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ASTRONOMY 
 
 WITHOUT MATHEMATICS. 
 
 i**- £i tSirmHva'r^C 
 
 EDMUND BECKETT DENISON, LL.D., Q.C., F.R.A.8. 
 
 AnTHOB OF 
 tna BUUmXHTAItY TBSATISE on clocks and watches. iSD BELUI, 
 ~~ I^OrnESBONCHDBCH BtriLDIKQ, ETC.- 
 
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 NEW YORK: 
 
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 18fi9. 
 
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 8TEBE0TTPED BT 
 
 DENNIS BRO'S JE THORNE, 
 
 AUBUIIN, N, T. 
 
NOTE TO THE AMERICAN EDITION". 
 
 The great popularity of Denison's " Astronomy without Math- 
 ematics" has induced the American publishers to prepare an 
 edition specially adapted for general circulation in the United 
 States. The only changes which have been deemed advisable, 
 are : 1. A few verbal alterations, in cases of accidental errors or 
 inadvertences of style or statement. 3. The addition of an 
 Appendix, to which references are made by foot-notes in the 
 body of the work. 
 
 The care bestowed by the author in the preparation and ar- 
 rangement of his materials, the general accuracy of his tezt, the 
 simplicity of his explanations, his judicious presentation of inter- 
 esting topics, and the valuable information embodied in the notes 
 of the American editor, will commend the book to teachers as 
 well as to private readers, and we believe that it will also be foimd 
 worthy of a place in the libraries of professional astronomers. 
 
PREFACE TO THE THIRD EDITIOK 
 
 The sale of 3000 copies of this book in little more 
 than a year, many of them to persons of more educa- 
 tion than I originally contemplated, has induced me to 
 enlarge it considerably, going rather deeper into the 
 subject, and adding some explanations which I did not 
 venture on before. The additions chiefly relate to 
 meteors, nebulae, and stars, the moon's acceleration and 
 other disturbances, the tides, and the calculations for 
 Easter in all ages ; and a fuller account of the meth- 
 ods of weighing the sun, moon, and planets. The chap- 
 ter on telescopes also has been greatly enlarged ; for 
 astronomy lives by them, and I do not know where a 
 popular explanation of the theory of telescopes is to 
 be found. 
 
 I keep to the plan of using as few ' diagrams,' and as 
 few words, as will serve the purpose, because I am 
 satisfied that such explanations are both easier to read 
 and to remember — provided of course they are expla- 
 nations (see p. 100 n). A book is not a lecture. There 
 it may be prudent to say things several times over, in 
 diiferent ways, and to illustrate them as well as you 
 can ; for the hearers must not stop to think, or they 
 will be left behind. I have not scrupled to draw fig- 
 ures wherever I thought they would be useful. 
 
 \ repeat the warning of the former editions, that 
 this book only aims at making astronomy as easy as it 
 can be made if difficulties and the reasons of things 
 are really to be explained, and not evaded in vague 
 
Preface. 6 
 
 language which leaves people as ignorant as before. 
 It is idle to suppose that anything can be learnt of 
 astronomy as a science of causes and effects, without 
 some study and power of thought, and some natural 
 capacity for geometrical conceptions. But those who 
 cannot always follow the reasoning may still read the 
 results, treating the book as one of ' descriptive astro- 
 nomy' only, though it is really an introduction to 
 ' physical astronomy,' or the astronomy of causes and 
 effects. 
 
 Though no mathematical knowledge is required of 
 the reader, I do not profess anything so absurd as to 
 rebuild the ISTewtonian system without mathematics. 
 We soon come to a point in explanation where we must 
 either stop and disclose no more, or else bridge over 
 the chasm by adopting some simple result, or perhaps 
 rather difficult calculation ; such for instance as these 
 two : that a sphere, but not a spheroid, attracts as if 
 it were all condensed into its centre; and that the 
 time of performing an elliptic orbit is the same as of 
 the circle which contains it. All the arithmetical cal- 
 culations here are founded on propositions as simple as 
 these ; and yet these require geometry, algebra, trigo- 
 nometry, conic sections, and differential and integral 
 calculus (or else other obsolete contrivances) to prove, 
 though not to use them. 
 
 I nave taken no small pains to avoid mistakes, but 
 I cannot expect to have entirely succeeded ; for even 
 great astronomers occasionally commit them, from 
 haste of writing or imperfect recollection ; and some- 
 times they have to recant absolute mistakes of reason- 
 ing. I do not pretend to be an astronomer at all ; but 
 having been pressed into this service by a kind of ac- 
 cident, I have done the best I could for it. 
 
 E. B. D. 
 
 33 Queen Anne Street, W., 
 FeTjruary, 18S7. 
 
OOI^TEli^'TS. 
 
 CHAPTER I. 
 
 The Easth. 
 
 EAIU.T notions of its shape, g. Proof thatltiBroand.ii. Measuring it, 13. Is 
 a splieroid, 15. Latitude and longitude, 16. Cause of earth's ellipticity, 
 18. Different kinds of maps, 30. Land and water, and earth's surface and 
 volume, 23. Weighing it, 24. Law of gravitation, 27. Cavendish's and 
 other experiments, 30. Proofs of the earth's rotation, 3X. Its motion 
 round the sun, 33. Velocity and distance, and loss of gravity at equator, 
 36. Elliptic orbits and definition of ellipse, 38. Eccentricity, 39. Motion 
 of perihelion, 41. Change of seasons and glacial epochs, 42. Cause of the 
 seasons, 44. Trade Winds, 49. Length of the year and precession of tlie 
 equinoxes, 50. Kutation, 54. Sidereal and solar time, 57. Equation of 
 time, 58, 
 
 CHAPTER II. 
 
 The Sun. 
 
 Its diameter, and greatest and least distance, 61. Weight and density, 63, 
 Specific gravity in general, 64. Sun's parallax. 64. Gravity there, 65. 
 Spots and composition of the sun, 65. Atmosphere and photosphere, 67. 
 The willow leaves, 69. Meteoric theory of heat, 70. Its heat, 70. The 
 zodiacal light, 71. Latent and specific heat, 74. Heat is force, 76. Natui-e 
 and velocity of light, 77. Undulatory theoiy, 78. Light, sound, and elec- 
 tricity, 80. Colors and invisible rays, 81. Kesisting medium, 82. New 
 dimensions of the solar system, 3s. Masses of sun and planets reduced, 
 87. Attraction Is inetautaneous, 89. 
 
Contents. . 7 
 
 CHAPTER III. 
 
 TSE Moon. 
 
 Her size and distance, 91. Surface, 92. Botation, 93. Moves the earth, 97. 
 Shape of her orbit round the smi, 99. Orbit round the earth a revolving 
 ellipBe, 100. Nodes recede, 101. Libratlous, 102. The moon^s shape, 104. 
 Light and heat there, 105. Phases, 106. Periods, 107. Harvest moon, 109. 
 Moonlight in winter, no. Eclipses, in. Moon never totally eclipsed, 116. 
 Chaldsean Saros of eclipses, 118. Metonlc cycle and rules for Easter, 119. 
 Eeform of the calendar, 121. Gregorian calendar wrong. 122. The Golden 
 numbers, 124. To find the days of the week in all times, 127. To find 
 Easter, 128. The tides, 131. The moon^s disturbances, 140. Secular accel- 
 leration, 141. Annual equation, 142. Betardationofthe earth, 143. Calcu- 
 lation of disturbing forces, 146. Variation, 150. Parallactic inequality, 151. 
 Advance of apsides, 152. Change of eccentricity, 133. Evection, 155. Ke- 
 ccssion of nodes, 153, Moon is disturbed by Venus,iS9. 
 
 CHAPTER IV. 
 
 The Planets. 
 
 The Ave old planets, i6i. Copemican theory, 163. Mercury, 164. Venus, 166. 
 Mars, 167. Spheroidicity of planets, 169. Asteroids, 170. Jupiter, 171. 
 Saturn, 173. The long inequality, 174. TJranus, 177. Discovery of Nep- 
 tune, 179. Vulcan, 184. Bode's law of distances, 185. Betrogradation of 
 planets, 186. Saturn's ring, 189. His moons, 193. TJranus's, 194. Jupi- 
 ter's, 194. Effect of his oblateness, 196. Velocity of light and longitude 
 found by Jupiter's moons, ig8. Longitude by other methods, 200. Tran- 
 sits of Venus and Mercury, and finding sun's distance, 202. Parallax of 
 sun and moon, 207. Of Mars, 208. Aberration, 211. Befraction of the air, 
 313. Its effect on suu and moon, 215. Twilight, 217. Meteors of August, 
 218. Of November, 319. Aerolites, 222. 
 
 CHAPTER V. 
 
 The Laws or Planetart Motion, Comets, NEBTitiB, ahd Staes, 
 
 Kepler's laws, 223. Gravitation and inertia, 223. The planets move the sun, 
 228. Newton's laws of motion, 231. Momentum vis viva, and moment of 
 inertia, 233. Elliptic orbits, 237. Centrifugal force measured, 240. Gravity 
 at the equator, 241. Weighing of the sun, 243 ; of the moon, 249, 251. Law 
 oftimo and distance, 249. Absolate masB of earth and sun, 251. ■Weighing 
 
Contents. 
 
 of Jupiter and Saturn, 2 J2. Absolute measure of gravity, xsj. Three laws 
 of Biabllity, 155. Conic sections, zs6. Comets, 260. Perihelion velocity, 
 261. Hemarkable comets, 262. Comets' tails, 264. Kebular hypothesis of 
 formation of the solar system, 266. Nebulas, and the Milky Way, 270. 
 Time of rotation of nebulae, 275. The'stars, 277. Magnitudes and distan- 
 ces, 278, Time of light coming from stars, 281 . Spectrum analysis of stars, 
 283. WeighiDg and measuring of double stars, 285. Size of Sirins, 286. 
 Bevolution of all the stars, 287. Possible time of it calculated, 28S. The 
 celestial globe, 290. Kight ascension and declination, 292. Celestial and 
 heliocentric longitude, 292. Altitude, Azimuth, and Dip of horizon, 293. 
 Orreries, 29$. 
 
 CHAPTER VI. 
 
 Oh Telescopies. 
 
 Invention of telescopes, 296. Friar Bacon and Galileo, 296. Space-penetra - 
 ting power, 298. Loss of light in telescopes, 298. Masjniflcation, 299. Mi- 
 croscope, 301. General view of astronomical telescope, 301. Lawofreflrac- 
 tion, 302. Internal reflection, 305. Focal length of lenses, 307. Centrical 
 and eccentrical pencils, 308. Magnifying glass, 308. Spectacles, 310. As- 
 tronomical telescope and Galilean (figures), 312. Magnifying power, 311, 
 Galileo's telescope, 313, Field of view, 314- Brightness, 31 S. Color disper- 
 sion, 316. Aerial and acromatic telescopes, 317, 318. Secondary colors, 
 319. Spherical aberration, 321, Huyghens's eye piece, 323. Hamsden's, 
 324. Micrometers, 325. Keflecting Telescopes, 326. Largest retracting 
 telescope, 326. Beflection of concave mirrors, 327. Gregory, Cassegrain, 
 Newton, Herschel, and Lord Eosse'a telescopes, 329. Figure of Newton's 
 telescope, 330. Speculum metal and silvered speculum, 331. Helioscopes, 
 332. Transit and mural circle. 334, Altazimuth, 337. Equatorial, 338, Be- 
 cordlng barrel, 337. Note on the Great Pyramid, 341. Appendix, 341. 
 
ASTRONOMY 
 
 WITHOUT MATHEMATICS. 
 
 CHAPTEE I. 
 
 THE EAETH. 
 
 Foe several thousand years people supposed that the 
 world was a large platform, and that if you went far 
 enough you would come to the edge everywhere, as 
 you do at the sea shore. They thought the sun set in 
 the sea and rose out of it ; at any rate the old Greek 
 poets said so, and invented wonderful contrivances for 
 carrying the sun round or under the earth in the night 
 from west to east again.* 
 
 But you may ask, does not David in the Psalms (i8 
 and 39) speak of the ' round world ? ' No, he does 
 not. There is no such word as ' round ' in the original 
 Hebrew ; nor in the old Greek translation called the 
 Septuagint, said to have been made by seventy-two 
 learned men about three centuries before Christ ; nor 
 in the Bible version of the Psalms. Its appearance in 
 the Prayer book is a curious result of successive trans- 
 lations. That version was translated from the Latin 
 
 ^- ^ : ^ 
 
 * See Sir G. C. Lewis's ■' Astronomy of tHe Ancients.' 
 1* 
 
10 The Emih. 
 
 one called the Ynlgate, which alone was used until the 
 Eeformation ; and St. Jerome, who made the Yulgate 
 Psalter from the Septuagint, translated the Greek word 
 for ' the inhabited world ' into that which had become 
 the common name for the world with the Latin writers, 
 who did know that the earth was round,* viz., orHs 
 terra/rum. And so Cranmer, 'whose image will be 
 seen reflected on the calm surface of the liturgy while 
 the Church of England remains,'t not unnaturally 
 followed it, and translated almost IHqicbXLj fvmda/menta 
 orbis terrarum into ' the foundations of the round 
 world.' But the translators of the Hebrew. Bible after- 
 ward did not. 
 
 It wonld have been contrary to the habit of the 
 Bible to anticipate and reveal a scientific discovery, 
 which men would make for themselves in time, and 
 which was of no consequence to their religious faith 
 and life. It is not contrary to the habit of the Bible, 
 nor at all superfluous, to declare continually, wherever 
 there is an opportunity (as we may say), that the sun 
 and the moon and the stars, and the earth and all that 
 is in it, did not grow of themselves, as some people 
 fancy, but were created, or made out of nothing by 
 the word of God. For all the science in the world 
 could never prove that ' in the beginning God created 
 the heaven and the earth : ' we can only know that by 
 revelation. ' Through faith (not science or observation 
 and reasoning) we understand that the worlds were 
 framed by the word of God, so that things which are 
 
 * Cicero, ' De Natnra Deoram,' ii. 19, etc. 
 t Prottae'9 ' History of England,' t. 391. 
 
Its Roundness. 11 
 
 « 
 
 seen were not made of things which do appear ' (Heb. 
 xi. 3.) And all experience fihows that men who dis- 
 believe that, believe nothing else that is revealed. 
 This has nothing to do with the proper interpretation 
 of the particular words in which the successive acts 
 of creation are described in the first chapter of the 
 Bible. Nearly all learned men now agree at least 
 in this, that the word translated ' day ' there does 
 not always mean a day of twenty-four hours, but may 
 mean periods of enormous length, each ended by some 
 marked division of time or epoch. 
 
 Thales, who was called one of the wise men of 
 Greece, is commonly said to have discovered that the 
 world is a round globe, about 600 e.g.* Consequently, 
 men and water And all things stand all round it with- 
 out falling off; and what we call ' upright ' only means 
 upright with reference to the surface of the earth or 
 of water where we are, or in a line toward the centre 
 of the earth ; toward which all things fall or press ; so 
 that they fall in opposite directions here and in the 
 Indian Ocean, which you may see is exactly opposite 
 to the United States, in that model of the earth with 
 the countries painted on it, which they call a terresfnoL 
 globe. 
 
 You may ask how we know all this. "We know fij'st 
 that the earth is round, in some general sense of the 
 
 * Sir G. C. Lewis says there is no evidence in support of this tradition. It is 
 of very little consequence ; but on the other hand there seems a strong proha- 
 hility of the Chaldseans having known far more aho^t astronomy than the mere 
 roundness of the earth, ages before Thales. See a chapter on Chaldsean as- 
 tronomy in Mr. Proctor's book on Saturn, and the late John Taylor's book on 
 tho Great Pyramid. 
 
12 Rovmdness of the Ewrth. 
 
 word, by finding that we can actually go quite round 
 it by sea or land in every direction, except where we 
 are stopped by ice or mountains. Secondly, we find 
 that the sea is nowhere flat, but rises everywhere like 
 a low round hill, so that the masts of distant ships are 
 always seen before their hull or body. So great plains, 
 which are level like the sea, are not flat like a table, 
 but rise visibly between two people at a distance, as 
 the sea rises between the ships. And such plains, and 
 level surfaces of water all over the world, rise the same 
 height in the same distance, viz., 8 inches in a mile, 
 and not i6 but 32 or 4x8 inches in 2 miles, 9x8 
 inches or 6 feet in 3 miles, and so on for a considerable 
 distance, exactly as it would upon a globe according 
 to geometrical rules. So that two very tall men stand- 
 ing 6 miles apart on a level plain can only just see 
 each other's heads with telescopes, and what we call 
 the 'visible horizon,' or boundary of sight by the level 
 ground or sea, is everywhere 3 miles from an eye 6 
 feet high. All this can only happen on a globe of a 
 certain size. And what we call level only means flat 
 when the surface is not large enough for the curve to 
 be distinguished. It means really the surface which 
 fluids take at rest, to which a plumb line is upright, 
 and which is equidistant everywhere from the centre 
 of the earth — subject to a small correction which you 
 will see at page 19. 
 
 Again we see the outline or shadow of the earth it- 
 self upon the moon in eclipses, and that is always 
 round, whatever part of the earth may face the moon 
 just then. Indeed, as eclipses were observed and pre- 
 
Sise of the EaHTi. 13 
 
 dieted and recorded as important astronomical events 
 long before distant voyages at sea were made, the 
 roundness of the world was very likely first considered 
 to be -proved by them, though the notion of its being a 
 platform may have been given up before. For a body 
 which always casts a round shadow can be nothing 
 but a globe, as you may easily see if you hold up things 
 of other shapes before the sun in different positions. 
 
 Measuring the Earthi — After it was found that the 
 earth is a globe, it was natural to try to measure it ; 
 but it was long before that could be done accurately. 
 It may indeed be done approximately from the figures 
 just now given ; for it may be proved that if the earth 
 is a globe, its diameter is to the distance of the visible 
 horizon as that is to the height of your eye above the 
 plain ; which you will find gives 7920 miles, for a 
 height of two _yards and a distance of the horizon of 
 3 miles or 5280 yards. But this method admits of no 
 great accuracy, and these figures are rather the result 
 than the source of really accurate knowledge of the 
 earth's size ; for the rays of light near the ground are 
 irregularly bent or refracted by the air, so that you do 
 not in fact see straight, and cannot distinguish where 
 the visible horizon for really straight lines of sight 
 would be ; and a very small error in the distance of 
 the horizon will make a very large one in the size of 
 the earth. 
 
 It has now been measured by other means which I 
 will describe presently ; and it is found to be 24,907 
 miles round the equator / which is a circle round the 
 middle of the earth at an equal distance from the 
 
14 Size of the Ea/rth. 
 
 north and south poles. The poles are the two ends of 
 that imaginary axis round which the earth turns every 
 day. All circles round the earth and going through 
 the poles ai"e called meridians; and so everyplace has 
 its own meridian, which runs from north to south, and 
 the sun crosses that circle at the noon or mid-day of 
 that place. All circles which divide any globe equally 
 are called great circles, because no greater can be 
 drawn. Any straight cut or section through a globe, 
 which does not divide it equally, makes a small circle. 
 The shortest road between any two places on a globe 
 is by the great circle which passes through them both : 
 hence comes what is called ' great circle sailing.' The 
 diameter of a globe is necessarily also the diameter (or 
 line through the centre) of every great circle ; and you 
 should remember that the circumference of every circle 
 is very nearly 3^ of its diameter; that is, if the diame- 
 ter is seven feet or miles the circumference may be 
 called 22; or more exactly, circumference = 3' 141 6 
 diameters very nearly ; but no number of figures can 
 express the exact proportion. The radius is half the 
 diameter. 
 
 The greatest equatorial diameter is 7926"6 niiles. 
 Some measurers of the earth make it nearly two mile? 
 less than this through 104° east and 76° west longi- 
 tude ; but later calculations seem to make this doubt- 
 ful (see p. 169). At any rate we may practically treat 
 the earth as round at the equator, and at all the small 
 circles parallel to it, which are called ^araZZeZs of lati- 
 ivde. 
 
 The polar axis may be called 7899^ miles, or 500 
 
The Ea/rth is a Spheroid. 15 
 
 millions of inclies about a thousandth longer than our 
 inch (a quite insensible difference), or 20 millions of 
 the ' sacred ' cubit of the Jews, according to Newton's 
 estimate thereof (see p. 341). Sir J. Herschel remarks 
 that the French metre, 39'37i inches, the newest cmd 
 worst measure in the world, differs much more than 
 that from the fraction it pretends to be, a 40 millionth 
 of a far more uncertain quantity, the length of a me- 
 ridian. A few minor scientific men fancy that by 
 Avriting in French measures they can make the world 
 follow them into adopting this inconvenient, inaccu- 
 rate, unstridable measure, and its subdivisions. They 
 only make the greatest part of the world not follow 
 them, in another sense.* 
 
 The polar axis being thus about 26^ miles less than 
 the mean equatorial one, the earth is not quite a globe 
 or sphere, but what is called a spheroid / which means 
 something like a sphere. There are two kinds of 
 spheroids, one flattened at the poles, and fatter round 
 the equator, as the earth is, which is called an oblate 
 spheroid, and is formed by turning an ellipse round its 
 smallest diameter ; and the other, formed by turning 
 an ellipse round its greatest diameter, is thinner at the 
 equator, and drawn out at the poles, like an egg with 
 two small ends, which is called a prolate spheroid. 
 The spheroidicity of the earth or any other planet is 
 usually called its ellipticity : which means the propor- 
 tion between the difference of the two axes or semi-axes 
 of an ellipse, and the greater of them ; or the propor- 
 tion of EB to AC in the figure at p. 39. 
 
 * See Appendix, Note I. 
 
16 Latitude cmd Longitude. 
 
 Consequently every meridian of the earth is an el- 
 lipse, and not strictly a great circle ; though for ordi- 
 nary purposes it may be called one, as the ellipticity, 
 or the proportion of 26J miles to 7926, is only one 
 298th, so little that you could not perceive it on any 
 globe that could be made. An ellipse is shorter than 
 the circle containing it by very nearly half the ellip- 
 ticity, so long as that is small : when it is not, the relar 
 tion between them is complicated. Therefore a meri- 
 dian is a 596th shorter than the equator. In giving the 
 velocity of the earth and planets in their orbits, which 
 are all elliptical, I shall treat them as circles for sim- 
 plicity, as their ellipticity is very small, and nothing 
 turns on the precise amount of the velocity. 
 
 As I have had to mention meridians and latitude, I 
 had better explain at once what latiimde and longitude 
 are. The circumference of every circle may be di- 
 vided into 360 equal parts, called degrees ; and again, 
 every degree (1°) contains 60' (minutes), and every 
 minute (i') contains 60" (seconds), which have nothing 
 to do with minutes and seconds of time. That is the 
 way that parts or arcs of circles are always measured, 
 and angles also, or the opening between the two 
 straight lines called radii, reaching from the centre to 
 the circumference of any circle, whether the circle is 
 actually drawn or only imagined to be drawn. For 
 instance, the angle 90° means the opening at the centre 
 of the circle between two lines drawn to the two ends 
 of a quadrant of the circumference ; and 90° is called 
 a right angle, and lines at right angles are also said to 
 he perpendicular to each other; for ' perpendicular ' in 
 
Gaiise of the Earth's Elliptieity. 17 
 
 mathematics does not always mean upriglit, though an 
 upright stick is of course perpendicular to a piece of 
 level ground or the surface of water, which is always 
 level. If you stick a pencil with its flat end upon a 
 globe, it is perpendicular to the surface of the globe, 
 or upright. 
 
 All the meridians of the earth then (treating them 
 as circles, notwithstanding their slight ellipticity) are 
 divided into 360°, and those degrees are measured from 
 the equator toward each pole, and every such degree 
 is called a degree of latitude, and therefore each pole is 
 at latitude 90°. Again, the distance of any meridian, 
 measured in degrees on the equator and on all the 
 small circles of latitude as well, from any other meri- 
 dian which is used as a standard or o (called zero), is its 
 longitude, and is the longitude of all the places on it. 
 A degree of longitude contains 69" 17 miles at the 
 equator ; but as you go further north or south, the me- 
 ridians come closer together, and so a degree of longi- 
 tude measures fewer miles the further you get from the 
 equator. In England a degree of longitude measured 
 east or west from the meridian of Greenwich Observa- 
 tory, which is our zero, is about 43 miles, whereas a 
 degree of latitude measured on any meridian, is about 
 69 miles everywhere — that is, 69*4 in high latitudes, 
 and 68"8 near the equator, for the reason which you 
 will see presently. 
 
 Long before anybody attempted to measure the 
 difference between the equatorial and polar diameters. 
 Sir Isaac ]?Tewton, who was born on Christmas Day, 
 1642, and died in 1727, calculated what it ought to.be ; 
 
18 Centrifugal Force at the Equator. 
 
 though calculation will not make it quite right, ii-om our 
 ignorance of the density of different parts of the earth. 
 It was probably once all fluid, like the lava from vol- 
 canoes, with all the water hanging over it as steam ; and 
 even now it gets i ° hotter for every 90 feet down a mine, 
 and water is hotter as it comes from greater depths.* 
 It would then take tlie shape of a globe, like a drop 
 of rain, or melted lead in making shot, because the 
 mutual attractions of the particles balance themselves 
 in that shape only. When it began to spin it would 
 swell at the equator, and shrink at the poles, as a large 
 elastic hoop will do if you spin it quickly round its 
 diameter. Newton calculated how much extra weight 
 laid on the equator would balance the loss of weight or 
 gravitation to the centre there, by reason of the cen- 
 trifugal force arising from the spinning ; which increases 
 as the square of the velocity of rotation ; f i.e., it would 
 be four times as great if the earth turned twice as fast, 
 and if it turned round in an hour and 25' min. people 
 could not stand at the equator, but would be thrown 
 off. Besides that, he had to calculate how much the 
 attraction to the centre is altered by the alteration of 
 the shape from a sphere to a spheroid, and the result is 
 compounded of those two calculations (see p. 37). 
 
 If you wish to know how the circumference and the 
 polar and equatorial diameters of the earth are mea- 
 sured, it is done thus. As the earth turns once round 
 or 360° in 24 hours, two places whose meridians are 
 
 * Appendir, Note II. 
 
 t Centrifugal force also increases as the aiamefer, tnt bo does the attraction 
 Of a globe upon its surface, wMch counteracts it (see p. 29). 
 
The Em-th. 19 
 
 crossed by the same star at an interval of 4 minutes, 
 are 1° of longitude apart ; and the equatorial circum- 
 ference (treated as a circle) is 360 times the measured 
 distance of two such places on the equator. But the 
 rotary motion of the earth is of no use for measuring 
 latitude ; and besides that, meridians of the earth are 
 not circles but ellipses, and the plumb line, or a line 
 perpendicular to the surface of water, does not point 
 quite to the earth's centre, in consequence of the sphe- 
 roidicity, though we popularly say it does, exisept at the 
 equator and the poles. ! Places are i" apart when their 
 plumb lines make an angle of 1° with each other, which 
 can be measured by the stars, as they are so far off that 
 they may be used as fixed points in the great sphere of 
 the heavens with its centre at the centre of the earth. 
 It is found by measuring in this way that 1° of latitude 
 in Sweden is •§■ mile longer than at the equator, and 
 100 yards longer in Scotland than in the south of 
 England. And thus, by taking a few degrees at 
 different latitudes, a whole meridian, or section of the 
 earth through the poles, can be made up and measured 
 both in shape and size, and the difference of the equa- 
 torial and polar diameters ascertained. A nautical or 
 geometrical mile or Tcnot is i' of longitude at the equa- 
 tor, or about one-sixth longer than a common mile. 
 
 MAPS. 
 
 One consequence of the earth being round is that no 
 map of any large part of it can be correct. You can- 
 not make a large piece of paper lie close upon a globe 
 without crumpling the edges. Therefore if the middle 
 
20 Ma^ps of th6 Em-th: 
 
 of a countiy is drawn on the map as it would be on the 
 globe, the outsides would be drawn too large, and vice 
 versa / and the larger the country is, the more some 
 parts of it must be enlarged beyond others, or distorted. 
 Maps are made on various plans, some distorting the 
 couatry in one way, and some in another. The com- 
 mou ' map of the world,' in two flat circles, makes the 
 equator only twice as long as the diameter of the earth, 
 instead of 3^ as long. And each of those two circles, 
 which stand for the hemispheres, or half the surface of 
 the globe, only show half as much surface as a hemis- 
 phere of that diameter really has. 
 
 That mode of ' projecting ' a hemisphere or any part 
 of it on a plane is called the orthographic, because it 
 shows the surface as it would be seen straight by paral- 
 lel lines of sight from an infinite distance. It repre- 
 sents the middle of the country tolerably right, but the 
 outsides are crowded, and very much so toward the 
 edges of the hemisphere. 
 
 - If you suppose the globe transparent and the eye at 
 the cmtipodes, or the opposite end of a diameter to the 
 country looked at, it is BQem^vo^Qcted st&reograpJiicaUy 
 on any plane at right angles to that diameter, such as 
 a plane touching the sphere at the other end of the 
 diameter, or a glass plate parallel to that and nearer 
 to the eye. This is in fact a perspective view of the 
 country from the middle of its antipodes ; for perspec- 
 tive only means transparent, and the glass plate theory 
 is the foundation of all perspective drawing. But the 
 country must be reversed for the map, as it is seen from 
 the inside instead of the outside of the globe. This 
 
Or, Projections of the Sphere. 21 
 
 method has the advantage of preserving the shape of 
 eveiy part, but the middle parts are now more crowded 
 than the outsides, though not so much as the converse 
 in the orthographic projection. 
 
 The gnomonic projection has the eye at the centre of 
 the transparent globe instead of the surface, and is seen 
 upon a plane touching it, or on any transparent plate 
 parallel to that plane. This too is sufficiently accurate 
 for countries near the point of contact, but at any con- 
 siderable distance from it the outsides are very much 
 exaggerated, as you may easily see if you draw a circle 
 touching a straight line, and divide it into some equal 
 parts and draw lines from the centre through those di- 
 visions to that line, which represents the tangent plane. 
 It has the advantage that all great circles through the 
 point of contact are opened out into straight lines cross- 
 ing there, and all small circles parallel to the plane are 
 projected into circles with that point for their common 
 centre. This is frequently used for star maps, the plane 
 of projection touching at the pole, so that all meridians 
 are straight lines crossing there ; but circles of declina- 
 tion (which correspond to parallels of latitude on the 
 terrestrial globe) are not equidistant, as they are in — 
 
 The equidistamb projection, which has the eye above 
 the sphere at a height=half the chord of a quadrant, 
 or 707 of the radius, looking straight down through 
 the globe upon a tangent plane at the opposite pole. 
 This makes the latitude or declination circles for a 
 considerable distance nearly equidistant, and the meri- 
 dians straight lines as before; and so this is the best 
 kind of map for the regions round the poles. . 
 
22 Maps of the Earth. 
 
 In Meroator's prelection, which is a favorite one for 
 maps, the globe is supposed to be stretched out on the 
 inside of a cylinder which touches it all round the 
 equator, and the cylinder is then cut and opened out 
 flat or ' developed.' But besides that, since paa-allels 
 of latitude in England would be stretched wider to fit 
 the cylinder in about the proportion of 43 to 69 (p. if), 
 therefore degrees of latitude or the length of pieces of 
 the meridian are drawn in such maps wider than those 
 near the equator in the same proportion, in order to 
 keep the true proportions between the length and 
 breadth of each division of the map ; and so the dimen- 
 sions increase rapidly towards the poles. For this rea- 
 son it is unfit for maps near the poles ; and the maps 
 of countries of high latitude must be made on a difier- 
 ent scale from those near the equator, or rather, as if 
 they had been developed from a different globe, in or- 
 der to get them on anything like the same scale. 
 
 There is yet another, very convenient for some pur- 
 poses, called the conical projection. Suppose you want 
 to map a country in the latitude of England. A hol- 
 low cone is supposed to be dropped over the globe, of 
 such an angle that it will touch it all round at latitude 
 52°, and therefore the top of the cone will be vertically 
 over the north pole. Then the country is drawn as it 
 would appear on the inside ot this cone to an eye at 
 the centre of the earth, and the cone is ' developed.' 
 Consequently, the meridians are all straight lines con- 
 verging toward a point which was the top of the cone, 
 and the parallels of latitude are nearly equidistant, 
 and in fact are drawn quite so for convenience. In 
 
Tlie JEartNs Surface and Volume. 23 
 
 this there is scarcely any distortion for a moderate 
 breadth of country from north to south, or a zone be- 
 tween two parallels of latitude near to the circle of 
 contact with the cone. This also is used for star maps, 
 and so, indeed, are all the projections, except Merca- 
 tor's and the orthographic, and they all have their ad- 
 vocates.* 
 
 Sir J. Herschel remarks that London is very nearly 
 the centre of that hemisphere of the globe which con- 
 tains more dry land than a hemisphere described round 
 any other place as its pole. Those who have read a 
 little of Greek history know that a certain place, Del- 
 phi, was called the navel of the world, being then sup- 
 posed to be the middle. The real one, you see, is not 
 in Greece, but England. In order to see this, take a 
 terrestrial globe, and elevate the north pole 51-^° above 
 the north side of the wooden horizon, and bring Lon- 
 don up to the brass meridian : then all above the hori- 
 zon is the hemisphere with London for its pole or high- 
 est point, and it includes all Europe and Africa, and 
 all Asia except a few promontories, and all North 
 America and most of South, leaving only the rest of 
 it, and Australia and some islands, to the other hemis- 
 phere. 
 
 The following proportions of land and water over 
 the globe, and the north and south hemispheres, and 
 the five continents, with their islands, have been ascer- 
 tained by weighing paper patterns of them taken from 
 
 * See Enc. Brit. ' Geography (mathematical),' and a chapter on this subject 
 in Proctor's ' Handbook of the Stars,' a book chiefly of tables. There are other 
 projections of the sphere, but these are the principal. 
 
24 The EaHh. 
 
 a globe. All tlie water 145 million square miles, and 
 all the land 52 ; water north of equator 59, land 39; 
 water south of equator 86, land 13 : land in Asia 18, 
 Africa 12, North America 8, South 7, Europe 3^, and 
 Australia 3^^. 
 
 The earth's surface is four times the area of one of 
 its great circles, or 3'i4i6 times that of a square sur- 
 rounding it ; only that has to be reduced about a 400tli 
 for the eUipticity of a 298th ; and the result is 197 
 million square miles. The surface of any zone, or 
 band round the earth between two parallels of latitude, 
 is proportionate to its thickness, neglecting the eUip- 
 ticity. Therefore the surface of each hemisphere is 
 divided equally at 30° of latitude ; for a cut through 
 there cuts the axis half way between the pole and the 
 equator. 
 
 The solid content of a globe is '5236, or a little more 
 than half of the surrounding cube. And the bulk (but 
 not the surface) of any spheroid is to that of the sphere 
 which touches it all round as their different axes. 
 Therefore the earth is a 298th less than the sphere 
 which would contain it. But a sphere as large as the 
 earth would only have a diameter of 791 7-2 miles ; for 
 the cube of that = the polar axis of 7899' 5 x the 
 square of the mean equatorial diameter 7926, if these 
 figures are right (see p. 169 note). Therefore the earth 
 contains 259,845 million cubic miles. 
 
 "WEIGHING THE EARTH, AOT) LAW OF ATl^EACTION. 
 
 But a far more difficult thing has been done with 
 the earth than measuring, and that is weighing it ; if it 
 
The Schehallien JEayperiment. 25 
 
 can be said to have been done yet with certainty. Sir 
 Isaac Newton, by what Sir John Herschel well calls 
 ' one of his astonishing divinations,' hit upon the very 
 weight for the earth, which is nearly the average of all 
 the modern experiments and calculations, viz., that it 
 is about 5^ times as heavy as if it were all made of 
 water, or half as heavy as lead : which is expressed by 
 saying that its average density or specific gravity is 
 51^, that of water being alwaj'S taken as i, except in 
 speaking of airs or gases. The earth is on the whole 
 about twice as heavy as if it were all made of the 
 hardest and heaviest stones ; but beyond that, we know 
 nothing of its composition. • "Whatever the inside is 
 made of, it must be squeezed together with tremendous 
 pressure by the weight of all above it, and so it may 
 be a great deal denser than any materials of the same 
 kind near the surface. 
 
 I can only give you a very general account of the 
 different contrivances for finding the weight of the 
 earth. One is, by seeing how much the plumb bob is 
 pulled aside by the attraction of a mountain near it, 
 aud'comparing that, by calculations known to philoso- 
 phers, with the effect of the attraction of the whole 
 earth ; which can be most accurately measured by the 
 time of vibration of a pendulum, as you will see in a 
 later part of this book, where we shall weigh the sun. 
 This is called the Schehallien experiment, because it 
 was first made at the mountain of that name in 
 Scotland. 
 
 The nature of it is this. If two plumb liiies are 
 hung lOO feet apart, they make an angle of i"with 
 3 
 
26 Weighing the Earth. 
 
 each other, because each is pointing, we may "say, to 
 the centre of the earth ; and therefore at 6000 feet 
 apart, or rather more than a mile (5280 feet), the 
 plumb lines are inclined i' to each other. But if there 
 is a great mass of mountain rising between them, that 
 will attract each of the plumb bobs, and draw them 
 nearer together, because all matter attracts all other 
 matter, as you will hear more fully as we go along. 
 The. mass of the mountain can be calculated with tol- 
 erable certainty from its size, and weighing specimens 
 of the rocks which it is composed of; for you must 
 know that mass in mathematics means not merely size, 
 but size and density together ; in fact, what we com- 
 monly call weight; only there are reasons for keeping 
 the two words distinct in some mathematical calcula- 
 tions. It is possible to calculate how much the moun- 
 tain ought to draw the plumb bobs aside, and make 
 them converge more than i', if the whole earth were of 
 the same density as the mountain. But in fact the 
 mountain never does attract them so much as it ought 
 on that supposition. Therefore that supposition of the 
 mountain being as dense as the average density of the 
 earth is wrong; and they can calculate how much 
 wrong, or how much the average density of the whole 
 earth exceeds the ascertained density of the mountain, 
 and they find that the earth must be on the whole 
 about twice as dense or heavy as if it were all made 
 of the same rocks as the Schehallien, and about 5^ 
 times as heavy as water. 
 
 Another mode of weighing the earth is to see how 
 much faster a pendulum goes at the bottom of a deep 
 
La/uo of Gramitation. 27 
 
 mine than at the top. If the earth were all the way 
 through of the same density as the rocks near the sm*- 
 face, the pendulum would go slower, as I will explain 
 presently. But it does go faster ; and that proves that 
 the earth gets much denser. The only experiment of 
 that kind that has yet been made however, by the pres- 
 ent Astronomer Eoyal in 1854, has given a density so 
 much beyond all the other methods, that very little 
 weight can be given to it ; especially as still later ex- 
 periments of the Sehehallien kind, made by Sir H. 
 James, the superintendent of the Ordnance survey, at 
 Arthur's Seat, near Edinburgh, have given a density 
 rather below than above the old amount of 5^ times 
 that of water. 
 
 The mine experiment has been altogether unfortu- 
 nate : once before, the instruments gob on fire, and 
 another time a great piece of rock slipped and stopped 
 the operations. I will now explain the reason of what 
 I said about the pendulum losing or gaining according 
 to the density of the earth. We can hardly stir a step 
 in ' physical asti-onomy,' or the astronomy of causes 
 and effects, without having the law of gravitation 
 forced upon us, and therefore you had better learn at 
 once what it is. 
 
 Law of Gravitation, — Newton assumed, what all the 
 results prove to be true, that every atom of matter in 
 the universe attracts every other with a force which in- 
 creases as the square of the distance decreases : i.e., at- 
 traction is 3 X 3 or 9 times as strong at a third of the 
 distance, and so on. Also n, body composed of 3 atoms 
 must attract three times as strongly as one. Therefore 
 
28 Attraction of Globes. 
 
 the law of gravitation is, that the attraction of one 
 body A upon another B varies as the mass, or quantity 
 of matter, or what is commonly called weight, of A, 
 and inversely as, or as i divided by, the square of the 
 distance between each atom of A and of B, or be- 
 tween their centres of attraction. What is called ' the 
 attraction of A upon B ' does not depend at all upon 
 the mass of B, or the reciprocal attraction of B upon 
 A, but means the accderatmg force of A upon B ; 
 which is measured by the velocity -with which A's at- 
 traction makes B move, without any reference to A's 
 own motion. No alteration of the mass of B makes 
 any difference in its motion under A's attraction, so 
 long as A itself is free to move. For although doub- 
 ling B doubles the attraction between them, or their 
 relative velocity of approach, or the force with which 
 they would compress a spring separating them, yet B's 
 inertia or resistance to motion will be doubled also, and 
 therefore its absol/iote velocity will remain the same as 
 before ; but A's absolute velocity, and the accelerating 
 force of B upon A, will be doubled by doubling B's 
 mass. And so it is true, though it looks like a paradox, 
 that the earth's attraction moves the sun just as much 
 as it would move a pea at the same distance. The at- 
 traction of the earth, or of the sun, is measured by the 
 velocity with which things fall or move toward them : 
 for it is only the resistance of the air that makes feath- 
 ers or grains of sand fall slower than a lump of lead. 
 For the next step you must accept the following 
 things as proved by mathematics : (i) a solid globe of 
 uniform density attracts everything anywhere outside 
 
Weighing the Earth down a Mine. 29 
 
 it, as if its whole mass were condensed into its centre : 
 (2) the same is true if the globe is made up of a set of 
 shells or coats, like an onion, each shell having a differ- 
 ent density from the others, provided only each shell 
 has the same density throughout : (3) a spherical or a 
 spheroidal shell exerts no attraction in one direction 
 more than another upon a body inside it ; or the body 
 would float in the air anywhere indifi'erently inside of 
 such a shell : (4), which follows from the first, the at- 
 traction of two globes of equal density on bodies at 
 their siirfaee is simply in the proportion of their diam- 
 eters or radii ; for their attractions are directly as the 
 masses, which are as the cubes of the radii (p. 24) ; but 
 the attractions at the surface are also inversely as the 
 squares of the radii ; and dividing the cubes by the 
 squares, you have the result that the attractions of the 
 globes at their surface are as their radii. 
 
 Then if the earth were of the same density through- 
 out, the attraction on things at the bottom of a mine, 
 say a mile deep, would evidently be less than it \*ras at 
 the surface, exactly in proportion to its depth ; because 
 the shell a mile thick all round the earth goes for 
 nothing when you have got inside it by going down the 
 mine, and the attraction becomes that of an earth of a 
 mile less radius. Attraction of the earth near its sur- 
 face is only another name for gravity, and it is gravity 
 which makes a pendulum swing, and the weaker gravi- 
 ty is the slower it will swing. 
 
 But if this shell a. mile thick is of very light stuff, 
 say for simplicity of calculation, of no weight at all, the 
 result of going down the mine will be very different ; 
 
30 The Cmendish Experiment 
 
 for at the bottom of the mine there will be now the 
 whole mass or weight of the earth attracting as before, 
 and the pendulum will be brought a mile nearer to the 
 centre, from which all the attraction has to be measured, 
 by our first and second rules. Therefore the attraction 
 will be greater in the proportion of the square of what 
 you may call 4000 miles to the square of 3999, or will 
 be one 2000th more than at the surface. And as it is 
 also a mathematical fact that the quickness (the con- 
 verse of 'time') of a pendulum increases in proportion 
 to the square root of the force of gravity, it comes back 
 to this, that the pendulum would gain one second in 
 4000, or nearly a minute in three days, down the mine 
 of such an earth, instead of losing as in the earth of 
 uniform density. Between these two extremes, of the 
 outside of the earth being quite as dense as the inside, 
 and being of no density at all, there is of course some 
 medium condition in which the force of gravity down 
 the mine would be exactly the same as at the surface ; 
 and there is some other condition between that medium 
 and that of no outside density, in which the pendulum 
 would gain, not a minute in 66 hours, but a second in 
 ten hours ; which was the result of the experiments 
 in the Harton colliery. But the actual density of the 
 shell all round the globe has to be estimated before 
 the mean density of the earth can be calculated from 
 it and the observed gain of the pendulum ; and it 
 is pretty clear that that has not been done accurately 
 yet. 
 
 The most remarkable of all the earth-weighing ex- 
 periments is that which goes by the name of Mr. Cav- 
 
For Weighing the Earth. 31 
 
 endisli, who first performed it in the year 1798. It 
 has since been done again many times over, with every 
 possible provision for accuracy, by the late Francis 
 Baily and by Dr. Reich abroad. Sir J. Herschel warns 
 us that some of the professed explanations of it are 
 radically wrong ; and I doubt if anything beyond 
 the following general description can be made intel- 
 ligible to ordinary readers. This is founded on Cav- 
 endish's own account in the Philosophical Transac- 
 tions. The account of Baily's experiments fills the 
 whole of the 14th volume of the Astronomical Society's 
 Memoirs. 
 
 It is a contrivance for weighing the earth against a 
 globe of lead of 6 inches radius. And that is done by 
 comparing the known effects of the earth's attraction on 
 pendulums at 4000 miles from its centre, with the ex- 
 perimental attraction of two such globes, at 9 inches 
 from their centres, on two balls balanced on a long rod 
 hung from its middle by a wire, which gently resists 
 being twisted. When the globes are brought sideways 
 near the balls they move them a little against the re- 
 sistance of the wire, and make them oscillate very 
 slowly about their new position ; which was 14', or one 
 2 50th of the length of each radius, from the old position ; 
 and each vibration took about 420 seconds, with the 
 wire generally used. For simplicity let each radius or 
 half of the rod, with its ball, be as long as would beat 
 seconds if hung as a pendulum, under the earth's at- 
 traction. Then it may be proved that the earth's den- 
 sity is to that of lead, as 250 x 36 x the square of 
 420, is to 81 -x the earth's diameter in feet, that of the 
 
32 Proofs of the Earth's Botation. 
 
 globes being i foot.* If a stiffer wire is used tbe balls 
 move through a less angle, but the square of the time 
 of vibration decreases in the same proportion, and so 
 the result is the same ; and all the performers of the 
 experiment have agreed in finding the earth's density 
 between 5*44 and 5-67 times that of water. The Sche- 
 hallien experiments, and another at Mount Cenis, made 
 it rather under 5, the Edinburgh one 5*3, and Mr. 
 Airy's mine experiments 6-57. But whatever the ab- 
 solute weight of the earth may be, the proportion be- 
 tween it and the sun and moon and planets will remain 
 the same, as they ai'e all calculated from it. So the 
 earth's diameter is the standard by which all the solar 
 system has to be measured, as you will see hereafter. 
 A mistake in the received size of the earth stopped 
 !N'ewton's belief in his own discoveries for some years, 
 until it was remeasured and corrected. 
 
 MOTIONS OF THE EAETH. 
 
 As soon as it was discovered that the earth is a globe, 
 it could not require much philosophy to conclude that 
 day and night and the visible motion of the stars were 
 caused by its rotation, rather than by that vast number 
 of heavenly bodies all revolving together round it, 
 while they kept their own distances from each other as 
 if they were fixed in a frame. 
 
 The two following direct proofs of the earth's rota- 
 tion were invented a few years ago by M. Foucault. 
 
 * A fuller account of this may te found in Mr. Alry's LecturoB (lately repul)- 
 lisliecT with the title of Popular Astronomy) ; hut even that assumes some 
 mathematical results, as this does. See also p. ajj. 
 
Bs devolution Hound the Sun. 33 
 
 If a heavy ball is liung by a long string from the ceil- 
 ing and set and kept s^vinging, taking care to make it 
 oscillate in one plane and not revolve, it will be seen 
 after some time to be svi^inging across the floor in a 
 different direction from that in which it started. The 
 reason is that the floor has revolved under it with the 
 rotation of the earth. If such a pendulum were swung 
 at the north or south pole, the floor would revolve 
 under it in 24 hours ; at the equator it would not re- 
 volve at all ; and at intermediate places, such as Eng- 
 land, it revolves slower than at the poles, but still 
 enough to be visible in an hour or so. The same thing 
 may be shown in the machine called the gyroscope, 
 where a heavy disc or wheel, turning on pivots set in a 
 ring which itself turns on other pivots at right angles 
 to the disc pivots (like the gimbals of a ship compass), 
 will keep spinning in the same plane and with its axis 
 pointing to the same star, while the frame which car- 
 ries the ring moves round it with the rotation of the 
 earth. This is the more complete experiment, and 
 may be performed anywhere. For if the wheel is spun 
 in any latitude, with its axis in any direction at right 
 angles to the earth's axis, the force of rotation of the 
 wheel will keep the plane of the ring directed to the 
 same stars, while the outer frame turns round it on 
 the other pivots, which will be parallel to the earth's 
 axis. 
 
 From that date of 600 b.c. to about 1 500 a.d. we 
 
 hear of only one suggestion that the sun does not go 
 
 round the earth, but the earth round it. Archimedes 
 
 tells us in a book of his own, that another astronomer, 
 
 3* 
 
34 Copernicus, Galileo, Newton. 
 
 Aristarchns, about 280 b.c, held the opinion that the 
 earth goes round the sun in a circle, and that the size 
 of that circle is quite insignificant compared with the 
 distance of the stars. Unfortunately for the credit of 
 Archimedes, he entirely disbelieved it ; as we shall see 
 that much more modern astronomers have discredited 
 other people's discoveries which were equally correct. 
 It is true that nothing of the kind appears in the only 
 small book of Aristarchus himself which has come 
 down to us, ' On the sizes and distances of the sun and 
 moon ; ' but there is nothing there contradictory to it, 
 and he may have discovered the motion of the earth 
 after he had written his book on its distance from the 
 sun.* One can hardly suppose that a man like Ar- 
 chimedes would take the trouble to combat the opinion 
 of another mathematician on such an important ques- 
 tion without knowing that he held it. It is quite cer- 
 tain, however, that Aristarchus knew the rotation of the 
 earth, and that it is the earth's shadow that eclipses 
 the moon ; and that he had a tolerably correct idea 
 how much further off the sun is than the moon, though 
 he did not know the real size, and therefore the real 
 distance, of either of them. Some writers have given 
 Pythagoras credit for teaching that the earth goes 
 round the sun, about 500 b.o., but apparently with no 
 good reason. 
 
 But if Aristarchus satisfied himself that the earth 
 moves round the sun, the world itself and even the 
 best astronomers were not moved into believing it for 
 
 * I take these statcaienta from the ' Lives ' of these two philosophers, atid 
 •Xnotatious therein : I flo not profess to have read their books. 
 
The Ea/rtKs Orbit round the Srni. 35 
 
 nearly 2000 years more; that is until the time of 
 Copernicus, a German priest, who was born in 1472, 
 a few years before a still more celebrated reformer of 
 old opinions, Lnther. And for a good while the astro- 
 nomers and everybody else refused to believe him any 
 more than Aristarchus, and invented all sorts of inge- 
 nious theories to account for the visible motions of the 
 planets. But the more those motions were examined, 
 the more impossible it was found to reconcile them 
 with any theory except that of Copernicus, which 
 treated the earth and the planets as all revolving round 
 the sun. Then came Galileo, a still greater astronomer 
 of Florence, wlio died the year JTewton was born, and 
 invented the telescope, which gave him the power of ex- 
 amining their motions still more accurately; and he 
 found other proofs of Copernicus's theory, and (as is 
 well known) was imprisoned for three years by the 
 Eoman Inquisition for publishing them. Afterward, 
 about the year 1680, Newton made the far greater dis- 
 covery of the reason why the earth and the planets 
 and the moon all move, and must move for ever as 
 they do, founded on that law of gravitation which I 
 have already stated. I will explain afterward how 
 that law affects them ; for the present we will go on 
 with their motions as they are. 
 
 The earth then goes round the sun, so as to see the 
 same stars again in a line with him, in a year of about 
 365^ days ; a day meaning the time of one rotation of 
 the earth on its own axis from noon to noon, or the 
 average time of the sun's twice passing the same meri- 
 dian. But we sliall have to consider the length of the 
 
36 Loss of Grmiiy at the Equator. 
 
 year more exactly afterward, and also of different 
 kinds of days wliicli are dealt with in astronomy. 
 
 If the world were habitable all round, and not di- 
 vided by the Pacific Ocean, which is opposite to Eu- 
 rope, it would be impossible to avoid a sudden break 
 of time somewhere, which would make the same day 
 March 20 on one side of the boundary and March 2 1 
 on the other side. For the days begin and end later 
 as you go west, and earlier as you go east ; and if a 
 man could sail round the earth westward he would 
 find that he had lost a day in his reckoning by the sun 
 when he came home, and that he had gained one by 
 sailing round the globe eastward, as the earth turns 
 from west to east. 
 
 The earth's mean distance from the sun (that is, the 
 average between the greatest and least distances) is 
 91,404,000 miles, according to the latest calculations,* 
 and therefore the whole length of the earth's path or 
 orMt is 574,310,000 miles. If you work it out you 
 will find that that comes to a rate of travelling through 
 space of 65,5 18 miles an hour, or i8"2 miles in a second, 
 — or 80 times as fast as sound, or an ordinary cannon- 
 ball, goes through the aix'. 
 
 The reason why we neither feel moving with this 
 enormous velocity, nor are thrown off by the centrifu- 
 gal force of the earth's rotation, is that both motions 
 are steady, and motion without shaking is as easy as 
 rest. We carry the air with us, as in a railway 
 carriage, and therefore feel no wind. Also the attrac- 
 tion of the earth is 289 times greater than the centri- 
 
 * Appendix, Note III. 
 
Loss of Oramty at the Equator. 37 
 
 fngal force even at the equator where it is greatest. 
 Nevertheless things are lighter there than in high lati- 
 tudes. A spring balance which weighs rightly at the 
 poles would mark 289 pounds as weighing 288 at the 
 equator, from the centrifugal force alone (pp. 16,241). 
 
 Things weigh less at the equator from another cause 
 besides. An oblate spheroid attracts less there than 
 at its poles (but see p. 197). If the earth's density 
 were uniform, the difference would be only a fifth of 
 the ellipticity or j^ ; but the inside is much denser 
 than the outside (p. 26). If the outside had no density 
 at all, the equatorial attraction would be to the polar 
 invei'sely as the squares of the two radii, or a 149th 
 less; for when quantities differ by a small fraction, 
 their squares differ by twice that fraction. In fact 
 the equatorial attraction from this cause is a 590th less 
 than the polar. Therefore taking both losses together, 
 194 pounds at the poles will only weigh 193 in the 
 same spring balance at the equator ; and between 
 London and the equator 1000 pounds lose about 3, and 
 a clock pendulum loses 2\ minutes a day. 
 
 Another objection may occur to you, as it did to 
 those who imprisoned Galileo, that in several places in 
 the Psalms it is said that ' the earth shall never move,' 
 and so forth. But that has been answered by the late 
 Dr. McCaul, who showed that the Hebrew word which 
 is translated ' move,' really means to shake or totter ;* 
 and so those passages of the Bible, instead of contra- 
 dicting the truth, were only waiting to confirm it, as 
 soon as the truth itself was discovered by the advance 
 
 * Aids to Faith, p. 219. 
 
38 Ellvptic Orbits. 
 
 of science ; for the stability of the universe against 
 shocks and permanent disturbances is now proved to 
 be a consequence of the law of gravitation. The same 
 may be said of the famous passage, ' Let there be light,' 
 which people used to admire for its poetic grandeur, 
 but they had no idea till lately that no other words 
 would have been equally correct ; for it is now ascer- 
 tained that light is not a thing to be created, like 
 water, but rather a state of things, like fire or noise. 
 
 ELLIPTIO CEBIT OF THE EAETH. 
 
 Hitherto I have spoken of the earth going round the 
 sun in a circle, as it does very nearly, but not quite ; 
 for the earth's orbit is'an ellipse, and not a circle. It 
 is remarkable that another ancient astronomer, Hip- 
 parchus, about 150 e.g., found out that the (apparent) 
 orbit of the sun round the earth was not quite a circle, 
 though he stopped short of the greater step which 
 Aristarchus had probably made for himself 100 years 
 before, and missed observing that the apparent orbit 
 of the sun round the earth is the same as the real 
 orbit of the earth round the sun. This figure of the 
 ellipse is so important in astronomy, that you had 
 better learn at once what it is ; for every oval or figure 
 like a flattened^ circle is not an ellipse. Ovals for 
 picture frames are often made out of four pieces of 
 circles, two of a large one and two of a small put 
 together ; but no pieces of any circles will make a real 
 ellipse. The simplest way to make one is to take a 
 piece of thin string with a loop at each end ; stick a 
 pin through each loop into the table through a sheet 
 
Eccentricit/y and ElUpticity. 
 
 39 
 
 of paper, leaving the string quite loose between them ; 
 put a pencil in to stretch the string out, and run it 
 along, always keeping the string tight : then the 
 pencil T/ill describe an ellipse. 
 
 Here is a figure of an ellipse with the circle contain- 
 ing it. SPII is the string : 
 SH being the places of 
 the pins, each of which is 
 & focus of the ellipse ; C is 
 the centre of both the el- 
 lipse and the circle ; ACD 
 Is the axis Tnajor or the 
 greatest diameter, which 
 evidently = the length of 
 the string or SP-I-HP, or 
 twice SB. BCF is the axis ininor. The nearer together 
 the foci are, the more the ellipse apjjroaches a circle, 
 or the less eccentric it is : the proportion of CS to CA 
 (not OS alone) is called the eccentricity, which is there- 
 fore expressed by a fraction, either a vulgar fraction or 
 a decimal, as may be most convenient ; and you see it 
 
 is much greater than the ellipticity -^ (p. i S) : when 
 
 the ellipse is very nearly a circle the ellipticity may 
 be called half the square of the eccentricity. We speak 
 of the eccentricity of the planets' orbits, because we 
 are not concerned with their centre but their focus, 
 where the sun always is. On the other hand we have 
 nothing to do with the focus of a meridian of the earth, 
 and so we speak of its ellipticity. SB or AC is also 
 the ' mean distance. 
 
40 Eooentrioity of the EartKs Orbit. 
 
 If you want to find the minor axis of a planet's orbit 
 from the major axis and eccentricity (which are always 
 the things given) you may do it by this rule : the 
 square of the linear eccentricity, i.e., SC^the sum of 
 the semi-axes x their difference ; the reason of which 
 will be evident to any one with a little knowledge of 
 mathematics. Taking SP alone, which is called the 
 radius vector, it evidently varies more or faster the 
 more eccentric the ellipse is. 
 
 But though the string method is the best for draw- 
 ing an ellipse, there is another definition of it which it 
 is important to understand. An ellipse is the oblique 
 or perspective view of a circle. For if the circle is 
 turned a little on its diameter, it will cover from your 
 eye the elliptic space ABDF, and all the lines at right 
 angles to that diameter, such as OF and y (which let- 
 ter is always used in mathematical books for those 
 lines, called ordinates), will be less, in proportion to 
 the ellipticity EB, considering AC invariable. But 
 the centre of the perspective ellipse does not coincide 
 with that of the circle, or fall upon the line of sight 
 from the eye to the centre of the circle, except when it 
 is seen so far off that all the lines of sight may be con- 
 sidered parallel, as in looking ' at any of the heavenly 
 bodies. 
 
 The ellipse is the shape of the orbit of all the plan- 
 ets, and of the moons round any planets which have 
 any. The linear eccentricity of the earth's orbit is 
 •0168, or one 60th, of its semi- axis major or mean dis- 
 tance of 91,404,000 miles : and therefore the earth is 
 about three million miles nearer the sun at one time 
 
Motion of Earth! a Perihelion. 41 
 
 than another. You may very likely think this is the 
 cause of the diflference between winter and summer ; 
 but it is no such thing. On the contrary, the earth 
 happens to be nearest to the sun in the middle of our 
 winter — the ist of January, though it has not always 
 been so, and will cease to be so again. For the whole 
 ellipse turns round, going forward, or in the same di- 
 rection as the earth itself moves, ii"'8 a year, or i° 
 in 308 3'ears ; and at the same time the equinoctial 
 points, on which the times of all the seasons depend, 
 as you will see presently, go backward at the rate of 
 So"'i a year, or completely round iii 25,868 years. 
 And as one goes one way and the other the other way, 
 it is the same as if the places of perihelion and aphe- 
 lion, or nearest and furthest distances from the sun, 
 went forward at the rate of 61" -g a year, or completely 
 round in 20,984 years, relatively to the equinoxes, 
 from which all celestial measures are taken, though 
 the time of absolute or sidereal revolution of the peri- 
 helion is 109,830 years. 
 
 The time the earth takes to return to perihelion is 
 called the anomalistio year, because the distance of a 
 planet from perihelion, or of the moon from perigee or 
 point of nearest approach to the earth, is called its trito 
 anomaly ; and the distance it would have gone in the 
 same time if it moved uniformly, or in a circle instead 
 of an ellipse, is its mean anomaly ; and their differ- 
 ence is called the equation of the centre: all these 
 being measured by the angles described by the radius 
 vector round the sun — or earth in the case of the moon. 
 The anomalistic year is 25m. longer than the equinoc- 
 
42 Former Cold of North Hemisphere. 
 
 tial year, in consequence of the advance of the perihe- 
 lion. But this is only a fact, and not a period used for 
 calculation. 
 
 Ton see from the above rate of advance of the peri- 
 helion that the earth was nearest to the sun in summer, 
 and furthest off in winter of the northern hemisphere, 
 about 3600 years before the creation of Adam. And 
 then it was much hotter in summer and colder in win- 
 ter ; Sir John Herschel calculates, no less than 23°, a 
 very serious difference indeed, and other calculations 
 make it more. To some extent that is so now in the 
 southern hemisphere, and the summer is much hotter 
 in Australia and South Africa than equally far north 
 of the equator ; but it would be much worse when all 
 the land of the north hemisphere was exposed to 23° 
 more heat and cold, whereas now it is chiefly the sea 
 that receives it, on which heat makes less impression. 
 
 Moreover it is calculated that the eccentricity of the 
 earth's orbit was formerly much greater than it is now, 
 in the following proportions: 
 
 Present eccentricity •0168 
 
 50,000 years ago -oiji 
 
 100,000 " " • '0473 
 
 150,000 " " -0332 
 
 200,000 " " '0569 
 
 aio,ooo maximum "0575 
 
 250,000 " " -0258 
 
 300,000 '' " '04^ 
 
 350,000 " " -0195 
 
 400,000 same as now '0170 
 
 and then the eccentricity increased again. Whenever 
 in that long period of great eccentricity, from 80,000 
 
Olacial Epochs. 43 
 
 to 300,000 B.O., the earth was at aphelion in the 
 northern winter it was much colder than it is now. 
 Let us see how much. When the eccentricity was 
 ■oS7S;the sun's greatest distance was nearly 97 million 
 miles, against our present nearly 90 in winter. Our 
 average winter temperature is called 39°. Bat this is 
 only 39° above an arbitrary zero, which is of no use for 
 measuring the power of the sun. We must reckon 
 from the absolute zero, or the heat of no sun at all, 
 which is estimated from various experiments to be 490° 
 below our zero : so that our winter heat is really 529° 
 on the absolute scale, or the sun raises the temperature 
 so much above what it would be if there were no sun. 
 And the heat is inversely as the square of the sun's dis- 
 tance. Therefore the average heat of the winter of 
 
 Europe about 210,000 years ago was ■ — ^ " = 
 
 9409 
 
 45 S°! 01" 74° below the present heat, whenever the north- 
 ern winter happened at aphelion.* And as the aphe- 
 lion revolves relatively to the equinoxes in 21,000 years, 
 there must have been ten of these northern winters 
 in aphelion during the period of great eccentricity : 
 not all so intense as the one 210,000 years ago, but all 
 far beyond our present cold ; and nearer the pole of 
 course it was colder still. 
 
 Every such time was probably a glacial epoch, as it 
 is called by geologists, when all Europe was covered 
 with ice, which the heat of summer had not time to 
 
 * See Mr. James CroU'B paper in the Philosophical Journal of October, 1865, 
 and a much fuller one in February, 1867, confirming these conclusions by many 
 moj^e reasons ; and Tyndall on heat, p. 79. 
 
44 CoAise of tlve Seasons. 
 
 melt, being also obstructed by the evapoi-ation from the 
 melting snow ; and which slid down our valleys like 
 the glaciers in the Alps, and as icebergs slide into the 
 Arctic seas. Moreover it is thought that the weight of 
 that ice was enough to shift the centre of gravity of the 
 earth and keep down most of the land of the northern 
 hemisphere below the level of the sea, as geologists say 
 it has been. Then also the winter was nearly a month 
 longer than the summer, as it is now a week shorter, 
 because the earth moves quickest when it is nearest to 
 the sun ; though the heat received by the whole earth 
 is the same in each i8o° of its revolution, the longer 
 time of the distant half of the orbit making up for its 
 greater distance : but that is by no means the case for 
 each hemisphere separately. The heat received in a 
 given time by any pianet is inversely as the area of the 
 orbit, or as 3"i4i6 x the product of the semi-axes. 
 But the axis major is practically constant, and the 
 gradual increase of our semi-axis minor, which is now 
 only 12,800 miles less than the major, is insignificant 
 for this purpose, though not for another (see p. 141). 
 
 The SeaSOnSi — The real cause of summer and winter 
 is that the earth's axis does not stand upright in her 
 orbit round the sun, which is called the ecliptic, but 
 one pole always leans 23° 28' toward what we call the 
 north of the heavens or fixed stars, and the other pole 
 leans as much to the south. Consequently, when the 
 earth is on the south side of the sun, the north pole, 
 and the northern hemisphere generally, are turned 
 toward the sun, and the south pole away from him, 
 and it is summer in the north and winter in the soutli. 
 
Ga/me of the Seasons. 45 
 
 Six months after, the earth having gone half round the 
 sun, the north hemisphere is turned away from him, 
 and the south hemisphere then looks toward the sun, 
 and so it is winter in the north and summer in the 
 south. 
 
 Summer is not only the time of warmth, but also of 
 longest days ; and to explain that we must consider 
 the earth's rotation as well as revolution. You had 
 better take a terrestrial globe and elevate the north 
 pole 23^° above the wooden horizon^ which we may 
 take to represent the boundary of light ahd darkness, 
 assuming the sun to stand i-ight above it; for of course 
 there is no such thing really as ' above and below ' in 
 the heavens, and we only use these terms for conveni- 
 ence. Then as you spin the globe round for its daily 
 rotation, you will see that nothing within the a/rctio 
 circle, 23^° from the north pole, ever goes below the 
 horizon or into darkness ; that is, the sun never goes 
 below the horizon of the people within that circle in 
 the northern midsummer. At the same time nothing 
 within the anta/rctic circU, 23^° from the south pole, 
 comes into the light at all, and so those people have 
 no daylight in the middle of their winter. The con- 
 verse of all this evidently takes place at the opposite 
 time of the year. But half way between those times, 
 when the earth is either east or west of the sun, the 
 two poles are equidistant from the sun ; and so the 
 light received by the two hemispheres is equal. And 
 to measure the length of days at those times you must 
 lay the poles level with the horizon or boundary of 
 light, and you will see that every part of the globe is 
 
46 Equinoxes omd Solstices. 
 
 just as long above as below it, or the days and nights 
 are equal. 
 
 Therefore those times ai'e called the equinoxes, which 
 occur on March 21 and September 23. Midsummer 
 and midwinter ai-e called the solstices, because the sun 
 then stays at the same distance from the equator for a 
 few days, and the days remain of the same length 
 before they begin slowly to get shorter or longer again. 
 From the solstices to the equinoxes the two poles grad- 
 ually approach to an equal distance from the sun ; and 
 so the difference of days and nights gradually dimin- 
 ishes to o, and after the equinoxes increases again. 
 At the poles they have a winter of six months in which 
 they never see the sun, and a summer (though a very 
 cold one) in which the sun never sets. The nearer 
 you go to the poles the greater is the difference of days 
 and nights at all times except the equinoxes. Even in 
 the north of England the difference is visibly greater 
 than in London and the south. 
 
 At the equinoxes the sun appears on the equator, 
 which means in astronomy not merely a circle round 
 the earth equidistant from the poles, but the plane of 
 that circle extended to the heavens. And as the sun 
 is always in the ecliptic, the equinoxes are the places 
 where the equator and ecliptic cross each other. For 
 all these purposes we may properly talk of the eartli 
 as turning on a fixed axis, and the sun moving round 
 the earth in the ecliptic. For their relative motions 
 are just the same as if they did so ; and if there were 
 no other bodies in the universe to measure by, no 
 human being could ever have found out that the 
 
Tropics mid Zones. 47 
 
 earth does not stand still with the sun revolving round 
 it. 
 
 The equator is necessarily as much inclined to the 
 ecliptic as the axis of the earth is to that line perpen- 
 dicular to the ecliptic which is called the axis of the 
 ecliptic, whether it is in the sun or the earth. There- 
 fore two circles, each 23° 28' from the equator, are the 
 boundaries of the sun's journey to the north and the 
 south of the equator. They are called the tropics, 
 which means the turning places of the sun ; the north- 
 ern one is the tropic of Cancer, which the sun touches 
 at our midsummer, and the southern is called the 
 tropic of Capricorn, which the sun reaches in our win- 
 ter and the southern midsummer. The band between 
 the tropics is called the torrid zone, the two arctic cir- 
 cles the frigid zones, and the spaces between are the 
 temperate zones : and they cover respectively "4, "oS, 
 and "52 of the earth's surface. 
 
 But you may ask why should the torrid zone be al- 
 ways hot, since the sun is nearly 47° away from each 
 tropic when he is at the other, while he is almost di- 
 rectly overhead to parts of each temperate zone when 
 he is at the tropic nearest to it. The reason is that 
 places within the torrid zone get a greater quantity of 
 sunshine nearly or quite direct in the whole year than 
 any places can outside of it ; and the heat is accumu- 
 lated, or as it were bottled up in the earth, and stays 
 tliere after the sun has left that place or latitude. The 
 heat received anywhere depends on the directness of 
 the sun's rays, or its apparent verticality overhead; 
 for a square foot or a square mile of surface evidently 
 
4:8 Signs of the Zodiac. 
 
 catches more or less rays from a fire or the sun accord- 
 ing as it faces them directly or is turned obliquely to- 
 ward them. Now every place within the tropics has 
 the sun directly over it not only once, but twice a year, 
 and has in fact two summers, one as the sun is going 
 from the equator to the tropic and another as he re- 
 turns over the same latitude toward the equator. In 
 that way a quantity of heat is accumulated which no 
 place beyond the torrid zone can get. Still it is some- 
 times hotter in the low latitudes of the temperate 
 zones than it is at other times within the torrid zone. 
 More heat is gained in long sunny days than radiates 
 away in their short nights, and the excess accumu- 
 lates and makes July and August hotter than June 
 and May. 
 
 The equinoctial points, where the planes of the equa- 
 tor and ecliptic cross each other, are of great impor- 
 tance in astronomy, because nearly all the celestial 
 measures are reckoned from the point of the vernal 
 equinox, which is called the first point of Aries T ; the 
 autumnal one being the first point of Libra =ii=. Aries 
 and Libra are the names of two clusters of stars or 
 constellations, which were imagined by the ancients to 
 represent a ram and a pair of scales ; and they are also 
 still kept as the names of two of the divisions of the 
 ecliptic into twelve parts called signs of the zodiac, 
 which are these: — Aries T, Taurus y, Gemini n, 
 Cancer © (of which the first point is the summer sols- 
 tice, from which the northern tropic is named), Leo ^, 
 Virgo Ti]i, Libra =ii=, Scorpio M., Sagittarius :^ , Capri- 
 corn -V3 (the winter solstice is at the tropic of Capri- 
 
The Trade Winds. 49 
 
 corn), Aquarius ^, and Pisces )-(. But they are now 
 seldom used in astronomical books. About 2200 years 
 ago the sun used to appear entering the constellation 
 Aries when he also entered that first point of the sign 
 Aries, or the equinoctial point. But now the sign 
 Aries has left that constellation, for the reason you 
 will see presently, and is in the constellation Pisces. 
 
 Trade WindSi — The heat of the torrid zone and its 
 velocity of rotation produce the trade winds, which 
 blow constantly in the same directions in the same lat- 
 itudes on the great oceans ; though not so constantly 
 on land, on account of variations in heat and other 
 causes of disturbance. The heat expands the air and 
 makes it rise from the equatorial regions, and then the 
 denser air from cooler latitudes comes in, and would 
 make a constant north wind (in this hemisphere) if the 
 earth were either stationary or cylindrical. For if the 
 earth had no rotation the air would have no east or 
 west motion ; and if it were a cylinder all the air would 
 be carried round with it from west to east with the 
 same velocity, and would be no more felt as a wind 
 than the air you carry with you in a railway carriage, 
 though it moves as fast as a very high wind. But the 
 earth's surface moves 1040 miles an hour at the equa- 
 tor, goo at latitude 30°, only 520 at latitude 60°, and 
 less still toward the poles, where its velocity from ro- 
 tation sinks into nothing. 
 
 Therefore while the air comes south (in our hemi- 
 sphere) it is always coming to a place which moves 
 faster eastward than the place it came from ; and so 
 that north wind becomes N". E. relatively to the surface 
 3 
 
50 Precession of the Equinoxes. 
 
 of the earth : just as a weathercock would point N". E. 
 in a north wind if you carried one with you running 
 east. This is the principal trade wind, blowing from 
 N. E. in the northern hemisphere and from S. E. in the 
 southern, up to about latitude 30°. Near the equator 
 its eastern character is lost, because there is no material 
 increase of velocity in the earth as you get very near 
 the equator. Also the north and south winds meet 
 there, and make a calm : but the line of greatest heat 
 and calm is a little north of the equator. 
 
 The air which rose from the equator must go some- 
 where, and it goes in an upper current toward the 
 poles, and begins to fall again when it gets cool, to fill 
 up the space left by the air coming to the equator. 
 And as that air from the equator started for the north 
 with an eastward velocity of 1040 miles an hour, and 
 comes down again on latitudes which move much 
 slower, it is felt there as a S."W. wind in this hemisphere, 
 and IT.W in the other. This secondary or cmU-trade 
 wind prevails from about 30° to 60° latitude at sea, 
 and makes ships sail from North America to England 
 nearly twice as fast as from England to America. 
 
 PBECESSION OF THE EQUINOXES AHD LENGTH 
 OF THE TEAB. 
 
 I said at p. 35 that the earth goes round the sun, so 
 as to see the same star again in a line with him, or in 
 conQv/nctwn, in about 365 J mean solar days. The ex- 
 act time is 365-2563 days (omitting farther decimals), 
 and that is called a sidereal year. But the important 
 thing for all practical purposes is the year of seasons, 
 
Length of the Yeaur. 51 
 
 called the tropical or equinoctial year ; and that is a 
 little shorter than the sidereal for this reason. The 
 equinoctial points T and =£i= recede among the stars. 
 If that imaginary line where the plane of the equator 
 cuts the plane of the ecliptic points this year to any 
 two given stars va, the east and west, next year it will 
 point 5o"t away from them; so that the sun will 
 reach the spring equinox, or cross the equator from 
 south to north, 2o|- minutes before it comes again into 
 conjunction with the same star as before. Conse- 
 quently, if we reckoned by sidereal years, the seasons 
 would get sensibly wrong in no very long time. In 
 fact the equinoctial points are known to have moved 
 30°, which is equivalent to a month, or from the con- 
 stellation Aries into Pisces, since the early days of as- 
 tronomical records 2200 years ago. This is called the 
 precession of the equinoxes, because it makes them 
 precede their sidereal time ; and was discovered by 
 Hipparchus about 1 50 b.c. 
 
 The length of the equinoctial or tropical year is now 
 settled by astronomers — at least by the English ones — 
 to be 365'2422i6 mean solar days, or 365d. 5h. 48m. 
 47^. ; and the French measure is practically the same. 
 It is reckoned from the time when a mean sun going 
 at the average speed determined from the longest expe- 
 rience would pass through the mean f ; for that also 
 has its variations, and the motion of 50"- 1 a year is 
 not quite uniform in each year, as you will see presently. 
 But first let us try to realize what kind of motion the 
 earth goes through to produce this effect. 
 
 Get a common ' celestial globe,' and set the axis of 
 
62 Precession of tlie Equinoxes, 
 
 rotation upright, and consider the ' wooden horizon ' 
 to represent the ecliptic or plane of the sun's orbit. 
 Then the axis of the globe will be the axis of the eclip- 
 tic, though it is not. so on the globe ; but we want the 
 poles of the ecliptic as marked on the globe to repre- 
 sent the poles of the equator or of the earth, for our pres- 
 ent purpose. Now let our new north pole of- the earth 
 lean toward the north side of the room : and you may 
 consider the sun as going round the earth from right to 
 left, or west to east (through south). If there were no 
 precession the pole would always point the same way, 
 and the sun would always be among the same stars at 
 the same seasons. But the poles of the earth twist 
 olowly round the poles of the ecliptic westwards, or the 
 opposite way to the sun, keeping about 23^° from them, 
 and going quite round in 25,868 years. Turning our 
 globe round its upright axis from left to right, you will 
 see the equinoctial points recede along the wooden 
 ecliptic : which makes the equinoctial year those 20m. 
 20s. shorter than the sidereal. 
 
 The pole star, or the equinoctial stars, of any epoch 
 are thus an index to it in the great cycle of precession : 
 a Draeonis, now 25° off the pole, was only 3° 42' when 
 the Great Pyramid probably was built, about 2170 b. 
 0., a few centuries after the Flood. Its ^trance pas- 
 sage is 3° 42' inclined to the earth's axis, and looks dufe 
 north : therefore that pole star then looked straight 
 down that long narrow passage exactly at midnight on 
 the shortest day, and at its lower transit every day. 
 And at the same time the south meridian was crossed 
 by the Pleiades in y , where the equinoctial point was, 
 
Caused ly the Shape of the Earth. 53 
 
 and whicli are associated with the beginning of the year 
 by traditions and customs all over the world, including 
 the well-known eastern worship of the bull.* 
 
 Tou would not suppose that this precession of the 
 equinoxes was caused by the protuberance of the equa- 
 tor. ISTewton discovered that it is. The protuberance 
 of the equator is like a heavy ring round the earth. 
 The attraction of the sun acts more strongly on the 
 nearest than on the furthest part of the ring, and would 
 immediately pull the equator straight into the plane 
 of the ecliptic, if the spinning of the earth did not 
 resist it by always carrying off that part of the equator 
 which is nearest to the sun and most attracted for the 
 moment. The result is that the whole plane of the 
 ring is twisted backward, but its inclination to the 
 ecliptic is not altered (see p. 159). The motion is not 
 uniform, but less as the sun approaches the equator ; 
 for when he is in the plane of the equator (at the equi- 
 noxes) he can exert no force to disturb it. 
 
 Por the purpose of explaining the cause of preces- 
 sion I have only mentioned yet the sun's attraction ; 
 but in fact the moon contributes to it in the same way, 
 and even more : for she also moves nearly in the eclip- 
 tic, and attracts the front of the ring or the part which 
 is nearest to her, more than the back parts which are 
 furthest off. ' Not only is the front of the ring attracted 
 more than the middle, but the middle is attracted more 
 than the back : and so it is on the whole the same as 
 
 * This is a more complete coincidence than Sir J. Herschel pointed out : see 
 his ' Astronomy,' p. jo6 ; and vole. ii. and iii. of Mr. Fiazzl Smyth on the Pyra- 
 mid. 
 
54: Cause of Precession. 
 
 if the sun and moon pulled the front of the ring or of 
 the equator up or down toward the plane of the eclip- 
 tic, and also pushed the back part dowd or up. More- 
 over, as the effect of the sun and moon on precession 
 is all due to the difference of attraction on the near 
 and far parts of the earth's protuberance, the moon 
 really does more toward it than the sun, though her 
 attraction upon the whole earth is very little compared 
 •with the sun's, who is enormously larger and heavier ; 
 but then she is very much nearer ; and the difference 
 between the back and front of the earth is a 30th of 
 the moon's distance, but hardly a 12,000th of the sun's, 
 and their differential force is inversely as the cubes of 
 their distances (see p. 133). The result is, that giving 
 the moon the benefit of her nearness, and the sun the 
 benefit of his greater weight, the moon does above 
 twice as much as the sun in producing the precession ; 
 as we shall see afterward that she also does in produc- 
 ing the tides by the same difference of attraction on 
 the opposite sides of the earth. 
 
 The lunar part of the precession varies like the solar, 
 according as the moon is near or far from the equinoc- 
 tial points ; and so it may be least when the solar pre- 
 cession is greatest ; or they may both be at their maxi- 
 mum or minimum together. The 50"- 1 is the average 
 or mean precession in a year. 
 
 Nutation. — There is yet another irregularity in the 
 lunar precession. In consequence of the moon being 
 sometimes a little above, and sometimes below the 
 ecliptic, she does not pull quite on a level with the 
 Bun, and so produces a sort of nodding of the pole from 
 
Nutation of the EariKs Aads. 55 
 
 its average motion in a circle round the pole of the 
 ecliptic; and that is called nutation. The poles of 
 the earth, or of the equator, go like a man walking 
 between the rails round a race course, but with a wavy 
 motion from one side to the other, instead of walking 
 along the middle. But the word ' nutation ' is used to 
 comprehend all the variations of precession, both for- 
 ward and sideways. g"'2 is the extent of the nutation 
 on each side of the middle or average course of the 
 pole in its circle of 23" 28' radius round the pole of ^he 
 ecliptic ; and the length of each wave, or rather half 
 wave, from one crossing of the middle of the course 
 to another, is 3' 10.", corresponding to 9*3 years, or 
 half the time of one revolution of the moon's nodes, 
 or places where she crosses the ecliptic (p. loi). You 
 must understand that this 3' 10" is not measured 
 round the pole of the ecliptic as a centre (in which 
 case it would be 9'3 x 5o"'i), but as an arc on the 
 great sphere of the heavens, with the earth's centre for 
 its centre, as all the celestial measures are. 
 
 The amount of the nutation, or disturbance of the 
 earth's axis by the moon when acting in a different 
 direction from the sun, is one of the means used for 
 calculating her power of attraction, or mass, compared 
 with the sun's. Another is the comparison of their 
 effects upon the tides (p. 133). The protuberance of 
 the earth disturbs the moon in return, as you will see 
 at p. 197 that Jupiter's oblateness disturbs his moons. 
 
 Before we leave precession, I should tell you of a 
 curious use that has been made of it, toward settling 
 the question whether the earth is a thin shell of rocks 
 
66 Change of Oiliguity of the EeUpUo. 
 
 full of melted lava or other fluid inside, as some per- 
 sons supposed. The late Mr. Hopkins of Cambridge 
 calculated that the precession would be greater than 
 it is if the earth is not solid for at least iocx3 miles 
 deep. For the sun and moon would twist the axis of 
 the earth rather more if the protuberance under the 
 equator were fluid ; which may be roughly illustrated 
 thus i — A pendulum with a bob made of a glass globe 
 filled with quicksilver swings rather faster than a pen- 
 dulum similar and equal in all respects except in the 
 bob being solid. The reason is that the hollow globe 
 does not stick to the mercury and hold it fast, but 
 slides round it as it turns a little in swinging, and so 
 the mercury does not turn with it ; whereas the whole 
 of a solid bob has to be turned as well as swung at 
 eveiy vibration, which uses up more of the force of 
 gravity, on which the quickness of the pendulum de- 
 pends. 
 
 Not only do the points of crossing of the equator 
 and ecliptic recede 50" a year along the ecliptic, but 
 the obliquity of the eolijitic itself, or its inclination to 
 the equator, decreases about half a second a year, and 
 will go on decreasing until it has got nearly down to 
 22° from the present 23^°, when it will increase again. 
 The reason is that the whole ecliptic or plane of the 
 earth's orbit is slowly tilted by the attractions of the 
 other planets. Moreover, as it neither turns on the 
 line of equinoxes, as a hinge or axis through the sun, 
 nor directly across it, it makes the annual precession a 
 little less than it would be otherwise, and also lets it 
 increase a V0ry little, so that the tropical year, or year 
 
Sidereal a/nd Solar Time. 67 
 
 of seasons, is 4 seconds shorter than it was 2000 years 
 ago, though the absolute time of the earth's revolution 
 round the sun has not altered. 
 
 SIDBEEAL AND SOLAE TIME. 
 
 We must now consider the different measures of a 
 day more exactly. And first we may remark that the 
 day of astronomers begins at the noon after the mid- 
 night when our common day begins, and has no a. m. 
 or p. M., bat simply 24 hours. Thus 1 1 a. m., i Janu- 
 ary, 1867, was 23 o'clock 31 December, 1866, in astron- 
 omical almanacs. But there is also a time called side- 
 real, which is still more different. The sidereal day 
 here begins and ends when the equinoctial point T 
 crosses the meridian of Greenwich. If the sun is on the 
 meridian at the same time (as he is only at the vernal 
 equinox) he will not have quite got there again by the 
 time f is there again, or at the end of that sidereal day ; 
 because the earth has meanwhile moved on a day's jour- 
 ney in her orbit and passed the sun a little, and so has to 
 turn a little more than quite round for Greenwich to face 
 the sun again, since she rotates in the same direction as 
 she revolves round the sun, from west to east, like a 
 small wheel of 8 teeth rolling round a very large one of 
 2922. For that small wheel would turn 365!- times 
 relatively to the large one in one revolution round it, 
 but 366.J times absolutely, or relatively to one side of 
 the room : which may help to clear some people's ideas 
 about the rotatioli of the moon (p. 94). 
 
 A sidereal day then is practically the time of one 
 absolute revolution of the earth, or the time between 
 3* 
 
58 Equation of Time. 
 
 two traftsits of the same star. For the precession of 
 the equinoxes makes no sensible difference in a day, 
 and it is the same thing whether we call a sidereal day 
 the time between two transits of T or two transits of 
 the same star. Not only is the daily precession 366 
 times less than the annual, but the motion of T 
 through 50"' I only makes a difference of 3^ seconds 
 of time in the time of the clock, or the arrival of T 
 at tlie meridian at the end of a year, though it makes 
 20m. 20s. difference in the time of the sun (or earth) 
 reaching <f again, as I said at p. 51. There is neces- 
 sarily one more sidereal than solar days in a year, or 
 the sidereal day is one 366th shorter than the solar. 
 Moi-e exactly, a sidereal day is '9973 of a mean solar 
 day, or 3 (solar) m. SS'iis. shorter; or a solar day is 
 I "002738 of a sidereal one ; and the hours and minutes 
 are in the same ratio. The sidereal day never begins 
 at 12 o'clock, except at noon the 21st of March, and 
 at midnight of the 23rd of September. It also reckons 
 by 24 hours, not 12 ; and its length is invariable (but 
 see p. 144). 
 
 Equation of Timei — We have yet to inquire what a 
 solar day really is. I call the equinoctial year 365 ■- 
 242216 mean solar days. A true sun dial solar day 
 is the time between two transits of the sun over the 
 meridian. But such days are by no means of equal 
 length, and so far different that in November the true 
 solar time is a quarter of an hour before mean time, 
 which is clock time, and in February a quarter of an 
 hour behind it. That is the reason why the afternoon 
 light appears to be so much longer after Christmas 
 
Equation of TiTne. 59 
 
 than at the corresponding time before Christinas. The 
 causes of the inequality of solar days are the unequal 
 velocity of the sun in his elliptical (apparent) orbit, 
 and still more, the unequal motion of the sun in the 
 direction parallel to the equator, in consequence of the 
 obliquity of the ecliptic to the equator. Even if he 
 moved quite uniforiply, say i° a day for simplicity, in 
 the ecliptic, you will see if you look at a celestial 
 globe, that meridians through eacli degree of the eclip- 
 tic will not go through each degree of the equator, 
 which is the hour circle of the globe. The mean solar 
 day then is the average of all these variable solar days 
 throughout the year. And the thing called the equa- 
 tion of time is the difference between the time of day 
 by the clock of mean time and the time by the sun 
 dial. It is the column headed in the almanacs ' Clock 
 before sun,' or ' Clock after sun.' It varies a little 
 from one leap year to another ; but a fixed table will 
 do very well for common use with a sun dial, such as 
 is given at p. 12 of the 4th edition of the Eudiinentary 
 Treatise, on Clocks, with the rule for correcting it from 
 year to year. It is practically the same every fourth 
 year, like the time of sunrise and sunset. 
 
 Until a few years ago the clocks in the west of Eng- 
 land used to be behind those of London, or Greenwich 
 Mean Time: at Oxford 5 minutes behind, Bristol 10, 
 Edinburgh 1 3, Exeter i4, Glasgow 17, and so on. The 
 reason of that was that mean solar time, or the average 
 time of noon throughout the year, varies (no less than 
 true solar time) with the longitude of each place ; for 
 a place in the east comes to face the sun sooner than one 
 4* 
 
60 Equation of Time. 
 
 in the west, as the earth turns from west to east. 
 When railways brought all places in England in a 
 manner together, the inconvenience o'f these separate 
 local times became intolerable ; and the city of Oxford, 
 which was the last to retain it in the west, has at last 
 given it up, except at the cathedral. Some places in 
 the eastern counties still absurdly keep their eastern 
 time in advance of Greenwich. This difference between 
 local time and G.M.T. (as it is called) has to be borne 
 in mind in taking the time from a sun dial or any more 
 accurate meridian instrument. It has to be added to 
 ' clock after sun ' in west longitude, and subtracted 
 from ' clock before sun ; ' and the contrary in east. 
 
CHAPTEK 11. 
 
 THE SUN. 
 
 In considering ' tlie solar system,' whicli means the 
 sun and all the planets and their moons or satellites, it 
 is important to get a clear idea of the size and weight 
 of the sun, as it is his attraction that keeps them all in 
 order, moving in their proper times and distances, be- 
 sides his other equally important business of sending 
 light and heat to them — if they all have inhabitants 
 to use it. The sun's diameter is 845,220 miles, or 
 io6'64 times that of the earth. But that gives you no 
 idea that he is more than a million times as big as the 
 earth ; which he is. If you take two bricks and lay 
 them lengthwise, and then put two more alongside, and 
 then put four more on the top of those four, you will 
 have made a thing of the same shape as a brick, but 
 twice as long, twice as broad, and twice as high, and 
 therefore eight times as big altogether ; if you treated 
 three bricks in the same way you would make one 
 twenty-seven times as big. And the same is true what- 
 ever the shape is ; for a body of any shape may be 
 made up of an infinite numberof infinitely small cubes : 
 and if each of them is doubled or trebled in every di- 
 mension, the body will be doubled or trebled in every 
 dimension, and will therefore be increased eightfold or 
 twenty-sevenfold in bulk. This is expressed by saying 
 that the bulk, or volume, or solid content, and (if the 
 
62 The Sun's Size cmd Distamce. 
 
 densities are equal) the mass varies as the cuhe of the 
 diameter. Therefore the sun is nearly 107 x 107 x 107 
 or 107^ (as they write the cube of 107) times as large 
 as the earth ; the actual proportion is 1,217,000, allow- 
 ing for the earth's spheroidicity as at page 24. 
 
 The sun's distance from the earth varies from 
 89,860,000 to 92,950,000 miles. The mean between 
 the greatest and least is called the mean dista/nce ; and 
 that is now considered to be 91,404,000 miles, .though 
 you will find it in older books called 95 and 95^ mil- 
 lions, because it was believed to be so until lately.* 
 You will hear more of that con-ection as we go on, and 
 also how the diameter and distance of the sun are both 
 measured. It is also io8'2 times the sun's diameter, 
 and 23,064 times the earth's radius. 
 
 As nearly all distances in the solar system are mea- 
 sured by millions, and nobody ever counted a million 
 of anything, it is worth while to stop a little to under- 
 stand what it is, by the help of a few specimens. A 
 million days are 2730 years ; so there have been not 
 much more than two million days since the creation 
 of Adam, and rather more than one million since the 
 time of Solomon. A railway train going thirty miles 
 an hour and never stopping would take nearly four 
 years to go a million miles, and eleven years to go once 
 round the sun, and three hundred and sixty years to go 
 from here to the sun. If you had a million shillings to 
 count one by one, and did it as fast as you could for 
 ten hours a day, it would take a fortnight ; and the 
 million shillings would weigh nearly six tons, or be a 
 
 * Appendix, Note III. 
 
The Sun's Mass and Density. 63 
 
 heavy load for a railway truck. A million is a thou- 
 sand thousands, and a billion is a million millions, ac- 
 cording to the English method of notation, or a thou- 
 sand millions according to the French method. The 
 French method is usually taught in "the schools of the 
 United States. 
 
 But instead of describing the size of the sun by these 
 immense figures, you will have a better idea of it by 
 comparing it with something nearer its own size. 
 There is no solid body that we know of at all to be 
 compared to it, and so let us make one out of a dis- 
 tance that we know. A globe large enough to contain 
 the whole orMt of the moon, or one with the earth for 
 its centre and the moon rolling on its surface, would 
 only be one-sixth as big as the sun. The sun is quite 
 round, not spheroidal like the earth. 
 
 But though the sun is above a million times as big 
 as the earth, he is only 316,560 times as heavy, and is 
 consequently made of much lighter materials. His 
 density or specific gravity is only "26 of the earth's, or 
 not much greater than water, and rather less than if 
 he were made of coal. You must not suppose that we 
 can measure the sun's density except through his size 
 and his weight, which has to be calculated from his ef- 
 fect upon the earth's motion, as I shall show you to- 
 ward the end of the book. The density of anything 
 is in direct proportion to its weight and in inverse pro- 
 portion to its bulk ; and as we can calculate that the 
 sun is only a quarter as heavy as he ought to be ac- 
 cording to his size in proportion to the earth, we know 
 that his average density is only a quarter of the earth's. 
 
64 Ths SurHs Parallax. 
 
 The way to find the density or specific gravity of any- 
 thing heavier than water, which is always taken for 
 the nnit, is to weigh it first out of the water and then 
 hanging in water by a string as thin as possible, and 
 the specific gravity is the weight in air divided by the 
 difierence of the two weights. It is easy to remember 
 that 36 cubic feet of water weigh a ton ; or a tank sis 
 feet square and six feet deep will hold six tons of wa- 
 ter. A cubiofoot of water contains 6-23 gallons, and 
 therefore a gallon weighs almost exactly 10 lbs., which 
 it is convenient to remember. 
 
 In the table at the end of the book I have given the 
 specific gravities of the sun and planets, assuming the 
 earth's to be 5'5, which is the greatest of them all but 
 Mercury, and very much greater than all but Mercury 
 and Venus. Saturn is as light as deal and the light 
 woods (not coi-k as Sir J. Herschel says, for that is only 
 •24) ; and the other three great planets and the sun 
 are about the same as the very heavy woods and coal. 
 Mars and the moon are about as heavy as diamonds 
 and heavier than stones, which are generally 2"j, but 
 much lighter (as the earth is too) than any of the 
 common metals except aluminium, which is only 2'25 
 or the same as glass. All these densities, remember, 
 are only measured through the earth's, which depends 
 on the experiments described in the first chapter. 
 
 I may as well say here, that I have calculated afresh 
 all the quantities in that table which depend upon the 
 m^easure used by astronomers called the sun's parallax : 
 which is the apparent size of half the greatest diameter 
 of the earth as it would be seen from the sun's centre. 
 
Grwoify on the Sun. 65 
 
 or half the angle between two lines drawn from the 
 sun to the outsides of the earth at the equator ; which 
 manifestly varies inversely as the sun's distance. You 
 will see hereafter how all the dimensions of the solar 
 system (except the moon's) depend on that. I now 
 take the parallax at 8"'g4 3* according to the latest 
 average of observations, instead of the old 8"'57 corre- 
 sponding to the certainly erroneous sun's distance of 
 95,383,000 miles. The distance of two bodies always 
 means in astronomy the distance of their centres ; and 
 the diameter of a planet means its greatest diameter. 
 
 From the diameter and mass of the sun, compared 
 with those of the earth, we can calculate their com- 
 parative attractions and the force of gravity at their 
 surfaces. For by the law of gravitation (p. 29) that is 
 directly as the mass and inversely as the square of the 
 distance from the centre. Therefore 
 
 gravity on the sun's surface_3i6,56o_ 
 gravity on the earth's surface"" io6-6^ ' 
 
 or a man on the sun would feel as heavy as if he had 
 nearly 27 others laid upon him, and would be squeezed 
 flat by his own weight. 
 
 Sun Spots. — Probably the solid sun is a smaller body 
 inside the luminous shell which we see, and which is 
 now called the photosphere, or light-giving sphere of 
 the sun. And the spots which sometimes appear are 
 holes in the photosphere, through which we catch a 
 glimpse of the dark and solid sun. Their depth is es- 
 timated at about 30,000 miles ; and they have enabled 
 us to measure the time of the sun's absolute or sidereal 
 rotation on his own axis, in 25d. 7h. 48m., and to see 
 
 * Appendix, Note IV. 
 
66 Sim Spots. 
 
 that it leans 7° 20' from a line perpendicular to the 
 ecliptic ; and such a line as that, through any body in 
 the solar system, is called the axis of the ecliptic. 
 Therefore the sun's equator is inclined 7° 20' to the 
 ecliptic : which means that that is the angle between 
 them. The sun's north pole leans most toward the 
 earth on September 13 and his south pole on March 
 II. Therefore spots move apparently in straight but 
 oblique lines across his -face for about the first three 
 weeks of June and December, but at other times in 
 very flat semi-ellipses. 
 
 Some sun spots are nearly 50,000 miles wide, and 
 are quite visible without telescopes when the sun is 
 dimmed by a fog. You may wonder how we can say 
 that they are holes in the photosphere rather than dark 
 clouds or patches upon it. If you look at a church 
 with deep windows, the ' splays ' or jambs of the win- 
 dows which directly face you look equally wide on 
 both sides ; but the jambs of the windows which you 
 see obliquely appear wider on the far side than the 
 near. And the spots on the sun have sides, less dark 
 than the spots themselves, which appear of equal width 
 while the spot is about the middle of the sun, but 
 wider on the far side and narrower on the near when 
 the spot is away from the middle. These varying 
 sides then are almost certainly the jambs or sloping 
 sides of the spots, which are deep holes in the photo- 
 sphere. But they are not permanent holes like the 
 volcano craters in the moon ; for in some years there 
 are none at all. They have a maximum every 11 
 years ; and they get smaller as they come opposite to 
 
The SurOs Atmosphere. 67 
 
 Yentis, and to Jupiter in a less degree, but apparently 
 the earth does not affect them:* the reason of all which 
 is yet unknown. The theory which prevailed, of their 
 coinciding with ' magnetic storms,' or spontaneous dis- 
 turbances of the magnetic needle, seems to be aban- 
 doned.* They are confined to two zones from near 
 the sun's equator to about 30° of latitude, answering 
 to the region of our trade winds. 
 
 The sun is also estimated to have an atmosphere, 
 reaching 72,000 miles above the photosphere, because 
 ' flames ' or brilliant clouds are seen as high as that 
 when the sun is darkened by the moon in a total 
 eclipse. Moreover the middle of the sun looks brighter 
 than the edges, because the outer rays come to us 
 obliquely, through a greater length of sun's atmos- 
 phere than those which come straight through it from 
 the middle of the sun ; and that is not the case with 
 the moon which has no atmosphere. Bright patches 
 or clouds, called yacwZflB, are also seen on the left side 
 of spots, as if they were left behind : the spots are 
 called maculcB. The earth's atmosphere decreases so 
 rapidly in density, that half its mass is within 3^ miles 
 above the sea ; and at 80 miles high there can be prac- 
 tically little atmosphere.f Cold increases upward, as 
 you get further from the heated earth, at the rate of 
 1° for 400 feet. 
 
 The only thing like a real discovery as to the com- 
 position of the sun is that the photosphere contains 
 some of the same materials as the earth, in an incan- 
 descent state. For when a ray of sunlight is spread 
 
 * Appendix, Note V. + Appendix, Note VI. 
 
68 Specbrum Analysis. 
 
 out into a spectrum, or band of rainbow colors, by 
 sending it through a prism, there are to be seen in it 
 a multitude of dark lines of various thickness and at 
 irregular distances, but always the same in every solar 
 spectrum through the same kind of prism ; they are 
 called Fraunhofer's lines, because he first completely 
 investigated them, though they had been observed 
 before by "Wollaston. Again, when the light of any 
 inflamed metallic vapor is sent through a prism, its 
 spectrum consists of one or more bright colored bands 
 or lines, sometimes as many as sixty or seventy, as in 
 the spectrum of iron. But if you send the white light 
 of an electric flame or any white hot but not vaporised 
 metal through that same metallic vapor, those bright 
 lines are immediately turned into some of the dark 
 ones of the solar spectrum, which then lights up the 
 spaces left dark in the vapor spectrum with the usual 
 rainbow colors. And if the vapoi'S of any nuinber of 
 metals are burnt together, each will keep its own set of 
 lines, bright so long as there is no purer or more com- 
 plete light behind it, and comparatively dark as soon 
 as there is. Hence it is almost certain that Fraun- 
 hofer's lines in the solar spectrum are caused by the 
 vapors of some of our metals existing in either the 
 atmosphere or the photosphere of the sun. 
 
 Iron and some of the secondary metals are thus found 
 in the sun, but none of the other common or principal 
 metals. In like manner some of them appear in some 
 of the stars and others in other stars ; which may ac- 
 count for their different tinges of color. For every 
 color is due to the absence or stoppage of some of the 
 
'■ Willmo Lea/oes ' on ike Sim. 69 
 
 rays which go to make np a perfect, or pure, or white 
 light. Things look red which stop or absorb all or 
 most of the rays which fall on them, except the red 
 ones. Therefore the question which is sometimes asked, 
 whether colors exist in the dark, is nonsense ; and you 
 might as well ask, does light exist in the dark ? There- 
 fore also it is that white objects are more visible in a 
 dim light than dark or colored ones ; for a white sur- 
 face is one which reflects a good many, at least, of all 
 the rays which fall upon it, but a dark one reflects very 
 few, and a colored surface reflects only one kind of raj'S, 
 absorbing all the rest ; as you may easily see by feeling 
 them under a hot sun, when a white board reniains cool, 
 while a dark one is very hot, from the rays which it 
 has absorbed. 
 
 Mr. ISTasmyth, who has retired from engineering to 
 astronomy, with the fame of having added to the pow- 
 ers of man by inventing the steam hammer, and other 
 persons too, have observed that the sun is covered with 
 bright clouds of the form of willow leaves, which over- 
 lie each other in all directions and sometimes close up 
 over a spot. These bodies are also called ' rice-grains' 
 and ' pores,' according to the fancy of different observ- 
 ers, and they are now admitted to be of more irregular 
 forms than was represented in some of the early pic- 
 tures of them. Sir J. Herschel thinks that they are 
 not clouds in the sense of being any kind of vapor, 
 but collections of loose solid matter, for which I do not 
 know any better word than ' fluffy ;' and that they float 
 in some highly heated gas or atmosphere which makes 
 them so hot as to be luminous, though it is itself invisible. 
 
70 Speculations on the Svn^s Heat. 
 
 HEAT OF THE SUIT. 
 
 Yon will find in Herschel's Astronomy an account 
 of the curious speculations whicli liave been made as 
 to the generation of heat in the sun, and also some cal- 
 culations of the amount of heat which it sends out in 
 any given time. One is, that each square yard of sur- 
 face gives out every hour rather more than would be 
 got by burning six tons of coal ; and from that you 
 may deduce the following calculation. Six tons of coal 
 weigh about as much as seven cubic yards of the sun's 
 average substance : that is, if the sun were a great 
 coal he would have to bum down seven yards an hour 
 to give out the heat that he does; or 35 miles deep 
 would be burnt off him in a year, which is about a 
 40C)0th part of his whole bulk. The consequence is 
 that the sun would have been burnt away into ashes, 
 which would burn no more (though all his weight 
 would remain there), in 4000 years. We may be quite 
 sure then that the sun's heat is not kept up by the 
 burning of his own substance, at any rate. 
 
 Another mode of keeping up the sun's heat has 
 lately been imagined, which is remarkable for its in- 
 genuity if not for its probability. It is that the heat 
 is due to a constant shower of meteoric stones falling 
 all round the sun with the velocity they would fall 
 with, if they once got near enough for his attraction to 
 overpower their velocity in another direction. You 
 may soon satisfy yourself that blows will produce heat 
 by holding a nail in your fingers while you hammer it 
 well on an anvil: in fact it can be hammered hot 
 
The Zodiacal lAght. 11 
 
 enough to light a match. It is calculated that if stones 
 as heavy as granite fell all over the sun 12 feet thick 
 in a year with the greatest possible velocity (384 miles 
 a second) from the sun's attraction, they would main- 
 tain the actual heat ; or if they fell with the more 
 probable velocity of small planets dragged out of their 
 orbits sideways into the sun, or 270 miles a second, 24 
 feet of thickness a year would be required. And as 
 granite is about 2^ times as heavy as the average mat- 
 ter of the sun, that would be equal to an addition of a 
 mile to the sun's diameter every 50 years, or an in- 
 crease of the sun's weight or mass by about a jocxjth 
 part in the 2000 years since tolerably accurate astron- 
 omical observations began to be recorded. And that 
 is enough to have shortened the year by more than 
 half an hour, according to the law by which the length 
 of the year depends on the sun's mass and distance, as 
 you will see hereafter. But the year has not short- 
 ened ; for even the four seconds mentioned at p. 57 
 are no shortening of the absolute revolution of the 
 earth, but are only due to the tilting of the ecliptic. 
 
 This last objection certainly would fail, so far as the 
 earth is concerned, but would still hold as to Mercury 
 and Yen us, if all the meteors or little planets which 
 fall into the sun are equally scattered round the sun in 
 a sphere within the orbit of the earth ; because then 
 the attraction of the meteors and sun together, on 
 bodies outside them all, is -the same as if they were 
 concentrated in the sun (p. 30). Accordingly it is next 
 supposed that this store of meteors is a certain nebu- 
 lous and faintly luminous mass called the zodiacal light, 
 
72 The Zodiacal Light. 
 
 sometimes looking like the Milky Way, -which sur- 
 rounds the sun in the form of a lens, or very flat sphe- 
 roid (but not a sphere) extending nearly or quite as far 
 as the orbit of the earth. Then it is assumed that a 
 sufficient number of these asteroids are continually 
 missing their way (as we may say) and getting dragged 
 into the sun and swallowed up for food to maintain 
 his heat by their concussion. The cause of their so 
 missing their way is imagined to be a certain resisting 
 medium called CBther, which we shall see afterward 
 that there really is reason to believe fills all space, and 
 must in time gradually contract the orbit of everything 
 which moves in it. 
 
 Not that there is the least evidence of any solid body 
 in the universe having yet been at all affected in its 
 orbit by such resisting medium ; though one comet is 
 believed to have had its period shortened in that way. 
 You will see afterward how extremely unlike comets 
 are to granite or meteoric stones in density. Nor do 
 the comets themselves, though very sensitive to dis- 
 turbance by reason of their lightness, appear to be 
 ever disturbed by passing right through that mass of 
 zodiacal light, about 180,000,000 miles wide, full of 
 real or supposed meteors or asteroids. The crossing of 
 two streams of meteors by the earth every August and 
 November gives no help to the theory of the sun being 
 supplied from a store within the earth's orbit ; espe- 
 cially as they are now supposed to have orbits reaching 
 beyond Uranus, and not at all likely to fall into the 
 sun (p. 221). Every now and then there have been, 
 flashes of light in the sun, which may possibly have 
 
Meteorio Theory of Sun's Heat. Y3 
 
 come from some sucli concussion. Otherwise there are 
 no visible indications of this rain of stones upon the 
 sun ; but perhaps that could hardly be expected ; for 
 if a whole year's supply came iu one piece, it would be 
 smaller than the moon, which could not be seen at that 
 distance in the brightness of the sun. In this state of 
 things it is not surprising that the meteoric hypothesis 
 is not yet accepted by astronomers as the real explana- 
 tion of the sun's heat, though no more probable one 
 has been invented. 
 
 Leaving these speculations into unknown causes, 
 Sir J. Herschel gives us another measure which, has 
 been experimentally obtained of the quantity of the 
 sun's light and heat by himself and others ; viz., that 
 the light of any piece of his surface is 146 times greater 
 than that of an equal surface of lime under the oxy- 
 hydrogen flame, which gives the most intense light and 
 heat we can make ; and that the heat is enough to 
 melt fourteen yards thick of ice, laid all over the sur- 
 face of the sun, in a minute. From this you may 
 easily calculate by the rules at p. 24, that that shell of 
 ice round the sun would be about three-fifths of the 
 size of the earth ; and therefore the whole heat of the 
 sun is enough to melt an earth of ice in less than two 
 minutes. And it would boil all that water in two 
 minutes more, and turn it all into steam in a quarter 
 of an hour from the time it was first applied to the ice. 
 Of course the earth receives only a very small part of 
 all the heat sent off by the sun ; indeed less than the 
 .2,cxx) millionth part, and all the planets together only 
 a 237 millionth. So although we have so much diffi- 
 4 
 
74 Heat of the Sim. 
 
 culty in finding out how the sun's heat is kept up, he 
 can afford to waste (as far as we know) many million 
 times as much as all his satellites can use. The earth 
 receives as much heat in a year as would melt lOO feet 
 thick of ice, or boil 66 miles deep of water, all over the 
 globe. 
 
 Latent and Specific Heat. — These figures require a little 
 explanation. It takes nearly as much heat to turn ice 
 into ice-cold water as to raise that water afterward to 
 212° or boiling : or a pound of boiling water will melt 
 very little more than a pound of ice. The exact heat 
 required to melt the ice is 143°. Therefore water is 
 said to contain 143° of latent heat, which is merely 
 employed in keeping it fluid, and does not disclose 
 itself by the thermometer. And again boiling water 
 will swallow up 980° more heat before it is all turned 
 into steam, and so that is the latent heat of steam. 
 Conseqiiently it takes nearly half as much heat to melt 
 a piece of ice as to boil it, and just nine times as much 
 to boil it all away into steam as to melt it, or five and 
 a half as much as to make ice-cold water boil. But 
 for this remarkable law of nature rivera and ponds and 
 cisterns would often be frozen to the bottom in a single 
 night, and remain so as long as the air is a single de- 
 gree below freezing ; and as soon as it rose the least 
 above 32° all the ice and snow that had fallen would 
 melt suddenly, and produce destructive floods; and 
 steam would be unmanageable and almost useless. 
 "Whereas, if you melt a pot of lead it falls into fluid 
 nearly all at once, not like a lump of ice put into warm 
 water ; and in like manner lead, or cast iron, or mercu- 
 
Latent cmd Specifio Heat. 75 
 
 ry (at 40° below zero), get solid nearly all at once, be- 
 cause tbey have so much less latent heat than water : 
 that of lead is only 9°.* 
 
 Two other causes prevent water, especially water 
 only open at the top, from freezing too quickly. One 
 is that it takes much more heat to raise a pound of 
 water 1° of a thermometer than a pound of anything 
 else : thirty times as much as mercury or lead, solid or 
 fluid ; twice as much as oil, and half as much again as 
 spirits of wine. Therefore they call the speaific heat 
 of water thirty times that of mercury, and so on. And 
 again, the steam of water expands far more than any 
 other steam. The other reason is that water (like an- 
 timony, f iron, and a few other things which crystallize 
 in freezing) expands instead of contracting before it 
 freezes, as well as when it is heated, and so the ice and 
 very cold water keep at the top. Besides that, it con- 
 ducts heat very slowly. The hot water in boiling rises 
 bodily, and in that way heat is soon eon/veyed through 
 the mass. Therefore the sea gets heated slowly and 
 gives out a great deal of heat in cooling, and so equal- 
 izes temperature. ' 
 
 You may have read in the life of George Stephenson 
 how he astonished another scientiiic man by telling him 
 that his railway engines wei-e driven by the sun's heat 
 ' bottled up ' in the earth from perhaps millions of years 
 ago. Probably he did not know that the same idea 
 had been published before, in one of the early editions 
 
 * See Chamljers' Chemistry. In some books latent heat is confounded with 
 specific, 
 t Types could not be cast sharp enough to use till that was discovered. 
 
76 Heat and Force Identical. 
 
 of Herschel's Astronomy. But it is a fact which should 
 be borne in mind while we are reflecting on the busi- 
 ness of the sun in the order of nature, that every plant 
 and tree and bit of grass that grows is raised by the 
 sun's heat. And as coal is known to be the condensed 
 vegetables of former ages, chiefly of gigantic kinds of 
 ferns, which could only grow in a veiy hot and damp 
 air, it is certainly true that every bit of coal or wood 
 which we burn now owes its condition and power of 
 giving out heat again to the heat which it borrowed 
 from the sun, whether twenty years ago or twenty 
 millions. 
 
 Morepver you should understand that every force in 
 the world, that is, every power to lift a weight, from 
 you}' own foot off the ground to every ton lifted by a 
 steam engine, and all the millions of tons of leaves and 
 wood added to the trees every year, and the clouds ftiU 
 of rain lifted from the sea and rivers by evaporation, 
 as much as the steam off a boiling pot, is all due to 
 heat ; that is, ultimately to the sun. In short, heat and 
 force are now well known to be convertible into each 
 other. The only force which may seem an exception 
 to that rule is the force which lifts the tide or makes 
 the great tidal wave keep rolling round the earth after 
 the moon and sun, as I shall explain afterward ; but 
 that also comes from the sun's and moon's attraction, 
 and the original impulse with which the earth and 
 moon were first set gojng, both rotating and revolv- 
 ing in their orbits ; for the planets could no more set 
 themselves originally in motion than they could create 
 themselves out of nothing. Those who wish to pursue 
 
Seat and Light. 77 
 
 this subject further should read Mr. Grove's ' Correla- 
 tion of Forces,' which will open to them new views of 
 the ultimate identity of all the forces in the universe ; 
 but still leaving the great force of all, the attraction of 
 all matter for all other matter, standing out as an in- 
 dependent law of nature, imposed at the creation to 
 keep the universe in order. 
 
 The sun's rays heat the air very little, and dry air 
 not at all ; and the rays which have come through the 
 slightly damp atmosphere heat glass also very little, 
 though glass stops a good deal of the heat from a fire, 
 and damp air keeps heat from radiating away. But 
 for the moisture of the atmosphere we should be ex- 
 posed every night to the cold of space, 490° below 
 zero, except so far as the heat retained in the earth 
 helped us ; and that would rapidly radiate away. A 
 mirror which reflected all the rays it receives would not 
 be warmed at all. "When the sun's rays fall on ice and 
 snow they are chiefly employed in melting it, and so 
 the heat becomes latent and is not given out into the 
 air, which is chiefly warmed by contact with the earth. 
 The sun's rays part with about one-third of their heat 
 in passing through the atmosphere, and probably one- 
 sixth of it in the last 4 miles. 
 
 KATUEE AND VELOCITY OF LIGHT. 
 
 Although we do not know how light and heat are 
 generated in the sun, we know that they ai-e to a great 
 extent identical and come together ; and we know how 
 long they take to come ; and that they are not any 
 substance, or fluid, or gas, emitted from the. sun, like 
 
Y8 Tha Zfndulatory Theory. 
 
 tlie atoms or vapors of scent which are thrown off from 
 odoriferous bodies without diminiBhing their weight 
 sensibly ; but vibrations, or undulations, or little waves, 
 generated somehow by the sun or a fire, in what is 
 called, for want of a better name, the lumin^erous 
 oether, which is now believed with good reason to per- 
 vade all space, and perhaps even the interstices be- 
 tween the atoms of which solids and fluids are com- 
 posed. This grand discovery of the uncbulatory theory of 
 light was published by the late Dr. Thomas Young of 
 Cambridge in 1802 ; and after being denounced as pal- 
 try, absurd, contradictory, mathematically impossible, 
 and many other things, by the Edinburgh Eeview,* is 
 now universally received as the only theory that ex- 
 plains all the knowii phenomena of light, and is said by 
 Sir J . Plerschel to ' suffice, if it stood alone, to place its 
 author in the highest rank of scientific immortality.' 
 
 It is fair to mention, however, that this theory had 
 been suggested, but not proved, long before by 
 Huyghens, and rejected by ISTewton for the emission 
 theory. I 'should remind you also, that the undula- 
 tions of aether which make light and heat have this 
 material difference from the undulations of common 
 air which make sound, that they can only travel in 
 straight lines ; except when they are diverted by 
 passing from one transparent medium into another of 
 
 * And that twice over, in articles in the ist and jth vols, of that Eeview, well 
 known to he Lord Brougham's. Dr. Yonna; also completed the theory of the 
 tides, which had heen left imperfect by Newton, Laplace, and other great mathe- 
 maticians, and discovered the mode of interpreting Egyptian hyeroglyphics; ho 
 was also a consideraMe scholar. He died in 1819, fifty-six years old, two years 
 after Laplace, who was seventy-eight. See his Life and Writings, edited by 
 Dean Peacock. 
 
Light, Sound, and Electricity. 79 
 
 diflferent density, or when they are reflected by a 
 mirror. The reason at the bottom of this is that light 
 vibrates across the line in which it travels, like the 
 waves you send along a rope by shaking it at one end; 
 but those of sound go backward and forward, as the 
 waves run along a field of corn under the wind, and 
 also spread all round like those from throwing a stone 
 into a pond. But light vibrates in all directions across 
 every line in which it travels, and not only in one 
 plane like the vibrations of the rope ; and those lines 
 radiate in every direction from each luminous point, so 
 that the faces of the waves advance like a sphere con- 
 tinually expanding. But it is impossible to explain 
 here how those results follow.* 
 
 Eight is itself invisible except by its effect in illumi- 
 nating things on which it falls. If you look into one 
 end of a blackened tube with the other end closed, and 
 a slit or hole in the side for light to come through, you 
 see no stream of light, but only a little faint reflection 
 from the black surface. But if you stick a white ball 
 or a bright wire in, so that the light falls on it, it will 
 shine as usual. What is called a sunbeam shining 
 through a hole in the shutter in a dark room is not 
 light which you see, but the motes of dust shining in 
 the light. The full moon stands in a flood of sunlight, 
 which is dark all round her and invisible, except where 
 it illuminates her surface. 
 
 Here let us stop a moment to reflect on the impor- 
 tance of this fact, or law of nature, that light will 
 
 * Tou may read with advantage the explanation .of light in Sir J. Herschers 
 ' Familiar LecturcB,' lately puMished. 
 
80 lAght, Sound, and Electricity. 
 
 only travel in straight lines (except when it is artifi- 
 cially diverted), while sound can turn round corners, 
 only rather weakened by it. Not merely the whole 
 science of measuring, bat the power of judging of 
 sizes, distances, and positions of everything we cannot 
 touch, depends on the lines' of sight going straight. 
 If the case had been reversed, we should have lost a 
 great deal by every sound being stopped the moment 
 anything came between it and us, and gained nothing ; 
 while if we did not see straight the whole world would 
 be in confusion, and it is hardly too much to say that 
 the human race could not exist. 
 
 There is also a vast difference between light and 
 sound in their rate of travelling. The waves of sound 
 go only 377 yards in a second, while the earth itself 
 goes i8"2 miles, and light ten thousand tinies faster 
 than that ; while electricity (which again is probably 
 another kind of vibration, of the solid atoms of bodies, 
 and certainly not a fluid) runs along a wire about half 
 as fast again as light.* So if the earth were a cannon 
 ball, shot at the sun from its present distance, with the 
 velocity it now travels with, and the moment of explo- 
 sion . telegraphed to the sun, they would get the tele- 
 gram there in about S minutes, and see the earth com- 
 ing in 8 minutes, and would have nearly two months 
 to prepare for the blow, which they would receive 
 about I ^years before they heard the original explosion. 
 This is merely taking the sun as a target to be shot at, 
 without regard to his power of attracting the earth at 
 the final rate of 384 miles a second. 
 
 * Appendix, Note VII. 
 
The Invisible Bays of Light. 81 
 
 You see then that ' rays of light,' or a ' beam of 
 light,' which we still speak of for convenience, are not, 
 as they used to be thought, emanations of anything in 
 mathematical straight Hnes of length without breadth ;' 
 but are merely the straight course in which vibrations 
 run along the asther in every direction ; and a lumin- 
 ous body is one which somehow or other has the power 
 to excite those vibrations of light — and heat, which (as 
 far as we know) always attends it. 
 
 The general connection of light and heat is obvious 
 enough : it required a great philosopher to separate 
 them as they come together from the sun. It is not 
 merely that things can be moderatelj hot without be- 
 ing luminous (though not very hot), or that some lights 
 have little heat ; but Sir William Hersehel discovered 
 that every ray of sunlight can be split up into visible 
 and invisible rays ; the visible ones consisting of the 
 seven colors of the rainbow, as Newton had discovered 
 long before, and the invisible rays consisting of heat, 
 even hotter than the red or hottest end of the visible 
 spectrum or band of colors, which appear when light 
 passes through a prism or wedge-shaped piece of glass. 
 Besides these there are other invisible rays beyond the 
 violet or cool end of the spectrum, the effects of which 
 can be made visible by chemical contrivances, as in 
 photography ; as the invisible Lot rays can be concen- 
 ti-ated by a lens or burning glass, eo as to set things on 
 fire. 
 
 And they all come out in this order, beginning with 
 the most refracted : chemical rays (invisible), violet, 
 indigo, blue, green, yellow, orange, red, and then hot 
 
82 The lywrnmiferous JEiher, 
 
 rays invisible. Those invisible hot rays would be hot- 
 ter still, but that they are the rays which are chiefly 
 stopped by our atmosphere and go to warm iti From 
 an electrical light the invisible rays are much hotter 
 than the visible ones, not having many miles of air to 
 pass through.* "We shall see more about this refraction 
 of light into colors in the chapter on telescopes. 
 
 Resisting Medium, — But now comes a surprising as- 
 tronomical result of this theory of light consisting of 
 undulations of some aether or medium filling all space, 
 but far thinner than the most rarefied air we can pro- 
 duce, so thin that we can neither weigh it nor test its 
 presence by anyjpositive evidence as yet. Still, if it 
 exists it must have some density ; and then space is 
 not absolutely emptj', and the earth and planets do not 
 move through space absolutely without friction and 
 resistance, any more than a stone or a pendulum moves 
 through the air without friction, only that the resistance 
 of the sether is immeasurably small. The only one of 
 the heavenly bodies which is even probably suspected 
 to have been sensibly obstructed by this resisting me- 
 dium, is a small comet called Encke's, which returns 
 every 3^ years, and whose period shortens about 2,\ 
 hours in every revolution : and therefore its distance 
 from the sun is shortening too. Sir J. Herschel how- 
 ever suggests another possible cause for this, in the 
 comets leaving part of theii* tails behind them every 
 time they pass the sun, as some of them manifestly do. 
 The tail matter is repelled, not attracted by the sun, 
 and so there is more attraction left on the remainina; 
 
 * See Tyndall oa Heat, § 308, etc. 
 
Is probably a Resisting Medium. 83 
 
 matter of the comet, which makes it move in a smaller 
 orbit, and therefore faster. The matter which after- 
 ward forms the tail appears not to be emitted on the 
 far side, and therefore cannot produce a recoil toward 
 the sun, or a diminution of the orbit. Encke's comet 
 however has no tail. The general opinion of astrono- 
 mers is in favor of the resisting medium, though more 
 from the phenomena of light than from the accelera- 
 tion of a single comet. 
 
 Acceleration by obstruction will seem strange ; but 
 you must remember that a body may perform a smaller 
 orbit in less time though it goes fewer miles per hour ; 
 and the only velocity visible among the heavenly 
 bodies is their velocity of revolution. The resistance 
 does diminish the actual or linear speed, and thereby 
 diminishes the centrifugal force which resists the sun's 
 attraction, and so the orbit is contracted and the time 
 of revolution quickened. And though comets are so 
 thin and light that we can see through their bodies, 
 and they can pass among the moons of Jupiter without 
 sensibly disturbing them, still they are composed of 
 matter and obey the same laws as the planets ; and if 
 the friction of the luminiferous aether can drive one 
 comet a mile nearer to the sun in any time whatever, 
 it is absolutely certain that in some time or other, 
 though it may be an inconceivable number of years off, 
 the earth itself will fall upon the sun — and be again 
 melted by the blow : as its original state of fusion is 
 conjectured to have come from the sudden bringing 
 together of its atoms.* 
 
 * Appendix, Note VHt. 
 
84: PossiUe Mid of the Solar System. 
 
 We need not be alarmed at this conclusion. AH 
 the heavenly bodies had a beginning, for they were 
 created, and they may as naturally have an end. The 
 earth, we know, will have an end, 'and the ele- 
 ments shall melt with fervent heat.' By whatever 
 means the sun's heat is maintained, no philosopher 
 doubts that it is by the using up of something, or the 
 destruction of some motion, for heat and force are uni- 
 versally convertible.* And so, although the law of 
 the stability of the universe is true, and all the bodies 
 in the solar system can recover from the temporary 
 disturbances'^hich are always taking place by their 
 mutual attractions, it is almost equally certain that in 
 the lapse of ages permanent changes take place also. 
 Some conspicuous stars have gone out since men began to 
 observe them, and the stars are not little things like the 
 earth, but many of them probably as big as the sun, and 
 some certainly much bigger, as we shall see hereafter. 
 
 The ultimate destruction of the solar system, hoAV- 
 ever remote it may be, is a tremendous consequence 
 to follow from such an apparently minute fact as that 
 light consists of very, small vibrations of an immeasur- 
 ably thin aether. But thus are the laws of nature 
 linked together. Perhaps there may be some counter- 
 acting and preserving force in operation not yet dis- 
 covered. And now leaving these contemplations of 
 the future, and returning to the actual velocity of light, 
 we shall find that its latest measure confirms the late 
 discovery that all the dimensions in the solar system 
 had been rather overrated by astronomers. 
 
 " Appenflix, Note IX. 
 
Measuring the Veloeity of Light. 85 
 
 NEW DIMENSIONS OF THE SOLAE STSTEM. , 
 
 From 1769 to 1862 the sun's mean distance was con- 
 sidered to be 95^ million miles, and so you will find it 
 still in most books of astronomy. Mr. Airy, the As- 
 tronomer Royal, said in his lectures delivered in 1856, 
 ' there is no probability of an error of half a million 
 of miles ' in this distance. Yet within six years it be- 
 came almost certain that there was an error of four 
 millions. The error was even then suspected by Pro- 
 fessor Hansen, of Gotha, from the 'parallactic in- 
 equality ' of the moon appearing to be wrong ; and 
 the suspicion was confirmed by certain observations of 
 Mars in 1862, which will be explained at p. 209. 
 
 For you must understand that the measuring of the 
 actual distances in miles is difficult, though measuring 
 the relative distances is not. It is easy enough for as- 
 tronomers to make out, for instance, that the distances 
 of Mercury, Yenus, and the earth from the sun are 
 nearly in the proportions of 4, 7, and 10 ; but the dif- 
 ficulty is to find a scale to measure any one of the dis- 
 tances by. A convenient scale is the diameter of tlie 
 earth itseK; but that is so exceedingly small compared 
 with the smallest of those distances from the sun and 
 other' planets, that it is very difiicult to apply it accu- 
 rately. I will explain hereafter how it is done ; just now 
 we are only inquiring into the velocity of light. About 
 the same time that astronomers began to suspect that the 
 dimensions of the solar system had been overrated, some 
 new experiments on the velocity of light were made in 
 France with machines on the principle of one which 
 
86 New Dimensions of the Solar Bystmi. 
 
 had been invented by M. Foucault or Mr. "Wheatstone 
 for electricity ; and it was found that the velocity for- 
 merly accepted, of 192,000 miles in a second, had been 
 overrated to about the same extent. Then came fur- 
 ther astronomical observations, which have all tended 
 to confirm those of 1862 ; and it seems now to be tol- 
 erably clear that the velocity of light cannot be more 
 than 184,000 miles a second, and the sun's distance 
 about 91,404,000 miles,* or just one 24th less than 
 the old amount. 
 
 On the 8th of December, 1874, and again in 1882, 
 will occur an event of great astronomical importance, 
 which has not happened since 1761 and 1769, when 
 some mistake was evidently made, and will not happen 
 again for more than another century, that is, a transit 
 of Venus as a small black spot over the face of the 
 sun. For that phenomenon is considered to afford 
 more accurate means than any other of measuring the 
 sun's distance compared with the diameter of the earth : 
 not that a single observation of that kind can prevail 
 over the concurrence of evidence that now exists in 
 favor of the reduced distance ; but any confirmation 
 thereby will be decisive. 
 
 "With the distance of the sun, every distance and 
 diameter in the solar system (except of the moon, 
 which is measured independently, being tolerably near), 
 and even the weights or masses of the sun And planets 
 have to be altered. First of all, their real diameters 
 depend at once upon their distance : for all that we 
 can say of them is that they appear a certain width ; 
 
 * Appendix, Note HI, 
 
Why the Sun''s. Mass has to he Altered. 87 
 
 that is, the two lines of sight from their opposite sides 
 to our eye enclose a certain angle ; but a shilling near 
 your eye looks as big as a crown further oflf"; and so 
 we have to wait till we know whereabouts the sun has 
 to be placed between the two lines of sight which 
 measure his apparent diameter before we can say what 
 real diameter it corresponds to. Therefore it is plain 
 that the diameters have to be reduced, with the dis- 
 tauces, by a 24th of the old amount ; and as the bulk 
 varies with the cube of the diameter (as I explained 
 at p. 24) the bulks have to be reduced in the propor- 
 tion of 24? to 23* ; which you will see, if you calcu- 
 late it, makes a reduction of "12 or nearly one 8th in 
 the old bulk of the sun and all the planets except the 
 earth. 
 
 The reason for altering their masses or weights is not 
 so evident ; for you must remember that we know 
 nothing of their density except through their size and 
 weight ; and so we cannot say that the weight is to be 
 reduced an 8th because the diameter is reduced a 24th. 
 Still it has to be done for quite another reason, and as 
 it happens, in just the same proportion. One of the 
 things proved by Newton was that the mass of the sun 
 must bear a fixed proportion to the cube of its distance 
 from the earth and the square of the time of the earth's 
 revolution, as you will see at p. 250. Strictly speaking, 
 it is the mass of the earth and sun together, but the 
 earth is so small in comparison that it may practically 
 be left out of the calculation. And as there can be no 
 considerable mistake about the length of the year, we 
 may say that the mass of the sun bears a fixed pro- 
 
88 Advcmtage of Vulgar Fractions. 
 
 portion to the cube of its mean distance. Therefore, 
 when we find the distance has been overrated by a 
 23rd, we have to reduce the mass of the sun in the 
 proportion of 24' to 23^, or 100 to 88, or nearly 8 to 7. 
 
 Here I stop for a moment to remark how much 
 better tlie old, and the mathematical, habit of using 
 vulgar fractions in small figures is than the modern 
 vulgar habit of jper-centing everything. It Is very easy 
 to remember, when you have once learnt it (which few 
 people seem to do), that a reduction of a 24th corre- 
 sponds to an increase of a 23rd, a reduction of one- 
 third to an increase of one-half, and so on ; or that 
 any two numbers, such as 2 and 3, which differ by one- 
 half of the smaller number, differ by one-third of the 
 larger. But no such rule can be expressed in decimals. 
 People nowadays, especially commercial ones, will not 
 even tallv of a half or two-thirds of anything, but of 
 ' 50 per cent.,' and ' 66 and two-thirds per cent. ; ' and 
 sometimes they talk of a thing costing 100 per cent, less, 
 as if it was the converse of costing 100 per cent, more, 
 instead of being nonsense ; for they do not mean that 
 it costs nothing, but half as much. If they would 
 condescend to use vulgar fractions whenever they can 
 be expressed in small figures, and especially when the 
 numerator is i, they would save a great many words 
 and make fewer mistakes. In some cases decimals are 
 best. - 
 
 Then comes the question, why the masses of the 
 planets are to be reduced for their new distances. 
 Jupiter and the others which have moons affect their 
 moons or satellites as to time and distance just as the 
 
Reduction of the Plcm.ets' Masses. 89 
 
 sun affects the earth ; and therefore, as the satellites, 
 distances have to be reduced with the rest, the masses 
 of their attractors or jprvma/ries must be reduced like 
 the sun's. As to Mercury, Venus, and Mars, which 
 have no moons, their masses were calculated from their 
 powers of disturbing, that is, attracting their neigh- 
 bors and comets, and therefore their masses also have 
 to be diminished as the cubes of their distances. 
 
 But how is that reduction to be measured or ex- 
 pressed ? By the only unit or standard of weight that 
 we have for all the solar system, that is, the earth. It 
 is much easier, as I shall show you afterward, to weigh 
 the sun in ea/rths, than you have already seen it is to 
 weigh the earth in tons, or to compare it with an equal 
 bulk of lead or water. Therefore if we keep the old 
 figures for the proportions between the masses of the 
 sun and planets, while we reduce the sun from the old 
 figure of 359,500 to 316,560 times the earth, we shall 
 have reduced them all. Still you may ask one more 
 question : Why is not the proportion of the sun to 
 Jupiter to be reduced as the proportion of the sun to 
 the earth is ? Because the mass of the sun was not 
 obtained from Jupiter, nor that of Jupiter from the 
 sun. And that would be not reducing Jupiter at all ; 
 for he would be first reduced one 8th with the sun, and 
 then increased the same one 7th again, to reduce the 
 sun's proportion to his. Neither the sun nor Jupiter 
 is the unit of mass' for the solar system, but the 
 earth is. 
 
 Attraction is Instantaneous.— As I have given the 
 times which electricity and light would and do take to 
 
90 AtbracUon is Instcmtcmeous. 
 
 reach the earth from the sun, I ought to add that the 
 force of gravitation alone seems to be instantaneous.* 
 All the calculations of the motions of the most distant 
 bodies, no less than of the nearest, assume that it is, 
 and the results justify the assumption. See the note 
 on this at the end of Sir J. Herschel's lecture on the 
 sun. 
 
 * Appendix, Note X, 
 
CHAPTEE III. 
 
 THE MOON. 
 
 Ip you ask half a dozen people how big the moon 
 looks, you wiU probably get as many different answers, 
 varying perhaps from a shilling to a large round table. 
 The truth is that the question has no meaning, because 
 a shilling near the eye looks as large, or makes the 
 same angle with the eye, as a plate further off, or as 
 the moon further still. But then some people say the 
 moon looks bigger than the sun, others smaller, and 
 others about the same size ; and that has a meaning, 
 and can be wrong or right ; for we can compare the 
 angle which each of them makes with the eye, and say 
 which has the real ap^pearance (quite a different thing 
 from the reality) of being the largest. Without any 
 measuring at all, there happens to be a ready way of 
 knowing that they really appear about the same size, 
 sometimes one a little larger, and sometimes the other. 
 For the moon quite covers the sun in a total eclipse, 
 but leaves a bright' ring all round it in an annular 
 eclipse. In the former she appears at least as large as 
 the sun, being at her nearest to the earth : in the lat- 
 ter a little less, being further off. 
 
 But as soon as we take the distance of the sun and 
 moon into account, we begin to see what a little thing 
 the moon is. Her distance from us varies from 225,- 
 720 to 251,940 miles; for she moves in a more eni[)ti- 
 
92 The MoovUs Distcmce and Size. 
 
 cal orbit round the earth than the earth does round the 
 sun, the eccentricity being one i8th, or above three 
 times the earth's. The mean distance is 238,830 miles, 
 or 30- 1 367 times the earth's diameter. The moon's 
 diameter is only 2165 miles ; therefore the earth is 49 
 times as big, and the sun more than sixty million times 
 as big as the moon. If you take a ball of an inch di- 
 ameter for the earth, the moon will be little more than 
 a quarter of an inch, or about the size of a pea 2J feet 
 off, and the sun a globe of 9 feet diameter standing 
 320 yards off. Moreover the density of the moon is 
 only -6 of the earth's, or 2-4 of the sun's density; for 
 the earth is 81 '5 times, and the sun nearly 26 million 
 times as heavy as the moon ; and gravity there is 
 only one-sixth of what it is here, although her sur- 
 face is only about one-fourth as far from her centre as 
 ours is. 
 
 It is curious that the sun's distance is nearly as much 
 greater than the moon's (383 times) as his diameter is 
 (390 times) ; also that the moon's distance is 1 10 times 
 her diameter, and the sun's 108 times his diameter, 
 which again is 107 times the earth's diameter. 
 
 The moon is so near that we can see the nature of 
 its surface to be extremely different from the earth's 
 surface. It is covered in many places with craters of 
 old burnt-out volcanoes, far larger than any on the earth, 
 and with the heaps of ashes which were thrown out 
 forming mountainous rings around them. Indeed it 
 seems that all volcanic action is not extinct, as some 
 persons have supposed that one crater called Linne, 7 
 miles wide, has been lately raised from a hollow into a 
 
Tile Moon^s Surface. 93 
 
 hill.* Tliere are also wide plains called seas : but not 
 of water, for there are no signs of any ; nor of any at- 
 mosphere affecting the light near the edges of the 
 inoon,f as there are in the snn and planets. The reg- 
 ularity and distance of the rings from the craters, and 
 the height of the mountains, would depend on the vol- 
 canic force and the inclination at which the matter was 
 thrown ou% as there was no wind to stop or spread the 
 ashes ; and that force would be practically greater on 
 account of the weakness of gravity and the quicker 
 cooling and contraction, from the moon's smallness and 
 the want of any atmosphere to check radiation away 
 of the heat into space. 
 
 A telescope which magnifies looo times shows the 
 moon as we should see it 233 miles off without a tele- 
 scope ; for we have to measure between the two sur- 
 faces, not the centres. A town a mile wide, at that 
 distance, would look nearly half as wide as the moon 
 does, and a building of the size of York Minster would 
 look larger than Yenus. :j: "We see the shadows of the 
 mountains clearly when they are turned sideways to 
 the sun. 
 
 Rotation of the Moon. — It is one of the peculiarities of 
 the moon, and (as far as we can see) of the moons be- 
 longing to other planets also, to keep the same face al- 
 ways toward us, as if it were set fast upon a stick 
 reaching to the earth. This is accurately described by 
 saying that the moon turns on her axis exactly in the 
 same time as she revolves round the earth. Tou may 
 very likely think that the moon's keeping the same face 
 
 * Appendix, Note XI. + Appendix, Note XH. % Appendix, Note XIII 
 
94 The Mom's Botation. 
 
 to us proves that she does not turn on her axis at all ; 
 and every now and then somebody undertakes to prove 
 that all the astronomers are wrong in saying that the 
 moon rotates or turns upon her axis ; as others under- 
 take to prove that all the mathematicians in the world 
 are wrong in saying that the diameter of a circle bears 
 no exact proportion to its circumference or area.* I 
 know it is as hopeless to convince them as it was the 
 man who insisted that his head had been twisted round. 
 But for the information of those who wish to learn I 
 will give a few of the innumerable proofs that can be 
 given of the moon's rotation : see also p. 57. 
 
 First, then, the deniers of it forget, or do not know, 
 that if they were on the sun, or anywhere else in space 
 except just here inside the moon's orbit, they would 
 see o;^posite sides of her every fortnight ; and the pro- 
 position that she does not turn would then appear as 
 absurd to them as the contrary does now, and with 
 much better reason. Perhaps the mistake arises from 
 people thinking of the moon's axis as a wire stuck 
 through it, and forgetting that if it were, the question 
 would remain whether the wire is turning on its own 
 mathematical axis or not. If you put a wire upright 
 through a ball with one side painted white, and bend 
 it into a horizontal tail which you can hold always 
 pointing to one side of the room, while you carry the 
 ball round you with the white side always toward you, 
 
 * I mean no proportion measurable by any finite number of figures, or by 
 any geometrical construction with rule and compasses. Matbemaiicians know 
 very well that it can be done mechanically by a cycloid, or the curve traced by 
 any point In the circumference of a wheel rolling on a straight rail ; for the base 
 of the cycloid evidently — the circumference of the wheel. 
 
Proofs of the Moon's Rotation. 95 
 
 it will plainly enough turn on the axis, because the 
 axis has been kept from turning. But if you keep the 
 tail of the wire always toward you, the ball does not 
 turn on that axis because it has turned itself just as 
 much. 
 
 Or you may lay a book on a revolving table, at some 
 distance from the centre. The denierg of the moon's 
 rotation say the book does not turn on its own axis be- 
 cause it keeps the same face to the centre of the table 
 when you turn it round. But that can have nothing 
 to do with its distance from the centre. Keep gently 
 moving it nearer to the centre while the table revolves, 
 and at last it will be visibly turning on its own axis 
 and nothing else. Then at what moment did it sud- 
 denly start from no rotation at all into a rotation at the 
 same rate as the table ? 
 
 Once more, set a cotton reel with a hole bored across 
 it, upright and fast on a nail at the end of a stick 
 which can revolve round a pin stuck in a table, as the 
 moon revolves round the eaa-th ; and put a string in 
 the hole as if you were going to wind it round the 
 reel. Then turn the stick round, holding the string in 
 your other hand : it will wind once round the reel for 
 every revolution of the stick, just as much as if it had 
 stood still and you had turned it round between your 
 fingers. 
 
 So much for the fact of the moon's rotation. But 
 you may prove also that a force of rotation must have 
 been originally impressed upon it to make it rotate 
 even at that slow rate of otice a month. Hang a globe 
 or anything else, so marked as to distinguish the sides. 
 
96 Supposed Gause of it. 
 
 by a thin string from the end of a stick held in your 
 hand, till it has got quite steady : then turn round, 
 carrying the stick with you so as not to shake the 
 globe. Tou will find that by the time you have gone 
 half round, the side which was facing you at first has 
 got furthest ofl^, but still faces the same side of the 
 i"oom. Why ? Because there has been no force im- 
 pressed on it to make it turn, and nothing ever moves 
 except in obedience to some force. 
 
 Some astronomers believe that the reason why the 
 moon keeps the same side toward the earth is that her 
 far side is heavier, or ' her centre of gravity 33 miles 
 beyond her centre of figure ; ' which it is said would 
 keep the heavy side furthest from the earth by centrir 
 fugal force, as the heavy end of a stick will keep furth- 
 est off if you swing it round by a string tied to its 
 middle, not to its^ centre of gravity.* But that anal- 
 ogy seems to me to fail ; for there is nothing to make 
 the moon tui-n upon her centre of figure : on the con- 
 trary, a body that is free to turn will always choose an 
 axis through its centre of gravity to turn upon ; and 
 if you swing the stick by its c. g. the heavy end will 
 not keep furthest off any more than the light one. It 
 is remarked truly that the c. g. being nearest to the 
 invisible side of the moon would make all the water 
 go there, if there is any, and so account for its non- 
 appearance : but there seems to be no reason for be- 
 lieving that there is any.f 
 
 Again it is said in other books that when the moon 
 
 * Hersohers Onllines of Astronomy, p. 287 (ed. 7 and 8)^ 
 t Appendix, Note XIV. 
 
Centre of Oravity of Ea/rih and Moon. 97 
 
 was fluid the difference of the earth's attraction on the 
 near and far sides would make her egg-shaped, as the 
 moon's attraction raises the water of the tides slightly 
 into that shape. And so it would — but not to the ex- 
 tent of 30 miles, as I shall show you at p. 1 34. If she 
 really has that shape, some better cause for it has to 
 be discovered yet, which must apply to the moons of 
 Jupiter and Saturn also. Mr. Adams does not con- 
 sider it proved by any of the calculations which are 
 supposed to prove it. But it may still be open to other 
 proofs, and I shall have to mention one at p. 104, if the 
 observations there stated are correct ; and if so, the c. 
 g. being nearer the far side will account for that side 
 keeping furthest off, not by centrifugal force but for 
 another reason. 
 
 Another illustration will be useful for another pur- 
 pose. If a man is Y^'ltzing with a little girl, each 
 turns round the other, and on their own axes too, fac- 
 ing: all the four walls of the room in succession ; but 
 besides that, the man being the heaviest makes a smaller 
 circle round the girl than she does round him : that is, 
 he swings her round him. And so the earth goes 
 round the moon as truly as the moon round the earth, 
 only in a circle as much smaller as the earth is heavier : 
 that is, 8 1 '5 times. The point between them which 
 remains fixed, if we forget their joint motion round 
 the sun, is theii- centre of gravity, which is 81*5 times 
 nearer the centre of the earth than of the moon ; and 
 as the distance of the moon is only sixty times the 
 earth's radius of 3963 miles, the centre of gravity of 
 the earth and moon is 2918 miles froin the earth's cen- 
 
98 '■ ReSmcing the Earth to Rest^ 
 
 tre. So if you fix two balls on the ends of a stick five 
 feet long, one four times as heavy as the other, and 
 whirl it into the air, they will turn round the centre 
 of gravity, four feet from the light ball and one foot 
 from the heavy on.Oj, 
 
 The cause of the moon keeping her distance from 
 the earth is simply the centrifugal force of both of 
 them. If the light ball just now described can slide 
 along the stick, but is connected with the other by a 
 string, and you whirl them into the air, they will keep 
 the string stretched by their centrifugal force. The 
 same is the reason why the earth and all the planets 
 keep their distance from the sun, subject to the varia- 
 tions due to elliptic motion, as I shall explain here- 
 after. 
 
 Calculations of the moon's motion are simplified by 
 ' reducing the earth to rest,' as it is called ; which might 
 be done by assuming another equal moon at the same' 
 distance on the opposite side of the earth, so that the 
 common centre of gravity and of motion would always 
 be the earth's centre. Bat this hypothesis would make 
 the mass of the whole system too great, and also increase 
 the time of revolution ; and the more correct way of 
 reducing the earth to rest is to suppose the mass of 
 the moon transferred to the earth, leaving her only an 
 empty shell ; for the attraction on, but not of, a body is 
 independent of its own weight or mass (p. 28). It is 
 important to attend to this in calculations of the moon's 
 period round the earth, and in weighing the sun there- 
 by, as we shall do hereafter. But whenever we have to 
 deal with the moon's atti-action, as on the equatorial 
 
Eeal Shape of the Moon^s Orbit. 99 
 
 protuberance of the earth in precession and nutation, 
 or on the water of the earth in the tides, we must of 
 course deal with her real mass, though we may still 
 consider the earth at rest, except in rotation, because 
 that alone affects those disturbances. 
 
 The Moon's Orbiti — The motion of the moon in 
 space is harder to understand than its motion round 
 what we call for convenience the earth, meaning that 
 point in the earth which is their centre of gravity. 
 You may think it is not so very difficult to see that the 
 moon describes a kind of spiral path with loops in it, 
 round the earth, which goes on in its orbit while the 
 moon turns round it in the same direction from west 
 to east ; and the pictures in some books represent it so. 
 But it does nothing of the kind, as I will show you by a 
 thing which you may see any day. If a long pencil 
 were stuck into a railway wheel at any point within 
 the circumference, so as to mark its own path on a wall 
 at the side of the railway, you would see that it is 
 nothing but a wavy line, with the lower bends rather 
 smaller than the upper.* And the moon's path round 
 the sun — and the earth's too in a less degree, is only the 
 regular elliptic orbit of their common centre of gravity 
 with 25 places in it, for full and new moon, a little more 
 and a little less alternately curved toward the sun ; 
 for none of the bends are convex, but all only more or 
 less concave, toward the sun. 
 
 This is difficult to conceive and impossible to draw 
 
 * That curve is called a troclwH ; but when the tracing point is on the cir- 
 cumference of the wheelit hecomes aa/cloid, which is rather like the long half 
 of an ellipse, hut very different in its mathematical properties, which are rc- 
 markahle in various ways. 
 
100 JRevohing MKpse. 
 
 on a small scale, but easy to prove by a little calcula- 
 tion of which I will give you the result. The c. g. 
 of the earth and moon moves through 14° 33' in its 
 orbit round the sun from one half moon to another ; and 
 such an arc described with a radius of 9 if million miles 
 bends outward 684,000 miles beyond the chord or 
 straight line joining its two ends. But the new moon 
 is only (say) 240,000 miles nearer the sun than the two 
 half moons, and therefore it is still 444,000 beyond the 
 chord that joins them : that is to say, the path of the 
 new moon through space is concave toward the sun to 
 that extent. It is something like a piece of a circle de- 
 scribed with a radius of 150 instead of gif- million 
 miles, though it is not really circular. In the same 
 way you would find that the path of the full moon re- 
 sembles an arc of about 60 million miles radius, evi- 
 dently more concave toward the sun. 
 
 But in all calculations of the moon's visible place, her 
 orbit may be considered an ellipse, with the earth stand- 
 ing in the focus, and the sun revolving round them 
 both. And this elliptic orbit, or its major axis, or line 
 of apsides, 01: perigee, and ajpogee, or least and greatest 
 distances from the earth, revolve in 8 years and 3 iO"57S 
 days, going forward ; whereas the earth's perihelion 
 takes 1 10,880 years to revolve, independently of the 
 precession of the equinoxes, which go the other way. 
 
 You may easily see a revolving ellipse if you hang a 
 weight by a long string to a hook in the ceiling and 
 send it swinging in any direction, except straight across 
 like a pendulum or quite round in a circle, which indeed 
 you would find it difiicult to do. If you stand near the 
 
Hecession of the Nodes. 101 
 
 narrow part of the ellipse where the weight passes by 
 you, and stay there, the ellipse will wheel round and the 
 weight will hit you before long. The only difference is 
 that the centre of force in this ellipse is not in the 
 focus, but the centre, the force being the tendency of 
 the weight toward the vertical or lowest point ; and the 
 force is nearly in direct proportion to its distance from 
 that point, instead of the inverse proportion of the square 
 of the distance. This ellipse wheels forward, or in the 
 same direction as the weight, because the force does 
 not quite vary as the distance ; and the lunar ellipse 
 similarly wheels forward because the moon's attraction 
 to the earth is a little diminished by the sun, as will be 
 explained at p. 152. 
 
 The moon's axis, like the earth's, does not stand up- 
 right to her orbit, but it is much less inclined than the 
 earth's, leaning only 6° 39' from the poles of her orbit, 
 instead of 23° 28' ; that is, the moon's equator is so 
 much inclined to the plane of her orbit. And again 
 the moon's orbit round the earth is inclined 5° 9' to the 
 apparent orbit of the sun round the earth, or the ecliptic; 
 So we have now got one more plane and one more in- 
 clination to consider than we had in dealing with the 
 earth's equator. It is however simplified by the fact 
 that the crossing of the moon's orbit and equator always 
 coincides with the crossing of her orbit and the ecliptic ; 
 though that line, whicli is called the line of nodes, re- 
 volves backward 19° 19' a year, making a sidereal, 
 revolution in i8-6 years, or 6793-391 days, or 223 cal- 
 endar months, just as the equinoctial points '=p and :^, 
 the nodes of the earth's equator and ecliptic, revolve 
 
102 The Maori's Lilration. 
 
 once round in 25,868 years (p. 52). The node at whicli 
 the moon goes up to the north of the ecliptic is called 
 the ascending node SL, and the one where she goes 
 down again to the south is the descending node ^5 . 
 
 Moreover it is remarkable that the ascending node 
 of her orbit is the descending node of her equator : or 
 at the moment when she rises in her orbit from the 
 ecliptic at the inclination of 5° 9', her equator is in- 
 clined I ° 30' to it the other way. Therefore the ecliptic 
 always lies between the plane of the moon's orbit and 
 the plane of her equator, making the angle 5° 9' with 
 the former and 1° 30' with the latter. You will se6 
 presently the importance of understanding clearly the 
 inclination of the moon's orbit, and the recession of the 
 nodes: its cause will be explained at p. 156. 
 
 Libratiotli — It is not strictly true that the moon al- 
 ways keeps exactly the same face to us, as I said she 
 did ; for she apparently rolls a little in her orbit, so as 
 sometimes to show a little more on the right side, and 
 sometimes on the left. That is called libration / and 
 it has been ingeniously used to take stereoscopic photo- 
 graphs of the moon, which show the heights of the 
 mountains and the depths of the volcanoes. For a 
 stereoscopic view of anything is got by taking two 
 pictures of it, one seeing a little more of the right side 
 and the other of the left, just as your two eyes do, as 
 you may see by winking them alternately while you 
 look at something not far off; and that is the way the 
 eyes judge of distance, when it is not too great for any 
 sensible difference in the view which each eye takes. 
 
 The reason of the moon appearing so to roll is, that 
 
The MoovUs Libration. 103 
 
 she does not move in her elliptical orbit with uniform 
 velocity, but does turn on her axis uniformly, and so 
 the two motions do not exactly keep pace with each 
 other. She is sometimes 6° 18' before, and sometimes 
 6° 18' behind the longitude she would have if she moved 
 uniformly : that being her greatest ' equation of the 
 centre,' or difference of true and mean ' anomaly ' or 
 distance from perigee (p. 41). And the libration is 
 the same as if the moon stood still rotating uniformly, 
 and the earth went round her and at the same pate on 
 the average, but sometimes 6° 1 8' before and behind its 
 mean place. Consequently that is the amount of the 
 Uhration in longitude, each way ; and we see on the 
 whole 12° 36' beyond the half of the moon's surface ; 
 and indeed nearly 16° by reason of the other inequali- 
 ties in her motion which will be described hereafter. 
 
 But the polar axis also leans 6° 39' from the perpen- 
 dicular to her orbit, and sometimes one of her poles 
 leans toward us and sometimes the other ; and so we 
 see also 6° 39' beyond each pole alternately. This is 
 called the Uhration in latitude. Now the area of a 
 lune, or slice of surface of a globe between two meridi- 
 ans, is evidently measured by their difference, of longi- 
 tude ; and therefore the surface disclosed by libration 
 in longitude bears the same proportion to the whole 
 moon's surface as 15° 50' does to 360°, and that by 
 libration in latitude as 13° 18'. Therefore (except that 
 they overlap a little) we may say that we gain by all 
 the four librations together, ^ of the otherwise invisible 
 half of the moon, or we see foiu- 7ths of her whole 
 surface at one time or another. 
 
104 The MomCs True Shape. 
 
 Shape of the Moon, — Some of the stereoscopes of the 
 moon taken at librations certainly present the appear- 
 ance of a prolate spheroid pointing toward the earth. 
 This may be an optical illusion or defect ; but other 
 observations appear to show that she actually is like 
 an egg with the small end toward the earth and the 
 large end behind. If you look straight endways at an 
 egg, with one eye, you cannot tell that it is not spheri- 
 cal, but you can if you see it a little obliquely. By a 
 carefal measuring of angles you might find its exact 
 dimensions, and the more obliquely you can see it the 
 better. ISTow we can get a view of the moon's small 
 end (if it is one) even more oblique than the libration 
 of 8°, by looking at her from the east side of the earth, 
 or at moonrise, when the libration is toward the west, 
 and in that way gain an angle = 4000 miles divided 
 b}' 239,000, or about 1° more. It appears (as I under- 
 stand the description) that the oblique profile of the 
 moon does show her to be egg-shaped ; and so much; 
 that unless her density is irregulai-, her centre of 
 gravity is full 30 miles nearer the back than the front ; 
 though yoiT can hardly call the middle of the length 
 of an egg its ' centre of figure ' or ' of symmetry.' 
 
 And then this consequence would follow : the aver- 
 age mass of the long half of the moon is further from 
 her c. g., round which she must rotate (p. 97), than 
 the mass of the short half; and the earth's attraction 
 on each half, being inversely as the square of its dis- 
 tance from them, will be greatest when the long half 
 points to the earth : which will tend to keep it there ; 
 though not strongly enough to prevent the libration of 
 
Effect of it on her Rotation. 105 
 
 a few degrees, -which is due to the moon's force of ro- 
 tation. But how she got that shape is a question yet 
 unsolved. 
 
 Light and Heat there^ — Though -we see only one side 
 of the moon, the sun sees or shines all round her in 
 the course of her month of 29^ days, from one new 
 moon to another ; and one of the consequences of there 
 being no water, and no clouds or vapor or air round 
 the moon, is that the side facing the sun is probably 
 hotter than boiling water, and the dark side for the 
 time is colder than the hardest frost on the earth ; in 
 fact as cold as space (p. 42), except so far as it is 
 warmed by the heat remaining from the sunny fort- 
 night. So if there be any living creatures there, they 
 must be able to bear those extreme changes every fort- 
 night, besides doing without air and water. 
 
 The moon has no light of her own, and only reflects 
 the sun's. Dr. WoUaston made out that full moon- 
 light is 800,000 times weaker than sunlight, which he 
 found equal to 5563 wax candles at the distance of a 
 foot, while the moon was only the 144th of a candle.* 
 Therefore, as the moon fills about the 240,000th of the 
 visible hemisphere, we should get much less light from 
 a whole sky full of moons than we do from the un- 
 clouded sun. 
 
 JUEoonshine gives no sensible heat, even when con- 
 centrated by a very large concave mirror into the great- 
 est possible intensity on a thermometer ; and it causes 
 cold indirectly ; for a full moon is found to drive away 
 clouds, and a light night is always colder than a cloudy 
 
 * Philosophical Transactions of i8z9, p. zo. 
 5* 
 
106 The Moon gives no Meati 
 
 one (when other things are equal), because the heat of 
 the earth radiates away into space instead of being kept 
 iu by the clouds.* 
 
 Phases of the Moorii — The moon is full when the 
 earth is between it and the sun, and therefore we 
 see all the bright face ; except when there happens to 
 be an eclipse of the moon, of which I shall have more 
 to say presently. The new moon is when the moon 
 is between the sun and the earth, so that only the dark 
 side faces us, and we see no moon at all. The phases 
 of the moon are nothing but the Greek word for her 
 faces or appearances. She increases in width, or waxes, 
 from new to first half moon in the form called a cres- 
 cent, with the horns turned to the left, and wcmes from 
 second half moon to new as a crescent with the horns 
 turned to the right. From fbrst half to full moon, and 
 from full to second half moon, in the second and third 
 ' quarters,' one side is a semi-circle and the other side 
 is half an ellipse, which the hollow side of the cres- 
 cent also is. That shape is called giVbous. The ellipse 
 in each case is the oblique view of half of that me- 
 ridian of the moon which divides the light from the 
 darkness. If you turn back to the figure of an ellipse 
 within the circle of which it is the perspective view 
 (p. 40), the shaded part represents the crescent, and 
 the unshaded the gibbous phase of the moon. 
 
 A day or two before or after new moon you may sefe 
 what is sometimes called ' the new moon with the old 
 one in her arms,' or a bright narrow crescent, with the 
 rest of the circle just light enough to be seen. That is 
 
 * Appendix, Note XV. 
 
Eourtlhshme on the Moon. lOY 
 
 the reflection of the earthshine back from the moon. 
 "When it is further from new, the earthshine appears 
 less and the proper moonshine stronger, and so the 
 earthshine is too weak to be seen. 
 
 As the moon has no light of her own, but only reflects 
 the sun's light to us, so we do the same to Iier. If there 
 were any men in the moon they would see correspond- 
 ing phases of the earth, which would be full to them 
 when the moon is new to us ; and the earth would ap- 
 pear to them nearly thirteen times as large as the 
 moon does to us : it would be sixteen if the diameter of 
 the earth were quite four times that of the moon ; for 
 the apparent size of a globe, which is called its disc^ 
 being a great circle of the globe, varies as the square of 
 the diameter, while the real size or solid content varies 
 as the cube, as I explained before. 
 
 Periods of the Moon. — 'Period ' means the time of 
 performing a journey round. The exact time from 
 one new or fall moon to another is 29d. I2h. 44m. 2-878., 
 or 29-5306 days, or 708-734 hours. But to reach a 
 second new moon, or the line between the earth and 
 sun, the moon has to go rather more than once round 
 the earth, because the earth has not been standing still, 
 but by the time the moon has got once absolutely 
 round the earth we and the moon together have gone 
 forward about 44 million miles in our joint annual 
 journey round the sun. And as the moon goes round 
 the earth in the same direction as the earth round the 
 sun, or what we call from west to east, she has to travel 
 further round the earth to get again into aline between 
 it and the sun. If the earth stood still, the solar pe- 
 
108 Periods of the Moon. 
 
 riod of the moon would be only z/d. 7I1. 43m. ii"5s., 
 or 27-32i66 days, or 65572 hours, which is its sidereal 
 or absolute period with reference to the stars. The 
 other, of 29J days, is called the synodAcal period, or 
 simply, a lunation. 
 
 The synodical period of two bodies revolving round 
 a third, really or apparently, is the time of their all 
 three coming a second time into the same relative posi- 
 tion. If the two go the same way, their relative or 
 synodical velocity is evidently the difference of their 
 separate velocities : if they go opposite ways, the sum 
 of them ; and velocities of revolution evidently vary in- 
 versely as the periods, or directly as the reciprocals of 
 the periods ; for i divided by anything is called its re- 
 ciprocal. Therefore the reciprocal of the synodical pe- 
 riod is the difference, or the sum, of the reciprocals of 
 the two absolute or sidereal periods. From which it 
 follows by a common sum in fractions that the synodi- 
 cal period of two bodies going the same way, like the 
 sun and moon, is the product of their several periods 
 divided by the difference ; or by the sum of them if 
 they go opposite ways, like the moon and her nodes. 
 
 A lunation then is the time the moon takes to come 
 again into the same longitude with the sun, called coti- 
 junction, or to get 360° in advance of the sun, or ap- 
 parently once round the earth, measuring by the sun. 
 Therefore the moon advances 12° 11' 27" more than the 
 sun daily. And as the earth turns in the same direc- 
 tion, it has to turn that 12° 11 J' more than once round 
 for the moon to cross the same meridian again ; which 
 makes the lunar day 2411. 49m. Twelve lunations, or 
 
Time of Moonrise. 109 
 
 354'367 days, are called a hina/r yea/r. Therefore the 
 moon is neai-ly 1 1 days older (until the excess is more 
 than 29^) at the beginning of every year on the aver- 
 age. The moon's age at that time is called the epact 
 of the year. 
 
 Harvest Moorii — But the time of moonrise is by no 
 means uniformly 49 minutes later every day. It va- 
 ries for the same reason as the time of sunrise, and 
 even more, because the moon goes 5° further from the 
 equator than the sun (see p. loi). The sun rises at the 
 same time for several days at each solstice, but the 
 moon nearly at the same time for several days in every 
 month, from a different cause, in high latitudes like 
 this. To understand this you had better take a globe, 
 and elevate the north pole about 52° above the south 
 horizon. Then if you turn the globe round from right 
 to left, as it really moves, you will see that the ecliptic 
 at one of its nodes or crossings of the equator rises 
 nearly flat, or a good deal of it seems to rise at once, 
 and at the other node it rises at a great angle to the 
 horizon. And you vs'ill see that the node which so 
 rises flat is the one with the mark T upon it, or the 
 node of the vernal equinox, which is the ascending 
 node of the sun. If you stick two or three wafers on 
 the ecliptic there, about 12° apart, they will represent 
 the moons of successive nights, which will evidently 
 rise nearly at the samfe time. And that happens to 
 some hind of moon every month, as the moon goes 
 through all the signs of the ecliptic in 27-32 days. 
 But it only happens to one full moon in the year, viz., 
 to that one which comes at or nearest to T, which is 
 
110 The ' Harvest Moon: 
 
 when the sun is at the opposite node =£2=, or the autum- 
 nal equinox. So the moon rises about full at six o'clock 
 or sunset for several nights together, always within a 
 fortnight of September 23 ; and that is called the har- 
 vest moon. 
 
 This uniformity of time of moonrise for a few days 
 in every month is also greatest when the moon's as- 
 cending node coincides with T , for then the moon's 
 orbit is 28|° inclined to the equator, and still nearer 
 to the colatitude of England, which is about 38°. On 
 the other hand, when the moon's ascending node coin- 
 cides with the sun's descending node =i:, the inclina- 
 tion of the moon's orbit to the equator is only i8|-°, 
 and therefore the ' harvest moon ' effect is considerably 
 less. 
 
 Moonlight in Winter. — There is much more moonlight 
 in winter than in summer. I do not mean merely that 
 there are longer nights for the moon to shine in ; but 
 that in winter the brightest fortnight of the moon 
 comes when the northern hemisphere of the earth is in 
 the best condition for it, being turned toward the full 
 moon, and so having it highest and longest above the 
 horizon. You will see that in a minute, if you remem- 
 ber that the sun is lowest above the horizon in winter, 
 because the north pole then leans away from him ; and 
 as the full moon is opposite to the sun, the north pole 
 must lean toward the full moon when it leans away 
 from the sun. Therefore each hemisphere has as much 
 longer full-moonlight than darkness in winter as it has 
 longer sunlight than darkness in summer. The nights 
 of short moon in winter are also the pights of new 
 
Time of Greatest Moonlight. Ill 
 
 moon, when there is the least moon to lose. And the 
 contrary of all this holds in summer, -when the moon 
 is less wanted. 
 
 In consequence of the moon's orbit being 5° inclined 
 to the ecliptic or sun's apparent orbit, the moon may 
 be 5° higher above our horizon than the sun ever is; 
 or it may be so much lower. I remember a woman 
 saying that the night when she saw a man stealing 
 some fowls was the brightest she had ever seen. I had 
 the curiosity to look whether there was any real ground 
 for the remark, and was surprised to find there was ; 
 for it happened to be the night of full moon at Christ- 
 mas, and just at the time (which comes once in nine- 
 teen years) when the moon's ascending node coincided 
 with the sun's ascending node =p , and so the moon really 
 was the highest that she ever is in latitude 52^^°, viz : 
 38^° for the colatitude (or diiference between 90° and 
 latitude) + 23^° for the inclination of the equator to 
 ecliptic -F 5° for the moon's extreme elevation above 
 the ecliptic, = 6y°, or three quarters of the height 
 from the horizon to the zenith, or highest point of the 
 heavens. 
 
 ECLIPSES. 
 
 An eclipse of the moon is the moon passing through 
 the shadow of the earth from the sun, which darkens 
 the moon, as the shadow of a tree darkens the ground, 
 only much more, because there is nothing to reflect and 
 spread the light over the moon in shadow (subject to 
 what is said at p. 116), as sunlight is spread by the' 
 air in some way that no other light is. ^^ You know that 
 
112 Eclipses of Sim and Moon : 
 
 the shadow of the tree stands still, and is equally visible 
 to you wherever you stand, if you are near enough to 
 see it at all. And so you never read in the almanac 
 there is going to be an eclipse of the moon visible in 
 Spain, or l^orway, but not here ; except that of course 
 it is only to be seen by the people on that side of the 
 earth which faces the moon just then. 
 
 But eclipses of the sun are always announced to be 
 visible at certain places, and astronomers travel into 
 distant countries to see remarkable eclipses of the sun ; 
 that is, either total eclipses or annular, which I have 
 described already (p. 91). If you put a small screen 
 between you and the iire, it will only hide the fire while 
 you are in one place ; that is, the eclipse of the fire is 
 only visible to you there, and if you move to another 
 part of the room you see no eclipse. So the new moon 
 may hide the sun from England while we can see him 
 here. And that is why there are more eclipses of the 
 moon than of the sun in any one country, though there 
 are in the whole world more eclipses of the sun. 
 
 But now comes the question why is there not an 
 eclipse of the sun and moon at every new and full 
 moon ? If the moon were always in the plane of the 
 orbit of the sun or earth (whichever we like to call it) 
 there would be, and that is why that plane is called 
 the ecliptic. But she is not : the sun and earth and 
 moon could not be represented by balls floating on the 
 same water. The moon's orbit is inclined to the eclip- 
 tic 5° 9' ; and therefore the moon is only in the eclip- 
 tic, or on a level with the earth and sun, twice in each 
 lunation, when she is just rising above or going below 
 
Why not at every New and Full Moon. 113 
 
 the ecliptic at either of the two points called the nodes ; 
 and therefore eclipses can only happen when the new 
 or full moon is at or near a node. 
 
 Moreover, the nodes do not stay in the same place, 
 but keep moving backward, so that each node is nearly 
 47' less than half a circle or i8o° from the previous 
 one; or they recede 19" 19' a year, as I said at p. loi. 
 This twisting motion of the plane of the moon's orbit 
 backward is quite distinct from the revolution of the 
 line of apsides of the elliptical orbit forward in about 
 half the time. The recession of the nodes might be 
 represented by dipping a sheet of tin into -the water, 
 inclined at 5° to it, and twisting it round from left to 
 right, always keeping the same inclination ; while the 
 advance of the apsides would be represented by an 
 elliptical plate fixed loosely to the tin sheet by a pin 
 through the focus, and turned from right to left, keep- 
 ing the focus on the water. 
 
 If you put two candles close together, and hold a 
 stick between them and the wall at a suitable distance, 
 it will cast a narrow black shadow on the wall, bounded 
 by two paler ones. Within the black shadow both 
 candles are eclipsed, but in the pale shadow only one.* 
 So the earth in a lunar eclipse casts a black shadow on 
 the moon, called the unibra, within which the sun is 
 totally eclipsed to the moon, surrounded by & penumbra 
 
 * This enables ns to compare two candles ; for that one is the brightest which 
 most illuminates the shadow of the other ; therefore the right-hand candle is 
 the best if the left shado* is the darkest. A lamp with a very wide flame would 
 represent the sun more accurately, but two candles are easier to manage. 
 When the sun is narrowed by an eclipse the fringes of shadows are narrower 
 than usual on the corresponding sides of objects. 
 
114 8ola/r Eclipses. 
 
 which is a pale shadow on the moon to us, and a par- 
 tial eclipse of the sun to those parts of the moon which 
 the penumbra covers. In like manner the moon hides 
 the whole sun from the earth wherever the umbra or 
 full shadow of the moon falls in a solar eclipse ; and 
 those parts of the earth which see the sun partially 
 eclipsed would appear covered with a pale shadow to 
 the people in the moon if there were any. 
 
 But the moon being much smaller than the sun, her 
 umbra is a cone which runs to a point at 230,180 miles 
 beyond the moon when the sun is nearest, and 238,010 
 when he is furthest off; and therefore falls short of the 
 earth's surface if the new moon is near apogee, and 
 reaches beyond it at perigee, and almost exactly to 
 it at mean distances of the sun and moon (p. 92). 
 When the umbra does not reach the earth it is only 
 touched by the penumbra, and the parts of the earth 
 which are swept over by the middle of the penumbra 
 then see a partial but annular eclipse of the sun : for 
 an annular eclipse is only a particular kind of partial 
 one. Of course every eclipse begins and ends as a par- 
 tial one. The places a little way off that middle line 
 see a partial but not an annular-eclipse.* If the moon 
 is near enough for the umbra to reach the earth there 
 is a total eclipse of the sun at all the places which it 
 
 * Some pictures of eclipses in books are likely to give an erroneous idea of 
 au annular eclipse, by making tlie middle of tlie penumbra black likS the umbra, 
 ■which is contrary to the fact. No picture seems to me likely to make the mat- 
 ter clearer : in fact I had drawn one, and written a description of eclipses to 
 suit it ; but It was longer than this instead of shorter, and harder rather than 
 easier, so far as I could judge. There is a good account of the jSrincipal ' solar 
 eclipses ' under that head in the English Cyclopedia. 
 
Lwnar Eclipses. 115 
 
 covers. It can only cover a spot 147 miles in diameter, 
 or move over a band of that width upon the earth, and 
 the totality never lasts as much as 5 minutes at one 
 place. 
 
 But the earth is so large that its umbra in a lunar 
 eclipse reaches far beyond the moon, and may be as 
 much as 5950 miles wide where the moon crosses it, 
 and is never less than 5650. And as the moon is only 
 2165, there can never be anything like an annular 
 eclipse of the moon. A total eclipse may last i h. 
 45 m., and it is an hour more than that from the first 
 contact to the last, since the moon takes about an hour 
 to move across her own width. The earth's penumbra 
 is 10,200 miles wide at the mean distance of the mooUi 
 These things cannot be calculated without some geom- 
 etry, so I only give the results. Those who wish and 
 are able to do it must remember that the earth (or sun) 
 is moving as well as the moon, and therefore we must 
 take the synodical period of 708734 hours in reckon- 
 ing her' velocity with reference to the earth's shadow. 
 By that means the shadow may be treated as station- 
 ary ; and the velocity of the moon is then 2234 miles an 
 hour at perigee, where she goes fastest, and 2001 at 
 apogee, and her moan velocity 21 17; while her mean 
 sidereal velocity belonging to her absolute period of 
 655719 houi-s is 2288 miles an hour. The mean ve- 
 locity per hour is of course the length of the orbit 
 divided by the hours m which it is described ; and the 
 velocity at perigee or apogee x those distances = the 
 mean velocity X the mean distance, by one of Kepler's 
 laws which will be explained at p. 224. 
 
116 Some Phenomena of Eclipses. 
 
 The moon goes appai'ently, as well as really, east- 
 ward througL. the shadow or over the sun, because she 
 goes round the earth I3"37 times faster than the sun 
 or the shadow, and" the earth's rotation eastward * 
 only makes them both appear to move westward 
 equally. 
 
 The Moon is never Totally Eclipsed. — A lunar eclipse 
 differs from a solar one in the boundaries of the umbra 
 and penumbra being much less clearly marked. "We 
 either positively see or do not see the whole or part of 
 the sun, and the part which is not eclipsed remains as 
 bright as usual. For a solar eclipse is simplified by 
 the moon having no atmosphere to refract the sun's 
 rays as they pass by her. The only violation of the 
 sharply defined boundary of light and darkness on the 
 sun's disc is the unexplained phenomenon of Balm's 
 heads, which are beads of light appearing between the 
 edges of the sun and moon at the moment when an 
 annular eclipse opens and closes. But the penumbra 
 of the earth is only a gradually darkening shadow, 
 into which the moon cannot be seen definitely to 
 entei-. 
 
 Moreover the umbra itself is invaded and colored red 
 by rays which would pass by the earth and beyond the 
 umbra altogether if they were not refracted and turned 
 inward by our atmosphere, which they have to pass 
 through as obliquely as possible in skirting the edges 
 of the earth. And they are bent inward so much that 
 they cover the whole umbra at the distance where the 
 moon crosses it, leaving the undiluted umbra a shorter 
 
 * Appendix, Note SVI. 
 
Possible Eclipses in a Year. 117 
 
 cone which does not reach the moon. In other woi-ds, 
 tliei-e is no such thing as a total eclipse of the moon, 
 or no time when she is quite invisible and does not 
 receive some rays of the sun through the refraction 
 of the earth's atmosphere ; except that the rays are 
 sometimes stopped instead of refracted by the earth's 
 atmosphere being cloudy where they pass, and then 
 the eclipsed moon is quite invisible, and stars passing 
 behind her disappear suddenly without any visible 
 cause. 
 
 Unless a new moon'happens within 17° of a node, the 
 sun escapes being eclipsed to any part of the earth ; and 
 unless a full moon happens Avithin 11° 21' of a node, it 
 clears the earth's shadow and is not eclipsed : passing 
 through the penumbra only is not reckoned an eclipse 
 of the moon. These limits are for mean distances, and 
 are rather wider when the moon is nearer or the sun 
 further. But the line of nodes must be crossed by the 
 sun in his motion round the earth twice a year, and 
 so there are at least two opportunities for an eclipse of 
 each kind ; and there may be more. There are never 
 less than two eclipses of the sun every yeai", and there 
 can easily be four partial ones. For there are 34° at 
 each node within which one can happen ; and if the 
 moon gives the sun one partial eclipse 17° before 
 reaching the node, she will be in time to give another 
 at the next new moon, as the sun will only have moved 
 about 29° in that time. Indeed there can be five solar 
 eclipses, as there can be 13 new moons in a year, or 
 1 3 full moons, though not 1 3 of both. Therefore if there 
 is an eclipse, lunar or solar, before January 11, there 
 
118 ChaMascm Cycle of Eclipses. 
 
 may be a similar one at the end of December, if the po- 
 sition of the nodes is favorable, 'which it is the more 
 likely to be from their receding, so that the sun reaches 
 the same node again in 3 54' 72 days, or very nearly 12 
 lunations (p. 108). A total or an annular eclipse must 
 be close to a node. There can be only thi-ee lunar 
 eclipses in the year, at intervals of 177 days or 6 luna- 
 tions, as the moon can only be eclipsed once within the 
 22° at each node, and she may escape altogether. So 
 there are never more than 7 eclipses in a year, for if 
 there are 5 solar there cannot be 3 lunar, and vice 
 versa. 
 
 CYCLES OF THE MOON, AND EASTEE. 
 
 There are two cycles of the moon so nearly of the 
 same length that they are liable to be confounded, 
 though they are perfectly distinct. The first is the 
 Chaldsean SaroSj or cycle of eclipses. If the sun, 
 moon, and one of the nodes are in conjunction to-day, 
 they will all three meet again in 18 common years and 
 10 or II days. For 223 lunations take 6585'32 days, 
 and 19 synodical periods of the sun and the moon's 
 nodes take 6586'28 days; which are not quite 20 min- 
 utes short of 18 common years and 10^ or 11 -J- days 
 according as they happen to include 5 or 4 leap years. 
 Therefore after that the eclipses will recur in the same 
 order : not of course on the same days of the month, 
 but all 10 or 11 days later. There are generally 29' 
 eclipses of the moon and 41 of the sun in that time. 
 But they recur, you see, nearly 8 hours later in the 
 day ; which may make all the difference of the eclipse 
 
The Metonic Gyde. 119 
 
 being visible or not, and may transfer it from one clay- 
 to another. Therefore the Chaldseans made a more 
 perfect cycle of three Saroses, which are 54 years and 
 31 or 32 days, within an horn*. 
 
 The Metonic Cycle is so called because it was discov- 
 ered by Meton, an Athenian, e.g. 433. It has nothing 
 to do with eclipses, but is of far more importance than 
 the Saros, as the time of Easter has been fixed by it as 
 long as it has been fixed by any rule at all ; for many 
 centuries without any modification, and latterly with 
 some ; but still that cycle is the rule : — 235 lunations 
 take 6939'69 days, or so little less than 19 years of 365 J 
 days that they only differ by a day in 322 years. There 
 is indeed a whole day's difference according as the 19 
 years include 4 or 5 leap years ; and the complete cycle 
 is 4 X 19 or 76 years. But the world has always been 
 content to use only ig patterns of years at once for 
 finding the Easter moon : only they are now shifted in 
 the calendar 7 times in 1200 years. ISTeglecting for the 
 present that difference between leap and common years, 
 which corrects itself every 4 years, and the moon's com- 
 ing a day sooner in 322 years, we may say that the new 
 and full moons come again on the same days of the 
 same months every 19th year. Therefore there are 
 only 19 out of the 30 days after the vernal equinox on 
 which the equinoctial full moon can fall; and the 19 
 golden numhers prefixed to those days in the Prayer 
 book calendar mean that the day against the golden 
 number of the year is the day of Paschal moon, or full 
 moon next before Easter Sunday. 
 
 For by the rule which has existed over all the world 
 
120 The Rules fw East&f. 
 
 since the first Council of Nice in 325, Easter is the 
 Sunday after the fall moon next after the 20th of 
 March. The Council left the moon to be found as it 
 might be ; and further disputes arose on that ; which 
 were ended by Pope Hilarius in 463 ordaining what has 
 ever since been the law of church and state, that the 
 Paschal moon should not be the actual fuU moon to be 
 found by astronomers (if it ever was so), but the 14th 
 day of the moon by the Metonic cycle. 
 
 Consequently the Paschal moon often differs from the 
 true equinoctial moon by a day or two, and Easter may 
 be a week or even five weeks earlier or later than it 
 would be if it followed the real moon. Indeed, unless 
 an ' Easter meridian ' were agreed on for the whole 
 world, it might still differ five weeks in different 
 places, even if it were fixed by astronomers and there- 
 fore made incalculable either forward or backward 
 by anybody else. For if there is full moon in London 
 very early in the morning of Saturday, March 21, Eas- 
 ter would be the next day thei^e : but that same full 
 moon may be on Eriday night of Mai'ch 20 at Exeter 
 or Oxford by true time, and therefore would not be the 
 Paschal moon there, which would be on Sunday, April 
 19, and Easter not till April 26 — ^which it never is 
 now. 
 
 The correction of the Metonic cycle for the purpose 
 of keeping the ecclesiastical full moon tolerably near 
 the real one on the average involves the whole subject 
 of the reformation of the calendar, or the change from 
 old to new style, which is intimately connected with 
 astronomy. It occupies many pages in the books 
 
Reform of the Calendar. 121 
 
 which treat of it fully, but I hope to explain all that is 
 important of it in a short compass. The most complete 
 treatise ou it is Mr. De Morgan's in the ' Companion 
 to the Almanac ' for 1845 and 1846, and his article on 
 Easter in the English Ojclopedia.* There is also a 
 very clear paper about it in the Philosophical Transac- 
 tions of 1750 by the Lord Macclesfield of tliat time, 
 .which prepared the way for the Act of Parliament, 24 
 G. II., cap. 23 (1751), for changing the style in Sep- 
 tembe-, 1752. But none of these contain a table for 
 old style as well as new, nor do the old Prayer books, 
 though they had one of a different kind for old style 
 only. 
 
 THE CALENDAR. 
 
 It was known very early that a year is no exact 
 number of days ; and different nations had different 
 plans fbr occasionally adding or intercalating days, to 
 make up for the fraction lost in each year. It is not 
 worth while to go into the history of these contrivances. 
 The present scheme of three years of 365 days and a 
 fourth of 366 was invented for Julius Csesar by Sosi- 
 genes of Alexandria, b.o. 45, and lasted without altera- 
 tion (except a temporai-y mistake corrected by Augus- 
 tus) until the time of Pope Gregory XIII. It was then 
 found that the real equinox fell 10 days before the 
 nominal one of March 21, and that Easter had got four 
 
 * Yon wfllfiiid there, and in Eees's Cyclopedia, a great deal also abont the 
 epact, or age of the (ecclesiastical) moon at the beginning of the year. But we 
 can do as well without it, and the subject wants no superfluous complication. 
 Mr. De Morgan also published a ' Book of Almanacs,' containing the 35' patterns 
 of years for the 35 possible days of Easter, according to the rules : if they fol- 
 lowed the real moon there would be 36 almanacs. 
 6 
 
122 The Gregoriom Oalenda/r is Wrong. 
 
 days wrong besides, from the error in the Metonie cycle. 
 Ten days were accordingly struck out of the calendar 
 between the 4th and isth of October, 1582; and to 
 prevent the error for the future it was decreed that 
 every looth year should lose its leap day except those 
 divisible by 400. 
 
 This fs the Gregorian or !N"ew Style, which was in- 
 vented by Olavius, a Jesuit, and adopted in all the Ko- 
 man Catholic countries before the end of 1582, and by 
 the Protestant German states in 1 700, but not by us 
 till 1752, and Sweden the year after, and not yet by 
 Eussia and the Greek Church, which are therefore now 
 12 days wrong. In 175 1 Parliament enacted that the 
 day after 2 September, 1752, should be September 14 : 
 dropping thus 1 1 days, because we had got one day 
 more wrong by allowing 1 700 to be a leap year. Of 
 course due provision was made that nobody should lose 
 or gain 1 1 days' interest on their debts ; nevertheless 
 the Act caused riots among the common people, who 
 cried out ' Give us back our eleven days,' as if they 
 would die so much the sooner for the loss of them. 
 Fortunately they were not masters. 
 
 Moreover you must remember in reading old books 
 that the years legally began on March 25 until the 
 change of style ; and therefore such an event as the 
 execution of Charles I. on "Tuesday, 30 January, 1649, 
 as we call it now, is said in old books to have been in 
 1648. A 3'ear of the average length of 365^ days is 
 called a Julian year ; but the years of the Julian era 
 are reckoned from i Januaiy, 4713 B.C., an arbitrary 
 epoch which was invented for reasons of no consequence 
 
Better Plans for a Calenda/r. 123 
 
 now ; so 1867 was the 658otliof the ' Julian era.' For 
 there was no a.d. o, and the 19th century of our era 
 (which probably begins 4 years after the real birth of 
 Christ) began on i January, 1801, not 1800. 
 
 To turn old style into new add 1 1 to the day of the 
 month in the century beginning 18 Feb., 1700; which 
 becomes i March, as that February would have no leap 
 day by !N". S. : add 10 days in the two centuries from 19 
 Feb., 1500, 9 from 20 Feb., 1400, and so on. 
 
 Now let us see what amount of error the Gregorian 
 calendar still leaves, after professing to correct itself 
 every 400 years. A million Julian years are 365,250,- 
 000 days; but a million equinoctial years are 365,242,- 
 216 days. Therefore the problem is how to drop 7784 
 leap days in a million years in some neat and simple 
 way : that is, one day in 1 28*47 years. But the Gre- 
 gorian plan drops a day in I33'333 years in the long 
 run. Therefore the error is 4'86 days in I28'47 x I33" 
 •333 years, or a day in 3524 years. Accordingly Sir J. 
 Herschel proposes to carry the cori'ection a step further 
 by making every 4000th year lose its leap day, though 
 it is divisible by 400 ; which would go for about 28- 
 000 years without an error of a day. 
 
 But I think the mere statement of the problem sug- 
 gests the best solution of it : 128=32 x 4 ; and that is 
 a number easy to remember, being formed by 7 succes- 
 sive duplications of i, or 2'. Therefore the simplest 
 of all plans would be just to let every 128th year lose 
 its leap day ; and that only makes a day Avrong in 35,- 
 440 years ; which is ten times more accurate than the 
 Gregorian scheme, and seven times better than another 
 
124: Oregoriom Rule for Easter.- 
 
 wMch is commended by Sir J. Herschel, for dropping 
 one day in 132 years (misprinted 128) by postponing the 
 leap day of every 32nd year to the 33rd ; which would 
 also make the leap years no longer all divisible by 4. 
 
 Even if reckoning by centuries is thought essential, 
 still the Gregorian scheme is not the best. It would 
 be both more correct and more symmetrical to drop 
 the leap day of every century except the fifth, instead 
 of the fourth.* For that makes leap years of 2500, 
 3000, etc., instead of 2400, 2800, 3200, and only accu- 
 mulates an error of a day in 4646 years : which again 
 could be rectified in 5000 years more completely than 
 the Gregorian error in 4000. 
 
 Correction for Easteri — The next question was how 
 Easter was to be set right, and the Metonic cycle modi- 
 fied for the future, to prevent the ecclesiastical and 
 real moons from getting wider and wider apart. For 
 that purpose they wanted a new set of pattern years, 
 to be indicated by the 19 golden numbers, and also a 
 contrivance for shifting them again when they had got 
 a whole day wrong permanently. The golden num- 
 bers of successive years have always run regularly 
 from I to ig without any dislocation. So the, first 
 thing was to put them against a new set of days be- 
 tween March 21 and April 18 (inclusive), and that ar- 
 rangement was to last till the year 1700. Then they 
 are to be shifted on this plan : at every century divisi- 
 
 * Priar Bacon In J167 urged Pope Clement IV. to reform the calendar on a 
 plan equivalent to this. He had found out somehow the length of the year 
 more exactly than anybody else then or for a long time afterward. But he was 
 put in' prison for ten years instead, a fact which has been merged in the greater 
 fame of Galileo. 
 
A^ustment of the Golden Numbers. 125 
 
 ble by neither 3 nor 4 they are all to.be advanced a 
 day in the calendar, and to be set back a day in every 
 century divisible both by 3 and 4, i.e., by 12, and not 
 altered in the others. 
 
 The reason of that contrivance was this. Suppose 
 the sun and moon to be right at some century divisible 
 by 1 2, which we may call o for this purpose. Then at 
 the year 100 the Gregorian year loses a (leap) day, and 
 so the moons of the next century will all come a day 
 later than if that day had not been dropped, and the 
 golden numbers must be advanced a day accordingly ; 
 and the same at 200. At 300 the year again loses a 
 day; but by that time the JMetonie cycle has put the 
 nominal moon nearly a day too forward, and therefore 
 ought to lose a day ; and so they balance without any 
 alteration. In 400 the year does not lose a da_y, and 
 the moon does not, and so again they are right ; and 
 thus you may go on till 1200, which drops no day ; but 
 the moon has then lost a day since 900, and so the 
 golden numbers have to be put back. But Clavius 
 supposed the moon to lose a day by the Metonic cycle 
 in 300 years more nearly than it does. 
 
 Tou may see the result of all this in the Prayer book 
 in II and III of the ' General tables for finding the 
 Sunday letter and the places of the golden numbers,' 
 for as long as the world and the Gregorian rules may 
 last. The small irregularity in the last three days of 
 possible Paschal moons, which you will see in Table 
 III, was introduced for an ecclesiastical reason not 
 worth explaining here, and not affecting the principle 
 of these calculations. The smaller numbers by the 
 side of the centuries in Table II show at once how 
 
126 Error in the Rules for Easter. 
 
 many days the golden numbers are advanced on the 
 whole since 1600, which will evidently be 5 days in 
 1200 years on the average, being set back once in that 
 time, and forward 6 times. 
 
 Now let us see how nearly this keeps Easter right, 
 assuming it to have been set right in 1600. Though 
 235 lunations take ih. 26m. less than 19 Julian years, 
 they take ih. 46m. more than 19 equinoctial years, so 
 that the moons of every 19th year, or of every year 
 with the same golden number, come a day later after 
 213^ years. In. other words, if our years were correctly 
 adjusted, by dropping two days iu 257 years, a new or 
 full moon of March 21 now would fall on March 22 in 
 any year of the same golden number after 213 years, 
 and on March 23 after 427 years, and so on. So that 
 instead of the golden numbers advancing at the rate 
 of 5 days in 1200 years, they ought to advance 5 days 
 in 1067 years. Again 235 lunations exceed 19 Gre- 
 gorian years (whose average length is 365*2425 days) so 
 much that the moon advances a day in 227J of those 
 years, or 5 days in ii37andnot 1200 years : so that 
 the Gregorian rule does not keep the moon right even 
 for the average Gregorian year, which is itself wrong. 
 
 The result of all this is that the rules for keeping 
 Easter neither keep it by the real equinoctial moon of 
 each year, nor by a moon which is right on the average 
 of a long period, either for the real equinox, or for the 
 artificial year which we adopt, and sometimes leave it 
 five weeks off the real time. Indeed it is very doubtful 
 whether Easter is not always from 2 to 36 days later " 
 than the real anniversary of our Lord's resurrection.* 
 
 * See a ftill dieciission of this point in Brown's Orclo Scedorum. 
 
To jmd the Days of the Week in History. 127 
 
 It seems that even Clavius the J esuit, who did the 
 astronomical work of reforming the calendar for Greg- 
 ory XIII., ventured to publish the suggestion that it 
 would be better to make Easter always the Sunday 
 after March 21 than to let all the great festivals and 
 holidays except Christmas wander over five weeks of 
 the calendar in the vain attempt to follow the moon. 
 Perhaps before the year 2000 the world will be in a 
 condition to revise the calendar and re-consider that 
 question without prejudice. 
 
 To find the Days of the Week. — But whatever is done 
 hereafter, Easter and the days of the week for given 
 days of the month can only be found for the past and 
 the present by the existing rules, and so it is important 
 to understand them. The table in the Prayer book 
 for finding the Sunday letter, or the day of the week 
 for any given day of the month, is silent until 1600, 
 and wrong from 1600 till September 1752. They cop- 
 ied the Gregorian tables into the Act of 1751 without 
 taking the trouble to adapt them to this country. The 
 proper table and rule are these : 
 
 A 1800. 
 
 1 G 14 Sep. 1752 to 1799. 
 
 2 F. 2500. 
 
 3 E 2400, 2300. 
 
 4 D 2200. 
 
 5 C every year of old 
 
 style in all countries. 
 
 6 B 2000, 1900. 
 
 Rule for Finding the Sunday Letter.— Add to the year 
 
 For countries which 
 changed their style 
 in 1582 it must be, 
 
 1 G 1700. 
 
 2 F 1583 to 1699 : 
 the rest as before. 
 
128 Rule for finding Easter. 
 
 its fourth part, omitting fractions, and also the num- 
 ber set opposite to it or to the last century before it in 
 this table, and then divide by 7 : the remainder over 
 indicates the Sunday letter. But until the end of 
 February in leap years the letter above the one so in- 
 dicated is the Sunday letter. When that is found you 
 have only to look at the Prayer book calendar, where 
 all the days marked with that letter are Sundays in 
 that year. The reason of the rule is that a common 
 year ends on the same day of the week as it begins, or 
 has one day over the 52 weeks, and a leap year two 
 days. Therefore starting from o, there are as many 
 days over beyond some entire weeks as the niunber of 
 the year + the number of leap years, and as many days 
 of the week over as the remainder after dividing those 
 days by 7. The arrangement of the letters in the cal- 
 endar, beginning with A for January i, is only arbi- 
 trary, and happens to require the constant addendum 
 of 5 to make the Sundays of all the years until the 
 change of style come right. After 1752 the di-opping 
 of leap day in every century not divisible by i dis- 
 places the Sunday letters of the next 100 years by one 
 day. 
 
 To Find EastePi — ^Either of the tables in the Prayer 
 book for this purpose is right since 1752. I have made 
 a similar one for old style, which is right for every 
 country until it changed, and therefore here till 1753 ; 
 for Easter of 1752 went by old style. First come all 
 „the possible days of the Paschal moon, and therefore 
 of the golden numbers, and a week more to reach a 
 Sunday when that moon is on Sunday, April 18. Then 
 
Rule for finding Easter. 129 
 
 come the Sunday letters of those days according to the 
 calendar. The column headed O. S. has the golden 
 numbers as they stood until the change of style. The 
 next has them up to 1899 (inclusive) from 1753 in 
 England, and from 1700 in Roman Catholic countries ; 
 and the following one from 1900 to 2199, unless the 
 rule is altered before then. The column headed 1583 
 to 1699 is only for the countries which changed in 1 582 
 (see English Cyc. Easter). 
 
 The way to use the table is this. Add i to the year 
 and divide by 19, and the remainder is the golden 
 number, no remainder corresponding to 19. Find that 
 in the column which the year belongs to, and run your 
 eye horizontally back to the day of the month opposite 
 to it, which is the Paschal moon, or the 14th day of 
 the moon according to the epaot (p. 121, n.). The 
 Sunday after it is found by the rule which I gave for 
 the Sunday letter, and that is Easter Sunday. 
 
 6* 
 
130 
 
 Table for finding Easter. 
 
 Possible days 
 
 
 1753 
 
 1900 
 
 1583 
 
 
 
 ' of Pull Moon 
 
 0. S. 
 
 to 
 
 to 
 
 to 
 
 C5 
 
 and Eastei 
 
 •■ 
 
 
 1899. 
 
 2199 
 
 1699. 
 
 
 <^k 
 
 March 21 
 
 c 
 
 16 
 
 14 
 
 
 3 
 
 
 
 2a 
 
 D 
 
 5 
 
 3 
 
 14 
 
 
 
 • . •« 
 
 ■ *3 
 24 
 
 B 
 F 
 
 13 
 
 II 
 
 3 
 
 II 
 
 fe 
 
 
 25 
 26 
 
 A 
 
 2 
 
 • • 
 
 1 1 
 
 19 
 g 
 
 
 
 
 19 
 
 
 H 
 
 s"? 
 
 27 
 
 B 
 
 10 
 
 g 
 
 19 
 
 
 28 
 
 29 
 
 
 D 
 
 \i 
 
 i6 
 
 8 
 
 16 
 
 S 
 
 ^1 
 
 30 
 31 
 
 E 
 F 
 
 7 
 
 5 
 
 16 
 
 s 
 
 13 
 
 
 
 
 
 April I 
 
 G 
 
 15 
 
 13 
 
 
 2 
 
 p 
 
 •§1^ 
 
 
 
 
 
 
 
 2 
 
 A 
 
 4 
 
 2 
 
 13 
 
 
 
 ^1^ 
 
 3 
 
 B 
 
 , , 
 
 
 2 
 
 10 
 
 
 4 
 5 
 
 
 B 
 
 12 
 I 
 
 
 
 
 
 
 I 
 
 10 
 
 18 
 
 
 
 6 
 
 E 
 
 
 I'g 
 
 
 7 
 
 
 
 7 
 
 P 
 
 9 
 
 7 
 
 18 
 
 
 
 
 
 
 
 
 
 
 
 ^1-3 
 
 8 
 9 
 
 G 
 A 
 
 17 
 
 15 
 
 7 
 
 IS 
 4 
 
 
 
 
 
 10 
 
 B 
 
 6 
 
 4 
 
 .15 
 
 
 
 , 
 
 II 
 
 
 
 
 
 4 
 
 12 
 
 
 ^"^ 
 
 12 
 
 D 
 
 14 
 
 12 
 
 
 I 
 
 ffl 
 
 !l§ 
 
 13 
 
 E 
 
 3 
 
 I 
 
 12 
 
 
 
 "i-= 
 
 14 
 
 P 
 
 .. 
 
 
 I 
 
 9 
 
 
 
 IS 
 
 G 
 
 II 
 
 9 
 
 
 
 
 bSi ' 
 
 16 
 
 A 
 
 
 
 9 
 
 17 
 
 <J 
 
 s 
 
 17 
 
 B 
 
 19 
 
 17 
 
 17 
 
 6 
 
 
 S « 
 
 ig 
 
 C 
 
 8 
 
 6 
 
 6 
 
 14 
 
 
 40 
 
 19 
 
 D 
 
 ^ 
 
 
 
 
 
 20 
 
 E 
 
 ^ 
 
 OS OS 
 OS OS . 
 
 
 
 >. 
 
 21 
 
 F 
 
 > . 
 
 00 ., 00 r!3 
 
 •Si 
 
 n 
 
 "■"2 9 
 
 22 
 23 
 24 
 
 G 
 A 
 B 
 
 03 03 
 
 
 
 
 
 »s 
 
 C 
 
 s 
 
 
 1^ 
 
 f§ 
 
The Forces which jpro&uce the Tides. 131 
 
 THE TIDES. 
 
 Most people know as much about the tides as this, 
 that they are somehow caused by the moon, and rise 
 about 50 minutes later every day because the moon 
 comes to the meridian so much later. But they are 
 due to the sun also in exactly the same way, only not 
 so much because of his greater distance. You know 
 that every particle of the earth and water is attracted 
 toward the suu with a force directly proportioned to 
 his mass, and varying inversely as the square of his dis- 
 tance (p. 28). Therefore he attracts the water on the 
 near side of the earth more, and that on the far side 
 less, than the solid mass of the earth, which may be 
 considered condensed at its own centre. And that 
 comes to the same thing as if the earth's attraction both 
 on the waters nearest the sun and farthest from him 
 were a little diminished. 
 
 But that is not all. If you pull two balls not far 
 apart with long strings of equal length held in one 
 hand, you wiU also pull them toward each other, with 
 a force which varies as the angle between the strings 
 (so long as it is a small one), or as the distance of the 
 balls apart divided by the length of the strings. In the 
 same way the sun's attraction tends to squeeze in all the 
 waters which lie at or near 90° from the point facing 
 the sun, or to increase the earth's attraction on them. 
 But this contractive force, or the resolved part of the 
 sun's force on the sides of the earth, toward the centre, 
 is only half the other sepai-ating or differential force, as 
 I will show you presently. Therefore if you call the 
 
132 Caloulation of tJidr Amoimt. 
 
 contractive force i, there is altogether a force of 3 tend- 
 ing to make the water facing the sun, and at the back 
 of the earth opposite to the sun, higher than the water 
 90° from those places. 
 
 Tlie moon does the same in all respects, and in a 
 greater degree ; for although the general attraction of 
 the moon on the earth is very small compared with 
 the sun's, yet her differential attraction and her con- 
 tractive force on the opposite sides of the earth are 
 greater, because she is so much nearer, and both these 
 forces depend on the proportion of the earth's radius 
 to the distance of the sun and moon respectively. If 
 you like to see the calculation of the actual amount 
 and effect of the tidal forces of the sun and moon, and 
 the proportion which they bear to gravity, or to the 
 earth's attraction on its own water, it can be done as 
 follows. 
 
 Calling the earth's radius and mass each i for short- 
 ness, the sun's distance is 23,064 (p. 62), and his mass 
 316,560. Therefore by the same kind of calculation 
 as at p. 65, his attraction on the earth's centre is 
 
 ? — ^^ -2- ^ ; and for his attraction on the near 
 
 23,064' 
 
 and the far sides of the earth we must put 23,063 and 
 23,065 respectively for his distance. Then his differ- 
 ential force on the water at each of those places is 
 the difference between his attraction there and at the 
 centre ; which you will- find, if you take the trouble to 
 work through the figures, in both cases very nearly= 
 
 316,560 gravity x 2 . , ,■,. ^ ^ . 
 
 - — — — 2--_,_^i . And as this 2 represents twice 
 
 23,064= 
 
OalculaUon of the Height of the Tides. 133 
 
 the earth's radius, the sun's differential force is to 
 gravity as twice the earth's radius x sun's mass is to 
 the cube of his distance. Similarly the moon's differ- 
 ential force is to gravity as twice the earth's radius 
 X moon's mass ('0123) is to the cube of her distance 
 (60O. 
 
 And to each of these half as much more has to be 
 added for the contractive force ; for that is the general 
 attraction of the sun or moon on the earth x the small 
 fraction which has the earth's radius or i for numer- 
 ator and the sun's or moon's distance for denominator. 
 Thus the same cubes of distance come in as before, but 
 not the 2 in the numerator ; and the whole tidal force 
 
 „ sun 316,560x3 , „ moon -0123x3 
 
 .of r-=- — ^ . ; and of 7—= — A-^- 
 
 gravity 23,004^ gravity 60^ 
 
 You would find if you worked out these figures, that 
 gravity is nearly 6 million times the moon's tidal force, 
 and 13 million times the sun's, at mean distances; or 
 the attraction of the earth on its own water is 4 million 
 times greater, and the centrifugal force at the equator 
 (p. 37) 13,800 times greater, than the tidal forces of 
 the sun and moon together. 
 
 If the earth were a fluid sphere of nearly 21 million 
 feet radius, and of uniform density = the present aver- 
 age density, the tidal force at every depth would be 
 the same 4 millionth of the central attraction, since 
 they both vary as the distance from the centre (p. 29). 
 And as the weight of a prolate spheroid is to the sphere 
 which it contains as their different axes, the tidal force 
 would pull out the sphere into a spheroid whose semi- 
 axis major exceeds the minor by Si feet and lies in 
 
134 The EartKa Tidal Force on the Moon. 
 
 the line ofsyzygy, pointing to the sun and moon. But 
 the inside density being fiye times that of the water 
 outside, attraction decreases inward less than the tidal 
 force, or proportionately increases ; and the result of 
 that and the solidity of the earth is that the tidal ellip- 
 ticity due to sun and moon together is only a 6 mil- 
 lionth, or the highest tide is 3f feet above the lowest 
 in the open sea. 
 
 You would find also that the moon's force on the near 
 side of the earth is rather more, and on the far side 
 rather less, than the fraction at page 133: in fact they 
 are about in the proportion of 21 to 20. There is no 
 such difference for the sun, because his distance is 
 23,064 times the earth's radius, instead of 60 times. 
 
 There is the same kind of excess in the force of the 
 earth on the near side of the moon ; but though the 
 earth is 81^ times heavier than the moon, yet the moon's 
 radius is only the 220th of the earth's distance, and the 
 excess of attraction on the near side of the moon over 
 the far side is only one 7Sth, and therefore quite inade- 
 quate to account for its supposed shape (p. 97), though 
 enough to have made her a slightly prolate spheroid 
 when she was fluid. The whole tidal force of the earth 
 on the moon is about 120 times that of the moon on the 
 earth, being greater as the earth's mass and gravity on 
 the surface exceed those of the moon, and less in the 
 proportion of their diameters. 
 
 If you cannot follow these calculations you may ac- 
 cept it as proved that the tidal forces of the sun and 
 moon are as their masses directly and the cubes of their 
 distances inversely. And their distances vary enough 
 
Weight of the Moon Calculated. 135 
 
 to make a considerable difference in these proportions 
 at different times. "When the sun is at his nearest 
 and the moon at her furthest, he is 360 times further 
 off, and the cube of that is 46 millions ; but when she 
 is nearest and he is furthest his distance is 410 times hers, 
 of which the cube is 6g millions. So we have those two 
 cubes in favor of the moon to set against 26 millions 
 for the sun, who is so much the heaviest. Therefore 
 the sun's tidal force varies from rather more than half 
 to rather less than a third of the moon's. 
 
 Weighing the Moon by the Tides. — And on this an 
 important result is founded. We have been assuming 
 the moon's mass to be known and calculating the pro- 
 portions of the tidal forces from it. But in fact it was 
 just the contrary. The moon's mass was first ascei'- 
 tained from observations of the difference of the lunar 
 and solar tides, i.e., of spring and neap tides at various 
 places (which I will explain presently), though it has 
 since been done by other methods (see pp. 55, 248). 
 Newton from imperfect measures of the tides made the 
 earth 40 times as heavy as the moon, and Laplace 70 ; 
 Mr. Airy called it 80 in 1856. Mr. Adams's figure is 
 8 1 "5, making the moon '0123 of the earth, as I put it 
 just now, an easy figure to remember. Sir J. Herscher 
 lowers the moon to one 88th of the earth ; and you will 
 see some intermediate results at pp. 248, 250. The 
 moon's mass is not affected by the late alteration of the 
 mass and distance of the sun ; for the sun's tidal force 
 remains the same as before, his mass and the cube of 
 his distance being reduced equally. 
 
 The attraction of the earth then being diminished 
 
136 What makes the Tides Visible. 
 
 in the line of centres of the earth and moon, and in- 
 creased in all directions across that line (omitting the 
 sun for the present), the water takes the form of a very 
 slightly prolate spheroid pointing toward the moon : 
 the elevation of high above low water, or the ellipticity 
 of the spheroid, being just enough for the weight of the 
 elevated water to balance the loss of ordinary gravity 
 of the water by the tidal attraction. That is, it would 
 be a prolate spheroid if the earth were not itself a far 
 more oblate one, on which the tidal protuberance of 
 only a few feet is superimposed ; but in calculations 
 about the tides the earth is always assumed to be 
 spherical for simplicity. And the sun tends to form 
 another spheroid pointing toward him. 
 
 But all this might take place and yet hardly any tide 
 be visible, if the earth always kept the same face to- 
 ward the moon, as she does to the earth. There would 
 then be only a solar tide about a foot high, which would 
 also move so slowly round the earth that its effects 
 would be quite different from what we see now. The 
 ebb and flow of the tide, by which alone it is felt as a 
 great power over the world, depends upon the earth's 
 rotation within the water, while part of it is held up by 
 tlie tidal force. The easiest way to understand the 
 effects of the rotation is to suppose the earth iixed, and 
 the sun going round it from east to west in 24 hours, 
 and the moon in 24h. 49m. The moon then drags the 
 two opposite tidal waves after her at the rate of 1000 
 miles an hour at the equator, leaving the sun's action 
 out of the question for the present. 
 
 Not that the water itself moves at anything like that 
 
Imnar and Solar Tides. 137 
 
 speed, or that mucli of it is carried round the earth at 
 all, except in long periods. The thing that travels 
 with the moon is the two alternate states of elevation 
 and depression of the water at 90° apart. A wave is 
 the transmission of a state, not of a body. The water 
 is indeed moved to the very bottom of the sea, and a 
 good deal of it moves forward, and some back again 
 afterward, besides being lifted and let down again. 
 Although the tidal wave travels westward with the 
 relative motion of the moon, the tide itself moves toward 
 an eastern as well as a western shore, because that is the 
 necessary effect of the whole mass of water rising. And 
 when the advancing water is stopped by land it can only 
 dispose of itself by rising much higher than the 3 feet 
 of the open sea. Waves raised by a wind stir the 
 water to a very little depth, and not much water is 
 carried forward in them. They ' break ' on a shore 
 because the friction of the ground stops the bottom of 
 the water from going as fast as the top, which therefore 
 tumbles over. 
 
 Neap and Spring Tides.— The sun has his two tidal 
 waves as well as the moon, but of less than half the 
 size ; and therefore it is best to consider the tide as 
 mainly belonging to the moon and modified by the 
 sun, as follows : At half moons, or, guadraiure, when 
 the sun is 90° or 6 hours from the moon, they pull 
 across each other, and the sun tries to make high water 
 where the moon is making low water. The moon's tidal 
 force being more than twice the strongest, prevails, but 
 the tide is only due to the difference of the two forces, 
 and so rises and falls least, and that is called n-ea^ tide. 
 
138 The Tides do not quite follow the Moon. 
 
 When the moon is past quadrature and has not 
 reached syzygy, or the line of new and full moon, the 
 tide is kept in advance of it by the sun; but after 
 syzygy the tide lags behind the moon, being kept back 
 by the sun. Consequently the tide of any place is not 
 regularly 49 minutes later every day, as if it obeyed 
 the moon only, but sometimes as much as an hour later, 
 and sometimes only 38 minutes. This is called the 
 jpriming and lagging of the tides. But when the sun 
 and moon are in syzygy, either in conjunction or op- 
 position, they augment each other's tidal force and 
 produce spring tides, which are the sum of the lunar 
 and solar tides, and rise the highest and fall the lowest. 
 And these again are greatest at the equinoxes, because 
 then the sun is on the equator and the moon must be 
 within 5° of it, and so they are in the best position for 
 drawing the water from the sides to the front or back 
 of the earth. For if you wanted to pull a sluggish 
 globe round, you would wrap a string round it at the 
 equator and pull in the plane of the equator. 
 
 The top of the tidal wave however does not really 
 point to the moon at spring tides, but 45° or 3 hours 
 behind it in the open sea, and much more where it is 
 obstructed by land. For the inertia and friction of the 
 water takes some time to overcome, and so the effect is 
 always behind the cause. Spring tides are also a day 
 or two after new and full moon, because the tidal force 
 keeps accumulating for several days while the sun and 
 moon are near together, and there is a greater amount 
 of it in the four days with syzygy in the middle than 
 in the four days before syzygy. So the hottest and 
 
Variations of the Tide Tyy Land. 139 
 
 coldest weather is after and not at the solstices. The 
 more the tide is impeded by laud the longer it natu- 
 rally is behind the proper astronomical time : in Lon- 
 don it is two days behind. Sometimes it has to come 
 round islands, and is divided into two streams : conse- 
 quently there are places where two tides come by roads 
 of different lengths, and so rise and fall 4 times a day ; 
 and others where the low tide by one road neutralizes 
 the high one by the other. 
 
 In running up gradually narrowing channels it rises 
 much higher tlian on the sea shore ; as high as 50 feet 
 above low water at Bristol, and in some parts of the 
 world 100, though it is only about 12 feet generally on 
 an open shore. Sometimes the tide rolls up a river 
 which gets gradually narrower, when the wind helps 
 it, with a face like a wall and the velocity of a railway 
 train, upsetting everything in its way. This is called 
 the l)ore in the Severn and Avon and some other rivers, 
 and the eager in the Humber, and it is far greater in 
 ^ome American and Asiatic rivers. On the other hand, 
 when the tide has to make its way into a large sea 
 through a narrow passage, like the Straits of Gibraltar 
 into the Mediterranean, it is unable to produce any 
 sensible rise and fall over such a sea. 
 
 But the tide sweeps rapidly over wide and level 
 sands, so as to overtake and drown people sometimes, 
 because a rise of a few inches then runs over a great 
 area of sand ; and it becomes soft under the water, like 
 a bog, or ' quick,' because moving water lifts and car- 
 ries sand and stones along with it, according to their 
 smallness and (probably) the square of its velocity. For 
 
140 DisUi/rbcmces of the Moon. 
 
 the weight of the stones increases as the cube of their 
 diameter, but the surface only as the square, and the 
 power of the stream to move them varies directly as 
 their surface and inversely as their weight ; and there- 
 fore varies inversely as their diameter, or as the cube 
 root of the weight, among stones of the same specific 
 gravity and general shape. A river goes on rising for 
 some time after ' slack water,' when things cease to 
 float upward, because the natural flow of the river 
 downward balances the tidal flow upward, but both 
 raise the water. 
 
 DISTrjEBANOES OF THE MOOIT. 
 
 There is scarcely one element of the orbits of the 
 planets or their moons that is not subject to continual 
 disturbance, by the attraction of every other body which 
 is large enough and near enough to afiect them sensi- 
 bly. All these disturbances in one way or other ulti- 
 mately compensate themselves : some of them first 
 moving the body, or its orbit, a little in one direction, 
 and then an equal distance the other way ; others pro- 
 ducing recessions or advances of nodes or apsides, 
 which in time work i-ound ; and there is one remarka- 
 ble acceleration of the moon, which has such. aiiJfenor- 
 mously long period that it may be said to/'int^fease 
 perpetually, though the time will come for it to change. 
 It is quite beyond the scope of an elementary book like 
 this to describe all the inequalities (as they are called) 
 of the moon alone, to say nothing of the planets. 
 Here and there I must notice a few of them, as I have 
 already precession and nutation — a disturbance upon 
 
Moon^s Secular Acceleration. 141 
 
 a disturbance of the earth. I will only select the most 
 important of them for explanation, leaving you to pur- 
 sue the inquiry in Sir J. Herschel's Astronomy, or Mr. 
 Airy's Gravitation, which I think generally clearer in 
 its treatment of this difficult subject ; or in Newton's 
 Principia, where the problem was first solved. One 
 thing will be noticed here which was not known when 
 those books were written. 
 
 Moon's Secular Acceleration. — 'We saw at p. 43 that 
 the minor axis of the earth's orbit has been long 
 increasing ; and it will increase for 24,000, years yet, 
 while the major axis remains permanently u^naltered. 
 If so, the sun's average (not mean) distance from the 
 earth and moon of coui-se increases, and his power to 
 disturb the moon decreases. Now let us see what that 
 disturbance does. The sun attracts the new moon 
 more than the earth, and the full moon less, because of 
 their difierence of distance : therefore at both syzygies 
 the sun's differential force practically diminishes the 
 earth's attraction on the moon. When they are equi- 
 distant from the sun he draws them closer togethei', as 
 you would two separated balls by pulling them with 
 strings of equal length. But this contracting force 
 may be proved to be only half as great as the differen- 
 tial force, as in the similar case of the tides (p. 1 3 1). 
 At intermediate places both forces are evidently less ; 
 and at certain points nearer quadrature than syzygy 
 they balance each other. Therefore on the whole the 
 differential force greatly preponderates, and weakens 
 the earth's attraction, and so enlarges the moon's 
 orbit, and therefore her time of performing it is longer 
 
142 The MoorCs Annual Equation. 
 
 than if there were no sun. But as his power of thus 
 retarding the moon slowly decreases with the increase 
 of his average distance, she is compa/ratwely accelerat- 
 ed ; and also comes gradually nearer to the earth — ■ 
 about 4 inches a year. 
 
 As this retarding force is greatest in winter, when 
 the sun is nearest (p. 41), the moon fallsmost behind 
 her mean place in April, after half a year's excess of 
 retardation, and similarly advances 11' 12" before it 
 in October. This is called the annual equation ; but 
 the mean place here referred to is the mean elliptical 
 place, which itself is found by applying the equation 
 of the centre (p. 41) to the mean place she would have 
 if she moved in a circle. Its greatest amount is 6° 18', 
 which is equivalent to I2h. 24m. of time. 
 
 But now comes a most remarkable result of the lat- 
 est investigation of this small advance of the moon of 
 only 12" in a century; the meaning of which is that 
 the moon is 12" past the line of syzygy at the time 
 when she would just be there if she had kept her mean 
 velocity of 100 years before. Or turning seconds of 
 space into time, as she moves almost exactly 1" in 2 
 seconds, you may say that the full moon comes 24 sec- 
 onds too soon at the end of every century. This was 
 first ascertained by Halley in 1693, from a comparison 
 of old observations, but not accounted for until Laplace 
 explained, it, as I have described, nearly a century after- 
 ward. And all the astronomers who followed him 
 considered his calculations complete, though calcula- 
 tions of this kind are only approximate, and made 
 on the principle of taking into account all the quanti- 
 
Correction of the Secular Acceleration. 143 
 
 ties which are not too small to be appreciable. But 
 Mr. Adams taking up the matter afresh in 1853 dis- ' 
 covered that they had all disregarded something which 
 turned out by no means too small to be appreciable 
 when properly examined ; for it was large enough to 
 reduce the accelerating effect of the increase of the 
 minor axis of the earth's orbit by one-half. Thus half 
 of the known acceleration of tlie moon was again left 
 unaccounted for ; and where is it to come from ? 
 
 That question has been answered mathematically by 
 M. Delaunay, as it had been generally by Mayer and 
 others before ; and his explanation is now received even 
 by our Astronomer Koyal, who wrote an elaborate 
 paper which he said proved Delaunay's three principal 
 conclusions to be wrong ; but he sent an '■addendum'' 
 to it afterward, saying he found that they were right.* 
 It is remarkable that Le Yerrier, Hansen, and other 
 foreign astronomers, altogether rejected for a good 
 while Mr. Adams's correction of the liinar acceleration. 
 But mathematical truth can afford to wait for recogni- 
 tion : it may lie a long time at the bottom of the well, 
 but once brought up it never sinks again. 
 
 The theory now is that the earth's rotation is get- 
 tins slower from the friction of the tides over the 
 earth, and also from the drag between the moon and 
 the earth and water in keeping up the tidal wave con- 
 tinually in a fresh place. For the earth turns east- 
 ward while the tides stay behind looking at the moon, 
 and more water moves westward than eastward over 
 the earth (p. 136); and all the friction of the water 
 
 * See Astronomical Society's notices for April, 1866. 
 
144 Heta/rdation of the Ea/rth iy the Tides: 
 
 moving westward uses up some force of the earth's ro- 
 tation. 
 
 Even supposing no water to move over the ground, 
 the mere elevation of the tidal wave does not run as 
 freely as if the water had no internal friction and did 
 not cling to the earth at all. An east wind sending 
 Avaves along a field of corn transmits some force to the 
 ground, and would move the field very slowly west- 
 ward if it were afloat. And if the earth had had no 
 original rotation, the moon di'agging the tidal wave 
 round it would by this time have given it some rota- 
 tion.* Therefore the i-otation eastward is diminished, 
 and some of the force of rotation is used up in main- 
 taining the tide. 
 
 This has been otherwise explained by saying that 
 the tidal protuberance, like a mountain always stand- 
 ing out about 45° eastward of the moon (p. 138), serves 
 as a lever or handle for the tidal attraction to take 
 hold of and retard the earth's rotation, ' the water of 
 the ocean being partly dragged over the earth as a 
 brake.' But this either suggests the idea of the force 
 and the work to be done being much greater than it is, 
 or else leaves it unexplained how any drag at all is 
 transmitted through the water to the earth, apart from 
 the direct friction spoken of before. If the moon flew 
 away the tides would go on nmning round the earth 
 or rising and falling everywhere, for a time, but would 
 gi-adually wear out ; and the foixje we are considering 
 13 that which keeps them up. Moreover it must recip- 
 rocally act upon the moon as a tangential force, dimin- 
 
 * Appendix, Note XVII. 
 
Is part of Moon's Appa/rent AoceloraUon. 145 
 
 ishing her velocity and centrifugal force and radius of 
 orbit, and therefore shortening her period a Httle, 
 (P- 83). 
 
 Consequently the earth gets so much slower that any 
 meridian, treated as a hand of the earth clock (and all 
 our clocks only represent the earth's rotation), would 
 be 12 seconds slow at the end of 100 years compared 
 with a clock which had gone all that time at the rate 
 of the earth's rotation 100 years before. If you wish to 
 know how much a day the earth must lose to produce 
 this result, you must remember that the daily loss 
 accumulates by ' arithmetical progression ' into these 12 
 seconds in a centui-y ; and that represents a loss of not 
 quite the 66th of a second in the length of the day in 
 2500 years. So also you would find that a lunation is 
 only half a second shorter than it was 2500 years ago.* 
 
 The real and apparent acceleration of the moon from 
 both causes together has accumulated to i^°, or 3 
 times her own diameter, or 3 hours in time, in 2500 
 years. And that is quite enough to make a material 
 difference in the places where a solar eclipse was visi- 
 ble, and therefore in some important dates of ancient 
 history. For if the old accounts of a great battle say 
 that it was stopped by an eclipse, and we can calculate 
 that no great eclipse of the sun was visible there at 
 any time but one, at all near the received date, we may 
 be sure that was the time of the event. In this view, 
 and also for finding the longitude at sea, as will be 
 
 * The sum of an ' arithmetical series ' is the nuinher of its terms (here the 
 days In 1500 years) + the sciuare of that number, all X half the common differ- 
 ence (here the daily retardation). 
 
14:6 Old Dates fixed hy Eclipses. 
 
 explained hereafter, all the little disturbances of the 
 moon are important, because her exact place can only 
 be calculated by taking them into account. 
 
 Mr. Croll has pointed out another permanent effect 
 of the tides on the moon herself. The solar tide wave 
 must retard the motion of the earth round the centre 
 of gravity of the earth and moon, in the same way as 
 the lunar tide retards the motion of the earth round 
 its own centre of gravity, and must therefore gradu- 
 ally diminish the distance of the moon. For if the 
 earth turned in a month, the lunar tide would only be 
 a stationary and therefore invisible elevation of the 
 water in one place ; but the solar wave would move 
 round it in the month in consequence of the earth's 
 monthly revolution round the joint centre of gravity ; 
 and that must destroy some of the force of that motion, 
 or of the earth's centrifugal force round that c. g. 
 The moon is not directly affected thereby, but the 
 earth is brought nearer to the c. g., and therefore their 
 distance is diminished, and their orbit round the joint 
 c. g. made smaller and therefore quicker. I do not 
 know that any calculation has been made of the 
 amount of these disturbances, but it must be much less 
 than the others which we have been considering.* 
 
 Measure of the Disturbing Forces,— We can easily 
 calculate without mathematics the proportion which 
 the differential and contractive forces bear to the or- 
 dinary earth-force on the moon, at the places where 
 they are each greatest, i.e., at syzygies and quadratures 
 respectively. The earth's force on the moon is the 
 
 * See Mr. Croll's paper in the Philosophical Journal of August, 1866. 
 
Calculation of the MoorCs Distwrhances. 147 
 
 mass of earth + moon divided by the square of their dis- 
 tance, as we want to consider the earth at rest (p. 98) ; 
 and we will take their mean distance for the unit of dis- 
 tance, to avoid large figures. The sun's distance from 
 the earth will then he 383, and his distance from the 
 new moon 382 ; and his attraction on them respect- 
 ively = ^^ and -^. The differential force is 
 383 Z^z' 
 
 the difference of these ; which by a common sum in 
 
 fractions = ^ J* B which again you vnll 
 
 146,689x145,924 ° •' 
 
 ^'^^ — r-s — r,- Biit the sun is 312,720 times as 
 (382-5/ 
 
 heavy as the earth + moon ; and if you substitute that 
 
 figure you will find the differential force = — ? — , or 
 
 89-5 
 
 is that fraction of the earth's force on the moon, which 
 
 force is i according to the assumptions we made. 
 
 But the force at full moon is the difference between 
 
 sun ■, sun 767 sun 2 sun , . , 
 
 383' 384= 383^x384' (383-5)' 
 
 with the same figure for the sun's mass = 
 
 ^ 90-2. 
 
 You will see the consequence of this excess at new 
 
 moon presently. But you have here the proof that the 
 
 differential force varies inversely as the cube of the 
 
 sun's distance very nearly, and directly as twice the 
 
 difference of distance of earth and moon from the sun ; 
 
 for the 2 in the numerator represents that : all which 
 
 IS very like the case of the tides. Similarly we should 
 
 find that the contractive force, which is half the dif- 
 
lis Lam of the Distwrlmg Forces. 
 
 ferential when they are both at their maximum, ia 
 measured by the actual distance of the moon sideways 
 from the line of syzygies, divided by the cube of the 
 sun's distance. These last, observe, are general values 
 of the forces, for all positions of the moon, which can 
 easily be worked out (though at greater length) on the 
 principle of the calculation which I gave just now. 
 
 It may seem odd, but it is the fact, that the magni- 
 tude of the disturbing forces on the moon depends on 
 the proportion of the length of the year to her sidereal 
 period of 27'32 days ; which is I3'37. This cannot be 
 proved here beyond observing that iyi7^ or 179 is 
 twice the denominator of the fraction which represents 
 the proportion of the sun's disturbing force to the 
 earth's attraction (p. 147). And this, like the tidal 
 force, is not affected by the late correction of the sun's 
 mass ; for the mass and the cube of the distance had 
 to be altered equally, as the length of the year de- 
 pends on both of them. Moreover the moon's period 
 is lengthened by an eighth of that fraction, or a 716th ; 
 as it would be if the earth's mass were reduced a 358th. 
 The mean distance is also increased a 1432nd, for a 
 reason which you will see at p. 224. It is convenient 
 to mention here, that where a quantity is increased by 
 a small fraction, its square and cube practically increase 
 by twice and thrice that fraction, and its square and 
 cube roots by half and a third of it, i. e., by doubling 
 or trebling its denominator. 
 
 Tangential and Radial Forces.— Now let us see what 
 the differential and contractive forces do, besides pro- 
 ducing that long acceleration of the moon which I 
 
Mow they Act on the Moon. 149 
 
 have spoken of, and which is only a very small part of 
 their effects. As both forces are acting at every part 
 of the orbit except syzygy and quadrature, where they 
 alternately vanish, they must combine to produce a 
 resultant force in some direction between them, as two 
 winds blowing across each other would send a ship or 
 a ball in some diagonal course between them. As the 
 differential force always acts from the line of quadra- 
 tures, QQ in the figure at p. 153, and is proportional 
 to twice the moon's distance therefrom, it always ac- 
 celerates her from quadrature to syzygy and retards her 
 from syzygy to quadrature. And as the contractive 
 force always acts toward the line of syzygies SS, and 
 is proportional to the moon's distance therefrom, it 
 also accelerates her toward syzygy and retards her after 
 syzygy. Thus a part of both these forces, whenever 
 they Tioth exist, is always resolved into a tangential 
 force, which accelerates before syzygy and retards after 
 it ; and the rest is resolved into a radial force, which 
 acts with the earth's attraction for 35° on each side of 
 quadrature, and more strongly against it for 55° on 
 each side of syzygy. At 55° from syzygy they balance 
 each other, and the radial force vanishes ; but the tan- 
 gential force is greatest half way between syzygy and 
 quadrature, at the places called octcmts ; and there it 
 amounts to J of the differential force at the adjacent 
 syzygy : but this cannot be proved here. 
 
 Tou will easily see that the sun is really a little fur- 
 ther from the moon than the earth at true quadratures 
 or 90° from syzygy. But the difference is only 4^ ', 
 the angle corresponding to half the moon's distance 
 
150 The MooTkS Variation. 
 
 divided by the sun's ; which is much too small to affect 
 any calculations that can be given here ; and so is the 
 inequality arising from the sun's force in the plane of 
 the moon's orbit being rather less vsrhen she is not also 
 in the ecliptic— as she never is quite, except at the 
 nodes. In the figure at p. 153 I have marked the for- 
 ces with arrows according to their directions, and I 
 have given the radial force at syzygies two arrows, 
 because it is double of that at quadratures, except so 
 far as they all vary with the moon's distance from the 
 earth : e.g., the tangential force at A (say) 50° after Sa 
 exceeds the tangential force at P 50° after Si as much 
 as EA exceeds EP. 
 
 Variation! — This constant acceleration up to syzygy 
 and retardation after it, makes the moon alternately 35' 
 42", or rather more than her own width, before and 
 behind her mean longitude ; and this is called her va- 
 riaidon. You would probably expect it to carry her 
 further away from the earth at syzygy than quadrature. 
 But it does just the contrary. For the moon going 
 fastest at syzygy, from a cause different fi'om the earth's 
 attraction, is least drawn out of her forward course by 
 the earth and goes further on toward quadrature ; and 
 so the orbit becomes an oval with its sides at syzygy 
 audits ends at quadrature, and the minor axis a 70th 
 less than the major (supposing the undisturbed orbit to 
 be a circle). But you must not confound this secondary 
 oval with the much more elliptical general orbit of the 
 moon, of which this is only a small disturbance. 
 
 Parallactic Inequality.— We saw just now that both 
 the diiferential and contractive forces, and therefore 
 
The Parallactic Inequality. 151 
 
 their resultant tangential force, are a little greater on 
 the near side of the orbit than on the far side. Con- 
 sequently the ' variation ' at new moon exceeds that at 
 full moon by the small amount of 2' 6". If you work 
 out the calculation at p. 147 completely, keeping the 
 moon's distance as i, but increasing the sun's to 400, 
 as it used to be reckoned, and increasing his mass to 
 357,050 in proportion to the cube of the distance, you 
 will find the average disturbing force the same as be- 
 fore, but the difference between its two extremes a lit- 
 tle less. And with some trouble you might find that 
 this difference of the differential force = 6 x sun's mass 
 (keeping that the same) divided by his distance* (or 
 distance x cube of distance). Eut since the mass bears 
 a fixed proportion to the distance', that leaves this in- 
 equality to vary inversely as the sun's distance, as his 
 parallax does. Hence it is called the parallactic 
 inequality ; and this alone of all the disturbances 
 gives any measure of the sun's distance. In fact its 
 observed excess over the amount due to the sun's old 
 distance raised the first suspicion in 1854 of that dis- 
 tance being too great, as I said at p. 85. 
 
 Subject to this small difference of 2', the ' variation ' 
 compensates itself in opposite halves of the orbit — pro- 
 vided the two halves are alike in the long run. But 
 they are not : for there is a gradual decrease in the 
 length of the radius vector, as explained at p. 142 ; 
 and Professor Adams found that the consequence is 
 that the ' variation ' does not quite correct itself every 
 fortnight oji the average, as Laplace and everybody 
 else had concluded that it did ; and that it produces 
 
152 Advance of the Mootv's Apsides. 
 
 a secular retardation about half as great as the accele- 
 ration due to the other cause, as I have already said. 
 
 The Advance of the Apsides may be shortly proved as 
 follows, though it requires a long investigation to trace 
 all its causes through all the positions of the moon and 
 of her orbit. It is only necessary to remember first, 
 that the sun's disturbing force on the whole weakens 
 the attraction toward the earth. 
 
 The imdisturbed apogee is the place where the moon 
 would begin to move toward the earth if there were 
 no sun ; but if the earth's attraction is weakened there, 
 it cannot pull the moon round the corner so quickly, 
 and she will carry the apse along with her a little. 
 At perigee she begins to leave the earth again ; but 
 if the earth's attraction is weakened there, she will 
 begin to leave sooner than she would otherwise; or 
 that apse comes sooner, or recedes. But these opposite 
 effects by no means balance each other ; for the differ- 
 ential force varies as the moon's distance from the 
 earth, which is about a 19th greater at apogee and a 
 19th less at perigee than at mean distance. Besides 
 that, the earth's attraction is itself about a 9th less at 
 apogee and a 9th greater at perigee than at mean dis- 
 tance : and we found at p. 147 that the mean differ- 
 ential force is about a 90th of the earth's mean at- 
 traction. Therefore the actual differential force will 
 diminish the earth's actual attraction at apogee by 
 
 — X — X — = — , but at perigee only I_ x — x 
 19 9 90 82 19 9 
 
 — = — whenever the apses are in syzygy and the 
 90 107 ^ ■^^'^ 
 
Three different Oomses of it. 153 
 
 effect of the forces in disturbing them is greatest. The 
 effects of the contractive force must be opposite to 
 those of the differential force ; but as I said at pp. 148, 
 149, only half as great, and lasting much less time. 
 Therefore on the whole the advance of the apses pre- 
 ponderates. 
 
 The sun then would drive the apses forward even if 
 he stood still ; for we have said nothing yet about his 
 motion. But he does go round the earth the same 
 way as the apses, and therefore drives them faster. He 
 also stays in company with a progressing apse, keeping 
 up its progress, longer than with a receding apse which 
 he only meets, and thus makes them progress still 
 more. By these two causes the advance of the apses 
 is made twice as great as it would be without them. In 
 the same way the apses of the earth's orbit are carried 
 round by the attraction of the planets, the greatest 
 weight of them by far being outside the earth's orbit ; 
 but in 1 10,880 years instead of 9, as their disturbing 
 force is much weaker than the sun's upon the moon. 
 
 Change of Eccentricity.— The sun disturbs the eccen- 
 tricity of the moon's orbit 
 so much that it is worth -Ao!i'!;!-5il!£5'''». 
 
 while to attempt the ex- ■*■ ^ 
 
 planation of that also. In 
 this figure, which I have 
 described already, P and 
 A are perigee and apogee 
 for this one position of 
 the orbit, or of syzygy and 
 quadrature with respect to perigee and apogee : other 
 
154 Ohcmge of Eccentricity of Orbit, 
 
 positions would want other figures, but this will serve 
 our purpose. 
 
 1 . When the moon is at the syzygy Si she is approach- 
 ing perigee, or the radius vector is decreasing ; but the 
 radial force there acts against it and tends to keep the 
 radius from shortening so much, and therefore makes 
 the orbit less eccentric (p. 40). At the opposite syzygy 
 Ss the radius is lengthening, and the radical force tends 
 to lengthen it still more, or to increase the eccen- 
 tricity. "We must see then which prevails. At 82 the 
 moon is further aS, and going slower ; and so the dis- 
 turbing force is both greater and has a longer time to 
 act (p. 153): therefore the increase of eccentricity pre- 
 vails thus far. 
 
 2. At Qi the radius is lengthening, but the contrac- 
 tive radial force acts against it, and therefore dimin- 
 ishes eccentricity. At Q2, after apogee, the radius is 
 shortening, and the contractive force helps to shorten it 
 and therefore increases eccentricity. And QjE is 
 greater than QiE, and therefore again the increase 
 prevails. At these places there is no tangential force. 
 
 3. The tangential force, retarding from syzygy to 
 quadrature, diminishes the moon's velocity at P, and 
 therefore diminishes her centrifugal force, or power to 
 fly further off from perigee, or to increase her radius 
 vector ; and so the eccentricity is diminished by the 
 tangential force at P. But it is increased by the tan- 
 gential force retarding the moon at A and weakening 
 her power to resist the earth's attraction, which short- 
 ens her radius faster than if she moved quicker and 
 had more centrifugal force there. And the force is 
 
Is One OoMse of the Momi's M^ection. 155 
 
 both greater and acts longei* at apogee than perigee, 
 as before ; so again the eccentricity is increased. At 
 intermediate places near Qj and S^ the efiects of the 
 tangential force balance each other pretty nearly. 
 
 The result is that all the disturbing forces increase 
 the eccentricity when the moon has to pass through the 
 apses before quadrature and after syzygy. You may 
 easily infer that the eccentricity is diminished when 
 she has to pass the apses after quadrature and before 
 syzygy. And when they lie in either syzygy or quad- 
 rature the forces balance each other and do not disturb 
 the eccentricity. Nevertheless it is greatest when the 
 apses are in syzygy and least when they are in quad- 
 rature. For those places move round the earth with 
 the sun in a year, while the apses take nearly 9 years 
 to revolve in the same direction ; therefore Si is ap- 
 proaching P whenever they are in the position of this 
 figure, in which the eccentricity is increasing ; and it 
 goes on increasing till syzygy has reached the apse, and 
 consequently it is greatest then. Similarly it is least 
 with the apses in quadrature. And on the whole it 
 varies so much as to be half as great again with the 
 major axis in syzygy as in quadrature. 
 
 Evectiotli — The variation of eccentricity and the ir- 
 regular motion of the apses produce together the larg- 
 est and earliest observed of all the displacements of the 
 moon ; which was called Evection, or the carrying 
 away of the moon fi-om her mean longitude (not her 
 mean anomaly but her" mean elliptical place) by as 
 much as 1° 20' alternately backward and forward; 
 making her oscillate through 5 times her own width 
 7* 
 
156 Why the Moon^s Ixodes Recede, 
 
 ill the time of the sun's passing perigee twice, or about 
 a year and six weeks. Consequently it depends on, or 
 is & function of, the difference of longitude of sun and 
 moon, and also of the true anomaly or distance from 
 perigee, on which the motion of the apses depends. 
 
 Recession of the Nodes. — In all these cases of dis- 
 turbance the orbit that we speak of as if it were a ring 
 capable of being moved, is the instantaneous ellipse, 
 which the moon would go on describing thereafter if 
 the disturbances were stopped. The recession of the 
 moon's nodes is caused by the attraction of the sun on 
 the moon, exactly as the precession of the equinoxes, or 
 nodes of the equator and ecliptic, is by the attraction 
 of the sun and moon on the equatorial protuber- 
 ance of the earth, which may be considered a ring of 
 satellites stuck together. "We need only consider the 
 motion of one satellite to explain them both. 
 
 But the effect of the sun's and moon's attraction on 
 the equatorial ring or ' elliptical excess ' is much less 
 than it would be if there were not a sphere many times 
 heavier inside, which has to be dragged round with it 
 in giving the twisting motion to the earth's axis. The 
 bulk of the elliptical excess is a 149th of the whole 
 earth (an oblate spheroid and the sphere within it being 
 in the proportion of the squares of their different axes) : 
 but the outside is not half as dense as the inside (p. 26) ; 
 and therefore the mass of the whole earth is probably 
 300 times that of the elliptical excess. Again that is 
 not really concentrated into a ring, but spread over the 
 v/hole surface, from 13 miles thick at the equator down 
 to nothing at the poles. Moreover the forces which 
 
And the EartKs Equinoxes. 157 
 
 produce all tlie disturbances vary as the distance be- 
 tween the two attracted bodies, and the moon is 60 
 times farther from the earth's centre than the equator 
 is. Erom all these causes together, and the absence 
 of the moon of course in disturbing her own nodes, the 
 earth's nodes or equinoctial points recede 1390 times 
 slower than the moon's. 
 
 r. As the sun occupies all sorts of positions with 
 respect to the nodes during a year (or rather less, since 
 the nodes revolve backward in nearly 19 years) we 
 must consider them in succession. First let us take 
 the nodes in quadrature. Then the tangential force 
 urges the moon forward as she rises to syzygy and her 
 greatest latitude from the ecliptic. Besides that, there 
 is always a force toward the ecliptic, as the sun's dif- 
 ferential force is always trying to pull the near side 
 and push the far side of the moon's orbit down to the 
 ecliptic, except when he is himself in the line of nodes 
 and therefore in the plane of the moon's orbit. This 
 is called the resolved force of the sun toward the 
 ecliptic. 
 
 Now if the moon's apparent path in the heavens, 
 rising from the ecliptic and coming down to it again, 
 is opened out into a flat picture, it will look like the 
 path of a stone or a ball shot from the ground at an 
 angle of 5°, and coming down to it again at the same 
 angle, the place of falling corresponding to the next 
 node. But that disturbing force of the sun which acts 
 toward the ecliptic is the same as if a wind blew down 
 upon the ball ; and the effect of that would manifestly 
 be to make it reach the ground sooner, or the node to 
 
158 Why the MoorCs Nodes Recede. 
 
 recede. Also while the ball is rising, the downward 
 force would evidently keep diminishing the angle of 
 inclination of its course ; but would increase it while 
 falling; and so the inclination would end as it began j 
 though the moon or the ball has not risen so high above 
 the ecliptic or the earth, nor gone so far, as if the dis- 
 turbing force had not acted. 
 
 The tangential force, which always urges the moon 
 forward from quadrature to syzygy, tends to postpone 
 her arrival at the next node, and also to make her 
 course flatter, or the inclination of the orbit less ; but 
 the same force acts the contrary way from syzygy to 
 the next quadrature and node, and so those two balance 
 each other. 
 
 2. Next let the nodes be in the line of syzygy. As 
 the sun is then in the plane of the moon's orbit, he 
 plainly can do nothing toward pulling or pushing the 
 moon out of her orbit, or altering either the nodes or 
 the inclination. And from these two cardinal posi- 
 tions of the nodes we may conclude, that as they re- 
 cede through the whole lunation in one case, and never 
 advance in the other, they must on the whole recede, 
 even if they advance a little in some intermediate 
 position. 
 
 3. But we may as well complete the inquiry by see- 
 ing what happens when they are neither at quadra- 
 tm-e nor syzygy; and first, let each node be after 
 quadrature. Then as the moon comes down to node 
 from quadrature, toward the sun, he pulls her forward 
 out of the course she is taking, and so makes the node 
 advance, i. e., makes her reach the ecliptic later. In 
 
The Moon is disimrbed h/ Venus. 159 
 
 the opposite quarter of the orbit, the force acts simi- 
 larly, and there also makes the node advance. But in 
 the rest of the orbit the disturbing force is toward the 
 ecliptic as before, and therefore makes the nodes recede. 
 
 4. Lastly, let the nodes be before quadrature and 
 after syzygy. Then while the moon goes up from node 
 to quadrature, leaving the sun, he also pulls her back, 
 which makes her course less parallel to the ecliptic, as 
 a head wind would make the course of a ball, while 
 rising, still less parallel to the earth : and that post- 
 pones her arrival at the next node, or makes it advance. 
 And the same thing happens in the opposite quarter, 
 as usual. But in the rest of the orbit the force is 
 downward, or toward the ecliptic, and so the nodes 
 recede. Therefore on the whole, as the nodes cannot 
 advance through more than two quarters of the orbit 
 in any lunation, or through more than half that quantity 
 on the average, and recede in all the rest, our former 
 conclusion was right, that the recession greatly pre- 
 ponderates. 
 
 If we followed the inclination also through the last 
 two cases, we should find that it is diminished when 
 the nodes are in the third position, but increased when 
 they are in the fourth, and therefore is not altered in 
 a complete set of lunations, when the nodes and sun 
 have gone all round each other, except by the minor 
 disturbances beyond the scope of this book. 
 
 Two very small disturbances of the moon have been 
 discovered by Professor Hansen, both due to Yenus. 
 "We shall see at p. 167 that she retards the earth, and 
 therefore enlarges its orbit or distance from the sun. 
 
160 TJie Moon is disturbed iy Venus. 
 
 for 120 years (whicli Mr. Airy discovered), and there- 
 fore increases the moon's secular acceleration for that 
 time ; and then diminishes it for another 120 years : 
 the effects accumulating to 23" in that time. The 
 other is of a more complicated kind, and accumulates 
 to 27" in 1 36 years. Sir J. Herschel says these alone 
 were wanting to account for all the observed inequali- 
 ties of the moon's motioDj of which there are more 
 than 50 altogether. The first idea of the 'lunar 
 theory,' or of the moon's disturbances being chiefly 
 caused by the sun, was conceived by Jeremiah Hor- 
 rocks, who died at 21, about a year after the transit of 
 Venus in 1639, which he alone predicted. 
 
CHAPTEE IV. 
 
 THE PLANETS. 
 
 The earth is by no means the only body that goes 
 round the sun. In the earliest times of astronomy it 
 was observed that there were five stars unlike all the 
 rest in their behavior: apparently going round the 
 earth (independently of their daily rising and setting) 
 in longish periods, though with some irregular motions 
 backward and forward, two of them taking less than a 
 year to go round, and the other three taking nearly 2, 
 13, and 30 years. These five wamdering stars neither 
 kept the same distance from each other nor from the 
 other stars, which are called fixed because they do not 
 visibly change their places, except to the very small 
 amount which I shall describe hereafter ; and they 
 were therefore called planets. The ancients either 
 named them after some of their gods, or their gods 
 after them : for it is by no means agreed which came 
 first, the heathen mythology and worship of false gods, 
 or the belief in the planets influencing the bodies and 
 fortunes of men : the study of which is called astrology. 
 I have no doubt that came first, for reasons which are 
 not material to state here. The planets have still kept 
 these old names, and no doubt always will. For it 
 may be observed that no civilized nations, nor perhaps 
 any nations now, can invent 'proper names,' i. e., 
 names for persons or places : they can only copy or 
 compound old ones. 
 
162 Ancient Worship of the Planets. 
 
 The names of those five old planets are Mercuiy, 
 Yenns, Mars, Jupiter, and Saturn ; and the days of the 
 week are still named after them, with the addition of 
 the sun and moon ; either directly, as Saturday, Sun- 
 day, Monday, or through the Saxon names for the 
 others : thus Tuesday is the day of Mars (Tuisco), 
 Wednesday of Mercury (Woden), Thursday of Jupiter 
 (Thor), and Friday of Yenus (Friga). 
 
 These five planets with the sun and moon were also 
 the principal characters in that ' host of heaven ' which 
 the idolaters of old worshipped long before the Greeks ; 
 some of whom, you remember, took Paul and Barna- 
 bas for Jupiter and Mercury, and others worshipped 
 an image of Diana, the goddess of the moon. The first 
 Greek historian Herodotus says that the first Greek 
 poet Homer, who probably lived about the time of the 
 prophet Elisha, borrowed the names of Jupiter and 
 most of the other Grecian gods from the Egyptians, 
 who also practiced astrology or divination by the plan- 
 ets, as the Chaldseans did (Herod. II., 4, 50, 53, 82). 
 Baal and Ashtoreth were the gods and idols of the Sun 
 and Yenus, and Msroch probably of Saturn. He is 
 represented in Assyrian sculptures encircled by a ring, 
 which was the symbol of Time, the Greek name of 
 Saturn ; also as a human eye, whicli resembles the ob- 
 lique view of Saturn and his Eing. They represented 
 Yenus with a^ crescent (p. 202), which can no more be 
 seen without a telescope than Saturn's Ring * (see Mr. 
 Proctor's Saturn, p. 197). 
 
 Indeed no one who has inquired into both subjects 
 
 * Appendix, Note XVIII. 
 
The Planets go Round the Sun. 163 
 
 can doubt that Pagan idolatry was closely connected 
 with the belief in the influence not only of the sun, 
 but of the moon and planets, on the bodies and affairs 
 of men.* There is reason to believe that the things 
 translated groves in some passages of Scripture, which 
 were ' built ' and ' set up on every high hill and under 
 every green tree,' and carried out of the house of the 
 Lord by the good king Josiah and burnt, and there- 
 fore certainly not groves of trees, were wooden machines 
 representing the planets and their apparent motions,f 
 and used as images of the powers then supposed to 
 rule the world, rather than the Lord who made the 
 heavens and all the host of them, and will one day 
 ' make new heavens and a new earth wherein dwelleth 
 righteousness.' 
 
 When Copernicus found out that the earth goes, 
 round the sun, he found the same of the planets also ; 
 in other words, he discovered that the earth is one of 
 the planets. And as I said before, people were at last 
 driven to accept his theory or explanation of the plan- 
 ets' motions by finding that no other would account for 
 them ; for the planets would not appear where they 
 ought according to any of the other theories. "When 
 we come to the aberration of light (p. 211) you will see 
 a direct proof of the earth's motion in the apparent 
 annual motion of all the stars, though that was not 
 observed till long after Copernicus. 
 
 It was long before the reason of their motions was 
 discovered, or the full number of the planets. Sir Isaac 
 
 * See Faber's ' Origin of Pagan Idolatry,' etc. 
 t See Landseer's ' Sabsean Eesearohee.' 
 
164r Their EllvpUc Motion discovered hy Kepkr. 
 
 Newton knew of no more than the five old ones and 
 the earth, because the telescopes of his time were not 
 large enough, that is, did not take in enough planet- 
 light at a mouthful, to show the smaller planets and 
 the more distant ones which haye been discovered 
 within the time of people now living. Copernicus did 
 not even get so far as to discover that the planets de- 
 scribe ellipses, although it was known to Hipparchus 
 1700 years before that the sun is not always at the 
 same distance from the earth, because his disc is larger 
 at some times than at others. The elliptic motion 
 was discovered by Kepler soon after the year 1600, 
 together with two other remarkable laws of planetary 
 motion, which I will explain hereafter. But he, like 
 Copernicus, only found that the planets observed these 
 laws of motion as a fact. Newton found a reason for 
 them, and proved that every planet round the sun, and 
 every moon or satellite round a planet, must observe 
 them. 
 
 The dimensions, weights, and motions of the whole 
 solar system, which means the sun with its planets and 
 their moons, are now considered to be as follows : 
 
 I. The first planet is Mercury ^ , 35 million * miles 
 from the sun at mean distance, and going round him 
 in nearly 88 days. By ' days ' I mean our days, and 
 not the planet's own days ; for if we want to compare 
 their periods or times of going roimd the sun, we must 
 measure them all by days of the same length ; and you 
 
 * I give the more precise figures In tlie table generally . The distances in pro- 
 portion to the earth's mean distance are Independent of the accuracy of that, 
 and of course the periods ; and so are the masses in proportion to the sun, but ' 
 not to the earth. 
 
Mercury. 165 
 
 will soon see that there is no known relation whatever 
 between a planet's period or year and the length of 
 his day or time of rotation. 
 
 His diameter must now be called 3050 miles ; and 
 so he is not quite 3 times as large as the moon, but 
 more than five times as heavy, because he is rather 
 denser than the earth (I'liS ), which is 15*4 times as 
 heavy and 17-5 times as large as Mercury. The sun is 
 nearly 5 million times as heavy. But though Mercury 
 is so small, he turns on his axis slower than the earthy 
 his sidereal day being 5-^ minutes longer than ours. 
 He is always so close to the sun, never more than 29° 
 off, that he is difficult to observe accurately, and can 
 never be seen except as a ' morning or evening star,' 
 just before sunrise or after sunset. 
 
 Mercury's orbit is the most inclined to the ecliptic of 
 any (7°) and also much the most elliptical, the eccen- 
 tricity being "2056 ; and therefore his greatest distance 
 is '4666 and his least "3075 times our mean distance. 
 But even this great eccentricity only makes the axis 
 minor about one joth less than the major (see p. 39). 
 I said that the earth moves through space 65,500 miles 
 an hour ; but Mercury goes much faster, viz., 105,000, 
 or nearly 30 miles a second on the average, but 35 at 
 perihelion which is 28 million miles from the sun, and 
 23 at aphelion which is 42 millions, calculated as I 
 explained for the moon at p. 115. 
 
 The apparent size of- the sun's disc to a planet, and 
 the light and heat received there, vary inversely as the 
 square of the distance, but the apparent diameter of 
 the disc varies inversely as the distance only. And as 
 
166 Venv^. 
 
 the sun's apparent diameter here is 32', you will easily 
 calculate that at Mercury it varies from i"38' to 2°29', 
 and that the apparent size of the sun, and the light and 
 heat, are from 5 to nearly 1 1 times as much as they are 
 here. The apparent diameter of Mercury to us of 
 course varies still more, viz., from 5" when he is 
 beyond the sun, to 12" when he is between the sun 
 and us. The eccentricities of the other planets' orbits 
 are so small that I shall not notice these distinctions 
 between their greatest and least distances. 
 
 2. Venus ? J the next of the planets, has the most 
 circular orbit of them all, its eccentricity being only 
 ■007, and the semi-axes minor only 1600 miles less than 
 the major ; which is 66 million miles, or "7233 of our 
 distance from the sun. Her period is 224*7 ^^js, and 
 her sidereal day 39 minutes less than ours. The orbit 
 is inclined to the ecliptic 3° 23'. Consequently she 
 travels nearly 77,000 miles an hour, and the apparent 
 diameter of the sun there is 44', or half as wide again 
 as he appears to us. The diameter of Venus is 7770 
 miles, or a little less than the earth's ; but she weighs 
 one-fifth less than the earth, and is ^^ as heavy as 
 the sun, her density being only "836 of the earth's. 
 She gets twice as much light and heat as we do ; and 
 in fact her brightness prevents her from being as well 
 observed as some of the moi-e distant planets. Tt has 
 been ascertained lately by experiments that Yenus is 
 ten times as bright as the brightest part of the full 
 moon ; which is probably owing to the atmosphere of 
 Venus being a much better reflector than the rough 
 surface of the moon without an atmosphere (p. 93). 
 
The '■Inequality'' of Venus and the Ea/rth. 167 
 
 From that cause we haye not yet ascertained how much 
 her axis leans, or the amount of her spheroidicity. 
 Probably it is much the same as the earth's ; but the 
 axis is thought to lean very much more. Gravity at 
 her surface is less in the same proportion as her mass, 
 the distance of the surface from the centre being prac- 
 tically the same as here. Her apparent diameter is as 
 little as 9f" when she is on the other side of the sun, 
 and as much as 6i" when she is nearest to the earth ; 
 and she exhibits phases like the moon (see p. io6). 
 
 Venus, like Mercury, never appears far from the sun : 
 47° is the greatest angle, or Apparent distance, called 
 the elongation, ever made by the lines of sight from us 
 to the sun and Venus. Consequently she never appears 
 but as a morning or an evening star, rising a little be- 
 fore or setting a little after the sun ; but she is some- 
 times visible by day, even without a telescope. Her 
 transits over the sun will be spoken of hereafter. 
 Thirteen years of Venus agree with 8 of the earth 
 within a day, and in that time they have 5 conjunctions, 
 or 5 synodical periods ; which produces that disturbance 
 of both their orbits which I referred to at page 159, 
 accelerating the earth and retarding Venus a little 
 for 120 years and then reversing the effects for 120 
 years more, as I will explain for the much larger ine- 
 quality of the same kind between Jupiter and Saturn. 
 
 3. The next planet is the Earth ®, of which I have 
 said enough in Chapter I. 
 
 4. The first planet beyond the earth, or the first of 
 what are called the superior planets, is Mars $, ■ I* 
 would have been better to call them exterior, leaving 
 
168 Mms. 
 
 the tei-m ' superior ' for the much larger but less dense 
 ones which come after Mars, and after a great gap in 
 the system, and rotate more than twice as fast not- 
 withstanding their size, as you will see presently. His 
 mean distance from the sun is 139 million miles, or 
 I '5237 of the earth's distance, and he performs his 
 circuit in 687 days, or a little less than two years : and 
 therefore his Telocity is about 53,000 miles an hour, 
 and the sun's apparent diameter there is 21'. The 
 eccentricity of his orbit is one nth, and it is inclined 
 1° 51' to the ecliptic. He again is small, his diameter 
 being only 4155 miles ; and his density must be only 
 •65 of the earth's or a little more than the moon's, as 
 the earth is 8^ times as heavy, but only 5^ times as 
 large; and the stin is 2,680,337 times as heavy. 
 
 Mars gives us better opportunities of seeing him 
 than any of the other planets : better than, his supe- 
 riors, because the nearest of them is never less than 8 
 times as far off as he is sometimes ; and better than 
 the two inferiors, because they are too near the sun. 
 Tou will see by adding and subtracting our solar dis- 
 tance to and from his, that he is at one time only 48 
 million miles from us, and at another 230. Conse- 
 quently his diameter appears nearly 5 times larger, 
 and his whole disc 25 times larger, at one time than 
 the other, and his apparent diameter varies from 4" to 
 18". Astronomers have been able to observe, what 
 they have not in Venus or Mercuiy, the inclination of 
 his equator to his orbit, and find it 28° 42', or not 
 much more than ours, and its inclination to our eclip- 
 tic 3° 18'. His day is as much longer than ours as 
 
The Ellipticity of Mars. 169 
 
 Venus's is shorter, 39 minutes, or as other authorities 
 say, 37- 
 
 There are appearances of snow at his poles, which 
 decreases in their summer ; and there is something in 
 his composition which makes him generally look red ; 
 but some parts look green, which are therefore thought 
 to be water or vegetation. There are clear indications 
 of an atmosphere in all the planets, though few,- if 
 any, in the moon. His heat from the sun is less than 
 half of ours, and gravity on his surface about the same 
 as on Mercury, and less than half of what it is here. 
 
 The spheroidicity of Mars is given in most books as 
 a 60th, or 5 times greater than the earth's ; and Mr. 
 Main's Greenwich observations in 1862 made it as 
 much as a 38th. But even the smaller of these amounts 
 is hardly possible. The oblateness does not depend on 
 the size, for the reason given in the note to p. 9, but 
 varies as the square of the velocity of rotation, and in- 
 versely as the density or attraction of the globe on its 
 own matter. As I^ewton divined long before anybody 
 proved it, the oblateness is proportional to the centri- 
 fugal force at the equator divided by the equatorial 
 attraction of the globe (p. 241). It also decreases 
 (down to a certain limit) with an increase of density 
 toward the centre, like the tidal ellipticity (p. 134). 
 If the earth's density were uniform, its ellipticity would 
 be a 230th instead of a 298th.* 
 
 * I believe the most complete treatise on the difficult problem of ' the flgrare 
 of the earth ' is Archaeacon Pratt's. And in the Phil. Mag. for Feb., 1867, he 
 comes to the concluBion that the polar diameter is 7899-74 miles, and that of the 
 equator (which he does not believe to be elliptical) 7926-6, and therefore the 
 ellipticity a 296th, and the diameter of an ecLual sphere 7017-6 (see p. 24). 
 
lYO Ths Asteroids. 
 
 Therefore Mars having '65 of the earth's average 
 density, and rather less velocity of rotation, ought to 
 have an ellipticity of a 205th if its density varies like 
 the earth's; which could hardly be seen. In order 
 that such a globe may be 8 times more oblate than the 
 earth, either its density must be very much greater 
 outside than inside, which is almost impossible ; or it 
 must have turned very much faster when it was fluid 
 than it does now (which again is very unlikely, see p. 
 234), and have become solid in that shape before it 
 began to turn slower ; or else it must be much denser 
 at the poles than at the equator for the matter at the 
 poles to balance the greater bulk at the equator : 
 which is equally unlikely. And on any of those sup- 
 positions all the water must run from the equator to 
 the poles, to get nearer the centre of attraction : which 
 does not accord with the appearance of the planet; 
 and it is satisfactory' to find that later observers agree 
 that Mars has no sensible ellipticity. 
 
 S- AsteroidSi — After Mars there is a great gap among 
 the old planets, as you see from the distance which I 
 mentioned just now of the nearest of them, and there 
 is no such gap in the distances beyond. But on the 
 first day of this century, i January, 1801, began the 
 discovery of a batch of little planets, or fragments of 
 one which had been blown to pieces, which have now 
 reached more than 100 in number, after standing for a 
 good many years at 4. They are called the asteroids, 
 which means things like stars, but should rather have 
 been caR&di plcmetoids. The first foiu' and rather the 
 largest are named Vesta, Ceres, Pallas, and Juno. The 
 
Jv/pite/r. 171 
 
 others have almost exhausted the names of all the 
 heathen goddesses, and they are now generally indicat- 
 ed by mere numbers enclosed in a circle, as ®. They 
 are all very small, the first two being under 230 miles 
 in diameter, and some too small to measure. As the 
 largest is jij as large as Mercury, and the moon 
 would make 706 of it, it is evident that all of them to- 
 gether would only make a very insignificant planet. 
 They lie scattered about between the distances of 240 
 and 300 million miles from the sun, and their periods 
 accordingly vary from 3^ to S^ years, by one of Kep- 
 ler's laws, which I M'ill explain afterward. Some of 
 them have orbits much more inclined to the ecliptic 
 than any of the regular planets, which further helps 
 the supposition that they are bits of a planet blown to 
 pieces ; and so does the fact that some of them appear 
 not to be round. Gravity there must be so small that 
 a man could jump many times his own height. 
 
 6. After these little asteroids comes Jupiter %, a 
 planet of a very different order from any we have seen 
 yet, 1246 times bigger than the earth, and about ^ 
 as large as the sun. But he, like the sun, is made of 
 something not much heavier than water, or "242 of the 
 earth's density ; for he is only 302 times as heavy as 
 the earth, and ~ as heavy as the sun. The sun's 
 diameter is nearly ten times Jupiter's, and Jupiter's 
 eleven times the earth's. Notwithstanding his great 
 size he turns round on his axis in five minutes under 
 ten hours, and consequently the centrifugal force is so 
 great that his equatorial diameter exceeds his polar 
 axis by one i6th. If his density were uniformly what 
 
172 Range of Heat in some of the Plcmets. 
 
 it is on the average, only a quarter of the earth's, his 
 ellipticity would be about one 9th. Or if it increased 
 inward at the same rate as the eai-th's (whatever that 
 may be) his ellipticity would be one 13th, according 
 to what I said respecting Mars. As it is only a 17th, 
 Jupiter's density must increase inwai'd even more than 
 the earth's, and his outside is much lighter than water. 
 That being his ellipticity, his bulk is a 17th less than 
 a sphere of his equatorial diameter of 86,936 miles, and 
 about an eighth more than a sphere of his polar diame- 
 ter (pp. 24,156). 
 
 Jupiter's mean distance from the sun is 475^- million 
 miles, or 5 '2028 times the earth's, and his periodic 
 time or year nearly twelve of ours, or 4332'6 days. 
 Consequently his rate of travelling through space is 
 27,000 miles an hour, and the sun's apparent diameter 
 there is only 6' 6" ; and his average heat a 38th of 
 ours. As his density is nearly the same as the sun's, 
 the force of gravity on his surface bears nearly the 
 same proportion to that on the sun's surface as their 
 diameters do (see p. 65), and is 2-7 times as much as 
 on the earth ; or a man on Jupiter would feel nearly 
 three times as heavy as on the earth. 
 
 Jupiter stands nearly upright in his orbit ; that is, 
 liis equator is only inclined 3° 4' to it, and only 1° 19' 
 to the ecliptic. The former is the inclination which 
 affects the seasons, except so far as they depend on the 
 changes of distance from the sun ; and it is too little 
 to make any sensible difference between summer and 
 winter. But as the eccentricity of his orbit is "048, he 
 is nearly a tenth of his mean distance, or 46 million 
 
Saturn. 1T3 
 
 miles, nearer the sun at perihelion than at aphelion ; 
 and as the heat varies inversely as the square of the 
 distance, he gets a fifth more heat at perihelion than 
 at aphelion; which is the same proportionate differ- 
 ence as if our summer was i io° hotter than winter : 
 remembering that we must reckon from the ' absolute 
 zero' of probably 522° below freezing, as at p. 43. 
 The change of seasons in Mars, with an eccentricity 
 of '093, must be still greater ; its perihelion heat ex- 
 ceeding that of aphelion as 1-37 to i, or more than the 
 difference of freezing and boiling water ; independ- 
 ently of his axis being more oblique than ours. And 
 in Mercury it is 2-23 times hotter at perihelion than 
 aphelion ; or they have a change nearly = the differ- 
 ence between frozen quicksilver and melted lead every 
 six weeks.* 
 
 Jupiter not only is but looks considerably larger than 
 any other planet, except Venus when she is nearest to 
 the earth, his diameter varying from 30" to 46", or 
 about one fiftieth of the sun's and moon's. His disc is 
 seen in telescopes to have some dark bands or belts 
 round it, which are always parallel to the equator, and 
 are supposed to be clouds carried round with him ; and 
 sometimes darker spots are seen, which may be open- 
 ings in the clouds. But Jupiter has a far more impor- 
 tant characteristic than these, in his four moons or sat- 
 ellites, of which I will say more after we have gone 
 through the planets themselves. 
 
 7- Saturn ^ isnotmuchsmallerthan Jupiter, but less 
 than a third of his weight, being made of something 
 
 * In all these estimates the influence of atmospheric vapors is disregajflea. 
 
174 The '■Long In&quality.'' 
 
 only •124 as dense as the earth, and only two-thirds as 
 heavy as water. His diameter is 73,590 miles, or 
 rather more than nine times the earth's ; but though 
 he is 730 times as large as the earth, he is only 90 times 
 as heavy, and the sun is 3502 times his weight. He is 
 even more spheroidal than Jupiter, his polar diameter 
 being one eleventh less than his equatorial, though his 
 density is much less and his velocity of rotation rather 
 less ; for heiturns in loj hours. His equatorial parts 
 are supposeoPtp be drawn out further by the attraction 
 of the Ring, which I will mention presently. His ap- 
 parent diameter is. generally 18", as he is too far off 
 for the diameter to be much affected by the earth being 
 in one part of her orbit or another. His mean distance 
 from the sun is 872 million miles, or 9*539 times the 
 earth's, and his year is 10,759 ^^J% or 29|- of ours (the 
 same number as the days of a revolution of the moon) ; 
 consequently he moves through space about 20,000 
 miles an hour, and the sun appears only 3' 20" wide 
 there. His light and heat are only one 90th as great 
 as the earth's, and gravity on his surface is very little 
 more than on the surface of the earth. Saturn stands 
 very differently from Jupiter, with his equator inclined 
 26° 50' to his orbit and 28° 10' to the ecliptic, to which 
 his orbit is inclined 2° 29'; and its eccentricity is 
 •056. .. 
 
 Long Inequality of Jupiter an| Saturn.— I have now 
 to describe a disturbance of these two planets by each 
 other, of which it is diflEicult to know how much to 
 attempt. When I tell you that Mr. Airy says (' Gravi- 
 tation,' p. 1 50) that ' the calculations necessary to dis- 
 
The ' Great Inequality' of J'wpiter and Saturn. 1Y5 
 
 cover tlie effect of it are probably the most complicated 
 that physical science has ever required,' you will see 
 that nothing beyond a very general account can be 
 expected here. If you wish to follow the subject as far 
 as it can be carried without mathematics, you will find a 
 longer explanation of it in the book aforesaid, and 
 later ones in Sir J. Herschel's Astronomy, and in Mr. 
 Proctor's book on Saturn ; * but you must not expect 
 to find any complete explanation of such a subject 
 easy. 
 
 The mean angular velocities of Jupiter and Saturn, 
 being inversely as their periods, are nearly in the pro- 
 portion jof 5 to 2, or more exactly, 72 to 29. It fol- 
 lows that every conjunction falls 242° 42' beyond the 
 last ; or as if they fell on the angles^of an equilateral 
 triangle which itself revolves, the same way as the 
 planets, at the rate of 2° 42' in their synodical period 
 of I9'86 years. If the angles are marked A B C in 
 the direction of motion, the conjunctions fall in the 
 order A C B A, coming round again to the same place 
 at the third conjunction, except that it has then moved 
 forward 8° 6'. The real motion is more irregular than 
 this, because the velocity of the planets varies in differ- 
 ent parts of their elliptic orbits. But if that M'ere all, 
 the variations would compensate each other, and all 
 that would hajppen would be this : — as Jupiter ap- 
 
 * I think it may be a public benefit to inform those who have to employ en- 
 grayers that their habit of making the letters in diagrams and architectural 
 drawings as small as they can has made some of the plates in Mr. Proctor's 
 valuable book totally useless to any common eyes, and even short-sighted eyes 
 can hardly make them out. The present fashion of pale and thin printing is 
 equally stupid. 
 
176 Its Greatest Amount. 
 
 proaches every conjunction lie would be pulled for- 
 ward a little by Saturn, who would be himself pulled 
 back, and the contrary as they leave conjunction ; and 
 the eifects of the disturbances would not accumulate. 
 
 But besides the iinequal velocities, the eccentricity 
 of the orbits makes the distances also not quite the 
 same before and after conjunction ; and therefore the 
 disturbances at any series of conjimctions may or may 
 not compensate each other. Moreover the orbits are 
 not in the same plane, and that causes another varia- 
 tion of their mutual attraction ' resolved ' in the plane 
 of either of them ; also the nodes where they cross are 
 continually shifting, as our equinoxes do in precession. 
 Mr. Airy says it is impossible to do more than state 
 that the mathematical result of all this is that for 
 about 460 years the major axis of one planet's orbit is 
 getting lengthened, and therefore its period (which 
 always depends on the major axis only) is lengthened, 
 and that of the other shortened ; and then for another 
 460 years the effect is reversed. Though the alteration 
 is exceedingly small in one synodical period, yet by 
 the end of the 460 years it accumulates, like the lunar 
 acceleration (p. 146), into something considerable, viz., 
 an alteration of Saturn's longitude by 48' and of the 
 heavier Jupiter's by 21'. The eccenti-icities of the two 
 orbits are also disturbed, and the perihelion of each is 
 made to advance and recede alternately for 425 years. 
 
 Similar coincidences of periods exist, as I said, be- 
 tween Venus and the earth, with 5 points of conjunc- 
 tion in the circle, instead of 3 ; and others not so ex- 
 act between other pairs of planets. But none of them 
 
JJrcmus. 177 
 
 produce nearly so large a distui-bance as this of Jupi- 
 ter and Saturn, which accumulates for such a long 
 time, and is therefore rightly called either the great or 
 the long inequality. Tou will see hereafter that a 
 similar, and relatively a greater, disturbance exists 
 among three of Jupiter's satellites, whose conjunctions 
 recur at only one place in the orbit of each pair. 
 
 Saturn also has eight moons, which I postpone like 
 Jupiter's for the present. His far more important 
 characteristic is the Eing, which revolves round him 
 like a continuous circle of satellites, as indeed it is 
 probably ; and therefore that also may better be de- 
 ferred till we have gone through the planets them- 
 selves. 
 
 8. Uranus. — These are all the planets that were 
 known until the year 1781, when a new one was dis- 
 covered by the late Sir W. Ilerschel, who was once an 
 organist at Doncaster, and afterward the greatest 
 astronomer of his time, and by a piece of rare good 
 fortune the father of another not inferior to himself, 
 and we may now add, the grandfather of a third astro- 
 nomer. He, like Newton and Gralileo, invented and 
 made telescopes of his own, larger and more powerful 
 than had been ever made before ; which have been 
 since copied on a still larger scale by Lord Eosse. 
 
 With one of his smaller telescopes Herschel found a 
 new planet further off than Saturn, and too far to be 
 seen without a strong telescope, though it afterward 
 appeared that it had been seen before, but not discov- 
 ered to be a planet. At first it was called the Geor- 
 gian star, after the king in whose time and neighbor- 
 8* 
 
178 Orbits of TJrcmus's Moons. 
 
 hood (viz., at Slough near Windsor) it was discovered, 
 and by whom Hersohel had been liberally assisted. 
 But the public preferred to name it after its diseoverei", 
 until at last they both gave way to another heathen 
 god Uranus, the father of Saturn and grandfather of 
 Jupiter, and whose name in Greek means the heaven 
 itself, beyond which it was supposed there was nothing 
 further to be found. 
 
 This planet Uranus ip or $ is 1753 million miles 
 from the sun, or 19' 1824 times the earth's distance, 
 and takes 84 years and 7 days to' go round him, at the 
 rate of 14,500 miles an hour. He also ranks as a large 
 planet ; for his diameter is nearly half Saturn's, or 
 33,836 miles, and he is 'jo times as large as the earth, 
 but only i5|- times as heavy, his density being -22 or 
 rather less than Jupiter's. He is 20,470 times lighter 
 than the sun. His orbit is the nearest to the ecliptic 
 of them all, being only inclined 46' 29"; and its eccen- 
 tricity "046. He has four, and some think seven or 
 eight moons, which behave differently from all others 
 in the solar system, having their orbits as nearly per 
 pendicular to the ecliptic as 79°, and nearly circular, 
 and moving the opposite way to all the other moons 
 and planets. His apparent diameter is only 4"; and 
 consequently his inclination and time of rotation are 
 not yet ascertained. But his spheroidicity is thought 
 to be as great as Saturn's ; and it is most likely that 
 his equator is nearly perpendicular to the ecliptic, as 
 the orbits of Jupiter's and Saturn's moons nearly coin- 
 cide with the equator of their planet. This also ac- 
 counts for his appearing sometimes spherical and some- 
 
N^twne. 179 
 
 times spheroidal, according as his pole or equator is 
 presented to us. The light and heat there must be 330 
 times less than here; gravity one-seventh less; and 
 the apparent diameter of the sun i' 42". 
 
 9- NeptunCi^Still the solar system was not exhausted, 
 as it was supposed to be when Uranus was so named. 
 All these planets were discovered by being seen, and 
 seen to move ; but one more was proved to exist with- 
 out being seen, and afterward found by looking for it 
 where the discoverers said it would be found. The 
 history of this diseoveiy is so remarkable, and at the 
 same time is given so imperfectly, and sometimes so 
 unjustly, in larger books than this, that I shall relate 
 it more fully than would otherwise be necessary, taking 
 my account of it from the most authentic source, viz., 
 the correspondence and statement of the Astronomer 
 Eoyal, in vol. xvi. of the Astronomical Society's Me- 
 moirs : which very few writers of the English language, 
 on either side of the Atlantic, and of course no French 
 ones, have apparently taken the trouble to read before 
 publishing their own versions of the transaction. 
 
 Just forty years after the discovery of Uranus, astro- 
 nomers began to complain that he did not appear in 
 his proper place, as calculated from the earlier obser- 
 vations, by which his orbit was supposed to be as well 
 ascertained as that of any other planet. And some 
 people went so far as to doubt whether Newton's law 
 of attraction might not be subject to variation at so 
 great a distance from the sun. You may be curious to 
 know what sort of error it was in the motion of Uranus 
 that caused so much uneasiness. By the year 1830 his 
 
180 Tlie Discovery of Neptune, 
 
 longitude, or distance from the equinoctial point T, 
 had got wrong by 30", which would naake him appear 
 wrong in his time of crossing the meridian of any ob- 
 servatory by two seconds of time ; and he had got 2' or 
 8 seconds wrong in 1845. This does not seem much 
 to be disturbed about in a planet 175 3 million miles oif, 
 or to make the laws of the universe suspected by some 
 astronomers, and the existence of an unseen disturber 
 of the peace of Uranus by others. But so it was ; and 
 it may give you some idea of the accuracy now expected 
 in astronomy. 
 
 ; The first person who appears to h ave openly suggested 
 the idea of Uranus being disturbed by a more distant 
 planet was the Rev. T. J. Hussey, of Hayes, who wrote 
 to the Astronomer Royal to that effect in JSTovembei", 
 i 834 ; and he said that two foreign astronomers, A. 
 Bouvard and Hansen, agreed with him'. But Mr. Airy 
 answered that he ' did not think the irregularity of 
 Uranus was in such a state as to give the smallest hope 
 of making out the nature of any external action on the 
 planet,' — if there was any, which he doubted ; and 
 preferred supposing that the earlier observations had 
 been wrong. In 1837 E- Bouvard, the nephew of A. 
 Bouvard, again wrote to Mr. Airy suggesting the same 
 cause ; who again answered (in substance) that he did 
 not believe it, and added — -'if it be the effect of any 
 unseen body, it will be nearly impossible ever to find 
 out its place.' By 1842 Bessel and other eminent as- 
 tronomers seem to have avowed the same opinion as 
 Dr. Hussey, but without convincing the English As- 
 tronomer Royal, who still appears to have had no 
 
hy Mr. Adcmis first. 181 
 
 solution of his own, except his guess at the inaccuracy 
 of observations, although the attention of astronomers 
 had now been directed to it for twenty years, and the 
 error had been getting worse. 
 
 But suggesting that there must be a planet somewhere 
 was a very different thing from setting to work to cal- 
 culate whereabouts in all space it must be, with a strong 
 presumption only, from the distances of the others, 
 that it would be nearly twice as far from the sun as 
 Uranus, and near the ecliptic like the rest. In 1844 
 Professor Challis, the head of the Cambridge Observa- 
 tory, wrote to ask the Astronomer Koyal for some 
 Greenwich tables of Uranus for a ' young friend of his, 
 Mr. J. 0. Adams, of St. John's College, who was at 
 work upon the theory of Uranus.' Mr. Airy of course 
 sent them ; and in September, 1845, Professor Challis 
 wrote again to say that ' Mr. Adams had completed 
 his calculations of the perturbations of Uranus by a 
 supposed ulterior planet.' In October,. 1845, ^i"- ■^^- 
 ams himself left at the Greenwich Observatory what 
 Mr. Airy justly called afterward, ' the important pa- 
 per,' giving the result of his calculations and the place 
 where the new pla/net woxild frobdhly he found. 
 
 Still Mr. Airy could not believe that a young man, 
 who had only taken his degree (of senior wrangler) the 
 year before had actually found the place of a planet 
 which he believed not to exist at all, and to be ' nearly 
 impossible to find ' if it did. So instead of encouraging 
 Mr. Adams, or taking steps to get the planet looked 
 for by the best telescopes of %'arious observatories, he 
 sent him a question which he called an experimentiim 
 
182 Mr. Adams mid M. Le Yerrier. 
 
 cruois, or what is popularly called a posing question. 
 He afterward expressed his ' deep regret,' not at hav- 
 ing done so, but that Mr. Adams did not answer it ; 
 and the world has not been informed what delayed the 
 answer : whether it required time to answer it as com- 
 pletely as Mr. Adams wished ; or whether the maker 
 of what has been called the greatest astronomical dis- 
 covery since Newton, felt that his announcement of it 
 might have been better received by the public repi'e- 
 sentative of English astronomical science. 
 
 But while Mr. Airy was waiting to believe in the 
 discovery, an eminent foreign astronomer stepped into 
 the field and confirmed it ; for in June, 1846, M. Le 
 Verrier gave to the French Academy his own independ- 
 ent calculations for a new planet, which nearly agreed 
 in their result with those which Mr. Adams had given 
 in 1845. As soon as they came here, Mr. Airy's doubts 
 vanished, although M. Le Verrier had no more an- 
 swered his question than Mr. Adams; and then he 
 confessed (what indeed his question showed before) 
 that he had doubted the accuracy of Mr. Adams's inves- 
 tigations until he received M. Le Terrier's confirma- 
 tion of them ; which does not mean that he considered 
 the calculations wrong in any definite way, but simply 
 that he doubted any man's ability to make them. 
 
 Then he did set to work to get the planet looked for 
 in the place indicated ; and Professor Challis at once 
 undertook the search with the great Cambridge tele 
 scope, and soon found what turned out to be the planet, 
 and he noted it as ' appearing to have a disc,' which 
 only planets have. But imfortunately he had no star 
 
Mr. Airy cm the Discover^y of Neptune. 183 
 
 map to compare his observations with ; and also delay- 
 ed comparing his own successive observations with each 
 other, until Dr. Galle of Berlin had not only found the 
 planet, but found that he had found it, on 23 Septem- 
 ber, 1846. Not that the finding of a planet where you 
 are rightly told to look for it is any great feat, or at all 
 parallel to Herschel's finding of an unsuspected planet. 
 Only one more sentence need to be quoted from Mr. 
 Airy's certainly candid statement to complete the story, 
 and to show how Mr. Adams and this country lost the 
 undivided credit of this great discovery, which he un- 
 questionably first made and first disclosed. Mr. Airy 
 says, ' I consider it quite within probability, that a 
 publication of the elements (of the planet's orbit) ob- 
 tained in October, 1845 i^"^^ given to him by Mr. Ad- 
 ams then), might have led to the (telescopic) discovery 
 of the planet in November, 1845,' seven months before 
 M. Le Yerrier disclosed his calculations. 
 
 The name of Neptune ^ was soon given to the new 
 planet, on the same principle as the others ; only they 
 were obliged to go back to the brother of Jupiter, as he 
 had no more ancestors in the Pagan mythology. 
 Neptune was afterward found to be 2745 million miles 
 from the sun, or 30-0363 times the earth's distance, and 
 to have a period of 60,127 ^^7^^ ^^ 1^41' years; and 41 
 of his revolutions take the same time as 81 of Uranus's. 
 He only goes 10,500 miles an hour, or just i as fast as 
 Mercury; and the sun appears there only i' in diame- 
 ter, or no larger than Yenus sometimes does to us. 
 Neptune's diameter is believed to be a little larger than 
 Uranus's, viz., 38,133 miles, and his bulk consequently 
 
184 N&plMne and Ms Satellite. 
 
 115 times the earth's. The present estimate is that 
 his weight is seventeen times the earth's, or that the 
 sun is about 18,780 times his weight. His orbit is 
 almost as circular as Yenus's, and i" 46' inclined to 
 the ecliptic. Nothing is known of his time of rotation, 
 or the inclination of his axis ; but though his apparent 
 diameter is only 2', he is said to be visibly spheroidal, 
 which implies a quick rotation. The light and heat there 
 cannot be above a thousandth of what they are here, and 
 gravity about a quarter less than on the earth's surface. 
 
 One satellite of ITeptune has been already discovered, 
 by Mr. Lassell, revolving in 5d. 2ih., and going more 
 distinctly retrograde, or opposite to the usual direction, 
 than the satellites of Uranus, because its orbit is only 
 29° inclined to the orbit of the planet. It will be 
 curious to ascertain whether his rotation is retrograde 
 also. It also turns out that the planet had been occa- 
 sionally seen before as a very faint star ; but as it was 
 never seen twice in the same place, the observations 
 had been hastily treated as mistakes ; a warning to all 
 men never to disregard any new fact, until they are 
 quite sure that it is not one, or is really unimportant : 
 beside that other warning to men in high places, which 
 the history of the discovery clearly enough proclaims. 
 
 10. There is yet one more planet suspected to exist", 
 a very small one, only 14 million miles from the sun, 
 and going round him in igf days, in an orbit 13" in- 
 clined to the ecliptic. It was discovered (if at all) by 
 a French physician named Lesearbault in 1859, and 
 M. Le Verrier seems to consider its existence not im- 
 probable. The name of Yulcan was assigned to it. 
 
Sujyposed Planet Vulcan. 185 
 
 But its exiBtence becomes more improbable every year 
 that passes without its being seen again. 
 
 Bode's Law of Distances. — I have several times spoken 
 of the distances at which the new i)lanets of this cen- 
 tury were expected to be found, from the proportionate 
 distances of the older ones ; and you will see in run- 
 ning over them that there is a rough approximation to 
 a successive doubling of the distances of all beyond 
 the iirst. If you divide them all by 9 millions, to get 
 rid of a great number of figures, they will come in 
 pretty nearly the following proportions : Mercury 4 
 Yenus 7, Earth 10, Mars 16, Ceres and Pallas and 
 some of the other asteroids 28, Jupiter 52, Saturn 100, 
 Uranus 196, IsTeptune 305. Or Earth is twice as far 
 beyond Mercury as Yenus is. Mars 4 times as far, As- 
 teroids 8, Jupiter 16, Saturn 32, Uranus 64 ; but ISTep- 
 tune is not 128 times as far, but only 102 ; that is, he 
 is 700 million miles too near according to this rule, 
 which is called Bode's law, from its discoverer. And 
 so the only thing like a rule which there was to guess 
 l^^eptune's distance by, turned out wrong. !N"everthe- 
 less it had been useful as a first approximation ; for it 
 is a common practice in astronomy, when several things 
 are unknown together, to assume a probable value for 
 one of them, and then correct thena backward after 
 some of them are found. Moreover the direction in 
 which JSTeptune was to be looked for was rightly cal- 
 culated, which was the main point. The error in dis- 
 tance afiected his apparent place very little, as he was 
 then nearly in a line with the sun and earth, and his 
 distance is 30 times the earth's. 
 
186 Bodis Lcmjo of Distomce's. 
 
 You must Tinderstand that this rule of Bode's is 
 only errypirical, which means founded upon trial or ex- 
 perience — more or less extensive as the case may be — 
 and has nothing to do with that great law of universal 
 attraction or gravitation which keeps the planets in 
 their orbits, moving with a certain velocity, and makes 
 their j^ears bear a certain proportion to their distances, 
 and enables their weights to be calculated from their 
 disturbances of each other and of their moons, and an 
 unknown planet to be found by calculation, and raises 
 the tides, and prevents the whole earth and the sea, 
 and everything that is in them, from flying to pieces 
 like the wringings of a mop, and running off in 
 straight lines into infinite space for ever. 
 
 Retrogradation of Planets, — Although the planets seen 
 from the sun would appear to go round him, as they 
 do, with vei'y little variation in their pace, they appear 
 from the earth to go very difierently and irregularly 
 in speed, and sometimes actually to go backward. The 
 angular velocities round the sun are inversely as the 
 periods : so the angular velocity of the earth is always 
 nearly 165 times Neptune's, and 84 times Uranus's, 
 and 2g^ times Saturn's, and 12 times Jupiter's, and 
 twice Mars's, and two-thirds of Venus's, and a quarter of 
 Mercury's. But the angular velocities round the earth 
 follow no such rule, and indeed no rule at all that can 
 be expressed without a great deal of calculation : all 
 we can do is to explain why both the interior and ex- 
 terior planets sometimes appear to us to retrograde in 
 their motion. 
 
 First take an exterior planet, Jupiter, and let him 
 
Why Plwnets appear Retrograde sometimes. 187 
 
 and the sun be on opposite sides of the earth, which is 
 called ' Jupiter in opposition ; ' and remember we have 
 nothing to do with the daily rotation of the earth for the 
 present. Then how does Jupiter appear to move ? He 
 is really going, like the earth, from west to east (looking 
 south) or from right to left, or opposite to the way the 
 hands of a clock go ; but he goes more slowly than the 
 earth — so slowly that he would fall behind the line of 
 equinoxes if he had started on that line ; which you 
 may consider to run through the earth and to be car- 
 ried along with it through the heavens, moving over 
 the infinite plane of the ecliptic, and always keeping 
 parallel to itself, but for the trifling change of the 
 ' precession.' Consequently Jupiter will then appear 
 to go backward, or in the opposite direction to the sun ; 
 not merely slower than the sun, but so much slower as 
 to appear retrograde in longitude : which is reckoned 
 forward up to 360° from the equinoctial point T for 
 celestial bodies, along the ecliptic, not along the equa- 
 tor from an arbitrary meridian like Greenwich, as ter- 
 restrial longitude is. So a slow-sailing ship appears to 
 go backward from a faster one which is passing it. Ju- 
 piter and all the other exterior planets accordingly are 
 ' retrograde ' for a few months nearly every year while 
 the earth is passing them. 
 
 The two inferior or interior planets do the same, 
 when they are between the earth and sun, which the 
 exteriors of course never are. It will be easier to un- 
 derstand if you disregard the curvature of the orbits, 
 and consider Yenus and the earth moving in parallel 
 lines eastward past the sun, as they do for a short time 
 
188 RdrogradaUcyn of Plcmets. 
 
 when Yenus is between the earth and sun. The eartli 
 moves eastward nearly 66,000 miles an hour; and 
 therefore the sun appears to move at that rate west- 
 ward, by the stars, though not by what we call ' east 
 and west ' on the earth ; or measuring by longitude in 
 the ecliptic, he advances 2^', the angle corresponding 
 to an arc of 66 on a radius of 91,404. But Yenus 
 moves 77,000 miles an hour eastward, and therefore 
 appears to move 1 1,000 the opposite way to the sun, or 
 retrograde ; and as she is then 25,292,000 miles from 
 us, she apparently recedes in longitude through the 
 angle corresponding to 11 divided by 25,292, or i^' an 
 hour, for some days every tim-e she passes between the 
 sun and earth, or at every synodical period (see p.108 
 and table at the end). On the other hand all the plan- 
 ets appear to go faster than they do on the opposite 
 side of the sun ; and at some intermediate places they 
 cannot be seen to move at all, or appear stationary. 
 
 SATTIEn's EING and SATELLirES. 
 
 Saturn's King, treating it first as single, is very thin 
 and flat, like a ring stamped out of a card, having an 
 outside diameter rather more than twice as wide as the 
 planet, or 166,000 miles.* Its inside diameter is 109,- 
 130, or only 17,660 miles from Saturn's equator, which 
 is 73,590 miles across ; and therefore the breadth of 
 the ring itself is 28,335 miles. It is so thin that it is 
 difficult to say what its thickness is ; but it is now con- 
 
 * I reduce all the measures of the rings given in the latest hooks, in the same 
 proportion as the diameter of Saturn, to suit the new sun's parallax. Different 
 hooks give rather different measures, and nothing turns upon their accuracy. 
 
SabwrTCs Ring. 189 
 
 sidered to be not more than loo miles. And the 
 strangest thing about it perhaps is that it seems con- 
 tinually to get thinner and wider, and sometimes breaks 
 out into fresh divisions apparently, besides the long 
 recognized and well marked division into two; of 
 which the outer ring is 9450 miles wide, the inner 17,- 
 300, and the space between them 1690. 
 
 If Saturn stood like Jupiter, with his equator (and 
 ring) nearly in the plane of the ecliptic, we should have 
 known next to nothing about the ring ; for we should 
 have seen nothing but two bright lines like handles, 
 each of them about as long as Saturn's radius. Some- 
 times the satellites appear like beads tlireaded on the 
 thin line of the ring. Sir W. Herschel found, by ob- 
 serving the motion of certain lumps or inequalities on 
 the edge of the ring, that that part of it at any rate 
 revolves round Saturn in loh. 32m. 15s. or 3 minutes 
 longer than his own'time of rotation. But as Saturn's 
 equator and ring are now* inclined 28° 10' to the 
 ecliptic, w© sometimes see it in oblique perspective, 
 and therefore as an ellipse, getting a very good sight 
 of the whole width of the rings and the spaces between 
 them each and the body of Satiu'n. If you want to 
 realize this, get somebody to hold a globe in the ordi- 
 nary wooden frame at some distance from you ; when 
 your eye is in the plane of the wooden horizon you can 
 
 * The inclinations of all the planets' equators to the ecliptic depend (1) on 
 their inclinations to their own orhits, (3) on the inclination of those to the 
 ecliptic, and (3) on the position of their nodes or equinoctial points ; but all the 
 nodes revolve bo slowly, and the inclinations of the orbits are so small, that I 
 do not complicate the descriptions with any further notice of these changes ; 
 but I have given the present longitudes of perfhelion in the table at the end. 
 
190 SatMrn'a Rvng. 
 
 only see its edge ; but when it is inclined a great deal, 
 you can see the whole of it except the part behind the 
 globe, and you see also the space between them. The 
 minor axis of the perspective ellipse of the ring when 
 most elevated is "47 of the major axis. 
 
 But the sun also must be elevated above the plane of 
 the ring, and on the same side as the earth, though not 
 necessarily as much, to illuminate the side of the ring 
 facing the earth, to enable us to see it. You will find 
 in Mr. Proctor's book pictures of Saturn and the rings 
 in all possible phases, with a vaj'iety of interesting de- 
 scriptions of them which it would be out of place to give 
 here. If Saturn has inhabitants they enjoy an eclipse 
 of the sun by the ring for about 1 5 years, or half Sat- 
 urn's period, over a considerable width of his surface.* 
 
 The mass of the ring has been calculated as the 1 1 8th 
 part of that of Saturn, from its [effect in disturbing 
 some of the satellites. And that agrees pretty well 
 with the estimate of 100 miles of thickness, on the as- 
 sumption that its average density is the same as Saturn's 
 own. There have been various speculations as to its 
 composition. Laplace proved that it could not be in 
 one solid flat piece, but must be at least divided into 
 two ; and the same kind of reasoning has made it 
 necessary to carry the division still further. For the 
 outer and inner parts of a ring of anything like that 
 width require different velocities and periods to pre- 
 serve their equilibrium, according to well-known laws 
 of motion which will be explained in the next chapter. 
 If the inner part only went as fast as the outer, its cen- 
 
 * Appendix, Note XIX. 
 
Its PrdbabU Constitution. 191 
 
 trifugal force would not be enough to keep it from be- 
 ing dragged into Saturn by attraction. For the ring 
 itself is far too thin and weak to be able to hold itself 
 together by its own cohesion or internal attraction, 
 against such a force as that tending to pull it in pieces. 
 Moreover the outer edge revolves in the proper time for 
 a satellite at that distance ; but too slow for one at the 
 inside, or even at the middle of the breadth of the ring. 
 But a far more serious objection to the rings being 
 rigid at all has been made by Prof Peirce, of Harvard 
 University.* Mr. J. 0. Maxwell, of Trin. Coll., Cam- 
 bridge, received the Adams Prize Essay of 1857 for 
 the disciission of a special case which was embraced in 
 Prof. Peirce's previously published and exhaustive paper, 
 viz., that, in order to preserve its equilibrium in revo- 
 lution, each ring must be so uneven in density that its 
 centre of gravity may be more than 9 times further 
 from the light side than the heavy one, if it is rigid ; 
 which is completely at variance with observation — un- 
 less there is that enormous and improbable latent dif- 
 ference in the opposite sides. Moreover the rings 
 woidd then be much nearer the planet on one side 
 than the other, which they are not, though they are a 
 little. You may ask how a ring can be stronger for 
 being limrvp than rigid, provided it is of the proper shape, 
 whatever that may be. The answer is that there is no 
 proper rigid shape, not even that defined by Mr. Max- 
 well : which certainly does not exist ; and if it did, it 
 could not maintain itself against disturbances. Conse- 
 quently a set of rigid narrow rings is no less impossible 
 than a single wide one. 
 
 * Appendix, Nole SX. 
 
192 Saturn^s Bmgs. 
 
 The idea has been therefore entertained that the 
 rings might be fluid, so that all their parts could move 
 along each other as they pleased. But to this also it 
 is objected that the constant changes of motion would 
 cause waves, which would break the rings in pieces, for 
 the effect of waves lasts long after the force has passed 
 away which raised them. Moreover the appearance of 
 the rings contradicts that theory. For beside the one, 
 or we must now say two, very, evident divisions, there 
 are indications of more, and especially of an inner 
 ring, discovered by Prof. Bond of Harvard University, 
 darker than the others, but semi-transparent, as if com- 
 posed of bodies near together but not too close to see 
 through. And in fact that is now taken by astrono- 
 mers to be the constitution of all the rings of Saturn. 
 Each ring, of only one satellite in width everywhere, 
 whether distinctly or indistinctly separated from the 
 rest, is thought to be a vast number of satellites mov- 
 ing in one orbit, and therefore in the same time; as 
 everybody has read lately there is reason to believe 
 that several rings of meteors travel round the sun in 
 orbits which cross ours at certain days in the year. 
 
 Another fact quite inconsistent with rigidity, is that 
 the rings as a whole have been getting thinner, and 
 about one-sixth wider, the inner parts coming nearer 
 to the planet, but the outer not altering, in the 200 
 years since Huyghens first discovered and jneasured it ; 
 and more rapidly in the 80 years since Sir "W. Herschel's 
 time than in the 120 years before. Galileo saw the 
 ring but could not make it out : he did distinguish 
 Jupiter's satellites with his second telescope in 16 10. 
 
Satwrn^s Moons. 
 
 193 
 
 Saturn's Moons are far less interesting than his rings. 
 It is enough to say here that the eighth and last in the 
 order of discovery, but the seventh in distance and a 
 very small one, was only found .in 1848, and oddly 
 enough, by Mr. Lassell in England and Mr. Bond in 
 America on the same night. The sixth is much the 
 largest, and so it has been called Titan in Sir J. Her- 
 sehel's re-naming of them all byname sinstead of num- 
 bers — Mimas, Enceladus, Tethys, Dione, Ehea, Titan, 
 Hyperion, lapetus. But though some of these were 
 the ' giants ' of the ancient world who fought with 
 Jupiter, they are very small satellites of Saturn now, 
 some too small to be seen except with the best telescopes, 
 lapetus is one of the larger ones, and much further 
 off than the rest, being 32 of Saturn's diameters distant 
 from his centre. Our moon's distance is 30 times the 
 earth's diameter ; but then Saturn's diameter is nine 
 times ours. Their distances and periods, which is all 
 that seems accurately known of them, are expressed 
 shortly in the following table. The first five follow 
 Bode's law pretty well, but not the others. 
 
 
 Distance j-eom Satubk. 
 
 Period. 
 
 Discovered. 
 
 
 In miles. 
 
 lu radii of ^ 
 
 I Mimas 
 
 123,633 
 158,590 
 196,480 
 251,680 
 352,390 
 814,840 
 985,380 
 2,367,600 
 
 3-36 
 4-31 
 5-34 
 6-84 
 
 9-55 
 
 22-14S 
 
 26-78 
 
 64-36 
 
 d. h. m. 
 
 22 37 
 
 1 8 S3 
 
 1 21 18 
 
 2 17 41 
 4 12 25 
 
 15 22 41 
 21 7 8 
 29 7 54 
 
 j. 1789 
 ■ 1684 
 
 2 Enceladus 
 
 3 Tethys 
 
 5 Khea 
 
 1672 
 
 6 Titan 
 
 1655 
 
 7 Hyperion 
 
 8 lapetus 
 
 1848 
 1671 
 
194 Jujpit&r's Moons. 
 
 The periods of Uranus's moons run from 2-53 to 
 1077 days, and their distances from 128,800 to 1,570,- 
 000 miles; the last three distances nearly doubling 
 each other, and the first five differing by about 50,000 
 miles each. But two at least of them are doubtful. 
 
 Jupiter's Four Satellites are of far more importance 
 than Saturn's eight. They are all of a substantial size : 
 the first, second, and fourth about as large as our moon, 
 but a great deal lighter, having one-third, two-thirds, 
 and half the moon's density ; and the third having a 
 diameter and weight about half as much again as the 
 moon, and three times her bulk. That satellite also 
 has two-thirds of the density of the moon, or two-fifths 
 of the earth's density, and half as much again as Jupi- 
 ter's own. These variations of density among the 
 planets and their satellites are remarkable, following 
 no apparent law whatever. 
 
 Their distances from the centre of Jupiter are respec- 
 tively about 6, 9^, 1 5f , and 27 times his radius ; which 
 do follow Bode's law very well, the differences being 
 successively about 3, 6, 12. Consequently the first ap- 
 pears to people on his surface (if there are any) about 
 as large as our moon does to us, the second and third 
 abont half as wide, and the fourth a quarter as wide, 
 or one-sixteenth as lai-ge. But the largest of Jupiter's 
 moons has only the 11,300th of his mass, while our 
 moon is nearly one-eightieth of the earth. Their ele- 
 ments are more fully given in the table at the end of 
 the book. 
 
 They all move so nearly in the plane of Jupiter's 
 orbit (which also nearly coincides both with his equa- 
 
Their Mutual Distwrbam.ces. 195 
 
 tor and our ecliptic) that they are eclipsed every time 
 they pass, except that the fourth escapes sometimes. 
 And they, like our moon, always show thd same face 
 to their planet. The largest of them looks rather less 
 than Neptune, and the apparent diameter of the small- 
 est, at our mean distance from them, is under i". 
 
 The period of the first is 42^ hours, of the second 
 85i, of the third I72f, and of the fourth i6d. i6^h. ; 
 so that the periods of the first three are successively a 
 very little more than double of each other. And from 
 that coincidence of periods several remarkable conse- 
 quences follow. Whenever the second and third are 
 in conjunction, the first is exactly opposite to them ; 
 and the place for conjunction of the first and second 
 will be opposite to that for the second and third. Con- 
 sequently they cannot all be either eclipsed from the 
 sun, or hidden from us by Jupiter, or made invisible 
 by being in front of his disc with the sun shining on 
 both it and them. But two may be eclipsed or hidden 
 while the other is so made invisible ; and as the fourth 
 may be anywhere, that may also be invisible at the 
 same time in any of those three ways. One of these 
 very rare Tion-pJienomena took place on 21 August, 
 1867, from 10.4 to 10.49 ^- ^• 
 
 Another consequence is, that all the three orbits are 
 mutually afiected by the disturbances, or strongest 
 mutual attractions, constantly recurring at the same 
 place for the first and second, and at the opposite place 
 for the second and third (which aggravates the efiect 
 upon the second). The effect is to make the first and 
 second orbits ellipses with the apses dways on the line 
 
196 Effect of a Plcmefs Oblateness 
 
 of conjunctions ; and a kind of secondary major axis 
 or pair of apses of the third orbit also : only that hap- 
 pens to have a larger independent eccentricity of its 
 own in another direction. But \}a.Q jperijove of the first 
 and the apojove of the second are at their place of con- 
 junction ; and the second's perijove and the third's 
 •apojove are at their conjunction ; or the three orbits 
 are pushed away from each other at the two places of 
 conjunction. 
 
 But the periods are not exactly double of each other ; 
 and that makes the line of conjunctions and of apses 
 revolve slowly backward as each conjunction comes a 
 little before the place of the last ; and it takes 486^ 
 days for them to work round, which makes 68 periods 
 of the third satellite, and 137 of the second, and 275 
 of the first. This is something like the ' great inequal- 
 ity' of Jupiter and Saturn (p. 174): indeed this is re- 
 latively greater, for it takes many more periods of the 
 disturbed bodies to work round. Moreover here every 
 conjunction falls near the same place, but only every 
 thwd conjunction of Jupiter and Saturn, and the two 
 intermediate ones rather counteract than aggravate 
 the effect. This subject is worked out at greater length 
 in Mr. Airy's G-ravitation than we can afford to it here ; 
 and it is not an easy one. , 
 
 The satellites of Jupiter, and Saturn too, suffer 
 another disturbance from the great oblateness of those 
 planets, and from Saturn's ring. Let us see how the 
 attraction of a sphere on a satellite in the plane of its 
 equator is altered by shaving pieces off the poles and 
 laying them round the equator, so as to make it into 
 
On its Satellites. 197 
 
 an oblate spheroid. The piece laid on the side nearest 
 to the moon will attract more than it did at the pole, 
 because it is both brought nearer and also into the 
 direct line of attraction, or the line of centres. The 
 piece laid on the far side loses force by being put fur- 
 ther off, though not quite so much as the otfeer gained 
 by being put nearer (see p. 134) and gains by being 
 brought into the line of centres. The attraction of the 
 pieces which are moved from the poles to the equidis- 
 tant sides of the equator is not sensibly altered. There- 
 fore on the whole the equatorial attraction of the sphe- 
 roid is greater than of the sphere, on a moon at the 
 same distance from the planet's centre, though it is less 
 on his own equator, being further off (p. 37). 
 
 The distance of the first moon being 6 times Jupi- 
 ter's radius, his attraction on it would be thus increased 
 by nearly a 24th of his ellipticity, or a 408th, and 
 would shorten its period an 81 6th, if his outside were 
 as dense as his inside : which it is not (p. 1 72), and there- 
 fore the effect is less. (This calculation does not hold 
 for attraction on the surface, nor at small distances). 
 The effect is less on the fourth moon, which is 27 radii 
 off, in the proportion of 6^ to 27^, or not one 20th as 
 much. Therefore also each moon is less accelerated at 
 apojove than perijove ; and that corwpa/rative loss of 
 attraction at apojove makes their apses advance, as the 
 greater loss of earth's attraction at apogee than peri- 
 gee makes the apses of our moon advance (see p. 152). 
 
 The spheroidicity of the earth produces similar 
 effects on the moon, but too small to be appreciable in 
 her period, because the earth is much less oblate than 
 
198 Time of Light coming from the Sun, 
 
 Jupiter, and our moon's distance is 60 times the earth's 
 radius. But on the other hand, she goes much higher 
 aboTe the equator than his moons do, viz. 28J° ; and 
 consequently she suffers another disturbance in return 
 for her disturbing our polar axis by nutation (p. 55). 
 For the polar attraction of an oblate spheroid at a 
 given distance falls short of that of an equal sphere 
 twice as much as the equatorial attraction exceeds it, 
 though the attraction at the pole is greater (p. 37). 
 Therefore the earth''s attraction on the moon is greatest 
 wlien she is on or near the equator ; and thus she is 
 disturbed both in latitude and longitude by the earth's 
 ellipticity : which can be calculated from the amount 
 of these disturbances, and is foimd to agree with the 
 result obtained by other means. 
 
 The eclipses of Jupiter's moons are seen on his right 
 or left side according as the earth is right or left of the 
 line from the sun to Jupiter, which prolonged is the 
 axis of his shadow. Besides the eclipses by his shadow 
 and OGGultations behind him, the satellites may trcmsit 
 over his face, either as dark or bright spots, according 
 to the position of the sun, and according as they pass 
 over a dark belt of Jupiter or a bright part of his face ; 
 or they may cast their shadows as dark spots upon him 
 while their bodies appear bright, either on his disc or 
 beyond it. 
 
 Velocity of Light Discovered, — Some years after Gali- 
 leo had discovered the satellites, their eclipses were 
 observed always to come too soon when Jupiter Was 
 at his nearest to the earth, and too late when he was 
 furthest off. The extreme difference between the early 
 
<mdj Longitvde found hy J'wpUer's Moons. 199 
 
 and the late eclipses was i6m. 26s., a variation far too 
 great to be tolerated or attributed to mistakes even in 
 the early days of astronomy. Accordingly Romer, a 
 Danish astronomer, in 1675 hit upon the solution that 
 the light itself takes some definite time to come, and 
 spends those 16J minutes in coming across the whole 
 width of the earth's orbit, or the difiereiace between 
 the nearest and the furthest distances of Jupiter and 
 his moons from us. Therefore the light takes SJ min- 
 utes to come here from the sun, which is at half tliat 
 difiierence of distances. This does not tell us what the 
 velocity of light is until the sun's distance is ascer- 
 tained, which we shall see how to do presently ; or if 
 the velocity of light is found by other means (see p. 86), 
 this will tell tis the sun's distance. But we have not 
 quite done with Jupiter's moons yet. 
 
 Finding the Longitude.— The longitude of a place is 
 simply the difference between the local clock time and 
 the clock time of Greenwich, turned into degrees at 
 the rate 4 min. to 1° (p. 19). The time of anyplace 
 is that of a clock which is at 12 when the sun is on the 
 meridian of the place, subject to the correction called 
 the equation of time, which is the same for all places, 
 and is given in the almanacs for every day (see p. 58). 
 The time of noon at any place may be found from the 
 fact that the sun is equally high above the horizon, and 
 shadows are of equal length, at equal times before and 
 after noon; besides other methods involving more 
 mathematics, and by the stars as well as the sun. The 
 beginning or end of an eclipse of a satellite of Jupiter, 
 which happens very often, may be used as a common 
 
200 Finding the Longitude. 
 
 signal to be observed at Greenwich and the place in 
 question, as much, as if it were a rocket visible to them 
 both, or the local time of one telegraphed to the other. 
 Instead of the eclipses being observed at Greenwich, 
 they are calculated for Greenwich time, and published 
 several years beforehand in the ]!^autical Almanac; 
 and so when you have observed the local time of an 
 eclipse anywhere, you can at once see its difference 
 from the Greenwich time. But after all this method 
 is little used in navigation, because these eclipses can- 
 not be observed accurately enough on board ship, and 
 it is superseded by that of 
 
 Longitude by Lunar Distances.— The apparent dis- 
 tance of the moon from the sun, or any convenient 
 star, can be measured by the seaman's instrument, 
 called a sextant, first invented by ISTewton, in which 
 you see one object coinciding with a second reflection 
 of the other from two small mirrors,, when one of them 
 is turned through the proper angle, which you can 
 then read off; and a man can hold it in his hands 
 steadily enough.* This apparent distance has next to 
 be turned into true distance, which means the apparent 
 distance as it would be seen from the earth's centre 
 (see p. 21 1), by calculations for which tables are pro- 
 vided : and the local time of the observation must be 
 taken. The true distance of the moon from the sun 
 and some suitable stars is given in the Nautical Alma- 
 nac for every 3 hours of Greenwich time tlirough the 
 year ; from which the Greenwich time corresponding 
 to the observed distance (reduced to true) can easily be 
 
 * See Newton's paper in Ph. Tr. vol. 43, and Sextant in English Cyc. 
 
Finding the LaUlmde of a Place. 201 
 
 found ; and its difference from the local time of the 
 observation is the longitude of the place. 
 
 If you have some chronometers keeping Greenwich 
 time, no other observation is wanted but that of the 
 local time of noon, or of any star transit whose Green- 
 wich time is in the almanac. It is not safe to rely on 
 one watch, and if two differ you do not know which is 
 most right ; but three give a tolerably safe average, 
 if they are good ones ; and of course the more you have 
 the better. Sometimes a great many are used for fix- 
 ing the longitude of important places, and ai-e carried 
 backward and forward between them and Greenwich. 
 The parliamentary reward of £20,000 was given to 
 John Harrison in 1767, for making the first chrono- 
 meters that would find the longitude within 30 miles 
 after a long voyage ; and they have been very much 
 improved since then. The makers send chronometers 
 to the Koyal Observatory for six months' trial, and the 
 Admiralty buys some of the best of them every year. 
 But the trials are too much confined to variations of 
 temperature, far beyond what the chronometers are 
 practically exposed to, even in Arctic expeditions : see 
 the Rudimentary Treatise on Clocks, p. 299. 
 
 Finding the latitude is a more straightforward opera- 
 tion, and there are several ways of doing it. For the 
 latitude of a place is the angular distance of the equa- 
 tor from the zenith, along the meridian ; which is equal 
 to the sun's zenith distance at noon, added to his 
 declination or distance from the equator, or 90° — his 
 pola/r distance, which is in the almanac for eveiy 
 day. 
 
202 Measurmg the Sun's Distcmce 
 
 TEA^SITS OF VENUS AND MEEOUET, AND MEASUEINQ 
 
 OF sun's distance. 
 
 Though Mars and Yenus have no moon, they some- 
 times have the shape of one themselves, and Mercury 
 too ; only he is too small and too near the sun for his 
 shape to be often seen. But Mars never has the cres- 
 cent shape, because that can only be the phase of a 
 body between the earth and sun, which Yenus is some- 
 times, just like our moon, only much further off. Mars 
 can appear no narrower than gibbous (seep. io6), as the 
 moon does in the second and third quarters, or the 
 fortnight with the full moon in the middle. I have 
 already said that the sun's distance has lately been 
 corrected by cei-tain observations of Mars, but the de- 
 termination of it from the transits of Yenus is consid- 
 ered to be more accurate, and is one of the most im- 
 portant and curious problems in astronomy, though 
 Yenus is very seldom available for it. Mr. Airy calls 
 it ' a noble problem, deserving attention as well for its 
 ingenuity as the certainty of its conclusions ; ' but it is 
 now certainly concluded that a mistake was made by 
 one of the observers the last time it was done. He 
 adds that ' it is one of the most dilficult subjects for a 
 lecture that he knows.' Nevertheless we will try what 
 can be done toward making it intelligible, by an ex- 
 planation somewhat different from his, though of course 
 ending in the same result. 
 
 Iliad better first tell you that 'Ohq ^opcrrUonate dis- 
 tances of the sun, earth, and Yenus, to the diameter of 
 the sun can be easily ascertained without waiting for 
 
h/ a Tra/nsit of Vmus. 203 
 
 the rare phenomenon of transits. The proportion be- 
 tween the sun's diameter and distance is nothing more 
 than his apparent diameter, 32' ; for that one figure 
 expresses to those who understand how to reduce de- 
 grees and minutes into numbers, that the sun's diame- 
 ter is to his distance as the length of an arc of 32' is to 
 an arc equal in length to the radius of the circle, or 
 57° 18' ; and the numerical value of 32', or the sun's 
 diameter divided by his distance, is '00935. From that 
 you may easily calculate that the radius (= i) is io8*2 
 times the length of 32', and therefore we are cer- 
 tain that the sun's mean distance is io8'2 times 
 his diameter, whatever that may be. Next the ap- 
 parent or proportionate distance of Yenus' from the 
 sun is found, as I have said before, by observing the 
 angle between Venus and the sun when they are widest 
 apart ; and that being done on various occasions, when 
 she is at different distances from the sun, her whole 
 orbit is known ; and therefore her proportionate dis- 
 tance from the earth and sun is known at the time when 
 the transit is going to happen. And we may say, in 
 round numbers, that the sun is then twice and a half 
 as far from Venus as the earth is, though we yet know 
 nothing of the actual length of any of their distances. 
 A transit of Venus over the sun is exactly the same 
 in principle as an eclipse of the sun by a very small 
 moon, or what is the same thing to us, a very distant 
 one ; and like our eclipses, it can only happen when 
 Venus is at or very close to one of the nodes where 
 her orbit crosses our ecliptic, exactly at the same time 
 that she is crossing between the earth and sun, which 
 
204 Measwing the Sim's Distance 
 
 is called inferior oonjv/nction: (Superior conjunction 
 is when the planet passes behind the sun, but we have 
 nothing to do with that). Now if two men stand be- 
 fore a post with a wall behind it, they will see different 
 places on the wall eclipsed or hidden by the post ; and 
 if the post is as far from the two eclipsed places as it 
 is from the men, the two eclipse^4vill be exactly as far 
 apart as the two meii are ; if the wall is twice as far 
 from the post, the two eclipses will be twice as far 
 apart, and so on. 
 
 Therefore two people on the earth, as far apart as 
 they can conveniently get for them both to see the 
 transit of Yenus from beginning to end, will see at the 
 same time the two transit spots twice and a half as far 
 apart in real distance on the sun as the observers are 
 distant from each other. Suppose they are 7200 miles 
 apart (measuring through the earth the shortest way) 
 then the two transit spots will be 18,000 miles apart 
 on the sun ; and we have only one step more to take in 
 order to find the diameter of the sun in miles ; and 
 that. is, to get an accurate map made of the disc of the 
 sun with the exact positions of the two spots at the 
 same time ; for then we can measure their distance on 
 the map and see what proportion it bears to the diame- 
 ter, and we know that 18,000 miles bears that same 
 proportion to the real diameter of the sun, and the 
 business is done. 
 
 The real difficulty is to get this sun-map made accu- 
 rate enough to measure from, or to get the exact dis- 
 tance of the spots at the same moment, remembering 
 that the two observers are nearly half way round the 
 
lyy a Transit of Venus. 205 
 
 earth from each other. For that purpose the following 
 contrivance is adopted. Instead of observing the tran- 
 sit at one moment only, each man observes the whole 
 path of Venus across the sun ; or rather in reality he 
 observes the exact time it takes / for they can observe 
 the first and last contact of the spot far more accurately 
 than they can measure distances on the bright face of 
 the sun ; and it is not necessary that they should see 
 anything but the beginning and the end of the transit. 
 The places on the earth are so chosen that the paths 
 may appear not only parallel, but at the widest dis- 
 tance possible apart, forming two chords across the sun, 
 parallel to the diameter which Yenus would pass along 
 if she was exactly in the ecliptic and seen from the 
 centre of the earth. The two paths may be on differ- 
 ent sides of the sun's centre if Venus is exactly at a 
 node, but they are more likely to be on the same side, 
 in which case their difference of length is greater, and 
 the observations more likely to give an accurate result. 
 For the accuracy of the map depends on this : you 
 have a circle of known diameter to start with, because 
 the time Venus would take to cross the middle of the 
 sun is known from the proportion which his diameter 
 bears to the orbit of Venus, and the time she takes to 
 perform it. So if that time were known to be 6 hours 
 we might draw a circle of 6 inches diameter for the sun ; 
 and if one observer reported his transit to have lasted 5 
 hours, we should find the place where a chord 5 inches 
 long will exactly fit ; and if the other transit lasted 5i 
 hours, we should put in another chord si inches long, 
 parallel to and near the former. (The real lengths 
 
206 Transits of Venus and Mercufry. 
 
 could not be exactly these, but that does not signify). 
 The distance between'two chords of 5 and 5 J inches in 
 a circle 6 inches wide can be calculated with the ut- 
 most accuracy, and also the proportion of that distance 
 to the diameter, which is the proportion of the 18,000 
 miles to the real diameter of the sun, the thing we 
 wanted. 
 
 I have said nothing about the rotation of the earth 
 during the time the transit lasts ; bat of course due 
 allowance has to be made for that by methods known 
 to astronomers. What I have told you embraces all 
 the principles of the calculation ; which was first sug- 
 gested by James Gregory in 1663 (who made the first 
 reflecting telescopes, different from Newton's) for both 
 Mercury and Venus. But Mercury is more difiicult to 
 observe, and it was long before it could be applied to 
 Venus : for her last transit had been in 1639, the year 
 Gregory was born, and they only come in pairs of 8 
 years separated by io5f and 121^ alternately. The 
 last were in June 1761 and 1769, and the next will be 
 in December 1874 and 1882. No doubt provision will 
 be made both by England and France for having the 
 observations made at several pairs of distant places, in 
 case any of them should be clouded, with greater ac- 
 curacy than appears to have been obtained in 1769, 
 when the celebrated Captain Cook w:ent to Otaheite 
 for the purpose at the expense of George the Third ; 
 and other people went to Lapland, where the begin- 
 ning of the transit was observed in the evening of a 
 short night, and the end the next morning : the mis- 
 take is supposed to have been made there, and not at 
 
Parallax of Sim and Moon. 207 
 
 Otaheite. In 1761 the observation at St. Helena was 
 spoilt by clouds. 
 
 Transits of Mercury are much more frequent, com- 
 ing at intervals of 7 and 1 3 years ; but the same cause 
 which makes them more frequent makes them less use- 
 ful, viz., Mercury's being nearly twice as near the Sun 
 as Yenus is, and twice as far from us ; which makes 
 the breadth between the two transit paths four times 
 as narrow as for Yenus, and much more than foiu* 
 times less likely to give a correct measure of the sun's 
 diameter and distance. 
 
 sun's paeallax, xnb moon's. 
 
 Another thing which is found as part of the same 
 operation in a transit of Yenus, is the apparent diame- 
 ter of the earth as seen from the sun ; half of which 
 angle is called the sun's parallax ; and that is merely 
 the earth's equatorial radius of 3963 miles divided by 
 the sun's distance of 91,404,000, which represents an 
 angle of 8"'943. Until lately it was considered 8"-57, 
 but it is increased one 23d with the diminution of the 
 sun's distance by one 24th, as stated at page 87. This 
 parallax, rather than the sun's distance, is usually taken 
 as the standard measure for the whole solar system, 
 though of course they go together. The parallax of 
 Yenus, or the apparent radius of the earth seen from 
 Yenus at her nearest to us, is about five times the sun's 
 parallax, being in the inverse proportion of their dis- 
 tances from the earth. All the measures of the solar 
 system may be calculated from the parallax of Yenus 
 or. Mars, or any other of the planets, as the proportion- 
 
208 Parallax of Sun and Ma/rs, 
 
 ale distances of them all are known independently, and 
 only one real distance is wanted to supply the scale for 
 measuring them all. 
 
 The parallax of the sun or moon then is the converse 
 of their apparent semi-diameter, being the earth's ap- 
 parent semi-diameter to them I'espectively. The moon's 
 mean parallax is 57' 2", and her mean semi-diameter 
 15' 33"} ^^<i ttie sun's mean semi-diameter 16'. 
 
 As I have said that the moon's distance is measured 
 independently of the sun and any planet, I ought to 
 tell you how it is done — in the best way, for there are 
 several, as you may see in Mr. Airy's lectui-es and 
 other books. Two observers as far apart as they can 
 be for them both to see the moon at once, will see her 
 apparently covering two different places among the 
 stars behind her, as a post covers different places in the 
 wall behind it from different observers; that is, she 
 will appear to each of them at a different distance from 
 the same star pretty near her ; which distance is mea- 
 sured, as usual, by the angle between the two lines of 
 sight from the observer to the star and to the moon's 
 centre. When all proper cori'ections have been made, 
 the difference of those two angles will be the apparent 
 distance of the two observers from each other as seen 
 from the moon's centre. Then that differential angle 
 must be a little increased, in proportion as their real 
 distance falls short of the diameter of the earth ; and 
 half the angle so increased will be the apparent radius 
 of the earth as seen from the moon, and that is the 
 moon's parallax. 
 
 You may ask why the sun's parallax should not be 
 
ly Olservations m 1862. 209 
 
 got in the same way, without calling in the aid of Ve- 
 nus. Because stars near the sun cannot be seen, even 
 with a telescope, and stars far from tlie sun will not 
 do, for various reasons. Ifi sunlight all objects are un- 
 steady, from the variation of the density of the air, as 
 will be explained with Eefraction, and as you may see 
 in a very hot day, and still better near a fire in the 
 open air ; and the unsteadiness is magnified by tele- 
 scopes, so that the sun's distance from a star, either far 
 or near, could not be accurately measured. Other 
 methods have been used, but they are not accurate 
 enough for modern astronomy. 
 
 The parallax of Mars however can be got with con- 
 siderable accuracy in the same way as the moon's, 
 though he is 200 times further ofil When Mars is in 
 conjunction with the earth, or opposition to the sun 
 (shortly called ' Mars in opposition ') he is only 48 
 million miles off, and may be under 44, if he is in op- 
 position and perihelion together, as the eccentricity of 
 his orbit is '093. (The earth's aphelion is so situated 
 that we cannot get the benefit of the eccentricities of 
 both orbits at once.) This happens every 8 years, and 
 it was a little before the last opposition at perihelion, 
 in 1862, that suspicion had begun to attach to some of 
 the transit of Venus observations of 1769, and to tlie 
 received distance of the sun. Advantage~^as taken 
 of that favorable position of Mars to get his parallax 
 by a series of observations at places as far off as possi- 
 ble in latitude and near the same meridians, such as 
 Greenwich, and Melbourne in Australia, Pulkowa, 
 and the Cape of Good Hope, where an observatory 
 
210 Pwrallaas in General. 
 
 has been established for observing the southern hemis- 
 phere. 
 
 The result was that all the observations agreed in 
 giving that parallax, and therefore distance, for Mars 
 which makes (by proportion) the sun's distance nearly 
 91^ million miles : which is curiously about midway 
 between the distance obtained by similar observations 
 of Mars in 1672, and those of Venus a century after- 
 ward. It must be remembered that although a single 
 observation of Mars' parallax is not equal to one of a 
 transit of Yenus, properly taken, we are not limited to 
 one or two of Mars in a century, but a great many may 
 be taken at each opposition. No less than 60 were got 
 in Australia in the two or three months while Mars 
 was conveniently situated in 1862. And it may now 
 be said for the first time in the history of astronomy 
 that there are both many and independent proofs of 
 the sun's distance being what is received by astrono- 
 mers ; for several others beside these and the velocity 
 of light (p. 86) have since been added, and transits of 
 Venus are no longer of the importance that they were ; 
 though it is highly desirable that the late corrections 
 should be tested by this, which is undoubtedly the best 
 single method of all. 
 
 ' Parallax ' is also used for other things, both in the 
 heavens and in the earth. The parallax of the fixed 
 stars is always referred, not to the radius of the earth, 
 but to the radius of the earth's orbit, of which I shall 
 have more to say hereafter. Parallax may be defined 
 to be the apparent displacement of a body from shifting 
 your place of observation, and you may say that it va- 
 
Aberration. 211 
 
 ries inversely as the distance, when the distances are 
 great. Thus if you see a fire at some unknown dis- 
 tance to the north, and if it still looks north when you 
 have gone a good way east or west, you may be sm-e it 
 is a long way oif, because its parallax is small. So too 
 the long hand of a church clock does not really point 
 to 1 5 or 45 minutes when it seems to do so, but a little 
 below them, because the hand is not close to the face, 
 and so is displaced by parallax from the position you 
 would see it in if you were on a level with it. 
 
 All astronomical observations, except those for find- 
 ing a parallax, have to be corrected for the earth's 
 parallax, or as it is called shortly — parallax, or the dis- 
 tance of the observers from the earth's centre. Other- 
 wise the observations made and recorded at one ob- 
 servatory would be unintelligible and useless at any 
 other. By that correction they are ' reduced ' to the 
 earth's centre as a common point for all observations 
 alike. We have now to consider two other corrections 
 which have to be made before any observation can be 
 considered complete, and in a state fit to be recorded 
 for future use. 
 
 ABEEEATION AND EEFEACTION. 
 
 The stars and sun and planets are apparently dis- 
 placed, or are seen in wrong places, from another cause 
 called dberraUon. The discovery of it by Bradley in 
 1727, the year that JSTewton died, was a consequence 
 and confirmation of the previous discovery, that light 
 takes a definite and measurable time to come from the 
 sun and planets ; not that the planets have any light 
 
212 Aberration. 
 
 of their own, but only reflect the sun's, as the moon 
 does. "We may explain abeiTation thus : If you are 
 running when the rain comes down straight without 
 any wind, you get wet in front and not behind, and the 
 rain beats against you as it would if you were standing 
 still and the wind blowing in your face. And if you 
 carry an empty telescope tube pointed straight up, the 
 rain will not fall through it, but will strike against the 
 back inside : if you want the rain to fall through, you 
 must slope the tube forward, more or less according to 
 your velocity forward compared with that of the rain 
 downward. Then for rain substitute light, and the 
 motion of the earth for your own running, and you 
 know what aberration is. 
 
 Therefore whenever we are moving directly across 
 the light from any star, it appears before its true place 
 by 20" j which = '00001, or the proportion of the ve- 
 locity of the earth to that of light (pp. 36, 86). When 
 we are approaching a star, or receding straight from it, 
 there is evidently no aberration. Therefore stars near 
 the ecliptic oscillate 40" backward and forward every 
 year apparently, in consequence of the real motion of 
 the earth in its orbit. But stars near the poles of the 
 ecliptic are carried round a whole circle of aberration 
 40" in diameter, neglecting the insignificant ellipticity 
 (not eccentricity) of the earth's orbit. All other stars 
 have their aberration circle foreshortened into a per- 
 spective ellipse, wilii a major axis of 40" lying across 
 the line to the earth, and the minor axis diminished 
 according to the star's latitilde or distance from the 
 ecliptic, where the ellipse sinks into the line of oscilla- 
 
Eefraction of the Air. 213 
 
 tion. But aberration always diminislies the sun's lon- 
 gitude by 20"; for we see him by the light which left 
 him 8J minutes ago when he was 20" behind his pres- 
 ent place. 
 
 The November meteors, which I shall describe pres- 
 ently, afford a much stronger case of aberration. 
 They go the opposite way to the earth, and at rather 
 greater speed, in an orbit inclined only 17° to the eclip- 
 tic. Therefore they meet the earth with quite as much 
 velocity as if it stood still and they went twice as quick 
 or twice as far forward in a second from any given 
 height above the ecliptic ; that is, at the apparent in- 
 clination of 8° or 9° instead of 17°. 
 
 Refraction. — There is yet another correction to be ap- 
 plied to most telescopic observations before they can 
 be said to have given the true place of a star, as it 
 would appear from an earth no bigger than a point, 
 quite still, and without an atmosphere to bend the rays 
 out of a straight line. For this is what the atmosphere 
 does, and it is called refraction. If you hold a straight 
 stick obliquely in a trough of water, the part in the 
 water appears to be bent upward, or further from up- 
 right, with an elbow at the surface of the water, and 
 the trough itself looks less deep than it does when it is 
 empty ; so that if you stand where you cannot quite 
 see the bottom of the trough empty, you will be able 
 to see it when it is full. And conversely, if your head 
 were under water all things outside of it would appear 
 elevated. 
 
 The reason is, first, that you see everything in the 
 direction of the rays as they at last reach the eye, by 
 
214 Refraction of the Air. 
 
 whatever road they have come, as things seen in a mir- 
 ror appear to be behind it ; and secondly, the rays of 
 light are always bent toward the perpendicular in the 
 denser of two mediums which they pass through 
 obliquely. Consequently anything in the water is seen 
 by rays in the air more oblique than those which 
 started from it in the water, and so it appears lifted. 
 Again, as air is denser than empty space, though very 
 much thinner than water, the rays which come to us 
 from the stars are bent toward the pei-pendicular, and 
 so they appear higher than they are, except when they 
 are already as high as possible, or in the zenith. And 
 the sun and moon always do, since they are never in 
 our zenith ; and the lower they are tlie more they are 
 raised by refraction. For the law of refraction is, that 
 the distance of any point of a ray from the perpendi- 
 cular to the surface in the second of the two given 
 transparent mediums always bears a fixed proportion 
 to what the distance of the same point would have 
 been if the ray had not been bent aside or refracted. 
 This is shortly expressed to those who know a little 
 trigonometry by saying that the sine of the angle of 
 incidence bears a fixed proportion to the sine of the 
 angle of refraction. I shall have to say more about 
 this in the chapter on telescopes. 
 
 Besides this, the density, and therefore the refraction 
 of the air decreases upward, as you may see by taking 
 a portable barometer up a mountain ; so much that, 
 although an atmosphere of uniform density and of the 
 known weight of 15 lbs. on the square inch would 
 only reach 5 miles high, it does in fact reach about 
 
Refraction of the Air. 215 
 
 80.* Consequently the rays are not bent with a single 
 elbow, like the image of the stick in water, but into a 
 curve, continually getting more upright as they approach 
 the earth, and you see the star in the direction in which 
 the rays at last reach the eye. Moreover, the lower the 
 star is, the more obliquely the rays enter the air, and 
 the more they are bent. Eefraction is also diminished 
 by heat, as that expands the air and makes it thinner, 
 and it increases with a rise of the barometer, which 
 indicates increased density in the air, and so it can only 
 be found and applied by tables which have been pre- 
 pared from long experience. Mr. Airy calls ' refrac- 
 tion the bane of astronomers,' because it cannot be 
 calculated with certainty, like aberration and paral- 
 lax. 
 
 The average amount of it for objects about half way 
 up from the horizon to the zenith is about i', but at 
 the horizon it is as much as 33', which is rather more 
 than the apparent diameter of the sun or moon. Con- 
 sequently tliey have really set, or have not risen, when 
 they appear to be just above the horizon, being lifted 
 their whole height by refraction ; and the moon may 
 be totally eclipsed with the sun apparently above the 
 horizon, though the earth is really straight between 
 them. From the same cause the French cliffs, and 
 even ships on the sea near them, can be seen from the 
 English coast in some states of the atmosphere. There 
 is no I'efraction sideways, and it increases so rapidly 
 toward the horizon, that the lower edge (or limb, as 
 they call it) of the sun or moon is lifted rather more 
 
 * Appendix, Note ATI. 
 
216 Swn and Moon look SmaUer nea/r Horizon. 
 
 than the tipper, and therefore they do not appear quite 
 round, but visibly broader than high when they are 
 just rising or setting, and yet not exactly elliptical, but 
 more flattened at the bottom than the top. 
 
 Their appearing larger at the horizon to the naked 
 eye is only an optical delusion ; for in fact they appear 
 smaller, as the vertical diameter is diminished by 
 refraction and the horizontal one is not increased. The 
 delusion is generally attributed to our being able to 
 compare them at the horizon with things on the earth. 
 But I doubt if this is the proper explanation ; for it is 
 equally the case when it is too dark to see anything 
 but the moon itself just rising, and at sea, where there 
 is nothing else to be seen. Most people have observed 
 how much larger a man looks against the horizon on 
 the top of a hill than when you are at the top and he 
 is at the bottom. I believe the reason is that when 
 things are on the horizon we compare their linear di- 
 mensions with the length of horizon which the eye 
 takes in, but that in the middle of the sky or earth we 
 compare the area they cover with the area the eye 
 takes in. Assuming the eye to see distinctly over 30° 
 of apparent width, the moon on the horizon covers one 
 60th of that ; but when it is high up it only fills the 
 3600th of the area of sky which the eye sees all round 
 it. Between these two extremes some compromise is 
 involuntarily made, and a different one by different 
 people's eyes. 
 
 Another theory has lately been propounded, that the 
 enlargement of the sun and moon on the horizon is due 
 to a peculiar efifect of the red rays on the eye, which 
 
Cause of Twilight. 217 
 
 then evidently preponderate, from the greater absorp- 
 tion of the other colors by the atmosphere. But 
 whether right or wrong, that theory seems yet in a 
 very imdeveloped state. 
 
 From another cause the moon really measures alto- 
 gether less when she is rising or setting ; that is, when 
 we are just coming into or going out of sight of her. 
 For the parts of the earth to which the moon is just 
 rising may be further off by half the earth's diameter 
 than the places full in front, which have the moon on 
 their meridian ; and the earth's radius is one 6oth of 
 the moon's distance. Therefore her diameter will ap- 
 pear one 6oth less, and her whole disc about one 30th 
 less to the places furthest off, which can just see her, 
 than to the places nearest. But the difference of the 
 earth's radius is practically nothing in the distance of 
 the sun, and therefore you cannot say that the sun ap- 
 pears smaller on the horizon than on the meridian, as 
 the moon does, except from refraction. 
 
 Twilight is also caused by the air, but not in the same 
 way as refraction. "When the sun is not more than 
 about 15° below the horizon his rays are a little re- 
 fleeted down to us from the vapors and other small 
 partieles of matter in the air. The more obliquely the 
 sun goes down the longer he takes to get as low as 15" 
 below the horizon, and consequently twilight lasts 
 longer in high latitudes than within the tropics, where 
 the sun's path may be vertical or through the zenith at 
 noon, and is quite so in some latitude there every day. 
 The blueness of the sky is also due to the reflection 
 of the violet rays by the air, from some unknown 
 10 
 
218 Meteors of August. 
 
 cause; and 
 and greate 
 altogether. 
 
 cause ; and is therefore greater when the sun is low, 
 and greatest at night, when he is below the horizon 
 
 METEOKS OE SHOOTING 8TAES. 
 
 As these bodies are of the nature of very small plan- 
 ets, they may properly be considered next. Early in 
 the morning of the 14th of November, 1866, '■'Can great 
 November star shower ' came according to prediction. 
 Every year at that time there is a shower of meteors 
 or shooting stars ; but every 33 years it has a maximum. 
 Prof. H. A. Newton, of Tale College, first propounded 
 what may be called a meteoric theory, which Mr. Ad- 
 ams has lately confirmed, as you will see presently.* 
 
 Besides the November shower there is another on 
 August 10 every year, which I will speak of first, be- 
 cause its period is more simple, agreeing within a min- 
 ute, from the earliest Chinese Annals (in which they 
 are recorded nearly 1000 years ago) with the length 
 of the sidereal year, or absolute revolution of the earth 
 without regard to the precession of the equinoxes ; and 
 therefore these August meteors come a day later in 71 
 equinoctial years. It does not appear that they have 
 a maximum of 33 years or any other time. Their 
 eflfect would be produced either by a continuous ring 
 of meteors revolving round the sun in any period and 
 any orbit which crosses the earth's orbit at the place 
 where it now is on the loth of August : or by a mere 
 cluster of meteors revolving in a similar orbit with ex- 
 actly the sidereal period of the earth ; for if such a 
 
 ♦Appendix, Note XXI. 
 
Tlie November Meteors. 219 
 
 cluster once meets the earth where their orbits cross, 
 it must always do so, until the revolution of apses of 
 the orbits throws them out of contact. 
 
 If there is a continuous ring of meteors the earth 
 must fall in with them every time it crosses the ring. 
 If it were not only a ring, but a plate of meteors as 
 wide as the earth's orbit, as the zodiacal light (p. 71) is 
 thought to be, then we should fall in with it again 
 about February 10: which is not the case. Even a 
 continuous ring would be crossed again in February, 
 unless it is an ellipse differing so much from the earth's 
 orbit that they only intersect once, and miss each other 
 where they pass at the opposite half of each orbit, which 
 is probably the case. 
 
 AU the meteors of each shower appear to radiate 
 from one point, or more correctly, from one small cir- 
 cle or ellipse. The l^ovember meteors radiate from 
 such a point in the constellation Leo. But this is only 
 the effect of perspective. The earth drives through, 
 and nearly at them ; for they go the opposite way to 
 the earth, and cross the orbit so obliquely, that we 
 practically meet them, and with more than twice the 
 earth's velocity. If a quantity of rockets were shot 
 toward you from the far end of a street, they would all 
 appear to radiate outward from the veiy small space 
 which the width of the far end of the street fills in 
 your eye ; except the few which came straight at you, 
 and they would appear to stand still ; and so do a few 
 of the meteors. So they appear to radiate from that 
 point of the meteoric ring, or the stars beyond it, 
 toward which the earth is going just then, or whose 
 
220 Their ^Radiant Fomf 
 
 longitude is 90° less than the sun's. The height of the 
 meteoric shower will not occur until we have come in 
 sight of that point in the sky toward which the earth 
 is moving, i.e., until those stars have risen; which 
 must be near midnight, just as the moon rises at mid- 
 night when she is 90° behind the sun. It begins about 
 1 5 minutes sooner in the south hemisphere than here, 
 because that part of the earth touches the inclined plane 
 of the meteoric ring first. 
 
 All the phenomena of the ' November ' showers ' ap- 
 peared to suit Mr. Newton's hypothesis of a thin ring 
 of meteors, with a longish lump in it, all revolving the 
 opposite way to the earth in 3S4'62 days in a nearly 
 circular orbit rather smaller than ours, inclined 17° to 
 it and crossing it where the earth is on November 14 
 now ; — 34 such periods =33 years ; and so we fall in 
 with the great shower every 33 years, and sometimes 
 the next year also on account of the length of the 
 lump, but in other years we cross the thin part only. 
 The nodes recede with reference to the direction of the 
 meteors, or advance in longitude, 52"'4 a year : which, 
 with the precession of our equinoxes 5i""i, makes the 
 meteors a day later nearly every 34 years (only an ac- 
 cidental concurrence with the other 34). 
 
 But Mr. Adams has calculated that the nodes of 
 such an orbit would not recede 52" a year under the 
 disturbances of the planets, but only 21"; and that the 
 orbit which does satisfy all the conditions makes the 
 period of each meteor those same 33*25 years which 
 have been the average period of the great shower ever 
 since 13 October, 902 (= 18 Oct. N. S.), with an ec- 
 
The Meteors are Burnt in the Air. 221 
 
 centricity 54 times greater than the earth's orbit, and 
 a major axis iO"34 times greater, reaching therefore 
 beyond Uranus and crossing our orbit near the meteoric 
 perihelion. Consequently the thickest part of the ring 
 crosses our orbit once in the 3 3 '2 5 years instead of 34^ 
 times (though we meet it only once) according to the 
 other theory. As the great shower lasts two hours and 
 the ring is 17° inclined to the ecliptic, it must be about 
 35,000 miles thick. And from the necessary velocity 
 in such an orbit that thick part of the ring must be 
 above 800 million miles long near perihelion (p. 261), 
 as we sometimes cross it two years running; and the 
 whole orbit is 4420 million miles long. This is very 
 like some of the comets' orbits ; and the August mete- 
 ors are now thought to have a much longer orbit even 
 than this. Astronomers have already identified the 
 orbits of two known comets with these two meteoric 
 rings.* 
 
 The meteors of all the meteoric days have much the 
 same character. They are mostly very small, Mr. 
 Herschel says few above a pound in weight, and many 
 only a few grains. Their average height above the 
 earth when we first see them appears to be about 75 
 miles, and they generally disappear at 50 miles, in 
 consequence of being burnt up by the heat they gene- 
 rate by friction in passing through the air with a 
 velocity of 2500 miles a minute. That is at once the 
 cause of their brilliancy and the reason why so few of 
 them are found ; and those that are found bear marks 
 of intense heat and of their outsides having been melted. 
 
 * SeeE. A. S. Monthly Notices of April, 1867. 
 
322 Different Classes of Meteors. 
 
 Some of them may pass by the earth altogether and 
 continue their orbit round the sun. 
 
 The number of meteors in one of the really great 
 showers is enormous: the one of November, 1866, had 
 only about 8000 at Greenwich, which is quite scanty 
 compared with some that are recorded, in which 
 300,000 were said to be reckoned at the same place ; 
 and they are so near the earth that observers at places 
 a good way apart must see a different set of meteors. 
 
 There is a different class of much larger stones called 
 aerolites and fireballs, which come singly and at no 
 fixed times, and generally in the afternoon, it is said, 
 while the meteor showers are chiefly after midnight. 
 Accounts of the doings of such stones in various ages 
 of the world have been published lately. Not a few 
 people have been killed by them, and buildings set on 
 fire. Their roaring noise, like their heat, is only 
 caused by their rushing through the air. They gene- 
 rally contain a great deal of iron and bismuth, and 
 none of them any substance unknown upon the earth. 
 
CHAPTER Y. 
 
 THE LAWS OF PLAlvTETAET MOTION. 
 
 "We will now return to the planets and their satel- 
 lites, to consider the laws by which they all move, an^ 
 may move for ever, by virtue of certain consequences 
 of their laws of motion, and the small eccentricity of 
 their orbits, which enables them to recover from every 
 disturbance, and makes all theinequalities only period- 
 ical, and the equilibrium of the system stable. 
 
 Kepler's lawSi — I have ah-eady mentioned one of those 
 three laws which Kepler found by observation that all 
 the planets follow, and which Newton afterward proved 
 by mathematics that they must follow, under the uni- 
 versal law of gravitation. The one I mentioned was, 
 that the planets move in ellipses round the sun, in one 
 focus of the ellipse. 
 
 The second is that the radius vector from the sun to 
 any planet sweeps over equal areas in equal times in 
 the same orbit, though not in different orbits ; therefore 
 going faster when the planet is near the sun than when 
 it is further off, as I said about the moon and Mercury 
 at pp. 115, 165. This is called the lavf of conservation 
 of OA^eas ; and it is only the same thing in other words 
 as saying that the angular velocity in any given orbit 
 varies inversely as the square of the distance. 
 
 For the area of any narrow sector of the orbit, be- 
 
224: Conservation of Areas. 
 
 tween two radii so near together that we may consider 
 the bit of orbit a straight line, is half the numerical value 
 of the small angle between them x the product of the 
 two radii — or x the square of either of them, as they 
 do not sensibly differ when close together ; and as that 
 small angle described in an unit of time is the angular 
 velocity, and varies (as may be proved by mathematics) 
 inversely as the square of the radius vector, it follows 
 that the area described in an unit of time, and there- 
 fore in any time, is invariable in the same orbit ; but 
 not in different orbits, under different original forces of 
 projection. In fact the central force disappears in the 
 calculation of the area, which is left depending only on 
 the original distance and velocity across the radius 
 vector. Therefore the radial disturbing force (p. 149) 
 does not affect the areal velocity of the moon, though 
 it increases her distance and diminishes her angular 
 velocity. But the tangential force, so far as it is not 
 balanced on opposite sides of the orbit, does affect the 
 areal velocity ; and its gradual decrease, with the in- 
 crease in the sun's average distance, retards the moon 
 in the way discovered by Mr. Adams (p. 1 5 1). 
 
 Although this law of angular velocity looks so like 
 the law of gravitation in another form, it has nothing 
 at all to do with it, and would be equally true if the sun 
 attracted at the tenth power of his distance, or in any 
 other way. Therefore finding that the planets follow 
 that law proves nothing more than that the sun is their 
 centre of attraction according to some law or other. 
 Kepler's two other laws do depend on the attraction 
 varying inversely as the square of the distance. 
 
Kepler's Law of Time cmd Distcmce. 225 
 
 The third of them is, that if you take the distances 
 of any two planets from the sun, and cube them, and 
 square the number of days or hours in their periodic 
 times, the two cubes will bear the same proportion to 
 each other as the two squares. Thus, omitting frac- 
 tions, the cube of Mercury's distance in millions of 
 miles is 35'= 42,875 ; and of Yenus's 66^— 287,496, 
 which is about 6f times 42,875. Then the square of 
 Mercury's period in days is 88''=7744, and of Venus's 
 225^=50,625, which again is 6|- times 7744. And so 
 you would find it with any of the other planets, or with 
 Jupiter's or Saturn's moons. 
 
 Gravitation and Inertia. — We do not know at all what 
 the force called gravitation is, by which all bodies at- 
 tract each other, any more than we know what mag- 
 netism or electricity is, by which pieces of iron in a 
 certain state attract each other still more visibly and 
 strongly at short distances. What we do know is that 
 that force is universal, and acts even on invisible gases, 
 such as common air or steam, just as it does on either 
 the planet or the metal Mercury, or makes an apple 
 fall to the ground because its stalk has got too weak to 
 resist the earth's attraction. If you squeeze more air 
 into a bottle by an air pump it weighs heavier in a 
 balance ; and weight (of things on the earth) is their 
 mass multiplied by the attraction of the earth. One 
 thing is heavier than another because it contains more 
 of what is called ' matter,' which only means the stuff 
 that gravitation acts upon, and which by its inertia 
 resists disturbance (see p. 28). For it is quite an inde^ 
 pendent law of nature, that the inertia of bodies, or 
 10* 
 
226 Gravity and Inertia. 
 
 their power to resist attraction, is exactly proportionate 
 to their quantity of attraction ; or that their mass mea- 
 sured by either test is the same. For anything we 
 know, attraction might have been in proportion to the 
 surface, or something else which does not vary as the 
 inertia, as electrical attraction perhaps is : that certainly 
 does not vary as the mass. 
 
 The air on the top of a mountain is notoriously thin- 
 ner and lighter, or there is less of it in a cubic foot, or 
 in your lungs at every breath, than at the bottom. 
 Why? Because the air near the earth is squeezed 
 closer by the weight of all the air above it, just as a 
 great heap of feathers is denser at the bottom than at 
 the top, and also by the greater attraction nearer to the 
 centre of the earth. I have already said (p. 67) that 
 half of all the mass or weight of the air is contained 
 within the height of some of the highest mountains, 
 and that it decreases in density as it gets higher, until 
 it becomes so thin that its force of expansion as a gas 
 is balanced by the attraction of the earth. For the 
 force which resists gravity in the air, and prevents it 
 from being all squeezed down flat upon the earth, is 
 the strong disposition of all airs or gases to expand and 
 keep themselves loose. It is that which makes gun- 
 powder explode, when the solid powder is suddenly 
 changed by heat into a gas which wants to fill 4000 
 times the space, and has strength enough to burst open 
 anything that resists it. Water turned by heat into 
 steam acts in just the same way, only it is done more 
 gradually. , 
 
 There can be no doubt that the theory of gravity 
 
How Gra/oitation Acts. 22T 
 
 varying as the inverse square of the distance was sug- 
 gested to Newton, and indeed to Kepler who guessed 
 at it before him, by wliat takes place with light and 
 heat, and generally with all emanations in rays or 
 straight lines. For the rays spreading out in all direc- 
 tions will evidently cover a space twice as wide and 
 twice as high, when they have got to twice the distance 
 from the centre from which they radiate. And as the 
 space covered by the rays is four times as large, it must 
 be covered only one-fourth as thickly as an equal space 
 at half the distance. 
 
 But though this analogy was enough to suggest the 
 law, and make it worth trying with the prodigious 
 labor which Newton bestowed upon it, to see if it 
 would explain all the motions of the planets, the 
 analogy is a long way from explaining the cause or ac- 
 tion of the law itself. For you observe that this law 
 of the effect of emanations varying inversely as the 
 square of the distance depends upon the size and posi 
 tion of the body which is to receive them. If you 
 turn a board obliquely to the fire it gets less heat, ex- 
 actly as a circle is diminished into an ellipse in perspec- 
 tive ; or if you double up the same board or a plate of 
 metal into a narrower one, it will also get less heat in 
 the same position . But gravitation cares nothing about 
 shape or position : every bit of matter in the universe 
 has found out every other bit of matter, and is now 
 attracting it according to its distance. Of all this 
 there is nothing more to say but that it is the univer- 
 sal law of nature. All matter was created under this 
 general condition, just as thg various kinds of matter. 
 
228 How Attraction is Measured. 
 
 lead, iron, air, water, trees, corn, and all other things, 
 whether simple or compound, were created with special 
 conditions or laws of nature of their own, which they 
 always follow without hesitation or variety. 
 
 But though nobody knows why all bodies attract 
 each other, or what gravitation is, we know very well 
 the laws by which it acts, viz., that the attraction of 
 one body on another varies as the mass of the body 
 which we treat for the time as the attracting one, and 
 inversely as the square of the distance of their centres 
 of gravity if they are far apart, whether they are 
 spheres or not. The attraction which we deal with in 
 astronomy is measured by the velocity which tlie 
 attracting body produces in a second of time in the at- 
 tracted one. But the proportion in which each moves 
 the other absolutely, or with reference to their common 
 centre of gravity, if they are both free to move, is the 
 inverse proportion of their masses : that is, the earth 
 moves the moon eighty times as much as the moon 
 moves the earth ; and the common point between them, 
 round which they both really turn, is their centre of 
 gravity, which is eighty times nearer the earth's centre 
 than the moon's, as I explained at page 86. 
 
 So all the planets really move the sun, but in an in- 
 sensible degree, as the weight or mass of them all is 
 only about a 700th of the sun, and omitting Jupiter, 
 the sun is 2500 times as heavy as all the rest. If they 
 all piilled together in the same line, the c. g. of the 
 whole solar system would still fall within the sun, as 
 that of the earth and moon is within the earth. And 
 Kepler's law of times and distances, to be stated quito 
 
Correct La/m of Time cmd Distmice. 229 
 
 correctly, say for Mercury and Venus, requires a doixble 
 rule of three sum, such as this : — The square of Venus's 
 period is to the square of Mercury's as the cube of 
 Yenus's distance multiplied by mass of Sun and Mer- 
 cui-y is to the cube of Mercury's distance multiplied 
 by mass of Sim and Yenus. We might omit the mul- 
 tiplication if Mercury and Yenus were equal, which 
 tliey are not ; or if they were both insignificant com- 
 pared with the sun, which they are ; or even if their 
 difference was insignificant compared with the sun, 
 which it is still more. So Jupiter's moons are too 
 small to affect their own periods sensibly, but our moon 
 is not ; and so we have to ' reduce the earth to rest ' in 
 order to calculate the moon's motions, as described 
 at p. 98. 
 
 But it may occur to you to ask, why does not a thing 
 which weighs a pound at a yard from the earth weigh 
 only a quarter of a pound at two yards, if attraction 
 varies as the inverse square of the distance ? Because, 
 as I told you at page 29, the attraction of the earth on 
 anything outside it does not depend on the thing's dis- 
 tance from the surface, but from the centre, where the 
 whole mass of the earth may be supposed to be con- 
 densed for all purposes of attraction, except where its 
 spheroidal shape has to be taken into account. And 
 no distance that we can move to makes any sensible 
 difference in the distance from the earth's centre, 
 except by such delicate tests as the time of clock pen- 
 dulums, as in the experiments for weighing the earth. 
 
 I will now try to explain, as far as it can be done 
 without mathematics, how gravity keeps the planets in 
 
230 Original Prelection of the Plwnets. 
 
 their orbits, instead of either letting them run away 
 beyond the power of the sun to pull them back again', 
 or gradually pulling them closer and closer, till they 
 fall into the sun. 
 
 Besides the creation of the sun and planets and their 
 moons, and of this law of gravity, one thing more must 
 have been done for them, since they certainly could 
 not do it of themselves. They must have been projected 
 or set going, both turning on their axes with the ve- 
 locity they do turn with, and moving forward with 
 the velocity which enables them to resist the force 
 which tries, to drag them into the sun. I say moving 
 forwa/rd, because the only motion which they needed 
 to start with v/as projection forwards in a straight line 
 across the direction in which the sun attracts them. 
 The laws of motion could do all the rest. For that 
 was enough to start the contest between the two forces 
 which are always contending against each other — the 
 force of original projection, under which they are al- 
 ways trying to run off in a straight line, and the force 
 of attraction which is always drawing them away from 
 that line, and bending their course toward the sun. 
 
 "We tails: of planets being ' projected,' ' set spinning,' 
 and so forth, not becaluse any astronomer believes they 
 were suddenly started by the Creator in that way, or 
 that their motions are not the result of some very sim- 
 ple original disposition of the matter of the universe 
 and the laws ordained for it ; but merely because the 
 present condition of things is what it would have been 
 if the planets had been so projected and set spinning 
 under the known laws of nature. 
 
First and Second Laws of Motion. 231 
 
 THE LAWS OF MOTION, AND ELLIPTIC OEBITS. 
 
 The first of l^ewton's tliree fundamental laws of 
 motion, on which the science of dyna/mics is fonnded 
 (which word means the effect of force in producing 
 motion), is that a body once in motion will go on for 
 ever in the same straight line and with the same ve- 
 locity, till some new force turns it aside. So the force 
 of original projection keeps a planet always moving on 
 across the line between it and the sun, and produces 
 what is known as the centrifugal force ; and attraction 
 is the second force, which is always drawing it aside 
 from a straight line into a curve. Tou see the same 
 thing in a small way in every stone you throw : it 
 wants to go forward in a straight line, by the first law 
 of motion, but is constantly drawn out of it by the at- 
 traction of the earth ; and so it moves in a curve, and 
 at last falls to the earth because the attraction is too 
 strong for any velocity we can give. it. If you throw a 
 ball across the wind, the wind also bends its course 
 into another curve sideways ; so there you have three 
 forces acting, and producing a motion compounded of 
 them all, in a ' curve of double curvature.' 
 
 The second law of motion explains the action of 
 these double forces, or of any number of forces. That 
 law is, that a second force draws a body aside from the 
 straight line which it is taking under the action of the 
 first, just as much in any given time as if it had not 
 been moving at all, or had been moving with any other 
 velocity. The consequence is, that at the end of the 
 time it is found just where it would have been if the 
 
232 Second and Third Laws of Motion. 
 
 first force had first sent it to its place in the first 
 straight line, and then the second had suddenly carried 
 it off from there in another straight line ; or conversely, 
 as if the first force had sent it forward in no time at 
 all, and the second had then moved it in its own time. 
 And that again comes to the same thing as if the body 
 moved along the diagonal of a parallelogram (or a four- 
 sided figure with its opposite sides parallel) of which 
 one side is the course it would have taken in the given 
 time under the first force alone, and either of the two 
 cross sides represents the course it would have taken 
 under the second force alone. This is called the com- 
 position or the parallelogram of forces. The effect of 
 any third force is added in the same way, taking the 
 diagonal which represents the resultant of two of the 
 three forces, to form one side of another parallelogram, 
 and the third force as the cross side ; and so on. But 
 here we have no occasion to consider more than two. 
 
 The truth of this second law of motion may be 
 shown by a machine which is so contrived as to shoot 
 one ball forward by a spring at the same time that it 
 drops another ; and though one ball of course travels 
 much further than the other, you may hear them both 
 fall on the floor together. So if you drop anything in a 
 railway carriage going forty miles an hour, it falls as fast 
 as it would if the carriage were standing still, though 
 it has really moved in a curve combining the forward 
 motion of the carriage with the downward motion un- 
 der gravity. 
 
 Newton's third law is, 'Action and reaction are 
 equal and opposite ; ' or the planets attract the sun as 
 
Momentvm, <md Vis Viva. 283 
 
 much as he attracts them, though they do not move 
 him as much. It has been proposed to substitute for 
 this the following : ' The velocity generated in a given 
 mass by any force in an unit of time is proportional to 
 the force.' But that is really part of Newton's second 
 law and has no relation to the third. 
 
 What are called ' moving ^forces ' are measured by 
 the product of the mass moved x the velocity gene- 
 rated in the unit of time : which is a second. That 
 product is also called momentum,. "Velocity means 
 the rate at which the body would go on moving uni- 
 formly if no more force wei'e applied to it, and is al- 
 ways measured in feet in calculations of forces. 
 
 This measure of moving force, or the force a body 
 moves with (as if we were calculating the force with 
 which the earth would strike the sun) differs from the 
 measure of gravity or attraction at page 28, which 
 disregards the mass of the body moved. That is the 
 measure used in astronomy, and in most problems of 
 motion ; for we generally want to find the motion of a 
 body in a given time, and not its momentum. Attrac- 
 tion ietween two bodies, measured either as a moving 
 force, or as the pressure they would produce on a spring 
 between them, is the product of the two masses, divid- 
 ed by the square of their distance ; while the acceler- 
 ating force of attraction is the mass of the body whose 
 motion we are not considering, divided by the square 
 of their distance. If we want to treat one of them as 
 fixed, instead of the centre of gravity of the two, we 
 must reckon the accelerating force on the other as tlie 
 s^im of the two masses divided by the square of their 
 
234 Moment of Inertm. 
 
 distance, as I explained at p. 98 about the earth and 
 moon, and in all the calculations of periods. 
 
 The momentum of a moving mass is also called its 
 ' quantity of motion.' But the square of the velocity 
 X the mass, or more conveniently, x half the mass, 
 is called its ' force of motion ' or vis viva / which is 
 the measure of the work done or capable of being done 
 by such a weight moving with such a velocity, Jsefore 
 it is all used up. For that vis viva of a body falling 
 may be proved to = its weight (or mass x the force 
 of gravity) x the height it has fallen ; which would of 
 course raise another equal weight through the same 
 height, as a measure of the work done. Again the vis 
 viva of rotation, as of a fly-wheel or of the earth in 
 raising the tides (p. 144) is half the mass of each parti- 
 cle X its own velocity", or x its distance" from the axis 
 x the velocity" of rotation : which therefore depends 
 on the shape and density as well as the weight ; being 
 evidently greater the more the body spreads out from 
 the axis of rotation. 
 
 And as ' forces of translation,' which merely drive a 
 body along, are resisted by the inertia of its mass, so 
 rotatory forces are resisted by a rotatory inertia called 
 moment of inertia, which is the sum of the masses of each 
 particle x its distance" from the axis. So that in each 
 ease the vis viva or force of motion is the proper inertia 
 x velocity" of translation or rotation. That is why a 
 large thin grindstone is both harder to drive and harder 
 to stop than a small thick one of the same weight. 
 The moment of inertia of a sphere or spheroid is | of 
 what it would be if its mass were all condensed into a 
 
Centrifugal = Centripetal Force in a Circle. 235 
 
 thin band round its equator ; and therefore varies as 
 the sqiiare of the equatorial radius. The time of rota- 
 tion of a shrinking globe also decreases as the square 
 of that radius ; because each particle obeys the law of 
 conservation of areas (p.223). Therefore the day would 
 shorten an hour if the earth contracted a 48th. But 
 the day has certainly not shortened since the earliest 
 recorded eclipses ; and therefore the earth has not 
 shrunk in that time at any rate. 
 
 If a stone could be thrown fast enough, the attraction 
 of the earth would not be able to bring it to the earth, 
 but only toward it, pulling it aside from the straight 
 line in which it would go through space if there were 
 no attraction ; and so it might go round the earth for 
 ever, like the moon. 
 
 You know that centrifugal force is that which stretches 
 the string by which anything is swung round. But the 
 word centrifugal, or centre-avoiding, gives some peo- 
 ple an incorrect idea of what that force is. It is not 
 really a force which makes things fly away from any 
 centre, but makes them go straight forward whenever 
 they are set at liberty. If you let the stone go out of 
 a sling when it is at the highest or the lowest part of 
 its swing, it does not go straight up or down, but for- 
 ward, in the direction of a tangent to the circle in 
 which it was swinging. So what is called centrifugal 
 force is really due to nothing but the desire the body 
 has to move on in a straight line M'ith the velocity 
 which it has at each moment, according to the first law 
 of motion: we shall calculate its amount presently. 
 The force which draws a planet away from the straight 
 
236 Circulwr Orbits Possible. 
 
 line in which it desires to fly ofij brings in the second 
 law of motion, and is called the centripetal, or centre- 
 approaching force ; which of course is the attraction 
 of the sun. 
 
 Therefore if a planet is ' projected,' or at any time 
 finds itself moving, exactly at right angles to its radius 
 vector, as a circle is everywhere at right angles to its 
 radius, and with a velocity producing a centrifugal force 
 which exactly balances the sun's attraction at that 
 distance, it will go on moving in that circle until some 
 other force comes to disturb it. If that had been so, 
 and if there were no disturbances instead of many, we 
 need say no more ; and then we might never have 
 ' known what the law of gravitation is ; for planets could 
 revolve in circles undisturbed under any conceivable 
 law of force. But their state of equilibrium might be 
 such that the smallest disturbance would either send 
 them into the sun, or else away from him for ever. We 
 know that none of the planets do move in circles, and 
 that their velocity and direction are never both together 
 such as a circular orbit would require ; and therefore 
 we must donsider what sort of orbit they will describe in 
 those circumstances. 
 
 Let us take the most simple case of a planet found 
 moving at any distance from the sun which does not 
 vary for a short time, but with a velocity too great for 
 a circular orbit of that size. Then the distance must 
 inevitably begin to increase very soon, just as a weight 
 swung round by an elastic string will stretch it and fly 
 further out if you increase the velocity. But will it 
 merely describe a larger circle, as the weight does,? 
 
Besolution of Forces. 237 
 
 No : because the force of the sun diminishes as it gets 
 further off, but the elastic force of the string increases ; 
 and if the velocity were too great for the attraction in a 
 small circle, it will be still more too great for the 
 wealier attraction in a larger circle. The next ques- 
 tion is, will it go further and further off as the attrac- 
 tion gets weaker, and so describe a spiral widening 
 out into infinity ? That question cannot be answered 
 without mathematics, beyond merely saying that it will 
 not describe such a spiral under the existing law of 
 gravitation, though it would under another not very 
 different. We can however show in a general way 
 how the planets escape running off into spirals, and 
 describe ellipses instead, and how they manage to get 
 away again from the sun after approaching nearer and 
 nearer to him with his attraction increasing. We can- 
 not get beyond this without mathematics ; but you will 
 not be surprised at that when you know that Newton 
 had to invent a new method of calculation before he 
 could prove that the orbits must be elliptical. 
 
 Elliptic OrbitSi — I have several times spoken of a 
 force being ' resolved ' into some direction oblique to 
 itself. As two forces can be compounded into one, 
 which is called their resultant (p. 232), so one force 
 can be divided or resolved into two in any directions 
 we please ; except that there can be no resolved part 
 of any force in a direotion at right angles to its action. 
 That is the reason why we need not consider the reso- 
 lution of forces in circular motion ; for the motion in a 
 circle is always at right angles to the radius, or to the 
 direction of the central force. But in an ellipse it is 
 
238 EUiptioal Orbits. 
 
 not so, except at the two apses. If you look at this 
 figure of an ellipse, you see the motion of the planet 
 
 P in either direction, 
 toward T or # (both in 
 the tangent to the el- 
 lipse), is not perpen- 
 dicular to PSj the line 
 to the sun. So that 
 if it is going in the 
 direction PT, part of 
 the sun's force goes to 
 increase the Telocity in 
 that direction, while 
 the other part is pulling it out of that direction. The 
 sun's force then may be resolved into two forces, one 
 in the tangential direction PT, and the other in the 
 direction TS perpendicular to PT ; and these two forces 
 are proportionate to those two lines. On the other 
 hand, if the planet is going away from the sun toward 
 t, part of the sun's force resolved in the same way goes 
 to diminish the planet's velocity in the direction P#, 
 and the other part toward drawing it aside out of the 
 direction P^. 
 
 Then while it approaches the sun S, or while the 
 radius vector SP diminishes, the force represented by 
 PT continually augments the velocity, till at last at 
 some point A the velocity becomes so great that the 
 centrifugal force first balances and then beats the cen- 
 tripetal ; and so that point becomes an apse, or a place 
 where the radius vector changes from decreasing to in- 
 creasing, or vice versd. So we may say that the sun 
 
Spiral Orbits, and Ellipse Round Centre. 239 
 
 enables the planets to run past him at perihelion by 
 having made them run faster toward perihelion — only 
 not straight toward him. The converse of this takes 
 place when the planet is approaching aphelion, in a 
 direction like Pa (but on the lower side of the figure, 
 or reversed). Then the sun has been gradually reduc- 
 ing the velocity by that part of his force which is re- 
 solved into the direction PT, now acting against the 
 increase of the radius vector. At last the velocity is 
 reduced so much that attraction prevails over centrifu- 
 gal force, and the radius vector can increase no longer, 
 but begins to diminish again at a the further apse, and 
 tlien the planet begins to approach the sun again, and 
 so the orbit is completed. But why the apses recur at 
 the same places, under a force varying exactly as the 
 inverse square of the distance, and why the orbit is an 
 ellipse and not some irregular or egg-shaped oval, can- 
 not possibly be explained without mathematics ; and 
 even they will only prove that it is so. 
 
 I showed you in the moon's disturbances (p. 152), 
 that if the central force is less at aphelion than it would 
 be according to the square of the distance, that is if it 
 varies in a rather higher ratio, the apses do not recur 
 at the same places, but advance, and the orbit is a kind 
 of spiral returning into itself, that is, a revolving ellipse. 
 1 now add that if the central force varied in as high a 
 ratio as the inverse cube of the distance (as disturbing 
 forces do), the centrifugal force and the force of attrac- 
 tion could never balance each other again if either of 
 them once got the best of it; and so the orbit would 
 become a complete spiral never returning into itself 
 
240 Measv/re of Centrifugal Force. 
 
 again, and the planets would either run at last into the 
 sun, or else further and further off into infinity, accord- 
 ing to which force had accidentally preponderated for 
 a moment. 
 
 The only other law of force of which we have any 
 experience, and that only as an indirect result of the 
 universal law of gravitation in some special cases, is 
 when the force varies directly as the distance, as in the 
 case of a pendulum bob which is attracted to the cen- 
 tre of its vibration by a force very nearly in proportion, 
 to the distance from it, whether the pendulum is 
 vibrating in one plane, as in a clock, or revolving as a 
 conical pendulum ; and in a mass of stars, as we shall 
 see afterward. It is singular that in that case also an 
 ellipse is described, though the law of force is so ex- 
 tremely different ; onlj' the centre of force is then in 
 the centre instead of the focus of the ellipse. The rea- 
 son why the apses of the pendulum -ellipse revolve, as 
 stated at p.ioi, is that the force does not increase quite 
 as fast as the distance ; and so both apses are in the 
 condition of the moon's apogee at p. 152. 
 
 Centrifugal Forcei — ^We have so much to do with 
 csntrifugal force in these matters that you may as well 
 see how it is measured. It is the force which resists 
 deflection from a straight line into a circle, if the body 
 moves in one, or into that circle which agrees for a 
 moment with whatever curve the body moves in, and 
 which is called the ' circle of curvature ' there ; and it 
 is measured by twice the amount of such deflection in 
 the unit of time, which is a second : that being the 
 velocity ^e?" second which such a force would produce 
 
Measwre of Angular Velocity. 241 
 
 in a second, according to calculations which you must 
 take for granted. Now the deflection of any small arc 
 from the tangent at the end of it — the square of its 
 length divided by twice the radius of the circle, as I 
 said of the roundness of the earth at p. 13. But that 
 small arc represents the velocity of the body in its orbit. 
 Therefore centrifugal force = the square of the linear 
 velocity divided by the radius of curvature. 
 
 Again as linear velocity = radius x angular velocity, 
 when the radius is not changing, centrifugal force also 
 = the radius vector x square of angular velocity, in a 
 circle, or near the apses of an ellipse. Centrifugal 
 force is not a force at all in the proper sense of being 
 a cause of motion, for it is only a consequence of mo- 
 tion ; but it is a very convenient measure of the force 
 which balances it, and which truly is one, whether it 
 is the tension of a string or the attraction of the sun ; 
 for in either case it pulls the body out of the straight . 
 line in which it would move otherwise. 
 
 But you may ask, how is angular velocity of degrees 
 and minutes to be reduced to common numbers ? 
 Simply by the fact that an arc of 180° is 3 •141 6 times 
 the length of the radius ; and therefore the angle of 1 80° 
 = 3"i4i6, which is usually denoted by the Greek letter 
 7z for shortness. Consequently 4° nearly = -07, and 
 57° 17' 45"= I. Thus the earth's angular velocity 
 of rotation is expressed by 360° X 1-0027 (p. 58) or 
 6-3005, divided by 86,400, the seconds in a solar day. 
 And the centrifugal force at the equator is the square 
 of this angular velocity X the earth's radius of 3963 X 
 5280 feet. You will find this — -11 127; which is a 
 10 
 
24:2 Linear Velocity in Ellijptio Orbit. 
 
 289th of the force of gravity at the equator, which 
 would be 32-2 if it were not diminished by the centri- 
 fugal force to 32"09 (p. 246). 
 
 "When the radius vector changes, linear velocity no 
 longer bears such a simple relation to angular; but 
 this rule holds always : whatever curve a body moves 
 in, the linear velocity varies inversely as a perpendicu- 
 lar drawn from the sun, or the place which the radius 
 vector is measured from, to a tangent of the curve at the 
 point where the body is, or to the direction of its mo- 
 tion for the moment, in which it would go on if the at- 
 traction were suddenly withdrawn. The tangent TP< 
 of an ellipse (p. 238) is easily drawn from the fact that 
 it makes equal angles with the two focal distances SP, 
 HP ; and ST, the perpendicular to it, always cuts it in 
 the circle which contains the ellipse. No planetary 
 orbit is anything like so elliptical as I have made the 
 figure for distinctness, and therefore ST, the perpen- 
 dicular on the tangent, is really much nearer to SP, the 
 radius vector. The linear velocity then varies inversely 
 as ST, but the angular inversely as SP^ And that is 
 true whichever way the planet is going, whether to- 
 ward T or t. At the apses the perpendicular ST coin- 
 cides with the radius vector, and the former rule holds. 
 
 A planet then goes slower the further it is from the 
 sun, but not so much slower in linear as in angular 
 velocity. What I said before about the increase of 
 velocity from aphelion to perihelion, and the decrease 
 from perihelion to aphelion, referred to linear velocity 
 in miles ; for it is actual velocity or motion by which 
 all forces are measured, and which the laws of motion 
 
EUiptio Period same as in a Circle. 243 
 
 govern. Therefore taking our example of Mercuiy 
 again, its linear velocity at perihelion is greater than 
 at aphelion in the proportion of 3 to 2, while the an- 
 gular velocity is as 9 to 4 (p. 224). And the linear 
 velocity is always made up of that due to the original 
 , impulse (which never wears out), increased and dimin- 
 ished from time to time by the resolved or tangential 
 part of the attraction of the sun. 
 
 WEIGHING OF THE SUN AND MOON. 
 
 "When I said that the sun is 316,560 times as heavy 
 as the earth (p. 63), I promised to prove it before we 
 finished the book. It can be done independently of 
 the moon, and in another way through the moon 
 (which is the usual one), by common arithmetic and 
 nothing harder than a few sums in fractions ; suppos- 
 ing you understand them, and remember that the same 
 quantity may be divided out of a numerator and de- 
 dominator, or out of the numerator of one fraction and 
 multiplied into the denominator of another which is 
 equal to it, and that two equal fractions or sums with 
 the mark = between them are called an equation, and 
 that both sides of an equation maybe squared, or have 
 their square root taken, or be multiplied or divided or 
 increased or diminished, by the same quantity, with- 
 out affecting their equality. 
 
 "We are now in a condition, first to weigh the sun 
 against the earth; secondly, to prove the necessary 
 truth of Kepler's law of the square of the periodic time 
 varying as the cube of the distance ; and thirdly, to 
 carry it farther, and show tlie relation of the time of 
 
24i Weighing the Sun By the Ea/rth. 
 
 any planet by itself to its own mean distance from the 
 sun, which is a very different thing from comparing the 
 times of two planets with each other. Kepler's law 
 would have had no meaning at all if there had been no 
 other planets than the earth, and -would have been 
 worth very little if there had been only two or three : 
 indeed we have seen that Bode's law of distances holds 
 (pretty nearly) from Yenus up to Uranus, but fails both 
 for Mercnry and Neptune. But Newton's law of time 
 and distance is as good for one planet as a hundred ; 
 being a necessary consequence of the law of gravitation, 
 that the attraction of a globe (or anything else if very 
 distant) varies as the mass divided by the square of 
 the distance from the centre. 
 
 As we must begin somewhere, with some proposition 
 already proved by mathematics, let us start with this 
 very simple one, which will also simplify our calcula- 
 tions considerably. The time of performing an ellip- 
 tical orbit round the sun in the focus is independent of 
 the eccentricity ; and therefore it is the same as if the 
 eccentricity were nothing at all, or the ellipse a circle 
 with the sun in the centre, and that circle the one 
 which contains the ellipse. Therefore also in calculat- 
 ing the whole period of revolution we have no need to 
 trouble ourselves about the variations of velocity in 
 different parts of the orbit, and we may (for this pur- 
 pose) treat the orbit of the earth or any other planet as 
 not almost, but absolutely circular. 
 
 As we are going to weigh the sun in earths we must 
 remember that we know nothing of the earth's mass 
 except through its power of attraction, as in the experi- 
 
The Measure of Gravity. 245 
 
 ments for weighing it. And we must see liow that is 
 measured before we can go further. Like all other 
 forces, it is measured by the velocity it generates in the 
 agreed unit of time, a second : and a foot is the agreed 
 unit of length, and therefore of velocity. By a machine 
 called Attwood's (of which you may see a picture in 
 the English Cyclopedia) an accelerating weight can be 
 made to act for a second on one of two others, which 
 are balanced on a string over a light pulley with as 
 little friction as possible. After the accelerating weight 
 is stopped or thrown off, the others are found to move 
 on with a momentum equivalent to that of the accel- 
 erating weight moving with a uniform velocity of about 
 32 ieeiper second. The momentum of the accelerating 
 weight shows that it would have fallen only 16 feet in 
 the first second, but would have gone 64 feet in the 
 second, 16x9 in the third, and so on, if it had been 
 left falling without impediment. But independently 
 of that, which involves mathematics, here you have the 
 experimental fact that gravity generates a velocity of 32 
 feet per second ; which is therefore the figure by whicli 
 it is always represented, subject to small corrections. 
 
 A more accurate way of finding it, free from the im- 
 pediments of friction, but involving some mathematics, 
 is by measuring the length of a pendulum which vi- 
 brates in a second. The length is not measured to the 
 centre of gravity, but to a point a little lower, called 
 the centre of oscillation, which depends on its shape. 
 Taut can be got very accurately by a well-known con- 
 trivance, described in the Eudimentary Treatise on 
 Clocks (p. 45, ed.'4). I can only tell you here that 
 
246 Weighing the Sun hy the Earth, 
 
 gravity = the length of the one second pendulum x 
 It'' (tt being the symbol commonly used for 3*i4i6); 
 That length in England is 3-26 feet (for we must reckon 
 in feet), and therefore gravity = 9-86 x 3"26 = 32'i4. 
 But that is rather too little on account of the centrifugal 
 force of rotation (p. 242), and gravity may be taken at 
 32"22 feet on the surface of the earth as a non- rotating 
 sphere with a radius of probably 3958"6 miles (p.24).' 
 
 Now let us see what proportion the sun's force must 
 bear to that, in order to balance the centrifugal force 
 of the earth's revolution (not rotation). Centrifugal 
 force in a circle = linear velocity' divided by radius 
 (p. 241). The radius of the earth's orbit is 91,404,000 
 X 5280 feet: its velocity is very nearly i8'2 miles (p. 
 36), and the velocity' is 33i'2 x 5280" feet. Therefore 
 cent, force , 331-2 x 5280' _ i . 
 
 gravity ~ 32.23 x 91,404,000 x 5280 ~ i684'2 
 * Cent, force ' may stand for either centrifugal or cen- 
 tripetal, for they must evidently balance each other. 
 Therefore gravity is 1684 times the sun's attraction 
 here ; and if the earth weighed 1684 tons and was 
 held by a chain instead of the sun's attraction, the 
 chain would only have to bear a strain of one ton. 
 
 Now for the next step, we know that the central 
 
 force is the sun's mass divided by his distance', aiid 
 
 that gravity is the earth's mass divided by its mean 
 
 radius', and that the latter is 1 684*2 times the former. 
 
 ^ earth - 1684-2 sun ,, „ 
 
 Ur ■ — — — - = — 3 -• therefore 
 
 3958-6' 91,404,000' 
 
 sun 
 
 = 91,404 ^00' ^ g 5 . 
 
 earth 3958-4' X 1684-2 
 
omd. Tyy the Moon. 247 
 
 "which is very near indeed to the 3 16,560 whicli I gave as 
 the result of other calculations. These two sums might 
 have been done in one, but I -wanted to show you the 
 proportion between the sun's attraction here and gravity. 
 
 If you have a fancy to see the weight of the sun in 
 tons, you must turn the earth into tons from its bulk in 
 cubic miles (p. 24), and its density = 5^ times that of 
 water, of whicli 36 cubic feet weigh a ton. Then mul- 
 tiply by 316,560, and you will find the tons in the sun 
 to be about 1850 with 24 cyphers after it, or 1850 
 septillions, as the earth is 5842 with 18 cyphers after 
 it, or 5842 quintillious of tons. 
 
 The following is the usual way of weighing the sun ; 
 
 but it requires the moon's weight to be known first to 
 
 do it accurately. The mean radius of the earth's orbit 
 
 is 38276 times that of the moon round the earth, and 
 
 the moon's mean angular velocity is I3'37 times the 
 
 earth's, being inversely as their sidereal periods (p. io8). 
 
 Therefore the moon's mean linear velocity round the 
 
 earth reduced to rest (p. 98) is to the earth's round the 
 
 sun, as I3'37 to 38276. The sun's force on the earth 
 
 is to the earth's force on the moon as their deflections 
 
 from a straight line per second, which are as the squares 
 
 of their velocities divided by the diameter of each orbit, 
 
 as I explained for centrifugal force at p. 241 ; and those 
 
 diameters are as 38276 to i. Therefore the deflections 
 
 are as 38276 to I3■37^ But the sun must be 38276* 
 
 times heavier than if he were at the distance of the 
 
 moon, to produce the effect he does here. Therefore 
 
 1. ,1 sun 38276' ^o 
 
 altogether z = - — '-^ = 313,580, 
 
 ^ earth + moon 13-37= 
 
248 Weighing the Moon hy Verms. 
 
 which makes the sun = 317,430 times the earth alone, 
 adding one 8i*5th for the weight of the moon. 
 
 But you see this result is wrong, exceeding the for- 
 mer one considerably, though it is near enough for a 
 general explanation. Let us see if we cannot get a 
 little nearer. I said at p. 148 that the sun's disturb- 
 ance lengthens the moon's period as much as if the 
 earth's mass were reduced a 358th. Therefore its real 
 proportion to the sun is so much greater than it appears 
 to be by calculation from the moon's actual period. 
 And if you reduce 317,430 by a 3 5 8th, the result is 
 316,550; which is again very near the figure I have 
 used throughout, deduced from the sun's mass as given 
 in books of good authority for the old parallax (pp. 
 64, 86). 
 
 This same calculation evidently affords another mode 
 of weighing the moon, if the sun has been first weighed 
 independently ; for the moon must be such a fraction 
 of the earth's mass as will give the right result for the 
 sun in the above calculation. In most popular astrono- 
 mies the distinction between the earth and earth + 
 moon is disregarded in weighing the sun ; which makes 
 a difference of nearly an 80th in his weight, besides 
 misleading people as to the principle of measuring 
 forces round a moveable centre, which I have several 
 times mentioned. Moreover the following method of 
 weighing the moon without the sun is founded on that 
 very distinction. 
 
 It is not the earth's centre E, but G the centre of 
 
Proof of Kepler's Third Law. 249 
 
 gravity of earth and nioon, that really describes what we 
 call the earth's orbit. Therefore the earth is before 
 its mean place at first half moon, and behind it at second 
 half moon by the distance EG, which is now the thing 
 we have to find, for we only knew it at p. 97 because we 
 then assumed the moon's weight to be known. Now 
 when Venus is at her nearest or ' inferior ' conjunction, 
 she is apparently displaced at half moons first one way 
 and then the other, by a parallax or angle which = EG 
 divided by the distance of Yenus from us ; and as that 
 is known, and twice that angle is quite large enough to 
 measure, being nearly i', therefore EG can be found. 
 Then we know that the moon is to the earth as EG is to 
 MG, and therefore the mass of the moon is found, as 
 EM is known. 
 
 Mr. Airy said in 1856 that all the methods of weigh- 
 ing the moon make the earth ' about 80 times ' her 
 weight. If that were so then, this must make it aboiit 
 83 now, since the distance of Yenus from the earth is 
 reduced a 24th, with all the planetary distances, and 
 EG is increased in the same proportion while MG is 
 not sensibly altered, being very much larger. You will 
 see presently that another method gives the same result. 
 
 Law. of Time and Distance.— We can easily prove also 
 Kepler's third law independently, or rather that more 
 complete form of it given by Newton, which deter- 
 mines the period of each planet from its own distance 
 only. Centrifugal force = distance x angular ve- 
 locity^, still treating the orbit as circular (pp. 241, 
 244). Angular velocity per second is 360° or 2n 
 or 6-2832 divided by the planet's period in seconds. 
 
250 Absolute Mass of Earth and Sun. 
 
 r^, n .„ , ~ 4^' distance t> ^ ^i • 
 Therefore centrifugal force = r-ri — • -t>ut tnis 
 
 STin's mass 
 
 must — the central force or ,. — : 47r' = 39'48. 
 
 _, , . ,, 39-48 distance ' 
 
 Therefore period' = ^ , . 
 
 ^ sun s- mass 
 
 Strictly speaking, the denominator is the mass of 
 the sun and planet together, for the reason given at p. 
 233. But as the sun is 1048 times heavier than the 
 largest planet, we may practically disregard their 
 weight, and consider the central force to he that of the 
 sun's mass only. 
 
 But what is the meaning of the ' sun's mass ' in these 
 fractions, and by what figures is it to be represented ? 
 If we say it is 316,560 times the earth's mass, we have 
 only to put the same question about that. Generally 
 we have to talk of masses only by comparison, and 
 then we measure them by their three dimensions, of 
 length, breadth, and height, and their density com- 
 pared with water or the earth ; or else we say that 
 they contain as much matter as will weigh so many 
 pounds or tons under the action of gravity here ; or 
 that the weight here = the length x breadth x height 
 X density x 32'2. But 'density' has no absolute 
 meaning, only a relative one, and all these figures to- 
 gether do not tell us how much such a mass will move 
 another in a second at the distance of a foot or a mile. 
 We still want to know how to measure the attracting 
 or ' accelerating ' effect of any given mass ; and we can 
 only do that by referring it ultimately to the attractive 
 force of the earth at the unit of distance from its cen- 
 
Weighing the Moon Indej>endently. 251 
 
 tre, supposing it to be all condensed there ; which we 
 can measure easily. 
 
 For this absolute force of the earth (as it is sometimes 
 called) is evidently 3958-6' x 5280'' X 32-22. Or the 
 earth's mass at the unit of distance has 14,076 trillions 
 of those things of which gravity is 32-22. Conse- 
 quently the sun has 316,560 times as much, or has 
 nearly 4456 quintillions of units of force at the unit 
 of distance, which (remember) is a foot. 
 
 Weighing the IVIoon Independently. — "We can now also 
 weigh the moon without the aid of Yenus, or the tides, 
 or the still more delicate measurement of nutation, from 
 nothing but her period and distance and the known 
 measure of gravity. For the mass which determines 
 her period is that of earth + moon : which must there- 
 fore = 39-48 X the cube of her mean distance, divided 
 by the square of the number of seconds in her sidereal 
 period of 655-716 hours, with a 7 17th taken ofi' for the 
 addition of a 716th by the sun's disturbance (p. 
 148). You will find that gives the absolute mass of 
 
 , 39-48 X 238,830' X 5280' 
 
 earth + moon = ^^-i -^ ' "^ „ = 
 
 2,357,300' 
 
 14,246 trillions. But we saw just now that the abso- 
 lute mass of the earth alone is 14,076 trillions, or 170 
 less than this earth + moon. The difference, or mass 
 of the moon alone, is -012 of the earth's mass; which 
 comes very near to Mr. Adams's proportion of '0123, 
 and still nearer to the present result of the weighing 
 of the moon by Yenus at p. 248 ; for this makes the 
 earth 82-8 times the moon; and as I said at p. 135, 
 other authorities are inclined to make it rather more. 
 
252 Weighing of Jupiter, Saturn, Etc. 
 
 It seems odd that ^ve should be less certain of the 
 weight of tlie moon than of the sun and Jupiter, 
 which are nearly 400 and 2000 tirCies further off. But 
 it is easier to find the mass of a heavy body from its 
 attraction on light ones than of a light one from its 
 disturbance of heavy ones. 
 
 You may think however that we have been only 
 reasoning in a circle, because we have reduced the 
 moon's period to meet its increase by the sun's disturb- 
 ance, and yet tliat calculation of the disturbing forces 
 at p. 147 assumed the moon's weight to be known. 
 But there is nothing in that objection ; for if you omit 
 the moon's mass altogether in calculating the disturb- 
 ances, putting 316,560 instead of 312,720 for the sun's 
 mass, it makes a quite insignificant difference in this 
 i-esnlt. I should mention that I have not also reduced 
 the moon's distance, though it is increased a little by 
 the sun's disturbance (p. 148), because the increase of 
 the period by a 716th is on the assumption of the dis- 
 tance not being increased ; if that is taken into account, 
 the period is increased (and must therefore now be 
 diminished) rather more, and so the result would be 
 the same, both here and at p. 248.* 
 
 The mass of Jupiter, or any planet with satellites, is 
 found from the distance and period of any of them, by 
 a calculation like that which I gave just now. For 
 Jupiter's absolute mass, or force at the unit of distance, 
 = 39*48 X the cube of any moon's distance (in feet) 
 divided by the square of its period (in seconds), which 
 you will find in the table at the end. And then di- 
 
 *Se<! Archdeacon Pratt's Mechanical Philosophy, § 299. 
 
Ahsolute Measure of Chravity. 253 
 
 viding the result by the 14076 trillions for the earth's 
 mass, you have the proportion of Jupiter to earth. 
 
 Or the mass of Jupiter may be compared with the 
 sun, through any of his moons, as we compared the sun 
 with the earth through our moon ; except that Jupiter 
 is too heavy to be neglected in comparison with the 
 sun, and his moons may be neglected in comparison 
 with him — ^just the converse of the earth. The result is. 
 
 Sun _ 2j: & 0's dist ance' X (£'s period' 
 
 Jupiter ~ li & (["s distance' X 2£'s period' ' 
 
 the I meaning only that Jupiter's mass is not neglect- 
 ed. This ought to come 1248 or nearly so, and 302 
 for the proportion of Jupiter to the earth, as stated at 
 p. 171. For Jupiter's mass has been also verified sev- 
 eral times from his disturbances of some of the as- 
 teroids, and all the calculations give nearly that re- 
 sult. 
 
 The distances of Jupiter's moons are given in all the 
 books only in terms of his equatorial radius, as in the 
 third column of the table at the end. But those reduced 
 to miles, by the diameter which is generally given, all 
 produce a mass about a 17th too great, by either of the 
 calculations ; therefore the distances are nearly a 50th 
 too great ; which is far too large an error to be ac- 
 counted for by the mutual disturbances of the satellites 
 or by Jupiter's oblateness (p. 197). I have therefore 
 given such distances in miles as will agree with this as- 
 certained mass, neglecting the disturbances (as they af- 
 fect the distances very little) and leaving the propor- 
 tionate distances as I find them in other books; ob- 
 serving only that astronomers are not quite agreed 
 
264 Absolute Measure of AUroGUon. 
 
 about the proportion of Jupiter's diameters, and there- 
 fore about the length of one of them. It is not worth 
 ■while to go through the same process for Saturn's 
 moons; and Uranus's are too uncertain yet to say 
 more about them than I have at p. 194. 
 
 All these satellites are too small compared with their 
 primaries to be weighed through their own periods by 
 the method given for our moon just now. The masses 
 have to be found from their mutual disturbances. And 
 so the planets without moons have to be weighed from 
 their disturbances of each other, and of comets. All 
 that belongs to very high mathematics ; but I have now 
 shown you how the dimensions and weights of all the 
 solar system are measured ultimately by a foot rule and 
 the vibrations of a pendulum. 
 
 And you can now appreciate the delicacy of the 
 Cavendish experiment for weighing the earth (p. 30), 
 though it looks easy at first sight to try how much one 
 mass attracts another. If you divide the weight of the 
 earth, 5842 quintillion tons (p. 247), by its absolute 
 force, you will find that it takes a mass of 415,000 tons 
 to exert a force of i foot per second at the distance of a 
 foot from its centre. Such a globe made of platinum, 
 the heaviest thing there is, would be no feet in di- 
 ameter ; and therefore you could not put anything to be 
 attracted by it nearer to its centre than 56 feet ; and 
 its force at that distance would be only the 3136th of a 
 foot per second. Or to illustrate it by what is called 
 statical force, as in the Schehallien experiment, suppose 
 such a globe to be brought within a foot of the bob of 
 a long pendulum (hung, not balanced as in Cavendish's 
 
Three Laws of StoMUty. 255 
 
 experiment), it would only draw the pendulum out of 
 the vertical by the angle whose numerical value is 
 
 -p— : or such a pendulum would have to be 
 
 3136 X 32-2' ^ 
 
 I "6 miles long for its bob to be moved an inch by the 
 attraction of that mass of platinum put close to it. So 
 that the force of attraction, which produces such vast 
 effects throughout the universe, is almost immeasura- 
 bly small, except upon a very great scale. 
 
 Laws of Stability! — I will finish this account of the 
 laws of motion of the solar system by stating three 
 remarkable results of that permanent stability or equi- 
 librium of the system which I have several times men- 
 tioned. The first is, that if you multiply the mass of 
 each planet (taking the earth, or any other, for the 
 unit) by the square root of its mean distance from the 
 sun, and by the square of the eccentricity of its orbit, 
 the sum of all those products is invariable, however 
 each eccentricity may vary. The second is, that if you 
 multiply each mass by the square root of its mean 
 distance, and by the square of the numerical value 
 (p. 241) of the inclination* of its orbit to a certain 
 fixed plane near the ecliptic, the sum of those products 
 also is constant. 
 
 The third is the most striking of all. The sum of 
 all the areas swept over by the radius vector of every 
 planet, satellite, and comet in the whole system, in any 
 given time, is invariable, though they all may be made 
 to deviate a little from the law of the conservation of 
 
 • strictly it is the trigonometrical tangent of the inclination, btit the angles 
 are too amall for the distinction to be important. 
 
256 Oiroular and Elliptio Conio Sections. 
 
 areas (p. 223) by their mutual disturbances. The area 
 so described by any planet in a second, being constant 
 in the same orbit, is the area of the whole orbit divided 
 by the number of seconds in its year ; and that area is 
 3' 141 6 X the product of the two semi- axes, or x the 
 square of the mean distance if the orbit maybe treated 
 as a circle. From this, and the law that the square of 
 the period varies as the cube of the distance, you may 
 easily deduce another, that the areas described iu dif- 
 ferent orbits in any time round the same central mass 
 are as the square roots of the distances. Therefore the 
 areal velocity of distant planets is greater than of near 
 ones, though both their angular and linear Telocity is 
 less. 
 
 THE TOUE CONIO SECTIONS. 
 
 There are two other curves beside circles and ellip- 
 ses, in which the planets could, and some of the comets 
 do move, under the law of gravity varying inversely as 
 the square of the distance. They are ihQ parabola uni. 
 the hyperbola. A parabola is the curve described by 
 a stone thrown into the air any way but straight up, or 
 by water rushing out of a hole in the side of a pretty 
 full cask, except that the resistance of the air spoils the 
 accuracy of the curve. You may see a hyperbola in 
 the shadow on the wall from a round shade over a lamp 
 set near the wall. But the remarkable thing is that 
 these four curves, the circle, the ellipse, the parabola, 
 and the liyperbola, which are the only possible ones* 
 for the heavenly bodies to describe under the existing 
 laws of nature, are all produced by cutting straight 
 
 * Appendix, Note XXII. 
 
Parabola and Ellipse. 257 
 
 through a cone in different directions. Yon may have 
 heard the term conic sections -vvithont knowing what it 
 means, or why whole books of mathematics ai-e written 
 about them. There is no part of geometry with so 
 many curious and elegant problems in it ; and the 
 reason of its importance is that all the heavenly mo- 
 tions are performed in conic sections. I must therefore 
 explain what they are, as far as I can without mathe- 
 matics. 
 
 You probably know that a cone is a roimd pyramid, 
 or a thing with a circle for its bottom and a point for 
 its top, and straight sides. "WTien the top is vertically 
 over the centre of the circle it is called a rigJit cone, 
 and AVhen it is not, an oblique one. Consequently the 
 outline or any vertical section of a right cone is a tri- 
 angle with two equal sides. If you cut a few cones 
 out of some soft stuff such as a turnip, or get them 
 turned in wood to be sawn through afterward, you 
 will be able to get a better idea of the different conic 
 sections than from any pictures, and therefore I shall 
 not give any. First of all it requires no cutting to see 
 that all sections parallel to the base of the cone, or at 
 right angles to its axis, must be circles ; and a circle 
 may be considered an ellipse of no eccentricity, or witli 
 its two foci run together. Next, any other section of 
 the cone which comes out at both sides is an ellipse, 
 as you will see by cutting it through in that way. And 
 I should tell you that the cone is considered to be ex- 
 tended to any length required for the elliptical section 
 to come out again lower down on one side than the 
 other. You would hardly believe without trying it, 
 
258 Parabola and Hyperbola. 
 
 that the lower end of the section which comes out 
 where the cone is broad is no broader than the higher 
 end where it is narrow ; and yet so it is. 
 
 An ellipse is also produced by any oblique section 
 of a cylinder, as a circle is the section directly across a 
 cylinder. But in fact a cylinder is only a cone of 
 infinite length, so that its sides may be considered par- 
 allel, as when we look at the stars, which are some of 
 Ihem certainly much bigger than the sun, and yet look 
 like points. This is the explanation of what I said at 
 p. 40, that the oblique view of a circle is an ellipse, 
 whether near or far off: if near, the lines of sight form 
 a cone with the eye at the point ; if very far off, they 
 may be considered to form a thin cylinder. 
 
 A parabola is a very different looking thing, though 
 it is produced by a very little deviation from the sec- 
 tion of a cone which makes an ellipse. It cannot be 
 cut out of a cylinder. It is like half — or rather, some 
 indefinite part less than half, of an ellipse of infinite 
 length, or one whose eccentricity = i, or whose centre 
 is so far from its foci that SO or HO (p. 39) may be 
 considered = AC. It has no minor axis, for its centre 
 is at an infinite distance, that is, nowhere. It is made 
 by cutting through a cone anywhere by a cut parallel 
 to the opposite ' slant side.' Therefore there can be 
 only one parabola, as there is only one circle, at any 
 given distance from the top of the cone, which must 
 now be considered of infinite extent downward. The 
 two legs of a parabola. are always getting more parallel 
 to its axis, but never become quite so, however far they 
 are extended. 
 
Sections of an OHlique Cone. 269 
 
 A hyperbola is made by any section of the cone 
 which is neither parallel to its slant side nor comes out 
 of it at both sides. It does not diifer mnch in appear- 
 ance from a parabola, only spreading out wider. The 
 slightest deviation of the parabolic cut toward the ver- 
 tical turns the parabola into a hyperbola ; but there 
 may be any number of hyperbolas at any point in the 
 cone. In some popular astronomies a vertical section 
 only is said to be a hyperbola, but that is a mistake. 
 Its legs continually approach two straight lines called 
 asymptotes, which are in fact the outline of the cone 
 itself, but never reach them, because they approach 
 with a continually decreasing curvature : just as a 
 series of weights of a pound, half a pound, a quarter, 
 and so on, approaches o, and their sum approaches 
 two pounds, but the limit is never reached in either 
 case. 
 
 A parabola and hyperbola can theoretically be de- 
 scribed by strings fi"om the focus or foci (for a complete 
 hyperbola has two, being two hyperbolas set back to 
 back), but not practically. The focus of all the four 
 conic sections is the point where they are touched by 
 a sphere which also touches the hollow cone all round, 
 like a ball put into a wine glass : if you draw the out- 
 line of the cone and the axis of any conic section such 
 a sphere is represented by a circle. 
 
 The only thing I have to say about oblique cones is 
 to reconcile what is said about the shape of countries 
 being correctly represented in a stereographic map 
 (p. 20) with the fact that eveiy perspective view of a 
 circle is an ellipse; and one perspective view of G,\(iv^ 
 
260 Comets. 
 
 ellipse is a circle. The explanation is, that an oblique 
 cone allows two sets of circular sections, leaning oppo- 
 site ways with respect to the axis of the cone ; and in 
 the stereographic projections of a small circle of the 
 earth upon the plane of the equator seen from one of 
 the poles, the oblique cone from the eye to that circle 
 cuts the equator or any parallel plane in a smaller cir- 
 cle turned the other way ; and therefore the shapes of 
 countries in such a map are truly represented, though 
 some of them must be on a larger scale than others in 
 any flat map of a round, world. 
 
 COMETS. 
 
 These singular bodies in several respects lie midway 
 between the planets and the stars and nebulae not re- 
 solvable into stars. For they have orbits with the sun 
 in their focus, like the planets, and yet some of them 
 probably run off among the stars, though they must 
 take millions of years to get there. But they are not 
 solid like the planets, being thin enough to see through 
 in their densest part, and therefore nebulous, or com- 
 posed of some kind of gaseous matter whose expansive 
 force overcomes its mutual attraction and prevents it 
 from coalescing into a solid lump. Most of them de- 
 scribe ellipses far more eccentric than the planets ; but 
 some describe parabolas and even hyperbolas, and those 
 will probably never* pay another visit to the solar sys- 
 tem through all eternity, but will run off till they are 
 caught by the attraction of some other sun among the 
 stars. The great majority of comets move in the same 
 
 * Appendix, Note XXHI. 
 
Perihelion Velocity in different Orbits. 261 
 
 direction as all the planets, and in orbits not very ob- 
 lique to the ecliptic. The periods seem to be divided 
 into two classes, one from 3 to 7 years and the other 
 mostly above 70 years. 
 
 They take their name, which means hairy, from their 
 tails, though many comets have no tails, and to those 
 which have it is only a temporary appendage shot out 
 in the direction from the sun, when they get within a 
 certain — or rather uncertain distance from perihelion, 
 for they vary in that respect also. The head of a 
 comet is a round, bright, transparent, nebulous kind 
 of mass, sometimes much larger than any of the plan- 
 ets, and even half as wide as the sun, but so thin that 
 small stars can be seen right through the nucleus or 
 bright spot in the middle of the head ; and so light 
 that they pass among the satellites of Jupiter without 
 disturbing one of them, though such comets have been 
 themselves completely thrown out of their orbits by 
 Jupiter's attraction. Their less violent disturbances 
 can be calculated beforehand, and their motions pre- 
 dicted accordingly, like those of the planets and the 
 moon, when their regular orbit has been once deter- 
 mined, as it can be by as few as three observations, fix- 
 ing their velocity as well as their places, or by only 
 two observations at perihelion, the sun being always in 
 the focus of the orbit. Consequently there is not the 
 smallest doubt that they perform their journey round 
 the sun in obedience to the universal law of gravi- 
 tation, however little matter they may have for it to 
 act on. 
 
 The greatest or perihelion velocity, either of planets 
 
262 Some Bemd/rhahle Comets. 
 
 or comets, follows a simple rule according to the na- 
 ture of the orbit. In a parabokj velocity' at peri' 
 helion = 39-48 X sun's absolute mass (p. 252) divided 
 by perihelion distance and multiplied by 2. In an 
 ellipse it is that same quantity multiplied by i + the 
 eccentricity, which is necessarily less than the 2 of the 
 parabola. In a hyperbola the rule is the same as in 
 the ellipse, but the eccentricity must exceed i to make 
 a hyperbola ; which cannot be explained without mathe- 
 matics. A circle is only an ellipse whose eccentricity 
 = o. Therefore the velocity in a parabolic orbit is 
 4/2 or I '4 1 times greater than in an elliptic orbit of 
 very small eccentricity, at the same perihelion dis- 
 tance. 
 
 Consequently the whole orbit of a comet can be laid 
 down in the map of space from only two observations 
 at perihelion, with their exact tim.es. For they give its 
 least distance and velocity, and that gives the eccen- 
 tricity, and the least distance and eccentricity give the 
 mean distance (see p. 39) ; and the plane of the orbit 
 must go through the two observed places and the mid- 
 dle of the sun, and that gives its inclination to the eclip- 
 tic and the longitude of its nodes. But though the 
 linear velocity is least in a circle for a given least dis^- 
 tance, it is greatest for a given inean distance ; for a 
 circular orbit is performed in the same time as any 
 ellipse which it contains (p. 244), and which of course 
 is a shorter orbit. 
 
 About 36 comets have been recognized as return- 
 ing ; some in moderately short orbits, others in very 
 long ones, corresponding to their periods, according to 
 
So7ne Jiemarkaile Comets. 263 
 
 the same rule of time and distance which all bodies 
 going in elliptic orbits round the sun must follow. The 
 comet of 1843 went nearest to the sun of any that is 
 known, within 80,000 miles of his surface, or within 
 one-third of the moon's distance from the earth. It 
 swung past him with a velocity of 366 miles in a 
 second, or 20 times as fast as the earth moves, and 1000 
 times as fast as any cannon ball, and was exposed to a 
 heat four times as great as the melting heat of agates 
 and rock crystal, the most refractory of all things, and 
 yet came away none the worse, as far as we could see, 
 being already in a state of gas. And it was bright 
 enough to be seen in full sunshine in America. 
 
 It is curious that the grandest comet ever known, 
 which shot out a tail of 60 million miles in two days,* 
 and twice as long afterward, appeared in 1680, six 
 years before N'ewton published the Principia. That 
 also went much nearer to the sun than the moon is to 
 us, and travelled 20,000 miles a minute, and was ex- 
 posed to a heat 25,600 times as great as the hottest 
 sunshine in the tropics. The . earth only escaped a 
 collision with Biela's comet by a month in 1832 ; so it 
 is pleasant to reflect on their tenuity. That same 
 comet suddenly split into two, which travelled amicably 
 together and reappeared so in 1852. Since that time 
 they have not been seen. Halley's comet, which fol- 
 lowed Newton's by two years, and has a period of 
 nearly 76 years, subject to delays by Jupiter and Sat- 
 urn which can be calculated, in 1835 swelled to at 
 least 74 times its size in 17 days after perihelion, tiUit 
 
 ~ * Appendix , Note XXIV. 
 
264 Comets' Tails. 
 
 got too thin for its ligM to be seen, and so vanished ; 
 but no doubt it gathered itself together again, and will 
 reappear in due time. 
 
 Comets seem to have a habit of apparently shrinking 
 near the sun; but Sir J. Herschel thinks that that is 
 really swelling so much that their outer parts become 
 dissipated, or made too thin to be visible, as the whole 
 comet was in 1835. I have already spoken of the 
 gradual acceleration of Encke's comet, which returns 
 in 3-J- years, and the theory founded upon it, at p. 83. 
 Mercury's mass was calculated from the disturbance 
 which it caused in this comet. The comet with the 
 iinest tail of modern times, was that called Donati's, 
 which most of us saw in the autumn of 1858, and which 
 nobody will see again for 2100 years, as it has to go 80 
 times as far off as Neptune. Its tail was 30 million 
 miles long, and 90,000 wide just behind the head, and 
 much wider beyond. Ton may see a fuller account of 
 the strange behavior and peculiarities of the most fa- 
 mous comets in Sir J. Herschel's Astronomy, and in 
 his ' Familiar Lectures.' 
 
 The tails are generally like a long brush in shape, 
 but some spread out short and wide, like a ' bat's wing ' 
 burner gas flame : the story of one with 6 tails in 1744 
 is perhaps fabulous. Some tails are short and thin. But 
 very nearly all of them agree in being directed from 
 the sun, and projected with such velocity that it is evi- 
 dent that they differ from all other matter in being 
 violently repelled instead of attracted by the sun, 
 though not by their own matter or that of the comet 
 whose attraction holds them together, until they are 
 
How Comets' Tails a/re Fo'rmed. 265 
 
 first exploded by the sun's heat and then projected 
 backward by his repulsive force. The first explosion 
 of tail matter does not take place backward, but rather 
 forward, or toward the sun ; and the tall is a little 
 open just behind the comet, as it would be if the mat- 
 ter was shot out forward and sideways, and at the same 
 time blown away backward by a strong wind or force 
 from the sun. The apparent wheeling of a comet's 
 tail round it, so as always to keep pointing from the 
 sun, is to be explained like the appearance of the 
 smoke or steam from a locomotive chimney, which 
 seems to be carried along with it of a nearly constant 
 length, and yet is really continually dissipated and be- 
 coming invisible at the far end, and continually re- 
 newed by emission from the chimney. A comet is 
 continually shedding the far end of its tail by dissipa- 
 tion, and growing it afi'esh by emission. 
 
 Again the bend in the tail away from the direction 
 of motion is due to the farther parts of the tail-smoke 
 being, as it were, left behind by the near ones and by 
 the head. For in order to keep pace apparently with 
 the comet, the smoke ought to have a greater velocity 
 forward when it has reached a larger orbit, than it 
 started with: which it has no means of acquiring. 
 Probably the tail-smoke is lost in space, and never gath- 
 ered in again, as there is reason to believe the mere 
 expansion of the head is. The nucleus, or bright spot 
 in the middle of the head, generally remains visible, 
 though stars can be seen through it. 
 
 It appears to be proved by two distinct methods that 
 comets' tails are not self-luminous, though the heads 
 13 
 
266 The Light of Comets atid their Tails. 
 
 are. Polarization of light cannot be explained farther 
 here than by saying that light reflected at an angle of 
 33° from a non-metallic surface, such as glass with a 
 black cloth behind it, is polarized, or made to vibrate 
 (see p. 78) in one plane only, and will not reflect again 
 from another similar mirror set directly across the first, 
 but is all stopped as completely as if the second glass 
 were itself a black cloth. The light of comets' tails is 
 found to be polarized, and therefore is concluded to be 
 reflected light. 
 
 Mr. Hnggins and Professor Miller have gone a step 
 farther by the spectrum analysis (p. 68) and have ascer- 
 tained that the light reflected is not of the same kind 
 as vrould be reflected from the nucleus, which is self- 
 luminous by the same tests; and therefore the tail- 
 light must be reflected from the sun. The spectrum 
 of an inflamed gas is not a continuous band of colors, 
 but consists of a few bright lines at intervals ; and the 
 nucleus of a comet, as of some of the flxed nebulee, ap- 
 pears to be only such a gas. There have been however 
 very few opportunities yet of examining comets since 
 this great invention of the spectrum analysis; and 
 it is possible they may not be all alike. The extract^ 
 dinary brightness of the comet of 1843 while very near 
 the sun looks like the sun having at least a share in the 
 effect.* 
 
 THE NEBULAE HYPOTHESIS. 
 
 Astronomers have naturally speculated what was 
 the rudest and simplest state of the matter of the uni- 
 verse from which the present condition of the solar 
 
 * AppentUs, Note XXV. 
 
The Nebular Hypotliesis. 267 
 
 system may have been evolved by the known laws of 
 nature and one original impulse from the hand of 
 God. No one can reflect on all the motions of the sys- 
 tem (except of a few moons and the November meteors) 
 taking place in one direction, both of revolution and ro- 
 tation, and all very nearly in one plane, without feeling 
 inclined to attribute them to some common cause. And 
 when we see, besides, the multitude of complicated 
 motions, such as those of the moon, which are all un- 
 questionably due to one law of gravitation, the proba- 
 bility becomes still greater of some very simple origin 
 of the jiresent state of things. 
 
 Gravitation is quite incompetent by itself, either to 
 set a single body in revolution or rotation, or to deter- 
 mine the directions in which they should move — except 
 to run all into a heap together. For anything that 
 gi'avitation coixld do to prevent it, the 8 large plan- 
 ets and the lOO little ones, to say nothing of all the 
 moons and rings, might divide the whole i8o° of pos- 
 sible inclinations equally or unequally among them, 
 and any number of them might go in opposite direc- 
 tions to the others. The chances against their all com- 
 ing within such narrow limits and in the same direc- 
 tion by what is called accidental coincideHce have been 
 calculated to be above a trillion to one. They must 
 either have been all deliberately put down where they 
 are, and projected with the velocities they have, or else 
 their motions must be the result of some much simpler 
 state of things ; and if that is possible under the exist- 
 ing laws of nature, nobody can doubt that it is more 
 probable than the other alternative. 
 
268 The NebuLa/r Hypothesis 
 
 Laplace and others have come to the conclusion that 
 it is possible ; and that if an enormous globe of nebu- 
 lous matter were once set slowly whirling in the gene- 
 ral direction, which we call from west to east, it would 
 gradually contract by cooling, and revolve faster (p. 
 235) and then throw off rings, and then each ring 
 might break and gather itself into a globe, which 
 would partake of the original rotation besides revolv- 
 ing in the general direction ; and each globe itself 
 might afterward throw off smaller rings, which would 
 either stay as such, like Saturn's, or run together into 
 moons. 
 
 The distances of the planets therefore represent the 
 force of expansion of the original nebula against the 
 attraction of its parts. Indeed whether this nebular 
 theory is true or not, the distance of everything from 
 everything else represents all the force which has been 
 expended from ' the beginning in separating ' them, or 
 the vis viva they would acquire in coming together by 
 attraction. And its increase as they approach is no 
 ' creation of force,' as has been imagined ; nor its de- 
 crease as they recede, a ' destruction of force : ' it 
 only re-appears in one case and becomes latent in the 
 other. 
 
 The explanation devised for the abnormal motions 
 of the moons of Ui'anus and Neptune is that those 
 planets had their axes turned out of the perpendicular 
 to their orbits before they threw off the rings which 
 became their moons. But I have seen no suggestion 
 to explain how the earth's axis could be so disturbed 
 after it had become a rotating globe (which resists 
 
For the FwrnaUon of the /Solar System. 269 
 
 such disturbance) and thrown off a moon ; or how the 
 meteors come to revolve in the opposite direction. 
 
 It must be added however that experiments can be 
 made to exhibit the process, and to show that a mass 
 of fluid, and therefore of nebulous matter, set rotating, 
 will throw off i-ings when its velocity is increased 
 enough to overcome its cohesion ; and that if the ring 
 breaks it will gather itself up into a globe by the at- 
 traction of its particles. Plateau's experiment is this : 
 water is lightened with spirits of wine till it has such 
 a specific gravity that oil will lie anywhere within it. 
 Some oil is put in, and immediately becomes a globe 
 by its own attraction of cohesion (p. i8). The whole 
 inclosed in a glass box is set spinning, with an axis 
 through the oil to make it spin too : the oil globe first 
 spreads into an oblate spheroid : then as it is whirled 
 faster it throws off a ring, which revolves round it ; 
 after a time the ring breaks and gathers itself up into 
 a smaller globe, which rotates besides revolving round 
 the large globe ; and then another ring is thrown off 
 as the velocity is again increased, and so on. 
 
 The experiment is defective in neither being able to 
 represent the attraction nor the contraction of the 
 globe, on which the whole theory depends, for the 
 reasons given at p. 169. The ring is thrown off when 
 the centrifugal force exceeds the equatorial attraction, 
 which decreases as the oblateness increases (p. 37) ; and 
 that depends also on the proportions of inside and out- 
 side density. English mathematicians seem to consider 
 this theory too uncertain to be worth expounding yet ; 
 but French ones have calculated from the present sun's 
 
270 Illusi/ration of the Nebula/r Theory. 
 
 rotation, that when it was as wide as each planet's or- 
 bit, it would turn in nearly the present period of that 
 planet : and the same of the planets and their moons. 
 
 And it is easy to see that if the solar mass did tnrn 
 at that rate, there would be the same equilibrium as 
 now between the centrifugal and attractive forces ; for 
 the attraction on any planet, and therefore on the sun's 
 equator, if it swelled into a globe as wide as the plan- 
 et's oi'bit, would be the same as it is now (p. 29) ; and 
 we shall see at p. 275 that a nebulous globe of any 
 uniform density can all rotate together. But if the 
 density then increases inward, the ' moment of inertia ' 
 decreases, and the outside will turn faster and fly far- 
 ther out, increasing the oblateness and perhaps separat- 
 ing as a ring. 
 
 An impulse given to a planet a little beyond its 
 centre would produce both the present motions of ro- 
 tation and translation, as you may see by striking a 
 floating ball in that way ; but it is diflScult to see where 
 such an impulse was to come from, acting in the same 
 direction on all the planets. 
 
 NEBULJE. 
 
 Bright patches in the sky, like a shining mist of 
 definite size, are called nebulae, whether they are large 
 or small, resolvable into stars by telescopes or not. 
 The success of Lord Kosse's great telescope in resolv- 
 ing into stars some nebulae which had resisted all 
 others, led a good many people to jump to the conclu- 
 sion thai all nebulae consists of stars, and are only not 
 resolved because they are too remote for the power of 
 
Many UTebulce consist of Stwrs. 271 
 
 any telescopes yet made ; just as a distant fliglit of 
 birds, or a plague of locusts in the East, "which looks 
 like a black cloud, can be distinguished either by com- 
 ing nearer or being looked at through a telescope which 
 magnifies the spaces between them. But we shall see 
 reasons presently for rejecting this conclusion, and ac- 
 cepting the other, that some nebulse are mere masses 
 of stars, and that others are as truly nebulous as comets 
 or their tails. 
 
 The grandest of all the nebulse, -the Milky Way, as 
 it was named by the Greeks and Romans, is that band 
 of strongish light which stretches over the whole sky in 
 clear nights without a moon. That is resolvable into 
 stars ; and onr sun with all his planets is only one of 
 them. For astronomers have come to the conclusion 
 that this band of light is nothing but a mafs of stars 
 laid thicker together than they are in the rest of space 
 in the form of a great grindstone, split and opened out 
 in on ' dirfection into something live the shape of a Y- 
 If we were placed in the middle of such a mass, it 
 would evidently look brightest in the direction in which 
 there are most stars ; as when you stand in a narrow 
 plantation of fir trees, it looks darkest in the direction 
 of its length, because the trees are dark while the stars 
 are light. And as they are all so distant that they 
 appear equidistant, they look like a bright band all 
 round us, of the width of the ' grindstone,' and as far 
 off as p.ll the stars appear to us. 
 
 The next most remarkable nebulae of the same kind, 
 but a vast deal smaller, are the things called the Ma- 
 gellanic Clouds in the southern hemisphere, though 
 
272 Milky Way and Magella/nic Clouds. 
 
 they are hardly nebulae exclusively. Sir. J. Herscliel, 
 who erected an observatory at the Cape of Good Hope 
 in order to complete his father's survey of the northern 
 hemisphere of stars by adding to it the southern, de- 
 scribes these ' clouds ' as consisting of visible stars, of 
 nebulse resolvable into stars, and of nebulae irresolva- 
 ble, though for convenience the whole of each mass 
 may be called a nebula. The larger of the two is 
 about 6° wide, or has 12 times the apparent diameter 
 of the moon, or 144 times her apparent size. And from 
 this is deduced a remarkable argument against the hy- 
 pothesis of all nebulae consisting of stars too far off to 
 be distinguished. Unless the clouds are like pillars or 
 very long ellipsoids turned endways to us, or if the 
 depth is anything like the same as the width, the depth 
 of the largest one corresponds to an apparent width of 
 6°, which means a width of about one-tenth of its dis- 
 tance from us ; therefore the back parts are only so 
 much farther off than the front. But it contains visi- 
 ble stars as well as nebulae ; and a nebula of stars ir- 
 resolvable by such telescopes as were usedj must be, 
 not one-tenth farther off than the visible stars, but many 
 thousand times as far : therefore those nebulae cannot 
 consist of stars — ^^if the visible stars are really in the 
 nebulae, and not in front of it, and therefore only op' 
 tically within it. 
 
 Another argument for the existence of purely nebu- 
 lous nebulae is founded on the spiral shape of some of 
 thein, and on that shape coming out not less but more 
 distinctly with increased telescopic power. Such a 
 shape, like that of a bunch or mop of long thin featheis 
 
All WebuloB do iiot consist of Stars. 273 
 
 or wisps whirled round, necessarily implies rotation of 
 a mass with some kind of continuity in it, almost as 
 certainly as if we saw it revolving, though it revolves 
 too slowly to have been yet observed ; and their size is 
 probably enormous, and their density very small in- 
 deed ; which, I will show you presently, implies a very 
 slow rotation. If the 'nebular hypothesis' is right, 
 these spiral nebulje may be other solar systems in the 
 course of formation.* 
 
 Here again the great modern test of the spectrum 
 analysis comes in with a decisive answer to the question 
 whether all the nebulae consist of stars, as some people 
 too hastily concluded when some new ones were re- 
 solved. Mr. Huggins and Professor Miller have found 
 about one-third of the 60 nebulae they have examined, 
 to be merely gaseous like the heads of comets (p. 261) ; 
 and have even gone so far as to ascertain that nitrogen 
 gas is the principal element in the composition of some 
 of them. Other nebulae give a continuous spectrum, 
 like the sun or a solid white hot body, and some have 
 an opaque nucleus within a gaseous envelope. 
 
 Some of them have other curious shapes implying a 
 definite constitution, such as the Crab and the Dumb- 
 bell nebulae ; and several have two heads like the di- 
 vided comet of Biela and like double stars, each of the 
 
 * I take these arguments from the ' Plurality of Worlds,' a book well knoivn 
 to have heen written hy the author of the Bridgewater Treatise on Astronomy, 
 and which (whether right or wrong) was very superior to the answers to it ; 
 thou<'h of course the whole subject is one of speculation. It is now almost 
 certain that the time the earth has been occupied by men is a mere drop in the 
 ocean of its history as a planet ; and the ' Plurality ' only dealt with men as in- 
 habitants of other worlds. It did not notice the great variations of heat in the 
 game planet described at p. 172. 
 13* 
 
274 '■ Plometary'' Neimlm. 
 
 heads being m-anifestly a kind of focus of the nebulous 
 iijatter which extends over them both. Some small 
 round ones, of uniform density or brightness, are called 
 planetary nebulae, a very inappropriate name, it seems 
 to me ; for they do not wander, which * planetary ' 
 means : they are certainly self-luminous, which planets 
 are not : they are (as yet) unresolvable, and certainly 
 not solid ; and if they should turn out to be resolvable 
 into 'star dust,' then they are still less like planets : 
 their only resemblance is that they are round ; for the 
 smallness is only the eifect of their distance. Though 
 the largest of them has an apparent diameter not much 
 greater tlian Neptune's, it has no visible parallax, 
 which means (for stars and nebulae) that the diameter 
 of the earth's orbit is as nothing compared with its 
 distance. If it had as much parallax as the nearest 
 star, that apparent diameter implies a real one nearly 
 twice as great as that of Neptune's orbit. Sir. J. Her- 
 schel says it must be far less brilliant than the sun, as 
 one might expect of a nebula. A few nebulae are an- 
 nular, and one of those has been resolved into stars, as 
 Saturn's ring is probably resolvable into a vast number 
 of small satellites. 
 
 It is worth while to notice a remarkable result of the 
 law of gravitation on the separate members of a large 
 and uniform mass of either stars or nebulous matter. 
 I said at p. 29 that the shell composed of the outer 
 parts of a globe exerts no attraction one way more than 
 another upon those within the shell. "We proved at p. 
 250 that the square of the time of revolution varies as 
 the cube of the distance divided by the mass of the at- 
 
Time of Rotation of a Webula. 275 
 
 tracting sphere. But if the sphere is uniformly dense 
 or full of stars or nebulous matter, the mass itself varies 
 as the cube of its radius, or distance of the outer parti- 
 cles from the centre. Consequently the time of revo- 
 lution of all .the particles is independent of their dis- 
 tance, and depends on nothing but their density or 
 closeness together; and the time varies inversely, or 
 the velocity of revolution varies directly, as the square 
 I'oot of that density. And it may easily be proved that 
 this holds not merely for the particles revolving in 
 great circles, or in a plane through the centre of the 
 sphere, but also for those revolving in small circles like 
 parallels of latitude round a polar axis. 
 
 So we have this remarkable result, that a sphere of 
 matter equally dense throughout, whether its particles 
 are free as in a fluid globe, or freer still as in a nebu- 
 lous one, can all revolve together, with equilibrium 
 between the centrifugal and attractive forces, without 
 any part wanting to lag behind the rest as they would 
 in a flat ring like Saturn's (p. 190). Then if the inner 
 parts alone contract, and become denser than the outer, 
 they will revolve faster, for the reason given at p. 234 ; 
 but the outer parts will revolve at the same rate as be- 
 fore; for the mass of the sphere within remains the 
 same, and attracts as if it were all condensed into its 
 centre, though the density is no longer uniform. Con- 
 sequently the outside of a nebulous globe densest in 
 the middle must revolve slower than the inner parts, 
 if equilibrium is to be preserved. Also the particles 
 of a revolving nebula which is contracting must evi- 
 dently move in spirals toward the centre, and get 
 
276 Rotation of a Nebulous Globe 
 
 denser as they get nearer ; wHch is the aspect of the 
 spiral nebulse. But mere rotation round a polar axis 
 would not prevent the globe from collapsing into a flat 
 spheroid by the attraction of its particles toward the 
 equator. They can only be kept apart by the expan- 
 sive force, or else by revolving in great circles in all 
 directions. 
 
 Another consequence of the law of time and distance 
 and mass at p. 251 is that a dense globe of stars or 
 loose matter must turn faster than a thin one of the 
 same size to preserve equilibrium. Tliis may seem 
 strange, but you must not judge of bodies which contain 
 their own force by anything we can deal with on the 
 earth. "VVe saw that the time varies inversely as the 
 square root of the mass, and therefore of the density of 
 a mass of given size ; and therefore the velocity of ro- 
 tation varies directly as that same square root. So if 
 the earth's orbit were nearly filled up with a sun of 
 that size, and of the same density as the present sun, 
 the earth would have to go roimd it in 2h. 45 ra. instead 
 of a year, to, avoid falling upon it and being stuck there. 
 And a globe of little planets or any other loose matter 
 of that average density, filling up the orbit of the earth, 
 must rotate with that velocity. But on the other hand, 
 if the sun were spread out into a sphere large enough 
 to fill the earth's orbit, it would take a year to rotate 
 (p. 270). None of the spiral nebulae turn fast enough 
 for their motion to have been perceived yet, though 
 some double stars revolve in 30 years : which is there- 
 fore a proof of the extreme thinness of these nebulse, if 
 they do rotate. Of course a solid or a fluid globe, whose 
 
Must Increase with, its Density. 277 
 
 internal attraction can compress it no more, may tiu'n 
 as slowly as it likes, though not too fast, or it will break, 
 and throw off its equator. The earth might turn 17 
 times faster than it does before it would throw off its 
 equator, which now only swells out 5 miles beyond 
 the radius of an equal sphere (pp. 24, 37). 
 
 THE STAES. 
 
 I mentioned the parallax of the stars just now ; and 
 I explained at p. 208 that the parallax of the sun, or 
 moon, or any planet, is the apparent radius of the earth 
 as it would be seen from those places ; and that it is 
 found by observing the difference of apparent position 
 of those bodies from opposite ends of the earth's diam- 
 eter. But it is of no use to look at the stars from op- 
 posite sides of the earth for the purpose of measuring 
 their distance. A very feAV of them can just be per- 
 ceived to change their places with reference to the rest, 
 when looked at from opposite ends, not of the earth, 
 but of the earth's orbit, or at intervals of half a year. 
 That therefore is our scale for measuring their distan- 
 ces ; and their parallax is the apparent radius of our 
 orbit, as it would look to them, or the angle whose nu- 
 merical value is our sun's distance divided by their 
 distance. There is only one star, a Centauri, so near 
 that our orbit would appear to it as Avide as the planet 
 Neptune does to us, its parallax being o"-976 ; the 
 next, 61 Cygni, is nearly twice as far : two stars called 
 21,238 Lalande and 1830 Groombridge, si and 4 
 times ; 70 Ophiuchi, a Lyrse, and Sirius, all above 6 
 
278 Distance of the Nea/rest Stars. 
 
 times as far ; Arctnrus and two or three more have a 
 just sensible parallax. 
 
 Now let ns see what this means in actual distance. 
 The numerical value of o"'g'j6, bj the method I ex- 
 plained at p. 241, is '0000047; or the nearest star is 
 21 1,340 times as far off as the sun, or about 19-^ trillion 
 miles. And as light takes 8J minutes to come here 
 from the sun, you may easily calculate that it takes 3-J 
 years to come from the nearest star, 6J years from the 
 next nearest (as far as we know), and 2 1 from Sirius 
 the brightest star ; and of course much longer from the 
 fainter stars, which there is good reason to believe are 
 on the average more distant than the bright ones, as 
 we shall see presently. 
 
 Magnitudes of the Stars,— The stars are proverbially 
 ' innumerable,' like the sands on the sea shore. Even 
 the number visible with telescopes is beyond all count- 
 ing. It is estimated . roughly, by dividing them into 
 districts and counting a few, that not less than 5 mil- 
 lions can be seen with a telescope far below the largest ; 
 24 of the brightest ai-e called ' stars of the first magni- 
 tude,' 60 of the second, 200 of the third, and about 
 15,000 of the smaller magnitudes above the eighth. 
 These divisions are merely arbitrary ; but now that 
 their brilliancy can be compared by experiments, Sir 
 J. Herschel and other astronomers have proposed that 
 some more definite scale of magnitude shall be agreed 
 on. But their brightness or apparent magnitude bears 
 no definite relation to their distance, or even to the 
 intrinsic brightness of the few stars whose distance has 
 been measured. Sirius is 4 times as bright as a Cen- 
 
Magnitudes of Stars. 279 
 
 tauri, though 6J times as far off; therefore it must be 
 170 times (or 4X 6-5') as bright at the same distance; 
 and Sirius gives out nearly 400 times as much light as 
 the sun. It has been estimated that the light of sun- 
 shine here is a hundred million times as much as we re- 
 ceive from all the stars. 
 
 The principle which Sir J. Herschel advocates for 
 classifying the stars by magnitudes, i.e., by apparent 
 brightness, and to which the received classification does 
 approximate, is that stars of the second magnitude 
 should be those which are on the average twice as far 
 off as those of the first, as far as we can judge ; third 
 magnitude stars three times as far as the first, and so on. 
 But how is that to be done ? It can only be done on 
 the assumption that the stars generally give out the 
 same absolute qiiantity of light, or would all look equal- 
 ly bright if they were at the same distance ; which 
 means that the intrinsic or specific brightness x the 
 real size of the disc of each star is the same ; which it 
 certainly is not individually, as we saw just now, but 
 maybe nearly enough for a general average. We must 
 also assume for this purpose what is rather improbable, 
 that no star light is worn out or lost in its passage 
 through the luminiferous aether, as sound is by the air 
 which carries it, especially when there is any wind 
 against it : a fact which no philosopher has yet at- 
 tempted to explain, as far as I know. Bells which are 
 heard 6 miles off with the wind cannot be heard a mile 
 against a wind which you can hardly feel. 
 
 Assuming these two things, the light of the stars here 
 will vary inversely as the square of their distance ; and 
 
280 8ta/r Magnitudes and Distances. 
 
 therefore a star of the second 'magnitude ' must be one 
 ■whose apparent light is only a quarter of those of the 
 first magnitude; the light of third magnitude stars is 
 one ninth of the first, and so on. The proportions of 
 two lights can be pretty accurately measured by con- 
 trivances for cutting off so much of the brighter one as 
 will reduce it to an equality with the dimmer one ; 
 which the eye can judge of, though it cannot compare 
 proportions of light without such aid. In this way then 
 the distances of the stars visible by the eye can be ap- 
 proximately measured in terms of. the distance of the 
 nearest, a Centauri, which is measured by its parallax 
 of nearly i ", or ias multiples of 20 trillion miles. 
 
 But we can go a great deal farther than this with 
 our measures. An eye pupil twice as wide as usual 
 would take in four times as many rays of light from the 
 same star at a given distance, or would take in as many 
 rays from a star at distance 2 as a common eye at dis- 
 tance I, since they spread out four times as wide at 
 twice the distance. You will see in the next chapter 
 that telescopes only penetrate into space by virtue of 
 their being a very large eye, or bringing into our eye 
 much more light from a star than it can collect for it- 
 self. Therefore if a telescope which penetrates into 
 space 200 times farther than we can see without it, 
 brings into view a new star, making it as bright (say) 
 as one of the 8th magnitude, it must be 200 times as far 
 off as the average of 8th magnitude stars, upon the as- 
 sumptions we have made, or 1600 times the 20 trillion 
 miles which is the unit for star magnitudes or distances. 
 And as the light takes 3-^ years to come here from the 
 
Time of IHght coming from the Stars. 281 
 
 nearest stars, it would take 5 300 years from such a star 
 as I have described, of which there are multitudes be- 
 yond counting, and others far beyond them. 
 
 ' As the stars recede into distance, time recedes with 
 them ; and there are stars from which Noah might be 
 seen stepping into the ark, and Eve listening to the 
 temptation of the serpent.'* How far they do re- 
 cede we cannot tell : all we know is that no telescope 
 has yet reached the outside of them ; for every in- 
 ci'ease of telescopic power only brings more stai-s into 
 view. 
 
 If light is wasted or worn out in its passage through 
 space, as it is through a fog, these calculations fail to 
 this extent, that the light decreases faster than the 
 square of the distance increases, and we do not know 
 how. much faster. In that case stars beyond a certain 
 distance can never be seen in any number of years, 
 because the undulations of tlieir light would be all 
 gone before they reached our distance. Indeed by the 
 ordinary law of emanations according to the inverse 
 square of the distance, the rays from any single star 
 may become too thinly spread to make any distinct 
 impression on the eye, even when gathered in by any 
 possible telescope ; but a multitude of stars then com- 
 bine their light and produce a luminous haze or nebula, 
 which may or may not be resolvable by telescopes. 
 
 If all the stars above the horizon at night, visible 
 and invisible together, cover an apparent space equal 
 to the sun, and are intrinsically as bright, their light 
 does waste prodigiously in coming. Otherwise the 
 
 * Fronde's Lecture on The Science of History. 
 
282 Stars ha/oe no Visible Size. 
 
 night would be as bright as the day, for the reason given 
 at p. 165. And if the sky is so full of stars that we 
 nowhere see right through, them, the earth would get 
 455,600 times the light and heat it does from the sun, 
 unless they waste in coming.* The question whether 
 light is lost depends on whether the luminiferous aether 
 is perfectly elastic, i.e., whether its particles inove 
 among themselves without friction or viscidity : a wave 
 will go farther in water (though even that has some 
 friction) than in treacle. 
 
 As a globe as wide as the earth's orbit would have a 
 scarcely measurable diameter to any of the starsj it is 
 not to be expected that any single star should have a 
 measurable size to us. Every now and then people 
 have fancied that they could discern an apparent disc 
 of some very bright star, but it has always turned out 
 to be what is called the spurious disc with rings round 
 it ; which is one of the consequences of the undulatoly 
 theory of light, and is larger with a small object glass 
 than a large one. A planet looks larger in a tele- 
 scope, because even a disc of 2", like Neptune's, can 
 be magnified into an angle 1000 or 2000 times greater ; 
 but a telescope only shears a star of the rays which 
 give it a false appearance of size to the eye. The 
 twinkling of the stars also cannot be explained without 
 a fuller account of the imdulatory theory of light than 
 can be given here. Again, when the dark edge of the 
 moon comes over a star it is extinguished suddenly ; 
 which not only proves that the moon has little if any 
 atmosphere, but that all the rays from the star to our 
 
 * Appeudis, Note XXVI. 
 
Double md Treble Stars. 283 
 
 eye are practically close together, as if it were a point, 
 even at the distance of the moon. 
 
 Many stars are not single, but consist of two or three 
 so close together that they require a higli magnifying 
 power to separate them, though their distance apart 
 must be enormous to show even then ; and they revolve 
 round each other. About 600 brilliant double stars 
 are known, besides the more numerous fainter ones ; 
 and in 120 of these pairs the companions have differ- 
 ent colors, besides different degrees of brightness. 
 And both the brightness and the color sometimes 
 change during their revolution, as if their opposite 
 sides were different. Many single stars change also. 
 Sirius (which however is double) was called a red star 
 1700 years ago, in the time of the astronomer Ptolemy, 
 the author of the erroneous theory of the solar system 
 which is called after him ; but it is now a white one. 
 Several others change periodically, and a few rotate 
 like the sun. In 1866 a star in Corona Borealis 
 rose quickly from the 9th to the 2d magnitude, and 
 soon after faded again, as othei'S have done before. 
 Everybody has heard of the ' lost Pleiad,' by which 
 those once conspicuous ' seven stars ' are reduced to 
 six. And many others have come in and gone out 
 within the time of astronomical history. 
 
 Spectrum Analysis of Stars.— The star which started 
 into temporary splendor in May, 1866, and again from 
 August till November, was analyzed by Messrs. Hug- 
 gins and Miller by the method I have several times 
 mentioned, and its spectrum showed a great outburst 
 of hydrogen gas. A good many other stars have been 
 
284 Spectrimi Anah/sis of States. 
 
 examined in the same way, and are found to contain 
 sodium (oi- the base of common salt), magnesium, mer- 
 cury, antimony, bismuth, hydrogen, and some other 
 less common materials, including a few not known 
 upon the earth. Their light, like the sun's, comes from 
 some solid incandescent matter surrounded with an at- 
 mosphere containing the vapors of these elements. The 
 same investigators say that blue or purple stars are 
 always found in connection with a stronger red one ; 
 and there is no doubt that the difference of color in tlie 
 stars is due to chemical differences of composition, and 
 temporary changes of color always show a change in 
 the condition of their atmosphere. 
 
 Double Stars, — The question whether the second and 
 paler of a pair of stars is a planet or a star, means 
 simply, is it self-luminous, or does it only shine by re- 
 flection ? For in all other respects there is no differ- 
 ence, although we are accustomed to think of planets 
 as much smaller companions of a sun than the com- 
 panions of some of the stars appear to be. If any star 
 is found to have a satellite or companion which is peri- 
 odically brighter and dimmer, and quite invisible at 
 that part of its orbit which we call inferior conjunction 
 for Yenus and Mercury, when they are between the 
 sun and us, that would be a planet. The author of 
 the ' Plurality of Worlds ' remarks that the periodical 
 obscuration of the star Algol does not prove the pas- 
 sage of a planet over it ; for the planet would have to 
 be one-sixth as wide as its own orbit to produce the 
 observed effect ; which is altogether impossible. But 
 it seems now tolerably certain that Sirius has a compan- 
 
Revolution of Double Stars. 285 
 
 ion or satellite : which may be called a planet from 
 its darkness, hut is very unlike one in its mass ; for its 
 existence was discovered by its causing a large nuta- 
 tion backwai'd and forward in the mean motion of Sir- 
 ius, which I shall speak of presently ; and the planet 
 has been since observed, and its distance measured. 
 
 In this case, and in all the revolving pairs of 'stars, 
 -whenever they are within measurable distance, and 
 there has been time to see the period of their revolu- 
 tion, we can also ascertain their joint masses, and some- 
 times even their separate ones. For by the law of 
 time and distance, of which I liave said so much, the 
 period of two bodies revolving roimd each other de- 
 pends only on their mean distance and their joint 
 mass. So if their distance can be got, clear of fore- 
 shortening from an oblique position of their orbit, that 
 and their period will give their joint mass. And then 
 the proportion of the masses to each other can be 
 found in the way described at p. 251 for the earth and 
 moon, provided we can measure also how much either 
 of the two stars oscillates about the mean place or 
 common centre of gravity. 
 
 Let us illustrate this by performing the operation on 
 Sirius, assuming that the facts stated are correct. His 
 companion is said to be at least 47 times as far off as 
 we are from the sun, or above half as far again as 
 Neptune, and the period is about 49 of our years. 
 Then if you look back at p. 250 you will see at once 
 
 that '^^^ must = ^ = 43-25 ; remembering that 
 sun 49 
 
 ' Sirius ' here means the mass of himself and his com- 
 
286 Elements of some Double Spars. 
 
 panion together, as ' sun ' means really sun + earth. 
 But astronomers have also made out that Smus oscil- 
 lates, or nutates, on each side of the joint c. g. i6J 
 times our mean distance ; and therefore the joint mass 
 has to be divided between them in the proportion of 
 nearly 31 to 16; and we may say that Sirius alone = 
 
 31 X 43 X sun „ „!.• 1 • J.- 
 ^"i = 28 suns. Ihis, however, is very lar 
 
 47 _ \ _ _ _ 
 
 short of the measure given by their relative quantities 
 of light ; for that proportion of mass (if the densities 
 are equal) gives a diameter only 3 times, and a disc 9 
 times the sun's, instead of 400, which is found to be the 
 proportion of their lights ; but Sirius may be made of 
 materials lighter (in both senses) than the sun. 
 
 Similarly the elements of three other double stara 
 have been found, and we may arrange them as follows, 
 making the sun the unit of mass, and our year and 
 mean distance the units of period and distance between 
 the two associates. I have not the data (if they exist) 
 for separating the masses of any but Sirius, as we did 
 just now. 
 
 
 Joint mass. 
 
 Periofl. 
 
 Distance. 
 
 Parallax. 
 
 Sirius . . 
 
 • . 43-15 
 
 49 
 
 47 
 
 "•IS 
 
 70 OpUiuchi . 
 
 . ■ 3-1 
 
 96 
 
 30'5 
 
 •16 
 
 a Centauri . 
 
 • • -55 
 
 78 
 
 15 
 
 •98 
 
 61 Cygni . . 
 
 . . "I 
 
 514 
 
 29-3 
 
 •54 
 
 Motion of the StarSi — There remains the question, 
 are the stars generally in motion ; and if so, what kind 
 of motion ? That can be answered d priori thus : — 
 Nothing can be at rest where attraction is universal. ' 
 Not rest but motion is essential to the stability of the ' 
 
Revolution of all the Stars. 287 
 
 universe ; or rather we must say, it is a choice of mo- 
 tions ; either the stars are all moving to destruction 
 and chaos at a common centre, or else they keep their 
 distance only by revolving round it. Whether they 
 extend an infinite or a finite distance into space, vi^hich 
 must needs be infinite, as time must run into eternity, 
 can make no difierence ; for each successive shell of 
 stars does nothing to help those within it against their 
 mutual attraction to the common centre, since a spher- 
 ical shell exerts no attraction within it (p. 29). And 
 so the whole starry sphere, whether we like to consider 
 it a large or a small one, must revolve under the condi- 
 tions I explained before for any single nebula or globe 
 of stars. That assumed indeed that they are scattered 
 uniformly through the sphere : which they are not, for 
 some parts of the sky are much more starry than others, 
 especially the Milky Way; but that only varies the 
 forces, and affects the direction and velocity of the mo- 
 tions, and not the argument. 
 
 But the question has been answered by observation 
 also. Sir W. flerschel discovered that one important 
 star, the sun, moves toward the constellation Hercules, 
 at a rate which is now estimated as about the diameter 
 of the earth's orbit in a year. And lately a general 
 motion of them all, but one, is thought to have been 
 ascertained. That one is Alcyone or tj Tauri of the 
 Pleiades (p. 52), which is therefore assumed to be tlie 
 centre of the universe. But Sir J. Herschel thinlcs tliat 
 unlikely, because Alcyone is 26° away from the Milky 
 Way, which can hardly be conceived according to 
 dynamical laws to revolve in any direction but in its 
 
288 Possible Time of Revolution 
 
 own plane, as a grindstone does, and therefore not 
 round a centre so far off. 
 
 ]S"ot that the stars of the Milky Way need revolve in 
 the same time like the particles of a grindstone, or are 
 likely to revolve round any polar axis. For a mass of 
 stars so revolving would have sunk into the plane of 
 the equator through the common centre of gravity long 
 ago by their mutual attraction, unless indeed they keep 
 vibrating across it. Each star probably moves in an 
 independent plane passing through the common c. g., 
 as the planets move in planes through the sun all with- 
 in the narrow band of the zodiac. Indeed if the solar 
 system and the other stars did revolve in the same time 
 round any polar axis, we could no more see that any 
 of them moved at all than one part of the earth can 
 see the rotation of the other parts. But the theory 
 of the motion of the stars is in a very imperfect state 
 as yet. 
 
 It is worth while to calculate the kind of period 
 which a revolving mass of stars would require for equi- 
 librium between attraction and centrifugal force, on 
 some simple assumption of the average density of the 
 mass. Let us assume then that all the stars are as 
 heavy as the sun on the average, and generally as near 
 each other as the nearest of them is to us, i.e., 20 tril- 
 lion miles off: which we will write 2 ® for shortness, 
 meaning thereby 2 with 1 3 cyphers, which only have 
 to be doubled and trebled for squares and cubes. This 
 assumption is probably too favorable to the average 
 density of the universe, but it will answer well enough 
 for the purpose of illustration. But how close can the 
 
of the Universe of Stars. 289 
 
 stars be packed, or can they be packed at all, so tliat 
 all the intervals may be the same ? The first of these 
 questions is not so easy as it looks, and I can only give 
 you an answer which is the result of some trigonome- 
 try. 
 
 If you want to see how the stars must be arranged, 
 to be at equal distances, get some bullets and lay three 
 together in a triangle, or four in a rhombus (not a 
 square), and go on from that making as large a heap 
 as you like. In this way each star stands in the mid- 
 dle of a sphere of 12 others equidistant from it and from 
 each other. If any number of balls are so packed in 
 a rhomboidal box which exactly fits them, it may be 
 proved that each ball occupies on the average 707 of 
 the cube of its own diameter ; while the ball itself is 
 only "523 of that cube, as I said at p. 15. Therefore 
 in order to find the average density of the universe, we 
 must consider the matter of each star to be expanded 
 into the bulk of 707 x 8 ® cubic miles. 
 
 But we saw at p. 270 that the sun would rotate in a 
 year if his present matter filled the earth's orbit, or 
 were expanded into a globe of "523 X914' ®' cubic 
 miles ; and that the time of rotation of nebulous globes 
 is inversely as the square root of their densities. Con- 
 sequently the time of rotation of a globe of the average 
 density of the universe is to a year as the square root 
 of the larger of the above masses is to that of the 
 smaller ; which you will find, if yon take the trouble 
 to work it out, is a little more than 119 millions to i. 
 Therefore that is the number of years which the whole 
 
 mass of stars would take to revolve, on the assumption 
 18 
 
290 H'aming of the Stars. 
 
 we have made : and still more if the average density 
 is less. And as 360° = 1,296,000", this amounts to a 
 revolution of i" in nearly 92 years. 
 
 The central attraction on a particle of a very flat 
 spheroid of uniform density also varies as its distance 
 from the centre, as in a sphere ; though the amount of 
 attraction is of course less than at the same distance 
 from the centre of the sphere, because the attracting 
 mass is less in the proportion of the minor axis to the 
 major. But the atti-action is 2'2 as much as it would 
 be according to that proportion. Therefore if the 
 Milky "Way is equivalent to a spheroidal mass of the 
 density we assumed, and with a major axis =10 times 
 the minor, its internal attraction will be "22, and the 
 velocity of revolution ./■22 or about half of what we 
 calculated for the sphere. 
 
 THE CELESTIAL GLOBE. 
 
 The thing called a celestial globe is a picture on a 
 globe of all the stars as they appear in the sky. Prop- 
 erly of course such a globe should be hollow, and we 
 in the inside of it. And in fact all the pictures a/re 
 reversed from what they appear in the sky. Take the 
 well-known figure of the Great Bear or Charles's Wain, 
 which is much more like the outline of a saucepan 
 with a bent handle. As we see it in the sky, and in 
 star maps, this handle, or the bear's tail of three stars, 
 is on the left hand (except at that time of night when 
 it is turned wrong side up) and the two stars which 
 form the outer edge of the saucepan, and are the point- 
 ers to the north pole star, are on the right hand. But 
 
What Celestial Globes JSepresent. 291 
 
 it is drawn the opposite way on the ontside or convex 
 surface of the celestial globe. The celestial globe is 
 covered all over with pictures of those Bears, Lions, 
 Virgins, Scorpions, and other things, into which the 
 ancients fancifully divided the groups of stars and called 
 them constellations. The particular stars are usually 
 denoted by Greek letters attached to the Latin name 
 of the constellation, as a Alpha Lyrse- (the brightest 
 northern star), a Centauri (the nearest), ^ Beta Leonis, 
 y Gamma Draconis, and some by numbers in a con- 
 stellation, as 6 1 Cygni, etc. A few have names of 
 their own, as Sirius, Alcyone, Polaris, now the pole 
 star within i° 24', but once 46° off in consequence of 
 precession ; and it will be as near as 26' in 360 years. 
 Another thing which a celestial globe has on it is the 
 Zodiac, which is a band of 8° on each side of the eclip- 
 tic, or 16° altogether, and on which are constellations 
 of the 12 Signs of the Zodiac, and within which all 
 the planets have their orbits, except some of the Aste- 
 roids. 'But as I explained in speaking of the preces- 
 sion of the equinoxes (p. 51), the signs have left their 
 constellations nearly 30° within the time of astronomi- 
 cal records. Therefore you see on the celestial globe 
 the two crossings of the equator and ecliptic, °P and =2=, 
 lying toward the left hand of the picture of the Fishes 
 and of the lady called Yirgo, the Astrsea of the poets, 
 who holds the scales of justice (Libra) in her hand. 
 Celestial globes, like terrestrial, are made to turn upon 
 the poles of the equator, not on the poles of the eclip- 
 tic. One would be just as right as the other for a 
 celestial globe, only they represent different things ; 
 
292 Bight Ascension and Dedination. 
 
 turning the globe on the poles of the equator represents 
 the daily rotation, which produces the rising and set- 
 ting of the sun, moon, and stars : turning it on the 
 poles of the ecliptic would represent the motion of the 
 moon round us in 2gf days, and the apparent motion 
 of the sun round us in a year. 
 
 Right Ascension and Declination. — I have already said 
 at p. 48, that longitude of the ieavenly bodies is mea- 
 sured from f on the ecliptic, and not on the equator 
 from the meridian of Greenwich : which indeed would 
 be nonsense, as Greenwich is changing its position 
 with reference to every star every minute. Perhaps it 
 might as well have been measured on the equator, as 
 another name has had to be invented for the measur- 
 ing of the stars along the equator from that point f 
 where all the celestial measures of that kind begin. 
 ^That other name is Bight Ascension, commonly writ- 
 ten K. A. ; and there is no east and west in it : it rims 
 on from o to 360° on the celestial globe, and from left 
 to right, to correspond to the earth's moving as it does 
 through space from right to left. The thing which 
 corresponds to terrestrial latitude for the stars is called 
 declination : that is their distance from the equator. 
 E. A. is also measured by time, like terrestrial longi- 
 tude, 15° 15' 15" being = ih. im. is. 
 
 There is also heliocentric longitude, which is the an- 
 gle between two planes through the sun's centre and 
 perpendicular to the ecliptic, one through the planet 
 and the other through T ; while geocentric or common 
 celestial longitude is the angle between two such planes 
 through the earth's centre. So the terrestrial longi- 
 
N.P.D., Altitude and Azimuth. 293 
 
 tude of any place is the angle between the two planes 
 called meridians, which both go through the earth's 
 centre, intersecting all along the polar axis, one through 
 the place in question and the other through Greenwich 
 observatory, or whatever place each nation takes for 
 its zero or o° of longitude. The declination great cir- 
 cles for E. A. also intersect in the polar axis, only the 
 zero is at '=p ; but th* quasi-meridians for celestial 
 longitude intersect in the poles of the ecliptic. Per- 
 haps I should remind you that the angle between (or 
 the inclination of) two planes contains the same num- 
 ber of degrees, minutes, and seconds as the arc which 
 the planes cut out of any circle of any radius drawn 
 round their line of intersection as an axis, such as 
 small circles of latitude no less than the equator. Of 
 course the larger you make the circle the larger the arc 
 is — in magnitude, but not in degrees, as the minutes 
 on a clock face are larger than on a watch. 
 
 Polar Distance is the comjilement of declination, or 
 the difference between the declination and 90°. This 
 is so much used that it is abbreviated in almanacs for 
 our hemisphere into N.P.D. So the zenith distance 
 of a star is the complement of its altitude above the 
 horizon. The distance of the pole, or pole star (if it 
 were exactly there) above the horizon, or the distance 
 of the zenith from the equator, is the latitude of the 
 place where you are ; and the distance of the pole 
 from the zenith is the colatitude, or the difFerence of 
 the latitude from 90°. 
 
 The ' horizon ' in astronomy does not mean the acci- 
 dental boundary of your sight by earth or sea, but a 
 
294 Cdestial LaUiude and Longitude. 
 
 plane through the middle of the earth parallel to the 
 ' level ' surface of water or mercury where you are, or 
 to which the plumb line is perpendicular (see p. 12). 
 The dip of the horizon is the angular distance that you 
 can see below the hemisphere of sky of which the base 
 is the level plane through your eje ; and it is evidently 
 greater the higher you are above the earth. The low- 
 est point, opposite to the zenith, is called nadir / both 
 of which words came from Arabia, like the whole con- 
 trivance of our common numbers, without which the 
 science of arithmetic and mathematics must almost 
 have stood still. 
 
 The word Azimuth also came from there ; which is 
 to Altitude what Longitude is to Latitude, and what 
 E. A. is to Declination ; viz., the distance between two 
 great circles like meridians, passing through the zenith 
 instead of the pole ; and it is reckoned from the north 
 point of the horizon; but the Prime Yertical goes 
 through the zenith and the east and west points. Ce- 
 lestial latitude is measured from the ecliptic, as terres- 
 trial latitude is from the equator. Formerly the lon- 
 gitude of the planets was, and in some almanacs still 
 is, given by the signs, as =£= 25", or K 6°, but more fre- 
 quently now by the con-esponding number of degrees, 
 205°, or 336°. Ancient dates are sometimes fixed by 
 certain stars having been said to rise heliacally then ; 
 which means rising just before the sun. 
 
 An ecliptic is generally drawn on terrestrial globes, 
 crossing the equator at 30° west and 1 50° east longi- 
 tude : but it has no meaning there. On the celestial 
 globe it has a meaning and a proper place ; and the 
 
Orreries. • 295 
 
 great circle half way between T and =^, which passes 
 through the poles of the equator and ecliptic and the 
 places of the solstices, is called the solstitial colure. 
 
 Orreries — so named after a Lord Orrery who bought 
 a plcmetarium 1 50 years ago — are of great antiquity, 
 and may be called working models of the solar system. 
 In old times, of course, they made the earth its centre ; 
 but they were made by Huyghens and Eomer to illus- 
 trate the Copernican system and the motions of Jupi- 
 ter's moons, and much later by Dr. Young and other 
 astronomers. By a proper combination of wheels, of 
 which I gave a simple illustration at p. 57, they can 
 be made to show all the regular motions of the earth, 
 moons, and planets, the changes of seasons, and the 
 conditions of eclipses. But an orrery with a moon 
 . only a quarter of an inch wide and an earth of an inch 
 ought to have a Jupiter of 1 1 inches, a Saturn's ring 
 of 21, a sun of 9 feet, and a platform of 1 1 miles diam- 
 eter to take in the .orbits of all the planets on their 
 proper scale. 
 
CHAPTER YI. 
 
 ON TELESCOPES. 
 
 Books on astronomy generally give some account of 
 the modes of mounting and using telescopes for differ- 
 ent purposes, while the explanation of the principles 
 or ' theory ' of the telescope itself is left to treatises on 
 optics.* But for that simple looking tube, with only 
 a large magnifying glass at one end and a small one at 
 the other, we should hardly know more of astronomy 
 than the Chaldseans did, or at any rate Copernicus, 
 who only guessed but could not prove that the sun is. 
 the centre of the solar system ; nor should we be able 
 to measure a single celestial distance, or even the 
 earth itself accurately, or navigate the great seas with 
 certainty of reaching any given place, and the civili- 
 zation of the world would stand still. Therefore I 
 propose to give a fuller explanation of the theory of 
 telescopes than in the former editions of this book, so 
 far as it can be given without resorting to mathematics. 
 
 I said at p. 35 that Galileo was the first inventor of 
 telescopes; and so he was for astronomical purposes, 
 though it appears that they had been made above three 
 centuries before in England by the famous Friar Eoger 
 Bacon, the greatest philosopher from his own time to 
 
 * There ia a clear and concise treatise on optica in Chambers' Edacatlonal 
 Course, requiring very little knowledge of geometry. 
 
Iwuentors of the Telesccype. 297 
 
 that of his more celebrated namesake Francis Bacon, 
 Lord Yerulara and St. Albans, commonly called Lord 
 Bacon, which he never was. Friar Bacon also invented 
 gunpowder, and so many other things that he was ac- 
 cused of magic, like Dr, Faustns, one of the inventors 
 of printing, and imprisoned by the Pope for ten years 
 — much longer than Galileo, and only released when 
 he was 74, after which he lived six years, and died at 
 Oxford in 1294. But the invention perished with him, 
 and was lost till it was made again by Jansen and 
 Lippershey in Holland, in what form is not knoAvn, 
 and very soon after independently by Galileo in 1609, 
 in the form which still survives in opera glasses. 
 
 That was however soon superseded by what is called 
 the ' astronomical telescope,' as the best kind of watches 
 are called chronometers, though all watches are or 
 profess to be chronometers or time-measurers.. Kepler 
 seems to have suggested this construction first, though 
 he did not execute it, nor perceive all its advantages. 
 One Father Scheiner was the first who did, and it was 
 afterward improved by Father Eheita ; and still more 
 in 1656 by the celebrated Huyghens, who discovered 
 the true law of refraction, and also invented a compound 
 eye glass which is still for some piirposes the best. For 
 the present I only say that Galileo's telescope is a short 
 tube with a large convex lens, or one thickest in the 
 middle, at one end, called the object glass, and- a small 
 concave one called the eye glass at the other end. The 
 ' astronomical ' differs from it in having a convex eye 
 glass at the end of a longer tube. In both, the object 
 glass must be flatter than the eye glass to produce any 
 13* 
 
298 A Telescope is a Lcvrge Eye. 
 
 magnification : indeed the amount of it depends en- 
 tirely on the proportion of the curvature of the eye 
 glass to that of the object glass. 
 
 Space-penetrating Power.— But before we talk of 
 magnifying there is another part of the business of tele- 
 scopes, which is much more simple, though less un- 
 dei'stood by most people, who are satisfied with know- 
 ing that a telescope is a combination of glasses for mak- 
 ing distant things look nearer by magnifying them. 
 It is not by means of magnification that telescopes en- 
 able us to see millions of stars beyond what the eye alone 
 can see : or as it is called, to penetrate farther into 
 space. The space-penetrating power depends simply on 
 a telescope being a very large eye, and bringing into 
 our eye as many more of the rays from each point of a 
 distant object as the area of the object glass exceeds the 
 area of the pupil of the eye. It is true that a telescope 
 cannot bring into the eye all the rays it receives unless 
 the magnifying power is at least equal to the width of 
 the object glass divided by the width of the pupil ; 
 but the magnification of powerful telescopes far exceeds 
 that. 
 
 To measure that power we must remember that we 
 should want an eye four times as large, or twice as wicU, 
 to see a star just visible no'w, if it were removed twice 
 as far ; since the rays are spread out four times as thin 
 at twice the distance. Besides that, at least one-eighth 
 of the light is lost in passing through" the glasses ; 
 which is the same as if the area of the object glass 
 were only |, or its width yl or -935 of what it really 
 is. Therefore the space-penetrating power of a tele- 
 
Space-penetrating Power. 299 
 
 scope is to that of one eye as '9 of the width of the 
 object glass is to the width of the pupil — nearly a 
 quarter of an inch. 
 
 By far the largest telescopes that are made have 
 concave mirrors instead of object glasses, as I shall ex- 
 plain afterward; and their space-penetrating power 
 would be greater in the same proportion as their width ; 
 but much more light is wasted by reflection than by 
 passing through glasses : indeed nearly half generally 
 instead of one-eighth. Therefore the space-penetration 
 of Lord Kosse's telescope, whose mirror or speculum is 
 
 6 feet wide, is x 72 x -^ or 144 times that of a 
 
 4/2 |/2 
 
 pair of eyes ; for the power of two eyes is 4/2 times that 
 of one,* on the same principle as just now stated. This 
 means that the telescope will penetrate a sphere of 
 stars of 144 times the radius that we can see without 
 it ; and therefore, if the stars are scattered equally 
 thick all through what we can only call infinite space, 
 we can see 144' or nearly 3 million times the usual 
 number with such a telescope, except so far as they hide 
 each other. The penetration of a refracting telescope, 
 i.e., one with an object glass instead of a mirror, 15 
 inches wide, the largest yet completed, is about 47 
 times that of a pair of eyes by the same calculation. 
 
 Magnification! — In both cases the whole of the rays 
 received by the telescope are made to converge by the 
 mirror or object glass in just the same way as the rays 
 from the sun converge into a small bright spot or image 
 of the Sim in a common burning glass. But tliis is con- 
 
 * Appendix, Note XXVH. 
 
300 What Distinat Vision depmds on. 
 
 nected -with tlie otlier part of the business of telescopes, 
 •whicli requires a good deal more explainiug. Magnifi- 
 cation, either by a single convex lens or by any com- 
 bination of glasses, consists in the rays which come to 
 the eye from the outsides of an object being made to 
 come at a wider angle with each other than they do 
 naturally. But that is not all : if it were, a common 
 2 inch magnifying or bui'ning glass which makes a 
 bright spot at 3 inches from the glass would magnify 
 the sun from 32', its apparent diameter to the naked 
 eye, to 37°, or nearly 70 times : that being the angle 
 at which the outside rays converge to a focus through 
 such a glass. 
 
 Indeed if you could look at the sun through such a 
 glass with yom* eye at the focus — and you may at the 
 moon — it would appear magnified enormously; but 
 then it is all in confusion, because the rays from each 
 point of the sun or moon are not brought to a point 
 again at the back of the eye, where a distinct (though 
 inverted) picture of everything seen must be formed if 
 it is to be seen distinctly. The picture of the last 
 thing seen actually remains there for a time after 
 death, as Scheiner discovered long ago. The condition 
 for distinct vision is that the bundle or pencil of rays 
 which come from each point of the object to the glass 
 shall be brought to the eye in a parallel state, like a 
 thin stick of straight wir^ (or rather diverging for 
 short-sighted eyes), but tlie pencils from the various 
 points of the object must make some sensible angles 
 with each other if it is to have any apparent size ; and 
 the wider those angles are the more it is magnified. 
 
Effect of Magnifioation. 301 
 
 It may occur to you to ask what is tlie use of apply- 
 ing telescopes to the stars when no magnifying power 
 will make them look any larger (p. 283). The use is 
 to measure their distances from each other and from, 
 the meridian or the equator or any other circle ; for 
 those of course are magnified by the telescope, and are 
 measured by the time the star takes to move over the 
 distance, or by the angle which the telescope has to be 
 moved through from one star to another. 
 
 Again you may say that terrestrial objects seen 
 through a telescope do not appear larger than usual, 
 but only nearer. But that is exactly the same thing ; 
 for, as I have said several times, the apparent size of 
 anything has no meaning except with reference to its 
 distance ; and it is only because we know by experi- 
 ence the size of houses, trees, and men, that we do not 
 think of them as appearing larger when seen near than 
 far off. "When we come to mountains or buildings 
 above the usual size we are constantly deceived, and 
 generally take them to be smaller and nearer than they 
 are. On the other hand very small things do appear 
 very large through a microscope, which is only a tele- 
 scope adapted for near objects, because we can never 
 see them so large with the naked eye, in consequence 
 of the eye not being able to see anything distinctly 
 nearer than 5 or 6 inches ; for then the rays from each 
 point diverge too much to be brought to points again 
 by the lenses of the eye, and to form a distinct picture 
 on the retina. 
 
 The short explanation of an astronomical telescope 
 is this : see the figure at p. 312. Eays of light spread 
 
302 
 
 Astronomical Telesco>pe. 
 
 from the point A of tke object ? in all direction?. A 
 certain number or pencil of them is received by the 
 object glass OPQ all over it: after passing thj-ougli 
 that convex lens tbey converge to a point or focus /. 
 In like manner tbe rays from another point B come to 
 the object glass, crossing the A rays without at all in- 
 terfering with them, and are made to converge to an- 
 other point i at the same distance as f,' and in that 
 way an inverted image y* of the object AB is formed 
 there, which is as much smaller than the object as Of 
 is less than OA, and very near the eye glass abo. Then 
 the rays that came from A diverge a little from/ to a, 
 and the B rays to b, and there both pencils are refracted 
 and made to converge into the eye at E, making a much 
 wider angle dEb with each other than they did coming 
 from the object : which is the increase of apparent size 
 or magnification. Besides that, the rays in each pencil 
 are changed by the eye glass from diverging into par- 
 allel ones, which is the state the eye requires for dis- 
 tinct vision. 
 
 Law of Refraction. — But tho^igh this is enough as a 
 
 general description of a 
 telescope, I have given 
 you no reasons for any- 
 thing that the lenses 
 do, nor explained what 
 the magnifying power 
 is for any given pair of 
 glasses. If you want 
 to understand these 
 things we miist begin 
 
Refraction hy Glass. 303 
 
 again -with refraction, and see how tliey all come 
 from one tolerably simple law- — not to go farther 
 back into the causes of that, which are far more com- 
 plicated. For this purpose we must use a figure. A 
 BC is a prism or wedge-shaped piece of glass, and Sa 
 a ray of light entering it at a from the air, and XaY 
 perpendicular to the surface. Then instead of the ray 
 going on straight, it is bent aside at a into the direc- 
 tion ab nearer the perpendicular, unless it happens to 
 be itself perpendicular, in which case only it goes 
 straight. 
 
 So long as the obliquity is small, as it always is with 
 the rays which fall on telescope lenses, the angle S«X 
 in the air may be considered to bear the constant pro- 
 portion of |- to the angle iaY in the glass ; which 
 number is therefore called the index of refraction for 
 glass ; as i is for water (compared with air) and 2*44 
 for diamonds. At & the ray emerges again out of glass 
 into air, and follows the same law, turning off into an 
 angle G5Z which is \ of the angle aiY in the glass, 
 Y5Z being the perpendicular at that surface. The an- 
 gle in the thinner medium is generally called the ' angle 
 of incidence ' and that in the dense one the ' angle of 
 refraction ; ' but they are just the same whichever way 
 the ray is going, and it is better to think of them as 
 the air angle and the glass angle. 
 
 If the two surfaces of glass were parallel, it is evi- 
 dent that the second refraction at h would just balance 
 the first at a, and the ray would emerge parallel to its 
 first direction, only a little pushed aside, according to 
 the thickness of the glass and the obliquity of the in- 
 
304: General Law of Refraction. 
 
 cidence on either surface. But if the surfaces are in- 
 clined, the emerging ray (whether it is Sa or G5) devi- 
 ates from its original direction, as you see in the figure, 
 and is brought downward toward the thickest part of 
 tlie glass, or away from the angle C, called the refract- 
 ing angle, where the two refracting surfaces meet. It 
 may be proved by a little geometry that the deviation 
 is half the re&acting angle of the prism ABC ; i.e., if 
 Sa is parallel to BAG or any other line in that sort of 
 position, the angle JGA will be half the angle G, pro- 
 vided the angles at a and h are small ones ; and they can- 
 not both be small unless C is. 
 
 Though this is not a treatise on optics, yet to prevent 
 mistakes I had better tell you that when the angles of 
 incidence and refraction are not small the relation be- 
 tween them is not quite so simple. If you draw any 
 
 line SX at right angles to aX, the fraction -— is 
 
 called the sine 6i the angle SaX. (In old books SX 
 
 would be called ' the sine of SaX to radius Sa,' but the 
 
 other is the most convenient definition and the least 
 
 liable to mistake.) And the true relation between the 
 
 two angles, either at a or 5, is that the sine of the air 
 
 angle is | of the sine of the glass angle. Yon will find 
 
 in books of mathematical tables the sines of aU angles 
 
 SX 
 up to 90°, when the -^ evidently becomes i, and they 
 
 can go no farther, but decrease again ; but so long^ as 
 the angles are small, the sines increase in very nearly 
 the same proportions as the angles themselves, which 
 simplifies telescope calculations materially. 
 
Internal Heflection. 305 
 
 Internal Reflection. — On the other hand when you get 
 up to 41° 48', the sine of that is | ; and therefore no 
 glass angle beyond that can have any corresponding 
 air angle, as there is no angle with a sine greater than 
 I. And then conies the remarkable consequence that 
 a ray trying to emerge from glass at any greater ob- 
 liquity than that, which is called the critical angle, 
 cannot get out there at all, but is reflected back again, 
 more completely than if the back of the glass were 
 silvered for a miiTor. Thus if the angle obY in the 
 figure were as much as 42°, the ray would not go out 
 of the glass, but would be reflected to g on the third 
 side of the prism, and there go out to E with thp usual 
 refraction, unless it arrived at g also too obliquely to 
 get out, when it must try again at the side BO. The 
 critical angle of water is 48^°, and of diamonds only 
 28° : which is the reason of the brilliancy of diamonds, 
 and of dewdrops which stay on certain leaves unbroken, 
 viz., that a great deal of the light falls on them too ob- 
 liquely to pass through their back surface, and so it is 
 reflected. 
 
 Reflecting Prism. — Therefore if light falls directly 
 (i.e., perpendicularly) on one of the narrow faces of a 
 prism whose angles are 90°, 45° and 45°, it reaches the 
 wide face at the same angle 45" ; and as that exceeds 
 42°, it is all reflected out directly at the other narrow 
 face, at right angles to its original direction ; and sucli 
 a prism is a far better reflector than a flat mirror set at 
 the angle 45" to reflect at right angles. You will see 
 at p. 332 the use which is now made of this in New- 
 ton's telescope. A little light indeed is reflected at 
 
306 Refraction ly a Lens. 
 
 every change of surface, which is the reason of the loss 
 of light in passing through the glasses of telescopes, 
 but only a little until the critical angle is reached. 
 You will see how this also is turned to account in tele- 
 scopes for looking at the sun (p. 333). 
 
 If you find it difficult to believe that light cannot 
 pass in any direction through a transparent medium, 
 you may verify it by hanging something a little under 
 water on the far side of a trough or basin, and looking 
 at it while you bring your eye down to the water on 
 the near Jside. At a certain distance it will vanish, 
 although you can see the far side of the trough below 
 and beyond it. The reason is that the rays which 
 ought to come from it to your eye reach the surface of 
 the water too obliquely to come out, and are reflected 
 down again internally, as you would see if your head 
 were under the water instead of over it. 
 
 Thus far there is no material difference between a 
 thin prism of which ABC is a section and a convex 
 lens ; for though its faces are segments of spheres 
 '(either alike or not), a ray passes through it exactly as 
 it would through a prism touching it at the points », S, 
 as I have drawn the figure. ' But when many rays fall 
 on different parts of the surface, a lens will manifestly 
 affect them all differently; and those which fall near 
 the edge will be more refracted than those near the 
 middle, because the surfaces are more inclined to each 
 other near the edge than near the middle, where they 
 are parallel. And it fortunately happens that all the 
 raj's from any one point, far or near, after passing 
 through a convex lens, whether directly or obliquely, 
 
Focal Length of a Lens. 307 
 
 converge again (very nearly) to another point or focus 
 somewhere in a straight line through the first point 
 and the centre of the lens ; which line is called the 
 axis of that pencil: as you may see with the rays 
 spreading from A or B over the lens OPQ in the as- 
 tronomical telescope at p. 312, and converging again 
 to/" and * in the axis of each pencil. 
 
 Focal Lengthi — If the rays come from a point so far 
 off that they are practically parallel, the point of con- 
 vergence is called the focus of the lens, and its distance 
 from the lens is called the focal length. Conversely if 
 rays emanate from a point at the focal length, they 
 will emerge from the lens parallel ; but if from a point 
 nearer, they will not converge at all, but only diverge 
 less than before : if from a point beyond the focal 
 length, they converge to a focus somewhere beyond the 
 focal length on the other side. These two points of 
 divergence and convergence are called conjugate foci, 
 and are related by a very simple rule : if the distance 
 of one is u and of the other v and the focal lengthy, then 
 v is the product of/w divided by their difference, u be- 
 ing greater thany,- if it is less, then v is the distance 
 beyond u from which the rays appear to diverge after 
 passing through the glass. 
 
 The focal length of a glass lens bears a very similar 
 relation to the radii of curvature of its two surfaces. 
 For it is twice their product divided by their sum ; or 
 by their difference, if one side is concave but the lens 
 effectively convex or thickest in the middle, which is 
 called a meniscus. Therefore the focal length of an 
 equi-convex lens is the radius of curvature of each face. 
 
308 Centrical and Eccentrical PeiioUs. 
 
 and of a plano-convex it is twice tlie radius of curva- 
 ture of the one spherical face. Another practical rule 
 is that the focal length is the square of half the width 
 of the glass divided by its effective thickness or excess 
 of the middle over the edges. But these rules only 
 hold for glass, whose refractive index is \. 
 
 Concave lenses also have a focal length, which may 
 be calculated by the same rules, but it means a differ- 
 ent thing. When parallel rays fall on such a lens they 
 are made to diverge as if from a point much nearer 
 than the object, and the distance of that point is called 
 the focal length. Kays really coming from that point, 
 or an}' other, by no means emerge parallel, but are 
 spread out wider by a concave lens. 
 
 A pencil of raj's may pass either ' centricallyj' through ■■ 
 a lens, like the pencils AO/" and all the others through 
 the object glass of the telescopes (p. 312), or 'eccen- 
 trically ' as they do at a and J in the eye-glass. The 
 surfaces across the centre of a lens are practically par- 
 allel to each other, or perpendicular to the axis of any 
 pencil of rays through the centre ; and therefore cen- 
 trical pencils are not refracted at all, though the rays 
 which compose them are refracted to a focus some- 
 where in the axis of the pencil. But eccentrical pen- 
 cils are wholly refracted, and bent inward, and so made 
 to converge to the eye E beyond the lens. 
 
 That also shows how a common magnifying glass or 
 convex lens magnifies things near the focal distance. 
 All the points in the object, except the middle one, are 
 seen by eccentrical pencils, which are refracted and 
 made to converge to the eye at wider angles with each 
 
Images in Telescopes. 309 
 
 other than the pencils by which those points are seen 
 without the glass. "Without the glass the apparent size 
 of S would be/E^ (drawing lines /E, lE to the eye) ; 
 with the glass it is dEb, a much larger angle. If the 
 object is much neai'cr the glass than its focal length, 
 the rays of each pencil diverge too much for the glass 
 to make them parallel enough for distinct vision ; and 
 besides that, the angle /"E* is then nearly the same as 
 fflE5, and so there is little magnification. On the other 
 hand, if it is too far off, the rays of each pencil con- 
 verge into the eye and again produce indistinctness ; 
 and the pencils also become so nearly centrical that 
 there is no sensible magnifying. The magnifying 
 power of a glass for objects near its focus evidently 
 varies inversely as the focal length. 
 
 ImageSi — But a magnifying glass has another and a 
 very different effect, much less known, on which the 
 action of a telescope depends. If you fix a common 
 long-sighted spectacle glass in the open window, and 
 recede from it, any object which you see through it 
 becomes more and more confused, until it suddenly 
 reappears inverted and smaller. In fact it is no longer 
 the object itself that yoii see, but its image formed as I 
 described at page 303. If the object is so far off that the 
 rays of each pencil may be called parallel, the image will 
 evidently be at the focus of the lens ; and the width 
 of the image is to that of the object as the focal length 
 is to the distance, or = the numerical value of the ap- 
 parent diameter x the focal length. If the object is 
 a little beyond the focal length, the image is farther 
 off and largei', as in microscopes. So the transparent 
 
310 How a Tdeacope oats. 
 
 pictures put upside dpwn into a magie lantern, a little 
 beyond the focus "of a convex lens, cast a magnified 
 image of themselves upright on the wall, or on a wet 
 sheet if you want to see them from the other side. On 
 the other hand the small bright spot under a burning 
 glass is the image of the sun, taking his shape reversed 
 in an eclipse, or that of the crescent moon. 
 
 SpectacleSi — Short-sighted eyes recjuire the rays from 
 each point to come in a diverging state, as they do 
 when the object is very near, though the pencils from 
 the different points then come more converging, and so 
 the object looks larger than average eyes can ever see 
 it. Therefore concave spectacles suit short sight, 
 which make the rays diverge. Long-sighted eyes re- 
 quire parallel rays, and therefore cannot distinctly see 
 things near, without convex glasses to make the .di- 
 verging rays parallel, not really to magnify the objects, 
 though they do so. Holding a book far off makes the 
 rays of each pencil come parallel or 'nearly bo, but it 
 also makes the letters appear smaller and send less 
 light into the eye ; so that it is the same as reading 
 smaller print by a worse light ; therefore it is better 
 to use spectacles. 
 
 "When we talk of parallel rays, it always means the 
 rays of one pencil and not the rays of different pencils ; 
 only the stai'S are so far off that the pencils of rays 
 from their outsides come to us at no angle that can be 
 measured by the utmost magnifying power, and there- 
 fore practically as parallel as the rays of each pencil ; 
 and that is why a star in the largest telescope looks 
 nothing but a point, though Neptune or the satellites 
 
Astronomical Telescope. 311 
 
 of Jupiter can be magnified into discs of visible size. 
 For the rays of a pencil diverge no more than the width 
 of the glass divided by the distance, while the extreme 
 rays or pencils from a planet make an angle = the 
 width oi the planet divided by its distance. 
 
 Astronomical TelescopOi — You can now understand 
 the operation of an astronomical telescope, and how its 
 glasses must be placed, and how much it magnifies. 
 The image being formed at the focus of the object glass 
 differs from an ordinary small picture in the fact that 
 the rays can only go from it in the directions in which 
 they are sent by the object glass. Consequently the 
 eye glass must be exactly at the proper place to receive 
 them and bring them in a proper state for vision 
 to the eye : that is, for ordinary eyes it must have its 
 own focus coinciding with the image or the focus of 
 the object glass, and be moved a little nearer to it for 
 short sighted eyes which require diverging rays. 
 
 Magnifying Poweri — Then for the magnification : the 
 pencils osE, 5E are parallel to /c, ic, the lines from ,/ 
 and i through the centre of the eye glass, because _/" 
 and i are at the focal distance ; and so the apparent 
 size of the image a'Eb =foi, instead of AOB the nat- 
 urally apparent size of the object, which is evidently 
 the same as .fOi. Those angles being small are in- 
 versely as the distances of the image _/* from and o 
 (the angles are immensely exaggerated in the figure for 
 distinctness) ; or the magnification is the focal length 
 of the object glass divided by that of the eye glass. 
 And that is the case in all telescopes, except two of the 
 reflecting ones Avhich I shall mention afterward. 
 
312 Astronoinical Td6S0(ype; GalUecm. 
 
GaWLeam Telesco^pe, or Opera Glass. 313 
 
 You see then that the business of the object glass is 
 not to magnify the object, but to form an image of it 
 to be magnified by the eye glass ; and the flatter the 
 object glass (provided it has some convexity), or the 
 greater its focal length, the larger is the image that it 
 forms. The average diameter of Jupiter being nearly 
 40" or "0002, the size of its image from a glass of 10 
 feet focal length is "024 inches ; and with an eye glass of 
 I inch focal length, the apparent diameter of li will be 
 magnified 120 times, or into 80' or nearly 3 times that 
 of the sun or moon, and Jupiter will look 7 times as 
 big as the moon, and Saturn nearly as big as the moon. 
 
 But magnifying power is reckoned by diameter and 
 not by disc, which varies as the square of the diame- 
 ter.* 
 
 Galileo's Telescope (see figure at p. 312) has a concave 
 eye glass placed at its own focal length before the locus 
 of the object glass instead of behind it. Therefore it 
 makes the rays of each pencil come out parallel, but 
 the pencils themselves diverge into the eye at an in- 
 creased angle lea, instead of converging to E. But 
 that produces a widened image in the eye just as well. 
 Here no image is formed in the telescope, and there is 
 no inversion ; for though the pencils Aa, B5, cross at 
 O as before, the A pencil from the top of the object 
 comes into the eye in the direction lea, as from the top 
 of a larger ? , and the B pencil Fc5 comes from the 
 bottom. The crossing of the apparent directions does 
 
 * This does not apply to advertisements of microscopes magnifying drops of 
 Trater many tlioneand times : they evidently call a magnification of two hun- 
 dred, 40,000. 
 14 
 
314: Field of View in Tdescopes. 
 
 not signify. In the astronomical telescope the rays 
 cross in a different way at E where they enter the eye, 
 and a diaphragm or plate with a hole in it is fixed there 
 to guide the eye to the proper place. 
 
 Field of Viev/. — But the capacity or field of view in 
 the Galilean telescope is limited by the size of the pupil 
 of the eye, as the pencils diverge into it ; and for the 
 same reason very high magnifying powers cannot be 
 used. I have drawn all the parts of both telescopes 
 out of all real proportion, or else the small angles made 
 by the rays would not be visible. The field of view 
 AOB, or the largest apparent diameter that can be seen, 
 evidently = the angle aOh or the width of the pupil 
 divided by the length of the Galilean telescope, even 
 if the eye is close to the eye glass : which is a reason 
 for keeping them short. In the astronomical telescope 
 that angle is not the width of the eye, but the width 
 of the eye glass, divided by the length of the telescope, 
 measuring by the axis of the two extreme pencils, which 
 however allows half of their rays to slip over the edges 
 of the eye glass. And the field is limited to that by a 
 diaphragm fixed across the focus to stop any pencils 
 more oblique than those whose axes fall just within the 
 eye glass. Glasses of higher power or shorter focal 
 length are also smaller, and so the field of view is less 
 the higher the power that is used in the same tele- 
 scope. 
 
 Long telescopes with high magnification often have 
 & finder set upon their back, which is a short and small 
 telescope of less power, in which the eye glass is much 
 larger in proportion to the length, and therefore the 
 
Telescopes Diminish Brightness. 315 
 
 field of view greater. This also was an invention of 
 Newton's. 
 
 Another serious defect of the Galilean telescope for 
 astronomy is, that, as the rays never reach a focus, there 
 can be no cross wires set there to mark the transit of a 
 star, or to define its exact position, which is a most 
 important part of the business of telescopes. The 
 inversion by the astronomical telescope does not signify. 
 In land telescopes of that construction another lens, or 
 more, is introduced to reverse the image back again, 
 and then it is far better than the Galilean. 
 
 Brightness! — Although a telescope gathers much 
 more light than the eye, and would act as a burning 
 glass, it does not make objects look brighter, but dim- 
 mer. For if all the light received by it enters the eye, 
 it can only exceed that received without the telescope 
 as the area of the object glass exceeds that of the pupil, 
 or as the squares of their widths. And if you draw a 
 pencil of rays going centrally through both glasses you 
 will see that its width at the object glass (which is the 
 width of the glass itself) is to its width at the eye glass 
 (and therefore to that of the pupil, if the whole pencil 
 is to enter it) as their focal lengths, or as the magnify- 
 ing power. But the rays are spread out wider and 
 thinner over the magnified object or image in propor- 
 tion to the square of the magnifying power, and there- 
 fore (under the above condition) as the area of the 
 object glass is to that of the pupil. So the magnifica- 
 tion diminishes the brightness at least as much as the 
 size of the object glass increases it ; and the brightness 
 is reduced still more, either if the pencil does not fill 
 
316 Ohrojnatic Dispersion. 
 
 the eye by reason of the magnification, or if it is too 
 large to be all taken in. Beside that, there is the loss 
 of light in the glasses, as stated at page 298. So the 
 image in a telescope can never be as bright as the 
 object. 
 
 Color DispersioHi — But both kinds of refracting tele- 
 scopes, so far as I have described them yet, have a 
 serious defect, which made Newton and others abandon 
 them for reflecting ones. If you look through a strong 
 magnifying glass, or a common cheap telescope, you 
 see things fringed with colors and indistinct ; as is still 
 more visible in the magnified images of a magic lantern. 
 The reason is that the seven colors which make up a 
 ray of white light are not refracted equally by any sub- 
 stance ; or every ray of white light is split up by re- 
 fraction into seven colored ones, each of which wiU not 
 divide again. It is not very easy to see it at one re- 
 fraction, but it is evident enough with two, as when a 
 ray emerges from a prism. Turning back to p. 302, 
 the whole emergent ray does not come in the direction 
 5G, but only the middle or green part of it, while the 
 red is refracted through the smaller angle ZbR, and the 
 violet as much as ZbY : the whole spectrum consisting, 
 first of invisible cool and chemically acting rays, most 
 refracted, then violet, indigo, blue, green, yellow, 
 orange, red, and then invisible hot rays, as described 
 at p. 81. Those at the violet end are comprehensively 
 called blue, and those at the other end of the spectrum, 
 red. 
 
 Consequently a lens cannot bring any pencil of rays 
 really to a focus ; but a blue image of every point of 
 
Chromatic Dispersion. 81Y 
 
 the object is formed nearer to the lens than the red 
 one, and a green one somewhere between the two, 
 which is overlapped by the red rays before they reach 
 their focus and by the blue diverging again after 
 they have passed their focus. Therefore the smallest 
 image of each point in the object is a little circle about 
 half way between the red and the blue foci ; and that 
 circle is never less than one-fiftieth of the width of the 
 lens. Consequently large object glasses could not be 
 used until this defect Avas cured by the invention of 
 compound aGhromatic or colorless glasses, which I will 
 describe presently. 
 
 Not only does the space-penetrating power depend 
 on the size of the object glass, but the power of ' divid- 
 ing ' double stars, in consequence of the ' spurious 
 disc ' being less with a large glass than a small one 
 (p. 282 ; and see E. A. S. Monthly Notices, 1867, p. 
 
 235)- 
 
 Moreover the color dispersion is increased to the eye 
 by increasing the magnifying power of the eye glass, 
 or using one of very short focal length ; and as the 
 magnifying power of the telescope is the focal length 
 of the object glass divided by that of the eye glass, the 
 only way of getting a highly magnifying telescope was 
 to have an object glass of very great focal length. Ac- 
 cordingly they were sometimes made from 100 to 600 
 feet long, so that the object glass had to be set upon a 
 pole, without any tube, and with strings and levers to 
 pull it into the right direction for sending the rays into 
 the eye glass. Such telescopes were called aerial. 
 "We ai'c I'eturning to them in one respect ; for tubes are 
 
318 Aerial Telesa/pes. 
 
 being superseded by open frames for very large tele- 
 scopes, but the glasses are of course fixed in them. 
 
 Achromatic Telescopes.— By an unlucky mistake in 
 an experiment Newton missed the discovery which was 
 not made use of till nearly a century afterward, by 
 DoUond the optician, though it was made before by 
 Mr. Hall of Worcester, that glasses of different densi- 
 ties have different powers of dispersion into colors for 
 the same amount of general refraction. A lens of flint 
 glass, which contains lead and has a specific gravity 
 y6, spreads the blue rays farther away from the red^ 
 though it may refract the red or the green equally with 
 a crown glass of the same convexity, whose specific 
 gravity is only 2'5. Therefore a slightly concave flint 
 lens may refract the blue rays outward just enough to 
 correct the excessive refraction of them inward by a 
 very convex crown lens, while it leaves a general bal- 
 ance of refraction over, because the two lenses together 
 are on the whole convex or thickest in the middle. Or 
 the same thing may be done by putting a double con- 
 cave flint lens between two convex crown lenses. I 
 have drawn the object glass of the GalUean telescope 
 in this way, and that of the astronomical with only two 
 glasses, in the pictures at p. 312. 
 
 The rule for determining their proportions is simply 
 that the focal length of the flint glass must bear the 
 same proportion to that of the crown glass (or of the 
 pair of them) as their respective dispersive powers, or 
 as -068 to -033. For the width of the spectrum of a 
 flint glass prism, or the angle between the extreme red 
 and violet rays, is -30 of its mean angle of refraction, 
 
AchromatiG Lenses. 319 
 
 or the refraction of the green ray, while the ratio of 
 dispersion to mean refraction by a crown glass is gen- 
 erally "17, or about half as much as by the flint glass. 
 
 Secondary Colors. — Still the correction is imperfect, 
 by reason of what is called the irrationality of disper- 
 sion ; which means that the ratio of dispersion to the 
 mean refraction is different for the different colors by 
 any two kinds of glass. A flint glass prism which dis- 
 perses the blue rays as far from the red as a given 
 crown glass (of a greater refracting angle) does not 
 disperse the green rays quite as far from the red as the 
 crown glass does. Consequently when the prisms are 
 put together, turned opposite ways, they bring out the 
 red and blue rays together, but leave a little green 
 behind ; and so the image through a compound lens 
 of that kind is still fringed with green, and has not a 
 sharp outline ; or the definition is imperfect. And al- 
 though this secondary dispersion is only a 6oth of the 
 primary, and therefore a 3000th of the width of the 
 object glass, it is enough to leave a sensible advantage 
 on the side of reflecting telescopes, which .make.no 
 colors. Even if a third glass of some different compo- 
 sition is used, none has yet been discovered which 
 completely annihilates the colors. 
 
 Dr. Blair indeed did effect it in a temporary way by 
 fluid lenses (of course enclosed between glasses), which 
 had the requisite action on the different colors: but 
 they could not be made to last. Certain oils and gums 
 refract green light much less than even flint glass ; and 
 a long table of them, in the invei'se order of their effect 
 on green light, is given in Sir D. Brewster's treatise on 
 
320 Secondary Colors. 
 
 Optics in the Encyclopaedia Britannica. Mr. Grove 
 17 years ago made some solid lenses of resin and castor 
 oil,* and also of hardened Canada balsam and castor 
 oil, interposed in a meniscus form between the crown 
 and flint glass lenses, which he said very nearly de- 
 stroyed all secondary color ; and he advised opticians 
 to pursue the subject on that footing, as they alone have 
 generally the means of doing it. Mr. Wray, an opti- 
 cian, has laXelj patented a ' semi-fluid cement ' of oil of 
 cassia and castor oil, or Canada balsam, to be inter- 
 posed as a meniscus lens, ' hermetically sealed ' between 
 the crown and flint lens, the crown glass being made 
 flatter at the back than usual, to leave I'oom for the 
 meniscus of cement, as Mr. Grove's was. . 
 
 This is said to answer extremely well for the achro- 
 matism, but great doubt is expressed as to the possi- 
 bility of making any but a solid composition permanent, 
 under the changes of temperature to which telescopes 
 are exposed. For an observatory must keep nearly the 
 same temperature as the air outside, or the refraction . 
 is disturbed (p. 215). The glasses of a compound ob- 
 ject glass are indeed often united by a transparent ce- 
 ment of the same refractive index as glass, as nearly 
 as possible ; since that prevents internal reflection (p. 
 305), by making the two glasses as it were continuous, 
 instead of leaving a film of air between them. But 
 such cement is of the same thickness throughout, and 
 as thin as possible ; while these dispersive cement len- 
 ses are thickest in the middle, and therefore do not 
 expand and contract equally at the middle and the 
 
 * See Ast. Soc. Monthly Notices of 18^3, and Phil. Mag., March, 1S67. 
 
Spherical Aierration. 321 
 
 edges, and are difiScult to keep hermetically sealed be- 
 tween glasses if they are fluid. The principle of these 
 inventions is, that the oils and resins refract green 
 light less than glass ; and therefore if the convex lens 
 is in efi'ect made partly of glass and partly of these 
 things, they produce together a convex dispersion of 
 all the colors in such a ratio that the concave dispersion 
 of the flint lens can correct it : or that the two oppo- 
 site dispersions become 'rational' instead of irra- 
 tional. 
 
 Spherical Aberration. — Independently of color disper- 
 sion, a lens with spherical faces does not bring the rays 
 of any pencil or of any color absolutely to a point ; 
 for those which come through the edges are brought to 
 a focus rather sooner than those near the middle of the 
 lens. The amount of this spherical aberration, as it 
 is called, depends on the shape of the lens, and even 
 on which side of it faces the light, if the two sides are 
 different ; and it is always much less than the chro- 
 matic dispersion even of a single lens. The shape 
 which gives the least aberration (of those which can 
 be used) has the front face 6 times as convex as the 
 back ; which opticians call a crossed lens. The diame- 
 ter of the least circle of aberration, or the image of a 
 point in such a lens, is only the 2160th of its width, 
 when the focal length is 10 times the width. And it 
 increases as the cube of the width, but inversely as tlie 
 square of the focal length ; so in that 'respect also the 
 long aerial telescopes had an advantage. 
 
 Theoretically this defect may be cured by making the 
 faces of the glass not quite spherical, but practically 
 14* 
 
322 Oompov/nd Eye-Pieces. 
 
 they cannot be ground in any other form. And fortu- 
 nately the aberration can be corrected in another way, 
 by selecting proper degrees of curvature for the separate 
 lenses of a compound object glass, without interfering 
 with their achromatism, as that depends only on their 
 foca,l lengths, not on their total convexity and con- 
 cavity. Various distributions of the curvature will 
 answer, and the Kev. C. Pritchard, P.E.A.S., has calcu- 
 lated a table of corresponding curvatures to suit vari- 
 ous focal lengths, so that any glasses may be matched 
 at once. The best general forms are like those in 
 which I have sketched the two object glasses at p. 312 
 with the curvatures very much exaggerated. The 
 crown glass in front has its front face more convex 
 than the back, and the flint glass behind fits it, and has 
 its back convex, but less so than either of the crown 
 glass faces, so as to be effectively concave (which is 
 called a 'reversed meniscus '), while the whole object 
 glass is of course effectively convex. It is not worth 
 while to go through the similar arrangements for treble 
 glasses. 
 
 Compound Eye-pieceSi — Hitherto I have spoken of 
 the eye-glass as a single lens : which it still is for very 
 high magnifying powers. But where distinctness or 
 dejmition is chiefly aimed at, the eye-glass also must 
 be corrected for color, for spherical aberration, and for 
 something else. For suppose a centrical pencil of rays 
 to be brought exactly to a focus by the object glass : 
 it goes on to the eye-glass eccentrically, except of 
 course the one pencil which keeps the axis of the tele- 
 scope ; and then instead of being refracted into a 
 
Hvygh&ns's Eye Piece. 323 
 
 round and colorless stick or pencil of parallel rays, it 
 is spread out into a colored oval one, of rays not par- 
 allel in one direction (that of the plane of the paper in 
 the figures at p. 312) but diverging. Moreover all the 
 points of the image ought to be at the focal length of 
 the eye-glass, or it ought to be concave toward the eye- 
 glass : but in fact it is rather concave the other way, 
 being foi-med by the object glass. Consequently the 
 rays from the outside of the image come from rather 
 too far oif and are more turned inward by the eye-glass 
 than if they came exactly from its focal distance. 
 
 It is found that the best way to correct all these 
 errors is to divide the work of the eye-glass between 
 two less convex lenses separated by a space equal to 
 the mean of their focal lengths, or 2 inches if the focal 
 length of the first is 3, and of the second (next the eye) 
 I inch. The first is called the field glass, because the 
 field of view now depends on that (see p. 314), and it 
 stands at half its own focal length before the focus of 
 the object glass ; so that the image is now formed be- 
 tween the field glass and the eye glass. The best shape 
 for them is a menisous field glass with radii of curva- 
 tui-e as 4 to 1 1, and a ' crossed lens' {i.e., of radii i and 
 6) for the eye-glass, both with their convex faces to- 
 ward the light. 
 
 This is Huyghens's eye-piece, except that he had two 
 plano-convex lenses ; and it is also called the negative 
 one, because the image is behind one glass, but before 
 the other. And this involves a serious^defect for some 
 purposes. The image no longer has the same propor- 
 tions as tlie object, but the outer parts of it are con- 
 
324 Bamsden^s Bye Piece. 
 
 tracted, because the pencils near tlie edges of tlie field 
 glass are most refracted or bent inward. The image 
 of a square would thus have its corners bent in or the 
 sides curved out, and a piece of netting would appear 
 to have the outer meshes smaller than the middle 
 ones. Consequent^ it will not do to put cross wires 
 there for measuring an object or its position. 
 
 For that purpose the positive eye piece was invented 
 by Kamsden, a famous instrument maker about 1780. 
 In it both the glasses are beyond the focus of the ob- 
 ject glass, where the wires are, and so they and the 
 image are equally magnified by . the eye piece. The 
 glasses are two equal plano-convex lenses with their 
 convexities facing, at a distance = f of each of the 
 focal lengths ; and the field glass is a quarter of its own 
 focal length beyond the focus of the object glass. It is 
 not quite achromatic, but it has less spherical aberra- 
 tion and distortion tlian the other. 
 
 The principle of both the positive and negative eye 
 pieces is this. An achromatized pencil from the ob- 
 ject glass reaching the field glass eccentrically, its blue 
 rays are there most refracted, and are sent to the eye 
 glass nearer its centre than the red rays which are 
 least refracted. But those which strike the eye glass 
 nearest the centre, or least obliquely, are least refracted 
 by it ; and so the less refraction there balances the 
 greater refraction before, and the blue and the red 
 come out parallel, and then the lenses of the eye unite 
 them into a colorless point. The same kind of compen- 
 sation takes place also between the rays which are most 
 refracted and least refracted by the field glass by reason 
 
Micrometer a/iid HeUometer. 325 
 
 of spherical aberration, and the convexity of the image 
 being turned the wrong way ; and so these compound 
 eye pieces correct both defects. 
 
 The inverted image is turned into an erect one in 
 land telescopes by an eye-piece generally made of four 
 glasses; but we are not concerned with them. All 
 eye pieces are made adjustable for different eyes by the 
 eye glass sliding nearer to the object glass for short- 
 sighted eyes, which require the rays of each pencil to 
 diverge a little instead of being parallel. 
 
 MicrometerSi — The measuring wires which I have 
 several times spoken of are stretched across a small 
 frame which slides a little across the telescope with a 
 screw, of which the head is graduated, so that you can 
 measure exactly how far any wire has to move to bring 
 it into contact with a star, to measure its distance from 
 any other star or the meridian or zenith. The same 
 screw apparatus is also used outside some telescopes to 
 adjust or measure their position, with microscopes at- 
 tached to it, and in each case it is called a micrometer, 
 or measurer of small distances. The wires are fixed in 
 and parallel to the meridian for transit purposes, and 
 horizontally for measuring altitudes and polar distances. 
 The wi^es in the reading off micrometers are set so that 
 they may be brought to coincide with the graduations 
 of the circle they are to read off. 
 
 Since each pencil of rays spreads over the whole 
 object glass, half a leas makes the same image as the 
 whole one, only not so bright. If the two half lenses 
 slide along the common diameter two images will be 
 visible whenever the two centres do not coincide. 
 
326 The Largest Telescopes. . 
 
 And the distance from the fii'st contact of the images 
 to the last, measured by the tm-ns of the screw which 
 moves one half lens along the other, is evidently twice 
 the width of each image. This makes the best mi- 
 crometer for measuring diameters of the sun and plan- 
 ets and distances of double stars, and is called a hdAo- 
 •meter. 
 
 EEFLEOTING TELESCOPES. 
 
 In order to escape from the inconveniences of either 
 color dispersion or long aerial telescopes, JN^ewton and 
 Gregory and Cassegrain and Herschel invented their 
 several forms of the reflecting telescope, which produces 
 no colors, except in the eye glass, which Huyghens 
 had corrected long before compound object glasses 
 were made. And they are still used because mirrors 
 or specul/umos can be made much larger than object 
 glasses in the present state of art ; and I have shown 
 you already that the space-penetrating power of a tele- 
 scope depends on its size, or the quantity of rays it can 
 talce in. 
 
 Herschel's largest telescope had a speculum 4 feet 
 wide, and Lord Eosse's 6, with 53 feet focal lengtli 
 weighing 4 tons ; whereas the largest lens thatjias yet 
 been completed is only a little more than a foot wide, 
 with a focal length of 12 feet. So that it greatly ex- 
 ceeds any refracting telescope both in penetration and 
 magnifying power ; for in reflectors as well as refractors 
 the magnification varies as the focal length of the ob- 
 ject mirror. But a telescope is now in progress of 
 which the compound object glass is 25 inches wide. 
 
Reflection of a Concave Mirror. 327 
 
 It has been cast by Messrs. Chance of Birmingham and 
 made into a telescope by Mr. Cooke of York, and thus 
 far there is every reason to believe it is successful. It 
 has a steel tube 30 feet long. 
 
 The theory of reflection is much simpler than of 
 refraction, and is all deduced from this one law of na- 
 ture, that rays are always reflected from any surface at 
 the same obliquity as they fall upon it. Consequently 
 an object appears as far behind a flat mirror as it really 
 is in front of it, the apparent distance being determined, 
 as the real distance is, by the different view which each 
 eye takes, or else by moving one eye (p. 102). There- 
 fore you can see your whole body in a mirror half as 
 wide and half as high as you are. 
 
 Reflection from curved surfaces is not so simple as 
 from flat ones ; but, as with lenses, it is much simpli- 
 fied by the mirroi's of telescopes being only small seg- 
 ments of their spheres, and by the rays only falling on 
 them at a very small obliquity. I had occasion to tell 
 yon at p. 242 that the tangent at any point of an ellipse 
 (or a straight line which coincides with the ellipse there 
 for a moment) makes equal angles with the two focal 
 distances SP, HP (p. 238). Consequently by the law 
 of reflection rays of light emanating from one focus of 
 a prolate spheroidal miiTor are all reflected to the 
 other. Again a parabola is only part of an ellipse with 
 one focus at an inflnite distance from the other (p. 258) ; 
 and therefore rays coming from an infinite distance, i.e., 
 parallel rays, are all reflected by a parabolic mirror to 
 its focus ; and rays from a lamp at the focus are all re- 
 flected parallel to the axis. Moreover it may be proved 
 
328 Images m Conoa/ve Mirrors. 
 
 that a parabola for a short distance near the apse coin- 
 cides with a circle whose radius is twice the distance 
 of the focus from the apse ; and therefore a spherical 
 mirror reflects parallel rays (not far from its axis) to a 
 point half way along the radius of curvature, subject to 
 the spherical aberration, which is due to the slight de- 
 viation of the circle from the parabola. 
 
 Fortunately that is much less than from a lens of the 
 same size, as there are no means of correcting it. The 
 diameter of the least circle of aberration or smallest 
 image of a point is the cube of half the width of the 
 mirror divided by the square of 4 times the focal length, 
 or by the square of the diameter of the sphere of which 
 the mirror is a part. So that it is only the 12,800th of 
 the width of a mirror whose focal length is 10 times its 
 width ; and in Lord Eosse's 6 feet speculum of 53 feet 
 focal length the image of a point is a spot '006 inches 
 wide, subject to magnification by the eye glass of the 
 telescope. 
 
 A concave mirror then deals with rays which fall 
 upon it like an equi-convex lens whose radii of curva- 
 ture are half that of the mirror, or a plano-convex of 
 the same curvature as the mirror ; but that is only true 
 of glass whose refraction is ^ (p. 308). 
 
 All reflecting telescopes have a concave mirror as 
 wide as the tube, placed at its lower end, which re- 
 ceives the rays from the object and sends them to a 
 focus, and forms an image, exactly like an object glass, 
 only without colors. Its. diameter again = the focal 
 length x^ the numerical value of the angle represent- 
 ing the apparent diameter of the object, or its real 
 
Four 'Kinds of Befleoting Telescopes. 329 
 
 diameter divided by its distance. . Thus the image of, 
 Jupiter in Lord Eosse's telescope is about an 8 th of an 
 inch wide ; and the spherical aberration or image of 
 each point is spread over a 20th of the width and a 
 400th of the area of the whole image, whether it is 
 magnified much or little by the eye glass. You see 
 this is considerably less than the chromatic aberration 
 of a lens only a foot wide with 10 feet focal length, 
 even when it is corrected as much as possible (p. 319). 
 A concave mirror, like a convex lens, magnifies things 
 if your eye is near it, but diminishes and inverts them 
 if you are far enough off to see the image and not the 
 object in the glass : a convex mirror, like a concave 
 lens, diminishes without inverting. 
 
 There are four ways of dealing with the rays after 
 reflection from the great mirror. In Gregory's tele- 
 scope they were allowed to come to a focus, making a 
 first image there, and then were reflected again by a 
 smaller concave mirror set in the middle of the tube 
 to another focus which is also the focus of the eye 
 glass, passing to it through a hole in the middle of the 
 great mirror; and so the first image is reversed back 
 again at the second. In Oassegrain's telescope the 
 second mirror is convex, and receives tlie rays before 
 they come to a focus, acting like a concave lens, only 
 sending the rays backward instead of forward, less 
 converging than before, to a focus which again coincides 
 with that of the eye glass. 
 
 In ^Newton's telescope the second mirror is a flat 
 one, which makes no difference in. the convei'gence, 
 and also receives the rays before they reach a focus, 
 
330 
 
 Wewtonicm Telescope. 
 
 might send them to the eye glass thi'ough a hole 
 in the great mirror like those 
 other two telescopes. But instead 
 of that, the small mirror is set 
 obliquely in the tube, at an angl« 
 of 45° to the axis, and so it re- 
 flects the rays out sideways at 
 right angles to the axis. Hersehel's 
 telescope dispenses with the second 
 mirror altogether; for the great 
 one is set a little askew at the 
 bottom of the tube and sends the 
 rays to an eye piece fixed just with- 
 in the edge of the tube at the other 
 end. This can only be done in 
 very large tubes, where the observ- 
 er's head in the mouth of the tele- 
 scope only cuts off a comparatively 
 small part of the rays. 
 
 The action of Newton's tele- 
 scope is shown in this figure, in 
 which you may easily trace the rays 
 and pencils of rays from the points 
 AB of the object to the large 
 mirror OPQ and the small one M, 
 and thence to a and i, converging 
 to E the eye, at the wide angle 
 «E5. Cassegrain's telescope has 
 one advantage over Newton's, that 
 the small mirror being convex 
 tends to correct the spherical ab- 
 
JSpeoulum Metal. 331 
 
 erration of the large one. Gregory's on the contrary, 
 being concave, aggravates it. The magnifying power 
 of them both may be said to be the square of the focal 
 length of the great mirror divided by the product of 
 the focal length of the small one and of the eye glass. 
 The magnifying power of ISTewton's and Herschel's is 
 the same as in the two refracting telescopes, i.e., the 
 focal length of the great mirror divided by that of the 
 eye piece. 
 
 Sir J. Herschel expects the l!^ewtonian telescopes to 
 supersede all others now that a method has been 
 invented, which Newton himself again divined as 
 possible, of silvering concave glass speculums instead 
 of casting those heavy metal ones which also have to be 
 reshaped every time they are repolished. Lord Rosse's 
 speculum weighs 4 tons and is supported at the back 
 on many points by a complicated system of levers to 
 balance the pressure ; for an inequality of the thick- 
 ness of a thread bends a heavy speculimi enough to 
 destroy all convergence to a focus. It is not the 
 quicksilver back of the glass that reflects, as in a look- 
 ing glass, but a thin film of silver chemically deposited 
 on the front or hollow face, which can be repolished, 
 and even renewed when necessary, the glass being 
 once for all ground to the proper shape. If this can 
 only be made parabolical, so as to get rid of spherical 
 aberration, the days of refracting telescopes probably 
 are numbered. It is said that a good speculum of 
 this kind reflects nearly as much light as is refracted 
 through an object glass, while a bell-metal speculum 
 loses one-third of the light. The only difference be- 
 
332 Swn Telescopes, 
 
 tween speculums and bells is that bells are (or should 
 be) made of 1 3lbs. of copper to 4 of tin, while speculums 
 are 128 to 59, for reasons given in the Treatise on 
 Clocks and Bells. 
 
 All the reflecting telescopes but Gregory's make an 
 inverted image at the focus of the eye piece ; and tliey 
 all require compound eye pieces like refracting tele- 
 scopes. Newton's has this farther advantage over all 
 other telescopes, that the reflection out sideways may 
 be made to take place through the hollow cross axis on 
 which the telescope is balanced (if the great speculum 
 is not a heavy one) and so the observer may sit still in 
 one position and has neither to look upward, which is 
 fatiguing, nor to mount into a box high in the air to 
 look downward as with Herschel's. Lord Eosse's 
 telescope is on the Herschelian principle. ' 
 
 Gregory's and Cassegrain's telescopes are almost 
 abandoned because of the quantity of light which they 
 lose by the second direct reflection. Much less is lost 
 by the oblique reflection in Newton's, especially when 
 a reflecting prism is substituted for the second mirror, 
 as explained at p. 305. 
 
 Helioscope. — There is yet one other kind of telescope 
 which ought to be described now that observations of 
 the sun's surface are becoming an important part of 
 astronomy. The sun is far too bright to be examined 
 through any common telescope ; for though it does not 
 increase the apparent brightness, as I explained at p. 
 316, still it acts as a burning glass concentrating all 
 the light from the object glass into the eye. Conse- 
 quently it is necessary to reduce the quantity of light 
 
or JEfeUosoqpes. 333 
 
 very much, and yet to keep the other advantages of 
 magnification and of gathering a large pencil of rays 
 from every point. For it does not answer to reduce 
 the light by using a very small object glass, or covering 
 Tip all of it except a small hole in the middle. It is 
 necessary first to gather a large bundle of rays, and 
 then as it were to filter them by some contrivance 
 which will only let a small part of the light come into 
 the eye. The common expedient of smoked or colored 
 glass will not do either; for such a glass soon gets 
 heated by the rays which it stops, and a colored image 
 is an imperfect one. 
 
 The plan generally adopted is to make them fall 
 obliquely on one of the wide faces of a thin prism, a 
 little before they reach the focus. That is a very dif- 
 ferent thing from sending them directly into ajiarrow 
 side of a right angled prism, as in the Newtonian tele- 
 scope, which is intended to reflect them all internally 
 from the wide face. This thin prism reflects a little 
 of the rays from its first face, by virtue of the property 
 which all transparent substances have of reflecting 
 some of the light which falls on them while they trans- 
 mit all the rest. Here about one 30th of the rays are 
 reflected, and the rest refracted into and through the 
 prism, and sent away as not wanted through the end 
 of the telescope which is left open, while the re- 
 flected rays come out sideways into an eye piece 
 as in jSTewton's. The reason for using a prism in- 
 stead of a flat piece of glass is that the internal re- 
 flection from the second surface also would interfere 
 with the reflection from the first surface if they were 
 
334: Trwmit Circle. 
 
 parallel, whereas the prism sends it off in another 
 direction. 
 
 This gathering and filtration of the rays is sometimes 
 carried farther, by using a large double concave object 
 glass, unsilvered, as a reflector of some of the rays 
 which fall on its first surface, leaving all the rest to be 
 dispersed through it into the air behind. The rest of 
 the construction is the same as before. But M. Fou- 
 cault, the inventor of the pendulum experiment for 
 proving the earth's rotation (p. 33), has lately discov- 
 ered that an object glass covered outside with a thin 
 film of silver or gold burnished bright will reflect away 
 the greatest part of the rays, while it is transparent 
 enough to transmit some, which are fit for vision and 
 examination of the sun, but only slightly colored blue. 
 Probably this helioscope will supersede the others, ex- 
 cept that it requires a distinct object glass and practi- 
 cally a distinct telescope, while the first only requires 
 a prismatic eye piece to be used instead of the common 
 one, which is movable. 
 
 "We have yet to consider the different ways of fixing 
 telescopes for different kinds of observations. 
 
 Transit Circle.^ — This most important telescope is 
 mounted on a cross axis lying east and west, so that 
 the tube can only move in the plane of the meridian. 
 It is now made so as to combine the work for which 
 two telescopes were formerly used, one called a trcmsit 
 instrument for observing transits across the meridian, 
 and the other called a mv/ral circle because it was set 
 against a wall for observing distances from the equator 
 or the pole. The Green wich transit circle tube and its 
 
Altitudes hy Reflection. 335 
 
 cross axis are of cast iron, instead of brass as iisiial. 
 The axis or its pivots until lately were always set on 
 what were called Vs, being bearings of that shape ; for 
 a pivot cannot be made to lie steady in a semi-circular 
 bearing. But at last this rude method of obtaining a 
 steady bearing has been superseded by the simple de- 
 vice of cutting a wide notch or piece out of the bottom 
 of a semi-circular block, so that the pivot still presses 
 only on the sides, but with a much wider bearing than 
 on Vs, which the axis only touches in a line on each 
 side, and often wears away unevenly. 
 
 Moreover the new compound of copper and alumi- 
 nium, which will probably be called al-hronse (as bronze 
 is a softer bell metal), is coming into use for telescope 
 bearings, as it is four times as strong as brass. It was 
 hoped that it would do instead of silver bands or circles 
 for graduation ; but though it spoils with the air less 
 than brass, it is far inferior to silver. It is evidently 
 essential to accurate observation that the bearings 
 should be not only level, but as firm as possible, and 
 they are consequently laid on piers built deep into the 
 ground. The instrument is used to observe the transit 
 of a star across the meridian, by pointing it to the proper 
 height to catch the star, and then looking at it as soon 
 as it comes into the field of view, and observing the 
 time by the clock when the image of the star crosses the 
 middle micrometer wire, or rather all the wires in suc- 
 cession. The difference between the times of two stars 
 crossing the meridian is the difference of their right 
 ascension, which is now generally given in tables of 
 stars in time and not in degrees (p. 292.) 
 
336 Mural Circle. 
 
 Again, distances from tlie zenith are found by mea- 
 suring the angle which the telescope makes with the 
 vertical when a star on the meridian is seen exactly on 
 the wire which is set across the focus horizontally, or 
 at right angles to the transit wire. And that is done 
 in a remarkably neat way, by first looking at the star 
 itself, and then at its reflection in a basin of mercury 
 which forms an artificial horizon : sometimes a plate 
 of glass is laid upon it to stop any tremor of the surface 
 and to keep it clean. By the law of refiection (p. 327) 
 the telescope always makes the same angle with the 
 horizon when pointed at the star and at its reflection, 
 only in one case it is looking upward and in the other 
 downward. Therefore the altitude of the star, or 90° 
 — the zenith distance — -is half the angle moved thi'ough 
 by the telescope between the two observations. And 
 the declination, or distance of the star from the equa- 
 tor, or what is more commonly used, its north polar 
 distance (N.P.D.) is easily got from its zenith distance 
 and the colatitude of the observatory, which is the con- 
 stant distance of its zenith from the pole, as the latitude 
 is the distance of the zenith from the equator. 
 
 For this purpose the transit circle has a large wheel 
 fixed to one end of its cross axis with a graduated rim, 
 and the graduations are read off by several of those 
 microscopes called micrometers {t^. 325) which are fixed 
 at convenient places on the pier near the rim of the 
 circle. In the old mural circles the graduated circle 
 is fixed on the wall, and the telescope carries an index, 
 with micrometers for reading off accurately. The 
 telescope need not be so accurately in the meridian for 
 
AMazimuili. • 837 
 
 observing N.P.D. as for transits; and so setting it 
 against a wall as near N. and S. as possible was enough 
 without all the precautions which a transit instrument 
 requires.- For [the bearings of a transit instrument 
 must be exactly level and of the same size, and their 
 centres exactly east and west, and the pivots which 
 ride in them exactly round and exactly alike, and the 
 cross axis exactly at right angles to the telescope axis, 
 or the line through the centres of all the glasses. These 
 things require constant watching and testing, and math- 
 ematical formulae have to be calculated for the neces- 
 sary corrections. Most of them can be tested by lifting 
 the axis over and reversing the pivots, and looking 
 through the telescope opposite ways ; but for heavy 
 cast iron tubes other methods equally accurate are used 
 at Greenwich. These details however are beyond the 
 scope of this book. 
 
 Altazimuthi — ^A transit circle being only able to move 
 in the meridian of course can only see stai'S-when they 
 come close to the meridian. But if the horizontal cross 
 axis of the telescope is can-ied by a frame which itself 
 turns round on a vertical axis, or on a horizontal bed, 
 it can be directed anywhere ; and it will measure alti- 
 tudes above the horizon by moving the telescope up 
 and down, and azimuths by turning the frame ; and so 
 that is called an Altazitmith. The level plate at the 
 bottom of the frame is graduated, and there is a fixed 
 index with micrometers against it to show the degrees 
 of azimuth through which you turn it. 
 
 The zenith sector is only another form, or rather a 
 part of the same instrument, being a telescope moving 
 15 
 
338 • Equatorial Telescope and Clodlc. 
 
 only over a moderately small graduated arc fixed at the 
 zenith for observing stars near it, and therefore taking 
 less room for its size than a complete altitude instru- 
 ment with its vertical circle of degrees. 
 
 Equatoriali — Again if the frame in which the tele- 
 scope is set turns on an axis not vertical, but parallel 
 to the earth's axis, that also gives an universal motion ; 
 the telescope can be pointed to a star and then clamped 
 or screwed fast to the frame, and if the frame is turned 
 round from east to west at the same rate that the earth 
 turns the opposite way, the star will stay in the tele- 
 scope. It is called an Equatorial, because each point 
 in it then moves parallel to the equator. That is the 
 instrument used for gazing at the planets or moon, and 
 sweeping the sky for new planets or comets, besides 
 other purposes independent of the meridian. There- 
 fore the largest object glasses that can be made perfect 
 are generally used for equatorial. Herschel's great 
 telescope was set in a frame which also turned azim- 
 uthally ; and Lord Eosse's larger one, though it stands 
 between stone piers, is capable of some sideway motion 
 so as to be used equatorially, and can be kept pointed 
 to a star for some time by machinery for the pur- 
 pose. 
 
 In those heavy telescopes the speculum end of the 
 tube stays on the ground. The lighter telescopes 
 which are mounted equatorially are balanced on their 
 middle : and they are often connected with a peculiar 
 kind of clock which turns the frame at the proper rate, 
 and so keeps the telescope pointed to the star without 
 the observer having anything to do except liimself to 
 
Recording Barrel. 339 
 
 follow it. THese clocks do not go by beats of a vibrat- 
 ing pendulum, wliich would be magnified by the tele- 
 scope into leaps of the star ; but they have either a 
 revolving pendulum, or balls like the 'governor ' of a 
 steam-engine, or some equivalent contrivance to make 
 the motion continuous. 
 
 This plan has been adopted lately for recording ob- 
 servations at once.* Another continuous motion clock 
 drives a barrel covered with paper, turning round (say) 
 once in ten minutes, and at the same time advancing 
 endways, by its pivot being made into a screw, so that 
 a pencil fixed against it would draw a spiral line round 
 the barrel. "When the observer sees his transit he 
 touches a pin, which by the common electric telegraph 
 machinery makes a pencil strike the barrel and make 
 a dot ; which marks its own time, as the time at which 
 every point on the barrel comes under the pencil is de- 
 fined. The time of every observation, except mere 
 gazing at the disc of a planet or moon or comet, is an 
 essential part of it, and a clock is a necessaiy compan- 
 ion of almost every telescope ; but clock making is 
 farther removed from astronomy than telescopes, and I 
 have written about that elsewhere. / 
 
 There are many other minute contrivances for secur- 
 ing accuracy of position, observation, and measure- 
 ment, which cannot be described here. You will have 
 some idea of the accuracy required from this : — a good 
 observer can tell when a star crosses the wire of his 
 telescope within the tenth of a second by the clock, 
 which is always placed so that he can both see and 
 
 » Appendix, Note XXVIII. 
 
340 Oonoluding Observations. 
 
 hear it, and count the beats from the minute and sec- 
 ond which he has looked at. 
 
 And now that you have finished this book you must 
 not fancy that you have learned astronomy : you have 
 only been introduced to it, by a short and easy road. 
 Even when you have learned all that is yet known of 
 it, you may well remember the saying of Sir Isaac 
 Newton, who at the end of his long life, after having 
 opened more of the knowledge of the universe to 
 human view than any man before or since, said that he 
 felt like a boy who had been picking up a few pebbles 
 on the sea-shore while the ocean of truth lay before 
 him. He knew, like all true philosophers, lovers of 
 true knowledge, that however far they range into the 
 mysteries of nature, as into the 'star-sown deep of 
 space,' they will find a wider space beyond, which they 
 must wait to penetrate, until the time when ' no good 
 thing shall be withheld ' from those on whom the reve- 
 lation already given has not been thrown away, and 
 they shall no more ' see through a glass darkly.' 
 
Missing Page 
 
APPENDIX 
 
 Note on tub Geeat Pykamid. 
 
 Besides the astronomical theory of the Great Pyramid, noticed 
 nt p. 5i, the Astronomer Royal of Scotland propounds another, 
 which is shortly this. The capacity of the famous porphyry 
 coffer, which holds just 4 English quarters of corn and is certainly 
 no mere stone coffin, expresses the earth's specific gravity 57 x * 
 tenth part of the cube of a double cubit of 25-025 inches, the 20 
 millionth of the earth's axis (p. 15) ; and each side of the 13 acre 
 base of the Pyramid was 365^ such cubits. 
 
 That earliest and largest of all the Pyramids exhibits such proofs 
 of mathematical design, and such astonishing precision of exe- 
 cution throughout, that we need not be surprised at any new geo- 
 metrical relations of its parts being brought to light. But it re- 
 quires an unusual amount of faith to accept this ingenious thepiy 
 of Mr. Fiazzi Smyth's on the present evidence. Nothing more is 
 known of the 'sacred cubit' of the Jews than that it was some- 
 where about 23 inches, as Newton calculated. And if some of 
 Noah's descendants were unconsciously inspired to choose exactly 
 the 20 millionth of the earth's axis for their sacred cubit, no such 
 measure appears in the Pyramid, except that two diagonals of the 
 four comer-stone sockets cut in the rock are 100 and 200 inches. It 
 is not to be found in the coffer, except by virtue of that hypotheti. 
 cal multiplier 57 ; which is not the earth's density according to any 
 of the earth-weighing experiments (p. 32), or any probable mean 
 of them. And if it is, how were the builders of the Pyramid to 
 know it? Mr. Smyth admits only by revelation: a revelation 
 of the earth's density 36 centuries before even the motion of the 
 earth was known ; unless that also was revealed and forgotten. 
 
34r2 Ajtpendia;. 
 
 Lastly, the base of the Pyramid is itself still uncertain by at least 
 two cubits, for reasons given in his book ; and one unknown 
 quantity can hardly be asserted with confidence to be 365JI times 
 another. 
 
 But some things in the Pyramid are known. Newton per- 
 ceived, from the frequent multiples of 2075 inches that occur in 
 the passages and chambers, that that was the ' working cubit' of 
 the builders. He did not know the capacity of the coffer, or he 
 would have found that it just — a double working cubit cubed. 
 And if he had known the angle of the slopes, 51° 51', he would 
 
 TT II 
 
 have seen that it makes the width = - or — of the height, and the 
 
 2. 7 
 
 area of each face almost exactly — the square of the height. I 
 wonder that John Taylor, who first perceived these things, did 
 not see also that the builders having fixed on that slope, and 
 being evidently fond of multiples of 10, were almost sure to take 
 440 working cubits for the base: that being the only multiple of 
 10 and 1 1 at all near that size, and making the height also a mul- 
 tiple of 10 and 7, or 2gq : all which figures stared them in the face 
 by the rule on which they worked. And 44° such cubits, or 761 
 feet, are within a few inches of the same mean of the four latest 
 measures of the base as Mr. Smyth himself adopts. I venture to 
 propound this as a more probable theory of ' Pyramid metrology ' 
 than one which assumes a revelation of the density and diameter 
 of the earth to Cheops or his architect. 
 
 Note I., Page 15.— Although national prejudice is plainly in- 
 dicated by the author's language, his objections to the French 
 metre are well grounded. But the advantages of a uniform 
 standard of measurement are so great, that the French plan bids 
 fair to become general, in spite of the inaccuracy of its basis. 
 
 Note II., Page i 8. — Some recent observations on the temperature 
 of deep wells have led the observers to suppose that the increase 
 of temperature below the surface soon reaches a limit. Theo- 
 retical discussions of the tidal action of the sun and moon have 
 also been published to controvert the hypothesis of a fluid 
 nucleus. 
 
 Note III., Page 36. — The present leslimates of the earth's mean 
 distance from the sun vary from 91,000,000 to 93,500,000 miles. 
 
A^endis. 343 
 
 Prof. Newcomb, of the Washliigton Observatory, has re-examined 
 the observations of the transit of Venus in 1769, and after allow- 
 ing due weight to more recent observations, he concludes that 
 they indicate a distance of 92,383,000 miles, with a probable error 
 of 136,000 miles. M. Liais, from the observations upon Mars in 
 i860, estimated the distance at 93,416,000 miles. Some considera- 
 tions connected with the mean atmospheric inertia seem to in- 
 dicate a distance of about 92,300,000 miles. 
 
 Note IV., Page 65. — The probable parallax fixed upon by Pro- 
 fessor Newcomb is 8"'85. 
 
 Note V., Page 67. — Professor Wolff considers that Venus, the 
 Earth, and Jupiter, all contribute sensibly by their attraction 
 toward the formation of sun spots. I do not know on what 
 ground the author asserts that the theory of a connection with 
 magnetic disturbances seems to be abandoned. 
 
 Note VI., Page 67. — There is no sensible refraction in the 
 atmosphere above the height of about 50 miles, but some me- 
 teoric and meteorologic observations indicate an aerial altitude of 
 more than 200 miles. 
 
 Note VII., Page 80. — The experiments that indicate a greater 
 velocity for electricity than for light, were conducted with wires, 
 under circumstances that may possibly have been modified by 
 induction. The eoMct coincidence of observed magnetic disturb- 
 ances with disturbances in the solar photosphere, seems to render 
 it probable that the normal velocity of light and electricity may 
 be precisely the same. If a metallic wire could be made trans- 
 parent there is no reason for supposing that it would transmit a 
 wave of electricity more rapidly than a wave of light. 
 
 Note VIII., Page 83. — The nebular hypothesis assumes that the 
 solar system was in a fused and vaporous condition before the 
 " bringing together of its atoms commenced." 
 
 Note IX., Page 84.— If " heat and force are universally con- 
 vertible," conversion is not the same as destruction. Matter and 
 force are undergoing continual changes of form, but we have no 
 reason for believing that either is ever destroyed. 
 
 Note X., Page 90. — La,place estimated that the force of gravity 
 must be transmitted with a velocity at least 6,000,000 times as 
 great as that of light. 
 
344 Appendix. 
 
 Note XI., Page 93. — The supposed eruption of Linne was not 
 confirmed by subsequent careful observations, but other phe- 
 nomena have been noticed upon the lunar surface which may be 
 owing to volcanic action. 
 
 Note XII., Page 8.— The bright border that surrounds the 
 " lumiere cendree," or darker portion of the moon's disc, in the 
 new moon, has been attributed to refraction by the lunar atmos- 
 phere. Some of the photographs of the solar eclipses of August, 
 1868 and 1869, exhibit a degree of illumination of the moon's 
 surface for which refraction would afford a ready explanation. 
 
 Note XIII., Page 93. — The town and building are supposed 
 to be at the distance of 233 miles, and viewed in the same manner 
 as the moon and Venus. 
 
 Note XIV., Page 96. — Sir John Herschel fancied that the lu- 
 nar plains presented alluvial appearances, and Prof Frankland 
 thought that he could perceive traces of glacial action and mo- 
 raines. 
 
 Note XV., Page 106. — The action of the full moon in dispelling 
 clouds is still the subject of much discussion. The experiments 
 of Melloni have established the fact that the moon's rays have a 
 sensible amount of heat, but any lunar influence on the weather is 
 probably more dependent on tidal attraction than on heat radia- 
 tion. My own discussions of the rain-fall for forty years, at Phila- 
 delphia, and at Surrey, England, showed a tendency at each sta- 
 tion to diminished rain-fall both at new and at full moon. 
 
 Note XVI., Page 116. — Confusion is sometimes occasioned by 
 a failure to discriminate between the different relations of motion. 
 The actual orbital motion of the earth at noon is in the same di- 
 rection as the sun's apparent motion, and in an opposite direction 
 to the motion of rotation at that time, or toward the point of 
 compass that we then call west. But at midnight the motions of 
 daily rotation and of annual revolution coincide in direction, each 
 being toward the point of compass that we then call east. A stone 
 let fall at noon moves almost horizontally westward, with a ve- 
 locity of more than 18 miles per second, but a stone falling at mid- 
 night moves eastward with a velocity slightly greater than at noon. 
 
 Note XVII., Page 144.— The author, through an evident in- 
 advertence, stated that " if the earth had had no original rotation. 
 
Appendix. 345 
 
 the moon dragging the tidal wave round it looo miles an hour 
 (p. 121) would by this time have given it some rotation westward." 
 The tidal wave he refers to is due to the earth's rotation ; if the 
 earth were at rest, the equatorial tidal wave would follow the 
 moon in its monthly revolution eastward, at the rate of about 34 
 miles an hour. 
 
 Note XVIII., Page 162. — Some distinguished observers have 
 seen the crescent shape of Yenus with the naked eye. But the 
 crescent, as a symbol of light that regularly waxes and wanes, 
 may have been connected with the representations of Venus, 
 without any actual observations of its literal, as well as figurative, 
 appropriateness. 
 
 Note XIX., Page 190. — If the ring is composed of meteorites, 
 they do not, perhaps, wholly eclipse the sun. 
 
 Note XX., Page 191. — Mi\ Denison credited the determination 
 of the structure of. Saturn's rings to Mr. Maxwell, a claim which, 
 I believe, Mr. Maxwell himself has never made. Prof. Peirce's 
 results were published in the proceedings of the American Scientific 
 Association, and Mr. Maxwell's essay is, on its very face, a prize 
 essay written to confirm those results. 
 
 Note XXI., Page 218. — Prof. Newton proved that the meteors 
 could have but one out of five well marked orbits, and stated that 
 a computation of certain perturbations ought to decide which one 
 of the five was the true one. Prof. Adams made the computa- 
 tions, and obtained one of the neatest demonstrations possible 
 that only the 33}^ years period could answer the conditions of the 
 problem. [See Monthly Notices of the Koyal Astronomical Society 
 (vol. 27, p. 247) for April, 1867.] 
 
 Note XXIL, Page 256. — A body colliding with a larger one 
 might, possibly, describe an angular orbit. But the angle also is 
 one of the conic sections. 
 
 Note XXIII., Page 260. — A comet, after leaving the solar sys- 
 tem in a parabolic or hyperbolic orbit, might pass around one or 
 more stars in a similar orbit, and finally return to our system. 
 Such an occurrence would, however, be extremely i mpossib le. 
 
 Note XXIV., Page 263. — Professor Peirce has shown that this 
 apparent rapid increase of tail may have been owing to an opti- 
 cal illusion. 
 
346 Apipendix. 
 
 Note XXV., Page 266. — The most satisfactory explanation of 
 cometary plienomena has been given by Professor Peirce, in a 
 communication presented to the French Academy, and published 
 in the Comptes Rendus. 
 
 Note XXVI., Page 282.— Struve, and other astronomers, have 
 supposed that a large portion of the light which is emitted from 
 the stars is lost by the interference of the various luminous undu- 
 lations with each other. 
 
 Note XXVII., Page 299.— The author has overlooked the 
 physiological considerations which are involved in the question. 
 It may well be doubted whether a man with two eyes can see 2 
 times as far as a man with one. 
 
 Note XXVIII., Page 339.— The first suggestion of the chrono- 
 graph, for employing the magnetic telegraph to record astro- 
 nomical observations, was probably made by Professor A. Dallas 
 Bache. Professor Sears C. Walker planned an instrument which 
 was constructed by Dr. John Locke of Cincinnati, and attached to 
 a clock in the observatory of the Philadelphia High School. 
 Professor Wm. C. Bond of Harvard University, afterward in- 
 vented the Spring Governor, which has effected a marked im- 
 provement in the accuracy of minute observations. 
 
Il^DEX. 
 
 Aberration of liglit, 211 
 
 • of teleBCopes, 321. 
 
 Absolute mass of earth and sun, 250. 
 Acceleration of comets, 82. 
 
 by obstruction, how, 83, 144. 
 
 — ■ moon^a secular, 141. 
 Accelerating force, 28, 233. 
 Achromatic glasses, 318. 
 Adams, Mr., discovered. Neptune, 181. 
 
 on meteors, 220. 
 
 on moon^s acceleration, 143. 
 
 shape, 97. 
 
 -weight, 135. 
 
 ASrolites, 222. 
 
 .dSlther, luminiferous, 82. 
 
 Air. height of, 67, 315, 343. 
 
 colder upwards, 67. 
 
 how much heated by sun, 77. 
 
 refraction of, 13, 213. 
 
 Airy, Mr., Gravitation, 141, 174 ig6, 
 
 Lectures, 32, 85, 208, 249. 
 
 inectnality of Veuns, 160. 
 
 on discovery of Neptune, 183. 
 
 on distance of sun, 85. 
 
 on disturbances, 141. 
 
 on 'long inequality,' 176. 
 
 on moon's acceleration, 143. 
 
 weight, 249. 
 
 Alcyone centre of universe, 287. 
 
 Algol a double star, 284, 
 
 Altazimuth, 337. 
 
 Altitude, 293, 336. 
 
 Aluminium bronze, 335. 
 
 Angles, how measured, 16, 203, 241. 
 
 Angular velocity, law of, z86, 224. 
 
 how measured, 241. 
 
 Annual equation of moon, 142, 
 
 parallax of stare, 387. 
 
 Annular eclipse, 91, 114. 
 
 Anomalistic year, 41. 
 Anomaly, mean and true, 41. 
 Antipodes, 20. 
 Aphelion, 41 : winter at, 43. 
 Apogee, 100 : attraction at, 152. 
 
 velocity at, 115. , 
 
 Apses, advance of earth's, 42, 153. ' 
 
 moon's, loi, 152. 
 
 Archimedes, 33. 
 
 Arcs and angles, 16, 241, 293. 
 
 Arctic circle, 45. 
 
 Area of circle, 24. 
 
 of ellipse, 44, 256. 
 
 of a lune, 103. 
 
 zone^ 24. 
 
 conservation of, 223, 256. 
 
 Areal velocity of moon, 224. 
 
 of planets, 256. 
 
 Aries, 48, 58, m : motion of, 51. 
 
 measures from, 58, 292. 
 
 Aristarchus, 34. 
 
 Arithmetical progression, 14s. 
 
 Ascension, right, 292. 
 
 Asteroids, 170. 
 
 Astrology, i6z. 
 
 Astronomical telescope, 297, 302, 31a. 
 
 Atmosphere, see air. 
 
 of BUU, 67. 
 
 moon's, 93, 282. 
 
 planets, 169. 
 
 Attraction, law of, 228, 28, 225. 
 differential, 53, 131, 141, 147. 
 
 is insl-antaneous, 90. 
 
 of a sphere, 29. 
 
 a spheroid, 290, 37, ig6. 
 
 Attractive force, absolute, 251. 
 Attwood's machine, 245." 
 Axis major of orbit, 39. 
 earth's does not vary, 44. 
 
348 
 
 Index. 
 
 Asis minor does, 141, 
 
 moon's axia leans, 102. 
 
 earth's 44 : sun's, 66. 
 
 length of earth's, 14. 
 
 Azimuth, 294, 337. 
 
 BAcnE, Pro.f. A. D., 346. 
 Bacon, Francis, 297. 
 
 Friar Roger, 124, 296. 
 
 Baily's beads, 116. 
 
 Bear, great, 290. 
 
 Bell metal, 332. 
 
 Biela's comet, 263. 
 
 Blair's achromatic lenses, 319. 
 
 Bode's law of distances, 194, 185, 193. 
 
 Bond, ^r., 192, 193, 346, 
 
 Bradley's aberration, 211. 
 
 Brougham, Lord, 78. 
 
 Bulk of globe, 24 ; spheroid, 133, 156. 
 
 of planets altered, 87. 
 
 Calendab, reform ot, 122. 
 Candles, test of, 113. 
 Cassegrain's telescope, 329. 
 Cavendish experiment, 30, 254. 
 Celestial globe, 290. 
 
 latitude, 294. 
 
 longitude, 187, 292. 
 
 Centre of gravity of moon, 96, 104. 
 Centreof gravity of earth and moon, 
 
 97> =51- 
 
 of solar system, 228. 
 
 Centrifugal force, 18, 37, 235, 240. 
 
 of moon, 92. 
 
 Centripetal force, 235. 
 Chaldsean astronomy, ix. 
 
 Saros of eclipses, 118. 
 
 Challis, Mr., sees Neptnne, 182. 
 Chronograph on Eecording Barrel, 339, 
 
 346. 
 Chronometers for longitude, 201. 
 Circles are a kind of ellipse, 257, 262. 
 
 a conic section, 258. 
 
 great, 14 : sailing by, 14. 
 
 how divided, 17. 
 
 of curvature, 240, 327. 
 
 perspective of, 40, 258. 
 
 transit, instrument, 334. 
 
 Circular orbits, 236. 
 Circumference and diameter, 14. 
 Clavins reformed the calendar, 122, 127. 
 Clock telescope-driving, 338. 
 time of, 59, 201. 
 
 Coal represents sun's heat, 76. 
 
 Colatitude, ixi, 293. 
 
 Cold increases upwards, 67. 
 
 of the moon, 105. 
 
 of north and south, 49. 
 
 of space, 43. 
 
 Colors, 81. 
 
 in telescopes, 316. 
 
 Comets, 260. 
 
 Encke's, retarded, 82. 
 
 - nature of their tails, 83, 264. 
 
 Cones, oblique, 259. 
 Conic sections, 256. 
 Conical projection maps, 22. 
 pendulum, 240. 
 
 Conjunction, 50, 108. 
 
 inferior, 204, 249, 2S4. 
 
 Conjugate foci, 307. 
 Conservation of areas, 223, 235, 356. 
 Constellations, 48, 291. 
 Continents, land in, 23. 
 Contractive force, 131, 141. 
 Cook, Captain, 206. 
 Copernicus, 33, 163. 
 Correlation of forces, 77. 
 Crescent moon, 106. 
 
 Venus, 162, 202. 
 
 Croll, Mr., 43, 146, 
 Cube of distance, 54, 133. 
 Cubic measure, 24, 61, 
 Cycles of moon, ii3. 
 
 DAT has not shortened, 233. 
 has lengthened, 143. 
 
 - sidereal and solar, 35, 57. 
 
 - of week, how named, 162. 
 - how to find, 127. 
 
 Decimals and vulgar fraction^, 83. 
 Decimation, 201, 242. 
 Delaunay on retardation of earth, 143. 
 Density or specific gravity, 63. 250. 
 
 of earth, 25 to 32.- 
 
 moon, 92, sun, 63. 
 
Index. 
 
 349 
 
 Density planets, see them. 
 Diameters of earth, 14, 24, 169 n. 
 
 of sun and planets altered, 86. 
 
 apparent, how measured, 87, 
 
 203. 
 Differential force, 53, 131,141, 147. 
 Dimensions of solar system, 85. 
 Dip of horizon, 294, 
 Disc of a globe, 107, 165. 
 
 spurious, of stars, 2S2. 
 
 Dispersion of light, 81, 316. 
 Distance means that of centres, 65. 
 
 represents original force, 268. 
 
 and time, law of, 225, 228. 
 
 Bode's law of, 185, 193, 194. 
 
 of solar system altered, 86. 
 
 proportion of sun and moon's, 
 
 92. 
 
 see planets, stars, etc. 
 
 Disturbances -of moon, 140. 
 
 Jupiter's moons, 195. 
 
 of planets, 180. 
 
 comets, 82, 264. 
 
 DoUond, 318, 
 Double stars, 283, 284. 
 
 Eabth, 9. 
 
 density of, 32. 
 
 diameters of, 14, 24, 169. 
 
 Earth, retardation by tides, 143, 
 
 revolution, time of, so. 
 
 — — has not altered, 57. 
 
 - round, how proved, 12. 
 
 - not qnite round, 15. 
 
 - rotation proved, 33. 
 
 - rotation is from west to eSat, 36, 
 
 57. =!i6- 
 vis viva of, 234. 
 
 - shape, how found, 19. 
 
 - size and surface measured, 19, 
 
 24. 
 
 - size seen from moon, 107. 
 
 - spheroidicity, 15, 18, i6g. 
 effects of, 37, 53, 197. 
 
 - time of rotation constant, 58. 
 - not quite, 143. 
 
 - displaced by moon, 97. 
 . distances from sun, 62. 
 - how found, 209. 
 
 - effect of sailing round, 36. 
 
 - effect of striking the sun, 83. 
 
 - eUipticlty of, 15. 
 
 its effect on moon, 197. 
 
 - fluid once, 18 ; why, 83. 
 
 - how deep solid, 56. 
 
 - inclination of axis, 44, 56. 
 
 - measuring of, 13, x8. 
 
 - moves eastward, 36, 188. 
 
 - nearest sun in winter, 41. 
 
 - old ideas about, 9. 
 -orbit elliptical, 38. 
 
 and revolves, 40, 153. 
 
 eccentricity of, 41. 
 
 size of, 36. " 
 
 - ' reduced to rest,' 98, 147, 247. 
 
 — ■ — velocity of, 36, 146. 
 
 weighing of, 24. 
 
 weight in tons, 247. 
 
 Earthshine on moon, 107. 
 Easter moon, 120, 125. 
 
 table for finding, 130. 
 
 Eccentricity of ellipse, 39 . 
 
 of earth's orbit, 40. 
 
 decreases, 42, 141. 
 
 moon's, 92. 
 
 its variation, 153. 
 
 planets, see them, 
 stability of, 255. 
 Eclipses, III. 
 
 Chaldsean cycle of, ii3. 
 how many a year, 117. 
 of moon, 12, X15. 
 sun, 91, 114. 
 dates fixed by, 146, 
 Jupiter's moons, 195, 198. 
 
 longitude found by, 199. 
 
 Ecliptic, 44, 47. 
 
 axis of, 47, 52, 66. 
 
 marked on globes, 294. 
 
 Obliquity of, 56 ; decreases, 56. 
 
 Egypt, astronomy in, 52, i6z, 341. 
 Electricity, speed of, 80, 343. 
 
 observations recorded by, 339, 
 
 346. 
 Ellipse, IS, 39. 
 a conic section, 257. 
 
350 
 
 Index. 
 
 Ellipse, force in centre of, 239. 
 
 Instantaneous, 156. 
 
 orbit of all planets, 40, =23. 
 
 velocity in, 262. 
 
 perspective of circle, 40, 212, 25S. 
 
 revolving, 41, 100, 239. 
 
 EUipticity. 16, 39 (see spheroidicity). 
 Elliptic motion explained, 237. 
 
 discovered by Kepler, 164. 
 
 Elongation, 167. 
 
 Emanations, law of, 227. 
 
 Empirical rules, z86. 
 
 Bncke's comet, 82. 
 
 English inch, oldest measure, is- 
 
 Epact, 109, 129. 
 
 Equation, annual, of moon, 142. 
 
 of centre, 41, 103, 142. 
 
 of time, 58, 199. 
 
 Equations, meaning of, 243. 
 Equator, size of, 13, 24, 169. 
 inclination of, alters, 56. 
 
 is hour circle of globe, 59- 
 
 loss of gravity at, 37, 197, 242. 
 
 moon^s. 102 ; sun's, 66. 
 
 see Jupiter, Saturn, Uranus. 
 
 Equatorial gravity, 37, 197, 242. 
 
 telescope, 338. 
 
 Equidistant projection maps, 21. 
 
 Equilibrium of solar system, 38, 140. 
 
 of Saturn's ring, igo. 
 
 of mass of stars, 288. 
 
 Equinoctial points, 48 ; recede, 41, 100. 
 
 year, 51, 58, 123. 
 
 Equinoxes, 46. 
 
 precession of, 50, 157. 
 
 Bvection of moon, 155. 
 
 Exteiior planets, 167, 187. 
 
 Eye-piece of telescopes, 297, 322. 
 
 erecting, 325. 
 
 Huyghens's negative, 323. 
 
 Kamsden's positive, 324. 
 
 FiNDEK telescope, 314. 
 
 Flint glass, 318. 
 
 Focus of ellipse, 39, 259. 
 
 sun is in, 223, 260. 
 
 of parab(da and hyperbola, 259. 
 
 of telescope, 307. 
 
 Focal length, 307. 
 
 Force, absolute, of earth, 251. 
 
 accelerating, 28, 233. 
 
 centrifugal, 18, 37, 233, 240, 246, 
 249. 
 
 centripetal, 235, 246. 
 
 composition of, 232. 
 
 differential, 53, 131, 141, i47. '5i- 
 
 • moving, 233. 
 
 • how measured, 146, 233, 245. 
 ■ equivalent of heat, 76. 
 
 - of motion, or vis viva, 234. 
 
 - radial and tangential, 148, 238. 
 
 - resolution of, 237. 
 
 - varying as distance, 240. 
 -as inverse square, 27, 65, 147, 
 
 246. 
 
 cube, 54, 133, 147. =39- 
 
 Foucault's experiments, 32, 334. 
 Fractions, vulgar, advantage of, 88. 
 Frankland, Prof, 344. 
 Fraunhofer's lines, 68. 
 Freezing of water, 73, 75. 
 Friction of air melts meteors, 221. 
 retards pendulums, 82. 
 
 of tides on earth, 143. 
 
 of luminiferous tether, 82. 
 
 Gauleo, 35, 297. 
 
 discovered Jupiter's moons, 192. 
 
 his telescope, 312. 
 
 Galle, Dr., finds Neptune, 183. 
 Geographical mile, 19. 
 George HI., King, 177, 206. 
 Gibbous moon, 106 : Mars, 202. 
 Glacial epoch, 43. 
 
 action, on moon, 344. 
 
 Globe, apparent disc of, 107, 165. 
 
 celestial, 290. 
 
 terrestrial, 11. 
 
 Gnomonic projection maps, 21. 
 Granite, 71. 
 
 Gravitation, law of, 27, 225. 
 see Airy. 
 
 Gravity, how measured, 245. 
 
 on sun and planets, see them. 
 
 absolute force of, 253. 
 
 • Increases down mines, 29. 
 
Index. 
 
 351 
 
 Gravity lees at eqnator, 37, 243, 
 
 velocity of, 343. 
 
 Greenwlcli time, 57, 59, 200. 
 
 meridian of, 17, 
 
 time, longltade by, 200. 
 
 Gregory, James, 206, 329. 
 
 Pope, XIII.'b calendar, 121. 
 
 Grove, Mr., 77, 320. 
 Groves in the Bible, 163. 
 Gunpowder, expansion of, 226. 
 Gyroscope, 33. 
 
 Hallst found moon's acceleration, 
 142. 
 
 Mb comet, 263. 
 
 Hansen, Professor, 85, 143, 159. 
 Harrison's chronometers, 201. 
 Harvest moon, 109. 
 Heat, annual quantity of, 44. 
 
 connection with light, 81. 
 
 - — — increases down mines, 18. 
 
 decreases upwards, 67. 
 
 invisible rays of, 81. 
 
 is force, 76. 
 
 ■ latent and specific, 74. 
 
 sun's, 70 to 77. 
 
 in moon, 105. 
 
 none in moonlight, 105. 
 
 of north and south hemispheres, 
 
 42. 
 
 of sun, 68, 70 : in planets, 73. 
 
 received by earth, 44, 73. 
 
 is inversely as distance^, 43, 165. 
 
 Heliacal rising, 294. 
 
 Heliometer, 326. 
 
 Helioscope, 332. 
 
 Hemisphere In maps too small, 20. 
 
 ■ division of, 23. 
 
 north, former cold of, 43. 
 
 Herschel, Mr. A.., on meteors, 221. 
 Herschel, Sir J.'s, Astronomy, 23, 64, 
 69, 76, 96, 14X, 274, 287. 
 
 Lectures, 79, 90. 
 
 on change of seasons, 42. 
 
 metre and yard, 15. 
 
 moon's alluvial plains, 344. 
 
 moon's mass, 135. 
 
 sun's heat, 73. 
 
 Herschel, Sir J.'s, reform of calendar, 
 124. 
 
 star magnitudes, 279. 
 
 Herschel, Sir W., found Uranus, 177. 
 
 on Saturn's ring, 189. 
 
 separated light and heat, 81. 
 
 his telescope, 330. 
 
 Herodotus and Homer, 162. 
 Hipparchus, 38, 51, 164. 
 Hopkins, W., on precession, 56. 
 Horizon, astronomical, 293. 
 
 mercurial reflecting, 336. 
 
 visible, 12 : dip of, 294. 
 
 Horrocks, Jeremiah, 160. 
 Huggins, see spectrum analysis. 
 Hnyghens, 78, 192, 29s, 323. 
 Hyperbola, 259, 260, 262. 
 
 Inclination defined, 66, 293. 
 
 of equator to ecliptic, 47. 
 
 variation of, 56. 
 
 - of moon's nodes, loi, 159. 
 
 - of moon's equator, 102, 
 
 - sun's equator, 66, 
 
 - planets, see them. 
 - stability of, 255. 
 
 Inequalities of moon, 140, 160. 
 
 of Earth and Venus, 167. 
 
 Jupiter and Saturn, 174, 
 
 Jupiter's satellites, 196. 
 
 moon's parallactic, 150. 
 
 Inertia, law of, 28, 225. 
 
 moment of, 234. 
 
 Inferior conjunction, 204, 284. 
 Interior planets, 187. 
 Instantaneous ellipse, 156. 
 attraction is, go. 
 
 Jaues, Sir H., weighs the earth, 27. 
 Jansen, an inventor of telescopes, 297. 
 Julian year and era, 121. 
 Jupiter, 171. 
 
 long inequality, 174. 
 
 spheroidicity, effect of, 196. 
 
 satellites, 194. 
 
 ■ their distarbances, 19s. 
 
 use made of theln, 198, 199. 
 
352 
 
 Index. 
 
 Espleb's laws, 223, 22B, 244, 249. 
 
 telescope, 297. 
 
 guesBed at gravitation, 227. 
 
 Knot, or nautical mile, 19. 
 
 LAI7D and water on glolie, ^4. 
 Laplace, 142, 268, 343. 
 Lassell, Mr., 184, 193. 
 Latitude, terrestrial, 14, x6. 
 
 ■ ■ how found, 201. 
 
 celestial, 294. 
 
 Laws of motion, Newton's, 231. 
 
 Kepler's, 223, 229, 244, 249. 
 
 of gravitation, 27, 223. 
 
 time and distance, 223, 229. 
 
 proved, 249. 
 
 Bode's, 185, 193, 194. 
 
 of reflection, 327. 
 
 of refraction, 302. 
 
 - of stability, 255. 
 
 Lead, speciflc heat of, 74. 
 
 twice as heavy as earth, 25, 31. 
 
 Leap year, 123. 
 Lens, 306. 
 
 compound, 317, 318: crossed, 
 
 321. 
 Lescarbaulf 3 planet, 184. 
 Level, 12, 294. 
 
 Le Verrier, second discoverer of Nep- 
 tune, 182. 
 
 on moon's acceleration, 143. 
 
 Liais, M,, 343. 
 Libra, 48, no, 291. 
 Libration of moon, 102. 
 Light, nature of, 38, 77. 
 
 is invisible, 78. 
 
 polarization of, 266. 
 
 undulatoiy theory, 78. 
 
 velocity, 80, 85, 198. 
 
 of moon, 105, and Venus, 166. 
 
 of stars, 279. 
 
 time it takes, i8o. 
 
 see refraction, and telescopes. 
 
 LinnI, 93, 343. 
 Local time, 59. 
 
 how found, 199. 
 
 Locke, Dr. J., 346. 
 
 Loudon middle of the world, 23. 
 
 London, tides at, 139. 
 Longitude, 16. 
 
 modes of finding, 199. 
 
 celestial, 187, 294. 
 
 heliocentric, 293. 
 
 measured westward, 187, 294. 
 
 " Lumiere cendree," 93, 344. 
 Lunar distances, 200. 
 
 -rainfall, 344. 
 
 volcanoes, 93, 343. 
 
 year, 109. 
 
 Lunation, length of, 108. 
 
 shorter than formerly, 145. 
 
 Lunc, area of, 103. 
 
 McCaul, Dr., 37, 
 
 Macclesfleld, Lord, on calendar, 121. 
 
 Magellanic clouds, 271. 
 
 Magnifying power, 300, 313, 331. 
 
 Magnitudes of stars, 278. 
 
 Maps, different kinds of, 20. 
 
 Mars, 167 ; spheroidicity, 169. 
 
 parallax of, 209. 
 
 phases of, 202. 
 
 Mass, what, 26, 28, 250. 
 
 absolute, of earth, and sun, and 
 
 moon, 250. 
 -.— — ■ of earth, 26 to 32. 
 
 of moon, 135. 
 
 how found, 55, 135, 248, 250. 
 
 of sun and planets, altered, 87. 
 
 — —see planets, each. 
 
 of sun, how found, 243, 247. 
 
 of Jupiter, 253. 
 
 Maxwell, Mr., on Saturn's rings, 191. 
 Mayer on earth's retardation, 143. 
 Mean diameter of earth, 24, 169, 246. 
 —^ distance, 36, 39. 
 
 not average distance, 141. 
 
 see planets, meteors, and moon. 
 
 time, 58. 
 
 Measuring the earth, 13, 19. 
 
 sun's distance, 202. 
 
 moon's, 2o8. 
 
 planets', 85, 203. 
 
 Melloni, Sig., 344. 
 Meniscus lens, 307, 322. 
 Mercator's projection, 22, 
 
Index. 
 
 353 
 
 Mercurial horizon, 336. 
 Mercury (quicksilver), 56, 
 
 planet, 164. 
 
 transits of, 206. 
 
 velocity of, 243. 
 
 Meridians, 14, x6, 293. 
 
 how measured, 19. 
 
 Metale in sun and stars, 68, 283. 
 
 in meteoric stones, 222. 
 
 Meteoric theory of heat, 70. 
 
 Meteors, 218. 
 
 Metonic cycle, 119. 
 
 Metre, French, a had measure, 15. 
 
 Micrometer, 323. 
 
 Microscope, 301. 
 
 Midsummers, two, in tropics, 48. 
 
 Mile, nautical, etc., 19. 
 
 Milky "Way, 271, 287. 
 
 Miller, sm spectram analysis. 
 
 Million, hillion, trillion, etc., 63. 
 
 Mine experiments, 29. 
 
 Mirror, 77, 328 ; sextant, 200. 
 
 of telescopes, 327, 332. 
 
 Momentum, 233. 
 
 of inertia, 234. 
 
 Moon, 91. 
 
 acceleration of, 141 to 146. 
 
 advance of apsea, 152. 
 
 annual equation, 142. 
 
 . apparent and real size, gi, 
 
 centre of gravity of, 96, 104. 
 
 - earth and moon, 97, 249. 
 
 - cycles of, 118. 
 
 - density, 92. 
 
 - differential force of, 54, 133. 
 
 of sun on, 147, 149. 
 
 of earth on, 135. 
 
 - diameter, 92 ; apparent, 208. 
 
 - distances, 392. 
 tow found, 208. 
 
 - disturhances, 140 to 160, 
 
 - eclipses, m to iiS. 
 
 - evcction, 155. 
 
 - eccentricity, 92. 
 variation of, 153, 
 
 197. 
 
 - gravity on, 92. 
 
 - greatest elevation, m, 198. 
 -harvest, no. 
 
 Moon, heat and cold of, 105. 
 
 inclination of equator, loi. 
 
 of orbit, 102. 
 
 - influence on rain, 344. 
 
 - instantaneous ellipse, 136. 
 
 - keeps same face to us, 93. 
 
 - lihration, 102. 
 -light, 105, III. 
 
 moat in winter, no. 
 
 - longitude found by, 200. 
 
 - mass of, 135, 251. 
 
 - mass found, 55, 135, 249, 251. 
 
 - new and full moon, 106, 
 
 - never totally eclipsed, 116. 
 
 - no air or water, 93, 282, 344. 
 
 - nodes recede, loz, 113, 158. 
 
 - nutation caused by, 54. 
 
 - orbit, shape of, 99. 
 a revolving ellipse. 
 
 152. 
 
 - parallax, 208. 
 
 - parallactic inequality, 85, 150. 
 
 - perigee and apogee, 100. 
 
 - periods, 107, 148, 247. 
 
 - phases, 106. 
 
 - precession caused by, 53. 
 
 - recession of nodes, loi, 158. 
 
 - refraction, its effect on, 217. 
 
 - rotation, 93. 
 
 - rises later every day, log. 
 
 - stereoscopic views of, 104. 
 
 - supposed shape of, 96, 104, 
 
 - shine cold, 105. 
 
 surface, 92. 
 
 telescopic view of, 93. 
 
 tides caused by, 131 to 138. 
 
 tidal force of earch on, 134. 
 
 * variation,' 150. 
 
 velocity, 115. 
 
 Venus disturbs, 159. 
 
 weighed, 13s, 2471 24S, 251. 
 
 Motion, laws of, 231. 
 
 force of, or vis viva, 234. 
 
 Motions, celestial, are eastward, 52, 
 
 116, 187 ; westward, 220, 268. 
 Moving force, 233. 
 Mural circle, 334. 
 
364 
 
 Index. 
 
 Nadib, 294. 
 
 Nasmyth, Ur,, 6g. 
 
 Nebular hypotliesiB, s66. 
 
 Nebulse, 270. 
 
 Neptune, discovery of, 179. 
 
 does not follow Bode's law, 185. 
 
 Newcomb, Prof. S., 343. 
 Newton, Prof. H. A., 218, 34s. 
 Newton, Sir Isaac, bom and died, 17. 
 
 gravitation, law of, 27, 325. 
 
 calculated earth's shape, 18. 
 
 moon's disturbances, 141. 
 
 guessed earth's density, 25. 
 
 knew six planets only, 164. 
 
 mistake about achromatism, 318. 
 
 lawa of motion, 231. 
 
 published Principia, 263. 
 
 telescope, 330. 
 
 theory of light wrong, 78. 
 
 Nice, Council of, 120. 
 
 Nodes, recession of, loi, 113, 158. 
 
 synodical period with sun, 119. 
 
 of meteors recede, 220, 
 
 Nutation, 54 ; of Sirius, 283. 
 
 Objisct glass, 297. 
 
 largest yet made, 326. 
 
 achromatic, 318. 
 
 Oblate spheroid, 15, 169, 171. 
 
 mass of, 24, 156. 
 
 attraction of, 37, ig6, 290. 
 Oblique cones, 259, 
 Obliquity of ecliptic, 56. 
 Octants, 149. 
 Opposition, 187, 209. 
 Orrery, 295. 
 
 Orthographic maps, 20. 
 Oscillation, centre of, 245. 
 
 Parabola, 257. 
 
 comots move in, 260.. 
 
 Parabolic mirror, 331. 
 Parallactic Inequality, 8s, 150. 
 Parallax generally, 210. 
 
 sun's, 64, 207, 343. 
 
 Mars' 8, 209. 
 
 moon's, S08. 
 
 Parallels of latitude, 14. 
 
 Peirce, Prof. B., 191, 34s, 346. 
 Pendulum, time of, 245. 
 ■- earth weighed by, 30. 
 
 - force on, 240. 
 
 experiment for rotation, 32. 
 
 fluid bob, 56. 
 
 gravity found by, 246. 
 
 gravity makes it swing, 29. 
 
 - stopped by friction, 82. 
 
 - loses at equator, 37. 
 
 - revolving, 100, 240. 
 
 - time depends on gravity, 30, 246. 
 
 Perigee, 100; attraction at, X52, 
 
 revolves, 152. 
 
 velocity at, 115. 
 
 Perihelion, 41 : revolves, 41, 153. 
 
 velocity in different orbits, 262. 
 
 Periods of moon, 107, 148,, 247. 
 
 of planets, see them and Time. 
 
 in ellipse same as in circle, 243. 
 
 Perspective, 20, 40, 258, 259. 
 Phases, xo6, 202. 
 Philadelphia rain-fall, 344. 
 Photosphere of sun, 65. 
 Planetary motion, laws of, 223. 
 Planets, 161 to 207, see each. 
 
 disturb each other, 161, 175, 180. 
 
 masses found thereby, 
 254- 
 the earth's apses, 153. 
 
 - heathen worship of, 162. 
 -laws of their motion, 223. 
 of distance, Bode's, 185. 
 
 - move the sun, 228. 
 the ecliptic, 56. 
 
 - only reflect light, 166, 212, 284. 
 
 - relative periods of, 186. 
 
 - retrogradation, 187. 
 
 - total weight of, 228. 
 
 Plumb line, 17, 19, 26, 
 ' Plurality of worlds,' 273, 284. 
 Polar distance, 201, 293. 
 Polarization of light, 266. 
 Poles of earth or equator, 14, 45. 
 
 revolve in precession, 52. 
 
 of ecliptic, 52. 
 
 moon's, how seen, 103. 
 
 sun's, inclined, 66. 
 
Index. 
 
 355 
 
 Pole star, 52, 290. 
 
 Pratt, Archdeacon, 169, 252. 
 
 Preceasion of equinoxes, 50, 157. 
 
 proves earth solid, 56. 
 
 Prime vertical, 294. 
 
 Printing and engraving, 175 n. 
 
 Fiism, 81, 303. 
 
 reilecting, 305. 
 
 Pritchard, Kev. C. , 322. 
 Proctor, Mr., u, 23, 175, 190, 
 Projections of sphere, 21. 
 
 of planets, 230, 235. 
 
 Prolate spheroid, iSi 133. 
 
 formed by tides, 133. 
 
 Proportions of land and water, 24, 
 
 of planets'" distances, 85, 203. 
 
 Pyramid, great, 52, 341. 
 
 QuADRATUKB, or half moon, 137, 146, 
 
 *i5o. 
 Quicksands, 139. 
 
 Kaut, tidal, 344. 
 Ramsden^s eye-piece, 324. 
 BecesBion of nodes, zox, 220. 
 
 cause of, 156. 
 
 Recording Barrel or Chronograph, 339, 
 
 346. 
 Reflection, law of, 327. 
 Reflecting telescopes, 326, 
 Refraction, law of, 302, 
 
 of the air, 13, 213, 320. 
 
 raises sun and moon, 215. 
 
 Refracting telescopes, 302 to 326. 
 Resisting medium, 82. 
 Petardation of earth, 144. 
 Retrogradation of planets, 186. 
 Revolving ellipse, 41, 100, 239. 
 
 why its apses advance, 
 
 240. 
 
 pendulum, 100, 239. 
 
 Right ascension, 292. 
 
 Ring, see Saturn, Meteors. 
 
 Rfimer, 199, 295. 
 
 Rosye's, Lord, telescope, 270, 299, 331. 
 
 Rotation of earth, 32, ^34. 
 
 ' — retarded, 144. 
 
 of moon, 93, 
 
 Rotation of nehulse, 273, 275. 
 
 of mass of stars, 290. 
 
 Roundness of earth, ix : sun, 63. 
 
 Sailinq round the earth, 14, 36. 
 SaroB of eclipses, xx8. 
 Satellites of Saturn and Uranus, 193, 
 194. 
 
 Jupiter's, 194. 
 
 Neptune's, 184. 
 
 all keep same face to planet, 93. 
 
 Saturn, 173. 
 
 * long inequality,' 174. 
 
 moons, 193. 
 
 lings, 188, 295. 
 
 effect of spheroidicity, ig6. 
 
 Schehallien experiment, 25, 254. 
 
 Seasons, 44. 
 
 Septuagint, 9. 
 
 Sextant, 200. 
 
 Shooting stars or meteors, 218. 
 
 Sidereal time, 57. 
 
 day, 58. 
 
 year, 50. 
 
 sun's mass depends on, 
 
 87. 
 Signs of zodiac, 48, zgi, 
 Sirius, 278, 283, 285. 
 Sky, why it is blue, 217. 
 Smyth, Mr. Piazzi, 53, 341. 
 Solar time, 57. 
 
 system, 61, 
 
 its motion in space, 287. 
 
 new dimensions of, 64, 83. 
 
 possible end and origin, 84, 266. 
 
 Solstices, 46, 139. 
 
 Solstitial colure, 295. 
 
 Sound, difference from light, 80, 
 
 effect of wind on, 279. 
 
 velocity of, 80. 
 
 Space, cold of, 43. 
 
 not empty, 82. 
 
 penetrating power, 280, 298. 
 
 Specific gravity, 25, 64 : heat, 74, 
 
 of planets, see ' density.' 
 
 Spectacles, 310. 
 
 Spectrum, 81, z66. 
 
 analysis, 63, 266, 273, 283. 
 
356 
 
 Index. 
 
 Speculum of telescopes, 331. 
 Spiere, attraction of, 30. 
 
 of Btars, motion of, 287. ■ 
 
 Spherical ghell, attraction of, 30. 
 
 aberration, 321, 329. 
 
 Spheroid, mass of, 133, 156, 
 
 attraction of, 37, 196, 290. 
 
 oblate, IS, 24 : prolate, 15, 133. 
 
 Spheroidicity, effects of, 37, 196. 
 
 of earth, 17, 197. 
 
 Mars and Jupiter, 169, 172. 
 
 Spiral orhits possible, 239. 
 
 Spots on snn, 65, 343. 
 
 Spring Governor, 346. 
 
 Stability of solar system, 38, 84, 140. 
 
 three laws of, 255. 
 
 Stars, 277. 
 
 distance of nearest, 277. 
 
 double, 284, 317. 
 
 magnitudes, 278. 
 
 mass of severalj 286. 
 
 motion of, 286. 
 
 occultation by moon, 282. 
 
 ' parallax of, 277. 
 
 ^ spectrum analysis of, 283. 
 
 time of revolution, 285, 288. 
 
 used as fixed points, 208. 
 
 Stereographic maps, 20, 259. 
 Stereoscopes of moon, 104. 
 Struve, 346. 
 
 Style, old and new, 120. 
 Summer and winter, 44. 
 
 hotter in south hemisphere, 42. 
 
 longer than winter in north, 44. 
 
 Summers, two in torrid zone, 48. 
 Sun, 61. 
 
 ~ apparent diameter, 20S. 
 
 to each planet, see them. 
 
 atmosphere, 67. 
 
 diameter, size, etc., 61. 
 
 distance, new and old, 6s, 8s, 
 
 343- 
 
 disturbs moon, 141 to is8. 
 
 eclipses of, 112. 
 
 gravity on surface, 6s. 
 
 - heat and light of, 68, 70, 73. 
 
 - mass in earth's, 62 ; old, 89. 
 how found, S43, 248. 
 
 Son, absolute mass, 25a. 
 
 motion In space, 287, 
 
 moved by the planets, 228. 
 
 parallax, 64, 208. 
 
 how found, 207. 
 
 photosphere, 66. 
 
 precession caused by, S3. 
 
 rotation and spots, 6s. 
 
 tides caused by, 131. 
 
 Sun, weighing of, 63, 243, 247. 
 
 willow-leaves on, 69. 
 
 Sunday letter, rule for, 127. 
 Superior planets, 167, 187. 
 Surface of globe and zone, 24. 
 
 of a lune, 103. 
 
 of a spheroid, 24. 
 
 Surrey rain-fall, 344. 
 Synodical periods, 108. 
 Syzygy, 134. 146, 148. 
 
 Tahgentiai, force, 144, 148, 224, 238. 
 Telescopes, 296. 
 
 achromatic, 318. 
 
 astronomical, 302, 311. 
 
 Invented by Kepler, 297. 
 
 Cassegrain's, 329. 
 
 Galileo's, 297, 312, 314. 
 
 Gregory's, 329, 331. 
 
 Herschel's, 177, 326, 331. 
 
 Newton's, 330. 
 
 Lord KoBse's, 270, 299, 331. 
 
 field of view, 314. 
 
 invention of, 296. 
 
 loss of light in, 298. 
 
 of brightness, 31s. 
 
 magnification by, 299, 313, 331. 
 
 penetrate space, 280, 298. 
 
 refracting (see astronomical). 
 
 reflecting, 326. 
 
 speculums, 331. 
 
 modes of mounting, 33s. 
 
 - see ecLuatorial, transit, etc. 
 
 Telescopic size of moon, 93, 
 Thales, n. 
 Tides, 131. 
 
 moon weighed by, 135, 
 
 retard the earth's rotation, 143. 
 
 tidal force of earth on moon, 134 
 
Index. 
 
 357 
 
 Time and distance, 225, 229, 249. 
 
 Greenwich mean, 60. 
 
 Bidereal and solar, 57. 
 
 of pendulums, 245. 
 
 of revolution of stars, 288. 
 
 of rotation of nebulae, 275. 
 
 Torrid zone, heat of, 48. 
 Trade winds, 49. 
 Transit circle, 334. 
 
 of Mercury, 206. 
 
 of Venus, 86, 202. 
 
 of Jupiter's moons, 198. 
 
 Tropical year, 51, 123. 
 Tropics, 47. 
 
 two summers in, 48. 
 
 Twilight, 217. 
 
 Tyndall, Dr., on heat, 43, 82. 
 
 Uhbba and pennmhra, 113 to 115. 
 XJndulatory theory, 78. 
 TJranus, 178 : moons, 194. 
 distnrhed by Neptune, 180. 
 
 ' Vabiation ' of the moon, 150. 
 Velocity measures force, 233. 
 
 angular and linear, 186, 241. 
 
 of earth, 36, 246. 
 
 in different orbits, 262. 
 
 light, sound, and electricity, 80. 
 
 planets, and moon, see them. 
 
 comets, etc., at perihelion, 262. 
 
 stars, 288. 
 
 Venus, 162, 166. 
 
 disturbs the moon, 139. 
 
 inequality of, 167. 
 
 phases, 162, 202, 345. 
 
 transits, 86, 202. 
 
 Vis viva, 234. 
 
 Volcanoes in moon, 92, 344. 
 
 Vulcan, 184. 
 
 Vulgar lections, value of, 88. 
 
 Vulgate version, 9. 
 
 Waikbh, Prof., S. C, 346. 
 Water, proportions to land, 24. 
 
 freezes slowly, 74. 
 
 latent and specific heat of, 74. 
 
 weight of, 64. 
 
 Waves, 78, 137. 
 Week, days of, 162. 
 
 finding in past times, 127. 
 
 Weighing of the earth, 24, 
 
 of sun, 63, 243, 247. 
 
 moon, 133, 247, 248, 251. 
 
 Weight, what, 28, 230. 
 
 loss of, at eiiuator, 37, 24a. 
 
 of earth, 247. 
 
 Bun, 63, 247. 
 
 moon, 135, 231. 
 
 of all the planets, 338. 
 
 of each, see them. 
 
 Winter cold formerly, 42. 
 
 moonlight in, no. 
 
 Wolff, Prof., 343. 
 WoUaston, Dr., 68, 105. 
 World, the round, 9. 
 Wray's object glasses, 320. 
 
 Tear, sidereal, 50. 
 
 ' sun's mass depends on. 
 
 87. 
 and moon's disturbances, 
 
 147. 
 
 equinoctial or tropical, 51, 133. 
 
 apparently shortens, 57. 
 
 - not really, 71. 
 
 Toung, Dr., his discoveries, 78. 
 
 Zenith, m, 2x4, 293. 
 
 distance, 201, 293, 336. 
 
 sector, 337. 
 
 Zodiac, signs of, 48, 291. 
 Zodiacal light, 71. 
 Zones, the, 47. 
 area of, 24.