The Elements of Navigation and Nautical Astronomy

The Elements of Navigation and Nautical Astronomy A TEXT-BOOK OF NAVIGATION AND NAUTICAL ASTRONOMY CHARLES First revison BY H. COTTER Ex.C., B.Sc. (London), M.Sc. (Wales), PH.D. (London), F.R.I.N. HIMADRI Revised by 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 1 THE MATHEMATICS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter 1 Plane Trigonometry 1 Para. 1 Introduction 3 2 Complementary Angles 4 3 Trigonometrical Functions as Straight Lines 5 4 The Signs of the Trigonometrical Ratios of Angles between 90 0 and 180 0 6 5 The Standard Formulae 7 6 Special Angles 8 Chapter 2 Circle Measure 10 Para. 1 The Radian 10 2 Trigonometrical Ratios of Small Angles 12 Chapter 3 The Traverse Table and the Solution of Plane Right-angled Triangles 14 Para. 1 Introduction 14 2 Plane Right-angled Triangles and their Traverse Table Solutions 14 Chapter 4 Compound Angles 20 Para. I Trigonometrical Ratios of the Sum of Two Angles 20 2 Trigonometrical Ratios of the Difference of Two Angles 21 3 Products as Sums and Differences 21 4 Sums and Differences as Products 22 Chapter 5 Oblique-angled Triangles and their Solutions 23 Para. I Introduction 23 2 The Sine Formula 23 3 The Cosine Formula 25 4 The Haversine Formula 26 5 The Tangent Formula 28 ix

x THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter 6 Spherical Trigonometry 3I Para. I The Geometry of the Sphere 3I 2. Propertise of a Spherical Triangle 33 3. Spherical Trigonometry 33 Chapter 7 The Stereo graphic Projection and the Graphical Solutions of Spherical Triangles 34 Para. I Introduction 34 2 The Stereographic Projection 34 3 The Principles of the Stereographic Projection 34 4 To Project a Great Circle about a Given Point as Pole 36 5 To Project a SmaIl Circle about a Given Point as Pole 36 6 To Find the Locus of Centres of all Great Circles which pass through a Given Point 37 7 To Project a Great Circle through Two Given Points 38 8 To Measure a Given Arc of a Projected Great Circle 39 9 IO To Measure a Projection of a Spherical Angle Examples 39 39 I I Figure Drawing 43 Chapter 8 The Trigonometrical Solutions of Spherical Triangles 44 Para. I The Spherical Sine Formula 44 2 The Spherical Cosine Formula 44 3 The Spherical Haversine Formula 46 4 The Four Parts Formula 47 Chapter 9 Napier's Rules 49 Para. I Napier's Rules for Solving Right-angled Spherical Triangles 49 2 Napier's Rules for Solving Quadrantal Spherical Triangles 53 3 The Solution of Oblique Spherical Triangles by Napier's Rules 54 PART2 THE SAILINGS Chapter 10 The Shape and Size of the Earth 59 Para. I The Earth 59 2 Describing a Terrestrial Position 60 3 The True Shape of the Earth 62 4 The Nautical Mile 63 5 Reduction of the Geographical Latitude 64 6 The Geographical Mile 65

CONTENTS xi Chapter 11 The Rhumb Line" 67 Para. 1 Introduction 67 2 The Sailings 67 3 Parallel Sailing 68 4 The Parallel Sailing Formula 68 5 Plane Sailing 70 6 Proof of Plane Sailing Formulae 70 7 Traverse Sailing 71 8 The Departure Position 72 9 Current 72 Chapter 12 The Mercator Chart 76 Para. 1 The Navigator's Chart 76 2 Features of a Mercator Chart 77 3 The Defects of a Mercator Chart 77 4 Distortion of the Mercator Projection 77 5 Meridional Parts 79 6 Meridional Parts for the Terrestrial Spheroid 82 7 Constructing Mercator Charts 82 Chapter 13 Mercator Sailing and Middle Latitude Sailing 86 Para. 1 Introduction 86 2 Mercator Sailing 86 3 Rhumb Line Sailing when Course Angle is Large 88 4 Middle Latitude Sailing 90 5 Crossing the Equator 90 6 The Day's Run 91 7 The Day's Work 92 Chapter 14 Great Circle Sailing 96 Para. 1 Introduction 96 2 The Gnomonic Chart 97 3 Practical Great Circle Sailing 98 4 Composite Great Circle Sailing 103 PART3 INTRODUCTION TO CHARTWORK Chapter 15 Introduction to Chartwork 109 Para. 1 Coastal Navigation 109 2 Charts 109 3 The Natural Scale of a Chart 110 4 Description of a Chart 111 5 Chart Abbreviations and Symbols 112 6 Hints when using Charts 115

xii THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter 16 Correcting the Course 118 Para. 1 The Three Norths 118 2 The Earth's Magnetism 118 3 The Three Courses 119 4 The Three Bearings 119 5 Compass Error 120 6 Variation 120 7 Deviation 121 8 The Deviation Card or Table 122 9 Leeway 124 Chapter 17 The Position Line 128 Para. 1 Fixing by Cross-Bearings 128 2 The Cocked Hat 129 3 Transits 129 4 Relative Bearings 129 5 Fixing by Bearing and Angle 129 6 Fixing by Bearing and Sounding 130 7 Fixing by Sector Light 130 8 Choosing Marks for Fixing 130 9 Angle of Cut 131 Chapter 18 The Transferred Position Line 133 Para. 1 Introduction 133 2 Transferring a Position Line 133 3 The Running Fix 134 4 Additional Use of a Position Line 135 5 Doubling the Angle on the Bow 135 6 7 The Four-Point Bearing Problem with Leeway and Current Special Angles 138 136 Chapter 19 Position Line by Vertical Angle: Distance of the Horizon 140 Para. 1 Distance off by Vertical Angle 140 2 Distance of the Theoretical Horizon 142 Chapter 20 Position Line by Horizontal Angle 146 Para. 1 Geometrical Principles 146 2 Application to Fixing 147 3 The Horizontal Danger Angle 148 4 Fixing by Horizontal Angles 148 5 Reliability of the Horizontal Angle Fix 148 6 Method of Recording a Fix by Horizontal Angles 149 7 Examples of Fixes by Horizontal Angles 149 8 9 Use of Tracing Paper for Fixing by Horizontal Angles The Station Pointer 151 151

CONTENTS Chapter 21 The Three-Bearing Problem 153 xiii Para. 1 Principles 153 2 Practice 153 3 Examples 154 Chapter 22 The Three Positions: Current Sailing 156 Para. 1 The Three Positions 156 2 Other Navigational Terms 157 3 Examples 157 4 Current Sailing 158 Chapter 23 Position Line by Radio Beating 162 Para. 1 Introduction 162. 2 Convergency of the Meridians 162 3 The Half-Convergency Correction 164 Chapter 24 Tides 166 Para. 1 The Tide 166 2 The Equilibrium Theory of the Tide 167 3 Effect of Earth's Rotation on Tides 167 4 The Moon's Effect 168 5 The Sun's Effect 169 6 The Luni-Solar Tide 169 7 The Progressive Wave Theory of the Tide 170 8 The Standing Wave Theory of the Tide 170 9 Priming and Lagging of the Tides 171 10 Tidal Streams 171 11 Practical Tide Problems 172 PART4 GENERAL ASTRONOMY Chapter 25 The Universe 185 Para. 1 The Stars 185 2 The Solar System 187 3 Kepler's Laws of Planetary Motion 190

x THE ELEMENTS OF NAVIGATION AND NAUTICAL ASTRONOMY Chapter 6 Spherical Trigonometry 31 Para. 1 The Geometry of the Sphere 31 2. Propertise of a Spherical Triangle 33 3. Spherical Trigonometry 33 Chapter 7 The Stereo graphic Projection and the Graphical Solutions of Spherical Triangles 34 Para. 1 Introduction 34 2 The Stereographic Projection 34 3 The Principles of the Stereo graphic Projection 34 4 To Project a Great Circle about a Given Point as Pole 36 5 To Project a Small Circle about a Given Point as Pole 36 6 To Find the Locus of Centres of all Great Circles which pass through a Given Point 37. 7 To Project a Great Circle through Two Given Points 38 8 To Measure a Given Arc of a Projected Great Circle 39 9 To Measure a Projection of a Spherical Angle 39 10 Examples 39 11 Figure Drawing 43 Chapter 8 The Trigonometrical Solutions of Spherical Triangles 44 Para. 1 The Spherical Sine Formula 44 2 The Spherical Cosine Formula 44 3 The Spherical Haversine Formula 46 4 The Four Parts Formula 47 Chapter 9 Napier's Rules 49 Para. 1 Napier's Rules for Solving Right-angled Spherical Triangles 49 2 Napier's Rules for Solving Quadrantal Spherical Triangles 53 3 The Solution of Oblique Spherical Triangles by Napier's Rules 54 PART2 THE SAILINGS Chapter 10 The Shape and Size of the Earth 59 Para. 1 The Earth 59 2 Describing a Terrestrial Position 60 3 The True Shape of the Earth 62 4 The Nautical Mile 63 5 Reduction of the Geographical Latitude 64 6 The Geographical Mile 65

Chapter 11 The Rhumb Line' 67 CONTENTS Para. 1 Introduction 67 2 The Sailings 67 3 Parallel Sailing 68 4 The Parallel Sailing Formula 68 5 Plane Sailing 70 6 Proof of Plane Sailing Formulae 70 7 Traverse Sailing 71 8 The Departure Position 72 9 Current 72 xi Chapter 12 The Mercator Chart 76 Para. 1 The Navigator's Chart 76 2 Features of a Mercator Chart 77 3 The Defects of a Mercator Chart 77 4 Distortion of the Mercator Projection 77 5 Meridional Parts 79 6 Meridional Parts for the Terrestrial Spheroid 82 7 Constructing Mercator Charts 82 Chapter 13 Mercator Sailing and Middle Latitude Sailing 86 Para. 1 Introduction 86 2 Mercator Sailing 86 3 Rhumb Line Sailing when Course Angle is Large 88 4 Middle Latitude Sailing 90 5 Crossing the Equator 90 6 The Day's Run 91 7 The Day's Work 92 Chapter 14 Great Circle Sailing 96 Para. 1 Introduction 96 2 The Gnomonic Chart 97 3 Practical Great Circle Sailing 98 4 Composite Great Circle Sailing 103 PART 3 INTRODUCTION TO CHARTWORK Chapter 15 Introduction to Chartwork 109 Para. I Coastal Navigation 109 2 Charts 109 3 The Natural Scale of a Chart 110 4 Description of a Chart 111 5 Chart Abbreviations and Symbols 112 6 Hints when using Charts 1I5