CORNELL;. UNIVERSITY LIBRARY V
BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 189I BY
HENRY WILLIAMS SAGE
Cornell University Library
QB 43.M927I6 1916
An introduction to astronomy,
3 1924 012 499 756
AN INTRODUCTION TO ASTEONOMY
THE MACMILLAN COMPANY
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Cornell University Library
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AN INTRODUCTION
TO
ASTRONOMY
BY
FOREST RAY MOULTON, Ph.D.
PKOFESSOR OP ASTRONOMY IN THE UNIVERSITY OF CHICAGO
RESEARCH ASSOCIATE OF THE CAKNBGIE INSTITUTION
OF WASHINGTON
NEW AND REVISED EDITION
Weto goris
THE MACMILLAN COMPANY
1916
All rights reserved
OOPYEIGHT, 1906 AND 1916,
By TBE MACMILLAN COMPANY.
Set up and electrotyped. Published April, 1906. Reprinted November, 1907; July, igo8; April, 1910; April, 1911; September, 1912; September, 1913; Optober, 1914.
New and revised edition November, igi6.
J. S. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A.
PREFACE
The necessity for a new edition of " An Introduction to Astronomy" has furnished an opportunity for entirely re- writing it. As in the first edition, the aim has been to pre- sent the great subject of astronomy so that it can be easily comprehended even by a person who has not had extensive scientific trainiag. It has been assumed that the reader has no intention of becoming an astronomer, but that he has an interest in the wonderful universe which surrounds him, and that he has arrived at such a stage of iatellectual development that he demands the reasons for whatever conclusions he is asked to accept. The first two of these , assumptions have largely determined the subject matter which is presented ; the third has strongly influenced the method of presenting it.
While the aims have not changed materially since the first edition was written, the details of the attempt to accomplish them have undergone many, and in some cases important, modifications. For example, the work on reference points and liues has been deferred to Chapter IV. If one is to know the sky, and not simply know about it, a knowledge of the coordi- nate systems is indispensable, but they always present some difficulties when they are encountered at the beginning of the subject. It is believed that the present treatment prepares so thoroughly for their study and leads so naturally to them that their mastery will not be found dif&cult. The chapter on telescopes has been regretfully omitted because it was not necessary for understanding the remainder of the work, and because the space it occupied was needed for treatrag more vital parts of the subject. The numerous discoveries in the sidereal universe during the last ten years have made it neces- sary greatly to enlarge the last chapter.
VI PREFACE
As now arranged, the first , chapters are devoted to a discus- sion of the earth and its motions. They present splendid examples of the characteristics and methods of science, and amply illustrate the care with which scientific theories are established. The conclusions which are set forth are bound up with the development of science from the dawn of recorded history to the recent experiments on the rigidity and the elas- ticity, of the earth. They show how closely various sciences are interlocked, and how much an understanding of the earth depends upon its relations to the sky. They lead naturally to a more formal treatment of the celestial sphere and a study of the constellations. A familiarity with the brighter stars and the more conspicuous constellations is regarded as important. One who has become thoroughly acquainted with them will always experience a thrill when he looks up at night into a cloudless sky.
The chapter on the sun has been postponed until after the treatment of the moon, planets, and comets. The reason is that the discussion of the sun necessitates the introduction of many new and difficult topics, such as the conservation of en- ergy, the disintegration of radioactive elements, and the prin- ciples of spectrum analysis. Then follows the evolution of the solar system. In this chapter new and more serious de- mands are made on the reasoning powers and the imagination. Its study in a measure develops a point of view and prepares the way for the consideration, in the last chapter, of the tran- scendental and absorbingly interesting problems respecting the organization and evolution of the sidereal universe.
Lists of problems have been given at the ends of the prin- cipal divisions of the chapters. They cannot be correctly answered without a real comprehension of the principles which they involve, and in very many cases, especially in the later chapters, they lead to important supplementary results. It is strongly recommended that they be given careful consideration.
The author is indebted to Mr. Albert Barnett for the new star maps and the many drawings with which the book is illus- trated, with the exception of Tigs. 23 and 30, which were
PREFACE VU
kindly furnished by Mr. George Otis. He is indebted to Professor David Eugene Smith for photographs of Newtcta, Kepler, Herschel, Adams, and Leverrier. He is indebted to the Lick, Lowell, Solar, and Yerkes observatories for a large amount of illustrative material which was very generously furnished. He is under deeper obligations to his colleague, Professor W. D. MacMillan, than this brief acknowledgment can express for assistance on the manuscript, on the proofs, and in preparing the many problems which appear in the book.
F. E. MOULTON. The Uniteksitt of Chicago, September 25, 1916.
CONTENTS
CHAPTER I Preliminary Considerations
1. Science
2. The value of science
3. The origin of science ...
4. The methods of science
5. The imperfections of science
6. Great contributions of astronomy to science
7. The present value of astronomy .
8. The scope of astronomy
PAGE
1
2
4
6
10
14
16
19
CHAPTER U
THE EARTH
I. The Shape op the Earth
9. Astronomical problems respecting the earth 10, 11. Proofs of the earth's sphericity 12, 14, 15. Proofs of the earth's oblateness 1-3. Size and shape of the earth
16. The theoretical shape of the earth
17. Different kinds of latitude .
18. Historical sketch on the shape of the earth
26 27 31 33 38 39 40
II. The Mass of the Earth and the Conditioi« of its Interior
19. The principle by which mass is determined .... 43
20. The mass and density of the earth . . . . . .45
21-23. Method? pf determining the density of the earth ... 46
CONTENTS
24. Temperature and pressure in the earth's interior
25, 26. Proofs of the earth's rigidity and elasticity .
27. Historical sketch on the mass and rigidity of the earth
PAGE
51 52 62
III. The Earth's Atmosphere
28. Composition and mass of the earth's atmosphere 29-31. Methods of determining height of the atmosphere
32. The kinetic theory, of gases ....
33. The escape of atmospheres
34. Effects of the atmosphere on climate .
35. Importance of the constitution of the atmosphere
36. R61e of the atmosphere in life processes
37. Kefraction of light by the atmosphere
38. The twinkling of the stars . . . .
64 65
71 72 74 74 76
CHAPTER III
THE MOTIONS OF THE EARTH I. The Rotation of the Earth
39. The relative rotation of the earth
40. The laws of motion
41-43. Proofs of the earth's rotation
44. Consequences of the earth's rotation .
45. Uniformity of the earth's rotation
46. The variation of latitude
47. The precession of the equinoxes and nutation
77 79 82 85 87 89 92
II. The Revolution of the Earth
48. Relative motion of the earth with respect to the sun 49-52. Proofs of the revolution of the earth
53. Shape of the earth's orbit
54. Motion of the earth in its orbit .
55. Inclination of the earth's orbit .
56. The cause of the seasons .... .57. Relation of altitude of pole to latitude of observer
58. The sun's diurnal circles
59. Hours of sunlight in different latitudes
60. The lag of the seasons
61. Effect of eccentricity of earth's orbit on seasons
62. Historical sketch of the motions of the earth
96 98 102 103 105 107 108 109 HI 112
u;j
115
CONTENTS
XI
ARTS.
63. 64. 65. 66. 67. 68.
CHAPTER IV
EErBRENCE Points and Lines
Object and character of reference points and lines
The geographical system .
The horizon system .
The equator system .
The ecliptic system .
Comparison of systems of coSrdinates 69, 70. Finding the altitude and azimuth 71, 72. Finding the right ascension and declination 73. Other problems of position ....
PAGE
121 122 123 125 127 127 130 133 135
CHAPTER V
The Constellations
74. Origin of the constellations 138
75. Naming the stars 138
76. Star catalogues 141
77. The magnitudes of the stars 142
78. The first-magnitude stars 143
79. Number of stars in first six magnitudes 145
80. Motions of the stars 145
81. The Milky Way, or Galaxy 146
82. The constellations and their positions (Maps) .... 148
83. Finding the pole star 149
84. Units for estimating angular distances 150
85-101. Ursa Major, Cassiopeia, Locating the equinoxes, Lyra,
Hercules, Scorpius, Corona Borealis, Bootes, I^eo, An- dromeda, Perseus, Auriga, Taurus, Orion, Canis Major,
Canis Minor, Gemini 150
102. On becoming familiar with the stars 167
CHAPTER VI
Time
103. Definitions of equal intervals of time 169
104. The practical measure of time . 170
105. Sidereal time 171
Xll
CONTENTS
AET8. ■"*««
106. Solar time l'i'2
107. Variations in length of solar days . . ... 172
108. Mean solar time 175
109. The equation of time l'?6
110. Standard time ... 177
111. Distribution of time 179
112. Civil and astronomical days 181
113. Place of change of date 181
114-116. Sidereal, anomalistic, and tropical years .... 183
117. The calendar 184
118. Finding the day of week on any date 185
CHAPTER VII
The Moon
119. The moon's apparent motion among the stars
120. The moon's synodical and sidereal periods
121. The phases of the moon .
122. The diurnal circles of the moon 123 The distance of the moon ....
124. The dimensions of the moon
125, 126. The moon's orbit with respect to earth and sun
127. The mass of the moon ....
128. The rotation of the moon ....
129. The librations of the moon
130. The density and surface gi-avity of the moon
131. The question of tl\e moon's atmosphere .
132. Light and heat received from the moon .
133. The temperature of the moon . 134-138. The surface of the moon . 139. Effects of the moon on the earth 140-142. Eclipses of the moon and sun .
188 189 190 192 194 196 197 198 200 201 202 203 204 205 207 217 218
CHAPTER VIII
THE SOLAR SYSTEM
I. The Law of Gravitation
143. The members of the solar system
144. Relative dimensions of the planetary orbits
226
227
CONTENTS
xui
AETS. Page
145. Kepler's laws of motion 229
146, 147. The law of gravitation 230
148. The conic sections 234
149. The question of other laws of force ....;. 236
150. Perturbations 237
151. The discovery of Neptune 238
152. The problem of three bodies 241
158. Cause of the tides 242
154. Masses of celestial bodies 244
155. surface gravity of celestial bodies 245
11. Orbits, Dimensions, and Masses of the Planets
156. Finding the dimensions of the solar system
157. Elements of the orbits of the planets (Table) .
158. Dimensions and masses of the planets (Table)
159. Times for observing the planets
160. The planetoids
161. The question of undiscovered planets
162. The zodiacal light and the gegenscheiu
246 248 252 255 257 261 262
CHAPTER IX
THE PLANETS
I. Mercurt and Venus
163. Phases of Mercury and Venus 266
164. Albedoes and atmospheres of Mercury and Venus . . . 268
165. Surface markings and rotation of Mercury .... 269
166. The seasons of Mercury 270
167. Surface markings and rotation of Venus 271
168. The seasons of Venus 272
II. Mars
169. The satellites of Mars
170. The rotation of Mars
171. The albedo and atmosphere of Mars
172. The polar caps and temperature of Mars
173. The canals of Mars ....
174. Explanations of the canals of Mars .
273 274 276
277 283 285
XIV
CONTENTS
III. Jupiter
176, 176. Jupiter's satellite system . . . .
177, Discovery of the velocity of light
178, 179. Surface 'markings and rotation of Jupiter . 180. Physical condition and seasons of Jupiter
PAGE
289 291 292 296
IV. Saturn
181. Saturn's satellite system .... 182-184. Saturn's ring system . 186. Surface markings and rotation of Saturn . 186. Physical condition and seasons of Saturn .
. 297
299-304
. 305
. 306
V. Uranus and Neptune
187. Satellite systems of Uranus and Neptune .
188. Atmospheres and albedoes of Uranus and Neptune
189. Periods of rotation of Uranus and Neptune
190. Physical conditions of Uranus and Neptune
306 307 307 308
CHAPTER X COMETS AND METEORS
I. Comets
191. General appearance of comets .
192. The orbits of comets .
193. 194. The dimensions and masses of comets 196. Families of comets
196. The capture of comets
197. On the origin of comets
198. Theories of comets' tails
199. The disintegration of comets .
200. Historical comets ....
201. Halley's comet
. 311
. 313,
316, 317
318 . 320
322 . 323 . 327
328 . 332
II. Meteors
202. Meteors, or "shooting stars '
203. The number of meteors
204. 205. Meteoric showers
337 338 339
CONTENTS XV
ARTS.
'"""■ PAGE
206. Connection between comets and meteors . . : . . 341
207. Effects of meteors on the solar system 34.3
208. Meteorites . . 343
Theories respecting the origin of meteors 345
209
CHAPTER XI
THE SUN I. The Sun's Heat
210. The problem of the sun's heat 349
211. Amount of energy received from sun .... 349
212. Sources of energy used by man 351
213. Amount of energy radiated by sun 353
214. The temperature of the sun 354
215. Principle of the conservation of energy . ... 355
216. 217. Theories of the sun's heat 356-359
218. Fast and future of sun on contraction theory .... 360
219. The age of the earth 360
w
II. Spectkum Analysis
220. The nature of light .... . .365
221. On the production of light 366
222. Spectroscopes and the spectrum 369
223-226. The laws of spectrum analysis 371-375
III. The Constitution of the Sun
227. Outline of the sun's constitution 378
228. The photosphere 379
229-231. Sunspots, distribution, periodicity, and motions . 381-384
232. The rotation of the sun 388
238. The reversing layer 390
234. Chemical constitution of reversing layer . 392
235, 236. The chromosphere and prominences . . . 394, 395
237. The spectroheliograph 398
238. The corona 401
239. The eleven-year cycle , . , 404
XVI
CONTENTS
CHAPTER XII EVOLUTION OF THE SOLAR SYSTEM I. General Considerations on Evolution
ARTS.
240. Essence of the doctrine of evolution ....
241. Value of a theory of evolution
242. Outline of growth of doctrine of evolution
PAOB
407 408 410
II. Data of Frorlem or Evolution of Solar System
243. General evidences of orderly development . . . 413
244. Distribution of mass in the solar system . ■ . . 414
245. Distribution of moment of momentum 416
246. The energy of the solar system 419
III. The Planetesimal Theory
247. Outline of the planetesimal theory .
248. Examples of planetesimal organization
249. Suggested origin of spiral nebulse
250. The origin of planets
251. The planes of the planetary orbits .
252. The eccentricities of the planetary orbits
253. The rotation of the sun
254. The rotation of the planets
255. The origin of satellites
256. The rings of Saturn
257. 258. The planetoids and zodiacal light
259. The comets ...
260. The future of the solar system .
421 422 424 431 433 434 436 437 440 441 442 442 443
IV. Historical Cosmogonies
261. The hypothesis of Kant .
262. The hypothesis of Laplace
263. 264. Tidal forces and tidal evolution
265. Effects of tides on motions of the moon
266. Effects of tides on motions of the earth
267. lldal evolution of the planets .
. 446 . 449 452, 454 . 456 . 456 . 460
CONTENTS
XVil
CHAPTER XIII I. The Apparbjst Distbibution of the Stars
ABT8. piGE
268. On the problems of the sidereal universe 463
269. Number of stars of various magnitudes . ... 464
270. Apparent distribution of the stars 470 '
271. Form and structure of the Milky Way 473
II. Distances and Motions op the Stars
272. Direct parallaxes of nearest stars 476
273. Distances of stars from proper motions and radial velocities . 481
274. Motion of sun with respect to stars 482
275. Distances of stars from motion of sun 484
276. Kapteyn's results on distances of stars 486
277. Distances of moving groups of stars 487
278. Star streams 490
279. On the dynamics of the stellar system 491
280. Runaway stars 498
281. Globular clusters 500
III. The Stars
282. Double stars 505
283, 284. Orbits and masses of binary stars 507
285, 286. Spectroscopic binary stars 510
287-293. Variable stars of various types 515
294. Temporary stars 523
295. The spectra of the stars 527
296. Phenomena associated with spectral types .... 530
297. On the evolution of the stars . , 532
298. Tacit assumptions oif theories of stellar evolution . . . 534
299. Origin and evolution of binary stars 543
300. On the infinity of the physio^ universe in space and in time . 548
301. 302. 303. 304.
IV. The Nebdl^
Irregular nebulse 550
Spiral nebulae 554
Ring nebulsB . . "^^
Planetary nebulae 560
LIST OF TABLES
HO.
I. The first-magnitude stars
II. Numbers of stars in first six magnitudes
m. The constellations
^ IV. Elements of the orbits of the planets .
V. Data on sun, moon, and planets
VI. Dates of eastern elongation and opposition
VII. Jupiter's satellite system .
VIII. Saturn's satellite system
IX. Saturn's ring system .
X. Rotation of the sun in different latitudes
XI. Elements found in the sun
XII. Distribution of moment of momentum in solar system
Xm. Distances of ejection for various initial yelocities
XIV. Numbers of stars in magnitudes 5 to 17
XV. Distribution of the stars with respect to the Galaxy
XVI. Table of nineteen nearest stars ....
XVII. Distances of stars of magnitudes 1 to 15
XVIII. Binary stars whose masses are known
PAGE
144 145 147 249 254 256 290 298 300 389 393 417 428 466 471 478 486 509
LIST OF PHOTOGRAPHIC ILLUSTRATIONS
NO. PAGE
1. The Lick Observatory, Mt. Hamilton, Cal. . . frontispiece
2. The Yerkes Observatory, Williams Bay, Wis. . . facing 1
3. The moon at 8.5 days (Ritchey; Yerkes Observatory) . . 20
24. Orion star trails (Barnard ; Yerkes Observatory) . . 77
25. Circumpolar star trails (Ritchey) 78
54. The 40-inch telescope of the Yerkes Observatory . . 138
55. The Big Dipper (Hughes; Yerkes Observatory) . . 149
57. The sickle in Leo (Hughes; Yerkes Observatory) . . 157
58. Great Andromeda Nebula (Ritchey ; Yerkes Observatory) . 158
59. The Pleiades (Wallace ; Yerkes Observatory) .... 161
60. Orion ( Hughes ; Yerkes Observatory) 163
61. Great Orion Nebula (Ritchey ; Yerkes Observatoi-y) . 164 68. The earth-lit moon (Barnard ; Yerkes Observatory) . . 192 75. Moon at 9| days (Ritchey ; Yerkes Observatory) . . . 208
77. The Crater Theophilus (Ritchey ; Yerkes Observatory) . . 210
78. Great Crater Clavius (Ritchey ; Yerkes Observatory) . 212
79. The full moon (Wallace ; Yerkes Observatory) . .215
86. Johann Kepler (Collection of David Eugene Smith) \ . . 229
87. Isaac Newton (Collection of David Eugene Smith) . . . 232
90. William Herschel (Collection of David Eugene Smith) . 239
91. John Couch Adams (Collection of David Eugene Smith) . 240
92. Joseph Leverrier (Collection of David Eugene Smith) . . 240 99. Trail of Planetoid Egeria (Parkhurst ; Yerkes Observatory) . 259
103. Mars (Barnard ; Yerkes Observatory) 275
108. Mars (Mount Wilson Solar Observatory) ..... 286
113. Jupiter (E. C. Slipher ; Lowell Observatory) . . . 295
117. Saturn (Barnard ; Yerkes Observatory) 301
119. Brooks' Comet (Barnard ; Yerkes Observatory) . . . 312
124. Delavan's Comet (Barnard ; Yerkes Observatory) . . . 325
125. Encko's Comet (parnard ; Yerkes Observatory) . . . 329 1^6. Morehouse's Comet (Barnard ; Yerkes Observatory) . . 333 128. Halley's Comet (Barnard ; Yerkes Observatory) . . .335
133. Long Island, Kan., meteorite (Farrington) . . . .344
134. Canon Diablo, Ariz., meteorite (Farrington) . . . .346
xxii LIST OF PHOTOGRAPHIC ILLUSTRATIONS
PAGE
135. Durango, Mexico, meteorite (Farrington) . . . .345
136. Tower telescope of the Mt. Wilson Solar Observatory . . 348
141. Tlie sun (Fox ; Yerkes Observatory) 376
144. Sun spot, July 17, 1905 (Fox ; Yerkes Observatory) . 382
146. Sunspots with opposite polarities (Hale; Solar Observatory) . 386
147. SolarObservatory of the Carnegie Institution, Mt. Wilson, Cal. 387
149. Solar prominence 80,000 miles high (Solar Observatory) . . 396
150. Motion in solar prominences (Slocum ; Yerkes Observatory) . 397
152. Spectroheliogram of sun (Hale and EUerman; Yerkes Observa-
tory) . . . . . . 400
153. Spectroheliograms of sun spot (Hale and EUerman ; Solar Ob-
servatory) . ... . . 401
154. The sun's corona (Barnard and Ritchey) .... 402 157. Eruptive prominences (Slocum ; Yerkes Observatory) . . 426
159. Great spiral nebula M. 51 (Ritchey ; Yerkes Observatory) . 429
160. Great spiral nebula M. 33 (Ritchey ; Yerkes Observatory) 430 162. Laplace (Collection of David Eugene Smith) . . . 449
165. Milky Way in Aquila (Barnard ; Yerkes Observatory)
166. Star clouds in Sagittarius (Barnard ; Yerkes Observatory)
167. Region of Rho Ophiuchi (Barnard ; Yerkes Observatory) 171. Hercules star cluster (Ritchey ; Yerkes Observatory)
173. Spectra of Mizar (Frost ; Yerkes Observatory)
174. Spectra of Mu Orionis (Frost ; Yerkes Observatory)
180. Nova Persei (Ritchey ; Yerkes Observatory) .
181. The spectrum of Sirius (Yerkes Observatory) .
182. The spectrum of Beta Geminorum (Yerkes Observatory)
183. The spectrum of Arcturus (Yerkes Observatory) . . 529
184. The Pleiades (Ritchey ; Yerkes Observatory) . . 537
187. Nebula in Cygnus (Ritchey ; Yerkes Observatory) . 651
188. Bright and dark nebulae (Barnard ; Yerkes Observatory) 554
189. The Trifid Nebula (Crossley reflector ; Lick Observatory) 555
190. Spiral nebula in Ursa Major (Ritchey ; Yerkes Observatory) . 556
191. Spiral nebula in Andromeda (Crossley reflector ; Lick Observ-
atory) 657
192. Great nebula in Andromeda (Ritchey ; Yerkes Observatory) . 559
193. Ring nebula in Lyra (Sullivan ; Yerkes_Observatory) . . 660
194. Planetary nebula (Crossley reflector ; Lick Observatory) . 561
462 472 474 501 511 513 525 527 528
AN INTKODUCTION TO ASTEONOMY
A:N^ mTRODUCTIOJN^ TO ASTROIN^OMY
CHAPTER I PRELIMINARY CONSIDERATIONS
1. Science. — The progress of mankind has been marked by a number of great intellectual movements. At one tinie the ideas of men were expanding with the knowledge which they were obtaining from the voyages of Columbus, Magel- lan, and the long list of hardy explorers who first visited the remote parts of the earth. At another, milUons of men laid down their lives in order that they might obtain toleration in religious behefs. At another, the struggle was for poUtical freedom. It is to be noted with satisfaction that those movements which have involved the great mass of people, from the highest to the lowest, have led to results which have not been lost.
The present age is known as the age of science. Never before have so many men been actively engaged in the pursuit of science, and never before have its results con- tributed so enormously to the ordinary affairs of life. If all its present-day appUcations were suddenly and for a con- siderable time removed, the results would be disastrous. With the stopping of trains and steamboats the food supply in cities would soon fail, and there would be no fuel with which to heat the buildings. Water could no longer be pumped, and devastating fires might follow. If people es- caped to the country, they would perish in large numbers
2 AN INTRODUCTION TO ASTRONOMY [ch. i, 1
because without modern machinery not enough food could be raised to supply the population. In fact, the more the subject is considered, the more clearly it is seen that at the present time the lives of civihzed men are in a thousand ways directly dependent on the things produced by science.
Astronomy is a science. That is, it is one of those sub- jects, such as physics, chemistry, geology, and biology, which have made the present age in very many respects altogether different from any earlier one. Indeed, it is the oldest science and the parent of a number of the others, and, in many respects, it is the most perfect one. For these rea- sons it illustrates most simply and clearly the characteristics of science. Consequently, when one enters on the study of astronomy he not only begins an acquaintance with a subject which has always been noted for its lofty and un- selfish ideals, but, at the same time, he becomes famihar with the characteristics of the scientific movement.
2. The Value of Science. — The importance of science in changing the relations of men to the physical universe about them is easy to discern and is generally more or less recognized. That the present conditions of life are better than those which prevailed in earlier times proves the value of science, and the more it is considered from this point of view, the more valuable it is found to be.
The changes in the mode of Hving of man which science has brought about, will probably in the course of time give rise to marked alterations in his physique; for, the better food supply, shelter, clothing, and sanitation which have recently been introduced as a consequence of scientific dis- coveries, correspond in a measure to the means by which the best breeds of domestic animals have been developed, and without which they degenerate toward the wild stock from which they have been derived. And probably, also, as the factors which cause changes in living organisms become better known through scientific investigations, man will consciously direct his own evolution.
CH. I, 2] PRELIMINARY CONSIDERATIONS 3
But there is another less speculative respect in which science is important and in which its importance will enor- mously increase. It has a profound influence on the minds of those who devote themselves to it, and the number of those who are interested in it is rapidly increasing. In the first place, it exalts truth and honestly seeks it, wherever the search may lead. In the second place, its subject matter often gives a breadth of vision which is not otherwise ob- tained. For example, the complexity and adaptability of living beings, the irresistible forces which elevate the moun- tains, or the majestic motions of the stars open an intellectual horizon far beyond that which belongs to the ordinary af- fairs of life. The conscious and deUberate search for truth and the contemplation of the wonders of nattire change the mental habits of a man. They tend to make him honest with himself, just in his judgment, and serene in the midst of petty annoyances. In short, the study of science makes character, as is splendidly illustrated in the hves of many celebrated scientific men. It would undoubtedly be of very great benefit to the world if every one could have the dis- cipUne of the sincere and honest search for the truth which is given by scientific study, and the broadening influence of an acquaintance with scientific theories.
There is an important possible indirect effect of science on the intellectual development of mankind which should not be overlooked. One of the results of scientific discoveries has been the greatly increased productivity of the. human race. All of the necessities of life and many of its luxuries can now be supplied by the expenditure of much less time than was formerly required to produce the bare means of existence. This leaves more leisure for intellectual pm-suits. Aside from its direct effects, this is, when considered in its broad aspects, the most important benefit conferred by science, because, in the final analysis, intellectual experiences are the only things in which men have an interest. As an illustration, any one would prefer a normal conscious life
4 AN INTRODUCTION TO ASTRONOMY [ch. i, 2
for one year rather than an existence of five hundred years with the certainty that he would be completely unconscious during the whole time.
It is often supposed that science and the fine arts, whose importance every one recognizes, are the antitheses of each other. The arts are beUeved to be warm and human, — science, cold and austere. This is very far from being the case. While science is exacting in its demands for pre- cision, it is not insensible to the beauties of its subject. In all branches of science there are wonderful harmonies which appeal strongly to those who fully comprehend them. Many of the great scientists have expressed themselves in their writings as being deeply moved by the aesthetic side of their subject. Many of then! have had more than ordinary taste for art. Mathematicians are noted for being gifted in music, and there are numerous examples of scientific men who were fond of painting, sculpture, or poetry. But even if the common opinion that science and art are opposites were correct, yet science would contribute indirectly to art through the leisure it furnishes men.
3. The Origin of Science. — It is doubtful if any impor- tant scientific idea ever sprang suddenly into the mind of a single man. The great intellectual movements in the world have had long periods of preparation, and often many men were groping for the same truth, without exactly seizing it, before it was fully eonapreh ended.
The foundation on which all science rests is the principle that the imiverse is orderly, and that all phenomena succeed one another in harmony with invariable laws. Consequently, science was impossible until the truth of this principle was perceived, at least as appUed to a hmited part of nature.
The phenomena of ordinary observation, as, for example, the weather, depend on such a multitude of factors that it was not easy for men in their primitive state to discover that they occur in harmony with fixed laws. This was the age of superstition, when nature was supposed to be con-
CH. I, 3] PRELIMINARY CONSIDERATIONS 5
trolled by a great number of capricious gods whose favor could be won by childish ceremonies. Enormous experience was required to dispel such errors and to convince men that the universe is one vast organization whose changes take place in conformity with laws which they can in no way •alter.
The actual dawn of science was in prejiistorie times, probably in the civilizations that flourished in the valleys of the Nile and the Euphrates. In the very earliest records of these people that have come down to modern times it is foimd that they were acquainted with many astronomical phenomena and had coherent ideas with respect to the mo- tions of the sun, moon, planets, and stars. It is perfectly clear from their writings that it was from their observations of the heavenly bodies that they first obtained the idea that the universe is not a chaos. Day and night were seen to succeed each other regularly, the moon was found to pass through its phases systematically, the seasons followed one another in order, and in fact the more conspicuous celestial phenomena were observed to occur in an orderly sequence. It is to the glory of astronomy that it first led men to the conclusion that law reigns in the universe.
The ancient Greeks, at a period four or five hundred years preceding the Christian era, definitely undertook to find from systematic observation how celestial phenomena follow one another. They determined very accurately the number of days in the year, the period of the moon's revolu- tion, and the paths of the sun and the moon among the stars ; they correctly explained the cause' of eclipses and learned how to predict them with a considerable degree of accuracy; they undertook to measure the distances to the heavenly bodies, and to work out a complete system that would represent their motions. The idea was current among the Greek philosophers that the earth was spherical, that it turned on its axis, and, among some of them, that it revolved around the sun. They had true science in the modem
6 AN INTRODUCTION TO ASTRONOMY [ch. i, 3
acceptance of the term, but it was largely confined to the relations among celestial phenomena. The conception that the heavens are orderly, which they definitely formulated and acted on with remarkable success, has been extended, espe- cially in the last two centuries, so as to include the whole universe. The extension was first made to the inanimate world and then to the more comphcated phenomena asso- ciated with Uving beings. Every increase in carefully recorded experience has confirmed and strengthened the belief that nature is perfectly orderly, until now every one who has had an opportunity of becoming familiar with any science is firmly convinced of the truth of this principle, which is the basis of all science.
4. The Methods of Science. — Science is concerned with the relations among jfcenomena, and it must therefore rest ultimately upon observations and experiments. Since its ideal is exactness, the observations and experiments must be made with all possible precision and the results must be carefully recorded. These principles seem perfectly obvious, yet the world has often ignored them. One of the chief faults of the scientists of ancient times was that th^y indulged in too many arguments, more or less metaphysical in charac- ter, and made too few appeals to what would now seem ob- vious observation or experiment. A great Enghsh philoso- pher, Roger Bacon (1214-1294), made a powerful argument in favor of founding science and philosophy on experience.
It must not be supposed that the failure to rely on obser- vations and experiment, and especially to record the results of experience, are faults that the world has outgrown. On the contrary, they are still almost imiversally prevalent among men. • For example, there are many persons who be- lieve in dreams or premonitions because once in a thousand cases a dream or a premonition comes true. If they had written down in every case what was expected and what actually happened, the absurdity of their theory would have been evident. The whole mass of superstitions with
CH. I, 4] PRELIMINARY CONSIDERATIONS 7
which mankind has burdened itself survives only because the results of actual experience are ignored.
In scientific work great precision in making observations and experiments is generally of the highest importance. Every science furnishes examples of cases where the data seemed to have been obtained with greater exactness than was really necessary, and where later the extra accuracy led to important discoveries. In this way the foundation of the theory of the motion of the planets was laid. Tycho Brahe was an observer not only of extraordinary industry, but one who did all his work with the most painstaking care. Kepler, who had been his pupil and knew of the excellence of his measurements, was a computer who sought to bring theory and observation into exact harmony. He foimd it impossible by means of the epicycles and eccentrics, which his predecessors had used, to represent exactly the observa- tion of Tycho Brahe. In spite of the fact that the discrep- ancies were small and might easily have been ascribed to errors of observation, Kepler had absolute confidence in his master, and by repeated trials and an enormous amount of labor h'i finally arrived at the true laws of planetary motion (Art. 145). These laws, in the hands of the genius Newton, led directly to the law of gravitation and to the explanation of all the many peculiarities of the motions of the moon and planets (Art. 146) .
Observations alone, however carefully they may have been made and recorded, do not constitute science. First, the phenomena; must be related, and then, what they have in common must be perceived. It might seem that it would be a simple matter to note in what respects phenomena are similar, but experience has shown that only a very few have the ability to discover relations that are not already known. If this were not true, there would not be so many examples of new inventions and discoveries depending on very simple things that have long been within the range of experience of every one. After the common element in the observed
8 AN INTRODUCTION TO ASTRONOMY [ch. i, 4
phenomena has been discovered the next step is to infer, by the process known as induction, that the same thing is true in all similar cases. Then comes the most difficult thing of all. The vital relationships of the one class of phenomena with other classes of phenomena must be discovered, and the several classes must be organized into a coherent whole.
An illustration will make the process clearer than an extended argument. Obviously, all men have observed moving bodies all their Uves, yet the fact that a moving body, subject to no exterior force, proceeds in a straight Une with uniform speed was not known until about the time of GaHleo (1564-1642) and Newton (1643-1727). When the result is once enunciated it is easy to recall many confirmatory experiences, and it now seems remarkable that so simple a fact should have remained so long undiscovered. It was also noted by Newton that when a body is acted on by a force it has an acceleration (acceleration is the rate of change of velocity) in the direction in which the force acts, and that the acceleration is proportional to the magnitude of the force. Dense bodies left free in the air fall toward the earth with accelerated velocity, and they are therefore subject to a force toward the earth. Newton observed these things in a large number of cases, and he inferred by induc- tion that they are universally true. He focused particularly on the fact that every body is subject to a force directed toward the earth.
If taken alone, the fact that bodies are subject to forces toward the earth is not so very important; but Newton used it in cormection with many other phenomena. For example, he knew that the moon is revolving around the earth in an approximately circular orbit. At P, in Fig. 3, it is moving in the direction PQ around the earth, E. But it actually moves from P to R. That is, it has fallen toward the earth through the distance QR. Newton perceived that this motion is analogous to that of a body falling near the surface of the earth, or rather to the motion of a body which
CH. r, 4] PRELIMINARY CONSIDERATIONS 9
has been started in a horizontal direction from p near the surface of the earth. For, if the body were started hori- zontally, it would continue in the straight line pg, instead of curving downward to r, if it were not acted upon by a force directed toward the earth. Newton also knew Kepler's laws of planetary motion. By combining with wonderful insight a number of classes of phenomena which before his time had been supposed to be unrelated, he finally arrived at the law of gravitation — " Every particle of matter in the universe attracts every other par- ticle with a force which is directly pro- portional to the product of their masses and inversely proportional to the square of their distance apart." Thus, by per- ceiving the essentials in many kinds of Fig. 3. — The motion phenomena and by an almost unparal- ?* ^^ moon from p
, Z . , . . , to " around E is smu-
leled stroke or gemus m combmmg them, lar to that of a body he discovered one of the relations which Projected horizontally
from p.
every particle of matter in the universe has to all the others. By means of the laws of motion (Art. 40) and the law of gravitation, the whole problem of the motions of bodies was systematized.
There is still another method employed in science which is often very important. After general principles have been discovered they can be used as the basis for deducing particular conclusions. The value of the particular conclu- sions may consist in leading to the accomplishment of some desired end. For example, since a moving body tends to continue in a straight line, those who build railways place the outside rails on curves higher than those on the in- side so that trains will not leave the track. Or, the knowledge of the laws of projectiles enables gunners to hit invisible objects whose positions are known.
The value of particular conclusions may consist in ena-
10 AN INTRODUCTION TO ASTRONOMY [ch. i, 4
bling men to adjust themselves to phenomena over which they have no control. For example, in many harbors large boats can enter or depart only when the tide is high, and the knowledge of the times when the tides will be high is valuable to navigators. After the laws of meteorology have become more perfectly known, so that approaching storms, or frosts, or drouths, or hot waves can be accurately foretold a considerable time in advance, the present enormous losses due to these causes will be avoided.
The knowledge of general laws may lead to information regarding things which are altogether inaccessible to obser- vation or experiment. For example, it is very important for the geologist to know whether the interior of the earth is solid or liquid ; and, if it is solid, whether it is elastic or viscous. Although at first thought it seems impossible to obtain reliable information on this subject, yet by a number of indirect processes (Arts. 25, 26) based on laws established from observation, it has been possible to prove with cer- tainty that the earth, through and through, is about as rigid as steel, and that it is highly elastic.
Another important use of the deductive process in science is in drawing consequences of a theory which must be ful- filled in experience if the theory is correct, and which may fail if it is false. It is, indeed, the most efiicient means of testing a theory. Some of the most noteworthy examples of its application have been in connection with the law of gravitation. Time after time mathematicians, using this law as a basis for their deductions, have predicted phenom- ena that had not been observed, and time after time their predictions have been fulfilled. This is one of the reasons why the truth of the law of gravitation is regarded as having been firmly estabUshed.
5. The Imperfections of Science. — One of the char- acteristics of science is its perfect candor and fairness. It would not be in harmony with its spirit to attempt to lead one to suppose that it does not have sources of weakness.
CH. I, 5] PRELIMINARY CONSIDERATIONS 11
Besides, if its possible imperfections are analyzed, they can be more easily avoided, and the real nature of the final conclusions will be better understood.
It must be observed, in the first place, that science con- sists of men's theories regarding what is true in the universe about them. These theories are based on observation and experiment and are subject to the errors and incompleteness of the data on which they are founded. The fact that it is not easy to record exactly what one may have attempted to observe is illustrated by the divergence in the accounts of different witnesses of anything except the most trivial occurrence. Since men are far from being perfect, errors in the observations cannot be entirely avoided, but in good science every possible means is taken for ehminating them.
In addition to this source of error, there is another more insidious one that depends upon the fact that observational data are often collected for the purpose of testing a specific theory. If the theory in question is due to the one who is making the observations or experiments, it is especially diflBcult for him to secure data uninfluenced by his bias in its favor. And even if the observer is not the author of the theory to which the observations relate, he is very apt to be prejudiced either in its favor or against it.
Even if the data on which science is based were always correct, they would not be absolutely exhaustive, and the inductions to general principles from them would be sub- ject to corresponding uncertainties. Similarly, the general principles, derived from various classes of phenomena, which are used in formulating a complete scientific theory, do not include all the principles which are involved in the particular domain of the theory. Consequently it may be imperfect for this reason also.
The sources of error in scientific theories which have been enumerated are fundamental and will always exist. The best that can be done is to recognize their existence and to minimize their effects by all possible means. The fact that
12 AN INTRODUCTION TO ASTRONOMY [ch. i, 5
science is subject to imperfections does not mean that it is of little value or that less effort should be put forth in its cultivation. Wood and stone and brick and glass have never been made into a perfect house ; yet houses have been very useful and men will continue to build them.
There are many examples of scientific theories which it has been found necessary to modify or even to abandon. These changes have not been more numerous than they have been in other domains of hmnan activities, but they have been, perhaps, more frankly confessed. Indeed, there are plenty of examples where scientists have taken evident satisfaction in the alterations they have introduced. The fact that scientific theories have often been found to be imperfect and occasionally positively wrong, have led some persons who have not given the question serious consideration to suppose that the conclusions of science are worthy of no particular respect, and that, in spite of the pretensions of scientists, they are actually not far removed from the level of superstitions. The respect which scientific theories deserve and the gulf that separates them from superstitions will be evident from a statement of their real nature.
Suppose a person were so situated that he could look out from an upper window over a garden. He could make a drawing of what he saw that would show exactly the relative positions of the walks, shrubs, and flowers. If he were color bUnd, the drawing could be made in pencil so as to satisfy perfectly all his observations. But suppose some one else who was not color blind should examine the drawing. He would legitimately complain that it was not correct because the colors were not shown. If the colors were correctly given, both observers would be completely satisfied. Now suppose a third person should look at the drawing and should then go down and examine the garden in detail. He would find that the various objects in it not only have positions but also various heights. He would at once note that the heights were not represented in the drawing, and a little
CH. I, 5] PRELIMINARY CONSIDERATIONS 13
reflection would convince him that the three-dimensional garden could not be completely represented in a two-dimen- sional drawing. He would claim that that method of trying to give a correct idea of what was in the garden was funda- mentally wrong, and he might suggest a model of suitable material in three dimensions. Suppose the three-dimen- sional model were made satisfying the third observer. It is important to note that it would correctly represent all the relative positions observed by the first one and all the colors observed by the second one, as well as the additional in- formation obtained by the third one.
A scientific theory is founded on the work of one or more persons having only hmited opportunities for observation and experiment. It is a picture in the imagination, not on paper, of the portion of the universe under consideration. It represents all the observed relations, and it is assumed that it will represent the relations that might be observed in all similar circumstances. Suppose some new facts are discovered which are not covered by the theory, just as the second observer in the garden saw colors not seen by the first. It will be necessary to change the scientific theory so as to include them. Perhaps it can be done simply by adding to the theory. But if the new facts correspond to the things discovered by the third observer in the garden, it will be necessary to abandon the old theory and to construct an entirely new one. The new one must preserve all the rela- tions represented by the old one, and it must represent the new ones as well.
In the light of this discussion it may be asked in what sense scientific theories are true. The answer is that they are all true to the extent that they picture nature. The relations are the important things. When firmly established they are a permanent acquisition; however the mode of representing them may change, they remain. A scientific theory is a convenient and very useful way of describing the relations on which it is based. It correctly represents
14 AN INTRODUCTION TO ASTRONOMY [ch. i, 5
them, and in this respect differs from a superstition which is not completely in harmony with its own data. It implies many additional things and leads to their investigation. If the impUcations are found to hold true in experience, the theory is strengthened ; if not, it must be modified. Hence, there should be no reproach in the fact that a scientific theory must be altered or abandoned. The necessity for such a procedure means that new information has been obtained, not that the old was false.'
6. Great Contributions of Astronomy to Science. — As was explained in Art. 3, science started in astronomy. Many astronomical phenomena are so simple that it was possible for primitive people to get the idea from observing them that the universe is orderly and that they could discover its laws. In other sciences there are so many varying factors that the uniformity in a succession of events would not be discovered by those who were not deliberately looking for it. It is sufficient to consider the excessive complexities of the weather or of the developments of plants or animals, to see how hopeless would be the problem which a people with- out a start on science would face if they were cut off from celestial phenomena. It is certain that if the sky had al- ways been covered by clouds so that men could not have observed the regular motions of the sun, moon, and stars, the dawn of science would have been very much delayed. It is entirely possible, if not probable, that without the help of astronomy the science of the human race would yet be in a very primitive state.
Astronomy has made positive and important contribu- tions to science within historical times. Spherical trigo- nometry was invented and developed because of its uses in determining the relations among the stars on the vault of the heavens. Very many things in calculus and still higher
' The comparison of scientific theories with the picture of the objecta seen in the garden is for the purpose of making clear one of their particular features. It must be remembered that in most respects the comparison with so trivial a thing is very imperfect and unfair to science.
_CH. I, 6] PRELIMINARY CONSIDERATIONS 15
branches of mathematics were suggested by astronomical problems. The mathematical processes developed for astro- nomical applications are, of course, available for use in other fields. But the great science of mathematics does not exist alone for its applications, and to have stimulated its growth is an important contribution. While many- parts of mathematics did not have their origin in astro- nomical problems, it is certain that had it not been for these problems mathematical science would be very different from what it now is.
The science of dynamics is based on the laws of motion. These laws were first completely formulated by Newton, , who discovered them and proved their correctness by con- sidering the revolutions of the moon and planets, which describe their orbits under the ideal condition of motion in a vacuum without any friction. The immense importance of mechanics in modern life is a measure of the value of this contribution of astronomy to science.
The science of geography owes much to astronomy, both directly and indirectly. A great period of exploration fol- lowed the voyages of Columbus. It took courage of the highest order to sail for many weeks over an unknown ocean in the frail boats of his time. He had good reasons for think- ing he could reach India, to the eastward, by saDing west- ward from Spain. His reasons were of an astronomical nature. He had seen the sun rise from the ocean in the east, travel across the sky and set in the west ; he had ob- served that the moon and stars have similar motions ; and he inferjed from these things that the earth was of finite ex- tent and that the heavenly bodies moved around it. This led him to believe it could be circumnavigated. Eelying upon the conclusions that he drew from his observations of the motions of the heavenly bodies, he maintained control of his mutinous sailors during their perilous voyage across the Atlaptic, and made a discovery that has been of immense consequence to the human race.
16 AN INTRODUCTION TO ASTRONOMY [ch. i, 6
One of the most important influences in modern scientific thought is the doctrine of evolution. It has not only largely- given direction to investigations and speculations in biology and geology, but it has also been an important factor in the interpretation of history, social changes, and even religion. The first clear ideas of the orderly development of the uni- verse were obtained by contemplating the relatively simple celestial phenomena, and the doctrine of evolution was cur- rent in astronomical hterature more than haK a century before it appeared in the writings of Darwin, Spencer, and their contemporaries. In fact, it was carried directly from astronomy over into geology, and from geology into the biological sciences (Art. 242).
7. The Present Value of Astronomy. — From what has been said it will be admitted that astronomy has been of great importance in the development of science, but it is commonly believed that at the present time it is of little practical value to mankind. While its uses are by no means so numerous as those of physics and chemistry, it is nevertheless quite indispensable in a number of human activities.
Safe navigation of the seas is absolutely dependent upon astronomy. In all long voyages the captains of vessels frequently determine their positions by observations of the celestial bodies. Sailors use the nautical mile, or knot, which approximately equals one and one sixth ordinary miles. The reason they employ the nautical mile is that this is the distance which corresponds to a change of one minute of arc in the apparent positions of the heavenly bodies. That is, if, for simplicity, the sun were over a meridian, its altitude as observed from two vessels a nautical mile apart on that meridian would differ by one minute of arc.
Navigation is not only dependent on simple observations of the sun, moon, and stars, but the mathematical theory of the motions of these bodies is involved. The subject is so difficult and intricate that for a long time England and
CH. I, 7] PRELIMINARY CONSIDERATIONS 17
France offered substantial cash prizes for accurate tables of the positions of the moon for the use of their sailors.
Just as a sea captain deternaines his position by astro- nomical observations, so also are geographical positions located. For example, explorers of the polar regions find how near they have approached to the pole by observations of the altitude of the sun. International boundary lines in many cases are defined by latitudes and longitudes, instead of being determined by natural barriers, as rivers, and in all such cases they are located by astronomical observations.
It might be supposed that even though astronomy is essen- tial to navigation and geography, it has no value in the ordinary activities of life. Here, again, first impressions are erroneous. It is obvious that railway trains must be run ac- cording to accurate time schedules in order to avoid confusion and wrecks. There are also many other things in which accurate time is important. Now, time is determined by observations of the stars. The miUions of clocks and watches in use in the world are all ultimately corrected and controlled by comparing them with the apparent diurnal motions of the stars. For example, in the United States, observations are made by the astronomers of the Naval Observatory, at Washington, on every clear night, and from these observa- tions their clocks are corrected. These clocks are in elec-* trical connection with more than 30,000 other clocks in various parts of the country. Every day time signals are sent out from Washington and these 30,000 clocks are automatically corrected, and all other timepieces are directly or indirectly compared with them.
It might be inquired whether some other means might not be devised of measuring time accurately. It might be supposed that a clock coiild be made that would run so accurately as to serve all practical purposes. The fact is, however, no clock ever wd,s made which ran accurately for any considerable length of time. No two clocks have been made which ran exactly alike. In order to obtain a satis-
0
18 AN INTRODUCTION TO ASTRONOMY [ch. i, 7
factory measure of time it is necessary to secure the ideal conditions under which the earth rotates and the heavenly bodies move, and there is no prospect that it ever will be possible to use anything else, as the fundamental basis, than the apparent motions of the stars.
Astronomy is, and will continue to be, of great importance in connection with other sciences. It suppUes most of the fundamental facts on which meteorology depends. It is of great value to geology because it furnishes the geologist information respecting the origin and pre-geologic history of the earth, it determines for him the size and shape of the earth, it measures the mass of the earth, and it proves impor- tant facts respecting the condition of the earth's interior. It is valuable in physics and chemistry because the imiverse is a great laboratory which, with modem instruments, can be brought to a considerable extent within reach of the investigator. For example, the sun is at a higher tempera- ture than can be produced by any known means on the earth. The material of which it is composed is in an incan- descent state, and the study of the light received from it has proved the existence, in a number of instances, of chemical elements which had not been known on the earth. In fact, their discovery in the sun led to their detection on the earth. It seems probable that similar discoveries will be made many times in the future. The sun's corona and the nebulse contain material which seems to be in a more primitive state than any known on the earth, and the revelations afforded by these objects are having a great influence on physical theories respecting the ultimate structure of matter.
Astronomy is of greatest value to mankind, however, in an intellectual way. It furnishes men with an idea of the wonderful universe in which they live and of their position in it. Its effects on them are analogous to those which are produced by travel on the earth. If a man visits various countries, he learns many things which he does not and can- not apply on his return home, but which, nevertheless,
CH. I, 8] PRELIMINARY CONSIDERATIONS 19
make him a broader and better man. Similarly, though what one may learn about the millions of worlds which occupy the almost infinite space within reach of the great telescopes of modern times cannot be directly applied in the ordinary affairs of life, yet the contemplation of such things, in which there is never anything that is low or mean or sordid, makes on him a profound impression. It strongly modifies the particular philosophy which he has more or less definitely formulated in his consciousness, and in harmony with which he orders his Ufe.
8. The Scope of Astronomy. — The popular conception of astronomy is that it deals in some vague and speculative way with the stars. Since it is obviously impossible to visit them, it is supposed that all conclusions respecting them, except the few facts revealed directly by telescopes, are pure guesses. Many people suppose that astronomers ordinarily engage in the harmless and useless pastime of gazing at the stars with the hope of discovering a new one. Many of those who do not have this view suppose that astronomers control the weather, can tell fortunes, and are very shrewd to have discovered the names of so many stars. As is true of most conclusions that are not based on evidence, these conceptions of astronomy and astronomers are absurd.
Astronomy contains a great mass of firmly established facts. Astronomers demand as much evidence in support of their theories as is required by other scientists. They have actually measured the distances to the moon, sun, and many of the stars. They have discovered the laws of their motions and have determined the masses of the principal members of the solar system. The precision attained in much of their work is beyond that realized in most other sciences, and their greatest interest is in measurable things and not in vague speculations.
A more extended preliminary statement of the scope of astronomy is necessary in order that its study may be entered on without misunderstandings. Besides, the relations among
20
AN INTRODUCTION TO ASTRONOMY [ch. i, 8
the facts with which a science deals are very important, and a preliminary outUne of the subject will make it easier to place in their proper position in an organized whole all the various things which may be set forth in the detailed discussions.
The most accessible and best-known astronomical object is the earth. Those facts respecting it that are determined
entirely or in large part by astronomical means are properly regarded as belong- ing to astronomy. Among them are the shape and size of the earth, its average density, the condition of its interior, the height of its atmos- phere, its rotation on its axis and revolu- tion around the sun, and the climatic con- ditions of its surface so far as they are determined by its re- lation to the sun.
The nearest celes- tial body is the moon. Astronomers have found by fundamentally the same methods as those which surveyors employ that its distance from the earth averages about 240,000 miles, that its diameter is about 2160 miles, and that its mass is about one eightieth that of the earth. The earth holds the moon in its orbit by its gravi- tational control, and the moon in turn causes the tides on the earth. It is found that there is neither atmosphere nor water
Fig. 4. — The moon 1.5 days after the first quarter. Photographed with the 40~inch telescope of the Yerkes Observatory.
CH. I, 8] PEELIMINARY CONSIDERATIONS 21
on the moon, and the telescope shows that its surface is covered with mountains and circular depressions, many of great size, which are called craters.
The earth is one of the eight planets which revolve around the sun in nearly circular orbits. ' Three of them are smaller than the earth and four are larger. The smallest, Mercury, has a volume about one twentieth that of the earth, and the largest, Jupiter, has a volume about one thousand times that of the earth. The great sim, whose mass is seven himdred times that of all of the planets combined, holds them in their orbits and lights and warms them with its abundant rays. Those nearest the sun are heated much more than the earth, but remote Neptune gets only one nine-hundredth as much light and heat per unit area as is received by the earth. Some of the planets have no moons and others have several. The conditions on one or two of them seem to be perhaps favorable for the development of life, while the others cer- tainly cannot be the abode of such life as flourishes on the earth.
In addition to the planets, over eight hundred small planets, or planetoids, and a great number of comets circu- late around the sun in obedience to the same law of gravita- tion. The orbits of nearly all the small planets lie between the orbits of Mars and Jupiter ; the orbits of the comets are generally very elongated and are unrelated to the other members of the system. The phenomena presented by the comets, for example the behavior of their tails, raise many interesting and puzzling questions.
The dominant member of the solar system is the sun. Its volume is more than a million times that of the earth, its temperature is far higher than any that can be produced on the earth, even in the most efficient electrical furnaces, and its surface is disturbed by the most violent storms. Often masses of this highly heated material, in volumes greater than the whole earth, move along or spout up from its surface at the rate of several hundreds miles a minute.
22 AN INTRODUCTION TO ASTRONOMY [ch. i, 8
The spectroscope shows that the sun contains many of the elements, particularly the metals, of which the earth is com- posed. The consideration of the possible sources of the sun's heat leads to the conclusion that it has supplied the earth with radiant energy for many milUons of years, and that the supply will not fail for at least a number of milUon years in the future.
The stars that seem to fill the. sky on a clear night are Sims, many of which are much larger and more brilliant than our own sun. They appear to be relatively faint points of light because of their enormous distances from us. The nearest of them is so remote that more than four years are required for its hght to come to the solar system, though Ught travels at the rate of 186,330 miles per second; and others, still within the range of large telescopes, are certainly a thousand times more distant. At these vast distances such a tiny object as the earth would be entirely invisible even though astronomers possessed telescopes ten thousand times as powerful as those now in use. Sometimes stars appear to be close together, as in the case of the Pleiades, but their apparent proximity is due to their immense distances from the observer. There are doubtless regions of space from which the sun would seem to be a small star forming a close group with a number of others. There are visible to the unaided eye in all the sky only about 5000 stars, but the great photographic telescopes with which modem observatories are equipped show several hundreds of millions of them. It might be supposed that telescopes with twice the light-gathering power would show proportionately more stars, and so on indefinitely, but this is certainly not true, for there is evidence that points to the conclusion that they do not extend indefinitely, at least with the frequency with which they occur in the region around the sun. The visible stars are not uniforinly scattered throughout the space which they occupy, but form a great disk-like aggregation lying in the plane of the Milky Way.
CH. I, 8] PRELIMINARY CONSIDERATIONS 23
Many stars, instead of being single isolated masses, like the sun, are found on examination with highly magnifjdng telescopes to consist of two suns revolving around their common center of gravity. In most cases the distances between the two members of a double star is several times as great as the distance from the earth to the sun. The existence of double stars which may be much closer together than those which are visible through telescopes has also been shown by means of instruments called spectroscopes. It has been found that a considerable fraction, probably one fourth, of all the nearer stars are double stars. There are also triple and quadruple stars; and in some cases thousands of suns, all invisible to the unaided eye, occupy a part of the sky apparently smaller than the moon. Even in such cases the distances between the stars are enormous, and such clusters, as they are called, constitute larger and more wonderful aggregations of matter than any one ever dreamed existed before they were revealed by modern instruments.
While the sun is the center around which the planets and comets revolve, it is not fixed with respect to the other stars. Observations with both the telescope and the spec- troscope prove that it is moving, with respect to the brighter stars, approximately in the direction of the brilliant Vega in the constellation Lyra. It is found by use of the spectro- scope that the rate of motion is about 400,000,000 miles per year. The other stars are also in motion with an average velocity of about 600,000,000 miles per year, though some of them move much more slowly than this and some of them many times faster. One might think that the great speed of the sun would in a century or two so change its relations to the stars that the appearance of the sky would be entirely altered. But the stars are so remote that in comparison the distance traveled by the sun in a year is neghgible. When those who built the pyramids turned their eyes to the sky at night they saw the stars grouped in constellations almost
24 AN INTRODUCTION TO ASTRONOMY [ch. i, 8
exactly as they are seen at present. During the time cov- ered by observations accurate enough to show the motion of the sun it has moved sensibly in a straight line, though in the course of time the direction of its path will doubtless be changed by the attractions of the other stars. Similarly, the other stars are moving in sensibly straight lines in every direction, but not altogether at random, for it has been found that there is a general tendency for them to move in two or more roughly parallel streams.
In addition to learning what the imiverse is at present, one of the most important and interesting objects of astron- omy is to find out through what great series of changes it has gone in its past evolution, and what will take place in it in the future. As a special problem, the astronomer tries to discover how the earth originated, how long it has been in existence, particularly in a state adapted to the abode of life, and what reasonably may be expected for the future. These great problems of cosmogony have been of deep inter- est to mankind from the dawn of civilization ; with increasing knowledge of the wonders of the universe and of the laws by which alone such questions can be answered, they have become more and more absorbingly attractive.
I. QUESTIONS
1. Enumerate as many ways as possible in which science is beneficial to men.
2. What is the fmidamental basis on which science rests, and what are its chief characteristics ?
3. What is induction ? Give examples. Can a science be de- veloped without inductions ? Are inductions always true ?
4. What is deduction? Give examples. Can a science be de- veloped without deductions ? Are deductions always true ?
5. In what respects may science be imperfect ? How may its im- perfections be most largely eUminated ? Are any human activities perfect ?
6. Name some superstition and show in what respects It differs from scientific conclusions.
7. Why did science originate in astronomy ?
CH. I, 8] PRELIMINARY CONSIDERATIONS 25
8. Are conclusions in astronomy firmly established, as they are in other sciences ? . ■
9. In what fundamental respects do scientific laws differ from civil laws ?
10. What advantages may be derived from a preliminary outUne of the scope of astronomy ? Would they hold in the case of a sub- ject not a science ?
11. What questions respecting the earth are properly regarded as belonging to astronomy? To what other 'sciences do they re- spectively belong ? Is there any science which has no common ground with some other science ?
12. What arts are used in astronomy? Does astronomy con- tribute to any art ?
13. What references to astronomy in the sacred or classical htera- tures do you know ?
14. Has astronomy exerted any influence on philosophy and rehgion ? Have they modified astronomy ?
CHAPTER II THE EARTH
I. The Shape of the Eakth
9. Astronomical Problems respecting the Earth. — The
earth is one of the objects belonging to the field of astronom- ical investigations. In the consideration of it astronomy has its closest contact with some pf the other sciences, par- ticularly with geology and meteorology. Those problems respecting the earth that can be solved for other planets also, or that are essential for the investigation of other astronom- ical questions, are properly considered as belonging to the field of astronomy.
The astronomical problems respecting the earth can be divided into two general classes. The first class consists of those which can be treated, at least to a large extent, with- out regarding the earth as a member of a family of planets or considering its relations to them and the sun. Such prob- lems are its shape and size, its mass, its density, its interior temperature and rigidity, and the constitution, mass, height, and effects of its atmosphere. These problems will be treated in this chapter. The second class consists of the problems involved in the relations of the earth to other bodies, partic- ularly its rotation, revolution around the sun, and the con- sequences of these motions. The treatment of these prob- lems will be reserved for the next chapter.
It would be an easy matter simply to state the astronom- ical facts respecting the earth, but in science it is necessary not only to say what things are true but also to give the reasons for believing that they are true. Therefore one or more proofs will be given for the conclusions astronomers have reached respecting the earth. As a matter of logic
26
CH. II, 10]
THE EARTH
27
one complete proof is sufficient, but it must be remembered that a scientific doctrine consists of, and rests on, a great number of theories whose truth may be more or less in ques- tion, and consequently a number of proofs is always desir- able. If they agree, their agreement confirms belief in the accuracy of all of them. It will not be regarded as a burden to follow carefully these proofs ; in fact, one who has arrived at a mature stage of intellectual development instinctively demands the reasons we have for believing that our conclu- sions are sound.
10. The Simplest and most Conclusive Proof of the Earth's Sphericity.^ — Among the proofs that the earth is
round, the simplest and most concliisive is that the plane of
the horizon, or the direction of the plumb line, changes by an
angle which is direcRy proportional
to-tlhe distance the observer travels
along the surface of the earth,
whatever the direction and distance
of travel.
It will be shown first that if
the earth were a true sphere the
statement would be true. For
simphcity, suppose the observer
travels along a meridian. If the
statement is true for this case,
it will be true for all others,
because a sphere has the same
curvature in every direction.
Suppose the observer starts from
Oi, Fig. 5, and travels northward
to O2. The length of the arc
O1O2 is proportional to the angle
a which it subtends at the center of the sphere. The planes
of the horizon of Oi and O2 are respectively OJIi and OoBi.
' The earth- is not exactly^ound, but the departure from sphericity will be neglected for the moment.
Fig. 5. — The change in the di- rection ■ of the plumb line is proportional to the distance traveled along the surface of the earth.
28 AN INTRODUCTION TO ASTRONOMY [ch. ii, 10
These lines are respectively perpendicular to COi and CO2. Therefore the angle between them equals the angle a. That is, the distance traveled is proportional to the change of direction of the plane of the horizon.
The plumb lines at Oi and O2 are respectively OiZi and O2Z2, and the angle between these hues is a. Hence the dis- tance traveled is proportional to the change in the direction of the plumb Une.
It will be shown now that if the surface of the earth were not a true sphere the change in the direction of the plane of the horizon would not be proportional to the distance traveled
on the surface. Suppose Fig. 6 represents a plane section through the non- spherical earth along whose surface the ob- server travels. Since the earth is not a sphere, the curvature of its surface will be different at differ- ent places. Suppose that
Fig. 6. -If the earth were not spherical, ^lO^ IS One of the flatter equal angles would be subtended by arcs regions and O3O4 is one of different lengths. r .1
01 the more convex ones. In the neighborhood of O1O2 the direction of the plumb Une changes slowly, while in the neighborhood of O3O4 its direc- tion changes more rapidly. The large arc O1O2 subtends an angle at Ci made by the respective perpendiculars to the surface which exactly equals the angle at C3 subtended by the smaller arc O3O4. Therefore in this case the change in direction of the plumb line is not proportional to the dis- tance traveled, for the same angular change corresponds to two different distances. The same result is true for the plahe of the horizon because it is always perpendicular to the plumb line.
Since the conditions of the statement would be satisfied
CH. n, lO]
THE EARTH
29
in case the earth were spherical, and only in case it were spherical, the next question is what the observations show. Except for irregularities of the surface, which are not under consideration here, and the oblateness, which will be dis- cussed in Art. 12, the observations prove absolutely that the change in direction of the plumb line is proportional to the arc traversed.
Two practical problems are involved in carrying out the proof which has just been described. The first is that of measuring the distance between two points along the sur-
FiG. 7. -
- The base line A\Ai is measured directly and the other distances are obtained by triangulation.
face of the earth, and the second is that of determining the change in the direction of the plumb line. The first is a refined problem of surveying; the second is solved by observations of the stars.
All long distances on the surface of the earth are deter- mined by a process known as triangulation. It is much more convenient than direct measurement and also much more accurate. A fairly level stretch of country, Ai and A2 in Fig. 7, a few miles long is selected, and the distance between the two points, which must be visible from each other, is measured with the greatest possible accuracy.
30 AN INTRODUCTION TO ASTRONOMY [ch. ii, 10
This line is called the base line. Then a point A3 is taken which can be seen from both Ai and A^. A telescope is set up at Ai and pointed at A2. It has a circle parallel to the surface of the earth on which the degrees are;.marked. The position of the telescope with respect to this circle is recorded. Then the telescope is turned until it points toward A3. The difference of its position with respect to the circle when pointed at A2 and at A3 is the angle A2A1A3. Similarly, the telescope is set up at A2 and the angle A1A2A3 is meas- ured. Then in the triangle A1A2A3 two angles and the in- cluded side are known. By plane geometry, two triangles that have two angles and the included side of one respectively equal to two angles and the included side of the other are exactly alike in size and shape. This simply means that when two angles and the included side of the triangle are given, the triangle is uniquely defined. The remaining parts can be computed by trigonometry. In the present case suppose the distance A2A3 is computed.
Now suppose a fourth point A 4 is taken so that it is visible from both A2 and A3. Then, after the angles at A2 and A3 in the triangle A2A3A4 have been measured, the line A3A4 can be computed. This process evidently can be con- tinued, step by step, to any desired distance.
Suppose Ai is regarded as the original point from which measurements are to be made. Not only have various dis- tances been, determined, but also their directions with respect to the north-south line are known. Consequently, it is known how far north and how far east A 2 is from Ai. The next step gives how far south and how far east A3 is from A2. By combining the two results it is known how far south and how far east A3 is from Ai, and so on for succeeding points.
The convenience in triangulation results partly from the long distances that can be measured, especially in rough country. It is sometimes advisable to go to the trouble of erecting towers in order to make it possible to use stations separated by long distances. The accuracy arises, at least
CH. 11, 12] THE EARTH 31
in part, from the fact that the angles are measured by in- struments which magnify them. The fact that the stations are not all on the same level, and the curvature of the earth, introduce little difBculties in the computations that must be carefully overcome.
The direction of the plumb Una at the station Ai, for example, is determined by noting the point among the stars at which it points. The plumb line at A2 will point to a dififerent place among the stars. The difference in the two places among the stars gives the difference in the directions of the plumb lines at the two stations. The stars apparently move across the sky from' east to west during the night and are not in the same positions at the same time of the day on different nights. Hence, there are here also certain cir- cumstances to which careful attention must be given in order to get accurate results.
11. Other Proofs of the Earth's Sphericity. — There are many reasons given for believing that the earth is not a plane, and that it is, indeed, some sort of a convex figure ; but most of them do not prove that it is actually spherical. It will be sufficient to mention them.
(a) The earth has been circumnavigated, but so far as this fact alone is concerned it might be the shape of a cu- cumber. (6) Vessels disappear below the horizon hulls first and masts last, but this only proves the convexity of the surface, (c) The horizon appears to be a circle when viewed from an elevation above the surface of the water, This is theoretically good but observationally it is not very exact. (d) The shadow of the earth on the moon at the time of a lunar eclipse is always an arc of a circle, but this proof is very inconclusive, in spite of the fact that it is often men- tioned, because the shadow has no very definite edge and its radius is large compared to that of the moon.
12. Proof of the Oblateness of the Earth by Arcs of Latitude. — The latitude of a place on the earth is deter- mined by observations of the direction of the plumb line
32 AN INTRODUCTION TO ASTRONOMY [ch. ii, 12
with respect to the stars. This is the reason that a sea cap- tain refers to the heavenly bodies in order to find his loca- tion on the ocean. It is found by actual observations of the stars and measurements of arcs that the length of a degree of arc is longer the farther it is from the earth's equator. This proves that the earth is less curved at the poles than it is at the equator. A body which is thus flattened at the poles and bulged, at the equator is called oblate.
In order to see that in the case of an oblate body a degree of latitude is longer near the poles than it is at the equator, consider Fig. 8. In this figure E represents a plane section
of the body through its poles. The curvature at the equator is the same as the curvature of the circle Ci, and a degree of latitiide < on S at its equator equals a degree of latitude on Ci. The curvature of E at its pole is the same as the curvature of the circle d, and a degree of lati-
""-- .--' tude on E at its pole equals a
I^G. 8. — The length of a degree degree of latitude on C2. Since
t^:^d"ii^t^:ttt1he';or" C^ i« greater than C, a degree
of latitude near the pole of the oblate body is greater than a degree of latitude near its equator.
A false argument is sometimes made which leads to the opposite conclusion. Lines are drawn from the center of the oblate body dividing the quadrant into a number of equal angles. Then it is observed that the arc intercepted between the two fines nearest the equator is longer than that intercepted between the two lines nearest the pole. The error of this argument lies in the fact that, with the exception of those drawn to the equator and poles, these fines are not perpendicular to the surface. Figure 9 shows an oblate body with a number of lines drajwn perpendicular
CH. 11, 13] THE EARTH 33
to its surface. Instead of their all passing through the center of the body, they are tangent to the curve AB. The line AE equals the radius of a circle having the same curvature as the oblate body at E, and BP is the radius of the circle having the curva- ture at P.
13. Size and Shape of the Earth. — The size and shape of the earth can both be determined „ „ „ ,. ,
. r *i°- »• — Perpendiculars to the surface of
trom measurements ot an oblate body, showing that equal arcs
arcs. If the earth were subtend largest angles at its equator and
' , smallest at its poles.
sphencal, a degree of arc
would have the same, length everywhere on its surface, and its circumference would be 360 times the length of one de- gree. Since the earth is oblate, the matter is not quite so simple. But from the lengths of arcs in different latitudes both the size and the shape of the earth can be computed.
It is sufficiently accurate for ordinary purposes to state that the diameter of the earth is about 8000 miles, and that the difference between the equatorial and polar diameters is 27 miles.
The dimensions of the earth have been computed with great accuracy by Hayford, who found for the equatorial diameter 7926.57 miles, and for the polar diameter 7899.98 miles. The error in these results cannot exceed a thousand feet. The equatorial circumference is 24,901 .7 miles, and the length of one degree of longitude at the equator is 69.17 miles. The lengths of degrees of latitude at the equator and at the poles are respectively 69.40 and 68.71 miles. The total area of the earth is about 196,400,000 square miles. The volume of the earth is equal to the volume of a sphere whose radius is 3958.9 miles.
D
34 AN INTRODUCTION TO ASTRONOMY [ch. ii, 14
14. Newton's Proof of the Oblateness of the Earth. —
The first proof that the earth is oblate was due to Newton. He based his demonstration on the laws of motion, \the law of gravitation, and the rotation-of the earth. It is therefore much more compUcated than that depending on the lengths of degrees of latitude, which is purely geometrical. It has the advantage, however, of not requiring any measurements of arcs.
Suppose the earth, Fig. 10, rotates around the axis PP'. Imagine that a tube filled with water exists reaching from
the pole P to the center C, and then to the sur- face on the equator at Q. The water in this tube exerts a pressure toward the center because of the attraction of the earth for it. Consider a unit voliune in the part CP at any distance D from the center ; the pressure it exerts toward the
Fig. 10. — Because of the earth's rotation Center equals the earth's
around PP' the column CQ must be attraction for it because
longer than PC. .
it is subject to no other forces. Suppose for the moment that the earth is a sphere, as it would be if it were not rotating on its axis, and con- sider a unit volume in the part CQ at the distance D from the center. Because of the symmetry of the sphere it will be subject to an attraction equal to that on the corre- sponding unit in CP. But, in addition to the earth's at- traction, this mass of water is subject to the centrifugal force due to the earth's rotation, which to some extent counter- balances the attraction. Therefore, the pressure it exerts toward the center is less than that exerted by the corre- sponding unit in CP- If the earth were spherical, all units
CH. II, 15] THE EARTH 35
in the two columns could be paired in this way. The result would be that the pressure exerted by PC would be greater than that exerted by QC ; but such a condition would not be one of equihbrium, and water would flow out of the mouth of the tube from the center to the equator. In order that the two columns of water shall be in equihbrium the equatorial column must be longer than the polar.
Newton computed the amount RQthy which the one tube must be longer than the other in order that for a body hav- ing the mass, dimensions, and rate of rotation of the earth, there should be equihbrium. This gave him the oblate- ness of the earth. In spite of the fact that his data were not very exact, he obtained results which agree very well with those furnished by modern measurements of arcs.
The objection at once arises that the tubes did not actually exist and that they could not possibly be constructed, and therefore that the conclusion was as insecure as those usually are which rest on imaginary conditions. But the fears aroused by these objections are dissipated by a little more consideration of the subject. It is not necessary that the tubes should run in straight lines from the surface to the center in order that the principle should apply. They might bend in any manner and the results would be the same, just as the level to which the water rises in the spout of a teakettle does not depend on its shape. Suppose the tubes are deformed into a single one connecting P and Q along the surface of the earth. The principles still hold ; but the ocean connection of pole and equator may be considered as being a tube. Hence the earth must be oblate or the ocean would flow from the poles toward the equator.
15. Pendulum Proof of the Oblateness of the Earth. — It seems strange at flrst that the shape of the earth can be determined by means of the pendulum. Evidently the method cannot rest on such simple geometrical principles as were sufl&cient in using the lengths of arcs. It will be found that it involves the laws of motion and the law of gravitation.
36 AN INTRODUCTION TO ASTRONOMY [ch. ii, 15
The time of oscillation of a pendulum depends on the in- tensity of the force acting on the bob and on the distance from the point of support to the bob. It is shown in ana- lytic mechanics that the formula for a complete oscillation is
< = 27rV|7^,
where t is the time, x = 3.1416, I is the length of the pen- dulum, and g is the resultant acceleration ' produced by all the forces to which the pendulum is subject. If I is deter- mined by measurement and t is found by observations, the resultant acceleration is given by
g =
Consequently, the pendulum furnishes a means of finding the gravity g at any place.
In order to treat the problem of determining the shape of the earth from a knowledge of g at various places on its surface, suppose first that it is a homogeneous sphere. If this were its shape, its attraction would be equal for all points on its surface. But the gravity g would not be the same at all places, because it is the resultant of the earth's attrac- tion and the centrifugal acceleration due to the earth's rotation. The gravity g would be the greatest at the poles, where there is no centrifugal acceleration, and least at the equator, where the attraction is exactly opposed by the centrifugal acceleration. Moreover, the value of g would vary from the poles to the equator in a perfectly definite manner which could easily be determined from theoretical considerations.
Now suppose the earth is oblate. It can be shown mathe- matically that the attraction of an oblate body for a particle at its pole is greater than that of a sphere of equal volume and density for a particle on its surface, and that at its equator the attraction is less. Therefore at the pole, where
' Force equals mass times acceleration. On a large pendulum the force of gravity is greater but the acceleration is the same.
OH. II, 15] THE EARTH 37
there is no centrifugal acceleration, g is greater on an oblate body than it is on an equal sphere. On the other hand, at the equator g is less on the oblate body than on the sphere both because the attraction of the former is less, and also because its equator is farther from its axis so that the cen- trifugal acceleration is greater. That is, the manner in which g varies from pole to equator depends upon the oblate- ness of the earth, and it can be computed when the oblate- ness is given. Conversely, when g has been foimd by ex- periment, the shape of the earth can be computed.
Very extensive determinations of g by means of the pen- dulum, taken in connection with the mathematical theory, not only prove that the earth is oblate, but give a degree of flattening agreeing closely with that obtained from the measurement of arcs.
The question arises why g is determined by means of the pendulum. Its variations cannot be found by using balance scales, because the forces on both the body to be weighed and the counter weights vary in the same proportion. However, the variations in g can be determined with some approxima- tion by employing the spring balance. The choice between the spring balance and the pei^dulum is to be settled on the basis of convenience and accuracy. It is obvious that spring balances are very convenient, but they are not very accurate. On the other hand, the pendulum is capable of furnishing the variation of g with almost indefinite precision by the period in which it vibrates. Suppose the pendulum is moved from one place to another where g differs by one hundred-thousandth of its value. This small difference could not be detected by the use of spring balances, however many times the attempt might be made. It follows from the formula that the time of a swing of the pendulum would be changed by about one two-hundred-thousandth of its value. If the time of a complete oscillation were a second, for ex- ample, the difference could not be detected in a second ; but the deviation for the following second would be equal to
38 AN INTRODUCTION TO ASTRONOMY [ch. ii, 15
that in the first, and the difference would be doubled. The effect would accumulate, second after second, and in a day of 86,400 seconds it would amount to nearly one half of a second, a quantity which is easily measured. In ten days the difference would amount to about 4.3 seconds. The important point in the pendulum method is that the effects of the quantities to be measured accumulate until they be- come observable.
16. The Theoretical Shape of the Earth. — The oblateness of the earth is not an accident ; its shape depends on its size, mass, distribution of density, and rate of rotation. If
Fig. 11. ^ Oblate spheroid. Fig. 12. — Prolate spheroid.
it were homogeneous, its shape could be theoretically deter- mined without great difficulty. It has been found from mathematical discussions that if a homogeneous fluid body is slowly rotating it may have either of two forms of equi- librium, one of which is nearly spherical while the other is very much flattened like a discus. These figures are not simply oblate, but they are figures known as spheroids. A spheroid is a solid generated by the rotation of an ellipse (Art. 53) about one of its diameters. Figure 11 is an oblate spheroid generated by the rotation of the ellipse PQP'Q' about its shortest diameter PP'. Its equator is its largest circumference. Figure 12 is a prolate spheroid generated by the rotation. of the ellipse PQP'Q' about its longest diam- eter PP'. The equator of this figure is its smallest cir- cumference. The oblate and prolate spheroids are funda- mentally different in shape.
CH. II, 17] THE EARTH 39
Of the two oblate spheroids which theory shows are figures of equilibrium for slow rotation, that which is the more nearly spherical is stable, while the other is unstable. That is, if the former were disturbed a little, it would retake its spheroidal form, while if the latter were deformed a Uttle, it would take an entirely different shape, or might even break all to pieces. In spite of the fact that the earth is neither a fluid nor homogeneous, its shape is ahnost exactly that of the more nearly spherical oblate spheroid corresponding to its density and rate of rotation. This fact might tempt one to the conclusion that it was formerly in a fluid state. But this conclusion is not necessarily sound, because, in such an enormous body, the strains which would result from appreciable departure from the figure of equi- librium would be so great that they could not be withstood by the strongest material known. Besides this, if the con- ditions for equilibrium were not exactly satisfied by the solid parts of the earth, the water and atmosphere would move and make compensation.
The sun, moon, and planets are bodies whose forms can Hkewise be compared with the results furnished by theory. Their figures agree closely with the theoretical forms. The only appreciable disagreements are in the case of Jupiter and Saturn, both of which are more nearly spherical than the corresponding homogeneous bodies would be. The reason for this is that these planets are very rare in their outer parts and relatively dense at their centers. It is probable that they are even more stable than the correspond- ing homogeneous figures.
17. Different Kinds of Latitude. — It was seen in Art. 12 that perpendiculars to the water-level surface of the earth, except on the equator and at the poles, do not pass through the center of the earth. This leads to the defini- tion of different kinds of latitude.
The geometrically simplest latitude is that defined by a line from the center of the earth to the point on its surface
40
AN INTRODUCTION TO ASTRONOMY [ch. ii, 17
occupied by the observer. Thus, in Fig. 13, PC is the earth's axis of rotation, QC is in the plane of its equator, and 0 is the position of the observer. The angle I is called the geo- centric latitude.
The observer at 0 cannot see the center of the earth and cannot locate it by any kind of observation made at his station alone. Consequently, he cannot directly determine I.
All he has is the perpen^ dicular to the surface de- fined by his plumb Une which strikes the line CQ at A. The angle Ix be- tween this line and CQ is his astronomical latitude. The difference between the geocentric and astro- nomical latitudes varies from zero at the poles and equator to about 11' in latitude 45°. Sometimes the plumb line has an abnormal direction because of the attractions of neighboring mountains, or because of local excesses or deficiencies of matter under the surface. The astronomical latitude, when corrected for these anomalies, is called the geographical latitude. The astro- nomical and geographical latitudes rarely differ by more than a few seconds of arc.
18. Historical Sketch of Measurements of the Earth. — While the earth was generally supposed to be flat down to the time of Columbus, yet there were several Greek philoso- phers who believed that it was a sphere. The earliest phi- losopher who is known certainly to have mai'ntained that the earth is spherical was Pythagoras, author -of the famous Pythagorean proposition of geometry, who lived from about 569 to 490 B.C. He was followed in this conclusion, among others, by Eudoxus (407-356 b.c), by Aristotle (384-322
Fig. 13.
-Geocentric and astronomical latitudes.
CH. It, 18] THE EARTH 41
B.C.), the most famous philosopher of antiquity if not of all time, and by Aristarchus of Samos (310-250 B.C.). But none of these men seems to have had so clear convictions as Eratosthenes (275-194 b.c), who not only believed in the earth's sphericity but undertook to determine its dimensions. He had noticed that the altitude of the pole star was less when he was in Egypt than when he was farther north in Greece, and he correctly interpreted this as meaning that in traveling northward he journeyed around the curved sur- faqe of the earth. By very crude means he undertook to measure the length of a degree in Egypt, and in spite of the fact that he had neither acburate instruments for obtaining the distances on the surface of the earth, nor telescopes with which to determine the changes of the direction of the plumb line with respect to the stars, he secured results that were not surpassed in accuracy until less than 300 years ago.
After the dechne of the Greek civilization and science, no progress was made in proving the earth is spherical until the voyage of Columbus in 1492. His ideas regarding the size of the earth were very erroneous, as is shown by the fact that he supposed lie had reached India by crossing the Atlan- tic Ocean. The great explorations and geographical dis- coveries that quickly followed the voyages of Columbus con- vinced men that the earth is at least globular and gave them some idea of its dimensions.
There were no serious attempts made to obtain accurate knowledge of the shape and size of the earth until about the middle of the seventeenth century. The first results of any considerable degree of accuracy were obtained in 1671 by Picard from a measurement of an arc in France.
In spite of the fact that Newton proved in 1686 that the earth is oblate, the conclusion was by no means universally accepted. Imperfections in the measures of the French led Cassini to maintain until about 1745 that the earth is pro- late. But the French were, taking hold of the question in
42 AN INTRODUCTION TO ASTRONOMY [ch. ii, 18
earnest and they finally agreed with the conclusion of New- ton. They extended the arc that Picard had started from the Pyrenees to Dunkirk, an angular distance of 9°. The results were pubhshed in 1720. They sent an expedition to Peru, on the equator, in 1735, under Bouguer, Condamine, and Godin. By 1745 these men had measured an arc of 3°. In the meantime an expedition to Lapland, near the Arctic circle, had measured an arc of 1°. On comparing these measurements it was found that a degree of latitude is greater the farther it is from the equator.
In the last century all the principal governments of the world have carried out very extensive and accurate surveys of their possessions. The English have not only triangulated the British Isles but they have done an enormous amount of work in India and Africa. The Coast and Geodetic Survey in the United States has triangulated with unsurpassed pre- cision a great part of the country. They have run a level from the Atlantic Ocean to the Pacific. The names most often encountered in this connection are Clarke of England, Helmert of Germany, and Hayford of the United States. Hayford has taken up an idea first thrown out by the Eng- lish in connection with their work in India along the borders of the Himalaya Mountains, . and by using an enormous amount of observational data and making appalling com- putations he has placed it on a firm basis. The observations in India showed that under the Himalaya Mountains the earth is not so dense as it is under the plains to the south. Hayford has proved that the corresponding thing is true in the United States, even in the case of very moderate eleva- tions and depressions. Moreover, deficiency in density under the elevated places is just enough to offset the eleva- tions, so that the total weight of the material along every radius from the surface of the earth to its center is almost exactly the same. This theory is known as the theory of isostasy, and the earth is said to be in almost perfect iso- static adjustment.
CH. ii.aQ] THE EARTH 43
II. QUESTIONS
1. In order to prove the sphericity of the earth by measurement of arcs, would it be sufflcient to measure only along meridians? (Consider the anchor ring.) *
2. Do the errors in triangulation accumulate with the length of the distance measured ? Do tjie errors in the astronomical deter- mination of the angular length of the arc increase with its length ?
3. How accurately must a base line of five miles be measured in order that it may not introduce an error in the determination of the earth's circumference of more than 1000 feet ?
4. Which of the reasons given in Art. 11 actually prove, so far as they go, that the earth is spherical? What other reasons are there for believing it is spherical ?
5. The acceleration g in mid-latitudes is about 32.2 feet per second ; how long would a pendulum have to be to swing in 1, 2, 3, 4 seconds ?
6. Draw to scale a meridian section of a figure having the earth's oblateness.
7. Newton s proof of the earth's oblateness depends on the knowledge that the earth rotates ; what proofs of it do not depend upon this knowledge ?
8. Suppose time can be measured with an error not exceeding one tenth of a second ; how aecm-ately can g be determined by the pendulum in 10 days ?
9. Suppose the soMd part of the earth were spherical and per- fectly rigid ; what would be the distribution of land and water over the surface ?
10. Is the astronomical latitude greater than, or equal to, the geocentric latitude for all points on the earth's surface ?
11. What distance on the earth's surface corresponds to a degree of arc, a minute of are, a second of are ?
12. Which of the proofs of the earth's sphericity depend upon modern discoveries and measurements ?
II. The Mass of the Earth and the Condition of ITS Interior 19. The Principle by which Mass is Determined. — It is important to understand clearly the principles which are at the foundation of any subject in which one may be interested, and this appUes in the present problem. The ordinary method of determining the mass of a body is to weigh it.
44 AN INTRODUCTION TO ASTRONOMY [ch. ii, 19
That is the way in which the quantity of most commodities, such as coal or ice or sugar, is found. The reason a body has weight at the surface of the earth is that the earth attracts it. It will be seen later (Art. 40) that the body attracts the earth equally in the opfJosite direction. Con- sequently, the real property of a body by which its inass is determined is its attraction for some other body. The underlying principle is that the mass of a body is proportional to the attraction which it has for another body.
Now consider the problem of finding the mass of the earth, which must be solved by considering its attraction for some other body. Its attraction for any given mass, for example, a cubic inch of iron, can easily be measured. But this does not give the mass of the earth compared to the cubic inch of iron. It is necessary to compare the attrac- tion of the earth for the iron with the attraction of some other fully known body, as a lead ball of given size, for the same unit of iron. Since the amount of attraction of one body for another depends upon their distance apart, it is neces- sary to measure the distance from the lead ball to the at- tracted body, and also to know the distance of the attracted body from the center of the earth. For this reason the mass of the earth could not be found until after its dimensions had been ascertained. By comparing the attractions of the earth and the lead ball for the attracted body, and making proper adjustments for the distances of their respective centers from it, the number of times the earth exceeds the lead ball in mass can be determined.
Not only is the mass of the earth computed from its at- traction, but the same principle is the basis for determining the mass of every other celestial body. The masses of those planets that have satellites are easily found from their attractions for their respective satelUtes, and when two stars revolve around each other in known orbits their masses are defined by their mutual attractions. There is no means of determining the mass of a single star.
CH. 11, 20] THE EARTH 45
20. The Mass and Density of the Earth. — By applica- tions (Arts. 21, 22) of the principle in Art. 19 the mass of the earth has been found. If it were weighed a small quantity at a time at the surface, its total weight in tons would be 6 X W^, or 6 followed by 21 ciphers. This makes no appeal to the imagination because the numbers are so extremely far beyond all experience. A much btetter method is to give its density, which is obtained by divid- ing its mass by its volume. With water at its greatest density as a standard, the average density of the earth ^is 5.53.
The average density of the earth to the depth of a mile or two is in the neighborhood of 2.75. Therefore there are much denser materials in the earth's interior ; their greater density may be due either to their composition or to the great pressure to which they are subject. The density of quartz (sand) is 2.75, Umestone 3.2, cast iron 7.1, steel 7.8, lead 11.3, mercury 13.6, gold 19.3, and platinum 21.5. It follows that no considerable part of the earth can be com- posed of such heavy substances as mercury, gold, and plati- num, but, so far as these considerations bear on the question, it might be largely iron.
The distribution of density in the earth was worked out over 100 years ago by Laplace on the basis of a certain as- sumption regarding the compressibility of the matter of which it is composed. The results of this computation have been compared with all the phenomena on which the disposition of the mass of the earth has an influence, and the results have been very satisfactory. Hence, it is supposed that this law represents approximately the way the density of the earth increases from its surface to its center. Accord- ing to this law, taking the density of the surface as 2.72, the densities at depths of 1000, 2000, 3000 miles, and the center of the earth are respectively 5.62, 8.30, 10.19, 10.87. At no depth is the average density so great as that of the heavier metals.
46
AN INTRODUCTION TO ASTRONOMY [ch. ii, 21
21. Determination of the Density of the Earth by Means of the Torsion Balance. — The whole difficulty in deter- mining the density of the earth is due to the fact that the attractions of masses of moderate dimensions are so feeble that they almost escape detection with the most sensitive apparatus. The problem from an experimental point of" view reduces to that of devising a means of measuring ex- tremely minute forces. It has been solved most successfully by the torsion balance.
The torsion balance consists essentially of two small balls, bb in Fig. 14, connected by a rod which is suspended from
Fig. 14. — The torsion balance.
the point 0 by a quartz fiber OA. If the apparatus is left for a considerable time in a sealed case so that it is not dis- turbed by air currents, it comes to rest. Suppose the balls bb are at rest and that the large balls BB are carefully brought near them on opposite sides of the connecting rod, as shown in the figure. They exert sUght attractions for the small balls and gradually move them against the feeble resistance of the quartz fiber to torsion (twisting) to the pbsition b'b". The resistance of the quartz fiber becomes greater the more it is twisted, and finally exactly balances the attraction of the large balls. The forces involved are so small that several hours may be required for the balls to reach their final positions of rest. But they will finally be reached and the angle through which the rod has been turned can be recorded.
CH. II, 21] THE EAKTH 47
The next problem is to determine from the deflection which the large balls have produced how great the force is which they have exerted. This would be a simple matter if it were known how much resistance the quartz fiber offers to twisting, but the resistance is so exceedingly small that it cannot be directly determined. However, it can be found by a very interesting indirect method.
Suppose the large balls are removed and that the rod connecting the small balls is twisted a httle out of its posi- tion of equihbrium. It will then turn back because of the resistance offered to twisting by the quartz fiber, and will rotate past the position of equilibrium almost as far as it was originally displaced in the opposite direction. Then it will return and vibrate back and forth until friction de- stroys its motion. It is evident that the characteristics of the oscillations are much like those of a vibrating pendulum. The formula connecting the various quantities involved is
t = 2 xVi/f,
where t is the time of a complete oscillation of the rod joining h and h, I is the distance from A to h, and / is the resistance of torsion. This equation differs from that for the pendulum. Art. 15, only in that g has been replaced by/. Now I is measured, t is observed, and / is computed from the equation with great exactness however small it may be.
Now that / and g are known it is easy to compute the mass of the earth by means of the law of gravitation (Art. 146). Let E represent the mass of the earth, R its radius, 2 B the mass of the two large balls, and r the distances from BB to 66 respectively. Then, since gravitation is propor- tional to the attracting mass and inversely as the square of its distance from the attracted body, it follows that E^ 2B ^
In this proportion the only unknown is E, which can there- fore be computed.
48
AN INTRODUCTION TO ASTRONOMY [ch. ii, 22
22. Determination of the Density of the Earth by the Mountain Method. — The characteristic of the torsion balance is that it is very delicate and adapted to measuring very small forces ; the characteristic of the mountain method is that a very large mass is. employed, and the forces are larger. In the torsion balance the balls BB are brought near those suspended by the quartz fiber and are removted at will. A mountain cannot be moved, and the advantage of using a large mass is at least partly coimterbalanced by this disadvantage. The necessity for moving the attracting
body (in this case ^' ^ A the moimtain) is
obviated in a very ingenious manner.
For simpUcity let the oblateness of the earth be neg- lected in explaining the mountain method. In Fig. 15, C is the center of the earth, M is the mountain, and Oi and O2 are two stations on opposite sides of the moun- tain at which plmnb lines are suspended. If it were not for the attraction of the mountain they would hang in the directions OiC and O2C. The angle between these fines at C depends upon the distance between the stations Oi and Oi. The distance between these stations, even though they are on opposite sides of the moun- tain, can be obtained by triangulation. Then, since the size of the earth is known, the angle at C can be computed.
Fig. 15.
- The mountain method of determining the mass of the earth.
CH. II, 22] THE EARTH 49
But the attraction of the mountain for the plumb bobs causes tHe plumb lines to hang in the directions OiA and OiA. The directions of these lines with respect to the stars can easily be determined by observations, and the difference in their directions as thus determined is the angle at A.
What is desired is the deflections of the plumb line pro- duced by the attractions of the mountain. It follows from elementary geometry that the sum of the two small deflec- tions COiA and CO2A equals the angle A minus the angle C. Suppose, for simplicity, that the mountain is sym- metrical and that the deflections are equal. Then each one equals one half the difference of the angles A and C. There- fore the desired quantities have been found.
When the deflection has been found it is easy to obtain the relation of the force exerted by the mountain to that due to the earth. Let Fig. 16 represent the g, plumb line on a large scale. If it were not for the mountain it would hang in the direc- tion OiBi ; it actually hangs in the direction OiB'i. The earth's attraction is in the direc- tion OiBi, and that of the mountain is in the direction BiB'i. The two forces are in the same ratio as QiBi is to BiB\, for, by the law p^^ jg _ ^^^ of the composition of forces, only then would deflection of a the plumb line hang in the direction OiB'i. ^""
The problem of finding the mass of the earth compared to that of the mountain now proceeds just Uke that of find- ing the mass of the earth compared to the balls BB in the torsion-balance method. The mountain plays the r61e of the large balls. A mountain 5000 feet high and broad would cause nearly 800 times as much deflection as that produced by an iron ball a foot in diameter. The advantage of the large deflection is offset by not having very accurate means of measuring it, and also by the fact that it is neces- sary to determine the mass of a more or less irregular shaped mountain made up of materials which may lack much- of
50 AN INTRODUCTION TO ASTRONOMY [ch. ii, 22
being uniform in density. In spite of these drawbacks this method was the first one to give fairly accurate results.
23. Determination of the Density of the Earth by the Pendulum Method. — It was explained in Art. 15 that the pendulum furnishes a very accurate means of determining the force of gra'^ity. Its delicacy arises from the fact that in using it the effects of the changes in the forces accumulate indefinitely; no such favorable circumstances were present in the methods of the torsion balance and the mountain.
Suppose a pendulum has been swung at the surface of the earth so long that the period of its oscillation has been accu- rately determined. Then suppose it is taken at the same place down into a deep pit or mine. The force to which it is subject will be changed for three different reasons, (a) The pendulum will be nearer the axis of rotation of the earth and the centrifugal acceleration to which it is subject will be diminished. The relative change in gravity due to this cause can be accurately computed from the latitude of the position and the depth of the pit or mine. (6) The pendu- lum will be nearer the center of the earth, and, so far as this factor alone is concerned, the force to which it is subject will be increased. Moreover, the relative change due to this cause also can be computed, (c) The pendulum will be below a certain amount of material whose attraction will now be in the opposite direction. This cannot be computed directly because the amount of attraction due to a ton of matter, for example, is imknown. This is what is to be found out. But from the time of the oscillation of the pen- dulum at the bottom of the pit or mine the whole force to which it is subject can be computed. Then, on making cor- rection for the known changes (a) and (6), the unknown change (c) can be obtained simply by subtraction. From the amount of force exerted by the known mass above the pendulum, the density of the earth can be computed by essentially the same process as that employed in the case of the torsion-balance method and the mountain method.
CH. II, 24] THE EARTH 51
24. Temperature and Pressure in the Earth's Interior. —
There are many reasons for believing that the interior of the earth is very hot. For example, volcanic phenomena prove that at least in many localities the temperature is above the melting point of rock at a comparatively short distance below the earth's surface. Geysers and hot springs show that the interior of the earth is hot at many other places. Besides this, the temperature has been found to rise in deep mines at the rate of about one degree Fahrenheit for a de- scent of 100 feet, the amount depending somewhat on the locality.
Suppose the temperature should go on increasing at the rate of one degree for every hundred feet from the surface to the center of the earth. At a depth of ten miles it would be over 500 degrees, at 100 miles over 5000 degrees, at 1000 miles over 50,000 degrees, and at the center of the earth over 200,000 degrees. While there is no probabiUty that the rate of increase of temperature which prevails near the surface Seeps up to great depths, yet it is reason- ably certain that at a depth of a few hundred miles it is several thousand degrees. Since almost every substance melts at a temperature below 5000 degrees, it has been supposed until recent times that the interior of the earth, below the depth of 100 miles, is hquid.
But the great pressure to which matter in the interior of the earth is subject is a factor that caimot safely be neg- lected. A cyhnder one inch in cross section and 1728 inches, or 144 feet, in height has a volume of one cubic foot. If it is filled with water, the pressure on the bottom equals the weight of a cubic foot of water, or 62.5 pounds. The pressure per square inch on the bottom of the column 144 feet high having the density 2.75, or that of the earth's crust, is 172 pounds. The pressure per square inch at the depth of a mile is 6300 pounds, or 3 tons in round numbers. The pressure is approximately proportional to the depth for a co-nsiderable distance. Therefore, the pressure per square
52 AN INTRODUCTION TO ASTRONOMY [ch. ii, 24
inch at the depth of 100 miles is approximately 300 tons, and at 1000 miles it is 3000 tons. However, the pressure is not strictly proportional to the depth, and more refined means must be employed to find how great it is at the earth's center. Moreover, the pressure at great depths depends upon the distribution of mass in the earth. On the basis of the Laplacian law of density, which probably is a good approximation to the truth, the pressure per square inch at the center of the earth is 3,000,000 times the atmospheric pressure at the earth's surface, or 22,500 tons.
It is a familiar fact that pressure increases the boiling points of hquids. It has beeh found recently by experiment that pressure increases the melting points of solids. There- fore, in view of the enormous pressures at moderate depths in the earth, it is not safe to conclude that its interior is molten without further evidence. The question cannot be answered directly because, in the first place, there is no very exact means of determining the temperature, and, in the second place, it is not possible to make experiments at such high pressures. There are, however, several methods of proving that the earth is solid through and through, and they will now be considered.
25. Proof of the Rigidity and Elasticity of the Earth by the Tide Experiment. — Among the several Hnes of attack that have been made on the question of the rigidity of the earth, the one depending on the tides generated in the earth by the moon and sun has been most satisfactory ; and of the methods of this class, that devised by Michelson and carried out in collaboration with Gale, in 1913, has given by far the most exact results. Besides, it has settled one very important question, which no other method has been able to answer, namely, that the earth is highly elastic instead of being viscous. For these reasons the work of Michelson and Gale will be treated first.
The important difference between a solid and a liquid is that the former offers resistance to deforming forces while
CH. II, 25] THE EARTH 53
the latter does- not. If a perfect solid existed, no force what- ever could deform it ; if a perfect liquid existed, the only re- sistance it would offer to deformation would be the inertia of the parts moved. Neither perfect sohds nor absolutely perfect hquids are known. If a solid body has the property of being deformed more and more by a continually appUed force, and if, on the appHcation of the force being discon- tinued, the body not only does not retake its original form but does not even tend toward it, then it is said to be viscous. Putty is a good example of a material that is viscous. On the other hand, if on the application of a continuous force the body is deformed to a certain extent beyond which it does not go, and if, on the removal of the force, it returns absolutely to its original condition, it is said to be elastic. While there are no soUd bodies which are either perfectly viscous or perfectly elastic, the distinction is a clear and important one, and the characteristics of a solid may be described by stating how far it approaches one or the other of these ideal states.
In order to find how the earth is deformed by forces it is , necessary to consider what forces there are acting on it. The most obvious ones are the attractions of the sun and moon. But it is not clear in the first place that these at- tractions tend to deform the earth, and in the second place that, even if they have such a tendency, the result is at all appreciable. A ball of iron attracted by a magnet is not sensibly deformed, and it seems that the earth should be- have similarly. But the earth is so large that one's intui- tions utterly fail in such considerations. The sun and moon actually do tend to alter the shape of the earth, and the amount of its deformation due to their attractions is measurable. The forces are precisely those that produce the tides in the ocean.
It will be sufficient at present to give a rough idea, cor- rect so far as it goes, of the reason that the moon and sun raise tides in the earth, reserving for Arts. 263, 264 a more
54 AN INTRODUCTION TO ASTRONOMY [ch. ii, 25
complete treatment of the question. In Fig; 17 let E rep- resent the center of the earth, the arrow the direction toward the moon, and A and B the points where the line from E to the moon pierces the earth's surface. The moon is 4000 miles nearer to A than it is to E, and 4000 miles nearer to E than it is to S. Therefore the attraction of the moon for
Fig. 17. — The tidal bulges at A and B on the earth produced by the moon.
a unit mass at A is greater than it is for a unit mass at E, and greater for a unit mass at E than it is for one at B. Since the distance from the earth to the moon is 240,000 miles, the distance of the moon from A is fifty-nine sixtieths of its distance from E. Since the attraction varies inversely as the square of the distance, the force on A is about one thirtieth greater than that on E, and the difference between the forces on E and B is only slightly less.
It follows from the relation of the attraction of the moon for masses at A, E, and B that it tends to pull the nearer material at A away from the center of the earth E, and the center of the earth away from the more remote material at B. Since the forces are known, it is possible to compute the elongation the earth would suffer if it were a perfect fluid. The result is two elevations, or tidal bulges, at A and B.
CH. II, 25] THE EARTH 56
The concentric lines shown in Fig. 17 are the lines of equal elevation. A rather difficult mathematical discussion shows I that the radii EA and EB would each be lengthened by about four feet. Since the earth possesses at least some degree of rigidity its actual tidal elongation is somewhat less than four feet. When it is remembered that the uncertainty in the diameter of the earth, in spite of the many years that have been devoted to determining it, is still several hundred feet, the problem of finding how much the earth's elonga- tion, as a consequence of the rapidly changing tidal forces, falls short of four feet seems altogether hopeless of solution. Nevertheless the problem has been solved.
Suppose a pipe half filled with water is fastened in a hori- zontal position to the surface of the earth. The water in the pipe is subject to the attraction of the moon. To fix the ideas, suppose the pipe lies in the east-and-west direction in the same latitude as the point A, Fig. 17. Suppose, first, that the earth is absolutely rigid so that it is not deformed by the moon, and consider what happens to the water in the pipe as the rotation of the earth carries it past the point A. When the pipe is to the west of A the water rises in its eastern end, and settles correspondingly in its western end, because the moon tends to make an elevation on the earth at A . When the pipe is carried past A to the east the water rises in its western end and settles in its eastern end. Since the earth is not absolutely rigid the magnitudes of the tides under the hypothesis that it is rigid cannot be experimen- tally determined ; but, since all the forces that are involved are known, the heights the tides would be on a rigid earth can be computed.
Suppose now that the earth yields perfectly to the disturb- ing forces of the moon. Its surface is in this case always the exact figure of equilibrium. Consider the pipe, which is attached to this surface, when it is to the west of A. The water would be high in its eastern end if the shape of the surface of the earth were unchanged. But the surface to
56 AN INTRODUCTION TO ASTRONOMY [ch. ii, 25
the east of it is elevated and the pipe is raised with it. More- over, the elevation of the surface is, under the present hypothesis, just that necessary for equiUbrium. Therefore, in this case there is no tide at all with respect to the pipe.
The actual earth is neither absolutely rigid nor perfectly fluid. Consequently the tides in the pipe will actually be neither their theoretical maximum nor zero. The amount by which they fall short of the value they would have if the earth were perfectly rigid depends upon the extent to which it yields to the moon's forces, and is a measure of this yield- ing. Therefore the problem of finding how much the earth is deformed by the moon is reduced to computing how great the tides in the pipe would be if the earth were absolutely rigid, and then comparing these results with the actual tides in the pipe as determined by direct experiment. After the amount the earth yields has been determined in this way, its rigidity can be found by the theory of the deformation of solid bodies.
In the experiment of Michelson and Gale two pipes were used, one lying in the plane of the meridian and the other in the east-and-west direction. In order to secure freedom from vibrations due to trains and heavy wagons they were placed on the grounds of the Yerkes Observatory, and to avoid variations in temperature they were buried a number of feet in the ground. Since the tidal forces are very small, pipes 500 feet long were used, and even then the maximum tides were only about two thousandths of an inch.
An ingenious method of measuring these small changes in level was devised. The ends of the pipes were sealed with plane glass windows through which their interiors could be viewed. Sharp pointers, fastened to the pipe, were placed just under the surface of the water near the windows. When viewed from below the level of the water the pointer and its reflected image could be seen. Figure 18 shows an end of one of the pipes, S is the surface of the water, P is the pointer, and P' is its reflected image. The distances of P and P'
CH. II, 26] THE EARTH 67
from the surface iS are equal. Now suppose the water rises ;
since P and P' are equidistant from S, the change in their
apparent distance is twice the change
in the water level. The distances
between P and P' were accurately / ■
measured with the help of perma- / | ,
nently fixed microscopes, and the ' — ^
variations in the water level were
determined within one per cent of
their whole amount.
In order to make clear the accuracy T, ,„ t, j , •
V.J. ^i°- 18. — End of pipe in
oi the results, the complicated nature the Micheison-Gaie tide of the tides must' be pointed out. experiment. Consider the tidal bulges A and B, Fig. 17, which give an idea of what happened to the water in the pipes. For simpKcity, fix the attention on the east-and-west pipe, which in the ex- periment was about 13° north of the highest latitude A ever attains. The rotating earth carried it daily across the merid- ian of A to the north of A, and similarly across the meridian of B. When the relations were as represented in the dia- gram there were considerable tides in the pipe before and after it crossed the meridian at A because it was, so to speak, well on the tidal bulge. On the other hand, when it crossed the meridian of B about 12 hours later, the tides were very small because the bulge B was far south of the equator. But the moon was not all the time north of the plane of the earth's equator. Once each month it was 28° north and once each month 28° south, and it varied from hour to hour in a rather irregular manner. Moreover, its distance, on which the magnitudes of the tidal forces depend, also changed continuously. Then add to all these complexities the cor- responding ones due to the sun, which are unrelated to those of the moon, and which mix up with them and make the phenomena still more involved. Finally, consider the north- and-south pipe and notice, by the help of Fig. 17, that its tides are altogether distinct in character from those in the
58 AN INTRODUCTION TO ASTRONOMY [ch. ii, 25
east-and-west pipe. With all this in mind, remember that the observations made every two hours of the day for a period of several months agreed perfectly in all their char- acteristics with the results given by theory. The only dif- ference was that the observed tides were reduced in a con- stant ratio by the yielding of the earth.
The perfection of this domain of science is proved by the satisfactory coordination in this experiment of a great many distinct theories. The perfect agreement in their charac- teristics of more than a thousand observed tides with their computed values depended on the correctness of the laws of motion, the truth of the law of gravitation, the size of the earth, the distance of the moon and the theory of its motion, the mass of the moon, the distance to the sun and the theory of the earth's motion around it, the mass of the sun, the theory of tides, the numerous observations, and the lengthy calculations. How improbable that there would be perfect harmony between observation and theory in so many cases unless scientific conclusions respecting all these things are correct!
The extent to which the earth yields to the forces of the moon was obtained from the amount by which the observed tides were less than their theoretical values for an unyielding earth. It was found that in the east-and-west pipe the ob- served tides were about 70 per cent of the computed, while in the north-and-south pipe the observed tides were only about 50 per cent of the computed. This led to the astonish- ing conclusion, which, however, had been reached earher by Schweydar on the basis of much less certain observational data, that the earth's resistance to deformation in the east- and-west direction is greater than it is in the north-and-south direction. Love has suggested that the difference may be due indirectly to the effects of the oceanic tides on the general body of the earth.
On using the amount of the yielding of the earth estab- lished by observations and the magnitude of the forces exerted
CH. II, 26] THE EARTH 59
by the moon and sun, it was found by the mathematical processes which are necessary in treating such problems, that the earth, taken as a wl)ole, is as rigid as steel. That is, it resists deformation as much as it would if it were made of solid steel having throughout the properties of ordinary good steel.
The work of Michelson and Gale for the first time gave a reliable answer to the question whether the earth is viscous or elastic. It had almost invariably been supposed that the earth is viscous, because it was thought that even if the enormous pressure keeps the highly heated material of its interior in a solid state, yet it would be only stiff like a soHd is when its temperature approaches the melting point. In fact, Sir George Darwin had built up an elaborate theory of tidal evolution (Arts. 265, 266), at the cost of a number of years of work, on the hypothesis that the earth is viscous. But the experiments of Michelson and Gale prove that it is very elastic.
If the earth were viscous, it would yield somewhat slowly to the forces of the moon and sun. Consequently, the tilting of the surface, which carries the pipes, would lag behind the forces which caused both the tilting and the tides in the pipes'. There is no appreciable lag of a water tide in the pipe only 500 feet long, and consequently the observed and computed tides would not agree in phase. On the other hand, if the earth were elastic, there would be agreement in phase between the observed and computed tides. It is more difficult practically to determine accurately the phase of the tides than it is to measure their magnitudes, but the obser- vations showed that there is no appreciable difference in the phases of the observed and computed tides. These results force the conclusion that the elasticity of the earth, taken as a whole, cannot be less than that of steel, — a result ob- viously of great interest to geologists.
26. Other Proofs of the Earth's Rigidity. — (a) There is a method of finding how much the earth yields to the forces
60 AN INTRODUCTION TO ASTRONOMY [ch. ii, 26
of the moon and sun which is fundamentally equivalent to that of measuring tides in a pipe. It depends on the fact that the position of a pendulum depends upon all the forces acting on it, and, if the earth were in equihbrium, the line of its direction would always be perpendicular to the water- leyel surface. Consequently, if the earth yielded perfectly to the forces of the moon and sun, a pendulum would con- stantly remain perpendicular to its water-level surface. But if the earth did not yield perfectly, the pendulum would undergo very minute oscillations with respect to the solid part analogous to those of the water in the pipes. A modi- fication of the ordinary pendulum, known as the horizontal pendulum, was found to be sensitive enough to show the oscillations, giving the rigidity of the earth but no satis- factory evidence regarding its elasticity.
(6) The principles at the basis of the method of employ- ing tides in pipes apply equally well to tides in the ocean. Longer columns of water are available in this case, but there is difficulty in obtaining the exact heights of the actual tides, and very much greater difficulty in determining their theo- retical heights on a shelving and irregular coast where they would necessarily be observed. In fact, it has not yet been found possible to predict in advance with any considerable degree of accuracy the height of tides where they have not been observed. Yet, Lord Kelvin with rare judgment in- ferred on this basis that the earth is very rigid.
(c) Earthquakes are waves in the earth which start from some restricted region and spread all over the earth, diminish- ing in intensity as they proceed. Modem instruments, depending primarily on some adaptation of the horizontal pendulum, can detect important earthquakes to a distance of thousands of miles from their origin. Earthquake waves are of different types; some proceed through the surface rocks around the earth in undulations like the waves in the ocean, while others, compressional in character like waves of sound in the air, radiate in straight Unes from their sources.
CH. II, 26] THE EARTH 61
The speed of a wave depends upon the density and the rigidity of the medium through which it travels. This prin- ciple appUes to earthquake wavqs, and when tested on those which travel in undulations through the surface rocks there is good agreement between observation and theory. Con- sider its application to the compressional waves that go through the earth. The time required for them to go from the place of their origin to the place where they are observed is given by the observations. The density of the earth is known. If its rigidity were known, the time could be com- puted ; but the time being known, the rigidity can be com- puted. While the results are subject to some uncertainties, they agree with those found by other methods.
(d) The attraction of the moon for the equatorial bulge slowly changes the plane of the earth's equator (Art. 47). The magnitude of the force that causes this change is known. If the earth consisted of a crust not more than a few hundred miles deep floating on a liquid interior, the forces would cause the crust to slip on the liquid core, just as a vessel con^ taining water can be rotated without rotating the water. If the crust of the earth alone were moved, it would be shifted rapidly because the mass moved would not be great. But the rate at which the plane of the earth's equator is moved, as given by the, observations, taken together with the forces involved, proves that the whole earth moves. When the effects of forces acting on such an enormous body are con- sidered, it is found that this fact means that the earth has a considerable degree of rigidity.
(e) Every one knows that a top may be spun so that its axis remains stationary in a vertical direction, or so that it wabbles. Similarly, a body rotating freely in space may rotate steadily around a fixed axis, or its axis of rotation may wabble. The period of the wabbling depends upon the size, shape, mass, rate of rotation, and rigidity of the body. In the case of the earth all these factors except the last may be regarded as known. If it were known, the rate of wab-
62 AN INTRODUCTION TO ASTRONOMY [ch. ii, 26
bling could be computed ; or, if the rate of wabbling were found from observation, the rigidity could be computed. It has recently been found that the earth's axis of rotation wabbles slightly, and the rate of this motion proves that the rigidity of the earth is about that of steel.
27. Historical Sketch on the Mass and Rigidity of the Earth. — The history of correct methods of attempting to find the mass of the earth necessarily starts with Newton, because the ideas respecting mass were not clearly formu- lated before his time, and because the determination of mass depends on the law of gravitation which he discovered. By some general but inconclusive reasoning he arrived at the conjecture that the earth is five or six timps as dense as water.
The first scientific attempt to determine the density of the earth was made by Maskelyne, who used the mountain method, in 1774, in Scotland. He found 4.5 for the density of the earth. The torsion balance, devised by Michell, was first employed by Cavendish, in England, in 1798. His result agreed closely with those obtained by later experi- menters, among whom may be mentioned Baily (1840) in England, and Reich (1842) in Germany, Cornu (1872) in France, Wilsing (1887) in Germany, Boys (1893) in England, and Braun (1897) in Austria. The pendulum method, using either a mountain or a mine to secure difference in elevation, has been employed a number of times.
Lord Kelvin (then Sir William Thomson) first gave in 1863 good reasons for beheving the earth is rigid. His con- clusion was based on the height of the oceanic tides, as out- fined in Art. 26 (b). The proof by means of the rate of transmissioii of earthquake waves owes its possibifity largely to John Milne, an EngUshman who long lived in Japan, which is frequently disturbed by earthquakes. His interest in the character of earthquakes stimulated him to the inven- tion of instruments, known as seismographs, for detecting and recording faint earth tremors. The change of the posi-
CH. II, 27] THE EARTH 63
tion of the plane of the earth's equator, known as the pre- cession of the equinoxes, has been known observationally ever since the days of the ancient Greeks, and its cause was understood by Newton, but it has not been used to prove the rigidity of the earth because it takes place very slowly. The wabbling of the axis of the earth was first established observationally, in 1888, by Chandler of Cambridge, Mass., and Kiistner of Berlin. The theoretical applications of the rigidity of the earth were made first by Newcomb of Wash- ington, and then more completely by S. S. Hough of Eng- land. The first attempt at the determination of the rigidity of the earth by the amount it yields to the tidal forces of the moon and sun was made unsuccessfully in 1879 by George and Horace Darwin, in England. Notable success has been achieved only in the last 15 years, and that by improvements in the horizontal pendulum and by taking great care in keeping the instnmaents from being disturbed. The names that. stand out are von Rebeur-Paschwitz, Ehlert, Kortozzi, Schweydar, Hecker, and Orloff. The observations of Hecker at Potsdam, Germany, were especially good, and Schweydar made two exhaustive mathematical discussions of the subject.
III. QUESTIONS
1. What is the difference between mass and weight ? Does the weight of a body depend on its position? Does the inertia of a body depend on its position ?
2. Can the mass of a small body be determined from its inertia ? Can the mass of the earth be determined in the same way ?
3. What is the average weight of a cubic mile of the earth ?
4. Discuss the relative advantages of the torsion-balance method and mountain method in determining the density of the earth. Which one has the greater advantages ?
5. What is the pressure at the bottom of an ocean six miles deep ?
6. Discuss the character of the tides in east-and-west and north- and-south pipes during a whole day when the moon is in the posi- tion indicated in Fig. 17, and when it is over the earth's equator.
64 AN INTRODUCTION TO ASTRONOMY [ch. ii, 27
7. What are the advantages and disadvantages of a long pipe in the tide experiment ?
8. If a body i» at A, Fig. 17, is its weight greater or less than normal as determined by spring balances? By balance scales? What are the facts, if it is at B?
9. Enumerate the scientific- theories and facts involved in the tide experiment.
10. List the principles on which the several proofs of the earth's rigidity depend. How many fundamentally different methods are there of determining its rigidity?
III. The Eaeth's Atmosphere
28. Composition and Mass of the Earth's Atmosphere. —
The atmosphere is the gaseous envelope which surrounds the earth. Its chief constituents are the elements nitrogen and oxygen, but there are also minute quantities of argon, hehum, neon, krypton, xeon, and some other very rare con- stituents. When measured by volume at the earth's sur- face, 78 per cent of the atmosphere is nitrogen, 21 per cent is oxygen, 0.94 per cent is argon, and the remaining elements, occur in much smaller quantities.
Nitrogen, oxygen, etc., are elements; that is, they are substances which are not broken up into more fundamental units by any physical or any chemical changes. The thou- sands of different materials that are found on the earth are all made up of about 90 elements, only about half of which are of very frequent occurrence. The union of elements into a chemical compound is a very fundamental matter, for " the compound may have properties very unlike those of any of the elements of which it is composed. For example, hydrogen, carbon, and nitrogen are in almost all food, but hydrocyanic acid, which is composed of these elements alone, is a deadly poison.
Besides the elements which have been enumerated, the atmosphere contains some carbon dioxide, which is a com- pound of carbon and oxygen, and water vapor, which is a compound of oxygen and hydrogen. In volume three
CH. II, 29] THE EARTH 65
hundredths of one pfer cent of the earth's atmosphere is carbon dioxide ; but this compound is heavier than nitrogen and oxygen, and by weight, 0.05 per cent of the atmosphere is carbon dioxide. The amount of water vapor in the air varies greatly with the position on the earth's surface and with the time. There are also small quantities of dust, soot, ammonia, and many other things which occur in variable quantities and which are considered as impurities.
The pressure of the atmosphere at sea level is about 15 pounds per square inch and its density is about one eight- hundredth that of water. This means that the weight of a column of air reaching from the earth's surface to the limits of the atmosphere and having a cross section of one square inch weighs 15 poimds. The total mass of the atmosphere can be obtained by multiplying the weight of one column by the total area of the earth. In this way it is found that the mass of the earth's atmosphere is nearly 6,000,000,000,000,000 tons, or approximately one miUionth the mass of the sohd earth. The total mass of even the carbon dioxide of the earth's atmosphere is approximately 3,000,000,000,000 tons.
29. Determination of Height of Earth's Atmosphere from Observations of Meteors. — Meteors, or shooting stars as they are commonly called, are minute bodies, circulating in interplanetary space, which become visible only when they penetrate the earth's atmosphere and are made incandescent by the resistance which they encounter. The great heat developed is a consequence of their high velocities, which ordinarily are in the neighborhood of 25 miles per second.
Let m, Fig. 19, represent the path of a meteor before it encounters the atmosphere at A. Until it reaches A it is invisible, but at A it begins to glow and continues luminous until it is entirely burned up at B. Suppose it is observed from the two stations Oi and O2 which are at a known dis- tance apart. The observations at Oi give the angle AO1O2, and those at O2 give the angle AO2O1. From these data the
66 AN INTRODUCTION TO ASTRONOMY [ch. n, 29
other parts of the triangle can be computed (compare Art.
10). After the distance OiA has been computed the perpen- dicular height of A from the sur- face of the earth can be computed by using the angle AOiOs. Similarly, the height of B above the surface
Fig. 19. — Determination of the height of meteors. O^ ^'^^ earth can
be determined. Observations of meteors from two stations show that they ordinarily become visible at a height of from 60 to 100 miles. Therefore, the atmosphere is sufficiently dense to a height of about 100 miles to offer sensible resistance to meteors. Meteors usually disappear by the time they have descended to within thirty or forty miles of the earth's surface.
30. Determination of Height of Earth's Atmosphere from Observations of Aurorse. — Aurorae are almost certainly electrical phenomena of the very rare upper atmosphere, though their nature is not yet very well understood. Their altitude can be computed from simultaneous observations made at different stations. The method is the same as that in obtaining the height of a meteor.
The southern ends of auroral streamers are usually more than 100 miles in height, and they are sometimes found at an altitude of 500 or 600 miles. Their northern ends are much lower. This means that the density required to make meteors incandescent is considerably greater than that which is sufficient for auroral phenomena.
31. Determination of Height of Earth's Atmosphere from the Duration of Twilight. — Often after sunset, even to the east of the observer, high clouds are briUiantly illuminated by the rays of the sun which still fall on them. The higher
CH. II, 31]
THE EARTH
67
the clouds are, the longer they are illuminated. Similarly, the sun shines on the upper atmosphere for a considerable time after it has set or before it rises, and gives the twihght. The duration of twilight depends upon the height of. the atmosphere. While it is difficult to determine the instant at which the twihght ceases to be visible, observations show that under favorable weather conditions it does not disap- pear until the sun is 18 degrees below the horizon.
In order to see how the height of the atmosphere can be determined from the duration of the twilight, consider Fig. 20. The sun's rays come in from the left in hnes that are sensibly par- allel. The ob- server at 0 can see the illumin- ated atmosphere at P ; but if the atmosphere were much shallower, it would not be visible to him. The region P is midway between 0 and the sunset point. Since 0 is 18 degrees from the sunset point, it is possible to compute the height of the plane of the horizon at P above the surface of the earth. It is found that 18 degrees corresponds to an altitude of 50 miles. That is, the atmosphere extends to a height of 50 miles above the earth's surface in quantities sufficient to produce twilight.
The results obtained by the various methods for determin- ing the height of the atmosphere disagree because its density decreases with altitude, as is found by ascending in balloons, and different densities are required to produce the different phenomena. It will convey the correct idea for most appli- cations to state that the atmosphere does not extend in ap- preciable quantities beyond 100 miles above the earth's sur-
FlG.
20. — Determination of the height of atmosphere from the duration of twilight.
the
68 AN INTRODUCTION TO ASTRONOMY [ch. ii, 31
face. At this altitude its density is of the order of one four- millionth of that at the surface. When the whole earth is considered it is found that the atmosphere forms a relatively- thin layer. If the earth is represented by a globe 8 inches in diameter, the thickness of the atmosphere on the same scale is only about one tenth of an inch.
32. The Eonetic Theory of Gases. — It has been stated that every known substance on the earth is composed of about 90 fundamental elements. A chemical combination of atoms is called a molecule. A molecule of oxygen con- sists of two atoms of oxygen, a molecule of water of two atoms of hydrogen and one of oxygen, and similarly for all substances. Some molecules contain only a few atoms and others a great many ; for example, a molecule of cane sugar is composed of 12 atoms of carbon, 22 of hydrogen, and 11 of oxygen. As a rule the compounds developed in connec- tion with the life processes contain many atoms.
The molecules are all very minute, though their dimen- sions doubtless vary with the number and kind of atoms they contain." Lord Kelvin devised a number of methods of determining their size, or at least the distances between their centers. In water, for example, there are in round numbers 500,000,000 in a Une of them one inch long, or the cube of this number in a cubic inch.
In solids the molecules are constrained to keep essentially the same relations to one another, though they are capable of making complicated small vibrations. In liquids the molecules continually suffer restraints from neighboring molecules, but their relative positions are not fixed and they move around among one another, though not with perfect freedom. In gases the molecules are perfectly free from one another except when they collide. They move with great speed and collide with extraordinary frequency ; but, in spite of the frequency of the collisions, the time during which they are uninfluenced by their neighbors is very much greater than that in which they are in effective contact.
CH. II, 33] THE EARTH 69
The pressure exerted by a gas is due to the impact of its molecules on the walls of the retaining vessel. To make the ideas definite, consider a cubic foot of atmosphere at sea-level pressure. Its weight is about one and one fourth ounces, but it exerts a pressure of 15 pounds on each square inch of each of its six surfaces, or a total pressure on the surface of the cube of more than six tons. This imphes that the mole- cules move with enormous speed. They do not all move with the same speed, but some travel slowly while others go much faster than the average. Theoretically, at least, in every gas there are molecules moving with every velocity, however great, but the number of those having any given velocity diminishes rapidly as its difference from the aver- age velocity increases. The average velocity of molecules in common air at ordinary temperature and pressure is more than 1600 feet per second, and on the average each mole- cule has 5,000,000,000 colUsions per second. Therefore the average distance traveled between coUisions is only about asoVoo of an inch.
From the kinetic theory of gases it is possible to deter- mine how fast the density of the air diminishes with increase of altitude. It is found that about one half of the earth's atmosphere is within the first 3.5 miles of its surface, that one half of the remainder is contained in the next 3.5 miles, and so on until it is so rare that the kinetic theory no longer apphes without sensible modifications.
33. The Escape of Atmospheres. — Suppose a body is projected upward from the surface of the earth. The height to which it rises depends upon the speed with which it is started. The greater the initial speed, the higher it will rise, and there is a certain definite initial velocity for which, neg- lecting the resistance of the air, it will leave the earth and never return. This is the velocity of escape, and for the earth it is a little less than 7 miles per second.
The molecules in the earth's atmosphere may be con- sidered as projectiles which dart in every direction. It has
70 AN INTRODUCTION TO ASTRONOMY [ch. ii, 33
been seen that there is a sm'all fraction of them which move with a velocity as great as 7 miles per second. Half of these will move toward points in the sky and consequently would escape from the earth if they did not encounter other molecules. But in view of the great frequency of collisions of molecules, it is evident that only a very small fraction of those which move with high velocities can escape from the earth. However, it seems certain that some molecules will be lost in this way, and, so far as this factor is concerned, the earth's atmosphere is being continually depleted. The process is much more rapid in the case of bodies, such as the moon, for example, whose masses and attractions are much smaller, and for which, therefore, the velocity of escape is lower.
It should not be inferred from this that the earth's at- mosphere is diminishing in amount even if possible replenish- ment from the rocks and its interior is neglected. When a molecule escapes from the earth it is still subject to the attrac- tion of the sun and goes around it in an orbit which crosses that of the earth. Therefore the earth has a chance of acquiring the molecule again by collision. The only excep- tion to this statement is when the molecule escapes with a velocity so high that the sun's attraction cannot control it. The velocity necessary in order that the molecule shall escape both the earth and the sun depends upon its direction of motion, but averages about 25 miles per second and cannot be less than 19 miles per second. But besides the molecules that have escaped from the earth there are doubtless many others revolving around the sun near the orbit of the earth. These also can be acquired by collision. The earth is so old and there has been so much time for losing and acquir- ing an atmosphere, molecule by molecule, that probably an equilibrium has been reached in which the number of mole- cules lost equals the number gained. The situation is analogous to a large vessel of water placed in a sealed room. The water evaporates until the air above it becomes
CH. n, 34] THE EARTH 71
SO nearly saturated that the vessel acquires as many mole- cules of water vapor by collisions as it loses by evaporation.
The doctrine of the escape of atmospheres imphes that bodies of small mass will have limited and perhaps inappre- ciable atmospheres, and that those of large mass will have extensive atmospheres. The imphcations of the theory are exactly verified in experience. For example, the moon, with a mass ^ that of the earth and a velocity of escape of about 1.5 miles per sfecond, has no sensible atmosphere. On the other hajid, Jupiter, with a mass 318 times that of the earth and a velocity of escape of 37 miles per second, has an enormous atmosphere. These examples are typical of the facts furnished by all known celestial bodies.
34. Effects of the Atmosphere on Climate. — Aside from the heat received from the sun, the most important factor affecting the earth's cUmate is its atmosphere. It tends to equaUze the temperature in three important ways, (a) It makes the temperature at any one place more uniform than it would otherwise be, and (&) it reduces to a large extent the variations in temperature in different latitudes that would otherwise exist. And (c) it distributes water over the surface of the earth.
(a) Consider the day side of the earth. The rays of the sun are partly absorbed by the atmosphere and the heating of the earth's surface is thereby reduced. The amount absorbed at sea level is possibly as much as 40 per cent. Every one is famihar with the fact that on a mountain, above a part of the atmosphere, sunlight is more intense than it is at lower levels. But at night the effects are reversed. The heat that the atmosphere has absorbed in the daytime is radiated in every direction, and hence some of it strikes the earth and warms it. Besides this, at night the earth radiates the heat it has received in the daytime. The at- mosphere above reflects some of the radiated heat directly back to the earth. Another portion of it is absorbed and radiated in every direction, and consequently in part back
72 AN INTRODUCTION TO ASTRONOMY [ch. ii, 34
to the earth. In short, the atmosphere acts as a sort of blanket, keeping out part of the heat in the daytime, and helping to retain at night that which has been received. Its action is analogous to that of a glass with which the gardener covers his hotbed. The results are that the variations in temperature between night and day are reduced, and the average temperature is raised.
(6) The unequal heating of the earth's atmosphere in various latitudes is the primary cause of the winds. The warmer air moves toward the cooler regions, and the cold air of the higher latitudes returns toward the equator. The trade winds are examples of these movements. Their im- portance will be understood when it is remembered that wind velocities of 15 or 20 miles an hour are not uncommon, and that there is about 15 pounds of air above every square inch of the earth's surface.
One of the effects of the winds is the production of the ocean currents which are often said to be dominant factors in modifying cUmate, but which are, as a matter of fact, relatively unimportant consequences of the air currents. A south wind will often in the course of a few hours raise the temperature of the air over thousands of square miles of territory by 20 degrees, or even more. In order to raise the temperature of the atmosphere at constant pressure, over one square mile through 20 degrees by the combustion of coal it would be necessary to burn ten thousand tons. This illustration serves to give some sort of mental image of the great influence of air currents on climatic conditions, and if it were not for them, it is probable that both the equatorial and polar regions would be uninhabitable by man.
35. Importance of the Constitution of the Atmosphere. — The blanketing effect of the atmosphere depends to a con- siderable extent on its constitution. Every one is famiUar with the fact that the early autumn frosts occur only when the air is clear and has low humidity. The reason is that water vapor is less transparent to the earth's radiations than
CH. II, 36] THE EARTH 73
are nitrogen and oxygen gas. On the other hand, there is not so much difference in their absorption of the rays that come from the sun. The reason is that the very hot sun's rays are largely of short wave length (Art. 211) ; that is, they are to a considerable extent in the blue end of the spec- trum, while the radiation from the cooler earth is almost entirely composed of the much longer heat rays. Ordinary glass has the same property, for it transmits the sun's rays almost perfectly, while it is a pretty good screen for the rays emitted by a stove or radiator.
The water-vapoT content of the atmosphere varies and cannot surpass a certain amount. But carbon dioxide has the same absorbing properties as water vapor, and in spite of the fact that it makes up only a very small part of the earth's atmosphere, Arrhenius believes that it has important climatic effects. He concluded that if the quantity of it in the air were doubled the climate would be appreciably warmer, and that if half of it were removed the average temperature of the earth would fall. Chamberhn has shown that there are reasons for believing that the amount of carbon dioxide has varied in long oscillations, and he sug- gested that this may be the explanation of the ice ages, with intervening warm epochs, which the middle latitudes have experienced.
If the effect of carbon dioxide on the cHmate has been correctly estimated, its production by the recent enormous consumption of coal raises the interesting question whether man at last is not in this way seriously interfering with the cosmic processes. At the present time about 1,000,000,000 tons of coal are mined and burned annually. In order to burn 12 pounds of coal 32 pounds of oxygen are required, and the result of the combustion is 12 -|- 32 = 44 pounds of carbon dioxide. Consequently, by the combustion of coal there is now annually produced by man about 3,670,000,000 tons of carbon dioxide. On referring to the total amount of carbon dioxide now in the air (Art. 28), it
74 AN INTRODUCTION TO ASTRONOMY [ch. ii, 35
is seen that at the present rate of conabustion of coal it will be doubled in 800 years. Consequently, there are grounds for believing that modern industry may have sensible climatic effects in a few centuries.
36. Role of the Atmosphere in Life Processes. — Oxygen is an indispensable element in the atmosphere for all higher forms of animal life. It is taken into the blood stream through the lungs and is used in the tissues. Its proportion in the atmosphere is probably not very important, for it seems probable that if it had always been much more or much less, animals would have become adapted to the dif- ferent condition. But if the earth's crust had contained enough material which readily unites with oxygen, such as hydrogen, silicon, or iron, to have exhausted the supply, it seems certain that animals with warm, red blood could not have developed. Such considerations are of high impor- tance in speculating on the question of the habitability of other planets.
The higher forms of vegetable matter are largely composed of carbon and water. The carbon is obtained from the car- bon dioxide in the atmosphere. The carbon and oxygen are 2 separated in the cells of the ^ plants, the carbon is retained, and the oxygen is given back to the air. 37. Refraction of Light by the Atmosphere. — When hght passes from a rarer to a denser medium it is bent toward the perpendic- ular to the surface between the two media, and in general the Fig. 21 . — The refraction of greater the difference in the densi-
light.
ties of the two media, the greater is the bending, which is called refraction. Thus, in Fig. 21, the ray I which strikes the surface of the denser medium at A is bent from the direction AB toward the perpendicular to the surface AD and takes the direction AG.
CH. II, 37]
THE EARTH
75
Now consider a ray of light striking the earth's atmosphere obliquely. The density of the air increases from its outer borders to the surface
of the earth. Conse- ' ■^'
quently, a ray of Ught is bent more and more as it proceeds down through the air. Let I, Fig. 22, represent a ray of hght coming from a star S to an observer at 0. The star is really in the direc- tion OS", but it appears to be in the direction OS' from which the light comes when it strikes the observer's eye. The angle between OS" and OS' is the angle of refraction. It is zero for a star at the zenith and increases to a httle over one-half of a degree for one at the horizon. For this reason a
Fig. 22. — Refraction of light by the earth's atmosphere.
f IG. 23. — The sun is apparently flattened by refraction when it is on the
horizon.
celestial body apparently rises before it is actually above the horizon, and is visible until after it has really set. If the sun or moon is on the horizon, its bottom part is apparently raised more than its top part by refraction, so that it seems to be flattened in the vertical direction, as is shown in Fig. 23.
76 AN IlsrTRODUCTION TO ASTRONOMY [ch. ii, 38
38. The Twinkling of the Stars. — The atmosphere is not only of variable density from its highest regions to the sur- face of the earth, but it is always disturbed by waves which cause the density at a given point to vary continually. These variations in density cause constant small changes in the refraction of light, and consequently alterations in the direction from which the light appears to come. When the source is a poirft of hght, as a star, it twinkles or scintil- lates. The twinkling of the stars is particularly noticeable in winter time on nights when the air is cold and unsteady. The variation in refraction is dffierent for different colors, and consequently when a star twinkles it flashes sometimes blue or green and at other times red or yellow. Objects that have disks, even though they are too small to be discerned with the unaided eye, appear much steadier than stars because the irregular refractions from various parts seldom agree in direction, and consequently do not displace the whole object.
IV. QUESTIONS
1. What is the weight of the air in a room 16 feet square and 10 feet high?
2. How many pounds of air pass per minute through a windmill 12 feet in diameter in a breeze of 20 miles per hour ?
3. Compute the approximate total atmospheric pressure to which a person is subject.
4. What is the density of the air, compared to its density at the surface, at heights of 50, 100, and 500 miles, the density being deter- mined by the law given at the end of Art. 32 ? This gives an idea of the density required for the phenomena of twilight, of meteors, and of aurorsB.
5. Draw a diagram showing the earth and its atmosphere to scale.
6. The earth's mass is slowly growing by the acquisition of meteors ; if there is nothing to offset this growth, will its atmosphere have a tendency to increase or to decrease in amount ?
7. If the earth's atmosphere increases or decreases, as the case may be, what will be the effect on the mean temperature, the daily range at any place, and the range over the earth's whole surface ?
8. If the earth's surface were devoid of water, what would be the effect on the mean temperature, the daily range at any place, and the range over its whole surface ?
CHAPTER III THE MOTIONS OF THE EARTH
I. The Rotation of the Eakth
39. The Relative Rotation of the Earth. — The most casual observer of the heavens has noticed that not only the sun and moon, but also the stars, rise in the east, pass across the sky, and set in the west. At least this is true of those stars which cross the meridian south of the zenith. Figure 24 is a photograph of Orion in which the telescope was kept fixed while the stars passed in front of it, and the horizontal streaks are the images traced out by the stars on the photo- graphic plate.
The stars in the north- ern heavens describe circles around the north pole of the, sky as a center. Two hours of observation of the posi- tion of the Big Dipper will show the character of the motion very clearly. Figure 25 shows circumpolar star trails secured by pointing a fixed telescope toward the pole star and giving an exposure of a little over an hour. The conspicuous streak a little below and to the left of the, center is the trail of the pole star, which therefore is not exactly at the pole of the heavens. A comparison of this picture with
77
Fig. 24. — • Star trails of brighter stars in Orion (Barnard).
78
AN INTRODUCTION TO ASTRONOMY [ch. hi. 39
the northern sky will show that most of the stars whose trails are seen are quite invisible to the unaided eye.
Since all the heavenly bodies rise in the east (except those so near the pole that they simply go around it), travel across
the sky, and set in the west, to reappear again in the east, it fol- lows that either they go around the earth from east to west, or the earth turns from west to east. So far as the simple mo- tions of the sun, moon, and stars are concerned both hypotheses are in perfect harmony with the observations, and it is not pos- sible to decide which of them is correct without additional data. All the apparent motions prove is that there is a relative motion of the earth with respect to the heavenly bodies.
It is often supposed that the ancients were unscientific, if not stupid, because they believed that the earth was fixed and that the sky went around it, but it has been seen that so far as their data bore on the question one theory was as good as the other. In fact, not all of them thought that the earth was fixed. The earliest philosopher who is known to have believed in the rotation of the earth was Philolaus, a Pythagorean, who lived in the fifth century b.c. His
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Fig. 25. — Circumpolar star trails (Ritchey) .
CH. Ill, 40] THE MOTIONS OF THE EARTH 79
ideas were more or less mystical, but they seem to have had some influence, for they were quoted by Copernicus (1473- 1543) in his great work on the theory of the motions in the solar system. Aristotle (384-322 B.C.) recognized the fact that the apparent motions of the stars can be explained either by their revolution around the earth, or by the rota- tion of the earth on its axis. Aristarchus of Samos (310- 250 B.C.) made the clearest statements regarding both the rotation and the revolution of the earth of any philosopher of antiquity. But Hipparchus (180-110 B.C.), who was the greatest astronomer of antiquity, and whose discoveries were very numerous and valuable, beheved in the fixity of the earth. He was followed in this opinion by Ptolemy (100-170 A.D.) and every other astronomer of note down to Copernicus, who believed the earth rotated and revolved around the sun.
40. The Laws of Motion. — One method of attacking the question of whether or not any particular body, such as the earth, moves is to consider the laws of motion of bodies in general, and then to answer it on the basis of, and in harmony with, these laws. The laws of nature are in a' fundamental respect different from civil laws, and it is un- fortunate that the same term is used for both of them. A civil law prescribes or forbids a mode of conduct, with pen- alties if it is violated. It can be violated at pleasure if one is willing to run the chance of suffering the penalty. On the other hand, a law of nature does not prescribe or compel anything, but is a description' of the way all phenomena of a certain class succeed one another.
The laws of motion are statements of the way bodies actually move. They were first given by Newton in 1686, altholigh they were to some extent understood by his prede- cessor Gahleo. Newton called them axioms although they are by no means self-evident, as is proved by the fact that for thousands of years they were quite unknown. The laws, essentially as Newton gave them, are :
80 AN INTRODUCTION TO ASTRONOMY [ch. hi, 40
Law I. Every body continues in its state of rest, or of uni- form motion in a straight line, unless it is compelled to change that state by an exterior force acting upon it.
Law II. The rate of change of motion of a body is directly proportional to the force applied to it and inversely propor- tional to its mass, and the change of motion takes place in the direction of the line in which the force acts.
Law III. To every action there is an equal and oppositely directed reaction ; or, the mutual actions of two bodies are al- ways equal and oppositely directed.
The importance of the laws of motion can be seen from the fact that every astronomical and terrestrial phenomenon involving the motion of matter is interpreted by using them as a basis. They are, for example, the foundation of all mechanics. A little reflection will lead to the conclusion that there are few, if indeed any, phenomena that do not in some way, directly or indirectly, depend upon the motion of matter.
The first law states the important fact that if a body is at rest it will never begin to move unless some force acts upon it, and that if it is in motion it will forever move with uniform speed in a straight Une unless some exterior force acts upon it. In two respects this law is contradictory to the ideas generally maintained before the time of Newton. In the first place, it had been supposed that bodies near the earth's surface would descend, because it was natural for them to do so, even though no forces were acting upon them. In the second place, it had been supposed that a moving body would stop unless some force were continually applied to keep it going. These errors kept the predecessors of Newton from getting any satisfactory theories regarding the motions of the heavenly bodies.
The second law defines how the change of motion of a body, in both direction and amount, depends upon the ap- plied force. It asserts what happens when any force is act- ing, and this means that the statement is true whether or
CH. Ill, 40] THE MOTIONS OF THE EARTH 81
not there are other forces. In other words, th6 momentary
effects of forces can be considered independently of one
another. For example, if two forces, PA and PB in Fig.
26, are acting on a body at P, it will move in the direction
PA just as though PB
were not acting on it, / ^^^-'^^
and it will move in the / ^ /'
direction PB just as / ^^^--^'''''^
though PA were not /^--^^^
acting on it. The result p ^'■^
is that when they are Fig. 26. — The parallelogram of forces.
both acting it will go from P to C along PC. Since PACB is a parallelogram, this is called the parallelogram law of the composition of forces.
The first two laws refer to the motion of a single body; the third expresses the way in which two bodies act on each other. It means essentially that if one body changes the state of motion of another body, its own state of motion is also changed reciprocally in a definite way. The term " action " in the law means the mass times the rate of change of motion (acceleration) of the body. Hence the third law might read that if two bodies act on each other, then the product of the mass and acceleration in one is equal and opposite to the product of the mass and acceleration in the other. This is a complete statement of the way two bodies act upon each other. But the second law states that the product of the mass and acceleration of a body is propor- tional to the force acting on it. Hence it follows that the third law might read that if two bodies act on each other, then the force exerted by the first on the second is equal and opposite to the force exerted by the second on the first. This statement is not obviously true because it seems to contradict ordinary experience. For example, the law states that if a strong man and a weak man are pulling on a rope (weight of the rope being neglected) against each other, the strong man cannot pull any more than the weak man. The
82 AN INTRODUCTION TO ASTRONOMY [ch. hi, 40
reason is, of course, that the weak man does not give the strong one an opportunity to use his full strength. If the strong man is heavier than the weak one and pulls enough, he will move the latter while he remains in his tracks. This seems to contradict the statement of the law in terms of the acceleration; but the contradiction disappears when it is remembered that the men are subject not only to the forces they exert on each other, but also to their friction with the earth. If they were in canoes in open water, they would both move, and, if the weights of the canoes were included, their motions would be in harmony with the third law.
Since the laws of motion are to be used fundamentally in considering the motion of the earth, the question of their truth at once arises. When they are applied to the motions of the heavenly bodies, everything becomes orderly. Be- sides this, they have been illustrated millions of times in ordinary experience on the earth and they have been tested in laboratories, but nothing has been found to indicate they are not in harmony with the actual motions of material bodies. In fact, they are now supported by such an enormous mass of experience that they are among the most trustworthy con- clusions men have reached.
41. Rotation of the Earth' Proved by Its Shape. — The shape of the earth can be determined without knowing whether or not it rotates. The simple measurements of arcs (Art. 12) prove that the earth is oblate.
It can be shown that it follows from the laws of motion and the law of gravitation that the earth would be spherical if it were not rotating. Since it is not spherical, it must be rotating. Moreover, it follows from the laws of motion that if it is rotating it will be bulged at the equator. Hence the oblateness of the earth proves that it rotates and deter- mines the position of its axis, but does not determine in which direction it turns.
42. Rotation of the Earth Proved by the Eastward Devi- ation of Falling Bodies. — Let OP, Fig. 27, represent a
CH. Ill, 42] THE MOTIONS OP THE EARTH
83
Fig. 27. — The eastward deviation of falling bodies proves the eastward rotation of the earth.
tower from whose top a ball is dropped. Suppose that while the ball is falling to the foot of the tower the earth rotates through the angle QEQ'. The top of the tower is carried from P to P', and its foot from 0 to 0'. The distance PP' is somewhat greater than the distance 00'. Now consider the falling body. It tends to move in the direction PP' in accord- ance with the first law of motion be- cause, at the time it is dropped, it is carried in this direction by the rotation of the earth. Moreover, PP' is the dis- tance through which it would be carried if it were not dropped. But the earth's attraction causes it to descend, and the force acts at right angles to the Une PP'. There- fore, by the second law of motion, the attraction of the earth does not have any influence on the motion in the direction PP'. Consequently, while it is descending it moves in a horizontal direction a distance equal to PP' and strikes the surface at 0" to the east of the foot of the tower 0'. The eastward deviation is the distance O'O". The small diagram at the right shows the tower and the path of the falling body on a larger scale.
The foregoing reasoning has been made on the assumption that the earth rotates to the eastward. The question arises whether the conclusions are in harmony with experience. The experiment for determining the deviation of falling bodies is complicated by air currents and the resistance of the air. Furthermore, the eastward deviation is very small, being only 1.2 inches for a drop of 500 feet in latitude 40°. In
84 AN INTRODUCTION TO ASTRONOMY [ch. hi, 42
spite of these difficulties, the experiment for moderate heights proves that the earth rotates to the eastward. Father Hagen, of Rome, has devised an apparatus, having analogies with Atwood's machine in physics, which avoids most of the dis- turbances to which a freely falling body is subject. The largest free fall so far tried was in a vertical mine shaft, near Houghton, Mich., more than 4000 feet deep. In spite of the fact that the diameter of the mine shaft was many times the deviation for that distance, the experiment utterly failed because the balls which were dropped never reached the bottom. It is probable that when they had fallen far enough to acquire high speed the air packed up in front of them until they were suddenly deflected far enough from their course to hit the walls and become imbedded.
43. Rotation of the Earth Proved by Foucault's Pendulum. — One of the most ingenious and convincing experiments
for proving the rotation of the earth was devised in 1851 by the French physicist Foucault. It de- pends upon the fact that accord- ing to the laws of motion a freely swinging pendu- lum tends con- stantly to move in the same plane." Suppose a pendulum suspended at 0, Fig. 28, is started swinging in the meridian OQ. Let OF be the tangent at 0 drawn in the plane of the meridian. After a certain interval the meridian OQ will have rotated to the position O'Q'. The hne O'V is drawn parallel to the line OV. Conse- quently the pendulum will be swinging in the plane EO'V.
Fig. 28. — The Foucault pendulum.
CH. Ill, 44] THE MOTIONS OF THE EARTH
85
The tangent to the meridian at 0' is O'F. Consequently, the angle between this line and the plane in which the pendulum will be swinging is V'O'V, which equals OVO'. That is, the angle at V between the meridian tangents equals the apparent deviation of the plane of the pendulum from the meridian. For points in the northern hemisphere the devi- ation is from a north-and-south direction toward a northeast- and-southwest direction. The angle around the cone at V equals the total deviation in one rotation of the earth. If 0 is at the earth's pole, the daily deviation is 360 degrees. If 0 is on the ea,rth's equator, the point V is infinitely far away and the deviation is zero.
Foucault suspended a heavy iron ball by a steel wire about 200 feet long, and the deviation became evident in a few minutes. The experiment is very simple and has been re- peated in many pla)ces. It proves that the earth rotates eastward, and the rate of deviation of the pendulum proves that the relative motion of the earth with respect to the stars is due entirely to its rotation and not at all to the motions of the stars around it.
44. Consequences of the Earth's Rotation. — An itripor- tant consequence of the earth's rotation is the direction of air currents at considerable dis- tances from the equator in both northern and southern lati- tudes. Suppose the unequal heat- ing of the atmos- phere causes a certain portion of it to move north- ward from 0, Fig. 29, with such a velocity that if the earth were not rotating, it would arrive at A in a certain
|
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/y. |
X7 |
/!■ |
|
^ 1 |
/ |
||
|
/ ^ |
7 |
' / |
|
|
-^ |
t' |
||
|
JSi- |
|
Fig. 29. — The deviation of air currents.
86 AN INTRODUCTION TO ASTRONOMY [ch. hi, 44
interval of time. Suppose that in this interval of time the meridian OQ rotates to the position O'Q'. Hence the mass of air under consideration actually had the velocities OA and 00' when it started from 0, the former with respect to the surface of the earth and the latter because of the rotation of the earth. By the laws of motion these motions, being at right angles to each other, are mutually independent, and the air will move over both distances during the interval of time and arrive at the point A", which is east of A'. Con- sequently, the mass of air that started straight northward with respect to the surface of the earth along the meridian OA will have deviated eastward by the amount A'A".
The deviation for northward motion in the northern hemisphere is toward *the east ; for southward motion, it is toward the west. In both cases it is toward the right. For similar reasons, in the southern hemisphere the devia- tion is toward the left.
The deviations in the directions of air currents are evi- dently greater the higher the latitude, because near the poles a given distance along the earth's surface corresponds to an almost equal change in the distance from the axis of rotation, while at the equator there is no change in the dis- tance from the earth's axis. It might be supposed that in middle latitudes a moderate northward or southward dis- placement of the air would cause no appreciable change in its direction of motion. But a point on the equator moves eastward at the rate of over 1000 miles an hour, at latitude 60 degrees the eastward velocity is half as great, and at the pole it is zero. If it were not for friction with the earth's surface, a mass of air moving from latitude 40 degrees to latitude 45 degrees, a distance less than 350 miles, would acquire an eastward velocity with respect to the surface of the earth of over 40 miles an hour. The prevailing winds of the northern hemisphere in middle latitudes are to the northeast, and the eastward component has been found to be strong for the very high currents.
CH. Ill, 45] THE MOTIONS OF THE EARTH 87
Obviously the same principles apply to water currents and to air currents. Consequently water currents, such as rivers, tend to deviate toward the right in the northern hemisphere. It has been found by examining the Missis- sippi and Yukon rivers that the former to some extent, and the latter to a much greater extent, on the whole scour their right-hand banks.
All the proofs of the earth's rotation so far given depend upon the laws of motion. There is one independent reason for believing the earth rotates, though it falls a little short of proof. It has been found by observations involving only geometrical principles that the sun, moon, and planets are comparable to the earth in size, some being larger and others smaller. Direct observations with the telescope show ' that a number of these bodies rotate on their axes, the re- mainder being either very remote or otherwise unfavorably situated for observation. The conclusion by analogy is that the -earth also rotates.
45. The Uniformity of the Earth's Rotation. — It follows from the laws of motion, and in particular from the first law, that if the earth were subject to no external forces and were invariable in size, shape, and distribution of mass, it would rotate on its. axis with absolute uniformity. Since the earth is a fundamental means of measuring time its rotation cannot be tested by clocks. Its rotation might be compared with other celestial phenomena, but then the question of their uniformity would arise. The only re- course is to make an examination of the possible forces and changes in the earth which are capable of altering the rate of its rotation.
The earth is subject to the attractions of the sun, moon, and planets. But these attractions do not change its rate of rotation because the forces pulling on opposite sides balance, just as the earth's attraction for a rotating wh^el whose- plane is vertical neither retards nor accelerates its motion.
88 AN INTRODUCTION TO ASTRONOMY [ch. hi, 45
The earth is struck by milhons of small meteors daily coming in from all sides. They virtually act as a resisting medium and sHghtly retard its rotation, just as a top spin- ning in the air is retarded by the molecules impinging on it. But 'the mass of the earth is so large and the meteors are so small that, at their present rate of infall, the length of the day cannot be changed by this cause so much as a second in 100,000,000 years.
The moon and the sun generate tides in the water around the earth and the waves beat in upon the shores and are gradually destroyed by friction. The energy of the waves is transformed into heat. This means that something else has lost energy, and a mathematical treatment of the sub- ject shows that the earth has suffered the loss. Conse- quently its rotation is diminished. But as great and irre- sistible as the tides may be, their energies are insignificant compared to that of the rotating earth, and according to the work of MacMillan the day is not increasing in length from this cause more than one second in 500,000 years.
Before discussing the effects of a change in the size of the earth or in the distribution of its mass, it is necessary to explain a very important property of the motion of rotating bodies. It can be shown from the laws of motion that if a body is not subject to any exterior forces, its total quantity of rotation always remains the same no matter what changes may take place in the body itself. The quantity of rotation of a body, or moment of momentum, as it is technically called in mechanics, is the sum of the rotations of all its parts. The rotation of a single part, or particle, is the product of its mass, its distance from the axis of rotation passing through the center of gravity of the body, and the speed with which it is moving at right angles to the Une joining it to the axis of rotation. It can be shown that in the case of a body rotating as a solid, the quantity of rotation is proportional to the product of the square of the radius and the angular velocity of rotation, the angular velocity of
CH. Ill, 46] THE MOTIONS OF THE EARTH 89
rotation being the angle through which the body turns in a unit of time.
Now apply this principle of the conservation of the mo- ment of momentum to the earth. If it should lose heat and shrink so that its radius were diminished in length, then the angular velocity of rotation would increase,, for the product of the square of the radius and the rate of rotation must be constant. On the other hand, if the radio-active sub- stances in the earth should cause its temperature to rise and its radius to expand, then the rate of rotation would de- crease. Neither of these causes can make a sensible change in the rotation in 1,000,000 years. Similarly, if a river rising in low latitudes should carry sediment to higher lati- tudes and deposit it nearer the earth's axis, then the rate of rotation of' the earth would be increased. While such factors are theoretically effective in producing changes in the rotation of the earth, from a- practical point of view they are altogether negligible.
It follows from this discussion that there are some influ- ences tending to decrease the rate of the earth's rotation, and others tending to increase it, but that they are all so sinall as to have altogether inappreciable effects even in a period as long as 100,000 years.
46. The Variation of Latitude. — It was mentioned in connection with the discussion of the rigidity of the earth (Arts. 25, 26), that its axis of rotation is not exactly fixed. This does not mean that the direction of the axis changes, but that the position of the earth itself changes so that its axis of rotation continually pierces different parts of its surface. That is, the poles of the earth are not fixed points on its surface. Since the earth's equator is 90 degrees from its poles, the position of the equator also continually changes. Therefore the latitude of any fixed point on the surface of the earth undergoes continual variation. The fact was discovered by very accurate determinations of latitude, and for this reason is known as the variation of latitude.
90 AN INTRODUCTION TO ASTRONOMY [ch. hi, 46
The pole wanders from its mean position not more than 30 feet, corresponding to a change of latitude of 0.3 of a second of arc. This is such a small quantity that it can be measured only by the most refined means, and accounts
, Fig. 30. — The position of the pole from 1906 to 1913.
for the failure to discover it until the work of Chandler and Kiistner about 1885.
In 1891 Chandler took up the problem of finding from the observations how the pole actually moves. The varia- tion in its position is very complicated, Fig. 30 showing it
CH. Ill, 46] THE MOTIONS OF THE EARTH 91
from 1906 to 1913. Chandler found that this compHcated motion is the result of two simpler ones. The first is a yeady motion in an ellipse (Art. 53) whose longest radius is 14 feet and shortest radius 4 feet; and the second is a motion in a circle of radius 15 feet, which is described in about 428 days. More recent discussions, based on observa- tions secured by the cooperation of the astronomers of several countries, have modified these results to some extent and have added other minor terms.
The problem is to account for the variation of latitude and for the different periods. Unless a freely rotating ob- late rigid body is started turning exactly around its shortest axis, it will undergo an oscillation with respect to its axis of rotation in a period which depends upon its figure, mass, and speed of rotation. Hence it might be supposed that the earth in some way originally started rotating in this manner. But since the earth is not perfectly rigid and un- yielding, friction would in the course of time destroy the wabbling. In view of the fact that the earth is certainly many millions of years old, it seems that friction should long ago have reduced its rotation to sensible uniformity around a fixed axis, and this is true unless it is very elastic instead of being somewhat viscous. The tide experiment (Art. 25) proves that the earth is very elastic and suggests that perhaps the earth's present irregularities of rotation have been inherited from greater ones produced at the time of its origin, possibly by the falling together of scattered meteoric masses. But the fact that the earth has two dif- ferent variations of latitude of almost equal magnitude is opposed to this conclusion. The one which has the period of a year is probably produced by meteorological causes, as Jeffreys infers from a quantitative examination of the ques- tion. The one whose period is 428 days, the natural period of variation of latitude for a body having the dynamical properties of the earth, is probably the consequence of the other. In order to understand their relations consider a
92 AN INTRODUCTION TO ASTRONOMY' [ch. hi, 46
pendulum which naturally oscillates in seconds. Suppose it starts from rest and is disturbed by a small periodic force whose period is two thirds of a second. Presently it will be moving, not like an undisturbed pendulum, but with one oscillation in two thirds of a second, and with another oscillation having an approximately equal magnitude, in its natural period, or one second.
Euler showed about 1770 that if the earth were absolutely
rigid the natural period of oscillation of its pole would be 305
' days. The increase of period to 428 day's is due to the fact
that the earth yields partially to disturbing forces (Art. 25).
Many parts of the earth have experienced wide variations in climate during geological ages, and it has often been sug- gested that these great changes in temperature were pro- duced by the wandering of its poles. There are no known forces which could produce any greater variations in latitude than those which have been considered, and there is not the slightest probability that the earth's poles ever have been far from their present position on the surface of the earth.
47. Precession of the Equinoxes and Nutation. — There is one more phenomenon to be considered in connection with the rotation of the earth. In the variation of latitude the poles of the earth are slightly displaced on its surface; now the changes in the direction of its axis with respect to the stars are under consideration.
The axis of the earth can be changed in direction only by forces exterior to itself. The only important exterior forces to which the earth is subject are the attractions of the moon and sun. If the earth were a sphere, these bodies would have no i effect upon its axis of rotation, but its oblateness gives rise to very important consequences.
Let 0, Fig. 31, represent a point on the equator of the oblate earth, and suppose the moon M is in the plane of the meridian which passes through 0. The point 0 is moving in the direction OA as a consequence of the earth's rotation. The attraction of the moon for a particle at 0 is in the di-
CH. Ill, 47] THE MOTIONS OF THE EARTH
93
rection.OJf. By the resolution of forces (the inverse of the parallelogram of forces law) the force along OM can be re- solved in two others, one along OE and the other along the hne OB perpendicular to OE. The former of these two forces has no effect on the rotation ; the latter tends to move
Fig. 31. — The attraction of the moon for the earth's equatorial bulge causes the precession of the equinoxes.
the particle in the direction OB, and this tendency, combined with the velocity OA, causes it to move in the direction OC (the change is greatly exaggerated). Therefore the direc- tion of motion of 0 is changed; that is, the plane of the equator is changed.
The moon, however, attracts every particle in the equatorial bulge of the earth, and its effects vary with the position of the particles. It can be shown by a mathematical discus- sion that cannot be taken up here that the combined effect on the entire bulge is to change the plane of the equator. It is evident from Fig. 31 that the effect vanishes when the moon is in the plane of the, earth's equator. Therefore it is natural to take the plane of the moon's orbit as a plane of reference. These two planes intersect in a certain line whose position changes as the plane of the earth's equator is shifted. The plane of the earth's equator shifts in such a way that the angle between it and the plane of the moon's orbit is constant, while the line of intersection of the two planes ro-
94 AN INTRODUCTION TO ASTRONOMY [ch. hi, 47
tates in the direction opposite to that in which the earth turns on its axis.
The plane in which the sun moves is called the plane of the ecliptic, and the moon is always near this plane. For the moment neglect its departure from the plane of the ecliptic. Then the moon, and the sun similarly, cause the line of the intersection of the plane of the earth's equator and the plane of the ecUptic, called the line of the equinoxes, to rotate in