The element of navigation and nautical

The Elements of Navigation and Nautical Astronomy A TEXT-BOOK OF NAVIGATION AND NAUTICAL ASTRONOMY First revison BY CHARLES H COTTER Ex.C., B.Sc (London), M.Sc (Wales), PH.D (London), F.R.I.N Revised by HIMADRI K LAHIRY Master Mariner, B.Sc (Nautical Science) (Liverpool), M.A., B.Mus., M.B.A (Stirling), M.N.I., M.R.I.N., M.1 Mar (Tech), F.R.Met.S, M.C.I.T Senior Lecturer in the Department of Maritime Studies, The Nautical College, Fleetwood GLASGOW BROWN, SON & FERGUSON, LTD., NAUTICALPUBLISHERS 4-10 DARNLEYSTREET,G41 2SD What is beyond the horizon? Let us discover PREFACE THIS new edition includes numerous changes and additions Various steps involving calculations and procedure have been included in worked out examples particularly in navigational sight calculations All calculations are worked out into metric (S.I.) units Although the plan of this present book follows closely of the original, the new features include the latest methods of Tidal calculation using 1987 Tide Tables All information relating to chartwork has been updated Other new features are the inclusion of a chapter in Satellite Navigation System and Omega Many new diagrams have been added This revised edition and enlarged version provide the basic groundwork for the professional examination for Class I, Class II and Class III Certificate of Competency in Navigation and Nautical Astronomy and aimed specially to cadets preparing for B/TEC National Diploma and Higher Diploma in Nautical Science This book brings a fresh approach to the study of Navigation, and the emphasis is on the understanding of principles as well as on practical applications It provides a thoroughly comprehensive and logically arranged scheme of studies in all fields of Navigation and Nautical Astronomy The book is splendidly produced in large format, and among its attractive features are numerous clear line drawings, a wide range of worked examples and a large number of exercises, making it ideally suitable for classroom tution as well as for students working independently My share of the labour involved in producing the revised version was greatly reduced by Mrs Manjusha Lahiry, my wife, for her patient and invaluable assistance with proof reading It is a pleasure for me to record my sincere appreciation I have great faith and hope that the students and officers in Merchant Navy from all parts of the world will benefit from this book HIMADRI K LAHIRY FLEETWOOD, 1987 vii CONTENTS PART THE MATHEMATICS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter Plane Trigonometry Para Chapter Introduction Complementary Angles Trigonometrical Functions as Straight Lines The Signs of the Trigonometrical Ratios of Angles between 900 and 1800 The Standard Formulae Special Angles Circle Measure 10 Para The Radian 10 Trigonometrical Ratios of Small Angles Chapter 12 The Traverse Table and the Solution of Plane Right-angled Triangles Para Introduction 14 Plane Right-angled Triangles and their Traverse Table Solutions Chapter Para I Chapter Para I Compound Angles 20 Trigonometrical Ratios of the Sum of Two Angles 20 Trigonometrical Ratios of the Difference of Two Angles Products as Sums and Differences 21 Sums and Differences as Products 22 Oblique-angled Triangles and their Solutions Introduction 23 The Sine Formula 23 The Cosine Formula 25 The Haversine Formula 26 The Tangent Formula 28 ix 23 21 14 14 x THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter Spherical Trigonometry 3I Para I The Geometry of the Sphere 3I Propertise of a Spherical Triangle Spherical Trigonometry 33 Chapter The Stereo graphic Projection Triangles 34 33 and the Graphical Solutions of Spherical Para I Introduction 34 The Stereographic Projection 34 The Principles of the Stereographic Projection 34 To Project a Great Circle about a Given Point as Pole 36 To Project a SmaIl Circle about a Given Point as Pole 36 To Find the Locus of Centres of all Great Circles which pass through a Given Point 37 To Project a Great Circle through Two Given Points 38 To Measure a Given Arc of a Projected Great Circle 39 To Measure a Projection of a Spherical Angle 39 IO Examples 39 I I Figure Drawing 43 Chapter The Trigonometrical Solutions of Spherical Triangles 44 Para I The Spherical Sine Formula 44 The Spherical Cosine Formula 44 The Spherical Haversine Formula 46 The Four Parts Formula 47 Chapter Napier's Rules 49 Para I Napier's Rules for Solving Right-angled Spherical Triangles 49 Napier's Rules for Solving Quadrantal Spherical Triangles 53 The Solution of Oblique Spherical Triangles by Napier's Rules 54 PART2 THE SAILINGS Chapter 10 The Shape and Size of the Earth Para I 59 The Earth 59 Describing a Terrestrial Position 60 The True Shape of the Earth 62 The Nautical Mile 63 Reduction of the Geographical Latitude The Geographical Mile 65 64 xi CONTENTS Chapter 11 The Rhumb Line" 67 Para Introduction 67 The Sailings 67 Parallel Sailing 68 The Parallel Sailing Formula 68 Plane Sailing 70 Proof of Plane Sailing Formulae 70 Traverse Sailing 71 The Departure Position 72 Current 72 Chapter 12 The Mercator Chart Para 76 The Navigator's Chart 76 Features of a Mercator Chart 77 The Defects of a Mercator Chart 77 Distortion of the Mercator Projection 77 Meridional Parts 79 Meridional Parts for the Terrestrial Spheroid Constructing Mercator Charts 82 82 Chapter 13 Mercator Sailing and Middle Latitude Sailing Para Introduction 86 Mercator Sailing 86 Rhumb Line Sailing when Course Angle is Large Middle Latitude Sailing 90 Crossing the Equator 90 The Day's Run 91 The Day's Work 92 Chapter 14 Great Circle Sailing Para 86 88 96 Introduction 96 The Gnomonic Chart 97 Practical Great Circle Sailing 98 Composite Great Circle Sailing 103 PART3 INTRODUCTION TO CHARTWORK Chapter 15 Introduction to Chartwork Para 109 Coastal Navigation 109 Charts 109 The Natural Scale of a Chart 110 Description of a Chart 111 Chart Abbreviations and Symbols 112 Hints when using Charts 115 xii THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter 16 Correcting the Course Para The Three Norths 118 The Earth's Magnetism 118 The Three Courses 119 The Three Bearings 119 Compass Error 120 Variation 120 Deviation 121 The Deviation Card or Table 122 Leeway 124 Chapter 17 The Position Line Para 118 128 Fixing by Cross-Bearings 128 The Cocked Hat 129 Transits 129 Relative Bearings 129 Fixing by Bearing and Angle 129 Fixing by Bearing and Sounding 130 Fixing by Sector Light 130 Choosing Marks for Fixing 130 Angle of Cut 131 Chapter 18 The Transferred Position Line Para 133 Introduction 133 Transferring a Position Line 133 The Running Fix 134 Additional Use of a Position Line 135 Doubling the Angle on the Bow 135 The Four-Point Bearing Problem with Leeway and Current Special Angles 138 Chapter 19 Position Line by Vertical Angle: Distance of the Horizon Para Distance off by Vertical Angle 140 Distance of the Theoretical Horizon 142 Chapter 20 Position Line by Horizontal Angle Para 146 Geometrical Principles 146 Application to Fixing 147 The Horizontal Danger Angle 148 Fixing by Horizontal Angles 148 Reliability of the Horizontal Angle Fix 148 Method of Recording a Fix by Horizontal Angles 149 Examples of Fixes by Horizontal Angles 149 Use of Tracing Paper for Fixing by Horizontal Angles 151 The Station Pointer 151 136 140 xiii CONTENTS Chapter 21 The Three-Bearing Problem 153 Para Principles 153 Practice 153 Examples 154 Chapter 22 The Three Positions: Current Sailing Para The Three Positions 156 Other Navigational Terms Examples 157 Current Sailing 158 156 157 Chapter 23 Position Line by Radio Beating 162 Para Introduction 162 Convergency of the Meridians 162 The Half-Convergency Correction· 164 Chapter 24 Tides Para 10 11 166 The Tide 166 The Equilibrium Theory of the Tide 167 Effect of Earth's Rotation on Tides 167 The Moon's Effect 168 The Sun's Effect 169 The Luni-Solar Tide 169 The Progressive Wave Theory of the Tide 170 The Standing Wave Theory of the Tide 170 Priming and Lagging of the Tides 171 Tidal Streams 171 Practical Tide Problems 172 PART4 GENERAL ASTRONOMY Chapter 25 The Universe 185 Para The Stars 185 The Solar System 187 Kepler's Laws of Planetary Motion 190 x THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter Spherical Trigonometry 31 Para The Geometry of the Sphere 31 Propertise of a Spherical Triangle Spherical Trigonometry 33 Chapter The Stereo graphic Projection Triangles 34 Para 10 11 33 and the Graphical of Spherical Introduction 34 The Stereographic Projection 34 The Principles of the Stereo graphic Projection 34 To Project a Great Circle about a Given Point as Pole 36 To Project a Small Circle about a Given Point as Pole 36 To Find the Locus of Centres of all Great Circles which pass through a Given Point 37 To Project a Great Circle through Two Given Points 38 To Measure a Given Arc of a Projected Great Circle 39 To Measure a Projection of a Spherical Angle 39 Examples 39 Figure Drawing 43 Chapter The Trigonometrical Solutions of Spherical Triangles Para Solutions The The The The 44 Spherical Sine Formula 44 Spherical Cosine Formula 44 Spherical Haversine Formula 46 Four Parts Formula 47 Chapter Napier's Rules 49 Para Napier's Rules for Solving Right-angled Spherical Triangles 49 Napier's Rules for Solving Quadrantal Spherical Triangles 53 The Solution of Oblique Spherical Triangles by Napier's Rules 54 PART2 THE SAILINGS Chapter 10 The Shape and Size of the Earth Para 59 The Earth 59 Describing a Terrestrial Position 60 The True Shape of the Earth 62 The Nautical Mile 63 Reduction of the Geographical Latitude The Geographical Mile 65 64 CONTENTS xi Chapter 11 The Rhumb Line' 67 Para Introduction 67 The Sailings 67 Parallel Sailing 68 The Parallel Sailing Formula 68 Plane Sailing 70 Proof of Plane Sailing Formulae 70 Traverse Sailing 71 The Departure Position 72 Current 72 Chapter 12 The Mercator Chart Para 76 The Navigator's Chart 76 Features of a Mercator Chart 77 The Defects of a Mercator Chart 77 Distortion of the Mercator Projection 77 Meridional Parts 79 Meridional Parts for the Terrestrial Spheroid Constructing Mercator Charts 82 82 Chapter 13 Mercator Sailing and Middle Latitude Sailing Para Introduction 86 Mercator Sailing 86 Rhumb Line Sailing when Course Angle is Large Middle Latitude Sailing 90 Crossing the Equator 90 The Day's Run 91 The Day's Work 92 Chapter 14 Great Circle Sailing Para 86 96 Introduction 96 The Gnomonic Chart 97 Practical Great Circle Sailing 98 Composite Great Circle Sailing 103 PART INTRODUCTION TO CHARTWORK Chapter 15 Introduction to Chartwork Para I 109 Coastal Navigation 109 Charts 109 The Natural Scale of a Chart 110 Description of a Chart 111 Chart Abbreviations and Symbols 112 Hints when using Charts 1I5 88 448 THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Exercises to Chapter 20 (continued) 10 220° X 2·3 ml 11 1·6 ml or 4,7 ml 12 6,3 ml Exercises to Chapter 21 070° 332° 1'1 ml Dist = 6·2 ml.; Set = 060°; Co = 350°; Drift = 4,5 ml Rate = 1'1 kn Exercises to Chapter 22 184° 122°; A/C 23° to starboard 158°; 1314kn W Magnetic Co 255°; Current: 288° X 13 ml Co = 159°; Time = hr 45 m.; Dist 280° = Exercises to Chapter 23 090%° Exercises to Chapter 24 13 14 15 16 17 7: approx 2·7 m above cd 22·2 m 1·3 m U.K.C 1005 hours 0604 hours Exercises to Chapter 25 (2Y2)5 (2Y2)27 Exercises to Chapter 27 11° 31W N.; 54° 21W E (a) 00° 00'; 90° 00' E (b) 23W N.; 270° 46° 16' N.; 323° 43"8 = 17·4 ml ANSWERS TO EXERCISES Exercises to Chapter 31 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 34° 36° 43° 49° 52° 32° 24° 54° 36° 56° 34° 42° 36° 47° 62° 79° 64° 41° 48° 57'·1 39"8 37"1 57'·1 35',2 28'9 54',3 29"5 49"3 46"2 22"9 54"6 19',6 40"6 16"4 26"2 09',2 21"4 14',6 Exercises to Chapter 32 (i) (ii) (iii) (i) (ii) 23° 23° 56° 23° 56° 10 15° II 14° 12 23° 13 23° 00° 20',3 00° 23"5 00° 10"5 00° 10"0 23° 13"9 20"5 N.; 15"3 N.; Il'·3 N.; 15',9 N.; 18"3 N.; 58"6 S.; 34"8 S.; 16',0 N.; 20',0 S.; S.; 28° 02"8 E N.; 16° 47',4 W S.; 178° 05"0 E N.; 175° 40"4 W N.; 139° 00"5 E 00° 07"2 E 25° 34"1 E 39° 26"8 E 175° 10',5 E 39° 31"3 W 165° 04"7 W 6° 53"7 W 174° 32',0 E 45° 20',5 W Exercises to Chapter 33 10 II 12 13 14 (i) 10 h 00 m.; (ii) 22 (i) 13 h 22 m.; (ii) 10 (i) 15 h 58 m.; (ii) 00 (i) 09 h 44 m.; (ii) 03 63° 06' N 48° 34' S 090°-270° through Lat 090°-270° through Lat 090°-270° through Lat h h h h 01 m.; 16 ill.; 51 m 59 m.; (iii) 00 h 3~ m (iii) 22 h 44 m (iii) 23 h 31 m 33° 45"0 S Long 78° 45' E 44° 51"8 N Long 29° 45' W 50° 00"3 N Long 160° 45' W 449 450 THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Exercises to Chapter 33 15 090°-270° through 16 090°-270° through 17 090°-270° through 18 090° -270° through 19 090°-270° through 20 090°-270° through 21 090°-270° through (continued) Lat 32° 34"0 S Long 156° 00' E Lat 56° 53'-9 N Long 53° 15' W Lat 50° 32"9 N Long 70° 00' W Lat 29° 39"1 S Long 75° 00' E Lat 4° 42"0 N Long 152° 00' E Lat 68° 25"0 N Long 120° 00' W Lat 00° 12"2 S Long 23° 00' W Exercises to Chapter 34 14 15 16 17 18 19 20 21 22 23 24 25 P.L P.L P.L P.L P.L P.L P.L P.L P.L P.L P.L P.L 000°-180° Intercept 10,3 ml Towards Azimuth 090° 144\12°_324\12° Intercept 8,7 ml Towards Azimuth 234W 053°-233° Intercept 4,4 ml Away Azimuth 324° 179° -359° Intercept 4'6 ml Towards Azimuth 089° 159W-339W Intercept 5'8 ml Away Azimuth 069\12° 073\12°-253\12° Intercept 6'8 ml Towards Azimuth 343\12° 138W-318\12° Intercept 32·3 ml Away Azimuth 228\12° 029°-209° Intercept 2'5 ml Away Azimuth 299° 061\12°-241\12° Intercept 6,3 ml Towards Azimuth 151\12° 006W-186\12° Intercept 7·2 ml Away Azimuth 096W 042°-222° Intercept 3'5 ml Towards Azimuth 132° 136W-316W Intercept 23'4 ml Away Azimuth 226\12° Exercises to Chapter 35 10 11 12 13 14 15 16 17 40° 25° 44° 60° 51° 40° 39° 36° 10' N., 46° 13' W 10' S., 120° 25\1.1'W 20' N., 155° 26' W 2912' S., 30° 21' W 00' N., 43° 55' W 30' S., 123° 4412' W 52' N., 00° 58' W 25' N., 09° 18' W Exercises to Chapter 36 Alioth (32) will cross North Gacrux (31) will cross South with a small altitude Gienah (29) will cross South Nunki (50) has crossed North Canopus (17) will cross South with a low altitude Betelgeuse (16) will cross South Alnilam (15) has just crossed South Elnath (14) has crossed South Bellatrix (13) has crossed South Capella (12) has crossed North Rigel (11) has crossed North Antares (42) will cross South Atria (43) will cross South Eltanin (47) will cross North ANSWERS TO EXERCISES Exercises to Chapter 37 10 45° 26"1 N 42° 21"2 N P.L 089Yz-269Yz° through Lat 38° 27',2 N Long 33° 00' W P.L 089-269° through Lat 27° 49"5 N Long 75° 00' W 7°'5 W 2°·8 W 2°'0 W Exercises to Chapter 40 942"5 per hour 12 m 36 s 13 m 26 s 451 INDEX PAGE PAGE ABBREVIATIONS on chart 112 ABC tables 100,297, 324, 326 Abnormal refraction 144 Absolute magnitude 185 Abscissa •.•.•.••.•.•.•.• •.••.••.•.• 204 Aclinic line 350 Adjustmentsof compass •.•.•.•.•.••.•.••••••••• 352 of sextant 376 AdmiraltyList of Radio Signals 165,381 Tide Tables 172 Age of Moon 169,227,229 A and K Tables 332 Almanac 222 Alternating light 112 Altitudeapparent 237 circle of equal 257 corrections 243 maximum 342 meridian 261 observed 237 of Polaris 318 parallel of 205 rate of change 341 sextant 237 true 237 Ambiguity (180°) 388 Ambigudous case 24 Amphidromicpoint 171 system 17l Amplitudedefinition 297 formula 297 tables 297 uses 326 Angle(s)back 249 complementary compound 20 Greenwich hour 204 horizontal 148 horizontal danger 148 hour 203, 216 local hour 206 of cut 130,148 of elongation 185 sidereal hour 204 spherical 32 vertical sextant 140 Angular acceleration 359 ••••••••••••• Annular eclipse 231 Anomalistic year 222 Anshutz 356 Antarctic circle 199 Antipodes 31 Aphelion 191, 194,231 Apogee 226 Apparentaltitude 237 annual motion of Sun 194 diurnal motion 209 magnitude 186 solar day 194 time 221 Applied couple 359 Apse line 192 Apsides 192 Arc of Sextant 374 Arctic circle 199 Aries, First Point of 205 Arm of couple 357 Arrow up (down) 354 Artificialhorizon 2SO magnet 349 satellite 187 Ascending note - 226 Asteroid 187 Astronomicallatitude - 65 mean sun 218 position line 235,153 refraction 238 telescope 357 triangle 207 twilight 335 Astronomygeneral 183 nautical 253 Atomicclock 223 time 223 Augmentation 240 Autumn 196 Autumnal equinox 196,227 Axial motion of Earth 197 Axis of great circle 31 Azimuthdefinition 205, 295 limiting 341 mirror 354 tables: ABC 100, 297, 324 tables: Burdwood 297, 328, 334 453 454 THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY PAGE AZlmutht.ables: Da~is AZimuth oscillation BACK angle Ballisticdeflection tilt Barycen~r~' 297, 328, 334 368 249 " 372 372 168 Base line : : : : : :: : : : : : : : : : : : : : : :: :: : :: :: Base line extension Bearing(s)and angle " and sounding check compass cross definition fixing by four point magnetic Mercatorial open radio relative' : : : : : :: :: : : : : : : : : : : : : : : :: : : : : :: rhumb line true Bellini-Tosi D.F Bissextile year " Bissextus Bow (doubling angle on) Boyle's Law Brown S.G Bubble attachment 250, Burdwood azimuth tables 297, 328, 393 393 CABLE Caesium atom Calculated zenith distance CalendarChristian definition ecclesiastical Gregorian Julian Old Style Roman Calends Calibration of D.F Cancer, Tropic of Capricorn, Tropic of Cardinal points Cartesian co-ordinates Catalogue of hydrographic charts Cautionary notices C correction Celestialconcave equator hemisphere horizon 64 223 296 , , 129 130 128 119 128 205 129 136 119 162 134 12 129 162 119 390 224 224 136 383 356 324 334 224 224 224 223 223 223 223 223 390 199 199 60 20 I 110 116 303 194 206 206 205 PAGE Celestial latitude longitude meridian po~e .••• sp ere Cent~fugal force Centnng error Change (New) Chart- abbreviatIOns catalogue coastal : compans?n umt const~ctlOn correctIOns datu~ descnp~lOn of ¥nomn.lc Iso~omc lattice M.ercator ~Iscellaneous pan plotting route scale symbols time zone title triangle wind and current work Charteddepth " height range Check b eari ng Chermk f ee Choosmg marks for fixing ~~osen latitude ;ono.m~ter escnptlOn Journal u~e o.f wmdmg Clr~le (of)dIUrnal equal altitude great illumination latitude position prime vertical small vertical Circular measure Circumpolarbody star Civil twilight A- 206 205 206 206 194 191 378 189 112 III 110 401 83 III 166 III 98 350 392 74, 76, 96 109 109 304 109 110 112 221 112 86 110 109 112 114 112 128 38 151 278 380 376 381 380 209 257 31 196 204 259 210 31 205 10 209, 210 210 336 455 INDEX PAGE PAGE 109 Coastal navigation ll5 Coast Radio Station 129 Cocked hat 378 Collimaton error 224 Common year 396 · •.••.•• Comparison frequency •.•.•.• Compass352 adjustments 352 bearing 352 correction 1I9 course 121 deviation 350 dry card 120, 296 error 356 gyro 60, 350, 356 instrument 350 liq~id 350 magnetic 1I8 north 371 period Compensator weights 369 Complementary angle 103 Composite great circle 20 Compound angles 392 Confocal hyperbolae Conjunction 190 inferior 190 superior 359 Conservation of angular momentum 115,394 Consol 186 Constellation 34 Conventional projection 162 Convergency Co-ordinated Universal Time 224 204 Co-ordinates 170 Co-range lines Correction 243-248 of altitudes 1I8 of courses 303 of longitude Cosecant Cosine 25 Cosine formula Cotangent 170 Co-tidal lines 358 Couple Course96 angles 1I9 compass 1I8 correction 1I9 magnetic 1I9 true 190 Crescent 128 Cross bearings 91 Crossing the Equator 209 Culmination 73,92 Current 158 Current sailing DAMPINGdefinition 368 Damping368 torque 142,148 Danger angle 221 Date line •.• 296, 333 Davis' Azimuth tables Day 214 apparent solar 213 lunar 214 mean solar 214 sidereal 194,213 solar 197 Daylight and darkness 91 Day's run 90, 92 Day's work Dead reckoning72 definition 156 position 256 "d" correction Decca397 fix 397 lane 396 lattice chart 396 navigator 397 zone 206 Declination 397 Decometer 383 Deep sea lead line 68 Departure 72 Departure position 199, 227 Descending node 1I5 Detector light Deviation122 card 121 definition 122 table 1I5 D.F station lIS Diaphone Difference of61 latitude 61,68 longitude 79 meridonial parts Dip239 effect of refraction on magnetic 350 349 needle 237 of sea horizon 143 Dipping range 388 Direction finder 202 Direct motion Distance96 great circle 237 of sea horizon 140 off by vertical angle 77 on Mercator chart 67 rhumb line 257 zenith 391 Distance difference systems Diurnal209 circles 166 tides 225 Dominicalletter ••• 456 THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY PAGE PAGE 405 Doppler shift Doubling the angle on the bow 136 72 Drift (current) 362 Drift (gyro) 364 Drifting 72,156 D.R position 166 Duration of tide 218 Dynamical Mean Sun EARTH197 annual motion 197 axial motion 349 magnetism 359 -Moon system 59 radius 194 revolution 197 rotation 403 satellite 59 shape 59 size 62 true shape 225 Easter 171 Ebb tide 225 Ecclesiastical calendar 384 Echo-sounder Eclipse 231 annular of Moon 230 230 of Sun 231 partial Ecliptic195 definition 195 obliquity of 204 system 62 Ellipticity of the Earth 188 Elongation angle 92 Engine distance 229 Epact 216,217 Ephemeris time Equation of time 217 Equator 91 crossing the definition 60 magnetic 350 206 Equatorial system Equilibrium theory of the tide 167 196 Equinoctial point Equinoxautumnal 196 196 definition 196 spring 196 vernal 394 Equisignal 59 Eratosthenes 375 Erecting telescope Error(s)119, 296 compass 368 gyro compass 303 in altitude 302 in longitude 368 latitude Error(s)latitude, course and speed rolling Estimated position "." """" "J" 72, Euler, L " •••• " Evening star Ex-meridianaltitude by intercept method by Napier's rules formula observation problem tables Extra-galactic nebulae Extreme range 369 369 156 403 189 311 312 313 312 311 316 314 185 143 FALSE echoes Family of confocal hyperbolae Figure drawing 199, 204, First Point of Aries Fix (by)bearing and angle bearing and sounding cross bearings Decca four point bearing horizontal angles reliability of running transit bearings vertical angles Fixed light Fixing, choosing marks for Flamsteed, J Flashing light Flinders' bar Flinders, M Flood stream Fogbell detector light diaphone gong gun horn nautophone whistle Force Foucault's pendulum Four parts formula Four point bearing Free gyroscope Frigid zone 401 392 43 213 GALACTIC arch Galaxy Geocentric latitude Geographical latitude 185 185 63 129 130 128 397 136 146 148 134 131 140 112 130 403 113 352 352 171 115 115 114 115 115 115 115 115 357 356 47 136 365 198 64 458 THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY PAGE International date line 221 International System (Sol.) Units 222 Interpolation tables 256 Interval between maximum and meridian altitudes 343 Irradiation 243 Island universe 185 Isoclinic lines 350 Isogonal 120 Isogonic chart 120,350 JULIAN calendar o •• , 223 •••••••••••••••••••• KAUFMAN, G 76 KelvinLord 352,403 sounding machine 383 sounding tube 383 187 spheres 190 Kepler, J Kepler's Laws 190 Knot 64 171 LAGGING of tides Lalande, J Jo 403 Land effect 390 Latitudeastronomical 63 261, 263 by meridian altitude 318 by Polaris celestial 204 204 circle of 368 error geocentric 63 geographical 63 mean 90 middle 90 of Geographical Position 253 60 parallel of reduced 63 reduction to 63 Lattice charts 392 Lead linedeep sea 383 hand 383 223 Leap second 224 Leap year 124 Leeway Libra, First Point of 204 Librations 230 Lightalternating 112 112 characteristics dipping 113 fixed 113 flashing 113 fog detector 115 113 group flashing Morse Code 112 occulting 112 '" PAGE Lightrising sector year : JA Limiting azimuth Line of apsides Lines of force Liquid compass Liquid gravity control Local hour angle Locus Logclock definition Longitudeand time celestial definition general remarks method of sight reduction for of Geographical Position Looming Loop aeriel Loran Loxodromic curve Lubber line Lucifer Lunarday method month tide Lunation Luni-solar tide 000000000000000000 • •••••••••••••••••••• MACROBIUS Magnetartificial corrector Magneticbearing compass course dip equator field lines of force meridian north poles variation MagnetismEarth's indices permanent ship's terrestrial Magnitudeabsolute apparent 143 112 185 341 191 349 350 366 206 146 385 385 220 218 60 279 277 253 147 388 395 67 351 189 213 235 228 169 228 169 224 349 352 119 350 119 349 350 349 349 118 118 350 120 118 351 351 351 349 186 186 459 INDEX PAGE PAGE 170 Main stream of flood Map projection97 gnomonic 76 Mercator 34 stereo graphic 286 Marcq St Hilaire method 357 , Mass , 403 Mayer, T 342 Maximum altitude Mean113,140 H.W.S 112, 140 , latitude 194,213 solar day 214,217 Sun 166 tide level to middle latitude correction 90 Mercator chart76 advantages 77 defects 74, 76, 96 description 77 distortion 78 exaggeration of meridians 77 exaggeration of parallels 80 principle 34 projection 86 sailing 162 Mercator bearing , Meridian261 altitude 206 celestial 309 celestial bodie~ near the 162 convergency of 60 definition 118 magnetic 205 observer's inferior 205 observer's superior 209, 265 passage 347 virtual 261 zenith distance 78 Meridional parts 82 Mer Pts for spheroid 224 Meton 224 Metonic cycle 82 Micrometer 90 Middle latitude sailing 337 Midnight sun Milegeographical , 65 63 nautical 185 Milky Way 187 Minor planet 109 Miscellaneous charts 292 Modified formula 358 Moment of couple Momentum 357 Moon 229 age of altitude corrections for 243 240 augmentation 227 change Mooncrescent declination of distance of , eclipse of ,1, first quarter ~ full , gibbous harvest horizontal parallax of hunter's last quarter librations of new orbit of phases of retardation of rise (set) waxing and waning winter and summer fun Morning star Morse code light Motiondirect retrograde Myerscough and Hamilton tables 228 227 226 230 228 228 228 229 242 229 228 230 228 226 227 226 235 228 229 188 112 202 202 332 205 NADIR Napier's Rules 313 for ex-meridian problem 54 for oblique spherical triangles 53 for quadrantal spherical triangles 49 for right angled spherical triangles 110,111 Natural scale Nautical 222, 235,240, 250, 254, 258 Almanac 253, 318, 334 astronomy 63 mile 335 twilight 115 Nautophone Navigationalinstruments (see Instruments) 189 planets 409 satellites 185 stars 169 Neap tide 228 New Moon 357 Newton's Laws of Motion 390 Night effect 376 Non-adjustable errors of sextant 227 Node 91 Noon position North118 compass 118 magnetic 118 pole 118 true 401 North-up display 111 Notices to mariners 225 Number (Golden) , INDEX PAGE Radiotime signal Radiusof curvature of Earth , of gyration vector Ramark Range of tide Rates of change Reckoning Recording fix by Horizontal Angles Reduced latitude Reduction to the latitude Reduction to the meridian '" Reduction to soundings '" Reed Refraction Relative bearing Relative display Retardation Retrograde motion Retrogressive curve Rhumb linedefinition sailing Rising range Rising (setting) phenomena Rising piece Rolling error Rotational times Running fix ••••••••••••••••••••••••• "0 236 63 62 359 191 401 166 339 72 149 64 64 314 17l 115 238 129 401 226 202 202 67 67 144 333 374 369 223 134 SAILINGSapproximate great circle 96 composite great circle 103 current 158 direction , 110 great circle 96 Mercator 86 Middle latitude 90 parallel 68 plane 70 traverse 71 Satellites 187,403 Scalenatural 110,112 on Mercator chart '" 77 Schuler tuning 411 Sea clutter 401 Sea horizondip of 237 distance of 402 Search coil 389 Seasons , 196 Secant , Secondary great circle 31,204 Sector light 113 240 Semi-diameter Semi-diurnal tide 166 Sense aerial 389 461 PAGE Sense finding unit 389 Set (of current) 72 Sextantadjustments 269 altitude 237 care of ••••.• JIir 379 description of 374 errors 376 principles 375 shades 374 telescopes , 375 use of 379 Shade error 379 Ship magnetism 351 Shore effect 390 Short method tables 330, 389 Side error 377 Siderealday 220 hour angle 207 period 226 time 216 year 222 Sine Sine formula 23 Six o'clock hour circle 210 Sight 276 Sight reduction 276 Sight reduction tables 329 Slip 91 Small angles '" 12 Small circle 36 Small corrections 115 Soft iron 315 Solarday 194, 220 system '" 187 tide 169 time 213 Solstice 196 Sosigenes 224 Sound, speed in water 384 Sounding machine 383 Special angles 8, 32, 138 Speed error 369 Sperry E B 356 Spherecelestial , 194 31 geometry of , Sphericalangle 44 cosine formula haversine 46 hyperbolae 392 sine formula , 44 triangle 33 31 trigonometry Spinning top 361 Springequinox 226 tide 169 462, THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY PAGE PAGE Standard formulae nautical mile standard port time Standing wave theory of the tide Star magnitudes Stars Station pointer Steaming time Stereo graphic projection Summerseason solstice Sunastronomical mean dynamical mean mean rise (set) true Sunday letter Superior transit Synchronism Synodic period 64 In 222 170 186 185 151 92 34 196 196 218 218 217 333 333 225 225 170 228 TABLES- 332 100, 297, 324 296 amplitude 312 azimuth 297, 328, 334 Burdwood's 332 Comrie's 296 Davis' 332 Dreisonstok's 332 Gingrich's 303 longitude correction 332 Myerscough and Hamilton AandK ABC 324 nautical astronomical 330 Ogura's 318, 319, 321 Pole Star 329 sight reduction 332 Smart and Shearme 14,69 traverse 386 Taffrail log Tangent 28 Tangent formula 374 Tangent screw 199 Temperate zone 142 Theoretical horizon 333 Theoretical sunrise (set) 153 Three bearing problem 60 Three letter points 166 Tidal curve 171 Tidal streams Tidediurnal 166 ebb 171 167 effect of Earth's rotation on 167 equilibrium theory of height of In 171 lagging of Tide169 luni-solar Moon's effect on 168 169 neap 17l priming of " JJir 170 progressive wave theory of ' 166 semi-diurnal 170 standing wave theory of , Sun's effect on 169 Tilt 362 364 Tilting 368 Tilt oscillations Time220 and longitude 215 apparent solar 215 at any instant 221 at sea 223 atomic 219 comparison of 392 difference systems 223 ephemeris 217 equation of 216 mean solar 223 rotational 220 sidereal 236 signals 222 standard 213 units 222 zones 380 Tipsy 199 Torrid zone Total correction table247 Moon 244 star 245 Sun Tracing paper for fix by horizontal 151 angles 133 Transferred position line 129, 201, 209 Transit (marks in) 404 TRANSIT (satellite navigation) 69,n Traverse sailing 14,71 Traverse table Trigonometrical5 function ratio Trigonometry 149 Tropics 222 Tropical year 237 True altitude 63 True latitude 401 True motion display Twilight 335 astronomical 335 civil 335 definition 335 nautical 115 Tyfon UNIVERSAL law of gravitation Universal time Universe 191 223 185 ... 76° The secant of an angle may, therefore, be defined as the reciprocal of the cosine of the angle Similarly, the cosecant of an angle may be defined as the reciprocal of the sine of the angle The. .. lengths of the sides A C and BC are functions of the angle A (and angle B), and of the length of the hypotenuse AB Values of the sides A C and BC are tabulated in the traverse table against values of. .. printed at the bottoms of the pages, such that the sum of the angles at the top and bottom of the page is 9()0 Thus, at the top of the page which is labelled 25°, the angle 65° -the complement of 25°_