NAVIGATION AND NAUTICAL ASTRONOMY INCLUDING THE THEORY OF COMPASS DEVIATIONS A TREATISE ON NAVIGATION AND NAUTICAL ASTRONOMY INCLUDING THE THEORY OF COMPASS DEVIATIONS PREPARED FOR USE AS A TEXT-BOOK AT THE U. S. NAVAL ACADEMY BY COMMANDER W. C. P. MUIR, U. S. NAVY Head of the Department of Navigation U. S. Naval Academy THIRD EDITION Revised and En'argeu ANNAPOLIS, MARYLAND THE UNITED STATES NAVAL INSTITUTE 1911 COPYRIGHT, 1906 COPYRIGHT, 1908 COPYRIGHT, 1911 BY PHILIP R. ALGER Secretary and Treasurer U. S. NAVAL INSTITUTE ot& (gafcttnore BALTIMORE, MD., U. 8. A. M^ PEEFACE. In this volume, the endeavor has been made to place under one cover the allied subjects of Navigation, Theory of Com- pass Deviations, and Nautical Astronomy; and, though the book has been written primarily for the use of midshipmen, it is believed that the various subjects have been so presented that any zealous student with only a slight knowledge of trigonometry may be able to master any method given. An effort has been made to include in this work not only the results of a large practical experience at sea, but informa- tion gleaned, during tours of duty in this department, from a study of the best English and American authorities. Much attention has been given to a description of the various navigational instruments, their uses, and errors; and to the principles involved in the construction of charts as well as to an account of the work usually performed on them. The Theory of the Deviations of the Compass has been presented in the popular way for the practical man, and from a mathematical standpoint for more advanced students. In Part II enough of Theoretical Astronomy has been in- corporated to enable any one without a previous knowledge of that science to pursue the study of the practical part of Nautical Astronomy. In the chapter on Time, I have gone much into detail and have illustrated the theory by the solutions of many examples, because an experience of years as an instructor has shown 272623 vi PREFACE that beginners usually find it an intricate subject. In this chapter, as in all other parts of the book, practical illustra- tions follow immediately the theory on which they are based. In a consideration of " lines of position," much space has been given not only to the theories and practice of Sumner, but also to the later adaptation of those theories by Johnson and Marcq Saint-Hilaire. All these methods are worthy of close examination, and each will be found to have its special advantages. However, if the student is pressed for time, he is advised to confine his attention to what will be described as the " chord method " which embodies the present practice of the United States Naval Service. A chapter on " Tides " and one on " The Identification of Heavenly Bodies " have been included ; a knowledge of both these subjects is essential to the modern navigator. The text contains no reference to " lunars," that method of finding longitude being regarded as obsolete in these days of excellent chronometers. It is believed that the time is not far distant when " lunar distances " will disappear from the Nautical Almanac. Acknowledgments are due to Lt.-Comdr. W. V. Pratt, IT. S. N., and to Lt. C. P. Snyder, IT. S. N. ? for assistance in proof-reading; to Midshipman H. G. Knox, U. S. N., for work on Tables II and III ; and to the following-named firms for the loan of certain electrotypes : E. S. Eitchie & Sons of Boston, Keuffel & Esser, T. S. & J. D. Negus, and John Bliss & Co. of New York. In conclusion, I desire to express my gratitude to Lt.- Comdr. B. W. Wells, U. S. N., and to Lt.-Comdr. G. K. PREFACE vii Marvell, U. S. 1ST., for valuable criticism of the original manuscript and for assistance in eliminating errors from the finished book. W. C. P. MUIR. DEPARTMENT OF NAVIGATION, U. S. NAVAL ACADEMY, MAY 1, 1906. NOTE TO SECOND EDITION. Chapter XX has been rewritten and enlarged, and, besides a few minor changes which have been made in certain parts of the text where deemed desirable for greater clearness, the following additions have been made : a new method of equal altitudes suggested by Mr. G-. W. Littlehales, Hydrographic Engineer, U. S. Navy Department; plates X to XVI; and four appendices. This opportunity is taken of thanking those officers of the service and friends outside the service who have kindly ex- pressed appreciation of my efforts,, or who may have offered suggestions with reference to this edition. W. C. P. M. JULY 1, 1908. NOTE TO THIRD EDITION. A complete revision of this book has been made wherever changes have been found necessary to make the text and ex- amples correspond with the new method of estimating course and azimuth, an innovation due to the recently adopted grad- uation of the compass. Some new matter has also been added. For*suggestions as to this revision, my thanks are due Com- manders S. S. Eobison and J. H. Sypher, U. S. N., and to Commander Geo. R. Marvell, U. S. N., who relieved me as Head of Department of Navigation, U. S. Naval Academy, on my detachment from active duty, August, 1909. W. C. P. M. SHELBYVILLE, KY., MARCH 15, 1911. LIST OF WOEKS CONSULTED. Spherical and Practical Astronomy, Chauvenet. General Astronomy, Young. Elements of Astronomy, White. The Heavens, Guillemin. Navigation and Nautical Astronomy, Coffin. Navigation, Asa Walker. American Practical Navigator, Bowditch. Navigation and Nautical Astronomy, Martin. Navigation, Merrifield & Evers. Nouvelle Navigation Astronomique, Villarceau. Modern Navigation, Hall. Wrinkles in Practical Navigation, Lecky. Finding a Ship's Position at Sea, Sumner. Finding Latitude and Longitude in Cloudy Weather, A. C. John- son. How to Find the Stars, Rosser. Handy Book of the Stars, Whall. Marine Surveying, IL, S. N. A. Notes on Navigation, U. S. N. A. Practical Problems and Compensation of the Compass, Diehl. British Admiralty Manual of Deviations. Mathematical Theory of the Deviations of the Compass, Howell. Manual of Deviatons, F. J. Evans. Instructions for Care of Chronometers and Watches, Bureau of Equipment, Navy Department. Proceedings U. S Naval Institute. Method of Least Squares, Merriman. CONTENTS. PAET I. NAVIGATION AND COMPASS DEVIATIONS. CHAPTER I. PAGE General definitions: Navigation, pilotage, nautical astronomy, observations on the form and size of the earth. Axis. Poles. Equator. Meridians. The Prime meridian. Par- allels of latitude. Latitude, Difference of latitude. Middle latitude. Longitude. Difference of longitude. Geographical and nautical miles. Rhumb line or loxo- dromic curve. Course. Distance. Bearing of an object or place 1-5 CHAPTER II. List of navigational instruments and books usually provided. Speed measures: Revolution of screw; patent logs; log chip, line and sand glass. Sounding apparatus: Hand, coasting, and deep sea leads; Thompson's and Tanner's sounding machines. Use of sounding data. Charts: description, construction, and use of polyconic, polar', gnomonic, and Mercator charts; advantages and disadvantages of each system of projection; conventional notation and hydrographic signs; plotting and taking off positions, laying courses, and measuring distances; cor- rection of charts; arrangement and stowage on board. The 3-arm protractor and its use. The " Three-point problem " and rules governing the selection of objects to be angled on 5-44 CHAPTER III. Section I. The compass. The U. S. Navy compensating bin- nacle. The azimuth circle. The pelorus. The illumi- nated dial pelorus. The use of a pelorus to determine a magnetic heading 44-55 Section II. The earth's magnetism and the elements of that magnetism. Magnetic poles. Magnetic equator. Mag- netic meridian. Magnetic latitude. Relation of true and magnetic meridians. Variation. Deviation. Correction of bearings. Leeway. Correction of courses. Local at- traction 55-66 CONTENTS PAGE Section III. Finding the deviation by reciprocal bearings, by bearings of a distant object, by time azimuths, by ranges. Description, construction, and use of Napier's diagram . 66-77 Section IV. " Hard " and " soft " iron. Magnetic induction. How a ship becomes magnetic. Magnetic forces acting on a compass needle in an iron or steel ship and the effect of each in producing deviation. Semicircular de- viation. Quadrantal deviation. Constant deviation. Causes and characteristics of each kind of deviation. The approximate equation for deviation. Determination of the approximate coefficients by inspection of a devia- tion table. Heeling error. Mean directive force.. 77-104 CHAPTER IV. Section I. Mathematical theory of compass deviations. Consideration of the various forces acting on a compass needle in an iron or steel ship. Finding the components of each force in certain definite directions through the compass and then the resultant of all the forces in each direction. Representation of the effect of the soft iron of the ship by nine soft iron rods. Symmetrical and un- symmetrical soft iron. The fundamental equations . . . 104-116 Section II. Transformation of the fundamental equations. Forces of earth and ship to head, to starboard, to mag- netic North, and to magnetic East. Formulae for com- puting deviations. Subdivisions of the deviation and a consideration of the various coefficients. The ship's polar force and starboard angle. The " Gaussin error ". . . . .116-128 Section III. Method of least squares applied to the determi- nation of coefficients. Formation of normal equations. Equations for exact coefficients in terms of the approxi- mate coefficients 128-136 Section IV. Analysis of deviations and the determination of exact coefficients 136-142 Section V. Observations for horizontal and vertical forces ashore and on board. Determination of X and fi 142-150 Section VI. Determination of 33, (, and 2) by observations in one quadrant, in one semicircle. Determination of S3 and ( from observations of deviation and horizontal force on one heading; of 39, GT, 3D, X, a, and e from observations on two headings. Determination of the forces of hard and soft iron causing semicircular deviation. Placing a Flinders bar. Computation of deviations from coeffi- cients 150-165 Section VII. Heeling error. Change in the fundamental equations due to the ship's heeling. The " Heeling co- efficient." Methods of determining heeling error. Cor- rection of heeling error by vibrations, by using the heel- ing adjuster, by the tentative method 165-178 CONTENTS xi PAGE Section VIII. Compensation of the compass: (1) when devia- tions are known, (2) when deviations are unknown. De- termination of magnetic courses when the deviations are unknown, using the pelorus or azimuth circle. Given 93, (, and S), to compensate on one heading 178-189 Section IX. The dygogram: its construction and use. To construct a dygogram when the exact coefficients are known. To construct a dygogram when the deviations and horizontal force are known (1) for two opposite mag- netic courses, (2) for two magnetic courses not oppo- site o 189-204 CHAPTER V. Piloting. The bearing of an object. A line of position. A line of bearing. A position point. Fixing ship's position near land (1) by sextant angles; (2) by cross bearings; (3) by a bearing and distance; (4) by change of bearing of a single object, using tables or the graphic method; (5) by doubling the angle on the bow. Distance of pass- ing an object abeam.: Horizontal and vertical danger angles. Danger bearings. Lights as danger guides. Fog signals 204-220 CHAPTER VI. The sailings. Preliminary definitions. Plane sailing. Us6 of traverse table. Traverse sailing. Graphic explanation of traverse sailing. Sources of data. Preparation of the traverse form and data. Parallel sailing. Middle lati- tude sailing. When not advisable to use middle latitude sailing. Current sailing: solutions by construction, by trigonometry, and by the traverse table. Mercator sail- ing. Graphic illustration of the theory of Mercator sail- ing. When not advisable to use Mercator sailing. Cor- rection to the middle latitude. Day's work by D. R.. .220-263 CHAPTER VII. Great circle sailing. Comparison of rhumb and great circle tracks. Preliminary definitions. Great circle course and distance by (1) computation, (2) azimuth tables, (3) great circle charts, (4) graphic approximation. Finding the vertex and point of maximum separation. Use of ter- restrial globe. Graphic chart methods. Composite sail- ing by (1) gnomonic charts, (2) computation, (3) graphic methods . . 263-285 xii CONTENTS PART II. NAUTICAL ASTRONOMY. CHAPTER VIII. PAGE General definitions: Nautical astronomy, the celestial sphere, axis, poles, celestial equator, horizons, zenith, nadir, ce- lestial meridian, ecliptic, equinoctial and ecliptic points, the colures. Consideration of spherical coordinates. The ecliptic system: celestial latitude and longitude. The equinoctial system: declination, polar distance, parallels of declination, hour circles, transit, hour angle, solar time, sidereal time, relation between solar and sidereal days, right ascension, relation of hour angle and right ascen- sion. The horizon system: celestial horizon, vertical circles, prime vertical, azimuth, amplitude, altitude, zenith distance. Proof that latitude equals the altitude of the elevated pole. The astronomical triangle. Projec- tions on planes of the meridian, the equator, and hori- zon , . . 287-303 CHAPTER IX. ' The sextant. The optical principle of the sextant and its ap- plication in the measurement of angles. The vernier. Reading the sextant. Excess of arc. Constant and acci- dental errors of the sextant. Errors of graduation and eccentricity. Prismatic effect of mirrors and shade glasses. Adjustment of the sextant. Index error and its determination. Using a sextant to observe altitudes of the sun, or a star, at sea. Measuring horizontal angles. General care of sextant. Resilvering mirrors. The arti- ficial horizon, its care and preparation for use, its ad- vantages, its theory. Method of observing with artifi- cial horizon 303-327 CHAPTER X. Description of chronometers. Reception and stowage on board. Winding and comparison. Cleaning and oiling. Transportation. Effect of change of temperature. Hart- nup's law. General equation. Temperature curves. In- structions for management and use. Comparing, stop, and torpedo-boat watches. Winding, general care, and prepa- ration for shipment 327-340 CHAPTER XL Comparison of sidereal and tropical years. The calendar. The Gregorian correction. The sidereal year. Relation of solar and sidereal time.. ..340-343 CONTENTS xiii CHAPTER XII. PAGE Time and its measurement. Sidereal day. Apparent solar day. Mean solar day. Equation of time. Relation of local sidereal time, the hour angle, and right ascension of a body. Astronomical and civil time. Standard time. Conversion of arc into time and vice versa. Relation between the local times at two places. Finding Green- wich time. Gain or loss of time with change of position. Crossing the 180th meridian 343-362 CHAPTER XIII. The Nautical Almanac. Finding a required quantity for a given time. Use of second differences. Mean time of moon's transit. Mean time of a planet's transit 362-380 CHAPTER XIV. Interconversion of apparent and mean times. Formulae for the interconversion of mean and sidereal time intervals. Conversion of mean time into sidereal time and vice versa. Conversion of apparent time into sidereal time and vice versa. Relation of time, hour angles, and right ascensions, and problems involving them. To find the local mean time of upper transit of a particular heavenly body, also the time of lower transit. To find the watch time of transit of the sun, of a star, the moon, or a planet. To find the hour angle of any heavenly body at a given time and place. To find what stars will cross the merid- ian between two given times 380-411 CHAPTER XV. Corrections to an observed altitude, using (1) a sea horizon, (2) an artificial horizon. Refraction. Parallax. Dip of the horizon. Error of dip. Distance of sea horizon. Range of visibility at sea. Dip of a point nearer than the sea horizon. Apparent semi-diameter. Augmentation of the moon's semi-diameter. Theoretical and practical methods of correcting altitudes. Correction of altitude for run 411-432 CHAPTER XVI. Solution of the "astronomical triangle." Finding the true altitude. Altitude and time azimuth of a heavenly body. Altitude-azimuth of a heavenly body. The haversine formula for azimuth. Amplitudes. Use of azimuths and amplitudes in finding compass error. To determine when an error in altitude, or an error in latitude, will have the least effect on the azimuth. True bearing of a ter- restrial object. Hour angle and local time from an ob- served altitude. Haversine formula for hour angle. xiv CONTENTS PAGE Conditions of observation. Sunrise or sunset sights. Time of sunset. Duration of twilight. Hour angle of any heavenly body when in the horizon. Length of day and night. To determine when an error in altitude, or an error in latitude, will have the least effect on the hour angle, and hence the best time to observe for longitude. Hour angle of a heavenly body when on or nearest to the prime vertical 432-488 CHAPTER XVII. Latitude from meridian altitudes above or below the pole. The constant. Finding latitude by observations of bodies out of the meridian: (1) The 0"0' method, (2) an ap- proximate method involving both latitude and longitude, (3) by reduction to the meridian, (4) by altitude of Po- laris, (5) Chauvenet's method, (6) Prestel's method. " Angle of the vertical " or " reduction of the lati- tude " 488-548 CHAPTER XVIII. Chronometer error. Distinction between error and correc- tion. To find the rate. Sea rate. Irregular rate. Finding error and rate by transits; by time signals; single or double altitudes; equal altitudes of a fixed star, the sun, or a planet. Rating chronometers by telegraph or wireless signals. The U. S. system of time signals. To correct the middle time in equal altitudes for a small difference of altitude. Methods of observation. Com- parison of equal and double altitudes. Longitude ashore by electric signals; ashore or afloat by equal altitudes, Bingle altitudes, double altitudes. An approximate meth- od of equal altitudes of the sun for longitude at sea. A method of finding longitude at sea by equal altitudes of the sun when the ship is proceeding at high speed. . .548-589 CHAPTER XIX. Sumner's method. A heavenly body's geographical position and its coordinates. Circles of equal altitude. A line of position. Curves of equal altitude on a Mercator chart. Determination of points on the curve. Double alti- tude problem. Simultaneous observations. Advantages of simultaneous over double altitude observations. Re- lation between circles of equal altitude and the astro- nomical triangle. Method of determining a line of posi- tion. Graphic illustrations of the various ways in which a line of position may be used at sea or near the coast. To allow for uncertainties in time or altitude. Defini- tion of longitude and latitude factors. Plotting lines by the chord method, by the tangent method. Position by CONTENTS xv PAGE simultaneous observations, one body on the meridian, one on or near the prime vertical. Position by the " mu- tual correction " method. Computing the intersection of two lines determined (1) by the chord method, (2) by the tangent method 589-636 CHAPTER XX. The new navigation or the method of Marcq Saint-Hilaire. 636-661 CHAPTER XXI. A day's work at sea, rules, form, and solution of an ex- ample 661-670 CHAPTER XXII. The tides. Definitions relating to tides. Causes of tides and the daily inequality of tides. Effect of the sun. Priming and lagging. Luni-tidal intervals. Establishment of the port. General laws of tides. Tidal currents. Times of high and low water and current data from the Tide Tables. High or low water by computation 670-680 CHAPTER XXIII. Distinction between planets and fixed stars. Distinction of the principal planets. Grouping and classification of stars. List of the navigational stars. Constellations of reference. Stars referred to the " Dipper " (Ursa Major), to Orion, to the Southern Cross. Identification in cloudy weather 680-696 CHAPTER XXIV. General observations as to the compasses, the sextant, the chronometers, and the charts. General duties of a navi- gator before going to sea or entering pilot waters. Discrepancy in a. m. and p. m. sights when abnormal refraction exists. Error of a ship's position. Coefficient of safety. Advisability of keeping landmarks in sight when possible. General duties of navigator going in or out of port. Using the seconds of data 696-708 Tables and extracts from Nautical Almanac 708-738 APPENDICES. Appendix A. Description of submarine-bell system 747 Appendix B. First compensation of a compass before pro- ceeding to sea and procedure in special cases when com- pensating the compass, on one heading 748 Appendix C. Use of azimuth tables in finding Z, M, t, and a great circle course 756 Appendix D. Description .of Dr. Pesci's nomogram and its use to the navigator 759 Appendix E. Table of compass points and degrees from N. (to the right) 764 xvi CONTENTS PLATES - Plates I and II. Illuminated dial pelorus 53 Plate III. Representation of the nine soft iron rods 113 Plates IV and V. Mercator charts with mercator and great circle tracks between two given places 269 and 274 Plate VI. The principal stars around the North celestial pole, d > 30 684 Plate VII. The principal stars of declination less than 30 and R. A. O h to XIP 688 Plate VIII. The principal stars of declination less than 30 and R. A. XIP to XXIV h 689 Plate IX. The principal stars around the South celestial pole, d > 30 691 Plate X. Conventional signs and symbols, U. S. Hydro- graphic Office charts 739 Plate XI. Hydrographic signs, U. S. C. and G. Survey charts. 740 Plates XII and XIII. Topographic signs, U. S. C. and G. Survey charts 741 and 742 Plate XIV. General abbreviations on charts 743 Plate XV. Circles of equal altitude on the mercator chart.. 744 Plate XVI. The variation of the compass for 1910 745 PART I. NAVIGATION AND THEORY OP THE DEVIATIONS OF THE COMPASS. NAVIGATION, THEORY OF THE DEVIATIONS OF THE COMPASS, AND NAUTICAL ASTRONOMY. CHAPTEE I. DEFINITIONS AND GENERAL OBSERVATIONS. Article 1. Navigation is the science of determining the posi- tion of a ship at sea, and of .conducting a ship from one position on the earth to another. 2. There are three general methods of locating a ship: (1) When near the coast by bearings, or bearings and dis- tances, of known objects on charts constructed by various methods or projections to represent the earth's surface; (2) by course and distance made good from a known position, involving the principles of plane trigonometry; (3) by obser- vations of heavenly bodies, involving the principles of spher- ical trigonometry. The first may be called pilotage, the second dead reckoning, the third nautical astronomy all independ- ent in theory, but all used practically in the course of a voyage from one port to another distant port. 3. As a ship is located by the latitude and longitude of her position, it is proper to begin here with the elementary geographical definitions. The earth is an ellipsoid of revolution, the equatorial radius being 3963.307 miles, the polar radius 3949.871 miles. Hence the meridians are ellipses, though the parallels of latitude are circles. For the general purposes of navigation the earth is assumed to be a sphere. 2 NAVIGATION The axis of the earth is that diameter passing through the poles of the earth and about which the earth daily revolves from west to east. The earth's equator is a great circle of the earth whose plane is perpendicular to the axis at its middle point. The plane of the equator divides the earth into two hemispheres, the one containing the north pole being called the northern hemisphere, the one containing the south pole being called the southern hemisphere. Every point of the equator is equi- distant from the poles. Terrestrial meridians are great circles of the earth passing through the poles. The meridian of a place on the earth is that meridian passing through the place. The prime meridian is that meridian from which the longi- tude of places on the earth is measured. The meridian of Greenwich is almost universally accepted as the prime meridian. Parallels of latitude are small circles of the earth whose planes are perpendicular to the axis. The latitude of any place on the earth's surface is its angu- lar distance from the plane of the equator north or south, measured from to 90, on the meridian passing through the place. The middle latitude of two places in the same hemisphere is half the sum of their latitudes. The term is not strictly applicable where the places are situated on opposite sides of the equator. The longitude of any place is the inclination of its merid- ian to the meridian of some fixed station known as the prime meridian, and is measured by the arc of the equator included between these two meridians. Longitude is usually reckoned from to 180 east or west of the prime meridian (usually that of Greenwich). It is thus apparent that any point DEFINITIONS AND GENERAL OBSERVATIONS 3 whose latitude and longitude are known can be located on the globe or chart representing the earth's surface. The difference of latitude of any two places is the portion of a meridian included between the two parallels of latitude passing through the places. When both places are on the same side of the equator, their difference of latitude is found by subtracting the smaller from the larger latitude, and when the two places are on opposite sides of the equator, the differ- ence of latitude is found by adding the two latitudes ; when a ship in any latitude sails towards the pole of that hemisphere she increases her latitude, when she sails away from the pole she decreases her latitude; the difference of latitude being called N. or S. to indicate the direction of the change. The difference of longitude of any two places is the angle at the pole, or the corresponding arc of the equator, between the meridians passing through the two places. When the two places are in longitudes of a different name, their difference of longitude is found by taking the sum, or 360 the sum. The difference of longitude is called E. or W. to denote the direction of change. In other words, in combining latitude and difference of latitude, also longitude and difference of longitude, the operation must be performed algebraically, the terms N. and S. being considered as opposite signs, likewise the terms E. and W. The geographical and nautical miles. The geographical mile is the length of a minute of arc of the equator; the nautical or sea mile is the length of a minute of arc of a circle having a radius equal to the radius of the curvature of the meridian in the latitude of the place considered. The meridians being ellipses flattened at the poles, the linear length of 1' of the meridian is slightly different for different latitudes; is least at the equator and greatest at the poles, its mean value of 6080.27 feet being taken as the 4 NAVIGATION length of the nautical mile. For navigational purposes, the geographical and nautical miles may be considered the same. The rhumb line or loxodromic curve is a line on the surface of the earth which makes a constant angle with each succes- sive meridian. If a ship sails on a loxodromic curve, the constant angle made by this line with the meridian is called the " true course." For trigonometric computations, the course is meas- ured in degrees from North or South toward East or West, according to the data; though, in practice, navigators consider it as estimated, in both hemispheres, from the North point, around to the right, from to 360. The distance between two places, or the distance run by the ship on a course, is the length of the loxodrome joining the two places measured in nautical miles. 4. Sailing a certain distance on a given true course, the distance North or South from the place left, measured on a meridian, is the difference of latitude, and the distance East or West on a parallel is the departure for that latitude, both expressed in sea miles. Should the course be due East or West on the equator, the distance would be difference of lon- gitude. Later on the relation between departure and difference of longitude will be shown to be that departure equals the differ- ence of longitude multiplied by the cosine of the latitude. 5. The hearing of an object or place is the angle which the great circle passing through the object (or place) and observer makes with the meridian. It may be expressed as true, mag- netic, or per compass, according as the meridian is true, magnetic, or per compass. Bearings, like courses, are ex- pressed practically by modern navigators, from the North point, around to the right, from to 360. CHAPTEE II. NAVIGATIONAL INSTRUMENTS. Description and Use of Logs, Leads, Sounding Machines, Charts and Protractors. 6. Besides being provided with the usual book outfit con- sisting of a log book, a treatise on navigation, one on deviation of the compass, useful tables, azimuth tables, nautical almanac for the current year, also tide and sunset tables, corrected buoy lists, light lists and sailing directions, a file up to date of notices to mariners, and an outfit of charts for the regions to be sailed over, a navigator must be provided with a com- pass, azimuth circle, pelorus, sextant, protractor, parallel rulers, dividers, chronometers, artificial horizon, mercurial barometer, a wet and dry bulb thermometer, a log and line (preferably a patent log), hand and deep-sea leads and lines, and, if possible, Sir Wm. Thompson's or Captain Tanner's sounding machine, a good binocular and long glass. The more important of the instruments used in applying the principles of navigation will be considered in this and the succeeding chapter; whilst the sextant and chronometer, belonging properly to the subject of nautical astronomy, will be considered in Part II. 7. Speed measurers. The distance traversed by a ship on any course being dependent on her speed, the accurate deter- mination of this speed is a matter of great importance to the navigator. Revolution of screw. The author has found in his expe- 6 NAVIGATION" rience on ships of various classes that the revolutions of the screws furnish a most convenient and accurate log. Having made runs of known distances, in given times, under favor- able conditions, the speed being uninfluenced by currents, and revolutions carefully noted, it is easy to find the coefficient of revolutions per minute for one knot and to tabulate the revo- lutions per minute to make any desired speed. A little ex- perience should teach the navigator what allowance to make for adverse winds and seas, and for any unusual trim of the ship. Patent logs. There are many mechanical contrivances, of as many various forms but embodying the same general prin- ciples, called patent logs, which, under normal conditions, are very fair registers of speed. However, they are far from ac- curate and need careful watching, even when in good working order. If correct at one speed they are not liable to be so at a faster or slower speed, and they register differently in a head or following sea. The error of each patent log should be ascertained under varying conditions of wind and sea, at different speeds and draft for every run between well-deter- mined points, provided the speed is not affected by tide ; each rotator and register should be lettered and a record kept of their errors. 8. General description, The most successful type may be said to consist of ( 1 ) the rotator, a hollow but enclosed coni- cal shaped piece of brass with small vanes, towed astern by a specially made line, and caused to revolve more or less rapidly according to the ship's speed, by the pressure of water; (2) the register, located on the rail aft, in which cyclometer gear is worked by the rotator through the agency of the line, and the miles and tenths of a mile run thus indicated on the dial plate; (3) the specially made line, the length of which is an important factor in correct registering; experience can best decide this length for different speeds; a high speed requires NAVIGATIONAL INSTRUMENTS 7 a greater length than a low speed ; under ordinary conditions it is advisable to use the length of line issued with the log by the manufacturer, about 400 feet. FIG. 1. Negus Taffrail Log. It is well to have two patent logs, each of a different manu- facture, one on each quarter, and the error of each should be carefully determined by readings taken at times when the position of the ship has been accurately found, and on runs unaffected by currents. 8 NAVIGATION The Bliss and Negus logs are perhaps the most reliable ones on the market. They are shown in Figs. 1, 2, and 3. The mechanism of the patent log requires care and frequent oiling. In use, the lines must he watched to prevent being fouled by each other, by seaweed, cleaning rags, barrels, or debris carelessly thrown overboard. Instruments usually accompany patent logs for changing the pitch of the rotator blades to correct an error in register- FIG. 2. Bliss Star Log. ing; but, if the error is small, it is better to leave it uncor- rected, and apply it to the record of speed. 9. Patent electric log. This log is the same as the or- dinary patent log except that the gearing registering the knots and tenths of a knot closes an electric circuit every time a tenth is turned. The circuit thus closed magnetizes a sole- noid, which in turn attracts a bar. This bar, by means of suitable levers, moves a train of gearing which registers the tenths of a knot and knots. This electrically controlled regis- ter is placed on the bridge, or in the pilot house, where it can be easily read by the officer of the deck. 10. Log chip, line and sand glass. The speed of sailing NOTE. The Nicholson Ship Log includes a clock, speed dial, counter, record drum and chart. It is operated by floats in the load level and speed pipes, the height of water in speed pipe depending on the vessel's speed ahead. It shows the speed and mileage sailed, and records the speed on a chart for every minute of run. NAVIGATIONAL INSTRUMENTS ships, before the patent log came into vogue, was determined by the use of the log chip, log line, and sand glass. The log chip was a wooden quadrant about 5 inches in diameter, weighted with lead on the circular edge to make it float up- right, joined by a three-legged bridle to the log line wound on the reel. The two legs of the bridle to the lower corners were joined to a pin which fitted into a socket secured to the leg attached to the upper corner. The first 15 or 20 fathoms, called the stray line, was indicated by a piece of red bunting, and as this bunting went over the rail (the chip being well clear of the ship), the sand glass was turned at the order " Turn/' and at the order " Up " the line was held, and by a sharp jerk the chip was untpggled. The order " Up " was given when the sand had run through, the length of line out at that instant, indicated by knots and tenths of a knot (marks on the line), gave the speed of the ship. The line was subdivided into lengths of 47 feet and 3 inches called knots, and marked by short pieces of fish line, thrust through the strands, and having one, two, or three knots, etc., tied in them according to the number of lengths from the stray line mark. Each knot was subdivided by pieces of white rag into lengths of two-tenths of a knot each. In marking the line, the distance between knots was gotten from the pro- portion, " length of knot in feet is to one sea mile in feet as 28 seconds are to the number of seconds in an hour." The glass itself, being a 28-seconds glass, was for speeds of four knots and under; for higher speeds a 14-seconds glass FIG. 3. Bliss TaffrailLog. 10 NAVIGATION was used, and to get the ship's speed per hour, using this glass, the knots and tenths, as shown by the line run out, were doubled. Much depended on the manner of heaving the log, and errors of line and glass were hard to guard against. It did not afford a continuous record of the speed, and was not to be depended on for speeds over 10 knots. 11. Sounding has for its object the measuring of the depth of water and ascertainment of the character of the bottom; the former being shown on charts in fathoms or feet, accord- ing to the depth, and the latter noted, where known, as mud, sand, ooze, coral, etc. 12. Sounding apparatus. The depth of water is ascer- tained by the sounding machine or the lead. There are several kinds of leads used, according to the depth of water. Hand lead. On entering or leaving port and in shallow water, generally speaking in less than 20 fathoms, casts are taken by the hand lead, a cylindrical lead, weighing from 7 to 14 pounds, attached to a line of from 20 to 30 fathoms in length, properly marked, and that too when wet. The coasting and deep-sea leads. The coasting lead, weighing from 25 to 50 pounds, is used in depths from 25 to 100 fathoms, beyond which depth the deep-sea lead, weighing from 80 to 100 pounds, becomes necessary. The coasting and deep-sea leads are hollowed out at the bottom to receive an arming of tallow to bring up specimens of the bottom. The lines are marked at 10 fathoms with one knot, at 20 fathoms with two knots, and so on, and at every intermediate five fathoms with small strands; at 100 fathoms it is marked with a piece of red bunting. It is necessary to reduce the speed of the ship when getting casts with either the coasting or deep-sea lead, and a loss of time ensues; however, those disadvantages are obviated if the ship is supplied, as every ship should be, with a sounding machine, of which class of instruments Sir Wm. Thompson's NAVIGATIONAL INSTRUMENTS 11 and Commander Tanner's are certainly the best. Very accu- rate results can be gotten, without loss of time or reduction of speed, in depths up to 100 fathoms of water with these ma- chines. 13. Sir Wm. Thompson's sounding machine consists of a wooden or metal frame bolted to the deck, and carrying a drum of about one foot in diameter, on which is wound about 300 fathoms of seven-stranded flexible galvanized steel wire rope; the drum is con- trolled by a brake, and is provided with handles for heaving in, and a dial plate to record the amount of wire run out. To the wire is attached about 9 feet of log line, and an elongated sinker; between the wire and sinker, fast to the line, is a small copper tube closed by a cap with a bayonet joint at the top though perforated at the lower end. This tube is fitted to carry, when sounding, a glass tube hermetically closed at its upper end but open at its lower- end. The glass tube is coated on the inside with chromate of silver. As the lead sinks, the sea water is forced up the tube in obedience to well- known physical laws, and chemical action of the salt water changes the coating into chloride of silver and its color from light salmon to a milky white, and this FIG. 4a. 121 NAVIGATION change takes place as far up as the water ascends in the tube. A graduated boxwood scale to which the glass tube is applied shows the depth to which it descended, and the arming on the sinker shows the character of the bottom. In heaving in, after sounding, care must be taken to keep the glass tube upright and prevent water from running into the upper part of the tube. Chemically coated tubes which have been used, in shallow water may be used again in water known to be deeper, and hence tubes dis- colored for only a fraction of their length should be saved for future use. The depth recorder. Instead of using the glass tubes the depth recorder may be used. This is a metallic tube in which a piston is forced up by water pressure against the tension of a spring; as the sinker descends the piston is forced up, carry- ing with it a small marker; on being hauled up after sounding the piston descends, but the marker remains and indicates on a graduated scale the depth to which the recorder had been. The marker must be set at zero and the valve screwed up just before use ; after each cast unscrew the nut, slacken the valve, and turn the recorder upside down to drain out the tube. Fig. 4b. Directions for use. A pamphlet containing full directions for using the machine is issued by Messrs. John Bliss & Co., of New York, the Ameri- can agents, and accompanies each machine. The Thompson machine is shown in Fig. 4a. 14. Error of machine. The sounding machine FlQ 4b is not reliable within the field of action of the hand Depth lead, and, always after a short use, has an error Recorder. ^^ ^ navigator g^ould determine. This can be easily done when a ship is stationary in fairly deep water NAVIGATIONAL INSTRUMENTS 13 I FIG. 5. 14 NAVIGATION by comparing depths gotten by the machine and by use of the coasting lead. 15. The Tanner machine consists of a metal frame in three parts, a column of steel surmounted by two brass discs joined at their peripheries. The discs carry a shaft, a drum with V-shaped flanges on which is wound the sounding wire, cranks, compressor arms, brake lever, register, and correction table, at the same time forming a guard to prevent slack turns from flying off the drum. The wire consists of 300 fathoms of 7-stranded flexible gal- vanized steel wire rope. The brake mechanism is simple and direct acting, and being in full view of the operator is easily' controlled. The drum is held by moving the lever in one direction, and released by the reverse movement. The cranks are not to be unshipped from the machine. They are provided with automatic locking bo*lts which act when pre- paring for action ; when these bolts are withdrawn, the cranks fall down each side of the column, and the handles are thrust into the friction scores where they are securely held, and at the same time exert a slight friction on the shaft, almost counteracting the inertia of the drum while sounding. The register, which shows approximately the amount of wire out, is directly beneath the shaft, on the port side of the machine as it is set up on the deck. The correction table at- tached to the top of the machine shows the number of fathoms out corresponding to the dial register. Experience has shown that at 10 knots speed with a depth of more than 50 fathoms, the ratio of wire out to actual depth is about two to one; at depths greater than 50 fathoms, and with speed increasing to 15 knots, the proportion is three to one; and in a heavy sea from four to one. The sinker, which weighs about 18 pounds, has the appear- ance of an ordinary coasting lead with an iron rod projecting NAVIGATIONAL INSTRUMENTS 15 ' from its upper end. The sounding wire secures to an eye in the end of the rod. However, a length of stray line made of cotton cod line or signal halliard stuff, long enough to reach the machine when the sinker is up, is often preferred on account of its flexibility. The glass tube, which may be either the Tanner-Blish or Thompson tube, when in use with this machine, is carried in a brass shield seized to the wire or stray line above the sinker. The Tanner combination lead is sometimes used instead of the sinker; it weighs 30 pounds, carries a shield and sounding tube within a central tube in the lead itself. On account of the delicate sounding tube, care must be taken not to let the lead strike the ship's side when reeling in. 16. The Tanner-Blish tube. Commanders Tanner and Blish, U. S. N". have patented a tube designed to be used continuously if not accidentally broken. It is a glass tube 24 inches long of small uniform bore, the walls of which are ground or clouded. They are translucent when dry, but clear when wet, and the line of demarcation between the clear and the translucent part is used to determine the depth in the same manner as with the chemical tube. The bore is ground spirally to counteract the effect of capillary attraction. In the act of sounding, the upper end of the tube being closed by a rubber cap and the lower end remaining open, the column of air will be compressed by the water which will enter it in proportion to the depth to which the tube descends, the divid- ing line between the clear and translucent glass indicating the height to which the water entered the tube. One end of the tube is left unground, and the rubber cap should be placed always on the other end. When its interior is dry the tube will indicate the depth without fail ; if not dry, no results will follow the cast. Free circulation of air is essential in drying the bore, and to this end the cap must be removed. In rainy or foggy weather the 16 NAVIGATION tube may be dried in the engine room. The bore may be cleaned at any time by allowing a few drops of alcohol to run through it, back and forth, several times, and then rinsing the bore in fresh water. The tube should not be used more than a dozen times without being rinsed in fresh water. Before sounding see the tube dry and translucent : If any part of it is clear, put that end to the lips and draw dry air through it with long inhalations filling the lungs, repeating the process until the whole glass is translucent; then put on the cap and proceed with the sounding. Among the articles furnished with the machine is a small air pump for drying the tubes. 17. The machine in action. The quartermaster of the watch and two men are required for the efficient operation of the machine. The sounding tube is inserted in the shield, open end down, and the shield is seized to the wire or stray line; the sinker is armed, bent to the wire or stray line, and lowered over the stern, the wire or line passing over the roller of the stern leader. At the machine the slack turns are reeled in by one man, brake applied, and cranks thrown out of action, the handles thrust into friction scores, and the pointer noted at zero. When ready, the quartermaster, with hand on the brake, eases down the sinker till it is near the water, then allows the wire to run freely till the sinker reaches the bottom, or the designated amount of wire has run out; as the wire runs out it is checked only as required to prevent slack turns, and a bent metal rod or hardwood stick, called the finger pin, is pressed lightly on the running wire, and indicates that the sinker has reached bottom by suddenly approaching the deck as the wire momentarily slackens; at this moment the brake is applied, both cranks thrown into action, and the wire reeled in by the two assistants. The quartermaster, watching over the stern, regulates the speed of reeling in, and signals NAVIGATIONAL INSTRUMENTS 17 4 ' stop" when the sinker is up, hauls it on board, examines the arming, notes the character of the bottom, the brakes having been applied, and handles thrown out of action. The sounding tube is removed from the shield, applied to the scale and depth read, after which the arming of the lead is renewed and the tube dried preparatory to another cast. If the Tanner lead is used instead of a sinker, the shield (with tube) is inserted in the central tube of the lead before or after it is suspended over the stern. The Tanner machine is shown in Fig. 5. 18. Use of sounding data. It is often possible to ascertain, and frequently possible to verify, a ship's position by sound- ings. In thick and foggy weather, soundings may prove the only safeguard. The U. S. Naval Eegulations are very stringent in their requirements of a navigator on this subject. How- ever, great care must be exercised in trusting to soundings alone. Several taken at random will seldom locate a ship; in fact may, by misleading, invite disaster. When approaching land and on soundings, with no known marks in sight from which the position of the ship can be gotten, keep the lead going; note the nature of the bottom as evidenced by the arming of the lead, the depth and time of each sounding, and the course and distance to the next one. Take a piece of tracing paper or muslin sufficiently large to cover the area likely to include the ship's position during the runs considered; rule it with the meridians of the chart in use. Mark an assumed position in such a part of the tracing as to have sufficient room for the courses and distances from that position, write near it the depth and character of bottom at the first sounding. Lay down to the scale of chart the course and distance from the first position to that of the sec- ond cast, note as before the depth and character of bottom at that cast. Having a traverse of several positions and sound- 18 NAVIGATION ings, move the tracing up and down, and from side to side, keeping the ruled meridians parallel to those of the chart, until the soundings and character of bottom on the tracing correspond in close agreement with those on the chart. In this way a fair location of the ship on the chart may be got- ten, even though exact correspondence of data is not found. 19. Charts are representations of certain parts of the earth's surface upon a plane surface, in accordance with some one of several definite systems of projection, an effort being made to satisfy the conditions that any two distances from the center of the chart shall have the same ratio as the corresponding distances on the earth's surface, that the ratio of the areas of definite limits on the chart and the ratio of the same limits on the earth shall be the same, and that the ratio of all cor- responding angles shall be unity. However, since the earth is an ellipsoid, and the representation of it is on a flat surface, i t is evident there must be distortion, and the effort should be to make this a minimum. Charts are made primarily for the use of seagoing men, and show meridians and parallels of latitude, the details of the coast, light houses, life saving stations, mountains and promi- nent hills near the coast, soundings, dangers and shoals, nature of bottom, light ships, fog signals, buoys, beacons, tidal and current data, variation, etc., all of which may assist the navi- gator in making a successful voyage. Charts are divided into general, sheet, and harbor charts; the elaborateness of detail depending on the scale, and the character depending on the purposes to be served. General charts comprise an entire ocean, or a large part of it, or a considerable extent of coast line with adjacent waters. In addition to information referred to above, general charts should show the principal sailing routes. A sheet chart is a detached portion of a general chart and is made on a larger scale. It usually gives the information NAVIGATIONAL INSTRUMENTS 19 first referred to, and enables a navigator to use the channels for entering the bays and large harbors. A harbor chart is one of a harbor or a harbor and its ap- proaches; the curvature of the earth is not considered, but owing to the small extent included in the chart, there is no distortion. It is made by assuming an observation spot, the latitude and longitude of which are determined, measuring a base line, cutting in signals by sextant or theodolite, filling in the detail, and running cross lines of soundings. When the approaches to a harbor are of any extent, as for the U. S. harbors, the charts made by the Coast Survey are on the polyconic projection. 20. Systems of projection. There are three principal sys- tems of projection used in chart making, (a) Polyconic, (b) G-nomonic, (c) Mercator, a so-called projection by which rhumb lines appear as right lines on plane surfaces. 21. The polyconic projection. In this the earth's surface is developed on a series of cones tangent to the earth, a differ- ent one for each parallel, the parallel forming the base of the cone, the vertex of which is on the axis of the earth produced. The parallels of latitude are developed as arcs of circles, but being from different centers and with different radii they are not parallel; the meridians, except the middle one, are curved and converge toward the pole. The degrees of latitude and longitude are projected, prac- tically in their true length, in consequence there is no distor- tion at the middle meridian, and very little anywhere, if the limits of the chart in longitude are narrow. As the minutes of latitude are practically of the same length, one scale of distances may be used for any part of the chart. The geodesic line between two places (the shortest distance on the spheroid) will be projected practically as a straight line in its true length, but the loxodrome will be projected as 20 NAVIGATION a curved line and the true course will change from the begin- ning to the end of the voyage. For this reason practical sea- going people find it an inconvenient projection. Fortunately, however, the course may be taken as a straight line on a chart of large scale. Certain meridians and parallels are subdivided in different parts of the chart; and whenever it is desired to plot the ship's position, the subdivisions nearest to the position must FIG. 6. be used, and the same rule must be followed in taking off the latitude and longitude of a position. 22. Equations for the coordinates of a polyconic chart. If, as in Fig. 6, N is the normal terminating in the minor axis, and L the angle it makes with the major axis; a, the a, equatorial radius ; e, the eccentricity ; then N = {]L~~ G sin Jj) and r = N cot L is the slant height of the tangent cone and the radius of the developed parallel, the developed par- allel being a circle. Since in practice it would be incon- NAVIGATIONAL INSTRUMENTS 21 venient to describe the arcs with radii, they can better be drawn by constructing them from their equations,, and it will be found convenient to have x and y f the rectangular co- ordinates of a point, whose latitude is L and whose longitude differs from that of the middle meridian by n, expressed as functions of the radius of the developed parallel and the angle the radius makes with the middle meridian. Let 6 be this angle (Fig. 7), the origin being taken at L, the point of intersection of any parallel with the middle meridian; the middle meridian as axis of Y, the perpendicular through L as axis of X, then the coordinates of any point P whose latitude is L and longitude is n from the middle meridian, will be x = r sin = N cot L sin (1) y r (1 cos 6} = N cot L versin 6 (2) where is some function of n . To determine the relation of n and 0, it is only necessary to remember that the parallels are projected with their true length, in other words, "the distance LP, Fig. 7, equals the distance between L and P on the spheroid, measured on the parallel passing through L and P, therefore angles at the centers of the two arcs will be in inverse proportion to the radii, or N cot L X = N cos L X n> ; therefore, e = n sin L. (3) These three equations are sufficient to project any point of the spheroid given by its latitude and the number of degrees of longitude from the middle meridian. In tables prepared by the U. S. Hydrographic Office, the elements of the terrestrial spheroid, and the coordinates of curvature, x and y, are tabulated in meters. 23. Construction of a polyconic chart (Fig. 8). Draw a straight line LL' for the middle meridian; using Table IV of the projection tables referred to, take out for Lat. L from column D m (degree of Lat.) the distance which is laid off 22 NAVIGATION from L according to the scale of the chart. It is equal to Lm^ and locates the parallel for Lat. m x at the middle me- ridian; lay off m m 2 , taken out of the tables for Lat m , and locate Lat. m 2? continue this till the rectified arc of the me- DIAGRAMMATIC DRAWING OF A POLYCONIC CHART. Distorted for illustration. FIG. 8. x and y the coordinates for latitude L and for n of longitude each side of the middle meridian. ridian LL' is completed. Through the points thus found, draw the perpendiculars KLK, K v m^K^ etc., to represent the axis of X in each case. NAVIGATIONAL INSTRUMENTS 23 On these perpendiculars, set off to E. and W. of the middle meridian the abscissas x, and on lines at right angles towards the pole the ordinates y. The coordinates are taken from Table II, or Table III, of " The Projection Tables/' accord- ing to the detail required, and laid down, according to the given scale, for each parallel of latitude and each required longitude. The diagrammatic sketch of a polyconic chart, Fig. 8, will serve to illustrate the distortion at meridians removed from the middle one. The series of cones divides the surface to be projected into a series of zones, each zone tangent to those adjacent to it above and below 'only at the middle meridian, and separating from them to the eastward and westward. To complete the tangency and make the chart continuous, s'n 2 , s 2 n ly etc., should be stretched, so that the lower edge of the zone n 2 m 2 n 2 will coincide with the upper edge S 2 m 2 s 2 of the lower zone along a middle curve, even then producing slight distortion which would increase with the longitude of points from the middle meridian. 24. Gnomonic projection. In this system, the earth's sur- face is projected by rays from the center upon a plane tangent to the earth's surface at a given point, so it is apparent that all great circles will be projected as straight lines. The great circle track is represented as a straight line and for this reason such charts are often called great circle charts. Except when the point of tangency is at the pole, the parallels will be conies. The U. S. Hydrographic Office issues a series of charts to cover the various cruising grounds of the world, and on these are diagrams with full explanations for their use in finding the great circle course and distance between two points. The polar chart. The simplest form of the gnomonic chart is the polar chart, Fig. 10, in which the tangent plane is tangent at the pole; on such a chart great circles are 24 NAVIGATION straight lines, the meridians are right lines radiating from the pole, whilst the parallels are projected as circles whose center is the pole; though accurate in high latitudes, this projection would give a distorted chart for low latitudes. 25. Construction of a polar chart. In Fig. 9, let AB be the plane tangent at the pole 8, pp' the parallel to be pro- jected, p-ip\ will be the diameter of the projected parallel which will be a circle. Let R = earth's radius, x the radius FIG. 9. of projected parallel; therefore, x = R cot L. The radius of the parallel of 45 Lat., after projection, is R since cot 45 is unity, and this radius is called the radius of the chart; and to find the radius of any other parallel on the chart, we have x = R cot L, where R is the number of units of the scale in the radius of the projected 45th parallel. To find the length of a degree of longitude on any parallel we have 360 Eeferring to Fig. 10, let us construct a polar chart to com- prise the earth's surface from 45 N. Lat. to the pole. Par- allels at intervals of 5, meridians at intervals of 15, R = 36 NAVIGATIONAL INSTRUMENTS millimetres; taking cotangents of latitude to nearest third decimal place, we have the following values of x\ A POLAR CHART. FIG. 10. Scale : Radius lat. 45= 36 millimetres. For radius of 50, x 36 X -839 = 30.204 millimetres. 55, x = 36 X .700 = 25.200 " 60, x = 36 X .577 = 20.772 65, x 36 X .466 = 16.776 70, x 36 X .364 = 13.104 " 75, x 36 X .268= 9.648 " 80, x 36 X .176= 6.336 With these different radii and also R = 36 mm., draw the concentric circles, number the parallels properly, and with a 26 NAVIGATION ds dy dm protractor divide the outer circumference into 15 divisions, drawing radii through the points. Mark one of these me- ridians and the others as indicated on the chart; W. longi- tude to right, E. longitude to left. Use of the polar chart. The navigator, by drawing a straight line between the two required points, can see at a glance whether it is practicable to follow the great circle route ; take off, if desirable so to do, the latitude and longitude of the vertex, and of other points along the track, transfer them to the mercator chart, and then lay courses on the mercator chart from point to point of this transferred track. For instance, suppose it is de- sired to go by great circle route from a point off Sable Island, Lat. 45 K, Long. 60 W. to a point in the English Channel, Lat. 50 N"., Long. 0. Use the polar chart, Fig. 10. Draw the straight line A B and then PV perpendicular to it from P. The position of V which is the vertex, the point nearest the pole, is readily seen by inspection; or measure the distance PV in millimetres, divide it by 36, the scale of the chart, and the result is the natural cotangent of the Lat. of V. For the longitude measure with a pro- tractor the angle between the meridian of V and the one adjacent to it, applying the angle properly. 26. The loxodrome. Before taking up the subject of the mercator chart and its construction, it is desirable to con- sider mathematically the loxodromic curve and to find the equation expressive of the difference of longitude, reckoned from the point at which this curve intersects the equator in NAVIGATIONAL INSTRUMENTS 27 terms of any given latitude and the constant course C. The deduction following is taken principally from Walker's Navi- gation by permission of the author of that excellent treatise. In Fig. 11, let P be any point on the earth's surface, situ- ated on the meridian PE and on the loxodrome PQ, and let EE' be the equator. Denote the equatorial radius by a, the radius of the parallel of P by x, and the radius of curvature of the meridian PE at point P by p. Denote the latitude of P by L, its longitude by A, the course by (7, the earth's me- ridional eccentricity by e } the longitude at which the loxo- drome crosses the equator by A . If we let y represent the number of sea miles along the parallel of P included between the prime meridian and the meridian of P, and as the longi- tude of P is A, we shall have, when A is in minutes of arc representing sea miles. -^ = the circular measure of x a the angle between the planes of the two meridians. Let ds denote the rate of the point P along the tangent to the loxo- drome, and dm and dy be its rectangular components in the tangent plane at P. Then, Now, since the element of the terrestrial meridian at its intersection with any parallel of latitude is equal to the product of the radius of curvature and element of latitude at that point, in accordance with the principle that the radius of curvature varies inversely as the angle between consecu- tive normals, or, in this case, as the element of latitude, we have dm = P dL, therefore, tan C = 3fL = -^L (5) am apaL But from differential calculus we have the following well- known expressions for the properties p, x, and e, of the ter- restrial spheroid considered as an ellipsoid of revolution: 28 NAVIGATION a (1 e 2 ) _ a cos L ~~ (l-e 2 sin 2 Z)^ ~ (1 e z sin 2 c = 0.003407562 the compression of the earth, hence by substitution in (6), ill - a ^ an ^(1 g2 ) ^ m "(I e 2 sin 2 L) cos* By integrating (7) between the limits A and A, and L, we shall have A A , the difference of longitude required. = a tau g but r o cos i(l e 2 siu' Z) * (1 2 )cosirfi o coW(l-^ein^) (8) f [^^^^ - , \ L -^ os , Ld L "I (9) LJo cos 2 i Jo 1 'sm-'Lj cos 2 L Jo 1 sin 2 L T i i_ 1 4- sinZ 1 + sinL 1 sin Z/ 3 1 smL e cos L e cos L [ L _e^o^LdL_ _ ( L ,f ecosL e cos L \ ^^ Jo 1 e 2 sin 2 ^"~Jo \l + esinj6 1 e sin L) = log J+eBinL & 1 e Bin L hence } log = log tan + also l +eB inX = esinZ g ^Z +g 'sin'X + 5 1 e sin X 3 5 whence by substitution in (9), A _ ; = a tan (7 flog tan ^ + -^-] -.'(sin L + ** + ^^ + etc.)] (10) or if we put e sin L = sin <, we shall have, NAVIGATIONAL INSTRUMENTS 29 In this the logarithm is Naperian, so to reduce to common logarithms divide by the modulus m = .434294482 ; then in- troducing the value of a, the equatorial radius, 3437.74677 minutes of equatorial arc, or nautical miles, we have, A A = D = tan G ~ 7915. 704 flog u tec (^ + | -*lo gl . tan or D = 7915/704 tan log loT = ' (12) 27. The mercator chart, At the equator on the spheroid a degree of longitude equals a degree of latitude, but as the poles are approached the length of a degree of longitude becomes less, and finally zero at the poles, while the degrees of latitude undergo but slight change. On the mercator chart, which owes its origin to one Gerard Mercator, who lived in Flanders from 1512 to 1574, the meridians are drawn parallel to each other and perpendicular to a straight line representing the earth's equator, the distance apart on the chart being the distance between them on the spheroid, in minutes of arc on the equator, multiplied by the scale of the chart; thus the departure on the various parallels of latitude is increased and made equal to the difference of longitude. As a compensation, and in order to preserve the proportion that exists between degrees of latitude and longitude at differ- ent parts of the earth's surface, and to maintain the relative position and direction of objects charted, the infinitesimal divisions of a meridian in the latitude of any parallel must be increased in the same ratio as the departure on that par- allel. Eegarding the earth as a sphere this ratio would be as 30 NAVIGATION sec L to 1, though allowance is usually made for the merid- ional eccentricity. The series of parallels will, therefore, appear as a series of right lines parallel to, and at such in- creasing distance from the equator as to maintain the re- quired equality of angles and make the loxodromic curve a straight line. Let D denote the difference of longitude between the meridians marking the intersections of the loxodrome, first with the equator, and second with any parallel of latitude L; and let M denote the augmented latitude for latitude L on the chart, we then have, D = M tan C; but this D and this are the same ones that appear in equa- tion (12) above, therefore, tan M= 7915/704 log! Ei (13) The value of M in nautical miles for any latitude L is known as the meridional parts for that latitude. The merid- ional parts have been computed and are found tabulated in various works ; in Bowditch's Navigator and in " The Useful Tables" they are found in Table 3 from to 79 59', at intervals of V . The value of the compression used in com- P utin 8 Table3was 29085- Since the meridional parts for any latitude L are the num- ber of nautical miles, or 1' of longitude, in the meridional distance from the equator to the parallel of latitude L on the Mercator chart, the meridional difference of latitude for Lats. L and L' is the difference of meridional parts for those latitudes if of the same name, or the sum of the meridional parts for Lats. L and L' if of a different name ; in other words, NAVIGATIONAL INSTRUMENTS 31 it is the algebraic difference in either case, represented by m, or ra = M 2 ~ M ^ . If in equation (12) we regard e as zero, in other words, if the earth be considered as a sphere instead of as a spheroid, we shall have, D = 7915'.704 tan C Iog 10 tan f 45 + \ . (14) 28. Construction of a Mercator chart. The method of construction depends on whether the chart is to include the equator, and if so, the position of equator on the chart; and also whether the scale is to depend on the extent of paper in the direction of the meridians, or at right angles to them. By the term scale is meant the actual length on the chart of 1' of arc of longitude on the earth's surface. If the chart includes the equator, the values of M as taken from Table 3 are to be measured off directly from the equator in the proper direction. If the chart does not include the equator, then the lowest parallel to be represented on the chart is taken as the origin of parallels, and the distance from it to any other parallel is the meridional difference of latitude, as explained in the pre- ceding paragraph. If the extent of paper between the upper and lower parallel is limited, this distance is measured and divided by the merid- ional difference of latitude for the two parallels, and the result is the length of V of arc of longitude or the scale of the chart. Multiply this by 60 for the length of one degree, and the length of one degree by the number of degrees of longitude to be charted, to obtain the distance between the Eastern and Western neat lines or bounding meridians of the chart. In case the paper is limited in an East and West direction, draw a line near the bottom of the paper to represent the low- est parallel ; divide this line into as many equal parts as there are degrees of longitude to be represented on the chart ; then 32 NAVIGATION the length of one of these divisions divided by 60 gives the scale of the chart. This scale multiplied by the meridional difference of latitude for the parallel representing the origin and any other parallel L' will give the actual distance between the two parallels. FIG. 12. As an illustration, construct a Mercator chart to include latitudes 45 K to 49 N. and longitudes 141 W. to 145 W., scale 14.4 mm. = 1 longitude, and subdivide each degree of both latitude and longitude into six divisions of 10' each (see Fig. 12). NAVIGATIONAL INSTRUMENTS 33 In the center of the paper draw a vertical line to represent the middle meridian 143 W. Near the lower edge of the sheet erect a perpendicular to this line which is the southern inner neat line of the chart, or 45th parallel of latitude. From the intersection of these two lay off distances of 14.4 mm. and 28.8 mm. on the parallel, both Eastward and Westward, and through these points draw lines parallel to the middle me- ridian; the two outer lines will be the extreme meridional neat lines of the chart, or the meridians of 141 W. for the Eastern limit, and 145 W. for the Western limit. It now remains to locate and draw in the parallels; their distances from the origin of parallels is determined from the following self-explanatory table : Latitude. Mer. Pts., M. Mer. Diff. Lat. or m. Multi- plier. Distance to parallels in Millimetres. The Multi- 14.4 plier is -^-be- 45 3013.4 14.4 cause 1 =14.4 46 3098.7 85.3 60 20.472 ram. .*. 1' = 47 3185.6 172.3 m 24 41.328 14.4 48 3274.1 260.7 62.568 -- mm. Al- 49 3364.4 351. 84.240 ou ways check the work as indi- cated. Check 869. 2 x .24 = 208.608 On the right or Eastern neat line lay off the distances in mil- limetres, as shown in the table above, from the lowest parallel ; 20.472 mm. to the 46th, 41.328 mm. to the 47th, 62.568 mm. to the 48th, and 84.240 mm. to the 49th parallel. Through the points thus determined, rule right lines perpendicular to the meridians and these will be the various parallels required. Check the rectangularity of the construction by measuring the diagonals which should be equal. Draw the outer neat lines of the chart at distances desired, extend to them the meridians and parallels. Subdivide the degrees of latitude and longi- tude between the inner and outer neat lines by using proper- tional dividers or by a geometrical process. 34 NAVIGATION 29. Advantages and disadvantages of the different projec- tions. The polyconic chart has practically no distortion along the middle meridian, is well adapted to all latitudes, shows areas in their proper relation as to magnitudes, and permits the use of a single scale of distance anywhere. However, the meridians and parallels are curved, the rhumb line is curved, and there is distortion as the longitude departs from the mid- dle meridian. The gnomonic chart is useful simply for find- ing the great circle course and distance; for navigational pur- poses it is useful in high latitudes where the Mercator projec- tion fails. It gives a distorted idea of the earth's surface at points some distance from point of tangency of plane of pro- jection, and on it the rhumb line is curved. On the special form known as the polar chart the rhumb line is spiral. For navigational purposes the Mercator chart is by far the most convenient. The shapes of small areas are but little distorted; latitudes and longitudes may be laid down easily and accurately. The ship's track is a straight line, and the angle this line makes with any meridian is the course. How- ever, it cannot be used in very high latitudes advantageously, the expansion being too great. The relative areas of land or bodies of water in different latitudes cannot be compared by the eye. The first objection is obviated by using a polar chart for those regions, the second is unimportant to mariners. 30. Conventional notation, and hydrographic signs. Soundings are in feet or fathoms, as indicated under the title, and refer to the plane of mean low water for Atlantic Coast charts, that of the mean of lower low water of each tidal day for Pacific Coast charts, issued by U. S. Coast Survey. On British Admiralty charts the plane of reference is low water of ordinary spring tides. Upon harbor and bay charts of the United States, the con- tour lines, or lines of equal depth, are traced for every fathom NAVIGATIONAL INSTRUMENTS 35 up to three fathoms. Within the three-fathom mark the chart is shaded, the shading being lighter for each fathom; beyond the three-fathom line there is no shading. On section charts of the coast, contours of 10, 20, 30, and 100 fathoms are shown. Only the latter curve is given on large general charts. No bottom, for instance, at 50 fathoms, . Nature of the bottom. The material of the bottom is ex- pressed by capital letters, M for mud, G for gravel, S for sand, etc.; colors or shades by two small letters, yl., yellow, gy., gray, etc. ; other qualities by three small letters, as brk., broken, sml., small, etc. . A combination of these placed by a sounding shows at once the material, color, and nature. Buoys. These are indicated thus: B., black; E., red; H. S., horizontal stripes, black and red, danger buoy; V. S., vertical stripes, black and white, channel buoy. 1ST means a nun buoy, C a can buoy, S a spar buoy. On enter- ing a harbor, black buoys are left on the port hand, red on the starboard hand. Black buoys have odd, red buoys even numbers. Buoys with perch and square, or with perch and ball, are often found at turning points. There are also bell and whistling buoys, lighted (gas or electric) buoys, and white anchorage buoys. Yellow buoys are used to mark quar- antine grounds or stations. Dangers. Eock awash at low water, * ; sunken rock, +. Dangers of doubtful existence, marked E. D. ; if known, but of doubtful position, marked P. D. Anchorage, HK ; .a wreck, J^ or 1 1 1 ; light ship, T T Lights. Light houses are indicated by a yellow spot with a red or black dot, or as shown in Plates X and XI, end of book. Visibility is for a height of eye of 15 feet above the sea level. NOTE. Symbols on charts vary according to the origin of the charts. See Plates X and XI. 36 NAVIGATION Character of light. Indicated by abbreviations: Lt. F. W. A fixed steady light, white. Lt. Fig. E. Short flashes, longer intervals, color red. Lt. Int. R. Long flashes, short intervals, color red. Lt. Eev. W.- Intensity gradually increasing and decreas- ing, color white. Lt. F. and Fig. Combined fixed and flashing. Currents. These are indicated by feathered arrows point- ing in general direction of set, with figures to indicate drift in knots per hour; current flood, by an unfeathered arrow with one, two, or three cross marks for 1st, 2d, or 3d quarter of flow, with figures to indicate velocity in knots per hour; current ebb, as for flood, using an unfeathered half arrow; 31. Use of charts. Spread the chart out before you on the chart board with the North direction away from you ; in this way no readings will be upside down. In connection with the chart a navigator requires the use of a pair of parallel rulers, a pair of dividers, a sharp pencil, a reading glass, and sometimes a course protractor. The parallel rulers are used to transfer a course or bearing from the compass rose so as to pass it through a given point, or to transfer a line passing through a given point to the compass rose in order to ascer- tain the true or magnetic bearing or course; the dividers are used for taking off and measuring distances, whilst both are used in plotting or taking off the latitude and longitude of a point. To find the latitude of a place on a Mercator chart. Bring the edge of the parallel rulers to pass through the place par- allel to a parallel of latitude; where it cuts the graduated meridian on the chart's side is the latitude. To find the longitude. Bring the edge of ruler to pass through the place parallel to a meridian; where it cuts the graduated parallel at top or bottom of chart is the longitude. NAVIGATIONAL INSTRUMENTS 37 To plot a given latitude and longitude on a Mercator chart. Place edge of ruler along the parallel of latitude nearest given latitude, move ruler parallel to itself till edge passes through given latitude on the graduated meridian, hold it firmly to prevent slipping; with dividers take from upper or lower graduated margin the distance of given longitude from nearest meridian, and lay it off from the same meridian along the edge of the parallel rulers. Or, in the absence of dividers, with a pencil point draw a light line along edge of ruler across approximate longitude; then lay the ruler par- allel to the meridian, the edge cutting the longitude scale at the proper longitude, and cros^ the above line along the ruler's edge; the intersection is the plotted position. On a polyconic chart, positions are plotted, or taken off, less accurately; the graduated parallel and meridian of that graduated subdivision nearest the position being used. To measure a distance between two points on a Mercator chart. In whatever way the distance may run, take off the distance with a pair of dividers and measure it along the grad- uated meridian or latitude scale, so that the middle of the line will be in the middle latitude between the two points; for instance, on chart, Fig. 12, the line AB should be measured so that its middle point g will be over h. In case the distance runs E. and W. on a parallel, then the distance should be measured equally each side of the parallel; for instance, on the same chart as above (Fig. 12), the distance BC should be so applied to the latitude scale that its middle point would be over Tc. In case the distance is too great to be conveniently included between the points of the dividers, take with the dividers a convenient unit from the latitude scale so that the middle latitude will be about midway between the points of the dividers, then step off this unit along the distance to be measured, turning the dividers alternately to right and to left, counting the number of times the unit is contained in 38 NAVIGATION the distance. The unit, which may be 5, 10, or any number of miles, multiplied by the number of times it is stepped off, plus any fraction of the unit (measured in its own middle latitude) to the end of the line, will give the required distance. In making the above measurements the middle parallel is never drawn but is assumed by inspection. In measuring distances on a polyconic chart, reference is made not to the margins of the chart but to the single scale of distance under the title of the chart. To find the course from one point to another on the Mer- cator chart. Lay down the ruler so its edge passes through the two points, and draw a line if desired. Now move the ruler parallel to itself till the same edge passes through the nearest compass rose and read the course or bearing from the diagram. Or, having drawn the line in the first place, meas- ure with a protractor the angle it makes with any meridian. When the diagram is constructed with reference to the true meridian, its readings indicate the true course, otherwise the magnetic course. Exactly the same method of procedure is followed in find- ing the course on a polyconic chart, but, from the nature of the projection, it is evident that this straight line is not a rhumb line, and that the course must be changed after a time, on account of the angle between the meridians ; the length of the time depending on the general bearing between the points, on the distance, on the latitude, and on the scale of the chart. The course and distance run by a ship on the rhumb line from a given point being known, to find the ship's position on the mercator or polyconic chart. Place the edge of the parallel rulers so as to pass through the center of the com- pass rose and the reading of its circumference representing the course (true or magnetic). Move the ruler parallel to itself till the same edge passes through the given point. Draw a light line in the desired direction and lay off the NAVIGATIONAL INSTRUMENTS 39 distance run from the given point on this line, or, along the edge of the ruler, if the line is not drawn, and the ship's place is determined. The distance is taken from the proper scale as explained in previous paragraphs for both the mercator and polyconic charts. To plot the ship's position by cross bearings. Correct each bearing for the deviation of the compass due to the direction of the ship's head when bearing was taken ; the magnetic bear- ings are thus obtained. Place the edge of the parallel rulers over a magnetic compass rose, the edge passing through the center and reading of the circumference representing the magnetic bearing. Move the ruler parallel to itself till the same edge passes through the proper object, draw a light line through the approximate position of the ship. This line is a line of bearing and the position of the ship is somewhere on it. In the same way draw the line of bearing corresponding to the second object. The ship being on both lines will be at their intersection on the chart. To obtain good cuts, these lines should make angles not less than 30, the best cuts, of course, being given when the lines are at right angles to each other. If the compass rose is a true and not a magnetic rose, the bearings must be corrected for the variation as well as the deviation. 32. Correction of charts. 1 Charts, to be of any service, should be reliable, and to be reliable they must be kept cor- rected to date. The information for this purpose can be got- ten from " Notices to Mariners/' bulletins published weekly by the Hydrographic Office of the Navy Department, and the Light-House Board; also from the branch hydrographic offices at our important sea ports. 33. Arrangement and stowage of charts. The U. S. Hy- drographic Office issues to ships of the navy Hydrographic Office (H. 0.), Coast Survey (C. S.), and British Admiralty (B. A.) charts. Eegardless of the publication or chart num- 1 The U. S. Naval Wireless Telegraph Stations on the seaboard transmit daily at 6 a. m., 2 p. m., and 10 p. m., standard time, from the Hydrographic Office to vessels at sea, information as to obstructions that are dangerous to navigation. 40 NAVIGATION ber, all charts issued are arranged as far as practicable in geo- graphical sequence, numbered consecutively, and divided into portfolios, each portfolio containing about 100 charts. The consecutive numbers in each portfolio begin with the even hundred; a chart whose consecutive number is, for instance, 520, will be found in portfolio No. 5. General charts of the ocean will be found in portfolio No. 1. General charts of the station for the use of the commander-in- chief will be found in portfolio No. 10; those for the use of the wardroom officers in portfolio No. 11. Each portfolio should have a separate drawer, in a nest of drawers, built in the pilot house and convenient to the chart table. 34. The three-arm protractor. In determining the position of a ship by sextant angles between known objects along a coast, the three-arm protractor will prove itself an invaluable instrument. It consists of a graduated brass circle having three arms, the straight edges of which all pass through the center of the circle. The center arm is fixed and the zero of graduation is coincident with its straight edge. The other two arms are movable and both are fitted with clamp screws and tangent screws. As the movable arms turn away from the central arm, the angles gradually increase, and when the arms are clamped, a vernier, with reading microscope, gives the angle to the least count of the vernier. Extension pieces are provided for each arm. It is impossible to shut the right arm close home, as the beveled straight edge of the fixed arm is on its left side, so if the" right arm can not be set for a small right observed angle, set the left arm for it; then swing the right arm around and set it for the sum of the two observed angles, reading from zero to the left. To plot a vessel's position with a three-arm protractor. Select three objects that can be seen and reflected, that are well located on the chart, and so situated with reference to each other that the observer's position will be well determined. NAVIGATIONAL INSTRUMENTS 41 Get simultaneously the angle between the middle object and the right one (called the right angle), and the angle between the middle object and the left one (called the left angle). The lateral arms of the protractor having been set to their proper angles, and the same verified, the instrument is placed on the chart, the edge of the central arm passing through the The Three-Arm Protractor. middle object and kept there whilst the instrument is moved around till the edges of the lateral arms also pass through their respective objects. The center of the instrument is at the point of observation which is lightly marked upon the chart by pencil or the spring point of the center punch. Tracing paper or linen with angles laid off and properly numbered may be used as a substitute. The diagram (Fig. 13) will illustrate the different cases 42 NAVIGATION that may be met with in practice. A, B y C are the three ob- jects forming the triangle called the great triangle, the circle through which is called the great circle. The position of the observer is at the intersection of the circles of which the sides of the great triangle are chords, the position of the centers of these circles, and hence of their intersection, depends on the observed angles. The nearer these secondary circles inter- sect at 90, the better the " fix/' In cases in which the cen- ters of the circles are near each other, and near the center of the great circle, the position is more or less indeterminate, and such angles are called " revolvers." NAVIGATIONAL INSTRUMENTS 43 CASE 1. The two angles observed are >180; the position of observer is within the triangle and is well determined. CASE 2. The sum of the two angles = 180 ; the observer is on one side of the great triangle, and the position is well determined. CASE 3. A range and one angle; a good determination of position. CASE 4. The middle object is nearer than the other two; the position can be determined very well, but A should not be so close as to make angles too small, small angles making position uncertain. CASE 5. Using three objects in line or nearly so, as in the case of objects 5, A', and 0. An excellent arrangement; the larger the angles, the more reliable the " fix/' CASE 6. Where the sum of the observed angles is the supplement of BAG; the position is indeterminate as it may be anywhere on the great circle. CHAPTEE III. NAVIGATIONAL INSTRUMENTS. The Compass and Pelorus. Compass Error. Theory of Deviations. SECTION I. 35. The mariner's compass is one of his most important and essential instruments, showing him how he is steering, enabling him to direct his ship on a desired course, or to get bearings of objects in sight from which to determine his po- sition. It consists essentially of a needle, or a series of needles, of strong and powerful magnetism, attached to a properly graduated card which is mounted at its center on a pivot in the center of the compass bowl, and has free movement in the horizontal plane. The bowl is made of copper, hemispheri- cal in shape, is heavy as well as ballasted, and swings on knife edges in gimbals, thus enabling the card to maintain a hori- zontal position even in a seaway. Inside the bowl are painted two vertical black lines 180 apart, the one towards the head of ship being called the lub- ber's line. The bowl is so mounted that a line through the pivot and the lubber's line is parallel to the keel line of the ship, so that this lubber's line indicates the course, or the direction of the ship's head per compass. 'The compass card is divided into 360; the graduation be- ginning with at North runs around to the right and is numbered at every fifth degree. The card is also divided into points and quarter points (see Appendix E). NAVIGATIONAL INSTRUMENTS 45 The two general classes. There are two gerieral classes of compasses in use, the dry and the liquid. In the latter, the bowl is filled with liquid which, together with the hollow card, gives a certain amount of buoyancy to the card and hence regulates its pressure on the pivot and ease of movement, and also, through its inertia, tends to prevent or reduce vibrations due to the ship's motion. The liquid compass is used in the U. S. Navy, and, accord- ing to the purposes it serves, a compass is designated as service, conning tower, or boat compass. Though made on the same general principles, they embody different degrees of excellence and have cards of different sizes. The service com- pass has a card 7-J inches, conning tower 5 inches, boat com- pass 4 inches in diameter.* The service comDass is further designated, according to its use and location on board, as standard, steering, manoeuvring, battle, auxiliary battle, top and check compass. Location of standard. The standard is the compass by which the ship should be navigated, all others being regarded as auxiliaries, as for the use of helmsmen, etc. It should be placed in the midship line of the ship, at a position where the mean directive force is a maximum, if pos- sible; as far removed as practicable from considerable masses of iron, especially if vertical, the influences of the dynamos or electrical currents, stands of arms, or other iron or steel subject to occasional removal. It should be mounted at least five feet from an iron deck or beams, in a compensating bin- nacle, easily accessible at all times, conveniently near the steering compass, and so located that all around bearings of land or heavenly bodies can be observed. 36. The service or 7^" liquid compass. This compass con- sists of a tinned brass skeleton card 7-J inches in diameter. It is of a curved annular type, the outer ring convex on the upper and inner side, graduated to read to quarter points, * The compasses in submarines are of special types, usually furnished by the con- tractors to suit the special conditions. As a rule they are transparent and set in the deck so as to be read either from inside or outside of the boat, reflecting prisms and lenses being used where necessary. 46 NAVIGATION with the outer edge divided to half degrees, and figured at each fifth degree from at North, numbering to the right through 360. The card has a concentric spheroidal air vessel, to assist in giving buoyancy to the card and magnets, so that the pressure on the pivot at 60 F. will vary between 60 and 90 grains. The air vessel has a hollow cone, open at the lower end, carrying a sapphire cap at the apex, by which the card is supported on the pivot. The magnets, four in number, consist of cylindrical bundles of steel wires, each .06 of an inch in diameter, strongly mag- netized, put into a sealed cylindrical case and secured to the card parallel to its North and South diameter. The cases of two of the magnets, each magnet 5J" long, pass through the air vessel to which they are soldered, and have their ends se- cured to the bottom of the card ring, like ends on chords of nearly 15 passing through their extremities. The other two cases containing magnets, each 4f inches long, are placed par- allel to the longer magnets, on chords of nearly 45 of a circle through the extremities, and the ends are secured to the bottom of the card ring. The card is mounted in a bowl, made of cast bronze, on a bell-metal pivot fastened to the center of the bottom of the bowl by a flanged plate and screws. Through this plate and the bottom of the bowl are two small holes which communicate with a metallic self-adjusting expansion chamber located just beneath the bowl. These holes permit a circulation of liquid between the bowl and expansion chamber, and it is the func- tion of the latter to keep the bowl full of liquid without show of bubbles, or undue pressure that might be caused by change in the volume of the liquid due to changes of temperature. The liquid used is composed of 45 per cent pure alcohol and 55 per cent distilled water, and remains liquid at a tempera- ture lower than 10 F. The inside of the bowl is painted white with a paint insoluble in the above liquid. An enam- eled plate is secured on the inside of the bowl, and on this plate a lubber's line is drawn. The bowl is fitted with a glass cover, the edge of which is NAVIGATIONAL INSTRUMENTS 47 closely packed with rubber, completely preventing leakage or evaporation of the liquid, which at all times fills the bowl. The rim of the compass bowl is made rigid and its outer and upper edges turned accurately, that the service azimuth circle when in use may be properly seated. The bowl has a false bottom containing a leaden weight as ballast to keep the bowl horizontal. FIG. 14. United States Navy Standard Compass. As made by E. S. Ritchie & Sons. The compass is mounted in gimbals with knife-edge bear- ings in its binnacle.* 37. The IT. S. Navy standard compensating binnacle. The binnacle stand includes, in a single brass casting, the circular base, cylindrical pedestal, conical magnet chamber, cylindrical compass chamber, and the graduated arms for the quadrantal correctors at right angles to the keel line of the binnacle (see Fig. 15). * A new compass, having both card and bottom of bowl transparent, with electric illumination from below, is to be issued. The 5%" card is much smaller than bowl. The lubber's point is a pointer from side of bowl to card. Besides expansion chamber and weight, the bowl has an annular ring containing oil as an oscillation damper. 48 NAVIGATION The hood is spun of stout polished brass, has a hinged plate- glass front opening upwards, and a sliding door opposite the glass to permit bearings to be taken in wet weather without removal of the hood. Clips on the binnacle take over the rounded edge of the binnacle hood and hold it on. The hood carries in its center a lamp provided with a prism for reflect- ing on the compass dial the light of a 5 c. p. electric light. The plug for this light is in the pedestal below. The rectangular method * of compensation is used in cor- recting the semicircular deviation. The correcting magnets are mounted on trays which can be raised or lowered, indepen- dently of each other, by a screw moved by beveled gears, and so constructed that they will pass each other in any position, the mechanism permitting an extent of travel of 12 inches. The semicircular magnets are held in their receptacles by a spring-closing device, each carrier or tray having a group of three magnets each side of the vertical axis, making six in all, the tubes being horizontal, one over the other. The quadrantal correctors are removable soft iron spheres, secured to the brackets by screw bolts, the centers of the spheres lying in the same horizontal plane as the compass needles. They are capable of motion towards or from the compass, the distance from whose center is indicated by a scale in inches and quarter inches on each arm. The heeling corrector consists of a cylindrical magnet having a hook in each end to which is attached a chain. Cen- trally, in the vertical axis of the binnacle, there is a hollow brass tube extending the entire depth of stand from the bot- tom of compass chamber. By removing the compass, the heeling corrector attached to its chain may be lowered into this tube, and held at the proper height by the chain which passes over a roller at the top of the tube and secures to a cleat, or by a set screw in the magnet chamber. * By fore and-aft and athwartship magnets (Art. 81, 82.) NAVIGATIONAL INSTRUMENTS 49 38. The gyro-compass. This consists essentially of a gyroscope, its axis being horizontal when started, mounted below an annular or ringed-shaped bowl of mercury, in which the float carrying the gyroscope, with compass card attached above, is suspended. The bowl is supported by gimbal rings so hung within the binnacle as to prevent vibration. The gyroscope being spun to about 20,000 revolutions per minute by a motor, the rotor of which is the gyro, gives a very strong directive force. Briefly stated, this arrangement is first a pendulum tending to hang vertically, and second, when the gyro is rotating, it is a gyroscope tending to maintain its fixed plane of rotation. It is the combination of these two efforts that causes it to point to the North. This may be readily understood by considering its action as follows : The gyro being set in motion with the pendulum hanging vertically, there is at first no force acting to turn it from its original plane of rotation. But should the axis of rotation be in any other direction than North and South, suppose East and West, the rotation of the, earth would in a few minutes put the pendulum out of plumb, provided the gyro were able to retain its original plane of rotation. The pull of gravity endeavoring to bring the pendulum vertical, and the pull of the gyro endeavoring to retain its original plane, cause a new movement about a vertical axis, which turns the axis of rotation of the gyro into the meridian. Being independent of the magnetism of both earth and ship, the gyro-compass is well adapted to use in locations behind armor, in close proximity to magnetic material and stray magnetic fields, and hence its special use is as a battle compass. By means of proper transmitting mechanism, the indications of a " Master Compass " located, say in the central station, or wherever most convenient, without regard to surrounding mag- netic conditions, may be shown by " Repeater Compasses " elec- trically connected with it and installed in the conning tower, plotting room, steering engine room and fire-control tower. The gyro-compass must be compensated in any given latitude for deviations caused by speed and course, also for errors due to marked changes of latitude. Tables giving the correction in each case are supplied with the instrument which is fitted with a movable lubber's point for compensating purposes. As the frictional resistance to rotation about a vertical axis is made a minimum, the axis of the gyro does not of itself steady on North, but oscillates each side of the meridian. A special device is used to damp the oscillations, which is not effectually done in less than one hour and a half after first starting. This fact and the further fact, that on a marked change of course, the gyro does not settle down for several minutes on the new course, are the greatest drawbacks to its use as a battle compass. However, for all purposes of navigation, when it is not necessary for the personnel to remain behind armor, the magnetic compass, mounted in the open and properly compensated, is superior. FIG. 15. U. S. Navy Standard Compensating Binnacle. As made by Keuffel & Esser Co. NAVIGATIONAL INSTRUMENTS 51 39. Azimuth circle. This consists of a composition ring turned true to fit the compass bowl. At one extremity of a diameter is a curved mirror hinged to move around a horizon- tal axis and facing at the other extremity a prism completely encased in a brass case except for a narrow vertical slit. The sun's rays reflected by the mirror upon the slit appear as a thin pencil of light on the graduations of the card circle. A level on the rim serves to show when it is horizontal. FIG. 16. U. 8. Navy Azimuth Circle. (To fit Navy Standard Compass No. 1.) As made by E. S. Ritchie & Sons. A second set of vanes, at the extremities of a diameter at right angles to the first, is used for direct bearings of distant objects, of stars,- or of the sun when partially obscured. At one end is a mirror reflecting the image of the sun or star, and a prism reflecting the card circle and vane simultaneously to the eye at the other end (Fig. 16). 40. The pelorus (Fig. 17). This is an instrument located on board ship at some point where a clear view can be ob- tained for taking bearings. It is most convenient to have one at each end of the flying bridge. It consists of a circu- NAVIGATION lar plate mounted in gimbals whose knife edges rest in the Y's of a vertical standard rigidly secured in place. The circular plate has a raised flange on its periphery with heavy marks 90 apart to correspond with the fore-and-aft and athwartship lines of the ship. Concentric with the plate and each other are a dial plate and an alidade, each capable of independent movement in azi- muth. The dial plate is simply a dumb compass card of metal, graduated to quarter points and single degrees, whose upper surface is flush with the raised periphery, and is provided with a clamp. The alidade is fitted with a level, folding sight vanes, hinged re- flector, and a sliding peep sight with neutral glass. The line of sight through the vanes passes through the vertical axis of the instrument and is indi- cated on the dial plate by an in- dex at each end of the alidade. The alidade is also fitted with a clamp. A heavy balance weight is attached to the lower center of the plate. It may be used to eliminate the compass error from observed bearings by setting the alidade to a reading which is the compass course corrected for the error, and, as long as the ship is on that particular heading for which the dial is set, all bearings by the pelorus will be true; if correction is made for deviation only, then the bearings will be magnetic. FIG. 17. PLATE I. THE ILLUMINATED DIAL PELOKUS. PLATE II. TUB ILLUMINATED DIAL PELORUS- CHAMBER AND BOWL. 54 NAVIGATION 41. Illuminated dial pelorus (Plates I and II). The pelorus standard consists of three parts, the base, supporting column, and pelorus chamber; on the top edge of the latter are scored the fore-and-aft and athwartship marks. The pelorus bowl is mounted on two trunnions in a gimbal ring which pivots on the athwartship diameter of pelorus chamber. The pelorus bowl is built up of a top bowl ring and a shell of sheet brass. A seat is turned in the top bowl ring to re- ceive the assembled pelorus card, the principal part of which consists of a disk of clear plate glass 9 inches in diameter by 3^-of an inch thick. The card is graduated in degrees near its outer edge, every fifth degree accentuated, and every tenth degree is marked in figures. Graduations run from at N". to 360 around to right, and, inside these degree marks, the card is graduated to quarter points. An azimuth circle marked in degrees is permanently secured to the bowl ring. The pelorus card has unobstructed rotation except when clamped by a clamping screw, which is at the 90 graduation of the azimuth circle. There is an alidade capable of free revolution either way, or of being securely clamped ; it is pro- vided with a level, folding sights, hinged reflector, and peep sight. This pelorus is designed to be used at night without a navigator's lantern, the transparent dial being illuminated by an electric light placed beneath it in the standard. 42. The use of pelorus to determine a magnetic heading. To place the ship's head on any magnetic point by the pelorus : (1) With the known latitude of place and declination of body say the sun find from the azimuth tables the sun's true bearing for certain selected local apparent times. From these true bearings find the sun's magnetic bearings for the same times by applying the variation of the locality, easterly variation being applied to the left, westerly variation to the right of the true azimuth. (2) Shortly before the earliest time selected, set that point COMPASS CORRECTIONS 55 of the pelorus corresponding to the magnetic heading desired on the forward keel line or indicator and clamp the plate ; set the sight vanes to correspond with the sun's magnetic bearing at the selected time and clamp the vanes to the plate. (3) By the use of engines and helm, bring the sight vanes on the sun and keep them there, being careful not to disturb the clamps of plate or vanes, noting at the instant of the selected local apparent time the heading per compass. That heading per compass corresponds to the magnetic direction desired. SECTION II. 43. Compass corrections. The compass needle seldom points to the true North, so in order to obtain a true course or a true bearing, certain corrections must be applied to the compass course or the compass bearing, as the case may be. They consist, according to circumstances, of one or more of the following; i. e., variation, deviation, leeway. Each of these terms will be explained at the proper time. 44. The earth's magnetism. The earth is a huge, natural but irregular magnet, having a resultant pole in each hemi- sphere which, however, are not coincident with the geograph- ical poles, and a magnetic equator which does not coincide with the geographical equator. The North magnetic pole is approximately in Lat. 70 N, Long. 96| W, the South mag- netic pole in Lat. 73J S, Long. 147J E. Eecognizing the laws of attraction and repulsion between two bar magnets, and the analogy that exists between the magnetic character of the earth and a bar magnet, it is evident that the mag- netism of the North magnetic pole is of an opposite kind to that of the North-seeking end of a magnetized needle; there- fore, if we regard the magnetism of the North seeking end of the needle as North magnetism, we must consider the North magnetic pole as having South polarity and the South mag- netic pole as having North polarity. 56 NAVIGATION However, physicists do not agree as to which shall be called North magnetism, that of the North-seeking end of the needle or that of the North magnetic pole, so it is convenient to dis- tinguish them by colors, calling the first red and the second blue. The general effect of the earth's magnetism is to draw the North end of the needle towards the North and the South end towards the South ; but, with the exceptions noted further on, a freely .suspended magnetized needle, affected only by terrestrial influences, generally speaking, neither points in the direction of the true meridian, nor lies in a horizontal plane, nor occupies the same relative position in two different places. Line of force and dip. The direction in which the needle does point at any given place is the " line of total magnetic force " at that place, the inclination of which below the hori- zontal plane is called the " magnetic dip." The line of total force is spoken of as the " line of force." The magnetic poles. Those two positions at which the line of total force is vertical are known as the magnetic poles ; a freely suspended magnetized needle would be vertical at the poles, with the North end down at the North magnetic pole and the South end down at the South magnetic pole. The magnetic equator. The line joining all those posi- tions on the earth's surface at which the " line of force " is horizontal is known as the magnetic equator, which is to the northward of the geographical equator in the Indian Ocean and the western half of the Pacific Ocean, and to the south- ward of it in the Atlantic and eastern half of the Pacific Ocean. As we have magnetic poles and a magnetic equator, analo- gously we have magnetic latitude; all points of the earth in North magnetic latitude have South or blue magnetism, and all points in South magnetic latitude have North or red magnetism. THE EARTH'S MAGNETISM 57 The dip increases from at the magnetic equator to 90 at the magnetic poles; the "total force" increases from a minimum at the magnetic equator to a maximum at the " magnetic foci," of which there are two in each hemisphere, located about in 52 N., 92 W.; and 70 N., 120 E.; 70 S., 145 E. ; and 50 S., 130 E. The total force at the foci is between two and three times that at the magnetic equator. The magnetic meridian. The magnetic meridian of any place is that great vertical circle in the plane of which the " line of force " lies. The variation. Excepting at points along two lines, one at present passing through Brazil and the eastern part of the United States, and the other through Australia, the Arabian Sea, and the Black Sea, called "lines of no variation," the magnetic meridian nowhere corresponds with the true merid- ian, but inclines to the East or West of it, making with it at a given place an angle called the " variation " at that place. As a compass needle is constrained by its mode of suspension to move only in a horizontal plane, variation may be defined as the angle through which the compass needle is deflected from true North by terrestrial magnetism alone. 45 Elements of the earth's magnetism. The distribution of the earth's magnetism at any place may be indicated by its three elements : (a) The variation. (b) The dip. (c) The total force or magnetic intensity. (a) and (c) are found by means of the magnetometer and (b) by the dip circle, but for the purpose of representing the amount and direction of the earth's force on the needle, in- stead of considering the total force on the needle, it is more convenient to consider the components of that force, viz. : (1) The horizontal force, or that component in the direc- tion of a tangent to the earth's surface, and in the plane of the magnetic meridian. 58 NAVIGATION (2) The vertical force, or that component acting down- wards and at right angles to the above. Relation of dip and the forces of the earth's magnetism. Letting 6 = the magnetic dip, T = the total force, H = the horizontal force, Z = the vertical force, we shall have (Pig. 18), HT cos 0. Z T sin 6. Magnetic Meridian FIG. 18. 46. Charts of the earth's magnetic elements. The TJ. S. Hydrographic Office (also corresponding offices in foreign countries) issue variation charts, charts of magnetic dip and also of the horizontal intensity of the earth's magnetism. The tide tables, issued annually by the U. S. C. and G. Survey, give the variation at most of the world's seaports. The variation chart. This shows by lines of equal value, called isogonic lines, drawn at convenient intervals, the amount and direction of the variation over the surface of the globe. Generally speaking the variation is westerly over the Atlantic Ocean, the Mediterranean, and the Indian Ocean ex- cepting the Bay of Bengal ; easterly over the Bay of Bengal, the Pacific Ocean, the Gulf of Mexico, and the Caribbean Sea. The chart of magnetic dip. This shows by lines of equal RELATION OF MERIDIANS 59 value, called isoclinal lines, drawn at intervals of one degree, the magnetic dip over the surface of the globe. In the re- gions of northerly dip, where the North end of the needle is drawn downward, these lines are full; in the regions of southerly dip, where the South end of the needle is drawn downwards, they are broken lines. The annual rates of change of dip are expressed in minutes of arc by numbers in the re- gions where they are placed ; a plus sign indicating increasing, a minus sign decreasing dip. Taking the horizontal force at the magnetic equator as unity, the increase of dip with magnetic latitude, as shown by this chart, is approximately in accordance with the formula tan dip = 2 tan mag. Lat. The chart of horizontal force. This shows the horizontal intensity expressed in C. G. S. units by lines of equal value. An inspection of this chart will show that the horizontal force is a maximum near the magnetic equator and dimin- ishes as we approach the magnetic poles where it is zero. Since the horizontal intensity is the directive force on the needle, it is plain that a disturbing influence would have greater effect on the needle when the value of H is less and vice versa ; and, therefore, at places in high magnetic latitudes, the needle is less reliable than at places near the magnetic equator, when subjected to the same influences antagonistic to that of the earth. As a knowledge of the value of H in the locality of the ship is often necessary in compass work, the navigator will find this chart a useful one. With the value of H from this chart and the value of from the chart of magnetic dip, the value of Z ', the vertical component of the earth's total force at a place, may be found. Charts of Z may also be used. 47. Relation of true and magnetic meridians. From what has preceded, it is plain that a compass needle, constrained by mechanical arrangements to move in a horizontal plane 60 NAVIGATION and installed on board a perfectly stable and non-magnetic ship, will lie in the magnetic meridian, at an angle called the variation with the true meridian, to the eastward or westward of it, depending on the geographical position, and "will pos- sess a directive force depending on the magnetic latitude. The variation is not only different in different localities, ex- cept at those places on the same isogonic line, but it is differ- ent in the same locality at different times, owing to a small but gradual and constant motion of the magnetic poles. The " line of no variation/' which now passes through the Arabian and Black seas, was a little to the westward of the PIG. 19. FIG. 20. Azores in 1492, and it is recorded that Columbus, on his westward voyage, noted the change in the compass bearing of the pole star; the needle at first pointing to the eastward of the pole star, then directly at it, and finally to the westward of it as the voyage progressed. Besides the regular and periodic annual change, the needle is subject to a slight diurnal change, moving gradually back and forth through a very small arc. Variation is shown on all navigational charts at the com- pass rose and also by isogonic lines. All magnetic courses and bearings are estimated from the magnetic meridian, but it is often desirable and necessary to find the corresponding angles from the true meridian, which EULES FOR VARIATION 61 may be found by applying the variation of the place to the magnetic course or bearing in the proper way. In Figs. 19 and 20, let ON be the true meridian; OM the magnetic meridian ; V = NOM the variation of the place ; OH the keel line or direction of the ship's head ; then NOH is the true course and MOH the magnetic course, considered positive to the right. In Fig. 19 the North end of the needle is drawn to the eastward and The true course = the magnetic course -f- variation. In Fig. 20 the North end of the needle is drawn to the west- ward and The true course = the magnetic course variation. 48. Rule for naming variation. Mark it East (E) or +, if the North end of the needle is drawn to the right, the observer considered as at the center of the compass and looking in the direction of that end of the needle; mark it West (W) or , if the North end of the needle is drawn to the left, observer as before. When the North point of the needle is drawn to the right, the magnetic meridian is to the right of the true meridian, and the magnetic bearing of a fixed object is to the left of its true bearing by the amount of the variation; similarly when the North point of the needle is drawn to the left, the mag- netic meridian is to the left of the true meridian, and the magnetic bearing of a fixed object is to the right of its true bearing by the amount of the variation. Rule for applying variation. Hence, when applying varia- tion to magnetic courses or bearings to obtain the correspond- ing true courses or bearings, looking from the center of the compass toward the compass rhumb, apply variation to the right when E. or + ; to the left when W. or . Or, if the magnetic course or bearing is in degrees, add the variation if E. or +, subtract it if W. or . To find the magnetic courses (or bearings) from the true courses (or bearings) do the reverse. 62 NAVIGATION EXAMPLES, VARIATION EAST. 49. Given the following- magnetic courses, to find the cor- responding true courses : Magnetic Course . 85 137 230 353 Variation +15 +15 +15 +15 True Course 100 152 245 8 EXAMPLES, VARIATION WEST. Given magnetic courses, to find the true courses: Magnetic Course . 85 137 230 3 Variation 15 15 15 15 True Course 70 122 215 348 50. local attraction. There is cause of disturbance of the compass needle, when the ship is in certain localities, due to the fact that the mineral substances in the land under the water possess magnetic properties, especially in shallow waters of volcanic regions. Well-authenticated observations show that the navigator must be on his guard against the dangers of this attraction on the coasts of Iceland, off Cape St. Francis, in Odessa Bay, off the coast of Madagascar, off the volcanic islands near Java, at the Isles de Los, and especially near Cossack in North Australia. 51. Deviation. So far we have considered the compass as if installed on board an absolutely non-magnetic ship, and as affected only by terrestrial magnetic influences, with variation as its only correction or error. However, when that same compass is mounted on board an iron or steel ship, it is sub- ject to further error in its indications. Besides having a directive force in the magnetic meridian given it by H, the earth's horizontal intensity, the North end of the needle is acted upon by the general magnetism induced in the iron or RULES FOR DEVIATION 63 steel of the ship by the earth's inducing forces, with the result that the needle assumes a resultant direction, the angle be- tween which and the magnetic meridian is called " deviation/' When the deviation is zero, the ship's force acts in the magnetic meridian, increasing or diminishing the earth's di- rective force. The deviation of a compass varies with the ship in which it is mounted ; varies according to position in the same ship ; and for a given position, under like circumstances, varies in amount and direction according to the heading of the ship. It also varies with change of ship's position on the earth's surface. For these reasons the deviations of every compass, mounted on board and used for navigating or steering the ship, should be determined for every 15 compass rhumb at a time when the vessel is on an even keel, in her normal sea- going condition, with projectiles, guns, davits, cranes, re- movable masses of iron, etc., secured as if for sea. The deviations of the standard compass, which alone must be used for navigating the ship, should be tabulated and a corrected copy of the table should be kept on deck for the use of the officer of the deck and the navigator. Such a table is needed for finding the magnetic course from the compass course steered; for correcting the compass bearings of fixed objects on shore; or for obtaining the compass course to be steered to make good a certain magnetic course. 52. Rule for naming deviation. Mark it East (E) or + if, under the influence of the ship's magnetism, the North end of the needle is drawn to the eastward, or to the right of the magnetic meridian; mark it West (W) or if the North end of the needle is drawn to the westward, or to the left of the magnetic meridian. When the North point of the needle is drawn to the right, the observer at the center and looking in the direction of that point, the compass meridian is to the right of the magnetic 64 NAVIGATION meridian and the compass bearing of a fixed object is to the left of its magnetic bearing by the amount of the deviation; similarly when the North point of the needle is drawn to the left, the compass meridian is to the left of the magnetic merid- ian, and the compass bearing of a fixed object is to the right of its magnetic bearing by the amount of the deviation. Kule for applying deviation. When applying deviation to compass courses, or bearings, to obtain magnetic courses, or bearings, looking as if from the center of the compass toward the compass rhumb, apply the deviation, due to the ship's heading at the time, to the right when E. or -\- ; to the left when W. or . Or, if the compass course or bearing is in degrees, add the deviation if E. or -)-? subtract it if W. or . To find the compass course from a given magnetic course do the reverse. In this connection, attention is particularly called to the fact that all bearings are to be corrected for the deviation due to the direction of the ship's head at the moment they were taken. 53. Compass error. When variation and deviation are to be allowed for at one time, add them algebraically, giving the name of the greater to the result which is known as " compass error," generally written C. E. To obtain the true course or bearing from a given compass course or bearing, apply the C. E. to right if E. or + ; to the left if W. or , looking from the center of the compass toward the compass rhumb. Or, if the compass course or bearing is in degrees, add the compass error if E. or -(-, subtract it if W. or . To obtain a compass course from a given true course do the reverse. For office work and in examples similar to 4, 5, 6, and 8 the signs -f- and are' preferable to the terms E. and W. The use of the latter, however, will be illustrated in examples 1, 2, 3, and 7. Example 1. Given Var. = 13 W, Dev. on North (p. c.) COURSE CORRECTIONS 65 2 E, on NE (p. c.) 1 W, and on East (p. c.) 4 W; find the true courses corresponding to the above compass courses. Var. 13 W Course (p. c.) North Dev. 2 E C. E. 11 W C. E. 11 W True Course 349 Var. 13 W Course (p. c.) 45 Dev. 1 W C. B. 14 W C. E. 14 W j True Course 31 Var. 13 W Course (p. o). Dev. 4W C. E.17W C. E.1,W True Course 73 54. Leeway. With sailing ships, the wind, besides driving the ship in the direction of her keel, frequently forces her bodily to leeward, so that the course through the water is to leeward of the one steered. This angle between the course and the direction the ship is actually moving, as indicated by the ship's wake, is the leeway. Being always from the wind, as a correction it is marked East when the ship is on the port tack, West when the ship is on the starboard tack. Here also, East is + and West is . Given a course (p. c.), to find the true course. Ex. '2. A schooner sails 28 (p. s. c.), Dev. 6 E, Var. from chart 21 E, wind SE, leeway 11. Find the true course. Var. 21 E. Course (p. c.) 28 Dev. 6 E. Correction 16 E. Leeway 11 W. True course 44 Correction 16 E. Given the true course, to find the compass course. Ex. 3. The true course to destination from the ship's posi- tion is 22 30', Var. 15 W, Dev. 6 E. The ship will be on the port tack, probable leeway 6. Eind the course to be steered. Var. 15 W. True course 22 30' Dev. 6 E. Reversed correction 3 E. Leeway 6 E. Compass course 25 30' Correction 3 W. 66 NAVIGATION The correction in this example being 3 W is applied the reverse way, or, as if it is easterly. The word correction is used here because leeway is not an error of the compass. Strictly speaking variation is not an error and cannot be compensated for; deviation only is an error. SECTION III. 55. Finding the deviation. For reasons that are now ap- parent it is essential that a table of deviations should be ob- tained for all compasses mounted on board as soon as possible after a vessel is commissioned, that the table for the standard compass should be checked from day to day, and a new one made out after any marked change of magnetic latitude. For a new vessel built of iron or steel, observations should be made on the 24 equidistant 15 rhumbs before compen- sation; after compensation the residual deviations may be found by observing on 12 equidistant headings, though in both cases, if possible to do so, it would be better to swing with both helms and to take the mean of the two deviations on each heading as the correct deviation for that heading. As the ship is steadied on each heading and observations for deviation are made at the standard, the ship's head should be noted by observers at the steering and pilot-house com- passes ; then from the headings and deviations of the standard, the magnetic heading of the ship at each observation may be found. A comparison of each magnetic heading with the cor- responding heading by each of the compasses will give the deviation for the heading of the compass compared. Before an observation is taken on any heading, the ship should be steadied on it for three or 'four minutes, in order that the needle may be at rest and under magnetic influences normal for that heading at the time of observation. The ship itself should be steady, or its motion a minimum, when the observer takes his observations. EECIPROCAL BEARINGS 67 The deviation may be obtained by any one of four methods : (1) By reciprocal bearings; (2) By bearings of a distant object; (3) By azimuths or amplitudes of a celestial body; (4) By ranges of known magnetic bearing; or, by two or more of the above combined. 56. (1) By reciprocal bearings. This method is available when the ship is in a basin or a smooth harbor, and the com- passes are free from all disturbing influences except the ship's own magnetism and that of the earth; and when there is a suitable position on shore for mounting a compass where there are no local magnetic influences, above or below ground, to disturb its readings. A careful observer is sent ashore with a spare compass on a tripod which is placed where it can be seen distinctly from the ship with the naked eye, in a spot absolutely free from all local magnetism. The requisite warps having been prepared, the ship is swung around so as to bring her head, per standard compass, upon each heading on which observations for deviation are to be taken; of course, if circumstances permit, it is advisable to observe on each of the 24 equidistant 15 rhumbs. Then, by means of prearranged signals, the mutual bearings of this shore compass and the standard compass on board are observed at the moment when the ship's head is steady, and has been steady at least three minutes, on each of the required compass headings. To guard against mistakes, the time of each bearing should be observed, both on board and ashore, by compared watches; and it is advisable for the shore ob- server to mark the time and bearing of the standard from the shore compass at each observation on a blackboard pro- vided for the purpose, so that in case of an apparent incon- sistency, the observations can be immediately repeated and the necessity obviated for again swinging the ship. NOTE. Whenever bearings are taken with the azimuth circle, it should be hori- zontal with the bubble of the level centered. Celestial bodies should be observed for deviation when on or near the P. V. and at a low altitude (see Art. 222). 68 NAVIGATION The bearing of the standard from the shore compass at a given instant, reversed, is the correct magnetic bearing of the shore compass from the standard at that instant, and the difference between this magnetic bearing and the bearing taken at the standard on board at the same time will be the devia- tion due to the particular heading of the ship at the moment of observation. This deviation is marked according to the rule given in Article 52. The results of the swinging are recorded as in the form used in the following example solved on page 69. Ex. 4- Having swung a Monitor for deviations of the standard and battle compasses by method of reciprocal bear- ings, find the deviations of standard on 24 rhumbs, and of battle compass for the magnetic headings. Data as in form. 57. (2) By bearings of a distant terrestrial object. This method is convenient when the ship is at anchor in a harbor, or roadstead, with the object so far distant that the magnetic bearing will not alter sensibly as the ship heads on the various headings say about eight to ten miles for a ship swinging, anchored at short stay. This method may be used at sea, the ship steaming around an entire circle, provided the object is so far distant that the parallax does not exceed 30', the paral- lax being the angle whose tangent equals the radius of the circle in which the ship is swinging divided by the mean dis- tance of the object. At sea, even under the most favorable conditions, it involves more or less error; and, if the ship is in the locality of tides or currents, this method should not be used with the ship underway. By this method, a distant but distinctly visible object, as a clearly defined point of a distant peak, a light-house, or other mark, is observed as the ship at anchor swings slowly to tide, is steamed around, or swung at her moorings, but steadied sufficiently long on each heading to allow the magnetism of the ship to settle down. EECIPROCAL BEARINGS 69 : CO T- HCO^^TH | | | | | OOOOO OTHCOO(M goo O CO OOOtfiOOOOlOOlOoOOOOlO OlO OrH iHTHTHrHi-HTH '9ca[(; peppB eq no '8 OS 'va g jo Times of observation by watch. a'cOl^-i (t ^H'!O^Ht'- XS^^iOOOi^f^ and tion on of compass lars of compe 74 NAVIGATION entered in column 9. Bearing in mind the fact that the + sign is given to easterly errors, variation, and deviation, and the sign to westerly errors, variation, and deviation, and also that deviation equals compass error variation, the sign of the deviation should be apparent. Easterly deviation may be marked either -f- or E., westerly deviation ( ) or W. The magnetic azimuth of the ship's head may be found by applying the deviations of the standard compass to the head- ings per standard. At the instant of observing the sun's azimuth per standard, the ship's head by all other compasses on board should be noted. The readings of these compasses compared with the corresponding magnetic azimuths of the ship's head will give the deviations of the compasses on their particular headings, the deviations being marked as per rule Article 52. 59. (4) By ranges. Eanges, whose magnetic bearings are known, may be found in various localities, having been speci- ally laid out, or formed under natural conditions. The data concerning a number of such ranges have been published in a pamphlet by the U. S. C. and G-. Survey. When steaming across these ranges on various headings, the compass bearing of the range may be taken. The deviation for any heading will be the difference be- tween the compass bearing of the range on that heading and the known magnetic bearing, marked easterly when the mag- netic bearing is to the right of the compass bearing, westerly when the magnetic bearing is to the left of the compass bearing. 60. Napier's diagram. This is a graphic representation of deviations on either compass or magnetic headings, and it furnishes a ready method of finding the magnetic course cor- responding to a given compass course, and vice versa. It consists of a vertical line of convenient length divided into 24 equal parts representing the 24 15- rhumbs of the A GRAPHIC METHOD 75 Compass courses on dotted lines. Magnetic courses on solid lines,. FROM NORTH TO 1OO SOUTH DEVIATION FROM 1OO" SOUTH TO 36O NORTH DEVIATION DEVIATION EAST Curve of Deviations, Napier's Diagram. 76 NAVIGATION compass; beginning at the top, these are numbered in order, 0, 15, 30, 45 up to 360 from North around to the right. The line is also divided into 360 equal parts representing de- grees, numbered at every fifth degree around to the right. Usually the curve is shown in two parts as on page 75. The vertical line is intersected at 15 rhumbs by two lines, a plain line inclined upward and to the right, a dotted line inclined upward and to the left, each making an angle of 60 with the vertical. To construct a curve. Take on the vertical line the com- pass course for which the deviation has been obtained ; lay off this deviation, to the scale of the vertical line, on the dotted line which passes through the course, or in a direction paral- lel to the dotted lines, to the right if the observed deviation is easterly, to the left if westerly, and mark the point so ob- tained with a dot. Having done this for each observed devia- tion, trace a fair curve through the points, and this will be the deviation curve. The deviations should be obtained on eight or more rhumbs equally distributed around the compass, but these need not be 15 rhumbs. If not possible to get the deviations on more than four rhumbs, these should be as near the quadrantal points as possible. Rule I. From a given compass course to find the corre- sponding magnetic course. Take the compass course on the vertical line; move thence on or parallel to the dotted lines till the curve is intersected, thence on or parallel to the plain lines till the vertical line is intersected. This point in the vertical line will be the re- quired magnetic course. Rule II. From a given magnetic course to find the corre- sponding compass course. Find the magnetic course on the vertical line; from this point move on or parallel to the plain lines till the curve is intersected, thence on or parallel to the dotted lines meeting the vertical line in the required compass course. COMPASS DEVIATIONS 77 SECTION IV. THE THEORY OF COMPASS DEVIATIONS. 61. Soft iron and hard iron. Preliminary to investigating the causes of deviations, it is essential to consider the char- acter of the iron used in building a ship and the influence of the earth's magnetism on that iron. Considering its physical characteristics, iron may be desig- nated magnetically by the terms " soft " and " hard." Soft iron is iron which, under the influence of a magnetic force, will instantly acquire magnetism by induction, but will as quickly lose it when that force is removed. In other words its magnetism is transient induced magnetism. Hard iron is less susceptible to magnetic induction, but, when once magnetized, it retains a large part of its mag- netism permanently. Furthermore, the greater the hardness and the less easily it can be magnetized, the greater the amount of magnetism it is capable of retaining and the longer it will retain it. 62. Effect of earth's magnetism on a soft iron rod. If a rod of soft iron be held in the direction of the " line of force," it will instantly become magnetic. If in North latitude, the lower end will have induced in it North or red magnetism, and will repel the North end of the compass needle; and the upper end will have induced in it South or blue magnetism, and will attract the North end of the compass needle. If inclined to the "line of force," its induced magnetism will be proportional to the cosine of the angle of inclination; therefore, at 90, or at right angles with the "line of force," the rod will be in a neutral condition; beyond 90 the mag- netism will be reversed, that end which was at first of North polarity will have South polarity, and the intensity of the magnetism induced will increase till at 180 it will again be a maximum. 78 NAVIGATION If instead of being held in the direction of the "line of force," the rod is moved in a horizontal plane, it will be sub- ject to induction only by the horizontal component of the total force, so that if placed East and West magnetic, the bar, being at right angles to the inducing force, will be in a neutral condition and have no effect on either end of the com- pass needle. If held in any other position in the horizontal plane, its South end will attract, and its North end will repel the North end of the compass needle with a force proportional to the horizontal intensity multiplied by the cosine of the rod's mag- netic azimuth. If a soft iron rod be held in a vertical position, it will have magnetism induced in it by the vertical component of the earth's magnetism. In North latitude, the upper or South end will attract the North end of the compass needle; at the magnetic equator, being perpendicular to the line of force, it will be in a neutral condition; and in South magnetic lati- tude the polarity will be reversed, the upper end then having North polarity will repel the North end of the needle. If a rod, held in a position favorable to induction, is ham- mered, twisted, bent, or otherwise subjected, to mechanical violence, the amount of magnetism it will receive is increased. This magnetism diminishes more or less rapidly in the first few weeks but a portion of it is retained for months, perhaps for years, unless removed by similar mechanical violence applied in an opposite way. This condition is known as one of subpermanent magnetism. A plate of iron has magnetism induced in it in a similar way, the magnetism being divided into regions of opposite polarity by a neutral plane at right angles to the direction of the earth's total force; and its permanency is dependent on the character of the iron and the treatment given it. The law of induction, as explained for rods and plates, ex- MAGNETIC INDUCTION 79 tends to bodies of a third dimension, whether of regular or irregular shape ; the line connecting the induced poles, called the magnetic axis, lying in the direction of the line of force, with a neutral area whose plane is perpendicular to that axis. 63. Magnetic induction in an iron or steel ship. Apply- ing this law of induction to bodies of even the varied and complex form of an iron or steel ship, it is easy to understand how such a vessel should receive magnetism by induction and have it partially fixed in the course of construction by the processes of bending, twisting, hammering, or riveting to which the various parts are subjected. Since the facility with which the induction takes place and the ability of the iron to retain the magnetism induced de- pend both on the character of the iron and the treatment it receives, it is convenient to consider separately the earth's effect on the soft and hard iron, and also their effects on the compass needle. The iron in which only temporary magnetism is induced consists of the kind denominated as " soft iron/' and in a ship this is either horizontal or vertical, or if not so, it may be resolved with components in those planes so that the earth's effect on soft iron will be : (1) Transient magnetism induced in horizontal soft iron, or that developed in the horizontal soft iron of the ship by the inductive action of H, the horizontal component of the earth's total force. It is transient in character and as it depends for its force upon H'j which varies with the cosine of the dip, its force will be zero at the magnetic poles and a maximum at the magnetic equator. (2) Transient magnetism induced in vertical soft iron, or that developed in the vertical soft iron of the ship by the in- ductive action of Z, the vertical component of the earth's total force. It is transient in character and as it depends for 80 NAVIGATION its force upon Z, which, varies with the sine of the dip, its force will be zero at the magnetic equator and a maximum at the magnetic poles. Subpermanent magnetism induced in the ship while build- ing. The remainder of the ship's iron, consisting of that denominated "hard iron" and of that of a character inter- mediate between hard and soft iron, when acted upon by the earth's inducing forces in the process of building, assumes the character of a large magnet, more or less permanent, whose distribution of magnetism depends on the place of building and the azimuth of the ship's head at the time. Whilst build- ing, the ship's polar axis and neutral plane respectively cor- respond, more or less, to the direction of the earth's total force and a plane at right angles to it. The magnetism thus de- veloped is known as subpermanent magnetism, as it is not entirely permanent; suffering a diminution, after launching of the vessel and with change of direction from that in which the ship was built, for a lapse of several years till its mag- netism settles down to practically a permanent condition. This state of affairs is in no wise due to induction in soft iron, and is modified if the vessel is launched before its hull is practically completed. 64. Forces acting on a compass needle in an iron or steel ship. It is evident then that a compass needle, besides being acted upon by the earth's horizontal force which tends to keep the needle in the magnetic meridian, is subject to three distinct disturbing influences derived from the ship itself : (1) Subpermanent magnetism ; (2) Transient magnetism due to vertical induction in ver- tical soft iron; (3) Transient magnetism due to horizontal induction in horizontal soft iron ; and that the resultant of these three forces, when not acting in the plane of the magnetic meridian, will deflect the needle, SUBPERMANENT MAGNETISM 81 producing the total deviation for the particular heading of the ship. 65. Effect in producing deviation of each of the ship's disturbing forces when acting on the compass needle. (1) Effect of subpermanent magnetism. We have seen that the location of the poles of this subpermanent magnetism depends upon two things: 1st, the magnetic azimuth of the ship's head while building; 2d, the direction of the line of force at the place of building; hence the North (or red) pole will be in that part of the ship which was North in building and the South (or' blue) pole will be in that part which was South in building. The repulsion of the North pole of the ship simply doubles the attraction of the South pole for the North end of the compass needle, therefore it may be laid down as a general rule that, under the influence of the sub- permanent magnetism, the North end of the compass needle will be attracted to that part of an iron or steel ship which was South in building; hence in an iron ship built head North, the North end of the needle will be attracted toward the stern. Heading N. or S. there will be no deviation; in the former case the directive force is diminished, in the latter case, increased. As the ship swings in azimuth from these neutral points the needle is deflected ; ' toward the East for westerly headings with a maximum of deviation about West, toward the West for easterly headings with a maximum about East. In an iron ship built head South, the North end of the com- pass needle will be attracted toward the stem or head of the ship, and results just the reverse of the above will be ob- tained. In an iron ship built head East, the North end of the compass needle will be attracted to the starboard side; head- ing East or West there will be no deviation, in the former case the directive force is diminished, in the latter case it is 82 NAVIGATION increased. As the ship swings in azimuth from these neutral points, the compass needle will be deflected toward the East for northerly headings with a maximum about North, toward the West for southerly headings with a maximum about South. In an iron. ship built head West, the North end of the needle will be attracted toward the port side, and results the reverse of those in a ship built head East will be obtained. The accompanying diagrams will illustrate the distribution of magnetism in ships built head N"., E., S., and W. (mag- netic) at a place where the magnetic dip is 68 30', and the SEMICIRCULAR DEVIATION 83 character of the deviations due to the subpermanent magnetism of these four ships is illustrated in the following curves. In reading these curves, the azimuths are taken on the vertical line and the deviations on the ordinates perpendicular thereto, the curve being to the right of the vertical line when the deviations are easterly, to the left when westerly. Estimating the azimuth of the ship's head from the neutral points, these curves are " curves of sines " ; they show the deviation to be the same in amount, but of opposite sign, on points differing 180 in azimuth, and the neutral points to correspond to those points of the compass on which the ship's head and stern were in building. Semicircular deviation. The above is called a semi- circular deviation because it is easterly in one semicircle and westerly in the other, as the ship's head moves around a com- plete circle in azimuth. If the compass is principally disturbed by the magnetic in- fluences of the hull of the ship, the neutral points, or points of no deviation, will be opposite to each other and will cor- 84 NAVIGATION respond to those points of the compass towards which the ship's head and stern were directed in building; and the de- viation, or more exactly the sine of the deviation, in each semi- circle, is proportional to the sine of the azimuth of the ship's head measured from the neutral point, the azimuth being that shown by the disturbed compass. Since the force due to subpermanent magnetism is con- stant for all latitudes, and its effect in producing deviation is inversely as H, the directive force of the earth, the semi- circular deviation due to subpermanent magnetism varies with change of latitude. 66. (2) Effect of transient magnetism due to vertical in- duction in vertical soft iron. The vertical component of the earth's force, Z, induces magnetism in the vertical soft iron and fittings of the ship, producing a resultant pole of South polarity towards which the North end of the compass needle is attracted. As the vertical inducing force remains the same at a given place, the magnetism induced by it does not vary as the ship turns in azimuth; therefore, it produces a semi- circular deviation following the same law as that caused by subpermanent magnetism, with the exception that the effect produced in this case, being directly proportional to Z and in- versely proportional to H, varies as the tangent of the mag- netic dip, or as tan 6. The semicircular deviation caused by induction in ver- tical soft iron is the kind formerly found in wooden ships; the neutral points being North and South, and the deviation easterly in the Eastern semicircle, westerly in the Western semicircle. It constitutes the smaller part of the semicir- cular deviation of iron or steel ships. The ship's polar force and the starboard angle. As the two forces we have just considered, those due to subpermanent magnetism and transient magnetism induced in vertical soft iron, produce the same kind of deviation, it is convenient to SEMICIRCULAR DEVIATION 85 take them jointly and to consider the North point of the needle as acted upon by their resultant force, known as the ship's polar force; its horizontal component makes with the fore and aft line of the ship an angle, which measured from ahead around to the right (from to 360) is known as the starboard angle and designated by the letter a. The semicircular component forces and the coefficients of semicircular deviation. Now this resultant polar force of South polarity, attracting the North end of the compass needle toward a certain point in the ship, may itself be re- solved into two component forces, one acting in the fore-and- aft line of the ship, the other in the athwartship line through the compass. Let 93 represent the semicircular force acting in the fore-and-aft line and & the semi- circular force in the athwartship line; the former is marked + ^ acting toward the ship's head, ( ) if toward the stern ; the lat- ter is marked + ^ acting to star- board, ( ) if to the port side. The signs of these forces are de- pendent on the value of the star- board angle a as indicated in Fig. 25. 93 and ( are known as the exact coefficients of the semi- circular deviation; but +' 93 is the ship's polar force to head and + ( is the ship's polar force to starboard, both expressed in terms of the " mean value of the force of earth and ship to magnetic North" as a unit. This fact will be apparent from a study of section II, chapter IV. Let be the position of a compass ; OS,. , OS* , OS S , or OSi be a ship's polar force represented in Fig. 25 as acting FIG. 25. 86 NAVIGATION respectively from the starboard bow, starboard quarter, port quarter, or port bow, and in the horizontal plane. For the force 08^ or 4 , 23 = Om, for the force OS a or 08 S , 3$ = On; the sign of 23 is + when acting to head, ( ) when acting to stern as indicated in the figure. For the forces 08^ , 08 2 , 08 3 , and <9S 4 , the values of < are represented by mS^, nS 2 , nS s , and ra 4 respectively; the sign of ( is + when acting to starboard, ( ) when act- ing to port as indicated in the figure. For the force OS ,%is + , (is + , and a is < 90, OS 2 , $ is (), is + , and a is > 90 and <180, OS S , % is ( ), ( is ( ), and a is >180 and <270 , 0& 4 , 23 is + , ( is ( ), and a is >270 and <360 , all values of a being measured from ahead around to the right, as shown by circles in the figure. It is convenient to find the angle x = tan- 1 -J- and then to Jo take a = x, 180 x, 180 + x, or 360 --a, according as the polar force acts from the starboard bow, starboard quar- ter, port quarter, or port bow respectively. For illustration, consider the two components positive. The forces 33 and ( exert each an attraction on the North end of the compass needle similar to that of a permanent magnet, producing semicircular deviation as the ship swings in azi- muth. The force + 23 causes no deviation on the heading North, but as the ship swings toward the East, the needle deviates toward the East, and the deviation increases with the azimuth by constant increments till the ship heads about East, when the force is at right angles to the direction of the needle and produces its maximum effect ; then as the swinging continues the deviation diminishes by constant decrements till, when the ship heads South, the deviation is zero and the needle is again in the magnetic meridian. If the swinging SEMICIRCULAR DEVIATION 87 is continued through the western semicircle, the effect is re- peated except that the deviation is westerly and a maximum in amount at West. Letting the maximum deviation produced be represented by B (which is approximately the snr 1 23), the deviation on any other heading due to 23 will be a fraction of B, the amount and sign depending on the azimuth of the ship's head per NW FIG. 28. FIG. 29. compass. B is known as the approximate coefficient of semi- circular deviation due to the force 93. If as in Fig. 26, the azimuths are read off on the vertical line, and if, at the points representing the azimuths of the ship's head per compass, ordinates are erected perpendicular to the vertical line and representing according to a given scale of parts the corresponding deviations, the curve drawn through the extremities of the ordinates, showing the devia- tion to be zero at North and South and a maximum at East and West, will be a curve of sines ; and the value of the devia- 88 NAVIGATION tion due to the fore-and-aft semicircular force 23 for any heading of the ship z' per compass will be B sin z'. Had the force 23 attracted the North end of the needle to the stern, or if the sign of 23 had been ( ), a curve similar to Fig. 27 would have resulted, the deviations being westerly for those headings on which the force + 23 produced easterly deviations and easterly for those on which + %$ produced westerly deviations. The force -(- & causes a maximum easterly deviation when heading North which diminishes as the ship swings in azi- muth to the eastward till, when heading East, the force is in the magnetic meridian and produces no deviation. As the swinging is continued, the North end of the needle deviates to West and the amount of the deviation increases by con- stant increments till, when the ship heads South, the deviation is again a maximum in amount but westerly in sign. If the swinging is continued, this effect is repeated, the sign of the deviation being opposite to that of the corre- sponding point 180 distant in azimuth. Letting the maximum deviation be represented by C (which is apprximately sin' 1 (), the deviation on any other heading due to + ( will be a fraction of (7, the amount and sign depending on the azimuth of the ship's head per compass. C is known as the approximate coefficient of semicircular de- viation due to + > "the athwartship semicircular force. As seen in Fig. 28, the curve is one of cosines, and the value of the deviation on any heading (z f ) per compass due to the force + will be C cos z', and the deviation on that compass heading due to the result- ant of the semicircular forces + 23 and + (, or V^3 2 + & 2 > will be B sin z' -\- C cos z'. TRANSIENT MAGNETISM 89 67. (3) Effect of transient magnetism due to horizontal induction in horizontal soft iron. Since the magnetism in- duced in horizontal soft iron by the earth's horizontal force varies as the cosine of the inclination of the iron to the direc- tion of the earth's horizontal force, it is evident that we must now deal with an induced force which, unlike the forces producing semicircular deviation, is a variable one. That the force induced may be a function of the azimuth of the ship's head, the horizontal soft iron of the ship should be considered as lying either in the fore-and-aft, or the athwart- ship direction through the compass. As a greater part of the horizontal soft iron is so situated, and as soft iron at intermediate angles may be represented by components parallel to those directions, the effect of all the soft iron in the ship, magnetized by the earth's horizontal component and acting in the horizontal plane through the compass, may be replaced by the effects of two systems of horizontal iron, one placed fore and aft and one athwartships. It will be shown later on that when the soft iron is symmetrically distributed on each side of the fore-and-aft line of the ship, and the ship is on an even keel, that the forces due to the induced mag- netism of the two systems may be replaced by those of a single fore-and-aft and a single athwartship rod; the force due to the induced magnetism of the first rod will act in the fore-and-aft horizontal line, and the force due to the induced magnetism of the second rod will act in the athwartship hori- zontal line through the compass. The character of each force, whether attracting or repell- ing the North end of the compass needle, will depend on the position of the rod ; if entirely forward or abaft, to starboard or to port, of the compass, one end of the rod will act ; if continuously extending above or below the compass needle, the opposite end will act. If a rod is entirely on one side of the compass, a similar rod, similarly situated but distant 180, will simply double the effect of the first rod. 90 NAVIGATION Depending on the location of the compass on board, the effect of the symmetrical horizontal soft iron may be that of one, or another, of the arrangements of the rods as shown in Figs. 30, 31, 32, and 33. Taking the first rod (1) for consideration and regarding the magnetism induced in the rod on the various headings as represented in Fig. 34, we see that with the* ship heading North no deviation is produced but the directive force is in- creased; as the ship swings to the eastward, the North point of the needle deviates to the East (or to the right) ; at NE., the maximum easterly deviation is caused, since at that angle with the meridian, the induced magnetic force, though equal only to H cos 45, has a greater proportionate effect in caus- ing deflection of the needle. As the ship continues to swing O-^- -e 1 - 2 3 FIG. 30. FIG. 31. FIG. 32. FIG. 33. to the eastward, the rod gradually loses its magnetism, and the attraction on the needle diminishing it gradually returns toward the meridian. When the ship heads East, the rod is at right angles to the line of force and has no effect on the needle which is again in the meridian. For the NE. quadrant the curve of deviations is as represented in Fig. 35 from North to East. The ship continuing to swing, the end of rod (1) nearest the compass, having North magnetism induced in it, will repel the North end of the compass needle, and the curve of deviations traced will be exactly the same as in the NE. quad- rant except that it will be on the opposite side of the vertical line. When the ship heads South there is no deviation, but the directive force is again increased. From South to West we QUADRANTAL DEVIATION 91 will have the same curve (the deviations being the counter- part in amount and sign) as from North to East, at West the needle fails of effect. From West to North the deviations and the curve are the same as in the SE. quadrant. duadrantal deviation. The deviation here illustrated is known as quadrantal deviation, and is so called because it is easterly and westerly alternately in the four quadrants as the ship's head moves around a complete circle in azimuth. Its zero points coinciding very nearly with the cardinal points, N.E. s w. S.E. and the points of maximum deviation being at the quadrantal points, 'each characteristic occurring twice in a semicircle, the amount of the deviation (or more properly the sine of its amount) is proportional to the sine of twice the azimuth of 92 NAVIGATION the ship's head measured (as will be seen later) from a line half way between the magnetic North and the compass North. The quadrantal deviation usually found on board iron or steel ships is of the type represented by the curve of Fig. 35, easterly in the NE. and SW. quadrants, westerly in the SE. and NW. quadrants, or the type usually spoken of as a positive quad- rantal deviation. The horizontal component of the earth's total magnetic North N.E. S.W. East S.E. South East S.E South West N.W. North FIG. 35. West N.W. North FIG. 36. North N.E. force is the directive force acting on the compass needle, the magnetism induced in the horizontal iron by the earth's hori- zontal force is the disturbing force acting on the needle, and the ratio between the two is constant; therefore the quad- QUADRANTAL DEVIATION 93 rantal deviation may be expected to remain unchanged in all magnetic latitudes, and even by lapse of time, so long as the distribution of the ship's horizontal soft iron remains un- changed. Quadrantal force and coefficient of quadrantal deviation. Let 2) represent the quadrantal force, the sign of the force being + when producing a positive quadrantal curve as shown by Fig. 35; and letting D (which is approximately sin' 1 3)) represent the maximum deviation, or. that on the quadrantal points, the deviation on any other heading due to the force + 3) will be a fraction of D and a function of twice the azimuth; in other words D is the approximate coefficient of quadrantal deviation of the type represented by the curve (Fig. 35). (See Art. 79.) If D-L represents the coefficient of quadrantal deviation due to the induced magnetism of rod (1), then the quadrantal deviation on any heading z' per compass due to the effect of rod (1) will be D! sin 2z'. Taking the rod (2) under consideration and regarding the magnetism induced in the rod on the various headings as rep- resented in Fig. 34, it is seen that when the ship heads North no effect is produced, the rod being at right angles to the line of force; as the ship swings to eastward, polarities are developed as shown and the North end of the needle is re- pelled until, when the ship heads NE. ? there is a maximum westerly deviation. When the ship heads East, the rod is in the meridian, and its induced magnetism increases the directive force on the needle, but produces no deviation. It is thus seen that the effect of the magnetism induced in a rod lying wholly on one side of the compass, as the ship swings from North to East, is, in the case of an athwartship rod, just the reverse of that in a fore-and-aft rod (1). If the swinging is continued till the ship again heads North, the deviations on the various headings may be represented by a curve simi- 94 NAVIGATION lar to Fig. 36, which shows for any azimuth of the ship's head a deviation of the same kind, but of the opposite sign, to that shown by the curve of Fig. 35 as in the case of rod (1). When the ship heads West the directive force is increased as was the case when the ship headed East. The quadrantal deviation caused by rod (2), being westerly in the N"E. and SW. quadrants and easterly in the SE. and NW. quadrants, is known as a negative quadrantal deviation. In this case, the case of a negative quadrantal deviation, if ( ) D 2 is the quadrantal coefficient, the quadrantal devia- tion on any heading z f per compass due to the induced mag- netism of an athwartship rod (2) will be D 2 sin 2z'. The combined quadrantal deviation due to an arrangement of iron similar to that illustrated in Fig. 30 will be (+!>! + ( D 2 ) ) sin 2z r = D sin 2z'. The deviation being easterly or westerly in the NE. and SW. quadrants (and hence westerly or easterly in the SE. and NW. quadrants), according as the effect of rod (1) is greater or less than the effect of rod (2). In cases where the rod (2) equals rod (1), an arrangement of iron similar to Fig. 30 will increase the directive force without causing deviation. A rod (3), Fig. 31 and Fig. 33, continuously extending above or below the compass in a transverse plane, will have an effect directly the opposite of rod (2), diminishing the direc- tive force at East and West and producing a positive quad- rantal deviation as represented by Fig. 35. If D z be the coefficient of quadrantal deviation due to rod (3), then the quadrantal deviation on any heading z' per compass will be D 3 sin 2z', and the combined quadrantal deviation due to an arrangement of iron similar to Fig. 31 will be (>! + D 3 ) sin 2z f = D sin 2z'. QUADRANTAL DEVIATION" 95 A rod (4), Fig. 32 and Fig. 33, continuously extending above or below the compass, in the fore-and-aft plane, will exert on the North poin.t of the compass needle an effect just the opposite to that exerted by rod (1) in the same plane but wholly on one side of the compass, diminishing the directive force at North and South and producing a negative quad- rantal deviation as represented by Fig. 36. If Z> 4 be the coefficient of the quadrantal deviation due to rod (4), then the quadrantal deviation on any heading z' per compass due to induced magnetism of rod (4) will be (_) D 4 s in 2z', and the combined quadrantal deviation due to an arrangement of iron similar to Fig. 32 will be ( D 2 + ( Z> 4 ) ) sin 2z' = D sin 2z f , and that due to an arrangement similar to Fig. 33 will be (Z> 3 + ( Z> J ) sin 2z' = D sin 2z', the sign of D depending on which is the greater, the mag- netism due to rod (3) or that due to rod (4). In this arrangement when rod (3) equals rod (4), the directive force will be diminished, but no deviation will be produced. Hence, we have for the general expression representing quadrantal deviation, on any heading z' per compass, due to horizontal induction in horizontal soft iron symmetrically sit- uated, when D is the approximate coefficient, the term D sin 2z'. The quadrantal deviation as represented by D is usually caused by the action of rods (1) and (3), Fig. 31, or excess of effect of (3) over (4), Fig. 33, since it is usually positive and the directive force on the needle is diminished. 68. In case the soft iron is unsymmetrically situated in the horizontal plane through the compass, an additional force due to horizontal induction in fore-and-aft iron may act on the needle, and it may be represented as that of a fore-and-aft 96 NAVIGATION rod to starboard or port of the compass, rod (5), Fig. 37; and an additional force due to horizontal induction in athwartship iron may also act on the needle, this force being represented as that of an athwartship rod forward or abaft the compass, rod (6), Fig. 37. s.w, S.E. Eegarding the magnetism induced in rod (5) as represented in Fig. 37 for the various headings per compass, it is seen that with the ship heading either North or South, the North point of the compass needle will have a maximum deflection to eastward, being most strongly attracted in the first case by the induced South pole, and equally repelled in the second case by the induced North pole of the rod (5). QUADRANTAL DEVIATION 97 When the ship heads either East or West, the rod (5) has no effect. At the quadrantal points the deviation is the same fraction of the maximum and also easterly. The curve rep- resenting the deviations is similar to the plain curve of Fig. 38. From Fig. 37, it is seen that when the ship heads North or South rod (6) has no effect. When the ship heads East or West, the North point of the needle will have a maximum deflection to the westward, being most strongly repelled in the first case by the North pole of (6) and equally attracted in the second case by the South pole of (6). At the quad- rantal points the deviation is the same fraction of the maxi- mum and is also westerly. The curve of deviations in this case is similar to the dotted curve of Fig. 38. When the two rods (5) and (6), Fig. 37, act together, rod (5) causes a maximum easterly deviation when rod (6) 98 NAVIGATION has no effect, and rod (6) causes a maximum westerly devia- tion when (5) has no effect; at the quadrantal points the easterly deviation caused By (5) neutralizes the westerly de- viation caused by (6) ; the resultant curve of deviation, as represented in Fig. 39, shows maxima when the ship heads on the cardinal points and minima when the ship heads on the quadrantal points. This deviation, due to horizontal induction in soft iron unsymmetrically distributed about the compass, is of the quadrantal type and is in general very small. Let ( represent the force producing this particular kind of deviation, -+- when it causes an easterly deviation between North and NE. as shown in Fig. 39 ; let E (which is approxi- mately sin' 1 (?) be the maximum deviation, or that on the cardinal points, then the deviation on any other heading z' per compass due to the force + ( will be a fraction of E and will be found from the expression E cos 2z'. E is known as the approximate coefficient of the quadrantal deviation due to horizontal induction in horizontal soft iron unsymmetrically distributed about the compass. ( See Art. 80. ) If the rods (5) and (6) are in the port bow (Fig. 40.), the deviation due to their effect on any azimuth will be the .same in amount but of the opposite sign to that produced in the case when they were in the starboard bow, the coefficient will be ( ) Ej and the deviation on any heading z f per compass will be E cos 2z'. If the rods are both in one quarter, the effect will be the same as if they were in the opposite bow; the effect of having one of the rods in the bow and one in the opposite quarter is the same as if both rods were in that bow, or both in that quarter. So the general expression for the' quadrantal deviation on CONSTANT DEVIATION 99 any heading z per compass due to horizontal induction in soft iron unsymmetrically distributed about the compass is E cos 2sf. 69. Constant deviation. When the soft iron is not sym- metrically distributed on each side of the fore-and-aft line through the compass, or the compass is not in the midship line, a constant term may be noted in the deviation. It is usually very small and is called constant because it is the same in amount and direction on all headings ; it is marked East or (+) when the easterly deviation is in excess, West or ( ) when the westerly deviation is in excess. If due to the unsymmetrical soft iron of the ship, it is known as a real constant deviation which will not vary with change of latitude, as the ratio of the force induced in the iron and the directive force on the compass needle is a constant. The FIG. 40. FIG. 41. FIG. 42. force producing constant deviation is represented by 91, the approximate coefficient or deviation itself by A. (See Art. 80.) If rods (5) and (6) are situated as in Fig. 41, (5) will produce the plain curve of Fig. 38 when the ship is swung, and (6) will produce a curve similar to the dotted curve of Fig. 38 except of the opposite sign; that is to say (5) will cause a maximum of deviation on North and South, a mini- 100 NAVIGATION mum on East and West, all being easterly; (6) will cause a minimum of deviation on North and South, and a maximum on East and West, also easterly. In other words, as the effect of one rod increases, that of the other decreases, their result- ant effect as the ship swings in azimuth being a constant easterly deviation. If, however, rods (5) and (6) are situated as in Fig. 42, their resultant effect as the ship swings in azimuth will be a constant westerly deviation. A real value of A is rare on board ships with well-located compasses; more often there is an apparent value due to a badly placed lubber's line, index or other instrumental error, or an error in the assumed direc- tion of the magnetic North. Other errors of a like nature may exist, but, whatever the causes, they may all be repre- sented by the approximate coefficient of constant deviation A. 70. If 8 represents the sum total of the deviations due to the various forces considered for the heading z' per compass we shall have 8 = A + B sin z' + C cos z' + D sin 2z' + E cos 2z', (16) A, B, C, D, and E being known as the approximate coefficients of the deviation they respectively represent. (See Art. 77.) Though the curve of total deviations, as shown on Napier's diagram (Art. 60), is irregular and unsymmetrical, the curves due to the separate forces considered are in themselves per- fectly symmetrical, the irregularity arising from the super- position of all of them, the combined curve representing the resultant effect of all the forces. 71. Determination of coefficients by inspection. The co- efficients A, B, G, D, and E may be found from the .deviations observed with the ship's head on the 24 equidistant 15 com- pass rhumbs; also on any 12 or 8 of them if equidistant, as will be indicated in Art. 89, the process there followed being known as the " analysis of deviations " ; but A, B, C, and E APPROXIMATE COEFFICIENTS 101 may be approximately determined by inspection, from the de- viations on the four cardinal points, and D may be also approximately obtained from the deviations on the four quad- rantal points. Using formula (16), Art. 70, and paying particular atten- tion to the signs of functions of the azimuths, we have equa- tions expressing the deviations on the eight principal points as follows, letting 8 3 be the sine or cosine of 45 : + + E + C8 3 + D C8 3 D -C +E -C8 3 + D E + C8 3 -D and it is apparent from these equations that, using the word mean in its algebraic sense, A is the mean of the deviations on the four cardinal points, or any four or more equidistant compass headings. B is approximately the deviation at East, or the deviation at West with the sign changed ; but more accurately the mean of these two values. C is approximately the deviation at North, or the deviation at South with the sign changed; but more accurately the mean of these two values. D is approximately the mean of the deviations at NE. and SW., or the mean of the deviations at SE. and NW. with the sign changed; but more accurately the mean of both these means. E is the mean of the deviations on the four cardinal points of the compass after the signs of the deviations on East and West have been reversed. On North So = A NE. S 3 = A + BS, East S 6 A + B SE. So = A + B8; South 8 12 = A SW. 8 18 = A B8. West Sis A B NW. 8 21 A B8. 102 NAVIGATION Ex. 7. Given the following deviation table, find by inspec- tion the nearest approximation to the values of A, B, C, D, and E. North 3 12' E NE 14 24 E East 12 00 E SE 5 24 E South 1 36' E SW 7 00 W West 14 24 W NW 12 12 W To find A. To find B. To find C. At North 3 12' E At East 12 00' E At North 3 12' E East 12 00 E West ( ) 14 24 W South ( ) 1 36 E South 1 36 E West 14 24 W 2)26 24' E 2) 1 36' E Algebraic Sum 2 24' E B = 13 12' E C = 48' E A= 2 24' E = QO 36> E To find D. To find E. At NE 14 24' E At SE 5 24' E At North 3 12' E SW 7 00 W NW 12 12 W East ( ) 12 00 E South 1 36 E 2)7 24' E 2)6 48' W West ( ) 14 24 W 1st value of D = 3 42' E 2d value D 3 24' W 2d value of D( ) 3 24 W 2)7 06' E D = 3 33' E 72. Heeling error. So far only those forces acting on the North point of the compass needle to produce deviation when the ship is upright have been considered, and it is necessary to consider other forces when an iron ship is inclined from the vertical. There are certain forces which, acting only vertically and producing no deviation in the former case, will have in the latter case lateral components tending to draw the North point of the needle to one side or the other. Such are the vertical component of the ship's subpermanent magnetism and the vertical component of the magnetism induced in vertical soft iron. If they act vertically downward when the ship is on an even keel, the North end of the needle will go to windward when the ship heels, otherwise to leeward. In addition, the horizontal deck beams and all other horizon- tal transverse iron become more or less magnetized by the earth's vertical inducing force, and the South polarity of their MEAN DIRECTIVE FORCE 103 upper or weather ends will attract the North end of the needle to windward. The resultant effect of these forces, when the ship is in- clined, is known as the heeling error, the direction of which, de- pending on circumstances, may be to windward or to leeward. The heeling error will vary with change of latitude because that part due to the vertical component of the subpermanent magnetism varies as -jf, and the parts due to magnetism in- duced by the earth's vertical force vary as tan 0. This error is a maximum on northerly or southerly courses, a minimum on easterly or westerly courses, and, for inter- mediate headings, varies practically as the cosine of the azi- muth of the ship's head. The causes and effects of heeling error may be better under- stood after a careful study of the next chapter. 73. Mean directive force. The ship's forces acting on the compass have been considered primarily as causing deviation, but they have an additional effect, increasing or diminishing the earth's directive force on the various headings as the ship swings in azimuth. This can be easily seen by resolving any force so that one component will be in the direction of the undisturbed needle and one at right angles to it in the hori- zontal plane. Whilst the latter component acts to produce deviation, the former increases or diminishes the directive force according as it draws the North point of the needle to the northward or southward. The force of earth and ship to magnetic North will vary with the azimuth of the ship's head, and its mean value for equidistant headings will be the mean directive force acting on the needle which, experience shows, is less than unity (H being unity) in nearly all iron or steel ships. Other conditions being the same, the best location for a compass is that position where it will have the greatest mean directive force. (See Art. 76.) CHAPTER IV. MATHEMATICAL THEOKY OF THE DEVIATIONS OF THE COMPASS. SECTION I. 74. Mathematical theory of the deviations of the compass. By considering all the iron of a ship as magnetically either hard or soft, it has been shown that on board an iron or steel ship, a freely suspended needle is acted upon by (1) the earth's total force; (2) the force due to the subpermanent magnetism induced in the ship in building; (3) those forces due to the transient magnetism induced in soft iron by the earth's force. At a given place the force (1) is a constant force, though it does not draw the North point of the needle towards the same point in the ship on all azimuths of the ship's head. The force under (2) is a constant force and attracts the North point of the needle towards the same point in the ship for all azimuths. The forces under (3) are constant or variable, de- pending on whether the inducing force is the vertical or hori- zontal component -of the earth's force. In investigating mathematically the theory of compass de- viations, it is necessary first to find the components of the various forces acting on the North point of the needle in certain definite directions through the compass, and the resultant of the components in each direction. Let these directions be the fore-and-aft horizontal line, the transverse horizontal line, and the vertical line through the point of suspension of the needle, the length of which is THEORY OF DEVIATIONS OF COMPASS 105 regarded as infinitely small when compared with the distance of the nearest iron, or, what amounts to practically the same thing, the North point of the needle is considered the origin of coordinates. Examining first the earth's force, let 0, Fig. 43, represent the North point of a magnetic needle on board an absolutely FIG. 43. non-magnetic ship ; let it be the origin of a system of rectan- gular coordinates of which OX the horizontal fore-and-aft line, OY the horizontal transverse line, and OZ the vertical line through the point are the axes respectively denominated X, Y, and Z ; the directions to head, to starboard, and verti- cally downward being regarded as positive. The South end of the needle is not considered, since an attraction or repulsion of the North end would be a repulsion or attraction of the south end, and the effect on the South end simply doubles that on the North end without changing the direction of the needle. 106 NAVIGATION COMPASS DEVIATION Therefore, we may confine the investigation to the action on the North point of the needle of the earth's total force which, for purposes of illustration, may be represented in intensity and direction by OT in the figure. The vertical plane of OT is that of the magnetic meridian OM', the angle M'OT is the magnetic dip, OM represents in intensity and direction the earth's horizontal inducing force, and OC the intensity of the earth's vertical force through the origin of coordinates (see equation 15, Article 45). If the needle is a freely suspended needle, it will point in the direction OT; if a compass needle, it will point in the direction OM; in other words, OM is the directive force acting on the compass needle, and is in the horizontal plane of XY. Since a force represented by a vector or line of given length and direction, such as OM, can be resolved into two com- ponents at right angles to one another, all being in the same plane, then the force OM, or H f can be resolved into the forces OA and OB. Therefore, letting M'OX be the magnetic azimuth of the ship's head, measured positively around to the right from the magnetic meridian, then, if in Fig. 43, 's 0AM, BOA, and OM T are right angles, we shall have the components of the earth's total force in the direction of the three axes as follows : OA = H cos z, force of the earth's magnetism drawing the North point of the needle at towards the ship's head; OB = H cos (90 + z) = H sin z, the force draw- ing it to starboard ; OC = Z, the force drawing it vertically downward. Let the component OA in the axis of X be called X, the component OB in the axis of Y be called Y t the. component OC in the axis of Z be called Z THEORY OF DEVIATIONS OF COMPASS 107 Now, if the earth's total force be disregarded, and forces equivalent to X, Y, and Z be supposed to act in their proper axes on the North point of a freely suspended magnetic needle at 0, it will take the direction of OT '; a compass needle under the same forces would take the direction OM. If instead of being mounted on board an absolutely non- magnetic ship, the magnetized needle is mounted on board an iron or steel ship in the same geographical locality, the forces due to the ship's magnetism will act on the North point of the needle in addition to the earth's force; a dipping needle will take the resultant direction of all the forces, and a compass needle the direction OM". The angle M'OM" is the deviation due to the ship's magnetic forces for the particular heading per compass z' ' , and is marked East or -f- when OM." is to the right of OM', that is, when the North point of the compass is drawn to the Eastward ; otherwise, West or ( ) . As the force due to the subpermanent magnetism of the hard iron of the ship alters neither its intensity nor its direc- tion as the ship swings in azimuth, the components of this force in the three axes will be constant. They are represented by P when drawing the North point of the needle towards the ship's head, Q when drawing the North point of the needle to star- board, R when drawing the North point of the needle vertically downward. Expressed in terms of the earth's horizontal force, we have P O 7? abstract quantities -^-, f>jj- The greater portion of the soft iron on board ship lies in one, or another, of the three axes considered, and since the effect of soft iron ^ng in intermediate directions may be represented by other iron parallel to the three axes, the inves- 108 NAVIGATION COMPASS DEVIATION tigation of the effect of soft iron may be confined to the effects of iron parallel to those axes, and may properly begin with a consideration of the fore and aft system. It is plain that the northern portions of this iron will always have induced in them North polarity and the southern portions South polarity as the ship swings through a complete circle in azimuth,, that the intensity of the force due to the induced magnetism will vary with the azimuth of the ship's head, and that the polari- ties will be reversed when the ship's head passes through an azimuth of 90 or 270. If the ship heads North magnetic, the force induced in the fore and aft soft iron will be a maxi- mum and will bear a certain specific ratio to the earth's hori- zontal force. If that ratio be 1 9 then the force induced will be IH. With the ship on any magnetic azimuth' z, the in- duced force will be IH cos z. As X = H cos z, the force on the heading z will be the same fraction of X that the possible maximum is of H, and as in this case it is desirable to express a force with a suggestion of the axis in which the iron may be, it is convenient to represent the force induced in the fore- and-aft iron as IX, or as >a fraction of the fore-and-aft com- ponent of the inducing force; furthermore, the variations in the force induced and the reversal of polarities referred to will occur in the same order and at the proper time even though X is taken as the inducing force in the axis of X. Therefore, it is mathematically correct to assume that the soft iron, lying in the direction of any one of the axes, has magnetic force induced in it by the earth's resolved com- ponent in the same axis and by that component only. The fore-and-aft component X will induce magnetism in the fore-and-aft iron which may practically be considered as a fore-and-aft system of parallel magnets attracting the North point of the needle with a force IX towards a point or pole in the system, I being a constant dependent only on the soft iron in the ship. THEORY or DEVIATIONS OF COMPASS 109 This force will act in the fore-and-aft vertical plane through the compass only when the iron is symmetrically situated with reference to that plane, and in the fore-and-aft horizontal line only when the iron is symmetrically situated with refer- ence to the horizontal plane through the compass. Such is not the usual case, and this force, instead of acting towards the ship's head, acts in some other direction. The components of IX will be aX to the ship's head, dX to starboard, gX vertically downward. y , y, and I are the direction cosines of IX; a, d, and g are not forces but are constant ratios, and do not change with azimuth or geographical position. These values depend only on the amount, arrangement, and capacity for induction of the soft iron of the ship. Thus a is the ratio between the component X and the com- ponent in the same axis of the force induced in fore-and-aft soft iron; d is the ratio between the component X and the component to starboard of the force induced in fore-and-aft soft iron and is zero when that iron is symmetrically situated with reference to the fore-and-aft vertical section through the compass; g is the ratio between the component X and the component downward of the force induced in fore-and-aft soft iron. In the same way it may be shown that the magnetism of the transverse iron is induced in it by Y and only by F, and that the force so induced is a specific fraction of Y with com- ponents bY to the ship's head, eY to starboard, hY vertically downward. Also, that the earth's vertical force Z, and that only, will 110 NAVIGATION COMPASS DEVIATION induce a force in the vertical soft iron which will be a specific fraction of Z , and the components of this force in the three axes will be cZ to ship's head, fZ to starboard, TcZ vertically downward. I, e, li, c, f, and Tc t like a, d, and g, are abstract quantities or constant ratios that do not vary with change of azimuth or geographical position, and are similarly denned; ~b, f, and h become zero when the transverse and vertical soft iron is sym- metrically situated with reference to the fore-and-aft vertical plane through the compass. If the transverse iron is so situ- ated, and extends across the ship, there will be induced a pole of North polarity on one side of the compass and a pole of South polarity on the opposite side at an equal distance from the compass and the vertical fore-and-aft section through it. One pole will repel and the other will attract the North point of the needle with an equal force, the components of which in the fore-and-aft line will be equal and of opposite sign; in other words, b will be zero. If the transverse iron does not extend across the ship, but is broken, the nearer pole on one side will be of one polarity and the nearer pole on the other side will be of the opposite polar- ity, and if the iron is symmetrical with reference to the fore- and-aft section through the compass, & will reduce to zero for the reasons above given. In like manner it may be shown that under the same cir- cumstances the components in the vertical direction will have equal values with contrary signs; in other words, h will re- duce to zero. It is evident that / will be zero when the vertical iron is symmetrically distributed on each side of the vertical fore-and- aft plane through the compass, since the pole of the system will lie in that plane. THEORY OF DEVIATIONS OF COMPASS 111 Fundamental Equations. It follows then that a compass needle on board an iron or steel ship is acted on by the follow- ing forces whose components are given for the axes of X, Y, and Z, in order : (1) The earth's magnetic force whose components in the three axes are X, Y, and Z. (2) The resultant pole of the ship's subpermanent mag- netism whose components are P, Q, and R. (3) The magnetic force due to transient induced magnetism in fore-and-aft soft iron whose components are aX, dX, and gX. (4) The magnetic force due to transient induced magnetism in transverse soft iron, whose components are IT, eY, and hY. (5) The magnetic force due to transient induced mag- netism in vertical soft iron whose components are cZ, fZ, and TeZ. Therefore, if X f , Y', and Z' represent respectively the com- bined forces due to the magnetism of earth and ship in the axes of X, Y , and Z acting on the North point of the needle, then X' = X + aX + IY + cZ + P. (17) Y' = Y + dX + eY + fZ + Q. (18) Z' = Z + gX + hY+JcZ + R. (19) These equations, first used by M. Poisson and now known by his name, form the groundwork of all equations used in the mathematical theory of the deviations of the compass. From the above equations it is plain that for the subper-' manent magnetism of the ship one or more permanent mag- nets, P, Q, and R, may be substituted, and for the soft iron 112 NAVIGATION COMPASS DEVIATION of the ship, whatever may be its amount or direction, nine soft iron rods, a, ~b, c, d, e, f, g, h, and lc, illustrated in Plate III, may be substituted ; the magnetism induced in the rods a, ~b, and c will produce a force in the axis of X, + if to head, ( ) if toward the stern ; the magnetism induced in d, e, and / will produce a force in the axis of Y, + if to starboard, ( ) if to port; and the magnetism induced in g, h, and Tc will produce a force in the axis of Z t + if vertically downward, ( ) if upward. Parameters. The quantities a, 6, c, d, e, f, g, h f ~k, P, Q, and E are called parameters and are constant; the first nine depending on the amount, arrangement, and capacity for in- duction of the soft iron of the ship, and the last three on the amount and arrangement of the hard iron. The parameters a, ~b, c, d, e, f, g, h, and Je are ratios though physically represented by rods. They are not forces but become forces only when multiplied by the inducing forces acting on the rods they represent. Thus the force induced in the rod a by the inducing force X is aX. A distinction must be drawn between the case in which the coefficient of a rod is '+, and that in which it is ( ) . These parameters are ratios between two forces and their signs depend upon the signs of the two forces. The components of the earth's force in the axes are taken as the inducing forces in those axes, and since the cosines of angles between 90 and 270 are negative, the inducing force in any axis is negative if the azimuth of that axis is between 90 and 270. Thus, if the ship heads NE., the inducing force from starboard is H cos (90 + z) = .707T, or a force equal to .707H" acts from the port beam. Now the force induced in a rod is posi- tive if it draws the North point of the needle to head, to star- ' board, or vertically downward. The following cases will serve to illustrate the points dis- cussed. In Fig. 44, the ship heading North, the inducing PLATE III. o -b" VJ +(7 ( DIAGRAM SHOWING THE POSITIONS OP THE NINE SOFT IRON ROBS WHICH REPRESENT THE WHOLE OF THE SOFT IRON OF A SHIP AS REGARDS ITS ACTION ON THE COMPASS. 114 NAVIGATION COMPASS DEVIATION force X from head is positive ; the near end of the rod a has South polarity and draws the North end of the needle to head and the force induced aX is positive, therefore the ratio = , v .= + a I n Fig. 45, the rod is acted upon by a negative force from ahead (H cos 180 = X, or E =( )Z), or the inducing force acts from the stern ; the near end of the rod has North polarity and repels the North end of the needle to the stern and the force induced aX is negative, therefore aX the ratio = seen to be + a. = -\- a. The coefficient of this rod is thus FIG. 44. In the case of an a rod extending continuously through the compass, the signs of the induced forces would be the reverse of the above, whilst the signs of the inducing forces would be unaltered, and the ratio in both cases would be a. In Fig. 46, the ship heads NE., and the azimuth of the rod e is SE. The inducing force, (H cos (90 + z) = F), is Y, and the starboard end has South magnetism induced in it. The force induced draws the needle to starboard and is 4-eY -\- eY, therefore the ratio = = e. In Fig. 47, the ship heads NW. The inducing force is + Y. The starboard end of the rod has induced in it North polarity which repels THEORY OP DEVIATIONS OF COMPASS 115 the North point of the needle, the induced force is eY ' , the _ ratio = , Y = e - ^ ne coefficient of an athwartship rod extending on both sides of the compass is thus seen to be - e. In the case of an e rod entirely on one side of the compass the induced forces would have exactly the opposite effect, whilst the inducing forces would remain unchanged in sign and the ratio would be -f- e. The signs of the coefficients of any of the remaining rods may, in a similar way, be proven to be as indicated in Plate III, remembering that it is convenient to consider the action on the N.W N.E. FIG. 46. FIG. 47. North point of the needle of only the nearer end of a rod which is represented as entirely on one side of the compass, but that in the cases in which the rod continuously extends from one side of the compass to the other, as in the cases of a f e, and Tc f either or both ends may be considered as acting. Five of the rods, a, c, e, g, and Tc f are symmetrically placed with reference to the fore-and-aft midship section; &, d, -f, and h are not so placed. If the soft iron of a ship may be supposed symmetrically distributed with reference to that plane, ~b, d, f, and h may each be considered equal to zero. 116 NAVIGATION COMPASS DEVIATION SECTION II. 75. Transformation of the fundamental equations. To adapt Poisson's equations to the various uses to which a navi- gator may apply them, they must first be expressed in terms of quantities usually given or desired, that is, in terms of H, the horizontal force of the earth ; H', the horizontal force of the earth and ship ; 0, the magnetic dip ; z, the magnetic course, or the azimuth of the ship's head measured eastward from the correct magnetic North ; z f , the compass course, or the azimuth of the ship's head measured eastward from the direction of the com- pass North; S = z ~ z', the deviation of the compass due to the compass heading z or the magnetic heading z. Eeferring again to Fig. 43, if m is the resultant pole of the ship's magnetic force, attracting the North end of the needle, Od will be the force in the horizontal plane through the compass, and the components of Om in the three axes X, Y, and Z respectively will be Oa, Ob, and Of. The earth's direc- tive force being OM, the ship's disturbing force Od, both in the horizontal plane, the needle will be acted upon by the resultant of these, Or = H', and the components of Or in the axes of X and Y respectively will be X' = On and Y' nr. (For z' in Fig. 43, nr is to port, and hence is negative.) Bearing in mind the definitions previously given, we have : X = H cos %, X' = H' cos z', Y =H cos (90 +z) = H sin z, Y' = II' cos (90 + *0 = H' sin z r , Z H tan 0. Substituting these values in equations (17), (18), and THEORY OF DEVIATIONS or COMPASS 117 (19), dividing (17) and (18) by H and (19) by Z, we have: TTl r> Sr cos z' = (1 + a) cos z & sin z + c tan + (20) XI XI ^ sin z' = d cos 3 (1 + e) sin z + f tan + -(21) H H ~ = ?-;; cos z - -A- sin 2 + 1 + + ~. (22) ^ tan0 tan0 Z Since .ZT is the force of the earth and ship in the direction of the disturbed needle, and since a force can be resolved into two components, in axes at right angles, all in the same plane, by multiplying it by the cosine of the angle its direction makes with each axis, and since the denominator in each equation denotes the unit of measure, equation (20) gives the force of earth and ship to head, equation (21) the force of earth and ship to starboard, each in terms of the earth's horizontal force as unit; equation (22) the force of earth and ship downward, in terms of the earth's vertical force as unit. As the azimuth of the ship's head may be anything from to 360, it is de- sirable to have these forces in two fixed directions, in the magnetic meridian and at right angles to the meridian. 76. Force of earth and ship to magnetic North. To find the components, in the direction of magnetic North, of the forces of earth and ship acting to head and to starboard, mul- tiply (20) by cos z; and (21) by cos (90 + z), or what amounts to the same thing, by ( sin z) ; and take the alge- braic sum of the results. This will give the force of earth and ship to magnetic North. Performing the operations indicated, we have : TTf JT cos z' cos z = (1 + a) cos 2 z & sin z cos z p -f- c tan 6 cos z + -^- cos z. Also, TTt _ sin z' sin z = d cos z sin z 1 + (1 + e) sin 2 z f tan 6 sin z -? sin z. 118 NAVIGATION COMPASS DEVIATION And by algebraic addition, TTt j^ (cos z' cos z + sin z' sin z) = (d + 6) sin z cos z + (1 + a) cos 2 z + (1 + e) sin 2 z + (c tan + TT ]cos 2! (/tan -f ^ J sin z. From plane trigonometry, cos z' cos z -f- sin z' sin 2 = cos (z ~ z') = cos 8, cos< , = 1 + os ^ sin 2 z = 1 -? ga ' and sin 2 cos * = A Therefore, by substitution, lcoB* = - (d + *) 5*** + (1 + a) Collecting the terms involving the same function of the azimuth z, we have the force of earth and ship to magnetic North in terms of the earth's horizontal force as unit, expressed by the equation, ^cos (5 = 1 + ^ + (c tan + ~) cos z - (/tan + ^ sin + 5LzJ cos 2^ (23) \ -"/ * . So Force of earth and ship to magnetic East, and hence of ship alone, as the force of earth to East is always zero. In a simi- lar way, we may find the components in the direction of mag- netic East of the forces of earth and ship acting to head and to starboard, multiplying (20) by cos (90 z) = sin z, and (21) by cos z, and adding the results algebraically. THEORY OF DEVIATIONS OF COMPASS 119 Performing these operations, making trigonometric substi- tutions and collecting the terms as before, we have the force of the ship alone to magnetic East in terms of the earth's hori- zontal force as unit, expressed by the equation, l^sin d = fL_l + (c tan + ?\ sin z + //tan + ^\ cos z a ~ e sin 2 z + d + b cos 2z. (24) If observations be made on four or more equidistant azi- muths, the mean of results from (23) will be the mean value of the force of earth and ship to magnetic North; and the mean of results from (24) will be the mean value of the force of the ship to magnetic East. The mean values from each equation will be represented by the constant terms, since the sum of two, four, or more equi- distant values of sine or cosine is zero, and terms having sine or cosine as a factor will disappear. Therefore, the mean value of the force of earth and ship to magnetic North in terms of H as unit is 1 + s 5 this quantity, called A, is generally less than unity in iron ships, a fact which indicates a mean directive force on the needle less than the earth's directive force in such ships. The mean value of the force of ship to magnetic East in terms of H as unit is -~ This quantity is + when the easterly deviations are in excess of the westerly deviations, ( ) when the westerly deviations are in excess. This term reduces to zero when the soft iron of the ship is symmetrically distributed with reference to the fore-and-aft vertical section through .the compass. Letting 1 + 2-+J be represented by X, and dividing (23) 120 NAVIGATION COMPASS DEVIATION and (24:) by A, we have, and + /tan + cos 2 + -sin 2z (26) , 1 d+ I + j o cos 2 z. \Hj which is the mean value of H' cos 8, or of the force of earth and ship to magnetic North,, will be referred to hereafter as " the mean force to North." To simplify these expressions, let their constant terms and the quantities connected with the various functions of the magnetic course be represented by the old English capital let- ters as follows = * ; 1 (/tan, Then by substituting these in (25) and (26), we have: TJI Yg cos 8 = 1 + $8 cos z ( sin z + cos 2z (5 sin 2z. &- sin 8 = 51 + '93 sin z + ( cos + 3) sin 2^ + @ cos 22. Equation (27) gives the combined force of earth and ship to magnetic North and equation (28) the force of the ship to magnetic East, both in terms of the " mean force to North " as unit of measure. The mean value of (28) is ST, the mag- nitude and sign of which will indicate the excess of easterly over westerly deviations, or the reverse. THEORY OF DEVIATIONS OF COMPASS 121 77. Formulae for computing the deviations. Dividing (28) by (27), we obtain 3T + 33 sin z + < cos z + S) sin 2z + ( cos 2z tan 8 = L _J _J ! /on } 1 + 33 cos s ( sin 2 + 2) cos 2z & sin 2z { which will give the deviation (8), by means of its tangent, on any magnetic course z when the five coefficients 31, 33, (, 2), and @ are known. Substituting ^-^ for tan 8 in (29) and clearing of frac- tions, we have : sin 8 -\- 3} cos z sin 8 ( sin 3 sin 8 + cos 22 sin 8 ( sin 22 sin 8 51 cos 8 + ^3 sin 2 cos 8 + @- cos z cos 8 + 2) sin 2z cos 8 + @ cos 2s cos 8. Transposing and collecting terms, sin 8 = 31 cos 8 + 33 (sin z cos 8 cos z sin 8) -}- ( (cos 2 cos 8 -\- sin 2 sin 8) -j- 2) (sin 22 cos 8 cos 22 sin 8) + ( cos % z cos ^ + s i n % z s ^ $)> but from plane trigonometry, since z' = z 8 and 22' = 2 (z 8), or 2 (z f + ~] = (22 8) = (22' + 8), we shall have sin 8 = 31 cos 8 + 33 sin z f + ( cos z' + 3) sin (22 r + 8) + (cos (22' + 8). Equation (30) gives the deviations by means of its sine nearly, though not entirely, in terms of the compass course z'. When the deviations are of moderate amount, say not more than 20, equation (30) may be written with sufficient ac- curacy, 8 = A + B sin z 9 + C cos z f + D sin 2z' + E cos 2z' (31) in which the approximate coefficients, A, B, C, D, and E may be given in degrees and minutes and the deviation (8) deter- 122 NAVIGATION COMPASS DEVIATION mined for the various compass courses in degrees and min- utes, without introducing a greater error than 25' in the com- puted deviation, even when near the agreed upon maximum limit of 20. In formula (30), the coefficients Sf, SB, <, 2), and (, called the exact coefficients of the various deviations produced, are in reality the forces producing those deviations expressed in terms of the "mean force to North," (A~ff), as unit; they are very nearly the natural sines of the angles represented by the approximate coefficients A, B, C, D, and E of equation (31), if 8 is of moderate amount! When the soft iron is symmetrically arranged on each side of the fore-and-aft vertical plane through the compass, b = 0, d = 0, / = 0, h = Q, and therefore both 51 and ( reduce to zero, and equations (29) and (31) become respectively, 33 sin z + G cos z + si 11 && . " 1 + 93 cos z ( sin z + cos 2z and 8 = B sin z' + G cos z' + D sin 2s'. (33) Equation (30) will become, by developing the terms in the second member, sin 8 = 21 cos 8 + 33 sin z' + ( cos 3' + 3) sin 2s' cos 8 +. S) cos 2s' sin 8 + ( cos 2z f cos 8 sin 2z' sin 8. In this expression, 8 being small, we can let cos 8 equal unity without material error, and the above becomes, sin 8 = 51 + 33 sin z' + ( cos z' + $ sin 2z' + ( cos 2/ + ) cos 2/ sin 8 @ sin 2z' sin 8. ( is very small and sin 8 is very small, therefore the last term @: sin 2z f sin 8 is very small and may be neglected, hence, by transposing cos 2z' sin 8 and dividing through by the expression (1 S) cos 2z'), we have : sn = (34) 1 The correct angular measure corresponding' to any arc value is obtained by multiplying said arc by the radian 57. 3; thus D = S3 x 57.3. THEORY OF DEVIATIONS OF COMPASS 123 which is very nearly exact and gives the deviations in terms of the exact coefficients and the compass courses. When the soft iron is symmetrically distributed 51 and ( will disappear in (34). 78. Subdivisions of the deviation. In (31), the several parts of the deviation are: A, the constant deviation due to transient magnetism in- duced in soft iron represented by parameters d and ~b, or to constant errors of observation, etc. ; B sin z r +. cos z', the semicircular deviation due to sub- permanent magnetism and the transient magnetism in- duced in vertical soft iron; D sin 2z' + E cos 2z', the quadrantal deviation. The first or larger part, D sin 2z' 9 is due to transient magnetism in horizontal soft iron symmetrically situated with reference to the fore-and-aft vertical plane through the compass. The second, or smaller part, E cos 2z' 9 is due to transient magnetism induced in horizontal un- symmetrically situated soft iron. 79. Relation between A and ). Both these coefficients de- pend for their value on the parameters a and e. The coeffi- cient , which equals - a "~ e , represents the force in terms A til of \H which produces the larger part of the quadrantal devia- tion, the force itself being XffS). This part is generally posi- tive, being due to -(- a and e f or the excess of e over a. The horizontal force at the compass is that of the earth, the ship's subpermanent magnetism, and that induced in soft iron combined and acting in the direction of the compass needle, this resultant force being represented by H'. The component to magnetic North is H' cos 8 and the mean value of this com- ponent for an entire revolution of the ship's head is X5", which is generally less than unity, H itself being considered as unity. In the complete revolution of the ship's head, the increase of 124 NAVIGATION COMPASS DEVIATION directive force due to the subpermanent magnetism in one semicircle exactly equals the decrease due to the same cause in the other semicircle, so the diminution in the value of XH is not due to subpermanent magnetism but arises from hori- zontal induction in soft iron represented as constituent parts of A, therefore A is generally less than unity and as it equals 1 + a T" e , the diminution is due to a and e, or the 3 x + M s = E s etc., etc. Squaring both members of each equation and adding, we have To find the most probable value of x, we must make the -sum of the squares of the errors a minimum, or, what amounts to the same thing, make the left hand member of equation (40) a minimum. Differentiating it with respect to x and placing the first differential equal to zero, after dividing through by the com- mon factor 2 dx, we have (6, x + M,) I, + (& 2 x + M 2 ) Z> 2 + (6. x + M 3 ) I, + . . . = which gives the most probable value of x. The equation (41) is called the normal equation in x and equals the sum of the equations in group (37) taken after each equation has been multiplied by the coefficient of x in that equation. THEORY OF DEVIATIONS OF COMPASS 131 In the same way, letting N i9 N 2 , N s , etc., represent in the equations of group (38) all the terms independent of y, we will have the normal equation in y (c 2 y + N 2 ) c 2 which gives the most probable value of y. This normal in y equals the sum of the equations in group (37) taken after each equation has been multiplied by the coefficient of y in that equation. Similarly, we may find the normal equation in z, and in all the other unknown quantities. The normal equations will be the same in number as the unknown quantities, and the value of these quantities ob- tained therefrom will be the most probable value. Comparing the normal equations with the equations of con- dition, the following rule for the formation of the normals is evident. "Multiply each equation of condition by the coefficient of each unknown quantity in that equation taken with its sign. The sum of the resulting equations in which the coefficients of x are multipliers will be the normal equation in x } and simi- larly for the others/' 86. Let 8 , 8 , 8 2 , etc., be the deviations obtained from observations when the ship heads on the 15 rhumbs per com- pass represented by z' , z\, z' 2 , etc., then (31) will become sin z' -\-C cos z' +D sin 2z' +E cos 2z' * sin z\+C cos z\+D sin 2z\+E cos 2z\ 8 2 =A+B sin z' 2 +C cos z' 2 +D sin 2z' 2 +E cos 2z' 2 .(43) =A+# sin z' 23 +C*cos z' 23 +D sin 2z' 2S +E cos 2z' 2 132 NAVIGATION COMPASS DEVIATION In the above the sines and cosines may be replaced by the symbols S , S lf S 2 , etc., representing the sines of angles of 0, 15, 30, etc., remembering, however, that the cosine of an angle is the sine of its complement, that is, cos z = siii z 5 , cos z 2 = sin z 4 , etc., also that sin 2z^ = sin z 2 , or sin 2z t = 8 2 , etc. For uniformity's sake, the symbols 8 Q and 8 6 which are respectively zero and unity will appear as multipliers. Strict attention must be paid to the signs of the functions on the azimuths used. Making the proper substitutions we have 8 = A + B8 + C8 Q + D8 + E8 Q 8 = A. + BSi + C8 5 +.D8 2 + E8, S 2 = A+BS 2 + C8 + D8, + ES 2 (44) 8 23 = A B8i + C8 5 DS 2 + E8 which are the 24 observation equations, or equations of con- dition, from which the five normal equations must be found by the method of least squares as expressed in the rule of Article 85, before the coefficients A, B, C, D, and E can be found. To form the normal equation in A. Multiply each equa- tion of group (44) by the coefficient of -A in it; the coefficient in each case being unity, no change will be made in any term. Then adding all members on each side of the equality sign, we shall have. 8,, = 24A. (45) Since the sines and cosines of courses differing by 180 are numerically the same with opposite signs, their sum is zero, THEORY OF DEVIATIONS OF COMPASS 133 and the summation of sines, or cosines, on an even number of equidistant azimuths is zero; hence terms involving B, C, D, and E do not appear in (45). Equation (45) may be put in the following form : * (*9 + *2l) >~ f = 6 ^, < 46 > + * U (4 + 1G> + i (10+ ) I" + H * (5 + 5 17> + * (11+ S 23 ) }" J which explains the formation of columns (5) and (11) of the analysis sheet, Article 89, and the steps leading to the find- ing of A; for instance, J(8 + 8 12 ) will give the mean value of A and E on N. and S. ; \ (8 6 + 8 18 ) will give the mean value of A and E on E. and W. ; the A will have the same sign in both cases, the E will have the contrary sign in each case; therefore, i -{ i(8 + 12) + 4(8 + 8 is) f will give the mean value of A on four rhumbs, and as each term in a bracket rep- resents a mean value of A for four rhumbs, we shall have 6 A on the opposite side of the equality sign in (46). To form the normal in B. Multiply each equation of group (44) by the coefficient of B in that particular equation, having due regard for the signs ; in other words multiply the first equation by S , the second by S 19 etc., then add results on both sides of the equality sign. On the right hand side all the coefficients except B will disappear as the summation of the resulting multipliers for them will consist of two parts, each part containing identical terms with opposite signs; therefore, .S, = B \ 2 (# * + # 6 2 ) + 4 = B -{ 2 + 4 (1 + 1 + i) }- = 12B and the normal in B is, for observations on 24 15-rhumbs, 80^0 + 81^1 + 82^2 ---- 8 28 #1 = 125. (47) The grouping of the deviations in the analysis sheet, the finding of column (6) marked semicircular deviation, and 134 NAVIGATION COMPASS DEVIATION the use of the divisor 6 in finding B from observations on 24 headings is thus explained; i. e., since S is multiplied by the same quantity numerically as S 12 , S x the same as 8 13 , etc., but with a different sign, and since semicircular deviation is the algebraic difference of the deviations on opposite rhumbs di- vided by 2, if the results in (47) be grouped according to the multipliers and divided by 2, then (47) will become ..) (52) + 4 -I 4(8 4 + 5 ie) - 4(8io+ 5 ia) } fi 1 * + 4 U (8 6 + 8 U ) - i( 5 n+ *2s) }- S 2 J which explains the formation of columns (5) and (12) and the steps leading to the determination of D in the analysis sheet. The normal equation in E. By the same rule and methods similar to those pursued in the case of D, we have the normal equation in E : 80^6 + 81^ + 82^2 +Z 23 8 4 = 12E. (53) If equation (53) is grouped by multipliers and divided twice by 2, we shall have J- 8 6 +4-1 i(S 1 +Si 3 )-4(S T +*w) r S* "] l- S 2 +4U(83+i6)-4(9+a)}- ^o ^=3^ (54) H - fia K4-I 4(6+8ll)-4(*U+*88) } "I -#4 U which explains the steps leading up to the determination of E in the analysis sheet. If observations are taken on only 12 equidistant headings, the divisors in the analysis sheet will be for A, E, and Q, 3 instead of 6, and for D and E, f instead of 3. If taken on only the eight principal rhumbs, the divisors for A, B, and C will be 2, and for D and E unity. 87. The expression S = A + B sin z' + C cos z' + D sin 2z' + E cos 2' considers the deviation as composed of only the constant, semi- circular, and quadrantal deviations, while in fact terms of a sextantal'type, octantal type, etc., may exist. This is shown by the following equation : S = A + B sin 2' + C cos z' + D sin 2z' + E cos 2.z' + jP sin 3z' + cos 82' + H sin 4z' + K cos 4s' (55) + L sin 5s' + Jf cos 5z' + tf sin 6z'. 136 NAVIGATION COMPASS DEVIATION which is obtained from (30) by a series of expansions, sub- stitutions, and eliminations,, and which is exact to terms of the third order inclusive. 88. Determination of exact coefficients. For all practical purposes the deviation is expressed with sufficient accuracy by the five approximate coefficients ; if, however, greater accuracy is required, as when the deviation much exceeds 20, it should be expressed by the exact coefficients. These may be deter- mined by the method of least squares, or from the following equations involving the approximate coefficients : 2T = sin4. (56) 33 = sin B (1 + sin D + T V versin B J versin C) + sin X sin #. (57) & = sin C (1 ^ sin D J versin B -f- T V versin C) + 4 sin B X smE. (58) = sin D (1+ $ versin D). (59) ( = sin E sin A X sin D. (60) SECTION IV. 89. Analysis of deviations and the use of the form. The deviations entered on this form should be for the 15 compass rhumbs, and, if the observations were not on these rhumbs, plot the deviations on a Napier diagram, and take off the total deviation on each 15 compass rhumb and transfer it to column 2 or 4, Table I, of the form. The following ex- ample will illustrate the process : Ex. 8. Compute the approximate and exact coefficients from the deviation table found in Ex. 4, Art. 56. Find also the starboard angle a. (1) Write down the observed deviations in columns (2) and (4), opposite the proper rhumbs, prefixing the sign -f- "to the easterly deviations, the sign ( ) to the westerly devia- tions. THEORY OF DEVIATIONS OF COMPASS 137 (2) Form column (5) by taking half the algebraic sum of columns (2) and (4). Since the constant and quadrantal deviations have the same sign and the semicircular has the opposite sign on azimuths differing 180 , this process elimi- nates the semicircular deviation, and column (5) records the constant and quadrantal deviation on the equidistant 15 rhumbs per compass from to 165 inclusive, or from 180 to 345 inclusive. (3) Form column (6) by taking half the algebraic differ- ence of columns (2) and (4) ; or what is the same thing change mentally the signs of the quantities in column (4), then take half the sum of columns (2) and (4), entering re- sults in column (6). This process eliminates the constant and quadrantal deviations, column (6) being the semicircular de- viation on the equidistant 15 rhumbs per compass from to 165 inclusive, or from 180 to 345 inclusive, if we con- sider the signs changed. The correctness of columns (5) and (6) may be proved by adding them algebraically; the sum of the quantities opposite any heading should equal the quantity in column (2) oppo- site the same heading. (4) Since the semicircular deviation on any compass azi- muth of the ship's head z' is B sin z' + C cos z' y the quan- tities in column (6) are multiplied by the multipliers set opposite them in columns (7) and (8) to form respectively the products of columns (7) and (8), the first set of multipliers represented by S , 8 lf S 2 . . . . 3 being the natural sines of the rhumbs 0, 15, 30 .... 90 respectively, and the sec- ond set represented by S 6 , 8 5 . . . . 8 being the natural cosines of the same rhumbs, S being and S Q unity. The multiplication is facilitated by Table IV of this book, or by Table X of " Diehl's Compensation of the Compass." When the angle is greater than the tabulated arc, as 42 10', it may be divided into two parts, each part to come within the 138 NAVIGATION COMPASS DEVIATION limit of the table, as 30 and 12 10', and' the sum of the results for the two parts taken. In these multiplications careful attention must be paid to the rule of signs; that is, + multiplied by +> or ( ) multi- plied by ( ) gives -}-, and + multiplied by ( ) gives ( ) . The algebraic sum of the products in each of the columns (7) and (8) divided by 6, 3, or 2, according as the obser- vations were taken on 24, the 12 or 8 principal compass rhumbs, will give from column (7) the approximate coefficient B and from column (8) the approximate coefficient C. An approximate check on B may be obtained by taking the mean of the deviations at East and at West, the sign of the latter being changed; an approximate check on C may be obtained by taking the mean of the deviations at North and South, the sign of the latter being changed. (5) Proceed to find A, V, and E, following in Table II a process similar to that followed in finding B and C. Write down the upper half of column (5) in column (9) and the lower half of column (5) in column (10). From columns (9) and (10), form columns (11) and (12) in the same way in which we formed columns (5) and (6) of Table I, and prove their correctness in the same manner. It will be readily seen that by this process we have separated the constant and quadrantal deviations; column (11) is the constant part of the deviation, each of the eight values being derived from the deviations on four rhumbs of the compass 90 from each other, being the mean of the deviations as repre- sented in the brackets of the left-hand member of equation (46). (6) The sum of the quantities in column (11) divided by 6, 3, or 2 according as the observations were taken on the 24, 12, or 8 principal rhumbs will give the value of A. An ap- proximate check on A may be obtained by taking the alge- braic mean of the deviations on the 4 cardinal points. ANALYSIS OF DEVIATIONS 139 Column (12) is the quadrantal deviation on 15 rhumbs from to 75 and from 180 to 255; or, with the signs changed, the quadrantal deviation from 90 to 165, or from 270 to 345. Each of the eight values in column (12) is derived from the deviations on four rhumbs of the compass 90 apart, as shown in equations (52) and (54). (7) Since the quadrantal deviation on any compass head- ing of the ship z f is D sin 2z' + E cos 2z', the quantities in column (12) are multiplied by the multipliers set opposite them in columns (13) and (14) to form respectively the prod- ucts of columns (13) and (14), the first set of multipliers being the natural sines of twice the azimuth of the ship's head, the second set being the natural cosines of twice the same azimuths. In these multiplications careful attention must be given to the signs. The algebraic sum of the products in each of the columns (13) and (14) divided by 3, f , or unity, according as the observations were taken on 24, the 12 or 8 principal compass rhumbs, will give from column (13) the approximate coeffi- cient D and from column (14) the approximate coefficient E.' An approximate check on D is the mean of the deviations on the quadrantal points, the signs of the deviations on SE. and NW. being changed before the mean is taken. An approximate check on E. is the mean of the deviations on N., S., E., and W., the signs of the latter two being changed. 140 NAVIGATION COMPASS DEVIATION = sin D [1 +J versin D] = + . 05001 [1 + .00042] = + . 0500 6 = sin E sin A sin D = -.00378 +.0092 X -05001 = (-) .0033 < _ .0863 log. 8 . 93601 tan x = -- = + .2197 log. 9.34183 X = 21 26'44" log. tan. 9 . 59418 a = 338 3316". SECTION V. 90. Determination of A. This coefficient is the one which expresses the proportion of the mean horizontal force north- ward of the earth and ship to the horizontal force on shore and may be found from equation (27) written as follows, jj' cos 8 ' IT 1 + $ cos z ( sin z + cos 2z @ sin 2z ^ 61 ^ when the horizontal force and the deviation for the magnetic azimuth z are known in addition to the exact coefficient 23, (, 3), and @. In the above equation H' = the horizontal force of earth and ship combined ; H = the horizontal force of earth, considered as unity ; 77' y 7 * - = -^73-, T being the time of n vibrations (say 10) of a small horizontal needle, 3 to 4 inches long, on shore in a place free from local magnetic disturbances; T' that of the same number of vibrations of the same needle, the center of which is in the same place exactly as that occupied by the center of the compass needle, when the compass is in place. OBSERVATIONS FOR HORIZONTAL FORCE 143 Great care should be exercised in taking the vibrations, and the mean of a number of determinations should be used, since the error of a single set might be comparatively large. 77' T 2 The equation ~ = - is true only for infinitesimal arcs of vibration, but may be taken as sufficiently exact for all practical purposes if the arcs do not exceed 20. However, the amplitude of arc should be as small as possible consistent with obtaining 10 well-defined vibrations. The place on shore where the needle is vibrated should be free from local attraction, a fact that may be determined in the following way, namely: place a compass on its tripod and set up a staff about 50 yards distant, note the bearing of the staff per compass ; interchange tripod and staff and again note the bearing of staff. Do the same thing on a line per- pendicular to the first line. If the bearing and reverse bear- ing in each case differ by 180, the locality may be assumed free from magnetic local influences. The horizontal force instrument. This instrument is used in finding the ratio of the horizontal force on board ship in the position of the compass to that on shore. It consists of a cylindrical brass case, with a removable glass cover, mounted upon a rectangular base which is provided with levels and leveling screws. The case contains a horizontal circle graduated to degrees, and in the center a pivot which supports a small lozenge- shaped magnetic needle fitted with an adjustable sliding weight to counteract the dip and capable of vibrating freely in the horizontal plane. Observations for horizontal force ashore. Find a level spot free from local attraction, level the horizontal force instru- ment and orientate it. By means of a small magnet draw the needle aside about 20, quickly removing the magnet to a proper distance. Then as the needle passes the zero line the 144 NAVIGATION COMPASS DEVIATION first time " mark the time " or start the stop watch, as the needle passes the zero line the second time count " one/' at the next passage "two/' and so on till the count "ten/' when the time is again noted or the stop watch stopped. The interval of time will be the time required by the needle to make 10 vibrations. Observations on board. Observations are similarly taken on board, the center of the horizontal force needle occupying the exact place usually occupied by the center of the compass, which with all correctors must be removed to a safe dis- tance. The horizontal force instrument is leveled on a brass table in the compass chamber, the spindle of the table entering the central vertical tube of the binnacle. The magnetic azimuth and deviation may be determined by any one of the usual methods. The coefficients should have been determined as accurately as possible on equidistant compass courses ; 24, 12, or 8 equi- distant headings. If the observations be taken on four equidistant magnetic azimuths we will have A = iS^cos8 (62) because, the summation of the sines and cosines being zero, the exact coefficients will disappear. NOTE: In the service compass, the plane of the needles is three- fourths of an inch below the bottom of the wyes in which the compass rests when placed in the binnacle, and it may be located by placing in the wyes a straight edge at the center of which is pasted a piece of paper projecting vertically downward % of an inch. COMPUTATION OF A 145 o -*- o 1 ,0 M ! I 05 S o S 5 2 < a . cJ O II 2 S *S* o 23 Q .a PQ S & S 5s ^ ^ . II ^ GQ m I i cj =P1 * H O 5 o ^ d 5 ^ -.2 b O CT 1 *"* ' O tQ 15 0) ~ O +~ CO . s co n w o i! i sl v ii Pltd 146 NAVIGATION COMPASS DEVIATION Ex. 10. It is required to find A, from the following observa- tions for horizontal force made ashore and on board a moni- tor in the position of the standard compass, the magnetic courses and deviations being found by interpolation in the Standard's table of deviations, Art. 55. Magnetic heading. Deviations. Horizontal vibrations. North 4 35' T 14 S .60 T 15.66 East + 12 00 16.28 Sputh + 4 54 18.17 West 14 09 17.80 For Head, North. For Head, East. 15 S .66 log 1.19479 15 S .66 log 1.19479 14 .60 log 1.16435 16 .28 log 1.21165 0.06088 9.96628 8 + 4 35' cos 9.99861 8 2 + 12 00' cos 9.99040 ^'cos 84 1.1468, log 0.05949 $ cos 8, .9051, log 9.95668 a. n. For Head, South. For Head, West. 15 3 .66 log 1.19479 15 S .66. ., log 1.19479 18 .17 log 1.25935 17 .80 log 1.25042 9.87088 9.88874 54' ____ cos 9.99841 . 8 4 14 09'. . .cos 9.98662 S* cos 8 3 .7401, log 9.86929 jf cos 8 4 =.7505, log 9.87536 H " 1.1468 + .9051 + .7401 + .7505 A,= - - = .0000 DETERMINATION OF ^ 147 91. Determination of a, e, ~b, and d, given 51, (, 3), and A,. .Zfo. .n. Given the following coefficients (Exs. 8 and 10), 9t = .0092, @ = .0033, = .0500, A = .8856, it is required to find a f e, 6, and d. See Art. 79 and Art. 80. a = /l(l + 3))-l = .8856 x 1.05 - 1. = .9299 1. = (-).0701 = A (1 - 5)) - 1 = .8856 x .95 1 = .8413 - 1 = (-J.1587 d b = 2291 = 2 x .8856 x .0092 = .0163 d + 6 = 2A( = 2 x .8856 x .0033= .0058 2d= - .0221 ... d = -.0110 26= + .0105.-. & = + .0052 92. Determination of parameters g and ft and the vertical force of the earth and ship. In equation the vertical force of the earth and ship is expressed in terms of the earth's vertical force as a unit of measurement. 7' The mean value of -~ on two or more equidistant azimuths (J? \ 1 -|- Ic -f- - } of the second member. Z' Letting p be the mean value of ^, or the mean force down- ward of the earth and ship in terms of the earth's vertical force as unit, then ^=1 + ^+ f ; therefore, From (63), the value of /*, -9- , and -r - are derived tantf' tan0 from observations on 4, 8, 12, or 24 equidistant courses, using similar tabular forms to those used in finding the approximate coefficients. 148 NAVIGATION COMPASS DEVIATION As with the horizontal force instrument, the times of, n vibrations of a dipping needle may be observed on board and on shore, the vibrations being made in a plane perpendicular to the compass meridian on board and perpendicular to the magnetic meridian ashore. The dipping needle is correctly placed for vibrations when, at rest, the magnetic axis of the needle is vertical. If T be the time of say 10 vibrations of this needle on shore, and T' that of the same number of vibra- tions on board, the center of the needle being in the exact position occupied by the center of the compass needle, then >7t /7T2 * = . The times of vibrations are obtained in the same way as with a horizontal needle, the needle being deflected from the zero point about 10 in this case. The magnetic azi- muth of the ship may be obtained at the same time in any one of the usual ways. Owing to the fact that the sine and cosine of angles differing Z* 180 have opposite signs, //, will be the mean of -= observed on two opposite magnetic headings. Eegarding h as zero, if g is known, we may find ^ from one Z f observation of-^-; if one observation is made on magnetic j Z' East or West, g will disappear and /*, will equal - . Z' If observations for -^- are made on four equidistant mag- netic headings, then fjL=lS.. (64) If the observations are made on N"., E., S., and W., mag- netic, then JJL may be found from all four, g from the observa- tions at N". and S., li from those at E. and W., a fact that is apparent from a consideration of equation (63). DETERMINATION OF 149 Ex. 12. It is required to find //, g, and h from the following observa- tions for vertical force made ashore and on board a " Monitor " in the position of the standard compass, tan 6 being 2.86. Mag. heading. North East South West For Head, North. 19 S .75 log 1.29557 18.43 log 1.26553 -p= 1.1484... log 0.06008 /v For Head, South. 19 S .75 log 1.29557 19.08 log 1.28058 0.01499 2 Vertical vibrations. T' 18 S .43 T 19 S .75 18.94 19.08 19.00 For Head, East. 19 S .75 log 1.29557 18.94 log 1.27738 0.03004 0.01819 2 2 Z = 1.0874... log 0.03638 For Head, West. 19 S .75 log 1.29557 19.00 log 1.27875 0.01682 2 ~? = 1.0715. . .log 0.02998 ^ = 1.0805. . .log 0.03364 + 1.0874 + 1.0715 + 1.0805 4 On K, ^ = 1.1484 On West, ~ On S.,^f = 1.0715 On East, : 2)0.0769 2) 0.0069 --= 0.0384 5-^-27 = - 0034 tan 6 tan g .0384 X 2.86 = .1098 h .0034 X 2.86 = ( ) .0097 9 . being the value of ^ at N". the value of -^ at S. tan Z Z ^ h being the value of -' at W. the value of ^L at E. tan e Z Z 150 NAVIGATION- COMPASS DEVIATION SECTION VI. 93. Other methods of finding the exact coefficients 93, (, and 2). In the case of a compass well located on board ship, 3t and ( are either zero, or very small, and for all practical purposes may be neglected without appreciable error. The equations for deviations then become very simple on the two cardinal and the intercardinal points of any quadrant. From observations made on such points the compass may be fairly well compensated, and, in the case of one already compen- sated, a very good residual curve may be obtained by sub- stituting the resulting values of 93, (, and S) in equation (34), as illustrated in Art. 97. This is a good method whether at sea or at anchor in port swinging to tide. Even if at a dock, a vessel's head may be sprung around sufficiently to get the required observations. When practicable, choose that quadrant in which the direc- tive force on the needle is strong. In NE. quadrant. Letting 51 and @? be zero, we shall have for compass courses North, NE., and East, from (30), the fol- lowing : (1) sin 8 = < + sin 8 .'. < = sin 8 (1 3)). (2) sin 8, = 33 3 + <5 8 + 3) cos 8 8 . (3) sin S 6 = $ sin S 6 .'. 93 = sin 8 6 (1 + ). Substituting value of 33 and ( in (2), sin 8 3 = 3 [sin 8 (1 )+ sin 8 C (1 + $)]+ cos 8 3 . + -j cos 8 3 Si (sin 8 sin 8 6 ) } sin 8 8 cos 8 3 # 3 (sin 8 sin 8 6 ) OBSERVATIONS IN ONE QUADRANT 151 Multiplying through by $ 3 , and as # 3 2 = -J, we have for the NE. Quadrant, _ 3 sin 8, & (sin 8 + sin 8 6 ) S 3 cos 8 3 i(sin 8 sin 8 6 ) 23=(l + S))sin8 6 <=(! >)sin8 ( = (1 2))sin8 12 And for the SW. Quadrant, #3 sin 8 15 | ( sin 8 12 -f- sin '18 (65) Similarly for the SE. Quadrant, - 8 3 sin 8 9 + (sin 8 6 + sin 8 12 ) i(sin8 6 sin8 12 ) , ,, ? 8 cos8 1B - 4(sin8 12 sin8 18 ) L ^ And for the NW. Quadrant, _ 8 3 sin 8 21 -\- % (sin 8 18 -\- sin 8 ) # 3 cos 8 21 i(sin 8 18 sin 8 ) , ( . 93 = (l + 3))sin8 18 = (1 S))sin8 If observations have been made on the three cardinal points and two intercardinal points of one semicircle, consider each quadrant of that semicircle separately, find the values of 58, (, and 3), from each, and take the mean of the two determinations of each coefficient; or, combine the formulae in the proper quadrants before proceeding with the computation. 152 NAVIGATION COMPASS DEVIATION Ex. 13. A distant object, the magnetic bearing of which was 326 45' bore (p. s. c.) respectively 355, 335, and 328, as the ship headed (p. s. c.) successively South, SW., and West. Eequired 33, (, and 3). Deviation on South = 8 12 = 28 15', sin 8 12 = .473, cos 8 12 + .881. Deviation on SW. = S 15 = 8 15', sin 8 15 = .143, cos 8 15 = + .990. Deviation on West = S 18 = 1 15', sin 8 18 = .022. _ .707 x (-.143) _ i (-.473 - .022) ~ .707 x .990 \ \ .473 (.022) [- -.1011 +.2475 .1464 + .1582 .6999 +.2255 .9254 5)3 _ _ (1 + .1582)( .022) _ 1, 1582 x. 022= +.0255 ( = (1 .1582)( .473)= +.8418 x .473 = +.3982 94. Determination of 33 and ( from observations of devia- tion and horizontal force on one heading. Regarding 51 and @? as zero, that is & and d as zero, we have from (20) and cos z' = (1 +a) cos z + c tan + or, jp cos '=(! + a) cos z + A$. (69) |^sin z' = (1 + e) sin z + / tan + Q. H n. or ^sin z' = (1 + e) sin z + AS. (70) H Substituting the values of (1 + a) and (I -\- e) from (35) and (36), transposing and dividing through by A, we have, 33 = -^cos z f (1 + 2)) cos -z. (69a) A/2 S = -^sin z' + (1 >) sin a. (70a) x/z 2' is the azimuth of the ship's head per compass; z is the magnetic azimuth of the ship's head and may be determined by OBSERVATIONS ON Two HEADINGS 153 a time azimuth of the sun, from the bearing of a distant object of known magnetic bearing, or by reciprocal bearings ; after the above data have been obtained, remove the compass to a suffi- cient distance and take vibrations of a horizontal needle in the exact place of the compass needle, calling T' the time of 10 vibrations. Take vibrations of the same needle ashore, H' T 2 calling T the time of ten vibrations there, then -77-= 777,2. 1 J- Therefore, if A and can be obtained; 23, (, and a may be found. These coefficients, A and ), are so nearly the same for com- passes in similar positions in similar ships that, in the absence of any better values, they may be taken as the same as those in a sister ship, or assumed. With the approximate values of 23 and ( and the assumed value of 3D, the compass may be roughly corrected when in dry dock, moored to a wharf, or when it is impossible to get ob- servations on more than one heading, provided, however, that there is no other iron vessel, nor other causes of disturbance, sufficiently near to exercise magnetic influence. If the com- pass should not be compensated, then a table of approximate deviations may be made by the formulae of Art. 97. Such observations may be valuable in determining the loca- tion of compasses for ships while still on the stocks. 95. Determination of 23, (, 3), A, a, and e from observations of deviation and horizontal force on two headings, 51 and ( heing neglected. Let z\ and z' 2 be the two compass headings; z^ and z 2 TT' fTJ2 the two magnetic headings ; -==* = - 7 = >r the horizontal force of H -/i TT' rpi earth and ship on the first heading in terms of H; -i 7^ JJ. -L a the same on the second heading. Then we have from (69) and (70) for the two headings : 154 NAVIGATION COMPASS DEVIATION = cos z( (l + a} cos z l (71) jti = Jj* cos z 2 ' jfZ 01 (1 + fl) = cos z a (72) H' cos Zi ^= 2 cos zi TT' n^ si f (COS Zi COS Z 2 ) (1 + e) sin z x (74) (1 + e) sin z 2 (75) or i (sin z l sin z 2 ) From (73) and (76), (35) and (36), A = i{(l -f-^) + (1 + e)} Adding (71) and (72) and dividing by A, 1 f 1 (H[ , H^ \ % = + 7- 1 TJ- ^ ^ cos z t + -^-cos z 2 1 (1 + 0) [_-^-(cos Zi + cos z 2 )J I Adding (74) and (75) and dividing by A ? i f i /# , . , ^r; . A -T\T^ 811] ^+^ sm ^ - (1 + e) r^- (sin zi + sin z 2 ) J | ( tan a = -' (73) (76) (77) (78) (79) (80) (81) Since the sines and cosines of angles differing 180 have opposite signs, if the observations are taken on magnetic OBSERVATIONS ON Two HEADINGS 155 courses diametrically opposite, (1 + a) in (79) and (1 + e) in (80) will disappear, J(cos z cos z z ) will become cos % and -J (sin z^ sin z z ) will become sin z v ; and equations (73), (76), (79), and (80), being much simplified, will become If T-T' TT' ~\ 1 **i i -"i t ( /oo^ 1. + a = - < -rfCQS Z^ -TT COS Z 2 f V^*^ COS Z t 1 + e = o-( -^-sin z[ -rr sin ; } (83) = T { T ( sin Zx ^OOS^+^COB*;)]- (84) (85) This method is strongly recommended, and most excellent results may be obtained when the vibrations are carefully taken. In the steering compasses of some of the battleships, the ship's force exceeds that of the earth and it is impossible to obtain a curve of deviations for such compasses by swinging ship, and resort must be had to vibrations. However, care must be exercised in selecting the rhumbs, as the formulae will fail if the magnetic azimuths are equally distant from any one of the cardinal points; for if equally distant from North or South, (1 + a) reduces to the form ; and if equally distant from East or West, (1 + e) takes that form. As a general rule, select rhumbs on or near opposite quadrantal points, or on or near two adjacent cardinal points. This method is valuable in locating compasses on board new ships and in determining beforehand the forces of the ship, and, if desired, a deviation table. Select the different places 156 NAVIGATION COMPASS DEVIATION where, for other reasons, the compass might be located ; obtain the compass and magnetic headings and the time of 10 vibra- tions of the horizontal needle in the exact position to be occu- pied by the compass needle when the ship is on the first heading; change the position of the ship and do the same on the second heading; note the time of 10 vibrations of the same horizontal needle ashore. Proceed with the computa- tion, and all other considerations being equal, select that spot as the best location for the compass, where A is greatest. The following form not only facilitates the solution but indicates the data. OBSERVATIONS ON Two HEADINGS 157 rH rH t- Z> rH t* N O o ^ o a t- t- s . 1 + T I> J> ^ N o o o o t- 1, CO O B 1 ^ o I + T < Si o t O CM w ^_^ rH g CO OS s-^ TH '53 ^ ^_* ^ -^ ^ ^N i + + *L % O> o TH 0? rH 00 I> t- ^ o <* T- cc b- "1 *~ ^^ . * ^.^^ * > ' ^ ^ ?. Si J*i _ + I J, o ^ 1 .5 00 * CO JO CO OS CO t~ a CQ 50 5 1 v^ II S M . CQ ^ + Hn HM rji -s og 1 CO CO 00 C co os c s II d 5 ^ co u 4- ii TH t- O5 TH TH TH C 5 -i - i rtj [^ d TH + + ^. 1- r 2 II II II 1 1 II II II p f ft, CO* *' CO CO III ig a a 5 Z> CQ CQ ; x g'|b^w Q ft r rH |r< rH | 1 "* ii II II bo T~ ^ f & ft te CO CO n" V OS " 00 s II 00 + + ti o T^ T-J 03 03 ti "7 ' l > and T* A A A A (2) What were the values of 23 and ( at New York, = .185, tan 0=3.14? (3) What was the total force at New York due to subper- manent magnetism, and what was the total force due to tran- sient magnetism induced in vertical soft iron ? THE Two PARTS OF 23 AND 161 (4) A Flinders bar having been placed while the ship was at New York, it is required to find the direction in which it was placed, and the compass heading after correction, if be- fore, the vessel headed 48 30' (p. c.), dev. ~H 41 30', the azimuth of the ship's head remaining the same. By substitution in (86) we have .219 X .697 = 4- + T- X 2.605 X .219 .153 = .128 - + .433-- .031 = .137 -f- X .393 = ~ + - x 1.393 X .311 .122=4 + - = .024 .219 X (- .119) = -f- + ^ X 2.605 X .219 .311 X ( .08) = -J- + ^- X 1.393 X .311 (-) .026 = -- + .570 -f (-) .025 = -^ + .433 (-) .025 = (-) .001 = .137 4- (-).025 = -5 .003 - A L =(-).022 -f=(-) .007 (2) To find the value of 93 and ( at New York, P c , + ((-) -007 X 3.14) = y + -226 X 3.14 (S"" = .130 + .710 = + .840 ( = ( ) .119 .022 = ( ) .141 (3) + .130 = force in keel line due to subpermanent mag- netism at New York. ( ).119 = force transverse to keel line due to subper- manent magnetism at New York. + .710 = force in keel line due to vertical soft iron at New York. ( ) .022 = transverse force due to vertical soft iron at New York. 162 NAVIGATION COMPASS DEVIATION tana.., ^(^^ B (-)-JlM therefore a s ^ = 317 31' 45" tan a, = ^=y|^ = (-) .03099 therefore a v = 358 14' + . Total force due to subpermanent magnetism V(-130) 2 + ( .119) 2 = .176. Total force due to vertical soft iron + ( .022) 2 = .710. (4) The Flinders bar is placed in the angle a v = 358 14'+ with its lower end on a level with the compass needles, or in the angle a v + 180 = 178 14' + with its upper end on a level with the compass needles, and at such a distance as to neutralize the deviation due to the vertical soft iron. In case the corrector is to be at a fixed distance from the compass, then increase or decrease the number of rods till the desired effect is produced. The deviation resulting from the equation tan sin z + -*r tan cos z tan*= -jr- (a) 1 -f- -- tan cos z L- tan sin z A A is the deviation due to vertical soft iron on the magnetic head- ing 2. In the example z r = 48 30', dev. = + 41 30', and there- fore z is East. Substituting the values of j tan and -^- tan found in part (2) of this example, we have * _ + -710 X 1 + (- .022) X .710 = 1+.710 xO (-.022 X 1) "" 1.022 = .6947 .'. d = + 34 47'. DEVIATIONS FROM THE COEFFICIENTS 163 The deviation due to vertical soft iron being + 34 47' on the given heading, that amount should be removed by the Flinders corrector; therefore, after the correction has been made, the compass heading should be 83 17'. 97. Computation of deviations from the coefficients. Vari- ous methods have been explained for obtaining new values of the coefficients, especially the changing ones ; having obtained these it may be desirable to compute a deviation table; or knowing the values of j, -j, Q and y, it may be necessary to determine the deviations for certain localities to be visited where there may be no opportunities for swinging ship. From the approximate coefficients the deviations may be obtained from the equation 8 A + B sin z' + C cos sf -f D sin 2sf + E cos 2z'; and from the exact coefficients, they may be found for mag- netic azimuths from equation % + % sin g + (cos s + g) sin 2z + ( cos 2z ~ I + 23 cos z ( sin z + SD cos 2z ( sin 2z' then by use of Napier's diagram the deviations may be found for the compass headings. However, deviations are desired for compass headings and may be found for such from equation (34), - ~ The following form facilitates the computation. 164 NAVIGATION COMPASS DEVIATION 00 03 o 1J j 3 -8558S8S8S Ot-COt-t-tr-Ol-lO O + a 1 1 1 1 + + + 1 (3 ^ d^ 1 "** 03 2> N Pa- ll} CQ S 1 ! 1 1 + + + 1 T3 0) ,c +a ^ ! illiilli T3 {3 1-1 5 J - ^^, eg CJ OQ iilosiil 2 O + 1 + 1 V N ~T~ v?t II % rH O r-t O i 1 TS a o3 | rT s | i i 1 1 1 O i l 00 1 1 1 1 + + + 1 + II & o a 1 1 1 1 1 1 i 1 to, OQ + 1 + 1 CO "i ' o II ^ rl N O TH O i i O T d 'S 1 K Ml tf i "N 1 f l' + + + 1 w 3 T to ^ w T| II (2) i ,-H OQ CC --I 1 II o II 8P a ^ a lilillll Bj *3 e III + + + o a 1 1 S t- *-< II s? a t^ rH CC H .-a DO 1*3 '^ O' x a 03 o oojogjog^o^o 'C P< K P< ^ ~ * C* ** CO HEELING ERROR 165 It may be noted that the second halves of columns 33 sin z' and & cos z' are the same as the upper halves with a change of signs. Column 2) sin 2z' is the same in the lower half as in the upper half, no signs changed; the same is true for the column cos 2z'. SECTION VII. Heeling Error. 98. In Art. 74 it was shown that, with the ship upright, the magnetic forces acting on the compass needle to head, to star- board, and vertically downward were expressed by Poisson's equations, in which X, Y, and Z were the components of the earth's force in the three directions called respectively the axes of X, Y, and Z, the first two being in the horizontal plane; that magnetism was induced in the parameters o> 6, c . . . . & by the earth's component parallel to the direction in which the parameters lay, the induced force in each case being a linear function of the inducing force. When the ship heels, the transverse and vertical iron alter their directions and make with the axes of Y and Z, respec- tively, an angle equal to the angle of heel ; for Poisson's equa- tions to express, under the new conditions, the forces to head, in the inclined transverse direction of the deck, and in the inclined direction of the keel, these directions must be taken as new axes, and the earth's force resolved parallel to them. Let the resolved components of the earth's force be YI in the inclined transverse direction of the deck, Zt in the inclined direction of the keel, the force X to head being unchanged by heeling. The force induced in the fore-and-aft iron will be the same as before the ship was heeled, the force induced in the trans- verse iron will be the same linear function of Yt as it formerly was of Y, and that induced in the iron formerly vertical will 166 NAVIGATION COMPASS DEVIATION be the same linear function of Zi as it formerly was of Z, because, the axes and the iron being parallel, the ratio of the earth's component in the axis and the force induced in the iron of that axis will not be changed by heeling, and hence the values of the parameters in the equations for the new axes remain unchanged. In other words, the rods a, d, and g will be magnetized by force X; ~b, e, and Ji by force YI; c, /, and Jc by force Z t ; whilst the components of the subpermanent magnetism remain un- changed. FIG. 52. Therefore, letting X', Yj, and Zi represent respectively the forces of earth and ship in the new axes to head, to star- board, and to keel, Poisson's equations become : X ' = X + aX + lYi + cZi + P. (87) Y; = Yi + dX + eYi + fZi + Q. (88) Zi =Zi + gX + hYi + JcZi + R. (89) The next step is to express the forces represented by equa- tions (87), (88), and (89), in terms of the components X, Y, and Z ; to do which it will be necessary to substitute the values of Yi and Zi in terms of those quantities. HEELING ERROR 167 In Fig. 52 , let OY and OZ be the transverse and vertical axes, ship upright; OY i and OZi the corresponding axes, ship heeled i. OA = Y, the horizontal component of earth's force to star- board ; OB = Z, the vertical component of earth's force ; then T is that component of the earth's total force that acts in an athwartship plane through the compass, on the North point of the needle at 0, which changes neither in direction nor intensity when the ship heels; and hence OR = Yi is the component of the earth's force in the new axis to starboard, and OL = Zi is the component of the earth's force in the new axis to keel. But from the figure OR = ON + NM + MR = OA cos i + (AM + MT) sin i = OA cos t + OB sin i orYi Y cos t + Z sin i, (90) and OL = OK LK = OK FB = OB cos i BT sin i or Zi Z cos * T sin i. (91) Substituting (90) and (91) in (87), (88), and (89), and collecting the terms with common factors, we have X' = X + aX + ( I cos i c sin i) Y + (b sin i + c cos i) Z + P, (92) Y'i = (cos i + e cos i / sin i) Y ~f- dX + (sin i + e sin i + / cos i) Z + Q, (93) Z'i = (cos i -{- h sm i -\- Je cos i) Z -\- gX (sin i h cos i + k sin t) Y -\- R, (94) which are the forces acting still in the new axes, though in terms of X, Y, and Z. As the compass needle is constrained to move in the hori- zontal plane, to obtain an expression for deviation due to the 168 NAVIGATION COMPASS DEVIATION above forces, we must obtain their components in the hori- zontal plane. Since the ship is heeled about the axis of X, the force X' in equation (92) is already acting in the horizontal plane, and it is only necessary to obtain the horizontal component of earth and ship to starboard represented by F'. Eef erring again to figure 52, Let OP represent, in intensity and direction, that component of the total force of earth and ship which acts in an athwartship plane through the compass after the ship has heeled i, then OR" is the component of that force in the new axis to star- board, and Y f = OA" is the component of that force in the horizontal plane to starboard. But from the figure: OA." = OA' A" A' = OA' R" 8 OR" cos i R"P sin i OR" cos i OL" sin i; or T - OA" Y'i cos t Z' sin t. (95) Substituting in (95) the values of Y'i and Z' t from (93) and (94), we have Y' = (cos 2 i + e cos 2 i / sin i cos i) Y + d cos i X + (sin i cos i -\- e sin i cos i -f- / cos 2 i) Z -(- Q cos i (sin i cos i -\- h sin 2 i + fc sin i cos i) Z g sin i X -f- (sin 2 i Ti sin i cos i -j- k sin 2 i) Y R sin i. Y* = { sinH'-f-cos 2 i \ Y-}- \ dcosi gsini } X -f \ e cos 2 i (/ -f- h) sin i cos * + & sin 2 1 } F -j- \ /cos 2 1 -f (e &) sin * cos i h sin 2 i\ Z -f- cos 4 R sin t. Since sin 2 i -f- cos 2 t = 1, and by substituting 1 sin 2 i for cos 2 i t we have P 7 = Y+ \ d cos i g sin t }> JT-f { e (/-f A) sin t cos i "I _ (g--^)sinH' } Y+ \ f + (e fysinicoai f (96) (/ + A) sin** } ^+ cost jBsin*. J HEELING ERROR 169 Equations (92) and (96) are of the form X' = X + diX + biY + aZ + Pi , (97) Y' = Y + diX + e,Y + fiZ + Q t , (98) if di = 0,. bi= b cos i c sin i. a = b sin i -\- c cos i. di = d cos i a sin i. I /QQ^ e = e (/ + ft) sin i cos i (e- Jc) sin 2 i. ' ^ ' fi = f -f- (e A;) sin t cos i (f + ft) sin 2 i. Qt = Q cos i E sin i. 99. The coefficients when ship is heeled. As the values of the parameters and magnets have changed in consequence of the ship's heeling, so have the magnetic coefficients which de- pend on them. Therefore, if A* , 2Ii , 95i , (* , < , (* , respectively, repre- sent the altered 1 values of the coefficients A, 2C, SB, (, 2), (, due to the heeling of the ship through i 9 by substitution in the equations for the exact coefficients, we have A< = A ^ Q sin i cos i sin 2 i. (100) Ai5ti = A5T cos i + ~~^ s ^ n * (101) = ASB + -{ 5 sin i c versin i } tan 0. (102) 7? 1 = A(S + { (e A;) sin t cos i - sin i ft) sin 2 O tan fi-wrint l^^ = A + i^ sin t cos i + i^ sin 2 i. (104) = A@ cos i t-^ sin t. (105) Approximate values of the coefficients when the ship is heeled. If the soft iron is symmetrically arranged on each 170 NAVIGATION COMPASS DEVIATION side of the fore-and-aft plane through the compass, 1), d, f, h, 21 and ( will be zero; and as a steady angle of heel would be small, we may without much sacrifice of accuracy replace sin i by i f letting cos i = 1, versin i = 0, and sin 2 i = 0. Equa- tions (100) to (105) will then give \i = A. 8 = 9. R =<+|(*-*-f.)tanrf. : K106) ( = & -f Ji, if J = 1. 1 c X Sh =. 2). 100. Deduction of the equation expressing heeling devia- tion. If 8 represents the deviation for a given compass course z' when the ship is on an even keel, Si the deviation for the same compass course when the ship heels i to starboard, then equation (34) becomes in each case, the approximate heeling coefficients being substituted, 8 (1 _ 2) C os 2z') = t + $ sin 2' + ( cos z' -^ sin 2z* + ( cos 2z' 8, (1 & cos 2z r ) = +^jri + % sin /+(( + ft) cos s' + S) sin 2' cJ r9 i cos 2s'. 2A Therefore, since by the hypothesis 21 and ( are zero, (8, _ 8) (i_S) cos 2/) = ^=^ i + Jt cos it Substituting cos 2 2' sin 2 2' for cos 22', multiplying XI ^^__ >y A,- i by (sin 2 2' + cos2 z ')? rearranging the terms in the HEELING ERROR 171 right member of the equation,, also regarding (8* 8) 2) cos 2z f as zero, and transposing 8 we have Si = 8 + Ji cos /+ ~ i sin 2 z' -| t cos 2 z', (107) which will give the heeling deviation on the compass heading z' when 8, J , c, and g are known. It has already been shown how to find 8 (Art. 55), also c and g (Arts. 96 and 92). The method of finding J will be explained in Art. 103. 101. General effect of ship's heeling on the deviation of the compass. Equation (107) shows that the effect of heel- ing, besides altering the term ( by the expression Ji, is to in- troduce a constant term and a quadrantal term of the @? type ; iron which is symmetrical with the ship upright becoming unsymmetrical when the ship heels. It is readily seen that c introduces a ( &) = ci, and g t a (+ d) = gi, when the ship heels, and that these cause the 2-U and @^. c represents vertical soft iron in the midship line, as the smokestack; the effect of c depends on the proximity of the compass to the pole of c, and is a minimum when the ship heads North or South, a maximum when the ship heads East or West, -y is generally A -f- and seldom exceeds .100 for the usual positions of compasses- The parameter g represents soft iron parallel to the axis of X, as the keel or propeller shaft; the effect of g depends on the proximity of the compass to the pole of g, being greatest when the compass is well forward or well aft and the ship's head North or South. The effect of g is a minimum so far as location of the compass is concerned when the compass is at equal distances from each end of the ship, and disappears in all cases when the ship heads East or West, -j is generally + and seldom exceeds .100. The effects of both c and g may be neglected in the ordi- nary cases in navigation. 172 NAVIGATION COMPASS DEVIATION The term Ji cos z', however, cannot be neglected, as it is often large in amount. When J is ( ) , as it usually is for compasses on the upper deck, it represents a deviation of the North point of the needle to windward; when +? as it may be for a compass on the main deck, it represents a deviation to leeward. ( ) J is called the heeling coefficient to windward, ( ) Ji cos z' being the heeling error to windward, a maximum on North and South courses and a minimum on East and West courses. J is generally a fractional number and indicates the heeling deviation, for each degree of heel, arising from a change in the value of & due to heeling, on N. and S. courses (p. c.). 102. General effect of the ship's heeling on the coefficients determined when the ship is upright. An inspection of equa- tions in group (106) shows that the coefficients depending on fore-and-aft action, 93 and 2), are unaltered ; that 21 and ( undergo a slight change; and that & is considerably altered. As & has its maximum effect when the ship heads North or South and its minimum effect when the ship heads East or West, it is apparent that the heeling error is a maximum or a minimum under the same circumstances. 103. Different ways of expressing the heeling coefficient and "1 / 7? \ the use of each. The expression J = le ~k -~\ tan B may be written ( ) J = -- ( _ e _|_ & _|_ ** J tan 6, and then, if the right hand member of the equation is posi- tive, it indicates (as is usual in North magnetic latitude) a deviation of the North end of the needle to windward. y Since tan 6 = -T, e = X (1 ) 1, and /x 1 = ~k 7? 1 / ~2 , the expression J = j f may HEELING COEFFICIENT 173 be put in the forms below, each of which will be shown to serve a special purpose : (a)J= x( /M J- ~ (c] J = (%) + 1- l) tan o + p ~-^- tan 0. Form (a) shows the changes which may be expected in ( )J on change of magnetic latitude ; \H is always -f- ; tan 6 is + in the northern, ( ) in the southern hemisphere ; at the usual position of a standard compass on a ship built in North mag- netic latitude, ( ) e, + Tc t and + R are positive. ( ) e and + fc will change sign in South latitude, + R will not, there- fore the heeling error will be to windward unless the ship is so far South in the southern hemisphere that ( e ~\- Jc) tan r> is greater than -^ . Form (6) shows how the heeling deviation is caused; ( ) g i expresses the effect due to vertical induction in horizontal soft iron represented by the rod ( e) , inclined at an angle i to the horizontal plane, and acting against a di- rective force \H ; the effect being a heeling error to wind- ward in North magnetic latitudes, to leeward in South mag- netic latitudes. - : i expresses the combined effect of vertical induction in vertical soft iron represented by the rod -f- & and the component to keel of the subpermanent mag- netism represented by + R, both acting at an angle i from the vertical and against the directive force XH. The effect of this part is a heeling error to windward or leeward according as the resultant force of JcZ + R is + or ( ) . 174 NAVIGATION COMPASS DEVIATION Form (c) is useful for computing separately the heeling deviation due to (1) vertical induction in horizontal trans- verse soft iron, (2) vertical induction in vertical soft iron and vertical subpermanent magnetism ; f 3) + -j 1 J tan eZ j fj. l . JcZ+R = -Try- expressing the first part, and { ^ tan = Jl expressing the second part. Form (d) is the most convenient form for computing the heeling coefficient ; the value of 3) having been determined by analysis of a table of deviations or from observations on two headings, p and A by vertical and horizontal vibrations on board and ashore, and 6 taken from a chart of magnetic dip. In order that there may be no semicircular heeling error ( ) J must be zero ; therefore, [JL - = 6. That is to say Mean vertical force of earth and ship /*= i Vertical force of earth and Mean vertical force of ship Vertical force of earth e. DETERMINATION OF HEELING ERROR 175 Ex. 18. With the following data from examples 11 and 12, viz., A = .8856, 3) = + .05, tan 2.86, /x = 1.0969, g .1098, c being neglected, it is required to find ( ) J and the total deviation on courses South and NW. (p. c.) when the ship is heeled 1st 10 to starboard, 2d 10 to port. Deviation, when ship is upright, on South + 4 30', on NW. 16. J= f& + l\ tan = (.05 + ^-ggg 1J2.86 = 0.8254 and + J= 0.8254 mi ,.-,-, j. . . ,:\ + if heeled to starboard The ship heels 10, therefore t = 10 [ ' . , , J if heeled to port. At South, Ji cos z' *L i cos 2 z' = ( ) .8254 x 10 x (1) .124 x 10 x 1 A = + 8. 254 1.24 = + 7. 014 = + 7 01'. At NW., Ji cosz' *L i cos 2 z' = ( ) .8254 x 10 x .707 .124 x 10 x .5 A (_) 50.886 0.620 = ( ) 6.456 = ( ) 6 27'. Ship heeled to starboard, on South, 8 = + 4 30' + ( + ) 7 01' = + 11 31'. Ship heeled to port, on South, 8 _p 4 30' ( + ) 7 01' = ( ) 2 31'. Ship heeled to starboard, on NW., 8 = ( ) 16 + ( ) 6 27' = ( ) 22 27'. Ship heeled to port, on NW., 8 = ( ) 16 + ( + ) 6 27' = ( ) 9 33'. 104. Determination of heeling error by listing and then swinging the ship. The deviations may be determined by swinging the ship, first, upright, then heeled i. The dif- ference on any compass azimuth of the two results will be the heeling error for that angle of heel on that course. The results of this practical method no doubt would be more satisfactory after the work was done ; it is a tedious pro- cess, however, and the heeling error is usually determined 176 NAVIGATION COMPASS DEVIATION theoretically from observations already shown to be of a simple character, when not corrected by the tentative method. If swinging takes place both before and after listing, c may be found from the observations at East or West, as then 8i = S -j j- i; and g may be determined from the obser- vations at North or South, as then 8* = 8 + ^ y *> ^ an d J having been computed. 105. Correction of heeling error by vibrations. As e is minus and less than unity when the quadrantal deviation has not been corrected, it is thus evident, in order that there may be no heeling deviation under such circumstances, that the mean vertical force at the position of the compass must be less than that on shore, and that the time of n vibrations of a vertical needle at the position of the compass represented by T' must be greater than the time of the same number of vi- brations of the same needle ashore represented by T. Eegarding Ti as zero, (63) becomes jr- = fir, = ** + g cot cos * aQd r = T (108) COt 0COS2 but when the heeling error is corrected /u= A. (1 %)) =l-\- e. T T ~~ */% (! $)) + # cot 6 cos z ~~ */l + e + gcotOcoaz' When g is unknown, the ship's head may be placed on East or West magnetic, If the spheres are in position there will be a new value of A and perhaps a residual value of S). Let the altered values be A 2 and 2 , HEELING ADJUSTER 177 Then with the spheres in place equations (109) and (110) become T ' It is thus seen that the heeling error may be corrected by so altering the vertical force that the vertical vibrations of a dip- ping needle shall take place in the proper time found, accord- ing to the circumstances, from equations (109), (110), (111), or (112). The vertical force is altered as desired by the vertical move- ment of a vertical magnet in the binnacle tube below the cen- ter of the needle. It is customary now to correct the heeling error by what is known as the heeling adjuster, or by the tentative method (Art. 108 (5)). 106. Correction of heeling error by using the heeling ad- juster. The heeling adjuster is a small brass box provided with levels and leveling screws, mounting on a horizontal axis a needle which is free to vibrate in the vertical plane, its ten- dency to dip being counteracted by a small sliding platinum weight whose distance from the axis of suspension may be measured by a scale on the glass cover. There is a small glass window in each end provided with an index line to mark the horizontal plane. Without the small weight, the needle before being magnetized was exactly balanced, so the weight is in- tended to balance the vertical magnetic force ashore or on board. Letting b and a, respectively, denote the distance be- tween the weight and the center of the needle when the needle is exactly balanced on board and ashore, the heeling adjuster b Z' being properly placed in the magnetic meridian, then = ~^> and, when the ship's head is East or "West magnetic, = p.. 178 NAVIGATION COMPASS DEVIATION In order that there may be no heeling error we must have p=l + e=-A(l ). Therefore, I = a\ (1 ) (113) before the quadrantal spheres are placed. If Ao , 2 9 f*2 be the altered values of A, , and /* after the spheres are in place, then fji 2 A 2 (1 2 ) or & = aX 2 (1 S> 2 ). (114) To correct the error. Place the weight at reading & from (113) or (114), according as the spheres are off or on the brackets, head ship East or West magnetic, put the adjuster on the brass table provided for this purpose, in the exact posi- tion of the compass needle, the adjuster properly placed in the meridian. If the needle remains horizontal there is no heeling error. If one end dips, place the heeling corrector magnet in the tube, proper pole up; raise or lower it till the adjuster needle is horizontal. The heeling error is then cor- rected. SECTION VIII. COMPENSATION OF THE COMPASS. 107. Principles and object of compensation. It has been shown that each kind of deviation is due to certain forces, either of attraction or repulsion, acting on the North point of the compass needle, and it is evident from the known laws of magnetic action that these forces can be neutralized and hence deviation reduced to zero, by introducing other forces of the same magnitudes, but such as to act in the opposite directions. By compensation, the deviations are not only reduced to zero, or to convenient amounts, so that a change in azimuth of the ship's head is represented by a similar apparent movement of the compass, but the directive force of the needle is equalized on the different headings. After compensation, all cor- rectors should be secured in place and their positions noted in the Compass Journal. COMPENSATION OF THE COMPASS 179 108. Order of compensation. Since the correctors, when in place, exert a mutual action on each other and thereby create forces additional to those of the ship, it is essential that the semicircular correction, which is the largest and most im- portant one, should be made when the magnetic conditions approximate as nearly as possible to those when the compen- sation is complete; therefore, the quadrantal and heeling correction should precede the semicircular correction. The quadrantal spheres, when in place, correct a portion of the heeling error and for this reason it is desirable that the spheres should be in place before the heeling correction is made. However, if the values of A. and , before the spheres are in place, are known by computation, the heeling correc- tion may properly be made by the method of Art. 106, as the first correction, the distance used for the position of the weight on board being b = a\ (1 S)). If the correction should be made by this method after the spheres have been placed, we must find the distance & from the equation & = aX 2 (1 S) 2 ), A 2 and 2 being altered values of A, and 5D. The heeling error may, however, be corrected by the tenta- tive method and, in that case, the following will be the order of compensation : (1) Correction for quadrantal deviation (approximate). (2) Correction for heeling error (approximate). (3) Correction for semicircular deviation. (4) Correction for quadrantal deviation. (5) Correction for heeling error. (6) Swing for residuals. It being assumed that the ship is on an even keel; all movable local masses of iron or steel in the vicinity of the compass secured in their normal positions for sea; and the binnacle of Type VI stripped of all correctors, which are placed at a safe distance ; we will proceed to compensate the standard compass. 180 NAVIGATION COMPASS DEVIATION From data by computation. Having obtained a curve of deviations for the standard compass by any of the methods previously referred to, take from the Napier's diagram the compass headings corresponding to North, NE., and East magnetic, then head the ship successively on those rhumbs, steadying on each at least four minutes. Note carefully the reading of the steering compass when the ship is steadied on those rhumbs. Then proceed with (1) The approximate correction of the quadrantal devia- tion. If the value of S) is known or can be estimated, place the spheres on the brackets according to Table III of " Diehl's Compensation of the Compass/' or Table V of this book. If 3) is unknown, place them at a mean position of 13.5 inches for the 7-inch spheres. If spheres of this size overcor- rect at the outer limit, use smaller ones, remembering that one sphere will correct half as much as two of the same size. (2) The approximate correction of the heeling error. Having no means to determine A 2 and SD 2 , place the ship's head East or West magnetic by means of the steering compass. The needle of the heeling adjuster having been made hori- zontal on shore, with the weight in a given position, must be made nearly horizontal on board, position of weight un- changed, by means of the vertical correcting magnet in the central tube, the non-weighted end inclined perceptibly up- wards. In case no observations were made ashore, place the heel- ing magnet in its tube, North end up in North magnetic latitude unless there is reason to know that the ship's verti- cal force acts upward, and lower it to the bottom. (3) To correct the semicircular deviation. Neglecting the values of 51 and (, it is apparent from the equation S3 sin z + & cos z + 2 s i n %z i. i. ** tan 8 = ! ! , m which 2 is 1 + 23 cos z ( sin z + 2 cos 2z To NEUTRALIZE 33 AND ( 181 the coefficient of quadrantal deviation left uncorrected, that when the ship heads magnetic North or South, tan 3 = _-=^__-_- that the forces 33 and 2 act in the meridian ; that the transverse force ( is the only one acting to produce deviation; and that in order to reduce the deviation on those headings to zero, it is only necessary to neutralize (, which may be done by introducing an equal but opposing force in the transverse line. When the ship heads East or West magnetic, the above [ rtl equation becomes tan 8 = 1 =P( SB 9 ^ e ^ orces an ^ SD 2 act in the meridian, and the fore-and-aft force 33 is the only one acting to produce deviation; to reduce the deviation on those headings to zero^ it is on^y necessary to neutralize 33, which may be done by introducing an equal but opposing force in the fore-and-aft line. Therefore, To neutralize the force (. Head the vessel North magnetic and keep it steady by the steering compass. Eun the athwart- ship carrier down. If the compass shows easterly deviation, the force ( draws the North point of the needle to starboard ; enter one or more magnets on each side of the athwartship carrier, North or red ends to starboard; move the carrier up or down until the compass points North magnetic. If the compass shows westerly deviation, the force ( draws the North point of the compass needle to port; enter the athwartship magnets with North or red ends to port; raise or lower the carrier till the compass points North magnetic. Or head the vessel South magnetic, enter the athwartship magnets with North or red ends to port if the deviation is easterly, or to starboard if the deviation is westerly; raise or lower the carrier till the compass points South magnetic. To neutralize the force 33. Head the vessel East magnetic 182 NAVIGATION COMPASS DEVIATION and keep it steady by the steering compass. Kun the 'fore- and-aft carrier down. If the compass shows easterly deviation, the force 23 draws the North point of the needle to head; enter one or more magnets on each side of the fore-and-aft carrier, North or red ends forward; move the carrier up or down till the compass heading of the ship is East. If the compass shows westerly deviation, the force 23 draws the North point of the needle to stern ; enter the fore-and-aft magnets with North or red ends aft and raise or lower the carrier till the compass heading is East. Or head the vessel West magnetic; enter the fore-and-aft magnets with North or red ends aft, if the deviation is east- erly, or forward if the deviation is westerly ; raise or lower the carrier till the compass heading is West. (4) To correct the quadrantal deviation. With the semi- circular forces neutralized there remains only S) 2 to cause de- viation, and when the ship heads NE., SE., SW., or NW., mag- netic, 2z being 90 or 270, the equation for deviation becomes tan 8 = 5D 2 , and to reduce the deviation on those headings to zero, it is only necessary to neutralize the force $D 2 , which may be done by introducing an equal but opposing force. The quadrantal deviation is usually positive, and hence is corrected by placing the quadrantal spheres to starboard and port of the compass in which position they produce a negative quadrantal deviation, the soft iron of the sphere having the effect of the a and + e rods ; therefore, To neutralize the remaining quadrantal force. Having corrected the semicircular deviation, head the vessel NE. (or SE., SW., NW.) magnetic and keep it steady by the steering compass. If any deviation is shown, move the spheres on the side brackets in or out until the compass heading is NE. (SE., SW., NW.). If the spheres over-correct at the outer limits of the CORRECTION OF HEELING ERROR 183 brackets, use smaller ones; if they undercorrect when placed at the inner limits, use larger ones. (5) To correct the heeling error. In case the heeling cor- rector has not been placed by shore observations, and is in the- bottom of the central tube of the binnacle, if there is sufficient sea on to give a moderate roll on a North or South course, steer North or South per compass and observe the vibrations of the card as the ship rolls from side to side. These will be greater than those due to the ship's real motion in azimuth when the heeling error is material, therefore raise the heeling corrector slowly till the vibrations almost disappear, leaving an amplitude of 1 or 2 to avoid over-correction. It must not be forgotten, however, that the correction once made is for that particular latitude only and the position of the heeling corrector must be changed for any considerable change of magnetic latitude. In the case of a vessel heeling steadily on a North or South course, the deviations observed when heeled may be compared with those when the ship is upright on the same course, and the difference removed by raising or lowering the corrector. If the conditions are not favorable for the final placing of the heeling corrector, reserve it for a future time. (6) Swing ship on the sixteen principal rhumbs and obtain a table of residual deviations; either readjust the correctors, proceeding as in the original correction, or use the residual deviations to run on. In case of re-compensation, 'the vessel must be again swung for a final table of deviations. The -ship may now be placed with its head on any two adja- cent cardinal points magnetic by the standard compass, and the other compasses corrected for semicircular deviation ; then on the intercardinal point magnetic by the standard compass, and the others corrected for quadrantal deviation. 184 NAVIGATION COMPASS DEVIATION 109. Determination of the magnetic courses when compen- sating compasses whose deviations are unknown. Select ahead of time the locality, the date, and the limits of local apparent time between which the observations must be made. Take from the Nautical Almanac the sun's declination for the instant midway between the time limits, and, by the method of Art. 58, find from the azimuth tables the sun's true bearing at intervals, say, of ten minutes of time, for the known latitude and declination. Apply the variation for the locality and obtain the magnetic bearings. Make a table with a column of magnetic bearings opposite a column of local apparent times, or construct a curve ; the ordinates representing local apparent times at intervals of ten minutes ; the abscissae, the correspond- ing magnetic azimuths for the given latitude and declination. On the date selected proceed to the locality, set the watch to local apparent time, and shortly before the first selected time, set that rhumb of the pelorus corresponding with the desired magnetic heading to the ship's head and clamp the plate. Set the sight vanes to correspond with the sun's magnetic bearing at the selected time and clamp the vanes. Working the engines slowly and using the helm, bring the sight vanes to bear on the sun and keep them there till the watch shows the selected local apparent time when the ship heads on the desired magnetic point, and the ship's head per standard should be noted, also the ship's head per steering compass. Let it be assumed that we have obtained the head- ings by the steering compass corresponding to any two adja- cent cardinal points and the intervening quadrantal point, all magnetic, and that a careful record has been made of the same. Then to compensate, it is only necessary to proceed as explained in Art. 108. HEADING MAGNETIC COURSES 185 Ex. 19. Having decided to compensate the compass on April 18, 1905, off Cape May, in latitude 39 N"., longitude 74 30' W., between the hours of 8 a. m. and 10 a. m., local apparent time, it is required to make a table of magnetic bearings of the sun at intervals of ten minutes between the limits named, and to determine the compass readings corres- ponding to magnetic North, NE., and East. Variation 8. L. A. T. of middle instant 9 00 00 O's declination. H. D. Longitude of locality 4 68 OOW. At G. A. noon. N. 10 41' 12" N. 52".46 G. A. T. of middle instant 1 58 00 Corr. 1 43 1.97 or April 18th. 1.97 Dec. = N. 10 42' 65" 103".36 = N.10.7 Lat. 39 N. I On page 90, azimuth tables. Dec. 10 1ST. f L.A.T. 9 h J Z = N.I13 32' E. For Dec. 11 N. Z = N. 112 34 E. Diff. for 1 of Dec. ( ) 58' Diff. for 0.7 of Dec. ( ) 41' Hence we have for Lat. 39 K and Dec. 10. 7 N". as follows : L. A. T. Sun's true azimuth. Sun's magnetic azimuth. 8 h 00 m a. m. 100 55' 108 55' 8 10 a.m. 102 44 110 44 8 20 a.m. 104 36 112 36 8 30 a.m. 106 33 114 33 8 40 a.m. 108 34 116 34 8 50 a.m. 110 40 118 40 10 00 a.m. 128 32 136 32 To head magnetic North. The ship being on the station ahead of time before 8 a. m., local apparent time, set the North point of the pelorus to correspond with the ship's head, 186 NAVIGATION COMPASS DEVIATION clamp the plate; set the sight vanes to the magnetic bearing of the sun 108 55' (by table or curve) and clamp the vanes. So manoeuvre the ship by the engines and helm that the sight vanes will point directly toward the sun. By helm and en- gines, keep the sight vanes on the sun till the watch set to local apparent time indicates 8 a. m. At that instant the ship heads North magnetic. The standard compass reads 6 (for example). The steering compass reads 8 (for example). Note carefully the heading by the steering compass at this time. To head NE. magnetic. Say it is desired to be on this heading at 8 h 20 m a. m. Set the NE. point of the pelorus to the ship's head, clamp the plate; set the sight vanes to the magnetic bearing of the sun 112 36' (by table or curve) and clamp the vanes. Proceed as before, keeping the vanes on the sun till 8 h 20 m a. m., when the ship heads NE. mag- netic and The standard compass reads 38 30' (for example). The steering compass reads 22 30' (for example). Note carefully the heading by steering compass. To head magnetic East. Let 8 h 40 m a. m. be the selected time. A short time before this clamp the pelorus plate with the East point on the forward keel line or indicator and clamp the vanes to indicate magnetic bearing of the sun 116 34' (by table or curve). Proceed as before, keeping the vanes on the sun till the watch set to local apparent time shows 8 h 40 m a. m. At that instant the ship will be heading East magnetic and The standard compass reads 78 00' (for example). The steering compass reads 56 00' (for example). Again note carefully the heading by steering compass. To head a magnetic course by azimuth circle. Knowing the magnetic bearing of the sun for a given instant at the place of observation, or of a distant object, set the direct sight COMPENSATION ON ONE HEADING 187 vanes of the azimuth circle to the right or left of the ship's head by compass by an angle equal to that which the sun or object is to the right or left of the magnetic heading desired at the selected instant. By using helm and engines bring the sight vanes on the sun, keeping them on it till the watch shows the selected time, when the ship will be on the desired mag- netic heading. In the case of a distant object the time is not considered. 110. To compensate on one heading, as when riding to a tide, in a dry dock, or when alongside a wharf, etc.* Having obtained the exact coefficients 23, (, and S) by any of the methods already referred to, also the magnetic heading, and knowing the compass heading and deviation, compute the deviation due to each coefficient by substituting that coefficient alone in the equation, 23 sin z + & cos z + S) sin 2z ~ 1 + 33 cos z ( sin z -j- cos 2z then find what should be the compass heading as each amount is successively neutralized. If the value of A. is known, make observations with the heel- ing adjuster ashore and on board, finding b =. a\ (1 ) ; and, neglecting g, place the heeling corrector magnet in place (Art. 106). However, if the values of A 2 and S) 2 may be determined after the quadrantal spheres are in place, first put the spheres on the brackets and correct the quadrantal devia- tion; then, finding & = a\ 2 (1 S) 2 ), place the heeling cor- rector. Move the quadrantal correctors in or out, keeping them equally distant from the compass needles, till the amount of deviation due to 2) is corrected. Then correct the amount of deviation due to that force, $8 or (, which, for the ship's heading, is more nearly at right angles to the direction of the compass needle. Thus, if the ship's head is more nearly North or South, eliminate the de- * For procedure in special cases, as when heading North (E., S. or W.), mag- netic, etc., see Appendix B. 188 NAVIGATION COMPASS DEVIATION viation due to ( by means of the athwartship magnets first, and then eliminate that due to 33. If the ship's heading is more nearly East or West, reverse this procedure, eliminating first the deviation due to 33 and then that due to Gj When the forces 33, (, and 2) have been neutralized, com- pass and magnetic courses should be the same (31, (, zero). Ex. 20.' In example 14, Art. 95, the following coefficients were found for a standard compass by observations of de- viation and horizontal force on two opposite headings, viz.: 33 = ( ) .0747, ( = ( ) .3142, S> = + .1211. It is required to find the deviation due to each force when the ship heads 199 30' (p. s. c.), dev. + 25 30', and the compass heading per standard as each is successively neutral- ized, the ship being kept steady on the corresponding magnetic heading by the steering compass. To find the deviation due to 2), ta ^=Tff^= TTcr- 1 = + - 1211and { ,=+6.64<16". To find deviation due to 33, To find deviation due to Therefore, note the corresponding heading by the steering compass and keep the ship steadied on that heading, in case the vessel is not secured in dock or to a wharf. Then, (1) By means of the spheres neutralize the force + SD> making the ship's head per standard compass 206 24' 16". (2) By means of the athwarthship magnets, North or red ends to port, neutralize the force ( ) (, making the compass headings 222 20' 51". (3) By means of the fore-and-aft magnets, North or red ends aft, neutralize the force ( ) 33, making the compass heading 225, which is the magnetic heading. As 8 + 2 52' 28", the amount of error is only about 13'. 1 It found, by neutralizing SB and in a certain order, that the elimination of one force leaves the other at a small angle with the needle, a condition unfavorable for its elimination, consider the effect of a reversal of that order with a view to improving conditions. THE DYGOGRAM 189 111. Values of A and E to be left unconnected. In all cases of the compensation of the compass, when A or E or their algebraic sum is as much as 1, the amount should be left uncorrected. A has a constant value and sign on all head- ings, the quadrantal deviation represented by E varies as cos 2z' and changes sign at East and West. Therefore, if compensating semicircular deviation on North or South, the amount to be left uncorrected for A and E would be the algebraic sum of the amounts due to their signs by analysis ; if on heading East or West, it would be the algebraic sum of the amounts due to A with sign unchanged and to E with sign changed from that by analysis. SECTION IX. 112. 1 The Dygogram; Its Construction, Description, and Use. The dygogram is one of the graphic methods of repre- senting the deviations of the compass for magnetic headings; it also shows the horizontal components of the magnetic force acting on the compass needle, the directions in which they act, and the deviations produced by each component as well as the total deviation for any magnetic heading. The word " dygogram " is a contraction for " dynamo-gonio- gram," meaning a " force and angle diagram." It is a geomet- rical construction fulfilling the conditions of the general ex- pression _ 2t + 33 sin z + & cos z + sin 2z + ( cos 2z I -f- $8 cos z : ( sin z + S) cos 2z (5 sin 2z ' as will be shown further on. To construct the Dygogram when 21, 23, (, >, and & are known. Navigators of the U. S. naval service have blank Art. 112 1 taken from an article by Comdr. John Gibson, U. S. N., in Proceedings of U. S. Navai Institute, Vol. XX, No. 3. 190 NAVIGATION COMPASS DEVIATION forms supplied upon which there is a vertical scale, OP, rep- resenting unity, which is divided into 100 equal parts, and by estimate into 1000 parts; and an arc, with as a center and radius equal to OP, divided into degrees, upon which devia- tions may be read off. When no blank is at hand, a similar scale may readily be constructed. In all cases the line OP is equal to unity and is vertical, and at P there is a horizontal line. Example. Let %= + .053, 23 = + .222, ( = + .220, S) = + .226, @ = + .063. For reference, see Fig. 53. From P lay off PA = ^ to the right if ^ is + , to the left if . " A AE = $ " " " ( " +, " . ^ " ED' =<) upwards "+, (as it usually is). 7)' " IXJ?' = 58 " " 33 " + , downwards if . jj/ t< j?/jyr = ( to the right ( +, to the left if -. With A as a center, and a radius equal to AD' = describe a circle, called the " generating circle." From 2V draw a straight line through D' and produce it until it inter- sects the generating circle a second time, which point mark Q. The point Q is called the " pole " of the dygogram and is one of the necessary points to have. From D' produce ND' for the distance D'S equal to D'N. Take a straight-edge of paper of sufficient length and lay it down on .ZV$; mark, on the edge of the paper, dots opposite the points N, D', and 8; move the paper around so that the center dot moves on the circum- ference of the generating circle and with the edge always passing through the pole Q; by means of pencil dots opposite the end marks on the paper-edge, a sufficient number of points may be obtained for drawing in free hand the curve of the dygogram. To mark the dygogram for magnetic headings, lay a pro- tractor on the line NS, its center at Q, and dot off the head- ings required (usually every 15 rhumb) ; through each of ILLUSTRATING CONSTRUCTION 191 30' FIG. 192 NAVIGATION COMPASS DEVIATION these points and through Q draw a line and extend it across the dygogram. Where the lines cut the dygogram are the points required; the first cut to the right of N. (looking from S. to N.) is, say, 15, the 2d, 30, the 3d, 45, the 4th, 60, the 5th, 75, etc. Draw small circles around the points and mark each one correctly. To construct a table of deviations for magnetic headings when only the exact coefficients are known. Having pro- ceeded so far as to find the required points of the curve and marked each correctly, as explained above, draw a line from each point to (or until it intersects the graduated arc) ; the deviation then for each magnetic heading is shown by the angle which the line drawn from that point makes at with the vertical graduated line, and is read off from the graduated arc; if to the right of the vertical line the deviation is East or -}-, and if to the left, West or . It is usual to record the deviations in a tabular form as follows (as example, case of Fig. 53 is taken) : Magnetic Heading. Deviation. Magnetic Heading. Deviation. +13 00' 180 5 30' 15 +20 10 195 2 15 30 +26 30 210 45 45 +31 55 225 1 00 60 +34 00 240 4 30 75 +31 30 255 8 45 90 +21 00 270 12 00 105 + 4 00 285 13 15 120 7 30 300 11 15 135 11 50 315 7 00 150 11 30 330 1 00 165 9 00 345 +5 50 To construct a table of deviations for compass headings when only the exact coefficients are known. In practice it is necessary to have a table of deviations for " compass head- ings," and to get it when only the exact coefficients are known, proceed as explained for constructing the dygogram and the table of deviations for magnetic headings. Then construct the Napier's curve by laying off on the Napier's diagram along THEORY OF THE DYGOGEAM 193 the "full lines" the deviations for the magnetic headings; draw in the curve and then take off the deviations along the " dotted lines " for the " compass headings/' and record them in tabular form, one column being " Compass Headings " and the other one " Deviations." To show that the dygogram satisfies the conditions of the general expression. _ 21 + 33 sin z + ( cos z + sin 2z + ( cos 2z ~ 1 + $ cos z ( sin z + & cos 2z ( sin 2z For reference, see Fig. 54. Construct the dygogram, as previously explained, and take any point R of the dygogram corresponding to the magnetic heading z. The position of the different coefficients, or the lines representing the forces, as laid down in constructing the dygogram, are for a magnetic heading North; for any other heading z the lines and different triangles remain of the orig- inal size, but assume new positions in regard to the center A. As the ship swings through the magnetic azimuth z, the keel line DD'B' swings around D as a center, through the angle z, in the new position DD"K, cutting the generating circle at the point D". According to the construction of the dygogram a line, QD"R, making an angle z at Q with the line NS, will cut the dygogram at the point for the azimuth z; this line will pass through D" because the angles D'QD" and D'DD" are each equal to z, and, as both Q and D are on the circum- ference of the circle, the angles are each measured by half the same arc, D'D". According to the construction, D"R = D'N = V93 a + a ,and by geometry, the angle B'D'N = KD"R = a; therefore, a perpendicular let fall from R upon DD" produced will cut at K such that D"K -= and KR = <. Thus it is seen that, in swinging through an azimuth z, the triangle of polar forces, D'B'N, has assumed the position D"KR. The triangle A ED' will revolve around A as a center in 194 NAVIGATION COMPASS DEVIATIO' such a manner that while the ship turns through an angle z, the triangle AED' will turn through an angle 2z. Above it was seen that half the arc D'D" measured the angle D'QD"=z; therefore, the angle at the center, measured by the same arc, would be equal to 2z; that is, D'AD" = 2z, and therefore the other sides of the triangle, AE and ED', will turn through the angle 2z; or EAE f 2z and E'D"D" f 2z (D m being verti- cally below D" on the line PBC). The forces, as represented by the coefficients, have kept their original values or strength, but now act in new directions to produce deviation and to affect the directive force of the needle. PA = $1 remains constant; AE' = (, but acts at the angle 2z with its former position; E'D" = acts at an angle 2z; D"K = 33 and KB (, but each acts at the angle z with its former position. From each of the points E', D", K, and E let fall perpen- diculars upon the two axes having P as an origin. As R is the point of the dygogram for azimuth z, the de- 7" /? viation 8 = FOR; then, tan 8 = Q-J* in which LR = force of earth and ship to magnetic east in terms of mean force to "N. as unit ; OL = force of earth and ship to magnetic north in terms of mean force to 1ST. as unit. By referring to the figure it is seen that the angles NQR, D'DD", D'QD", DD"D" f , D"KB, KRL are all equal to each other and to the azimuth z. It may also be seen that D'AD" = EAD' EAD"; E'AS = E'AD" EAD"; or EAD" = E'AD" E'AS; hence D'AD" = EAD' E'AD" + E'AS. But EAD'=E'AD">; .'. D'AD"=E'AS = 2z = MD"D'" = HE'S. Now, LR = PC = PA + AS + SD'" + D'"B + EG; but PA =%;AS = cos2z. THEORY OF THE DYGOGRAM = 4- .040 5b = .246 S = + .430 J) = 4- .230 -' f .114 FIG. 54. GEOMETRICAL DEMONSTRATION OP THE DYGOGRAM. 196 NAVIGATION COMPASS DEVIATION Let E'M = r and MD" s; then r + 5 = ; SD'" = SM + MD'" = r sin 2z + s sin 2z; ... ##'" = (r + s) sin 20 = S) sin 2s. ZTB = 93 sin 2; 5(7 = ( cos z; .-. # = 51 + 93 sin 3 + cos z +. sin 2z + @ cos 22. Again, OL OP + (WG FP) + (GJff #), being the point where the horizontal line from D" cuts the vertical axis. But OP 1 ; WG = E'S + D'"D" =r cos 20 + 5 cos 2z = (r -\- s) cos 2z = cos 2z. FP = @ sin 2 2 ; OH = 93 cos 0; LH = & sin z; .-. OL = 1 + 93 cos z sin 2 + S) cos 20 sin 2z. 51 + 93 sin z + ( cos z + sin 2z + cos 2z " The line iV^ although taken as the zero line for laying off the magnetic headings, does not represent the direction of the keel line of the ship for magnetic North or South. The ver- tical line represents the keel line for magnetic North or South (magnetic meridian) and the direction for any other heading z is represented by drawing from D (vertically below E) 9 a line DD"K making an angle z = D'DD" with the vertical. As 93 is laid off to head and & to starboard (opposite, if negative), the angle B'D'N = KD"E = a (the starboard angle). It must be seen that the points marked on the curve of the dygogram are not really directions of the ship's head, but are the points on the curve which show, for the headings desig- nated, the deviations of the compass and the position of the forces in regard to the meridian, for those headings. To represent the direction of the ship's head on the dygo- gram for any designated point of the curve, join the point on the curve with Q and note the intersection of this line with the circumference of the generating circle; a line drawn from D through this point of intersection will give the keel line of the THEORY OF THE DYGOGRAM 197 ship. Confusion may be avoided by drawing around the point of intersection with the generating circle the outlines of a ship with its head in the proper direction. OP = unity = mean force to North = mean directive force in the compass needle = \H. Where A is unity, the mean force to North becomes H, the horizontal force of the earth at that place. 1 H' OL = -j- -g~ cos 8 = force of earth and ship to magnetic North, in terms of mean force to North as unit (for any par- ticular azimuth z of ship's head) = directive force of needle. 1 ft* LR =--y sin S = force of earth and ship to magnetic East (in terms of XH, the mean force to North as unit, for any particular azimuth z) = force tending to draw the needle from the magnetic meridian, thus causing deviation. 1 H r OR =-v- TT-= force in the direction of the disturbed needle; the needle being drawn by the force to east (LR), out of the meridian, through the angle FOR 8 for that particular azi- muth z. The angle POA = deviation due to constant force 5t (same for all headings). AOE' = deviation due to induced force in unsymmetrical soft iron, represented by coefficient (. E'OD" = deviation due to induced force in symmetrical soft iron, represented by coefficient . D"OK = deviation due to polar force to head, represented by coefficient 23. KOR = deviation due to polar force to starboard, repre- sented by coefficient (. Of course, the sum of any two or more of these angles is equal to combined deviation caused by the combined forces designated. Thus the deviation for magnetic azimuth z caused by the forces represented by 5T, (, and S) is POD" = POA +AOE' + E'OD". 198 -NAVIGATION COMPASS DEVIATION A correct idea of what the dygogram is may be obtained from the following, viz. : Suppose a compass needle pivoted at (see Fig. 54), its half length when equal to OP being considered as unity, that is, equal to the mean force to North, XH. Suppose the needle capable of assuming a length pro- portional to the force in the direction of its length, for each heading. From an inspection of the dygogram, it is seen that (-I TTf \ OR = ~j- ~ j varies in amount or length as the ship swings in azimuth. Now, as the ship swings in azimuth, through a complete circle, the end of the needle will trace out the curve of the dygogram, its end, at any azimuth z being at the point R of the dygogram, showing a deviation 8 = FOR. Various cases might be given where a knowledge of the dygogram would be of great assistance; such as where the values of 2t, 23, (, >, and ( are all known, and it is desired to compensate on any heading while at the dock ; in which case the deviation due to ST, (, 23, and ( has to be left uncorrected in compensating the quadrantal deviation, and that due to 51 and @: left uncorrected in compensating that due to 23 and (. Fig. 55 shows the manner in which the various forces re- volve, by which the final curve of the dygogram is traced out, when all the forces have appreciable values as represented by the exact coefficients. OP is equal to unity = \H. PA is the dygogram due to the constant force represented by .ST. The inner circle is the dygogram due to the induced force represented by @?, standing to one side of the meridian line on account of the constant force 21 ; the circle is properly marked. The next circle, having a radius equal to V 2 + 2 > i g ^ ne dygogram due to both induced forces, represented by ( and , standing to one side of the meridian line on account of the constant force 21, and if 21 is zero its center is at P. THEORY OF THE DYGOGEAM 199 200 NAVIGATION COMPASS DEVIATION The small shaded triangle is the triangle of induced forces producing the quadrantal deviation, and revolves around A as center, the rate of revolution being double that of the ship in swinging. The large shaded triangle is the triangle of polar forces pro- ducing the semicircular deviation; it revolves on the circum- ference of the quadrantal circle, its apex continually touching that of the inner triangle, the center of revolution being at the point D (see Figs. 53 and 54) and its rate of revolution being the same as that of the ship in swinging. The final curve, that traced out by the outer corner of the triangle of polar forces, is the curve of the dygogram. If both 21 and @: are zero, the center of the second circle becomes P and its radius 3). If 51, @?, and 2) are all zero, the dygogram for the semi- circular forces is a circle whose center is P and radius V 33 2 + 2 ; if then either S3 or ( becomes zero, the dygogram of the remaining force will be a circle whose center will be P and whose radius will be the remaining force. 113. Given the deviations and the horizontal force on two courses, regarding 51 and @ as zero, to find A, $, (, and $ by construction. Let Z-L and Z 2 be the two magnetic courses, 8^ and 8 2 the TTt TTt corresponding deviations, -rr and -~^ be the horizontal forces in terms of H. 1 H' Eeferring to Fig. 54, it is seen that if OR = y -go then OP = 1, OL = 1 + $ cos z & sin z '+ > cos 2z @ sin 2z f and LR = 51 + SB sin z + ( cos z + %) sin 2z + cos 2z. iff Now, if OR is taken as ^- , we shall have OP = \, OL A + A$ cos z A( sin z + A cos 2z A sin 2z, and LR = A5C + A33 sin 3 + AS cos a + A sin 2z + A( cos 2z. OBSERVATIONS ON Two COURSES 201 Points on a dygogram corresponding to these last values of OL and LR as coordinates may be found thus : For R^ lay off the angle POR = 8 and take OR,=^ for R 2 lay off the angle POR 2 = 3 2 and take OR 2 = ?. Call R, and R 2 datum points (Figs. 56 and 57). FIG. 57. There are two different constructions under the above gen- eral heading. (1) When the two magnetic courses are diametrically op- posite (Fig. 56). Find the datum points R. L and R 2 and draw RJ& 2 , bisecting it in Q-, a point of the generating circle. Through G draw AD, parallel to the magnetic direction of the keel line, and intersecting OP in D. Draw OD' perpendicular to AD inter- secting OP in D'. Bisect DD' at P and drop a perpendicular 202 NAVIGATION COMPASS DEVIATION from R i and R z on AD. Then OP A, GA == GA' A93, AS,. = A'R 2 = A, and PD' = AS). By drawing the outline of a ship about G, the signs of A93 and AS become apparent. Having these quantities, find 93, (, and and construct the dygogram. If the two magnetic courses are !NT. and S., G will be at D'. and we can not determine D, and hence neither A nor S). If they are E. and W., G will be at D, and we can not determine D r , and hence neither A nor 3). (2) When the two magnetic courses are not diametrically opposite. It has been shown in Art. 112 that a point of the dygogram, for example R, Fig. 54, may be found by laying off the angle z == D'DD" 9 then measuring off D"K = 93 and KR = (. From Fig. 57 it is seen that R may be found by laying off 93 D'B (=. D"K), making the angle MD'B = z, and DC = & at right angles to the course line D'B, and then completing the parallelogram. Now for another course MD'B' lay off D'B r 93 in the direction of the course line and DC' = & at right angles to it, completing the parallelogram and finding the datum point R 2 . Hence if R is a datum point corresponding to one course MD'B, CR is a line parallel to and R^B a line perpendicular to the course line through that datum point. In the same way C'R 2 and R 2 B' are lines through R 2 respec- tively parallel to and perpendicular to the second course line. Let u be the intersection of the two lines through R and R 2 parallel to the course lines, that is of CR and C'R 2 ; and let v be the intersection of lines perpendicular to them, that is of R B and R 2 B' ; then from geometry it is plain that if we draw vD' and uD J they will intersect at w, a point' of the gen- erating circle DD', that vD' bisects BvB' and uD bisects CuC', and further that w is on the circle passing through u, v, and the datum points R^ and R 2 . Therefore, to construct a dygogram when the two courses are OBSERVATIONS ON Two COURSES 203 not diametrically opposite, find the datum points R and R 2 as in case (1). Through R^ and R 2 draw lines parallel to the keel lines meeting in u and lines perpendicular thereto meet- ing in Vj, the keel line through R corresponding to the course Z-L and that through R 2 to the course z 2 . Through v draw a line parallel to one bisecting the angle between course lines, in other words bisecting the angle RuR 2 . This line cuts the vertical line in D'. From u drop a perpen- dicular on vD', intersecting it at w and the vertical line at D. Bisect DD' at P. From D drop perpendiculars DO and DC', respectively, on the 1st and 2d course lines ; from If, perpen- diculars D'B and D'B', respectively, on lines at right angles to said course lines. Then OP = A, and DD' is the diameter of the generating circle. PD = AS), D'B = D'B' = A33, DC == DC' = Ad. Having found these values, find the coefficients, and con- struct the dygogram. The construction fails when the magnetic courses are equally distant from any cardinal point ; for, if equally distant from 1ST. or S., RJ) R 2 D and RJ)R 2 the difference of magnetic azimuths = R^wR 2 , the point v will be in the vertical line, w will be at D, and it will be impossible to determine D' ; if equally distant from East or West, w will coincide with D' t and it will be impossible to determine D* 113a. To find 23 and ( from observations on one heading TJf (see Art. 94). Having determined 8, z, and ^|L and assuming A and , let OP = unity represent the direction of magnetic North (Fig. 56). Describe a circle, center P, radius 3), cut- ting OP in D. Draw DA, making PDA == z and cutting the circle in G. Lay off POR 8 (the given deviation 8), to right if +, to left if ( ) ; take OR^ --\-^'> dr P R i A perpendicular to DA; then GA = 33 and AR = S. An outline of a ship around G, heading properly, will make the CHAPTEE V. PILOTING. FIXING SHIP'S POSITION NEAR LAND. DANGER ANGLE. DANGER BEARINGS. FOG SIGNALS. 114. Piloting in its broad sense is the act of conducting a ship where navigation is dangerous, as, when coasting, passing through channels, and into harbors. Before reaching pilot waters, a navigator should study the charts and sailing direc- tions of the region, know that they are up-to-date, be con- versant with landmarks and aids to navigation, and the state of tides and currents of the locality at the time when he may navigate the waters. After reaching pilot waters, a keen look- out must be kept for dangers as well as aids to navigation, soundings should be taken and depths and character of bot- tom obtained compared with indications of the chart ; in shoal water the hand lead should be kept going. Advantage should be taken of the first opportunity to locate the ship's position by bearings of known objects, and having laid a course clear of all dangers, the ship's position must be frequently plotted by the most convenient of the methods herein explained. Before proceeding to explain them it will be well to define general terms of frequent use in navigation. The bearing of an object from a ship is the angle between the meridian and the great circle which passes through the object and the observer on board, and it indicates the direc- tion in which the object is seen from the ship. It is called true, magnetic, or compass according as the meridian considered is that passing through the geographical FIXING POSITION NEAR LAND 205 poles, the magnetic meridian, or the direction of the compass needle. On board ship bearings, by compass or peloms, are measured from North, to the right, from to 360. A line of position is any line, straight or curved, on which the ship's position is known to be. It is obtained from ob- servations of either celestial bodies, or terrestrial objects. ' A line of bearing. When a line of position is obtained from a bearing, to make it more distinctive and to indicate ' its origin, it is called a line of bearing. Position point. Any point on either a line of position or a line of bearing at which the ship's position may be assumed is a position point; the actual position of the ship, or a fix, is determined by the intersection of two lines of bearing, two lines of position, or one of each. 115. To fix the position of the ship near land when two or more landmarks of known position are in sight. (1) By sextant angles. Select three objects so as to give well-conditioned circles (see Art. 34). Generally speaking, the angles should, if possible, be over 30 and the objects in line, or the middle one nearest the observer. Observe by a sextant, whose I. C. is known, the angle between the middle and right objects, and at the same time the angle between the middle and left objects, which are known respectively as the right and left angles ; set the right and left arms of a station pointer, or 3-arm protractor, for their respective angles, place protractor on the chart, move it over chart till the beveled edge of each arm passes simultaneously each through its own object. The center of the instrument locates the ship's posi- tion on the chart. A special application of this method is when two of the objects are in range and but one angle is taken. This is by far the preferable method from the standpoint of accuracy, as the position is independent of compass errors, 206 NAVIGATION speed errors, or current effect. Angles can be taken from any part of the ship whilst bearings must be taken from the standard compass or pelorus. The geometrical theory of the method will be understood from Fig. 58 which, by construction, embodies a method that can be used in the absence of a 3-arm protractor or tracing paper on which the angles could be ruled. Let A., B, and C, Fig. 58, be the three known positions. The observer at P measures angle x between A and B, and angle y between B FIG. 58. and C. Using the two objects A and B and angle x alone, the observer may be anywhere on the segment APB which be^ comes a line of position. If x is < 90, the line of position is greater than a semicircumference, if x = 90, the line of position is 180, if x is > 90, the line of position is < 180, and the position point may be anywhere on the arc, since all angles on the same segment of a circle equal each other. Using the two objects B and C and the angle y alone, the observer may be anywhere on the segment BPC which becomes a second line of position, the length of which is governed by rules as already explained for the first line of position ABP t BY CROSS BEARINGS 207 and the position point is somewhere on the arc BPC, and. being also on ABP, is at their intersection P; in other words, P is the fix. There are three cases in this method: (1) Both angles < 90, (2) both > 90, (3) one < 90 and one > 90. When an angle is < 90, the center of the circle passing through the observer's position and the two objects is on the same side of the line joining the two points as the observer ; in this case lay off AD and BD making angles 90 x with AB, D will be the center of the first circle. When an angle is > 90, the center of the circle passing through the observer's position and the two objects is on that side of the line join- ing the two objects remote from the observer ; in this case lay off BE and CE making angles y 90 with BC, E will be the center of the second circle. Hence, we have the general rule : take the complement of the observed angle; if -{-, the center of circle will lie on the same side of line joining observed ob- jects as the observer; if ( ), the center of circle will be on the opposite side. The indeterminate case is when the observer and the three objects are on the same circle, or nearly so, or when the two centers nearly coincide. To avoid such a condition see Art. 34. In case no protractor is at hand it is only necessary to measure the distance AB, erect a perpendicular at its middle point M, and lay off MD = MB tan (90 x) to find the center D, then describe the circle with radius DB. In a similar way find E and with radius EB describe the second circle, the intersection giving the fix P. (2) By cross bearings. This consists in finding the fix by two or more lines of bearing. The bearings per standard compass of two points of land, or objects, whose positions are projected on the chart, having been obtained, are corrected for the deviation due to the ship's head at the instant the bear- ings were taken. Lay the parallel rulers on the nearest com- 208 NAVIGATION pass rose, the edge passing through the center and the degree on the circumference representing the magnetic bearing of a given object, transfer the edge of rulers parallel to itself till it passes through the given object. Draw a light line along the edge. This is a line of bearing and the position of the ship is somewhere on it. In a similar way draw a second line of bearing through the second object, and the ship being also somewhere on this line, the fix is at the intersection of the two. If the compass rose had been true instead of magnetic, the compass bearing would have been corrected for variation as well as for the deviation of the compass. The difference of bearings should be as near 90 as possible for best results; if the difference is small, 15 to 20, a small error in the' bearing will make the fix uncertain. The position determined from the bearings of only two objects may be in error, even when the angles of intersection are good, due to an error in the assumed deviation or even an error of the chart ; for these reasons a third line of bearing should be obtained, if a third object is available. Should these three lines of bearing form at their intersection only a small equilateral triangle, its center may be regarded as the fix. In finding a fix by cross bearings, take first the bearing of that object nearest the fore-and-aft line, ahead or astern, since such an object will change its bearing less than one more nearly abeam, in the interval between bearings. If one ob- ject is ahead or abeam, knowing the course, its bearing is taken mentally, and it is only necessary to get a bearing of the second object. A bearing of one object and a bearing of a range, or a bearing of one object in connection with a sextant angle to another object, provided the angle to the second object is sufficiently large, will give a good fix. 116. (B) When one object only is available. Having taken a compass bearing of the single object, we have a line of bearing which is true or magnetic according as the com- To FIND DISTANCES . 209 pass bearing is corrected for compass error or for deviation alone. If true, it is laid down from a true rose ; if magnetic, from a magnetic rose. The ship is somewhere on this line. A fix on this line may be gotten from its intersection with a line of position found at the instant, or from one brought up to the instant of getting a bearing by the run, or by knowing latitude or longitude, or by knowing the distance of the object. The distance of the object may be estimated; or gotten from its angular altitude, if its height is known; or by Buck- ner's method when available; or by an accurate range-finder.* Knowing the distance, the ship is somewhere on a line of position whose center is the object and whose radius is the distance, and, being also on the line of bearing, the ship is at their intersec- P IG< 59. tion. An estimated dis- tance may be often verified by a cast of the lead where sound- ings vary considerably. Suppose in Fig. 59, AB = h = the height in feet of an object whose angular altitude at C= a . Then if ABC = 90 and BC d, d = . Now as a is very small and ex- tan a pressed in minutes of arc, tan a = a sin 1'; therefore, d = - ? , and to express d in nautical miles, divide sec- a sin V ond member by 6080.27; substituting value of sin V we have d (in sea miles) _ h (in feet) h^ 1 __ 557 _^ ~ a (6080.27) X .00029 = " a ' 1.76328 " a ' Ex. 21. A lighthouse 140 feet high subtends an angle of 16'; find the distance in nautical miles. 140 d = -jg X .567 = 4.96 nautical miles. * The Barr and Stroud range-finder is sufficiently accurate for navigational pur- poses at distances varying between 800 and 7000 yards, and its use is practicable on board ship. 210 NAVIGATION Table 33 of Bowditch gives the distance, by vertical angle, at intervals of one-tenth of a mile up to 5 sea miles for ob- jects whose heights vary from 40 to 2000 feet. In this method B, the foot of the object, should be seen and the angle ABC should be 90 ; for this reason the observer's eye should be as low down as possible. The error due, how- ever, to a slight height of the eye is inappreciable, but that due to the visible shore line B', not being at B the foot of the object, might be material. In other words, B' should be at B and C" at C. A Dip D FIG. 60. Buckner's method is one often used for finding the distance of a target, and may be used for finding the distance of an isolated object beyond which the sea horizon can be seen. In Fig. 60, AB is the height of eye above sea-level; EAC, the dip of sea horizon C due to height h = AB; D, isolated object ; a, the angle between the object and sea horizon beyond; and h (in feet) a (in yards) = 3 tan (a-j-Dip) Table 34 of Bowditch gives the distance in yards for various values of angle a observed at heights of observer's eye varying from 20 to 120 feet. Lecky's off-shore distance tables are more extended in their application, likewise Von Bayer's diagram. To find the distance of an object of known height, just vis- ible on the sea horizon to an observer's eye of a given height, Fix BY CHANGE OF BEARING 211 it is sometimes useful to use Table 6 of Bowditch which gives the distance of visibility of objects at sea for different heights ; the distance, owing to the uniform curvature of the sea, de- pending on the heights both of object and observer's eye. Ex. 22. From a height of 45 feet, a light, whose height from the light list is 160 feet, is seen to disappear below the horizon ; find its distance in nautical miles. For 45 feet, distance 7.7 miles. For 160 feet, distance 14.5 miles. Kequired distance = 22.2 miles. Table 6 of Bowditch is calculated from the formula (to be deduced later on) d = 1.148 V K li = height of object in feet, d = distance in nautical miles of the object just visible in the horizon, the observer's eye being at the surface of the earth. Now, if li' = the height of eye in feet, d =1.148 (V^+ V^')- Ex. 23. An observer, height of eye 36 feet, sees in the sea horizon the top of a lighthouse known to be 121 feet high. What is the distance in nautical miles ? d = 1.148 (V36 + V121) = 1-148 X 17 = 19.55 miles. 117. (C) By two bearings of a single object and the course and distance run in the interval. (1) This involves the solution of a plane triangle, given one side and the two adjacent angles. The ship is steering the course AB (Fig. 61). At first observation, ship is on the line of bearing AC and the patent log is read. At second bearing, ship is on the line of bearing BC and patent log is again read. The dif- ference of readings of patent log gives the distance AB, from the course and bearings the angles A and B are known, and C = 180 (B + A). 212 NAVIGATION FIG. 61. (1) By plane trigonometry. . ,-. sin B . A -r> , pj.o, sin A. . AG = imtf X AB and S0 = IhH7 X The distance of passing abeam CD = BC sin B. Tables have been compiled giving factors by inspection; the factor of the first column multiplied by distance run be- tween bearing lines gives the distance of object at second bearing, the factor of the second column multiplied by the distance run gives the distance when the object was abeam. The arguments in these tables are the difference be- tween course and first bearing, differ- ence between course and second bear- ing ; the difference of bearings in Table 5 A, Bowditch, being at intervals of quarter points, in Table 5B at intervals of two degrees; the factors in each table being for a distance run of one mile. So far as the factors are concerned, it is immaterial whether the course and bearings are per compass, magnetic, or true, provided they are all from the same meridian. Ex. 24- A ship heading 292 (p. c.), var. + 12, dev. -(-9, had a lighthouse bearing 234 (p. c.) ; after a run of 10 miles, the same light bore 166 (p. c.). Find the -distance at second bearing, also when light was abeam. Difference between course and first bearing 58. Difference between course and second bearing 126. Table 5B, factor from first column = 0.91 and distance at second bearing = 9.1 miles. Table 5B, factor from second column = 0.74 and distance abeam = 7.4 miles. It is to be understood that the course and distance are DOUBLING ANGLE ON THE Bow 213 those over the ground, and due allowance must be made for currents. A simple application of this method is what is known as the " bow and beam bearing." The bearing of an object is taken when it is 45 on the bow, and patent log noted; an- other bearing is taken when the object is abeam and patent log again read. The distance run between the times of the two bearings is the distance of the object when abeam, pro- vided the course and distance have not been influenced by currents or bad steerage. If the first bearing was taken abeam and the second when the object was on the quarter, the distance run multiplied by 1.4 will give the distance of the object at second bearing, the distance run being the distance of object when abeam. (2) The graphic solution of this method is easier and, cer- tainly, more general than the factor solution. By means of parallel rulers, draw the first line of bearing AC (Fig. 61) on the chart. When the bearing has changed sufficiently draw the second line of bearing on the chart. Cut these lines by one representing the course, transferred from the compass rose by parallel rulers, so that the distance intercepted between the two lines of bearing shall equal the distance run. If the course used was the course made good, and no current affected the run, the points of intersection of the course line with the lines of bearing will be the positions of the ship at the times of taking the bearings. (3) Doubling the angle on the bow. In this case note the patent log when the object has a certain bearing on the bow, or, in other words, when there is a certain angle between the ship's head and the object; again note the patent log when this bearing on the bow, or angle, has doubled ; the dif- ference between the two readings of the patent log, or the distance run, is the distance of the object at the second bear- ing. 214 NAVIGATION This is shown graphically in Fig. 62. Since DBG = WAO and also equals DAG + BCA, BOA = DAC and BC = AB distance, run. Hence, the rule " when the angle on the bow is doubled, the distance of the object at the second bearing equals the distance run " ; provided there is no current. A special application of this method, and one universally used, is the four-point bearing, the double angle being 90, FIG. 62. FIG. 63. and the distance run the distance when abeam. A com- mon application is when the first bearing is two points on the bow, giving the distance on the bow as distance run; again when the first bearing is 60 on the bow, the distance run will be the distance of object when it bears 30 abaft the beam. Distance of passing an object abeam. In case the first angle on the bow is 26 and the second angle is 45, the distance run between the two bearings will be the distance of passing the object abeam, if course and distance are un- affected by current. A knowledge of this distance is of im- portance as the point is approached. DANGER ANGLE 215 118. Danger angle. When sailing along a coast and it is desired to avoid sunken rocks, or shoals, or dangerous obstruc- tions at or below the surface of the water, and which are marked on, the chart, the navigator may pass these at any de- sired distance by using what is known as a danger angle, of which there are two kinds, a horizontal and a vertical danger angle; the former requires two well-marked objects projected on the chart, lying in the direction of the coast, and suffi- ciently distant from each other to give a fair-sized horizontal angle; the latter requires a well-charted object of known height. In Fig. 63, let AMB be a portion of a coast along which a vessel is sailing on course CD f A and B two prominent objects projected on the chart; S and S' are two outlying shoals, reefs, or dangers. To pass at a given distance outside the center of danger 8'. With the middle point of danger as a center, and the given distance as a radius, describe a circle; pass a circle through A and B tangent to the seaward side of the first circle. To do this practically, it is only necessary to join A and B, and draw a line perpendicular to the center of AB, then ascertain by trial the location of the center of circle EAB. Measure the angle a = AEB f set sextant to this angle, and remember- ing that AB subtends the same angle at all points of the arc AEB, the ship will be outside the arc AEB, and clear of danger $', as long as AB does not subtend an angle greater than a to which the sextant is set. Now, should it be desired to pass a certain distance inside of a danger S with the middle point of danger as a center and the desired distance as a radius describe a circle; pass a second circle through A and B tangent to this circle at G. Measure the angle BGA = < with a protractor. Then, as long as the chord AB subtends an angle greater than <, the ship will be inside the circle A GB and clear of danger 8. 216 NAVIGATION Should both, dangers exist and it be desired to pass between them with a margin of safety already referred to, steer on course CD so that the angle subtended by AB shall be < a but > . 119. Vertical danger angle. Practically the same general principle is involved in this as in the horizontal danger angle ; however, only one object is used and that one must be of known height. FIG. 64. In Fig. 64, draw circles around the dangers 8 and 8' with radii representing a safe margin of safety. Let A B be an object of known height. With A as a center draw circles tangent at E and G. Measure the distances AE and AG; find from Table 33, Bowditch, or by computation, the angular altitude of AB of known height for distance AE, let it be a; also for distance AG, let it be . To pass outside of and clear of 8' the angular height of A B must be < a. To pass inside and clear of 8 the angular altitude of AB must be > . These are the limits for ensuring a safe passage between 8 DANGER BEARING 217 and '8'. In observing a vertical danger angle, the observer should be as near the water line as possible to minimize errors due to height of eye, and the angle at A should be 90. 120. Danger bearing. A bearing, properly taken and used, may often be of great use in keeping a ship out of danger. Suppose C and C" to be shoals or rocks near the coast (Fig. 65). A ship is passing on course efg. Lay down on the chart the tangents AD and Ag with any desired margin of safety. With A as a center describe the arcs of circles to in- clude dangers C and C'. Note the magnetic bearing DA and gA and find what should be the compass bearings for the given course. Now before the distance of A is reduced to the radius of the circle's arc en- closing C, the bearing of A must be to the left of the danger bearing DA and kept so to avoid danger. The bearing of A must not get to the left of gA till the distance of A is greater than the radius of the arc enclosing C". It may often be pos- sible to find a danger bearing on a range; for instance, if A and B are in range on the danger bearing !M,-the object B must be kept open to the right to ensure safety, and the guarantee is better than a compass bearing would give. Lights as danger guides. Lights of lighthouses may be used to give warning of danger as the object A was used above. Many lights, showing white over safe waters, show red over sectors embracing areas of rocks, shoals, or depths over which the approaches would be dangerous, and it is the duty of the navigator to keep out of the danger sectors. The magnetic bearings showing the limits of the sectors are given from sea- FIG. 65. 218 NAVIGATION ward, and the bearing of the light, taken frequently, should not be allowed tu get within the danger sector unless the dis- tance of the light is known to be greater than the radius enclosing the dangers. A navigator is furnished with charts, light lists, and sailing directions, all of which give details as to color, character, and visibility of navigational lights, and the navy regulations make it his duty to become thoroughly conversant with these details before coming within their range of visibility. It may sometimes, in fact, has often happened, that one light is sighted when the run indicates the ship to be in the region of another light for which a lookout has been kept; hence, the rule, on sighting a light, is to compare its visible character- istics with those laid down in the light lists, and, if appar- ently not a fixed light, the duration of its periods must be noted by watch. It must not be forgotten, however, that abnormal atmos- pheric conditions may increase the range of visibility of a light, whilst mist may decrease it and is often found to make white lights appear red. When a fixed light is first sighted, especially under fair conditions, and there is any question as to whether it is a lighthouse or a vessel's light, simply descend a short distance, and again look for the light. A vessel's light is of limited intensity and, if seen at all, can be seen at any height; a navigation light can be seen, as a rule, as far as the curvature of the earth will permit and this distance, de- pending on the height of eye and of the light, will be lessened by the observer going lower down. A descent of a few feet may cause a navigation light to disappear. 121. Fog signals. A navigator should make himself famil- iar with all fog signals of the locality in which he is cruising; in foggy weather and in the neighborhood of signals, the closest attention must be paid in an effort to hear them and locate their direction. When heard, their periods should be FOG SIGNALS 219 timed and a comparison made with those recorded in the light lists to ensure identification. However, it must not be for- gotten that atmospheric conditions affect the transmission of sound and at times cut it off entirely, producing a silent zone in a locality where the signal could be heard distinctly at other times; hence, if dependent on aerial signals, in a fog, slow the ship, navigate with extreme caution, keep the lead going on soundings, make every effort to guard against over- logging, and, as a last resort, if the depth of water will permit, anchor the ship. Ordinarily the sound of an aerial fog sig- nal, if it is to leeward, will be heard sooner from aloft; if to windward, from a point nearer the water. Fortunately, the cases are now rare in which the navigator has to depend on aerial signals, for the submarine-bell has been generally adopted by maritime nations as a means of making fog signals (see Appendix A). In using this system, listen with the starboard receiver when the bell is known to be on the starboard side, otherwise use the port receiver; at all events if the bell is heard in both receivers its direction will be shown as being to starboard or port by the greater intensity of sound in the starboard or port receiver, and the correct bearing of the bell may be obtained by so changing the course as to bring it directly ahead, at which position the intensity of sound becomes the same in both receivers, provided the listener can hear equally well in both ears; otherwise there will be an error of direction, the amount being dependent on the difference of hearing in the listener's ears. The submarine signal is more reliable than the aerial signal since it can be heard on board vessels fitted with receivers at greater distances, these distances varying according to condition of instruments, attention, and delicacy of hearing of listener; its direction can be ascertained with fair accuracy, and the sound is not subject to the silent zone, though there is always a possibility that the bell's mechanism may be deranged. An approximate fix may be obtained from the bearing of the bell combined with a sounding or with a line of position (if recent and reliable) brought up to the instant of locating the direction of the bell; also from two bearings of the bell and the course and distance run in the interval. However, in a fog, even when within hearing of a sub- marine-bell whose direction may be fairly well determined, the navigator must be watchful and cautious, should reduce speed and make a judicious use of both log and lead, bearing in mind the fact that the effect of cross currents encountered will be increased as speed is reduced. CHAPTER VI. THE SAILINGS. 122. The position of a ship at sea, at a given moment, is denned by its latitude and longitude. This position is con- nected with one left, or with one to which the vessel is bound, by the true course and distance between them. A course and distance can be resolved into difference of latitude and departure, and this departure converted into difference of longitude; so that knowing the course and dis- tance sailed from a given position, the latitude and longitude of the position arrived at can be found ; or, when desired, the course and distance between two given positions may be found. The various methods of solution of these problems are termed Sailings, and include, Plane, Parallel, Middle Latitude, and Mercator Sailings. The term " Dead Reckoning " includes all calculations to determine a ship's position, given only the true courses and distances run from a given point of departure. It involves the principles of the various sailings explained below. The latitude and longitude determined by " Dead Reckon- ing" are noted thus, "Lat. by D. R.," "Long, by D. R." The position by D. R. is liable to error due to bad steering, improper logging of the distances run, faulty allowance for leeway, effects of wind, currents, etc., and the exact position of the ship, out of sight of land, can be determined only by celestial observations. The term " Day's Work," though frequently applied so as to embody only dead reckoning, the finding of the and d made by D. R. from the point of departure, and the C and d by D. R. from position arrived at to destination, should properly include results arising from a knowledge of the ship's true TAKING THE DEPARTURE 221 position obtained either by bearings or celestial observations as indicated in Art. 310. Particular attention must be paid to plane and parallel sailings as a full understanding of their principles is essential to an understanding of middle latitude sailing which is gen- erally employed for short distances, as in a day's run; for longer distances it will be better to use Mercator sailing, which is a method based on the principles already explained in the articles on the Mercator chart. The methods of laying down a ship's run and finding the position graphically have been explained in Chapter II. 123. Taking the departure. On leaving port, at the begin- ning of a voyage, the ship's position is fixed by some one of the methods explained in the chapter on " Fixing positions near land," the best method available at the time, of course, being used; and from a last position thus obtained, the succeeding traverse, as laid down graphically on the chart, takes its com- mencement. This final position may be taken from the chart and be considered the point of departure from which future positions may be calculated in the navigator's work book. However, it is frequently the custom to take from the last position at which objects can be distinctly seen, the bearing and distance of some fixed point of land, lighthouse, light vessel, or beacon, whose latitude and longitude are known, and to consider its reversed true bearing and distance as a true course and distance sailed by the vessel, thus taking its posi- tion as the point of departure. If the distance is not known, it must be estimated. This is what is known as taking a departure. The navigator must be careful to correct the compass bear- ing of this point for the variation, and the deviation of the compass due to the ship's heading at the time of taking the departure, then to reverse the true bearing to get the true course the ship is assumed to have sailed. This reversed true bearing is called the departure course, and it equals the true bearing 180. 222 NAVIGATION When a departure is thus taken the departure course and distance appear in the record of the first day's work, in the proper columns of the tabulated form, and are treated like any other course and distance. Noon position as a point of departure. In the succeeding part of the voyage, each noon position by observation is taken as a new point of departure. Latitude left and longitude left. In dead reckoning, these terms refer to the latitude and longitude of the point of departure. Latitude in and longitude in. These terms in dead reck- oning apply to the latitude and longitude arrived at, and are marked " by D. K." when by account, or " by obs." when by observation. Course and distance made good. For a given interval of time the course and distance from the last point of departure to the position by observation at the end of that time are the course and distance made good. 124. The following notation will be followed in this book whenever it may be necessary to represent the quantities re- ferred to below: C will represent the course measured from the North or South towards East or West. CN " " " course measured from North around to the right from to 360. L " " " latitude. A " " " longitude. L! " " " latitude of place left. A! " " " longitude of place left. L 2 " " latitude of place arrived at. A 2 " " longitude of place arrived at. l=L 2 ~L " " " difference of latitude. DX^^ " " " difference of longitude. p " " departure. d " " distance sailed on course C. L Q " " " the middle latitude = PLANE SAILING 223 Latitude is North, or South according as the place is in the Northern or Southern hemisphere. Longitude is East or West according as the place is on a meridian East or West of Greenwich. In the solutions by computation of the triangles of plane, middle latitude and Mercator sailings, the course C is an interior angle of the triangle and is not greater than 90. Its general direction is determined by I and p or m and D. Having been found, C should be expressed as (7 N for practical purposes (Exs. 39, 52, and 53). If the data includes the course as (7 N , express it as C and indicate its proper direction before proceeding with the com- putation (Ex. 38). In solutions by inspection, as in dead reckoning, the course should be retained in the form of (7 N as the traverse tables are tabulated for courses up. to 360 (Ex. 25). PLANE SAILING. 125. For small distances at sea, the curvature of the earth may be neglected, and the small portion of the earth passed over may be regarded as a plane surface, on which the meridians are parallel right lines perpendicular to the equator, the parallels of latitude are right lines paral- lel to the equator, and the length of a de- gree is assumed the same whether meas- ured on the equator, meridian, or parallel. Though this assumption is not strictly correct, the results obtained by plane sail- ing may be considered sufficiently exact F IG . 66. for any ordinary day's run. The relations existing between the parts that enter into plane sailing are indicated in a right triangle in which C is the course, I the difference of latitude, p the departure in the latitude left or that arrived at. From an application of 224 NAVIGATION the principles of plane trigonometry, the relations are from Fig 66, Z = d cos ai p = dsmC,[ (115) p = I tan 0. j The solution of the above equations is facilitated by the use of Tables 1 and 2, Bowditch (which are tables for the solu- tion of any right triangle), calling d the hypothenuse, Z the side adjacent, and p the side opposite the course C. Table 1 gives the courses in quarter points and distances up to 300 for each unit; Table 2 gives courses in degrees and distances to 600. Should the distance for which I and p are desired be greater than the limit of the table, subdivide the distance into two or more parts, finding the I and p for these separate parts and adding; thus the cliff, of lat. for 1340 miles, course 10, will be the diff. of lat. for 600 + cliff, of kt. for 600 + diff. of lat. for 140, course 10, and the departure may be found in the same way; or 1340 may be di- vided by 4, giving 335, then I and p may be found for 335, course 10, and multiplied by 4; simi- larly any other factor, as 5 or 10, might be used. Since I = d cos and p = d sin C, it is apparent that the I and p for any course are re- spectively the p and I for the complement of the course, as shown in the tabulated form. So it is apparent that all ques- tions involving C, d, I, and p can be solved by inspection by using Any expression involving sines, Table 2. Diif . of Lat. and dep. for 10 (170, 190, 350) Dist. Lat. Dep. I 1.0 0.2 2 2.0 0.3 3 3.0 0.5 4 3.9 0.7 5 4.9 0.9 6 5.9 1.0 7 6.9 1.2 8 7.9 1.4 9 8.9 1.6 10 9.8 1.7 Dist. Dep. Lat. 80 (100, *80 3 , 280) Tables 1 and 2, Bowditch. PLANE SAILING 225 cosines, secants, tangents, or cotangents of the following forms may be referred to the traverse table. Thus x 30 sin 60, ] Also 35 = x tan 50, If p=d sin 60, L If p = /tan 50, When d = 30, find p. J When p 35, find L A triangle may be solved by the "Rule of Sines" in the same way. When the distance sailed is so great that the curvature of the earth cannot be neglected. In Fig. 67, let P be the ele- vated pole of the earth. and A two places on the surface connected by the loxodrome CA. Let C'A! be an arc of the equator intercepted between the meridians of G and A. Consider the distance CA to be divided into a very large number of equal distances; each distance form- ing with its corresponding differ- ence of latitude and departure a right triangle. All these triangles are similar, two angles of each tri- angle, the right angle and the course C, being equal to the corresponding angles in the other triangles. Each triangle is so small that it may be taken as a plane right triangle. Such would be the triangle dbc for the small distance ca. Now letting d , d 2 , d s . . . . d n be the small distances into which CA is divided; Z , 1 2 , l s l n , their corresponding differences of latitude for the course C; Pi, p 2 , p 3 p n , the corresponding departures for the same course, each departure being measured in the latitude of its own triangle ; 226 NAVIGATION we have ?j_ di cos C, 1 2 = d 2 cos C ... .l n d n cos (7; p =: t? sin (7, p 2 = tZ 2 sin (7 . . . . p n = d n sin (7. Therefore, k + Z 2 + Zs - - - k ' = K + 4 + d 3 . . . . d n ) cos (7. Fi + P2 + ^s ^n = (di + d 2 + d s d n ) sin C. Now since the parallels of latitude through (7 and A are the same distance apart on all meridians, the difference of latitude of C and A is the sum of the partial differences of latitude, and, as the total distance is the sum of the partial distances, we have 1 = k + h + k - - - * = K + d 2 + d, . . . d n ) cos C, ) or I = d cos C. \ ( And if each partial departure is measured in the latitude of its own triangle, the sum of these partial departures will represent the true departure in the triangle CAB, and hence, P = P i + p 2 + Ps'-Pn=(d i +d 2 + d 3 .. d n ) sin C, ) or p =. d sin C. ) So that I and p are calculated by the same formulae whether the curvature of the earth is, or is not, considered. However, the sum of the partial departures is less than the distance be- tween the meridian left and the meridian arrived at measured in the lower latitude, and greater than that measured in the higher latitude, and is approximately equal to the departure of the parallel midway between the two. When both points of departure and arrival are on the same side of the equator the latitude of the parallel midway between is known as the middle latitude, and is equal to the half sum of the two latitudes; in other words, L = - x ~j~ 2 . (118) From what has been said above, it is evident that a ship sailing due North or South (true) remains on the meridian, changes her latitude only, and the distance sailed is simply a TRAVERSE SAILING 227 difference of latitude, is either " Northing " or " Southing/' and must be so entered in the tabulated form of work. When a ship sails due East or West (true), she remains on her paral- lel, does not change her latitude, and the distance sailed is either " Easting " or " Westing," and must be so entered in the form for work, this departure to be later converted into its proper difference of longitude. When a ship sails due East or West (true), on the equator, the distance East or West is itself difference of longitude. When a ship sails on a loxodrome, at an acute angle with the meridian, she alters both her latitude and longitude. TRAVERSE SAILING. 126. If a ship sails on several courses instead of a single course, she makes an irregular track, called a traverse, and it is the function of traverse sailing to find the single course and distance that' would have taken the ship to the position arrived at, in other words, the resultant course and distance as well as the corresponding difference of latitude and de- parture. If C , C 2 . . . . C n be the different courses, and d , d 2 . . . . d n the corresponding distances, then, Z = d cos C lf l 2 = d 2 cos C 2 .... l n = d n cos C n ; Pi = di sin C 19 > 2 d 2 sin (7 2 .... p n = d n sin C n ; and, as before, I = Z x + 1 2 + 1 3 . - . . In , P = P + P 2 + Ps - Pn , p being measured along the parallel . 1 ~^~ 2 , or, as in the case a of a single course, "the whole difference of latitude is equal to the sum of the partial differences of latitude, and the whole departure is equal to the sum of the partial departures." The word sum is used in its algebraic sense, that is to say, if 1 N is the sum of the northerly differences of latitude and l s 228 NAVIGATION the sum of the southerly differences of latitude, then I = 1 N ~ l s and is of the same name as the greater ; and if p w is the sum of westerly and p E of easterly departures, p = p w ~p E and is of the same name as the greater. The traverse table referred to under plane sailing greatly facilitates the computation. Having found the resultant I and p, the course and dis- tance made good, or the resultant course and distance, are gotten from the formulae or by inspection from Table 2. The traverse may run irregularly, and into higher lati- tudes than the latitudes of ] <--*-) the extremities of the dis- tance made good, so that the departure of this course and distance made good may not be the same as the sum of the partial departures; hence, if necessary, separate the trav- erse into two or more parts, and calculate for each part separately. However, if the traverse does not go into too high latitudes, the error, like- ly to arise, may be considered immaterial in an ordinary day's run. Graphic explanation of traverse sailing. To further i .1 i j? explain the principles of traverse sailing, let W'E'gb represent a portion of a Mer- cator chart; let A, located on a meridian NS and a parallel GRAPHIC EXPLANATION 229 of latitude WE', be a place sailed from, and F a place arrived at, after sailing successively from A to B, to C (directly East), to D (directly North), to E, and to F (Fig. 68). Let merid- ians and parallels be drawn through each point of the traverse ; the triangles thus formed and the difference of latitude and departure corresponding to each distance are indicated in the figure. . _ C Diff. of Lat. = Ac. For distance AB \ _. , _ | Dep. = cB = lib. n ( Diff. of Lat. = 0. For distance BC 4~ ^ n , ( Dep. BC Im. nr . f Diff. of Lat. = CD = cd. For distance CD \ ~ (Dep. = 0. -n T x T-> T7 ( Diff- f Lat. = Dn = de. For distance DE 1 ^ J Dep. = nE = mg. Ei J'4. T?T? f Diff f Lat ' = E 9 ~ eJl ' For distance EF \ _ { Dep. = ^. The course made good is hAF, distance made good AF, the corresponding difference of latitude Ah, and departure JiF. Eegarding directions towards the top of page and to the right hand as positive, differences of latitude towards the top of page (in this case North) are +, towards the bottom (or South) are , departures to the right (or East) are +, to the left (or West) are . It will be seen, by examining the figure, that All and JiF are respectively the algebraic sum of all the differences of lati- tude and departures corresponding to the several courses and distances sailed. Proof : I = + Ac + (cd de) + eh = + Ac + ce + eh = + Ah. p = Jib + bm + mg gF = Jim + mF = + JiF. Sources of data. -The navigator will find in the ship's log book the latitude and longitude of the point of departure, the 230 NAVIGATION" compass courses and distances sailed, and correction for lee- way, if any; then taking from the chart or tables the varia- tion of the locality, and from the deviation table the devia- tions for the various compass courses steered, he will have the data for working the traverse. If in a region of known currents, he must allow for the set and drift as explained. later (see Arts. 129-131). The preparation of the traverse form and data. (1) In case a departure course enters into the computation, the com- pass bearing of the point of departure is corrected for varia- tion, and for the deviation due to ship's head when the bearing was taken, and the reversed true bearing thus obtained is entered in the column of true courses, the distance in the column of distances, thus forming the first course and distance of the tabulated form. (2) Each compass course is corrected for variation, devia- tion, and leeway, and the result entered in the form under the head of true course, the sum total of distances run on each true course being placed opposite that course in the distance column. (3) Enter Table 2, Bowditch, find each true course (7 N at top or bottom of page, and for each true course and distance find the corresponding differences of latitude and departure, placing them in their respective columns, opposite the courses ; the difference of latitude being placed in the "N." column when the course is northerly, in the " S." column when south- erly; the departure being placed in the "E." column when the course is easterly, in the "W." column when westerly. When distances are in miles and decimals, multiply by 10 or by 100, take out for the new whole number the desired quan- tities, and divide them by the multiplier just used. Thus, for 29.3 take out I and p for 293 and divide by 10. Where the course is between two given degrees, first find Z and p for each and interpolate. TRAVERSE FORM AND DATA 231 Add up the " diff . of lat." and " departure " columns ; take the difference between the northings and southings to which give the name of the greater; do the same for the E. and W. departures; these resulting differences are, respectively, the difference of latitude and departure of the resultant course. (4) Then look in Table 2 and turn to that page on which can be found coincidently in the lat. and dep. columns the above mentioned resultant difference of latitude and departure. The angle at the top or bottom of page, as the case may be, will be the course made by D. K., estimated from the North point to the right, the particular quadrant being determined by a consideration of the resulting differences of latitude and de- parture. The course (7 N will be taken out in the 1st, 2d, 3d, or 4th quadrant according as the co-ordinates / and p show it to be in the general direction of NE., SE., SW., or NW., re- spectively. On the same line with the difference of latitude and de- parture, in the Distance Column, will be found the distance made by D. E. (5) When the exact values of / and p are not found together on any page, turn to that page where they are found to agree most closely, and interpolate between the degrees of this and an adjoining page for the course, and also between the dis- tances involved till close approximations to the real values of C and D are reached. (6) The compass being graduated from at North, around to the right through 360, attention is called to the fact that the 1st, 2d, 3d, and 4th quadrants are respectively the NE., SE., SW., and NW. quadrants ; and, for the proper marking of 7 and p, that the course (7 N is northerly in the 1st and 4th quad- rants, southerly in the 2d and 3d, easterly in the 1st and 2d, and westerly in the 3d and 4th. 232 NAVIGATION Ex. 25. On April 3, 1905, at 1 p. m., took departure, Cape Henry Lighthouse (Lat. 36 55' 35" N., Long. 76 00' 27" W.) bearing (p. s. c.) 293, distant 10 miles, ship's head East (p. s. c.) ; deviation + 3, variation from chart 6. Thence ran till 8 a. m. next day as follows. Required the course and distance by D. R. from the lighthouse, and the latitude in. Courses (p. s. c.). Distance. Dev. Leeway. Wind. 73 60 + 3 3 Nly 118 20 + 6 3 Nly and Ely 160 10 + 3 3 Ely 319 10 -6 3 Nly and Ely 26 28 + 3 3 Sly and Ely To illustrate the application of the various corrections, each course will be considered separately, and corrections applied one at a time, though in practice the algebraic sum is applied mentally. Departure Course. Bearing of Lt. (p. c.) 293 Deviation +3 Course (p. c.) Leeway Course thro' water Deviation Magnetic Course Variation True Course 1st Course. 2d Course. 73 + 3 118 + 3 76 + 3 121 -1-6 Magnetic Bearing 296 Variation 6 79 - 6 127 -6 True Bearing of Lt 290 Departure Course ... .110 73 121 Course (p c ) 3d Course. 4th Course. 5th Course. 160 + 3 319 -3 26 -3 Leeway Course through Water. . . . Deviation 163 + 3 316 6 23 + 3 Magnetic Course 166 6 310 -6 26 6 Variation True Course 160 304 20 FORM OF WORK 233 E -3 o ft .2 . * S| G H "S 5 O 2 .2 I? 3 70. in the same latitude, to find the departure; though sometimes the latitude of the parallel may be required, the difference of longitude and corresponding departure between the two places being known. CASE I. Given p, to find D. Ex. 26. K ship in latitude 49 35' K and longitude 22 30' W. sails due South (true) 65 miles, then due East (true) 120 miles; find latitude and longitude in. By computation: L, = 49 35' N fZ 2 = 48 30 sec 10.178741 ^ = 22 30' W I = 1 05 SJ p = 120 log 2.07918 ID = 3 01.1 E 1 =48 30 / N[z> =181.1 log 2.25792J^ 2 =19 28'.9W By inspection: Enter table 2 with latitude as a course, find p -in the latitude column, and opposite in the distance col- 236 NAVIGATION unm is P. It may be necesary to interpolate as in the ex- ample. The Lat. of the parallel is 48 30' IsT. For 48, j9 = 119.8, Z) = 179 For 49, p = 119.4, Z) = 182 p = 120.4, .0 = 180 p- 120.1, D=183 jt? = 120, D= 179.33 p = 120, D = 182.86 Hence for Lat. 48 30' K, and p = 120, D 181.095. CASE II. Given D f to find p. Ex. 27. A ship sails on a parallel of latitude 41 30' S. from A in longitude 18 30' E. to B in longitude 2 10' W. Find the distance sailed in nautical miles. Long, of A 18 30' E L = 41 30' ____ cos 9.87446. " B 2 10 W D = 1240 ..... log 3.09342 D = 20 40' W p= 928'.7 log 2.96788 = 1240' Ans. 928.7 miles. By inspection : Enter table 2 with Lat. for the course, find D in the distance column, and, opposite in the latitude column, will be found p. When D is greater than any tabulated distance, pursue either of the following methods which are given in full to illustrate the use of the traverse tables. Divide D by 10, find corresponding p in latitude column, then multiply by 10, thus : For L 41, D 124; p = 93.6 1 By interpolation and L = 42, D = 124; p = 92.1 j multiplication, For L = 41 30', D = 1240; p = 928.5. A closer result may be gotten by inspection by considering D in three parts, 600 + 600 + 40. For Lat. 41, D = 600, p = 452.8 D 600, p = 452.8 D 40, p = 30.2 D = 1240, p = 935.8 EXAMPLES FOR EXERCISE 237 For Lai 42, D= 600, p = 445.9 D = 600, p = 445.9 P = 40, p = 29.7 P = 1240, p = 921.5 By interpolation, for Lat. 41 30' S., D = 1240; p = 928.65. CASE III. To find the latitude, given p and D. Ex. 28. A.ship in longitude 45 10' W. sails due W. (true) 186.8 miles, and is then in longitude 48 58' W. Find the latitude. By computation : cos L =. A 2 = 48 58' W p= 186.8 ....... log 2.27138 A! = 45 10 W D 228 ........ log 2.35793 D 348'W L = 3459'N cos 9.91345 = 228 W By inspection: Turn to that page of table 2 on which D = 228 is found in the distance column and p = 186.8 in the diff. of lat. column, opposite D. The angle at the top or bottom of page, as the case may be, will be the latitude. If exact coincidence of D and p are not found on a given page, then the angle must be found by interpolation. In this example Lat. is found to be 35 N". Examples Under Parallel Sailing. Ex. 29. A ship in latitude 38 N. sailed due West till she changed her longitude 5. What distance did she sail? Ans. 236.4 miles. Ex. 80. A ship in Lat. 40 K, Long. 160 W., sails due East till her longitude is 150 30' W. Find by inspection the distance sailed. Ans. 436.6 miles. 238 NAVIGATION Ex. 31. How far must a ship sail due East in Lat. 60 N. to change her longitude 5 ? Ans. 150 miles. Ex. 32. From a place in Lat. 30 N., Long. 50 20' W., a ship sails due West 240 miles, then due N". 240 miles, and due E. 240 miles. Find Lat. and Long, in by inspection. Ans. Lat. 34 K; Long. 50 07'.6 W. Ex. 33. A ship in Lat. 60 N. sails due West 75 miles. How much does she change longitude? Ans. 2 30' W. Ex. 34. A ship in Lat. 38 K, Long. 159 10' W. sails due E. 405 miles. Find Lat. and Long. in. Ans. Lat. 38 K; Long. 150 36' W. Ex. 35. Two ships in Lat. 35 N"., distant from each other 150.7 miles, sail due North at the same speed for 300 miles. Find by inspection how much closer they are at the end of run. Ans. 9.7 miles. Ex. 36. Find by inspection in what latitude the length of a degree of longitude will be 46 miles. Ans. 40. Ex. 37. Two ships are steaming due East at the same speed. B changes longitude twice as fast as A, who is in the 20th parallel of 1ST. latitude and to southward of B. Find B's latitude by computation. Ans. 61 58' 31" K MIDDLE LATITUDE SAILING. 128. In plane sailing, the assumption was made that the earth was an extended plane, and, though this assumption was false, the errors for small distances were considered imma- terial. Were the earth an extended plane, the departure would be the same in both latitudes left and arrived at. It has been shown that the departure is approximately that of the middle latitude. In parallel sailing, the earth was regarded as a sphere and a relation was established between departure and difference of longitude. MIDDLE LATITUDE SAILING 239 Now, combining the principles' of plane and parallel sail- ing, we have middle latitude sailing, which finds the difference of longitude corresponding to a departure measured in the middle latitude, and, by partially nullifying the false assumptions of plane sailing, gives a nearer approxi- mation to true results. A still nearer ap- proximation to the truth may be gotten by applying from Bowditch a correction to the middle latitude, and considering the departure measured on this corrected parallel. However, if necessary to do this, it would be better, after finding C and L 2 by plane sailing, to find D by Mercator sailing, explained later on. The relations of the quantities involved in middle latitude sailing are shown in Fig. 71 by combining the triangles of plane and parallel sailings, regarding the departure as meas- ured in the latitude of the middle parallel. FIG. 71. Let L n = the middle latitude, then I = d cos C, p = d sin 0. D = p sec L d sin C sec L I tan C sec Z/ . tanC = D cos L n (120) Lo= A_A, L 2 = A + I A 2 = A, + D. When not advisable to use M. L. sailing. The results got- ten by using middle latitude sailing are more accurate in low latitudes, less so in high latitudes, and the inaccuracy is greater the greater the difference of latitude, or for a given distance sailed, the smaller the course. Hence when the lati- tudes are high (over 50 N. or S.), or course small with a large distance producing large differences of latitude, it is 240 NAVIGATION better not to use middle latitude sailing, but to use Mercator sailing. It is not advisable to use middle latitude sailing when the places left and arrived at are on different sides of the equator, unless the two parts of the track on opposite sides of the equator are treated separately, except in the case where the distance each side is so small that the departure is practically equal to the difference of longitude. In ordinary practice examples under middle latitude sail- ing come under one of the two following cases. For other variations, however, it is only necessary to draw a figure show- ing the relation of the parts, and to use that formula which will give the unknown from the known parts. Case I. Given the course and distance sailed from a place of known latitude and longitude, to find the latitude and longitude arrived at. Ex. 38. A ship in Lat. 36 40' S., Long. 48 40' W. sailed 38 (true) 150 miles. Find Lat. and Long. in. Solution by computation : o / // d 150 log 2.17609 log'2.17609 L l 36 40 S tf N38E.. ..cos 9.89653. ... sin 9.78934 Z r= 1 58 12 N Z=118'.2N log 2.07262 Z 2 34 41 48 S p log 1.96543 L Q = 35 40 54 S L = 35 40' 54" S secO.09030 ^ = 48 40 W 113'. 69 E ..................... log 2.05573 D 1 53 41 E By inspection: Enter Table 2 with course 38, opposite 150 in distance column, find /= 118.2 in Lat. column and p=92.3 in Dep. column. Then with the middle latitude 35f as a MIDDLE LATITUDE SAILING 241 course find, by interpolation, opposite p in Lat. column, the D in distance column. For Lat. 35, ^ = 92.3, = 112.7 I . for Lat ^ y ^ = 92.3, D = 113.7 " 36, p 92.3, D z= 114.1 | o / // o / // L! = 36 40 S A! = 48 40 00 W Zrz 1 58 12 N D 1 53 42 E 2 = 34 41 48 S A 2 = 46 46 18 W A, = 35% S Case II. To find the course and distance between two posi- tions of known latitude and longitude. Ex. 39. Find the course and distance from Lat. 43 03' 24" K, Long. 5 56' 30" E. to Lat. 39 26' 42" N., Long. 23' 00" W. By computation: O I It O I ff O I It LI= 43 03 24 N AI = 5 56 30 E LI = 43 03 24 N I/ 2 = 39 26 42 N A 2 = 23 00 W I/ 2 = 39 26 42 82 30 06 I = 216'.7 = 3 36 42 S D = 379'.5 = 6 19 30 W L = 41 15 03 N D = 379.5 log 2.57921 I/o = 41 15' 03" cos 9.87612 p log 2.45533 . . log 2.45633 I = 216.7 log 2.33586 log 2.33586 0= S 52 47' W (tfN=232 47') tan 0.11947 sec 10.21837 d = 358.28 miles log 2.55423 By inspection. Enter Table 2 with L = 41J as C, find corresponding to D 379.5 in distance column, p = 285.3 in Lat. column, thus For Lat. 41, D 379.5; p 286.4 | Therefore For Lat. 42, D 379.5; ,p:=282. )Z = 41%' , D = 379.5, ^ = 285.3 Then find corresponding to I 216.7 and p = 285.3 the course and distance thus: For p 285.3, I 222.9; d 362. C N = 232 | Therefore by p 285.3, Z = 215.; d 357.2 C N = 233j" interpolation For ^ = 285.3, I 216.7; rf = 358.4 <7 N 232% 242 NAVIGATION Again attention is called to the fact that/if the difference in latitude is large, the assumption that the departure is properly measured in the middle latitude is not strictly correct ; and, if greater accuracy is desired, a correction from Bowditch must be applied to the middle latitude to obtain the proper parallel on which to take the departure (see Art. 133). However, it is just as easy and more correct to use Mercator sailing. When the two places considered are on opposite sides of the equator, no sensible error will be made in the case of an ordi- nary day's run, which will seldom exceed 400 miles, by taking the difference of longitude equal to the departure. If the distance is great, use Mercator sail- ing, except when the course is large (more nearly East or West), in which case use middle latitude sail- ing (see Art. 132). However, when the distance be- tween two places, one in North lati- tude and the other in South latitude, is great, and it is desired to use middle latitude sailing in finding the difference of longitude, the two portions of the track on different sides of the equator may be treated sepa- rately. Thus in Fig. 72, let the coordinates of the place A be L! , A! , and those of the place C in the opposite hemi- sphere be. L 2 , X 2 . The track A C is divided by the equator EQ into two parts, AB and EC. For AB we have FIG. 72. i L : tan (7, and neglecting AL X , = D, = Pl sec ^ = L, tan sec^ 1 EXAMPLES FOR EXERCISE For BC we have 243 p 2 = ( ) L 2 tan C, and neglecting &L 2 , BE = D 2 = p 2 sec - = ( ) L 2 tan C sec -^, whence QE or D = D + D 2 . Therefore, for this case we have the following formulae: I = d cos C, = LI tan C sec -^ (121) D 2 = ( ) L 2 tan C sec \ \ I T\ Ao A! ~pX/ Instead of the middle latitude L and ^ L 2 , we may for greater precision use ( L + A .L ) and (J L 2 -f- A L 2 ). Examples in Middle Latitude Sailing. (By inspection.) Ex. 40. From 7^ 49 28' 30" K, A t 03' 15" B., sailed 312 (p. s. c.) 36 miles, variation 20, deviation 2. Find by D. E. L 2 and A 2 . Ans. 2 = 49 40' 48" N. A 2 = 48 51 W. Ex. 41. From 48 20' 29" K, A 5 07' 48" W., sailed 257 (p. s. c.), 22.2 miles, variation 20% deviation 3, thence 232 (p. s. c.), 216.5 miles, variation 20, deviation -1. Find L 2 and A 2 . Ans. L 2 = 45 01' 55" K A 2 =: 8 16 30 W. Ex. 42. A ship leaving Lat. 49 50' N., Long. 10 16' W., sails to the southward and westward till her departure is 188 244 NAVIGATION miles and the latitude reached is 47 28' N". Find the course, distance, and longitude in. fCourse C N = 233. AnsA Distance 236. [\ 2 = 15 00' 42" W. Ex. 43. A ship sails from L 24 23' S., \^ 100 30' E., and, by observations the next day, finds her position to be 25 43' 12" S., 104 52' 38" E. What was her true course and distance? Ans. <7 N = 108.6. Distance 251.9 miles. Ex. 44- Find by computation the true C and d from L l 23 00' N"., A! 109 55' W., to L 2 35 30' K, \, 139 45' E.. (without correcting middle latitude). Ans. N = 277 25' 12". Distance 5807.5 miles. CUBBENT SAILING. 129. A current may be defined as a body of water moving steadily in one direction. The set of a current is its course, or the direction in which it is moving. The drift is the distance the current sets a ship in the time considered. Thus the drift in 20 hours being 10 miles, the drift per hour, -J mile, would be more properly called the rate. When the rate per hour is known, the drift for any given time is easily found. When a ship sails directly with or directly against a current, her motion is increased or retarded by the amount of the drift in the interval. When a ship sails obliquely to a current, her motion may be accelerated, or retarded, according to the angle between the course of the ship and the set of current ; and the distance made good is the diagonal of a parallelogram of which one side is the distance made in the direction of the keel and the CURRENT SAILING 245 other side the distance the ship is carried by the current in the direction of the set, in the same interval of time. This resultant direction is in accordance with Newton's first and second laws of motion. Current sailing. Current sailing may, therefore, be defined as the means of finding the course and distance made good when a ship's motion is affected by tides or currents, or a course to be steered to make good a given course. Problems in current sailing. There are two general cases in practice. Case I. Given a course steered and distance run, to find course and distance made good through a current of known set and rate. Ex. 45. In Fig. 73, let MM' be the meridian of a place A in North latitude and let it be assumed that a ship, leaving A, steers 210 (true) 8 knots per hour, through a current setting her to the eastward (true) 2 miles per hour. Lay off AB = 8 miles in the direction 210*, the speed per hour of the ship on her course. Lay off AC = 2 miles in the direc- tion East, the drift and set of cur- rent in the same interval of time. Complete the parallelogram by drawing BD and CD, and join AD. By the principle of "the composi- tion of forces," the ship at the ex- piration of one hour will be at D } having been moved along the diagonal AD under the joint action of two forces, her own propelling force and that of the current. The result under the joint forces is the same as if each force had acted in succession, that is, as if the ship had gone from A to B FIG. 73. 246 NAVIGATION tinder her own propelling force uninfluenced by current, had then stopped, and been swept from B to D by the current. In the particular diagram (Fig. 73) AD is the distance made good; MAD, the course made good (from IS]", to right). Solution by construction. Having a Mercator chart, it is only necessary to lay off AB and AC from the known position A in the proper directions, complete the parallelogram as above explained, then measure the angle MAD and the distance AD. Solution by trigonometry. To solve by trigonometry, make a rough sketch to show the conditions. Referring to Fig. 73, we have the angle ABD = 60, AB 8, BD = 2, and it is required to find L BAD and AD. Then the course made good OT/.MAD = 210 BAD = 210 A. Since ABD = 60, A+D . 180 60 120 _ fino :__ a = BD, From plane trigonometry we have, as ^ & = AD, d = AB, D A d a , B 6 , o AO tan H = T cot -x = -- cot 30. 6 log 0.77815 80 log cot 10.23856 10 ar. co 9.00000 -^46 06' 07" tan 0.01671 * ,4 ~ 13 5o Do To find AD = b, b~d- ^^= 8 sin 60 cosec 106 06' 07" o t it 8 log 0.90309 210 60 log sin 9.93753 A 135353 106 06' 07" log cosec 10.01738 MAD 196 06 07 6 7.211 log 0.85800 The course made good is C N = 196 06' 07". Distance made good per hour = 7.211 miles. CURRENT SAILING '247 Solution by the traverse table. It is a simpler plan, how- ever, to consider the course sailed and the set of the current as two separate courses in a traverse as below. Though ap- proximate, results will be sufficiently correct. Diff. lat. Departure. True Courses. Dist. N S E W 210 8 6.9 4.0 90 2 .... . . . 2.0 6.9 2.0 4.0 2.0 2J) With Z = 6.9 ) Course made good C7 N =zl96 Zr=6.9) p = 2.0 J Distance made good = 7. 2 miles. In the example worked above, the course and distance of the ship, and set and drift of the current were given for one hour only, but the principle holds good for any example in which the set and drift of current for a given time may be taken as a course and distance in a traverse. 248 NAVIGATION Ex. 46. A ship in Lat. 40 30' N., Long. 48 05' W., at noon on Jan. 10, sailed till noon Jan. 11, 240 (p. s. c.) 223 miles. Var. 24, dev. 2. A current sets 95 (true) 0.75 of a mile per hour. Find (7 N and d made good, Lat. and Long. in. Course (p.c.) Var. Dev. True Course Dist. 3 E W 240 24 2 214 223 184.9 124.7 Current 95 18 1.6 17.9 / ff L^ = 40 30 00 N I = 3 06 30 S / tf \ = 48 05 00 W D - 2 17 30 W 186.5 Z = 186.5 17.9 124.7 17.9 ^=fcl06.8 L 3 = 37 23 30 N L = 38 56 45 N \ = 50 2^ 30 W D = 137.5 W ~ . ,. ( Course made good C f N =209. 8. By inspection j Digtance mad j good 2U 9 mile8> Having found I and p, the course and distance might be gotten from the formulas tan C = d = I sec G. However, the result by inspection is sufficiently close. Case II. Given the ship's speed per hour and the bearing of a port or destination, find the course to be steered, through a current of known set and rate, in order to keep that port, or point of destination, on the same bearing. Solution by construction. Let N8 (Fig. 74) be a meridian passing through the place A, the point of departure on the chart. Let AB be the bearing of the port, or the direction to be made good. Draw AB on the chart. Draw AC to rep- resent the set and hourly rate of the current. With extremity C of the current line as a center, and with a radius equal to CURRENT SAILING 249 the ship's speed per hour, taken in the same units as the rate of current, describe an arc cutting AB in D. Then AF, drawn parallel to CD, will be the direction the ship should be steered to keep the port on the same bearing AB. Ex. 47. What will be the magnetic course from a point off Key West to make Morro Light House at Havana, the mag- netic bearing of which from chart is 197 ? The passage is across the Gulf Stream, setting 75 (mag.) 2 miles per hour. Speed of ship 12 knots. By construction. Pro- ceed as just explained. In Fig. 75, AB is the magnetic bearing of the port 197. Lay off AC = 2 miles in the direction 75. FIG. 74. PIG. 75. to represent the rate and set of current. With C as a center- and radius of 12 miles, strike an .arc cutting AB in D. Draw AF parallel to CD. NAF is the course to steer. By trigonometry. By a rough sketch show the actual state of affairs, as in Fig. 75. From G, the extremity of the cur- 250 NAVIGATION rent line, drop a perpendicular CK on the bearing line, then ^ sin(75 17)= 2 sin 58. From the traverse table, p = d sin C, and as KG = 2 sin 58, therefore, 1.7 = 2 sin 58 = 12 sin x. 1.7 = 12 sin x. Course to steer = L SAF + 180 = 17 + 8 + 180 = 205. Magnetic course to steer, 205. Or, since L NAC = 75 and L SAD = 17, / DAC = 122, and we have, 2:12 = sin x: sin 122. Hence by logs., x = 8 08' ; therefore, course = 205 08'. Solution by the traverse table and traverse sailing. Reverse the direction of the current and consider the ship to sail from C to A, and then from A to D, finding the course from C to D. Traverse Table. Current ( Reversed 1 Courses (Mag.). Dist. Diflf. of Lat. Departure. N S E W 197 255 12 2 11.5 0.5 3.5 1.9 1=12 p = 5A Course Cy = 205 (Magnetic) This solution by traverse sailing is sufficiently correct for all practical purposes, but is theoretically in error, as the dis- tance sailed in the direction of the port AB is not 12 miles, but less than 12 miles, which is the distance sailed per hour in the direction of the course. Ex. 48. Steaming at the rate of 8.5 knots per hour, one wishes to make good a course 76 magnetic, through a cur- CURRENT SAILING 251 rent setting 12 magnetic 3.5 miles per hour. What must be the magnetic course? Ans. C N = 97 43' 16". Ex. 49. A port bears from a ship 359 (mag.) distant 127 miles. Steaming at 16 knots per hour, through a current that sets 320 (mag.) at the rate of 3 miles per hour, find the compass course to make the port, deviation 2, var. - 7. Find also the time occupied in making the voyage. Ans. Compass course <7 N = 7 46' 34". Distance good per hour, 18.22 miles. Time, 6 h .97. 130. Current from noon positions. To current is usually attributed the discrepancy between the noon positions at sea by observation and by dead reckoning, or, at any instant, the difference between the position by dead reckoning and one obtained by bearings of known landmarks. The distance between the two positions divided by the num- ber of hours elapsed since leaving a position, assumed to be correct, will give the hourly rate of the current; the bearing of the position by observation from that by dead reckoning being the set, or direction of the current. It must not be forgotten, however, that the current, thus computed and so called, may be due to careless steering, im- proper logging or determination of the speed, or to errors of observation, rather than to any real motion of the waters of the sea. Ex. 50. On April 10, a vessel's noon position by observa- tion was Lat. 40 44' K, Long. 47 12' 30" W.; by D. K., Lat. 40 37' K, Long. 46 51' 48" W. Find set and drift of current since preceding noon. Lat. by obs. 40 44' 00" N" Long, by obs. 47 12' 30" W " D. E. 40 37 00 N " D. R. 46 51 48 W 1= r N D = 20'.7:=20'42"W = 40$ N p = 15'.7 W (Set, 294. | Drift, 17.2 miles in 24 hours. 252 NAVIGATION In the, above example, since the position by observation is to the northward and westward of that by dead reckoning, or account, it is evident that the ship was set to the northward and westward by the current, therefore mark I as 1ST., and D and p as W. For the middle latitude 40f , considered as a course, find D in the distance column of Table 2, and opposite D, take p out of the latitude column. With I and p find the corresponding C and d, or, in other words, the set and drift of the current. Ex. 51. At noon on Jan. 10, the ship's position by observa- tion was Lat. 25 43' 12" S., Long. 104 52' 38" E. The posi- tion by D. E. from previous noon was Lat. 25 52' 48" S., Long. 104 30' 24" E. Find the set and drift of current. Lat. by obs. 25 43' 12" S Long, by obs. 104 52' 38" E " D.E. 25 52 48 S " D. E. 104 30 24 E I 9'.6 = 9' 36" N D 22'.23 = 22' 14" E L = 25J S p 20'.02 E f Set, 64.4. ' j Drift, 22.2 miles in 24 hours. In this example the true position is to the northward and eastward of that by account, therefore the ship was set to northward and eastward, and we must mark I, N. ; D and p, E. 131. Tidal currents. The navigator should pay careful attention to the subject of tidal currents, and shape his course, or work his reckoning, to make due allowance for the pos- sible set and drift, in all localities where such currents have been investigated. Much information may be found on charts and in sailing directions. Finding from the tide tables the times of high and low waters at places along a coast, it may often be possible to make allowance, during a run at" such times, for a set towards or from that coast MERCATOR SAILING 253 When the wind has been strong and steady from tme direc- tion for any length of time, a current may be produced setting directly to leeward, or if already existing, its rate may be greatly increased. The navigator should anticipate and en- deavor to allow for its effect. MERCATOR SAILING. 132. It has been shown that the methods of middle latitude sailing are sufficiently exact for short distances, a day's run for instance, but for finding the difference of longitude be- tween two places widely separated in latitude, or for finding the course between two such places, it is liable to great error. To avoid such errors resort is had to Mercator's sailing, which is based on principles fully explained in Art. 26, and applied in the construction of the Mercator chart, and which furnish the formula that gives practically correct results. On the Mercator chart, the meridians are drawn parallel to each other and perpendicular to the equator and parallels of latitude, so arcs on parallels are represented as equal to the corresponding arcs of the equator, or differences of longitude ; in other words, expanded in a certain ratio. In order that the rhumb line on the chart may make the same angle with each meridian, each infinitesimal element of latitude must be expanded in the same ratio in which each infinitesimal ele- ment of the parallel has been expanded. If the earth were a perfect sphere, this ratio would be as the secant of the latitude, but as the earth is a spheroid, its eccentricity must be con- sidered. The formula from Art. 27, D M tan C, or D = tan tf [7915'.704 (lo glo tan ( + f .)- , lo glo tang+|))] gives the relation existing in Mercator sailing between the constant course C, the latitude L of a point on the loxodrome, 254 NAVIGATION and the difference of longitude of that point and the longitude in which the loxo'drome crosses the equator. M in the equa- tion is the augmented latitude, or the length of the line on the Mercator chart indicating the latitude, expressed in nau- tical miles, according to the scale of the chart. If it is desired to find the difference of longitude between two places in two different latitudes L t and L 2 , substitute in the equation successively the values L and L 2 , letting M x be the augmented latitude corresponding to // ; M 2 be the augmented latitude corresponding to L 2 ; " DI be the difference of longitude from A (Fig. 76), where track crosses the equator, to the first point in latitude L^ ; " D 2 be the same to second point L 2 . LI i E 1 FIG. 76. FIG. 77. In Fig. 76, let EE' be the equator, L the parallel of 1st latitude, L 2 the parallel of 3d latitude, C the constant course, then, Z> = M tan C. D 2 = M 2 tan C. D = D 2 Di= (M 2 MJ tan C = m tan C. (122) m equals the meridional difference, or augmented difference of latitude between, LJ. and L 2 , and is the length of the line on the Mercator chart which represents the true difference of latitude between- L and L 2 , expressed in nautical miles, ac- cording to the scale of the chart. MERCATOR SAILING 255 Table 3 of Bowditch is a table of meridional parts at inter- vals of one minute of arc up to 80, compression having been taken as . In case L and L 2 are of different names, as in Fig. 77, where EE' equals the equator, lf and M 2 are of different names, and the algebraic difference M 2 M^ be- comes M 2 + -^i- Therefore, D=D 2 + D 1 = (M 2 + MJ tan C. Graphic illustration of the theory of Mercator sailing. Let C'E' represent an arc of the equator, and CA represent a distance sailed on a rhumb line from C in Lat. _L to A in Lat. L 2 , shown on the spheroid in Fig. 78, and on the Mer- cator chart of the corresponding limits in Fig. 79. Conceive this distance to be subdivided into a large number of small parts, and the elementary triangles to be formed of which the corresponding differences of latitude ^ , 1 2 , etc., are represented in Fig. 78 by Cn, oi, etc., and the departures by no, ir, etc. Each partial departure of Fig. 78 is represented in Fig. 79, a section of a Mercator chart, as an expanded arc equal to the corresponding arc of the equator, no equal to C'G, tr to GK, etc. ; so that the departure on the spheroid, being equal to the 256 NAVIGATION sum of the partial departures, is expanded into the corres- ponding difference of longitude on the Mercator chart. But, in order that the angle C on the chart shall remain constant and equal to that on the spheroid, and, that the simi- larity of the corresponding elementary triangles may be main- tained, the ratio of increase of each partial difference of lati- tude must be the ratio of expansion of each partial departure, and the true difference of latitude CB (Fig. 78) be repre- sented by CB on the chart (Fig. 79). The triangles of Mercator and plane sailing. The parts involved in Mercator sailing may be represented by a right triangle CEF, CE being the augmented difference of latitude m, representing the true differ- ence of Lat. C A = /; if AB is drawn parallel 'to.EF, ABC will be the tri- angle of plane sailing, AB the de- parture, and CB the true distance of which the expansion on the Mercator chart is CF, since the ratio between / FIG. 80. and m is the same as that between p and D. It is thus seen that the tri- angle CEF furnishes the formula for converting departure into difference of longitude without making the false assump- tions of middle latitude sailing. From Fig. 80 all the formulae necessary for Mercator sail- ing can be deduced. From triangle CEF, D = m tan C. ) From triangle ABC, d I sec C. \ Various problems under Mercator sailing may be solved by the above formulse, but those of actual practice may be said to be: MERCATOR SAILING 257 Case I. Required the C and d between two places of known position. Formulae : tan C = -, d=l sec C. m Case II. Required the latitude and longitude in, after sailing a true C and d from a place of known position. Formulas : I = d cos C, L 2 = L + I* D = m tan C, A 2 = A x + D. As the results by Mercator sailing and by middle latitude sailing do not differ sensibly for small distances, the use of Mercator sailing comes principally under Case I, when the two places are far apart. When not to use Mercator sailing. Since in Mercator sail- ing the difference of longitude is found from a formula in- volving tan C, and tangents of angles near 90 change very rapidly, it is seen that any error in m, the meridional differ- ence of latitude, is greatly increased as an error in difference of longitude when the course approaches 90 or 270. In such cases use middle latitude sailing. Use of traverse table, Problems in Mercator sailing can be solved by the traverse table; the difference of longitude and meridional difference of latitude, being respectively the sides opposite and adjacent in a right triangle, should be -looked for in the dep. and diff. of lat. columns, respectively. In using this table where long "distances are involved, the quantities given may all be reduced by a common divisor till within the limits of d, I, and p as tabulated, and the results afterwards correspondingly enlarged. This, however, will in- volve some error in results. Graphic solution of problems in Mercator sailing are made in every-day navigation, when the reckoning is kept by con- struction on the Mercator chart as fully explained in Art. 31. 258 NAVIGATION Ex. 52. Find the true course and distance by Mercator sailing from L = 10 36' N \ L 2 = 36 30' N A 1= =56 34WJ t0 A 2 =15 22 W By computation. o / o / LI = 10 36 N Jfj = 635.4 ^ =56 34 W X 2 = 36 30 N Jf 2 = 2341.3 2 2 =15 22 W Z = 25 54 N m = 1705.9 J) =41 12 E = 1554' N = 2472' E Z = 1554' N .................................. log 3.19145 D = 2472' E ............ log 3.39305 m = 1705.9 ............ log 3.23195 G = N 55 23' 28" E ..... tan 10.16110 .......... sec 10.24568 d = 2736.1 miles .............................. log 8.43713 A ( Course, (7 N = 55 23' 28". *' (Distance, 2736.1 miles. By inspection. Enter Table 2. Turn to that page where will be found the nearest coincidence, D in dep. col. and m in diff. lat. col. Now, by interpolation, D = 247.2, m = 173.05; = 55 D = 247.2, m = 166.66; C 56 D = 247.2, ro = 170.59; C = 55. 38 Therefore, by inspection, course is (7 N = 55. 38. Now with the course and I = 155.4 find d. For C N = 55, 1= 155.4 ; d = 271 C N = 56, I 155.4; d = 277.8 Therefore, for C N = 55.38, Z = 155.4; (2 = 273.58. Having used a divisor of 10 originally, the true distance by inspection is 2735.8 miles. Of course, the above interpolation, recorded for illustration, is supposed to be done mentally. MIDDLE LATITUDE CORRECTION 259 Ex. 53. Find true C N and d from Brisbane to Acapulco. o / // o / // Li = 27 27 32 S M l = 1703.7 8 \ = 153 01 48 E 2 = 16 49 10 N M 2 = 1017.2 N 2 2 = 99 55 50 W I = 44 16 43 N m = 2720.9 N D = 107 02 23 E = 2656'. 7 N = 6422'.37 I = 2656.7 ................................... log 3.42434 D 6422'.37 ............ log 3.80770 m = 2720.9 ............. log 3.43471 C = N 67 02' 24"E ____ tan 10.37299 .......... sec 10.40884 d = 6810.5 miles ................ . ............. log 3.83318 f Course N = 67 02' 24". {Distance 6810.5 miles. 133. To find the value of the correction to the middle latitude. In middle latitude sailing, it was stated that the formula D = p sec L = Z tan C sec L was not strictly cor- rect, but that it would be correct, if, to L , was applied a cor- rection AL, such that the formula D = li&nC sec (L + AL) would give the same result as D = m tan C. From these two may be gotten - = cos (L + AL) = 1 - 2 sin AL = 2 *vr* m r~L . (124) Values of this correction have been tabulated where the arguments are the middle latitude and the difference of lati- tude. It has already been stated that for small values of l f it is unimportant; and that in those cases where its use might be desirable, it would be better to use Mercator sailing. 260 NAVIGATION Examples in Mercator Sailing. (By computation.) Ex. 54- Find true course and distance from L 50 53' N., A 156 46' E., to L 2 12 04' S., A 2 77 14' W. JCourse C N = 119 25' 28". [Distance 7688.35 miles. Ex. 55. Find true course and distance from Lj 42 20' N., A! 31 30' W., to L 2 56 40' K., A 2 20 40' W. Ans fCourse C N = 25 59' 04". * (Distance 956.7 miles. Ex. 56. Find true course and distance from L t 45 02' S., A 20 19' W., to L 2 65 20' S., A 2 18 37' W. Ans J Course CN ~ 177 19 ' 46 "' [Distance 1219.4 miles. . Ex. 57. A. ship sails from Lat. 15 20' K, Long. 24 20' W., 135 (true) a distance of 2500 miles. Find Lat. and Long. in. , f 2 = 14 07'48"S. 1*4= 5 15 48 E. Ex. 58. Find the true course and distance by Mercator sailing from a point in Lat. 35 30' K, Long. 140 52' E. (off Cape Inaboye, Japan), to a point in Lat. 33 S., Long. 71 49' W. (off Valparaiso). See Ex. 63 and Plate V. Course ^ N Ans Distance 9300.55 miles. DAY'S WORK BY D. E. 134. In most works on navigation, the subject of "Day's Work " follows the sailings, and is considered without refer- ence to positions by observation; as these are an essential part of the data used in the daily work of a navigator, this DAY'S WORK BY D. E. 261 general subject is reserved till after the chapters on latitude and longitude by observations have been studied and under- stood. However, it must be recalled that all the calculations enter- ing into the daily dead reckoning itself have been made, and the methods used have been treated, under the head of pilotage, or, of the several sailings. Such are the various methods of fixing the ship's position near land after leaving port; taking departure; use of departure course and distance as a course and distance of the traverse; correction for lee- way, variation, and deviation, of the various courses indicated in the ship's log book; entry of the true courses and the dis- tances sailed on each in the proper columns of the tabulated form ; consideration of the set and drift of a known current as a separate course and distance of the traverse; finding the resultant difference of latitude and departure; the resultant course and distance; conversion of departure into difference of longitude; finding by D. E. the latitude and longitude at end of run, and the course and distance to port of destination. It has been shown that the noon position by observation is the true place from which to begin the dead reckoning of the following day ; and, in case of a discrepancy between it and the position by D. E., that this discrepancy, if not due to inci- dental errors of navigation, may be attributed to current, the set and drift of which correspond to the course and distance from the noon position by D. E. to that by observation. For the solution of a day's work in which positions by obser- vation are used, see Chapter XXI. In the following example, a day's work by D. E. is illus- trated. 262 NAVIGATION o o ""! ^ S ^ oj : ye "? *r^2 fo 1 H _fl ^ rt HI Wt-^HCOO -COOS . i-liO tO ^ CD jj p .2 P H "SSS :-3 : S ?iS ^^ WWW li^st S cc ::::::::: ft ^S5 85 3j O -3 1 I S5 Pg2 (M 1 ""^ ^? O J> fl rQ -2 o II II II II II * O5 ^ 02 to ^^^i (CDCOt"MrH(M C^ 4J 8, II 2^1 1| Q fc t-'(MiGiOW3i-(t-(MiC ^ if a 3 bfl 4^ 1-1 . ^ v 'm CO S fl fl S ^* O5 ^ ^ i O -iH _O 'o ^3 OS * d 5 .2 SJSS*SSS bo be p a 33 to o S S ^2 I-J -M O fl P ~J2 'S J3 o5 3 C3 Irt X3 "*"" p | fc fcfc S O O" 02 ~*^ S H grj O .22* 1 O 5 L- L- 16 C^l 1- S X O O$ CO rt ** r( CQ sllll i?7 l|l H ! "'I 8 II II II s ^ *-* 13 * '"S ^ I 5 4 oo o o~r3 >d a ^o * ^ g 2 i ||o i0oioiaiocoio 0) S & S 1.2 1 M o "cS "S to* W . . -M jj" ^ > & ^^ p C3 cd u ^ . ^ > c II |o,-;|^ P sTTT++TT+ W ^ cJ ^3 Cu ^ -1-3 ^ O5 *^ 26 ^ * w ' 5 2 ^ H) P' " co 1 |ri C3 ^2 g " 12 1-. S -M ^^ v' > siTTTTTTT II II 1 3 CJ TH _ o r'ft 33 ^ w S ^^ o fQ 3 ^ ** sfli ^ ^ ^ i o /< /^ /^1 T3 T? n-4 ^Q QQ CHAPTEK VII. GREAT CIRCLE SAILING. COMPOSITE SAILING. GRAPHIC METHODS. 135. Great circle sailing is the method of solving problems of navigation that arise from a ship following a great circle track from point of departure to destination. The arc of the great circle passing through two places is the shortest dis- tance between these places, so that a rhumb line is always longer than the great circle distance, except when it coincides with a meridian or the equator. The rhumb line has long been used by navigators because of the constancy of the course, the ease with which it can be laid down on or taken from a Mercator chart, and the sim- plicity of the calculations it involves. However, steam vessels, unlike sailing vessels, are independent of winds and currents, and are capable of following that route which means a saving of distance and of time; so it is fair to presume that in the future the great circle will be followed as closely as possible unless it goes into latitudes too high, meets lands, or passes through regions of ice and dangerous navigation. Comparison of tracks. The difference in distance and, hence, the saving of time is less when the loxodrome ap- proaches a great circle, as is the case between places near the equator, or near the same meridian ; the contrary holds, how- ever, for places in high latitudes, especially when differing much in longitude; a case most remarkable for saving of dis- tance and great divergence between the two tracks occurs when the two places are on the same parallel, but differ 180 in longitude. The Mercator course is either East or West, 264 NAVIGATION while the great circle course is North or South, across the ele- vated pole and 90 away from the former. A rhumb line laid down on a Mercator chart passes directly through the point of destination ; the great circle track plotted on the same chart will be a circuitous path, nearer the pole than the Mercator track, and often going into higher lati- tudes than is practicable for safe navigation. The great circle track between two places in different hemispheres has a double curvature when plotted on the Mercator chart, the curve in each hemisphere being the same in its entirety. However, a ship following the rhumb line, steers the same course, makes the same angle with each successive meridian, but never heads directly for the port till it is in sight, or till the end of the voyage ; another ship following the great circle track always heads for the port, but does so by steering a constantly changing course. Definitions. The great circle course is the angle which the great circle passing through a place makes with the me- ridian of that place. The initial course is the angle which the great circle through points of departure and destination makes with the meridian of point of departure; the final course is the angle which it makes with the meridian of desti- nation. The distance is the length of the arc of the great circle which forms the path of the ship between the two points, ex- pressed in nautical miles. Vertices. In accordance with geometrical principles, the equator bisects the great circle passing through the two places, and that point of the circle in each hemisphere which is farthest from the equator is the vertex in that hemisphere ; in other words, the vertices are the points of highest latitude. Only one vertex is considered, and that is in the hemi- sphere whose pole determines the course. The vertex .may or may not be between the two places. If the initial and final GREAT CIRCLE SAILING 265 courses are both less than 90, the vertex falls between the two places; if one is greater than 90, the vertex falls on the arc produced 180 from this course. Points of maximum separation. Since all points of the great circle track betwen two places in the same hemisphere, except those of departure and destination, are nearer the pole than the rhumb line, it follows that there must be some one point where the meridian distance between the two tracks is greatest, and this is the point of maximum separation. It is apparent that at this point the courses on the two tracks are equal. Hence, knowing the rhumb course, it is only neces- sary to find that point of the arc where the great circle course would equal it. Finding great circle course and distance. There are four general methods for solving the great circle problem : (1) By computation. (2) By azimuth tables. (3) By great circle charts. (4) By graphic approximation. By computation. The problem consists in the solution by spherical trigonometry of a spherical triangle, formed by the meridians passing through the two places and the great circle arc forming the ship's track. The lengths of the two sides are known, being equal to the co. latitudes of the two places; the included angle at the pole is the difference of longitude of the two places; so the triangle, having two sides and the included angle given, may be solved by Napier's Analogies, but preferably by Napier's Eules. To plot the curve, not only the vertex, but a series of points along the curve should be determined by their coordi- nates; and, having been plotted on the Mercator chart, the curve may be traced through them. In Fig. 81, let A be the point of departure, B the point of destination, AB the great circle passing through them, V its 266 NAVIGATION vertex, m 1? ra 2 , etc., points along the curve, P the elevated pole. Position of A, Lat. L , Long. A ; therefore, AP = 90 . Position of B, Lat. L 2 , Long. A 2 ; therefore, PB = 90 L 2 . APB = difference of longi- tude of A and B A 2 ^ A . (7 X is the initial course ; C 2 , the final course, d = dis- tance A 5. Drop a perpendic- ular BO (= fc) on J.P, so that one triangle P0 shall include two of the known parts, and the triangle ABO shall in- elude the required C and 6?. Also, dividing AP into two parts, PO = < and OA = 90 (L + ). To find the initial course <7 . Applying Napier's Eules to triangle P05^ cos (A 2 ^ A ) = tan tan L 2 . or, tan = cos (A 2 ^ A ) cot L 2 , (1^5) also, sin = cot (A 2 ^ Ai) tan fc; ) to triangle AOB, cos (L + ^>) = cot ^ tan A;; J therefore, cot = cot (A 2 ^ A ) cos (L + ) cosec . (126) To find the distance d = AB. Proceeding as above, sin L 2 = cos cos Tc f cos d = sin (L + <^>) cos fc, therefore, cos 6Z = sin (L + ) sin L 2 sec < . (127) The distance is found in degrees, minutes, etc., of a great circle which will be reduced to minutes for distance in nauti- cal miles. PV is the arc of a meridian perpendicular to the great circle track ; therefore PVA and PVB are right angles. V is the vertex ; Lat. L v , Long. X v . The vertex lies between A and B, unless either (7 X or (7 2 is > 90. GREAT CIRCLE SAILING 267 To find the latitude and longitude of the vertex. In the right triangle APV f PV = 90 L v and A'PV = \ v ~ A , therefore, by Napier's Rules, cos L v = cos L-L sin (7 , (128) sin LI = cot <7 cot (A* ~ A^, ) , 129) cot (Au ^ A ) = sin LI tan C . J To find the latitude and longitude 6f other points of the curve. Assume meridians Pm , Pm 2 , etc., differing in longi- tude 19 2 , etc., (say 5 or 10, if desired) from the longi- tude of vertex, and solve the right triangles thus formed by Napier's Eules. Therefore, tan L mi = cos tan L v , , ) ( . , -.- f\ i T etc. > ( .LoU) tan L m2 = cos 2 tan L v , j Each of these last formulae will give a position each side of the vertex, or two points of the curve; thus from first equation, Lat. m^ , Long. (X v ) and Long. (Au + ), from second equation, Lat. m 2 , Long. . (X v 2 ) and Long. (At, + 2 ) . It must not be forgotten that the course is ever varying, and that the course to be steered at any meridian is the angle which that meridian makes with the track. By the solution, the angle will be found from the elevated pole towards the East or West, but, if it is found to be greater than 90, as when the vertex is, or has been left behind, it may be convenient to name the course as from the depressed pole, or the supplement of the angle found by computation in the triangle of which the elevated pole is one point. To find the course in any longitude A^. In Fig. 81, PVG is a right triangle, PV = 90 - - L v , VPG = A, ~ A^ . Let C g be the course in Long. A^ , and C q be the course at the equator ; then, by Napier's Eules, 268 NAVIGATION cos C g sin L v sin (\v ~ X g ), (131) and for the final course C 2 in Lat. L 2 , Long. A 2 , cos (7 2 = sin Lt; sin (A^ ~ A 2 ). (132) At the point of crossing the equator sin (X v ~ \q) = sin 90, therefore, at the equator, C q = Co.L v , (133) and Ag = ^90. (134) Precautions. In solving the triangle, let the elevated pole (the pole of that hemisphere in which lies the position L^) be at one angle of the triangle, regard L^ as positive and L 2 , when of a different name from L , as negative. Strict regard must be had to the signs of the functions. may be taken out as positive up to 180 ; or, if tan (/> is negative, instead of taking in the second quadrant, it may be regarded as nega- tive, the foot of the perpendicular falling the other side of the pole. The angle GI will be found with its correct value, if attention is paid to the signs. The course is from the elevated pole, East or West, as B is East or West of A. C g and C 2 are reckoned from the elevated pole when ap- proaching the vertex and from the depressed pole when going away from it, toward East or West according as the ship is proceeding eastward or westward. The fact of the vertex being ahead or astern is determined by a comparison of its longitude with that of the meridian from which the course is taken. Though, when considering the above mentioned courses as angles of a spherical triangle, the method given is the proper one to pursue in the solution of that triangle ; still, as soon as found, the course, for practical purposes, should be expressed in the more convenient form of CN which is measured from North, around to the right, from to 360. C N will be simply z, 180 z, 180 + z, or 360 z, according as the course by solution is N. x E., S. x E., S. x W., or N. x W., respectively. PLATE IV, 270 NAVIGATION c* 2 ^ d d ft 8 -H O 03 -^ tS o 1 ^-* ^ 'V -o o p* U .2 ^. a I> o 5 CO ^ < 1 ) c 5 CO t- o o j | IO Tj o I 1 D c 1 ) d a c t ) c i QQ c I H > t 1 C5 10 CO CO CO CO 1 05* d o 05 ft T-H 1 1 1 d g d CO OQ 3 o o T-l CO CO IO CO CO CO ^ i CO O5 d d TH r-l OJ 1 O 4) -M 03 o -t-J o o CO CO * t> o 03 CO i 1 O r-l 00 O l> CO CO j i O5 CO CO * 05' d d . CO i i . co' 1 1 OQ -M (3 : ; 1 1 O -2 I II N ^ IO OQ w ^ o ^ tt co o o 1-H O IO OJ OS t> 00 r-l CO r-1 CO Tji CO CO IO CO CO 10 St, 00 CO O5 05 CO 1O iO T-l CO CO r-l T-l s ^ i ii ii D O 5 . o "S FORM AND EXAMPLE 271 Ex. 61. In the above example find the coordinates of two points of the curve, one in a longitude 40 to eastward of the vertex, the other 40 to the westward. Find also the course at this western point and the final course (see Fig. 81). Here = 40, \ ~ \ d 40 X 2 = 82 37' 18", Formulas tan L m = tan L v cos 0. cos = 82 37' = 40 = 72 34 67 43 N 52 10 N 18 52 ?, -s \ =s in L v sin (A^ ^ in LU sin (A^ ' -A,). -A 2 ). 18" sin 9.99639 cos 9.88425 sin 9.80807 20 S tan 10.50319 sin 9.97959 sin 9.97959 00 S tan 10.38744 22 W cos 9.78766 54 W cos 9.97598 . COOrdmate6oP lnt3 East of vertex Lat. 67 43' 00" S., Long. 81 07' 18" E. West of vertex Lat. 67 43 00 S., Long. Course at Western Point <7 N = 307 49' 38". Final conrse (7 N = 341 07' 06". 1 07 18 E. 272 NAVIGATION *i is s* Hi -T* od M a* o" P w "j ^o _/, H -3 ri i ^ u II fl ef^ri ~ fl J. DQ * M CO 1C O eo rt rj II "* " 4- M kf 03 42 ^ & 4 C P" .^ ^ H &0 O O "7? -+J H t3 S "O k +J 03 PI 02 "o i: ^ ^ S o^ sir ^0305 a d t- 71S ill OOOOOOOlO ^rH^r-I^OOCOOO -* CO t- t- 0202020202000200 GO T I O CO CO CO Oi O o' o' o* o o o' o' o' "* oost-cooooosco o os' os' os' os os* os' os OS OS OS OS OS OS OS OS -H -H -H-H -H-H -H FORM AND EXAMPLE 273 To' find the point of maximum separation. The point of maximum separation between the great circle and Mercator track is the point where the two courses are the same. FIG. 82. In Fig. 82, A B is the Mercator track, A SB the great circle route, on which V is the vertex and 8 the point of maximum separation. To find this latter point it is only necessary to solve the right triangle VPS for PS (= 90 L m8 ) and Z VPS (= \v ~ A ms ), where L ms is the Lat. and A TOS the longitude of the point, having given PV = 90 L v and PSV the Mercator course. The point is one of but little practical value. PLATE V. 1 s O o E x o ^ 7 ?7 /// '? s s FORM AND EXAMPLE 275 & & H 4J ^-N faJD ^3 O fl 3 ^ he o* Ills 3 o g * *" 'So S _ ^ 5 2 I S-l Sj . 41 t i O) r- H CO s 1 05 05 OS* 05 e d s C c 3 1 1 co 1 * o: o o: as 1 ^ 1 4 S ^ 'S o: C 1O CO t- * a 05 o s s g 1 1 TH rH oc r o o> CT C t -M o o 02 o j^ CO -* oo os o iH TH O 1O l~ CQ a! CO (M 00 TH OS rH rH \ | II 04 ^ -^ i-isi +* * S o5 a s ~~v O O + ,:! V Ed OS 1 ^ 00 h _ TH oc Tj ) o o CO o- ir tH OQ TH IT ^ i- 1 O5 lO o t- CO IT r- TH II II II 1 1 | * 1 * 1 r? - - f^ ? -\ || ) ^ K^ -e- ^ r-t ^ r cT t ! k: ^ =f r-T 276 NAVIGATION (b) To find the points of maximum separation, find the coordinates of the point of the curve in each hemisphere where the great circle course equals the Mercator course 46' 28". L v ............ 35 43' 07" ...... cos. 9.90950 ...... cosec. 10.23373 Her. C ........ =863 46 28 W .. .cosec 10.04718 ........ cos 9.64533 Aw ........... = 25 10 .......... cos 9.95668 A, a*. ....... = 49 11 45 ........................... sin 9.87906 *, in North Lat. = 133 36' 25" E \ in South Lat. . . =46 23' 35" W 49 11 45 ~ ......... = 49 11 45 JU,. in North Lat.= 177 11 50 W 1 j ^ in South Lat. = 95 35 20 W Lat.. = 25 ION J 1 Lat =25 10 S (c) The results of the solution of this subdivision of the example will be found tabulated on the opposite page. Finding the great circle course from azimuth tables. The azimuth of a heavenly body is tabulated in the tables with the arguments L, d, and t. Now, from the observer's position the azimuth of a heavenly body is the same as the great circle course to that terrestrial position having the same heavenly body in its zenith ; so, to use the azimuth tables for finding the great circle course to a place, it is only necessary to substitute for the body's declination the latitude of destination, for the body's hour angle the difference of longitude between the places expressed in time, and to consider the latitude of de- parture as that of the observer. The rules for marking the azimuth apply for marking the course. Solution by gnomonic charts, or great circle sailing charts. This subject has already been considered under the head of gnomonic charts (Art. 24) . Especial reference is made to the gnomonic charts issued by the IT. S. Hydrographic Office, on which are provided the means of determining the great circle FORM AND EXAMPLE 277 - I v cocococoeocococococo x. eoeocoeooocococococo COCOCOCOOO'.OCOSO COOO'.O t-i>eoO O5t-COOOOOSCOCO OS OS OS c OS OS* OS* OS* O5OSOSOSO5OSOSOSOSOS S3 OQ 'M CO Q 2 C3 H -! FH ^ 278 NAVIGATION course and distance directly, without transferring positions to a Mercator chart. Eeference is also made to the polar chart (Art. 25), which is available for either hemisphere. Any meridian may be taken as that of Greenwich, and the two places, between which the great circle course and distance are desired, having been plotted, join them by a straight line. This line is the great circle track from which any number of coordinates may be transferred to a Mercator chart. The polar chart is especially available in the Southern hemisphere where great circle sailing possesses so many advantages. Solution by graphic methods; Use of terrestrial globe. Locate the two places on the globe, move it till both places coincide with the upper edge of the horizon circle. Draw a line between the two points along the edge of the horizon circle. This will be the required great circle distance which can be measured by the scale on the horizon circle, and, when reduced to minutes of arc, will be the distance in nautical miles. Take off the latitudes and longitudes of as many points as may be desired, transfer them to a Mercator chart, and trace in the arc. The courses and distances from point to point on this arc may be gotten directly from the chart ; or, by computation, using middle latitude or Mercator sailing. Graphic chart methods. Various methods are now used to lay down a great circle track on a Mercator chart. These ob- viate the calculations which, by some people, may be consid- ered laborious. Towson's method permits a track to be laid down on a Mer- cator chart with a great degree of accuracy. His linear index gives the latitude and longitude of the vertex, whilst the accompanying tables give the true course at every degree of longitude from the vertex. Airy's method. The following method, proposed by Profes- sor Airy, when Astronomer Eoyal, lays down a curve which is a very close approximation to the great circle arc : SOLUTION BY GRAPHIC METHODS 279 (1) Join the two places on the chart by a straight line. Erect a perpendicular at its middle point, on the side next to the equator, producing the perpendicular beyond the equator, if necessary. (2) Find the middle latitude between the two places, and with this middle latitude enter the table below and take out the corresponding parallel. The intersection of this parallel with the perpendicular will be the center of the required arc. Middle Latitude. Name. Corresponding Parallel. Middle Latitude. Name. Corresponding 1 Parallel. ' 20 Opposite 81 13 52 Opposite 11 33 22 78 16 54 6 24 24 74 59 56 1 13 26 71 26 58 Same 4 28 67 38 60 9 15 30 63 37 62 14 32 32 59 25 64 19 50 34 55 05 66 25 09 36 50 36 68 30 30 38 46 70 35 52 40 41 18 72 41 14 42 36 31 74 46 37 44 31 38 76 52 1 46 26 42 78 57 25 48 21. 42 80 62 51 50 16 39 An approximate great circle track may be thus laid down : Compute the initial and final great circle courses between the two places A . and B. Join AB on the chart, erect a perpen- dicular at its middle point. Find the differences between the Mercator and the two computed great circle courses. Lay off the angles DAB and EAB equal to these differences. Erect perpendiculars to AD and AE f cutting the first perpendicular in p' and p. The point c midway between p and p' will be the center of the required arc whose radius will be cA (Fig. 83). 280 NAVIGATION 136. Composite sailing. Whenever the great circle track passes into higher latitudes than it is practicable or desirable to go, some of the advantages may be secured without going into regions of ice and danger, by following a composite track, a form of sailing first proposed by Mr. Tow- son. Decide on the parallel above which it is inadvisable to go, sail on the arc of a great circle which passes through the point of departure and has its vertex on this limiting paral- lel, proceed along the parallel till there is met a second great circle which, passing through the point of destination, has its vertex also on the limiting parallel; then follow this arc to destination. There are three general methods used in composite sailing : (1) By gnomonic charts. (2) By computation. (3) By graphic methods. By gnomonic charts. Draw lines from points of departure and destination tangent to the limiting parallel. In the case of the great circle sailing charts of tfie U. S. Hydrographic Office, find the track from point of dopjirture to point of tangency and from second point of tangency to point of des- tination, the intervening distance being found along the par- allel from a Mercator chart or by parallel sailing. On a polar chart (see Art. 25) tangents are drawn in the same way to the limiting parallel. Suppose it is desired to find the com- posite track from L = 45 N., A = 150 E., to L 2 = 47J N., X 2 = 130 W., the limiting parallel being 50 N. From the first position draw CE (Fig. 10) tangent to the parallel of 50, and DF tangent to the same parallel; C and D being, respectively, the points of departure and destination. Transfer any desired number of points, including points of tangency, COMPOSITE SAILING 281 from the track on the gnomonic chart to the Mercator chart, by the coordinates of latitude and longitude. Then sail from point to point of the first great circle till the parallel is reached, along the parallel, and from point to point of the second great circle to destination. FIG. 84. By computation. In Fig. 84, let BE be the limiting par- allel, AB and EF the great circles which pass, respectively, through the points of departure A and destination F, and have their vertices on the limiting parallel. The composite track will be ABEF. Since the limiting parallel furnishes the L v in each case, PB = PE = CoL v . Letting the differ- ent elements that enter be represented as indicated in Fig. 84, by Napier's rules, we have, In triangle ABP, sin (7 = cos L v sec L^. cos D = cot L v tan L . cos d : = cosec L v sin L . In triangle FEP, sin C 2 = cos L v sec L 2 . cos D 2 = cot L v tan L 2 . -(135) cos d 2 = cosec L v sin L 2 < In triangle BEP, D = (\ 2 ~ AJ (Z^ + D 2 ), p = D cos L v . Composite distance = d = d-^ + p + d 2 . 282 NAVIGATION Ex.64. From example 60 (Art. 135), we have L 35 40' S., A! 118 06' 07" E., L 2 22 15' S., A 2 41 30' W. ; it is required to find the shortest possible route by composite sailing when the 60th parallel of South Lat. is the limiting parallel. Find the distance, initial and final courses. Solution of the eastern triangle Fig. 84. O / It LI = 3540008 sec 10.09022 tan 9.85594 sin 9.76572 L v = 60 S cos 9.69897 cot 9.76144 cosec 0.06247 8 37 59 03 W sin 9.78919 = 65 31 15 cos 9.61738 d 1 = 47 40 48 2860'. 8 cos 9.82819 Solution of the western triangle Fig. 84. o t n 2 = 2215 S sec 10.03360 tan 9.61184 sin 9.57824 L v = 60 S cos 9.69897 cot 9.76144 cosec 10.06247 sin 9.73257 .... cos 9.37328 d^ 64 04 20 3844'.33 cos 9.64071 D = a a ~ 3,,) - (A + 2 ) D = 1064.62 log 3.02719 D = 17 44' 37" L v 60 cos 9. 69897 D 1064'.62 p ~ 532.3 log 2.72616 Initial course <7 N = 217 59' 03" d t 2860.8 Final course (7 N = 327 18' 06" d y 3844.33 - See plate (IV). d p + d l + d^ 7237.43 miles. Comparing the great circle and composite distances in this example, it is seen that the great circle distance is 7136.86 miles, the composite distance 7237.43 miles, or that there is a difference of 100.57 miles in favor of the great circle. Graphic methods. In graphic methods, use may be made of the terrestrial globe, or the track may be laid down on a Mercator chart approximately as follows : GRAPHIC METHODS 283 Decide on a limiting parallel. Join the two places on the chart by a straight line, at whose center erect a perpendicular and prolong it till it meets the limiting parallel. Through this point of intersection and the two given points pass a circle; then sail from point to point on this circular route, by middle latitude or Mercator sailing, till the limiting parallel is reached; along that parallel to the second point of intersec- tion with the circle ; then from point to point of the remainder of the circle, by middle latitude or Mercator sailing, till des- tination is arrived at. Examples Under Great Circle Sailing. Ex. 65. Find the great circle initial course and distance from Melbourne in Lat. 37 49' 53" S., Long. 144 58' 42" E., to Callao in Lat. 12 03' 53" S., Long. 77 08' 20" W. Also Lat. and Long, of the vertex. N = 132 55' 36". , d = 6984.37 miles. '* L v = 54 40' 00" S. A* = 158 25' 27" W. Ex. 66. Find the great circle initial course and distance from San Francisco in Lat. 37 47' 30" N., Long. 122 27' 49" W., to Sydney in Lat. 33 51' 41" S., Long. 151 12' 39" E. Also position of vertex. <7 N =r 240 17' 10". . , d 6445.25 miles. .,= 46 39' 32" S. \ v = 100 30' oi" E; Ex. 67. (a) Find the great circle initial course and dis- tance from Cape Vanderlind, Urup I., Lat. 45 37' N., Long. 149 34' E.,, to Pt. Eeyes Lt. Ho. on the coast of California, Lat. 37 59' 39" K, Long. 123 01' 24" W. Also Lat. and Long, of vertex. 284 NAVIGATION (&) Find the longitudes of intersection of the great circle with the 50th parallel of North latitude, and the course at first intersection. (c) Not wishing to go further North than the 50th parallel, on account of the Aleutian Islands, find the increase of dis- tance by pursuing the 50th parallel from the 1st to the 2d intersection, instead of following the great circle entirely. '0 N =62 46' 15". ^ = 3738 miles. } ^L V = 51 32' 26" N. A,, = 174 40' 45" W. : Long. of West intersection, 166 30' 46" E. Long, of East intersection, 155 52' 16" W. Course at West intersection,^ = 75 22' 19". (c) Increase of distance, 14.7 miles. Ex. 68. Find the great circle initial course and distance from Brisbane, Australia, Lat. 27 27' 32" S., Long. 153 01' 48" E., to Acapulco, Lat. 16 49' 10" N., Long. 99 55' 50" W. Also Lat. and Long, of the vertex. J/ N =82 04' 28". d = 6748.63 miles. L v = 28 29' 44" S. = 136 13'45"E. Ex. 69. Find the great circle initial course and distance from a point off Cape Agulhas in Lat. 34 55' S., Long. 20 01' E., to a point off Java Head in Lat. 6 55' S., Long. 105 02' E. Also Lat. and Long, of the vertex. fC' N =92 51' 32". d = 4918.4 miles. A U = 2500'11"E. PART II. NAUTICAL ASTRONOMY. CHAPTER VIII. GENERAL DEFINITIONS. THE VARIOUS SYSTEMS OF SPHERICAL CO-ORDINATES, AND CORRELATED TERMS. 137. Nautical astronomy is a special application of practi- cal astronomy to the needs of seagoing people, who, by obser- vations of the heavenly bodies, are enabled to determine the latitude and longitude at sea, and the error of their principal navigational instruments, the chronometer, the compass, and the sextant. The heavenly bodies are the fixed stars, and those bodies constituting what is known as the solar system; namely, the sun, the planets and their satellites, comets, and meteors. The fixed stars, numbering many millions, are situated at immense distances beyond the limits of the solar system. Of all these heavenly bodies, only the following need be considered for navigational purposes : the sun, the moon, four planets (Mars, Venus, Jupiter, and Saturn), and about 30 fixed stars. Astronomy teaches that the planets revolve about the sun, from West to East in elliptical orbits, at varying rates of speed, according to their positions in their orbits as well as their distances from the sun, and at the same time rotate on their- axes. The period of a complete revolution, or time required to move through 360 in its orbit, constitutes the planet's sidereal period or year ; and the period of a complete rotation on its axis is a planet's sidereal day. The earth is one of the planets of the solar system ; its orbit is in a plane inclined about 23 27 y to the plane of the equi- 288 NAUTICAL ASTRONOMY noctial. The form of this orbit is elliptical, the sun being at one of the foci. The advance of the earth in its orbit is irreg- ular, being the most rapid near perihelion, about January 1, and slowest near aphelion, about July 1. 138. The celestial sphere. To an observer on the earth's surface all the heavenly bodies appear to lie upon the concave surface of a sphere of indefinite radius of which only half is visible, the other half being cut off by the horizon. Owing to the insignificant ratio of the earth's radius to that of this assumed sphere, the eye of the observer may be considered as being at 0, the earth's center. This sphere is called the celes- tial sphere. However, these bodies, like r, i, u, v, w, etc. (Fig. 85), are not at the same distance from the observer, and, being projected on the celestial concave, their apparent posi- tions depend on their directions only and not on linear dis- GENERAL DEFINITIONS 289 tances. In like manner, the various fixed points and circles of the terrestrial globe defined in Chapter I may be projected on the celestial sphere, the point of sight considered to be at the center of the earth, in Fig. 85. The axis of the celestial sphere is the indefinite prolonga- tion of the earth's axis intersecting the celestial sphere in two points called the celestial poles, corresponding to and named like the North and South poles of the earth. That pole above the horizon at any place is known as the elevated pole, the one below the horizon as the depressed pole. Axis PP' (Figs. 85 and 86). The celestial equater, also called the equinoctial, is the great circle of the celestial sphere in which the plane of the terrestrial equator, indefinitely extended, meets the celestial sphere; EQ (Fig. 85), EDWC (Fig. 86). Horizons. A plane passed tangent to the earth's surface at the feet of the observer will be his sensible horizon, H'H f (Fig. 85) ; a second plane parallel to this through the center of the 290 NAUTICAL ASTRONOMY earth will be his rational horizon, N8 (Fig. 85) ; and these two planes indefinitely extended intersect the celestial sphere in practically one great circle called the celestial horizon. The point Z directly over the observer's head is the Zenith, the point N a , directly opposite and under his feet, is the Nadir. The celestial meridian is the great circle of the celestial sphere passing through the poles of the heavens, the zenith and nadir. It intersects the horizon in the North and South points, the North point being the one nearer the North pole. It is the great circle of the celestial sphere cut out by the indefinite extension of the plane of the terrestrial meridian. That semicircle which lies on the same side of the axis as the zenith is the upper branch; the other semicircle is the lower branch of the meridian. In Fig. 85, PQP'E is the meridian (in this particular figure it is also the solstitial colure), PQP' is the upper branch, PEP' the lower branch of the meridian. In Fig. 86, PZP'D is the me- ridian, PZP' being the upper branch. Ecliptic. Though the earth in reality moves around the sun, completing its revolution of 360 in one sidereal year, the sun's center apparently de- scribes a circle in the opposite direction on the celestial sphere, and this great circle is the ecliptic, CC' (Fig. 85) ; also OTC".(Fig. 87), a projection on the plane of the horizon. Angle of planes of ecliptic and celestial equator. The celestial equator is inclined to the ecliptic at the same angle that the earth's equator is inclined to the earth's orbit, about 23 2Vy. Equinoctial and ecliptic points. The two opposite points FIG. 87. GENERAL DEFINITIONS 291 of intersection of the equinoctial and ecliptic are practically fixed points on the celestial sphere. The sun's center crosses the equator twice a year, once about March 21, once about September 21, and these being times of equal day and night, are called equinoxes, and the points of crossing, equinoctial points. The point known as the first point of Aries is the point of the equinoctial occu- pied by the sun in passing from the southern to the northern hemisphere, on or about March 21 ; hence it is called the vernal equinoctial point. The other point is occupied by the sun's center on or about September 21, and is called the autumnal equinoctial point. Though now about 30 distant respectively from the constellations of Aries and Libra, in early ages they defined the western limits of those signs in which the cor- responding constellations lay, and hence were designated as the first points of Aries and Libra. Owing to the precession of the equinoxes, the constellation Aries has passed from the sign of Aries into that of Taurus, but the vernal equinoctial point, designated by the sign T, is still called the " first point of Aries." The points of the ecliptic 90 from the equinoctial points are called solstitial points, as at these points the sun reaches its greatest declination, occupying the northern one about June 21, and the southern one about December 21; in other words, the obliquity of the ecliptic equals the sun's greatest declination, North or South. The hour circle passing through the solstitial points is called the solstitial colure, PQP'E, (Fig. 85). The hour circle passing through the equinoctial points is called the equinoctial colure, POP' (Fig. 85), also PT (Figs. 86 and 87). The sign T stands for the vernal equinoctial point; the term vernal equinox refers to the time of the sun's passing through that point, but, as custom sanc- tions its use to represent the point, the term " vernal equinox" will in future be applied to the point, and its symbol will be T . 292 NAUTICAL ASTSONOMY 139. Determination of a point of the celestial sphere. The position of any point on the surface of the celestial sphere is determined when its angular distances are given from any two great circles on that sphere, whose positions are known. The equinoctial and ecliptic are fixed great circles on the celestial concave, and the vernal equinox is practically a fixed point on the equinoctial, having a motion of only 50 ''2 a year to the westward due to precession. Each of these great circles is used as the primary of a system of coordinates in fixed observatories; but at sea altitudes are measured above the visible horizon, and then referred to the celestial horizon, so that for seagoing people a system in which the horizon is the primary becomes necessary. Hence three systems are in use, each named after its primary, (1) Ecliptic, (2) Equinoctial, (3) Horizon Systems. The Ecliptic System and Correlated Terms. 140. The ecliptic system. The primary circle of this sys- tem is the ecliptic which has already been defined; the sec- ondaries are great circles passing through the poles of the ecliptic called circles of latitude, the one passing through T, the vernal equinox, being the principal one, #T (Fig. 88). Celestial longitude is the arc of the ecliptic intercepted between the vernal equinox and the circle of latitude passing through the body, reckoned positively towards the East, from to 360. The celestial latitude of a body is the angular distance from the plane of the ecliptic measured on a circle of latitude pass- ing through the body. In Fig. 88, CO' is the ecliptic; THE EQUINOCTIAL SYSTEM 293 the celestial longitude; and KS, the celestial latitude of the heavenly body 8. The coordinates in this system are unaffected by diurnal rotation ; hence it is a convenient system at fixed observatories, especially when considering the motions of the sun and bodies composing the solar system. It is not used at sea. The Equinoctial System and Correlated Terms. 141. The equinoctial system. In this system, the primary is the equinoctial which has already been defined, and the sec- ondaries are the great circles passing through the poles of the equinoctial. The solstitial colure is a secondary common to this and the ecliptic system. The secondary of this system passing through the zenith of a place is called the celestial meridian, and that one passing through a heavenly body is called a declination circle. The declination of a heavenly body is its angular distance from the plane of the equinoctial, measured on the declination circle passing through the body. It is given in degrees, min- utes, and seconds, and is marked N". or S., according as the body is North or South of the equinoctial (BA, Fig. 86). The polar distance of a heavenly body is its angular distance from the pole (usually from the elevated pole), and, being measured on a declination circle, it equals 90 the dec- lination; but, if the declination is negative (of an opposite name from the latitude), the polar distance equals 90 + the declination. Parallels of declination are small circles whose planes are parallel to that of the equinoctial. The rotation of the earth is always performed in the same interval of time, a sidereal day, which is -divided into 24 side- real hours, and gives to the fixed stars an apparent movement in planes parallel to the equinoctial, through 360 in the same interval of time. From the time of apparent rising in 294: NAUTICAL ASTRONOMY the East till the time of apparent setting in the West, the stars maintain their relative positions with reference to each other. This apparent motion, being due to the daily rotation of the earth, is called apparent diurnal motion of the heavens, and the path of any one star during its complete revolution is called its diurnal circle. Right sphere. To an observer at the equator, stars will rise and set vertically and their diurnal circles will be bisected by the horizon, so that the stars will be 12 hours above and 12 FIG. 89. hours below the horizon; the planes of the diurnal circles being at right angles to the observer's horizon, the celestial sphere in this case is called a right sphere. Parallel sphere. Could an observer be at the North pole, he would see the stars of North declination sailing around, maintaining a constant altitude above the horizon, never ris- ing and never setting. Stars of South declination would be invisible. The planes of the diurnal circles being parallel to the horizon, the celestial sphere would in this case be called a parallel sphere. THE OBLIQUE SPHERE 295 Oblique sphere. To an observer at some point between the equator and the pole, say the North pole, the stars will rise and set at an oblique angle with the horizon. This applies to any heavenly body, whose declination will permit of any part of its diurnal circle coming above the horizon. A body of declination will rise in the East point, set in the West point, and be the same length of time above and below the horizon; EqW (Fig. 89) is the diurnal circle of such a body. A body of North declination will rise and set to northward of the East and West points, and be above the horizon more than 12 hours. In North latitude, stars of South declination, if visible at a place in North latitude, rise and set to south- ward of the East and West points, and will be above the hori- zon less than 12 hours. Since the declination of the sun in summer time is of the same name as the elevated pole, the sun is then above the horizon more than 12 hours; in other words, summer days are longer than winter days. Those stars whose polar distance is less than the altitude of the elevated pole, which is the radius of the circle of perpetual apparition, NK (Fig. 89), never set, but revolve around the elevated pole of the heavens. Those whose diurnal circles lie within the circle of perpetual occultation, RS (Fig. 89), never rise, and hence are invisible. This aspect of the heavens is known as the oblique sphere. Hour circles. In this apparent revolution of the heavenly bodies around the earth, their declination' circles are continu- ously describing angles around the poles, which are called from the divisions of time hour angles, and, analogously, the declination circles are called hour circles; hence hour circles are defined as great circles passing through the poles of the heavens. PB (Fig. 86) is the hour circle of the body A. As a star, for example A (Fig. 86), moves in its diurnal path about the pole, a point B of its hour circle moves uni- 296 NAUTICAL ASTRONOMY formly over the equinoctial through 360 of arc in 24 sidereal hours, 15 of arc in one hour, 15' of arc in one minute, and 15" of arc in one second of time, thus establishing a relation between arc and time. What is said here about the apparent movement of a star's hour circle will apply to the movements of the hour circle of any heavenly body whose increase of right ascension is uni- form ; and, as time in any system used is the angle at the pole, measured by an arc of the equinoctial, all time, however meas- ured, is converted into arc at the rate of 15 of arc to one hour of time. See Art. 178. Transit or culmination. The passage of a celestial body across the meridian of a place is called its transit or culmina- tion ; the upper transit occurs when it crosses the upper branch of the meridian, and the lower transit when it crosses the lower branch of the meridian. When a body's diurnal path is within the circle of perpetual apparition, both transits occur above the horizon, the upper one above the pole, the lower one below it; whilst those bodies, whose diurnal circles lie within the circle of perpetual oceultation, are never visible at the given place. Hour angle. The hour angle of a heavenly body, or of any point of the sphere, is the inclination of the hour (or declina- tion) circle passing through the body, or point, to the celestial meridian, and is measured by the arc of the equinoctial inter- cepted between these two circles. Hour angles are properly reckoned from the upper branch of the meridian, positively toward the West, and are usually expressed in hours, minutes, and seconds of time from O h to 24 h . However, for conven- ience in practical work, it is better, in fact it is usual in the American naval service, to regard the hour angle as minus when the body observed is East of the meridian up to 12 h . In Fig. 86, ZPA is the hour angle of the body A and it is measured by the arc CB. SOLAR AND SIDEREAL TIME 297 Solar time. The hour angle of the sun is called solar time, there being 24 hours of solar time in the interval between two consecutive upper transits of the sun over the same meridian, and this interval is called a solar day. Sidereal time. The hour angle of the 1st point of Aries, or vernal equinoctial point, is called sidereal time, there being 24 hours of sidereal time in the interval between two consecutive upper transits of the 1st point o>f Aries over the same meridian, and this interval is called a sidereal day. Owing to the fact that the 1st point of Aries is practically a fixed point of the equinoctial, the sidereal day is the time of revolution of the earth on its axis, or, in other words, of the apparent revolution of the celestial sphere through 360. Relation between solar and sidereal days. Owing to the angular movement of the sun in its apparent orbit to the east- ward (this apparent motion of the sun being due to the move- ment of the earth in its orbit about the sun), the sun comes to the meridian each day on an average about 3 m 5 6 s . 555 of sid- ereal time later than on the previous day ; therefore, the solar day is longer by that amount than the sidereal day. Right ascension. The right ascension of a heavenly body is the inclination of its hour circle to that passing through the vernal equinox, or the arc of the equinoctial intercepted be- tween these two hour circles. It is measured from the vernal equinox positively to the eastward from hours to 24 hours. For body A (Fig. 86), ^PB, measured by the arc T#, is the right ascension. The fixed stars are at such immense distances as to be un- affected by the earth's change of position in its orbit; the co- ordinates of this system, however, declination as well as right ascension, are slightly affected by the precession of the equi- noxes. Relation of H. A. and R. A. From the preceding defini- tions of hour angle and right ascension it is evident from Fig. 298 NAUTICAL ASTRONOMY 86, in which. A is a heavenly body, T the 1st point of Aries or vernal equinox, PZN a the meridian, PB and PT hour circles, that the local sidereal time which equals the right ascension of the meridian is the angle ZPT, measured by T C; the hour angle of the heavenly body A is the angle ZPA, measured by CB; its right ascension is rPB, measured by the arc TP, and T<7 = CB + TB f or (1) the local sidereal time at a given instant always equals th& algebraic sum of the hour angle and the right ascension of the same ~body at that instant. When the hour angle is zero>, the heavenly body is on the meridian, and its right ascension then equals the local sidereal time at that instant, or (2) the right ascension of the meridian at a given instant equals the local sidereal time. These are two facts that must be fully realized and under- stood by every navigator ; and it follows from the first propo- sition that when two of the angles are given, the third can be easily found. The right ascension and declination of heavenly bodies are determined at fixed observatories, and tabulated in the Nau- tical Almanacs ; knowing these, the position of a heavenly body is easily determined in this system, the right ascension being reckoned along the equinoctial to the eastward from the vernal equinox in a manner similar to the reckoning of longi- tude from the prime meridian on the terrestrial sphere; and the declination is reckoned North or South of the equinoctial along the declination circle, as latitude is reckoned North or South of the terrestrial equator along a terrestrial meridian. This system is the most convenient one for representing the motions of the fixed stars, owing to the very slight changes in coordinates. The Horizon System and Correlated Terms. 142. The horizon system. The primary circle of this sys- tem is the celestial horizon ; the secondaries are great circles THE HORIZON SYSTEM of the celestial sphere passing through the zenith and nadir; their planes being perpendicular to the horizon, they are called vertical circles. The principal secondary is the celestial meridian which intersects the horizon in the North and South points, each of which is named from the nearest pole. The celestial meridian is the secondary common to hoth the horizon and equinoctial sys- tems; SZNNa (Fig. 90). The prime vertical is the vertical circle pasing through the E. and W. points of the horizon ; its plane is, therefore, perpendicular to that of the celestial meridian. ZWN a E (Fig. 90), is the prime vertical. The azimuth of a heavenly body is the angle at the zenith, measured by the arc of the celestial horizon, between the me- ridian and the vertical circle passing through the body ; PZK, (Fig. 90), for body A. Though the azimuth, as an angle of the astronomical tri- angle, is reckoned from the elevated pole towards the East or West, according as the body is East or West of the meridian, and though so estimated when tabulated in azimuth tables, still navigators of the present day reckon azimuth in 'both hemispheres more conveniently from the North point of the horizon, around to the right, from to 360. If the angle Z found by solution is x, navigators will con- sider the azimuth, or Z N , simply as x, 180 - x, 180 + x or 360 x, according as the bearing of the body by solution is K x E., S. x E., S. x W. or N x W., respectively. The amplitude of a heavenly body is the angular distance of the body, when in the horizon, from the prime vertical. It is 300 NAUTICAL ASTRONOMY reckoned from the East point when the hody is rising, and from the West point when setting; towards North or South according as the body is North or South of the prime vertical. The true altitude of a heavenly hody is its angular distance from the plane of the celestial horizon, measured on a vertical circle passing through the body from to 90 ; KA (Fig. 90), for body A. The zenith distance is the angular distance of the body from the zenith, measured on its vertical circle, and equals the com- plement of the altitude; ZA (Fig. 90), for body A. From what has been said it follows that in this system the position of a body is given by its altitude and azimuth, the coordinates determined by navigators at sea, so observed posi- tions are referred to this system ; but as tabulated elements to be used by navigators all over the world must be referred to a system unaffected by the position of the observer, the value of the equinoctial system becomes apparent. The predicted positions according to this latter system are found in the American Ephemeris and Nautical Almanac, as well as in other publications. Referring to Fig. 90, let be the observer, QQ f intersection of planes of celes- Z the zenith, tial equator and meridian. tf a the nadir, A a heavenly body, P the elevated or N. pole, ZAK its vertical circle, SZNNa the celestial meridian, AK its altitude, NESW the celestial horizon, AZ its zenith distance, EZ WN a the prime vertical, PZK its azimuth, N the North point of the horizon, PO JVthe altitude of the elevated S the South point of the horizon, pole, PP f the axis of the sphere, Q l Z the declination of the zenith. To prove that latitude equals the altitude of the elevated pole. The arc of the meridian SQ' (Fig. 90), intercepted be- tween the planes of the celestial equator, QOQ', and celestial horizon, SON, measures the inclination of the planes of these THE ASTRONOMICAL TRIANGLE 301 two great circles to each, other; this inclination is also meas- ured by the arc ZP intercepted between their poles, but ZP = 90 PN and SQ' = 90 Q'Z; therefore, PN = Q'Z, or, PON Q'OZ. Terrestrial latitude has been, denned in Art. 1 as the angular distance of a place measured on its meridian N". or S. of the equator. ' As the zenith is the projection of a place and the equinoctial the projection of the terrestrial equator on the celestial sphere, the latitude of a place is the declination of the zenith; therefore, Lat. = Q'OZ = PON, or, latitude equals the altitude of the elevated pole. 143. The astronomical triangle. The spherical triangle 'PZA (Fig. 86), formed by arcs of the celestial meridian, and the vertical and hour circles passing through the body A, is called the astronomical triangle, and it is this triangle that the navigator solves in working for latitude or longitude, re- membering that when the observed body is on the meridian the triangle reduces to a straight line. The angles are : ZPA the hour angle, PZA the azimuth, and PAZ the position angle ; the sides of the triangle are PZ, the co-latitude of the place of observation, AP the polar distance, and AZ the zenith distance of the body. The position angle is not used, but when any three of the other five parts are given, the re- maining two can be found by spherical trigonometry. By definition the co-latitude and zenith distance can never be greater than 90. If the declination is of the same name as the latitude, it is regarded as positive and the polar distance equals 90 minus the declination; if of a different name from the latitude, the declination is regarded as minus and the polar distance equals 90 plus the declination. In studying the astronomical triangle diagrams will be 302 NAUTICAL ASTRONOMY found most useful, and the most appropriate are those found by stereographic projections in which the point of sight is at one pole of the primitive circle. In Figs. 91, 92, and 93, PZM is a projection of the astronomical triangle. FIG. 91. FIG. 92. On the plane of the meridian, the point of sight is at the E. or W. point, and both Z and P are on the primitive circle (Fig. 91), in which M is a heavenly body West of the meridian. On the plane of the equator, the point of sight is at the de- pressed pole. P is at the center of the primitive circle, me- ridian, and declination or hour circles are projected as straight lines (Fig. 92). On the plane of the horizon, the point of sight is at the nadir. Z is at the center of the primitive circle. All vertical circles, and hence the celestial meridian, are projected as FIG. 93. straight lines (Fig. 93). CHAPTEE IX. THE SEXTANT, THE VERNIER, AND THE ARTIFICIAL HORIZON. METHODS OF OBSERVING HEAVENLY BODIES. 144. The sextant. The sextant is a small portable instru- ment used for measuring the angles between two bodies or objects, whether or not one or both are celestial or terrestrial, and for measuring the altitudes of heavenly bodies or terres- trial objects above the visible horizon. Its principal use is at sea, where the use of fixed instruments would be impossible, in measuring altitudes for finding the latitude and longitude. The octant is a similar instrument, and is used for the same purposes, but the length of its limb is only about one-eighth of a circle. As the name implies, the arc or limb (c) of the sextan i (Fig. 94) is equal to about one-sixth of a circle, or 60 of arc, though graduated, as will be explained later on, so that eacli degree of the limb is really divided into two degrees of gradu- ation, the subdivisions of the degrees being frequently as close as 10' of arc, on an arc of silver, gold, or platinum. The limb and its supporting frame are of brass. A brass index arm (o), pivoted at the center of the circle whose arc forms the limb, is movable, carrying at the movable end a vernier (d) and magnifying glass (g) to read subdivisions of the gradu- ated arc, and at the pivoted end a silvered mirror (a) whose plane must be perpendicular to that of the index arm. and frame. This mirror, called the index glass, moves with the 304 NAUTICAL ASTRONOMY index arm. A second glass (&), called the horizon glass, one- half transparent and one-half silvered, the dividing line being parallel to the plane of the instrument, is fixed and should also be perpendicular to the plane of the limb. The graduations of limb and vernier should be such that the zero of one will be in coincidence with the zero of the other when the index and horizon glasses are parallel. A telescope (i) which directs the line of sight through the THE SEXTANT 305 horizon glass and parallel to the plane of the instrument, is carried in a ring capable of movements at right angles to- the plane of the instrument,, shifting the axis of telescope from the silvered to the transparent part of the horizon glass, or vice versa. Colored glasses (h) of different shades are fitted for use before both index and horizon glasses. The index arm is fitted with a clamp (e) for securing it to FIG. 95. the limb, and a tangent screw (/) for giving it small motions after clamping. Besides the telescope (i), the sextant box is usually fitted out with a star or inverting telescope (k), a plain or sighting tube (Z), and neutral glasses or caps (n) for the telescopes. The use of these caps obviates the necessity for the use of the colored shade glasses. The box also contains a screw driver, adjusting keys, a magnifying glass, and spare mirrors. 306 NAUTICAL ASTRONOMY 145. The optical principle of the sextant, The optical principle of the construction of the sextant is thus stated: " The angle between the first and last directions of a ray of light, which has suffered two reflections in the same plane, is equal to twice the angle which the two reflecting surfaces make with each other " To prove this, let M and m be the two reflecting mirrors of a sextant whose planes are perpendicular to the plane of the sextant, in this case the plane of the paper (Fig. 95). Let B be a body whose ray falling on M is reflected to m and by m to the eye at E; then BEm will be the angle between the first and last directions of ray BM, after having been reflected twice in the same plane. The angle between the mirrors is equal to the angle between lines perpendicular to them, pp f being perpen- dicular to M and mp f to m, and it is required to prove that BEm or h = 2a. Since the angle of incidence equals the angle of reflection, BMp = pMm, and Mmp' = p'mE ; by geometry from kMp'm, x = y -(- a . ' . 2x = %y -\- 2a from &MEm, 2x = 2y + h therefore h = 2a 146. Application of the principle in measuring angles. Suppose it is desired to measure the angular distance between two bodies, B and H, H sufficiently distant that the rays H'M and Hm are sensibly parallel. The instrument is held so that its plane passes through both objects, the object H being seen directly through the telescope and horizon glass. Now let the index arm be so placed and clamped that the two glasses are parallel to each other; then will the ray H'M be reflected by the two glasses parallel to itself, and the observer's eye at E will see both direct and reflected images in coincidence. Sup- pose this position of the index arm is MI, then for the given position of the horizon glass, / should be the zero of gradua- tions of the limb. Now move the index bar, and with it the THE VERNIER 307 fixed mirror M, to the position MI', so that a ray from the second object B shall be reflected in the direction mE; the observer looking directly at H through the transparent part of the horizon glass, sees the reflected image of B in coincidence with the direct image of H. The angle h is the angle meas- ured, but h is twice the angle between the mirrors, or Ji = 2a ; and, since a = IMI' 9 h is twice the angle through which the index bar has moved, that is, twice the difference of the read- ings I and /'. To avoid doubling the angle, every half degree of II', and in fact of the whole limb, is marked as a whole degree, and the observer, reading directly from the limb, has only to subtract the reading at I from that at I', to get the angular distance between H and B. If the instrument is in proper adjustment, the reading at / is zero, that is, the limb is graduated from I as an origin. If this point of reference, I, does not coincide with the zero of graduation, the sextant has an error, called index error, which affects all angles ob- served with it at the time. The degrees of the limb are further subdivided, those of the finest sextants being divided inttf six equal parts, each part 10' of arc, and in order to read fractions of these divisions, recourse is had to the vernier. 147. The vernier. This is a graduated scale (Fig. 96) to slide along the divisions of a graduated limb to facilitate the readings to fractions of a division of the limb. It is so con- structed that the length of the vernier is exactly the length of a certain integral number of divisions of the limb, and is divided into one more or less divisions than that certain num- ber ; the fraction of a division of the limb is indicated by the division of the vernier which is in coincidence with a division of the limb, as will be explained later. The most usual method of construction is to make the number of divisions on the vernier one more than on the corresponding arc of the limb, and the explanation of this type follows. 308 NAUTICAL ASTEONOMY To explain the working of a vernier, let AB (Fig. 96) be the arc of a limb, each division 20' ; CD the vernier, the length of which is taken as 19 times the length of a division of the limb and is divided into 20 equal parts, thus each division of the vernier comprises 19' of arc or is less by 1' of arc than any division of the limb. The first line of the vernier is the zero line, and the reading of the limb is determined by the posi- tion of this zero. If this zero coincides with any division of the limb, the division line of the vernier marked 1 falls short of the next division of the limb by 1', the next division line of the vernier marked 2 falls short of the next line of limb by 2', and so on until the line marked 20 of the vernier coincides with a line of the limb; hence, if the vernier is advanced FIG. 96. . through 1' of arc, the line marked 1 of the vernier will coincide with a division of the limb, if we advance it through 2' of arc, the line marked 2 will coincide with a division of limb, and so on, and if the nth line of the vernier is found to be in coincidence with a division of the limb, it will be evident that the zero of the vernier has advanced n minutes. General rule for navy sextants. The general rule fol- lowed in the construction of verniers for the U. S. Navy is to take the length of the vernier exactly equal to the length of a certain integral number of divisions of the limb and di- vide, the vernier length into equal parts, the number of which HEADING THE SEXTANT 309 must be greater by one than the number of the divisions of the limb. Let I = value of a division of the limb, v = value of a division of -the vernier, n = number of parts into which the vernier is divided, n 1 = number of parts in the corresponding length of the limb, n n n> Least count of vernier = I - Of course, the graduations of the limb and the vernier must be in the same unit. Ex. 70. The limb of a sextant is divided to 10' of* arc. Construct a vernier to read to 10" of arc. The least count being 10", a division of the limb 10' = 600", a division of the vernier is 590". Therefore, 600 (n1) = 590 n f 60 n 59 n = 60, n = 60. Take 59 divisions of the limb for the length of the vernier, and divide it into 60 equal parts. Ex. 11. A sextant limb reads to 15' of arc; the vernier is taken in length as 44 divisions of the limb. What is the least count of the vernier ? 148. Reading the sextant. First note the position of the zero of the vernier, then read the limb up to the division line immediately to the right of the zero of the vernier; this will be a certain number of degrees, or a certain number of degrees plus a certain number of the divisions of a degree. Say the sextant limb is graduated to 10' of arc, and suppose the near- est division referred to is 33 40'; now follow the arc of the vernier until a vernier line is found in coincidence with any 310 NAUTICAL ASTRONOMY line of the limb. Suppose the least count of the vernier is 10" of arc, and the reading of the vernier at the coincident line is 2' 20", then the angle is 33 42' 20". All angles measured on the limb are spoken of as " on the arc." Excess of arc. The limb of a sextant is generally grad- uated not only to 120, but the limb is often of such extent as to be graduated up to 150. This part of the limb is the arc proper, but, in all sextants, the limb and graduated arc are continued to the right beyond the zero for a short distance, and this arc is called the "excess of arc," and angles meas- ured on it are spoken of as " off the arc." These angles are read from zero to the right, or backwards, and the vernier must also be read backwards. If a division of the limb is n minutes, then the vernier is marked to read n minutes; so read the vernier directly and subtract its reading from n min- utes to get the vernier reading to be added to the reading of the limb, off the arc, immediately to the left of the zero of the vernier. Thus if a sextant limb reads to 10', the vernier to 10", and the zero of the vernier falls between 3 10' and 3 20' off the arc, and the vernier, if read directly, shows 2' 50", then the vernier read backwards would be 7' 10", and the angle 3 17' 10". 149. Errors. Sextants are subject to two general classes of errors. The first class comprises what are known as con- stant errors, and, though they usually arise from defects in construction, they may at times be occasioned by injuries re- ceived in legitimate use, or to abuses due to ignorance or carelessness. These errors should be eliminated, or ascer- tained and tabulated by the maker. In a high-grade instru- ment from a maker of good reputation, this class of errors should not exist. The second class, known as the adjustment class, comprises those errors that should be removed by the navigator himself. Constant errors. (1) The centering error is due to the ERRORS or SEXTANT 311 fact that the axis of the index bar is not at the center of the limb nor perpendicular to its plane. ~No sextant should be bought without careful inspection and not until after tests as to the centering error have been made.* If the eccentricity is found to be greater than .005 of an inch, the instrument should be rejected. (2) Error of graduation. The limb may not be a plane surface, and graduations of both limb and vernier may be inex-act. There may be flexure of limb due to varying tem- perature, or accidental blows, producing great errors in angles. (3) Prismatic effect of mirrors and shade glasses, due to a want of parallelism between the two faces. The above are all faults of construction. The combined total errors of eccentricity and graduation can be ascertained together by measuring known angles with the sextant ; the error can be found for a number of positions of the index bar, and then for other intermediate angles by interpolation. The known angles referred to may consist of angles laid off by a theodolite at intervals of 10 to 20 to cover the range of the sextant. The combined error can also be ascertained hy a series of artificial horizon observations, observing stars of nearly equal altitudes N". and S. of the zenith. Half the difference of latitudes resulting from each star will be the error for that altitude. The correction will be minus, if the latitude from the star on the polar side of the observer is greater than that from the star on the equatorial side of the observer ; and plus, if vice versa. As this error varies on different parts of the arc, and generally increases with the angle, it would require many observations to determine it with any degree of satis- faction. The determination of this error at sea is an entirely differ- ent proposition ; theoretically it can be done by measuring the angular distance between two stars, and comparing this with the angular distance ascertained by computation. This, how- * Navigating sextants issued to the U. S. Navy should bear a certificate of in- spection from the U. S. Naval Observatory, giving the correction, for eccentricity at intervals of 10" of arc. 312 NAUTICAL ASTRONOMY ever, is not practicable on account of the complications due to refraction and aberration. However, since sextants are liable to accidents at sea, it may be desirable to ascertain this error, if only approximately. Now, if we can observe the angular distance between stars on the same vertical circle, the question of refraction will become a very simple matter, as the altitudes may be either observed (the horizon being clear), or computed for the instant of measurement of the arc. Those stars that have practically the same right ascension, or right ascensions differing by 12 hours, will be on the meridian, and hence on a vertical circle at the same time. There are many groups of such stars whose right ascensions do not differ more than either 30 minutes, or 12 h 30 m , and these might be used without much error. However, the right ascensions of the following groups are practically the same a Aurigse ) 77 Ursae Majoris 1 ft Orionis \ a Virginis j a Scorpii | a Pavonis s Ophiuchi J y Cygni and the right ascensions of the groups below differ by prac- tically 12 hours, a Ursas Minoris j a Tauri j y Geminorum ) a Virginis J a Trianguli Australis J a Lyrse J and the stars of each pair are on the meridian at practically the same time. Now the true distance, within an error of a few seconds of arc only, neglecting aberration, between the stars in each of the first four groups, that is, stars of the same right ascension, is the difference of their polar distances at the instant of meridian passage; for stars of the second groups, that is, for stars whose right ascensions differ by 12 hours, the true distance will be the sum of their polar distances. If any stars, paired off as above, are visible, measure the PRISMATIC EFFECTS 313 arc between them, when on the meridian; either observe or compute their altitudes, and take from tables the correspond- ing refractions. If both stars are on the same side of the zenith, add the difference of refractions to the observed arc; if the stars are on opposite sides of the zenith, add the sum of refractions to the observed arc ; the result is the corrected sextant distance; the difference between this and the true distance obtained from the polar distances is the total error for that angle. Knowing the index error, the error due to eccentricity and graduation may be found. Knowing the right ascension of a star, it is easy to find the ship's time of its transit, and hence the time for measur- ing the arc. However, for many apparent reasons, even this method is ordinarily impracticable, so that at sea the sextant should be guarded carefully against all possible injuries. Graduations. Examine carefully the graduations of both limb and vernier. If the zero of the vernier is in coincidence with one division of the limb, an inspection of the divisions of the vernier should show an increasing separation between the divisions of vernier and limb in a direction towards the zero of vernier till the last division of the vernier is reached, when it should be coincident with a division of the limb. By shifting the position of the vernier, the divisions of the limb are tested for equality, whilst for magnitude they may be tested by measuring known angles of various sizes. Faults of graduation, if developed on inspection before buying, should cause the rejection of the sextant. Prismatic effect of index glass. This can be tested by ob- serving a large angle, say, 120 to 130, between two objects, and again measuring the angle with the index glass upside- down. If measurements agree, the sextant having been ad- justed in both cases, there is no prismatic error. If they do not agree, reject the mirror. When a reflected image, the angle being large, is not clearly defined, or there seems to be 314 NAUTICAL ASTRONOMY a fainter outline on a clearer image, it is evident that rays reflected from the inner and outer faces of the index glass are not parallel, and that the glass is prismatic. Prismatic effect of horizon glass. The want of parallel- ism of the two faces of the horizon glass is not a matter of great importance, as all angles and the index correction are affected alike. Prismatic effect of shade glasses. A want of parallelism in shade glasses, when used in front of the index glass, will affect the index correction, which should be determined with and without them. The index error should be determined also for each combination of shade. These shade glasses, when known to be prismatic, should be discarded; and, if thought to be prismatic, colored caps should be put on the telescopes and the use of shade glasses discontinued. Imperfections of shade glasses between the eye and horizon glass, or in the colored cap of the telescope, affect the object and the reflected image alike, so that the angle between them is unaffected. 150. Adjustment of the sextant. It is the duty of the navigator, or of the person using a sextant, to keep it in ad- justment; in other words, to see that the index and horizon glasses are both perpendicular to the plane of the instrument and parallel to each other when the zeros of the. vernier and limb are in coincidence, and that the line of sight of the tele- scope is parallel to the plane of the instrument. To adjust the index glass. Hold the sextant in the left hand, place the index bar near the center of the limb, and, with the index glass nearest the eye, the eye being near the plane of the instrument, look into the mirror so as to see the reflected image of the limb. If the image and the arc as seen direct form a continuous line, the adjustment is correct. If they form a broken line, the mirror inclines forward or ADJUSTMENT OF SEXTANT 315 backward, according as the image rises or droops. Adjust the glass by means of screws at the back of the glass till the arc and its image appear perfectly continuous. To perfect the adjustment, it may be necessary in some cases to tilt the mirror so much as to require the use of a liner of blotting paper under one edge. To adjust the horizon glass. The horizon glass may pro- duce two kinds of error, a lateral error and an index error. Place the zeros of vernier and limb in coincidence, and look through the telescope at a star. If the two images coincide, the adjustment is correct. If the reflected image is to the right or left of the direct image, there is lateral error due to the fact that the horizon glass is not perpendicular to the frame; if the reflected image is above or below the direct image, there is index error due to the fact that the mirrors are not parallel. The adjusting screws for this glass are sometimes back of the glass, at other times below and to one side. Move the arm and bring the mirrors parallel so as to have the reflected image on the same line but to right or left of the direct one. By proper screws remove the lateral error, so that, as the arm is moved, the reflected image passes directly over the direct image. Now place the zeros in coincidence, and, by the proper screws, make the two images coincide, thereby eliminating the index error. However, these two errors are so intimately connected that in an effort to remove one, the other is affected; so it is better to adjust by the "halving method." Place the zeros in coincidence, remove half the lateral error, then half the index error, and so on till adjustment is perfect. The sea horizon may be used to test the adjustment of the horizon glass as follows: hold the instrument vertically and make the reflected and direct image of the horizon a con- tinuous line. Then incline the instrument so that its plane makes but a small angle with the plane of the horizon. If 316 NAUTICAL ASTRONOMY the true and reflected horizons are in perfect continuation, each of the other, the glass is perpendicular to the plane of the instrument, the reading of limb and vernier being the index error. If the reflected horizon appears above the true one, the glass leans too much inward, otherwise outward. The sun may be used in the same way as the star was used, but, owing to its size and brightness, perfection of adjustment is not so easily reached. To make the line of sight of telescope parallel to the plane of the instrument. Screw the inverting telescope into the collar, turn the eye tube till the two wires at its focus are parallel to the plane of the instrument. Place the sextant upon a table in a horizontal position, look along the plane of the limb, and make a mark upon a wall, or other vertical surface, at a distance of about 20 feet; draw another mark above the first at a distance equal to the height of the axis of the telescope above the plane of the limb; then so adjust the telescope that the upper mark, as viewed through the telescope, falls midway between the wires. The adjustment is made by tightening or loosening one of the two adjusting screws on the collars, doing the reverse with the other screw. Index error. Before using a sextant to make observations of any kind, the sextant being otherwise in adjustment, it is necessary to find the point of the graduated arc where the zero of the vernier falls when the two mirrors are parallel to each other, or at the time when the reflected image of a distant object is found to be coincident with the direct image of the object itself. If this point is not coincident with the zero of the limb, the sextant has an index error which affects all angles taken at the time. These angles, as measured, will be too small or too large, according as the zero of the vernier falls to the right or to the left of the zero of the arc ; in other words, the reading of the vernier is subtractive, if on the arc, INDEX CORRECTION 317 additive, if off the arc. The error applied in this way is known as the index correction, and is represented by the let- ters I. C. Should it be desired to eliminate the index error, place the zeros of vernier and limb in coincidence, and, by means of the proper adjustment screws, turn the horizon glass about an axis perpendicular to the plane of the instrument till the reflected and direct images of a star or distant object are in coincidence. Be careful, however, not to disturb the perpendicularity of the horizon glass. It is not advisable to try to keep the index error at zero, but it should be deter- mined before every observation under any circumstances ; and the knowledge that there is one, even though small, makes its determination necessary at the time of any set of observations. To determine the index correction. By a star. Bring the reflected image of the star into coincidence with its direct image, then read the arc and vernier. The reading is the index correction ; -f- if off the arc, if on the arc. By the sea horizon. Hold the instrument vertical, and make the true and reflected horizons continuous. The read- ing of the arc and vernier is the correction, -|- or as before. By the sun. Bring upper limb of reflected image of sun tangent to the lower limb of the sun seen directly, read the sextant, +. if off the arc, if on the arc; then bring the lower limb of the reflected image tangent to the upper limb of the sun seen directly, read the sextant, + if off, if on the arc. The index correction will be one-half the algebraic sum of the two readings. If well taken, the result can be checked by finding from the Almanac the sun's semidiameter which should equal one-fourth ftie algebraic difference of the two readings. However, since refraction acts in the vertical plane, and affects the vertical diameter of the sun, the amount of course depending on the altitude, it is preferable at low altitudes to use the horizontal diameter in finding the index 318 NAUTICAL ASTRONOMY correction; but, as the difference of the refractions for the upper and lower limbs of the sun, for altitudes over 30 , amounts to only a few seconds of arc, there is no practical advantage in using the horizontal diameter at the higher altitudes. 151. Using a sextant. To observe at sea an altitude of the sun, in other words, the angular distance along a vertical circle from the sun to the horizon: adjust the telescope to distinct vision by looking through the tele- scope at the horizon, moving the eye piece in or out till the horizon is clearly and distinctly seen, then screw it into its collar; see the instrument in adjustment and ascertain the index correction; put down the necessary shade glasses before the index glass, place the index bar near the zero of the limb, and see the tangent screw in mid position ; hold the instrument in the right hand by the handle so that its plane shall be vertical, and direct the line of sight 'to a point of the horizon directly below the sun. Now move the index bar with the left hand, and, if the sextant is held properly when its reading is near the altitude of the sun, its reflected image will be seen to descend. Make an approximate contact with the horizon and clamp the index bar. Now rotate the sex- tant slightly around the line of sight as an axis, making the reflected image skirt along the horizon; and, by means of the tangent screw, find one point at which the sun is just tangent to the horizon ; the reading of the sextant at that time is the altitude of the sun's lower limb. Just before the altitude is taken tell the assistant to " stand by," and at the instant when the altitude is taken, say " mark." * The assistant notes the seconds, minutes, and hours of his watch, recording opposite the time the degrees, minutes, and seconds of the altitude. At the time "mark/' the sun should be at the lowest point of its arc, just tangent to the horizon and in the center of the *An experienced observer should be able to note the time of his own observations. OBSERVING ALTITUDES 319 field of view. If by inclining the sextant, the sun is moved from this center to points nearer the plane of the instrument, or farther off, the angle, as read on the arc, will be too great. If there is much glare around the horizon, as is frequently the case when the sun is observed, especially at low altitudes, shade glasses should be turned down in front of the horizon glass as may be necessary. The amount of light and, hence, the brightness of the reflected object can be varied by moving the telescope towards or from the plane of the sextant. To observe the altitude of a star. In observing a star, the observer can use the inverting telescope, which has greater magnifying power than the ordinary direct one, or he can use the plain tube. Place the zero of the vernier on the zero of the limb, look at the star, hold the instrument vertical, move the index bar outward, keeping the reflected image in sight till the image of the star is just below the horizon; clamp, and whilst rotating the instrument, use the tangent screw and find the lowest point of the swing just on the hori- zon. It is advisable not to use any telescope or tube until able to observe well without it. In bringing down and ob- serving a star with a tube or the unassisted eye, keep both eyes open. This method is essential to avoid bringing down the wrong star, and it might be used for the sun with begin- ners, though it would be very trying on the eye. Sometimes when latitude is approximately known, and an observation of a star on the meridian is to be made, the instru- ment can be advantageously set to the approximate altitude. An observer can determine his personal error in measuring altitudes of stars by taking several altitudes of Polaris for latitude at a place whose latitude is accurately known. To measure the angle between two visible objects. Direct the line of sight (or the telescope which should be and can be easily used after a little practice) toward the left hand object, if both are nearly in the same horizontal plane, or 320 NAUTICAL ASTRONOMY toward the lower one, if one is much above the other. With this object in direct view, through the plain part of the hori- zon glass, move the index arm until the image of the other object, after reflection by the index glass, is seen in the sil- vered portion of the horizon glass. Having made a partial contact of the two images, clamp the instrument, screw in the telescope (if not already in its collar), perfect the contact with the tangent screw, and then read the limb. If, for any reason, it should be desirable to point the tele- scope to the right hand object, hold the sextant upside down, with the handle in the left hand and above. 152. Care of sextant. Keep your sextant in your own hands, or in its box which should not be put on a table from which it may be thrown off, nor in a roomy drawer wherein it may slide. Do not allow any one else to use it. Never put it away damp, as the moisture will surely cloud the mirrors and rust the metal. Wipe off the mirrors and arc with chamois or silk, but permit no polishing of the arc of limb or vernier. In adjusting the instrument when screws work against each other, be sure to loosen one as the other is tightened. When in adjustment, do not tinker with the screws even to remove an index correction, which, if small, can be allowed for. Keep tangent screw in mid position. Keep the arc clean by occasionally applying a drop of ammo- nia, and do not use oil except on screw threads. 153. Resilvering mirrors. It often happens that mirrors are injured by dampness or other causes, especially when doing hydrographic work, and then they require resilvering, which may be done in the following way : Take an unbroken piece of tin foil about one-quarter of an inch larger in all dimensions than the mirror to be resilvered, lay it on the clean surface of a pane of glass about five inches square, smooth out the foil, being careful not to tear it, put a drop of mercury on the foil spreading it carefully with the THE ARTIFICIAL HORIZON 321 finger over the surface, put on another drop and repeat the operation, and continue the process till the surface is fluid, being very careful that no mercury gets under the tin foil. Lay on the supporting glass a piece of tissue paper so that its edge shall cover the edge of the foil; having cleaned the glass to be silvered, lay it on the tissue paper, and transfer it slowly and carefully to the mercury surface, keeping a gentle pressure on the glass to prevent the formation of bubbles. FIG. 97a. ARTIFICIAL HORIZON. Place the mirror face downward and slightly inclined, to allow any surplus mercury to run off, and let it remain so till the following day, when the tin foil should be trimmed off flush with the edge of the mirror, and a coating of varnish made from spirits of wine and red sealing wax applied. 154. The artificial horizon (see Fig. 97a). This consists of a small shallow basin about 8 inches long, 4 inches wide, and f inch deep, containing mercury or any other fluid whose surface will reflect a heavenly body. The surface must 322 NAUTICAL ASTRONOMY be horizontal, and free from products of oxidation or other matter diminishing the reflecting power. The basin or receptacle, made of wood or iron, is covered by a roof consisting of two pieces of plate glass in a frame to protect the surface of the mercury from dust or wind. An iron bottle is furnished to contain the mercury when not in use ; this bottle is provided with a screw stopper and a funnel to prevent loss of mercury in handling ; these articles complete the artificial horizon outfit. In the absence of mercury, molasses or oil may be used; but, with oil, the receptacle or basin must be blackened on the inside. Reflecting horizons of black glass, plane and accurately ground, made level by levelling screws, are sometimes used, though not recommended. Care of and preparation of artificial horizons for use. The artificial horizon finds its principal use on shore in rating chronometers, and then should be at its best and used under the most favorable conditions. The surface of the mercury must be clean and free from dust and the roof perfectly dry. Scum and impurities may be removed from mercury by gently drawing over its surface the straight edge of a piece of blotter cut to the length of the basin, pressing it below the surface of the mercury, and in- clining it so that it may act like a scoop. If the mercury is alloyed, wash it with sulphuric acid, then with water, and filter it through muslin. Before pouring the mercury into the basin, see the basin well cleaned and dry, remove the funnel and stopper, and screw on the funnel, so that, by passing through a greatly con- tracted passage, the mercury may be cleansed. Place a finger over the opening, shake up the bottle, then invert it, and let any scum rise to the surface. Hold the bottle inverted over the basin, remove the finger, and let the mercury run into it. THE ARTIFICIAL rfoRizoN 323 When the basin is full, put the finger over the aperture and reverse the bottle; it is better to do this before all the mer- cury is out, or nearly so, in order to keep back any scum or impurities. The roof should be placed over the basin for a few moments to allow any moisture in the imprisoned air to be deposited on the glass surface of the roof, which is then lifted, wiped off, and replaced. A piece of cloth for the basin to rest on, and large enough to receive the edges of the roof, will keep out moisture. A roof should be used whose glass has no prismatic effect, but this can be eliminated, where it exists, by reversing the FIG. 97b. roof in each set of observations. However, in observing stars on both sides of the zenith, since the mean of results will be taken, and the prismatic effect is of an opposite sign in each case, the observer should keep the same end of the roof towards himself in each set of observations. Advantages of the artificial horizon. By using an arti- ficial horizon, and halving the angle, the errors of observation, whether of instrument or observer, are also halved ; and the correction for dip, which depends on both the height of eye and refraction, is obviated, as the artificial horizon furnishes a horizontal plane. Its use, however, is limited to shore observations. 324 NAUTICAL ASTRONOMY Theory of the artificial horizon (Fig. 97b). A ray from B is reflected from the basin H to the eye, making BHH* = Z.EHH"=Z.H'HB'. A ray from B is also reflected by the sextant mirrors, making the sextant image coincident with the basin image. Now ERE' HEB + Z, HBE, but the body B is so far off that the ray BR is practically parallel to the ray BM, so that the angle ERE' L BEH angle read from the arc 2BHH', or twice the altitude of B. To take an observation of the sun, using an artificial hori- zon.-r-Select an observation spot so that the basin may be evenly placed on a solid foundation, in a sheltered position undisturbed by breezes, or any movements or jars in the vicinity which might ripple its surface. In case equal alti- tudes are to be observed, the spot should be so selected that, if the sun is observed at one altitude on one side of the me- ridian, its view may not be shut out by houses,,trees, hill tops, or other obstructions, when at the same altitude on the other side of the meridian. Clean the basin and put it with its length nearly in line with the direction of the sun, but a little in advance ; pour the mercury into the basin and see its surface cleared of scum and impurities. Wipe the glass of the roof and cover the basin. Determine the I. C. and put the tangent screw in mid position. The observer should sit on a low stool or on the ground, with his back supported, if possible ; assuming the most com- fortable position possible under the circumstances, so as not to tire himself; and, at the same time, so placing his eye that he may see the image of the sun reflected from the center of the mercury. Turn down the necessary shade glasses before both the index and horizon glasses, and, without putting in the tele- scope, direct the line of sight to the sun, and bring it down till the lower limb of the image reflected by the mirrors over- USE OF ARTIFICIAL HORIZON 325 laps the upper limb of the image reflected by the mercury. Screw in the telescope with a colored cap on the eye piece, throw back the colored shade glasses, and proceed to observe the altitude. The sextant reading corrected for I. C. gives double the apparent altitude of the limb observed. Half the result, corrected for S. D. (-(- for the lower limb, for the upper limb), parallax, and refraction gives the true alti- tude of the center. If observing in the forenoon, the suns will separate; have the time marker " stand by " and as they separate, the lower limb of the apparent sextant image being just tangent to the upper limb of the horizon image, say " mark." The assistant notes the time, records it, and also the angle. It is usual to take the altitudes at equal intervals in arc, setting the sextant at the next division after one observation, and waiting for tangency or contact. The interval should be sufficient to permit care and accuracy in reading and ob- serving. If not afraid of the prismatic effect of the shades, different colored shade glasses may be used before the index and hori- zon glasses, giving different colors to the two images of the sun, and making it easier to distinguish them. If used, these shade glasses give sufficient protection to the eye, and the colored cap of the telescope is not used. Determining the limb observed. If immediately after con- tact the two images of the sun were observed to separate in the forenoon, or close in on each other in the afternoon, then the limb observed was the lower limb ; otherwise, the observed limb was the upper limb. To take an observation of a star, using an artificial hori- zon. The observer places himself so as to see the image of the star reflected in the mercury of the basin ; this image will seem as far below the surface of the mercury as the real star seems above it. 326 NAUTICAL ASTRONOMY Before screwing in the tube or the inverting telescope, place the zero of the vernier on the zero of the limb, direct the line of sight to the star, both eyes open, keep the plane of the sex- tant vertical, and move the index bar, keeping the star's reflected image in sight, till the image reflected by the mir- rors (the sextant image) is in coincidence with that reflected by the mercury and seen directly through the center of the tele- scope collar and the horizon glass. The sextant reading cor- rected for I. C. gives double the star's apparent altitude. Half the result, or the apparent altitude, corrected for refraction, gives the true altitude of the star. If intending to use the tube or inverting telescope, screw it in as soon as the star has been brought down, and proceed with the observations, saying "mark" to the assistant when the two images are in coincidence. CHAPTER X. CHRONOMETERS AND TORPEDO-BOAT WATCHES. STOP AND COMPARING WATCHES. CHRONOMETERS. 155. Definition. The term chronometer is applied to a portable timepiece of superior workmanship, furnished with special mechanism consisting of compensation balance, bal- ance spring, and escapement, so constructed as to obviate changes in its rate due to expansion or contraction of its mechanism, through effect of heat or cold. The term chro- nometer, however, is more generally applied to one adapted for use on board ship.' A marine chronometer should beat half seconds. Its special function is to furnish the time of the prime meridian, almost universally taken as that of Greenwich. Mean or sidereal chronometer. A chronometer may be regulated to keep mean or sidereal time ; if to keep mean time, it is called a mean time chronometer and its units are those of mean solar time; if to keep sidereal time, it is called a sidereal chronometer and its units are those of sidereal time. (See Art. 171.) The mean time day at any place commences when the mean sun is on the upper branch of the meridian at that place, and the theory of a mean time chronometer, keeping the time of that place, is that it shows O h O m s when the mean sun is on the upper branch of the meridian. Practically every chronometer has an error and a daily 328 NAUTICAL ASTRONOMY rate, gaining or losing, so that in order to have a chronometer regulated to a local or to Greenwich mean time, its error on that time must be known, and also its daily rate, i. e., the daily gain or loss. Both are plus, if the chronometer is fast CHRONOMETER. and gaining; minus, if slow and losing. It is customary, however, to use the error as a correction, which is the quan- tity to be applied to a chronometer reading to reduce it to the correct time at that instant. This is plus when the chronometer is slow, and the daily change is plus when the chronometer is losing. If the correction of a given mean CHRONOMETERS 329 time chronometer is on G. M. T., the resulting time will be GL M. T. The letters C. C. usually represent this correction. Since a sidereal day at a place begins when the vernal equi- nox is on the upper branch of the meridian at that place, the theory of the sidereal chronometer is that it should show Qh Qm QS when the vernal equinox is so situated, and its error on any sidereal time (the amount it is fast or slow on that time) and its daily rate must be known in order to say the chronometer is regulated to that time. Bating. The question of rating a chronometer will be considered elsewhere in this work. Number for safety. The U. S. Naval vessels carry at least three good chronometers. In this way, each may be a check on the other, and irregularity on the part of one will be made evident by a comparison of 2d differences as recorded each day, it being assumed that all will not have the same kind of daily rate. There is no particular advantage in having two chronome- ters, except for the possibility of injury to one, for, if they begin to differ widely, after a period of regularity, it would be difficult to determine which is the good or faulty one, in the absence of means to check their indications. Stowage on board. The chronometer, swung in gimbals in its own case, is placed in the chronometer box as soon as re- ceived on board. This box has as many compartments as there are chronometers, and each compartment is well padded with hair and lined with baize cloth to prevent sudden changes of temperature and to reduce shocks and tremors as much as possible. The box is surrounded by a strong casing suffi- ciently large to admit of a clear space of at least two inches all around. Before reception of the chronometers o*n board, the box and its casing should be secured to a solid block of wood, bolted to the beams of the deck, in a part of the ship that is to be the permanent abode of the instruments; as low 330 NAUTICAL ASTRONOMY down in the ship as possible where the temperature may be equable, and where gun-fire may have its least effect; amid- ships and so as near the center of motion as convenient, suffi- ciently far forward as not to be affected by vibrations of the screw; removed from influences of masses of iron, especially vertical iron, dynamos, electric wiring, or magnets of any description. The chronometers, when stowed, should be al- lowed to swing freely in the gimbals, should constantly occupy the same relative position, with the Xll-hour mark towards the same part of the ship, and should not be removed except for necessity. Designation. Instead of designating chronometers by their numbers or maker's name, it is customary to denote them by the letters of the alphabet, A, B, C, D, etc. The standard, called A, made by a maker of well-known reputation, should have a first-class record, a clear, distinct beat to half seconds, and a small stable rate, the stability of rate being of more importance than its amount. It should occupy a central position among the others. Of course, a record should be made in the chronometer journal of the number and maker of chronometers to which the above letters may be assigned. 156. A maximum and minimum thermometer should be kept in the case with the chronometers, and recorded at the time of winding ; it should be kept in a vertical or nearly ver- tical position. A small horse-shoe magnet is used to reset the indices. This magnet should be kept with keeper on outside the box, and never brought near the, chronometers. Should the mercury column become divided, move the indices well away from the column, then holding the thermometer verti- cal, bulb end up, give it a quick movement downward, bring- ing it to an abrupt rest. Repeat this process, if necessary, till the break in the column disappears, being careful to keep the indices clear of the mercury. 157. Winding. Some chronometers are made to run 8 CHRONOMETERS 331 days and are wound but once a week; however, it is believed they would run better if wound every day. Our service chronometers will run for 56 hours, and should be wound every 24 hours, at regular and stated times, in order to bring into play each day the same part of the spring, and thereby contribute to regularity and stability of rate. The time for winding chronometers depends on the commanding officer, and may be just before 8 a. m. or just before noon; at all events, they should be reported wound to the captain at 8 a. m. or noon, as the case may be. To wind. Place the left hand over the face, turning slowly and carefully the chronometer bowl in its gimbals; then, holding the bowl firmly in the left hand, rotate the valve with forefinger or thumb of left hand, according to direction of rotation, or as most convenient, until the key hole is uncov- ered; place the key in position with right hand, wind slowly to the left the required number of half turns, usually 7 or 8, counting the half turns as the chronometer is wound, to avoid too much force at the last one, though be careful to wind to a full stop. After winding, remove the key, and the valve should close automatically; if it does not, close it to keep out dust and dampness; then the chronometer is eased to its original position, the Xll-hour mark pointing as before. It is well to note after winding that the indicator on face of dial, which shows the number of hours since winding, then reads O h . After winding, compare chronometers, fill in the columns of chronometer comparison book, reset maximum and minimum thermometer, and stow the magnet away so as not to be any- where near the chronometers. \ When run down. When run down, a chronometer will not start on winding; however, the balance wheel can be set in vibration by a quick, but not a violent, horizontal circular motion. ,)32 NAUTICAL ASTKONOMY Resetting hands. Should it be desirable to reset the hands, unscrew the glass cover from the face, place the winding key on the projecting stem, and turn the hands in one direction only, which should always be ahead. If practicable to do so, avoid altering the position of the hands by starting the chro- nometer at the time indicated. Never touch the hands nor turn them backwards. 158. Comparison. The following method of comparing is generally followed, it being better that a single observer should alone make the comparison. The observer determines upon a certain time by standard for the comparison, and, holding the comparison book in the left hand, enters that time for A. Opening the glass top of A, but only the wooden top of B, so as not to hear the ticking of the latter, he takes a beat from A, say 5 seconds, before the second hand is to reach the comparing mark. Casting his eye upon the dial plate of B, listening intently to the beats of A, he counts by ear the beats which elapse before the second hand reaches the comparing mark. At that instant he reads the seconds, minutes, and hours of B. A second comparison will verify the first, or indicate an error. Enter the reading of B below that of A, note the difference and 2d differences in their proper places in the comparison book. Compare C, and all the other chronometers in the same way with A, recording carefully the hours, minutes, and seconds of each comparison in the proper places. It is desirable to consider the standard fast of the other chronometers as well as fast on the Greenwich mean time, whatever the actual indications of A may be, so that by sub- tracting the comparison of any chronometer with the stand- ard from the error of the standard (adding 12 hours to the latter, if necessary), the error of that particular chronometer, fast on Greenwich time, is obtained. To illustrate, suppose A is fast of G. M. T. O h 10 m 10 s and A B = ll h 10 m 50 s . CHRONOMETERS The first may be written thus A G. M. T. - O h 10 m 10 s and A B =11 10 50 Therefore, B is fast of G. M. T. 59 m 20 s , or B G. M. T. = ; O h 59 m ^n Comparison of a mean and sidereal time chronometer. hi the comparison of a mean time chronometer with a sidereal time chronometer, the difference between the two can be ob- tained within a very small fraction of a second by watching for the coincidence of their beats. Since the second of sider- eal time is shorter than that of mean time, or 1 s of sidereal time =: O s .99?27 mean time, the sidereal chronometer gains on the mean time chronometer O s .00273 in 1 s , and therefore gains one beat, or s . 5, 'in 183 seconds. Hence, once in about every three minutes the two chronometers beat together, and, as the observer, when watching one, and counting the beats of the other, fails to note any difference in the beats, he re- cords the corresponding half seconds of the two chronometers and notes the minutes and hours of each. 159. Cleaning and oiling. Chronometers should be cleaned and oiled every three or four years; when the oil becomes dried, or thickened, the rate will be irregular. Besides, the mechanism should be cleaned occasionally to prevent or re- move any rust that may follow exposure to dampness. 160. Transportation. Whenever chronometers are to be transported, even for short distances, clamp the catch of gim- bal ring, and carry them in their transportation boxes, or in a handkerchief which is passed under the box, through handles, and square knotted on top. Great care must be taken to give them no shock or circular movement; when a chronometer is carried in a boat, it should be kept suspended by the hand. Chronometers may be transported while run- ning in the following way : remove the bowl from the gimbal ring, being careful not to let it fall or be jarred when un- 334 NAUTICAL ASTRONOMY screwing the pivot screw of the ring; wrap the bowls in soft paper, place them, dials up, in circular paste-board boxes with corrugated paste-board packing (if these boxes are not on hand, wrap bowls in cotton) ; place these boxes, tops up, wrapped in cotton, or hair, in a large rectangular basket, as far from the center as possible; put down and secure top of basket, and carry it by its handles, protecting it from jars or jerky movements. For transportation to a distance. Chronometers should, whenever possible, be sent by hand; but if necessary to send by express, as when sending to a long distance, and also for repairs, they should be allowed to run down, be dismounted, and the balance stayed with clean cork at diametrically oppo- site points. Place the gimbals in the bottom of the case, over which put a pad of cotton wrapped in soft paper to form a seating for the bowl. The chronometer bowl is wound around with rolls of cotton in paper, seated on the cotton pad in the bottom of the case, and then covered by a similar pad. See the case tightly packed and closed, put into its transport- ing box, which is then securely closed and itself wrapped in a thick padded covering, and marked " Delicate Instrument, Carry by Strap." To stay the balance with cork. Unscrew the bezel, leav- ing the chronometer movement free in the bowl. Turn over the bowl on the left hand, supporting it by the fingers around the dial edge; lift the bowl, uncovering the movement; stop the balance by touching it very lightly with a dry piece of paper, and stay it by two dry and clean strips of cork placed gently under the outer rim at points diametrically opposite, so as not to cover oil holes or touch any other parts 1 of the mechanism. Replace the movement in the bowl and screw on the bezel. 161. Effect of change of temperature. When marked changes of climate are encountered, as when on a long voyage, HARTNUP'S LAW 335 the chronometer rate will change, and this change is univer- sally recognized as due to the changes of the temperature ex- perienced by the chronometer; Hartnup's law, governing the peculiarities developed under such circumstances, as usually stated, reads as follows : " Every chronometer goes fastest in some certain temperature, called the temperature of compen- sation, and this can be calculated for each chronometer from rates determined in three fixed temperatures. As the tem- perature varies either side of this temperature of compensa- tion, the chronometer goes slower, and the rate varies as the square of this variation in degrees." This should be the mean temperature to which chronome- ters are subjected; and considering their actual use, for navy chronometers, it is approximately 69 F. General equation. If be the temperature of compensa- tion, r the chronometer rate at that temperature, z the tem- perature constant or change of rate for one degree of tempera- ture either side of 0, 0^ the temperature for which the cor- responding rate is r x , then the effect of temperature alone is expressed by the general equation : r 1 = r + z(0 O 2 - This equation serves for temperatures between 45 F. and 90 F., but outside these limits, the change of rate is propor- tional to a higher power than the square. The quantities involved in this equation differ for every chronometer. For the same chronometer r will vary, but so long as the temperature compensation is maintained the same, that is, so long as the compensating balance is not changed, 8 and z will remain practically constant. The equation is that of a parabola. The requirements of the equation are satisfied by the general equation of a para- bola, 336 NAUTICAL ASTRONOMY Taking the axis of X in the line representing the tempera- ture of compensation, if the ordinate y is the variation of temperature in degrees from the temperature of compensation, the abscissa x is the change in rate for this variation, then y and x = r^ r; and for the value of y = unity, After having found 9, z, and r, any number of points on the parabola may be found, by solving the general equation, assuming at intervals of 5, and finding the correspond- ing r . To find 6, z, and r. At the U. S. Naval Observatory all chronometers are subjected to two tests in the temperature room, the variation being from 90 F. to 50 F. in the first; and from 50 F. to 90 F. .in the second. The chronometers are exposed in each test for one week to certain predetermined temperatures under certain fixed rules; the errors being de- termined at the beginning and end of each week, the daily rates for the several temperatures are obtained from them. The data found at the mean temperatures of 55, 70, and 85 F. are used in the general formula for the determina- tion of 0, z, and r, and from these values, a curve is constructed for each chronometer. This curve with rates plotted up to date, known as Form No. 1, or "Bate Curve for Tempera- ture at Observatory/' accompanies a chronometer when issued. The same record sheet contains Form No. 2, or " Rate Curve and Observations on Board Ship," and Form No. 3, or " Ob- served Errors and Mean Daily Rates and Temperatures." These, with Form No. 4, " Record of Daily Comparisons and Memoranda," should receive the navigator's close attention. 162. Sea temperature curve. It is not at all likely that the chronometer will have the same rate on board as at the observatory, though a new curve, if determined, may prove very similar to that found at the observatory ; therefore, when a rate has been determined on board, and is m> ^ j anuar y 6, find the Greenwich time. d h m s Local astronomical time Jan. 5, 20 48 17 Longitude in time West + 4 41 28 Greenwich astronomical time Jan. 6, 1 29 45 Ex. 82. In Long. 103 58' E., the local time being Febru- ary 15, 7 h 35 m 40 s a. m., find the Greenwich time. d h m s Local astronomical time Feb. 14, 19 85 40 Longitude in time East ( ) 6 55 52 Greenwich astronomical time Feb. 14, 12 39 48 Ex. 83. In Long. 135 15' E., the local time being Jan- uary 20, 5 h 10 m 30 s p. m., find the Greenwich time. d h m s Local astronomical time Jan. 20, 5 10 30 Longitude in time East ( ) 9 01 00 Greenwich astronomical time Jan. 19, 20 09 30 By pursuing a course just the reverse of the above, sub- tracting West longitudes from/ and adding East longitudes to given Greenwich times expressed astronomically, the local To FIND GREENWICH I)ATE 355 astronomical times may be found and converted into local civil times. 180. To find the Greenwich date and mean time from the time data of an observation. Navigators are supplied with chronometers from which to obtain the Greenwich mean time of their observations, and for this time the various ele- ments involved, such as declination, right ascension, etc., are taken from the Nautical Almanac. Having the error on G. M. T. and the daily rate before leaving port, the error of the chronometer at any given in- stant can be found. To the watch time of an observation add the C W, or the difference between the chronometer and watch, obtained by comparison just before or after the observation, to get the corresponding chronometer time. To this chronometer time, apply its error on Greenwich mean time, adding if the chro- nometer is slow, subtracting if it is fast. The result, or the result plus 12 hours, will be the G. M. T. This ambiguity in the number of hours arises from the fact that chronometer dial plates are graduated like watches from hours to 12 hours, instead of from hours to 24 hours, and it is neces- sary to know whether the chronometer is a. m. or p. m. in order to fix the true Greenwich time and date. However, there need be no trouble or ambiguity, if approximate Green- wich time and date are gotten from the approximate local time by the rules just given in Art. 179, and compared with the result obtained as above from the watch time and chro- nometer comparison. The following examples will show how to decide whether the G. M. T., in a given case, is the corrected chronometer face or this quantity plus 12 hours. 356 NAUTICAL ASTRONOMY Ex. 84- October 31, about 5 a. m., the time data of an observation were, W. 7 h 25 m 12 s , C W l h 44 m 17 s , chronom- eter fast on G. M. T, 27 m 31 s , Long. 8 h E. Find G. M. T. !h m a W. 7 25 12 C W 1 44 17 C. 9 09 29 C. C. ( ) 27 31 Greenwich ast. time Oct. 30, 9 approx. I [G. M. T. Oct. 30, 8 41 58 These results are so close as to remove all doubt. Ex. 85. November 17, about 10 h 53 m a. m., in Long. 2 h 20 m 10 s E., the time data of an observation were W. 4 h 15 m 27 s , C W 4 h 07 m 20 s , chronometer slow on G. M. T. 10 m 15 s . Find G. M. T. Civil time Nov. 17 d h m s h m s W. 4 15 27 C-W 4 07 20 (a.m.) j 53 a PP rOX ' Local ast. time Nov. 16, 22 53 approx. Long. East (-) 2 20 10 | 8 23 47 Greenwich ast. ) Noy 16 2 32 50anDrox ' ' ' time. J I G. M. T. Nov. 16, 20 33 02 It will be noticed that in the above example 12 hours must be added to the chronometer time and the date of the pre- vious day taken to make the two Greenwich times and dates more nearly agree. It may sometimes be the case that the watch by which ob- servations are taken is not regulated to local time, in fact, may be far out, and that the approximate time of observation is known no closer than by the words a. m. or p. m. (as in the case of set examples). But the result may be reasoned out correctly thus : Apply the C W and C. C. to the W. T. of observation. The corrected chronometer reading, or this reading plus 12 hours, will be the required G. M. T. To* determine which proceed thus : Express the given ap- proximate local civil time astronomically and find the as- To FIND GREENWICH DATE 357 tronomical date and hour-limits between which the true local time should lie, these limits being determined by the words a. m. or p. m. ; thus to illustrate, for Sept, 15, a. m., the limits of astronomical date and time would be Sept, 1-1, 12 h to 24 h , and for Oct. 10, p. m., they would be Oct. 10, O h to 12 h . Apply the longitude (adding if West, subtracting if East) to the local astronomical date and hour-limits, and the result will be the Greenwich astronomical date and hour-limits be- tween which the true Greenwich time must lie. " Then, if the corrected chronometer reading falls between the Greenwich limits, it is the correct G. M. T.; if not, add 12 h to the corrected chronometer reading and the result, fall- ing between the limits, will be the G. M. T., the date in either case being that of which the hours are a part. Ex. 86. November 15, a. m. time, in Long. 10 h W., W. T. of an observation was 7 h 50 m 30 s , C W 6 h 10 m 20 s , C. C. + 4 m 20 s . Find the G. M. T. h m 8 W. 7 50 30 C-W 6 10 20 C. 2 00 50- C. C. +4 20 Corrected chro. J.2 05 10 reading. Approximate local civil time Nov. 15, a. m. Local astronomical date and hour limits. Longitude West +'10 10 Greenwich astronomical date and hour limits. Nov. 14 d 22h to 15'' 10 The corrected chronometer reading falls between the limits, the hours are of the 15th, .-. G. M. T. Is Nov. 15, 2" 05 m 10 8 . Ex. 86 (a). July 11, p. m. time, in Long. 150 E., the time data of an observation were as follows: W. = 9 h 12 m 17 s , C W 8 h 15 m 14 s , C. C. + 10 m 20 s . Find G. M. T. h m s W. 9 12 17* C W 8 15 14 C. 5 27 31 C. C. + 10 20 Corrected chro. V 5 37 51 reading. Approximate local civil time July 11, p. m. Local astronomical date and hour limits. Longitude East Greenwich astronomical date and hour limits, } July al } July ll d O h to 10 12> 10 U h to ll d 2 1 ' 358 NAUTICAL ASTRONOMY The corrected chronometer reading does not fall between the above limits, so adding 12 h to it gives l? h 37 m 51 s . which falls between the limits; the hours are a part of the 10th; and .'. G. M. T. = July 10, 17 h 37 m 51 s . In case the local time is given as an exact time and the longitude as merely East or West, the same method holds good. Ex. S7. Jan. 10, 8 a. m., in West longitude, given the fol- lowing data: W. T. 9 h 40 m 30 s , C W 5 h 20 ra 20 s , C. C. + 5 m 10 s . Find G. M. T. W. c-w h m s 9 40 30 5 20 20 C 3 00 50 C. C. + 5 10 Corrected "V chro. l3 06 00 reading, j Local civil time Jan. 10, 8 a. m. Local astronomical time Jan. 9 rt 20 h -Longitude West + to Greenwich astronomical ) date and hour limits, j Jan. 9> to 10' 1 8> The corrected chronometer reading falls between the limits, the hours being of the 10th. .'.G. M. T. Jan. 10. 3 h 06 m 00 s . If both time and longitude are given as within a twelve-hour range ; i. e., time only as a. m. or p. m. and longitude only as E. or W., the limits of the approximate Greenwich astronom- ical time will be 24 hours apart and two solutions will result. Ex. 87 (a). Aug. 10, a. m. time, in West Long., given the following data : W. T. 8 h 00 m 10 s , C W 6 h 40 m 20 s , C. C. + 5 m 30 s . Find G. M. T. h m s 8 00 10 6 40 20 W. C W C C. C. Corrected ) chro. I 2 46 00 reading, j 2 40 30 + 5 30 Approximate local civil tinle Aug. 10, a. m. dale Longitude West Greenwich astronomical date and hour limits + Oi> 12> O h 12>> al | . . } S " 12HolO d GAIN OR Loss OF TIME 350 Here both the corrected chronometer reading and that read- ing -f 12 hours fall between the limits, and hence we have a double solution : G. M. T. = Aug. 9, 14 h 46 m 00 s , = Aug. 10, 2 h 46 ra 00 s . Examples for practice. Find G. M. T., given, W. T. of Obs. C W C. C. Approx. Local date. Long. Answers. h m s h m s m s h m s (88) 3 23 43 6 11 33 ( ) 7 18 April 23, a. m. 90 W April 22, 21 27 58 (89) 7 53.26 4 38 56 ( ) 9 27.6 Jan. 19, a. m. 30 W Jan. 19, 22 54.4 (90) 11 49 33 3 59 30 ( )30 22 Oct. 22, a. m. 120 E Oct. 21, 15 18 41 (91) 5 20 21 3 16 24 ( )25 21 Nov. 5, p. m. 135 E Nov. 4, 20 11 24 (03) 7 35 10 10 20 17 + 10 04 Nov. 9, 8 a. m. 30 E Nov. 8, 18 05 31 181. Gain or loss of time with change of position. Cross- ing the 180th meridian. It has been shown that local noon at any meridian is when the sun is on the upper branch of that meridian, that at all places East of that meridian at that in- stant it is past noon, or time is fast of that of the given merid- ian ; at all places West of that meridian it is not yet noon, or time is slow of that of the given meridian. Hence it is evident that if a navigator travels East, carrying a watch regulated to the time of the meridian departed from, and if he desires to set the watch to the time of a meridian to the eastward, he must set it ahead at the rate of 1 hour for 15 change of longitude, or 24 hours for every 360 ; in other words, in going east- ward around the world, or through 360 of longitude meas- ured in an easterly direction, he gains a day in his reckoning of time. 360 NAUTICAL ASTRONOMY In the same way, if sailing westward around the world, or through 360 of longitude measured in a westerly direc- tion, he loses a day in his reckoning of time. So that if he leaves the given meridian and goes around to the eastward, keeping his time regulated to each succes- sive local meridian, his reckoning of time on return to his point of departure will be one day ahead of the local reckon- ing ; in other words, he would think it, say, Thursday when in reality it was Wednesday. Had he gone around to the westward, he would have logged his return as Tuesday, if the day in reality was a Wednesday. To avoid such misconceptions and to keep accurate run of dates, when crossing the meridian of 180, going eastward, repeat one day; when crossing it, going westward, drop one day from the calendar; at the same time changing the name of the longitude. Illustration. Suppose a ship, going eastward, crosses the 180 meridian at local apparent noon, April 10; find the corresponding Greenwich time and date. Then, from this result, considering the longitude as of opposite name to that first used, find the local time and date. h m a Local apparent time 00 00 00 April 10 Longitude 180 East 12 00 00 East Greenwich apparent time 12 00 00 April 9 Longitude 180 West 12 00 00 West Local apparent time 00 00 00 April 9 In other words, going eastward, and crossing the 180 ( meridian, repeat a day. 361 Suppose the conditions of the illustration to be as except the ship is going westward, and the chango from West to East longitude, then, h m a Local apparent time 00 00 00 April 10 Longitude 180 West 12 00 00 West Greenwich apparent time 12 00 00 April 10 Longitude 180 East 12 00 00 East Local apparent time 00 00 00 April 11 In other words, going westward and crossing the 180 ( meridian, omit one day from the calendar. CHAPTEE XIII. NAUTICAL ALMANAC AND SUBOKDINATE COMPUTA- TIONS. 182. The Ephemeris and Nautical Almanac* published by authority of Congress is subdivided into three general parts, but all the information required by navigators is contained on pages 2-177, 233-510, and 650-697 (edition of 1912). Pages 2-145 of Part I comprise what is called the Calendar giving the data under the heads of the several months; each month has assigned to it 12 pages numbered by the Roman numerals I to XII. In this book reference will be made only to the contents of those pages of use to the navigator. Page I gives for Greenwich apparent noon of each day the sun's apparent right ascension, declination, and equation of time, with hourly differences in adjoining columns. The hourly differences themselves are for the instant of apparent noon at Greenwich, and, when great accuracy is required, cor- rections should be made for second differences. The sun's semi-diameter and the sidereal time of the semi-diameter pass- ing the meridian are also given. The chief use of page I is for observations when the sun is on the meridian, as for lati- tude, in which case longitude is the Greenwich apparent time. Page II contains for Greenwich mean noon of each day the sun's right ascension, declination, equation of time, and side- real time of mean noon (R. A. M. O ) with their hourly dif- ferences. Where great accuracy is desired second differences should be used. Page IV contains the moon's semi-diameter and equatorial * A Nautical Almanac, also issued, contains the Calendar, the geocentric ephem- erides of the five visible planets, mean places of 200 fixed stars, a monthly list of stars for navigators, tables giving approximate times of meridian transit for certain stars, * and h of stars on P. V., a star map, and Tables I to VII of the " Ephemeris and Nautical Almanac." NAUTICAL ALMANAC 363 horizontal parallax for each mean noon and midnight of Greenwich mean time and the hourly changes of the horizontal parallax. The mean time of the moon's upper transit to tenths of a minute and the moon's age are given on this page. Pages V to XII contain the moon's right ascension and declination for each day and hour of Greenwich mean time with the differences for one minute opposite each hour. Pages 146-177 contain the right ascensions and declinations of the seven major planets for the instant of Greenwich mean noon; also their times of meridian passage at Greenwich to tenths of a minute. Pages 233-486 give the right ascension and declination of 825 principal stars, their mean places being on pp. 233-250. Signs. In the Nautical Almanac a + sign before the hourly difference of declination means the heavenly body is moving toward the North ; a sign, under like circumstances, means the body is moving toward the South ; a -f- sign before the declination of a planet or star means North, a sign means South ; in all other cases + means increasing, means diminishing numerically. In the examples of this book, declinations will be charac- terized by the letter N. when North, by letter S. when South ; and hourly differences by letter N. if the body is moving toward the North, otherwise by the letter S. 183. Greenwich time essential. Before using the Almanac, the correct Greenwich time must be obtained, as the elements likely to be used are tabulated for that time, except in the cases of the apparent places of fixed stars which are tabulated for Washington time ; so, when taking out the right ascension and declination of fixed stars for a given instant, first find the Washington time corresponding. Work from nearer noon. Good judgment tells us to work always from the nearer Greenwich noon, and, if the Green- wich time of a given date is greater than 12 hours, it is better 364 NAUTICAL ASTRONOMY to regard it as less than 12 hours, and a minus quantity of the next day. For example, if G. M. T. = 17*. 3 March 3, regard it as ( ) 6 h .7 March 4, and proceed accordingly, working backwards. 184. Second differences. Ordinarily first differences will do in taking out the various elements in such cases change in the element is regarded as proportional to the small intervals of time employed; but, if great precision is required, as in taking out the sun's declination and equation of time for use in equal altitudes for chronometer error, the reduction for second differences should be introduced, and the hourly differ- ence interpolated for the middle instant between the Almanac date and the given time, thus using the mean rate of change during the interval. However, it must not be forgotten that the quantities in the Almanac are approximate, given only to a certain decimal, and that it is useless to interpolate to a lower order than said decimal. Besides, at sea, where the time may be in error, excess of refinement in making corrections will not contribute to accuracy. As all the examples of this work are meant to be practical, second differences will be used only in the cases of the sun's declination and equation of time as referred to above. Letting H^ be the hourly difference for the Greenwich noon preceding the given Greenwich time, H 2 be that for the following noon, t be the number of hours of Greenwich time after the first date, for which the value of the quantity is required, TT _ rr 4- then #! rb 2 cM X will be the mean hourly difference. In case t equals the hours of Greenwich time before the second date, and the value of the quantity is required for that instant, then HI ^F CM x ^r w ^ ^e ^ ne mean hourly difference. NAUTICAL ALMANAC 365 E Xf g$ f On April 2, 1905, the hourly difference of declina- tion of the sun at Greenwich apparent noon was X. 57".73. At the same time on April 3, it was N. 5 7". 51. Find the mean hourly rate at local apparent noon in Long. 75 W. April 2. Here ff 1 = 57". 73 57". 51 9 - ff, = -0".22 At local apparent noon in longitude 75 W. G. A. T. = t (in formula) = + 5 h , and__ = + ff l + x = N. 57". 73 + (".01) (2>>.5) = N. 57". 73 ".025 = N. 57". 705 Mean hourly difference = N. 57". 705. Ex. 94. On April 2, 1905, at Greenwich apparent noon the sun's H. D. of declination was N. 57".73; at the same time April 3, it was N. 57".51. Find the mean hourly rate and the correction of the declination for local apparent noon April 3, at a place in longitude 45 E. As before,^ = 57". 73 #., = 57". 51 0".01 "At local apparent noon in longitude 45 E. G. A. T. = t (in formula) = 3^ and - = I*. 5, - =N.57".51 = N. 57". 51 + ".015 = N. 57". 525, .Mean hourly rate = N. 57". 525. The correction = /JT 2 + H * ~ ^ x -M t = N. 57". 525 x ( 3 h ) = S. 172". 575 = S. 2' 52". 575. The mean hourly rate, when working forward, being H! H *^ A Hl X t y and the correction being -j H l H *~ Hl X [ t* 0. /& ( /&4: fy ) we see that the expression for the correction corresponds to that for uniformly accelerated or retarded motion in mechanics, V representing the initial velocity or change of the element; 366 NAUTICAL ASTRONOMY a, the acceleration or retardation, or the increase or decrease of the change per unit of time next smaller than that for which V is given; and S f the correction. This formula is general in its application, but it must be remembered that, if, as in the case of the sun, V is a difference for one hour, given in the Almanac for each day, and taken at the nearest Greenwich date, a will be -fa of the change in V for 24 hours, in other words, the hourly change of V ; and, if, as in the case of the moon, V is the difference for one minute given in the Nautical Almanac for each hour, a will be -g-V of the change in V for 60 minutes, or, in other words, the change in V for one minute of time. In case the given time is nearer to a subsequent date than a preceding Almanac date, the formula may be used, working backwards; V will be the quantity for the subsequent date in the Almanac and t the time before this date. Taking the first of the two preceding examples to find the correction, 8 = correction = j N. 57".73 + ( 0".01 X |) } 5 = K 288".525 = N. 4' 48".525. Taking the second one, we have 8 correction = J K 57".51 + ( 0".01 X ( )) i ( 3) = K 57".525 X ( 3) = S. 2' 52".575. 185. To find from the Almanac a certain element for a given mean time at a given place. (1) Find the Greenwich mean time corresponding to the local mean time, as previously explained. (2) Take from the Nautical Almanac, for the nearest mean time date preceding the given Greemvich mean time, the re- quired quantity, and the corresponding " difference for 1 hour'' or " difference for 1 minute," noting the name and sign NAUTICAL ALMANAC 367 of each. Multiply the "difference for 1 hour" by the hours and decimals of an hour, or the " difference for 1 minute " by the minutes and decimals of a minute of the remaining Green- wich mean time. Apply this quantity algebraically according to sign to the quantity first taken out. Or, take out the quan- tity for the nearest subsequent date, and the proper difference. Multiply this difference by the fraction of time from the given Greenwich date to the Almanac date, then subtract the product algebraically. To take out the R. A. M. O or " sidereal time of mean noon." Find what it is for the Greenwich mean noon pre- ceding the given Greenwich mean time ; at the bottom of the column will be found the difference for 1 hour = 9 s . 85 6 5. This, multiplied by the hours and decimals of an hour of the Greenwich mean time, will give the correction to be added to the quantity first taken out. Table III at the end of the Nau- tical Almanac for converting a mean solar into a sidereal time interval should be used for finding this correction. The quantities given in the Ephemeris for Washington mean time may be taken out in the same way, first finding the Washington mean time corresponding to the given local mean time. Ex. 95. For a L. M. T. January 8, 1905, 8 h 16 m 54 s a. m., in longitude 80 31' 30" W., find the sun' right ascension and declination, also semi-diameter, equation of time, and right ascension of mean sun, using first differences only. First find the Greenwich mean time. h in s Local astronomical mean time Jan. 1, 20 16 54 Longitude from Greenwich West + 5 22 06 Greenwich mean time Jan. 8, 1 39 00 Jan. 8, P.65 For mean time use page II. NOTE. All the data from the Nautical Almanac of 1905 necessary for the solution of examples in this book may be found on pages 729-738, 368 NAUTICAL ASTRONOMY To take out the sun's right ascension. Sun's R. A. H. D. h in a Jan. 8, at Greenwich mean noon 19 16 10.62 10 S .925 H. D. 10 8 .925 X l h .65 + 18.03 G. M. T. l h .65 Required R. A. of the sun 19 16 28.65 Corr. 18 S .03 To find the sun's declination. Sun's Dec. H. D. o / " Jan. 8, at Greenwich mean noon S 22 17 39.3 N 19".85 H. D. N 19".85 X l h .65 N 32.75 G. M. T. l h .65 Required declination of the sun S 22 17 06.55 Corr. N 32".75 To find the sun's semi-diameter. The change of sun's semi-diameter is so small, even in 24 hours, that it is tabulated only for Greenwich apparent noon on Page I. In actual practice it would only be taken out to the nearest second of arc. January 8, sun's S. D. = 16' 17".75 or 16' 18". To find the equation of time. Eq. of time. H. D. Jan. 8, at Greenwich mean noon 6 42.54 + l s .068 H. D. 1 8 .068 X l h .65 ' + 1.76 G. M. T. l h .6b Required equation of time 6 44.30 Corr. + 1 9 .76 ( ) to mean time. To find the right ascension of the mean sun. R. A. M. o H. D. h in a Jan. 8, at Greenwich mean noon 19 09 28.08 9 8 .8565 H. D. 9 8 .8565 X l h .65 (or using Table III) 16.26 G. M. T. l h .65 Required R. A. M. Q 19 09 44.34 Corr. 16 8 .26 NAUTICAL ALMANAC 369 Ex. 96. Find the right ascension, decimation, semi-diam- eter, and horizontal parallax of the moon for 1905, January 3, L. M. T. 10 h 10 m 06 s p. m. in longitude 45 East; also the right ascension and declination of the planet Jupiter. h m s Local astronomical mean time Jan. 3, 10 10 06 Longitude from Greenwich East, ( ) 3 00 00 Greenwich astronomical mean time Jan. 3, 7 10 06 = 7 h 10 m .l = 7M7 Moon's R. A. M. D. Jan. 3, at 7 hrs. of G. M. T. 17 12 25.60 2 S .3443 M. D. 2 S .3443 X 10 m .l = + 23.68 10 m .l Jan. 3, at 7 h 10 m 06 s of G. M. T. R. A. = 17 12 49.28 + 23 S .68 Moon's Declination. M. D. o / // Jan. 3, at 7 hrs, of G. M. T. S 17 56 59.3 S 3".156 M. D. 3".156 S X 10 m .l = S 31.88 10 m .l Jan. 3, at 7 h 10 m 06 s of G. M. T. Dec. = S 17 57 31.18 S 31".88 The moon's semi-diameter for G. M. T. 7 h 10 m 06 s Janu- ary 3. The moon's semi-diameter is tabulated only for noon and midnight; therefore, if the given G. M. T. is less than 12 hours, take from the noon column ; if G. M. T. is greater than 12 hours, take from the midnight column. In the former case, divide the difference between the semi-diameters at noon, and at midnight by 12, and multiply quotient by the hours and decimals of G. M. T. for the change. In the latter case, divide the difference between the semi-diameters at midnight and following noon by 12, and multiply quotient by hours and decimals in excess of 12 hours for the change. 370 NAUTICAL ASTRONOMY For moon's semi-diameter. i // i a Jan. 3, at noon S. D. 15 44.4 At midnight S. D. 15 40.6 Decrease in 7M7 = 2.3 At noon S. D. 15 44.4 S.D. at given G.M.T. = 15 42.1 Decrease in 12 h = 3.8 " l h = 0.32 71*17 2.29 Moon's horizontal parallax for G. M. T. 7 h 10 m 06 s January 3. Jan. 3, at noon H. P. 57 40 Diff. for 1 hour ( ) 1".13 Correction ( ) 8.1 G. M. T. 7M7 H. P. at given G. M. T. 57 31.9 Correction ( ) 8."10 It will be noticed that the moon's H. P. is taken out for noon or midnight, according as the G. M. T. is less or greater than 12 hours, and corrected by the difference for one hour, multiplied by the remaining hours and decimals of the Green- wich mean time. To find the right ascension of the planet Jupiter. Jupiter' sR. A. H. D. h in s Jan. 3, at Greenwich mean noon 1 18 55.07 + O s .571 H. D. + 8 .571 X 7M7 = + 4.09 G. M. T. 7 h .17 Required R. A. of Jupiter 1 18 59.16 Corr. 4 8 .094 To find the declination of Jupiter. Jupiter's declination. H. D. o / // Jan. 3, at Greenwich mean noon N 6 57 41.2 N 4".2G H. D. N 4".26 X 7M7 = N 30.5 G. M. T. 7M7 Required declination of Jupiter N 6 58 11.7 Corr. N 30".54 NAUTICAL ALMANAC 371 For a given mean time to find the right ascension and decli- nation of a star. Ex. 97. Let these elements of the star Arcturus (a Bootis) be required for the L. M. T., 1905, January 19, l h 48 ra 15 s a. m.., at a place in longitude 40 15' W. A mean place table (N. A. for 1905, pp. 304-312) is used as an index, and shows on page 308 for Arcturus an approxi- mate E. A. 14 h ll m 19 s . 7. The apparent right ascension and decimation are found on page 368, in a table, pp. 324 to 400, in which fixed stars are tabulated in order of right ascension for Washington mean time at intervals of ten days. The right ascension and declination of Arcturus are there found for W. M. T. January 20 d .8 as follows : E. A. 14 h ll m 19 S .29, change for 10 days + O s .32. Dec. K 19. 40' 31".l, change for 10 days S. 2".0. To find the Washington mean time. Local civil mean time Jan. 19 (a. m.) Local astronomical mean time Jan. 18 Longitude West G. M. T. corresponding Jan. 18 Longitude of Washington West Washington mean time Jan. 18 = Jan. 18 d .47. h m 8 1 48 15 13 48 15 + 2 41 00 16 29 15 ) 5 08 15.78 11 20 59.22 Diff. of Washington time. Star's Right Ascension. Tabulated time, Jan. Given time, Jan. d 20.8 18.47 h m s 14 11 19.29 Corr. .075 s Change in l d + 0.032 Interval ( ) 2*. 33 3 Correction ( ) 0.075 Interval, ( ) 2.33 R. A. 14 11 19.215 Star's Declination. / // N 19 40 31.1 Corr. N 0.5 it Change in Id S 0.2 Interval ( ) 2d.83 Correction N 0.466 Dec. N 19 40 81.6 372 NAUTICAL ASTRONOMY Except in the case of Polaris, it is not usual to take out the E. A. and decimation of stars with such precision; the ele- ments as tabulated for the nearest day being used in observa- tions of the stars at sea. The above method is given for use in those cases where extreme precision might be required. 186. To find from the Almanac a certain element of the Gun for a given local apparent time. (1) Find the corres- ponding Greenwich apparent time; to do which express the local . apparent time astronomically, applying to it the longi- tude, plus if West, minus if East, as already explained for finding G. M. T. The elements are then to be taken from page I of the Al- manac where they are given for apparent noon. 187. To find a certain element of the sun when it is on the meridian of a given place or at local apparent noon. Proceed exactly as explained above. The most common use of this problem is when finding the sun's declination in the case of a meridian altitude of the sun, of an altitude near noon, or in the case of finding declina- tion of sun and equation of time in equal altitudes for chronometer error. At the instant of apparent noon, the local apparent time is Qh 0m Q S> Therefore, if in longitude 60 W. on January 5, we have h m 9 Local astronomical apparent time Jan. 5, 00 00 Longitude West + 4 00 00 Greenwich ast. apparent time Jan. 5, 4 00 00 But if in longitude 60 E. at local apparent noon on Janu- ary 5, we would have h m s Local astronomical apparent time Jan. 5, 00 00 Longitude East ( ) 4 00 00 Greenwich apparent time Jan. 5, ( ) 4 00 00 Or Jan. 4, 20 00 00 NAUTICAL ALMANAC 373 From the above it is clear that in longitude West, the G. A. T. of local noon is equal to the longitude, or it is after noon of the same date by the number of hours in the longi- tude; but that in East longitude at local apparent noon the G-. A. T. is before the noon of local date by the number of hours in the longitude, or G. A. T. = ( ) longitude. Hence enter Page I and take out, for Greenwich noon of the same date as the local civil date, the required quantities; mul- tiply the hourly difference by the hours and decimals of longi- tude; apply the correction for a time after noon if longitude is West, for a time before noon, if longitude is Bast, noting whether the quantities are increasing or decreasing for times after or before noon, and applying the corrections accordingly. Ex. 98. Find the sun's declination and equation of time for local apparent noon at a place in longitude 5 h .l W. on January 2, 1905. Times. Sun's declination. H. D. Eq. of T. H. D. o * it At G. A. noon Jan. 2, S 22 57 16.3 N 13.13 Corr. N 5.1 At L. A. noon Jan. 2, S 22 56 09.34 Corr. 66".96 111 fl 8 4 00.41 + 1.176 h Corr. + 6.00 A = + 5.1 4 06.41 Corr.+ 5.998 + to A pp. T. 374 NAUTICAL ASTRONOMY Ex. 99. January 11, 1905, in longitude 96 08' 51" W., find the sun's declination and equation of time at local appar- ent noon, using 2d differences. Here longitude 6 h 24 m 35 S .4 W. = 6 h .41 W. = G-. A. T. of local apparent noon. Times. Sun's Dec. H. D. at G. A. Noon and Change. At G. A. noon Jan. 11, S 21 51 50 Corr. N 2 28.97 At L. A. noon Jan. 11, S 21 49 21.03 Jan. 11, H. D. Jan. 12, " Change in 24 hours Change in 1 hour Change in -^-hrs. = Jan. 11, H. D. Mean H. D. G. A. T. = A W N 23.10 > 24.16 (+) 1.06 (+) 0.0442 3h.2 W(+) 0.141 N 23.10 N 23.241 = 6h.41 Corr. N 148" .97 Times. Eq. of T. H. D. at G. A. Noon and Change- m s At G. A. noon Jan. 11, 7 57.14 Corr. + 6.38 Jan. 11, H. D. Jan. 12, " Change in 24 hours Change in hrs. Jan. 11. H. D. Mean H. D. G. A. T. = A W = + 0.998 + 0.972 At L. A. noon Jan. 11, 8 03.52 + to A pp. time (-) 0.026 (-) 0.003 + 0.998 1 + 0.995 Corr. + 6.38 When longitude is East, the Greenwich apparent time of local noon equals ( ) longitude of the local civil date. NAUTICAL ALMANAC 375 Ex. 100. April 4, 1905, in longitude 10 h 04 m 49 8 .6 East, find the sun's declination and equation of time at local ap- parent noon, using 2d differences. Here the G. A. T. of local apparent noon equals ( ) 10 h .08. Times Sun's Dec. H. D. at G. A. Noon and Change. / If At G. A. noon April 4, N 6 33 35.6 Corr. 8 9 37.87 April 3, H. D. April 4, " Change in 24 hours Change in 1 hour Change in -^ hrs. April4,H.D. Mean H. D. G. A. T. = ( ) X = Correction n N 57.61 N 57.28 (-) 0.23 (-) 0.0096 ( + ) 0.048 N 57.28 At L. A. noon April 4, N 5 22 57.73 N 57.328 (-) IQh .08 S 577".87 Times. Bq. of T. H. D. at -G: A. Noon and Change. m B At G. A. noon April 4, 3 10.82 Corr. + 7.43 April 3, H. D. April 4, " Change in 24 hours Change in | hrs. April 4, H. D. Mean H. D. G. A. T. = (-) A = Correction ( ) " 0.742 ( ) 0.736 At L. A. noon April 4, 3 18.25 (+) 0.006 () 0.001 ( ) 0.736 ( ) 0.737 (-)10h.08 + 78.43 To find the sun's declination and equation of time at local apparent midnight, proceed as in the above examples, using for G. A. T. in the first of the two preceding examples (12 hrs. + A) 18 h .41 January 11, and in the second (12 hrs. A) = 12 hrs. 10 h .08 = l h .92 April 4. 188. To find the local mean time of transit of the moon over a given meridian on a given date, and the moon's right ascension, declination, semi-diameter, and horizontal parallax at that instant. The Nautical Almanac, page IV of each month, contains the Greenwich mean time of each transit of 376 NAUTICAL ASTRONOMY the moon over the meridian of Greenwich. This time is the hour angle of the mean sun when the moon is on the meridian, and, therefore, equals the difference of the right ascensions of the moon and mean sun. Both the moon and mean sun in- crease their right ascensions daily, but the increase for the moon is greater than that for the mean sun, so that each day the moon gets further and further to the eastward of the mean sun, and in the diurnal revolution comes to the meridian later each day than on the preceding day; the number of minutes varying with the moon's motion, but approximating an average value of 50 minutes. This retardation, represented by ~R, occurs during a passage of the moon over 24 hours of longitude, and for any longitude r> X hrs. the retardation will be X A, so it is easily seen that if there were no retardation whatever, the local time of the moon's meridian passage in any longitude would be the same as that at Greenwich, but there is retardation, and the local mean time of transit over a meridian is gotten from the Greenwich mean time of Greenwich transit by computing the amount of retardation corresponding to the number of hours and decimals of an hour of longitude, and applying it to the Greenwich mean time of Greenwich transit, adding that amount for west longitude, subtracting for east longitude; west longitude being regarded as -J-, east longitude as ( ) . r> The value of ~ , or the hourly retardation, is given in its Z4. appropriate column opposite the time of transit on page IY. The reduction for longitude is tabulated in table 11, Bowditch, the arguments being " longitude/' and " daily variation of the moon's passing the meridian." The times given in the Almanac are for the astronomical date, and care must be exercised in finding the meridian pas- sage on a given civil date; hence the rules: NAUTICAL ALMANAC 377 (1) Take the time of the moon's meridian- passage from the Nautical Almanac for the given civil date when the time tabulated plus the correction for retardation is less than 12 hours, because then the astronomical date is the same as the civil date. (2) When it is seen that the sum of the tabulated time of passage plus the correction for retardation on the given civil date will be greater than 12 hours, take out the time of pas- sage for the day before, since in this case the astronomical date is one day less than the civil date. (3) Multiply the " Diff. for 1 hour" by the longitude in hours and decimals, adding the product to the G. M. T. of meridian passage at Greenwich when longitude is West, sub- tracting if longitude is East. The result will be the local mean time of local transit (see Art. 196 (c) ). Ex. 101. In longitude 100 30' W. find the time of me- ridian transit of the moon for 1905, January 19 (civil date), then the corresponding Gr. M. T. Meridian Transit of the Moon. Retardation. o r Long. 100 30 W or Long. 6h 42m w = 6h.7 W G. M. T. of Gr. transit Jan. 19, Corr. for longitude West L. M. T. of local transit Jan. 19, Longitude West G. M. T. of local transit Jan. 19, h m 10 53.8 + 15.88 11 09.68 + 6 42 For l h , 2 m .37 \ = + 6h.7 Corr.+ 15m.88 17 51.68 Ex. 102. In longitude 100 30' E. find the time of me- ridian transit of the moon for 1905, January 22 (civil date), then the corresponding G. M. T. Meridian Transit of the Moon. Long. 100 30' E G. M. T of Gr. transit Jan. 21, 12 49m or Corr. for longitude East (-) 16.01 Long. 6h 42 m E L. M. T. of local transit Jan. 21, 12 32.99 = 6h.7 E Longitude East ( ) 6 42 G. M. T. of local transit Jan. 21, 5 50.99 Retardation. For lh,2m.39 Corr. 16m .01 378 XAUTICAL ASTRONOMY The times of transits at Greenwich are given only to the nearest tenth of a minute, and the resulting local time of local transit will be only approximate, though sufficiently exact for navigators. A more exact time may be found by first finding the approximate L. M. T. of local transit and then the approxi- mate G. M. T. of local transit for which the moon's right ascension may be taken out. This right ascension is the local sidereal time of the moon's local transit and the local mean time corresponding may be found (see Ex. 132). If then other elements are desired at the time of the moon's local transit, find the G. M. T. corresponding to the L. M. T. just found, and take out for this G. M. T. the required ele- ments. 189. To find the local mean time of transit of a planet over a given meridian on a given date, and the correspond- ing G. M. T., also the planet's right ascension and declination at that instant. The mean time of each meridian transit for the meridian of Greenwich is given in the Almanac for each of the seven major planets. On certain dates there may be retardation, on others acceleration, in the times of return to the meridian. In the case of a retardation, the time of local transit is found as in the case of the moon; in the case of acceleration, the sign of the reduction for longitude is re- versed. Or, considering the hourly retardation +, the hourly acceleration ( ) , west longitude +, and east longitude ( ) , the rule of signs will determine the sign of the reduction. Having found the L. M. T. of local transit, deduce the G. M. T. by applying the longitude, and take out for this G. M. T. the planet's right ascension and declination (Art. 185, Ex. 96). Having found the right ascension of a planet when it is on the meridian, take this as local sidereal time and find the corresponding local mean time ; the result will be closer than NAUTICAL ALMANAC 379 the time tabulated which, however, is sufficiently exact for navigators. It will be noticed that, while in the case of the moon the retardation was given for one hour, in the cases of planets, the retardation or acceleration will be obtained for 24 hours by taking the difference of the times of Greenwich transit for the given day and the day following when in West longitude, the difference of those times for the given day and day preced- ing when in East longitude. The change for 1 hour will then be one twenty-fourth of the difference for 24 hours. This hourly change multiplied by the hours and decimals of an hour in the longitude will be the change for longitude. Ex. 103. In longitude 75 W. find the L. M. T. of transit of Jupiter, 1905, January 4, civil date; also right ascension and declination of Jupiter for that instant. Meridian Transit of Jupiter. Difference. Long. 75 W = 5'' W. At Gr. noon Jan. 4, Correction for longitude W. L. M. T. of local transit Jan. Longitude West G. M. T. of local transit Jan. h m 6 24.5 - 0.77 h m for 24 = 3.7 for 5 = 0.77 4, 6 23.73 + 5 4, 11 23.73 Times. Jupiter's R.A. H. D. Jupiter's Dec. H. D. At Gr. noon Jan. 4. Corr. for G. M. T. At local transit h m s 1 19 09.14 + 6.85 + Os.601 G.M.T. 11M 1 It N 6 59 25.5 Corr. N 50.5 N 4".43 G.M.T. 11M 1 19 15.99 Corr. + 68.85 N 700 16 Corr. N 60".50 CHAPTER XIV. RELATION OF MEAN, APPARENT, AND SIDEREAL TIMES. CONVERSION OF TIME. RELATION OF TIME, HOUR ANGLES, AND RIGHT ASCENSIONS, AND A CONSIDERATION OF PROBLEMS INVOLVING THEM. FINDING LOCAL AND WATCH TIMES OF A BODY'S TRANSIT, ETC. 190. To interconvert apparent and mean time. The equa- tion of time being the difference between the hour angles of the true and mean suns, or, in other words, between apparent and mean times, when one is given, the other is obtained by applying the equation of time with its proper sign of appli- cation to the given time. Thus, if for the same instant, t m represents local mean time, t a represents local apparent time, E represents equation of time with positive sign of applica- tion to apparent time, Hence for the given local time (apparent or mean), ex- pressed astronomically, find the Greenwich time (apparent or mean). Take out of the Nautical Almanac for the Green- wich instant the equation of time (from page I of the required month when apparent time is given, or page II when mean time is given). The reduction then is made by applying the corrected equation of time to the given time, with the proper sign as shown at the top of the column in which it is found, MEAN AND APPARENT TIMES 381 The equation of time found on page I, Nautical Almanac, is the mean time of apparent noon at Greenwich, and, if cor- rected for longitude, it is the mean time of local apparent noon. The equation of time found on page II, Nautical Almanac, is the apparent time of mean noon at Greenwich, and if cor- rected for longitude it is the apparent time of local mean noon. Ex. 104. January 2, 1905, in longitude 75 30' W., find the local apparent time corresponding to a local mean time 8 h 10 m 10 s p. m. hms m s h m s L. M. T. 8 10 10 Jan. 2. Eq.t.atOh=4 0.33 H D +K176 L-M.T.= 8 10 10 Long. 5 2 00 W Corr. = + 15.62 G. M. T. W>.2 Eq. t. 4 15.85 G.M.T. 13 12 10 Jan. 2. Eq. t. =4 15.85 Corr. + 15>*.52 L.A. T. 8 05 54.15 = 13h.3 ( ) to M. T. Ex. 105. April 3, 1905, in Long. 100 45' E., find the local mean time corresponding to 5 h 10 m a. m., local apparent time. hms ms hms L.A.T. 17 10 00 April 2. Eq. t. at Oh = 3 46.46 H.D 0".748 L.A.T. 17 10 00 Long. 6 43 00 E Corr. = 7.82 G.A.T.10M5 Eq. t . 3 38.64 G.A.T. 10 27 00 April 2. Eq. t. =3 38.64 Corr.-7.82 L.M.T. = 17 13 38.64 + to App. T. or April 3, a. m., 5 13 38.64 191. Formulae for the intercon version of mean and side- real time intervals. Since a sidereal year contains 365.25636 mean solar days, or 366.25636 sidereal days, each unit of v\ mean solar time will contain o^'f^.o^ sidereal units of the o65. 25636 same denomination, or each unit of sidereal time will contain units f mean time of the same denomination. Since both are uniform measures of time, any interval of time expressed either in mean solar or sidereal units may be expressed in units of the other denomination. Thus, if any interval of time be represented by t if ex- pressed in mean solar time, by s if expressed in sidereal time, s 366.25636 _ whence s = t + .0027379/, (137) t s .00273045, (138) 382 NAUTICAL ASTRONOMY and by these formulae any interval of the one kind of time can be converted into an interval of the other kind of time. The reduction is facilitated by the use of Table II of the Nautical Almanac for converting sidereal intervals into mean solar time intervals, which contains for each second of s the value .0027304^ expressed in minutes and seconds; also by Table III, for converting a mean solar time interval into a sidereal time interval, which contains for each second of t the value .002 73 79 expressed in minutes and seconds. Tables 8 and 9 of Bowditch are for the same purpose. If t and s are in units of hours, the above formulae become s = t (1 + 9 S .8565) = t +' 9-.8565$, (139) t = s (1 9 S .8296) s 9 s .8296s, (140) so that in the absence of the above mentioned Tables the re- duction may still be conveniently calculated. Acceleration and retardation. If in (137) t = 24 hrs., s will equal 24 h 3 m 56 S .5553; or in a mean solar day sidereal time gains on mean time 3 m 56 S .5553, and this is called the acceleration of sidereal on mean time. If in (138) s = 24 hrs., t = 24 h minus 3 m 55 S .9094, or in a sidereal day mean time loses on sidereal time 3 m 55 S .9094, and this is the retarda- tion of mean solar on sidereal time. Examples on the conversion of a mean solar time interval into a sidereal time interval. Ex. 106. Express 10 hours of mean solar time in sidereal time. Taking formula s h = t h (l + .0027379), we have s 10 h .027379 = 10 h Ol m 38 3 .564. Taking formula s h = ^ h (l + 9 S .8565), we have s 10 h Ol m 38 8 .565. Using table III, Nautical Almanac, we have t = a mean solar time interval, 10 h 00 m 00 s From table III, reduction to a sidereal interval -j- 1 38 .565 The required sidereal time interval 10 h Ol m 38 3 .565 MEAN AND AITAUKNT TIMES 383 Ex. 101. Express 15 b 33 m 29 s of mean time in sidereal time. t =. a mean solar time interval 15 h 33 m 29 s From table III, reduction to a sidereal interval -\- 2 33 .347 The required sidereal time interval 15 h 36 m 02 S .347 Express in sidereal time : Ex. 108. 7 h 29 m 30 S .5 of mean time. Ans. 7 h 30 m 44 8 .342 sidereal time. Ex. 109. l h 14 m 03 s of mean time. Ans. l h 14 m 15 s . 164 sidereal time. Ex. 110. 23 h 15 m 10 s of mean time. Ans. 23 h 18 m 59M90 sidereal time. Examples on the conversion of a sidereal interval into a mean solar time interval. Ex. 111. Express 10 h 30 m 00 s of sidereal time in mean solar time. Taking formula t h = s h (l .0027304), we have t = io h .5 O h .0286692 10 h .471331 = 10 h 28 m 16 3 .79. Taking formula t h = s*(l 9 S .8296), we have t = 10 h .5 (1 9 8 .8296) = 10 h 30 m l m 43 9 .2108 = 10 h 28 m 16 3 .789. Using table II, Nautical Almanac, we have s = a sidereal time interval = 10 h 30 m 00 s From table II, reduction to a mean time interval 1 43 .210 The required mean solar time interval 10 h 28 m 16 S .790 Express in mean solar time : Ex. 112. II* 04 m 12 S .94 of sidereal time. Ans. ll h 02 m 24M25 mean time. Ex. 113. 15 h 08 m 33 S .37 of sidereal time. Ans. 15 h 06 m 04 3 .525 mean time. Ex. 114. 19 h 13 m 36 S .65 of sidereal time. Ans. 19 h 10 m 27'.659 mean time. 384 NAUTICAL ASTRONOMY 192. Having the mean time at any place, to find the cor- responding sidereal time. Let A represent the longitude of the place expressed in time, + when West, ( ) when East. t the hour angle of the mean sun expressed positively and, therefore, the local mean time. (-|-A) the G. M. T. or elapsed mean time interval since Greenwich mean noon. 8 the hour angle of T and hence the local sidereal time. (8 + A) the Greenwich hour angle of T or Green- wich sidereal time. a the right ascension of the mean sun (E. A. M. O ) at Greenwich mean noon. If a mean time interval since Greenwich mean noon is (^-|-.A) h , the corresponding sidereal time interval will be ( + A) h (1 + .0037379). Having now the sidereal in- terval since Greenwich mean noon and the sidereal time of Greenwich mean noon, or a- , the Greenwich sidereal time will be (8 + A) h = fl + (t + *) h (1 + .0027379) 8 + \ = a +.\ + t + (t + \) (.0027379) S. T. } = ^ + t + (t + X) (.0027379) (141) The right-hand column of page II of the Almanac contains a for each Greenwich mean noon under the head " Sidereal Time" or "Eight Ascension of Mean Sun." As t + A is the G. M. T., a should be taken out for Greenwich mean noon of the given Greenwich date, and corrected for the hours, min- utes, and seconds of Greenwich mean time, using Table III of the Almanac. Hence the rule : Express the local mean time astronomically and find the G. M. T. and date. Then to the local astronomi- cal mean time add th& sidereal time or the right ascension of CONVERSION OF TIME 385 the mean sun taken from the Nautical Almanac for noon of the Greenwich date, and also the reduction from Table III for the hours, minutes, and seconds of the Greenwich mean time. The sum, if less than 24 hours, will ~be the local sidereal time (L. 8.T.). If the sum is greater than 24 hours, reject 24 hours and the remainder will be the L. 8. T. Since the sidereal time (R. A. M. O ) at Greenwich mean noon, corrected for the G. M. T. corresponding to the given L. M. T., is the right ascension of the mean sun at the instant of the given L. M. T., the above equation (141) is simply an algebraic expression of what has already been proven, namely. " The sidereal time at a given place is equal to the right ascen- sion of the mean sun plus the local mean time" (Art. 173). It is usual to keep the solar day; but should it be desired to state the sidereal day, prefix to a the sidereal day at the instant of Greenwich mean noon, which is the same as the astronomical day for six months after the vernal equinox, one day less for six months before the vernal equinox. At the instant of the vernal equinox,, the sidereal time and mean solar time coincide. Before that time the mean sun transits before the vernal equinox ; after that time, it transits after the vernal equinox. Examples on the conversion of local mean time into local sidereal time. Ex. 115. January 18, 1905, in longitude 55 15' W., the local mean time is 8 h 06 m 29 S .5 p. m. Find the local sidereal time (see rule in this Article). h m 8 The local astronomical mean time Jan. 18, 8 06 29.5 Longitude from Greenwich West + 3 41 00 The Greenwich mean time Jan. 18, 11 47 29.5 h m s R. A. M. O Jan - 18, at Greenwich mean noon 19 48 53.64 Reduction for G. M. T., Table III, or 9 8 .8565 X ll h .7915 1 56.223 Add the local astronomical mean time 8 06 29.5 The required local sidereal time (rejecting 24 hrs.) 3 57 19.363 386 NAUTICAL ASTRONOMY Ex. 116. January 10, 1905, in Long. 137 35' E., the L. M. T. is 5 h 17 m 30 s a. m. Find the L. S. T. h m s Local astronomical mean time Jan. 9, ' 17 17 30 Longitude from Greenwich East 9 10 20 Greenwich mean time Jan. 9, 8 07 10 h m a R. A. M. Q at Greenwich mean noon Jan. 9, 19 13 24.64 Reduction for G. M. T. Table III 1 20.029 Add the local astronomical mean time 17 17 30 Required local sidereal time (rejecting 24 hrs.j 12 32 14.669 Ex. 111. April 16, 1905, the Greenwich mean time is 9 h 10 m 30 s a. m. Find the Greenwich sidereal time. Greenwich astronomical mean time April 15, 21 10 30 R. A. M. O at Greenwich mean noon April 15, 1 31 53.76 Reduction for G. M, T. Table III 3 28.711 Required Greenwich sidereal time 22 45 52.471 Ex. 118. January 20, 1905, at the U. S. Naval Academy, when the 75th meridian mean noon signal was received, a sidereal clock read 20 h 23 m 19 S .5. Shortly after the receipt of this signal a comparison of this clock with a mean time chronometer was: sidereal clock, 20 h 43 m 29 s ; mean time chronometer, 5 h 24 m 16 s . Find the error of chronometer on G. M. T. (see example 125, Art. 193). Note that here the error of the sidereal clock is not given. h m s At 75th meridian mean noon sidereal clock reads 20 23 19.5 At time of comparison sidereal clock reads 20 43 29 Sidereal interval since 75th mer. mean noon 20 09.5 Reduction to a mean time interval Table II 3.303 Mean time interval since 75th mer. mean noon 20 06.197 Longitude of 75th meridian West + 5 00 00 Greenwich mean time of comparison 5 20 06.197 Chronometer time of comparison 5 24 16 Error of mean time chronometer, fast on G. M. T. 4 09.80? CONVERSION OF TIME 387 Ex. 119. At Cebu I. Plaza, Lat. 10 17' 30" K, Long. 123 54' 18" E., April 10, 1905, L. M. T. = 5 h 45 m 30 s a. m. Find L. S. T. h m Local astronomical mean time April 9, 17 45 30 Longitude East from Greenwich 8 15 37.2 Greenwich mean time April 9, 9 29 52.8 b m R. A. M. Q at Greenwich mean noon April 9, 1 08 14.45 Reduction for G. M. T. 1 33.616 Local astronomical mean time 17 45 30 Local sidereal time 18 55 18.066 Ex. 120. April 1, 1905, in longitude 2 h 13 m 20 s West, the L. M. T. is 9 h 48 m 06 s p. m. Find first the G. S. T., then the L. S. T. h m g April 1, the local astronomical mean time is 9 48 06 Longitude from Greenwich West + 2 13 20 April 1, Greenwich mean time 12 01 26 R. A. M. Q April 1 -at Greenwich mean noon 36 42.03 Reduction for G. M. T. Table III 1 58.513 Greenwich sidereal time 12 40 06.543 Longitude from Greenwich West ( ) 2 13 20 Local sidereal time 10 26 46.543 Ex. 121. January 25, 1905, at the Naval Academy, Annap- olis, Md., in longitude 5 h 05 m 56 S .5 W., when the time signal was received from Washington indicating jnoon of 75th merid- ian West longitude, mean time, a sidereal clock read 20 h 15 m 388 NAUTICAL ASTRONOMY 09 9 . Kequired the error of the sidereal clock on local sidereal time. h m a 75th meridian mean time at 75th mer. mean noon 00 00 Longitude of 75th meridian West -f 5 00 00 Jan. 25, G. M. T. of 75th mer. mean noon 5 00 00 Longitude of Naval Academy West ( ) 5 05 56.5 L. M. T. at instant of 75th mer. M. N. 23 54 03.5 h m a Jan. 25, R. A. M. Q at G. M. noon 20 16 29.54 Reduction for G. M. T. of 75th mer. mean noon (5 hrs.) 49.282 Local ast. mean time at Naval Academy 23 54 03.5 Local sidereal time at Naval Academy 20 11 22.322 Reading of sidereal clock at the instant 20 15 09 Error of sidereal clock on L. S. T., fast, 3 46.678 193. Having the sidereal time at any place, to find the local mean time. Since A, the longitude, is the G. M. T. of local mean noon, or of the instant when the mean sun is on the upper branch of the local meridian, according to the notation of Art. 192, a + .0027379A, will he the local sidereal tjme at local mean noon. 8 (a -f- .0027379A) will be the sidereal interval since noon as it is L. S. T. the sidereal time of local mean noon. [8 (a + .0027379A)] [1 .0027304] will be the mean time interval since noon, and to find L. M. T. it is only necessary to add the astronomical day to this mean, time interval. Hence the rules: (1) Take from ihe Nautical Almanac for Greenwich mean noon of the given local astronomical day the right ascension of CONVERSION OF TIME 389 the mean sun; apply to this the reduction for longitude (which is the change in the mean sun's right ascension for that num- ber of hours) taken from Table III, Nautical Almanac; add- ing for West, subtracting for East longitude. The result will be the right ascension of the mean sun at local mean noon, or sidereal time at that instant (local hrs. of mean time). (2) Subtract this from the given L. S. T. (adding 4 hrs. to the L. S. T. if necessary for subtraction) and the result will be the sidereal interval from local mean noon. (3) Apply to this the reduction of a sidereal to a mean time interval taken from Table 11 ' , Nautical Almanac, which is always subtractive. The result, after prefixing the given astronomical day f is the required local mean time. In the absence of Tables, the reduction may be made by using the formulas (139) and (140) of Art. 191. Caution. It is much better to convert a given L. S. T. and afterwards, if desired, find the G. M. T., than to 'first find G. S. T. and then convert it into G. M. T., for the reason that the right ascension of the mean sun must be taken out for the given astronomical day. To convert G. S. T. the Greenwich astronomical date must be known, and as this may or may not be the same as the local astronomical date, an error might result. As a little thought can easily determine the Greenwich date, this caution may seem unnecessary to those thoroughly familiar with the subject; to others, however, it is most im- portant. In cases where the G. M. T. is known in addition to the L. S . T., the method of reduction is very simple. From formula (141), t = S K + (t + A) (.0027379)] (142) 390 NAUTICAL ASTRONOMY Rule: For the G.M.T. (t -f A) take out the right ascenr sion of the mean sun., subtract it from the given L. 8. T., and the result will be L. M. T. of the given astronomical date. So also from Art. 173, it is plain that local apparent time equals L. S. T. minus the apparent right ascension after cor- rection for Gr. M. T., if taken from page II, or for G. A. T. if taken from page I of the Nautical Almanac. Examples on the conversion of time ; L. S. T. into L. M. T. Ex. 122. January 8, a. m., 1905, at Eoyal Observatory, Lisbon (Long. O h 36 m 44 S .68 W.), local sidereal time is 10 h 44 m 30 s . Find the local mean time. First find the astronomical day, which is January 7. b m 9 R. A. M. Q or sidereal time at Greenwich mean noon Jan. 7, 19 05 31.52 Reduction for longitude West + 6.036 The sidereal time of local mean noon 19 05 37.556 The given local sidereal time ( + 24 hrs. for the subtraction) 10 44 30 The sidereal interval from noon 15 38 52.444 Reduction of a sid. to a M. T. interval Table II 2 33.812 The required astronomical mean time Jan 7, 15 36 18.632 Or civil time Jan. 8, (a. m.) 3 36 18.632 Ex. 123. April 15 (civil date), 1905, in Long. 129 30' 45" E., the local sidereal time is 23 h 56 m 30 s . Find the local mean time. The above example does not say whether it is a. m. or p. m., but the astronomical date must be known before taking out the E. A. M. O . To determine this look up the approximate R. A. M. Q, which is found to be about 1J hours. Subtract- ing this from the L. S. T. leaves an approximate astronomical CONVERSION OF TIME 391 mean time of over 22 hours ; the civil time is, therefore, a. m., and hence the local astronomical date is April 14. ii in a R. A. M. O or sidereal time at Greenwich mean noon April 14, 1 27 57.21 Reduction for Long. (8 h 38 m 03 s East) Table III 1 25.102 The sidereal time of local mean noon 1 26 32.108 The given local sidereal time 23 56 30 The sidereal interval from noon 22 29 57.892 Reduction of a sid. to a M. T. interval Table II 3 41.159 The required astronomical L. M. T. April 14, 22 26 16.733 Or civil date April 15, (a. m.) 10 26 16.733 Ex. 121f-. On January 11, astronomical time, 1905, the sidereal clock time of transit of a Leonis (Regulus) over the middle wire of a transit instrument at the U. S. Naval Acad- emy was 10 h 05 m 2 2 s . 5. Later a comparison of the sidereal clock and a mean time chronometer was : Sid. clock, 10 h 30 m 20 S .5 ; M. T. chro., 8 h 05 m 10 s . Find the error of chronometer on Gk M. T. Longitude of Naval Academy, 5 h 05 ra 56 8 .5 West. R. A. of >f< a Leonis at transit equals the L. S. T. Reading of sidereal clock at star's transit Error of sidereal clock on L. S. T. fast Reading of sidereal clock at comparison L. S. T. at instant of comparison R. A. M. Q or sidereal time at G. M. noon Jan. 11, Reduction for longitude West (5 h 05 m 56 S .5) Sidereal time of local hrs. The given L. S. T. at comparison Sidereal interval from noon Reduction of a sid. to a M. T. interval Table II The local mean time at instant of comparison Longitude of Naval Academy West G. M. T. at instant of comparison Reading of M. T. chronometer at comparison Error of chronometer (dropping 12 hrs.), slow on G. M. T. h m s 10 03 19.54 10 05 22.5 2 02.96 10 30 20.5 10 28 17.54 h m g 19 21 17.75 50.258 19 22 08.008 10 28 17.54 15 06 09.532 2 28.452 15 03 41.08 + 5 05 56.5 20 09 37.58 8 05 10 4 27.58 392 NAUTICAL ASTRONOMY The following example worked under Art. 192 without first finding the error of the sidereal clock, by considering only the sidereal interval from noon to time of comparison, and finding the corresponding mean time interval from noon and then the Gr. M. T., will now be worked by finding the clock error on L. S. T. as indicated in the solution. Ex. 125. January 20, 1905, at IT. S. Naval Academy, when the 75th meridian mean noon signal was received, a sidereal clock read 20 h 23 m 19 S .5. Shortly after the receipt of this signal a comparison of this clock with a mean time chronometer was: Sid. clock, 20 h 43 m 29 s ; M. T. chro., 5 h 24 m 16 s . Find the error of the clock on L. S. T. and the error of the chronometer on G-. M. T. (see example 118, Art. 192). h m s The 75th mer. mean time of 75th mer. mean noon 00 00 Longitude of 75th meridian West + 5 00 00 G. M. T. of 75th mer. mean noon Jan. 20, 5 00 00 R. A..M. or sidereal time G. M. noon Jan. 20, 19 56 46.76 Correction for G. M. T. 49.282 G. S. T. of 75th meridian mean noon 57 36.042 Longitude of Naval Academy West 5 05 56.5 L. S. T. of 75th meridian mean noon 19 51 39.542 Sidereal clock time of 75th meridian noon 20 23 19.5 Error of sidereal clock on L. S. T. fast 31 39.958 Sidereal clock time of comparison 20 43 29 L. S. T. at instant of comparison 20 11 49.042 h m a R. A. M. O or sidereal time at G. M. noon Jan. 20, 19 56 46.76 Reduction for longitude (5 h 05 m 56 8 .5 W.) + 50.258 Sidereal time of local hrs. 19 57 37.018 Given local sidereal time 20 11 49.042 Sidereal interval from noon 14 12.024 Reduction of a sid. to a M.T. interval Table II 002.327 Required L. M. T. at instant of comparison 14 09.697 Longitude of Naval Academy West + 5 05 56.5 The G. M. T. at instant of comparison 5 20 06.197 M. T. chronometer reading at comparison 5 24 16 Error of M. T. chronometer on G. M. T., fast 4 09.803 CONVERSION OF TIME 393 194. Relation between apparent time and sidereal time. From Art. 173 it is seen that local sidereal time is equal to the true sun's right ascension plus the local apparent time, so that, having a given local apparent time, to find the local sidereal time: (1) Find the Greenwich apparent time and date. (2) Take out from the Nautical Almanac left-hand column of page I (of the proper month), the apparent right ascen- sion and correct it for the hours and decimals of an hour of G. A. T., using the tabulated hourly difference. (3) To the local astronomical apparent time add the above corrected apparent right ascension. The sum, if less than 24 hours, will be the local sidereal time (L. S.T.) if the sum is greater than 24 hours, reject 24 hours, and the remainder will be the L. S. T. Or the following method may be pursued : (1) Find the G. A. T. and date. (2) For this G. A. T. take out the equation of time from page I, N. A. Apply the equation of time with its proper sign to the local apparent time (L. A. T.), obtaining the cor- responding local mean time (L. M. T.) which can be converted into L. S. T. as before explained. Ex. 126. On January 8, 1905, in Long. 135 15' E. ? the local apparent time is 5 h 10 m 30 s a. m. Find the local side- real time. h m s Jan. 7, local astronomical apparent time 17 10 30 Longitude from Greenwich East ( ) 9 01 Greenwich apparent time Jan. 7, 8 09 30 h m R. A. App. O at Greenwich apparent noon Jan. 7, 19 11 49.31 Corr. for G. A. T. (10 8 .949 X 8M6) 1 29.34 Add local astronomical apparent time 17 10 30 Required local sidereal time 12 23 48.65 195. Relation of time, hour angles, right ascensions, and a consideration of problems involving them. 394 NAUTICAL ASTKONOMY A great many problems arise in everyday practical naviga- tion which, involve the consideration of hour angle and right ascension. Such problems are readily solved if the definitions of these terms and of local sidereal time are well understood. Some will be illustrated in the following articles. 196. To find the local mean time of transit of a particular heavenly body across the meridian of a given place, the longitude of the place, or G-. M. T., being known. (a) In case the L. M. T. of transit of the sun is desired, it is only necessary to remember that the instant of transit of the true sun is apparent noon, and at this instant the equa- tion of time taken from page I of the Almanac and corrected for longitude (which is the G. A. T. of the instant) is the hour angle of the mean sun. If the equation of time is addi- tive to apparent time, the L. M. T. of the sun's transit is the equation of time, and the local date is the given astronomical date; if the equation of time is sub tractive from apparent time, the L. M. T. of the sun's transit is 24 hours the equa- tion of time, and the local date is that of the day preceding the given astronomical day. Ex. 127. January 27, 1905, in Long. 52 30' W., find the local mean time of upper transit of the true sun, or of local apparent noon. A = G. A. T. of sun's transit = 3h 30 L. M. T. of local apparent noon equals the equation of time, or Jan. 27, Oh 12^ 638.01 (p. m.) Equation of Time. At G. A. noon 12 51.21 H.D. + Os.514 G.A.T. 3h.5 T , 1 - 8U Corr Eq. of T. =12 53.01 1 + to Apparent time. Ex. 128. April 27, 1905, in Long. 52 30' W., find the local mean time of the upper transit of the true sun, or of local apparent noon. \ = G. A. T. of sun's transit = 3'' 30'" L. M. T of local apparent noon equals the equation of time. b m it or April 27, ( ) 02 23.05 or April 26, 23 67 36.95 or April 27, 11 67 36.95 (a. m.) Equation of Time. I H. D. At G. A. noon Corr. Eq. of T. ~~ 8 ~l +08.408 G. A. T. 3h.6 + ls.43 ( ) to Apparent time. TRANSIT OF A GIVEN STAR 395 If the G. M. T. of local apparent noon is given, the equation of time should be taken from page II of the Nautical Almanac. When any heavenly body is on the upper branch of the meridian of a place, its right ascension is the right ascension of the meridian, or the local sidereal time (Art. 173), and to find the L. M. T. of transit it is only necessary to obtain from the Nautical Almanac the right ascension of the body at that instant, and, remembering that this is the local sidereal time, reduce it to L. M. T. (Art. 193). The time of transit of the sun across the meridian may be found in this way, which, however, is a longer way than the method used on page 394. (&) To find the L. M. T. of transit of a given star across the upper branch of a given meridian. The American Ephemeris and Nautical Almanac contains the apparent right ascension and declination of more than 825 of the principal stars for the upper culmination at Wash- ington. These right ascensions may be taken as the right ascensions for the upper culmination at any other meridian, except in the cases of a few circumpolar stars whose right ascensions may be reduced by interpolation for differences of longitude, if desired. In one table, the mean places of these stars are given for the beginning of the Besselian fictitious year 1912, that is, for the moment when the sun's mean longitude is 280 (January l a .006, 1912, at Washington). See footnote, page 362. In the following table, the apparent places are given for every tenth upper transit at Washington for all except the 25 circumpolar stars; for these latter the apparent places are given for every upper transit. Knowing the name of a star, its approximate right ascen- sion is found in the former table, and it may then be con- veniently looked up in the second table referred to above. In both tables the stars are arranged in the order of their right ascensions. 396 NAUTICAL ASTRONOMY Having the longitude of the place and the right ascension of the star which is the L. S. T. at the instant of the star's upper transit of that meridian, we find the L. M. T. of transit by the method explained in Art. 193. Ex. 129. April 12 (civil date), 1905, in longitude 5 h 05 m 56 S .5 W., find the local mean time of the upper transit of the star a Scorpii (Antares) whose E. A. is 16 h 23 m 36 S .43. An examination of the approximate E. A. M. O and E. A. of the star shows the astronomical date to be the llth. h m a April 11, R. A. M. Q at G. M. noon 1 16 07.55 Reduction for longitude West, Table III + 50.258 Sidereal time of local hrs. 1 16 57.808 The given L. S. T. = >|<'s R. A. 16 23 36.43 The sidereal interval from noon 15 06 38.622 Reduction of the sid. to a M. T. interval, Table II ( ) 2 28.531 The required L. M. T. of -)f' s transit April 11, 15 04 10.091 Or civil date April 12, (a. m.) 3 04 10.091 Ex. 130. January 5, 1905, at the Naval Academy, Annap- olis, Md. (Long. 5 h 05 m 56 S .5 W.), find the 75th meridian West longitude mean time of the upper transit of the star a Canis Majoris (Sirius) across the local meridian. h m s Jan. 5, R. A. M. Q at G. M. noon 18 57 38.41 Reduction for longitude West, Table III + 50.258 The sidereal time of local hrs. 18 58 28.668 The L. S. T. at time of transit = star's R. A. 6 40 58.86 The sidereal interval from mean noon 11 42 30.192 Reduction of sid. to a mean time interval, Table II ( ) 1 55.088 L. M. T. of transit of Sirius Jan. 5, 11 40 35.104 Diff. of longitude of local and 75th meridians + 5 56.5 The 75th meridian time of local transit 11 46 31.604 TRANSIT OF THE MOON 397 (c) To find the L. M. T. of transit of the moon across the upper branch of a given meridian. In the case of the moon the L. M. T. of transit may be found from page IV of the Nautical Almanac, where the mean time of the upper transit at Greenwich is given to tenths of a minute ac- companied by a column of differences for one hour of longitude. These hourly differences express, for each day, the mean hourly increase of the moon's right ascension. The number of minutes in this column multiplied by the hours of longi- tude will give a correction, + for West, ( ) for East longi- tude, to be applied to the time of Greenwich transit to give the L. M. T. of local transit. In taking out the time of the meridian passage of the moon., it must not be forgotten that the result will be astronomical time and not civil time. When the Almanac time of passage, after correction for longitude, gives a time greater than 12 hours of a given astronomical day, it is plain that this is not the time of passage on the civil day of the same date. Hence, if the time of passage over the meridian* is desired for a given civil date, and it is seen by inspection that the tabulated time after correction for the longitude will be greater than 12 hours, then it will be necessary to take out the Greenwich time of Greenwich transit for the day before (see Art. 188). Ex. 131. Find the time of the meridian passage of the moon over the upper branch of the meridian in longitude 60 30' East for January 25 (civil date), 1905. Since an inspection of page IV of the Nautical Almanac shows the astronomical time of transit will be greater than 12 hours, the meridian transit of the preceding astronomical date, January 24, must be used in order to make the civil date January 25. m 8 Appro*. M. T. of transit of moon over j ^ the mer. of Greenwich Jan. 24. ) Corr. or retardation for Long. East 8 57 A 4 h .03 L. M. T. of local transit Jan. 24 15 26 03 Corr. -8. 95 Or civil date Jan. 25, (a. m.) 3 26 03 398 NAUTICAL ASTRONOMY The above method is sufficiently accurate for all purposes of everyday practical navigation, but if a more accurate time of transit of the moon is desired, find the above L. M. T. of local transit, apply the longitude to obtain the G. M. T. of local transit, and for this G. M. T. take out the moon's right ascen- sion which is given in the Nautical Almanac, pp. V-XII, for each hour of G. M. T. with corresponding minute differences. The moon being on the upper branch of the meridian, its right ascension is the L. S. T., which can be reduced to L. M. T. (Art. 193). In the solution on page 399 will be shown the method of re-correcting the L. M. T. of transit of the moon for the differ- ence between itself and the approximate L. M. T. as found above. ( d) To find the L. M. T. of transit of a planet across the upper branch of a given meridian. In the case of a planet, the Nautical Almanac gives the G. M. T. of the transit over the Greenwich meridian for the nearest tenth of a minute for each day of the year. The dif- ference of the times for two successive days will give the daily retardation or acceleration. This divided by 24 and the result multiplied by the number of hours of longitude, + for West longitude, ( ) for East longitude, will give the retarda- tion or acceleration to be applied to the Greenwich time of Greenwich transit to give the L. M. T. of local transit. As in the case of the moon, if the sum of the approximate time of transit of the Greenwich meridian and the retardation (or acceleration) is less than 12 hours, the time of the transit of the planet should be taken out of the Nautical Almanac for the given civil date; if that sum is greater than 12 hours, the time of transit must be taken for the day before the given civil day. TIME OF MOON'S TRANSIT 399 a g g ft O o o g a 05 CO lO 7i O OS *H O a ** s 43 7? "* d S rt H ll s d* O ^ rt -M 43 li "2 w I *> W CO SfS 1-8 ^1 60 d pprox. M. T. of transi the meridian of Gre etardation for Long. ] . M. T. of local trans: ong. East 1 ^0 "o 4 M nlnl o d 12 hours ; therefore, the astronomical date corresponding to the time of transit April 5, civil date, is April 4. Acceleration. B. A. M. Q h m h m s Approx. M. T. of Gr. Tr. April 4 14 41.3 For 24h, 4m.l! At G. M. N. 48 31.69 Acceleration for A = 3^ .5 W - 0.6 For IN 0.17 Corr. G. M. T. 2 69.174 Approx. L. M. T. of local transit 14 40.7 For A. W, 0.695 61 30.864 Longitude West + 3 30 G. M. T. of local transit ) = 18 10.7 April 4 (approx.) h m s s April 4, R. A. of Mars at G. M. noon 15 33 18.18 April 4, H. D. 0.259 Correction for G. M. T. (3* diff.) 5.60 April 5, H. D. - 0.389 R. A. of Mars on meridian = L. S. T. 15 33 13.58 Change in 34*> 0.130 R. A. M. corrected for G. M. T. 61 30.864 " " l^ 0.0054 L. M. T. of transit of planet Mars, April 4, 14 40 41.716 "9.1h 0.049 or civil date April 5, (a. m.) 3 40 41.716 April 4, H. D. 0.259 Mean H. D. 0.308 G. M. T. 18'> .18 Correction 5 8 .6 197. To find the time of transit of the moon, a planet, or of a given star across the lower branch of a given meridian. To find the time of a body's lower culmination, the L. S. T. is taken as 12 hours plus the right ascension, or, what amounts to the same thing, 12 hours may be added to the longitude of the place. The latter method is preferable when finding the approximate times in case of the moon and planets. W. T. OF SUN'S TRANSIT 401 Ex. 134. Find the L. M. T. of the lower culmination of the star a Argus (Canopus) in longitude 60 East on April 4, a. m., 1905. In this case (12 hours + *'s K. A.) = 18 h 21 m 50 8 .46 = L. S. T. at the instant of lower culmination. h m a April 3, R. A. M. Q at G. M. noon 44 35.13 Reduction for A 60 E, Table III 39.426 The sidereal time of local hrs. 43 55.704 The L. S. T. of lower culmination 18 21 50.46 The sidereal interval from mean noon 17 37 54.756 Reduction of sidereal to a M. T. interval, Table II 2 53.313 The L. M. T. of lower culmination April 3, 17 35 01.443 Or civil time April 4, (a. m.) 5 35 01.443 198. To find the watch time of transit of a given heavenly body across the upper branch of a given meridian. The simplest and most practical way of observing the me- ridian altitude of a heavenly body is to calculate beforehand its watch time of transit, and then to observe the altitude when the watch indicates that time. (a) Watch time of sun's transit. In the case of the sun, the a. m. longitude brought up to noon by means of the run in longitude from the time of a. m. sight to noon, expressed in time, is the G. A. T. of noon of the given astronomical date, if in West longitude ; or, if in East longitude, it is a negative, or ( ), G. A. T. of the given astronomical date. For this G. A. T. take out the equation of time, and find the G. M. T. of noon ; apply the chronometer correction with the sign of application reversed, and get the C. T. of noon from which, by subtracting the C W, find the watch time of local apparent noon. Every navigator should do this be- fore going on deck to observe his meridian altitude. Another 402 NAUTICAL ASTRONOMY way of arriving at the same result is to obtain from his fore- noon sight the watch error on L. A. T., and apply to this error, the difference in longitude for the run from sight to noon. Ex. 135. April 4, 1905, in Long. 85 30' W., given the C W = 5 h 52 m 05% chronometer fast on G. M. T. 5 m 03 S .38, find the W. T. of local apparent noon. Long. = G. A. T. of local ) J " Equation of time apparent noon April 4 f At G. A. H. j .., 1Q g2 n D _ m Equation of time + 3 06.62 April 4 j G. M. T. of local apparent noon 5 45 06.62 Corr. - 4.20 G. A. T. 5.7 Chronometer fast on G. M. T. + 6 03.38 Eq. of T. 3 06.62 Corr. 4.20 C. T. of local apparent noon 5 60 10 + to App. T. C W 6 63 05 W. T. of local apparent noon 11 58 05 Ex. 136. January 20, 1905, in Long. 132 15' E., if the C W is 3 h 17 m 30 s , and the chronometer is slow on G. M. T. 6 m 19 s . 2 9, what is the watch time of local apparent noon ? h m Equation of time Long. = G. A. T. of noon Jan. 20,(-) 8 49 00 At G A N or G. A. T. is Jan. 19, 15 11 00 T 'n > 11 05.79 H.D. + .739 Equation of time + 10 69.29 ** G. M. T. of apparent noon 15 21 69.29 C <> rrectlon 6.5 G. A.T.-8.8 Chronometer slow ( ) 6 19.29 C. T. of local apparent noon 8 15 40 C-W 8 17 30 Eq. of T. 10 59.29 Corr. - 6.5 + to App. T. W. T. of apparent noon 11 58 10 (&) Watch time of a star's transit. In the case of stars, the right ascension at the instant of upper transit is the L. S. T. Knowing the longitude, find the corresponding G. M. T. of local transit ; apply the chronometer correction and C W as in Exs. 135 and 136 and get the watch time of transit. Remember that at the instant of lower transit the L. S. T. equals the right ascension plus 12 hours. Ex. 137. January 10, 1905, in longitude 5 h 32 m 15 s West, find the watch time of upper transit of the star a Aurigse W. T. OF TRANSIT OF MOON OR PLANET 403 (Capella) if the C W is 5 h 35 m 10 s and the chronometer slow on G-. M. T. 2 m 04 S .018. The star's E. A. = 5 h 09 m 41 S .68 = L. S. T. at transit. Jan. 10, R. A. M. Q at G. M. noon 19 17 21.19 Reduction for Long. (5 h 32 m 15 s W), Table III + 54.58 The sidereal time of local hrs. 19 18 15.77 The given L. S. T. = star's R. A. 5 09 41.68 The sidereal interval 9 51 25.91 Reduction Table II 1 36.892 L. M. T. of star's local transit 9 49 49.018 Longitude West + 5 32 15 G. M. T. of local transit 15 22 04.018 Chronometer slow on G. M. T. ( ) 2 04.018 C. T. of star's local transit 3 20 00 C W 5 35 10 W. T. of transit of star Capella 9 44 50 (c) Watch time of the transit of the moon or a planet. For the moon or planets, find from the Nautical Almanac the G. M. T. of local transit to the nearest tenth of a minute (Art. 188 and Art. 189), apply the chronometer correction and C W as above and find the W. T. of transit. 199. To find the hour angle of any heavenly body at a given time and place. (a) In the case of the sun, the hour angle reckoned posi- tively from the upper meridian towards the West is the L. A. T. If the sun is East of the meridian, the hour angle is negative and is equal to 24 hours the apparent time. Having then a given mean time or sidereal time, the longi- tude or G. M. T. being known, the L. A. T. may be found by Art. 190, Art. 193, or Art. 194. 404 NAUTICAL ASTRONOMY Ex. 138. April 10, a. m., 1905, Long. 129 30' 45" E., L. M. T. 10 h 25 m 19 s , find the true sun's hour angle. Local ast. mean time April 9, Longitude East G. M. T. April 9, or April 10, ( 10. 21) L. M. T. (astronomical) April 9, Equation of time L. A. T. = sun's H. A. April 9. or April 10, Equation of time At G.M.N. I m s April 10 f 1 28.25 Os.685 Corr. + C.99 10^.21 Eq. of T. 1 35.24 + 6<>.99 (-) to M. T. h m 8 22 25 19 1 35.24 + 23 23 43.76 - 1 36 16.24 Ex. 139. April 6, 1905, a. m., Long. 162 49' 15" W., L. S. T. = 18 h 42 m 10 s , find the H. A.s of mean and true suns. Reduction for longitude Sidereal time of local O h The given L. S. T. The sidereal interval Reduction to a M. T. interval L. M. T. April 5. I (H. A. Mean Sun) f Longitude W. G. M. T. April 6, h m a )on 62 28.24 + 1 46.99 Equation of time ( ) to M. T. in B 54 15.23 18 42 10 2 35.86 H. D. 8 .721 - 3.32 G. M. T.+ 4h.6 17 47 54.77 - 2 54.951 2 32.54 Corr. - 3s.32 17 44 59.819 + 10 51 17 L. M. T. April 5, 17 44 59.819 Eq. of time - 2 32.54 4 36 16.819 Vprn5 =H ' A 'f 174327.279 or April 6, (-) 6 17 32.721 Ex. 140. January 3, 1905, a. m., in Long. 150 Q 09' 54" W., the W. T. of obs. of the sun was 8 h 04 m 35 s , C W 10 h 07 m 15 s , chronometer fast on G-. M. T. 7 m ll s .5. Find the true sun's H. A. h m e W = 8 04 35 C-W 10 07 16 C. 6 11 60 C. C. ( ) 7 11.5 G.M.T. ' Jan 604 38.5 Equation of time H. D. At G. M. noon 4 28.38 -f ls.161 Corr. G. M. T. + 7.06 G. T. 6h.Q8 Eq. of T. 4 35.44 Corr.+ 7.06 ( ) to M. T. h m s G. M. T. 6 04 38.5 Long. W 10 00 39.6 L. M. T. 20~03~58T|< Sirius ( ) 2 34 15.68 L. S. T. R. A. >|< Achernar H. A. >|< Achernar 3 10 30 f 2 19 30 5 30 00 h m 8 55 21.94 50.19 3 10 30 4 06 42.13 4 06 42.13 1 34 07.75 +T32 34.38 Attention is called to the fact that for the G. A. T. the equation of time might have been taken from page I and ap- plied to the G. A. T. to obtain G. M. T. and then the L. S. T. found from the G. M. T. as in previous examples. H. A. OF POLARIS 407 g! H t S O C g CO a g .-r o o | -2 Us * . " ^ s? 03 O S .2 ^ -> .PH 8s +j 00 | _ ) i p s *ii j DQ iO 2 o o 5 el H ' ' ^ ~& ^ < &.a d< ii a H . o fl CO W Oc^ SS 05 o o H CO d"g, o" 408 NAUTICAL ASTRONOMY 200. To find the local time when the hour angle of a par- ticular heavenly body and the Greenwich time are known, or when the hour angle of a fixed star and the longitude are known. In the case of the sun, its hour angle reckoned westward is the L. A. T. of the given astronomical day ; if the sun is East of the meridian, the L. A. T. is 24 hours the hour angle, and the date is that of the preceding astronomical day. This L. A. T. may be reduced to mean or sidereal time (Art. 190 or Art. 194), as required, the Greenwich time or longitude being known. In the case of any other heavenly body, find its right ascension for the Greenwich instant. This, added algebraic- ally to the hour angle, will give the L. S. T. Subtracting from this the right ascension of the sun (true or mean), taken from the Nautical Almanac for the Greenwich instant, the remainder will be the hour angle of the sun (true or mean), and the hour angle will be the local time (apparent or mean), or 24 hours minus the local time (apparent or mean), according as the hour angle is -j- or ( ) . If, in finding the hour angle by subtracting the right ascen- sion from the L. S. T., it is seen that the L. S. T. is less than the right ascension, and it is desired to express the hour angle positively, add 24 hours to the L. S. T. before performing the subtraction. Ex. 145. April 16, 1905, G. M. T. 10 h 18 m , the moon's hour angle is 2 h 30 m East of the meridian of a certain place. Find the L. M. T. Right Ascension of the Moon, L. S.T. and L. M.T. R. A. M. April 16 at IQh H 20 03.43 M.D. + 2.3708 April 16 at G. M. noon f 1 35 50.32 Corr. for 18 minutes +42.674 18n Corr. for G.M. T. 141. Corrected R. A. C 11 20 46.104 Corr. + 428.674 Moon '8 H. A. East 2 30 00 L. S. T. 8 50 46.104 Corrected R.A.M.Q 1 37 31.842 L. M. T. April 16, 7 13 14.263 R. A. M. 1 37 3U STARS TO CROSS MERIDIAN 409 Ex. 11+6. April 9, astronomical time, 1905, at a given in- stant the hour angle of the star a Canis Minoris (Procyon) at Annapolis, Md., was 3 hours West of the meridian. At the same instant the hour angle of the star a Leonis (Regulus) was 1 hour East of a second meridian. Find the L. M. T. at each meridian. h m s h m 3 R. A. of star Procyon 7 34 20.18 R. A. of star Regulus 10 03 19.95 H. A. do. do. + 3 00 00 H. A. do. do. 1 00 00 L. S. T. (at Annapolis) 10 34 20.18 L. S. T. (2d meridian) 9 03 19.95 April 9, R.A.M. at GL M. noon 1 08 14.45 L. S. T. (Annapolis) 10 34 20.18 Reduction for Long. W, Tab. Ill 4 60.258 Diff . longitude 1 31 00.23 Sidereal time local Q h 1 09 04.708 L. M. T. at Annapolis 9 23 42.868 The given L. S. T. 10 34 20.18 L. M. T. (2d meridian) 7 52 42.638 Sidereal interval 9 25 15.472 or April 9, (p. m.) 7 52 42.638 Reduction Tab. II - 1 32.604 L. M. T. at Annapolis 9 23 42.868 or civil time April 9, (p. m.) 9 23 42.868 201. Given two mean times or two apparent times at a given place, to find what bright stars will cross the upper branch of the meridian between those two times. Since the right ascension of a body on the meridian is the right ascension. of the meridian, or, in other words, the L.S.T., it is only necessary to find the local sidereal times correspond- ing to the two given times. Any star whose right ascension lies between the two L. S. Ts. thus determined will cross the upper branch of the meridian between the two given times,, and any star whose right ascension lies between the two local sidereal times increased by 12 hours of sidereal time will pass the lower branch of the meridian between the two given times. The " mean place catalogue " of stars in the Nautical Al- manac is the more convenient one to use for this purpose. The visibility of a star at the time of its transit over any meridian will depend on the latitude of the place and the de- clination of the star, which determine whether the star is above or below the horizon. 410 NAUTICAL ASTRONOMY At sea, the mean or apparent time of transit of a heavenly body for a certain meridian is obtained with the idea perhaps of observing the body's altitude when on the meridian. It must not be forgotten that the ship's clock was regulated to apparent time at noon, and that the navigator must learn his watch error on the local time of the meridian over which he is to observe a transit. If his watch was correct at noon, it will be too fast on the local time of a meridian to the westward, too slow on the local time of a meridian to the eastward of the noon meridian by four minutes of time for each degree difference of longitude. It would be well, how- ever, for the navigator to carry a watch regulated to G. M. T., and, having found the Greenwich mean time corresponding to the required transit, to observe by that watch. Ex. 147. What stars of a magnitude greater than the second magnitude crossed the upper branch of the meridian of Annapolis, Md., above the visible horizon, between the hours of 8 p. m. and 12 midnight t)f 75th meridian West longitude, mean time, January 18, 1905. h m s h m s 75th meridian mean time 8 00 00 12 00 00 Long, of 75th meridian W 5 00 00 5 00 00 G. M. T Jan. 18, 13 00 00 17 00 00 Longitude of Annapolis West 5 05 56.5 5 05 56.5 Local astronomical mean times 7 54 03.5 11 54 03.5 R. A. M. Jan. 18 at G. M. noon 19 48 53.64 19 48 53.64 Correction for G. M. T. + 208.134 + 247.56 L. S. Ts. = limits of R. As. 3 45 05.274 7 45 44.70 All stars of a greater magnitude than the second whose right ascensions fall between the above limits and whose South declination is <51 01' 07" S. a Tauri. e Orionis. 8 Canis Majoris. a Aurigae. a Orionis. a 2 Geminorum. (3 Orionis. a Canis Majoris. a Canis Minoris. 13 Tauri. Canis Majoris. ft Geminorum. CHAPTER XV. CORRECTIONS TO AN OBSERVED ALTITUDE. 202. The observed altitude of a heavenly body above the sea horizon, at a given place, is the altitude of the body as indicated by the reading of the sextant with which the obser- vation was made, after correction for the index error (I. C.) previously explained. The true altitude of the body, at the given place, is the altitude of its center observed above the celestial horizon/ the eye of the observer supposed to be at the center of the earth. This point is selected as the common point to which to refer observations made at the surface, when combining them with the tabulated elements from the Nautical Almanac, in the solution of the astronomical triangle. To reduce an observed altitude of a heavenly body to a true altitude it is necessary to apply the following correc- tions: Dip, refraction, parallax, semi-diameter. Theoretic- ally they should be applied in the above order; following that order would give: after applying dip, (1) the apparent altitude of the limb; after applying refraction and parallax, (2) the true altitude of the limb; after applying semi-diameter, (3) the true altitude of the center. When an artificial horizon is used, the observed and ap- parent altitudes are the same ; in other words, there is no cor- rection for the dip. As already explained under the head 412 NAUTICAL ASTRONOMY ^ of artificial horizon, when a body is observed, the artificial horizon being used, the reading of the sextant is first cor- rected for I. C., and the corrected reading divided by 2 to get what is known as the observed altitude. In case of fixed stars, owing to their great distances, the semi-diameter and parallax are inappreciable, so that the only corrections to be applied are I. C., dip, and refraction. In case of planets, for sea observations, parallax and semi- diameter may be disregarded; however, if the observation is made with a telescope so powerful that the limb can be dis- tinguished, the semi-diameter should be applied. For refined observations ashore, both these corrections should be applied. For the ordinary sea observations of a planet, it will be sufficient to correct the altitude for I. C., dip, and refraction. In ordinary nautical practice, it is unnecessary to follow the theoretical order, except that in the case of the moon, it is essential to find, first, the apparent altitude of the moon's center, and for this to take out the correction for parallax and refraction combined. (Bowditch, Table 24.) 203. Refraction. It is a fundamental law of optics that a ray of light, when passing obliquely from one medium into another of different density, is bent towards, or from, a normal to the separating surface at the point of entrance, according as it passes from a lighter into a denser medium, or the reverse. The ray before entering the second medium is called the incident ray, after entering it the refracted ray. The inci- dent ray makes with the normal what is called the angle of incidence, the refracted ray makes with the normal the angle of refraction, and the difference between these two angles is called the refraction. Astronomical refraction. A ray of light from a heavenly body must pass through the atmosphere before reaching the observer. REFRACTION 413 The earth's atmosphere may be considered as formed of concentric spherical strata, that nearest the surface of the earth being of greatest density, and each succeeding stratum decreasing in density as its distance from the surface in- creases till the upper limit of the atmosphere is reached at a height of perhaps 50 miles from the surface. If the space between the upper limit of the atmosphere and a star be regarded as a vacuum, or filled with a medium which exerts no sensible effect on the direction of a ray of light, its FIG. 101. path will be a straight line till it meets the upper limit of the atmosphere. At this upper limit, the effect of refraction is very small, but, as the ray continuously passes through the atmosphere whose density increases by insensible degrees from stratum to stratum of infinitely small thickness, its path is a curve concave to the surface of the earth ; the plane of its path being in the plane of the normals which meet at the center of the earth. The last direction of a ray, or that at which it enters the eye of the observer, is in a tangent to the curve at this point 414 NAUTICAL ASTRONOMY and indicates the direction in which a body appears to the observer; so it is apparent that the effect of the bending of the rays is to apparently increase the altitude of the body without altering its azimuth. Astronomical refraction, then, which is the difference of direction between the ray that enters the eye of an observer and of the same ray before entering the atmosphere, making as it does an altitude appear greater than it really is, must be subtracted from an observed altitude of a body. The ray from a star 8 f entering the atmosphere at B (Fig. 101) is bent into the curve BA. The observer at A apparently sees 8 in direction AS'. The angle EBS is the angle of incidence; ZAS', the angle of refraction; and the ratio of their sines is a constant at a given place for a given at- mospheric condition. Refraction equals EBS EDS', be- cause the angle between AS' and B8 is equal to the difference of the angles that these lines make with any straight line cut- ting both. The refraction for a mean state of the atmosphere, that of a height of barometer of 30 inches and temperature of 50 F., can be found in Table 20A, Bowditch. A rise of temperature, or a fall of barometer, indicates a decrease of density of the atmosphere and hence a diminution of refraction. A fall of temperature, or a rise of barometer, would indicate the reverse (see Tables 21 and 22, Bowditch). In Table 20B, Bowditch, will be found, in the case of the sun only, the value of combined parallax and refraction. Refraction is zero when the body is in the zenith, about 36' when it is in the horizon, and for intermediate altitudes may be said to vary as the tangent of the zenith distance of the body, provided the zenith distance does not exceed 80. Owing to the irregularity of refraction at low altitudes it is advisable not to observe, at sea, altitudes of less than 10. The oval form of the sun and moon after rising and before PARALLAX 415 setting is due to the difference of refraction for the altitudes of the lower and upper limbs. Kefraction affects the dip, decreasing.it by about -j^th of the whole. 204. Parallax. In general, parallax may be denned as a change in direction of an object due to a change of the point of view. In astronomical observations, the observer is on the surface of the earth, and it is desired to reduce observations to what they would be if the observer were at the center of the earth. It is by the appli- cation of parallax that obser- vations are so reduced. Geocentric parallax is the angle at the body subtended by that radius of the earth which passes through the observer's position at the surface. When the heavenly body is in the horizon at H (Fig. 102), this angle has its greatest value and CAR is a right angle. Let- FIG. 102. ting this angle, called the hori- zontal parallax, be represented by P, the earth's equatorial radius by R, the distance of the body by d, we have sin P = R d The value of P from this formula is given in the Nautical Almanac for the sun, moon, and planets. Parallax in altitude. When a heavenly body is observed in any position other than in the horizon, the parallax to be applied is known as parallax in altitude. 416 NAUTICAL ASTRONOMY In triangle CAS, p is the parallax in altitude, ZAS the apparent zenith distance = z' = 90 - - h', ZC8 the true zenith distance of body = z, d the distance of the body, and we have sin sin z' sin p = sin P cos h'-. Since p and P are small angles, they are proportional to their sines; therefore, p = Pcosh'. (143) Parallax is additive to the observed altitude. 205. Dip of the horizon. The visible sea horizon is the small circle where tangents from the observer's eye meet the sea. The sensible and celestial horizons have already been defined (Art. 138). The dip of the horizon is the angular depression of the visible below the celestial hori- zon, and is due to the elevation of the observer's eye above the surface. In. Fig. 103, let BR be a portion of the earth's surface, C the center, CO a radius pro- longed to A, the eye of an ob- server; CB and CB' 9 radii making angles of 90 respec- tively with AB and AB', tan- gents to the surface. If this figure be revolved about AC, HH' will generate a plane par- allel to the celestial horizon and either A B or AB' will gener- ate a cone tangent to the earth at the visible horizon. Letting R be the earth's radius in feet, h the height of FIG. 103. DIP 417 observer's eye in feet, and D the dip, since D = HAB = ACS, we have R cos D = R I ^ ; but cos D " * 2 sin 2 -J D, sn = / 7}"\ 2 As Z) is small, sin 2 ^ D" = ( ) sin 2 1", and as h is very V * / small in comparison with R, justifying the assumption that R -f- h is sensibly equal to R, we have /)" - 1 l^h -sml"YlT The value of the mean radius in feet being 20,902,433 feet, Z>" = 63".803 yh, (144) D' = 1'.063 V^ ( 145 ) However, the value of the dip, as found above, is affected by refraction which raises the visible horizon, increases the distance at which an object in the horizon can be seen, and lessens the dip, so that when the effect of refraction is to be considered a change must be made in the formula. For a mean state of the atmosphere, barometer 30 inches, thermometer 50 F., it has been computed that the value of the dip, considering refraction, is given by the formula, = 58".801 ^ h (146) sin J. D r ' = 0'.98 \/ h (147) Application of dip. Table 14, Bowditch, gives the dip for various heights of the eye, computed so as to allow for the effect of the refraction of the atmosphere under normal con- ditions. Dip is one of the corrections to be applied to an observed altitude of a heavenly body to obtain the true alti- tude, and is subtractive to the observed altitude, as the visible horizon is below the celestial horizon. 418 NAUTICAL ASTRONOMY Error of dip. The position of the visible horizon, and hence the amount of dip, depends on the relative temperature of sea and air. The horizon is depressed below its mean position, and the dip is increased over the tabulated amount, when the sea is warmer than the air ; the reverse is true, when the air is warmer than the sea. Hence it is easily understood that tabulated dip for given conditions may be in error, and that this error will affect all altitudes observed under those conditions. The error of posi- tion thus caused may be considerable, especially in the Eed Sea and in regions of the Gulf stream. For this reason, the navigator must be cautious and, as experience shows that the error decreases with the height of the observer's eye, it would be well for him to observe from elevated positions. Chauvenet gives the following formula from which to find a correction, always subtractive to D r " : 24021" (* * ) , Corr. = - 1-Q when D r t is the temperature of air, and t that of water, using a Fahrenheit thermometer. 206. To find the distance of the visible horizon for a given height h of observer's eye. It has been seen that refraction reduces the angle of dip and increases the distance of the visible horizon, so that the distance of the visible horizon from an altitude In, when the dip is affected by refraction, may be considered to be the same that it would be from an alti- tude h + x, provided there was no effect of refraction. Let- ting d be the distance of the visible horizon from the height of eye of h + & feet and as before R the radius or the earth in feet, refraction not being considered, d= ^(R + h + x) 2 R 2 2Rh 2Rx 2hx RANGE OF VISIBILITY 419 Since (h -}- x) 2 is very small in comparison with 2R (h + x), let d = \/2R (h + x) y a; is a side of a triangle which, with- out appreciable error, may be considered as right angled, and the angle opposite x may also, without appreciable error, be taken as D D r ; therefore, x d sin (D D r ), x = sin (D D r ) V2# (h + x) ; but D D r = 5".002 hence x = 5.002 \A V^5 (ft + x) sin 1" z 2 = 50.045 (h 2 + for) sin 2 I" sin 2 1" + (25.02) *R 2 h 2 sin 4 1" = 50.045ft 2 sin 2 1" + (25.02) 2 5 2 ft 2 sin 4 1" a _25.025ft sin 2 1" = A V50.045 sin 2 V + (25.02) 2 5 2 sin* I" x =25.Q2Rh sin 2 1" __ h V50.04B sin 2 l /r + (25.02) 2 E 2 sin 4 1" Whence, since R = 20902433 feet, 25.025 sin 2 1" = .01229. 50.045 sin 2 1" = . 02458 1 = (25.02) 2 5 2 sin 4 l /r = .00015 J x = .16955&, and d y2R(h + z) = -\/2.3391Rh in feet. d (in nautical miles) - = 1.15V*- c?= 1.15V^> <^ i n nautical miles 1 , . , (148) d = 1.324 V^, d in statute miles J (149) 207. Kange of visibility at sea. If an observer whose eye is at A, height h feet, sees in his horizon at T (Fig. 103) a light or object of known height h', then since AB' = 1.15V* B'T = 1.15 V*', 420 NAUTICAL ASTRONOMY the distance of the light in nautical miles will be and in statute miles, d = 1.324 ( yh + Table 6, Bowditch, gives the distance of visibility of objects at sea in both nautical and statute miles for a given height of eye. Entering this table with heights of observer and object, respectively, the sum of the corresponding distances will be the distance of the object from the observer. Ex. 148. A light 121 feet above the level of the sea is just visible from a bridge of a steamer 49 feet above the water. Eequired the distance of the light in nautical miles. d = 1.15 ( Vm + V49) = 1.15 (11 + 7) = 1.15 X 18 = 20.7 miles. 208. To find the dip or de- pression of a point nearer than the horizon, as of a land hori- zon. In Fig. 104, x represents the height of an observer, hav- ing in sight a shore horizon B, which corresponds to the visible sea horizon of a height y on the perpendicular through the eye of the observer, and d equals the known distance of B. But by a previous article d = 1.15 Vy (i n nautical miles), therefore V# T^TK> ^ u ^ the dip at height y after correction for refraction is D r " = 58".801 yy = 58 "-8Qld = 51".13d. 1.15 The refracted rays Bx and By make with each other a small angle which represents, without appreciable error, the FIG. 104. SHORE HORIZON 421 difference of dip, or depression of B, as seen from the heights # and y; letting " represent this angle in seconds of arc, since Bxy is nearly right angled at y, we have Tan " =. - x y and, as " is a small x y 6080.27 d tan 1" = 6080.27d tan 1" x d 6080.27^ tan 1" (1.15) 2 6080.27 tan 1" |_25".651d. Eemembering what " is, and knowing that the dip at height y equals D r " = 51". 13d, to find the dip of B at height x, represented by D rx ", we have only to add <" to D r " at height y. Therefore D rx " = 25".479d + 33".924 |. For practical purposes it is only necessary to use the for- mula to the nearest tenth, or f x in feet, in nautical miles, n _ 9 K// K/7 .r QQ// Q a; n nauca , - 3 ' 9 31 DrJ' dip in seconds of arc, ( 160 > corrected for refraction. Shore horizon. When sailing near shore, or when in a harbor at anchor, an observer may be forced to use an alti- tude from a shore horizon. The dip may be calculated by the above formula, or taken out of Table 15, Bowditch. 209. Apparent semi-diameter. The apparent semi-diameter of a body is the angle subtended by its radius at the place of the observer, and for the same body varies with the distance of that body from the observer. The value given in the Nautical Almanac is the angle at the center of the earth sub- tended by the radius of the body. 422 NAUTICAL ASTRONOMY At sea, in sextant observations of the sun and moon, the upper or lower limb is brought into contact with the sea horizon; in observations on shore, when using an artifi- cial horizon, the opposite limbs of direct and reflected images are made tangent to each other. Since the altitude of the center of the body is re- quired, the angular semi-di- ameter of the heavenly body observed must be applied, plus or minus, according as the altitude observed was that of the lower or upper limb ; and as the observation of the limb is reduced, by FlG 10 5. the application of parallax, to what it would be if taken at the center of the earth, it is necessary to find the apparent semi-diameter of a heavenly body as it would be seen at that point. In Fig. 105, let M be the body, }i its apparent altitude, its true altitude, or 90 ZCM. its apparent zenith distance, its true zenith distance, its distance from A, its distance from the center of the earth, 8 = MCB, apparent S. D., as viewed from C, the center of the earth, S' MAB f , apparent S. D., as viewed from A, the -ob- server's position on the surface, li z r z at d APPARENT S. D. 423 R = CA = earth's radius, r = MB = MB' = linear radius of body. From the right-angled triangle MOB, sin 8 = T -T. When the body M is in the horizon of A, AM and CM are sensibly equal and, hence, the angle 8 is called the horizontal semi-diameter. 7? 7? It has been shown that sin P = -=- or d = . . d sin P Therefore, sin 8 =-5- sin P. Since 8 and P are small, they are proportional to their sines, hence 8 ~ P. ~ is a ratio, constant for any particular body, and, repre- senting it by C, we have log 8 = log C + log P. (151) For the moon, ^ = .272, so that having the moon's hori- zontal parallax, its semi-diameter may be gotten by multiply- ing it by .272 ; however, it is just as easy to take it out from the Almanac for the given Greenwich mean time. The Nautical Almanac gives the semi-diameter, also the horizontal parallax, of the sun, moon, and planets. To find the apparent semi-diameter as viewed from the ob- server's position on the surface : From the right-angled triangle AB'M (Fig. 105) sin S' = ~ , also, from AMC, f = 5BJL = 5* and d' = d B * d sin zf cos h cos h Substituting value of d r in expression for sin 8' 9 we have, sin 8' ~ COS ^ ; but ~ = sin 8, d cosh ' d i t therefore, sin 8' = sin 8- 424 NAUTICAL ASTRONOMY Now S' and 8 are small angles and proportional to their sines ; therefore 8' = 8 cos f ( 152 ) cos h Prom this formula S f may be found when 8, h', and h are known. As h is greater than h', cos h is less than cos h' ; therefore S' is greater than 8 t or the semi-diameter increases with the altitude of the body. This excess is called the aug- mentation, but is of appreciable value only in the case of the moon, for which body it is tabulated in Table 18, Bowditch. 210. To find the augmentation of the moon's semi-diameter. Let A$ be the augmentation, ' cos ^' < cos ^' cos ^ cos h' cos h = 2 sin (h' h) sin (h' + h) 2 sin \ (h h') sin i (h f sin i (ft 7 O sln 2 (^ cos /& Now, since & h' p = parallax in altitude and is very small, 2 sin $(h h') = 2 sin= p sin 1" = P cos h' sin 1". ^ As AS is small, | (^ + Ti) may be taken as h' and cos &' may be substituted for cos h; . . (P cos ft' sin 1") sin K' therefore AS = 8 1 _ - - cos h &8 = 8P sin h' sin \" =~ 8 2 sin h' sin 1"; T p but sin 1" is a constant for any one body, and may be rep- resented by K, therefore, A# = K8? sin h'. In the case of the moon, ^- = 3.6646 and log K 5.2496. A more rigorous formula may be found in Chauvenet's Astronomy, but the above will not involve an error greater than -". CORRECTING ALTITUDES 425 211. The following symbols may be used: Inst. m (or _Q.) Instrumental altitude of sun's lower limb. Obs. .Q. The above corrected for I. C. Inst. T-7 Instrumental altitude of sun's upper limb. Obs. 7!7 The above corrected for I. C. -0- True altitude of sun's center. 2 O Twice the altitude of sun's lower limb. IT Altitude of moon's upper limb. Jt, Altitude of moon's lower limb. -- Altitude of moon's center. Alt. * Altitude of star. 212. Theoretical and practical methods. The various cor- rections to an instrumental altitude are applied as indicated below in the examples given. Sometimes the theoretical and practical methods may give slightly differing results, which is a matter of no importance at sea. Bowditch's Tables are used. Tables II and III in the back of this book were not used in the examples given as illustrations, as the examples were solved before their construction. Ex. 149. January 3, a. m., 1905, the instrumental alti- tude of the sun's lower limb was 23 42' 00" I. C. + 1' 20". Height of eye 45 feet. Find the true altitude of the center of the sun. THEOBETICALLY. PRACTICALLY, AT SEA. o / // o r ft / // Instrumental 23 42 00 I. C. -f l 20 Corr. 23 42 00 + 8 58 S. D. -f I. C. -f 16 18 1 20 Observed 23 43 20 Dip (Tab. XIV) - 6 36 -e- 23 50 58 D. - p. & R. - 6 36 2 04 Apparent 23 36 44 Corr. -I- 8 58 p. & R. (Tab. XXB) - 2 04 True 23 34 40 S. D. + 16 18 True-0- 23 50 58 426 NAUTICAL ASTRONOMY Ex. 150. January 19, 1905, the instrumental altitude of star Arcturus was 29 22' 10". I. C. + 2'. Height of eye 45 feet. Find the true altitude. THEORETICALLY. PRACTICALLY. Oil! o / // i it inst. alt. * 29 22 10 Inst. alt. * 29 22 10 I. C. + 2 00 I. C. Observed alt. * Dip (Tab. XIV) + 2 00 Corr. True alt. 6 19 D. R. Corr. - 6 36 - 1 43 29 24 10 - 6 36 * 29 15 51 6 19 Apparent alt. * Ref. True alt. * 29 17 34 1 44 29 15 50 Ex. 151. January 8, 1905, the altitude of the lower limb of the sun, as observed with an artificial horizon, was 34 36'. I. C. 2'. Eequired the true altitude. THEORETICALLY, o / // Instrumental 20 34 36 00 I. C. - 2 00 PRACTICALLY. 20 34 36 00 S. D. I. C. - 2 00 p. & R. / /; + 16 18 - 2 57 Observed 20 34 34 00 Observed" 17 17 00 p. & R. (Tab. XXB) - 2 57 2)34 34 00 Corr. -f 13 21 Obs. 17 17 00 Corr. "" + 13 21 -0- 17 80 21 True 17 14 03 S. D. + 16 18 True -0- 17 80 21 To correct an instrumental altitude of the moon. Owing to the rapid change of the moon's semi-diameter and hori- zontal parallax, they must be reduced for the Greenwich mean time of observation; also, as the moon is nearer the observer at all altitudes than when in the horizon, the semi-diameter must be corrected for augmentation (Table 18, Bowditch). The combined correction for parallax and refraction will be found in Table 24, Bowditch, in which the arguments are the horizontal parallax at the top and apparent altitude of the moon's center at the side, the parallax being at intervals of CORRECTING ALTITUDES 427 one minute, the apparent altitude at intervals of ten minutes of arc. For seconds of parallax, enter the table abreast the approximate correction where the arguments are tens of sec- onds at the side and units at the top ; opposite the former and under the latter is the correction to be added. The addi- tional correction for minutes of altitude will be found in a table on extreme right of page, to be applied as there di- rected. Hence the rules : (1) For the given instant find the corresponding G. M. T. for which correct the moon's semi-diameter and horizontal parallax; also find from Table 18 the augmentation of S. D. (2) To the instrumental altitude apply the first correction, consisting of the algebraic sum of the I. C., dip, and aug- mented semi-diameter. The result will be the apparent alti- tude of the moon's center. (3) With the horizontal parallax and the apparent alti- tude find from Table 24, Bowditch, the second correction (for parallax and refraction combined) which, added to the ap- parent altitude of the center, will give the true altitude of the center. Ex. 152. January 19, 1905, in .longitude 100 30' W., at ll h 09 m 41 s local mean time, the instrumental altitude of the moon's lower limb was 61 04' 52" bearing N. I. C. +0'. Height of eye 30 feet above the sea level. Kequired the true altitude. L. M. T. Jan. 19 11 09 41 Instrumental alt._C_61 04 62 S. D. + 16 03.6 H. P. Longitude W + 6 42 00 1st correction" + 10 56.2 Aug. + 14.6 , G. M. T. Jan. 19 17 51 41 Am>flrent fllt . - SliT^ L\ a + _ 58 80.6 h m s 11 0941 + 6 4200 1 II Instrumental alt._C_61 04 62 1st correction*' + 10 56.2 S. D. + 16 03.6 Aug. + 14.6 i.e. + 000 Dip. - 628 1st I corr./+10 68.2 17 51 41 17h 51m.68 Apparent alt. -e-6l 15 48.2 Par. & Ref . (Tab. 24) + 27 46 True Alt. >6- 61 43 34.2 To be strictly accurate the H. P. found from the Nautical Almanac should be further corrected for the latitude of the place of observation (Table 19, Bowditch). However, it is usually disregarded, as in the above example. 428 NAUTICAL ASTRONOMY Correction of a planet's altitude. Theoretically a planet's altitude should be corrected for I. C., dip, refraction, parallax, and S. D. ; practically, at sea, only the first three are applied. The parallax is gotten from Table 17, Bowditch, the argu- ments being altitude and horizontal parallax. Ex. 158. April 3, 1905, observed the lower limb of planet Mars 32 15'. I. C. + 1'. Height of eye 20 feet. Kequired the true altitude. THEORETICALLY. Instrumental alt. of Mars Index correction Obs. alt. of lower limb Dip Apparent alt. of lower limb H. P. 12". 4: Par + 10'M Ref. 1' 32"/ True alt. of lower limb Semi-diameter True alt. of center The only difference between the use of planets and fixed stars for navigational purposes at sea lies in the fact that the E. A. and declination of a planet must be corrected for G. M. T., the corrections of the altitudes being practically the same, I. C., dip, and refraction. 213. Correction of altitude for run. A ship at sea seldom remains stationary between observations; as the ship moves, the zenith of the observer describes an arc on the celestial sphere, and the number of minutes in the arc are equal to the number of nautical miles run by the ship. A heavenly body, if observed simultaneously from the two ends of the ship's run, would have the two zenith distances and hence the two altitudes differing by an amount called, "the correction for the run." This must be found whenever it is desired to re- duce an observed altitude for a change of the observer's posi- tion. PRACTICALLY, AS AT SEA. O I II o r n / // 32 15 00 1 00 $ 32 15 00 Corr. 4 55 i.e. Dip Ref. Corr + 1 00 -4 23 -1 32 32 16 00 4 23 (-J 1 32 10 05 -4 55 32 11 37 1 22 32 10 15 7 32 10 22 CORRECTION FOR KUN 429 Now suppose an observer whose zenith is Z, observes a heavenly body M; after running a certain distance ZZ 1 (Z f being in one of several different positions Z\ , Z' 2 . . . Z' 6 ), he again observes the same body. To compare the two obser- vations, one must be reduced to what it would have been if observed at the other place. Let M be the position of the heavenly body, at the first observation, supposed fixed during the run, or as the observer's FIG. 106. zenith shifts from Z to Z* '; ZZ', the distance sailed in sea miles; C = NZZ', the course sailed estimated from the ele- vated pole ; and Z = NZM, the body's azimuth. If the course is directly towards the body, Z shifts to Z\ , and the body's zenith distance is lessened, or altitude in- creased by a number of minutes of arc equal to the run ZZ\ , in sea miles. If the course is directly away from the body, Z shifts to Z' 2 and the altitude is diminished. 430 NAUTICAL ASTRONOMY If the zenith shifts to any other position, by regarding tlio triangle ZZ'M as a spherical triangle (which it is), the re- duction may be obtained by a rigorous formula from Chau- venet : Afc = d cos (C Z) 4 d 2 sin 1' tan h sin 2 (CZ). (153) Now, with M as a center, and radius MZ', describe an arc cutting the first bearing line in d, that is from Z' 3 in d 3 , from Z\ in d 4 , from Z\ in 90 , Afe is if (C ~ Z) is 90 . Since A/i is zero when (C ~ Z) equals 90, or is smaller as (C ^ Z) approaches 90, it is better to reduce that alti- tude for which the difference of course and azimuth is nearer 90. (154) CORRECTION FOR RUN 431 If the second altitude is to be reduced, then C in the formula is the course reversed. If a single course and distance are run, (C ~ Z) may be by compass, but if a traverse is made from Z to Z r then the magnetic (or true) course and bearing should be used. The traverse table may be used by taking (C ~ Z), or 180 (C ~ Z) if (C ~ Z) is >90, for the course, the run as a distance, and looking for the correction in the difference of latitude column. This, in minutes and tenths of a minute of arc, is to be added to the first altitude if (C ~ Z) is <90, subtracted if (C ~ Z) is >90. Rules in finding correction for run. (1) Take a bearing of the heavenly body at each observa- tion. (2) For the elapsed time between observations find the course made good and distance run. (3) For the distance in sea miles find the altitude cor- rection Aft, in minutes and decimals of a minute of arc, to be applied as already explained. Ex. 154. On January 18, 1905, in 1ST. Lat. and W. Long., a. m. time, the sextant altitude of sun's lower limb, bearing K 137 E. (true), was 22 18' 20". I. C. + 2' 30". Height of eye 21 feet. After running NE.. (true) 33 miles, observed a p.m. altitude of sun's lower limb 21 28' 50". I. C. + 2' 10". Height of eye 21 feet. Reduce first altitude to what it would have been had it been observed at the same time at the second place. Course N 45 E Azimuth N 137 E o tt f (C~Z) =92.. cos 8.54282 At 1st place O 22 18 20 d = 33' log 1.51851 Aft 109 Ah = V 09" = I'.iS 'log 0.06133 At 2d place Q~li2 17 11 CHAPTER XVI. SOLUTION OF THE ASTRONOMICAL TRIANGLE. 214. In a consideration of the astronomical triangle, or any of the parts thereof, in this and succeeding chapters, the following notation will be used : L = latitude. h s = the sextant altitude of the heavenly body observed. h = the true altitude of the heavenly body. h r = the apparent altitude of the body. h r= the meridian altitude of the body. z = the true zenith distance of the body. z' = the apparent zenith distance of the body. Z = the meridian zenith distance of the body. d = its declination. p = its polar distance = 90 d (algebraically). Z = its azimuth measured from the elevated pole towards the East or West, from to 180. Z*$. = its azimuth measured from North around to the right, from to 360. i = its hour angle. 'A = its amplitude. M = its position angle. 215. Many of the various problems that confront a naviga- tor are solved by a solution of the astronomical triangle whose sides are 90 Ji f 90 L, and 90 d or p, and whose angles are t, Z, and M. Fig. 107 represents this triangle projected on the plane of the horizon. By spherical trigonometry, when any three of the parts are known, the others can be found. The posi- tion angle M is of no importance to the navigator, and it will not be further considered. So that for practical purposes FINDING TRUE ALTITUDE 433 there are given three to find the remaining two of the quanti- ties h f Z, t, L, and d. As the latter quantity is tabulated in the Nautical Almanac, the methods of finding the other quantities will be considered in the order given above. 216. The parts of the triangle, when used in the solutions as given data, are thus found. The latitudes and longitudes of places on shore, at which observations may be taken, are found from charts, tables of maritime positions as Appendix IV, Bowditch, or from sailing di- rections. At sea, they are found from previous determinations brought forward, or from subse- quent determinations carried back by dead reckoning; or from practically simultaneous observa- tions where one body is so favor- ably located that an error in time will not affect the resulting lati- tude to be used in a sight for time taken simultaneously, or where one body is near the prime vertical and an error in the latitude will not affect the resulting longitude to be used in a sight taken simultaneously for latitude. Having on board a chronometer regulated to Greenwich mean time, and having a comparison of the deck watch with the chronometer, and the watch time of observation of a given heavenly body, the G. M. T. of this observation is found and for it the body's declination is taken from the Nautical Almanac; the polar distance, which equals 90 d algebraic- ally, will be less than 90, when the name of the declination is the same as that of the latitude; greater than 90, or 90 + d, when its name is different from that of the latitude. FIG. 107. 434 NAUTICAL ASTRONOMY The instrumental altitude of the heavenly body is reduced to a true altitude (Art. 212), and if the hour angle is to be one of the given parts, it can be found by the methods of Art. 199 when the Greenwich mean time and longitude are both known. The True Altitude. 217. To find the true altitude of a heavenly body at a given time and place, when its azimuth is not required. Here the latitude and longitude are given; the Greenwich time is found from the local time and the longitude, and for this Greenwich time the body's declination is taken from the Nautical Almanac. In the case of the sun, the declination may be found from page I or page II of the Almanac depend- ing on whether the Greenwich time is apparent or mean time. The hour angle t of the body is next found. In the case of the sun West of the meridian, the H. A. will be the local apparent time; when East of the meridian, the H. A. will be 24 hours the local apparent time considered astronomically. When the body is the moon, a planet, or a star, it will be necessary to find in order the L. S. T., the body's right ascen- sion, and then its hour angle (Arts. 192 and 199). Applying to the triangle of Fig. 107, the fundamental for- mula from spherical trigonometry cos a = cos & cos c + sin & sin c cos A, we have sin h = cos(L^d) 2 cos L cos d sin 2 %t, (155) or cos z = cos(L^d) 2 cos L cos d sin 2 %t. (156) versinez 1 cos # . , /..-/.x M1 As haversme x = ~ -- = - ^ --- = sin 2 \x (156) will become by substitution, etc., haver z haver (L~d) + cos L cos d haver t. Defining 6 by haver 6 =cos L cos d haver t f We have haver z = haver (L~d) + haver "1 . j NOTE. When L and d are of different names (L~d) becomes numerically (L + d). The hour angle t is usually given in units of time, whereas the above formulae require t in arc. FINDING TRUE ALTITUDE 435 ft & - i ^i U 3 d W Q * ^J 1C CO 3 o c ) fi ,3 l> o O J-t tl 1 H a OJ < ) ^ ,- > t > C (5 O 5 1C , ^ o ^H ^ r^ GQ IO CO r -1 CO " T^ CO r H ^* a d J! Jj 4 Cfc 'S 1 II O * 2 -u o N O5 O5 05 O5 CO rH rH HH t- O5 t- oj-^CO O1C iCCO rH CQrH -*(M rHCO 1C 1C o'i i aCOCO rHCX 05rH J5 rH T^ to S Ttl * CO CQ 1C CM 50 o rt Cl CO TJI CO O5 S), and by Napier's rules, tan < = cot d cos t (1)" sin h = sin (L + ) sin d sec (2) cotZ = cot t cos (L + <#>) cosec (3)^ Following Chauvenet's methods, the above can be put into a more convenient form. If = 90 <', the above become tan <' = tan d sec t (V) cos '- sin (158) sn = sin cot t sin 0080' (2) (3) (159) In (159), ' is taken out in the same quadrant as i and is given the same sign as the declination ; that is, if the declina- tion is of the same name as the latitude, it is +, and $' is 438 NAUTICAL ASTRONOMY marked + ; if the declination is of a different name from the latitude, it is ( ), and ' is marked ( ). The mere fact of t being E. or W. has no influence on the signs of the functions sec t and cot t. If t is E. or ( ), the body is East of the meridian and the azimuth is marked East ; if Hs W. or +> the body is West of the meridian, and the azimuth is marked West ; in other words, the azimuth, being restricted to 180, is reckoned from the elevated pole (or the North point of horizon in North latitude, the South point in South latitude) towards the East or West according as the body is East or West of the meridian as indicated by the hour angle. Again, for emphasis, let it be repeated that a t's mark E. or W. does not affect the sign of its function in the above for- mulae. However, the signs of functions of other quantities must be followed, and care must be exercised to do so. When t = 6 hours, (ft' = 90, and the formula for Z (159) becomes of an indeterminate form. However, cos t =te** T and cot t = *|5-* tan (ft tan is so small, cos < is near unity; therefore, Z = p sin t sec (L + <) approximately. Example 157, and Ex. 158 worked first for North latitude, then for South latitude, will illustrate the method (formulae 159). A TIME AZIMUTH OF SUN 439 1 u> rH 00 00 00 00 * rH rH ^ ja' ' 00 2 >g CO Q rH ^ JO rH CC ^ ~f~ E^ oi o os* OS 49 W O ^ " E 1 1 1 rH o *p 1 1 O 3 JJ o d ^ I 00 CO * s s Q CO O rt< ** a 03 oo 1C CO 1C H + ^ g 2 g Q, t OS os t- CO r^ S Jj 00 00 C^J 1C rH 00 s ^ ^ OS o OS OS QO Q 5z f-K TH H 5 1 **" V 1 ' 1 + + * S d 1 a S > .5 o o 02 O O .2 '3 b^ t- oo CO 0 os o iC ^ rH rrj d 1 1 1 o O) G d o d rH 3 as 2 rH g 05 ^ 1C 00 t- -2 o ^ || iJ *^ * co* co' rH -2 * (M (M N* " cJ " O g H ^ II ob > d 1 03 55 l CO * ** ^ ro ^ od rl '~ o P ^'a 1 H 1 + T Hli 03 a' * ao J il 2 <* A * o *-- -o- N Jd 1C" * 03 w o b / O ^ fl - S I| ii w ^ . . H | II "o 2 ^ .2 2 g ~ v^ v N d o .2 r-l .2 Tfl -^1 lO rH IO HH rt r o m 10 10 10 CO r 6 "^ g co t- OS OO CO O COO O'-J ^'^' ^OlOCO CSCMO^TflCXlCO^. 1 ft co gll> "*CO r-l CO COCO COrH rfl 60 03 *5 - W -*rHO Or-I^COCXlr-lT^^ o d IH . rt OS 05 O5 05 05 OS O O ^ CO O 'S r E ^ to H ^ ^ ts ^^ . ~~ .. OS ^ H^ HS H^ 1 + 2 ^j Or^ft^r^^OQ^ ii ii .rH s ^Q C)ddr< r4 r45f V TIME AZIMUTHS OF SUN 441 Now work the above example for Lat. 28 30' S. o t n d = 19 40 28.8 tan 9.55334 sin 9.52722 t = E. 56 08 37.6 sec 10.26406 cot 9.82664 $' = ( ) 32 41 32 tan 9.80740 cosec 10.26750 sec 10.07490 L = 2830 S ' L = ( ) 61 11 32 cos + 9.68293 sin 9.94263 h = 172846 sin+ 9.47765 Z - S 124 56 05 B (Z N =55 03' 55") cot 9.84417 Here latitude is S. and +> declination is N. and ( ) ; signs should be followed as indicated. The azimuth is reck- oned from the elevated or S. pole, and to eastward. The above formula for azimuth is of great importance in finding the deviation of the compass : Suppose that the navi- gator about 8 h 08 m 45 s a. m., January 3, 1905, in Lat. 7 28' 06" N., Long. 150 09' 54" W., had observed the bear- ing of the sun per compass to be Z^ = 108 -J, ship's head 45 (p. c.), and it is required to find the deviation (see Ex. 157). Working the time azimuth we have Sun's bearing (true) 119 04' 40" Sun's bearing (p. c.) 108 30 Compass error = +10 34' 40" Variation = + 8 For 45 (p. c.), deviation = + 2 34' 40" Time azimuths may be obtained as above by the solution of the astronomical triangle, from azimuth tables, or by graphic methods from azimuth diagrams. Instead of deducing the above formula for Z by Napier's rules, the third and fourth of Napier's analogies may be used. These, applied to the astronomical triangle, give tan \ (Z + M ) = cot It cos J (L d) cosec (L+d) J The azimuth Z, however found, should be expressed for practical purposes in the form of Z^ which is measured from North, around to the right, from to 360. 442 NAUTICAL ASTRONOMY The Altitude-Azimuth Method. 219. To find the azimuth of a heavenly body from its ob- served altitude at a given place. Noting the time of observation by a watch compared with a chronometer regulated to Greenwich mean time, the G-. M. T. of observation is found, and for this the declination of the body is taken from the Nautical Almanac. Knowing the latitude, and reducing the sextant altitude to a true altitude, the three sides of the astronomical triangle are known. By spherical trigonometry, cos 2 *A= * 8 Bin (8 -a) sin b sin c in which a, b> and c represent the three sides of the triangle, Applying this formula to the astronomical triangle PZM, (Fig. 107), A = Z = the azimuth of the heavenly body ; a = p = the polar distance of body ; I = 90 L the co. latitude; c = 90 h = the co. altitude of the body. o__# + 5 4- c_ Qn o L + h p -~~ ~- Therefore, cos* J Z = cos cos Now letting s = % (L + h + p), then \ (L + h p) = s p, and cos 1 Z = coss cos ( g ~ ? cos L cos h L- (162) = \/ cos s cos (s p} sec L sec h. } ALTITUDE-AZIMUTH 443 We also have from spherical trigonometry orf I A = dn(fl-8)gin(fl-c) where sin b sm c In triangle PZM (Fig. 107), a = 90 d, b = co. L, c = 90 h. fl + I + c h + d co L cos Therefore, sin 2 J Z = cos L cos h Now letting s' = % (h + d + co. L), s' d = h co.L d rzz / V ssn s- (163) cos I/ cos J (162) is preferred when Z is >90, (163) when Z is <90. This problem is known as the altitude-azimuth. Formula (162) is more convenient for use in connection with the problem of finding the hour angle of a heavenly body and is more generally used. If the bearing of the heavenly body is observed by compass at the time of getting its altitude, or if the bearing at this time is interpolated for from previous and subsequent bear- ings per compass, the error of the compass can be found. It has already been shown that compass error is the differ- ence between the true and compass bearings of a heavenly body at the same instant, and is marked E. when the true bearing is to the right of the compass bearing, W. when the true bearing is to the left of the compass bearing. 444 NAUTICAL ASTRONOMY The above formulae are adapted for use with almost any tables, but certain tables, like Inman's Tables, are in use and contain log haversines and ^ log haversines (the term haver- sine <, meaning yer ^ m ^ or sin 2 4 ), and the formula for fO azimuth in the following form can be most conveniently used by those navigators furnished with Inman's Tables. From trigonometry sin J A = /sin(ff ft) sm(ff c) Y sin b sin c in which 8 - **c>_ 90 o h + L-p 2 ~2~ ' Therefore, sin- JZ = - og co s haversine Z = ' (h " cos L cos h log hayer ^= J log haver (p(liL)) + log haver (p -J- (Ji L)*) + log sec L + log sec h. (164) The solution by formula 162 is illustrated in Ex. 159 on page 445. The points to be noted in that example are the fol- lowing : The G. M. T. being > 12 hours, the declination is taken out for the next date and corrected backwards. The de- clination being of a different name from the latitude, the polar distance is >90. The true azimuth is marked from the South point of horizon because latitude is South; and, East as the body is East of the meridian. Amplitudes. 220. The amplitude of a heavenly body is its angular dis- tance from the East or West point when in the true horizon, and is marked N. or S. according as it is N". or S. of that ALTITUDE- AZIMUTH OF SUN 445 | H o g OS O O> 01 CO 01 S \\ -*J ' " J3* V* . S 3 ^r o V j ko * 10 10 * rt 'fl * 5 w ^ p QQ * 1 E . a H 3 :& s tr "S. "*".* I *d 1 rO O O ^ tn . o o ~ o, ^ CO CO S " oi o oi ci *-" +3 CO CO O O "S o" 10 _. d s rf& * & 3* _o a CO C3 Ol ^ lO iO *C OS & 33 O 03 c3 S | H 1 p ^ * II -|-i I s*!: & ^ O O g^ g | o 3 ^389^18 2 d os o O w > H * 01 |0 ^ t*. 2 d 1-1 1 H ^> O * * - -*2 | 2 3 55 o * "* bb ** d 3 *~* ^~* C^l ^ ^" ^" j jgjj p + + _! + coo cooi oo ^ oo 10 o CO 05 CO 00 SO CO o of*' '? h 12 r^^SSf2 Ci ocooi x-*.: oo x fl &* 1 s s 1 O o .^^ SS' ^ OS' OS OS* -^^ tjQ |_| P p, O x-v v_* -^ >-^ lijsil V* ft? CQ "*- 3 CO o" iON c. H H II II -H 000 88 oo o ^COOCO OO'O OO OlO OTt< "" H-> tn | ** O CO O CO ^ rH O ^H r o? ^j S S'5 Jill (^)lr^V vcooiioibt-t- cot- WIO | rH^oJOiOOl Olo n OJOSiOl>COrH iO o 1010 OJOIOS-*I>01 lOrH Is s 1 SS'S I ^ ^ S> -3 g O rH (O CO t~ ft) < 2 & T-H IO O oa Q5 ^COl>+CO ?*(v" 1 1 - - rf B||l -s ^ a, & H~ N ^ - . CO ~ ^ g!*il s S bo oJ ** lalgg li * rf ** ^ H *C ca ^ S rt j3 fe *i!j ^ ^ . P, .2 PH oJ 03 s -s N a a p, * Ot C? 83 H ^'."^t 1113! ^ouoS H O ^r P 446 NAUTICAL ASTRONOMY paint. In other words, it is the complement of the azi- muth when the body is in the true horizon. In Fig. 108 and Fig. 109, let PZM be a projection of the astronomical triangle on the plane of the horizon, the body M being in the horizon, and in both cases just rising. Let NZS be the celes- tial meridian, WQE the celestial equator, WZE the prime vertical, and EM = A = amplitude = 90 NM = 90 Z. In Fig. 108, the latitude and declination are of the same name, PM = 90 d,PN = Lai, NM = 9QA=Z, and in the triangle PNM, the angle PNM is a right angle. FIG. 108. In Fig. 109 the latitude and declination are of a different name; therefore, PM = 90 + d f PN = Lai, NM = 90 ^ A = Z, and, as before, PNM is a right angle. Applying Napier's rules to the triangle PNM, cos PM = cos PN cos NM, cos (90 d)= cos L cos (90 A) , sin d = cos L sin A , sin A =.sin d sec L. (165) It is evident from the two figures that a body will rise and TIME FOR AN AMPLITUDE 447 set to the northward or southward of the prime vertical ac- cording as its declination is N. or S. Amplitudes, computed by formula (165), are tabulated in Table 39, Bowditch, for which the arguments are declination at the top and latitude in the side column; the true ampli- tude is found under the former and opposite the latter. The azimuth tables give not only the azimuth, which is the complement of the amplitude when the body is in the horizon, but also the times of rising and setting. Time for observing an amplitude. This problem sup- poses the body to be in the true horizon, that is, the true alti- tude of the center to be 0. If h is a true altitude, In! an observed altitude of the center, then h = h f D R + p, or h r = h + D + R p, but when the sun's center is in the true horizon h = 0, D = dip depending on height of eye, R z= 36' 29", p = 9". Therefore, as observed, the altitude of the center will be 36' 20" + dip above the visible horizon; hence the rule, in taking an amplitude of the sun, is to observe the bearing per compass of its center when its center is about one sun's diameter, or the lower limb a semi-diameter, above the visible horizon. Note at the same instant the ship's head (p. s. c.), angle and direction of heel, and the time by a watch compared with a chronometer regulated to G. M. T. Or, the bearing of the center in the visible horizon may be obtained by taking the mean of the bearings of the upper and lower limb of the sun when rising or setting, and by applying a correction for the vertical displacement from Table 40, Bowditch, the observed amplitude may be reduced to what it would have been, if taken when the sun's center was in the true horizon. Stars are not often available for amplitudes, except in the cases of very bright stars or planets before setting, and then the altitude should be 36' 29" + dip above the horizon. If observed in the visible horizon, the correction from Table 40, Bowditch, must be applied. In the cases of the sun, a star, 448 NAUTICAL ASTRONOMY or a planet, this correction is applied to the right at rising in North, or setting in South latitude; otherwise to the left. The moon should not be observed for an amplitude, be- cause when it has its center in the true horizon, the center is not visible, due to the excess of the parallax over the refrac- tion. When the moon's center is seen in the visible horizon, or h' = 0, the true altitude of the center is ( -f- H. P. refraction dip) ; now, as the H. P. averages about 58', the refraction about 36', the dip being dependent on the height of the eye, when the moon's center is just seen, rising or set- ting, on the visible horizon, it is in reality 22' dip, or about one of its semi-diameters above the true horizon ; for this reason the moon should not be considered available for ampli- tudes. Amplitudes of the Sun. Ex. 160. At sea, in Lat. 40 20' N., Long. 60 15' W., about 6 h 20 m p. m. local apparent time, on April 5, 1905, the bearing per standard compass of the sun's center at the time when the center was estimated to be one diameter above the visible horizon was W. 10 30' N. Eequired the com- pass error. It is first necessary to find the Greenwich time and then the declination. If the approximate time had not been given, it could have been found from sunset or azimuth tables; the latter, however, would also give the azimuth = (90 the amplitude) . L. A. T. A West G. A. T. or April 5, h = 6 4 in s 20 00 01 Dec. Corr. Dec. N N N O t ft 5 55 27.4 9 50.3 H.D. Corr. N N 57".03 lOh.35 690".26 = 10 IQi 21 00 '.35 6 05 17.7 By Computation. By Inspection (Table 39 Bow.) L = 40K N d = 6 1 N o i n , _ , n ,, ^ \ o L = 40 20 N sec 10.11788 * \ Tr* fl ( True Amp. = W 7.93 Dec.= 60518N sin 9.02553 . _ A =W75947N sin 9.K341 s error o / ;/ True Amplitude = W 7 59 47 N ' " w Amplitude (p. s. c.) W 10 30 N Compass error 2 30 13 W _ AMPLITUDES 449 Ex. 161. At sea, in Lat. 38 S., Long. 85 E., at sunrise (L. A. T. about 4 h 46 m a. m.), January 9, 1905, the bearing (p. s. c.) of the sun's center at the time when the center was estimated to be one diameter above the visible horizon was E. 20 S. Eequired the compass error. h m L. A. T. of sunrise = 16 46 00 Jan. 8. A East = 540 G. A. T. Dec. S 22 17 37.1 H. D. N 19".86 Corr. N 3 40.4 G. A. T. llh.l 11 06 Jan.^8. Dec. S 22 13 56.7 Corr. N 220".4 By Computation. L = 38 S sec 10.10347 d =S2213'57" sin 9.67791 A =E28 41 44 S sin 9.68138 Q I II True Amplitude E 28 41 44 S Ampl. (p. s. c.) E 20 S Compass error 8 41 44 E By Inspection (Tab. 39 Bow.) TrueA-np. Amp. (p. s. c.) C. E. = 8 40' 48" E = E20 S 8.68 E Amplitude of a Star. Ex. 162. At sea, in Lat. 40 K, Long. 30 W., the bear- ing (p. s. c.) of star Sirius when in the visible horizon, at setting, was W. 2 S. The star's declination was S. 16 35 ; 20". Find compass error. By Computation. L = 40 N sec 10.11575 d = S 16 35' 20" Sin 9.46561 True Amplitude W 21 62 56 S sin 9.57136 Observed Amp. W 2 S ) Tab. 40 Corr. left 0.6 S f Of* Comp. Amp. = W 3.6 S = W 2 36 00 S True Amplitude =W 21 62 66 S Compass error 19 16 66 W By Inspection. o 21.92 S L 40 N I True Amp. d 16.6 S' Amp. (p. c.) W 2.6 S C. E. = 19 19' 12" W = 19.32 W Amplitude of a Planet. Ex. 163. On January 6, 1905, about 12 h 40 m a. m. L.M.T., in Lat. 35 K, Long. 150 15' E., Jupiter's bearing (p. s. c.) 450 NAUTICAL ASTRONOMY when in the visible horizon, at setting, was W. 7 30' S. Re- quired the compass error. L. M. T. of bearing Jan. 5 A. East (-) G. M. T. Jan. 5, h m 12 40 10 01 Dec. N Corr. N Dec. N / /( 701 14 12.2 H. D. N 4".61 G. M. T. 2h.65 Corr. N 12".22 ,2 39 2 h .65 7 01 26.2 By Computation. By Inspection. L = 35 N . . . . sec 10.08664 L = 35 N \ True AmT> w ao A w d = 7 01' 26" N . . . sin 9.08734 d = 7 N f 1] ae Amp< W True Amplitude W 8 35' 05" N sin 9.17398 Amp. (p. s. c.) W jB S C. E. = 16.6 E Obs. Amp. W 7 30' S Tab. 40 Corr. left _30 o / Compass Amp.= W 8S = W 8 00 00 S True Amplitude = W 8 35 05 N Compass error 16 35 05 E Azimuth Tables. 221. The azimuth tables issued by the Navy Department embody two separate publications, No. 71 and No. 120. In both, the azimuth is given at intervals of ten minutes of hour angle, the arguments being latitude, declination, and hour angle. In No. 71, the latitude runs from to 61 at intervals of 1, the declination from to 23 at inter- vals of 1. No. 71 is especially adapted to the case of the sun, though applicable to the cases of all bodies of a declina- tion less than 23 North or South. The hour angle is given in the p. m. column, 12 hours H. A. in the a. m. col- umn, the H. A. in case of the sun being local apparent time. When the body is in the true horizon, the hour angle is the time of sunset and 12 hours H. A. the time of sunrise. The azimuth is also given when the body is in the true horizon. No. 120 is intended for use with the stars, planets, and the moon. It is tabulated for latitudes from to 70 and de- clinations from 24 to 70 (see Appendix C). To use the Azimuth Tables. Take each argument to its next lower tabulated amount and find the azimuth corresponding from the tables, placing AZIMUTH TABLES 451 it on the first line in each column of the tabulated form following. Now consider two of the above arguments unchanged and the third to be of the next higher denomination till each argu- ment has been successively changed; find from the azimuth tables the azimuth corresponding to each set of arguments, placing the result in the column whose name at the top in- dicates the changing argument, and just below the azimuth first taken out. Then, having the change in azimuth for an interval of one argument, find the change for the given fraction of that interval. The algebraic sum of the changes for fractions of all intervals is a correction to be applied by sign to the azimuth first taken out. Having found Z, express it in the form of Z N . Though this may be done mentally, the example and form below will indicate the method of solution. Ex. 164. In latitude 30 30' N. find the azimuth of a heavenly body whose declination is 21 10' N. and whose H. A. is + 4 h 13 m . Arguments. Differences for 10 Min. of Hour Angle. Lat. 30 N Dec. 21 N H. A. + 4 h 10 N 83 28' W 82 23' lof Declination. N 83 28' W 82* 21' lof Latitude. N 83 28' W 84 08' Change for 10"" of H. A 65' 1 ot Dec. 1 of Lat. Change, therefore, for 3 m of H. A. " " ' 10' of Dec. " 30' of Lat. -67' + 40' Corr. 10.7 N 83 28' W Required azimuth- = N 83 17'. 3 W and Z N = 276 42'. 7 452 NAUTICAL ASTRONOMY 222. In both an altitude azimuth and a time azimuth the declination may be regarded as accurately known; in the former the altitude and latitude, in the latter the time and latitude are liable to error. Therefore, it is necessary to consider the effect on the resulting azimuth of small errors in data, and the determination of the most favorable position of a heavenly body for observations for Z. (1) In an altitude azimuth to find the variation in Z due to a variation in h. Taking the fundamental trigonometric formulae cos a = cos & cos c + sin & sin c cos A, sin A sin & = sin B sin a, and substituting A = Z, B M, a = 90 d, I = 90 L, c = 90 h f we have sin d = sin L sin h + cos L cos h cos Z "1 sin Z cos L = sin M cos d L (160) sin Z cos h = sin t cos d By differentiation, h and Z variable, = sin L cos lidli cos L cos Z sin h dh cos L cos h sin ZdZ dZ sin L cos h cos L cos Z sin h , x __- = . . ( 1 D 7 ) ah cos L cos h sin Z From trigonometr}', sin a cos B cos & sin c sin & cos c cos A. By substituting the above quantities in this formula, we have cos d cos M = sin L cos h cos L sin h cos J. Substituting cos d cos M and the value of cos L from (166) in (167), we have dZ cos d cos M , ir 7 /^/>o\ ^_ - = cot M sec h. (168) rtA sin 3f cos d cosec Z cos h sin Z EFFECT ON Z OF ERRORS IN h AND L 453 Also, by substituting cos d cos M and the value of cos h from (166) in (167), we have dZ = -- COB d cos M c08MsQcLcosecL (169) all cos L sin cos d cosec ^ sin Js (168) shows ^f to be least when M is 90 and h = 0. w/ (169) shows ^ to be least when M is 90, L is and i is 6 hours. (2) To find the variation in Z due to a variation in L. Differentiating, sin d = sin L sin h + cos L cos h cos Z; regarding L and Z as variables, we have = sin /i cos LdL cos 7t cos Z sin LdL cos h cos L sin ZdZ dZ sin h cos L cos h sin Jv cos Z dZ~~ cos h cos L sin Z By trigonometric substitution as in the previous case, cos d cos t = sin h cos L cos h sin L cos 2T. Substituting cos d cos and the value, of sin Z in terms of M from (166) in (170), we have cos d cos cos h cos .L sin M cos d sec cos t cos t sec cosec M. cos sin Substituting cos d cos / and the value of sin Z in terms of t from (166) in (171), we have C IZ cos d cos t dL cos /i cos L sin cos d sec h = cottsecL. (172) (171) shows ^? to be least when M is 90, h is 0, and t is 6 hrs. rJ *7 (172) shows - rr to be least when Hs 6 hrs. and L is 0. (I x/ 454 NAUTICAL ASTRONOMY In a Time Azimuth. (3) To find the variation in Z due to a variation in t. Taking the trigonometric formula cot A sin C = sin & cot a cos & cos C, and as in the first and second cases above, substituting A = Z, C = t, a 90 d, & = 90 L, we have an expression involving only those quantities used in the solution of a time azimuth, namely, cot Z sin i = cos L tan d sin L cos t. By differentiation, Z and t regarded as variables, sin t cosec 2 ZdZ -\- cot Z cos tdt = sin .L sin id/, cos Z cos i sin L sin i sin Z dt sin i cosec Z From trigonometry, cos 5 = cos A. cos C -j- sin A sin C cos &. By making the same substitutions as before, we have cos M = cos Z cos i + sin Z sin i sin L, or, cos If = (cos Z cos / sin L sin i sin Z} . j & , ______ , pr\Q lyT Therefore, ~= - = cos M sin Z cosec t. (173) at cosec Z sin t From (166), sin If cosec / = cos d sec 7i; therefore, -^ = cos M cos d sec h. (174) $ (173) shows ^? to be least when M is 90 and t = Q lira. (174) shows ^f to be least when M is 90 and 7t = 0. dt Conclusions. It is thus seen that the ideal circumstances for observations in the determination of the azimuth of a heavenly body, and hence of deviation of the compass, would be when the ob- server is on or near the equator, and the heavenly body is on the prime vertical in the true horizon, rising or setting, its ASTRONOMICAL BEARING 455 position angle being 90. However, in the determination of the deviation, azimuths can be taken at any time, provided the change of azimuth is not too rapid, or the altitude so great as to make important the errors arising from the want of verticality of the sight vanes of the azimuth circle or pelorus. True Bearing of a Terrestrial Object. 223. In the survey of a harbor, it is necessary to know the azimuth of at least one of its lines, that is the true bearing of some one station from another, that other lines may be laid off in their proper directions, and a meridian line drawn upon the chart. It may often be desirable to determine the true bearing of a distant peak or point in finding compass error. If a terrestrial object, whose true bearing has been deter- mined from a shore position, be observed from the same posi- tion by compass, the difference between the two bearings will give the variation of the locality. The azimuth of a terrestrial object may be found by com- bining in the proper way the angle between the terrestrial object and a heavenly body with the azimuth of the same heavenly body determined at the same instant. 224. First method. The angle between the two objects may be determined by using the azimuth circle of the standard compass or pelorus on board; but ashore, where more refined observations would be needed, it may be measured by a theo- dolite. This instrument is brought to bear on the heavenly body, and the time is noted or its altitude measured by a sec- ond observer simultaneously with the reading of the circle. In the absence of a second observer, the altitude of the heavenly body may be observed before and after the circle is read ; and from the times noted and their corresponding altitudes, by in- terpolation, the altitude at the instant of reading the circle 456 NAUTICAL ASTRONOMY may be obtained. Turn the telescope in azimuth, bringing it to bear on the terrestrial object and read the circle again. The difference of the two readings of the circle will be the dif- ference of azimuths of the two objects, which being applied to the true azimuth of the heavenly body found by (1) the time- azimuth method or (2) the altitude-azimuth method will give the true azimuth of the terrestrial object. When the heavenly body has an appreciable diameter, as in the case of the sun, both limbs must be observed thus : Bring FIG. 110. FIG. 111. the vertical wire of the telescope tangent to one limb of the sun, the neutral glass being used on the eye piece. Note the time and reading of the circle, then quickly bring the same wire tangent to the other limb of the sun. Note the time and reading of the circle. The mean of the times and read- ings will be those corresponding to an observation of the center. Then turn the instrument in azimuth and read the circle when the line of sight is on the terrestrial object. In case only one limb is observed, a correction must be ap- plied to the reading of the circle to reduce the bearing to that of the center. In Fig. Ill, this correction is the angle SZS' where 88' =. x, the sun's semi-diameter. ASTRONOMICAL BEARING 457 The triangle SZS' being right angled at S' and 90 ZS being equal to li, the true altitude, sin SS' sin S8' sin x sin tjtjto ; ~T? - 7~ sin ZS cos h cos Since the correction and semi-diameter are small, corr. = x sec h f (175) the sign of the correction depending on the limb observed. 225. Second method. In this case the true azimuth of the heavenly body is found as before, the astronomical triangle being solved for Z by either the time-azimuth or the altitude- azimuth method. Let Fig. 110 represent the bodies projected on the plane of the horizon and the triangles involved ; the heavenly body's true place is S f its apparent place M. The apparent place of the terrestrial object is 0, and MO is the observed angular distance of the object from the heavenly body's center (that is the sextant reading corrected for instrumental errors, and in the case of the sun for semi-diameter). PZ8 is the astronomical triangle, PZ the co-latitude, PS polar distance, ZS co-altitude, and PZ8 the azimuth. If Z is not gotten from a time azimuth, it is gotten from an altitude azimuth, the altitude being the true altitude of M found by observation when arc M is measured. Measure with a sextant the angular distance MO between the bodies. At the time of measuring the arc M 0, note the time or measure the altitude of M; also measure the altitude of 0. Correct the altitudes of both M and for instrumental errors and dip, thus getting the apparent altitudes of M and 0. The correction for dip is taken from Table 14, Bowditch, in case of a free horizon; from Table 15, in case of an obstructed horizon. ' In the latter case this correction may be computed by formula (150), Art. 208. 458 NAUTICAL ASTRONOMY When the true altitude is found from a time azimuth, it is reduced to an apparent altitude by adding the refraction and subtracting the parallax. Letting h' be the apparent altitude of M, Q ' " " 0. there are given in the triangle MZO the three sides: ZM= .90 h', ZO = 90 -- Q, MO = D, the corrected distance. To find MZO = , the difference of azimuths. From trigonometry, sin MZO = cos Q cos h D + 'Q + h' and letting -- - - = s, we have cos cos also from trigonometry, cos /t cos Q and letting - - - = s, we have : cos s cos (5 D) (179) eo.vco.0 *- (180) COS Formula (177) is preferable when is < 90, (179) when greatly exceeds 90. When the body is in the true horizon, Q = 0, that is the observed altitude is equal to the dip; in Fig. 110, aM is the ASTRONOMICAL BEARING 459 corrected distance, the triangle &Mm is right angled at m and by Napier's rules, cos D = cos li' cos aw, cos am = cos = cos D sec li r . (181) or in (180), if Q = 0, tan J= V [tan 4 (D + V) tan J (D V)] (182) If the observed object is exactly in the water line, the ap- parent altitude is equal to the dip and is negative, or Q = (-) dip. The most favorable conditions for observation are when the heavenly body is on the prime vertical at a low altitude and the distance MO approximates 90; the ideal condition being when both bodies are in the true horizon, or = D. When the terrestrial object presents a vertical line to which the sun may be brought tangent, the sun's diameter through the point of contact will not be in the direction of the dis- tance OM f but perpendicular to the vertical circle through the terrestrial object, ZO, and a correction must be applied to the measured distance to obtain D. It is obtained from the formula corr. = 8 sin M OZ where 8 = sun's semi-diameter. The altitude of is very small anyhow, and by consid- ering its altitude as zero, M OZ equals MsiZ, so that M OZ is found with sufficient accuracy from the right triangle Mma, taking Ma equal to the uncorrected distance, that is, the sex- tant arc corrected only for I. C. Letting this uncorrected distance be D', we have by Napier's rules, sin Ji' sin D' sin M sun = sin 7X cos MaZ 9 or, sin 7i' = sin D' cos MOZ f cos M OZ = sin h' cosec D'. (183) 460 NAUTICAL ASTRONOMY 8 - I- CO ^ CO ri fc 010 c3 2 T3 fl "S H O ^ C ic o o -oooo ! "- 1 ^ - 3 W S g 1 | ifr| > | n3 w ,-. ^ ^ " " "3 -~ 3 a f O ."S 2 4-.: i 1 5 ** "o M a" (-H ^2 1 Q* 1 To o 1 -2 a a O oarjCLP^ 1 ^ ^* ^CO s 'C d 5 < "o Jc* OnJO ^ | I| a n "^ .2 *M O c^ *^ O5 r>- ^ X "^V^*'.^^ "^ ,5 iH * CO W Os' (35* tn K 5 " Q ^ a 8 O2 05 Ifi w b -So o a o -a ^ 3 o + T SS S .5 2 * * 05 O' 05 a s -s _, > 2 w So o + + CO l> O O O fill 1^-1 ic co o o O5 iC OS * ^ O O OS CO t> CO CO ^5 OCXt T *'l'5'~ ( W jC^ sf r 1 9 | t: cd co" o ^0'-'COC?^g o o' os* o* os' os ~ * 1 j 00 "* O *H O ^ + + T ^ cT 3 1 * H *fi T-( ^ hO "S d CD O *3 ^H rH - 5 > ,0 ^ QQ fcfcfc V ^ V- < N ] wo a c3 fl <0 ' 13 *'- P 5 " . 5 . . ? i ^ . a o p? O '""' *" ^ ^ LJ' * (^ & "tf ee e B ^ ., s a> .. &q ^ S 'C ^ T . . H ^ II II -ftbC^abCt^O T" ^ bC *"* 5<5a a ^ . ^-M-K. "S '33 S ^ A *H HOUR ANGLE AND LONGITUDE 461 Hour Angle, Local Time, and Longitude. 226. To find the hour angle of a heavenly body at a given place, and thence the local time, the altitude of the body and the Greenwich time being known. Noting the time of observ- ing the body's altitude by a watch compared with a chro- nometer regulated to Green- wich mean time, the G-. M. T. of observation is found, and for this the declination of the body is taken from the Nauti- cal Almanac. Knowing the lati- tude and reducing the observed FIQ S n2 to a true altitude, the three sides of the astronomical triangle are known. By spherical trigonometry, sin A J! sin (8 b) sin (8 c) sin b sin c in which a, b, and c represent the three sides of the triangle Applying this formula to the astronomical triangle PZM (Fig. 112), and letting A = t, the hour angle of the heavenly body ; a = 90 h f the complement of its true altitude; b = 90 d = p, its polar distance; c =. 90 L, the complement of the latitude ; then s _ a + & + c = 90 h -f p + 90 L __ 9()0 __ 8- c= 90 - - - (90 -L) = 462 NAUTICAL ASTRONOMY Therefore, = lr if L pBnp -i cos sin p Now letting 5 = $(L + h + p), then %(L + p h) = sh, and sin t = I ["cosssm ( s ~ h ) "] (184) \ L cos L sin ;? J In like manner may be deduced from an application to the sin 8 sin (8 a) triangle of the formula cos 2 $ A = : = : - sm b sm c the following: cos t = I r sm ( g ~~ )cos(g ffjn ^ 8g ^ \ L cos L sm p where 5 = J (L + ft + ^)- The above formulae are adapted to use with any tables, but for those navigators supplied with " Inman's Tables " or any tables of log haversines and one-half log haversines, the fol- lowing deduction and formula will be found of interest. It is largely used in the British Navy, and has many advantages. From trigonometry, J A - a = z sm & sm c I 5 = 90 rf, L-\-d z 2 J = 90 -' ~ :i: _ (90 4- rf) = - e = 90 - L^J - (90 - ) = L + * ~ 2 8n . Therefore, sin 2 -| ^ ^ - -T j-y cos a cos L HOUR ANGLE AND LONGITUDE 463 Now n 2 t = haversine t, sin : ^4^ - =v'haver ( z (L d)), and sin = ^ haver (+ (L flf)). A Therefore, using logs : log haver tf =. J log haver (2 (L d} ) + J log haver (2 + (L d) ) (186) + log sec d + log sec When t greatly exceeds 6 hours, as is often the case in high latitudes, it should be found from formula (185). When L = 90, the zenith is at the pole; in (18 4) p + h = 90 and cos L Therefore, in very high latitudes it is impracticable to find with exactness the local time as the formulae for hour angles then approach the indeterminate form. The formulae also reduce to the indeterminate form when d = 90, at which time the star would be at the pole and, therefore, its altitude would be the latitude ; for this reason when working for time avoid stars of very large declination. For time, bodies should be observed when on or near the prime vertical (Art. 237), and the desirability of this position increases as the latitude increases. In latitudes beyond 66 30' an error of 1' in the altitude will cause an error of at least 10 s of time in the longitude. Using formula (184), it is not necessary to take out t in arc, then multiply it by 2, and convert it into time. In Table 44, Bowditch, t may be taken directly from the p. m. column corresponding to log sin J t in the sine column, In taking out hour angles, take them from the p. m. column, 464 NAUTICAL ASTRONOMY marking them + when the body is West of the meridian, ( ) when the body is East of the meridian. When the body is the sun, t its hour angle is local appar- ent time when the sun is West of the meridian; but if the sun is East of the meridian, its H. A. is ( ) i, and the local apparent time is 24 hours i. This is astronomical L. A. T. Thus, if the sun's H. A. is + 4 hours, the L. A. T. is 4 p. m. ; if the sun's H. A. is ( ) 4 the L. A. T. is 20 hours astro- nomically, or 8 a. m. civil time, that is to say, if the sun's H. A. is ( ) 4 hours, the sun will not be on the meridian for 4 hours. Having the H. A. of the sun which is L. A. T. or 24 hours - L. A. T., to obtain local mean time, the equation of time must be taken out of the Nautical Almanac for the Greenwich instant and applied with its proper sign. Then the difference between this local mean time and the corresponding Greenwich mean time will be the longitude; West if the Greenwich time is the greater, otherwise East. Conditions of observation. A little further on (Art. 237), it will be shown that altitudes for time, whether of the sun, moon, a planet, or a star should be taken when the body is on or near the prime vertical, and certainly more than 45 and less than 135 in azimuth. The altitude should be sufficiently high to eliminate errors of refraction, say above 10, and especially so when refraction may be affected by fog or mist. It should be a rule, when observing heavenly bodies, to take several altitudes in quick succession; and the mean of 3 or 5 altitudes, thus taken and so selected that the differences of altitude vary with the differences of time, should be used in preference to a single observation. Whenever the sun is ob- served for time, its compass bearing should be observed for compass error and the heading of the ship per compass also carefully noted. TIME SIGHT OF SUN 465 Such sights worked for time and longitude are known as " time sights." 227. Rules for working a time sight of the sun. (1) Find the Greenwich mean time and date. It is shown in the col- umn marked " Times," in the- form for work following, how the G. M. T. of observation is obtained. Applying the chro- nometer comparison (C W) to the watch time of observa- tion gives the chronometer time of observation, and if to this is applied the chronometer correction on G. M. T. when leav- ing port brought up to date for daily loss or gain, the result will be the G. M. T. of observation, but care must be taken to see that this time is astronomical time, and that the date is correct. (2) Reduce the sextant altitude to the true altitude of the center, and take from the Nautical Almanac for the Green- wich mean noon of the given astronomical day the sun's de- clination, H. D. of declination, equation of time, H. D. of the equation of time, and correct both declination and equa- tion of time for the G. M. T. If the declination is of the same name as the latitude, find p = 90 d; if the declina- tion is of a different name from the latitude, find p =. 90 + d. Note whether the equation of time is + or ( ) to apparent time. (3) Combine h, L and p as required in the equation (184) and as per form illustrated in example on page 467. Having found the log sin -J t, look for it in the column of sines (Table 44, Bowditch), and take out the corresponding time from the a. m. or p. m. column according as the sight is an a. m. or p. m. sight. This quantity is the local civil apparent time. The time from the p. m. column is also astronomical time, but 12 hours must be added to the reading from the a. m. column to reduce it to astronomical time. Applying the equa- tion of time, with the proper sign, to the local apparent time gives the local mean time of observation. i66 NAUTICAL ASTRONOMY (4) The difference between the local and Greenwich mean times of observation is the longitude in time; West if the local time is less than the Greenwich time, otherwise East (Art. 179). (5) Or, the G. A. T. may be found by applying the equa- tion of time, with the proper sign, to the G. M. T.; then the longitude will be the difference between the G. A. T. and L. A. T., West if G. A. T. is the greater, East if the L. A. T. is the greater. 228. Time sights of the moon, a planet, or a star. When the hour angle determined by the formulae of Art. 226 is that of any other heavenly body than the sun, that is, of the moon, a planet, or a star, the right ascension of the body must be taken out of the Nautical Almanac for the Greenwich in- stant (in the case of stars, for the Washington instant) ; then the algebraic sum of the hour angle and right ascension of this body will give the local sidereal time. Having con- verted the G. M. T. into the corresponding G. S. T. (Art. 192), the difference between the G. S. T. and L. S. T. will be the longitude, West if the G. S. T. is the greater; East, if the L. S. T. is the greater (Art, 179). A. M. TIME SIGHT OF SUN 467 .Is CO 00 t- t- J,' ^- *H n i 1 io 1? w 1 5 ^ ."M ^ S . S ^ O O 2 \ . Sg S H & OJ ^ ^ o ic d ^ g g . ^M a^ -| eo ^ o* & ^ H !? W ^<1 w 5 H ^3 O5 O Oi W 00 1 ^ Q H - J-l Q b- J2 O^COOt-00 ^ m . H S ' M * 10 h^ * * 1 - CO GO "? . . fr-lTtl ,0 C_I cJ pH rH f< tO A, O5 oo < =c O3 CX> . H . . O ^ ^* >c 4J T " W ^ d S ^^ ^ E^ lO O CO .- ^ ^ f g ""CO r- TH CO g] ^J ^ o rn O T-H rH ^O GO **< "S o WrHiCOO*"* JD^5 .S fl 00 rflXlt-^toJ, H'^ D t-, a + ^."1 T- 1 ^ B w ^ o ^ S !i r^odddo ^-*^o H W 468 NAUTICAL ASTRONOMY 2 oo a) in O o * -^ ^ C? Q - i S 'S rH P* 1C CQ . O 1C W a ^ O5 _ -* o CO SrH + ^H CO ^. " r-l 05 o '0 CO *C CO "* rH O o 1 o2 55 N d a, woo r* ^ oo oo - - t" .H g> 2 P. 4) .2 a S S a FH CJ O H PQ O ) Q PH ^ OS Z> 0) 0:5 1C O t- o O? O3 C- T* 01 r> * O3 CO oo o? 1C ^ ^i | HH bfi 2 rH1 O _,., o *"" o TIME SIGHT OF A STAR 469 229. Rules for working a time sight of a star. (1) When observed for time, a star should be on or near the prime verti- cal to give the best results. The observation should be made when the horizon is well defined during twilight, or when the moon is shining. (2) Find the correct G. M. T. and date. (3) For this G. M. T. take from the Nautical Almanac the R. A. M. O (Art. 185), and convert the G. M. T. into G. S. T. (4) Look for the star's approximate right ascension in the mean place table of fixed stars; then in the apparent place table find the star's R. A. and declination to the nearest sec- ond of arc. For use in a time sight they require no correction. (5) Reduce the sextant altitude to a true altitude. Sub- stitute h, L, and p in the formula, as per form following, and having found the log sin \ t, look for it in the log sine col- umn of Table 44, Bowditch, and abreast of it in the column marked " Hour p. m." will be found the star's H. A. or t in hours, minutes, and seconds of time. The exact value of t may be found by interpolation or by using the table of pro- portional parts at the foot of the page. (6) If the star is East of the meridian, mark t (); if the star is West of the meridian, mark t +. To the hour angle t apply the star's R. A.; the algebraic sum will be the L. S. T., the difference between which and the G. S. T. will be the longitude, West if the G. S.T.is> L. S. T., or East if G. S.T.is< L. 8. T. 470 NAUTICAL ASTRONOMY i oo oo oaS ^ S H 4J s 10 V C 1 s a 5 5 O ^* bfi-^ * r3 (^ fl s- ^ 02 S o . JJ * ^2 ^ OH s* u. w 05 Q d * *j m -^- i 02 * ^ -X- ^f 5^ I- C- ^ g CO H O .2 d * * j S o o .S a S *! S o g . -S H < - o P3 '^02 4J ^ O CO CO gs-g + l|| | fl J 08 2 ^ r" ^ Pr5 13 H *** co cr . Q U Q,H S W ^ ""* *> 02 o O ^StfcS oox^o^i. 1 " 1C 00 "O t~ CO CO HHO S 02 ^ -f* (- 5gfi ^ -*COCO -*05rJH t- ftrG ^3 a OSTHO^t-O*"* 1 | a * ^ < . rfj TH TH 03 S a) .^ II X m S 3 # o 5f i oOoOOQOQao t- t- eo o * * J rt j ^ 1^! COOOIOCOW03 ^A- ( _]o^<5 5W 1 -' gO5l> OCO CO OS CO w O3 ift O3 rH TH CO ^ tfl *J ^^ fl "H (X CD i * OC *>gTH -S :-s^ a ^ H rf cS . 4 3 at "-^ *O H loS ' fl5 ^ .^ S^ s s*s g tjgj Hi ^flg ^ s o a S 8" 1 do S 5S |g cSod I co SO ^o do do d SS ' 1SS3 88 | 9 -a-- 'a i ^ II I! II II II II i i 3s* d o o "73 m< m ^ co 2 * S=b H - o o S BS~ ^:i ?2? H * * Wo 2 O -M' 3 w CO <* 11 ii i 02 . J ^ O OO **< ^ O5 W 'O COCO ^-t fH M O Q) O 8 & 3 d O o c3 c3 c8 H b r 'So o ^3 C8 -S 8 jj 53 5 S fci) * ' .s ^ ** S ^ 1 feoo 22 ""^ n ^ S * g sa-3 L3 H ^' rt a fl o o Is ++! 1 + OO O t~ ^*O _, j-s^j _u?k HH ^ o^!55 St J2 > ' "^ w S S ts + ^ Q ' cJtt ^ 1 ^ s p ad^SftS ^9 -2 a p r . N 5 a >H o CO O 10-^1 8 1 1 H ' H - ^"O 1 N t^K^ ^ +T 11 2 c5 " 4J fH r- t^ N N HH O W ^i w ^ ^ s 2 ' o i ^^ 5 ^ 2, v, o _c1 en 10 ^ a5 14^ ^ "^ o fl oo o 5 -^ a - ? H H fl c8 . O 10 OB 1-5 . ^ . 1 S EH^ EH a a > dS^" a fl S M ^o dd dw 6 NAUTICAL ASTRONOMY 2 S d fi , s " H S * v in 2*'* d a S8 - S | oj ^^ o S o "S5 ft "* Kg 5 a P? , S 5 00 < "3 'S '""' : k- ~J ~ , ' *" (4 ^ ' d H Sd 11 1 W HH o ^ ^* o 2| a> o -d j ^d 3' . 0 -^ . >-l Tj< I-( t- 3? 1 & 1 1 1 1 w . ^N g Q . ^ 2 ^SS SS 83 o W^H o c-> 55 i-^lO i-H i-H iO O c3 ^ "7^ rtt-lO ^^ ^H O N s|& a 00 0) 1 Sill *^ "rf ^ 1 fc ss 474 NAUTICAL ASTRONOMY Ill liP S2I5 Q 0* ^ S he 5 H ^ H + "3 -3 H L* tJ a s s w s 4 cj -- do ti o 3 v PH .-! N i- S 4. ^ -^sp g a a 02 ~ 1C H | 2 j I- It- i ^ 1-1 ii-i H ^j W ^ ^_ fc feT a fl'S (ti R 55 ?ll. is OH o sj o S5S ^ a' ^' 8 * | ' ^ w w oQ 13 OQ yl ' gj 0>1 T-ll . d Q -9 05 ^ -S P * ae g w o g SSS 1 " S^'S + ^ | 11 II II II II H O -" H ^3 05 0< *l^ ^ 5 ^ H "** < EH EH O !*1tfE of- ^S^Ot^l^ o h . fl " ^ fl t*- ?fc "i\ O S rd ? 1 c* S 1^ ** * ^ Q) fl H^H ^ d S fc* J *M >^ S b" i d o 6 d i TIME OF SUNSET 475 233. Sunrise or sunset time sights. When the sun's cen- ter is in the visible horizon, East or West, its true altitude or h equals (refraction + dip parallax). The watch time of this instant may be found by noting the watch times when the lower and upper limbs are in the visible horizon, just appearing at sunrise or disappearing at sunset, and tak- ing their mean. Having the watch time of the instant, the C W, and the chronometer error on G. M. T., proceed to work a time sight of the sun with a negative altitude which numerically equals (refraction + dip parallax), the refraction and parallax being for an altitude of and the dip depending on the height of the eye. However, owing to the difficulty of noting the times at contact of limbs with the horizon and the uncertainties of refraction, the result should be regarded as only approximate; and this method should be used only when fog or cloudy weather has prevented, or may prevent, the navigator from getting more reliable observations of the sun or stars. 234. Time of sunset. The instant of sunset is when the sun's upper limb is just disappearing below the visible hori- zon, or when h == (refraction + dip + S. D. paral- lax). For this altitude, a given latitude, and declination, the hour angle of the sun may be found by the time-sight formula. This hour angle will be the civil local apparent time of sun- set; (12 hours the H. A.) will be the L. A. T. of sunrise. The L. M. T. of sunset may be found by applying the equa- tion of time for the instant to the L. A. T. of sunset. As the declination and equation of time are tabulated for Greenwich time, to find the declination at the instant of sun- set, it will be necessary to either assume, or take from azimuth or sunset tables, an approximate time of sunset, apply the longitude, and obtain an approximate Greenwich time of sun- set for which the declination mav be found. NAUTICAL ASTRONOMY For the equation of time it is better to proceed thus : Having found by computation in the time sight the L. A. T. of sun- set, find the correct G-. A. T. by applying the longitude, and for this G-. A. T. take from the Nautical Almanac the equa- tion of time which, if applied to the L. A. T. of sunset, will give the required L. M. T. The L. M. T. of sunrise and sunset may be found in Table 10, Bowditch, or in the Tide Tables issued by the U. S. C. and G-. Survey. 235. To find the duration of twilight.- Twilight lasts till the sun has sunken 18 below the visible horizon, and there will be continual light so long as the sun at its lower transit is not more than 18 below the visible horizon at the place. The difference of hour angles of the sun obtained by time sights, using altitudes of ( ) (ref. -f- dip + S. D. -- p) and of ( ) (18 + ref. +' dip + S. D. p) will give the duration of twilight. 236. To find the hour angle of a heavenly body when in the horizon. In Fig. 113, H is the body in the horizon, its H. A. is HPZ t and L HPN 180 -- i, PN L, PH = 90 d, and L PNH = 90. By Napier's rules, cos HPN = tan PN cot PH, or cos t tan L tan J, (187) and if Hs < 6 hrs., 2t will be < 12 hrs. ; also if J is > 6 hrs., 2t will be > 12 hrs. From the above it is apparent that bodies of positive decli- nation (same name as latitude) will be above the true hori- zon for more than 12 hours, bodies of negative declination will be above the horizon less than 12 hours. This interval FIG. 113. EFFECT OF ERRORS IN DATA 477 2t is for the sun an interval of apparent time; for a fixed star, an interval of sidereal time. Length of day and night. As the length of the day is de- termined by the length of time the sun is above the horizon of a place, it is evident that since cos t = tan d tan L, at the equator where L = 0, cos t = and t = 6 hours, so that at the equator the day .is 12 hours long in all seasons. At the equinoxes d = 0, tan d = 0, and 2t = 12 hours, so that when the sun is at the equinoxes^ the day is everywhere 12 hours. Within the Arctic Circle, if d + 23J and L = 66J, cos t = 1, 2t = 24 hours, and there is no night; this would be the case of midsummer in latitudes beyond 66J. If d = 23 and L = + 66J, cos t will be + 1, 2t will be hours, and there will be no day, only night, as in the case of midwinter in latitudes beyond 66 J. Thus it is plain that 12 hours is the average length of a day throughout a year; on a given date when the sun's declination is positive, the day is > 12 hours, and on a day six months later, when the sun's declination is numerically the same but negative, the day will fall equally short of 12 hours. 237. Effect of small errors in data. In the solution of the astronomical triangle for time, and hence for longi- tude, the elements involved are the declination from the Nautical Almanac which may be regarded as accurately known, a measured altitude which is affected by errors of observation, errors of the instrument, and errors of refraction, and a latitude by observation or dead reckoning which depends not only on accuracy of original determination by observation but on the correctness of course and distance run, etc. Thus it is apparent that both the altitude and latitude are liable to error, and it is desirable to consider the effect on the resulting longitude of (1) a small error in alti- tude, (2) a small error in latitude; (3) to find the position of a body when its altitude changes most rapidly, and then to 478 NAUTICAL ASTRONOMY determine the most favorable position of a heavenly body for observations for time or longitude. By substitution in the trigonometric formula cos a = cos 1} cos c + sin b sin c cos A, letting t = A, 90 h = a f W d = I, and 90 L = c. we have the fundamental equation sin li r= sin L sin d + cos L cos d cos . By differentiation, h and tf variables, cos hdh = cos L cos d sin tdt, dt___ cos ft dh cos L cos d sin ' but sin t = sin Z cos A sec d; therefore, * = - - -= -4^| = -. = - sec L cosec Z. (188) ^A cos L cos d sin ^ cos h sec d Differentiating the fundamental equation, L and t variables, dt sin d cos L cos d cos t sin 7v we have -ry = = : dL cos d cos L sin tf From trigonometry, sin a cos B = cos & sin c sin & cos c cos A. If * = A, B = Z, a = 90 h, I = 90 'd,c = 90 L, by substitution, we have cos h cos Z = sin d! cos L cos J sin L cos , and, by rule of sines, sin t = sin Z cos h sec d. Making these substitutions in equation for -=-= we have * J^Af^- ,= cot Z seel (189) dL cos d cos L sin Z cos /i sec d (188) shows H to be least when L and Z = 90. (189) shows J^,to be least when L = and Z = 90. EFFECT OF ERRORS IN DATA 479 To find the position of a body in azimuth when its altitude changes most rapidly. In Art. 222, by differentiation, we found formula (167) dZ _ si 11 L cos ^ cos L sin h cos Z dh cos L cos h sin Z 4 =tan L cosec Z tan h cot Z. an Therefore, dh = z = J^r ( 19 ) tan L cosec Z tan h cot ^ (190) shows dh, or the change in altitude, to be greatest when Z =. 90; or, in other words, when the heavenly body is on the prime vertical. Conclusions. ( 1 ) Considering the effects of errors in h and L, sights for longitude are best in low latitudes. (2) An error in altitude, or an error in latitude, will pro- duce the least change in the hour angle when the heavenly body is on the prime vertical. (3) The motion of a heavenly body in altitude is most rapid when it is on the prime vertical; its altitude can be taken with greater accuracy; and when the diurnal circle of a body corresponds to the prime vertical, the change of alti- tude is directly proportional to the change in time, and the mean of a number of altitudes will correspond to the mean of their times of observation. For these reasons it is better to observe a heavenly body when on or near the prime vertical when observing for time or longitude. When the latitude and declination are of the same name, there is no difficulty in observing the body near the prime vertical and at an altitude sufficiently great to eliminate the uncertainties of refraction. In low latitudes, when L and d are of the same name, the body may be on the prime vertical when only a few minutes from the meridian, in the case of the sun near noon, and still be available for time observations. 480 NAUTICAL ASTRONOMY These remarks should emphasize the fact that the suita- bility of heavenly bodies for time observations depends more on the azimuths than on the hour angles of such bodies. When L and d are of different names, the diurnal circles do not cross the prime vertical above the horizon, and, under such circumstances, bodies are nearest to the prime vertical when in the horizon; and such bodies should be observed, if necessary to observe them for time, as soon as the altitude is sufficiently high to be unaffected by errors of refraction that is at least 10. By the time the sun has reached a proper altitude for ob- servations in winter time, it is so far from the prime vertical that any error in altitude or latitude will produce a larger one in longitude, and this error will increase with the latitude. However, during twilight or moonlight, in winter, there need be no difficulty in finding suitable stars on or near the prime vertical from which to obtain reliable determinations of longi- tude. An inspection of the azimuth tables will indicate for a given latitude and declination the hour angle of a body when on, or nearest to, the prime vertical, and from it the local time may be found. Since when on the prime vertical, a heavenly body is fall- ing or rising most rapidly, and the changes of altitude are proportional to the changes of time, the effect of an error of 1' in the altitude on the resulting hour angle, and hence longi- tude, can be gotten by dividing the difference of any two of the recorded times of observation by /the difference of the cor- responding altitudes in minutes of arc. The result will be in minutes or seconds of time as the differences of times are in minutes or seconds of time. 238. The practical way of finding the effect of an error of 1' in the latitude, on the resulting longitude, is to find the longitudes for two latitudes differing, say 10' or 20', then di- WHEN ON OR NEAREST P. V. 481 vide the difference of longitude by the difference of assumed latitudes in minutes of arc. A study of Sumner's method will make this plain. In Table I of this book will be found tabulated the changes in longitude for a change of 1' of latitude, the arguments being the observer's latitude and the body's true azimuth. 239. To find the hour angle of a heavenly body, its true altitude, and azimuth, when nearest to or on the prime ver- tical, that is nearest in azimuth to 90 (Figs. A and B). There are seven cases that may be considered: (1) + d > L. It is evident that the azimuth will be greatest when the vertical circle is tangent to the diurnal circle as at M in PZM and at M in PZM^; M = 90, MI =. 90, cos t = tan L cot d, sin h = sin L cosec d f sin Z =. cos d sec L. The body is circumpolar when d equals or is > co. L as in the case of a body whose diurnal circle is NM. (2) + d = L. Here the diurnal circle is tangent to the prime vertical at the zenith. At the point sought, Z = 90. The body is on the meridian and t = 0. The body is in the zenith and h = 90. (3) + d < L. The diurnal circle crosses the prime ver- tical, and, at the moment of crossing, Z = 90 in triangle PZM S ; cos t = cot L tan d f sin K = sin d cosec L. (4) d = 0. Here the diurnal circle is in the plane of the equinoctial and passes through the East and West points of the horizon, as does also the prime vertical; those are the points sought. Triangle PZM 4 , Z 90, cos t = cot L tan d = 0, and t = 6 hours; sin h = sin d cosec L = 0, and h = 0. (5) d < L. In this case, d is of a different name from L. The diurnal circle intersects the prime vertical below the horizon. In triangle PZM 5 (Fig. B), Z 90, cos t = cot L tan ( d), cos t is negative, and t is > 6 hours; sin h cosec L sin ( d}, h is negative, and the body is below the horizon. 482 NAUTICAL ASTRONOMY FIG. B, WIIKX ON OR NEAREST P. V. 483 The nearest point to the prime vertical at which the body is visible is IP 5 (Fig. A), when the body is in the horizon. From the triangle PNM' 5 (Fig. A), we have cos t = tan L tan ( d), or cos t = tan L tan d and is < 6 hours ; cos NM' K = cos Z sec L sin ( d), cos Z is negative and Z > 90. (6) d = L. In this case, the diurnal circle is tangent to the prime vertical at the nadir. The triangle is PZM n (Fig. B), Z 90, cos t cot L tan (d) = 1, cos t 1 and t = 12 hours, sin li = cosec L sin ( d) - 1, or h = 90. The nearest point to the prime vertical at which the body is visible is M' Q (Fig. A), where the body is in the horizon. From, the triangle PNM' Q , cos t tan L tan ( d), or cos t = tan L tan d, t is < 6 hours; cos NM' Q = cos Z =.. sec L sin ( d), cos Z is negative, and Z is > 90. (7) - - d > L. Triangles PZM 7 and PZM a (Fig. B), Z > 90 from the elevated pole. At M 7 and M 8 , % is a minimum, estimated from the depressed pole, at elongation. The vertical circle is tangent to the diurnal circle. M 7 90, M 8 = 90. Therefore, in triangles PZM, and PZM S , right angled at M 7 and M 8 , cos t = tan L cot ( d), cos t is negative, and t is > 6 hours; sin h = sin L cosec ( d) and h is negative ; sin Z = sec L cos d, and should bo taken in the second quadrant. If d equals or is > co. L, the body is never visible as in triangle PZM 8 (Fig. B) . If the body is visible, then d is < co. L and a point in the horizon will be the nearest point to the prime vertical at which the body is visible, as J/' 7 , (Fig. A). In the triangle *PNM' 79 t is < 6 hours and Z is > 90, as in cases (5) and (G) when the body is visible. Recapitulation for hour angle when the body is visible and on or nearest to the prime vertical, 484 NAUTICAL ASTRONOMY When d is of the same name as L. If d is < L, use formula cos t = cot L tan d. (194) . If d is > L, use formula cos t tan L cot d. (195) ' When d and L are of different names, and if the body is ever above the horizon, it will be nearest to the prime vertical on rising or setting, then use formula cos t =. tan L tan d. (196) When + d = L, the diurnal circle passes through the zenith, so that when the declination is equal to, or nearly equal to, the latitude, observations for time may be made within a few minutes of meridian passage, the mean of altitudes cor- responding to a mean of the times. However, such is not the case when the body is near the meridian in azimuth, at which time the changes of altitude are proportional to the squares of the hour angles. The azimuth tables may be used for finding the time when a body is on or nearest to the prime vertical, and will give results sufficiently accurate for all practical purposes. In finding the t corresponding to a given log cos, in ex- amples 171 to 173, when cos t is -{-, use the p. m. column, dividing through by 2. When cos t is , use the a. m. column, adding 12 hours to the reading and then dividing by 2 to obtain t. For a body whose decimation and right ascension are changing sufficiently rapidly to require correction for G.M.T., the approximate time of being on the prime vertical must be known in order to get these elements for use, the declination for substitution in the proper formula, and the right ascen- sion to be applied to the hour angle in finding the L. S. T. to be converted into L. M. T. To get the approximate time, when the body is on the prime vertical, assume the hour angle or take it from the azimuth tables and apply it to the L. M. T. of local transit from the Nautical Almanac. A STAR'S TIME OF EISING OR SETTING 485 M co s II 03 OS rH OS CO CO l> t- iO CM rH Tfl CO rH OS OS' 0* 04 N CO rH A JO CX> rH -H co ca O rH O I- t- co' o co' CXJ CM rH ii CO t- l- 00 rH rn w oe O CM CM vV O rH g c o O * OS C? ?O O 7 j Jj fl fl 55 1 1 ** CNJ O t- H i! 0* g 'u Pi rH ^H ^ o o O i ^ a os < o coo q o c OT ^ -a o c g ^ ^5 o"~^ a . || DO a -a o 1 g a 5 3 |l ^ If ^ H ' a !" |H 2 ^ ,. a *. . .0 'd Q ' T3 2^o 11 ^ * Sr^l S( ^ 4:86 .NAUTICAL ASTRONOMY d 05 " CO r-l rH ^wA^x 0> S 5S o *" 03 00 O5 1C j^, ^j rd 03 00 rH J5 2 r"o 1C * O OS o ^ t 00 00 Z> W ^ -^ ^3 CO CO 05 g 1C rH W -^ 'rrt -M S OO 1 Cx r "* ^> .p-l d 1 ? ^ 2 ^05 | N 05 1 fe 03 oS ^ ,_H ^*-A^ ^x g S | Pj 1 d H . . . , .^5 t- O5 ^ GO rH - t^ 05 00 O 1C O rH 111 cd ^ O " l "" > d CO rH 00 1C O5 CO rH rH CO TiH O rH rH OS rH rH ^ ^ >' -^ t~+-*~*** 2 51 CO O * " * m CO* CO* co W rH . 00 O Tt< i> JO CO C^ 111 1 13 *j O rH 10 rH rH CO rH rH o co P 5 H .0 5 ^ d * W 10 05 1C rH goo 1C 1C *1 Tl 00* ^ g 5 /^N ! ^3 * CO CO A OS oj * ^ rH rH ,d O5 I O5 CO rH rH 1 JC^ 02 a 1 1 r 2 ft !2 1 e . 5 1 p> . ' co CQ SH B ^ w W S d B a v O d *w ^ Tft fl ^ * ol O rt I-H ft r t3 *" O O 10 O ' t> <5 CO 41 + " S 5 a * H "1 H |H |g H 0? ^ ^ II ^ -* . o d d "g ^ ^ 'C * -X- J 050 o5 h4 S P5 ij TIME WHEN PLANET is ON P. V. 487 e? 8' ^ ci S 3 a ct OS rH i |g | ft ,a & P O o a -* 3 2 > w H O OS OS _j '2^9 r^ ^ CO -o o * c o5 o> co o S CO CO 3S CO O 3 CO 53 00 * H ~ ^ 2 d OS* T-1* CO rH ,-J 9 M eo' o 1C O M CO Or CO O CO OS CO H rH CO > ft o *- S ft IS 00 rH O "* - rH rH 1 i-l > * 4-3 o Er* o3 ""* S o'ii CO (M CO ^ ^ ^rH ' OOS' OS* W ^ fl | 1 * 5 a ^3 PH* O CO 1 * & 1 ^ d * A ^ O O ^ 1 ~ . 03 .' u - oo" t ill 1 1 1 p **H 1 II fl - -2 0*2, * 5 -2 ^ ^ 1 " 1 4 1 W a 1 * *< n ,Si S r*5 fe || ll^l OS "^ n . co co ^ ; 3 ^ oj ? ^H'|oH 3) ill! ^ O rH CO o t- f SSS2 ^ J ^ fl *3 3 . tpH . ^ S OQr4 S H$ ""~? M '^ ^. H 1 O} nr ^3 ^* ,d^ rH O fl o o S 4-^> "*^ ^ rt E"^ ?-, S J3 | W 1 1 P H Ctt , << "* Pt JS ^* ^ d 2 S TJ k^ Q to OQ lull! 1 O ^ ^ oj ^ ^ 1-3^5^2 ^ CO CO 75 O 000 * .H w -^ A O ^ rH (M j; ! 1 f J t aH rH r-l a fl P-J oo" oo" a ^ ^ "S H -t fl r ^ sl^ll a 1 s - ^5 ;| | | H H gS H co o o S 1 p. r4W y S ^ $ d S ^ < M M . s? ^ (^ ^ Q ^ S 22 -5 ."^J o be -" "^ ^ *^ S ft fl ft 13 * ft ft ft ft CHAPTEK XVII. LATITUDE. 240. By definition, the latitude of a place is its angular distance North, or South of the equator, measured on the meridian passing through the place. From Art. 142 we know that latitude is the declination of the zenith of the place or the altitude of the elevated pole at the place, and the astronomical work of finding the latitude consists in finding one or the other of these arcs, the position of the body determining which arc should be found. In working for latitude the elements involved are Ji, d, and t, and the relation between them is shown in the fundamental trigonometric equation, sin h = sin L sin d + cos L cos d cos t. Now, if t = 0, as when the heavenly body is at its upper transit, sin h = cos z =sin L sin d + cos L cos d, cos z = cos (L d), z = L d and L z -f- d. L = z + d is the general formula for latitude from alti- tudes of bodies on the meridian. The value of L depends on the values of z and d and the method of their combination. This method of finding latitude by observations of bodies on the meridian is the simplest as well as the most accurate one, the results being independent of the time for. all practical purposes (except in the cases of the moon and planets where an error in time affects the declination), and having no MERIDIAN ALTITUDES 489 M, FIG. 114. greater errors due to altitude than the error in the altitude itself. The declinations of the sun and stars do not change rapidly, hence latitude from ob- servations of these bodies is but little affected by errors in longi- tude or time. When a heavenly body is on the upper branch of a meridian, its declination, zenith distance, and latitude are all measured by arcs of the same great circle, the proper combination of any two arcs to produce the third being shown in the illustrations of the four cases considered. Let Fig. 114 be a projection of the celestial sphere on the plane of the meridian. NS is the horizon ; N the North, 8 the South point. Z the observer's zenith. QQ f the equator. PP r the axis of the sphere. P the elevated pole. P' the depressed pole. QZ the declination of the zenith equals the latitude. NP the altitude of the elevated pole equals the latitude. Let M ! , M 2 , M 3 , MI be the four positions of the body to illustrate the four cases. (1) Case of M! whose declination is of a different name from the latitude, or negative. QM^ is the declination. M^Z is the zenith distance = 90 h = z. Then QZ = M 1 Z QM 1 , L = z d. (197) 490 NAUTICAL ASTRONOMY (2) Case of M 2 , decimation + and < L. L z + d. (198) (3) Case of M 3 , declination + and > L. L = d z. (199) In all these cases L is +, d is -f- or ( ) as it is of the same or a different name from the latitude. In (1) and (2) d is < L and the body bears towards the depressed pole, but in both cases z is + ; therefore it is to be marked the opposite of the bearing of the body. In (3) d is > L and the body bears towards the elevated pole, but in the formula z is ( ) ; therefore in this case also mark z the opposite of the body's bearing. In other words, give d its proper mark N. or S. If the body bears North, mark z S.; if it bears South, mark z N. The latitude will be the algebraic sum, with the name of the greater, N. or S. (4) Case of M , a heavenly body at its lower culmination. Then 'PN MN + PM 4 = M+N + (90 Q'MJ L = h + p = h + 90 d. (200) Formula (199) is also correct for this case, provided we use 180 --d instead of d. Writing (200) thus, h = L p, it is evident that the polar distance of a body must be less than the latitude of the place in order that the body may be visible on the meridian below the pole. A body visible at its lower transit in any latitude is termed circumpolar for that latitude. The sun's maximum decimation is about 23, or the polar distance a minimum of about 66 J from the North pole in June, or the South pole in December ; so that the latitude of an ob- server must be in excess of about 66 J to see the sun at its lower transit, in North latitude, at the time of the sun's MERIDIAN ALTITUDES 491 nearest approach to the North pole, as in June; or in South latitude, at the time of the sun's nearest approach to the South pole, as in December. On the supposition that alti- tudes under 10 are not reliable, owing to the uncertainties of refraction, the latitude would- have to be at least 75 and of the same name as the declination to justify meridian ob- servations of the sun below the pole. However, stars are available in all latitudes above 10, for observations under favorable conditions at their lower transit, and, in North latitude, the pole star is available at all times when visible and if of sufficient altitude. Since a heavenly body cannot be seen at its lower transit, unless its declination is positive, or of the same name as the latitude, it follows that the latitude resulting from an observa- tion of a body crossing the lower branch of the meridian will be of the same name as the declination. From the formula in each case it is apparent that an error in a meridian altitude produces an equal error in the result- ing latitude. In all meridian observations the declination of the heavenly body must be taken from the Nautical Almanac for the in- stant of transit; in the case of the sun, at upper transit the declination is corrected for the Greenwich apparent time of noon, which in West longitude equals -f- A of the given date, and in East longitude equals X of the given date or (24 hours A) of the day before; at lower transit the de- clination must be corrected for (12 hours -f- A) if A is West, or for (12 hours A) if A is East, in either case for the instant of local apparent midnight. In the case of stars, the declinations do not change with sufficient rapidity to require corrections, and hence when con- ditions are favorable, as in morning or evening twilight, ob- servations of stars on or near the meridian are desirable. In the case of the moon, the time should be accurately 492 NAUTICAL ASTRONOMY known, owing to the very rapid changes of declination; for this reason the moon is not so well adapted for observation as the sun or stars. In the case of any body observed on the meridian for lati- tude, the sextant altitude must be reduced to the true alti- tude of the center (Art. 212). Polaris is on the meridian when (Mizar) Ursse MajoriS; the second star in the handle of the "Dipper," is vertically above or below the pole star, and since Polaris changes its altitude very slowly when crossing the meridian, such times are the best for observation of that star. However, the lati- tude may be found without appreciable error from Polaris at any time when the conditions for observation are favorable (see Art. 254). 241. Work preparatory to observing a meridian altitude of the sun. It is customary at sea to find the watch time of local apparent noon (Art. 198), to begin observations 10 to 15 minutes before and to take continuous observations till the watch shows noon, the altitude at that time being taken as the meridian altitude. However, observations should be taken till the sun ceases to rise, or dips, in case it is not sta- tionary when the watch indicates noon, the maximum alti- tude being the meridian altitude, subject to the remarks of Art. 246. Before going on deck the navigator should have found the value of A /i; observing the sun by watch at 15, 10, or a few minutes before apparent noon, and applying the correction for A 7t 2 (see Art. 251), he knows very closely, minutes ahead of time, what the meridian altitude and the latitude will be. Ho should also have prepared a constant which, if properly ap- plied to the meridian altitude by sextant, will give the lati- tude at once. The latitude and longitude being approxi- mately known by D. R., find in order the declination and the approximate altitude, for which take out the parallax in alti- tude and refraction. CONSTANT FOR LATITUDE 493 Calling the algebraic sum of the I. C., dip, refraction, parallax, and semi-diameter c, the sextant meridian altitude h 8 , we have: For MI , dec ( ), L = (90 d c) h f . YorM 2 , d + and < L, L (90 + d c) h 8 . For M 3 , d + and > L, L = h 8 (90 d c). For M 4 , lower transit, L = h 8 -\- ( 90 d + c) or/i.+ (p + c). The quantities in brackets are called constants, are com- puted beforehand, and entered in the navigator's note-book. The constant is applied to the sextant altitude as indicated for each particular case. 242. To find the latitude from the sun's meridian altitude : (a) at upper transit; (b) at lower transit. (a) The local apparent time of transit is O h O m s of the day at the ship. Find the G. A. T. of local noon by apply- ing the longitude ; + if West, if East. (2) Take the sun's declination from page I of the Almanac for the proper month and correct it for the G. A. T., marking it, as it should be, N". or S. (Art. 185). (3) Eeduce the sextant altitude to the true altitude of the center (Art. 202). Subtract the latter from 90 to get the zenith distance. If the body bears N., mark the zenith dis- tance S.; if it bears S., mark the zenith distance N. (4) The latitude is the sum of the declination and zenith distance, if they are of the same name, and is marked like them; the difference of the two, if of different names and marked with the name of the greater. (b) At lower transit the local apparent time is 12 hours. Find the corresponding G. A. T. and for it the declination of the sun, then the polar distance = 90 d. (2) Reduce the sextant altitude as above explained. (3) Add the altitude and polar distance; the result is the latitude marked like the declination. 494 NAUTICAL ASTRONOMY d 2 g CO oT % P C? 01 ^ 1 00000 i O to r-l * CO TO 'o ^ >- '-13 'S a ^OCOCOt-tol-GO o to o to to 10 P J I 05 ?> CO TH ^ PEj fe c o 1 8 01 CO CO 8 1 o 0| i co" co* 'rt ^ ^j rH O5 ft d fl S ^ j,. to ff _r3 'g .2 M * 1 ^ Q P 1 g * -S o"i 7? ^ ^ ^o M fl "S S d i 02 Jz; CO d rt C o O QJ o3 0^0 002 h^ g "o ^ CO O 01 CO CO O P rSi "* T-( O O1 to rH rtl 03 ^ CO r-l to 05 o d n4 3 03 o' tf - ^6 ^ 73 I o Q ft Q^ ti ^ CO O O5 CO N OQ Hi C5 <1 p, Q - rH C 7 >- -I 1 5 w ^ CO r-l 10 05 t- ^ 3 0) II + 1 + 1 H ^ 1 CO CO ico'cdo OCO^OTH THO5to T-l to TH O 1 1 | H 2 ^ Jx C005COOCO to to o to o to ' * ^ ^ < oo? oo?> toto^-i W 05 Ci CO ^ 05 02 M (5 ft o 03 ,a 02 53 02 ft 02 5 02 fe "^ <$ ^?< ^ t- t- to 5 to o to 10 CO to -+J a 1 * ' 1 o 43 o1 -|- ?; s^ 5 3 m a " 1 o 1 Altitudes 01 ** % o .2 2$S "3 t ,-^ -*v S ; ^ nd the co constant. fl ^ *' 1 01^ CO V * S< . . ^^ 5 IT ^o * j* O 03 % H ^: <> .0) ' O . t a ** r; " I 1 3 o g -g o II -^ '><) p 'OMg' as . ,2 . . a i i ^ S* 4-3 KM P g c 020 3 w t-i CM H fl ^ NQ i c 03 fi * g | J | J | | g 0oo i fl J pj o + 4- | + 1 - 1 o : r4 1 . . ig - O O CO CO * tO 03 O O O be B a5 o i ifi + + 1 + o tJD | Pd ft ^^ C ^ 4-3 -M . CU GO O 2 OQ HH Q <5 (i ^ J bfl 5 PH' CS rH CO 05 + c. oco og g 53 QJ ^00 g^ o | 2 Hr ^ h^ fc p-' 1C O O * O r- II 1 d *" N rt| ^ "2 o .-> B g co : ~ c- -t^ S3 o 2 ^ a> -) : }H 056 p. p. p. >_ OQ ^ r-( 3* << M o eo ^ o o oo - *> T*I 111 OJ h-i H P, O O CO H Oi O! S | - e 5 p I-. _ Ho O I-. u_, S5 S OS t~ o: SS 01 +, 03 S I a, s I! Hi O o O O S o! O OS O QoQ ^ i o eo g 1 1H * OQ kl IH 05 6 a ^ MERIDIAN ALTITUDES 497 1! d 5 CO CO K's declination 35 18 15 S 54 41 45 N 57 43 33 S Corr. - 3 45 Latitude 3 01 48 S Ex. ISO. April 19, 1905, the sextant altitude of the star a Aurigae (Capella) on the meridian below the pole was 11 20'. I. C. + 2'. Height of eye 40 feet. Find the latitude. Altitudes, &c. Altitude Corrections. #'s Dec. and Polar Dist. ^s sextant alt. Corr. to alt. o / tt 11 20 00 8 55 i it I. C. + 2 00 Dip 6 12 Ref. - 4 43 o t n *'sd= N 45 54 04.3 *'sp= 440565.7 ^:'s True alt. r^'s polar distance 11 11 05 44 05 56 Corr. 8 65 Latitude 55 17 01 N The latitude has the same name as the star's decimation, otherwise the star would not be visible on the meridian below the pole. 244. To find the latitude by the meridian altitude of the moon. The moon being comparatively so near to the earth, its changes of declination and semi-diameter are more rapid than in the case of other bodies ; besides, its parallax is quite large. These elements require careful correction, and the great lia- bility to error, due to error of time, render observations of the moon less desirable than those of other bodies. When the moon is near the equator its declination changes most rapidly, and at such times the maximum altitude may differ consider- ably from the meridian altitude. A movement of the zenith 500 NAUTICAL ASTRONOMY due to high speed of observer's ship, especially in the direc- tion of the meridian, will intensify such a discrepancy (Art. 246) ; hence it is better to calculate the time of meridian pas- sage of the moon and consider the altitude observed at that time as the meridian altitude. Rules. (I) Find the Greenwich mean time and date of the moon's local meridian passage (Art. 188). (2) For this G. M. T. find the moons declination S. D. andH.P. (Art. 185). (3) Reduce the sextant altitude to the true altitude of the center (Art. 212). (4) After which., proceed as with the sun. 245. To find the latitude by the meridian altitude of a planet. (1) Find from the Nautical Almanac the G. M. T. of Greenwich transit. To this apply the retardation or accelera- tion for the longitude, and the result will be the L. M. T. of local transit (Art. 189), the retardation or acceleration per hour being one-twenty-fourth of the difference of times of transits on two successive days as indicated in the Nautical A Imanac. (2) From the L. M. T. of local transit and the error of ob- server's watch on L. M. T. find the watch time of l^cal transit and observe the planet's altitude at that time. Reduce the sextant altitude to a true altitude by applying the I. C., dip, and refraction, neglecting for sea observations 8. D. and par- allax. (3) To the L. M. T. of local transit apply the longitude and obtain the G. M. T. of local transit. For this G. M. T. find the planet's declination; after which> proceed as in the case of the sun. MERIDIAN ALTITUDES 501 3 8 3$ ^a q q 8ss CO M-OQ SS5 ? + OQ Jgj 00 fe t. 85 tf|j ft + d w ^5 "* b M **> 2 & S g s d o O 4> a-g H ""' 2.3 *S _, -4-J o s a? . eo oj CC S * O * IM CO I I + I I ft Sd I M 96 g LLS 351 & U . i i * .' la ^ ^' 0. .2 H I b a ff * 5aj3d 502 NAUTICAL ASTBQNOMY o o CO .LJ & o s * CO oo 5 OO to 1-rfl i t-' lolco "" O O? rjt - r-t CO o M Q o co c5 2 t ' 1 O i O and Zm = <'. By Napier's rules, we have tan < = cot d cos t, cos ' = cos sin h cosec d, \- (206) L = 90 (<#> ') FIG. 115. ' METHOD 509 (207) Following Chauvenet's methods, the above can be put into more convenient form. If for < in the above, 90 " be substituted, then tan <" = tan d sec t, cos <' = sin " sin h cosec d, L = " zp <'. Case of M , d + and > L. From the figure it is seen that $ Pm, the polar distance of m, the foot of the per- pendicular ; <" Qm, the declination of m, the foot of the perpen- dicular ; <' mZ, the zenith distance of m, the foot of the per- pendicular ; L = QZ, the declination of the zenith. But QZ Qm mZ or L = " <'. FIG. 116. As shown in the figure (115), the declination of M is posi- tive and > L, and finding the latitude in this case resolves itself into the finding of the declination and zenith distance on the meridian of the foot of the perpendicular and combin- 510 NAUTICAL ASTRONOMY ing them by the rules applying to a similar case in finding latitude from the meridian altitude of a body. Case of M 2 , d -f- and < L. In triangle PZM 2 (Fig. 116), <" = Qm 2 and ' = m z Z. QZ = Qm 2 + m 2 Z or L = <" + <'. Case of If 3 , d is negative. In triangle PZM 3 (Fig. 116), 0" = Qm 3 and 0' = w 3 ^. QZ = m 3 Z Qm 3 or L = f ". Case of M^ , t > 6 hours, J +. In triangle PZM 4 , <" = Qm and <' = m^Z. QZ Qm mZ = " '. In this case of M 4 , t is > 6 hours and d is + ; therefore, Qm^ is taken out +, same sign or name as d, but in the second quadrant or same quadrant as t. Now, 0ra 4 = " = 90 + Pm^ = 90 + p, mZ = ' = 90 Nm 4 = 90 7t, ~~li = L. Therefore this case corresponds to that of a body observed on the meridian below the pole. As <' is found from its cosine it may be either -f- or , thus giving two values of L, differing largely from each other, un- less ' is small. However, the latitude is approximately known, and no trouble need be experienced in determining how to mark cf>'. The following rules, closely attended to, will prevent any error in the proper marking, or the method of combining <" and ' to obtain the latitude. (1) The mere fact of t being +. (W.) f or (E>), has no influence on the signs of the functions. If t is > 6 hours sec t is ( ) and " is in the second quadrant. Therefore, (/>" ' ^i i TII oi) 511 (2) In formula (207) " is taken out in the same quad- rant as t and is marked N. or S. like the declination. ' f being the zenith distance of m, is marked like the zenith dis- tance of the body in a meridian observation for latitude; that is, if the body bears northerly, mark the zenith distance 8., if it bears southerly, mark the zenith distance N. Then combine " and f algebraically according to their names. The result will be the latitude. Under the following conditions this method is not condu- cive to accuracy, or fails entirely. (1) When ' is very small, that is, when Z is near 90 and the body is near the prime vertical, it cannot be found accu- rately from the cosine. (2) When d is 0, <" is 0, <' is indeterminate, and the lati- tude cannot be found by this method. Observations of the sun, planets, or fixed stars, worked for latitude by this method give excellent results; owing to the very rapid changes of the moon's elements, and the uncer- tainties of the hour angle due to the uncertainties of longi- tude, observations of the moon are not recommended. Rules for Working a "<' Sight. (1) Find the G. M. T. of observation for which in the case of the sun, take from the Nautical Almanac the declination and equation of time; or, in the case of any other body, its right ascension and declination and the right ascension of the mean sun, and also, if the moon has been observed, its semi- diameter and horizontal parallax; and reduce the sextant alti- tude to the true altitude of the center. (2) Find the body's t, then having t, d, and h, proceed by substitution in formula (207) to find $" and ', paying par- ticular attention to the rules preceding regarding the sign of t, and the naming and combining <" and <' to obtain the lati- tude. 512 NAUTICAL ASTRONOMY g o o o altitn l()m 06 li S H P-H *> 03 . H i "* d II 22 ^ OS -^ O II + I I l t CO M 5 P. 016 'S o a o fl 42 S5 8 S 58 ^ 8 II II II II II "S 8 ^S w ss 1-1 I , . ^ 00 O-< W O i-5 H) " <' SIGHT 513 .rt 33 5 *r I tOD "' method. Limits of Table 27. The limits of this table are the limits within which the method may be used with a fair degree of confidence in the accuracy of results. The use of the table depends on the object sought; for in- stance, in determining latitude, when surveying, the hour angles of bodies observed should be so small that the value of A/& itself should not exceed 1' ; on the high seas, the reduction obtained under conditions in which even the limits of the table are used will be sufficiently exact.' Again the table may be used at sea beyond its limits in the following way, if this use 522 NAUTICAL ASTRONOMY is desired: Suppose A ft = 4".0 and t = 24 minutes; since the table does not give Aft for t beyond 21 minutes, find the reduction for 12 minutes and multiply it by 4 to obtain Aft for 24 minutes. Therefore, the required Aft = (9' 36") X 4 = 38' 24". Proof of this is seen from the fact that 4. Hence, if the hour angle is greater than the tabulated one for the given value of A ft, take out the correction Aft for one- half the H. A. and multiply it ~by 4; the result will be the required Aft. Eestrictions in the tropics. In the tropics, where at transit the body's altitude may approach 90, the factor cosec (L ~ d) will be so large as to make Aft too great for the assumption made in the deduction, that sin J Aft = Aft sin 1" For such cases the value of A ft is not tabulated. In those regions, therefore, in summer time, this method is not applicable; however, it is not much needed owing to the strong probability of the sun being visible when on the meridian. To Find the Declination and Latitude. The declination to be combined with the meridian alti- tude ft should properly be corrected for the Gr. M. T. of ob- servation ; in the case of the moon this is essential ; in the cases of a planet or the sun, it is sufficiently accurate to use the de- clination at the instant of meridian transit, except when the hour angle is large, and in the case of the sun, therefore, the declination may be corrected for the longitude at upper tran- sit and for (12 hours -f A) at the lower transit. Having found ft and d f the latitude is found as in the case of a meridian altitude (Art. 240). It must not be forgotten that the latitude thus found is for the instant of observation, and that the latitude at the time "REDUCTION" TABLES 523 of transit (or in the case of a sun sight, near the upper merid- ian, the latitude at noon) may be found by applying the run in the interval. Errors in H. A. The effect of errors in H. A. may be min- imized by observing the body at practically the same altitude and with small hour angles on opposite sides of the meridian, reducing the latitudes found to noon and taking the mean of results for the true noon latitude. Various "Reduction" or "Ex-Meridian Tables." The Tables of Bowditch and of Brent, Walter, and Williams, which are practically identical, have been referred to. The argu- ments in these Tables are L, d, and t; so the navigator, having set his watch to L. A. T., may have in his note-book the cor- rections to be applied to altitudes to be observed at certain times by the watch to obtain the meridian altitudes, in fact have ready a constant allowing for the run to noon (see Art. 253), so that the noon latitude may be found at once by apply- ing this constant to the observed altitude. Besides, the above- mentioned tables are applicable to bodies of a declination as large as 63. Towson's Tables are also issued to the American navy. They are not applicable to bodies whose declination may be greater than 23 20'. The arguments used in these tables are d, h, and t, so that the correction is taken out after the altitude has been observed; a matter of delay if not of in- convenience. There have been various graphic and automatic methods for finding the value of the " Reduction to the Meridian," the best of which perhaps is the invention of Wm. Hall, Naval Instructor, R. N., and which consists of two calculating slides for automatic calculation. It is known as "Hall's Nautical Slide Rule/' Rules. (1) Find the watch time of transit (Art. 198), and the H. A. from the meridian, remembering it is to be in sidereal time for a fixed star, and that for the sun the 524 NAUTICAL ASTRONOMY mean time interval may be used. (2)Take from the Nautical Almanac the declination for the instant of transit,, or in the case of the sun, for local apparent noon (if sun was observed near lower transit, for local apparent midnight). (3) From Table 26, Bowditcli, take out A /t and from Table 27, A/i. (4) Reduce the sextant altitude to a true altitude of the center and apply Ji; adding, when the body was observed near upper transit, subtracting, for an altitude near lower transit. The result will be the meridian altitude. (5) Then proceed to find latitude as in Art. 240. Attention is called to the fact that formula (212) is made applicable to the case of a body near lower transit either by substituting 180 d for d, or, by substituting L for L since the lower transit of the meridian in a given latitude is the upper transit of the same meridian at the antipode; hence for a body below the pole take A /i from that part of Table 26 in which L and d are of different names (see bottom of last three pages of Table 26). KEDUCTION TO MERIDIAN 535 fsl & " ** w S ^ , a O O O5 O O CO 05 ^ "0 5 . CO | C * 3 I 00 ?O 1 CO * e S S d S a; fl rt 4^ ^^10 t^ it E-* S 33 5j a .2 M ^ 5 c^ HS ^ 5 o - M < no -g OQO - * < # # HH je r* OCO M CO CO CO CO CO CO O5 ^ S O O O OJ * W 3 JO O rJH CO rH c >O JO to O O C > rH rH CO* CO O -H O> OS T > OO OrH OO5 050** d S ^ 1 CO rH t- O5 O C * 05 CO CO C > )OO5 I>>O rHO U51O.O > CO CO O CO O5 03 a -a o CO t- 1 C- ^ rH rH 1 rH H CO _1_ rH OO O >O I 1 G H M A kH H ~ ^ -M ^5 '3 ' q N 2 J d g ^ ^ o 2 S g-2 b 2; oj 5~ -j a> >5? H ,-s II 5 * S k ^3 S fl ^ ^ .S ^ . W "* ^i a S s la 1. o i |i l^ <0 IB S ^ " . ** ijlll^ *.*. -5 * i-i ^ c "1 1 i! Si ? o 5 6 H ' ^ H H ,j CL ^ ^ a * ts H ^ c ^ _rt 1 * * ^r *C^ <1 OQ -X- S h^^ OQOO^^ "** <3 528 NAUTICAL ASTRONOMY 252. To find the latitude from a number of altitudes of a heavenly body observed very near the meridian, the longi- tude and Greenwich times being known. Very near the meridian, the change of altitude varies nearly as the square of the hour angle, so that the mean of the altitudes cannot be taken as corresponding to the mean of the times, but each altitude may be reduced to the meridian by the principles of Art. 251, and the mean of these used in finding the latitude, hence the term Circummeridian Altitudes. Let 7& , h 2 ,Ji z ...... Ji n be the several true altitudes ; /! , t 2 , tj ....... t n be the corresponding hour angles in minutes of time at the times of observations ; &Ji, A 2 7z,, A 3 7t . . . A n ft be the several reductions to the meridian ; where A^ = &oht\ , A 2 fe = A /^ 2 2 , and A n /t = A /i 2 n . Then for n observations, the mean value of the meridian alti- tude will be n n Substituting the values of the reductions as above, ,+ft, ...... k n+ (f. +<.+*. ..... <.) ^ (2U) From this value of h , the meridian altitude, the latitude is computed. The principles involved in this method suppose the declination not to change from the time of observation till the meridian passage, and as the declination for that in- stant is wanted to combine with what would be the meridian altitude, it is better to find the Greenwich time of passage (see Art. 196), and for this take out the body's declination. Then find the watch time of passage (Art. 198), the differ- ences between which and the watch times of observation will ClECUMMERIDIAN ALTITUDES 529 be the hour angles of the body expressed in mean time. The mean time interval differs from the apparent time interval only by the change in the equation of time in the interval ; so in the case of the sun the mean time intervals need not be reduced. In the case of a fixed star, they must be converted into sid- ereal time intervals. The values of refraction and parallax for the various alti- tudes will differ so slightly, that it will be sufficient to re- duce the mean of the sextant altitudes to a true altitude, to which the reduction will be applied to give the true meridian altitude. When possible, altitudes at about the same hour angles should be taken on both sides of the meridian in order to eliminate errors due to the time. The best results are gotten when two stars culminating at about the same altitude are observed on opposite sides of the zenith; for, by taking a mean of the two latitudes thus ob- tained, personal and instrumental errors, if the instruments are used in the same way and under like conditions, are elimi- nated. In using this method on shore, if prismatic effect is suspected in the roof of the artificial horizon, it would be better to take two sets of observations, the roof being reversed between the sets. At sea, single observations near the meridian are sufficient and A & from the tables are accurate enough; but for refined determinations of latitude on land, it is better to take a num- ber of observations near transit, on both sides of the meridian, using a bright star in preference to the sun, and computing the value of A 7L The barometer and thermometer should be noted during observations ashore, and a correction (Tables 21 and 22, Bow- ditch), dependent upon the instrumental indications, should be applied to the mean refraction. 530 NAUTICAL ASTRONOMY 73 00 O rfJ bfi s I g III *fi -r a o a? .S Mi - rH kH O g ^ d a o * S a r?g *2Ii d 1 3 * I i 2 " S 60 A pC) C3 O r4 'S3 5 I 80 *# = S8 o i I cf ^-M g + -g58SSSSS o 8S CO CO O 00OrH COrHTjl 5 OSOrH CX) ^ a "* 1 O O & 60 7 1 gg S S S CO + CO + COOr^ c pi ' *" S I2> o l 2 010 ^ S 1 tff ^ ^ M H e8 08 08 jj o *^ ^ r-OOSO 10 Oi^O SS ji ?*, J 60 "s S N^J * O r-> CO rH N * O Jo *3 d _o ^ " < CO O rH CO N O OS CM rj ^ *^j ^ M'l be > CO W CM a .2 O CM &CM ^ 1C CO rH CO o 5 10 CM o CM CM rH rH 1-1 rH ^ .2 ^ | sgs OS o + + I + I + I d p, ft s 05 ril l--* O5 CO - s O o o 08 o ft ^. W ft O S T ^ - 8 13-a -e H M - + 2 +2 S a s & So S 2 ^ OS o ^ O) *5 ill uJLfc CO CO CO CO CO CO CO c co" 10 os" w to* co' O5 * O rji O Ui rH OS 00 O to CO *-Or-(O5O5O5 OS OS OS OS OS OS OS ^OSOT^05rHOO I I I I I I + v prt 03 O J S 1 536 NAUTICAL ASTRONOMY 254. Fourth method. By altitude of the pole star. The given quantities are I, d, and h, the required one is L. Formulas (206) apply here, but owing to a very small polar distance in the case of Polaris, they can be simplified, tan = cot d cos i, cos <' = cos sin h cosec d f 90 L = <#>'. As before, is the polar distance of foot of perpendicular, ' is the zenith distance of foot of perpendicular. Now and p are so small, that having substituted 90 p for d, 90 z for h f we may consider cos < and sec p each unity, also tan < = tan 1' and tan p = p tan 1' ; hence the above will become $ p cos t, cos <'=cos z or <]>'=z, 90 =<(/>'=z= Lli^ or L=180 Therefore, L = h pcos t, (215) where ft is the true altitude of Polaris; p, the polar distance of the star at the instant of ob- servation ; i t its hour angle. Close attention must be given to the sign of cos i as it affects the sign of application of <. If t < 6 hours or > 18 hours, cos t is -f ; if t > 6 hours and < 18 hours, cos t is , and L = h + p cos t. The second value L = 180 h is inadmissible as it exceeds 90. Since by definition the latitude equals the altitude of the elevated pole equals PN 9 in position a, L=N& P& = h 1 p; at position b, L = PN N\> + bP = h 2 + p. The mean of these two will give excellent determinations, that is, the mean of the latitudes from observations at upper and lower transits. (See Fig. 117.) LATITUDE BY POLARIS 537 Let M be any position of Polaris when t is < 6 hours. Let ZM = Zd. Let Mm be a perpendicular to the meridian, and regard PMm as a plane triangle, then is the polar dis- tance of m and equals p cos t. By the above formulae L = h p cos t; in other words, Nm is assumed equal to Nd or HM, the star's altitude. For any other position as M^ , when t is >6 hours, L = h + p cos t and -ZVra! is assumed equal to Nd^ or H^M^ . Though these assumptions are a source of a slight error, the above method is sufficiently ex- act for all nautical purposes. It is available at all times when the horizon is distinctly seen, and the star Polaris is visible and of sufficient altitude to eliminate the errors of re- fraction. Its application is limited to the northern hemi- sphere. Table IV of the Nautical Almanac gives the value of p cos t at intervals of 5 minutes of hour angle, computed for a mean value of the right ascension and polar distance of Polaris for the current year. This correction, applied ac- cording to its sign, will give the latitude, which is not so accurate as that computed from the apparent right ascension and polar distance from the Almanac, except when these are near the values used in computing the table. In Art. 176 of Chauvenet's Astronomy, a rigorous formula is deduced, from which the latitude by altitude of the pole star may be found with great accuracy. It is L = h pcoa t + Jjo'sin I" sin 8 1 tan h | i p* sin 2 1" cos t sin 2 1 + i p' sin 8 1" sin 4 1 tan 8 h ] ( FIG. 117. 538 f.l I" 3 | - ( II CO O O -V-3 CO 03 S 43 II 3 d NAUTICAL ASTRONOMY S8 S to : ^3 & SH -P S * 88S ^ (N 10 T-I O HH'fi - ss ^ CO O I a 5f 8 g -P j O BH g S rf d S d * 5( LATITUDE BY POLARIS 539 1* S o o 11 5 "3) 1 5 o I? ss 2 a o co o M d -& . M a o 5fc SS sJ.lJl S ^ ^ S eo 2 I I si $r s . . ft o . f f f 00 Stf C5-3 J4(- ^-^- 540 NAUTICAL ASTRONOMY The sum of the last three terms in equation 216, page 537, represents dm in Fig. 117, also dm^ , etc. Table 28 of Bowditch is computed from that formula. The last two terms may he omitted with no greater error in the latitude than 1". If p cos t is the only correction ap- plied, the error will amount to only about 1' when t = 6 hours and li = 54, and a maximum of 3' when t = 6 hours and h = 68 30'. 255. Fifth Method, called Chauvenet's Method. This con- sists in finding the latitude by two altitudes near the meridian when the time is not known. It frequently happens that the time is uncertain, or the deck watch has not been compared with the chronometers, enabling the navigator to get the correct hour angle at obser- vation; under such circumstances this method is of great use to the practical navigator. Let h^ and Ji 2 be the true altitudes of the body at the first and second observations; TFj and W 2 be the corresponding watch times of ob- servation ; x and y be the unknown hour angles of the body, re- spectively, at the first and second observations; T be the interval of time between the observations, then! 7 = W 2 F ; x ~ y be the difference of hour angles of the body at the two observations. For the sun, it is an inter- val of apparent time and without error may be represented by T. For a star x ~ y is an inter- val of sidereal time which equals T when T is reduced by Table III to a sidereal interval. CHAUVENET'S METHOD 541 As in Art. 251, let 7i represent the true meridian altitude of the body, and A /t the change in altitude in l m from the meridian. Then, by formula (213), Taking the half sum of the above equations, we have = *!_+_* + *+jC AA . ' (817) but +jf = (^y\* 4 .(*y\*-_(*+y\*i(T\*. * ( . /jv>^~\^/- \T/ therefore, Taking the difference of the same equations, we have (x y) 2 A O and 7t-A ._ (219) / Substituting this value of ^in (218), Therefore, to obtain the meridian altitude by this method, two corrections must be added to the mean of the body's two true altitudes. The first is of the form of the reduction to 542 NAUTICAL ASTRONOMY the meridian, using one-half the elapsed time in place of the hour angle. The second is the square of one-fourth the dif- ference of the altitudes divided by the first correction, care being taken to have both terms of this fraction in the same unit, usually seconds of arc. The second correction is the larger of the two, as a gen- eral thing, and, as this depends largely on the difference of altitudes, the accuracy of the resulting latitude will depend on the precision with which the altitudes have been measured, since errors due to the tabulated dip, refraction and constant instrumental errors affect both altitudes alike. Having found Ti Q> proceed as in Art. 251 to find the latitude. When Ji 2 = Ti lf the second correction reduces to zero; therefore, the most favorable case is that of equal altitudes observed on each side of the meridian. The value of the hour angles may be obtained approxi- mately thus, From (218), X * &* ~ &i) _^(h, h l ). T |A 2 f Restrictions. The restrictions of this method are the same as those limiting the reduction of a single altitude to the meridian. It must be remembered, however, that the obser- vations in this method are not made at the same place. A CHAUVENET'S METHOD 543 slight change of the observer's zenith, which would result from a small interval between observations, would produce but a slight error and especially so when the course is at a right angle to the bearing of the body. When the interval is com- paratively large, and the distance run also of consequence, the first altitude must be reduced for the run (see Art. 213) to what it would have been, if observed at the same time at the second position. The value of T will not change. The latitude found will be that at the instant of the second observation; and to obtain the noon latitude, allowance must be made for the change in latitude during the run from the time of the second observation to noon. It is not necessary to reduce each altitude to a true altitude and then take the mean. It will be sufficiently accurate for practical purposes at sea to take the mean of the sextant alti- tudes, and reduce it to the true altitude of the center. 514 NAUTICAL ASTRONOMY ES S | 1 1 P o O II 2-rffi ^ 1C 00 * H S fl O _/ ^ . ^fl 05 I I M* ^ a JS S |j w S fc 1 QQ CO o o 5 02 05 05 O O5 in TH 1> CO CO iH bJO bfl bJD O O O - - ^ . T-l O SO t- ^ CO CO OQ w D p, O 00 (M SO d 05 * >o T* oo H ^ ! 3 SS< ^oo - rH iH T s'i II II II cT" L' f -r-r- E- H PRESTEI/S METHOD 545 256. Sixth method.- To find the latitude by the rate of change of altitude near the prime vertical (called Prestel's method). In Art. 237, by differentiation of the fundamental formula of the astronomical triangle, sin h = sin L sin d -)- cos L cos d cos t, regarding h and t as variables, we found formula (188), from which, expressing dt in time, we have dt = jg- sec L cosec Z, in which dh is a small change of altitude in seconds of arc, occurring in a very brief interval in seconds of time. If the altitude is increasing, as when the body is East of the meridian, the hour angle is diminishing or dt is ( ) ; if the altitude is diminishing, as when the body is West of the meridian, the reverse holds true. Let w i be the noted time when the body is at the alti- tude &! , w 2 that when the body's altitude is Ji 2 ' } then T = (w w 2 ) = *-= - 1 sec L cosec Z, and T = w 2 w * .. , ' l sec L cosec Z. cos L = ^^ cosec Z. (223) When Z is near 90, its cosecant varies very slowly and when Z = 90 we have cos L = h ^p l * ( 224 ) The accuracy of this method depends on the precision with which the altitudes are measured and the care with which times are noted. As the latitude is found from its cosine, the method is more precise in high than in low latitudes. Though the re- 546 NAUTICAL ASTRONOMY suit may be only approximate, it may be useful in restricting the ship's position to a limited portion of a Sumner line. The time when a body is on the prime vertical can be found from the azimuth tables, or from Art. 239, or sufficiently near by compass if its error is known, or by Table C of N". A. In case the body is within 2 of the P. V., measure the alti- tudes and note the times carefully, not letting T be > 8 m ; use formula (224:) and for high latitudes the result may be found within a limit of error that would still make it desirable. However^ only an emergency will justify the use of this method. Chauvenet recommends bringing one reflected limb of the sun into contact with the sea horizon, the time being noted ; then, keeping the sextant clamped, note the time when contact of the other limb occurs ; beginning in the forenoon with the upper limb, in the afternoon with the lower limb; dh will be the sun's diameter in seconds from the Almanac. In case the body is more than 2 from the P. V., use for- mula (223). Ex. 198. April 24, 1905, in' approximate latitude 43 20' N. 9 longitude 30 10' W., about 5 p. m., the sun bearing true West, the sun's reflected lower limb was brought tangent to the horizon. W. T. 4 h 59 m 03 s . The sextant being kept clamped, when the upper limb made contact with the horizon, the watch read 5 h Ol m 58 s . Find the latitude. Formula i **~*i h ^ 15 T= 175 L - 43 16 3 log 3.28133 colog 8.82391 colog 7.75696 15 T fcj^ Aj = sun's diameter = (15' 55".66) X 2 = 1911".32 r 15" N cos 9.86220 257. Reduction of latitude. In the previous articles of this chapter, we have assumed the earth to be a sphere, im- plying that a plumb-line at any point of the earth's surface, "KEDUCTION OF THE LATITUDE" 547 if extended, passes through the earth's center, and that the altitude of an observed body, after the usual corrections have been applied, is referred to the center. The earth is not a sphere but a spheroid, and the vertical line at any point of the surface as O'L in Fig. 118, which corresponds exactly with the normal drawn at that point, doe? not coincide with the earth's radius passing through the same point excepting at the equator and at the poles. The point Z where the ver- tical line O'L prolonged meets the celestial sphere is the geo- graphical zenith and the angle ZO'Q is the geographical lati- tude of the point L, as deter- mined by observations at sea. The point Z' in which the radius OL prolonged meets the celestial sphere, is the geocen- tric zenith and the angle Z'OQ is the reduced or geocentric latitude. The geocentric latitude is smaller than the geographical latitude at all places except at the equator and poles where they are equal ; the difference between the two being the angle OLO' called the " angle of the vertical " or the " reduction of the latitude." Though necessary in certain refined observations ashore, it is not necessary to consider the reduction at sea where ex- treme- precision is unattainable. FIG. 118. CHAPTEE XVIII. CHRONOMETER ERROR, CORRECTION, AND RATE. LONGITUDE ASHORE AND AT SEA. 258. It has already been shown in Chap. X that the chro- nometer is the navigator's means of getting the Greenwich mean time of any desired instant or observation. Though constructed with the greatest care and at much expense it is far from perfect, seldom indicating the exact time of the prime meridian and seldom running with regularity for any length of time. However, a sidereal or a mean time chronometer is said to be regulated to local or Greenwich time, when its error on that particular time, the amount by which it is fast or slow of that time, and its rate, or daily gain or loss, are known. Both the error and rate are positive, if the chronometer is fast and gaining; otherwise, negative; the sign of the error being the sign of application to the correct standard of time to get the chronometer reading. It is preferable, however, to regard the error as a correction to be applied to the chro- nometer reading to obtain the desired true time, and to con- sider the rate as a daily change. Both are positive or plus when the chronometer is slow and losing. 259. To find the rate. The rate is found by taking the algebraic difference (that is, the numerical difference when of the same name, the numerical sum when of a different name) of the errors on two different days and dividing it by the elapsed time in days and decimals of a day. The interval should be at least 5 to 7 days. When the errors are SEA KATE 549 determined at two different places, the times of observation should be reduced to one (say Greenwich) meridian and the interval found from the two reduced times. The rate will be gaining when both errors are fast and the last one is the greater, when both errors are slow and the last one the lesser of the two, or when the error changes from a slow to a fast one; otherwise, the rate will be a losing one. Ex. 199. The error of a given chronometer on G. M. T. on April 15 at noon was -|- 5 m 32 S .5; at noon on April 25 it was + 5 m 35 S .8. Eequired the daily rate. Error at noon April 15, + 5 m 32 S .5 Error at noon April 25, + 5 35 .8 Change for 10 days + 3 3 .3 Daily rate + .33 260. To find the error on a given date, knowing the error on another date and the daily rate. Multiply the daily rate by the number of days elapsed since the determination of the error and this, applied with proper sign to the original error, will give the error on the required date. Ex. 200. With the data of the above example, find the error of the same chronometer on G. M. T. at noon April 30. Error April 25, + 5 m 35 S .8 Daily rate + O s .33 Change + 1 .65 No. of days 5 Error April 30, + 5 m 37 S .45 Change + l s .65 261. Sea rate. Ordinarily the error and rate of a chro- nometer are determined entirely from shore determinations, and that error is brought up by its rate to the instant of later observations at sea in working for longitude. Now this rate found in port may be very different from the actual sea rate, even at the same temperature. In case an error is determined just before leaving port and again after return to the same port, the difference of errors divided by the elapsed time will give the sea rate. 550 NAUTICAL ASTRONOMY Again, a vessel on a voyage may stop at many places whose longitudes are well known, these having been determined per- haps by direct or indirect telegraphic connection with Green- wich or some place of known longitude. Say the error of the chronometer on G. M. T. is found at place A, of known longitude; by applying the longitude to the local time of determination of the error, the Greenwich time of the determination is gotten. At place E, obtain the same data, the chronometer error on G. M. T., and the Green- wich instant corresponding. The algebraic difference of the errors, divided by the elapsed Greenwich time, will be the sea rate, the rate being regarded as uniform. Ex. 201. On April 2, 1905, at Southampton, in longitude 1 18' W., a time ball was dropped at O h 00 m 00 s L. M, T. At this instant the error of a chronometer A was found to be slow 14 m 52 s on G. M. T. On April 28, 1905, at Lisbon, in longitude 9 12' W. by single altitudes in the forenoon, using artificial horizon, the error of the same chronometer was found to be slow of G. M. T. 8 m 50 S .005. C. T. of observation 9 h 15 m 36 S .2. Eequired the sea rate. hms hms ms April 2, L.M.T. 00 00 April 28, C. T. 9 15 36.2 Error April 2, slow 14 52 Long. W 613 C.C. +850 Error April 28, slow 8 50.005 G.M.T. April 2, 05 12 April 27, G.M.T. 21 24 26.2 Gain = 6 01.995 April 2, G.M.T. 05 12 = 361.996 Elapsed time 25 o s s sa K- O r ^ ? S O fl *J ^ .S o s 1 ^ * S - - & s a oJ M fl ^ S S ^ o s 1 O iO iO O IO CO b- O "08 -*J S . PH O fl. 03 . S <1 g g I 1 7 I H fcj ^j r> t- CO t^ ^ CO W o o IO OS - tH O 1O O 00 CO ^ -S! J ^ iH WO rj "T~ o S fl OS . & H 1 S rH H CO j*cU H ^ S J5 O* r : ^'EH cJ IH 3 tA . o CO t~ 0) ^ ^ go. r -43 eo d ^ 05* W ^ CO CO o ^'fe: "^ ^' J g'd ^ rrt CO (V) _o 33 ^ t- l-lr< l-sfc] OOOO w O S rH / O J"" 1 rt {H q ro 05 = 05 * rH OS CO CO ^ * CO 2 *tg1 OS a ^ 2 0'-3 ^ rH 1C ^ 05 ^ 1 C^j CO CO W C O O CO C 2 o' o' os' c TH rH CO rH M * rH OS" - o r- is 03 rt ^5 f^ +J O S + Q5 + oT 5||| 2 211 ^ 2' ' CQ ft O i-*' Cfl CT 1 5 ^ - 00 -H rH CM C* o |i fl Q & -*J "aj .10 "P | ;T T^t-COO ^.rHOCOl>00^ A JO rHrH CO ^CttCOO OJOCO Tt l> ^OOl>COrHT-l COrH-f-rH COiOrHTjHoO? . ^ ^ a O^COWCOOOS j" \ w - A. <&*(*' + *") tan C'C ^ JH ^^?5 s ? HEH H 1 | 0*0 ^ ^ wii drill w'wl I '^g I! l! <| W Q TO H n4 2s OOO5-*1O sa|| i i rt "" ^ O O ^- w ^ 2 r 88 O CO OJ ^0000 rH + ' S5P I S3 2 EHEH .11 2^-2 M ^ "^ &H O- 0^ P J * si^ >5?M O . bD o S-5- Sg to . EH . H C3 S I . rr O< . O 1 O A <)u co o o odd I I Ifcfc ^o ^H 1 c -a ^ hfl . rtltt o O S Sri M .1C C [ rH 1C ] OS 00 -< O OS <> CO O O O rH rH 'C ft ft I ft* CO CO t- O5 1C t- CO " co' do -=1 r-l CO HH do 0H . o CO tH Tjl O CO tH 1C CO t- rH I> 1C ,J2S TH O O CO o bo bfibeSS o o o o -I r5 O O H H ? g * o -. S 3 s'! 5- ~ -I + I EQUAL ALTITUDES 571 p J* 'M d o B 1 > d > ^. CO CO o CMt-i 85 IS! ? s'v; rs |fl S^l 5^ OQ -3O z B.S.SSSS :: ii + - _^ aj a* ** "" P S 22Ssa .. |||1 fi Q gi ftW gggM ^ < W H'h4 SPP 4- =^ cot + , = n latitude column, ,, ,._,, 10KXT , and S* 37- or 12* - (since eMs "*%%?* g -),wefindJf=4635';there- < = f-3764 log for'e by substitution in (226a), . .. 15 12s 33 " Ol g we find tdt as shown in the col- *Z!?3-*S g umns to right. The sign of application of tdt to the middle chronometer time may be found as above by following the signs of the quantities involved,, or by using the following simple precept : " For values of the position angle less than 90 , tdt should be added when the polar distance is increasing and subtracted when the polar distance is decreasing; and for values of the position angle greater than 90 , the reverse is the case/' When the first observation is west of the meridian. It is evident when equal zenith distances are observed in a latitude L, in this case, that their supplements may be considered as equal zenith distances observed at the antipode in latitude L on the same meridian; and that in the triangle to be con- sidered, which includes the elevated pole and the antipode, the angle at the body will be 180 M instead of M. Hence, we shall obtain the equation for noon at the antipode or for EQUAL ALTITUDES 575 midnight at the place of observer by substituting 180 M for M in (226a) ; therefore, tdt = r- sec d cot M. (ma) d and dd are taken from the Almanac for local apparent mid- night and marked as in Art. 269. The sign of application of tdt to the middle C. T. is found by following the signs of the quantities involved in (227a), or, by a reverse application of the precept given for a. m. and p. m. observations. When taking out M from the azimuth tables, it must not be forgot- ten that the hour angle from the upper meridian in this case is taken as the supplement of half the elapsed time or 12 h t. (See Appendix C.) 271. To correct the middle time for a small difference of altitude. The altitude at the second observation may differ slightly from that at the first observation through a change in refrac- tion which may be learned by noting the barometer and ther- mometer during both observations; through a change in the index correction; or through interference of clouds or other unexpected causes. -. coshdh ( dh in arc, /rtOOX From Art. 237, dt = - 15 cos cos d sin ; { dt in tim ;. (233) In this formula, dh is negative when dt is positive, since as hour angle increases, altitude decreases and vice versa. If dh represents the difference between the altitude ob- served, and the one that should have been observed, dt will be the corresponding difference in hour angles. This being the change during the whole elapsed time, \dt will be the correc- tion to be added to the middle chronometer time when the western . altitude is the greater ; to be subtracted, when the western altitude is the smaller. Or, if desired, take the difference between two readings of the sextant representing double angles by artificial horizon, and the difference of corresponding times. Find the change in time due to 1' or 1" of double altitude and multiply it by the known inequality of altitudes. This result will be the correction to the middle chronometer time, to be added when the western altitude is the greater ; otherwise, subtracted. 576 NAUTICAL ASTRONOMY 'e at- o CO TH HH do o> oo to o 1 1 ft ft ^ r< | an d the difference of times represented by iy will be D' = T e (T w + x). If, however, the signal is sent to A by the observer at B at the local time T w of the western station, the corresponding time recorded at A will not be T e but will be T e + x and the difference of times represented by D" will be T w , , m ^' ^ = T e T w which equals D. A . | * rST III II H H -3 -5 fl - C <4_,-8 K . . o . fip a a Q ..33 . ^ M ^ fc K S & -e ^ 05 rHOO flCO -f- S I ! 111! | ii ii H SS o t> o EHEH rh ' go SS $$ ^A 73 .O* . O o ^ OH O H) 586 NAUTICAL ASTRONOMY Such an approximate application of " equal altitudes " is only available in the tropics under conditions named. The method of equal altitudes for longitude has a more extended application when stars are used, as suitable ones can be found in any latitude. (b) Method of equal altitudes for longitude when the positions of ship and body change. When the body observed is on or near the prime vertical and the change of latitude is small, the error involved through neglecting this change will be small; however, if it is desired to correct for change of position, it may be done very closely in one of the following ways: (1) The correction may be made approximately by reset- ting the sextant at the second observation, so that the second altitude will be increased by the number of minutes of arc equal to the number of sea miles in the difference of latitude, when the vessel sails toward the sun ; or decreased in the same ratio when she sails away from the sun. The mean of the times of observations will then be without appreciable error the time of transit. (2) The mean of the times of equal altitudes of a heavenly body corresponds to the time of the maximum altitude, so that if we find the hour angle of the sun at its maximum alti- tude (Art. 246), that is, the interval of time between maxi- mum altitude and meridian passage, and apply it to the mean of the Greenwich mean times of observation, we will have the Greenwich mean time of local apparent noon. Applying to this the equation of time, we will obtain the G. A. T. of local apparent noon or longitude West. Should this be greater than 12 hours, subtract it from 24 hours; the remainder will be the longitude East. Remember that t, the H. A. of the sun at maximum alti- tude is easterly when the sun and zenith are separating, west- erly when approaching (Art. 246) ; and, if easterly, that t is LONGITUDE BY EQUAL ALTITUDES 587 additive to the mean of chronometer times, if westerly it is subtractive from that mean to give the C. T. of local appar- ent noon. In other words, when the ship and sun are approaching, the H. A. at maximum altitude is subtracted from the mean of chronometer times to give the chronometer time of merid- ian passage ; when the ship and sun are separating, the reverse rule holds. Ex. 212. On April 22, 1905, latitude by D. K. 26 00' K, longitude by D. E. 46 03' W., observed from the bridge of a vessel steaming 315 (true) 20 knots per hour equal altitudes of sun's lower limb as follows: In the forenoon HX 75 45'. W. ll h 48 m 12 s . C W 3 h Ol m 28 s . Chronom- eter slow of G. M. T. 2 m 05 s . I. C. + 1'. Height of eye 45 feet. After the lapse of about 20 minutes, the same limb of the sun was observed at the same altitude, W. 12 h 08 m 14 s . C W 3 h Ol m 27 s . The chronometer error, I. C., and height of eye as before. Required the longitude at noon. The sun's declination corrected for longitude is N. 12 06' 16", the H. D. N. 50".59, Eq. of T. corrected for longitude l m 28 s (+ to M. T.), and from Table 26, Bowditch, A /t = 7".15. Course. I Distance. IN I \V I L 26 07' 03" N 315 I 20' I 14'.1 I 14M I D = 15'.71 W 4 Hourly change of Long. = D = 62 8 .84 W expressed as time. // Observer's change in Lat. 14'.1 N per hour or 14.1 N per minute Change in sun's dec. 50".59 N per hour or 0.843 N per minute Ac (Art. 246) combined velocity of separation = 13.257 N per minute From formula (203), t= -*~ = ^|~ = 0-.927 = 0< 55" .62. As observer's zenith and the body are separating, t f the H. A. of maximum altitude is easterly, and as the ship changes longitude to the westward at the rate of 62 S .84 per hour, or O s .97 in O m .927, the corrected H. A. is O m 56 S .59 and the time 588 NAUTICAL ASTRONOMY of maximum altitude is O m 5 6 s . 59 before the instant of upper meridian transit. A. M. Times. P. M. Times. Time of Apparent Noon. h m Y. 11 48 12 :-W 3 01 28 h m a W. 12 08 14 C-W 3 01 27 G.M.T. of noon 3 02 42.09 Eq. of T. +1 28 3. C. +2 05 C. C. + 2 05 G.A.T. of noon 3 04 10.09 i. M. T. 2 51 45 P. M. G. M. T. 3 11 46 A. M. G. M T. 2 51 45 Long. = 46 02' 31".3 West Mid. G. M. T. 3 01 45.5 t East 56.59 G. M. T. of ap- parent noon 3 02 42.09 Single and Double Altitudes. 278. What has been said about the general subject of single and double altitudes, their advantages, uses, limiting condi- tions, etc., under the head of chronometer error (Art. 268), applies to the subject of longitude when these methods of ob- servation are used, either afloat or ashore. The finding of longitude by these methods has been fully explained in the chapter on " Solutions of the Astronomical Triangle" (Arts. 226-232). The question of finding lon- gitude at sea will be further amplified under the head of Sumner lines. CHAPTEE XIX. STJMNER'S METHOD. SUMNER LINES OR LINES OF POSITION. 279. A ship approaching the entrance to Chesapeake Bay, with Cape Charles light in sight on the starboard side and Cape Henry light on the port side, at a given moment, may be located at one of two points on a Mercator chart, without bearings having been taken, if the navigator knows the dis- tance from each lighthouse. Say the ship is p miles from Cape Charles and q miles from Cape Henry; with a pair of dividers and a radius of p miles, describe a circle on the chart with Cape Charles light as a center. Being p miles distant from that lighthouse, the ship is somewhere on that circle which is a line of position, passing, as it does, through the position of the ship. If a bearing, sounding, or other deter- mining factor can be gotten, a fix may be obtained. If a second circle be described, with a radius of q miles, from Cape Henry light as a center, we shall have a second line of position, at some point of which also the ship is located. Being on both circles at the same instant, the ship must be at one of the two intersecting points. If the ship's position is further restricted by a sounding, by latitude or by longitude, to the vicinity of one point, the other one, as a position, is eliminated. 280. Lines of position and how determined. As previously defined, a line passing through a position of the ship, whether a position by D. E. or by observation, is a line of position; it 590 NAUTICAL ASTRONOMY may be straight or curved, and it may be determined from celestial bodies as well as terrestrial objects. To Captain Sumner, an American shipmaster, is due the credit for first defining a ship's position upon a line, which he called a circle of equal altitudes, from the altitude of a heavenly body and its corresponding G. M. T. ; and also for determining the ship's position at one of the two intersecting points of two such circles. 281. A heavenly body's geographical position. Every heavenly body is at a given instant of time in the zenith of some point on the earth's surface ; this point is the geographi- cal position of the body ; for the sun, it may be called the sub- solar point, for any other heavenly body, the subastral point. The theory being the same for all bodies, the method, as applicable to the sun, will be described. 282. The sun's circle of equal altitudes. The sun being in the zenith of a given place, one-half of the earth will be illuminated (neglecting refraction), and the other half will be in darkness; the dividing line, called the circle of illumi- nation, will be everywhere 90 from the subsolar point. To observers anywhere on the circle of illumination, the sun will be in the horizon ; at the subsolar point, the sun will be in the zenith, and therefore its altitude will be 90. If the observer is at any intermediate point between the circle of illumination and the subsolar point, he will have the sun above his horizon and at an altitude less than 90. If a plane be passed through this intermediate position, parallel to the circle of illumination, its intersection with the earth's surface will cut out a small circle, at every point of which, at the given instant, the sun will have the same altitude. This circle is called a circle of equal altitudes with respect to the sun, and the sun's zenith distance, at the given instant, is the same at all points of the circle. LINES OF POSITION" 591 Observed zenith distance as radius of a circle of position. Since the distance of the observer's zenith from the heavenly body, in minutes of arc, is the same as the observers distance in sea miles from the body's geographical position, and in the case of the sun from the subsolar point, when an observer measures the altitude of the sun, the complement of which is its zenith distance, he actually finds his distance in sea miles from a known spot on the earth, and hence locates him- self on a circle of position exactly as did the observer, referred to in Art. 279, who found himself on a circle of position around Cape Charles or Cape Henry. 283. Coordinates of the geographical position of a heav- enly body. The geographical position of a heavenly body is located like any terrestrial point by its latitude and longitude ; the latitude being the body's declination, the longitude the body's Greenwich hour angle. In the case of the subsolar point, the latitude equals the sun's declination, and the longitude the Greenwich apparent time. Use of a terrestrial globe in connection with a Simmer circle. If the subsolar point be located on a terrestrial globe and a circle, whose radius equals the observed zenith distance of the sun, is drawn on the globe, with the subsolar point as a center, the observer will be somewhere on the circumference of this circle; since the subsolar point bears in a given direc- tion from him, his position is in the opposite direction from the subsolar point, so that the sun's azimuth at the time of the observation indicates the part of the circle on which the observer is situated; and his position would be fixed, if, having the above data, he should find his latitude or longitude, or a second circle of equal altitudes, projected from observa- tions of a second heavenly body, or from observations again of the sun after a lapse of sufficient interval of time, the obser- ver's position remaining unchanged. However, this graphic method cannot be used for the reason 592 NAUTICAL ASTRONOMY that it is impracticable to carry a globe of such, dimensions as to admit of accurate results. 284, On a Mercator chart. The circles of equal alti- tude will appear on this chart as shown in Plate XV, end of book, being drawn out towards the North and South points for reasons apparent to anyone familiar with the theory of the Mercator projection. These curves are called " Curves of equal altitudes." Fig. 121, right-hand side, shows the curves at intervals of 10 , in which $ is the geographical position of the body observed ; all these curves belong to the same system which Sumner called a system of illumination. It will be noticed that all these curves cut the parallels of latitude and meridians of longitude at different angles. Near the North and South points, the curves run about East and West with the parallels of latitude, and a large error in longitude makes but a slight error in latitude ; near the East and West points, the curves run with the meridians and a large error in lati- tude makes but a slight error in the resulting longitude; at intermediate points, the curves cut the parallels and meridians at varying angles, so that the error in longitude due to a given error in latitude depends on the body's azimuth. These facts can be regarded as additional proofs that bodies should be observed for latitude when on or near the meridian, and for longitude when on or near the prime vertical. Determination of points on the curve. If an observer has a given altitude, different assumed latitudes, within the limits of the curve, will give him different longitudes, and vice versa. Each latitude will give two points, one for an altitude East, one for an altitude West of the meridian through the observed body. By assuming a sufficient number of coordi- nates, the whole curve may be plotted. 285. (a) Double altitude observations. Suppose that on April 16, 1905, p. m. time, an observer at sea on the North LINES OF POSITION 593 Atlantic Ocean, observes the true altitude of the sun's center to be 50, the G. M. T. of observation being l h 15 m 30 s . Taking the required data from the Nautical Almanac (1905) for the given G. M. T., the sun's declination is found I30120 110 10090 80 70 60 50 4030 20 10 10 20 30 40 50 W es,L ong. N / ^ *-> ^x \ East \ Lor g. 60 50 40 30 20 10 10 20 30 40 50 60 70 / < "" / / X \ N ^ ^ \ \ / / / / X \ "~^ N \ \ , ' 1 / < " ^ ^N w \ \ 1 / / v N^ \ So ~7 -S h \ \ X p. / \ \ \ \ *> / / / / / / \ \ \ x^ J ^ 7 / / ^ \ \ \ \ ^ Y / ^ / / ^ X s> -^ \ X [ s ^ / / f x X ^ / FIG. 121. to be 10 N., the equation of time O m 06 s , additive to mean time, and the G. A. T. l h 15 m 36 s . Hence the latitude of the subsolar point 8 is 10 K, and its longitude l h 15 m 36 s West, or 18 54' W. From this point 8 as a center, -the curve of equal altitudes witn radius of 40 will be the right-hand curve TVXS' on the 594 NAUTICAL ASTRONOMY Mercator chart (Fig. 121). This curve tells us nothing more than the bare fact that the observer is somewhere on its cir- cumference. However, if the bearing of the sun at the instant of sight is given, the quadrant containing the position is indicated. If, in example, the sun bears southward and eastward, the ship is in the N~W. quadrant. Having obtained the curve and bearing of the sun, if either latitude or longi- tude is given, the ship's position is determined. Again, suppose that after the lapse of 3 h 24 m 24 s a second observation shows the sun's altitude to be 40. During this interval, the sun in its diurnal path will have passed to the westward at the rate of 15 of longitude per hour, carrying with it its geographical position and its system of curves of equal altitudes. The G-. A. T. becomes 4 h 40 m , or the longi- tude of the subsolar point S 2 is 70. The declination is K 10 03' 01", or the latitude of S 2 is K 10 03' 01", the zenith distance is 50, the curve on the Mercator chart is N'XS' 9 and the ship is somewhere on this curve. What we know now is that at the first observation the ship was on the right-hand curve NXS, and at the second observation she was on the left-hand curve N'X8' 9 therefore, if the ship did not change her position in the interval between the observations, she was at one or the other of the two points (X or Y) 9 in which the curves intersected, and the one which was the observer's position depends on the sun's bearings at the times of observation. (b) In case the observer changes his position between the observations. If the observer, whose position is somewhere on the right-hand curve, can be supposed to make an instan- taneous change of position, through a distance of N sea miles directly towards, or directly away from the geographical position S 19 on a great circle passing through S 19 the sun's altitude will be increased or diminished by N minutes of arc ; if the course is kept at right angles to the bearing of the sun, LINES OF POSITION 595 he will keep on his original curve of altitude ; if the course is at intermediate angles, the altitude will be changed propor- tionally (the change being expressed in Art. 213 by for- mula Aft d cos (C~Z)), and the observer will be on another circle of equal altitudes, belonging, however, to the same system of circles, that is, the system having 8^ as a center. In going then from a point on one circle of equal altitudes, which was the ship's position at the first altitude, on a certain course and distance made good, the observer arrives at a point on another circle of equal altitudes, of the same system, how- ever. The altitude at this latter circle corresponds to what would have been observed there at the instant of the first observation on the original circle. The circle of position can then be found after a run to the place of a second observation, either by reducing the first altitude to what it would have been at the place of the second observation, at the time of the first observation (see Art. 213) ; or by taking a point in the first curve, representing the ship's approximate position, laying off the course and distance made good in the interval from it to a second point, and then drawing through this second point a curve parallel to the cor- responding part of the first curve of altitude. The intersection of this transferred curve with the curve of altitude of the second observation will give the ship's position at the time of the second observation. The general method described in the two sections of this article is known in modern navigation as " Sumner's double altitude method." Ordinarily, it is impracticable to plot circles of position on a Mercator chart without previous calculations for many apparent reasons, but under certain circumstances this may be done, and from two circles the fix may be found with con- siderable accuracy. In certain cases, as may happen in the tropics, the sun may 596 NAUTICAL ASTRONOMY be observed when close to the zenith, say within a degree or so; the subsolar point located by its latitude and longitude (the latitude being the sun's declination and longitude the G-. A. T. of observation) ; and that portion of the circle near the ship's D. E. position drawn with the true zenith distance as a radius. After the sun's azimuth has altered from 25 to 30, and under such circumstances it will do so in a very short time, draw a second arc as the result of a second observation. Transfer the first arc for the run between sights, and the intersection of the transferred arc with the arc corresponding to the second observation will give the fix With a fair degree of accuracy, and this without any of the usual calculations. In using the method of double altitudes, there should be a change of bearing of the sun between observations of at least two points; of course, the nearer the change is to 90, the more nearly the resulting lines of position run at right angles to each, other, and the better the cut. Rapidity of change of azimuth dependent on L and d. The rapidity of change of the sun's azimuth will depend on the values of L and d, and the time an observer has to wait for that body to undergo a desired change of bearing may be found by inspection of the azimuth tables. The greater the difference between the values of L and d, the smaller will be the elapsed interval for a given change; for this reason, the interval in winter months will be smaller for observations of the sun. When the latitude and declination are nearly the same, it will be impossible to work the double altitude problem from observations on the same side of the meridian; however, having obtained an a. m. sight, if the meridian observation is lost, it will not take long for the sun after crossing the meridian to alter the first azimuth 90, so that lines giving LINES OF POSITION 597 excellent cuts may be obtained by combining observations on both sides of the meridian. 286. Simultaneous observations. The principles of the Sumner double altitude method, as explained in the case of the sun in the preceding article, apply as well to any other heavenly body; but, as a general thing, when one star may be observed, others are available, so that two (or more) may be observed at one time and the ship may be located at one of the two intersecting points of the resulting circles of position. Suppose two stars are observed at the same mo- ment at a given place; that one, whose subastral point is $ (Fig. 121), has an altitude of 50 and bears southward and eastward, and that the other, whose subastral point is S 2 , has an altitude of 40 and bears southward and westward. The result of the first observation locates the ship on the NW. arc of the circle of position NXS, the result of the second on the NE. arc of the circle N'XS', and, therefore, at their northern intersection X. Such observations are known as simultaneous observations; for these observations, bodies should be so selected that the resulting circles or lines of position will cut at good angles, not less than 30. Advantages of simultaneous over double altitude observa- tions. The former are preferred to the latter as the position of the ship may be obtained at once without an interval of waiting and the errors of the run when there is a change of position; besides, a third line may be obtained and the fix from two lines either verified or disproved, the fix being veri- fied when the three lines have practically the same point of intersection. It is probable that the navigator may have such a number of first or second magnitude stars to select from that good observations may be obtained in all latitudes. 287. Relation between circles of equal altitude and the astronomical triangle. Let Fig. 122 represent a projection 598 NAUTICAL ASTRONOMY on the horizon of a point 8 the geographical position of a heavenly body; PQ, the meridian of that point; PG, the meridian of Greenwich ; PZ and PZ 2 , the meridians of places on the earth's surface having at the same instant of time the same altitude of the body 8. Z- L Z 2 Z 3 is a Sumner curve or a circle of equal altitudes with respect to the body 8; QS, the latitude of the geographical position equals the body's de- clination; GPS, the Greenwich hour angle of the body, is the FIG. 122. longitude of 8. The triangles ZfS and Z 2 P8 are projections of astronomical triangles. The angle ZfS is the hour angle -ofjhe body at a place Z on the circle of equal altitudes; ZfS, the hour angle of the same body at the same instant at the place Z 2 , also on the same circle. Since the Greenwich hour angle, the declination, and the altitude are the same at Z i9 Z 2 , and other places on the same circle of altitude, the astronomical triangles Z^PS, Z 2 P8, etc., have two sides of one equal to two sides of the others, one side equal to 90 h and the other equal to 90^ but they differ in the values of the third side (PZ , PZ 2 , etc.), which side is the complement of the latitude, and also in the values of LINES OP POSITION- 599 the local hour angles, Z^PS, Z 2 PS, etc. The hour angles of the body at the local meridian, and hence the longitudes, are dependent on the different assumed values of the latitude when solving the astronomical triangle with given values of h and G. M. T. For the sun, the hour angles ZJ?8, Z 2 PS f etc., are the local apparent times, or 24 hours rthose apparent times, according as the times are less or greater than 12 hours'; GPS is the Greenwich apparent time, or 24 hours that time, according as the apparent time at Greenwich is less or greater than 12 hours ; and GPZ 2 and GPZ^ are the longitudes from Greenwich, respectively, of Z 2 and Z . By assuming latitudes and finding the corresponding longi- tudes, or by assuming longitudes and finding the correspond- ing latitudes, any number of points of the curve may be found and the whole circle projected. 288. Kule for assuming coordinates. The rule for assum- ing coordinates, based on what has been said as to the varying angles at which the curve of equal altitudes cuts meridians and parallels of latitude (Art. 284), and the demonstra- tions (see Arts. 237 and 248) as to the best times to observe for latitude or longitude, is as follows: assume latitudes and solve for longitudes when the body's Z N lies between 45 and 135, or 225 and 315; otherwise, assume longitudes and solve for latitudes. 289. Actual sea practice and method of determining the line, In actual practice at sea, it is never necessary to de- termine more than a small portion of the circle of equal altitudes, since the observer's position is generally known, to be within certain limits, both of latitude and of longitude. This small portion is the only part to be considered; it is called a line of position, and is at right angles to the heavenly body's true bearing. If three or more coordinates are as- sumed, especially if they are far apart or the body's altitude is great, the line may be a curved line. It is customary, 600 NAUTICAL ASTRONOMY however., when working for longitudes, to assume two lati- tudes differing say ~by 20' , or when working for latitudes to assume two longitudes differing by two minutes of time or 30' of arc; in loth cases, the dead reckoning position should be between the assumed coordinates. For such short dis- tances the chord thus obtained is practically equal to the included arc of the circle. This is known as the " method of chords " and for years has heen the practice of the officers of the U. S. Navy. It is evident that, in the case of a line thus determined, its angle with the meridian may be found by either middle latitude or Mercator sailing, and thence the true azimuth of the body whose bearing is at right angles to it. Since the circle of equal altitudes is at right angles to the true bearing of the body, a tangent to the circle at a given point, and for short distances either side, may be taken as practically equal to the arc between the same limits. There- fore, to determine a line, assume a latitude and find the corre- sponding longitude, or assume a longitude and find the corre- sponding latitude, both assumptions within the limits of the curve, thus determining one point of the circle of equal altitudes. The true azimuth of the body for the instant of observation having been determined in one of three ways (1) from the azimuth tables (Art. 221) or an azimuth diagram; (2) by observation, the compass bearing being corrected for variation and deviation of the compass; (3) by solution of the astro- nomical triangle (Arts. 218 and 219); (2) and (3) not being, however, the usual practice at sea; a line is drawn through the determined point at right angles to the body's true bearing. This line is a line of position, and this method of determining it is known as the " method of tangents." 290. To define a line of position. From what has been said in previous articles, a line may be defined in one of two LINES OF POSITION 601 wa ys when determined by the chord method, it is defined by its two points A 19 A 2 , thus : 18 50' S. . ( 19 10' S. 2 46' 24" W. 2 2 37' 31" W. j. t'O v T \s k/v/j.AJ.1. When determined" loj the tangent method, it is defined loj its one position point A and its direction thus : f!9 00' S fAzimuth of body Z N =6710' Position Pt. A \ no -\t Ke/fw" J obtained from azimuth tables, I A L 58 W. < g iyen L= igoQ^ r^8 h 01 m 52 8 , Line of Position 337 10' I d = N 11 50'. From the data of line A^A^ by middle latitude sailing, the direction of the line is found to be 33 7. 2 and the body's true bearing, Z N , 67. 2. In other words, the line has the same direction, however determined. Coordinates "computed" and "by observation." In this work, wherever they occur, the terms Lat. and Long, by observation will be taken as applying only to the ship's posi- tion or fix; the term computed latitude, as referring to that obtained from a sight by using the D. E. longitude or an assumed longitude; and the term computed longitude, as re- ferring to that obtained by using a D. E. latitude or an assumed latitude. The method of determining the line and the methods of finding the intersection of two lines will be considered in Arts. 295-310. 291. Uses of a Sumner line. If the G. M. T. and altitude of the heavenly body are correct, the line determined from the data passes through the position of the ship, and if the line, or the line produced, passes through a lighthouse, point of land, or a danger, the direction of the line gives at once the bearing of that particular object; if desiring to make that light or point of land, the navigator knows the course to steer ; to avoid the object, if a danger, it is only necessary to run at 602 NAUTICAL ASTRONOMY right angles to the direction of the line for a safe distance, and then,, by changing the course not more than 90 from this last course, the ship will go clear. If in Fig. 123, A^A^ is a line of position passing through FIG. 123. FIG. 124. a lighthouse or point of land B, the course to be steered to make B is the direction of the line towards B. If B is a danger, run the course and distance represented by cd, or, if safe, run a proper distance in the direction ce; then the direction A 2 A 2 will clear the danger. If the direction of the line is into the port of destination, the course in will be known; if its direction is towards a point to one side of the entrance, A 1 A 1 (Fig. 124), draw a line A 2 A 2 on the chart from the entrance EF, parallel to the line of position, shape a course cd at right angles to A^A , or, if safe, a course in direction ce till the vessel arrives on the parallel line A 2 A 2 - then steer in its direction for CHAKT INTERSECTIONS 603 the entrance. If the line runs parallel to the coast (BE, Fig. 124), the distance off shore will be known. A fix may be obtained by a verified sounding or by a bear- ing of an object of known position on the chart; and in this connection, when in the vicinity of dangers, attention is again called to the fact that, if a line is obtained from an observa- tion of a body on the prime vertical, the longitude will be well determined; if from an observation of a body on the meridian, the latitude will be well determined; even though the other coordinate may be somewhat in error. Owing to the fact that a line is always at right angles to the bearing of the body, it is often possible, especially at night when heavenly bodies in all directions are available, to get a line running in any desired direction, so as to show the bearing of land, distance of coast, etc. If an observation of a heavenly body is taken when bearing directly abeam* and the opportunities are many for so observ- ing not only stars but the sun the resulting line of position will be in the direction of the course; and if the line leads clear of danger the navigator may keep his course, if towards danger he may run off 90 for a safe distance, then resume his course, clearing the danger. Two lines intersecting at angles of not less than 30 (90 preferred) will give good fixes. When possible, it is better to verify this fix by a third line. During morning or evening twilight, or moonlight, when stars are visible, several may be observed at the same time, and they may be so selected that the corresponding lines will cut at excellent angles, and hence give excellent fixes. GRAPHIC OR CHART INTERSECTIONS. 292. Finding the noon position on a Mercator chart and the intersection of a line with another moved parallel to itself for the run between observations. The parallel of the 604 NAUTICAL ASTRONOMY latitude found at noon is nothing more nor less than a line of position obtained when the body is observed on the meridian, and the noon position is the intersection of this 44N FIG. 125. parallel with the a. m. line moved for the run between obser- vations Let Fig. 125 be a section of a Mercator chart with Sum- ner lines plotted thereon. Suppose the a. m. observation gave a line A^A^ (Fig. 125), and that the run to noon was 120 (true) 15 miles (in the direction of and equal to cd), and the noon latitude was that of the parallel MM; then the intersection of MM with A 2 A 2 , the forenoon line moved up for the run, was the noon position. UNCERTAINTY IN DATA 605 Again, in Fig. 125 c^ is a line of position obtained by working a "' sight of the sun for latitude, having as- sumed longitudes 8 30' W. and 9 W. After sailing 211 true 21.4 miles (c^d) and 205 10.8 miles (dc 2 ), a second line did-L was found from an observation of the sun worked as a time sight; it is required to find the ship's position at the second observation. Having plotted on the Mercator chart the first line c^ by its coordinates, it is apparent that the ship is somewhere on the line at the time of the observation, though the exact point is unknown. From any point of this line CjC-u lay off the true courses and distances run (in this case in the directions c^d and dc 2 ) and through the de- termined point draw c 2 c 2 parallel to c^; the ship at the time of the second observation is on the line C 2 c 2 . The second line by observation is plotted by its coordinates and intersects c 2 c 2 in y which is the position of the ship at the time of the second observation. The data for the two lines and for the ship's run of the second case in this article are given below. First line C& A' 8 30' W ) A/ 2 9 00' W) L\ 45 04' 03" N j L' 2 45 22' 12" N J Second line d& L\ 44 26' 40" TX\L" 2 44 46' 40" N" ) A" 8 47' 44" WJ A" 2 8 52' 26" W J Run between lines 211 (true) 21.4 miles. 205 (true) 10.8 miles. 293. Uncertainty in G. M. T. If there is an uncertainty in the Gr. M. T., parallels may be drawn on either side of the line, at a distance in longitude equal to the amount of the uncertainty, so that the true position will then be restricted in the case of a single line to a belt instead of a line, and in case of two lines to the area of a small parallelogram. 294. Uncertainty in altitude. If there is an uncertainty 60G NAUTICAL ASTRONOMY in altitude, parallels to the line may be drawn each side, at a perpendicular distance from it in nautical miles equal to the number of minutes of error in altitude. For a given uncertainty, to illustrate say V of altitude, when one body is on the meridian and the other on the prime vertical, the position may be anywhere in a square, with a maximum uncertainty of 2' both as to latitude and longitude. Thus if aa' and W (Fig. 126), represent the two lines of posi- tion, the observer would be at 0, provided there was no error ; but to allow for a possible error of V of altitude, + or , lines must be drawn on each side at the perpendicular distance of one sea mile. The observer may be at 1, 2, 3, or 4, or any- where within the square, making the limits of uncertainty two miles for both latitude and longitude. FIG. 126. For a difference of azimuth of eight points, where neither body is on the meridian or prime vertical, the rectangle will be shifted, and the uncertainty in latitude and longitude will increase till when the lines run NE. and NW. (SW. and SE.), each becomes a maximum; the position may vary in latitude 2.8 sea miles, and in longitude the same amount (Fig. 127). For a difference of azimuth greater or less than eight points. When the difference of azimuth of the two lines is INTERSECTION OF LINES 607 greater or less than 90, an error in altitude, + or ( ), will affect the possible position of the ship so as to make the un- certainty in latitude greater and in longitude less, or vice versa, according to the direction in which the parallelo- gram is elongated. For instance, if the difference of azi- muth is small and the mean azimuth is near East or West, as in Fig. 128, where aa' and W are the lines of position, it is seen that the possible variation in longitude (1 to 3) is FIG. 127. FIG. 128. small compared to the variation in latitude (2 to 4). So that, whilst longitude may be determined when the mean azimuth is near E. or W., the latitude may be far out when the difference of azimuths is small. Exactly the reverse is true when the mean azimuth is near N. or S. and the differ- ence of azimuth is small. Finding the intersection of Sumner lines. 295. The intersection of lines of position may be found : (1) By plotting, on a Mercator chart of the locality in which the ship may be, the lines determined (a) by the chord method, (b) by the tangent method. (2) By computation. 608 NAUTICAL ASTRONOMY 296. (a) The plotting of lines determined by the chord method. If a line of position is determined from celestial observations by assuming two latitudes and finding the corre- sponding longitudes, or, by assuming two longitudes and find- ing the corresponding latitudes, the assumed coordinates being about equally distant each side of the dead reckoning position, it is plotted on a Mercator chart by locating the two points thus determined and drawing a straight line between them. If a second line is plotted in the same way, the observer will be at the intersection of this second line with the first, provided there has been no change of his position in the interval between observations; however, if there has been a change, then the observer's position will be at the intersection of the second line with the first line after having been moved parallel to itself for the run in the interval. The principles involved are shown in Fig. 125. In case the observation is of a body on the meridian, the line of position becomes a parallel of latitude; if the body is observed on the prime vertical, the line will be a meridian. 297. (b) The plotting of lines determined by the tangent method. In this method, take the D. R. latitude and deter- mine from the given observations, by solution of the astro- nomical triangle, the corresponding longitude, calling it computed longitude ; or take the D. R. longitude and find the corresponding latitude, 'Calling it computed latitude. The point thus determined will be one point of the line and is plotted on the chart. Through this point draw a line at right angles to the bearing of the body for the instant of observa- tion, this bearing being found from the azimuth tables having given L, d, and t, or from an azimuth diagram. The line drawn will be a line of position. Considering the azimuth of the body less than 90 , the direction of the line is easily obtained from the true bearing of the body by reversing either LONGITUDE FACTOR 609 letter of the bearing and talcing the complement of the angle. Thus, if the body bore 8. 30 E., the corresponding line of position runs N. 60 E. or 8. 60* W. * The second line having been plotted in the same way, the observer's position will be at the intersection of the two lines, as explained in Art. 292. 298. Before explaining the methods of finding the inter- section of Sumner lines by computation, it is desirable to give a few definitions. Definition of longitude factor. The longitude factor of a line of position, represent- ed by the letter F, is the L T change in lorigitude due to 1' change in latitude. In the case of a line determin- ed by the chord method, it is found directly by divid- L ing the difference of the longitudes of the two points by the difference of their corresponding latitudes, or F = j* ~^ v = -^ , where AL is a change in latitude due to a change of AA in longitude, and vice versa (Fig. 129). Definition of latitude factor. The latitude factor of a line of position, represented by /, is i 9 or the change in latitude due to a change of 1' of longitude. In this method of defining a line, f=-^~^ 1 = ^(Fig. 129). When a line is determined by the tangent method, F equals 4^ and / equals A^ , as before ; but in this case it is neces- A.L A A sary to investigate and ascertain the relation between AL and AA and the determining quantities of a tangent line, namely latitude and the body's azimuth. * If the true bearing of the body is given in the form of ZTH, simply add or sub- tract 90 to obtain the true direction of the line as estimated from 0" at North around to the right. FIG. 129. 610 NAUTICAL ASTRONOMY Values of F and / and where they may be found. By ref- erence to Art. 237, it is seen that, by differentiation of the general equation of the astronomical triangle, -ry = sec L cot Z, dt being the change in longitude due to a change of dL in latitude. Now, if dL represents one minute of latitude, dt = sec L cot Z is the change in longitude due to a change of 1' of latitude, and, therefore, the longitude factor F sec L cot Z (236). This may be shown graphically from Fig. 130 and Fig. 131, in which the azimuth is considered as less than 90. In Fig. FIG. 130. FIG. 131. 130, the body bears either in the NE. or SW. quadrants and the direction of the line of position is NW. or SE. In Fig. 131, the sun bears in the NW. or SE. quadrants and the direction of the line is in the NE. or SW. quadrants. In both cases, Ap = AlcotZ; Ap = AA cos L ; AA cos L = AL cot Z, AA = AL sec L cot Z ; but therefore, and but therefore, F = sec L cot Z. (236) and the quantity sec L cot Z is the longitude factor. It is tabulated in Table I of this book, the arguments being L and LATITUDE FACTOR 611 Z. This factor may be found in Table C of Lecky, in Inman's Tables, and in Table II of A. C. Johnson's most excellent work on finding latitude and longitude in cloudy weather. Since / l_j / may be found from the above tables by first finding F from them the value of F and taking its reciprocal. It is now understood that if F is known and either AL or AA also known, the other may be found from the expression AX = AL X F, (237) and also that AL = AA X y , or AL = AA X f. (238) Rule for naming AL and AA when the azimuth of the body is given. Regarding the azimuth of the body as less than 90 and as estimated from either the North or South point of the horizon towards East or West, we have the following obvious rule: "If the change in latitude AL is of the same name as the first letter of the bearing, the change in longitude AA is of the contrary name to that of the second letter, and vice versa" Thus, if the body bears S. 45 E., if AL is 8., AA is W.; and if AL is N., AA is E. INTERSECTION BY COMPUTATION. 299. We are now prepared to find the intersection of lines of position by computation. The simplest case to be dealt with occurs when one line runs due N". or S., as when a body is observed on the prime vertical, and another line runs due E. or W., as when a body is observed on the meridian. The longitude of the ship is that of the first line, the latitude is the parallel of the second line. 612 NAUTICAL ASTRONOMY 0^ to co ^ ^ CO 00 ' 5* ^ ,2 * oo a ^ <-" T-l 05 bC w ^3 S " . -* CO S t"" 1 jj 3.3 S- ^^^' 3 5| ^J i a> * "^ o ^8SS S ^Sg If o* Q .| + I 1 l ^B ^ f-l t*,*p< 5 ?-t "rH ..J ^ O -^ =3 5^3 * H M S o ^ S < 3 '^ /~s 0) ^ O "S <*> t/1 g IH n pa. o 1? -c W .S o fc a'S _a * oo * co a p\ . g, 9 i S "02 OJ H 0) 3 2 E fl 1 1 B II t O CO 9 ^ M ^ P3 CG rs rt 1 1 o 2 ^ ** O * r .S -' o #p -x- ^J CO CO i-5 .0 S H r'S?o o^ aSi 8.* t-- c^i CO "* 8 o is i B s i o o o io co c^i co ^ 00 05 OS 05 OJ 15 rt ,^ ^ (M N t^ " ^ H pM V* r- ^ r-i A< O g)| | 2 Q a O CO CQ rt rt ^^ 1 1 S i 1 3^s I s EH^ O ji*j OD 1 fc . g^ S . H b. > W .^ ^ OQ - S 2 2 ^ o o D IS pa o S S i s ^ jl^ , ^ > * a ft'C ,5 & -ii:; i^T -,0 --*cooocO' : 3coto <=> "^ no* ^ ^m f-( a> in w -g ,3 ^H O THE CHORD METHOD 613 300. The next case is where the line of position (original or transferred for run), runs at an angle with both parallels and meridians, and is intersected by a line running due E. and W. ; this last line being a parallel of latitude from the meridian altitude of a body. This is a case of simultaneous observations, say of two stars; one on the meridian, the other off the prime vertical at the time of observation. As in example 213 the latitude from the meridian observation, being well determined, is used to work the time sight. Again, a case under this heading occurs when finding the noon position from an a. m. observation, a run to noon, and latitude from a meridian altitude of the sun. In the latter case, there are two ways of finding the intersection, according as the a. m. line is determined by the chord or tangent method. 301. (1) The chord method. Assume two latitudes, about 10' each side of the D. R. latitude, work a line of position, ob- taining the longitudes corresponding to the two assumed lati- tudes and hence two points of the line. Divide the difference of the computed longitudes by the difference of the two as- sumed latitudes. The result is the longitude factor F. Correct each position of the line for the run to noon, obtaining the corresponding points of the line at noon. The difference between the latitude of one point of this line, after being moved for the run, as an origin, and the latitude by meridian altitude is AZ/ but AA = AL X F. Knowing which way the line of position runs, the sign of application of AA is apparent. The result obtained by apply- ing AA to the longitude of the point taken as an origin will be the noon longitude by observation. Rule for naming AL and A A when the direction of the line of position is given. Regarding the direction of the line as an angle less than 90, and as estimated from either the North or South point of the horizon, towards the East or West 614 NAUTICAL ASTRONOMY point, we have the following rule: "If AL is of the same name as the first letter of the direction, AA is of the same name as the second letter; and vice versa." Thus, if the line runs N. 30 E., and the change of latitude is to the northward, the change of longitude will be to the eastward. This rule applies to the " chord method," and the rule in Art. 298 to the " tangent method." If the longitude by observation is desired at the time of the a. m. sight, having found the a. m. line, it is only necessary to run the noon latitude back to the time of a. m. sight by applying the run in latitude from sight to noon backward, thus getting the true latitude at the time of the a. m. sight. The difference between this true latitude at the time of a. m. sight and the latitude of one point of the line in its a. m. position as an origin, multiplied by the longitude factor F, gives the correction in longitude, or AA, to be applied to the longitude of the same point. The result will be the longitude by observation at the time of the a. m. sight. If, to this, the run in longitude from the time of sight till noon is applied, the result will be the longitude at noon by observation, which should agree with that obtained as in the previous article. 302. (2) The tangent method. Work up the dead reckon- ing to the time of a. m. longitude sight. Work the time sight with the D. E. latitude, calling the resulting longitude computed longitude. With the latitude, declination, and L. A. T. from the sight, find the sun's true azimuth from the azimuth tables or an azimuth diagram. The azimuth must be considered as less than 90 ; so, if that from the tables exceeds 90, estimated from one pole, use its supplement and reckon it from the opposite pole. With the D. E. latitude and the sun's azimuth, find from Table I (or Table C in Lecky, Table 38, Bowditch, or from Inman's Tables) the longitude factor F; write the value of F THE TANGENT METHOD 615 in the form for work, and near it the direction of the line thus: F = a./ , meaning that the variation in longitude for V of latitude is a and the line of position runs NE d . and SW d ., or F = a. \ in case the line runs NW d . and SE d . ; the direction of the line being obtained from the bearing (regarded as less than 90) ~by changing either letter of the bearing and taking the complement of the angle. To this D. E. latitude and computed longitude apply the run to noon, obtaining at noon a D. E. latitude and a computed longitude. The line at noon remains parallel to its direction at time of sight, and F has, of course, the same value. With the computed longitude, work the meridian altitude sight and find latitude at noon by observation. The difference between the latitude at noon by D. E. and by observation is AL, or the error in latitude. As before, AA = AL X F. In the absence of Table I, the correction AA may be found thus: Enter Table 2 of Bowditch's Useful Tables with the complement of the bearing as a course, find AL in the latitude column, and take the corresponding departure from the de- parture columns. This departure, converted into difference of longitude, will be AA. Having found AA, its sign of application may be found from rule of Art. 298, if the azimuth of the sun is considered ; or, rule of Art. 301, if the direction of the line is considered. Apply AA to the computed longitude for noon, the result will be the longitude at noon by observation. Should it be desired to find the true longitude at time of sight, run the noon latitude back to the time of the a. m. sight by applying the run in latitude from sight to noon back- wards, getting the true latitude at time of sight. The differ- ence between this latitude and the D. E. latitude at sight is AL; then AA = AL X F is the correction in longitude to be applied to the computed longitude at time of sight to give the longitude by observation at that time. 616 NAUTICAL ASTRONOMY M > ^ CO 5J 5 a a *- a fi (X) ^ 'd ts w 1 1 + "p ^5 C d __, i 1 f IS EH' J c 00 ' * - 2 ? S 30 t*" O O 1C 00 Tt< C" O <1 05 w d o' o 05' oi oJ ja * -S a, o Sco CO ^^ J_ . ~&fe ; H + + S *^ ' ^ fO PI CQ ^ rl r-i rsT* jg o d ,_ *> 05 j* ss II O ^j P fc" ^ K H f-f ^ "^ txj ^ rH o cxi C5 W CO O r^ ^ 5 ^ rH * CD O o oo cxj ^- j r> CO ^ fc 1 02 t CO CO 1 TH 05 0005^ i ~A s; d _o 00 CO "* CO 5 "^ rHOOOO'OCO dO5CC 05 ^-OSlClCr-iCOrH PcO *f r " ' "^ CO ^o ^5b coco H 7 S 5 1 " 1 05 *" rH X O rHO5O5CO c OlC rE ' T -( Ui TH -f O5 O5 ^0 h- s S 1C fl-\ TJ d ^ O O ic ^ _fO 05 K^ oj H d Q . ^ >> * S & 02 * <** +1 ^ ^ H : Sjf jS s ^ ' O < -00, O b- CO c HHK ^ ^ fcb ^ J K " H-1 O J ^ ^3 ^3 JX) fl ^3 _0 ii ffii rH ' l 1 +3 O T | H 3 ^ | | 38 *! A3 Tf^ rH 1C O O5 rH PL,<^ t3 o o c oo -* j> ^1?-I OQrH( 3 0, O O O5' CJi' 35* O B .o|i3 -- O 00 * o ic O g o 14 '- ^S r-* CO *2J ITT 03 G O *" TH g o fl fl w S S S -a -a f rH ^ -' g S g" -g rH + 00 TH HrH 02 -t^ TH ^ -*J -t^ 4J O5 O5 d ^ ' ^ Fj 1C o C 00 O? OO 00 H rO T , , Q^ 0i5 CD -ooocorHOOT w ^^ 1 ^1CO-*^O5OJ '"'CO . t^ ^ o^; ob " 1 5 ^ -^ ^rHOOO OiCCO SM C O500 O1CO1C1CC5O 1C O5 "* coco g| O5 H O i>D II 4 . M 03 co co 05 S 1C S d 2 0001CTT<0005TH ^2 w ^ rHO5O5COOiC 1C 1C O5 ^J ^ TH rH SO CO ' r_ t 1 Q CJ CO TH *'** TH CO CO ^ CH H TH THrH rf-kfft, ^^ . H rH H'EH hH ^ fab , t 618 NAUTICAL ASTRONOMY T? co 5J fl *: * co rH A w i V + H H g^ o K ^ * t- CO CO CO o B CO ^ lO 1O OS #*, <* CO ,d I- ^ 1 ii i T ^ d g o 1 P^ o d d o ^ H MUTUAL CORRECTION METHOD 619 When one Observation is of a Body within 45 of the Prime Vertical and the other of a Body within 15 of the Meridian. 303. The mutual correction method. This method applies to the following: (1) A case of simultaneous observations in which one body is observed near the prime vertical for time and one near the meridian for latitude; (2) a case of double altitudes of the sun with an intervening run the first alti- tude observed within 45 of the prime vertical for time, the second altitude observed within 15 of the meridian and worked as a " reduction to the meridian " sight. Having determined the first line, by either the chord or tangent method,, and corrected the coordinates of one point for the run, and having found the value of F , it is not un- usual to consider the latitude obtained by "reduction to the meridian" (using the computed longitude at the instant of observation in finding H. A.), as sufficiently exact for all practical purposes, and for this latitude to find, as in examples 214 and 215, the longitude of fix. Then the noon position is found by applying to the latitude and longitude of fix the run from the time of the second observation till noon. However, as the body at the second observation is not on the meridian and the resulting line of position is not East and West in direction, and as the sight is worked with a longitude which may be in error, the latitude obtained may be in error. For more precise results, the " mutual correction method " may be used, correcting the longitude from the time sight by the formula AX = Al^ X ^ , in which ALj. is the difference between the latitude of a posi- tion point in the first line corrected for the run to second observation and the computed latitude at the second observation; 620 NAUTICAL ASTRONOMY FI is the longitude factor of the first line ; and AA is the correction to be applied to the computed longitude at the second observation to give the longitude of fix. Then the latitude from the "reduction to the meridian" sight should be corrected by the formula AL 2 = AA -=- F 2 ; in which F 2 is the longitude factor of the second line ob- tained from Table I, knowing the latitude and sun's azimuth at second observation ; and AL 2 is the correction to be applied to the computed latitude at the second observation to give the latitude of fix. This method of "mutual correction " is applicable only where one observation is a time sight and one a sight near the meridian, the latitude from this latter sight being nearly correct. The following rules are given for tto second case under this heading, that of double altitudes of the sun; modifica- tions necessary to make them fit the first case, that of simul- taneous observations, will be apparent. Rules. Work the time sight by either the chord or tan- gent method; using the D. K. latitude in the latter case, or assuming latitudes about 10' each side of the D. K. latitude in the former case. Find the longitude factor of first line F^ . To the coordinates of one position point of the first line apply the run to the instant of second observation and obtain a D. E. latitude and a computed longitude; and, with this longitude, work the second sight by the "reduction to the meridian" method, obtaining a computed latitude. With the computed latitude and azimuth at the second ob- servation, find the longitude factor of the second line F 2 . Take AL equal to the difference between the D. E. latitude and the computed latitude at the second observation and find MUTUAL CORRECTION METHOD 621 AAj = AL X FI', a pply AAj. to the computed longitude at the second observation and obtain the longitude of fix. From AA find AL 2 = AAi -f- F 2 ; apply AL 2 to the com- puted latitude at the second observation and obtain the lati- tude of fix. Ex. 216. About 7.45 a. m., January 1, 1905, from an ob- servation of the sun in latitude 16 21' 34" N. by D. E. found the computed longitude to be 63 15' 30" W. True azimuth of sun = Z N I20f . Ran thence till ll h 40 m a. m. 315 (true) 30.7 miles when, by the "reduction to the meridian" method, the latitude was found to be 16 45' 04" K True azimuth of sun Z N = 172.8. Ean thence till noon 315 (true) 2.7 miles. Required the noon position. D. R. between sights. True Course. I Distance. Diff. Lat. Dep. I Diff. Long. 21.7 W 22'.64 W 315 I 30.7 I 21.7 N o / ii o i it Lat. by P. R. at 7.45 a.m. 16 21 34 N Long, computed at 7.45a.m. 63 15 30 W >iff. of Lat. to 11.40a.m. 21 42 N Diff. Long, to 11.40a.m. 22 38.4 W Lat. by D. R. at 11.40 a.m. 16 43 16 N Computed Long, at 11.40 a.m. 63 38 08.4 W Lo = 16 32 25N o / n At 11.40 a. m. computed Lat. 16 45 04 N At 11.40 a. m. Lat. by D. R. 16 43 16 N A LI = I'.S N = 1 48 N To correct the longitude from the a. m. time sight. Lat. 16.36 N, * 1= S 59J E, J\ = .62 Computed Long, at 11.40 a. m. 63 38 08.4 W AX 1 =AL 1 Xl ;T i=1.8X.62=lM16 AAj j 7 E AA 1 =1M16=1' 07" Long, of fix 11.40 a. m. 63 37 01.4 W Aij is northerly, Z l is southward and eastward; therefore, A^! is easterly. To correct the latitude obtained from the sight near noon. Lat.l6.75N,^ 2 =S7.2E, F 2 =8.29 Computed Lat. at 11.40 a. m. 164504 N AL 2 =OM3=7".8 Lat> of flx 1L40 a> m> 16 45 11.8 N AAj is easterly, Z^ southward and eastward; therefore, Ai 2 is northerly. To find the noon position. D. R. 11.40 a. m. to noon. True Course. 315 Lat. of flx 11.40 a. m. 16 Diff. of Lat. to noon Lat. in at noon 16 1 Distance. 1 2.7 i it 45 11.8 N 1 54 N 47 06 N 1 Diff. Lat. 1 Dep. I 1.9 N 1 1.9 W Long, of fix 11.40 a. m. Diff. of Long, to noon Long, at noon o / n 63 37 01.4 W 200 W 633901 W 622 JNAUTICAL ASTRONOMY 304. To determine the intersection of two lines running at an angle with both meridians and parallels, when position points having a common latitude are known, one for each line. Two lines with position points on a common parallel may be considered when we have simultaneous observations of two bodies favorably situated for finding time, the Z N of each being from 45 to 135 or 225 to 315; also when a line from a time sight is combined with one from a "' Dissimilar" M C FIG. 133. sight previously taken, the computed latitude from the latter, after correction for the run in the interval, being used in the time sight. Two cases occur under this head: (1) when both lines are in the same or opposite quadrants, being then called similar; (2) when the two lines run in adjacent quadrants, being then called dissimilar. The chord method. Let A and B be two points of a line determined by assuming latitudes L and L 2 , and C and D two points of a second line having the same coordinates; the two lines being the results of simultaneous observations of two different bodies; or, of observations taken at different times, whether of the same or different bodies, one line, how- ever, being brought up to the time of the second observation for the run in the interval. Let \\ and A' 2 be the longitudes of A and B; \'\ and A" 2 , of C and D respectively. ASSUMING LATITUDES 623 Let AB be the first line, F its longitude factor found as in Art. 298. Then, if P is the point of intersection, AL = PM, and the corresponding difference of longitude from A is AA = AM; therefore, AA t = AL X FI . If CD is the second line, its longitude factor is F 2 and, in the same way as above, CM AA 2 and AA 2 = AL X F 2 . In both figures, AC is the known difference of longitude of both lines for one assumed latitude. In Fig. 132, AC = AM CM = PM (F F 2 ), or AM CM = AL (F F 2 ) ; but AA AM = AL X F , and AA 2 = CM = AL X F 2 . In this figure, where the lines are similar, both AA and AA 2 are applied in the same direction, both East or both West, so as to make the resulting longitudes the same. Then, knowing the name of the correction in longitude for either line and the direction of the line, the name of the cor- rection in latitude is apparent, or is found by the rule of Art. 301, in which the direction of the line, instead of the bearing of the body, is considered. In Fig. 133, AC = AM + CM = PM (F + F 2 ) = AL (F + F 2 ) and AA = AM = AL X F and AA 2 = CM = AL X F 2 In this figure, where the lines are " dissimilar" AA t and AA 2 are applied in the opposite directions, westerly to the more easterly longitude and easterly to the more westerly lon- gitude, so as to make both resulting longitudes the same, which must be the case; otherwise an error has been made. Then, knowing the name of the correction in longitude for either line, and the direction of the line, the name of the correction for latitude is found. Apply the correction in latitude to the latitude of the parallel used as origin to find the latitude of fix. 624 NAUTICAL ASTRONOMY T3 Oi ~ S ^ <^'o3 J o3 0 3 g S *j I CQ 02 CQ i8 'S * 8 ";' -8SS o . sr oo o5 o5 o IN d a ^ *EH 2j bf) a 00 W fl O JP O fH ^. n4o H!. R., is used. The longitude factor of each line is gotten from Table I, as explained in Art. 298, and the direction of each line is ob- tained as shown in Art. 297. From this point on, the mode of procedure and the rules of the chord method apply. THE TANGENT METHOD 627 628 NAUTICAL ASTRONOMY 00 OQ DQ I HW 3 S>0 05 OS OJ CO Q >> rH" *3 03 O W W 7! rH ^ rH X X 00 00 M ) ^ J II 3 ^ t> r p." Px, ft, 'So x x a 3 * 1 : s r-( T3 pr^l in the co but A' t 5 oo a be '3 d .2 '3 * d 9 S 2 . s * M a ^, -"rs: O ?< < Hi ASSUMING LONGITUDES 629 306. To determine the intersection of two lines running at an angle with both meridians and parallels, when position points having a common longitude are known, one for each line. In previous articles we have considered a position point in each line with a common latitude. It may be necessary to con- sider two lines with position points on a common meridian, as in the case of two simultaneous observations worked for lati- tude by the <"<' method; or, in the case of a line from a "<' sight combined with one from a time sight, the com- FIG. 134. FIG. 135. puted longitude, from the latter, after correction for the run in the interval, being used in the "<' sight. The latitude factor. In this case, instead of using the variation in longitude for 1' of latitude, or the factor ]?, we use the variation in latitude for 1' of longitude, or the latitude factor / (which equals ). The chord method. Let AB and CD (Fig. 134) and (Fig. 135) be two lines of position obtained by working sights with the same assumed longitudes, the longitudes of the meridians of AC and BD; or, let AB be a first line moved for the run between observations; A and B being the coordinates of the 630 NAUTICAL ASTRONOMY first line at the instant of the second observation, the longi- tudes of A and B are used to work the sight for the second line CD. Let L\ and L' 2 be the latitudes of A and B; L\ and L" 2 , of C and D respectively. AB being the first line, / (found as in Art. 298) is its latitude factor; then, if P is the point of intersection of AB with the second line, taking the meridian of AC as origin, we have AA = PM and the corresponding AL from parallel of A is AM, but AM = PM X fa therefore, AL = AA X / . If CD is the second line, its latitude factor is f z , and, in the same way as above, CM = PM X / 2 > or AL 2 AA X fz In both figures, AC is the known difference of latitude of the position points of the two lines on the common meridian AC. In Fig. 134, AC AM CM = PM(f -- f 2 ), or the difference of latitude AC = AA (f f 2 ), then AM = AL A A X f i and CM = AL 2 = AA X f 2 In this figure, where the lines are "similar' 3 both correc- tions to the latitude, AL and AL 2 , are applied in the same way, either to N. or to S. so as to make the resulting latitudes the same. Knowing the name of the correction in latitude for either line, the name of the correction in longitude for the same line is apparent, or is obtained by the rule of Art. 301. In Fig. 135, AC = AM + CM = PM(f '-f f ), or the known difference of latitude AC = AA(f + f 2 ) ; then AM = AL = AA X f i and CM = AL 2 = AA X f 2 In this figure, where the lines are " dissimilar/' the cor- rections for latitude are applied in the opposite directions, northerly to the more southern latitude and southerly to the more northern latitude, so as to make both resulting latitudes the same. The name of the correction in longitude is found by rule (Art. 301). The correction in longitude, applied ASSUMING LONGITUDES 631 with its proper sign to the longitude of the common meridian, will give the longitude of fix. Rules for chord method, using a common longitude. (1) From the coordinates of the two points of each line, find the latitude factor f^ for the first line, and f 2 for the sec- ond line. (2) Divide the difference of the computed latitudes of the position points on the common meridian 'by the difference or sum of the latitude factors,, according as the lines are "simi- lar " or ee dissimilar" The result is the correction in longi- tude. (3) The correction in longitude, multiplied by the latitude factor of a line, gives the correction in latitude to be applied to the latitude of that line's position point on the common me- ridian. For " similar " lines, the latitude corrections are ap- plied the same way, for " dissimilar " lines the opposite way, to the computed latitudes, so as to malce the resulting latitude the same and to give the latitude of fix. (4) Knowing the name of the latitude correction, and the direction of the line of position corresponding, find by the rule (Art. 301) the name of the longitude correction. Apply this with its proper sign to the longitude of the common me- ridian; the result will be the longitude of fix. 632 NAUTICAL ASTRONOMY 5>. *& c? t f-j f 21 i o 55 ^ i ^. P + + 03 . ^ w fl w M H* 1 ^ 1 s g tt> *H ^. g sg oo ^S x^; a M g-o H H "g ^ g ~*l|Sg s as + s f>! * ' a H ^ o 1 O h O ^OteS^I" H T 8 CO P fe -M ?sp *~ ^ fl ~ c 2 p S iS ^ go 1-1 p* o g 5 w ^ tj ^ x -M IS ^ 'H ^ w ^. " lororo ts li |g* - 1 S ^S .15 H H 'g 08 jg ^ CO 83 iO |f < ^, -2 "^ d "" ^* S CT 3 S | |o 3 1 ^S S ^^^ r*o ""* -- ^ o .^f .a S S ^ "*" T3 "oS * "o 02 fc 02 j f 5--s rij?" (D P fH O o W E -^ "*~ "~ O ^ b^ ^ * cc - S 8 c S 8 02 Xl C) " hH O * fl _r *c. 2 *-" -0.2 ^ r-< -J ' w fi t& 3-u t _L 1 i _ .^ ^ OD* tl|} |3 u "T" 1 ^ "t" ^ ^ - 1 1 & 1 o3 . O o3 .O ^| Qda-2g' || 02' M P P, O 00 ^ ^ i-* f-( t> - S S S . ( C^l "- 1 C ^ ^ k. GO CO *O ** ob * fl ^ 3 * - ; " CO s F< >^ 4J t O ^ W J^ ^ r& ^3 "I 08 ^ ^* ^ A O ^ si , o S p ^ ^ r oio 9 Jl5f'*^| OQJOjg 00 ggj gg ^ lOiOm M a ^ g S Q or. ass & N ss s o sss g 6 w ^ , ""i "i " So S S ^ S rf s Oi ^ ^ fl? rH rj e * o fci 5 *; G i 4 Si**!* H^ Ej ** o o i " H ^ o s HA), then AB is the "altitude difference," IBcd is the circle of altitude passing through the place of observation, and B is one point of this circle which has certain attributes that make it very prominent in the methods of the so-called " New Navigation." This point is always nearer to the real position of the ship than the D. E. position, except when coincident with it, as may be seen by reference to Fig. 136, in which ahc, a circle described about A as a center with a radius equal to the pos- sible error and called " a circle of error," includes the ship's position which, being also on the circle of altitude bBcd f FIG. 136. 638 NAUTICAL ASTRONOMY must be- on the arc IBc. Since ABS is perpendicular to this arc at its middle point B it is evident that B is nearer than A to the ship's real position, and as B occupies the mean of the probable positions, it is less likely to be in error than other points of the arc. The circle of altitude might be drawn on a globe if the lat- ter should have sufficient dimensions (Art. 283) ; or it might be practicable to draw it on a Mercator chart in the case of a body of small declination observed at a high altitude, in other words, in case of many observations made in the tropics, by first locating the body's geographical position and then draw- ing a circle from that point as a center with a radius equal to the body's observed true zenith distance (Arts. 281-286). The point B would then be determined by construction at the intersection of this circle and the great circle passing through the D. E. position of the observer and the body's geographical position. The computed point. As the use of a globe is impracti- cable, and since circles of altitude may be represented by circles on a Mercator chart only under special circumstances, it is ordinarily necessary to find by computation the co-ordi- nates of the point B through which the circle of altitude passes, therefore, in the practice of the " New Navigation," this point becomes the first desideratum and will be referred to hereafter as the " computed point." Having determined the " altitude difference," this point may be found with suffi- cient accuracy in practical navigation by laying off this dif- ference on a loxodrome through the D. E. position instead of along the great circle bearing, the error produced by this sub- stitution, owing to the small size of the " altitude difference," being inappreciable, even under the most unfavorable conditions. Line of position. Since the D. E. position so limits that portion of the circle of altitude on which the observer may THE NEW NAVIGATION 639 be, it is necessary in practice to consider only a small arc and this will not differ materially within certain limits from a straight line drawn through the computed point at right angles to the body's bearing regarded as a loxodrome, except when the body observed is very near the zenith, the limits of coincidence depending on the value of the altitude difference as well as the altitude of the observed body. Formulae. The computed altitude and azimuth may be found from the formulae (159), the form for arrangement of work in finding h and Z being as shown in the solution of examples 157 and 158. Or, as is the better method, the computed altitude may be found from the formula sin h = cos (L ~ d) 2 cos L cos d sin 2 %t (239) and the true azimuth Z may then be taken from the azimuth tables (Art. 221), from an azimuth diagram, or found from a simultaneous compass bearing corrected for variation and deviation, provided the conditions are such as to admit of an accurate bearing being taken. For a body observed on the meridian formula (239) re- duces to sin h = cos (L~ d) or z = L ~ d, the same formula as in the direct method (240), and in this case the "New Navigation " has no advantages. Should the navigator be provided with Inman's Tables, or tables of haversines and log haversines, the following formulae , obtained from (209) and (210) by substituting (L~d) for z , are strongly recommended: haver z = haver (L~d)-\- haver 0,1 where z = 90 li and is defined by (240) haver 6 = cos L cos ? haver t. A line determined by the Marcq Saint-Hilaire method may be used in the same way as one determined by any other 640 NAUTICAL ASTRONOMY method (Art. 291) ; it may be combined to determine a fix with a terrestrial bearing or with a line found from a sight worked by some one of the direct methods. Conditions of observation. Owing to errors of refraction at low altitudes, and to the small limits within which the circle of altitude and its tangent at the computed point are coincident at very high altitudes as well as the practical diffi- culties of observation under such circumstances, it is desirable that heavenly bodies be observed, if possible at altitudes not less than 10 nor greater than 86. Advantages of the Marcq Saint-Hilaire method. The great advantage of this method of obtaining a line lies in the fact that since the formulce make it available practically without limitations as to azimuth, altitude, or hour angle, it furnishes one method equally applicable to all conditions, whether these conditions would otherwise require the formulce of a time- sight, a "' sight, or that of a body observed near the merid- ian. Except when finding latitude by meridian altitude, by reduction to the meridian, or by Polaris, or when finding longitude by the time-sight (tangent) method, the process of solution above described is simpler than any other method for either latitude or longitude. Rule for the determination of a single line. With the latitude and longitude of the given D. E. position, compute the true altitude of a heavenly body for the instant at which the observed true altitude is known, or will be known, and find from the azimuth tables the body's true azimuth Z for the same instant. Subtract the computed true altitude from the observed true altitude, calling the remainder the " altitude difference " and designating it by the letter a. Then run by dead reckoning, or lay down on the chart from the assumed D. E. position, the distance of a sea miles on a Mercator course equal to the observed body's azimuth if the observed altitude is the greater; on a course equal to the azimuth -j-180 , if THE NEW NAVIGATION 641 the observed altitude is less than that by computation. The point thus determined is the computed point, and a line (which may be drawn on the chart) through this point at right angles to the body's bearing line will be the line of posi- tion. In Fig. 137, AB is the "altitude difference," B the computed point, and BB' the line of position from an obser- vation of the body S t made at the D. R. position A. Double altitudes. If having determined the computed point B of one line BB' (Fig. 137), the run of the ship is laid off from B to C, at which point another altitude of the same, or of a different body, bearing in the direction C8 2 is both observed and computed, the hour angle from the meridian of C being used in the computation, then a second line of position DD' may be obtained by laying off the " alti- tude difference " at the second observation as before explained, the point C being the D. R. position and D the computed point of this second line. The intersection of DD' with CO' drawn through C parallel to the first line (CC f being the first line transferred for the run of the ship during the inter- val between observations) will be the fix F. These lines and the intervening run may be laid down on a Mercator chart (Fig. 137) and the fix found by construction (see Art. 292). Anticipating the work. Should the navigator, assuming the position A for a later instant, compute the true h and Z of a heavenly body to be observed at that future time and place, he may immediately plot that position on the chart and lay down thfe body's bearing line; then there will be nothing to do but to lay off the "altitude difference" and draw in the line of position when the actual altitude has been observed at the predetermined instant of G. M. T. The fact that the preliminary computation may be made, perhaps hours in advance of the actual measurement of the altitude, makes the Saint-Hilaire method most useful to any navigator desiring to anticipate his work. If for any reason the obser- 642 XAUTICAL ASTRONOMY FIG. 137. THE XEW NAVIGATION 643 vation should not be made at the exact G. M. T. used in the computation, the line plotted as above should be shifted so as to allow for the error (Art. 293), remembering that if the G. M. T. of observation is greater than that of computation the line must be shifted to the westward, otherwise to the eastward. The same result may be accomplished by laying off the " altitude difference " from a new position point found by so altering the longitude of the assumed position (to the westward or eastward as indicated above) that the hour angle corresponding to the G. M. T. of observation may be the same as that used in the solution, thus making the computa- tion for the altitude still hold good (see Ex. 222 and Fig. 145).. For any probable difference between the G. M. T. of observation and that of computation the changes in the ele- ments used for the observed body would be so slight as to produce only an inappreciable error in the results. So long as the longitude is changed as above to make the hour angle the same for the G. M. T., both of observation and computa- tion, no other adjustment will be necessary, as for instance for any change in the assumed position due to fleet maneu- vers. A line from a previous observation, however, brought up to the instant of the second observation, must be trans- ferred for the exact run in the interval ; then the intersection of the two lines will give the " fix " at the instant of second observation. Intersection by Computation-Double Altitudes. Eeferring to Fig. 137, let AB be the first altitude difference = a x , CD be the second altitude difference = a 2 , Z^ be the azimuth of the body at the first observation, Z 2 be the azimuth of the body at the second observation. Then the position of A being that by dead reckoning at the 644 NAUTICAL ASTRONOMY first observation, the position of B is obtained by running a distance a t in the direction of the first azimuth, or the oppo- site direction, as required by the conditions. The position of C (the D. R. position used in the solution of the second observation) is obtained from the position B and the dead reckoning between observations. If desired the run a and the run BC may be combined in one traverse, thus permitting C to be found directly from A. The position of F, or fix, is obtained by running a distance CF in a direction parallel to that of the first line (see, Ex. 220 and Fig. 137). As the line runs in two opposite direc- tions from C, it is only necessary to know that the general direction of CF is that of CD; the direction of CF cannot differ as much as 90 from that of CD, and hence that direc- tion from the point C is considered which fulfills this requirement. The distance CF = CD cosec CFD = a 2 cosec (Z t ~ Z 2 ) and is easily found by computation, or by using the traverse tables, entering the tables with (Z^^Z 2 ) as a course and taking out the distance CF in the distance column directly opposite the value a 2 found in the departure column (see Art. 125). The angle CFD, which equals the difference of the azimuths of the body, or bodies, at the two observations, or Z^ ~ Z 2 , is always acute if the same body is observed in both observa- tions, and is never greater than 90 in case the observations are of two different bodies. Intersection by Computation-Simultaneous Observations. When the position of the ship does not change between the sights, and in the case of simultaneous observations, the run BC of Fig. 137 is zero; therefore, in such cases, as shown in THE NEW NAVIGATION 645 Fig. 138, when the " fix " is to be determined by computation, use the D. K. position A in the solution of one observation, apply the altitude difference a^ to the position A and find the computed point B of the first line BB' ; then use this com- puted point B in the solution of the other observation, apply the altitude difference a 2 to the same point B and find the computed point D of the second line DD'. The intersection F of the two lines will be the " fix " (see Ex. 221). FIG. 138. Intersection by Construction. When the ship's position changes between observations, the " fix " by construction should be found by first determining the lines and then plotting them on the chart as indicated in Fig. 137. Should the ship's position not change between sights, or in the case of simultaneous observations, the computed point B of the first line may be used in the solution of the second sight and the " fix " by construction found as indicated in Fig. 138. However, it might be advisable to work both sights by using the co-ordinates of one and the same point A, 646 NAUTICAL ASTRONOMY laying off from that one position both altitude differences, each in its proper direction, for the determination of the computed points; then the intersection of the position lines, 20 10-- 50- 40' 20' -I G.M.T. 12^1 6 m April 26, 1905 76 50 W. 40' 30' 20' 7610' -10 20 - -50 FIG. 139. when drawn, will be the "fix" F (see Fig. 139). In this connection attention is called to the method of laying off courses and distances on the Mercator and polyconic charts (see Art. 31). THE NEW NAVIGATION 647 309. Special cases. The following cases are specially re- ferred to in order that the student may learn how to com- bine, perhaps with advantage under certain circumstances, the direct methods of Chapters XVII and XIX with the indirect method of Marcq Saint-Hilaire : (1) When one of the two observed bodies is on the merid- ian (Figs. 140 and 141). Let A be the D. K. position; B the computed point of the line BB' (or that line transferred for FIG. 140. FIG. 141. run to the instant of the second observation). Then whether the observation of the body on the meridian is solved by the Saint-Hilaire method, using the computed point B of the line BB' and the altitude difference a 2 , or by the direct method (Art. 240), the result will be same; in the one case BD = 0-2 and in the other it is the difference of latitude be- tween B and D. In either case BF = BD cosec BFD = BD cosec (Z^ ^ Z 2 ) = BD cosec Z\ , but it is unnecessary to find BF as the latitude is well determined and the longi- tude alone in doubt. Find the Departure DF = a^ cot Z t , then the difference of longitude which applied to the longi- tude of B will give the longitude of fix. 648 NAUTICAL ASTRONOMY In this particular case, however, it must not be forgotten that the direct method of Arts. 299 and 300, as illustrated in Ex. 213, is equally as simple. (2) When one of two observations is to be solved as a time- sight. Many navigators are averse to giving up the time-sight (tangent) method when conditions justify its use, at the same time preferring that of Saint-Hilaire to the <"<' method. This procedure may find application in simultaneous observa- tions of stars or in forenoon observations of the sun; the intersection may be found graphically on the chart or by computation as indicated be- low and in Fig. 142. Let BB' (Fig. 142) be one line from a time-sight of a body (not on the P. V.), whose azimuth is Z^ , trans- ferred for run to the instant '^Zj of observation of another body whose azimuth is Z 2 ; or a line from a time-sight simultane- ous with the observation of the body whose azimuth is Z 2 . The point B having been used in the solution of the other observation by the Saint-Hilaire method, D is the computed point of the line DD' and F the fix. BFD Z 1 ~ Z 2 ; FBR = 90 ~ Z ; BE = I and RF = p between the positions of B and F. Entering the traverse tables with Z l ~ Z 2 as a course, look for a 2 in the dep. column and find BF in the dis- tance column; with the direction of BF, that is 90 ~ Z^ , as a course and BF as a distance, take out the corresponding I and p; then find values of L n and D and, from the co- ordinates of B, the latitude and longitude of " fix " F. THE XEW NAVIGATION 649 (3) If the time sight is of a body observed on the prime vertical, then, as indicated in Figs. 143 and 144, the line of position BB' will run due north and south, the longitude of " fix " will be well determined and it will be necessary to determine only its latitude. This may be found from the latitude of B and the difference of latitude between B and F. In this case BF is this difference of latitude and BF = a 2 cosec (Z l ~ Z 2 ) = a 2 cosec (90 ^ Z 2 ) = a 2 sec Z 2 . zr *-a T H FIG. 143. FIG. 144. However, attention is called to the fact that by the direct methods the longitude would be well determined from the observation on the prime vertical and that an excellent " fix " would then result from using this longitude in the solution of the <'>' sight. (4) If the Marcq Saint-Hilaire method is applied to the first sight worked and the latitude of the computed point of this first line is used in working a time-sight (tangent method), the intersection, if not found by construction, may be found as explained in Art. 305. The ship's most probable position. From each one of sev- eral simultaneous observations, a line of position may be 650 NAUTICAL ASTRONOMY obtained and from n observations n lines will result; in case there have been no errors of observations, or otherwise, these lines should pass through one and the same point. However, there are always errors which may be due to the imperfection of the instrument itself or its adjustment, to error of the tabulated dip or refraction, to incorrect time, or the per- sonal equation of the observer, etc., and, in consequence, gen- erally speaking, there will be more than one point of inter- section, and there may be for n lines as many as ft points. It is evident that the ship is not at all points; the " theory of the probability of errors " shows that the most probable position is that point from which if perpendiculars are drawn to the lines of position, the sum of their squares shall be a minimum. The navigator, in his effort to check a " fix " from two lines by means of a third line of position, will often find that the three lines make a plane triangle; and, in such cases, though the most probable position may easily be found by construction, the practical navigator, re- garding this procedure as more a matter of theory than of practical value, will assume the ship's position at. the center of said triangle, especially if it is small and equilateral. Use of Table 44, Bowditch. This table has opposite t in the p. m. column the log sin \t in the sine column; so if using formula (239), look for t expressed in time in the p. m. column and from the sine column, directly abreast, take out the log sin \t which, multiplied by 2, will be the log sin 2 \t. This method is illustrated in Ex. 221, and the method of considering the half-angle in degrees is illustrated in Ex. 220. THE NEW NAVIGATION 651 S s o ^ fl H O O 00 ' - 3 2 3 o S j, S A 1 stjiii d ^ ,0 co a 3 d 5 * 1 * r :i gij i* a, S * x 1 i I * O * g *> TH gs s d o a od I I *> a o ^ O r-l T-! "a co & : a o S ^S *s s% ? g 3 + 1 a tJ g. - - 00 _ ^ 2 go OJ g o' e : d .d O 33 o ? 03 , -*J > d OQ * ^ o fl L- " P e3 O IS III OQ Altit Correct 1 1 X ' I I . M P P, O o ; t- + t- Oil o os Oi 11 II C-J >0 II II ts gs rt oo' es 88 652 NAUTICAL ASTRONOMY H ft CO 10 II II be J OQ OQ o fl s: H be fl II t * I S O H *^ o -H ss - fl ^ W rH fcfc ' - CO 1-* iH (M i. * s SA . 2 CO 1 ,0 8 o S 5 O g CO rt Is OB 3'- g 8 ,O T c 2 t> 08 s : rse les, u gSo 2 O - o g rt 0) In 2 PH o> fl "g .2 a 2 H C 'C THE NEW NAVIGATION 653 1* 00 t- CO 00 t- CO O OJ OS CO i B* ^ * O 1C CO CO OS Q O . 4_ co co o; os 1C ^4 ** CO' OS OS GO* + 00 a EH >c oo TO : - bfi "^ w * o o o -" CX j? H ^_; oo ^ *t *. ! CM ^ OJ *"* OJ ' 1 3 SOS OS P ^* 00 i w **t* & !ii . I 1 TH CC CO 00 o *c O JO C i O ^ - CO O 5] OS* C o o ! ^ T 1 TH s i .2 ^ OJ t 4* 03 P OQ 02 z o o ^ t^ O t> T* O a' - r-4 -o_o ^ ?O tH 1C TH O '"" 1 1 V ll .+ ^ "^ o Q O p, 1 ^ fc s aQ M -sp,a - g g ^; s i CO o OJ ^ ^ 01 o CO ^ 30 rH 10 CO CO TH T* CO -d CO ri 00 TH r-> 1C | , CO OJ 1C CO CO + CO *tt ^ ^ I & CM ^ '(M kq t3 ^ .^|H o o 1C TH SS T^S' g r-' 00 OS 05 i 0So OJ t- 05 OS OS Z> 1C 1C I rt CO ^ TH TH " Ol CO CJ TH w w II II II II a N " S > k, I O ^ CO 00 g 32 's - 00 OS co' fl .3 * co CH H ? fl s ^ s s 5 CO ^1J3 ss^ |:e ^Sq OHO a 654 NAUTICAL ASTRONOMY 2 * ^ 3 9 O O oo a S'S S w r-r ^5 ^a * O &X) co * ST fS ' -0 i' ^ . d o ' ^ Q OJ rH CM d ..h :liif bJDo O K StP * * COT^rHCO r-l - >>> CO -* ^h ^> ja eft r l> {> 2 I Ol-^OS O r-ITj^CM j^ ^. COCOOSOS CO ^i^ O O-* o o | ' ,-H r-< ' ^' O CO OOSOSOS OS ^ co^^ d ja o lN CM CM II W co r-l rH Ho CO S GQ t>C 'S w tc R ^ o o o o >rH ^ CM r^ ~>^ . 4 S tib o |l d || 5 c *?* Pi tJ OJ V. 00 3 *. S J ^ "* "* efQ CO CO 1C O 1C i> o co t- P2 , 00 OS t- r-i .,-H OQ " rH t 1 2 s=:-f; .2 1 N " z . *c3 -r> O ^ -2 rrt ^ d <5 O PH ^ ^ CM -*-i VJ i TJ 3-5 * O CD GO " O CO CM * rH :O O ** H 1 e " CM ^ H ^' ^_ ; . N - J .2 . 1 u ^ ^'-'dS'^r/Jcfl 02 u. * . P- o . . - ? -*- >* o ^ Q O C, D<1i-5rHPHrH-X--)^ W W PH THE NEW NAVIGATION 655 . o 2 CO 00 O rH O eo ic -^ rH rH rH O O 1C "* O5 CO * OS 05 O Os' OS OS o I- CO 2 1 j ~ o ;o " O CO - rH O + I U.O, fc OQ CO CO 5 ^ 00 - O iO * rH CO ^ O 00 0 bo SJ2 5 bJ0.2 --' W CQQ 1C iC CO O 5> * * 6 05 a> Q . 2 pS 2 05 3 Q S ^< CO LJ >H QQ (M rH SO CCJ Q II *g .-r _S) S *g e^f a bo w , a _.2 S ^ P .2 o a? ||1 * Z. ^;s -d a fci O o bp Q, - a S o 9 o o eq ; ^ o -4- c3 655A NA.UTICAL ASTRONOMY -aco"o "Hco a p Ij TH 1 r-i ... 00 o o o iC rH Q p d* H H *' t> CO ^ O O * . (MOW T* CO x> t- CO u t- t- OS a? a) o> ^; co > > > ^ 2^ + o 1 ~ l OS CO - r-i "^Ji O " fj H ^^^ 1 ^ * Ss* l i+ ^CQ^; ^ ^ 'So^o of ?25 'a ' t W'5b . ^ . ^ 2 < COOrH ;-*cOCM|| ** -g .bC SJ . 'O .: it' v , ' i r-3 r-5 "So PdOOO.QOOO^t^ HHr-5 5, about 7. 30 a. DD s lower limb wa of G. M. T. 5* 4] he haversine forn Altitudes. V O CO v 10 ^ 00 !> O o 00 i TH T ^ ^1^ :g SS ^ .2 Jl II II II II II II II Q ^ ^1 '- 1 rO O X3 ^ a) ~M CM PH alS*" * . "g 232 S 5"o-s d _ co eo * CM So o 60 rH l>CM OOJQrH y-*-* > > . W . o t- t- i& O5 CO O O OJ oS rt s3 O eo ea ''''*< oo . . K^. H 03 O5 O5 OS 5 5 525 ^ o* 2 fab II bb g CD CO 05 a Q o "S ^ ^ r^> s V^Wd " CO CO HH O .... 3 ^ rf S lii s fc o . . . . SJ O O5 H HC Q G ^ >j ^ O5O5O1O5 OTHTjHlO O ^ t^ rff< s s I - s ^ M "B O *2?l S "co? -. o_i r-H *O O O^JOQ - Q ^o^ o ^3 S oooo ^iccco "J| "J| .ti i_J Q PH O s ^ hSiS ^ O CQ 00 - "* TH O O 1C C OO TH co -^v-~ ic jo co o w :o - o ^ 1 * " * JO COCOCO O1C-* p,^- ." TH tH CO CO O? t CQ o ^ +3 w II II II ' II II II II g ^ "h^TS^iTSN^JQ . .X en - \ i ' . "3 r^ *^* t-4 "^ K^ ,Q -4^ OJ o s ? ^- .5 *j PH x-o -x- te" ^ , .-. w t- .t- j> eo - Q a CO CO t- O 00 t- < ^ d ^ v* ^ pH ^ ^H . ^^ ^O do c 656 NAUTICAL ASTRONOMY Ex. 222. April 7, 1905, a, m., by an observation of the sun taken at the G. M. T. O h 40 m 54 s April 8, 1905, a ship was faund to be on a line of position GM determined by a posi- tion point G in Lat. 39 27' N. and Long. 69 18' W., and by the sun's true azimuth Z N = 104 58'. Expecting the ship to maintain for several hours her course 320 (true) and speed 10 knots per hour, the navigator decided to anticipate as far as possible the work for a line from an observation of the sun to be taken three hours later, or at the G. M. T. 3 h 40 m 54 s . Assuming that the D. R. position at that time would be Lat. 39 50' K and Long. 69 43' W., he found for that time and place the sun's computed true altitude to be 54 37' 33" and its true azimuth or Z 2 to be Z* 153 42'. The above D. R. position and bearing were immediately laid down on the chart (see A, Fig. 145). The navigator failed to get his sight at the exact G. M. T. used in the computation, but 16 seconds later, or at the G. M. T. 3 h 41 m 10 s , he observed the sextant altitude of (D 54 27' 40"; I. C. +2'; height of eye 26 feet. It is required to find at the time of the second observation a position point and line by the Marcq Saint- Hilaire method and the " fix " by construction. Solution. The sun's true azimuth at second sight from tables, or Z 2J is Z N = 153 42', the computed true altitude of the sun's center is 54 37' 33", the observed true altitude is 54 40' 03", and hence & 2 is 2' 30" to be laid off in the direc- tion of Z y Since the G. M. T. of observation was 16 seconds later than that used in the computation, the longitude of the D. R. position of the second line must be so changed as to give for the G. M. T. of observation the same hour angle as that used in the computation; in this case the longitude must be increased by 16 seconds of time, that is by 4 minutes of arc. Therefore, from A', which is 4' of longitude directly to westward of A (see Fig. 145), lay off a, = 2' 30", in this THE LITIGATION G57 case, toward the sun and B will be a position point of the second line BB', and, as AM' is the first line brought up for 50 40 30 9 50 W. 40' 30' 69 W, FIG. 145. the run, ^ in Lat. 39 49' K, and Long. 69 43' W., by con- struction, will be the " fix." It is apparent that a 2 could have been laid off from A to 0, and the line then shifted to B in order to allow for the error of time (Arts. 31 and 293). 658 NAUTICAL ASTRONOMY Littlehales' graphic solution. The methods of the " New Navigation" have been applied by Mr. G. W. Littlehales of the Hydrographic Office, Navy Department, to the graphic solution of the astronomical triangle by means of a stereo- graphic projection on the plane of the observer's meridian. Knowing the latitude and longitude by D. R., and having found the observed body's hour angle and declination for the FIG. 146. instant of observation, the navigator may find graphically the values 9f Z and h for the given D. R. position, and, from these and the measured true altitude of the same body the computed point and line of position, as previously explained. The theory embodied may be briefly explained as follows: let Fig. 146 be the projection referred to above on a reduced scale; P, the north pole; Z, the observer's zenith; M, the observed body located by its hour angle ZPM and its decli- nation RM ; then PZM is the astronomical triangle in which THE NEW NAVIGATION 659 the known parts are PZ the co-L, PM the co-d, and ZPM the hour angle t. In this method PZ is considered as < 90 or > 90 according as L is north or south; PM is < 90 or > 90 according as d is north or south. The required parts of the triangle are PZM or Z and ZM the co-h, and they may be easily determined if referred to a system of co-ordinates which, like the equinoctial system, admits of permanent graduations. This is accomplished by revolving the astronomical triangle about the central point of the projection with the side PZ kept in coincidence with the bounding meridian till Z falls where P originally was and P and M are revolved respectively into the positions P' and M' so that the unknown parts PZM and ZM, respectively equal to P'PM' and PM', may be measured from the graduations of the projection ; PZM is reckoned from the left hand bounding meridian and P'PM from the right hand bounding meridian. It is apparent that M has described an arc of a circle whose radius is OM and which subtends an angle equal to PZ or co-Zr, hence to obviate actual revolution of the triangle, a series of equally spaced concentric circumferences and a series of equally spaced radial lines are drawn to facilitate identifi- cation, the former numbered from the center outward, the latter numbered so as to indicate the number of minutes of arc estimated from 08 and around to the right. After having plotted the body by its hour angle and declination, it is only necessary to note the number of the circumference and the number of the radial line passing through the position M', add to this latter number the distance ZP or co-L expressed in minutes of arc; the point where the radial whose number is the latter sum intersects the noted circumference will be the point M' whose hour angle and declination, read from the graduations of the projection, will be respectively the required Z and h of the body M. 660 NAUTICAL ASTROXOMY A stereographic projection has been constructed for a sphere 12 feet in diameter which is on such a scale as to admit of sufficient accuracy for practical navigation and to admit of convenient spacing of the various meridians, paral- lels, circumferences, and radials. Each quadrant is subdivided into 92 overlapping sections, making 868 in all; the plates representing them form a book of convenient size, each plate bearing the same number as the corresponding section of a small projection called the index- plate. The point M is roughly plotted on the index-plate and the circumference, radial line, and square are each noted by its number ; M' is then roughly plotted and the square in which it falls is also noted. Knowing the numbers of the squares and turning to them in the book of plates, the positions of M and M' are success- ively plotted, and the values of Z and h are taken from the second square. For plates and further information see Mr. Littlehales' book " Altitude, Azimuth, and Geographic Position." Solution by nomography. Lt. Radler de Aquino, Brazil- ian Navy, has suggested a method of finding li and Z by using a nomogram constructed by Dr. Pesci; this method, with certain modifications introduced by the author, will be found explained in Appendix D. Solution by Tables.* Navigators who find logarithmic work laborious may find a position point and a line of position by the Marcq St. Hilaire method by using tables from which 7i and Z can be taken for a position of assumed latitude and longitude, the computed point being then found by laying off the altitude difference from this assumed position in the direc- tion of Z or 180 -f- Z, as conditions may require. * See " Altitude and Azimuth Tables," by Lt. Radler de Aquino, Brazilian Navy, and " Altitude or Position Line Tables," by Frederick Ball, R. N., both books pub- lished by J. D. Potter, London. CHAPTER XXI. DAY'S WORK AT SEA. 310. In the chapter on the sailings attention was called to the fact that the general subject of a day's work was reserved till after the student had studied and understood the methods of finding latitude and longitude by the observation of celes- tial bodies. These methods having been considered, that subject will now be taken up. In the course of his routine work, a navigator, besides determining his latitude at or near noon and obtaining lines of position from observations of the sun, both a. m. and p. m., would get positions by cross lines of stars or planets, when conditions proved favorable, as in evening or morning twilight or when moonlight renders the horizon sufficiently distinct. Polaris, being available in the northern hemisphere, should be observed when conditions are favorable, and latitude desirable. The reckoning is estimated from the point of departure (Art. 123), or from the noon position at sea till noon of the following day, or till arrival at port of destination, if the voyage ends before noon. Owing to the facility of getting the latitude at noon by observations and the fact that longitude can be determined by observation within a few hours before noon and brought up to that time without appreciable error, it is convenient to compare the run by dead reckoning and by observation from noon to noon, and to regard the difference between the noon NAUTICAL ASTRONOMY positions by dead reckoning and by observation as due to current, though, as a matter of fact, it may be due to other causes as bad steering, faulty logging, etc. The navigator must report to the commanding officer at noon each day : (1) Latitude and longitude by D. E. at noon. (2) Latitude and longitude by observation at noon. (3) Course and distance made good. (4) Set and drift of current. (5) The deviation of the compass (on the course at time of a. m. sight perhaps). (6) Course and distance to destination. To attain these results, the following rules are laid down for a minimum of work. (1) Find the D. E-. positions at time of a. m. sight for longitude and at noon by working the traverse from the pre- vious noon or point of departure. (2) Find an a. m. line of position by either the chord or tangent method and the deviation of the compass when the sun is favorably situated for finding time. Plot this line on a Mercator chart and find graphically its intersection with another line, if possible. (3) Find the latitude at noon by observation from a me- ridian altitude of the sun, or by reduction to the meridian ; or bring up to noon a latitude obtained from the intersection of a (j>"cf>' and longitude lines. (4) Take the difference between the latitude by observation and latitude by D. E. at noon, mark it North or South as the former is to the northward or southward of the latter. This discrepancy may or may not be due to current, though usually so considered in the computation. Its value being for 24 hours, or from the time of departure, a proportional part for one hour, and hence for the interval between the forenoon sight and noon, may be obtained. THE DAY'S WORK 663 (5) Run the noon latitude back to the time of sight, cor- recting backwards for both the run from sight to noon and the proportion of current in latitude for that time. The re- sult will be the true latitude at time of sight. Find the longitude by observation at time of sight by find- ing the position point of the line corresponding to the true latitude at time of sight. (6) The difference between the longitude by observation and by D. E. at the time of a. m. sight is a discrepancy which may or may not be due to current, though usually so con- sidered in the computation. Its value being from noon of the previous day, or time of departure, to the time of a. m. ob- servation, a proportional part for one hour, and hence for the interval to noon, may be found. It is marked E. or W., according as the longitude by observation is to the eastward or westward of that by D. R. (7) Run the longitude by observation at time of sight up to noon by applying the run in longitude from time of sight to noon, and also the current in longitude for the same time, each with its proper sign. The result will be the longitude by observation at noon. (8) The course and distance from the noon position of the previous day, or point of departure, to the noon position by observation arrived at, will be the course and distance made good. (9) The course and distance from the noon position ar- rived at by D. R. to that by observation will be the set and drift of the current, so-called (Art. 130). (10) The course and distance from the noon position by observation arrived at to point of destination by middle lati- tude or Mercator sailing, will be the course and distance by that sailing to point of destination. Reference is made to chapter VI for manner of working dead reckoning and to chapters XVI and XVII for working of sights. 664 NAUTICAL ASTRONOMY The following problem will illustrate the points involved: Ex. 223. On January 2, 1905, at noon, a ship's position by observation was latitude 7 05' 42" N., longitude 148 19' W. Sailed thence until about 8 a. m. next day the following courses and distances; wind, variation, and deviation as in- dicated. Wind. Course (p. c.). Var. Pev. Leeway. Distance. Sly and Wly do do Nly and Wly do do do 301 285 276 256 233 212 220 CO CO 00 00 CO CO 00 +++++++ -3 -4 -5 -3 +1 +1 ;i 6 3 3 3 22.8 Miles. 31.6 34.5 17.9 16.1 It. 9 12.6 At about 8 a. m. observed an altitude of the sun's lower limb, 23 42'. I. C. (plus) 1' 20". Height of eye 45 feet. Watch 8 h 08 m 45 s . C W 10 h 03 m 15 s . Chronometer fast of G. M. T. ? m 2P.5. Sun's center bore (p. s. c.) 115, ship's head 291, variation + 8. Work a line of position, using latitudes 7 N. and 7 20' N. Work an altitude-azimuth with latitude 7 20' X. and find deviation. (The azimuth may be taken from tables.) Ean thence to noon 291 (p. s. c.), 39 miles, when observed meridian altitude of sun's lower limb, bearing South, 59 26' 10". I. C. (plus) 1' 20". Height of eye 45 feet. 1. Find latitude and longitude by D. E. at 8 a. m. 2. Work a line for longitude, and find deviation. 3. Find latitude and longitude by D. E. at noon ; true lati- tude at noon, and current in latitude; true latitude at a. m. sight. 4. Find true longitude at 8 a. m., current in longitude, and true longitude at noon. 5. Find (7 N and d made good, and set and drift of current. 6. Find (7 N and d to Guam by Mercator sailing using trigo- nometrical formulae. THE WORK 665 H *S P* i M H a H S 3 3 . I W bo O C3 |E OQ g COOOr^OO 3 . -P' : -* 10 CC T-^ 1 1 ( I I II + + 5o co co co ao oo oo +++++++ >> *? n ^ GGG NAUTICAL ASTRONOMY THE DAY'S WORK 667 eo ^ 3 1 eo ! " 3 3 i 1 02 PS 3 O * Q Q ' jj H 1 O TH 1 eS ^ . cf rd *> O i 3 g r-5 d . W *J a o CO 55 rH > fe. 1 ?? 1} 5 si CO ^ O CO* CO' ss W ^bc d' bi) ^ d ,0 TH (M 05 . rH ^l OS ^S * a ^' 3 IO TH ^! rfl ^ ' X c3 C 3* ^.^frJ ^ "" .2 I " a GO i . - CO O O rH rH rH (M JO CO CO ? ^ 1 > rH rH T I II Q 03 B + + 1 1 + k^ * 1 5 I T7 1 1 T 55 55 55 !5 ^ +3 - PH' CO CO (M -S +3 C3 3 i Q O p,=g fc rH O ^SrH 5? tJ . 3 05 M ' Q p, O ^ CO 10 CO rH oQ rf ^ CO CO _s .^ fc-l -* . O rH rH O5 CO* 1C rH CO rH rH W O oQ J S ^ . > Q ^3 . ^ I " -*a d P 1-5 oil $ ,^ 3^ 11 668 NAUTICAL ASTRONOMY o be bo oo o h- r-> 00 U5 to to bO bO rt 5 5 -2 " l! CO b- 00 CO THE DAY'S WORK 669 Had the tangent method been used to work the above time sight, we should have used the latitude by D. E. at the time of a. m. sight, 7 08' 42" N"., calling the resulting longitude com- puted longitude. With the latitude by D. K., the declination, and the L. A. T. from the sight, the sun's true azimuth, regarded as less than 90, would have been taken from the azimuth tables; and, with this azimuth and the latitude, the value of F found in Table I. The value of F and the direction of the line would have been written in the form for work thus : F = a X . Having found the true latitude at the time of a. m. sight, the difference between it and the latitude by D. E. at that time would have given AL, and, as before, we should have had AA = AL X F. Applying AA to the computed longitude at the time of sight, we should have had the true longitude at the time of sight. The procedure from this point would have been the same as in the chord method fully illustrated in Ex. 223. Marcq Saint-Hilaire line. Had the Marcq Saint-Hilaire method been used in working the a. m. sight, we should have used the D. E. position at the time of sight, Lat. 7 08' 42" K, Long. 150 30' 24" W., in finding the computed point of the position line, the arrangement of work in finding a position point of a line as shown on pages 651 and 652 being substi- tuted for that on page 666. Then with the azimuth and the latitude of the computed point, we should have found the longitude factor F from Table I and the direction of the line. The difference between the true latitude at time of sight and that of the computed point would have been AL and, as before, we should have had AA = AL X F. Having applied AA to the longitude of the computed point we should have had the true longitude at the time of sight. The procedure from this point on would have been the same as in the chord method illustrated in Ex. 223. CHAPTER XXII. TIDAL WAVES, TIDAL CURRENTS, AND FINDING TIME OF HIGH WATER. 311. Closely related to the subject of the moon's meridian transit is the subject of the tides, which, though a very broad one for a work of this scope, may be presented, even in its elementary form, with advantage to the student; by applying general rules, he may approximate to the time of high water for those places not tabulated in the tide tables. 312. Definitions, The phenomena of tides, as usually ob- served in tide-water regions, are a periodic rise and fall and a recurrent flood and ebb of the water; the word tide or tide- wave, properly refers to the vertical movement only, the hori- zontal movement being characterized as tidal current. The maximum height to which the tide rises is called high water, the lowest level to which it falls low water; that moment at either high or low water when no vertical movement takes place is called stand, and the difference in height between low and high water is called range. Flood is the inflow of tide water from the general direction of the ocean, ebb its recession towards the sea; the set of a current is the direction towards which it is flowing, drift the distance through which it flows in a given time, rate its velocity per hour, and slack the term applied to the period between tidal currents when there is no horizontal motion. 313. Causes of the tides. The tides are caused by the difference of the attractions exerted by the moon, and, in a THE TIDES 671 less degree by the sun, upon the earth and waters of the earth. By the law of gravitation, the attractive forces of the sun and moon decrease as the square of the distance increases, and hence exert a greater force on the nearer surface and a less force on the farther surface, than on intermediate parts; the resultant effect being a tendency to recede from the center in the parts not only just under the attracting body, but in the parts diametrically opposite. For purposes of illustration. The earth may be considered as surrounded by a uniform envelope of frictionless water, and, as illustrated in Fig. 147, let M be the moon whose mean FIG. 147. attractive force on the solid part of the earth may be assumed as acting at the center E; therefore, the moon exerts a greater force on the waters at A, just beneath it, than on the earth at E; a greater force on the earth at E, than on the water at A' diametrically opposite. The water at any other position, as at L, though attracted by the moon less strongly than that at A, will have its gravity toward the center diminished, and a tendency to go toward A, due to that component of the force along LM, which acts in the direction of the tangent at L; while the water at L' will have a tendency to go toward A f . The waters of the entire envelope, being free to yield to a similar tendency, will assume a spheroidal shape with the longer axis toward the moon, and thus two tidal waves, called lunar waves, will be formed at the points A and A', These 672 NAUTICAL ASTRONOMY will be points of high water, and midway between these ele- vations will be depressions of the water level, called low water, as at B and B f . Number of alternations. Ordinarily there are two princi- pal alternations of high and low water at a given place in a lunar day; and it may be observed at all places, except at the poles and on the equator, that the two daily high tides differ in height. This daily inequality is due to the inclination of the plane of the moon's orbit to that of the equator, and to the rotation of the earth on its axis. FIG. 148. In Fig. 148, let PP' be the earth's axis, P the North pole, QW the plane of the equator ; let M be the moon whose declina- tion is North, and equals the angle MEW; let L be a place on the earth's surface having the moon in its zenith. The tidal wave at L is the superior wave, its height may be represented by La, but at a place L' in the same latitude, and distant 180 in longitude, the height of the tide will be represented by L'af; owing to the revolution of the earth on its axis, these two places will change situations with respect to the moon in about 12 hours, and the height of the tide at L will then be equal to what it was at L' 12 hours before. This will be known as the inferior wave at L. This alternation of high water would theoretically occur at an interval of 12 hours, if the moon . remained at rest; but EFFECT OF THE SUN 673 owing to its advance to the eastward in its orbit, thereby arriv- ing at the same branch of the same meridian later each day by a mean amount of 50 minutes, the inferior wave, or tide of the lower culmination, will follow the superior wave, or tide of the upper culmination, by the average time of 12 hours and 25 minutes. 314. Effect of the sun. The attraction of the sun causes in the same way two solar waves at diametrically opposite points, which reinforce or diminish the lunar waves accord- ing to the relative positions of the sun and the moon in their respective orbits. Owing to the sun's great distance, the inequality of its attractions on the earth and waters of the earth is small, and the mean force of the moon in causing tides is about 2J times as great as the sun. When the sun and moon are in conjunction or opposition, they act together in producing the tidal wave, and the maxi- mum high and minimum low water of the month called spring tides result, with maximum tidal range; unusually high tides would result should the sun and moon happen to be, respect- ively, at perihelion and perigee at the time of new or full moon. At the first and third quarters of the moon, the sun and moon act at right angles to each other, and the effect of the solar wave is to diminish the height of the lunar wave; the minimum high and maximum low tides of the month, called neap tides, result with a minimum tidal range. Priming and lagging. When the moon is in the first and third quarters, the solar wave is to the westward of the lunar wave, and there is an acceleration in the time of high water called priming of the tides. When the moon is in the second and fourth quarters, the solar wave is to the eastward of the lunar wave, and there is a retardation in the time of high water, called lagging of the tides. 674 NAUTICAL ASTRONOMY 315. Luni-tidal interval. The theoretical assumptions in the preceding article are not fully justified by facts ; the earth is not entirely covered with water, and the water is not fric- tionless. Owing to the rotation of the earth, the inertia of the water, the variable depth of the ocean bed, the obstruc- tions offered by land, the general contour of the bottom, and the direction of channels, etc., high tide is not coincident with the moon's meridian transit, and the interval of time between the moon's meridian transit and the following high water is not the same for each day of the month. These intervals are known as luni-tidal intervals. The mean of these intervals on days of new and full moon is called the vulgar or common establishment of a port. It is frequently spoken of as the time of high water on full and change days, being found in the tidal data of charts as H. W. F. &C. The mean of all the luni-tidal high-water intervals observed throughout at least a lunar month, is called the corrected establishment of the port, and, when known, should be used, in preference to the common establishment, in finding the time of high water. It will be found tabulated for many ports in Appendix IV, Bowditch. 316. Age of tide. The greatest effect of the sun and moon in producing the tidal wave occurs at new and full moon, and the interval of time from the instant of new or full moon to the highest subsequent tide at any place is known as the retard or age of the tide. This varies with the locality, being one day on the Atlantic Coast of North America, and as much as 2| days on the Coast of England. General laws. Though the subject of tidal waves is com- plicated by the fact that the sun, moon, and earth do not occupy the same relative position more than once in a period of about 18 years, and by the further fact that every tide is largely affected by local conditions, such as depth of water, TIDAL CURRENTS C75 configuration of the coast, and even by interference of differ- ent parts of the same wave ; still the following elementary laws may be laid down as general for the moon's effect. (1) Two high tides will occur daily at a given place. (2) When the declination of the moon is 0, the two daily tides at a given place will be equal ; the greatest will occur at the equator, the least at the poles. (3) When the moon's declination is not 0, the two daily tides at all places except the poles and equator will be unequal ; the greatest tides and greatest daily inequality will occur at places whose latitude numerically equals the moon's declina- tion ; and the higher of the two tides will follow the moon's upper transit, when the latitude of the place is of the same name as the moon's declination. (4) The time of high water occurs after the moon's upper transit a number of hours equal to the establishment of the port. The time of the following low water 6 hours and 13 minutes after high water, and the time of the next high water at a mean interval of 12 hours and 25 minutes after the first high water. Tidal currents. A distinction must be drawn between tidal waves and tidal currents, the former referring to the vertical oscillations of the water, the latter to the horizontal inflow and outflow caused by the interferences offered the tidal waves by local formations and the frictional resistances of the bot- tom and sides of shoal, narrow and contracting channels, etc. Whilst it is of importance to know the times of high water when about to enter or leave a harbor, it is of more practical importance in the navigation of a vessel to be able to antici- pate a probable set and drift of a current and to allow for the same. It must not be forgotten that the changes of tidal currents seldom correspond with high and low waters, perhaps never except on open coasts or in wide and shallow basins, certainly 676 NAUTICAL ASTRONOMY not in large bodies of water having a relatively contracted entrance to the sea, as in the cases of Delaware and Chesa- peake Bays. Furthermore, a current in certain localities may flow in the offing one to three hours after it has turned along the shore; such peculiarities may often be found described in the sailing directions of those regions and should be studied by the navigator. In the tide tables issued by the U. S. C. & G. Survey will be found current diagrams for Georges Bank, Boston Harbor, Nantucket and Vineyard Sound, New York entrance and East Eiver, Delaware and Chesapeake Bays, and current tables, restricted, however, to points on the Atlantic and Pa- cific Coasts of the United States. In the diagrams, the set and rate of the current are given for three hours before and after high water; in the tables, for each hour of the tide, at some given reference station. An examination of these, when a vessel may be in the locali- ties therein considered, will often point out the most favor- able conditions under which the current should be encountered. When lying in a port of which the tidal information is in- complete, and under circumstances that will admit of obser- vations, a navigator should make every effort to gather all possible information about the local currents. For the method of making tidal observations and a description of the instru- ments used, etc., the student is referred to any standard work on Marine Surveying. 317. Times of high and low water. The quickest, most accurate, and hence most satisfactory method of finding the times of high and low water is by taking this information from tide tables, which are furnished navigating officers of the navy. General tide tables published by various foreign governments may be purchased in almost any seaport; and the U. S. C. & G. Survey publishes annually in advance tables TIME OF HIGH WATER 677 containing, in addition to the current matter referred to in Art. 316 predictions as to the times and heights of every high and low water in the following year at certain principal ports of the world regarded as standard ports for tidal purposes. For these ports, the times of tides are arranged in the order of the occurrence of tides in one line, the corresponding heights above the plane of reference (which for the Coast Survey Charts is that of mean low water) in a second line, a comparison of the heights indicating which are high and which are low waters. These predictions are extended to over 1000 other places by applying to the data of the proper stand- ard port, the tidal differences and ratios corresponding for the places. High water by computation. When tide tables are not available, the times of high and low water may be found by applying the principles of rule 4 (Art. 316). (1) Find the local mean time of the moon's upper transit at the place. (2) Add to this the high water or low water luni-tidal in- terval from Appendix IV, Bowditch, according as the time of high water or low water is desired. The result will be the required time. The H. W. luni-tidal interval, as tabulated in Bowditch, is the corrected establishment of the port; it may be taken from the chart of the locality; or the common establishment found on the chart, as H. W. F. & C., may be used without appreciable error. The times given in the Nautical Almanac on page IV are for the astronomical date. When the establishment is added to the local time of local transit, the result will be in astronomical time; the corre- sponding civil time may be a day later, so if the time of high water is desired for a given civil date, and it is found that the sum of the establishment plus the local time of local tran- 678 NAUTICAL ASTRONOMY sit will be greater than 12 hours, take out the time of transit for the preceding date, since in this case the astronomical date is one day less than the civil date, and, when the time is converted into civil time, the civil date of the tide in question will result. Ex. 22Jf. Find the times of high and low waters occurring on January 16, a. m., 1905, at Portland, Me. Latitude 43 39' 28" N., longitude 70 15' 18" W. In this example, the sum of the time of moon's transit and the hmi-tidal interval is greater than 12 hours; therefore, take out the time of transit for January 15. h m G. M. T. of Greenwich transit Jan. 15, 7 24.9 H. D. l m .94 Corr. for Long. 4 h .68 W + 9.08 Long. W 4 h .68 Corr. + 9 m .08 L. M. T. of local transit Jan. 15, 7 33.98 H. W. luni-tidal Int. Appx. IV, Bowditch 11 06 L. M. T. of high water Jan. 15, 18 39.98 or Jan. 16, 6 39.98 a. m. L. M. T. of local transit Jan. 15, 7 33.98 L. W. luni-tidal Int. Appx. IV, Bowditch 451 L. M. T. of low water Jan. 15, 12 24.98 or Jan. 16, 24.98 a. m. Ex. 225. Find the time of the higher high tide that occurs next after noon of April 9, 1905, at Port Adelaide. Lati- tude 34 50' 25" S., longitude 138 26' 58" E. On April 9, 1905, the moon's declination is 1ST.; therefore, the higher high tide occurs after the lower transit of the moon, April 9, the time of which may he found as helow. G. M. T. of Gr. npper transit, April, 9 3 30.7 Mean H. D. 2 m .06 G. M. T. of Gr. upper transit, 10 4 20.3 Long. E 9 .23 2 | 19 7 51 Corr. 19701 G. M. T. of Gr. lower transit, April, 9 15 55.5 Corr. for Long. 9 23 m E 19.01 L. M. T. of local lower transit, April, 9 15 36.49 H. W. Lun. Int. Appx. IV,. Bowditch 4 04 L. M. T. of higher H. W., April 9, 19 40.49 or April 10, 7 40.49 a, m. TIME OF HIGH WATER 679 Ex. 226. Find the time of high water occurring next after noon on April 6, 1905, at Hong Kong. Latitude 22 16' 23" N., longitude 114 10' 02". E. Is this the higher or lower high tide of the day? h m G. M. T. of Greenwich transit, April 6, 1 13.2 H. D. l m .81 Corr. for long. 7 h . 61 East, - 13.77 Long. E 7 h . 61 Corr. 13 m .77 L. M. T. of local transit, April 6, 59.43 IT. W. luni-tidal Int. Appx. IV, Bowditch 9 20 L. M. T. of high water, April 6, 10 19.43 or April 6, 10 19.43 p. m. Latitude and declination being of the same name, this is the higher of the two daily high, waters. CHAPTEE XXIII. IDENTIFICATION OF HEAVENLY BODIES. 318. A navigator is fortunately not dependent on observa- tions of the sun either in locating the position of his ship or in determining the error of his compass. Planets and fixed stars, Vhen visible and favorably situated, are available for that purpose. Owing to the large number of stars of the first two magnitudes of differing right ascensions, it is probable that several may be found favorably situated for cross lines at all hours during twilight, or when the horizon may be made sufficiently distinct by moonlight. In these days of fast ocean steamships, stellar observations are essential and an observer with some practice and a clear horizon should get good re- sults from sights for position; such sights should be avoided, however, when the horizon is uncertain. When working for compass error, it is only necessary to see and to know the star, and to obtain its compass bearing, it being immaterial whether the horizon is clear or clouded. The method of ob- servation as well as the methods of working stellar sights have been fully explained. 319. Distinction between planets and fixed stars. The planets change their positions in the heavens not only with reference to each other but to the fixed stars; they have a perceptible disc and shine with a steady light; fixed stars do not change their positions relative to other fixed stars, and they appear in the most powerful glasses simply as luminous points shining with a twinkling light. GROUPING OF STARS CS1 320. Distinction between planets. The only planets that need be considered by the navigator are Jupiter, Venus, Mars, and Saturn. Both Jupiter and Venus are larger and brighter than Sirius ; when only one is visible, it may easily be taken for the other, but a comparison of the estimated right ascen- sion of the visible planet with the tabulated right ascensions of Jupiter and Venus will decide which it is. When both are visible, (1) the one to the eastward will be the one of greater right ascension as indicated by the tabulated right ascensions of the Nautical Almanac; (2) the motion of Venus in right ascension is more rapid than that of Jupiter and in consequence its change of position among the fixed stars is more noticeable; (3) as Venus is an inferior planet with a maximum elongation of about 47, it is easily seen that as morning or evening star, it cannot be visible before sunrise or after sunset more than three hours and eight minutes, whereas Jupiter may be visible at any hour of night depending on its elongation which, as with all superior planets, varies from to 180. Mars may be recognized by looking up its right ascension and declination ; it is larger than a fixed star, and shines with a reddish color, which has caused it to be known as the " Ruddy Planet." Saturn, owing to its great distance, changes its relative position among the stars very slowly, and by the naked eye may be taken for a fixed star. Estimating its right ascension, or the use of good night glasses, will distinguish it from fixed stars. The three planets first mentioned are more frequently used in practical navigation. 321. Grouping and classification of stars. From remote ages stars have been grouped in constellations, those of each constellation, as a rule, being arranged in order of brightness and distinguished by having Greek or Roman letters prefixed to the name of the constellation, or by numerals when both G82 NAUTICAL ASTRONOMY alphabets have been exhausted, the brightest star of the group being represented by the letter a. Specific names are usually given to the most conspicuous stars. Stars are found in nautical almanacs, arranged according to their right ascensions and classified by magnitudes or brightness, the lowest magnitude assigned to stars just visible to the naked eye being the sixth. Assigning to sixth magni- tude stars an average brightness of unity, and regarding the stars of one magnitude about 2J times as bright as those of the lower magnitude, the average brightness of first magnitude stars should be 100. . Of course, there are marked deviations from this rule, the most notable exception being Sirius, which is perhaps 500 times as bright as a star of the sixth magnitude. 322. Navigational stars. The twenty brightest stars: a Canis Majoris (Sirius), a Argus (Canopus), a Aurigae (Ca- pella), a Bootis (Arcturus), a 2 Centauri, a Lyrse (Vega), (3 Orionis (Eigel), a Eridani (Achernar), a Canis Minoris (Procyon), {3 Centauri, a Aquilae (Altair), a Crucis, a Orionis (Betelgeux), a Tauri (Aldebaran), a Virginis (Spica), a Scorpii (Antares), ft Geminorum (Pollux), a Piscis Aus- tralis (Fomalhaut), a Leonis (Regulus), a Cygni (Deneb), and perhaps a dozen more may be classed as navigational stars, and every navigator should be able to recognize these and to select the ones most favorably situated for his purposes. To do so, it is useless to make a study of the constellations based on the fanciful grouping of stars by the ancients; it is only necessary to know (1) one conspicuous constellation in the northern heavens about which to group stars of North declination; (2) one in the region of the equinoctial leading to a knowledge of -others in the same region, to some one of which, stars of either North or South declination up to certain limits may be referred; (3) one in the southern hemisphere that may assist in locating the stars adjacent to the South celestial pole. POINTS OF REFERENCE G83 323. Constellations of reference. The constellations rec- ommended for obvious reasons in carrying out the above plan are (1) Ursa Major or "The Dipper" ; (2) Orion; (3) the Southern Cross. The student having made himself familiar with the visible stars of these constellations, and having learned certain bright stars near them, should trace out others in one of three ways : (1) by bearings and angular distances; '(2) by prolonging a line (straight or curved) passing through two known stars till at a certain approximate dis- tance it may pass through a required star; (3) by the geometrical figures, which in many cases, three or more bright stars form with each other. The first method is unsatisfactory as the bearing of one star from another is a great circle bearing and should be noted when the known star is at its upper culmination and as near the zenith as possible conditions seldom governing. An in- spection of star maps on the Mercator projection would only confuse the student as the bearings there shown are not great circle bearings. The second and third methods in connection with Plates VI to IX will perhaps be found the best and most expeditious methods for indentifying stars when the surrounding heavens are visible. 324. Description of Plate VI. The plate shows the princi- pal stars in the northern hemisphere whose declination ex- ceeds 30. The Eoman numerals on the margin show the meridians of right ascension at intervals of one hour. As the right ascension of the meridian is the L. S. T., if the observer faces the North and holds the plate so that the numeral which represents the L. S. T. at the time of observation is upper- most, the stars^ in the upper part of the plate will be shown in the same relative positions as they appear in the heavens. If the observer faces the North and holds the plate so that PLATE VI. 1 T\ ^ \\ $ ^ . X Polaris'^ O Aurigae or Capella A J^ ffLyrae^f or Vega*- / THE PRINCIPAL STARS AROUND THE NORTH CELESTIAL POLE or A DECLINATION GREATER THAN 30 N. STARS IN PLATE YI 685 the name of a month found in the margin is uppermost, the plate will show the visible heavens around the pole as they ap- pear about 8.30 p. m. in that particular month; the number of stars in the lower part of plate cut off by the horizon depend- ing on the latitude of the observer. Ursa Major, commonly called the " Dipper " from its shape, one of the brightest and most conspicuous of the northern constellations, consists of seven principal stars. Beginning with the edge of the bowl they are (a) Dubhe, (ft) Merak, (y) Megrez, (8) Phecda, (e) Alioth, () Mizar, (^ Benet- nasch. The first two (a and ft) are the brightest and, point- ing to the pole star (Polaris), are known as the pointers. Polaris is the principal star of Ursa Minor which appears also in the shape of a smaller dipper, Polaris being in the extremity of the handle. Cassiopeia. About the same distance from Polaris as the "Dipper," but on the opposite side, is Cassiopeia's chair, whose five principal stars appear in the form of the letter M or W, according to the position of the constellation in its diurnal path. ft Cassiopeia. A line from y Ursse Majoris through Polaris, produced about 30, leads to (3 Cassiopeia. a Cassiopeia. A line from 8 Ursse Majoris through Polaris leads to a Cassiopeia called Schedir, the farthest one of the chair from the pole star. Square of Pegasus. A line from the pointers through Po- laris, produced beyond Cassiopeia, leads first to ft Pegasi (Scheat), then to a Pegasi (Markab), two stars in a notice- able figure resembling a square ; the other two being y Pegasi (Algenib) and a Andromedse (Alpheratz), the latter nearer the pole (Plates YI and YIII). a Lyrse or Vega. A line from y passing between 8 and Ursse Majoris leads to Yega, a very bright star of a decided 686 NAUTICAL ASTRONOMY blue tint, which is attended by five other stars, making, with Vega, two triangles. a Cygni or Deneb. A line from y through 8 Ursae Majoris extended passes between Vega and Deneb. Also a line from Algenib through Scheat (Plate VIII), continued to nearly twice its distance, leads to Deneb. a Aquilae or Altair. A line from Polaris midway between Vega and Deneb leads to Altair, which is further distinguished by having an attendant star each side of it and by proximity to the Dolphin which shows five stars, four of which form a small diamond (Plate VIII). Altair, Vega, and Deneb form a triangle nearly right angled at Vega (Plate VIII). a Aurigse or Capella. A line from y Ursa? Majoris passing between the pointers (a and ft Ursse Majoris) leads to Capella, a very bright star of a yellow tinge, attended by a small tri- angle of three stars to the southward of it called " the kids." A line from the middle star of Orion's belt through Orion's head and ft Tauri leads to Capella (Plate VII). Capella, Algol, and Aldebaran form an equilateral triangle (Plate VII). a Bootis or Arcturus. A line from Dubhe passing between y and 8 Ursse Majoris leads to Arcturus, and the handle of the dipper curves toward it. Arcturus is a very bright star with a reddish tint, is at- tended by a small triangle of three stars, is as far from the pointers on one side as Capella is on the opposite side; it forms bold triangles with Spica and Eegulus, also with Spica and Antares, both triangles nearly right angled at Spica (Plate VIII). a and ft Geminorum Castor and Pollux. A line from 8 Ursae Majoris passing between the pointers leads to Castor and Pollux, which are about as much one side of the Dipper as the Northern Crown is the other side. A line from the STARS IN PLATE VII 687 middle star of Orion's belt (Plate VII) through Betelgeux leads to Castor, which shines with a greenish light. Betel- geux, Procyon, and Pollux (Plate VII) form a triangle, right angled at Procyon. a Leonis or Regulus. A line from 8 Ursse Majoris passing between pi and y Ursse Majoris leads to Eegulus. This is a bright white star and, being in the handle of the so-called sickle or reaping hook, is a very prominent one. It forms a triangle with Spica and Arcturus, right angled at Spica (Plate VIII). 325. Description of Plates VII and VIII. The principal stars of a declination less than 45, North and South, are shown in these plates, those whose right ascensions are be- tween and XII hours in Plate VII, those of a right ascen- sion greater than XII hours in Plate VIII. If about 8.30 p. m. in a particular month, these plates be so held overhead that the feathered arrow points North whilst the Eoman numerals increase to the eastward, then the bright- est stars of the heavens near the meridian will be those stars in the plates whose right ascensions are* indicated by figures below the name of the given month. 326. Orion and the stars it leads to. Orion, the most beau- tiful constellation of the heavens, consists of a quadrilateral formed of three bright stars and one of lesser magnitude, the figure being longer in the North and South direction. The NE. star is a Orionis (Betelgeux) ; the NW., y Orionis (Bel- latrix) ; and the SW., /? Orionis (Rigel). Within the quadrilateral are three small stars, nearly equi- distant, and in a line nearly NW. and SE., forming what is known as Orion's belt. Nearly midway between the two northern stars and a little further to the northward are three small stars forming a triangle in the imaginary head of Orion. a Canis Majoris or Sirius. This, the brightest star of the heavens, shines with a scintillating white light. The three PLATE VII, a I I 2 I a PLATE VIII, 690 NAUTICAL ASTRONOMY stars of Orion's belt point southeastward to Sirius, which forms an equilateral triangle with Betelgeux and Procyon. Sirius, Rigel, and the triangle in Orion's head form a triangle right angled at Rigel. a Canis Minoris or Procyon. A line from Bellatrix through Betelgeux, curving to the southward and eastward, leads to Procyon, a star of a yellowish tint. A line from Arcturus through Denebola and Regulus leads to Procyon. a Tauri or Aldebaran. A line from Betelgeux through the three stars in Orion's head and extended to three times the distance leads to Aldebaran, which, shining with a decided reddish tint, is conspicuous as forming a V with four other stars. a Arietis or Hamel. A line drawn from Betelgeux through Aldebaran leads to Hamel, which may be known by two small stars southwestward of it. Hamel, Menkar, and a Tauri form a triangle nearly right angled at Menkar. /? Leonis or Denebola. A line from Procyon through Regu- lus leads to Denebola at a little over half the distance. For Regulus, see Art. 324. a Virginia or Spica. About 35 SE. from Denebola is Spica, a bright white star, which forms with Arcturus and Denebola an equilateral triangle. Four stars of the constel- lation Corvus form the shape of a " spanker," the gaff point- ing to Spica. a Scorpii or Antares. A line from the Dolphin through Altair leads to Antares which is a bright star of a decided reddish tinge, forming with adjacent stars the approximate figure of a hand glass, Antares at junction of glass and handle. It forms with a and (3 Librae a long triangle, a Libras being on a line between Spica and Antares. A line from Regulus through Spica extended to the same distance passes a little to the southward of Antares. A line from a 2 Crucis through (3 Centauri, produced three times its length, leads to Antares (Plate TX). PUTE IX. PRINCIPAL STARS AROUND THE SOUTH CELESTIAL POLE OT A DECLINATION GREATER THAN 30 S. G92 XAUTICAL ASTRONOMY a Piscis Australis or Fomalhaut. A line from Scheat through Markab extended about 45 leads to Fomalhaut, which forms with three other stars an irregular quadrilateral. It forms an equilateral triangle with a Pavonis and Achernar (Plate IX). The Pleiades. A line from midway between Algenib and Alpheratz through a Arietis, extended the same distance, leads to the Pleiades,, a remarkable cluster, of which six stars are visible to the naked eye. 327. Description of Plate IX. This plate shows the prin- cipal stars of the southern hemisphere whose declination ex- ceeds 30. What was said in Art. 324 about the Eoman numerals and najnes of months around the margin of Plate VI, apply to the numerals and months of this plate with this exception, that the observer faces the South. The Southern Cross. This is the most conspicuous constel- lation of the southern hemisphere, and is outlined by four bright stars; when the cross is above the pole, a 2 Crucis is the southernmost, /? Crucis the easternmost, y Crucis the north- ernmost, and 8 Crucis the westernmost star of the cross. As a line through a and /? Centauri points directly to the cross, those two stars are known as the pointers. a Argus or Canopus. This star is next to Sirius in bril- liancy, is midway betwen Eigel and the cross. Rigel, a Co- lumbae, Canopus, J3 Argus, and a 2 Crucis are at equal dis- tances apart in a very slightly curved line. a Eridani or Achernar. Is about midway between Canopus and Fomalhaut, forms an equilateral triangle with a Pavonis and Fomalhaut, also with {3 Argus and a Columbse. 328. In cloudy weather. In case the surrounding heavens are clouded and it is desired to ascertain the name of a single star that may be out, its altitude and azimuth having been taken, the name may be found in the Nautical Almanac, from its right ascension and declination obtained thus: NOTE. Having a star's observed h and Z the G. M. T. of observation, and observer's position, t and d may be found from Aquino's " Altitude and Azimuth Tables." The L. S. T. and t will give the R. A. which, with the declination, will identify the star. FINDING THE COORDINATES 693 (1) when the body is on the meridian, its right ascension is the L. S. T. of the instant of observation (Art. 173), and the declination may be found from the known latitude of the place and the measured true meridian altitude; (2) in case the body is not on the meridian, the approximate right ascension (the L. S. T. at transit) may be obtained from the local time of observation and the body's estimated hour angle, and the approximate declination from the known latitude and the estimated altitude at transit; (3) the star may be projected stereographically from its observed altitude and azimuth, and the right ascension and declination determined with sufficient accuracy to distinguish its name; (4) the coordinates may be obtained by the use of a celestial globe; (5) having deter- mined by observation the true altitude and azimuth of a heavenly body, its right ascension may be found by applying to the L. S. T. of observation the hour angle taken from the azimuth tables, as explained in Appendix C, and the declina- tion may then be found from the formula cos d = sin Z cos h cosec t* This formula will give the numerical value of the declination but will not determine the sign, about which, however, there should be no ambiguity except when the declination is very small. Having determined the numerical value of the decli- nation by computation, enter the tables with L, t and -\-d (that is of same name as latitude) and see if the azimuth found tabulated there agrees with that used in the computa- tion. If agreement is found, the declination is properly marked ; if not, the declination is negative and verification should be sought on that supposition. With the values of L, t and Z, an inspection of the tables in most cases will determine the value of the declination without the necessity of computation. When inspecting the Nautical Almanac or a star table in an effort to identify an observed heavenly body, through an agree- ment of tabulated coordinates with those obtained by any of the methods referred to above, the navigator should always consider the possibility of having observed a planet instead of a fixed star. *The use of this formula in this connection was first proposed by Lt.-Coradr. G. W. Logan, U. S. N., in The Proceedings of U. S. Naval Institute, No. 104. 694 NAUTICAL ASTRONOMY There are various graphic methods for determining the names of stars; those proposed by Admiral Sigsbee and Lt.- Comdr. Rust, IT. S. N., Mr. G. W. Littlehales of the Hydro- graphic Office, and Lt. Radler de Aquino of the Brazilian Navy, are among the best. For the details of Admiral Sigs- bee's method the student is referred to H. 0. chart No. 1560 ; for Comdr. Rust's method to The Proceedings of IT. S. Naval Institute Nos. 116, 123, and 124; for Mr. Littlehales' method to his admirable work " Altitude, Azimuth, and Geographical Position"; for Lt. Radler de Aquino's method to the nomo- gram explained in Appendix D. It is well, when navigating, to note at twilight the appoxi- mate bearings and altitudes of prominent stars whose names are known, whether desired for observations or not; then under circumstances above referred to, a single bright star peeping out from the clouds at a time when an observation is desired might be recognized from its approximate position, noted about the same time a night or two before, when the weather conditions were such as to make identification without question. Ex. 227. At sea, January 19, 1905, a. m., in latitude by D. R. 50 33' N. and longitude by D. R. 40 04' W., weather cloudy, a bright star was observed, through a break in the clouds, on the meridian bearing South ; star's sextant altitude 51 54' 10"; I. C. +3'; height of eye 36 feet; W. T. of obser- vation 2 h ll m 10 s ; C W 2 h 39 m '55 s ; chronometer slow on G. M. T. l m 10 s ; what was the name of the star ? h m a o I n in W. T. 2 11 10 * 1 8 \ 51 54 10 S . I. C. + 3 00 C-W 2 39 56 Corr. - 3 38 Dip - 5 53 C.C. + 1 10 ^ H 88 Ret. .-_OJ5 SAJ't 16 53 16 *'sz 88 09 88 N Corr. - 8 38 RA.M.0 194863.64 Latitude 60 83 00 N Corr. G. M. T. + 2 46.287 *'sd 12 23 32 N G. S. T. 12 43 54.927 Long. W. 2 40 16 10 03 38.927 An inspection of the mean place tables of the Nautical Almanac of 1905 shows the above star to have been a Leonis (Regulus) whose tabulated coordinates were: R. A. 10 b 03 m FINDING THE COORDINATES 695 18 S .838, dec. N. 12 25' 54'.'22. In practice, at sea, it will be unnecessary to correct the R. A. M. O. for the G. M. T., an approximate R. A. being generally sufficient for the identi- fication of the star. Ex. 228. April 5, 1905, about 7 h ll m 18 s ,. p. m., of local mean time, in latitude by D. R. 20 40' S. and longitude by D. R. 90 12' E., weather cloudy, observed a bright star through a break in the clouds; star's sextant altitude 25 58' 40"; I. C. +1'; height of eye 19 feet; star's bearing (p. s. c.) or Z N z=307, variation 8, ship heading East (p. s. c.), deviation -j-2. Required the hour angle and name of star ? h m s o / 11 in L. M. T. 7 11 18 *'s h a 25 58 40 I. C. +1 00 Long. E. 6 00 48 Corr. - 5 15 Dip - 4 16 R.M.o 06228.24 Corr. G. M. T. 11.581 G.S.T. 2 03 09.821 *'** "the ship's position may be any- where within a circle described from the determined position as a center with a radius of 2 miles, and courses should be shaped with this uncertainty in view. When desiring to lay a course, from a position determined at sea, to pass a danger, it would be prudent to multiply the assumed average error by a number (2, 3, or 4, according to circumstances), called a "coefficient of safety," and, con- sidering the result obtained as the limit of possible error, to describe a circle about the determined position with that limit as a radius; then to shape the course from a point on that side of the circle nearest the danger to be passed. The general principle embodied in the use of a " coefficient of safety " was more correctly applied in connection with Sum- ner lines as explained in Arts. 293 and 294, wherein a ship's position was shown to be somewhere within a parallelogram formed by drawing parallels on each side of each line of posi- tion and at such distances as to include errors of altitude, time, etc., which might be assumed by the navigator as prob- able under existing conditions. By using the parallelogram, the navigator is better enabled to see in which direction his position is the most in doubt (see Figs. 126, 127, 128). 336. The advisability of keeping landmarks in sight. Considering the uncertainty of positions at sea, it is prudent, when navigating coasts well charted, lighted, and buoyed, especially when there are outlying lightships, as in the case of the eastern coast of the United States, to make certain landmarks or lights in regular succession and at short inter- vals of time, being careful, however, to see that the ship is not set by currents into regions of possible danger. If by so doing a vessel is not taken too much out of the direct course to destination, it is always advisable, whenever 706 NAUTICAL ASTRONOMY possible, to sight a mark from which to take a fresh departure, whether making ready to close in with the land for the pur- pose of entering port, or to put out to sea in view of approach- ing fog or bad weather. In pilot waters, the ship's position should be located by observations of permanent landmarks if practicable, as buoys are frequently out of place and lightships are sometimes so, even when just replaced, on the station. In the vicinity of dangers, whether buoys are in sight or not, the use of the danger angle is advisable (see Arts. 118 and 119). In running a channel, where there are no well-defined land- marks, and as to which there may be some doubt, the ship may be steered through it by zig-zagging occasionally from side to side, keeping the lead going, and thus showing on which side of the channel the vessel may be and in which direction the course must be changed to find deeper water, care being taken, however, not to run into dangerously shoal water. In going in or out of port, try to pick up a range, ahead or astern as the case may be; and, as local currents may be uncertain, watch for any possible indications of their set and strength, such as the riding of buoys, the general heading of vessels at anchor, or the opening of the range on which the ship may be steering. A comparison of the courses and dis- tances sailed by compass and those made good as indicated by bearings will give the set and drift of the current. Before anchoring, the navigator should know not only the set of the tide but the maximum rise and fall to ensure having, at low tide, sufficient water under the bottom. When desiring to find an anchorage on two bearings, approach it upon that bearing which may be the most convenient one, reduce speed, and stop in sufficient time to let go the anchor when the ship is also upon the second bearing. CAUTION AS TO DATA 707 337. Disregarding the seconds of data when solving the astronomical triangle. The ship's position being subject to the errors enumerated in Art. 335, and hence uncertain, even under the most favorable conditions, some navigators believe themselves justified in using their data only to the nearest minute of arc in solving the astronomical triangle. If the errors due to such procedure were known to offset other errors, such theories would be tenable; but as it is equally probable that they would augment them, it seems advisable, in the absence of any definite knowledge as to the effect of neglect- ing the seconds, to exercise great care in obtaining data and then to use the values obtained, when working sights for either latitude or longitude. ros NAUTICAL ASTRONOMY ^ 32 S3 r5 ?P *$ poio6oiri-^ oocoeoc-ic-jc-Jc^rH I o wS5h?S?2cOrHOCO^?50^Sooof2S :* ffi CO 5>4r4r4rJ D O k r-(OCO>OOOC^CtOr-l.t~COC5t~lCOtHOOOCO'<*C.50 ( lOlO : * : 5!'*COeOe '* o ?S'~ lt ' coclin(: ' < = > *O*COt^Oo'OlOlO'*-*COCOC^C^C-lr-;rHrHTHr-;OOC50OO c> ci d cs d d d d d d d d d o* d d d d d d d d d d d d d o* oor-iiftOiTjcd'rf'*^eseoc^c<3frjc4ff4 X>- CO 1> CO iffl >O -* CO CO O O O O O O C5 O* O* O O C5 O C5 e a e a 8 8 fi s 8 I H r-i T-5 IH IH d d d d d d d d d d d d d d d d d d d d d d d d d d o' d d d d d d d d o' d d o* d d d d d d d d o' d d d d d d d d d d d d d d d d d o' d d OiHCOlOOOrH^OOC^COO^Oi^O^^C^^THCSCO^CQCit^lOO^O 5rHOO>0OOJ>t- rH rH rH IH d d d d d d o* o* d d d d d d o* o" o' d d d o' d d d MO^rojl>jWiOO^^CoScqC^MrHiHr-JrHOOO rHi-Hi-irHodo'ddddddddddddddddddddd r-i r-5 r-i d d d d d o' d d d d d d d o' d d d d d d d d d d 3SS&83fcg8888$S8S8aS3S8S3g8 r-i r-i r-i d d d d d d d d d d d d d d d d d d d d d d d o" d r-i r-5 r-i d d d d d o' o' d d d d d d d d d d d d d d d d d d r-i r-i d d d d d d d d d d d d d d d d d d d d d d d d d d o* d d d d d d d d d d d d d d d o' d d d o" d d d d 712 NAUTICAL ASTRONOMY 11 9 I H o '& * I 1 '" 5 w ung jo -q t^l>t~OOOOOOOOOiO5O5O>O5O5ClC5OOOOOOOOO S 5! 9 8 8 9 8 51 . S S S R . 8 * fi 8 8 8 CORRECTION OF SUN'S ALTITUDE 713 Q 3$ *3 QD 9$ t* O 06 O fft QQ 4 - i& 3B- M tt Q S L Qlft^OOtHi-HfHCt~J>J>>'OOoocoooceooooao 000OOOOOOOOOOOC5O5 d O r~ I C*J CO ^* S co eo rt CORRECTION OF SUN'S ALTITUDE 715 OOOOOOOOOOOOOOGOOO OGOOCOGQOOOQcOQQCQOQOQOQaOOQOQ o eo'to l01fliO OOOOSOSOSOi OlOlOsOJOJOSOOJ s g S S S R ooo OOOOO rH rH r-l rH rH ? s s O>O5O5OSO5O5OOSOiCSO5OSOOOOOOOOOO oooosioooi-Hiftoec^gocooosiHeoomosiooieo "- V. +7T |S 9- - a" " j" * 7tT a" " !- sss MA 1 1 1 r ' |" : Bo lot- 5^5^^ +++ 1 1 1 I 55 3" " 716 NAUTICAL ASTRONOMY ^ S 2- a l< 2 h | g " 2 K 3 K (MrHOO v OiOJOsoO5OSOOlO OOCi't^->*D5COO*~'*'rHM(MpOinOlOl-i " (Mrt-*COOSOSOOCOOOOOOOOOOOOO 4 ^!g53S^S5S3S8S^^S^SSgglS v O(Mcxjoooooocx)oooooooot^t^ 5 ommr-ie2ooOOOOOOOOOOOOt-t-t~t-t-^t^-t-t~ 1 9 S ft $ * 3 & g 8 ft S 8 9 $ 8 & 3 S 3 8 8 (MrHrH s S e)iHc> CORRECTION OF STAR'S ALTITUDE 710 OOOOOOCOOOOOCOCOGOOOOOCOQOCOOOQOGOt^-l^-l^t^-t^^ 333S3888SS38335SSg;23g8 ?$ 8 8 8 S 2 S 3 53 S3. 8 8 8 8 SI .53 rJ ?3 8 S 8 3 $ 58 8 8 720 NAUTICAL ASTRONOMY TABLE IV. USED FOB CALCULATION OF COEFFICIENTS B, C, D, AND E. PRODUCTS OF ARCS MULTIPLIED BY THE SINES OF 15 RHUMBS. ARCS. s a Sin. 15 S 2 Sin. 30" S 3 Sin. 45* S 4 Sin. 60 S 5 Sin. 75 ARCS. o / O 1 O 1 o / o / O f o / 010 3 5 7 9 010 010 020 5 010 014 017 019 020 030 8 015 021 026 029 030 040 010 020 028 035 039 040 050 013 025 035 043 . 048 050 1 016 030 042 *052 058 1 1 10 018 035 049 1 1 1 8 1 10 1 20 021 040 057 1 9 117 1 20 130 023 045 1 4 118 127 1 30 1 40 026 050 111 127 137 1 40 1 50 028 055 118 135 146 1 50 2 031 1 125 144 156 2 210 034 1 5 132 153 2 6 210 220 036 110 139 2 1 215 220 230 039 115 146 210 225 230 240 041 120 153 219 235 240 250 044 125 2 227 244 250 3 047 130 2 7 236 254 3 310 049 135 214 245 304 310 320 052 140 221 253 313 320 330 054 145 229 3 2 323 330 340 057 150 236 311 333 340 350 1 155 243 319 342 350 4 1 2 2 250 328 352 4 410 1 5 2 5 257 337 4 1 410 420 1 7 210 3 4 345 4 11 420 430 110 215 311 354 421 430 440 112 220 318 4 2 430 440 450 115 225 325 411 4 40 450 PRODUCTS or ARCS BY SINES 721 TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). ARCS. Si Sin. 15 S 2 Sin. 30 S 3 Sin. 45 S 4 Sin. 60 S 5 Sin. 75 ARCS. / f , / / O 1 o / 5 118 230 332 420 450 5 510 120 235 339 428 459 510 520 1 23 240 346 437 5 9 520 530 125 245 353 446 519 530 540 128 2 50 4 4 54 528 540 550 131 255 4 7 5 3 538 550 6 133 3 4 15 512 548 6 610 136 3 5 422 520 557 610 620 138 310 429 529 6 7 620 630 141 315 436 538 617 630 640 1 44 320 443 546 626 640 650 146 325 450 555 636 650 7 149 330 457 6 4 646 7 6 710 151 335 5 4 612 655 710 720 154 340 511 621 7 5 720 730 1 56 345 518 630 715 730 740 159 350 525 638 724 740 750 2 2 355 532 647 734 750 8 2 4 4 539 656 744 8 810 2 7 4 5 546 7 4 753 8 10 820 2 9 410 554 713 8 3 820 830 212 415 6 1 722 813 830 840 215 420 6 8 730 822 840 850 217 425 615 739 832 850 9 220 430 622 748 842 9 910 222 435 629 756 851 910 920 225 440 636 8 5 9 1 920 930 228 445 643 814 911 930 940 230 450 650 822 920 940 950 233 455 657 831 930 950 722 NAUTICAL ASTRONOMY TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). ARCS. B, Sin. 15 S 2 Sin. 30 S 3 Sin. 45 S 4 " Sin. 60 S 5 Sin. 75 ARCS. / o / / o / 1 t O 1 10 235 5 7 4 840 940 10 1010 238 5 5 711 848 949 1010 1020 240 510 718 857 959 1020 1030 243 515 725 9 6 10 9 1030 1040 246 520 733 914 1018 1040 10 5C 248 525 740 923 1028 1050 11 251 530 747 932 1038 11 11 10 253 535 754 940 1047 11 10 11 20 256 540 8 1 949 1057 11 20 1130 259 545 8 8 958 11 6 11 30 11 40 3 1 550 815 10 6 1116 11 40 1150 3 4 555 822 1015 1126 11 50 12 3 6 6 829 1024 1135 12 1210 3 9 6 5 836 1032 1145 1210 1220 312 610 843 1041 1155 1220 1230 314 615 850 1050 12 4 1230 1240 317 620 857 1058 1214 1240 1250 319 625 9 4 11 7 1224 1250 13 322 630 912 1116 1233 13 1310 324 635 919 11 24 1243 13 10 1320 327 6 40 926 1133 1253 1320 1330 330 645 933 1141 13 2 1330 1340 332 650 940 1150 1312 1340 1350 335 655 947 1159 1322 1350 14 337 7 954 12 7 1331 14 1410 340 7 5 10 1 1216 1341 1410 1420 343 710 10 8 1225 1351 1420 1430 345 715 10 15 1233 14 1430 1440 348 7 20 1022 1242 1410 1440 1450 350 725 1029 1251 1420 1450 PRODUCTS OF ARCS BY SINES TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). 723 ARCS. Si Sin. 15' S 2 Sin. 30 S 3 Sin. 45 S 4 Sin. 60 S 5 Sin. 75 ARCS. / O f o / o r o / o / / 15 353 730 1036 1259 1429 15 1510 356 735 1043 13 8 1439 1510 1520 358 740 1051 1317 1449 1520 1530 4 ] 745 1058 1325 1458 1530 1540 4 3 750 11 5 1334 15 8 1540 1550 4 6 755 1112 1343 1518 1550 16 4 8 8 1119 1351 1527 16 1610 411 8 5 1126 14 1537 1610 1620 414 810 1133 14 9 1547 1620 1630 416 815 1140 1417 1556 1630 1640 419 820 1147 1426 16 6 1640 1650 421 825 1154 1435 1616 1650 17 424 830 12 1 1443 1625 17 1710 427 835 12 8 1452 1635 1710 1720 429 840 1215 15 1 1645 1720 1730 432 845 1222 15 9 1654 1730 1740 434 850 1230 1518 17 4 1740 1750 437 855 1237 1527 1714 1750 18 440 9 1244 1535 1723 18 1810 442 9 5 1251 1544 1733 1810 1820 445 910 1258 1553 1743 1820 1830 447 915 13 5 16 1 1752 1830 1840 450 920 1312 1610 18 2 1840 1850 452 925 1319 1619 1812 1850 19 455 930 1326 1627 1821 19 1910 458 935 1333 1636 1831 1910 1920 5 940 1340 1645 1840 1920 1930 5 3 945 1347 1653 1850 1930 1940 5 5 950 1354 17 2 19 1940 1950 5 8 9-55 14 1 1711 19 9 1950 724 NAUTICAL ASTRONOMY TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). ARCS. fifi Sin. 15 S 2 Sin. 30 S 3 Sin. 45' S 4 Sin. 60 s, . Sin. 75 ARCS. , 1 / f , / / 20 511 10 14 9 1719 1919 20 2010 513 10 5 1416 1728 1929 2010 2020 516 1010 1423 1737 1938 2020 2030 518 1015 1430 1745 1948 2030 2040 521 1020 1437 1754 1958 2040 2050 524 1025 1444 18 3 20 7 2050 21 526 1030 1451 1811 2017 21 21 10 529 1035 1458 1820 2027 21 10 21 20 531 1040 15 5 1829 2036 21 20 21 30 534 1045 1512 1837 2046 21 30 21 40 536 1050 1519 1846 2056 2140 21 50 539 1055 1526 18 54 21 5 21 50 22 542 11 1533 19 3 2115 22 2210 544 11 5 1540 1912 2125 2210 2220 547 1110 1548 1920 2134 2220 2230 549 1115 1555 1929 2144 2230 2240 552 1120 16 2 1938 2154 2240 2250 555 1125 16 9 1946 22 3 2250 23 557 1130 1616 1955 2213 23 2310 6 1135 1623 20 4 2223 2310 2320 6 2 1140 1630 2012 2232 2320 2330 6 5 1145 1637 2021 2242 2330 2340 6 8 1150 1644 2030 2252 2340 2350 610 1155 1651 2038 23 1 2350 24 613 12 1658 2047 2311 24 2410 615 12 5 17 5 2056 2321 2410 2420 618 1210 1712 21 4 2330 2420 2430 620 1215 1719 2113 2340 2430 2440 623 1220 1727 2122 2350 2440 2450 626 1225 1734 2130 2359 2450 PRODUCTS OF ARCS BY SINES TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). 725 AECS. B, Sin. 15 S 2 Sin. 30 S 3 Sin. 45 S 4 - Sin. 60 S 5 Sin. 75 AECS. / / 1 t / O 1 - / 25 628 1230 1741 2139 24 9 25 2510 631 1235 1748 2148 2418 2510 2520 633 1240 1755 2156 2428 2520 2530 636 1245 18 2 22 5 2438 2530 2540 639 1250 18 9 2214 2448 2540 2550 641 1255 1816 2222 2457 2550 26 644 13 1823 2231 25 7 26 2610 646 13 5 1830 2240 2517 2610 2620 649 1310 1837 2248 2526 2620 2630 652 1315 1844 2257 2536 2630 2640 654 1320 1851 23 6 2545 2640 2650 657 1325 1858 2314 2555 2650 27 659 1330 19 6 2323 26 5 27 2710 7 2 1335 1913 2332 2614 2710 2720 7 4 1340 1920 2340 2624 2720 2730 7 7 1345 1927 2349 2634 2730 2740 710 1350 1934 2358 2643 2740 2750 712 1355 1941 24 6 2653 2750 28 715 14 1948 2415 27 3 28 2810 717 14 5 1955 2424 2712 2810 2820 720 1410 20 2 2432 2722 2820 2830 723 1415 20 9 2441 2732 2830 2840 725 1420 2016 2450 2741 2840 2850 728 1425 2023 2458 2751 2850 29 730 1430 2030 25 7 28 1 29 2910 733 1435 2037 2516 2810 2910 2920 736 1440 2045 2524 2820 2920 2930 738 1445 2052 2533 2830 2930 2940 741 1450 2059 2542 2839 2940 2950 743 1455 21 6 2550 2849 2950 726 NAUTICAL ASTRONOMY TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). ARCS. Si Sin. 15 S 2 Sin. 30 S a Sin. 45 S 4 Sin. 60' S 5 Sin. 75" ARCS. / 1 / / / / , 30 746 15 2113 2559 2859 30 3010 748 15 5 2120 26 8 29 8 3010 3020 751 1510 2127 2616 2918 3020 3030 754 1515 2134 2625 2928 3030 3040 756 1520 2141 2633 2937 3040 3050 759 1525 2148 2642 2947 3050 31 8 1 1530 2155 2651 2957 31 31 10 8 4 1535 22 2 2659 30 6 31 10 3120 8 7 1540 22 9 27 8 3016 31 20 31 30 8 9 1545 2216 2717 3026 31 30 3140 812 1550 2224 2725 3035 31 40 31 50 814 1555 2231 2734 3045 31 50 32 817 16 2238 2743 3055 32 3210 820 16 5 2245 2751 31 4 3210 3220 822 1610 2252 28 3114 3220 3230 825 1615 2259 28 9 3124 3230 3240 827 1620 23 6 2817 3133 3240 3250 830 1625 2313 2826 3143 3250 33 832 1630 2320 2835 3153 33 3310 835 1635 2327 2843 32 2 3310 3320 838 1640 2334 2852 3212 3320 3330 840 1645 2341 29 1 3222 3330 3340 843 1650 2348 29 9 3231 3340 3350 845 1655 2355 2918 3241 3350 34 848 17 24 3 2927 3250 34 3410 851 17 5 2410 2935 33 3410 3420 853 1710 2417 2944 3310 3420 3430 856 1715 2424 2953 3319 3430 3440 858 1720 2431 30 1 3329 3440 3450 9 1 1725 2438 3010 3339 3450 PRODUCTS OF ARCS BY SINES TABLE IV. Products of Arcs Multiplied by the Sines of 15 Rhumbs. (continued). 727 ARCS. S t Sin. 15 S 2 Sin. 30 S 3 Sin. 45 S 4 Sin. 60 S 5 Sin. 75 ARCS. / / o r / / 1 O 1 35 9 4 1730 2445 3019 3348 35 3510 9 6 1735 24 52 3027 3358 3510 3520 9 9 1740 2459 3036 34 8 3520 3530 911 1745 25 6 3045 3417 3530 3540 914 1750 2513 3053 3427 3540 3550 916 1755 2520 31 2 3437 3550 36 919 18 2527 3111 3446 36 3610 922 18 5 2534 3119 3456 3610 3620 924 1810 2541 3128 35 6 3620 3630 927 1815 2549 3137 3515 3630 3640 929 1820 2556 3145 3525 3640 3650 932 1825 26 3 3154 3535 3650 37 935 1830 2610 32 3 3544 37 3710 937 1835 2617 3211 3554 3710 3720 940 1840 2624 3220 36 4 3720 3730 942 1845 2631 3229 3613 3730 3740 945 1850 2638 3237 3623 3740 3750 948 1855 2645 3246 3633 3750 38 950 19 2652 3255 3642 38 38 10 953 19 5 2659 33 3 3652 3810 3820 955 1910 27 6 3312 37 2 3820 3830 958 1915 2713 3321 3711 3830 3840 10 1920 2720 3329 3721 3840 3850 10 3 1925 2728 3338 3731 3850 J 28 NAUTICAL ASTRONOMY TABLE V. DISTANCES PROM COMPASS AT WHICH QTTADRANTAL CORRECTORS SHOULD BE PLACED FOR VALUE OF D. FOB BINNACLES OF TYPE VI. Distance from V il 1 III/ UJ. U. Compass on Graduated Quadrantal Arms. For 7-Inch Spheres. For 9-Inch Spheres. For Filled Chain Boxes. Inches. o / t o t 11 12 00 21 15 11.5 9 15 19 00 10 30 12 7 45 17 00 8 45 12.5 6 45 14 15 7 30 13 5 45 12 00 6 15 13.5 4 45 10 00 5 15 14 4 00 8 45 4 20 14.5 3 30 7 30 4 00 15 3 00 To find D when $> is given, use formula : D = 2) X 67.3. EXTRACTS FROM N. A 1905 729 l P'-'E H ^ O H q^ jo ?30 NAUTICAL ASTRONOMY C OS O (M OS O O O O ,-( O-i I> O l> CO CO (M CO O O CO CO ' t od i> oo os rH I> 05 CO CO CO 10 OTHrHCO rH T^CO-^ O OS 00 00 t- JO O CO i-i rH W5 r* T* CO O5 T-Hr-li-lOOOOO -* OS CO i- 00 W^HT-I ^rH CO 05 rH CO O 1C t- CO * CO * t- rH 05 CXJ * ^ CO* q^uow apparent n t south decl ed the same as tha eclination indicates ter for mean noon fixed to the hourl I"I S'S o 0)0) EXTRACTS FROM N". A, 1905 731 732 NAUTICAL ASTRONOMY cl ^(MrHOCOTH-^Wr-IW S^ W CO CO C5 05 TH rH TH O O O fr- rH CO SO 00 OS t-i rH O TI r-l(McO^T-(C^Ttl O O rH TH rH rH rH Ol O5 O? I! si ' S3 a a J| 21 8 S^d 20 .3 g^3 s fl >> OT3 II 5; S.^2 OQ S s a) CD d ^^H r "a O 5^3 ft.S . O 5W TH O O Cl'-H CT>THTH T HO5O?OCOCO'*i OS Oi O5 OS O> O> O> OS OS ,r2 S P? O ^ COOCOrHOl Ttl>O5t>CO ccjioOSWJ>O5Tj5oJC TflCO OO5 oq^ jo N a S -2 a 3 h' a S EXTRACTS FROM N. A. 1905 733 734 NAUTICAL ASX-RONOMY H II Meridian o Greenwich O M s'l S'w fir-, TH ^- o o r- t --i j> co as i> ++++++++I 1 1 1 1 D O W r C5 !- Oi 00 CO Co l> J> CO 5 i EXTRACTS FROM N. A. 1905 "35 CO O? 5D r4 i (' i-H T I O CO CO OS * O> t- * CO t- OQ ^ ^ ^ fc ^ ^ OQ CO 02 03 CO * rt^ CO O> CO CO * -^ <* OT c? II - O * CO OiJ 0000 1* + ^ CO CO CO CO CO x %n 1 ^ OS OS OS OS CO CO CO CO a P g p S B J | II OS CXI CO 10 02 OS OS CO I- O Si H f 05 OS OS *" O CN! GO 1C 3, O i CO CO CO J> t~ W -s S CO s -1 OS OS < ^a OQ *>g PH 1 r-i O O O .s g i 1C c o * T3 "^ ^ 5 H 0} CO CO rH rH 00. r-i rH a gg ^ddr-Id T-H CO rH cdd CM CO *j H S ^ *i Pi OSOOlOOlCSWf-41- d .2 . -1 * SO CO -*^ r^ H 5- CO T*H 1C T*1 H 00 rH rH rH iO iO *O AC O 1O O "tj *& o o o o & , 3 o + ^ ^ ,44 ^ 1 o f -ii^^^i^^^rH^^ 88 - 10 o ic ic PH a ft d 03 s |f a 2 00 GO' os' d rH ss ^d d o' d la - la 3 i s ^ i o3 ft K-J <} ^ ^ EXTRACTS FROM .N. A. 1905 73' d d r-" 73 CO r+t _o o TH T- t- X If < ^ 1C "* CO' CO' rti |5 OS i Co' I-H CO Oi' CO CO 73 O2> "^ CO ^Oi 73 TH OJ '3 1 Oi O5 O O5 Oi TH rH O3 rH rH 5 ^H GO' Oi' TH TH TH rH CM 03 8 < cd *T3 10 ^ 1C* Tf rH rH "cd ^ lH i 73 CO' CO CO q M 1O 1C *C 1C jH "^ o o o o d o r + 73 73 03 73 r 1 ^ o o o o d 1 d a c o a .a'fn || f- GO "*! CO HjSi Oi CO rH o' o' o' d 73 73 73 73 a d +3 O bed t rH Oi CO 1C ajOi Oi o 73' O3* CO* CO i ^ | fl CO 'd ' ? gj 1C C CO CO 1 ^"iii S | i l! i ^ "C 'E 'E fl d o & 73 1C CO CO t- rj* Oi .2 CO 73 r^ "eS ^ ^^ J> Oi rH yi 00 CO* t^ "c^5 id - *' CO 73 CO* CJ "r^ 35 T t ^0000 5 o 35 | - 1C 1C 1C 1C 1C 1C 1C ^COCOCOCOCOCOCO r f - CO CO 5O CO o o o o s d a c 1 4, .2 J3 03 ttg CO CO Oi O CO rH rH CO W CO CO Oi CO Oi J> 1C GO CO GO GO t~- J>- t'- I il = CO CO CO O o^O CO CO CO CO CO' *' rf* ._, rH rH T i GO 5 oa. O -^ft 5j fl cS 153 d d d d ^ o d Sp ^0 rH JtH rH 73 CO CO r ~ 73 03 W rH -5 r^ Anchorage for small vessels. ...................... .f> Rock above -water ........ , ........... . ............ .,.-..,! Rock underwater ........ ........ _ ................... , Rock atvash at any stage of the tide.* (*} Rock -whose position is doubtful ._^...{ P.D Rock "whose* existence is doubtful ......... D Jfo bottom, at 50 fathoms 'Currents, velocity .2 knots .............. ,..._ ; (Flood it Taiots,. \Ebbl knot, ....... 2d. hour flood current ol . ,., 3d,, hour ebb current ....... ... " ,,, * The period of a tidal current and its direc- tion is sometimes denoted by I Qr., If Qr., etc*, or lh.,11 h..etc., on the, arrow thus: 3d* Quarter, flood current. ................. _' Q' , 1st. quarter ebb current ...................... ' Q r , 2d. hour flood current ...................... , _ 11 n t 4th hour ebb current w r> Tidal Currents Cities (according to scale of chart; Towns and "Tillages (according to scale,) J^ -tt o Single- houses t. - * sa Churches ~ -~ + E 4 Fort or Battery. + f* Windmill.- ... _ _..j5 8 Observation spot,. e Single trees and groups... i Cemetery.. ; - ^ " or Fences and Sedges ... ......... -- Triangulation station Flagstaff. .......... Semaphore- or Signal station Storm, signal station Dam, ...... Fish weirs. In locating the, symbol for lighthouses and lightvessels on charts the Ught dot is placed in the- geographical position assigned to the lighthouse or Ughtvessel. When, on a Ughtvessel there are more lights than one. either on the. same or on different masts, the middle of the line joining the centers of the first and last dots is placed in the geographical position assigned to the lightvcssel. When there is no explanatory note refer- ring to the buoys on a chart their color is indicated by "words or their abbrevuitions placed near the buoy The ring at .the end of all buoys, and the middle of tfiA base line of the symbol for beacons is placed in the geographical position assignr ed. to the, buoy or beacon. U. S. Coast and Geodetic Survey Charts Hydrographic Signs PLATE XI. Lighthouse r ^ Mighthoue on. small scale chart OlcLlight tourer ^Beacon, lighted, & IBeajcan,-not,'lighte.d A Spindle (or atcJte ) \ add word. Spindle if frpace allows . JLightship & Wreck -* Anchorage ."* Covering and. -uncovering rock. JRocTt a-wash, at low wcvter * Sunken, roclz -v Salving Station + L.S.S. (T) signifies connection, yrii. telegraphic system. 27b bottom at, 2O fathoms , buoy V or add, word, -white or yellow as buoy...~ ^Horizontally striped, buoy Perpendicularly striped, buoy * iiH J3uf>ys with perch, and, $cju.c.re jBuoys -witsL perch, and, bcJi Xighted buoy... ^ , in plaee of o _, a.s \ 2xu)orini buoy. < Landmark . o^s Cupola,, Standpipe , etc Whirlpool Ode rip 1. Current, not tidal , drift in Ttnots a.3 2.O " f flood,,Ttrat au&rter . drift in, Jcn&ts, CLS 0.^ l.O 0.3 bb ... , ^ other-wise bO^e. flood.. U S. Coast and Geodetic Survey Charts Topographic Signs, PLATE XII. 'Xtyw Water Rocky Ledges Roclty Bluff Eroded JBanh Sand and STungl* Palms OaJ* JDecidiu>ua and Undergrowth Pint Cactt U. S. Coast and Geodetic Survey Charts Topographic Signs. PLATE XIII Cypress Swamp Groat RiceDikea AIHtchcs Wooded JtfarsK Submerged Marsh = ^J^ =? Curves of equal elevation and intermediate curves JScl Grass EXp PLATE XIV. GENERAL ABBREVIATIONS ON HYDROGRAPHIC OFFICE CHARTS AND U. S. COAST AND GEODETIC SURVEY CHARTS. ABBREVIATIONS FOR KINDS OF BOTTOM. M Mud. S Sand. G Gravel. Sh Shells. P Pebbles. Sp Specks. Cl Clay. St Stones. Co Coral. Oz.. ..Ooze. bk Black. wh White. rd Red. yl Yellow. gy Gray. bu Blue. dk Dark. It Light. gn Green. br Brown. ABBREVIATIONS NEAR BUOYS. hrd Hard. sft Soft. fne Fine. crs Coarse. brk Broken. Irg Large. sml Small. rky Rocky. stk Sticky. stf.. ..Stiff. U. S. C. and G. S. Charts. C Can. N Nun. S Spar. Hydrographic Office Charts. B, bk.... Black. Y, yl Yellow, W,wh White. Ch, chec Checkered. R, rd Red. H.S . . Horizontal stripes. G, gn Green. V.S Vertical stripes. F Fig... Fl Fls... Fixed. Flashing. Flash. Flashes. ABBREVIATIONS FOR LIGHTS. Rev Revolving. V. E Eclipses. W White. R. . . . Red. . . . Varied by. Sec Sector. Bn. . . .Beacon. L. W Low water. kn Knots. H. W High water. H. W. F. & C High water at full and change of moon. L. S. S . . Life-saving station. P. D Position doubtful. E. D . . .Existence doubtful. A wireless station is indicated by a point in a circle and the legend "Wireless Station." The information as to a submarine bell is covered by the legend "Submarine Bell" at the spot, and, in case of a lightship, in the table. PLATE XV. (1) Pole without the circle, curve is closed; as CC'C", DD'D". (2) Pole within the circle, curve is sinusoidal ; as AA'A". (3) Pole on the circle, curve is open, branches meeting at infinity, asymptotos parallel to meridians ; as BB'B". APPENDIX A. DESCRIPTION OF THE SUBMARINE-BELL SYSTEM. The equipment furnished by the submarine-bell company to a vessel or station depends on the purposes for which intended, and may therefore be considered under the two general heads given below. The sending apparatus, consisting of the bell and accessories, varies according to the use made of the system. When installed on light ships and tenders, the outfit consists of a bell mounted on a case containing the striking mechanism which is operated by compressed air supplied, through a hose, from air tanks; a davit, with chain and windlass, for raising or lowering the bell over the ship's side; and a code ringer for so controlling the strokes as to make automatically the code number of the light ship. For use near certain dangers or turning points, the bell is hung on a tripod standing on the bottom, and the clapper is actuated by powerful magnets energized by a current sent from a shore power-house, through an armor-protected submarine cable. When suspended from buoys the bell has a mechanism consist- ing of a combination of ratchets and pawls through whose agency a spring is compressed to a certain point by the wave-action on the buoy, and then automatically released, causing the clapper to strike the bell. \ The receiving apparatus installed in vessels consists of two small tanks placed in the forward part of the vessel, well below the water-line, one against the starboard side, one against the port side. In each tank are two microphones immersed in liquid which receive the sounds, when the sound-waves strike the ship's side. These sounds are transmitted to the pilot-house, or other location of the direction-indicator. Wires are run from the tanks to the battery box which supplies the power, thence to the direc- tion-indicator which is a small round metallic case fastened to the wall with telephone receivers hung on each side, and bearing on its face a switch for connecting either starboard or port microphones with the receivers; a dial indicates the one connected. T-iS APPENDIX B. COMPENSATION OF COMPASSES AT A SHIP-YARD BEFORE PROCEEDING TO SEA.* Before a vessel is sent to sea for the first time from a dock-yard or a navy-yard, the navigator should, by a preliminary compen- sation, so reduce the deviations and equalize the' directive force of the compasses that they may be used to steer by until he shall be able to compensate them regularly and obtain a residual curve. This preliminary compensation should always be made, when possible, from data obtaine'd from observations and vibrations on two headings, assuming 51 and ( as zero, and determining j$, (f, and by the method of Art. 95 if computation is made, as it should be whenever construction would give acute angles of inter- section, or by the method of Art. 113 if the dygogram is used. Then compensation -should be made as explained in Art. 110; or, by using the indications of the dygogram, neutralizing first the quadrantal force and then in the proper order as shown by the dygogram the semicircular forces. As each corrector is placed, the deviation should be reduced to the amount indicated by a dygogram of the remaining force or forces only. Provided the compasses are uninfluenced by the presence of other vessels, structures or masses of steel or iron, the required observations may be obtained: (1) When the vessel is in drydock and also, at an earlier or later time, alongside a dock or sea-wall. (2) When moored alongside a dock or sea-wall and the ship can be either winded or sprung out to a suitable heading (see Arts. 95, 110, and 113). Whilst X, 23, d, and 2) may all be determined when observa- tions on two headings are possible, it may sometimes happen that these can be made on only one heading. Under such circum- stances X and must be assumed, and, for reasons given in Art. 95, they may be assumed as those of a similarly situated compass on a similar ship. Make the necessary observations and vibrations referred to in Art. 94 before the quadrantal spheres are placed and determine 23 and & by computation, using equations (69a) and (70a); or by * Ser- " Tlio First Compensation of a Vessel's Compasses," issued by Bureau of Equipment, Navy Department, IUOG. APPENDICES 749 construction as explained in Art. 113a. Then having 23, (, and , proceed to compensate as directed in Art. 110. If these values are found by construction, Art. 113a, the dygogram may be used when compensating to indicate the deviation and the changes in deviation as each force is successively neutralized. SPECIAL PROCEDURE WHEN THE SHIP IS ON CERTAIN HEADINGS. Special procedure for compensation of the compass, ship head- ing on a cardinal point magnetic (assuming 21 and & as zero). Having obtained the coefficients 23, (, and S> by any of the methods explained in Chapter IV, it may happen, when compen- sation takes place on one heading, as contemplated in Art. 94, that the vessel is unavoidably heading with the keel-line north and south magnetic or east and west magnetic. Again, the vessel may be on the stocks or alongside a dock with the keel-line as indicated above and compass coefficients unknown, when it be- comes necessary to assume X and & as those of a compass simi- larly situated on a sister ship, to determine 93 and (, and then to compensate the compass (see Arts. 94 and 113a). In such cases compensation for two of the forces according to Art. 110 may appear indeterminate and special procedure becomes necessary. (1) Let the heading be assumed as magnetic north. On this head- ing neither 93 nor is compensated, PD' shortens to zero, D'B and BG respectively assume the positions PB' and B'C' and the needle takes the position 00', a position it should assume when under the influences only of 93 and (. If 93 were then compensated, the line PB' would shorten to zero and B'C' would assume the posl- APPENDICES 751 tion PC", the needle would take the direction 00", and the deviation would become POC" (which angle equals tan-i () and be due to ( alone uninfluenced by any other of the ship's forces; therefore, after <> has been eliminated by placing the spheres according to Table V, and only the semicircular forces remain acting, ship heading north magnetic, place the fore-and- aft corrector-magnets so as to alter the deviation from POC' to POC", then place the athwartship corrector-magnets to reduce the deviation from POC" to zero; the needle will take the direction of OP and the compass will be compensated. (2) Let the heading be assumed as magnetic east. In this case, 23 produces the deviation, the amount of which, however, is in- fluenced by the forces ( and 2) acting athwartship and in the magnetic meridian; therefore, the procedure should be as follows: Place quadrantal spheres as indicated above, neutralizing $>; by means of the athwartship magnets so alter the athwartship forces that the compass will indicate a deviation of tan-i 93 (angle 752 APPENDICES FOB', Figs. 151 and 152), thus neutralizing (; then, by means of magnets properly placed in the fore-and-aft carrier, make the compass indicate the heading east magnetic, eliminating the force 83 and completing the compensation of the compass. The steps above taken when the ship headed east magnetic may be illustrated by the dygograms (Figs. 151 and 152); both ft and ( being + in Fig. 151 and in Fig. 152. When S) has been neutralized PD shortens to zero, DB and BC respectively assume the positions PB' and B'C', the deviation be- comes POC', and the needle takes the direction 00'. When & has been neutralized, B'C' shortens to zero, the needle lies in direction OB', the deviation becomes POB' (which angle equals tan-i 93) and is due to 23 alone uninfluenced by any other of the ship's forces. If this value is reduced to zero by fore-and-aft corrector- magnets properly placed, the compass needle should take the direction OP and the compensation be effected. (3) With S> eliminated and ship heading east or west per com- pass. if the heading is such that after the quadrantal force has APPENDICES 753 been eliminated, the ship should be heading east per compass when the deviation is easterly, or west per compass when the deviation is westerly, the athwartship force ( will lie in the vertical plane through the compass needle, and, as shown by the dygograms (Figs. 153 and 154), the corresponding corrector- magnets, if used first, would have no apparent effect when com- pensating. However, knowing the deviation and compass head- ing, we may easily find the magnetic heading and the amounts of deviation produced respectively by the forces 23 and ( on that heading; after which, the compensation by the method of Art. 110 is very simple, for the force 'ft, being more nearly at right angles with the needle, should be eliminated first and then the force (. If not wishing to compute the deviation due to these forces, nor to apply the method of Art. 110, we may construct a dygo- gram and use it in compensating. The dygograms (Pigs. 153 and 154) show that if the force 93 should be compensated, PB would shorten to zero, the needle would assume the position OC", the deviation would become POC', and the remaining force ( 754 APPENDICES would be left at a favorable angle with the direction of the needle for its elimination. Therefore, place the fore-and-aft 153. FIG. 154. corrector-magnets at such a height as to change the deviation from FOB to POC', compensating the force 5; then place the APPENDICES 755 athwartship corrector-magnets at such a height as to reduce the deviation from POC' to zero, compensating the force (, and thereby effecting the complete compensation of the compass. (4) On a heading of no semicircular deviation. If the heading of the ship is such that, after the quadrantal force has been elimi- nated, no deviation is shown when S3 and ( are known to have appreciable values, then it is evident that the semicircular forces are neutralizing each other on that particular heading. This O FIG. 155. state of affairs is indicated by the dygogram (Fig. 155), the compass needle there lying in the meridian OP. The method employed in Art. 110 for the elimination of S3 and ( may be followed in this case and the compensation of the compass effected without any difficulty. However, if not wishing to compute the deviations due to S3 and (, the dygogram may be used in compensating as indicated below; at all events, it will serve the useful purpose, as it does in all cases when used, of indicating which force it is preferable to eliminate first. For the particular values of S3 and ( indicated by Fig. 155, it 756 APPENDICES is evident that it would be better to eliminate first the force (, shortening BC to zero, causing the needle to lie in the direction OB, and leaving the remaining force 23 at an angle with the direction of the needle more favorable for its elimination than would have been the angle for the elimination of ( had the force 93 been neutralized first. Therefore, place the athwartship magnet-correctors at such a height as to produce a deviation equal to the angle FOB, com- pensating the force (; then place the fore-and-aft corrector- magnets so as to reduce the deviation FOB to zero, eliminating the force S3, and completing the compensation of the compass. APPENDIX C. GENERAL USE OF AZIMUTH TABLES. By the azimuth tables issued to the navy the azimuth (Z) is found when the hour angle (t), the declination (d), and the latitude (L) are given; in other words, one angle of a spherical triangle may be found when two sides and the included angle are given. Therefore, these tables may be used to find the position angle (M ) which may be desired for use in Littlehales' method of equal altitudes (Art. 270), the hour angle (t) of an unidentified heavenly body whose true altitude and true azimuth are known (Art. 328), and the great circle course from one given place to another given place (Art. 135). Let Fig. 156 represent the astronomical triangle lettered as shown; then, having given f, L, and d, to find Z, we have from Napier's analogies Tan* (Z ^f)=cot$fsini (L d) sec* (L-fd) ^ Tan} (Z + M) = cot if cos (L d) cosec i (L + d) / (a) these being the formulae by which the azimuth tables, now issued to the navy by the navy department, were computed. (A) Having given L, d, and t, to find #, we have the following rule: Enter the azimuth tables, in the given latitude; at the intersection of the horizontal line through the given hour angle and the vertical column under the given declination will be found APPENDICES 757 the required true azimuth, estimated from the elevated pole and reckoned from to 180 towards the east or west as the body is east or west of the meridian (see Art. 221). H. .O. Publication No. 71, in which the latitude runs from to 61 and declination from to 23, each at an interval of 1, has a different table for north and south latitudes. H. 0. Pub. No. 120, in which the latitude runs from to 70 and declination from 24 to 70, each at an interval of 1, has but one table for both cases; but, when the latitude and declination are of a differ- ent name, the tables are to be entered with the supplement of the hour angle and the supplement of the tabulated azimuth is to be taken for the required true azimuth. (B) Given t, L, and d, to find the position angle M In this case we will have from Napier's analogies TanHJf #) cot 4* sin J (d. L) seci (d + L) j m Tani (M + Z) = cot|cos (d L) cosec $ (d + L) f Comparing formulae (b) with formulae (a) there is noted an interchange of M and Z, and of L and d, and it is evident that the tables may be used to find M, having given t, L, and d; but it must be remembered that by following a rule similar to that for taking out the azimuth in the cases when L and d are of a different- name, we shall obtain the supplement of the required M instead of M itself. It must also be remembered, in the case of an observed heavenly body, that the position angle M will be greater than 90 only when the declination is greater than the latitude and of the same name and when the body is observed between the point of maximum azimuth and the upper meridian, and that it will not be greater than 90 when L and d are of a different name. Therefore, the following rules should be fol- lowed: (1) When L and d are of the same name, enter the azimuth tables with * in the hour angle column, using the given declination as latitude and the given latitude as declination, and take out the value of M from the tabulated azimuths. (2) When using H. O. Pub. No. 71, L and d being of a different name, follow the above rule and M will be the supplement of the angle taken from the tabulated azimuths. (3) When using H. O. Pub. No. 120, L and d being of a different name, enter the tables with 12 h t in the hour angle column, using the decimation as lati- tude and latitude as declination, and the angle taken from the tabulated azimuths will itself be the position angle M. 758 APPENDICES The t referred to here is the body's hour angle from the upper meridian; when finding M for use in the solution of equal alti- tudes, it will be sufficiently accurate to consider the hour angle as half the elapsed time when the first observation is east of the meridian, and. as the supplement of half the elapsed time when the first observation is west of the meridian (see solution of Ex. 207, Art. 270). (C) Given Z, L, and ft, to find t. In this case we will have from Napier's analogies Tan A (t M) = cotZsini (L ft) sec (L + ft) ^ ,. TanJ (t + M) =cot*Zcos (L ft) cosec \ (L + ft)| Comparing formulae (c) with formulae (a) there is noted an interchange of Z and t and a substitution of ft for d; so to find a heavenly body's hour angle- (t), having given the latitude (L) of the ship, the unknown body's true altitude (ft) and its true azimuth (Z) estimated from the elevated pole, we have the fol- lowing rule: Convert the azimuth into time and consider it as an hour angle; enter the azimuth tables in the given latitude, with the azimuth used as an hour angle and the altitude used as declination, and take from the tabulated azimuths the body's hour angle expressed in arc (see Art. 328 and Ex. 228, \ (D) To find the great circle course between two places. Enter the tables in the given latitude of the place of departure, with the difference of longitude between the two places expressed as time in the hour angle column, and the latitude of destination in the declination column, and take from the tabulated azimuths the great circle course named from the elevated pole, towards east or west as the place of destination is to eastward or westward of place of departure (see page 276). When the difference of longitude between the two places is greater than 6 hours and the value is not found tabulated, as it may not be in H. O. Pub. No. 71, enter the tables in the given latitude of departure, with the supplement of the difference of longitude expressed as time in the hour angle column, and the latitude of destination with name changed in the declination column, then the supplement of the tabulated azimuth will be the great circle course to be marked as before directed. APPENDICES 759 APPENDIX D. SOLUTION OF THE ASTRONOMICAL TRIANGLE BY NOMOGRAPHY.* The nomogram constructed by Dr. Pesci of the Royal Italian Naval School is a diagram for the graphic solution of equations of the form tan a tan & = sin c by the ingenious method of aligned numbered points. It has been adapted by Lt. Radler de Aquino, Brazilian Navy, to the solution of the astronomical triangle through similar equations given below in group (a), which are of the form cot a cot & = cos c. * See " Proceedings of the U. S. Naval Institute," No. 126, page 633. 760 APPENDICKS This nomogram consists of two parallel scales of equal length separated by a distance equal to that length and so joined by a diagonal scale as to form a figure similar to the letter N (Fig. 157). As adapted by Lt. Radler de Aquino, CD is a scale of consines, EF of cotangents (limited to angles greater than 45), and ED is so graduated that when the numbered points corresponding to any two given quantities of the equation cot a cot & = cos c, each taken on its proper scale, are aligned, the numbered point where the line intersects the remaining scale will correspond to the required quantity. The points corresponding to cosines, the right-hand functions of group (a), must always be taken on the right-hand scale CD; the points corresponding to co- tangents, the left-hand functions of group (a), may be taken from the other two scales (the left and diagonal) indiscriminately, provided that the left scale is used only for cotangents of angles greater than 45. This restriction on the use of the left- hand scale arises from the na- ture of the equation cot a cot & cos c, which shows that both FIG 158 a an( ^ ^ cannot be less than 45 at once; and, therefore, when either a or & is less than 45 it must be read on the diagonal scale, the remaining one being read on the left scale; and when both a and 6 are greater than 45, either may be read on either the left or diagonal scale. Any two points of the nomogram, not on the right-hand scale, aligned with of the right-hand scale, are marked by complimentary numbers and for this reason the graduation is easily checked. Let Fig. 158 be the astronomical triangle; P, the elevated pole; Z, the zenth of the observer; and M, the observed body whose altitude is h and declination is d; then PZ is the co-latitude. If the triangle is divided into two right triangles by a perpendicu- lar Mm and the parts be lettered as shown, a complete solution may be effected, through Napier's rules, by the following equations: APPENDICES '61 (a). cot b cot (90 d) = cos t cot B cot (90 ft) = cos Z cot cot ( 90 a) = cos 1) cot Z cot ( 90 a) = cos B The following precepts .are given by Lt. Radler de Aquino for the determination, without resort to signs, of the values of B and the quadrant of Z in the various cases dependent on the rela- tive values of L and d: f /&>L:B=(90+L) d and L of same name J 90" 90;Z<90 d and L of different name B 90 (L+6) ; Z>90 In other words, in the first two cases when d and L are of the same name and t < 6 h , 90 must be added to the smaller of the two quantities & and L and from the sum the greater should be subtracted. In the third and fourth cases & and L are always added together; if the sum is greater than 90, subtract 90 from it; if less than 90, it is subtracted from 90. The precepts given above for determining the value of B and the quadrant of Z might be re- placed by the much simpler and more covenient precepts of Art. 249, provided the equations in group (a) should be expressed in the nomenclature of that article, a nomenclature with which the American naval ser- vice is familiar. The student is referred to Art. 249 and to Figs. 115 and 116 therein representing the various positions of the heavenly body dependent on the values of L and d for a full explanation of the reasons given for the precepts suggested below. Let PMZ, Fig. 159, be the astronomical triangle projected on the plane of the horizon; then m is the foot of the perpendicular fc dropped from the position angle M on PZ. =Pm is the polar distance of m; and if " = QQ-~ " is determined from the formula L = " + 0'. 90 fc is determined by Z and 90 0'. is determined by 90 k and 0". 90 d (or d) is determined by 0" and t Ex. 230. April 5, 1905, p. m., in latitude 20 38' S. and longitude 90 10' E., weather cloudy, a bright star was observed, through a break in the clouds, bearing N. 61 E. (true) ; -)f' s h 25 10'; W. T. of obs. 7 h 20 m 40 s ; C W 5 h 51 m 30 s , chronometer fast of G. M. T. l m 40 s ; required the name of the star. Solution (Fig. 157): With 90 h and Z, find 90 0' = 44 (line IB) and 0' = 46 S. (marked S. as the body bears N.). From L = 0" + 0' we have 0" = 25 22' N. With Z and 90 0', find 90 fc = 37 40' (line IIB). With 90 k and 0", find t = 55 10' = +3 h 40 m 40 s , body being east of the meridian (line IIIB). With 0" and t, find 90 d = 74 50' (line IVB) and d = 15 10' N., d being of the same name as 0". From the other data we have L. S. T. = 8 h 03 m 50 s and therefore the -X-'s R. A. is ll h 44'" 30 s ; this with the declination of 15 10' N. identifies the star as 3 leonis. 764 APPENDICES ,.^ fe-.^ 800t-O>OtO^1-HQl^CPOCO(M'^OOOt-?OlOCO(yi^HOaD>^taiOCO(NtH ^COIMr-iOiO^SST-loS^KXMrtO^COWr-iOO^COT-lOiO^COC^r-l aDOCOO50-1lOt-O??tOO5 (-^tt-OCOO-W -^Jt o-^ i ^^ -: ' N W cs c-:rTco-*<-*-^>cioiousotoo MM N e M M K M N at M < M ec oimcS e m 8 ieeia - ?^ UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ASTRONOMY LIBRARY AUG 1 3 197J LD 21-100m-ll,'49(B7146sl6)476 72948