^■P^WHi^""^ ,>im ' 1 ^' «4 iitm 1 ^to .0 r /^^/" •o c o :5 £• ^ b« XI C u- c j; ^ Q> ■u o c o •? £ tu 4J ►= « « ^ o c -a -c c a = - Tl n M rr* C •3-^5 --/ •— <" u bo *^ ^ w a ^ to 3 n j3 - D E '- si a 3 CL O^ -J O* ti ?s O S" o- o- C O •5 .5 - " ' a; -a i> .t: re -=: E "-t- Ix fl» H cn d S fc/) to ^- O rt o ^ o .r o ^ 3 — I o ffi ■«-> , ^ I t> V »*- ..- , IB -a en 1} -0 n 3 u 60 n c 4> •0 u re •0 4 C 1 X c ^ >. -a 3 0, •0 c c •^-t "o w -f r t) 0) .^ >. c re E a; c •a "S w CQ a; c E 3 > ITi 0. re •0 c •a u ■*-* c re 5 '3 .2' c rt X c a 3 •5 re C C/l 4; ■0 XI > c 'c -5 •0 < m u > ]l H rt < •0 n C C 0^ (A E a v •0 3 •a 3 •< < v. re in £ UJ C tu U & X :^ *-< j= I H Ovil & Mechanical Engineer. SAN FRANCISCO, CA .T>. No. 9 ^IcQ 2. 25 American Practical Navigator An Epitome of Navigation and Nautical Astronomy By NATHANIEL BOWDITCH, LL. D., Etc. c n c 4) 5 o 1;; ^ o Id ?^ 4-) c 1) u I/! (LI J3 c o 1= _- o CO rt o *^ > i! "O •- "n -^ > ^ IJ S c) c *^ -a ^ "" .S "o 3^-5 O z w -J CQ < .55 •" •= — X X x'>* X X O Z Da < f- Q J O X I— I B) "T3 1 S -3 be o J o a; rt H o c o o O 4J 4-t S 1) 4-t C c ~ > '-^ -^ i. .2 •" •- -.3 .C > > ._:=:= > X X X X X X XXX XXX u V H » o a; T3 3 C 4; S 4-> o en c ttJ o JJ X •5 w C fl) aj in S 3 c -Q 2 .t: ■" -a C -13 E (/5 be o 4-> 3 < HI H o . u u a; (U u Xi >. 6 "S 3 c/i c S5 .2 Ji 13 c .•3 lU c _ I- c (U direction of the Bureau. The revision Consists principally in the substitution of the more concise and convenient methods of the present day for the obsolete methods of the past, and a complete reari-angement under proper chapters and paragraphs for read}- reference, keeping in view, however, the character of the work as a Practical Navigator. The revision having been completed, it was submitted to Capt. Ralph Chandler, U. S. Navy, for a final review, and having received a satisfactory report from that officer it has been accepted bj^ the Bureau and will hereafter bo substituted for the former editions of the work. William D. Whiting, Chief of Bureau. Bureau of Equipment, Navy JJe_partmetit, March 18, 190o. A revision of Bowditch's American Practical Navigator having become neces- sary, the work has been cooipletcd by Lieut. G. W. Logan, U. S. Navy, under the supervision of the Hydrographer to the Bureau of Equipment. The revision was approved by a Board consisting of Capt. Colby M. Chester, U. S. Navy, Commander C. J. Badger, U. S. Navy, and Lieut. Commander C. C. Rogers, U. S. Navy. It is directed that this revised edition be substituted for all former editions. R. B. Bradford, Chief of Bureau. 2 VK5BB YEDDING OF PROMINENT COUPLE THIS | '31 ^^^rnoon at First Presbyteri an Chf^rcb, ''^^ ~ik- ii't'.? PREFACE. The copyright of the New American Practical Navigator, by the late Dr. Bowditch, became the property of the United States Government under the provision of an act of Congress to establish a Hydrographic Office in the Navy Department, approved June 21, 1866. Under the direction of the Bureau of Navigation, at that time charged with such publications, the work was revised in 1880 by Commander P. H. Cooper, U. S. Navy, certain chapters being contributed by Lieuts. Richard Wainwright and Charles H. Judd, U. S. Navy, and the whole being reviewed by Capt. Ralph Chandler, U. S. Navj'. The object of this revision was to improve the general arrangement, and to introduce the more convenient and precise methods of navigation that had come into practice since the book was originally written. The progress that has been made in the science of navigation since 1880 has rendered necessary a second extensive revision, to take cognizance of the changes of methods and instruments that have accompanied the general introduction of high-speed vessels built of iron and steel. This work has been carried out, under the direction of the Bureau of Equipment, by Lieut. G. W. Logan, U. S. Navy, who was aided in the collection of data and preparation for publication by Lieut. T. A. Kearnej, U. S. Navj; the chapters on Winds and Cyclonic Storms were contributed by Mr. James Page, nautical expert, Hydrographic Office. There has been an extensive rewriting of the text, with the object of amplifying those matters that are of the greatest importance in the modern practice of navigation, and of omitting or condensing those of lesser importance; and the revision of the tables has proceeded along similar lines. This has involved, among other things, a much wider treatment of the subject of the compass; an extension of the traverse table for degi'ees to distances up to 600 miles; an improved table for reducing circum- meridian altitudes; the combination of the tables of maritime positions and tidal data; the omission of certain special methods for finding position by two observations; the addition of a series of annotated forms for the working of all sights, atid the intro- duction of a number of new tables of use to the navigator. The explanation of the method of lunar distances, with its accompanying tables, has been I'etained, in order to be available for use when required; but since this obser- vation is so rarely employed in modern navigation, everything pertaining thereto has been incorporated in an appendix, that it may be distinct from matter of every-day use to the navigator. For convenience in use the work has been divided into two parts, of which the first comprises the text and its appendices, and the second the tables. W. H. H. SOUTHERLAND, Commander, U. S. Navy, HydrograpJwr, Hydrographic Office, Bureau of Equipment, Navy Department, ' • Washington, D. C, March 19, 1903. M767913 X O T E Part I of this edition is a reprint of the revised edition of 1903 with typo- graphical errors corrected. Part II was revised and enhirfjed August 10, 1911 (see p. 503). Jonx J. Knapp, Captain, U. S. Navy, H;/ilrograph^r. Hydrographic Office. Bi REAi- OF Navigation, Xavv Department, Washin^on, D. C, January 15, 1912. P J^ R T I TEXT AND APPENDICES. "Th» oe««Ln !■ th« •lemeut which nrtffht eontaln such conditions. The ) oiigrinatlon of the oooan flta It to sus- tain life better tljan any other en- vironment, and the fundamental char- acteristics of the ocean also encourasfp •volution of active life. Can we not therefore assume that the ocean Is the Attest of known substances for the creation of the origin of life? "The orlsrln of lite itself depends upon stability. Onr present existence depends upon stability of climate and other elements of onr environments. Where conditions are most stable there you will find most active and progressive life. There Is one sub- stance that posspsses more stability than anythlnp else we know of and that Is the ocean. "In the ocean we see every degree of development from the highest to the lowest forms of life. It we are to be- lieve In the theory of evolution mny we not fro a step further and conclude that Inorganic life under certain con- ditions, such«aa those contained In the ocean, will develop organic lite?" OOISTTEN'TS OF P^RT I. Page. Orders relating to revision 2 Preface S Abbreviations 9 Chapter I. Definitions relating to Navigation M II. Instruments and Accessories in Navigation 13 III. The Compass Krror ?9 IV. Piloting 42 V. The Sailings 50 VI. Dead Reckoning 60 VII. Definitions relating to Nautical Astronomy 63 VIII. Instruments employed in Nautical Astronomy 66 IX. Time and the Nautical Almanac 74 X. Correction of Observed Altitudes 82 XI. The Chronometer Error 87 XII. Latitude 94 XIII. Longitude 103 XIV. Azimuth 109 XV. The Sumner Line 114 XVI. The Practice of Navigation at Sea 124 XVII . Marine Surveying 131 XVIII. Winds 142 XIX. Cyclonic Storms 147 XX. Tides 153 XXL Ocean Currents 158 Appendi.x I. Extracts from the American Ephemeris and Nautical Almanac for the year 1879, which have reference to examples for that year given in this work 163 II. A collection of Forms for working Dead Reckoning and various Astronomical Sights, with not«s explaining their application under all circumstances 171 III. Explanation of certain Rules and Principles of Mathematics of u.se in the Solu- tion of Problems in Navigation 178 IV. Maritime Positions and Tidal Data 190 V. Lunar Distances 288 Index 333 7 ABBREVIATIONS USED IN THIS WORK. Alt. (or A) Altitude. A. M Ante meridian. Amp Amplitude. App ..Apparent. App. t Apparent time. Ast Astronomical. Ast. t Astronomical time. Aug Augmentation. Az. (orZj Azimuth. C Course. C C Chronometer correction. C— W Chronometer mmif.? watch. Chro. t Chronometer time. Co. L Co. latitude. Col Column. Corr Correction. Coe Cosine. Cosec Cosecant. Cot Cotangent. d (or Dec. ) Declination. D (or DLo) Difference lorigitude. Dep Departure. Dev Deviation. Diff Difference. Dist Distance. DL Difference latitude. D. R Dead reckoning. E. , Ely Ea.st, easterly. Elap. t Elai)8ed time. I'xj. eq. alt Equation equal altitude" Kq. t Etjuation of time. G. (or Gr. ) Greenwich. G. A. T Greenwich apparent Jie. G. M. T Greenwich mean time. G. S. T. Greenwich sidereal time. h Altitude. H Meridian altitude. H. A. (or() Hour angle. H. D Hourly difference. H. P. (or Hor. par. )... Horizontal parallax. Hr-s Hour-s. H.W High water. I. C Index correction. L. (or IM. ) Latitude. L. A. T Local apparent time. L. M. T Local mean time. L. S. T Local sidereal time. Lo. (or Long) Longitude. Log Logarithm. Lun. Int Lunitidal interval. L. W Low water. m Meridional difference. Merid Meridian or noon. Mag Magnetic. M. D Minute's difference. Mid Middle. Mid. L Middle latitude. M. T Mean time. N. , Nly North, northerly. N. A. ("or Naut. Aim. ). Nautical Almanac. Np Neap. Obs Observation. p ( or P. D. ) Polar distance. p. c Per compass. P. D. (orp) Polar distance. P. L. (or Prop. Log.) .Proportional logarithm. P. M Post meridian. p. atitude is reckoned North and South, from the equator as an origin, through 90° to the poles; thus, the latitude of the point T is MT, north, and of the point T', M'T', north. The Difference of iMtitude ))etween any two places is the arc of a meridian intercepted between their parallels of latitude, and is called North or South, according to direction; thus, the difference of latitude between T and T' is Tn' or T'u, noVth from T or south from T'. The longitude of a place on the surface of the earth is the arc of the equator intercepted between its meridian and that of some place from which the longitude is reckoned. Longitude is measured East or West through 180° from the meridian of a designated place, such meridian being termed the lYtine Meridian; the prime meridian used by most nations, including the United States, is that of Greenwich, England. If, in the figure, the prime meridian be PCJQP', then the longitude of the point T is QM, east, and of T', QM', east. The Difference of Longitude lietween any two places is the are of the equator inter- cepted between their meridians, and is called East or West, according to direction; thus, the difference of longitude between T and T' is MM', east from M or west from M'. The Departure is the linear distance, measured on a parallel of latitude, between two meridians; unlike the various quantities previously defined, departure is reckoned in miles; the departure l)etween two meridians varies with the parallel of latitude upon which it is measured; thus, the departure between the meridians of T and T' is the number of miles correspomling to the distance Tn in the latitude of T, or to n'T' in the latitude of T'. 12 DEFINITIONS BELATING TO NAVIGATION. The curved line which joins any two places on the earth's surface, cutting all the meridians at the same angle, is called the Hhumb Line, Loxodromic Curve, or Equiangular Spiral. In the figure, tliis line is represented by TrT'. The constant angle which this line makes with the meridians is called the Course; and the length of the line between any two places is called the Distance between those places. The unit of linear measure employed by navigators is the Nautical or Sea Mile, or Knot. It is equal to one minute of latitude — that is, to the length of that portion of a meridian which subtends at the earth's center the angular measure of one minute; since, however, on account of the fact that the earth is not a perfect sphere, this distance is not exactly the same in all latitudes, a mean value is adopted for the length of the knot, and it is regarded as equal to 6,080.27 feet. For the purposes of navigation, the variation from this value in different latitudes is so small that it may be neglected, and the knot may be assumed equal to a minute of latitude in all parte of the earth; hence, when a vessel changes her position to the north or south by one nautical mile, it may always be considered that the latitude has changed 1'. Owing to the fact that the meridians all converge toward the poles, the difference of longi- tude produced by a change of position of one mile to the east or west will vary with the latitude; tlius a departure of one mile will equal a difference of longitude of I'.O at the equator, of I'.l in the latitude of 30°, and of 2'.0 in the latitude of 60". The Great Circle Track or Course between any two places is the route between those places along the circumference of the great circle which joins them. In the figure, this line is represented by 'R/T'. From the properties of a great circle (which is a circle upon the earth's surface formed by the inter- section of a plane passed through its center) the distance between two points measured on a great circle track is shorter than the distance upon any other line which joins them. Except when the two points are on the same meridian or when both lie upon the equator, the great circle track will alwavs differ from the rhumb line, and the great circle track will intersect each intervening meridian at a different angle. INSTRUMENTS AND ACCESSOBIES IN NAVIGATION. 13 CHAPTER II. INSTEUMENTS AND ACCESSORIES IN NAVIGATION. DIVIDERS OR COMPASSES. 7. This instrument consists of two legs movable about a joint, so that the ix>ints at the extremitiee of the legs may be set at any requirerl distance from each other. It is used to take and transfer dis- tances and to describe arcsand circles. When used for the former purpo.se it is termed diriders, and the extremities of both legs are metal points; when used for describing arcs or circles, it is called a cmn- pass, and one of the metal points is replaced by a jiencil or pen. PARALLEL RULERS. 8. Parallel riilerx are used for drawing lines parallel to each other in any direction, and are particu- larly useful in transferring the rhumb-line on the chart to the nearest compass-rose to ascertain the course, or to lay off bearings and courses. PROTRACTOR. 9. This is an instrument used for the measurement of angles upon paper; there is a wide variation in the material, size, and ."hape in which it may be made. ( For a description of the Three Armed Protractor, see art. 432, Chap. XVII. ) THE CHIP LOO. 10. This in.strument, for measuring the rate of sailing, consists of three parts; viz, the log-chip, the log-line, and the log-glags. A light substance thrown from the ship ceases to partake of the motion of the vessel as soon as it strikes the water, and will be left behind on the surface; after a certain inter- val, if the distance of the ship from this stationary object te measured, the approximate rate of sailing will l)e given. The log-chip is the float, the log-line is the measure of the distance, and the log-glau defines the interval of time. The log-chip is a thin wooden quadrant of aliout 5 inches radius, loaded with lead on the circular edge sufficiently to make it swim upright in the water. There is a hole in each comer of the log- chip, and the log-line is knotted in the one at the apex; at about 8 inches from the end there is seized a wooden socket; a piece of line of proper length, being knotted in the other holes, has seized into its bight a wooden jieg to fit snugly into the socket before the log-chip is thrown; as soon aa the line is checked this peg pulls out, thus allowing the log-chip to Ije hauled in with the lea.st resistance. The log-line is al)out 1.50 fathoms in length, one end made fast to the log-chip, the other to a reel upon which it is wound. At a distance ot from 1.5 to 20 fathoms from the log-chip a permanent mark of red bunting aV)out 6 inches long is placed to allow sufficient gtrau line for the log-chip to clear the vessel's eddy or wake. The rest of the line is divided into lengths of 47 feet 3 inches called knots, by pieces of fish-line thrust through the strands, with one, two, three, etc., knots, according to the number from stray-line mark; each knot is further subdivided into five equal lengths of two-tenths of a knot each, marked by pieces of white rag. The length of a knot depends upon tlie numterof seconds which the log-glass measures; the length of each knot must liear the same ratio to the nautical mile (^V ot a degree of a great circle of the earth or 6,080 feet) that the time of the glass does to an hour. In the United States Kavy all log-lines are marked f6r log glasses of 28 seconds, for which the proportion is: 3600 : 6080 = 28- : a:, X being the length of the knot. Hence, . a: = 47".29, or 47" 3'°. The speed of the sliip is estimated in knots and tenths of a knot. The Ujg-glam is a sand glass of the same shape and construction as the old hour-glass. Two gla-sses are used, one of 28 seconds and one of 14 seconds; the latter is employed when the ship is going at a high rate of speed, the number of knots indicated on a line marked for a 28-second glass being doubled to obtain the true rate of speed. 11. The log in all its parts should be frequently examined and adjusted; the peg must be found to fit sufficiently tight to keep the log-chip upright; the log-line shrinks and stretches and should often be verified; the log-glass should be compared with a watch. One end of the glass is stopped with a cork, by removing which the sand may be dried or its quantity corrected. 12. A ground log consists of an ordinary log-line, with a lead attached instead of a chip; in shoal water, where there are no well-defined objects available for fixing the position of the vessel and the course and speed are influenced by a tidal or other current, this log is sometimes used, its advantage being that the lead marks a stationary point to which motion may be referred, whereas the chip would drift with the stream. The speed, which is marked in the usual manner, is the speed over the ground, and the trend of the line gives the course actually made good by the vessel. 14 INSTRUMENTS AND ACCKS80RIES IN NAVIGATION. THE PATENT LOG. 1 3. This is a mechanical contrivance for registering the distance actually run by a vessel through the water. There are various types of patent logs, but for the most part they act upon the same principle, consisting of a registering device, a fly or rotator, and a log or tow line; the rotator is a small .spindle with a number of wings extending radially in such manner as to form a spiral, and, when drawn through the water in the direction of its axis, rotates about that axis after the manner of a screw pro- peller; the rotator is towed from the vessel by means of a log or tow line from 20 to 50 fathoms in length, made fast at its apex, the line teing of special make so that the turns of the rotator are transmitted through it to the worm shaft of the register, to which the inboard end of the line is attached; the regis- tering device is so constructed as to show upon a dial face the distance run, according to the numlx'r of turns of its worm shaft due to the motion of the rotator; the register is carried at some convenient point on the vessel's quarter; it is frequently found expedient to rig it out upon a small boom, so that tlie rotator will be towed clear of the wake. 14. Though not a perfect instrument, the patent log affords the most accurate means available for determining the ves-sel's speed through the water. It will usually be found that the indications of the log are in error by a constant percentage, and the amount of this erior should be determined by careful experiment and applied to all readings. Various causes may operate to produce ina(«uracy of working in the patent log, such as the bending of the wings of the rotator by accidental blows, fouling of the rotator by sea weed or refuse from the ship, or mechanical wear of parts of the register. The length of the tow-line has much to do with the working of the log, and by varying the length the indications of the instrument may sometimes be adjustee ascertained in order that proi)er allowance may be made therefor in taking soundings at night. 19. The deep-sea lead may be armed by filling with Tallow a hole hollowed out in its lower end, by which means a sample of the bottom is brought up. THE SOUNDING MACHINE. SO. This machine possesses advantages over tlie deep-sea lead, forwhich it is a substitute, in that soundings may be obtained at great depths and with rapidity and accuracy without stopping the ship. It consists essentially of a stand holding a reel upon which is wound the sounding wire, and which is controlled by a suitable brake. Crank handles are provided for reeling in the wire after the sounding has been taken. Attached to the outer end of the wire is the lead, which has a cavity at its lower end for the reception of the tallow for arming. Above the lead is a cylindrical case containing the depth- registering mechanism ; various devices are in use for this purpose, all depending, however, upon the increasing pressure of the water with increasing depths. 21. In the iMrd Kelvin machine a, slender glass tube is used, sealed at one end and open at the other, and coated inside with a chemical substance which changes color upon contact with sea water; this tube is placed, closed end up, in the metal cylinder; as it sinks the water rises in the tube, the contained air being compressed with a force dependent upon the depth. The limit of discoloration is marked by a clearly defined line, and the depth <]f the sounding corresponding to this line is read off from a scale. TuImjs that have been used in comparatively shallow water may be useil again where the water is known to be deeper. IKSTRUMENTS AND ACCESSORIES IN NAVIGATION. 15 22. A tube whose inner surface is (fround has been substituted for the chemical-coated tube, ground glass, when wet, showing clear. The advantage of these tubes is that they may be used an indefinite number of times if thoroughly dried. To facilitate drying,arubbercapi8 fitted to the upper end, which, when removed, admits of a circulation of the air through the tube. 23. As a substitute for the glass tubes a mechanical depth recorder contained in a suitable case has been used. In this device the pressure of the water acts upon a piston against the tension of a spring. A scale with an index pointer records the depth reached. The index pointer must be set at zero before each sounding. 24. Since the action of the sounding machine, when glass tubes are used, depends upon the com- pression of the air, the barometric pressure of the atmosphere must be taken into account when accurate results are required. The correction consists in increasing the indicated depth by a fractional amount according to the following table: Bar. reading. Increase. // 29.75 30.00 30.50 30.75 One-fortieth. One-thirtieth. One-twentieth. One-fifteenth. THE KAKINER'S COMPASS. 25. The Marinefg Compaes is an instrument consisting either of a single magnet, or, more usually, of a series of magnets, which, being attached to a graduated circle pivoted at the center and allowed to • Fio. 2. swing freely in a horizontal jdane, has a tendency to lie with its magnetic axis in the plane of the earth's magnetic meridian, thus affording a means of determining the azimuth, or horizontal angular distance from that meridian, of the ship's )ject8, terrestrial or celestial. 16 INSTRUMENTS AND ACCESSORIES IN NAVIGATION. 46. The circular tard of the compass (fig. 2) is divided on its periphery into 360°, numbered from 0° at North and South to 90° at East and West; also into thirty-two divisions of lip each, called yjoi'nte, the latter being further divided into half-poii)ts and quarter-points ; still liner sulxlivisions, eighth-points, are sometimes used, though not indicated on the card. A system of numbering the degrees from 0° to 360°, always increasing toward the right, is shown in the figure. This system is in use by the mariners of some nations, and its general adoption would carry with it certain undoubted advantages. 27. Boxing the Compass is the process of naming the points in their order, and is one of the first things to be learned by the young mariner. The four principal points are called cardinal points and are named North, South, East, and West; each differs in direction from the adjacent one by 90°, or 8 points. Midway between the cardinal points, at an angular distance of 45°, or 4 points, are the inter-cardinal points, named according to their position Northeast, Southeast, etc. Jlidway between each cardinal and inter-cardinal point, at an angular distance of 22J°, or 2 points, is a point whose name is made uj) of a combination of that of the cardinal with that of the inter-cardinal point: North-Northeast, East- Northeast, East-Southeast, etc. At an angular distance of 1 point, or 11|°, from each cardinal and inter- cardinal point (and therefore midway between it and the 22J°-division last described), is a point which bears the name of that cardinal or inter-cardinal point joined by the word hy to that of the cardi- nal point in the direction of which it lies: North by East, Northeast by North, Northeast by East, et(^ In boxing by fractional points, it is evident that each division may be referred to either of the whole points to which it is adjacent; fot instance, NE. by N. J N. and NNE. J E. would describe the same division. It is the custom in the United States Navy to box /j-ojji North and South toward East and West, excepting that divisions adjacent to a cardinal or inter-cardinal point are always referred to that point; as N. J E., N. by E. J E., NNE. J E., NE. J N., etc. Some mariners, however, make it a prac- tice to box from each cardinal and inter-cardinal point toicard a 22}°-point (NNE., ENE., etc.) ; as N. I E., N. by E. J E., NE. by N. J N., NE. J N., etc. The names of the whole points, together with fractional points (according to the nomenclature of the United States Navy), are given in the following table, which shows also the degrees, minutes, and seconds from North or South to which each division corresponds: N. toE. N. to W. S. toE. S. to W. Pts. Angular measure. North: N. JE N. JE N. IE N. byE N. byE. JE... N. byE. JE... N. byE. |E... NNE NNE. JE NNE. JE NNE. |E NE. bvN NE. iN NE. JN NE. t N NE NE. JE NE. JE NE. IE NE. byE NE. byE. J E.. NE. byE. JE.. NE. bvE. i E.. ENE ....: ENE. JE ENE. JE ENE. |E E. bvN E. JN E. JN E. JN East North: N. iW N. J W N. }W N.by W N. byW. iW. N. bvW. JW. N. by W. }W. NNW NNW. JW... NNW. J W... NNW. J W... NW. by N NW. JN NW. +N NW. i N NW NW. i W NW. J W NW. i W NW. by W NVV.byW.JW NW. by W. JW NW.byAV. |W WNW WNW. i W... WNW. J W... WNW. J W... W. byN W. JN W. J N W. IN West South: S. JE S. JE S. }E S.byE 8. byE. JE. S. byE. JE. S. by E. J E . SSE , SSE. JE SSE. JE SSE. JE.... SE. bvS SE. JS SE. JS SE. JS SE SE. JE SE. JE SE. JE SE. byE SE. by E. J E SE. by E. J E SE. by E. J E ESE ESE. JE.... ESE. JE.... ESE. JE.... E. byS E. JS E. JS E. JS East South: S. J W S. JW S. J W S. by W S. bvW. JW... S. bvW. JW... S. by W. JW... ssw i i 1 U li 2 SSW. JW '• 2J ssw. J W . SSW. J W SW. byS SW. J s . . SW. JS.. SW. JS.. SW 2J 2J 3 3J 3J 3J 4 4J SW. J w.... SW. JW.... SW. JW I 4| SW. byW I 5 SW. byW. JW. 5 J SW. hyW. JW.I 5J SW. bvW. JW.I 5J WSW....". 6 WSW. JW 6J WSW. JW ! 6 J WSW. JAV ! 6J w. by S . W. JS. W. JS. AV. J S . West 7 7J 7* 2 48 45 5 37 30 8 26 15 11 15 00 14 03 45 16 52 30 19 41 15 22 30 00 25 18 45 28 07 30 30 56 15 33 45 00 36 33 45 39 22 30 42 11 15 45 00 00 47 48 45 50 37 30 53 26 15 56 15 00 59 03 45 61 52 30 64 41 15 67 30 00 70 18 45 73 07 30 75 56 15 78 45 00 81 33 45 84 22 30 87 11 15 90 00 00 28. The compass card is mounted in a bowl which is carried in gimbals, thus enabling the card to retain a horizontal position whiie the ship is pitching and rolling. A vertical black line called the lub- ber's line is marked on the inner surface of the bowl, and the compass is so mounted that a line joining \\» pivot with the lubber's line is parallel to the keel line of the vessel; thus the lubber's line always indicates the compass direction of the ship's head. 2S. According to the purpose which it is designed to fulfill, a compass is designated as a Standard, Steering, Check, or Boat Compass. INSTRUMENTS AND ACCESSOKIES IN NAVIGATION. 17 30. There are two types of compass in use, the vet or Umiid and the dri/; in the former the bowl is filled with liquid, the card l)eing thus partially buoyed, with consequent increased ease of working on the pivot, and the liquid further serving to decrease the vibrations of the card when deflectetl by reason of the motion of the vessel or other cause. On account of its advantages the liquid compass is used In the United States Navy. 31. The Navy Service 7J-inch Liquid Compass. — This consists of a skeleton card 7J inches in diameter, made of tinned brass, resting on a pivot in liquid, with provisions for two pairs of magnets symmetrically placed. The magnet system of the card consists of four cylindrical bundles of steel wires; these wires are laid side by side and magnetized as a bundle Ijetween tlie poles of a jiowerfiil electromagnet. They are afterwards placed in a cylindrical ca.se, sealed, and secured to the card. Steel wires made up into a bundle were adopted because they are more homogeneous, can be more perfectly tempered, and for the same weight give greater magnetic power than a soliil steel bar. Two of the magnets are placed parallel to the north and south diameter of the card, and on the chords of 15° (nearly) of a circle passing througli their extremities. These magnets penetrate the air vessel, to which they are soldered, and are further secured to the Iwttom of tlie ring of the card. The other two magnets of the system are placed parallel to the longer magnets on the chords of 45° (nearly) of a circle passing through their extremities, and arc secured to the bottom of the ring of the card. The card is of a curved annular type, the outer riu" being convex on the upper and inner side, and is graduated to read to one-fourth point, a card circle lieing adjusted to its outer edge and divided to half-degrees, with legible figures at each 3°, for use in reading bearings by an azimuth circle or in laying the course to degrees. The card is ]irovi(le(l with a concentric spheroidal air vessel, to buoy its own weight and that of the magnets, allowing a ])res.sure of between 60 and 90 grains on the pivot at 60° F. ; the weight of the card in air is .3,060 grains. The air vessel has within it a hollow cone, ojieu at its lower end, and provided with the pivot bearing, or cap, containing a sapi)hire, which rests upon the pivot and thus sujiports the card; the cap is provided with adjusting screws foraccurately centering the card. The pivot is fastened to the center of the lK)ttom of the bowl by a flanged plate and scivws. Through this plate and the bottom of the bowl are two small holes which communicate with the expansiw1. The pivot is of gun metal with an iridium cap. The card is mounted in a bowl of cast bronze, the glass cover of which is closely packed with rubber, preventing the evaporatiim or leakage of the liquid, which entirely fills the Ixiwl. This liquid is com- posed of 45 per cent pure alcohol and 55 per cent distilleeing covered with an insoluble white paint. Beneath the bowl is a metallic self-adjusting expansion chamber of elastic metal, by means of which the lx)wl is kept constantly full without the show of bubbles or the development of undue pressure <'aused by the change in volume of the liquid due to changes of temperature. The rim of the compass l»wl is made rigid and its outer edge turned strictly to gauge to receive the azimuth circle. !J2. The Dky Compass. — The Lord Kelrin Cotnjyiim, which may l)e regarded as the standard for the nonliquid type, consists of a strong jiaper card with the central parts cut away and its outer edge stiffened bj- a thin aluminum ring. The pivot is fittearallel, measuring from the middle meridian toward each side, the distances of 0'".9, ]'".8, 2'". 7, 3'". 6, etc., in order to determine the points where meridians answering to full degrees cross the parallels drawn on the chart. Through the points thus found draw the meridians. Draw then the outer neat lines of the chart at a convenient distance outside of the inner neat lines, and extend to them the meridians and parallels. Between the inner and outer neat lines of the chart subdivide the degrees of latitude and longitude as minutely as the scale of the chart will ])ermit, the sulxlivisions of the degrees of longitude being found by dividing the degrees into equal parts, and the subdivisions of the degrees of latitude being accu- rately found in the same manner as the full degrees of latitude previously deseril)ed, though it will generally be found sufficiently exact to make even subdivisions of the degrees, as in the case of the fongitude. The subdivisions between the two eastern as well as those between the two western neat lines will serve for measuring or estimating terrestrial distances. Distances Vjetween points l)earing North and South of each other may be ascertained by referring them to the suV)divisions between the same paral- lels. Distances representeiderature of the mercurial column, and 62° F. as that of the metal scale; it is therefore important to make this reduction, as well as that for instrumental error (art. 56), in order to be enabled to compare the true barometric pressure with the normal that may be expected for any locality. The following table gives the value of this correction for each 2° F., Fig. 5. INSTRUMENTS AND ACCESSORIES IN NAVIGATION. 23 the plus sign showing that the correction is to be added to the reading of the ship's barometer and the minus sign that it is to be subtracted: Tempera- ture. Correction. Tempera- ture. Correction. Tempera- ture. Correction. Tempera- ture. Correction. ! o Inch. o Inch. o Iruih. Inch. 20 -1-0. 02 40 -0.03 60 -0.09 80 -0.14 22 -1-0. 02 42 -0.04 62 -0.09 82 -0.14 24 -1-0. 01 44 -0.04 64 -0.09 84 -0. 15 26 H-0. 01 46 -0.05 66 -0.10 86 -0.15 28 0.00 48 -0.05 68 -0.10 88 -0.16 30 0.00 50 -0.06 70 -0.11 90 -0.16 32 -0. 01 52 -0.06 72 -0.12 92 -0.17 34 - 0. 02 54 -0.07 74 -0.12 94 -0.17 36 -0.02 56 -0.07 76 -0.13 96 -0.18 38 -0.03 58 -0.08 78 -0.13 98 -0.18 As an example, let the observed reading of the mercurial barometer be 29.95 inches, and the tem- perature as given by the attached thermometer 74°; then we have: Observe — — . .. ^ ^ — ,«(- — — — ^^ ..J — — ., — .^.v. .. [^.jj. ! are three classes of thermometer, distinguished according to the method of graduating the Hows: the Fahrenheit, in which the freezmg point of water is placed at 32° and its boiling C (in feet) = .52, 494 I i_|_ if a centigrade thermometer is used, (in meters) = 16,000^1+?^^^^. THE THERMOMETER. 58. The Thermometer is an instrument for indicating tenii)eraturc. In its construction advantage is taken of the fact that bodies are expanded by heat and contracted by cold. In its most usual form the thermometer consists of a bulb filled with mercury, connected with a tube of very fine cross-sectional area, the liquid column rising or falling in the tube according to the volume of the mercury due to the actual degree of heat, and the height of the mercury indicating upon a scale the temperature; the mer- cury contained in the tube moves in a vacuum produced by the expulsion of the air through l)oiling tlie mercury and then closing the top of the tube by means of the blowpipe. There! scale as follows . point (under normal atmospheric pressure) at 212°; the' Ceiitif/mile, in which the freezing point is at 0° and the boiling point at 100°; and the Reaumur, in which these points are at 0° and 80°, respectively. The Fahrenheit thermometer is generally used in the United States and England. Tables will be found in this work for the interconversion of the various scale readings (Table 31). 59. The thermometer is a valuable instrument for the mariner, not only by reason of the aid it affords him in judging meteorological conditions from the temperature of the air and the amount of moisture it contains, but also for the evidences it furnishes at times, through the temperature of the sea water, of the ship's position and the probable current that is being encountered. 60. The thermometers employed in determining the temperature of the air ( wet and dry bulb) and of the water at the surface, should be mercurial, and of some standard make, with the graduation etched upon the glass stem; they should be Cf)mpared with a<"curate standards, and not accepted if their read- ings vary more than 1° from the true at any point of the scale. CI. The dry-bulb thermometer gives the tempera- ture of the free air. The wet-bulb thermometer, an exactly similar instrument the bulb of which is sur- rounded by an envelope of moistened cloth, gives what is known as the temperature of ernporntion, which is always somewhat less than the temperature of the free air. Froin the difference of these two temperatures the observer may determine the proximity of the air to saturation; that is, how near the air is to that )ioint at which it will be obliged to precipitate some of its moisture (water vapor) in the form of liijuid. With the envelojie of the wet bulb removed, the two ther- mometers should read precisely the same; otherwise they are practically useless. The two tliermoTneters, the wet and the dry bulb, shciuld be hung within a few inches of each other, and the surroundings should be as far as possible identical. In ]iractice the two tliermometers are generally in<'losed within a small lattice case, such as that shown in figure 6; the case should be placed in a position on deck remote froiii any source of artificial heat, sheltered from the t during the ])roc- ess of reading. The cloth envelope of the wet bulb should be a single thickness of fine muslin, tightly stretched over the bulb, and tied with a fine thread. Tlie wick which serves to carry the water from the cistern to the bulb should consist of a few threads of lampcotton, and should be of sufficient length to admit of two or three inches being coiled in the cistern. . The muslin envelope of the wet bulb should be at all times thoroughly moist, but iU)t dripping. When the temperature of the air falls to 82° F. the water in the wick freezes, the capillary action is at an end, the bulb in consequence soon becomes (piite dry, and the thermometer no longer shows the temperature of evaporation. At such times the bull) should be thoroughly wetted with ice-col£X^ .**---a-^ .^-^^. c^^r c^ £7^ / <- .^-c/ z:.-^ "S^i- li THE COMPASS ERROR. 29 ■^ CHAPTER III THE COMPASS ERROR. <.^ CAUSES OF THE ERROP. 7:1. When two magnets are near enough together to exert a mutual influence, their properties are such an to cause those poles whicli possess similar magnetism to repel, and those which possess magnet- ism of opposite sorts to attract one another. The earth is an immense natural m^net, having in each hemisphere a pole lying in the neighlx>r- •>^ ii< i\ of the geographical pole, though not exactly coincident therewith; consequently, when a magnet, ^^^such as that of a compass, is allowed to revolve freely in a horizontal plane, it will so place itself as to be parallel to the lines fif magnetic force in that plane created l)y the earth's magnetic poles, the end which we name north pointing to the north, and the south end in the opposite direction. The north end of the compass — north-seeking, as it is sometimes designated for clearness — will be that end which has o|)j>osite polarity to the earth's north magnetic pole, this latter possessing the same sort of -Vnagnetism as the so-called south pole of the compass. 74. By reason of tlie fact that the magnetic pole diffefsin position from the geographical pole, the ^t compass needle will not indicate true directions, but each compass point will differ from the corres[)ond- ing true point by an amount dependent upon the angle between the geographical and the magnetic pole at the position of the observer. The amount of this difference, expressed in angular measure, is the Variation of the Compass (sometimes callefl also the Derlination, though this term is seldom employed by navigatfirs). The variation not only changes as one travels from point to point on the earth, being different in different localities, but, as it has been found that the earth's magnetic poles are in constant motion, it " '«rgoes certain changes from year to year. In taking account of the error it produces, the navigator . therefore be sure that the variation used is correct not only for the place, but also for the lime ■r consideration. The variation is subject to a small diurnal fluctuation, but this is not a material conitKeration with the mariner. 75. Besides the error thus produced in the indications of the compass, a further one, due to Loral Attraction, may arise from extraneous influences due to natural magnetic attraction in the Vicinity of the ; vessel. Instances of this are quite common when a ship is in port, as she may be in close proximity to vessels, docks, machinery, or other masses of iron or steel. It is also encountered at sea in localities wfiere the mineral substances in the earth itself possess magnetic qualities — as, for example, at certain ])laces in Lake Superior and at others off the coast of Australia. When due to tlie last-named cause, it may be a source of great danger to the mariner, but, fortunately, the number of localities subject to local attraction is limited. The amount of this error can seldom, if ever, be determined; if known, it might jiroperly be included with the variation and treated as a part thereof. 76. In addition to the variation, the compass ordinarily has a still further error in its indications, which arises from the effect exerted upon it by masses of magnetic metal within the ship itself. This is known as the Di-riation of the Compass. For reasons that will be explained later, it differs in amount for each heading of the ship, and, further, the character of the deviations undergo modification as a vessel proceeds from one geographical locality to another. APPLYING THE COMPASS ERROR. 77. From what has been explained, it may be seen that there are three methods by which bearings or courses may be expressed: (a) tr>ie, whentheyreferto the angular distance from the earth's geographical meridian; (';) magnetic, when they refer to the angular distance from the earth's magnetic meridian, and must be corrected for variation to be converted into true; and (c) hy compass, when they refer to the angular distance from the north indicated by the compass on a given headmg of the ship, and must be corrected for the deviation on that heading for conversion to magnetic, and for both deviation and variation for conversion to true bearings or courses. The process of applying the errors under all circum- stances is one of which the navigator must make himself a thorougli master; the various problems of conversion are constantly arising; no course can beset nor liearing plotted without involving the applica- tion of this problem, and a mistJake in its solution may produce serious consequences. The student is therefore ui^ed to give it his most careful attention. 30 THE COMPASS ERROR. 78. When the effect of a compass error, whether arising from variation or from deviation, is to draw the north end of the compass needle to the right, or eastward, the error is named east, or is marked + ; when its effect is to draw the north end of the needle to the left or westward, it is named icest, ormarlied ~. Figures 7 and 8 represent, respectively, examples of easterly and westerlj^ errors. In both cases consider that the circles rep- resent theobserver's horizon, N and S being the correct nortli and south points in each case. If N' and S' repre- sent the corresponding points indicated by a com pass whose needle is deflected by a com- pass error, then in the first case, the north end of the needle being drawn to the right or east, the error will be easterly or positive, and in the second case, the north end of the needle being drawn to the left or west, the com- pass error will be westerly or negative. Considering ligure 7, if we assume the easterly error to amount to one point, it will be seen that if a direction of N. by W. is indicated by the compass, the correct direction should be north, or one point farther to the right. If the compass indicates north, the correct bearing is N. by E. ; that is, still one point to the right If we follow around the whole card, the same relation will be found in every case, the corrected bearing being always one point to the right of the compass bearing. Conversely, if we regard figure 8, assuming the same amount of westerly error, a compass bearing of N. by E. is the equivalent of a {'orrect bearing of north, which is one point to the left; and this rule is general throughout the circle, the corret'ted direction being always to the left of that shown by the compass. 79. Having once satisfied himself that the general rule holds, the navigator may save the necessity of reasoning out in feach case the direction in which the error must be applied, and need only charge his mind with some single formula which will cover all cases. Such a one Is the following: When the correct direction, is to the right, the error is east. The words correct-righl-east, in such a case, would be the key to all of his solutions. If he had a compass course to change to a corrected one with easterly deviation, he would know that to obtain the result the error must be applied to the right; if it were desired to change a correct course to the one indi- cated by compass, the error being westerly, the converse presents itself— the correct must be to the left— the uncorrected will therefore be to the right; if a correct bearing is to be compared with a com- pass bearing to find the compass error, when the correct is to the right the error is east, or the reverse. SO. It iiHist be remembered that the word east is equivalent to riyht in dealing with the compass error, and west to left, even though they involve an apparent departure from the usual rules. If a vessel steers NE. by compass with one point easterlv error, her corrected course is NE. by E. ; but if she steers SE., the corrected course is not SE. by E., but SE. by S. Another caution may be necessary to avoid confusion; the navigator should always regard himself as facing the point under consideration when he applies an error; one point westerly error on South will bring a corrected direction to S. by E.; but if we applied one point to the left of South while looking at the compass card in the usual .„j.ay— north end up — S. by W. would l)e the point arrived at, and a mistake of two points would be the result. . . 81. In the foregoing explanation reference has been made to " correct" directions and compass errors" without specifying "magnetic" and "true" or "variation" and "deviation." This has been done in order to make the statements apply to all cases and to enable the student to grasp the .^ubjei-t in its general bearing without confusion of details. Actually, as has already been pointed out, directions given may be true, magnetic, or bj; compass. By applying variation to a magnetic bearing we correct it and make it true, by applying deviation to a compass bearing we correct it to magnetic, and by applying to it the combined deviation and variation we correct it to true. Whichever of these operations is undertaken, and whichever of the errors is considered, the process of correction remains the same; the correct direction is always to the right, when the error is east, by the amount of that error. Careful study of the following examples will aid in making the subject clear: Examples: A bearing taken by a compass free from deviation is N. 76° K.; variation, 0° W.; required the true bearing. N. 71° E. A bearing taken by a similar compass is NW. bv W. i W.; variation, i pt. W.; require/ \ these lines, it must be at their intersection, the only point that fulfills •^/ \ c both conditions. — yKr i"* ^^ figure 1.3, if A and B are the objects and OA and OB the lines y^ \^ j passing through them in the observed directions, the ship's position / \^ 1 will be at O, their intersection. \^ 1 135. If it be possible to avoid it, objects should not be selected \^ \ for a cross bearing which subtend an angle at the ship of less than 30° N^ or more than 150°, as, when the lines of bearing approach parallelism, p^B a small error in an observed bearing gives a large error in the result. V\. For a similar reason objects near the ship should l)e taken in prefer- ^ > ence to those at a distance. Fig. 13. 1 36. When a third object is available a bearing of that may be , taken and plotted. If this line intersects at the same point as the other two (as the bearing 00 of the object C in the figure), the navigator may have a reasonable a.ssurance that his " fix " is correct; if it does not, it indicates an error somewhere, and it may have arisen from inaccurate observation, incorrect determination or application of the deviation, or a fault in the chart. PILOTING. 43 137. What may be considered as a form of thia method can be used when only one known object ia in sight by taking, at the same instant as the bearing, an altitude of the sun or other heavenly body and noting the time; work out the sight and obtain the Sumner line (as explained in Chapter XV), and the intersection of this with the direction-line from the object will give the observer's position in the same wav as from two terrestrial bearings. 138. Bearing and Distance op a Known Object. — When only one object is available, the ship's position may be found by observing its bearing and distance. Follow the preceding method in the mat- ters of taking, correcting', and plotting the bearing; then, on this line, lav off the distance from the object, which will give the point occupied by the observer. In figure 14, if A represents the object and AO the bearing and distance, the position sought will Tie at O. 139. It is not ordinarily easy to find directly the distance of an object at sea. The most accurate method is when its height 'is known and it subtends a fair-sized angle from the ship, in which case the angle may be measured by a sextant," and the distance computed or taken from a table. Table 3.3 of this work gives distances up to 5 miles, corresponding to various heights and angles. Captain Lecky's "Danger Angle and Offshore Dis- tance Tables" carries the computation much further. The use <5f this method at great distances must not be too closely relied upon, as small errors, such a.s those due to refraction, may throw •out the results to a material extent; but it affords an excellent approximation, and as this method of fixing positjon is employed only when no other is available the best possible approximation has to suffice. In measuring vertical angles, strictness requires that the observation should l)e so made that the angle at the foot of the object should equal 90° and that the triangle be aright triangle, as OMN, figure 15, where the line OjVl j; truly horizontal, and not as in the triangle O'MN, where the condition is not fulfilled. This error is inai)preciable, however, save at very close dis- tances, when it may be sufficiently corrected by getting down as low as possible on board the vessel, so that the eye is near the water-line. One condition exists, however, where the error is material — that shown in figure 16, where the visible shore-line is at M', a considerable distance from M, the point vertically below the summit. In this case there is nothing to mark M in the observer's eye, and it is essential that all angles be measured from a point close down to the water-line. If a choice of objects can be made, the best results will be obtained by observing that one which subtends the greatest angle, as small errors will then have the least effect. There is another meth(xl for determining the distance of an object, which is available under certain circumstances. This consists in observing, from a position aloft, the angle between the object and the line of the sea horizon beyond. By reference to Table 34 will be found the distance in yards corresponding to different angles for various heights of the observer from 20 to 120 feet. The method is not accurate beyond moderate distances (the table being limited to 5.000 vards) and is obviously only available for finding the distance of an isolated object, such as an islet, vessel, or target, over which the horizon may be seen. In employing this method the higher the position occupied bv the observer the more precise will be the results. 140. In observing small angles, such as those that occur in the methods just described, it is some- times convenient to measure them on and off the limb of the sextant. First look at the bottom of the object and reflect the top down into coincidence; then look through the transparent part of the horizon glass at the top and bring the bottom up by its reflected ray. The mean of the two readings will be the true angle, the index correction having been eliminated by the operation. 141. When the methods of finding distance by a vertical or a.horizon angle are not available, it must be obtained by such means as exist. Estimate the distance by the appearance; take a sounding, and note where the depth falls upon the line of bearing; at night, if atmospheric conditions are normal, consider that the distance of a light when sighted is equal to its maximum range of visibility, remem- bering that its range is stated for a height of eye of 15 feet; or employ such method as suggests itself under the circumstances, regarding the result, however, as an approximation only. 142. The Bearing op a Known Object and the Angle between two Known Objects. — This method is seldom employed, as the conditions always permit of cross bearings being taken, and the latter is generally considered preferable. Take a bearing of a known object by compass or pelorus and observe the sextant angle between some two known objects. The line of bearing is plotted as in former methods. In case one of the objects of the observed angle is that whose bearing is taken, the angle is applied, right or left as the case may be, to the bearing, thus giving the direction of the second object, which is plotted from the compass rose and parallel rulers. If the object whose bearing is taken is not one of the objects of the angle, lav off the angle on a three-armed protractor, or piece of tracing paper, and swing it (keeping the legs or lines always over the two objects) until it passes over the line of tearing, which defines the position of the ship; there will, except in special cases, be two points of intersection of the line with the circle thus described, and the navigator must know his position with sufficient closeness to judge which is correct. 143. Two Bearings of a Known Oriect. — This is a most aseful method, which is frequently employed, certain special cases arising thereunder teing particularly easy of application. The process aThe use of the sextant is explained in Chapter VIII. 44 PILOTING. is to take a careful bearing and at the same moment read the patent log; then, after running a convenient distance, take a second bearing and again read the log, the difference in readings giving the intervening run; when running at a known speed, the time interval will also afford a means for determining the distance run. The problem is as follows: In figure 17, given OA, the direction of a known object, A, at the first observation; PA, the direction at the second observation; and OP, the distance traversed between the two; to find AP, the distance at the second observation. Knowing the angle POA, the angular distance of the object from right ahead at the first bearing; OPA, the angular distance from right astern at the second bearing; and OP, the distance run; we have by Plane Trigonometry: PAO = ]80°-(POA + OPA); and AP=OP X sin POA sin PAO' Fig, If, as is frequently the case, we desire to know the distance of passing abeamr we have: AQ = AP X sin OPA. Tables 5A and 5B give solutions for this problem, the former for intervals of bearing of quarter points, the latter for intervals of two degrees. The first column of each of these tables gives the value of AP, the distance of the ship from the observed object at the time of taking the last bearing, for values of OP equal to unity; that is, for a run between bearings of 1 mile. The .second column gives AQ, the distance of the object when it bears abeam, likewise for a value of OP of 1 mile. When the run between bearings is other than 1 mile, the number taken from the table must be used as a multiplier of that run to give the required distance. Example: A vessel steering north takes a bearing of a light NW. J W.; then runs 4.3 miles, when the bearing is found to be WSW. Required the distance of the light at the time of the second bearing. Difference between course and first bearing, 4J pts. Difference between course and second bearing, 10 pts. Multiplier from first column. Table 5A, 0.88. 4.3 miles X 0.88 = 3.8 miles, distance at second bearing. Example: A vessel on a course S. 52° E. takes the first bearing of an object at S. 26° E., and the second at S. 2° W., running in the interval 0.8 mile. Required the distance at which she will pass abeam. Difference between course and first bearing, 26°. Difference between course and second bearing, 54°. Multiplier from second colunm, Table 5B, 0.76. 0.8 mile X 0.76 = 0.6 mile, distance of passing abeam. 144. As has been said, there are certain special cases of this problem whei'e it is exceptionally easy of application; these arise when the multiplier is equal to unity, and the distance run is therefore equal to the distance from the object. When the angular distance on the bow at the second bearing is twice as great as it was at tlie first bearing, the distance of the object from the ship at second bearinf; is equal to the run, the multiplier being 1.0. For if, in figure 18, when the ship is in the first position, O, the object A bears a°on the bow, and at the second position, P, 2a°, we have in the triangle APO, observing that APO = 180° - 2a, and POA = a: PAO = 180° -{POA + APO), = 180° —(a: + 180° — 2a:), = a. Or, since tli« angles at O and at A are equal to each other, the sides OP and AP are equal, or the distance at second bearing is equal to the run. This is known as doubling the angle on the bow. 145. A case where this holds good is familiar to every navigator as the bow and beam bearing, where the first bearing is taken when the object is broad on the bow (four points or 45° from ahead) and the second when it is abeam (eight points or 90° from ahead); in that case the distance at second bearing and the distance abeam are identical and equal to the run between bearings. 146. AVhen the first bearing is 263° from ahead, and the second 45°, the -pjj, -j^g distance at which the object idll be passed abeam yfill eqna.1 the run between bear- ings; this may be proved by computation or by reference to the tables and is a very convenient fact to remember, as it shows the navigator at once, if about to pass a point, how wide a berth he is going to give the offiving dangers. ' 147. There is 0. graphic method of solving this problem that is considered by some more convenient than the use of multipliers. Draw upon the chart the lines OA and PA (fig. 19), passing through the object on the two observed bearings; set the dividers to the distance run, OP; lay down the parallel rulers in a direction parallel to the course and move them toward or away from the observed object until some point is found where the distance between the lines of bearmg is exactly equal to the distance between the points of the dividers; in the figure this occurs when the rulers lie along the line PILOTING. 45 Fig. 19. OP, and therefore O represents the position of the ship at the first bearing and P at the second. For any other positions O'P', 0"P", the condition is not fulfilled. 148. Another graphic solution is given by the Distance Finder, devised by Lieut. J. B. Blish, U. S. Navy. This consists of a semicircle whose circumference is graduated in degrees. Two pieces of thread, made to swing about a pin-head at the center, are laid down to represent the lines of bearing, and ease in measuring distances is afforded by series of cross lines similar to those on a piece of profile ^ paper. 149. The method of obtaining position by two bearings of the game object is one of great value, l^y reason of the fact that it is frequently necessary to locate the ship when there is but one land- mark in sight. Careful navigators seldom, if ever, miss the oppor- tunity for a lx)W and beam bearing in passing a light-house or other well-plotted object; it involves little or no trouble, and always gives a feeling of added security, however little the position may be in doulrt. If about to pass an object abreast of which there is a danger — a familiar example of which is wlien a light-house marks a point off which are rocks or shoals — a good assurance of clearance should be obtained before bringing it abeam, either by doubling the angle on the bow, or by using the 26J°— 1.5° liearing; the latter has the advantage over the former if the object is sighted in time to permit of its use, as it may te a.ssumed that the 4h° (bow) bear- ing will always be observed in any event, and this gives the distance abeam directly, saving the necessity of plotting the position at second Ijearing (as obtained by doubling the angle) and then carry- ing it forward. 1 50. It must be rememiaered that, however convenient, the fix obtained by two bearings of the same object will be in error unless the course and distance are correctly estimate