Chapter XIII: The Nineteenth Century (4)

Mercury, I. 14-16; II. 25, 26, 45, 47, 51; III. 66; IV. 73, 75, 81,
86-89; VI. 121, 124; VII. 136 _n_, 139, 142, 144; IX. 185; XIII.
288, 290, 294, =297=. _See also_ the following headings

Mercury, mass of, XI. 248

Mercury, phases of, VI. 129

Mercury, rotation of, XIII. 297

Mercury, transit of, X. 199

Meridian, II. =33=, 39; III. 57; VI. 127; VIII. 157; X. 207, 218, 221

_Meteorologica_ (of Aristotle), II. 27

Meteors, XIII. 305

Meton’s cycle, II. 20

Metric system, XI. 237

Micrometer, VIII. =155=; XIII. 279, 281

Milky Way, II. 30, 33; VI. 120; XII. =258=, =260-262=; XIII. =317=

Mimas, XII. 255

Minor planets, XI. 250 _n_; XIII. 276, 281, 284, 288, =294=, 295, 297,
318

Minor planets, mass of, XIII. 294

Minute (angle), I. 7

Mira, XII. 266

Mongols, Mongol astronomy, III. 62

Month, I. 4, 16; II. 19-21, 40, 48; IX. 173; XI. 240; XIII. 293, 320.
_See also_ the following headings

Month, anomalistic, II. 40

Month, draconitic, II. =40=, 43

Month, empty, II. =19=, 20

Month, full, II. =19=, 20

Month, lunar, I. =16=; II. 19, 20, 40

Month, sidereal, II. 40

Month, synodic, II. =40=, 43

Moon, I. =1=, 4, 5, 11, 13-16; II. 19-21, 25, 28, 30, 32, 39, 43;
III. 68, 69; IV. 81, 86; V. 104, 105 _n_; VI. =119=, 121, 123, 129,
130, 133; VII. 145, 150; VIII. =153=; IX. 169, 180, 181, 188, 189;
X. 198, 204, 213, 215, =226=; XI 228, 235; XII. 256, 257, =271=;
XIII. 272, 292, 293, =296=, 297, 301, 320. _See also_ the following
headings

Moon, angular or apparent size of, II. 32, =41=, =43=, 46 _n_, 48;
IV. 73, 85, 90; V. 105 _n_

Moon, apparent flattening of, II. 46

Moon, atmosphere of, XIII. 296

Moon, distance of, I. 15; II. 24, 25, 30, =32=, =41=, 43, 45, 48,
=49=, 51; IV. 85, 90; V. 100, 103; IX. 173 185; X. 223; XIII. 293,
320

Moon, eclipses of. _See_ Eclipses

Moon, librations of, VI. =133=; X. 226; XI. =237=, 239

Moon, map of, X. 226; XIII. 296

Moon, mass of, IX. 188, =189= XI. 235

Moon, motion of, I. =4=, 8, 13, 15, 17; II. 20, 24-26, 28, 37, 39,
=40=, 43, 47, =48=, 51; III. 60; IV. 73, 81, 85, 89, 90; V. 111;
VI. 133; VII. =145=, 150; VIII. 156; IX. 169, =173=, =174=, 179,
=184=, 185, 189, 194, 195; X. =201=, 204, 213, =226=; XI. 235,
237, 248; XIII. 287, 290, 297, 320. _See also_ Lunar theory

Moon, origin of, XIII. 320

Moon, parallax of, II. =43=, =49=; IV. 85. _Cf. also_ Moon, distance
of

Moon, phases of, I. =4=, 16, 17; II. 19, 20, 23, =28=, 43, 48;
III. 68, 69; VI. 123

Moon, rotation of, X. 226; XI. 248; XII. 267; XIII. 297

Moon, shape of, II. 23, 28, 46; VI. 119; XI. 237

Moon, size of, II. =32=, 41; IV. 85

Moon, tables of. _See_ Tables, lunar

Moons. _See_ Satellites

Morning star, I. 14. _See also_ Venus

Morocco, III. 61

Motion, laws of. _See_ Laws of motion

Multiple stars. _See_ Stars, double and multiple

Mural quadrant, X. 218, 225 _n_

Music of the spheres, II. =23=; VII. 144

_Mysterium Cosmographicum_ (of Kepler), V. 108; VII. 136, 144

Nadir, III. 64

_Nautical Almanac._ _See_ Almanac, Nautical

Nebula in Argus, XIII. 307

Nebula in Orion, XII. 252, 259, 260; XIII. 311

Nebulae, X. 223; XI. 250; XII. 252, 256, =259-261=; XIII. =306-308=,
=310=, =311=, =317=, =318=, 319, 320

Nebulae, spiral, XIII. 310

Nebular hypothesis, XI. =250=; XIII. 318-320

Nebulous stars, X. 223; XII. 260, 261

Neptune, XIII. =289=, 295, =297=

Neptune, satellite of, XIII. 295

_New Almagest_ (of Kepler), VII. 148

=New Almagest= (of Riccioli), VIII. 153

New moon. _See_ Moon, phases of

New stars. _See_ Stars, new

New Style (N.S.), II. 22. _See also_ Calendar, Gregorian

Newton’s problem, XI. =228=, 229, 249

Newtonian telescope, IX. =168=; XII. 252, 253, 256

Night-hour, I. 16

Node, II. =40=, 43; V. 111; IX. 184; X. 213, 214; XI. =236=, 246

Nubeculae, XIII. 307

Nucleus (of a comet), XIII. 304

Nürnberg school, III. =68=; IV. 73

Nutation, X. 206, 207, =213-215=, 216, 218; XI. 232, 248; XII. 263

Νυχθήμερον, I. 16 _n_

Oberon, XII. 255

Obliquity of the ecliptic. _See_ Ecliptic, obliquity of

Observational astronomy, XIII. 272, 273

Occultations, I. =15=; II. 30

Octaeteris, II. 19

Olbers’s comet, XIII. 291

_Old Moore’s Almanack_, I. 18 _n_

Old Style (O.S.). _See_ Calendar, Julian

Opposition, II. =43=, 48 _n_; III. 60; IV. 87, 88; V. 111; VIII. 161;
XIII. 281, 284, 297

Opposition of Mars, VIII. =161=; XIII. =281=, 284, 297

Optical double stars, XII. 264

_Optics_ (of Gregory), X. 202

_Optics_ (of Newton), IX. 192

_Optics_ (of Ptolemy), II. 46

_Optics_ (of Smith), XII. 251

_Opus Majus_, _Minus_, _Tertium_ (of Bacon), III. 67

_Opuscules Mathématiques_ (of D’Alembert), XI. 233

Orion, nebula in, XII. 252, 259, 260; XIII. 311

_Oscillatorium Horologium_ (of Huygens), VIII. =158=; IX. 171

Pallas, XIII. 294

Parabola, IX. =190=; XI. 236 _n_; XIII. 276

Parallactic inequality, XIII. 282

Parallax, II. =43=, =49=; IV. 85, =92=; V. 98, 100, 110; VI. =129=;
VII. 145; VIII. =161=; X. 207, 212, 223, 227; XII. 257, 258, 263,
264; XIII. 272, 278, 279, 281-284

Parallax, annual, VIII. 161. _See also_ Parallax, stellar

Parallax, horizontal, VIII. 161

Parallax of the moon. _See_ Moon, parallax of

Parallax of the sun. _See_ Sun, parallax of

Parallax, stellar, IV. =92=; V. 100; VI. =129=; VIII. =161=; X. 207,
=212=; XII. 257, 258, 263, 264; XIII. 272, =278=, =279=

Parallelogram of forces, IX. 180 _n_

Parameters, variation of, XI. 233 _n_. _See also_ Variation of elements

Παραπήγματα II. 20

Partial eclipses, II. 43

Pendulum, pendulum clock, V. =98=; VI. =114=; VIII. =157=, 158, =161=;
IX. 180, 187; X. 199, 217, 221, 223; XI. 231. _See also_ Gravity,
variation of

_Pendulum Clock_ (of Huygens), VIII. =158=; IX. 171

Penumbra (of a sun-spot), VI. =124=; XII. 268

Perigee, II. =39=, 40, 48; IV. 85. _See also_ Apse, apse-line

Perihelion, IV. =85=; XI. 231. _See also_ Apse, apse-line

Periodic inequalities. _See_ Inequalities, periodic

Perturbations, VIII. 156; IX. =183=, =184=; X. 200, 204, 224, 227;
=XI.= _passim_; XIII. 282, 293, 294, 297

Phases of the moon. _See_ Moon, phases of

_Phenomena_ (of Euclid), II. 33

Phobos, XIII. 295

Photography, XIII. 274, 279-281, 294, 298, 299, 301, 306

Photometry, XIII. 316. _See also_ Stars, brightness of

Photosphere, XII. =268=; XIII. 303

Physical double stars, XII. 264. _See also_ Stars, double and multiple

Planetary tables. _See_ Tables, planetary

Planetary theory, II. 51, 52, 54; III. 68; IV. 86-90; XI. =228=, 230,
231, 233, =235=, =236=, =242-247=, 248; XIII. 286, =288-290=, 293,
_See also_ Planets, motion of

Planets, I. 13, =14=, 15, 16; II. 23-27, 30, 32, 51; III. 68; IV. 81;
V. 104, 105, 110, 112; VI. 119, 121; VII. 136, 144; VIII. =154=,
155; X. 200; XI. 228, 250; XII. 253, 255, 257, =267=, 271; XIII.
272, 275, 276, 281, 282, 294-296, =297=, 318, 320. _See also_ the
following headings, _and_ the several planets Mercury, Venus, etc.

Planets, discoveries of, XII. =253=, 254, 255, 267; XIII. =289=,
=294=, 295, 318

Planets, distances of, I. 15; II. 30, 51; IV. 81, 86, 87; VI. 117;
VII. 136, 144; IX. 169, 172, 173

Planets, inferior, I. =15=; IV. 87, 88. _See also_ Mercury, Venus

Planets, masses of, IX. 185; XI. 245, 248; XIII. 294. _See also_ under
the several planets

Planets, minor. _See_ Minor planets

Planets, motion of, I. 13, =14=, =15=; II. 23-25, =26=, =27=, 30, =41=,
45, 47, =51=, 52; III. 62, 68; IV. =81=, =86-90=, 92; V. 100, 104,
=105=, 112; VI. 119, 121, 129; VII. =139-142=, =144=, 145, 150, 151;
VIII. 152, 156; IX. 169, =170=, =172-177=, =181=, =183=, 194;
X. 199, 204; XI. 228, 229, 245, =250=; XIII. 275, 276, 282, 294.
_See also_ Planetary theory

Planets, rotation of, VIII. =160=; IX. 187; XI. 228, 250; XII. 267;
XIII. =297=

Planets, satellites of. _See_ Satellites

Planets, stationary points of, I. =14=; II. 51; IV. =88=

Planets, superior, I. =15=; IV. 87, 88. _See also_ Mars, Jupiter, etc.

Pleiades, VI. 120; XII. 260

Poles (of a great circle), II. 33 _n_

Poles (of the celestial sphere), I. =8=, 9, 10; II. 33, 35; IV. 78;
VI. 129; X. 207, 214; XIII. 285

Poles (of the earth), IV. 82; IX. 187; X. 220, 221; XIII. 285

Pole-star, I. 8, 9

Pollux, XII. 266

Pons-Brooks comet, XIII. 291

Postulates (of Ptolemy), II. 47

Postulates (of Coppernicus), IV. 76

Praesepe, XII. 260

Precession (of the equinoxes), II. =42=, =50=; III. 58, 59, 62, 68;
IV. 73, 83, =84=, 85; V. 104, 112; VI. 129; IX. =188=, 192;
X. 213-215, 218, 221; XI. 228, =232=, 248; XIII. 277, 280

_Prima Narratio_ (of Rheticus), IV. =74=; V. 94

Primum Mobile, III. 68

_Principia_ (of Descartes), VIII. 163

_Principia_ (of Newton), IV. 75; VIII. 152; IX. 164, =177-192=, 195;
X. 196, 199, 200, 213; XI. 229, 234, 235, 240

_Principles of Philosophy_ (of Descartes), VIII. 163

_Probabilités, Théorie Analytique des_ (of Laplace), XI. 238

Problem of three bodies. _See_ Three bodies, problem of

_Prodromus Cometicus_ (of Hevel), VIII. 153

Prominences, XIII. =301=, 302, 303

Proper motion (of stars), X. =203=, 225; XII. 257, =265=; XIII. 278, 280

Prosneusis, II. =48=; III. 60; IV. 85

_Prussian Tables_, V. =94=, 96, 97, 99; VII. 139

Pythagoreans, II. 24; IV. 75

Quadrant, V. 99; X. 218, 225 _n_

Quadrature, II. =48=; III. 60; V. 111

Quadrivium, III. 65

_Recherches sur différens points_ (of D’Alembert), XI. =233=, 235

_Recherches sur la précession_ (of D’Alembert), XI. 215

Reduction of observations, X. 198, =218=; XIII. 277

Reflecting telescopes, IX. =168=; XII. 251-255

Refracting telescopes, IX. 168. _See also_ Telescopes

Refraction, II. =46=; III. 68; V. 98, =110=; VII. =138=; VIII. 159,
=160=; X. =217=, 218, 223; XIII. =277=

Relative motion, principle of, IV. =77=; IX. 186 _n_

Renaissance, IV. 70

_Results of Astronomical Observations_ (of John Herschel), XIII. 308

Retrograde motion, I. 14

Reversing stratum, XIII. 303

Reviews of the heavens, XII. 252, 253

Revival of Learning, IV. 70

Rhea, VIII. 160

Rigel, III. 64

Right ascension, II. =33=, 39; X. 198, 218; XIII. 276

Rills, XIII. 296

Rings of Saturn. _See_ Saturn, rings of

Rotation of the celestial sphere. _See_ Daily motion

Rotation of the earth, sun, Mars, etc. _See_ Earth, Sun, Mars, etc.,
rotation of

Royal Astronomical Society. _See_ Astronomical Society, Royal

Royal Society, IX. 166, 174, 177, 191, 192; X. 201, 202, 206, 208;
XII. 254, 256, 259, 263; XIII. 292, 308

_Rudolphine Tables_, V. 94; VII. =148=, 151; VIII. 156

Ruler, I. 16

Running down of the solar system, XIII. 293, 319

_Saggiatore_ (of Galilei), VI. 127

Sappho, XIII. 281

Saros, I. =17=; II. 43

Satellites, VI. =121=, =127=, 129, 133; VII. 145, 150; VIII. =154=,
=160=, 162; IX. 170, 183-185; X. 210, 216; XI. 228, 248; XII. 253,
=255=, =267=; XIII. 272, 283, =295=, 296, 297, 318, 320. _See also_
Jupiter, Saturn, etc., satellites of

Satellites, direction of revolution of, XI. 250; XIII. 295, 318

Satellites, rotation of, XI. 250; XII. 267; XIII. 297

Saturn, I. 14-16; II. 25, 51; IV. 81, 87; V. 99; VI. 123; VII.
136 _n_, 142, 144; VIII. 154, 156; IX. 183, 185, 186; X. 204;
XI. 228, 231, 235, 236, 243-246; XII. 253, =267=; XIII. 288,
=297=. _See also_ the following headings

Saturn, mass of, IX. 185

Saturn, rings of, VI. =123=; VIII. =154=, =160=; XI. 228, =248=;
XII. 267; XIII. =295=, =297=

Saturn, rotation of, XII. 267; XIII. 297

Saturn, satellites of, VIII. =154=, =160=; IX. 184; XI. 228;
XII. 253, =255=, 267; XIII. =295=, 297, 307

Scientific method, II, 54; VI. =134=; IX. =195=

Seas (on the moon), VI. 119; VIII. 153; XIII. 296

Seasons, I. 3; II. =35=, 39; IV. =82=; XI. 245

Second (angle), I. 7

Secular acceleration of the moon’s mean motion, X. =201=;
XI. 233, 234, =240=, 242; XIII. =287=

Secular inequalities. _See_ Inequalities, secular

_Selenographia_ (of Hevel), VIII. 153

_Selenotopographische Fragmente_ (of Schroeter), XII. 271

Sequences, method of, XII. 266

Shadow of earth, moon. _See_ Eclipses

“Shining-fluid” theory, XII. =260=; XIII. 310, 311

Shooting stars. _See_ Meteors

Short-period comets, XIII. 291

Sidereal month, II. 40

Sidereal period, IV. 86, 87

Sidereal system, structure of, XII. 257, =258=, 259-262; XIII. =317=

Sidereal year, II. 42

_Sidereus Nuncius_ (of Galilei), VI. 119-122

“Sights,” V. 110; VIII. =155=; X. 198

Signs of the zodiac, I. 13

Sine, II. 47 _n_; III. 59 _n_, 68 _n_

Sirius, XIII. 316 _n_

Solar eclipse. _See_ Eclipse

Solar system, stability of, XI. 245; XIII. 288, 293

Solstices, I. =11=; II. 36, 39, 42

Solstitial points, I. 11

Space-penetrating power, XII. 258

Spanish astronomy, III. 61, 66

Spectroscope, XIII. 299. _See also_ Spectrum analysis

Spectrum, spectrum analysis, IX. 168; XIII. 273, =299-302=, 303,
=304=, 306, 309, =311-314=, 317, 318

_Sphaera Mundi_ (of Sacrobosco), III. 67

Sphere, attraction of, IX. 173, =182=; XI. 228

Sphere, celestial. _See_ Celestial sphere

Sphere, doctrine of the. _See_ Spherics

Spheres, celestial, crystal. _See_ Celestial spheres

Spheres, music of the, II. =23=; VII. 144

Spherical form of the earth, moon. _See_ Earth, Moon, shape of

Spherics, II. =33=, 34

Spica, II. 42

Spiral nebulae, XIII. 310

Stability of the solar system, XI. 245; XIII. 288, 293

Stadium, II. 36, 45, 47

Star-atlases, star-maps, I. 12 _n_; X. 198, 223; XII. 259, 266;
XIII. 280, 294

Star-catalogues, II. =32=, =42=, =50=; III. 62, =63=; IV. =83=;
V. =98=, =107=, 110, 112; VIII. =153=; X. =198=, =199=, 205,
=218=, =223-225=; XII. 257; XIII. =277=, =280=, 316

Star-clusters, VI. 120; X. 223; XII. 258, =259=, =260=, 261; XIII.
307, 308, 310, 311, 318

Star-gauging, XII. =258=; XIII. 307

Star-groups. _See_ Constellations

Stars, I. 1, 5, 7-10, 12-15, 18; II. 20, 23-26, 29, 30, 32, 33, 39,
40, 42, 45-47, 50; III. 56, 57, 62, 68; IV. 73, 78, 80, 86, 89, 92;
V. 96-100, 104, 105, 110; VI. 120, 121, 129; VIII. 155, 157, 161;
IX. 186 _n_; X. 198, 199, 203, 207-214, 218, 223; XI. 228; XII. 253,
=257-266=, 267; XIII. 272, 277-280, 283, 304, =306-318=, 320.
_See also_ the preceding and following headings

Stars, binary. _See_ Stars, double and multiple

Stars, brightness of, II. 42; XII. 258, 266; XIII. 278, 280, 316,
317. _See also_ Stars, variable

Stars, circumpolar, I. =9=; II. 35

Stars, colours of, XII. 263; XIII. 309

Stars, distances of, I. 7; II. 30, 32, 45, 47; IV. 80, =92=; V. 100;
VI. 117, =129=; XI. 228; XII. 257, 258, 265, 266; XIII. =278=,
=279=, 317. _See also_ Parallax, stellar

Stars, distribution of, XII. 257, 258. _See also_ Sidereal system,
structure of

Stars, double and multiple, XII. 256, =263=, =264=; XIII. 306-308,
=309=, =314=, 320

Stars, magnitudes of, II. 42; XII. 266; XIII. 280, 316. _See also_
Stars, brightness of

Stars, motion of. _See_ Stars, proper motion of, _and_ Daily motion
(of the celestial sphere)

Stars, names of, I. 12, 13; III. 64

Stars, nebulous, X. 223; XII. 260, 261

Stars, new, II. 42; V. 100, 104; VI. 117, 129; VII. 138; XII. 266;
XIII. 312

Stars, number of, I. 7 _n_; XIII. 280

Stars, parallax of. _See_ Parallax, stellar

Stars, proper motion of, X. =203=, 225; XII. 257, =265=; XIII. 278,
280

Stars, rotation of, XII. 266

Stars, spectra of, XIII. 311-314, 317

Stars, system of. _See_ Sidereal system, structure of

Stars, variable, XII. =266=, 269; XIII. 307, 312, =314=, =315=

Stationary points, I. =14=; II. 51; IV. =88=

Stjerneborg, V. 101

Summer solstice, I. 11. _See also_ Solstices

Sun, I. 1, 4, 10, 13, 14, 16; II. 21, 23-26, 28-30, 32, 35, 36, 40,
43, 45, 48, 51; III. 68, 69; IV. 73, 75, 77, 79-82, 85-90, 92;
V. 98, 103, 105, 110, 111; VI. 119, 121, 123, 124, 126, 127, 129,
132; VII. 136, 139-141, 144-146, 150; VIII. 153, 154, 156;
IX. 170, 172-175, 181, 183-186, 188-190, 194; X. 198, 200, 202,
205, 210, 213, 223, 227; XI. 228, 235, 236, 240, 243, 245, 250;
XII. 257, 265, =268=, =269=; XIII. 272, 278, 283, 288, 292-294,
297, =298-303=, 304, 305, =307=, =319=, 320. _See also_ the
following headings

Sun, angular or apparent size of, II. =32=, 38, 39, =41=, 43, 46 _n_,
48; IV. 73, 90; V. 105 _n_

Sun, apparent flattening of, II. 46

Sun, distance of, I. 15; II. 24, 25, 30, =32=, 38, =41=, 43, 45, 48,
=49=, 51; IV. 81, =85=, 86, 87, 90, 92; V. 111; VII. 144, 145;
VIII. 156, =161=: IX. 185, 188; X. 202, 205, 223, =227=; XI. 235;
XIII. 278, =281-284=

Sun, eclipses of. _See_ Eclipses

Sun, heat of, XII. 268, 269; XIII. 303, =307=, =319=

Sun, mass of, IX. 183, 184, =185=, 189; XI. 228; XIII. 282

Sun, motion of, I. 3, 5, 8, =10=, =11=, 13, 15-17; II. 20, 21, 24-26,
35, 37, =38=, =39=, 40, 42, 43, 47, 48, 51; III. 59; IV. 73, =77=,
=79=, 85, 86, 87, 92; V. 104, =105=, 111; VI. 121, 126, 127, 132;
VIII. 160; IX. =186=; X. 223; XI. 235; XII. =265=; XIII. 288

Sun, parallax of, II. 43; V. 98, 110; VII. 145; VIII. =161=; X. 223,
=227=; XIII. =281-284=. _See also_ Sun, distance of

Sun, rotation of, VI. =124=; VII. 150; XI. 250; XIII. 297, =298=, 302

Sun, size of, II. =32=; IV. 85; VII. 145; IX. 173; XIII. 319

Sun, tables of. _See_ Tables, solar

Sun-dials, II. 34

Sun-spots, VI. =124=, 125; VIII. 153; XII. =268=, 269; XIII. =298=,
300, 302, 303

Superior planets, I. =15=; IV. 87, 88. _See also_ Mars, Jupiter, etc.

Svea, XIII. 294

Synodic month, II. =40=, 43

Synodic period, IV. 86, 87

_Synopsis of Cometary Astronomy_ (of Halley), X. 200

_Systema Saturnium_ (of Huygens), VIII. 154

_Système du Monde_ (of Laplace), XI. =238=, 242 _n_, =250=

_Système du Monde_ (of Pontécoulant), XIII. 286

_Table Talk_ (of Luther), IV. 73

Tables, astronomical, III. 58, =60-63=, =66=, 68; IV. 70; V. =94=,
96, 97, 99, 110; VII. 139, =148=; VIII. 156, =160=; X. =216=, 217;
XIII. 277. _See also_ the following headings

Tables, lunar, II. =48=; III. =59=; X. =204=, 216, 217, =226=; XI.
=233=, 234, =241=; XIII. =286=, 290

Tables, planetary, III. =63=; V. 108, 112; VII. 142, 143; X. =204=,
216; XI. 235, =247=; XIII. =288=, 289, 290

Tables, solar, III. =59=; IV. =85=; V. 111; VIII. 153; X. =224=, 225,
226; XI. 235, =247=; XIII. 290

Tables, Alfonsine, III. =66=, 68; V. 94, 96, 99

Tables, Hakemite, III. =60=, 62

Tables, Ilkhanic, III. 62

Tables, Prussian, V. =94=, 96, 97, 99; VII. 139

Tables, Rudolphine, V. 94; VII. =148=, 151; VIII. 156

Tables, Toletan, III. =61=, 66

_Tables de la Lune_ (of Damoiseau), XIII. 286

_Tabulae Regiomontanae_ (of Bessel), XIII. 277

Tangent, III. 59 _n_, 68 _n_

Tartars, Tartar astronomy, III. 63

Tebbutt’s comet, XIII, 305

Telescope, III. 67; VI. =118-124=, 134; VII. =138=; VIII. =152-155=;
IX. =168=; X. 207, 213, 218; XII. 251, =252-258=, 260, 262, =271=;
XIII. =274=, 300, 301, 306, =310=, 317

_Theoria Motus_ (of Gauss), XIII. 276

_Theoria Motuum Lunae_ (of Euler), XI. 233

_Théorie de la Lune_ (of Clairaut), XI. 233

_Théorie ... des Probabilités_ (of Laplace), XI. 238

_Théorie ... du Système du Monde_ (of Pontécoulant), XIII. 286

_Theory of the Moon_ (of Mayer), X. 226

_Theory of the Universe_ (of Wright), XII. 258

Thetis, VIII. 160

Three bodies, problem of, XI. =228=, =230-233=, 235

Tidal friction, XIII. =287=, 292, 293, =320=

Tides, VI. 130; VII. 150; IX. =189=; XI. 228-230, 235, =248=;
XIII. =287=, =292=, 293, 297, =320=

Time, measurement of, I. 4, 5, 16. _See also_ Calendar, Day, Hour,
Month, Week, Year

Titan, VIII. 154

Titania, XII. 255

_Toletan Tables_, III. =61=, 66

Torrid zones, II. 35

Total eclipse, II. =43=; VII. 145; X. 205; XIII. =301=. _See also_
Eclipses

Transit instrument, X. 218, 225 _n_

Transit of Mercury, X. 199

Transit of Venus, VIII. 156; X. =202=, 205, 224, =227=; XIII. =281=,
282, 284

Translations, III. 56, 58, 60, 62, 66, 68

Transversals, V. 110 _n_

Trepidation, III. =58=, 62, 68; IV. 84; V. 112

Trigonometry, II. 37 _n_, 47 _n_; III. 59 _n_, 64 _n_, 68 _n_; IV. 74

Trivium, III. 65

Tropical year, II. 42

Tuttle’s comet, XIII. 291

Twilight, III. 69

Twinkling of stars, II. 30

_Two New Sciences_ (of Galilei), VI. =133=, 134 _n_; VIII. 152

Tychonic system, V. =105=; VI. 127

Umbra (of sun-spots), VI. =124=; XII. 268

Uniform acceleration, VI. 133. _See also_ Acceleration

Uraniborg, V. 101

_Uranometria Nova Oxoniensis_, XIII. 316

Uranus, XII. =253=, 254, 255, 267; XIII. 276, 288, 289, =297=

Uranus, rotation of, XIII. 297

Uranus, satellites of, XI. 250 _n_; XII. =255=, 267; XIII. 272, =295=

Variable stars. _See_ Stars, variable

Variation (of the moon), III. =60=; V. =111=; VII. 145

Variation of elements or parameters, XI. 233 _n_, =236=, 245

Variations, calculus of, XI. 237 _n_

Vega, III. 64

Venus, I. 14-16; II. 25, 26, 45, 47, 51; III. 68; IV. 75, 81, 86, 87;
V. 98, 100, 103; VI. 121, 123; VII. 136 n, 139, 142, 144; VIII.
=154=; IX. 181, 185; X. 223, 227; XI. 235, 245; XII. =267=, 271;
XIII. 282, =297=. _See also_ the following headings

Venus, mass of, XI. 235, 248

Venus, phases of, VI. =123=, 129

Venus, rotation of, VIII 160; XII. 267; XIII. 297

Venus, transits of. _See_ Transits of Venus

Vernal equinox, I. 11. _See also_ Equinoxes

Vernier, III. 69 _n_

Vertical, II. 33; X. 221; XIII. 285

Vesta, XIII. 294

Victoria, XIII. 281

Virtual velocities, XI. 237 _n_

Vortices, VIII. =163=; IX. 178, 195

Wave, wave-length (of light) XIII. 299, 300, 302

Weather, prediction of, II. 20; VII. 136

Week, I. 16

Weight, VI. 116, 130; IX. 180

Weights and Measures, Commission on, XI. 237, 238

_Whetstone of Witte_ (of Recorde), V. 95

Winter solstice, I. 11. _See also_ Solstices

Year, I. 3, 4, 16; II. 19-22, 42, 47; III. 66; V. 111

Year, sidereal, II. 42

Year, tropical, II. 42

_Zadkiel’s Almanack_, I. 18 _n_

Zenith, II. =33=, 35, 36, 46; III, 64; X. 221

Zenith-sector, X. 206

Zodiac, I. =13=; X. 224

Zodiac, signs of the, I. 13

Zodiacal constellations, I. 13

Zones of the earth, II. =35=, 47

FOOTNOTES:

[1] In our climate 2,000 is about the greatest number ever visible at once, even to a keen-sighted person.

[2] Owing to the greater brightness of the stars overhead they usually seem a little nearer than those near the horizon, and consequently the visible portion of the celestial sphere appears to be rather less than a half of a complete sphere. This is, however, of no importance, and will for the future be ignored.

[3] A right angle is divided into ninety degrees (90°), a degree into sixty minutes (60′), and a minute into sixty seconds (60″).

[4] I have made no attempt either here or elsewhere to describe the constellations and their positions, as I believe such verbal descriptions to be almost useless. For a beginner who wishes to become familiar with them the best plan is to get some better informed, friend to point out a few of the more conspicuous ones, in different parts of the sky. Others can then be readily added by means of a star-atlas, or of the star-maps given in many textbooks.

[5] The names, in the customary Latin forms, are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces; they are easily remembered by the doggerel verses:—

The Ram, the Bull, the Heavenly Twins,
And next the Crab, the Lion shines,
The Virgin and the Scales,
The Scorpion, Archer, and He-Goat,
The Man that bears the Watering-pot,
And Fish with glittering tails.

[6] This statement leaves out of account small motions nearly or quite invisible to the naked eye, some of which are among the most interesting discoveries of telescopic astronomy; see, for example, chapter X., §§ 207-215.

[7] The custom of calling the sun and moon planets has now died out, and the modern usage will be adopted henceforward in this book.

[8] It may be noted that our word “day” (and the corresponding word in other languages) is commonly used in two senses, either for the time between sunrise and sunset (day as distinguished from night), or for the whole period of 24 hours or day-and-night. The Greeks, however, used for the latter a special word, νυχθήμερον.

[9] Compare the French: Mardi, Mercredi, Jeudi, Vendredi; or better still the Italian: Martedi, Mercoledi, Giovedi, Venerdi.

[10] See, for example, _Old Moore’s_ or _Zadkiel’s Almanack_.

[11] We have little definite knowledge of his life. He was born in the earlier part of the 6th century B.C., and died at the end of the same century or beginning of the next.

[12] Theophrastus was born about half a century, Plutarch nearly five centuries, later than Plato.

[13] _Republic_, VII. 529, 530.

[14] Confused, because the mechanical knowledge of the time was quite unequal to giving any explanation of the way in which these spheres acted on one another.

[15] I have introduced here the familiar explanation of the phases of the moon, and the argument based on it for the spherical shape of the moon, because, although probably known before Aristotle, there is, as far as I know, no clear and definite statement of the matter in any earlier writer, and after his time it becomes an accepted part of Greek elementary astronomy. It may be noticed that the explanation is unaffected either by the question of the rotation of the earth or by that of its motion round the sun.

[16] See, for example, the account of Galilei’s controversies, in chapter VI.

[17] The =poles= of a great circle on a sphere are the ends of a diameter perpendicular to the plane of the great circle. Every point on the great circle is at the same distance, 90°, from each pole.

[18] The _word_ “zenith” is Arabic, not Greek: cf. chapter III., § 64.

[19] Most of these names are not Greek, but of later origin.

[20] That of M. Paul Tannery: _Recherches sur l’Histoire de l’Astronomie Ancienne_, chap. V.

[21] Trigonometry.

[22] The process may be worth illustrating by means of a simpler problem. A heavy body, falling freely under gravity, is found (the resistance of the air being allowed for) to fall about 16 feet in 1 second, 64 feet in 2 seconds, 144 feet in 3 seconds, 256 feet in 4 seconds, 400 feet in 5 seconds, and so on. This series of figures carried on as far as may be required would satisfy practical requirements, supplemented if desired by the corresponding figures for fractions of seconds; but the mathematician represents the same facts more simply and in a way more satisfactory to the mind by the formula s = 16 t^2, where s denotes the number of feet fallen, and t the number of seconds. By giving t any assigned value, the corresponding space fallen through is at once obtained. Similarly the motion of the sun can be represented approximately by the more complicated formula l = nt + 2 e sin nt, where l is the distance from a fixed point in the orbit, t the time, and n, e certain numerical quantities.

[23] At the present time there is still a small discrepancy between the observed and calculated places of the moon. See chapter XIII., § 290.

[24] The name is interesting as a remnant of a very early superstition. Eclipses, which always occur near the nodes, were at one time supposed to be caused by a dragon which devoured the sun or moon. The symbols ☊ ☋ still used to denote the two nodes are supposed to represent the head and tail of the dragon.

[25] In the figure, which is taken from the _De Revolutionibus_ of Coppernicus (chapter IV., § 85), let D, K, M represent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and let S Q G, S R E denote the boundaries of the shadow-cone cast by the earth; then Q R, drawn at right angles to the axis of the cone, is the breadth of the shadow at the distance of the moon. We have then at once from similar triangles

G K - Q M : A D - G K :: M K : K D.

Hence if K D = _n_. M K and ∴ also A D = _n_. (radius of moon), _n_ being 19 according to Aristarchus, G K-Q M: _n_. (radius of moon)-G K

:: 1 : _n_

_n_ . (radius of moon) - G K

= _n_ G K - _n_ Q M

∴ radius of moon + radius of shadow

= (1 + 1∕_n_) (radius of earth).

By observation the angular radius of the shadow was found to be about 40′ and that of the moon to be 15′, so that

radius of shadow = 8∕3 radius of moon;

∴ radius of moon

= 3∕11 (1 + 1∕_n_) (radius of earth).

But the angular radius of the moon being 15′, its distance is necessarily about 220 times its radius,

and ∴ distance of the moon

= 60 (1 + 1∕_n_) (radius of the earth),

which is roughly Hipparchus’s result, if _n_ be _any_ fairly large number.

[26] _Histoire de l’Astronomie Ancienne_, Vol. I., p. 185.

[27] The chief MS. bears the title μεγάλη σύνταξις or great composition though the author refers to his book elsewhere as μαθηματικὴ σύνταξις (mathematical composition). The Arabian translators, either through admiration or carelessness, converted μεγάλη, great, into μεγίστη, greatest, and hence it became known by the Arabs as _Al Magisti_, whence the Latin _Almagestum_ and our _Almagest_.

[28] The better known apparent enlargement of the sun or moon when rising or setting has nothing to do with refraction. It is an optical illusion not very satisfactorily explained, but probably due to the lesser brilliancy of the sun at the time.

[29] In spherical trigonometry.

[30] A table of chords (or double sines of half-angles) for every 1∕2° from 0° to 180°.

[31] His procedure may be compared with that of a political economist of the school of Ricardo, who, in order to establish some rough explanation of economic phenomena, starts with certain simple assumptions as to human nature, which at any rate are more plausible than any other equally simple set, and deduces from them a number of abstract conclusions, the applicability of which to real life has to be considered in individual cases. But the perfunctory discussion which such a writer gives of the qualities of the “economic man” cannot of course be regarded as his deliberate and final estimate of human nature.

[32] The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude, _a sin_θ + _b sin_ (2φ-θ), where _a_, _b_ are two numerical quantities, in round numbers 6° and 1°, θ is the angular distance of the moon from perigee, and φ is the angular distance from the sun. At conjunction and opposition φ is 0° or 180°, and the two terms reduce to (_a_-_b_) _sin_θ. This would be the form in which the equation of the centre would have presented itself to Hipparchus. Ptolemy’s correction is therefore equivalent to adding on

_b_ [_sin_θ + _sin_ (2φ - θ)], or 2 _b sin_φ _cos_ (φ-θ),

which vanishes at conjunction or opposition, but reduces at the quadratures to 2 _b sin_θ, which again vanishes if the moon is at apogee or perigee (θ = 0° or 180°), but has its greatest value half-way between, when θ = 90°. Ptolemy’s construction gave rise also to a still smaller term of the type,

_c sin_ 2φ [_cos_ (2φ + θ) + 2 _cos_ (2φ - θ)],

which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.

[33] Here, as elsewhere, I have given no detailed account of astronomical instruments, believing such descriptions to be in general neither interesting nor intelligible to those who have not the actual instruments before them, and to be of little use to those who have.

[34] The advantage derived from the use of the equant can be made clearer by a mathematical comparison with the elliptic motion introduced by Kepler. In elliptic motion the angular motion and distance are represented approximately by the formulae _nt_ + 2_e sin nt_, _a_ (1 - _e cos nt_) respectively; the corresponding formulæ given by the use of the simple eccentric are _nt + e′ sin nt_, _a_ (1 - _e′ cos nt_). To make the angular motions agree we must therefore take _e′_ = 2_e_, but to make the distances agree we must take _e′ = e_; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ become _nt_ + 2_e′ sin nt_, _a_ (1 - _e′ cos nt_), and _both_ agree if we take _e′ = e_. Ptolemy’s lunar theory could have been nearly freed from the serious difficulty already noticed (§ 48) if he had used an equant to represent the chief inequality of the moon; and his planetary theory would have been made accurate to the first order of small quantities by the use of an equant both for the deferent and the epicycle.

[35] De Morgan classes him as a geometer with Archimedes, Euclid, and Apollonius, the three great geometers of antiquity.

[36] The legend that the books in the library served for six months as fuel for the furnaces of the public baths is rejected by Gibbon and others. One good reason for not accepting it is that by this time there were probably very few books left to burn.

[37] The data as to Indian astronomy are so uncertain, and the evidence of any important original contributions is so slight, that I have not thought it worth while to enter into the subject in any detail. The chief Indian treatises, including the one referred to in the text, bear strong marks of having been based on Greek writings.

[38] He introduced into trigonometry the use of _sines_, and made also some little use of _tangents_, without apparently realising their importance: he also used some new formulæ for the solution of spherical triangles.

[39] A prolonged but indecisive controversy has been carried on, chiefly by French scholars, with regard to the relations of Ptolemy, Abul Wafa, and Tycho in this matter.

[40] For example, the practice of treating the trigonometrical functions as _algebraic_ quantities to be manipulated by formulæ, not merely as geometrical lines.

[41] Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such as MDCCCXCVIII.

[42] On trigonometry. He reintroduced the _sine_, which had been forgotten; and made some use of the _tangent_, but like Albategnius (§ 59 _n._) did not realise its importance, and thus remained behind Ibn Yunos and Abul Wafa. An important contribution to mathematics was a table of sines calculated for every minute from 0° to 90°.

[43] That of “lunar distances.”

[44] He did not invent the measuring instrument called the _vernier_, often attributed to him, but something quite different and of very inferior value.

[45] The name is spelled in a large number of different ways both by Coppernicus and by his contemporaries. He himself usually wrote his name Coppernic, and in learned productions commonly used the Latin form Coppernicus. The spelling Copernicus is so much less commonly used by him that I have thought it better to discard it, even at the risk of appearing pedantic.

[46] _Nullo demum loco ineptior est quam ... ubi nim’s pueriliter hallucinatur_: Nowhere is he more foolish than ... where he suffers from delusions of too childish a character.

[47] His real name was Georg Joachim, that by which he is known having been made up by himself from the Latin name of the district where he was born (Rhætia).

[48] The _Commentariolus_ and the _Prima Narratio_ give most readers a better idea of what Coppernicus did than his larger book, in which it is comparatively difficult to disentangle his leading ideas from the mass of calculations based on them.

[49] _Omnis enim quæ videtur secundum locum mutatio, aut est propter locum mutatio, aut est propter spectatæ rei motum, aut videntis, aut certe disparem utriusque mutationem. Nam inter mota æqualiter ad eadem non percipitur motus, inter rem visam dico, et videntem_ (De Rev., I. v.).

I have tried to remove some of the crabbedness of the original passage by translating freely.

[50] To Coppernicus, as to many of his contemporaries, as well as to the Greeks, the simplest form of a revolution of one body round another was a motion in which the revolving body moved as if rigidly attached to the central body. Thus in the case of the earth the second motion was such that the axis of the earth remained inclined at a constant angle to the line joining earth and sun, and therefore changed its direction in space. In order then to make the axis retain a (nearly) fixed direction in space, it was necessary to add a _third_ motion.

[51] In this preliminary discussion, as in fig. 40, Coppernicus gives 80 days; but in the more detailed treatment given in Book V. he corrects this to 88 days.

[52] Fig. 42 has been slightly altered, so as to make it agree with fig. 41.

[53] Coppernicus, instead of giving longitudes as measured from the first point of Aries (or vernal equinoctial point, chapter I., §§ 11, 13), which moves on account of precession, measured the longitudes from a standard fixed star (α _Arietis_) not far from this point.

[54] According to the theory of Coppernicus, the diameter of the moon when greatest was about 1∕8 greater than its average amount; modern observations make this fraction about 1∕13. Or, to put it otherwise, the diameter of the moon when greatest ought to exceed its value when least by about 8′ according to Coppernicus, and by about 5′ according to modern observations.

[55] Euclid, I. 33.

[56] If P be the synodic period of a planet (in years), and S the sidereal period, then we evidently have (1∕P) + 1 = 1∕S for an inferior planet, and 1 - (1∕P) = 1∕S for a superior planet.

[57] Recent biographers have called attention to a cancelled passage in the manuscript of the _De Revolutionibus_ in which Coppernicus shews that an ellipse can be generated by a combination of circular motions. The proposition is, however, only a piece of pure mathematics, and has no relation to the motions of the planets round the sun. It cannot, therefore, fairly be regarded as in any way an anticipation of the ideas of Kepler (chapter VII.).

[58] It may be noticed that the differential method of parallax (chapter VI., § 129), by which such a quantity as 12′ could have been noticed, was put out of court by the general supposition, shared by Coppernicus, that the stars were all at the same distance from us.

[59] There is little doubt that he invented what were substantially logarithms independently of Napier, but, with characteristic inability or unwillingness to proclaim his discoveries, allowed the invention to die with him.

[60] A similar discovery was in fact made twice again, by Galilei (chapter VI., § 114) and by Huygens (chapter VIII., § 157).

[61] He obtained leave of absence to pay a visit to Tycho Brahe and never returned to Cassel. He must have died between 1599 and 1608.

[62] He even did not forget to provide one of the most necessary parts of a mediæval castle, a prison!

[63] It would be interesting to know what use he assigned to the (presumably) still vaster space _beyond_ the stars.

[64] Tycho makes in this connection the delightful remark that Moses must have been a skilled astronomer, because he refers to the moon as “the lesser light,” notwithstanding the fact that the apparent diameters of sun and moon are very nearly equal!

[65] By transversals.

[66] On an instrument which he had invented, called the _hydrostatic balance_.

[67] A fair idea of mediaeval views on the subject may be derived from one of the most tedious Cantos in Dante’s great poem (_Paradiso_, II.), in which the poet and Beatrice expound two different “explanations” of the spots on the moon.

[68] _Ludovico delle Colombe_ in a tract _Contra Il Moto della Terra_, which is reprinted in the national edition of Galilei’s works, Vol. III.

[69] In a letter of May 4th, 1612, he says that he has seen them for eighteen months; in the _Dialogue on the Two Systems_ (III., p. 312, in Salusbury’s translation) he says that he saw them while he still lectured at Padua, _i.e._ presumably by September 1610, as he moved to Florence in that month.

[70] _Historia e Dimostrazioni intorno alle Macchie Solari._

[71] Acts i. 11. The pun is not quite so bad in its Latin form: _Viri Galilaci_, etc.

[72] _Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum eatur, non autem quomodo Coelum gradiatur._

[73] From the translation by Salusbury, in Vol. I. of his _Mathematical Collections_.

[74] The only point of any importance in connection with Galilei’s relations with the Inquisition on which there seems to be room for any serious doubt is as to the stringency of this warning. It is probable that Galilei was at the same time specifically forbidden to “hold, teach, or defend in any way, whether verbally or in writing,” the obnoxious doctrine.

[75] This is illustrated by the well-known optical illusion whereby a white circle on a black background appears larger than an equal black one on a white background. The apparent size of the hot filament in a modern incandescent electric lamp is another good illustration.

[76] Actually, since the top of the tower is describing a slightly larger circle than its foot, the stone is at first moving eastward slightly faster than the foot of the tower, and therefore should reach the ground slightly to the _east_ of it. This displacement is, however, very minute, and can only be detected by more delicate experiments than any devised by Galilei.

[77] From the translation by Salusbury, in Vol. I. of his _Mathematical Collections_.

[78] The official minute is: _Et ei dicto quod dicat veritatem, alias devenietur ad torturam_.

[79] The three days June 21-24 the only ones which Galilei _could_ have spent in an actual prison, and there seems no reason to suppose that they were spent elsewhere than in the comfortable rooms in which it is known that he lived during most of April.

[80] Equivalent to portions of the subject now called _dynamics_ or (more correctly) _kinematics_ and _kinetics_.

[81] He estimates that a body falls in a second a distance of 4 “bracchia,” equivalent to about 8 feet, the true distance being slightly over 16.

[82] _Two New Sciences_, translated by Weston, p. 255.

[83] The astronomer appears to have used both spellings of his name almost indifferently. For example, the title-page of his most important book, the _Commentaries on the Motions of Mars_ (§ 141), has the form Kepler, while the dedication of the same book is signed Keppler.

[84] The regular solids being taken in the order: cube, tetrahedron, dodecahedron, icosahedron, octohedron, and of such magnitude that a sphere can be circumscribed to each and at the same time inscribed in the preceding solid of the series, then the radii of the six spheres so obtained were shewn by Kepler to be approximately proportional to the distances from the sun of the six planets Saturn, Jupiter, Mars, Earth, Venus, and Mercury.

[85] Two stars 4′ apart only just appear distinct to the naked eye of a person with average keenness of sight.

[86] _Commentaries on the Motions of Mars_, Part II., end of chapter XIX.

[87] An ellipse is one of several curves, known as =conic sections=, which can be formed by taking a section of a cone, and may also be defined as a curve the sum of the distances of any point on which from two fixed points inside it, known as the =foci=, is always the same.

Thus if, in the figure, S and H are the foci, and P, Q are _any_ two points on the curve, then the distances S P, H P added together are equal to the distances S Q, Q H added together, and each sum is equal to the length A A′ of the ellipse. The ratio of the distance S H to the length A A′ is known as the =eccentricity=, and is a convenient measure of the extent to which the ellipse differs from a circle.

[88] The ellipse is _more_ elongated than the actual path of Mars, an accurate drawing of which would be undistinguishable to the eye from a circle. The eccentricity is 1∕3 in the figure, that of Mars being 1∕10.

[89] _Astronomia Nova_ αἰτιολογητος _seu Physica Coelestis, tradita Commentariis de Motibus Stellae Martis._ _Ex Observationibus G. V. Tychonis Brahe._

[90] It contains the germs of the method of infinitesimals.

[91] _Harmonices Mundi Libri V._

[92] There may be some interest in Kepler’s own statement of the law: “Res est certissima exactissimaque, quod proportionis quae est inter binorum quorumque planetarum tempora periodica, sit praecise sesquialtera proportionis mediarum distantiarum, id est orbium ipsorum.”—_Harmony of the World_, Book V., chapter III.

[93] _Epitome_, Book IV., Part 2.

[94] Introduction to the _Commentaries on the Motions of Mars_.

[95] Substantially the _filar micrometer_ of modern astronomy.

[96] Galilei, at the end of his life, appears to have thought of contriving a pendulum with clockwork, but there is no satisfactory evidence that he ever carried out the idea.

[97] In modern notation: time oπf oscillation = 2π√(_l_∕_g_).

[98] _I.e._ he obtained the familiar formula (_v^2_)∕_r_, and several equivalent forms for _centrifugal force_.

[99] Also frequently referred to by the Latin name _Cartesius_.

[100] According to the unreformed calendar (O.S.) then in use in England, the date was Christmas Day, 1642. To facilitate comparison with events occurring out of England, I have used throughout this and the following chapters the Gregorian Calendar (N.S.), which was at this time adopted in a large part of the Continent (cf. chapter II., § 22).

[101] From a MS. among the Portsmouth Papers, quoted in the Preface to the Catalogue of the Portsmouth Papers.

[102] W. K. Clifford, _Aims and Instruments of Scientific Thought_.

[103] It is interesting to read that Wren offered a prize of 40_s._ to whichever of the other two should solve this the central problem of the solar system.

[104] The familiar _parallelogram of forces_, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to the _Principia_, and was, by a curious coincidence, published also in the same year by _Varignon_ and _Lami_.

[105] It is between 13 and 14 billion billion pounds. See chapter X. § 219.

[106] As far as I know Newton gives no short statement of the law in a perfectly complete and general form; separate parts of it are given in different passages of the _Principia_.

[107] It is commonly stated that Newton’s value of the motion of the moon’s apses was only about half the true value. In a scholium of the _Principia_ to prop. 35 of the third book, given in the first edition but afterwards omitted, he estimated the annual motion at 40°, the observed value being about 41°. In one of his unpublished papers, contained in the Portsmouth collection, he arrived at 39° by a process which he evidently regarded as not altogether satisfactory.

[108] Throughout the Coppernican controversy up to Newton’s time it had been generally assumed, both by Coppernicans and by their opponents, that there was some meaning in speaking of a body simply as being “at rest” or “in motion,” without any reference to any other body. But all that we can really observe is the motion of one body relative to one or more others. Astronomical observation tells us, for example, of a certain motion relative to one another of the earth and sun; and this motion was expressed in two quite different ways by Ptolemy and by Coppernicus. From a modern standpoint the question ultimately involved was whether the motions of the various bodies of the solar system relatively to the earth or relatively to the sun were the simpler to express. If it is found convenient to express them—as Coppernicus and Galilei did—in relation to the sun, some simplicity of statement is gained by speaking of the sun as “fixed” and omitting the qualification “relative to the sun” in speaking of any other body. The same motions might have been expressed relatively to any other body chosen at will: _e.g._ to one of the hands of a watch carried by a man walking up and down on the deck of a ship on a rough sea; in this case it is clear that the motions of the other bodies of the solar system relative to this body would be excessively complicated; and it would therefore be highly inconvenient though still possible to treat this particular body as “fixed.”

A new aspect of the problem presents itself, however, when an attempt—like Newton’s—is made to explain the motions of bodies of the solar system as the result of forces exerted on one another by those bodies. If, for example, we look at Newton’s First Law of Motion (chapter VI., § 130), we see that it has no meaning, unless we know what are the body or bodies relative to which the motion is being expressed; a body at rest relatively to the earth is moving relatively to the sun or to the fixed stars, and the applicability of the First Law to it depends therefore on whether we are dealing with its motion relatively to the earth or not. For most terrestrial motions it is sufficient to regard the Laws of Motion as referring to motion relative to the earth; or, in other words, we may for this purpose treat the earth as “fixed.” But if we examine certain terrestrial motions more exactly, we find that the Laws of Motion thus interpreted are not quite true; but that we get a more accurate explanation of the observed phenomena if we regard the Laws of Motion as referring to motion relative to the centre of the sun and to lines drawn from it to the stars; or, in other words, we treat the centre of the sun as a “fixed” point and these lines as “fixed” directions. But again when we are dealing with the solar system generally this interpretation is slightly inaccurate, and we have to treat the centre of gravity of the solar system instead of the sun as “fixed.”

From this point of view we may say that Newton’s object in the _Principia_ was to shew that it was possible to choose a certain point (the centre of gravity of the solar system) and certain directions (lines joining this point to the fixed stars), as a base of reference, such that all motions being treated as relative to this base, the Laws of Motion and the law of gravitation afford a consistent explanation of the observed motions of the bodies of the solar system.

[109] He estimated the annual precession due to the sun to be about 9″, and that due to the moon to be about four and a half times as great, so that the total amount due to the two bodies came out about 50″, which agrees within a fraction of a second with the amount shewn by observation; but we know now that the moon’s share is not much more than twice that of the sun.

[110] He once told Halley in despair that the lunar theory “made his head ache and kept him awake so often that he would think of it no more.”

[111] December 31st, 1719, according to the unreformed calendar (O.S.) then in use in England.

[112] The apparent number is 2,935, but 12 of these are duplicates.

[113] By Bessel (chapter XIII., § 277).

[114] The relation between the work of Flamsteed and that of Newton was expressed with more correctness than good taste by the two astronomers themselves, in the course of some quarrel about the lunar theory: “Sir Isaac worked with the ore I had dug.” “If he dug the ore, I made the gold ring.”

[115] Rigaud, in the memoirs prefixed to Bradley’s _Miscellaneous Works_.

[116] A telescopic star named 37 Camelopardi in Flamsteed’s catalogue.

[117] The story is given in T. Thomson’s _History of the Royal Society_, published more than 80 years afterwards (1812), but I have not been able to find any earlier authority for it. Bradley’s own account of his discovery gives a number of details, but has no allusion to this incident.

[118] It is _k sin_ C A B, where _k_ is the constant of aberration.

[119] His observations as a matter of fact point to a value rather greater than 18″, but he preferred to use round numbers. The figures at present accepted are 18″·42 and 13″·75, so that his ellipse was decidedly less flat than it should have been.

[120] _Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre._

[121] The word “geometer” was formerly used, as “géomètre” still is in French, in the wider sense in which “mathematician” is now customary.

[122] _Principia_, Book III., proposition 10.

[123] It is important for the purposes of this discussion to notice that the vertical is _not_ the line drawn from the centre of the earth to the place of observation.