PLATE L 
 
 COMPARATIVE MAGNITUDES OF THE PLANETS, 
 
 1, MERCURY. 5. URANUS. 
 
 2, MARS. 6, NEPTUNE. 
 
 3, VENUS 7, SATURN. 
 
 4, EARTH 8. JUPITER. 
 
THE 
 
 ELEMENTS 
 
 THEORETICAL AND DESCRIPTIVE 
 
 ASTRONOMY, 
 
 the tt 
 
 BY 
 
 CHARLES J. WHITE, A.M., 
 / 1 
 
 ASSISTANT PROFESSOR OF MATHEMATICS IN HARVARD COLLEQB 
 
 FOURTH EDITION, REVISED. 
 
 PHILADELPHIA: 
 
 CLAXTON, EEMSEN & HAFFELFINGER. 
 
 1880. 
 
NA/S 
 
 Entered according to Act of Congress, in the year 1869, by 
 CHARLES J. WHITE, 
 
 ir> the Clerk's Office of the District Court of the United States for the District, of 
 Maryland. 
 
PREFACE TO THE THIRD EDITION. 
 
 first edition of this work was published in 1869, to meet 
 the requirements of the students of the United States Naval 
 Academy. In preparing it, I endeavored to present the main 
 facts and principles of Astronomy in a form adapted to the 
 elementary course of instruction in that science which is com- 
 monly given at colleges and the higher grades of academies. I 
 selected those topics which seemed to me to be the most important 
 and the most interesting, and arranged them in the order which 
 experience had led me to believe to be the best. The third edi- 
 tion of the book is issued with no change in the general plan, 
 and with only those changes which the advance of astronomical 
 knowledge in the last six years renders necessary. 
 
 In the descriptive portions of the work, I have endeavored to 
 give the latest information upon every topic which is introduced. 
 On not a few points the opinions of competent observers are by 
 no means the same ; and on these points I have endeavored to 
 give, as far as possible, the various opinions which now exist. 
 The distances and the dimensions of the heavenly bodies are 
 given to correspond with the value of the solar parallax which 
 is at present adopted in the American Ephemeris ; the recent 
 theories upon the connection of comets and meteors, the principles 
 of spectroscopic observation, and the conclusions concerning the 
 
IV PREFACE. 
 
 
 
 constitution and the movements of the heavenly bodies which 
 such observation induces, are given, it is hoped, in sufficient 
 detail. 
 
 No clear conception of the processes Dy which most of the 
 fundamental truths of Astronomy have been established can be 
 attained without some knowledge of Mathematics. I have en- 
 deavored, however, to confine the theoretic discussions within the 
 limits of such moderate mathematical knowledge as may fairly 
 be expected in those readers for whom the treatise is intended. 
 Certain definitions and formulae, with which the student may 
 possibly not be familiar, will be found in the Appendix ; and, 
 with this aid, I believe that every portion of the work can be 
 read without difficulty. 
 
 In the preparation of the work, many authorities have been 
 consulted ; the principal ones being Chauvenet's Manual of 
 Spherical and Practical Astronomy, and Chambers's Descriptive 
 Astronomy. The treatise has been used as a text-book in the 
 United States Naval Academy, the Massachusetts Institute of 
 Technology, Harvard College, and other institutions ; and I am 
 indebted to officers of these institutions for many valuable sug- 
 gestions as to errors and improvements. I trust that this new 
 edition will be found to be free from mistakes, and that it will be 
 useful, not only to the class of students for whom it is especially 
 prepared, but to others who may wish to know the general prin- 
 ciples and the present state of the science of Astronomy. 
 Harvard College, Cambridge, Mass., 1875. 
 
CONTENTS. 
 
 The Greek Alphabet PAGE ix 
 
 CHAPTER I. 
 
 GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS. THE 
 CELESTIAL SPHERE. 
 
 The heavenly bodies. Astronomy. Form of the earth. Diurnal mo- 
 tions of the heavenly bodies. Eight, parallel, and oblique spheres. 
 Definitions. Theorems. The astronomical triangle. Spherical co- 
 ordinates. Vanishing lines and circles. Spherical projections 11 
 
 CHAPTER II. 
 
 ASTRONOMICAL INSTRUMENTS. ERRORS. 
 
 The clock : its error and rate. The chronograph. The transit instru- 
 ment: its construction, adjustment, and use. The meridian circle. 
 The reading microscope. Fixed points. The mural circle. The 
 altitude and azimuth instrument. Method of equal altitudes. The 
 equatorial. The sextant. The artificial horizon. The vernier. 
 Other astronomical instruments. Classes of errors 28 
 
 CHAPTER III. 
 
 REFRACTION. PARALLAX. DIP OF THE HORIZON. 
 
 General laws of refraction. Astronomical refraction. Geocentric and 
 heliocentric parallax. The dip of the horizon 51 
 
 CHAPTER IV. 
 
 THE EARTH. ITS SIZE, FORM, AND ROTATION. 
 
 Measurement of arcs of the meridian by triangulation. Spheroidal 
 form of the earth. Its dimensions. Its volume, density, and 
 weight. Rotation of the earth. Change of weight in different 
 latitudes. Centrifugal force. The trade-winds. Foucault's pendu- 
 lum experiment. Linear velocity of rotation 59 
 
Vi CONTENTS, 
 
 CHAPTEK V. 
 
 LATITUDE AND LONGITUDE. 
 
 Four methods of finding the latitude of a place. Latitude at sea. 
 Reduction of the latitude. Longitude. Greenwich time by chro- 
 nometers, celestial phenomena, and lunar distances. Difference of 
 longitude by electric and star signals. Longitude at sea. Compa- 
 rison of the local times of different meridians PAGE 72 
 
 CHAPTEK VI. 
 
 TIIE SUN. THE EARTH'S ORBIT. THE SEASONS. TWILIGHT. THE 
 ZODIACAL LIGHT. 
 
 The eciiptic. Distance of the sun from the earth determined by transits 
 of Venus. Magnitude of the sun. The earth's orbit about the sun. 
 The seasons. Twilight. Rotation of the sun, and its constitution. 
 The zodiacal light 83 
 
 CHAPTER VII. 
 
 SIDEREAL AND SOLAR TIME. EQUATION OF TIME. THE CALENDAR. 
 
 The sidereal and the solar year. Relation of sidereal and solar time. 
 The equation of the centre. The equation of time. Astronomical 
 and civil time. The calendar 100 
 
 CHAPTER VIII. 
 
 UNIVERSAL GRAVITATION. PERTURBATIONS IN THE EARTH'S ORBIT. 
 ABERRATION. 
 
 The law of universal gravitation. The mass of the sun. The earth's 
 motion at perihelion and aphelion. Kepler's laws. Precession. 
 Nutation. Change in the obliquity of the ecliptic. Advance of the 
 line of apsides. Diurnal and annual aberration. Velocity of light. 
 Aberration a proof of the earth's revolution 107 
 
 CHAPTER IX. 
 
 THE MOON. 
 
 The orbit of the moon, and perturbations in it. Variation of the 
 moon's^meridian zenith distance. Distance, size, and mass of the 
 moon. Augmentation of the semi-diameter. The phases of the 
 moon. Sidereal and synodical periods. Retardation of the moon. 
 The harvest moon. Rotation ; librations and other perturbations. 
 Tlu- lunar cycie. General description of the moon 120 
 
CONTENTS. Vll 
 
 CHAPTEK X. 
 
 LUNAR AND SOLAR ECLIPSES. OCCULTATIONS. 
 
 Lunar eclipses. The earth's shadow. Lunar ecliptic limits. Solar 
 eclipses. The moon's shadow. Solar ecliptic limits. Cycle and 
 number of eclipses. Occultations. Longitude by solar eclipses and 
 occultations PAGE 135 
 
 CHAPTEK XI. 
 
 THE TIDES. 
 
 Cause of the tides. Effect of the moon's change in declination. General 
 laws. Influence of the sun. Priming and lagging of tides. The 
 establishment of a port. Cotidal lines. Height of tides. Tides in 
 bays, rivers, &c. Four daily tides. Other phenomena 146 
 
 CHAPTER XII. 
 
 THE PLANETS AND THE PLANETOIDS. THE NEBULAR HYPOTHESIS. 
 
 Apparent motions of the planets. Heliocentric parallax. Orbits of 
 the planets. Inferior planets. Direct and retrograde motion. Sta- 
 tionary points. Evening and morning stars. Elements of a planet's 
 orbit. Heliocentric longitude of the node. Inclination of the orbit. 
 Periodic time. Mercury. Venus. Transits of Venus. Superior 
 planets: their periodic times and distances. Mars. The minor 
 planets. Bode's law. Jupiter: its belts, satellites, and mass. Saturn 
 and its rings. Disappearance of the rings. Uranus. Neptune. 
 The nebular hypothesis 155 
 
 CHAPTEE XIII. 
 
 COMETS AND METEORIC BODIES. 
 
 General description of comets. The tail. Elements of a comet's orbit. 
 Number of comets and their orbits. Periodic times. Motion in 
 their orbits. Mass and density. Periodic comets. Encke's comet. 
 Winnecke's or Pons's comet. Brorsen's comet. Biela's comet. 
 D' Arrest's comet. Faye's comet. Mechain's comet. Halley's 
 comet. Remarkable comets of the present century. The great 
 comet of 1811. The great comet of 1843. Donati's comet. The 
 great comet of 18G1. Meteoric bodies. Shooting stars. The No- 
 vember showers. Height and orbits of the meteors. Detonating 
 meteors. Aerolites. Connection of comets and meteoric bodies 187 
 
Viii CONTENTS. 
 
 CHAPTEK XIV. 
 
 THE FIXED STARS. NEBULA. MOTION OF THE SOLAR SYSTEM. REAL 
 MOTIONS OF THE STARS. 
 
 Proper motions of the fixed stars. Magnitudes. Constellations. 
 Constitution of the stars. Distance of the stars. Bessel's differen- 
 tial observations. Heal magnitudes of the stars. Variable and tem- 
 porary stars. Double and binary stars. Colored stars. Clusters. 
 Kesolvable and irresolvable nebulae. Annular, elliptic, spiral, and 
 planetary nebulae. Nebulous stars. Double nebulae. The Ma- 
 gellanic clouds. Variation of brightness in nebulre. The milky way. 
 Number of the stars. Motion of the solar system in space. Real 
 motions of stars detected with the spectroscope PAGE 213 
 
 APPENDIX. 
 
 Mathematical definitions, theorems, and formulae 242 
 
 Kirkwood's law 247 
 
 Chronological history of astronomy 248 
 
 Sketch of the history of navigation 255 
 
 Table I. Elements of the planets, the sun, and the moon 256 
 
 II. The earth... 257 
 
 " III. The moon 257 
 
 " IV. Elements of the satellites 258 
 
 . " V. The minorplanets 259 
 
 " VI. Schwabe's observations of the solar spots 261 
 
 " VII. Elements of the periodic comets 261 
 
 " VIII. Transits of the inferior planets 262 
 
 " IX. Stars whose parallax has been determined 262 
 
 " X. The constellations 263 
 
 " XI. Examples of variable stars 266 
 
 " XII. Examples of binary stars 267 
 
 [NDEX .. 269 
 
THE GKEEK ALPHABET. 
 
 The following table of the small letters of this alphabet is 
 given for the use of those readers who are unacquainted with 
 the Greek language. 
 
 a 
 
 Alpha. 
 
 V 
 
 Nu. 
 
 P 
 
 Beta. 
 
 5 
 
 Xi. 
 
 v 
 
 Gamma. 
 
 
 
 Omicron. 
 
 $ 
 
 Delta. 
 
 7t 
 
 Pi. 
 
 e 
 
 Epsilon. 
 
 P 
 
 Rho. 
 
 ? 
 
 Zeta. 
 
 (7 
 
 Sigma. 
 
 n 
 
 Eta. 
 
 * 
 
 Tau. 
 
 $or 
 
 Theta. 
 
 U 
 
 Upsllon. 
 
 i 
 
 Iota. 
 
 <P 
 
 Phi. 
 
 X 
 
 Kappa. 
 
 z 
 
 Chi. 
 
 a, 
 
 Lambda. 
 
 $ 
 
 Psi. 
 
 P 
 
 Mu. 
 
 G) 
 
 Omega. 
 
ASTRONOMY. 
 
 CHAPTER I. 
 
 GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS. 
 
 1. The heavenly bodies are the sun, the planets, the satellites 
 of the planets, the comets, the meteors, and the fixed stars. 
 
 The planets revolve about the sun in elliptical orbits, and the 
 satellites revolve in similar orbits about the planets. The earth 
 is a planet, as we shall see hereafter, and the moon is its satel- 
 lite. The comets revolve about the sun in orbits which are 
 either ellipses, parabolas, or hyperbolas. Comparatively little 
 is known with any degree of certainty about the meteors; but 
 it is probable that they too revolve about the sun. 
 
 The sun, the planets, the satellites, and the comets constitute 
 what is called the solar system. The fixed stars are bodies which 
 lie outside of this system, and preserve almost precisely the 
 same configuration from year to year. 
 
 The heavenly bodies may be considered to be projected upon 
 the concave surface of a sphere of indefinite radius, the eye of 
 the observer being at the centre of the sphere. This sphere is 
 called the celestial sphere. 
 
 2. Astronomy is the science which treats of the heavenly 
 bodies. It may be 'divided into Theoretical, Practical, and De- 
 scriptive Astronomy. 
 
 Theoretical Astronomy may be divided into Spherical and 
 Physical Astronomy. 
 
 Spherical Astronomy treats of the heavenly bodies when con- 
 sidered to be projected upon the surface of the celestial sphere. 
 It embraces those problems which arise from the apparent diur- 
 
 ll 
 
12 FORM OF THE EARTH. 
 
 nal^inpiion pf^fejfeay.enly bodies, and also those which arise 
 from. anyichaiiges in. the apparent positions of these bodies upon 
 
 ffca ceiestiaL&phere. 
 Physical Astronomy treats of the causes of the motions of the 
 heavenly bodies, and of the laws by which these motions are 
 governed. 
 
 Practical Astronomy treats of the construction, adjustment, 
 and use of astronomical instruments. 
 
 Descriptive Astronomy includes a general description of the 
 heavenly bodies; of their magnitudes, distances, motions, and 
 configuration ; of their appearance and structure ; of, in short, 
 every thing relating to these bodies which comes from observa- 
 tion or calculation. 
 
 3. Form of the Earth. We may assume, at the outset, that 
 the form of the earth is very nearly that of a sphere. The fol- 
 lowing are some of the reasons which may be given for such an 
 assumption : 
 
 (1.) If we stand upon the sea-shore, and watch a ship which 
 is receding from the land, we shall find that the topmasts remain 
 in sight after the hull has disappeared. If the surface of the 
 sea were merely an extended plane, this would not happen ; for 
 the topmasts, being smaller in dimensions than the hull, would 
 
 in that case disappear first. 
 The supposition that the sur- 
 face of the sea is curved, 
 however, fully accounts for 
 this phenomenon, as may be 
 seen in Fig. 1. Let the curve 
 Fig.i. CBG represent a portion of 
 
 the earth's surface, and let A 
 
 be the position of the observer's eye : it is at once evident that 
 no portion of the ship, S, will be visible which is situated below 
 the line AH, drawn from A tangent to the earth's surface at C. 
 The same figure also shows why it is that when a ship is ap- 
 proaching land any object on shore can be seen from the top- 
 masts before it is seen from the deck. 
 
 (2.) At sea, the visible horizon everywhere appears to be a 
 circle. This also is easily explained on the supposition that the 
 
DIURNAL MOTION. 
 
 13 
 
 eartli is spherical in form; for if, in Fig. 1, the line AH is 
 turned about the point A, and is continually tangent to the 
 sphere, the points of tangency, C, D, E, &c., will form the visible 
 horizon of the spectator, and will evidently constitute a circle. 
 
 (3.) A lunar eclipse occurs when the earth is situated between 
 the moon and the sun. Now the shadow which the earth at 
 such a time casts upon the moon is invariably circular in form : 
 and a body which in every position casts a circular shadow 
 must be a sphere. 
 
 4. Diurnal motion of the heavenly bodies. Two things will be 
 noticed by an observer who watches the heavens during any 
 clear night. The first is, that all the heavenly bodies, with the 
 exception of the moon and the planets, retain constantly the 
 same relative situation ; and the second is, that all these bodies, 
 without any exception whatever, are continually changing their 
 positions with reference to the horizon. Let us suppose the 
 observer to be at some place in the Northern Hemisphere. A 
 plane passed tangent to the earth's surface at his feet will be his 
 
 Fig. 2. 
 
 sensible horizon, and a second plane, parallel to this, passed 
 through the centre of the earth, will be his rational horizon. If 
 
14 DIURNAL MOTION. 
 
 these two planes oe indefinitely extended in every direction, they 
 will intersect the surface of the celestial sphere in two circles; 
 but the radius of the celestial sphere is so immense in compari- 
 son with the radius of the earth, that these two circles will 
 sensibly coincide, and will form one great circle of the sphere, 
 called the celestial horizon. In other words, the earth, when 
 compared with the celestial sphere, is to be regarded as only a 
 point at its centre. 
 
 Let Fig. 2, then, represent the celestial sphere, at the centre 
 of which, 0, the observer is stationed. Let the circle HESW be 
 his celestial horizon, of which His the north point, S the south, 
 E the east, and W the west. 
 
 If he looks towards the southern point of the horizon, and 
 watches the movements of some star which rises a little east 
 of south, he will see that it rises above the horizon in a cir- 
 cular path for a little distance, attains its greatest elevation 
 above the horizon when it bears directly south, and then 
 descends and passes below the horizon a little west of south. 
 In the figure, abc represents the path of such a star. If he 
 notices a star which rises more to the eastward, as at d, for 
 instance, he will see that it also passes from the east quarter of 
 the horizon to the west in a circular path, attaining, however, a 
 greater altitude above the horizon than that which the star a 
 attained, and remaining above the horizon a longer time. A 
 star which rises in the east point will set in the west point, and 
 will remain twelve hours above the horizon. Turning his atten- 
 tion to some star which rises between the north and the east, as 
 the star g, for instance, he will find that its movements are simi- 
 lar to the movements of the stars already noticed, and that it 
 will remain above the horizon for more than twelve hours. 
 Finally, if he turns towards the north, he will see certain stars, 
 called circumpolar stars, which never pass below the horizon, but 
 continually revolve about a fixed point in the heavens, very near 
 to which point is a bright star called the Pole-star, which, to the 
 naked eye, appears to be stationary, though observation shows 
 that it also revolves about this same fixed point. In the figure, 
 Im represents the orbit of a circumpolar star. 
 
 If the same course of observations be repeated on the follow- 
 
DIURNAL MOTION. 15 
 
 ing night, the observer will find that the situations of the stars 
 with reference both to each other and to the horizon are the 
 same that they were when he first began to examine them ; the 
 motions which have already been described will be repeated, the 
 circumpolar stars will still revolve about the same point in the 
 heavens, and, in short, all the phenomena of which he took 
 note will again be exhibited. 
 
 5. Inferences. Three important truths are proved by a series 
 of observations similar to that which we suppose to have been 
 made. 
 
 (1.) The points at which the path of each star intersects the 
 horizon remain unchanged from night to night, as long as the 
 geographical position of the observer remains the same. 
 
 (2.) All the stars, whether they move in great or in small 
 circles, make a complete revolution in identically the same in- 
 terval of time; that is to say, in twenty-four hours. 
 
 (3.) If we call that point about which the circumpolar stars 
 appear to revolve the north pole of the heavens, and call the 
 right line drawn from this point through the common centre of 
 the earth and the celestial sphere the axis of the celestial sphere, 
 the planes of the circles of all the stars are perpendicular to this 
 axis. 
 
 Whether, then, the earth remains at rest, and the celestial 
 sphere rotates about its axis, as above defined, or the celestial 
 sphere remains at rest, and the earth rotates within it on an axis 
 of its own, one thing is certain : the axis of rotation preserves 
 in either case a constant direction. If the celestial sphere 
 rotates about its axis, this axis always passes through the same 
 points of the earth's surface ; and if the earth rotates within the 
 celestial sphere, its axis of rotation is constantly directed to the 
 same points on the surface of the celestial sphere. 
 
 We shall see hereafter the reasons which have led to the 
 adoption of the theory that these apparent motions of the stars 
 are really due to the rotation of the earth upon its own axis. 
 At present, for the sake of convenience in description, we shall 
 Consider the earth to be at rest, and shall speak of the apparent 
 motions of the heavenly bodies as though they were real. 
 
 6. Farther observations. Let us now suppose that the observer 
 
16 DIURNAL MOTION. 
 
 leaves the place where he has hitherto been stationed, and travels 
 in the direction of the point about which the circumpolar stars 
 have appeared to revolve, and which we have called the north 
 pole of the heavens. The general character of the phenomena 
 which he observes will not be changed ; but he will notice a 
 change in this respect : the elevation of the north pole above 
 the horizon will continually increase as he travels towards it, 
 and the planes of the circles of the stars, remaining constantly 
 perpendicular to the axis of the celestial sphere, will become 
 less and less inclined to the plane of the horizon. The conse- 
 quence of this will be that stars which are near the southern 
 point of the horizon will remain a shorter time above the hori- 
 zon, and will finally cease to appear ; while in the northern 
 quarter of the heavens the number of stars which never pass 
 below the horizon will continually increase. Finally, if we sup- 
 pose the observer to go on until the north pole is directly above 
 his head, the stars which he sees will neither set nor rise, but 
 will continually move about the sphere in circles whose planes 
 are parallel to the plane of the horizon. In such a situation, it 
 is evident that half of the celestial sphere will be perpetually 
 invisible to him. Referring to Fig. 2, such a state of things is 
 represented by supposing the line OP to be moved up into coin- 
 cidence with OZ, the planes of the circles abe, def, &c., still re- 
 maining perpendicular to OP. The stars a and d will lie con- 
 tinually below the horizon, and the stars g and I continually 
 above it. 
 
 Such a sphere as this just now described, where the planes of 
 revolution are parallel to the plane of the horizon, is called a 
 parallel sphere. 
 
 If the observer, instead of travelling towards the north pole, 
 travels directly from it, its elevation above the horizon will con- 
 tinually decrease, and the obliquity of the planes of the circles 
 to the plane of the horizon will continually increase, until he 
 will at length reach a point at which the north pole will lie in 
 his horizon and the planes of ths circles will be perpendicular to 
 its plane. Referring again to Fig. 2, the line PO will in this 
 case coincide with OH, and the arcs abc, def, &c., will have 
 their planes perpendicular to the plane of the horizon. It is 
 
.DEFINITIONS. 17 
 
 evident that at this point every star in the celestial sphere will 
 come above the horizon once in twenty-four hours, and that half 
 of every circle will lie above the horizon, and half below it. 
 
 The geographical position which the observer has now reached 
 is some point on the earth's equator (Art. 7). Such a sphere as 
 this, where the planes of the circles are perpendicular to the plane 
 of the horizon, is called a right sphere. Besides the right and 
 the parallel sphere, we have also the oblique sphere, where the 
 planes of the circles are oblique to the plane of the horizon. 
 Such a sphere is represented in Fig. 2. 
 
 If the observer travels still farther in the same direction, the 
 north pole will sink below his horizon, and the other extremity 
 of the axis of the celestial sphere, called the south pole, will 
 rise above it. There will be circumpolar stars revolving about 
 this pole, and, in brief, all the phenomena which the observer 
 noticed while travelling towards the north pole will be repeated 
 as he travels towards the south pole. 
 
 DEFINITIONS. 
 
 7. We are now prepared to define certain points, angles, and 
 circles on the earth and on the celestial sphere. 
 
 The axis of the celestial sphere is, as we have already seen, an 
 imaginary line drawn from the north pole of the heavens through 
 the common centre of the earth and of the celestial sphere, 
 and produced until it again meets the surface of the celestial 
 sphere in the south pole. The points where this axis meets the 
 surface of the earth are called the north pole and the south pole 
 of the earth. 
 
 That pole of the heavens which is above the horizon at any 
 place is called the elevated pole at that place; the other is called 
 the depressed pole. 
 
 The axis of the earth is that diameter which passes through 
 the poles of the earth. 
 
 Thf earth's equator is a great circle of the earth, whose plane 
 is perpendicular to the axis. This circle is, of course, equidis- 
 tant from the two poles, and divides the earth into two hemi- 
 spheres. That hemisphere which contains the north pole is 
 
18 DEFINITIONS. 
 
 called the northern hemisphere, the other is called the southern 
 hemisphere. 
 
 Parallels of latitude are small circles of the earth whose planes 
 tire perpendicular to the axis. 
 
 Terrestrial meridians are great circles of the earth passing 
 tli rough the poles. 
 
 The latitude of any place on the earth's surface is its angular 
 distance from the plane of the equator. This angle is measured 
 by the arc of the meridian included between the place and the 
 equator. Latitude is reckoned, either north or south, from 
 to 90. 
 
 The longitude of any place is the inclination of its own meri- 
 dian to the meridian of some fixed station, and is measured by 
 the arc of the equator included between these two meridians. 
 Longitude is usually reckoned east or west of the fixed meri- 
 dian, from (P to 180. The meridian of Greenwich, England, 
 is most commonly taken for the fixed meridian, though the 
 meridians of Washington, Paris, and other places are also taken 
 for the same purpose. The fixed meridian is called the prime 
 meridian. 
 
 The arc of a parallel of latitude included between any two 
 meridians is the departure between those meridians for that lati- 
 tude. It is evident that if the departure between any two meri- 
 dians is taken on two different parallels of latitude, the depart- 
 ure at the greater latitude will be the smaller. 
 
 Upward motion at any place is motion from the centre of tho 
 earth. Downward motion is towards the same point. 
 
 The celestial horizon has already been defined (Art. 4). Lines 
 drawn perpendicular to the plane of the horizon are called 
 vertical lines. The vertical line at any place, if indefinitely 
 prolonged, meets the celestial sphere in two points. The upper 
 point is called the zenith, the lower point the nadir. 
 
 The celestial meridian of any place is the great circle in which 
 the plane of the terrestrial meridian of that place, when indefi- 
 nitely produced, meets the surface of the celestial sphere. The 
 axis of the sphere divides the celestial meridian into two semi- 
 circumferences : that which lies on the same side of the axis as 
 the zenith is called the upper branch of the meridian, und tho 
 
DEFINITIONS. 19 
 
 other is called the lower branch. The points where the celestial 
 meridian and the celestial horizon intersect are called the north 
 and the south point of the horizon, the point which is the nearer 
 to the north pole being the north point. The line in which 
 the planes of these same two great circles intersect is called the 
 meridian line. 
 
 Vertical circles are great circles of the sphere which pass 
 through the zenith and nadir. That vertical circle the plane 
 of which is perpendicular to the plane of the celestial meridian 
 is called the prime vertical. The points in which the prime ver- 
 tical cuts the horizon are called the east and the west point of the 
 horizon ; and the line which joins these two points is the east 
 and west line. 
 
 The celestial equator, also called the equinoctial, is the great 
 circle of the sphere in which the plane of the earth's equator, 
 indefinitely produced, meets the celestial sphere. 
 
 The altitude of a heavenly body is its angular distance above 
 the plane of the celestial horizon, measured on a vertical circle 
 passing through that body. The zenith distance of the body is 
 its angular distance from the zenith, and is evidently the com- 
 plement of the altitude. 
 
 The azimuth of a celestial body is the inclination of the verti- 
 cal circle which passes through the body to the celestial meri- 
 dian, and is measured by the arc of the celestial horizon in- 
 cluded between this vertical circle and the celestial meridian. 
 Azimuth may be reckoned from either the north or the south 
 point of the horizon, and towards either the west or the east. 
 Navigators usually reckon it from the north point in north lati- 
 tude and the south point in south latitude, and east or west as 
 the body is east or west of the meridian, thus restricting it 
 numerically to values less than 180. Astronomers, on the con- 
 trary, usually reckon from the south point, to the right hand, 
 from to 36(T. 
 
 The amplitude of a heavenly body is its angular distance from 
 the prime vertical when in the horizon. It is reckoned from 
 the east point when the body is rising, and from the west point 
 when the body is setting, towards the north or the south as the 
 hotly i& to the north or the south o r the prime vertical. 
 
20 DEFINITIONS. 
 
 The vernal equinox is a certain fixed point upon the equinoc- 
 tial. It is also called the first point of Aries. 
 
 Hour circles, or circles of declination, are great circles of the 
 sphere passing through the poles of the heavens. 
 
 The right ascension of a heavenly body is the inclination of 
 its hour circle to the hour circle which passes through the vernal 
 equinox ; or it is the arc of the equinoctial intercepted between 
 these two hour circles. Right ascension is usually reckoned in 
 hours, minutes, and seconds (an hour being taken equal to 15 
 of arc), and is always reckoned to the eastward, from Oh. to 24h. 
 
 The declination of a heavenly body is its angular distance 
 from the plane of the celestial equator, measured on the circle 
 of declination passing through the body. It is reckoned in 
 degrees, minutes, &c., to the north and the south. The polar 
 distance of a body is its angular distance from either pole, mea- 
 sured on its hour circle. Usually, however, when we speak of 
 the polar distance of a body, we mean its angular distance from 
 the elevated pole. 
 
 If, in Fig. 2, with the arc HP, which measures the altitude 
 of the elevated pole, as a polar radius, we describe a circle about 
 the pole as a centre, it is evident that the stars whose circles lie 
 within this circle will never set. This circle is called the circle 
 of perpetual apparition. It is equally evident that stars whose 
 circles lie within a circle of the same magnitude, described about 
 the depressed pole as a centre, will never come above the horizon. 
 This circle is therefore called the circle of perpetual occupation. 
 
 The passage of a celestial body across the meridian is called 
 its transit or culmination. When the body is within the circle 
 of perpetual apparition, both transits occur above the horizon, 
 one above the pole, the other below it. These are called the 
 upper and the lower transit. For all bodies outside this circle, 
 and not within the circle of perpetual occultation, the upper 
 transit occurs above the horizon, the lower below it. For all 
 bodks whatever, the upper transit occurs when the body crosses 
 the upper branch of the meridian, and the lower transit when 
 it crosses the lower branch. 
 
 The hour angle of a heavenly body is the inclination of the 
 circle of declination which passes through the body to the 
 
CELESTIAL SPHERE. 
 
 21 
 
 celestial meridian, and is measured by the arc of the celestial 
 equator included between these two circles. Hour angles are 
 reckoned positively towards the west, from the upper culmina- 
 tion, from to 360, or Oh. to 24h. 
 
 The hour angle of the sun is called solar time, and that of the 
 first point of Aries sidereal time. The interval of time between 
 two consecutive upper transits of the sun is called a solar day, 
 and the interval between the upper transits of the first point of 
 Aries is called a sidereal day. The celestial sphere apparently 
 makes one revolution about the earth in a sidereal day. The 
 solar day is, on the average, about 3m. 56s. longer than the 
 sidereal day. 
 
 8. Some of the preceding definitions are illustrated in the 
 diagram, Fig. 3. In this figure is the position of the observer, 
 HESW his celestial horizon, Pp the axis of the heavens, P 
 the elevated and p the depressed pole, Zthe zenith, and A^the 
 nadir. The circle HZSN is the observer's celestial meridian. 
 It may be noticed that this circle is at once a vertical and an 
 
 hour circle. The circle ECWD is the equinoctial, and the circle 
 EZWN, perpendicular to the meridian, is the prime vertical, 
 
22 THIXJKEMS. 
 
 cutting the horizon in E and W^ the east and the west point of 
 the horizon. The equinoctial, being also perpendicular to the 
 meridian, passes through the same points. If P is supposed to 
 be the north pole, if is the north point of the horizon, and 8 the 
 south point. 
 
 Let A denote some celestial body. GA is its altitude, ZA its 
 zenith distance, H G its azimuth as reckoned by navigators, and 
 SG its azimuth as reckoned by astronomers, all these elements 
 of position being determined by the arc of a vertical circle, ZG, 
 passed through A. Let an arc of an hour circle, PB, be also 
 passed through A. Then is AB its declination, PA its polar 
 distance, and if V be taken to denote the position of the vernal 
 equinox, VB is the right ascension of A. The right ascension 
 may also be represented by the angle VPS, which the arc VB 
 measures. The angle ZPB is the hour angle of A, and ZPV is 
 the hour angle of the vernal equinox, or the sidereal time. 
 This angle may also be designated as the right ascension of the 
 meridian. 
 
 The circle KH, drawn about P with the. radius PH, is the 
 circle of perpetual apparition. The star whose path is repre- 
 sented by Im never passes below the horizon : I is its upper, m 
 its lower culmination. SR represents the circle of perpetual 
 occultation, and the stars whose paths lie, like or, within this 
 circle, never come above the horizon. 
 
 9. Theorem. The sidereal time at any place is always equal to 
 the sum of the right ascension and the hour angle of the same body. 
 This is an. important astronomical theorem, and is readily proved 
 in Fig. 3, in which ZPV, the sidereal time, is the sum of ZPB, 
 the hour angle, and VPB, the right ascension of the celestial 
 body A. Any two of these three angles, then, being given, the 
 third is readily obtained. 
 
 Corollary. When a celestial body is at its upper culmination 
 at any place, its right ascension is equal to the sidereal time at that 
 place. This is evidently true, because the hour angle of a body 
 when at its upper culmination is zero, and hence, from the theorem, 
 the right ascension of the body is equal to the sidereal time. For 
 example, in Fig. 3, if I is a body at its upper culmination, VPZ is 
 both its right ascension and the right ascension of the meridian. 
 
THEOREMS. 
 
 Fig. 4. 
 
 10. Theorem. The latitude of any place on the earth's surface 
 is equal to the altitude of the elevated pole at that place. Let L 
 (Fig. 4) be some place on the 
 
 earth's surface, Pp the earth's 
 axis, and EQ the equator. The 
 line HR y tangent to the earth's 
 surface at L, is the horizon, and 
 Z the zenith, of L. According to 
 the definition already given, LOQ 
 is the latitude of L. Let the 
 earth's axis be indefinitely pro- 
 longed, and at L let the line LP", 
 parallel to the earth's axis, be also 
 indefinitely prolonged. Owing 
 to the immensity of the celestial sphere when compared with 
 the earth, these two lines will sensibly meet at a common point 
 on the surface of the celestial sphere, and this common point 
 will be the elevated pole. The elevated pole, then, to an ob- 
 server at L, will lie in the direction LP", and P'LH will be its 
 altitude. 
 
 Now we have, HLZ = POQ 
 
 ZLP" = ZOP f 
 .'.P"LH=LOQ 
 which was to be proved. 
 
 11. Theorem. The latitude of a place is equal to the declination 
 of the zenith of that place. This is easily seen in Fig. 4 : for the 
 declination of any body or point is its angular distance from the 
 plane of the celestial equator, and hence ZOQ in the declination 
 of the zenith. 
 
 Either of these theorems might be deduced from the other, 
 but it is better to consider them as independent propositions. 
 
 12. The Astronomical Triangle. The spherical triangle PZA 
 (Fig. 3) is called the astronomical triangle. It is formed by the 
 arcs of the meridian of the -place, and of the vertical circle and 
 the hour circle passing through some heavenly body, which are 
 included between the zenith of the observer, the elevated pole, 
 and the position of the body as projected on the surface of the 
 celestial sphere. The three sides are: %P, the co-latitude of the 
 
24 DIURNAL CIRCLES. 
 
 place; PA, the polar distance of the body; and ZA, its zenith 
 distance. The three angles are: ZPA, the hour angle of the 
 body; PZA, its azimuth; and PAZ, an angle which is rarely 
 used, and which is commonly called the position angle of the 
 body. 
 
 The co-latitude and the zenith distance can evidently never 
 be greater than 90. The polar distance is equal to 90 minm 
 the declination: and it is less than 90 when the body is on the 
 same side of the celestial equator as the observer, but becomes 
 numerically 90 plus the declination when the body is on the 
 opposite side. In the former case the declination of the body 
 is said to have the same name as the latitude, in the latter case, 
 to have the opposite name. 
 
 13. Diurnal Circles. We have already seen that the apparent 
 daily motions of the stars are performed in circles, the planes 
 of which are perpendicular to the axis of the sphere. These 
 circles are called diurnal circles. The phenomena which have 
 been observed with reference to these circles (Arts. 4 and 6) are 
 explained in Fig. 3. The circles of all the stars which rise in 
 the arc of the horizon, ES, are evidently divided by the horizon 
 into two unequal parts, the smaller of which in each case lies above 
 the horizon. Hence these stars are above the horizon less than 
 twelve hours. On the other hand, the greater portions of the 
 circles of those stars which rise between E and H lie above the 
 horizon, and the stars themselves are above it for more than 
 twelve hours. 
 
 Any body, then, whose declination is of the same name as the 
 elevated pole will be above the horizon more than twelve hours, 
 while a body whose declination is of a different name from that 
 of the elevated pole will be above the horizon less than twelve 
 hours. And further: those bodies which are in north declination, 
 in other words, to the north of the celestial equator, will rise to 
 the north of east and set to the north of west; while bodies in 
 south declination will rise and set to the south of the east and 
 the west point. 
 
 When a star has no declination, that is to say, is on the celestial 
 equator, it will rise due east and set due west, and will remain 
 twelve hours above the horizon. 
 
SPHERICAL CO-ORDINATES. 25 
 
 1-1 Right and Parallel /Spheres. When an observer travels 
 towards the elevated pole, the radius of the circle of perpetual 
 apparition, being equal to the altitude of the pole, continually 
 increases, and the number of stars which never set increases in 
 like manner. The number of stars which never rise also increases. 
 Finally, if he reaches the pole, the celestial equator coincides 
 with the horizon, the east and the west point disappear, and the 
 bodies which are on the same side of the equator with the ob- 
 server are perpetually above the horizon, and revolve in circles 
 whose planes are parallel to it, while the bodies which are on 
 the opposite side of the equator never rise. As he travels to- 
 wards the equator, the circles of perpetual apparition and occul- 
 tation alike diminish, the diurnal circles become more and more 
 nearly vertical, and when he reaches the equator, the equinoctial 
 becomes perpendicular to the horizon and coincides with the 
 prime vertical, and the horizon bisects all the diurnal circles. 
 At the equator, then, every celestial body comes above the hori- 
 zon, and remains above it twelve hours. 
 
 15. Spherical Co-ordinates. The position of any point on the 
 surface of a sphere is determined, as soon as its angular distances 
 are given from any two great circles on that sphere whose po- 
 sitions are known. Thus the geographical position of any point 
 on the earth's surface is known when we have determined its 
 latitude and longitude; in other words, when we know its an- 
 gular distance from the equator and from the prime meridian. 
 In like manner we know the position of any point on the sur- 
 face of the celestial sphere when either its altitude and azimuth, 
 or its right ascension and declination, are given. In the first 
 of these two systems of co-ordinates the fixed great circles are 
 the celestial horizon and the celestial meridian, the origin of co- 
 ordinates being either the north or the south point of the horizon. 
 In the second system the fixed great circles are the equinoctial 
 and the circle of declination which passes through the vernal 
 equinox, and the origin of co-ordinates is the vernal equinox. 
 In Fig. 3, if we know the arcs HG and GA, or the arcs VB 
 and BA, we evidently know the position of A. 
 
 16. Vanishing Points and Vanishing Circles. Every one knows- 
 that as h? increases the distance between himself and any object, 
 
26 VANISHING POINTS AND CIRCLES. 
 
 the apparent magnitude of the object decreases ; and that, if he 
 recedes far enough from it, it witl be reduced in appearance to 
 a point. Every one also knows that when he looks along the line 
 of a railroad track the lines appear to converge, and that, if 
 the track is straight, and the curvature of the earth does not 
 limit his vision, the rails will ultimately appear to meet. 
 
 These familiar illustrations will serve to show what is meant 
 by a vanishing point. The actual distance between the rails of 
 course remains the same ; but the angle at the eye which this 
 distance subtends decreases as the eye is directed along the 
 track, until at last it ceases to subtend any appreciable angle at 
 the eye, and the rails apparently meet. This point where the 
 rails appear to meet is called the vanishing point of the two 
 lines; and, in general, the vanishing point of any system of pa- 
 rallel lines is the point at which they will appear to meet, when 
 indefinitely prolonged. We have already seen (Art. 10) that 
 the pole of the heavens is the vanishing point of lines drawn 
 perpendicular to the equator, and the same may be said of the 
 poles of any circle on the celestial sphere. For instance, the 
 poles of the horizon at any place are the zenith and the nadir; 
 and any system of lines perpendicular to the horizon will appa- 
 rently meet, when prolonged indefinitely, in these two points. 
 And again, the east and the west point of the horizon are the 
 poles of the meridian, and lines drawn perpendicular to the 
 meridian will have these points for their vanishing points. 
 
 The same principle holds good when applied to any system 
 of parallel planes. They will appear to meet, when indefinitely 
 extended, in one great circle of the sphere, and this circle is 
 called the vanishing circle of that system of planes. The celes- 
 tial horizon is, as has already been stated (Art. 4), the vanishing 
 circle of the planes of the sensible and the rational horizon, and 
 indeed of any number of planes passed parallel to them. The 
 celestial equator is the vanishing circle of the planes of all the 
 parallels of latitude, and, in short, every circle of the celestial 
 sphere may be regarded as the vanishing circle of a system of 
 planes passed perpendicular to the line which joins the poles of 
 that circle. 
 
 17. Spherical Prnject'&ns. The point? and circles of either the 
 
SPHERICAL PROJECTIONS. 27 
 
 earth or the celestial sphere, or of both, may be projected upon 
 the plane of any great circle of either sphere. The plane on 
 which the projections are made is called the primitive plane, and 
 the circle which bounds this plane is called the primitive circle. 
 Several distinct methods of projection will be found in treatises 
 on Descriptive Geometry. Of these, the most common are the 
 orthographic, the stereographic, and Mercator's projection. 
 
 In the orthographic projection, the point of sight is taken in 
 the axis of the primitive circle, and at an infinite distance from 
 that circle. All circles whose planes are perpendicular to the 
 primitive plane are projected into right lines; all circles whose 
 planes are parallel, to the primitive plane are projected into 
 circles, each of which is equal to the circle of which it is a pro- 
 jection ; and all other circles are projected into ellipses. 
 
 In the stereographic projection, the point of sight is at either 
 pole of the primitive circle, and its distance from that circle is 
 finite. In this projection every circle is projected as a circle, 
 unless its plane passes through the point of sight, in which case 
 it is projected into a right line. 
 
 Mercator's projection is employed in the construction of charts 
 representing the earth's projection. In this projection the 
 parallels of latitude are represented by parallel right lines, and 
 the meridians are also represented by parallel right lines, per- 
 pendicular to the equator. The meridian projections are equi- 
 distant, but the distance between the successive latitude projec- 
 tions increases as we recede from the equator. The advantage 
 offered by this projection to navigators is that the ship's track, 
 as long as the course on which it sails is unaltered, is represented 
 on the chart by a straight line, and that the angle which this 
 line makes with each meridian is the course, 
 
28 INSTRUMENTS. 
 
 CHAPTER II. 
 
 ASTRONOMICAL INSTRUMENTS. ERRORS. 
 
 18. THIS chapter will be devoted to a general description of 
 the common astronomical instruments, of the class of observa- 
 tions to which each is adapted, and of the manner in which such 
 observations are made. No attempt will be made to describe 
 the elaborate mechanism by which, in many cases, the usefulness 
 of the instrument is increased and its manipulation is facilitated ; 
 but enough, it is hoped, will be said to enable the student to 
 form a clear conception of the prominent features of each instru- 
 ment which is described. There is no lack of excellent treatises 
 on Astronomy, in which those who wish to investigate this subject 
 more thoroughly will find all the details, which the limits pre- 
 scribed to this book will not permit to enter here, clearly and 
 elaborately presented. 
 
 THE ASTRONOMICAL CLOCK. 
 
 19. The astronomical clock is a clock which is regulated 
 to keep sidereal time, and is an indispensable companion to the 
 other astronomical instruments. It is provided with a pendulum 
 so constructed that change of temperature will not affect its 
 length. The sidereal day at any place commences, as has al- 
 ready been stated, when the vernal equinox is on the upper 
 branch of the meridian of that place, and the theory of the 
 sidereal clock is that it shows Oh. Om. Os. when the vernal equi- 
 nox is so situated. Practically, however, it is found that every 
 clock has a daily rate; that is to say, it gains or loses a certain 
 amount of time daily. In order, then, that a clock may be re- 
 gulated to sidereal time, it is necessary to know both its error 
 si nd its dally rate; the error being the amount by which it is 
 fast or slow at any given time, and the daily rate being the 
 
THE CLOCK. 29 
 
 amount which it gains or loses daily; and knowing these, it 
 is evidently in our power to obtain at any desired instant 
 the true sidereal time from the time shown by the face of the 
 clock. 
 
 It is to be noticed further, that, except as a matter of con- 
 venience, a small rate has no advantage over a large one; but 
 it is very important that the rate, whether large or small, shall 
 be constant from day to day; so that, of two clocks, one of which 
 has a large and constant rate, and the other a small and varying 
 one, the preference is to be given to the former. 
 
 Clocks may be regulated to keep either local sidereal time or 
 Greenwich sidereal time, or both. 
 
 20. Error of the Clock. To obtain the error of a clock on the 
 local sidereal time at any observatory, we make use of the pro- 
 position, already demonstrated (Art. 9), that the right ascension 
 of any celestial body, when at its upper culmination on any 
 meridian, is equal to the sidereal time at that meridian. The 
 Nautical Almanac gives the right ascensions of more than a 
 hundred stars which are suitable for observations for time. By 
 means of an instrument, properly adjusted, we determine the 
 instant when any one of these stars is on the meridian, and the 
 time which the clock shows at that instant is noted. This is the 
 clock time of transit, and the right ascension of the star, taken 
 from the Almanac, is the true time of transit; and a comparison 
 of these two times will evidently give us the amount by which 
 the clock is fast or slow on local sidereal time. 
 
 21. Daily Rate. If, in a similar manner, we obtain the error 
 of the clock on the next or on any subsequent day, the difference 
 of these two errors will be the gain or loss of the clock in the 
 interval; and hence, if we divide this difference by the number 
 of the days and parts of days which have intervened between 
 the two observations, the quotient will be the daily gain or 
 loss. 
 
 22. Chronograph. The accuracy of astronomical observations 
 id much enhanced by recording the times of the observations by 
 means of an electric current. A cylinder, about which a roll 
 of paper is wound, is turned about on its axis with a uniform 
 motion by the use of appropriate machinery. A pen is pressed 
 
30 
 
 THE TRANSIT INSTRUMENT. 
 
 Fig. 5. 
 
THE TRANSIT INSTRUMENT. 31 
 
 down upon the paper, and is so connected with a battery that 
 whenever the circuit is broken a mark of some kind is made upon 
 the paper. The wires of this battery are connected with the 
 sidereal clock in such a way that every oscillation of the pendu- 
 lum breaks the circuit. Every second is thus recorded upon 
 the revolving paper. The observer also holds in his hands a 
 break-circuit key, with which, whenever he wishes to note the 
 time, he breaks the circuit, and thus causes the pen to make its 
 mark upon the paper. The line which the pen describes upon 
 the paper will be something like this : 
 
 A BO 
 
 L i_l l_l _L_ 
 
 The equidistant marks, a, b, c, &c., are the marks caused by the 
 pendulum, and the marks J, B, C, are the marks which the pen 
 makes when the circuit is broken by the observer. The distance 
 from A to 6, B to c, &c., can be measured by a scale of equal 
 parts, and the time of an observation can thus be obtained within 
 a small fraction of a second. 
 
 The cylinder is also moved by a fine screw in the direction 
 of its own length, so that the pen records in a spiral. 
 
 This instrument is called a Chronograph. There are several 
 varieties of the chronograph in use by different astronomers, but 
 the main principle in all of them is similar to that of the instru- 
 ment just now described. 
 
 THE TRANSIT INSTRUMENT. 
 
 23. The transit instrument is used, as its name implies, in 
 observing the transits of the heavenly bodies.. Fig. 5 represents 
 a transit instrument. It consists of a telescope, TT, sustained 
 by an axis, AA, at right angles to it. The extremities of this 
 axis terminate in cylindrical pivots, which rest in metallic sup- 
 ports, FF, shaped like the upper part of the letter Y, and hence 
 called the Ys. These Ys are imbedded in two stone pillars. In 
 order to relieve the pivots of the friction to which the weight 
 of the telescope subjects them, and to facilitate the motion of the 
 
32 THE TRANSIT INSTRUMENT. 
 
 telescope, there are two counterpoises, WW, connected with 
 levers, and acting at XX, where*there are friction rollers upon 
 which the axis turns. When the instrument is properly adjusted, 
 the telescope, as it turns about with the axis AA, will continu- 
 ally lie in the plane of the meridian ; and, in order to effect this, 
 the axis A A should point to the east and the west point of the 
 horizon, and be parallel to its plane. There are therefore screws 
 at the ends of the axis, by which one extremity of the axis may 
 be raised or depressed, and may also be moved forward or 
 backward. 
 
 24. The Reticule. In the common focus 
 of the object-glass and the eye-glass is placed 
 the reticule, a representation of which is given 
 in Fig. 6. It consists of several equidistant 
 vertical wires (usually seven) and two hori- 
 zontal ones. If the instrument is accurately 
 
 adjusted to the plane of the meridian, the ig - 6- 
 
 instant that any star is on the middle wire is the instant of its 
 transit. These wires are also called the cross-wires. 
 
 25. Adjustment. The axis of rotation of the instrument is an 
 imaginary line connecting the central points of the pivots. 
 
 The axis of collimation is an imaginary line drawn from the 
 optical centre of the object-glass, perpendicular to the axis of 
 rotation. 
 
 The line of sight is an imaginary line drawn from the optical 
 centre of the object-glass to the middle wire. 
 
 The transit instrument is accurately adjusted in the plane of 
 the meridian, when the line of sight of the telescope lies con- 
 tinually in that plane, as the telescope revolves. Three things, 
 then, are readily seen to be necessary : the axis of rotation must 
 be exactly horizontal ; it must lie exactly east and west ; and the 
 line of sight and the axis of collimation must exactly coincide. 
 Practically, these conditions are rarely fulfilled; but they can, by 
 repeated experiments, be very nearly fulfilled, and the errors 
 which the failure rigorously to adjust the instrument causes in 
 '.he observations will be constant and small, and can be accu- 
 rately determined. 
 
 20. Application. The principal application of the transit 
 
THE MERIDIAN CIRCLE. 
 
 33 
 
 Fig 7 
 
34 THE MERIDIAN CIRCLE. 
 
 instrument in observatories is to the determination of the right 
 ascensions of celestial bodies. Knowing the constant instru- 
 mental errors just now mentioned, and the error and the rate of 
 the clock, we can easily obtain the true sidereal time at the 
 instant of the transit of a celestial body, which time is at once, 
 as we have already seen, the right ascension of that body. 
 
 THE MERIDIAN CIRCLE. 
 
 27. The meridian circle is a combination of a transit instru- 
 ment, similar to the one above described, and a graduated circle, 
 securely fastened at right angles to the horizontal axis, and 
 turning with it. A meridian circle which is set up at the United 
 States Naval Academy, Annapolis, Maryland, is represented in 
 Fig. 7. The horizontal axis bears two graduated circles, CC, 
 C'C'y the first of these circles being much more finely graduated 
 than the second, the latter being only used as a, finder, to set the 
 telescope approximately at any desired altitude. R and R re- 
 present two of four stationary microscopes, by which the circle 
 CCis read; LL is a hanging level, by which the horizontally of 
 the axis is tested. The cross-wires are illuminated by light 
 which passes from a lamp through the tubes AA, and through 
 the pivots which are perforated for this purpose, and is reflected 
 towards the reticule by a metallic speculum which is set within 
 the hollow cube M. The quantity of light admitted is regulated 
 by revolving discs with eccentric apertures, which are placed 
 between the Ys and the tubes AA, and are moved by cords car- 
 rying small weights, JSS. 
 
 The object in having so many reading microscopes is to dim- 
 inish the errors arising from the imperfect graduation of the ver- 
 tical circle. When any angle is to be measured, the readings of 
 all four microscopes are first taken. The telescope, carrying the 
 circle with it., is then moved through the angle whose value is 
 required, and the new readings of the microscopes, which have 
 remained stationary, are taken. We thus obtain four values, 
 one from each microscope, of the angle measured. Theoretically, 
 these values should be identical, and if they are not, their mean 
 is taken a^ the true measure of the angle observed. 
 
THE MICROSCOPE. 
 
 35 
 
 Fig. 8. 
 
 28. The Reading Microscope. The reading microscope ia 
 represented in Fig. 8. 
 The observer, placing 
 his eye at A, sees the 
 image of the divisions 
 of the graduated circle 
 MN, formed at D, the 
 common focus of the 
 glasses A and C. He 
 will also see a scale 
 of n6tches, nn, and 
 two intersecting spider 
 threads, as shown in 
 Fig. 9. These threads 
 are attached to a sliding 
 frame, aa, which is 
 moved by means of a 
 fine screw, ce, the head 
 of which, EF y is gra- 
 duated. The scale of 
 notches is immovable, 
 and is so constructed 
 that the distance be- 
 tween the centres of any 
 two consecutive notches 
 is equal to that between 
 the threads of the screw, thus making the number of teeth passed 
 over by the spider threads equal to the number of complete re- 
 volutions made by the screw. The central notch is taken as the 
 point of reference, and is distinguished by a hole opposite to it. 
 There is a fixed index at i, to which the divisions on the head of 
 the screw are referred. When any division of the limb does not 
 coincide with the central notch, the spider threads are moved 
 from the central notch to the division, and the number of revo- 
 lutions and fractional parts of a revolution which the screw 
 makes is noted. If now we suppose the value of each division 
 of the graduated circle, MN, to be 10', and that ten revolutions 
 of the screw suffice to carry the spider threads across one of these 
 
 
 F 
 
 3- - 
 
 3 * 
 
 
 
 
 :^j 
 
 
 a 
 
 ' 
 
 \ ! \^ 
 
 \Wmmmmm\ n \ 
 
 iiilMJP 
 
 -8! 
 
 / \ aivw" 
 
 
 : -r> 
 
 
 
 Fig. 9. 
 
36 FIXED POINTS. 
 
 divisions, then will one revolution of the screw correspond to an 
 arc of 1' ; and if we further suppose that the head of the screw 
 is divided into 60 equal parts, then each division on the head 
 will correspond to an arc of V . In such a case, the complete 
 reading of the limb is obtained to the nearest second. By in- 
 creasing the power of the microscope, the fineness of the screw, 
 and the number of the graduations on the screw-head, the read- 
 ing of the limb may be obtained with far greater precision. 
 
 29. Fixed Points. The meridian circle, being also a transit 
 instrument, may be used as such; but the object for which it is 
 specially used is the measurement of arcs of the meridian. In 
 order to facilitate such measurement, certain fixed points of re- 
 ference are determined upon the vertical circle. The most im- 
 portant of these points are the horizontal point, by which is 
 meant the reading of the instrument when the axis of the tele- 
 scope lies in the plane of the horizon ; the polar point, which is 
 the reading of the instrument when the telescope is directed to 
 the elevated pole ; the zenith point, and the nadir point. 
 
 30. The Horizontal Point. As the surface of a fluid, when at 
 rest, is necessarily horizontal, and as, by the laws of Optics, the 
 angles of incidence and reflection are equal to each other, the 
 image of a star reflected in a basin of mercury will be depressed 
 below the horizon by an angle equal to the altitude of the star 
 at that instant. If, then, we take the reading of the vertical 
 circle when a star which is about to cross the meridian is on the 
 first vertical thread of the reticule, and then, depressing the 
 telescope, take the reading of the circle when the reflected image 
 of the star crosses the last vertical thread, and, by means of 
 small corrections, reduce these readings to what they would have 
 been, had both star and image been on the meridian when 
 the observations were made, the mean of these two reduced 
 readings will be the horizontal point. The horizontal point 
 having been thus determined, the zenith point and the nadir point, 
 being situated at intervals of 90 from it, are at once obtained. 
 Knowing the horizontal and the zenith point, we are able to 
 measure the meridian altitude or the meridian zenith distance 
 of any celestial body which comes above our horizon. And 
 further, as the latitude is equal to the altitude of the elevated 
 
NADIR POINT. 
 
 37 
 
 Fig. 10. 
 
 pole, if tlie latitude of the place is accurately known, we can at 
 once obtain the polar point by applying the latitude to the 
 horizontal point. 
 
 31. Nadir Point. The nadir point may be independently 
 obtained in the following manner. Let the 
 
 telescope, represented in Fig. 10 by AJ3, be 
 directed vertically downwards towards a basin 
 of mercury, CD. The observer, placing his 
 eye at A, will see the cross-wires of the tele- 
 scope, and will see also the image of these 
 wires reflected into the telescope from the 
 mercury. By slowly moving the telescope, 
 the cross-wires and their reflected image may 
 be brought into exact coincidence, and the 
 reflected image will then disappear. The 
 line of sight of the telescope is now vertical, 
 and the reading of the vertical circle' will be 
 the nadir point, from which the other points can readily be found. 
 There is a variety of methods by which each of these points 
 can be obtained, without reference to any other; and by com- 
 paring the results which these different independent methods 
 give, the errors to which each result is liable may be very con- 
 siderably diminished. 
 
 32. Use of the Meridian Circle. The meridian circle may be 
 used in connection with the sidereal clock, to find the right 
 ascension and declination of any celestial body. The telescope 
 is directed towards the body as it crosses the meridian, and the 
 time of transit as shown by the clock, and the reading of the 
 vertical circle, are both taken. We have already seen how the 
 right ascension of the body is obtained from the clock time of 
 transit. The difference between the reading of the circle, which 
 we suppose to have been taken, and the polar point, is the polar 
 distance of the body, the complement of which is the declina- 
 tion. Or we may obtain the declination still more directly by 
 previously establishing the equinoctial point of the instrument, 
 the reading, that is to say, of the vertical circle when the tele- 
 scope lies in the plane of the equinoctial.* 
 
 * As the direction in which a star appears to lie is not, owing to refruc- 
 4 
 
38 MURAL CIRCLE. 
 
 On the other hand, if we ma ke the same observations upon a 
 Btar whose right ascension and declination are known, we can 
 determine the latitude and the sidereal time of the place of 
 observation. 
 
 THE MURAL CIRCLE. 
 
 33. The mural circle is, in construction, adjustment, and use, 
 essentially a meridian circle. The only important difference 
 between the two instruments is in the manner in which they are 
 mounted. The horizontal axis of the mural circle, instead of 
 being supported at both extremities, is supported only at one, 
 which is let into a stone pier or wall. Owing to the lack of 
 symmetrical support, and also to the fact that the instrument 
 does not admit of reversal (which is an important element in the 
 adjustment of the meridian circle, and consists in lifting it out 
 of the Ys, and turning the horizontal axis end for end), the 
 mural circle can be regarded only as an inferior type of the 
 meridian circle. 
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 34. The general principles on which the altitude and azimuth 
 
 instrument is constructed are seen 
 in Fig. 11. Through the centre of 
 a graduated circle, C'6", and per- 
 pendicular to its plane, is passed an 
 axis, A A. At right angles to this 
 axis is a second axis, one extremity 
 of which is represented by B. This 
 second axis carries the telescope, TT, 
 and also a second graduated circle, 
 (7(7, whose plane is perpendicular to 
 that of the circle C" C' '. The tele- 
 scope admits of being moved in the plane of each circle, and 
 
 tion and other causes, which will be explained in Chap. III., the direction 
 in which it really lies, certain small corrections must be applied to the 
 reading of the circle to obtain the reading which really corresponds to the 
 direction of the star. 
 
ALTITUDE AND AZIMUTH INSTRUMENT. 39 
 
 microscopes or verniers are attached to the instrument, by means 
 of which arcs on either circle can be read. 
 
 If this instrument is so placed that the principal axis, AA, 
 lies in a vertical direction, we shall have an altitude and azimuth 
 instrument, sometimes called an altazimuth. The circle C'C f 
 will then lie in the plane of the horizon, and the axis AA, in- 
 definitely prolonged, will meet the surface of the celestial sphere 
 in the zenith and the nadir. The circle (7(7, as it is moved 
 with the telescope about the axis AA } will continually lie in a 
 vertical plane. 
 
 35. Fixed Points. Altitudes maybe measured on the vertical 
 circle, when we know the horizontal or the zenith point, the de- 
 termination of which has already been described in Arts. 30 and 
 31. In order to measure the azimuth of any celestial body, we 
 must, in like manner, establish some fixed point of reference on 
 the horizontal circle, as, for instance, the north or the south 
 point, by which is meant the reading of the horizontal circle 
 when the telescope lies in the plane of the meridian. 
 
 36. Method of Equal Altitudes. One of the most accurate 
 methods of obtaining the north or the 
 
 south point of the horizontal circle is 
 called the method of equal altitudes. 
 Let Fig. 12 represent the projection of 
 the celestial sphere on the plane of the 
 celestial horizon, NESW. Z is the 
 projection of the zenith, P of the pole, 
 and the arc AA' the projection of a 
 portion of the diurnal circle of affixed 
 star, which is supposed to have the 
 same altitude when it reaches A', west 
 of the meridian, which it had at A, east of the meridian. 
 
 Now, in the two triangles PAZ, PA'Z, we have PZ common, 
 PA' equal to PA (since the polar distance of a fixed star re- 
 mains constant), and ZA equal to ZA', by hypothesis ; the two 
 triangles are therefore equal in all their parts, and hence the 
 angles PZA and PZA' are equal. But these' two angles are the 
 azimuths of the star at the two positions A and A f . We may 
 
40 EQUATORIAL. 
 
 say, then, in general, that equal, altitudes of a fixed star corre- 
 spond to equal angular distances from the meridian. 
 
 Now, let the telescope be directed to some fixed star east of the 
 meridian, and let the reading of the horizontal circle be taken. 
 When the star is at the same altitude, west of the meridian, let 
 the reading of the horizontal circle again be taken ; the mean of 
 these two readings is the reading of the horizontal circle when 
 the axis of the telescope lies in the plane of the meridian. 
 
 37. Use of the Altitude and Azimuth Instrument. This instru- 
 ment is chiefly used for the determination of the amount of re- 
 fraction corresponding to different altitudes. Refraction, as will 
 be seen in the next chapter, displaces every celestial body in a 
 vertical direction, making its apparent zenith distance less than 
 its true zenith distance. At the instant of taking the altitude 
 of a celestial body, the local sidereal time is noted, from which, 
 knowing the right ascension of the body, we can obtain its hour 
 angle from the theorem in Art. 9. We shall then have in the 
 astronomical triangle, PZA (Fig. 3), the hour angle ZPA, the 
 side PA, or the polar distance of the body, and the side PZ, the 
 co-latitude of the place of observation. We can, therefore, com- 
 pute the side ZA, which is the true zenith distance of the body 
 observed ; and the difference between this and the observed zenith 
 distance, (corrected for parallax, [Art. 55,] and instrumental 
 errors,) will be the amount of refraction for that zenith distance. 
 
 The construction of the instrument enables us to follow a 
 celestial body through its whole course from rising to setting, 
 measuring altitudes and noting the corresponding times to any 
 extent that we choose; and the amount of refraction correspond- 
 ing to each altitude can afterwards be computed at our leisure. 
 
 THE EQUATORIAL. 
 
 38. The equatorial is similar in general construction to the 
 altitude and azimuth instrument. It is, however, differently 
 placed, the plane of the principal graduated circle, C' C', coin- 
 ciding, not with tKe plane of the horizon, but with that of the 
 celestial equator, from which peculiarity of position comes the 
 
EQUATORIAL. 4 1 
 
 name of equatorial. The circle C'C', when thus placed, is called 
 the hour circle of the instrument, and the axis A, at right angles 
 to it, is called the hour or polar axw. It is evident that the axis 
 is directed towards the poles of the heavens. The circle CO, 
 the plane of which is perpendicular to the plane of the circle 
 C" C' t will lie continually in the plane of a circle of declination, 
 as the instrument is turned about the polar axis, and is hence 
 called the declination circle. The axis on which this latter circle 
 is mounted is called the declination axis. 
 
 39. Use of the Equatorial. The equatorial is employed prin- 
 cipally in that class of observations which require a celestial 
 body to remain in the field of view during a considerable length 
 of time. The manner in which this 
 requirement is met is explained 
 in Fig. 13. Let A A' be the polar 
 axis of an equatorial, directed to- 
 wards the pole of the heavens, P. 
 Let ss*s" be the diurnal circle in 
 which a star appears to move 
 about the pole. Suppose the tele- 
 scope, TT, to be turned in the di- 
 rection of the star when at , and Fig. is. 
 
 to be moved until the intersection of the cross-wires and the star 
 coincide, and then clamped. Now, if the instrument is made to 
 revolve about the axis AA' t with an angular velocity equal to 
 that of the star about the pole, it is plain that, since the angle 
 which the axis of the telescope makes with the polar axis remains 
 unchanged, and is continually equal to the angular distance of 
 the star from the pole, the coincidence of the cross-wires and the 
 star will remain complete. 
 
 A clock-work arrangement, called a driving-clock, is now usually 
 connected with large equafHrials, by which the instrument may 
 be moved uniformly about its polar axis, at the required rate, so 
 that the observer has ample time to measure the angular diameter 
 of a celestial body, to measure the angular distance between two 
 stars which are near each other, and to make other micrometric 
 observations of a similar character. 
 
42 
 
 SEXTANT. 
 
 THE SEXTANT. 
 
 40. The sextant is an instrument by which the angular dis- 
 tance between two visible objects may be measured. It is used 
 chiefly by navigators ; but its portability gives it great value 
 wherever celestial observations are required. The angles for 
 the measurement of which it is used are the altitudes of celestial 
 bodies, and the angular distances between celestial bodies or 
 terrestrial objects. 
 
 Fig. 14 is a representation of the sextant. Its form is that 
 
 of a sector of a circle, the arc of which comprises 60. A movable 
 arm, CD, called the index-bar, revolves about the centre of the 
 sector. This bar carries at one extremity a vernier, D. At the 
 other extremity of the index-bar, and revolving with it, is placed 
 a silvered mirror, C, the surface of which must be perpendicular 
 to the plane of the instrument. This glass is called the index- 
 glass. Another glass, N, called the horizon- glass, is attached to 
 the frame of the instrument, and only its lower half is silvered. 
 
SEXTANT. 43 
 
 This glass is immovable, and its surface must be perpendicular 
 to the plane of the instrument. T is a telescope, directed to- 
 wards the horizon-glass, with its line of sight parallel to the 
 plane of the instrument. F and E are two sets of colored 
 glasses, which may be used to protect the eye when the sun is 
 observed. M is a magnifying glass, to assist the eye in reading 
 the vernier. G is a tangent screw, which gives a slight motion to 
 the index-bar, and is used in obtaining an accurate coincidence 
 of the images. 
 
 41. Optical Principle of Construction. The sextant is con- 
 structed upon a principle in Optics which may be stated thus : 
 The angle between the first and the last direction of a ray which 
 has suffered two reflections in the same plane is equal to twice the 
 angle which the tivo reflecting surfaces make with each other. 
 
 To prove this: In Fig 15, 
 let A and B be the two re- 
 flecting surfaces, supposed to 
 be placed with their planes 
 perpendicular to the plane of 
 the paper. Let SA be a ray 
 of light from some body, S, 
 
 which is reflected from A to 
 
 B, and from Bin the direction 
 BE. Prolong SA until it meets Fig 15 
 
 the line BE. Then will the 
 angle SEB be the angle between the first and the last direction 
 of the ray SA. At the points A and B let the lines AD and 
 jBObe drawn perpendicular to the reflecting surfaces, and pro- 
 long AD until it meets BC. The angle DCB is equal to the 
 angle which the two surfaces make with each other. We have 
 then to prove that the angle SEB is double the angle DCB. 
 
 Now since the angle of incidence always equals the angle of 
 reflection, SAD and DAB are equal, and so are ABC and CBE. 
 We have, by Geometry, 
 
 SEB = SAB ABE 
 
 = 2 (DAB ABC), 
 
 = 2 DCB. 
 
 42. Measurement of Angular Distances. Suppose, now, that 
 
44 
 
 SEXTANT. 
 
 Fig. 16. 
 
 we wish to measure the angular* distance between two celestial 
 bodies, A and B (Fig. 16). The instrument is so held that its 
 plane passes through the two bodies, and the fainter of them, 
 
 which in this case we suppose to 
 be B, is seen directly through the 
 horizon-glass and the telescope. 
 B is so distant that the rays B' C 
 and Bm, coming from it, may be 
 considered to be sensibly parallel. 
 Let ab and CI be the positions of 
 the index-glass and index-bar when 
 ~D the index-glass and the horizon- 
 glass are parallel. Then will the 
 ray B'C be reflected by the two 
 glasses in a direction parallel to 
 itself, and the observer, whose eye is 
 at D, will see both the direct and the 
 
 reflected image of in coincidence. Now let the index-bar be 
 moved to some new position, CI', so that the ray from the second 
 body, A, shall be finally reflected in the direction of ml). The 
 observer will then see the direct image of B and the reflected 
 image of A in coincidence; and the angular distance between 
 the two, bodies is evidently equal to the angle between the first 
 and the last direction of the ray A C, which angle has already 
 been shown to be equal to twice the angle which the two glasses 
 now make with each other, or to twice the angle ICI'. If, then, 
 we know the point Jon the graduated arc at which the index-bar 
 stands when the glasses are parallel, twice the difference between 
 the reading of that point and that of the point /' will be the 
 angular distance of the two bodies. 
 
 To avoid this doubling of the angle, every half degree of the 
 arc is marked as a whole degree, when the graduation is made ; 
 so that, in practice, we have only to subtract the reading of / 
 from that of /' to obtain the angle required. 
 
 43. Index Correction. The point of reference on the arc from 
 which all angles are to be reckoned is, as we have already seen, 
 the reading of the sextant when the surfaces of the index-glass 
 and the horizon-glass are parallel. This point may fall either 
 
ARTIFICIAL HORIZON. 45 
 
 at the zero of the graduation, or to the left or to the right of it; 
 and to provide for the last case, the graduation is carried a short 
 distance to the right of the zero, this portion of the arc being 
 called the extra arc. The reading of this point of parallelism 
 is called the index correction, and is positive when it falls to the 
 right of the zero, and negative when it falls to the left. Suppose, 
 for instance, that the instrument reads 2' on the extra arc when 
 the glasses are parallel : all angles ought then to be reckoned 
 from the point 2', instead of from the zero point; in other words, 
 2' is a constant correction to be added to every reading. 
 
 There are several methods of finding the index correction. 
 One method, which can readily be shown from Fig. 16 to be a 
 legitimate one, is to move the index-bar until the direct and the 
 reflected image of the same star are in coincidence, and then 
 take the reading, giving it Its proper sign according to the rule 
 above stated. Another method, generally more convenient, in 
 which the sun is used, may be found in Bowditch's Navigator, 
 and in most treatises on Astronomy : where also may be found 
 the methods of testing the adjustments of the sextant. 
 
 44. The Artificial Horizon. In order to obtain the altitude 
 of a celestial body at sea, the sextant is held in a vertical posi- 
 tion, and the index-bar is moved until the reflected image of the 
 body is brought into contact with the visible horizon seen through 
 the telescope of the sextant. The sextant reading is then cor- 
 rected for the index correction; arid corrections must also be 
 applied for parallax, refraction, and the dip of the horizon, as 
 will be explained in the next Chapter. If the body observed is 
 the sun or the moon, either its upper or its lower limb is brought 
 into contact with the horizon, and the value of its angular 
 semi-diameter (given in the Nautical Almanac) is subtracted or 
 added. 
 
 On shore, use is made of the artificial horizon, already alluded 
 to in Art. 30. This commonly consists of a shallow, rectangular 
 basin of mercury, the surface of which is protected from the 
 wind by a sloping roof of glass. The observer so places himself 
 that he can see the image of the body whose altitude he wishes 
 to measure reflected in the mercury. He then moves the index- 
 bar of the sextant until the image of the body reflected by the 
 
46 
 
 VERNIER. 
 
 sextant is in coincidence with Jhat reflected by the mercury. 
 .The sextant reading is then corrected for the whole of the index 
 correction. Half of the result will be, as shown in Art. 30, 
 the apparent altitude of the body, to which must be applied 
 the corrections for parallax and refraction to obtain the true 
 altitude. When the sun or the moon is observed, the upper or 
 the lower limb of the image reflected by the sextant is brought 
 into contact with the opposite limb of the image reflected by the 
 mercury, and the correction for semi-diameter also is applied. 
 
 45. The Vernier. ---The vernier is an instrument by which, as 
 by the reading microscope previously explained, fractions of a 
 
 division of a limb may be read. 
 In Fig. 17, let AB be an arc of 
 a stationary graduated circle, 
 and let CD be a movable arm, 
 carrying another graduated 
 arc at its extremity. The value 
 of each division of the limb 
 A B is one-sixth of a degree, or 
 10'. The arc on the arm CD 
 is divided into ten equal parts, 
 and the length of the arc be- 
 tween the points and 10 is 
 Fi - 17 - equal to the length of nine di- 
 
 visions of the arc AB. This arc, which the lirnb CD carries, is 
 called a vernier. Since the ten divisions of the vernier equal in 
 length nine divisions of the limb, it follows that each division 
 of the vernier comprises 9' of arc ; in other words, any division 
 of the vernier is less by V of arc than any division of the limb. 
 The reading of any instrument which carries a vernier is al- 
 ways determined by the position of the zero point of the vernier. 
 If, now, the zero point of the vernier exactly coincides with a 
 division of the limb, the point 1 of the vernier will fall V behind 
 the next division of the limb, the point 2 will fall 2' behind the 
 next division but one, and so on; and if, such being the case, the 
 vernier is moved forward through an arc of 1', the point 1 will 
 corne^nto coincidence with a division of the limb; if it is moved 
 forward through an arc of 2', the point 2 will come into coiiici- 
 
VERNIER. 47 
 
 dence with a division on the limb ; and, in general, the number 
 of minutes of -arc by which the zero point of the vernier falls 
 beyond the division of the limb which immediately precedes it 
 will be equal to the number of that point of the vernier which 
 is in coincidence with a division of the limb. If, then, the zero 
 point falls between any two divisions of the limb, as 11 20' and 
 11 30', for example, and the point 2 of the vernier is found to 
 be in coincidence with any division of the limb, we know that 
 the zero point is 2' beyond the division 11 20', and that the com- 
 plete reading for that position of the vernier is 11 22'. 
 
 46. General Rules of Construction. In the construction of all 
 verniers similar to the one above described, the same rules of 
 construction must be followed : the length of the arc of the ver- 
 nier must be exactly equal to the length of a certain number 
 (no matter what) of the divisions of the limb, and the arc must 
 be divided into equal parts, the number of which shall be greater 
 by one than the number of these divisions of the limb. Following 
 these rules, and putting 
 D the value of a division of the limb, 
 d " " " " " vernier, 
 n = the number of equal parts into which the vernier is divided, 
 
 we have D d = as a general formula. 
 
 The difference D d is called the least count of the vernier. 
 
 If, in Fig. 17, we take the length of the vernier equal to 59 
 divisions of the limb, and divide it into 60 equal parts, we shall 
 have 
 
 which is the least count on most of the modern sextants. 
 
 Verniers are sometimes constructed in which the number of 
 equal parts on the vernier is less by one than the number of the 
 
 D 
 divisions of the limb taken. In this case we have d D = ~ 
 
 ')!/ 
 
 and the only difference between this class of verniers and the 
 class above described is that the graduations of the limb and 
 the vernier proceed in this clas& in opposite directions. 
 
48 SPECTROSCOPE. 
 
 OTHER ASTRONOMICAL INSTRUMENTS. 
 
 47. The zenith telescope, the theodolite, and the universal instrn' 
 ment are, in general principle, only modified forms of the port- 
 able altitude and azimuth instrument. 
 
 The octant (sometimes improperly called the quadrant} is 
 identical in construction with the sextant, excepting only that 
 its arc contains 45. 
 
 The prismatic sextant carries a reflecting prism in place of the 
 ordinary horizon-glass, and the graduated arc comprises a semi- 
 circumference. 
 
 The reflecting circle is still another modification of the sex- 
 tant, in which the graduated arc is an entire circumference, and 
 the index-bar is a diameter of the circle, revolving about the 
 centre, and carrying a vernier at each extremity. Sometimes 
 the circle has three verniers, at intervals of 120 of the gradu- 
 ated arc. 
 
 The spectroscope is an instrument which is used, as its name 
 indicates, -in the examination of the spectra both of terrestrial 
 substances and of the heavenly bodies. Its use as an instrument 
 of astronomical research is comparatively recent, but it has 
 already led to many interesting and remarkable discoveries con- 
 cerning the constitution of the heavenly bodies. It consists 
 essentially of three parts: a tube, a prism (or a set of prisms), 
 and a telescope. Rays of light from either a celestial body or 
 an artificial flame are made to enter the tube through an ex- 
 tremely narrow slit at its extremity. These rays pass through 
 the tube, and fall upon the prism. If necessary, lenses may be 
 placed within the tube, so that the rays, as they issue from it, 
 shall fall upon the prism in parallel lines. These rays are dis- 
 persed by the prism, and a spectrum is formed. This spectrum 
 is then examined by means of the telescope. There is also an 
 arrangement by .which rays of light from two substances or 
 bodies can be introduced through the slit without interfering 
 with each other, so that their spectra can be formed simul- 
 taneously, one above the other, and the points of resemblance or 
 difference between them can be accurately noted. 
 
 It is well known that the solar spectrum contains a large 
 
KRRORS. 49 
 
 number of dark and narrow parallel lines, which are called 
 Fraunhofer's lines. The spectra of the stars and of artificial 
 lights also contain similar series of lines, differing from each 
 other, each series, however, being constant for the same body or 
 light. The spectra of chemical substances also present certain 
 peculiarities, so that each spectrum indicates with certainty the 
 substance which produces it. Hence, by a comparison of the 
 spectra of the heavenly bodies with those of known chemical 
 substances, the existence of many of those substances in the 
 heavenly bodies has been definitely established. Nor is this all ; 
 the inspection of any spectrum suffices to tell us whether the 
 light which forms it comes from a solid or a gaseous body, and 
 whether, if the light comes from a solid body, it passes through 
 a gaseous body before it reaches us. 
 
 The results of these investigations will be noticed when we 
 come to the description of the heavenly bodies ; and the method 
 of investigation will be further illustrated in the Article on the 
 constitution of the sun. (Art. 102.) 
 
 ERRORS. 
 
 48. However carefully an instrument may be constructed, how- 
 ever accurately adjusted, and however expert the observer may 
 be, every observation must still be regarded as subject to errors. 
 These errors may be divided into two classes, regular ami -irre- 
 gular errors. By regular errors we mean errors which remain 
 the same under the same combination of circumstances, and 
 which, therefore, follow some determinate law, which may be 
 made the subject of investigation. Among the most important 
 of this class of errors are instrumental errors: errors, that is to 
 say, due to some defect in the construction or adjustment of an 
 instrument. If, for instance, what we call the vertical circle of 
 the meridian circle is not rigorously a circle, or is imperfectly 
 graduated; or if the horizontal axis is not exactly horizontal, 
 or does not lie precisely east and west ; any one of these imper- 
 fections will affect the accuracy of the observation. The observer, 
 however, knowing what the construction and adjustment of the 
 instrument ought to be, can calculate what effect any given im- 
 .5 
 
50 ERRORS. 
 
 perfection will produce upon his observation, and can thus de- 
 termine what the observation would have been had the imper- 
 fection not existed. Regular errors, then, may be neutralized 
 by determining and applying the proper corrections. 
 
 Irregular errors, on the contrary, are errors which are not 
 subject to any known law. Such, for example, are errors pro- 
 duced in the amount of refraction by anomalous conditions of 
 the atmosphere ; errors produced by the anomalous contraction 
 or expansion of certain parts of the instrument, or by an un- 
 steadiness of the telescope produced by the wind ; and, more par- 
 ticularly, errors arising from some imperfection in the eye or the 
 touch of the observer. Errors such as these, being governed 
 by no known law, can never be made the subject of theoretic 
 investigation; but being by their very nature accidental, the 
 effects which they produce will sometimes lie in one direction 
 and sometimes in another ; and hence the observer, by repeating 
 his observations, by changing the circumstances under which he 
 makes them, by avoiding unfavorable conditions, and finally by 
 taking the mean, or the most probable value of the results which 
 his different observations give him, can very much diminish the 
 errors to which any single observation would be exposed. 
 
 NOTE. For complete descriptions of the various astronomical instru- 
 ments, the student is referred to Chauvenet's Spherical and Practical As 
 tronomy; Loomis's Practical Astronomy; and Pearson's Practical Astronomy 
 (published in England). 
 
REFRACTION. 51 
 
 CHAPTER III. 
 
 REFRACTION. PARALLAX. DIP OF THE HORIZON. 
 REFRACTION. 
 
 49. When a ray of light -passes obliquely from one medium 
 to another of different density, it is bent, or refracted, from its 
 course. If a line is drawn perpendicular to the surface of the 
 second medium at the point where the ray meets it, the ray is 
 bent towards this perpendicular if the second medium is the 
 denser of the two, and from it if the first medium is the 
 denser. 
 
 In Fig. 18, let A A, BB, represent two NEC 
 
 media of different density, the density of 
 BB being the greater. Let CD be a 
 ray of light meeting the surface of BB 
 at D. At D erect the line ND perpendi- / 
 
 cular to the surface of BB, and prolong <? ^ 
 
 it in the direction DM. The ray CD is Fi s- 18 - 
 
 called the incident ray, and the angle ND C the angle of inci- 
 dence. When the ray enters the medium BB, it will still lie in 
 the same plane with CD and ND, but will be bent towards the 
 line DM, making with it some angle GDM, less than the angle 
 ND C. To an observer whose eye is at G, the ray will appear 
 to have come in the direction EG, which is therefore called the 
 apparent direction of the ray. DG is called the refracted ray, 
 and the angle GDM the angle of refraction. The angle EDC, 
 the difference between the directions of the incident and the 
 refracted ray, is called the refraction. 
 
 . It is shown in Optics that, whatever the angle of incidence 
 may be, there always exists a constant ratio between the sine 
 of the angle of incidence and that of the angle of refraction, 
 as long as the same two media are used and their densities are 
 unchanged. We have, then, in the figure, 
 
REFRACTION. 
 
 sin NDQ __ 
 sin GDM = ' 
 
 k being a constant for the two media A A and BB. 
 
 If the second medium, instead of being of uniform density, is 
 composed of parallel strata, each one of which is of greater 
 density than the one immediately preceding, as is represented 
 c _ in Fig. 19, the path of the ray through 
 these several strata will be a broken 
 line, Dabc; and if the thickness of 
 ~B each of these successive strata is sup- 
 posed to be indefinitely small, this 
 broken line will become a curve. 
 
 In the figures above used, the media are represented as sepa- 
 rated by plane surfaces; but the same phenomena are noticed, 
 and the same laws hold good, if the media are separated by 
 curved surfaces. 
 
 50. Astronomical Refraction. It is determined by experiment 
 that the density of the air gradually diminishes as we ascend 
 above the surface of the earth, and it is estimated that at a 
 distance of fifty miles above the surface the upper limit of the 
 air is reached ; or, at all events, that the density of the air is so 
 small at that distance that it exerts no appreciable refracting 
 power. We may, therefore, consider the air to be made up of 
 a series of strata concentric with the earth's surface, the thick- 
 ness of each stratum being in- 
 definitely small, and the den- 
 sity of each stratum being 
 greater than that of the stra- 
 tum next above it. Now, in 
 s Fig. 20, let the arc BD repre- 
 sent a portion of the earth's 
 surface, and the arc MN a por- 
 tion of the upper limit of the 
 atmosphere. Let 8 be a ce- 
 lestial body, and SA a ray of 
 light from it, which enters the 
 atmosphere at A. Let the 
 normal (or radius) AC be 
 
REFRACTION 53 
 
 drawn. As the ray of light passes down through the atmos- 
 phere, it is continually passing from a rarer to a denser 
 medium, so that its path is continually changed, and becomes 
 a curve AL, concave towards the earth, and reaching the earth 
 at some point L. Since the direction of a curve at any point 
 is the direction of the tangent to the curve at that point, the 
 apparent direction of the ray of light at L will be repre- 
 sented by the tangent LS f , and in that direction will the body 
 S appear to lie, to an observer at L. If the radius OL'be 
 indefinitely prolonged, the point Z, where it reaches the celestial 
 sphere, will be the zenith of the observer at L, the angle ZLS' 
 will be the apparent zenith distance of the body 8, and the 
 angle which the line drawn from the body to the point L 
 makes with the line LZ will be the true zenith distance of S. 
 The effect, then, of refraction is to decrease the apparent zenith 
 distances, or increase the apparent altitudes of the celestial 
 bodies. Since the incident ray SA, the curve AL, and the 
 tangent LS' all lie in the same vertical plane, the azimuth of the 
 celestial bodies is not affected. 
 
 51. General Laws of Refraction. By an investigation of the 
 formulae of refraction, and by astronomical observations already 
 described (Art. 37), the amount of refraction at different alti- 
 tudes has been obtained, and is given in what are called "tables 
 of refraction." The following general laws of refraction will 
 serve to give the student some idea of its amount, and of the 
 conditions under which it varies: 
 
 (1.) In the zenith there is no refraction. 
 
 (2.) The refraction is at its maximum in the horizon, being 
 there equal to about 33'. At an altitude of 45 it amounts 
 to 57". 
 
 (3.) For zenith distances which are not very large, the re- 
 fraction is nearly proportional to the tangent of the zenith 
 distance. When the zenith distance is large, however, the ex- 
 pression of the law is much more complicated. No table of re- 
 fraction can be trusted for an altitude of less than 5. 
 
 (4.) The amount of refraction depends upon the density of the 
 air, and is nearly proportional to it. The tables give the re- 
 fraction for a mean state of the atmosphere, taken with the 
 
54 PARALLAX. 
 
 barometer at 30 inches and the thermometer at 50. If the 
 temperature remains constant, and the barometer stands above 
 its mean height, or if the height of the barometer is constant, 
 and the thermometer stands below its mean height, the density 
 :)f the atmosphere is increased, and the refraction is greater 
 than its mean amount. Supplementary tables are therefore 
 given, from which, with the observed heights of both barometer 
 and thermometer as arguments, we may take the necessary cor- 
 rections to be applied to the mean refraction. 
 
 (5.) Since the effect of refraction is to increase the apparent 
 altitudes of the celestial bodies, the amount of refraction for 
 any apparent altitude is to be subtracted from that apparent 
 altitude, or added to the corresponding zenith distance. 
 
 52. Effects of Refraction. The apparent angular diameter of 
 the sun and of the moon being about 32', and the refraction iu 
 the horizon being 33', it follows that when the lower limb of 
 either body appears to be resting on the horizon, the body is m 
 reality below it. One effect, then, of refraction is to lengthen 
 the time during which these bodies are visible. Still another 
 effect is to distort the discs of the sun and the moon when near 
 the horizon : for since the refraction varies rapidly near the 
 horizon, the lower extremity of the vertical diameter of the 
 body will be more raised than the upper extremity, thus appa- 
 rently shortening this diameter, and giving the body an ellip- 
 tical shape. When the body comes still nearer the horizon, its 
 disc is distorted into what is neither a circle nor an elljpse, but 
 a species of oval, in which the curvature of the lower limb is 
 less than that of the upper one. The apparent enlargement of 
 these bodies when near the horizon is merely an optical delu- 
 sion, which vanishes when their diameters are measured with an 
 ii.strument. 
 
 PARALLAX. 
 
 53. The parallax of any object is, in the general sense of the 
 word, the difference of the directions of the straight lines drawn 
 to that object from two different points: or it is the angle at 
 the object subtended by the straight line connecting these two 
 points. In Astronomy, we consider two kinds of parallax : yco- 
 
PARALLAX. 
 
 55 
 
 centric parallax, by which is meant the difference of the directions 
 
 of the straight lines drawn to the centre of any celestial body from 
 
 the earth's centre and any point on its surface, and heliocentric 
 
 parallax, or the difference of the directions of the lines drawn to 
 
 the centre of the body from the cen- 
 
 tre of the earth and the centre of the 
 
 sun. The former is the angle at the 
 
 body subtended by that radius of 
 
 the earth which passes through the s f 
 
 place of observation : the latter the 
 
 angle at the body subtended by the 
 
 straight line joining the centre of 
 
 the earth and that of the sun. 
 
 54. Geocentric Parallax. In Fig. 
 21, let Obe the centre of the earth, 
 and L some point on its surface, of which Z is the zenith. Let 
 S- be some celestial body. The geocentric parallax of the body 
 is the angle CSL. Let S> be the same body in the horizon. 
 The angle LS' C is the parallax of the body for that position, 
 and is called its horizontal parallax. If we denote this horizontal 
 parallax by P, the earth's radius by E, and the distance of the 
 body from the earth's centre by d, we have, by Trigonometry, 
 
 Fig. 21. 
 
 To find the parallax for any other position, as at S, we repre- 
 sent the angle LSC by p, and the apparent zenith distance 
 of the body, or the angle ZLS, by z, the sine of which is equal 
 to the sine of its supplement SLC. We have from the tri- 
 angle LSC, since the sides of a plane triangle are proportional 
 to the sines of their opposite angles, 
 
 sin p R 
 
 sin z d 
 Combining this equation with the preceding, we have, 
 
 sin p = sin Psin z. 
 
 Since P and p are small angles, we may consider them propor- 
 tional to their sines, and thus have, finally, 
 
 p P sin z. 
 The parallax, then, is proportional to the sine of the zenith 
 
56 PARALLAX. 
 
 distance, and may be found for any altitude when the hori 
 zontal parallax is known. It evidently decreases as the alti- 
 tude increases, and in the zenith becomes zero. 
 
 55. Application of Parallax. In order that observations 
 made at different points of the earth's surface may be com- 
 pared, they must be reduced to some common point. Geocen- 
 tric parallax is applied to reduce any altitude observed at any 
 place to what it would have been had it been observed at the 
 earth's centre. We see from Fig. 21 that parallax acts in a 
 vertical plane, and that the zenith distance of the body as ob- 
 served from the earth's centre, or the angle ZC8, is less than the 
 observed zenith distance ZLS, by the parallax CSL. Parallax, 
 then, is always subtractive from the observed zenith distance, 
 and additive to the observed altitude. 
 
 The parallax above described is, strictly speaking, the paral- 
 lax in altitude. There is also, in general, a similar parallax in 
 right ascension, and in declination, formulae for deriving which 
 from the parallax in altitude are given in other works. 
 
 56. Heliocentric Parallax. It may sometimes happen that we 
 wish to reduce an observation from what it was at the centre 
 of the earth to what it would have been if it had been made 
 at the centre of the sun. Fig. 21, and the formulae obtained 
 from it, will apply equally well to this case, by making the 
 necessary changes in the description of the figure and in the 
 names of the angles. 
 
 Let S be still a celestial body, but let C be the centre of the 
 sun, and L that of the earth. The angle p will then represent 
 the heliocentric parallax, and the angle SLCihe angular dis- 
 tance of the body from the sun, as measured from the earth's 
 centre, or, as it is called, the body's elongation. The angle P 
 will be the greatest value of the heliocentric parallax, taken 
 when the body's elongation from the sun is 90, and is called 
 the annual parallax. We shall then find, from the formulae 
 of Art. 54, that the annual parallax has for its sine the ratio 
 of the distance of the earth from the sun to that of the body 
 from the sun, and that the parallax for any other position is 
 the product of the annual parallax by the sine of the body's 
 elongation. 
 
DIP OF THE HOEIZON. 
 
 57 
 
 Fig. 22. 
 
 DIP OF THE HORIZON. 
 
 57. The dip of the horizon is the angular depression of the 
 visible horizon below the celestial hori- B 
 zon. In Fig. 22, let HG be a por- 
 tion of the earth's surface, and C 
 the earth's centre. Let a radius of 
 the earth, CA, be prolonged to some 
 point D, beyond the surface, and let 
 an observer be supposed to be at the 
 point D. At the point D let the line 
 BD be drawn perpendicular to the line 
 CD, and also the line DH, tangent to 
 the earth's surface at some point H. If these two lines be revolved 
 about the line CD, DB will generate the plane of the celestial 
 horizon (since we have seen that all planes passed perpendicular 
 to the radius will, when indefinitely extended, mark out the 
 same great circle on the celestial sphere), and DH will gene- 
 rate the surface of a cone, which will touch the earth in a 
 small circle. If we disregard for the present the effect of the 
 earth's atmosphere, this small circle will be the visible horizon 
 of the observer at D, and the angle BDHvfill be the dip. DA 
 is the linear height of the observer at D. 
 
 Now let a radius, CH, be drawn to the point of tangency H. 
 The angles BDH and HCD, having their sides mutually per- 
 pendicular, are equal. Represent the angle HCD by D, the 
 earth's radius by J2, and the observer's height, AD, by h. ID. 
 the triangle DHC, right-angled at H, we have, 
 
 since D is small, it is better to use the formula, 
 
 cos D = 1 2 sin 8 * D : 
 hence we have, 
 
 =1 28in f iD: 
 
58 
 
 DIP OF THE HORIZON. 
 
 As the angle D is small, we mav take 
 
 sin \ D = J D sin 1" : 
 
 and as h is very small in comparison with JR, we may also 
 assume (E -{- h~) to be sensibly equal to E. Making these 
 changes in the above equation, and finding the value of D, we 
 have 
 
 Substituting in this expression the value of the earth's radius 
 in feet, we have, finally, 
 
 D = 63".8 y~h^ 
 h being expressed in feet. 
 
 The dip, then, for a height of one foot is 63".8 ; and for 
 other heights it is proportional to the square root of the number 
 of feet in the height. 
 
 58. Effect of Atmospheric Refraction. 
 If the effect of atmospheric refrac- 
 tion is taken into consideration, the 
 G line HD must be a curved line, as is 
 represented in Fig. 23. The point H 
 will then appear to lie in the direc- 
 tion DH f , to the observer at D, and 
 the dip will be the angle BDH '. The 
 Fi s- 23 -* effect, then, of refraction is to decrease 
 
 the dip, the amount by which it is decreased being about -j-^th 
 of the whole. 
 
 59. Application of Dip. Tables have been computed in 
 which may be found the proper dip for different heights above 
 the surface of the earth. Dip constitutes one of the correc- 
 tions which are to be applied at sea to the observed altitude of 
 a celestial body to obtain its true altitude: its altitude, that is, 
 above the celestial horizon. Since the visible horizon lies below 
 the celestial horizon, this correction is evidently subtractive. 
 
 The curved line HD is tangent to the earth's surface at H. 
 
MEASUREMENT OF THE EARTH. . r >9 
 
 CHAPTER IV. 
 
 THE EARTH. ITS SIZE, FORM, AND ROTATION. 
 
 60. HAVING seen what are the construction and the adjustments 
 of the principal astronomical instruments, to what uses each is 
 adapted, and what corrections are to be applied to the observa- 
 tions which are taken, we are now ready to proceed to the solu- 
 tion of some of the many questions of interest which the study 
 of Astronomy opens to us. And first of all, let us see how as- 
 tronomical observations will help us to a knowledge of the size 
 and the form of the earth. The question is one of the first im- 
 portance; for upon the determination of the size of the earth 
 depends in a great measure, as we shall see further on, the deter- 
 mination of the magnitudes and the distances of the other 
 heavenly bodies. Having seen the facts which seem to point to 
 the conclusion that the earth is spherical in form, w r e will start 
 with the assumption that this conclusion is correct, and proceed 
 to determine the magnitude of the earth, regarded as a sphere. 
 
 Now, we know from Geometry what is the ratio between the 
 radius of a sphere and the circumference of any great circle of 
 that sphere; and, therefore, if we can obtain the length of the 
 circumference of any great circle of the earth, of a meridian, 
 for instance, we can at once determine the radius of the earth. 
 And more than this : if we can measure the length of any known 
 arc of this meridian, of one degree, for instance, we can compute 
 the length of the entire circumference. The determination of 
 the earth's radius, then, depends only on our ability to satisfy 
 these two conditions: 
 
 (1.) We must be able to measure the linear distance on the 
 earth's surface between two points on the same meridian. 
 
 (2.) We must be able to measure the angular distance between 
 these same two points. 
 
 61. First Condition. A reference to Fig. 24 will explain how 
 
00 
 
 MEASUREMENT OF THE EARTH. 
 
 Fig. 24. 
 
 the first of these two conditions may be satisfied. Let A and G 
 represent the two points on the same meridian the 
 distance between which we wish to measure. We 
 have already seen (Art. 36) how an altitude and 
 azimuth instrument may be so adjusted that the 
 sight-line of the telescope will lie in the plane of 
 the meridian. Let an instrument be so adjusted 
 at the point A. Let some convenient station B be 
 taken, visible at A, and let the distance AB be 
 carefully measured. This distance is called the 
 base-line. Now, by means of the telescope, adjusted 
 to the plane of the meridian, let some point C be 
 established on the meridian, which shall also be 
 visible from B, and let the angles CAB and ABC 
 be measured. We now know in the triangle ABC 
 two angles and the included side, and can compute the distances 
 AC and CB. We now take some suitable point D, measure 
 the angles DCB and DBC, and knowing CB, we can obtain the 
 distance CD. The instrument is now taken to the point C, and 
 again established in the plane of the meridian, either by the 
 method of Art. 36, or by sighting back, as it is called, to A, or by 
 both methods combined. A third point on the meridian, E, 
 visible from D, is then selected, the angles ECD and ED C are 
 measured, and the distances EC and ED computed. This pro- 
 cess is continued until the whole distance between A and G has 
 been obtained. 
 
 This method of measurement is called the method of triangu- 
 lation. The base-line, AB, is purposely taken under circum- 
 stances which favor its accurate measurement, and the rest of 
 the work consists in the determination of horizontal angles, 
 which presents no special difficulty, and in the solution of tri- 
 angles by computation. 
 
 Two things are to be noticed in reference to these triangles. 
 The first is that in selecting the points B, C, D, &c., care must be 
 taken so to choose them that the triangles ABC, BCD, &c., shall 
 be nearly equiangular; since triangles in which there is a great 
 inequality of the angles (ill-conditioned triangles, as they are 
 called) will be much more likely to cause some error in the work. 
 
MEASUREMENT OF THE EARTH. 61 
 
 The second thing to be noticed is that these triangles are really 
 spherical triangles, and must therefore be solved by the formulae 
 of Spherical Trigonometry. If, for any reason, we are forced 
 to take any of the points C, E, &c., off the meridian, the cor- 
 responding distances can be reduced to the meridian by appro- 
 priate formulae. 
 
 The correctness of the result may be tested by measuring the 
 distance GF, and comparing its measured length with that ob- 
 tained by computation. The marvellous accuracy of this method 
 of measurement is shown by the fact that in an arc of the meri- 
 dian measured by the French at the close of the last century, 
 and which was several hundred miles in length, the discrepancy 
 between the measured and the computed length of the second 
 base-line was less than twelve inches. 
 
 62. Second Condition. The second condition requires that 
 the angle at the centre of the earth, subtended by the arc of the 
 meridian measured, shall be obtained. This angle is evidently 
 the difference of latitude of the two extremities of the arc, and 
 therefore all that is needed to satisfy this condition is that the 
 latitude of each extremity shall be determined by appropriate 
 observations. 
 
 Instead of determining the latitude of each place independently 
 of the other, we may, if we choose, obtain the difference of 
 latitude directly, by observing at each place the meridian zenith 
 distance of the same celestial body. In Fig. 25, let A and G be 
 the two extremities of the arc, C the centre 
 of the earth, and S the celestial body on 
 the meridian. If Z' is the zenith of the 
 point A, the meridian zenith distance of 8 
 at A, reduced to the centre of the earth, is 
 the angle Z' CS. In the same manner the 
 true meridian zenith distance of S at G is 
 the angle ZCS. The difference of these two 
 zenith distances, or the angle ZCZ\ is evi- 
 dently the difference of latitude of G and A. Fi s- 25 - 
 
 If the celestial body crosses the meridian between the l\vo 
 zeniths, as at $', the difference of latitude is numerically the 
 sum of the two meridian zenith distances. 
 
62 FORM OF THE EARTH. 
 
 63. Results. By the process ajbove described, or by processes 
 of a similar character, arcs of different meridians, and in differ- 
 ent latitudes, have been carefully measured. The sum of the 
 arcs thus measured is more than 60, and the length of a degree 
 of the meridian has been found to be, on the average, 69.05 miles. 
 Multiplying this by 360, we obtain 24,858 miles for the circum- 
 ference of a meridian, and dividing this circumference by -, 
 (3.1416) we find the length of the earth's diameter to be 7912 
 miles. 
 
 64. Spheroidal Form of the Earth. One remarkable fact is 
 noticed when we compare the lengths of the degrees of the meri- 
 dian, measured in different latitudes ; and that is, that the length 
 of the degree is not the same at all parts of the meridian, but sen- 
 sibly increases as we leave the equator. The length of a degree 
 at the equator is found to be 68.7 miles, whilst at the poles it is 
 computed to be 69.4 miles. The conclusion drawn from this 
 fact is that the figure of the earth is not rigorously that of a 
 sphere, since a spherical form necessarily implies an absolute 
 uniformity in the length of a degree in all parts of a great circle. 
 In order to determine the exact geometrical figure of the earth, 
 we must bear in mind that the curvature of a line is always pro- 
 portional to the change in the direction of the tangents drawn 
 at successive points of that line. Now, since the altitude of the 
 elevated pole at any place is equal to the latitude of that place, 
 it follows that an advance towards the pole of one degree in lati- 
 tude is accompanied by a depression of one degree in the plane of 
 the horizon. If, therefore, in order to effect a change of one de- 
 gree in our latitude, we are forced to advance a greater number 
 of miles at the pole than at the equator, we conclude that the cur- 
 vature of the meridian is less at the pole than at the equator. Now, 
 
 this same inequality in its curvature is 
 also a peculiarity of the ellipse : and hence 
 we infer that the form of the earth's me- 
 ridians is not that of a circle, but that 
 of an ellipse, as represented in Fig. 26. 
 The axis of the earth, Pp, corresponds 
 to the minor axis of an ellipse, at the 
 extremities of which the curvature is the 
 
DENSITY. 63 
 
 least; and the equatorial diameter of the earth, EQ, corresponds 
 to the major axis of an ellipse, at the extremities of which the 
 curvature is the greatest. 
 
 The form of the earth, then, is that of the solid which would 
 be generated by the revolution of an ellipse about its minor 
 axis, which solid is called in Geometry an oblate spheroid. A 
 more common but less accurate name given to the form of the 
 earth is that of a sphere, flattened at the poles. 
 
 65. Dimensions of the Earth. The following are the dimen- 
 sions of the earth, when its spheroidal form is taken into con- 
 sideration. The determination is that of Sir G. B. Airy, the 
 Astronomer Royal of England. 
 
 Polar diameter ......................... 7899.170 miles. 
 
 Equatorial diameter ................... 7925.648 miles. 
 
 These values are believed to be within a quarter of a mile of the 
 true values. They differ from the results obtained by the astro- 
 nomer Bessel by only about o^th of a mile. 
 
 The compression, or oblateness, of an oblate spheroid is the 
 ratio of the difference between the major and the minor axis 
 of the generating ellipse to its major axis. The compression of 
 
 the earth is therefore ' . : which is about th. 
 
 If a and b represent the semi-major and the semi-minor axis 
 of the generating ellipse, the expression for the volume of the 
 oblate spheroid is $xa' 2 b. Substituting in this expression the 
 values of a and b, we find the earth's volume to be about 260 
 billions of cubic miles. 
 
 66. Density of the Earth. There are various methods of de- 
 termining the mean density of the earth. The following is a 
 brief summary of the method of determining it by means of the 
 torsion balance. This balance consists of a slender wooden rod, 
 supported in a horizontal position by a very fine wire at its 
 centre. To the extremities of this rod are attached two small 
 leaden balls. If left free to move, this horizontal rod will of 
 course come to rest when the supporting wire is free from 
 rorsion. Two much larger leaden balls are now brought near 
 the two suspended balls, and on opposite sides, so that the 
 attractions of both balls may combine to twist the wire in 
 
64 ROTATION. 
 
 same direction. The smaller balls will be sensibly attracted 
 by the larger ones, and the horizontal rod will change the 
 direction in which it lies. The amount of this deflection is 
 very carefully measured, and from it is computed the attraction 
 which the large balls exert on the small ones. But we know 
 the attraction which the earth exerts on the small balls, it being 
 represented by their weight: and we know also the volumes 
 of the earth and the attracting balls. Finally, we know the 
 density of lead: and from these data it is possible to compute 
 the mean density of the earth. 
 
 A series of over 2000 experiments of this nature was con- 
 ducted in England, in 1842, by Sir Francis Baily. The mean 
 density of the earth, obtained from these experiments, was 5.67 : 
 the density of water being the unit. Other methods of deter- 
 mining the density of the earth have been employed, the main 
 principle in each method being the comparison of the at- 
 traction exerted by the earth upon some object with that ex- 
 erted by some other body, whose density can be ascertained, 
 upon the same object. The results of these experiments do not 
 differ materially from the results of the experiments with the 
 torsion balance. 
 
 The volume and density of the earth being known, what is 
 commonly called its weight can be computed. It is found to 
 be about six sextillions of tons. 
 
 ROTATION OF THE EARTH. 
 
 67. Up to this point we have assumed the earth to be at 
 rest, and the apparent diurnal motions of the heavenly bodies 
 to be real motions. By careful observation of the sun, the 
 moon, and the most conspicuous of the planets, astronomers 
 have demonstrated that each of these bodies rotates upon a 
 fixed axis. Analogy, therefore, points to a similar rotation of 
 our own planet: and besides this, there are many phenomena 
 which are inexplicable if the earth is at rest, but which are 
 fully accounted for on the supposition that it rotates upon an 
 axis. We will now examine the principal of these phenomena. 
 
 68. The weight of the same body is not the same in different 
 latitudes. Careful experiments made in different latitudes show 
 
CENTRIFUGAL FORCE. 65 
 
 that the weight of the same body is not constant at all parts 
 of the earth's surface, but increases with the latitude. A body 
 which weighs 194 pounds at the equator will weigh 195 pounds 
 if taken to either pole ; that is to say, the weight of any body is 
 increased by j^th of itself when carried from the equator to 
 the pole. This experiment cannot be made with the ordinary 
 balances in which bodies are weighed : since it is obvious that 
 the same cause, whatever it may be, which affects the weight 
 of the body will also affect that of the weights by which it is 
 balanced, and by the same amount, so that the scales will still 
 remain in equilibrium. If, however, we test the weight of a 
 body (the force, that is to say, with which it tends to the earth's 
 centre) by the effect which it has in stretching a spring, the 
 increase of weight will be found to be as stated above. 
 
 Part of this increase of weight is due to the spheroidal form 
 of the earth, since a body when at the pole is nearer the centre 
 of the earth than when at the equator. The amount of increase 
 due to this cause has been calculated to be about ^ih; hence 
 the difference between j^th and g^o^ n J which is 2^-gth, still re- 
 mains to be accounted for. We shall now see how it is com- 
 pletely accounted for by the supposition that it is the effect of 
 the centrifugal force which is induced by a rotation of the earth 
 upon its polar axis. 
 
 69. Centrifugal force. The tendency which a body has, when 
 revolving about any point as a centre, to recede from that centre, 
 is called its centrifugal force. The formula for the centrifugal 
 force may be found in any treatise on Mechanics, and is as 
 follows : 
 
 in which / is the centrifugal force, r the 
 radius of the circle of revolution, and t the 
 veriodic time, or the time in which the 
 revolution is performed. Now, in Fig. 27 
 let the earth be supposed to rotate about 
 its polar axis, Pp, once in every sidereal 
 day, which, as we have already seen 
 (Art. 7), is 3m. 56s. less than the mean 
 
66 CENTRIFUGAL FOIICE. 
 
 solar day, and therefore contains 86,164s. Substituting thi. 
 value of t in the formula given al>ove, and substituting for r the. 
 value of the earth's equatorial radius in feet, and computing 
 the value of/, we shall find that the centrifugal force at the 
 equator is .1113 feet. Now the actual force of gravity at the 
 equator is found, by Mechanics, to be 32.09 feet. If the earth 
 were at rest, the force of gravity at the equator would evidently 
 be 32.09 + -1H3 feet. Hence the diminution of gravity at 
 the equator, due to centrifugal force, (in other words, the loss 
 of weight), is equal to y^f^, or 5 |^th. 
 
 Since the periodic time (t in the formula) is constant for all 
 places on the earth's surface, it is evident, from the formula, that 
 the centrifugal force at any place L is to the centrifugal force 
 at the equator as the radius of revolution at L, or LM, is to CQ 
 But we have in the figure, 
 
 ML ML 
 
 -QQ~ = ~QL cos <& C cos Latitude. 
 
 Denoting, then, the centrifugal force at the equator by C, 
 and that at L by c, we have, 
 
 c = C cos L: 
 or the centrifugal force varies with the cosine of the latitude. 
 
 The centrifugal force at L acts in the direction of the radius 
 of revolution ML. Let its amount be represented by LB, 
 taken on LM prolonged. This force may be resolved into two 
 forces : LA, in the direction from the centre of the earth, and AB, 
 at right angles to LA. The force LA, being directly opposed 
 to the attraction of the earth, has the effect of diminishing the 
 weight of bodies at L, and may therefore be taken to represent 
 the loss of weight at L. 
 
 Denoting the loss of weight by w, and the centrifugal force 
 at L by c, as before, we have, from the triangle ABL, 
 w = c cos L. 
 
 But we have already, 
 
 c = C cos L : 
 /. w = C cos 2 L. 
 
 Now, at the equator, as is evident from the figure, the whole 
 effect of the centrifugal force is exerted to diminish the weight 
 of bodies, and C therefore also represents the loss of weight at 
 
TRADE WINDS. 07 
 
 the equator. We. have then, finally, that the loss of weight of a 
 body at any latitude, due to centrifugal force, is equal to the pro- 
 duct of -.j^th of the weight multiplied by the square of the cosine 
 of the latitude. 
 
 70. Spheroidal Form of the Earth due to Centrifugal Force. 
 We see, then, that the supposition that the earth rotates upon its 
 axis fully explains the observed difference in the weight of the 
 same body in different latitudes. But this is not all : for if we 
 assume that the particles of matter of which the earth is com- 
 posed were formerly in a fluid or molten condition, and there- 
 fore free to move, the spheroidal form of the earth is itself a 
 proof of the earth's rotation. Numerous experiments may be 
 made to show that, for a fluid body at rest, the form of equili- 
 brium is that of a sphere : and that, for a fluid body which 
 rotates, the form of equilibrium is that of a spheroid, the oblate- 
 ness of which increases with the velocity of rotation. Knowing 
 the volume and the density of the earth, and assuming the time 
 of rotation to be twenty-four sidereal hours, it is possible to 
 calculate the form of equilibrium which a fluid mass under 
 these conditions will assume : and this form is found to be that 
 of a spheroid, with an oblateness very nearly identical with the 
 known oblateness of the earth. 
 
 This tendency of a fluid mass to assume a spheroidal form 
 under rotation may also be shown in Fig. 27. The centrifugal 
 force LB was resolved into the two forces LA and AJB, the 
 former of which forces has already been discussed. The effect 
 of the latter force, AB, is evidently a tendency in the particle 
 L to move towards the equator EQ; and a similar force acting 
 upon all the particles of matter on the earth's surface, excepting 
 those at the poles and at the equator, will cause them all to 
 move in the direction of the equator, and thus give a spheroidal 
 form to the mass. 
 
 71. Trade Winds. The trade winds are permanent winds 
 which prevail in and sometimes beyond the torrid zone. These 
 winds are northeasterly in the northern hemisphere and south- 
 easterly in the southern hemisphere. The air within the torrid 
 zone being, in general, subject to a greater degree of heat than 
 the air at other portions of the earth's surface, rises, and its 
 
68 PENDULUM EXPERIMENT. 
 
 place is filled by air which comes in from the regions beyond the 
 tropics. If the earth were at res*t, these currents of air would 
 manifestly have simply a northerly and a southerly direction. 
 Now, we all know that, when we travel in any direction on a 
 still day, or even when the wind is moving in the same direc- 
 tion with us, but with a less velocity, the wind seems to come 
 from the point towards which we are going. We see from Fig. 
 27 that, if the earth is rotating upon its polar axis, the linear 
 velocity of rotation decreases as the latitude increases. Hence, 
 the air from beyond the tropics, having at the start only the 
 linear velocity of the place which it leaves, will, as it moves 
 towards the equator, have continually a less velocity than that 
 of the surface over which it passes, and will seem to come from 
 the quarter towards which those places are moving. If, then, 
 the earth is rotating from west to east, these currents of air will 
 have an apparent motion from the east, which motion, when 
 compounded with the motion from the north and the south, 
 before mentioned, will give us the northeasterly and south- 
 easterly winds which we call the Trades. 
 
 72. The Pendulum Experiment* The last and decidedly the 
 most satisfactory proof of the earth's rotation which we shall 
 notice, is that which comes from the apparent rotation of the 
 plane of a freely-suspended pendulum, when made to vibrate 
 at any point on the earth's surface except the equator. 
 
 It is an established law in Mechanics that a pendulum, freely 
 suspended from a fixed point, always vibrates in the same plane ; 
 and also that if we give the point of support a slow movement 
 of rotation about a vertical axis, the plane of vibration will still 
 remain unchanged. If, for instance, we suspend a ball by a 
 string, and, having caused it to vibrate, twist the string, the ball 
 will rotate about the axis of the string, while the plane in which 
 it vibrates will not be affected. 
 
 Now, let us suppose that a pendulum is suspended at the north 
 pole, and is made to vibrate: and let us further suppose that the 
 earth rotates from west to east, once in 24 hours. The line in 
 which the plane of vibration intersects the plane of the horizon 
 
 * This is called FoucaulCs experiment. A full discussion of it is given in 
 the American Journal of Science, 2d series, vols. XII-XIV. 
 
PENDULUM EXPERIMENT. 69 
 
 will move about in the plane of the horizon, in a direction oppo- 
 site to that in which the earth is rotating, and with an equal 
 velocity, thus completing one revolution in 24 hours. In Fig. 
 28, let AGED be the horizon of the ob- 
 server at the north pole, and let the earth 
 rotate in the direction indicated by the ( \\y \\ 
 
 arrows. Let the pendulum at P be set 
 swinging in the direction of some diameter, 
 AB, of the horizon. At the end of an hour, 
 the rotation of the earth will have carried 
 this diameter to some new position A'B', 
 at the end of the next hour to some new position A"J3", &c. : 
 while the pendulum will still swing in the original direction AB. 
 To the observer, then, unconscious of the earth's rotation, the 
 plane of vibration, which really remains unchanged, will appear 
 to rotate in a direction opposite to that in which the earth is 
 rotating. 
 
 At the south pole, under the same suppositions, a similar phe- 
 nomenon will be noticed, except that the plane of vibration will 
 apparently move in the opposite direction. Thus, if at the north 
 pole the apparent motion of the plane is like that of the hands 
 of a clock, as we look on its face, the apparent motion at tho 
 south pole will be the opposite to this. 
 
 Again, if a pendulum is made to* vibrate in the plane of a 
 meridian at the equator, there will be no apparent change in the 
 plane of vibration, since it will always coincide with the plane 
 of the meridian, and hence the pendulum will continue to 
 swing north and south during the entire period of the earth's 
 rotation. The condition that the pendulum shall here swing in 
 the plane of a meridian is entirely unnecessary, and is made 
 only for the sake of illustration ; for there will be no apparent 
 change in the plane of vibration, whatever may be the direction 
 in which the pendulum is made to vibrate. 
 
 The apparent rotation, then, of the plane of vibration of the 
 pendulum is 360 in 24 hours at the poles, and nothing at the 
 equator. At places lying between the equator and the poles, 
 the apparent angular motion of the plane of vibration will be 
 between these two limits; in other words, less than 360 in 
 
70 LINEAR VELOCITY OF ROTATION. 
 
 24 hours. Appropriate investigations show that the apparent 
 angular motion of the plane of* vibration at any place in any 
 interval of time is equal to the angular amount of the earth's 
 rotation in that time, multiplied by the sine of the latitude of 
 the place.* Thus, at Annapolis, we have for the angular motion 
 in one hour, 
 
 15 sin 38 59' = 9 26': 
 
 so that the plane of vibration will make one apparent rotation 
 at Annapolis in 38h. 09m. 
 
 Such is the theory of the pendulum experiment. Now, nume- 
 rous experiments have been made in different latitudes, and in 
 every case an apparent rotation of the plane of vibration from 
 east to west has been observed, with a rate agreeing very closely 
 with that demanded by the theory ; and the conclusion is irre- 
 sistible that the earth rotates on its polar axis, from west to east, 
 once in every sidereal day. 
 
 73. Linear Velocity of Rotation. Taking the equatorial cir- 
 cumference of the earth to be 24,900 miles, we have a linear 
 velocity of over 1000 miles an hour, and over 17 miles a minute. 
 This is the velocity at the equator. The linear velocity at other 
 points on the earth's surface is less than this, since the circum- 
 ferences of the parallels of latitude are less than the circumfer- 
 ence of the equator. Since the circumference of any parallel is 
 
 * This formula may be obtained by the principles of the resolution of ro- 
 tation, given in treatises on Mechanics. Thus, in the 
 figure, the rotation of the point L about the axis of 
 the earth, PO, may be resolved into two rotations, 
 one about the radius LO, and the other about the 
 radius MO, drawn perpendicular to LO. If v re- 
 presents the angular velocity of L about the axis 
 PO (or 15 in one hour), and v f and v" the angular velocities about the 
 axes LO and MO, we have, from Mechanics, 
 
 v' = v cos LOP, and v" v cos POM. 
 
 NOAV, the rotation about the axis OM will have no effect in changing the 
 apparent position of the plane of vibration of the pendulum, since it is 
 analogous to the case at the equator considered in the text ; while the rota- 
 tion about the axis LO, being analogous to the case at the pole, will pro- 
 duce a similar effect. The apparent angular motion, then, of the plane 
 of vibration will be v cos LOP, or v sin Lat. 
 
LINEAR VELOCITY OF ROTATION. 71 
 
 to that of the equator as the radius of the parallel is to the 
 radius of the equator, the linear velocity will diminish as we 
 leave the equator in the same ratio that the radii of the succes- 
 sive parallels diminish : in the ratio, that is, of the cosine of the 
 latitude, as was proved in Art. 69. For instance, the cosine of 
 60 being J, the linear v o looHv nt thst latitude is on'y 8 
 a minute. 
 
72 LATITUDE. 
 
 CHAPTEK V. 
 
 LATITUDE. LONGITUDE. 
 LATITUDE. 
 
 74. THE latitude of any place on the earth's surface has been 
 proved, in Articles 10 and 11, to be equal to either the altitude 
 of the elevated pole or the declination of the zenith at that 
 place. We shall now proceed to explain the principal methods 
 by which either one or the other of these arcs may be found. 
 
 75. First Method. Let Fig. 29 represent a projection of the 
 
 celestial sphere on the plane of the celestial 
 meridian, RZHN, of some place. HR is 
 the celestial horizon at that place, Z the 
 zenith, P the elevated pole, and EQ the 
 equator. Let s represent some circumpolar 
 star, whose declination is known, at its 
 lower culmination. Let its meridian alti- 
 tude be observed, and corrected for instru- 
 mental errors and refraction. (For all celestial bodies except the 
 sun, the ( moon, and the planets, the corrections for parallax and 
 semi-diameter will be inappreciable.) To this corrected altitude 
 add the star's polar distance, the complement of the star's known 
 declination. The sum is the altitude of the elevated pole, or 
 the latitude. 
 
 If the circumpolar star is at its upper culmination, as at $', 
 the polar distance is to be subtracted from the corrected altitude. 
 If Ti r and h denote the corrected altitudes at the upper and 
 the lower culmination, p' and p the corresponding polar dis- 
 tances, and L the latitude, we have evidently 
 L = h'p' 
 L = h + /; : 
 whence L = 2 (k' -f //.) -f } (/> p)- 
 
LATITUDE. 1 
 
 In this formula the value of the latitude does not depend on the 
 absolute value of either polar distance, but merely on the chart ye 
 of the polar distance between the two transits, which is usually 
 s?o small as to be neglected. This method, then, is free from any 
 error in the declination, and is used at all fixed observatories. 
 
 76. Second Method. When the star is at its upper culmina- 
 tion, it will, in general, be more convenient to find the declina- 
 tion of the zenith from the meridian zenith distance of the star. 
 Taking the star s', for instance, and denoting its meridian zenith 
 distance by z, and its declination by d, we have 
 
 L = ZQ = Qs' Zs' ^d z. (a) 
 
 For the star s", we have 
 
 L = Zs" + Qs" = z + d, (b) 
 
 and for the star s" 
 
 L = Zs" Qs" = z d. (c) 
 
 From these three formulae a general rule may be deduced, appli- 
 cable to the upper culmination of every star. We notice that 
 in the formulae (a) and (6), where d is positive, the stars s' and 
 s" are on the same side of the equator with the elevated pole; 
 that is to say, their declinations have the same name as the ele- 
 vated pole; while in the formula (c) the declination has the 
 ^.opposite name. We also notice that in the formulae (6) and (r,} t 
 where z is positive, the stars are on the opposite side of the zenith 
 from the elevated pole; in other words, their bearing has the op- 
 posite name to that of the pole : while the bearing of the star ', 
 in the formula for which z is negative, has the same name as the 
 elevated pole. The general rule, then, for all these stajs will be 
 the following: If the star bears south, mark the zenith distance 
 lorth; if it bears north, 'mark the zenith distance south; mark 
 the declination north or south, as the star is north or south of 
 vJie equator, and combine the zenith distance and the declina- 
 tion, thus marked, according to their names. 
 
 77. Third Method. A very successful adaptation of the pre- 
 ceding method is made by using two stars which culminate at 
 nearly the same time, but on opposite sides of the zenith, as s' 
 and s" in Fig. 29. These two stars are so selected that the dif- 
 ference of theii zenith distances is very small, and can be mea- 
 
74 LATITUDE. 
 
 sured directly by means of a micrometer. By the formulae of 
 the preceding article we have for a', 
 
 L^d z, 
 
 and for s", denoting its meridian zenith distance and declinatioii 
 by z' and d', 
 
 L = d' + z', 
 whence we have, 
 
 L = J (d + d") -f i- (z r z\ 
 
 The determination of the latitude is thus made free from any 
 error in the graduations of the vertical circle, and depends only 
 on the known declinations of the two stars, and on the difference 
 of their zenith distances. Errors in the refraction are also very 
 nearly eliminated. 
 
 This is the principle of what is called Talcott's Method, a 
 method very commonly used by the United States Coast Survey. 
 The instrument employed is the zenith telescope, a modification 
 of the altitude and azimuth instrument. The two stars are so 
 selected that the difference of their zenith distances is less than 
 the breadth of the field of the telescope. The instrument is set 
 in the plane of the meridian to the mean of the two zenith dis- 
 tances, and for the star which culminates first. When this star 
 crosses the meridian, it is bisected by the micrometer wire, and 
 the micrometer is read. The instrument is then turned 180 in 
 azimuth, and the process is repeated with the second star. The 
 difference of the zenith distances is then obtained from the dif- 
 ference of the two micrometer readings, and added to the half 
 sum of the two declinations, according to the formula. 
 
 78. Fgurth Method. When the local time (either solar or 
 sidereal) is known, the latitude may be obtained from altitudes 
 which are not measured on the meridian. Let Fig. 30 be a pro- 
 jection of the celestial sphere on the plane 
 of the horizon. Z is the zenith of the place, 
 P the elevated pole, PZthe co-latitude, and 
 S a star, whose altitude is measured. SPZ 
 is the hour angle of the star, which can be 
 obtained from the local time noted at the 
 instant the altitude is observed. PS is the 
 star's known polar distance. In the triangle 
 
REDUCTION OF THE LATITUDE. 
 
 SPZ, we have the sides ZS and SP, and the angle SPZ, and can 
 therefore compute the value of the co-latitude, PZ, by the for- 
 mulae of Spherical Trigonometry. 
 
 An analytical investigation of the formulae by which this pro- 
 blem is solved shows that errors in the observed altitude and the 
 time have the less effect upon the result the nearer the body is 
 to the meridian. 
 
 79. Methods of Finding the Latitude at Sea. The second and 
 the fourth of the methods above described are the methods most 
 commonly employed in finding the latitude at sea. The sun is 
 the body which is generally used, its altitude above the sea 
 horizon being measured with a sexta*nt or an octant. The time of 
 noon being approximately known, the observer begins to measure 
 the altitude of the lower limb of the sun a few minutes before 
 noon, and continues to measure it until the sun ceases to rise, or 
 "dips," as it is called. The greatest altitude which the sun 
 attains is considered to be the meridian altitude, although, rigor- 
 ously speaking, it is not. The proper corrections for index-error, 
 dip, refraction, parallax, and semi-diameter are next applied to 
 the sextant reading, and the result is the sun's true meridian 
 altitude, from which the latitude is obtained by the rule given in 
 Art. 76. 
 
 When cloudy weather prevents the determination of the meri- 
 dian altitude of either the sun or any other celestial body, an 
 altitude obtained within an hour of transit, on either side of the 
 meridian, maybe used to find the latitude by the fourth method, 
 Art. 78. Bowditch's Navigator contains special tables by which 
 the computation, particularly when the sun is observed, may bo 
 greatly facilitated. 
 
 80. Reduction of the Latitude. 
 Owing to the spheroidal form of the 
 earth, the vertical line at any point 
 of the surface, as Z' 0' in Fig. 31, 
 which corresponds exactly with 
 the normal drawn at that point, 
 does not coincide with the radius of 
 the earth, LO, passing through the 
 
 point, excepting at the equator 
 
76 LONGITUDE. 
 
 and the poles. It is necessary, Jhen, in refined observations, to 
 distinguish between the geographical zenith, Z 1 ', the point where 
 the vertical line, when prolonged, meets the celestial spheie, and 
 the geocentric zenith, Z, the point in which the radius meets the 
 sphere. Since there are two zeniths, there are also two lati- 
 tudes : Z' 0' Q, the geographical latitude, and ZOQ, the geocentric 
 latitude. The geographical latitude is evidently greater than 
 the geocentric, by the angle OLO', which is called the reduction 
 of the latitude. Formulae and tables for finding this reduction 
 are given in Chauvenet's Astronomy. It is only considered in 
 cases where the highest accuracy in the results is required. 
 
 LONGITUDE. 
 
 81. Let Fig. 32 represent a projection of the celestial sphere 
 on the plane of the equinoctial ABCG. P is the projection of 
 the elevated pole, and PG, PA, and PB are 
 projections of arcs of great circles of the 
 sphere passing through the pole. L?t PG 
 represent the projection of the meridian of 
 Greenwich, PA that of the meridian of some 
 other place on the earth's surface, and PJ> 
 that of the circle of declination passing 
 Fi - 32 - through some celestial body S. Then will 
 
 the angle CPA represent the longitude of the meridian PA 
 from Greenwich, GPU will represent the Greenwich hour angle 
 of the body S, and APB will represent its hour angle from the 
 meridian PA. The difference between these two hour angles is 
 evidently equal to the longitude of any place on the meridian 
 PA. The longitude, then, of any place on the earth's surface 
 is equal to the difference of the hour angles of the same celestial 
 Lody at that place and at Greenwich, at the same absolute instant 
 of time. When the Greenwich hour angle is the greater of these 
 two hour angles, reckoned always to the west, the longitude of 
 the place is west : when it is the smaller, the longitude is east. 
 
 If PB is the hour circle passing through the sun, the longi- 
 tude of the place is the difference of the solar times at the place 
 and at Greenwich: if it is the hour circle passing through the 
 
CHRONOMETERS. 77 
 
 vernal cqainox, the longitude is the difference of the two side- 
 real times. In order, then, to determine the longitude of any 
 place, we nmst be able to determine both the local and the 
 Greenwich i/me (either sidereal or solar) at the same instant. 
 
 There are- various methods of obtaining the local time, one 
 of which hhs already been described (Art. 20). It may be 
 noticed here that we are always able, by means of the Nautical 
 Almanac, to convert sidereal time into solar time, or solar into 
 sidereal (Art. 105). It remains, then, to determine the Green- 
 wich time, either sidereal or solar, to do which several distinct 
 methods may 6e employed. 
 
 82. Greenw.ck Time by Chronometers. If a chronometer is 
 accurately regulated to Greenwich time, that is to say, if the 
 amount by which it is fast or slow at Greenwich on any day, 
 and its daily .gain or loss, are determined by observation, the 
 chronometer can be carried to any other place the longitude 
 of which is desired, and the Greenwich time which the chrono- 
 meter gives can be directly compared with the time at that place. 
 This would be a perfectly accurate method, if the rate of the 
 chronometer remained constant during the transportation; but, 
 in fact, the rate of a chronometer while it is carried from place 
 to place is very rarely exactly the same that it is while the 
 chronometer is at rest. By using several chronometers, however, 
 and by transporting them several times in both directions be- 
 tween the two places, and finally by taking a mean of all the 
 results, the error may be reduced to a very minute amount. 
 For instance, the longitude of Cambridge, Mass., was deter- 
 mined by means of fifty chronometers, which were carried three 
 times to Liverpool and back, and from them the longitude was 
 obtained with a probable error of only 4th of a second of time. 
 
 83. Greenwich Time by Celestial Phenomena. There are cer- 
 tain celestial phenomena which are visible at the same absolute 
 instant of time, at all places where they can be seen at all. Such 
 are the beginning and the end of a lunar eclipse ; the eclipses 
 of the satellites of the planet Jupiter by that planet ; the 
 transits of these satellites across the planet's disc, and their nc- 
 cultations by it. The Greenwich times at which these various 
 phenomena will occur arc computed beforehand, and are pub- 
 
78 LUNAR DISTANCES. 
 
 Jished in the Nautical Almanac. . The observer, then, to obtain 
 his longitude, has only to note the local time at which any one 
 of these phenomena occurs, and to compare that time with the 
 corresponding Greenwich time given in the Almanac. The dif- 
 ficulty of determining the exact instant at which these phe- 
 nomena occur, however, diminishes to some extent the accuracy 
 of the results. On the other hand, the times of solar eclipses 
 mid of occultations of stars by the moon, although not identical 
 at different places, can be very accurately determined : and 
 iicnce these phenomena are often employed in obtaining longi- 
 tudes. (Art. 164.) 
 
 84. Greenwich Time by Lunar Distances. By the lunar distance 
 of a celestial body is meant its true angular distance from the 
 centre of the moon, as it would be seen at the centre of the 
 earth. The lunar distances of the sun, of the four brightest 
 planets, and of nine bright stars are given in the Nautical 
 Almanac, computed for every third hour of Greenwich solar 
 time. An observer, then, who wishes to determine his longitude, 
 measures the apparetd angular distance of the moon from some 
 one of these bodies, and also notes the local time at which the 
 observation is made. He then finds from this apparent distance, 
 by means of appropriate formulae and tables, the true geocentric 
 angular distance, at the time of observation, between the two 
 bodies. He then enters the table of lunar distances in the 
 Almanac with this distance, and finds the corresponding Green- 
 wich time, from which, and the local time noted, he can deter- 
 mine his longitude. 
 
 85. Difference of Longitude by Electric Telegraph. When two 
 stations, the difference of longitude of which is desired, are 
 connected by an electro-telegraphic wire, the difference of longi- 
 tude may be determined by means of signals made at either 
 station, and recorded at both. Suppose, for instance, there are 
 two stations A and B, of which A is the more easterly, and 
 that each station is provided with a clock regulated to its own 
 local time. Let the observer at A make a signal, the time of 
 which is recorded at each station. Let A denote the difference 
 of longitude of the two stations, T the local time at A at which 
 the signal is made, and T' the corresponding time at B. Since 
 
ELECTRIC SIGNALS. 79 
 
 A is to the east of B, its time is greater at any instant tlinn 
 that of B. We have then, supposing the signal to be recorded 
 simultaneously at the two stations, 
 
 A^r r. 
 
 Experience proves, however, that the records of the signal 
 are not exactly simultaneous, since time is required for the 
 electric current to pass over the wire. In the example above 
 given, then, if we denote the time required by the electric fluid 
 to pass from A to B by x, the time recorded at B will evidently 
 be, not T', but T' -|- * ; so that the expression for the difference 
 cf longitude will be 
 
 l'=T T' x. 
 
 Now let us suppose that instead of the signal's being made 
 by the observer at .4, it is made by the observer at B, at the 
 time T'. The corresponding time recorded at A will not be I 7 , 
 but T -\- x. In this case, then, the expression for the difference 
 of longitude will be 
 
 r= T+ x r. 
 
 Taking the mean of the values of A' and /', we have 
 I ()! _|_ ;/') = - T T'=L 
 
 Any error, therefore, which is caused by the time consumed 
 by the electric current in passing between two stations is elimi- 
 nated by determining the difference of longitude by signals made 
 at both stations, and taking the mean of the results. 
 
 86. Difference of Longitude by "Star Signals." The " method 
 of star signals" is a modification of the method describe^ in 
 the preceding paragraph, which is extensively used in the 
 United States Coast Survey. The principle on which this 
 method rests is that, since a fixed star makes one apparent revo- 
 lution about the earth in exactly twenty-four sidereal hours, 
 the difference of longitude between two meridians is equal to 
 the interval of sidereal time in which any fixed star passes from 
 one of these meridians to the other. The clock by which this 
 interval of time is measured may be placed at either station, 
 or indeed at any place which is in telegraphic communication 
 with both stations. Two chronographs, one at each station, are 
 connected with the wire and the clock, and upon them are 
 
SO LONGITUDE AT SEA. 
 
 recorded, by breaks in the circuit as explained in Art. 22, the 
 successive beats of the clock. A transit instrument is adjusted 
 to the meridian at each station. As the star crosses the several 
 threads of the reticule of the transit instrument at the eastern 
 station, the observer, by means of a break-circuit key, records 
 the instants upon both chronographs. The same process is 
 repeated as the star crosses the wires at the western station. 
 Now, it is evident that the elapsed time between the transits at 
 the two meridians has been recorded upon each chronograph. 
 Each of these values of the elapsed time is to be corrected for 
 instrumental errors, errors of observation, and for the gain or 
 loss of the clock in the interval ; and the mean of the two 
 values, thus corrected, is taken as the difference of longitude of 
 the two places. 
 
 By making similar observations on several stars on the same 
 night, by repeating the observations on subsequent nights, by 
 exchanging observers and using different clocks, and, finally, 
 by taking a mean of the results, a very accurate determination 
 of the difference of longitude may be secured. 
 
 87. Method of Finding the Longitude at Sea. The method of 
 finding the longitude at sea which is usually employed is the 
 method of Art. 82. The Greenwich time is given by chro- 
 nometers regulated to Greenwich time, and the local time is 
 obtained from the observed altitudes of celestial bodies. The 
 sun is the body the altitude of which is most commonly used 
 for this purpose; but altitudes of the most conspicuous of the 
 placets and the fixed stars may also be successfully employed. 
 Altitudes of the moon are to be avoided, except in cases where 
 no other body is available. At the instant when the altitude 
 of any celestial body is observed, the time shown by a watch is 
 noted. This \vatch, either shortly before or after the observa- 
 tion, is compared with the Greenwich chronometer, and by 
 means of this comparison the Greenwich time of the observa- 
 tion is obtained from the time given by the watch. The neces- 
 sary corrections are applied to the sextant reading to obtain 
 the body's true altitude. We shall then have, in the triangle 
 PZS, Fig. 33, the side ZS, the zenith distance of the body, PS 
 its polar distance, obtained from the Nautical Almanac, and 
 
COMPARISON OF LOCAL TIMES. 81 
 
 PZthe co-latitude of the place of observation ; 
 
 the latitude being determined by some one of 
 
 the methods already given (Art. 79), and being 
 
 reduced to the time of observation by the run z>| 
 
 of the ship given by the log. In the triangle 
 
 PZS, then, having the three sides given, we 
 
 can compute the angle SPZ, which is the hour 
 
 angle of the body. From this hour angle the Fig 33- 
 
 local time can be readily found, from which, and the Greenwich 
 
 time already obtained, the longitude may be determined. 
 
 In case there is no chronometer on board, the method of 
 lunar distances is the only regularly available method of deter- 
 mining the Greenwich time. At the present day, however, 
 lunar distances are mainly employed as checks upon the chro- 
 nometer, since any change in the rate of a chronometer will 
 cause a discrepancy between the Greenwich time shown by the 
 chronometer and that deduced from observation. 
 
 It can be shown, by proper methods of investigation, that 
 an error in the assumed latitude, or in the body's altitude, 
 causes the less error in the resulting hour angle the nearer 
 the body is to the prime vertical. It is best, then, in observing 
 the altitude of any celestial body for the purpose of obtaining 
 the local time, to observe it w T hen the body bears nearly east or 
 west, provided the altitude is not so small as to be sensibly 
 affected by errors in the refraction. It may also be shown that 
 in selecting celestial bodies for observations of this character, it 
 is best, if the other conditions are satisfied, to take those bodies 
 which have the smallest declinations. 
 
 88. Comparison of the Local Times of Different Meridians. 
 Since the local time, either solar or sidereal, is the greater 
 at the more easterly of any two meridians, it follows that a 
 watch or chronometer which is regulated to the time of any 
 one meridian will appear to gain when carried to the west, 
 and to lose when carried to the east: the amount of gain or 
 loss in any case being the difference of longitude, in time, 
 of the two meridians. A watch, for instance, which gives the 
 correct solar time at Boston will, even if it really is running 
 accurately, appear to gain nearly twelve minutes when taken 
 
82 COMPARISON OF LOCAL TIMES. 
 
 to New York. If, then, a watch which is regulated to the solar 
 time of any meridian is carried to the east, the difference of 
 longitude in time between the meridian left and that arrived 
 at must be added to the reading of the watch, to obtain tho 
 time at the second meridian: if it is carried to the west, th<> 
 difference of longitude must be subtracted. 
 
ECLIPTIC. 
 
 THE SUN. 
 
 CHAPTER VI. 
 
 THE EARTH'S ORBIT. THE SEASONS. 
 THE ZODIACAL LIGHT. 
 
 TWILIGHT. 
 
 89. The Ecliptic. IF a great circle on any globe is assumed 
 to represent the celestial equator, and any point of that circle 
 is taken to represent the vernal equinox, the relative positions 
 of all bodies, the right ascension and declination of which are 
 known, can be plotted upon this globe, and we shall have a 
 representation of the celestial sphere. The poles of the great 
 circle will represent the poles of the celestial sphere, and all 
 great circles passing through these poles will represent circles 
 of declination. We have seen, in the chapter on Astronomical 
 Instruments, in what manner the right ascension and the decli- 
 nation of any celestial body can be determined at any time by 
 observation. If we thus determine the position of the sun from 
 day to day, and mark the corresponding points upon our celes- 
 tial globe, we shall find that the sun appears to move in a great 
 circle of the sphere from west to east, completing one revolution 
 in this circle in 365d. 6h. 9m. 9.6s. of our ordinary solar time. 
 This interval of time is called the sidereal year. The great circle 
 in which the sun appears to move is called the ecliptic, and the 
 two points in which it intersects the celestial equator are called 
 the vernal and the autumnal equinox. 
 
 Let Fig. 34 be a representation of 
 the celestial sphere. EA Q Fis the equi- 
 noctial, Pp is the axis of the sphere, and 
 P the north pole. The circle A CVD re- 
 presents the ecliptic, V the vernal, and 
 A the autumnal equinox. The sun is 
 at the vernal equinox on the 21st of 
 March It thence moves eastward and 
 northward, and reaches the point C, 
 
84- DISTANCE OF THE SFX. 
 
 where it has its greatest northern declination, on the 2 1st of June. 
 This point is called the northern summer solstice. From this point 
 it moves eastward and southward, passes the autumnal equinox 
 A on the 21st of September, and reaches the point D, called the 
 northern winter solstice, on the 21st of December. It thence 
 moves towards V, which it reaches on the 21st of t March. 
 
 The obliquity of the ecliptic to the equinoctial is the angle 
 CVty, measured by the arc CQ. This angle or arc is evidently 
 equal to the greatest declination, either north or south, which 
 the sun attains, and is found by observation to be about 23 27'. 
 
 90. Definitions. The latitude of a celestial body is its angular 
 distance from the plane of the ecliptic, measured on a great circle 
 passing through its poles, and called a circle of latitude. In Fig. 
 34 the arc Ks is the latitude of the body s. The longitude of a 
 celestial body is the arc of the ecliptic intercepted between the 
 vernal equinox and the circle of latitude passing through the 
 body. Thus VK is the longitude of the body s. Longitude is 
 properly reckoned towards the east. 
 
 The hour circle which passes through the solstices, the circle 
 DHCB, is called the solstitial colure. The hour circle which 
 passes through the equinoxes is called the equinoctial colure. 
 
 91. Signs. The ecliptic is divided into twelve equal parts, 
 called signs t which begin at the vernal equinox, and are named 
 eastward in the following order: Aries, Taurus, Gemini, Cancer, 
 Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, 
 Pisces. Hence the vernal equinox is called the first point of 
 Aries. 
 
 The Zodiac is a zone or belt on the celestial sphere, extend- 
 ing about 9 on each side of the ecliptic. 
 
 DISTANCE OF THE SUN FROM THE EARTH. 
 
 92. Relative Distances of the Earth and Venus from the Sun. 
 It is found by observation that the mean value of the sun's 
 angular semi-diameter remains constant from year to year, 
 being always 16' 2". Since any increase or decrease in the 
 distance of the earth from the sun will evidently be accom- 
 panied by a corresponding decrease or increase in the sun's 
 angular semi-diameter, we conclude that the mean distance of 
 
DISTANCE OF THE SUN. 
 
 85 
 
 the earth from the sun is also constant from year to year. The 
 distance of tne earth from the sun is obtained by determining 
 the sun's horizontal parallax from certain observations made 
 upon the planet Venus. This planet revolves in a nearly cir- 
 cular orbit about the sun, in a plane only 3 inclined to the 
 plane of the ecliptic. Its distance from the sun is less than 
 that of the earth from the sun, and hence it sometimes passes 
 between the earth and the sun, and is seen apparently moving 
 across the sun's disc. This phenomenon is called a transit of 
 Venus. As a preliminary to the determination of the earth's 
 distance from the sun from one of these transits, it is neces- 
 sary to obtain the relative distances of Venus and the earth 
 from the sun. To do this, in Fig. 35 let S 
 be the sun, E the earth, and W V" V" the 
 orbit of Venus about the sun. It is evident 
 that the greatest angular distance (or elon- 
 gation) of Venus from the sun, the greatest 
 value, that is, of the angle VES, will occur 
 when the line from the earth to Venus is 
 tangent to the orbit of Venus, as repre- 
 sented in the figure. The orbit of Venus is 
 not really a circle, but an ellipse, and hence 
 the distance VS is slightly variable. So, 
 also, is the distance SE; hence the greatest 
 elongation is also variable, being found to lie between the limits 
 of about 45 and 47. Assuming its mean value to be 46, we 
 have in the right-angled triangle VSE, 
 
 VS = SE sin 46 .72 SE. 
 
 Neglecting the inclination of the orbit of Venus to the plane 
 of the ecliptic, we shall have, at the time of a transit, when 
 Venus is at V' 
 
 V'E = .28 SE. 
 
 Hence at the time of a transit the distance of Venus from the 
 sun is to that of Venus from the earth as about 72 to 28.* 
 
 * If we know the periodic time of Venus and that of the earth, the ratio 
 of the distances of these two planets from the sun can be obtained by Kepler's 
 Third Law (Art. 117), that "the squares of the periodic times of any two 
 planet* are proportional to the cubes of their mean distances from the sun." 
 
 8 
 
86 DISTANCE OF THE SUN. 
 
 93. Transit of Venus. In Fig. .36, let S denote the centre of 
 the sun, and CADN its disc : let V be Venus, and E the centre 
 
 Fig. 36. 
 
 of the earth. Let HK be that diameter of the earth which is 
 perpendicular to the plane of the ecliptic, and let an observer be 
 supposed to be stationed at each extremity. In order to simplify 
 the explanation, let us neglect the rotation of the earth during 
 the observation, and suppose Venus to move in the plane of the 
 ecliptic. To the observer at If, Venus will appear to move 
 across the sun's disc in the chord CD, and to the observer at JT, 
 in the chord AB. Regarding VHK and VFG as similar tri- 
 angles, we have, by Geometry, 
 
 FG:HK= GV: 7ff = 72:28 
 
 =- HK. 
 
 Again, we can obtain the angle which the line FG subtends 
 at the earth's centre in the following manner. Let the observer 
 at H note the interval of time in which the planet crosses thf 
 sun's disc in the chord CD, and the observer at K HIQ interval 
 in which it moves through the line AB. Since there are tables 
 which give us the angular velocity, as seen from the earth, both 
 of the sun and of Venus, we can deduce the angles at the earth's 
 centre subtended by the chords FB and GD, and knowing also 
 the angular semi-diameter of the sun, in other words, the angle 
 at the earth's centre subtended by SB or SD, we can compute 
 the angles at the earth's centre subtended by FS and GS t and, 
 finally, the angle subtended by FG. 
 
 We have now determined the angle subtended by the line FG, 
 at a distance equal to that of the earth from the sun, and also 
 the ratio of FG to the earth's diameter. It is evidently easy to 
 obtain from these values the angle at the sun subtended by the 
 
.MAGNITUDE OF THE SUN. 87 
 
 earth's radius, which angle is the sun's horizontal parallax, ;u* 
 we have already seen in Art. 54. 
 
 Although we have assumed in this discussion that the two ob- 
 servers are stationed at the extremities of the same diameter, it 
 is really only necessary that they shall be at two places whose 
 difference of latitude is large. The earth's rotation and other 
 things which we have here neglected must be taken into con- 
 sideration in the practical determination of the sun's parallax. 
 
 94. Distance of the Earth from the Sun. The last two transits 
 of Venus were in 1769 and 1874, and from observations in 1769, 
 the sun's horizontal parallax was determined to be 8". 6. Later 
 observations of a different character have given a horizontal 
 parallax of 8".848,* which is here used. The results of the exten- 
 sive observations in 1874 cannot yet be given. 
 
 From Art. 54, we have for the distance in miles of the earth 
 from the sun, 
 
 d = E cosec P = 3962.8 cosec 8".848 = 92,400,000 miles. 
 
 95. Magnitude of the Sun. 
 The length of the sun's radius 
 can be at once obtained as soon 
 as we know its distance from 
 the earth. Thus, in Fig. 37, 
 
 let S be the centre of the sun, ri s- 37 - 
 
 and E that of the earth. The angle AES is the apparent semi- 
 diameter of the sun, which we obtain by observation, its mean 
 value being, as already stated, 16' 2". We have then, in the 
 right-angled triangle AES, 
 
 SA = 92,400,000 sin 16' 2" = 431,000 miles. 
 The sun's linear radius, then, is equal to nearly 109 of the earth's 
 radii; and since the volumes of spheres are proportional to the 
 cubes of their radii, the volume of the sun bears to that of the 
 earth the enormous ratio of 1,286,000 to 1. 
 
 By observations and calculations which will be described in 
 fhe Chapter on Gravitation (see Art. 114), the mass of the sun is 
 found to be about 327,000 times that of the earth; or about 670 
 times the sum of ihe masses of all the planets of the solar system. 
 
 * Determination of Professor Simon Newcomb, Tjnited States Navy. TU 
 value obtained by Leverrier is 8 x/ .95. 
 
88 
 
 ORBIT OF THE EARTH. 
 
 THE EARTH'S ORBIT. 
 
 96. Revolution of the Earth about the Sun. Up to this point 
 we have spoken of the apparent animal motion of the sun in the 
 ecliptic from west to east, as though the earth were really at rest, 
 and the sun revolved about it in its orbit. But when we take 
 into consideration the immense mass of the sun compared with 
 that of the earth, we are almost irresistibly led to conclude that 
 the apparent annual revolution of the sun is the result, not of the 
 actual revolution of the sun about the earth, but of that of the 
 earth about the sun. Such a revolution of the earth, from west 
 to east,* would give to the sun precisely that apparent motion 
 
 in the ecliptic which has been ob- 
 served. This may be seen in Fig. 
 38. Let S be the sun, EE'E" the 
 earth's orbit, and the outer circle 
 S'S"S'" the great circle in which 
 the plane of the ecliptic, indefi- 
 nitely extended, meets the celestial 
 sphere. When the earth is at E, 
 the sun will be projected in $'; 
 when the earth is at E', the sun 
 will be projected in S", &c. ; that 
 is to say, while the earth moves 
 
 Fig. 38. 
 
 * Whatever the absolute motion of any celestial body moving in a circle 
 or an ellipse may be, the appearance presented in that motion will be re- 
 versed if the spectator moves from one side of the plane in which the 
 motion is performed to the other. Thus, the apparent daily motion to the 
 westward of any celestial body is the same as the motion of the hands of a 
 clock as we look upon its face, to an observer who is on the north side of 
 the plane of the diurnal circle in which the body moves, as is seen in any 
 latitude north of 23 27' in the motion of the sun ; while the same west- 
 ward motion presents the opposite appearance if the ohserveristo the south 
 of the plane of motion, as may be seen in these latitudes in the case of the 
 Great Bear. The appearance presented by a motion from west to east is 
 of course the reverse of this ; hence when we say that the earth or any 
 other body moves about the sun from west to east, we mean that, to an ob- 
 server situated to the north of the plane of motion, the body appears to movein a 
 direction opposite to that in which the hands of a dock move. 
 
ORBIT OF THE EARTH. 89 
 
 about the sun in the direction EE'E", the sun will apparently 
 move about the earth in the same direction, S'S"S'". 
 
 Against this theory, then, of the earth's revolution there is 
 nothing to urge; and analogy gives us a strong argument in 
 favor of it. Almost every celestial body in which any motion 
 at all can be detected is found to be revolving about some other 
 body, larger than itself. The moon revolves about the earth; 
 the satellites of the planets revolve about the planets ; and the 
 planets themselves, some of which are much larger than the 
 earth, and at a much greater distance from the sun, revolve 
 about the sun. Henceforward, then, we shall include the earth 
 in the list of planets, and consider the sidereal year to be -the 
 interval of time in which the earth makes one complete revolu- 
 tion about the sun. 
 
 97. Linear Velocity of the Earth in its Orbit. The number of 
 miles in the circumference of the earth's orbit, considered as a 
 circle, is obtained by multiplying the radius of the orbit by 2*. 
 If we then divide this product by the number of seconds in a 
 year, we shall have, in the quotient, the number of miles through 
 which the earth moves about the sun in a second of time. It 
 will be found to be about 18.4 miles. 
 
 98. Elliptical Form of the Earth's Orbit. Although, as has 
 already been stated, the mean value of the sun's angular semi- 
 diameter remains constant from year to year, careful measure- 
 ments of the semi-diameter show that it varies in magnitude 
 during the year, being greatest about the first of January, and 
 least about the first of July. The evident conclusion from this 
 fact is that the distance between the earth and the sun also varies 
 during the year, being greatest when the sun's semi-diameter is 
 the least, and least when it is the greatest. The truth of this 
 conclusion may be seen in Fig. 37, in which we have 
 
 AS 
 
 sm 
 
 As AS of course remains constant, ES will vary inversely as sin 
 AE/S, or since the sines of small angles are proportional to 
 the angles themselves, inversely as the angle AES itself. The 
 angular semi-diameter of the sun is 16' 17". 8, the least 
 
90 ORBIT OF THE EARTH. 
 
 is Jo' 45".5: hence the ratio of tfce greatest to the least distance 
 is that of 16' 17".8 to 15 45".5, or of 1.034 to 1. 
 
 Let us now assume any line, SA in Fig. 39, for instance, as our 
 unit of measure, and prolong it until 
 SH is to SA as 1.034 is to 1. Then if S 
 denotes the sun, SA and SH will repre- 
 sent the relative distances of the earth 
 from the sun on about the first of Janu- 
 ary and the first of July. On certain 
 
 ^ ^ D days throughout the year, let the advance 
 
 Fi r 39. of the sun in longitude since the time 
 
 when the earth was at A be determined, and let the angular 
 semi-diameter of the sun on each of these days be measured. 
 Lay off the angles ASB, ASC, &c., equal to these advances in 
 longitude. Since, as may readily be seen in Fig. 38, the appa- 
 rent advance of the sun in longitude is caused by the advance 
 of the earth in its orbit, and is equal to it, the angles ASB, ASC t 
 &c., will represent the angular distances of the earth from the 
 point A on the days when the different observations were made. 
 Let us next take the lines SB, SC, &c., of such lengths that 
 each line may be to SA in the inverse ratio of the corresponding 
 semi-diameters. If, then, we draw a line through the points 
 A, B, C, &c., we shall have a representation of the orbit of the 
 earth about the sun. The curve is found to be an ellipse, the 
 sun being at one of the foci. The point A, where the earth is 
 nearest to the sun, is called the perihelion, the point H, the aphe- 
 lion; and the angular distance of the earth from its perihelion 
 is called its anomaly. 
 
 The eccentricity of the ellipse, or if is the centre of the 
 ellipse, the ratio of OS to OA, is evidently equal to about T ?oV?> 
 or -g-^th. A more accurate value of it is .0167917. This eccen- 
 tricity is at present subject to a diminution of .000041 a 
 century ; but Leverrier, a French astronomer, has proved that 
 after the eccentricity has diminished to a certain point it w'H 
 begin to increase again. 
 
 THE SEASONS. 
 
 99. The change uf seasons on the earth is caused by the 
 
SEASONS. 
 
 91 
 
 inequality of the days and nights, and this inequality is a result 
 of the inclination of the plane of the equinoctial to that of the 
 ecliptic. The relative positions of the sun and the earth at dif- 
 ferent parts of the year are represented in Fig. 40. S represents 
 
 Fig. 40. 
 
 the sun, and ABCD the orbit of the earth. Pp is the axis of 
 rotation of the earth, and EQ the equator. The plane of this 
 equator is supposed to intersect the plane of the ecliptic in the 
 line of equinoxes A C. Since, as we have already seen, the sun 
 appears to be on this line on the 21st of March and the 21st of 
 September, the earth itself must also be on this line at the same 
 time. Suppose, then, the earth to be at A on the 21st of March. 
 The sun will evidently lie in the direction AS, and will be pro- 
 jected on the celestial sphere at the vernal equinox. Now, since 
 a line which is perpendicular to a plane is perpendicular to every 
 line in that plane which is drawn to meet it, the axis Pp at A, 
 being perpendicular to the equator, is also perpendicular to the 
 line AS, which is common to both the plane of the equator and 
 
92 SEASONS. 
 
 the plane of the ecliptic. Half of each parallel of latitude on 
 the earth will therefore lie in light and half in darkness ; and 
 hence, as the earth rotates on the axis Pp, every point on its 
 surface will describe half of its diurnal course in light and 
 half in darkness : in other words, day and night will be equal 
 over the whole earth. Since the direction of the axis of rota- 
 tion remains unchanged, the same condition of things will occur 
 when the earth is at C, on the 21st of September. Let the 
 earth be at B on the 21st of June. Here we see that, as the 
 earth rotates on its axis Pp, every point on its surface within 
 the circle ab will lie continually in the light, and will hence 
 have continual day, while within the corresponding circle a'b' 
 the night will be continual. We see also that at the equator 
 the days and nights will be equal, and that every point between 
 the equator and the circle ab will describe more of its diurnal 
 course in light than in darkness, and will thus have its days 
 longer than its nights ; while between the equator and the circle 
 ab' the nights will be longer than the days. Similar phenomena 
 will occur when the earth is at D, on the 21st of December, 
 except only that it will then be the southern hemisphere in 
 which the days .are longer than the nights, and the southern 
 pole at which the sun is continually visible. 
 
 Such, then, is the inequality of the days and nights caused 
 by the inclination of the plane of the equinoctial to that of the 
 ecliptic. As the sun apparently moves from either equinox, 
 the inequality of day and night continually increases, reaches 
 its maximum when the sun arrives at either solstice, and then 
 continually decreases as the sun moves on to the equinox : the 
 day being longer than the night in that hemisphere which is 
 on the same side of the equator with the sun. Now, any point 
 on the earth's surface receives heat during the day and radiates 
 it during the night : and hence, when the days are longer than 
 the nights, the amount of heat received is greater than the 
 amount radiated, and the temperature increases ; while, on the 
 contrary, when the days are shorter than the nights, the tempe- 
 rature decreases : and thus is brought about the change of sea- 
 eons on the earth. 
 
 Another fact, depending on the same cause, and tending to 
 
SEASONS. 93 
 
 the same result, must also be taken into consideration; and 
 that is that the temperature at any place depends on the ob- 
 liquity of the sun's rays : on the altitude, in other words, which 
 the sun attains at noon. Now we have, from Art. 76, 
 
 z = L d: 
 
 from which we see that, the latitude remaining constant, the sun 
 attains the greater altitude, the greater its declination when it 
 has the same name as the latitude, and the less its declination 
 when it has the opposite name : so that the nearest approach to 
 vertically in the sun's rays will occur at the same time that 
 the day is the longest. An exception, however, must be noticed 
 to this general rule, in the case of places within the tropics : 
 since at these places, as may be seen from the formula, the sun 
 passes through the zenith when its declination is equal to the 
 latitude, and has the same name. 
 
 100. Effect of the Ellipticity of the Earths Orbit on the Change 
 of Seasons. The elliptic form of the earth's orbit has very 
 little to do with the change of seasons. For although the earth 
 is nearer to the sun on the 1st of January than on the 1st of 
 July, yet its angular velocity at that time is found by observa- 
 tion to be greater, and to vary throughout the whole orbit in- 
 versely as the square of the distance. Now it may readily be 
 shown that the amount of heat received by the earth at different 
 parts of its orbit also varies, other things being equal, inversely 
 as the square of the distance : so that equal amounts of heat 
 are received by the earth in passing through equal angles of 
 its orbit, in whatever part of its orbit those angles may be 
 situated. Still, although the change in distance does not mate- 
 rially affect the annual change of seasons, it does affect the 
 relative intensities of the northern and the southern summer. 
 The southern summer takes place when the earth's distance is 
 only about f|jths of what it is at the time of the northern 
 summer: hence, the intensity at the former period will be to 
 that at the latter in the ratio of about (|$) 2 to 1, or about }| to 
 1 : in other words, the intensity of the heat of the southern sum- 
 mer will be y^th greater than that of the heat of the northern 
 summer. 
 
94 TWILIGHT. 
 
 TWILIGHT. 
 
 101. If the earth's atmosphere did not contain particles of 
 dust and vapor, which serve to reflect the rays of light, the 
 transition from day to night would be instantaneous, and the 
 intermediate phenomenon of twilight would have no existence. 
 
 This phenomenon 
 is explained in 
 Fig. 41, in which 
 ABC represents 
 
 B<^ / x \ a portion of the 
 
 earth's surface, 
 
 Fig- 4i. & and EDF a por- 
 
 tion of the atmosphere. Let the sun be supposed to lie in the 
 direction AS, and to be in the horizon of the place A. All of the 
 atmosphere which lies above the horizontal plane SD will then 
 receive the direct rays of the sun, and A will receive twilight 
 from the whole sky. The point B will, on the contrary, be illu- 
 minated only by the smaller portion of the atmosphere included 
 within the planes EB and AD and the curved surface ED; and 
 at the point C the twilight will have wholly ceased. Strictly 
 speaking, the lines AS, BE, &c., should be slightly curved, owing 
 to the effects of refraction, but the omission involves no change 
 in the explanation. 
 
 It is computed that twilight ceases when the sun is about 18 
 below the horizon, measured on a vertical circle. The more 
 nearly perpendicular to the horizon is the diurnal circle in 
 which the sun appears to move, the more rapid will be the sun's 
 descent below the horizon ; hence, the length of twilight dimin- 
 ishes as we approach the equator and increases as we recede 
 from it. Furthermore, we see in Fig. 2 that the greater the 
 declination of the sun, the smaller is the apparent diurnal circle 
 in which it moves, and the greater will be the length of time 
 required for the sun to reach the depression of 18 below the 
 horizon. The shortest twilight, therefore, occurs at places on the 
 equator, when the sun is on the equinoctial, and its length is 
 then Ih. 12m. Near the poles the length of twilight is at times 
 very great. Dr. Hayes, in his last expedition towards the North 
 
APPEARANCE OF THE SUN. 95 
 
 P^le, wintered at latitude 78 18' N., so far above the circle ab, 
 Fig. 40, that the sun was continually below the horizon from the 
 noddle of October to the middle of February; but at the begin- 
 ning and the end of this interval twilight lasted for about nine 
 hours. At the poles twilight lasts nearly a month and a half. 
 
 GENERAL DESCRIPTION OF THE SUN. 
 
 102. When the sun is observed with a telescope, spots are 
 noticed upon its surface. These spots appear to cross the sun's 
 disc from east to west, and with different rates, the rate of 
 motion of spots at the sun's equator being the greatest. We 
 therefore conclude that these spots have not only an apparent 
 rsotion, caused by the sun's rotation, but also a proper motion 
 of their own. By appropriate investigation of these motions, 
 it is found that the sun rotates from west to east upon a fixed 
 axis, in a plane inclined at an angle of about 7 to the plane of 
 the ecliptic. The period of this rotation is about 25 days. 
 
 Much uncertainty exists as to the nature of these spots. Un- 
 til recently, it was held that the sun is surrounded by two 
 atmospheres, of which only the outer one (called the photo- 
 sphere} is luminous, and that the spots are rents in these 
 atmospheres through which the solid body of the sun is seen. 
 These spots are for the most part confined to a zone, extending 
 about 35 on each side of the sun's equator. They differ widely 
 iii duration, sometimes lasting for several months, and some- 
 times disappearing in the course of a fe^ hours. They are 
 sometimes of an immense size. One was seen in 1843, with 
 a diameter of nearly 75,000 miles : it remained in sight for 
 a week, and was visible to the naked eye. In 1858, a much 
 larger one was seen, its diameter being over 140,000 miles. As 
 a general thing, each dark spot, or umbra, as it is called, has 
 within it a still darker point, called the nucleus, and is sur- 
 rounded by a fringe of a lighter shade, called the penumbra. 
 Sometimes several spots are inclosed by the same penumbra; 
 and occasionally spots are seen without any penumbra at all. 
 
 On the theory of two atmospheres, the existence of the 
 penumbra is explained by supposing the aperture in the outer 
 and luminous stratum to be wider fb;.n that in the inner one, 
 
96 APPEARANCE OF THE SUN. 
 
 and that portions of the inner ^tratum, being subjected to a 
 strong light from above, are rendered visible: the umbra itself 
 being, as already remarked, the solid body of the sun seen through 
 both strata. According to Mr. J. N. Lockyer, "sun-spots are 
 cavities qr hollows eaten into the photosphere, and these different 
 shades [the penumbra, umbra, and nucleus] represent different 
 depths." [See Note, page 99.] 
 
 One very curious and interesting discovery in relation to these 
 spots is that of a periodicity in their number. This discovery 
 was made by Schwabe, of^Dessau, whose researches and obser- 
 vations on this subject covered a period of more than twenty-five 
 years. The number of groups of spots which he observed in a 
 year varied from 33 to 333, the average being not far from 150.* 
 He found the period from one maximum to another to be about 
 ten years. Professor Wolf, of Zurich, after tabulating all the ob- 
 servations of spots since 1611, decided that the period varied from 
 eight to sixteen years, its mean value being about eleven years. 
 Recent investigations show that this periodicity is in some way 
 connected with the action of the planets, of Jupiter and Venus 
 particularly, upon the sun's photosphere. It is a curious fact that 
 magnetic storms and the phenomenon called Aurora or Northern 
 Lights have a similar period, and are most frequent and most 
 striking when the number of the solar spots is the greatest. 
 
 Still other phenomena which are seen upon the sun's disc are 
 the faculce, which are streaks of light seen for the most part in 
 the region of the spots, and which are undoubtedly elevations or 
 ridges in the photosphere: and the luculi, which are specks of 
 light scattered over the sun's disc, giving it an appearance not 
 unlike that of the skin of an orange, though relatively much less 
 rough. The cause of these lucuii is unknown. 
 
 At the time of a total eclipse of the sun by the moon, the disc 
 of the sun is observed to be surrounded by a ring or halo of light, 
 which is called the corona. The breadth of this corona is more 
 than equal to the diameter of the sun. Many theories have been 
 advanced to explain this phenomenon, one of which is that it is 
 due to the existence of still another atmosphere, exterior to the 
 
 * A table of Schwnbe's observations is given in the Appendix. 
 
CONSTITUTION OF THE SUN. 97 
 
 photosphere. Another theory is that this corona consists of 
 streams of luminous matter, radiating in all directions from the 
 sun. Rose-colored protuberances, sometimes called red flamea, 
 are also seen, which are usually of a conical shape, and are 
 sometimes of great height. In the total eclipse of August 17th, 
 1868, one was observed with an apparent altitude of 3', corre- 
 sponding to a height of about 80,000 miles. These protuberances 
 were formerly supposed to be similar in character to our terres- 
 trial clouds; but Dr. Jannsen, the chief of the French expedi- 
 tion sent out to the East to observe the total eclipse of August, 
 1868, examined their light with the spectroscope, and found them 
 to be masses of incandescent gas, of which the greater part was 
 hydrogen. Dr. Jannsen also made the interesting discovery that 
 these protuberances can be examined at any time, without wait- 
 ing for the rare opportunity afforded by a total eclipse. He 
 observed them for several successive days, and found that great 
 changes took place in their form and size. Mr. Lockyer, of 
 England, who has since examined them, pronounces them to be 
 merely local accumulations of a gaseous envelope completely 
 surrounding the sun : the spectrum peculiar to these protuber- 
 ances appearing at all parts of the disc. 
 
 It has already been stated (Art. 47) that the spectroscope en- 
 ables us to establish the existence of certain chemical substances 
 in the sun, by a comparison of the spectra of these substances 
 with that of the sun ; or, more precisely, by a comparison of the 
 lines, bright or dark, by which these different spectra are dis- 
 tinguished. The number of the parallel dark lines in the solar 
 spectrum which have been detected and mapped exceeds 3000; 
 and careful examination also shows that some of these are double. 
 3ome of the more prominent of these lines have received the 
 names of the first letters of the alphabet; D, for example, is a 
 very noticeable double line in the orange of the spectrum. When 
 certain chemical substances are evaporated, either in a flame or 
 by the electric current, the spectra which they form are also 
 characterized by lines, which, however, are not dark, but bright. 
 If, for instance, sodium is introduced into a flame, its incan- 
 descent vapor produces a spectrum which is characterized by a 
 brilliant double band of yellow; and it is especially noticeable 
 
98 CONSTITUTION OF THE SUN. 
 
 that this yellow band coincides, exactly in position with the 
 dark line D of the solar spectrum. In the same way the 
 spectrum of zinc is found to contain bands of red and blue- 
 that of copper contains bands of green: and, in general, the 
 spectrum of each metal contains certain bright bands or lines, 
 peculiar to itself, and readily recognized. We may therefore 
 conclude that an incandescent gas or vapor emits rays of a certain 
 refrangibility and color, and those rays only. 
 
 Again, it is proved by experiment that if a ray of white light 
 be allowed to pass through an incandescent vapor, the vapor 
 will absorb precisely those rays which it can itself emit. If, for 
 instance, a continuous spectrum be formed by a ray of intense 
 white light from any source, and if the vapor of sodium be intro- 
 duced in the path of this ray, between the prism and the source 
 of light, a dark band will appear in the spectrum, identical in 
 position with the bright yellow band which we have already 
 noticed in the spectrum of sodium, and which we found to be 
 identical in position with the dark line D of the solar spectrum. 
 
 We are now ready to apply the principles established by 
 these experiments to the case of the sun. The sun is, as we 
 saw above, a sphere surrounded by a vaporous envelope. This 
 sphere would of itself emit all kinds of rays, and therefore 
 give a continuous spectrum ; but the photosphere which sur- 
 rounds it absorbs those of the sun's rays which it can itself 
 emit. The dark line D of the solar spectrum shows, as in the 
 experiment above described, that sodium has been introduced 
 in the path of the sun's rays : in other words, that sodium is in 
 the sun's photosphere. In the same way, Professor Kirchhoff, 
 to whom we owe this remarkable discovery, has established the 
 existence in the photosphere of iron, calcium, magnesium, chro- 
 mium, and other metals. In the case of iron, more than 450 
 bright lines have been detected in its spectrum : and for every 
 one of these lines there is a corresponding dark line in the solar 
 spectrum. 
 
 We also see, from the preceding experiments, how the presence 
 of bright lines in the spectrum of the rose-colored protuberances 
 could prove to Dr. Jannsen that these protuberances were not 
 masses of cloud.?, reflecting the light of the sun, but masses of 
 
ZODIACAL LIGHT. 99 
 
 ircandescent vapor. We shall see another instance of the same 
 description when we come to examine some of the nebulae. 
 
 THE ZODIACAL LIGHT. 
 
 103. At certain seasons of the year a faint nebulous light, 
 not unlike the tail of a comet, is seen in the west after twilight 
 has ended, or in the east before it has begun. This is called 
 the Zodiacal Light. Its general shape is nearly that of a cone, 
 the base of which is turned towards the sun. The breadth of 
 the base varies from 8 to 30 of angular magnitude. The apex 
 of the cone is sometimes more than 90 to the rear or in advance 
 of the sun. According to Humboldt, it is almost always visible, 
 at the times above stated, within the tropics: in our latitudes it 
 is seen to the best advantage in the evening near the first of 
 March, and in the morning near the middle of October. 
 
 Of the many theories proposed to account for the zodiacal 
 light, the one which seems to be most widely accepted is that it 
 consists of a ring or zone of rare nebulous matter encircling 
 the sun, which reaches as far as the earth, and perhaps extends 
 beyond it. According to another theory, it is a belt of meteoric 
 bodies surrounding the sun. A very interesting and valuable 
 series of observations upon the. Zodiacal Light was made by 
 Chaplain Jones, United States Navy, in the years 1853-5, in 
 latitudes ranging from 41 N. to 53 S, The conclusion which 
 he drew from his observations was that the light was a nebulous 
 ring encircling the earth, and lying within the orbit of the moon. 
 
 NOTE. The surface of the sun is now the object of assiduous observa- 
 tion and study. The shining surface of the sun, whence come our light 
 and heat, is called the photosphere. Outside of this is the chromosphere, to 
 which the red flames belong, and which is largely composed of hydrogen 
 and the vapors of metals. Outside of the chromosphere lies the corona. 
 Modern research has disproved much that was formerly believed concern- 
 ing the physical constitution of the sun and its envelope; but it has by no 
 means established what that constitution really is. 
 
SIDEREAL AND SOLAR TIMES. 
 
 CHAPTER VII. 
 
 SIDEREAL AND SOLAR TIME. THE EQUATION OF TIME. THE 
 CALENDAR. 
 
 104. Sidereal and Solar Days. IT is important to distin- 
 guish between the apparent annual motion of the sun in the 
 ecliptic, from west to east, and the apparent diurnal motion 
 towards the west, which the rotation of the earth gives to all 
 celestial bodies and points. A sidereal day is the interval of 
 time between two successive transits of the vernal equinox over 
 the same branch of the meridian. A solar day is the interval 
 between two similar transits of the sun. But the continuous 
 motion of the sun towards the east causes it to appear to move 
 more slowly towards the west than the vernal equinox moves. 
 The solar day is therefore longer than the sidereal day, the 
 average amount of the difference being 3m. 55.5s. And fur- 
 thermore, in the interval of time in which the sun makes one 
 complete revolution in the ecliptic, the number of daily revo- 
 lutions which it appears to make about the earth will be less 
 by one than the number of daily revolutions made by the 
 equinox. The sidereal year, then (Art. 89), which contains 
 865d. 6h. 9m. 9.6s. of solar time, contains 366d. 6h. 9m. 9.6s. 
 of sidereal time. [See Note, page 154.] 
 
 105. Relation of Sidereal and Solar Times. Since the side- 
 real day is shorter than the solar day (and, consequently, the 
 sidereal hour, minute, &c., than the solar hour, minute, &c.), it 
 is evident that any given interval of time will contain more 
 units of sidereal than of solar time. The relative values of the 
 sidereal and the solar days, hours, &c., are obtained as follows: 
 We have from the prec2ding article, 
 
 366.25G36 sidereal days = 365.25636 solar days: 
 
 one sidereal day = (5.99727 solar day, 
 one sidereal hour = 0.99727 solar hour, &c. 
 
EQUATION OF TIME. 101 
 
 Having, therefore, an interval of time expressed in either solar 
 or sidereal units, we may easily express the same interval in 
 units of the other denomination. This is called the conversion 
 of a solar into a sidereal interval, and the reverse : and tables 
 for facilitating this conversion are given in the Nautical Al- 
 manac. 
 
 Again, knowing the sidereal time at any instant, the hour- 
 angle, that is to say, of the vernal equinox, the corresponding 
 solar time, or the hour-angle of the sun, is readily obtained by 
 subtracting from the sidereal time the sun's right ascension. 
 This is indeed a corollary of the theorem proved in Art. 9, 
 from which we see that the sum of the sun's right ascension 
 (which can always be found in the Nautical Almanac), and 
 its hour-angle, is the sidereal time. Either of these times, then, 
 may be converted into the other. \ "* / 
 
 THE EQUATION O 
 
 106. Inequality of Solar Days. Observation shows that the 
 length of the solar day is not a constant quantity, but varies at 
 different seasons of the year, and, indeed, from day to day. A 
 distinction must therefore be made between the apparent or 
 actual solar day, and the mean solar day, which is the mean 
 of all the apparent solar days of the year. A uniform measure 
 of time may be obtained from the apparent diurnal motion 
 with reference to our meridian of a fixed celestial body or 
 point. It may also be obtained from the apparent diurnal 
 motion of a celestial body which changes its position in the 
 heavens, provided that two conditions are satisfied ; first, the 
 plane in which the body moves must be perpendicular to the 
 plane of the meridian: and second, its motion in that plane 
 must be uniform. Both these conditions are so very nearly 
 satisfied by the motion of the vernal equinox, that any two 
 sidereal days may be considered to be sensibly equal to each 
 other; but neither condition is satisfied by the motion of the 
 sun. It moves in the ecliptic, the plane of which is not, in 
 general, perpendicular to the plane of the meridian : and its 
 motion in this plane is not uniform. We have, therefore, two 
 
102 EQUATION OF TIME. 
 
 causes of the inequality of the soltir days, the effect of each of 
 which we will now proceed to examine. 
 
 107. Irregular Advance of the Sun in the Ecliptic. Observa- 
 tion shows that the sun's motion in longitude is not uniform. 
 The mean daily motion is, of course, obtained by dividing 360 
 by the number of days and parts of a day in a year, and is 
 59' 8."3. But the daily motion about the first of January is 
 61' 10", while about the first of July it is only 
 57' 12". In Fig. 42, let the circle AM'M" 
 represent the apparent orbit of the sun in 
 the ecliptic about the earth E, and let the 
 sun be supposed to be at the point A where 
 its daily motion is the greatest, on the first 
 of January. Let us also suppose a fictitious 
 ^si^n (which we will call the first mean sun) 
 to niqye in the ecliptic with the uniform rate of 59' 8". 3 daily, 
 iik.H'l t ; > )>o ul, the. jximj.4 a ^ tne sanie time that the true sun is 
 there. On the next day the mean sun will have moved eastward 
 to some point M, while tli3 true sun, whose daily motion is at this 
 time greater than that of the mean sun, will be found at some 
 point T, to the east of M. The true sun will continue to gain on 
 the mean sun for about three months, at the end of which time 
 the mean sun will begin to gain on the true sun, and will finally 
 overtake it at the point B, on the first of July. During the 
 second half of the year the mean sun will be to the east of the 
 true sun, and at the end of the year the two suns will again be 
 together at A. 
 
 The angular distance between the two suns, represented in 
 the figure by the angles TEN, T'EM', &c., is called the Equa- 
 tion of the Centre. It is evidently additive to the mean longi- 
 tude of the sun while it is moving from A to B, and subtractive 
 from it while it is moving from B to A. Its greatest value is 
 about 8 minutes of time. 
 
 Since the rotation of the earth gives to both these bodies a 
 common daily motion to the west, it is plain that from January 
 to July the mean sun will cross the meridian before the true 
 sun, and that from July to January the true sun will cross the 
 meridian before the mean sun. 
 
EQUATION OF TIME. 103 
 
 108. Obliquity of trie Ecliptic to the Meridian. Even if the 
 sun's motion in the ecliptic were uniform, equal advances of the 
 sun in longitude would not be accompanied by equal advances 
 in right ascension, in consequence of the obliquity of the 
 ecliptic to the meridian. The truth of this may be seen in 
 Fig. 43. Let this figure represent 
 the projection of the celestial sphere 
 on the plane of the equinoctial co- 
 lure PApH. A and H are the 
 equinoxes, P and p the celestial 
 poles, AeH the equinoctial, and 
 A EH the ecliptic. Let the ecliptic 
 be divided into equal arcs, AB, BC, 
 &c., and through the points of divi- 
 sion, B, C, &c., let hour-circles be 
 drawn, meeting the equinoctial in 
 the points b, c, &c. Now, since all great circles bisect each 
 other, AEH is equal to AeH, and if Pep is the projection of an 
 hour-circle perpendicular to the circle PApH, AE and Ae are 
 quadrants, and equal. The angle PBC is evidently greater 
 than PAB, PCD is greater than PBC, &c. : in other words, the 
 equal arcs AB, BC, &c., are differently inclined to the equi- 
 noctial. The effect of this is that the equinoctial is divided 
 into unequal parts by the hour-circles Pb, PC, &c., be being 
 greater than Ab, cd than be, &c. It is to be noticed, further, 
 that the points B and b, being on the same hour-circle, will be 
 on the meridian at the same instant of time: and the same is 
 true of C and c, D and d, &c. 
 
 Now, if A is the vernal equinox, the first mean sun, moving 
 in the ecliptic with the constant daily rate of 59' 8". 3, will 
 pass through that point on the 21st of March. Let another 
 fictitious sun (called the second mean sun) leave the point A at 
 the same time, and move in the equinoctial with the same uni- 
 form daily rate. Since BAb is a right-angled triangle, Ab is 
 less than AB. Hence, when the first mean sun reaches B, the 
 second mean sun will be at some point m, to the east of b: 
 when the first mean sun is at C, the second mean sun will be to 
 the east of c, &c. : and the second mean sun will continue to 
 
104 EQUATION OF TIME. 
 
 lie to the east of the first mean sun until the 21st of June (the 
 summer solstice), when both suns will be at the points E and e 
 at the same instant of time, and will therefore come to the 
 meridian together. In the second quadrant, the second mean 
 sun will lie to the west of the first mean sun, and both suns 
 will reach H, the autumnal equinox, on the 21st of September. 
 The relative positions in the third and the fourth quadrant will 
 be identical with those in the first and the second. 
 
 From the 21st of March, then, to the 21st of June, the second 
 mean sun, being to the east of the first mean sun, will come 
 later to the meridian ; and the same will also be true from the 
 21st of September to the 21st of December. In the two other 
 similar periods the case will be reversed, and the second mean 
 sun will come earlier to the meridian than the first mean sun. 
 The greatest difference of the hour-angles of these two mean suns 
 is about 10 minutes of time. 
 
 109. Equation of Time. It is by means of these two fictitious 
 suns that we are able to obtain a uniform measure of time from 
 the irregular advance of the sun in the ecliptic. The second 
 mean sun satisfies the two conditions stated in Art. 106, and 
 therefore its hour-angle is perfectly uniform in its increase. 
 This hour-angle is the mean solar time of our ordinary watches 
 and clocks. The hour-angle of the true sun is called the ap- 
 parent solar time: and the difference at any instant between the 
 apparent and the mean solar time is called the equation of time. 
 
 Let Fig. 44 be a projection of the 
 celestial sphere on the plane of the 
 horizon. Z is the zenith, P the 
 pole, EVQ the equinoctial, CL the 
 ecliptic, and Fthe vernal equinox. 
 Let T be the position of the true 
 sun in the ecliptic, and M that of 
 second mean sun in the equinoctial. 
 The angle TPM is evidently the 
 equatioi of time. This angle is mea- 
 sured by the arc AM, or VM VA : 
 
 the difference, that is, of the right ascensions of the true and 
 tbe second mean sun. But since the angular advance of the 
 
JALEKDAR. 105 
 
 second mean sun in the equinoctial is, by hypothesis, as shown 
 in the previous article, equal to the angular advance of the first 
 mean sun in the ecliptic, it follows that the right ascension of 
 the second mean sun is always equal to the longitude of the 
 first mean sun, or, as it is usually called, the true sun's mean 
 longitude. The equation of time, then, is the difference of the 
 sun's true right ascension and mean longitude; and thus com- 
 puted is given in the Nautical Almanac for each day in the 
 year. It reduces to zero four times in the year, and passes 
 through four maxima, ranging in value from 4 minutes to 16 
 minutes. 
 
 110. Astronomical and Civil Time. The mean solar day is 
 considered by astronomers to begin at mean noon, when the 
 second mean sun (usually called simply the mean sun) is at its 
 upper culmination. The hours are reckoned from Oh. to 24h. 
 The mean solar day, so considered, is called the astronomical day. 
 
 The civil day begins at midnight, twelve hours before the 
 astronomical day, and is divided into two parts of twelve hours 
 each, called A.M. and P.M. 
 
 We must, therefore, carefully distinguish between any given 
 civil time and the corresponding astronomical time. For in- 
 stance, January 3d, 8 A.M., in civil time, is the same as Janu- 
 ary 2d, 20h., in astronomical time. 
 
 THE CALENDAR. 
 
 111. Owing to causes which will be explained further on, the 
 position of the vernal equinox is not absolutely stationary, but 
 moves westward along the ecliptic, with an annual rate of about 
 50". 2. The sun, then, moving eastward from the equinox, will 
 reach it again before it has made one complete sidereal revo- 
 lution about the earth. This interval of time in which the sun 
 moves from and returns to the equinox is called a tropical year, 
 and consists of 365d. 5h. 48m. 47.8s. The Julian Calendar 
 was established by Julius Csesar, 44 B.C., and by it one day was 
 inserted in every fourth year. This was the same thing as as- 
 suming that the length of the tropical year was 365d. 6h., 
 instead of the value given above, thus introducing an accumu- 
 lative error of llm. 12s. overy year. This calendar was 
 
106 CALENDAR. 
 
 adopted by the Church in 325 A.p., at the Council of Nice, and 
 the vernal e juinox then fell on the 21st of March. In 1582, 
 the annual error of llm. 12s. caused the venial equinox to fall 
 on the llth of March, instead of the 21st. Pope Gregory XIII. 
 therefore ordered that ten days should be omitted from the year 
 1582, and thus brought the vernal equinox back again to the 
 21st of March. Furthermore, since the error of llm. 12s. a year 
 amounted to very nearly three days in 400 years, it was decided 
 to leave out three of the inserted days (called intercalary days) 
 every 400 years, and to make this omission in those years which 
 were not exactly divisible by 400. Thus of the years 1700, 
 1800, 1900, 2000, all of which are leap years according to the 
 Julian calendar, only the last is a leap year according to the 
 reformed or Gregorian calendar. By this calendar the annual 
 error is only 24 seconds, and will not amount to a day in much 
 less than 4000 years. 
 
 This reformed calendar was not adopted by England until 
 1752, when eleven days were omitted from the calendar. The 
 two calendars are now often called the old style and the new style. 
 For instance, April 26th, O.S., is the same as May 8th, N.S. In 
 Russia the old style is still retained, though it is customary to 
 give both dates ; as 1868, ^?. All other Christian countries 
 have adopted the new style. 
 
UNIVERSAL GRAVITATION. 107 
 
 CHAPTER VIII. 
 
 LAW OF UNIVERSAL GRAVITATION. PERTURBATIONS IN THE 
 EARTH'S ORBIT. ABERRATION. 
 
 112. The Law of Universal Gravitation. The earth, as we 
 have seen in Chapter VL, revolves about the sun in an elliptical 
 orbit, with a linear velocity of eighteen miles a second. At every 
 point of its orbit the centrifugal force induced by this revolution 
 must create in the earth a tendency to leave its orbit, and to go 
 off in the direction of a tangent to the orbit at that point. To 
 counteract this centrifugal force, there must constantly exist a 
 centripetal force, by which the earth is at every instant deflected 
 from this rectilinear path which it tends to follow, and is drawn 
 towards the sun ; and in order that the orbit of the earth may 
 remain unchanged in form, as observation shows that it does 
 remain, these two forces must be in constant equilibrium. Ad- 
 mitting, then, the existence of such a centripetal force, it remains 
 to determine the nature of the force, and the laws under which 
 it acts. 
 
 The force is believed to be identical in nature with that force 
 which causes all bodies, free to move, to tend towards the earth's 
 centre, and which we call the force of gravity. At whatever 
 height above the surface of the earth the experiment may be 
 made, this attractive force of the earth is found to exist ; and 
 there is no good reason for assuming any finite limit beyond 
 which this force, however much its effects may be lessened by 
 other and opposing forces, does not have at least a theoretic 
 existence. And, furthermore, as the sun and the other heavenly 
 bodies are all masses of matter like the earth, there is every 
 reason for concluding that they too, as well as the earth, possess 
 this power of attracting other bodies towards their centres. Nor 
 is this attractive power a characteristic of large bodies alone : for 
 
108 UNIVERSAL GRAVITATION. 
 
 we have already seen in the experiment with the torsion balance, 
 described in Art. 66, that small globes of lead exert a sensible 
 attraction upon still smaller globes. We may therefore assume 
 that what is true of each of these masses, large and small, as a 
 whole, is no less true of the separate particles of which it is com- 
 posed, and that every particle of matter in the universe has an 
 attractive power upon every other particle. 
 
 In order to determine the laws under which this attractive 
 power is exerted, we have only to assume that the laws which 
 are shown by experiment to obtain at the earth's surface hold 
 equally good throughout the universe; so that whatever the 
 masses of bodies may be, or whatever the distances by which 
 they are separated from each other, the forces with which any 
 two bodies attract a third will be directly proportional to the 
 masses of the two attracting bodies, and inversely proportional 
 to the squares of their distances from the third body. 
 
 This, then, is Newton's Law of Universal Gravitation. Every 
 particle of matter in the universe attracts every other particle, 
 ivith a force directly proportional to the mass of the attracting 
 particle, and inversely proportional to the square of the distance 
 between the particles. In applying this general law to the par- 
 ticles which compose the masses of the heavenly bodies, Newton 
 has demonstrated that the attraction exerted by a sphere is pre- 
 cisely what it would be if all the particles in the sphere were 
 collected at its centre, and constituted one particle, with an at- 
 tractive power equal to the sum of the powers of these different 
 particles. 
 
 113. Verification of the Law in the Case of the Moon. The 
 moon is shown by observation to revolve about the earth in a 
 period of 27.32 days, at a mean distance from the earth of 238,800 
 miles. If we take the formula for centrifugal force given in 
 Art. 69, 
 
 and substitute for r the moon's distance in feet, and for t its 
 period of revolution in seconds, we shall find for the centrifugal 
 force, 
 
 /= 0.0089 feet: 
 
) 
 
 MASS OF THE SUN. 109 
 
 that is to say, in one second the earth tends to give the moon a 
 velocity towards itself of 0.0089 feet. Now the force of gravity 
 on the eartn's surface at the equator is 32.09 feet; and if the law 
 of gravitation is assumed to be true, the force of gravity at the 
 
 32 09 
 
 distance of the moon will be - feet, since the distance of 
 
 (oO.Zo7) 
 
 the moon from the earth is equal to 60.267 of the earth's radii. 
 The value of this expression is found to be 0.0088 feet. The two 
 results vary by only joio^h of a foot: and it is therefore fair to 
 conclude that the centrifugal force of the moon in its orbit is 
 really counteracted by the earth's attraction. 
 
 In whatever way the law of gravitation is tested in connection 
 with the observed motions of the heavenly bodies, the facts which 
 come by observation are always found to be in close agreement 
 with the results which the law demands ; and it is safe to say 
 that the truth of this law is as satisfactorily demonstrated as is 
 that of the laws of refraction, of the laws of sound, or of the 
 many other natural laws which depend upon observation and 
 experiment for their ultimate proof. 
 
 114. The Mass of the Sun. Let A denote the attraction ex- 
 erted by the sun on the earth, and a that exerted by the earth 
 on a body at its surface. Let If denote the mass of the sun, m 
 that of the earth, r the radius of the earth, and R the radius 
 of the earth's orbit. We have, then, by the law of gravitation, 
 
 A M r 2 
 
 a m R 2 
 But A must equal the earth's centrifugal force in its orbit, or 
 
 . 8 in which t is 365.256 days, reduced to seconds, and R is 
 
 expressed in feet. We have also a equal to 32.09 feet. Substi- 
 tuting these values, we have, 
 
 m ~ 32.09f r 2 
 
 Substituting the known values of the different quantities, we 
 shall have 
 
 = 327,000: 
 m 
 
 or the- mass of the sun is equal to that of 327,000 earths. 
 Jo 
 
110 MOTION OF THE EARTH 
 
 The. density of the sun, compared with that of the earth, being 
 
 327,000 
 equal to the mass divided by the volume, is ro 
 
 density is therefore equal to about Hh of that of the earth, and 
 to about f ths of that of water. 
 
 115. Weight of Bodies at the Surface of the Sun. The weight 
 of the same body at the surface of the earth and at that of the 
 sun will be directly as the masses of the two spheres and inversely 
 as the squares of their radii. We shall find that the weight 
 of a body at the sun is about 27.7 times its weight at the earth; 
 so that a body which exerts a pressure of 10 pounds at the earth 
 would exert a pressure at the sun equal to that of 277 of the 
 same pounds; and a man whose weight is 150 pounds would, 
 if transported to the sun, be obliged to support in his own body 
 a weight equivalent to about two of our tons. 
 
 116. The Earth's Motion at Perihelion and Aphelion. We 
 have already seen that the angular velocity of the earth in its 
 orbit is the greatest at perihelion, when the earth is the nearest to 
 the sun, and is the least at aphelion, when the earth is the farthest 
 from the sun. This irregularity of motion is a consequence of 
 the attraction exerted by the sun on the earth, as may be seen in 
 
 Fig. 45. In this figure, let S be the sun, 
 P the perihelion of the earth's orbit, and 
 A the aphelion. Let the earth move 
 from A to P, and suppose it to be at the 
 point E. The attraction of the sun on 
 the earth, along the line ES, may be re- 
 solved into two forces, one of which, 
 acting in the direction of the tangent EB, 
 B will evidently tend to increase the velo- 
 
 Fig. 45. city of the earth in its orbit. At the 
 
 point E", on the contrary, where the earth is moving toward A, 
 the effect of the sun ? s attraction is to diminish the earth's velocity. 
 lu general, then, the earth's velocity will increase as it moves 
 from aphelion to perihelion, and decrease as it moves from peri- 
 helion to aphelion. 
 
KEPLER'S LAWS. Ill 
 
 KEPLER S LAWS. 
 
 117. In the early part of the seventeenth century, more than 
 tifty years before the announcement by Newton of the law of 
 universal gravitation, the astronomer Kepler, by an examination 
 of the observations which had been made upon the motions of the 
 planets, and which had shown that the planets revolved about 
 the sun, discovered the following laws : 
 
 (1.) The orbit of every planet is an ellipse, having the sun at 
 one of its foci. 
 
 (2.) If a line, called a radius vector, is supposed to be drawn 
 from the sun to any planet, the areas described by this line, as 
 the planet revolves in its orbit, are proportional to the times. 
 
 (3.) The squares of the times of revolution of any two planets 
 are proportional to the cubes of their mean distances from the sun. 
 
 These laws were verified by Newton in his Principia, in a 
 course of mathematical reasoning, the foundation of which was 
 his theory of universal gravitation. With regard to Kepler's 
 first law, he showed that any two spherical bodies, mutually at- 
 tracted, describe orbits about their common centre of gravity, 
 and that these orbits are limited in form to one or another of the 
 four conic sections, the circle, the ellipse, the parabola, and the 
 hyperbola. For instance, in the case of the earth and the sun, 
 the earth does not describe an ellipse about the sun at rest, but 
 both earth and sun revolve about their common centre of gravity. 
 It is shown in Mechanics that the common centre of gravity of 
 any two globes is at a point on the straight line joining their in- 
 dependent centres of gravity, so situated that its distances from 
 the centres of the two globes are inversely as the masses of the 
 globes. Hence the distance of the common centre of gravity 
 
 of the earth and the sun from the centre of the sun is ' ' 
 
 oJ7,U()0 
 
 miles, or only about 280 miles ; so that the sun may practically 
 bo considered to be at rest. 
 
 With regard to Kepler's second law, Newton further showed 
 that the angular velocity with which the radius vector moves 
 must be inversely proportional to the square of its length. 
 
112 PRECESSION. 
 
 Finally, he proved also the tru^h of the third law, provided 
 only that a slight correction is introduced when the mass of the 
 planet is not so small as to be inappreciable in comparison with 
 that of the sun. If t and t' denote the times of revolution of 
 any 'two planets, m and m' their masses (the mass of the sun 
 being unity), and d and d' their mean distances from the sun, 
 Newton showed that we shall always have the following pro- 
 portion : 
 
 I + m' 1 + m' 
 
 If m and m' are so small that they may be omitted without 
 sensible error, this proportion becomes identical with Kepler's 
 third law. 
 
 
 PERTURBATIONS IN THE EARTHS ORBIT. 
 
 118. Precession. Although the absolute positions of the 
 planes of the ecliptic and the equinoctial in space, and their re- 
 lative positions to each other, remain very nearly the same from 
 year to year, there are nevertheless certain small perturbations 
 in these positions which are made evident to us by refined and 
 extended observations. The principal of these perturbations is 
 called precession. 
 
 The latitudes of all the fixed stars remain very nearly the 
 same from year to year, and even from century to century: and 
 we therefore conclude that the position of the ecliptic with 
 reference to the celestial sphere remains very nearly unchanged. 
 But the longitudes of the stars are all found to increase by an 
 annual amount of 50".2: and hence the line of the equinoxes 
 must have an annual westward motion of the same amount. 
 This westward motion is called the precession of the equinoxes. 
 Since the ecliptic remains stationary in the heavens (or at least 
 so nearly stationary that the latitudes of the stars only vary 
 by half a second of arc in a year), this precession must be con- 
 sidered to be a motion of the equinoctial on the ecliptic. 
 
 In Fig. 46, let EQ represent the equinoctial, and LC the eclip- 
 tic. .BFis the line of the equinoxes, which moves about in the 
 plane of the ecliptic, taking in course of time the new position 
 B'V. Perhaps the clearest conception of this motion is ob- 
 
PRECESSION. 
 
 113 
 
 tained by considering P, the pole of 
 the equinoctial, to revolve about A, 
 the pole of the ecliptic, in the circle 
 PG (the polar radius of which, AP, 
 is 23 27'), moving westward in this 
 circle with the annual rate of 50".2, 
 and completing its revolution in 
 25,817 years. 
 
 A general explanation of the 
 cause of precession may be given by 
 means of Fig. 47. The earth may 
 be regarded as a sphere surrounded by a spheroidal shell, as 
 represented in the 
 figure by EPQp, 
 and the matter in 
 this shell may be 
 considered to form 
 a ring about the 
 earth in the plane 
 of the equator, as 
 shown in E'FQ'p'. 
 It is to the attrac- 
 tion of the sun and 
 the moon on this 
 ring, combined with 
 the earth's rotation, 
 that the precession 
 of the equinoxes is 
 due. In the figure 
 let S be the sun, the 
 circle ABV the ecliptic, and E"Q" this equatorial ring of the 
 earth. Let J.CTbe the plane of the equinoctial, meeting the 
 plane of the ecliptic in the line of equinoxes A V. This plane 
 ij> by definition determined at each instant by the position of 
 the earth's equator. The attraction exerted by the sun upon 
 the different particles of the ring in that half of it which is 
 nearer the sun (the particle E", for instance), may be resolved 
 into two forces, one acting in the plane of the equator, and the 
 
 Fig. 47. 
 
114 PRECESSION. 
 
 other in a direction perpendicular 4;o that plane, or in the direc- 
 tion E"d. The sun's attraction upon the nearer half of the 
 ring, then, tends to draw the plane of the ring nearer to the 
 plane of the ecliptic. On the other hand, the sun's attraction 
 upon the farther half of the ring tends to bring about the 
 opposite result ; but since, by the law of attraction, the latter 
 effect is less than the former, we may consider the whole result 
 of the sun's attraction upon the ring to be a tendency in the 
 plane of the ring to come nearer to the plane of the ecliptic. 
 Therefore, if the ring did not rotate, the plane of the earth's 
 equator would ultimately come into coincidence with the plane 
 of the ecliptic. 
 
 But the ring does rotate, about an axis perpendicular to its 
 own plane ; and the combined result of this rotation and of the 
 rotation about the line of the equinoxes, above described, is 
 that the plane of the equinoctial, while it preserves constantly 
 its inclination to the plane of the ecliptic, moves about in a 
 westerly direction: the line of intersection of the two planes 
 also moving about in the same direction, and thus giving rise to 
 the precession of the equinoxes. 
 
 Similar results will evidently follow if S represents the moon 
 instead of the sun. Owing to the greater proximity of the 
 moon to the earth, however, the results of its attraction are 
 more than double those of the attraction of the sun. There is 
 still another perturbation in the position of the line of the equi- 
 noxes which is a result of the mutual attraction between the 
 earth and the other planets. This attraction tends to draw the 
 earth out of the plane of the ecliptic, without affecting in any 
 way the position of the plane of the equinoctial. The result is 
 an annual movement of the equinoxes towards the east. This 
 perturbation is exceedingly minute, being only about yth of a 
 second of arc in a year. The value 50". 2 is the algebraic sum 
 of all these perturbations. 
 
 119. Results of Precession. One result of precession is to 
 make the interval of time between two successive returns of the 
 sun to the vernal equinox less than the time of one sidereal 
 revolution, by the time required by the sun to pass over 50". 2, 
 which is 20m. 21.8s. Hence we have the tropical year, to which 
 
NUTATION. 115 
 
 reference has already been made in Art. 111. Another result 
 is that the signs of the Zodiac (Art. 91) no longer coincide with 
 the constellations after which they are named, but have retreated 
 towards the west by about 28, or nearly one sign : so that the 
 constellation of Aries now lies in the sign of Taurus. Still 
 another result is that the same star is not the pole-star in dif- 
 ferent ages. Referring to Fig. 46, the pole of the heavens, P, 
 will have revolved about A to the position G, in the course of 
 about 13,000 years; and a star of the first magnitude, called 
 Vega, which is now about 51 from the pole, will at that time be 
 less than 5 from the pole, and will be the pole-star. 
 
 120. Nutation. Since precession is the result of the tendency 
 of the sun to change the position of the plane of the equator, it 
 is evident that there will be no precession when the sun is itself 
 in the plane of the equator, in other words, at the equinoxes, 
 and that the precession will be at its maximum when the sun is 
 the farthest from the plane of the equator: that is to say, is at the 
 solstices. The amount of precession due to the influence of the 
 moon is subject, to a similar variation, being the greatest when 
 the moon's declination is the greatest. The result is that the 
 pole of the heavens has a small oscillatory motion about its 
 mean place. This motion is called nutation. If the effect of 
 nutation could be separated from that of precession, the pole 
 would be found to move in a very minute ellipse, having a 
 major axis of 18". 5, and a minor axis of 13". 7, the period of 
 one revolution in this ellipse being about nineteen years. Since, 
 however, these two perturbations co-exist, the result is that the 
 pole of the heavens revolves about the pole of the ecliptic, not 
 in a circle, but in an undulating curve, as 
 
 represented in Fig. 48 : the amount of the 
 deviation being very much exaggerated in 
 the figure. 
 
 121. Change in the Obliquity of the Eclip- 
 tic. It was stated in Art. 118 that the lati- 
 tudes of the stars were found to vary from 
 year to year by a very minute amount. 
 
 This change in the latitudes is due to a change in the position 
 of the plane of the ecliptic, involving a change in the obliquity 
 
116 ABERRATION. 
 
 of the ecliptic. The obliquity .of the ecliptic in 1878 was 
 23 27' 18": and the annual amount of diminution to which it 
 is now subject is 0".46. Mathematical investigations show that 
 after certain moderate limits have been reached this diminution 
 will cease, and the obliquity will begin to increase. The arc 
 through which the obliquity oscillates is about 1 21', and the 
 time of one oscillation is about ten thousand years. 
 
 122. Advance of the Line of Apsides. The line connecting 
 the earth's perihelion and aphelion is called the line of ajisides. 
 This line revolves from west to east, with an annual rate of 11". 8; 
 a perturbation due to the attraction exerted on the earth by the 
 superior planets. The time in which the earth moves from peri- 
 helion to perihelion is called the anomalistic year (from anomaly, 
 Art. 98). It is evidently longer than the sidereal year, and is 
 found to contain 365d. 6h. 13m. 49.3s. 
 
 ABERRATION. 
 
 123. The apparent direction of a celestial body is determined 
 by the direction of the telescope through which it is observed. 
 In consequence of the motion of the earth, and the progressive 
 motion of light, the telescope is carried to a new position while 
 the light is descending "through it, and therefore the apparent 
 direction of the body will differ from its true direction. 
 
 \? In Fig. 49, let OF be the posi- 
 
 tion of the axis of a telescope at 
 the instant when the rays of light 
 from the star S reach the object 
 glass 0. The rays, after passing 
 through the glass, begin to con- 
 verge towards a fixed point in 
 space, with which, at this instant, 
 the intersection of the cross-wires 
 Fig. 49. coincides. Let the earth be moving 
 
 in the direction FA. Since the transmission of light is not 
 instantaneous, time is required for the light to pass from to 
 the fixed point in space, and in that time the earth will carry 
 the axis of the telescope to some new position 0'F f . The cross- 
 wires will then be at F' t while the rays, whose motion in space 
 
ABERRATION. 117 
 
 is en f irely independent of any motion of the telescope, will tend 
 to meet at the point F. In order, then, to have the image of the 
 star coincide with the intersection of the wires, the telescope 
 must be so moved that its axis will lie in the position O'F. The 
 star will then appear to lie in the direction FSi, while its true di- 
 rection is of course FS: and the angle which these two directions 
 make with each other, or the angle F' O'F, is called the aberra- 
 tion. Representing this angle by A, and the angle OFF' , the 
 angle between the apparent direction of the star and the direc- 
 tion in which the earth is moving, by I, we shall have, 
 
 sin 4: BinI=FF f : O'F'. 
 
 But the ratio FF' : O'F' is the ratio between the velocity of 
 the earth and that of light : so that the sine of the aberration is 
 equal to the ratio of the velocity of the earth to that of light, 
 multiplied by the sine of the angle J. 
 
 124. Diurnal Aberration. Aberration causes the celestial 
 bodies to appear to be nearer than they really are to that point 
 of the celestial sphere towards which the motion of the earth 
 is directed at the instant of observation. As a correction, then, 
 it is to be applied in the opposite direction. There is evidently 
 no aberration when the motion of the earth is directly towards 
 the star, and the greatest amount of aberration occurs when 
 the direction of the earth's motion is at right angles to the 
 direction of the star. 
 
 Aberration is of two kinds, corresponding to the daily and 
 the yearly motion of the earth. The diurnal aberration tends 
 to displace all bodies in the direction in which the earth is 
 carrying the observer : that is to say, in an easterly direction. 
 It evidently varies -with the linear velocity of the observer, and 
 is therefore the greatest at the equator and zero at the poles. 
 Owing to the minuteness of the velocity of any point of the 
 earth's surface about the axis in comparison with the velocity 
 of light, the diurnal aberration is extremely small, its greatest 
 value being less than Jd of a second of arc. 
 
 125. Annual Aberration. The displacement of a star occa- 
 sioned by the motion of the earth in its orbit about the sun is 
 called the annual aberration. The effect which it has on the mo- 
 tion of any body will depend on the relative situation of that body 
 
118 
 
 ABERRATION. 
 
 to the plane of the ecliptic, as ma^r be seen in Fig 50. In thia 
 figure /S represents the sun, ABCD the orbit of the earth, and K 
 the pole of the ecliptic. Suppose a star to be at K. As the earth 
 moves through A, in the direction indicated by the arrow, the 
 star will be displaced from Kiv a; as the earth moves through 
 
 B, the star will be seen 
 at b, &c. Since the 
 direction of this star is 
 always at right angles 
 to the direction in 
 which the earth is 
 moving, the aberra- 
 tion will continually 
 be at its maximum, 
 as shown in the pre- 
 vious article, and the 
 star will describe a 
 circle about its true 
 place as a centre. If 
 the star is in the plane 
 of the ecliptic, as at s, there will be no aberration when the 
 earth is at A or (7, and the aberration will be at its maximum 
 when the earth is at B or D. The star will therefore during 
 the year describe the arc b'd', equal in value to twice the maxi- 
 mum of aberration, and having the true place of the star at its 
 middle point. If the star is situated between the pole and the 
 plane of the ecliptic, it will describe an ellipse, the semi-major 
 axis of which is the maximum of aberration, and the semi- 
 minor axis of which increases with the latitude of the star. 
 
 126. Velocity of Light. The maximum value of aberration is 
 the same for all bodies, and may be obtained by observing the 
 apparent motion of a fixed star during the year. Its value 
 has thus been obtained, and is 20". 4. Now, since the maximum 
 of aberration occurs when the angle J, in the formula in Art. 
 123, is 90, we shall have, denoting this maximum by A' t 
 
 A> FF ' 
 
 sin A - - Q,p,- 
 But FF' is the velocity of the earth in its orbit, or 18.4 miles 
 
ABERRATION. H9 
 
 a second. Hence, the velocity of light in a second is 18.4 
 miles multiplied by the cosecant of 20".4, which will be found 
 to be 185,600 miles. Experiments of a totally different character 
 have given almost precisely the same result ; and it is believed 
 that this estimate is within a thousand miles of the true velocity. 
 
 If we divide the disr.uice of the earth from the sun by this 
 velocity, we find that it requires 8m. 18s. for light to pass over 
 that distance. When we look at the sun, therefore, we see it, 
 not as it is at the time of observation, but as it was 8m. 18s. 
 previously; and in the same way every other celestial body 
 appears to be in a different position from that which it really 
 occupies at the instant we observe it. It may be well to notice 
 here, that the time required for light to pass from any celestial 
 body to the earth is an element in the computation of the appa- 
 rent place of that body given in the Nautical Almanac. In 
 the case of a body which changes its actual position on the celes- 
 tial sphere, as a planet, for instance, allowance must also be made 
 for what is called planetary aberration ; since even if the earth 
 were stationary, the apparent position of the body would be 
 behind its true position by the amount of its motion in the time 
 required for light to come from the body to the earth.* 
 
 127. Aberration a Proof of the Earth's Revolution about the 
 Sun. The existence of the phenomenon of aberration, as de- 
 scribed in Art.. 125, is a matter of undoubted observation : and 
 when the close agreement of the velocity of light obtained in 
 the preceding article with the velocity obtained by independent 
 philosophical experiments is taken into consideration, it is fair 
 to regard the existence of aberration as a strong direct proof of 
 the revolution of the earth about the sun. Another proof, simi- 
 lar in many respects to this, will be noticed when we come to 
 the subject of the eclipses of Jupiter's satellites. 
 
 * Herschel suggests (Outlines of Astronomy, % 385) that this might be 
 called the equation of light, in order to prevent its being confounded with tho 
 real aberration of light. 
 
120 ORBIT OF THE ITOCXN. 
 
 CHAPTER IX. 
 
 THE MOON. 
 
 128. The Orbit of the Moon. While the moon, in common 
 with all the celestial bodies, has the apparent westward motion 
 which is due to the rotation of the earth, it also changes its rela- 
 tive position to the other bodies, and is continually falling behind, 
 or to the east of them, in this diurnal motion. In other words, 
 it has an independent motion, either real or apparent, from west 
 to east. This eastward motion is so rapid, that we only need to 
 observe the relative situations of the moon and some conspicuous 
 star, during a few hours on any favorable night, to notice a per- 
 ceptible change in their angular distance. If the right ascension 
 and the declination of the moon are determined from day to day, 
 precisely as the same elements of the sun's position were deter- 
 mined (Art. 89), and the corresponding positions are laid down 
 upon a celestial globe, we shall find that the moon makes a com- 
 plete revolution in the heavens, about the earth as a centre, in 
 an average period of 27d. 7h. 43m. 11.5s. We shall also find 
 that the plane of the moon's orbit intersects the plane of the 
 ecliptic at an angle whose mean value is 5 8' 44", and in a line 
 which, like the earth's line of equinoxes, is continually revolving 
 towards the west : so that the apparent orbit of the moon is not 
 a circle, but a kind of spiral. This revolution is much more 
 rapid, however, in the case of the moon, the amount of retro- 
 gradation being about 1 27' in a month, and the complete revo- 
 lution being effected in 18.6 years. 
 
 The movement of the moon in its orbit is represented in Fig. 
 51. Let E be the earth, and the circle MANC the plane of the 
 ecliptic. Let the moon be at J/at any time. Then will MN be 
 the line in which the plane of the moon's orbit intersects the 
 plane of the ecliptic. This liiie is calk d the line of the nodes. That 
 
ORBIT OF THE MOON. 121 
 
 extremity of the line through which the moon passes in moving 
 from the southern to the northern 
 side of the ecliptic is called the 
 ascending node, the other the de- 
 scending node. Let the moon 
 move on from M in the arc MB. 
 When it descends to the ecliptic, 
 it will meet it, not at the point N, 
 but at some point N'. and the line 
 of the nodes will take the new posi- 
 tion N'M'. The moon moves on to the other side of the ecliptic, 
 passes through the arc N'D, and when it again returns to the 
 ecliptic, will meet it, not at M f , but at some point M", and the 
 line of the nodes will take the position M"N". The revolution 
 of the line of the nodes is evidently in an opposite direction to 
 that in which the moon itself revolves, and is therefore from 
 east to west. 
 
 129. Cause of the Retrogradation of the Nodes. This retro- 
 grade movement of the moon's nodes is similar in character to 
 the precession of the equinoxes, and is due to the attraction 
 which the sun exerts upon the moon. Since the plane of the 
 moon's orbit is inclined to the plane of the ecliptic, the attrac- 
 tion of the sun will, in general, tend to draw 7 the moon out of its 
 orbit towards the ecliptic. The only exceptions to this rule will 
 occur when the moon is at one of its nodes, and is therefore i.i 
 the plane of the ecliptic, and also when the line of the moon's 
 nodes passes through the sun, at which time the attraction of the 
 sun is exerted along this line, and consequently in the plane of 
 the moon's orbit. In Fig. 51 let the moon be at B, and the sun 
 anywhere in the ecliptic except on the line of the nodes. As the 
 moon moves on, the sun is continually drawing it down to the 
 ecliptic, and it will hence meet the ecliptic, not at N, but at jY'. 
 The same effect will be seen at every other position of the moon 
 in its orbit, with only the exceptions already mentioned. 
 
 130. Change in the Obliquity of the Moon's Orbit. It is evi- 
 dent from the same figure that the angle which the arc BN' 
 makes with the plane of the ecliptic is greater than the angle 
 
 which an arc drawn through B and N would make. The obli- 
 11 
 
122 ORBIT OF THE MOOX. 
 
 quity of the plane of the moon's orbit is therefore increased a* 
 the moon approaches the node. It may be shown in the same way 
 that the obliquity is diminished as the moon recedes from the node. 
 The extreme limits which this angle attains are 5 20' 6" and 
 4 57' 22". 
 
 131. Elliptical form of the Moon's Orbit. The angular dia- 
 meter of the moon varies at different points of its orbit, while 
 its mean value remains the same from month to month ; we there- 
 fore conclude, as we concluded in the case of the sun, that its 
 distance from the earth is not constant, the greatest distance cor- 
 responding to the least diameter, and the least distance to the 
 greatest diameter. If we neglect the retrogradation of the 
 moon's nodes, and represent graphically the moon's orbit by a 
 method identical with the method employed in representing the 
 earth's orbit (Art. 98), we shall find the orbit to be an ellipse, 
 with the earth at one of the foci. The eccentricity of the ellipse 
 is 0.0549, or very nearly -j^th. 
 
 132. Line of Apsides. That point in the moon's orbit where 
 it is the nearest to the earth is called the perigee, and that point 
 where it is the farthest from the earth, the apogee. The line 
 connecting these two points is called the line of apsides. It is 
 also the major axis of the moon's orbit. This line revolves in the 
 plane of the moon's orbit from west to east, making a complete 
 revolution in very nearly nine years. 
 
 The following description of the moon's orbit, and of the 
 changes to which it is subject, is given by Herschel in his Out- 
 lines of Astronomy. " The best way to form a distinct conception 
 of the moon's motion is to regard it as describing an ellipse about 
 the earth in the focus, and at the same time to regard this ellipse 
 itself to be in a twofold state of revolution; 1st, in its own 
 plane, by a continual advance of its own axis in that plane; 
 and 2dly, by a continual tilting motion of the plane itself, 
 exactly similar to, but much more rapid than, that of the earth's 
 equator." 
 
 133. Variation in the Moons Meridian Zenith Distance. From 
 the formula in Art. 76 we have, 
 
 z--=L d 
 At any place, then, the latitude remaining constant, the least 
 
DISTANCE OF THE MOON. 
 
 123 
 
 meridian zenilh distance will occur when the moon's declination 
 has the same name as the latitude, and is at its maximum ; and 
 the greatest will occur when the declination has the opposite 
 name, and is also at its maximum. Since the plane of the 
 moon's orbit is inclined, at the most, 5 20' to the plane of the 
 ecliptic (Art. 130), the greatest value of the declination, either 
 north or south, is 5 20' -f- 23 27'. The variation in the meri- 
 dian altitude will therefore be double this amount, or 57 34'. At 
 Annapolis, in latitude 38 59' N., the greatest altitude is 79 48', 
 the least 22 14'. There is an exception to this general rule in 
 the case of those places whose latitude is less than 28 47': since 
 at those places the greatest altitude occurs when the moon is in 
 the zenith, or, as is evident from the formula, when the declina- 
 tion is equal to the latitude, and has the same name. 
 
 The new moon is in the same part of the heavens that the sun 
 is in (Art. 139), and the full moon is in the opposite part. Since 
 the sun attains its least altitude in winter and its greatest in 
 summer, new moons will run low in winter and full moons will 
 run high : while in summer the opposite of this will take place. 
 
 DISTANCE, SIZE, AND MASS OF THE MOON. 
 
 1 34. Since the sine of the moon's horizontal parallax is the 
 
 ratio of the radius of the earth to the distance of the nioon from 
 the earth, it is evident that we can determine this distance as 
 soon as we obtain the horizontal parallax. The horizontal 
 parallax may be found in the following manner: In Fig. 52, let 
 be the centre of the earth, EQ its equator, and A and B the 
 positions of two observers on the same meridian, whose zeniths 
 
124 DISTANCE OF THE MOON. 
 
 are Z and Z '. Let M be the mood's position when crossing the 
 meridian. The apparent zenith distance at A, corrected for 
 refraction, is the angle ZAM, the geocentric zenith distance is 
 ZOM, and the difference of these two angles, the angle A MO, 
 is the parallax in altitude. In the same way OMB is the pa- 
 rallax at B. Represent the parallax at A by p, that at B by 
 p, the horizontal parallax by P, the apparent meridian zenith 
 distance at A by z, and that at B by z'. 
 We have, by Geometry, 
 
 p = z A M, 
 p' = z' BOM, 
 and, consequently, 
 
 " p + p f = z + z' AOB. 
 We have also, from Art. 54, 
 
 p = P sin z, 
 p' = P sin z', 
 and, therefore, 
 
 p -f- p' = P (sin z -f sin z'). 
 
 Combining the two equations, and finding the expression for P, 
 _ g + z' A OB 
 
 sin z -j- sin z' 
 
 But A OB is evidently the difference of latitude of A and B. 
 We have, then, as our method of finding the moon's horizontal 
 parallax, to subtract the difference of latitude of the two places 
 from the sum of the apparent zenith distances, and to divide 
 the remainder by the sum of the sines of the two zenith dis- 
 tances. 
 
 It is important that the two places of observation shall 
 differ widely in latitude. It is not, however, necessary that 
 they shall be on the same meridian, since, either from tables 
 of the moon's motion, or from actual observation on successive 
 days before and after the time of observation, we can obtain 
 the change of meridian zenith distance corresponding to any 
 known difference of longitude, and thus reduce the two observed 
 zenith distances to the same meridian. 
 
 135. The Moon's Horizontal Parallax. By observations simi- 
 lar to those above described, the mean value of the moon's 
 equatorial horizontal parallax is found to be 57' 3". The mean 
 
MAGNITUDE AND MASS. 125 
 
 distance of the moon from the earth is therefore 3962.8 miles 
 multiplied by the cosecant of 57' 3", or 238,800 miles. The hori- 
 zontal parallax varies between the limits of 61' 32" and 52' 50", 
 and the distance between 257,900 and 221,400 miles. It must 
 be noticed that by the mean value of the horizontal parallax 
 given above is not meant the half sum of the two extreme 
 values, but the value which the parallax has when the moon is 
 at its mean distance from the earth. 
 
 136. Magnitude of the Moon. The angular semi-diameter of 
 the moon at its mean distance from the earth is found to be 
 15' 33."5. Its linear semi-diameter is therefore obtained by 
 multiplying the mean distance by the sine of 15' 33". 5, and 
 is found to be 1080.8 miles, or about T 3 T ths of the radius of the 
 earth. The volumes of two spheres being to each other as the 
 cubes of their radii, the volume of the moon will be found to 
 be about ^th of that of the earth. 
 
 137. Mass of the Moon. The mass of the moon is obtained 
 by the following considerations. If the sun does not affect the 
 gravity of the moon to the earth, and the mass of the moon is 
 inappreciable in comparison with that of the earth, then the 
 centrifugal force of the moon in its orbit ought exactly to equal 
 the attraction of the earth on the moon. But if the moon has 
 a sensible mass, it will, by the law of gravitation, attract the 
 earth, and its centrifugal force must be sufficient to counter- 
 balance the sum of the mutual attractions of the earth and 
 the moon. Now, if we refer to what was demonstrated in Art. 
 113, we find that the moon's actual centrifugal force is greater, 
 by -g^th, than the attraction of the earth at the distance of the 
 moon. This would make the mass of the moon -^gth of that of 
 the earth. But it is found that the sun diminishes sensibly the 
 gravity of the moon to the earth. This, therefore, must also be 
 taken into account, and the resulting value of the mass of the 
 moon is found to be about -g-'jSt of that of the earth. 
 
 The density of the moon, being directly as the mass, and 
 inversely as the volume, will be |f, or about |ths of the density 
 of the earth. 
 
 138. Augmentation of the Moon's Semi-Diameter. If at any 
 lime we measure the angular semi-diameter of the moon, we 
 
126 PHASES. 
 
 shall find that it increases with the moon's altitude, being least 
 when the moon is in the horizon and greatest when in the 
 zenith. This increase is explained in Fig. 53. Let E be the 
 
 centre of the earth, and 
 M that of the moon. With 
 the distance between jand 
 M as a radius, describe the 
 semi-circumference AM' B. 
 When the moon is in the 
 horizon of the point C, its 
 distances from C and from 
 E are very nearly equal. 
 53. But as the moon rises, the 
 
 distance CM continually decreases, while EM, the distance of the 
 moon from the earth's centre, remains sensibly constant. When 
 the moon is in the zenith, or at M', the distance CM' is less than 
 EM' by the radius of the earth. Now, the angular semi-diame- 
 ter of the moon will increase, as shown in Art. 98, very nearly 
 as the distance of the moon from the observer decreases. But 
 the earth's radius is about ^ O th of the distance of the moon 
 from the earth's centre: therefore the semi-diameter of the 
 moon in the zenith will be greater than the semi diameter in 
 the horizon by -g-^th of itself, or by about 15". This increase 
 is called the augmentation of the moon's semi-diameter. 
 
 THE MOON'S PHASES. 
 
 139. Two bodies are said to be in conjunction when they have 
 the same longitude. They are said to be in opposition when 
 their longitudes differ by 180 ; and in quadrature when their 
 longitudes differ by either 90 or 270. 
 
 The moon is an opaque body, which is rendered visible to 
 us by the rays of light which it reflects from the sun. The 
 phases of the moon are due to the different relative positions 
 to the sun and the earth which it has while revolving about 
 the earth. 
 
 In Fig. 54 let #be the earth, and the circle ACFH the orbit 
 of the moon. Since the inclination of the plane of the moon's 
 orbit to the plane of the ecliptic is only a few degrees, we may 
 
PHASES. 
 
 127 
 
 neglect it in this case, and suppose the two planes to coincide. 
 Let the sun lie in the direction ES. Since the distance of the 
 
 Fig. 54. 
 
 sun from the earth is about 387 times the distance of the moon 
 from the earth, the lines JES, ff/S, J3S, &c., drawn to the sun 
 from different points of the moon's orbit, may be considered to be 
 sensibly parallel. Let us first suppose the moon to be in con- 
 junction with the sun at the point A. Here only the dark 
 portion ot the moon is turned towards the earth, and the moon 
 is therefore invisible. This is called new moon. As the moon 
 moves on towards B, the enlightened part begins to be visible, 
 and when it reaches C, 90 in longitude from the sun, half the 
 enlightened part is visible, and the moon is at its first quarter. 
 When the moon is at F, in opposition to the sun, all the illumi- 
 nated part is turned towards the earth, and the moon is full. 
 The moon wanes after leaving F, passes through its laat quarter 
 at ff, and finally becomes again invisible. 
 
 Between A and C the moon is crescent, as represented at L, 
 and between C and F it is gibbous, as represented at N. The 
 same terms are also applied to the appearance of the moon be- 
 tween H and A and between F and H. 
 
 140. Phases of the Earth to the Moon. It is evident from Fig. 
 54 that the earth presents phases to the moon identical in cha- 
 racter with those presented by the moon to the earth, although 
 
128 
 
 SIDEREAL AND SYNODICAL PERIODS. 
 
 similar phases are Dot presented by each body at the same time. 
 Thus at the time of new moon tlie earth is full to the moon : 
 and the light which it then reflects to the moon renders the 
 unenlightened part of the moon faintly visible to the earth. As 
 the moon moves on to its first quarter, the earth reflects less and 
 less light to it, until finally the unenlightened portion disappears. 
 
 SIDEREAL AND SYNODICAL PERIODS. 
 
 141. The sidereal period of the moon is the interval of time 
 in which it makes one complete revolution in its orbit about 
 the earth. The synodieal period (or lunation) is the interval 
 between two successive conjunctions or oppositions. Owing to 
 the earth's revolving about the sun, and carrying the moon with 
 it, the synodieal period is longer than the sidereal period, as may 
 be seen in Fig. 55. 
 
 Let S be the sun, E the 
 earth, and MANB the or- 
 bit of the moon. Let the 
 moon be at M, in conjunc- 
 tion with the sun. As the 
 moon moves about E in 
 the curve MANB, the 
 earth also moves about the 
 sun in the direction EE'. 
 The next conjunction will 
 therefore not occur until the 
 moo;n reaches M". Now, 
 if through E' we draw the 
 line M'N' parallel to MN t the sidereal period of the moon is 
 completed when the moon reaches M'. The synodieal period is 
 therefore greater than the sidereal period by the time required 
 by the moon to pass through the angle M'E'M". This angle is 
 evidently equal to the angle ESE', which is the angular advance 
 of the earth in its orbit in the period of one synodieal revo- 
 lution of the moon. In one lunar month, then, the angular ad- 
 vance of the moon in its orbit is greater by 360 than the angu- 
 lar advance of the earth in its orbit. If, therefore, we denote 
 the moon's sidereal period in days by P, its synodieal period by 
 
 Fig. 55. 
 
SIDEREAL AND SYNODICAL PERIODS. 129 
 
 S, and the earth's sidereal period, or one sidereal year, by T, we 
 shall have, 
 
 360 
 
 - = the earth's daily angular velocity, 
 
 360 
 
 p = the moon's daily angular velocity, 
 
 360 
 
 - = the moon's daily angular gain on the earth, 
 
 Hence we shall have, 
 
 360 360 360. 
 
 > 
 
 P T S 
 
 ST 
 
 ~ S+T 
 
 The sidereal period of the moon is therefore obtained by mul- 
 tiplying the sidereal year by the moon's synodical period, and 
 dividing the product by the sum of the sidereal year and the 
 synodical period. 
 
 142. Values of the Synodical and Sidereal Periods. The value 
 of the synodical period is not constant, but varies from month 
 to month. A mean value may, however, be obtained by divid- 
 ing the interval of time between two oppositions, not conse- 
 cutive, by the number of synodical revolutions in that interval. 
 Now, the day, the hour, and even the probable minute, at which 
 an opposition of the moon occurred in the year 720 B.C., were 
 recorded by the Chaldseans; and by comparing this time with 
 the results of recent observations, an extremely accurate value 
 of the mean synodicaT period is obtained It is found to be 
 29d. 12h. 44m. 3s. We have, then, for the value of the side- 
 real period, by the formula in the preceding article, 
 
 _ 365.256 X 29.53 
 ~ 3657256 -r- 29.53 JS 
 whence we obtain the value already given in Art. 128. 
 
 143. Retardation of the Moon, and the Harvest Moon. The 
 mean daily motion of the moon towards the east is about 13, 
 while that of the sun is, as we have already seen, about 1: 
 hence the moon is continually falling to the rear of the sun in 
 apparent westward motion, and the interval of time between any 
 two successive transits of the moon is greater than the similar 
 
1<50 ROTATION OF THE MOON. 
 
 interval in the case of the sun. r he moon, therefore, rises later, 
 and sets later, day by day. This is called the retardation of the 
 moon. Its amount varies considerably in value, but is on the 
 average about fifty minutes. 
 
 The less the angle which the plane of the moon's orbit makes 
 with the plane of the horizon, the less does the advance of the 
 moon carry it with reference to the horizon, and, consequently, 
 the less is the retardation of the moon in rising. Now, since 
 the moon's orbit very nearly coincides with the ecliptic, the 
 retardation in rising will in general be the least, when the 
 ecliptic makes the least angle with the horizon. By reference 
 to a celestial globe, it will be seen that the ecliptic makes the 
 least angle with the horizon when the vernal equinox is in the 
 eastern horizon. The least retardation in rising, therefore, 
 occurs in each month when the moon is near the sign of Aries. 
 This least retardation is especially noticeable when it occurs at 
 the time of full moon. Now, when the moon is in Aries, and 
 full, the sun must be in Libra, or near the autumnal equinox. 
 This occurs about the 21st of September. About the time, then, 
 of the full moon which occurs near the 21st of September, the 
 moon will rise, for two or three nights, only about half an hour 
 later each night. Usually this small retardation is noticed at 
 the times of two full moons, one in September and the other in 
 October. The first is called the Harvest Moon, the second the 
 Hunter's Moon. All this relates to the Northern Hemisphere. 
 
 ROTATION. LIBRATIONS AND OTHER PERTURBATIONS. 
 
 144. Rotation of the Moon. By observation of the spots upon 
 the disc of the moon, it is found that very nearly the same 
 surface of the moon is turned continually towards the earth. 
 The conclusion drawn from this fact is that the moon rotates 
 upon an axis in the same time in which it revolves about the 
 earth, or in 27.3 days. The plane in which this rotation is 
 performed makes an angle of about 1 32' with the plane of the 
 ecliptic. 
 
 If there are any inhabitants of the moon, their day will be 
 equal in length to about twenty-nine of our days, and their 
 ni^ht to about twenty-nine of our nights. Since the plane of 
 
LIBRATIONS OF THE MOON. 
 
 131 
 
 the moon's equator is so nearly coincident with the plane of tho 
 ecliptic, there will hardly be any sensible change of seasons : 
 or if there is, the lunar day will be the lunar summer, and the 
 night the winter. To the inhabitants of one hemisphere the 
 earth will be perpetually invisible, while to the inhabitants of 
 the other hemisphere it will present the appearance of a. body 
 very nearly stationary in their sky, exhibiting phases similar 
 to those which we see in the moon, with a radius nearly four 
 times that of the moon, and a surface about thirteen times that 
 of the moon. 
 
 145. Librations. By libration is meant an apparent oscilla- 
 tory movement of the moon, which enables us, in the course of 
 its revolution, to see something more than an exact hemisphere. 
 
 The libration in longitude is due to the fact that the moon's 
 rotation on its axis is perfectly uniform, while its motion about 
 the earth is not. Hence the line drawn from the centre of the 
 
 Fig. 56. 
 
 earth to that of the moon does not always intersect the surface 
 of the moon at the same point, and we are able at times to look 
 
132 OTHER PERTURBATIONS. 
 
 a few degrees, east or west, beyond the mean visible border. If, 
 in Fig. 56, ABCD represents the earth, E its centre, and R 
 the centre of the rnoon, the dotted lines at iV denote the limits 
 between which, as the moon revolves about the earth, the visible 
 border may deviate from its mean position. 
 
 The libration in latitude is due to the fact that the axis of the 
 moon, remaining constantly parallel to itself, is not perpen- 
 dicular to the plane of the moon's orbit, but is inclined to it at an 
 angle of about 83 19'. We are therefore able at certain times 
 to see about 6 41' beyond the north pole of the moon, and at 
 other times the same amount beyond the south pole. Thus in 
 Fig 56, when the moon is at M, we can see beyond the pole P, 
 and when the moon is at 0, beyond the pole p: since in each 
 case we can see nearly that portion of the moon which lies be- 
 tween the earth and the circle ab, whose plane is perpendicular 
 to the plane of the moon's orbit. 
 
 The diurnal libration is due to the difference between that 
 hemisphere of the moon which is turned towards the centre of 
 the earth and that which is turned towards any point on the 
 surface. When, for instance, the moon is at Z/, an observer at 
 C will see the same hemisphere which is turned towards the 
 earth's centre, while an observer at G will see a different one. 
 The hemisphere which is turned towards any observer when 
 the moon is rising will also be different from the one which is 
 turned towards him when the moon is setting. It is evident in 
 the figure that the amount of this libration varies with the angle 
 ERG; that is to say, with the moon's parallax. 
 
 Notwithstanding all these librations, we are able to see in all 
 only about fV ths of the moon's surface, according to Arago: 
 the remainder being continually concealed from our view. 
 
 146. Other Perturbations. Besides these librations, and the 
 perturbations already mentioned (Art. 132), there are other per- 
 turbations in the moon's longitude of which only a very brief notice 
 can here be given. The greatest of these perturbations is called 
 evection, and was discovered by Ptolemy in the second century. 
 It arises from the variation in the eccentricity of the moon's 
 orbit, and from the fluctuations in the general advance of the 
 line of the apsides. By it the moon's mean longitude is alternately 
 
LUNAR CYCLE. 133 
 
 increased and decreased by about 1 20'. Another perturbation 
 in the moon's motion is called variation. It depends solely on 
 the angular distance of the moon from the sun, and its maximum 
 is 37'. The annual equation depends on the variable distance 
 of the earth from the s'in, and amounts to 11'. The secular acce- 
 leration is an increase in the moon's motion which has been going 
 on for many centuries, at the rate of about 10" a century. This 
 perturbation is partly due to the diminution of the eccentricity 
 of the earth's orbit; and from what has been said on that subject 
 in Art. 98, it is evident that this inequality will at some distant 
 day become a secular retardation. [See Note, page 154.] 
 
 All of these perturbations are satisfactorily explained by the 
 investigation of what is known as the problem of the three bodies, 
 in which two bodies are supposed to revolve about their common 
 centre of gravity, according to the law of universal gravitation, 
 and the effects of the attraction exerted by a third body upon 
 the motions of these two bodies are made the object of mathe- 
 matical examination. 
 
 THE LUNAR CYCLE. 
 
 147. If we multiply the number of days, hours, &c., in a 
 synodical period of the moon (Art. 142) by 235, the product 
 will be 6939d. 16h. 27m. 50s. Now, in a period of nineteen civil 
 years there are either 6939 days, or 6940 days, according as there 
 are four or five leap years in that period. If, then, in any year, 
 new moon occurs on any particular day of the month, the first 
 of January, for instance, it will occur again on the first of Janu- 
 ary (or at all events within a few hours of its end or beginning), 
 after an interval of nineteen years ; and all the new moons and 
 the other phases will occur on very nearly the same days through- 
 out the second period of nineteen years on which they occurred 
 during the first period. This period is called the Lunar Cycle. 
 It is also called the Metonic Cycle, having been originally dis- 
 covered, B.C. 432, by Meton, an Athenian mathematician. The 
 present lunar cycle began in 1862. 
 
 This cycle is used in finding Easter: Easter being the first 
 Sunday after the full moon which occurs either upon or next 
 after the 21st day of March. 
 
134 APPEARANCE OF THE MOON. 
 
 The golden number of any ye%r is the number which marks 
 the place of that year in the cycle. It may be found for any 
 year by adding 1 to the number of that year, and dividing the 
 sum by 19 ; the remainder (or 19, if there is no remainder) is 
 the golden number. 
 
 Four lunar cycles, or seventy-six civil years, constitute what 
 is called the Callippic cycle. 
 
 GENERAL DESCRIPTION OF THE MOON. 
 
 148. When viewed through powerful telescopes, the surface 
 of the moon is found to be made up of mountains, valleys, and 
 plains, similar in general appearance to those that exist on the 
 earth. As a whole, however, the surface of the moon is much 
 more uneven than that of the earth. The heights of over 1000 
 lunar mountains have been measured, and some of them have 
 been found to exceed 20,000 feet. Many of these mountains 
 bear the appearance of having been at one time volcanoes, far 
 surpassing in size and activity those on the earth. The common 
 belief among astronomers seems to be that these lunar volcanoes 
 are now extinct. Messrs. Beer and Madler, two Prussian astron- 
 omers who have made the moon their special study, have de- 
 tected no signs of activity in any of the volcanoes which they have 
 examined. A few years ago, certain phenomena were noticed 
 which seemed to show that one at least of these volcanoes, 
 named Linne", is not extinct : but later observations do not confirm 
 this suspicion. 
 
 There are no signs of the existence of water on the moon. 
 Certain large dark patches are seen, which were formerly con- 
 sidered to be oceans, gulfs, &c., and were so named ; but increased 
 telescopic power shows that they are dry plains. 
 
 It seems to be still an open question whether or not the moon 
 has an atmosphere. If there is an atmosphere, it must be of an 
 extremely minute height and density ; for we see no clouds and 
 no twilight, and there is nothing in the phenomena of the occul- 
 tations of stars by the moon which shows the existence of even 
 the rarest atmosphere. Some observers, however, and among 
 them Messrs. Beer and Madler, believe that they have detected 
 signs of the existence of a very slight atmosphere. 
 
PLATE II 
 
 TOTAL ECLIPSE OF THE SUN, OF JULY 18, 1860, 
 showing the Corona and the Red Flames; as observed by Dr. 
 Feilitzsch, at Castellon de la Plana 
 
LUNAR ECLIPSE. 135 
 
 CHAPTER X. 
 
 LUNAR AND SOLAR ECLIPSES. OCCULTATIONS. 
 
 149. Eclipses. The obscuration, either partial or total, of the 
 light of one celestial body by another is in astronomy termed an 
 eclipse. When the earth comes between the sun and the moon, 
 the light of the sun is shut off from the moon, and we have a 
 Mjnar eclipse. A lunar eclipse can occur only at the time when 
 the moon is in opposition to the sun, that is to say, at the time 
 :>f full moon. When the moon comes between the earth and the 
 sun, the light of the sun is shut off from the earth, and we have 
 a solar eclipse. A solar eclipse can occur only at the time of 
 new moon. An eclipse of a star or a planet by the moon is called 
 an occultation. 
 
 If the orbit of the moon lay in the plane of the ecliptic, a 
 lunar and a solar eclipse would occur in every month. Owing, 
 however, to the inclination of the plane of the moon's orbit to 
 the plane of the ecliptic, the latitude of the moon is usually too 
 great to allow either kind of eclipse to take place; and it is only 
 in special cases, when the moon is in or near the plane of the 
 ecliptic at the time of conjunction or opposition, that an eclipse 
 of the sun or the moon is possible. 
 
 LUNAR ECLIPSE. 
 
 150. In Fig. 57 let S be the centre of the sun, and E that of 
 
 Fig. 57. 
 
136 LUNAR ECLIPSE. 
 
 the earth. Draw the lines BH. and GG, tangent to the two 
 spheres. These lines will meet at some point A, and AHEG 
 will be a section of the shadow cast by the earth. 
 
 The whole shadow is of a conical shape, the vertex of the cone 
 being at A ; and a lunar eclipse will occur whenever the moon is 
 within this shadow. Draw the tangent lines BG and CH. 
 KDL is a section of a second cone whose vertex is at Z>. The 
 earth's shadow is called the umbra, and that portion of the second 
 cone which lies outside of the umbra is called the penumbra. 
 Thus KHA and A GL are sections of the penumbra. It must 
 be noticed, in regard to the construction of this figure, that since 
 only one tangent can be drawn to the circumference of a circle 
 at any one point, the lines BG and CH do not touch the two 
 circumferences at precisely the same points at which BH and 
 CG touch ; and that, furthermore, in all these figures the relative 
 size of the sun should be immensely greater than it is. 
 
 Now let M'MM" represent a portion of the moon's orbit at the 
 time of a lunar eclipse. As soon as the moon passes within the 
 line DK, some of the rays of the sun will be cut off from it by 
 the earth, and its brightness will begin to decrease. The whole 
 disc, however, will still be visible. As soon as the moon begins 
 to pass within the line HA, the disc will begin to disappear, and 
 when the whole disc has passed within the cone, the eclipse will 
 be total. 
 
 151. Different Kinds of Eclipses. When the moon's orbit 
 is so situated that only a part of the moon enters the umbra, 
 we have a partial eclipse. When the moon does not enter the 
 umbra, but merely touches it, we have an appulse. When the 
 centre of the moon coincides with the line which connects the 
 centre of the earth and that of the sun, the eclipse is cen- 
 tral. A central eclipse occurs very rarely, if indeed it occurs 
 at all. 
 
 152. The Semi-Angle of the Umbral Cone. The semi-angle of 
 the umbral cone is the angle EA G, Fig. 58. Now we have, by 
 Geometry, 
 
 SEC= ECG + EAG. 
 But SEC is the sun's angular semi-diameter, and EGG is its 
 
LUNAR ECLIPSE. 137 
 
 horizontal parallax. Putting S for SEC, and P for ECG, we 
 have, 
 
 EAG = S P. 
 
 Fig. 58. 
 
 153. The Angular Semi-Diameter of the Shadow at the Distance 
 of the Moon. The angular semi-diameter of the shadow at the 
 distance of the moon is the angle MEM'. We have, by Geometry, 
 
 EM' G = MEM' + EA G. 
 
 Now EM' G is the moon's horizontal parallax, which we will 
 represent by P', and the value of EA G has been obtained in 
 the preceding article. We therefore have, 
 
 MEM' = P'+P S. 
 
 Observation shows that the earth's atmosphere increases the 
 apparent breadth of the shadow by about its one-fiftieth part : 
 hence in practice the angular semi-diameter of the shadow is 
 taken equal to f J (P' -f- P S). If we substitute in this expres- 
 sion the least values of P f and P, and the greatest value of JS t 
 from the table given in Art. 155, we shall find that the least 
 value of the angular semi-diameter of the shadow is about 37' 25" : 
 so that the entire breadth of the shadow is always more than 
 double the greatest diameter of the moon. 
 
 154. Length of the Earth's Shadow. The length of the shadow, 
 or the line EA, can be computed from the right-angled triangle 
 EA G, in which we have, 
 
 EA=JEQco8ec(8 P). 
 
 The mean value of this length is 858,000 miles, or more than 
 three times the distance of the moon from the earth. 
 
 155. Lunar Ecliptic Limits. We see from Fig. 58 that a lunar 
 eclipse can occur only when the moon's geocentric latitude at 
 the time of opposition, (or at full moon,) is less than the sum 
 of the angular semi-diameter of the shadow and the semi-diameter 
 
138 
 
 LUNAR ECLIPSE. 
 
 of the moon. If we represent the. moon's semi-diameter by 
 the expression for this sum is 
 
 If the moon's geocentric latitude at the time of opposition is 
 greater than the greatest value which this expression can attain, 
 no eclipse can possibly occur : if it is less than the least value of 
 the expression, an eclipse is inevitable. These two values of 
 this expression are called the lunar ecliptic limits. Now, we have 
 by observation the following values of P, P' } &c. : 
 
 
 MAXIMA. 
 
 MINIMA. 
 
 P' 
 
 61' 32" 
 
 52' 50" 
 
 p 
 
 9 
 
 9 
 
 S' 
 
 16 46 
 
 14 24 
 
 s 
 
 16 18 
 
 15 45 
 
 In order to find the greatest value of the expression, we sub- 
 stitute in it the greatest values of P, P and $', and the least value 
 of S. The result is 1 3' 37": and no eclipse will occur when the 
 moon's latitude exceeds this limit. The least value of the expres- 
 sion is 51' 49": and when the moon's latitude at opposition is less 
 than this, an eclipse cannot fail to occur. There are some con- 
 siderations, however, which have not been taken into account, 
 which may increase each of these limits by about 16". 
 
 When the moon's latitude at opposition is within these limits, 
 an eclipse is possible, but not necessarily certain. In order to 
 determine whether in such case it will or will not occur, the 
 actual values which P, P', S and S' will have at that time must 
 be substituted in the expression, and the result compared with 
 the corresponding latitude of the moon. 
 
 156. Since a lunar eclipse is caused by the moon's entering 
 the earth's shadow, it will be seen at the same instant of time by 
 every observer who has the moon above his horizon : and the 
 character of the eclipse, whether total or partial, will be every- 
 
SOLAR ECLIPSE. 139 
 
 where the same, As the moon's motion towards the east is more 
 rapid than that of the earth (and consequently of the shadow), 
 the eclipse will begin at the eastern limb of the moon. A total 
 eclipse of the moon may last for nearly two hours. Even when 
 totally eclipsed, however, the moon does not, in general, disappear 
 from view, but shines with a dull reddish light. This phenomenon 
 is caused by the earth's atmosphere, which refracts the rays of 
 light from the sun which enter it near the points G and H, Fig. 
 57, and turns them into the cone. The rays which pass still 
 nearer to these points are probably absorbed by the atmosphere, 
 thus giving rise to the observed increase of the shadow mentioned 
 in Art. 153. 
 
 SOLAR ECLIPSE. 
 
 157. In Fig. 59, let S represent the sun, E the earth, and M 
 the moon, at the time of a solar eclipse. HAK will be a section 
 of the moon's umbra, and GHA and AKD, sections of its pen- 
 umbra. 
 
 Fig. 59. 
 
 To an observer situated within the umbra, at any point of the 
 arc dby the eclipse will be total ; while to one situated within the 
 penumbra, as at L, for instance, the eclipse will be partial. 
 Beyond the penumbra no eclipse whatever will be seen. Hence 
 the geographical position of the observer determines the charac- 
 ter of the eclipse : a condition different from that in the case of 
 a lunar eclipse, which we have seen is the same to all observers. 
 
 158. Length of the Moon's Shadow. It is evident that, to an 
 observer at the apex of the shadow A, the angular semi-diametera 
 of the sun and the moon would be equal. Now, the mean angular 
 semi-diameters of these two bodies as seen from the earth's centre 
 
140 SOLAR ECLIPSE. 
 
 are nearly equal ; hence the mean position of the apex does not fall 
 
 very far from the earth's centre E. An approximate value of the 
 
 length of the shadow may be thus obtained. We have in Fig. 59, 
 
 . _.,_ HM CS 
 
 8mHAM =AM=AS' 
 
 But we have just now seen that HAM is the sun's angular 
 genii-diameter, as seen from A ; and as AE is small compared 
 with AS, we may consider the angle HAM to be the sun's geo- 
 centric semi-diameter. Denoting this by S, we have, 
 
 sin = . 
 AM 
 
 Now, if S f represents the moon's geocentric semi-diameter, we 
 have, 
 
 HM 
 
 Combining these two equations, and finding the value of AM, 
 we have, 
 
 Knowing, then, the distance of the moon from the earth's centre, 
 and the semi-diameters of the sun and the moon, we may find 
 the length of AM. When the two semi-diameters are equal, we 
 have AM equal to EM, and the apex is at the earth's centre. 
 When the semi-diameter of the moon is greater than that of 
 the sun, the apex falls beyond the earth's centre : when it is 
 less, the apex does not reach the centre. Appropriate calcula- 
 tions will show that when both sun and moon are at their mean 
 distances from us, the apex falls short of the earth's surface : 
 and that when the moon is at its least distance from the earth, 
 and its shadow is the longest, the apex falls about 14,000 miles 
 beyond the earth's centre. 
 
 159. Different Kinds of Eclipses. When the shadow falls 
 beyond the earth's surface, the eclipse is total, as we have 
 already seen, within the umbra, and partial within the penumbra. 
 When the apex just touches the earth, the eclipse is total only 
 at the point where it touches. When the apex falls short of 
 the surface, there will be no total eclipse ; but at the point in 
 which the axis of the cone, prolonged, meets the earth, the 
 
SOLAR ECLIPSE. 141 
 
 observer will see what is called an annular eclipse, the moon 
 being projected upon the disc of the sun, but not covering it. 
 
 160. Solar Ecliptic Limits. In Fig. 60 let S represent the 
 sun, E the earth, and M the moon. No eclipse of the sun can 
 occur unless some part of the moon passes within the lines BG 
 B 
 
 G 
 
 Fig. 60. 
 
 and GD, drawn tangent to the sun and the earth : that is, unless 
 the moon's geocentric latitude is less than the angle MES. 
 Now we have, 
 
 MES = MEA + AEB + BES, 
 and, also, 
 
 AEB= CAE CBE. 
 
 BES is the sun's semi-diameter, MEA that of the moon, CAE 
 the moon's horizontal parallax, and CBE the sun's : hence, 
 using the notation already employed in Art. 155, we have, 
 MES = 8 + 8 + P. P. 
 
 The greatest value of this expression is found by employing 
 the greatest values of S, $', and P', and the least value of P, 
 as given in Art. 155, and is 1 34' 27" : and there will be no 
 eclipse if the moon's latitude at conjunction is greater than 
 this amount. The least value is 1 22' 50" ; and if the latitude 
 at conjunction is less than this an eclipse is inevitable. These 
 two values, which, owing to certain considerations omitted in 
 this discussion, should both be increased by about 25", are called 
 the solar ecliptic limits. In order to determine whether an 
 eclipse will occur when the moon's latitude at conjunction falls 
 within these limits, we must substitute in the expression the 
 values which the different quantities will really have at that 
 time, and compare the result with the corresponding latitude 
 of the moon. 
 
 161. General Phenomena. Since the moon moves towards 
 
142 CYCLE OF ECLIPSES. 
 
 the east more rapidly than the sun, a solar eclipse will begin 
 at the western side of the sun. For the same reason the moon's 
 shadow will cross the earth from west to east, and the eclipse 
 will begin earlier at the western portions of the earth's surface 
 than at the eastern. The moon's penumbra is tangent to the 
 earth's surface at the beginning and the end of the eclipse, so 
 that the sun will be rising at that place where the eclipse is 
 first seen, and setting at the place where it is last seen. A solar 
 eclipse may last at the equator about 4J hours, and in these 
 latitudes about 3.\ hours. That portion of the eclipse, however, 
 in which the sun is wholly concealed can only last about eight 
 minutes: and in these latitudes, only about six minutes. 
 
 The darkness during a total eclipse, though subject to some 
 variation, is scarcely so intense as might be expected. The sky 
 often assumes a dusky, livid color, and terrestrial objects are 
 similarly affected. The brighter planets and some of the stars 
 of the first magnitude generally become visible ; and sometimes 
 etars of the second magnitude are seen. The corona and the 
 rose-colored protuberances described in Art. 102 also make 
 their appearance. When the sun's disc has been reduced to a 
 narrow crescent, it sometimes appears as a succession of bright 
 points, separated by dark spaces. This phenomenon bears the 
 name of Baily's beads. The dark spaces are supposed to be 
 the lunar mountains, projected upon the sun's disc, and allowing 
 the disc to show between them. 
 
 Occasionally the moon's disc is faintly seen, shining with a 
 dusky light. This is caused by the rays of the sun, reflected 
 back to the moon by that portion of the earth's surface which 
 is still illuminated by the sun : just as at the time of new moon 
 its entire disc is rendered visible. 
 
 CYCLE AND NUMBER OF ECLIPSES. 
 
 162. Cycle of Eclipses. In order that either a solar or a 
 lunar eclipse shall occur, it is necessary, as we have seen, that 
 the moon shall be near the ecliptic (in other words, near the 
 line of nodes of its orbit), at either conjunction or opposition. 
 It is evident that when the moon is near the line of nodes at 
 such a time, the sun also must be near the same line. The 
 
NUMBER OF ECLIPSES. 143 
 
 occurrence of eclipses, then, depends on the relative situations 
 of the sun, the moon, and the moon's nodes, and is only pos- 
 sible when they are all in, or nearly in, the same straight line. 
 AVe have already seen (Art. 128) that the line of nodes is con- 
 tinually revolving to the west, completing a revolution in about 
 18.6 years. The sun, then, in its apparent path in the ecliptic 
 will move from one of the moon's nodes to the same node again 
 in less than a year. This interval of time may be called the 
 synodical period of the node, and is found to be 346.62 days. 
 
 Now, we have, 
 
 19 X 346.62d. = 6585.8d: 
 
 and, the lunar month being 29.53 days, we have also, 
 223 X 29.53d. = 6585.2d. 
 
 If, then, the moon is full and at its node on any day, it will 
 again be full, and at the same node, or very nearly at it, after 
 an interval of 6585 days : and the eclipses which have occurred 
 in that interval will occur again in very nearly the same order. 
 This period of 6585 days, or 18 years and 10 days, is called the 
 cycle of eclipses. It was known to the Chaldsean astronomers 
 under the name of Saros. Care must be taken not to confound 
 this cycle with the lunar cycle described in Art. 147. 
 
 163. Number of Eclipses. Since the limit of the moon's lati- 
 tude is greater in the case of a solar eclipse than in the case 
 of a lunar eclipse, there are more solar eclipses than lunar 
 eclipses. Usually 70 eclipses occur in a cycle, of which 41 
 are solar and 29 are lunar. Since we know that a solar eclipse 
 is inevitable when the moon is so near the line of nodes at 
 conjunction that its latitude is less than 1 23' 15", we can com- 
 pute the corresponding angular distance of the sun at the same 
 time from this line ; and having computed this, we may also 
 determine the length of time required by the sun in passing 
 through double this angle, or, in other words, the time required 
 in passing from one of these limits to the corresponding limit on 
 the other side of the same node. If we do this, we shall find 
 that the sun cannot pass either node of the moon's orbit without 
 being eclipsed : and therefore there must be at least two solar 
 eclipses in a year. The greatest number that can occur is five. 
 The greatest number of lunar eclipses in the year is three, and 
 
144 OCCULTATIONS. 
 
 there may be none at all. The greatest number of both kinds 
 of eclipses in a year is seven ; the usual number is four. 
 
 Although the annual number of solar eclipses throughout the 
 whole earth is the greater, yet at any one place more lunar eclipses 
 are-visible than solar. The reason of this is that a lunar eclipse, 
 when it does occur, is visible over an entire hemisphere, while 
 the area within which a solar eclipse is visible is very much 
 more limited. 
 
 OCCULTATIONS. 
 
 164. An occultation of a planet or a star will occur whenever 
 the planet or star is so situated in latitude as to allow the moon 
 to come in between it and the earth. In order to determine the 
 limit of a planet's latitude within which an occultation of the 
 planet is possible, let us refer to Fig. 61. In this figure, E is the. 
 
 A 
 
 Fig. 61. 
 
 centre of the earth, P that of a planet, and M that of the moon. 
 An occultajion will occur when the moon comes between the 
 tangent lines GB and AH. Let EC be the plane of the ecliptic. 
 PEC is then the geocentric latitude of the planet, and MEG 
 that of the moon. 
 
 We have, 
 
 PEC = PEG + GET) + DEM + MEC, 
 and also, 
 
 GET) = EDB EGD. 
 
 Now, PEG is the planet's semi-diameter, EGD its horizontal 
 parallax, DEM the moon's semi-diameter, EDB its horizontal 
 parallax, and MEC, as above stated, its latitude. The value 
 
OCCULT ATIONS. ". 15 
 
 of PEC, therefore, can very readily be obtained. If P, instead 
 of representing a planet, represents a star, the distance PE 
 becomes so great that AH and BG are sensibly parallel, and 
 the star's parallax and semi-diameter reduce to zero. In this 
 case the greatest value of PEC, within which an occupation 
 can occur, will be the sum of 5 20' 6", 61' 32", and 16' 45", 
 which is 6 38' 23". 
 
 Since the moon moves from west to east, the occultation 
 always takes place at its eastern limb. From new moon to full, 
 the dark portion of the moon is to the east, as may be seen in 
 Fig. 54, and from full moon to new, the bright limb is to the 
 east. When an occultation occurs at the dark edge, particu- 
 larly if the moon is so far on towards its first quarter that the 
 dark portion is invisible, the disappearance is extremely strik- 
 ing, as the occulted body appears to be extinguished without 
 any visible interference. 
 
 As already stated in Art. 83, a solar eclipse, or an occultation 
 of a planet or star, although not visible at different places at 
 the same absolute instant of time, may still be made the means 
 of very accurately determining the longitude of a place, or the 
 difference of longitude of two places. For instance, in the case 
 of a solar eclipse, we may deduce, from the local times of the 
 beginning and the end of the eclipse, as observed at any place, 
 the time of true conjunction of the sun and the moon: the time 
 of conjunction, that is to say, as seen from the centre of the earth. 
 If, then, we compare the local time of true conjunction with the 
 Greenwich time of true conjunction, it amounts to comparing 
 the local and the Greenwich timje corresponding to the same 
 absolute instant: and the difference of these two times will evi- 
 dently be the longitude of the place of observation from Green- 
 wich. 
 
 13 
 
146 
 
 TIDES. 
 
 CHAPTER XL 
 
 THE TIDES. 
 
 165. THE surface of the ocean rises and falls twice in the 
 course of a lunar day, the length of which is, as we have already 
 seen (Art. 143), about 24h. 50m. of mean solar time. The rise 
 of the water is called flood tide, and the fall 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. 
 
 166. Cause of the Tides. The tides are due to the inequality 
 of the attractions exerted by the moon upon the earth and the 
 waters of the ocean, and to a similar but smaller inequality in 
 the attractions exerted by the sun. 
 
 In order to examine the phenomena of the tides, we will con- 
 sider the earth to be a solid globe, surrounded by a shell of 
 water of uniform depth. The centrifugal force induced by the 
 rotation of the earth would tend to give a spheroidal form to 
 this shell of water; but as we wish simply to examine the effects 
 of the attractions exerted by the moon and the sun, we will dis- 
 regard the notation of the earth, and consider it to be at rest. 
 
 In Fig. 62, then. letABCD represent the earth, and the dotted 
 
 Fig. 62. 
 
 line GHIKihe surrounding shell of water. Let Mloe the moon. 
 The attraction of the moon on the solid mass of the earth is the 
 
TIDES. 147 
 
 same that it would be if the whole mass were concentrated at 
 the point E. Now since, by the law of gravitation, the at- 
 traction of the moon on any two particles is inversely as the 
 square of the distances of the two particles from the moon, the 
 attraction exerted upon the particle of water at G will be greater 
 than that exerted upon the general mass of the earth, supposed 
 to be concentrated at E. The particle G will therefore tend to 
 recede from the earth : that is to say, its gravity towards the 
 earth's centre will be diminished, although, as is plain, it will not 
 move. The same result will follow at the opposite point /. 
 The moon will exert a greater attractive power upon the mass 
 of the earth than upon a particle at /, and will tend to draw the 
 earth away from the particle: so that the gravity of the particle 
 at I towards the earth's centre will also be diminished. Since 
 the ratio of the distances ME and MG is very nearly equal to 
 the ratio of the distances MI and ME, the amount of the decrease 
 of gravity at G and at I will be nearly the same. 
 
 Let us next examine the effect of the moon's attraction at 
 some point L, not situated vertically under the moon. The at- 
 traction of the moon at this point is less than that at the point 
 G, since the distance ML is greater than the distance MG; and 
 since the attraction exerted on the mass of the earth is, of course, 
 the same for both points, the difference of the attractions exerted 
 on the earth and the water is less at the point L than at the 
 point G. At the point L, however, this inequality of attraction 
 is not wholly counteracted by gravity : for if the force with which 
 the moon tends to draw a particle at L along the line ML be re- 
 solved into two forces, one in the direction of the radius EL, 
 and the other in the direction of the tangent LT, the latter force 
 will cause the particle to move towards the point G. The same 
 result will follow at any other point of the arc HGK: so that 
 all the water in that arc tends to flow towards the point G, and 
 to produce high water there. 
 
 In the same way it may be shown that the water in the arc 
 HIK tends to flow towards the point /, and to produce high 
 water at that point. 
 
 The result, then, of the attraction of the moon, exerted under 
 the suppositions which we made at the outset, is to give to the shell 
 
148 
 
 TIUES. 
 
 N 
 
 of water a spheroidal form, as shown in the figure, the major 
 axis of the spheroid being directed towards the moon. Suitable 
 investigation shows that the difference of the major and the minor 
 semi-axis of this spheroid is about fifty-eight inches. 
 
 167. Daily Inequality of the Tides. The rotation of the earth, 
 and the inclination of the plane of the moon's orbit to the plane 
 of the equator, produce in general an inequality in the two daily 
 tides at any place. In order to show this, we will suppose that 
 the spheroidal form of the water is assumed instantaneously in 
 each new position of the earth as it rotates. In Fig. 63, let E 
 
 M be the centre of the 
 earth, surrounded, 
 as in Fig. 62, by 
 a spheroidal shell 
 of water, the trans- 
 verse axis of the 
 spheroid lying in 
 the direction of the 
 moon M. Let Pp 
 be the axis of rota- 
 tion of the earth, 
 and CD the equator. 
 The angle MED is 
 the moon's declination. The water is at its greatest height, as be- 
 fore, at the points A and B, and the height at other points dimi- 
 nishes as the angular distance of those points from the line GH 
 increases. Let / be a place having the same latitude that A has, 
 but situated 180 from it in longitude. The height of the tide 
 at /is represented by IK. In a little more than twelve hours 
 the rotation of the earth will have caused I and A to change 
 places with reference to the moon. I will then be where A is 
 in the figure, and will have a tide with the height A G, while 
 A will be where /is now, and will have a tide with the height 
 IK. It is not necessary to prove that IK is less than A G. We 
 see, then, that at both A and / the two daily tides are unequal, 
 the greater of the two occurring at each place at the time of the 
 moon's upper culmination at that place, or being, at all events, 
 the one which occurs next after that culmination. The same 
 
 Fig. 63. 
 
TIDES. 149 
 
 daily inequality of tides may be shown to exist at any other 
 points on the earth's surface, as, for instance, at L and 0. At 
 the equator, however, and also at the poles, the two daily tides 
 are sensibly equal, as may readily be seen from the figure. 
 
 168. General Laws. As far as the influence of the moon is 
 concerned in causing tides, the following general laws may be 
 deduced from what has been shown in the preceding articles. 
 
 (1.) When the moon is in the plane of the celestial equator, or, 
 in Fig. 63, when EM coincides with ED, the tides are greatest 
 at the equator, and diminish at other places as the latitude in- 
 creases ; and the two daily tides at any place are sensibly equal. 
 
 (2.) When the moon is not in the plane of the celestial equa- 
 tor, the two daily tides at any place except the poles and the 
 equator are unequal. The greatest tides, and the greatest ine- 
 quality of tides, occur at those places whose latitude is numeri- 
 cally equal to the moon's declination. If the place is on the 
 same side of the equator as the moon, the greater of the two 
 daily tides occurs at or next after the upper culmination of the 
 moon; if the place is on the opposite side of the equator, the 
 greater tide occurs at or next after the lower culmination of the 
 moon. 
 
 (3.) Owing to the retardation of the moon (Art. 143), there is 
 a similar retardation in the occurrence of high water. The 
 length of the lunar day being on the average 24h. 50m., the aver- 
 age interval of time between two successive tides is 12h. 25m. 
 
 169. Influence of the Sun in Causing Tides. All that has been 
 said in the preceding articles with regard to the influence of the 
 moon in creating tides is equally true with regard to the influ- 
 ence of the sun. The mass of the sun being so immense in 
 comparison with that of the moon, it might be supposed that 
 the influence of the sun over the tides would be greater than 
 that of the moon, even although its distance from the earth is 
 much greater than that of the moon. But such is not the case 
 in fact. The height of the tide produced by either body is not 
 so much due to the absolute attraction which that body exerts, 
 as to the relative attractions which it exerts on the solid mass of 
 the earth and on the water : and the moon is so much nearer to 
 the earth than the sun, that the difference of its attractions on 
 
150 PRIMING AND LAGGING. 
 
 the earth and the water is greatej than the corresponding dif- 
 ference in the case of the sun. It is computed that the effect 
 of the moon in creating tides is about 2i times that of the sun. 
 
 170. Combined Effects of both Sun and Moon. Since each 
 body, independently of the other, tends to raise the surface of the 
 water at certain points, and to depress it at certain other points, 
 the tides will evidently be higher when both bodies tend to raise 
 the surface of the water at the same time, than when one tends 
 to raise and the other to depress it. At new and full moon the 
 two bodies act together, while at the first and the last quarter 
 they act in opposition to each other. The tides at the former 
 periods will therefore be the greater, and are called spring tides ; 
 and the tides at the latter periods are called neap tides. The ratio 
 of the spring to the neap tide is that of (2i -f- 1) to (2 1), 
 or of 5 to 2. 
 
 The height of the tide is also affected by the change in the 
 distance of the attracting body. For instance, when the moon 
 is in perigee, the tides tend to run higher than when the moon is in 
 apogee; and when the moon is in perigee, and also either new or 
 full, unusually high tides will occur. 
 
 171. Priming and Lagging of the Tides. Each of these bodies 
 may be supposed to raise a tidal wave of its own, and the actual 
 high water at any place may be considered to be the result of 
 the combination of the two waves. When the moon is in its 
 first or its third quarter, the solar wave is to the west of the lunar 
 one, and the actual high water will be to the west of the place 
 at which it would have been had the moon acted alone. There 
 is therefore at these times an acceleration of the time of high water, 
 which is called the priming of the tides. In the second and the 
 fourth quarter, the solar wave is to the east of the lunar one, 
 and a retardation of the time of high water occurs, which is 
 called the lagging of the tides. 
 
 172. Although the theory of the tides, on the supposition that 
 the earth is wholly covered with water, admits of easy explana- 
 tion, the actual phenomena which they present are very much 
 more complicated, and must be obtained principally from obser- 
 vation. The lunar wave mentioned in the preceding article 
 being greater than the solar wave, we may consider the two to- 
 
ESTABLISHMENT. 151 
 
 gether to constitute one great tidal wave, which at every moment 
 tends to accompany the moon in its apparent diurnal path to- 
 wards the west, raising the waters at successive meridians, but 
 giving them little or no progressive motion. This tidal wave 
 would naturally move westward with an angular velocity equal 
 to that of the moon, so that at the equator its motion would he 
 about 1000 miles an hour ; but the obstructions offered by the 
 continents, the irregularity of their outlines, the uneven surface 
 of the ocean bed, and the action of winds and currents and fric- 
 tion, all combine not only to diminish the velocity of the tidal 
 wave, but also to make it extremely variable. 
 
 173. Establishment of a Port. The interval of time between 
 the moon's transit over any meridian and high water at that 
 meridian varies at different places, and varies also on different 
 days at the same place. This interval of time is called the 
 luni-tidal interval. The mean of the values of this interval on 
 the days of new and full moon is called the common establish- 
 ment of a port. The mean of all the luni-tidal intervals in the 
 course of the month is called the corrected establishment. These 
 establishments are obtained by observation, and are given in 
 Bowditch's Navigator, and also in other works. Thus the estab- 
 lishment of Annapolis is 4h. 38m., and that of Boston llh. 12m. 
 The time of transit of the moon over any meridian on any day 
 can be obtained from the Nautical Almanac: and the sum of 
 this time of transit and of the establishment of any port will be 
 the approximate time of that flood tide which occurs next after 
 the transit. Suppose, for instance, the time of high water at 
 Annapolis, on January 11, 1867, is desired. We have from the 
 Almanac the time of the moon's transit, 4h. 33m. ; adding to this 
 the establishment, 4h. 38m., the sum is 9h. llm. This is, in this 
 case, the time of the evening tide. The time of the morning tide 
 may be obtained by subtracting 12h. 25m. from this time, which 
 will give us 8h. 46m. A.M. A more accurate result in this 
 last case might be obtained by taking from the Almanac the 
 time of the preceding lower transit, and adding the establish- 
 ment to it; but practically this would be a needless refinement, 
 for the two results would only vary by about two minutes. 
 
 The time of transit which the Almanac gives is in astronomical 
 
152 COTIDAL LINES. 
 
 time: hence the resulting time of J^igh water will also be in astro- 
 nomical time, and it will frequently happen that the time which 
 we find, when turned into civil time, will fall on the civil day 
 subsequent to the day for which the time is desired. Take, for 
 instance, January 23, 1867, at Annapolis. The time of transit is 
 January 23, 15h. 34m. : hence the time of high water is Janu- 
 ary 23, 20h. 12m., or, in civil time, January 24, 8h. 12m. A.M. 
 If, then, we wish the time of high water on the morning of the 
 civil day January 23d, we must take from the Almanac the time 
 of transit for the astronomical day of January 22d. 
 
 There are other tables given in Bowditch's Navigator, and in 
 the United States Coast Survey Reports, by which the time of 
 high water can be obtained with greater accuracy than by the 
 method given above. 
 
 174. Cotidal Lines. If the tidal wave were everywhere uni- 
 form in its progress, it would come to all points on the same 
 meridian at the same time. But, owing to irregularities induced 
 by local causes, such is not the case, and places on different 
 meridians often have high water at the same instant of time. 
 Charts are therefore published on which are drawn lines con- 
 necting places where high tides occur at the same instant: and 
 these lines are called cotidal lines. These lines are usually ac- 
 companied by numerals, which indicate the hours of Greenwich 
 time at which high tides occur on the days of new and full moon 
 along the different lines. 
 
 175. Height of Tides. At small islands in mid-ocean the height 
 of the tides is not great, being sometimes less than one foot. When 
 the tidal wave approaches a continent, and the water begins to 
 shoal, the velocity of the wave is diminished, and the height of 
 the tide is increased. When the wave enters bays opening in 
 the direction in which the wave is moving, the height of the tide 
 is still further increased. 
 
 The eastern coast of the United States may be considered to 
 constitute three great bays : the first included between Cape Sable, 
 in Nova Scotia, and Nantucket, the second between Nantucket 
 and Cape Hatteras, and the third between Cape Hatteras and 
 Cape Sable, in Florida. In each of these bays the tides, in gene- 
 ral, increase in height from the entrance of the bay to its head. 
 
HEIGHT OF TIDES. 153 
 
 For instance, in the most southern of these bays, the tides at 
 Cape Sable and Cape Hatteras are not more than two feet in 
 height; while at Port Royal, at the head of this bay, they are 
 about seven feet. The same thing is noticed, in general, in 
 smaller bays and sounds. For instance, in Long Island Sound, 
 the height of the tide is two feet at the eastern extremity, and 
 more than seven feet at the western. This increase of height is 
 particularly noticeable in the Bay of Fundy, in which the height 
 is eighteen feet at the entrance, and fifty and sometimes seventy 
 feet at the head. 
 
 There are in some cases, however, special causes which create 
 exceptions to this general rule of increase of the tides between 
 the entrance and the head of a bay. In Chesapeake Bay, for 
 instance, which is wider at some places than it is at the entrance, 
 and which lies about north and south, the tides in general dimin- 
 ish in height as we ascend the bay. 
 
 176. Tides in Rivers. The same general principle holds good 
 in the tides of rivers. When the channel contracts or shoals 
 rapidly, the height of the tide increases: when it widens or 
 deepens, the height decreases. In a long river, then, the tides 
 may alternately increase and decrease. For instance, at Tivoli, 
 on the Hudson, between West Point and Albany, the tide is 
 higher than it is at either of those two places. 
 
 177. Different Directions of the Tidal Wave. The tidal wave 
 naturally tends to move towards the west ; but the obstructions 
 offered by the continents and the promontories, and the irregular 
 conformation of the bottom of the ocean, materially change the 
 direction of its motion. Sometimes its direction is even towards 
 the east. From a point about one thousand miles southwest of 
 South America there appear to start two tidal waves, which travel 
 in nearly opposite directions, one towards the west and the other 
 towards the east. 
 
 178. Four Daily Tides. At some places the tides rise and 
 fall four times in each day. This is ascribed to the existence of 
 two different tidal waves, coming from opposite directions. This 
 phenomenon occurs on the eastern coast of Scotland, where one 
 wave comes into the North Sea through the English Channel, 
 while a second wave comes in around the northern extremity of 
 
154 TIDES IN LAKES. 
 
 Scotland. At places where these two waves arrive at different 
 times, each wave will produce two daily tides. 
 
 179. Tides in Lakes and Inland Seas. If there is any tide in 
 lakes and inland seas, it is usually so slight as to be scarcely 
 measurable. A series of careful observations has demonstrated 
 the existence of a tide in Lake Michigan, which is at its height 
 about half an hour after the moon's transit. The average height 
 which it attains, however, is less than two inches. 
 
 180. Other Phenomena. Along the northern coast of the Gulf 
 of Mexico there is only one tide in the day, the second one being 
 probably obliterated by the interference of two waves. An ap- 
 proximation to this state of things is noticed on the Pacific coast, 
 where at times one of the daily tides has a height of several feet, 
 and the other a height of only a few inches. A very curious 
 statement is made by missionaries in reference to the tides at the 
 Society Islands. They say that the tides there are uniform, not 
 only in the height which they attain, but in the time of ebb and 
 flow, high tide occurring invariably at noon and at midnight : so 
 that the natives distinguish the hour of the day by terms de- 
 Bcriptive of the condition of the tide. 
 
 It is now generally admitted that one result of the friction of 
 the tides is a diminution of the velocity of the earth's rotation ; and it is 
 possible that the moon's secular acceleration (page 133) is partly due, not 
 to an increase in the moon's orbital velocity, but to this same diminution 
 of the earth's rotation. The amount of the diminution is, however, so very 
 small that all attempts to compute it have been thus far unsuccessful. 
 
PLANETS. 15. r ) 
 
 CHAPTER XII. 
 
 THE PLANETS AND THE PLANETOIDS. THE NEBULAR 
 HYPOTHESIS. 
 
 181. The Planets and their Apparent Motions. There are other 
 celestial bodies besides the sun and the moon, which, while they 
 share the common diurnal motion towards the west, appear to 
 change their relative positions among the stars. These bodies are 
 called planets, from a Greek word signifying wanderer. Some of 
 them are visible to the naked eye, and some only become visible 
 by the aid of a telescope. In some of them the change of position 
 among the stars becomes apparent from the observations of a 
 few nights : while in others even the annual change of position 
 is so small that they have for ages been considered to be fixed 
 stars.* 
 
 This change of position is determined, as it was determined in 
 the case of the sun and the moon, by observations of their right 
 ascensions and declinations. When such observations are made, 
 the apparent motions of the planets are found to be very irregular. 
 Sometimes they appear to move towards the east, and sometimes 
 towards the west, while at other times they appear to be station- 
 ary in the heavens. Such irregularity in the direction of their 
 motion is at once seen to be incompatible with the supposition 
 
 * The times of meridian passage of all the planets which ever become 
 conspicuously visible are given in the American Ephemeris. The altitude 
 which any planet has when on the meridian is obtained from the corre- 
 sponding zenith distance, and this is found at any pla^ce whose latitude 
 is known, from the formula 
 
 *=: d: 
 
 the declination being also given in the Ephemeris. It must be remembered 
 that (L d) becomes numerically (Zr + d) when L arid d have different 
 names ; that z in any case has the same name as (L d) ; and that when a 
 is north, the bearing of the body is south, a.nd conversely. 
 
156 PLANETS. 
 
 that they are, like the moon, satellites of the earth, revolving 
 about it as a centre. The next supposition that will naturally 
 be made is that they may revolve about the sun. 
 
 182. Heliocentric Parallax. In order to test the correctness 
 of this second supposition, we must first, from the apparent 
 motions which are observed from the earth, deduce the corre- 
 sponding motions which would be seen by an observer stationed 
 at the sun. Fig. 64 will serve to show how, by means of a body's 
 heliocentric parallax (Art. 56), the position which that body 
 would have if seen from the sun's centre in other words, its helio- 
 centric position may be determined from its geocentric position. 
 In this figure let S represent the sun, AB CE the earth's orbit, 
 the plane of which intersects the celestial sphere in the circle 
 
 Fig. 64. 
 
 , and P the position of a planet when projected upon the 
 plane of the ecliptic. Let the vernal equinox be supposed to lie 
 in the direction EV or SV, which two lines must be supposed 
 sensibly to meet when prolonged to the celestial sphere. Draw 
 the lines EP and SP. Since the distances of the planet from the 
 sun and the earth are finite, these lines will not lie in the same 
 direction, and the angle EPS which they make with each other 
 will be the difference of the directions in which the planet is seen 
 from the earth and the sun : in other words, the planet's helio- 
 
ORBITS OF THE PLANETS. 157 
 
 centric parallax in longitude. Through S draw SK parallel to 
 ED. The angle VEP, or its equal, VSK, is the planet's geo- 
 centric longitude, and is obtained by observation. The sum of 
 this angle and the angle EPS is the angle VSP, or the planet's 
 heliocentric longitude. Provided, then, we know the angle EPS, 
 we can readily obtain the angle VSP. Now, in the triangle PES 
 we know from Kepler's Third Law the ratio of the sides SP 
 and ES, which are the distances of the planet and the earth from 
 the sun; and the angle PES, the planet's angular distance from 
 the sun, or its elongation (Art. 56), can be obtained by observa- 
 tion. The angle EPS may then be readily computed. 
 
 By a similar method the planet's heliocentric latitude may be 
 determined from its geocentric latitude, and the heliocentric 
 place of a planet may thus be obtained at any time. 
 
 183. Orbits of the Planets. When the motions of the planets 
 as they would be seen from the centre of the sun are thus deduced 
 from their observed motions with reference to the earth, all the 
 apparent irregularities of motion disappear. The planets are 
 found to revolve from west to east in ellipses about the sun in 
 one of the foci, the eccentricity of the ellipses diminishing, as a 
 general rule, as the magnitude increases. The planes of the 
 orbits are found to be nearly coincident with the plane of the 
 ecliptic, and Kepler's and Newton's laws are exactly fulfilled in 
 the case of each planet. The line in which the plane of each 
 planet intersects the plane of the ecliptic is called the line of 
 nodes, and the terms perihelion and aphelion have the same 
 signification that they have in the case of the earth. 
 
 184. Inferior and Superior Planets. The planets are divided 
 into two classes : inferior and superior planets. The inferior 
 planets are those whose distances from the sun are less than the 
 distance of the earth from the sun, and whose orbits are therefore 
 included within the orbit of the earth. The inferior planets are 
 Mercury and Venus.* The superior planets are those whose 
 
 * There are some astronomers who are inclined to suspect the existence 
 of a third inferior planet, Vulcan, distant about 13,000,000 miles from the 
 sun. Two American observers are convinced that they saw such a planet 
 during the total solar eclipse of Julv 29, 1878. 
 14 
 
158 
 
 INFERIOR PLANETS. 
 
 distances from the sun are greater than that of the earth, and 
 whose orbits therefore include the orbit of the earth. The 
 superior planets are Mars, Jupiter, Saturn, Uranus, and Neptune. 
 There is besides these a group of small planets, called minor 
 planets, planetoids, or asteroids, situated between Mars and 
 Jupiter. Up to October, 1878, 192 of these minor planets had 
 been discovered. The earth is also a planet, lying between 
 Venus and Mars. It is therefore a superior planet to Mercury 
 and Venus, and an inferior planet to the other planets. Its 
 sidereal period is greater than the periods of the inferior planets, 
 and less than those of the superior planets. 
 
 INFERIOR PLANETS. 
 
 185. In Fig. 65 let S represent the sun, pp'p"p"" the orbit 
 of an inferior planet, the plane of which is supposed to coincide 
 
 Fig. 65. 
 
 with the plane of the ecliptic, ABDEt\\Q orbit of the earth, and 
 the circle KGH the intersection of the plane of the ecliptic with 
 the celestial sphere. Suppose the earth to be at E. When the 
 planet is at P, between the earth and the sun, or at P"' 9 on the 
 opposite side of the sun to the earth, it has the same geocentric 
 
DIRECT AND RETROGRADE MOTION. 159 
 
 longitude as the sun, and is in conjunction with it. The position 
 at P is called the inferior conjunction, and that at P'" the superior 
 conjunction. 
 
 The greatest angular distance of the planet from the sun will 
 evidently occur when the line connecting the planet and the earth 
 is tangent to the orbit of the planet ; that is to say, when the planet 
 is at P" or P"". The position at P" is called the greatest western 
 elongation of the planet, that at P"" the greatest eastern elongation 
 of the planet. We have already seen in Art. 92 that the rela 
 tive distances of the planet and the earth from the sun are at 
 once obtained when we have measured the greatest elongation. 
 
 186. Direct and Retrograde Motion. The motion of the infe- 
 rior planets is always in reality from west to east, or direct, as it 
 is called; but when the planet is near its inferior conjunction, 
 its motion is apparently from east to west, or retrograde. This 
 apparent retrograde motion is explained in Fig. 65. Let the 
 planet be at its inferior conjunction at P, and let both the earth 
 and the planet move on about the sun in the direction EABD. 
 The angular and the linear velocity of the planet about the sun 
 being greater than they are in the case of the earth, when the 
 earth arrives at E', the planet will be at some point P', and will 
 lie in the direction E'G; the sun, on the other hand, will lie in 
 the direction E'S'. While, then, the earth is advancing from E 
 to E', the sun and the planet appear to move away in opposite 
 directions from the point (7, on which both were projected when the 
 earth was at E. But the apparent motion of the sun is invariably 
 towards the east ; hence the planet has apparently moved tow'ards 
 the west. It may also be shown that the same apparent retro- 
 grade motion occurs when the planet is approaching inferior 
 conjunction, and is within a short distance of it. 
 
 187. Stationary Points. When the planet is at P"", it is 
 moving directly towards the earth in the direction P""E, and 
 the motion of the earth in its orbit gives the planet an apparent 
 motion in advance. The same must also be the case when the 
 planet is at P". Since then the motion of the planet is direct 
 at the greatest elongations, and retrograde at inferior conjunction, 
 there -must be a point in the orbit between inferior conjunction 
 and each elongation at which the planet neither advances nor 
 
160 ELEMENTS OF A PLANET'S ORBIT. 
 
 recedes, but appears stationary 4n the heavens. These points 
 are called the stationary points. 
 
 At all other parts of the orbit except those which have been 
 discussed, the apparent motion of the planet is direct ; but the 
 velocity with which it moves is subject to great variation. It 
 was on account of this irregularity, both in the direction and the 
 amount of their apparent motions, that these bodies were called 
 wandering stars by the ancient Greeks. 
 
 188. Evening and Morning Stars. Except at the times of 
 conjunction, an inferior planet is either to the east or to the west 
 of the sun. When it is to the east of the sun it will set after 
 the sun has set, and when it is to the west of the sun it will rise 
 before the sun has risen. In certain parts of its orbit the planet's 
 elongation from the sun is sufficiently great to carry the planet 
 beyond the limits of twilight, or 18 (Art. 101); it will then be 
 an evening star if to the east of the sun, and a morning star 
 if to the west. It is only to Venus, however, that these terms 
 are commonly applied, at least so far as the inferior planets are 
 concerned : since Mercury is so near to the sun that it is seldom 
 visible, and even when it is visible, it appears like a star of only 
 the third or the fourth magnitude. 
 
 189. Elements of a Planet's Orbit. In order to compute the 
 position in space at any time of either an inferior or a superior 
 planet, we must be able to determine: 
 
 1st. The relative position of the plane of the planet's orbit to 
 the plane of the ecliptic ; 
 
 2d. The position of the orbit itself in the plane in which it 
 lies; 
 
 3d. The magnitude and the form of the orbit; and 
 
 4th. The position of the planet in its orbit. 
 
 These four conditions require the knowledge of seven distinct 
 quantities. The first condition is satisfied if we know (1) the 
 position of the line in which the plane of the orbit intersects the 
 plane of the ecliptic, or, what amounts to the same thing, the 
 longitude of the planet's nodes, and (2) the inclination of the 
 two planes to each other. The second condition is satisfied if 
 we know (3) th longitude of the perihelion. The third condi- 
 tion is satisfied if we know (4) the semi-major axis, or the 
 
LONGITUDE OF THE NODE. 
 
 161 
 
 ]>lauet's mean distance from the sun, and (5) the eccentricity of 
 the orbit. Finally, the last condition is satisfied if we know 
 (6) the time in which it makes one complete revolution about 
 the sun, or its periodic time) and (7) the time when the planet 
 is at some known place in its orbit, as, for instance, the peri- 
 helion. These seven quantities are called the elements of the orbit. 
 190. Heliocentric Longitude of the Node. A planet is at its 
 nodes when its latitude is zero; and if the heliocentric longitude 
 of the planet at that time can be determined, it will also be the 
 heliocentric longitude of the node, since the line of nodes of 
 every planet passes through the sun. But the heliocentric longi- 
 tude of a planet when at its node differs from the geocentric 
 longitude (which may be obtained directly from observation), 
 excepting only in case the earth itself happens at that time to 
 be on the line of nodes. . We must therefore be able to deduce 
 the heliocentric longitude of the planet from the geocentric. 
 When the planet's distance from the sun is known, this can be 
 done by the method explained in Art. 182; and when this dis- 
 tance is not known, the following method can be used. 
 
 N 
 
 In Fig. 66, let 8 be the sun, POHK the orbit of a planet, 
 CDEE' the orbit of the earth, and NM the line of nodes of the 
 
162 INCLINATION OF A PLANET'S ORBIT. 
 
 planet. Let the vernal equinox lie in the direction EV or SV. 
 Let E be the position of the earth when the planet is on the line 
 of nodes at P. The elongation of the planet, or the angle PES, 
 and the geocentric longitude of both planet and node, or the 
 angle VEP, can be obtained by observation. Suppose both 
 planet and earth to move on in their orbits, and the earth to be 
 at E' when the planet again reaches the same node, and let the 
 planet's elongation at this time, or the angle SE'P, be observed. 
 Now, since the earth's orbit, although represented in the figure 
 by a circle, is raally an ellipse, ES and E'S will not in general 
 be equal to each other. The value of each, however, can be 
 readily obtained from the solar tables. The same tables will 
 also give us the angle ESE', which is the angular advance of the 
 earth in its orbit in the interval. In the triangle ESE', then, 
 knowing two sides and the included angle, we can compute the 
 sid EE', and the angles SEE' and SE'E. The angles PES 
 and PE'S having been obtained by observation, we can find the 
 angles PEE' and PE'E. Then in the triangle PEE', knowing 
 two angles and the included side, we can compute the side EP. 
 Finally, in ths triangle PES, knowing an angle and the two in- 
 cluding sides, we can obtain PS, or the planet's distance from 
 the sun, and the angle EPS, thi planet's heliocentric parallax, 
 from which, and the planet's observed geocentric longitude, we 
 can obtain the planet's heliocentric longitude, as in Art. 182. 
 This is, as we have already seen, the heliocentric longitude of 
 the node. 
 
 The nodes of every planet are found to have a westward 
 movement, similar in character to the precession of the equi- 
 noxes. It is a very slight movement, however, being only 70' a 
 century in the case of Mercury, and being less than that in the 
 case of the other planets. 
 
 191. Inclination of the Planet's Orbit to the Ecliptic. The 
 line of nodes of any planet being, as we have just now seen, a 
 nearly stationary line in the plane of the ecliptic, the earth 
 must pass it once in very nearly six months in its revolution 
 about the sun. The inclination of the plane of any planet's 
 orbit to the plane of the ecliptic may be determined by obser- 
 vations made when the earth is on the line of nodes. In Fig. 
 
PERIODIC TIME. 163 
 
 67 let E be the earth, and EN the line 
 
 of nodes of a planet. Let AEN be the 
 
 plane of the ecliptic, and P the position 
 
 of a planet projected on the surface of 
 
 the celestial sphere. With EP as a 
 
 radius, let the arc PA be described, per- 
 
 pendicular to the plane of the ecliptic, ri s- 67 - 
 
 and also the arcs PN and NA. In the spherical triangle PNA, 
 
 right-angled at A, the arc PA, which measures the angle PEA, 
 
 is the geocentric latitude of the planet; AN, or the angle AEN, 
 
 is the difference between the geocentric longitudes of the planet 
 
 and the node; and the angle PNA is the inclination of the plane 
 
 of the orbit to the plane of the ecliptic. In the triangle PNA, 
 
 we have, 
 
 tan PA 
 
 It may be noticed in this formula that, since the sine of a small 
 angle varies more rapidly than the sine of a large angle, an. 
 error in AN will affect the result the less, the greater AN itself 
 happens to be at the time of observation : that is to say, the 
 farther the planet is from the node. 
 
 192. The Periodic Time of an Inferior Planet The time in 
 which an inferior planet makes one complete revolution about 
 the sun, or its periodic time, may be found by taking the interval 
 of time between two successive passages of the planet through 
 the same node. The accuracy of this method is however dimin- 
 ished by the small inclination (less than 7) of the planes of the 
 orbits to the plane of the ecliptic, which renders it difficult to 
 determine the instant when the latitude of the planet is zero. 
 It is found to be better to determine the planet's synodical period, 
 or the interval between two successive conjunctions of the same 
 kind, and from it to compute the sidereal period. The condi- 
 tions of this problem, and the method by which it is solved, are 
 identical with those in the case of the synodical and the sidereal 
 period of the moon, Art. 141. The formula there given was, 
 
 ST 
 
 ~8+ T' 
 In applying this to the case of an inferior planet, F and 3 
 
164 MERCURY. 
 
 denote the sidereal and the synodical period of the planet, and 
 T denotes, as before, the sidereal*year. 
 
 Instead of using the interval between two conjunctions as 
 the synodical period, we may take the interval between two 
 greatest elongations of the same kind. A very accurate mean 
 synodical period is obtained by taking two elongations separated 
 by a long interval, and dividing this interval by the number 
 of synodical periods which it contains. 
 
 193. It is hardly necessary to describe the methods by which 
 the other elements given in Art. 189 are determined. It will 
 readily be understood that the distance of the planet from the 
 sun, obtained as in Art. 190, or by Kepler's Law, will enable 
 us to obtain the form and the magnitude of the ellipse in which 
 the planet moves. The method by which the longitude of the 
 perihelion is obtained, although not intrinsically difficult, is too 
 elaborate for this work. Finally, when the longitude of the 
 perihelion is obtained, the time of the planet's perihelion passage 
 may evidently at once be determined. The perihelion of Venus 
 has a very minute retrograde motion : the perihelia of .all the 
 other planets have an eastward motion, similar to that of the 
 earth's line of apsides. 
 
 MERCURY. 
 
 194. Mercury revolves about the sun at a mean distance of 
 about 36,000,000 miles, the eccentricity of its orbit being about 
 4th. Its synodical period is about 116 days, and its sidereal 
 period 88 days. Its real diameter is about 3000 miles.* Its 
 mass is a matter of considerable uncertainty, and quite different 
 values of it are given by different astronomers. It may be 
 assumed to be approximately equal to y^th of the mass of the 
 earth. 
 
 The greatest elongation of Mercury is only about 28 20' : 
 
 * The distances and the diameters of all the celestial bodies except the moon 
 depend for their accuracy upon the accuracy with which the solar parallax 
 is determined. An error of I" in this parallax would affect the sun's dis- 
 tance from us to the amount of several millions of miles, and propor- 
 tionally also the distances and the diameters of the other celestial bodies ; 
 the values given must therefore be considered to be only approximate. 
 
VENUS. 165 
 
 anrl hence the planet is rarely visible to the naked eye, and is 
 never a conspicuous object in the heavens. It is not generally 
 believed that it has any atmosphere. It exhibits phases similar 
 to those of the moon, and due to the same causes. It is asserted 
 by some observers that it rotates on an axis in about 24 hours, 
 though the truth of the assertion is by no means unchallenged. 
 If it has any compression, it is extremely small. 
 
 VENUS. 
 
 195. The mean distance of Venus from the sun is about 
 67,000,000 miles, its synodical period is 584 days, and its sidereal 
 period 225 days. The eccentricity of its orbit is small, being 
 only about T 7 oths. Venus is nearly as large as the earth, its 
 diameter being 7,600 miles. Its mass is about Jths of the mass 
 of the earth. It has no perceptible compression. 
 
 The greatest elongation of Venus from the sun amounts to 
 about 47 15', and hence it is often visible as an evening or a 
 morning star. At certain times its brightness is so great that 
 it can be seen in broad daylight with the naked eye, while at 
 night shadows are cast by the objects which it illuminates. It 
 exhibits phases similar to those of Mercury. 
 
 It seems to be generally admitted that Venus has an atmos- 
 phere the density of which is not very different from that of 
 the earth's atmosphere. Observations of spots upon the disc 
 go to show that the planet rotates upon an axis in a period of 
 about 23^ hours; but this conclusion is by no means certain. 
 
 The existence of a satellite of Venus was formerly suspected, 
 but no satellite was seen at the transit in December, 1874. 
 
 196. Transits of Venus. We have already seen (Art. 93) 
 how a transit of Venus across the sun's disc is employed in de- 
 termining the distance of the earth from the sun. If the plane 
 of the orbit of Venus coincided with the plane of the ecliptic, 
 a transit would occur whenever the planet came into inferior 
 conjunction, or once in every 584 days. Owing to the inclina- 
 tion of the planes to each other, however, it is evident that at 
 the time of inferior conjunction the planet m^y have too great a 
 latitude to touch any part of the sun's disc. Now the phenomenon 
 of a transit of Venus is analogous to a solar eclipse, and there- 
 
166 TRANSITS OF VENUS. 
 
 fore if in Fig. 60 we suppose M to be Venus, the formula ob- 
 tained in Art. 160 will apply equally well to the limits of the 
 geocentric latitude of Venus within which a transit is possible. 
 These limits are the sum of the semi-diameters of the sun and 
 the planet and of the parallax of the planet, diminished by 
 the parallax of the sun. The greatest value of the limits will 
 be found to be about 17' 49". When, therefore, the latitude of 
 Venus is more than 17' 49" at the time of inferior conjunction, 
 no transit will occur : and as Venus in every sidereal revolution 
 attains a latitude of over 3 23', it is at once evident that a 
 transit is only a rare occurrence. 
 
 197. Intervals between Transits. Since the latitude of Venus 
 is so small when a transit occurs, it is plain that the planet must 
 be either at or very near one of its nodes. Now, let us suppose 
 that Venus is at its node at the time of inferior conjunction, 
 under which circumstances a transit will, of course, take place. 
 The sidereal period of Venus is 224.7 days. Now, we have, 
 
 224.7d. X 13 = 2921.11d.; and, 
 
 365.256d. X 8 = 2922.05d. 
 
 At the end of eight years, then, Venus will be very near the 
 same node at the time of inferior conjunction, and a transit will 
 probably occur. Again, we have, 
 
 224.7d. X 382 = 85835.4d. 
 365.256d. X 235 = 85835.16d. ; 
 
 so that transits at the same node also occur every 235 years. In 
 the same way transits may also occur at the other node: and the 
 intervals between transits at either node are found to be 8, 105<}, 
 8, 12H, 8, &c. years.* The longitude of the ascending node of 
 Venus is 75 20', and a transit at that node must occur, when it 
 occurs at all, at the time when the earth is near that point, which 
 is about the 6th of June. For a similar reason transits at the 
 descending node occur about the 6th of December. 
 
 The last two transits occurred in June, 1769, and December, 
 1874. The next two will occur in December, 1882, and June, 
 2004. [See Table VIII., Appendix] 
 
 1 
 
 * Thus the years of transits at the ascending node are 1761, 1769, and 
 2004: at the descending noJe, 10^9, 1874, and 1882. 
 
SUPERIOR PLANETS. 
 
 167 
 
 SUPERIOR PLANETS. 
 
 198. The superior planets are, as they have already been de- 
 fined to be, planets whose orbits include the orbit of the earth. 
 They have, like the inferior planets, superior conjunction, but 
 can evidently have no inferior conjunction. Their elongation 
 from the sun, eastern and western, can have all values between 
 and 180. When their elongation is 90, they are said to be in 
 quadrature, and when it is 180, in opposition. Much that has al- 
 ready been said in this chapter in reference to the inferior planets 
 is equally true in reference to the superior planets. The elements 
 of the orbits are in both instances the same, and so are the me- 
 thods by which the heliocentric longitudes of the nodes and the 
 inclinations of the orbits are determined. In some other points 
 there is a difference between the two classes of planets; and these 
 points we shall now proceed to examine. 
 
 199. Retrograde Motion. The superior planets, like the infe- 
 rior planets, have at times an apparent retrograde motion, which 
 
 N 
 
 Fig. 68. 
 
 occurs at or near the time of opposition. The explanation of 
 this retrogradation will be seen by a. reference to Fig. 68. S is 
 
168 PERIODS OF A SUPERIOR PLANET. 
 
 the sun, EE'E"E" r the orbit of. the earth, and C GDP the orbit 
 of a superior planet, the plane of which is supposed to coincide 
 with the plane of the ecliptic, and to meet the celestial sphere in 
 the circle ANBM. Let the earth be at E, and the planet at P, 
 180 in geocentric longitude from the sun. The planet will ap- 
 pear to be projected upon the celestial sphere at the point M. 
 Let both earth and planet revolve in their orbits in the direction 
 indicated by the arrow. When the earth has reached the point 
 E r , the planet, whose angular velocity is less than that of the earth, 
 will have reached some point P', and will lie in the direction E'M': 
 in other words, it has apparently retrograded. If, on the other 
 hand, the earth is at E" and the planet at P, or in superior con- 
 junction, the apparent motion of the planet is at that point 
 direct, and its angular velocity appears to be greater than it 
 really is ; for when the earth is at E" r , and the planet at P', the 
 latter, having in reality moved through the arc MM" since con- 
 junction, appears to have moved through the arc MM'". 
 
 The apparent motion, then, of a superior planet is direct in 
 all cases except when it is at or near its opposition. The ap- 
 parent motion of an inferior planet has been shown to be retro- 
 grade at and near the time of inferior conjunction (Art. 186). 
 Now, since the earth is a superior planet to an inferior planet, 
 and an inferior planet to a superior planet, we see, by comparing 
 the two cases, that the retrograde motion occurs in each class 
 of planets at and near the time when the inferior planet conies 
 between the sun and the superior planet. 
 
 The stationary points in the orbit of a superior planet are 
 identical in character with those in the orbit of an inferior 
 planet, and occur when the retrograde motion is changing to the 
 direct motion, or the direct to the retrograde. 
 
 200. Synodical and Sidereal Periods. The synodical period 
 of a superior planet is the interval of time between two suc- 
 cessive conjunctions or two successive oppositions. When a planet 
 is in conjunction with the sun, it is above the horizon only in 
 the day time; but when it is in opposition, it is above the horizon 
 during the night, and can therefore be readily observed. Hence 
 in obtaining the synodical period it is better to employ the in- 
 terval of time between two oppositions: and by determining the 
 
DISTANCE FROM THE SUN. 1H!) 
 
 times of two oppositions which are not consecutive, and div^ling 
 the interval between them by the number of synodical revo- 
 lutions which it contains, we may obtain a mean value of tK*? 
 synodical period. By using times of opposition which were ob- 
 served and recorded before the Christian era, a very accurate- 
 value of the mean synodical period may be obtained. 
 
 The method of deducing the periodic time from the synodica-' 
 period is the same that was used in the case of the inferio; 
 planets (Art. 192), with the important exception that in the 
 present case it is the earth that gains 360 upon the planet in the 
 course of a synodical revolution, and not the planet that gains it 
 upon the earth. If, therefore, we denote the periodic times of 
 the earth and the planet by Taud P, and the synodical period of 
 the planet by S, we shall have (Art. 141), 
 360 _ 360 _ 36<T 
 T JP "IT 
 
 8 T ' 
 
 which gives the value of any planet's sidereal period in terms of 
 its synodical period and the sidereal year. 
 
 201. Distance of a Superior Planet from the Sun. The distance 
 of a superior planet from the sun may be obtained by the method 
 of Art. 190: or it may be obtained from observations made at 
 the time it is in opposition. In 
 Fig. 69, let S be the sun, E the 
 
 earth, and P a superior planet 
 at the time of opposition, the 
 planes of the two orbits being p 
 
 supposed to coincide. At the 
 
 end of a short interval, let the earth have moved to E f , and the 
 planet to P'; the angle E' OE will be the amount of the apparent 
 retrogradation of the planet in that time. The periods of the 
 earth and the planet being known, we can compute the angular 
 advance of each planet in the given interval, thus obtaining the 
 angles E'SE and P'SP. The radius vector of the earth's orbit, 
 SE', can be found from the Nautical Almanac. Then, in the 
 triangle E'SP', we know the side E'S, the angle E'SP', which is 
 the angular gain of the earth on the planet, and the angle E'P'S, 
 
170 MARS. 
 
 which is the sum of P'SO, or tjie advance of the planet, ami 
 P' OS, or the apparent retrogradation of the planet. We can 
 therefore compute the side SP', which is the distance required. 
 
 If we suppose the real distance of the sun from the earth not 
 to be known, this method will still give us the ratio between 
 this distance and the distance of the planet from the sun. And 
 furthermore, if we can determine the real distance of the planet 
 from the earth by observations of its parallax at the time of 
 opposition (when it is nearest to the earth), by obtaining its 
 displacement in right ascension when far east and far west of the 
 same meridian, we can readily obtain the real distance of the 
 earth from the sun. Such observations have been made upon 
 the planet Mars, and the distance of the earth from the sun 
 has been deduced, as already given (Art. 94). 
 
 202. Evening and Morning Stars. The angular velocity of a 
 superior planet towards the east is less than that of the earth, 
 and consequently also less than the sun's apparent angular 
 velocity in the same direction. After conjunction, therefore, the 
 planet will lie to the west of the sun, and its elongation will 
 continually increase. When this elongation exceeds about 30, 
 the planet will begin to be visible as a morning star, and will 
 so continue until it has fallen 180 to the west of the sun, and 
 is in opposition, It will then rise about sunset and set about 
 sunrise. After the time of opposition it will lie more than 180 
 to the west of the sun, or, what is the same thing, less than 180 
 to the east of it, and will rise before sunset. It will therefore 
 
 jvenjng star from opposition to conjunction. 
 
 MARS. 
 
 203. The synodical period of Mars is 780 days, and its side- 
 real period 687 days. Its mean distance from the sun is 
 141,000,000 miles, and the eccentricity of its orbit about -Jyth. 
 Its diameter is 4,100 miles. Different values are given to its 
 compression, varying between the limits of -g^th and ^th. Its 
 mass is about Jth of that of the earth. It has two very small 
 satellites, discovered by Prof. A. Hall, U. S. Navy, in 1877. 
 
 204. Phases. At opposition and conjunction the same hemi- 
 sphere is turned towards both the sun and the earth, and con- 
 
MINOR PLANETS. 171 
 
 sequently the planet appears full. At quadrature it appears 
 slightly gibbous. It is the only one of the superior planets 
 which exhibits any sensible phases, excepting possibly Jupiter. 
 
 Mars shines with a red light, and at opposition is a very con- 
 spicuous object, sometimes equalling Jupiter in brilliancy. 
 
 205. Rotation, &c. When examined in a telescope, the sur- 
 face of Mars is seen to be covered with patches of a dull red- 
 dish color, which are supposed to be land, interspersed with 
 spots of a bluish or greenish hue, which are supposed to be 
 water. By observation of these spots Mars is found to rotate 
 upon an axis once in about 24* hours. The axis of rotation 
 is inclined at an angle of 61 18' to the plane of the ecliptic, 
 and hence there must be a change of seasons not very different 
 from the change which takes place on the earth. White spots 
 are seen near the poles, which decrease in the Martial summer 
 and increase in the winter. These spots are supposed to be snow. 
 The existence of an atmosphere of a moderate density is gene- 
 rally admitted by astronomers. 
 
 THE MINOR PLANETS. 
 
 206. Bode's Law. In 1778 the astronomer Bode, of Berlin, 
 announced (though he did not discover) the following curious 
 relation between the distances of the different planets from the 
 sun. The statement of this relation usually goes by the name 
 of "Bode's law." If we take th'e series of numbers 
 
 0, 3, 6, 12, 24, 48, 96, 192, 384, 
 
 each of which, except the second, is double the preceding one, 
 and add 4 to each of these numbers, the resulting series, 
 
 4, 7, 10, 16, 28, 52, 100, 196, 388, 
 
 will approximately represent the relative distances of the planets 
 from the sun.* Thus Mercury is 36,000,000 miles from the 
 tiun, and Venus 67,000,000 miles; and these distances are to each 
 other nearly in the ratio of 4 to 7. There was however (the 
 
 * The last two numbers were not in the series as originally announced 
 by Bode, since Uranus and Neptune had not then been discovered. The 
 real distance of Neptune is one-fourth less than it should be, if this law 
 were anything more than a coincidence. 
 
172 MINOR PLANETS. 
 
 minor plants Demg tken undiscovered) a break in the series, 
 there being no planet corresponding to the number 28; and 
 Bode ventured to predict that another planet might be found to 
 exist at that point of the series: that is to say, between Mars and 
 Jupiter. A similar prediction was made by Kepler, about the 
 beginning of the seventeenth century. 
 
 207. Discovery of the Minor Planets. In 1800, six European 
 astronomers formed an association for the express purpose of 
 searching the heavens for new planets; and within the next six 
 years four minor planets were discovered. These were named 
 Ceres, Pallas, Juno, and Vesta. No more were discovered until 
 the end of 1845, but since that time some have been discovered 
 in nearly every year. The number discovered up to October, 
 1878 (including Ceres, &c.), was 192. 
 
 The mean distances of these bodies from the sun vary from 
 200,000,000 to 300,000,000 miles. They are all very small, 
 the largest being probably not over 300 miles in diameter, and 
 many of the others being too small to admit of measurement. 
 Vesta is the only one which is ever visible to the naked eye, and 
 its visibility is very rare. Some of the others are so small that 
 they can scarcely be seen with the strongest telescope, even at 
 opposition. Their total mass is about one-third of the earth's. 
 
 The French astronomer Leverrier has concluded that the 
 mass of these minor planets is by no means sufficient to produce 
 the perturbations in the orbit of Mars and in that of Jupiter 
 which are believed to be due to the attractions of this group. 
 It is therefore extremely probable that many other planets, 
 hitherto undiscovered, belong to the same cluster. 
 
 208. Olbers's Theory. Shortly after the discovery of the first 
 four minor planets, Dr. Olbers advanced the theory that these 
 planets were fragments of a single planet, which had been broken 
 in pieces by volcanic action or by some other internal force. This 
 theory is still in favor with many astronomers ; others, however, 
 object to it on the ground that if these bodies did formerly con- 
 stitute one single body, their orbits ought to have a common 
 point of intersection, which is very far from being the case. There 
 are, however, certain striking resemblances in their orbits, which 
 seem to argua something common in their origin. 
 
JUPITER. 173 
 
 JUPITER. 
 
 209. Jupiter is the largest planet of our system. At times it 
 surpasses Venus in brilliancy, and even casts a shadow. It re- 
 volves about the suflTat a mean distance of 481,000,000 miles. 
 The eccentricity of its orbit is about 2 ' th. Its synodical period is 
 399 days, and its sidereal period 4333 days, or about 11.9 years. 
 Its diameter is about 89,000 miles, and its volume is about 1400 
 times that of the earth. It rotates on an axis in a little less 
 than ten hours, and has a compression of -j^th. Its phases are 
 so slight as to be scarcely perceptible. 
 
 210. Belts. When examined through a telescope, the disc of 
 Jupiter is seen to be streaked with dark belts, lying nearly 
 parallel to the plane of its equator. With powerful telescopes 
 these belts are found to have a gray or brown tinge. They are 
 sometimes nearly permanent for several months, and sometimes 
 they change their shape materially in the course of a few minutes. 
 There are usually one broad and several narrower belts on each 
 side of Jupiter's equator. 
 
 It is generally supposed that Jupiter is s.urrounded by a dense 
 atmosphere, and that these belts are fissures in this atmosphere, 
 through which the dark body of the planet is seen. The distri- 
 bution of the atmosphere in lines so nearly parallel to the equator 
 is supposed to be due to currents in the atmosphere, similar in 
 character to our trade-winds, but having a more decided easterly 
 and westerly tendency, from the more rapid rotation and the 
 greater size of the planet. A point on Jupiter's equator rotates 
 with a velocity of about 28,000 miles an hour, while a point on 
 our own equator rotates with a velocity of only about 24,000' 
 miles a day. 
 
 211. Satellites. Jupiter is attended by four satellites or moons, 
 revolving about it from west to east. They are distinguished' 
 from each other by the numbers 1., II., III., and IV., the first sat- 
 ellite being the nearest to Jupiter. The second satellite is about 
 as large as our moon, and the others are somewhat larger. They 
 are not usually visible to the naked eye, though a few instances 
 to the contrary are on record. The distance of the first satellite 
 from Jupiter is 270,000 miles, and that of the fourth is 1,200,000 
 
,1/S. SATELLITES OF JUPITER. 
 
 miles. The first revolves about Jkipiter in a period of 42 hours, 
 a/id thu fourth in a period of 16d. 17h. 
 
 212. Phenomena Presented by the Satellites. The satellites, in 
 the course of their revolution about their primary, present four 
 distinct classes of phenomena, which are shown in Fig. 70. In 
 this figure let S be the disc of the sun, and EE'E"E'" the orbit 
 
 Fig. 70. 
 
 of the earth. Let J be Jupiter, and ABDG the orbit of one of 
 its satellites. Since the planes of all the orbits very nearly 
 coincide with the plane of the ecliptic, we may consider ABDG 
 to lie in that plane. Suppose the earth to be at E, and, in order 
 to simplify the case, suppose it also to remain at that point 
 during the short time required by the satellite to revolve about 
 Jupiter. 
 
 An eclipse of the satellite will occur when it passes through 
 the arc MN, since it is then within the shadow formed by lines 
 drawn tangent to the disc of the sun and that of Jupiter. It may 
 readily be calculated that the length of the shadow, from J to C\ 
 is about 55,000,000 miles, so that the shadow extends far beyond 
 the orbit of the fourth satellite. In extremely rare cases this 
 satellite, owing to the inclination of its orbit to the ecliptic, may 
 fail to be eclipsed. 
 
 An ocGultation of the satellite will occur when it passes through 
 the arc AB, since it is then within the cone formed by lines 
 drawn from E tangent to the disc of Jupiter. 
 
 A transit of the shadow will occur when the satellite passes 
 through the arc GH, its shadow being then cast upon the disc 
 of Jupiter, and moving across it as a small round spot. 
 
VELOCITY OF LIGHT. 175 
 
 , a transit of the satellite will occur when it passes 
 trough th3 arc KL. 
 
 It will evii^ntly depend on the relative situations of the sun, 
 the eaxth, anii Jupiter, whether all these phenomena will be 
 observed or not When the earth, for instance, is at E' or E'" f 
 it is plain that 01. ly an occultation and a transit will occur. 
 
 The relative situations of the satellites to each other and to 
 their primary are constantly changing. Sometimes all are on 
 the same .side of the primary : sometimes only one is visible, 
 and sometimes, though very rarely, all are invisible. The times 
 at which these different phenomena will occur are computed 
 beforehand, and are given in the American Ephemeris, the time 
 used being that of the me.ridian of Washington. The longitude 
 of any place can therefore be obtained, at least approximately, 
 by observations of these phenomena. 
 
 213. Velocity of Light. If the transmission of light is not in- 
 stantaneous, it is evident tha,t the same phenomenon, if observed 
 both at E' and E'" (Fig. 70), vill not occur at the same absolute 
 instant of time at both places, but will occur later at E'" by 
 the time required for light to cross the orbit of the earth, a dis- 
 tance of 185,000,000 miles. And such is actually the case. 
 This peculiarity was first noticed by Romer, a Danish astrono- 
 mer, in 1675, who found that the times at which the phenomena 
 occurred were earlier by about eight minutes at E' , and later by 
 the same amount at E'", than the times computed for the mean 
 distance of Jupiter from the sun. The time required by light in 
 passing from E' to E'" has been found by observation to be very 
 nearly 16m. 27s.: whence the velocity of light is calculated to be 
 187,000 miles a second, a result agreeing very closely with the 
 velocity obtained from the constant of aberration, discussed in 
 Chapter VIII. 
 
 214. Mass of Jupiter. The mass of Jupiter is much more 
 accurately known than the mass of any of the planets which 
 have hitherto been described. The reason is that Jupiter is 
 attended by satellites whose distances from their primary, and 
 whose periods of revolution, can be obtained by observation. 
 We are thus enabled to compare directly the attraction which 
 Jupiter exerts on one of its satellites with the attraction which 
 
176 SATUKN. 
 
 ihe sun exerts on Jupiter; and a^, by the law of gravitation, the 
 ratio of these two attractions is directly as the ratio of the masses 
 of the two attracting bodies, and inversely as the square of 
 the ratio of the distances through which these attractions are 
 exerted, it is evidently within our power to obtain the ratio of the 
 two masses. Since the attraction of the sun on Jupiter is equal to 
 the centrifugal force of Jupiter in its orbit, if we denote the dis- 
 tance of Jupiter from the sun by D and its sidereal period by T, 
 we have, by the formula of Art. 69, the expression for the sun's 
 
 4- 2 Z> 
 attraction on Jupiter equal to ; In the same way, denoting 
 
 the distance of a satellite from Jupiter by d, and its period by t, 
 
 we have for the attraction of Jupiter on the satellite, ' : so 
 
 t 
 
 D? 
 
 that the ratio of the two attractions is . Finally, denoting 
 
 d J. 
 
 the mass of the sun by M, and that of Jupiter by ra, we shall 
 have, 
 
 D? M^ tf 
 
 df*~ m X Z>' ; 
 
 M m* 
 
 m 
 
 By this formula an approximate value can be obtained of the 
 mass of any planet which is attended by a satellite. 
 
 The mass of Jupiter is found to be ju^th of that of the sun, 
 or about 312 times that of the earth. 
 
 SATURN. 
 
 215. Saturn is, next to Jupiter, the largest planet of our sys- 
 tem, and may fairly be considered to be the most interesting. 
 It revolves about the sun at a mean distance of 881,000,000 
 miles, in an orbit whose eccentricity is about T ^tli. Its synodical 
 period is 378 days and its sidereal period 29.45 years. Its 
 diameter is 72,000 miles, and it has a compression of about Jth. 
 It is attended by eight satellites, the planes of whose orbits, with 
 one exception, very nearly coincide with the plane of its equator. 
 Two of these satellites are rarely visible in any but the strongest 
 
RINGS OF SATURN. 177 
 
 telescopes. Their distances from the planet range from 121,000 
 miles to 2,300,000 miles, and their sidereal periods from 22 hours 
 to over 79 days. With one exception, they appear to be smaller' 
 than our moon. 
 
 216. Rotation, (Sec. Saturn rotates upon an axis which is in- 
 clined at an angle of about 62 to the plane of the ecliptic, in a 
 period of 10 J hours. Belts are seen on the body of the planet, 
 similar to those of Jupiter, although less marked. Other indi- 
 cations of the existence of an atmosphere have also been observed. 
 The mass of the planet, determined by the motions of its satellites, 
 is about 3sVo^ n of that of the sun, or about 93 times that of the 
 earth. 
 
 217. Rings of Saturn. When observed through a telescope, 
 Saturn is seen to be surrounded by a marvellous system of lumi- 
 nous rings, lying one within another in the plane of the planet's 
 equator, and very nearly concentric with the planet. Although 
 the planet itself was known to the ancients, the existence of these 
 rings was not suspected until the seventeenth century. They were 
 then supposed to be two rings, one within the other ; but later obser- 
 vations, with improved instruments, have made it almost certain 
 that the exterior of these two rings is itself composed of two, and 
 it is probable that similar subdivisions also exist in the interior 
 ring. A third independent ring, lying within the others, was 
 discovered in 1850, by Professor Bond, at the Observatory of 
 Harvard College. 
 
 Considering only the two rings which were first discovered, 
 and neglecting the subdivisions that may exist in each, the ap- 
 proximate dimensions of the rings are as follows : 
 
 Outer diameter of exterior ring 170,000 miles. 
 
 Breadth of exterior ring 10,000 miles. 
 
 Distance between the two rings 2,000 miles. 
 
 Outer diameter of interior ring 146,000 miles. 
 
 Breadth of interior ring 17,000 miles. 
 
 Distance of interior ring from Saturn 20,000 miles. 
 
 218. The thickness of the rings is very small. Sir John 
 Herschel estimated it at not more than 250 miles, while Professor 
 Bond considered it to be only about 40 miles. The rings appear 
 to rotate about the planet from west to east, the period of rotation 
 
178 RINGS OF SATURN. 
 
 being about 1(H hours, according toHerschel. Various theories 
 have been advanced as to their composition. The prevailing 
 theory is that they are a collection of meteoric bodies, revolv- 
 ing about the planet precisely as similar rings or groups of 
 meteoric bodies are found to revolve about the sun. (Chap. XIII.) 
 A second theory is that they are composed of nebulous matter, 
 and still a third is that they are solid. According to Professor 
 Bond, however, they are in a fluid state; and there are many 
 considerations which favor this view. 
 
 The main cause of the stability of the rings in reference to 
 the planet is undoubtedly the centrifugal force induced by the 
 rapid rotation above mentioned. Professor Peirce, in developing 
 the theory proposed by Professor Bond, maintains that the per- 
 manence of the equilibrium is finally dependent upon the attrac- 
 tions exerted by the satellites upon the rings. Herschel says 
 that unless they were originally adjusted in their present position 
 with the minutest precision, they must have been gradually 
 formed under the influence of all the existing forces. 
 
 219. The rings must present a magnificent spectacle to the 
 inhabitants of the planet. To an observer on Saturn's equator 
 they will appear as an arch, passing through the zenith, and 
 through the east and the west point of the horizon. To such an 
 observer only the edge of the rings is visible. As he moves 
 away from the equator, the altitude of the rings decreases, and 
 the side of the rings becomes visible, presenting an appearance 
 not unlike the familiar one of the rainbow. Under the most 
 favorable circumstances of position, the rings will be projected 
 against the sky as an arch with the enormous angular breadth 
 of about 15, which is about 30 times the diameter which the 
 sun presents to us. 
 
 220. Disappearance of the Rings. As Saturn revolves about 
 the sun, the plane of its rings remains, like the plane of the 
 earth's equator, fixed in space, and intersects the plane of the 
 ecliptic in a line which is called the line of nodes of the rings. 
 In Fig. 71, let S be the sun, ABCD the orbit of the earth, and 
 EHLN the orbit of Saturn. Let HN be the line of nodes of 
 the rings, and draw the lines GO and KM parallel to HN, and 
 tangent to the earth's orbit. When the planet is at H, the plane 
 
RINGS OF SATURN. 
 
 179 
 
 of the rings passes through the sun, and only the edge of the 
 rings is illuminated. In such a case the rings will disappear, or 
 at all events will only be seen, in very powerful telescopes, as 
 an exceedingly narrow line. Furthermore, if, while the planet 
 is within the lines GO and KM, the earth encounters the plane 
 
 of the rings, they will again disappear. And thirdly, if, while 
 the planet is within the same limits, the plane of the rings 
 passes between the earth and the sun, the dark side of the rings 
 will be turned towards the earth, and they will disappear- 
 When the planet is beyond these limits, it is evident "that the 
 rings will always be visible, and will present an elliptical appear- 
 ance, as represented at E. 
 
 Now, we can readily compute the length of time which Saturn 
 requires in passing through the arc GK. For in the triangle 
 CSK, right-angled at C, we know the sides OS and KS, or the 
 distance of the earth from the sun and that of the planet, and can 
 therefore obtain the angle CKS. It will be found to oe about 6 V. 
 This angle is equal to the angle K8H\ and therefore double 
 this angle, or 12 2', is the angle through which Saturn moves 
 about the sun in passing through the arc GK. Now we know 
 that Saturn makes a complete revolution about the sun in 
 
180 URANUS. 
 
 10,759 days; and therefore \ve may find by a simple proportion 
 the time which it requires to pass through 12 2'. This time 
 is found to be 359.6 days, or very nearly a sidereal year; so that 
 the earth makes very nearly one complete revolution about the 
 sun while Saturn is passing through the arc OK. 
 
 221. Number of Disappearances. Since Saturn's period of 
 revolution is 29.45 years, these disappearances will occur at in- 
 tervals of a little less than 15 years. Since the time during 
 which the planet remains within the limits GO and KM is only 
 six days less than a year, and since the earth may encounter the 
 plane of the ring at any point in its orbit, it is quite certain 
 that one such meeting will occur, and under certain circum- 
 stances there may be three. Suppose, for instance, that the 
 earth is at a when Saturn is at G, and that both bodies move 
 about the sun in the direction EHLN. The earth will meet 
 the plane of the rings somewhere in the arc aA, and the 
 rings will disappear. The rings will continue to be invisible 
 for some time, since their plane will lie between the earth 
 and the sun. The earth will overtake the plane before Saturn 
 reaches the point H, and after that time the rings will be visible 
 until the planet is at H, when the plane passes through the sun, 
 and the rings again disappear. The earth will now be near the 
 point b, and the rings will continue to be invisible, since their 
 dark side is turned to the earth. The earth, passing through C, 
 will again meet the plane somewhere in the arc CD, and after 
 that time the rings will be visible. No more disappearances 
 will occur for about fifteen years, at the end of which interval 
 the planet will pass through the arc MO. 
 
 The last disappearance took place in 1878; the next will take 
 place in 1892. 
 
 URANUS. 
 
 222. All the planets which have thus far been described, 
 except of course the minor planets, were known to the an- 
 cients; but the last two are among the comparatively recent 
 discoveries of astronomers. Uranus was discovered by Sir 
 William Herschel, in 1781, by pure accident. Herschel, as 
 well as other astronomers whose attention was directed to it, 
 
URANUS. 181 
 
 at first supposed it to be a comet; and it was only after 
 several months of observation that it was found to be a planet. 
 Several names were suggested for it, but the name of Uranus 
 was finally adopted. The astronomical symbol for it which 
 the English have adopted is formed from the initial letter of 
 Herschel's name. 
 
 Upon searching the star catalogues and other astronomical 
 records, it was found that the planet had been observed no less 
 than twenty times in the preceding 90 years, and had been con- 
 sidered to be a fixed star, its daily motion being so slight as to 
 have escaped notice. Indeed its period is so great that even its 
 annual change of position is only a few degrees. 
 
 223. These previous records, however, were of great assistance 
 to astronomers in the determination of the elements of the 
 planet's orbit. The synodical period of the planet is 370 days, 
 and the sidereal period 30,687 days, or nearly 84 years. Its dis- 
 tance from the sun is 1,800,000,000 miles, and its diameter 33,000 
 miles. It is barely visible to the naked eye at opposition. 
 
 It is attended by four satellites, which are only visible in the 
 most powerful telescopes; and by means of their movements the 
 mass of the planet is found to be about 13 times that of the 
 earth. One very remarkable point about these satellites is that 
 their motion about their primary is retrograde, or from east to west, 
 in planes inclined about 79 to the plane of the ecliptic, while 
 the motions of all the other satellites which have hitherto been 
 described are from west to east, and in planes making very small 
 angles with the plane of the ecliptic. Sir Win. Herschel be- 
 lieved that he had discovered four other satellites; but recent 
 observations, with powerful telescopes, do not confirm his belief. 
 
 Scarcely more can be said of the physical appearance of 
 Uranus than that it is uniformly bright. It exhibits neither 
 spots nor belts, and therefore nothing can be determined as to 
 any axial rotation, which is a question of special interest in the 
 case of this planet, since one of the supports on which what is 
 called " the nebular hypothesis" rests (Art. 228) is the assump- 
 tion that the satellites of every planet revolve about it in the 
 same direction in which it rotates on its axis. 
 16 
 
182 NEPTUNE. 
 
 NEPTUNE. 
 
 224. Early in the present century the conviction forced itself 
 upon the minds of many astronomers that there must exist still 
 another planet, exterior to Uranus. The circumstance which 
 led to this conclusion was the existence of irregularities in the 
 orbit of Uranus, over and above the irregularities which were 
 due to the attractions exerted by the planets then known. The 
 first systematic attempt to deduce the elements of the orbit of 
 this unknown planet from these irregularities seems to have been 
 made by Mr. Adams, of England, in 1843-5. The position which 
 he assigned to the planet was in heliocentric longitude 329 19', 
 but this determination was not then made public. The same 
 intricate problem was also solved by M. Leverrier, of Paris, in 
 1845-6, and the longitude which he obtained was 326. During 
 the summer of 1846 search was made for the planet in England, 
 but without success, owing to the want of a proper star-map. 
 The observatory at Berlin, however, was better supplied; and on 
 the night of September 23d, in compliance with a request made 
 in a letter received that day from Leverrier, Dr. Galle at once 
 detected, in longitude 326 52', what was apparently a star of 
 the eighth magnitude, though it was not laid down on the map. 
 Subsequent observations showed that this body was really a 
 planet, and it was agreed to give it the name of Neptune. 
 
 "Such," in the words of Hind, "is a brief history of this most 
 brilliant discovery, the grandest of which astronomy can boast, 
 and an astonishing proof of the power of the human intellect." 
 
 225. The synodical period of Neptune is 367 days, and its 
 sidereal period 60,127 days, or about 164 years. Its mean dis- 
 tance from the sun is 2,800,000,000 miles, and its diameter 37,000 
 miles. It is attended by one satellite, and some astronomers 
 suspect the existence of a second. The mass of Neptune is about 
 seventeen times that of the earth. The planet is not visible to 
 the naked eye. 
 
 Nothing is yet determined as to the physical appearance or 
 the axial rotation of the planet. A remarkable circumstance in 
 connection with the satellite is that, like the satellites of Uranus, 
 it moves about its primary from ea*t to west. 
 
NEBULAR HYPOTHESIS. 183 
 
 226. )i may help us in our conception of the immense distance 
 *t Neptune Irom us, even when it is in opposition, to consider 
 thai light, with its velocity of 186,000 miles a second, requires 
 four hours to come from the planet to the earth. If there are 
 any inhabitants of Neptune, the sun will to them have an appa- 
 rent diameter of only ^th of what it has to us, since the distance 
 of Neptune from the sun is about thirty times that of the earth. 
 It will therefore appear to them only about as large as Venus 
 appears, to us, under the most favorable circumstances. Saturn, 
 Jupiter, and Uranus may possibly be visible to them as extremely 
 small bodies, but it is very doubtful if any of the other planets 
 of our system are visible, even with the strongest telescopes. 
 We shall see hereafter, in the Chapter on Fixed Stars, that, with 
 a base-line of even 185,000,000 miles (the diameter of the earth's 
 orbit), attempts to determine the distances of the stars are, as a 
 general rule, wholly unsuccessful ; but it is very likely that similar 
 attempts made by observers on Neptune, with a base-line thirty 
 times as long as ours, may give more satisfactory results. 
 
 227. Relative Sizes and Distances of the Planets. The relative 
 distances of the planets Irom the sun, their relative magnitudes, 
 as well as other numerical data concerning them, will be found 
 in tables in the Appendix. In Plate I. will be seen a represen- 
 tation of their relative magnitudes, as they would appear to an 
 observer stationed at the sanitf distance from all of them. 
 
 THE NEBULA 2, HYPOTHESIS. 
 
 228. Points of Resemblance in the Planetary Phenomena. The 
 light of the planets and the satellites, when examined in the 
 spectroscope, produces only the ordinary spectrum of reflected 
 solar light. While, therefore, the spectral analysis of the light 
 of the sun, the stars, and other heavenly bodies which shine by 
 their own light, enables us to determine to some extent the ele- 
 ments of which they are composed, a similar experiment tells 
 us nothing of the constitution ot the planets or the satellites. 
 What we do know, however, of their form, their appearance, 
 their mass, and their density, leads us to conclude that they are 
 bodies not dissimilar to the earth in general constitution. There 
 are, besides, certain remarkable coincidences in the various phc- 
 
184 NEBULAR HYPOTHESIS. 
 
 nomena exhibited by the sun, the planets, and the satellites, 
 which seem to point to a common origin of the whole solar sys- 
 tem. The principal of these coincidences are the following: 
 
 (1.) All the planets revolve about the sun in the same direction 
 in which the sun rotates upon its axis: that is to say, from west 
 to east. 
 
 (2.) The planes of the planetary orbits nearly coincide with 
 the plane of the sun's equator. 
 
 (3.) The satellites of each planet, as far as known, revolve 
 about their primary in the same direction in which the primary 
 rotates upon its axis. The satellites of Uranus and Neptune 
 may or may not form an exception to this rule, for these planets 
 are so distant that observation fails to detect any axial rotation 
 in them. 
 
 (4.) The planes of the orbits of the satellites of each planet 
 approximately coincide with the plane of that planet's equator. 
 
 (5.) Both planets and satellites revolve in ellipses of small 
 eccentricity. 
 
 229. The Nebular Hypothesis. The idea of the nebular hypo- 
 thesis seems to have presented itself at about the same time to 
 both Sir William Herschel and Laplace. The principal points 
 in it are the following. All the matter which now composes the 
 sun, the planets, and the satellites once existed as a single nebu- 
 lous mass, extending beyond the present orbit of Neptune, and 
 rotating on an axis from west to east. In the progress of ages 
 this nebulous mass slowly contracted and condensed, from the 
 loss of the heat which it radiated into space, and from the 
 gravitation of its particles towards the centre. As its dimen- 
 sions became less, its velocity of rotation became greater, ac- 
 cording to the laws of Mechanics: since any particle. moving in 
 a circle of any radius with a certain linear velocity would, as it 
 approached the centre, move in a smaller circle with nearly the 
 same linear velocity, and would therefore have a greater angular 
 velocity. Finally, the centrifugal force generated by this increased 
 velocity at the surface of the equator of the mass exceeded the 
 attraction towards the centre, and a nebulous zone was detached, 
 which revolved independently of the interior mass, just as the 
 rings of Saturn have been seen to revolve about that planet. This 
 
NEBULAR HYPOTHESIS. 185 
 
 gone, by concentration at certain points within itself, broke up 
 into separate masses ; and these masses, either from slight differ- 
 ences of velocity or from the preponderating attraction of sotne 
 fraction larger than the others, eventually formed one body, re- 
 volving about the central mass. And, furthermore, as these 
 separate masses came together, a motion of rotation was com- 
 municated to the combined mass, just as a whirlpool or an eddy is 
 formed when two streams of water meet; and this rotating mass, 
 condensing and contracting in its turn, threw off from itself a 
 second zone, which underwent all the changes above described. 
 Thus were formed a planet and its satellite, each revolving about 
 its primary in the direction of that primary's axial rotation : and 
 by a continuation of the process the whole system of planets and 
 satellites was evolved. 
 
 230. Necessary Conditions. It is a necessary condition of the 
 truth of this hypothesis, that the planets shall revolve (as they do 
 revolve) about the sun in the same direction in which it rotates. 
 It is also necessary that each satellite or system of satellites shall 
 revolve about its primary in the same direction in which that 
 primary rotates. It is not, however, absolutely necessary that 
 the outer planets shall rotate in the same direction in which they 
 revolve; although such a coincidence might be expected, since 
 the revolution of the outer particles from which a planet was 
 formed would be more rapid than that of those which were 
 nearer to the sun. 
 
 If we assume this hypothesis to be true, the rings of Saturn are 
 to be considered as rings which did not form satellites after they 
 were thrown off from the planet; while in the case of the minor 
 planets the ring broke up into separate masses, which have con- 
 tinued to revolve in independent orbits about the sun. 
 
 231. Experiment in Support of the Hypothesis. The possible 
 truth of the nebular hypothesis is supported by an ingenious ex- 
 periment devised by M. Plateau.* A mass of olive-oil was im- 
 mersed in a mixture of alcohol and water, the density of the mix- 
 ture being made exactly equal to that of the oil. In this way 
 
 * See Annales de Chimie, vol. xxx. (1850). The experiment is also de- 
 Bcribed in Carpenter's Mechanical Philosophy, &c., one of the volume!:; of 
 Bonn's Scientific Library (London). 
 
186 NEBULAR HYPOTHESIS. 
 
 the mass of oil was practically withdrawn from the influence oi 
 gravitation. When made to rofate, the mass assumed a sphe- 
 roidal form, and finally, when the velocity of rotation was suffi- 
 ciently great, a ring of matter was thrown off in the equatorial 
 region. This ring subsequently broke up into independent 
 masses, each of which assumed a globular form, rotated on an 
 axis of its own, and continued to revolve about the central mass : 
 thus presenting precisely the successive phenomena which are 
 assumed in the nebular hypothesis to have occurred in the for- 
 mation of the solar system. 
 
 232. The truth of the nebular hypothesis is by no means uni- 
 versally admitted by astronomers and other scientific men ; and 
 it is difficult to say what is the predominant belief about it at 
 the present time. The high scientific reputation of those who 
 originated it, and of those who have since supported it, is suffi- 
 cient justification for giving it a place in this treatise; but it 
 must not be forgotten that its truth is still very emphatically an 
 open question, and that many great minds are numbered with 
 its opponents. 
 
 Sir William Herschel was led to the adoption of the nebular 
 theory by his examination of that class of celestial bodies called 
 nebulae, some of which presented in his day, and present now, 
 the appearance of masses of nebulous matter. Recent spectro- 
 scopic examinations of some of these nebulae (Art. 286) go to 
 show that they are really what they seem to be, masses of incan- 
 descent vapor; and this discovery gives a new interest to the ne- 
 bular hypothesis. Mr. Lockyer, in his Elementary Lessons in 
 Astronomy, says that "it may take long years to prove or dis- 
 prove this hypothesis; but it is certain that the tendency of 
 recent observations is to show its correctness." 
 
 Fresh doubts are thrown upon the truth of the nebular hy- 
 pothesis by the discovery of the satellites of Mars (203); since 
 the angular velocity of the inner satellite appears to be three 
 times as great as the rotation of the planet. 
 
 A statement of "Kirk wood's Law," which may have some 
 bearing on the.nebular hypothesis, will be found in the Appen- 
 dix. 
 
PLATE III, 
 
 COMETS. 
 
 1, BIELA'S COMET 
 
 2, ENCKE'S COMET. 
 
 3. GREAT COMET OF 1861, 
 
 4, DONATI'S COMET, 1858. 
 
COMETS. 187 
 
 CHAPTER XIII. 
 
 .V^lS AM) METEORIC BODIES. 
 COMETS. 
 
 233. Ger^tc\ Description of Co, nets. A comet is a body of 
 nebulous appeal ance and irreguUi shape, revolving in an orbit 
 about the sun. Comets have usually been considered to con- 
 sist for the most part of nebulous matter; but the theory has 
 lately been advanced that they are collections of minute meteoric 
 bodies. 
 
 Comets differ widely from each other in appearance, and no 
 description of them can be given to which thsre will not be 
 many exceptions. Generally speaking, a comet consists of three 
 parts: the nucleus, the coma, und the tail. The nucleus and the 
 coma together form the head. The nucleus is a blight point, 
 like a star or a planet, which may be either a solid mass, or a mass 
 of nebulous matter of a density greater than that of the rest of 
 the comet. The diameter of the nucleus varies considerably in 
 different comets: that of the comet of 1845 (in)* was about 
 8000 miles, while that of the comet of 1806 was only 30 
 miles. The average value is not over 500 miles: and in many 
 comets no nucleus whatever is perceptible. 
 
 The coma is a mass of cloud-like matter, mere or less nearly 
 globular in form, which surrounds the nucleus. The nucleus, 
 however, as a general thing, is not situated at the centre of the 
 coma, but lies towards that margin which is the nearer to the sun. 
 The diameter of the coma is different in different cornets: thai 
 of the comet of 1847 (y) was only 18,000 miles, while that of tKwi 
 comet of 1811 (i) was over 1,000,000 miles. Usually, however.. i\i 
 
 * The number (in) means that this was the third comet which appeared 
 it the course of the year. 
 
188 COMETS. 
 
 is less than 100,000 miles. It is frequently noticed that the coma 
 decreases in apparent diameter as the comet approaches the sun, 
 and increases as the comet recedes from it. On the supposition 
 that the coma consists of vaporous matter, this phenomenon is 
 explained by the assumption that the intense heat to which the 
 comet is subjected as it approaches the sun is sufficient to rarefy 
 this vaporous matter to such an extent that some of it becomes 
 invisible. 
 
 The tail is a train of cloud-like matter attached to the head, 
 which usually lies in a direction nearly opposite to that in which 
 the sun lies from the head. The tail is usually very small when 
 the comet first appears, and sometimes is not even perceptible. 
 As the comet approaches the sun, the length of the tail increases, 
 and sometimes becomes enormous. In the comet of 1811 (l), for 
 instance, the length of the tail was 100,000,000 miles ; and in 
 that of 1843 (i) it was 200,000,000 miles. 
 
 The angular length of the tail depends not only on its abso- 
 lute length, but also on its distance from the earth, and on the 
 direction in which the axis of the tail lies. There are six comets 
 on record of which the tails subtended angles of over 90 ; and 
 one of these, that of 1861 (n), had a tail of 104 in length, as 
 observed at some places. 
 
 234. Diversity of Appearance. The description above given 
 may be considered to apply to comets taken as a class ; but, as 
 already remarked, important exceptions are often noticed in 
 individual comets. Indeed, it is hardly possible to compare any 
 two comets without finding marked points of difference in them. 
 Some comets are not visible at all, except by the aid of powerful 
 telescopes, and are hence called telescopic comets ; while others, 
 again, are so conspicuous as to be visible to the naked eye in full 
 daylight. Some comets have more than one tail; the comet of 
 1823, for instance, had a tail turned towards the sun, in addition 
 to the usual one turned from it. The comet of 1744 is reported 
 to have had six tails, spread out like an immense fan, through 
 an angle of 117; but the truth of the record is not above 
 suspicion. 
 
 Not only do comets differ thus widely from each other in ap- 
 pearance, but even the same comet changes its appearance from 
 
TAIL OF THE COMET. 189 
 
 day to day. Sometimes the nucleus decreases in diameter as it 
 approaches the sun : sometimes its brightness increases, and jets 
 of luminous matter are thrown off from it in the direction of the 
 sun. The length of the tail often increases with marvellous 
 rapidity ; in the case of the Great Comet of 1843 (i), the increase 
 was estimated to be about 35,000,000 miles a day, after the 
 comet had passed its perihelion. There are some instances on 
 record of a comet's having separated into two distinct comets. 
 This is asserted in the Greek records of a comet which appeared 
 in 370 B.C. : and Biela's comet presents an indubitable instance 
 of this kind. This comet was observed in 1826 and 1832, and 
 was determined to be a comet with a period of nearly seven years. 
 Its return in 1839 was not observed. It again appeared in 1846, 
 and then presented the extraordinary appearance of two comets, 
 moving side by side, at a distance apart of over 150,000 miles. 
 
 235. The Tail. The general form of the tail is that of a trun- 
 cated cone, the larger base being at the extremity of the tail. It 
 is noticed that the tail is always brighter near the borders than 
 along the middle, from which it is inferred that it is hollow: 
 since only on such a supposition would the line of sight pass 
 through more luminous matter when directed to the edges than 
 when directed to the middle. With regard to the formation of 
 the tail, the most generally accepted theory seems to be that the 
 matter of which the nucleus is composed is excited and dilated 
 by the action of the sun's rays, as the comet approaches the sun, 
 and that particles of vaporous matter are thrown off from it ; and 
 that these particles are driven to the rear by some repulsive force 
 exerted by the sun, and thus form the tail. What this repulsive 
 force exerted by the sun is, has not yet been determined ; but the 
 general situation which the tail of a comet has with reference to 
 the sun seems to justify the inference that some such force does 
 exist. Nor has it yet been determined what is the force which 
 originally detaches these vaporous particles from the nucleus: it 
 may be the same repelling force which drives them to the rear, it 
 may be a force generated in the nucleus itself, or it may be a 
 combination of both these forces. If we adopt the theory of *ne 
 meteoric structure of these bodies, the tail is to be considered as a 
 cloud of minute particles of matter, held together by their mu- 
 
190 
 
 TAIL OF THE COMET. 
 
 Fig. 72. 
 
 tual attraction, or by the attraction exerted upon them by the 
 denser mass which constitutes the head. 
 
 236. Curvature of the Tail. The tail of a comet is usually not 
 straight, but is concave towards that part of space which the 
 comet is leaving. If we assume the existence of a solar repul- 
 sive force, similar to that mentioned in the preceding article, 
 this peculiarity of shape may be thus explained. In Fig. 72, let 
 8 be the sun, and GCD 
 a portion of the orbit of a 
 comet. When the nucleus 
 is at A, let a particle be 
 driven from it in the di- 
 rection SA, with a force 
 sufficient to carry it to L 
 in the time in which the 
 nucleus moves from A to C. 
 When the nucleus reaches 
 C, this particle, still retain- 
 ing the motion which it had in common with the nucleus, will be 
 found at some point M. In the same way a particle driven from 
 the nucleus when it is at B will be found at some point K, when 
 the nucleus reaches C: and, in general, when the nucleus is at C 
 the tail will not lie in the direction SN, but in the direction of 
 the curve CKM, as shown in the figure. 
 
 237} Elements of a Comet's Orbit. A comet is identified at its 
 successive returns, not by its appearance, which is liable, as we 
 have already seen, to serious changes, but by the elements of its 
 orbit. In consequence of the comparative ease with which the 
 elements of a parabola can be calculated, astronomers are in the 
 habit of using that curve to represent at first the approximate 
 form of a comet's orbit. The elements of a parabolic orbit are 
 five in number, and are as follows : 
 
 (1.) The inclination of the orbit to the plane of the ecliptic: 
 
 (2.) The longitude of the ascending node: 
 
 (3.) The longitude of the perihelion: 
 
 (4.) The time at which the comet passes its perihelion: 
 
 (5.) The distance of the comet from the sun at perihelion. 
 Tables and formulae have been constructed by which these 
 
NUMBER OF COMETS. 191 
 
 elements can be computed from the results of three distinct ob- 
 servations of the position of the comet: and these three observa- 
 tions may all be made, if necessary, within the space of 48 hours. 
 The parabolic elements having thus been obtained, the catalogues 
 of comets are searched to see if these elements are similar to 
 those recorded of any previous comet. As it is highly impro- 
 bable that the elements of any two Comets will coincide through- 
 out, the presumption is a strong one, if two comets, visible at 
 different times, move in the same orbit, that they are one and the 
 same comet: and the more often the coincidence is repeated, the 
 more nearly does the presumption approach to a demonstration. 
 
 238. Number of Comets, and their Orbits. The number of 
 comets which have been recorded since the Christian era is 
 over 770: and there are about 80 recorded as observed before 
 that date. Of these 850 appearances of comets, some may un- 
 doubtedly have been only reappearances of the same comet: 
 and, indeed, in some cases comets have been identified with 
 other comets previously observed; but this can hardly be the 
 case with the majority of these bodies. Besides these comets 
 thus recorded, there must have been many others so situated as 
 to be above the horizon only in the day-time: and such comets 
 would become visible only in case of the occurrence of a total 
 solar eclipse. A coincidence of this kind is recorded by Seneca 
 as having occurred 62 B. c., when a large comet was seen in close 
 proximity to the sun during a solar eclipse. The improvement 
 of telescopes in recent years has greatly increased the number 
 of comets which become visible, and 204 were observed between 
 the years 1800 and 1876. We are justified, therefore, in conclud- 
 ing that the comets which have really come within our system 
 since the Christian era are to be reckoned by thousands. Two 
 centuries and more ago, Kepler made the remarkable statement 
 that " there are more comets in the heavens than fishes in the 
 ocean." 
 
 The orbit in which a comet moves may be either an ellipse, a 
 parabola, or an hyperbola. The orbits of 329 comets have been 
 subjected to mathematical investigations, and the results of these 
 investigations may be thus tabulated : * 
 
 * The list of these comets, and the facts known concerning them, are 
 given in (r. F. Chamliers's D&tct'iptilt A^trououu/ (Oxford, Kng.). 
 
192 PERIODIC TIMES OF COMETS. 
 
 Comets with elliptical orbits* : 20; 
 
 Subsequent returns of these comets 66; 
 
 Comets with elliptical orbits, which have not returned 43 ; 
 
 Comets with parabolic orbits 194; 
 
 Comets with hyperbolic orbits 6. 
 
 A comet whose orbit is either a parabola or an hyperbola will 
 not return to our system ; provided, at least, that the attraction 
 of other bodies does not alter the character of the orbit. It 
 must be noticed, however, that some of the orbits which are 
 called parabolic, may really be ellipses of an eccentricity so 
 great as to render their elements undistinguishable from those 
 of parabolas. In whatever conic section a comet may move, 
 the sun is always at the focus. 
 
 239. Periodic Times. The sixty-three comets which have been 
 found to move in elliptical orbits differ widely from each other 
 in the length of their periods. Among the twenty comets whose 
 returns have been observed, there are seven with short periods, 
 lying between three and fourteen years. There is no doubt that 
 these seven comets are periodic ; but there is some uncertainty with 
 regard to some others of the remaining thirteen. Two elements of 
 such uncertainty are the unsatisfactory character of the records 
 of the observations made in the earlier ages, and the length of 
 time which the periods embrace, being often several hundred 
 years. Halley's comet, however, with a period of about seventy- 
 five years, is unquestionably to be added to the list of periodic 
 comets about which there is no doubt. There are five other 
 comets, with periods not very different from that of Halley's, 
 which have been discovered within the present century, and 
 which have as yet made no return. With regard to the remain- 
 ing comets to which elliptical orbits and periods have been 
 assigned, little more can be said than that these periods embrace 
 hundreds and even thousands of years. 
 
 240. Motion of Comets in their Orbits. The motions of comets 
 in their orbits about the sun are not performed in the same 
 direction, the number of those whose motion is retrograde being 
 about the same as the number of those whose motion is direct. 
 According to Chambers, an examination of the motions of the 
 various comets shows "that with comets revolving in elliptic 
 
MAf.:S OF THE COMETS. 193 
 
 orbits there is a strong and decided tendency to direct notion. 
 The same obtains with the hyperbolic orbits : with the parabolic 
 orbits there is a rather large preponderance the other way ; and 
 taking all the calculated comets together, the numbers are too 
 nearly equal to afford any indication of the existence of a general 
 law governing the direction of motion." 
 
 The angles which the planes of the orbits make with the 
 plane of the ecliptic have values ranging from to 90 ; but 
 " there is a decided tendency in the periodic comets to revolve 
 in orbits but little inclined to the plane of the ecliptic :" and 
 if we take all the comets into consideration, "w r e find a decided 
 disposition in the orbits to congregate in and around a plane 
 inclined 50 to the ecliptic." 
 
 Owing to the great eccentricity of the orbits, some of the 
 comets approach very near to the sun at the time of perihe- 
 lion passage: the comet of 1843 (i), for instance, came within 
 100,000 miles of the sun's surface. For the same reason the 
 distances to which some of the comets with elliptical orbits recede 
 from the sun are immense ; thus the comet of 1844 (n) receded 
 to a distance of 400,000,000,000 miles, over 130 times the dis- 
 tance of Neptune from the sun. The velocity of the comets at 
 perihelion is sometimes enormous ; this same comet of 1843 swung 
 about the sun through an arc of 180 in only two hours, and 
 moved with the velocity of 350 miles a second. 
 
 241. Mass and Density of the Comets. The minuteness of tli3 
 mass of the comets is proved by the fact that they exert no per- 
 ceptible influence on the motions of the planets or the satellites 
 although they sometimes pass very near to them. Thus Lexell's 
 comet, 1770 (i), in its advance towards the sun, became entan- 
 gled with the satellites of Jupiter, and remained near them for 
 five months, without sensibly affecting their motions. The effect 
 of Jupiter's attraction on the comet, however, was very striking. 
 The comet had a period of about five years, and yet it never 
 appeared after 1770. It was found, by computation, that at 
 its first return after that date it was so situated as not to become 
 visible; and that in 1779, before its second return, it came 
 nearer to Jupiter than Jupiter's fourth satellite : and the pre- 
 sumption is that its orbit was so changed and enlarged that the 
 
194 LIGHT OF THE COMETS. 
 
 comet no longer comes near enough to the earth to become viable. 
 This same comet came within about 1,400,000 miles of the 
 earth in 1770 : near enough, had its mass been equal to that 
 of the earth, to increase the length of the year by nearly three 
 hours ; but no sensible effect was produced. 
 
 The mass, then, of the comets being so small, and their 
 volume so large, the density of the matter of which they are 
 composed must be exceedingly rare. Indeed, it must be vastly 
 more rare than that of the lightest gas or vapor of which we 
 have any knowledge: for stars of the smallest magnitude are 
 distinctly seen, and usually, too, with no -perceptible diminution 
 of brightness, through all parts of the comets excepting perhaps 
 the nucleus; and this too in cases where the volume of nebulous 
 matter has a diameter of 50,000 or 100,000 miles. 
 
 242. Light of the Comets. The question whether or not 
 comets shine by their own light does not seem to be satisfac- 
 torily decided. The existence of phases would of course prove 
 that they shine by reflected light ; but although in one or two 
 cases the statement has been made that phases have been 
 detected, the truth of the statement has in no case been univer- 
 sally accepted. Undoubtedly the distance of some comets is 
 so great that phases might exist and still escape observation, 
 as in the case of the superior planets ; but, on the other hand, 
 some comets come so near to the earth that there seems to be 
 no good reason why phases, if any exist, should not be noticed. 
 Observations upon the light of the comets have been made, 
 both with the polariscope and with the spectroscope. The ob- 
 servations made with the polariscope seem to establish the fact 
 that the comets shine, partly at all events, by the reflected 
 light of the sun : as, for instance, the observations of Airy and 
 others on Donati's comet in 1858. Mr. Huggins, of England, to 
 whom we owe so many interesting discoveries made with the 
 Bpectroscope, has recently examined the light of several comets 
 with that instrument. In Brorsen's comet he found that the 
 nucleus and part of the coma shone by their own light. In 
 Tempers comet the nucleus shone by its own light, and the 
 coma by the reflected rays of the sun. In some comets the 
 light from the nucleus resembled that which conies from the 
 
-PERIODIC COMETS. 
 
 195 
 
 gaseous nebulse. In one comet, there was a remarkable resem- 
 blance in the spectrum produced by its light to that produced 
 by carbon, not only in the position of the bands, but in charac- 
 ter and relative brightness. 
 
 Altogether, the question as to the light of the comets may 
 fairly be regarded as still an open one, to be decided, perhaps, 
 by future observations. 
 
 PERIODIC COMETS. 
 
 243. It has already been stated, in Art. 239, that there are at 
 least eight comets which are undoubtedly periodic, seven of these 
 being comets with short periods, and the eighth being Halley's 
 comet. The following table contains a list of these comets. 
 
 NAME. 
 
 PERIOD IN 
 YEARS. 
 
 NUMBER OF 
 APPEARANCES 
 OBSERVED. 
 
 LAST AP- 
 PEARANCE. 
 
 Encke, 
 
 3.3 
 
 22 
 
 1878 
 
 WinneckeorPons, 
 
 5.5 
 
 4 
 
 1875 
 
 Brorsen, 
 
 5.6 
 
 5 
 
 1879 
 
 Biela, 
 
 6.6 
 
 6 
 
 1852 
 
 D' Arrest, 
 
 6.6 
 
 4 
 
 1877 
 
 Faye, 
 
 7.4 
 
 5 
 
 1873 
 
 Mechain or Tuttle, 
 
 13.7 
 
 3 
 
 1871 
 
 Hal ley, 
 
 76. 
 
 5 
 
 1835 
 
 ENCKE 8 COMET. 
 
 244. On November 2fi, 1818, a small and ill-defined telescopic 
 comet was discovered in the constellation Pegasus, by the 
 astronomer Pons, at Marseilles. It remained visible for seven 
 weeks, and many observations were made upon it. Professor 
 Encke, of Berlin, finding that the elements of the orbit did not 
 agree with those of a parabola, determined to subject them to a 
 rigorous investigation, according to the method proposed by 
 
196 PERIODIC COMETS. 
 
 Gauss. This investigation shewed that the orbit was elliptical, 
 and that the period of the comet was about 3 J years. He fur- 
 ther identified the comet with the comets of 1786 (i), 1795, and 
 1805, and predicted that it would return to perihelion on May 
 24, 1822, after being retarded about nine days by the influence 
 of Jupiter. 
 
 " So completely were these calculations fulfilled, that astrono- 
 mers universally attached the name of ' Encke' to the comet of 
 1819, not only as an acknowledgment of his diligence and success 
 in the performance of some of the most intricate and laborious 
 computations that occur in practical astronomy, but also to mark 
 the epoch of the first detection of a comet of short period ; one 
 of no ordinary importance in this department of science." 
 
 The comet has since been observed at every reappearance, the 
 appearance in 1878 being the twenty-second on record. In 
 1835, it passed so near to the planet Mercury as to show con- 
 clusively that the generally received value of that planet's mass 
 must be far too great: since the planet exerted no perceptible 
 influence on the comet's orbit. 
 
 The comet is sometimes visible to the naked eye. It usually 
 appears to have no tail ; but in 1848 it had two, one about 1 in 
 length, turned from the sun, and the other of a less length and 
 turned towards it. At perihelion the comet passes within the 
 orbit of Mercury: while at aphelion its distance from the sun is 
 nearly equal to that of Jupiter. 
 
 One very curious feature in connection with this comet is that 
 its period is steadily diminishing, by an amount of about 2 
 hours in every revolution, the period having been nearly 1213 
 days in 1789-92, and only about 1210 days in 1862-65. Encke's 
 own theory to account for this diminution is that the space 
 through which the comet moves is filled with some extremely 
 rare medium, too rare to obstruct the motions of the planets, but 
 dense enough to offer sensible resistance to the progress of the 
 comets. The effect of this diminution of velocity is to diminish 
 the comet's centrifugal force, so that the comet is drawn nearer 
 to the sun, and its orbit becomes smaller. But as the orbit 
 becomes less, the angular velocity of the comet is increased, and 
 its period of revolution is decreased. 
 
PERIODIC COMETS. 197 
 
 This theory of Encke's concerning the existence of a resisting 
 medium is by no means universally accepted by astronomers. 
 
 WINNECKE'S OR PONS'S COMET. 
 
 245. The second comet in the list was discovered by Pons, on 
 June 12, 1819. Professor Encke assigned to it a period of 5 
 years, but the comet was not seen again until March 8, 1858, 
 when it was detected by Winnecke, at Bonn. He was at first 
 inclined to consider it a new comet, but soon identified it with 
 the one previously discovered by Pons. Its distance from the 
 sun at perihelion is about 70,000,000 miles, and its distance at 
 aphelion 520,000,000 miles. It also appeared in 1869 and 1875. 
 
 BRORSEN'S COMET. 
 
 246. This comet was discovered by M. Brorsen, at Kiel, on 
 February 26, 1846. The orbit was found to be elliptical, with a 
 period of about 51 years, and its return to perihelion was fixed 
 for September, 1851 ; but its position at that time was so un- 
 favorable for observation that it was not detected. It was seen 
 at its next return to perihelion, on March 29, 1857. It again 
 escaped detection in 1862, but was seen in this country on May 
 11, 1868. It was seen also in 1873, and 1879. 
 
 Its perihelion distance is 60,000,000 miles, and its aphelion 
 distance, 530,000,000 miles. 
 
 BIELA'S COMET. 
 
 247. This comet was discovered by M. Biela, an Austrian 
 officer, at Josephstadt, Bohemia, on February 27, 1826. It was 
 observed for nearly two months, and was identified with comets 
 which had previously been seen in 1772 and 1805. 
 
 Its next return to perihelion was fixed for November 27, 1832 ; 
 and the comet passed perihelion within twelve hours of that .time. 
 On October 29, 1832, it passed within 20,000 miles of the earth's 
 orbit : but the earth did not reach that point of its orbit until a 
 month afterwards. No little alarm was created, however, outside 
 of the scientific world, when it became generally known how 
 near to the earth's orbit the comet would approach. 
 
 At its return in 1839 it was not observed, owing to its close proxi- 
 
198 PERIODIC COMETS. 
 
 mity to the sun. It was again Detected on November 28, 1845, 
 and by the end of the year it was found to have separated into 
 two parts, and to present the extraordinary appearance of two 
 comets, moving side by side, at a distance apart of over 150,000 
 miles. It again returned in 1852, and presented the same ap- 
 pearance; but the distance between the parts had increased to 
 over one million of miles. Since that time the comet has never 
 been seen, unless perhaps in 1872, as a meteoric shower. 
 
 Two theories have been advanced to account for this singular 
 separation. One is that the division may have been the result 
 of some internal repulsive force, similar to that which forms 
 the tails of comets; the other is that it may have been the result 
 of collision with some asteroid. At perihelion the comet passes 
 within the orbit of the earth, and at aphelion it passes beyond 
 that of Jupiter. 
 
 D'ARREST'S COMET. 
 
 248. This comet was discovered by Dr. D'Arrest, at Leipsic, 
 on June 27, 1851. It remained in sight for about three months, 
 and its period was determined to be about 6? years. Its return 
 in November, 1857, was accordingly predicted, and the predic- 
 tion was verified; although, owing to the comet's great southern 
 declination, it was only observed at the Cape of Good Hope. 
 The unfavorable situation of the comet in 1864 prevented its 
 being seen; but it was seen on its returns in 1870 and 1877. 
 
 Its perihelion distance is about 100,000,000 miles, and its 
 aphelion distance more than 500,000,000 miles. 
 
 FAYE'S COMET. 
 
 249. This comet was discovered by M. Faye, at the Paris 
 Observatory, on November 22, 1843. It had a bright nucleus and 
 a short tail, but was not visible to the naked eye. The elements 
 of its orbit were investigated by Leverrier, who predicted that 
 it would return to perihelion on April 3, 1851 ; and it returned 
 within about a day of the time predicted. It has since made 
 three returns, viz., in 1858, 1865, and 1873. The dimensions of 
 its orbit are nearly the same as those of D' Arrest's comet. 
 
PERIODIC COMETS. 
 
 199 
 
 ME*CHAIN'S OR TT ITLE'S COMET. 
 
 250. This comet was discovered by Mechain, at Paris, on 
 January 9, 1790. Its period was calculated to be less than 14 
 years; but the comet was not seen again until January 4, 1858, 
 when it was detected by Mr. H. P. Tuttle, at the Harvard Col- 
 lege Observatory. Its third appearance was in 1871. 
 
 H ALLEY'S COMET. 
 
 251. In the latter part of the seventeenth century, Sir Isaac 
 Newton published his Principia. In that great work he as- 
 sumed that the comets were analogous to the planets in their 
 revolutions about the sun, although no periodic comet had then 
 been discovered. He explained the methods of investigating 
 the orbits of the comets, and invited astronomers to apply these 
 methods to the various comets which had been observed. Hal- 
 ley, a young English astronomer, and afterwards the second 
 Astronomer Royal, after a careful investigation, identified the 
 comet of 1682 with comets which had appeared in 1531 and 
 1607: the period of the comet being about 75 years. The 
 fact that the interval of time between the first and the second 
 of these appearances was not exactly equal to that between the 
 second and the third seemed at first to offer some difficulty; 
 but Halley, "with a degree of sagacity which, considering the 
 state of knowledge at the time, cannot fail to excite unqualified 
 admiration," advanced the theory that the attractions of the 
 planets would exert some influence on the orbits of the comets. 
 Having thus decided that this comet was a periodic comet, Hal- 
 ley predicted the return of the comet about the beginning of 
 the year 1759 ; and the comet passed its perihelion on March 12, 
 in that year. The comet again appeared in 1835, and its 
 next appearance will be in 1912. 
 
 The comet is a very conspicuous one, with a tail sometimes 
 30 in length and sometimes 50. The comet has been traced 
 back through the astronomical records, with more or less cer- 
 tainty, to 11 B.C., the number of appearances being about seven- 
 teen. It is not impossible that it was this comet which appeared 
 in 1066, when it is recorded that a large comet excited dread 
 
200 REMARKABLE COMETS. 
 
 throughout Europe, and was in England considered to presage 
 the success of the Norman invasion. It is also probably iden- 
 tical with the comet )f 1456, which had a splendid tail 60 in 
 length. 
 
 Halley's comet at perihelion is nearer to the sun than Venus, 
 while at aphelion it recedes beyond the orbit of Neptune. 
 
 REMARKABLE COMETS OF THE PRESENT CENTURY. 
 THE GREAT COMET OF 1811. 
 
 252. The comet of 1811 (i) was discovered on March 26, 1811, 
 and was visible about seventeen months. It was very conspi- 
 cuous in the autumn of 1811, remaining visible throughout the 
 night for several weeks. Sir William Herschel states that the 
 nucleus was well defined, with a diameter of about 428 miles; 
 that it was of a ruddy hue, while the surrounding nebulous matter 
 had a bluish-green tinge. The tail was about 25 in length, 
 and 6 in breadth. Its aphelion distance from the sun is 14 
 times that of Neptune, and its period, according to Argelander, 
 is 3065 years, with an uncertainty of 43 years. 
 
 THE GREAT COMET OF 1843. 
 
 253. The comet of 1843 (i) was first seen in the southern 
 hemisphere in February, and became visible in the northern 
 hemisphere the next month. It was decidedly the most won- 
 derful comet of the present century. Its nucleus and coma 
 shone with great splendor, and its tail was a luminous train of 
 about 60 in length. On the day after its perihelion passage, 
 and when only 4 distant from the sun, it was seen in broad 
 daylight in some parts of New England, and its distance from 
 the sun was measured with a sextant. It is described as having 
 been at that time as well defined, in both nucleus and tail, as the 
 moon is on a clear day. The comet is remarkable for its small 
 perihelion distance, which was only about 540,000 miles ; so that 
 the comet came within 100,000 miles of the sun's surface. The 
 intensity of the heat to which the comet must then have been 
 subjected is almost inconceivable. Since 540,000 miles is about 
 i- 70 th of the distance of the earth from the sun, and the inten- 
 
REMARKABLE COMETS. 201 
 
 sity of heat varies inversely as the square of the distance, the 
 heat to which the comet was subjected must have been about 
 29,000 times as intense as the heat which prevails at the earth's 
 surface: a heat nearly twenty times that required, as shown by 
 experiments with powerful lenses, to melt agate and carneliau. 
 For some days aftei this, the tail had a fiery red appearance; and 
 its enormous length of over 200,000,000 miles, and the mar- 
 vellous rapidity with which it was formed, were undoubtedly the 
 results of the heat which it endured. 
 
 The rapidity with which this comet moved about the sun has 
 already been noticed (Arc. 240). The period has been computed 
 to be about 175 years. 
 
 DGtfATl's COMET. 
 
 254. This comet, 1858 (vi), was discovered on June 2d, by Dr. 
 Donati, at Florence. It was then only discernible with a tele- 
 scope, but became visible to the naked eye about the last of 
 August. Indications of a tail began to be noticed about the 20th 
 of August, and in a few weeks the tail assumed a noticeable 
 curvature, which subsequently became one of the most interesting 
 points connected with the comet. The comet passed its perihelion 
 on September 29th, and was at its least distance from the earth 
 on October 10th. Its tail subtended an angle of 60, and 
 had an absolute length of 51,000,000 miles. It disappeared 
 from view in the northern hemisphere in October, but was seen 
 in the southern hemisphere until March, 1859. 
 
 This comet was not as large as some others of the comets, but 
 it was particularly noted for the intense brilliancy of its nucleus. 
 The nebulosity surrounding the nucleus was also peculiar in its 
 appearance. It consisted of seven luminous envelopes, parabolic 
 in form, and separated from each other by spaces comparatively 
 dark. These envelopes were detached in succession from the comet's 
 nucleus, at intervals of from four to seven days. They receded 
 from the nucleus with the daily rate of about 1000 miles. Per- 
 fectly straight rays of light, or "secondary tails," were also seen. 
 
 The comet has a period of about 2000 years. A magnificent 
 memoir of this comet, by Professor G. P. Bond, is contained in 
 the second volume of the Annals of the Harvard Observatory. 
 
202 METEORIC BODIES. 
 
 TIIE GREAT CO*MET OF 1861. 
 
 255. This comei;, the second of the year, was discovered in the 
 southern hemisphere on May 13th, but was not seen in England 
 until June 29th, about two weeks after its perihelion passage. 
 The nucleus was round and unusually bright, and the tail at one 
 time attained the length of over 100. The comet remained in 
 sight for about a year. 
 
 "In a letter published at the time in one of the London papers, 
 Mr. Hind, an English astronomer, stated that he thought it not 
 only possible, but even probable, that in the course of Sunday, 
 June 30th, the earth passed through the tail of the comet, at a 
 distance of perhaps two-thirds of its length from the nucleus." 
 Mr. Hind also stated that on Sunday evening there was noticed, 
 by both himself and others, a peculiar illumination in the sky, 
 like an auroral glare; and a similar phenomenon seems to have 
 been noticed outside of London. 
 
 According to the observations of Father Secchi, the light of 
 the tail, and that of the rays near the nucleus, presented evidences 
 of polarization, while the nucleus itself at first presented no 
 evidences whatever ; afterwards, however, the nucleus presented 
 decided indications of polarization. Secchi states that he thinks 
 this "a fact of great importance, as it seems that the nucleus on 
 the former days shone by its own light, perhaps by reason of the 
 incandescence to which it had been brought by its close proximity 
 to the sun." 
 
 METEORIC BODIES. 
 
 256. Under the general head of meteors are included three 
 classes of bodies : 1st. The ordinary shooting stars, some of which 
 ean be seen rushing across the heavens on almost any clear night ; 
 2d. Detonating meteors, which are shooting stars, commonly of an 
 A nusual size, whose disappearance is followed by a sound like that 
 fan explosion ; 3d. Aerolites, which, after the flash and the ex- 
 plosion with which they are generally accompanied, are precipi- 
 tated to the earth in showers of stones and metallic substances. 
 It is oijsy within recent years that the decided attention of astro- 
 
 b.as been directed to these bodies, and comparatively 
 
SHOOTING STARS. 
 
 203 
 
 little is known with certainty about them ; but the ger-^ral belief 
 is that they are all essentially of the same nature, differing from 
 each other rather in size and density than in other mor^ impor- 
 tant respects. 
 
 257. Shooting Stars. Scarcely a clear night passes during 
 which shooting stars are not seen. The a\erage number of th">?* 
 which can be seen at any place by one observer, on a cloudlet 
 moonless night, is estimated to t>e about six an hour. There is 
 however, an hourly variation in the number observed, the mini 
 mum occurring about 6 P.M., and the maximum about 6 A.M 
 According to a French writer on this subject, the mean numbro 
 of meteors observed is given in the following table : 
 
 HOURS 
 
 7-8 
 
 9-10 
 
 11-12 
 
 1-2 
 
 3-4 
 
 5-6 
 
 
 P.M. 
 
 
 
 A.M. 
 
 
 
 MEAN NUMBER.. 
 
 3.5 
 
 4 
 
 5 
 
 6.4 
 
 7.8 
 
 8.2 
 
 .It is further estimated that the number seen at any one place 
 by a number of observers sufficient to watch the whole hemi- 
 sphere of the heavens is 42 an hour, on the average, or about 
 1000 daily: and that the number which could be seen daily over 
 the whole earth, under favorable circumstances, is more than 
 8,000,000. This is the number of those large enough to be vis- 
 ible to the naked eye : it is simply impossible to estimate the 
 number of those -which could be seen with the aid of telescopes. 
 
 It is further noticed that there are more shooting stars observed 
 in the second half of the year than in the first. At certain sea- 
 sons of the year, either in consecutive years or after the lapse of 
 a certain number of years, there are unusually brilliant displays 
 of these meteors, which are called star showers. The number >f 
 recognized star showers now exceeds fifty ; and prominent among 
 them are the shower of August 9-11 and that of November 
 11-13. 
 
 Professor Harkness, of the Washington Observatory, after an 
 elaborate investigation of the quantity of matter in the ordinary 
 shooting star, concludes that it is not far from one grain. 
 
 258. The November Shoi'cr. There are several historical 
 
204 NOVEMBER SHOWER. 
 
 notices of brilliant displays of meteors which occurred in the 
 early centuries of the Christian era: and ten of these, occurring 
 between the years 902 and 1698, took place in October or 
 November. The first display, however, of which we have any 
 detailed account, occurred in 1799, on the morning of the 13th 
 of November, and was visible over nearly the whole of the 
 western continent. Humboldt witnessed it in South America, 
 and thus describes it : " Towards the morning of the 13th we 
 witnessed a most extraordinary scene of shooting meteors. 
 Thousands of bodies and falling stars succeeded each other 
 during four hours. Their direction was very regular, from north 
 to south. From the beginning of the phenomenon there was not 
 a space in the firmament equal in extent to three diameters of 
 the moon which was not filled every instant with bodies or fall- 
 ing stars. All the meteors left luminous traces or phosphor- 
 escent bands behind them, which lasted seven or eight seconds." 
 
 Similar showers also occurred on the same day of the month 
 in the years 1831, 1832, and 1833, the last one being the most 
 splendid on record. It lasted from ten o'clock on the night of 
 the 12th to seven o'clock on the morning of the 13th, and was 
 visible over nearly the whole of North America. The display 
 reached its maximum about four A.M. An observer at Boston 
 about six o'clock counted 650 shooting stars in a quarter of an hour. 
 Large fireballs with luminous trains were also seen, some of 
 which remained visible for several minutes. Even stationary 
 masses of luminous matter are said to have been seen : and one 
 in particular is mentioned as having remained for some time 
 in the zenith over the Falls of Niagara, emitting radiant streams 
 of light. 
 
 The November shower was witnessed again in 1866, both in 
 this country and in Europe; but the display was much more 
 brilliant in Europe. The maximum seems to have taken place 
 about two A.M. on the 14th, when nearly 5000 meteors were 
 counted in an hour at Greenwich. At half-past one, 124 were 
 counted in one minute. The case was reversed with the shower 
 of 1867, the display being more brilliant in this country than in 
 Europe. The report on the shower from the United States 
 Observatory at Washington states that as many as 3000 were 
 
HEICHT OF METEORS. 205 
 
 counted in one hour. The most magnificent phase seems to 
 have occurred about half-past four A.M. on the 14th. Professor 
 Loomis states that at New Haven about 220 a minute were 
 counted at this time. Many others were undoubtedly rendered 
 invisible by the light of the moon, which was then very nearly 
 full. Most of the brighter meteors left trains of phosphorescent 
 light, which remained visible for several seconds, and in some 
 cases for several minutes. 
 
 In 1868, the display began somewhat before midnight on the 
 13th and continued until daybreak on the 14th. Professor 
 Eastman, of the Washington Observatory, says in his report, 
 that " considering the number and brilliancy of the meteors, 
 their magnificent trains, and the magnitude of the meteoric 
 group through which the earth passed, this shower was unques- 
 tionably the grandest that has ever been witnessed at this Obser- 
 vatory." Over 5000 meteors were counted, and it was estimated 
 that at five A. M. on the 14th the number falling in the whole 
 heavens was about 2500 an hour. Several very brilliant meteors 
 were observed. One in particular was brighter than Jupiter. 
 It was at first of a deep orange color, afterwards green, and finally 
 light blue. It left a train of 7 in length, which passed through 
 the same changes of color, and remained visible for half an 
 hour. The paths of 90 meteors were traced upon a chart, and 
 were found in nearly every instance to start from a point in the 
 constellation Leo. 
 
 259. Height, &c. of the Meteors. Concurrent observations 
 were made at Washington and Richmond, in November, 1867, 
 for the purpose of determining the parallax of the meteors, and 
 thence their distance. It was found that they appeared at an 
 average height of 75 miles, and disappeared at the height of 
 55 miles. The velocity with which they moved relatively to 
 the earth was 44 miles a second. Other observations have given 
 nearly the same results. 
 
 The light of the meteors is probably due to the intense heat 
 generated by the resistance of the air to the progress of these 
 bodies. Notwithstanding the extreme rarity of the air at the 
 height of the meteors, it is still believed that the heat resulting 
 
 from such immense velocity is sufficient to fuse any known sub- 
 is 
 
206 ORBITS OF METEORS. 
 
 stance. A body moving with tlys velocity at the earth's surface 
 would acquire a temperature of at least 3,000,000. An exami- 
 nation of the light of the meteors with the aid of the spectro- 
 scope, by Mr. A. Herschel, showed that some of the meteors 
 were solid bodies in a state of ignition, but that most of them 
 were gaseous. 
 
 260. Orbits of the Meteors. It is noticed that the November 
 meteors, or at all events the great majority of them, seem to 
 come from the same point in the heavens, a point in the con- 
 stellation Leo. So also the August meteors come from a point 
 near the head of the constellation Perseus. Such points are 
 called radiant points. Other showers have also other radiant 
 points, situated in various parts of the heavens. The number 
 of such points now recognized is more than 60. The paths in 
 which meteors having the same radiant point move during the 
 instant of time that we see them, are really parallel straight 
 lines, the apparent convergence of the paths being merely the 
 result of perspective; in other words, the radiant point is the 
 vanishing point (Art. 16) of these parallel lines. 
 
 Knowing the direction and the velocity with respect to the 
 earth of the motion of a meteor, it is easy to compute the same 
 elements of its motion with reference to the sun. The results of 
 such computation, together with the existence of the radiant 
 points and the periodic recurrence of showers, have led to the 
 theory that the November meteors are collected in a ring, or in 
 several rings, or possibly in a series of clusters or groups, which 
 revolve about the sun ; and that the showers occur when the 
 earth encounters these rings or groups. The theory of one ring is 
 exemplified in Fig. 73.* Let ABCD represent the orbit of the 
 earth, and A GBE a ring of meteors revolving about the sun. 
 If A and B are the points at which the earth enters the ring, 
 there will be displays of meteors, when the earth is at these 
 points, similar to the August and the November shower. Unless 
 the plane of the ring coincides with the plane of the ecliptic, 
 there will be no showers at any other points of the earth's orbit. 
 If we imagine the ring to be broken, or to be of unequal 
 
 * Phipson's Meteors, Aerolites, and Falling Stars. London, 1867. 
 
ORBITS OF METEORS. 207 
 
 thickness, or to consist of a series of groups, we may account for 
 the irregularities which exist in the annual showers ; and by 
 
 Fig. 73. 
 
 supposing the ring to have a period of about 331 years, we may 
 account for the extraordinary displays of meteors which happen 
 in about that interval of time. The duration of a shower, and the 
 known velocity of the earth in its orbit, enable us to obtain an 
 approximate value of the breadth of the ring. Professor East- 
 man estimates that the breadth of that portion of the ring through 
 which the earth passed in November, 1868, could not have 
 been less than 115,000 miles. The breadth of the stream in 
 1867 was less than this, but more densely packed with meteors. 
 
 Leverrier, a French astronomer, has computed the elements 
 of the orbit of the November meteors. He finds the major 
 semi-axis to be 10.34, the perihelion distance 0.989 (the radius 
 of the earth's orbit being unity), and the eccentricity 0.9044. 
 This would carry the aphelion beyond the orbit of Uranus, if 
 both orbits were projected upon the plane of the ecliptic. 
 
 According to Professor Loomis, the relative situations of the 
 orbit of the November meteors and the orbits of the earth and 
 the other planets, are represented in Fig. 74 ; the orbit of the 
 meteors being a very eccentric ellipse, the aphelion of which 
 lies beyond the orbit of Uranus, and the period of the me- 
 teors being 331 years. According to the same authority, the 
 August meteors revolve in a similar but much more eccentric 
 ellipse, of which the aphelion lies far beyond the orbit of Nep- 
 tune. The theory has also been advanced that meteors (or, at 
 all events, some of them) are to be regarded as satellites of 
 
208 
 
 DETONATING METEORS. 
 
 ^Orlnt of Uranus? 
 
 Orbit of Jupif er 
 
 the earth rather than of the sum. On this subject Sir John 
 Herschel says, in his Outlines of Astronomy : "It is by no means 
 
 inconceivable that the earth, 
 approaching to such as dif- 
 fer but little from it in direc- 
 tion and velocity, may have 
 attached them to it as per- 
 manent satellites, and of these 
 there may be some so large 
 as to shine by reflected light, 
 and to become visible for a 
 brief moment; suffering, after 
 that, extinction by plunging 
 into the earth's shadow." 
 
 261. Detonating Meteors. 
 The height and the velocity 
 of these bodies are not essen- 
 tially different from those of 
 the ordinary shooting stars. 
 They are, however, generally 
 of an unusual brilliancy, and 
 their appearance is followed 
 by an explosion, or a series 
 of explosions, the intensity 
 of which is sometimes terrific. 
 Records of more than eight 
 hundred detonating meteors 
 are to be found in scientific journals. The phenomena connected 
 with the appearance of these bodies are, however, so nearly iden- 
 tical in character, that one instance may suffice to exemplify all. 
 "On the 2d of August, 1860, about 10 P.M., a magnificent fire- 
 ball was seen throughout the whole region from Pittsburg to 
 New Orleans, and from Charleston to St. Louis, an area of 900 
 miles in diameter. Several observers described it as equal in 
 size to the full moon, and just before its disappearance it broke 
 into several fragments. A few minutes after the flash of the 
 meteor there was heard throughout several counties of Ken- 
 tucky and Tennessee a tremendous explosion, like the sound of 
 
AEK3LITES. 209 
 
 distant cannon. Immediately another noise was heard, not 
 quite so loud, and the sounds were re-echoed with the prolonged 
 roar of thunder. From a comparison of a large number of 
 observations, it has been computed that this meteor first became 
 visible over Northeastern Georgia, about 82 miles above the 
 earth's surface, and that it exploded over the southern boundary 
 line of Kentucky, at an' elevation of 28 miles. The length of 
 its visible path was about 240 miles, and its time of flight eight 
 seconds : showing a velocity relative to the earth of 30 miles per 
 second. It is hence computed that its velocity relative to the 
 sun was 24 miles per second." 
 
 The explosions are probably due to the sudden compression 
 and shocks to which the air is subjected as the meteor rushes 
 through it, as happens when a gun is fired ; or to the rushing 
 of the air into the vacuum which the body creates in its rear. 
 The appearance of these bodies is so sudden, and their velocity 
 so great, that it is almost impossible to obtain any definite value 
 of their magnitude. The diameters of some of them are stated 
 to have been several thousand feet in length, but the estimate 
 must be taken with considerable caution, particularly as it is 
 impracticable to distinguish between the meteor itself and the 
 blaze of light which surrounds it. 
 
 262. Aerolites. Although the ordinary shooting stars some- 
 times appear to break in pieces, there is no evidence that any 
 part of them falls to the earth. But occasionally solid masses 
 of stone or of metallic substances do fall to the earth, their fall 
 being usually preceded by the flash and the discharge of a deto- 
 nating meteor. There is no doubt whatever about the authen- 
 ticity of most of these cases, and the record of them extends 
 far back into ancient history. A fall of meteoric stones near 
 Home, 650 years before Christ, is mentioned by the historian 
 Livy; and a large block of stone is said to have fallen in 
 Thrace, near what is now called the Strait of Dardanelles, 465 
 years before Christ. The entire number of aerolites of which 
 we have any determinate knowledge is more than 400 ; and more 
 than twenty falls of aerolites have occurred in the United States 
 since the beginning of the present century. The British Mu- 
 seum contains a large collection of aerolites, one of which 
 
210 AEROLITES. 
 
 weighs 8287 pounds ; and many other similar specimens are to 
 be found in the cabinets of colleges and museums, both in this 
 country and in Europe. The following are instances of falls 
 Which have occurred since 1800. 
 
 In 1807, on the morning of December 14th, a brilliant meteor, 
 with an apparent diameter equal to about one-half of that of 
 the moon, was seen moving over the 'town of Weston, in the 
 southwestern part of Connecticut. After its disappearance, 
 three loud explosions were heard, followed by a continuous 
 rumbling. Fragments of stone were precipitated to the earth 
 within an area of a few miles in diameter.. The entire weight 
 of these fragments is estimated to be about 300 pounds. One 
 fragment, weighing 36 pounds, is preserved in the museum 
 at Yale College. The specific gravity of the aerolite was about 
 3-}, and among its components were silex, oxide of iron, magne- 
 sia, nickel, and sulphur. 
 
 On the 1st of May, 1860, about noon, there was a number 
 of explosions over the southeastern part of Ohio. Stones were 
 seen to fall to the earth, and in some cases they penetrated the 
 earth to a distance of three feet. About thirty fragments were 
 found, the largest of which weighs 103 pounds, and is to be 
 found in the cabinet of Marietta College. The combined weight 
 of these thirty fragments is not far from 700 pounds, and the 
 specific gravity and the composition are very similar to those of 
 the Weston aerolite. 
 
 Another phenomenon of this character occurred in Piedmont, 
 on February 29, 1868, about the middle of the forenoon. There 
 was a heavy discharge like that of artillery, followed, after a 
 short interval, by a second discharge. A mass of irregular 
 shape was seen in the air, enveloped in smoke, and followed by 
 a long train of smoke. Other bodies, similar in appearance to 
 meteors, were also seen. The analysis of the stones which fell 
 showed the existence in them of the components mentioned 
 above, and also of copper, manganese, and potassium. 
 
 Other aerolites have been subjected to chemical analysis. 
 Of the 65 elementary substances known, 24 at least have been 
 found in aerolites, and no new elements have been discovered. 
 A meteoric shower usually consists of meteoric iron and meteoric 
 
CONNECTION OF COMETS AND METEORS. 211 
 
 Ftone ; the iron is an alloy of which the principal part is nickel, 
 and which also contains cobalt, tin, copper, manganese, and 
 carbon ; the stone contains chiefly those minerals which are 
 abundant in lava and trap-rock. The proportions in which 
 these ingredients enter into the composition of different aerolites 
 differ greatly : sometimes an aerolite contains 96 per cent, of 
 iron, sometimes scarcely any iron at all. A substance called 
 schreibersite, which is a compound of iron, nickel, and phos- 
 phorus, is always found in these bodies. 
 
 The explosion of an aerolite may be due either to the intense 
 heat generated by its rapid motion, or to the pressure to which 
 it is subjected by the resistance of the atmosphere. 
 
 263. Origin of Aerolites. Many theories have been advanced 
 to explain the origin of these bodies. One theory is that they 
 may be formed in the atmosphere by the aggregation of minute 
 particles drawn up from the surface of the earth ; but one objec- 
 tion to this theory is that it does not account for the nearly 
 horizontal direction in which they move, and for the great 
 velocity of their motion. A second theory, that they are thrown 
 from terrestrial volcanoes, is open to the same objection. A 
 third theory, that they may be ejected from the volcanoes of the 
 moon, is weakened by the fact that observation shows no signs 
 (or at least almost no signs) of activity in the lunar volcanoes. 
 The most probable theory is that they are, like the planets and 
 the comets, satellites of the sun, revolving about it in orbits 
 which intersect the orbit of the earth, and that their fall to the 
 earth's surface is either the direct result of their own motion, 
 or is due to the resistance of the atmosphere, and the attraction 
 exerted upon them by the earth. 
 
 264. Possible Connection of Comets and Meteoric Bodies. The 
 facts which have been presented in this chapter in relation to 
 comets and meteoric bodies point to one certain conclusion : 
 that space, or at least that portion of space through which the 
 earth moves, must be considered to be filled with a countless 
 number of comparatively minute bodies, the aggregate mass of 
 which cannot fail to be very great. In 1848, Dr. Mayer, of 
 Germany, advanced a theory that the light and the heat of the 
 sun are caused by the incessant collision of meteoric bodies with 
 
212 CONNECTION OF COMETS AND METEORS. 
 
 its surface. In connection with.this subject, Professor William 
 Thompson states that if the earth were to tall into the sun, th 
 amount of heat generated by the shock would be equal to that 
 which the sun now gives out in 95 years ; and that the planet 
 Jupiter, under similar circumstances, would generate an amount 
 of heat equal to that given out by the sun in 32,000 years. 
 
 There is a striking similarity between the elements of the 
 orbit of TempePs comet and those of the orbit of the Novem- 
 ber shower, mentioned in Art. 260. There is also a similar 
 coincidence in the orbit of the August shower and that of the 
 Great Comet of 1862 ; and the opinion is gaining ground with 
 astronomers, not only that each of these comets leads the group 
 with the elements of whose orbit its own elements so nearly 
 coincide, but also that there is a close connection, generally, 
 between comets and meteoric bodies. The following statement 
 of this new theory is taken from an article by Professor Simon 
 Newcomb, in the North American Review of July, 1868. 
 
 "The planetary spaces are crowded with immense numbers 
 of bodies which move around the sun in all kinds of erratic 
 orbits, and which are too minute to be seen with the most power- 
 ful telescopes. 
 
 "If one of these bodies is so large and firm that it passes 
 through the atmosphere and reaches the earth without being 
 dissipated, we have an aerolite. 
 
 " If the body is so small or so fusible as to be dissipated in 
 the upper regions of the atmosphere, we have a shooting star. 
 
 " A crowd of such bodies sufficiently dense to be seen in the 
 sunlight constitutes a comet. 
 
 "A group less dense will be entirely invisible, unless the 
 earth happens to pass through it, when we shall have a meteoric 
 shower." 
 
 It is not impossible to conceive that the planets themselves 
 may have been formed by the aggregation of these minute 
 bodies, a method of formation exactly the opposite of that 
 which is set forth in the nebular hypothesis. An article in 
 which such an origin is suggested for the planets, the comets, 
 and Saturn's rings, will be found in the North American Review 
 for 1864. 
 
FIXED STARS. 213 
 
 CHAPTER XIV. 
 
 THE FIXED STARS. NEBULA. MOTION OF THE SOLAR SYSTEM. 
 
 265. We have now examined the motions and the orbits of 
 all the known members of the solar system. We have seen that 
 the planets and their satellites, besides their apparent diurnal 
 motion towards the west in orbits whose planes are perpen- 
 dicular to the axis of the celestial sphere, have also independent 
 motions of their own in elliptical orbits, the sun being at the 
 common focus of the orbits of the planets, and each planet being 
 at the common focus of the orbits of its satellites. Besides these 
 bodies, there is a vast number of other bodies visible in the 
 heavens, the phenomena presented by which are radically differ- 
 ent from those which have hitherto been noticed. Continuous 
 observations have been made upon the stars from year to year, 
 and even from century to century ; and it has been found that, 
 after the results of these observations have been freed from the 
 effects of precession and nutation (which, by shifting the position 
 of the points or the planes of reference, may give the stars an 
 apparent motion), the real change of position of the stars is 
 extremely small. Sirius, the brightest star in the whole heavens, 
 has an annual motion of V ; a Centauri, the brightest star in 
 the southern hemisphere, has an annual motion of nearly 4" ; 61 
 Cygni and s Indi, both small stars, have annual motions of 
 5" and 1" respectively. In only about 30 stars, however, has 
 the amount of this change of position been found to be greater 
 than 1" a year ; and in the others, if any motion at all is de- 
 tected, it is only that of a few seconds in a century. These 
 motions are called proper motions, to distinguish them from 
 those which are only apparent, and the stars are called fixed 
 stars : a term which must be understood to imply, not that 
 they have no motion in space, but that whatever motion they 
 
214 FIXED STARS. 
 
 have makes no perceptible alteration in their position upon the 
 celestial sphere. 
 
 When a star moves obliquely to the line joining the earth and 
 the star, its motion in its orbit can be resolved into two motions: 
 one along the line of sight, either directly towards the earth or 
 directly from it, and the other at right angles to that line. This 
 latter motion will cause the star to shift its apparent position 
 upon the celestial sphere, and will be the proper motion above 
 described ; or, as it may be called, the transverse proper motion. 
 Now, it is evident that in order to obtain the real motion of any 
 star in space we must be able to determine, not only its trans- 
 verse motion, but also its motion towards or from the earth. As 
 long as the detection of such a motion depended upon our ability 
 to detect either an increase or a diminution of the star's bright- 
 ness, its immense distance from us rendered the task a hopeless 
 one ; but very recently the spectroscope has afforded us the means 
 of solving this problem. The details of this method will be 
 given at the end of this Chapter. 
 
 266. The Number of the Fixed Stars. The number of stars in 
 the entire sphere which are visible to the naked eye is between 
 6000 and 7000, according to the largest estimate ; but the number 
 of those visible at any one time at any place is less than 3000. 
 By means of telescopes, thousands and even millions of other 
 stars are brought into view, in such numbers as almost to defy 
 any attempt at computation. 
 
 267. Magnitudes. The fixed stars are classified arbitrarily by 
 astronomers according to their relative brightness, the different 
 classes receiving the name of magnitudes, and the first magnitude 
 comprising those stars which are the brightest. Different astro- 
 nomers, however, sometimes assign different magnitudes to the 
 same star. According to Argelander's classification, there are 
 20 stars of the first magnitude, 65 of the second, 190 of the third, 
 425 of the fourth, &c., the numbers in the following magnitudes 
 increasing very rapidly. It is estimated that there are at least 
 20,000,000 stars in the first fourteen magnitudes. Those stars 
 which are visible to the naked eye are comprised in the first six 
 magnitudes ; and stars of the twentieth magnitude are detected 
 with the most powerful telescopes. 
 
CONSTELLATIONS. 215 
 
 There is, however, a great difference in the brightness of stars 
 which belong to the same magnitude. Sirius, for instance, is 
 fifteen times as bright as some other stars of the first magnitude.* 
 
 268. Constellations. In order to facilitate the formation of 
 catalogues of the stars, they are separated into groups, called 
 constellations. Ptolemy, in the second century, enumerated 48 
 constellations: 21 northern, 12 zodiacal, and 15 southern. The 
 twelve zodiacal constellations have the same names that the signs 
 of the zodiac bear, which are given in Art. 91 ; indeed, the signs 
 really took their names from the constellations. Owing, how- 
 ever, to the precession of the equinoxes, the signs and the con- 
 stellations no longer coincide (see Art. 119), the constellation of 
 Aries being in the sign of Taurus, &c. 
 
 Since the time of Ptolemy, about sixty other constellations 
 have been added to the list. Not all of these, however, are ac- 
 cepted by astronomers, and the list of those constellations which 
 are generally acknowledged comprises only about 86 : 29 north- 
 ern, 12 zodiacal, and 45 southern. 
 
 269. Remarkable Constellations. The most remarkable of the 
 northern constellations is that called Ursa Major, or the Great 
 Bear, often called " The Dipper" from the well-known appearance 
 presented by its seven conspicuous stars. The two of these seven 
 
 * Any one ?f the most prominent stars may be identified when on the 
 meridian, in the following manner. The right ascension of the star, given 
 in the Ephemeris, is, by Art. 9, the local sidereal time of the star's transit ; 
 and from this sidereal time the local mean solar time can be obtained, as 
 proved in Art. 105, by subtracting from it the right ascension of the mean 
 sun. When only a rough estimate is desired, the quantity given on page 1 I. 
 of each month in the Ephemeris, in the last column on the right, may be 
 taken as the mean sun's right ascension. This will give the time of transit 
 within four minutes. The more rigorous process is to apply to the quan- 
 tity taken from page II. a correction taken from Table III. in the Appendix 
 to the Ephemeris, using the longitude of the place as an argument, and 
 adding the correction if the longitude is west. The result is then subtracted 
 from the sidereal time as above, and the remainder is diminished by a 
 correction taken from Table II., with the remainder itself as an argument. 
 (See example in the latter part of the Ephemeris.) 
 
 The star's meridian altitude is found by the method given in the note 
 to Art. 181. 
 
216 CONSTELLATIONS. 
 
 stars which are the most remote from the handle of the dipper 
 are called the pointers, since the right line joining them will 
 always, when prolonged, pass very nearly through the pole-star. 
 This constellation contains fifty-three stars of the first five magni- 
 tudes, including one of the first and three of the second. 
 
 There is another "Dipper," much less conspicuous, consisting 
 also of seven stars, the pole-star being at the extremity of the 
 handle. These stars form a part of the constellation Ursa Minor, 
 which contains, in all, twenty-three stars of the first five magni- 
 tudes. The pole-star itself is of the second magnitude. 
 
 The constellation Orion is a magnificent one, and was fan- 
 cifully supposed by the ancients to bear some resemblance to a 
 giant. It contains two stars of the first magnitude, four of the 
 second, and thirty-one of the next three magnitudes. The three 
 stars, situated nearly in a straight line, which form the giant's 
 belt, are a very conspicuous part of the constellation. This 
 constellation, in the northern hemisphere, bears south about 9 P.M. 
 in the early part of February ; its altitude at that time at any 
 place being nearly equal to the co-latitude of that place. Sirius, 
 the brightest star in the heavens, is also seen at that time to the 
 left of this constellation, and a little below it. 
 
 The constellation Pegasus contains forty-three stars of the 
 first five magnitudes. Four of these stars are of the second 
 magnitude, and nearly form a square. This square bears south 
 about 9 P.M. in the early part of October, with an altitude, in 
 the northern hemisphere, about 15 greater than the co-latitude 
 of the place of observation. 
 
 The constellation Gemini, or the Twins, takes its name from 
 two bright stars, nearly of the first magnitude, called Castor 
 and Pollux. They are situated near each other, and bear south 
 about 9 P.M. in the latter part of February, in the northern 
 hemisphere, their altitude being about 30 greater than the co- 
 latitude of the place of observation. 
 
 270. Stars of the Same Constellation. Stars of the same con- 
 stellation are distinguished from each other by the letters of the 
 Greek or the Roman alphabet, or by numerals. The Greek 
 letters were first used by Bayer, a German astronomer, in 1604, 
 who called the brightest star in a constellation a, the next 
 
NAMES OF STARS. 217 
 
 brightest/3, &c. Thus tne poie-srar bears the astronomical name of 
 a Ursx Miiioris, and the pointers of the dipper are called and 
 p 1 Ursse Majoris. Owing, however, either to carelessness on the 
 part of Bayer or to changes in the brightness of some of the stars, 
 this alphabetical arrangement does not in all cases accurately 
 represent the relative brilliancy of the stars in a constellation. 
 
 Tne entire number of stars now catalogued amounts to several 
 hundreds of thousands. Three catalogues published by Arge- 
 lander, in 1859-62, contain over 320,000 stars observed at Bonn. 
 In large catalogues, the stars are usually numbered from be- 
 ginning to end, in the order of their right ascensions. 
 
 271. Stars with Special Names. Some of the stars, particularly 
 the more conspicuous ones, have special names, which were given 
 
 *a Eridani Achernar. 
 
 a Tauri Aldebaran. 
 
 a Aurigse Capella. 
 
 (3 Orionis Rigel. 
 
 *a Argus Canopus. 
 
 a Canis Majoris Sirius. 
 
 a Canis Minoris Procyon. 
 
 p Geminorum Pollux. 
 
 a Leonis Regulus. 
 
 a Virginia Spica. 
 
 a Bootis Arcturus. 
 
 a Scorpii Antares. 
 
 a Lyrae Vega. 
 
 a Aquilae Altair. 
 
 a Piscis Australis Fomalhaut. 
 
 to them by ancient astronomers. Instances are given in the 
 accompanying table, all the stars contained in it being commonly 
 considered to be of the first magnitude. About 90 stars are 
 
 thus named, though most of these names are no longer used. 
 1!) 
 
218 CONSTITUTION OF STAHS. 
 
 .All of these stars, excepting; those marked with an asterisk, 
 come above the horizon throughout the United States. The 
 position of each, in right ascension and declination, can be 
 found in the Ephemeris for any day in the year. Besides the 
 stars in the preceding table, there are three others of the first 
 magnitude: a Crucis, a Centauri, and ft Centauri. They are all 
 stars of large southern declination, and do not come above the 
 horizon in any part of the United States excepting the southern 
 parts of Texas and Florida. 
 
 272. Constitution and Diversity of Brightness. The spectra 
 of stars are found to contain dark lines, similar in character to 
 those by which we have seen that the solar spectrum is distin- 
 guished (Art. 102). These systems of lines differ from the system 
 of lines in the solar spectrum, and they are also different in dif- 
 ferent stars. This difference, however, consists in the absence 
 of certain lines seen in the solar spectrum, and not in the pre- 
 sence of new ones: nearly every line observed having its 
 counterpart in the solar spectrum. The examination of these 
 spectra, and the comparison of the dark lines which they con- 
 tain with the bright lines found in the spectra of terrestrial 
 substances, enable us to establish the presence of certain of 
 these substances in the stars, precisely as was done in the case 
 of the sun. In Aldebaran, for instance, the presence of sodium, 
 magnesium, tellurium, calcium, antimony, iron, bismuth, and 
 mercury, has been detected ; in Sirius, of sodium, magnesium, 
 iron, and hydrogen ; in a Orionis, of magnesium, sodium, cal - 
 cium, and bismuth. 
 
 The diversity of brightness in the stars may be due either to 
 a difference in their distances from us, or to a difference in their 
 actual magnitudes, or to a difference in the intrinsic splendor 
 with which they shine. Probably all these causes exist ; but it 
 is fair to conclude that, as a general rule, the brightest stars are 
 the nearest to us. Observations made for the purpose of deter- 
 mining the distances of the stars go to justify such a conclusion ; 
 although they also show that the rule is not an absolute one, 
 since some of the fainter stars are found to be nearer to us than 
 some of the brighter ones. 
 
 273. Distance of the Fixed /Stars. "N~o perceptible difference 
 
DISTANCE OF THE STARS, 219 
 
 is detected in the position of a star when observed at places of 
 widely different latitudes. The conclusion drawn from this 
 fact is, that the stars are so distant that lines drawn from any 
 two points on the earth's surface to the same star are sensibly pa- 
 rallel : in other words, that the stars have no geocentric parallax. 
 To determine the distance of the stars, then, we must have re- 
 course to their heliocentric parallax. In Fig. 75 let S be the 
 sun, AE'BE the orbit of the earth, s the position of a fixed 
 
 Fig. 75. 
 
 star, supposed to lie in the plane of the ecliptic, and NM a por- 
 tion of the celestial sphere. From s draw lines sE, sE', tan- 
 gent to the earth's orbit, and also prolong them beyond s, until 
 they meet the arc of the celestial sphere NM. Draw the radii 
 vectores SE' and SE to the points of tangency. If we suppose the 
 star to be at rest, it will lie in the direction E's, when the earth 
 is at E', and in the direction Es, when the earth is at E, and 
 the motion of the earth about the sun will give the star an ap- 
 parent oscillatory movement over the arc ee'. The true helio- 
 centric direction in which the star lies is Sa' ; and the difference 
 of the directions in which the star lies at any time from the suri 
 and the earth is its heliocentric parallax. This difference of 
 direction is evidently at its maximum when the earth is at E' or 
 E; and this maximum, or the angle SsE, is called the annual 
 heliocentric parallax, or simply the annual parallax. 
 
 Numerous attempts have been made to determine the annual 
 parallax of the stars, by comparing observations made when the 
 earth is at E and E'. The nicest observation, however, has failed 
 to detect in any star a parallax as great as 1"; and in only 12 
 stars has the slightest appreciable parallax been discovered. 
 
220 DISTANCE OF THE STARS. 
 
 274. The distance of the fixed stars, then, is so great that the 
 radius of the earth's orbit, 92,400,000 miles, does not subtend 
 an angle of even I" at that distance. If, in the triangle SsE, 
 we suppose the angle SsE to be equal to 1", we shall have, 
 
 Sa = 92,400,000 cosec 1": 
 
 which will be found to be about nineteen trillions of miles. This 
 is only the inferior limit of the distance of the stars: that is to 
 say, whatever the distance may be, it cannot be less than this ; 
 but how much greater it may be, particularly in the case of those 
 numerous stars in which no movements whatever of parallax 
 can be detected, it is impossible to calculate. 
 
 275. Immensity of this Distance. It is hardly possible to obtain 
 a clear conception of a distance of nineteen trillions of miles. 
 Perhaps the nearest approach to such a conception is made by con- 
 sidering that light, moving with a velocity of 186,000 miles a 
 second, and passing from the sun to the earth in a little more than 
 eight minutes, would consume about 3^ years in accomplishing 
 such a distance: so that when we look at the brightest stars in 
 the heavens, we see them, not as they are now, but as they were 
 3 years ago; and if any one of them were to be destroyed at 
 any instant, we should continue to see its image for three years 
 and more after that time. 
 
 Before such distances, the dimensions of the solar system 
 shrink to the insignificance of a mere point in space. Neptune, 
 the most distant of the planets, is nearly three billions of miles 
 from the sun ; and yet, if Neptune and the sun could both be 
 seen from the nearest fixed star, the angular distance between 
 them would never be greater than about 30", which is only about 
 6 l th of the angle which the sun subtends to us. 
 
 276. Differential Observations. In dealing with so small a 
 quantity as the annual parallax of the stars, it is important to 
 avoid all circumstances by which even the most minute errors 
 may be entailed upon the observations-. The apparent position 
 of a star is affected, not only by parallax, but by precession, 
 nutation, aberration, and the star's own proper motion. The 
 laws of precession and nutation enable us to decide what amount 
 of the apparent motion is due to them; and the effect of a star's 
 proper motion is also readily separated from that of parallax, 
 
DISTANCE OF THE STARS. 
 
 221 
 
 since the former changes the position of the star from year to 
 year, while the latter only changes its position during the year, 
 causing it to lie now on one side and now on the other of its 
 true position, but giving it no annual progressive motion. But 
 it is not so easy to separate the effects of aberration and paral- 
 lax. Aberration, as we have already seen (Art. 125), causes a 
 star to describe a circle, an ellipse, or an arc, about its true po- 
 sition as a centre, according to its situation with reference to 
 the plane of the ecliptic ; and it is easy to see that the paral- 
 lactic movement of a star is of precisely the same character. 
 Thus we have already seen, in Fig. 75, that a star situated in 
 the plane of the ecliptic will oscillate by parallax over the arc 
 ee ; and if the star is not in the plane of the ecliptic, lines 
 drawn from all points of the earth's orbit to the star, and thence 
 prolonged to the celestial sphere, will evidently meet the sphere 
 in a circle if the star is at the pole of the ecliptic, and in an 
 ellipse if it is not at the pole. 
 
 In order to separate the eifects of parallax and aberration, 
 the following method was adopted by the astronomer Bessel. 
 Instead of attempting to determine by direct observation the 
 change of position of the star whose parallax was sought, he 
 selected another star of much less magnitude, and therefore sup- 
 posed to be at a much greater distance, which lay very nearly in 
 the same direction as the first star, and observed the changes in 
 the distance between these two stars and in the direction of the 
 line joining them, during the year. Fig. 76 will serve to ex- 
 plain the general principle of this 
 method. Let S be the position of the 
 star whose parallax is sought, and s 
 the position of the smaller star, both 
 being projected on the surface of the 
 celestial sphere. By the motion of the 
 earth in its orbit the star $ will describe 
 the parallactic ellipse ADBC, and the 
 star s, the ellipse adbc. When S ap- 
 pears to be at A, s will appear to be at 
 a; when S is at D, s will be at d, &c. 
 It is evident that Aa and Bb will lie 
 
222 MAGNITUDE OF THE STARS. 
 
 in different directions, and that d will be greater than Cc: and, 
 therefore, by observing the different directions of the line joining 
 the two stars, and also its different values, during the year, we 
 may obtain the difference of parallax of the two stars, and, ap- 
 proximately, the parallax of S. 
 
 277. Results. This method was applied by Bessel to the star 
 61 Cygni. For the sake of greater accuracy he made use of 
 two very small stars, situated very near to that star, whose ab- 
 solute parallaxes he assumed to be equal. The parallax which he 
 obtained for this star was 0".35. Other observations make it 
 0".56: and the mean of these two values, or 0".45, corresponds 
 to a distance of about forty-two trillions of miles, a distance 
 which light would require seven years to traverse. The parallaxes 
 of eleven other stars have been obtained with more or less accu- 
 racy. The star about whose parallax there is the least doubt is a 
 Centauri, which is probably the nearest to us of all the stars. Its 
 parallax is 0".92: corresponding to a distance of about twenty- 
 one trillions of miles. , A table of these stars is given at the end 
 of this book. 
 
 According to the Russian astronomer Peters, the mean paral- 
 lax of the stars of the first magnitude is 0".21, corresponding to 
 a distance which light would traverse in 15s years. Another 
 Kussian astronomer, Struve, concludes that the distance of the 
 most remote stars which can be seen in Lord Rosse's great tele- 
 scope is about 420 times the distance of the stars of the first 
 magnitude: from, which the marvellous inference is drawn that 
 the distance of the most remote telescopic stars from the earth is 
 only traversed by light in 6500 years. 
 
 A knowledge of the distance of a star, and of its proper mo- 
 tion, enables us to estimate the amount in miles of its transverse 
 motion. Sirius, for instance, has a proper motion of l".2o a year, 
 and its parallax is 0".23. Hence its annual transverse motion is 
 equal in amount to the radius of the earth's orbit multiplied by 
 i^_5 . w hi c h is a motion of about sixteen miles a second. This is 
 only the projection upon the celestial sphere of its real motion, 
 which may be much greater. 
 
 278. Real Magnitudes of the Stars. Hitherto, when we have 
 determined the distance of a celestial body, we have been able to 
 
VARIABLE STARS. 223 
 
 compute its real diameter by means of observations made upon 
 its angular diameter. But this method fails when we attempt to 
 make use of it in obtaining the magnitude of the stars, since 
 they do not present any measurable disc. It is true that with 
 the better class of telescopes some of the stars appear to have a 
 sensible disc; but this disc is really what is called a spurious 
 one. This is proved by the fact that when a star is occulted by 
 the moon, the size and the shape of the apparent disc remain un- 
 altered up to the time of occultation, and its disappearance is then 
 instantaneous. In the case of a solar eclipse, however, or of the 
 occultation of a planet, the disappearance of the disc is gradual. 
 We can, however, obtain some idea of the probable magni- 
 tude of a star by comparing the light which it emits with that 
 which is emitted by the sun. This comparison is made by 
 means of the light of the moon. The ratio of the light of the 
 sun to that of the full moon, and the ratio of the light of the full 
 moon to that of some of the stars, have been obtained by appro- 
 priate experiments; and, from a comparison of the results of 
 these experiments, it is inferred that if the sun were removed to 
 a distance from the earth equal to that of the nearest fixed star, 
 it would appear only as a star of the second magnitude. The 
 probability, then, is that unless there is a marked difference in 
 the intensity of the light which these different bodies emit, the 
 sun is not so large as most, and perhaps all, of the stars of the 
 first magnitude. There is, however, as might be expected from 
 the delicacy of the observations, some discrepancy in the results 
 of these various photometric experiments : the light of Sirius, for 
 instance, is said by some observers to be one hundred, and by 
 others to be four hundred, times as great as the light of our sun 
 would be, were it removed to a distance from us equal to that 
 of Sirius. 
 
 VARIABLE AND TEMPORARY STARS. 
 
 279. Variable Stars. There are certain stars which exhibit 
 periodic changes in their brightness, the periods being in some 
 cases only a few days in length, and in other cases embracing 
 many years. The star o Ceti, called also Mir a, is an example 
 of this class of stars. AVhen brightest, it is a star of the second 
 
224 TEMPORARY STARS. 
 
 magnitude. It remains in this* state for about two weeks, and 
 then begins to diminish in brightness, becoming wholly invisible 
 to the naked eye in about three months, and appearing in tele- 
 scopes as a star of the ninth or the tenth magnitude. After about 
 five months it again appears, and in three months again reaches 
 its maximum of brightness. The period in which these changes 
 occur is 33H days: at least, that is its mean value: its extreme 
 values being 25 days more and 25 days less than this. It is also 
 noticed that the rate of its increase and decrease of brightness is 
 not always the same, and that one maximum of brightness is 
 not always equal to another. 
 
 Algol) or ft Persei, is another remarkable variable star. It 
 remains for about sixty-one hours as a star of the second magni- 
 tude. At the end of that time it begins to decrease in bright- 
 ness, and becomes a star of the fourth magnitude in less than 
 four hours. After about twenty minutes its brightness begins 
 to increase, and another period of less than four hours brings it 
 up again to a star of the second magnitude. 
 
 The whole number of variable stars is between 150 and 200. 
 
 280. Several theories have been advanced in explanation of 
 this periodicity of brightness in the variable stars. One theory 
 is that the surfaces of these stars are not uniformly luminous, and 
 that therefore, in rotating upon their axes, they may present at 
 one time the lighter portions of their surfaces to the earth, and 
 the darker portions at another. Such a variation of luminosity 
 might be caused by the presence of spots on the surfaces of the 
 stars, similar to the spots on the sun, but of much greater extent. 
 A second theory is, that nebulous bodies may revolve as satellites 
 about these stars, and may intercept their light by coming in be- 
 tween them and the earth. The irregularities noticed in the pe- 
 riods of these stars, however, seem to constitute an objection to this 
 second theory, while they may be allowed by the first theory, since 
 analogous fluctuations in the periods and the magnitudes of the 
 sun's spots have been observed. Arago suggests that, if it is 
 true, as has been asserted by some astronomers, that these stars 
 when at their minimum are surrounded by a kind of fog, the 
 diminution of light may be due to the interference of clouds. 
 
 281. Temporary Stars. There are stars which have appeared 
 
DOUBLE STARS. 225 
 
 at times in different parts of the heavens, and have afterwards 
 disappeared. Such stars are called temporary stars. In compar- 
 ing recent catalogues of stars with the catalogues of ancient as- 
 tronomers, it is found that some stars which were formerly visible 
 are no longer to be seen, while others which are now visible to 
 the naked eye are not mentioned in the ancient catalogues. 
 Some of these cases may be due to errors of observation, but 
 hardly all of them. Moreover, similar instances have occurred 
 in modern times. For example, it is recorded by the Danish 
 astronomer, Tycho Brahe, that in November, 1572, a brilliant 
 star suddenly blazed forth near the constellation Cassiopeia, and 
 remained in sight about 17 months. When at its brightest 
 phase, it equalled Venus in splendor, and was visible in broad 
 daylight. It disappeared in 1574, and has never since been 
 seen. Similar temporary stars are recorded as having appeared 
 in or near the same place, in 945 and 1264. It is therefore 
 possible that this may be really a variable star, and that it may 
 reappear in the latter part of the present century. 
 
 In May, 1866, a star of the ninth magnitude, in the constel- 
 lation Corona Borealis, suddenly increased in brightness, and 
 then rapidly decreased. On the 12th of the month it was of the 
 second magnitude ; on the 14th, of the third ; and the rate of 
 decrease in its brightness was for some time about half a mag- 
 nitude a day. Lockyer says that there is good reason to believe 
 that this sudden increase of brilliancy was due to the ignition 
 of hydrogen in the star's atmosphere. 
 
 It is very probable that these temporary stars are really in 
 no respect different from variable stars, except in the length of 
 their periods. Sir John Herschel says, with reference to these 
 stars, that it is worthy of notice that all of them which are on 
 record have been situated in or near the borders of the Milky 
 Way. 
 
 DOUBLE AND BINARY STARS. 
 
 12 Lyncis. e Lyrae. 
 
 282. Double Stars. Many of the stars which appear single 
 
226 .BINARY STARS. 
 
 to the naked eye, are found, "wjien examined in telescopes, to 
 consist of two stars, apparently very near to each other. These 
 are called double stars. Only four were known to exist until near 
 the close of the last century, when Sir William Herschel dis- 
 covered about 500. The whole number of double stars now 
 known exceeds 9000. In some cases the two stars are nearly 
 of the same magnitude, but more frequently one of them is a 
 large star, and the other a small one. Castor is an instance of 
 the former class, and Sirius, Vega, and the pole-star are instances 
 of the latter class. Some stars are found to consist of three, four, 
 five, or more stars, and are called triple, quadruple, &c. stars. 
 The star e Lyrse, for instance, appears in ordinary telescopes to 
 consist of two stars, but with telescopes of greater power each 
 of these stars is resolved into two others. 
 
 283. Binary Stars. The question arises with reference to these 
 combinations : are the stars which compose them really con- 
 nected, as are the sun and the planets ; or is their appearance 
 merely an optical illusion, arising from the fact that" the stars 
 in any one combination happen to lie in the same direction 
 from the earth, although they may at the same time be at an 
 immense distance from each other? The chances, at all events, 
 are very much against the latter supposition. The astronomer 
 Struve has calculated that there is only about one chance in 9570 
 that, if the stars of the first seven magnitudes were scattered at 
 random in the heavens, any two of them would fall within 4" 
 of each other; and yet more than 100 such cases have been 
 observed. He has further calculated that there is only about one 
 chance in 200,000 that three stars would accidentally fall within 
 30" of each other, so as to form a triple star; and at least four 
 such cases are to be found. 
 
 The chances, then, are that the stars in these various combi- 
 nations are physically connected. But more than this : it was 
 announced by Sir William Herschel in 1803, after twenty-five 
 years of observation upon many of the double stars, that in 
 each double star which he had examined, the two stars of which 
 it was composed revolved about each other in regular orbits, 
 and in fact constituted a sidereal system. Subsequent observa- 
 tions bj other astronomer? have fully verified this conclusion, 
 
BINARY STARS. 227 
 
 anrl about 600 double stars have been found to consist of stars 
 revolving about each other, or rather about their common centre 
 of gravity, according to the Newtonian law of gravitation. 
 Such double stars are called binary stars, to distinguish them 
 from other double stars, the components of which have not as 
 yet been found to be physically connected. There are also 
 other double stars, the components of which, while as yet they 
 do not seem to revolve about each other, have constantly the 
 same proper motion : thus- showing that they are in all probability 
 moving as one system through space. Triple, quadruple, &c. 
 stars, whose constituents are found to be physically connected, 
 are called ternary, quaternary, &c. stars. 
 
 The orbits of the binary stars are found to be ellipses of con- 
 siderable eccentricity. The periods of their revolutions have 
 also been approximately determined, and extend over a very 
 wide range. There are only about twelve stars whose periods 
 are less than 100 years, and only about 150 whose periods are 
 less than 1000 years. 
 
 284. Alpha Centauri. The star a Centauri, a star of the first 
 magnitude in the southern hemisphere, is found to consist of 
 two components, one of the first magnitude and the other of the 
 second. The relative positions of these two components have 
 been carefully noted durirg the last 40 yeacs. 
 In Fig 77, A represents one of those compo- 
 nents, and B B B" the apparent path of the 
 other about it. The major axis of the orbit 
 is about 40'', and the period 75 years. Its 
 eccentricity is 0.63. 
 
 Since the distance of a CeiJtauri from the 
 earth is approximately known, we can ob- 
 tain some idea of the dimensions of the 
 orbit. If E in the figure represents the earth, 
 we shall have, in the right-angled triangle 
 CO E, the angle E equal to 20", and the side ? 
 
 EO equal to 21 trillions of miles. Hence, Fig. 77. 
 
 CO = OEx tan 20" = 2,000,000,000 miles : 
 which is about equal to the distance of Uranus from the sun, or 
 to 22 times the radius of the earth's orbit. 
 
228 
 
 COLORED STATES. 
 
 Again, knowing the radius o/ the orbit and the period, we 
 can obtain an approximate value of the mass of a Centauri 
 from the formula in Art. 214. It will be found to be about 
 twice the mass of the sun. 
 
 COLORED STARS. 
 
 285. Many of the stars, both isolated and double, shine with 
 a colored light. The isolated colored stars are usually red or 
 orange : blue or green stars being very uncommon. The num- 
 ber of red stars now recognized is at least 300, about 40 of 
 which are visible to the naked eye. Among the most con- 
 spicious of the red stars are Aldebaran and Antares ; and it is 
 worthy of notice that nearly all the variable stars are of this 
 color. According to Mr. Ennis's observations, Capella, Kigel, 
 Procyon, and Spica are blue; Sirius, Vega, and Altair are green; 
 and Arcturus is yellow. 
 
 The components of the double stars are often of different co- 
 lors; blue and yellow, or green and yellow; and, less frequently, 
 white and purple, or white and red. The following table con- 
 tains a few of the many instances of such stars which might be 
 given : 
 
 Name. 
 
 Magnitude 
 of 
 Components. 
 
 Color of the Larger. 
 
 Color of the Smaller. 
 
 7} Cassiopeiae 
 a Piscium 
 
 4 7 
 5 6 
 
 Yellow. 
 Pale Green. 
 
 Purple. 
 Blue. 
 
 i Cancri 
 e Bootis 
 C Coronae 
 pCygni 
 
 5 8 
 3 7 
 5 6 
 3 7 
 
 Orange. 
 Pale Orange. 
 White. 
 Yellow. 
 
 Blue. 
 Sea Green. 
 Purple. 
 Blue. 
 
 When the colors of the components are complementary, and the 
 components are of very unequal size, it is possible that only one 
 of the colors may really exist; the other being, according to a 
 law of Optics, merely the result of contrast. Such an opinion is 
 held by some astronomers ; but the objection is raised to it by 
 others that, if one of these colors is only accidental, it ought to 
 disappear when the eye is shielded from the light of the star 
 
PLATE IV. 
 
 NEBULJE. 
 
 1. ANNULAR NEBULA, 57 M LYR/. 
 
 2. PLANETARY NEBULA, 3614 H VIRGINIS. 
 
 3. NEBULOUS STAR, i ORIONIS. 
 
 4. SPIRAL NEBULA, 99 M VIRGINIS. 
 
 5. CRAB NEBULA IN TAURUS. 
 
NEBULAE. 229 
 
 which has the other color: which, however, is very far from 
 Jteing the case. Another objection is that a similar phenomenon 
 ought to be seen in all colored stars whose components are of 
 miaqual sizes: whereas many double stars are found in which 
 both components have the same color, sometimes red and some- 
 times blue. In one of Struve's catalogues, out of 596 double 
 stars, there were: 
 
 295 pairs, both white: 
 118 pairs, both yellowish or reddish: 
 63 pairs, both bluish : 
 120 pairs of totally different colors. 
 
 In a few instances stars have changed their color. Sirius, 
 for instance, is now green; but both Ptolemy and Seneca ex- 
 pressly state that in their day it had a reddish hue. Capella, 
 which is now blue, was formerly red. Some observers state that 
 seventeen stars of the first magnitude are colored, and that seven 
 of these have changed their color. This change of color is par- 
 ticularly .noticeable in the variable stars. In that of 1572 (Art. 
 281), the color changed from white to yellow, and then to a de- 
 cided red; and it is observed generally in the variable stars that 
 the redness of the light increases as its intensity diminishes. 
 
 CLUSTERS AND NEBULAE. 
 
 286. If we examine the heavens on any clear night when the 
 moon is below the horizon, we shall find here and there groups 
 of stars, which present a hazy, cloud-like appearance. These 
 groups are classified into clusters and nebulce, although it is diffi- 
 cult to establish any precise distinction between the two classes. 
 When the different members of a group/ or at all events some 
 of them, can be separated from each other with the naked eye, the 
 group is called a cluster. A nebula, on the other hand, is either 
 wholly invisible to the naked eye, or, if seen at all, presents, as 
 the name implies, only an ill-defined, cloudy appearance. The 
 number of nebulae which have been discovered is over 5000. 
 Some of these nebulae are resolved, with the aid of the telescope, 
 into separate stars: while others, even when examined in the 
 most powerful telescopes, preserve their cloudy appearance, and 
 
230 
 
 give no trace of stars. The former are called resolvable nebuja;, 
 the latter irresolvable nebulae. 
 
 As every increase of telescopic power has rendered resolvable 
 some of the nebulse previously classified as irresolvable, the com- 
 mon opinion of astronomers, until within the last few years, was 
 that there was no distinction between these nebulse more funda- 
 mental than that of distance, magnitude, or intensity of light. 
 Recently, however, the light of some of these nebulas has been 
 subjected to spectroscopic analysis, and the results seem to show 
 the existence of decidedly different classes of nebulse. The re- 
 searches of Mr. Huggins show that the light of some of these 
 nebulse gives spectra, which are distinguished, not by dark lines, 
 but by bright ones : which shows, as we have already noticed 
 (Art. 102), that the light does not come from solid bodies in a 
 state of ignition, but from incandescent vapor or gas. Some at 
 least, then, of the nebulse are to be considered as bodies of a 
 gaseous nature ; and there are indications that the gases of which 
 they are composed are hydrogen and nitrogen. The Great Ne- 
 bula in Orion is an instance of this class of nebulse. 
 
 Th(> nebulse are very unequally distributed over the heavens. 
 They congregate especially in a zone crossing the Milky Way at 
 right angles, and they are especially abundant in the constella- 
 tions Leo, Ursa Major, and Virgo. 
 
 287. Examples of Clusters. The most noticeable cluster is the 
 well-known group called the Pleiades. This group contains six 
 or seven stars which are visible to the naked eye, the brightest 
 star being of the third magnitude : and glimpses of many more 
 can be obtained by examining the group with the eye turned 
 sideways. With the aid of a telescope between one hundred and 
 two hundred stars can be detected. 
 
 The Hyades are a group in Taurus, near the star Aldebaran, in 
 which from five to seven stars can be counted. This group was 
 supposed by the ancients to have some influence upon the rain. 
 
 Prsesepe, or the Bee-hive, in Cancer, is visible to the naked eye 
 as a luminous spot. With a telescope of moderate power, more 
 than forty conspicuous stars are seen within a space about 30' square, 
 together with many smaller ones. Before the invention of the te- 
 lescope, this group was probably one of the few recognized nebulse. 
 
NEBULAE. 231 
 
 288. Different Forms of Nebulce. The groups which usually 
 go by the name of nebulae present, as might be expected, almost 
 every variety of form ; and most of these forms are too irregular 
 to admit of any classification or description. Such is not the case, 
 however, with all of them : and some of the different varieties of 
 shape are exemplified in the following list: 
 
 1. Annular Nebulae; 
 
 2. Elliptic Nebulas; 
 
 3. Spiral Nebulise; 
 
 4. Planetary Nebulas; 
 
 5. Nebulous stars. 
 
 There are also individual nebulae whose names indicate the 
 general appearance which they present : such are the " Horse-shoe 
 Nebula," the "Crab Nebula," the "Dumb-bell Nebula," &c. 
 
 289. Annular Nebulae. Of annular or ring-shaped nebulae, 
 the heavens present four examples. The most remarkable one is 
 situated in the northern constellation Lyra. Sir John Herschel 
 says of it: "It is small, and particularly well-defined, so as 
 to have more the appearance of a flat, oval, solid ring than of* a 
 nebula. The axes of the ellipse are to each other in the pro- 
 portion of about four to five, and the opening occupies rather 
 more than half the diameter. The central vacuity is filled iu 
 with faint nebulas, like a gauze stretched over a hoop. The 
 powerful telescopes of Lord Rosse resolve this object into ex- 
 cessively minute stars, and show filaments of stars adhering to 
 its edges." 
 
 290. Elliptic Nebulce. There are several instances of elliptic 
 or oval nebulas, of various degrees of eccentricity. The very 
 conspicuous nebula called " the Great Nebula in Andromeda" is 
 an example of this class. This object is distinctly visible to the 
 naked eye, and is sometimes mistaken for a comet. When viewed 
 with a telescope of moderate power, it has an elongated oval 
 form ; but in the largest telescopes its aspect is very different 
 from this. Professor G. P. Bond traced it through a length of 
 4, and a breadth of 2^, and also discovered in it two curious 
 black streaks, lying nearly parallel to the major axis of the 
 oval. He also succeeded in detecting in it evidence of a stellar 
 constitution. 
 
232 NEBULAE. 
 
 Some elliptic nebulae are remarkable for having double stars 
 at or near each of their foci. 
 
 291. Spiral Nebulas. Observations with the Earl of Rosse's 
 telescope have led to the discovery of nebulae which consist 
 of spiral bands proceeding from a common nucleus, and some- 
 times from two nuclei. The most remarkable of these spiral 
 nebulae is situated near the extremity of the tail of the Great 
 Bear. It consists of nebulous coils diverging from two luminous 
 centres, about 5' apart, and gives evidence of stellar composition. 
 
 292. Planetary Nebulce. Planetary nebulae received their name 
 from Sir William Herschel, on account of the resemblance in form 
 which they bear to the larger planets of our system. The out- 
 lines of some of them are well defined, while those of others are 
 indistinct; and some of them shine with a blue light. About 
 twenty-five have been discovered, most of them being situated 
 in the southern hemisphere. One of these nebulae is situated 
 near ft Ursse Majoris, and has a diameter of 2' 40". It was dis- 
 covered by Me*ehain, in 1781, and is described as "a very singular 
 object, circular and uniform, which after a long inspection looks 
 like a condensed mass of attenuated light." Perforations and a 
 spiral tendency have been detected in it by the Earl of Rosse. 
 
 These planetary nebulae can hardly be globular clusters of 
 stars, as they would in that case be brighter in the middle than 
 at the borders, instead of being, as they are, uniformly bright 
 throughout. Some astronomers suppose that they are assem- 
 blages of stars, arranged either in cylindrical beds, or in the 
 form of hollow spherical shells. 
 
 293. Nebulous Stars. Nebulous stars are stars which are sur- 
 rounded by a faint nebulous envelope, usually circular in form, 
 and sometimes several minutes in diameter. In some cases this 
 envelope terminates in a distinct outline, while in others it gra- 
 dually fades away into darkness. According to Hind, nebulous 
 stars "have nothing in their appearance to distinguish them from 
 others entirely destitute of such appendages ; nor does the nebu- 
 lous matter in which they are situated offer the slightest indica- 
 tions of resolvability into stars, with any telescopes hitherto 
 constructed." 
 
 294. Double Nebulce. M. D'Arrest, of Copenhagen, mentione 
 
VARIATION OF BRIGHTNESS. 233 
 
 fifty double nebulae whose components are not more than 5' apart, 
 and estimates that there may be as many as 200 such double 
 nebulse. The two components are generally of a globular form. 
 It is possible that these components may in some cases be 
 physically connected. In one instance there seem to have been 
 changes in the distance and the relative position of the two mem- 
 bers during the last eighty years, which indicate the possibility 
 of a revolution of one of the members about the other. 
 
 295. Magellanic Clouds. These two nebulse, called also nube- 
 cula major and nubecula minor, are situated near the southern 
 pole of the heavens. They are visible to the naked eye at night, 
 when the moon is not shining, and have an oval shape. One 
 of them covers an area of forty-two square degrees, the other an 
 area of ten square degrees. Sir John Herschel found them to 
 consist of swarms of stars, clusters, and nebulse of every 
 description. 
 
 296. Variation of Brightness in the Nebulce. In the case of a 
 few of the nebulse a change of brightness has been discovered, 
 which is to some extent analogous to what has already been 
 noticed in connection with the variable and the temporary stars. 
 In 1852, Mr. Hind discovered a small nebula in the constella- 
 tion Taurus. This nebula was observed from that time until 
 the year 1856; but in 1861 it had entirely disappeared. In 
 1862 it was seen in the telescope at Pulkowa, and for a short 
 time appeared to be increasing in brightness; but towards the 
 end of 1863, Mr. Hind, upon searching for it with the telescope 
 with which it was originally discovered, was unable to find any 
 trace of it. It is a curious coincidence that a star situated very 
 near to this nebula has changed, in the same interval of time, 
 from the tenth magnitude to the twelfth. 
 
 There are four or five instances on record of similar changes 
 in the brightness of nebulse. The nebula known as 80 of 
 Messier's Catalogue, is said to have changed, in the months 
 of May and June, 1860, from a nebula to a well-defined star 
 of the seventh magnitude, and then to have resumed its ori- 
 ginal appearance. The change was, however, so rapid and so 
 decided, that some astronomers are inclined to ascribe it to the 
 existence of a variable star between the nebula and the earth, 
 
234 MILKY WAY. 
 
 rather than to a variation in tflie nebula itself. The Great Ne- 
 bula in the constellation Argo, when observed by Sir John Her- 
 schel in 1838, contained in its densest part the star y, which was 
 then of the first magnitude. Observations made in 1863, how- 
 ever, showed that this star had become entirely free from its 
 nebulous envelope, and had also been reduced to a star of the 
 sixth magnitude. The outline of certain parts of the nebula 
 was also found to be different from what it had been represented 
 to be by Herschel. More recent observations show still further 
 changes in this nebula. 
 
 297. The Galaxy, or Milky Way. The Milky Way, that well- 
 known luminous band which stretches through the heavens, may 
 be considered to be a continuous resolvable nebula. If we follow 
 the line of its greatest brightness, and neglect occasional devia- 
 tions, we find its course to be nearly that of a great circle of the 
 heavens, inclined to the equinoctial at an angle of about 63. 
 At one point a kind of branch is sent off, which unites again 
 with the main stream at a distance of about 150 from the point 
 of separation. The angular breadth of the belt varies from 6 
 to 20, the average breadth being about 10. When examined 
 with the telescope, this band is found to be made up of thousands 
 and millions of telescopic stars, grouped together with every 
 degree of irregularity. Of the 20,000,000 stars of the first four- 
 teen magnitudes, about nine-tenths are in or near the Milky Way. 
 At some points the stars are so close to each other that they 
 form one bright mass of light; while at other points there are 
 dark spaces containing scarcely a star which is visible to the 
 naked eye. A very noticeable break of this kind occurs in that 
 part of the Milky Way which lies nearest to the south pole. 
 This dark space is *about 8 in length and 5 in breadth. It 
 contains only one star visible to the naked eye, and that is a very 
 small one. Early southern navigators gave to this vacancy the 
 name of the Coal /Sack. 
 
 298. Number of the Stars. Scarcely more can be said of the 
 probable number of the stars, than that it is as inconceivably 
 great as are the distances by which they are separated from the 
 earth and from each other. Sir William Herschel states that 
 on one occasion, while observing the stars in the Milky Way, he 
 
MOTION OF THE SOLAR SYSTEM. 235 
 
 estimated that 116,000 stars passed through the field of his tele- 
 scope in fifteen minutes; and that, on another occasion, nearly 
 260,000 stars passed in forty-one minutes. It may assist us in 
 correctly appreciating the magnitude of these numbers to re- 
 member that the number of stars visible to the naked eye in 
 the whole heavens is less than 7000 (Art. 266). Results still 
 more astonishing follow from an examination of the nebulae. 
 Take, for instance, the " Great Nebula in Andromeda," which, 
 by observations made with the telescope at Cambridge, Mas- 
 sachusetts, within the last few years, has been found to give 
 evidence of stellar composition. According to Professor G. P. 
 Bond, its length is 4, and its breadth 2J. Now, if a telescope 
 just suffices to resolve a nebula into separate stars, it is safe to 
 assume that the angular distance between any two contiguous 
 stars in it cannot be greater than 1". If, then, we multiply to- 
 gether the number of seconds in the length of this nebula, and 
 the number of seconds in its breadth, we obtain over 129,000,000 
 for an approximate estimate of the number of stars in this one 
 nebula. 
 
 MOTION OF THE SOLAR SYSTEM IN SPACE. 
 
 299. We have now accustomed ourselves to consider the earth 
 to be a planet, rotating upon a fixed axis once in every sidereal 
 day, and revolving about the sun once in every sidereal year; but 
 we have up to this point considered the sun and the solar sys- 
 tem to be at rest in space. There are, however, observations 
 which tend to show that such is not, in truth, the case, but 
 that the whole solar system is in rapid motion through space. 
 Analogy certainly supports such a conclusion. "VVe have already 
 seen that the solar system, immense as it may seem to be in 
 itself, must be considered as scarcely more than a point in com- 
 parison with the entire celestial system (Art. 275); and we 
 have also seen reasons for concluding (Art. 278) that there are 
 many bodies among the stars which are very much larger 
 than the sun. There is, then, no reason for supposing that the 
 solar system is the centre of the celestial system; and there is 
 nothing violent in the conclusion that, just as Jupiter revolves 
 about the sun, and carries its satellites with it, so the sun may 
 
236 
 
 MOTION OF THE SOLAR SYSTEM. 
 
 revolve about some other body or'point, and carry its system of 
 planets and satellites with it. 
 
 300. The Apparent Motiom of the Stars. The proper motions 
 of the stars, already mentioned, may be explained in three dis- 
 tinct ways. First, they may be what they seem to be, real 
 motions of the stars, performed in orbits of immense radii; or, 
 secondly, they may be only apparent motions, caused by a change 
 of the position of the solar system in space; or, thirdly, they 
 may be the results of both these causes existing together. The 
 last of these theories was advanced by Sir William Herschel, 
 in 1783 ; and although doubts were afterwards expressed as to 
 its truth, later observations have tended to support it, and it is 
 now generally accepted by astronomers. If we suppose for a 
 moment that the solar system is approaching any given point 
 in the heavens, the pole-star, for instance, the result will be that 
 the stars which lie about that point will appear to separate 
 slowly from it, while the stars which lie about the diametrically 
 
 opposite point will appear to close up around that point ; and in- 
 deed all the stars will apparently move in an opposite direction 
 to that in which we suppose the solar system to be moving, those 
 
MOTION OF THE SOLAR SYSTEM. 237 
 
 stars having the greatest motion which lie in a direction at right 
 angles to the direction of the motion of the solar system. In 
 Fig. 78, let E be the position in space of the earth at any time, 
 and let P,A,B,D, and G be stars supposed to be situated at equal 
 distances from the earth. Let the earth, by the motion of the 
 whole solar system in space, be carried to some point E', in the 
 direction of the star P. It is evident that the angular distance 
 of the star A from P, when the earth is at E', or the angle PE'A, 
 is greater than the angular distance between A and P when the 
 earth is at E, or the angle PEA. In other words, while the 
 earth has moved from E to E', the star A has apparently receded 
 from P. So also the star D has apparently approached the star 
 C. The stars B and G have botli receded from P, the retro- 
 gradation of G, or the angle EGE', being greater than the 
 retrogradation of B, or the angle EBE'. 
 
 301. Direction and Amount of this Motion. The elements, 
 then, of the motion of the solar system through space are deter- 
 mined from what are called the proper motions of the stars, 
 llecent observations and calculations have not only confirmed 
 Herschel's theory that such a motion really exists, but have also 
 very nearly confirmed his conclusion as to the point towards which 
 the motion is directed. Different astronomers give different 
 positions to this point; but they all agree in the general conclu- 
 sion, that the solar system is at present moving in the direction 
 of a point in the constellation Hercules, situated in about 17 
 hours of right ascension, and 35 of north declination. 
 
 The calculations of M. Otto Struve lead him to the conclu- 
 sion that for a star situated at right angles to the direction of 
 the motion of the solar system and at the mean distance of the 
 stars of the first magnitude, the annual angular displacement 
 of the star due to that motion is 0."34 : that is to say, the dis- 
 tance through which the system moves in one year subtends at 
 the star an angle of that amount. Now, the mean parallax of 
 the stars of the first magnitude, or the angle subtended at the 
 mean distance of those stars by the radius of the earth's orbit, is 
 estimated tobeO."21 (Art. 277); hence the annual motion of the 
 solar system is the radius of the earth's orbit multiplied by Jf, 
 or about 150,000,000 miles: a little less than five miles a second. 
 
238 ORBIT OF THE SOLAR SYSTEM. 
 
 This is the estimate commonly accepted by astronomers. Mr. 
 Airy, however, deduces a motion of about twenty -seven miles a 
 second. 
 
 302. The Orbit of the Solar System. Although the motion of 
 the solar system through space appears to be rectilinear, it does 
 not follow that such is actually the case: since it may move in 
 an elliptical orbit, with a radius vector so immense that years 
 and even centuries of observation will be needed to show any 
 sensible change of direction. The probability is that the sun, 
 with its planets and their satellites, revolves about the common 
 centre of gravity of the group of stars of which it is a member; 
 and that this centre of gravity is situated in or near the plane 
 of the Milky Way. If we suppose the orbit in which our 
 system is moving to be an ellipse with a small eccentricity, then 
 the centre of this ellipse will lie in a direction which makes an 
 angle of about 90 with the direction in which the system is now 
 moving. Miidler concluded that Alcyone, the brightest star in 
 the Pleiades, was the central sun of the celestial sphere, while 
 Argelander, believing that this central point must lie in the 
 principal plane of the Milky Way, places it in the constellation 
 Perseus. It may be noticed, in connection with this subject, that 
 Sir William Herschel was inclined to believe that some of the 
 more conspicuous of the stars, such as Sirius, Arcturus, Capella, 
 Vega, and others, were in a great degree out of the reach of 
 the attractive power of the other stars, and were probably, like 
 the sun, centres of systems. 
 
 303. Years, and, in all probability, ages of observation will 
 be needed to determine the elements of this most stupendous 
 orbit. Madler's speculations have led him to the conclusion 
 that the period of this revolution is 29,000,000 years : a period 
 in comparison with which the years through which astronomical 
 observations have extended are utterly insignificant. The stu- 
 dent who is curious to know more of this subject will find the 
 details fully set forth in Grant's History of Physical Astronomy. 
 Grant himself says, in reference to the subject: "It is manifest 
 that al\ such speculations are far in advance of practical astro- 
 nomy, and therefore they must be regarded as premature, how- 
 ever probable the supposition on which they are based, or how- 
 
REAL MOTIONS OF THE STARS. 239 
 
 ever skilfully they maybe connected with the actual observation 
 of astronomers." 
 
 DETERMINATION OF THE REAL MOTIONS OF THE STARS. 
 
 304. We have already seen (Art. 265), that in order to deter- 
 mine the real motion of a star in space, we must be able to 
 determine, not only its transverse motion, which is indicated by 
 a change in its apparent position upon the celestial sphere, but 
 also its motion directly to or directly from the earth. Now, a 
 star situated at the nearest distance of the fixed stars, and 
 moving towards the earth with a velocity equal to that of the 
 earth in its orbit (18 miles a second), would diminish its dis- 
 tance from us by only about - 3 ^th in a thousand years. The 
 detection of any such motion by a change in the star's apparent 
 brightness is, therefore, utterly out of the question. The spectro- 
 scope, however, gives us quite another means of detecting such 
 a motion. 
 
 305. Analogy Between Light and Sound. A clear conception 
 of the principle upon which this method rests may be more 
 readily obtained if we first notice the analogy between the con- 
 duction of light and that of sound. Sound is the result of a 
 series of waves or pulses in the air, produced by the vibrations 
 of a sonorous body ; light is the result of a similar series of 
 pulses or waves in the luminiferous ether, caused by the vibra- 
 tions of the particles of a luminous body. The more rapidly 
 a sonorous body vibrates, the greater will be the number of 
 pulses or waves which it communicates to the air in the unit of 
 time, and, consequently, the higher will be the pitch of the tone 
 produced. In the case of a luminous body, the greater the 
 number of waves in the luminiferous ether which the vibrations 
 of any particle cause in the unit of time, the greater will be the 
 refrangibility of the ray produced ; in the solar spectrum, for 
 instance, the violet rays are the most refrangible of all the rays 
 which we can see, and the number of waves in the unit of time 
 necessary to produce them is greater than the number necessary 
 to produce rays of any other color. The refrangibility of a ray 
 is therefore analogous to the pitch of a tone. 
 
 306. Change, of Tone or Pefrnngibilihj. It is proved by ex- 
 
240 REAL MOTIONS OF THE STARS. 
 
 periment that if the distance between a sonorous body, producing 
 a tone of constant pilch, and the hearer, is diminished by the 
 motion of either, with a velocity that has an appreciable ratio to 
 that of sound, the pitch of the tone will appear to be heightened ; 
 and that if, on the contrary, the distance between the two is in- 
 creased with sufficient rapidity, the pitch of the tone will appear 
 to be lowered. So, too, in the case of light: if the luminous 
 body and the observer approach each other, the refrangibility 
 of the rays of light will be increased ; and if they separate, it 
 will be diminished. If, then, we can detect any change of re- 
 frangibility in the light of a heavenly body, we may conclude 
 that the distance of that body from the earth is changing. 
 
 307. Change of Refrangibility Detected. Father Secchi, in a 
 communication to the Comptes Rendus in the early part of 1868, 
 after stating the principles of this method, announced that he 
 had subjected the light of Sirius and of other prominent stars to 
 an examination with the spectroscope, but had detected no evi- 
 dence of motion. Since then, Mr. Huggins, an English observer, 
 who has devoted himself especially to spectroscopic investiga- 
 tions, has succeeded in detecting such a motion in Sirius. There 
 is a certain dark line F in the blue of the solar spectrum, which 
 corresponds to a bright line in the spectrum of hydrogen ; and 
 this same line also appears in the spectrum of Sirius. Now, 
 if Sirius has no motion either towards or from the earth, the 
 line F in its spectrum will coincide in position with the corres- 
 ponding line in the spectrum of hydrogen, when the two spectra 
 are compared by means of the spectroscope: while if, on the 
 contrary, Sirius has such a motion, its whole spectrum, lines 
 and all, will be shifted bodily, and the line F will no longer 
 coincide with the corresponding line in the spectrum of hydrogen. 
 
 Mr. Huggins, using a spectroscope of large dispersive power, 
 and carefully comparing the spectrum of Sirius with that of 
 hydrogen, finds that the line F in the spectrum of Sirius is 
 displaced, by about ^th of an inch. This displacement is 
 towards the red end of the spectrum, and indicates that the 
 refrangibility of the light of Sirius is diminished : since the 
 red rays are the least refrangible of all the colored rays of the 
 spectrum. Sirius is therefore receding from the earth. 
 
REAL MOTIONS OF THE STARS. 241 
 
 308. Amount of the Real Motion of Sirius. The amount of 
 recession corresponding to a displacement of this extent, when 
 observed in a spectroscope whose power is equal to that of the 
 one used by Mr. Huggins, is computed to be about 41 j miles 
 a second. But it happens that when the observation was 
 made, the earth was so situated in its orbit that it was receding 
 from Sirius, by its revolution in its orbit, at the rate of about 
 12 miles a second. The motion of the solar system in space, 
 computed to be five miles a second (Art. 301), must also be 
 taken into consideration ; and the point towards which this 
 motion is at present directed is almost exactly opposite to the 
 position of Sirius on the celestial sphere. The earth was there- 
 lore moving away from Sirius at the rate of about 17 miles a 
 second. If, then, we diminish the whole amount of the increase 
 of distance between Sirius and the earth in one second, by the 
 amount of the motion of the earth through space in one second, 
 or 17 miles, we find that Sirius is moving through space, away 
 from the earth, at the rate of about 24A miles a second. 
 
 By combining this motion with the transverse motion of 
 Sirius, we can obtain an approximate value of its real motion. 
 In Art. 277, the transverse motion of Sirius was computed to 
 be about 16 miles a second. The resultant of these two motions 
 is a motion of about 29 miles a second : or, 900,000,000 miles 
 a year. 
 
 Mr. Huggins further concludes that five stars of the Dipper 
 are receding from us, and that the bright star Arcturus is ap- 
 proaching us. 
 
 309. The numerical results of the preceding article may not 
 be beyond criticism ; but the grand fact remains, that in all 
 probability the motions of these distant bodies, which have so 
 long seemed to be wrapped in hopeless mystery, are soon to 
 come within the reach of our observation. The knowledge of 
 these motions will be a powerful auxiliary in the determination 
 of the law which undoubtedly binds all the heavenly bodies 
 together in one great system ; and it is not presumptuous to 
 expect that at some future day no one can say how distant or 
 how near this law will be revealed to us. 
 
 21 
 
242 APPENDIX. 
 
 APPENDIX. 
 
 MATHEMATICAL DEFINITIONS AND FORMULA. 
 
 PLANE TRIGONOMETRY. 
 
 1. The complement of an angle or arc is the remainder ob- 
 tained by subtracting the angle or arc from 90. 
 
 2. The supplement of an angle or arc is the remainder ob- 
 tained by subtracting the angle or arc from 180. 
 
 3. The reciprocal of a quantity is the quotient arising from 
 
 dividing 1 by that quantity : thus the reciprocal of a is 
 
 4. In the series of right triangles ABC, AB'C', AB"C", &c., 
 
 Fig. 79, having a common angle A, we have by 
 
 Geometry, 
 
 BC __ B'C' _ B"C" 
 AB ~ AB' ~ = AB"' 
 BC __ B'C' B"C" 
 AC~ AC' r ~ AC"' 
 
 41L4K _ AB " 
 
 ~AC~ AC' ~~ AC*' 
 
 The ratios of the sides to each other are there- 
 Fig. 79. ore ^ ne same j n a n right triangles having the 
 same acute angle : so that, if these ratios are known in any one 
 of these triangles, they will be known in all of them. These 
 ratios, being thus dependent only on the value of the angle, 
 without any regard to the absolute lengths of the sides, have 
 received special names, as follows : 
 
 The sine of the angle is the quotient of the opposite side 
 
 BC 
 divided by the hypothenuse. Thus, sin A = -T-TV 
 
 The tangent of the angle is the quotient of the opposite side 
 
 BC 
 
 divided by the adjacent side. Thus, tan A = 
 
 -A \j 
 
TRIGONOMETRY. 243 
 
 The secant of the angle is the quotient of the hypothenuse 
 divided by the adjacent side. Thus, sec A = 
 
 5. The cosine, cotangent, and cosecant of the angle are respec- 
 tively the sine, tangent, and secant of the complement of the 
 angle. Now, in Fig. 79, the angle ABC is evidently the com- 
 plement of the angle BA C. Hence we have, 
 
 . p AC 
 cos A = sin B = -T-= ; 
 
 AC 
 BC 
 AB 
 
 cot A = tan B = j^^, 
 
 cosec A sec B = 
 
 6. Since the reciprocal of y- is , we see, from the preceding 
 
 o a 
 
 definitions, that the sine and the cosecant of the same angle, the 
 tangent and the cotangent, the cosine and the secant, are re- 
 ciprocals. 
 
 7. If two parts of a plane right triangle in addition to the right 
 angle are given, one of them being a side, the triangle can be 
 solved: that is to say, the values of the remaining parts can be 
 obtained. This solution is effected by means of the definitions 
 above given. 
 
 8. When an angle is very small, its sine and its tangent are 
 both very nearly equal to the arc which subtends the angle in 
 the circle whose radius is unity. Hence, to find the sine or the 
 tangent of a very small angle approximately, we have, if a; is a 
 small angle expressed in seconds, 
 
 sin x = tan x = x sin 1". 
 If x is expressed in minutes, 
 
 sin x = tan x = x sin 1'. 
 
 9. If x and y are any two small angles, we have from the pre- 
 ceding article, 
 
 sin x x sin 1" _ x . 
 
 sin y y sin V y ' 
 
 that is, the sines (or tangents) of very small angles are propor- 
 tional to the angles themselves. 
 
 10. Cos x = 1 2 sm 2 J x. 
 
244 
 
 APPENDIX. 
 
 11. The sides of a plane triangle are proportional to the sinea 
 of their opposite angles. 
 
 12. In order to solve a plane oblique triangle, three parts must 
 be given, and one of them must be a side. 
 
 SPHERICAL TRIGONOMETRY. 
 
 13. A spherical triangle is a triangle on the surface of a sphere, 
 formed by the arcs of three great circles of the sphere. 
 
 14. In a spherical right triangle, the sine of either oblique 
 angle is equal to the quotient of the sine of the opposite side, 
 divided by the sine of the hypothenuse. Thus, in the triangle 
 
 B ABC, right-angled at C, Fig. 80, we have, 
 
 sin EC 
 sin AB ' 
 
 15. The cosine of either oblique angle is 
 equal to the quotient of the tangent of the 
 adjacent side, divided by the tangent of the 
 Thus, 
 
 tan A C 
 
 sin A =. 
 
 Fig. 80. 
 
 hypothenuse. 
 
 cos A = 
 
 tan AB 
 
 16. The tangent of either oblique angle is equal to the quo- 
 tient of the tangent of the opposite side, divided by the sine of 
 the adjacent side. Thus, 
 
 tan BC 
 
 tan A -. T-^ 
 
 sin AC 
 
 17. A spherical right triangle can be solved when any two 
 parts in addition to the right angle are given. The solution is 
 effected by means of the formula} given in the preceding articles. 
 
 ANALYTIC GEOMETRY. 
 
 18. The circle, the ellipse, the hyperbola, and the parabola are 
 
 often called conic sections; since each 
 curve may be formed by intersecting 
 a right cone by a plane. 
 
 19. An ellipse is a plane curve, in 
 which the sum of the distances of each 
 point from two fixed points is equal to 
 a given line. Thus, in Fig. 81, the 
 
ANALYTIC GEOMETRY. 
 
 245 
 
 sum of the distances of the point M from the two fixed points F 
 and F' is always constant, wherever on the curve ACBD the 
 point M may be situated. 
 
 The two fixed points are called the foci of the ellipse, and the 
 middle point of the line which joins them is called the centre. 
 
 A line drawn from either focus to any point of the curve is 
 called a radius vector. 
 
 A line drawn through the centre, and terminating at each ex- 
 tremity in the curve, is called a diameter. That diameter which 
 passes through the foci is called the transverse or major axis: 
 and that diameter drawn perpendicular to the transverse axis is 
 called the conjugate or minor axis. Thus, AB is the transverse 
 axis, and CD the conjugate axis. The transverse axis is equal 
 to the constant sum of the distances FM and F'M. 
 
 The eccentricity of the ellipse is the quotient of the distance 
 from the centre to either focus, divided by half the major axis. 
 
 OF 
 
 Thus, -y= is the eccentricity. 
 
 20. The solid generated by the revolution of an ellipse about 
 its major axis is called a prolate spheroid: that generated by its 
 revolution about its minor axis is called an oblate spheroid. 
 
 The expression for the volume of an oblate spheroid is ^xa^b : 
 in which a and b denote the semi-major and the semi-minor axis 
 of the generating ellipse, and x the ratio of the circumference 
 of a circle to its diameter, or 3.1416. 
 
 The compression, or oblateness, of an oblate spheroid is the 
 ratio of the difference between the major and the minor axis of 
 the generating ellipse to the major axis. 
 
 21. The hyperbola is a plane curve, in which the difference of 
 the distances of each point 
 
 from two fixed points is 
 equal to a given line. 
 Thus, in Fig. 82, the dif- 
 ference of the distances of 
 any point M of the curve 
 from the two fixed points 
 F and F' is equal to a 
 given line. 
 
246 
 
 APPENDIX. 
 
 The two fixed points are called the foci, and the middle point 
 of the line joining them is called the centre. 
 22. The parabola is a plane curve, every 
 point of which is equally distant from a 
 fixed point, and from a right line given in 
 the plane. Thus, in Fig. 83, in which AB 
 is the given line, and F the given point, the 
 distances FM and GM are equal to each 
 other, for any point M of the curve. 
 Fig. ss. The fixed point is called the focus. 
 
 MECHANICS. 
 
 23. The resultant of two or more forces is a force which singly 
 produce the same mechanical effect as the forces themselves 
 
 jointly. 
 
 The original forces are called components. In all statical in- 
 vestigations the components may be replaced by their resultant, 
 and vice versa. 
 
 24. If two forces be represented in magnitude and direction 
 by the two adjacent sides of a parallelogram, the diagonal will 
 represent their resultant in magnitude and direction. Thus, in 
 
 Fig. 84, if a force act at A 
 in the direction AD', and a 
 second force act at A in the 
 direction AB', these two forces 
 being represented in magni- 
 tude by the lengths of the 
 lines AD and AB respectively, 
 the resultant of these two 
 forces will be a force in the 
 direction A C f , and of a magnitude represented by the length 
 of the line AC. The parallelogram is called the parallelogram 
 of forces. 
 
 25. Conversely: any given force may be resolved into two 
 component forces, acting in given directions. Thus, in Fig. 84, 
 let A C be the given force, and AB' and AD' the given directions. 
 From C draw CB parallel to AD' } and CD parallel to AB'. 
 
247 
 
 AB and AD will be the two components acting in the given 
 directions. 
 
 26. The force which must continually urge a material point 
 towards the centre of a circle, in order that it may describe the 
 circumference with a uniform velocity, is equal to the square of 
 the linear velocity divided by the radius. This proposition is 
 expressed in the following formula ; 
 
 v being the space passed over in the unit of time, r the radius of 
 the circle, and / the magnitude of the force. 
 
 27. The force which constantly urges a body towards the 
 centre of its circular path is called a centripetal force. The 
 tendency which the body has to recede from the centre, in con- 
 sequence of its inertia, or the resistance which it offers to a de- 
 flection from a rectilinear path, the resistance being estimated 
 in the direction of the radius, is called a centrifugal force. These 
 two forces are in equilibrium as long as the body moves in the 
 same circular path, and the same expression serves for the mea- 
 sure of each force. 
 
 28. Let t be the periodic time, or the time of one revolution. 
 We shall evidently have, 
 
 vt = 2-r; 
 
 Substituting this value of v 2 in the formula in Art. 26 above, we 
 shall have, 
 
 KIRKWOOD'S LAW. 
 
 In 1848, Daniel Kirkwood, of Pennsylvania, announced the 
 discovery of an analogy in the distances, masses, periods of ro- 
 tation, and periods of revolution of the principal planets. This 
 analogy is known under the name of Kirkwood' s Law. The 
 
248 APPENDIX. 
 
 original statement of it will be found in the American Journal 
 of Science, Vol. ix., Second Series, and is as follows: 
 
 " Let P be the point of equal attraction between any planet 
 and the one next interior, the two being in conjunction: P' that 
 between the same and the one next exterior. 
 
 "Let also D = the sum of the distances of the points P, P' 
 from the orbit of the planet, which I shall call the diameter 
 of the sphere of the planet's attraction : 
 
 " jy = the diameter of any other planet's sphere of attraction 
 found in like manner: 
 
 "n = the number of sidereal rotations performed by the 
 former during one revolution around the sun: 
 
 " n f = the number performed by the latter ; then it will be 
 found that 
 
 n 2 : n /f = D 5 : D' 3 ; or n = ri , 
 
 That is, the square of the number of rotations made by a planet 
 during one revolution round the sun, is proportional to the cube 
 
 n 
 of the diameter of its sphere of attraction: or j^ is a constant 
 
 quantity for all the planets of the solar system." 
 
 This analogy, when first announced, created much interest 
 among scientific men, many of whom considered that, if its truth 
 were established, it would be a powerful argument in favor of the 
 nebular hypothesis. There is so much uncertainty attending the 
 determination of the masses and the periods of rotation of many 
 of the planets, that the statement of this analogy has been ex- 
 cluded from the main text of this book. 
 
 ASTRONOMICAL CHRONOLOGY. 
 
 The science of Astronomy seems to have been cultivated in 
 very early ages by almost all the Eastern nations, particularly 
 by the Egyptians, the Persians, the Indians, and the Chinese. 
 Records of observations made by the Chaldseans as far back as 
 2234 B.C. are said to have been found in Babylon. The study 
 of Astronomy was continued by the Chaldseans, and in later times 
 
ASTRONOMICAL CHRONOLOGY. 249 
 
 by the Greeks and the Romans, until about 200 A.D. After that 
 time it was neglected for about six centuries, and was then re- 
 sumed by the Eastern Saracens after Bagdad was built. It was 
 brought into Europe in the thirteenth century by the Moors of 
 Barbary and Spain. A full account of the rise and the progress 
 of the study of Astronomy, is given in Vinee's Astronomy, 
 Vol. IT. 
 
 The instrument principally used by the ancient astronomers 
 seems to have been a sort of quadrant, mounted in a vertical 
 position. Ptolemy describes one which he used in determining 
 the obliquity of the ecliptic. The Arabian astronomers had one 
 with a radius of 21 f feet, and a sextant with a radius of 57 f feet. 
 
 The following dates, with scarcely any change or addition, 
 have been taken from George F. Chambers's Descriptive Astro- 
 nomy (Oxford, 1867), in which many other interesting astrono- 
 mical dates, here omitted, may be found^ [See Note, page 254.] 
 
 B.C. 
 
 720. Occurrence of a lunar eclipse, recorded by Ptolemy. It 
 seems to have been total at Babylon, March 19, 9ih. P.M. 
 
 640-550. Thales, of Miletus, one of the seven wise men of Greece. 
 He declared that the earth was round, calculated solar 
 eclipses, discovered the solstices and the equinoxes, and re- 
 commended the division of the year into 365 days. 
 
 585. Occurrence of a solar eclipse, said to have been predicted 
 by Thales. 
 
 580-497. Pythagoras, of Crotona, founder of the Italian Sect. 
 He taught that the planets moved about the sun in elliptic 
 orbits, and that the earth was round; and is said to have 
 suspected the earth's motion. 
 
 547. Anaximander died. He asserted that the earth moved, and 
 that the moon shone by light reflected from the sun. He 
 introduced the sun-dial into Greece. 
 
 500. Parmenides, of Elis, taught the sphericity of the earth. 
 
 490. Alcmreon, of Crotona, asserted that the planets moved from 
 west to east. 
 
 450. Diogenes, of Apollonia, stated that the inclination of the 
 earth's orbit caused the change of seasons. Anaxagoras is 
 said to have explained eclipses correctly. 
 
250 APPENDIX. 
 
 B.C. 
 
 432. Melon introduced the luni-solar period of 19 years. He 
 observed a solstice at Athens in 424. 
 
 384-322. Aristotle wrote on many physical subjects, including 
 Astronomy. 
 
 370. Eudoxus introduced into Greece the year of 3651 days. 
 
 330. Callippus introduced the luni-solar period of 76 years. 
 Pytheas measured the latitude of Marseilles, and showed 
 the connection between the moon and the tides. He also 
 showed the connection of the inequality of the days and 
 nights with the differences of climate. 
 
 320-300. Autolychus, author of the earliest works on Astronomy 
 extant in Greek. 
 
 306. First sun-dial erected in Rome. 
 
 287-212. Archimedes observed solstices, and attempted to mea- 
 sure the sun's diameter. Aristarchus declared that the 
 earth revolved about the sun, and rotated on its axis. 
 
 276-196. Eratosthenes measured the obliquity of the ecliptic, 
 and found it to be 20 2 degrees. He also measured a degree 
 of the meridian with considerable exactness. 
 
 190-120. Hipparchus, called the Newton of Greece. He dis- 
 covered precession : used right ascensions and declinations, 
 and afterwards latitudes and longitudes : calculated eclipses : 
 discovered parallax : determined the mean motions of the sun 
 and the moon; and formed the first regular catalogue of the 
 stars. 
 
 50. Julius Caesar, with the Egyptian astronomer Sosigenes, under- 
 took to reform the calendar. 
 
 A.D. 
 
 100-170. Ptolemy, of Alexandria, author of the Ptolemaic Sys- 
 tem of the Universe, in which the earth is the centre. He 
 wrote descriptions of the heavens and the Milky Way, 
 and formed a catalogue giving the positions of 1022 fixed 
 stars. He appears to have been the first to notice the re- 
 fraction of the atmosphere. 
 
 642. The School of Astronomy, at Alexandria, founded ten 
 centuries previously by Ptolemy the Second, was destroyed 
 by the Saracens under Omar. 
 
 762. Rise of Astronomy among the Eastern Saracens. 
 
ASTRONOMICAL CHRONOLOGY. 251 
 
 A.n. 
 880. Albatani, the most distinguished astronomer between Hip- 
 
 parchus and Tycho Brahe. He corrected the values of pre- 
 cession and the obliquity of the ecliptic, formed a catalogue 
 of the stars, and first used sines, chords, &c. 
 
 1000. Abul-Wefa first used secants, tangents, and cotangents. 
 
 1080. The use of the cosine introduced by Geber, and also some 
 improvements in Spherical Trigonometry. 
 
 1252. Alphonso X., King of Castile, under whose direction cer- 
 tain celebrated astronomical tables, called the Alphomine 
 Tables, were compiled. 
 
 1484. Waltherus used a clock with toothed wheels. (The earliest 
 complete clock of which there is any certain record was 
 made by a Saracen in the thirteenth century.) 
 
 1543. Publication of Copernicus's De Revolutionibus Orbium Ce- 
 lestium, setting forth the Copernican System of the Universe. 
 
 1581. Galileo determined the isochronism of the pendulum. 
 
 1582. Tycho Brahe commenced astronomical observations near 
 Copenhagen. 
 
 1594. Gerard Mercator, author of Mercator's Projection. (The 
 
 date is doubtful, and may have been as early as 1556.) 
 1576. Fabricius discovered the variability of o Ceti. 
 
 1603. Bayer's Maps of the Stars published. 
 
 1604. Kepler obtained an approximate value of the correction 
 for refraction. 
 
 1608. Jansen and Lippersheim, of Holland, are said to have 
 invented the refracting telescope, using a convex lens. It 
 is, however, a disputed point as to who really invented the 
 telescope. The use of the lens seems to have been known 
 about fifty years before this. 
 
 1609. Kepler announced his first two laws. 
 
 1610. Galileo, using a telescope with a concave object-lens, dis- 
 covered the satellites of Jupiter, the librations of the moon, 
 the phases of Venus, and some of the nebulae. 
 
 1611. Spots and rotation of the sun discovered by Fabricius. 
 1614. Napier invented logarithms. 
 
 1618. Kepler announced his third law. 
 
 . The first recorded transit of Mercury, observed by Gas- 
 sendi. The vernier invented. 
 
252 APPENDIX. 
 
 A.D. 
 
 1633. Galileo forced to deny the Copernican theory. 
 
 1639. First recorded transit of Venus, observed by Horrox aiil 
 Crabtree. 
 
 1640. Gascoigne applied the micrometer to the telescope. 
 1646. Fontana observed the belts of Jupiter. 
 
 1654. The discovery of Saturn's rings by Huyghens. (Galileo 
 had previously noticed something peculiar in the planet's 
 appearance.) 
 
 1658. Huyghens made the first pendulum-clock. (The discovery 
 is also ascribed to Galileo the younger.) 
 
 1662. Royal Society of London founded. 
 
 1 663. Gregory invented the Gregorian reflecting telescope. 
 
 1664. Hook detected Jupiter's rotation. 
 
 1666. Foundation of the Academy of Sciences at Paris. 
 1669. Newton invented the Newtonian reflecting telescope. 
 
 1673. Flamsteed explained the equation of time. 
 
 1674. Spring pocket-watches invented by Huyghens. (Said also 
 to have been invented, somewhat earlier, by Dr. Hooke.) 
 
 1675. Transmission of light discovered by Romer. Transit In- 
 strument used to determine right ascensions by Romer. 
 Greenwich Observatory founded. 
 
 1687. Newton's Principia published. 
 
 1690. Ellipticity of the earth theoretically determined by Huy- 
 ghens. 
 
 1704. Meridian Circle used by Romer. 
 
 1711. Foundation of the Royal Observatory at Berlin. 
 
 1725. Compensation pendulum announced by Harrison. 
 
 1726. Mercurial pendulum invented by Graham. 
 
 1727. Aberration of light discovered by Bradley. 
 1731. Hadley's Quadrant invented. 
 
 1744. Euler's Theoria Motuum published, the first analytical 
 work on the planetary motions. 
 
 1745. Nutation of the earth's axis discovered by Bradley. 
 1750. Wright's Theory of the Universe published. 
 
 1765. Harrison rewarded by Parliament for the invention of the 
 
 Chronometer. 
 
 1767. British Nautical Almanac commenced. 
 1769. Transit of Venus successfully observed. 
 
ASTRONOMICAL CHRONOLOGY. 253 
 
 A.D 
 
 1774. Experiments by Maskelyne to determine the earth's den- 
 sity. 
 
 1781. Uranus discovered by Sir William Herschel. Messier's 
 catalogue of Nebulae published. 
 
 1783. Herschel suspected the motion of the whole solar system. 
 
 1784. Researches of Laplace on the stability of the solar system. 
 
 1786. Publication of Herschel's catalogue of 1000 nebulae. 
 
 1787. Herschel began to observe with his forty-feet reflector. The 
 Trigonometrical Survey, of England commenced. 
 
 1788. Publication of La Grange's Mecanique Analytique. 
 
 1789. The rotation of Saturn determined by Herschel, and a 
 second catalogue of 1000 nebulae published. 
 
 1792. Commencement of the Trigonometrical Survey of France. 
 1796. French Institute of Science founded. 
 1799. Laplace's Mecanique Celeste commenced. 
 1801-7. The minor planets Ceres, Pallas, Juno, and Vesta dis- 
 covered. 
 
 1802. Publication of Herschel's third catalogue of Nebulae. 
 
 1803. Publication of Herschel's discovery of Binary Stars. 
 
 1804. Proper motions of 300 stars published by Piazzi. 
 
 1805. Commencement of attempts to determine the parallax of 
 the stars. 
 
 1812. Troughton's Mural Circle erected at Greenwich. 
 
 1820. Foundation of the Royal Astronomical Society of London. 
 
 1821. Observatory at Cape of Good Hope founded. 
 1823. Cambridge (England) Observatory founded. 
 1835. Airy determined the time of Jupiter's rotation. 
 
 1837. Value of the moon's equatorial parallax determined by 
 Henderson. The East Indian arc of 21 21' completed. 
 
 1838. Parallax of 61 Cygni determined by Bessel. 
 
 1839. Parallax of a Centauri determined by Henderson. Im- 
 perial Observatory at Pulkowa (Russia) founded. 
 
 1840. Harvard Observatory founded. 
 
 1842. Washington Observatory founded. Mean density of the 
 earth determined by Baily. 
 
 1843. Periodicity of the solar spots detected by Schwabe. 
 
 1844. Taylor's catalogue^of 11,015 stars. Transmission of time 
 by electric signals commenced in the United States. 
 
254 APPENDIX. 
 
 A.D. 
 
 1845. Discovery of the fifth minor planet, Astrsea. (187 othera 
 have since been discovered.) 
 
 1846. Discovery of the planet Neptune. 
 
 1847. Lalande's catalogue of 47,390 stars republished by the 
 British Association. 
 
 1851. Foucault's pendulum experiment to demonstrate the earth 'a 
 rotation. Completion of the Russo-Soandinavian arc. 
 
 1854. Difference of longitude of Greenwich and Paris deter- 
 mined by electric signals. 
 
 1855. Commencement of the American Ephemeris. 
 
 1858. De La Rue obtained a stereoscopic view of the moon. 
 (The first photograph of the moon was made by Dr. J. W. 
 Draper, of New York, in 1840.) 
 
 1859. Publication of Section I. of Argelander's " Zones", con- 
 taining 110,982 stars. Suspected discovery of the planet 
 Vulcan, lying within the orbit of Mercury. Completion of 
 the Berlin Star Charts commenced in 1830. 
 
 1861. Appearance, in June, of a comet with a tail of 105 (the 
 longest on record). Publication of Section II. of Argelan- 
 der's " Zones", containing 105,075 stars. 
 
 1862. Section III of Argelander's "Zones", containing 108,129 
 stars. Notes on 989 Nebulae, by the Earl of Rosse. Bond's 
 Memoir on Donati's Comet of 1858, published in the Annals 
 of the Harvard Observatory. 
 
 1863. Announcement by several computers that the received 
 value of the sun's parallax is too small by about 0".3. 
 Spectroscopic observations of celestial objects, by Huggins 
 and Miller. 
 
 1864. Publication of Sir John Herschel's great catalogue of 
 5079 nebulae. 
 
 1806. Magnificent display of shooting-stars on the morning of 
 November 14th. 
 
 .NOTE. The third edition of Chambers' s Descriptive Astronomy (Oxford, 
 J877) contains an exceedingly interesting and elaborate summary of As- 
 tronomical Chronology. 
 
.NAVIGATION. 255 
 
 NAVIGATION. 
 
 THE earliest accounts of Navigation appear to be those of 
 Phoenicia, 1500 B.C. Long voyages are mentioned in the earliest 
 mythical stories ; but the first considerable voyage of even pro- 
 bable authenticity seems to be that of the Phoenicians about 
 Africa, 600 B.C. The Roman navy dates from 311 B.C., and 
 that of the Greeks from a much earlier time. After the decay 
 of these nations, commerce passed into the hands of Genoa, 
 Venice, and the Hanse towns ; from them it passed to the Portu- 
 guese and the Spanish ; and from them again to the English and 
 the Dutch. 
 
 The attractive po^ser of the magnet has been known from 
 time immemorial ; but its property of pointing to the north was 
 probably discovered by Roger Bacon in the thirteenth century. 
 When first used as a compass, the needle was placed upon two 
 bits of wood, which floated in a basin of water ; and the method 
 of suspending it dates from 1302. The variation of the compass 
 was discovered by Columbus, in 1492. The compass-box and 
 the hanging-compass were invented by the Rev. William Bar- 
 lowe, in 1608. 
 
 Plane charts were used about 1420. The discovery that the 
 loxodromic curve is a spiral was made by Nonius, a Portuguese, 
 in 1537. The log is first mentioned by Bourne, in 1577. Loga- 
 rithmic tables were applied to Navigation by Gunter, in 1620; 
 and middle-latitude sailing was introduced three years after- 
 wards. Other dates relating to the subject of Navigation are 
 given in the Astronomical Chronology. 
 
256 
 
 APPENDIX. 
 
 TABLE I. 
 
 THE PLANETS, THE SUN, AND THE MOON. 
 
 
 
 Inclina- 
 
 Eccentri- 
 
 Greatest 
 
 Least dis- 
 
 Mean dis- 
 
 
 Name. 
 
 Symbol. 
 
 the 
 
 
 city 
 
 of 
 
 dista 
 
 nee 
 
 tance f 
 
 rom 
 
 tance 
 
 from 
 
 Mean distance 
 
 
 
 Ecliptic. 
 
 Orbit. 
 
 from Sun. 
 
 Sun. 
 
 Su 
 
 a. 
 
 from Suu in miles. 
 
 
 
 6 t 
 
 ! 
 
 
 
 
 
 
 
 
 
 
 Mercury 
 
 
 
 70008 
 
 0.20560 
 
 0.46669 
 
 0.30750 
 
 0.38710 
 
 35,760,000 
 
 Venus .. 
 
 9 
 
 3 23 31 
 
 0.00684 
 
 0.72826 
 
 0.71840 
 
 0.72333 
 
 66,820,000 
 
 Earth ... 
 
 rjy or Q 
 
 
 
 0.01679 
 
 1.01679 
 
 0.98321 
 
 1.00000 
 
 92,380,000 
 
 Mars 
 
 rf 
 
 1 51 05 
 
 0.09326 
 
 1.66578 
 
 1.38160 
 
 1.52369 
 
 140,800,000 
 
 Jupiter. 
 
 "4 
 
 1184C 
 
 0.04824 
 
 5.45378 
 
 4.95182 
 
 5.20280 
 
 480,600,000 
 
 Saturn... 
 
 h 
 
 2292S 
 
 0.05560 
 
 10.07328 
 
 9.00442 
 
 9.53885 
 
 881,200,000 
 
 Uranus.. 
 
 (?) or $ 
 
 46 3C 
 
 0.04658 
 
 20.07612 
 
 18.28916 
 
 19.18264 
 
 1,772,000,000 
 
 Neptune 
 
 W 
 
 1465S 
 
 0.00872 
 
 30.29888 
 
 29.77506 
 
 30.03697 
 
 2,775,000,000 
 
 Moon.... 
 
 
 
 5084C 
 
 0.05491 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 Name. 
 
 Sidereal Period 
 in days. 
 
 DailyHe- 
 
 liocentric 
 Motion. 
 
 Synodic 
 Period 
 in days. 
 
 Max. Diam- 
 eter from . 
 
 Min. Diam- 
 eter from 
 
 Diam. at 
 mean 
 distance. 
 
 Diameter 
 in miles. 
 
 
 
 o t n 
 
 
 
 tt 
 
 
 ff 
 
 
 tt 
 
 
 Mercury 
 
 87.969 
 
 40533 
 
 115 
 
 877 
 
 12.9 
 
 
 04- ^ 
 
 06.7 
 
 3,000 
 
 Venus... 
 
 224.701 
 
 13608 
 
 583.921 
 
 1' 06.3 
 
 
 09'.7 
 
 17.1 
 
 7,600 
 
 Earth ... 
 
 365.256 
 
 5908 
 
 
 
 
 
 
 
 
 7,925.6 
 
 Mars 
 
 686.980 
 
 3127 
 
 779 
 
 936 
 
 30.1 
 
 
 04.1 
 
 11.1 
 
 4,100 
 
 Jupiter. 
 
 4,332.585 
 
 459 
 
 398 
 
 884 
 
 50.6 
 
 
 30.S 
 
 37.2 
 
 89,000 
 
 Saturn... 
 
 10,759.220 
 
 201 
 
 378 
 
 092 
 
 20.3 
 
 
 
 16.1 
 
 72,000 
 
 Uranus.. 
 
 30,686.821 
 
 42 
 
 369.656 
 
 04.3 
 
 
 O3.c 
 
 03.9 
 
 33,000 
 
 Neptune 
 
 60,126.722 
 
 22 
 
 367 
 
 488 
 
 02.9 
 
 
 02.6 
 
 02.7 
 
 37,000 
 
 Moon.... 
 
 27.322 
 
 
 29.531 
 
 33' 31 
 
 28' 48 
 
 31 '07 
 
 2,161.6 
 
 Sun 
 
 
 
 
 
 32 35.6 
 
 31 
 
 31 
 
 32 03.6 
 
 861,400 
 
 
 
 Volume. 
 
 Mass. 
 
 Density. 
 
 Rotation. 
 
 Inclination 
 
 App. Diam. 
 
 Name. 
 
 of A> 
 
 tis to 
 
 of Sun 
 
 
 Ecliptic. 
 
 fromPlanet. 
 
 
 
 
 
 
 
 
 d. h 
 
 m. 
 
 a. 
 
 
 
 t tt 
 
 Mercury 
 
 0.052 
 
 Jffn7J(TTT 
 
 2.02 
 
 24 05 
 
 30 
 
 
 
 82 49 
 
 Venus... 
 
 0.851 
 
 ^oo'c 
 
 "Fff 
 
 0.90 
 
 23 21 
 
 23 
 
 73 
 
 32' 
 
 44 19 
 
 Earth ... 
 
 1.000 
 
 32Ti 
 
 
 1 
 
 00 
 
 23 56 
 
 04 
 
 66 
 
 33 
 
 32 04 
 
 Mars 
 
 0.239 
 
 
 
 0.50 
 
 24 37 
 
 23 
 
 61 
 
 18 
 
 21 02 
 
 Jupiter . 
 
 1,387.431 
 
 To X 
 
 f 
 
 0.23 
 
 < 
 
 ) 55 
 
 21 
 
 86 
 
 55 
 
 6 10 
 
 Saturn... 
 
 746.898 
 
 ri 
 
 J-Q 
 
 0.13 
 
 10 14 
 
 24 
 
 62 
 
 
 3 22 
 
 Uranus.. 
 
 72.359 
 
 Tn 
 
 OU 
 
 
 
 18 
 
 
 
 
 
 
 1 40 1 
 
 Neptune 
 
 98.664 
 
 
 OfJ 
 
 
 
 18 
 
 
 
 
 
 
 1 04 
 
 Moon.... 
 
 0.020 
 
 2T7J fj TToTT 
 
 0.63 
 
 27 07 43 
 
 11 
 
 88 
 
 30 
 
 
 Sun 
 
 1,284,000 
 
 
 
 
 0-25 
 
 25 07 48 
 
 82 
 
 30 
 
 
TABLES. 257 
 
 Too much confidence must not be placed in the absolute accuracy of all 
 the elements above tabulated. The necessity of this caution will become 
 obvious to any one who will compare the values of these elements as they 
 are given by different authorities. The relative distances, the apparent 
 diameters, and the periods of the planets, being matters of direct observa- 
 tion, are known to a great degree of accuracy ; but the absolute distances 
 and diameters, and the volumes, depending as they do upon the distance 
 of the earth from the sun, must be considered to be only approximately 
 known. The masses, too, of some of the planets are uncertain : and so also 
 must be the densities, which depend upon the masses. 
 
 TABLE II. 
 
 THE EARTH. 
 
 ~ 
 
 Density 5.67 (water being 1) 
 
 Polar Diameter 7899.170 miles. 
 
 Equatorial " 7925.648 
 
 Compression 0.00334 
 
 Length of sidereal year 365d. 06h. 09m. 09.6s. 
 
 " tropical " " 05 48 47.8 
 
 " "anomalistic" " 06 13 49.3 
 
 " "sidereal day 23 56 04.09 
 
 Eccentricity of orbit 0.0167917 
 
 Inclination " " in 1868 23 27' 23^' 
 
 Annual diminution 0".4645 
 
 " precession 50".24 
 
 " advance of line of nodes 11". 8 
 
 TABLE III. 
 
 THE MOON. 
 
 Distance from earth in earth's radii 60.267 
 
 Mean distance from earth 238,800 miles. 
 
 Greatest " " " 257,900 " 
 
 Least " " " 221,400 " 
 
 Sidereal revolution 27d. 07h. 43m. 11.5s. 
 
 Synodical " 29 12 44 03 
 
 Mean daily geocentric motion 13 10' 36" 
 
 Eevolution of nodes 6793.43 days. 
 
 " perigee 3232.58 " 
 
 Mean horizontal parallax 57' 03" 
 
 Greatest " " 61 32 
 
 Least " " 52 50 
 
 Radius in terms of earth's radius 0.2727 
 
 Mass " " " " mass j\ 
 
 Density" " " " density f 
 
253 
 
 APPENDIX. 
 
 TABLE IV. 
 
 SATELLITES. 
 SATELLITES OF MARS. 
 
 Names. 
 
 Inclination of 
 orbit to orbit of 
 Primary. 
 
 Instance from 
 Primary in 
 miles. 
 
 Sid. Period. 
 
 Diameter in 
 miles. 
 
 Magni- 
 tude. 
 
 I. Phoblis.... 
 II. Deimos 
 
 [They move very 
 iifHriyin the plane 
 oi'Mars'sequator.] 
 
 5,800 
 14,500 
 
 d. h. m. 
 07 39 
 
 1 06 18 
 
 [Not yet de- 
 termined, but 
 very small.] 
 
 11 
 
 12 
 
 SATELLITES OF JUPITER. 
 
 I Jo 
 
 O 1 tt 
 
 3 05 30 
 3 04 25 
 
 3 00 28 
 
 2 58 48 
 
 208,000 
 426.000 
 ($79,000 
 1,200,000 
 
 1 18 28 
 3 13 15 
 7 03 43 
 16 16 32 
 
 2400 
 2100 
 3500 
 3000 
 
 7 
 7 
 6 
 
 7 
 
 II. Enropa. -. 
 III. Ganvinede 
 IV. Callisto.... 
 
 SATELLITES OF SATURN. 
 
 I. Mimas 
 II. Knee lad us 
 III Tethvs.. . 
 
 [The first 7 move 
 
 122.000 
 156,000 
 193,000 
 248,000 
 347,000 
 805,000 
 1,018,000 
 2,334,000 
 
 22 37 
 
 1 08 53 
 1 21 18 
 2 17 41 
 4 12 25 
 15 22 41 
 21 07 07 
 79 07 54 
 
 1000 
 
 500 
 500 
 1200 
 3300 
 
 1800 
 
 17 
 15 
 13 
 
 12 
 10 
 8 
 17 
 9 
 
 IV. Dion'e 
 V. Ehea 
 VI. Titan 
 VII. Hyperion. 
 VIII. lapetii.s.... 
 
 plane of tlie plan- 
 et's equator : the 
 8th in a pl.-me in- 
 clined about 120 
 to that plane J 
 
 SATELLITES OF URANUS. 
 
 I. Ariel 
 Il.Umbriel... 
 III. Titania 
 IV.Oberon 
 
 [The motion of 
 the satellite.-, is 
 retrograde, in a 
 plane inclined 790 
 to the plane of the 
 ecliptic.] 
 
 124,000 
 173,000 
 284.000 
 380,000 
 
 2 12 48 
 4 03 27 
 8 16 56 
 13 11 07 
 
 
 
 SATELLITE OF NEPTUNE. 
 
 I. 
 
 290 to ecliptic; 
 motion retrograde 
 
 220,000 
 
 5 21 08 
 
 
 14 
 
TABLES. 
 
 259 
 
 TABLE V. 
 
 THE MINOR PLANETS. 
 
 No. 
 
 Name. 
 
 Year 
 of Dis- 
 cov- 
 ery. 
 
 Discoverer. 
 
 Inclina- 
 tion. 
 
 Eccen- 
 tricity. 
 
 Period 
 Years. 
 
 Distance 
 in Radii 
 of Earth's 
 orbit. 
 
 Diam- 
 eter in 
 miles. 
 
 1 
 
 Ceres 
 
 1801 
 
 Piazzi .... 
 
 10 36' 
 
 080 
 
 46 
 
 2 77 
 
 227 
 
 9 
 
 Pallas 
 
 1802 
 
 Olbers 
 
 34 42 
 
 240 
 
 46 
 
 2 77 
 
 172 
 
 3 
 
 Juno 
 
 1804 
 
 Hard in " 
 
 13 03 
 
 0256 
 
 44 
 
 2 67 
 
 112 
 
 4 
 
 Vesta 
 
 1807 
 
 Olbers 
 
 7 08 
 
 090 
 
 36 
 
 2 36 
 
 228 
 
 5 
 6 
 
 Astrsea 
 Hebe 
 
 1845 
 1847 
 
 Hencke 
 Hencke 
 
 5 19 
 14 46 
 
 0.190 
 0201 
 
 4.1 
 
 3 8 
 
 2.58 
 2 43 
 
 61 
 
 100 
 
 7 
 8 
 
 Iris 
 Flora 
 
 
 Hind 
 Hind 
 
 5 27 
 5 53 
 
 0.231 
 157 
 
 3.7 
 33 
 
 2.39 
 2 20 
 
 96 
 60 
 
 q 
 
 Metis 
 
 1848 
 
 Graham . 
 
 5 36 
 
 123 
 
 37 
 
 2 39 
 
 76 
 
 10 
 11 
 12 
 13 
 
 Hygeia 
 Parthenope 
 Victoria.... 
 Egeria 
 
 1849 
 1850 
 
 De Gasparis. . 
 De Gasparis . 
 Hind 
 
 De Gasparis.' 
 
 3 47 
 4 36 
 8 23 
 16 32 
 
 0.101 
 0.099 
 0.219 
 088 
 
 5.6 
 
 3.8 
 5.6 
 4 1 
 
 3.15 
 2.45 
 2.33 
 
 2 58 
 
 111 
 62 
 41 
 73 
 
 14 
 
 Irene 
 
 1851 
 
 Hind 
 
 9 07 
 
 165 
 
 42 
 
 259 
 
 68 
 
 15 
 
 Eunomia... 
 
 
 De Gasparis. 
 
 11 44 
 
 188 
 
 43 
 
 264 
 
 12 
 
 16 
 17 
 
 Psyche 
 Thetis 
 
 1852 
 
 De Gasparis.. 
 Luther 
 
 3 04 
 5 35 
 
 0.136 
 127 
 
 5.0 
 3 9 
 
 2.93 
 247 
 
 93 
 52 
 
 18 
 10 
 
 Melpomene 
 
 
 
 Hind 
 Hind 
 
 10 09 
 1 32 
 
 0.217 
 158 
 
 3.5 
 
 3 8 
 
 2.30 
 2 44 
 
 54 
 61 
 
 20 
 21 
 
 Massilia.... 
 Lutetia ... 
 
 
 De Gasparis. . 
 Goldschmidt. 
 
 41 
 3 05 
 
 0.144 
 162 
 
 3.7 
 3 1 
 
 2.41 
 244 
 
 68 
 40 
 
 22 
 
 Calliope. 
 
 
 Hind 
 
 13 44 
 
 104 
 
 50 
 
 291 
 
 96 
 
 23 
 24 
 25 
 
 Thalia 
 Themis 
 Phocea 
 
 1853 
 
 Hind 
 De Gasparis. 
 Chacornac. 
 
 10 13 
 48 
 21 34 
 
 0.232 
 0.117 
 253 
 
 4.3 
 
 5.6 
 37 
 
 2.62 
 3.14 
 240 
 
 42 
 36 
 31 
 
 26 
 
 Proserpine 
 
 
 Luther 
 
 3 35 
 
 0088 
 
 43 
 
 266 
 
 47 
 
 27 
 ?8 
 
 Euterpe .... 
 Bellona 
 
 1854 
 
 Hind 
 
 Luther . . 
 
 1 35 
 
 9 21 
 
 0.173 
 150 
 
 3.6 
 46 
 
 2.35 
 
 2 78 
 
 39 
 59 
 
 29 
 30 
 31 
 
 Amphitrite 
 Urania 
 Euphrosyne 
 
 
 Marth 
 Hind 
 Ferguson 
 
 6 07 
 2 05 
 26 25 
 
 0.072 
 0.127 
 0216 
 
 4.1 
 3.6 
 5.6 
 
 2.55 
 2.36 
 3 16 
 
 83 
 51 
 50 
 
 32 
 33 
 34 
 
 Pomona .... 
 Polyhymnia 
 Circe 
 
 1855 
 
 Goldschmidt. 
 Chacornac. .. 
 Chacornac.. 
 
 5 29 
 1 56 
 5 26 
 
 0.082 
 0.338 
 0110 
 
 4.2 
 4.8 
 44 
 
 2.58 
 2.86 
 268 
 
 35 
 
 38 
 29 
 
 So 
 
 
 
 
 '8 10 
 
 214 
 
 5 2 
 
 301 
 
 25 
 
 36 
 
 37 
 
 Atalanta ... 
 Fides 
 
 
 
 Goldschmidt. 
 Luther 
 
 18 42 
 3 07 
 
 0.298 
 175 
 
 4.6 
 
 43 
 
 2.75 
 
 264 
 
 20 
 41 
 
 38 
 
 Leda 
 
 1856 
 
 Chacornac .... 
 
 6 58 
 
 156 
 
 45 
 
 274 
 
 29 
 
 30 
 
 Ltetitia 
 
 
 
 10 21 
 
 111 
 
 46 
 
 277 
 
 87 
 
 40 
 41 
 4 9 
 
 Harmonia. 
 Daphne .... 
 Isi<s 
 
 
 
 Luther 
 Goldschmidt. 
 
 4 15 
 16 45 
 8 35 
 
 0.046 
 0.270 
 0226 
 
 3.4 
 4.6 
 38 
 
 2.27 
 
 2.77 
 244 
 
 
 43 
 
 
 1857 
 
 Pogson . . 
 
 3 27 
 
 168 
 
 33 
 
 220 
 
 
 44 
 
 
 
 Goldschmidt. 
 
 3 41 
 
 149 
 
 38 
 
 242 
 
 
 45 
 
 46 
 
 47 
 
 Eugenia.... 
 Hestia 
 Melete 
 
 
 
 Goldschmidt. 
 Pogson 
 Gold a chmidt 
 
 6 34 
 2 17 
 8 01 
 
 0.082 
 0.162 
 237 
 
 4.5 
 4.0 
 42 
 
 2.72 
 2.52 
 2 60 
 
 
 L 
 
 
 
 
 
 
 
 
 
260 
 
 APPENDIX. 
 
 TABLE V. Continued. 
 
 No. 
 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 61 
 62 
 63 
 64 
 65 
 66 
 67 
 68 
 69 
 70 
 71 
 72 
 73 
 74 
 75 
 
 8 
 
 78 
 79 
 80 
 81 
 82 
 83 
 84 
 85 
 86 
 87 
 88 
 89 
 90 
 91 
 
 Name. 
 
 Year 
 of Dis- 
 cov- 
 ery. 
 
 Discoverer. 
 
 Inclina- 
 tion. 
 
 Eccen- 
 tricity. 
 
 Period 
 in 
 
 Years. 
 
 Distance 
 in Radii 
 of Earth's 
 orbit. 
 
 Diam- 
 eter in 
 Miles. 
 
 Aglaia ... 
 
 1857 
 
 
 5 00' 
 6 29 
 3 08 
 2 47 
 10 14 
 7 24 
 5 07 
 11 47 
 7 20 
 15 04 
 5 02 
 18 17 
 8 36 
 2 12 
 3 34 
 5 45 
 1 19 
 3 28 
 3 04 
 5 59 
 8 28 
 7 58 
 11 39 
 5 24 
 23 19 
 2 25 
 3 59 
 5 00 
 2 02 
 2 28 
 8 39 
 4 37 
 8 37 
 7 56 
 2 51 
 5 02 
 9 22 
 11 56 
 4 48 
 
 5 09 
 15 13 
 
 0.128 
 0.076 
 0.238 
 0.287 
 0.063 
 0.004 
 0.213 
 0.199 
 0.139 
 0.108 
 0.041 
 0.163 
 0.119 
 0.170 
 0.185 
 0.127 
 0.125 
 0.120 
 0.154 
 0.184 
 0.175 
 0.186 
 0.183 
 0.120 
 0.174 
 0.044 
 0.238 
 0.307 
 0.188 
 0.136 
 0.205 
 0.195 
 0.200 
 0.212 
 0.226 
 0.084 
 0.238 
 0.194 
 0.205 
 0.081 
 0.167 
 0.205 
 0.148 
 
 4.9 
 5.5 
 5.4 
 4.3 
 3.7 
 5.5 
 4.2 
 4.6 
 4.6 
 5.6 
 4.4 
 5.1 
 4.5 
 5.5 
 3.7 
 3.7 
 4.4 
 6.7 
 4.3 
 3.8 
 5.2 
 4.6 
 4.2 
 3.4 
 4.6 
 4.3 
 4.6 
 4.4 
 6.2 
 4.4 
 4.2 
 3.8 
 3.5 
 4.S 
 4.6 
 3.8 
 3.6 
 4.3 
 5.4 
 6.5 
 4.6 
 4.0 
 5.5 
 
 2.88 
 3.10 
 3.09 
 2.65 
 2.38 
 3.10 
 2.61 
 2.71 
 2.77 
 3.16 
 2.70 
 2.97 
 2.71 
 3.13 
 2.39 
 2.40 
 2.68 
 3.42 
 2.65 
 2.42 
 2.99 
 2.77 
 2.61 
 2.27 
 2.76 
 2.66 
 2.78 
 2.67 
 3.39 
 2.67 
 2.62 
 2.44 
 2.30 
 2.86 
 2.76 
 2.43 
 2.37 
 2.66 
 3.09 
 3.49 
 2.75 
 2.53 
 31.2 
 
 
 
 Goldschmidt. 
 Goldschmidt. 
 Ferguson 
 Laurent 
 Goldschmidt. 
 Luther 
 
 Pales 
 
 
 Virginia.... 
 Nemausa.... 
 
 1858 
 
 Calypso 
 
 
 Alexandra.. 
 Pandora . 
 
 
 
 Goldschmidt. 
 Searle 
 
 Mnemosyne 
 Concordia... 
 Daniie 
 
 1859 
 1860 
 
 Luther 
 
 Luther 
 
 Goldschmidt. 
 Chacornac 
 
 Olympia.... 
 Erato 
 
 
 
 Echo 
 
 
 Ferguson 
 De Gasparis.. 
 Tern pel 
 
 Ausonia 
 Angelina.... 
 Cybele 
 
 1861 
 
 
 Maia 
 
 
 T uttle 
 
 
 
 Pogson 
 
 Hesperia.... 
 Leto 
 
 
 
 Schiaparelli.. 
 
 Panopea . .. 
 
 
 Goldschmidt. 
 C.H.F. Peters 
 
 Luther . . 
 
 Feronia 
 
 
 Niobe 
 
 
 Clyde 
 
 1862 
 
 Tuttle 
 
 Galatea.. .. 
 
 Tempel 
 Peters 
 
 Eurvdice.... 
 Preia 
 
 
 D' Arrest 
 Peters 
 
 Friar o-a 
 
 
 
 Diana 
 
 1863 
 i'864 
 
 Luther 
 
 Eurynome.. 
 Sappho 
 Terpsichore 
 Alcmene.... 
 Beatrix .... 
 
 Watson 
 
 Pogson 
 
 Xempel.... 
 
 1865 
 
 Luther 
 
 De Gasparis. . 
 Luther 
 
 Clio 
 
 Io 
 
 
 Peters 
 
 Seraele . 
 
 1866 
 
 Tietjen 
 
 Svlvia 
 
 
 Thisbe 
 
 
 Peters 
 Stephan 
 Luther 
 Stephan 
 
 Julia . 
 
 
 Antiope 
 
 
 jEsfina. .. 
 
 
 
 
 The number of minor planets discovered up to October, 1878, was 192. 
 The diameters are derived from photometric experiments made by Pro- 
 fessor Stampfer, of Vienna. They are probably only relatively correct. 
 
TABLES. 
 
 261 
 
 TABLE VI. 
 
 SCHWABE'S OBSERVATIONS OF THE SOLAR SPOTS, 
 
 Ysar. 
 
 Number 
 of days 
 of obser- 
 vation. 
 
 New Groups. 
 
 Days on 
 which the 
 Sun was free 
 from spots. 
 
 Year. 
 
 Number 
 of days 
 of obser- 
 vation. 
 
 New Groups. 
 
 Days on 
 which the 
 Sun was free 
 from spots. 
 
 1826 
 
 277 
 
 118 
 
 22 
 
 1846 
 
 314 
 
 157 
 
 1 
 
 7 
 
 273 
 
 161 
 
 2 
 
 7 
 
 276 
 
 257 
 
 
 
 8 
 
 282 
 
 225(Max.) 
 
 
 
 8 
 
 278 
 
 330(Max.) 
 
 
 
 9 
 
 244 
 
 199 
 
 
 
 9 
 
 285 
 
 238 
 
 
 
 1830 
 
 217 
 
 190 
 
 1 
 
 1850 
 
 308 
 
 186 
 
 2 
 
 1 
 
 239 
 
 149 
 
 3 
 
 1 
 
 308 
 
 151 
 
 
 
 2 
 
 270 
 
 84 
 
 49 
 
 2 
 
 337 
 
 125 
 
 2 
 
 3 
 
 247 
 
 33(Min.) 
 
 139 
 
 3 
 
 299 
 
 91 
 
 3 
 
 4 
 
 273 
 
 61 
 
 120 
 
 4 
 
 334 
 
 67 
 
 65 
 
 5 
 
 244 
 
 173 
 
 18 
 
 5 
 
 313 
 
 79 
 
 146 
 
 6 
 
 200 
 
 272 
 
 
 
 6 
 
 321 
 
 34(Min.) 
 
 193 
 
 7 
 
 168 
 
 333(Max.) 
 
 
 
 7 
 
 324 
 
 98 
 
 52 
 
 8 
 
 202 
 
 282 
 
 
 
 8 
 
 335 
 
 188 
 
 
 
 9 
 
 205 
 
 162 
 
 
 
 9 
 
 343 
 
 205 
 
 
 
 1840 
 
 263 
 
 152 
 
 3 
 
 1860 
 
 332 
 
 211(Max.) 
 
 
 
 1 
 
 283 
 
 102 
 
 15 
 
 1 
 
 322 
 
 204 
 
 
 
 2 
 
 307 
 
 68 
 
 64 
 
 2 
 
 317 
 
 160 
 
 3 
 
 3 
 
 312 
 
 34(Min.) 
 
 149 
 
 3 
 
 330 
 
 124 
 
 2 
 
 4 
 
 321 
 
 52 
 
 111 
 
 4 
 
 325 
 
 130 
 
 4 
 
 1845 
 
 332 
 
 114 
 
 29 
 
 1865 
 
 307 
 
 93 
 
 25 
 
 TABLE VII. 
 
 PERIODIC COMETS. 
 
 Name. 
 
 Last perihelion 
 passage observed. 
 
 Longi- 
 tude of 
 Asc.Node 
 
 Inclina- 
 tion. 
 
 Eccentri- 
 city. 
 
 Semi- 
 major 
 axis. 
 
 Period 
 in 
 Days. 
 
 Motion. 
 
 Encke's 
 Winnecke's... 
 Brorsen's 
 
 1878 
 1875, Mar. 12 
 1879, Oct. 12 
 1 Sf>i>, Sept. 23 
 1877 
 1873, JnlvlS 
 1871, Dec. 1 
 1835, Nov. 16 
 
 o t 
 
 334 31 
 113 31 
 101 46 
 245 57 
 
 148 27 
 209 40 
 268 54 
 55 10 
 
 1 
 
 13 05 
 10 48 
 29 45 
 12 34 
 13 56 
 11 22 
 54 32 
 17 45 
 
 0.847 
 0.755 
 0.802 
 0.756 
 0.660 
 0.558 
 0.830 
 0.967 
 
 2.22 
 3.14 
 3.14 
 3.50 
 3.44 
 3.81 
 6.03 
 17.99 
 
 1210 
 2020 
 
 2037 
 2415 
 2366 
 2715 
 4986 
 28000 
 
 Direct. 
 Direct. 
 Direct. 
 Direct. 
 Direct. 
 Direct. 
 Direct. 
 Ketro. 
 
 D' Arrest's.... 
 leave's 
 
 Mechain's 
 Halley's 
 
262 
 
 APPENDIX. 
 
 TABLE VIII. 
 
 TRANSITS OF THE INFERIOR PLANETS. 
 
 Mercury. 
 
 Venus. 
 
 1802. 
 
 November 8. 
 
 1639. 
 
 December 4. 
 
 1815. 
 
 November 11. 
 
 1761. 
 
 June 5. 
 
 1822. 
 
 November 4. 
 
 1769. 
 
 June 3. 
 
 1832. 
 
 May 5. 
 
 1874. 
 
 December 8. 
 
 1835. 
 
 November 7. 
 
 1882. 
 
 December 6. 
 
 1845. 
 
 May 8. 
 
 2004. 
 
 June 7. 
 
 1848. 
 
 November 9. 
 
 2012. 
 
 June 5. 
 
 1861. 
 
 November 11. 
 
 2117. 
 
 December 10. 
 
 1868. 
 
 November 4. 
 
 2125. 
 
 December 8. 
 
 1878. 
 
 May 6. 
 
 2247. 
 
 June 11. 
 
 1881. 
 
 November 7. 
 
 2255. 
 
 June 8. 
 
 1891. 
 
 May 9. 
 
 2360. 
 
 December 12. 
 
 1894. 
 
 November 10. 
 
 2368. 
 
 December 10. 
 
 TABLE IX. 
 
 LIST OF STARS WHOSE ANNUAL PARALLAX HAS BEEN 
 
 COMPUTED. 
 (All these are doubtful.) 
 
 Stars. 
 
 Parallax. 
 
 Distance in radii 
 of the Earth's 
 orbit. 
 
 Observer. 
 
 o. Centauri 
 
 0.92 
 
 224,000 
 
 Maclear. Henderson 
 
 61 Cygni 
 
 / 0.35 
 
 589,000 
 
 Bessel. 
 
 21258 Lalande 
 17415 Oeltzen 
 
 (.0.56 
 0.27 
 025 
 
 368,000 
 764,000 
 825 000 
 
 Auvers. 
 Auvers. 
 Kriiger. 
 
 1830 Groombridge.. 
 70 Ophiuchi 
 a Lyne 
 
 0.23 
 0.16 
 155 
 
 897,000 
 1,289,000 
 1 331 000 
 
 C. A. Peters. 
 Kr tiger. 
 W. & O. Struve. 
 
 Sirius 
 
 0150* 
 
 1 375,000 
 
 Henderson. Peters. 
 
 i Ursse Majoris.. 
 A returns 
 Polaris 
 
 0.133 
 0.127 
 007 
 
 1,550,000 
 1,624,000 
 2 950,000 
 
 C. A. Peters. 
 C. A. Peters. 
 C. A. Peters. 
 
 Capella 
 
 005 
 
 4,130,000 
 
 C. A. Peters. 
 
 . 
 
 
 
 
 Another determination gives 0".23. 
 
TABLES. 
 
 263 
 
 TABLE X. 
 
 THE PRINCIPAL CONSTELLATIONS. 
 Those found in Ptolemy's Catalogue (137 A.n.) are in Italics. 
 - THE NORTHERN CONSTELLATIONS. 
 
 No. 
 
 Name. 
 
 Coordinates of Centre. 
 
 R.A. D. 
 
 Name of 
 Principal Star of 
 1st or 2d magnitude. 
 
 Number 
 of stars of 
 1st mag. 
 
 Number 1 
 of stars 
 of first 
 ftve mag- 
 nitudes. 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 
 29 
 
 
 h. m. 
 1 
 1 10 
 
 2 
 3 30 
 5 45 
 6 
 7 55 
 10 05 
 10 40 
 12 40 
 13 
 14 35 
 15 
 15 40 
 15 40 
 16 45 
 17 20 
 17 50 
 18 10 
 18 40 
 19 30 
 19 40 
 20 
 20 20 
 20 20 
 20 40 
 21 
 21 40 
 22 25 
 
 
 
 35 
 
 60 
 32 
 47 
 68 
 42 
 50 
 36 
 58 
 26 
 40 
 30 
 78 
 30 
 10 
 27 
 66 
 5 
 15 S. 
 35 
 10 
 18 
 25 
 42 
 44 
 15 
 6 
 65 
 15 
 
 Alpheratz (a) 
 
 Mirfak (a) 
 Capella (a) 
 
 Dubhe (a) 
 
 Cor Caroli (a) 
 Arcturus (a) 
 Polaris (a) 
 
 Unnkalhay (a) 
 Rasalgeti (a) 
 Thuban (a) 
 
 Vega (a) 
 Altair (a) 
 
 Deneb (a) 
 Markab (a) 
 
 1 
 
 1 
 1 
 1 
 
 1 
 1 
 
 18 
 46 
 5 
 40 
 36 
 35 
 28 
 15 
 53 
 20 
 15 
 35 
 23 
 19 
 23 
 65 
 80 
 6 
 4 
 18 
 33 
 5 
 23 
 67 
 13 
 10 
 5 
 44 
 43 
 
 
 
 
 Camelopa.rd.us 
 
 Aurioci 
 
 
 
 UTSCL J\fctjor 
 
 Coma Berenicis 
 
 Bootes 
 
 
 Corona Borealis 
 
 
 Draco 
 
 TaurusPoniatowskii 
 Clypeus Sobieskii... 
 
 
 
 Vulpecula et Anser. 
 
 
 
 
 
 
 
 Total 
 
 
 
 
 6 
 
 827 
 
 
264 
 
 APPENDIX. 
 
 TABLE X. Continued. 
 
 THE ZODIACAL CONSTELLATIONS. 
 
 No. 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 
 12 
 
 Name. 
 
 Coordinates oi Centre. 
 
 K.A. P. 
 
 Name of 
 Principal Star of 
 1st or 2d magnitude. 
 
 Number 
 of stars of 
 1st laag. 
 
 Number 
 of stars 
 of first 
 five mag- 
 nitudes. 
 
 Aries 
 
 h. m. 
 2 30 
 
 4 
 7 
 
 8 40 
 10 20 
 13 20 
 15 
 16 15 
 18 55 
 21 
 22 
 20 
 
 18 N. 
 18 
 
 25 
 
 20 
 15 
 3N. 
 
 15 S. 
 26 
 32 
 20 
 9S. 
 ION. 
 
 Hamal (a) 
 Aldebaran (a) 
 f Castor (a) 
 \ Pollux (0) 
 
 Regulus (a) 
 Spica (a) 
 Zubenelg (a) 
 An tares (a) 
 
 SecundaGiedi(a) 
 Sadalmelik (a) 
 
 1 
 1 
 
 1 
 1 
 
 1 
 
 17 
 
 58 
 
 28 
 
 15 
 
 47 
 39 
 23 
 34 
 38 
 22 
 25 
 18 
 
 Taurus 
 
 
 Cancer 
 
 
 
 Libra 
 
 Scorpio 
 
 Sagittarius 
 
 
 
 JPisces 
 
 
 Total 
 
 
 
 
 5 
 
 364 
 
 
 THE SOUTHERN CONSTELLATIONS. 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 
 n 
 
 18 
 19 
 20 
 21 
 
 Apparatus Sculptoris. 
 
 20 
 1 
 2 
 2 20 
 2 40 
 3 15 
 3 40 
 4 
 4 40 
 4 40 
 5 20 
 5 25 
 5 25 
 5 25 
 5 30 
 6 45 
 7 
 7 25 
 7 40 
 7 40 
 10 
 
 32 
 50 
 12 
 30 
 70 
 57 
 30 
 62 
 62 
 42 
 75 
 35 
 55 
 20 
 
 24 
 2 
 5 
 50 
 68 
 
 
 Menkar (a) 
 Achernar (a) 
 
 Phact (a) 
 
 Arneb (a) 
 Rigel (ft) 
 Sirius (a) 
 
 Procyon (a) 
 Canopus (a) 
 
 1 
 
 2 
 1 
 
 1 
 2 
 
 13 
 32 
 32 
 6 
 25 
 11 
 64 
 11 
 17 
 6 
 9 
 15 
 17 
 18 
 37 
 27 
 12 
 6 
 133 
 9 
 3 
 
 Qetus 
 
 Fornax Chemica . ... 
 Hvdrus 
 
 Horologiuni 
 
 
 Reticulus Rhomboidalis .. 
 
 Csjla Sculptoris 
 
 J^fons Mensae 
 
 Columba Noachi . ... 
 Equuleus Pictoris... 
 Lspus 
 
 Orion 
 
 
 Monoceros . ... 
 
 Canis Minor 
 
 
 Piscis Volaiis 
 Sextans 
 
 
TABLES. 
 
 265 
 
 TABLE X. Continued. 
 
 THE SOUTHERN CONSTELLATIONS. 
 
 No 
 
 Name. 
 
 Coordinates of Centre. 
 
 R.A. D. 
 
 Name of 
 Principal Star of 
 1st or 2d magnitude. 
 
 Number 
 ofsta.rsof 
 1st mag. 
 
 Number 
 of stars 
 of first 
 five mag- 
 nitudes. 
 
 22 
 23 
 24 
 J25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 33 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 
 
 h. m. 
 
 10 
 10 
 10 50 
 11 20 
 12 15 
 12 20 
 12 25 
 13 
 15 
 15 20 
 15 25 
 15 40 
 16 
 17 
 17 
 18 30 
 18 40 
 19 20 
 20 40 
 21 
 21 
 21 40 
 22 20 
 23 45 
 
 
 
 10 
 35 
 78 
 15 
 60 
 18 
 68 
 48 
 64 
 76 
 45 
 65 
 45 
 54 
 
 40 
 53 
 68 
 37 
 55 
 80 
 32 
 47 
 66 
 
 Alphard (a) 
 Algorab (a) 
 
 Rasalague (a) 
 Fomalhaut (a) 
 
 1 
 
 2 
 1 
 
 49 
 7 
 17 
 9 
 10 
 8 
 7 
 54 
 2 
 7 
 34 
 11 
 12 
 15 
 40 
 7 
 8 
 27 
 5 
 15 
 16 
 16 
 11 
 21 
 
 Antlia Pneumatica.. 
 Cliamselcon 
 
 Crater 
 
 Crux 
 
 Corvus 
 
 Musca Australia 
 
 Circinus - 
 
 A pus 
 
 
 TriangulumAustrale 
 Norma 
 
 Ara 
 
 
 Corona Australls 
 Tel6scopium 
 
 Pavo 
 
 
 Indus ... ... 
 
 Octans 
 
 Piscis Australia 
 
 Toucan 
 
 
 45 
 
 86 
 1 
 
 Total 
 
 
 
 
 11 
 
 911 
 
 
 
 
 
 
 22 
 
 2102 
 
 
 23 
 
266 
 
 APPENDIX. 
 
 TABLE XL 
 
 VARIABLE STARS. 
 
 Name. 
 
 Coordinates, 1870. 
 
 R.A. D. 
 
 Period in 
 Days. 
 
 Change of 
 Magnitude. 
 
 1 
 
 Authontj. 
 
 
 h. m. s. 
 
 o t 
 
 
 
 
 a Cassiopeise. 
 
 33 09 
 
 55 49.4 N 
 
 79.1 
 
 2 to 2.5 
 
 Birt, 1831 
 
 o Ceti ... . 
 
 2 12 47 
 
 3 34.1 S 
 
 331.34 
 
 2 " 12 
 
 D. Fabricius 1596 
 
 /?Persei 
 
 2 59 43 
 
 40 27.2 N 
 
 2.87 
 
 2.5" 4 
 
 Montanari, 1669 
 
 A Tauri 
 
 3 53 29 
 
 12 07.3 N 
 
 3.95 
 
 4" 4.5 
 
 Baxendell, 1848 
 
 e Aurigae 
 
 4 52 38 
 
 43 37.7 N 
 
 250. 
 
 3.5" 4.5 
 
 Heis, 1846 
 
 fj- Doradus.... 
 
 5 05 47 
 
 61 58.4 S 
 
 Long. 
 
 5 " 9 
 
 Moesta, 1865 
 
 a Orionis 
 
 5 48 08 
 
 7 22.8 N 
 
 196. 
 
 1 " 1.5 
 
 J. Herscliel, 1836 
 
 C Gerainorum 
 
 6 56.24 
 
 20 45.5 N 
 
 10.16 
 
 3.8" 4.5 
 
 Schmidt, 1847 
 
 a Hydrse. 
 
 9 21 12 
 
 8 05.9 S 
 
 55. 
 
 2.5" 3 
 
 J. Herschel, 1837 
 
 R Leonis 
 
 9 40 34 
 
 12 01. 8 N 
 
 331. 
 
 5 "11.5 
 
 Koch, 1782 
 
 77 Argus 
 
 10 40 02 
 
 59 0.1 S 
 
 46yrs. 
 
 1 " 4 
 
 Burchell, 1827 
 
 R Hydra? 
 
 13 22 37 
 
 22 36.4 S 
 
 447.8 
 
 4 "10 
 
 Maraldi, 1704 
 
 Z Virginia.... 
 
 13 27 45 
 
 1232.7 8 
 
 ? 
 
 5 " 8 
 
 Schmidt, 1866 
 
 T Coronae 
 
 15 54 04 
 
 26 17.5 N 
 
 ? 
 
 2.5" 9.8 
 
 Bermingham, 1866 
 
 30 Herculis... 
 
 16 24 22 
 
 42 10.1 N 
 
 106. 
 
 5 " 6 
 
 Baxendell, 1857 
 
 Nov. Ophiu... 
 
 16 52 13 
 
 12 41.4 S 
 
 ? 
 
 4.5 " 13.5 
 
 Hind, 1848 
 
 a Herculis.... 
 
 17 08 43 
 
 14 32.4 N 
 
 ? 
 
 3.1" 3.9 
 
 W. Herschel, 1795 
 
 R Clyp. Sob. 
 
 18 40 33 
 
 5 50.5 S 
 
 89. 
 
 5 " 9 
 
 Pigotl, 1795 
 
 (3 Lyra? 
 
 18 45 17 
 
 33 12.7 N 
 
 12.91 
 
 3.5" 4.5 
 
 Goodricke, 1784 
 
 13 Lyra 
 
 18 51 23 
 
 43 46.6 N 
 
 46. 
 
 4.2" 4.6 
 
 Baxendell, 1855 
 
 ^ 2 Cygni 
 
 19 45 34 
 
 32 35.2 N 
 
 409.2 
 
 5 "13 
 
 Kirch, 1686 
 
 T? Aquilse 
 
 19 45 51 
 
 040!4N 
 
 7.18 
 
 3.6" 4.4 
 
 Pigott, 1784 
 
 34 Cygni 
 
 20 13 
 
 37 37.8 N 
 
 18yrs. 
 
 3 " 6 
 
 Jansen, 1600 
 
 RCephei 
 
 20 23 41 
 
 8844. N 
 
 73 yrs. 
 
 5 "11 
 
 Pogson, 1851. 
 
 fj- Cephei 
 
 21 39 31 
 
 58 11.1 N 
 
 5 or 6yrs. 
 
 4 " 6 
 
 W. Herschel, 178S 
 
 6 Cephei 
 
 22 24 21 
 
 5745. N 
 
 5.37 
 
 3.7" 4.8 
 
 Goodricke, 1784 
 
 j3 Pegasi 
 
 22 57 28 
 
 27 22.', N 
 
 ? 
 
 2 " 2.5 
 
 Schmidt, 184. 1 
 
 
 
 
 
 
 1 
 
TABLES. 
 
 267 
 
 TABLE XII. 
 
 BINARY STARS. 
 
 Name. 
 
 Co-ordinates, 1870. 
 
 K.A. D. 
 
 Magnitude 
 of Com- 
 ponents. 
 
 Semi- 
 major 
 Axis. 
 
 Eccen- 
 tricity. 
 
 Period 
 in 
 
 Years. 
 
 Calculator. 
 
 Herculis. ... 
 
 h. m. a. 
 
 16 36 
 
 O f 
 
 3151 N 
 
 3 6 
 
 n 
 1.25 
 
 0.45 
 
 36.3 
 
 Villarceau. 
 
 rj CoronaeBor. 
 
 15 18 
 
 3047 N 
 
 6 6J 
 
 0.95 
 
 0.28 
 
 43.6 
 
 Winnecke. 
 
 f Cancri 
 
 8 04 45 
 
 18 02.4 N 
 
 6-7-7 
 
 1.29 
 
 0.23 
 
 58.9 
 
 Madler. 
 
 f UrsaeMaj... 
 
 11 11 15 
 
 3216 N 
 
 4 -6J 
 
 2.45 
 
 0.39 
 
 63.1 
 
 J. Breen. 
 
 a Centauri.... 
 
 14 30 48 
 
 60 17.9 S 
 
 1 2 
 
 15.50 
 
 0.95 
 
 80.0 
 
 E.B.Powell. 
 
 w Leonis ...... 
 
 9 21 
 
 939 N 
 
 6-7 
 
 0.85 
 
 0.64 
 
 82.5 
 
 Madler. 
 
 r Ophiuchi... 
 
 17 56 
 
 811 S 
 
 5 6 
 
 0.82 
 
 0.037 
 
 87. 
 
 Madler. 
 
 70 Ophiuchi.. 
 
 17 58 52 
 
 2 22.5 N 
 
 4 7 
 
 4.19 
 
 0.44 
 
 92.8 
 
 Madler. 
 
 A Ophiuchi... 
 
 16 24 
 
 217 N 
 
 4 6 
 
 0.84 
 
 0.477 
 
 95. 
 
 Hind. 
 
 7 Coronas Aus. 
 I Bootis 
 
 18 57 38 
 14 45 22 
 19 41 
 
 37 14.7 S 
 19 38.5 N 
 4448 N 
 
 6 6 
 3i 9 
 
 2.54 
 12.56 
 1.81 
 
 0.60 
 0.59 
 0.60 
 
 100.8 
 117.1 
 
 178.7 
 
 Jacob. 
 J.Herschel. 
 Hind. 
 
 6 Cygni 
 
 ri Cassiopeiae . 
 
 41 09 
 
 57 07.8 N 
 
 4 -7* 
 
 10.33 
 
 0.77 
 
 181. 
 
 E.B.Powell. 
 
 7 Virginia... . 
 
 12 35 05 
 
 044.2S 
 
 4 4 
 
 3.58 
 
 0.87 
 
 182.1 
 
 J.Herschel. 
 
 o CoronseBor. 
 
 16 10 
 7 26 18 
 21 01 04 
 
 3412 N 
 32 10.4 N 
 38 05.1 N 
 
 6 6J 
 3 3J 
 5 6 
 
 2.71 
 8.08 
 15.4 
 
 0.30 
 0.75 
 
 195.1 
 252.6 
 452. 
 
 Jacob. 
 J.Herschel. 
 
 61 Cygni 
 
 fi Bootis 
 
 15 19 35 
 
 3750 N 
 
 4 8 
 
 3.21 
 
 0.84 
 
 649.7 
 
 Hind. 
 
 y Leonis 
 
 10 12 47 
 
 20 30.1 N 
 
 2 4 
 
 
 
 1200. 
 
 
INDEX. 
 
 [The references are to the pages.] 
 
 Aberration of light, 116; diurnal and 
 annual, 117; separated from par- 
 allax, 221. 
 
 Acceleration, secular of the moon, 133. 
 
 Aerolites, 209. 
 
 Algol, 224. 
 
 Altazimuth, 39. 
 
 Altitude, 19. 
 
 Altitude and azimuth instrument, 38; 
 use of, 40. 
 
 Altitudes, method of equal, 39. 
 
 Amplitude, 19. 
 
 Andromeda, nebula in, 231. 
 
 Annular eclipse, 141. 
 
 Anomaly, 90. 
 
 Aphelion, 90. 
 
 Apogee, 122. 
 
 Appulse, 136. 
 
 Apsides, of earth, 116; of moon, 122; 
 of planets, 164. 
 
 Arc of meridian measured, 59. 
 
 Argo, nebula in, 234. 
 
 Aries, first point of, 20, 84. 
 
 Ascension, right, 20 ; related to sidereal 
 time, 22. 
 
 Asteroids, 171; table of, 260. 
 
 Astronomy, 11; chronology of, 248. 
 
 Atmosphere, height of, 52. 
 
 Attraction, law of, 108. 
 
 Axis, of the heavens, 15; of the earth, 
 17; of rotation and collimation, 32. 
 
 Azimuth, 19. 
 
 Base-line, 60. 
 Bode's law, 171. 
 Branches of meridian, 18. 
 
 Calendar, 105. 
 
 Centauri, alpha, distance of, 222; a 
 
 binary star, 227. 
 Centrifugal force, 65, 247. 
 Ceres, discovery of, 172. 
 
 Chronograph, 29. 
 
 Chronometer, Greenwich time given 
 by, 77. 
 
 Circle, vertical, 19 ; hour, 20 ; of per- 
 petual apparition, 20; diurnal, 24; 
 of latitude, 84. 
 
 Circle, meridian, 34; mural, 38; re- 
 flecting, 48. 
 
 Clock, astronomical, 28; driving, 41. 
 
 Clusters of stars, 229. 
 
 Coal sack, 234. 
 
 Collimation, axis of, 32. 
 
 Colures, 84. 
 
 Comets, 187; diversity of appearance, 
 188; tail, 1 89 ; orbits, 191 ; periods and 
 motion, 192 ; mass and density, 193 ; 
 light, 194; periodic, 195; Encke's, 
 195 ; Winnecke's or Pons's,197 ; Bror- 
 sen's, 197; Biela's, 197; D'Arrest's, 
 198; Faye's, 198; Mechain's, 199; 
 Halley 's, 199 ; Great, of 1 811 , 200 ; of 
 1843, 200 ; Donati's, 201 ; of 1861, 202 ; 
 connection with meteors, 211; ele- 
 ments of periodic, 261. 
 
 Compression, 63. 
 
 Conjunction, 126; inferior and supe- 
 rior, 159. 
 
 Constellations, 215; list of, 263. 
 
 Co-ordinates, spherical, 25. 
 
 Corona, 96. 
 
 Count, least, 47. 
 
 Crescent, 127. 
 
 Cross-wires, 32. 
 
 Culmination, 20. 
 
 Cycle, lunar, 133; of eclipses, 142. 
 
 Cygni, 61, its distance, 222. 
 
 Day, solar and sidereal, 21 ; inequality 
 of solar, 91; astronomical and civil, 
 105; intercalary, 106. 
 
 Declination, 20. 
 
 Degree of meridian, 62. 
 
270 
 
 INDEX. 
 
 departure, 18. 
 
 Dip of the horizon, 57. 
 
 Dipper, 215. 
 
 Disc, spurious, of stars, 223. 
 
 Distance, zenith, 19,' polar, 20. 
 
 Earth, general form of, 12; spheroidal 
 form, 62; dimensions, 63; density, 
 
 Gibbous, 127. 
 Golden -number, 1 34. 
 Gravitation, universal, 107. 
 Heliocentric, parallax, 56, 156; motion 
 
 of planets, 157. 
 Hemisphere, 17. 
 
 Horizon, 13 ; points of, 19 ; artificial, 45. 
 Horizontal point, 36. 
 
 63; linear velocity of rotation, 70; Hour angle, 20; relation to sidereal 
 
 revolution about the sun, 88 ; orbital I time, 22. 
 
 velocity, 89; orbit, 89; motion at Hyades, 230. 
 
 perihelion and aphelion, 110; revo- Hyperbola, 245. 
 
 lution proved by aberration, 119; 
 
 phases, 127 ; elements, 257. 
 Eccentricity of an ellipse, 90. 
 Eclipses, 135; lunar, 135; solar, 139; 
 
 total, 142; cycle of, 142; number of, 
 143; of Jupiter's satellites, 174. 
 
 Ecliptic, 83. 
 
 Ecliptic limits, lunar, 137; solar, 141. 
 
 Elements, of planetary orbit, 160; of 
 ^ometary, 190. 
 
 Ellipse, 244. 
 
 Elongation, 56 ; greatest eastern and 
 western, 159. 
 
 Equation, of centre, 102; of time, 104; 
 annual, 133. 
 
 Equator, 17; celestial, 19. 
 
 Equatorial, 40. 
 
 Equilibrium of centrifugal and centri- 
 petal forces, 107. 
 
 Equinoctial, 19. 
 
 Equinox, vernal, 20, 83. 
 
 Error of clock, 28. 
 
 Errors of observation, 49. 
 
 Establishment of port, 151. 
 
 Evection, 132. 
 
 Evening star, 160, 170. 
 
 Faculae, 96. 
 
 Finder, 34. 
 
 Flames, red, 97. 
 
 Forces, centrifugal and centripetal,247. 
 
 Foucault's experiment, 68. 
 
 Galaxy, 234. 
 Gemini, 216. 
 
 Geocentric, parallax, 55; motion of 
 planets, 155, 158, 167. 
 
 Incidence, angle of, 51. 
 Index correction, 44. 
 
 Jupiter, 173; mass of, 175. 
 
 Kepler 's laws, 111. 
 Kirkwood's law, 247. 
 
 Latitude, 18; equal to altitude of pole, 
 23 ; methods of determining, 72 ; at 
 sea, 75; reduction of, 75; celestial, 
 84. 
 
 Level, hanging, 34. 
 
 Librations, 131. 
 
 Light, analysis of, 48 ; of sun, 97; velo- 
 city of, 118 ; of planets, 183 ; of stars, 
 218; of nebulse, 230. 
 
 Line of sight, 32. 
 
 Longitude, 18; how determined, 76; by 
 telegraph, 78; by star signals, 79; 
 at sea, 80; celestial, 84; by eclipses 
 and occupations, 145. 
 
 Luculi, 96. 
 
 Lunar distance, 78. 
 
 Lunation, 128. 
 
 Magellanic clouds, 233. 
 
 Magnitudes, 214. 
 
 Mars, 170. 
 
 Mercury, 164; transits of, 262. 
 
 Meridian, 18; prime and celestial, 18; 
 
 line, 19. 
 Meteors, 202 ; showers, 203; height and 
 
 velocity, 205 ; orbits, 206 ; detonating, 
 
 208. 
 Microscope, reading, 35. 
 
INDEX. 
 
 271 
 
 Milky way, 234. 
 
 Minor planets, 171 ; list of, 260. 
 
 Mira, or o Ceti, 223. 
 
 Moon, orbit of, 120 ; nodes, 120 ; obli- 
 quity of orbit, 121 ; form of orbit, 122; 
 line of apsides, 122; meridian zenith 
 distance, 122; distance, 123; magni- 
 tude and mass, 125; augmentation 
 of diameter, 125; phases, 126; side- 
 real and synodic periods, 128; retar- 
 dation, 129 ; harvest, 130 ; rotation, 
 130; librations, 131; other pertur- 
 bations, 132; general description, 
 134; elements, 257. 
 
 Morning star, 160, 170. 
 
 Motion, diurnal, 13; upward and 
 downward, 18; west to east, 88 
 (note); direct and retrograde, 159, 
 167. 
 
 Mural circle, 38. 
 
 Nadir, 18; point, 37. 
 
 Navigation, sketch of, 255. 
 
 Nebulae, resolvable and irresolvable, 
 230; annular and elliptic, 231 ; spiral 
 and planetary, 232 ; nebulous stars, 
 232; double, 232; variation of bright- 
 ness, 233. 
 
 Nebular hypothesis, 183. 
 
 Neptune, 182; immense distance of, 
 183. 
 
 Nodes, of moon's orbit, 120; heliocentric 
 longitude of planet's, 161. 
 
 Noon, 105. 
 
 Nubeculce, 233. 
 
 Nutation, 115. 
 
 Obliquity of ecliptic, 84, 115. 
 Occultation, 135, 144; of Jupiter's sa- 
 tellites, 174. 
 Octant, 48. 
 Gibers' s theory, 172. 
 Opposition, 126. 
 Qrion, 21 6. 
 
 Pendulum experiment, 68. 
 
 Penumbra, of solar spots, 95 ; of eolipsa 
 136. 
 
 Perigee, 122. 
 
 Perihelion, 90. 
 
 Perturbations, in earth's orbit, 112; in 
 moon's, 130. 
 
 Phases, of moon, 126 ; of earth, 127; of 
 Mercury and Venus, 165; of Mars, 
 171. 
 
 Photosphere, 95. 
 
 Planetoids, 171; list of, 260. 
 
 Planets, 155 ; orbits of, 157 ; inferior, 158j 
 stationary points, 159,168; elements 
 of orbit, 160; heliocentric longitude 
 of node, 161 ; inclination of orbit, 
 102; periods, 163,168; superior, 167 j 
 distance, 169 ; elements, 256. 
 
 Plateau's experiment, 185. 
 
 Pleiades, 230. 
 
 Points, fixed, 36. 
 
 Pointers, 216. 
 
 Poles, of the heavens, 15, 17; of the 
 earth, 17. 
 
 Pole-star, 14. 
 
 Position angle, 24. 
 
 Prcesepe, 230. 
 
 Precession, 112. 
 
 Problem of three bodies, 133. 
 
 Projections, spherical, 27. 
 
 Proper motions, 213. 
 
 Quadrant, 48. 
 Quadrature, 126. 
 
 Radiant points, 206. 
 
 Rate of clock, 28. 
 
 Refraction, 51; astronomical, 52; ge- 
 neral laws, 53; effects of, 54. 
 
 Resisting medium, 196. 
 
 Reticule, 32. 
 
 Reirogradation, 159, 167, 
 
 Rings of Saturn, 177; disappearance 
 of, 178. 
 
 Parabola, 246. 
 
 Parallax, geocentric and horizontal. 55: 
 
 heliocentric. 5G, 156; annual, 21<). 
 Pcqaxus, 216 
 
 os, 143. 
 Satellites, elements of, 258. 
 Saturn, 170; rings of, 177. 
 \SeaftonSf yo. 
 
272 
 
 INDEX. 
 
 Sextant, 42 ; prismatic, 48. 
 
 Shadow, of earth, 136; of moon, 139. 
 
 Signs of zodiac, 84. 
 
 Sirius, light of, 223 j orbit of, 240. 
 
 Solar system, 11; orbit of, 238. 
 
 Solstices, 84. 
 
 Spectroscope, 48; use of, 97. 
 
 Sphere, celestial, 11; parallel, 16; right 
 and oblique, 17. 
 
 Spheroid, oblate, 63, 245. 
 
 Spots, solar, 95; observations of, 261. 
 
 Star signals, 79. 
 
 Stars, circumpolar, 14; fixed, 213; 
 number of, 214,234 ; magnitudes,214 ; 
 of first magnitude, 217 ; constitution, 
 218; distance, 218; differential ob- 
 servations, 220; real magnitudes, 
 222; variable and temporary, 223; 
 double and binary, 225 ; colored, 228; 
 examples of variable, 266; of binary, 
 267. 
 
 Stationary points, 159, 168. 
 
 Style, old and new, 106. 
 
 Sun, distance of, 84; magnitude, 87; 
 rotation, 95; constitution, 98; irreg- 
 ular advance, 102; first mean, 102; 
 second mean, 103; mass and density, 
 109; size compared with stars, 223; 
 motion in space, 235; elements, 256. 
 
 Synodical revolution, of moon, 128; of 
 planets, 163, 168. 
 
 Talcotfs method of finding latitude, 74. 
 
 Telegraph, used in determining longi- 
 tude, 78. 
 
 Telescopic comets, 188. 
 
 Tempel's comet, leads November 
 shower, 212. 
 
 Theodolite, 48. 
 
 Tides, 146; daily inequality, 148; ge- 
 neral laws, 149; influence of sun, 
 149; spring and neap, 150; priming 
 
 and lagging, 150; tidal wave, 151 j 
 establishment, 151; cotidal lines, 
 152; height, 152; four daily, 153,- 
 in lakes, 154. 
 
 Time, solar and sidereal, 21 ; sidereal 
 and right ascension, 22 ; Greenwich, 
 77 ; local time at different meridians, 
 81; astronomical and civil, 105. 
 
 Torsion balance, 63. 
 
 Trade-winds, 67. 
 
 Transit instrument, 31. 
 
 Transit, 20; of inferior planets, 262. 
 
 Triangle, astronomical, 23. 
 
 Triangulation, 60. 
 
 Twilight, 94. 
 
 Umbra, of solar spots, 95 ; of eclipses, 
 
 136. 
 
 Universal instrument, 48. 
 Uranus, 180. 
 Ursa, major, 215; minor, 216. 
 
 Vanishing points and circles, 26. 
 
 Variation, 133. 
 
 Venus, relative distances from sun and 
 earth, 84; transit of, 86, 165, 262; de- 
 scription of, 165. 
 
 Vernier, 46. 
 
 Vertical, lines, 18; prime, 19. 
 
 Vulcan, 157 (note). 
 
 Weight, in different latitudes, 64; on 
 the sun, 110. 
 
 rear, sidereal, 83; tropical, 105,114; 
 anomalistic, 116. 
 
 Zenith, 18; geographical and geocen- 
 tric, 76. 
 
 Zenith telescope, 48; use of, 74. 
 Zodiac, 84. 
 Zodiacal light, 99. 
 
 THE END. 
 
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