THE PRINCIPLES AND PRACTICE OF NAVIGA TION The Principles and Practice of Navigation BY A FROST Master Mariner, M.RIN GLASGOW BROWN, SON & FERGUSON, 4-10 DARNLEY STREET LID Copyright in all countries signatory to the Berne Convention All rights reserved LIST OF CONTENTS CHAPTER I The earth-its shape The figure of the earth The measurement of position on the earth's surface Latitude and longitude Geocentric and geographical latitude Change of position on the earth's surfacedifference of latitude Difference of longitude The measurement of distance on the earth's surface-the nautical mile The geographical mile The measurement of direction Courses and bearings Variation and deviation First Edition Revised Revised Reprinted Reprinted Reprinted - 1978 1983 1988 1993 1994 1997 CHAPTER The mercator chart The rhumb line The scale of distance on a mercator chart Meridional parts The construction of the mercator chart Natural scale CHAPTER The Loxodrome Parallel sailings Plane sailing Mercator sailing The middle latitude and mid lat sailing I CHAPTER Great circles The vertex Great circles on a mercator chart Convergency The curve of constant bearing Great circle sailing Composite great circle sailing The gnomonic chart Making good a great circle track CHAPTER The celestial sphere The measurement of position on the celestial sphere The apparent motion of the sun on the celestial sphere-the ecliptic Greenwich hour angle and local hour angle The Nautical Almanac Altitude and azimuth Celestial latitude and celestial longitude Conversion between co-ordinate systems The PZX triangle CHAPTER Correction of altitudes Dip Refraction Formulae for dip and refraction Semi-diameter The augmentation Parallax Its reduction for latitude Parallax in altitude Total correction tables CHAPTER Time The solar day The sidereal day Variation in the length of the solar day Mean solar time The equation of time Universal time Atomic time and co-ordinated universal time Sidereal time Calculations on time Precession of the equinox nutation The year The civil calendar ISBN 85174 542 ISBN 85174 444 (Revised First ~dition) ©1997-BROWN, SON & FERGUSON, LTD., GLASGOW, G41 2SD Printed and Made in Great Britain CHAPTER The earth-moon system The motion of the moon on the celestial sphere in SHA and in declination The phases of the moon Retardation in the meridian passage of the m~on Retardation in moonrise and moonset The moon's rotation Librations of the moon Eclipses The ecliptic limits The recurrence of eclipses v I 20 30 42 70 88 108 128 vi THE PRINCIPLES AND PRACTICE OF NAVIGATION CHAPTER Planetary and satellite motions Universal gravitation Artificial satellites The solar system Relative planetary motion Phases of planets Retrograde motion of planets The relationship between relative motion of planets and the 'v' correction CHAPTER 10 Figure drawing The stereo graphic projection The equidistant projection Sketch figures to illustrate navigational problems The solution of theoretical problems by spherical trigonometry 142 161 THE PRINCIPLES CHAPTER 11 The motion of the heavens The earth's motion within the solar system and its effect on the apparent motion of the heavens The effect of a change of latitude on the apparent motion of the heavens The length of daylight to a stationary observer The seasons The effect of the earth's orbital motion on the apparent motion of the heavens-the change of declination of the sun Rates of change of hour angle Rates of change of altitude Twilight Variation in the length of twilight Finding the times of sunrise and sunset and the limits of twilight by solution of the PZX triangle CHAPTER 12 The celestial position line Methods of obtaining a position through which to draw the position line The Marcq St Hilaire method The longitude by chronometer method The meridian altitude method CHAPTER 13 The calculation of the position line The elements of the PZX triangle The Marcq St Hilaire method The longitude by chronometer method Noon position by forenoon sight and meridian altitude The ex-meridian problem Ex-meridian tables CHAPTER 14 Meridian altitudes Finding the time of meridian passage The longitude correction Finding the latitude by meridian altitude Lower meridian passage Maximum and meridian altitudes CHAPTER 15 The pole star problem Pole star tables Latitude by pole star 182 199 218 238 256 CHAPTER 16 The azimuth problem The ABC tables Compass error by ABC tables Amplitudes The observed altitude at theoretical rising and setting The amplitude formula Compass error by observation of the amplitude 262 APPENDIX Specimen practical navigation papers Specimen principles of navigation papers 274 ANSWERS TO EXERCISES 279 INDEX 289 EXTRACTS FROM THE Nautical Almanac 295 AND PRACTICE OF NAVIGA TION For All Courses Leading to Department of Trade Certificates of Competence The earth, the measurement of position, direction, and distance The Mercator Chart and its use in navigation The sailings Great circles-great circle sailing and the gnomonic chart The celestial sphere, the measurement of position on the sphere -the nautical almanac Correction of altitudes Time and its measurement The earth moon system Planetary motion 10 Figure drawing 11 The motions of the heavens 12 The plotting of position lines 13 The reduction of sights 14 Meridian observations 15 The pole star problem, pole star tables 16 Amplitudes and azimuths CHAPTER I THE EARTH The earth is a flattened sphere, which is rotating about one of its diameters, referred to as the axis of rotation The two points where the axis meets the surface of the earth are called the poles of the earth The circle drawn around the earth midway between the poles so that every point on it is equidistant from each pole is called the Equator The flattening is around the poles, and is caused by the tendency of the mass of the earth to fly off the surface at a tangent to the circle which it describes about the axis This causes an acceleration away from the centre of the circle around which any mass is moving Thus in Figure 1.1 a mass M tends to move along a direction M M' Any mass on the equator therefore is accelerated away from the centre of the earth, C A mass m at some point off the equator, tends to move along a direction m m' and is therefore accelerated away from L, the centre of its rotation This acceleration can be resolved into two directions one directed away from the centre of the earth, and the other at right angles to this direction along the surface of the earth towards the equator Thus any mass not on the equator has tendencies to move away from the centre of the earth and towards the equator This means that the earth's rotation is causing a THE PRINCIPLES lliEMRlli AND PRACTICE OF NAVIGATION shifting of mass towards the equator and a bulging outwards of the equatorial mass away from the earth's centre The earth is therefore distorted into an oblate spheroid, which is the solid formed by rotating an ellipse about its minor axis Any cross section of the earth taken through the poles therefore will be an ellipse If we imagine all the irregularities of the land surfaces planed off so that we have a sea level earth, it is this figure that would be the ellipsoid This is given the name of the geoid Describing the geoid as an ellipsoid is an oversimplification In fact any cross section of the geoid departs from a perfect ellipse The ellipse to which this cross section approximates to is called the reference ellipse The amount by which the geoid departs from the reference ellipse is small but measurable by modern gravimetric readings In recent years much has been learned about the true shape of the earth by the study of perturbations in artificial earth satellites The Measurement of Position on the Earth's Surface ~ Great Circle This is a circle on the surface of a sphere, whose plane passes through the centre of the sphere It is, therefore, the largest circle that can be drawn on a sphere of given radius Between any two points on the surface of the sphere there is only one great circle that can be drawn, except if the two points are at opposite ends of a diameter In this case there is an infinite number of great circles that can be drawn through them The shortest distance between two points on the surface lies along the shorter arc of the great circle between them Poles of a great circle These are the points on the sphere which are 90° removed from all points on the great circle Each great circle will have two poles, the line joining which will be perpendicular to the plane of the great circle Small Circle This is any circle on the surface of a sphere which is not a great circle The plane does not pass through the centre of the sphere and the circle therefore does not divide the _sphere into two equal halves Secondary great circles Any great circle which passes through the poles of another great circle is said to be secondary to that circle, which is then referred to as its primary Thus it could be said that the great circles that pass through the poles of the earth's rotation are secondaries to the earth's equator It does not specifically refer to this special case however It is a general term which may be used with reference to any great circle on a sphere and those great circles that cut it at right angles, hence passing through its poles To define a position on any plane surface we can assume two axes of reference at right angles to each other The definition of any point is obtained by stating the distance of the point from each of the two axes of reference In mathematics the axes are usually called the x-axis and the y-axis, and the distances of the point from these lines are called the co-ordinates of the point So defined the position is unambiguous On a spherical surface such as the earth the two axes of reference are two great circles, and instead of linear distance we use angular distances The co-ordinates used to define a position are called LATITUDE and LONGITUDE LATITUDE The axis from which this co-ordinate is measured is the equator, the plane of which is perpendicular to the earth's axis of rotation Every point on this great circle will be at an angular distance of 90° from each of the earth's poles A parallel of latitude This is a small circle on the surface of the earth whose plane is perpendicular to the earth's spin axis, and therefore parallel to the plane of t.he equator The latitude of any point can therefore be defined as the arc of a secondary to the equator which is contained between the equator and the parallel of latitude through the point being considered It is measured 0° to 90° North or South of the equator in degrees minutes and seconds of arc Thus all positions on the same parallel of latitude have the same latitude The latitude of the equator is 0° and that of each pole is 90° N or S LONGITUDE The axis from which this co-ordinate is measured is a semi-great circle which runs between the two poles of the earth and passes through an arbitrary point in Greenwich This line is a secondary to the equator and is called the Prime Meridian THE PRINCIPLES AND PRACTICE OF NAVIGATION There are an infinite number of semi-great circles that can be drawn between the poles Each one of these is called a meridian Given a position on the earth there is one meridian that passes through it The meridian that passes through the antipodal point of the position is called the anti-meridian of that position A meridian and its anti-meridian together form a great circle which is a secondary to the equator The longitude of any point can be defined as the lesser arc of the equator or the angle at the pole, between the meridian of Greenwich and the meridian through the point being considered It is measured from 0° to 180° on either side of the prime meridian and named east or west In the figure let L be the centre of curvature of the meridian at O LO is therefore the radius of curvature of the meridian at O It will cut the earth's surface at in a right angle and is therefore the vertical at O The geographical latitude is angle OLE' This will equal angle OFE The geographical latitude of an observer can be defined therefore as the angle between the vertical at the observer and the plane of the equator Geocentric Latitude Geographical Latitude The fact that the earth is not a true sphere means that the definition of latitude given must be modified The geographical latitude is the latitude of a position as observed This assumes that the earth is a sphere with radius the same as the radius of curvature of the meridian at the position being measured As the earth is an oblate spheroid the shape formed by a meridian and its anti-meridian is an ellipse The radius of curvature of the ellipse will be greatest at the poles and least at the equator This is the angle at the centre of the earth between the line joining the earth's centre to the observer and the plane of the equator In the diagram the geocentric latitude will be angle OCE The term latitude in navigation means geographical latitude or latitude as observed The difference between the geographical and the geocentric latitudes is zero at the equator and the poles and maximum in 45° N and S The difference here will be about II' of arc The geocentric latitude is given approximately by the formula: 4>-11'6 sin 24> where 4> = the geographical latitude Thus in geographical latitude 60° the geocentric latitude becomes: 60° - 11·6 x sin 120° = 60° -(11·6 = 60° - 10,04' = 59° 49'96' x 0,866) THE PRINCIPLES AND PRACTICE OF NAVIGATION THE EARTH the numerical difference between the longitudes, i.e the greater minus the smaller If they are on opposite sides of the Greenwich meridian, i.e if the longitudes are of opposite name, then the d long will be found by the sum of the two longitudes If however the d long found by this means is more than 180°, as the d long is defined as the LESSER arc of the equator between two positions, then the d long is obtained by subtracting the result from 360° The d long is named East or West according to the direction travelled D lat is named according to the direction travelled, North or South The d long between any two positions on the earth's surface is the lesser arc of the equator contained between the two meridians which pass through the two positions This is illustrated in Figure 1.5 From the figure it can be seen that if the two positions are on the same side of the Greenwich meridian then the d long will be Note D lats and d longs are usually required in minutres of arc The number of degrees is therefore multiplied by 60 and the odd minutes added on to express them in this manner To get from A to B a vessel must sail to the south and to the west D lat is therefore named S., and d long W For both d lat and d long the rule in this case is 'same name take the difference' THE R THE PRINCIPLES AND PRACTICE Find the d lat and d long between the two positions 20° 10,4' N 13° 04-5' W and 5° IS'O' S So40'S' E Pas A Pas B d lat Note Latitudes and longitudes are of the opposite name and the d lat and the d long therefore are obtained by the sums The direction travelled in is south and east D lat is therefore named S., and d long E Find the d lat and d long between the two positions 10° 00,0' S 30° 15,0' E and 67° 40,0' N 70° 30,0' W 10° 00,0' S 30° 15,0' E 67° 40,0' N 70° 30,0' W 77° 40,0' N d long 100° 45,0' W = 4660,0' N = 6045,0' W 5° 15,6' S 17° 56,0' N 23° 11,6' N = 1391·6' N I 16SO15,0' E 12So 16,5' W 296° 31-5' = 63° 2S-5' E = 3S0S-5'E Note The d long found by adding the longitudes of opposite name is more than IS0° It is therefore subtracted from 360° Note that the direction of travel is east across the IS0th meridian A vessel steaming north and east makes good a d lat of 925'S' and a d long of 1392,6' If the initial positions was 25° 20,7' N 46° 45·2' W find the position at which the vessel arrived initial position d.lat final position 25° 20,7' N 15° 25'S' N 40° 46-5' N 46° 45·2' W 23° 12,6' E 23° 32·6' W EXERCISE IA Find the d lat and d long between the following positions: S 10 40° 10,6' N 9° 25·2' W 35° 15,6' N 22° 12,4' W 10° 12,6' N 50 03'S' E 20040,0' S 170009,1' E 30° 03,3' N 152043.3' W 11031.7' N 17S000,0' E So42·6' S 1620 41·7' W 15° 20,0' S 130° 35,4' E 52°10·7'S.171°0S·0'E 60040-5' S 151° 23-5' W Final position 47° 15,7' N 50 25,9' N 50 IS·7' S 13006,5' N 42° 24,0' N 5° 14,9' S 7° 53'S' N 33° 10,5' N 27°02·3'S 10° 57,7' S u 21° 14,3' W 11 37,7' W 70 IS'S' W 17So 51·1' E 174° 01'S' W 177° OO'S'W 135° 27,9' E 155° 40,0' W 34°02·3'E 92° 47,6' W EXERCISE IB Find the d.lat and d.long between the two positions 5° 15,6' S 16So15-0'E.and 17° 56·O'N.12So 16·5'W Pas A Pas B d.lat initial position 20° 10,4' N 13° 04-5' W 5° IS'O' S So40'S' E 25° 2S'4' S d long 21° 45,3' E = 152S-4'S = 1305,3' E Pas A Pas B d.lat EARTH OF NAVIGATION Given initial position 20° 50,5' S 17So49,7' E., d lat 330 14,0' N d long 15° 37,7' E Find the final position Given initial position 390 40,6' N 9° 21'S' W., d lat 30 57' N., d long 27° 07,0' E Find the final position If a vessel's arrival position is 300 10,6' S 4040,3' E., and the d lat and d long made good was 720 IS'S' S and 3S0 54,7' E respectively, what was the initial position? A ship steered a course between north and east making good a d lat of 3So 55,5' and a d long of 200 41·S' If the position reached was 21° 10-4'N 16So IS·7' W., what was the initial position? The Measurement of Distance The unit of distance used in navigation is the nautical mile Subunits are the cable which is 0·1 of a nautical mile, and the fathom which is 0-001 of a nautical mile In navigation calculation of position is made in units of arc, degrees and minutes It is convenient therefore to use as a unit of distance, the length of a minute of arc of a great circle upon the surface of the earth Thus the nautical mile is taken as the length of a meridian which subtends an angle of one minute at the centre of the earth This definition however assumes a perfectly spherical earth which is not the case It can be modified such that one minute of geographical latitude is equal to one nautical mile in any given latitude 10 THE PRINCIPLES AND PRACTICE OF NAVIGATION Thus redefined with reference to a spheroidal earth the nautical mile is: The length of a meridian between two parallels of latitude whose geographical latitudes differ by one minute Consider the diagram THE EARTH II but in different latitudes is equal to the d lat between the two places The unit of speed at sea is the knot This is a speed of one nautical mile per hour The Geographical Mile This is the length of one minute of arc of the equator, or the length of the equator which subtends an angle of one minute at the centre of the earth The equator is the only true great circle on the reference ellipsoid, and the centre of the equator is the centre of-the earth The geographical mile therefore is a constant length of 1855·3 metres It will be equal to the length of one minute of longitude at the equator by definition The Measurement of Direction The geographical latitude of A will be angle ACE If this angle is 0° l' then the geographical latitude of A will be 0° l' N and AE will be the length of a nautical mile at the equator C is the centre of curvature of the meridian at the equator The geographical latitude of B is angle BC'E', and that of B' is angle B'C'E' If the difference, i.e angle B'C'B is one minute then the length of BB' is the nautical mile in that latitude The centre of curvature of the meridian at B is C', and the radius of curvature Be', is greater than the radius of curvature at the equator AC Therefore the length of arc BB' is greater than the length of the arc AE The length of the nautical mile as defined varies as the latitude At the equator the length is 1842·9 metres At the poles it is 1861·7 metres In practice a value of 1853 metres (6080 ft) is used and this is considered a standard nautical mile The true length of the nautical mile in any latitude is given by the formula: 1852·3- 9-4 cos (2 x Latitude) The variation in the length of the nautical mile has no significance in practical navigation Any units of d lat are taken as units of distance, and the distance between two places on the same meridian The three figure notation - The observer is considered to be at the centre of his compass, the plane of which represents tl).e plane of the horizon The direction ofthe meridian through the observer towards the north geographical pole is taken as the reference direction and called 000° The circumference of the compass card representing the horizon is divided into 360 degrees and any direction from the observer is expressed as the angle measured clockwise from the reference direction of 000° Thus the direction of east in 3-figure notation is 090° (never 90°) Thus the direction of south in 3-figure notation is 180° Thus the direction of west in 3-figure notation is 270° Thus the direction of north in 3-figure notation is 360° or 000° The 3-figure notation is used to express: Course The direction of movement of the observer Bearing The direction of an object from the observer Any instrument which is designed to measure these quantities is called a compass and to measure direction correctly the reference or zero mark on the compass card must be aligned with the direction of 000° on the horizon If this is not the case then it is necessary to find the true direction in which the compass zero points in order that a correction may be applied to find the true direction of north The gyro compass Gyroscopic compasses are liable to small errors which should not exceed one or two degrees If the north point of the compass card points to the left (to the west), of the true direction of the meridian, then all indications of direction taken from the compass will be greater than the true value 290 INDEX Diurnallibration motion of the heavens Dynamical mean sun 136 182 112 E Earth, figure of poles of I rotation of 1,74,108,182-187 shape of I Earthshine 132 Eclipses, annular 137 lunar 138 partial 137, 138 penumbral 138 recurrence of 139 solar 137 Ecliptic, definition of 70 obliquity of 72 Ecliptic limits 138 Ellipse, reference Elongation, of planets 150 of the moon 132 Equation of time 113, 119 Equator, celestial (see Equinoctial) terrestrial I Equatorial gnomonic projection 66 horizontal parallax 101 Equidistant projection 169 Equinoctial, definition of 71 projection of 163 change of position of 122, 125 Equinoxes 186 Errors of the compass 14 calculation of 264 272 Evening star 151 Ex-meridian problem 86, 230 tables 233 F equidistant Figure drawing, projection stereographic projection Figure of the earth First point of Aries (see Aries) of Libra 130, Full moon G Geocentric latitude parallax Geographical latitude mile position Geoid Gibbous moon planet Gnomonic chart projection Gravitation PAGE Great circle, definition of equation on a mercator chart poles of secondary to Great circle sailing composite Greenwich apparent time hour angle, definition of extraction from almanac meridian Gregorian calendar Gyroscope Gyroscopic inertia precession I Inferior conjunction planet Intercept, calculation of definition of Intercept terminal position 134 132 Kepler's Knot II 73 132 151,156 65 65 142 (lTP) 150 150 219 201 202 J Julian calendar Jupiter 98,101 43 3 50 58 115 74 77 126 120 120 121 H Harvest moon 134 Haversine formula 196, 220, 224, 231 Horizon, dip of 89 rational 80,88, 161, 183 sensible 88 visibJe 88 Horizontal parallax 100 Hour angle, cause of change of 182, 189 Greenwich 74 local 78 rates of change of J88 sidereal 73 Hour circles (see Celestial meridians) Hunter's moon 134 169 161 291 INDEX PAGE 126 148, 155 K laws L Latitude, celestial byex-meridian by meridian altitude by pole star definition of difference of geocentric geographical limiting middle parallel of ','C- 143 II 86 230 246 256 58 40 PAGE PAGE 1\6, 118 Leap second 126 year 183-187, \94 Length of daylight 193, 195 twilight 130, 134 Libra, first point of 135 Librations of the moon 93 Light refraction of 138 Limits, ecliptic 85, 199 Line of posi tion 218 calculation of 108 Local time 80 hour angle, calculation of 78 definition of 182-189 cause of change of 188 rate of change of 114 mean time 241 of apparent noon 239 of meridian passage Longitude, 206, 224 by chronometer 242 correction definition of difference of 74 of the GP 156 Loop of regression 251 Lower meridian passage 30 Loxodromic curve 138 Lunar eclipse 132 day 131 phases 129, 131 Lunation M 12 Magnetic meridian 12, 15 north 201 Marcq St Hilaire method 219 calCulation of 202 plotting of 179 Maximum azimuth 150,151 elongation 58 latitude 124 Mean precession 113,115 Mean solar time 112 sun, astronomical 112 dynamical 117 Measurement of time 20-28 Mercator projection 210,227,238 Meridian altitude 72,163 Meridian, celestial definition of 210,227,238 passage 251 lower prime 23 Meridional parts, definition of 24, 38 difference of 40 Middle latitude II Mile, geographical nau tical 128 Month, sidereal 129 synodic Moon, age of eclipses of its effect on precession librations of motion in declination motion in S.H.A nodes of the orbit phases of regression of the nodes retardation in transit in rising and setting rotation of sidereal period of synodic period of Moonshine Morning star 1·32 138 123 135 129 128 129, 136 131 129 132 133 135 128 129 132 151 N 81 Nadir 26 Natural scale 10 Nautical mile, definition of 10 formula for 10 standard 193,196 Nautical twilight 147 Navigational satellites 131 New moon Newton, Sir Isaac, his deductions 143 from Kepler's laws 129 Nodes of the moon's orbit 129 regression of 210, 227 Noon position II, 15 North 124 Nutation, causes of 125 effects of Obliquity of the ecliptic Opposition, of planets of the moon Orbital motion of satellites of the moon Orthomorphism 72 154 131 145, 147 128, 132 22 p Parallactic angle Parallax, annual definition of geocen tric horizontal in altitude Parallel of latitude Parallel sailing Penumbral eclipse Perigee Perihelion Perpetual day noon Phases of planets of the moon 219 101 99 98 100 102 31 138 137 110 185, 187 33 151, 156 131 292 INDEX PAGE Photographic zenith tube II X Plane of projection 20, 161 Plane sailing 34 Planetary motion 142 Planets, brightness of 152, 156 meridian passage of 244 retrograde motion of 152, 155 Plotting of position lines, by exmeridian 236 by longitude by chronometer 206 by Marcq St Hilaire 202 by meridian altitude 210 Point of projection 161 Polar distanee 219 gnomonic chart 65 variation 115 Poles, of a great circle of the earth I of the ecliptic 87, 122 of the equinoctial 71, 182 Pole star problem, 256 tables 257 Posi tion circle 85, 199 Position, geographical 73 lines (see also plotting of position lines) 85, 200 measurement of on the celestial sphere 71 measurement of, on the earth Precession of the equinox 120 of a gyrose ope 121 Prime meridian vertical , 163,268 Projections 161 equidistant 169 gnomonic 65 mercator 20 stereographie 161 Perturbations of satellites 146, 147 PZX triangle 83,218 Quadrature Q 154 R Rate of change, of altitude 191 of hour angle 188 Rational horizon , 80,88,99,161 Recurrence of eclipses 139 Reference ellipse Refraction, effect on rising and setting 91,267 definition of 93 formula for 94 terrestrial 96 Relative motion of planets 150 of the sun 70 Retardation in meridian passage 132, 242 in rising and setting 133 I 293 INDEX Retrograde motion of planets Rhumb line Right ascension Rising and setting, calculation of times of theoretical visible Rotation of earth of the moon S PAGE 152, 155 20, 30 117 194, 268 I X2, 194 193,26X I, 182 135 Sailings, the 30 eomposite great circle 58 grea t circle 50 mercator 37 mid lat 40 parallel " 31 plane 34 Satelli tes, artificial 145 navigational 147 of planets 128 Scale, natural 26 Seasons 186 Secondary great circle Semi-diameter 96 Sensible horizon ' 88 Shape of the earth I Sidereal day 109,116 hour angle 73 moon's change of 128 sun's change of 109,110,191 month 128 period, of earth 125 of planets 150 of the moon 128 Sidereal time 116 year 125 Small circle Solar day 109 eclipse 137 system 148 time 109 Solstices III, 186 Standard nautical mile 10 Stars, circumpolar 176, 185 diurnal motion of 182 hour angle of ,C 76 meridian passage of 244 annual parallax of 101 Stationary point 153 Stereographic projection , 161 Summer solstice , 186 Sun, annual motion of 70,186,187 astronomical mean 113 distance of 148 dynamical mean 112 motion in declination of 188 motion in S.H.A, of 70,109, 110, 189 PAGE PAGE 100 parallax of variation in change of S.H.A of 110 Sunrise and sunset, 194, 268 times of 270 calculation of bearing at 150 Superior conjunction 148,154 planets 14X Synchronous orbit 135 rotation , 129 Synodic month 129 period of moon , 150 of planets T Terrestrial refraction pole Theoretical rising and setting Time, apparent calculations on co-ordinated universal , defined as an angle equation of Greenwich local mean measurement of of rising and setting sidereal solar universal Total correction of altitudes eclipse, lunar solar Transferred position line Transits of Venus Tropical year True sun (see Sun) , 96 182,267 108,114 118 116 114 113, 119 115 108 113 117 194,268 116 109 115 105 138 137 204 151 126 Twilight, astronomical civil length of nau tical perpetual 193 193 195 193 196 U Universal time U.T.C 142 115 116 gravitation V 12 Variation 76, 157 'v' correction 143 Velocity, angular of planets 14X, 15 L 152 Venus IX6 Vernal (spring) equinox 42 Vertex, of great circle 81, 162 Vertical circle 163, 26X prime 88 Visible horizon 268 rising and setting 268 times of W Waxing and waning of the moon Winter solstice Year, anomalistic civil leap sidereal tropical Y , 131 186 126 126 126 125 126 Z Zenith distance tube , 81 82 131 ... CHAPTER 11 The motion of the heavens The earth's motion within the solar system and its effect on the apparent motion of the heavens The effect of a change of latitude on the apparent motion of the. .. The definition of any point is obtained by stating the distance of the point from each of the two axes of reference In mathematics the axes are usually called the x-axis and the y-axis, and the. .. (to the west), of the true direction of the meridian, then all indications of direction taken from the compass will be greater than the true value 12 THE PRINCIPLES AND PRACTICE OF NAVIGATION THE