NAVIGATION FOR SCHOOL AND COLLEGE BY A c GARDNER AND W G CREELMAN Puhlished hy BROWN, SON & FERGUSON, LTD 4-10 DARNLEY STREET NAVIGATION FOR SCHOOL COLLEGE AND BY A Copyright c GARDNER AND W G CREELMAN in all countries signatory to the Berne Convention All rights reserved CONTENTS INTRODUCTION 1 THE TERRESTRIAL PROPERTIES SPHERE AND ITS Great and Small Circles Spherical Angles Spherical Triangles Equator, Poles, Parallels and Meridians, Latitude and Longitude, D lat, D long, Nautical Mile and Knot, Statute Mile, Geographical Mile, Kilometre, Departure, the Rhumb Line First Edition /965 Second Edition /976 Reprinted /986 DIRECTION ON THE EARTH'S SURF ACE 18 The earth as a magnet, Magnetic Poles, Magnetic Equator, Variation, Isogonic Lines The ship as a magnet, and how it is magnetized Deviation and Error of the Compass The Mariners' Compass in Points, Quadrants and Degrees The correction of courses and bearings Basic principles of Magnetic and Gyro Compasses THE "SAILINGS" 43 The Parallel Sailing Formula and its applications Plane Sailing and Middle Latitude Sailing The Traverse Table The Day's Work The Mercator chart Meridional parts and D.M.P Mercator Sailing The Gnomonic chart Great Circle Sailing ISBN 85174 2~6 © 1986 X & FERGUSON, LTD., GLASGOW Printed and Made in Great Britain BROWN, SON G41 2SD TERRESTRIAL POSITION LINES The methods of obtaining terrestrial position lines Plotting position lines on squared paper, allowing for "run", and current WIT bearings, their correction and use i 86 THE SOLAR SYSTEM True motion of earth and planets round the Sun, and of the Moon round the earth Kepler's First and Second Laws of planetary motion The Seasons THE CELESTIAL SPHERE The apparent motion of the Sun, Moon, Stars and Planets Celestial Poles, Celestial Meridians, Ecliptic, First Point of Aries and First Point of Libra, Equinoctial The position of heavenly bodies in the celestial sphere, viz: Declination, Right Ascension, Sidereal Hour Angle, Greenwich Hour Angle and Local Hour Angle, Zenith, Rational Horizon, Polar Distance, Zenith Distance, True Altitude Circumpolar bodies The Geographical Position of a heavenly body TIME 105 110 THE RISING AND SETTING OF HEA VENL Y BODIES THE TIDES THE MAGNITUDE OF STARS 212 Theoretical and Visible Sunrise and Sunset Twilight The Moon's phases and their effect on the Tides Spring and Neap Tides Reduction of Soundings to Chart Datum The Magnitude of stars and the varying magnitude of planets APPENDIX I 226 Explanation of trigonometrical formulae used APPENDIX II 230 Extracts from Nautical Almanac and Tide Tables The year and the day The Apparent Solar Day, the Mean Solar Day, the Sidereal Day and the Lunar Day Apparent Time, Mean Time, and the Equation of Time, the relationship between time and longitude Zone Time and Standard Time The Nautical Almanac and its uses THE SEXTANT AND ALTITUDES Optical principles of the Sextant Its adjustment and use Nonadjustable errors of Sextant The Sextant Altitude of a Heavenly Body Index Error, Dip, Refraction, Semi-diameter and Parallax True Altitude and True Zenith Distance The astronomical position circle FIGURE DRAWING The construction of scale diagrams of the celestial sphere in the plane of the celestial meridian and in the plane of the celestial horizon, the determination by scale drawing of the Latitude, Altitude, Azimuth and Hour Angle of a heavenly body 10 ASTRONOMICAL CALCULATIONS 122 APPENDIX III 238 Abridged Nautical Tables, viz: Traverse Table, Meridional Parts Table, Mean Latitude to Middle Latitude Conversion Table, Altitude Correction Tables, Natural Haversines Table ANSWERS TO EXERCISES 253 INDEX 261 142 162 169 Latitude by Altitude of sun and star when on the meridian above and below the pole Verification by scale diagram Calculating the Azimuth and Amplitude of Sun and Star to find the Error and Deviation of the compass-Solution by Spherical Haversine formula and by tables Verification by scale diagram The Marc St Hilaire method of determining a position line 11 ASTRONOMICAL POSITION LINES The plotting of astronomical position lines, and their combination with position lines obtained by other means, taking into account the ship's "run" The use of single position lines 205 11 iii INTRODUCTION If! recent years a number of schools and colleges have introduced navigation into their curricula because of the subject's broad educational value, and the examining bodies for the General Certificate of Education and the Scottish Certificate of Education have been setting examination papers in navigation for some considerable time This book was first published in 1965 to meet the needs of students preparing for these examinations, and it has now been revised and brought up to date with the introduction of metric units where necessary Chartwork is not included in the book as this subject is dealt with in several other publications It is particularly well covered by Captain W H Squair's "Modern Chartwork", which also deals with radio and electronic aids to navigation A set of ordinary four-figure mathematical tables is required to work the examples for exercises that are given in the book All other tables that are used are contained in the Appendices For those readers who possess volumes of Nautical Tables, such as those published by Norie, Burton or Inman, answers are also given to five-figure accuracy, where required The authors wish to acknowledge their indebtedness to the publishers of Norie's Nautical Tables for permission to reproduce certain items contained in those Tables; namely part of the table of Meridional Parts, the Mean Latitude to Middle Latitude conversion table and some of the Altitude corrections Acknowledgements are also due to Her Majesty's Hydrographic Department for permission to reproduce a page from the 1974 European Tide Tables and to Her Majesty's Stationery Office for permission to reproduce certain extracts from the 1958 Nautical Almanac There has been no significant change in the layout of the daily pages of the Almanac since 1958, and the examples and exercises, based on the extracts from that year which are given in this book, have withstood the test of time 1976 A.e.G W.G.C CHAPTER THE TERRESTRIAL SPHERE AND ITS PROPERTIES The earth, or the terrestrial sphere, as it is sometimes called, is not quite a true sphere The scientists whose work it is to study the exact shape of the earth and whose branch of science is known as geophysics, continue to make fresh discoveries from time to time, but their general conclusions appear to be that the earth is an "oblate spheroid" This means that it is a sphere-shaped body, slightly flattened at the poles, its polar diameter being about 23 miles less than its equatorial diameter It will be seen later how this slight irregularity in the shape of the earth affects navigation, with particular reference to the definition of the nautical mile as a unit of distance But in most problems of navigation the earth is treated as if it were, in fact, a true sphere; and for this reason it is essential that certain basic properties of the sphere should be clearly understood at the outset The shape of a true sphere may be defined as that shape which is created by the rotation of a semi-circle about its diameter; or alternatively, a sphere may be defined as a body, every point on the surface of which is equidistant from its centre Great and small circles Circles may be described on the curved surface of a sphere that are similar in appearance to those described on any flat or plane surface But whereas, on a plane surface, only one kind of circle can be drawn, those drawn on the surface of a sphere can be of two quite different types These are known as Great Circles and Small Circles respectively In Fig 1, ABDE is a great circle and abde is a small circle, each drawn on the surface of a sphere of centre C The important point to observe is that, if a knife is used to cut through the sphere along the circumference of the circle ABDE, the knife blade will pass through the centre C and the sphere will be cut in half Whereas, if a knife is used to cut through the sphere along the circumference of the circle abde, the knife blade will not pass through C, and the sphere will not be cut in half A great circle, therefore, is defined as any circle on the surface of a sphere, the plane of which passes through the sphere's centre And a small circle is any circle on the surface of a sphere the plane of which does not pass through the sphere's centre Spherical angles On a plane or flat surface, an angle is formed by the intersection of two straight lines On the surface of a sphere, a spherical angle is formed by the intersection of the arcs of two great circles Spherical triangles On a plane or flat surface, a plane triangle is formed when an area is bounded by three straight lines On the surface of a sphere a spherical triangle is formed when an area is bounded by the arcs of three great circles In Fig 2, angle A, angle B and angle D are spherical angles because each angle is formed by the intersection of the arcs of two great circles on the surface of the sphere with centre C Similarly the triangle ABD is a spherical triangle because it is an area on the surface of a sphere bounded by the arcs of three great circles The dimensions of a spherical angle, i.e the number of degrees and minutes that it contains, is determined by the tangents to the great circles at their point of intersection For example, the value of the spherical angle B is determined by the number of degrees and minutes contained between the tangents BT and BT at the point B In spherical triangles, the sides, as well as the angles are measured in degrees and minutes The reason for this IS that each side is the arc of a great circle which subtends a certain angle at the centre of the sphere For example, the side BD is an arc of a great circle that subtends the angle BCD at the centre of the sphere Similarly AD subtend~ the angle ACD and AB subtends the angle ACB So if, in Fig 2, angle BCD = 90°, then side BD = 90° Similarly, if angle ACB = 90°, then side AB = 90° and if angle ACD = 150°, then side AD = 150° From this it will be seen, incidentally, that the sum of the three sides of the spherical triangle ABD amounts to 330° Sum of the sides of a spherical triangle If the size of the spherical triangle ABD is contracted by bringing its sides closer and closer together, the triangle will eventually become so small that it will be a mere dot The sum of three sides will then be zero If, on the other hand, the size of the triangle ABD is expanded, by pushing its sides further and further apart until the area bounded by the arcs of the three great circles is as large as possible, then NAVIGATION FOR SCHOOL AND COLLEGE the enclosed area will become a hemisphere, and the sum of its three sides will be the circumference of the sphere, i.e 3600 Therefore, the sum of the sides of a spherical triangle can vary from 0° to 360°, and in calculations, each side must be calculated separately THE TERRESTRIAL SPHERE The geographical poles On the surface of the earth there are two points known as the aeographical poles These are define~ as the two points where the axis of rotation of the earth passes through the earth's surface In Fig 3, PN and Ps are the geographical poles Sum of the angles of a spherical triangle In a plane triangle, the fact that the sum of the three angles amounts to 180° can often be used to assist in calculations; but this is not so in a spherical triangle If the spherical triangle ABD is contracted in size by bringing its sides closer and closer together, a time will come when the triangle occupies a very small area on the surface of the sphere When this point is reached, the triangle may be considered to be flat or plane, because a small area on a sphere is flat or plane In other words, the triangle has become a plane triangle, and the sum of its angles is 180°, as in any other plane triangle If, on the other hand, the size of the triangle ABD is expanded as before, each of the angles A, Band D will eventually become 180°, and the sum of the three angles will amount to 5400 Therefore, the sum of the angles of a spherical triangle can vary from 180°to 540°, and in calculations, each angle must be calculated separately Some terrestrial definitions Meridians A meridian, or meridian of longitude, is a semi-great circle on the surface of the earth extending from pole to pole All meridians meet at the North and South Poles In Fig 3, PNAjmPs and PNlkBPs are meridians of longitude passing through the positions A and B respectively The prime meridian This is the meridian which passes through Greenwich and all other meridians are numbered with relation to it It is the meridian of longitude 0° In Fig the point G indicates the position of Greenwich, and PNGhPs is therefore the prime meridian, because it passes through Greenwich The equator The equator is a great circle on the surface of the earth midway between the two poles Every point on the equator is 90° of arc removed from each pole, this measurement being made along the arc of any meridian from equator to pole In Fig 3, QjhkQ is the equator and PNAj, Psmj, PNlk, PsBk are all arcs of 90 • Parallels of latitude A parallel of latitude is a small circle on the surface of the earth parallel to the equator Parallels of latitude are numbered with relation to the equator, which is in latitude 00 In Fig 3, aAla is the parallel of latitude on which A is situated and bmBb is the parallel of latitude on which B is situated Note: At this stage, it should be noted that the area bounded by the arcs PNA, PNI and Al is not a spherical triangle PNA and PNI are arcs of great circles, but Al is an arc of a small circle Therefore, by the definition of a spherical triangle, the area bounded by these three arcs, although triangular in appearance, is not a NAVIGATION FOR SCHOOL AND COLLEGE spherical triangle The significance of this point will be appreciated later The latitude of a place The latitude of any given place (such as A in Fig 3) is the arc of any meridian contained between the equator and the given place It is also the corresponding angle at the centre of the earth In Fig 3, the latitude of A is the arc jA or the arc kI It is also the angle ACj or the angle ICk Latitudes are measured in degrees and minutes, North or South of the equator B is in South latitude, and its numerical value is given by the dimensions of the arc jm or the arc kB The North Pole is in latitude 90° North and the South Pole in latitude 90° South The longitude of a place The longitude of any given place (such as A in Fig 3) is the arc of the equator contained between the prime meridian and the meridian passing through the place It is also the corresponding angle at the centre of the earth In Fig the longitude of A is the arc hj or the angle hCj Since A lies to the westward of Greenwich, its longitude is expressed as so many degrees and minutes West The longitude of B is the arc hk or the corresponding angle hCk, and this is expressed in degrees and minutes East of Greenwich Longitudes, i.e the numbers attached to the meridians, range from 0° to 180° East and from 0° to 180° West, the 180° meridian being named both East and West Difference of latitude The difference of latitude, or d.lat, as it is usually called, between two places is the arc of any meridian contained between the parallels of latitude on which the two places are situated In Fig 3, the d.lat between A and B is the arc Ajm or the arc Bkl In numerical calculations it should be apparent that if one place is North of the equator and the other place is South of the equator, i.e if their latitudes are of "opposite names", the d lat between the two places is found by adding their latitudes together If a ship is sailing from A to B the d.lat made good is the sum of the two arcs jA and jm and it is named South, because the ship is sailing in a southerly direction If a ship is sailing from B to A, the d.lat THE TERRESTRIAL SPHERE is named North, because the movement is in a northerly direction When two places are both North or both South of the equator, i.e when their latitudes are of the "same name" then the d lat between them is found by subtracting the lesser latitude from the greater, but it is still named according to the direction of the ship's movement, either North or South Difference of longitude The difference of longitude, or d long, as it is usually called, between two places is the lesser arc of equator contained between the meridians passing through the two places In Fig 3, the d long between A and B is the arc jk If the ship is sailing from A to B, the d long is named East, but if she is sailing from B to A it is named West, i.e according to the direction of movement In numerical calculations it should be apparent that when one place is East of Greenwich and the other place is West, the d long between them is found by adding their longitudes together Thus, the d long between A and B is the sum of the two arcs hj and hk When the two places are both East or both West of Greenwich, the d long between them is found by subtracting the lesser longitude from the greater, but the d long is still named according to the direction of movement Crossing the 180° meridian When finding the d long between two places which are on opposite sides of the 180° meridian and each less than 90° of longitude from it, some care must be taken For instance, if a ship sails from long 165°E to long 170°W the d long will be 25° East, and if a ship sails from long 150°W to long 160°E then the d long will be 50° West The reader should reason out for himself why this is so, bearing in mind that the d long is named according to the direction of the ship's movement A globe of the earth, with meridians marked on it, may be used to obtain a clearer understanding of these examples Examples l(a) Find the d.lat and d.long made good when a ship sails from A in lat 40° W N long 30° IS' W to B in lat 30° 20' N long 42° 30' W 10 NAVIGATION FOR SCHOOL AND COLLEGE A B d.lat or d.lat lat 40° 10' N lat 30° 20' N 9° 50'S 60 = 590' S long 30° 15' W long 42° 30' W d.long 12° 15' W 60 or d.long = 735' W Note that d.lat and d.long are often expressed in minutes of latitude and minutes of longitude, respectively Find the d.lat and d.long made good by an aircraft, flying from C in lat 20° 10'N long 3° 20' W to Din lat 5° 20' S long 6° 15'E C D d.lat or d.lat lat 20° 10' N lat 5° 20' S 25°30'S 60 = 1530' S long 3° 20' W long 6° 15'E d.long 9° 35' E 60 or d.long = 575' E THE TERRESTRIAL SPHERE M in lat N in lat C in lat Din lat Pin lat Q in lat E in lat Fin lat Kin lat L in lat 18°51' S 01041'S 16°23' S 07° 18' N 17° 19' N 07° 49' S 46° 24' S 32° 53' S 29° 47' N 29° 47'S long long long long long long long long long long 11 24°47' E 06° 39' E 14° I7'W 22° 28' E 162° WE 153°27' W 140° 18'W 171°46' E 18°59' W 18°59' E The nautical mile The nautical mile is the unit of distance used by navigators at sea and in the air For most practical purposes it is considered to be a length of 6080 ft In point of fact, however, the nautical mile varies in length, and how this comes about is explained as follows Find the d.lat and d.long made good on a voyage from San Francisco to Sydney San Francisco is in lat 37° 48' N long 122° 27' W and Sydney is in lat 33° 52' S long 151° 13' E San Francisco lat 37° 48' N Sydney lat 33° 52' S d.lat 71° 40'S 60 or d.lat = 4300' S long 122° 27' W long 151° 13'E 273° 40' 360 00 d.long 86° 20' W 60 d.long = 5180' W 180°(XY 122° 27' Diff = 57° 33' 180°00' 151°13' Diff = 28° 47' d long 86° 20' W 60 d.long = 5180'W It is most important that the d.lat and d.long should be "named" correctly, i.e N or Sand E or W respectively EXERCISE I(a) Find the d.lat and d.long mad.e good between the following positions Assume that, in each case, the ship or aircraft is proceeding from the first position to the second position The d.lat and d.long must be correctly named I A in lat 33° 42' N long 23° 17' W Bin lat 46° 18' N long 64° 56' W X in lat 47° 39' N long 86° 43' W Yinlat 18°16'N long 36°06'W The earth, as shown in Fig 4, is an oblate spheroid It is flattened at the poles Therefore the curvature of the earth's surface along a meridian at the poles is less than the curvature of the earth's surface along a meridian at the equator Thus Cp is the centre of curvature of the arc ab, near the pole, and CQ is the centre of curvature of the arc de, near the equator The length of a nautical mile in any' given latitude is defined as "the arc of a meridian subtended by an angle of one minute at the centre of curvature of the meridian for the given latitude" Thus, in Fig 4, if the angle at CQ is min, then the arc de is the length of a nautical mile at the equator Similarly, if the angle 12 NAVIGATION FOR SCHOOL AND COLLEGE THE TERRESTRIAL SPHERE at Cp is min, then the arc ab is the length of a nautical mile at the pole; ab is greater than de The Admiralty Manual of Navigation gives a formula for determining the length of a nautical mile in different latitudes It is: Length of nautical mile = (6077'1- 30,7 cos 2¢) ft, where ¢ latitude At the equator ¢ is 0°, ' 2¢ is 0° But the cosine of 0° = .' the length of the n mile at the equator is 6077'1 - (30' x 1) = 6046,4 ft At the pole ¢ is 90°, 2¢ = 180° But the cosine of 180° = -1 .' the length of the n mile at the pole is 6077'1- (30'7 x -1) = 6077'1 + 30,7 = 6107'8 ft = the If one uses this formula to calculate the length of the nautical mile in other latitudes, it will b~ found that the "accepted" figure of 6080 ft is the length of the nautical mile in the latitude of the English Channel It will also be observed that in the definition of the nautical mile given above, the expression "arc of the meridian" is used But the arc of a meridian subtended by an angle of at the centre of the earth is also of latitude It follows, therefore, that the nautical mile is, in fact, the same unit as of latitude Thus, if two places are in the same longitude, the difference of latitude between them, expressed in minutes, is the same as the actual distance between them, expressed in nautical miles The nautical mile in metric units The accepted figure of 6080 ft for the United Kingdom nautical mile equals 1·8532 km The International nautical mile equals 1·852 km or 1852 m T he knot The knot is a unit of speed One knot means "a speed of one nautical mile per hour" Ten knots means a speed of ten nautical miles per hour and so on It is therefore manifestly incorrect to say that the distance from one place to another is a hundred knots when, in fact, it is a hundred nautical miles It would be equally 13 incorrect, of course, to say that a ship's speed was "twelve nautical miles" The correct term here is "twelve knots", unless one adopts a rather unusual and cumbersome mode of expression, and describes the ship's speed as "twelve nautical miles per hour", which is perfectly correct The term "knot" is derived from one of the early methods of determining a ship's speed A small billet of wood, called a "log", was secured to the end of a line, called a "log-line", which was wound on to a reel Knots were tied in the line at certain specific di~tances apart, depending on the type of sand glass that was used in' conjunction with the line If the sand glass were a 14 sec glass, the knots on the line would be 23·64 ft apart, and if the glass were a 28 sec glass the knots would be 47·29 ft apart The method of use was as follows: the log was lowered over the stern and allowed to trail in the water at the end of a stray length of the log line, the first knot on the line being held firmly by the hand against the taffrail At a given signal, the sand glass would be turned, and the line released The log would float astern as the ship went ahead, the line would uncoil from its reel, and the san"d would run through the glass As the last grain of sand left the glass, the number of knots (except the first one) that had passed over the taffrail whilst the sand had been running was noted, and this gave the speed of the ship The mathematics involved in determing the spacing of the knots on the line is not difficult If a ship's speed is knot, she will move 6080 ft forward through the water in 3600 sec Then, in 14 sec, she will move forward 6080 x 14/3600 or 23·64 ft The statute mile The statute mile, or "land mile", is not generally used in navigation, but navigators may, on occasion, be required to convert nautical miles to statute miles or vice versa The statute mile is a unit of distance which was established by statute in the reign of Queen Elizabeth I of England It is 5280 ft in length Therefore the relationship it bears to the nautical mile is given by the ratio 5280/6080, which, when reduced to its lowest terms, becomes 66/76 Thus, to convert a given number of statute miles to nautical miles, the given number of statute miles must be multiplied by 66/76 Conversely, to convert a given number of nautical miles to statute miles, the given number of nautical miles must be multiplied by 76/66 When working such conversions, it is APPENDIX ABRIDGED III NAUTICAL TABLES Traverse table Meridional parts for the terrestrial spheroid table Mean latitude to middle latitude conversion table Altitude correction tables Natural haversines table Partly reproducedfrom Nories Tables (1973) by permission of Imray, Laurie, Norie and Wilson 238 254 (7) NAVIGATION FOR SCHOOL AND COLLEGE Table A (1) dev 6°E (2) 217°C (3) 284° C (4) dev 5° W (5) 245° C (6) 172°M var 20° W var 5° E 262° T var 15°E 230° T var 12° E (7) 346° C (8) 280° M (9) dev 3°E (10) 201°C (11) dev NIL (12) dev] W 348° M 275°T var 2SOW 175°T var 42° E var NIL Table B (1) error 20° W (2) dev 5° W (3) 183°C 026° T error 20° E var 5° W (4) error 4° E (5) 338° C (6) dev 12°W 307° T var 2° E error 20° W EXERCISE lII(a) (1) 6°33'I'E (2) 380'6 miles (3) 158°09'I'W (4) 338·9 miles (5) lat 23°30'N, distance apart 364'1 miles (6) distance steamed 974 miles, distance apart 199'9 miles (7) lat 39° 10' N, distance steamed 3206 miles (8a) 900 kt (8b) 602'2 kt (9) lat 00° 00' N, long 38° 56' W EXERCISE lII(b) (1) (a) course 053° 55', distance 147'6 miles (b) course 19I° 51', distance 234'1 miles (c) course 329° 09', distance 286·5 miles (2) (a) course 272° 20', distance 6371·5 miles (b) course 123°23', distance 1387 miles (c) course 261° 16', distance 2501 miles (3) lat 42° 51'1' N, long 32° 28,8' E (4) course 241°44', distance 2091 miles (5) distance 1436 miles, long 86° 58-5'E (6) parallels 34° 28' and 30° 02' Nor S course 141° 03' (7) course 099° 58', distance 2797 miles ETA I Ith May at 0100 hr G.M.T (8) alteration position lat 46° 26' N, long 38° 04' W 2nd course 112°38', distance 1600 miles ETA 17th August at 0200 hr G.M.T EXERCISE lII(c) (I) (a) d.lat 40'3 N, dep 38,9 E (b) d.lat 2'87N, dep 16·3W (c) d.lat 86'8S, dep 31·6E (d) d lat 47'8 S, dep 47,8 W (e) d.lat 49'8 N, dep 00,00 (f) d.lat 81'1 S, dep~15'6 W (2) (a) course 205°, distance 51 miles (b) course 0110, distance 86,5 miles (c) course 144to, distance 49,7 miles (d) course 305to, distance 49,7 miles (e) course 3151°, distance 93·2 miles (f) course 2601, distance 99·2 miles (3) (a) dep 67,9 W (b) dep.17·55'E (c) 3B'E (d) dep 9'0' E (4) (a) d long 22,9' E (b) d.long I,65' E (c) d.long 70,5' W (d) d.long 91SE (5) course 339~0, distance 40,5 miles (6) 1st ship: lat 20° 32'9' N, long 72° 37'6 E 2nd ship: lat 20° 43' N, long 73° 09,4' E (7) lighthouse lat 25° 23'6' S, long 39° 01·5' W (8) lat 46° 0703'N, long 178°53,9' W ANSWERS EXERCISE lII(d) (b) course 224~0, distance 851 miles (1) (a) course 099°, distance 80,6 miles (c) course o64io, distance 66,3 miles (2) course 215~0, distance 81·4 miles lat 35° 09' N, long 80° 13,7'W (3) course 339';'°, distance 65,5 miles lat 20° 14,5' S, long 137° 53,5' E (4) course I 69Y)° , distance 88·2 miles lat 65° 44'8'S, long 179° 40' 3' W (5) course 305';'°, distance 75,6 miles lat 20° 44,9' N, long 59° 19,8' W (6) course 045° , distance 86'4 miles lat 45° 27·2' N, long 120° 36,8' E (7) lat 20° 46,6' S, long 19° 23,7' W set 326°, Drift 31 miles (8) course 225Y)°, distance 78,3 miles lat 54° 22' N, long 15° 47,4' W bearing 315°, distance 29·7 miles EXERCISE IlI(e) (1) (a) course 068° SO'5',distance 1543·2 miles (b) course 317° 11'6', distance 1300·3 miles (c) course 287° 08'2', distance 1109·5 miles (2) lat 59°04'3'N, long 35°01'E (3) lat 31° 12·2' N, long 155°26,8' W (4) lat 21°43'2'N, distance 1565-6 miles (5) course 085° 23'7', long 106°29,8' E EXERCISE IlI(f) (I) initial course 091° 15,0', distance 3132 miles (G.c sailing) course I II ° 25', distance 3194 miles (Mercator sailing) (2) initial course 295°, final course 310°, distance 4567 miles (3) initial course 243° 36', final course 224° 42', distance 6746 miles (4) by great circle sailing lat 15°3B'S, long 34° 17·7'W by parallel sailing lat 24° 46' S, long 36° 39·9' W (5) lat 51°27'3'N, long 22°57'I'E (6) by great circle ETA 27th March at 0804 hr by rhumb line ETA 27th March at 1124 hr EXERCISE lII(g) (b) 0,617.'\ (2) (a) lat 6T 58-5'N or S (3) course 236° 51', distance 830,3 miles (mid lat sailing) course 236° 45', distance 827·9 miles (plane sailing) (4) course 086° 51'7', distance 2249·0 miles (Mercator sailing) course 086° 52', distance 2247·5 miles (mid lat sailing) (5) height 1107 mm, breath 1763 mm (6) parallels 59° 43' and 45° 49' (7) A 13°27,6' N or 25° 14,4'S B 25° 14,4'N or 13°27,6' S (8) distance 14·7 miles (9) d.long 41'17', nat scale 1/318,960 255 256 (c) declination 22°39'I'N O.H.A 82°52-3' (d) declination 12° 15,3'SORA 248° 58·6' (e) declination 23° IO'I'S ORA 82°22'2' (b) 0833 O.M.T on 20th September (17) (a) 1356 G.M.T on 19th September (d) 2346 O.M.T on 26th October (c) 1839 O.M.T on 7th June (e) 0002 O.M.T on 1st January 1959 (b) long 34° 16'E (18) (a) long 53° 36-1'W (19) ORA sun 341°28' (20) O.M.T 17h32m55s (b) 2hlOmJ4S (21) (a) 3h50ml4s (b) 21h12m12' h m s (22) (a) 15 52 l2 EXERCISE IV(a) (I) lat 45°07'2'N, long 13°4N'W (2) lat 54° 20·2' S, long 79° 50,8' E, distance off 5,4 miles (3) A OOT B 078° (4) lat 35° 40,5' N, long 85° 23-8' E (5) lat 44°42'2'S, long 65°32'3'E (6) lat 64° 33'6'N, long 168°02·7'W EXERCISE IV(b) (1) Ist bearing lat 35° 10,8'S, long 02° 10,6' E 2nd bearing lat 35° 04-5' S, long 02° 39,8' E (2) Course 256°T Time taken 13'35 hrs (3) Course 037° T Time taken 9-43 hrs (4) 1st bearing lat 45° 26,5' N, long 23° 28,8' W 2nd bearing lat 45° 53,0' N, long 23° 17,5' W (5) 1st bearing lat 44° 33-8'N, long 15° 59' W 2nd bearing lat 44°01'7'N, long 16°29'2'W (6) 1st bearing lat 49° 38,5' N, long 06° 13,6'W 2nd bearing lat 49° 44,5' N, long 05° 20·9' W (7) lat 49° 511' N, long 05° 05' W (8) 1st bearing lat 50 22'N, long OIOO7'W 2nd bearing lat 49°46'N, long 04°00'W EXERCISE VIII (4) index error 04' 30" "on", Mayor August (5) index error 0° 33' "on", distance off 6·48 miles (b) 27° 56,8' (6) (a) 46° 30,7' (e) 65° IH' (d) 18°37-4' (b) W 35·4' (7) (a) 53° 49·8' (e) 26° 22,2' (d) 59° 07,0' EXERCISE IV(c) (I) bearing 208°06' (3) bearing 237° 20' (2) bearing 117°07' (from ship) (4) 048° 46' EXERCISE VIta) (I) long 81° E (2) G.HA 221° 45' (4) GRA 322° (5) long 21°E (7) S.H.A 270°, G.H.A 167° 257 ANSWERS NAVIGATION FOR SCHOOL AND COLLEGE (3) G.H.A 'Y' 256° (6) R.A 55° (8) Long 105°E EXERCISE VI(b) (I) RA 4h40m, R.A 20h40m (2) declination 20° N, G.H.A 052° (4) lat 36°N, long 89° 14'E (3) lat 12° 18' S, long 4r 13' E (5) (a) Sunset at 2000 hrs (b) Sunset at 1600 hrs (6) (a) Lat 45° N, Decl 45° N (b) Lat 49° 39,9' S, Star Peacock (7) (b) Between 25°07-7' Nand 28°01,6' S EXERCISE VII (2) 2140 L.M.T (I) 17h52m08'G.M.T (4) L.AT 1401 GAT 1301 (3) 54 (5) Eq T - 12 (6) Eq T+II (b) 1145 (8) G.H.A sun = 002° 30' (7) (a) 1208 (9) Zone + G.M.T 1500 (10) Zone -8 Zone time 2000 hr (12) LAT 1456 (II) Eq T +4 (13) April 4th at 1900 standard time April 4th at 1904 L.M.T (14) March 10th at 2300 standard time (15) March 12th at 0300 standard time (16) (a) declination 1°33,7' N O.H.A 12°01·2' (b) declination 1°00·2' N G RA 171° 39' EXERCISE IX (b) altitude (I) (a) altitude 36!°, azimuth 259° (d) altitude (c) altitude 18°, azimuth 140° (b) altitude (2) (a) altitude 9°, L.H.A 321° (d) altitude (c) altitude 31°, L.H.A 64° (3) L.HA 289° and 071° O.M.T 0539 and 1507 (4) azimuth 088°, O.M.T 0424 (5) L.HA 325°, declination 18°N EXERCISE X(a) (1) 5r 11·4'N (4) 1°39·2' S (7) 29° 46,8' S (10) 68°02'1' S (13) 50° 12·6'N (16) 56° 39'1' S (19) 39° 40·9' N (22) 12° 15,0'N (25) 7°07,7' N (2) 31°39-5'S (5) 5°05,6' N (8) 21° 43,9' N (11) 27° 10,9' S (14) 10°28,8' S (17) 58°04'7'N (20) 26°40·5' N (23) 76°48,7' S (c) 65°00,6' (c) W 26,3' 20°, azimuth on~o 30°, azimuth 028° 43°, L.H.A 332° 58°, L.H.A 014° (3) 39° 12,7' S (6) 38° 06·6' N (9) 69° 54-5'S (12) 6° 07·2' S (15) 62° 13-3'N (18) 26° 14' N (21) 45° 32' S (24) 68° 57·9' S EXERCISE X(b) (Figures in brackets are those obtained A.B.c Tables, the errors and deviations by the use of five-figure Nautical Tables_ If the true azimuths have been found obtained should be within 0-25° or 0° 15' of the figures given here.) PZ (I) 35° 50' PX 89° 17-3' P 52° 28' W (2) 51 ° 20' 106° 25,6' 14"44-2'W (3) 50° 20' 67°20-1' 31°48'E (4) 54° 45' 98° 28,7' 61°19-8'E (5) 42° 50' 66° 47-8' 39° 56' W (6) 32° 25' 51°15-3' 37° 26-9' E (7) 42° 55' 66° 43-2' 74° 59' E (8) 39° 30' 29°49-5' 100° 49-2' E (9) 42° 10' I WO34-8' 30° 35-1' E 77° 24-6' 84° 13-9' E (10) 56° 27' True amplitude (II) E26°46'S (26° 49-5') (12) E 37° 44' N (37° 45') (Continued ZX 68° 29' (68° 29-3') 56° 47' (56° 48') 31° 50' (31 ° 49-4') 72° 24' (72°24') 39° 48' (39° 49') 30° 40' (3.0° 39-1') 63° 10' (63° 09-5') 52° 24' (52° 24-5') 73° 43-5' (73°42-7') 78°21' (78° 20-3) Z 121° 32' (121 ° 33-5') 163° 10' (163° 02') 112°44' (112° 45-6') 114° 26' (114"26') 112° 55' (112° 53') 'IW31' (I W 34') 83° 52' (83° 54') 38° 06' (38° 04') 150° 10' (150° 15') 82° 34' (82° 31') True bearing 116°49' Error 6°49'E Devil/tion 19°09'E 052° 16' 12°44'W 22°44'W Error Deviation 255° 32' 15° 32' E 21° 12'E 112° 17-9' S 270° 00' 2° 17-9'E 10°00' W IT 17-9'E 2° 40' W True bearing (13) W 14" 28'S (W28-5') (14) E 22° 17-9' S (15) 00° 00' EXERCISE X(c) (The figures given in brackets lines are given to the nearest Devil/tion 35° 28' E 26° 50' W 10° 30' E 22°44'E 7"44'E 15° 34' W 6° 56' E 2° 55' E 14"35'W 8° 29' W 8° 54' E - 6° 54' E WOIO'W 7° 31' E are those obtained by the use of five-figure (1) PX 68° 22-3' P 57° 44-6' (W) TZD 52° 25-9' (2) 69° 18' 93° 55-7' 30° 31-7' (E) 38° 38-5' (3) 42° 20' 87° 47-8' 44° 31-3' (E) 59° 30-7' (4) 43° 10' 112° 20-3' 34"31-1' (E) 75° 47-8' CZD 52° 29' (52° 29-5') 38° 46' (38° 46-3') 59° 27' (59° 27 -2') 75° 52' (75° 52') 53° 50' (53° 50'7') (5) 41 ° 28' 77° 24-4' 48° 41-5' (W) 54° 03-7' (6) 43° 16,7' 112° 19-1' 17° 53-4' (W) 70° 49-8' 70° 54-5' (70° 54-6') 49° 12-4' (W) 50° 04-9' 49° 58' (49° 58-2') 54 ° 44' (8a) 43° 14' (8b) (9a) longitude 44° 55' (9b) 45° 10' 77° 49-8' 66° 44-4' 55° 56-2' (E) at noon 32° 53-5' E 4T 19-3'(W) 92" 01' 81° 14-2' and longitude Nautical Tables The azimuths and position whole degree_) PZ 43° 30' latitude Error 8° 28' E on page 259) True amplitude (7) True azimuth 238° 28' (238° 26-5) 343° 10' (343°02') 112°44' (112° 45-6') 114"26' (I 14" 26') 292° 55' (292° 53') 111 ° 31' (IW34') 083 ° 52' (083 ° 54') 141° 54' (141° 56') 029° 50' (029° 45') 082° 334' (082° 1') 19° 35-7'(E) at time of star observation 50° 10' 50° 12' (50° 11-8') 63° 21-4' 63 ° 02' (63° 02-2') 39° 51' 40° 08-3' (39° 51-2') = 45" 14'N, 35° 32' W_ Int_ 3-1 towards (3-6 towards) 7-5 towards (7-8 towards) 3-7 away (3-5 away) 4-2 towards (4-2 towards) 13-7 away (13-0 away) 4-7 towards (4-8 towards) 6-9 away (6-7 away) 2-0 towards (1-8 towards) 19-4 away (19-2 away) 17-3 away (17-1 away) Azimuth 278° 054° 126° 147° P Line 008° / 188° 324 °/ 144° 216°/ 036° 057° / 237° 245° 335°/ 155° 198° 288°/ 255° 345°/ 082° 165° 172° / 352" 236° 146°/ 326° 149° 059° / 239° 108° by 260 NAVIGATION FOR SCHOOL AND COLLEGE EXERCISE XI (1) (a) lat 54°21'8'N, long 36°06'3'E (b) lat 20 40'6'S, long 173° 13'2'W (c) lat 35° 18·2'N, long 72° 49,9' E (2) 1st position lat 44°31'4'N, long 16°39'3'W 2nd position lat 44° 50,6' N, long 16° 19,4'W (3) lat 54° 28' S, long 06° 54·2' E (4) lat 20 08'N, long W09'4'E (5) 0900 lat 44°49'4'S, long 18°41'2'E 1030 lat 44° 38'9' S, long 18°20,9' E (6) 1st position lat 10°23' N, long 12° 14,5' W 2nd position lat 10 36'TN, long 11°59'1'W 0 EXERCISE XII (a) (1) (2) (3) (4) (5) sunrise 0542 L.M.T sunset 1804 L.M.T 0510 L.M.T sunrise 055H L.M.T sunset 17551 L.M.T 0432 L.M.T (a) Ih32m (b) Oh45m EXERCISE XII (b) (I) (2) (3) (4) (5) (a) (a) (a) (a) (a) 15·2m (49'8 £1) 12'1 £1(3'7 m) -4,7 m (15'5 ft) -4,2 m (13-6 £1) -4'8 m (15'8 £1) (b) (b) (b) (b) (b) 0912 G.M.T 1512 G.M.T 0809 G.M.T 1347 G.M.T 1432 G.M.T INDEX Courses compass 32 correction of 31-35 magnetic 32 true 32 A.B.C tables 188 Agonic line 26 Altitudes 151-158 apparent 152 corrections to 152-157 maximum 169 meridian 169 observed 152 of Polaris 180 sextan t 152 true \17,152,157 Amplitude 166,191,192 Aphelion 107 Apparent Sun 122 Apparent time 126 Aries, First Point of 111 Azimuth 184 Date line 133 Day, definition 122 apparent solar 122-123 lunar 124 mean solar 123 sidereal 123 Day's work 58-U3 Declination of heavenly body 112, 137 parallel of 112 14, 15,43 Departure Deviation 27-29 Difference of latitude of longitude Dip 152 Dynamical mean sun 128 Bearings compass 33 great circle 100-101 magnetic 33 mercatorial 101 true 33 Binnacle 22 Celestial Meridian 1\1 Celestial poles \11 Celestial sphere 110 et seq Chart datum 218 Circumpolar bodies \18 Civil year 125 Collimation, error of ISO Compass gyro 36 et seq magnetic 21-25 poin ts of 30,31 quadrantal notation 30,31 three figure or 360° notation Convergency 101 Earth, shape of 3, \1 Ecliptic 1\1,128 Equation of time 128- \31 Equator Equinoctial \11 Errors of sextant 144-151 First Point of Aries of Libra 30,31 \11 \11 Geographical poles Geographical position \19, 120 Grea t circle definition of vertex of 80 261 262 Great circle sailing 79-84 Greenwich apparent time 126 Greenwich hour angle 113-115 Greenwich mean time 127 Gregorian calendar 125 Gyroscope, free 36-37 Gyroscopic inertia 37 Height of tide 218 Horizon rational 117 sensible 151 visible 151 Hour angles 112-115 Index error 146-149 Index glass 143 In tercept 197 Isogonic lines 26 Julian Calendar Kilometre Knot 12 125 14 Latitude, definition by meridian altitude 169-180 by Pole star 180 Libra, First Point of III Liquid compass 22 Local apparent time 126 Local hour angle 113 Local mean time 127 Log 12-13 Longitude of a place Longitude, difference of Loxodrome 16 Lunar day 124-125 Lunation 216 Magnetic Magnetic Magnetic Magnetic compass 21-25 equator 20 needles 21,22 poles 18,19 Magnetism ships 27-29 terrestrial 18-21 Mean latitude 15,49 Mean time 127 Mercator chart 64-69 Mercator sailing 71-75 Meridians celestial III magnetic 25 terrestrial Meridian altitudes 169-180 Meridian passage 170 Meridional parts 69, 70 Middle latitude 16,49 Middle latitude sailing 50 Mile geographical 14 nautical 11-12 statute 13 Moon, effect on tides 216,217 phases 216,217 Nadir 117 Nautical Almanac, use of Nautical mile II, 12 Neap tides 218 Oblate spheroid Observed altitude 152 Orthographic projection Parallax 154-156 Parallel of latitude Parallel sailing 43-47 Perihelion 107 Plane sailing 47-55 Planets 105-106 Polar distance 117 Poles celestial III geographical magnetic 19 Precession of equinoxes 112 of gyroscope 39,40 Prime vertical 117 Projections equidistant 164-168 orthographic 162,163 stereographic 162,163 Quadrature 263 INDEX INDEX 217 136-138 152 Refraction 16,17 Rhumb line 112 Right ascension 106-109 Seasons Semi-diameter, correction Sensible horizon for 153,154 151 Sex tan t 142-151 errors of 144-151 123 Sidereal day 112, 113 Sidereal hour angle Sunrise Sunset 218-224 123,126,128,130 Time, apparent 128-131 equation of time 123, 127, 128, 130 mean time 132-136 standard time 131,132 zone time 55-58 Traverse table {i Triangles, spherical Twilight 124 Sidereal time 123 Solar day Small circles 218 Spring tides 132-136 Standard time Stars, magnitude of 224-225 13 Stray line 107 Summer solstice 214-216 25-26,31-33 Variation 108 Vernal equinox 80 Vertex of great circle 212-214 212-214 Tides lowest astronomical 162,163 Tides -continued range of tide 220 218 spring tides Tide tables, use of 122 et seq Time Range of tide 220 117 Rational horizon Vertical circle Visible horizon 117 151 Winter solstice 107 122, 125 Year 125 civil year 125 tropical year tide moon's effect 216,217 218 neap tides 219 117 Zenith Zenith distance 117,157 ... an applied force or torque 40 NAVIGATION FOR SCHOOL AND COLLEGE pressing down on the South end This force turns through 900 in the direction of rotation and then acts as the precessed force PF... 40°13'Wand on the equator, steams oooO(T )for 1200miles, 090 (T) for 1200miles, 180 (T) for 1200 miles, and 270 (T) for 1200 miles Find her finalposition " 0 PX is the meridian of 20° W, and A and. .. triangle ABD is expanded, by pushing its sides further and further apart until the area bounded by the arcs of the three great circles is as large as possible, then NAVIGATION FOR SCHOOL AND COLLEGE