The art of astrronmical navigation

The Art of Astronomical Navigation BY S M BURTON Fellow of the Institute of Navigation Master Mariner Revised by CHARLES H COTTER Fellote of the Royal Institute af Navigation Master Mariner BROWN, SON & FERGUSON, 52 DARNLEY STREET LTD • CopYl'ight in all countries signatory to the Berne Convention All rights reserved PREFACE First Edition 1955 Second Edition 1962 Third Edition 1975 © 1975 BROWN, SON & FERGUSON, LTD, GLASGOW G41 2SG Printed and Made in Great Britain TO THE FIRST EDITION THIS book first appeared in 1935 under the title of A Manual of Modern"Navigation: its purpose then being to hasten what seemed the overslow conversion of Merchant Navy navigators from older methods to position.line methods With or without the assistance of the Manual that conversion may now be thought of as complete In the meantime, however, the sun and the moon and the stars have not changed in their courses, and Sumner's discoveryand it was a discovery, be it noted, not an invention-remains forever with us Consequently the principles enunciated in the original book have not changed either As to the practices: they are still taught and used today, and will, no doubt, long continue to be taught, although the normal method of applying them may (or may not) come to be through the medium of precomputed tables instead of by logarithmic calculation Parts I and II of the book have therefore been carefully combed through and brought up to date where necessary (with a chapter on the 'Short Methods' added) and Part III entirely re·edited: the effect of which, it is hoped, will be to revive the book's former popularity as a guide and friend to the practising navigator, making it one which he will always like to possess, and keep within easy reach For the benefit of those who may not have been acquainted with the original book it might be eXplained that its main purpose, as developed in Parts I and II, might be described as being to give a good general overall view of the astro-navigational wood for the benefit of those who have at least some nodding acquaintance with its trees-a view such that the navigator is enabled first to choose his own route through the wood, and then to keep his chosen course so as to reach his objective on the other side Part III is an effort to collect and compress within a relatively small space all the formulae and other procedures used in or in connection with navigation, many of which, although of casionally needed, are not sufficiently used to keep them always fresh in the memory v PREFACE TO THE SECOND EDITION RECENT changes in The Nautical Almanac have made it desirable that a new edition of this book should be brought into being The chief alterations will be found on pp 107-9, necessitated by the discontinuance of Right Ascension as a navigational concept Other minor corrections and adjustments have been made as necessary S M BURTON March 1962 PREFACE TO THE THIRD EDITION THE late Stephen M Burton was well-known over many decades for his deep interest in the practical needs of navigators To this end his Nautical Tables, first published in 1936, and now well established as a thoroughly practical tool of navigation, was produced (as those who knew him will testify) with the fervour of a crusader His little manual on the Art of Astronomical Navigation, like his Nautical Tables, is standing up well to the test of time In this new edition the only major change from the Second edition of 1962 has been in Chapter XVIII, which now draws attention to the existence of several navigation tables which may still fall into the hands of navigators, but which are rapidly being made obsolescent by the splendid inspection tables now to be found on almost every sea-going ship The revised chapter concentrates, therefore, on modern inspection tables and their use CHARLESH COTTER CONTENTS PART CHAP I II III IV V VI VII VIII THE I PRELIMINARY REMARKS THE TRAVERSETABLE POSITION LINES THE TRANSFERREDPOSITION LINE THE AsTRONOMICALPOSITION LINE THE AsTRONOMICAL TRIANGLE THE FOUR STANDARDMETHODS THE POSITION LINE AT CLOSE QUARTERS THE XI XII XIII XIV XV XVI XVII XVIII V-VI PRINCIPLES PART IX X PAGE PREFACES 10 17 21 2434 II PRACTICE SUCCESSIVEOBSERVATIONS REMARKS UPON SUCCESSIVE OBSERVATIONS IN GENERALANDTHENOON POSITIONIN PARTICULAR SIMULTANEOUSOBSERVATIONS STARS THE ANGLE OF INTERSECTION HEIGHT OF ALTITUDES THE PARTICULARCASE OF VERYHIGH ALTITUDES THE USES OF A SINGLE POSITION LINE POSITION LINE BY D.F BEARINGS NAVIGATION TABLES · · · · January 1975 PART MISCELLANEOUS NOTES VB 56 60 6470 72 75 79 83 91 III AND FORMULAE MISCELLANEOUSNOTES AND FORMULAE VI 43 • 106 PART I THE PRINCIPLES CHAPTER II THE TRAVERSE TABLE only in the sailings (D.R navigation), but in every method of fixing a ship's position, either celestial or terrestrial, the rightangled triangle is continually turning up and demanding quick and easy solution Now, although it is quite generally known that inspection of the Traverse Table is the quickest and easiest way of approximately solving any right-angled triangle, it is a curious fact that the method is by no means quite generally used N evertheless, the navigator who will take the trouble to acquire the Traverse-Table-habit early in life will find it a valuable convenience for the rest of his days Being set down in terms of the Plane Sailing triangle, it is necessary to have that figure always present in the mind when referring any other triangle to the Traverse Table Under the circumstances it is hardly possible for any person having to with the navigation of ships to forget the Plane Sailing figure With a little trouble, however, the mind can be so trained that, so far from any effort being required to recall it, the merest suggestion of D'lat or Dep will have the effect of instantaneously conjuring it up into the mind's eye With the Plane Sailing triangle so fixed in the subconscious mind it is only necessary, when referring any other triangle to the Traverse Table, * to bear in mind the following facts: NOT (i) The angle considered being always represented by the Course, consequently, (ii) the side adjacent to this angle is represented by the D'lat., • * Most Traverse Tables now have 'Dep.' and 'D'long.' included in their headings, and some go even further and include 'Hyp.', 'Adj.' and 'Opp.' THE (iii) and (iv) The reader of it': ART OF ASTRONOMICAL NAVIGATION the side opposite to this angle is represented by the Dep., the hypotenuse is always represented by the Dist following few examples may be of some use in helping any not already conversant with the process to 'get the hang Angle 25°, opposite side 75, required hypotenuse (177'5) Angle 76°, adjacent side 143, required opposite side (573'4) Hypotenuse 297, adjacent side 72, required opposite side (288'4) Hypotenuse 426, angle 31°, required adjacent side (365'2) Opposite side 79, adjacent side 342, required angle (13°) Adjacent side 23, hypotenuse 165, required angle (82°) The commonest use to which the Traverse Table is put, other than that for which it was originally intended, is to inter-convert Dep and D'long This is done in accordance with the proportions shown in the adjacent (Parallel Sailing) triangle: that is to say, with the latitude of the required conversion as 'Course' the corresponding Deps and D'longs will be found respectively in the 'D'lat.' and 'Dist.' columns There should not arise any confusion as to which is which if it is borne in mind that the 'Dist.' column is the hypotenuse column and D'long always exceeds departure This same triangle also applies for Mid lat Sailing, the angle used being Mid lat instead of Lat CHAPTER III POSITION LINES IF the mind is brought to consider the matter for a moment it will be realised that the word 'position' when used in the sense of geographical locality, must mean relative position Again, by further consideration of the point it will also be realised that relative position is composed of two elements, namely direction and distance, and that either one of these elements considered separately resolves itself into a line Therefore it will be evident that an unknown position can only be fixed by means of lines In navigation these two kinds of lines when so used are given the common name of position lines * Position lines defined by direction (bearings) are, of course, derived from light waves, detected by the eye, or sound or radar waves, and are therefore always of a true or great circle nature This means that the principle of cross-bearings, which is that the mutual bearing of any two places on the earth's surface is reciprocal, is fundamentally untrue In practice, however, this seldom makes any appreciable difference for ordinary compass bearings by eye, as the following table will show In the case of radio position lines (and possibly radar ranges) the matter can be of more importance, owing to these bearings being sometimes used across relatively long distances, and we shall therefore touch upon this matter in another place (Chapter XVII) These maximum errors arise, of course, on east and west bearings The table on page 8, like other small tables in this book, is not intended for 'use' in the ordinary sense of that word It is only * Position lines can, in point of fact, be:derived from other things than simple lines of direction or distance There are what might be called 'conditional position lines' The arcs derived from horizontal sextant apgles or t.hf paraboli~ curves from certain types of radio beacons are instances See also p 85 for the 'curve of constant bearing', THE ART OF ASTRONOMICAL NAVIGATION POSITION inserted here to assist the reader to form an opinion for himself Position lines defined by distance are necessarily of a circular, or 'arciform' shape As they are less easily obtainable under all circumstances, we not use them so frequently in coastal naviTABLE SHOWING MAXIMUM NEGLECT ERROR IN BEARING OWING TO LINES dence of tangent and circumference, the only way to find the exact point of contact is to find that perpendicular to the tangent which passes through the centre of the circle It is important to notice that the larger the radius of the circle the longer the length of apparent coincidence of tangent and circumference OF GREAT CIRCLE .- Distance (M.) Lat 30° 40° 50° 60° 70° 20 30 40 50 1° ·1 '2 ,3 ,5 p '2 ,3 '4 ,7 '2° ,3 ,4 '5 '2° '4 ,5 '7 1·1 ,9 gation as those derived from bearings; but in astronomical positionfinding they are, always have been, and probably always will be the only kind that can be obtained, and the following points concerning them are therefore worth particular notice A circle is a line drawn at an equal distance round a fixed point called its centre; the obvious but important corollary of which is, that every point on this line (the circumference of the circle) is at an equal distance from its centre Some people may become indignant upon the discovery that they have bought this kind of information, but it is curious how often the point has been missed in circumstances in which its recollection might have been most enlightening There are, however, two other attributes of the circle which it is extremely important that the reader should recall to his memory They are the tangent and the chord A tangent is a straight line drawn so as to touch, without cutting, the circumference of a circle Two points of navigational interest to notice about the tangent are: (i) that if a line (radius) be drawn from the centre of the circle to the point of contact it will always be perpendicular to the tangent, and (ii) that a tangent always appears to coincide with the circumference of the circle for a short distance on each side of the point of contact Incidentally, these t~o facts are very closely associated with each other, because, owing to this apparent coinci- A chord is a straight line joining the ends of an arc, X Y in Fig If a chord be bisected by a line at right angles to itself, this line will pass through the centre of the circle to which the chord pertains It should be noticed that when a chord subtends a very small arc of a circle it amounts to much the same thing as a tangent AAN B THE TRANSFERREDPOSITION LINE CHAPTER IV THE TRANSFERRED POSITION LINE ALTHOUGHit will be generally conceded that the most accurate astronomical positiom that can be obtained at sea are derived from stellar observations, the fact remains that the sun is by far the most accessible of the heavenly bodies, and consequently the most used and the most useful Since, however, all astronomical positions are obtained from distances, it follows that the sun can only give one position line at a time, and therefore every actual position, or fix, by the sun must involve a 'transferred position line' This means that the principle of, and all principles associated with, the transferred position line are of vital importance to astronomical position-fixing, and should be examined with according due care Whether position lines are derived from terrestrial or celestial sources does not in any way affect the manner of their use, and for certain reasons it will be convenient to demonstrate upon the latter Let A B (Fig 2) be a position line (P.L.) Then at the time this P.L was obtained it is known that the ship was somewhere on it Let it be given that from the time this P.L was obtained the ship steamed the course and distance represented by the line c c' Then if she had been at c she would have arrived at c' Now let it be assumed that she had been at d In this case she would have steamed an equal distance on a parallel course and 10 II have arrived at d' Draw the line A' B' through c'd' Then because c c' and d d' are equal and parallel it follows from the law of parallelograms that A' B' is parallel to A B This means that wherever the ship may have been on A B, she will, after making good the given course and distance, be on the line A' B' A' B' is therefore a transferred position line To transfer a position line, therefore, we simply move any point in it the course and distance made good in the interval, and through the point so found draw a line parall~l to the original position line We may call this a 'hardy' principle, since it holds good even though the ship's course be altered, or, which is the same thing, it is necessary to allow for a current in the interval Figs 2a and 2b are intended to demonstrate this It will be seen that the reason why the principle holds good despite an alteration of course is because two (or more) separate and distinct parallelograms are formed N ow consider an everyday instance of the principle applied in practice A ship steaming on the course 025° takes a bearing of the lighthouse L (Fig 3) and finds it to be 050°, which bearing she draws on the chart (050° + 180° = 230° from L) After continuing on this course for miles she takes another bearing which she finds to be 085°, which she also draws on the chart (265° from L) Required the position Draw in a line 025° to represent the course line and mark off along it miles from the first bearing (A B in Fig.) Through the point so found draw a line parallel to the first bearing, and the ship must be at C, the intersection with the second bearing (Note that had any other line, such as the dotted line, been assumed as the course line, the same position must necessarily have resulted.) The case of two distances is of special interest as being the only means by which a position can be fixed by two altitudes of the same or different heavenly bodies successively obtained In practice, of course, we generally use approximate position lines in the shape of tangents, or, possibly, chords; but in addition to the fact that the following method can, under certain circumstances (radar ranges, for instance), be actually used in practice, it is also illustrative of an important principle • First, it must be understood that the only practicable way to THE TRANSFERREDPOSITION LINE 13 C' as centre and the same radius as the first position line, describe the arc A' B' Then A' B' is the transferred position line (e' represents a small portion of the position line transferred as a tangent ) Now an example in practice A ship steaming 010 takes a sextant angle or radar range of the lighthouse L (Fig 5), which gives her distance from same as miles After continuing her course for miles another sight places her miles off Required the position From L layoff 010 miles (run) to L' Then with L' as centre and miles as radius, describe an arc This will be the first position lines transfured With L as centre and miles as radius describe the second position line The intersection then shows the position THE ISOSCELES TRIANGLE.Whenever two bearings are SQ taken that the angle between the course line and the second bearing is double that between the COurse line and the first, an isosceles ARTIFICIAL HORIZONS The original type of artificial horizon, still the most accurate of all horizons when used ashore, consisted of a trough of mercury: but a little thick oil (or even treacle) in a saucer can be made to serve The observation in this case is made by bringing the heavenly body down to the image reflected in the liquid With the sun, bring the two limbs into contact It is generally considered more satisfactory to observe opening suns, which means using lower limbs a.m and upper limbs p.m., but experienced observers would not have difficulty with closing suns With this horj.zon the angle measured is double the altitude, so that the process of correction is to apply any index or centring error first, and then to halve the remaind,er and correct as for an ordinary altitudeexcept that there is NO DIP 118 THE ART OF ASTRONOMICAL NAVIGATION To check and cotrect (1) Place the index at about 50° and, holding the instrument horizontal with the arc away, look into the index glass and note whether the true and reflected arcs appear in one line If not they should be brought into line by the adjusting screw at the top of the back of the index glass To check and correct (2) With the index about zero hold the sextant horizontal and look through the telescope and horizon glass at the horizon, when the true and reflected horizons should appear in one line If not they should be brought into line by means of the adjusting screwatthetop of the back ofthe horizon glass To check and correct (3) Stand the sextant on a table with the telescope, preferably the long one, shipped Stand a straight-edge on edge on the table beside the sextant and parallel with the t~lescope From a direction at right angles to the straight-edge and telescope lower the eye until the straight-edge and telescope appear close together, when it can be judged whether or not they are parallel If not there is collimation error, which should be adjusted by means of the screws on the upper and lower sides of the collimation collar (Ease back one before screwing up the other.) To check and correct (4) Set the sextant reading at zero and, holding the sextant vertical, look through the telescope and horizon glass at the horizon, when the true and reflected horizons should appear as one line If they are not the horizon glass may be adjusted to parallel the index glass by means of an adjusting screw at the bottom of the back of the horizon, glass The effect of adjusting (4) will have the effect of creating side error (2) It will be necessary, therefore, to keep adjusting (2) and (4) alternately until the error on both disappear On the other hand, I.E is the same for all angles, and as it is not a good thing to make too much use of the adjusting screws the I.E may be noted and applied to all sextant readings by simply 'reading high' or 'reading low' by the amount of the I.E value To determine this value: when performing operation (4), if the true and reflected horizons are not in line they should be brought into line by the tangent screw, and the sextant reading then gives the I.E If reading off the arc it is a + correction, and if on the arc it is a - correction In 'official' text-books it is generally recommended that (2), (3) and (4) be corrected by means of stars, using the inverting telescope and its cross wires These processes may be more MISCELLANEOUS NOTES AND FORMULAE 119 accurate in theory: but the simple processes advised above are a lot more easy to in practice, and give sufficiently accurate results for navigational purposes (4) may also be checked by bringing the true and reflected sun limbs into contact on and off the arc, the I.E being half the difference of the readings, + if greater reading is off the arc, - if on Text-books point out that the accuracy of these observations may be checked by adding the on and off readings together and dividing by and comparing with the semi-diameter for the day If this is done when the sun is very low (less than toO) it is advisable to use the sextant horizontally, taking right and left limbs, as refraction will be distorting the sun's vertical diameter LOGARITHMS Every table of logarithms is worked out for a certain base, which base, in the case of the common logs used in navigation, is to Definition of Logarithm The logarithm of a number to a given base is the index of the power to which the base must be raised to equal the number The two parts of a logarithm are called the 'index' and the 'mantissa' The index is always one less than the number of integral places (whole numbers) in the number Thus, the index of 25=1, 2·5=0, ·25 = r or, alternatively, 9, ·025=2 " " The minus sign, when used, is placed over the figure because only the index is negative The mantissa remains positive To perform multiplication, add the logs To effect division, subtract the logs To perform involution (raise a number to a given power), multiply the log by the power required Log 23=3 X 0·30103=0·90309=Log To effect evolution (extract a given root), divide the log by the root required, 0,90309 • Log {l8 = =0·30to3=10g MISCELLANEOUS NOTES AND FORMULAE 125 The lat and long of the vertex being known, the lat arid long of any number of points along the G.C track can be obtained in a very simple manner, two logs only being required in each case, of which one is a constant The process consists in assuming a number of equidistant longitudes and calculating their corresponding latitudes This applies equally, of course, to both G.C and composite G.C sailing The general formula for the lats is as follows: tan lat (reqd.) = tan lat Vx cos D'long,* where D'long = D'long between V and assumed long CHARTS When using 'short methods' there is no particular difficulty about plotting on plain paper; except that a rather large space, such as the back of a chart, is generally necessary A spot is selected for the first chosen position and the parallel and meridian 'shot in' The D'long between the first and second chosen positions is then referred to the Traverse Table to get the necessary Dep to mark off the second chosen position: and thence proceed as in Method I Where, however, certain areas are fairly frequently used it may be thought worthwhile to construct a blank Mercator chart; for which purpose the table on page 126 may be found convenient This scale may be a little small, especially for the lower latitudes, but the area required to be charted seldom exceeds 2° of latitude, and it is an easy matter to increase by half, or double, or treble the scale The longitude would be graduated from a common ruler of some kind and it is a very quick job to divide up the latitude scale with dividers Another quick and easy way of constructing a Mercator chart is as follows: Draw in the selected base-line and mark off and erect the meridians in accordance with the scale decided upon From the point of intersection of one of the meridians with the base-line draw a line making an angle with the base-line equal to the middle latitude between the base line parallel and the next parallel to be • drawn in, this line to be produced to intersect the next meridian * See footnote on p 123 Obviously, there is no need to this graphically The meridionallengths can be calculated from the formula Meridional length = Scale X sec mid lat or they may be taken out of the Traverse Table: in which case (as can be seen from the figure) the mid lat is the course and the scale the D'lat., the required meridional length being taken from the Dist column Gnomonic Charts The printed gnomonic charts generally supplied for taking off great circle tracks are constructed for some focal (or 'tangential') point in the middle latitudes These 'oblique' gnomonics are extremely difficult to construct without special instruments The POLAR GNOMONIC chart, however (in which the tangential point is the pole) is extremely easy to construct; and as a very fair idea of the great circle track can be got from them a few words on the subject may be worth insertion here On the polar gnomonic chart the central point is the pole, through which all meridians are drawn as straight lines, and about which all parallels are drawn as circles The relative radii ~f successive parallels, therefore, increase as their distance from the pole MISCELLANEOUS NOTES AND FORMULAE 129 It will be remembered that courses and distances cannot be taken off a gnomonic chart The idea is to transfer positions from the gnomonic to the Mercator and then sketch in the curve on the latter The latitude in which the gnomonic track cuts the meridians drawn are the points generally selected for transfer It may be further pointed out that there is no particular point in trying to follow the great circle track with any degree of exactitude Generally, a handful of miles on one side or the other makes no appreciable difference to the distance Hence, for a particular voyage a smaller chart could be drawn, reducing the radii accordingly Thus, if a 6" scale is chosen it is only necessary to halve the radii in the table To construct Draw a horizontal base-line 2' in length and mark the centre P, for the pole With centre P and radii given sweep in the parallels needed, and through P draw the meridians needed, generally 10°, 15°, or 20° apart This gives a chart covering 180° of longitude, which is sufficient for any surface navigation (Only when planning flights across the polar regions would it be necessary to draw in the complete circle.) It might be mentioned that without the assistance of a beam compass it is not easy to make a good chart, as circles of such large radii are involved PLANS It is occasionally necessary to place on or take off plans positions expressed in latitude and longitude In such cases it is convenient to have a longitude scale on the plan To construct a longitude scale-from one end of the scale of latitude and distance (which appears on all plans) draw a line making an angle with this latitude and distance scale equal to the latitude of the place From the divisions on the latitude scale drop perpendiculars to thi~ line The result is a longitude scale AANK MISCELLANEOUSNOTES AND FORMULAE 131 much further off than the other two The long bearing at once shows how the error must be adjusted to close the cocked hat To construct a deviation table from a limited number of bearings, proceed graphically Thus, draw a vertical straight line and divide it into 36 sections and mark the divisions from 0° to 360° Using any scale mark off as dots east ( +) and west ( - ) of this line the deviations obtained: the dots being placed opposite the relevant ship's head on the vertical scale Draw a freehand curve through these points, and the deviations on all points will then be shown by the (horizontal) measurement from the ship's head line to the curve The deviations should be equal and opposite on opposite points of the compass, and although this is never exactly so, it must always be nearly so Even if a full round of bearings has been taken this graph will check it and show at a glance if any serious errors have been made NOTE The scale used to mark off the deviations need have no relation to that used for the ship's head line An 18 em vertical line (t em for 10°), and whole ems for degrees of deviation, will be found as convenient as any TIDES 1.2.3.3.2.1 formula The deep sea navigator would never trust exact calculations of the height of the tide; indeed, knowing what he should about the vagaries of tides the more exact the calculation the more cynical he is likely to be Where, however, some sort of estimate has to be made a good working system is to divide the range for the particular tide by twelve (12) and then assume that it rises or falls in successive hours from low or high 123321 water by 12 12 12 12 12 12 of the range ThIs assumes a·6-hour , , , , , duration of rise or fall which is seldom the case (6h 12m being the average) but is generally near enough when used with liberal caution At standard ports the duration of rise or fall could be checked from the tide tables Note Range = H.W.ht -L.W.ht Where the L.W height is not given one must tread warily, asthe tidal information on the chart refers to spring and neap rise, not range It is easy to become rusty on tidal expressions-and it co.uldbe dangerous The diagram on p.133 may be used to refresh the memory if occasion demands 132 THE ART SOME OF ASTRONOMICAL USES NAVIGATION OF THE TRAVERSE TABLE The expression 'compo bearing' (meaning complement of the body's bearing) used hereunder is to be taken to mean the difference between the bearing of the body and either 90° or 270°, and is thus always an acute angle Thus: Compo 62° = 28° Compo 242° = 28° ,,127°=37° ,,328°=58° (1) To express a sight worked by the St Hilaire process in terms of the Longitude method With compo bearing as course, and intercept as D lat take out the Dist Turn this dist into D long and apply to the D.R long (2) To find the St Hilaire intercept from a sight worked by the Longitude process Turn D long between observed and D.R longs into Dep With compo bearing as co and Dep as Dist the corresponding D.lat will give the intercept required (3) To find the most probable position of the ship resulting from a single sight, apply the St Hilaire intercept to the D.R position as a course and distance (4) To find the correction (A and B) for a longitude sight With compobearing as co and 100 miles as D.lat., take out the Dep Convert this into D long and shift the decimal point two places to the left The result will be the correction for one mile error of latitude (5) To find the error in longitude for given error of altitude With compo bearing as co and given error as D lat., take out the Dist This, converted into D long., will be the value required * (6) To find the error in position due to given error in one altitude at different angles of intersection of position lines With compo angle of intersection as course and given error as D lat., corresponding Dist will be value required To estimate distance a vessel will pass off a light from rising bearing * The reasons for these proceedings will become apparent by considering them in the light of Fig 16; altitude-difference and intercept being the same thing As a matter of fact, Fig 16 might very reasonably be called the 'characteristic figure for applied nautical astronomy' THE ART OF ASTRONOMICALNAVIGATION MISCELLANEOUSNOTES AND FORMULAE With angle on bow as co and distance as Dist., distance off abeam will be in Dep column When approaching or rounding an elevated shore-mark with a distance-angle clamped on the sextant it is as well to bear in mind that 134- To find co to steer to pass given distance off light from rising bearing With rising distance as Dist and required distance off as Dep., co angle will be angle on bow to which light should be brought The Traverse Table can also be used as a table of proportional parts To find distanct' run in a given time Against Dist 60 (for hr.) find the ship's speed in the Dep column The distance run for any other number of minutes will then be found in the Dep column opposite the minutes in the Dist column Example: Speed 16~ knots, required distance run in h 23 m Under Co 16°, Dist 60, Dep.=16·5, and Dist 83=Dep 22·9= distance required Even half knots from 12l to 22l will be found in the Traverse Table, but interpolation can be done at sight As a conversion table: To turn metres into English feet, and vice versa: Under co 17° D lat = feet Approximately, } { (Error, foot in 328) Dep metres To turn metres into fathoms, and vice versa: Under co 29° D.lat.=metres APproximately Dep = fathoms }{ (Error, 1·5 fathoms in 100) OFF-SHOREDISTANCEBYSEXTANTANGLE For distance off by sextant angle-Dist =h cot (J For angle (to set on the sextant for a given distance) dist h cot (J= or tan (J=-h dist Generally, of course, tables of sufficient accuracy for navigational purposes are at hand: but occasions sometimes arise when it is convenient to have a table of masthead angles for short distances Such a table can be very quickly compiled from the above formulae 135 OVERLAP means DANGER 'overlap' meaning overlap of direct and reflected images in the horizon glass In the absence of tables the distance off by vertical sextant angle can be calculated by the formula hx34 hx·565 DISt (M) = -(J-" - or DISt = (J' The above formulae are only to be used when the sea-level base point of the object is within the observer's horizon They can also (obviously) be used for horizontal angles when passing land-marks which are too close together to yield a fix by crossbearings REVOLUTION TABLE In fine weather, in a ship of any considerable size, the revolutions are as accurate a method of determining the speed through the water as any patent log This being so, every ship's chartroom should display a table of revolution speeds for various slips The following advice upon the making up of such a table may therefore not be altogether unacceptable The pitch being quoted in feet, and the revolutions as revs per minute, we have: · pitch X 60 E ngme spee d =rx h 6080 k nots, were r=revo Iut1~ns To find the speed for a certain slip (say per cent) we introduce a percentage fraction (as M) into the term We then reduce the whole term to a log constant, to which we add the log of each number of revolutions for which the speed is required Speed=rx EXAMPLE Pitch =18 feet, required speeds for per cent slip 18 X 60 X 92 621 =log 2·79309 6080 X 100 -3-8-0-0=log3'S7978 • 9·21331 =log const for per cent slip 144 Typhoons, INDEX region and season, 138 Vertex of great circle, to find, 123 Vertical angle, distance by, 134-5 Vertical circle, 25 \Vinds, storm and local, 138-9 'Nireless telegraphy, conversion bearings by, 83-90 of Zenith, 18, 21-2 distance, same value as geographical, 18-19 calculation of, 27-8 comparing for intercept, 27, 29 Zodiac, signs of, 140 Zone time, nature of, 107-8 of meridian transit, 109-10 of twilight, 110-11 ... draw the position line on the chart we have only to 'prick off' the position of the body, and then, with the zenith distance as radius, sweep an arc in the direction of the D.R position Theoretically... 29·5 This, of course, is because the rate of change of the bearing increases until the object is abeam It will be evident, then, that two bearings of an object 16 THE ART OF ASTRONOMICALNAVIGATION... any other fixed sum The nautical mile, for the purposes of navigation, is the length of an arc on the earth's surface which subtends an angle of one minute at its centre An observer on the earth's