A TRIM. LATTICE AL. Hurriel Day Leut Jerrm 1905. A TREATISE ON ASTRONOMY. CAMBRIDGE, PRINTED BY MEICALFE AND CO. LIMITED, TRINITY STREET. A TREATISE ON ASTRONOMY FOR THE USE OF COLLEGES AND SCHOOLS. BY THE LATE HUGH GODFRAY, M.A., ST. JOHN'S COLLEGE, CAMBRIDGE; MATHEMATICAL LECTURES AT TRINITY HALL. Fifth Edition. London: MACMILLAN AND CO. 1894. PREFACE TO THE SECOND EDITION. Astronomy is a science admitting of so many methods of treatment, that a few words will be useful to explain the scope and object of the present work. It is essentially a student's book. Prepared originally for the use of the Author's pupils in the University, it embraces all those branches of the subject which have, from time to time, been recommended by the Board of Mathematical Studies; but the easier, and by far the larger, portion, adapted to the first three days of the Examination for Honours, may be read by the more advanced pupils in many of our schools. It is difficult to be original in an elementary work, and on a subject which has occupied men's thoughts from the earliest times. The Author's aim has been rather to convey clear and distinct ideas than to affect originality; and it is hoped that where he has deviated from the beaten track it has been at no sacrifice of simplicity. Before the invention of clocks the work of the practical astronomer was of a most wearisome nature. Every observation to determine the vi PREFACE. position of a celestial body would require the measurement of its distances from other bodies, whose positions were known; and this, besides being liable to a variety of instrumental and other errors, would involve long and tedious calculations. The perfection to which clocks have been brought has so greatly simplified this, by enabling the observer to make use of the uniform rotation of the earth as an element of observation, that the clock is almost as important an instrument in the observatory as the telescope. It was, therefore, thought that a short description of its construction and of the principles on which its accuracy depends would not be out of place in a Treatise on Astronomy. In this second edition the whole work has been very carefully revised, and numerous alterations and additions have been made. Some are merely verbal, others more important; but the order of the chapters and the general arrangement have been preserved. The Author desires to express his thanks for the valuable suggestions for which he is indebted to the kindness of several friends. CAMBRIDGE, May 1, 1874. CONTENTS. CHAPTER I. ✗
ART. PAGE
1—2 First appearances—Fixed stars 1
2—5 Rough determination of apparent motion 2
6 Greatest elongation—Visible horizon 4
7—8 Horizontal plane—Dip of horizon 5
9—10 Bifects of a change of station—the earth a polid 6
11 Celestial sphere—Sphere of the earth 8
12 Circumpolar stars—Meridian—North point, etc. 9
13—14 Versions—Altitude—Azimuth—Zenith distance—Culmination 10
15—21 The celestial sphere—Celestial sphere—Catalogue of stars
Celestial globe—Constellations11
17—18Spherical time—Hour angle14
CHAPTER II. ✗
ART. PAGE
19—22Torcential meridian—Axis of the earth—Poles of the earth—Torcential equator—Equatorial sphere—Right sphere17
23—24Observer on south side of the equator19
25Diameter of the earth (approx.)20
26Magnitude of the earth (approx.)21
27Magnitude of the earth (approx.)46.
28—30Latitude and Geographical—Geocentric—Co-latitude—the altitude of the vertical axis of the earth's rotation (approx.)22
31Longitude (approx.)23
32—34A revolution of the earth would produce the same appearance as a rotation of the stars (approx.)45.
35Arguments in favour of the earth's rotation (approx.)25.
36



































































A celestial globe showing constellations and stars. 22 45. 25. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. GEOGRAPHICAL DATA - © 2023 - ALL RIGHTS RESERVED - ANY REPRODUCTION WITHOUT WRITTEN PERMISSION IS STRICTLY PROHIBITED - THIS DOCUMENT IS FOR PERSONAL USE ONLY - DO NOT SELL OR REPRODUCE IN ANY FORM - ANY VIOLATION OF THIS AGREEMENT WILL BE SUBJECT TO LEGAL ACTION - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILED WITH THE COPYRIGHT OFFICE - ANY CLAIMS MUST BE FILELED viii CONTENTS. CHAPTER III. THE OBSERVATORY. CLOCKS. ART. 40 Want of precision in ancient observations 41 Greater accuracy of modern instruments and modes of observation 42 Clocks 43 The "masonic" or "halonie" clock 45 Its defects 46 The "pendulum" clock--Note on cycloidal pendulum 47 The "spring" or "lever" for re-winding 48 The "anular" escapement 49 Compensating pendulum 50 The "mercurial" pendulum 51-52 Mercurial pendulum--Its advantages 53 Conical pendulum--Springing governor PAGE 34 35 46 48 49 50 51 52 53 54 55 56 57 58 Collimation error--Adjustment 59-60 Correction of level error 61-62 Deviation error--Adjustment 63 Setting circle 64-65 Refracting telescopes 66 Collimating eye-piece 67 Reduction of observations to the mean wire 68-80 Mean wire--Correction of intervals of the wire 70 Correction of an imperfect observation. 71 Parallax error. 72 Ridge-toe instrument. 73 Determination of uncorrected level error. 74-75 Correction of parallax error. 76 Division error, first method. 77-82 Division error, second method. 83 Parallax error--Accidental errors, &c. CHAPTER IV. THE OBSERVATORY CONTINUED. THE TRANSIT INSTRUMENT. OBJECT OF THE TRANSIT--General description Mean of the wires--Line of collimation Collimation error--Correction by the transit Errors of the transit Collimation error--Adjustment Correction of a level error by the transit Correction of level error by the transit. Deviation error--Adjustment. Setting circle. Refracting telescopes. Collimating eye-piece. Reduction of observations to the mean wire. Correction of intervals of the wire. Correction of an imperfect observation. Parallax error. Ridge-toe instrument. Determination of uncorrected level error. Correction of parallax error. Division error, first method. Division error, second method. CHAPTER V. THE OBSERVATORY CONTINUED. THE MEAL CIRCLE. DESCRIPTION OF THE MURAL.--Zenith point. Micrometer screw-head.--Graduation of the mural in any position. Line of collimation.--Micrometer wire--Determination of values of the screw-head graduations. CHAPTER VI. THE OBSERVATORY CONTINUED. THE TRANSIT INSTRUMENT. Description of the transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.--The transit.- CONTENTS. ix | AWF | PAGE | |---|---| | 87 | Clump-Tongued-screw | | 88 | Error of runa | | 89 | Errors of adjustment of mural telescope | | 90 | Distortion of the mirror, illumination and deviation errors | | 93 | Errors of centring--Graduation, etc., eliminated or diminished by means of a second mirror | | 94 | Mode of observing with the mural--To find the difference of declination of two stars | | 95 | To find the difference of altitude of a star | | 96 | Determination of zenith-point | | 97–98 | " Double observation " with the mural | | 99 | Zenith-point determined by estimating eye-piece | | 100 | Zenith-point | CHAPTER VI. THE OBSERVATORY CONTINUER. VARIOUS INSTRUMENTS. 101 The transit circle . 102–103 The equatorial 104 Its use for determining a star's declination--Connection with clock work 105 Adjustment of the equatorial 106 Chiefly used in differential observations--Excellency of North-South adjustment 107 Wire-Position micrometer 108 Finder 109 The e-astrum--The transit in the prime vertical 110 The ring-micrometer 111 The sextant 112–115 Double-image microscopes--Dolland's--Recon's CHAPTER VII. THE SUN. 116–117 Tracing the sun's path among the stars--Methods employed by the ancients 118 Sun's semi-diameter--Its changes 119 To find the sun's declination by observation 120 To find the distance of right ascension between the sun and any fixed star 121 Determination of the semiparallel path 122 Equinoctial points--Eccentric points--First points of zodi- cals Sublunary points--Sublunary cusp 123 Yearly variation of the sun's path 124 First case at a place between 23° 27' and 66° 32' of north latitude--Equinox--Solution | AWF | PAGE | |---|---| | 74 | 74 | | 75 | 75 | | 76 | 76 | | 77 | 77 | | 78 | 78 | | 79 | 79 | | 80 | 80 | | 81 | 81 | | 82 | 82 | | 83 | 83 | | 84 | 84 | | 85 | 85 | | 86 | 86 | | 87 | 87 | | 91 | 91 | | 93 | 93 | | 94 | 94 | | 95 | 95 | | 96 | 96 | | 97 | 97 | X CONTENTS. ART. 125-126 Second case: at a place whose north latitude exceeds 60° 23' 4" 127-129 Third case: at a place whose north latitude is less than 30° 27' 4" 130 The same phenomena explained by attributing an annual motion to the earth-Change of declination explained on this supposition 131 Change of right ascension explained on the same supposition 132 Arguments in its favour 133 Arguments which may be employed for convenience or simplicity of explanation PAGES 99 101 102 103 105 106 107 108 CHAPTER VIII THE SEASONS. THE SEASONS Their lengths at present Temperatures of different parts of the earth, as affected by the seasons Zones of the earth-Tropical zone-Tropics-Arcule and antarctic circles-Frigid zones-Temperate zones The effects of winds on the temperature-Ocean currents-The Gulf stream Trade winds CHAPTER IX FINDING THE LATITUDE BY OBSERVATION. PRECISION OF A FIXED OBSERVATORY NEITHER ATTAINABLE NOR NECESSARY FOR THIS PURPOSE CLASSIFICATION OF METHOD Meridian observations-First method. By a circular arc-race-measured by the sun or a star when the declination is known Ex-meridian observations-Second method. By simultaneous altitudes of the sun and the elapsing time Allowance for the change of the ship's position in the interval between observations REMARKS ON THE USE OF THE THIRD AND FOURTH METHODS Fifth method. By observations made with the transit in the prime meridian Circum-meridian observations-Sixth method. By a heavenly body moving in a circle round the pole EXTENSION OF THIS METHOD Seventh method. By the polar star CHAPTER X THE LATITUDE BY MEANS OF THE STARS. FIRST METHOD. BY MEANS OF THE SUN. SECOND METHOD. BY MEANS OF THE MOON. THIRD METHOD. BY MEANS OF THE ECLIPTIC PLANETS. FOURTH METHOD. BY MEANS OF THE ECLIPTIC STARS. FIFTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON. SIXTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN. SEVENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS. EIGHTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS. NINTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON. TENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN. ELEVENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS. TWELFTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS. THIRTEENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON. FOURTEENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND THE ECLIPTIC STARS AND THE MOON AND THE SUN. FIFTEENTH METHOD. BY MEANS OF THE ECLIPTIC STARS AND THE MOON AND THE SUN AND THE ECLIPTIC PLANETS AND The CONTENTS. CHAPTER X. THE ECLIPTIC. CHAPTER XL. FORM OF THE EARTH'S ORBIT.
485. Position of first point of Y determined—First method PAGE
158 130
150 Second method (Flamsteed's) 131
77 Third method—This determines the obliquity 132
161 Advantages of Flamsteed's method 133
163-164 Determination of the obliquity of the ecliptic 134
165-168 Determination of the obliquity by the mean nodal 135
167-168 Conversion of right ascension and declination into latitude and longitude, and vice versa 136
169 Cumulative consequences of the sun's rights ascension and declination 137
170 Rise and setting of the meridian 138
171-172 To determine the position of the ecliptic relatively to the horizon at a given instant of a given day—Ascending point—Descending point—Change in declination during one day 139
173 Limiting positions of the ascending point 140
175 Longitude of the mean node 141
176 Distance of the sun variable—Angular velocity also variable PAGE
177 Equable distance between sun and earth (mean law) 142
178 The orbit an ellipse (Kepler's first law) 143
179 Ferigoe-Apogee-Percellation-Apseion-Line of apsides 145
180-181 Determination of the eccentricity by observation—Orbital elements—Apogee—Perigee—Percellation—Apsis—Line of apsides—Eccentricity—Mean anomaly—True anomaly—Mean longitude—True longitude—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—True anomaly—Mean anomaly—but not true longitude but mean longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude—is used in this place because it is more convenient than true longitude-is XII CONTENTS.
A.B.T. Page
197 Maximum values of first part 156
198 Second part 157
199 Equatorial time 158
200 Different kinds of years 158
CHAPTER XIII. REDUCTION AND CONVERSION OF TIME, FINDING THE TIME BY OBSERVATION. 204-202 To find the time under any meridian corresponding to a given time at some other meridian. 203 Gain or loss of a day in going round the earth 204 Into a mean solar interval into its equivalent alerdal interval 205 Into an alerdal interval into time, and conversely 206 To convert mean solar time at a given meridian into the corresponding alerdal time, and conversely 207 To convert apparent time into alerdal time, and conversely 208 To find the Time by Observation. First method--By meridian observation 209 Second method--By a single altitude of the sun or a known star 210 Third method--By two altitudes of the sun and the chasped time, or by simultaneous observations of two stars 211 Fourth method--By the equal altitudes 212 Methods used at sea CHAPTER XIV. VARIOUS PROBLEMS CONNECTED WITH THE DIURNAL MOTION. 217-218 To determine the times of rising or setting of a known body 219 To find the time occupied by the sun in rising, 220-222 To determine the lengths of day and night--Correction for latitude 223 To find when a body has its greatest altitude 224-227 To find the duration of twilight 228 To find on what day twilight occurs Anacrita. 230 Variation of the compass--Deviation 231 To find the azimuth of the sun at sun-rise or sun-set 232 To find the sun's azimuth by observation of altitude 233 At a given time 234 To find when the motion of a star is vertical Sun-dials
Page
171
172
173
174
175
176
178
186
179
180
181
CONTENTS. Xiii CHAPTER XV. ASTRONOMICAL REFRACTION.
AUT. PAGE
235 Law of refraction--The earth's atmosphere 185
236-237 General differential equation 184
238 Simpson's formula 186
239 Bessel's formula 187
240 Cusin's formula 188
241 Correction for barometer and thermometer 189
242 Correction determined by observation 189
243 Terrestrial refraction 190
244-245 Effect of refraction on the dip and on the distance of the visible 191
246-248 Connection of the dices of the sun and moon by refraction 192
249 Effect of refraction on rising and setting of bodies 194
CHAPTER XVI. FIGURE OF THE EARTH. PARALLAX.
250-253 The earth an oblate spheroid--Its compression--Reduction of geographical to geocentric latitude 195
254 Parallax of a heavenly body 197
255 Parallax when the earth considered spherical 198
256 Parallax in declination and in hour angle 199
257-258 Parallax when the earth considered oblongum 199
259-261 Parallax determined by meridian observations 201
262-263 Distances and magnitudes of sun and moon 204
264-265 Annual parallax of star 206
266-267 " determined by observation 207
268 Distances of some of the stars 210
269-270 Solar parallax--Motion of the solar system--Central sun 210
CHAPTER XVII. ABERRATION.
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Velocity of light compared with that of the earth





























































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Statement of the phenomenon of aberration

Value of coefficient of aberration

Effect of aberration on the place of a star

Aberration in right ascension and declination

" right ascension and declination

" aberration of y. Perseus

Solar, lunar, and planetary aberration

Dermal aberration

Statement of the phenomenon of aberration

Value of coefficient of aberration

Effect of aberration on the place of a star

Aberration in right ascension and declination

" right ascension and declination

" aberration of y. Perseus

Solar, lunar, and planetary aberration

Dermal aberration

AUT.236
CHAPTER XIX. THE MOON.
PAGE
205 Period of a revolution 239
204 Lamination or synodic period—Conjunction—Opposition—Quad- rature—Osculation—Syzygies 239
205 Distance of the moon's distance from the earth Phases 240
206—207 Phases of light 241
209 Direction of the line joining the cusps Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
209—301 Magnitude of the phase 243
Cycles and occultations.
CONTENTS. XV CHAPTER XXI. FIXING THE LONGITUDE BY OBSERVATION.
AUT. PAGE
382 The problem depends on that of finding the time at Greenwich 217
383 To find the longitude by chronometers 204
384-386 By electric telegraph 209
387 By celestial signals 209
388-387 By proportional logarithms 201
389-390 By moon-eclininging stars 267
406 The longitude may be found by eclipses and occultations 209
CHAPTER XXII. THE PLANETS.
AUT. PAGE
352 Planets known to the ancients--modern discoveries 270
353 Apparent complexity of the planetary motions 271
354 Similitude of the motions of the planets--Distances and periods of the principal planets--Bode's law (note) 272
355 Heliocentric and geocentric places of a planet--Curate distances 273
356 Elliptical orbits of the planets 274
357 How these elements may be determined 275
358 To find the periodic time of a planet 276
359 Solar-moon conjunctions 46
461 Transit of Venus--How employed to determine the sun's parallax 279
462 Delisle's and Halley's methods 285
463 Mercurial transits--Transit of Mercury in 1761 and 1769 284
467 Recurrence of transit of Venus 46
468 Phases of the planets 286
469-470 When Venus appears brightest 46
APPENDIX. THE CALENDAR. | AUT. | PAGE | |------|------| | 371 Connection between the year of Rome and the Christian era | 919 | | 372 Civil year | 990 | | 373 Julian calendar--Gregorian calendar--Furman calendar (note) | 291 | | 374 Golden number--Morton's lunar cycle | 292 | | 375 Menz's age--Epact | 250 |
THE TIDES. The phenomenon described--The cause found in the attraction of the sun and moon | AUT. | PAGE | |------|------| | 577 Laplace's wave-theory and Bernoulli's equilibrium-theory | 295 | | 578 The effect of the tides on the equilibrium-theory | oh. | | 579 Spring tides--Neap tides--Tide-waves lagging | 297 | | 580 Establishment of the port--Cotidal line | 298 | | 581 High and flood--slack water | oh. | | 582 Relative effects of the sun and moon on the tides | oh. | | 583 The cause found in the attraction of the sun and moon | 301 |
A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and Tides. A diagram showing the relationship between the Earth, Moon, Sun, and XVI CONTENTS. GREAT-CIRCLE AND MERCATOR'S CHARTS. A.P.E. 368 Object of charts-The hododrome 384 Great-circle charts 385-387 Mercator's charts EXAMPLES AND PROBLEMS. Chapters I.-VI. III.--XIII. XIV. XV.--XVIII. XIX.--APPENDIX. For an elementary course, the student may confine his attention to the following articles: 1--35, 40--66, 71--73, 83--147, 158, 159, 161, 163, 165, 166, 170, 171, 176--184, 189, 190--195, 198--205, 204, 225, 234, 235, 240, 241, 246--248, 250, 254--256, 260--263, 271, 272, 273--275, 280--283, 290, 291, 292, 293--300, 311--317, 348--351, 353--359, 361--364, 366--368, 371--385. PAGE 302 203 8 306 308 311 315 318 **ASTRONOMY.** CHAPTER I. THE STARS. 1. Suppose an observer, on a clear winter evening, to station himself on some elevation whence he may have an unobstructed view of the heavens. He will see a large number of stars, of different degrees of brilliancy, scattered without apparent law over the celestial vault. He will imagine the shape of the vault to be spherical, and although the idea may be found difficult at first, it will probably be vague and even different at different times yet the parts over head will not seem so far off as those near the horizon, the effect being the same as if the visible portion were a segment of a sphere less than a hemisphere.* 2. After some time the observer will notice that a change has taken place; some of the stars have disappeared below the horizon on one side, and new ones have arisen in the * That this spherical boundary has no real existence we shall have full proof when we find that the sun, the moon, the planets, the stars, &c., which, to us all seem situated in the surface are at immensely different distances. The sun is not only not nothing but an imaginary background formed by the atmosphere that surrounds it, but also its light is reflected back to us from the parts near the horizon is an optical illusion, due probably to the fact that the greater part of our light comes from the sun itself and reflects some absolute more of their light, and gives them an amount of indistinguishable, such as we know to be the effect of greater distance in the case of those objects which are accessible to us. E 2 THE STARS. [CHAP. L] opposite quarter. But he will remark that there is no alteration in their relative positions; and if, by means of a sextant, or any other instrument for measuring angular distances, he determines the angle between any two stars, he will find it to remain constant, and this, not only throughout that night, but after any interval of months or years.* The motion is, therefore, a general one of the whole system of stars, and in consequence of their fixed relative positions they are called *fixed stars*, to distinguish them from the planeted, or wandering stars, a small number of bodies, whose positions among the others are continually changing. 3. To determine roughly the nature of this general motion, let the observer be furnished with a number of threads or wires of equal lengths, and having a common fixed extremity, and let one of these threads be directed towards any arbitrarily chosen fixed star and there made fast; after any interval of time let a second thread be directed to the same star; a third after another equal interval, and so on. It will be found that the ends of the * Another curious illusion may here be noticed. Anyone watching the night sky could not fail to perceive that the distance between two neighbouring stars appears greater when they are near each other than when they are far apart. This effect is much more striking, as observed in the case of the full moon, which seems very much larger when rising than when high up in the heavens. This illusion is only apparent, and depends upon the fact that we see the moon through its own atmosphere. We see the moon attached, as it were, to the spherical canopy that surrounds us and, consequently, when near the horizon occupying that part which appears darkened by our atmosphere. In reality the moon is at rest; but owing to this appearance, therefore, that its actual magnitude does not alter, we should look for a corresponding increase in the angle which it subtends, but contrary to expectation, it remains stationary. Hence it is easy to understand how one may resist the impression that the full moon near the horizon is a much larger body than when high up in the heavens. The same principle applies of course to the apparent change of distance between neighbouring stars. Some of the stars have been ascertained to have a slow "proper motion," but to determine whether this motion is uniform or not requires observations over a long interval of time; and there is every reason to believe that, as our methods of observation still further improve, it will be found that no star can be strictly called fixed. (Chap. xvi.) CHAP. L.] THE STARS. 3 threads will all be situated in the circumference of a circle, or, which is the same thing, the threads themselves will lie in the surface of a right cone with a circular base; the angle of the cone will be different for different stars, but whatever star may have been chosen, the axis of all the cones will always be found to have a common direction inclined to the horizontal plane at an invariable angle, so long as the observer does not change his place of obser- vation. 4. Suppose $OP$ to be the direction of this common axis above the plane, then those stars which are furthest from $OP$ will only just appear above the horizon ; others will shew but a small portion of the cone as $A$, and as we approach A diagram showing three cones with their axes $OP$, $OQ$, and $OR$. The cones are tilted at different angles relative to each other and to the horizontal plane. The points where the cones intersect the plane are labeled A, B, C, D, E, F. a larger portion of each cone is gradually obtained as at $B$, $C$, and some of the stars will not set at all, but will give a cone completely above the horizon as at $D$. In the case of those stars whose direction is at right angles to the axis $OP$, the cone becomes a plane as at $B$. These observations point to the conclusion that the stars have a motion of revolution round this axis $OP$. The re- turn of day will, it is true, arrest the observations before the circuits have been completed ; but night after night the 32 4 THE STARS. [CHAP. I. same stars will be found going over the very same courses, except that each night they begin a few minutes earlier than the night before; and if the observations be repeated a few months later, the stars will be then seen completing those parts of their orbits above the horizon which the day-light has previously hindered us from following. 5. Furthermore, it will be found that the consecutive threads belonging to the same star, provided they be fixed, as we have supposed, at equal intervals of time, are separated by equal times, and therefore the motion of each star is uniform. The angular velocity about the axis will also be the same for all the stars ; because, consistently with what has been said before of their unaltered relative posi- tions, they will all be found to accomplish their revolutions in precisely the same time. The observations here indicated are very rough, but more accurate methods will fully confirm the results, if proper allowance be made for the slight distortion of the path caused by refraction, as hereafter explained. Visible horizon, Dip of the horizon. 6. Before proceeding further it will be useful to ascer- tain, at least approximately, the shape of the earth, from which the observer has to make all his observations. From the place he occupies he can see only a very limited portion of the surface, but he will be able to remark that the apparent boundary line where the heavens and earth seem to meet, or the visible horizon, as it is called, is circular, except where broken by hills or other irregu- larities; and if he has a calm sea for a boundary line this will seem perfect. This appearance he may find to follow him in whatever direction, and however far, he may travel ; the visible horizon is circular in all places and for all heights above the sea-level. **GRAP. I.]** THE STARS. 5 Now a sphere is the only surface which has this pro- perty of always giving a circle as the curve of contact of a number of tangent lines drawn from any external point. Therefore, so far as this rough test goes, the earth is a sphere limited in extent.* Without leaving his first station, the observer would be able to infer from the disappearance of some stars below the horizon on one side, and their re-appearance the next night on the other, having evidently passed under the earth in the interval, that the earth is isolated in space. The fact also of its having been so often circumnavigated proves this, and proves its limited extent as well. 7. Let $DFC$ represent the earth regarded as accurately a sphere, $C$ its centre, $A$ the eye of the observer, $DEF$ the visible horizon. Let $AB$ be the direction of a plumb line, which will be normal to the surface of still water at $B$, and therefore will pass through the centre $C$. $ABC$ is called a vertical line at $B$. A plane at right angles to this vertical line through any point of it is called the horizontal plane at that point. When drawn through $B$ it is called sensible, and when through $C$ rational, but the distinction will not be of any importance. * We shall hereafter see that it is in reality an oblate spheroid, differing however very little from a sphere, the polar diameter being 7980 miles long, and the equatorial diameter 7996 miles. To explain more fully the nature of the surface may be added the way in which a ship, sailing in any direction, is gradually lost sight of as it recrosses from the observer : the hull first disappears, then the lower mast and only, after a short time, the topmast. The appearance of such a ship projected on the moon in a lunar eclipse, is that which would be produced by a globular body. A diagram showing a sphere with a plumb line AB passing through C (the center) and intersecting DEF (the visible horizon). Points D and F are on the circumference of the sphere. 6 THE STARS. [CHAP. I. 8. The angle which $AD$ makes with the horizontal plane through $A$ is called the depression or dip of the horizon. It varies with the height of $AB$ and is the complement of the angle $DAC$, and therefore equal to the angle at the centre $DCA$. If $AB = h$, $BC = r$, $ACD = \theta$, and the arc $BD = x$, we find $$\cos \theta = \frac{r}{r + h}, x = r\theta,$$ whence the dip and the distance of the visible horizon will be known when the height above the surface and the radius of the earth are known.* If $h$ is small, then approximately $$\theta = \sqrt{\left(\frac{2h}{r}\right)} \text{ and } x = \sqrt{(2rh)}.$$ Effects of a change of station. 9. Let us now return to the observations on the stars, and suppose that the observer proceeds to some new station, and there recommences his series of operations as in Art. 3. He may perhaps discover some new portions of the heavens containing new stars, or he may lose some which he could see before; but he will find, however far he may have travelled, that all the stars previously observed retain their relative positions, so that they appear of exactly the same magnitude as before, in precisely the same time as before; and that the only change is in the inclination of the common axis of those cones to the horizontal. * On account of refraction these results will not be exactly correct. Rays from a point beyond $D$ will be seen and reach it in a direction above $D$. The angle of depression $\phi$ will therefore be diminished, and the visible distance $x$ increased; $\phi - \psi$ will be found to be reduced value of the one, and $x + \psi$ (the increased value) of the other. See Art. 30. A diagram showing angles and distances related to star observations. CHAP. I.] THE STARS. 7 plane, which inclination may be either greater or less ac- cording to the direction in which he has gone. Therefore the axis about which the stars revolve seems to pass through every place on the earth's surface, and the stars preserve the same position relatively to it and to one another for every observer. 10. But this forces upon us another conclusion. If the angular distances between the stars do not alter, however much we may change our position on the earth, it must be that the earth is a mere point, compared with the dis- tances of the stars. For, if two stars A, B, seen under the same angle from several points Q, R, S, ... then the points Q, R, S, ... will be situated in the surface generated by the revolution of the segment of the circle $A\overline{P}A'$ about the chord $AA'$; and if a new star $A''$ be observed under the same conditions, the points Q, R, S, ... will be also in the surfaces similarly described on the chords $AA'', AA''$: the same for a fourth star $A'''$, and so on; but P is the only point common to all these surfaces; therefore it is impossible that the angles subtended by the stars should really be the same at all places. The explanation is that the distances of these places from one another are too small, compared with the actual distances of the stars, to produce changes in the angles large enough to be de- tected even with the most refined instruments, and we shall hereafter see that angular changes due to displac- ments many thousand times greater are still too minute for our observation. The earth is, then, a mere space in space as regards the distances of the fixed stars : from all points on or within its surface, lines drawn to the same fixed star at the same instant will be parallel, and all the axes about which the heavens seem to revolve will remain fixed and parallel to one another. 8 **THE STARS.** [CHAIR. L] **Celestial sphere.** 11. Let the observer now surround himself with an imaginary sphere, having a radius perfectly arbitrary—whether a hundred yards, or a hundred times the distance of the furthest star, is immaterial, but perhaps his conceptions will be easier with the smaller radius—this will be his **celestial sphere**, which we must suppose to accompany him in all changes of position in such a manner that he may always occupy the centre. To this sphere he will refer the places of the heavenly bodies at any instant, by the points where the lines joining them with the centre cross the surface.* Thus if $HJKL$ represent his celestial sphere at the station of his first observations, the centre $O$ coinciding with the vertex of the cones in which lie the paths of the stars which lie respectively in the surfaces of the cones $A$, $B$, $C$, $D$ will be represented by the intersections of these cones with the surface of the celestial sphere, that is, by the circles $AA'A',BB'B',\text{etc}$. * It has been generally customary to define the celestial sphere as one having an indefinitely large radius. The conclusions obtained will be just the same, but when dealing with finite space this infinite space have both to be represented in the figure the reasoning may vary. It is only for the convenience of substituting spherical triangles for solid angles that we consider a part of this sphere as representing a surface in the sky, and therefore this sphere does not represent anything, not even apparent half of the heavens, for that seems nearest to us in the portion near the zenith above our heads. It will be found that all problems of Astronomy, in which the Celestial sphere is used, concern only the plane and the dihedral angles which the objects subtend at its centre. A diagram showing a celestial sphere with four cones labeled A, B, C, D intersecting at points A', B', C', D' on a line through O (the center). The lines AA', BB', CC', DD' represent circles on the surface of the celestial sphere. CHAP. I.] THE STARS. 9 The axis of rotation $OP$, the common axis all the cones, will be perpendicular to the planes of these circles ; these planes are therefore all parallel to one another. The horizontal plane through $O$ will give the great circle $HA'B'C_R$, which is called the horizon. The point in the heavens vertically above the observer is called his zenith and the opposite point the nadir. They are given in the celestial sphere by the vertical line $ZON$. The points $P$, $P'$, where the axis meets the celestial sphere, are called its poles : that one which is above the horizon in these regions being called the north pole, and the other the south pole. The circles described by the stars are called parallels, that one $B'BP$ whose plane passes through the centre of the sphere making the name of equator. The equator is therefore the great circle whose plane is at right angles to the axis of the celestial sphere. 12. The portions of the celestial sphere situated above the horizon will correspond to the visible part of the heavens. The stars situated in the equator $BB'P$ have their paths bisected by the horizon, and are therefore as long above as below it; those whose paths are further from the elevated pole will be a shorter time above the horizon, those which are nearer to it a longer time; the duration in every case being the same fraction of the whole diurnal period that the arc of the parallel above the horizon is of the whole circumference of that parallel. Those stars whose parallels are entirely above the horizon, and which consequently never set, are called circumpolar stars. The great circle $HZPR$ which passes through the zenith $Z$ and the pole $P$ is called the meridian of the observer. The straight line $HOR$, in which the plane of the meridian meets the plane of the horizon, is called the meridian line, and the points $R$ and $H$ the north and south points. 10 THE STARS. [CHAP. 1.] The horizontal line which is perpendicular to the north and south line is called the *east* and *west* line, and its extremities the east and west points. Since the meridian plane contains $OP$ it will be perpendicular to the plane of the equator, and because it contains $OZ$ it will also be perpendicular to the plane of the horizon, and therefore $B'F$, the intersection of these planes, will be perpendicular to the meridian plane, and will coincide with the east and west line. Hence the intersections of the equator and horizon determine the east and west points. Altitude, Azimuth, Zenith distance, Culmination. 13. It is often necessary to describe the position which a star or other heavenly body occupies at a given instant on the observer's celestial sphere. This may be done by referring it to the meridian and horizon. Let the figure represent the upper half of the observer's celestial sphere; $Z$, $P$ the zenith and pole; $HOR$ the meridian line; $E$, $W$ the east and west points. Every plane through $OZ$ will be a vertical plane, and its intersection with the celestial sphere is called a vertical circle or, simply, a vertical. Of these, the vertical which passes through the east and west points, and is perpendicular to the meridian, is called the *prime vertical*. Let $ZST$ be a vertical passing through a star $S$, whose diurnal path is the circle $ASMA$. It is obvious that the position of the star will be known when we know the angle $MZZ$, between the meridian and the vertical, and the arc A diagram showing a celestial sphere with a star labeled S. The prime vertical ZST passes through S. The meridian line HOR intersects ZST at point T. Points E and W are labeled for east and west points respectively. CHAP. I.] THE STARS. 11 T/S which measures the angle TOS between the star and the horizon. The angle MZS, or its equivalent the arc HT, is called the azimuth, and is usually reckoned from the south towards the east or west when the north pole is above the horizon, and from the north in the other case. The other angle TOS is called the altitude of the star. The angle ZOS, the complement of the altitude, is called the zenith distance, and may replace the altitude as one of the coordinates of the star's position. 14. The zenith distance diminishes as the star approaches the meridian ; for, if SP be joined by an arc of a great circle, the spherical triangle ZSP will have the two sides SP, PZ constant while the angle P diminishes, therefore the third side ZS also diminishes, and the star attains its greatest altitude in the meridian. It is then said to culminate. Those stars which never set will cross the meridian again below the pole, and have there their least altitude. The culmination of a celestial body is the instant of its attaining its greatest altitude. In the case of a star this coincides with-the meridian passage, but for a body whose distance from the pole is changing the culmination may take place a little before or a little after. See Art. 215. Declination and Right Ascension. 15. The altitude and azimuth, which thus define the position of a heavenly body, refer to one particular instant and one particular place of observation. Not only are these elements undergoing constant and rapid changes, owing to the diurnal motions, but they also differ, as we shall * Sometimes the supplementary angle PES is called the Azimuth, but all ambiguity is avoided by specifying the point from which it is reckoned : thus N. 80° E. is the same as S. 100° W. * 12 THE STARS. [CHAP. L. presently find, at the same instant, for observers at different places. We may, however, by referring the places of the heavenly bodies to the equator instead of the horizon, obtain elements entirely independent of the observer's position ; and, as far as the fixed stars are concerned, nearly inde- pendent of the time, their very slow change, to which we shall have to refer hereafter, amounting to only a few seconds of arc in the course of a year. The declination of a star is its distance from the equator measured by the arc of great circle which passes through that star and the pole. This great circle is called the star's declination circle, and accompanies the star in its diurnal course. The declination is either north or south, according to the side of the celestial equator on which the star is situated. Thus if $EQ$ be the celestial equator, $P$ the north pole, $P$ the south pole, $A$, $B$ two stars whose declination circles are $PA\alpha P$, $PB\beta P$, then $\alpha A$, $\beta B$ are the declinations of these stars, the first being north and the second south. The polar distance is the complement of the declination, therefore $PA$ and $PB$ are the north polar distances of the two stars, the one being less, the other greater than 90°. 16. The right ascension of a star is the angle made by its declination circle with that of some determine point in the celestial equator. It is measured by the arc of the equator intercepted between them, reckoned eastward through 360°. Thus if $M$ be the chosen point, $PM'\alpha$ its declination circle, then the right ascension of the stars $A$ and $B$ will be $\alpha M$, $\beta M$ A diagram showing a celestial sphere with a star labeled "A" and another labeled "B". The line connecting "A" and "B" intersects a line labeled "P" (the north pole) at an angle. Another line labeled "Q" (the celestial equator) intersects this line at an angle. The declination circle for "A" is shown as an arc around "A", and similarly for "B". The declination circle for "A" is labeled as $\alpha A$, and similarly for "B" as $\beta B$. The polar distance for "A" is shown as an arc around "A", and similarly for "B". The polar distance for "A" is labeled as $\alpha M$, and similarly for "B" as $\beta M". CHAP. I.] THE STARS. 13 respectively. The difference of right ascension will be $ab$, which is independent of $M$. Since the right ascension and declination of a star are coordinates of its position at the time, a register of these, together with their annual change, if any be found, will enable us to identify a star once observed. Such a register is called a Catalogue of Stars, and its correctness is of the highest importance. Astronomy, by delicate instruments and means of observation, which we shall have to describe hereafter (Chaps. III., IV.), are especially intended for the determination and extension of this register. When very great accuracy is not required, the register may be a globe on which the circles corresponding to the equator and the different declination circles and parallels are traced at certain intervals apart, and the stars are marked on this globe in positions corresponding to those they occupy in the celestial sphere. Instead of a globe a plane surface may be used, on which the positions are mapped according to certain rules of projection. This, though a less faithful representation of the heavens than the globe, will be more convenient for many purposes. Many of the stars have received particular names, but their number precludes the possibility of doing this for all of them,—so the ancients very early divided them into groups, or constellations, as they are called, to which they attached names; but in some cases suggested by fanciful resemblances figures of men and animals, and in others in a very arbitrary and confusing manner. The stars of each constellation are named in order of brilliancy by letters of the Greek Alphabet attached to them, as $\alpha$ Tauri, $\beta$ Orionis, &c. We must leave the particular declination circle, whence the right ascensions are measured, to be chosen hereafter. For reasons to be then explained, it will be convenient to take it not passing through any particular star, but through 14 THE STARS. [CHAP. L a point which, though partaking of the general diurnal motion, is really altering its position with respect to them, although at a rate so slow that the whole amount of change is only a few seconds annually. Sidereal Day, Sidereal Time, Hour Angle. 17. The uniform revolution of the whole system of stars around the polar axis takes place from east to west, and is accomplished in what is called a sidereal day. This day is always longer than an ordinary day, and clocks adjusted to keep sidereal time are called sidereal clocks. The whole day is divided into 24 hours, and the sidereal clock reckons onward from 1.33... to 24. The sidereal clock is so adjusted as to mark 0h, 0m, 0s, when the selected point (Art. 16), from which the right ascensions are reckoned, crosses the meridian of the observer. This point is called the first point of Aries, so that the correct definition of a sidereal day is "the interval between two consecutive transits of the first point of Aries;" and the sidereal time at any instant is "the number of sidereal hours, minutes, &c., since the last preceding transit of the first point of Aries." It will be at once obvious that the different stars will succeed each other across the meridian in the order of their right ascension; and that, on account of the uniformity of the motion, the right ascension of each star, when reduced to time at the rate of 24 hours for 360° or 1 hour for 15°, will give the sidereal time when that star comes to the meridian. 18. The angle which the declination circle of a star makes with the meridian, viz. the angle $SPZ$, is called the hour angle. The hour angle $SPZ$ and the polar distance $SP$ may be CRAP. 1.] THE STARS. 15 employed as coordinates of the star's position on the celestial sphere of the observer instead of the altitude and azimuth. A knowledge of the hour angle of a star will, on account of the uniform rate at which it varies, at once give the time the star will take to reach the meridian if it be on the east side of it, or the time elapsed since it crossed the meridian if it be on the west side. When a star is in the meridian one-half of its visible path is accomplished. Thus if $AMA'$ be the parallel described by the star, $A$, $A'$ being the points of rising and setting respectively, $HZPR$ the meridian, the two spherical triangles $APH$, $A'PH$ are right-angled at $H$, have the side $PH$ common, and $PA = PA'$ since the star's polar distance does not change, therefore the hour angle $APM$ at rising equals the hour angle $A'PM$ at setting. A diagram showing a star's position on a celestial sphere with lines indicating rising and setting times. 16 THE STARS. [CHAP. I.] For the convenience of reference, most of the terms introduced in this first chapter are here collected, and their connection shown on a single figure. Observer's celestial sphere. A diagram showing the observer's celestial sphere with various lines and points labeled. O. Observer. REHW...Horizon. Z...Zenith. S...Star. AMA'...Parallel of star. OP...Polar axis. P...North pole. HZR...Meridian. HR...Meridian line. R...North point. H...South point. ZST...Vertical of star. WZE...Prime vertical. E...East point. W...West point. EQW...Equator PSK...Declination circle of star. Y..."First point of Aries." The position $S$ of a star at any instant may be determined by either of the three following systems of coordinates, viz. (i) $ST$, the altitude (or $SZ$, the zenith distance), and $HT$, the azimuth ; (ii) $SK$, the declination (or $SP$, the north polar distance), and $rAX$, the right ascension ; or (iii) $SP$, the north polar distance, and $SPZ$, the hour-angle. ( 17 ) CHAPTER II. THE EARTH. 19. We have already said that a change of inclination of the polar axis to the horizontal plane will usually accompany a change of the observer's station. We shall now enter more fully into the circumstances of these changes, showing how they are explained by the globular form of the earth, and also in what manner they will enable us to arrive at a knowledge of its size. Axis of the Earth, Terrestrial Meridian, Equator. 20. Let the large sphere in the accompanying figure represent the earth, $C$ the centre, and let $A$ be the first station of the observer, $pP$ the constant direction of the polar axis, and the small sphere round $A$ the observer's celestial sphere. $AP$ drawn parallel to $pP$ will determine the pole $P$, the tangent plane $HAR$ at $A$ will determine the horizon, and the zenith $Z$ will be found by producing $CA$ to meet the observer's celestial sphere. Let $HR$ be the meridian line, then $PAR$ is the angular elevation of the pole above the horizon, or the altitude of the pole, and $ZA\!P$ its zenith distance; and if we draw the diameter $SCN$ of the earth parallel to $pP$, the zenith distance of the pole at $A$ will also be measured by the angle $ACN$. Since the meridian plane at $A$ contains $AP$ and passes through $C$, it will contain the diameter $SCN$; its intersection A diagram showing a large sphere representing the Earth with a smaller sphere around point A representing the observer's celestial sphere. Lines AP and ZA represent directions on this sphere. c 18 THE EARTH. [CHAP. II. with the surface of the earth will therefore be the great circle $SAN$. A diagram showing a celestial sphere with a line connecting two points on the surface of the Earth to a pole. The diagram includes labels for various points and lines, such as $P$, $Z$, $B$, $C$, and $N$. There is also a label "21." near the bottom left corner. 21. Let the observer travel along this great circle towards the elevated pole to some second station $B$. The pole $P$ being determined as before by means of a parallel $BP$ to $\frac{p}{p}$, and the vertical line $CBZ$ giving the new zenith $Z$, we see that the zenith has approached the pole by an angle equal to the difference between $ZAP$ and $ZBP$, or between $ACN$ and $BCN$, that is, by the angle $ACB$. By an observer ignorant of, or not attending to, the curvature of the earth, the change would be attributed to a motion of the pole which, carrying with it the parallel of the stars, would seem to have moved towards the zenith by that quantity. This way of considering the motion is perfectly adequate to a mere description of the phenomenon, A diagram showing a celestial sphere with a line connecting two points on the surface of the Earth to a pole. The diagram includes labels for various points and lines, such as $P$, $Z$, $B$, $C$, and $N$. There is also a label "21." near the bottom left corner. CHAP. II.] THE EARTH. 19 and, with this understanding, we shall occasionally make use of it. If we refer to the figure (p. 8) we shall see that the effect of an approach of $P$ to $Z$ will be that the parallels of the stars will make a smaller angle with the horizon, the number of circumstellar stars will increase, and some of those which before remained a short time above the horizon will now never be seen. 22. That diameter $SN$ of the earth which is parallel to the constant direction of the line about which the stars revolve is called the axis of the earth, and the points $N$ and $S$ the poles of the earth. A terrestrial meridian is any great circle whose plane passes through the axis of the earth. This definition applies equally well to the earth as a sphere or as a figure of revolution; and in that case, the various stations $A$, $B$, etc., along the curve $ABMN$ will obviously have their meridian planes coincident with the plane $ABMN$. The following definition : " A terrestrial meridian is the locus of all points on the surface which have their meridian planes parallel." The curve will not necessarily be a great circle. The great circle $QQ'$, whose plane is perpendicular to the earth's axis, is called the terrestrial equator, and all small circles parallel to the equator as $AA'$ are called parallels. 23. Let the observer continue his journey along the same meridian till he reaches the pole $N$. There $Z$ and $P$ will coincide, and the diurnal paths of the stars will be perpendicular to the vertical line, and therefore parallel to the horizon ; the horizon itself will coincide with the equator. No star will ever rise or set, and the visible heavens will be confined to those portions which are on the same side of the equator as the elevated pole. The observer has then what is sometimes called a *parallel sphere*, the different circumstances of which will be obvious C 2 20 THE EARTH. [CHAP. IL from the examination of the figure. There is no definite meridian, since the zenith and pole coincide, or rather every great circle through the zenith is a meridian. There will be no east or west line, nor in fact any other direction but south, supposing $N$ to be the north pole. 24. If instead of advancing from $A$ (fig. p. 18) towards the elevated pole, the observer travels in the exactly opposite direction, a contrary effect will be produced,—the pole will gradually recede from the zenith, and when the traveller reaches the terrestrial equator at $Q$, the poles $F$, $F'$ will be in the horizon, and the zenith will be a point of the celestial equator. The parallels of the stars will all cut the horizon at right angles and be bisected by it, so that half their course will be above, and half below—every part of the heavens becoming visible in each revolution. The sphere is then called a right sphere. 25. Continuing his journey in the same direction along the meridian from $Q$ towards $S$, the other pole will now become the elevated pole, and the stars round it the circumpolar stars. The sequence of phenomena being precisely the same as on the other side of the terrestrial equator, the south pole gradually approaching the zenith, and the parallels A diagram showing a globe with a meridian marked "Meridian" and a parallel of latitude marked "Parallel". The poles are labeled "North Pole" and "South Pole". CHAP. II.] THE EARTH. 21 of the stars about the north pole disappearing one after the other below the horizon. 26. The observer has hitherto been supposed to move along the same terrestrial meridian. Let us now take him to a new station $A'$ on a different meridian $N'A'Q'S$, and suppose $A'$ to be on the circle through $A$ parallel to the terrestrial equator. The two lines $CA$ and $CA'$ make equal angles with the axis of the earth, and therefore the zeniths of the two places $A$, $A'$, though having different directions in space, will be equally distant from the pole of each observer's celestial sphere (Art. 20); but the phenomena of diurnal motion depend on the altitude of the pole ; and consequently the very same phenomena will be perceived at $A'$ as at $A$, though not at the same time ; a star which passes through the zenith of $A$ will have passed through the zenith of $A'$ some time before, and through the zeniths of all places in the parallel $AA'$ in succession. The horizons of the two places will also intercept different parts of the heavens at the same instant, but the very same portions will successively, and in the same order, present themselves. Magnitude of the Earth. 27. We have seen (Art. 21) that the angle $ACB$ subtended by the arc $AB$ of a meridian is equal to the difference between the zenith distances of this pole at the two stations. Hence, assuming the observer to have measured these zenith distances, and also the distance travelled, he will know the arc $AB$ of a meridian at the centre, and from these the magnitude of the radius may be calculated. Instead of the change of zenith distance of the pole he may take the change of meridian zenith distance of any star; for, since the star's angular distance from the pole is constant, A diagram showing two points A and A' on different meridians, with lines CA and CA' connecting them to a common point C. 22 THE EARTH. [CHAP. IL the alteration in the meridian zenith distance of the former will be equal to that of the latter.* The length of the earth's radius will by this means be found to be something less than 4000 miles. Latitude and Longitude. 28. The different places on the surface of the earth are distinguished from one another by their latitude and longitude. The geographical latitude or simply the latitude is the angle between the zenith and the celestial equator at the place, or, which is the same thing, the angle between the vertical line and the plane of the equator. Thus at $A$ (fig. p. 16) the latitude is the angle $ZAE$. The geocentric latitude of a place is the angle subtended at the centre of the earth by the arc of meridian intercepted between the place and the equator. Thus $AQ$ being the meridian of $A$, the geocentric latitude of $A$ will be the angle $AZQ$. Considering the earth as a sphere the angle $ZAE$ will be equal to the angle $ACQ$, and the geographical and geocentric latitudes will be the same; but the distinction will be essential if we find the spherical shape not to be the true one (see chap. xvi). 29. Places on the equator $QQ'$ have latitude 0°, and the latitudes increase from 0° to 90° on each side of the equator, being reckoned north or south, according as the place is towards the north or south pole. The latitudes of two places $A$ and $A'$ on the same parallel will be the same. 30. The angle $PAZ$, the complement of $ZAE$, is called the co-latitude. * This method was employed by Eratosthenes (300 B.C.), except that he used the meridian zenith distance of the sun instead of that of a fixed star. A diagram showing latitude and longitude. CHAP. II.] THE EARTH. 23 It will be useful to remark that $PAR$, being the comple- ment of $PAZ$, will be equal to $ZAE$; that is, the elevation of the pole above the horizon is equal to the latitude of the place. 31. The longitude of a place is the angle made by its meridian plane with some one fixed meridian plane arbitrarily chosen, and is measured by the arc of equator intercepted between these two meridians. The longitudes are reckoned from the fixed meridian through 180° on each side, one being east and the other west. The Phenomena of Diurnal Motion explained by a Rotation of the Earth. 32. We have so far supposed that the motion of the stars is a real motion, and that the earth is the fixed stationary body which it appears to us to be. We know, however, that the appearances of motion are frequently deceptive, and that it is often hard, when two things are in relative motion, to determine whether either of them, and if either, of them is absolutely at rest. Thus, when we are in a railway carriage moving smoothly along, the houses and trees appear to move in the opposite direction ; and it is only common sense and our reason which tell us that it is we, and not they, who are moving. But when we stand at a station whence other trains are departing, it is often difficult to say, when looking at those from our own carriage, whether we ourselves have stopped and the other trains have com- menced moving, or whether the motion is entirely ours, or partly ours and partly theirs. 33. Accordingly, let us now suppose the stars to be stationary, and examine what kind of motion must be at- tributed to the earth, in order that, on this hypothesis also, the same appearances may be produced as we have already 24 THE EARTH. [CHAP. IL observed and recorded on the hypothesis of a stationary earth and moving stars. Suppose the earth to turn uniformly about its axis $S Y$ from west to east, or in the opposite direction to that of the diurnal motion of the stars ; and suppose it to accomplish its revolution exactly in that period which we found to be common to all the stars. An observer at $A$ will, after a certain interval, be carried to $A'$. His radius $CA'$ will make with $CN$ the same angle as before, and therefore the zenith of his celestial sphere, determined by the prolongation of $CA$, will be at a constant angular distance from the point found by drawing $A'P$ parallel to $CN$. But $A'P$ will be in the meridian plane $NAQ'$, which, turning with the earth, always passes through the same places on its surface ; and therefore $A'P$ will not only always point to the same place in the heavens, but will also retain a fixed direction relatively to the observer and the terrestrial objects around him. This is exactly the phenomenon observed, and the points of the celestial sphere determined by this fixed direction we have called the celestial poles. Again, the stars which were in the meridian plane $NAQ$ of the observer when he was at $A$ will no longer be so when he has been carried to $A'$; but, on account of their immense distance, they will all be found in a plane through $A'P$ parallel to $NAQ$; and they will therefore be on the west of the observer's present meridian plane, which has become $NAQ'$, the angle between the meridian and the great circle in which they lie being precisely the angle $QQ'C$ through which the earth itself has turned; and, as we suppose the A diagram showing a celestial sphere with a line drawn from a point on the equator (point A) towards a point on the horizon (point C). The line AC is extended to intersect a line drawn from another point on the equator (point B) towards a point on the horizon (point D). The line AD is then extended to intersect a line drawn from another point on the equator (point E) towards a point on the horizon (point F). The lines AC, AD, AE are labeled with arrows indicating their directions. The diagram also shows a line drawn from a point on the equator (point G) towards a point on the horizon (point H). This line GH is extended to intersect a line drawn from another point on the equator (point I) towards a point on the horizon (point J). The lines GI, GH, GJ are labeled with arrows indicating their directions. Diagram showing celestial sphere with arrows indicating directions. CHAP. II.] THE EARTH. 25 earth to turn uniformly, the stars will appear to move towards the west at a rate which also is uniform. 34. Hence, whether we assume the stars to be in motion as one connected system about an axis through the earth at rest, or suppose the stars stationary and the earth to revolve in the opposite direction with the same angular velocity about this same axis, there is nothing in either supposition inconsistent with the appearances presented, and we have so far nothing to guide us in our choice of the explanation. If the stars move, their declination circles will, one after the other, cross the meridian of the observer. If it is the earth that moves, then the meridian plane of the observer, carrying his zenith and horizon with it, will travel from west to east across the stars, and coincide with each declination circle in turn, the zenith during revolution retaining the same angular distance from the pole, and therefore describing that small circle of the celestial sphere whose declination equals the latitude of the place. Although the solution of problems concerning the apparent positions of the heavenly bodies will be correctly obtained on either supposition, and convenience alone need influence our selection, we may proceed at once to show which of these two suppositions is the more probable. Arguments in favour of the Earth's Rotation. 35. 1". If we suppose the earth to turn about its axis in 24 hours, places at the equator will move through some- thing like 2500 miles in that time, or about $\frac{1}{8}$ of a mile in a second. This velocity is great, but the other hypothesis will give to the stars, which we have reason to suppose to be bodies of enormous magnitude compared with the earth, A diagram showing a star moving across a circle representing a meridian plane. 26 THE EARTH. [CHAP. IL velocities infinitely greater—velocities which can only be reckoned by millions of miles in a second. 2'. The uniform period in which all the stars perform their circuit is, as we have seen, easily explained if the earth revolve ; but the stars must have some rigid physical connexion if the motion be theirs, as it is beyond the limits of probability that an immense number of unconnected bodies should all describe circles of various dimensions in exactly the same time. That the stars are not so connected is shown by what are called double stars, which alter their relative position as if revolving one round the other ; moreover the sun, moon, and planets particle of this general diurnal motion and move independently of the stars. Simplicity of explanation is therefore manifestly in favour of the earth's rotation. 3'. An argument may also be drawn from the shape of the earth. Our rough measurements have told us that it is spherical ; but a more exact determination will shew it to be an oblate spheroid, whose equatorial diameter is some 261 miles longer than its polar diameter. This flattening at the poles is just the kind of effect that would be produced by rotation in a fluid, or semi-fluid mass, such as the earth is supposed to have originally been. 4'. The observations made on the sun and planets shew that all these bodies, several of which are much larger than the earth, have a motion of rotation about axes through their centre. Proofs of the Earth Rotation. 36. 5'. But we are not left to rely on probabilities only; proofs of the earth's rotation can be obtained by direct experiment. CHAP. II.] THE EARTH. 27 Let $OA$ be a radius of a revolving sphere, $B$ a point in $OA$ produced. Suppose a body let fall from $B$ to be attracted towards $O$. The body will reach the sphere at some point $C$, describing as it falls a curve $BC$, because at $B$ it has a velocity of projection perpendicular to $OB$. Let $B'$ be the point which it would have reached, in the same time, if not let fall. Join $OB$ cutting the sphere in $A'$ the new position of $A$. Then, the force being central, and the velocity and direction of motion the same at $B$ for both paths $BB'$ and $BC$, the area $OBC$ will be equal to the area $OBF$; therefore $C$ will fall beyond $A'$ - the magnitude of $AC$ depending on the height $AB$ and on the angular velocity. If then the earth be at rest, the vertical $BA$, as shown by a plumb line, is the direction which a falling body will pursue; but if there be any rotation of the earth from west to east, the body will fall to the eastward of the foot $A$ of the plumb line $BA'$. The preceding investigation applies to an experiment supposed to be made at the equator; for we have assumed the rotation to be about an axis perpendicular to the plane $OBF$. Let us next consider $A$ a place in north latitude, and let the rotation take place, as indicated by the arrow, from west to east. A will describe the parallel $AA'$, whereas the body let fall from $B$ in $OA$ produced will be moving at right angles A diagram showing a rotating sphere with points labeled \( O \), \( B \), \( B' \), \( C \), and \( A' \). The text describes how a body falling from \( B \) towards \( O \) follows a curved path \( BC \) due to centrifugal force, and how this path changes if the Earth rotates. The diagram also shows a plumb line from \( A \) to \( A' \), indicating the direction of gravity. 28 THE EARTH. [CHAP. IL to the meridian plane $NAH$, and be acted on by a force towards the centre of the earth; consequently its path $BC$ will be in the plane of the great circle $MAC$, which touches the parallel in $A$. Therefore $C$ should be, not only east of $A$ as before, but also south of it. But $BA$ and $BA'$ will no longer be the directions of the plumb line; for the rotation about $NO$ will cause the line to diverge in the direction $BP$, south of $BA$, in such a manner that the earth's attraction and the tension of the string acting on the plummet $P$ may have a resultant perpendicular to $NO$. If $\lambda$ be the latitude of the place, or the angular velocity of the earth, $r$ its radius, $f$ the resultant acceleration, $$f = r\omega^2 \cos \lambda,$$ and if $h$ be the height $BP$, and $m_2$ the tension of the string, $m_2$ being the mass of the plummet, $$AP = h \sin ABP = \frac{f \sin \lambda}{g} = \frac{r\omega^2 \sin 2\lambda}{2g}.$$ Now, let the meridian $NA'$ cut the great circle $AC$ in $Q$, then $A'Q$ is approximately the southerly deviation of the falling body ; and if $\tau$ be the time of falling, we have $ANQ = \omega\tau$. Therefore, from the right-angled triangle $NAQ$, $$\tan \frac{NA'}{r} \cot \frac{NQ}{r} = \cos \omega\tau,$$ $$\sin NQ - NA' = \frac{1 - \cos \omega\tau}{r},$$ $$\sin NQ + NA' = 1 - \cos \omega\tau,$$ $$\sin A'Q = \frac{\sin 2\lambda}{2},$$ $$A'Q = \frac{r\omega^2}{g} \sin 2\lambda \approx \text{approximately},$$ $$= AP, \text{ since } h = \frac{1}{2}gt^2.$$ CHAP. II.] THE EARTH. 29 therefore, when $B$ reaches $B'$, the plummet $P$ will be at $Q$; that is, there will be no southerly deviation of the falling body relatively to the actual vertical $BP$. The resistance of the air has been neglected, but when this is taken into account the conclusion is still the same.* If, on the one hand, the time of descent is increased which would tend to make the body fall nearer the equator, the impulses it receives from the particles of air, which are describing circles about the axis, tend to bring it back towards the pole, and one effect is found to counteract the other. The easterly deviation of the earth's axis can be very approximately obtained as follows : the area $OBC$ described by the falling body under the action of the central force at $O$ will be approximately equal to $OHB$. Take away the equals $AOA$ and $AQO$, and we have $$\text{area } BAA'B = QOC + ABC,$$ but the curve $BC$ is approximately a parabola, therefore $$h.AQ = \frac{1}{2}.QC + \frac{1}{2}.(AQ + QC),$$ $$QC(\frac{1}{2} + \frac{1}{2}) = \frac{1}{2}.AQ,$$ $$\frac{1}{r}.QC = \frac{1}{r}\sqrt{r}\cos\alpha,$$ neglecting $\frac{4h}{3r}$ compared with 1, $$QC = \frac{4h}{3r}\cos\alpha.$$ The experiments which have been made with the greatest care by different persons all tend to confirm the results here obtained, and the amounts of easterly deviation, though small, agree closely with the theoretical values calculated on the supposition that the earth revolves on its axis once in 24 hours. In some of the experiments a small * Vide La Place Mil. Cil. vol. i. p. 98. and vol. ii. p. 104. Also The Earth and Its Mechanism, by H. Warrn, F.R.S., E.G.H., which contains a full account of the various experimental proofs of the rotation of the earth.
Area \( BAA'B \) \( QOC + ABC \)
Curve \( BC \) (approximately a parabola) \( h.AQ = \frac{1}{2}.QC + \frac{1}{2}.(AQ + QC) \)
\( QC(\frac{1}{2} + \frac{1}{2}) = \frac{1}{2}.AQ \)
\( \frac{1}{r}.QC = \frac{1}{r}\sqrt{r}\cos\alpha \) neglecting \( \frac{4h}{3r} \) compared with 1,
\( QC = \frac{4h}{3r}\cos\alpha \)
30 THE EARTH. [CHAP. II. northerly and in others a small southerly deviation was observed ; but these, like the small differences between the observed and the calculated values of the easterly deviation, are due to errors which cannot be avoided, such as currents of air or slight vibrations of the ball when set free. 37. $\delta'$. Two experiments devised a few years ago by Mons. Foucault are still more satisfactory, because the effects are more observable. The fact is known as the pendulum experiment, and may be explained in the following manner : Assume the earth to be in motion with an angular velocity ($\omega$) from west to east ; and, first, suppose a pendulum suspended over the north pole of the earth, and there made to oscillate. There would be no force acting on the pendulum out of the plane of oscillation, and therefore that plane would retain a fixed position in space ; and, the earth revolving under it from west to east, the different meridians would, one after the other, coincide with the plane of oscillation ; the apparent effect to a person at the pole, and not aware of his own motion, being a gradual shifting of the plane of oscillation from east to west ; a complete revolution being accomplished in one day $\left(\frac{2\pi}{\omega}\right)$. Next, suppose the pendulum to be placed at the equator ; the bob of the pendulum before the oscillations begin will partake of the general motion of rotation round the axis of the earth parallel to the plane of the equator, and, in A globe showing Earth's rotation around its axis. CHAP. II.] THE EARTH. 31 whatever direction the pendulum be set in motion, there is nothing to cause a disturbance of the plane of oscillation relatively to the horizontal plane, since all parts of the horizontal plane in the immediate vicinity of the bob of the pendulum will also have the same common velocity or rotation. This will not be the case at a place off the equator, because there these parts of the horizontal plane just under the poles which are nearest to the pole will have less velocity than those near the equator, and an apparent motion of the plane of oscillation with respect to the plane of the meridian will be the consequence. Let $A$ be a place in the north latitude $\lambda$. The angular velocity $\omega$ about the axis $OP$ may be resolved into two: first $\omega \cos AOE$, or $\omega \sin \lambda$ about $OA$, and the other $\omega \cos POB$ about an axis $OB$ perpendicular to $OA$ (Routh's Rigid Dynamics, p. 120). Their effects may be considered separately: the rotation about $OB$ will produce no disturbance since $A$ would be like a point in the equator with respect to that axis; but the rotation about $OA$ will have the same effect as in the first case considered—that of a pendulum at the pole. The angular velocity being $\omega \sin \lambda$, the time of a complete revolution will be $\frac{2\pi}{\omega \sin \lambda}$, but $\frac{2\pi}{\omega}$ is one day; therefore A diagram showing a pendulum swinging in a vertical plane, with a line indicating its path on a sphere. The diagram includes labels for points A, B, C, D, and E. 32 THE EARTH. [CHAP. IL the time will be 24 cosec λ hours. In the latitude of Cam- bridge this will be about 304 hours. Now a pendulum suspended, in this manner will not oscillate long enough to make a complete circuit, but it will do so during a sufficient time to enable us to verify that there is a displacement of the plane of oscillation, and, by a simple proportion, we can ascertain that if it con- tinued at the same rate it would agree with the result obtained. If the earth did not rotate we should have no means of explaining this result. 38. The other experiment, imagined by Mons. Foucault, is independent of the earth's attraction. If a body, symmetrical with respect to an axis through its centre of gravity, be made to rotate about that axis, the centre of gravity must be kept supported and the axis free to move in such a manner round it may be shown that no change will take place in the direction of the axis in space, provided no force but gravity act upon the body (Routh's Rigid Dynamics). By an ingenious contrivance, called a gyroscope, Mons. Foucault obtained this permanent axis, and verified that only when parallel to the earth's axis did it retain a permanent direction with respect to surrounding objects. In all other positions it moved just as the stars seem to move, and would, in fact, if pointed towards any particular star, have continued to point to that star during the whole time that its rotation lasted. The stars therefore have permanent For the success of the experiment several precautions must be observed. The pendulum must be as long as possible in order that the angle of elevation may be small, and the motion should originate by an impulse on the bob in the position shown in fig. 156. This is effected by drawing the pendulum out of the vertical, and when perfectly steady, setting it on its back. The bob on the horizontal plane will be approximately a straight line, but really an elongated shape, whose major axis, however, would have the same to west displacement mentioned above.—Quarterly Journal of Meteorology (1855) A diagram showing a pendulum suspended from a fixed point and rotating around an axis through its center of gravity. CHAP. II.] THE EARTH. 33 directions in space, and thus we have another independent proof that the apparent diurnal motion of the stars belongs in reality to the earth. 39. We shall, however, continue to speak of the rising and setting of the heavenly bodies and of their crossing our meridian, although we now know that strict astronomical language would require us to say, in the one case, that the plane of the horizon is sinking below, or rising above, the body; and, in the other, that the plane of the meridian is crossing it. But, as already stated in Art. 34, when concerned with questions which involve only the apparent directions, either supposition may be made use of. D (34) CHAPTER III. THE OBSERVATORY. 40. We must interrupt our observations to explain the construction and use of some of the instruments by means of which the coordinates of the stars' positions may be ascertained with extreme precision. The ancient astronomers generally divided their instruments to 10°, and although some observations are found recording much smaller sub-divisions, they are not entitled to much confidence. The latitude of Alexandria, as determined by Ptolemy, differs about a quarter of a degree from that which is given at present.* Tycho Brahe (1570), whose instruments and methods of observation were greatly superior to those of former astronomers, carried the division of some of his instruments to every minute, and could still further estimate to 10". This required instruments of a very large size, which consequently became unwieldy and liable to derangement from their weight. The precision besides was only apparent, for the telescope was not yet known, and his only means of pointing was by plain sights or projecting pinnales of metal, having slits in them, through which the star was observed. The uncertainty from this cause very far outweighed the supposed accuracy of the division. It may be said, that before the invention of the telescope the positions of the stars could scarcely be ascertained to within 5". * Delambre, Astronomie Moderne, p. 27. † Flamentated found that the fixed stars in Tycho Brahe's catalogue were generally less than 6" in error, and in some instances even more. (Bally's Life of Flamsteed, p. 125.) CHAP. III.] CLOCKS. 35 41. The invention of the telescope (1609) did not at once get over the difficulty. The first telescopes made were all on the principle of Galileo's, and they offered no means of fixing the direction of a star, since there is in them no position where cross-wires could be placed so as to be seen directly at the same time as the star. The Astronomical telescope consisting of two convex lenses was suggested by Kepler (1611), but Gassendi (1640) was the first to perceive its value in Astronomical observa- tions. The common focus of the two lenses gave a place, within the instrument, where cross-wires of reference could be fixed and seen distinctly with the celestial object. He at once adapted telescopic sights to his instruments, and was the first who did so with success, by the help of these through which he could easily point the optical axis of the telescope with the greatest nicety. Improvements in the means of dividing and graduating the instruments have since then gone on at a pace commen- surate with the requirements of Theoretical Astronomy, and in the instruments now in use in all large observatories the graduation is, by the aid of the micrometer (also an invention of Gassendi's) and of the microscope, estimated to a fraction of a second.* Clocks—The 'Balance' Clock. 42. Another event which largely contributed to the accuracy of observations, and was second in importance only to the discovery of the telescope, was the improvement effected in the means of measuring time by the introduction of the pendulum as a regulator of clock mechanism. The idea of employing the pendulum for this purpose was suggested by Galileo, who had remarked the apparent * For a full and interesting account of the advances of practical astronomy, and of the labours of those eminent men to whom its rapid progress in modern times is due see Griss's *History of Physical Astronomy*, p. 344. D2 36 THE OBSERVATORY. [CHAP. III. isochronism of the vibrations ; but Huysghens (1656) first succeeded in constructing a clock on this principle. The element of time had seldom been employed by any observer, and by none with confidence ; the clocks previously in use being liable to constant sources of irregularity. 43. The best form then known was the *Balance* clock, constructed on the following principle : A cord wrapped round a horizontal cylinder supported a heavy weight. The action of the weight tended to turn the cylinder, so that, if not restrained, an accelerated motion would be produced. But the cylinder was connected with a train of wheelwork in such a manner that the last wheel, a vertical one, was obliged to turn 60 times faster than the cylinder. To this wheel, called a crown-wheel from its shape, was applied the check or restraining power in the following manner :—A vertical rod called a * verge*, placed in front of and nearly in contact with the wheel, was moveable about pivots at its extremities, and had two projecting pallets separated by a distance equal to the diameter of the wheel. Firmly fitted to the verge were two thin projecting arms carrying small weights, forming with the verge a kind of cross which was called the *balance*. These arms were placed near the top, so that when the verge turned upon its pivot they passed over the teeth. The pallets were placed so that their planes being 90° apart, that the highest tooth of the wheel met the upper pallet and, in its effort to turn, pushed it round, and with it the whole balance. As soon as the highest tooth had passed, the lowest tooth found itself opposed to the lower pallet, and the motion just before given to the balance was thus checked and stopped. A reverse motion now took place to allow this lowest tooth to pass, when the upper pallet again came into play, and so on alternately. Thus the wheel always turning the same way gave the balance an oscillating motion, which moderated
43. The best form then known was the Balance clock,
constructed on the following principle :
A cord wrapped round a horizontal cylinder supported a heavy weight.
The action of the weight tended to turn the cylinder, so that,
if not restrained, an accelerated motion would be produced.
But the cylinder was connected with a train of wheelwork in such a manner that the last wheel, a vertical one,
was obliged to turn 60 times faster than the cylinder.
To this wheel, called a crown-wheel from its shape,
was applied the check or restraining power in the following manner :
A vertical rod called a verge, placed in front of and nearly in contact with the wheel,
was moveable about pivots at its extremities,
and had two projecting pallets separated by a distance equal to the diameter of the wheel.
Firmly fitted to the verge were two thin projecting arms carrying small weights,
forming with the verge a kind of cross which was called the balance.
These arms were placed near the top,
so that when the verge turned upon its pivot they passed over the teeth.
The pallets were placed so that their planes being 90° apart,
that the highest tooth of the wheel met the upper pallet and,
in its effort to turn, pushed it round,
and with it the whole balance.
As soon as the highest tooth had passed,
the lowest tooth found itself opposed to the lower pallet,
and the motion just before given to the balance was thus checked and stopped.
A reverse motion now took place to allow this lowest tooth to pass,
CHAP. III. CLOCKS. 37 and regulated the velocity of the descending weight and of the cylinder. The revolutions of the cylinder served to measure time ; its axis projecting outside of the framework was connected by other wheels and pinions, with hands indicating hours and minutes on a clock face. By shifting the weights along the cross arms of the balance, the rate of the clock could be increased or diminished. Fig. 1. shows a balance clock which was put up in the palace of Charles V, of France, about the year 1480, by Henri de Vic, a Norman. The figure is taken from Berthault's "Histoire des horloges et les Horloges," Paris, 1892; but all those parts of the figure which do not refer to the action of the weight and balance have been omitted. It will be seen that the balance $B$ is suspended by a cord $M$, in order to diminish the friction on the pivots. $P$ and $Q$ are the two pallets on the verge $PQ$, forming with the crown-wheel $A$ what is technically called the escapement. The weight $W$ acting upon the cylinder $C$ turns the wheel $F$ of 64 teeth. This wheel acts upon the pinion $G$ which has 8 leaves, and therefore makes 8 revolutions for every one revolution of the cylinder. On the same axle with $G$ is fixed the wheel $I$, which therefore revolves at the same rate as $G$. The wheel $H$ has 60 teeth, and acts upon the pinion $K$ which has 8 leaves ; $K$ therefore turns $\frac{7}{8}$ times faster than $H$, or 60 times faster than the cylinder. Now, if the weights $a$, $b$ on the arms of the balance be so adjusted, by trial, that the crown-wheel $A$, which is on the same axle with $G$, makes 8 revolutions in a minute, the cylinder will make one in an hour ; and the axle $D$ of the cylinder can be made to communicate this motion by another train of wheelwork (not represented in the figure) to the hour and minute hands of a clock face. 44. When the weight has run down, and it is necessary to wind it up again, a simple artifice allows of this being 38 THE OBSERVATORY. [CHAP. III, done without disturbing the hands or any of the wheels or pinions which connect the cylinder with the crown-wheel. The wheel $F$ is rigidly attached to the axle $D$, but the cylinder is not. A ratchet-wheel (not given in fig. 1, but shown in fig. 2) turning with the cylinder and rigidly attached to it is in contact with $F$, and a click fastened to the face of $F$ catches in the teeth of this wheel when the cylinder is turned by the action of the weight, thus producing motion in the whole train. But when the clock is being wound up, and the cylinder is turned in the opposite direction, the click slips over the teeth of the ratchet-wheel, and the cylinder turns alone. 45. Ingenious as was this clock, there were several defects in it which rendered it unfit for the delicate purposes of Astronomy. In the first place, it is obvious that the least irregularity in the relative position of the teeth of $F$ which acted upon the pallets would cause them to slip by more rapidly than others, producing a corresponding irregularity in the velocity of the descending weight. Again, the rapidity with which the oscillations of the balance were performed would depend upon the force with which the wheel pressed against the pallets, and upon the distribution of the mass of the balance relative to its axis. This latter was affected by every change of temperature ; and the former, not only by changes of temperature, but also by the addition of the varying weight of the unwound portion of the cord to the descending weight, however slight the chord might be. Tycho had some of these clocks, but for the above reasons he never relied upon them as elements of observations. The Pendulum Clock. 46. Huyghens had the happy idea of substituting the pendulum for the balance in such a manner that the down- CHAP. III.] CLOCKS. 39 ward motion of the weight was regulated by the isochronous* oscillations of the pendulum. The mode of escapement was the same as in the balance clock, but the verge with the two pallets was horizontal, as also the wheel which acted upon them. This will be easily understood from fig. 3, which is taken from the same work as the former (some parts being omitted as before). The difference between the balance clock and that of Huyghens seems slight, but it is essential. In the balance clock the weight is the cause of, and to a considerable extent, annihilation of the pendulum's pendulum clock the weight has no longer the same influence; the length of the pendulum regulates the duration of the oscillations, and any variation in the descending weight, or in the action of the immediate wheel, may increase or diminish the extent of the arc of oscillation, but has only a very slight effect on its duration. The weight, or some other source of power, is still necessary however to turn the hands and wheels which measure the time, and also to maintain the vibration of the pendulum. The oscillations of a rigid pendulum are not strictly isochronous, as was at first imagined by Galileo. Huyghens saw that this was not so; he described a pendulum as having a cycloid, and not an arc of a circle, and his pendulum was constructed to produce this effect. The upper part terminated in two points, one above and one below, and these two points were connected by a cycloid (see fig. 4), and thus gave to the centre of oscillation its cycloidal motion. (See Parkinson's Mechanics.) In fig. 5 is shown how this idea was not found to give results so satisfactory as had been expected--the theory which showed the oscillations to be isochronous, contemplated a simple pendulum oscillating freely, and took no account of frictional resistance. It was therefore rejected for the common pendulum oscillating about a fixed axis. The oscillation of this kind would be isochronous if friction were absent; but it may be shown that the departure from isochronism will be inappreciable so long as the oscillations do not exceed 10 degrees; and it may be further shown that when such an amount of impulse is given to it each time to restore the loss of motion due to friction, &c., this tends to maintain the extent of oscillations nearly constant. The cycloidal pendulum was altogether abandoned after the invention of the anchor escapement described on p. 41. 40 THE OBSERVATORY. [CHAP. III. dulum. This is done by causing the slight pressure on the pallets to be transmitted to the pendulum by means of the fork $S$, between the prongs of which, at $T$, the pen- dulum passes, and whose other extremity $R$ is attached to the verge $PQ$. Without this, the pendulum would, sooner or later, be brought to rest by the resistance of the atmosphere and by friction. Instead of the cylinder, Huyghens substituted a pulley, over which the cord passed, with a large weight on one side and a small one on the other. The cord was hindered from slipping by covering the circumference of the pulley with sharp projecting points. 47. We have seen how, in the balance clock, the weight was wound up again without acting upon the train or the hands. The same artifice will clearly apply to the pendulum clock, but during the time of re-winding the clock stops, and though the error would be unimportant among the other irregularities of the balance clock, it could not be overlooked in one intended for astronomical purposes. To Huyghens is due the following simple means of winding up the clock without interfering at all with its regular and uniform progression. Supposing (fig. 5) to be the pulley of Huyghens' clock. The cord passing over it, instead of being attached immedi- ately to the two weights, is an endless cord which passes under two smooth moveable pulleys from which the weights are suspended, and then over another pulley $X$ fixed to the framework of the clock. This last pulley is rough like the first to hinder the cord from slipping. A click $Z$ catches into the teeth of a ratchet-wheel in the circumference of the pulley $X$, so as to hinder it from turning in the direction of the heavier weight, but allows motion in the opposite direction. When, by the slow and gradual turning of the pulley $V$, To face p. 40 THE BALANCE CLOCK. Fig. 1. Fig. 2. THE PENDULUM CLOCK. Fig. 3. Fig. 4. Metallurgie & Cie Limited. Fig. 5. HARRISON'S GRIDIRON PENDULUM. Fig. 7. Fig. 6. GRAHAM'S MERCURIAL PENDULUM Fig. 8. Mecalli Ltd Limited. CHAP. III.] CLOCKS. 41 the weight $W$ has descended, and requires to be again drawn up, the hand must be applied to the cord at $m$. By pulling downwards, the weight $W$ will be raised, and $w$ lowered, with little if any, alteration in the tensions of the strings on each side of $V$,* and therefore none whatever in the going of the clock. 48. Among the numerous improvements which have been made in the construction of clocks since that time, we shall only particularise a few of the more important. The 'Anchor' Escapement. The anchor escapement, so called from its shape, is a modification of the verge-and-pallet escapement, and was substituted for it by Mr. William Clement, a clockmaker of London, about 1680 though the celebrated Hooke claimed to have invented it as early as 1666. Fig. 6 will explain the action of this escapement. The verge $AB$ is horizontal, and the fork $S$ with its prongs $T$ are attached to it, and act upon the pendulum just as in Huyghens' clock ; but the escapement wheel is vertical, and the pallets $P$, $Q$ are attached to a curved piece projecting from the verge, and in the same plane with the escapement wheel, of which it embraces a greater or less arc. With each swing of the pendulum, one of the two pallets catches into a tooth and stops the wheel ; then, on the return swing, the tooth slips by and the wheel begins to turn, until stopped by the pallet on the other side, and so on alternately. The great advantage of this escapement was its requiring a much smaller arc of oscillation in the pendulum, and therefore securing much better the isochronism of the oscillations. (See note, p. 39). * Any slight alteration will be due to the want of uniform velocity in the hand ; for instance, at the beginning and end of each pull. 42 THE OBSERVATORY. [CHA.P. III. Compensating Pendulums. 49. The time of oscillation of the pendulum depends upon its length; it is obvious, therefore, that the expansions and contractions due to heat and cold must interfere with its uniform rate. Clocks will lose in summer and gain in winter. The amount of this loss or gain will vary with the material of which the pendulum is composed, because all substances have not the same expansibility; but the irregularity will exist in all. In Huyghens's clock, $U$ is a small weight, which, being moved along the pendulum, serves to correct the rate of the clock. Harrison's gridiron- pendulum and Graham's mercurial-pendulum, both invented about the year 1726, are ingenious and valuable means of counteracting the effect of changes of temperature. They are compensating pendulums constructed by taking advantage of the unequal expansibility of different substances, and so arranging combinations of them as to leave the distance from the centre of oscillation to the centre of suspension unaltered at all temperatures, the pendulum becoming thus self-adjusting. 50. The gridiron- pendulum consists of 5 steel and 4 brass rods, connected at top and bottom by cross pieces of brass $B$, $C$, $D$, $E$, $F$, as in fig. 7, where the steel rods are represented by dark lines. The coefficients of expansion of steel and brass, i.e. the quantities by which any length of these metals must be multiplied in order to obtain the expansion for an increase of temperature of 1° centigrade, are* steel $\quad$0000107912, brass $\quad$0000187821, or in the proportion of 4 to 7 very nearly. Now, on an increase of temperature, the cross piece $B$ will be lowered by a quantity proportional to the length * Biot's *Physique*, Paris, 1824, vol. I. p. 331. CHAP. III.] CLOCKS. 43 of the steel rod which connects it with the point of suspen- sion $A$. $C$ will descend by the same quantity, and, in addition, that due to the lengthening of the steel rods from $B$ to $C$. $D$ is carried down with $C$, but the expansion of the brass rods connecting them will bring $D$ up again through some space. The motion of $E$ downwards will be the same as that of $D$, plus that due to the steel rods join- ing them, and so on. The centre of the bob of the pendulum, which is connected with $F$ by a steel rod passing freely through the cross pieces of the frame, will therefore increase its distance from $A$ by a quantity equal to the lengthening of the steel rods, minus that of the brass rods. Therefore, if the total lengths of the steel rods and brass rods be as 7 : 4, or in the inverse proportion of their expansibilities, the distance between the point of suspension and the centre of oscillation will remain unchanged. 51. In Graham's mercurial-pendulum (fig. 8), a glass cylinder containing mercury is suspended by a steel rod, which supports the bottom of the cylinder. The coefficients of linear expansion for 1° centigrade are mercury $\quad000060601 = m,$ glass $\quad000085752 = g.$ Therefore the coefficient of expansion of mercury in a glass tube is $$b m - 2g = 000162659,$$ and that of steel is $$000107912,$$ *The centre of the bob is not necessarily, nor generally, the centre of oscillation ; but in a second pendulum, when the bob is heavy and the rods light, the distance between them will be small and nearly constant.* For further information see *Phil. Trans.* 1798. See also *Monthly Edmonton*. 3rd Edition. p. 70. The following table gives an approximation to the resistance given by the relative dimensions of the parts of a Graham's pendulum, when the mercury is enclosed in a lead-iron cylindrical jar, into the top of which an iron rod is screwed.
A diagram showing a Graham's pendulum with a glass cylinder containing mercury suspended by a steel rod. 44 THE OBSERVATORY. [CHAP. III.] These are in the ratio of 15 to 1 nearly, therefore if the cylinder of mercury have about $\frac{1}{3}$ the length of the rod, and a sufficient diameter to bring the centre of oscillation near the middle of the mass of mercury, the fall of the centre due to any expansion of the steel rod will be counteracted by its rise due to the expansion of the mercury. 53. In the gibbous and mercurial pendulums, the duration of an oscillation is made greater or less by moving the bob in the one or the cylinder in the other by means of screws which connect them with the rod; but as they may both be over-compensated or under-compensated for changes of temperature, Graham's will have an advantage over the other, because it admits of easy correction, which the astronomer himself can perform by merely withdrawing or adding more mercury. Conical Pendulum, Spring Governor. 53. It will be obvious that the motion of the hands produced by the successive beats of the pendulum is intermittent, the advance being by jerks and starts. The exact termination of each second is well marked by the beat (supposing the pendulum to be a seconds-pendulum), but the subdivisions of the seconds can only be obtained by estimation. The perfection to which Astronomy has now attained requires that even these fractions of a second should be given accurately. To obtain a continuous uniform motion, Huygens suggested the use of a conical pendulum,* but it is only within the last few years that it has taken a practical form. The Astronomer Royal employs the rotation of a large conical pendulum to govern the motion of a clock. The inclination of the pendulum to the vertical soon adjusts itself, so that * Horologium Oscillatorium, p. 137. CHAP. III.] CLOCKS. 45 the resistance of the air and the friction of suspension are exactly balanced by the force which maintains the motion. A motion without jerks and sensibly uniform is the result.* A few years earlier (1849), the Messrs. Bond, in America, invented the Spring Governor, "consisting of a train of clock-work connected with the axis of a fly wheel. It has an escapement wheel, into the teeth of which pallets play by the oscillation of a pendulum as in ordinary clocks, the wheel being so connected with its axis that it is able to allow any angle while the wheel is detained by the pallets." The contrivance produces continuous and very approximately uniform rotation of the axis. * For a detailed description, see Appendix to Greenwich Observations, 1856. † Loomis' Practical Astronomy, p. 73. A diagram showing a spring governor mechanism. ( 46 ) CHAPTER IV. THE OBSERVATORY CONTINUED. The Transit Instrument. 54. The transit instrument is one of the most important in an observatory. Its object is to determine the precise A diagram of a transit instrument, showing two vertical columns with horizontal crossbars and weights hanging from them. CHAP. IV.] THE TRANSIT INSTRUMENT. 47 instant at which a celestial body crosses the meridian, and for this purpose it consists of a telescope moveable about a fixed horizontal traverse axis of rotation. This axis points east and west, and the optical axis of the telescope is at right angles to it. As the whole instrument turns the optical axis never deviates from the plane of the meridian, and may be directed towards any point in it. The fixed axis of rotation is obtained by having a strong tubular piece of metal whose end terminate in cylindrical pivots exactly equal in size rectangularly. These sockets, called "pivots," are firmly fastened to two stone piers; but, for the purposes of adjustment, the one admits of a small vertical and the other of a small horizontal motion by means of fine screws. The geometrical axis of the two pivots is the fixed axis of rotation, and the framework of the telescope projects on each side from the middle part of the hollow tube. In order to diminish the friction and wear of the $Y$s, the weight of the instrument is almost wholly counterpoised by two weights (see the figure) acting on levers over the piers, care being taken that sufficient weight remains on the $Y$s to ensure that the direction of the axis shall be determined by them. The telescope used is an astronomical telescope, with an achromatic object-glass and a Ramsden's eye-piece. In the focus of the object-glass is placed a frame-work carrying, at equal intervals, five or even more slender wires in perpendicular directions, intersected by two horizontal lines between which the star is observed. The frame-work admits of various small motions for the sake of adjustment. To render the lines visible at night, the light of a lamp placed on one of the piers is admitted through the hollow A diagram showing a telescope mounted on two stone piers with a frame-work projecting from each side. The frame-work has several wires extending vertically and horizontally, with a central point where they intersect. A lamp is shown on one of the piers. 48 THE OBSERVATORY. [CHAP. IV. pivot, and directed to them by an elliptic ring reflector placed diagonally at the junction of the axis and telescope. The quantity of light admitted may be regulated by a moveable plate, which enlarges or diminishes the aperture. A bright field and dark lines will thus be obtained ; but when very faint stars have to be observed, it will be better to illuminate the lines and leave the field dark, which may be done by means of a small lamp fitted to an aperture in the telescope tube between the lines and the eye-piece. The eye-piece used is the positive or Ramsden*. This has the advantage, that it may be changed for one of a different power without disturbing the wires, which, being situated beyond the field glass, are entirely separate from the eye-piece. And the field glass, being plano-convex, offers a flat surface to the image, so that the wires and all parts of the field of view are distinct at the same adjustment. 55. The object of having several vertical wires is to secure greater accuracy in the observation. We may always expect a slight error in estimating the exact instant, as shown by the clock, when a star appears to pass behind any one of the wires; but, by noting the times across each of the seven wires, and taking the mean, we obtain the time over an imaginary wire nearly, if not exactly, coincident with the middle one, and called the mean of the wires. As the error of estimation across the different wires is likely to be in excess for some and in defect for others, the probable error of the mean will be much less than the error of any single wire. The line of collimation is the straight line joining the centre of the object glass, with the point of this imaginary vertical wire midway between the two horizontal ones. * See Parkinson's Optics. A diagram showing a telescope with a movable plate in front of it, allowing for regulation of light admitted. CHAP. IV.] THE TRANSIT INSTRUMENT. 56. Supposing the instrument perfect both in construction and in adjustment, the line of collimation will be at right angles to the east and west line about which the instrument rotates, and will exactly describe the plane of the meridian; so that at the instant when any star or other object is on the mean wire it will be crossing the meridian. The manner of the observation is as follows: The sextant is set so as to enter the star's field, the observer writes down the hour and minute shown by the clock; then taking a second from the clock face and following the beats, he notes the instant when the star appears on the first wire, writes it down, and, continuing the reckoning, does the same for each of the seven wires. The mean will be the time of transit as shown by the clock. If the star is not on one of the wires exactly at a beat of the clock, the observer must judge of the distances of the wire from the positions of the star, one on each side, when two successive beats are heard, and thus estimate the fraction of a second. A practised observer will estimate to tenths. Of late years a method of observing has been employed by the American astronomers, and is coming into general use, by which the instant of crossing each wire is more accurately determined: A cylinder or drum, covered with a sheet of paper, is placed upon a table, and its motion is regulated by a wheel without motion, by connecting it with clock-work having either a conical pendulum or a spring governor for its regulator (Art. 53). At the beginning of every second, the clock interrupts an electric circuit, and a corresponding dot or mark is made on the paper. The cylinder has a slow motion in direction of its length, so that the marks made, which are generally about an inch apart, arrange themselves without confusion in a spiral curve on the paper. When the instant arrives which the observer wishes to record, he presses a button near his hand, and an instantaneous mark is made on the paper, which, by its position relatively to the adjacent A diagram showing a cylindrical device with a rotating arm connected to a clockwork mechanism. 40 50 THE OBSERVATORY. [CHAP. IV. seconds' marks, enables him to measure the fraction with very great accuracy. As the mark made is a permanent record of the observation, the measures need not be taken till after the observation is wholly concluded ; the observer has therefore not to interrupt himself in order to write down the seconds and fractions ; thus the wires may be much closer to one another and the whole observation take less time. The observations consist of two items which happen to be in the field together can even be carried on without additional difficulty and without confusion. 57. The foregoing description of the transit instrument shews that to be in perfect adjustment it must satisfy the following conditions : 1st. The line of collimation must be perpendicular to the geometrical axis about which the instrument revolves. 2nd. This geometrical axis must be exactly horizontal. 3rd. It must point accurately east and west. By satisfying these three conditions the line of collimation describes a great circle ; the second makes this great circle vertical, therefore passing through the zenith ; and the third ensures its passing also through the pole and therefore coinciding with the meridian. When these adjustments are not perfect, which is seldom, if ever, the case, there will be consequent errors called respectively collimation, level, and deviation errors. It is useless attempting to get rid of these errors altogether : they must be reduced as much as possible by mechanical means as described below, and the residual uncorrected errors must be carefully determined so that their effect on the observations may be calculated and allowed for. * For a complete description of the process, see Loomis Practical Astronomy.* A diagram showing the transit instrument's components and how they relate to each other. CHAP. IV.] THE TRANSIT INSTRUMENT. 51 Collimation Error (Mechanical Correction). 58. Place a graduated scale horizontally, at a long distance off, in such a position that the telescope may be directed upon it; and, when it is fixed there, note the reading intercepted by the middle wire. Then lift the telescope upwards on its bearings and replace it with the axis reversed, the left pivot being now in the right $Y$ and the right pivot in the left $Y$. Direct again to the scale, and if the reading shown by the middle wire be the same as before, the adjustment is correct; if not, move the wires by the screws provided for the purpose, until the reading opposite the middle wire is the mean between the two previous ones. The error will now probably be corrected, but to verify it, repeat the operation by reversing the axis as before, and correct again if necessary until in both positions the same reading of the distant scale is obtained. If the middle wire and the imaginary mean wire are not coincident there will still be collimation error; therefore all that must be attempted in the previous adjustment is as close an agreement as can be obtained without too much tuggingering with the screws. The collimation error of each of the wires, and thence of the mean wire, must be determined and allowed for afterwards. (See Arts. 74 and 80.) Level Error (Mechanical Correction). 59. -To make the next correction we require a spirit level of sufficient length to reach from one extremity of the axis to the other, and furnished with legs terminating in inverted $Ya$ to ride on the cylindrical pivots. The tube of the spirit level is a small arc of a circular ring of very large radius placed with its convex side uppermost in a brass framework. It is nearly filled with spirits of wine or other quick fluid, a vacant space (called the bubble) A diagram showing a transit instrument with a graduated scale and a telescope mounted on a tripod. x2 52 THE OBSERVATORY. [CHAP. IV. being left, which, owing to the gravitation of the liquid, always occupies the highest part of the tube. A graduated scale attached to the tube allows us to ascertain the position of the bubble. 60. Place the level on the pivots, and read the gradua- tions at each end of the bubble. Half the sum of these readings, supposing the zero to be at one end of the scale, will give the graduation at the middle. Now reverse the level, i.e. place the east end on the west pivot, and the west end on the east pivot ; then, if the middle of the bubble settles to the same position as before, the axis is horizontal. If not, raise or depress that end of the axis which admits of vertical motion, until the middle of the bubble occupies the point midway between its two former positions. The axis will then be horizontal ; but, to ensure accuracy, the operation should be repeated. (See Arts. 78 and 79.) Deviation Error (Mechanical Correction). 61. Supposing the errors of collimation and level to have been corrected, the line of collimation will describe a vertical circle, and the remaining error—that of deviation—will be detected as follows : Note the times marked by the clock when a circumpolar star passes the mean vertical wire, first above the pole, then, some twelve hours later, below the pole, and again above the pole after another such interval. If the two intervals are exactly equal the line of collimation describes a vertical plane which bisects the path of the star, A diagram showing a graduated scale attached to a tube with a bubble in it. CHAP. IV.] THE TRANSIT INSTRUMENT. 53 and therefore coincides with the meridian plane ; but if not, it deviates from that direction, and the vertical circle which it traces out does not pass through the pole. The error is corrected by means of the screw which gives a horizontal motion to one end of the axis. 62. This method would not be available at places near the equator ; the few stars which would be circumpolar stars there would, in their lower transit, pass too near the horizon to be observable. A method, which will apply in all cases, will be given below. (Art. 77.) Setting Circles. 63. A small graduated circle is attached vertically to one side of the telescope tube near the observer's end, and has a moveable diameter which carries a spirit level. This is called the setting circle, and its use is to bring the telescope rapidly to point to that part of the meridian where a star of known declination is about to cross. The moveable diameter is previously adjusted to the declination of the star, then the graduated rim, then the motion of the traverse about its axis, which is used to bring the bubble to the middle of the level, will just point the telescope to a star which has the given declination. Sometimes there are two setting circles, one on each side of the telescope tube ; and more, if thought necessary, might be so attached, but two are generally found sufficient. They are useful when it is requisite to take observations of two objects which follow one another rapidly, but differ con- siderably in declination. Collimating Telescopes. 64. It was formerly considered essential in every obser- vatory to have a distant meridian mark fixed due south of the transit, to verify its adjustments at any time. A diagram showing a telescope with a graduated circle attached to one side of the tube. 54 THE OBSERVATORY. [CHAP. IV.] The Observatory of Cambridge was built so that the transit instrument might be due north of the tower of Grantham church, which happened to be in the required direction, and between two and three miles distant. The necessity for a distant mark is now dispensed with by the following contrivance: The transit being perfectly adjusted, a small telescope, having cross wires at its focus, is fixed in a permanent position either to the north or south of the transit, at any convenient distance, and in such a position that, looking into its object glass through this small telescope, we may see the cross (which must admit of being illuminated for the purpose) in exact coincidence with the middle wire of the transit. The rays from the cross wires will emerge from the small telescope in parallel directions, and, falling upon the object glass of the transit, will answer the purpose of a permanent mark at an infinite distance. 65. By using two such fixed telescopes, one on the north, the other on the south side, the adjustment for collimation may be made without the troublesome operation of reversing the transit, provided the two collimators be so placed that each may look into the other as well as into the transit. To ensure their doing so, without removing the transit from its supports, it must be turned to point to the zenith, and through two openings or doors managed in the sides of the tube, one collimator may be adjusted on the other. Then if the transit, after being adjusted on the south collimator, be turned towards the north one and found to be in adjustment with it also, the line of collimation will be at right angles to the axis of rotation. Collimating Eye-piece. 66. When the level error has been corrected, the collimation adjustment may be very simply and accurately made, by pointing the telescope vertically downwards to a vessel of mercury placed below it. The rays from any point of CHAP. IV.] THE TRANSIT INSTRUMENT. 55 the wires will proceed in parallel directions from the object glass, and, after reflexion at the horizontal surface of the mercury, will return to the object glass in parallel directions, making with the vertical, on the other side of it, the same angle as before. An inverted image of the system of wires is thus formed at the focus, and when the middle of the mean wire coincides with its image, there is no collimation error. (See Art. 74.) In order to see the wires, an eye-piece is used called the collimating eye-piece, which has a lateral opening between the two lenses. Through this opening, light is introduced and reflected towards the wires whose dark side being towards the mercury, the real wires will be seen bright, and their reflection dark. The light is reflected towards the wires by means of an elliptic ring reflector, or by a piece of plate glass which, without interfering with the direct vision, reflects sufficient light to render the wire visible. Reduction of Observations to the Mean Wire. 67. We have said that five or seven vertical wires are used, and the mean of the times taken for the time of transit. A passing cloud or other accidental circumstance may cause us to miss the star at the moment of its passage across some of the wires, but the observation will not be lost if the transit across one or more of the wires is secured, provided the time be known which the star would take to pass from each wire to the imaginary mean. 56 THE OBSERVATORY. [CHAP. IV. This interval will not be the same for all stars, those near the equator moving more rapidly than those near the pole ; but there will be a simple connection between them, so that when the equatorial intervals are known, those for a star of given declination will be readily obtained and may be tabulated for use. 68. A star, its image, and the centre of the object glass, are always in a straight line; hence, if a star moves, the straight line joining it with the centre of the object glass traces out, by its intersection with the plane of the wires, the path of the image across the field of view. Now, considering the centre of the object glass as the vertex of the cones in fig. p. 3, we see that for a star in the equator this path will be a straight line, but for all others it will be an arc of a circle having its concavity towards or from the north pole, according as the declination of the star is north or south. The telescope being an inverting one, let $a$, $b$, $a'$, represent the apparent path of a star crossing the meridian between the zenith and the equator. The arrow indicates the direction of motion in north latitude. Let $a$ be the point of intersection of the horizontal axis with the wire whose interval is required, $m$ being the mean wire. Let $A$, $M$, $A'$ be points of the celestial sphere corresponding to $a$, $m$, $a'$. $AM'A$ will be on a great circle perpendicular to the meridian $PM$, and the interval required is the same fraction of 24 hours that the angle $APM$ is of 360°. A diagram showing a star's apparent path across a telescope's field of view. The star is shown moving from north to south (or vice versa) depending on its declination. The lines $a$, $b$, $a'$ represent different paths for stars at different latitudes. CHAP. IV.] THE TRANSIT INSTRUMENT. 57 The right-angled triangle $APM$ gives $$\sin AM = \sin APM \sin PA;$$ but $AM$ depending on $w$ is constant for all positions of the telescope, and for a star in the equator $\sin PA = 1$; therefore $AM$ is the equatorial value of $APM$, that is $AM$ is the equatorial interval of that wire. Call it $w$, so that $$\sin w = \sin APM \cos \delta$$ will give the value of $APM$ when $w$ is known. When the star is on, or within a couple of degrees of, the equator, $\cos \delta$ will be very nearly 1, and $APM = w$. Except for stars near the pole, $w$ and $APM$ will be both small, so that $$w = APM \cos \delta$$ nearly. 69. To determine $w$: Observe the times of transit of a star of known declination $\delta$ across each of the wires, and take the mean. From this mean subtract the time of crossing the first wire; this will give the angle $APM$ corresponding to that wire, whence, by the above formula, the value of $w$ may be calculated. In the same manner determine $w_1$, $w_2$, ..., $w_n$ for the second, third, etc., wires. Those which precede the pole are positive, the others negative, and their sum is obviously zero. Delambre [Act. l. v. p. 416] seemed to give the pre-ference to stars on, or near, the equator; but, in order that an error in the observation of $APM$ may have the less influence on the value of $w$, it is usual to select a star near the pole ; for, although the probable error in the observed value of $APM$ increases with the declination, on account * Toutes les étoiles peuvent ainsi servir à déterminer l'intervalle équatorial de la plus proche étoile visible dans le ciel; en revanche, il n'est pas encore connu quels sont les avantages que procure cette méthode sur celle qui est déjà pratiquée par l'Observatoire de Potsdam.* Un ou deux degrés de déclinaison n'auparavant aucune différence sensible. Quelque Astronome on aura qu'il y ait de l'avantage à choisir les étoiles cir-cumpolaires pour qu'elles ne varient pas beaucoup plus que celles du Vierge-tingue que procède cette méthode est plus que détruit par l'inconvénient de l'obsolescence. 58 THE OBSERVATORY. [CHAP. IV.] of the more oblique and more sluggish motion of the star across the wire ; yet, except for stars quite close to the pole, this increase will be more than counterbalanced by the advantage of a decrease in the cosine of the declination. 70. The equatorial intervals of the wires having been determined as above from the mean of a large number of observations on different stars, we shall proceed to show how they may be applied to the correction of an imperfect observation. Suppose a star observed at the second, fifth, sixth, and seventh wires at the times $t_1$, $t_2$, $t_3$, $t_4$. The estimated time across the mean wire will, by adding the corresponding $A/P$ to each, be $$t_1 + w_1 \text{ sec } \delta \text{ from the first wire observed},$$ $$t_2 + w_2 \text{ sec } \delta \text{ second },$$ $$t_3 + w_3 \text{ sec } \delta \text{ third },$$ $$t_4 + w_4 \text{ sec } \delta \text{ fourth },$$ adding and dividing by the number of wires, we obtain for the time of transit $$\frac{t_1 + t_2 + t_3 + t_4}{4} = \frac{w_1 + w_2 + w_3 + w_4}{4} \text{ sec } \delta,$$ i.e. multiply the algebraical mean of the equatorial intervals, corresponding to the wires observed, by the secant of the declination ; the product will be the additive correction to be applied to the mean of the times. Determination and Effect of Residual Errors. 71. The errors of collimation, level, and deviation, having been almost completely corrected by some of the mechanical methods indicated in the preceding pages, we will now proceed to indicate the means of determining and allowing for the small residual errors ; but as the angles we shall have to consider are extremely small, we shall first explain the CHAP. IV.] THE TRANSIT INSTRUMENT. 59 means employed for measuring such small angles with great accuracy. **Spider Line Micrometer.** 72. The instrument of greatest value for this purpose is the spider line micrometer. It consists of a small rectangular frame about three or four inches in length, by one inch in breadth, carrying a wire or spider line, which can be moved by means of a screw-head. This work is placed at the common focus of the object glass and eye-piece of the telescope so that the spider line may be almost in contact with the fixed wires. The threads of the screw should be perfectly uniform and regular, so that each turn of the screw-head may carry the spider line over an equal space. The value of one turn of the screw, i.e. the angle, subtended at the centre of the object glass, by the distance through which the line is shifted, can be measured by the method explained for determining the equatorial intervals of the wires [Art. 69]. Make the spider line coincide with one of the wires, then several thousandths or more turns of the screw-head, and divide the corresponding equatorial interval by the number of turns. In this way also may the regularity of the screw be tested throughout, the head being divided into sixty or one hundred parts to indicate the advance of the thread corresponding to the sixtieth or hundredth part of a revolution. In order to check the number of turns, a row of teeth or notches, called a comb, is visible at one side of the field A diagram showing a spider line micrometer. A rectangular frame carries a wire or spider line. The frame is placed at the common focus of an object glass and eye-piece of a telescope. The threads of a screw-head are shown on either side of the frame. 60 THE OBSERVATORY. [CHAP. IV of view. The line passes one of these at each revolution of the screw-head, and every fifth one is cut deeper than the others. The central one of all is distinguished by a small circular aperture.* **Determination of Uncorrected Level Error.** 73. In the description of the spirit level (Art. 59) we said that a graduated scale is attached to the tube. If we raise or lower one end of the tube, the bubble will move along the scale, and the change of inclination corresponding to each division may be easily determined experimentally, and expressed in parts of a second. If we place the level on the pivot and take the reading of each end of the bubble, then half the algebraic sum of the readings will be that of the highest point. Let $A$, $B$ (fig. 1) be the two pivots, $MN$ the bubble, $P$ its middle point, $O$ the zero; of the scale, $S$ the point corresponding to $O$ at the other end of the scale (so that $OS$ is parallel to $AB$). If $A$ and $B$ were on the same level, $P$ would occupy the middle point of $OS$. The inclination will therefore be measured by the difference between $OP$ and $\frac{1}{2}OS$. To determine $OS$, turn the level so that the east end may be over the west pivot, and west end over the east. * Sometimes two parallel wires are used, each moveable by its own screw. For discussion of this method of using them, see Charpentier's Astronomy, and Leonard's Practical Astronomy. † The simplest way of doing this, is by attaching the level to a mural circle. The mural circle which has the level in use, as the bubble passes from one graduation to the next, is seen given by its name. Fig. 1 Fig. 2 CHAP. IV.] THE TRANSIT INSTRUMENT. 61 pivot, as in fig. 2 ; $S$ and $O$ interchange places, and the middle of the bubble being at $P$, the mean reading will now give $OP$, which is equal to $SP$ of fig. 2. Let $OP$, or $b$, be the reading of $P$ in the first position, $OP'$, or $b'$, be the reading of $P'$ secondly...; therefore $OS \cdot b + b' = P' - P$; and the inclination to the horizontal, that is the level error, is $(OP' - OS)$, that is $\frac{a}{2}(a-b)$, where $a$ is the value in seconds of each division of the scale.* Determination of Uncorrected Collimation Error. 74. Apply the collimating eye-piece and the micrometer to the transit. Then, turning the telescope vertically downwards to a trough of mercury, as described in Art. 64, measure the angular distance between each wire and its image. One half of this angle will be the inclination to the vertical of the line which joins the wire with the centre of the object glass. Correct this for any level error there may be, and the result will be the collimation error of the wire in question ; that is, the angle which the line joining the wire with the centre of the object glass makes with the line at right angles to the axis of rotation. Correct this again for the interval in seconds of arc between this wire and the mean of the seven (Art. 69). This will give a value of the collimation error of the mean wire. Repeat the operation on each wire and take the mean of the result for the collimation error of the transit. 75. If the level error be not allowed for, the preceding result will be the sum or difference of collimation and level errors. But if the transit be lifted off its feet and reversed, * The figure supposes $O$, the zero of the scale, to be near one end of the instrument, and therefore $O$ near the other. If $O$ were at, say, near, the middle, and $O$ and $S$ on the same side of $P$, OS would be $a - b$ and the level error $\frac{a}{2}(a+b)$. 62 THE OBSERVATORY. [CHAP. IV. and the observation repeated, the second result will be the difference or sum of the same two errors, and both errors may thus be determined. Determination of the Uncorrected Deviation Error. 70. First Method. By a known circumpolar star observed above and below the pole. Suppose the level and collimation errors to be known and allowed for, as well as the rate of the clock, then the upper and lower transit should be separated by an interval of twelve sidereal hours exactly (Art. 61). When this is not the case the instrument does not trace out the meridian. Let $ZDPH$ be the meridian, $ZBAH$ the vertical in which the transit moves, $CAID$ the small circle described by a star, $A$ and $B$ being its positions at the instants of apparent lower and upper transits. Let $x$ be the deviation error $\angle ZPA$, $z$ the known polar distance $PB$ of the star, $\phi$ the known latitude $PH$ of the place, and $180^\circ - \phi$ the observed sideral interval $BA$. From $P$ let fall $PM$ perpendicular on $ZBA$. The triangle $BPM$ gives $$tan PM = cos BPM tan BP = sin \frac{\theta}{2} tan \Delta.$$ The triangle $ZPM$ gives $$sin PM = sin ZPM sin PZM = cos \phi sin x,$$ and since $\theta$, $\phi$, and $PM$ are small, $$x = PM sec \phi - \frac{\theta}{2} tan \Delta sec \phi.$$ This method can only be employed with advantage in A diagram showing a star's position relative to a transit circle. The diagram includes labels for various points such as Z, D, P, H, A, I, C, B, X, Y, Z', B', M'. The diagram also shows angles and distances between these points. 9/22 CHAP. IV.] THE TRANSIT INSTRUMENT. 63 the winter, because an interval of 12 hours must elapse between the two observations; and a star must be chosen which culminates about 6 o'clock in the evening in order to be still visible at the next transit. At places near the equator the method fails, as the few circumpolar stars are too near the horizon to be observable at these lower transits; and besides, the length of the night, while it varies very little during the year, will not remain when diminished by the duration of the morning and evening twilight, seldom to leave more than 10½ or 11 hours during which the stars may be seen. 77. Second Method. By Transits of two known Stars. Let $A'B'ZP$ be the meridian, $ZNM$ the great circle described by the transit, $x$ the deviation error $A'ZM$, $\Delta$, $\Delta'$ the polar distances of two known stars $A$ and $B$; $AM'A$, $BN'B$ their parallels. From its position, assuming that the star $A$ will reach the apparent meridian $ZNM$ in time measured by the angle $APM$; the star $B$ will take a time measured by $BP.N$. The difference between these, that is, $AP.B - MP.N$ measures the time that elapses between the crossings of the two stars over the mean wire of the transit. But if the instrument were in perfect adjustment the interval would be $AP.B$, which is the known difference of their right ascensions. Let $\theta$ be the excess N.P.M. of the observed over the known true interval, $\phi = \text{known latitude} = 90^\circ - ZP$. A diagram showing a transit instrument with a star's path on a great circle labeled ZNM. Points A', B', P, M', N', and Z are labeled. 64 THE OBSERVATORY. [CHAP. IV. The triangle $ZPM$ gives $$\cot PM \sin PZ = \cot Z \sin P + \cos PZ \cos P,$$ $$\cot \Delta \cos \phi = -\cot x \sin P + \sin \phi \cos P,$$ but $x$ and $P$ are both small; therefore $$z = |\sin \phi - \cot \Delta \cos \phi|.$$ So the triangle $ZPN$ gives $$P - \theta = x |\sin \phi - \cot \Delta' \cos \phi|;$$ therefore $$\theta = x |\cot \Delta' - \cot \Delta| \cos \phi,$$ $$z = |\sin \Delta' \sin \Delta'|$$ 78. In selecting stars for observation by this method, which is always applicable, it will be desirable to take two whose right ascensions are not widely different; because the time between the two transits being short, the chance of error in the observed values of $\theta$, so far as it is due to any irregularity of the clock, will be diminished. Again, the larger the value of $\theta$ will an error in its value affect the value of $z$; but $\theta$ varies as $\cot \Delta - \cot \Delta'$, therefore the two stars should differ considerably in declination. Corrections to be Applied to the Observed Time of Transit. *79. Let $u$, $v$, $x$ be the values of the uncorrected errors of level, collimation, and deviation respectively, determined by some of the preceding methods. We will now calculate their effects on the time of transit of a star; and in doing this we shall suppose them to exist separately, the aggregate correction being the sum of the partial corrections when the errors are small. Effect of Uncorrected Level Error. Let $RZP$ be the meridian, and $RAI$ the great circle traced out by the transit, A diagram showing a star's position relative to a meridian and a great circle. CHAP. IV.] THE TRANSIT INSTRUMENT. 65 u the level angle or error at $R$ between the two circles, $\Delta$ the polar distance of a star seen in the transit at $A$, $\phi$ the latitude = 90° - $PZ$, $\theta$, the error $APZ$ in the time of transit. The triangle $PAR$ gives $$\cot PA \sin PR = cot R \sin P + cos PR \cos P$$ or $$cot \Delta \sin \phi = \frac{u}{u} \cos \phi,$$ $$\theta_{1} = u \sin (\Delta + \phi)$$ This being the correction to be added to the observed time of transit, we see that $u$ must be considered positive when the western pivot is too high, and negative when too low. For a transit below the pole the reverse would be the case. Effect of Uncorrected Collimation Error. 80. Let the transit trace out the small circle $BAH$ parallel to the meridian $RMPH$, and let $AM$ be a perpendicular from the star $A$ to the meridian. Join $AP$. Let $\varpi = AM$, the collimation error, $\theta_1 = APZ$, the error in the time, $$\sin AM = \sin AP \sin APM,$$ $$\theta_2 = \frac{\varpi}{\sin \Delta}$$ $\varpi$ is to be considered positive when the line of collimation points to the east of the meridian, and negative when it points to the west. A diagram showing a transit instrument with a small circle BAH parallel to the meridian RMPH. The line AB is drawn from A to H, and AP is drawn from A to P. The angle $\theta_1$ is shown as the error in time, and $\theta_2$ is shown as the collimation error. 66 THE OBSERVATORY. [CHA.P. IV. Effect of Uncorrected Deviation Error. 81. As in Art. 77, the triangle $ZPM$ will give the equation $$\cot P M \sin P Z = \cot Z \sin P + \cos P Z \cos P,$$ whence $$P = \frac{\sin (\phi - \Delta) \cos \Delta}{\sin \Delta},$$ or $$\theta_s = -\frac{x \cos (\Delta + \phi)}{\sin \Delta}.$$ The fig. of Art. 77 shows that $x$ is to be reckoned positive when the east pivot of the axis of the transit deviates towards the elevated pole, and negative when it deviates in the other direction. 82. For the complete correction, therefore, $$\theta_1 + \theta_2 + \theta_3 = x \frac{\sin (\Delta + \phi) + r - x \cos (\Delta + \phi)}{\sin \Delta},$$ where $r$, $x$, and $z$ are to be taken with their proper signs, as explained above ; and the result being expressed in seconds of arc will have to be divided by 15 to reduce it to seconds of time. Personal Equation and other Errors. 83. Besides the errors we have spoken of, which may be called errors of adjustment, there are several other sources of error to which all instrumental observations are liable. First, errors of construction arising from some imperfection of workmanship. These can only be detected by a thorough study of each individual instrument, and by making repeated observations under various circumstances ; but a knowledge of the theory of the instrument will often suggest modes of eliminating these errors ; as, for instance, in the case of the mural (Art. 93), where the using opposite microscopes in pairs corrects a possible error in the position of the centre of graduation. Secondly, accidental errors due to extraneous causes, such as sudden and unobserved changes of temperature, atmo- CHAP. IV.] THE TRANSIT INSTRUMENT. 67 spheric indistinctness, looseness of screws, &c. The observer has no means of correcting these errors, except by multiplying his observations, whereby he may hope that, as the errors are accidental, they will sometimes tend one way and sometimes the other, and therefore that the mean result will have a less error than any single observation. Thirdly, a kind of error to which the name of personal error, or personal equation, has been given. It is often found that two observers, though equally trained in observing, will differ by a fraction of a second in their estimation of the time of transit of a star; and that this difference remains pretty nearly constant for months or years. One of them is to show how this error may exist, but as such a rate of discordance exists which, though small, must not be neglected when we have to combine observations made by the two. In the same observatory it is usual to select some one of the observers as a standard of reference, and to reduce all observations made by the others to the standard, by allowing for their personal equation as obtained by comparison with him. * See Arago's Mémoires Scientifiques, tome 11. p. 320. F2 ( 68 ) CHAPTER V. THE OBSERVATORY CONTINUED. The Mural Circle. 84. The mural circle determines the meridian zenith distance of an object, and is a necessary complement of the transit. It consists of a large graduated circle placed in a vertical plane against the western face of a wall which runs north and south. This circle revolves on a pivot about a strong axis which penetrates right through the wall, and which, for purposes of adjustment, can be slightly moved by screws on the western face. An astronomical telescope is firmly attached to the face of the circle, so as to be parallel to a diameter, and when the whole instrument is in perfect adjustment the line of collimation (Art. 86) describes the meridian. The graduations of the circle are made on the thickness of the rim, and read continuously from $0^\circ$ to $360^\circ$ at intervals of $5^\circ$. These divisions pass successively behind a small projecting piece of metal attached to the wall, called the pointer; and the pointer-reading, when the telescope is turned exactly to the zenith, is called the zenith-point. The difference between the zenith-point and that star, is the meridional zenith distance of that star. 85. The graduations are, as we have said, at intervals of $5^\circ$. The intermediate minutes and seconds are obtained by means of six microscopes firmly fastened to the wall, placed round the circle $60^\circ$ apart and looking towards the CHAP. V.] THE MURAL CIRCLE. 69 graduated rim. One microscope would be sufficient, but six are used to counteract errors of graduation, centering, and unequal expansion (Art. 93). They are lettered and referred to as in the figure, A, B, C, D, E, F. Each microscope is furnished with a micrometer, similar to the spider-line micrometer described in Art. 72. It is convenient, though not essential, that the microscopes be so adjusted that, when the pointer coincides with a graduation, all the micrometer wires, supposing each to occupy its zero position at the middle of the field of view, shall coincide with graduations. If the circle be then turned through a fraction of a division, an equal portion of the limb will pass opposite to each microscope; this precise fraction of arc can be obtained with extreme accuracy from the number of turns and parts of a turn of the screw-head necessary to carry the spider-line from the zero position till it coincides with the image of the division that has passed away from it. A diagram showing a circular scale with a pointer and a spider-line micrometer attached to it. 70 THE OBSERVATORY. [CHAP. V. Five turns of the screw-head should carry the spider-lines from one division of the limb to the next, so that each turn may indicate one minute ; a notched comb of metal at the side of the field will check the number of turns or minutes. The screw-head being divided into sixty equal parts, the reading off will give the additional seconds, and the mean of the six readings, added to the integral number of $5^{\circ}$ shown by the pointer, will give the reading of the mural for that position. 86. At the principal focus of the telescope is a framework carrying one horizontal and five vertical wires—-the line of collimation is the line joining the centre of the object-glass, with the intersection of the horizontal wire by the mean of the five. All these wires are fixed ; but another horizontal wire, called the micrometer wire, moveable across the field of view, is necessary in certain observations for measuring small distances of celestial objects from the fixed horizontal wire. These measurements are made in minutes and seconds of arc, by means of a graduated screw-head, which gives motion to the micrometer wire, and whose graduations have a known value. The minutes and seconds so obtained, added to, or subtracted from, the mural reading given by the pointer on microscopes, will give what the mural reading would have been if the fixed wire had occupied the position of the micrometer wire. The values of the graduations for the micrometer wire may be found thus.—Direct the telescope towards some distant mark which happens to be, or is purposely placed, in the meridian of the mural. Move the telescope micro-meter, by giving the screw-head some exact number of revolutions from its zero position, say ten for instance. Then turn the mural gently till the micrometer wire just bisects the mark, and read off the pointer and the six CHAP. V.] THE MURAL CIRCLE. 71 microscopes. Repeat the operation with the micrometer wire placed at ten revolutions on the other side of the zero, and again read the circle. The difference of the two readings will be the value, in arc, of twenty revolutions of the screw. Hence the value of one revolution, and of its fractions, will be known. Clamp, Tangent Screw. 87. In making an observation with the mural—when it has been turned so that the star is in the field of view, and a very small additional motion is required to bring it to the fixed wire—it will generally be found that the hand is too abrupt in its movements, and that a slower and more delicate means is necessary. This is obtained by a clamp and tangent-screw, of which there are several, so that, in all positions, one may be within reach of the other. The clamp consists of two metal plates, separated between which the rim of the circle slides. These lips project from another piece attached to the wall, and, by one turn of a screw, they can be tightened on the circle so as to fix it. The tangent-screw is another screw, with very fine threads, which, acting on the piece that carries the clamp, moves it and the circle with it at a slow pace in the direction of the arc, thus modifying the too rapid motion of the hand. Error of Runs. 88. When a change of temperature, or any other cause, alters the distance of a microscope from the rim of the circle, the distance, at the focus, between the images of two consecutive graduations, will alter; and five turns of the screw-head will no longer carry the micrometer wire accurately from one division to the next. The error arising from this is called the error of runs, and a corresponding correction becomes necessary. A diagram showing a clamp and tangent-screw mechanism for adjusting a microscope. 72 THE OBSERVATORY. [CHAP. V.] Suppose the interval between two graduations at the microscope $A$ to require $a^{\prime}$ more than the five turns, the correction to be applied to a smaller interval, measured by the microscope, must be obtained from the proportion $$b + a^{\prime} : a^{\prime} : \text{measured interval} : \text{correction}$$ This correction must be subtracted when $a$ is positive and added when $a$ is negative. In the same way may the reading of each microscope be corrected, but instead of applying the correction to each, which would constitute six proportions, it will be sufficient to correct the average observed interval by means of the average error ; and, as $a$ is always a small number of seconds, we may further simplify the proportion by using 5' for the first term instead of $b + a'$. The errors of runs must be examined from time to time, and if they become too considerable, they may be mechanically reduced by moving the microscopes towards or from the circle. Errors of Adjustment of Telescope. 89. The line of collimation of the telescope should describe the meridian—the instrument is therefore, like the transit, liable to the three errors of collimation, level, and deviation. But these errors have not the same importance as in the case of the transit; for, as the object of the mural being the determination of altitudes, a very small deviation from the meridian will not produce an appreciable error in the result; and it will generally be sufficient to correct, as far as possible, by mechanical adjustment. 90. Level error may be detected by means of a microscope, projecting from the face of the mural, with its axis parallel to the arc of the rim and revolving with it. Let a fine plumb line, fastened to some point above, pass in front of the circle just opposite its vertical diameter, and at such A diagram showing a telescope with a plumb line attached to it. CHAP. V.] THE MURAL CIRCLE. 73 a distance from it that it may be viewed by the microscope when near its highest position. Then turn the mural half round, so as to bring the microscope to its lowest position, and the thread should again occupy the same position in the field of view. If it does not, we must correct the error by means of the adjusting screws at the back of the wall. 91. Supposing the level error corrected, we may proceed to correct the collimation error, if there be any. This will be done by moving the framework, which carries the wires of the telescope, so that a star near the zenith may pass the middle wire at the same moment that it passes the middle wire of the transit. 92. The deviation error is next corrected, by observing a star near the horizon, and moving the axis by means of the screws at the back of the wall, so that the star may cross the middle wire of the mural and that of the transit at the same instant. Errors of Centering, Graduation, &c. 93. The mural is essentially a differential instrument, requiring always two positions to complete an observation. The angle through which the telescope turns is the quantity we have to determine, and we can easily ascertain it, by using reading microscopes on pivots, which will eliminate any error which arises from the centre of graduation not coinciding with either of rotation, or from irregularity in the form of the pivots. Thus, let the dark line in the figure represent the first position of the graduated circle when $H$, $K$ are the points of the limb opposite the microscopes $A$, $B$, and $M$ the zero of graduation. Let the fine line represent the second position of the circle, $O$ its centre, the line $HK$ having come into the position $H'K'$, and $M$ moved to $M'$. The telescope will have 74 THE OBSERVATORY. [CHAP. V. turned through the same angle as any line rigidly connected A diagram showing a telescope with a circular eyepiece. The telescope is oriented such that the axis of rotation is perpendicular to the plane of the diagram. The telescope is shown in two positions, one before and one after rotation. The angle between the two positions of the telescope is measured by the difference in readings at points A and B on the eyepiece. The difference in readings at A gives \(hOH' = 2ChH'\), and the sum of these gives \(2(ChH' + Ch'H') = 2HCH'\). The half sum is the angle between the two positions of the telescope. with the circle ; therefore \(HCH'\) is the angle to be found. The two readings of \(A\) are respectively \((MH\) or \(M'H'\) and \(M'h\), \(B\) \(\ldots\ldots\ldots\ldots(MHK\) or \(M'HK'\) and \(M'Hk\). The difference of the readings at \(A\) gives \(hOH' = 2ChH'\), \(\ldots\ldots\ldots\ldots B \ldots kOK' = 2CH'h\). The sum of these is \(= 2(ChH' + Ch'H') = 2HCH'\), and the half sum is the angle between the two positions of the telescope. The centre of rotation, that is the point about which the instrument is supposed to have turned in passing from the first position to the second, is not shown in the figure; and the investigations will show that, even when the pivots are not cylindrical, and fixed, when consequently the centre of rotation is not a fixed point, the readings of each pair of microscopes will give a value of the angle free from errors of centering, and from irregularity in the form of the pivots. The mean of the three pairs will diminish the risk of errors of graduation and of unequal expansion. CHAP. V.] THE MURAL CIRCLE. 75 Observing with the Mural. Zenith-Point. Polar-Point. 94. The difference of declination of two stars, or which is the same thing, the difference of their meridional zenith distances, may be obtained as follows. --When the first star is about to pass the meridian, turn the telescope so that it may be in the field of view. Then close the eye, and, measuring on the tangent screen, draw the circle, so that the star may be on the fixed horizontal wire* at the moment that it crosses the middle vertical one. Now read the six microscopes, and add the mean of the six readings to the pointer reading. Repeat the operation on the second star and take the difference of the results. This will give the difference of meridian altitudes of the two stars.* 95. If, instead of a second star, we use the image of the same star as seen by reflection the next evening in a trough of mercury, we shall obtain the meridian altitude of that star. Thus, if $PQ$ be the direction of the telescope on the first night, when we observe the star $S$ directly, and $P'Q'$ its A diagram showing a telescope pointing towards a star $S$. The telescope's position on the first night is marked as $PQ$, and on the second night as $P'Q'$. A line $R$ represents a fixed horizontal wire. The diagram also shows a trough of mercury with a mirror reflecting light from $S$ onto a tangent screen. direction on the second night when we observe it by reflection * Some instruments, instead of a fixed horizontal wire, have two fixed wires separated by an interval of 10°, and the star is kept exactly in the middle of the space between the wires. † These results require correction for refraction. See Chap. XV. 1 76 THE OBSERVATORY. [CHAP. V. at $R$, the surface of the mercury, then $QOQ'$ is the angle given by the difference of the two mural readings. Now $O$ is parallel to $RS$, and if $OH$ be drawn horizontal, and consequently parallel to the surface $AB$ of the mercury, the angles $SRB$ and $ORA$ being equal, $QOH$ and $HQO$ will be equal, hence $\angle OHB = \frac{1}{2} QOQ'$ or, half the difference of the mural readings will give the altitude of the star. 96. It is also obvious, that half the sum of the readings will be the reading which the instrument would give if the telescope coincided with the horizontal $OH$. This horizontal reading, increased or diminished by 90°, as the case may require, gives the zenith reading or zenith-point. When the zenith-point is known, one observation of a star is sufficient to determine its altitude --- Taking the difference between the zenith-point and the mural reading corresponding to the direct observation of the star, we obtain its meridional zenith distance. 97. One objection to the preceding method of observation and of determination of the zenith-point is that, in the interval between the observations, changes of temperature, &c., may occur, altering the amount of refraction of the star, and therefore its apparent direction ; and there is also this inconvenience, that, in consequence of unfavourable weather, several days may intervene before a complete observation can be obtained. The following method, known as "a double observation with the mural," has been devised for observing both by direct and reflected vision at one and the same transit. Some time before the star (whose zenith distance is supposed approximately known) comes to the meridian, set the circle by the pointer, so that the image reflected from * If the observer were at a considerable distance from the mural, it might become necessary to take account of the angle between the normals at $O$ and $H$. CHAP. V.] THE MURAL CIRCLE. 77 the mercury may be sure to enter the field of view. Then clamp the circle and read the pointer and the six microscopes. When the star enters the field of the telescope, follow it with the micrometer wire so as to bisect it just before it crosses the middle vertical wire. Unclamp the circle and turn it till the star, seen directly, is again in the field of view (which can be rapidly done by the approximately known pointer reading). Then, using the nearest clamp and its tangent screw, direct the star by the fixed horizontal wire. Finally, read off the pointer, the six microscopes, and the micrometer of the telescope. The former reading of the mural corrected by adding, with its proper sign, the micrometer reading will reduce the reflection observation to the fixed horizontal wire; and we shall thus have two circle-readings corresponding to direct and reflection observations of the same star. Half the sum of these readings will give the horizontal point and half the difference the meridian altitude of the star. 98. We here suppose that the star is so near the meridian at the two moments when it is observed that the error of altitude is inappreciable. But, if the interval were considerable, for instance, if the first observation were made at the moment when the star is crossing the first vertical wire, and the second when it is crossing the last, it would be necessary to calculate the change of altitude to the meridian. The formula for this correction is given in the method of finding the latitude by circum-meridian observations. See Chap. ix. 99. The zenith-point may also be obtained independently and very accurately by means of the collimating eye-piece. We may, as was described in the case of the transit (Art. 60), make the line of collimation exactly vertical by pointing the telescope vertically downwards towards a trough of mercury, and bringing the reflected image of the fixed horizontal A diagram showing a telescope with a micrometer scale attached to its eyepiece, used for measuring altitudes and azimuths. 78 THE OBSERVATORY. [CHAP. V. wire to coincide with the wire itself. The corresponding mural reading will be the nadir-point, which, increased by 180°, will give the zenith-point. 100. To find the Polar-point, the reading of the mural when the telescope is directed to the pole. Observe a circumpolar star in its upper and lower transits, and note the readings of the mural in the two positions. Thus, if $$Za \cdot Po' = \text{be the meridian of the observer},$$ $$e \quad \text{the star at the upper culmination},$$ $$H \quad \text{the polar-point and } \Delta \quad \text{the polar distance},$$ then $$Po = Po' = \Delta,$$ and one of the readings will be $H + \Delta$, the other $H - \Delta$.* Therefore the polar-point is given by half the sum of the two readings. Con. 1. Half the difference of the two readings is the polar distance of the star, which, subtracted from 90°, gives the declination. Con. 2. The difference between the polar-point and the zenith-point is the colatitude $ZP$, which, subtracted from 90°, gives the latitude. * These readings must be corrected for refraction and for instrumental errors. A diagram showing a circle with a line segment labeled ZP connecting points on opposite sides of a smaller circle. (79) CHAPTER VI. THE OBSERVATORY CONTINUED. The Transit-Circle. 101. Although the transit and the mural have been thus described as the two important instruments in an observatory, it is now found that a single instrument, combining the functions of both, can with advantage be made to take their place. Theoretically, the mural would do so, if the adjustment in the meridian could be relied on; but its peculiar mounting, requiring the support to be all on one side, makes it unable to bear the weight of a large instrument, and renders it liable to deflections from the meridian—deflections which are unimportant in their effect on the altitudes, but which could not be tolerated in the determination of times of transit. A Transit-circle has been recently mounted at the Cambridge Observatory in the place of the former Transit instrument. It is symmetrically made and supported. The axis, resting on four pillars, carries, besides the telescope and graduated circles, one on each side, about three feet in diameter, and revolving with the instrument. Two other circles, firmly fixed to the piers, carry the reading-microscopes, four on each, besides a pointer-microscope, which replaces with advantage the metal pointer of the mural. The telescope has a focal length of about nine and a half feet and an aperture of eight inches. The instrument is reversible, but there is little need for this provision, because two collimating telescopes, with apertures of six inches, have 80 THE OBSERVATORY. [CHAP. VL. been fixed on the north and south sides, and the adjust- ments can be accurately made without reversion (Art. 65). The counterpoises are not seen, being under the floor, where they act on levers that thrust upwards and support the weight. The graduations opposite the reading-micro- scopes receive light by an ingenious system of glass prisms ; and various other devices, too technical to be mentioned here, combine to render this a most effective and valuable instrument. Our observation with a transit-circle gives the two results which, with the transit and mural, required two observers. There is thus a great saving of labour ; and one source of error, from which the separate instruments were not free when observing small stars thickly clustered, is altogether impossible in the transit-circle ;-the error, namely, of ob- serving one star for altitude and, by mistake, a different one for time, so as to create an imaginary star combining the elements of two real stars. The adjustments and corrections for level, collimation, and deviation are made as for the transit, and need not therefore be dwelt upon. The Equatorial. 102. The only observations which can be made with the instruments, so far described, are restricted to the plane of the meridian ; but there are many occasions, such as eclipses and occultations, where the phenomena to be ob- served occur out of that plane. It may also be desirable to have a means of fixing with accuracy the position of an object at any moment when visible, especially of those objects which, like comets, have a rapid motion among the others. We therefore require telescopes so mounted as to allow of being pointed in any direction, and an especial advantage is secured when the object can be followed with ease, and retained in the field CHAP. VI.] THE EQUATORIAL. 81 of view, so as to afford sufficient time for micrometer measurements and minute examination. 103. One of the instruments devised for the purpose consists of a telescope $AB$ in contact with a graduated circle A diagram showing a telescope $AB$ in contact with a graduated circle $CD$. The telescope is pointed towards a star, and the angle $EOB$ is measured by turning the plane of the declination circle $CD$, without altering the angle $EOB$. This allows the star during its diurnal course, and thus verify the accuracy of the statement which follows observations in Art. 5 suggested, viz., that the stars move uniformly in circles round the axis of the celestial sphere. $CD$, and turning about an axis through the centre $O$ perpendicularly to the plane of the circle. One of the diameters of this circle is produced, and constitutes an axis $EF$, whose direction coincides with the axis of the celestial sphere. About this axis the whole instrument turns, and the angles through which it turns are measured by a second graduated circle $GH$, perpendicular to the axis. 104. If the telescope be directed towards a star, $EOB$ will be the angular distance of that star from the pole; the polar distance, or the declination, may therefore be measured by the graduated circle $CD$, and by merely turning the plane of this declination circle, without altering the angle $EOB$, we may follow the star during its diurnal course, and thus verify the accuracy of the statement which follows observations in Art. 5 suggested, viz., that the stars move uniformly in circles round the axis of the celestial sphere. * Allowance must be made for refraction, as will be explained hereafter. G 82 THE OBSERVATORY. [CRAP. VI.] In most instruments the circle $GH$ is connected with clock-work, the motive power being a descending weight and the regulator a conical pendulum or governor. A uniform motion without jerks is thus communicated to the instrument, which, when properly adjusted, can retain in the field of view a star or other object, so long as no change be required within the interval of time of the observer. But the principal use of the instrument is for differential observations—to find the difference of declination or of right ascension of two neighbouring bodies, the diameters of planets, the distances of double stars, &c. These observations can all be made with the instrument in a fixed position, and are therefore independent of errors of adjustment. 105. The above is the principal of all equatorials. The mounting may be different in different instruments, but in every one there must be—a fixed line coincident with the polar axis, a means of measuring the angle which the telescope makes with this fixed axis, and also of measuring the angle described by the instrument in turning about the axis. There will therefore be six adjustments: (1) The polar axis must have the same altitude as that of the celestial sphere. (2) The declination circle must mark zero when the line of collimation is in the plane of the equator. (3) The polar axis must be in the plane of the meridian. (4) The line of collimation of the telescope must be at right angles to the declination axis (the axis through $O$ about which the telescope turns). (5) The declination axis must be perpendicular to the polar axis. (6) The graduated circle $GH$ must mark zero when the telescope is in the meridian. CHAP. VI.] THE EQUATORIAL. 83 For the means of performing these adjustments we shall refer to works on practical astronomy. Loomis' *Introduction to Practical Astronomy*, Challis' *Syllabus*, &c. 106. The necessity for these corrections, and the liability of a heavy instrument mounted in this inclined position to get out of adjustment, render the equatorial somewhat uncertain in the determination of absolute declinations and right ascensions; but the facility of pointing it in any direction makes it a valuable instrument of determining the difference of right ascension or of declination between objects which both pass through the field of view while the telescope is in a fixed position. There are, as we have stated, various ways of mounting the equatorial, but the differences are points of detail which can easily be understood; for instance, in the Northumberland equatorial at the Cambridge Observatory, there is no declination circle, but a straight rod—which, by sliding in a tube, can be shortened or lengthened at pleasure—connects the eye-end of the telescope with a point in the axis of the instrument, and thus determines the angle at which the telescope is inclined to the polar axis. The angle itself is shown by graduations on the rod. 107. At the focus of the object-glass of the equatorial is a system of spider lines or wires, similar to those of the transit, and respectively parallel and perpendicular to the plane of the declination circle. Besides these, there must be moveable wires for micrometer measurements, such as those of the spider-line micrometer, already described in Art. 72. Position Micrometer. It is often necessary when two objects are in the field of view at the same time, to find the angle which the line joining them makes with the declination circle passing through either of them and the pole. For this purpose the eye-piece is furnished with a o 2 84 THE OBSERVATORY. CHAP. VI. spider-line which, by means of and rack a screw, admits of circular motion in a plane at right angles to the line of collimation. If the telescope be pointed towards one of the objects and the spider-line be turned to cover both, then the angle it makes with the fixed central wire is the angle required, and will be given by the screw-head. 108. Finder of Equatorial. When the equatorial is a large instrument of great magnifying power, the portion of the heavens embraced by the field of view becomes so small that it is somewhat difficult to point the telescope. A smaller instrument called the finder whose line of collimation is parallel to that of the equatorial, is usually attached to the larger instrument; this embraces a much wider field, and a star brought to the centre of the finder will be in the field of the equatorial. Altitude-Azimuth—Transit in the Prime Vertical. 109. The altitude and azimuth instrument, or alt-azimuth, is another instrument intended, like the equatorial, to make observations on objects situated in any vertical plane. As its name implies, it gives both the altitude and the azimuth of a celestial body. It consists of two graduated circles--one fixed in a horizontal plane, the other in a vertical plane which admits of being turned about a vertical axis so as to coincide with any azimuth. A telescope, parallel to the plane of this vertical circle, turns with it in azimuth and admits also of motion in altitude. It is, in fact, an equatorial with the axis pointing to the zenith, instead of to the pole ; and if we hold the figure, p. 81., so that $FE$ may be vertical, we shall have an alt-azimuth. The circle $CD$ will measure the altitude, and the circle $GH$ the azimuth of any body towards which the telescope is pointed. An alt-azimuth has been in use at Greenwich since 1847, with the special object of observing the moon off the meridian. CRAP. VI.] THE RING-MICROMETER. 85 A detailed description will be found in the Greenwich Observations for that year. By clamping the horizontal circle, the alt-azimuth telescope can only describe a vertical circle. This vertical circle may coincide with the meridian, in which case it becomes a transit-instrument ; or, it may coincide with the prime vertical, and, in this position, observations of such peculiar value can be made by a special instrument called a Transit in the Prime Vertical. In 1836, a year, born in 1807 in the Pulkowa Observatory. For a description of it we shall refer to Chauvenet's Astronomy. The Ring-Micrometer. 110. All the instruments so far described require to be mounted in a special manner for the purpose of observing; but very good observations of differences of declination and right ascension can be made in a very simple manner with an ordinary telescope fitted with the contrivance known as a ring-micrometer. This consists merely of a flat ring of metal with very fine edges, which is made perfectly circular and fixed at the focus of the telescope. When the difference of declination of two stars is so small that both may pass through the field of the telescope in a fixed time, then the telescope must be so pointed that these passages may take place, and note the times when one of the stars passes behind the ring at $A$ and $C$, and reappears at $B$ and $D$. The mean of these times will be the instant when it passes the middle point of the chord; that is, it will be the instant when the star is in the declination circle which passes through the centre of the ring, for the path of a star will obviously cut every declination circle at right angles. A diagram showing a ring-micrometer attached to a telescope. The ring is marked with lines indicating different declinations. A chord is drawn across the ring, and points $A$, $B$, $C$, and $D$ are marked on it. 86 THE OBSERVATORY. [CHAP. VI. The same being done for the other star, care being taken not to disturb the telescope, the difference of these two means will be the difference of right ascension of the two stars. To determine the difference of declination, we require the angular diameter of the ring. This must be found previously by observing the number of seconds that an equatorial star takes to cross the diameter : let it be $n$, then $15n$ is the value of the diameter in seconds of arc. Now, if $A$ be the angle required by the stars to describe the chord $BC$ let $b$, and if $B$ be the known declination of one of the stars, the angular lengths of the chords will be approximately (Art. 68) $$15 \cos B \text{ and } 15\cos A.$$ The chords and diameter being known, we can easily obtain the angular distance of each chord from the centre ; and their difference or sum, according as they pass on the same or opposite sides of the centre, will be the difference of declination of the two stars. The Sextant. 111. The sextant does not properly belong to a fixed observatory ; but its simplicity, its small size, its accuracy, and, still more especially, its requiring no fixed support, render it of the highest value to the navigator. For a full description of this most useful instrument and its adjustments, we shall refer to Charpentier's "Astronomy," and to Harbord's "Glossary of Navigation." The sextant determines the angular distance between two visible points (as two stars), or the angular altitude of a body above the visible horizon. The principle of its construction depends on the known consequence of the law of reflection that, "when a ray is reflected in one plane at each of two plane mirrors its deviation is double the angle between the mirrors." One mirror is attached to the moveable radius CHAP. VL.] DOUBLE IMAGE MICROMETER. 87 of a circular sector and another to a fixed radius, both mirrors being perpendicular to the plane of the sector. Then, if the moveable arm be turned until the image of a star after two reflections be made to coincide with that of a star seen directly (one-half of the fixed mirror being left unvisited for the purpose), the angle between the mirrors, and hence the angle between the two stars, can be indicated by the position of the moveable arm among the graduations of the arc. Double Image Micrometers. 112. As the accurate measurement of small angular magnitudes is of the highest importance, we shall describe two other instruments for the purpose, based on principles entirely different from those of the spider-line micrometer described in Art. 72. 113. Dollond's Double Image Micrometer. When a telescope is turned towards a distant object, a circular disc for instance, an image is formed at the focus. Each element Fig. 1 Fig. 2 of the surface of the object-glass gives a complete image of the object, the brightness only increasing with the number of elements employed. If then the object-glass be cut in two, and the two parts kept in their proper places (fig. 1), only one image of the disc will be seen. But if one-half be made to slide along the other, as in fig. 2, two perfect images of the disc will appear, more or less overlapping or separated, according to the distance through which the half-lens has 88 THE OBSERVATORY. [CHAP. VI. moved. This sliding motion of the half-lens is given by a screw with a graduated head. Suppose the two images of the disc to be separated till they are just in contact (as $A$ and $B$), the one image will have moved from the other by a distance equal to the diameter of the disc; and if a new contact be established by passing the moveable image to the other side of the fixed one, the whole change, as recorded by the screw-head, will correspond to twice the diameter. Therefore, supposing the values of the screw-head gradations to have been established, by measuring discs of known sizes, placed at known distances, we shall be able to measure the angular breadth of other bodies, such as the sun and planets. The angular distance between two stars $a$, $b$, can be measured in the same way, but the object-glass must first be turned till the line of separation of its two halves is in the direction of the line joining the two stars. Two pairs of stars $a_{1}b_{1}$, $a_{2}b_{2}$ will become visible by turning the screw, and coincidences will be established between $a_{1}$ and $b_{2}$, and again between $a_{2}$ and $b_{1}$, the difference between the two readings will be twice the angular distance between the two stars. 114. Rochon's Double Image Micrometer. It is well known that when a pencil of common light passes through certain crystalline substances it undergoes a bifurcation, and two pencils are produced. In uniaxial crystals, such as Iceland spar, one of the pencils is refracted as it would be through ordinary transparent substances, and is called the ordinary pencil; the other, called the extraordinary pencil, undergoes a refraction which varies with its position relatively to a certain line called the axis of the crystal. In one case the A diagram showing two circles labeled A and B, with a third circle C in between them. CHAP. VL.] DOUBLE IMAGE MICROMETER. 89 penciledo not separate: this is, when incident perpendicularly on a surface which is itself perpendicular to the axis of the crystal. Now, if we take two triangular prisms of Iceland spar $ABCG$ and $BCDE$, which, by juxtaposition, form a rectangular parallelopiped $AE$, and if they be cut through a plane so that the axes of the two crystals may be parallel to $AB$ and $DE$ respectively, then the light, incident on the face $ACFG$, will be parallel to the axis of one crystal and perpendicular to that of the other. No separation, therefore, will take place in the first crystal; but, on entering the second, the two pencils will proceed in different directions inclined to one another at an angle depending on the angle of the prisms. 115. Let $A$ (fig. a) be the object-glass of a telescope, $PQ$ a distant object subtending a small angle, $pq$ its image formed at the principal focus. If between the object-glass and the image we introduce the system of crystals above described, with the first surface perpendicular to the axis, the ordinary pencils will proceed without sensible interruption. Rig. a Rig. b Rig. c 90 THE OBSERVATORY. [CHAP. VI. tion, and, provided the opposite faces be parallel, an image $p_1q_1$ (fig. b) will be formed at the focus as before; but, on the pencils entering the second prism, the bifurcation will take place, and a second image $p_2q_2$ formed by the extraordinary rays, will be obtained. The distance between the two images will increase or diminish as the prisms move from or towards the focus; and therefore by sliding the framework, which carries the crystals backwards or forwards, we can bring the two images to be exactly in contact (fig. c). It is clear then that there will be a connection between the position of the prisms, when the images are just in contact, and the magnitude of the image $pq$, and therefore also of the angle $PAQ$ which the distant object subtends. If then we observe discs of known sizes, at known distances, we may graduate the telescopic tube in such a manner that the position of the prisms may at once give the angle subtended.* * M. Arago has suggested a modification of Reichen's microscope to obviate some slight inconveniences which attended its use. (Arago, *de l'Optique*, vol. ii. p. 77.) ( 91 ) CHAPTER VII. THE SUN. The Ecliptic. 116. The observer may now be supposed to possess instruments which will enable him to make observations with any required degree of accuracy, and we shall proceed to examine the movements of the sun, and of those other heavenly bodies which, unlike the stars, have, or seem to have, a motion of their own, independent of the diurnal motion. This diurnal motion, common to all the bodies, is, as we have seen (Chap. II.), only apparent, and belongs to the earth alone. If we take a celestial globe (Art. 16) on which the stars are all marked in their proper relative places, as also the poles, the equator, and some of the parallels and declination circles, the paths of the sun and of the planets will be conveniently traced by marking their positions on this globe each day. To the ancients the determination of the exact position of the sun relatively to the stars, was a problem of considerable difficulty, because the strong light of the sun hindered the stars from being seen at the same time, and the stars became visible only after the sun has disappeared below the horizon. Hipparchus employed the moon as a link to connect the sun with the stars—While the sun was above the horizon he determined its position relatively to the moon, and as soon after sunset as the stars became visible, he, in the 02 THE SUN. [CHAP. VII. same manner, connected the moon with them; then, making allowance for the change of position of the moon during the interval, he was able to deduce the sun's place among the stars. Tycho Brahe substituted the planet Venus for the moon with great advantage, not only an account of its slower and more uniform motion, but especially because the greater distance of the planet considerably lessened the chance of error. With the transit, the mural, and the clock, we avoid all these difficulties, and obtain results not only with greater facility but also infinitely more accurate. 117. Very rough observations will soon shew that the sun's daily path is constantly changing. This will be seen by noting the points where he rises and sets, the time he is above the horizon, and the greatest height he attains. These vary day by day, and hence the sun's declination varies also; for otherwise, like the stars, he would rise and set always in the same points, and attain the same meridian altitude every day. That the sun has a progressive motion from west to east among the stars may be inferred from the following facts:—Those stars which are seen near the western horizon shortly after sunset on any evening will, each successive day, remain a less time above the horizon; and few will disappear altogether in the strong sunlight; other stars more to the eastward having taken their places. On the other hand, a star in the east, which rises a little before the sun, so as just to be visible, will, each succeeding morning, be seen for a longer time, and attain a greater altitude above the horizon, before the sun's rays overpower it and render it invisible. We shall see further on that this motion of the sun, like the diurnal motion of the heavens, is only apparent, and is also due to an actual motion of the earth—a motion A diagram showing a celestial sphere with various lines indicating positions of stars at different times. CHAP. VII.] THE SUN. 93 of translation—which carries the earth round the sun, inde- pendently of the earth's rotation on its own axis. Sun's Semi-diameter. 118. When we speak of the path of the sun, we mean the path of the centre of his disc, which is a definite point, whereas the disc itself is a circle subtending an angle of about half a degree. As there is no mark by which the centre can be recognised, we are obliged to observe some point in the perimeter, and allow for the difference between that and the centre. We, therefore, here require the sun's semi-diameter, that is, the angle subtended at the eye of the observer by two lines, one directed to the centre of the sun, the other to a point in the edge. This may be determined by observations with some one of the instru- ments for measuring small angles, the spider-line micro- meter or the double image micrometer, as described in Arts. 72 and 115. The magnitude of the semi-diameter is found to be the same in all directions round the centre,* which shows that the disc is circular; but if the observations be made at different times during the year, the values obtained will be different; a slow decrease taking place from the 31st of December to the 1st of July, and a slow increase during the second half of the year. The maximum value (on the 31st of December) is 16°17'5". The minimum value (on the 1st July) ... 15°45'5". The changes are continuous throughout the year, but at corresponding dates in different years the same values recur. These values have been observed with the greatest care, and tabulated so as to be available for future use. (Naut. Astr., p. ii. of each month). * Provided the sun have sufficient altitude to free it from the effects of the large refractions to which it is subject when near the horizon. A diagram showing a circle representing the Sun's disc with a line drawn from its center to a point on its edge, indicating an angle subtended by this line at an observer's eye. 94 THE SUN. [CHAP. VIL. To find the Sun's Declination by Observation. 119. At the moment when the sun's centre is crossing the meridian, let the mural telescope be turned to 1 so as to make either the lower or the upper limb run along the fixed horizontal wire. The corresponding reading of the circle, indicated or diminished by the semi-diameter, is the reading of the sun's centre. The difference between this and the polar-point (Art. 100) gives the polar distance of the sun, the complement of which is the declination. Instead of the polar-point, we may use the zenith-point when the latitude of the place is known. The difference between the reading of the sun's centre and the zenith-point gives the zenith-distance, and the difference between the latitude and zenith-distance when the zenith and pole are on the same side of the sun, or their sum when they are on opposite sides, will be the declination. To find the Difference of Right Ascension between the Sun and any fixed Star. 120. On reference to Arts. 16 and 17, it will be seen that the difference of right ascension of any two bodies is measured by the interval in time between their transits across the meridian, as given by the sidereal clock. Hence, if, by means of the transit and sidereal clock, both the times be observed when the first and second limbs of the sun cross the meridian, the mean of the two times will give the instant when his centre is on the meridian ; and if this be compared with the time when some chosen star is crossing, the interval, reduced to degrees, will be the difference of their right ascensions, that is, the angle between their declination circles. If the clock gain or loss uniformly, we must multiply the interval so found by $\frac{24}{24+x}$ where $24+x$ is the number of hours marked by the clock between two successive transits of the same star. CHAP. VIL.] THE SUN. 95 Determination of the Sun's Annual Path. Ecliptic. First Point of Aries. 121. Now, on the globe which contains the representation of the stary heavens, let us mark, day by day, the position of the sun at noon by means of his declination, and of the angle his declination circle makes with that of a known star, as determined by the preceding observations. Thus, supposing $\sigma$ to be the star selected, $P$ the pole, $EQ$ the equator, make an angle $\alpha PA$ equal to the first difference of right ascension and mark off $PA$ equal to the corresponding arc distance of the star. $A$ will represent the first position of the sun. Let $B$, $C$, $D$, &c., obtained in the same manner, be the successive positions on the second, third, fourth, &c., days of observation. It will be found that all these points will arrange themselves on a great circle cutting the equator in two opposite points $T$ and $\alpha$, and inclined to it at an angle ($\omega$) of about 28° 27' 30". 122. This great circle, the plane of which contains the sun's yearly path, is called the ecliptic, and the angle ($\omega$) it makes with the equator is the obliquity of the ecliptic. Its intersections with the equator are called the equinoctial points, one ($T$) the first point of aries, the other ($\alpha$) the first point of libra. The sun will be found in the first of these points about the 21st of March, and in the other about the 23rd of September, his declination being then 0° and his polar distance 90°. The two points $M$, $N$ of the ecliptic, equidistant from the 96 THE SUN. [CHAP. VII.] equinoctial points, are called solstitial points; and if $K$ be the pole of the ecliptic, the great circle through $K$ and $P$ will pass through the solstitial points, and be perpendicular both to the equator and to the ecliptic. It is called the solstitial colure. When the sun reaches the solstitial point, he has his greatest declination 23° 27' north or south. This occurs about the 22nd of June and the 22nd of December. Therefore, from the 21st of March, when the declination is 0°, to the 22nd of June, the sun has an increasing north declination ; from the 22nd of June to the 23rd of September, the north declination decreases to zero ; from the 23rd of September to the 22nd of December the declination is south and increasing, and it then decreases until the 21st of March. Variations in the Length of the Day. 123. The combination of this annual motion of the sun, with the diurnal motion of the earth, will explain all the phenomena of the seasons, and of variations in the length of the solar day at particular places. This we shall proceed to show. Let fig. $a$ be the globe on which are delineated the stars, the pole, the equator, and the ecliptic; and let fig. $b$ represent the celestial sphere of an observer. We shall now endeavour to use two distinct figures; on fig. $a$ we shall trace the annual motion, and on fig. $b$ the apparent diurnal paths due to the earth's rotation ; fig. $a$ will be common to all observers, but fig. $b$ will vary with the observer's latitude. If the sun were fixed among the stars, he would, by the effect of the earth's rotation, describe a parallel—higher or lower above the horizon, and visible for a greater or less time, according to his declination and to the position of the observer—but in his progress along the ecliptic he changes his declination continually; and not only is the CHAP. VII.] THE SUN. 97 apparent path of each day different from that of the day before, but it ceases to be circular ; and all the daily paths together, if we include those portions which are hidden below the horizon, form one continuous spiral curve. The sun describes 360° in 365 days—his advance along the ecliptic is therefore about 1° per day; and the daily change in his declination is still slower, since it only varies in the same time from 23½ south to 23½ north, and back again. In the following explanation we will be sufficient to assume that during each day, the sun's declination is constant, retaining the value which it has at sun-rise; that is, we shall consider his daily paths to consist of a series of parallels instead of a spiral. Fig. a Fig. b 124. The first case we shall consider will be that of an observer in the northern hemisphere, whose latitude is somewhere between 23° 27' and its complement 66° 33'. Fig. $b$ is adapted to this supposition : $Z$ is the zenith, $P$ the pole, $EQ$ the equator, and $HR$ the horizon. The latitude is $ZQ$ or, its equal, $PR$. Let us begin on the 21st of March, when the sun is at $T$ (fig. $a$) and his declination is $0°$. He will rise at $E$, the east point of the horizon (fig. $b$), his hour angle being then $EPZ$, or $90°$; during the day he will describe the equator $EQW$, and will set at $W$, the west point, where u 98 **THE SUN.** [CHAP. VII.] his hour angle will again be $90^\circ$. The day and night will then be of equal duration, whence the term equinox applied to this epoch of the year. As marking the commencement of spring, it is further distinguished as the vernal equinox. On that day the sun attains an altitude $HQ$ equal to the co-latitude. A few days later, the sun will have advanced to $S$ (fig. $a$), having a north declination $ST$ or $S$; and if the parallel $FP'$ be drawn (fig. $b$) at a distance $6$ from the equator on the north side, this will be the diurnal path of the sun on that day. At sun-rise the sun will be at $F$ to the north of east, his hour angle will be $FPZ$, which = $EPZ$, that is, $= 90^\circ$; therefore the day will be longer than the night. His altitude at noon will also have increased, being $HP$, which = co-lat + decl. The length of the day and the meridian altitude will thus go on increasing until the 22nd of June, when the sun reaches the solstitial point $M$ (fig. $a$). Here his declination = $ME = MTE = \omega$. If we make $GQ'$ equal to $\omega$ (fig. $b$), the parallel $GG'$ will represent the sun's path on that day, the meridian alt. = co-lat + $\omega$, the hour angle at sun-rise is $GPQ$, and the azimuth $HG$. These are their greatest values ; for, after this, as the sun continues his course along the ecliptic, fig. $a$ shows that his declination begins to decrease, and therefore his daily path (fig. $b$) will approach the equator. For several days before and after the maximum, the change of declination will be very slight, and during this time there will be no perceptible change in the diurnal path—to the eye, he will rise at the same point of the horizon, reach the same meridian altitude, and set at the same point. Hence the term solstice applied to this period of the year, and solstitial point to the point $M$ of the ecliptic. This is the summer solstice. During the next three months the sun will pursue his CHAP. VII.] THE SUN. 99 course from $M$ to $\alpha$ (fig. a), his distance from the equator gradually diminishing. His diurnal paths will therefore be a repetition, but in an inverse order, of his paths during the preceding three months. We thus reach the autumnal equinox, the sun being again in the equator, and the night and day of the same length. After this the sun will pass to the south side of the equator; his daily paths, such as $CC$, $BB'$ (fig. b) will be on the further side of the equator from the elevated pole, his points of rising and setting will move more and more recede from $E$ and $W$ towards $H$, the hour angle at sun-rise will be less than a right angle and decreasing, and the altitude at noon each day less than the preceding, until the sun reaches his furthest distance from the equator on the south side at the solstice $N$ (fig. c). If $BB'$ be the corresponding diurnal circle, the mean altitude $$HB = HQ - QB' = \alpha - \beta.$$ From the winter solstice, which occurs about the 22nd of December, this sun returns to its starting point $T$, the south declination gradually increasing, and the sequence of diurnal circles from the 22nd of December to the 21st of March being, in an inverse order, the same as from the 23rd of September to the 22nd of December. 125. Suppose we travel towards the north pole until our latitude exceeds $60^\circ 32'$. Let fig. $c$ be the celestial sphere adapted to our new position, fig. $a$ remaining the same as before. The pole will be here at a distance $ZP$ from the zenith less than $67^\circ 1'$ or $\alpha$. For the latitude $\phi$ is the complement of $ZP$. Therefore $ZP$ (which = $HQ$ or $RU$) > $\alpha$. At the vernal equinox the sun, being in the equator, will describe $EQ$, the hour angle $EPZ$ at sunrise being, as before, a right angle ; and, consequently, the night and the day of equal length. H 2 100 THE SUM. [CHAP. VII. The increase in the length of the day will be more rapid than at the previous station, and when the sun has reached a point $S'$ (fig. a) in T.M., where his declination is $90^\circ - \phi$ or RU (fig. c), he will not set at all, but at midnight will just graze the horizon at $R$, and then re-appear, rising continually above the horizon during the whole time it takes to move from $S'$ to the corresponding point $S''$ on the other side of the solstice M. At the solstice itself the path, during twenty-four hours, is the small circle GG', the greatest altitude being HG', or $\omega + \cos\alpha$, and the least RG', or $\omega - \cos\alpha$. After passing $S''$ the sun will again begin to set, and at $\alpha$ the day and night will be of equal length. The days will continue to decrease until the sun's south declination at $S''$ is $90^\circ - \phi$. Then his diurnal path will only just graze the horizon at H, and a period of continuous night will set in from $S''$ to $S'''$, corresponding to that of continuous day about the summer solstice. The subsequent re-appearance of the sun above the horizon, and the gradual increase of daylight, will be easily understood without further explanation. 126. The points $S'$ and $S''$, which have a declination $90^\circ - \phi$, will recede from M as $\phi$ increases, and therefore the period of perpetual day will increase with the latitude, attaining its maximum when the observer is at the pole; for there the equator and the horizon coincide, the sun remains visible while on the north side of the equator from the vernal to the antumnal equinox, and this day of six months is followed by six months of perpetual night. A diagram showing a circle representing a celestial sphere with various lines indicating different positions of a celestial body over a period of time. The diagram includes labels such as "Fig. e", "T.M.", "RU", "M", "S'", "S''", "S'''", "H", "G", "G'", "RG'", "HG'", "ω + cosα", "ω - cosα", and "α". The diagram illustrates how a celestial body moves from one position to another over a 24-hour period. CHAP. VII.] THE SUN. 101 127. If we travel towards the equator until our latitude $\phi$ is less than 23° 27'1", the phenomena will again be different. Fig. $d$, where $ZQ$ or $PR$ is this lower latitude, will be adapted to this case ; and we must, as before, take fig. $a$ in connection with it. At the equinox the sun will describe the equator $EQ$, and the day and night be twelve hours each as before. Also as the sun's declination increases, his meridian altitude will increase until he has reached that point $S$ of his orbit (fig. $a$) where his declination is equal to $\phi$ the latitude. On that day his diurnal path will pass through $Z$ the zenith, and the sun at noon will be vertical. On the next and subsequent days, his declination still increasing, his path will cross the meridian between the zenith and the pole ; the hour angle at sun-rise, and therefore the length of the day, will still increase, but his meridian altitude will decrease. The solstice $M$ being passed, the days will begin to decrease, and once again the sun will culminate in the zenith ; then, until he has his greatest south declination, his meridian altitude will decrease, and the days get shorter. The return to $T$ will be attended with corresponding variations, which need not be traced. 128. At the equator itself the pole will be in the horizon, and the hour angle at sun-rise a right angle ; therefore the days will always be twelve hours long ; and as the celestial equator passes through the zenith, the meridian altitude will always be the complement of the declination. A diagram showing a celestial sphere with a line representing the Earth's axis passing through its center. The Earth is shown as a tilted sphere with a line indicating its axis of rotation. The diagram includes labels for various points on the sphere such as ZQ, PR, S, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, T, U, V, W, X, Y, Z. 102 THE SUN. [CHAP. VIL. 129. It will be obvious, by an examination of figs. b, c, d, that the variations in the duration of daylight throughout the year will be greater the higher the latitude ; and it can also be easily inferred from these figures, that the longest summer day and the longest winter night, at any place, are of equal lengths (see Chap. xiv.). In the southern hemisphere the phenomena of night and day will be the exact counterpart of those in the northern. 130. The foregoing articles explain the gradual change in the diurnal phenomena observed at particular places on the earth's surface—the general explanation for the whole earth will be best understood by an examination of the accompanying figure, where A, B, C, D, E represent successive positions of the sun in his yearly path. A diagram showing the Earth's rotation and the Sun's apparent motion across the sky. The Earth is depicted as a sphere with a line indicating its axis of rotation. The Sun is shown as a circle moving around the Earth's center. The diagram includes labels for different times of day and seasons. The arrow represents the direction of the sun's motion, and the earth at the centre turns in the same direction about its axis. Once a day, each terrestrial meridian comes in its turn opposite to the sun, and it is noon at all places in that meridian. To one-half of the earth there is daylight, to the other, night. If the centres of the sun and earth be joined by a straight line, the point where this line meets the earth will at that instant have the sun in its zenith, and will obviously be the centre of the illuminated part of the earth; and the line of demarcation between light CHAP. VIL.] THE SUM. 103 and darkness will be approximately* the great circle, of which this central point is the pole. It will be easily seen, that at the time of the equinoxes, when the sun is at $A$ or $D$, the boundary passes through the two poles ; and that at the solstices when the sun is at $C$ or $E$, one pole with $23\frac{1}{2}$ around it in all directions will have continuous daylight, and the other continuous night. Annual Motion of the Sun replaced by a Motion of the Earth. 131. We shall, in the next place, proceed to show that this daily displacement of the sun--this shifting of his position among the fixed stars, which constitutes what we have called the summer--can only be explained by a corresponding motion of the earth round the sun at rest, provided that we assume the distance of the stars from the earth to be infinitely great in comparison with that of the sun. Let $E$ and $S$ represent the earth and sun. An observer A diagram showing three positions of a star relative to the Earth and Sun. The Earth is shown rotating around its axis, while the Sun appears to move across the sky. The observer sees different stars depending on their position relative to these celestial bodies. at $E$ sees the sun $S$ in the direction of some star $a$. If the sun move to $S'$, the observer will then see him in the direction of some other star $b$. But the very same star $b$ would also be the sun's direction, if we suppose the sun to have remained stationary at $S$, and the earth to have moved to $E'$, found by making $SE'$, in the line $bS$ pro- * We say approximately, because, on account of the sun being much larger than the earth, the cone which envelops the two bodies will touch the earth in a small circle beyond this great circle. The effect of refraction will still further diminish the dark portion. 104 THE SUN. [CHAP. VII. dood, equal to $S.E$. Moreover, if the stars are so distant that the two lines $ES$, $E'S$, which lead to the same star, may be considered parallel as well as equal, the path $EE'$ round $S$ will be exactly equal and similar to the path $SS'$. Hence, whichever supposition be the true one, it is obvious that the apparent motion may nevertheless be explained as a consequence of the other. If we adopt the hypothesis of the earth's motion—and we shall see presently the arguments in its favour—we must suppose that during its annual motion about the sun its axis remains parallel to itself, in order that the polar distances of the stars may remain constant. The figure below will exemplify this motion of the earth, and will account for the observed changes of right ascension and declination of the sun. Let $A.D.C...$ be the path which the earth would have
21 Mar. 22 Dec.
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to describe about $S$ the sun, in a plane inclined at $23^{\prime}27^{\prime \prime}$ to the earth's equator, $PP'$ the earth's axis, $P$ the north pole.
Let $A$ be the position of the earth on the $21st$ of March when the plane of the equator $QQ'$ passes through the sun. The line joining the centres of the two bodies at any time will, by its intersection with the surface of the earth, CHAP. VII.] THE SUN. 105 determine that place which has the sun in its zenith at that moment. This will be some place in the equator when the earth is at $A$, and it would happen in turn to every place on the equator as the earth revolved about its axis, if we neglected the motion of translation of the earth during the twenty-four hours. Neglecting this motion would be the exact equivalent of the supposition we made, for simplicity of explanation, in Art. 123, where we assumed the sun's declination to remain constant during twenty-four hours. Some few days or weeks afterwards suppose the earth to have arrived at $B$. The plane of its equator, which is always perpendicular to $PP'$, will now be below the sun, the line joining the centres will make an acute angle with the axis $BP$, and the sun will be vertical to a place north of the equator, whose latitude is equal to the declination of the sun being between $27\frac{1}{2}$ and $28\frac{1}{2}$. If $SC$ be drawn at right angles to $SA$, when the earth reaches $C$, the line $CS$ will make a greater angle with the plane of the terrestrial equator than in any preceding or succeeding position, and the sun's north declination will have attained its maximum $28\frac{1}{2}$. This will be at the summer solstice, about the 22nd of June. From $C$ to $D$ the declination will decrease, and at $D$ be zero, the plane of the equator again passing through $S$. This will be the autumnal equinox. $E$ will be the position of the earth at the winter solstice when the sun attains his greatest south declination. 132. The series of changes in the sun's declination is thus seen to be a consequence of the earth's motion, and it will also be easily seen that the same motion produces the gradual increase of night according to: At $A$ the earth sees the sun in the direction $AS$, and, in all subsequent positions, the plane of the earth's equator remaining parallel to itself, the lines of intersection with 106 THE SUN. [CHAP. VII.] the plane of the ecliptic, viz. $EY$, $CY$, &c., will all have parallel directions, and the sun will seem to have moved through the angles $YEB$, $YCS$, &c., which exactly correspond to the angles which the arcs $AB$, $AC$, &c., in the supposed motion, fig., p. 102, subtend at the centre of the earth. Arguments in favour of the Earth's Motion. 133. The following are the chief arguments in favour of the supposition that it is to the earth, and not to the sun, that the motion of translation belongs. (1) It furnishes a simple explanation of the stationary points and of the retrograde motion of the planets (see Chap. xxi). These may also be explained on the other hypothesis ; but the ancient astronomers, who assumed the earth to be stationary at the centre, were driven to a complex system of epicycles to account for the phenomena. (2) It is in accordance with observation that the distance of the sun and the length of the year have just that relation to one another which they should have if we suppose the earth to be a planet revolving round the sun and subject to the same laws as the other planets. (3) The strongest proof is furnished by the "aberration of light." This admits of easy explanation if we suppose the earth to have a motion of translation, but no explanation has been given on any other hypothesis (see Chap. xvii). 134. Although the annual motion really belongs to the earth, and not to the sun, *we shall sometimes find it con- * When we say not to the sun, we do not mean that the sun is absolutely fixed, for we shall have occasion to conclude that the whole solar system—sun with its planets—is in motion. But what we are considering in the text is the internal relative motion of the parts of the system.
106 THE SUN. [CHAP. VII.]
the plane of the ecliptic, viz. $EY$, $CY$, &c., will all have parallel directions, and the sun will seem to have moved through the angles $YEB$, $YCS$, &c., which exactly correspond to the angles which the arcs $AB$, $AC$, &c., in the supposed motion, fig., p. 102, subtend at the centre of the earth.
Arguments in favour of the Earth's Motion.
133. The following are the chief arguments in favour of the supposition that it is to the earth, and not to the sun, that the motion of translation belongs.
(1) It furnishes a simple explanation of the stationary points and of the retrograde motion of the planets (see Chap. xxi). These may also be explained on the other hypothesis ; but the ancient astronomers, who assumed the earth to be stationary at the centre, were driven to a complex system of epicycles to account for the phenomena.
(2) It is in accordance with observation that the distance of the sun and the length of the year have just that relation to one another which they should have if we suppose the earth to be a planet revolving round the sun and subject to the same laws as the other planets.
(3) The strongest proof is furnished by the "aberration of light." This admits of easy explanation if we suppose the earth to have a motion of translation, but no explanation has been given on any other hypothesis (see Chap. xvii).
134. Although the annual motion really belongs to the earth, and not to the sun, *we shall sometimes find it con-
106 THE SUN. [CHAP. VII.]
the plane of the ecliptic, viz. $EY$, $CY$, &c., will all have parallel directions, and the sun will seem to have moved through the angles $YEB$, $YCS$, &c., which exactly correspond to the angles which the arcs $AB$, $AC$, &c., in the supposed motion, fig., p. 102, subtend at the centre of the earth.
Arguments in favour of the Earth's Motion.
133. The following are the chief arguments in favour of the supposition that it is to the earth, and not to the sun, that the motion of translation belongs.
(1) It furnishes a simple explanation of the stationary points and of the retrograde motion of the planets (see Chap. xxi). These may also be explained on the other hypothesis ; but the ancient astronomers, who assumed the earth to be stationary at the centre, were driven to a complex system of epicycles to account for the phenomena.
(2) It is in accordance with observation that the distance of the sun and the length of the year have just that relation to one another which they should have if we suppose the earth to be a planet revolving round the sun and subject to the same laws as the other planets.
(3) The strongest proof is furnished by the "aberration of light." This admits of easy explanation if we suppose the earth to have a motion of translation, but no explanation has been given on any other hypothesis (see Chap. xvii).
134. Although the annual motion really belongs to the earth, and not to the sun, *we shall sometimes find it con-
CHAP VIL.] THE SUN. 107 venient to retain the latter supposition when it simplifies, without vitiating the validity of, the reasoning. This we shall do in Chap. xii., when explaining the "equation of time," where we shall attribute the annual motion to the sun and the diurnal motion to the earth. A page from a book with text discussing the Sun's motion. ( 108 ) CHAPTER VIII. THE SEASONS. 135. The two equinoctial and the two solstitial points divide the ecliptic into four equal parts, but, owing to inequalities in their velocity, they are not described in equal times. See Chap. X. These unequal periods are called Seasons, and are distinguished as Spring, Summer, Autumn, and Winter. Spring commences when the sun is at T (fig., p. 95), and lasts till he reaches the solstitial point $M$; the summer, while he goes from $M$ to $\phi$, autumn from $\phi$ to $N$, and winter from $N$ to $\gamma$. Their lengths are at present—
Spring 92 days 21 hours
Summer 93 14
Autumn 89 17$\frac{1}{2}$
Winter 89 1
136. The discussion in Art. 124 shews that for an observer in the northern hemisphere, between the parallels of $23\frac{1}{2}^\circ$ and $66\frac{1}{2}^\circ$, the days in spring and summer are longer than the nights, and the reverse in autumn and winter; but spring differs from summer, and autumn from winter, in the mode in which the duration of daylight changes-increasing in the first and last, decreasing in the other two. The four seasons are still further characterised by their temperatures. The amount of heat received from the sun depends on the time he remains above the horizon, and, A diagram showing the positions of the equinoxes and solstices on the ecliptic. CHAP. VIII.] THE SEASONS. 109 in a still greater degree, on the altitude he attains during the day. Hence spring and summer must be warmer than autumn and winter. But the heat of summer must be greater than that of spring, although the same altitudes and the same lengths of days occur ; because, for some time after the solstice, the quantity of heat received each day, though decreasing, will be in excess of the quantity lost or used up, and the temperature will go on increasing ; just as the greatest heat of any day is not until about two hours after noon. But only then is an equal quantity produced to make up the great loss. For a similar reason the winter quarter will be colder than autumn, just as the coldest hour of the night is about two or three hours after midnight.* All this is in exact agreement *with what is observed in England, and in all places between the parallels of 23° 27½' and 66° 32½' of north latitude. *187. If we go to higher latitudes (Art. 125), the rays of the sun impinge more obliquely upon the earth, their heating power is weakened, and the mean temperature of the year is lowered. During the summer months, the continual presence of the sun for many days above the horizon compensates to some extent for this loss of heating power, so that in some places there is a warm, though brief, summer. On the other hand, the long winter nights reduce the temperature so as to produce that intense cold which covers the seas with ice, and renders the land unfit for permanent habitation beyond the latitude of 70°. \textasciitilde{}138. Between the equator and the parallel of 23° 27½' we have a belt or zone where the conditions are very different. On the equator itself the days are always 12 hours long, but the sun at noon is never more than 23° 27½' from The differences therefore to be noticed in the sun's distance from the earth have but slight influence on the amount of heat we receive from him. * 110 THE SEASONS. [CHAP. VIII. zenith, and twice a year, that is at the equinoxes, is exactly vertical at noon. We have here the main conditions for a high temperature during the whole year, and the seasons, instead of being marked by contrasts of temperature, as in the regions distant from the equator, are marked by contrasts of humidity, and consist of two many reasons and two dry seasons. To places off the equator, but within 25° 27' from it, the same remarks will apply—twice a year, but at unequal intervals, the sun will be vertical, and his distance from the zenith will vary, even at midwinter, be less than 47°. The climate is characterized by extremely elevated temperature at all times. \textgreater 139. It is obvious that what has been here said of the northern hemisphere will be equally applicable to the southern hemisphere, except that the phenomena in southern latitudes, though precisely the same, occur at exactly opposite epochs of the year. When it is summer in the south it is winter in the north ; the beginning of autumn in one hemisphere is the beginning of spring in the other. Zones of the Earth. \textgreater 140. The part of the earth included between the parallels of 25° 27' south and north of the equator is called the Torrid Zone, as its bounding parallels are called Tropics: the Tropic of Capricorn on the north side and the Tropic of Capricorn on the south. The parallel of 66° 32' north latitude is called the Arctic Circle, and the corresponding one in south latitude the Antarctic Circle. The regions round the two poles, bounded by the arctic and antarctic circles respectively, are called the Frigid Zones. The two belts of the earth comprised between the limits of the torrid and the frigid zones are called the Temperate Zones. CHAP. VIII.] THE SEASONS. 111 The reasons for these names will be understood from the previous explanations, but it should be remembered that as we approach the limits of the temperate zone, the character of the climate differs less and less from that peculiar to the neighbouring torrid or frigid zone. Additional Remarks on the Temperature. § 141. The temperature has been spoken of as varying with the duration of the sun's presence above the horizon and with his meridian altitude ; but if these were the only causes of disturbance, we ought to find exactly the same temperatures at all places in the same latitude, which we know to be far from being the case. The sea is found to preserve a much more uniform temperature than the land, and therefore islands and places on the coasts, will have their extremes of heat and cold much nearer to each other than places in the interior of continents. The cold of winter and the heat of summer depend more on the greater or less distance from the ocean than on the latitude. Thus, London and Irkutsk (Siberia) are nearly in the same latitude, but while London has a mean range of only 24°3 Fahrenheit, Irkutsk has 61°.
Mean Temperature. Mean of Year.
Winter Summer Dif. Dif.
London 51°37'N 37°0 F 67°1 F 20°1 F 49°8 F
Irkutsk 54°3'N - 6°2' 60°8 61° 21°6
This great difference is almost wholly thrown on the winter temperature, for summer heat of London is only about 1° higher than at Irkutsk, whereas the winter cold of the latter place is 38° more severe than that of the former, and the mean temperature of the year is 18° lower. Probably the principal agent in the mitigation of the cold of our winter is to be found in the prevalent S.W. or Anti-Trade Winds which come to us charged with the 112 THE SEASONS. [CHAP. VIII. warmth and moisture they have taken up in the inter-tropical regions of the Atlantic. The condensation of the vapour in the form of rain or snow restores the heat which had been expended in raising the vapour in those distant seas. Ocean currents will also have a very powerful influence in modifying the temperature. To whatever cause it may be due, it appears certain that there is a constant interchange of water between the Tropics and the Arctic regions. The large number of observations made by Dr. Carpenter during the last few years have led him to adopt the doctrine of a general oceanic vertical circulation sustained by opposition of temperature alone.* He infers that there is a continual bottom outflow of Polar water, due to increase of density and pressure, along the deepest channels of communication with other Oceanic basins. And as a necessary consequence of the reduction of level which must result from this outflow in the Polar area, there will be a continual surface inflow, for its restoration ; and thus a movement will take place in the surface-water from the Tropics towards each Pole. The rotation of the earth from west to east must be considered in connection with this movement of the waters. The nearer the equator, the greater will be the eastward velocity ; and therefore, as the surface current will, therefore, as it moves towards the Poles, travel more slowly than the parts of the earth which it successively reaches, thereby acquiring an eastward direction ; while for a like reason, the cold Polar current will flow towards the equator and westward. A north-easterly drift of this character is found carrying the waters of the mid-Atlantic towards the British Islands, and its warming influence is felt even as far north as Spitzbergen and Nova-Zembla. A cold Arctic current on the * See an article by Dr. Carpenter in Good Words, Jan. 1873. A diagram showing a model of ocean currents. CHAP. VIII.] THE SEASONS. 113 other hand, with a westerly tendency, sweeps along the eastern coasts of Greenland and of North America as far south as 40°; so that while the harbour of Hammerfest (Norway), in 70° north latitude, is free from ice all the year round, the opposite coast of America is blocked up as far south as 50° north latitude during the whole winter, and great parts of early spring. The same remarks will apply to the west coast of America and the opposite shores of Asia, but in a much less marked degree, for the shape of the continent gives no outlet for the cold current from the Arctic Ocean through Behring's Straits, although the north-easterly warm current from Japan is felt along the west coast of British Columbia, and even beyond the Straits, further up on the north coast of America. The opposite east coast of Asia has, however, a higher temperature than the corresponding latitudes of the east coast of America. The northerly drift above spoken of does not begin near the equator. Between the parallels of 40° north and south of the equator, surface currents are found describing circuits of immense extent, with central expansions of quiescent water covered with marine vegetation, and undisturbed except by winds and tides. These equatorial currents are probably due to the prevailing action of the trade and anti-trade winds; but, whether they owe their origin to these combined with other forces, or force themselves to a conclusion as an enormous body of water. If we follow the equatorial current of the North-Atlantic, we find it travelling westward from the coast of Africa towards the north coast of South America, where, being driven by the conformation of the land into the Gulf of Mexico, it is turned northward, and the Gulf Stream, as it is now called, flows through the Florida channel and outside of Cuba, a warm current with a speed of from two to four miles an hour. Taking now a north-easterly course it gradually leaves the American shore, and, 4 114 THE SEASONS. [CHAP. VIII.] propelled perhaps by the anti-trade winds, a portion of it joins the N.E. drift spoken of before, and adds its warmth to the other causes which render our winter climate milder than that of the corresponding places in the same latitude. There is now reason to believe that the larger portion of the Gulf-stream, after reaching the meridian of the Azores, turns southward and completes the circuit by rejoining the current off the coast of Africa. In low latitudes, the immense expanse of water keeps the temperature much more nearly dependent on the latitudes alone. Trade Winds. 142. What has been said about the ocean currents will prepare for the explanation of the Trade Winds. The air in the torrid zone, being heated by the land and sea, expands and rises, forming an upward current which must be fed by the lower air of the higher latitudes, and a current therefore sets in towards the equator from the temperate zones. The warm air which has risen spreads over the other, and thus a constant interchange takes place, causing an upper current from the equator towards the poles; but this upper current being gradually cooled sinks again to the earth somewhat in high northern latitudes. There may therefore be, in the torrid zone, an under current of air setting towards the equator; but this air, having less Easterly velocity of rotation than the portions of the earth which it successively reaches, will lag behind, and the current which starts from the northern hemisphere as a northerly wind will, as it advances southwards, gradually pass into a north-easterly wind, and in some cases even become an east wind. This constant current, which blows nearly all the year round, is called the N.E. trade-wind; and a corresponding current, called the S.E. trade-wind, advancing from the CHAP. VIII.] THE SEASONS. 115 southern hemisphere towards the equator, veers gradually round, becoming more and more easterly. The upper current of air, which sets north and south from the equator, having the same easterly velocity as the equator, will continue where it again reaches the earth, the *anti-trade* wind, the prevailing northerly velocity of our latitudes, and a north-westerly wind in the southern hemisphere; but the position of the sun, mountain ranges, and accidents of country, will modify these general results.* 143. The trades do not, however, meet and combine in a common easterly direction near the equator. They are separated by a belt some 600 miles in breadth, characterised by heavy rains and calms. In the Atlantic, the N.E. trade-wind comes from about 25° to 8° of north latitude, and the S.E. trade-wind from about 2° to 2° of south latitude, but these limits vary with the season. At its southern limit the N.E. trade loses its easterly tendency, and the wind gets gradually more northerly; and so at the northern limit of the S.E. trade, the wind shifts into a more southerly instead of easterly direction. Basil Hall, who notices this fact (*Fragments of Voyages*, 2nd series, vol. I., p. 289), gives the following explanation: The friction with the surface of the water destroys the velocity of the trade-wind, so that its westward motion is diminished. The motion towards the equator would also diminish from the same cause, but the original pressure which produces this motion is still acting, and the particles progress towards the equator, and as the successive parallels of latitude do not now increase so rapidly as at first, the westerly motion will be less. This equatorial belt, where the winds from the north and the south meet and counteract one another, is not stationary, *The explanation of the Trade Winds, given above, was first proposed by Halley (Phil. Trans., 1686, p. 167).* 12 116 THE SEASONS. [CHAP. VIII, the interior limits of the trades shifting north or south as the sun's declination alters. In the Indian Ocean, the S.E. trade crosses the equator in the summer, and extends as a S.W. monsoon to the foot of the Himalayas. In the same way the southern limit of the N.E. trade, during the winter months of the northern hemisphere, not only crosses the equator, but crosses it and extends in the northern hemi- sphere the N.W. monsoon ("Devoe's " Law of Storms"). In the 6th vol. of the Smithsonian Contributions to Knowl- edge, Prof. Coffin has collected and discussed an immense number of observations on the winds of the northern hemi- sphere. From these it may be inferred that the proximity of large continents has considerable influence on the winds. Thus, in the North-Atlantic and in the North-Pacific the trade wind veers gradually round to the east, except close to the west coasts of Africa and of America, where the land draws as it were the wind to it, and gives it a more northerly direction. Along the coast of California, following the trend of the land, it even becomes north-westerly. A page from a book with text discussing winds and monsoons. ( 117 ) CHAPTER IX. FINDING THE LATITUDE BY OBSERVATION. 144. The determination of the latitude is a problem of primary importance in an observatory, and the greatest accuracy, both of calculation and of observation, must be brought to bear upon it. At sea it becomes a problem of daily necessity, but the same precision is neither requisite nor possible. The ship's place is constantly changing, and the delicate instruments and methods of observation in use in an observatory must be reduced by others adapted to the unstable position of the observer. A second of latitude corresponds to a distance of about thirty-four yards, and the latitude of a fixed observatory is determined to within a few tenths of a second; but this is after a series of observations extending probably over a number of years. At sea the latitude can seldom be relied on to within a quarter of a mile, and the nearest half-mile is generally considered sufficient. 145. The methods may be classed under three heads: meridian observations, ex-meridian observations, and circum-meridian observations. Of the following methods the first is the simplest, and in a fixed observatory the most accurate, but it is not adapted to general use at sea, on account of the twelve hours' interval between the observations, which will frequently require one of them to be made in the day-time, when only the powerful fixed telescope of an observatory can perceive the star. A page from a book with text discussing the determination of latitude by observation. 118 LATITUDE BY OBSERVATION. [CHAP. IX. The second method is that in common daily use on board ship. The third, fourth, sixth, and seventh methods are sea methods, especially useful when clouds or other circumstances interfere with observations by the sextant. The fifth method can only be practised in an observatory fitted with a transit in the prime vertical. Meridian Observations. 140. First Method. To find the latitude by observations of the two meridian altitudes of a circumpolar star. Let $Z$ be the zenith, $P$ the pole, HR the horizon, $a$ the altitude $AR$ of a star at upper transit, $\beta$ the altitude $BR$ of the same star at lower transit; then, $PR = \frac{1}{2}(AR + BR)$, or, $\quad lat. = \frac{1}{2}(\alpha + \beta)$. In this method no previous knowledge of the star's declination is required. 147. Second Method. To find the latitude by observation of the meridian altitude of the sun, a star, or other body, whose declination is known. Let $S$ be the sun or star in the meridian, $QE$ the equator, $\delta$ the known declination $SQ$, $a$ the observed meridian altitude $SH$, $x = 90^\circ - a$, the zenith distance $ZS$; then $ZQ = ZS + SQ$, or, $lat. = x + \delta$. CHAP. IX.] LATTITUDE BY OBSERVATION. 119 If the object crossed the meridian between $Q$ and $H$, the formula would be $lat. = x - b$; and if between $Z$ and $P$, $lat. = b - z$. These are all included in the single formula $lat. = x + b$, provided $x$ be reckoned + when the zenith is north of the object, and -- when south ; and similarly, north declination $\pm$, and south declination $-$. The latitude will be north or south, according as the result is positive or negative. If the altitude is known, when the object is crossing the meridian below the pole, that is, between $F$ and $R$, it will be easily seen that $lat. \times 180^\circ - z = b$, and that it must necessarily be of the same name as the declination. In the practice of this method at sea, it is impossible to ensure that the observation shall be made exactly in the meridian; but the change of altitude being then very slow, no practical difficulty arises. The altitude of the sun or other body is taken a quarter or half-an-hour before the meridian passage, and the observation repeated at short intervals until the maximum is passed, which maximum is then taken for the meridian altitude. The observations are of course made with a sextant, the image of the star or of the sun's limb being brought into coincidence with the visible horizon and the instrumental reading corrected for dip, refraction, &c. Ex-meridional Observations. §148. Third Method. To find the latitude by simultaneous observations of the altitudes of two known stars. Let $S$, $S'$ be the two stars, $z$, $z'$ their zenith distances, $\Delta$, $\Delta'$ their known polar distances, $\alpha$ the known difference $SP'S$ of their right ascensions. In the triangle $SP'S$, the two A diagram showing a celestial sphere with two stars labeled S and S', and a line SP'S connecting them. The angle SP'S is labeled as α. 120 LATITUDE BY OBSERVATION. [CHAP. IX.] sides $PS$, $PS'$ and the contained angle are known, whence the third side $SS'$ and the angle $PSS'$ may be calculated : $$\cos SS' = \cos \Delta \cos \Delta + \sin \Delta \sin \Delta \cos z - (i),$$ $$\sin PS' = \frac{\sin \Delta \sin x}{\sin SS'}...(ii).$$ The three sides of the triangle $ZSS'$ are now known, and the angle $ZSS$ may be computed : $$\sin \frac{1}{2}(ZSS') = \sqrt{\left(\frac{\sin x}{z} + z + SS'\right) \sin^2 x}...(iii),$$ then $PSZ = (PS - ZSS)$ will be known. Finally, in the triangle $ZPS$ we have two sides and the contained angle, and the third side $ZP$ co-latitude may be computed : $$\sin lat = \cos ZP = \cos \Delta \cos z + \sin \Delta \sin z \cdot cos PSZ...(iv).$$ This method is not so extensively used by seamen as it deserves to be. It is susceptible of great accuracy, because, during the morning and evening twilight, the horizon is generally well defined, and the altitudes of bright stars can be readily observed. Moreover, so long as the same two stars are used, the values of $SS'$ and $PS'$ are constant, and may be tabulated for certain pairs of principal stars, leaving only two quantities, $ZSS'$ and $ZP$, to be determined from the formulae (iii) and (iv).\footnote{This is done in the "Star Tables" published by Capt. Shadwell, R.N.} The observations must be simultaneous ; but when there is only one observer, proceed as follows ---Take the altitude of $S$, then the altitude of $S'$, and again the altitude of $S$, noting the corresponding instant of time ---in less than the few minutes that this will occupy---the changes of altitude may be supposed to be uniform, and therefore the altitude of $S$, corresponding to the instant of the observed altitude of $S'$, may be easily found. A diagram showing a triangle with vertices labeled P, S, Z and angles labeled PSS', ZSS', ZPS'. The sides are labeled PS, PS', SS', ZP. CHAP. IX.] LATITUDE BY OBSERVATION. 121 149. Fourth Method. To determine the latitude by two observations of the altitude of the sun and the elapsed time. This is in reality the same problem as the last, the declination of the sun at the two times of observation replacing the declinations of the stars, and the elapsed time reduced to degrees, at the rate of 360 for 24 hours, giving the angle $S'S$, which is the difference of right ascension of the two stars. A modification, however, arises in the case of the sun, from the fact that his declination changes very little in the interval between the observations, and may be considered the same at each as it is at the middle time between the two. The triangle $S'S$ thus becomes an isosceles triangle, and bisecting it by a perpendicular from $P$ (not drawn in the figure), the two formulae (i) and (ii) are replaced by $$\sin S'S = \sin \Delta \cdot \sin \frac{\pi}{2}$$ $$\cot PSS' = \cos \Delta \tan \frac{\pi}{2}$$ the remainder of the solution will proceed as before. The same solution will obviously apply to two altitudes of the same star and the elapsed time, provided this elapsed time be reckoned in sidereal hours. 150. At sea, when two observations are taken, as in the fourth method, separated by an interval of time, allowance must be made for the change of the ship's position during the interval. This is practically a very simple operation, because the distance run always subtends a small angle at the centre of the earth. Let $Z$ and $Z'$ be the zeniths of the two places, and $S$ the sun or star, whose zenith distance $ZS$ was observed at the first place of observation. We have to A diagram showing a line segment representing a distance run on a map with a small angle marked at its end. The angle is labeled as $\alpha$. A line segment from point Z to point Z' represents another distance run on a different map. The angle between these two lines segments is also marked as $\alpha$. The diagram illustrates how to calculate latitude by observing two points on a map. 122 LATITUDE BY OBSERVATION. [CHAP. IX. determine $Z'S$, the zenith distance which would have been observed at the same instant at the second place. Let the ship's course by compass, and distance run during the interval, be measured by the usual methods practised at sea. Also let the compass-bearing of $S$ at the first observation, be noted. The angle $\theta$, between the ship's course and the bearing of $S$, will be the angle $SZZ'$; and the distance run by the ship in nautical miles will be the number of minutes in the arc $ZZ'$. Draw $ZZ'$ parallel to $SZ$. Then $ZX$ will be very approximately the difference between $SZ$ and $SZ'$, and, the sides of the triangle $ZZ'X$ being small, $$ZX = ZZ' \cos \theta;$$ therefore $$SZ = SZ - ZZ' \cos \theta,$$ and the first observation is reduced to what it would have been if made at the second place. "151. In order to determine the most favourable circumstance for the success of the third and fourth methods, let us examine the graphical solution of the problem. We shall assume that the declinations and right ascensions of the two bodies, or the declinations and elapsed time in the case of two positions of the same body, are accurately known; and we wish to ascertain under what conditions of observations, errors in the altitudes will have the least weight in the determination of the latitude. Let $P$ be the pole; $S$, $S'$ the two bodies, or the two positions of the same body. With $S$ as pole and a spherical radius $SZ$, or $x$, the zenith distance of $S$, describe an arc $ZX$. Similarly, with $S'$ as pole, and a spherical radius $-x$, describe the arc $XZ$, cutting the former in $Z$; then $Z$ will be the zenith and $PZ$ the co-latitude. A diagram showing a spherical coordinate system with a point P (the pole), two points S and S' (the two bodies), and a line segment ZP (the co-latitude). The diagram also shows a line segment ZX (the zenith distance) and another line segment XZ (the co-latitude). CHAP. IX.] LATITUDE BY OBSERVATION. 123 When the two observations are made with different instruments, the errors of observation not only being unknown, but having no necessary connection, it will be best to select the time when the angle between the two arcs $ZX$, $ZX'$ approaches nearest to a right angle ; for, if the possible error in each observation be $\theta$, the greatest possible error in the value of the latitude will, when the angle is a right angle, be the diagonal of a square, whose side is $\theta$ ; whereas, if the two arcs cut each other obliquely, the greatest possible error will be the longest diagonal of the parallelogram, whose angle is the same as that between the arcs, and the perpendicular distance between the opposite sides of which equals $\theta$. The two bodies should therefore be observed when their azimuths differ by about 90°. But when the two observations are made with the same instrument, so that the errors of altitude, though unknown, are likely to be the same at both observations, and in the same direction, that is, both in excess or both in defect, let $RV$ and $RV'$ be two arcs, having $S$ and $S'$ for poles, and for spherical radii the erroneous zenith distances $z-\theta$ and $z'-\theta$ respectively, then $R$ will be the erroneous zenith so determined ; and it is obvious that, if the meridian $ZP$ be perpendicular to $ZR$, the co-latitude given by $PR$ will not sensibly differ from the true one $PZ$. Now $ZR$ and $Z'R$ being supposed equal, $ZR$ bisects the angle $ZPR$, therefore the most favourable condition of observation will be when this angle is bisected by a line on the same side of the meridian, and equally inclined to the prime vertical, the difference of azimuths being at the same time near 90° if possible. 152. Fifth Method. To determine the latitude by observations made with the transit in the prime vertical (Art. 109). A diagram showing two arcs ZX and ZX' intersecting at an angle θ. The diagonals of squares formed by these arcs are shown as well as their lengths. A parallelogram is also shown with its longest diagonal representing another length. The angles between these lines are noted as θ. 124 LATITUDE BY OBSERVATION. [CHAP. IX. Let $S$ be the star in the prime vertical $ZS$, $P$ the pole. Let the instant of the star's crossing the mean wire be noted by the sidereal clock. The difference between this and the right ascension of the star will be the angle $SPZ = \alpha$ suppose. Let $\delta$ be the polar distance of the star. The right-angled triangle $SPZ$ gives $\cos P = \tan PZ \cot SPZ$. $\cot lat. = \cos^2 \tan \Delta$. The nearer the star passes to the zenith, the less will be the effect of an error in the time of transit. 153. If the same star be observed in the prime vertical both on the east and on the west side of the meridian, we shall require neither the right ascension of the star nor the sidereal time, and shall thus be independent of the error of the clock provided its rate be known. The angle $\alpha$ of the formula will be one-half the elapsed sidereal time between the two observations. When a transit instrument admits of reversal on its bearings, all instrumental errors may be eliminated by observing a star on opposite sides of the meridian on one night; and, after reversing the instrument, again observing the same star on another night; but for a fuller discussion of the method and of the various refinements which render it valuable, we must refer to Chauvenet's Spherical Astronomy. Circum-meridian Observations. 154. Sixth Method. To determine the latitude by the altitude of the heavenly body observed very near the meridian. A diagram showing a celestial sphere with a star labeled 'S' at a certain point on its surface. A line extends from 'S' to a point labeled 'P', representing the pole. Another line extends from 'S' to a point labeled 'Z', representing a prime vertical. A smaller circle is drawn around 'S' and 'Z', indicating a mean wire. An arrow points from 'S' to 'Z', indicating right ascension. Another line extends from 'S' to a point labeled 'R', representing a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around 'S' and 'Z', indicating a polar distance. A smaller circle is drawn around 'S' and 'R', indicating a polar distance. A smaller circle is drawn around 'S' and 'P', indicating a polar distance. A smaller circle is drawn around CHAP. IX.] LATITUDE BY OBSERVATION. 125 Let $h$ be the hour angle ZPS of the sun or star, $\delta$ the declination = 90° - PS, $\phi$ the latitude = 90° - PZ, $x$ the observed zenith distance ZS; $x - x =$ the meridian zenith distance = PS - PZ, = $\phi - \delta$. A diagram showing a triangle with sides labeled ZPS, PS, and PZ. The triangle ZPS gives $$\cos ZS = \cos PS \cdot \cos PZ + \sin PS \cdot \sin PZ \cdot \cos ZPS,$$ $$= \cos (PS - PZ) - 2 \sin PZ \cdot \sin PS \cdot \sin^{1/2} ZPS,$$ $$\cos z = \cos (x - z) - 2 \cos \phi \cdot \cos z \cdot \sin^{1/2} A,$$ $$2 \sin (z - \frac{1}{2}x) = 2 \cos \phi \cdot \cos z \cdot \sin^{1/2} A,$$ $$\sin x = \frac{\cos z}{\sin z} = \frac{\cos z}{\sin (z - \frac{1}{2}x)},$$ and approximately, since $x$ is small, $$x = 2 \frac{2 \cos^2 \phi}{\sin^2 z} = 2 \frac{A}{\sin^2 z},$$ and $$lat. = x - x + 8.$$ It will be observed, that the value of $x$ is obtained in terms of $\phi$, the unknown latitude, but we may there replace $\phi$ by its approximate value $x + 8$. § 155. The following modification of the method would be found useful at sea. The observations being made within a short distance from the meridian, $h$ is small; we may therefore write $\frac{1}{2} h$ for $\sin^{1/2} A$, and, if we suppose $x$ expressed in seconds of a degree and $h$ in minutes of time, $$x \sin 1^\circ = 2 \frac{2 \cos^2 \phi}{\sin^2 z} (\frac{1}{4} h + 8),$$ $$x = 2 \frac{\cos^2 \phi}{\sin^2 z} (\frac{1}{4} h + 8).$$ 126 LATITUDE BY OBSERVATION. [CHAP. IX.] The multiplier of $d$ may be considered constant, and equal to its meridian value, which we shall call $C$; therefore $$C = \cos\phi \cos d \sin 15°$$ and $$= C \cdot A.$$ $C$ is the change of altitude during the first minute before or after passing the meridian, and, being a function of the latitude and declination, its values may be conveniently tabulated. These values change very slowly, so that we may use that which corresponds to the roughly estimated latitude. All that the observer has to do, therefore, is to multiply the square of the number of minutes which elapse between the observation and the meridian passage by the proper factor $C$, and the result, added to the observed altitude, will give the meridian altitude, whence the correct latitude may be found. The precision of the method is really much greater than might at first be supposed, especially when the body does not pass near the zenith. Thus, in latitude 50°, when the declination is 0°, the error will not exceed 1°, so long as the hour angle is less than 40°; it will be less than 30° when the hour angle does not exceed 35°; and for about a quarter of an hour on each side of the meridian the error will be less than 1° and consequently much within the probable errors of observation. Passing clouds may hide the sun at the very instant when it is exactly on the meridian, and yet allow very good observations to be made on each side. But even when no clouds interfere with the meridian altitude, the mean result of several observations taken near the meridian, * See Mendonça Rice's Tables for Navigation and Nautical Astronomy. See also Louis Bapst's *Précis de Navigation*, Third Edit., which contains methods analogous to the above. CHAP. IX.] LATITUDE BY OBSERVATION. 127 and corrected by this method,* will be of greater value than that of any single observation, even a meridian one. 1156. The method just given supposes the hour angle known with tolerable accuracy. When this is not the case, we may proceed as follows : Let two observations be made, and the interval $t$ be noted ; Let $x$, $x'$ be the two observed zenith distances, $x$, $x'$ the corrections to reduce them to the meridian, $h$, $h'$ the corresponding hour angles, $x = CM$; $x' = Ch'$; therefore $x - x' = x - x' = C(h - h') = C(h - h') \cdot (h + h')$ ; $x - x'$ is the change of altitude, and one of the two factors $h - h'$ or $A + K$ will be $t$, the known interval of time. This equation will, therefore, furnish the other factor, $h \pm h' = t$, $h \pm h' = \frac{x - x'}{Ct}$ ; therefore $h = \frac{1}{2} \left( t + \frac{x - x'}{Ct} \right)$, whence $x = CM$ ; and $x - x$, the meridian zenith distance, will be known.* * Each observation must be corrected separately, for the arithmetic mean of the altitudes will not correspond to the arithmetic mean of the times. † See Dolez's Course de Navigation et d'Hydrographie. Chauvenet's Spherical and Practical Astronomy. 1157. Seventh Method. To determine the latitude by an observation of the pole star. This method is especially valuable at sea, because the observation may be made at any time when the star is visible ; and, on account of the proximity to the pole, the 128 LATITUDE BY OBSERVATION. [CHAP. IX. motion is always slow, so that an error in the estimated time has little influence on the result. Let $c$ be the co-latitude of the place, $h$ the hour angle of the star, $\Delta$ its polar distance, $x$ the zenith distance = $90^\circ - a$, where $a$ is the observed altitude corrected for refraction, &c. A diagram showing a star at an angle $\Delta$ to the horizon, with a line representing the observer's position. The angle between this line and the horizon is $h$. The vertical distance from the observer to the star is $x$. A smaller circle around the star represents the Earth's radius. If $c = z$ = altitude - latitude = $x$, then $x$ and $\Delta$ will be small quantities. $$\cos x = \cos c \cos \Delta + \sin c \sin \Delta \cos h$$ $$= \cos (z + x) \cos \Delta + \sin (z + x) \sin \Delta \cos h$$ $$= (\cos z - x \sin h) \cos \Delta + (\sin z + x \cos h) \sin \Delta \cos h$$ $$+ (\sin x + x \cos h - x^2 \sin h + ...)(1 - \frac{1}{2}(\Delta^2 + ...)) \cos h.$$ We shall proceed to express $x$ in a series of ascending powers of $\Delta$. Dividing by $\sin x$, and neglecting the fourth and higher powers of small quantities, $$\cot x = \cot z - x + 2x^3 - 6x^5 - 2x^7 + ...,$$ $$\cot z = 1 - x^2 + 2x^4 - 6x^6 + 2x^8 - ...,$$ therefore $$x = \cot z - 1 - x^2 + 2x^4 - 6x^6 + 2x^8 - ...,$$ $$= 1 - x^2 + 2x^4 - 6x^6 + 2x^8 - ...,$$ whence $$x = \cot z - 1 - x^2 + 2x^4 - 6x^6 + 2x^8 - ...,$$ and substituting this value in the terms of the second order, $$x = \Delta \cot h - \frac{1}{2}\Delta^3 \cot h \sin h;$$ This second approximation being used in the terms of the second order, and the first approximation in those of the third order, we obtain $$x = \Delta \cot h - \frac{1}{2}\Delta^3 \cot h \sin h^3 + \frac{1}{4}\Delta^5 \cot h \sin h^5;$$ or, if $x$ and $\Delta$ be expressed in seconds, $$x = \Delta \cot h - \frac{1}{2}\Delta^3 \sin h^3 + \frac{1}{4}\Delta^5 \sin h^5.$$ **CHAP. IX.]** **LATITUDE BY OBSERVATION.** 129 The maximum value of the last term corresponds to $$3\cos^2\delta - 1 = 0,$$ and its value is then $$\frac{2}{2\sqrt(3)}\Delta'\sin^4\delta.$$ Now $\Delta$ is at present less than $1^\circ 30'$, and the value of this term is, therefore, less than $0^\circ 5'$. We may, therefore, neglect it, and we have $$lat.=a-\Delta\cos\delta + \frac{1}{2}\Delta'\sin^2\delta\tan\delta + \sin^2\delta.$$ The Nautical Almanac contains tables for facilitating the computation. The first table gives the values of $\Delta$ cos$\delta$, the argument* being the sidereal time of observation, which differs from $h$ by a constant quantity. This first correction applied to $a$ gives an approximate latitude $a-\Delta$ cos$\delta$. The second table gives the value of the term $$\frac{1}{2}\Delta'\sin^1\delta\tan\delta\tan\delta,$$ the arguments being the sidereal time and the altitude. A third table is also given, to allow for the difference between the actual value of $\Delta$ and that value which is employed in the construction of the first table. The arguments are the date and the sidereal time. * The argument of a table is the known quantity which serves to determine the value of some dependent function. X ( 130 ) CHAPTER X. THE ECLIPTIC. Determination of the First Point of Aries. -158. The first point of aries (γ)—that point of the celestial equator which the sun's centre occupies when crossing from the south to the north side—is the zero point from which all right ascensions are reckoned (Art. 16). On account of its importance, it is necessary that we should know with great accuracy its position among the fixed stars, or, which amounts to the same thing, the position of the stars relatively to it, as expressed by their right ascensions. (See Chap. viii., Arts. 121, 122.) First Method. To determine the instant when the sun is in γ. Let us find by observation the sun's declination at noon on two successive days (Art. 119), selected so that it may be south on the first and north on the second of the two, having crossed the equator in the interval. Let α be the south and β' the north observed declinations, let α' and β' be the corresponding differences of right ascension between the sun and some chosen fundamental fixed star (Art. 120), then $$\alpha + \beta' - \alpha' = \text{change of declination},$$ and $$\alpha - \beta' = \text{right ascension},$$ may be considered as taking place uniformly during the CHAP. X.] THE ECLIPTIC. interval of twenty-four hours ; so that, if $x$ be the change of right ascension between the first position and $\gamma$, $$\frac{x}{a-\beta} = \frac{\delta}{\delta+\delta'}$$ Therefore, the star's right ascension $$= x - x = a - \left(\frac{a-\beta}{\delta+\delta'}\right)$$ $$= \frac{a\delta + \beta\delta'}{\delta+\delta'}.$$ 159. Second Method. Flamsteed's. Let $TMS$ be the equator, $TSO$ the ecliptic, and $o$ a fixed star. Let the zenith distance $z$ of the sun be observed at noon on some day near the vernal equinox, and also the difference $\alpha$ between the time of its transit and that of the star $o$, and let $\delta$ be the declination of the sun. Thus, if $S$ be the sun's place at noon, $SM$ and $oH$ the declination circles of the sun and star, $$SM = \delta, MH = z.$$ Now, when the sun is approaching the autumnal point $\phi$, let $P$ and $Q$ be two of its positions at noon on successive days, when the zenith distances $z'$ and $z''$ are—the one less and the other greater than $z$. Suppose these to be then observed, and also the corresponding declinations $PH'=\beta$, $QH''=\gamma$, between the transits of the sun and the same fixed star $o$. We may assume that, during the motion of the sun from $P$ to $Q$, the changes of declination and of right ascension K 2 A diagram showing a star (o) on an equatorial coordinate system with a line connecting it to a point on the ecliptic (E). The diagram includes labels for "Equator," "Ecliptic," "Star," "Zenith Distance," "Declination Circle," "Sun's Place," "Declination Circle," "Sun's Declination," "Right Ascension," and "Declination." The text explains how to calculate the right ascension of a star based on its position relative to the sun's transit. 132 THE ECLIPTIC. [CHAP. X. proceed at a uniform rate ; so that if $S'$ be the sun's position when the declination $SM'$ equals $SM$, we have $$P'M' = \frac{PP' - SM'}{P'Q'} = \frac{\delta - \delta}{\delta - \delta},$$ or $$P'M' = \frac{\delta - \delta}{\delta - \delta}(\beta - \gamma) = \frac{x - z'}{x' - x'}(\beta - \gamma);$$ therefore $MM' = a + \beta + \frac{x - z'}{x' - x'}(\beta - \gamma)$, and sun's right ascension at $S$ is $$\tau M = 90^\circ - \frac{1}{2}MM'$$ $$= 90^\circ - \frac{1}{2}(a + \beta) - \frac{1}{2}\frac{x - z'}{x' - x'}(\beta - \gamma),$$ star's right ascension $$= a + \tau M = 90^\circ + \frac{1}{2}(a + \beta) - \frac{1}{2}\frac{x - z'}{x' - x'}(\beta - \gamma).$$ 160. Third Method. Both the right ascension and the inclination $\omega$ of the ecliptic to the equator, or the obliquity of the ecliptic, as it is called, may be determined by two observations of the sun's declination and of the change of right ascension during the interval. Let $S_1 S'$ be the two positions of the sun on the ecliptic 'TSS'; $\delta$, $\delta'$ the observed declinations, and $\tau M$, or $M$, the sun's right ascension at first observation, $\tau M'$, or $\tau M + a$..............second.............. $a$ being the change of right ascension, determined, as in the previous methods, by comparison of the times of transit of the sun and some star. The right-angled triangles $\tau MS$, $\tau M'S'$ give $$\sin R = cot\omega\tan\delta,$$ $$\sin(R + a) = cot\omega\tan\delta'.$$ **CHAP. X.]** THE ECLIPTIC. 133 therefore $$\frac{\sin(\beta + \alpha)}{\sin\beta} = \tan\delta,$$ $$\cos\alpha + \sin\alpha\tan\beta = \tan\delta\cot\delta,$$ whence $$\cot\beta = \tan\delta\cot\delta\cosec^2 - \cot\alpha.$$ This determines the sun's right ascension, and from it may at once be found the right ascension of the star. The obliquity will be given by $$\cot\omega = \sin\beta\tan\delta.$$ 161. This method is exact, but there are practical advantages in Flamsteed's which render it valuable, although only approximate. If we compare the formulae, we shall see that Flamsteed's does not require the absolute declinations of the sun, but only the changes of declination, or of zenith distance between the first and second, and between the second and third observations; so that any uncertainty in the latitude of the observer, or any instrumental or other errors* which would affect each observation equally, will not influence the result. The right ascension of a fundamental star being thus found, that of any other celestial body will be obtained by observing the interval between its transit and that of the star. 162. In the chapter on precession, we shall see that $\tau$ is not a fixed point as we have hitherto supposed it, but that it has a slow--very slow--motion among the stars. There will in consequence be a slow change in their co-ordinates, determined at different epochs. Tables of the $\beta$ and decl. of 100 of the principal stars are given in the Nautical Almanac for every tenth day, and those of the sun for every day at Greenwich noon. The rate of change during the intervals being nearly uniform, we may calculate *Such as uncertain values of parallax or refraction--because in this method the sun has nearly the same altitude at the different observations.* 134 THE ECLIPTIC. [CHAP. X.] the B or deal, for intermediate times. The $B$ is generally tabulated in hours, minutes, and seconds ; but it will be easy to convert it, if desired, into degrees at the rate of 15° for every hour. **Determination of the Obliviquity of Ecliptic,** 163. Perhaps the most accurate method of determining the obliquity ($\omega$) is by observations of the meridian zenith distance of the sun at the two solstices. At these epochs the sun's declination is exactly equal to the obliquity ; and if $x$, $z$ be the observed zenith distances at summer and winter solstices respectively, and $\lambda$ the latitude, we have $$\lambda - \omega = z,$$ $$\lambda + \omega = x,$$ therefore $$\lambda = \frac{1}{2}(x + z).$$ The same observations give us also the latitude of the place $$\lambda = \frac{1}{2}(x + z).$$ 164. We must remark, however, that the sun is not likely to be exactly 90° from T when he crosses the meridian ; because, as he occupies the solstitial point only for an instant, he may at that instant be far from the observer's meridian, if not actually below the horizon. A correction will be necessary to allow for the small change in declination in the interval, Let TS$'$S be the ecliptic, S the sun near the solstice, SM the decl.$= \delta - \omega - x$, TM the right ascension = $90^\circ - \alpha$, where $x$ and $\alpha$ are small quantities ; $$\sin(90^\circ \pm \alpha) = \tan \delta \cot \omega,$$ $$\cos \alpha = \frac{\tan \delta}{\tan (\delta + x)}.$$ CHAP. X.] THE ECLIPTIC. 135 $$\begin{array}{rlr} {1-\cos\alpha}&{=\frac{\tan(\delta+x)-\tan\delta}{1+\cos\alpha}}\\ {1+\cos\alpha}&{=\frac{\tan(\delta+x)+\tan\delta}{1-\cos\alpha}}\\ {\tan\frac{\pi}{2}}&{=\frac{\sin x}{\sin(28+x)}}\end{array},$$ this determines $x$, and it may be expanded in the form $$x = \tan\frac{\pi}{2}\sin 28 + \frac{1}{2}\tan\frac{\pi}{2}\sin 48 + \ldots,$$ where the first term will be sufficient when the observations are made within five or six days on either side of the solstice. Let $x$ be the observed zenith distance and $\lambda$ the latitude, then if $\lambda$ is accurately known, $\delta - \lambda - x$ will also be known, and thence $\omega - \delta + x$ may be obtained by observations near one solstice only. But any uncertainty in the value of $\lambda$ will attach to $\delta$ and, through it, to $\omega$. The value of $x$ will also be affected, but the error will not be appreciable, and we may consider $x - x$ as accurately known; therefore Similar observations at, or near, the winter solstice will give $$\lambda + \omega = (z' - z'),$$ whence $$2z = (z' - z') - (z - z'),$$ with an accuracy depending on that of $z'$ and $z'$. Celestial Latitude and Longitude. 165. The position of a celestial body may be referred to the ecliptic instead of the equator. Thus, if $K$ be the pole of the ecliptic, $KvN$ a great circle through a star $v$, meeting the ecliptic at $N$, the latitude of the star will be known when $Y.N$ and $Nv$ are given ; these are called the longitude and latitude respectively. The latitude is the arc of a great circle drawn from the body perpendicular to the ecliptic. A diagram showing a celestial sphere with a star labeled 'v', a line from 'v' to 'N', and another line from 'v' to 'Y'. The lines intersect at 'N', forming angles with the horizon. The diagram also shows a line from 'v' to 'K', which is perpendicular to the plane of the ecliptic. 136 THE ECLIPTIC. [CHAP. X.] The longitude is the portion of the ecliptic intercepted between this circle and the first point of Aries. 166. The latitude takes its name north or south from that pole of the equator which is on the same side of the ecliptic as the body. Those stars which are situated in the acute angles, formed by the ecliptic and the equator, will have their latitude and declination of opposite names; that is, one north, the other south; but for all other stars they will be of the same name. The longitude is measured eastward from $\varpi$ through $360^{\circ}$. The ancients, who always referred the places of bodies to the ecliptic, subdivided the $360^{\circ}$ of longitude into twelve equal parts called Signa, and to these they gave the names of the constellations which occupied those signs in the early days of astronomical science. Thus, the first $30^{\circ}$ from $\varpi$ was called Aries, the next $30^{\circ}$ Taurus, and so on, the names and symbols of the twelve signs being—
Aries Taurus Gemini Cancer Leo Virgo
$\varpi$ $\varpi + 30^{\circ}$ $\varpi + 60^{\circ}$ $\varpi + 90^{\circ}$ $\varpi + 120^{\circ}$ $\varpi + 150^{\circ}$
Lira Scoptus Sagittarius Capricornus Aquarius Pisces
$\alpha$ $\alpha + 30^{\circ}$ $\alpha + 60^{\circ}$ $\alpha + 90^{\circ}$ $\alpha + 120^{\circ}$ $\alpha + 150^{\circ}$
These have fallen into disuse, and the constellations have so far shifted, that the appropriateness of the names has been lost. The first sign is no longer in the constellation Aries, but in that of Pisces. This shifting we shall explain in Chap. XVIII. Transformation of Coordinates. 167. The obliquity of the ecliptic being known, we can readily convert $\alpha$ and decl. into latitude and longitude, and vice versa. Thus, let $\sigma$ be a star, $\sigma M$ and $\sigma N$ arcs of great circles perpendicular to the equator and ecliptic respectively; join $T_0$ by an arc of great circle. CHAP. X.] THE ECLIPTIC. 137 Let $Y M$ the rt. ascension = $\alpha$, $Mg$ ... declination = $\delta$, $Y N$ ... longitude = $\lambda$, $N \sigma$ ... latitude = $\lambda$, $NY M$ ... obliquity = $\omega$. First. Suppose $\alpha$ and $\delta$ given, then the right-angled triangle $oYM$ gives $$\cos oY M = \cos \alpha \cos \delta \ldots (a),$$ $$\cot oYM = \sin \beta \cot \delta \ldots (b).$$ Therefore in the right-angled triangle $oYN$, we know $Yo$ and $oYN = oYM - \omega$, hence $$\tan Yo = \tan YN = \tan (\alpha - \delta) \ldots (c),$$ $$\tan YN = \tan Yo \cos (\alpha - \delta) \ldots (d),$$ (a) and (b) give the auxiliary quantities $Yo$ and $oYM$, then (c) and (d) determine $\lambda$ and $\omega$. Secondly. Suppose $\lambda$ and $\gamma$ given. The steps will obviously be the exact counterpart of those we have just described, thus $$\cos oYN = \cos \lambda \cos \alpha \ldots (e),$$ $$\cot oYN = \sin \lambda \cot \alpha \ldots (f),$$ determine the auxiliary quantities $Yo$ and $oYN$, then $$\sin Yo = \sin YN \sin (\alpha - \omega) \ldots (g),$$ $$\tan YN = \tan Yo \cos (\alpha - \omega) \ldots (h)$$ determine the declination and right ascension. 168. In the case of the sun, whose latitude is zero, the connection between the right ascension, declination, longitude, * These formulae may be modified so as to require only one of the auxiliary quantities, viz. the angle $\gamma$. For this purpose, by (b), it may be easily shown that $$\cot o = \sin (\alpha - 2\delta),$$ $$\sin o = \sin (\alpha - 2\delta) - \sin 2\delta,$$ $$\tan o = \frac{\cos (\alpha - 2\delta)}{\sin 2\delta}.$$ and a similar modification may be applied to the second set of equations. 138 THE ECLIPTIC. [CHAP. X. and obliquity, is given by the solution of the right-angled triangle $STM$ (fig., p. 132), whence $$\cos \delta = \cos B \cos \delta,$$ $$\sin \delta = \sin \omega \sin l,$$ $$\tan \delta = \sin B \tan \omega,$$ $$\tan B = \cos \omega \tan l.$$ When two of the four quantities $B$, $l$, $\omega$ are known, these equations will determine the others. 169. From the last of the equations above, we get $$\sec^2 B \frac{dR}{dt} = \cos \omega \sec^2 \frac{dl}{dt};$$ therefore $$\frac{dR}{dt} = \cos \omega \sec^2 \frac{dl}{dt};$$ therefore corresponding small increments of the sun's $B$ and longitude will be in the ratio of $\cos \omega : \cos^2 \delta$. Right Ascension of the Meridian. 170. By the right ascension of the meridian is meant the $B$ of those stars which are in the meridian at the instant considered. It is, therefore, the sidereal time expressed in degrees (Art. 17), and is equal either to the hour angle of the first point of $T$, or to its defect from $24^\circ$, according as $T$ is on the west, or on the east side of the meridian. Thus, if $\sigma$ be a star in the meridian, then $M$ is $T - T'$, being on the west of the meri- dian, then $TM$ is the $B$ of the meridian, and is obviously the same as $T'PM$, which is the hour angle of $T'$. But if $T$ be on the east side of the meridian as at $V'$, A diagram showing a celestial sphere with a star labeled \( M \) and another labeled \( V' \). The line \( TM' \) represents the right ascension of the meridian. A diagram showing a celestial sphere with a star labeled \( M \) and another labeled \( V' \). The line \( TM' \) represents the right ascension of the meridian. CHAP. X.] THE ECLIPTIC. 139 the $\pi$ of the meridian would be the whole circumference minus $T M$, that is, the whole circumference minus the hour angle of $Y'$. The right ascension of the meridian may also be found as follows:---Take from the Nautical Almanac the sun's $\pi$ for the given time, and add to it the time past noon, deducting 24° from the result if it exceeds that sum, Position of the Ecliptic at a Given Instant. -171. If the equator were a bright visible band in the sky, it would occupy a fixed position, of which we may readily obtain an idea by means of some of its points, viz. the east point with its opposite west point, and a point in the meridian at a distance from the zenith equal to the latitude. The position of the ecliptic is not so readily conceived, on account of its constant shifting. We must determine the point where it crosses the eastern horizon, called the ascending point, which at the same time gives the opposite or descending point; those, with some third point or with the inclination of the ecliptic plane to the horizon, will be sufficient. When the sun is above the horizon, his centre, which is always in the ecliptic, will give us the third point we want; or, we may find the point called the culminating point, where the meridian is cut by the ecliptic; or, again, we may find the nadogeminal, which is 90° from the ascending point, and is the highest point of the ecliptic above the horizon. 172. Let the ecliptic, equator, and horizon be as represented in the figure on the next page, the meridian being ZCM. Let $T M$ be the right ascension of the meridian (calculated as in Art. 170), which we shall call $M$. A diagram showing an illustration of celestial coordinates. 140 THE ECLIPTIC. [CHAP. X.] Let $Z$ be the zenith, $P$ the pole, $K$ the pole of the ecliptic. Let $KZ$ meet the ecliptic in $N$ and the horizon in $N'$, then $N$ is the nonageimal, and $N'N$ its altitude is equal to the inclination of the ecliptic to the horizon. Let $ZM$, the latitude of the place, = $\phi$, In the triangle $YEO$, we know $$TE = 90^\circ + M,$$ $$LE = ME = 90^\circ + \phi,$$ $$LT = \text{obliquity of the ecliptic} = \omega,$$ whence $EO$ and the angle $O$ can be found. cot $EO = \cot TE - \cot LE = \cot TE \cos E,$ therefore cot $EO = \cos E \cdot \cos \phi \cdot \sec M - \tan M \sin \phi$.....(a), $$\cos O = -\cos T \cdot \cos E + \sin T \cdot \sin E \cdot \cos TE;$$ therefore $$\cos O = \cos T \cdot \sin E - \sin T \cdot \cos E \cdot \cos M;$$.....(b), (a) gives the ascending point, and (b) the inclination or its equivalent, the altitude of the nonageimal, and thus the position of the ecliptic is determined. 173. If $C$ be the culmination point of the ecliptic, we have from the right-angled triangle $TMC$, $$\tan MC = \sin C \cdot \tan O,$$ and $$ZC = \phi - MC,$$ whence the culmination point is known, and this, with the ascending point given by (a), will also fix the position of the ecliptic. 174. $E$ and $O$ being the poles of $ZP$ and $ZK$ respectively, $EO$ is equal to the angle $PZK$. In the course of twenty-four hours $K$ describes a small circle round $P$, and the angle $PZK$ varies between certain CHAP. X.] THE ECLIPTIC. 141 limits—the greatest value corresponding to the position when $ZK$ is a tangent to the small circle, and consequently the angle at $K$ a right angle. Then $$\sin PZK = \frac{\sin \omega}{\sin ZP} = \cos \phi.$$ Hence, the greatest value of $EO$ is $\sin^{-1}(\frac{\sin \omega}{\cos \phi})$, and the ascending point oscillates to this extent from one side to the other of the east point; but the time from the most northerly to the most southerly position will not be the same as the time back again; for the angle $ZPK$ is acute, and the ratio of the two times will be $\frac{ZPK}{ZKP}$. When $K$ is in the meridian above the pole, $w$ rises, and the three points $v$, $O$, and $E$ all coincide; also $KZ$ has its least value $\omega - 90^\circ + \phi$, which is therefore the least inclination of the ecliptic to the horizon. When $K$ is in the meridian below the pole, $\alpha$ rises at $E$, and the angle between the ecliptic and the horizon has then its greatest value $\omega^o - \phi + \alpha$. At a place within the frigid zone the result will be somewhat different. There, $Z$ will fall within the small circle described by $K$, and the point $O$ will travel completely round the horizon, the ascending suddenly becoming the descending point, and vice versa. The greatest and least values of the inclination will be $\omega + (90^\circ - \phi)$ and $\omega - (90^\circ - \phi)$. 175. The longitude of the nonacesimal is an element sometimes wanted in the calculation of eclipses. Its value is easily inferred from the triangle $YEO$, for $YN = YO - 90^\circ$, $$cot YE = cot E \sin T + cos T \cos Y,$$ therefore $\tan YN = tan \phi \sin \omega + tan M \cos \omega$. ( 142 ) CHAPTER XL. FORM OF THE EARTH'S ORBIT. 176. The orbit of the sun round the earth, or of the earth round the sun, lies, as we have seen, in one plane; but what particular curve is described in this plane we have not yet ascertained. We shall proceed to an examination of this question, which, until the time of Kepler, remained without a satisfactory solution. It is very easily perceived, that the distance of the earth from the sun is not a constant quantity. For, by taking accurate observations of his apparent diameter at different times,* we find that it varies from 31° 1' 5" on the 1st of July to 32° 35' 6" on the 31st of December. We cannot suppose the magnitude of the sun to vary periodically, we must infer that his distance changes, and that the earth's orbit is not a circle with the sun at the centre. The angles subtended by the sun being small, must be very nearly inversely proportional to the distances. Again, if the angular motion of the sun in his orbit be observed, that is, his angular motion in longitude, its daily value will be found to vary from 0° 57' 11"5 to 1° 1' 9"9-, these minimum and maximum values occurring at the same epochs of July 1st and December 31st. 177. In the first place we shall remark, that the change in angular velocity is greater than that in apparent diameter; * An instrument called a Helioscope is specially adapted to this purpose. It is a telescope with a divided object-glass on the principle of Dulong's double-image micrometer (Art. 119). See Charvatin's "Astronomy," vol. ii. CHAP. XL.] FORM OF THE EARTH'S ORBIT. 143 for, the change of the former is about $\frac{1}{5}$ of its mean value, that of the latter only $\frac{1}{20}$. Or, while the apparent diameter changes in the ratio of $1 : 1 + \frac{1}{20}$, the angular velocity ...................... $1 : 1 + \frac{1}{5}$, which latter is the same as $1 : (1 + \frac{1}{20})$ nearly, and, at whatever times the numerical values can be compared, it will always be found that angular velocity $\propto$ (appt. diam.)$^{\prime}$, $$\propto \frac{1}{(\text{dist.})^{\prime}}.$$ Now let $E$ be the earth, $S$ the sun at any time, and $ST$ the arc described in one day. Let $S'ET$ be also described once at any other time. Then sector $SET$: sector $S'ET'$ $$= \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = \angle SET \times (SE)' : \angle S'ET' \times (S'E)' = 180^\circ - 2\alpha,$$ therefore the sector $SET =$ sector $S'T'$. Or, the areas, described in equal times by a radius sector joining the earth and sun, are equal in all parts of the orbit. This is one of Kepler's famous laws discovered after long and laborious calculation. This law tells us how the orbit is described, but says nothing about its form. A78. Another of Kepler's laws, also the result of many years study and observation, is, that the orbit is an ellipse, of which the earth occupies one focus, and this we may verify in the following manner: Let the apparent diameter of the sun, and his corresponding angular distance from the position of the 31st of December, that is, his change of longitude, be observed day by day throughout the year. 144 FORM OF THE EARTH'S ORBIT. [CHAP. XL. Let $E$ be the earth, $A$ the sun's place on the 31st of December, $B$ ........................................ 1st of July, A diagram showing an ellipse with points labeled A, B, C, D, and E. The major axis AB is horizontal, and the minor axis AC is vertical. The angle between AC and AB is labeled as $\theta$. The point D is at the top of the ellipse. when it has described 180° of longitude. $$AE : EB = \frac{1}{52.55^{\circ}6'31'31''\theta},$$ $$AE : AB = 31'51''+64''6-8,$$ and if $$AB=2a,\quad AC=a,$$ we find $$AE=0.98321a,$$ $$EB=1.01679a;$$ therefore $$EC=0.01679a.$$ Then, if an ellipse be described with $AB$ for major axis, and $E$ for one focus, any radius vector $ES$, making with $AE$ an angle $\theta$, will be numerically calculated from the formula $$ES=\frac{a(1-e)}{1+e\cos\theta},$$ where $e=0.01679;$ therefore $$ES\propto 1+e\cos\theta,$$ hence, if the sun describes this ellipse, we ought to find apparent diam. $\alpha=1+0.01679\times cosAES,$ a relation which is verified by the observations. Therefore the sun describes an ellipse about the earth in one focus, /or, according to the more correct statement, the earth describes// an ellipse about the sun in one of the foci (Art. 133). CHAP. XI.] FORM OF THE EARTH'S ORBIT. 145 179. The points of the orbit where the sun is at his greatest and least distances are called respectively *Apogee* and *Perigee*, when, to conform our language to appearances, we attribute the motion to the sun; but, when we employ the more correct supposition and regard earth as the body moving round the sun in the figure, the corresponding points of the earth's orbit are called *aphelion* and *perihelion*, and the line joining them is the *line of Apesides*. 180. The laws of elliptic motion having been thus established by observation, it will now be easy to explain why the seasons should be of different lengths, as stated in Art. 135. Let us make the earth revolve about the sun, Let $S$ be the sun in the focus of its orbit, $AP$ the apse line. The earth will be at $P$ on the 21st of December, and at $A$ on the 1st of July. Let $CD$ be the line of equinoxes, $MN$ at right angles to it, the line of solstices. A diagram showing the Earth's orbit around the Sun. The Sun is at one focus (P) and Earth is at another focus (A). The line CD represents the line of equinoxes, while MN represents the line of solstices. The earth is at $N$ on the 22nd of December, only 9 days before reaching $P$. A simple inspection of the figure shows that the winter quadrant NSC, which contains the least radius vector $SP$, must be the smallest, and the opposite summer quadrant MSD, the greatest. Of the two others, the spring quadrant CSM is larger than the autumn one DSN. The areas of the quadrants being unequal, so also will be the times of describing them be. L 146 FORM OF THE EARTH'S ORBIT. [CHAP. XL 181. At present the lengths of the seasons are as given in Art. 135. We say—at present—because any change in the position of $AP$ relatively to $CD$ will obviously affect those lengths; and our observations will show that both the line of equinoxes and the line of apsides are in motion, and in opposite directions; so that the angle $PSC$ annually diminishes about 47". Where, after 2100 years, the line $SP$ bisects the angle $NSC$, summer and winter will have respectively their greatest and least possible values, and spring and autumn will be of equal lengths. When, after 2650 years longer, $SP$ coincides with $SC$, summer and autumn will be equal, and longer than winter and spring, which will also be equal to one another. The subsequent changes can be easily followed. Of the 61'47" of annual change in the angle $PSC$, 59°-22" are due to a retrograde motion of $SC$ (see Chap. XVIII.). The remaining 11°-25" are due to a progressive motion of the Apsis-line, which we proceed to determine. To determine the Position and Motion of the Apsis-line. 182. We may remark that when the sun is in perigee, its distance from the earth is least, and consequently its apparent diameter greatest; and then also it attains its greatest angular velocity. But the changes in these quantities occur so slow for several days before and after, that it is impossible to detect, by direct observation, the precise instant when this occurs. It may, however, be done with tolerable accuracy by determining points on each side of perigee or apogee, at which either the diameters or the angular velocities are equal; then the apsis line will bisect the angle formed by the radii drawn to these positions. 183. A more accurate method of determining the instant and the position of perigee consists in observing the interval of time between two positions of the sun separated by 180°. A diagram showing the Earth's orbit around the Sun with lines representing the line of nodes (N), line of apsides (S), and line of perigees (P). The Earth's axis is tilted at an angle relative to its orbital plane. CHAP. XL.] FORM OF THE EARTH'S ORBIT. 147 When this interval is exactly half the time required to describe $360^\circ$, the two positions must be perigee and apogee, since no other line through the focus but the apse line bisects the area.* By a comparison of the position of the apse line at present with those which it has occupied at different times separated by long intervals, it is found to have a progressive motion of $11\frac{1}{25}$ a year. Position of the Earth in its Orbit at any Time. 184. When the instant of perihelion is known, and also the time required to make a complete revolution in the orbit, it is clear that there must be some means of ascertaining the position of the body corresponding to a given time. This we proceed to explain : Let $S$ be the sun at the focus, $P$ the perihelion, $E$ the earth at time $t$, measured from perihelion, $PSE$ being the angle described. We have already said that the radius vector joining the sun and earth will now revolve at a uniform rate. Let us consider a uniformly revolving radius $SR$ to start with the actual radius $SE$ from perihelion, and let the angular velocity of $SR$ be such that the two radii accomplish their revolution in exactly the same time ; then they will coincide at perihelion and at aphelion, and there only. The angular velocity of $SR$ is called the mean angular velocity of the earth. * For other methods of determining the position of the line of apsides see Delambre's Astronomie, vol. ii. p. 138. L2 148 FORM OF THE EARTH'S ORBIT. [CHAP. XL. The angle $PSE$, described by the true radius from perihelion, is called the true anomaly - the angle $PSR$, described in the same time by the uniformly revolving radius, is called the mean anomaly; and the angle $ESR$ between them is the equation of the centre. Therefore To find anomaly = Mean anomaly + Equation of centre. There is yet another angle connected with the position of $E$ which we shall find it useful to consider. It is the angle $QCP$ at the centre of the ellipse, formed by the apse line $CP$ and the line joining $C$ with $Q$, the point of the auxiliary circle which corresponds to $E$. This is called the eccentric anomaly. To find the Relation between the Mean and the Eccentric Anomalies. \textgreater 185. Let $n$ be the mean angular velocity of the earth, so that if $T$ be the periodic time $$n = \frac{2\pi}{T}$$ Then, mean anomaly at time $t = PSR = nt$. And, if we suppose $SQ$ joined, we have, by a property of the ellipse, and by Kepler's law, $$\frac{\text{area of } PQS}{\text{area of } PSE} = \frac{l}{r} = \frac{nt}{2\pi}$$ but, $$\text{area of } PQS = \text{sector } PCQ - \text{triangle } SCQ$$ $$= l^2u - l^2e \sin u,$$ where $u$ is the eccentric anomaly, $a$ the semi-major axis, and $e$ the eccentricity; therefore $$nt = a - e \sin u.$$ To find the Relation between the True and the Eccentric Anomalies. \textgreater 186. Referring to the figure, we have, $\theta$ being the true anomaly, $$SE \cos \theta = CM - CS;$$ CHAP. XL.] FORM OF THE EARTH'S ORBIT. 149 therefore $$\frac{a(1-e)}{1+e\cos\theta}\cos\theta=a\cos u-ae,$$ $$(1-e)\cos\theta=(\cos u-e)(1+e\cos\theta);$$ whence $$\cos e=\frac{\cos u-e}{1-e\cos u},$$ $$1-\cos e=\frac{(1+e)(1-\cos u)}{(1-e)(1+\cos u)},$$ $$\tan\frac{\theta}{2}=\sqrt{\frac{1+e}{1-e}}\tan\frac{\theta}{2}.............(i).$$ 187. The excentric anomaly is thus, by means of (i) and (ii), a link between the mean and the true anomalies. When $\theta$ is given, $t$ can be readily found ; but the more usual problem is the inverse of this, and requires us to find the value of $e$ corresponding to a given value of $t$. This might be done by employing a succession of trials to approximate to the value of $u$ in (i) and then determining $\theta$ by (ii). But, the quantity $e$ being small, the plan adopted is to express $u$ and thence $\theta$ in a series of ascending powers of $e$, by Lagrange's Theorem (see Differential Calculus), and retain only the important terms. We shall refer for the investigation to Tait and Steele's Dynamics (Chap. vi.). The result to the third power of $e$ is $$\theta=mt+2e\sin t+3^{\frac{e}{2}}\sin2t+\frac{e^3}{12}(13\sin3t-3\sin nt),$$ and Equation of centre $$x=e^2\sin t+6^{\frac{e}{2}}\sin2t+8c.$$ When $\theta$ has been found, the value of $$SE=\frac{1-e^2}{a}=1+e\cos\theta$$ may be determined ; or it may also be expressed, in terms of the time, in a series of ascending powers of $e$, $$SE=\frac{1-e^2}{a}=1+e\cos t+\frac{e^3}{3}(1-\cos2t)+\frac{5e^5}{6}(\cos t-\cos3t).$$ 188. If the longitude of the apse be added to the true anomaly, we get the true longitude of the sun ; and if added 150 FORM OF THE EARTH'S ORBIT. [CHAP. XL- to the mean anomaly we get what is called the mean longitude, therefore True long. = Mean long. + Equation of centre. 139. Besides the two laws, given in this chapter, which were found by Kepler to hold in the case of each planet, he, a few years later, discovered a third law, which establishes a remarkable connection between the periodic times of the different planets. This law, which furnishes one of the arguments for the earth's being a planet (Art. 133, (2)), is that " The squares of the periodic times vary as the cubes of the semi-major axes. " Kepler's three laws, and the other results of this chapter, can be shewn to be necessary consequences of the law of universal attraction—Newton's great discovery ; but that method of investigation belongs to Physical Astronomy, whereas we are concerned with Astronomy considered as a science of observation. ( 151 ) CHAPTER XII. UNITS OF TIME. EQUATION OF TIME. 190. For astronomical purposes the *sidereal* day is one of the principal units of time. It begins at the instant when the first point of aries is on the meridian ; a correct sidereal clock should then mark $0^{\circ} 0^{\prime} 0^{\prime\prime}$, and at any other instant the *sidereal time* will be the hour angle of T reckoned westward from $0^{\circ}$ to $24^{\circ}$ (Art. 17). A *solar day* is the interval between two successive transits of the sun's centre over the meridian. The sun changes his right ascension, advancing eastward among the stars, at the rate of about $1^{\circ}$ a day ; therefore the earth will have to turn nearly $361^{\circ}$ about its axis to complete a solar day, which will consequently be about $4^{\circ}$ longer than a sidereal day. The solar time at any instant is the hour angle of the sun's centre reckoned westward from $0^{\circ}$ to $24^{\circ}$. This is called the *apparent*, *solar time*, and is the time indicated by a sun-dial. If the sun's motion in right ascension were uniform, the solar days would be all equal to one another, but this is not the case. In the first place, the sun's motion in its own orbit is not uniform ; and secondly, even if it were, the corresponding motion in right ascension would not be uniform on account of the inclination of the orbit to the equator. 191. The solar day, marking the recurrence of light and darkness, is obviously that on which man, in civil life, must regulate his time, although the want of uniformity mentioned above hinders us from employing it as a measuring unit. We may, however, obtain a uniform measure of time A diagram showing the Earth's rotation around its axis and its revolution around the Sun. 152 UNITS OF TIME. [CHAP. XII. depending on the sun in the following manner. Conceive an imaginary body, called the mean sun, to move along the equator with the mean angular velocity of the true sun. The days marked by this mean sun will be all equal, and exactly the average of all the solar days during the year. Therefore a clock, whose motion is necessarily uniform, may be regulated on the mean sun. To complete the connection between the two suns, we must establish the starting point of the mean sun ; and it will be convenient so to choose it that the mean solar time and the apparent solar time may never be widely separated. The following has been adopted : Conceive another imaginary body, say a star, to have the same uniform angular velocity as the mean sun, but to move along the ecliptic instead of the equator, and to pass through perigee at the same time as the true sun ; then the motion of the mean sun is so adjusted that it may pass through T at the same time as the star. By referring to Art. 188, it will be seen that the connection between the two suns may be expressed by saying that *the right ascension of the mean sun is equal to the mean longitude of the true sun*; because the mean longitude of the true sun is the longitude of the supposed star. 102. Mean noon is the instant when the mean sun is on the meridian ; and the mean time at any instant is the hour angle of the mean sun reckoned westward from $0^{\circ}$ to $360^{\circ}$. These twenty-four hours constitute the astronomical mean day ; but for our purposes, it is found more convenient to begin each day at midnight, complete it at the next midnight, dividing it into two periods of 12 hours each. The two reckonings only agree in the afternoon of each day, thus Aug. 26 at 9 P.M. civil time is Aug. 26 at $2^{\mathrm{nd}}$ astro. time, but Aug. 27 at 4 A.M. Aug. 26 at $21^{\mathrm{st}}$.............. 103. The equation of time is the difference between apparent and mean time at any instant ; it is usually considered CHAP. XIL.] EQUATION OF TIME. 153 as the correction to be applied to the former to obtain the latter, and is therefore called positive when mean noon precedes true noon, and vice versa. It is obvious that the equation of time is the value, expressed in time, of the angle between the declination circles of the sun and the mean suns. In order to examine the variation in the equation of time, we may consider separately the two causes to which it is due, and the algebraic sum of the two effects will be approximately that due to their joint action. 104. First. Neglecting the elliptic motion, let us suppose the sun to describe his orbit $T M \infty N$ with uniform angular velocity, the mean sun describing the equator $T M \infty N$ with the same velocity, the two passing through $T$ at the same instant, and the earth at the entire turning about its axis $PP'$ in the same direction (indicated by the arrow) once a day. When the true sun is at $B$, the mean sun will be at $C$, where $TC = TR$, and if the declination circle $BD$ be drawn through $B$, $CD$ will measure the angle between the declination circles of $B$ and $C$, and therefore the equation of time. It is obvious that $C$ and $D$ will coincide only at the equinoxes and solstices. From equinox to solstice, $C$ will be in advance of $D$, and behind it from solstice to equinox. Hence, while the sun is between $T$ and $M$, any given meridian will, as the earth revolves, overtake first the true sun $B$ and then the mean sun $C$, i.e., apparent noon will precede mean noon, and the equation of time will be subtractive. In the same way it may be shown to be additive from solstice to equinox. A diagram showing a celestial sphere with a line representing Earth's orbit around Sun labeled T-M-N. The Sun's position is indicated by point S. A line from S to C represents one of the mean suns' positions on Earth's orbit. Another line from S to B represents another mean sun's position on Earth's orbit. Lines from B and C to D represent declination circles for B and C respectively. 154 UNITS OF TIME. [CHAP. XIL 495. Secondly. Neglecting the obliquity of the ecliptic, let $PBA$ be the sun's elliptic path, $H$ the place of the true sun between perigee and apogee, $C$ the corresponding place of the star or mean sun, the two coinciding at perigee and apogee only. At perigee the true sun has its greatest velocity, and will, therefore, shoot ahead of the mean, the interval between them continuing to increase so long as the sun's angular velocity exceeds its mean value. This will be at somewhat more than 90° from perigee (Art. 198); after this the interval will begin to diminish and the two will again coincide at apogee. Therefore, from perigee to apogee the successive meridians of the earth will overtake the mean sun before the true, i.e. mean noon will take place before true noon, or the equation of time will be additive. A similar reasoning will show that from apogee to perigee the equation of time will be subtractive. This gives merely the general character of the equation as due to this cause; its actual value will be obtained by multiplying the angular interval by cos sec $\theta$, which is the factor requisite to reduce a small arc from the ecliptic to its corresponding projection on the equator (Art. 169). Since the multiplier varies with the declination, it follows that the greatest equation of time, due to this cause, will not necessarily correspond to the greatest equation of the centre. Moreover, the position of the apse line with respect to the first point of aries is (Art. 181) changing by about 61°-47' annually, and therefore the same equations of the centre will not recur with exactly the same declinations, and this also will slowly affect the maximum equation of time due to the ellipticity of the orbit. CHAP. XII.] EQUATION OF TIME. 156 106. It remains now to combine the two parts of the equation of time into one sum, and when this is done we find that it vanishes four times a year, on or about the following dates :—April 15th, June 15th, August 21st, and December 24th. The following graphical method of effecting the combination will perhaps show the changes, and separate the day, more forcibly than tabulated numbers alone would do : Draw a horizontal line—the upper one in the figure—to represent the time, the successive days being supposed represented by equal successive intervals. A graph showing the variation of the equation of time over a period of time. The x-axis represents time, with labels for April 15th, June 15th, August 21st, and December 24th. The y-axis represents the equation of time, ranging from -10 to +10. The curve shows a series of peaks and troughs. Let the curve $BBBB$ be drawn, so that its ordinates, corresponding to each day, may represent that part of the equation of time, on that day, which is due to the obliquity of the ecliptic, positive values being set off on one side of the line and negative values on the other ; the curve will cross the straight line at the equinoxes and solstices, the maximum value at intermediate times being about $10^{\circ}$ (see Art. 197). 156 UNITS OF TIME. [CHAP. XII.] Again, draw the curve $CCCC$ to represent in a similar manner that part of the equation of time due to the elliptic form of the orbit, and whose greatest value is now about $7^{\circ}$ (see Art. 198). This will cross the straight line on the 31st of December and on the 1st July. To avoid confusion, draw another horizontal line—the lower one—parallel to equal and parallel to the former, to represent times as before; and let the curve $DDDD$ be traced by making its ordinates equal to the algebraic sum of the ordinates of the two former curves; this curve will represent the equation of time due to the combination. The figure shows the four vanishing positions specified above, and likewise the intermediate maximum values, viz.: \[ +14^{\circ}5 \text{ min. about February 11th}, \\ -3^{\circ}9 \text{ May 14th}, \\ +6^{\circ}2 \text{ July 25th}, \\ -16^{\circ}3 \text{ November 1st}. \] 197. If we refer to Art. 169, we have \[ \frac{dR}{dt} = \cos \omega \sec l \frac{dl}{dt}; \] but, when $l - B$ is a maximum, \[ \frac{dR}{dt} = \frac{dR}{dt}; \] therefore \[ \cos \omega \sec l = 1, \] \[ \cos l = \sqrt{\cos \omega}, \] which value of $\delta$ will, by the formulae of Art. 169, lead to \[ \sin(l - B) = \tan \frac{\omega}{2}, \] and therefore the maximum value of this part of the equation of time is \[ l - B = 2^{\circ}28' = 10 \text{ min. in time nearly}. \] The separate values of $l$ and $B$ will be found $46^{\circ}14'$ and $43^{\circ}46'$ from the equations \[ \sin l = \cos B = \frac{1}{\sqrt{2}} \sec \frac{\omega}{2}. \] CHAP. XII.] EQUATION OF TIME. 157 198. Again, the equations in Art. 187 will enable us to obtain the maximum value of the equation of the centre; and this, when powers of $e$ above the second are neglected, is found $=3e$ nearly, the corresponding value of $\theta$ being about $90^\circ$. But $e=\frac{1}{\sqrt{3}}$ (Art. 187); therefore maximum equation of the centre is $\frac{1}{\sqrt{3}}=0.577$ nearly; and, as this takes place at present near the equinoctial point where $\delta=0$, the factor $\cos \omega.\sec \delta$ becomes cos $27^{\frac{1}{2}}27^{\frac{1}{2}}$, or $\frac{1}{4}$ nearly, and the corresponding part of the equation of time will be $\frac{1}{4}\cdot\frac{1}{\sqrt{3}}\cdot0.577=0.147$ min. in time. The equation of time should also take account of other disturbances, such as --those due to precession, which produce unequal changes in the longitude and right ascension of the sun (Chap. XVIII). But for this and for the effect of planetary disturbances we shall refer to Delambre's *Astronomic*, vol. II. Equinoctial Time. 199. In addition to the three kinds of time--sideral, apparent or solar, and mean solar--there is another, sometimes used by astronomers, and called equinocial time, which has the advantage of being independent of the observer's meridian. When a phenomenon is observed at different places, the time of its occurrence will be registered differently, although the absolute instant may be the same for all. This, as we know, is due to differences of longitude; noon being reckoned, and the astronomical day beginning, at each place, at that moment the mean sun crosses the meridian of that particular place. Therefore, in order to institute a comparison between the observations, it is necessary to specify the place as well as the time of observation. The equinocial time is in time, expressed in mean days, that has elapsed since the preceding mean vernal equinox. The equinocial time is therefore reckoned from an epoch which is common to all observers. 158 UNITS OF TIME. [CHAIR. XIL] Different kinds of Years. 900. A year is the period of the earth's revolution about the sun from some determinate position back again to the same. If the starting point be a star or some point fixed among the stars, the interval is called a sidereal year. If we start from the first point of T, which, as we have already stated, has a retrograde motion of $50^\circ 22'$ annually, moving as it were to meet the earth, the period will not be so long. This is called the tropical year, and, as it determines the commencement of the seasons and all the important phenomena of vegetation and life, it is the unit marked out by nature for the use of man. By observations, separated by a long interval, it is found to consist of 365$^{2}42216$ mean solar days. A third year is obtained by taking for our starting point the perihelion of the earth's orbit. Observation shows that the apex line has a progressive annual motion of $11^\circ 25'$ (Art. 18); the earth will therefore have to move through this quantity, in addition to the $360^\circ$, in order to complete this period, which is called the anomalistic year. The relative magnitudes of the three years will therefore be $$\frac{\text{tropical}}{360^\circ - 50^\circ 22'} = \frac{\text{sidereal}}{360^\circ} = \frac{\text{anomalistic}}{360^\circ + 11^\circ 25'}$$ and the tropical year having been found 365$^{2}42216$ days, the sidereal year will be .............. 365$^{2}42374$.............. the anomalistic year will be .............. 365$^{2}42944$.............. The civil year contains an exact number of days, either 365 or 366. Its adjustment and dependence on the tropical year will be explained hereafter (Chap. xxii.). A diagram showing the positions of the Earth's orbit around the Sun at different points in time. ( 159 ) CHAPTER XIII. REDUCTION AND CONVERSION OF TIME. FINDING THE TIME BY OBSERVATION. 201. As the earth turns uniformly on its axis, one meridian after the other is brought opposite to the sun, and the different places have their noons in succession, according to their longitude (Art. 31). The solar time at a given place being the angle made by the sun's declination circle with the meridian of that place, it follows that the difference between the solar times at two different places, at the same instant, will be exactly the angle between the meridians of those two places; that is, their difference of longitude. The same will be true of the mean solar times or of the sidereal times; and, generally, the difference of longitude will be equal to the difference of the hour angles of any, the same, celestial point at the same instant. 202. Therefore, "to find the time under any meridian corresponding to a given time at some other meridian," we must convert the difference of longitude into time, at the rate of $15^\circ$ per hour, and add to, or subtract from, the given time; bearing in mind that, as the earth turns from west to east, the more easterly meridian will have its noon first, and must therefore reckon a more advanced time. For example, the longitude of the Paris observatory is $2^\circ 20' 9'' 45'$ east; that of the observatory at Pulkowa is $30^\circ 18' 30'' 9'$ east; what is the mean time at Paris when it is $1^\circ 5^\circ 12'$ of September 3rd at Pulkowa? A page from a book with text about reduction and conversion of time. 100 CONVERSION OF TIME. [CHA.P. XIII] h. m. a. Long. of Pulkowa 20° 17' 39" 9 E. Mean time at Pulkowa, Sept. 3d 1 5 12 " Paris 2° 29' 8" 43 E. Diff. of Longitudes 17° 59' 45" to be subtracted, Pulkowa being the more eastern; there- fore mean time at Paris 151 26 08 Sept. 3d 28 18 19-97 203. A curious consequence of this difference of local times will be the gain or loss of a day to a person travelling right round the world. We see from the above example that, supposing it possible for a person starting from Paris at noon to reach Pulkowa the next day at the noon of Pulkowa, he would have completed a day according to his own estimation, the sun having returned to his meridian, whereas the inhabitants of Paris would only reckon $24$ as to have elapsed; the traveller would thus have gained $1^{\text{st}}$ $32$ on those he left behind, and if he continued to travel eastward, it is easy to see that this gain would go on increasing. The general explanation of the gain or loss of a day may be easily given as follows: Every time that a person is carried completely round the axis of the earth, relatively to the sun, he reckons one day as he has changed. Therefore, supposing him to start from Paris to travel eastward until he returns to his starting-point, whatever number of turns the earth may make in the interval, that is, whatever number of days may be reckoned by those persons who have remained stationary, he will have made one turn more by his own motion, and therefore reckon one day more. If, on the contrary, he travel westward, or in the direction opposite to the earth's rotation, he will, as it were, cancel one of the turns which the earth has made, and reckon one day less. For a similar reason, two ships starting from England, and meeting, say in New Zealand, the one having gone round Cape Horn, the other round the Cape of Good Hope, A diagram showing the relative positions of three ships: one starting from England (left), one going around Cape Horn (right), and one going around Cape Good Hope (center). The diagram also shows the relative positions of these ships with respect to New Zealand. CHAP. XIII.] CONVERSION OF TIME. 161 will also differ in their reckoning by one day. The practical inconvenience of using two different days in the same place would oblige the settlers there to select between the two; and it seems that in New Zealand, Australia, and Polynesia generally, that day has prevailed which was brought round the Cape of Good Hope, probably from the earliest settlers having gone by that route. Ships usually change their reckoning at some point in the Pacific between New Zealand and America. Equivalent Sidereal and Mean Solar Intervals. 204. The sun advances among the stars in the same direction—west to east—as the earth revolves about its axis; any given meridian, therefore, in the course of a tropical year, crosses the first point of aries exactly once oftener than it does the sun. Now, the number of mean solar days in a tropical year is $365\frac{24}{2216}$, Art. 200, therefore the number of sidereal days in the same time is $366\frac{24}{2216}$. Hence, if $M$ and $S$ be the measures of the same interval expressed in mean and sidereal times respectively, we have $$M = \text{given time} = \frac{S}{365\frac{24}{2216}}$$ whence, $$M = 365\frac{24}{2216} - 1 = k,$$ $$S = 365\frac{24}{2216} + 1 = M + 1 = k + 1,$$ $$M = 365\frac{24}{2216} - 1 = k - 00278791.$$ To facilitate the reduction, the Nautical Almanac contains tables which give the values of $M$ corresponding to any number of hours, minutes, seconds, tentha, and hundredths of $S$, and conversely. In some tables the values of $kS$ and $k'M$ are registered opposite to the values of $S$ and $M$ respectively, then $S-kS$ gives $M$, or $M+k'M$ gives $S$. A table showing conversions between mean solar time (S) and sidereal time (M). 162 CONVERSION OF TIME. [CHAP. XIII. One sidereal day contains 23° 58' 4"·0906 of mean time, one mean solar day ...... 24° 38' 50"·5554 of sidereal time. Conversion of Time. 206. 1'. To convert the apparent solar time at a given place into mean solar time, and conversely. Here the equation of time only is required. The Nautical Almanac gives this equation for the apparent noon, and also for the mean noon of each day at Greenwich, on pages I. and II. of the successive months. If then, by applying the longitude as in Art. 202, we obtain the Greenwich which corresponding to the given local time, and, according as it is apparent or mean time which is given, refer to the tables on page I. or page II. of the month, and make the necessary correction for the change since noon, we shall obtain the equation of time : this equation, applied with its proper sign to the local time, will convert it from apparent to mean, or from mean to apparent. 206. 2'. To convert the mean solar time at a given meri- dian into the corresponding sidereal time. Let $PZN$ be the given meri- dian, $S$ the mean sun, and $\gamma$ the first point of aries. Then, $NPT$, the hour angle of the mean sun reckoned west- ward, is the mean time ; $NPT$, the hour angle of $\gamma$, is the sidereal time ; and $TP'S$ is the right ascension of the mean sun. Hence $$\text{sid. time} = \text{mean time} + \text{mean sun's right ascension}...(\gamma).$$ Now the value of the mean sun's right ascension is registered for the instant of Greenwich mean noon, on page III. of each month in the Nautical Almanac, under the heading of "Sidereal Time." This we shall denote by $B_r$. CHAP. XIII.] CONVERSION OF TIME. 163 We must, therefore, determine the Greenwich mean time corresponding to the given local time, by Art. 202, and correct the value of $x_{p}$ for the change in the interval, by multiplying this interval by the factor $X$ of Art. 304, that is, by allowing a uniform change at the rate of $5^{\circ} 56' 55$ in 24 hours, or $8^{\circ} 565$ in 1 hour. The corrected right ascension, added to the local mean time, will, according to the above formula, give the sidereal time. If $M$ be the mean time at the given place ; $L \ldots$ longitude of the place in time ; $x_{p}$ mean sun's right ascension at previous Green- wich noon, from page ii. of Nautical Almanac, then $M+L=$ mean time at Greenwich, $x_{p} + k' (M+L) =$ mean sun's right ascension ; therefore, by (i), sidereal time, or $S = M + (x_{p} + k' (M+L)) \ldots(ii).$ Ex. At Madras, in longitude $80^{\circ} 14' 19''$ east, an observa- tion is made on September 6th, 1865, at $5^{\circ} 21' 12''$ mean time ; find the corresponding sidereal time. Mean time at Madras, September 6th Longitude in time, east Mean time at Greenwich, September 6th
$h.$ m.$9$ 21 12'
$-5$ 20 87'
$b.$ m.$4$ 0 15'
or sidereal time at noon, Sept. 6th. page ii. of Nautical Almanac Change in $4^{\circ} 0$ 15' Mean sun's $k'$
$h.$ m.$11$ 8 0' 92'
$+39$ 47'
$b.$ m.$9$ 21 12'
Sidereal time 20 24 13'72 207. $3^{rd}$. To convert the sidereal time at any meridian into the corresponding mean time. M 2 164 CONVERSION OF TIME. [CHAP. XIII.] The solution of this problem is at once given by equation (ii) of the last article. From it we obtain $$M = \frac{S - R_0 - kL}{1 + k}$$ $$= (1 - k)(S - R_0) - kL,$$ because $(1 + k) = \frac{1}{1 - k}$ $$= S - R_0 - k(S - R_0 + L);$$ where it must be remembered that $L$ is positive in west, and negative in east longitude. The factor $k$ produces a change of 3° 55' 3094 in 24 hours, or 8' 229 per hour. Ex. Suppose the sidereal time at Madras, September 6th, 1845, to be 20° 24' 15'27", find the corresponding mean time.
$S$ 20° 24' 15'27" $R_0$ 11 2 21'45" $S - R_0$ 9 21 52'27"=9 21 52'27" $L$ -5 20 57'3" $S - R_0 + L$ 4 0 54'97" which, multiplied by $k$, gives - 0 39'47" therefore mean time 9 21 12'8"
When a star is in the meridian, its right ascension is equal to the sidereal time ; -- the above rule will therefore enable us to determine the mean time of transit of a known star. *208. 4*. To convert the apparent time at any meridian into the corresponding sidereal time, and conversely. Following the method of the preceding articles, we may convert the apparent time into mean time, and then the mean time into sidereal time; or inversely. Or, we may proceed directly by using the formulae of Arts. 206 and 207, taking $M$ to represent apparent time, provided that, instead of $R_0$, we employ the apparent right ascension of the sun at apparent noon, as given on page t. of the month in the Nautical Almanac ; although the change CHAP. XIII.] TIME BY OBSERVATION. 165 for one hour is no longer constant, its value, as given in the Almanac, may be considered constant during that day. Finding the Time by Observation. 200. First Method. By meridian observations. In a fixed observatory, fitted with a transit instrument, the readiest and most accurate way of finding the time is by noting the time of transit of a known star. By observing it at the different wires, and making the corrections and reductions explained in Chap. iv., Art. 79–83, the exact instant marked by the clock when the star is in the meridian will be determined, and the known right ascension of the star will be the corresponding sidereal time, which may be reduced to mean time if necessary (Art. 297), and thus the error of the clock on sidereal or mean time will be known. If the clock error be found in this manner at two different dates, separated by a few days, the change in the error divided by the number of days elapsed will give the average daily rate during that interval. With a good clock this daily rate will remain uniform for a considerable time. Having thus found the error and rate of a clock, i.e. how much it was too fast or too slow on a given day, and how much it gains or loses daily, we may easily find the true time at any instant corresponding to a given clock time. 210. When the sun is observed, if the times of transit of both limbs be noted, the mean of these will be the time of transit of the centre, i.e. apparent noon ; but if only one limb be observed, allowance must be made for the time (given on page t. of the month in the Nautical Almanac) that the semi-diameter takes to cross the meridian. This will determine the error of the clock on apparent time, and thence the error on mean time, by applying the equation of time.
CHAP. XIII.] TIME BY OBSERVATION. 165
for one hour is no longer constant, its value, as given in the Almanac, may be considered constant during that day.
Finding the Time by Observation.
200. First Method. By meridian observations.
In a fixed observatory, fitted with a transit instrument,
the readiest and most accurate way of finding the time is by
noting the time of transit of a known star. By observing it at
the different wires, and making the corrections and
reductions explained in Chap. iv., Art. 79–83, the exact
instant marked by the clock when the star is in the meridian
will be determined, and the known right ascension of the
star will be the corresponding sidereal time, which may be
reduced to mean time if necessary (Art. 297), and thus
the error of the clock on sidereal or mean time will be
known.
If the clock error be found in this manner at two different
dates, separated by a few days, the change in the error divided
by the number of days elapsed will give the average daily
rate during that interval. With a good clock this daily rate
will remain uniform for a considerable time.
Having thus found the error and rate of a clock, i.e. how
much it was too fast or too slow on a given day, and how
much it gains or loses daily, we may easily find the true
time at any instant corresponding to a given clock time.
CHAP. XIII.]TIME BY OBSERVATION.165
for one hour is no longer constant, its value, as given in the Almanac, may be considered constant during that day.
Finding the Time by Observation.
200. First Method. By meridian observations.
In a fixed observatory, fitted with a transit instrument,
the readiest and most accurate way of finding the time is by
noting the time of transit of a known star. By observing it at
the different wires, and making the corrections and
reductions explained in Chap. iv., Art. 79–83, the exact instant marked by the clock when the star is in the meridian will be determined,
and the known right ascension of the star will be
the corresponding sidereal time, which may be reduced to mean time if necessary (Art. 297), and thus
the error of the clock on sidereal or mean time will be known.
If the clock error be found in this manner at two different dates,
separated by a few days, the change in the error divided by
the number of days elapsed will give the average daily rate during that interval.
With a good clock this daily rate will remain uniform for a considerable time.
Having thus found the error and rate of a clock,i.e. how much it was too fast or too slow on a given day,
and how much it gains or loses daily,we may easily find the true time at any instant corresponding to a given clock time.
CHAP. XIII.]                                                            TIME BY OBSERVATION. 165                                       
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
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CHAP. XIII.]
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CHAP. XIII.]
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CHAP. XIII.]
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CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.
CHAP. XIII.]
TIME BY OBSERVATION.

for one hour is no longer constant, its value, as given in the Almanac, may be considered constant during that day. Finding the Time by Observation. 200. First Method. By meridian observations. In a fixed observatory, fitted with a transit instrument, the readiest and most accurate way of finding the time is by noting the time of transit of a known star. By observing it at the different wires, and making the corrections and reductions explained in Chap. iv., Art. 79–83, the exact instant marked by the clock when the star is in the meridian will be determined, and the known right ascension of the star will be the corresponding sidereal time, which may be reduced to mean time if necessary (Art. 297), and thus the error of the clock on sidereal or mean time will be known. If the clock error be found in this manner at two different dates, separated by a few days, the change in the error divided by the number of days elapsed will give the average daily rate during that interval. With a good clock this daily rate will remain uniform for a considerable time. Having thus found the error and rate of a clock, i.e. how much it was too fast or too slow on a given day, and how much it gains or loses daily, we may easily find the true time at any instant corresponding to a given clock time. When the sun is observed, if the times of transit of both limbs be noted, the mean of these will be the time of transit of the centre, i.e. apparent noon; but if only one limb be observed, allowance must be made for the time (given on page t. of the month in the Nautical Almanac) that the semi-diameter takes to cross the meridian. This will determine the error of the clock on apparent time, and thence the error on mean time, by applying the equation of time.

for one hour is no longer constant, its value, as given in the Almanac, may be considered constant during that day. Finding the Time by Observation. 200. First Method. By meridian observations. In a fixed observatory, fitted with a transit instrument, the readiest and most accurate way of finding the time is by noting the time of transit of a known star. By observing it at the different wires, and making corrections and reductions explained in Chap. iv., Art. 79–83, the exact instant marked by the clock when the star is in the meridian will be determined, and the known right ascension of the star will be the corresponding sidereal time, which may be reduced to mean time if necessary (Art. 297), and thus the error of the clock on sidereal or mean time will be known. If clock error be found in this manner at two different dates, separated by a few days, the change in error divided by number of days elapsed will give average daily rate during that interval. With good clock this daily rate will remain uniform for considerable time. Having thus found error and rate of a clock, i.e. how much it was too fast or too slow on given day, and how much it gains or loses daily, we may easily find true time at any instant corresponding to given clock time. When sun is observed, if times of transit both limbs noted, mean these will be time transit centre i.e. apparent noon; but if only one limb observed, allowance must made for time (given on page t. of month Nautical Almanac) that semi-diameter takes to cross meridian. This will determine error clock on apparent time, and thence error on mean time, by applying equation of time.

for one hour is no longer constant, its value, as given in Almanac, may be considered constant during that day. Finding Time by Observation. 200. First Method. By meridian observations. In fixed observatory fitted with transit instrument, readiest most accurate way finding time is noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal time which reduce mean time if necessary Art 297 thus error clock sidereal mean time known. If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable time Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clock time When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparent time thence error mean apply equation thereof.

for one hour is no longer constant, its value as given in Almanac may be considered constant during that day Finding Time by Observation. 200 First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal time which reduce mean time if necessary Art 297 thus error clock sidereal mean time known If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable time Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clock time When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparent time thence error mean apply equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal time which reduce mean time if necessary Art 297 thus error clock sidereal mean time known If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable time Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clock time When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparent time thence error mean apply equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal time which reduce mean时间 if necessary Art 297 thus error clock sidereal mean时间 known If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable时间 Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clock时间 When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparent时间 thence error mean apply equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal时间 which reduce mean时间 if necessary Art 297 thus error clock sidereal mean时间 known If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable时间 Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clock时间 When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparent时间 thence error mean apply equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding sidereal时间 which reduce mean时间 if necessary Art 297 thus error clock sidereal mean时间 known If clock error found this manner two different dates separated few days change error divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerable时间 Having thus found error rate clock i.e how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will be transit centre i.e apparent noon but only one limb observed allowance made for (given page t month Nautical Almanac) that semi-diameter takes cross meridian This determine error clock apparenttime thence error mean apply equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus error clock siderealmeantime known If clock error found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding time noting transit known star noting different wires making corrections reductions explained Chap iv Art 79-83 exact instant marked clock when star meridian determined known right ascension star corresponding siderealtime which reduce meantime if necessary Art 297 thus eror clock siderealmeantime known If clock eror found this manner two different dates separated few days change eror divided number days elapsed give average daily rate during interval With good clock this daily rate remain uniform considerabletime Having thus found eror rate clock e.g how much was too fast too slow given day how much gains loses daily we easily find true at any instant corresponding given clocktime When sun observed if times transit both limbs noted mean these will b e Transit Centre I.E Apparent Noon But Only One Limb Observed Allowance Made For (Given Page T Month Nautical Almanac) That Semi-Diameter Takes Cross Meridian This Determine Error Clock Apparenttime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with transit instrument readiest most accurate way finding.time noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give Average Daily Rate During Interval With Good Clocked This Daily Rate Remain Uniform ConsiderableTime Having Thus Found Eror Rate Clocked E.g How Much Was Too Fast Too Slow Given Day How Much Gains Loses Daily We Easily Find True At Any Instant Corresponding Given Clockedtimetime Thence Error Mean Apply Equation thereof For one hour is no longer constant its value as given in Almanac may be considered constant during that day Finding Time by Observation First Method By meridian observations In fixed observatory fitted with Transit Instrument Readiest Most Accurate Way Finding Time Noting Transit Known Star Noting Different Wires Making Corrections Reductions Explained Chap IV ArTtartment Thus Eror Clocked SiderealmeanTime Known If Clocked Eror Found This Manner Two Different Dates Separated Few Days Change Eror Divided Number Days Elapsed Give 166 TIME BY OBSERVATION. [CHAP. XXII. - 211. Second Method. By a single altitude of a known star, or of the sun, moon, or planet. When the body is at some distance from the meridian, let its altitude be observed with a sextant or alt-zimuth instrument, and let the corresponding clock-time be noted. In the case of the sun, moon, or planet, the altitude observed is necessarily that of the lower or upper limb, and the semi-diameter (given in the Naut. Alm.) must be added or subtracted to get that of the centre. A star has no appreciable semi-diameter. Corrections are usually made for any known instrumental errors ; for refraction (Chap. XV.) ; for dip when the observation is made at sea with a sextant (Art. 8) ; and finally for parallax (Chap. XIV). In the case of a star there is no parallax correction. If $a$ be the corrected alt., $x$ the zenith dist.=00°-a, $\delta$ the decl.* (from Naut. Alm.) $\Delta$ the polar dist.+00°-$\delta$, $\phi$ the latitude, $c$ the co. lat.=90°-$\phi$, $h$ the hour angle of the star, $x$, $\Delta$, and $c$ being the three sides of the triangle $SPZ$, the value of $h$ will be given by $$\sin h = \sqrt{\left(\frac{\sin \frac{1}{2}(x+c-\Delta)}{\cos \frac{1}{2}(\phi+\Delta+a)}\right)^2 + \left(\frac{\sin \frac{1}{2}(\phi+\Delta-a)}{\cos \frac{1}{2}\sin \Delta}\right)^2}$$ $$= \sqrt{\left(\frac{\cos \frac{1}{2}(\phi+\Delta+a)}{\cos \frac{1}{2}\sin \Delta}\right)^2 + \left(\frac{\cos \frac{1}{2}(x+c-\Delta)}{\cos \frac{1}{2}\sin \Delta}\right)^2}$$ $$= \sqrt{\left(\frac{\cos \sin (\phi+\Delta+a)}{\cos \sin \Delta}\right)^2 + \left(\frac{\cos \sin (x+c-\Delta)}{\cos \sin \Delta}\right)^2},$$ where $s = \frac{1}{2}(\phi+\Delta+a)$. When the body observed is the moon, a planet, or a star, the Naut. Alm. will give its right ascension $A$, then $A+ \frac{A}{15}$ CHAP. XIII.] TIME BY OBSERVATION. 167 will be the *sidereal* time of observation, the upper or lower sign being used according as the star is west or east of the meridian. If the sun be the body observed, then the apparent time will be $\frac{h}{15}$ if on the west side, or 24 hours - $\frac{h}{15}$ if on the east side, of the meridian. Instead of a single altitude it will be best to observe a series of altitudes in quick succession, and consider the mean of the altitudes as corresponding to the mean of the times. If the altitudes do not change at a uniform rate, calculate the error of the clock corresponding to each altitude, and take the mean of the errors for the clock error. \textasciitilde 212. The more rapid the change of altitude, the less will an error in the observation affect the time deduced from it; let us therefore examine for what position of the body this change is quickest. We have $\cos x = \cos e \cos \alpha + \sin e \sin \Delta \cos h$, therefore, $h$ and $x$ being the variables, $$\frac{dz}{dt} = \sin e \sin \Delta \sin h,$$ $$\frac{dx}{dt} = \frac{\sin e \sin \Delta \sin h}{\sin x} = \sin e \sin A,$$ where $A$ is the azimuth $PZS$. Therefore the altitude changes most rapidly when $A$ is a right angle, i.e., when the object is on the prime vertical ; and the nearer the body is to the prime vertical the more favourable will be the position for determining the time, provided, however, that the altitude of the object be not less than $8^\circ$ or $10^\circ$, for otherwise, the uncertainty of the atmospheric refractions so near the horizon would more than counterbalance the previous advantage. 108 TIME BY OBSERVATION. [CHAP. XIII. 213. Third Method. By two altitudes of the sun and the elapsed time, or by simultaneous observations of two stars. This method will be available when the latitude is unknown. Proceeding, as in Art. 148, we shall determine the latitude and the angle $PSZ$. This angle, with the co-latitude $PZ$, and the zenith distance $ZS$, will determine the hour angle $SZ$: $$\sin \text{hour-angle} = \frac{\sin ZS \sin PSZ}{\sin PZ};$$ and from the hour angle we deduce the time, as in the second method. 214. Fourth Method. By equal altitudes. If the times $T$ and $T'$, marked by a clock, be noted when a star has the same altitude before and after crossing the meridian, then $\frac{1}{2}(T + T')$ will be the time of its meridian transit, and the error of the clock on sidereal time, or on mean time, may be found in the first method. One of the advantages of this method is, that any error of graduation of the clock, or other instrument with which the observations are made, will have no effect on the result, because the two altitudes are taken at the same graduation. By taking several pairs of observations, the mean result will be probably freed from other accidental errors. 215. If, instead of a star, we employ the sun, a slight correction will be required for the change of declination, Let $b$ ......... latitude of the place; $\Delta$ ...... polar distance at apparent noon, $\theta$ ...... small hourly decrease of polar distance, $T$ ...... half-sum, or mean, of the two clock-times, $t_1$ ...... elapsed time between the observations, $x$ ...... small correction to be subtracted from $T$ to obtain the clock-time of apparent noon; A diagram showing a star's position relative to a clock face. CHAP. XIII.] TIME BY OBSERVATION. 169 therefore $t-x$ is the hour angle before noon at first observation, $t+x$ after second. $\Delta+tb$ polar distance at first observation, neglecting $\theta$, $\Delta-tb$ second. $x$ and $\phi$ are the same at both observations, and if $h$ be the hour angle corresponding to a polar distance $\Delta'$, we have $$\cos x = \sin \phi \cos \Delta' + \cos \phi \sin \Delta' \cos h,$$ therefore differentiating, $$0 = (\sin \phi \sin \Delta' - \cos \phi \cos \Delta') \frac{d\Delta'}{dh} dh + \cos \phi \sin \Delta' dh,$$ but $$d\Delta' = -tb, dh = 2r,$$ whence $$x = \tan h \cdot \cot \Delta' t_0,$$ $\theta$ being expressed in seconds of arc, $x$ will be so too, but dividing by 15 will reduce it to seconds of time. We may also write $t$ for $h$ and $\Delta$ for $\Delta'$, and the clock-time of apparent noon will be approximately $$T = \frac{\tan h}{\sin h} \cdot \cot^2 \theta.$$ The corresponding mean or sidereal time may then be calculated, and the error of the clock on either of them determined.* The correction $x$ is called the equation of equal altitudes. It must be remembered that $\theta$ is the hourly decrease of polar distance, and therefore when the polar distance is increasing, $\theta$ becomes negative. *216. At sea, the first method is not applicable; the second, by a single altitude, is that in general use; but the fourth method, by equal altitudes, may be employed advan- * See Delambre's *Observations*, vol. i., p. 583, where the value of the quantities we have here neglected is given to the nearest tenth. See also Chauvenet's *Observations*, which contains tables facilitating the calculation of $x$. 170 TIME BY OBSERVATION. [CHAP. XIII. tageously, provided a correction be made for the change of the ship's place during the interval. This correction may be made in the same manner as in Art. 160, where the altitude of the body at the first station is reduced to what it would have been if made at the same instant at the second place; but, as the altitudes would no longer be equal, a further correction becomes necessary, and this is most simply effected by making, at the second place, a couple of observations in quick succession, so as to find the time of change of latitude; and then, at the time when the altitude is the same as the reduced altitude of the first station. The error of the watch is thus found on the local time of the second station. In the case of the sun, this may be done without moving the index of the instrument—and therefore without interfering with the principal feature of the method—by observing the times when the lower and upper limbs respectively attain the given altitude; the time the sun takes to move through an altitude equal to his diameter thus becomes known. Another method consists in finding the equation of equal altitudes and the error of the clock, as if there were no change of place, and then determining the correction of the time due to the change. The error of the clock found in this manner is the error on the local time at the meridian midway between two stations. For details see Chauvenet's *Astronomy*, vol. I., p. 80, and Rapier's *Nautical Almanac*. There are various other ways of determining the time by observation, but the preceding are the only ones in constant use. For an account of other methods the reader is referred to Delambre's *Astronomy*. A page from a book, likely an astronomy textbook. ( 171 ) CHAPTER XIV. VARIOUS PROBLEMS CONNECTED WITH THE DIURNAL MOTION. Time of rising or setting of a known body. 4217. This is really only a particular case of finding the time by a single altitude (Art. 211), but, on account of its simplicity, we shall give a separate investigation. $HSR$ being the horizon, $Z$ the zenith, $P$ the pole, and $S$ the body, the right-angled triangle $PSR$ gives $$\cos SFR = \cot SP \tan PR;$$ therefore $$\cos (\text{hour angle } SPZ) = -\tan (\text{decl.)} \tan (\text{lat.});$$ and the hour angle being known, the time, apparent or sidereal, may be found as in Art. 211. The hour angle of setting will obviously be the same as that of rising. 4218. We have here neglected the refraction which, as we shall see in the next chapter, has the effect of making objects appear higher than they really are, especially near the horizon, where its value amounts to more than $36^\circ$; so that, when seen in the horizon, the body is really below it by that amount. A diagram showing a right-angled triangle with sides labeled HSR, ZP, and PSR. 172 **VARIOUS PROBLEMS CONNECTED [CHAP. XIV.]** We have also neglected the dip (Art. 8), which still further increases the error, by depressing the visible horizon. The value of the dip, depending on the height of the eye, is variable. When we wish to take these into account, we must treat the problem as in Art. 211, taking for zenith distance $90^\circ +$ refraction + dip. In the case of the sun, moon, and planets, there will be a further correction for parallax, Chap. xvi., which, acting in the opposite direction, has to be subtracted. The hour angle thus found will be that of the visible or apparent rising or setting. The problem, however, is not one of much practical value, on account of the uncertainty of the refraction near the horizon; and it is generally be sufficient to consider the above equation of it, which determines what is called the true rising or setting. The true rising of the sun occurs when his lower limb is rather more than half his diameter above the visible horizon. In the case of the moon, the parallax so much exceeds the refraction, that a contrary effect is produced, and the true rising has already taken place even before the upper limb appears. As to the stars, their light is absorbed by the atmosphere, and they do not become visible until they have attained an altitude of from $5^\circ$ to $10^\circ$ above the horizon. **A 219. To find the time occupied by the sun in rising at a given place on a given day.** The sun's rising is accelerated by refraction, &c., but the times of rising of the upper limb and of the lower limb will be accelerated by nearly the same quantity, and the number of seconds occupied by the sun in its visible rising may be calculated with respect to the rational horizon. If $D$ be the diameter of the sun in seconds, $\lambda$ the hour CHAP. XIV.] WITH THE DIURNAL MOTION. 173 angle when the zenith distance is $z$, $\delta$ and $\phi$ the declination and latitude, $n$ the number of seconds in the interval, $$\cos z = \cos \sin \delta + \cos \delta \cos \delta \sin h,$$ $$\sin z = \cos \phi \cos \delta \sin h,$$ and $z = 90^\circ$ nearly, $\cos z = 0$, $\sin z = 1$; therefore $$\cos A = -\tan \delta \tan \delta,$$ $$dA = \frac{dz}{\cos \phi \cos \sin h}.$$ or $$15n = D$$ therefore $$n = \sqrt{\left[\cos(\phi + \delta) - \cos(\phi - \delta)\right]}$$ determines the number of seconds the sun takes to rise. Lengths of day and night. 4 220. If $A$, or $15t$, be the hour angle of sun-rise or sun-set, $\delta$ the declination, and $\phi$ the latitude, then (Art. 217), $$\cos h = -\tan \delta \tan \phi,$$ and $$2t = \frac{2A}{15}$$ will be the length of the day in hours, $$2(12-t) = \frac{180-A}{15}$$ night.............. 221. If the change of declination during the day be taken into account, the morning and the afternoon will be of different lengths : thus, suppose the declination between sun-rise and sun-set to change from $\delta$ to $\delta + d\delta$, and the hour angle from $A$ to $A + dh$, $$\cos h = -\tan \delta \tan h,$$ therefore $$\sin A dh = sec^2 \tan h d\theta,$$ approximately, or $$dA = \frac{sec^2 \tan h d\theta}{\sin A} = \frac{sec^2 \tan h d\theta}{\sqrt{1-\tan^2 h}}$$ $$= sec^2 h d\theta = \sqrt{\left[\cos(\phi + \delta) - \cos(\phi - \delta)\right]}.$$ 174 VARIOUS PROBLEMS CONNECTED [CHAP. XIV.] When $\theta$ is positive, i.e. from winter solstice to summer solstice, the afternoon will be longer than the morning by $$\frac{\theta}{\sqrt{(\cos \phi + \sin \phi) (\cos \phi - \sin \phi)}}$$ seconds of time, and shorter by that quantity during the remainder of the year. † 222. If the equation $\cosh h = -\tan \delta \tan \phi$ be discussed to determine the length of the day at different places and at different times, we shall find results in exact accordance with the statements of Art. 128, 129. Thus, on a place on the equator, $\phi = 0$, therefore $\cos \delta = 0$, and $t = 360^\circ$. Therefore $2\pi = 12$ hours for all values of $\delta$, or the days are always equal to the nights. At the time of the equinox, $\delta = 0$, therefore $\cosh h = 0$ for all values of $\phi$, and the day is then equal to the night all over the earth. When $\delta = 90^\circ - \phi$, $\cos \delta = -1$, $h = 180^\circ$, and the day is 24 hours long. When $\delta = -(90^\circ - \phi)$, $\cos \delta = 1$, $h = 0$, and the sun does not rise. When $\delta > 90^\circ - \phi$, $h$ is imaginary, and the sun neither rises nor sets, but remains entirely above the horizon; and so on for other cases. † 223. To find the hour angle of a body when it has its greatest altitude. If the declination be constant, the greatest altitude will be when the body is in the meridian; but when the declination increases, the fall, immediately after passing the meridian, will be more than counter-balanced by the rise due to the increase of declination, and a still greater altitude than the meridian altitude will be attained. If the declination decrease, the greatest altitude will precede the meridian passage. CHAP. XIV.] WITH THE DIURNAL MOTION. 175 Let $\delta$ be the meridian declination, $\alpha$ its hourly increase in seconds of arc from Nant. Alm., $\theta$ the circular measure of increase in one second of time, $\beta$ the circular measure of $1^{\circ}$, whence $$\alpha = 60^\circ \times 15 \cdot \frac{\theta}{\beta}$$ Let $z$ be the zenith dist. at $t$ seconds past meridian, $$\phi \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$$ when $z$ is least, $$\frac{dz}{dt} = 0,$$ $$\theta [\cos (\delta +\theta) -\sin (\delta +\theta)\cos (\delta +\theta)\cos Bt] = 0.$$ Expanding, and retaining the important terms, $$\theta [\cos (\delta +\sin (\delta -\sin (\delta +\theta)\cos Bt) = 0,$$ $$t = 60^\circ - 15^\circ (\tan (\delta -\tan (\delta)),$$ $$= 60^\circ - 15^\circ (\tan (\delta -\tan (\delta)),$$ $$= 60^\circ - 15^\circ (\tan (\delta -\tan (\delta)).$$ In the temperate and in the torrid zones this will always be a small quantity, except in the case of the moon, whose declination sometimes changes rapidly. Twilight. 224. When the sun disappears below the horizon, darkness does not come on instantaneously, because the rays of the sun, though not coming to us directly, still illuminate the atmosphere above us; and the light, reflected and scattered in all directions by the particles of vapour, &c., which are held in suspension, reaches us with an intensity which gradually diminishes as the sun sinks lower. 176 VARIOUS PROBLEMS CONNECTED [CHAP. XIV.] Observation has shewn that some portion of this diffused light is brought to the observer so long as the sun is not more than 18° below the horizon ; after that, darkness begins. A corresponding period also precedes sunrise. The sublunary light which thus separates night from day is known as **twilight**. § 225. The duration of twilight will vary with the latitude and with the declination. Within the tropics, twilight is always short; because the sun's diurnal path is nearly vertical, and the 18° of depression after sunset are soon attained. At the equator, an interval of 72 minutes separates daylight from complete darkness; but the impression conveyed is that night follows day almost immediately. In high latitudes, the sun's path is so inclined to the horizon that a long interval elapses after sunset before the depression reaches 18°; and at midsummer, in all latitudes exceeding 48°, it will not, even at midnight, be so much as 18° below the horizon, and there will be no real night. § 226. To find the duration of twilight is, therefore, to find the time the sun takes to alter his zenith distance from 90° to 108° in the evening; or from 108° to 90° in the morning. With the ordinary notation, $$\cos 108^\circ = \sin 8^\circ \sin \phi + \cos \phi \cos h,$$ determines $h$ the hour angle of the end of twilight, and $$\cos h' = -\tan h \tan \phi,$$ gives $h'$ the hour angle of sunset; whence $h - h'$ the duration of twilight will be known. § 227. If $8 > 72^\circ - \phi$, then $90^\circ - 8 < \phi + 18^\circ$, i.e. the polar distance of the sun = latitude + 18°; the sun at midnight will be less than 18° below the horizon, and there will be no real night. A diagram showing a circle representing Earth's surface with a line indicating the position of the Sun at different times of day. CHAP. XIV.] WITH THE DIURNAL MOTION. 177 Thus, at Cambridge, in latitude $52^\circ 13'$, there will be no night while the declination exceeds $72^\circ - 52^\circ 13' = 19^\circ 47'$ north, or from about the 19th of May to the 24th of July. \footnote{228. To find the time of the year when twilight is shortest at a given place. Let $Z$ be the zenith, $P$ the pole, $S$ the sun at the commencement of morning twilight, when $ZS = 108^\circ$. Instead of giving the diurnal motion to the sun, let us make the earth revolve; and, when twilight ends, let $Z$ have come to $Z'$, where $Z'S = 90^\circ$; then the meridian $PZ$ will have described the angle $ZZ'S$, which will measure the duration of twilight. If $ZZ'$ be joined by an arc of a great circle, a triangle $ZSZ'$ will be formed, of which two sides are respectively $90^\circ$ and $108^\circ$, and therefore the third side $ZZ'$ cannot be less than $18^\circ$. If, on any day, the three points $Z$, $Z'$, and $S$ prove to lie on this great circle, $ZZ'$ will be exactly $18^\circ$, and twilight will be short. Let $Z$, $X$, and $V$ be their positions on that day. Join $PV$, and draw $PW$ at right angles to $ZX$. $$PZ = P X = 90^\circ - \phi_1; PV = 90^\circ - \delta;$$ $$WZ = WX = 90^\circ - \delta; WV = 90^\circ.$$ The right-angled triangles $$P VW, PW Z,$$ give $$\cos PV = \cos W P \cos WV,$$ or $$\sin \delta = \cos W P \cos WV,$$ $$\cos P Z = \cos W P \cos Z W,$$ or $$\sin \phi = \cos W P \cos V,$$ therefore $$\sin \delta = \tan^2 90^\circ \sin \phi,$$ which determines $\delta$, and thence the time of the year when twilight is shortest. Also $$\sin WZ = \sin P Z \sin ZPW.$$} N 178 VARIOUS PROBLEMS CONNECTED [CHAP. XIV. or $$\sin \delta = \cos \phi \sin \frac{1}{2} ZPX,$$ determines $ZPX$, the duration of shortest twilight. **Azimuths.** 229. The determination of the sun's azimuth has always been of great importance at sea as a means of determining the error of the compass, but the introduction of iron ships has made it a problem of almost daily occurrence. These ships introduce new disturbing causes, and complicate the indications of the compass needle, by combining their own influence with that of the earth, in such a manner that the deviation of the compass not only changes as we pass from one place to another, and from one ship to another, but even in the same ship and at the same place the amount of deviation will often change by many degrees with the direction of the ship's head (see Raper's Navigation, and also Adams' Elements, by Com. F. J. Evans, R.N., F.R.S., and A. Smith, Esq., F.R.S.). If the sun's compass bearing be observed, and his true bearing calculated, the difference will be the correction or error of the compass. This error is the algebraical sum of the variation of the compass, or that part which is due to the earth's action, and of the deviation which is the part due to the accidental position of the compass with respect to neighbouring masses of iron, &c. 230. To find the azimuth of the sun at sun-rise or sun-set. In the triangle $SPR$ (fig. p. 171), right-angled at $R$, $SR = PR$ is the azimuth (Art. 13, note), $$\cos SP = \cos SR \cos PR,$$ or $\sin \delta = \cos A \cos \phi$; therefore $$\cos A = \frac{\sin \delta}{\cos \phi}$$ which determines $A$, the azimuth measured from the north in north latitudes, and from the south in south latitudes. CHAP. XIV.] WITH THE DIURNAL MOTION. 179 This is the azimuth at the true rising, that is, the azimuth which the sun has when his lower limb is about midway between his centre and the visible horizon (Art. 218). 231. To find the sun's azimuth by observation of his altitude. With the usual notation in the triangle $SPZ$, $$\cos SP = \cos ZC \cos ZP + \sin ZS \sin ZP \cos ZP,$$ $$\sin \delta = \cos x \cos \phi - \sin x \sin \phi \cos A,$$ $$\cos A = \frac{\sin \delta - \cos x \cos \phi}{\sin z \cos \phi},$$ or $$\sin^2 A = \sqrt{\left(\frac{\cos z (\phi + z)}{z} \sin (\phi + z - B)\right)^2 + \left(\frac{\cos z (\phi - z)}{z} \sin (\phi - z - B)\right)^2}.$$ 232. To find the sun's azimuth at a given time of a given day. The time reduced to apparent time determines the hour angle $SPZ$, or $h$, then $\cot PS \sin PZ = \cot PZ \sin SPZ + \cot PZ \cos SPZ$, whence $$\cot A = \tan h \cot x - \sin x \cot A,$$ which may be adapted to logarithms by assuming $$\tan x = \cot A \cos h,$$ then $$\cot A = \frac{\tan h}{\sin x}.$$ This method has the advantage of being free from errors of atmospheric refraction, and also of being available on many occasions when the altitudes cannot be observed on account of the indistinctness of the horizon. In actual practice at sea, it is sufficient to calculate to the nearest $\frac{1}{2}$ degree, because the azimuth by compass, with which we have to compare the calculated azimuth, can scarcely be ascertained to within a degree. On account of the importance of the problem, the author will probably be excused for referring here to his method 3 180 VARIOUS PROBLEMS CONSIDERED [CHAP. XIV. of solving it by means of a diagram, which he has called the Time-Azimuth-Diagram. This has been engraved by the Hydrographic Office, Admiralty, and is now used extensively at sea. It gives the azimuth—without calculation—to within one-eighth of a degree, in much less time than calculation would require, and with scarcely a possibility of error, as the operation is of the simplest character and has no variety of cases. 233. To find the azimuth and the hour angle of a star, when its motion is vertical, the declination being greater than the latitude of the observer. Since the declination is greater than the Latitude, the polar distance will be less than the co-latitude ; therefore the small circle described by the star will pass between the zenith and the pole. If, then, a vertical $ZSN$ be drawn to touch the diurnal circle of the star, the point of contact $S$ will be the position of the star at the moment when its motion is vertical. The triangle $ZSP$, right-angled at $S$, gives $$\sin P = \sin ZP \sin PZS,$$ or $\sin A = \cos \delta \sec \phi,$ $$\cos P = \cot PZ \tan PS,$$ or $\cos h = \cot \tan \phi,$ which determine the azimuth and the hour angle. There will be a similar position on the other side of the meridian. Sun-Dials. 234 The general explanation of the principle of dialing will be easily understood from the following construction, the idea of which is taken from Ferguson's Lectures, Edinb. 1806. A diagram showing a star (S) moving vertically above a meridian line (MN), with points labeled Z, P, S, and N. CHAP. XIV.] WITH THE DIURNAL MOTION. 181 Conceive a transparent cylinder, having an axis $AB$ parallel to the axis of the earth. On the surface of the cylinder let equidistant generating lines be traced 15° apart, one of them xii-xii being in the meridian plane through $AB$, and the others x-i, ii-ii, &c., following in the order of the sun's motion. Then the shadow of the line $AB$ will obviously fall on the line xii-xii at apparent noon, on the line x-i at one hour after noon, on ii-ii at two hours after noon, and so on. If now the cylinder be cut by any plane $MN$ representing the plane on which the dial is to be traced, this shadow of $AB$ will be intercepted by this plane, and fall on the lines Axii, Axi, Axi, &c. The construction of the dial consists in determining the angles made by $Ax$, Axi, &c. with Axii; the line Axii itself being in the vertical plane through $AB$ may be supposed known. Supposing a sphere to be described about $A$ as centre, there will always be sufficient data afforded by the position of the plane, and the latitude of the place, to enable us to solve the problem. A diagram showing a transparent cylinder with lines drawn around its surface. The lines are spaced 15° apart. One line (xii-xii) is shown falling on a plane (MN) that represents the plane on which a dial is to be traced. The shadow of another line (x-i) falls on a different plane. 182 PROBLEMS ON DIAL MOTION. [CHAP. XIV. Thus, if we wish to construct a dial at $B$ on the horizontal plane $PQ$. First, determine the xii o'clock line $BK$, and let a sphere described about $B$ cut the axis in $H$, and the $a$ o'clock line in $R$. The triangle $HKR$ is right-angled at $K$, $HK$ is the elevation of the pole or latitude of the place = $\phi$, $KHR$ (the angle between the planes through the axis of the cylinder and the lines xii-xii, n-n, respectively) = $n \cdot 15^\circ$, and $KR$ is the required angle $\theta$ between $Bxii$ and $Ba$, $$\sin HK = \tan KR \cot KHR$$; therefore $$\tan \theta = \sin \phi \tan n \cdot 15^\circ,$$ where, giving to $n$ the values 1, 2, 3, &c., in succession, we shall obtain the angles which the 1 o'clock, n o'clock, &c. lines make on the plane of the dial with the xii o'clock line. It scarcely requires to be stated, that the shadow is cast by a rod or style which occupies the position of the axis of the cylinder. For other positions of the plane, the investigation will proceed on similar principles, but there is no need to dwell on a problem which has lost nearly all its interest and value since the perfection and cheapness of clocks and watches have brought them into general use as measurers of time, A diagram showing a horizontal plane PQ with a sphere described around B cutting the axis at H and making an angle with a line RA at R. 183 CHAPTER XV. ASTRONOMICAL REFRACTION. § 235. We have, on several occasions in the foregoing chapters, had to refer to refraction ; and we have always supposed that our observations were freed from the errors arising from it, in those cases where it affected the observed values. We shall now proceed to examine what refraction is, and what corrections are necessary on account of it. It is a well-known principle of optics that a ray of light moves through a transparent medium in a straight line so long as the density of the medium remains uniform; but that in passing obliquely from one such medium to another its course will be bent at the point of incidence in such a manner as to satisfy the following conditions : the two directions before and after incidence will be in one plane with the normal to the surface at that point; the angles formed with the normal will have their sines in a constant ratio so long as the medium remains uniform. When the rays pass from vacuum into a medium, the constant ratio of the sines is called the coefficient of refraction for that medium ; and when the rays passes from one medium into another, it is easily shown that the sines of the angles will be inversely as the coefficients of refraction. Thus, if $A$ and $B$ be the two media, $\mu_1, \mu_2, \mu_3$-their coefficients of refraction, $\alpha, \beta, \gamma$-the angles made with the normal, then $$\frac{\sin \alpha}{\sin \beta} = \frac{\mu_1}{\mu_2}.$$ Now, the atmosphere which surrounds the earth may be considered as formed of a series of concentric layers, the 184 ASTRONOMICAL REFRACTION. [CHAP. XV. density being uniform throughout each layer, but diminishing rapidly as we recede from the surface. Any plane through the centre of the earth and through a star will intersect those layers in concentric circles, and radii drawn in this plane will all be normals to the surfaces of the layers. By the preceding laws, therefore, a ray $SPQR...$, which commences its path in this plane will, at any point of the increasing density near the surface, describe a broken line or curve concave to the centre of the earth. Let $A$ be the point where it meets the surface. To an observer there, the star will appear in the direction of the ray when it reaches the eye, that is, in the direction $AT$ of the tangent to the path at $A$. The Astronomical refraction is the angle between this apparent direction and that in which the star would be seen if there were no atmosphere. If $ST$, produced, meet the vertical $AZ$ in $M$, $SMZ$ will be the true zenith distance of the star, and $TAM$ the apparent zenith distance.* Refraction, therefore, diminishes the zenith distances of all celestial bodies, but does not affect their azimuths. General Differential Equation. \(\Delta 236.\) If $\phi$, $\phi'$ be the angles made by two consecutive elements $PQ$, $QR$ of the path with the normal $OQ$, whose * In strictness, we ought to join $A$ with the star, to obtain the true zenith distance; but, on account of the immense distance of all heavenly bodies, the line so drawn would be sensibly parallel to $MO$. CHAP. XV.] ASTRONOMICAL REFRACTION. 185 length, measured from the centre, is $x$, $\mu$, $\mu'$ the coefficients of refraction of the two strata, and $p$, $p'$ the perpendiculars from the centre on the two directions, then $$\mu : \mu' = \sin \phi : \sin \phi$$ $$: p : p' = p : x$$ therefore $$\mu p = \mu' p' = \mu' p'' = ... = constant,$$ a result which is independent of the thickness of the strata ; and, when the path becomes a curve, the perpendicular on the tangent at any point, multiplied by the coefficient of refraction at that point, is constant throughout the whole curve. This result may be written $$\mu \sin \phi = \mu_0 \sin z \ldots \ldots \ldots (A),$$ $\mu_0$, $a$, and $z$ being the values of $\mu$, $x$, and $\phi$ at the surface of the earth. Again, $\phi - \phi'$, the elementary deviation of the ray at $Q$, may be represented by $\delta r$, if $r$ be the whole refraction. Let $\mu' - \mu = 2\delta r$. Then $$\mu \sin \phi = (\mu + 2\delta r) \sin (\phi - 2\delta r)$$ $$= (\mu + 2\delta r) (\sin \phi - 2\delta r \cos \phi),$$ $$= 0.$$ $\sin \phi - \mu \cos \phi,$ $$\frac{dr}{dx} = \tan \phi,$$ $$\frac{d^2r}{dx^2} = -\frac{\mu}{\sqrt{(1-\mu^2)}}.$$ Eliminating $\phi$ between $(A)$ and $(B)$, we get $$\frac{d^2r}{dx^2} = -\frac{\mu}{\sqrt{(1-\mu^2)}}.$$ (C) the general differential equation of refraction. This equation, however, cannot be integrated, as we do not know the connection between $x$ and $\mu$. 4 237. All that can now be done, in order to complete the solution, is to assume hypothetically a relation between $\mu$ and $x$, then integrate the equation, and, having made a sufficient number of direct observations of refraction to determine the 186 ASTRONOMICAL REFRACTION. [CHAP. XV. constants involved, consider that hypothesis as most nearly representing the true state of the case which gives results most in accordance with all other observations. Bessel's hypothesis satisfies this condition, but the investigation is too intricate for an elementary work.* We shall consider Simpson's hypothesis, which is much simpler, and from it we shall deduce Bradley's formula, which gives very correct values so long as the zenith distance does not exceed 85°. Simpson's Formula. 238. Simpson assumed that some power of the coefficient of refraction varies inversely as the distance from the centre of the earth, or $$\mu^{-a} \cdot \frac{1}{x},$$ $$\mu^{-a} x = C,$$ where $a$ is a constant to be determined ; therefore $$\frac{n+1}{\mu} + \frac{1}{x} \frac{dx}{d\mu} = 0,$$ and from (4) $$\frac{1}{\mu} + \frac{1}{x} \frac{dx}{d\mu} + \frac{1}{\tan \phi} \frac{d\phi}{d\mu} = 0;$$ therefore $$\frac{1}{\mu} = \tan \phi \frac{d\phi}{dx},$$ whence, by (B), $$\frac{dr}{ds} = \frac{1}{n}.$$ To determine the limits of integration, we remark that the earth's atmosphere, at a few miles distance from the surface, becomes so rarified that its action on the path of the ray may be neglected, and we may consider it as bounded by a limiting sphere where the density is so small that the coefficient of refraction is 1. * We may refer the student to Charvot's *Astronomy*, vol. i., and to Brémont's *Physical Astronomy*. 238 CHAP. XV.] ASTRONOMICAL REFRACTION. 187 Let $\phi$ be the value of $\phi$ at the limiting sphere, and $x'$ the value of $x$; then $$x'\sin\psi=\mu_ksin x,\quad x'=a\mu_n^{-1},$$ whence $$\mu_n^{-1}\sin\psi=\sin x,$$ and $$r=\int_{+\infty}^{\frac{d}{n}}\frac{dx}{n}=\frac{x-\sqrt{\phi}}{n},$$ $$r=\frac{z-\sin^{-1}\left(\frac{\sin x}{\mu_n}\right)}{n};$$ this is Simpson's formula of refraction. Bradley's Formula. 239. From Simpson's formula we get $$\frac{\sin x}{\sin(x-nr)}=\mu_n^{-1};$$ therefore $$\frac{\sin z-\sin(z-nr)}{\sin z+\sin(z-nr)}=\frac{\mu_n^{-1}-1}{\mu_n^{-1}+1},$$ $$tan\frac{nr}{2}=\frac{\mu_n^{-1}-1}{\mu_n^{-1}+1}\tan\left(\frac{z-nr}{2}\right),$$ or, approximately, $$r=\frac{\mu_n^{-1}-1}{\mu_n^{-1}+1}\tan\left(\frac{z-nr}{2}\right),$$ $$r=z\tan\left(\frac{z-nr}{2}\right),$$ which is Bradley's formula. The numerical values given by him were $$r=57^\circ036' \tan (z-3r'),$$ corresponding to a mean state of the atmosphere, when the barometer stands at 29.6 inches, and the thermometer at 50° Fahr. For zenith distances not exceeding 45°, we have very approximately $$r=57^\circ036' \tan (z-3r').$$ 188 ASTRONOMICAL REFRACTION. [CHAP. XV. Cassini's Formula. § 240. A formula of refraction based on the supposition of a homogeneous atmosphere was obtained by Dominique Cassini. Although not representing the actual state of nature, it gives tolerably satisfactory results for zenith distances not exceeding $80^\circ$, and is interesting as being the first attempt to determine refractions theoretically. Cassini assumed that if the atmosphere were replaced by one of the same uniform density as at the surface of the earth, and of such a definite height as to produce the same pressure at the surface, the refractions would be approximately the same. On this supposition the ray is bent only at its entrance into the spherical shell, and the formula is easily obtained. Let $SPA$ be the ray bent at $P$, $OA = a$ the earth's radius, $OP = a(1 + n)$ where $n$ is small, $\phi' + r = SPX$ the angle of incidence, $\phi' = OPA$, $r = PAZ$ the apparent zenith distance. By the laws of refraction $$\sin(\phi' + r) = \mu \sin \phi',$$ or, $r$ being small, $$\sin \phi' + r \cos \phi' = \mu \sin \phi';$$ therefore $$r = (\mu - 1) \tan \phi';$$ and, from the triangle $OAP$, $$\sin \phi' = \frac{\sin x}{1 + n};$$ therefore $$r = (\mu - 1) \frac{\sin x}{\sqrt{(1 + n)^2 - \sin^2 x}}.$$ CHAP. XV.] ASTRONOMICAL REFRACTION. 189 $$r = \frac{(\mu - 1) \sin z}{\sqrt{\cos^2 x + 2n}} \text{ neglecting } n^4,$$ $$r = \frac{(\mu - 1) \tan z}{\sqrt{(1 + 2n \sec^2 z)}},$$ $$r = (\mu - 1) \tan z (1 - n \sec^2 z).$$ § 241. The preceding results have all been obtained without taking into account the changes in the state of the atmosphere, and the refraction has been considered as dependent only on the apparent zenith distance ; whereas the density of the air, which is continually changing, must have considerable influence. The values given by the formulae must be considered as applying to some mean state of the atmosphere, and we must make corrections for alterations in the height of the thermometer and barometer, by assuming that the refraction varies as the density. Let $r$ be the tabulated refraction corresponding to a zenith distance $x$, the barometer standing at some definite height $h$, the temperature of the mercury as shown by the attached thermometer being $t$, and the temperature of the air given by an external thermometer $T'$. Suppose that the refraction $r'$ is required when the same zenith distance $x$ preserves the values of the quantities being respectively $k$, $t$, $T'$. Let $d$ and $p$ be the density and pressure of the air for the tabulated values $k$, $t$, $T'$. Then $$r' : r : d' : d : p' : p : p' : p : k' : k : h' : h : t' : t : T' : T'$$ $$d' : d : 1 + ET' : 1 + ET'$$ (Besant's Elementary Hydrostatics) $$p' : p : 1 + et' : 1 + et'$$ therefore $$r' : r : i : (1 + ET') (1 + et') : (1 + ET') (1 + et')$$ 100 ASTRONOMICAL REFRACTION. [CHAP. XV. and $$r' = r \frac{h'}{h} [1 - E(T'-T) - e (t'-t)],$$ where $E$ is the coeff. of expansion of air for 1° Fahr. =-000356, $$e \quad mercury \quad =-0001001.$$ Coefficient of Refraction Determined by observation of a Circumpolar Star. \textsection 242. We shall consider that the refraction $r$ is of the form $ax$, $a$ being a known function of the observed zenith distance, and of the barometer and thermometer readings, and $a$ the constant coefficient of refraction. Observe a circumpolar star at its upper and lower transits ; and let $x$, $x'$ be the two zenith distances obtained, $c$ the co-latitude of the place. Then $$c = \frac{1}{2} [(z + z') + (z' + r')]$$ $$= \frac{1}{2} [(z + z') + a(u + u')]$$ A second circumpolar star will, in the same manner, give $$c = \frac{1}{2} [(z_2 + z'_2) + a(u_2 + u'_2)]$$ therefore $$x + x' + a(u + u') = z_1 + z'_1 + a(u_1 + u'_1),$$ $$a = \frac{(z_1 + z'_1) - (x + x')}{(u + u') - (u_1 + u'_1)}.$$ Terrestrial Refraction. \textsection 243. Let $A$, $B$ be two places on the surface of the earth, each visible from the other, and let $APB$ be the curved path of the ray which connects them, $AB$ the chord. Then if $AT$, $BT$ be the tangents to the path at $A$ and $B$, and $AZ$, $BZ'$ be verticals, the apparent zenith distances of the two places as seen one from the other will be A diagram showing a circumpolar star at different positions on the celestial sphere. CHAP. XV.] TERRESTRIAL REFRACTION. 191 $ZAT = z$, and $ZBT = z'$; and the refractions, due to the atmosphere, will be $TAB = r$, and $TBA = r'$. The arc $AB$ being a small portion of a curve of finite curvature, the angles $r$ and $r'$ are approximately equal; therefore, $$2r + x + z' = 180^\circ + C,$$ $$r = \frac{180^\circ + C - x - z'}{2} \ldots \ldots \ldots (i),$$ which determines $r$; $x$ and $z'$ being known by observation, and $C$ by measurement of the distance $AB$ and the known radius of the earth. According to Biot, $$r = aC,$$ where $a$ depends on the state of the atmosphere; therefore $$a = \frac{180^\circ + C - x - z'}{2C} \ldots \ldots \ldots (ii).$$ A large number of experiments has given 0.078 for the mean value of the coefficient $a$. The extreme values being 0.05 in summer and 0.13 in winter. J 244. In Art. 8 (note), it was stated that refraction diminished the dip, and increased the distance, of the visible horizon. Let $A$ be the observer at a height $AH = h$ above the sea level, $AD$ the curved ray by which the horizon is seen at $B$, $C$ the corresponding angle at the earth's centre, $r = aC$ the refraction at $B$ and $A$. Draw the straight line $BA$. Then $$ZBA = 90^\circ + r;$$ therefore $$ZAB = 90^\circ - r + C.$$ 192 **TERRESTRIAL REFRACTION.** [CHAP. XV.] and, calling $a$ the radius of the earth, $$\frac{a + h}{a} = \frac{\sin(30^\circ - r)}{\sin(90^\circ - r) - C} = \frac{\cos r}{\cos(C - r)},$$ $$\frac{h}{a} = \frac{\cos r - \cos(C - r)}{\cos(C - r)} = \frac{2 \sin^2 C \sin(\frac{1}{2}C - r)}{\cos(C - r)},$$ and $C$ and $r$ being small angles, $$\frac{2a}{a} = \frac{C(1 - 2r)}{1 - 2r(C - r)} = C(1 - \frac{2r}{C}),$$ $$\sqrt{\left(\frac{2h}{a}\right)} = C(1 - \frac{r}{C}) = C(1 - a).$$ If $C_1$ be the value of $C$, supposing no refraction, $$C_1 = \sqrt{\left(\frac{2h}{a}\right)};$$ therefore $C = C_1(1 + a) = C_1(1 + 0.76) = C_1 + \frac{76}{100}C_1$ approximately. 245. Let $D_1 D_2$ be the corresponding values of the depression : we find, $AT$ being the tangent to the visual ray at $A$, $$D = ZAT - 90^\circ = ZAB - r - 90^\circ = C - 2r,$$ $$= C(1 - 2a);$$ but $$D_1 = C(1 - a);$$ therefore $$D_1 = \frac{D_1}{1-a} = D_1(1-a) = D_1 - \frac{D_1}{1-a}.$$ There are, however, considerable irregularities in these values from day to day ; and errors in the altitude of the sun or of a star, at sea, may frequently be traced to an unknown, and therefore uncorrected, change in the dip of the sea-horizon from which the altitudes are measured. **Other Effects of Refraction.** > 246. If we examine a table of refractions, we shall find that with the barometer at 30 inches, and Fahrenheit's thermometer at 50°— A diagram showing the relationship between the angle of elevation (θ), the angle of depression (φ), and the refractive index (n). CHAP. XV.] **ASTRONOMICAL REFRACTION.** 193 At apparent zen. dist. 45° the refraction is 0° 58' 2", 80° 5° 19' 2", 85° 9° 52', 88° 18° 26", 90° 36° 29". The change is very rapid near the horizon, and a consequent contraction of the vertical diameters of the sun and moon takes place giving them a sensibly oval shape just after rising or before setting, the lower half being somewhat more flattened than the upper half. For example : suppose the true altitude of the sun's lower limb to be 5° and his diameter 32', we find

True altitude lower limb upper limb
Refraction 9° 52' 8° 52"
Apparent alt. 5° 9° 52' 5° 40° 52"
the difference of which gives an apparent vertical diameter of 31', or a contraction of 1'. When nearer the horizon 'the contraction may extend to 5° or 6'. § 247. To find the contraction, produced by refraction, of a semi-diameter which makes an apparent angle $\theta$ with the vertical. Let $ACA'$ be the horizontal diameter of the oval disk, $CBH$ the vertical one, $CP$ the semi-diameter making an angle $\theta$ with $CB$, $PM$ the ordinate of $P$. The contractions of the ordinates are approximately proportional to their magnitudes. Produce $CB$, $CP$, $MP$ to meet the auxiliary circle in $D$, $Q$, and $R$, $$PR : BD :: MP : CB,$$ :: $MP : CP$ approximately; o A diagram showing a circle with points labeled A, B, C, D, E, F, G, H, P, Q, R, M, N, O, P', Q', R', S'. The line AB is horizontal. Line AC is inclined at an angle θ with AB. Line CP is perpendicular to AC. Line PM is parallel to AC and intersects AB at point P. Line CBH is vertical. Line DQ is parallel to AB and intersects AC at point D. Line QR is parallel to AB and intersects AC at point Q. Line RS is parallel to AB and intersects AC at point R. Point P' is on line CP such that PP' = PC/2. Point Q' is on line QR such that QQ' = QR/2. Point R' is on line RS such that RR' = RS/2. Points M and N are on line PM such that MN = PM/2. Points O and P are on line OP such that OP = OP'/2. Points Q and Q' are on line OQ such that OQ = OQ'/2. Points R and R' are on line OR such that OR = OR'/2. 194 ASTRONOMICAL REFRACTION. [CHAP. XV. therefore $$PR = BD \cos \theta.$$ Again, the small triangle $PQR$ right-angled at $Q$ gives $$PQ = PR \cos \theta,$$ therefore $$PQ = BD \cos \theta,$$ or, the contraction of any semi-diameter is equal to that of the vertical semi-diameter multiplied by the square of the cosine of the included angle. 248. The horizontal diameter itself will be slightly diminished by refraction. This is obvious, without special investigation, seeing that the extremities of the horizontal diameter are equally raised by refraction, each in its own vertical, and these verticals meet in the zenith—therefore the breadth must contract. The contraction is nearly constant for all altitudes of the sun or moon, and is about 0°5. Effect on the Rising and Setting of Heavenly Bodies. 249. Another effect of refraction is to accelerate the rising, and to retard the setting, of all bodies. The exact amount of this effect may be calculated by solving the usual spherical triangle $SPZ$, where $SZ = 90^\circ + r$, and comparing the hour angle with that obtained when $SZ = 90^\circ$. In the same way, the azimuth at apparent rising and setting will place the body more to the north (in our hemisphere) than if there were no atmosphere; but the great irregularities in the values of $r$, for these low latitudes, render the investigation of little practical value. ( 195 ) CHAPTER XVI. FIGURE OF THE EARTH PARALLAX. § 250. The determination of the exact figure of the earth is a problem of considerable difficulty, but of great importance. As stated in Art. 27, if a small arc of a meridian be measured on the surface, and also the difference of latitude of the two extremities of the arc, or, which is the same thing, the change in the meridian zenith distance of a star, we shall thus have an arc and the angle it subtends at the centre of the earth, whence the radius may be found. This, however, supposes the earth to be spherical ; and a more correct statement would be, that the value obtained is the radius of curvature of the meridian at the middle point of the small arc. Now, by actual measurement of such arcs in different latitudes, it has been found that the radius of curvature increases, or, in other words, that the curvature diminishes, and the earth becomes more flattened, as we approach the poles. The figure of the earth is found to be very approximately an oblate spheroid formed by the revolution, about its minor axis, of an ellipse whose semi-axes are respectively $$a = 3962.8 \text{ miles},$$ $$b = 3942.6 \text{ miles},$$ the axis of figure coinciding with the polar axis of the earth. These values of $a$ and $b$ are mean values : for, different meridians present slight discordances,—so slight, however, that we may here neglect them. For an account of the o 2 106 **FIGURE OF THE EARTH.** [CHAP. XVI.] notices and precautions required in the measurement of the arcs and angles, and for an explanation of other methods which have been employed in the determination of the figure of the earth, we must refer to works on geodesy. \> 251. The fraction $\frac{a-b}{a}$ is called the **compression.** If we represent it by $c$, and the eccentricity by $e$, we shall have $$\frac{b}{a} = 1 - c,$$ $$e^2 = 1 - \left(\frac{b}{a}\right)^2 = 2c - e^2.$$ The value of $c$ is $\frac{3}{20}$ nearly, and $e = 0816$. \> 252. Let $BOA$ be the meridian of a place $O$ on the surface of the earth, $BCB'$ the polar axis, $CA$ the equatorial radius, and $OG$ the normal at $O$. Then (Art. 29), $OGA$ is the geographical latitude, or $\phi$; $OCA$ is the geocentric latitude, or $\phi_1$; and $COG$ is the reduction of the latitude = $\phi - \phi_1$. If $x$, $y$ be the coordinates of $O$, $$\tan \phi' = \frac{y}{x}, \quad \tan \phi = \frac{y}{x(1-e^2)};$$ therefore $$\tan \phi' = (1-e^2) \tan \phi = \frac{y}{x(1-e^2)},$$ is the connection between the geocentric and the geographical latitudes; $$\tan (\phi - \phi') = \frac{\tan \phi - (1-e^2) \tan \phi}{1 + (1-e^2) \tan \phi} = e^2 \sin \phi \cos \phi,$$ or, approximately, $$\phi - \phi' = c \sin 2\phi,$$ which gives the reduction. CHAP. XVI.] PARALLAX. 197 \sqrt{253}. We shall also determine the distance $CO$. From above $$y = \frac{b}{a} \tan \phi,$$ $$\frac{y}{b \sin \phi} = \frac{\frac{c}{a}}{a \cos \phi},$$ therefore $$y' = \frac{b \sin \phi}{a^2 \cos^2 \phi + b^2 \sin^2 \phi} = \frac{a^2 \cos^2 \phi}{a^2 \cos^2 \phi + b^2 \sin^2 \phi},$$ therefore $$CO' = a^2 \cos^2 \phi + b^2 \sin^2 \phi,$$ whence $$CO = a (1 - c \sin^2 \phi),$$ approximately. Parallax. \sqrt{254}. There are some bodies so remote from us that, whether seen from the centre or from any point on the surface of the earth, the two directions will be so nearly parallel that no instrument we at present possess can measure, or even detect, their inclination. There are others within, what may be termed, a measurable distance, in which the difference of direction is, although small, a quantity which may be determined. It is therefore essential, in order to make the registered right ascensions, declinations, &c., available to all persons, that they should be reduced to some definite point ; and it will obviously be advantageous to select the centre of the earth as such point of reference, because the apparent motions of those bodies, as seen from the centre, are of a much more simple character than as seen from any point of the surface, and also because we can much more readily reduce the observations from any point to the centre than to another point. The declinations, &c., of all bodies A diagram showing the relationship between the distances and angles involved in parallax calculations. 198 PARALLAX, [CHAP. XVL registered in the Nautical Almanac, are those they would have as seen from the centre of the earth ; and when we make an observation at any place we must, when there is need, know how to transform it to what it would be at the centre. This is called the correction for parallax. The parallax of a heavenly body is the angle between the directions of two lines drawn to it, one from the observer, the other from the centre of the earth. Thus, if $C$ and $O$ be respectively the centre and the observer, $S$ a distant object, the angle $CSO$ is the parallax of $S$. \footnote{255. We shall first consider the earth as a sphere; then $CO$ produced will pass through the zenith of $O$, and the effect of parallax will be wholly in the vertical plane $ZOS$; it will change the semi-diameter from $ZCS$ to $ZOS$, the difference between these being the parallax itself $CSO$.} If $a$ be the radius of the earth, $D$ the distance $ZO$, $x$ the zenith distance $ZOS$, $p$ the parallax $CSO$, $$\sin p : \sin x :: a : D,$$ $$\sin p = \frac{a}{D} \sin x.$$ When $x=90^\circ$, $p$ becomes the horizontal parallax; let us represent it by $\Pi$, then $$\sin \Pi = \frac{a}{D},$$ therefore $$\sin p = \sin \Pi \sin x.$$ CHAP. XVL.] PARALLAX. 109 Except in the case of the moon, whose parallax sometimes exceeds $1^{\circ}$, we may substitute the angles for their sines and write $$p = \Pi \sin z.$$ \footnote{256. The effect of parallax on any coordinate of the body's position is generally spoken of as parallax in that coordinate, thus ---parallax in declination, parallax in hour angle, &c. The foregoing article shows that parallax in altitude or zenith distance is the whole parallax itself, and that parallax in azimuth is zero.} To find the Parallax in Declination and Hour-angle. Let $P$, $Z$, $S'$ be the pole, the zenith, and the body, on the celestial sphere of the observer at the surface of the earth. Suppose the observer transferred to the centre, and let $S$ be the new position of the body in the vertical $ZS$, then $$SS' = \Pi \sin ZS;$$ and if $\alpha$ and $\beta$ be the hour-angle and declination, referred to the centre; $h + \alpha$, $\delta - \beta$ the same, referred to the surface; draw the arc $SR$ perpendicular to $PS'$, then $$SP'S' = a, SR = B.$$ $$a = \frac{SR}{\sin PS'} = \frac{SS'\sin SSR}{\sin PS} = \frac{\Pi \sin ZS \sin SSR}{\sin PS},$$ $$a = \frac{\Pi \cos ZP \sin ZPS}{\sin PS} = \frac{\Pi \cos \phi \sin (A + \alpha)}{\cos \delta},$$ $\phi$ being the latitude of the observer. A diagram showing a celestial sphere with a body at point S', a line SS' connecting S' to S, and lines SP' and PS'. A triangle SP'S' is formed by these lines. 200 PARALLAX. [CHAP. XV.] Again, in $SP$, produced if necessary, take $SQ = 90^\circ$; therefore $PQ = \delta - \beta$, then $$\beta = SR = SS \cos ZS \cos ZS Q = \Pi \sin ZS \cos ZS Q$$ $$= \Pi (\cos ZP \cos PQ + \sin ZP \sin PQ \cos ZPQ)$$ $$= \Pi [\sin (\delta - \beta) - \cos \phi \sin (\beta - B) \cos (\delta + a)].$$ $\text{Art.}~257$. When the spheroidal form of the earth is taken into account, the problem is only slightly altered. When the observer is transferred to the centre the apparent displacement $SS$ of the body will no longer take place towards the geographical zenith $Z$, but towards the geocentric zenith $Z'$ where $ZZ' = \phi - \phi'$ the difference between the geocentric and geographical latitudes (Art. 258). The value of $SS$ will be given by $$\sin SS = \frac{CO}{D} \sin Z'S,$$ as in Art. 255, $$= \frac{a}{D} (1 - c \sin^2 \phi) \sin Z'S$$ (Art. 253), since $a$ is the equatorial radius, $\frac{a}{D}$ is the sine of the equatorial horizontal parallax, which is the element registered in the Nautical Almanac, and the parallax $\Pi$ in the geocentric horizon of the place will be given by $$\sin \Pi = \frac{a}{D} (1 - c \sin^2 \phi).$$ The figure shews that, except when the body is in the meridian, there will be parallax in azimuth as well as in altitude; and a comparison with the figure in the previous article shews that these parallaxes may be determined. CHAP. XVL.] PARALLAX. 201 by means of the formula which give $\alpha$ and $\beta$ in that article, if we use $90^\circ - ZZ'$ for $\phi$ and the observed azimuth for $h + \alpha$. And again, the parallaxes in hour-angle and in declination will be given by the same formula, if we use the geocentric instead of the geographical latitude. \(\ddagger 258.\) The formula $\sin (\text{equat. horiz. par.}) = \frac{\alpha}{D}$, in which $a$ is the known equatorial radius of the earth, shows that the determination of the parallax of a body is the same problem as the determination of its distance. For all bodies, except the moon, the distance $D$ is very great as compared with $a$, and the parallax consequently very small ; so that all the refinements of which modern astronomy is capable are necessary to determine it by observation. So much is this the case that the parallax of the sun, which was supposed to have been determined with great accuracy by the transit of Venus in 1769, has recently been found to be in error by about $\frac{1}{30}$th of itself. \(\ddagger 259.\) To determine the parallax of a heavenly body by meridian observations. Let $A$, $B$ be two stations in opposite hemispheres, on the same meridian but in widely different latitudes, and $S$ the body whose parallax is required. We must first remark that in the figure, for the sake of distinction, the earth is immensely exaggerated in comparison with the distance of $S$, so that the lines $AB$, $BE$ are nearly parallel, and the apparently large angle $ASB$ is really so small as not exceeding $1^{\circ}$, except in the case of the moon where it may reach from $1^{\circ}$ to $2^{\circ}$. The small angle $ASB$ is obviously the change in the declination of $S$, as seen from $A$ and from $B$. We have to find its value with the greatest accuracy, but we cannot do this by a comparison of the absolute declinations deduced 202 PARALLAX. [CHAP. XV.] from observed zenith distances, because the observed directions of $S$ will be affected by refraction, and the correction from the refraction tables will always be somewhat uncertain. To avoid this source of error and also any errors of the divided circle, a star is selected whose declination is so nearly the same as that of $S$, that both may pass through the field of view of the mural or transit circle in a fixed position of the instrument, and the difference of declination may then be measured by means of the micrometer. The very small difference of refraction due to the slight difference of zenith distance of the two bodies may be accurately found and allowed for; and thus, both $S$ and the star will be affected with the same errors with the exception of parallax which does not affect the star, and the angle obtained will give true difference of their declinations as seen from that station. If at the other station similar observations be made, and the position of $S$ be compared with the same star on the same day and therefore at the same time, the difference of their declinations as seen from that station will also be found, and the change in that difference will be the angle $ASB$. Let $\phi$ be the value of $ASB$ thus obtained, $\phi$, $\phi'$, their geographical latitudes of $A$ and $B$, $r$, $r'$, their geocentric latitudes, $a$ the equatorial radius, $c$ the compression, $D$ the distance $CS$, $I_1$, the equatorial horizontal parallax, $x$, the observed zen. dist. $ZAS$, $ZBS$. A diagram showing a celestial sphere with a star labeled "S" and another star labeled "A". The line connecting "S" and "A" is labeled "AB". The diagram also shows lines from "S" to points on a plane parallel to the horizon at different distances, labeled "P", "Q", etc., representing different positions of "S" on its path across the sky. CHAP. XVL.] PARALLAX. 203 Produce $CA$, $CB$ to $V$, $V_1$, then $$\sin ASC = \frac{a}{D} \sin VAS = \frac{a}{D} (1 - c \sin^2 \phi) \sin VAS$$ $$= \sin \Pi_1 (1 - c \sin^2 \phi) \sin (x - \theta + \phi) \ldots (\text{[i]})$$ $$\sin BSC = \sin \Pi_2 (1 - c \sin^2 \phi) \sin (x - \theta + \phi) \ldots (\text{[ii]})$$ and $$ASC + BSC = m \ldots (\text{[iii]}).$$ Eliminating $ASC$ and $BSC$ between these three equations, the value of $\Pi_1$ will be obtained in terms of known quantities. Except in the case of the moon, we may replace the sines of the small angles by the angles themselves, $$\Pi_1 = \frac{(1 - c \sin^2 \phi) \sin (x - \theta + \phi)}{(1 - c \sin^2 \phi) \sin (x - \theta + \phi)}. m$$ 260. We have taken the two places $A$ and $B$ on the same meridian, but this condition could with difficulty be secured, and is moreover not essential. For, by taking account of the change of declination of $S$ during the interval between its transits over the two meridians, we can, by simply adding or subtracting this change, reduce the difference of declination between $S$ and the star, observed at either place, to what it would have been had the meridians coincided.* 261. By this method the parallax of the moon has been determined, and recently that of the planet Mars when in opposition. The sun and the other planets are too far away to allow us to apply the method directly to them, but by Kepler's third law (Art. 189), if the distance of one planet from the earth be known, that of the sun and of the other planets may be inferred. The planet Mars was in a favourable position for the determination of its parallax in 1862, and the result of the observations then made necessitated a change in the previously *For further development of this and other methods, we shall refer to Charitonius' *Astronomy*, vol. i., the Rev. R. Main's *Practical and Ephemerid Astronomy*, and Woodhouse's *Astronomy*. A diagram showing angles and lines related to parallax calculations. 204 PARALLAX. [CHAP. XVI.] received value of the sun's parallax, altering it from 8°57 to 8°93. The value 8°57 was founded by the transit of Venus in 1769, and any error must be attributed to the imperfection of the instruments, &c., then in use ; for, the method is so much better adapted than any other to the calculation of the sun's parallax, than the determination by observations of Mars would not have led to the rejection of the former value, if the new one had not been confirmed in a remarkable manner by independent observations made on the velocity of light. The transits of Venus, which took place in 1874 and 1882, afforded opportunities of further verifying the correctness of this new value, and the results then obtained have shown it to be somewhat too great ; but the discrepancies between these two values from the various observations of those two transits make astrophysicists hesitate to name any particular value as undoubtedly correct. It is clear, however, that the true value cannot be very far from 8°80 ; and this is taken by many astronomers as a convenient round number in calculating the various constants whose values depend upon that of the sun's parallax. We shall, in a future chapter, give a brief sketch of the method of determining the sun's parallax by the transit of Venus.* Distances of the Sun and Moon. \textit{262.} The parallax being known, we may deduce the distance by the formula (Art. 258), $$\frac{\text{distance}}{\sin(\text{hor. par.)}} = \frac{\text{radius of earth}}{\text{radius of earth}}$$ Sun's distance = $$\frac{206265}{\sin(8°)} = \frac{206265}{8°} (\text{radins})$$ = $$23430 (\text{earth's radius})$$ = $$92884000 \text{ miles}.$$ * See an article on "Celestial Measurings," by Sir John Herschel, in Good Words for June, 1864. CHAP. XVI.] PARALLAX. 205 In the case of the moon whose mean parallax is 57' 1'-8, mean distance = 60 (earth's equatorial radius), = 237800 miles. The moon's distance from the earth is therefore only $\frac{1}{40}$ of the sun's distance. Magnitudes of the Sun and Moon. -263. If we measure the angular semi-diameter of a heavenly body whose parallax is known, we can determine its magnitude, $$\sin(\text{semi-diam.}) = \frac{\text{radius of body}}{\text{distance}}$$ $$\sin(\text{parallax}) = \frac{\text{radius of earth}}{\text{distance}}$$ therefore $$\sin(\text{semi-diam.}) = \sin(\text{parallax}) \cdot \frac{\text{radius of earth}}{\text{radius of body}}$$ In the case of the sun, semi-diameter = 16', parallax = 8° 30', radius of sun = 109 (earth's radius) = 432000 miles. In the case of the moon, semi-diameter = 15° 39' 9", parallax 57' 1'-8', radius of moon = $\phi_0$ (earth's radius) = 1690 miles. Since the moon's distance from the earth is only sixty times the earth's radius, we see that the magnitude of the sun is such that, if it were concentric with the earth, it would include the moon's orbit and extend nearly as far again beyond.* Distances of the Stars. Annual parallax. † 264. The stars are too far off to allow of any measurement of their distances being made by the foregoing means. * The mass of the sun is determined as follows: If $a$ be the radius of the earth, $E$ its mass, and $g$ the acceleration of gravity at the surface; also, if $D$ be the distance of the sun, then $E$ must be equal to $F D^2 g$, where $F$ is Newton's constant, Sect. iii.1, therefore $$g = \frac{E}{a^2} F D^2 g$$ $$\Rightarrow E = \frac{g a^2}{F D^2}$$ whence $S$ may be expressed in terms of $E$, and is found about 929000E. The mass of the moon is obtained by its effect on the earth's nutation, and is estimated at about 54000E. 206 PARALLAX. [CHA.P. XVL] But we have now got the distance of the sun, and although a globe of 8000 miles diameter subtends no appreciable angle at the nearest star, we may expect that a displacement of 186,000,000 miles—which, being the breadth of the earth's orbit, is the distance that separates two positions of the observer at six months' interval—will be sufficient to cause a measurable change in the apparent direction of some of the stars. The maximum angle which the radius of the earth's orbit subtends at any star is called the annual parallax of that star. § 265. The annual parallax of a star being given, to determine the parallax in longitude and latitude, at a given time, that is, the difference between these coordinates, as observed from the earth, and as they would be, if observed from the sun. Let $E$ be the earth, $S$ the sun, $s$ a star whose annual parallax $\pi$ is known. About $E$ describe the observer's celestial sphere (Art. 11). Let $YMO$ be the ecliptic, $K$ its pole, $O$ and $o'$ the places of the sun and star. Draw $Es$ parallel to $Sa$, and therefore meeting the sphere in the great circle through $O$ and $o'$. Draw also the quadrants $KeM$, $Ke'M'$. Then $o'a'$ is the displacement of the star owing to parallax. $MM'$ will be its effect on the longitude, and $Mr \sim M'o'$, on the latitude of the star. A diagram showing a celestial sphere with a line drawn from the center to a point labeled "s" representing a star. Another line extends from this point to another point labeled "o'" representing the observer's position. A third line extends from "s" to "o'" and intersects a line drawn from "s" to "a", which represents the sun. The diagram illustrates how to calculate parallax based on given data. CHAP. XVI.] PARALLAX. 207 Let $$SE = r,$$ and $$Ss = \Delta,$$ $$\sin \sigma = \frac{r}{\Delta} \sin SEs,$$ or $$\sigma' = p \sin \sigma O,$$ where $$p,$$ or $$\frac{r}{\Delta},$$ is the annual parallax, i.e., the maximum value of $$\sigma'$$, corresponding to that position of $$E$$ where $$SEs$$ is a right angle. From this we obtain $$MM' = \cos M' - p \sin \sigma' \cos M',$$ since $$\sigma' = \sigma,$$ very nearly, $$= p \sin M' - p \sin (\gamma' - \vartheta'M'),$$ therefore parallax in longitude = $$\frac{p \sin (\gamma' - \vartheta'M')}{\cos M'}$$ (i), where $$\odot$$ is the longitude of the sun, and $\bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \\ Again, parallax in latitude\\ = -\sigma' + p' cos'\\ = -p' sin' O cos'\\ = -p' sin'M' cos'MO\\ = -p' sin(lat. of star) cos(\odot - e),$$ (ii). Equations (i) and (ii) show that, owing to annual parallax, the displacement of a star from a mean position will extend from $$-p + p'$$ in a direction parallel to the ecliptic, and from $$-p sin(lat.) + p sin(lat.)$$ in a perpendicular direction; the apparent path of the star, during the sidereal year, being a small ellipse which has the above displacements for major and minor axes respectively. 206. Such is the immense distance of the stars that, out of a large number that have been attentively examined, only a very few show any sensible parallax. For the nearest of these, the annual parallax $$p$$ is less than 1"; and 208 PARALLAX. [CHAP. XVI] the determination of such a minute quantity requires the best instruments and the greatest precision. On account of the interest of the subject, we shall here briefly describe the two methods by which this measurement has been effected. To determine the annual parallax of a fixed star by observation. **First Method.** Let the star's longitude be observed when that of the sun differs by about 90° from it; and let the observa- tion be repeated six months later, when, the earth having moved to the opposite side of its orbit, the sun's longitude again differs by 90° from that of the star. Equation (i) shows that the parallax in longitude at the two observations will be respectively $$p \sec (\text{star's lat.}) - p \sec (\text{star's lat.})$$ Therefore, if the two observed longitudes of the star differ by $2\delta$, we shall have $$2\delta = 2p \sec (\text{star's lat.})$$ therefore $$p = 2 \cos (\text{star's lat.})$$ As the accuracy of $\delta$ depends on the accuracy with which the longitude is determined at each observation, and this again depends on the perfection of the instruments, on their stability, and on the care with which all corrections are made and all sources of error guarded against or taken into account, it follows that only very imperfect observa- tions of the most delicate kind that a star can be asserted to have, or not to have, a measurable parallax. The star α Centauri, a bright star in the southern hemi- sphere, was, by direct observations made at the Cape of Good Hope during the years 1832 (by Mr. Henderson) and 1839 (by Sir T. Maclear), found to have a parallax which was estimated at 0°'98'. CHAP. XVL] PARALLAX. 209 § 267. A Second Method, due to Bessel, is free from many of the difficulties inherent to the first :--If two neighbouring stars differ very much in brightness, we may presume that they are at very different distances from us ; and if we assume the smaller star to be so remote as to have no appreciable parallax, we may attribute any observed changes in the angular distance between them to the parallax of the nearer star. If these stars change go through a yearly cycle according to the parallax of each, and from these changes, we may obtain the parallax itself. The comparison star must be so situated that both may be seen at the same time in the field of view, and the angular distance between them may then be measured with very great accuracy by using the double image micrometer (Art. 112). The great advantage of this method over the former is the fact of its eliminating all uncertainty of refraction, all errors from a want of stability of the instrument, and its avoiding a number of corrections which have to be made for the determination of the absolute longitude.* Bessel applied this method with success to the star 61 Cygni, not a very bright star, being only of the fifth magnitude, but having near it, at distances of about 8' and 12" respectively, two much smaller stars between the ninth and tenth magnitudes. These were well situated also as to direction, being from 61 Cygni nearly at right angles to one another. Being found to have the same parallax from each of those, and gave as the result of their observations o'-35. Messers. Anwers and Struve, with more perfect instruments, have repeated the observations, and the mean of their results, which agree very nearly, gives o'-04. This is unquestionably the best determined star parallax. It corresponds to a distance 382,000 times that of the sun. * For a full investigation of the method we may refer to Chauvenet's *Astro- logy*, vol. i. p. 693. P 210 PARALLAX. [CHAP. XVL] § 368. The following are some of the stars whose parallaxes have been determined by one or other of these two methods. These are our nearest neighbours, and to convey a better idea of the enormous distances which separate us from them, we have given the time that their light takes to travel to us; it being known that light travels at the rate of 184,000 miles in a second, and takes $8^{\circ} 19'$ to come from the sun to the earth (see Chap. XVII).
Parallax. Time the light takes to reach the earth.
a Centauri 0°'98 210,000 3-315 years.
b Cygni 0°'54 382,000 6-029
a Lyre 0°'26 793,000 12-518
Sirius 0°'15 1,375,000 21-706
Arcurtus 0°'127 1,624,000 25-637
Polaris 0°'106 1,946,000 30-720
Secular Parallax. § 369. Many of the stars are found to have a proper motion which must not be confounded with the displacements due to annual parallax, or to those other causes of disturbance which we have yet to speak of. The essential distinction is, that these disturbances are periodic and the proper motion is not. A star which has proper motion is carried by it among the other stars at a uniform rate in a definite direction. Sir W. Herschel found that these motions could, in nearly every case, be explained by supposing the sun, with its accompanying system of planets, &c., to be sweeping through space towards some distant point. It is obvious, that if such a motion of the solar system exist, the stars, in the region towards which it is moving, must seem to open out and separate more and more, while those in the opposite direction must close up. CHAP. XVII. PARALLAX. 211 The point determined by Herschel was near the star $\lambda$ Herculis, but, by the investigations of subsequent observers, and by employing a large number of stars, a more accurate, though not widely different, position of the point has been obtained. "The motion of the solar system in space is directed to a point in the celestial sphere, situated on the right line which joins the two stars of the third magnitude $\pi$ and $\mu$ Herculis, at a quarter of the distance between them measured from the former." § 270. Astronomers have carried their speculations on the character of the sun's motion still further, and, reasoning from analogy, have supposed that its path in space is not a straight line, directed towards the point above determined, but a curve to which this line is a tangent; and that, in countless ages, the solar system describes a gigantic orbit round some central position or body. Mädler, discussing the proper motions of the stars, assigns the centre of attraction to one of the group of the Pleiades. Many ages must elapse before the confirmation of this statement can be received with certainty. Supposing such a centre—then, the secular parallax of a star belonging to a yet more distant system would be the angle which the radius of this immense orbit would subtend at the star. *Struve.* P 2 ( 212 ) CHAPTER XVII. ABERRATION. § 271. The light of the celestial bodies is not brought to the observer by instantaneous transmission, but by successive propagation ; and this motion, combined with that of the earth in its orbit, is the cause of an apparent displacement of the bodies, to which the name of aberration has been given. The existence of aberration is, as we shall presently shew, a necessary consequence of the velocity of the earth being comparable with that of light; but the ratio of the former velocity to the latter is so small (about 1 : 10000), that the resulting aberration—whose magnitude depends on this ratio—had remained hidden and unsuspected until detected and explained by Bradley about the year 1729. § 272. The gradual propagation of light had been discovered by Roemer in 1675. He remarked that the eclipses of Jupiter's satellites always preceded their predicted times when the earth and Jupiter were on the same side of the sun, and happened later when on opposite sides. This he satisfactorily accounted for by supposing that light takes time for its transmission—the predicted times being determined from a large number of observations—would correspond to the mean distance of the planet from the earth; and therefore, in the first case, if the earth and planet were at their nearest distance, the extinction of the light would be known to the observer earlier than if he occupied his mean or average distance, by so much time as light would CHAP. XVIL.] ABBEATION. 213 take to move through the radius of the earth's orbit ; and, in the second case, when at their furthest distance apart, it would be just as much later. This important fact has been confirmed of late years by direct observation. Mons. Fizeau, by a most ingenious apparatus, has succeeded in measuring the actual velocity of light (see Arago's *Astronomy*, vol. iv.) ; and Mons. Foucault, with improved instruments, has repeated the observations and obtained results in close agreement with those obtained indirectly by other means. § 273. Bradley, about the year 1729, when observing certain stars, found apparent displacements which he could not account for by attributing them to any known cause. These displacements were periodical, and, as the periods were the same every year—for all the stars involved, it was obvious that the orbital motion of the earth was some means concerned in producing them. After some failures, he hit upon the only hypothesis that seems able to account for the phenomenon. The phenomenon itself is this : "All the stars seem to be displaced from their mean position towards that point of the heavens to which the direction of the earth's motion tends at the moment ; and the amount of the displacement varies as the sine of the angle between the earth's direction and the line joining the earth and star—the constant multi- plier, or coefficient of aberration, being the same for all stars." § 274. To explain how this effect is produced by the com- bination of the motions of the earth and of light, suppose $E$ to be the earth, $AEB$ a part of its orbit, $ES$ the true direction of a star. Draw $VET$ a tangent to the orbit at $E$. Take $ET$ to represent the velocity of the earth in magnitude ; and $EK$, in $SE$ produced, to represent the velocity of light on the same scale. 214 ABERRATION. [CHAP. XVII.] Now, the relative motion will not be altered if any common velocity be given both to light and to the earth. Let a velocity $EV$, equal and opposite to $ET$, be so applied. Then the earth will be brought to rest at $E$, and light will have a velocity compounded of $EV$ and $EK$, that is, a velocity $EW$, the diagonal of the parallelogram $V'K'$. Hence the star instead of being seen in its true direction $ES$ is seen in the direction $ES$, and the displacement $SES$ is the aberration.* § 575. The aberration takes place in the plane $SET$, and towards the point to which the earth tends. The triangle $WEK$ gives $$\sin SES : \sin SET = WK : EK,$$ or $$\sin (aberration) = \frac{\text{vel. of earth}}{\text{vel. of light}} \cdot \sin SET.$$ The aberration being small, we may write the circular measure for the sine, and also sin $SET$ for $\sin SET$. The angle $SET$ is called the earth's way, therefore $$\text{aberration} = \frac{\text{vel. of earth}}{\text{vel. of light}} \cdot k \sin (\text{earth's way}),$$ where $k$ is the constant of aberration. *The effect of aberration may also be illustrated in several ways: A man walking in a straight line on a road, his feet drop full vertically must hold his umbrella a little forward, the effect of his walking is to make him appear as if he were going upon him, and this effect is the greater the faster he walks. Again, suppose a ship moving at right angles to the line joining the ship and the battery, and to go out at the opposite side. The two shot holes will not be immediately opposite one another; the distance advanced by the ship during the passage of the shot through it will cause the shot to be farther ahead than the entrance, and a A diagram showing a ship moving at right angles to a line joining it with a battery. The ship is labeled "S", the battery is labeled "B", and there are two shot holes labeled "SH" on either side of the ship. The diagram shows that the shot holes are not directly opposite each other due to the movement of the ship. Watermark: This document contains sensitive information and should not be shared or distributed without proper authorization. CHAP. XVII.] ABERRATION. 215 § 276. The value of $k$ is found as follows : By comparing numerous observations of the eclipses of Jupiter's satellites, it is inferred that light takes $8^{\circ} 18'$ to travel from the sun to the earth ; and during that time the earth will have described an arc of its orbit--supposed circular--whose length will be $$\frac{8^{\circ} 18'}{365}\times2\pi R,$$ where $R$ is the radius of the orbit. Therefore $$k = \frac{498 \times 2\pi}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \frac{498 \times 20}{365 \times 360 \times 60 \times 60} = \\ or, if expressed in seconds of arc,\\ A = \\ k = \\ This mean value of $k$ is subject to small variations (not exceeding $o' - o''$) due to the different velocities of the earth at different points of its orbit. 277. The effect of aberration will be to make the stars, when referred to the celestial sphere, describe small ellipses about their true places. Let the accompanying figure represent the celestial sphere of the observer : $TOS$ the ecliptic, II its pole ; $A$ a star and $S$ the sun ; $O$ a point on the ecliptic $9^{\circ}$ behind $S$. Since the direction of the earth's motion is at right angles to the line joining it with the sun (neglecting the eccentricity of the orbit), it is obvious that $O$ is the point of the ecliptic to which the earth tends, and therefore $AO$ will be the earth's way for the star $A$. In $AO$ take $AA' = k\sin AO$, $A'$ will be the star's apparent place. person on board, not allowing for the ship's motion, but judging of the position of his head by the direction taken by the ship, would imagine it to be in advance of its true place. A diagram showing a celestial sphere with a star labeled "A" on an ecliptic labeled "E". The diagram includes a line from "A" to "O", representing the apparent path of a star on Earth's surface. Another line from "A" to "A'", representing the actual path of a star in space. The angle between these two lines is labeled as "AO". There are also labels for "TOS" (the ecliptic), "S" (the sun), and "O" (a point on the ecliptic). A small triangle labeled "M" is shown near "O". 216 ABERRATION. [CHAP. XVIL] If $AM$ be drawn perpendicular to ILAH, and if $AM = x$ and $AM = y$, then $$x = AA' \cdot A\sin AM \cdot \sin AO \cdot \sin AM \cdot \sin AO \cdot \cos AH \cdot \cos OI,$$ $$y = AA' \cdot A\cos AM \cdot \sin AO \cdot \cos AH \cdot \sin AO \cdot \cos OH,$$ therefore, $$\left(\frac{y}{x}\right)^2 + \left(\frac{y}{x} \cdot \tan AO\right)^2 = 1,$$ therefore the star describes an ellipse about its mean place, the semi-major axis being parallel to the ecliptic and equal to $20''45$, the semi-minor axis $20''45 \times \sin (O - I)$. 278. This may also be shown geometrically: Referring to the figure (p. 214), the line $KW$ is parallel and equal to $ET$, which represents the earth's velocity; therefore, neglecting the small variations of velocity, $W$ describes, round $K$ as centre, a circle parallel to the plane of the ecliptic, and $SEW$ describes an oblique cone on a circular base round the axis $SEK$. The intersection of this cone by the celestial sphere of the observer will be approximately a plane curve, and therefore an ellipse. 279. To determine the aberration of a star in latitude and longitude. Let $\tau S$ the longitude of the sun = 0 (fig. Art. 277), $\tau H$ the longitude of the star = 4, AH the latitude of the star = λ, then, as above, aberration in latitude = $- AM = -20''45 \sin AO \cos OH$ = $-20''45 \sin AO \cos (L - TO)$ = $-20''45 \sin AO \sin (O - I)$. Aberration in longitude = $\frac{- AM}{\sin IL} = -\frac{20''45}{\cos AH} \sin OH$ = $-20''45 \sec λ \cos (O - I)$. * The curve described by W is the hodograph of the earth's path, and therefore accurately a circle, but $A$ is not the circle except on the supposition in the text—that the motion is uniform. CHAP. XVII.] ABERRATION. 217 280. To determine the aberration of a star in right ascension and declination. Let $TOS$ be the ecliptic, $YRT$ the equator, $P$ its pole, $S$ the sum, $A$ the star, $O$ the 0° point behind $S$ as before. From $A$, the apparent place of the star, draw $AN$ at right angles to $PATS$. Join $PO$ cutting the equator in $R$, and in $AP$ produced make $AQ = 90°$, and join $QO$. Let $YT$ the star's right ascension = $\alpha$, $AT.................$, declination = $\delta$, $OTR$ the obliquity of the ecliptic = $\omega$. Then $AN' = AA' \sin A = k \sin O \sin P$ $k \cos OR \sin RT$ $k \cos OR (\sin AT \cos YR - \cos AR \sin YR)$ $k \sin AR \cos TO - k \cos AR \cos T \sin TO$; and aberration in right ascension = $- AN' \sec \delta$ $= -k \sec \delta (\sin AR \sin O + \cos AR \cos O)$. 281. Again, aberration in declination = $- AN$ $= -AA' \cos AN' k \sin O \sin OP = k \cos OQ$ $k (\cos OP \cos PQ - \sin OP \sin PQ) \cos OPA$ $k [\sin OQ \cos \delta - (OR \sin O + sin TO)]$ $k [\sin TO \sin TQ \cos \delta$ $\quad -\sin O (\cos OR \cos OR + sin AR cos OR sin YR)]$ $k [\sin TO \sin TQ \cos \delta - (\cos AR cos TO + sin AR cos O sin YO)]$ $= -k (\cos OQ \cos O + k \sin O (\cos AR sin O - sin AR cos O))$. 218 ABERRATION. [CHAP. XVII. 282. It was a change in the polar distance of the star $\gamma$ Draconis which first drew Bradley's attention to the subject, and ultimately led to the discovery of aberration. This star was most favourably situated, because it passed so near to the observer's zenith that he had not to fear any errors of refraction. Moreover, its right ascension was nearly $270^{\circ}$, so that, at the time of the vertical equinox, when $\odot = 0^{\circ}$, the aberration in declination is $-k \sin (\omega + \delta)$, and, at the time of the ascending node, when $\odot = 360^{\circ}$, it is $+k \sin (\omega - \delta)$. These results follow immediately from a figure by means of the expression for aberration " $k \sin$ (earth's path)." The polar distance of $\gamma$ Draconis is therefore greatest about the end of March, and least about the end of September; the variation amounting to $24 \sin (\omega + \delta)$, or, since $\omega + \delta = 75^{\circ}$ nearly, the change of declination is $$40^{\circ} - 9 \sin 75^{\circ} = 39^{\circ} - 5^{\circ}.$$ Solar, Lunar, and Planetary Aberration. 4283. On account of the motion of the moon and of the planets, the light by which each is seen comes from a point of space which the body no longer occupies when its rays reach the observer. Thus, if $PQ$ be a portion of the actual path of a planet in space, $Q$ its true place, and $B$ that of the observer at the same moment, the planet will not be seen by rays coming from $Q$, but from a point $P$ which the planet occupied at a time $t$ before the observation; $t$ being the time light takes to move from the planet to the observer. The rays therefore reach the earth in the direction $PB$, but the earth's own velocity causes aberration, as in the case A diagram showing two points P and Q on a line segment PQ. Point P is labeled with a small circle around it. Point Q is labeled with a large circle around it. The line segment PQ has a label indicating its length as "PQ". A smaller line segment PB is drawn from P to B. The angle between PB and PQ is labeled as "t". A small arrow is pointing upwards from B towards P. CHAP. XVII.] ABERRATION. 219 of the stars, and if $A$ be the position of the earth at the time $t$ preceding the observation, $$AB : PB = \text{vol. of earth} : \text{vol. of light};$$ therefore, completing the parallelogram $BAP'P$, $BP$ will be the apparent direction of the planet when its true direction is $BQ$. The angle between the two directions $BQ$ and $BP'$ is called the planet's aberration. J 284. Since $BP'$ is parallel to $AP$, and $A$ and $P$ were corresponding positions of the earth and planet at a time $t$ preceding the positions $B$ and $Q$, we have the following simple rule for determining by calculation the apparent place at a given instant, the true path being supposed known: find the time $t$ which the light will take to reach the earth (the ratio of the planet's distance to that of the sun is supposed known) and calculate the true place for a time $t$ preceding the given instant, this will be the apparent place at the given instant. By reversing this rule, if we wish to find by observation the true place at a given instant, wait till a time $t$ has elapsed and then observe the apparent place, this will be the true place required. Thus, if $r = \rho a$ be the distance of the planet from the earth, where $\alpha$ is the mean distance of the sun, the light of the planet will take $\rho (\frac{\pi}{360})^{\circ}$ to reach the earth, and the apparent geocentric direction of the planet at any instant will be the direction it actually had a $(\frac{\pi}{360})^{\circ}$ previously. In the case of the moon, $\rho = \frac{1}{2400}$, and the aberration will always be very small. The same rules will obviously apply to the sun; but, in this case, the body being fixed, $Q$ will coincide with $P$, and all rays which reach the earth come from the point which the sun really occupies, and are therefore affected with stellar aberration only. The sun's true place in the ecliptic is always in advance of its apparent place, and we may calculate the amount of aberration as follows: A diagram showing a parallelogram BAP'P with lines AB, BP', BP, BQ. $$AB : PB = \text{vol. of earth} : \text{vol. of light};$$ 230 ABERRATION. [CHAP. XVII. Let $v$ be the velocity of the earth, and $V$ that of light, $\theta$ the angle which the direction of the earth's motion makes with the radius vector $r$, then $$\text{aberr} = \frac{v}{p} \sin \theta = \frac{vr \sin \theta}{Vr} = \frac{\text{constant}}{r} (Newton, Sec. II. Prop. 1)$$ $$= \frac{20^\circ 45'}{p}, \quad \text{where } p \text{ is always near } 1,$$ $20^\circ 45'$ corresponds to the mean distance $a$. Diurnal Aberration. † 285. We have hitherto considered the velocity of the observer as being the same as that of the centre of the earth ; there will, however, be in addition another aberration due to the rotation about the axis. A person at the equator will describe $2\pi \times 3960$ miles in a sidereal day. During that time the earth will be carried $\frac{2\pi}{T} \times 92884000$ miles in its orbit. Therefore, if $k'$ be the coefficient of diurnal aberration at the equator, $$k' = \frac{3960 \times 364\frac{1}{4}}{92884000},$$ whence $$k' = 0^\circ 319'$$ For an observer in latitude $\phi$, we must use the coefficient of aberration $k' \cos \phi$. The east point of the horizon is that towards which the observer is being carried, and which therefore will replace the point $O$ of the previous general investigation (fig., p. 215). The effect of diurnal aberration is so small that it may generally be neglected, except for a star very near the pole, whose right ascension, when on the meridian, will be increased by $0^\circ 319' \cos \phi \sec \delta = 0^\circ 0213' \cos \phi \sec \delta.$ ( 221 ) CHAPTER XVIII PRECESSION AND NUTATION. Precession. 286. When catalogues of the true places of the stars, corrected for refraction and aberration, are formed at different epochs, as explained in Chap. I., Art. 16, it is found that the coordinates are all undergoing continual variations. Now, these variations may be due either to actual motions of the stars themselves, called their proper motions, or to a displacement of the planes and circles to which they are referred. The former cause would obviously produce special disturbances which would vary from one star to another, whereas any general displacement, affecting all the stars, must be attributed to the latter. Long continued observations have shown that the latitudes of all stars are very nearly constant, while their longitudes increase at a mean rate of $50^{\circ} \cdot 2$ per annum. From this we infer that the ecliptic is very nearly a fixed plane, and that the planets also move harmoniously with the motion which causes the first point of Aries to move backwards along the ecliptic at this mean rate of $50^{\circ} \cdot 2$ per annum. When the right ascensions and declinations are examined in a similar manner, their variations are found to lead to the same conclusion—a gradual shifting of the plane of the equator; but we learn moreover from them that the inclination of the plane of the equator to that of the ecliptic, or what we have called the obliquity of the ecliptic, has an almost invariable value. A diagram showing the precession of the equinoxes. 222 PRECESSION AND NUTATION. [CHAP. XVIII. 287. The investigations of physical astronomy shew that these results are accounted for by the action of the sun and moon on that portion of the earth's mass which, owing to its spheroidal shape, bulges out beyond the inscribed sphere. This attraction alone will affect the mean value of the obliquity, but cannot alter the line of the equinoxes to move backward on the fixed ecliptic, and thus to increase the longitudes of all the stars by a common quantity, the yearly value of which (50°-38) is called the *luni-solar-precession*. 288. The latitude of the stars have been stated above to be only "very nearly" constant. They have, in fact, a general change which indicates a motion of the plane of the ecliptic itself ; and here again physical astronomy furnishes an explanation, by showing that the attractions of the planets tend to disturb the earth's path, that is, to alter the plane of its orbit, but have no effect on the plane of the equator. The effect is a diminution of the obliquity amounting to 48' in a century, and a slow progressive motion of the first point of Aq along the equator, causing an annual decrease of the right ascensions of all the stars. This is called the *planetary precession*. Effects of Precession. 289. The luni-solar-precession and the planetary precession combined give the general precession 50°-2 yearly; but, in examining the effects of these changes, we shall neglect the planetary precession, and consider the plane of the ecliptic as fixed, and the 50°-2 due to the motion of the equator alone.* Let $P$ ... that of the equator, $T$ ... the point of intersection of the two circles. * For a complete investigation we may refer to Charpentier's *Astronom*, vol. I., p. 600. CHAP. XVIII.] PRECESSION AND NUTATION. 223 Precession, as we have said, carries $\Upsilon$ backward along the ecliptic without altering the obliquity; but the obliquity is measured by $H\Omega$ the distance between the two poles, therefore the effect of precession is to move the pole $P$ backwards along the small circle $PP'$ parallel to the ecliptic. If $\Upsilon$ be the position of the first point of Aries at the beginning of any tropical year, then $\Upsilon$ will be its position at the beginning of the next, $\Upsilon\Upsilon$ being $50^{\circ}2$ measured backward, i.e. in a direction opposite to the sun's motion; and a complete revolution will be accomplished in 25,800 years. It is obvious that, by this regression, the tropical year is shortened, and the return of the equinox takes place earlier than it would otherwise have done, whence the term *precession of the equinoxes*, and the name *precession* applied to this motion. Another effect of precession will be the gradual shifting of the constellations with respect to the equinoctial points (Art. 166). Hipparchus, about 120 B.C., discovered the precession and pointed out some of its necessary consequences. The first point of Aries was that which in his constellation whose name he gave to it, though it has retained the name, has shifted through nearly $30^{\circ}$, and is no longer in the constellation Aries but in Pisces. In the course of time, the stars seen at certain seasons will give place to others, our present winter constellations becoming summer ones, and vice versa, until 25,800 years have elapsed, when the pole and the first point of Aries will return to their present positions. The present polar star will, in about 13,000 years, be $46^{\circ}$ from the pole; and, long before that, will have lost its claim to the title. At present it is still approaching the pole, and will do so yet for about 150 years; but the small circle $PP'$, A diagram showing a celestial sphere with a line labeled "Ecliptic" running around it. The diagram also shows a line labeled "Obliquity" extending from a point on one side of a small circle labeled "Small Circle" to another point on the other side of the same small circle. 224 PRECESSION AND NUTATION. [CHAP. XVIII. which the pole describes, passes by other bright stars which will become the polar stars in turn. Precession will also produce a slight alteration in the lengths of the seasons, as explained in Art. 181, but this effect is not so striking as those just described. 200. When the right ascension and declination of a star are known at any epoch, we can calculate its latitude and longitude (Art. 167). Adding $50^\circ 27'$ to the longitude will give the new longitude after the lapse of $t$ years, owing to precession. With this new longitude, and the same latitude and obliquity, we can obtain the new right ascension and declination. For a complete investigation of these changes we shall refer to Woodhouse's *Astronomy*, vol. I., or Chan- vons's *Astronomy*, vol. II., where we find here given only the pre- cession in declination, obtained by differentiating the equation which connects it with the longitude. If $l$ be the longitude, $\lambda$ the latitude of a star, $\alpha$. . . its right ascension, $\delta$ its declination, and $\omega$ the obliquity, we have, from the triangle $NPS$, where $S$ is a given star, $$\sin \delta = \cos \omega \sin \lambda + \sin \omega \cos \lambda \sin l,$$ $\delta$ and $l$ are the only variables, and differentiating with respect to $t$, $$\cos \delta \frac{d\delta}{dt} = \sin \omega \cos \lambda \cos l \frac{dl}{dt},$$ $$= \sin \omega \cos \lambda t \cos l \frac{dl}{dt},$$ by formulae (a) and (c), Art. 167, $$\frac{d\delta}{dt} = \sin \omega \cos \lambda t \frac{dl}{dt};$$ therefore the annual precession in declination $$= 50^\circ 27' \sin \omega \cos \lambda.$$ A diagram showing a triangle NPS with angles representing right ascension, declination, and longitude. GRAP. XVIII.] PRECESSION AND NUTATION. 225 **Nutation.** 291. In treating of precession we stated that 50°-2 was the mean value of the yearly motion of $\tau$, and that the obliquity of the ecliptic was very nearly constant. Bradley, when discussing his observations after the discovery of aberration, found that the changes of declination of the stars could not all be accounted for by precession and aberration alone. His results led him to infer that there exists a small subordinate motion, by which the pole of the equator is carried sometimes before, and sometimes behind, the mean plane of the moon's orbit, and that this would have brought it; at the same time the distance from the pole of the ecliptic is sometimes more, and sometimes less, than the mean value; so that the true path of the pole is of a wavy form. He also found that these changes were periodic, and that they completed their cycle in about 10 years. Bradley found an intimate connection between these oscillations of the earth's axis, to which he gave the name of Nutation, and the inclination of the moon's orbit to the plane of the earth's equator. This inclination varies with the position of the moon's node, i.e. of the line where the plane of the moon's orbit crosses the ecliptic, and this line describes a complete revolution in 18 years 220 days. The researches of physical astronomy have fully confirmed Bradley's suggestion, that the cause of nutation is to be found in the variable action of the moon in causing precession. The complicated motion of the pole of the equator may be expressed as follows:—Suppose $pp'$ to be the small circle described by this mean pole $p$. Now let us construct a ellipse with a major axis $18^{\circ}$ directed towards $\Omega$ (the pole of the ecliptic), and a minor axis $13^{\circ}7$ towards $P$ (the small circle). Then, as the mean pole $p$ moves uniformly along * See Airy's Treatise "Precession and Nutation." 226 PRECESSION AND NUTATION. [CHAP. XVIII] the small circle in a retrograde direction, carrying the ellipse with it, the true pole $P$ will move along the circumference A diagram showing the movement of the true pole $P$ along the circumference of an ellipse. of the ellipse, completing a revolution in 18 years 220 days. The true first point of Aries being the pole of $N P$ will differ from its mean position which is the pole of $N P$, except when $T$ is at the extremities of the major axis of the moving ellipse. The difference between the true and the mean position of $T$ is called the equation of the equinoxes. 292. We cannot better conclude this chapter than with an extract from Woodhouse's *Astronomy*, giving an account of the means by which Bradley separated nutation from aberration : "The star $\gamma$ Draconis, passing the meridian very near the zenith of Bradley's observatory, and being consequently very little affected by refraction, was the chief star of his observations. At this time in May, 1735, it passed the south of the zenith by about 39° than it did in September. ...Other stars also changed their declinations. The changes of declination of a small star in Camelopardalis, with an opposite right ascension to that of $\gamma$ Draconis, were observed at the same time as those of the latter star; and, it was Bradley's argument, that, if these phenomena (changes of declination) arose from a real nutation of the earth's axis, the pole must have moved as much towards $\gamma$ Draconis as from the star in Camelopardalis; but this not being the case, the hypothesis of a nutation of the earth's axis would not account for the observed phenomenon : more strictly speak- A diagram showing the movement of a star around a fixed point on Earth's axis due to precession. CHAP. XVIII.] PRECESSION AND NUTATION. 227 ing, it would not completely account for it, for, in fact, some part of the observed changes of declination was due to the effect of nutation. "Bradley, as we have seen, solved the above phenomena by the theory of aberration. Now, if such theory, with the known one of precession, would account for all observed change of zenith distances, or, of north polar distances, then there could be no changes but what arose from precession and aberration. Hence, since the aberration is the same at the same season of the year, the distance of $\eta$ Draconis, in September, 1737, only by the annual precession in north polar distance ; the distance, in September, 1729, from the distance in September, 1727; by twice the annual precession in north polar distance; and so on. Such, however, was not the observed fact. In 1728, after the effect of precession had been allowed for, $\eta$ Draconis was nearer the north by about $0^{\circ}8$ than in 1727. In 1729, nearer than in 1727, by $1^{\circ}5$. In 1730, by $4^{\circ}5$. In 1731, by nearly $8^{\circ}$. Here then was a new phenomenon, a change of north polar distance, indicating an inequality yet not discovered. "Bradley observed other stars besides $\eta$ Draconis; amongst others, the star above mentioned $\alpha$ Camelopardalis, and it is not a little worthy of notice, this same star, which, in the case of the former inequality (that of aberration), directed him to reject the hypothesis of a nutation of the earth's axis, here determined him to adopt it. For within the space of two years (from 1728 to 1730) the distance of $\eta$ Draconis and of the star in Camelopardalis were equal and in contrary directions; that is, whilst the former through the years 1728, 1729, 1730, 1731, was approaching the zenith, and consequently the pole, the latter was, by equal steps, receding from the zenith, and consequently from the pole. These phenomena then of the changes in the north polar distances could adequately be explained by supposing a q 2 228 PRECESSION AND NUTATION. [CHAP. XVIII. mutation in the earth's axis towards $\gamma$ Draconis, and from the small star in Camelopardalis. "After 1731, Bradley observed contrary effects to happen; that is, $\gamma$ Draconis receded from the zenith and north pole, and the star in Camelopardalis, by equal steps, approached these points; and this continued till 1741, a period more than nine years; after which time the latter began again to approach the former, and the latter to recede from it. These phenomena then, between 1731 and 1741, could be adequately explained by supposing, during that term, a nutation in the earth's axis, from $\gamma$ Draconis and towards the small star in Camelopardalis." "By examining various and numerous observations, and by discriminating those that happened at particular conjunctures, Bradley found abundant confirmation of the truth of his two theories--aberration and nutation. During a period of more than twenty years, he accounted for the phenomena of observation, that is, the changes in the declinations of various stars, by making those changes or variations consist of three parts--the first due to precession, the second to aberration, and the third to nutation; the quantities and laws of the two latter being assigned on the principles and by the formulae of his theories. "We cannot sufficiently admire the patience, the sagacity, and the genius of this astronomer, who, from a previously unobserved variation not amounting to more than 40 seconds, extricated, and reduced to form and regularity, two curious and beautiful theories." ( 229 ) CHAPTER XIX. THE MOON. J 293. Next to the sun, the moon is, to the inhabitant of the earth, the most important of the heavenly bodies. Its size, its rapid motion among the stars, its influence both real and supposed, must, from the earliest times, have given it a prominent place in the observations of astronomers. Like the sun, it advances among the stars in a direction opposite to that of the diurnal motion, but about thirteen times faster ; a complete revolution being performed with respect to the fixed stars in ..... 27 d. 7 h. 43 m. 11-461 s.* ... first point of T ...... 27 d. 7 h. 43 m. 4-614 s. ... sun.................. 29 d. 12 h. 44 m. 2-87 s. The difference between the first and third of these periods is due to the advance of the sun itself. During the 27½ days of the moon's sidereal revolution, the sun will have moved through some 27°, which it will take the moon about 2 days & hours to gain. The difference between the first and second periods is accounted for by the small regress of the first point of T in 37 days. * These are the values at present, for comparison with ancient observations led Halley to the conclusion that the moon's mean velocity is being accelerated, and the period of a revolution shortened. La Place perceived theoretically that this acceleration is confined within very narrow limits, and will be followed by a stationation. A diagram showing the relative positions of Earth, Moon, and Sun during a lunar eclipse. 230 THE MOON. [CHAP. XIX. 294. The period of a revolution with respect to the sun is called a lunar month or a lunation, and also a synodical period; and the commencement of the period is the instant when the bodies have the same longitude. It is obvious that they have not necessarily the same right ascension at that instant. When two bodies have the same longitude they are said to be in conjunction, when their longitudes differ by 180° they are in opposition, and when by 90° they are in quadrature. The points distant 45° from these four positions are called octants. The two positions 'conjunction' and 'opposition,' when spoken of jointly, are called syzygies. 295. If the apparent diameter of the moon be measured at different times, it will be found to vary within certain narrow limits; its distance from the earth will therefore also vary in a corresponding, but inverse, manner. Observations of the parallax (Art. 262) show that the mean distance is about 238,000 miles, or 60 times the earth's radius, and that the variation is about $\frac{1}{2}$ of this mean value, making the distance sometimes 57 radii, and sometimes 63. The moon may, therefore, be considered as a companion of the earth in its orbit round the sun, and is, in fact, called the earth's satellite. 296. One of the most striking phenomena connected with the moon is the change of its visible outline, or its phases, as the successive appearances are called. The moon being an opaque spherical body, which receives its light from the sun, becomes visible to us by reflecting this light. It is obvious that the hemisphere, which is turned towards the observer, will not generally be all lighted up, and that consequently the appearance presented will seldom be a complete circular disc, but will vary with the relative positions of the sun, the moon, and the observer. The phenomenon of the phases will be understood from CHAP. XIX.] THE MOON. 231 the accompanying diagram, which represents one synodical revolution of the moon about the earth, the sun being supposed to remain stationary in the direction $T.A.S$. The distance of the sun from the moon being so great, in comparison with the dimensions of the bodies themselves, we may consider that one-half of the moon is always illuminated and the other half dark. When the moon is in conjunction at $A$, the dark side A diagram showing the phases of the moon. The diagram shows the moon's position relative to Earth and Sun. The positions are labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. is turned towards the earth, and no portion of it is then visible. It is new-moon. About a week later, the moon is at $C$, 90° from $A$; the plane of separation of its bright and dark parts passes through the earth, and the visible portion appears like a bright semi-circular disc as $c$. This is the first quarter. In intermediate positions, such as $B$, the illuminated portion appears as a crescent ($b$), gradually expanding into the semicircle for full moon. After passing $C$, a greater portion of the illuminated half becomes visible, and the bright disc swells out at the centre, and presents the appearance ($d$) which is called gibbous. A series of illustrations showing different phases of the moon. The first illustration shows a crescent ($b$). The second illustration shows a semi-circular disc ($c$). The third illustration shows a crescent ($b$) becoming larger. The fourth illustration shows a semi-circular disc ($c$) becoming larger. The fifth illustration shows a crescent ($b$) becoming larger. The sixth illustration shows a semi-circular disc ($c$) becoming larger. The seventh illustration shows a crescent ($b$) becoming larger. The eighth illustration shows a semi-circular disc ($c$) becoming larger. The ninth illustration shows a crescent ($b$) becoming larger. The tenth illustration shows a semi-circular disc ($c$) becoming larger. The eleventh illustration shows a crescent ($b$) becoming larger. The twelfth illustration shows a semi-circular disc ($c$) becoming larger. The thirteenth illustration shows a crescent ($b$) becoming larger. The fourteenth illustration shows a semi-circular disc ($c$) becoming larger. The fifteenth illustration shows a crescent ($b$) becoming larger. The sixteenth illustration shows a semi-circular disc ($c$) becoming larger. The seventeenth illustration shows a crescent ($b$) becoming larger. The eighteenth illustration shows a semi-circular disc ($c$) becoming larger. The nineteenth illustration shows a crescent ($b$) becoming larger. The twentieth illustration shows a semi-circular disc ($c$) becoming larger. 232 THE MOON. [CHAP. XIX. About 14½ days after conjunction, the moon will be in opposition at $E$, and, the illuminated face being wholly turned towards the earth, presents a complete circular disc ($e$) called full-moon. From full-moon the sequence of changes will be precisely the same, but in a reverse order ; the disc being gibbous at $F$, half-full at the third quarter $G$, less than half at $H$, then disappearing altogether at the next new-moon. § 297. During the first half of the month, the moon will be less than 180° east of the sun, and will cross the meridian between noon and midnight. The western limb is the bright limb during that period, as figured in the diagram, from $6$ to $e$. In the second half the moon will cross the meridian between midnight and the next noon, and the eastern limb becomes the bright limb (from $e$ to $h$). Although, strictly speaking, this crescent is formed as soon as the moon leaves $A$, it only becomes visible when at an angular distance of 30° or 60° from $A$; the thin line of light which it presents being overpowered by the strong light of the sun. For the recurrence of the moon's phases, in accordance with the days of the calendar month, see the explanation of the golden number, Chap. xxiii. § 298. A simple inspection of the figure will shew that the earth must present phases to the moon—the exact counterpart of those which the moon presents to the earth. Thus, when it is new moon, the illuminated side of the earth will be towards the moon, which will then have full-earth. When there is a crescent at $B$, the earth will appear gibbous, and so on. This will explain a phenomenon which is often observed : After sunset, when the moon is a crescent, the remainder of the circular disc is frequently visible, shining with a pale A diagram showing a crescent moon crossing a line representing a meridian between noon and midnight. CHAP. XIV.] THE MOON. 233 grey light.* This faint light is due to the strong earth-light which then falls on the moon, and which the moon reflects back again to the earth. As the moon increases its angular distance from the sun, the amount of earth-light received by it diminishes, and the effect disappears. † 299. The line joining the two cusps is a diameter of the circle which separates the dark from the bright hemisphere ; it is therefore perpendicular to the line which joins the centres of the sun and moon. Again, it is a diameter of the circle which separates the hemisphere turned towards the observer from the opposite one, and is therefore perpendicular to the line joining the observer with the centre of the moon. It is therefore perpendicular to the plane which passes through the observer and the centres of the sun and moon ; or, in other words, the great circle joining the centres of the sun and moon will bisect the line of cusps at right angles. The angle this great circle makes with the horizon is very variable, so that in corresponding positions of the crescent moon in different months the line joining the cusps will be very differently inclined, being sometimes nearly horizontal, and at others nearly vertical. ‡ 300. The bright portion of the visible hemisphere of the moon is bounded by two semi-circles ; but as the inner one is seen obliquely, it is projected into a semicircle, whose major axis is the diameter of the moon, and whose minor axis is constantly varying. The following investigation will * Called ' lunule conflué' by the French. The faintly luminous portion of the moon will appear as if belonging to a smaller sphere than the bright crescent does ; but this is an illusion due to revolution. The phenomenon itself is sometimes popularly spoken of as "the old moon in the arms of the new." A diagram showing a crescent moon with two cusps (points) on either side. 234 THE MOON. [CHAP. XIX. show how the magnitude of the phase is connected with the positions of the sun and moon. Let $O$ be the centre of the moon, $OT$ the direction of the observer, $BAC$ the plane through $O$ perpendicular to this direction; $OS$ the direction of the sun, and $BDC$ the plane perpendicular to $OS$; then the lune comprised between $BAC$ and $BDC$ constitutes the visible luminous portion; and if $BDC$ be projected orthogonally on the plane of $BAC$, the projection $BMC$ will be the semi-ellipse forming the inner boundary of the luminous disc. Now $$OM = OD \cos DOM$$ $$= OA \cos SOV,$$ where $OV$ is the prolongation of $TO$; therefore $$AM = OA - OM$$ $$= OA \operatorname{versin} SOV.$$ The area of the illuminated disc varies as $AM$, i.e. as the versine of the exterior angle of elongation $SOV$. \footnote{301. When $SOV = 90^\circ$, $BMC$ becomes a straight line, and conversely. Hence, if the angular distance $OTS$ between the sun and moon, as seen by the observer $T$, be measured at the moment when the moon's disc is just half illuminated, two angles of the triangle $OST$ will be known, and thence the ratio of $ST$ to $OT$ can be found, and the parallax of the sun determined in terms of that of the moon. This method of finding the solar parallax is not practically available, on account of the difficulty—we may say impossibility—of fixing upon the precise instant when the moon is dichotomised, as this phase is sometimes} A diagram showing a circle representing the moon's disc, with points labeled O (centre), A (sun), B (moon), C (observer), D (prolongation of TO), E (prolongation of OV). Lines connect these points: OA, OB, OC, OD, OE. CHAP. XIX.] THE MOON. 235 called.* The regularity of the inner boundary is disturbed by the lunar mountains, and, when examined with a telescope, the line is found broken and jagged, with detached points of light here and there encroaching on the darker part; these detached points are the mountain tops which receive the sun's light while the intervening valleys are still in the shade. J 302. Age of the Moon. We have said that new-moon is the instant when the centres of the sun and moon are in conjunction. The age of the moon, at any instant, is the time, expressed in days, that has elapsed since the previous new-moon. When only integral values are employed, the moon is said to be one day old when less than 24 hours have elapsed since new moon, two days old during the next 24 hours, and so on. The moon's age is given, in the Nautical Almanac, to the nearest tenth of a day for each Greenwich moon (see Chap. xxiii.). Moon's Orbit. Nodes. J 303. When observations are carried on for a long period in order to ascertain the path of the moon, as was done in the case of the sun (Chap. vi.), it is found that its motion is much more complex. The right ascensions and declinations being determined day by day, and the positions marked on the globe, it is found that the curve described during each revolution, though approximately, is not accurately, a great circle, nor even a plane curve. We shall refer the motion to the ecliptic, the results being simpler than when referred to the equator—If at any instant a great circle be drawn through the direction of the moon's motion, this great circle will intersect the * Aristarchus, employing this method, found that the line of separation was a straight line when the moon was 87° from the sun, whence he inferred that the sun's distance from this point was 21 times greater than that of earth; instead of 60 times, which modern observation by more exact methods have given.
CHAP. XIX.] THE MOON. 235
Age of the Moon. We have said that new-moon is the instant when the centres of the sun and moon are in conjunction. The age of the moon, at any instant, is the time, expressed in days, that has elapsed since the previous new-moon. When only integral values are employed, the moon is said to be one day old when less than 24 hours have elapsed since new moon, two days old during the next 24 hours, and so on. The moon's age is given, in the Nautical Almanac, to the nearest tenth of a day for each Greenwich moon (see Chap. xxiii.).
Moon's Orbit. Nodes. When observations are carried on for a long period in order to ascertain the path of the moon, as was done in the case of the sun (Chap. vi.), it is found that its motion is much more complex. The right ascensions and declinations being determined day by day, and the positions marked on the globe, it is found that the curve described during each revolution, though approximately, is not accurately, a great circle, nor even a plane curve.
We shall refer the motion to the ecliptic, the results being simpler than when referred to the equator—If at any instant a great circle be drawn through the direction of the moon's motion, this great circle will intersect
Aristarchus, employing this method, found that the line of separation was a straight line when the moon was 87° from the sun, whence he inferred that the sun's distance from this point was 21 times greater than that of earth; instead of 60 times, which modern observation by more exact methods have given.
236 THE MOON. [CHAP. XIX. ecliptic in two opposite points called the moon's nodes ; the ascending-node being that where the moon crosses from the south to the north side of the ecliptic, and the other the descending node. The observations show that these points are not stationary, but that, while the plane of the orbit itself remains inclined at a constant angle of about 5° 9' to the plane of the ecliptic, the nodes recede along the ecliptic at an average rate of 3' 10"6 per day, or 1° 27' in each sidereal revolution of the moon. In one year the node is carried about 19° 20' back, and in about 18°6 years returns to its first position. The greatest latitudes is therefore always 5° 9' in every revolution ; but the greatest declination will vary, according to the place, with the sum of nodes, from 18° 18', when the plane of the orbit lies between the equator and ecliptic, to 28° 36' when outside ; the first value being the difference, and the second the sum, of the obliquity of the ecliptic and the inclination of the orbit to the ecliptic. 4304. The variation of the moon's apparent diameter indicates changes in her distance from us. If we pursue the same method of observation as for the sun (Chap. x.), we shall find (disregarding the shifting of the plane of her orbit, which during one revolution is less than 1/4°) that the moon describes an ellipse, the earth occupying one of the foci. The eccentricity of the elliptic orbit is about $\frac{1}{2}$; the greatest and least distances being respectively 251,700 miles and 225,600 miles. When the moon is at her greatest distance, she is said to be in apogee, and when at her least distance in perigee. Those points of the orbit are jointly spoken of as apes, and the line joining them the line of apesides. 4305. We found in the case of the sun (Art 182) that the apes advance along the ecliptic. The moon's apes also Approximately constant,—there are small periodical fluctuations. CHAP. XIX.] THE MOON. 237 progrede, but much more rapidly, nearly 40° in one year, or 5° in each revolution, so that in about 1/4 years the perigee arrives where the apogee was before. There are many other fluctuations and disturbances to which the moon is subject. The largest of these were known to ancient astronomers, and the most important of the small irregularities had been detected by observation before their physical cause was known. A complete account of them, however, is beyond the scope of the present work, and the investigation of all the perturbations can only be effected by the most refined analysis of physical astronomy. Librations. J 306. The moon always presents the same, or very nearly the same, face to an observer ; the mountains and valleys which cover the surface of our satellite are seen occupying nearly constant positions relatively to the centre of the disc; and relatively also to the plane of the orbit. Hence, therefore, that the moon revolves about its axis nearly perpendicularly to this plane at a time of rotation about this axis and of a revolution round the earth must be very nearly equal, and that the average of a large number of these periods must be exactly the same as both.* A controversy seems to be periodically arising as to the proper words to be used in describing this motion of the moon. There is no question about the phenomenon itself : both describe it correctly. One describes it as a motion of the moon in space, and accordingly, if asked to represent it by a model, makes use of the very same contrivance ; but the one asserts that (neglecting libration) libration is due to a rotation of the moon about its own axis with respect to its orbit, while the other describes it as is done in the text, as a rotation about its own axis combined with a revolution round the earth. In either case, whether we consider only one motion, both modes of expression are correct; but the mathematician would adopt the latter form in preference to the former, because the small-angle approximation is not applicable when we have to determine how far a body is by determining, first, the motion of its centre of gravity, and next, the motion of certain lines fixed in the body and moving with it. This is necessary for determining simultaneously both motions; and since determination of one motion does not imply determination of another; and this separation of the two sets of equations becomes so familiar, that the mind may bring itself * A controversial seems to be periodically arising as to the proper words to be used in describing this motion of the moon. There is no question about the phenomenon itself : both describe it correctly. One describes it as a motion of the moon in space, and accordingly, if asked to represent it by a model, makes use of the very same contrivance ; but the one asserts that (neglecting libration) libration is due to a rotation of the moon about its own axis with respect to its orbit, while the other describes it as is done in the text, as a rotation about its own axis combined with a revolution round the earth. 238 THE MOON. [CHAFF. XIX.] The angular velocity of the moon about its own axis is uniform, but its angular velocity about the earth is not so; hence we shall sometimes see a little more of the eastern limb, sometimes a little more of the western. This constitutes the phenomenon called *libration in longitude*. Again, the axis about which the moon rotates is not quite perpendicular to the plane of the ecliptic, though very nearly so, and hence a little beyond each pole will come alternately into view. This is the *libration in latitude*. A third libration is the diurnal or parallactic libration which arises from parallax. When we said that the moon always presents the same face to the observer, it was understood that the observer was supposed at the centre of the earth. It is obvious that at the rising and setting of the moon, portions become visible beyond the upper limb which disappear as the moon's altitude increases. § 307. To the inhabitant of the moon, if any, the days and nights will be approximately equal, and each about 14 of our days. The earth is, however, a moon to the moon, and will, to one half of our satellite, present the remarkable appearance of a globe, about 2° in diameter, fixed (except for to look upon the corresponding physical facts—the translation of the centre of gravity and the relation about an axis through the centre of gravity—as the only normal expression of motion. But we may also proceed in a different manner; we may determine the motion of the instantaneous axis, and thus the angular velocity about that axis. Now, when the instantanous axis is moving with respect to some other body, it is not true to say that this body has no motion; and it is this view taken by those who hold that the moon rotates about the earth that leads them to suppose that there must be great deviations from a circular path, the instantaneous axis will be a fixed axis passing through the earth's centre. When this assumption has arisen, the error committed by both parties has been that of denying any other conception of the motion to be possible besides their own. Each, finding that his own idea would perfectly explain his phenomena, has assumed that all others were wrong. When the libration and departure from circular motion are taken into account, the actual phenomena can no longer be represented by a rotation about the earth's axis; whereas, they are still accurately shown by the other mode of representation. A diagram showing a circle representing Earth with a smaller circle representing Moon orbiting around it. The Moon's position changes slightly over time due to libration. CHAP. XIX.] THE MOON. 239 small librations) in the sky, and having phases like those of the moon, but, on account of its size, giving about 18 times more light. Path of the Moon round the Sun. §308. That motion of the moon, which we have hitherto considered, is relative to the earth, and would be her absolute motion in space if the earth were fixed ; but as the two bodies move together round the sun, the actual path of our satellite, relatively to the sun, will be due to a combination of her own monthly motion round the earth, and of the earth's yearly motion about the sun. The orbit of the earth is elliptical, almost circular, and as there are about 124 lunations in a year, the moon's path must-neglecting the small inclination of the two orbits-cross that of the earth about 25 times. We might, perhaps, from this be led to expect that the curve described by the moon would consist of a series of loops or waves, as in figs. (a) and (b). Such, however, is not the case. The moon's orbit is everywhere close to that of the earth. For this reason all calculations of physical astronomy are necessary to prove the correctness of this statement, but the following calculation will show its probability : Let the curve $ACB$ represent (a) A diagram showing a series of loops or waves in a hypothetical lunar orbit. (b) A diagram showing a part of the earth's orbit with a point at which it is crossed by the moon when coming within, and another point where it next intersects. a part of the earth's orbit, $A$ a point at which it is crossed by the moon when coming within, and $B$ the next intersection * See the Author's Elementary Treatise on the Lunar Theory, 4th Editt. p. 85. 240 THE MOON. [CHAP. XIX. when going outside. If the moon's path is anywhere convex, it should be between $A$ and $B$. Draw the chord $AB$ and the sagitta $CD$. $AB$ is about $\frac{1}{35}$ of the orbit, and therefore subtends an angle $\frac{2\pi}{25}$ at the sun; and, if $R$ be the radius of the earth's orbit, we have approximately, $$CD = R \left( 1 - \cos \frac{\pi}{25} \right)$$ $$= \frac{1}{25} \left( \frac{\pi}{25} \right)^2 R$$ $$= \frac{1}{400} R$$ nearly $$= 3$$ times the moon's distance from the earth. The moon's path, therefore, lies between the chord $ADB$ and the arc $ACB$, and we may conclude, with great probability, that it is everywhere concave to the sun, as represented by the dotted line. Lunar Mountains. 309. To Galileo is due the discovery that the surface of the moon is covered with mountains and valleys. He also concluded from his observations that several of the mountain rise to an altitude of between four and five miles above the surrounding plane, and his results have been confirmed by the researches of Meszar, Beer and Mädler (1837). As the moon's diameter is only $\frac{1}{4}$ of that of the earth, we see that her mountains are comparatively very much loftier. When viewed through a telescope, the lunar mountains project shadows on the side opposite to the sun, and the lengths of these shadows, when measured by a microscope, will serve to determine the heights of the mountains above the planes on which the shadows are cast, proper account being taken of the inclination of the sun's rays. This is the method pursued by Meszar, Beer and Mädler. CRAP. XIX.] THE MOON. 241 J 310. Another method, which we shall proceed to describe, consists in measuring the distance of the small detached points of light which sometimes appear on the discopetion of the moon's disc, from the line of separation of light and shade. These points of light are the mountain tops which catch the sun's beams over the edge of the spherical surface while the intermediate plane is still in shade. Let the figure represent the visible disc of the moon, $P$ the projection of the bright summit of a mountain peak seen on the dark portion at the time when the sun's light just reaches it. The distance of $P$ from the edge $M$ must be measured with a micrometer in a direction $PM$ perpendicular to the line of cusps and the distance $AB$ between the cusps must also be measured. Let $m$ be the ratio of these measures. Then $PM$ is the projection of a tangent to the sphere, parallel to the sun's rays, and therefore making with $PM$ an angle equal to the complement of the exterior angle of elongation $SOV$, see p. 234. Let $PM = c$, and $AB = 2r$, and let the height of the mountain be $x$. Then $x(2r + x) = \text{eq. of tangent}$ $\Rightarrow \text{eq. of line of which } PM \text{ is the projection}$ $c^2\cos^2O$ where $O$ is the exterior angle of elongation ; therefore $x = m^2\cosec^2O$, neglecting the fraction $\left(\frac{x}{2r}\right)^2$. Whence the height of the mountain will be found, since $r, m$, and $O$ are known. R 242 THE MOON. [CHAP. XIX. Harvest Moon. 4 311. The moon crosses the meridian about $50\frac{1}{4}$ minutes later every day. This value is obtained as follows : In one synodical revolution the sun crosses the meridian exactly once oftener than the moon, therefore $29\frac{1}{2}$ solar days are equivalent to $28\frac{1}{2}$ lunar days, and the average length of a lunar day $$=\frac{29\frac{1}{2}}{28\frac{1}{2}}\mathrm{~solar~days}=1^{\circ}0^{\prime}50\frac{1}{2}''.$$ If the moon moved along the equator, and at a uniform rate, we should find that the times of rising or setting would get daily later by precisely $50\frac{1}{4}$ ; but its path nearly coincides with the ecliptic, and this fact causes considerable variation in the daily retardation. These variations depend on the latitude of the place : at Cambridge the retardation may amount to $1^{\circ}15'$, and at other times be only 18 or 20 minutes. Now, it has been observed that at the full moon nearest to the autumnal equinox, the times of rising on three or four successive evenings will follow one-set at a small interval ; and the farmers, "not doubting that it had been so ordered on purpose to give them an immediate supply of moon-light for their greater convenience in reaping the fruits of the earth," gave the name of harvest moon to this particular full moon. To explain this phenomenon, let us remark, that if a body approaches the elevated pole without altering its right ascension, its stay above the horizon is increased, but the sidereal time of crossing the meridian is unaltered ; therefore the time of rising is accelerated, and the acceleration will be greater, the greater the change of declination. If, on the other hand, the right ascension be increased * Ferguson's Astronomy.* CHAP. XIX.] THE MOON. 243 without any change of declination, the time of rising will be retarded in proportion to the increase of right ascension. When both circumstances exist, the retardation due to the increase of right ascension will be modified by the change of declination. For simplicity, we shall suppose the moon's orbit to coincide with the ecliptic. Then, supposing its daily motion in the orbit to be uniform (about 13°), it will be easily seen that the change of declination is most rapid when crossing the equator, and that at the same time the change of right ascension is least. The effect of this is that when the declination changes very slowly, and the right ascension increases by considerably more than 13° in one day, since the arc described being nearer the pole will subtend an enlarged angle. In northern latitudes it will therefore be when the moon crosses T that the conditions for a small retardation will arise; and, as the moon passes through this point at each revolution, the phenomenon of a small retardation must recur every month. It is, however, only at the autumnal equinox that the effect is noticed; for, the sun being then in $\alpha$, the moon, when in $T$, will be full, and the rising take place near sunset. In other months the moon, when in $T$, is only partially illuminated, and rises either during the day or late at night. After two or three days the increase of declination is slower, and that of right ascension faster, and the daily retardation soon increases till it attains, and then exceeds, its mean value. In southern latitudes the same phenomena will take place at the other equinox, which also will correspond with their harvests. If we take the actual orbit of the moon, which is inclined 5° to the ecliptic, the effect will be greater or less in different years, according as the position of the nodes makes the angle between the orbit and the equator greater or less, the cycle being completed in 18-6 years (Art. 303). n 2 244 THE MOON. [CHAP. XIX. The accompanying figure will serve to illustrate the explanation we have given above :- Let $AB$ be the horizon, $A$ the position of the moon at its rising on any day, $PA$ the polar distance, and $PAM$ the angle made by $PA$ with the moon's orbit $AMQ$. The values of $PA$ and $PAM$ will vary from day to day. Draw the declination circle $PAM$, making $PA$ an angle equal to the average change of the moon's right ascension (about 13°) in a lunar day, and draw the line $AK$. Then if the moon changed its right ascension only it would move to $K$, and the next day the retardation of the moon's rising on that of the point $A$ would be about 54° measured by the angle $APK$. [The retardation with respect to the sun is only 50°, because the sun advances 1° in the same direction as the moon]. But the moon will move along its orbit to some point $M$ about 13° from $A$, and if the parallel $MP$ be drawn, the moon will rise at $B$, the retardation on the rising of $A$ being measured by the angle $BPM$. The difference between the values of $APK$ and $BPM$ will measure the difference between the average and the actual retardation on that day. A simple inspection of the figure shows that the nearer $AM$ is to $AB$ the smaller will be the angle $BPM$, and the smaller the daily retardation. In high latitudes, where $AB$ would come between $AM$ and $AK$, the moon would rise earlier on the second day than on the first. ( 245 ) CHAPTER XX. ECLIPSES AND OCCULTATIONS. §312. The moon, being nearer to us than any other celestial body, will often, in its monthly revolution, interpose itself between us and some of these. When the moon thus hides a star from our view, the star is said to be occulted, but when it comes between us and the sun, the sun is said to be eclipsed. Solar Eclipse. §313. An eclipse of the sun may be partial, total, or annular. The moon's diameter, as seen by us, subtends an angle which varies from $28^\circ 48'$ to $33^\circ 32'$, the sun's diameter varies from $31^\circ 32'$ to $32^\circ 36'$, their mean values being very nearly the same ; so that, although so much smaller than the sun, the moon may, on account of her proximity to us, some times subtend a larger angle. According, therefore, to the magnitudes and positions of the two discs relatively to the observer, the sun may be partially or wholly eclipsed ; or it may appear like a bright ring of light surrounding a dark centre. §314. It is obvious that an eclipse of the sun can only occur at, or near, conjunction. If the moon's orbit actually, or very nearly coincided with the ecliptic, solar eclipses would recur at every new moon ; but the inclination of the moon's orbit (about $0^\circ 9'$) gives to the moon a latitude which, at the instant of conjunction, is sufficient to keep the two discs apart, unless they, at the same time, happen to be near a node. 246 ECLIPSES AND OCCULTATIONS. [CHAP. X.] The magnitude and the general character of a solar eclipse vary with the position of the observer, because the moon's parallax being large (nearly 1"), and the sun's small, a change of place on the earth will alter the apparent place of the moon and scarcely affect the sun's. Thus, the sun may be totally or annularly eclipsed by one observer, and only partially so to another; while, at the same time, a large portion of the earth will have no eclipse at all. An eclipse may even be total to one observer and annular to another--this would happen if, during the progress of the eclipse, the two apparent diameters became exactly equal; then, two observers, to each of whom the eclipse happened to be central, the one before and the other after this instant of equality of the diameters, would have, the one a total, and the other an annular, eclipse. Lunar Eclipse. 315. Since we have no celestial neighbour nearer than the moon, it is obvious that an eclipse of the moon cannot arise from the same cause as an eclipse of the sun. But the moon's light is derived from the sun, and when the earth interposes itself between the two, it cuts off the sun-light, and thus a part, or the whole, of the moon becomes dark, and a partial, or total, eclipse of the moon takes place. The eclipse cannot be annular; for, the cone of shadow projected on the disc is always where the moon crosses, project three times as broad as the moon itself. The appearance of the edge of the shadow on the disc is an arc of this circular section of the cone. An eclipse of the moon can only take place at, or near, opposition, that is, at full moon ; but the latitude of the moon may be such as to enable it to pass the shadow without entering, which explains why there is not a lunar eclipse at every full moon. As in the case of solar eclipses, A diagram showing a lunar eclipse. CHAP. XX.] ECLIPSES AND OCCULTATIONS. 247 a lunar eclipse will occur only when the moon, being full, is near a node of its orbit. 316. An essential distinction between the solar and the lunar eclipses is this: --in the one, the luminary is merely hidden from us; in the other, it actually loses its light. So that, as stated above, the character of a solar eclipse will vary from one observer to another, whereas a lunar eclipse all parts of that hemisphere of the earth which is turned towards the moon will see the eclipse, and in precisely the same phase." Ecliptic Limits. 317. It is found that, in order that a solar eclipse may be possible, the angular distance of the sun's centre from the node, at the instant of conjunction, must not exceed $18^\circ 26'$; and that an eclipse will certainly happen if this distance be less than $13^\circ 42'$. These are called the solar ecliptic limits. Between these values the eclipse is doubtful. It is also found that there will certainly be a lunar eclipse, provided the distance from the node, at the moment of full moon, be less than $9'$; and that the eclipse will be impossible if this distance exceed $12\frac{1}{2}$. These are the lunar ecliptic limits. Synodic Revolution of the Moon's Node. 318. The moon's node has a daily retrograde motion of $3^\circ 10'\ 64$, and the mean motion of the sun is $59'\ 33'$. The relative motion is therefore $62'\ 19'$ daily, and the sun will return to the same node in $$\frac{360 \times 60}{62\frac{1}{2}} = 346\ 62\ \text{days}.$$ *The parallactic libration will produce a small change, but the want of definiteness about the edge of the shadow renders it unnecessary to consider this.* 248 ECLIPSES AND OCULTATIONS. [CHAP. XIX. Hence, 173 days after passing through one node, the sun will come to the other. If the line of nodes retained a fixed position in the ecliptic, the sun would return to it year after year at the same or at very slowly changing dates, and the eclipses of the sun and moon would recur in the same months for a very long period. The retrograde motion of the nodes hastens the return of the eclipses and causes a constant and rapid shifting of their dates, a complete circuit of the calendar taking place in about 18 years (Art. 305). Number of Eclipses at one Node. 319. In 144 days, which is the interval between new-moon and full-moon, the sun and node will separate by $14\frac{1}{2}\left(92^{\circ}19'\right)=15\frac{1}{2}$. If, then, a full-moon happen exactly at the node, the preceding and the following new-moons will happen at $15\frac{1}{2}$ from it, and therefore within the superior limits of a solar eclipse, so that three eclipses may occur at that node—two solar and one lunar. But, if a new-moon occur exactly at the node, the preceding and the following full-moons will be beyond the lunar ecliptic limits, and only one eclipse (a solar one) will take place at that node. The same results will follow if a full-moon or a new-moon happen, not exactly at the node, but within a couple of days on either side of it; the preceding and the succeeding syzygy will both be less than $18\frac{1}{2}$, and both more than $12\frac{1}{2}$ from the node. Hence, at every passage of the sun through a node, there will be at least one eclipse, and there may be three. Number of Eclipses in a Year. 320. The sun takes 173 days to pass from one node to the other, and six lunations occupy 177 days ; so that a lunar CHAP. X.X.] ECLIPSES AND OCCULTATIONS. 249 eclipse happening exactly at one node will give a lunar eclipse 4 days after the sun passes the next node, and therefore too far from it to produce three eclipses at that second node. If, however, the lunar eclipse at the first node happens two days before the sun reaches it, the lunar eclipse at the next node will be two days only after the sun has passed it, and there may be three eclipses at each of the nodes. The sun going on will meet the first node again, and another lunar eclipse will occur six days after passing through ; this second passage through the first node can, however, produce only two eclipses, viz. the lunar eclipse just spoken of and a solar eclipse at the preceding new moon. The solar eclipse occurs when the lunar eclipse is the first solar eclipse of the two groups of three, and 12 lunations occupy 354 days, so that these seven eclipses may all be comprised in the same year, provided the first of the seven occur early in January. 124 lunations occupy 368 days, and therefore the eighth eclipse, the lunar, cannot come in. In order to bring it in, it would be necessary to shift the whole system back some days, but then the first solar eclipse of the first group would find itself in the December of the previous year, and the number of eclipses would still be seven. Therefore, in one year there cannot be more than seven eclipses, five of the sun and two of the moon, or four of the sun and three of the moon. In a similar manner it may be shown that a single (solar) eclipse near one node may be followed by a single (solar) eclipse near the next node. Therefore, there cannot be fewer than two eclipses every year, both of the sum. \(\checkmark\) 321. Speaking generally, there are more eclipses of the sun than of the moon ; thus, in a period of eighteen years there are on an average seventy eclipses—forty-one of the 250 ECLIIPSES AND OCCULTATIONS. [CHAP. XX. sun and twenty-nine of the moon—that is, roughly, in the same ratio as their ecliptic limits. But more eclipses of the moon than of the sun are seen at any given place, for the reasons already stated, that a lunar eclipse is visible to a whole terrestrial hemisphere at once, whereas a solar eclipse is visible to only a small portion of it. Total or annular eclipses of the sun are phenomena of very rare occurrence in a given locality, although there are on an average twenty-eight for the whole earth in every period of eighteen years. In London, according to Halley, no total eclipse had been observed between the 20th of March, 1140, and the 22nd of April, 1715, a period of 675 years. The Saros of the Chaldeans. 322. The ancients, who had no correct tables of the sun and moon to enable them to predict the eclipses with the precision to which modern astronomy has arrived, had nevertheless discovered a method of extreme simplicity, by the use of which they foretold these phenomena with very considerable accuracy. This method is still used to determine at what new-moons, or full-moons, eclipses will occur—the strict application of modern methods being afterwards employed to calculate the exact times. From what has been stated in the previous articles, we infer, that when the sun, the moon, and the node, return to the same relative positions, the same eclipses must recur. Now 1 lunation occupies............ 29'53059 days; and 1 synod. revol. of node ....... 346'62 days ; therefore 223 lunations occupy........ 658'32 days, and 19 synod. revol. of node ....... 685'78 days. Hence after 223 lunations, that is, a period of 18 years 11 days, or 18 years 10 days, according as 4 or 5 leap years CHAP. XX.] ECLIPSES AND OCCULTATIONS. 251 are comprised, the sun, the moon, and the node will return approximately to the same relative positions. If, therefore, during one of these cycles of 18 years 11 days, a record be made of all the eclipses which occur, they will be found approximately to repeat themselves. This period was known to the Chaldeans and called *Saros*. Conditions necessary for a Solar or Lunar Eclipse. §323. We shall proceed to investigate the circumstances of solar and lunar eclipses in a more particular manner. Let $ABV$ be a cone which envelopes both the sun $S$ and the earth $T$, the vertex $V$ being beyond the earth; then the portion between $T$ and $V$ will receive no light from the sun. A diagram showing a cone with its apex at V, enveloping both the sun S and the earth T. The cone's base is labeled ABV. About $T$ as centre, describe a sphere passing through the centre $M$ of the moon and cutting the cone in two circles whose diameters are $EF$ and $HK$. There will be an eclipse of the sun at some place on the earth if any portion of the moon come within $EF$; and there will be an eclipse of the moon if it enter $HK$. The figure shows that $EF$ is greater than $HK$, and would lead us to expect, as we have already seen, a greater number of solar than of lunar eclipses. Let a new cone envelope both the sun and earth, but on opposite sides, having its vertex $V'$ between the two bodies, and let it cut the sphere about $T$ on the further side in a circle whose diameter is $PQ$; then, as soon as the moon enters $PQ$, it will receive light from a portion only of the A diagram showing a second cone with its apex at V', enveloping both the sun S and the earth T. The cone's base is labeled P'Q'. The sphere is shown intersecting this cone in two circles whose diameters are EF' and HK'. The figure shows that EF' is greater than HK'. 252 ECLIPSES AND OCCULTATIONS. [CHAP. XX. sun's surface. The space comprised between the two cones $PQ$ and $HK$ is called the penumbra, the dark part $CVD$ being the umbra or shadow. When the edge of the moon enters the penumbra, it is not eclipsed, but its light begins to diminish until it reaches the shadow, and the diminution is so gradual, that it is extremely difficult to ascertain by observation the exact instant of the commencement of the eclipse. \(\S 324.\) A solar eclipse will take place at, or near, conjunction, if the angular distance between the centres of the sun and moon, as seen from the centre of the earth, is less than the semi-diam. of $\varphi + \angle STE$, i.e. $<\varphi$'s semi-d. + $TEC+TVC$, $<\varphi$'s semi-d. + $TEC+ATC-TAC$, $<\varphi$'s semi-d.+ $\varphi$'s paral., $\sim (<\varphi$'s semi-d.-$\varnothing$'s paral.). A lunar eclipse will take place at, or near, opposition, if the angular distance between the centres of the moon and of the shadow is $<\varphi$'s semi-d. + $\angle HTV$, $<\varphi$'s semi-d. + $THC-TVG$, $<\varphi$'s semi-d.+ $\varphi$'s paral., $\sim (<\varphi$'s semi-d.-$\varnothing$'s paral.). \S 325. The values of the semi-diameters, parallaxes, &c., for any epoch may be calculated from solar and lunar tables,* but it is found necessary in calculating lunar eclipses to increase the diameter of the earth's shadow at the distance of the moon by about $\frac{1}{4}$th part, in order to make the result agree with observation. We may explain the *The Nautical Almanac is always calculated for about three or four years in advance, and was commenced in 1672. When the eclipse falls in a year for which the Nautical Almanac has been calculated, all observations made during that year are calculated for instances of time separated by regular intervals, sufficiently close to enable us to obtain the values for intermediate instants by simple rules of interpolation. CHAP. XX.] ECLIPSES AND OCCULTATIONS. 253 necessity for this, by remarking that $HK$ (fig., p. 25) is determined by lines $ACH$, $BDK$ which just graze the earth's surface; but such rays as $AC$ and $BD$ are probably absorbed by the lower strata of the earth's atmosphere, and the first rays which pass through and proceed to $HK$ will, as it were, touch a sphere somewhat larger than the earth. **To determine the Time, Duration, and Magnitude of a Lunar Eclipse.** √ 336. Let $SN$ represent a portion of the ecliptic, $S$ being the centre of the earth's shadow at the moon instant of and $SM$, or $\lambda$, the latitude of the moon opposition, and let the sum of the semi-diameters of $p$ and shadow = $c$. A diagram showing a lunar eclipse. The Earth is shown with its shadow on one side. The Moon is shown moving across the Earth's shadow. The diagram includes labels for various points and lines. Let $MBN$ be the $p$'s path, and $N$ the node, $MA$, or $m$, the $p$'s horary motion in longitude, $AB$, or $p$, latitude, $SS'$, or $\alpha$, the $\odot S$ longitude. On $AM$ set off $SS = SS'$, and draw $AB$ equal and parallel to $AB$, then $MBN$ will be the relative path of the $p$ —the earth's shadow being supposed stationary at $S$. Draw $SP$ perpendicular to $MV$, this will be the nearest approach of the two centres. And if, with $S$ as centre and radius = $c$, we describe a circle cutting $MV$ in two points 254 ECLIPSES AND OCULTATIONS. [CHAP. XII. Q, Q', these will be, on the relative orbit, the positions of the moon at the beginning and end of the eclipse ; the corresponding points on the actual path may be obtained by drawing parallels through Q and Q' to MA. 327. Let $\angle AMB = \theta$ and $A'MB = \theta'$, then $\theta$ and $\theta'$ will be known from the equations $$\cot \theta = \frac{m}{p}, \quad \cot \theta' = \frac{m - s}{p},$$ and $$SP = MS \cos MSP = \lambda \cos \theta,$$ If this value $\lambda \cos \theta$ be greater than $c$, there will be no eclipse. If $\lambda \cos \theta$ be less than $c$, the eclipse will be partial or total -- partial, when $-\lambda \cos \theta$ which expresses the breadth of the portion of the moon eclipsed is less than the whole diameter $d$, and total, when greater. If $\lambda \cos \theta = c$, there will be no eclipse, the disc of the moon will just graze the shadow, and the corresponding value of $NS$ will be $$NS = MS \cot Y = \lambda \cot \theta$$ $$= c \cot 45^\circ,$$ the greatest and least possible values of $NS$ will be the superior and inferior ecliptic limits. 328. At the time $t$ after opposition, the latitude of the $p$ will be $\lambda - pt$, and the difference of longitude of the shadow and $p$ will be $(m - s) t$. Therefore the distance of the centres will be $$r = \sqrt{(\lambda - pt)^2 + (m - s)^2 t^2}.$$ The different circumstances of the eclipse may be all deduced from this formula. If we make $e$ equal to $c$, the two values of $t$ obtained from this equation will give the times of the beginning and end of the eclipse. If the two values of $t$ are imaginary, there will be no eclipse ; if equal, the beginning and end of the eclipse CHAP. X.] ECLIPSES AND OCCULTATIONS. 255 will be simultaneous, or, which is the same thing, the shadow and the moon will just come into contact, and no eclipse will take place. If we make $r$ equal to the difference of the semi-diameters of the moon and shadow, we shall obtain the times of the beginning and end of the total eclipse; but if the two values of $t$ so obtained are imaginary, the eclipse will not be a total one. Any value being given to $r$, the middle of the eclipses will correspond to the half-sums of the two corresponding times, and will be given by the formula $$t = \frac{p\lambda}{(m - s)^2 + p^2} = \frac{\lambda}{p}\sin t\theta.$$ 432. Solar eclipse. The computation of a solar eclipse, with reference to the whole earth, will be of the same character as that of a lunar eclipse, using the semi-diameter of the section $EF$ of the cone (fig., p. 251) instead of that of $HK$ the shadow. We can thus determine the time of beginning and ending of the eclipse generally on the earth. But the problem becomes much more complicated when we wish to ascertain the different phases of the phenomenon, as seen at a particular place on the earth's surface. The following is a rough outline of one of the methods of proceeding.--Having found the times of beginning and end of the general eclipse, fix upon some instant, for instance, $t_0$, and, for this instant, determine (by means of tables or by the Nautical Almanac) the latitudes, longitudes, parallaxes, and semi-diameters of the two bodies. Calculate the effects of parallax on the latitudes and longitudes of each body as seen from the given place, and, by means of these reduced latitudes and longitudes, determine the apparent angular distances of the centres of the two luminaries. This distance, compared with their apparent semi-diameters 256 ECLIPSES AND OCULTATIONS. [CHAP. XX. (also corrected for parallax), will determine the magnitude of the eclipse at that instant. We must, however, refer to other works for the detail and various modifications of the calculation (see Appendix to Nautical Almanac for 1836, by Woodhouse). § 330. The occultation of a fixed star by the moon is determined by the very same methods as the solar eclipse, except that, the star having no parallax and no semi-diameter, the calculation is somewhat simplified. § 331. Every phase of a lunar eclipse is visible to all parts of that hemisphere of the earth which is turned towards the moon. When the Greenwich time of the beginning of the eclipse is known, we can find what terrestrial meridian will at that moment reckon midnight ; and that place on this meridian which has its north or south latitude equal to the south or north declination of the sun will have the sun in its nadir, and therefore the moon in its zenith. If we find this place on a terrestrial globe, and describe a great circle about this place as pole, the hemisphere so marked out will see the beginning of the eclipse. In the same manner, the hemisphere which sees the end may be determined, and the space common to the two hemispheres will see all the circumstances of the eclipse. 257 CHAPTER XXI. FINDING THE LONGITUDE BY OBSERVATION. 332. The latitude and longitude of a place on the earth's surface are the coordinates of its position. The importance of a correct determination of these elements is obvious, because, when they are known, the position of that place relatively to other known places becomes determined. The latitude is readily found by observation (Chap. ix.), but the longitude has always presented difficulties, which have only been overcome by the introduction of more delicate instruments and more refined modes of observing. In Art. 201 it was shown that the difference of longitude of two places is connected with the difference of the times reckoned at the two places at the same instant; and, although the problem of finding the longitude has been attempted in a great variety of ways, it will be found that they ultimately resolve themselves into finding what time is reckoned at some initial meridian (that of Greenwich for instance) at a moment corresponding to a known local time at the observer's place.* Now, the local time is easily found by observation (Chap. xxi.), but difficulty is to obtain the corresponding time at the other meridian. There are two ways in which this may be done:—The first consists in the observer's carrying with him the time of that other meridian, by means of a watch or chronometer, * There is an exception to a method practised at sea and known as "dead reckoning," where the true place is found by noting the course and distance run since leaving some known position. The result is, however, only rough approximations, and are neglected so soon as celestial observations can be made. 2 258 LONGITUDE BY OBSERVATION. [CHAP. XXL whose error and rate are known. The second method requires an observer at each station, who shall note the local time of the occurrence of some celestial phenomenon, or of a preconcerted signal which can be simultaneously seen at the two places ; a comparison of the two times will determine the difference of longitude. In the case of most of the celestial phenomena, the Greenwich time of their occurring may be calculated and tabulated beforehand, so that the observer, with the Nautical Almanac in his possession, has, as it were, the simultaneous observation of the Greenwich observer. To find the Longitude by Chronometer. \textbf{v33.} Let an observation be made for finding the local time by any of the methods of Chap. xii., and let the corresponding time marked by a chronometer be noted. Then, supposing the chronometer to have a known error and rate, that is, supposing that on some previous occasion it had been compared with Greenwich time and its error ascertained, and also its daily gain or loss, there will be no difficulty in obtaining its present error on Greenwich time, and thence the Greenwich time itself, corresponding to the local time of observation. The difference of these two times will be the longitude in time, which may be converted into degrees at the rate of 15° for every hour. The longitude will be less than 15° when as the local time is greater or less than the Greenwich time. The value of this method depends on the accuracy of the chronometer, which is assumed not to have changed its rate in the interval. As the best instruments are liable to irregularities, the danger is lessened by the employment of several chronometers, and the longitude is then deduced by taking a mean of the results—more or less weight being given to the indication of each instrument according to our previous experience of its accuracy. The method by chronometers is especially valuable at sea. A diagram showing a clock with hands indicating time. CHAP. XXL] LONGITUDE BY OBSERVATION. 259 By Electric Telegraph. 334. The difference of longitude between two places $P$ and $Q$ may be very accurately determined when these places are connected by an electric telegraph. Let a signal be made at $P$, the more easterly station, at the time $T_1$ of $P$; and suppose that it is received at $Q$ at the time $T'_1$ of $Q$. ($T_1$ and $T'_1$ being both solar, or both sidereal, times obtained from the clock times at the two stations by correcting for the errors of the clocks). If the transmission of the electric current were instantaneous, the difference of longitude would be $$\lambda = T'_1 - T_1$$ but if $\lambda$ be the time required for the transmission of the signal, $T'_1 + x$ will be the time at $Q$ corresponding to the time $T_1$ at $Q$, therefore the difference of longitude is $$\lambda = (T_1 + x) - T'_1 = \lambda + x.$$ To eliminate the unknown quantity $x$, let a signal be made at $Q$ at the time $T'_1$ and received at $P$ at the time $T'_2$; then $$\lambda = T'_2 - T'_1$$ would be the difference of longitude if no time were lost in the transmission. Assuming this time to be again $x$, $$\lambda = T'_2 - (T'_2 + x) = \lambda - x,$$ whence $$\lambda = \frac{1}{2}(\lambda_+ + \lambda_-)$$ is known. 335. In the practice of this simple method, when we wish to obtain from it the very great accuracy of which it is capable, we have to take into account certain small possible errors. Such are— The errors of the assumed clock corrections. The personal errors of the observer who gives, and of him who receives, the signal (Art. 83). The error due to neglecting the small fraction of time s 2 260 LONGITUDE BY OBSERVATION. [CHAP. XXL required to complete the galvanic circuit after the finger touches the signal key, &c. &c. For a full examination of these errors, and of the means of eliminating them, or of reducing them to a minimum, we shall refer to Chaunvett's *Astronomy*, vol. 1., where the student will also find a description of the method called "by star signals," which is a slightly modified way of employing the electric telegraph. 336. For places which are not connected by electric communication, the difference of longitude may be obtained, when the stations are not too far apart, by observing simultaneously the time marked by the clock at each station when some signal, such as a flash of gunpowder or the disappearance of a ship's light, is fixed either at one of the stations or at some intermediate point. The method may be extended to find the difference of longitude of distant places when they can be linked by intermediate stations. By Celestial Signals. \(\dagger\) 337. Some celestial phenomena are of such a character that they may be observed at the same instant of absolute time by different observers. Such are— 1. The eclipses of Jupiter's satellites, 2. The appearance and end of a lunar eclipse, 3. The bursting of a meteor. Any one of those being observed at two stations, the difference of the corresponding local times will give the difference of longitude. The exact instants of the disappearance of Jupiter's satellites into the shadow of the planet, and of their reappearance, are predicted in the Nautical Almanac for Greenwich time; so that an observer noting his local time of either phenomenon, has at once a means of inferring his CHAP. XXL] LONGITUDE BY OBSERVATION. 261 absolute longitude. But the method is not susceptible of very great accuracy; for, the disappearance of a satellite taking place gradually, the apparent instant of disappearance, and therefore the longitude inferred, will depend on the power of the telescope employed ; and, at sea, the motion of the ship renders it impossible to keep the planet within view of a telescope of sufficient power to show the satellites. The method by which we determine longitude is uncertain; for, the boundary line of the earth's shadow on the moon is so indefinite, that observers, at the same place, will differ by two or three minutes in their estimation of the time of beginning or ending of the eclipse. The bursting of meteors, or shooting stars, might be useful if it were possible to anticipate them, and also to identify them when observed. By Lunar Distances. \(\checkmark\) 338. The moon has a rapid motion among the stars, altering its position by more than 13" in 24 hours. Certain bright stars and some of the planets are selected, which lie almost directly in the moon's path, and the angular distances of the moon's centre from these and also from the sun (as they would appear to an observer at the centre of the earth), are given in the Nautical Almanac for every third hour of Greenwich mean time. It will be easy to understand how this is made available for finding the longitude, whether at sea or on shore --- Measure, with a sextant, the angular distance between the moon's bright limb and a star, and correct for instrumental errors. Then add or subtract the moon's apparent semi-diameter according as the bright limb is towards or from the sun ; this will give the apparent distance of the centre. When the sun is the body whose distance from the moon is measured, the apparent semi-diameters of both bodies are always to be added, because the bright limb of the moon is turned towards the sun, and the observed distance is that of the nearest limb. A diagram showing a lunar eclipse with a star near the edge of the moon. 263 LONGITUDE BY OBSERVATION. [CHAP. XXX. The apparent distance so found must be further corrected for the effect of parallax and refraction, as explained below, to determine the geocentric or true distance. This true distance, compared with the distances registered in the Nautical Almanac, will determine the time at Greenwich, and thence, by comparison with the local time of observation, the longitude of the place. § 339. Only one observation has been supposed made, that of the distance ; but we want the local time to compare with the Greenwich time, therefore an altitude also must be simultaneously observed (Chap. viii). Moreover, the operation of clearing the distance, as the computation of the true distance is called, requires both the altitudes to be known. There should therefore be three observers, one measuring the distance, and the others, at the same moment, the altitudes of the bodies. The local time may be obtained from either altitude. § 340. Let $Z$ be the zenith of the observer, $S$ the apparent place of the star or sun, and $M'$ that of the moon. We shall assume that the effects of parallax and refraction are wholly in the vertical planes $ZS$ and $ZM'$;† and that the true places are $S$ and $M$. The parallax of the moon exceeds its refraction, therefore $M'$ is When there is but one observer he may proceed as follows.—Take the altitudes of the sun and moon at different times during their passage between the luminaries, again the moon's altitude, and finally that of the sun or star. The intervals of time having been also noted by a watch, the observed changes of altitude will give us a mean rate of change per hour. This will enable him to calculate what the altitudes would have been at the moment when the distance was taken. § 341. Suppose that $S$ is a plane through the geocentric zeniths, and re- fraction in a plane through the astronomical zeniths, but the error introduced in very much less than the probable error of observation, and may therefore be easily neglected. A diagram showing two planes intersecting at a point labeled "S". One plane contains points labeled "Z" (zenith) and "M" (moon), while another plane contains points labeled "Z" (zenith) and "M'" (moon). A line segment connects "Z" to "M", and another line segment connects "Z" to "M'". The diagram illustrates how parallax and refraction affect observations. CHAP. XXL.] LONGITUDE BY OBSERVATION. 263 below $M$, and the contrary for the sun, a planet, or a star, therefore $S$ is above $S$. The apparent zenith distances $ZS$, $ZM$ are observed at the same time as the distance $SM$. From these the true zenith distances $ZS$, $ZM$ must be found by correcting for parallax and refraction. In the triangle $ZSM$, the three sides $ZS$, $ZM$ and the contained angle $Z$ will be known, and the true distance $SM$ may be obtained. 341. The solution of these two triangles is a simple application of the rules of Spherical Trigonometry. Let $m'$, $s'$ be the apparent altitudes of the moon and star, $$\begin{array}{lll} m', s' & \text{true} \\ d & \text{apparent distance of their centres}, \\ d & \text{true} \\ \end{array}$$ If the angle $MSS'$ be called $\phi$, we have $$\cos d = \sin m'\sin s + \cos m'\cos s\cos \phi,$$ $$\cos d = \sin m'\sin s + \cos m'\cos s\cos \phi,$$ whence $$\frac{\cos d - \sin m'\sin s}{\cos m'\cos s} = \frac{\cos d - \sin m'\sin s}{\cos m'\cos s}.$$ This determines $\phi$, but the equation is not in form adapted to logarithmic computation. To prepare it for this purpose, add 1 to each side, $$\frac{\cos d + \cos (m+s)}{\cos m'\cos s} = \frac{\cos d + \cos (m'+s')}{\cos m'\cos s},$$ $$(1-2\sin^2d) + [2\cos^2(m+s)-1] = \frac{2\cos d(m+s')}{\cos m'\cos s},$$ $$= \frac{2\cos d(m'+s'-d')}{\cos m'\cos s},$$ $$\sin^2d\sec^2\frac{1}{2}(m+s) = \frac{\cos d}{\cos m'}\sec^2\frac{1}{2}(m'+s'-d'),$$ 264 **LONGITUDE BY OBSERVATION. [CHAP. XXL]** Let $\sin^2\theta = \cos m \cos s - \cos \frac{m}{2} (\sin^2 d + \sin^2 e) \cos \frac{m}{2} (m + s - d)$ therefore $\sin^2 d = \cos^2 \frac{m}{2} (m + s) (1 - \sin^2 \theta)$ $\sin^2 d = \cos^2 \frac{m}{2} (m + s) \cos \theta$ The formulae $(B)$ and $(C)$ determine $d$, and are adapted to logarithms. The above is Borda's solution, which is perfectly general and requires no distinction of cases. 342. We shall give another solution which has found great favour with mariners on account of its simplicity. It requires a table of natural versed sines. Subtract 1 from each side of equation $(A)$: therefore $\cos m - \cos (m - s) = \cos d - \cos (m - s)$ Assume $\cos m \cos s = 2 \cos \theta$ ……………………$(D)$; therefore $\cos d = \cos (m - s) + 2 \cos \theta (\cos d - \cos (m - s))$ $= \cos (m - s) + \cos (d + \theta)$ $+ \cos (d - \theta) - [\cos (m - s) + \theta] - [\cos (m - s) - \theta]$ whence, $\text{vers } d = \text{vers } (m - s) + \text{vers } (d + \theta)$ $+ \text{vers } (d - \theta) - \text{vers } (m - s) - \text{vers } (d - s)$ ……………………$(E)$, 343. Many other transformations have been proposed, but the above are among the simplest. Instead of obtaining the true distance $d$ directly, which, on account of its being a very small quantity, would require extended tables and great care in working for the proportional parts, we may employ approximate methods which find the difference between $d$ and $d'$. As this difference will always be a small angle seldom exceeding 1°, the work will be less troublesome without being practically less correct. The value of $d - d'$ will be found by successive approximation, or by development in a series in which the smaller terms are neglected. CHAP. XX.] LONGITUDE BY OBSERVATION. 265 344. The true distance having been found, we have to compute the corresponding Greenwich time. Referring to the Nautical Almanac, we find the calculated distance comprised between some two of those registered there ; and, on the assumption of a uniform variation of the distance during those three hours, a simple proportion will give the correction to be added to the first hour. If $d$ be the true distance ; $d_1$, $d_2$ the distances, given in the tables, separated by three hours ; $x$ the correction in hours, $$\log x = \log 8 + \log (d - d_1) - \log (d_2 - d_1).$$ 345. Proportional logarithms. This calculation is very much simplified by the use of a special table, called table of proportional logarithms, the idea of which is due to Dr. Maskelyne, the Astronomer Royal, at whose suggestion the yearly publication of the Nautical Almanac began in 1767. The above proportion may be written $$\frac{3}{x} = \frac{d_2 - d_1}{d_2 - d_1} = \frac{3}{d_2 - d_1} + \frac{3}{d_2 - d_1}.$$ Now the table of prop. logs. is so constructed that opposite to any number $a$ stands $\log \frac{a}{b}$, therefore $$\text{prop. log } x = \text{prop. log } (d - d_1) - \text{prop. log } (d_2 - d_1).$$ The Nautical Almanac gives not only $d_1$ and $d_2$ but also in a column between them prop. log $(d_2 - d_1)$ ; therefore at the same time that we take out $d_2$, the distance which precedes the true distance $d$, we take out prop. log $(d_2 - d_1)$, and we only have two inspections of the logarithmic tables instead of four. 346. We have supposed the distance between the moon and a star to increase or decrease uniformly ; but this will 266 LONGITUDE BY OBSERVATION. [CHAP. XXL not generally be the case, and the result obtained has to be further corrected by the usual rules of interpolation. The Nautical Almanac contains a table which gives the necessary correction for second differences in a convenient form. See Boole's *Finite Differences*, Chap. III., on 'Interpolation.' 347. The horizon at night is frequently so ill-defined that there is a large probable error in the altitude of the star, and the local time determined from it will have a corresponding uncertainty, although the altitude would be sufficiently accurate for clearing the distance. But as all vessels engaged in distant voyages are furnished with one or more chronometers, we may employ the observed lunar distance to determine—not the longitude of the ship—but the error of the chronometer on Greenwich time at the instant of observation ; then, at any convenient time afterwards, determine the longitude by an observation of the sun and chronometer. We may even dispense with the observation of the altitudes altogether at the time of taking the distance ; because, the supposed Greenwich time, given by the chronometer and the estimated longitude, will be sufficiently accurate to determine the time at ship, and thence the hour angle, which, with the known latitude and the declinations of the star and moon, will lead to the altitudes with sufficient accuracy for clearing the distance. The reader who is interested in this problem may consult Delambre's *Astronomy*, Chap. xx.; Mackay on Longitude, Bk. 3, Chap. iv.; Chambers's *Astronomy*. The method of lunar distances for determining the longitude owes its value to the rapid motion of the moon among the other heavenly bodies. If it moved twice as fast, the accuracy would be twice as great. The inhabitants of Jupiter must therefore, with their rapidly revolving moons, A diagram showing a celestial sphere with various lines and points indicating positions of stars and planets. CHAP. XXL.] LONGITUDE BY OBSERVATION. 207 be able to determine their longitudes with as great accuracy as we can our latitudes. By Moon-Culminating-Stars. § 348. The preceding method for the determination of the longitude depends for its success on the rapid change of the moon's distance from certain fixed stars. The change in its right ascension, which is equally rapid and sometimes even more so, is the basis of another method, not available at sea, but which, on land, leads to very accurate determinations. A transit instrument being mounted in the meridian plane, the instant is noted by an astronomical clock when the bright limb of the moon comes to the meridian, and also the time of transit of a selected star, taken from the list of moon-culminating-stars in the Nautical Almanac. These moon-culminating-stars are chosen, for each day, those having nearly the same declination as the moon, and not to differ very widely from each other in right ascension. The difference between the two times so observed is the difference of the right ascensions of the star and of the moon's bright limb, at the moment when the latter is in the meridian of the observer. If similar observations be made at Greenwich, we shall have the difference of right ascensions of the star and of the moon's limb when the latter is in the meridian of Greenwich. We can thus find the change in the right ascension of the moon's limb while it passes from one to another; and if this change be divided by the variation corresponding to one hour of longitude*, which is nearly uniform, * The column is headed "Variation of Q's right ascension in one hour of longitude," but it is calculated for the height limb, and therefore includes the effect of a change of the semi-diameter. (See Nautical Almanac Explanation). A page from a book with text and some mathematical symbols. 268 LONGITUDE BY OBSERVATION. [CHAP. XXX and is given in the Nautical Almanac for the instant of the passage over Greenwich, we shall obtain the difference of longitude in time. 349. Instead of corresponding observations at Greenwich, we may take the right ascensions of the moon's limb and of the star as registered in the Nautical Almanac, the computed places being now given with such accuracy that they may be considered the same as if they had been observed. When the two places differ considerably in longitude, we must use for divisor, not the variation for one hour given in the table, but one obtained by interpolation corresponding to the middle of the interval between the two observations. Let $x$, $T_1$ be the observed clock times of transit of $\gamma$'s limb and star, $$A_1, A_2$$ be the registered right ascensions of same at Greenwich transit, $$C$$ be the registered change in one hour of longitude corresponding to the middle time. Then, longitude = $$\frac{(T_1 - T_2) \cdot (A_1 - A_2)}{C}$$ Both numerator and denominator must be expressed in seconds, and the result will bethe longitude expressed in hours and fractions of an hour--west, if positive ; east, if negative. 350. The advantage of selecting stars which have nearly the same declination as the moon is, that the instrumental errors will affect both bodies equally. The longitude so determined must yet be corrected for the known errors of the transit (Chap. iv. Art. 82) ; for, if we suppose the star to be observed $t'$ too soon, we are in reality observing the transits across the meridian of an observer $t'$ to the east of ours, and the longitude obtained above will be his longitude, which must therefore be increased by $t'$. A diagram showing a celestial sphere with a line representing a transit of a star across a meridian. CHAP. XXI.] LONGITUDE BY OBSERVATION. 269 351. The longitude of a place may also be found by an eclipse of the sun, or by an occultation of a star or planet, but the operation is much too intricate for an elementary work, and the rare occurrence of these phenomena renders the methods of less practical value than those we have here considered. For a complete investigation of these, and of some other methods of determining the longitude of a place, we shall refer to Chauvenet's *Astronomy*. A page from a book with text discussing the determination of longitude through astronomical observations. ( 270 ) CHAPTER XXII. THE PLANETS. 352. We have already, on several occasions, spoken of the planets—those wandering stars whose brilliancy and motions among the other stars must from the very earliest times have attracted attention. Five planets were known to the ancients—"Mercury, Venus, Mars, Jupiter, and Saturn;" and the motions of these five were all found confined within a narrow zone of less than 8° on each side of the ecliptic. To this zone they gave the name of zodiac. No additions were made to the planetary system before the end of the 18th century. On the 15th of March, 1781, Sir W. Herschel discovered Uranus, "in the course of a review of the heavens," which was then visible in a telescope of a certain power was brought under close examination, when the new planet was immediately detected by its disc, under a high magnifying power." On the 1st day of January, 1801, another planet Ceres was added. This was the first of a series of very small telescopic objects, the number of which, now exceeding a hundred, seems likely to go on increasing with the increase of magnifying power of our telescopes. They are not, like the other planets, confined to the zodiac, and it has been conjectured that they are probably fragments of some larger planet, blown to pieces by an explosion. A still more remarkable planetary discovery was made some years ago almost simultaneously by Professor Adams A small image of a celestial body. CHAP. XXII.] PLANETS. 271 and Mons. Le Verrier. The history of this belongs to physical astronomy, and we shall refer the reader to the full account given in Grant's valuable work ; but few words however, will explain the peculiar circumstances which led to the discovery, and which mark it as one of the most wonderful in the whole range of scientific inquiry. The planet Uranus discovered, as we have seen above, by Herschel, was carefully observed over a series of years ; and, these observations serving as basis, the subsequent path of the planet was calculated and registered, taking carefully into account all known causes of disturbance. In a few years it was found that the positions of the planet did not exactly agree with the calculated positions, and it was surmised that the unknown cause of disturbance was probably an undiscovered planet, Adams and Le Verrier proposed themselves, and successfully achieved, the solution of the inverse problem : "Given the unaccounted-for disturbances of the Planet Uranus, to find the position, &c., of the unknown disturbing body." The indications furnished by their calculations led to the discovery of the planet Neptune. 353. When a planet is attentively observed, and its position marked down on the celestial sphere or map, night after night, it is soon found that its path differs essentially in character from those of the sun and moon. These luminaries have a motion which, though not strictly uniform, departs but little from uniformity ; but, in the case of the planets, we find, not only a want of uniformity, but also a change of direction. The planet, after advancing for some time from west to east in the same direction as the sun and moon, gradually reverses its speed, then stops, and begins to retrograde. After retrograding, with a velocity at first increasing and then diminishing, it again stops, and then recommences * History of Physical Astronomy.* A diagram showing the retrograde motion of a planet. 272 PLANETES. [CHAP. XXII.] its onward motion. The time of regression is, however, much less than that of progression, and there is on the whole an advance from west to cast. The ancients, who started with the fixed idea of an immoveable earth, found these retrograde motions and stationary positions a source of much perplexity, and, to explain them, were driven to complicated systems of epicycles. The planet was supposed to describe a circle uniformly, the centre of this circle itself moving on another circle, and so on. But, however ingenuous these systems were, they could not completely reconcile the observations. 354. We shall not enter into a description of the various hypotheses which preceded that of Copernicus (1543). This celebrated astronomer shewed that all these different and complicated planetary phenomena could be easily and satisfactorily explained by supposing the earth to be itself a planet, circulating round the sun in the same direction as the other planets. According to this system, which has long ceased to meet with the antagonism which its supposed contradiction of experience and revelation originally excited, the sun occupies the centre; and the planets succeed one another in the following order, beginning from the sun outwards. The mean distance of the earth is taken for unit, and the period is expressed in days.
Mean Distance from Sun. Period.
Mercury 0-3871 87-969
Venus 0-7235 224-700
Earth 1-0000 365-256
Mars 1-5237 686-980
Minor Planets (average) 2-6 1531
Jupiter 5-2028 4332-855
Saturn 9-5389 10750-220
Uranus 19-1827 30686-821
Neptune 30-0370 60126-720
The mean distances of the other planets differ slightly from those given above. CHAP. XXII.] THE PLANETS. 273 The two planets, Mercury and Venus, whose orbits are within that of the earth, are called inferior planets, all the others are superior planets.* 355. By a careful and elaborate study of the motions of the planet Mars, Kepler was led to the discovery of his three laws (Chap. xii.), which proved that the idea of circular motion, with its attendant epicycles, must be discarded for the much simpler one of motion in a conic section round the sun in the focus ; and Newton confirmed Kepler's discoveries by shewing them to be necessary consequences of his law of universal gravitation. In order, then, to see the motions of the planets in their greatest simplicity, we must conceive the observer stationed at the sun. But it is obvious that the positions which the planets would then occupy, relatively to the stars, would be very different from those which they seem to have when seen from the earth. The two positions are known as the heliocentric and the geocentric places of the planet ; and, when the dimensions and positions of the elliptic orbits of each of the earth and of the planet are known, as also the places which bodies occupy in them at any assigned time, it will be a simple application of the rules of plane trigonometry to convert heliocentric into geocentric latitudes and longitudes, and vice versa. * The distances of the principal planets from the sun may be roughly obtained by a somewhat curious law, known as *Bode's law*, given by him in 1784. We shall here give only a few examples of this law. For instance, after the second, each in formed by doubling the preceding ; next, add to each and divide by 10. We get values approximately coinciding with the planetary distances.
Mars Venus Earth Mars Mercury Jupiter Saturn Uranus Neptune
4 7 19 16 28 52 109 195 388
When Bode's law was first given, there was a gap between Mars and Jupiter, which was filled up by a new planet discovered by Leverrier. This discovery of Neptune, however, disturbs the series so much as to allow us to attach no more value to the law, beyond that of being a simple means of remembering the distances. 2 274 THE PLANETS. [CHAP. XXIL Thus, suppose $S$ to represent the sun, $AB$ the plane of the ecliptic, $EFG$ the earth's orbit, and $E$ the earth's place; $NPQ$ a planet's orbit and $P$ the place of the planet at the same time; $ST$ and $ET$ parallel lines in the direction of the first point of Aries. Then, drawing $PM$ perpendicular to the plane of the ecliptic, $PSM$ and $TSM$ are the heliocentric lat. and long. of $P$; $PEM$ and $YEM$ are geocentric ................... The heliocentric longitude of the earth ($YES$) differs by 180° from the geocentric longitude of the sun ($YES$, measured in direction of the arrow). A diagram showing a right-angled triangle PSM with sides SP, PM, and SM. The angle ESM is labeled as $\theta$. The diagram also shows a line segment EM extending from M to E, and another line segment YEM extending from Y to E. The diagram includes labels for points A, B, C, D, E, F, G, P, Q, R, S, T, U, V, W, X, Y, Z. The heliocentric distance $SP$ and the latitude $PSM$ being known, the right-angled triangle $PSM$ will furnish $PM$ and the curtailed distance $SM$. $SE$ and $SM$ being known, and the angle $ESM$ (the difference of heliocentric longitudes of planet and earth), $EM$ and the angle $SEM$ may be obtained, thence the angle $YEM$, which is the geocentric longitude. The geocentric latitude will also be readily obtained from the right-angled triangle $FEM$, of which $FM$ and $EM$ have been calculated. In a somewhat similar manner may a geocentric position be converted into a heliocentric one. 356. The heliocentric place of a planet can be calculated by the rules of elliptic motion, when certain quantities are --- CHAP. XXII.] THE PLANETS. 275 known which determine the dimensions of the elliptic orbit ; its position with reference to the ecliptic, and the position of the planet itself in the orbit at some known instant. These necessary data are called the elements of the planet, and are six in number. 1. The semi-axis of the ellipse, 2. The eccentricity, 3. The heliocentric longitude of the node, 4. The inclination, 5. The longitude of the apse, 6. The epoch. The line of nodes is the intersection $NQ$ of the plane of the orbit and the ecliptic, the point $X$, where the planet passes from the south to the north side of the ellipse, being the ascending node ($s$), and $NQ$ is the descending node ($d$). The heliocentric longitude of the node is always the longitude ($\tau SN$) of the ascending node, therefore (3) and (4) fix the position of the plane of the orbit. The position of the apse line, given by (5), indicates the direction of the major axis ; the magnitude of the ellipse is completely determined by (1) and (2). The epoch (6) is the longitude of the planet at a certain definite instant, and this being known, we may, by the rules of elliptic motion [Chap. xi., Art. 184], obtain the position of the planet in the orbit at any subsequent instant. 357. Before the place of a planet can be predicted, it is therefore necessary to determine its elements. Three complete observations of right ascension and declination will be sufficient; for from each observation can be inferred the corresponding geocentric latitude and longitude, and these will furnish two equations connecting the elements of the orbit. But although theoretically sufficient, it will be found practically more accurate and more simple to make observations at particular times, when the planet occupies selected positions. x 2 276 THE PLANETS. [CHAP. XXXL specially favourable for finding each element in turn. We shall refer the reader to Delambre's Astronomy. 358. To find the planet's periodic time. Let $T$ be one year, and $t$ the periodic time of a planet. Let the synodical period $S$ be observed, that is, the interval of time from conjunction to conjunction, or from opposition to opposition, or, generally, from any one position relatively to the sun to the same position again. Then $\frac{2\pi}{T}$ and $\frac{2\pi}{t}$ are the true angular velocities of the earth and planet, and $\frac{2\pi}{S}$ their relative angular velocity, therefore $$\frac{2\pi}{T} \sim \frac{2\pi}{t} = \frac{2\pi}{S};$$ therefore $$\frac{1}{t} = \frac{1}{T} - \frac{1}{S}$$ for a superior planet, $$\frac{1}{t} = \frac{1}{T} + \frac{1}{S}$$ for an inferior planet, whence, the periodic time of a superior planet is $$\frac{ST}{S-T},$$ and of an inferior one $$\frac{ST}{S+T}.$$ Stationary Points. 359. We shall proceed to shew how the stationary positions and the retrograde motions of planets are necessary consequences of the earth's own motion. If we consider elliptic orbits, the problem of the determination of the stationary points becomes a very complicated one; but, as these points are now objects of interest only from the fact of the ancients having been so puzzled to explain them, it will be sufficient to investigate their positions, considering the orbits as circles uniformly described round the sun as a centre. CHAP. XXIL.] THE PLANETS. 277 Let $S$ be the sun, $P$ and $Q$ two planets in conjunction, the direction of motion being indicated by the arrows. A diagram showing two planets, P and Q, in conjunction with the sun S. The direction of their motion is indicated by arrows. By Kepler's third law, period of $P$ : period of $Q$ :: $(SP)^{4}:(SQ)^{4}$, and if $Pp$, $Qq$, be small arcs described in the same time, $$\frac{FP}{FQ} = \text{vol. of } P : \text{vol. of } Q$$ $$= SP \times \text{ang. vol. of } P : SQ \times \text{ang. vol. of } Q$$ $$= \frac{SP}{SQ}$$ periodic time of $P$ : periodic time of $Q$ $$= 1 : \frac{1}{SP} : \frac{1}{SQ}$$ therefore, $PP > QQ$. Let $P_{0}$, $Q_{0}$ be positions of $P$ and $Q$ before conjunction, at the moment when the line joining them is a tangent to the inner orbit, and let $P_{n}$, $Q_{n}$ be the corresponding positions after conjunction. First, take the case of an inferior planet. Let $P$ represent it, and let $Q$ be the earth. While the planet moves from $P_{0}$ to $P_{n}$, let the earth move from $Q_{0}$ to $Q_{n}$. Take a fixed point $O$ to represent the supposed fixed position of the observer, and draw $OA$ parallel to $QP_{0}$, $Oa$ parallel to $QQ_{0}$. The apparent direction of the planet's motion will therefore be $\angle Aa$, in the direction of the arrow, i.e., direct. When the planet arrives at $P$ and the earth at $Q$. 278 THE PLANETS. [CHAP. XXIL draw $OB$ to represent the apparent direction; then, since $Pp > Qq$, the direction will change to $Ob$ parallel to $qp$, and the apparent motion is retrograde. Again, in the position $QpQ$, to which $OC$ is parallel, the motion of the planet alone, from $Pp$ to $pq$, would cause no change in the direction, but the earth moving to $q$, at the same time determining an apparent direction $Oe$ parallel to $g_pPp$, and the motion is again direct. While the earth, as agent, is passing from $Q$ to $Q_1$, the apparent motion of the planet changes from retrograde to direct; at some intermediate point, therefore, as $M$, the retrograde motion ceases in order to become direct, and the line $MN$, joining the two bodies, moves parallel to itself—the great velocity of $N$ being compensated by its moving more obliquely. The planet is then stationary. In the same way, it may be shown, that there must be a stationary point before conjunction, between $Q_1$ and $Q$. Secondly, consider a superior planet, and take $P'$ (same figure) now to represent the earth, and $Q$ the superior planet. When $S$, $F$, $Q$ are, as in the figure, in one straight line, the planet $Q$ is in opposition. Make the same construction as before, and produce the lines $AO$, $BO$, $CO$ to $X$, $Y$, $Z$; then $OX$, $OY$, $OZ$ are apparent directions of the planet, and a simple inspection of the figure will show that at $Q$ the apparent direction of movement will be retrograde, but direct at $Q_1$ and $Q_2$; that is, the motion of the planet seen from the other, will be exactly the same at the same time—both retrograde, both direct, or both stationary. 360. We may determine the stationary position analytically as follows : Let $c$ be the time from the positions $Q$, $P$ to the stationary points $M$, $N$. Join $SM$ and $SN$ (these lines are not drawn in the figure). Let $SM = a$, $SN = b$, and let $\phi'$ represent A diagram showing a superior planet's apparent motion relative to Earth. CHAP. XXII.] THE PLANETS. 279 the angles $M$ and $N$ of the triangle $SMN$. Also let $\alpha$, $\beta$ be the angular velocities of the bodies, $$QSM = \alpha t, PSN = \beta t, MSN = (\beta - \alpha) t.$$ In a small time $t$, $MN$ moves to $MN'$ parallel to itself, $$MN' = MN \sin n \cdot MN,$$ or, $$axt \cos \phi = -b\beta t \cos \phi.$$ By Kepler's law $$a : \beta : b : a : t,$$ therefore $$b\beta \cos \phi = -ab \cos \phi.$$ Also $$a \sin \phi = b \sin \phi,$$ $$\cos \phi = \sqrt{\left(\frac{a^2 + ab}{a^2 + ab + b^2}\right)},$$ $$\sin \phi = \sqrt{\left(\frac{b^2}{a^2 + ab + b^2}\right)},$$ $$\cot \phi = \sqrt{\left(\frac{a^2}{b^2} + 1\right)}.$$ (i), $$\cos \phi = \sqrt{\left(\frac{ab + b^2}{a^2 + ab + b^2}\right)},$$ $$\sin \phi = \sqrt{\left(\frac{a^2}{a^2 + ab + b^2}\right)},$$ $$\cot \phi = \sqrt{\left(\frac{b^2}{a^2} + 1\right)}.$$ (ii), $$\cos (\beta - a) t = -\cos (\phi + \psi),$$ $$= \sqrt{\left(\frac{ab}{a^2 + ab + b^2}\right)}.$$ (iii). (i) and (ii) determine the elongation of each planet as seen from the other at the moment when they appear stationary, and (iii) gives the difference of their heliocentric longitudes and the time from conjunction. The angles $QSM$ and $PSN$ can of course be found when $t$ is known. Transit of Venus. 361. Since the orbit of the planet Venus is within that of the earth, it will sometimes happen that the planet will find itself immediately between the sun and the earth, and a phenomenon analogous to an annular eclipse of the sun will 280 THE PLANETS. [CHAP. XXIL take place, except that the planet will hide only a very small part of the sun, and will appear like a dark spot moving across the sun's disc. The motion also, contrary to that of the moon in a solar eclipse, will be from east to west, because the planet is then retrograding. This, which is called a transit of Venus, furnishes the most accurate method for the determination of the sun's parallax. The phenomenon, being caused by the interposition between us and the sun of a nearer opaque body, will, as in the case of a solar eclipse, present local features which will vary from one observer to another. Neither the beginning nor the end of the transit will happen at the same absolute instant of time for all observers, nor will the chord of the sun's disc, which Venus seems to describe, be the same for all. These variations are due to the different distances of the two bodies. Venus being then at only about 1/4ths of the sun's distance, a displacement of the observer will cause a much greater apparent displacement of the planet than of the sun. The non-simultaneous occurrence is the basis of Delisle's method of determining the parallax; the non-coincident path is the basis of Halley's. We shall give a slight outline of the principles of these two methods. The necessary calculations for strict investigations are long, being of the same character as those for a solar eclipse, and we shall refer the reader to Chauvenet's *Astronomy* or to Delambre's *Astronomy*, vol. ii.* 362. Delisle's Method. Consider a cone enveloping both the sun and the earth with its vertex beyond the earth. That generating line of the cone which is first touched * Mr. Proctor's valuable papers on the transit of Venus are in the highest degree suggestive, and show not only a complete mastery of the problem, but a power of putting it in a clear and simple light, which but few writers possess. Mr. Proctor's writings on astronomical subjects deserve, and will well repay, careful study. CHAP. XXL.] THE PLANETS. 281 by Venus will determine on the earth the place of first contact. Consider a second cone enveloping the sun, and with its vertex at the centre of the earth; the transit will begin for the centre of the earth when Venus first touches this cone. Lastly, consider a double cone envelooping the earth and sun, and having its vertex between them (the distance of this vertex from the earth will be only $\frac{1}{130}$ of the sun's distance); the first contact of Venus with this cone will determine that place on the earth's surface at which the beginning of the transit is most retarded. For intermediate stations the transit will begin at intermediate times, the ingress being accelerated for some and retarded for others. The interval between the earliest ingress and the latest will, when the transit is a central one, amount to about 12 minutes; in other cases it will be longer—in 1874 it was 25 minutes. During this interval the earth will have turned on its axis through only a small angle, and the places of most accelerated and of most retarded ingress will be nearly antipodes one of the other. At both places the sun will be just in the horizon. If now, at any two stations situated as near these points as will be most practicable and convenient, the instant of ingress* be observed, and if† the longitudes of these stations are accurately known then the difference of absolute times will be known. Since this difference is entirely owing to the excess of the parallax of Venus over that of the sun, it will be possible to calculate this difference and the ratio of the parallaxes is known from the known ratio of the distances, therefore the parallaxes themselves can be found. The parallaxes will also be similarly determined by * The contact here spoken of must be the internal contact of Venus and the sun; the external contact of ingress could scarcely be determined with accuracy, since Venus would be perceived only after having encroached on the sun's disc. 282 THE PLANETS. [CHAP. XXII. observations on the egress of the planet at places near where the ingress is most accelerated and where it is most retarded. 363. Halley's Method. Instead of considering the absolute interval between an accelerated and a retarded ingress, let us take two stations in opposite hemispheres, at each of which both the ingress and the egress can be seen, but such that the chords of the sun's disc which the planet seems to describe may be widely separated. These chords, having different lengths, will be described in different times ; and if the instants of ingress and of egress be observed at each station, the difference of duration of the two transits will be known. This difference is again owing to the difference of the parallaxes, and will furnish an equation for determining it, which, with their known ratio, will, as in the former method, lead to the parallaxes themselves. 364. The advantage of Halley's method consists in its requiring only the duration of the transit to be accurately measured, and an ordinary chronometer whose rate is known will do this ; so that no lengthened stay on what would probably prove an uninhabitable land is necessary, nor any delicate observation for the determination of the longitude. The advantage of Delisle's method consists in its requiring only the beginning or the end of the transit to be observed ; so that many more stations will be suitable to the application of the method, and the difficulty of ascertaining the longitude of each station with the necessary accuracy will probably be more than compensated by the multiplied observations which can be made. 365. The following additional remarks will explain a little more fully how the sun's parallax is deduced from the observations. We shall consider Delisle's method : A diagram showing two stations observing a transit of a planet. CHAP. XXIL.] THE PLANETS. 283 Let $T$ be the Greenwich time, approximately known, at which the first internal contact takes place for an observer at the centre of the earth. Suppose an accelerated ingress to have been observed at the time $T-t$ at some place whose position is accurately known. Let $\alpha$, $\beta$ be the differences of right ascension and declination of Venus and the sun at the time $T$, and let $\alpha$, $\beta$ be the relative horary changes, so that $a=\alpha t$, $b=\beta t$ are the values at the instant of observation. Let $\kappa$ be the sun's parallax, $\lambda$ that of Venus ; $k$ being the known ratio of the distances at the moment of observation. The effects of parallax on the right ascension and declination ($\delta$) of the sun for the given place at the instant of observation will be $\omega X$ and $\varphi Y$, where $X$ and $Y$ are certain functions of the hour angle, declination and latitude. Since the positions are nearly coincident, the corresponding corrections for Venus will be $\kappa \omega X$ and $\kappa \varphi Y$, and the apparent differences of right ascension and declination at the moment of observation are $$a + at - (k-1)\omega X, b + bt - (k-1)\varphi Y.$$ If $c$ be the difference of semi-diameters of Venus and the sun, $$a^{\prime} c \cos^2 b + b^{\prime} c \sin^2 b,$$ $$[a + at - (k-1)\omega X] \cos^2 b + [b + bt - (k-1)\varphi Y]^2 = c^2;$$ whence, subtracting, and neglecting squares and products of the small quantities $at$, $bt$, and $\omega$, $$t = \frac{aX \cos^2 b + bY}{ax \cos^2 b + bY} = Mw,$$ so $$t' = M'w,$$ $M'$ being the value of $M$ at the second station, where a retarded ingress is observed at the time $T+t'$; then $$w = \frac{t+t'}{M+M'}$$ will be known, 284 THE PLANETS. [CHAP. XXII. because $t + \tau$, the absolute time interval, is known from the observations, when the longitudes of the two places are accurately known. 366. There will also be transits of the planet Mercury, and they will recur much more frequently than those of Venus ; but, on account of Mercury's proximity to the sun, the intervals between the accelerated and the retarded ingress or egress will be much less, and the durations of transit be much more nearly equal than in the case of Venus ; and it is especially desirable to observe these intervals and on the differences of the durations that the accuracy of the methods depends, the transits of Mercury are comparatively valueless for the determination of the sun's parallax. 367. The transits of Venus take place at irregular and distant intervals. The last occurred in 1882, and only five have yet been seen, viz. in 1639, 1701, 1769, 1874, and 1882. That of 1769 was observed from a great number of different stations on the surface of the earth, and the results of the observations gave the parallax of the sun's $\approx$57°. This value, however, was shown by observations of an entirely different character, made some years ago, to be considerably too small, and astronomers of all nations looked forward with special interest to the transits of 1874 and 1882, as likely to afford a certain solution of the question. With this view, in addition to the present number of observations, the leading governments of the world established stations at widely separated points of the earth's surface, to which daily accredited observers were sent. Various modes of observation were adopted, all resolving themselves, however, into the two methods of determining the sun's parallax, to which reference has already been made. The results of the numerous and careful observations CHAP. XXIL.] THE PLANETS. 285 thus taken shew the parallax of the sun to be about 8°-80°; probably somewhat greater, and, with our present knowledge, it seems hardly possible to approximate more closely to the exact value. A transit does not occur at every conjunction, for the same reason that an eclipse of the sun does not take place at every new moon. The inclination of the orbit gives to the planet a latitude which places it beyond the sun's disc; and thereby avoids a transit, except when the conjunction happens near one of the nodes. The intervals between the transits are alternately a short one and a long one. The short ones are always of 8 years' duration, and the long ones alternately 121½ and 105½ years, as the following table from Delambre's Astronomie will shew :
Dec. 6, 1631 June 7, 2004. 121½ years.
Dec. 4, 1639... 8 years. June 5, 2002... 8
June 5, 1761... 121½ Dec. 10, 2117... 105½
June 3, 1769... 8 Dec. 8, 2125 8
Dec. 8, 1874... 105½ &c. &c.
Dec. 6, 1882... 8 &c. &c.
The list, continued according to this law, must be looked upon as giving the years when a transit may be expected ; but an exact investigation must be made of the positions of the Sun and Venus before we can assert that a transit will or will not happen.
To ascertain for the intervals which separate the transits :— We now know that the periodic times of Venus and of the earth round the sun are respectively 224½ days and 365-256 days, so that the synodal period $S$, obtained from the equation $$\frac{1}{S} = \frac{1}{224\frac{1}{2}} - \frac{1}{365\frac{256}{2}}$$ will consist of 584 days,
will consist of 584 days,
Professor Harkness of the U.S. Naval Observatory, in his Memoir "The Solar System", published in London, has made the only published attempt to summarize our present knowledge on the subject; and he gives, as the result of his calculations from all available data of every kind, the value $S$=803±607.
A table showing intervals between transits of Venus. * Professor Harkness of the U.S. Naval Observatory, in his Memoir "The Solar System", published in London, has made the only published attempt to summarize our present knowledge on the subject; and he gives, as the result of his calculations from all available data of every kind, the value $S$=803±607. 286 THE PLANETS. [CHAP. XXIL 5 synodical revolutions will therefore require 2920 days, 8 revolutions of the earth................. 2922 days ; therefore eight years after a transit another conjunction will recur in the same part of the heavens very nearly, and where the planet was at the first near its node. We might thus almost say that we have a transit of Venus every eight years ; but it is found that at the second of two such consecutive transits the latitude of Venus will differ from its former value by from 20° to 24°. Another interval of eight years will bring a conjunction, at which the latitude of the planet will differ by from 40° to 48° from the first. This being greater than the sun's diameter, no third transit can take place ; and we must wait until the conjunctions occur near the other node, which, it appears, will require a period of 105$\frac{1}{2}$ or 121$\frac{1}{2}$ years. Phases of the Planets. 368. Like the moon, the planets are opaque ; and the light we receive from them is reflected sunlight. They will, therefore, present phenomena analogous to those of the moon. It will be easily seen that in the case of the inferior planets Mercury and Venus, the extreme angle of elongation $SOV$ (fig. p. 284) may have all values from $0°$ to $180°$, and, therefore, the phases will range through all values from a full round disc to a thin crescent, which itself vanishes at inferior conjunction. But, in the case of the superior planets, the angle which the radius of the earth's orbit subtends at the planet will always be very acute, and the supplementary angle, on the versine of which the phase depends (Art. 300), will always be very obtuse, so that nearly the whole of the disc will at all times be illuminated. The inferior planets will present a half-disc when at their greatest elongation or angular distance from the sun, that is, when the line from the earth to the planet touches the CHAP. XXL.] THE PLANETS. 287 planet's orbit. The greatest elongation of Venus varies from $45^\circ$ to $47\frac{3}{4}^\circ$, that of Mercury from $16^\circ$ to $24\frac{3}{4}^\circ$.* 369. To find when Venus appears brightest. The variations in the brightness of Venus are due, not only to the alteration in the phase of the planet, but also to the change in its distance from us. When approaching the earth, the diminution of illuminated area tends to diminish the brightness, but the distance is decreasing, and as the brightness varies inversely as the square of the distance, this latter cause will, at first, more than compensate for the loss due to the former, and Venus is found to be brightest when between its greatest elongation and inferior conjunction, at about $49^\circ$ from the sun. We shall investigate the position of greatest brightness on the supposition that the orbits both of the planet and of the earth are circular. Let $S$, $E$, $V$ be the sun, the earth, and Venus. Let $SE$, or $a$, and $SV$, or $r$, be the radii of the two orbits. $EV$, or $x$, the variable distance of Venus from the earth. Then $$\text{brightness} \propto \frac{\text{area of phase}}{x^2} = \frac{1 + \cos V}{x^2} = \frac{1 + r^2 - x^2 - a^2}{2rx}$$ Therefore, equating to 0 the differential coefficient with respect to $x$, $$\frac{2}{x^2} + \frac{3(a^2 - r^2)}{2rx} - \frac{1}{2rx} = 0,$$ $$x^2 + 4rx - 3(a^2 - r^2) = 0.$$ *These variations are due to the elliptic forms of the orbits. The eccentricity of the orbit of Mercury is much greater than that of Venus.* (A) 288 THE PLANETS. [CHAP. XXII.] whence $$x = \sqrt{(3a^2 + r^2)} - 2r \ldots (B),$$ determines the distance of the planet from the earth when the brightness is a maximum. Again, substituting in $(A)$ for the sides the sines of the opposite angles, to which they are proportional, we have $$\sin(V + E) + 4 \sin(V + E) \cos E = 3 \sin(V - \sin E),$$ $$\sin(V + E) + 4 \sin E - 3 \sin(V - E) = 0,$$ $$-2 \sin V + 4 \cos V \tan E + 4 \tan E = 0,$$ $$2 \tan E = \frac{\sin V}{1 + \cos V} = \tan \frac{V}{2} \ldots (C),$$ an equation given by Halley. By means of this and $$\frac{\sin V}{\sin E} = \frac{a}{r},$$ we may deduce $$\cos^2E + \frac{4}{3} r \cos E = \frac{4}{3} \ldots (D),$$ which determines the corresponding elongation of the planet. 370. We shall not enter into a description of the different planets. They vary considerably in size, from Jupiter, whose equatorial diameter exceeds 30,000 miles, to the minor-planets, some of which are probably less than 100 miles in diameter. Some of the planets have several moons or satellites; others have none. The moon, in addition to eight moons, is surrounded with several flat rings; but for a full account of these and other interesting details we shall refer to The Solar System, by J. R. Hind, Herschel's Outlines of Astronomy, Lardner's Handbook of Astronomy, &c. A diagram showing the orbits of the planets around the sun. ( 289 ) APPENDIX. THE CALENDAR. 371. WHAT we mean by the date of any event is the interval of time that separates that event from some definite epoch or era. For more than five centuries after the birth of Christ, the era in use was still, as before, the foundation of Rome. In the year 502 a Syriac monk, Dionysius Exiguus, proposed that all Christians should henceforward take the Saviour's birth as the starting point for the expression of dates ; and, from the result of his researches, he inferred that the birth took place on the 25th of December in the year 753 c.c.; but, as the beginning of the Roman civil year was on the 1st of January, i.e. only seven days later, it was found more convenient to make the Christian year begin on the same day. It was, therefore, decided that the year 1 of the Christian era should coincide with the year 754 of the foundation of Rome. It must be remarked that there is no year 0 A.D., reckoned by chronologists—the year of the birth, which is, the year 753 c.c. being called by them the year 1 B.C.; the year preceding that, the year 2 B.C., etc.; so that to find the number of years between corresponding days of two years, the one B.C. and the other A.D., we must subtract 1 from the sum of the two. We must refer to other works for the history of the various kinds of years which have, at different times and by different nations, been adopted, and we shall confine our attention to the Julian and the Gregorian calendars. w 290 THE CALENDAR. [APPENDIX.] 372. For the measure of intervals of time connected with civil life there are two natural units—the solar day and the tropical year. The solar day as indicated by the successive returns of light and darkness, and the tropical year by the periodical changes of the seasons. But it is obviously an essential condition of a unit that it shall be invulnerable, i.e., the solar day is not so; and another essential condition when several units are employed, is that they shall be commensurable, which is not the case with the solar day and the tropical year. The first difficulty is got over by taking for our unit the average of all solar days, or the mean solar day, as it is called (Art. 191), which answers the condition of perfect uniformity, and which, from the fact of mean noon being never very distant from apparent noon, is also in accordance with the recurrence of light and darkness. The mean day, therefore, with its subdivisions into hours, minutes and seconds, is the chronometric unit for short intervals of time. The difficulty connected with the larger unit is of a different kind, and cannot be obviated in the same way. The tropical year is very nearly uniform; but unfortunately it does not contain an exact number of days; and it would be an obvious inconvenience that a fraction of one day should belong to one year and the remainder to one of the same day to another year. A greater length is adopted for the civil year; it must consist of an integral number of days, and yet, however, retain a close connection with the tropical year, which contains 365-24216 days. This can be managed by having civil years of two different lengths, the one less and the other greater than the tropical year ; and the object of the calendar is to give definite rules * It is found that the ecliptic changes its position in space under the influence of the planets. This change is very slow—about 48" in a century—and produces a slow variation in the length of the tropical year, which is now about 4214 shorter than in the time of Hipparchus. A diagram showing the relationship between solar days, mean solar days, and tropical years. APPENDIX.] THE CALENDAR. 291 for the order of succession of these two years of 365 and 366 days respectively, so that the connection may be preserved in the simplest manner. 373. Soisiguen, an Alexandrian astronomer, employed by Julius Caesar to correct the confusion into which the calendar was then daily falling, proposed the ingenious contrivance of blestilile or leap year, whereby every year of 365 days were to be followed by a year of 366 days, thus giving to the average civil year a value of 365-25 days, which is a little more than the tropical year, the difference '00784 days amounting to 1 day in about 128 years, or rather more than 8 days in 400 years. This important change came into operation in the 44th year b.c. The next correction was made in 1582 A.D. by Pope Gregory XIII., with a view to take into account this difference of 3 days in 500 years ; and, by this means, to avoid a change which was gradually bringing the festival of Easter more and more into the summer season, whereas the ecclesiastical regulations required that it should be celebrated just after the spring equinox. The Gregorian calendar, which is now adopted by all Christians, except the Russians and the Greeks, is established on the following principle—Three common years of 365 days are to be followed by a year of 366 days as in the Julian calendar (the lowest number divisible without remainder by 4 without remainder), except when the fourth year terminates a century, as 1800, 1800, &c., and then it becomes a common year; except again, when the hundreds are divisible by 4, as 2000, 2400, &c., when it remains a leap year, as the Julian calendar would make it. In 400 civil years, therefore, as determined by the Gregorian rule, there will be 97 leap years instead of 100, and the average length of the civil year will be $365\frac{2}{5}$ days or $365\cdot24216$ days. The tropical year contains $365\cdot24216$ days, u2 292 THE CALENDAR. [APPENDIX. so the Gregorian rule makes the average civil year too long by 0-000284 days, producing an error of 1 day in about 4000 years.* In the year 1582, when Pope Gregory made this reformation, he also omitted 10 nominal days of the month of October, the day after the 4th being called the 15th. This was done for the purpose of bringing back the vernal equinox to the 21st of March, which was the date of its occurrence in 325 A.D., while the Council of Nicaea had laid down a rule was framed on the observance of the festival of Easter. The new style, as it was called, was not adopted in England until the year 1752, when 11 days had to be omitted, and the month of September in that year contained only 19 days, which were numbered 1, 2, 14, 15, &c. In Russia the old style is still maintained, and the year 1800 has added another day to the difference of styles, so that the dates in Russia are now 12 days behind ours. Traces of the old style still linger with us in our ‘Old Christmas-day,’ ‘Old Lady-day.’ Golden Number. 374. In connection with the calendar and its arrangements, we cannot pass over Moton’s lunar cycle, which is the basis of the ephemerisical rule for the determination of movable feasts—Easter, Whitsuntide, &c. In the earlier stages of astronomical science, when the festivals connected with the worship of the gods were dependent on the lunar phases, the difficulties attending the prediction of the days on which these festivals should be * The Persian method of interpolation deserves to be mentioned. It was introduced in the 11th century by Al-Biruni, who followed Gregory's reformation, and is even more exact.—Three years of 361 days are followed by a year of 366 days, as in the Julian calendar; this does seven times in succession, but six times in alternation; hence there are three years of 366 days in every year of 864 days. Thus in 50 years there are less than long years. Therefore, the average length of the Persian civil year is 365.242436 days or 365.242436 days, which is an excess of one present length of the tropical year by 0-000284 days or 1 day in about 4000 years.
374. In connection with the calendar and its arrangements,
we cannot pass over Moton’s lunar cycle,
which is the basis of the ephemerisical rule for
the determination of movable feasts—Easter,
Whitsuntide, &c.
In the earlier stages of astronomical science,
when the festivals connected with the worship
of the gods were dependent on the lunar phases,
the difficulties attending the prediction of
the days on which these festivals should be
* The Persian method of interpolation deserves to be mentioned.
It was introduced in the 11th century by Al-Biruni,
who followed Gregory's reformation,
and is even more exact.—Three years of 361 days are followed by a year of 366 days,
as in the Julian calendar; this does seven times in succession,
but six times in alternation; hence there are three years of 366 days in every year of 864 days.
Thus in 50 years there are less than long years.
Therefore, the average length of the Persian civil year is 365.242436 days or 365.242436 days,
which is an excess of one present length of the tropical year by 0-000284 days or 1 day in about 4000 years.
APPENDIX.] THE CALENDAR. 293 celebrated, were very great. About 433 b.c., Meton, an Athenian astronomer, discovered a relation between the length of the lunar month and the tropical year, which gave a simple rule for the solution of the problem. He found that after nineteen years the phases of the moon will recur on the same days of the same months ; so that, without any calculation, it was only necessary to observe and to record the days of full moon during one of these periods in order to know those for succeeding periods of nineteen years. These dates were ordered to be inscribed in letters of gold upon the public monuments, and in the modern use of the cycle of nineteen years, the number which marks the rank of any one of the years in the cycle, is still called the golden number of that year. To show how closely the Metonic cycle brings back the same phase of the moon to the same day of the year :—We know that the mean length of a lunation is 29^5306 days, and therefore 235 lunations occupy $$235 \times 29^{\circ}5306\mathrm{~d.~}=6939^{\circ}69\mathrm{~days},$$ the length of the tropical year is 365^{\circ}242 days, and 19 tropical years consist of $$19 \times 365^{\circ}242\mathrm{~d.~}=6939^{\circ}60\mathrm{~days},$$ and the difference in 19 years is only about 2 hours. The first year of a cycle may be chosen arbitrarily, and it is found that the year 1 b.c. would begin the cycle now in use ; therefore, to determine the golden number : Add 1 to the date and divide by 19, the remainder is the golden number. When the remainder is zero, the golden number is 19. Thus, the golden number for 1894 is 14. Moon's Age. Epaet. 375. The moon's age at any time is the interval that has elapsed since the last new moon, that is, since conjunction (Art. 302). 294 THE CALENDAR. [APPENDIX.] The moon's age on the 1st of January is called the epace for that year. Now it happens, whether by accident or by design, that the first year of the cycle of golden numbers, that is, the year whose golden number is 1, has new moon on the first of January; therefore, the epoch is 0 when the golden number is 1. A tropical year consists of 365+24 days. 12 lunations occupy $29\cdot5808 \times 12$ or $854\cdot367$ days. Therefore a tropical year contains 12 lunations and 10\cdot875 days. At the beginning of the first tropical year of a golden cycle the moon's age will be............ 0\cdot000 days. At the beginning of the second it will be..... 1\cdot075 ...... At the beginning of the third...... 2\cdot750 ...... At the beginning of the fourth ..... 32\cdot625 or subtracting one lunation ..... 29\cdot531 and so on. If the civil year and the tropical year were of the same length and began together, we should have the following rule for finding the age of the moon at the beginning of each year: Subtract 1 from the golden number, multiply by 19\cdot875 and divide by 29\cdot531, the remainder would be the moon's age. But the lunations are not always of the same length, as we have here supposed; and besides, the necessity for making the lunar months equal in excess of days, and the consequent adjustment, by means of leap years, would interfere with this, and the following is the rule adopted: Diminish the golden number by unity, multiply by 11 and decide by 30. The remainder is the epoch. In some years of each cycle of 19, the difference between the epochs so found and the moon's age on the first of January may exceed one day; but the rule, besides being simple and sufficiently approximate, gives a definite law for the determination of the movable feasts of the calendar, which is now the only important use of the epoch, A table showing lunar cycles and their corresponding ages. APPENDIX.] THE TIDES. 295 THE TIDES. 376. The waters of the ocean, even in the calmest weather, and without any perceptible cause, rise and fall at nearly regular intervals; covering and uncovering broad expanses of sand along the coast, and otherwise changing the depth of the sea, in some instances by forty or fifty feet, and even more. This phenomenon is too striking and too important not to have, from the earliest times, attracted attention. The rise and fall takes place twice every day, or, rather, twice in every lunar day, which occupies about 29.5 days; and the obvious connection between the tides, as these phenomena are called, and the motion of the moon, led philosophers to attribute them to the action of the moon, long before the true nature of that action was known to them. The explanation of the tide has been found in the theory of universal gravitation, which proves them to be necessary consequences of the joint attraction of the sun and moon combined with the rotation of the earth on its axis; and it may be shewn that the variations in the character of different tides are fully and satisfactorily accounted for by changes in the distances and relative positions of those luminaries. But the prediction of the circumstances of high and low water at any particular place becomes a problem of extreme difficulty, owing to local causes of disturbance which are unknown, or which we know not yet how to introduce into the investigation. Such are—obstacles to the fluid motion in the shape of headlands and islands, the varying depths at the bottom of the sea, the friction of the particles of water among themselves and against the shores, &c. 377. It is a strictly hydrodynamical problem, and as such was treated by Laplace, who, the first, attempted a general A diagram showing a cross-section of a tidal wave passing over a sandbar. 296 THE TIDES. [APPENDIX] solution on mathematical principles; but the difficulties attending this method of investigation are so great that he could not solve it in its generality, and he limited it by starting "with two suppositions which are inapplicable to the state of the earth." These are—that the earth is covered with water, and that the depth of this water is the same through the whole extent of any parallel of latitude." The late Astronomer Royal, in his popular work "Tides and Waves" in the Encyclopaedia Metropolitana, has fully discussed Laplace's Wave-theory, and has further advanced the problem by considering the motion of the tidal wave in a canal, under a variety of different circumstances as to depth, friction, &c. This supposition cannot apply to all cases, and leaves the general theory still imperfect. Newton, in the Principia, Lib. 1., Prop. 66, Cor. 19, considered an equatorial belt of water forming a canal round the earth, and inferred that there would be low water under the moon. Sir G. Airy's investigation confirms this conclusion, but shews also that its truth depends on the depth of the canal; and that if the depth exceeded a certain amount (about 12 miles), there would be high water under the moon instead of low.* The problem of the tides is one of the most difficult in the whole range of mathematical subjects, and any further account of it would be foreign to, and beyond the scope of, the present work. We shall, therefore, in words what is known as the equilibrium-theory of Bernoulli, a theory confessedly inadequate and even based on erroneous principles (giving high water under the moon, irrespective of the depth); but yet, before Laplace's time, it was the only really successful attempt to determine the laws of tidal action and to reduce them to numerical computation. * See an interesting paper by Mr. D. D. Heath in the Phil. Mag., March, 1867. APPENDIX.] THE TIDES. 378. Bernouilli supposed the earth to have a spherical solid nucleus surrounded by a shell of water of uniform depth; then he demonstrated that if one of the bodies—say the moon—were always vertically over the same point of the surface, the waters would take the shape of a prolate spheroid pointing to the moon. With this supposition $AB$ represent the solid earth surrounded by a shell $HK$ of water, $AH$ being the part just under the moon $m$, and $BK$ the part most remote. The attraction of the moon, varying inversely as the square of the distance, will be greater on the parts of the earth A diagram showing a spherical Earth with a moon orbiting around it. The Earth is divided into hemispheres, and there are lines indicating the positions of high and low tides. and on the waters about $A$ than on the centre $C$, and this last greater again than on the parts about $B$. But the cohesion of the solid nucleus hinders any deformation, whereas the body of fluid may take a new form, and a disturbance of the water relatively to the earth will take place owing to the difference of attractions. The waters at $H$, and over all the hemisphere $IAG$, which is nearest to the moon, will be more attracted than the solid earth, and, flowing from these sides as far as the boundary $IG$ towards $B$, they pile themselves up just under the moon and produce high water. A similar effect will take place on the exactly opposite side of the earth at $B$, from the attraction on that half of the waters being less than on the earth itself, so that they are, as it were, left behind, and the waters flowing from $IG$ towards $B$ produce high water there at the same time as at $A$. 298 THE TIDES. [APPENDIX. The heaping up of the waters on two opposite sides of the earth is due, then, not to the absolute attraction of the moon, but to the inequality in the value of this attraction at different distances. There is, however, one point which deserves notice, because a misconception may easily arise ; it is, that although the attraction on the waters differs from that on the earth by a greater quantity at $A$ than at any other place, this attraction will not draw the water away from $A$. Its only effect will be to diminish in an imperceptible degree the gravity acting towards the earth ; and if a ridge of land separated the waters from those which cover the rest of the earth, that is, if a small lake existed at $A$, the waters of this lake would not be drawn from the earth, and there would be no high tide.* The high water at $A$ is due to the action of the moon on those parts of the hemisphere $IAG$, at a considerable distance from $A$, where the inequality of attraction mentioned above, although of less intensity than at $A$, acts in an oblique direction, and does not find itself entirely counteracted by gravity. Exerting a tangential action which has no opposing force, it causes the waters to flow towards $A$. The same remarks will apply to the high water at $B$. But the moon does not remain constantly over the same place. The rotation of the earth about its axis $PP'$, independently of the motion of the moon itself, changes the relative positions rapidly ; and the theory, in order to adapt itself to the actual case, is obliged to make the supposition that the spheroidal form of the waters is in- stantaneously assumed in each new position of the earth. So that an observer situated on any meridian will have * In the same manner, and for the same reason, the weight of a man is diminished as the moon approaches his months ; and the effect of the sun being of the same kind, though of less intensity, we may say that at the time of new and full moon, when the two luminaries are together, the weight of a man is less at noon and midnight than at midday or sunset. APPENDIX] THE TIDES. 299 high water when his meridian comes into the position $PAP'$, and again, some twelve hours after, when it comes into the position $PBP'$, having in the interval passed through a position of low water. This explains the two tides observed in each lunar day. Another consequence also, which is evident from the figure, is that, except when the moon is in the equator, the apex of the waters will pass nearer to the observer, and the tide, therefore, will be higher at one of these transits than at the other. As the earth rotates about its axis from west to east, the tide wave will obviously travel from east to west. 378. All that has been said about the effect of the attraction of the moon in raising the waters will equally apply to the sun, but the sun's action will be only five-elevenths of the moon's; the enormously greater mass of the sun being more than compensated by the increased distance. There will then be two solar tides in each solar day, but the lunar tides being so much more powerful will override the solar ones ; and the tides actually observed will be those due to the moon, hastened or retarded, increased or diminished, as the case may be, by combination with the solar ones. Thus, at new-moon the sun and moon are in the same meridian, and at full-moon in opposite meridians; and, in either case, the two tidal waves coincide and a joint tide is produced, which is the sum of the two separate ones. These are called spring tides. But in the quarters, the two bodies are 90° apart, and high water from the sun corresponds to low water from the moon, and vice versa. The tide, therefore, has a diminished height, being the excess of the lunar over the solar tide. These are called neap tides. From new-moon to the first quarter, and from full-moon 300 THE TIDES. [APPENDIX. to the third quarter, the crest of the solar wave will be to the westward of the lunar wave, and the combination will produce a crest a little to the west, that is, in advance of the lunar wave, and high water will be hastened. In the same way during the second and fourth quarters, high water will be more or less retarded. This advance and retardation of the time of high water on the mean or average interval is called the *priming* and *lagging* of the tides. 380. Whether we suppose the crest of the tidal wave to be formed under the moon, or 90° from it, or in any other relative position, its advance will be modified by the interruption of the land, by the varying depth of the sea, and by a variety of other causes which will delay the formation of the wave, and the crest will follow the moon at a greater or less interval. This interval is widely different at different places, and even in the same port it varies considerably, owing to the priming and lagging described above. The difference between the instant of the moon's transit across the meridian on the day of new or full-moon, and the subsequent high water, is called the vulgar establishment of the port. The name corrected establishment has been given by Dr. Whewell to the mean of all intervals taken every day, under all circumstances ; and this mean is found not to be the same as the interval taken on the day of new or full-moon alone. Cotidal-lines are curve lines which connect all those places which have high water at the same absolute instant of time ; that is, all those places through which the crest or ridge of the tide-wave passes at the same instant. 381. In the motion of the tidal wave, we must remember that it is no bodily translation of the waters, but only an alteration of their form which produces the wave. There will, however, be currents caused by the tide, especially in 2 APPENDIX.] THE TIDES. 301 narrow channels; and it is found that these currents change their direction at different states of the tide, but not generally at the same time that the waters change level and fall. Thus the current which flows forward at the time of high water continues to flow in the same direction for some two or three hours longer, and is called the flood. The return current, which accompanies low water and continues to flow back for two or three hours after, is called the ebb. The change from flood to ebb, or from ebb to flood, is called slack water. 382. We have said that the effect of the moon in producing tides is about 24 as great as the sun's. The following investigation will shew this, referring to the fig., p. 257. Let the mass of the moon be represented by $M$, that of the earth by $E$, and let $r$ be the radius of the earth, and $p \times r$ the distance of the moon from $C$. Then, $g$ being the acceleration of gravity at the surface of the earth, $$\text{attraction of moon on } C : g = \frac{M}{E} \cdot \frac{E}{p^2} \cdot \frac{r}{r}.$$ Therefore, attraction of moon on $C = \frac{M}{E} \cdot \frac{q}{p^2}$, so, $$\text{attraction of moon on } H = \frac{M}{E} \cdot \frac{q}{(p-1)^2},$$ hence, disturbing force of moon on waters = $\frac{M}{E} \cdot \frac{q}{p^2}$ approximately. Similarly, if $S$ be the mass of the sun, and $nr$ its distance from the centre of the earth, the disturbing force of sun on waters = $\frac{S}{E} \cdot \frac{2g}{n^2}$. Now $E = 80M$, and $S = 322000E$ (see note, p. 205), $p = 60$, and $n = 23000$ (see Art. 262); therefore action of moon = $\frac{(32000)}{(23000)} = 1.4$ action of sun = $\frac{80 \times 322000}{(60)^2} = 2.2$. A diagram showing the relationship between the masses of the moon and sun, their distances from Earth, and their effects on tides. 302 GREAT CIRCLE CHARTS. [APPENDIX.] *** GREAT-CIRCLE AND MERCATOR'S CHARTS. 383. At sea, the problem the sailor has constantly presented to him is,--how to shape his course so as to arrive at his destination. For this he must be furnished with maps or charts on which the coast line of continents and islands is correctly traced, and all rocks and dangers properly indicated, together with the meridians and parallels. Then, day by day, he must determine the position of his ship, by finding his latitude and longitude according to the rules of Nautical Astronomy, some of which have been given in Chapters ix. and xxi. He must mark the place so found on the chart, and then determine the course or courses to be steered from this point to the port to which he is bound, or to those intermediate places where he must pass in order to avoid obstacles and dangers. Of the various curves or lines by which he may go from one point to another on the surface of the sphere, there are two which deserve particular attention. One is the shortest track, that is, the arc of a great circle passing through the two points; the other is the curve called a loxodrome, which meets every meridian at the same angle. In the great circle track, the course or the angle which the great-circle makes with the various meridians is not constant, and the lengthy calculations necessary for finding these courses have always proved a drawback to its use among seamen; whereas the simplicity of keeping of one course, and the facility with which the loxodrome is drawn on the chart invented by Mercator, have caused the universal adoption of the method, notwithstanding the greater distance which it was well known the ships would have to go over. APPENDIX.] **MERCATOR'S CHARTS.** 303 Of late years, the great extension of our commerce, the introduction of steam vessels, and our increasing intercourse with Australia and other distant parts of the earth, have revived great-circle sailing, by the use of which the Australian journey can be shortened 1000 miles or more. 384. It is obvious that, since a spherical surface cannot be developed into a plane, a chart, however constructed, must give a distorted representation of the surface, some parts being unduly enlarged or contracted in proportion to others; and it is also obvious that the nature of the curve which represents the ship's intended track will depend on the character of the projection on which the chart is constructed. If we suppose the eye at the centre of the sphere and the projection to be made by drawing lines through each point of the surface to meet a tangent plane, then every great-circle will be projected into a straight line, and the shortest route from point to point upon this line, be the straight line itself, will be that chosen. There is evidently the simplest kind of charts for the purposes of great-circle sailing, but we must refer elsewhere for a full description of their construction and use.* 385. Mercator's chart is constructed on the principle that every loxodrome shall be represented by a straight line. Perhaps the simplest idea of the construction will be obtained as follows : Conceive a sphere representing the earth with all the meridians, parallels, coast-lines, &c., traced upon it, to be made of elastic material, and let a cylinder surround this sphere touching it along the equator. Now, imagine * See a paper by the author in the Transactions of the Cambridge Philosophical Society, Sec. 2, vol. viii., p. 167. This was designed for facilitating great-circle sailing. By means of this, the determination of the track, with its various courses and distances, is rendered as easy as Mercator's sailing. A diagram showing a sphere with meridians and parallels drawn on it, and a cylinder surrounding it touching along the equator. 304 MERCATOR'S CHARTS. [APPENDIX. the sphere to expand so as to fill up the cylinder, the expansion at each point ceasing as soon as that point comes into contact with the cylinder. Suppose also that the expansion is uniform in all directions; then, obviously, the relative proportions of all parts will be maintained, but the scale will increase rapidly towards the equator. Let all the lines which were traced on the sphere now impress themselves on those parts of the cylinder with which they are in contact, and then let the cylinder be unrolled into a plane; this will be a Mercator's chart. From the fact that each individual small element of the surface retains its form, we see that all angles will remain unaltered, and the meridians become parallel straight lines perpendicular to the equator, therefore a loxodrome, cutting all meridians at the same angle, will also become a straight line. So that the track of a ship which keeps on a constant course will be represented by a straight line, and this is the principle which renders Mercator's chart so useful to the navigator. 386. To find the distance from the equator to any parallel of latitude on the Mercator's chart: Let $a$ be the radius of the equator, $x \ldots\ldots$ radius of the parallel in latitude $\phi$, $s \ldots\ldots$ length of the arc of the meridian between the equator and that parallel, $x \ldots\ldots$ corresponding distance on the chart, $\Delta s$ and $\Delta x$ small increments of $s$ and $x$. By the method of expansion explained above, the circle, radius $x$, becomes a circle, radius $a$, therefore each small element of that parallel is increased in the ratio $x:a$. Hence $\Delta s$, which is changed to $\Delta x$, must ultimately be in the same ratio. Therefore $$\frac{dx}{ds} = \frac{a}{x}.$$ APPENDIX.] MERCATOR'S CHARTS. 305 The earth being a sphere, $x = a \cos \phi$, $s = a \phi$; therefore $$\frac{dx}{d\phi} = a \sec \phi,$$ $$z = a \int_0^{\phi} \sec \phi d\phi = a \log \left[ \tan \left( 45^\circ + \frac{\phi}{2} \right) \right].$$ 387. If we take into account the spheroidal shape of the earth, and call $e$ the eccentricity of the ellipse, then, by Art. 253, $$x = \frac{a \cos \phi}{(1 - e^2 \sin^2 \phi)}, y = \frac{a (1 - e^2) \sin \phi}{(1 - e^2 \sin^2 \phi)},$$ $$\frac{ds}{d\phi} = \sqrt{\left( \frac{(dx)^2}{(d\phi)^2} + \frac{(dy)^2}{(d\phi)^2} \right)} = \frac{a (1 - e^2)}{(1 - e^2 \sin^2 \phi)}. $$ As before, $$\frac{dz}{da} = \frac{a}{x},$$ $$\frac{dz}{d\phi} = \frac{a (1 - e^2)}{\cos \phi (1 - e^2 \sin^2 \phi)},$$ $$= a e^{-e} \frac{ae^2 \cos \phi}{\cos \phi (1 - e^2 \sin^2 \phi)},$$ whence $x = a \log (\tan (45^\circ + \frac{\phi}{2}) - ae^{-e} \log (\frac{1+e\sin\phi}{1-e\sin\phi}))$ $$= a \log (\tan (45^\circ + \frac{\phi}{2}) - ae^{-e}\sin\phi) approximately.$$ (306) EXAMPLES AND PROBLEMS. CHAPTERS I-VI. 1. Shew that if $\alpha$, $\beta$ be the angles subtended between a star and the horizon at two heights, $a$, $b$, on the same vertical line, the radius of the earth is $\sqrt{(2b - (a + b))}$ very nearly. 2. Given the right ascensions and declinations of two stars, to find their distance. 3. The least angle which can be made with the horizon, by any great circle passing through the place of a star at a given time, is measured by the star's altitude. 4. The sun is at the same altitude at equal intervals of time before and after its transit of the meridian, supposing no change of declination to have taken place during the interval. 5. Find the declination of a star which, in a given latitude, rises in the north-east point. 6. Two observation circles $P_1 P B$ make with each other a small angle $\epsilon$ at $P_1$ and from a point $A$ (declination $\delta$) in one of them an arc $AB$ of a great circle is let fall perpendicularly on the other. Shew that the difference of declination of $A$ and $B$ is sin$\epsilon$sin$\delta$ nearly. 7. Given the distances of a planet from two stars, and also their right ascensions and declinations are known, find the right ascension and declination of the planet. 8. If a ship be proceeding along a great circle, and the observed latitudes be $a_1$, $a_2$, the distance traversed between the observations being in each case $s$, shew that $$x = r \cos(\sin\lambda_1 \sin\lambda_2)$$ $r$ denoting the earth's radius; and shew that the changes of longitude may also be found in terms of the three latitudes. 9. Given the latitudes and longitudes of two places where the inclination of the magnetic needle is nothing; to find the point of the terrestrial equator, which is cut by the magnetic equator, supposing it a great circle of the earth. CHAP. I.—VII.] EXAMPLES AND PROBLEMS. 307 10. Two places in latitude $45^{\circ}$ and whose difference of longitude is $90^{\circ}$ are at two-thirds of the distance of places on the equator with respect to each other. 11. If the same two stars rise together at two places, the places will have the same latitude. And if they rise together at one place and set together at the other, the places will have equal latitudes, but different longitudes. 12. Prove that all stars which rise at the same instant at a place within certain limits of latitude will, after a certain interval, be in a vertical great circle and determine those limits. 13. Given any altitude of a known star when it is on the prime vertical, find the latitude of the place. 14. A known star rises in the north-east point, find from this circumstance its latitude and longitude. 15. Two stars, whose right ascensions and declinations are known, were observed to rise at the same moment. Required the latitude of the place of observation. 16. It is observed that two stars whose right ascensions and declinations are given pass the prime vertical at the same instant. Required the latitude of the place of observation. 17. Two stars, whose right ascensions and declinations are given, have the same altitude, and the altitude of one of them is known ; the latitude of the other is required. 18. In a given latitude and longitude a vertical plane declines $a^{\circ}$ from the south towards the west. Find the place to whose horizon the plane is parallel. 19. Two stars are seen at a given place $A$ on the same vertical when at another place $B$ they are rising together. Find the latitude and longitude of $B$. 20. Given the latitudes and longitudes of three places on the earth's surface, find the latitude and longitude of one equally distant from them all. 21. If $C_1$, $C_2$, $C_3$ be the lengths of the meridian shadows of three equal vertical rods on the same day at three different places on the same meridian, then prove that if these lengths are connected by the equation $$C_1(Q_2 - C_3) + C_2(C_3 - Q_1) + C_3(Q_1 - C_2) = 0,$$ where $Q_1$, $Q_2$, $Q_3$ are their respective longitudes, then $\tan Q_1 \cdot \tan Q_2 \cdot \tan Q_3 = 0$. 22. How would an increase in the earth's velocity of rotation affect the latitude of a given place, supposing the form of the earth to remain unchanged? 23. It has been inferred that the axis of the earth's rotation always coincides sensibly with the same line of particles in the earth, and that it does not rapidly change its direction in space? X 2 308 EXAMPLES AND PROBLEMS. [CHAPEL. VIL.—XIII.] 24. If the western pivot of a transit be $a'$ higher and $b'$ more to the north than the eastern, a star is unaffected whose north polar distance is $\cos \alpha$ and whose declination is $\sin \beta$. Find the angle between the axis of the telescope and the meridian. 25. If the head of the micrometer screw of the mural telescope be divided into sixty parts, and a whole turn of the screw carry it through 0.01 inches, what will be the correction of the reading when the head indicates 15, and the focal length of the object-glass is 5 feet? 26. In obtaining the zenith distance of a star with the mural by reflection, it was found that the mercury at the point of reflection is slightly inclined to the vertical through the axis; show that, on this account, the zenith-point, inferred from a double observation, is too great by the angle which the horizontal distance of the point from the axis makes at the mean centre, and the zenith-distance of the star deduced from the reflection-observation is less than that from the direct observation by twice that angle. CHAPTERS VIL.—XIII. 27. When does the sun set at the point of the horizon opposite to that at which he rose? 28. Prove that, in the course of the year, the sun is as long above the horizon during its passage over any given place as below it. 29. State some of the more striking differences in the phenomena presented by the sun on June 21st to observers in lat. 40° N. and 40° S. respectively. 30. The declination of the sun when, for a given place within the Arctic circle, the sun at mid-day just appears above the horizon. 31. If the equator were perpendicular to the ecliptic, describe the apparent diurnal motion of the sun throughout the year to an observer, first, at a pole, secondly, at the equator, and thirdly, in some intermediate position. 32. Which of the following great circles are fixed, and which of them would, if visible, appear fixed to an observer not aware of his own motion about an axis passing through his eye? 33. Orion's belt being in the equator, and having about 5 h. 33 m. right ascension, during what part of the night will it be visible at the vernal and autumnal equinoxes? 34. Given, then, the midnight altitude and his midnight depression below or above horizon, find the latitude of the place and the sun's declination. A diagram showing a celestial sphere with various lines and angles labeled. CHAPS. VII.--XIII.] EXAMPLES AND PROBLEMS. 309 35. At a certain place within the Arctic circle the sun did not set for two months, what was the latitude? 36. When the sun has a given north declination, show at what parts of the earth he is visible, (1) during 24 hours, (2) during 12 hours. 37. At a place (lat. g) in the Arctic circle, the sun will remain above the horizon at the summer solstice for $\frac{2\pi}{3}$ cos $g$ cosec $w$ days, neglecting the eccentricity of the earth's orbit. 38. If $x$ be the latitude of a place at which, a month before the summer solstice, the sun is at its greatest height at day at place in latitude $x$, show that tan $x$ tan $y = (\frac{1}{3}\sec w)$ approximately, $w$ being the obliquity of the ecliptic. 39. Given the latitudes of longitude of Cork and Rio Janeiro, show that on any day the sun is on the horizon of both places at the same instant. 40. Show that, neglecting the change of declination, the curve traced out by the extremity of the shadow of a vertical rod on a horizontal plane will be a parabola. 41. If the angle between the equator and the ecliptic were 18°, what fractional part of the earth's surface would be included in the torrid zones, that is, between 23° 27' and 66° 33' N. ? 42. The true zenith distance of the pole star when it first passes the meridian is 46° 50' 40"75', and at the second passage 50° 52' 50"30'. Required the latitude of the place. 43. The mean declination of the sun was found November 23d to be 62° 41' 10". The chronometer indicated 7 h. 43 m. 14 s. The sun's declination at the preceding moon at Greenwich (as given by *Nautical Almanac*) was 30° 27' 56", and his hourly motion in declination 30° 7". From these data find their difference. 44. At noon on March 20th, 1858, the sun's declination was $1^{\circ}42'29''$, and the differences of the right ascension between the sun and a star 18 h. 1 m. 49 s. At noon on September 18th, the sun's declina- tion was $1^{\circ}36'29''$, and that of a star 18 h. 1 m. 49 s., right ascension. At noon on September 19th, the declination of the sun was $1^{\circ}36'29''$, and the difference of right ascension i.h. $32$ m. $24''$, find the right ascension of the star, and that of the sun at the time of finding it. 45. If the right ascension of a star be equal to its latitude, prove that its declination must be equal to its longitude. 46. Indicate on a celestial sphere all stars which have equal brightness and equal declinations with respect to their declination. 47. By observation it is found that the predicted place of a planet is erroneous in $\beta$ and declination by the quantities $a$ and $\beta$. Find from these the error in longitude and latitude. A diagram showing a celestial sphere with various lines indicating right ascensions and declinations. 310 EXAMPLES AND PROBLEMS. [CHAPS. VII.-XIII. 48. Assuming the sun's motion in longitude to be uniform, show that if $\alpha$, $\beta$ be the horary increments in right ascension and declination (expressed in circular measure), and $'\alpha$, $'\beta$ the horary variations of $\alpha$ and $\beta$, then $\alpha - \beta = 2\pi a$, $\beta - \alpha = 2\pi b$. 49. Show that when the inclination of the ecliptic to the horizon of a given place find its maximum and minimum values, and show within what limits of latitude it can be a right angle. 50. Represent by a figure the position of the ecliptic relative to the meridian at any given time, at noon, on the autumnal equinox; under what circumstances, of place and time, would the ecliptic coincide with the prime vertical? 51. Assuming that the intensity of heat for different distances varies as the square of their distances, show that equal amounts of heat are received by the earth from the sun during one revolution of describing equal angles round its, in whatever part of the ellipse those angles may be situated. 52. The mean length of the four quarters of the year, beginning with the spring quarter, are approximately $Q \cdot A \sin (B + 1^\circ)$; $Q \cdot A \cos (B + 1^\circ)$; $Q \cdot A \sin (B - 1^\circ)$; $Q \cdot A \cos (B - 1^\circ)$; $Q$ being one-fourth of a year. But the angular distance of the solar perigee from the autumnal equinox is nearly $A : Q = 1 : 2$, so that $A : Q = 1 : e^2 : v$, where $e$ is the eccentricity of the earth's orbit. 53. Show that the time occupied by the sun in passing through the $n^{th}$ sign of the zodiac, reckoning Aries the first, is approximately $$M = \frac{60}{(1 + e \sin B) \cos(B + 1^\circ)}$$ where $M$ is the $n^{th}$ part of the year, $B$ the angular distance of the solar perigee from the autumnal equinox, and $e$ the eccentricity of the earth's orbit. 54. The times of the sun's rising and setting on November 1st are found from the tables to be 6 h. 56 m. and 4 h. 32 m. respectively; find approximately the equation of time. 55. Show that in any year there are exactly some days before, and the time of winter lasts some days after, the shortest day. 56. Assuming that the maximum amount of the equation of time due to obliquity exceeds the maximum of that due to the eccentricity, show that this difference does not exceed four times a year. How many times in the year would it vanish were the imponderables of the two maxima reversed? 57. If the earth's perihelion coincides with a solstice, prove that, assuming no sun's true anomaly to be equal to $\pi/2 + e \sin \alpha$, and neglecting the square of the eccentricity, the equation of time would vanish at the solstices, and when $$e \cos w + e \cos m \sin^2 w = 0.$$ **CHAP. XIV.** EXAMPLES AND PROBLEMS. 311 the true and mean suns being supposed to start with the same right ascension when the true sun is in a solstice. 68. When the sun's longitude is $t$, if $x$ be the equation of time (expressed in angle) due to the obliquity of the ecliptic ($w$) alone, shew that $cot x = cot 2\theta - 2cot^2 \theta cos^2 w$. 59. Supposing the earth to be a sphere, uniformly round the zon, the maximum value of the equation of time would be very nearly $\frac{1}{2}$ second, assuming the obliquity to be small and equal to $w$. 60. At the head of the column for equinoxes given in March in the Nautical Almanac for 1848 was given "adding" $+0.17604$ d., and for the days 21, 22, and 23, the days in the columns are marked 364, 0, 1. Explain this clearly. 61. At any time being 4 hours, find the corresponding sidereal time, having given the sun's mean daily motion $8^\circ 30'$, and the $A$t at the preceding mean noon $14^\circ$. 62. If the adiurnal motion mean was $16^\circ 20'$ m. $49$, what was the time at o'clock, when a sidereal clock was at 8 h. 21 m., the sun's mean motion in longitude being $99^\circ 33'$ in a mean solar day? 63. Find the sun's apparent right ascension at mean noon Greenwich time, on June 1st, 1880, was 4 h. 38 m. 18-96 s. and on June 2nd, 4 h. 42 m. 24-73 s. Find the sun's apparent right ascension at 11 h. 20 m. a.m. on June 2nd, at a place $0^\circ$ east. And find its apparent right ascension at mean noon on June 2nd be 4 h. 42 m. 24-94 s., find approximately the equation of time. 64. If the time be found by a single altitude, shew that a small error in the latitude will have no effect on the time when the body is in the prime meridian. 65. A ship leaves London at noon on a certain day, and arrives at New Orleans (90° W. long.) at noon, local time, on the thirtieth day afterwards. What is the actual time of passage? 66. The sun's right ascension when crossed the meridian of Greenwich at mean noon yesterday and to day respectively are $72^\circ 23' - 21' - 15$ and $73^\circ 5' - 29' - 53$; find the mean time at which a star whose right ascension is $73^\circ 5' - 29' - 53$ crosses the meridian. 67. If Jupiter revolves round the sun in $4320$ of our days and round his axis in $10$ hours, find by how much his mean solar day exceeds his sidereal day. CHAPTER XIV. J 68. Given the sun's altitude at 6 o'clock, and also when due east, find the latitude of the place. 312 EXAMPLES AND PROBLEMS. [CHAP. XIV. à 69. Given the sun's declination, and that he is due east when half the time between his rising and 12 o'clock has elapsed, find the latitude of the place. à 70. Find the latitude of the place at which the sun sets at 5 o'clock on the shortest day. 71. The sun, the one the other five feet high, are placed, on a given day, perpendicularly to the horizon, at a distance of 20 feet from each other. During the foregoing the extremity of the shadow of the first rod falls at the base of the second. In the afternoon the extremity of the shadow of the second rod falls at the base of the first. Determine the latitude of the place and the azimuth of one rod seen from the other. à 72. Given the point of the horizon at which the centre of the sun's disc rises, find its altitude; at what point it crosses the meridian; find the time of year when this happens. 73. Given the sun's altitude and azimuth, and the latitude of the place; to find the declination and the hour of the day. 74. If a given latitude determine the vertical in which the difference of the altitude and declination is greatest, and show that it may be a maximum. 75. The right ascension and declination of a star being given, and also the time of the year when it rises with the sun; find the latitude of the place. à 76. Find the latitude of the place at which the sun sets at 10 o'clock on the longest day. 77. Given the latitude of the place and the sun's declination; to find his azimuth at noon. 78. Given the latitude of the place and the sun's declination; find time when the hour angle from noon and the sun's azimuth from the south are equal. 79. Given the sun's altitude and declination, and the sum of the azimuth and hour angle; to determine the latitude. 80. Given the latitude of the place and the sun's declination; find at what time of day the azimuth of the sun increases most slowly. 81. At a certain place in a given north latitude sum of the sun's declination and altitude is known at a given hour in morning; find altitude and declination. 82. The length of the longest day at a given place is 14 hours, Find the latitude, supposing the obliquity of the ecliptic $23^\circ 28'$. log tan $23^\circ 28' = 9.6301$, log cos $23^\circ 28' = -9.0010$, log tan $23^\circ 28' = 9.6301$, log tan $23^\circ 28' = 9.6301$. 83. The latitudes of two places on earth's surface are comple- mentary to each other, and on a given day one rises (e) hours earlier at one place than another; determine their latitude each place. **CHAP. XIV.]** **EXAMPLES AND PROBLEMS.** § 84. A sphere of given radius is suspended in the air at a given place. Determine the figure of its shadow on a horizontal plane at a given day and hour, and find the length of the shadow when the sun's zenith distance is given. § 85. Given the latitude of the place and the declination of the sun, the former being less than the latter, find at what time of the day the shadow will be longest. § 86. Given the sun's diameter and the latitude of the place, determine the declination when the time the sun takes to rise is a minimum. § 87. The direction of a street $\theta$ to the west of south; given that the street is $a$ feet long, and that its shadow, when observed by the shadow of the block of houses between them, show that if the breadth of the first street be $a$, the height of the houses $b$, and the altitude and azimuth of the sun $\phi$ and $\psi$ respectively, $$\tan \alpha = \frac{\sin \theta}{\cos \phi}$$ and find the breadth of each street. § 88. There are two walls of equal known height at right angles to each other, and running in different directions, show how to find the sun's altitude and azimuth by observing the breadth of the shadows of the two walls at any given time. And prove that the sum of the squares of the lengths of all the shadows will be the same whatever be the direc- tions of the walls. § 89. A person starts at sunrise to travel round the earth's equator. The sun crosses his meridian at noon and he reaches the point of starting at sunset. Given his exact latitude, find his speed. § 90. Prove that the perpendicular ascent of a star is always quickest in the prime vertical. § 91. If the altitude of a star be taken at the same place, on the same day, when it is in the same vertical on opposite sides of the meridian, show that the sum of their tangents will be to one another their sines as the sine of their declination is to one of those sines of the place. § 92. Find the lowest latitude at which twilight lasts all night. Why is this lower than it would be in equatorial latitudes? When does it occur? and when is its duration shortest? § 93. Show that the duration of twilight at the equator is $$\frac{12}{\sin \theta} - \sin \theta (\text{sin}^2 18^\circ \text{sec } \theta)$$ 1 § 94. Show that the time at which the sun is southeast may be deter- mined by means of the expression $\frac{1}{2}(\theta - \sin \phi)(\cos \phi \cos \tan \theta)$, where $\phi$ is the sun's north declination, $\theta$ is the latitude of the place, and $\tan \theta = \sin g$. 314 EXAMPLES AND PROBLEMS. [CHAP. XIV. 95. The altitude of a star when it crosses the meridian and the prime-meridian of a place are $a$ and $a'$; show that if $\beta$ be the declination of the star, and $\phi$ the latitude of the place, then $$\cot^2 \beta = \cos^2 a - \tan a \cot \phi = \tan a \sec a \sin \phi.$$ 96. If two stars rise simultaneously in azimuths which are supplementary, show that one is as long above the horizon as the other is below it; and if $\beta$ be the difference of their azimuths, and 24 hours the difference between their times of setting, show that the latitude $\phi$ of the place is given by $$\sin \beta = \cot a \tan (72^\circ - \phi).$$ 97. To a spectator in the northern hemisphere, the sun, whose declination is $18^\circ$ south, rises just two hours before noon; prove that the latitude of the place of observation is $$\tan^{-1} (\sqrt{3}) \sqrt{\frac{1 + \frac{1}{2} \sqrt{3}}{1 - \frac{1}{2} \sqrt{3}}}.$$ 98. To a spectator on the deck of a vessel, sailing due west at the equator, at the rate of 10 miles an hour, the time occupied by the sun in rising over the horizon being 1 hour 4 minutes. Given that the angular breadth of the sun is $32^\circ$, find approximately the earth's radius. 99. Obtain the alteration in the length of the shortest day in latitude $a$, corresponding to an alteration in the inclination of the earth's axis to the plane of its orbit. 100. At A, place, the north latitude of which is $54^\circ$, and at a time of the year when the sun's north declination is $18^\circ$, show that if $A$ is the hour angle (Z.P.S.) which the sun makes with the meridian at the moment of sunrise, then $\cos A = \frac{\sqrt{3}}{2}$, and the sun rises shortly after 4 o'clock. 101. Give formulae for determining how long the sun shines on a south wall, and how long on a west wall, in a given latitude on the shortest day. 102. Show how to graduate a horizontal sun-dial, and find the limits beyond which it is useless to graduate it. 103. Find the length of the shadow of a man six feet high in latitude $60^\circ$ at 8 a.m., on the 21st of March; find also the direction in which the shadow points. 104. In latitude $45^\circ$, at the equinox, find the time occupied by the sun in rising, assuming his diameter to be 30". 105. Prove that in certain parts of the earth the ecliptic is perpendicular to the horizon every day, and find at what hour each day. A diagram showing a celestial sphere with lines representing different latitudes and longitudes. **CHAP. XV.—XVIII.] EXAMPLES AND PROBLEMS.** 315 106. If $A$ be the hour angle, and $A$ the azimuth of a star at the instant when its altitude is equal to the latitude ($p$) of the place of observation, show that $$\cos A = \tan p \sin (45^\circ - A),$$ $\sin A = \sec p \sin (45^\circ - A)$, $d$ being the declination. 107. Show that the velocity in azimuth at rising is the same for all stars at a given place. 108. At a place in latitude $\phi$, a wall of height $h$ has an azimuth of $a$ to the south; show that at the time of the equinox the wall casts no shadow at $A_1$ tan $(\phi + \tan a)$ hours before noon, and at noon the breadth of the shadow is $h$ tan $\sin a$. **CHAPTERS XIV.—XVIII.** 109. What would be the effect of refraction on a fixed star as seen by an observer on the moon's surface when the star is in the plane of the moon's orbit and passes behind the earth? The horizontal parallax being 4' 10". 110. If $r$ be the horizontal refraction, show that the point of the compass where the sun rises is shifted by $4r$ $$\sin \phi = \frac{\sqrt{(\cos (\varphi - r) \cos (\varphi + r))}}{r}$$ where $\phi$ is the latitude. 111. Calculate the aberration produced by refraction in the distance of two stars which have been taken by a micrometer. 112. It has been argued that since a degree of latitude increases in proceeding from the equator towards the poles, the whole circumference of the terrestrial meridian must be greater than 360 times a degree at the equator than at any other part of it; but this is only true for a quarter, and that consequently the earth must be elongated instead of flattened at the poles. Show the fallacy of this. 113. The moon's apparent declinations, as observed on the same day at stations separated by 180 degrees, when hemispheres respectively, were $D$ and $D'$, the computed parallax corrections were $p$ and $p'$, and the increment in the moon's geocentric declination between the observations was $d$. The previously determined equatorial horizontal parallax is slightly inaccurate; show that the real value of the constant : the value already obtained :: $D' - D + d : p + p'$. **316** EXAMPLES AND PROBLEMS. [CHAP. XV.—XVII.] 114. The annual parallax of a double star is 0°-30′, the apparent angular distance between its two parts 15′, and the mean angular annual motion of revolution of one star about the other is nearly that of the sun. 411. Why is it not strictly true that the azimuth of a heavenly body is unaffected by parallax? 116. At what distance is the distance of a body from the earth to be so great that the sine and circular measure of the parallax may be considered equal, show that the locus of all bodies which, at a given instant and place, have their parallax in right ascension equal will be in a circular right cylinder touching the plane of the meridian along the axis of the earth. 317. What will be about the distance of a star of which the parallax is 0°? 118. Find the right ascension of a star for which the aberration in right ascension vanishes at the summer solstice. 119. Find the coefficient of aberration in decimals of a second when objects are moving with a carriage travelling 60 miles an hour, the velocity of light being 200,000 miles per second. If two planets describe circles in the same plane, when will the aberration in the position of one, as seen from the other, be greatest and least? 120. Find those stars for which, on a given day, the time of rising is not affected by aberration. 121. The apparent longitude of a star is recorded at various seasons of the year. Show how to determine its true longitude by eliminating any changes from other causes, are due to aberration or to parallax? and if due to both, what are the parts of the change separately due to one and the other? Shew that the difference between apparent and true longitude arising from both causes combined, vanishes four times a year. 122. If $f(x,p) = 0$ be the equation to the earth's orbit, shew that $f\left(\frac{dx}{dt}, \frac{dp}{dt}\right) = 0$ will be the equation to the path which a star appears to describe in consequence of aberration. 123. Calculate the velocity of light where it is found that the difference between the declination of a star at the pole of the ecliptic in March and September is 25′. The mean orbit being 92 millions of miles, and the year consisting of 365 days. 124. If light moved but twice as fast as the earth in her orbit, in what part of the heavens would the pole star appear, at a place in north latitude 60°, at each moment of the year? 125. At what season of the year is the aberration of a star in the position of the first point of T greatest? CHAP. XV.--XVIII.] EXAMPLES AND PROBLEMS. 317 -120. If the aberration of a star in longitude be the same as its aberration in latitude, prove that $$\sin 2\lambda = 2 \cot (\varnothing - \phi)$$ where $\lambda$, $\lambda$ are the longitudes and latitudes respectively of the star, and $\varnothing$ is the longitude of the sun. 121. A man runs at starting with velocity $v$, the direction of the wind appears to make an angle $\alpha$ with his course. It then appears uniformly to veer round through an angle $\beta$ during the race. Now the wind blows throughout uniformly, at right angles to his course. Show that if $v$ is the length of the race, the length of the course is $$v = v_0 \log (2 \cos \alpha)$$ -128. Show that at any given time all stars which lie in a certain great circle have no aberration whatever. 129. Prove that at any instant of which the aberration is declination at any instant of which the intersection of the celestial sphere with a cone, of which the circular sections are parallel respectively to the equator and to the great circle drawn through the sun's plane perpendicular to the ecliptic. 130. What line is there to the position of a place in order that, at some time in the day, a star in the ecliptic may have its error of aberration in a vertical plane? 131. A star whose real position is unaffected by parallax, refraction and aberration. 132. The star $\gamma$ Draconis has rt. ascen., 143. 63 m., and N.P.D. 58° 30' 31" Campanelli, who made observations of these stars, was able to separate the effects of aberration and nutation, and refer them to their real causes. 133. If a small change $\Delta w$ in the obliquity be due to nutation, show that the corresponding changes in a star's right ascension and declination will be $$\Delta \alpha = -tan \cos \Delta w \cdot \Delta w$$ $$\Delta \delta = sin \Delta w \cdot \Delta w$$ 134. Explain why on account of precession the intervals between the passages of the meridian through the same star differ from a mean sidereal day. If the colatitude of the star be less than that of the pole, this difference will vanish when the difference of longitude of the pole and star is $$\cos^2(\tan(\text{colat.of pole})) - \tan(\text{colat.of star})$$ 135. What is the present longitude of a star which was the polar star in A.D. 66? 318 EXAMPLES AND PROBLEMS. [CHAP. XIX.--APP. CHAP. XIX.--APPENDIX. § 195. At what season of the year will the crescent moon, when setting soon after the sun, have the line joining her horns most, and least, inclined to the horizon? § 137. Show that the full moon remains above the horizon of a place during nearly the whole of the night. § 196. What is the mean age of the moon that she may be seen in the south at 7 o'clock in the morning? Will the convexity of the crescent appear to a spectator on his right hand or his left? § 139. When the moon is a quarter old, what will be the earth's appearance to a spectator at the centre of his disc? At what points of his sky will the earth and sun respectively be seen? and what time of day will this occur? § 140. How much should the moon's velocity of rotation about its axis be increased, in order that its whole surface might be seen in the course of an orbital revolution? § 141. What is the probable inclination to the horizon of the line joining the apogee and perigee of her orbit? § 142. The sidereal month being 27½ days, find approximately the time at which the moon will rise-to-morrow night if she rises to-night at 8 p.m. Find which side and, roughly, what proportion of her surface will be illuminated. § 136. Find the length of the sidereal period of the moon, having given the synodic period as 27½ days. § 144. If the moon's orbit be inclined at an angle $\alpha$ to the equator, show that when the moon rises at the same sidereal time, on two successive nights, the latitude is $90^\circ - \alpha$. § 145. Prove that when a lunar eclipse is most observable it will be accompanied by a central lunar eclipse. § 146. In what parts of the earth may it be observed that the moon's rising becomes sooner or its successive evenings be reversed as regards the harvest following? § 147. If $\omega$ be angular velocities of the moon about the earth, and of the earth about its own, in orbits supposed circular, and if $\alpha$ be the maximum elongation from the earth as it might be seen from the sun's disc that would correspond to the successive greatest elongations are alternately $\frac{\omega}{\omega + \alpha}$ and $\frac{\omega}{\omega - \alpha}$ § 148. In an eclipse, does the shadow begin on the eastern or western limb of the body eclipsed? In a solar eclipse, does the shadow of the moon move eastward or westward on the earth's surface? § 149. If the moon's period were shortened to half its present CHAP. XIX.—APP. J EXAMPLES AND PROBLEMS. 319 length, the other elements of her orbit remaining unchanged, what would be the result on the frequency and duration of eclipses? 140. Show roughly that the duration of a total eclipse of the sun, for a place at the earth's equator, is doubled by the diurnal rotation of the earth. J161. In the inferior ecliptic limits are $2\pi$ and $\frac{3}{2}\pi$, and the satellite revolves $n$ times as fast as the sun, and its node regresses $q$ every revolution it makes round its primary, prove that there cannot be more consecutive solar eclipses at one node than the integer next less than $\frac{2\pi}{n-q}$. J162. The angular diameters of the earth's shadow and of the moon, as seen from the earth's centre, being respectively $1^{\circ} 50'$ and $3^{\circ} 50'$ nearly, and the distance between them being about 180 million miles, find (1) the entire duration of a lunar eclipse, (2) the duration of the totality of the eclipse, the moon being in the node of her orbit when in opposition to the sun. 163. Find roughly how long an eclipse of the moon can have at oppositions when (a) the sun may take place; gives distances of the sun and moon 24000 and 460, their radii 100 and $\frac{1}{2}$, the earth's radius being 1. Having been given the moon's period 27 days, find the greatest duration of an eclipse for the above values of the distances. J164. If $\theta$ be the circular measure of the inclination of the moon's relative orbit to the ecliptic, $\alpha$'s angle between its two nodes and the axis of rotation of the earth, $\beta$'s angular diameter of the moon, and the section of the umbra; show that, roughly, the duration of the eclipse is $4\sqrt{r/r'}-p-\sqrt{p'}$ hours, and that $\left(\frac{r-r'}{r}-\frac{\alpha}{2\pi}\right)$ of the moon's diameter are eclipsed. J165. If $\bar{x}$ and $\bar{x}'$ be the semi-vertices of cones of shadow and penumbra, $\theta$ their apparent semi-diameter, then $$2\sin\theta=\sin x+\sin x'$$ 166. The angular distance of Aldebaran from the moon's centre was observed at a certain place at noon on June 14th at Greenwich, at noon and at 3 h. The distances of the same objects were 65° 9' 30", and 68° 41' 30" respectively; determine the longitude of the place. J167. Supposing that the earth and a planet describe circles about the sun with equal periods but different eccentricities of motion of the planet. Also find when this motion is greatest and when it vanishes. J168. If $\theta$ be the angle of elongation of an inferior planet when observed to be stationary from another planet, show that $cot\theta=\sqrt{(n-1)}$, where $n$ is half the distance of the superior from inferior from the sun to that of the inferior; also that of Jupiter being supposed circular and in the same plane. A diagram showing a circle representing Earth's orbit around Sun with a smaller circle representing Jupiter's orbit around Sun. A line connects these two circles at a point representing Jupiter's position. 320 EXAMPLES AND PROBLEMS [CHAP. XIX.—APP. 418. Supposing the paths to be circular in one plane, and the planet's distance to be 16 times that of the earth, find for how long its motion will be retrograde, having cos $^{\circ}A = \frac{7}{2}$ nearly. 419. Given that the periodic time of Venus is nearly two-thirds that of the earth, find roughly what time of the year is indicated in these consecutive extracts from a diary. First month. Venus is an evening star this month. Enters Libra. Next month. Venus is too close to the sun to be easily seen. 1811. When the superior planet is stationary, show that its angular distance from the sun is $w = \tan^{-1} \sqrt{\frac{a}{r}}$, where $w$ is the ratio of the radius of the orbit to that of the earth. 162. If $\theta$ be the angle between the earth by the sun and a stationary point of a planet's orbit, and $e$ be the greatest elongation of the planet, prove that $$2 cot \theta - e = \cos \theta$$ 163. Surrounding the orbits of two planets to be circular and to lie in the same plane, show that the longitude of the interior, as seen from the exterior, will change more rapidly at the superior than at the inferior conjunction, but that its angular distance from the sun will change more rapidly at the inferior than at the superior conjunction. 164. A comet moving in a parabolic orbit, and a planet moving in a circular orbit, are in syzygy when the comet is perihelion. Determine the ratio of their aphelion distance of the comet to the distance of the planet, in each case, when they appear stationary which seem from each other. 165. If the diameter of a planet's orbit, supposed circular, be to the diameter of the earth's orbit as $1 : 3$, find the synodic and sidereal periods. Compare the phases of the earth and planet when the difference of their heliocentric longitudes is 30°. Also when it is 90°. Show that their brightness of the planet in the first case is to that in the second as $4 : 3$. 166. The synodic period of Jupiter and the earth is 308 days. Find the periodic time of Jupiter. 167. Explain why Jupiter are evening stars, and stationary; and find which way they will begin to move. 168. A ship has to sail from a place at $A$ to a place $B$, which is due east of it. (a) Show that the shortest course will not be due east. (b) Find the direction in which the ship must sail to make the shortest course. (c) Ex.: The latitude being 40° and the difference of longitude 90°, CAMBRIDGE: PRINTED BY WYKALLEN AND CO., LIMITED, TRINITY STREET. A TPT1 u