orbit is about 2070 miles, which therefore we may consider as the hourly velocity with which the moon's shadow passes over the earth, or at least over that part of it on which the shadow falls perpendicularly; in every other place the velocity will be increased in the ratio of the sine of the angle which MV makes with the surface, in the direction of its ECLIPSES OF THE SUN. 169, motion, to radius. But the earth's rotation upon its axis will also affect the apparent velocity of the shadow, and, consequently, the duration of the eclipse at any point of the earth. If the point be moving in the direction of the shadow, its velocity in respect to that point will be diminished, and, consequently, the time in which the shadow passes over that point will be increased; but if the point be moving in a direction contrary to that of the shadow, as may happen at places within the polar circle, the relative velocity of the shadow will be increased, and the time diminished. 311. Different kinds of eclipses of the sun.-A partial eclipse of the sun is one in which a part, but not the whole, of the sun is obscured. A total eclipse is one in which the sun is entirely obscured. It must occur at all those places on which the moon's shadow falls. A central eclipse is one in which the axis of the moon's shadow, or the axis produced, passes through a given place. An annular eclipse is one in which a part of the sun's disc is seen as a ring surrounding the moon. The apparent discs of the sun and moon, though nearly equal, are subject to small varia- Fig. 6. tions, corresponding to their variations of distance, in consequence of which the disc of the moon is sometimes a little greater, and sometimes a little - less than that of the sun. If the centres of the sun and moon coincide, and the disc of the moon be less than that of the sun, the moon will cover the central portion of the -_ - sun, but will leave uncovered around it a regular ring or annulus, as shown in Fig. 86. This is called an annular eclipse. 312. Duration of total and annular eclipses.-The greatest value of the apparent radius of the moon, as seen from the earth's centre, is 1006", which may be increased by the moon's elevation above the horizon, Art. 215, to 1024"; and the least value of the radius of the sun is 945". Their difference is 79". The greatest 170 ASTRONOMY. possible duration of a total solar eclipse will be the time required for the centre of the moon to gain upon that of the sun twice 79", or 158", which would be about 5m. if the earth did not rotate upon an axis; but, allowing for the earth's rotation, the greatest possible time during which the sun can be totally obscured is 7m. 58s. This will be the duration at the equator. In the latitude of Paris, the greatest possible duration of a total eclipse is 6m. 10s. The greatest apparent radius of the sun being 978", and the least apparent radius of the moon being 881", the greatest possible breadth of the annulus, when the eclipse is central, is 97". The greatest interval during which the eclipse can continue annular is the time required for the centre of the moon to gain upon that of the sun twice 97", or 194", which would be about 7m. if the earth did not rotate; but, by the earth's rotation, this quantity may be increased to 12m. 24s. at the equator. In the latitude of Paris, the greatest possible duration of an annular eclipse is 9m. 56s. Since the visual directions of the centres of the sun and moon vary with the position of the observer on the earth's surface, an eclipse which is total at one place may be partial at another, while at other places no eclipse whatever may occur. Since the moon's apparent diameter increases as her elevation above the horizon increases, it sometimes happens, when the apparent diameters of the sun and moon are very nearly equal, that the apparent diameter of the moon, when near the horizon, is a little less than that of the sun, but becomes a little greater than that of the sun as it approaches the meridian; that is, an eclipse which is annMlar at places where it occurs near sunrise, may be total at places where it occurs near midday. 313. To compute the values of the solar ecliptic limits.-No eclipse of the sun can take place unless some part of the globe of the moon pass within the lines AC and BD, which touch externally A- ~ Fig. 87. X _ ECLIPSES OF THE SUN. 171 the globes of the sun and earth. The apparent distance, MES, of the moon's centre from the ecliptic at this limit is equal to AEF+AES+FEM. But AEF=EFC-EAC. Hence MES= EFC -EAC + AES+ FEM =p' -p +s'+ s; that is, the sum of the apparent semi-diameters of the sun and moon, plus the difference of their horizontal parallaxes. Taking the greatest and least values of these quantities, we obtain the major limit of MES=10 34' 14", and the minor limit=l1 24' 19". Computing the corresponding distances from the moon's node, as in Art. 289, we find that if, at the time of conjunction, the sun's distance from the moon's node is more than 18~ 20', an eclipse is impossible; and if its distance from the node is less than 15~ 25', an eclipse is inevitable. Between these limits an eclipse may or may not occur, according to the magnitude of the parallaxes and apparent diameters. Since, then, an eclipse can only take place within a few degrees of the moon's node, and the sun passes the two nodes of the moon at opposite seasons of the year, it is evident that if an eclipse occurs in January, one or more eclipses may be expected in July; but no eclipse, either of the sun or moon, could possibly happen in April or October of the same year. 314. Number of eclipses in a year.-There may be seven eclipses in a year, and can not be less than two. When there are seven, five of them are of the sun and two of the moon; when there are but two, they are both of the sun. A solar eclipse is inevitable if conjunction takes place within 15~ 25' on either side of the moon's node, comprehending an arc of longitude of 30~ 50'. Now, during a synodic revolution of the moon, the sun's mean motion in longitude is 29~ 6', and in this time the moon's nodes move backward 1~ 31'. Hence the sun's motion with reference to the moon's node, in one lunation, is 30~ 37', which is less than 30~ 50'. Hence at least one solar eclipse must occur near each node of the moon's orbit, and therefore there must be at least two solar eclipses annually. But it may happen that two solar eclipses shall occur near each node, and also one lunar eclipse; and this will happen if opposition takes place very near the moon's node. In this case the moon will be almost centrally eclipsed; and since the sun's motion in reference to the 172 ASTRONOMY. node during half a lunation is only 15~ 18', it is evident that, both at the previous and following new moons, the sun may be within the ecliptic limits from the node, and may therefore be eclipsed at each of these new moons. At the full moon, which occurs in a little less than six months after the former, the sun will be near the other node of the moon's orbit. Consequently, there must be a large eclipse of the moon, and there may be an eclipse of the sun both at the previous and following new moons. At the new moon which occurs five and a half lunations after this latter full moon, and therefore a little before the close of the year, the sun will be near the node again, and must therefore be eclipsed. Thus there may be two eclipses of the moon and five of the sun within a period of twelve months, and these may all be embraced in one calendar year. 315. In the space of eighteen years there are usually about 70 eclipses, 29 of the moon and 41 of the sun. These numbers are nearly in the ratio of two to three. Nevertheless, more lunar than solar eclipses are visible in any particular place, because a lunar eclipse is visible to an entire hemisphere, while a solar is only visible to a part. The next eclipse of the sun, which will be total in any part of the United States, will occur August 7, 1869, and will be total in Virginia. The next annular eclipse will occur in 1875, and will be annular in Massachusetts. See the list of eclipses, page 325. 316. Period of eclipses. —At the expiration of a period of 223 lunations, or about 18 years and 10 days, eclipses, both of the sun and moon, return again in nearly the same order as during that period. The time from one new moon to another is 29.53 days, and, consequently, 223 lunations include 6585.32 days. The mean period in which the sun moves from one of the moon's nodes to the same node again is 346.62 days, because the node shifts its place to the westward 19~ 35' per annum. This period is called the synodical revolution of the moon's node. Now 19 synodical revolutions of the node embrace a period of 6585.78 days. Hence, whatever may be the distance of the sun from one of the moon's nodes at any new or full moon, he must, at the end of 223 lunations, be nearly at the same distance from the same ECLIPSES OF THE SUN. 173 node. Hence, after a period of 6585.32 days (which is 18 years 11 days when {here are four bissextile years in the period, or 18 years 10- days when there are five), eclipses must occur again in nearly the same order as during that period. This period was known to the Chaldaan astronomers. It was by them called the Saros, and was used in predicting eclipses. On page 327 is given a list of eclipses, which will illustrate the period of the Saros, and also show that seven eclipses may occur within a period of twelve months. 317. Occultations.-When the moon passes between the earth and a star or planet, she must, during the passage, render the body invisible to some parts of the earth. This phenomenon is called an occultation of the star or planet. The moon, in her monthly course, occults every star which is included in a zone extending to a quarter of a degree on each side of the apparent path of her centre. From new moon to full, the moon moves with the dark edge foremost; and from full moon to new, it moves with the bright edge foremost. During the former period, stars disappear at the dark edge, and reappear at the bright edge; while during the latter period they disappear at the bright edge, and reappear at the dark edge. The disappearance of a star at the dark limb is very sudden and startling, the star appearing to be instantly annihilated at a point of the sky where nothing is seen to interfere with it. 318. Darkness attending a total eclipse of the sun.-During a total eclipse of the sun, the darkness is generally so great as to render the brighter stars and planets visible. Each of the five brighter planets has been repeatedly seen during the total obscuration of the sun; all the stars of the first magnitude have in turn been seen, and, on some occasions, a few stars of the second magnitude have been detected. During a total eclipse, the degree of dark. ness is therefore somewhat less than that which prevails at night in presence of a full moon; but the darkness appears much greater than this, on account of the sudden transition from day to night. This darkness, however, has little resemblance to the usual darkness of the night, but is attended by an unnatural gloom, which is sometimes tinged with green, sometimes red, and some 174 ASTRONOMY. times a yellowish-crimson. The color of the sky changes from its usual azure blue to a livid purple or violet tint.'The color of surrounding objects becomes yellowish, or of a light olive or greenish tinge; and the figures of persons assume an unearthly, cadaverous aspect. 319. Moon sometimes visible in an eclipse of the sun.-During a total eclipse of the sun, the moon's surface is sometimes faintly illumined by a purplish-gray light, spreading over every part of the disc, so that the light of the disc is quite noticeable to the naked eye. In the eclipse of May 3, 1733, lunar spots were distinctly observed by Vassenius at Gottenberg. This effect is produced by the sun's light reflected from the earth to the moon; for the side of the earth which at such times is presented to the moon is wholly illumined by the sun, and the light of the earth is about 14 times that of the full moon. 320. Bright points on the moon's disc.-During the total eclipse of June 24, 1778, about a minute and a quarter before the sun began to emerge from behind the moon's disc, Ulloa discovered, near the northwest part of the moon's limb, a small point of light, estimated as equal to a star of the fourth magnitude. This point gradually increased, and became equal to a star of the second magnitude, when it united with the edge of the sun, which at that instant emerged from behind the moon. This phenomenon was doubtless due to the sun's rays shining through a deep valley on the moon's limb, and the long continuance of this light was due to the moon's motion being nearly parallel to that portion of the sun's circumference. A similar phenomenon was seen by M. Valz, of Marseilles, during the eclipse of July 8, 1842. Again the same phenomenon was seen during the eclipse of July 18, 1860, in Algeria, by two French observers, one with the naked eye, and the other with a telescope. The bright point gradually increased, until it blended with the light of the sun's disc as it emerged from behind the moon. During the eclipse of May 15, 1836, about 25 seconds before the middle of the eclipse, Professor Bessel, with the Konigsberg heliometer, observed a faint point of light near the edge of the moon's limb. The point became brighter, and other similar points ap ECLIPSES OF THE SUN. 175 peared beside it, which soon united, and in this manner rendered visible the whole of the moon's border between the extremities of the sun's cusps. Analogous phenomena have been observed in the occultation of stars by the moon. When a star just grazes the northern or southern limb of the moon, it sometimes disappears behind a lunar mountain, and reappears through an adjacent valley, to disappear again behind the next mountain. Several such disappearances and reappearances have been observed within an interval of a few minutes. 321. ATe corona.-During the total obscuration of the sun, the dark body of the moon appears surrounded by a ring of light called the corona. This ring is of variable extent, and resembles the " glory" with which painters encircle the heads of saints. It is brightest next to the moon's limb, and gradually fades to a distance equal to one third of her diameter, when it becomes confounded with the general tint of the heavens. Sometimes its breadth is nearly equal to that of the moon's diameter. The corona generally begins 5 or 6 seconds before the total obscuration of the sun, and continues a few seconds after the sun's reappearance. Sometimes the corona is distinctly seen at places where the eclipse is not quite total. The color of the corona has been variously described. Sometimes it is compared to the color of tarnished silver. Sometimes it is described as of a pearl white; sometimes of a pale yellow; sometimes of a golden hue; sometimes peach-colored, and sometimes reddish. The intensity of the light of the corona is sometimes such that the eye is scarcely able to support it; but generally it is described as precisely similar to that of the moon. The corona generally presents somewhat of a radiated appearance. Sometimes these rays are very strongly marked; and long beams have occasionally been traced to a distance of 3~ or 4~ from the moon's limb. 322. Cause of the corona.-Some have maintained that this corona is caused by the diffraction of the sun's light in its passage near the edge of the moon. But the diffracted light, surrounding an opaque circular disc, consists of concentric rings exhibiting a 176 ASTRONOMY. regular succession of colors-pale blue, yellow, and red. If the corona seen in solar eclipses were due to diffraction, it ought to exhibit a series of concentric colored rings, like those seen surrounding the moon when obscured by a thin haze. Such is not the appearance actually observed. It is more probable that this corona is due to an atmosphere surrounding the sun, extending to a height of several thousand miles above its disc, and reflecting a portion of the sun's light. The radiated appearance of the corona is probably analogous to the rays which are frequently seen in the western sky after sunset, and which are caused by the shadows of clouds situated near, or perhaps below our visible horizon. In like manner, the clouds which float in the solar atmosphere intercept a portion of the light of the sun's disc, and the space behind them is less bright than that portion of space which is illumined by the unobstructed rays of the sun. 323. Baily's beads.-When, in the progress of the eclipse, the sun's disc has been reduced to a thin crescent, this crescent often rig. ss. appears as a band of brilliant points, separated by dark spaces, giving it the ap-;^jf \^Sl ~pearance of a string of brilliant beads. The same peculiarity is noticed in annular eclipses a few seconds previous to the formation, and again a few seconds previous to the rupture, of the annulus. This phenomenon was first clearly de-'~. ^_ I ~ scribed by Sir Francis Baily on occasion of the annular eclipse of May 15,1836, and it has hence acquired the name of Baily's beads. This appearance is generally ascribed to the inequalities of the moon's surface. The outline of the moon's disc is, not a perfect circle, but is full of notches; and these inequalities are easily seen when the moon's disc is projected upon that of the sun. Just before the commencement of the total eclipse, the tops of the lunar mountains extend to the edge of the sun's disc, but still permit the sun's light to glimmer through the hollows between the mountain ridges. These appearances are materially modified by the color of the glass through which the observations are made. They are most ECLIPSES OF THE SUN. 177 conspicuous through a red glass, and through certain colored glasses are scarcely noticed at all. This peculiarity is probably due to the unequal penetrating power'of the differently colored rays of the sun. The red rays of the sun are less readily absorbed than any other rays of the spectrum; and a glass which transmits only the red rays will allow the sun's light to appear through minute crevices in the edge of the moon, when rays of any other color would be entirely absorbed by the colored glass through which the observation is made. 324. Flame-like protuberances. — Immediately after the commencement of the total obscuration, red protuberances, resembling flames, may be seen to issue from behind the moon's disc. These appearances were noticed in the eclipse of May 3, 1733, and they have been re-observed during every total solar eclipse which has taken place since that time. They did not, however, attract much attention before the eclipse of July 8, 1842, when they were carefully observed and delineated in accurate diagrams. They were again made the subject of special study in the eclipse of July 28, 1851, and also in that of July 18, 1860. The forms of these protuberances are very various, and some of them quite peculiar. Many of them are nearly conical, the height being frequently greater than the breadth of the base. Others resemble the tops of a very irregular range of hills stretching continuously along one sixth of the moon's circumference. Some of these protuberances reach to a vast height, and show remarkable curvature. One has been compared to a sickle; a second to a Turkish cimeter; a third to a boomerang, with one extremity extending off horizontally far beyond the support of the base; while a fourth was of a circular form, entirely detached from the moon's limb by a space nearly equal to its own breadth. The size of these protuberances is very various. Some have been estimated to have an apparent height of 3', which wduld imply an absolute height of 80,000 miles; while others have every intermediate elevation down to the smallest visible object. The colors of these protuberances have been variously described. Some have been called simply reddish, while others have been characterized as rose-red, purple, or scarlet; and a few have been represented as nearly white. During the solar eclipses of 1842, 1851, and 1860, the largest M 178 ASTRONOMY. of these protuberances were seen by the unassisted eye. In 1860, some of them were observed several seconds before the total obscuration; and in 1842, as well as in 1851, some of them remained visible from 5s. to 7s. after the sun's emersion. 325. These protuberances emanate from the sun.-These protuberances emanate from the disc of the sun, and not from that of the moon. This is proved by the following observations made in 1851. The protuberances seen near the eastern limb decrease in dimensions from the commencement of the total eclipse to its close, while those near the western limb increase from the commencement to the close; indicating that the moon covers more and more the protuberances on the eastern side of the sun's disc, and gradually exposes a larger and larger portion of the protuberances on the western side. Again, during the eclipse of 1860, the astronomers who went to Spain to observe the eclipse obtained two excellent photographs, in which these flame-like protuberances were faithfully copied; and it was found that the protuberances retained a fixed position with reference to the sun as the moon glided before it; and they did not change their form, except as the moon, by passing over them, shut them off on the eastern side, while fresh ones became visible on the western side. See Plate III. 326. VNature of these protuberances.-That these protuberances are not solid bodies like mountains is proved by their peculiar forms, the tops frequently extending horizontally far beyond the support of the base; and they sometimes appear entirely detached from the sun's disc without any visible support. The same argument proves that they are not liquid bodies; and hence we must conclude that they are gaseous, or are sustained in a gaseous medium. These flame-like emanations seem to be analogous to the clouds which float at great elevations in our own atmosphere; and we are naturally led to infer that the sun is surrounded by a transparent atmosphere, rising to a height exceeding one tenth of his diameter; and in this atmosphere there are frequently found cloudy masses of extreme tenuity floating at various elevations, and sometimes rising to the height of 80,000 miles above the luminous surface of the sun. METHODS OF FINDING THE LONGITUDE. 179 CHAPTER XII. DIFFERENT METHODS OF FINDING THE LONGITUDE OF A PLACE. 327. Difference of time under different meridians.-The sun, in his apparent diurnal motion from east to west, passes successively over the meridians of different places; and noon occurs later and later as we travel westward from any given meridian. If we start from the meridian of Greenwich, then the sun will cross the meridian of a place 15~ west of Greenwich one hour later than it crosses the Greenwich meridian-that is, at one o'clock, Greenwich time. A difference of longitude of 15~ corresponds to a difference of one hour in local times. In order, then, to determine the longitude of any place from Greenwich, we must accurately determine the local time, and compare this with the corresponding Greenwich time. 328. Method of chronometers.-Let a chronometer which keeps accurate time be carefully adjusted to the time of some place whose longitude is known-for example, Greenwich Observatory. Then let the chronometer be carried to a place whose longitude is required, and compared with the correct time reckoned at that place. The difference between this time and that shown by the chronometer will be the difference of longitude between the given place and Greenwich. It is not necessary that the chronometer should be so regulated as neither to gain nor lose time. This would be difficult, if not impracticable. It is only necessary that its rate should be well ascertained, since an allowance can then be made for its gain or loss during the time of its transportation from one place to the other. The manufacture of chronometers has attained to such a degree of perfection that this method of determining difference of longitude, especially of stations not very remote from each other, is one of the best methods known. The most serious difficulty in 180 ASTRONOMY. the application of the method consists in determining the rate of the chronometer during the journey; for chronometers generally have a different rate, when transported from place to place, from that which they maintain in an observatory. For this reason, when great accuracy is required, it is customary to employ a large number of chronometers as checks upon each other; and the chronometers are transported back and forth a considerable number of times. This is the method by which the mariner commonly determines his position at sea. Every day, when practicable, he measures the sun's altitude at noon, and hence determines his latitude. About three hours before or after noon he measures the sun's altitude again, and from this he computes his local time by Art. 145. The chronometer which he carries with him shows him the true time at Greenwich, and the difference between the two times is his longitude from Greenwich. 329. By eclipses of the moon,.-An eclipse of the moon is seen at the same instant of absolute time in all parts of the earth where the eclipse is visible. Therefore, if at two distant places the times of the beginning of the eclipse are carefully observed, the difference of these times will give the difference of longitude between the places of observation; but, on account of the gradually increasing darkness of the penumbra, it is impossible to determine the precise instant when the eclipse begins, and therefore this method is of no value except under circumstances which preclude the use of better methods. 330. By the eclipses of Jupiter's satellites.-The moons of Jupiter are eclipsed by passing into the shadow of Jupiter in the same manner as our moon is eclipsed by passing into the shadow of the earth. These eclipses begin at the same instant of absolute time for all places at which they are visible. If, then, the times of the beginning of an eclipse be accurately observed at two different places, the difference of these times will be the difference of longitude of the places. Since, however, the light of a satellite diminishes gradually while entering the shadow, and increases gradually on leaving it, the observed time of beginning or ending of the eclipse must depend on the power of the telescope used, and also upon the eye of the observer. This method, therefore, is of no METHODS OF FINDING THE LONGITUDE. 181 value at fixed observatories, where better methods are always available. 331. By an eclipse of the sun or the occultation of a star.-The times of the beginning and end of an eclipse of the sun, or of the occultation of a star or planet at any place, depend on the position of the place. We can not, therefore, use a solar eclipse as an instantaneous signal for comparing directly the local times at two stations; but we may deduce by computation from the observed beginning and end of an eclipse,the time of true conjunction of the sun and moon-that is, the time of conjunction as seen from the centre of the earth; and this is a phenomenon which happens at the same absolute instant for every observer on the earth's surface. If the eclipse has been observed under two different meridians, we may determine the instant of true conjunction from the observations at each station; and since the absolute instant of this phenomenon is the same for both places, the difference of the results thus obtained is the difference of longitude of the two stations. This is one of the most accurate methods known to astronomers for determining the difference of longitude of two stations remote from each other. This is especially true when the moon crosses a cluster containing a large number of stars, as the Pleiades. 332. By moon culminating stars.-Certain stars situated near the moon's path, and passing the meridian at short intervals before or after the moon, are called moon culminating stars. The moon's motion in right ascension is very rapid, amounting to about half a degree, or two minutes in time, during a sidereal hour-that is, during the interval that elapses from the time a star is on the meridian of any place, till it is on the meridian of a place whose longitude is 15~ west of the former. Hence the intervals between the passages of the moon' and a star over the meridians of two places differing an hour in longitude must differ about two minutes; and for other differences of longitude there must be a proportional difference in the intervals. Hence, if the intervals between the passages of the moon and a star over the meridians of two places be accurately observed, the difference of their longitude may be found by means of the moon's hourly variation in right ascension. 182 ASTRONOMY. The chief disadvantage of this method consists in this circumstance, that an error in the observed increase of right ascension will produce an error nearly 30 times as great in the computed longitude. Hence, to obtain a satisfactory result by this method requires a series of observations made with the utmost care, and continued through a long period of time. 333. By lunar distances.-The Nautical Almanac furnishes for each day the distance of the moon from the sun, the larger planets, and several stars situated near the moon's path. These distances are given for Greenwich time, and are such as they would appear to a spectator placed at the centre of the earth. A mariner on the ocean measures with a sextant the distance from the moon to one of the stars mentioned in the Almanac. I-Ie corrects this distance for refraction and parallax, and thus obtains the true lunar distance as it would be seen at the centre of the earth. By other observations, he knows the local time at which this distance was measured, and, by referring to the Nautical Almanac, he finds the Greenwich time at which the lunar distance was the same. The difference between the local time and the Greenwich time represents the longitude of the place of observation from Greenwich. This method of finding the longitude may be practiced at sea, and in long voyages should always be resorted to as a check upon the method by chronometers. 334. By the electric telegraph.-The difference of the local times of two places may be determined by means of any signal which can be seen or heard at both places at the same instant. When the places are not very distant, the explosion of a rocket, or the flash of gunpowder, or the flight of a shooting star may serve this purpose. The electric telegraph affords the means of transmitting signals to a distance of a thousand miles or mdre with scarcely any appreciable loss of time. Suppose that there are two observatories at a considerable distance from each other, and that each is provided with a good clock, and with a transit instrument for determining its error; then, if they are connected by a telegraph wire, they have the means of transmitting signals at pleasure from either observatory to the other for the purpose of comparing their local times. For convenience, we will call the most eastern station E, METHODS OF FINDING THE LONGITUDE.. 183 and the western W. The following is one mode of comparing their local times. 335. Mode of comparing the local times.-The plan of operations having been previously agreed upon, the astronomer at E strikes the key of his register, and makes a record of the time according to his observatory clock. Simultaneously with this signal at E, the armature of the magnet at W is moved, producing a click like the ticking of a watch. The astronomer at W hears the sound, and notes the instant by his clock. The difference between the time recorded at E and that at W is the difference between the two clocks. A single comparison of this kind will furnish the difference of longitude to the nearest second; but to obtain the fraction of a second with the greatest precision requires many repetitions, and this is accomplished as follows: At the commencement of the minute by his clock, the astronomer at E strikes his signal key, and the time of the signal is recorded both at E and W. At the close of 10 seconds the signal is repeated, and the observation is recorded at both stations. The same thing is done at the end of 20 seconds, of 30 seconds, and so on to 20 repetitions. The astronomer at W then transmits a series of signals in the same manner, and the times are recorded at both stations. 336. The velocity of the electric fluid.-This double set of signals not only furnishes an accurate comparison of the two clocks, but also enables us to measure the velocity of the electric fluid. If the fluid requires no time for its transmission, then the apparent difference between the two clocks will be the same, whether we determine it by signals transmitted from E to W, or from W to E. But if the fluid requires time for its transmission, these results will differ. Suppose the true difference of longitude between the places is one hour, and that it requires one second for a signal to be transmitted from E to W. Then, if at 10 o'clock a signal be made and recorded at E, it will be a second before the signal is heard and recorded at W —that is, the time recorded at W will be 9 hours and 1 second; and the apparent difference between the two clocks will be 59 minutes and 59 seconds. But if a signal be made at W at nine o'clock, it will be heard at E at 10 hours and 1 second; and the apparent difference between the two 184 ASTRONOMY. clocks will be 1 hour and 1 second. Thus the differences between the two clocks, as derived from the two methods of comparison, differ by twice the time required for the transmission of a signal from E to W. Numerous observations, made on the longest lines and with the greatest care, have shown that the velocity of the electric fluid upon the telegraph wires is about 16,000 miles per second. The mean of the results obtained by signals transmitted in both directions, gives the true difference between the two clocks, independent of the time required in the transmission of signals. 337. How the clock may break the electric circuit.-The most accurate method of determining difference of longitude consists in employing one of the clocks to break the electric circuit each second. This may be accomplished in the following manner: Near the lower extremity of the pendulum place a small metallic cup containing a globule of mercury, so that once in every vibration the pointer at the end of the pendulum may pass through the mercury. A wire from one pole of the battery is connected with the supports of the pendulum, and another wire from the other pole of the battery connects with the cup of mercury. When the pointer is in the mercury, the electric circuit will be complete through the pendulum; but as soon as it passes out of the mercury, the circuit will be broken. When the connections are properly made, there will be heard a click of the magnet at each station, simultaneously with the beats of the electric clock. If each station be furnished with an ordinary Morse register, there will be traced upon the paper a series of lines, of equal length, separated by short intervals, thus: The mode of using the register for marking the date of any event is to strike the key of the register at the required instant, when an interruption will be made in one of the lines of the graduated scale; and its position will indicate not only the second, but the fraction of a second at which the signal was made. We now employ the same electric circuit for telegraphing transits of stars. A list of stars having been selected beforehand, and furnished to each observer, the astronomer at E points his transit telescope upon one of the stars as it is passing his meridian, and strikes the key of his register at the instant the star passes sue THE TIDES. 185 cessively each wire of his transit, and the dates are recorded, not only upon his own register, but also upon that at W. When the same star passes over the meridian of W, the observer there goes through the same operations, and his observations are printed upon both registers. These observations furnish the difference of longitude of the two stations, independently of the tabular place of the star employed, and also independently of the absolute error of the clock. CHAPTER XIII. THE TIDES. 338. Definitions.-The alternate rise and fall of the surface of the sea twice in the course of a lunar day, or of 24h. 51m. of mean solar time, is the phenomenon known by the name of the tides. When the water is rising it is said to be flood tide, and when it is falling, ebb tide. When the water is at its greatest height it is said to be high water, and when at its least height, low water. 339. Spring and neap tides.-The time from one high water to the next is, at a mean, 12h. 25m. 24s. Near the time of new and full'moon the tide is the highest, and the interval between the consecutive tides is the least, viz., 12h. 19m. Near the quadratures, or when the moon is 90~ distant from the sun, the tides are the least, and the interval between them is the greatest, viz., 12h. 30m. The former are called the spring tides, and the latter the neap tides. At New York the average height of the spring tides is 5.4 feet, and of the neap tides 3.4 feet. 340. The establishment of a port.-The time of high water is mostly regulated by the moon; and for any given place, the hour of high water is always nearly at the same distance from that of the moon's passage over the meridian. The mean interval between the moon's passage over the meridian, and high water at any port on the days of new and full moon, is called the establishment of the port. The mean interval at New York is 8h. 13m., and the difference between the greatest and the least interval occurring in different parts of the month is 43 minutes. 186 ASTRONOMY. 341. Tides at perigee and apogee.-The height of the tide is affected by the distance of the moon from the earth, being highest near the time when the moon is in perigee, and lowest near the time when she is in apogee. When the moon is in perigee, at or near the time of a new or full moon, unusually high tides occur. 342. Cause of the tides.-The facts just stated indicate that the moon has some agency in producing the tides. The tides, however, are not due to the simple attraction of the moon upon the earth, but to the difference of its attraction on the opposite sides - _ -~ - rFig. 89..- ~. of the earth. Let ACEG represent the earth, and let us suppose its entire surface to be covered with water; also, let M be the place of the moon. The different parts of the earth's surface are at unequal distances from the moon. Hence the attraction which the moon exerts at A is greater than that which it exerts at B and H, and still greater than that which it exerts at C and G; while the attraction which it exerts.at E is least of all. The attraction which the moon exerts upon the mass of water immediately under it, near the point Z, is greater than that which it exerts upon the solid mass of the globe. The water will therefore heap itself up over A, forming a convex protuberance-that is, high water will take place immediately under the moon. The water which thus collects at A will flow from the regions C and G, where the quantity of water must therefore be diminishedthat is, there will be low water at C and G. The water at N is less attracted than the solid mass of the earth. The solid mass of the earth will therefore recede from the waters at N, leaving the water behind, which will thus be heaped up at N, forming a convex protuberance, or high water, similar to that at Z. The sea is therefore drawn out into an ellipsoidal form, having its major axis directed toward the moon. THE TIDES. 187 343. Effect of the sun's attraction. The attraction of the sun produces effects similar to those of the moon, but less powerful in raising a tide, because the inequality of the sun's attraction on different parts of the earth. is very small. It has been computed that the tidal wave due to the action of the moon is about double that which is due to the sun. There is, therefore, a solar as well as a lunar tide wave, the latter greater than the former, and each following the luminary from which it takes its name. When the sun and moon are both on the same side of the earth, or on opposite sides, that is, when it is new or full moon, their effects in producing tides are combined, and the result is an unusually high tide, called spring tide. When the moon is in quadrature, the action of the sun tends to produce low water where that of the moon produces high water, and the result is an unusually small tide, called neap tide. 344. Effect of the moon's declination on the tides.-The height of the tide at a given place is influenced by the declination of the moon. When the moon has no declination, the highest tides should occur along the equator; and the heights should diminish from thence toward the north and south; but the two daily tides at any place should have the same height. When the moon has north declination, as shown in Fig. 90, the highest tides on the side of the earth next the moon will be at places having a corFig. 90. responding north latitude, as at B, and on the opposite side at responding north latitude, as at B, and on the opposite side at those which have an equal south latitude. And of the two daily tides at any place, that which occurs when the moon is nearest the zenith should be the greatest. Hence, when the moon's dec 188 ASTRONOMY. lination is north, the height of the tide at a place in north latitude should be greater when the moon is above the horizon than when she is below it. At the same time, places south of the equator have the highest tides when the moon is below the horizon, and the least when she is above it. This is called the diurnal inequality, because its cycle is one day; but it varies greatly in amount at different places. The great wave which constitutes the tide is to be considered as an undulation of the waters of the ocean, in which (except when it passes over shallows or approaches the shores) there is little or no progressive motion of the water. 345. WIhy the phenomena of the tides are so comnplicated.-The actual phenomena of the tides are far more complicated than they would be if the earth were entirely covered with an ocean of great depth. The water covers less than three quarters of the earth's surface, and a considerable part of this water is less than a mile in depth. Two great continents extend from near the north pole to a great distance south of the equator, thus interrupting the regular progress of the tidal wave around the globe. In the northern hemisphere, the waters of the Atlantic can communicate with those of the Pacific only by Behring's Strait, a channel 36 miles in breadth, which effectually prevents the transmission of any considerable wave from the Atlantic to the Pacific through the northern hemisphere. In the southern hemisphere, the American continent extends to 56~ of S. latitude, and in about latitude 60~ commences a range of islands, near which are indications of an extensive antarctic continent, leaving a passage only about 500 miles in breadth. Through this passage the motion of the tidal wave (as we shall presently see) is eastward, and not westward; whence we conclude that the tides of the Atlantic are not propagated into the Pacific. 346. Cotidal lines.-The.phenomena of the tides, being thus exceedingly complicated, must be learned chiefly from observations; and in order to present the results of observations most conveniently upon a map, we draw a line connecting all those places which have high water at the same instant of absolute time. Such lines are called cotidal lines. The accompanying map, Plate I., shows the cotidal lines for nearly every ocean, drawn at intervals of 3 hours, and expressed in Greenwich time. THE TIDES. 189 347. Origin of the tidal wave.-By inspecting this map, we perceive that the great tidal wave originates in the Pacific Ocean, not far from the western coast of South America, in which region high water occurs about two hours after the moon has passed the meridian. The wave thus formed, if left undisturbed, would travel, like ordinary waves, with a velocity depending upon the depth of water. When the breadth of a wave is very great in comparison with the depth of water, the velocity of its progress is equal to that which a heavy body would acquire in falling by gravity through half the depth of the liquid. The velocity of such a wave for different depths of the ocean is as follows: F 25 feet,' 19 miles per hour. 100 " 39 " " When the 250" 61" depth of the- 1,000 ", the velocity of 122 " depth of the 1,000" 122 the wave is 27 water is 5,000he wave is 273 20,000 " 547 ".50,000 " J 865" 348. Progress and velocity of the tidal wave.-Since the moon travels westward at the rate of 1000 miles per hour over the equator, it tends to carry high water along with it at the same rate. But the shallow water of most parts of the ocean does not allow the tidal wave to travel with this velocity. The wave of high water, first raised near the western coast of South America, travels toward the northwest through the deep water of the Pacific at the rate of 850 miles per hour, and in about ten hours reaches the coast of Kamtschatka. On account of more shallow water, the same wave travels westward and southwestward with less velocity, and it is about 12 hours old when it reaches New Zealand, having advanced at the rate of about 400 miles per hour. Passing south of Australia, the tidal wave travels westward and northward into the Indian Ocean, and is 29 hours old when it reaches the Cape of Good Hope. Hence it is propagated through the Atlantic Ocean, traveling northward at the rate of about 700. miles per hour, and in 40 hours from its first formation it reaches the shallow waters of the coast of the United States, whence it is propagated into all the bays and inlets of the coast. The wave which enters at the eastern end of Long Island Sound is about 4 hours in reaching the western end, so that the wave is 44 hours old when it arrives at New Haven. 190 ASTRONOMY. 349. Tides of the North Atlantic.-A portion of the great Atlantic wave advances up Baffin's Bay, and at the end of 56 hours reaches the latitude of 78~. The principal part of the Atlantic wave, however, turns eastward toward the Northern Ocean, and in 44 hours brings high water to the western coast of Ireland. After passing Scotland, a portion of this wave turns southward with diminished velocity into the North Sea, and thence follows up the Thames, bringing high water to London at the end of 66 hours from the first formation of this wave in the Pacific Ocean. 350. Velocity of the tidal wave in shallow water.-As the tidal wave approaches the shallow water of the coast, its velocity is speedily reduced from 500, or perhaps 900 miles per hour, to 100 miles, and soon to 30 miles per hour; and in ascending bays and rivers its velocity becomes still less. From the entrance of Chesapeake Bay to Baltimore the tide travels at the average rate of 16 miles per hour, and it advances up Delaware Bay with about the same velocity. From Sandy Hook to New York city the tide advances at the rate of 20 miles per hour, and it travels from New York to Albany in 9h. 9m., being at the average rate of nearly 16 miles per hour. From New York Bay the tidal wave is propagated through East River until it meets the wave which has come in from the Atlantic through the eastern end of the Sound. This place of meeting is only 21 miles from New York, showing that the velocity of the tidal wave through East River is only 7T miles per hour-a result which must be ascribed to the narrowness and intricacy of the channel. 351. Tidal wave on the western coast of South America.-The tidal wave which we have thus traced through oceans, bays, and rivers, has every variety of direction; in some places advancing westward, and in others eastward; in some places northward, and in others southward; but in each case it may be regarded as a continuous forward movement, and the change in its direction results from a change in the direction of the channel. But there is one exception to this general rule. We have traced the origin of the tidal wave to a region about 1000 miles west of the coast of South America. From this point high water is not only propagated westward around the globe, but also eastward toward Cape THE TIDES. 191 Horn. In this region the motion of the tidal wave appears to be similar to that of the wave produced by throwing a stone upon the surface of a tranquil lake, the wave traveling off in all directions from the first point of disturbance. 352. Is the tidal wave a free or a forced oscillation?-If the moon should suddenly cease its disturbing action upon the waters of the ocean, the tidal wave already formed would travel forward with a velocity depending solely upon the depth of water, and this would be called afree wave. Now the moon continually tends to form high water directly beneath it-that is, it tends to carry high water westward at the rate of 1000 miles per hour over the equator. Such a wave, if it could actually be formed, would be called a forced oscillation, because its velocity would be independent of the depth of water. Is, then, the great tidal wave a free or a forced oscillation? We may answer this question by observing the velocity of the tidal wave in the Atlantic Ocean, whose depth has been approximately determined. The velocity of the tidal wave in the North Atlantic, from the-equator to latitude 50~, is about 640 miles per hour, corresponding to a depth of 27,500 feet, which is somewhat greater than the average depth of the Atlantic. The velocity of the tidal wave in the Atlantic appears to be about one third greater than that of a free wave, and this excess of velocity is probably due to the immediate action of the sun and moon; in other words, the tidal wave is, to some extent, a forced oscillation, but its rate of progress appears to be determined mainly by the depth of water. 353. Height of the tides.-At small islands in mid-ocean the tides never rise to a great height, sometimes even less than one foot; and the average height of the tides for the islands of the Atlantic and Pacific Oceans is only 34 feet. Upon approaching an extensive coast where the water is shallow, the velocity of this tidal wave is diminished, the cotidal lines are crowded more closely together, and the height of the tide is thereby increased; so that while in mid-ocean the average height of the tides does not exceed 3- feet, the average in the neighborhood of continents is not less than 4 or 5 feet. According to theory, the height of the wave should vary inversely as the fourth root of the depth; that is, in water 100 feet deep, the wave should be twice as high as in 192 ASTRONOMY. water 1600 feet deep. Fig. 91 shows the change in the form of waves in approaching shallow water. Fig, 91.. ~ L i H K 354. Hieight of the tides modified by the conformation of the coast. Along the coast of an extensive continent the height of the tides is greatly modified by the conformation of the shore line. When the coast is indented by broad bays which are open in the direction of the tidal wave, this wave, being contracted in breadth, must necessarily increase in height, so that at the head of a bay the height of the tide may be several times as great as at the entrance. The operation of this principle is exhibited at numerous places upon the Atlantic coast. Thus, if we draw a straight line Fig. 92. from Cape Hatteras to the southern Ipa; 1 part of Florida, it will cut off a bay about 200 miles in depth. At Cape, I 4 b. Hatteras and Cape Florida the tide I,~ \, I rises only 2 feet; at Cape Fear and /.j. St. Augustine it rises 4 feet; while............. at Savannah it rises 7 feet. 355. Tides in the Bay of Fundy.-If we draw a straight line from Nantucket to Cape Sable, it will cut off a bay in which the phenomena of the tides are still more remarkable. At Nantucket the tide rises only 2 feet; at Boston it rises 10 feet; near the entrance to the Bay of Fundy, 18 feet; while at the head of the bay it sometimes rises to the height of 70 feet. This result is due mainly to the contraction of the channel through which the tidal wave advances. Fig. 93. ~?> Qt ^ ~~ i^ THE TIDES. 193 356. Tides of Long Island Sound, etc.-So, also, at the east end of Long Island Sound, the tide rises only 2 feet; but in its progress westward through the Sound the height continually increases, until at the west end the height is more than 7 feet. At the entrance to Delaware Bay the tide rises only 38 feet, while at New Castle it rises 6- feet. The tide from the North Atlantic is propagated through the Gulf of St. Lawrence, and thence through the River St. Lawrence, at the average rate of about 70 miles per hour, being 12 hours from the ocean to Quebec. This tide increases in height as it advances, being only 9 feet at the mouth of the St. Lawrence, while - it is 20 feet at Quebec. 357. Tides nodified by a projecting promontory.-A promontory, as A, projecting into the ocean, rig. 94. A Ad" ~ so as.to divide the tidal wave and ^/ \f/^ throw it off upon either side, not 33A,/C ~only causes the tide at B and C to rise above the mean height, but sometimes reduces the tide at A below the mean height. Thus, at Cape Hatteras, the tide rises less than 2 feet in height, while along the coast on either side the tide rises to the height of 5 or 6 feet. So, also, on the south side of Nantucket, the tides are less than 2 feet in height, while along the coast north of Cape Cod the tide rises 10 feet in height. 358. Tides on the coast of Ireland.-So, also, on the southwest Fig. 95, N ~ r~ea~NcJOl a~ ^ ^ ^ R ~ l i ^ J T _ ^ ^; f ^ ^^^ ^^^^~~~~GfiS~ 194 ASTRONOMY. coast of Ireland, where the tidal wave from the Atlantic first strikes the coast, the tide:is less than it is at a short distance along the coast either eastward or northward. In some cases the form and position of a promontory are such as to divert the tidal wave from some part of the coast, and leave it almost destitute of a tide. Such a case occurs on the east coast of Ireland. The wave from the Atlantic, being forced up St. George's Channel, is driven upon the coast of Wales, where the tide rises to the height of 36 feet, while it is almost wholly diverted from the opposite coast of Ireland, where the range of the tide is only 2 feet. 359. Tides of rivers.-The tides of rivers exhibit the operation of similar principles. In a channel of uniform breadth and depth, the height of the tide should gradually diminish, in consequence of the effect of friction. But if the channel contracts or shoals rapidly, the height of the tide will increase. There is, then, a certain rate of contraction, with which the range of the tides will remain stationary. If the river contracts more rapidly, the height of the tides will increase; if the channel expands, the height of the tides will diminish. Hence, in ascending a long river, it may happen that the height of the tides will increase and decrease alternately. Thus, at New York, the mean height of the tide is 4.3 feet; at West Point, 55 miles up the Hudson River, the tide rises only 2.7 feet; at Tivoli, 98 miles from New York, the tide amounts to 4 feet; while at Albany it rises only 2.3 feet. 360. The diturnal inequality in the height of the tides.-If the sun and moon moved always in the plane of the equator, and the earth were entirely covered with water to a great depth, the two daily tides should have nearly the same height; but when they are out of the equator, the two daily tides should generally be unequal. The moon sometimes reaches 280 north declination, in which case it tends to raise the highest tide at a station in latitude 28~ north, while the highest tide on the opposite side of the earth should be in latitude 28~ south. Hence the two tides which are formed in the northern hemisphere under opposite meridians must be of unequal heights-that is, the morning and evening tides at a given place should be unequal. The same would be true for the southern hemisphere, but on the equator there would be no such diurnal inequality. THE TIDES. 195 361. Diurnal inequality in the North Atlantic Ocean.-Along the Atlantic coast of the United States, when the moon has its greatest declination, the difference between high water in the forenoon and afternoon averages about 18 inches; but this difference almost entirely disappears when the moon is on the equator. On the coast of Ireland, the diurnal inequality, at its maximum, is only one foot, while the average height of the tides is nine feet. On some parts of the European coast the diurnal inequality is still smaller, and can with difficulty be detected in a long series of observations. 362. Diurnal inequality on the Pacific coast.-On the Pacific coast of the United States, when the moon is far from the equator, there is one large and one small Fig. 96. tide during each day. In [ _[.-, ~,i - o n or T * 4 6 S B) A 810 1in in -'i s 2is X I a-~ the Bay of San Francisco, EEET~ ~ ~ i. _ _' _ ~_ the difference between high 7 and low water in the fore- 6 noon is sometimes only two 5 inches, while in the afternoon\ of the same day the difference is 5- feet. When the moon is on the equator this inequality disappears, and the two daily tides are near-' ly equal. At other places on the Pacific coast this inequality in the two daily tides is still more re- Fig. 97. markable. At Port Town — HOR S. send, near Vancouver's Isl- I —FET 4 10 1 i 6 1 20 i 2 2 2, and, when the moon has its 7 greatest declination, there is e no descent corresponding to \ morning low water, but mere- ly a temporary check in the rise of the tide. Thus one of the two daily tides becomes obliterated; that is, we find but one tide in the 24 hours. o Similar phenomena occur at 196 ASTRONOMY. other places upon the Pacific coast, and also on the coast of Iamtschatka. 363. Cause of these variations in the diurnal inequality.-The tide actually observed at any port is the effect, not simply of the immediate action of the sun and moon upon the waters of the ocean, but is rather the resultant of their continued action upon the waters of the different seas through which the wave has advanced from its first origin in the Pacific until it reaches the given port, embracing an interval sometimes of one or two days, and perhaps even longer. During this period the moon's action tends sometimes to produce a large tide, and sometimes a small one; and in a tide whose age is more than 12 hours, these different effects are combined so as sometimes partly to obliterate the diurnal inequality, and sometimes to exaggerate it. This is probably the reason why the diurnal inequality is less noticeable in the North Atlantic than in the North Pacific. 364. Four tides in 24 hours.-In some places the tide rises and falls four times in 24 hours. This happens on the east coast of Scotland, where the form of the tidal wave is such as is repreF rig. 98. sentedby the annexed figure. This anomaly is ascribed to the superposition of two tidal waves, one traveling round the north of Scotland, and advancing southward through the North Sea, while the other passes through the English Channel, and thence advances northward into the same sea. At some places these two waves arrive nearly at the same hour, and are so superposed as not to be distinguished from each other; but at other places one arrives 2 or 3 hours behind the other, thus presenting the appearance of high water 4 times in 24 hours. 365. Small tides of the Pacific Ocean.-Near the middle of the Pacific Ocean, in the neighborhood of the Society Islands, from latitude 13~ to 18~ S., and from longitude 140~ to 176~ W., the tides are smaller than have been found in any other portion of the open sea, averaging less than one foot in height. At Tahiti TIlE TIDES. 197 (latitude 17~ 29' S., longitude 149~ 29' W.), the tides at full moon rise to the height of about 15 inches, and at the quadratures only about 3 inches. There are two high waters daily occurring near noon and midnight, being seldom earlier than 10 A.M., or later than 2- PM. 366. Cause of these peculiarities.-It is uncertain what is the cause of this small height of the tides, but it is believed that the following consideration will explain it, at least in part. The original tide wave, starting from the eastern part of the Pacific Ocean, reaches Tahiti about six hours after the moon's transit over that meridian. Hence, when the main tidal wave of the Pacific reaches that port, the immediate effect of the moon is to produce low water at the same hour; and the superposition of these two waves produces a nearly uniform level of the water. The occurrence of high water within about two hours of noon every day seems to indicate that the power of the sun to raise a tide is here nearly equal to that of the moon. In the Atlantic Ocean, the influence of the moon upon the tides is generally about double that of the sun; but this ratio appears to be a variable olle. 367. Tides of the Gulf of Mexico.-The Gulf of Mexico is a shallow sea, about 800 miles in diameter, almost entirely surrounded by land, and communicating with the Atlantic by two channels, each about 100 miles in breadth. It is by the Florida channel that the tidal wave from the Atlantic is chiefly propagated into the Gulf, but its progress is so much obstructed by the West India Islands that its height is very much reduced. Between Florida and Cuba the tidal wave advances slowly westward; but after passing the channel it moves more rapidly, and reaches the western side of the Gulf in seven hours, showing an average progress of 125 miles per hour. The tides in the Gulf are every where quite small. At Mobile and Pensacola the average height is only one foot. The diurnal inequality is also quite large, so that at most places (except when the moon is near the equator) one of the daily tides is well-nigh inappreciable, and the tide is said to ebb and flow but once in 24 hours. 198 ASTRONOMY. 368. Tides of the Mediterranean.-The tides of the Mediterranean are generally so small as not to be regarded by navigators. Their average height does not exceed 18 inches. In the neighborhood of the Strait of Gibraltar the tide rises from 2 to 4 feet; at Venice it rises from 18 inches to 4 feet; and at Tunis it sometimes rises to the height of 3 feet. The length of the Mediterranean is 2400 miles, or nearly one third the diameter of the earth; and the average height of the tides is here at least one third what it is in the open sea. 369. Tides of inland seas.-In small lakes and seas which do not communicate with the ocean there is a daily tide, but so small that it requires the most accurate observations to detect it. The existence of a tide in Lake Michigan has been proved by a series of observations made at Chicago in 1859. The average height of this tide is 13 inches; and the average time of high water is 30 minutes after the time of the moon's transit. The length of Lake Michigan is 350 miles, or Id of the earth's diameter, and its tide is about A-d of that which prevails in midocean. CHAPTER XIV. THE PLANETS -THEIR APPARENT MOTIONS. -ELEMENTS OF THEIR ORBITS. 370. Number, etc., of the planets.-The planets are bodies of a globular form, which revolve around the sun as a common centre, in orbits which do not differ much from circles. The name planet is derived from the Greek word 7rXavrTnC, signifying a wanderer, and was applied by the ancients to these bodies because their apparent movements were complicated and irregular. Five of the planets-Mercury, Venus, Mars, Jupiter, and Saturnare very conspicuous, and have been known from time immemorial. Uranus was discovered in 1781, and Neptune in 1846, making eight planets including the earth. Besides these there is a large group of small planets, called asteroids, situated between the orbits of Mars and Jupiter. The first of these was discovered in 1801, and the number known in 1867 amounts to 94, The orbits of Mercury and Venus are included within the orbit THE PLANETS, ETC. 199 of the earth, and they are hence called inferior planets, while the others are called superior planets. 371. The satellites.-Some of the planets are the centres of secondary systems, consisting of smaller globes, revolving round them in the same manner as they revolve around the sun. These are called satellites or moons. The primary planets which are thus attended by satellites carry the satellites with them in their orbits around the sun. Of the satellites known at the present time, four revolve around Jupiter, eight around Saturn, four around Uranus, and one around Neptune. The moon is also a satellite to the earth. 372. The orbits of the planets.-The orbit of each of the planets is an ellipse, of which the sun occupies one of the foci. That point of its orbit at which a planet is nearest the sun is called the perihelion, and that point at which it is most remote is called the aphelion. The eccentricity of a planetary orbit is the distance of the sun from the centre of the ellipse which the planet describes, expressed in terms of the semi-major axis regarded as a unit; or, in other words, it is the quotient of the distance between the centre and focus, divided by the semi-major axis. The eccentricities of most of the planetary orbits are so minute that, if the form of the orbit were exactly delineated on paper, it could not be distinguished from a circle except by careful measurement. 373. Geocentric and heliocentric places.-The motion of a planet as it appears to an observer on the earth is called the geocentric motion, while its motion as it would appear if the observer were transferred to the sun is called its heliocentric motion. The motions of the planets can not be observed from the sun as a centre, but from the geocentric motions, combined with the relative distances of the earth and planet from the sun, we may deduce the heliocentric motions by the principles of Geometry. The geocentric place of a body is its place as seen from the centre of the earth, and the heliocentric place is its place as seen from the centre of the sun. 374. Elongation, conjunction, and opposition of a planet.-The 200 ASTRONOMY. angle formed by lines drawn from the earth to the sun and a planet is called the elongation of the planet from the sun; and it is east or west, according as the planet is on the east or west side of the sun. A planet is said to be in conjunction with the sun when it has the same longitude, being then in nearly the same part of the heavens with the sun. It is said to be in opposition with the sun when its longitude differs from that of the sun 180~, being then in the quarter of the heavens opposite to the sun. A planet is said to be in quadrature when it is distant from the sun 90~ in longitude. A planet which is in conjunction with the sun passes the meridian about noon, and is therefore above the horizon only during the day. A planet which is in opposition with the sun passes the meridian about midnight, and is therefore above the horizon during the night. A planet which is in quadrature passes the meridian about 6 o'clock either morning or evening. An inferior planet is in conjunction with the sun when it is between the earth and the sun, as well as when it is on the side of the sun opposite to the earth. The former is called the inferior conjunction, the latter the superior conjunction. 375. Why the apparent motions of the planets differ from the real mnotions.-If the planets could be viewed from the sun as a centre, they would all be seen to advance invariably in the same direction, viz., from west to east, in planes only slightly inclined to each other, but with very unequal velocities. Mercury would advance eastward with a velocity about one third as great as our moon; Venus would advance in the same direction with a velocity less than half that of Mercury; the more distant planets would advance still more slowly; while the motions of Uranus and Neptune would be scarcely appreciable except by comparing observations made at long intervals of time. None of the planets would ever appear to move from east to west. The motions of the planets, as they actually appear to us, are very unlike those just described, first, because we view them from a point remote from the centre of their orbits, in consequence of which the distances of the planets from the earth are subject to great variations; and, second, because the earth itself is in motion, and the planets have an apparent motion, resulting from the real motion of the earth. APPARENT MOTIONS OF THE PLANETS. 201 376. The apparent motion of an inferior planet.-In order to deduce the apparent motion of an inferior planet from its real motion, let CKZ represent a portion of the heavens lying in the plane of the ecliptic; let a, b, c, d, etc., be the orbit of the earth; and Fig. 99. A, _2 / h \ \ i 2 iA' 1, 2, 3, 4, etc., the orbit of Mercury. Let the orbit of Mercury be divided into 12 equal parts, each of which is described in 71 days; and let ab, be, cd, etc., be the spaces described by the earth in the same time. Suppose Mercury to be at the point 1 in his orbit when the earth is at the point a; Mercury will then appear in the heavens at A, in the direction of the line a 1. In 71 days Mercury will have arrived at 2, while the earth has arrived at b, and therefore Mercury will appear at B. When the earth is at c, Mercury will appear at C, and so on. By laying the edge of a ruler on the points c and 3, d and 4, e and 5, and so on, the successive apparent places of Mercury in the heavens will be obtained. We thus find that from A to C, his apparent motion is 202 ASTRONOMY. from east to west; from C to P, his apparent motion is from west to east; from P to T it is from east to west; and from T to Z the apparent motion is from west to east. 377. Direct and retrograde motion.-When a planet appears to move in the direction in which the sun appears to move in the ecliptic, its apparent motion is said to be direct; and when it appears to move in the contrary direction, it is said to be retrograde. The apparent motion of an inferior planet is always direct, except within a certain elongation east and west of the inferior conjunction, when it is retrograde. If we follow the movements of Mercury during several successive revolutions, we shall find its apparent motion to be such as is indicated by the arrows in the preceding diagram, viz., while passing from its greatest western to its greatest eastern elongation, it appears to move in the same direction as the sun toward P. As it approaches P its apparent motion eastward becomes gradually slower, until it stops altogether at P, and becomes stationary. It then moves westward, returning to T, where it again becomes stationary, after which it again moves eastward, and continues to move in that direction through an arc about equal to CP, when it again becomes stationary. It again moves westward through an arc about equal to PT, when it again becomes stationary, and so on. The middle point of the arc of retrogression, PT, is that at which the planet is in inferior conjunction; and the middle point of the arc of progression, CP, is that at which the planet is in superior conjunction. These apparently irregular movements suggested to the ancients the name of planet, or wanderer. 378. Apparent motion of a superior planet.-In order to deduce the apparent motion of a superior planet from the real motions of the earth and planet, let S be the place of the sun; 1, 2, 3, etc., be the orbit of the earth; a, b, c, etc., the orbit of Mars; and CGL a part of the starry firmament. Let the orbit of the earth be divided into 12 equal parts, each of which is described in one month; and let ab, be, cd, etc., be the spaces described by Mars in the same time. Suppose the earth to be at the point 1 when Mars is at the point a, Mars will then appear in the heavens in the direction of the line 1 a. When the earth is at 3 and Mars at c, he will ap APPARENT MOTIONS OF THE PLANETS. 203 Fig. 100. C510 pear in the heavens at. When the earth arrives at 4 Mars will pear in the heavens at C. When the earth arrives at 4, Mars will arrive at d, and will appear in the heavens at D. While the earth moves from 4 to 5 and from 5 to 6, Mars will appear to have advanced among the stars from D to E and from E to F, in the direction from west to east. During the motion of the earth from 6 to 7 and from 7 to 8, Mars will appear to go backward from F to G and from G to H, in the direction from east to west. During the motion of the earth from 8 to 9 and from 9 to 10, Mars will appear to advance from H to I and from I to K, in the direction from west to east, and the motion will continue in the same direction until near.the succeeding opposition. The apparent motion of a superior planet projected on the heavens is thus seen to be similar to that of an inferior planet, except that, in the latter case, the retrogression takes place near inferior conjunction, and in the former it takes place near opposition. 204 ASTRONOMY. 379. Conditions under which a planet is visible.-One or two of the planets are sometimes seen when the sun is above the horizon; but generally, in order to be visible without a telescope, a planet must have an elongation from the sun greater than 30~, so as to be above the horizon before the commencement of the morning twilight, or after the close of the evening twilight. The greatest elongation of the inferior planets never exceeds 47~. If they have eastern elongation, they pass the meridian in the afternoon, and, being visible above the horizon after sunset, are called evening stars. If they have western elongation, they pass the meridian in the forenoon, and,being visible above the eastern horizon before sunrise, are called morning stars. A superior planet, having every degree of elongation from 0 to 180~, may pass the meridian at any hour of the day or night. At opposition the planet passes the meridian at midnight, and is therefore visible from sunset to sunrise. 380. Phases of a planet.-That hemisphere of a planet which is presented to the sun is illumined, and the other is dark. But if the same hemisphere which is turned toward the sun is not also presented to the earth, the hemisphere of the planet which is presented to the earth will not be wholly illumined, and the planet will exhibit phases. The inferior planets exhibit the same variety of phases as the moon. At the inferior conjunction, the dark side of the planet is turned directly toward the earth. Soon afterward the planet appears a thin crescent, which increases in breadth until at quadrature it becomes a half moon. From quadrature the planet becomes gibbous, and at superior conjunction it beomes a full moon. The distances of the superior planets from the sun are, with but one exception, so much greater than that of the earth, that the hemisphere which is turned toward the earth is sensibly the same as that turned toward the sun, and these planets always appear full. 381. Elements of the orbit of a planet.-There are seven different quantities necessary to be known in order to compute the place of a planet for a given time. These are called the Elements of the orbit. They are, 1. The periodic time. ORBITS OF THE PLANETS. 205 2. The mean distance from the sun, or the semi-major axis of the orbit. 3. The longitude of the ascending node. 4. The inclination of the plane of the orbit to that of the ecliptic. 5. Theeccentricity of the orbit. 6. The longitude of the perihelion. 7. The place of the planet in its orbit at a particular epoch. If the mass of a planet is either known or neglected, the mean distance can be computed from the periodic time by means of Kepler's third law, so that the number of independent elements is reduced to six. The orbits of the planets can not be determined in the same manner as the orbit of the moon, Art. 207,sbecause the centre of the earth may be regarded as a fixed point relative to the moon's orbit, but it is not fixed relative to the planetary orbits. The methods therefore employed for determining the orbits of the planets are in many respects quite different from those which are applicable to determining the orbit of the moon, and also that of the earth. 382. To find the periodic time. First method. —Each of the planets, during about half its revolution around the sun, is found to be on one side of the ecliptic, and during the other half on the other side. The period which elapses from the time that a planet is at one of its nodes, till it returns to the same node (allowance being made for the motion of the nodes), is the sidereal period of the planet. When a planet is at either of its nodes, it is in the plane of the ecliptic, and its latitude is then nothing. Let the right ascension and declination of a planet be observed on several successive days, near the period when it is passing a node, and let its corresponding longitudes and latitudes be computed. From these we may obtain, by a proportion, the time when the planet's latitude is nothing. If similar observations are made when the planet passes the same node again, we shall have the time of a revolution. Example.-The planet Mars was observed to pass its ascending node as follows: 1862, December, 5d. 22h. 17m. 1864, October, 22d. 21h. 58m. 206 ASTRONOMY. The interval is 686.986 days, which differs but a few minutes from the most accurate determination of its period. When the orbit of a planet is but slightly inclined to the ecliptic, a small error in the observations has a great influence on the computed time of crossing the ecliptic. A more accurate result will be obtained by employing observations separated by a long interval, and dividing this interval by the number of revolutions of the planet. 383. Second method.-The synodical period of a planet is the interval between two consecutive oppositions, or two conjunctions of the same kind. The sidereal period may be deduced from the synodical by a method similar to that of Art. 205. Let p be the sidereal period of a planet, p' the sidereal period of the earth, and s the time of a synodic revolution, all expressed in mean solar days. The daily motion of the planet, as seen from the sun, is 3600 360~ 3, while that of the earth is -7; and ifp be a superior planet, h360 360~ the earth will gain upon the planet daily, - But in a synodic revolution the earth gains upon the planet 360~; that is, 3600 its daily gain is. Hence we have the equation 360 360 360 P' s Hence sp-sp' =pp', or --,. s-p For an inferior planet, we shall find in like manner -sp' S +p 384. How to obtain the mean synodic period.-Since the angular motion of the planets is not uniform, the interval between two successive oppositions will not generally give the mean synodical period. But if we take two oppositions, separated by a long interval, when the planet was found in the same position relatively to some fixed star, and divide the interval by the number of revolutions, we may obtain the mean synodical period very accurately. ORBITS OF THE PLANETS. 207 Example.-The planet Mars was observed in opposition as follows: 1864, November, 30d. 17h. 58m. 1817, December, 8d. 9h. 15m. The interval is 17159.37 days, which divided by 22, the number of synodic revolutions, gives for the mean time of one synodic revolution 779.97 days. By comparing the observations of Ptolemy, A.D. 130, with recent observations, the time of one synodical revolution is found to be 779.936 days; from which, according to the formula given above, the mean sidereal period of Mars is found to be 686.980 days. And in the same manner the periods of the other planets may be found. The following table shows the time of a synodical, as well as of a sidereal revolution of the planets: Synodical Sidereal Revolution. Mean daily Motion. Days. Days. Mercury.. 115.877 87.969 or 3 months. 4~ 5' 32".6 Venus... 583.921 224.701 " 7- " 1 36 7.8 Earth.... 365.256 " 1 year. 0 59 8.3 Mars.... 779.936 686.980 " 2 years. 0 31 26.7 Jupiter.. 398.884 4332.585 " 12 " 0 4 59.3 Saturn. e 378.092 10759.220 " 29 0 2 0.6 Uranus.. 369.656 30686.821 " 84 " 0 0 42.4 Neptune.. 367.489 60126.722 " 164 " 0 0 21.6 385. To find the distance of a planet from the sun.-The mean distance of a planet, whose periodic time is known, can be computed by Kepler's third law. It can, however, be determined independently by methods like the following: The distance of an inferior planet from the sun may be determined by observing the angle of greatest elongation. In the triangle SEV, let S be the place of the sun, Fig. 101. E the earth, and V an inferior planet at the time of its greatest elongation. Then, since the angle SVE is a right angle, we have SV: SE:: sin. SEV: radius; or SV SE sin. SEV. If the orbits of the planets were exact circles, this method would give the mean distance of the planet from the sun; but since this is not the case, we must observe the greatest elongation in different parts of E 208 ASTRONOMY. the orbit, and thus obtain its average value. The average value of the greatest elongation of Venus is 46~ 20'; whence the mean distance of Venus is found to be.7233, the distance of the earth from the sun being called unity. 386. Distance of a superior planet.-The distance of a superior planet, whose periodic time is known, may be found by measurFig. 102. ing the retrograde motion of the _ —-,B planet in one day at the time of -s E M A opposition. Let S be the place of the sun, E the earth, and M the planet on the day of opposition, when the three bodies are situated in the same straight line. Let EE' represent the earth's motion in one day from opposition, and MM' that of the planet in the same time. The angles ESE' and MSM' are known from the periodic times. Draw E'B parallel to SM; join E'M', and produce the line to meet SM in A. The angle SAE', which equals AE'B, is the retrogradation of the planet in one day, and is supposed to be known from observations. In the triangle E'SM', the side E'S and the angle E'SM' are known, and E'M'S=M'SA+M'AS; from these we can compute SM'. If we only know the periodic time of the planet, we are obliged, in the first approximation, to assume the orbit to be a circle in order to compute the angle MSM'; but if we observe the retrograde motion at a large number of oppositions in different parts of the orbit, we may obtain the average value of the arc of retrogradation, and hence we may compute the mean distance. Example. The average arc of retrogradation of Mars on the day of opposition is 21' 25".7. If we take the mean daily motions of the earth and Mars, as given on page 207, we shall find the mean sin. 0' 348" distance of Mars to be isn.5' = -1.52369, the distance of the sin. 52' 52".4earth from the sun being called unity. The following table shows the mean distances of the planets from the sun, expressed in miles, and also their relative distances, the distance of the earth being called unity: ORBITS OF THE PLANETS. 209 Mean Distance from the Sun. Relative Distance. Mercury.. 37,000,000 miles. 0.387 Venus... 69,000,000 "0.723 Earth.. 95,000,000 " 1.000 Mars... 145,000,000 " 1.524 Jupiter... 496,000,000 " 5.203 Saturn.. 909,000,000 " 9.539 Uranus... 1,828,0000000 " 19.183 Neptune.. 2,862,000,000 " 30.037 387. Diameters of the planets. Having determined the distances of the planets, it is only necessary to measure their apparent diameters, and we can easily compute their absolute diameters in miles. The apparent diameters of the planets are of course variable, since they depend upon the distances which are continually varying. The following table shows the mean apparent diameters, and also the absolute diameters of the planets, as well as their volumes, that of the earth being called unity: Equatorial Diameters. ~______ ~_____~ - Volume. Apparent. In Miles. Mercury.. 7" 3,000 Venus... 17 7,700 Earth.. 7,926 1 Mars.. 7 4,500 Jupiter.. 38 92,000 1412 Saturn.... 17 75,000 770 Uranus.... 4 36,000 96 Neptune.. 2 35,000 90 388. To determine the position of the nodes of a planetary orbit.Let the longitude of a planet be determined when it is at one of its nodes; this longitude will be the geocentric longitude of the node. Also, by means of the solar tables, let the longitude of the sun and the radius vector of the earth be found for the time the planet is at the node. When the planet returns to the same node again, let its longitude be again determined, as also the longitude of the sun and the radius vector of the earth. From these data (the node in the interval being supposed to remain fixed) the position of the line of the nodes may be determined, and also the distance of the planet from the sun at the times of observation. Let S be the place of the sun, E the earth, and P a superior planet at its node; and let E' be the place of the earth after the 0 210 ASTRONOMY.; Fig. 103. - planet has made an entire revolution, and returned to the point P. Then from the solar tables we can determine SE and SE', as also the angle ESE'. Hence EE' can be computed, as also E nx^/ ~ the angles SEE', SE'E. Now, since the angles SEP, SE'P are determined by the observations, we can obtain the angles PEE', PE'E. Then, in the triangle PEE', having two angles and one side, we can compute PE. Hence, in X the triangle PES, we have two sides and the included angle, from which we can compute SP, and also the angle ESP, which, added to the longitude of the earth when at E, will give the heliocentric longitude of the planet when at its node. When observations of this kind are made at a considerable distance of time from one another, it is found that the nodes of every planet have a slow motion retrograde, or in a direction contrary to the order of the signs. The most rapid motion of the nodes is in the case of Mercury, amounting to about 70' in a century. 389. To determine the inclination of an orbit to the ecliptic.-Let the time at which the sun's longitude is the same Fig. 104. N -i>4' l as the heliocentric longitude of the node be found P/1 B by means of the solar tables, and let the longitude and latitude of the planet be determined at the same time. S Let NSE be the line of a planet's nodes, S the sun, E the earth, and P the planet's place in its E orbit. From E as a centre, with a radius PE, suppose a sphere to be described whose surface meets the line NE in B; and let PA be an arc of a great circle perpendicular to the ecliptic. Then PBA will be a spherical triangle right-angled at A; the angle PBA will measure the inclination of the plane of the planet's orbit to the ecliptic; PA will measure PEA, the geocentric:latitude of the planet; and AB will measure AEB, the difference between the longitudes of the sun and planet. Then, by Napier's rule, we have ORBITS OF THE PLANETS. 211 R x sin. AB = tang. PA cot. PBA; tang. PA or tang. PBA=-. A sin. AB3 that is, the tangent of the inclination of the orbit is equal to the tangent of the planet's geocentric latitude, divided by the sine of the planets elongation from the sun, the earth being in the line of the planet's nodes. If, at the time of observation, the elongation of the planet from the sun was 90~, its geocentric latitude wouldbe the inclination of its orbit to the ecliptic; and the results of this method will be the more reliable the farther the planet is from its node. The orbits of the planets have generally small inclinations to the ecliptic. The orbit of Mercury is inclined about 7~, while all the other planets (with the exception of the asteroids) are inclined less than 4~. Four of the asteroids have inclinations exceeding 20~, and one has an inclination of 34~. 390. To determine the heliocentric longitude and latitude of a planet. — When the place:of the ascending node and the inclination of the orbit of a planet are known, the heliocentric longitude and latitude of a planet, and also its radius vector, may be deduced from the geocentric longitude and latitude. Let S be the place of the sun, E Fig. 105. p the earth, P the planet, and NS the line of the nodes of the planet's or- bit. From P draw PB perpendicu- lar to the ecliptic, and let a plane pass through E, P, and B, intersecting the line of the nodes in N. With N as a centre, and NE as a radius, C let a sphere be described, cutting the planes PNS, ENS, and PNE in the right-angled spherical triangle AEC. The angle PEB will be the geocentric latitude of the planet, BES will be the difference between the longitudes of the planet and sun, and the spherical angle ACE will measure the inclination of the planet's orbit to the ecliptic. 1st. In the triangle NES, the angle NES is known, being the supplement of BES; also ESN can be derived from the solar tables when the place of the node is given, and ES is also known; hence we can compute EN, NS, and the angle ENS. 2d. In the spherical triangle AEC, right-angled at E, the angle 212 ASTRONOMY. ACE is given, and also EC, which measures ENC; hence AE, which measures ANE, can be computed. 3d. In the triangle PNE, we know NE, ENP, and NEP, the supplement of the planet's geocentric latitude; hence PN can be computed. 4th. In the right-angled triangle NPB, we know NP and the angle PNB; hence PB and NB can be computed. 5th. In the triangle BNS, NB, NS, and the angle BNS are known; hence we can compute SB and NSB, which is the difference between the heliocentric longitude of the planet and that of its node. Hence the heliocentric longitude of the planet is determined. 6th. In the right-angled triangle PBS, we know PB and BS, from which we can compute the angle PSB, the planet's heliocentric latitude, and also PS, its distance from the sun. 391. To determine the longitude of the perihelion, the eccentricity, etc.-Assuming the orbit of the planet to be an ellipse, if we determine, by Art. 390 or Art. 388, the length and position of three radii vectores of-the planet, we can determine the form and dimensions of the ellipse. ig. 06 A Let SB, SC, SD be three radii vectores of the planet, given in length and // \ position. Draw the lines BC, BD, and produce them, making SB SD:: BF: C ar \ DF; and SB: SC::BE:CE; then SB-SD: SB:: BD:BF= S. B SB -- SD and / as SB-SC: SB.::BC: BE — SC SB-SCG foIE th icG Then the straight line passing through the points E and F will be the directrix of the ellipse. For BH, CI, DK being drawn perpendicular to EF, the triangles BEH, CEI are similar; therefore BH: CI:: BE: GCE. Now, by construction, BE: CE.:SB: SC; hence BH: CI::SB:SC; or BH: SB::CI:SC; also BH: DK::BF: DF::SB: SD. Therefore the perpendiculars BH, CI, DK are always in the same proportion as the lines SB, SC, SD; consequently, EF is the directrix of the ellipse, passing through B, C, and D. (Geom., Ellipse, Prop. THE INFERIOR PLANETS, MERCURY AND VENUS. 213 22.) Through S draw ASG perpendicular to FE; take GA: AS::CI: CS,andGP: SP:: CI: CS; then CI+CS: CS:: GS:SP= SC x SG SC x SG and AS I-; then A and P will be the vertices CI+CS, CIf-CS' of the ellipse. The lengths of SP and SA can accordingly be computed; their sum gives the major axis; and their difference, MS, divided by the major axis, is the eccentricity of the ellipse. Also, in the triangle BSM, we know BS, SM, and BM=PA -SB; whence the angle BSA is determined, which gives the position of the major axis relatively to SB. CHAPTER XV. THE INFERIOR PLANETS, MERCURY AND VENUS.-TRANSITS. 392. Greatest elongations of Mercury and Venus.-Mercury and Venus having their orbits far within that of the earth, their elongation or angular distance from the sun is never great. They appear to accompany the sun, being seen in the west soon after sunset, or in the east a little before sunrise. Fig. 10. Let S be the place of the sun, MA the orbit of Mercury, E the place of the earth, and M the place of the planet when at its greatest elongation, at M _ a which time the angle EMS is a right angle. Since \\~ ] the distances of the planet and the earth from the sun both vary, the greatest elongation must also vary. The elongation will be the greatest possible when SM is greatest and SE is the least; that is, when Mercury is at its aphelion and the earth at perihelion. Combining the greatest value of SM with the least value of SE, we find the greatest possible value of Mercury's greatest elongation to be 28~ 20'. Combining the least value of SM with the greatest value of SE, we find the least possible value of Mercury's greatest elongation to be 17~ 36'. In a similar manner, we find the greatest elongation of Venus to vary from 45~ to 47~ 12'. 393. Phases of Mercury and Venus.-The planets Mercury and 214 ASTRONOMY. Venus exhibit to the telescope phases similar to those of the moon. At the greatest elongations eastward or westward, we see only half the disc illuminated, as in the case of our own satellite Fig. 10S. at first or last quarter. As they move toward the superior conjunction, at A, their form becomes gibbous, and the outline of the disc becomes more nearly circular the nearer they approach the superior conjunction. Owing to the intensity of the sun's light, we lose the planets for a little time before and after the conjunction, but on emerging from the sun's rays we find the form still gibbous. The illumined part diminishes as the planets approach their greatest elongation, near which time they again appear as a half moon; and as they advance toward the inferior conjunction, the form becomes more nearly that of a crescent, until it is again lost in the sun's rays at C. MERCURY. 394. Period, distance from sun, etc.-Mercury performs its revolution round the sun in a little less than three months; but its synodic period, or the time from one inferior conjunction to another, is 116 days. Its mean distance from the sun is 37 millions of miles. The eccentricity of its orbit is much greater than in the case of any other of the large planets. At perihelion Mercury is only 29 millions of miles from the sun, while in aphelion it is distant 44 millions, making a variation of 15 millions of miles, which is about one fifth of the major axis of the orbit. When between the earth and the sun, the disc of this planet subtends an angle of about twelve seconds of arc; but as the planet approaches the opposite part of the orbit, its breadth does not exceed five seconds. The real diameter of Mercury is about 3000 miles. THE INFERIOR PLANETS, MERCURY AND VENUS. 215 395. Visibility of Mercury.-Since the elongation of Mercury from the sun never exceeds 28~ 20', this planet is seldom seen except in strong twilight either morning or evening; and it does not ever appear conspicuous to the naked eye, although it sometimes shines with the brilliancy of a star of the first magnitude. Supposing the atmosphere clear, the other circumstances that favor its visibility are that the greatest elongation should occur at the season when the twilight is shortest; that it should then be near the aphelion of its orbit; and that its distance from the north pole should be several degrees less than that of the sun. 396. Greatest brightness of Mercury.-Mercury does not appear most brilliant when its disc is circular like a full moon, because its distance from us is then too great; neither when it is nearest to us, because then it appears as a thin crescent, and almost the entire illumined part is turned away from the earth. The greatest brightness must then occur at some intermediate point. Assuming the orbits of the planets to be circular, and that the quantity of light received at the earth varies directly as the area of the visible part of the planet, and inversely as the square of the distance from the earth, it has been computed that Mercury is brightest between its greatest elongation and superior conjunction, when the elongation from the sun is 22~. When the planet is seen after sunset, the greatest brightness occurs a few days before the greatest elongation; when it is seen before sunrise, the greatest brightness occurs a few days after the greatest elongation. 397. Rotation on its axis.-By observing Mercury with powerful telescopes, some astronomers think they have discovered indications of mountains on its surface, and by examining them at various times it has been concluded thatthe planet has a rotation upon its axis in 24h. 5m. 28s. Other astronomers, with equally good means of observation, have never remarked upon the planet's surface any spots by which they could approximate to the time of rotation. There is but little difference between the polar and equatorial diameters, the compression probably not exceeding -T O. VENUS. 398. Venus, the most brilliant of the planets, is generally called 216 ASTRONOMY. the evening or the morning star. The evening and morning star, or the Hesperus and Phosphorus of the Greeks, were at first supposed to be different. The discovery that they are the same is ascribed to Pythagoras. 399. Period, distance, and diameter.-Venus revolves round the sun in about 71 months; but its synodic period, or the time from one inferior conjunction to another, is 584 days, or about 19 months. Its mean distance from the sun is'69 millions of miles; and since the eccentricity of its orbit is very small, this distance is subject to but slight variation. The apparent diameter of Venus varies much more sensibly than that of Mercury, owing to the greater variation of its distance from the earth. At inferior conjunction its disc subtends an angle of about 64 seconds of arc, while at superior conjunction it is less than 10 seconds. The real diameter of Venus is about 7700 miles, or nearly the same as that of the earth. 400. Venus sometimes visible during the full light of day.-The greatest elongation of Venus from the sun amounts to 47~, and, on account of its proximity to the earth, it is, next to the sun and moon, the most conspicuous and beautiful object in the firmament. When it rises before the sun, it is called the morning star; when it sets after the sun, it is called the evening star. When most brilliant, it can be distinctly seen at midday by the naked eye, especially if at the time it is near its greatest north latitude. Its brightness is greatest about 36 days before and after inferior conjunction, its elongation being then about 40~, and the enlightened part of the disc not over a fourth part of the whole. At these periods the light is so great that objects illumined by it at night cast perceptible shadows. 401. Rotation on an axis.-Astronomers have frequently seen dusky spots upon Venus, which have been watched with the view of ascertaining the time of a rotation. It is concluded that this time is about 23h. 21m.; but these observations are exceedingly difficult on account of the glaring light of the planet. 402. Twilight on Venus.-By observing the concave edge of the crescent, which corresponds to the boundary of the illuminated TRANSITS OF MERCURY AND VENUS. 217 and dark hemispheres, it is found that there is a gradual fading away of the light into the darkness, caused probably by an at. mosphere illuminated by the sun and producing the phenomena of twilight. 403. Suspected satellite.-Several observers of the last two centuries concurred in maintaining that they had seen a satellite of Venus. But Sir W. Herschel perceived no traces of a satellite; neither did Schr6ter, though he was most assiduous in his observations of Venus. It is therefore probable that the supposed appearances recorded by former observers were illusive. TRANSITS OF MERCURY AND VENUS. 404. When either Mercury or Venus, being in inferior conjunction, has a distance from the ecliptic less than the sun's semi-diameter, it will appear projected upon the sun's disc as a black round spot. The apparent motion of the planet being then retrograde, it will appear to move across the disc of the sun from east to west, in a line sensibly parallel to the ecliptic. Such a phenomenon is called a transit of the planet. 405. When transits are possible.-Transits can only take place when the planet is within a small distance of its node. Let N be the node of the planet's orbit; S the Fig. 109. pi centre of the sun's disc on the eclip- r tic, and at such a distance from the node that the edge of the disc just touches the orbit, NP, of the planet. A transit can only take place when the sun's centre is at a less distance than NS from the node. The mean value of the sun's semi-diameter being 16', and the inclination of Mercury's orbit to the ecliptic being 7~, and that of Venus 3, we find that a transit of Mercury can only take place within 2~ 11' of the node, and a transit of Venus within 4~ 30'. 406. Transits of Mercury.-The longitudes of Mercury's nodes are about 460 and 226~, at which points the earth arrives about the 10th of November and the 7th of May. The transits of Mercury must therefore occur near these dates; those at the ascending node taking place in November, and those at the descending node in May. 218 ASTRONOMY. The following are the dates of the transits of Mercury for the remainder of the present century: 1868, November 4. 1891, May 9. 1878, May 6. 1894, November 10. 1881, November 7. 407. Intervals between the transits.-In each of these cases the interval between two transits at the same node is 13 years. The reason is that 13 revolutions of the earth are made in nearly the same time as 54 revolutions of Mercury. For 365.256 x13=4748.33. And 87.9692 x 54=4750.34. When, therefore, a transit has occurred at one node, after an interval of 13 years, the earth and Mercury will return to nearly the same relative situation in the heavens, and another transit may occur. Transits sometimes occur at the same node at intervals of 7 years, and a transit at either node is generally preceded or followed, at an interval of 31 years, by one at the other node. 408. Transits of Venus.-The longitudes of the nodes of Venus are about 75~ and 255~, at which points the earth arrives about the 5th of June and the 7th of December. The transits of Venus must therefore occur near these dates; those at the ascending node taking place in June, and those at the descending node in December. The following list contains all the transits of Venus, from that which took place in 1639 (the first that was ever known to have been seen by any human being) to the end of the present century: 1639, December 4. 1874, December 8. 1761, June 5. 1882, December 6. 1769, June 3. 09. Intervals between the transits.-The interval between two transits at the same node is either 8 or 235 years. The reason of the first interval is that 8 revolutions of the earth are accomplished in nearly the same time as 13 revolutions of Venus. For 365.256 x 8-2922.05. And 224.701x 13=2921.11. Hence a transit at either node is generally preceded or followed, at an interval of 8 years, by another at the same node. TRANSITS OF MERCURY AND VENUS. 219 The period of 235 years is still more remarkable. For 365.256 x 235 =85835.3. And 224.701 x 382=85835.7. Hence, after an interval of 235 years, during which time Venus has made 382 rtvolutions, the earth and Venus return almost exactly to the same relative situation in the heavens. 410. Sun's parallax and distance.-The transits of Venus are important from their supplying data by which the sun's distance from the earth can be determined with far greater precision than by any other known method. The transits of Mercury supply similar data, but much less reliable, on account of the greater distance of that planet from the earth. The relative distances of the planets from the sun may be computed by Kepler's third law, when we know their periods of revolution. In this manner we ascertain that the distances of the earth and Venus from the sun are in the ratio of 1000 to 723. Hence, when Venus is interposed between the earth and sun, the ratio of its distances from the earth and sun is that of 277 to 723. Fig. 110. Let the circle FHKG represent the sun's disc; let E represent the earth, and A and B the places of two observers supposed to be situated at the opposite extremities of that diameter of the earth which is perpendicular to the ecliptic; also, let V be Venus moving in its orbit in the direction represented by the arrow. At present we will disregard the earth's rotation; that is, we will suppose the positions A and B to remain fixed during the transit. The planet will then appear to the observer at A to describe the chord FG, and to the observer at B the parallel chord 1HK. Also, when to the observer at A the centre of the planet appears to be at D, it will to the observer at B appear to be at C. Now AB was supposed to be perpendicular to the plane of the ecliptic; and since the plane of the sun's disc is also very nearly 220: ASTRONOMY. perpendicular to the ecliptic, the line AB may be regarded as parallel to CD, and hence we have CD: AB:: DV: AV::723: 277:: 2.61:1. Therefore CD (expressed in miles) =2.61 AB. The apparent distance between the points C and D on the sun's surface may be derived from the observed times of beginning and ending of the transit at A and B. Let the observer at A note Fig. 11l. the time when the disc of the planet E/^~~^ ~ first appears to touch the sun's disc on the outside at L, and also the time when it first appears at M wholly within the 271laN 2 sun's disc. L is called the external, and / M the internal contact. Also, let both.~\ / ~ the internal and external contacts at N \x.. / and P be observed when the planet is ~ —-~ ~leaving the sun's disc. Then, since the planet's rate of motion as well as that of the sun is already accurately known from the tables, the number of seconds of a degree in the chord described by the planet can be ascertained. In the same manner, the number of seconds in the chord described by the planet as observed at B can be ascertained. Knowing the length of DG, which is the half of FG, and knowing also SG, the apparent radius of the sun, we can compute SD. In the same manner, from the length of the chord HK, we can compute SC. The difference between these lines is the value of CD, supposed to be expressed in seconds. But we have already ascertained the value of CD in miles. Hence we can determine the linear value of 1" at the sun as seen from the earth, which is found to be 462 miles; and hence the angle which the earth's radius subtends at the sun will be 3963 462,or 8".58. This angle is called the sun's horizontal parallax; and from it, when we know the radius of the earth, we can compute the distance of the earth from the sun. It is not necessary that the observers should be situated at the extremities of a diameter of the earth, but it is important that the two stations should differ widely in latitude; and allowance must also be made for the diurnal motion of the earth. The transit of Venus in 1769 was observed with the greatest care at a large number of stations, extending from Lapland, latitude 70~ 22' N., to Otaheite, latitude 17~ 25' S., and the value of THE SUPERIOR PLANETS. 221 the sun's parallax resulting from these observations (8".58) is that which, until recently, has generally been accepted by astronomers. The mean distance of the earth from the sun, resulting from this value of the sun's parallax, is 95,300,000 miles. An accurate knowledge of this distance is of the greatest importance, since it serves as our base line for estimating the distances of all bodies situated beyond the limits of our solar system. See Art. 551. As there is still some uncertainty respecting the exact value of this quantity, astronomers generally call the mean distance of the earth from the sun unity, and estimate all distances in the planetary system by reference to this unit. 411. Other determinations of the sun's parallax.-When Mars is on the same side of the sun with the earth, it approaches comparatively near to the earth, and has a large horizontal parallax. Observations on the position of Mars have repeatedly been made at various observatories, both in the northern and southern hemispheres, from which the parallax of this planet has been deduced; and hence the parallax of the sun is easily computed, since the relative distances of the earth and Mars from the sun may be determined from the times of revolution. The horizontal parallax of the sun which has been deduced from these observations is 8".95. Considerations derived from the known velocity of light have led to nearly the same result; and it seems therefore probable that the value of the sun's horizontal parallax deduced from the last transit of Venus, and which has hitherto been generally received, will require to be somewhat increased. The effect will be to diminish slightly all the distances and magnitudes of the bodies of the solar system, except such as refer to the earth and moon. CHAPTER XVI, THE SUPERIOR PLANETS.-THEIR SATELLITES. 412. How the superior planets are distinguished from the inferior. -The superior planets, revolving in orbits without that of the earth, never come between us and the sun-that is, they have no inferior conjunction; but they are seen in superior conjunction and in opposition. Nor do they exhibit to us phases like those 222 ASTRONOMY.; of Mercury and Venus. The disc of Mars, about the period of his quadratures, appears decidedly gibbous; but the other planets are so distant that their enlightened surfaces are always turned almost entirely toward the earth, and the gibbous form is not perceptible. MARS. 413. Distance, period, etc.-The mean distance of Mars from the sun is 145 millions of miles; but, on account of the eccentricity of its orbit, this distance is subject to a variation of nearly one tenth its entire amount. Its greatest distance from the sun is 158 millions of miles, and its least distance 132 millions. The distance'of this planet from the earth at opposition is sometimes reduced to 35 millions of miles, while at conjunction it is sometimes as great as 255 millions. Its apparent diameter varies in the same ratio, viz., 31" to 24". Mars makes one revolution about the sun in 687. days; but its synodic period, or the interval from opposition to opposition, is 780 days. The inclination of its orbit to the plane of the ecliptic is 1" 51'. The real diameter of this planet is 4500 miles, and its volume about one fifth that of the earth. 414. Phases, rotation, etc.-At opposition and conjunction, the same hemisphere being turned to the earth and sun, the planet Fig. 112. - appears like a full moon, as shown at MI and M5. In,/5x -IrY^ ^all other positions it appears slightly gibbous; but the deficient portion never exceeds about one ninth of a hemisphere. \ / A When viewed with a good ~\X~ M ^-6