OGg^illE BOARD fKAtlNWlON. B¥#ii!lNSLEY LIBRARY OF THE University of California. GIFT OF Class GUIDE BOOK LOCAL MAKINE BOARD EXi^M:i]SrA.TION. THE ORDINARY EXAMINATION. BY THOMAS L. AINSLEY, TEACHER OP NAVIGATION. Slj^irft^-wittl]^ (^Mtixrttt SOUTH SHIELDS: PBINTED AND PUBLISHED BY THOMAS L. AINSLEY, l6, MARKET PLACE, AND 86, BUTE STREET, CARDIFF. LONDON: SIMPKIN, MARSHALL, & Co., STATIONERS' HALL COURT; HAMILTON, ADAMS, & Co., PATERNOSTER ROW ; SAMPSON, LOW, MARSTON, & Co., FLEET STREET; R. H. LAURIE, 53, FLEET STREET ; CHARLES WILSON (Late Norie and Wilson), 157, LE^VDENHALL STREET; J. IMRAY AND SON, 89 AND 102, MINORIES ; J. D. POTTER, 31, POULTRY. LIVERPOOL: G. PHILIP AND SON, CAXTON BUILDINGS. EDINBURGH : J. MENZIES & Co. SYDNEY & MELBOURNE : GEORGE ROBERTSON. FRODSHAM AND KEEN, CHRONOMETER, oMixUh i' Hauttcul Instrument glahcrs, 9, St. George's Crescent, LIVERPOOL. SOUTH SHIELDS : THOMAS L. AINSLEY, PRINTER, 16, MARKET PLACE. PREFACE TO THE FIRST EDITION. This Work is intended as a Guide to the Officers of all grades of the Merchant Service, in the examinations they are required to undergo before the Local Marine Board. It will be issued in Two parts: — Part I containing what is termed the Ordinary Examination, and Part II containing the Extra Examination. The present volume, which relates to the Ordinary Examination, contains model solutions of examples in the various problems required of Candidates when under Examination, with numerous Exercises to each Problem, together with a variety of Examination Papers. It also contains all requisite information respecting the Deviation of the Compass ; Lights of the English, St. George's, and Bristol Channels, &c. ; Stowage of Cargoes ; Invoice; Charter Party ; Bottomry Bonds, &c. In the preparation of the articles on Seamanship, the following works have been consulted : — ''The Kedge Anchor," by "W. Brady, U.S.N. ; "The Seaman's Friend," by R. H. Dana, jun ; "The Sheet Anchor," by Darcy Lever, Esq. ; while my obligations to other works have been duly acknowledged. The works of Abbott, Lees, Steele, and M'Culloch, &c., are the authorities that have been consulted on the subjects of Charter Party, Bills of Lading, &c. T. L, A, South Shields, yulv loth, 1856. 235923 ADVEETISEMENT TO THE THIRTY-EiaHTH EDITION. In this Edition of the "Guide Book," such alterations and additions have been made in the work as were necessary to adapt it to the present requirements of the Examinations — con- siderable alterations in the Examination Papers having recently come into operation. T. L. A, South Shields, May 30/A, 1880. EEEATA ET COREIGENDA. Page 345, Ex. 9, for index correction 4- 2° 30" read + 2' 30", „ 349, Ex. 9, for time by chronometer 8^ 7™ 37^ read S^ 7"" 27'. „ 352, Ex. 12, for Feb. iS"" iz*' 25™ 35^ read Feb. 28"^ iz'' 26™ 30^ CONTEI^TS. Notices of Examinations of Masters and Mates Examination in Colour Blindness Places and Days of Examinations . . Explanation of Signs used Principles and Practice of Arithmetic Decimal Fractions On Logarithms Multiplication by Logarithms Division by Logarithms Trigonometrical Tables Natural Sines, (&c. .. Trigonometrical Ratios Logarithmic Sines, &c. .. . Navigation — Definitions of, &c. Preliminary Rules in Navigation — To I'ind Difference of Latitude .. ., To Find Meridional Difference of Latitude '3 14 41 58 73 76 81 81 84 85 97 106 107 To Find the Latitude in ., ., ., 108 To Find Middle Latitude .. . To Find the Difference of Longitude To Find the Longitude in The Compass Correcting Courses — Leeway Variation . . . . . . . . , , . . Deviation of the Compass Methods of Finding the amount of the Deviation Correction of Compass Bearings 108 109 no 112 I20 122 130 Napier's Diagram .. .. ,, .. ,, ,. 161 Fundamental Formula2 of Navigation .. On the Traverse Table . . . . . . . . . . , , , . Traverse Sailing . > , , . , . . , , . . , , . , , Parallel Sailing .. .. ., ., .. ,. ,, ,, Middle Latitude Sailing .. .. .. ., Mercator's Sailing .. ., .. ,, ,, ., ., ,, ,, 200 The Day's Work .. .. .. .. .. .. ., ,. .. 206 Preliminary Rules in Nautical Astronomy — The Conversion of Civil into Astronomical Time .. The Conversion of Astronomi- cal into Civil Time The Conversion of Longitude into Time The Conversion of Time into Longitude On Finding the Greenwich Date ToReduce the Sun's Declination To Find Polar Distance To Find the Equation of Time 238 Correction of Sun's Observed Altitude .. ,, ,, 242 182 189 194 197 223 224 225 227 230 237 Vlll CONTENTS. To Find the Latitude by Sun's Meridian Altitude ,, Variation by an Amplitude On Finding the Time of High Water, by Admiralty Tide To find the Rate of a Chronometer Greenwich Date by Chronometer . . . . , , , , To Find the Hour-angle .. On Finding the Longitude by Chronometer On Sumner's Method .. ,. ,. ., ., On Finding the Variation by an Azimuth .. On Finding the Latitude by Reduction to Meridian On Finding the Latitude by a Meridian Altitude of a Fixed Examination Papers . . , . . . . . . , Quadrant and Sextant .. Adjustments of the Quadrant, Sextant, &c How to Find the Course to Steer in a Known Current Soundings .. ,, On the Chart Mercator's Chart .. ., Practical Examination in the use of the Chart Answers Tables Star PAGE. . 244 250 . 258 268 . 271 . 277 287 . 321 324 ■ 379 384 . 389 389 ■ 394 394 398 409 EXAMINATION OF MASTERS AND MATES POtt OBHTIIT'IOATES OF COIMPETENOY" TTmlcr " The Merchant Shiiiping Act, 1S54," AND VOLUNTARY EXAMINATION IN STEAM. r. Under tho provisions of "Tho Merchant Shipping Act, 1854," no "Foreign-going Ship"* or "Ilomo Trade Pfissenger Ship"* can obtain a clearance or transire, or legally proceed to sea, from any port in the United Kiugdom, unless the blaster thereof, and in the case of a Foreign-going Ship, the First and Second Mates or Only JIate (as tho case may be), and in tho case of a " Ilome Trade Passenger Ship " the First or only Mute (as the caso may be), have obtained and possess valid Certificates, either of competency or Service, appropriate to their several stations in such ship, or of a higher grade; and no such ship, if of one hundred tons burden or upwards, can legally proceed to sea unless at least one officer besides the Master- has obtained and possesses a valid Certificate, appropriate to tho grade of Only Mate therein, or to a higher grade ; and every person who having been engaged to serve as Master, or as First or Second or Only Mate of any " Foreign-going Ship," or as Master or First or Only Mate of a " Home Trade Passenger Ship," goes to sea as such Master or Mate without being at tho time entitled to and possessed of such a Certificate as the Act requires, or who employs any person as Master, or First, Second, or Only Mate of any " Foreign-going Ship," or as Master or First or Only Mate of any " Home Trade Passenger Ship," without ascertaining that ho is at the time entitled to and possessed of such Certificate,/o>- each offence incurs a penaltij not cxcuding fifltj pounds. 2. Every Certificate of Competencij for a " Foreign-going Ship " is to bo deemed to be of a higher grade than the corresponding Certificate for a "Homo Trade Passenger Ship," and entitles the lawful holder to go to sea in tho corresponding grade in such last-mentioned Ship ; but no Certificate for a " Home Trade Passenger Sh'p " entitles th» Iwlder to go to sea ai Master or Mate of a '^Foreign-going Ship." 3. Certificates of Competency will bo granted to thoso persons who pass the requisite examinations, and otherwise comply with the requisite conditions. For this purpose examiners have been appointed, and arrangements have been made for holding examinations at the ports and upon the days mentioned in tho TaLlo marked A. appended hereto. The days for examination are so arranged for general convenience, that a candidate wishing to proceed to sea, and missing the day at his own port, may proceed to another port where an examination is coming forward. 4. Candidates for examination must give in their names to the Local Blarino Board if the place where they intend to be examined is a port where there is a Local Marino Board, on or before the day of examination (except in the caso of London f and Liverpool), and • By a "Poreign-going Ship" is meant one which is bound to some place out of the United Kingdom beyond the limits included bet^veen the liiver Elbe and Bie.it; and by a "Home Trade Passenger «hjp" is meant any Home Trade Ship employed in carrying rassengcrs ; and it is to bo observed tliat Foreign Steam Ships U'licn employed in carnjini/ J'cissciii/cr.i hrticcen places in the United Kimjdnm are subject to all the Provisions of the Act, as rcgards'Certificates of Masters, Ifates, and iliifiineers, to which British Steam Ships are subject : s. 291 of the Merchant Shipping Act, 1854, and 3. 5 of the Merchaat Shipping Acts, &c , Amend- ment Act, 1862. + At London applications for examination must be made on Fridays from 10 till 4, and on Saturdays frorn from 10 till 3. At Liverpool applications for examination must bo made on Tuesdays, Wednc-.days, Thursdays, and Saturdays, during office hours. Examination of Masters and Mates. must conform to any regulations in this respect -which may bo laid down by the Local Marine Board ; and if this bo not done, delay may be occasioned. 5. Testimonials of character, and of sobriety, experience, ability, and good conduct on board ship will be required of all applicants, and without producing them no person will be examined. As such testimonials may have to be forwarded to the cfTico of the Registrar- General of Seamen in London for verification before any certificates can be granted, it is desirable that candidates should lodge them as early as possible. The testimonials of servitude of Foreigners and of British Seamen serving in foreign vessels, which cannot bo verified by tho Registrar- General of Seamen, must be confirmed either by the Consul of the country to which the ship in which the candidate served belonged or by some other recognized official authority of that country, or by the testimony of some credible person on the spot having personal knowledge of the facts required to bo established. Upon application to the Superintendent of the INIercantile Marine Office candidates will bo supplied with a form (Exn. 2), which they will be required to fill up and lodge with their testimonials in the hands of the examiners. 6. Services which cannot be verified by proper Entries in tho Articles of the Ships in which the Candidates have served cannot bo counted. Thus, — for instance, A Man will state his Service to have been as Second or Only Mate, and to support his assertion will produce a Certificate of Discharge or of Employment by the Master stating that he served as Mate, when on reference to the Articles it appears that he has actually been rated as Boatswain ; the service in such a case will not be regarded as having been in the capacity of IMate. AVhenever a Man has, from any causo, been regularly promoted on a vacancy in the course of the Voyage from the rank for which he first shipped, and such promotion, with the ground on which it has been made, is properly entered in the Articles and in the Official Log Book, he will of course receive credit for his service in the higher grade for the period subsequent to his promotion. 7. The examinations will commence early in the forenoon on the days mentioned in Table A, appended hereto, and will be continued from day to day until all the candidates whose names appear upon the Superintendent's list on the day of examination are examined. 8. Where the Local Marine Board are in every respect satisfied with the testimonials of a candidate, service in the coasting trade may be allowed to count as service, in order to qualify him for examination for a Certificate of Competency for Foreign-going Ships as a Mate, and two j'ears' service as Mate in the coasting trade may be allowed to count as service for a Master's Certificate, provided the candidate's name has been entered as Mate on the Coasting Articles, and provided he has already passed an examination. QUALIFICATIONS FOE CERTIFICATES OF COMPETENCY FOE A "FOEEIGN-GOINCt SHIP." The qualifications required for the several ranks undermentioned are as follow: — 9. A SECOND MATE must be seventeen years of age, and must have been four years at sea. In Navigation.— He must write a legible hand, and understand the first five rules of arithmetic, and the use of logarithms. He must be able to work a day's work complete, including the hearing's and distance of the port he is bound to, by Mercator's method ; to correct the sun's declination for longitude, and find his latitude by meridian altitude of the sun; and to work such other easy problems of a like nature as may be put to him. Ue must understand tho use of the sextant, and be able to observe with it, and read off the arc. (See List A, page 10). ■. • • i, In Seamanship.— He must give satisfactory answers as to the rigging and unrigging of ships, stowing of holds, &c. ; must understand the measurement of the log-line, glass, and lead-line ; bo' conversant with tho rule of the road, as regards both steamers and sailing vessels, akd the lights and fog signals carried by them, and will also be examined as to hia acquaintance with " the Commercial Code of Signals for the use of all Nations." ^Icamination of IFaders and Mates. 10. An ONLY MATE must 'oo nineteen years of ag;e, and have been five years at sea. la Navigation.— In addition to the qualiGcation required for a Second JIutc, an Oi.ly Mate must be able to observe and calculate the amplitude of the sun, and deduce the varia- tion of the compass therefrom, and be able to find the longitude by chronometer by tho usual methods. He must know how to lay off the place of tho ship on the chart, both by hearings of known objects, and by latitude and longitude. lie must bo able to dntermino tbe error of a sextant, and to adjust it ; also lo find the time of high water from the known time at full and change. (See List A, pngc lo). In Seamanship. — In addition to what is required for a Second Male, he must know how to moor and unmoor, and to keep a clear anchor ; to carry out an anchor ; to stow a hold ; and to make the requisite entries in tho ship's log. IIo will also be questioned as to his knowledge of the uso and management of tho mortar and rocket Hues in tho case of tho stranding of a vessol, as explained in the official log-book. 11. A FISST HATE must be nineteen years of age, and have served five years at sea, of which one year must have been as either Second or G.'ily Mate, or as both.* In Navigation. — In addition to tho qualification required for an Only Mdte, ho must bo able to observe azimuths and compute the variation ; to compare chronometers and keep their rates, and find the longitudo by them from an obscrvalionof the sun; to work tho latitude by single altitude of the sun oIT the meridian; and be able to use and adjust the sextant by the sun. In Seamanship. — In addition to the qualification required for an Only Mate, a more extensive knowledge of seamanship will be required as to shifting large spars and sails, managing a sliip in stormy weather, taking in and making sail, shifting j-ards and masts, &c., and getting heavy weights, anchors, &c., in and out; casting a ship on a lee-shore; and securing the masts in the event of accident to the bowsprit. 12. A MASTER must be twenty-ono years of age, and have boon six years at sea, of •which at least one year must have been as First or Only Mate, and one vear as Second Mate. In addition to the qualification for a First Mate, he mnst be able to find the latitude by a star, &c. He will be asked questions as to the nature of the attraction of the ship's iron upon the compass, and as to the method of determining it. He will be examined in so much of the laws of the tides as is necessary to enable him to shape a course, and to compare his soundings with tbe dopth.T marked on the charts. He will be examined as to his com- petency to construct jury rudders and rafts ; and as to his resources for the preservation of the ship's crew in the event of wreck. He must possess a sufiicient knowledge of what he is required to do by law, as to entry and discharge, and the management of his crew, and as to penalties and entries to be made in the official log ; and a knowledge of tho measures for preventing and checking the outbreak of scurvy on board ship. He will bo questioned as to his knowledge of invoices, charter-party, Lloyd's agr-nt, and as to the nature of bottomry, and he must bo acquainted with the leading lights of the channel ho has been accustomed to navigate, or which he is going to use. (See List B, page lo). In cases where an applicant for a certificate as Master Ordinary has only served in a fore and aft rigged vessel, and is ignor.nnt of the management of a square-rigged vessel, ha may obtain a certificate on which the words ^'■fore and aft rigged vessel" will be written. This certificate docs not entitle him to command a square-rigged ship. This is not, how- ever, to apply to Mates, who, being younger men, are expected for the future to learn their business completely. (See also page 9). 13. An EXTEA MASTER'S EXAMINATION is voluntary and intended for such persons as wish to prove their superior qualifications, and are desirious of having certificates for tha highest grade granted by the Board of Trade. In Navigation.— As the vessels which such Masters will command frequently make long voyages, to the East Indies, the Pacific, &e., the candidate -will be required to work a lunar observation by both sun and star, to determine the latitude by the moon, by Polar star ofl' the meridian, and also hy double altitude of tho sun, and to verify tho result by Sumner's method. He must be able to calculate the altitudes of the sun or star when they cannt.l In • Service in a surerior capacity is in all cases to tc cfiuivrtlent to service ir. ar. inferior cai:;.c-Jv. Examination of Masters and Mates. observed for tho purposes of lunavs,— to find the error of a watch hy the method of equal altitudes, — and to correct tho altitudes observed with an artificial horizon. He must understand how to observe and apply tho deviation of the compass; and to deduce tho sot and rat^ of tho current from tho D. R. and observation. lie will be required to explain the nature of groat circle sailing, and Irnow how to apply practically that know- lodge, but ho will not bo required to go into tho calculations. Ho must be acquainted with tho law of storms, so far as to know how he may probably best escape those tempests common to tho East and West Indies, and known as hurricanes. In Seamanship. — The extra examination will consist of an inquiry into the competency of tho applicant to heave a ship down, in case of accident befalling her abroad ; to get lower masts in and out; and to perform such other operations of a like nature as the Examiner may consider it proper to examine him upon. QUALIFIOATIONS rOR CERTIFICATES OF COMPETENCY FOE A "HOME TEADE PASSENGEE SHIP." 14. A MATE must write a legible hand, and understand the first four rules of arithmetic. He must know and understand the rule of the road, and describe and show that ho under- stands the Admiralty regulation as to lights. He must bo able to take a bearing by compass, and prick oflF the ship's course on a chart. He must know the marks in the lead line, and bo able to work and heave tho log. 15. A MASTER must have served one year as a Mate in the Foreign or Home Trade. In addition to tho qualifications required for a mate, he must show that ho is capable of navigating a ship along any coast, for which purpose he will bo required to draw upon a chart produced by the Examiner tho courses and distances he would run along shore from headland to headland, and to givo in writing the courses and distances corrected for variation, and the bearings of tho headlands and lights, and to show when the courses should be altered either to clear any danger, or to adapt it to the coast. Ho must under- stand how to make his soundings according to the state of the tide. He will also bo questioned as to his knowledge of the uSe and management of the mortar and rocket lines in the case of the stranding of a vessel, as explained in the Official IjOg Book. A first-class Pilot may be examined for a Master's Certificate of Competency for Home Trade Passenger Ships, notwithstanding that ho may not have served in the capacity of Mate. aENEEAL EULES AS TO EXAMINATIONS AND FEES. 16. The candidates will be allowed to work out the various problems according to the method and the tables they have been accustomed to use, and will bo allowed five hours to perform the work; at the expiration of which time, if thoy have not finished, they will be declared to have failed, unless the Local Marine Board sco fit to extend the time. 17. The fee for examination must be paid to the Superintendent of the Ji-rcantile Marine Office (Shipping Master). If a candidate fail in his examination, half t.Mi fee he has paid will be returned to him by the Superintendent of the Mercantile Marine Offiib on his producing the Form Exn. 17, late HH, which will bo given him by the Examiner. The fees are as follow : — FOR <' FOREIGN-GOING SHIPS." Second Mate ;^i o o First and Only Mate, if previously possessing an inferior certificate 0100 If not . 100 Master, whether Extra or Ordinary . . . 200 Master, if previously in possession of a certificate for " fore and aft rigged vessels " . . . 100 N.B. — Any person having a Master's Certificate of Competency for Foreign-going Ships may go up for 071 extra examination without payment of any Fee, but if he fails in his first examination, hcUf a Master's Fee will be chargc^Lfor each subsequent examination. Kvaminafion of Jffasfcrs and iFates. FOR "HOME TRADE PASSENGER SHIPS." ]\riito . , _,/"o 10 o j\Iuster ,..i . , . . . . loo 1 8. If tlio applicant passes he will receive tho Form Exn. 16, lato GG, from the Examiner, which will entitle him to receive his Certificate of Competency from the Superintendent of tho Mercantile Marine Oflice, at tho port to which ho has directed it to ho forwarded. If his tcstimoniiils havo been sent to the Kegiatrar to bo verified, they will be returned with his certificate. 19. If nn applicant is examined for a higher rank, and fails, hut passes an csaminalion of a lo'wer grade, ho may roccivo u, certificate accordingly, hut no piit of the feo will bo returned. 20. In every case tho Examination, whether for Only Mate, First Mute, or Master, is to commence with the problems for Second Mate. 21. In all cases of failure the candidate must be re-examined de u.orn. If a candidate fails in Seamanship ho will not bo re-examined until after a lapse of Six Months, to givo him time to gain experience. If ho fails three times in Navigation ho will not be ro-cxamincd until after a lapse of Throe Months. 22. As tho examinations of Masters and Mates are made compiilsor}', the qualifications have been Lept as low as possible; but it must bo distinctly understood ihat it is tho intention of the Board of Trade to raise the standard fiom time to time, whenever, as will no doubt be the case, the general attainments of officers in the merchant service sh.iU re:ider it possible to do so without inconvenience; and oflicors are strongly urged to employ their leisure hours, when in port, in the acquirement of the knowledge necessary to enable them .to pass their examinations; and IMasters will do well to permit apprentices and junior oiSccrs to attend schools of instruction, and to afl;brd them as much time for this purpose as possible. EXAMINATION OF MASTERS AND MATES WITH EEFERENOE TO THE COMMEECIAL CODE OF SIGNALS FOR THE USE OF ALL NATIONS.— INSTRUCTIONS TO EXAMINERS. 23. In transmitting the accompanying copy of the latest edition of the Commercial Codo of Signals for the uso of tho Examiners, tho Board of Trade desire to direct attention to the principal points connected with this Code as to which Candidates for Examination should he questioned. 24. At the same time, as the subject is probablj' new to some of the Examiners them- selves, the Board recommend to them a perusal of tho Report of the Signal Committee of 1855 (which will be found at the commencement of the Signal Book), and also the first few pages of the Boole. The information therein given will be found sufficient to malce tho Examineis theoretically acquainted with tho characteristics of tho New Code, and the advantages it claims to possess over other Codes, and will enable them to appreciate and urge upon Candidates for Examination tho facilities which tho new System of Signalling aflords for easy and rapid communication. 25. The "comprehensiveness" and ''distinctness" of the Commercial Code are its chief recommen dat ions. 26. Tho form of tho Hoist generally indicates tho nature of the Signal made, so that an observer can at sight understand the character of the Signal ho sees flying. 27. Tho Examinations should tend to elicit a knowledge of tho distinctive features of tho Code above alluded to. With this object tho Examiners should make the 2, 3, and 4 Flag Signals on the Frame hoard which is furnished for the purpose {ahvngs talcing care first to show the Ensign arid the. Code Tcnnar.t at the Gaff)* questioning the Candidates as to the distinguishing Forms of the respective Hoists, which will be indicated according as a Burgee, or a Pennant, or a Square Flag, is uppermost. • The object of HiIs is, oi course, to distinguish the Signalarfrom those of another Code. Examination of Ilaslcrs and Mates. 28. Thfi Candidate ought to know how to find in the Signal Book the communication or the inquiry he desires to make^ and how to make 5> ^> 7' ^> 9> ^^^ ^^ absolute value of itself, whereas the auxiliary digit o has no such value ; and on this account the former are termed significant figures, in contra-distinction to the last. It will, moreover, have occurred to the reader that every one of these significant digits, in addition to its absolute value, which is fixed and certain, possesses also a local value dependent upon the situation in which it is placed ; thus, in the expres- sion of the number four thousand three hundred and twenty-one, which will be 4321, the i in the first place on the right hand retains its absolute value ; the second figure 2, in its situation, denotes two tens, or twenty ; the third is three hund/red, and the fourth is four thousand ; so that the local values of 2, 3, and 4, are respectively, ten times, a hundred times, and a thousand times, as great as their absolute values ; and it is the circumstance of assigning to each of the significant figures a local as well as an absolute value, which confers upon the system the immense power which it possesses. NUMERATION. 14. Def. 6. — Numeration is the art of reading or estimating the value of a number expressed by figures, and is, therefore, the reverse of Notation. 15. From the circumstance of every figure possessing a local as well as an absolute value, it follows that the value of each figure must be estimated by the place which it occupies ; hence, a figure standing by itself expresses so many units ; a figure in the second place so many tens ; a figure in the third place so many hundreds, and so on ; consequently, if we suppose any numerical expression to be divided into periods or portions, each consisting of three figures as far as they go, the figures of the period on the right will be p Principles and Practice of Arithmetic. units and tens and hundreds of units ; those of the next will be units, tens and hundreds of thousands ; those of the third will be units, tens and hundreds of millions, and so on. Thus, 1. 25 is twenty-five. 2. 304 18 three hundred and four. 3. 5287 is five thousand two hundred and eighty-seven. 4. 70639 is seventy thousand six hundred and thirty-nine. 5. 306583 is three hundred and six thousand five hundred and eighty three. 6. 1648305 is one million six hundred and forty-eight thousand three hundred and five. 7. 53024367 is fifty-three million twenty-foiir thousand three hundred and sixty-seven. 8. 257008005 is two hundred and fifty-seven million eight thousand and five. In each of these instances we conceive the expression to be separated into periods of three figures each as far as they go, beginning at the right hand, as in 257008005, we observe that 005 is the first period, 008 the second, and the third period is 257, denotes two hundred and fifty-seven millions 008 eight thousands and 005 five units. 16. The last article will be rendered more clear by the following scheme, called the Numeration Table : — ■S g 6 7 8 H 5 6 wherein the local value of every figure in each of the horizontal rows is pointed out by the name written upwards at the top of the whole ; thus, in the third horizontal line from the bottom the figures will be read nine hundred and eighty-seven, and in the second line from the top, ninety-eight millions, seven hundred and sixty -five thousand, four hundred and thirty -two. Examples for Praotiob. Express in words :- r. 43 9- 505 17. 87054 25- I 00000 I 33- 20084216 4r 202202200 2. 60 10. 550 18. 70707 26. 8047328 34- 5001860 42 lOOIOOIOI 3 12 II. 1000 19. 60880 27. 4090300 35- 8080808 43 275008005 4 21 12. 2020 20. 99404 28. 5210007 36. 55700005 44 I 000 I 000 I 5 I GO 13- 3303 21. 903756 29. 6030405 37- 76014059 45 79030184 6 lOI 14. 4004 22. 202202 30- 9009900 38. 6006606 46 408076032 7 no '5- 7707 23- 400400 31- 41041014 39- 56700505 47 401400056 8 500 16. 8880 24. 550550 32- 3000006 40. 120015015 48 908500060 Principles and Practice of Arithmetic. ADDITION. 17. Addition is the collecting together of two or more numbers, and the amount of all of them is termed their sum. The sign + (plus) is employed to indicate addition, as 7 + 2 signifies that 2 is to be added to 7. Also, the sign := (equal) signifies that the numbers between which it is placed are equal : thus, 8 + i = 9- The process of addition depends upon the principle that the sum of two numbers is equal to the sum of their respective parts. Thus, Let it be required to find the sum of two numbers, 1724 and 4638, and explain the process. 1724 = I thousand ■\- 7 hundreds -\- 2 tens -j- 4 units, 4638 = 4 thousands -j- 6 hundreds •\- 3 tens -\- 8 units, and as the sum of these two numbers is equal to the sums of their respective parts, that sum is 5 thousands -j- '3 hundreds -f- 5 tens 4- 12 units. To each of the four parts into which the first number is separated add the part of the second which is under it, beginning at the units. Thus, 8 units and 4 units are 1 2 units, that is, i ten and 2 units ; again, 3 tens and 2 tens are 5 tens ; 6 hundreds and 7 hundreds are 1 3 hundreds, or i thousand and 3 hundreds. Lastly, 4 thousands and i thousand are 5 thousands, hence the sum is either 5 thousands 1 3 hundreds 5 tens and 1 2 units, or 6 thousands 3 hundreds 6 tens and two units = 6362. Hence, 18. The rule for simple addition is as follows : EULE I. Write the numbers to he added together in vertical columns so that the units of all the numbers may be in one column, the tens in the second, the hmdredn in the third, and so on. Draw a line under the last number, and, beginning with the column of units, add successively the numbers contained in each column ; if the sum does not exceed nine, write it down under the line, but if it contains tens reserve them to he added to the next column, writing down only the units of each column, and under the last column put the entire sum, whatever it may be. If the sum of any column be an exact number of tens, write for the units and carry the tens to the next column. Examples. Ex. I. Let it be required to find the sum of 26389, 38127, 2815, 6497, 835, and 3745. Write the numbers as at the side, so that the figures of the same class shall be 26389 in the same vertical column ; then taking the sum of each class, we find there are 38127 38 units, 27 tens, 31 hundreds, 25 thousands, and 5 tens of thousands. Now 38 ^^'-5 units are 3 tens and 8 units, then writing 8 below the units column, carry the 3 1^^ tens to the 27 tens, which together make 30 tens, or 3 himdreds and o tens. -inTr "Write o below the column of tens and reserve the 3 hundreds to be added to the 31 hundreds ; this gives 34 hundreds, or 3 thousands and 4 hundreds, and writing 78408 4 below the column of hundreds, carry the 3 thousands to the 25 thousands, and we get 28 thousands, or 2 tens of thousands and 8 thousands. Writing the 8 below the column of thousands, carry the 2 tens of thousands, making the entire sum =: 78408. 20 Principles and Practice of Arithmetic. 19. Verification of Addition,— The usual verification is to add both upwards and downwards and see if the sums agree. This is generally sufficient. If more is required, or if the student cannot get a long column to cast the same way both up and down, he can cut it up and add each portion separately ; then add the sums. EXERCISES IN SIMPLE ADDITION. (0 (2) (3) (4) (5) (6) (7) (8) 3^413 543123 536123 123456 761284 657890 692387 876578 452734 234512 453215 234561 612874 278679 4956 495 1 30421 7 '3145 1234 345612 8719 5798 87658 54939 3718 104234 4231 456223 46759 67843 769378 8797 24561 36142 51234 561234 587999 488567 579° 358428 341323 3451 613254 612345 987678 (13) 37429 87958 (15) 768453 (9) (10) (") (12) (14) (16) 662593 846914 516398 425396 567453 169964 145673 197794 395266 415327 854627 674958 654359 435434 366535 543543 841923 723456 735829 827694 531769 744315 679654 765976 356627 674216 916358 731045 765453 476757 341345 415161 725983 328427 827146 556677 147954 496059 569765 954131 346783 736259 (18) 633289 (19) 889900 645679 (21) 695969 (22) 694313 (23) 643167 (17) (20) (24) 987825 916427 695024 986257 985626 372519 586372 148537 736349 625736 538426 427385 796842 463726 477754 697296 856925 346831 827836 514986 915638 298534 638831 526438 7343'6 857936 735985 726326 809274 851372 951490 723649 827842 735784 216515 915827 444444 319628 479291 859698 936736 426467 859827 734482 913258 738543 863748 852619 842625 849753 910756 386912 872364 497791 376546 419648 759519 358358 683625 219863 410698 345345 356633 777777 846325 647846 745841 391285 742367 679567 459681 999999 987846 386921 536606 842163 946208 161514 453148 555555 333445 666777 888999 615827 807609 I 3 1549 567963 724483 335445 666777 888999 736846 915827 761346 313499 952637 25- 26. Add together the addenda (1) under exercises (i), (9), and (17); (2) under (2), (10), and (18) ; (3) under (3), (11), and (19); (4) under (4), (12), and (20) ; (5) under (5), (13), and (21) ; (6) under (6), (14), and (22) ; (7) under (7), (15), and (23); and (8) under (8), (16), and (24). Add together three hundred and nine million, four hundred and seventeen thousand, and eighty- seven ; six hundred and seventy-five thousand, and forty-nine ; seven thousand and ninety-seven million, eight hundred and fourteen thousand, three hundred and five ; seventy-nine million, five hundred and four thousand, and forty- nine ; six thousand and seventy-eight million, four hundred and thirty-nine thousand^ six hundred and forty-seven ; seven thousand million, eight hundred and seventy-six thousand, four hundred and twenty-nine. SUBTRACTION. 20. The process of finding a number which shall be equal to the difference of two numbers is called subtraction. It is customary to call the quantity from which the subtraction is made, the minuend ; the quantity to be sub- tracted, the subtrahend; and the result of the subtraction, the difference. Thus, then, we have, minuend — subtrahend = difference. Principles and Practiee of Arithmetic. 2 1 We may also write this as minuend = subtrahend + difference, which shows the connection between subtraction and addition. The operation of subtraction is indicated or expressed by the sign — , which is read minus or less by, with the use of the sign == ; thus, the excess of 7 above 3 will be expressed in the form 7 — 3 := 4, which is read 7 minus 3 equals 4 ; where the sign — between 7 and 3 denotes the subtraction of the latter from the former, and the sign = between 3 and 4 shows the equality of the excess to 4. 2 1 . The process of subtraction involves two principles ; the one is the equal augmentation or diminution of the numbers. In either way, the difference of the two numbers will not be altered ; for if the greater number be either increased or diminished by 7, for example, and the less be increased by 7, the numbers themselves will be altered. The other principle is this : since 12 exceeds 7 by 5, and 8 exceeds 6 by 2, then 12 and 8 together, or 20 exceeds 7 and 6 together, or 12 by 5 and 2 together or 7. Let it be required to take 231 from 574. Write the numbers as in the margin, units under units, tens Hundreds. Tens. Units, under tens, and hundreds under hundreds ; then 4 units exceed 1574 unit by 3 units, 7 tens exceed 3 tens by 4 tens, 5 hundreds exceed 231 2 hundreds by 3 hundreds. Therefore, by the second principle, all the first column together exceeds all the second column by all 3 4 3 the third column together, that is, by 3 hundreds 4 tens 3 units, or 343 which is the dif- ference between 574 and 231. Again, let it be required to subtract 33957 from 802126. Hundreds of Tens of Thousands. Thousands. Thousands. Hundreds. Tens. Units. 802126 =: 8 o 2 I 2 6 23957 = 23957 Now here a difficulty immediately arises since 7 is greater than 6, and cannot be taken from it, neither can 5 be taken from 2, 9 from i, 3 from 2, nor 2 from o. To obviate this we must have recourse to the first principle, and add the same quantity to both these numbers, which will not alter their difference. Add ten to the first number, making 16 units, and add ten also to the second number, but, instead of adding ten to the number of units, add one to the number of tens, making 6 tens. Again, add ten tens to the first number, and one hundred to the second, then add ten hundred to the first, and one thousand to the second, and so on, adding equal quantities to each. In this way the numbers will be changed into the following : — .f Tens. Units. 12 16 _ _ _ _ _6 _7 7 7 8 I 6 9 and the difference 778169 is obtained in the usual manner. Hence, when the upper figure is the less we must auyimnt it by ten, and retain one to be added to the lower figure immediately to tJie left. 22. The rule for simple subtraction is as follows : — EULE II. 1°. Put the smaller number under the greater, talcing cwre, as in addition, that units shall he wider units, tens wnder tens, hundreds under hundreds, and so on. Hundreds of Thousands. Tens of Thousands. Thousands. Himdreds. 8 10 12 II I 3 4 10 ii, Principles and Practice of Arithmetic. 2°. Beginning at the units, take each figure in the lower line from the figure above it, if the lower figwre he not the greater of the two, setting down the remainder below it. (See the operation in Ex. i, page 17). 3°. Put if any figure in the lower line he greater than that above it, add 1 o to the upper one, and then take the lower figure from that sum, setting down the remainder, and carrying one (i.e. adding ij to the next lower figure, and with which proceed as before, and so on, till the whole is finished. (See Ex. 2, page 17). The following examples will illustrate this rule. EXAKPLBS. Ex. I. Let it be required to subtract 42571 from 76594. In the units eolumn, 2 from 4 leaves 2, set down 2. In the tens column, 7 from 9 leaves 2, set down a. In the hundreds eolumn, 5 from 5 leaves o, set down o. In the thousands column, 2 from 6 leaves 4, set down 4. In the ten thousands oolutnn, 4 from 7 leaves 3, set down 3. From 76594 minuend. Subt. 42572 subtrahend. Bern. 34022 difference. Ex. 2. Subtract 7495 from 9263. 9263 In the process adopted in practice the figures in the minuend are not, as in 7495 the second example No. 21, page 16, actually altered ; and perhaps we might more simply explain the practical process as follows : 1768 To subtract 5 from 3 is impossible, so separate i ten from the 6 tens, and adding it to the 3 units, say 5 from 13 leaves 8. Now we are supposed to have separated i ten from tho 6 tens, but as the figure really remains 6, we still have to take i from it ; also we have to take from it the 9 in the lower line, so instead of taking away first i and then 9 more, take a'lvay 10 at once ; but 10 from 6 being impossible, separate i from the place of hundreds, and adding it as 10 tens to the 6 tens, aay 10 from 16 leaves 6. As we have not r»jally taken i from the 2 hundreds, we have still i to take from it ; also we have to take the 4 in the lower line ; instead of taking first i and then 4, take away 5 at once ; but 5 from z bem-^ impoasible separate i from the place of thousands and add it as 10 hundrbda to the 2 hcaidxeds, a ad say 5 from 12 leaves 7. As we have not really diminished the fiijiire 9 lu the place of thousands, we have still i to take from it, and likewise we have to take away the 7 in the lower lino, so taking away 8 at once from 9 we have i left in the place of thousands, and the entire difference is 1768. Ex. 3. Let it be required to subtract 27385 from 64927. Then placing the former number under the latter (as in the 64927 minuend. margin), we proceed thus : 27385 subtrahend. 37542 difference. In the units column, 5 from 7 leaves 2, set down 2. In the tens column, 8 from (not 2) but 1 2 leaves 4, set down 4 and carry i . In the hundreds column, (3 -\- i, i.e. 3 + i carried) from 9=4 from 9 leaves 5, set down 5. In the thousands column, 7 from (not 4) but 14 leaves 7, set down 7 and carry i. In the ten tfiomands column, (2 -{• i, i.e. 2 -\- i carried) from 6 z= 3 from 6 leaves 3, set down 3. Subt. minuend. 86025704 subtrahend. Ex. 4. As another example, let 86025704 be subtracted from 13074 1392. Then, having arranged the numbers as in the margin, From 1 30741392 we proceed thus: 4 from 12, 8, carry i ; i and o, r, i from 9, 8 ; 7 from 13, 6, carry i ; i and 5, 6, 6 from 1 1, 5, carry i ; i and 2, 3, 3 from 4, i ; o from 7, 7 ; 6 from 10, 4, curry i ; i and 8, 9, 9 from 13, 4, carry i ; aii'l i carried being tak^ n from i, leaves o, therefore, tho remainder is 447 156X8. Rem. 44715688 difference. Princifies and Practice of Arithmetic. 23 23. Verification of Subtraction.— The best verification is to add the subtrahend and difference. This ought to give back the minuend, or original quantity from which the subtraction was made. EXERCISES IN SIMPLE SUBTRACTION. .(0 706105 84694 (^) 804601 265061 (10) 508000 129 (3) 980001 980000 (") 403000 26901 (18) OIOOOIIIOIOII lOIIIOOI lOIIO (4) 600501 600492 (12) 393436 219050 (19) IOIOII6 802c (5) 702001 26000 (6) 601002 46003 (7) 50 I 00 I 20106 (8) 602004 1 1906 (9) 701628 20449 (13) 321288 213788 (14) 345876 123457 (15) 206011 48605 47f 241 (16) 8500000 90909 (17) lOOOOOOIOOO I 7077070077 '0330 '9337 (20) lOIIOOI COIIO looinofoii (21) )8632i7896 :826424862 Take each subtrahend 1 2 times from its minuend in the following examples : — (22) (23) (24) (25) (26) (27) 7432326 6677298 7213545 4362579 6002109 8100630 157689 67527 57636 9873 45108 6156 28. 29. 30- 31- 32- 33- "What number remains when one million four hundred thousand six hundred and nineteen has been taken from one hundred millions and two ? What is the difl'erence between sixteen thousand and eighteen, and one million? If eighty-nine be added to fourteen thousand six hundred and forty-three, how many must be added to the sum to make it ten millions ? How many must be added to the sum of 99, 416305, and 2108, that it may excoed the difference between 19104 and 605 by 1143200 ? What mu.^t be added to 7543 to make it 16000? The sum of two numbers is 84207, the less is 12327, what is the greater? 24. By help of the plus (+) and minus ( — ) signs, we can easily connect together in a single row a set of numbers, of which some are to bo added and others subtracted ; thus 4 + 6 — 3 — 2 means that 4 and 6 are to be added, and 3 and 2 are to be subtracted, so that 4 + 6 — 3 — 2 = 5, Instead of subtracting first 3 and then 2, we may, of course, subtract 5 at once ; so that the above is the same as 10 — 5^5. In some cases where additive and subtractive quantities are indicated, there is a difficulty which may be stated in a simple form by means of au example. 2-7 -I-8-1 If we begin from the left-hand end, our first operation is to subtract 7 from 2, which cannot be done directly ; wo have recourse, therefore, to a different arrangement, namely, to take all the additions first, and all the subtractions last. We write it, therefore, as 2+8-7-1 After adding the 2 and the 8, we may either subtract the 7 and the i in succession, or we may add the 7 to the i and subtract their sum from tho 24 Principles and Practice of Arithmetic. sum of the z and the 8. The simplest way, therefore, of getting at the result of a set of mixed additions and subtractions is to add the additive quantities, and then (separately) add the subtractive quantities, and subtract the sums Thus, if we have "5 + 427 — 684 + 337 — 15 135 684 789 427 15 699 »37 789 699 90 result. Examples por Practice. I, 361 + 483 — 246 — 179 = 419. 3- 573 — '84 + 602 — 67 = 924. 3. 12064 + 700628 — 109641 -\- 637 — 2604 =: 601086. 4. 23596 — 625 4- 72311075 — 13758 — 3506x85 4- 6879 :;= 68820880. 5. I — 2-J-4 — 84-16 — 32 4-64 — 128 4- 256 — 5'2 + 1024 =; 2048 4" 4096 — 8192 4- 16384 — 32768 + 65536 — 13107a 4- 262144 — 524288 4- 1048576 — 2097152 4- 4194304 = 2796203. MULTIPLICATION. 25. Multiplication is a short method of finding the amount of a number repeated any number of times; thus, when 3 is multiplied by 4, the number produced by the multiplication is the sum of 3 repeated 4 times, which sum is equal to ^ + $ -{■ 3 -\- 3 ov 12. 26. The number which is repeated, or, in other words, is to be multiplied, is called the multiplicand; the number denoting the repetitions — i.e., the number by which it is to be midtiplied — is called the multiplier ; and the amount, or the number which is found by multiplying the former by the latter is called the product The operation of Multiplication ia expressed by the sign X, which is read into, or muitiplied by ; thus, 5 X 7 = 35- Sometimea a full stop at the bottom of the figure is used for this ; thus, 2 X 7 or 2.7, are both used to express twice seven. So again, 4 X 5 X 13 = 260, expresses the continued product of 4, 5, and 13. Multiplier x Multiplicand = Product, Multiplicand X Multiplier — Product 27. If more than two numbers are multiplied together, the result is cailod the continued product 28. The multiplicand and multiplier are termed factors of the product, because they are factors or makers of the product. 29. To perform the operation of multiplication readily, a tahle, called the vmltipUcation tahle, must first be learnt, and the result which arises from multiplying one number by another, provided neither be greater than 12, must be committed to memory ; it is one of the few operations in arithmetic where the memory of rules is indispensible. Prmciples and Practice of Arithmetic. THE MULTIPLICATION TABLE. 2J I 2 3 4 5 6 2 3 4 5 6 1 7 1 8 9 10 II 12 4 6 S 10 12 f4 .6 18 20 22 24 6 9 12 ij 18 21 24 27 30 33 36 8 lO 12 12 18 16 20 24 20 24 28 32 36 40 4+ 48 25 30 30 35 40 45 50 5S 60 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 i6 24 32 40 48 56 64 72 80 88 96 9 i8 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 II 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 In this table the first horizontal line consists of the first twelve nnmbers in order ; the second consists of the products when multiplied by 2 ; the third contains their products when by 3 ; the fourth when multiplied by 4, and so on; and the table is repeated in the following manner. Thus, to make use of the second line of figures, we say — twice I are 2 twice 5 are 1 twice 9 are 1 8 twice 2 are 4 twice 6 are 12 twice 10 are 20 twice 3 are 6 twice 7 are 14 twice ri are 22 twice 4 are 8 twice 8 arc 16 twice 12 are 24 ^ Note. — It should be noticed that the "product of any two nunibera is the same whichever of them is taken as the multiplier— fi..^., 7 times 8 -: 3 times 7. This is 'llustrated in the above Table by a double row of darli figures forming the diagoaal of the square. ; 30. When two numbers are to be multiplied together, it is a matter of indilTerence, so far as the product is concerned, which of them be taken as the multiplicand or multiplier; in other words, the product of the first multi- plied by the second, will be the same as the product of the second multiplied 26 Principles and Practice of Arithmetic. by the first. Thus, 4X5 and 5X4 express the same thing, namely, 20. This is best seen as follows : — Here there are 20 stars, or asterisks. It is evident that the total number — 20 — is not altered by the way in which wo choose to group them, in order to count them. Thus we may either say there are 5 in each line, and 4 lines, or that there are 4 in each column, and 5 columns. Now one of these ways of counting takes the 20 as 4 X 5, the other as 5 X 4. Hence these are simply two ways of arriving at the same result, or product. 32. The fundamental principles upon which the process of multiplication depends are these : — (i) If we separate any multiplicand into any number of parts, and multiply each part severally by any number and add the results, the ivhole multiplicand is thus multiplied, e.g., 15, which maybe separated into 8 and 7 is multiplied by 9, if 8 and 7 be each multiplied by 9, aud the results added together. (2) If the multiplier be separated into any number of parts, and the multiplicand be multiplied severally by each of these parts and the results added together, this is equivalent to multiplication by the wliole mulf-iplier, e.g., if it be required to naultiply 17 by 12, and we multiply 17 by 4, and 17 by 8, and add these reeQjlts, we have then taken 17 exactly, 4 + 8 times or 12 times. Let it be required to multiply 6739 by the single figure 8. Since the product 6739 by 8 is evidently equal to the sum of the products of all its parts, and 6739 = 6000 -\- 700 4* 3*^ 4" 9 we must, therefore, multiply each of these parts by 8, and add together the results ; we have, therefore, the following operation : — sands . Thousands. 6 Hundreds. 7 Tens. 3 Units. 9 6739 8 4 5 8 a- 7 4 3 72 = 240 = 5600 = 48000 = product of 9 by 8 {a) 30 ,, 8 (*) „ 700 „ 8 (c) „ 6000 „ 8 {d) __ — *-'», 1— >— 5 3 •9 I 2 539'2 = i> 6739 .) 8 Now 9 units multiplied by 8 gives 72 units («), 3 tens multiplied by 8 gives 24 tens (b), 7 hundreds multiplied by 8 gives 56 hundreds (c), and 6 thousands multiplied by 8 gives 48 thousands (d). Writing these results in the ordinary way, and adding them together, aa ^bove, the suiti, which is the required p-oduct^ is J3912. Principles and Practice of AriiJunetic. 27 In practice the partial products 72, 240, 5600, and 48000, are not written down, but combined mentally into one sum; thus, we say 8 times 9 units arc 72 units, that is to say, 7 tens and 2 units, we accordingly write down 2 in the units' placo, and 6739 carry 7 to the place of tens ; S times 3 tons are 24 tens, and the 7 tens carried make 8 31 tens, or 3 hundreds and i ten. Put the i in the tens' place and carry Iho 3 hundreds; 8 times 7 hundreds are 56 hundreds, and the 3 hundreds carried make 53912 59 hundreds, or 5 thousands 9 hundreds ; write the 9 in the place of hundreds and carry 5 to the thousands. We have next, 8 times 6 thousands are 48 thousands, and the 5 thousands carried are 53 thousands, which is the entire number of thousands, the whole product being 53912. The reason of the following rulo will now be evident. When the multiplier is not greater than 12. EULE III. i"'. Place the midtipUer under the multiplicand, units wider units, and draw a line under the multiplier to separate it from the product. 2°. Commencing at the unit's figure multiphj each figure of the multiplicand hj the multiplier. If the product of the multiplier and any figure of the multipli- cand is less than ten, set it down under that digit ; hut set down the right-hand figure only of the product whe7i it is a number of more than one figure, and carry, as in addition. Set down the last product in full. ExAMrLES. Ex. I. Multiply 6209748 by 7. To work out this sum, i;ommence by placing the multiplier under the multiplicand, units under units. Beginning at the right hand, and multiplying each figure in the multiplicand separately, the work is as follows : — 7 times i, 56, write down the 6 aud carry 5 ; 7 times 4, 28, which, with 5 62C9748 carried, make 33, set down 3 and carry 3 ; 7 times 7, 49, and 3 carried, are 7 52, set down 2 and carry 5 ; 7 times 9, 63, and 5 are 68, set down 8 and carry 6; 7 times o, o and 6 are 6, set down 6 ; 7 times 2 are 14, set down 4 and 4346S236 carry i ; 7 times 6 are 42, and i 43, set down 43. The following, worked like the above example, require no further explanation. Multiply 73826 ^073142 466052 531462 By 8 9 II 12 Product 590608 81658278 5060572 6377544 Examples for Practice. 1. 34264896 X 2 5- 91823740526 X 6 9. 987654321 X 10 2. 654321987 X 3 6. 65217347S2 X 7 10. 891237654 X ir 3. 376543198 X 4 7- 485S6S7S8 X 8 II. 647853291 X 12 4. 379865782X5 8. 573241789X9 12- 919273654 X 12 13. Multiply 7452341 six times successively by the following multipliers, 2, 3, 4, j, 6, 7, 8, 9, II, and 12. 32. To multiply any number by 10, we have only to remove each of the figures of the multiplicand one place to the left and their value will be increased ten times, or, in other words, any number is multiplied by 10 by annexing 07ie cypher, by ico by annexing two cyphers, by 1000 by annexing ihree cyphers, &c. ; e.g., 85 X 10 = 850, for, by annexing the cypher, the 5 Princiiyles and Practice of Arithneiic. units have become 5 tens, and tte 8 tens have become 8 hundreds, i.e., the several parts of the multiplicand have each received a tenfold increase, and, therefore, the whole number has been multiplied by 10. Again, 2376 x 100 = 237600, ■where the value of each figure is increased a hundred times by writing to the right of the multiplicand as many cyphers as there are in the multiplier. (a) When the significant figure of the multiplier is not a unit, as for example 30, 400, or 700. Since these multipliers are the same, as 10 468 times 3, 100 times 4, or 1000 times 7 ; the multiplicand is first multi- '^°° plied by the significant figure 3, 4, or 7, (by E,ule III), afterwards the 327600 product is multiplied by 10, 100, or 1000 as in Art. 32) by writing one, two, or three cyphers to the right of the product. Thus, to multiply 468 by 700, we have the operation in the margin. "When the multiplier consists of many figures, the process is as follows : — Example. Ex. r. Multiply 47 86 ty 2783. That is, to take 4786 2783 times and add them all together; or, to take it 2000 times, 700 times, 80 times, and 3 times, and the sums together; or, to multiply it by 2000, by 700, by 80, and by 3, aad add the products together. Ordinary form (B) I ^ i=a H W CH H> / M Having placed thn multiplier under tlio 4786 multiplicand, units under units, tc-u3 2781 under tens, &o., proceed thus : — ' £ 4786 X 3 = 14358 = 14358 4786 X 8 = 38288 4786 X 8 tens = 38288 tc7zs = 38288 4786 X 7 = 33502 4786 X 7 /«<»fi?«.= 33502 7jM«(fi.=: 33502 4786 X 2 = 9572 4786 X 2 thous. r= 9572 thorn, ziz g ^ "j 2 ,•.4786X2783 = 13319438 Now, by simple addition, the sum '-—- '■ of these products =:i33i9438 .•.4786 X 2783 = 13319438 If the ordinary method of performing the operation (see under B above) be compared with the detailed form (under A), it will be observed that by arranging the figures in the second line of multiplication one place to the left of those in the first, those in the third one place to the left of those in the second, and so on, we retain the figures in each line in their proper places without the addition of the cyphers at the end of each line. The reason of the following rule will now be evident. 33. When the multiplier is greater than 12, we proceed as follows : — EULE IV. 1° Place the multiplier under the multiplicand so that the units of the former may he under those of the latter, the tens under tens, Sfc, and draw a Urn under the whole. 4786 X 3= H35S 4786 X 80= 382880 4786 X 700= 3350200 4786 X 2000:= 9572000 (0 (S) And the sum is 13319438 (4) Principles and Practice of Arithmetic. 19 2°. TFrite down the product of the wliole multiplicand ly the unit's digit of the multiplier, as in rule. In like mayiner write down the product of the multiplicand hy each of the remaining figures of the multiplier, observing to place the right-hand figure of each line in the column under the figure of the multiplier /rowi ichich it came (see Ex. i). (a) ]f the multiplicand contains a cypher, treat it as if it xcere a number, recollecting that ox i:=o, 0X2 = 0, and so on. (b) If one or more of the figures (not final figures) of the multiplier lo o, the corresponding partial product, or products, will he o or cyphers, and the lines may he entirely omitted, recollecting to give its proper value to the product, arising from multiplying hy the :iext figure (see Ex. 2). 3°. Then add all these partial products together, and their sum will he the entire product of the factors. Examples. Ex. Multiply 821436 by 6-]2$i6. 821436 672576 4928616 5750052 4107180 1642S72 5750052 .,.928616 552478139136 ITcrc the first figure (6) of the first partial product is set below the figure 6 in the multiplier, tlio first figure (2) of the second partial product is set belo-y 7, the multiplying figure, the first figure (o) of the third partial product is sot Ijelow 5, tlie fiprure multiplied by, and so on ; the first or right-hanil figure on each row standing directly under the multiplying figure producing it. Ex. 2. Multiply 32407 by 6005. 32407 6005 162035 194442 194604035 Here the first figure (5) of the first partial product is placed immediately iDelow 5, the figure multiplied by, and since the next two figures of the multiplier being cyphers, the corresponding products will be o or cyphers, and are entirely omitted, and the multipli- cand is then multiplied by 6 and the right-hand figure of the product is set in the column directly below;the multiplier 6. Ex.3. Multiply 729 by 63S17 = 63817 X729- 63817 729 574353 127634 446719 46522593 Ex. 4. 700013 X 54836 = 54S36 X 700013 54836 700013 164508 54S36 383852 38385912868 The following example will illustrate a point of some utility connected with the position of the first figure obtained by multiplication. Ex. 5. Multiply 3421975 by 11912. In this example it will be observed that though there are five figures to multiply by, we have only three separate lines of products. The reason is that in the first place wo multiply hy 12, next by 9, and then by 1 1, which shortens the sum by two whole lines— a considerable saving. It is necessary, however, to be careful to jilace the first figure in each Case ohtained by multiplication directly under the fijinre hy which you are fnuliijplyiny. 3421975 11912 40063700 30797775 37641725 40762566200 30 Principles and Practice of Arithnetie. Examples for Peacticb. I- 7S954236 X 3+ 4- 67869578 X 903 7. 815085 X 20048 2. 98765240 X 57 5- 23589647 X 678 8. 6437063 X 5006701 3. 93876129 X 95 6. looiooi X 999 9. 958S66 X S04002 10. 378421896 X 5928578 12. 987654321 X 1234567890 11. 28814412 X 12345678 13. 4771213 X 602059999 461762 X 930937 X 972744 X 708421 9090909 X III 1 1 II X 999999 X loioioi 9999 X 99980001 X 999700029999 9999 X 9998 X 1000 X loooi X 10002 9999 X 9999 X 9999 X 9999 X 9999 X 9999 34. If the multiplier to any proposed multiplicand consists of any one or moro of the nine digits, followed by a cypher, or any number of cyphers, then multiply according to the following BULE V. 1*. Place the multiplier under the multiplicand, so thai the significant ^^wr^ of the multiplier shall stand under the unit's figure of the multiplicand, and mxdtiply the successive figures of the multiplicand hy the significant figure of the multiplier, according to Eule lY. 2°. Then, to the product thus oltained, place to the right the same number of cyphers as are contained in the multiplier. Example. Multiply 123456789 by 80 and 800000. Multiplicand 123456789 Multiplicand 123456789 Multiplier 80 Multiplier 800000 Product 9876543120 Product 98765431200000 In the first of these examples we multiply first hy 8, according to Rule m, then annex {i.e., join to) to the product one cypher, bocauso the multiplier contains one cypher, in order to preserve the product in its proper place, as the product of 8 tens. In the second example the same rule is followed, but five cyphers are annexed, because the multiplier contains Jive cyphers, in order to preserve the product in its proper place as the product of 8 hutidred of thousands. 35. If the multiplier and multiplicand both end with cyphers, we may omit them in the working, and proceed according to the following EULE VI. Multiply the significant figures of the factors, as directed in Rule IV. Then^ to the product, affix as many cyphers as have been omitted from the end of both the multiplier and multiplicand. The principle of this annexation has been already explained (No. 32). Principles and Practice of Arithmetic. 31 Examples. Thus, if 570 be multiplied by 3200, and 4076S00 by 307000. (0 (2) 570 4076800 3200 307000 114 285376 171 123304 1824000 1 25 1577600000 In tho first case, the 7 multiplied by 2 ia the same as 70 multiplied by 200; and 70 multiplied by aoo gives 14000. In the second the product of the Bigniflcant figures is 40768 X 307 = 12515776, to this _;?i'c cyphers must be annexed, because 100 X 1000 z=: ,>4ioooo; and 12515776 X 100000 = 1251577600000, Examples FOB PiUOTICE. I. 2. 3- 4. 5- 80437050 X 90 98311950 X no 4381792800 X 5600 41508473 X 7002800 42456008000 X 5400 6. 7- 8. 9- 10. 765321 X 760 325758 X 72000 567423 X 4080 49864023 X 708600470 30001000300 X 400100020000 36. It is sometimes advantageous to split up a multiplier -which is the product of two or mor^ numbers, and multiply by its factors ; thus, if we have to multiply by 36, it is easier to multiply in this case by 6 and 6 (6 x 6 = 36), or by 4 and 9 (4 x 9 = 36), than to multiply by long multiplication, that is, by 3 tevis and 6. In any case we have two rows of multiplication, but in the last case we have an addition into the bargain. Example. Multiply 57894362 by 48, Since 48 = 6 X 8 ; or, 4 X 12 = 48, then the answer is found as below. 57894362 5789436a 6 4 6 times the multiplicand = 347366172 4 times the multiplicand = 231577448 8 12 2778929376 y/««. 2778929376 ^«s. .'. (6 times 8 times :=) 48 times tho • ' • (4 times 12 times =) 48 times the multiplicand. multiplicand. Examples for Praotiob. I. 685732 X 15 4- 279819 X 72 7. 108143145 X 121 a. 356628 X 36 5- 3567'8 X 8r 8. 117974340 X 132 3. 434560 X 56 6. 29362983 X 84 9. 128699280 X 144 37. Verification of Multiplication. — I. By casting out nines. — Add together the figures of the multiplicand, multiplier, and product separately, not counting any 9 that may occur, rejecting also 9 whenever, in adding up, the sum amounts to 9 or morej note each result. Multiply the first two 32 Principles and Practice of Arithmetic remainders, i.e., the remainder arising from casting out nines in the multipli- cand and multiplier, retaining, as before, only what is left after the rejection of all the nines from this product, if the sum of the digits exceed nine ; then, if the remainder which thus arises is the same as that from the product of the two factors, the operation is likely to be correct, unless there be some compensation of errors, or some figures misplaced.* Thu3, in the annexed example, we say (omitting the 9) 3 and 7 Multipl)^ 9037 6.... 7 are 10; then i and 6 are 7, which write down. Again, 2 and 8 are 2083..., 4 10; then I and 3 are 4, which is also put down near the multiplier. L^(' result is 4. This product, therefore, is incorrect, and, upon revising the multi- 4 plier, we find that the 3, after the nought, should ha,vo bqea 2.. Principles and Practice of Arithmetio. 33. DIVISION. 38. The object of division is to find how many times one number is contained in another. The quantity to be divided is called the dividend, the quantity by which we divide is the divisor, the number of times is the quotient, an(?: what remains over (if any such there be) is called the remainder. Dividend = divisor x quotient + remainder. TL^ operation of Division is expressed by the sign -;-, which is road by or diinde by; thus, 42 J- 7 = 6, implies that the result of the division of 42 by 7 is 6. The number 42 which is divided is called the dividend, that which divides, i.e., 7 the divisor and the result 6, the quotient, 39. The first idea of obtaining the result is to use subtraction and count the timai'v we have to use it. Thus, ?>€> find how many times 8 is contained in 34. 34 J (I) 26 1 ^'^ 18 _8 (3) 10 _! ^^^ "Wi,-^8ee we can take 8 away from 34/02^^ times in succession, and then we leave 2. But if we had helped ourselves by the multiplication table (of eight times) we might have done it more shortly. For since 5X8 = 40, 8 wiU go 5 times into 40 exactly ; therefore, 8 will not go 5 times into 34. Again, 4 x 8 = 32, and thus 8 will go 4 times into 34 and leave something over. This " su'^nething over" is evidently 34 — 32, or 2. So long as the quotient is a small number the process of continued subtrac- tion may be employed, but when the dividend contains the divisor a large number of times, it would be necessary to abridge the operation by taking away as many times the divisor at once as we please, provided the number of times is marked at each step. For example, to divide 115 by 12 wo may take away 8 times 12 at once, and afterwards take away 12; therefore, 12 may be subtracted 9 times, and the remainder is 7. 96 =z 8 times 12 12 = once 12 7 = remainder. In order to avoid the labour of repeated subtractions, we may lay down the following principle, viz., that if we separate any dividend into any number 54 Principles and Practice of Arithmetic. of parts, and find how often the divisor may be subtracted from these part3 (or how often the divisor is contained^ as it is called, in each of these), we shall, by adding these results, obtain the correct quotient of the whole dividend divided by that divisor, because it is evident that the loliole dividend will contain the divisor as many times as its several parts together contain it. Let it be required to divide 3168 by 27. Here the quotient will consist of three digits, and therefore there will bo at least 3 separate subtractions Now the figure in the hundreds' place cannot be more ^168 than I, and if the product 27 hundreds, or 2700, be 2700= 100 times 27 subtracted from the total product 3 168, the remainder 468 468 must contain the products of the tens and units of ^'^° '° ^™^^ ^' the quotient multiplied by the divisor 27. We now 19^ ■1 P, • , ■ 1 , ,• • .r. 189= 7 times 27 enquire how oiten 27 is contained ten times m 468, and this is found to be only once ten times; then 9 subtracting the partial product 27 tens, or 270 from 468, tho remainder ia 198. Lastly, we have to divide 198 by 27, which gives 7 for a qi-iotiant and a remainder 9; and, therefore, 3168 contains 27, 100 + 10 + 7, or 117 times, leaving 9 for the remainder. It will be seen that as often as 27 is contained in 31, so many hundred times will it be contained in 3100, or in 3168 ; arl aT often as 27 is contained in 46, so many ten times it will bo contained in 460, or 468, and in this manner any quotient figure is just as readily obtained as the last or unit's figure of it. 40. The preceding articles contain the principles of division, and all that remains is to apply them in the most economical manner. Example. Suppose we have to divide 29S7618 by 3605. Operation with cyphers in full. Operation without annexing cyphers. 3605)2987618(800 4- 20 -f 8^ 3605)2987618(828 2284000 or 828 28840 103618 10361 72100 7210 3^518 3'5'8 28840 28840 2678 2678 41 . If the divisor be not greater than 12. EULE VII. 1°. Eet down the dividend with a line under it to separate it from the Juture quotient ; and write the divisor on the left of the dividend with a curved line between them. Principles and Practice of Arithmetic. 35 2°. By the mxdtiplication table find the greatest mimher of times the divisor is contained in the Qrst figure of the dividend, or, if necessary, the first two or first ilaxeQ figures of the dividend, and place the figure denoting the times directly under the figure divided, or tmder the last figure ?/more than one have been taken, and carry tvhat is over, that is, regard the figure tchich is over to be prefixed to the foWovring fig7ire of the dividend. 3°. Pivide this number by the divisor, set down the result as the next figure of the quotient, carry the remainder to the next figure of the dividefid; and if the divisor is not contained in any figure of the dividend, 2^lacc a cyplior in the quotient one? prefix this figure to the next one of the dividend as if it loere a remainder, and proceed in the same manner till all the figures of the dividend are exhausted. The number thus found is the quotient. Examples. Ex. r. Divide 25602 by 3. Placing the dividend and divisor (3) as in the margin, we proceed thus: — 3)25602 3 is contained in 2, no times, so that nothing is to bo placed under the 2 ; 3 is contained in 25, i.e., 25 thousands, 8 times, {i.e., 8 thousands) and i over, 8 and "534 carry r ; this i regarded as prc/i.vcd to the 6, gives the number 16, i.e., 16 hundreds, 3 is in 16, 5 times and i over; 3 is in 10 {i.e., i hundred) 3 times and i ovur; 3 is in 12, 4 times. Therefore, the quotient is 8534; and this is the complete quotient as there is no remainder Ex. 2. Divide 7804623 by 5. "We say, 5 in 7, i and 2 over; 5 in 28, 5 and 3 over; 5 in 30, 6 ; 5 in 4, 5)7804623 o and 4 over ; 5 in 46, 9 and i over ; 5 in 1 2, 2 and 2 over ; 5 in 23, 4 and ■ 3 over. As there is here a remainder, we annex it, with the divisor 5 under 1560924J it, to the figures of the quotient, and call 15609245- the complete quotient. Ex. 3. Divide 84111648 by 12. We say, 12 in 8, «o times, so that nothing is to be placed under the 8; 12)84111648 12 in 84 goes 7 times and o over; 12 in i, no times and i over, put down o under the i ; 12 in 11 o times and 11 over; this 11 regarded as prefixed 7009304 to the next figure of the dividend gives the number in ; therefore say, 12 in 11 1, 9 times and 3 over; 12 in 36, 3 times and o over; 12 in 4, o times and 4 over ; 12 in 48, 4 times. Examples for Practice. !• 135792695-^2 5. 400678493-^-6 9. 254096146-^10 2. 584697386-^3 6. 276586437-^-7 10. 1101182267 -;- II 3- 399345884-^4 7- 6947421006-^8 II. 1095137170-^-12 4. 398244760-^5 8. 2470263075 -;- 9 12. 59437055312-^-12 Divide each of the following dividends six times successively by each of the foUowin* diTisors— 2, 3, 4, 5, 6, 7, 8, 9, 11, 12. 1. 6154778 3. 4086791 5. 2000707 7. 9914060 2. 7000000 4. 8821000 6. 9370005 8. 4706009 42. If the divisor be greater than 13. EULE VIII. 1°. Place the divisor and dividend in the same line, separated by a small curve line, and on the right of the dividend draw another line of the same kind; thus, — divisor ) dividend ( quotient 3 6 Principles and^ Practice of Arithmetic. 2°. Marli off from the left-hand &ide of the dividend a numler of figures equal in tiumher to those of the divisor, or one more, if necessarij, and find the greatest number of times the divisor is contained in this number ; place the figure which denotes this number on the right as the first figure of the quotient. 3°. Multiply the divisor bg this number and place the product under the number marked off at the left of the dividend.- Subtract the said product from that part of the dividend under ivhich it stands. 4°. Bring down the next flgtcre of the dividend and place it to the right of the remainder, and if the number thus formed he greater than the divisor, find the greatest number of times the divisior is contained in it, and write this number as the BQ(:,i-ndL figure of the quotient; but if this number be less than the divisor bring down the next figure of the dividend, or more, until a number not less than the divisor, is form^'d, remembering to place a cypher w the quotient for every figure brought down, except the last ; find how often the divisor is contained in this number ; then multiply, subtract, and bring down, ^'c, as before, till all the fgures of the dividend are exhausted. The number thus obtained is the quotient required. Note. — When the divisor is large, the learner will find assistance in determining the quotient fig'ire, by finding how many-tiines the first one or two figures on the left hand of the divisor is contained in the first one or more of those of the dividend. This will give pretty nearly the rijbt figure. Some allowance must, however, 1d°i made for carrying ir<,nx the product of the other figures of the divisor, to the product of the first iTito the quotient figure. If any product be greater than the number which stands above it, '.he last figure in the quotient must be changed for one of smaller value ; but if any remainder be greater than the divisor, or equal to it, the last figure of the quotient must be changed for a greater. Examples. Ex. I. Let it be required to divide 256434 by 346. Looking at the leading figure of the divisor, and also at that 346)256434(741 quotient, of the dividend, with the view of seeing whether the latter con- 2422 tains the former, which it does not, 3 being greater than 2 ; we — — . therefore commence with the number 25, formed by the first ^^ two figures of the dividend, and seeing that 3 is contained in 25 _^__ 8 times, we should put 8 for the first quotient figure ; but 394 bearing in mind that when the whole divisor is multiplied by 34^ this 8 we must attend to the carryings ; we perceive that 8 is too " great, we therefore try 7, and find 7 times 346 to be 2422, a ^ number less than 2564 above it, so that we can subtract ; the remainder is 142, which, when the next figure of the dividend is brought down, becomes 1423. We now take this as a dividend, and looking at the leading figures in this new dividend and the divisor, we see that that latter will go 4 times, we therefore put 4 for the second quotient figure, and multiplyiu" and subtracting we get 39 for the second remainder, and, by bringing down another figure we get 394 for a new dividend; the divisor goes into this once, so that the quotient is 741, and the final remainder 48 ; this remainder must be annexed with the divisor underneath to the quotient figures, so that the complete quotient is 741 -^sj which is the 346th part of 256434- Principles and Practice of Arithmetic. i1 Ex. 2. Divide 108419716214. by 5783. 5783 ) 108419716214 ( 18748005 quotient. Quotient figure (i) 5783 (8). (7). (4). (8). (o), (o), (5). 50589 1 3t remainder with next figure. 46264 43257 ^nd 40481 46296 46264 3" 3221 32214 28915 3rd 4th 5tli 6tb 7 th 3299 final remainder. It must bo noticed that if any dividend formed by a remainder and a figure brought down Bhould be less than tho divisor, that the divisor will go no times in that dividend ; so that a o will be the corresponding quotient figure ; and that, then, a second figure must be brought down as in tho operation annexed. The stepn marked t^T are inserted merely to show the i)rineiple. In praotico we simply put do'ivn the two noughts in the quotient, and go at once to 32214 for the divisor. Ex. 3. Divide 6421284 by 642. 6i2 goes once into 642, and leaves no remainder. Bring down the Vext figure (i) of tho dividend, then 642 is in i no times, put o in tho .quotient after the i. Tho next figure of tho dividend (2) being "brought down to the right of the i forms the dividend 12, then 642 13 in 12 no times, put a cypher as the next quotient figure; bring down the next figure of the dividend (8), then 642 in 128 goes no times, write o as the f corresponding quotient figure ; but the next figure brought down (4) makes the quantity trought down 1284, which contains the divisor twice, and gives no remainder. 642)6421284(10002 642 1284 1. 983296-7-13 2. 800062 "7- 23 3. 20067690 -^ 37 ID. I471O962989869 -7- 1709 II. 2107791630-7. 3654 16. lOOOOOOOOO — lOOI 17. 1 00000000000000 -7- III II 18. 48423I57I3782 .7-570634 19. 815240906170-^-763054 21. 490002800004-^-900702 26. Divide 1000 (with places of figures in the quotient) by Examples for Practice. 4. 9500864 -r 43 5. 9943000 --r 78 6. 8904030 -j- 89 12. 1395243584-7-5678 13- 3855999705-7-6789 7. 56703264.7-123 8. 57 1 4 14 204 -7- 809 9. 94010610 -7- 987 14. 940870015 -f- 8764 15. lOOOOOOOOO -f- 9999 21. 6680943744279021-7-95400621 22. 721932631112635269-^987654321 23. 2000000018760631-^-31622777 24. 7922283322805843200 -7- 879510067 25. 296229611814587191480656 -f- 972744 as many noughts added as may be necessary to give ten 2302585093. 38 Principles and Practice of Arithvietic. 43. When the divisor is a composite number, that is, can be separated into two or more factors, the division can bo effected by the following rule : — EULE IX. 1°. Divide hj one factor, setting doivn the quotient and remainder. 2°. Divide the quotient thus obtained hy another factor, scttinrj down the quotient and remainder, and so on, till all the factors are employed. The last quotient will be the answer required. 3°. The proper remainder is formd, when the divisor is resolved into hut two factors, ly ninltiplying the second remainder ly the first divisor, afid to the product addi?iri the first remainder, but when more than t^-o factors are employed hy multi2')lying the remainders after cvrey line hy all the divisors except their own, and adding the results. Examples. Ex. I. Divide 569736869 by 15. Here, since 15 is the product of 3 and 5, it is obvious that tbe quotient may bo obtained from successive divisions by 3 and 5. ^5 Hero, the remainder 2 in the first quotient is 2 units of the upper line ; but the remainder 4 in tbe second line consists of 4 units of the second line; and as each unit in the second line is tJirec times as great as each unit ia the upper lino, the remainder 4 is equal to 3 X 4 units of the upper line, i.e., is ( qaal to 1 2 ordinary units, hence the whole remainder is 2 -j- 1 2, or is 14. 569736869 37982457.... 4 Ex.3. Divide 8327965 by 7z. Here, since 72 is the product of 8 and 9, the quotient is obtained by successive divisions by 8 and by 9 ; otherwise, since 6 X 12 = 72, we may first divide bj' 6 and then divide the resulting quotient by 12. 8327965 Ex. 3. Divide 8327965 by 99. In this instance 99 is tho product of 9 and 1 1 ; we divide by 9 and the quotient thus arising we divide by 11. 9 99 72 8327965 • 925329. ...4 1040995 84120.. ..9 1 15666. ... I To deduce the remainders which would have been left had the divisions been performed by 72 and 99 in the usuil way, we may observe that the first partial remainder 5 (Ex. 2) must be units ; but the second remainder must be regarded as so many collections of 8 units each and that the first partial remainder 4 (Ex. 3) must be units ; but tho second dividend bein" as so maiiy collections of 9 units each, the second remainder must, in this case, be regarded as so manj' collections of 9 units each ; hence tho true remainders in these examplea are respectively I X 8 + 5 = 13, and 9 X 9 + 4 = 85- Ex. 5. Divide 24533279 by 432. Ex.4. Divide 2671998 by 192. The factors for 192 are 4 X 6 X 8 = 192. /4 192 2671998 667999., III333-' X 4 + 2 = 6 13916....5 X6 X4 + 6= 126 DivitlinK by 4lhc remainder is 2, cliTi'ling by 6 tho rcmaiii'li'r is" 1, diviiliii^ l.y 8 tbe remainder is 5, ■wbii'h pives a tolal remainder 1^6, fuiind thu^ : lliu remiiinder 2 is two units, the second remainder is one 4 and 2, innldng 6, the third remainder 5 is five times 24 (4 X 6) ="120, because iiavin;; divided by 4x6 = 24, the quotient Hij3? are twenty-fours, so that any remainder must consist of twenty-fours ; 1 20 added to tlie previous reuiaiuder 6, gives 126. Thus the answer is— quotient 13916, remainder 126. Since 6 X 8 X 9 = 432, vre may d'ivide successively by these numbers. 6 432 24533279 40S8879....5 511109....7 X 6 + 5 = 47 56789. ...8 X 8 X 6 + 47 =r remainder 43 1 . The first remainder is 5, (he second remainderis 7, then 7 times tlic first divisor (= 6) + j (the first remainder) = 47. Tho third remainder is 8, then 8 times S (the second divisor) = 64, and 64 times the first divisor 6 = 384 + remainder 47, gives the final remainder 431. Principles and Practice of Arithmetic. 39 Examples poe Phactice. I. 6489275432689467 -r 14 2- 598432789648320758-^22 3- 56983475689268-^36 4. 59864326S5946896 -i- 63 5- 987654321012345-7- 66 6. 543210123456789 -f- 121 7- 3.59f32i39'62i9ii -^ 132 8. 4902550716552769-^-144 44. Division may also bo abridged wliero the divisor is terminated by a cypher or cyphers ; we proceed as follows : — EULE X. 1°. Cut off the cyphers from the divisor, and as many fijurcs from the right-hand of the dividend as there are cyphers so cut off. at the right-hand end of the divisor, then proceed with the remaining figures in the usual mafiner (Rule VII or VIII), and if there are anything remaining after the division annex those figures which are cut off from the dividend; othenoise, the figures cut off will be the remainder. Note. — The same rule applies when the divisor and dividend both terminate with cyphers. Examples. Ex. r. Divide 3704196 by 20. 2,0)370419,6 185209^5 In the first of those examples you mark off with a comma the cypher or o in the divisor, and the first figure 6 to the right in the dividend ; this is equiva- lent to dividing both divisor and dividend by 10. You next divide the remaining figures 370410, to the left in the dividend, hy the divisor 2, according to Rule VII ; thus is obtained tlie quotient 185209, and remainder 1 ; to this remainder you annex the figi:re 6, which was cut off, and you have the complete ivmainder 16. The quotient may now be correctly Represented thus, 1852091^. Ex.3. Divide 271830 by 30. 3.0)27183,0 9061 Ans. ist. Cut ofT a cypher from the divisor, and also one from the dividend. ind. Divide 27183, the remainder of the dividend, by 3. the remainder of the divisor ; the quotient 9061 is the answer. Kx. 2. Divide 31086901 by 7100. 71,00)310869,01(4378 U'ish- 284 268 213 497 599 568 31 In the second example you follow tho same rule ; that is, you cut off two cyphers in tho divisor and two figures in the dividend, and obtain tlie quotient in the usual way, which is .1 -,78, and remainder 31 ; to this 31 annex tho two figures cut off from tho dividend, and you have the complete remainder j 101. Examples for Pkaoticb. r. 9357864837986496-^-50 2. 674008694738425 -j- 700 3. 987654321670 -J- 3000 4. 483795864973206789 -f- 120 5. 6550000280034 -f- 6300 6. 26799534687-7-7890000 45 . Verification of Division. — ( i . ) Multiply the quotient by the divisor, or the divisor by the quotient, and to the product add the remainder, if there be one. The result ought to be the same as the dividend; because we are only adding the divisor the same number of times, as it was subtracted in the operation of division. (2.) Subtract the remainder, if any, from the dividend, and divide the difFerenoo BO obtained by the quotient. The result should be equal to the divisor, if tho working be correct. 4© Principles and Practice of Arithmetic. MISCELLANEOUS EXAMPLES. 1. Express in figures : Ten thousand and four. Four thousand and four. Forty-four thousand and four. 2. 29483 + 7648 + 32479 + 586 + 298364 + 98765 4- S97 + 7S9 + 567S + 99. 3. From 6794006897 take 3985160534, and from loooio take looii. 4. Multiply 94785830 by 78060, and 879510067 by 90076096000. 5. Divide 5688208152 by 594, and 1 00000000000000 by 967. 6. Express in figures : One hundred million one hundred thousand and one hundred. 7. Add together 90473, 9456, 268, 59, 45694, 5437, 87668497, 2837, 9865, 3652, 999, and 8. Find the difi'erence between looooooooooo and 87649786. Take one hundred thousand two hundred and twelve from four million one hundred and one. 9. Multiply 326904678 by 3060900. 10. Divide 236487698743 by 85409, and 5754054870008880 -J- 8009909. 11. Express in figures: One hundred and three million eighty thousand two hundred and seven. 12. Add together 69074, 6745, 723, 29, 931648, 9005, 76245, 54267, 47096, and 7777. 13. From 78600070000 take 6974208506. 14. Multiply 167409678 by 768900. How much greater is 678 X 76 than 54675 -f- 9 ? 15. Divide 60000007006490088805 by 98706543. 16. Express in words and in figures how much greater the value of one 5 is than the other in the number 658457. 17. Multiply 129847 by 468. If, in the process, you shift all the figures resulting from the multiplication of the multiplicand by 4 two places further to the left and then add, of what two numbers will the result be the product ? 18. What number subtracted from 850967 will leave 3946 ? The 365th part of a number is 10 1 00 1, what is the number ? 19. The digits in the units' and millions' places of a number are 4 and 6 respectively, what will be the digits in the same places when 99999 is added to the number? 20. "What number must be added to sixty-nine thousand four hundred and twenty seven to produce three hundred and twenty-five millions seven thousand and twenty-one ? 21. Find the sum, difference, and product of 12345678 and 2S8144412. Find the sum, difi'erence, and product of 1234567 and 4321089, 22. 15996 tons of coal are exported in 43 ships: how many tons does each ship on tho average carry ? 23. How many years of 365 days each in 46355 days ? 24. How often can you subtract 6 from 47 112 ? What number subtracted three times from 78467 will leave 41426 ? 25. How many ships, each carrying 673 men, can transport an army of 22882 men ? 36. By what number must you divide 7460020 in order that the quotient may bo 52907 and the remainder 133? 27. 2036809 divided by a certain number gives a quotient 2031 with a remainder of 1474 : find the dividing number ? 28. A ream of paper contains 20 quires of 14 sheets each ; on each page there is room for 34 lines of writing : how many may be written in the ream ? 29. What is the number of holes in a sheet of perforated zinc, containing 1519 squELce inches, if there be 85 in the square inch ? 30. What will remain after subtracting 2i3 aa often as possible from 83216 P 31. The product of two numbers is 1270374 and half of one of them is 3129 : what ifl the other number ? 32. Find the sum, difference, product, and quotient of 1653 125 and 13225. Decimal Fractions. 41 33. Find the sum, difference, product, and quotient of 9765625 and 78125. 34. What number subtracted three times from 78467 ■will leave 41426 ? 35. A number increased by 13 times itself amounts to twenty millions five thousand and six : find the number. 36. If the sum of 250 and 173 be multiplied by their difference, and the product divided by 33 : find the result. 37. Find the difference between the sum of 47 15 added to itself 398 times, and the sum of 2017 added to itself 408 times. 38. A man died in 1873, aged 94 ; his son died in 1827, aged 17 : how old was the father when the son was born ? 39. A gentleman being asked how old his son is, replied "Tn 23 j'ears ho will be as old as I was when he was bom, and I shall be 58 : " how old is the son ? 40. The Mariner's Compass was invented in Europe in the year 1302 ; how long was that before the discovery of America by Columbus, which happened in 1472 ? 41. The distance of one of Jupiter's moops from that planet is 490140 miles, and of another 652650 j find how much further one is from Jupiter than the other. 42. From the centre of Saturn to the outside of his inner ring is 71870 miles, to the inside of the same ring is 54560 : find the breadth of the ring. DECIMAL FRACTIONS. 46. Aritliinetical operations become lengthy and troubleaomo if they involve many vulgar fractions of different denominations ; it becomes neces- sary, therefore, to devise a method of expressing fractions in such a manner that they may be easily reduced to the same denomination. To effect this all fractions are reduced to others having for denominators 10, 100, 1000, (See. Such fractions are called Decimal Fractions. 47. Decimals occur so frequently in all computations relating to Nautical Astronomy, that it becomes absolutely necessary to have a knowledge of their application and their relation to Vulgar Fractions. 48. In the Notation of Integers or common numbers, the actual value of each figure depends upon its position with respect to the place of units, its value in any one position being one-tenth of what it would bo if it stood one place further to the left ; thus the number mi denotes one thousand, one hundred, one ten, and one unit, or 1000 + 100 + 10 + i, where the second unit beginning with the right-hand one is ten times the first, the third is ten times the second, the fourth ten times the third, and so on ; or beginning with the first on the left, the second is the tenth part of the first, the third the tenth part of the second, and so on, till we come down to the last unit, which is merely one ; or in other words, th« figures decrease in a tenfold ratio from left to right. 49. Now we may evidently extend this principle still further, and on the same plan may represent one-tenth of one, one-tenth of thift, or one-hundredth of one, one-thousandth of one, and so on, by simply putting some mark of separation between the integers and i\iQQQ fractions. The mark actually used a 4* Decimal Fractions. is a dot or full stop, and is called the decimal point, thus 1 1 1 1 • 1 1 1 1 .* The unit (or i) next the dot, on the left, is i; the unit one place from this on the left is lo; the next is loo; the next looo, and so on. In like manner, the unit next the decimal point, on the right, is -^, the next y^, the next x^oo, and so on. In other words, any figure one place to the right of the unit's place will be one-tenth of what it would be if it were in the unit's place, and will thus really denote a decimal fraction ; any figure two places to the right of the unit's place will be one-hundredth of what its value would be if it were in the unifs place, and so on for any number of figures, as in the following liable, which may be regarded as an extension of the numeration table. 7654321-2345678 50. This being agreed upon, it follows that a decimal may either be con- sidered as the sum of as many fractions as it contains digits, or as a single fraction ; thus : — 5^7 — To ~ 100 1^ 1000 — 1000* .-,,f^- _0 1 . 3 _1 O I . 5 __ _p305 ^i'-'S — To T^ 100 *^ 1000 *^ 10000 — 10000* 10 204 10 T^ To T^ To 0^1000 1000* 5 1 . Hence, a decimal is always equivalent to the vulgar fraction whose numerator* 18 the decimal considered ,as integral, that is, the number itself, when the decimal point is suppressed, and whose denominator is i followed ly as many cyphers as there are decimal places in it. 52. We generally speak of any figure in a decimal as being in such a place of decimals; thus, for instance, in 3-14159, we should say that the 5 is in the fourth place of decimals, th^ 9 in the fifth place, and so on, reckoning from left to right. 53. The figures i, 2, 3, 4, 5, 6, 7, 8, 9, in a decimal are sometimes called significant figures or digits; thus, in such a decimal as '0002345, we should say that 2 is a significant digit, because it is the first figure which indicates a number, the cyphers only serving to fix the place in which the 2 occurs. 54. Numbers made up of whole numbers and fractions, either vulgar or decimal, are called mixed numbers ; for instance, 368 "4 14 is a mixed number, the figures which precede the decimal point (!;he 3, the 6, and the 8) are whole numbers or integers, while those which follow the point (*4H) ^^^ decimals. * The decimal point should be put at the top of the line of figures, thus— 5*7, because 5.7 with a stop at the hottom is used in most works to mean 5 X 7 = 35- Becimal Fractions. 43 55. To read off, or express in words, decimal fractions, read the decimal figures as »/ whole numbers, and to the last figure add the name 0/ the order deter- mined by the place it occupies; thus, '734 is read seven hundred and thirty-four thousandths; 58-64327 is read fifty-eight, together with sixty -four thousand three hundred and twenty-seven hundred- thousandths ; -080905 is read eighty thousand nine hundred and five millionths. In reading decimals as well as whole numbers, the unit's place should always bo mado the starting point. It ia advisable for the learner to apply to every figure the name of its order, or the place which it occupies, before attempting to read them. Beginning at the unit's place he should proceed towards the right, thus — units, tenths, htmdredths, thousandths, &c., pointing to each figure aa ho pronounces the name of its order. In this way he will be able to readi dejimals with as much ease as he can whole numbers. 56. The value of the decimal figures depending entirely on the place they occupy with respect to the point which separates the units from the tenths, any number of cyphers on their right may be annexed or efi'aced, without altering the value of the significant figures. For instance, 0*7 is the same aa 0*70, because the number that expresses the decimal fraction becomes ten times gyeater while its parts become hundredths, and are therefore diminished ten times, ITina T- — "L — J'OO i^P inUS yiy y-^Tj Too^' °'^'» and hence it is evident that annexing cyphers to the right hand of decimals does not change their value, for we only multiply both numerator and denominator by 10, 100, &c., and consequently does not alter their value at cdi Again, take a decimal such as '56, which, as already explained, means ? teaths 6 hundredths, it will follow that -560 means 5 tenths, 6 hundredths, no thousandths; whence the addition of the cypher to the right-hand has made no alteration in the value of the decimal. In fact, •56 = ^«o and "560 = T^^'V = iVo- Similarly -23, -230, and '2300, are all of equal value, for expressed as fractions they are respectively ^Voj tVo^o* ^^^ ^'^°"- 57. But placing cyphers between the decimal point and the other decimal figures does alter the value of the decimal, because this alters the place of the significant digits, the valuo being diminished ten times for each cypher that is prefixed, thus '7 = -1^, '07 = TOO) '007 = Tours> and so on. We infer from this, that as the value of a decimal is decreased ten-fold for every cypher added to the loft-hand, we do in fact divide a decimal by 10, by 100, by 1000, &c., as we shift the decimal point one, two, three, &c., places to the left ; and that conversely ])y shifting the decimal point one, two, three, &c., places to the right, we multiply the decimal by 10, by 100, by 1000, &c. For instance, the expression 56-789 is divided by 10 if written 5*6789, is divided by 100 if written "5678, and is divided by 1000 if written -056789; whereas the expression -00723 is multiplied by 10 if written -0723, is multi- plied by 100 if -723, and is multiplied by 1000 if written 7-23. 44 Addition of Decimals. Examples for Practice. Express as decimals — J. -i"o, T^, rrhs, and i§ ; also i^, i^^J-, -^0% and i§M. 2» lio} TTkToo* Tulioff) looooOOOj T!?? T61S> ToW) ToocTO) and. To oo • - 3.01 ±2ooi isaoiift iiiaaoifii 4itJL__ _3 3i , ., ^a?. , 3- 10 > locr > lOoouiTj nuUiontlia.' thousandths' tenths' biHionths" r 3 014.2 72819 072B1& 67 28100 6. In the following mixed numbers wi^ite the fractional parts in decimals : — 7 1000) 43 TooiTO' 9 To'Sooooo^' ^ ioooooo> and 35 toooBoo' 7. Express as decimal fractions the following : — Seventj'-three thousandths ; one hundred and ninety-seven ten thousandths ; one millionth; two hundred and sixty-one hundred thousandths; one thousand and one ten millionths, 8. Express as decimals the following :— One, and fifty-four hundredths ; twenty-four, and seventy-nine thousandths ; three hundred and fifteen, eight thousandths, and fifty millionths ; eleven hundred millionths ; nine thousandths, and three hundred thousandths. 9. One tenth ; three Landredths ; five thousandths ; one hundred and five thousandllis ; two millionths' cixty millionths; fL.^ty-one and eight hundredths; one thouEarrT and one thousaudth ; thirty and six millionths ; one hundred thousandth ; two thousand three hundred and S3venty-5ve hundred millionths. 10. Express in words the following decimals and mixed numhere :— •283, -5321, •74895> -82:056, 27-8354, 34-0009, 43-101007, 23-75, 2"375> '2375. •00002375. 11. Express in words the following: — •6, -92, -5498, 7-07, 26-405, -oooooi, -00037, ii-ioiior, -0440308, '82344, '13236. 12. Write in words 9-0457 ; 4004-0000345 ; 3-400 ; 524000634-0008034 ; -000003705 ; •000024056; 7005-000000674; looooo-ooooooi ; lO'oox ; 9-000028; 1-0006003. 13. I'oooooi ; -loooooi ; -oooooooi ; 1-13004; 9-203167; 4-3008004; 27-4627350. 14. What is the effect of moving the decimal point backwards or forwards ? ADDITION. 58. Decimals, or integers and decimals mixed, may be added together precisely as in ■whole numbers, care being taken so to arrange the figures that all the decimal points fall exactly under one another. This will ensure that tenths fall under tenths, hundredths under hundredths, &c. The reason of this arrangement will appear from the following considera- tion : if this rule were not observed, tenths would fall under nundredths, or hundredths undo, thousandths, as the case might be; and we should be attempting to add together fractions which had not common denonjinators. But if we arrange the decimal points all exactly beneath one another, tenths fall under tenths, hundredths under hundredths, &c., in other words, by so arranging thorn wc at once bring the several fractions to a common denominator, and can proceed to add them together. The decimal point, in the answer, will fall exactly beneath the decimal points in the quantities to be added. When the sum of any figures exceeds 10, 20, &c., carrying to the next denominator will be performed exactly as in whole numbers, whither the given quantities are all decimals, or are mixed integers or decimals. For as the value of each figure decreases tenfold as we proceed from left to right, the ruled of ordinary addition are immediately applicable. Addition of Deeimats. 4^ We have, therefore, the following rule for addition : — 1°. line ; Ex. •78 •678 2-258 Ex. EULE XI. Place the quantities so that their decimal points shall he i?i the same vertical for then the quantities of the same denomination will stand together. Then proceed as in addition o/tvhole numbers. Examples. I. For instance, let it bo required to add together -8, -78, and -678. Where we see that after writing in the answer 8 in the place of thousancKha, that 7 hundredths and 8 hundredths added together make 15 hundredths ; hut 15 hun- dredths are i tenth and 5 hundredths, writing 5 in the pUice of hundredths, and carrying one to the place of tenths, we obtain 22 tenths ; but 22 tenths are properly written as 2 integers and a tenths. 3-007 42-6 •3975 Again, where integers and decimals are mixed. Where writing 5 in the place of ten thousandths, the sum of 7 thousandths and 7 thousandths is 14 thousandths ; writing 4 in the place of thousandths, and carrying i to the place of hundredths, we obtain 10 as the sura in the hundredths place; but 10 hundredths are i tenth, carrying i to the place of tenths, we have 10 tenths ; but as 10 tenths are i unit, we carry i to the place 46-0045 of integers, and write 6 in the place of units, and 4 in the place of tens Ex. 3. Add together 9-12. 0-35 47 "4 9'12 o'35. 47'4, 1234-6789, and 13. Ex. 4. Add together 23-628, 4-1056, •0137, and '0042. 33-628 4-1056 ■0137 ■0042 36-87 Ex. 5. Add together 170, -0054, -5, and .S7-I42. 1234-6789 13 170 •0054 •5 87'i42 Tf?l6» 27-7515 Ex. 6. Add together 66199-3226, '301, 54-5, -00632, 1000, -07, and 32745'8ooo8, 66199-3226 •301 54"5 •00631 1000 •07 32745-80008 Examples £'oa Piiactioe. i'ind the value of 1. '225, 3*o86, 12*17, ■°05'^> ^^d 729*54; 2*63, '263, '0263, and "000263. 2. 8-1, 40-652, 98-51, 695-7, and 43"97o6 ; 69-75, 0-97, 0-059, 673'5. 4"8, and 932-6. 3. 897*4, 63-18, 400-03, 7*9, 63-9, and 5-0079 ; -00162, -1701, 325, 2-7031, and 3*000701 4. 3608-26, 360-826, 36-0826, 3-60826, and •22314; 467-3004, 28-78249, 1-29468, and 3-78241. 5. 36-053, -0079, -000952, 417, 85-5803, -0000501. 6. 87-1 4-0-376 + -0056 4- 49 + 3-009 + -709; 293-0072, 89-00301,29*84567, 924*00369, and 72*39602. 7. -8 + -046 + 9*1 + 3*09 + 8-6409 + 32 ; 1*721341, 8-620047, 5i'720345, 2-684, and 62-304607. 8. I + *i + *oi + 'ooi + *ooox ; 4*07 + -6201 + -936 + 39*08 + 1*0101 + 7. 9. I + -2 + -03 + '004 + -0005 ; '7, 50-08, 312-907, -4093, 494-5, and 87-003. 46 SultraUion of Decimals. SUBTEACTION. 59. In subtraction of decimals, or of integers and decimals mixed, for reasons precisely similar, the decimal points must be arranged to fall exactly beneath one another, and then the smaller quantity can be subtracted from the larger in the same manner as in whole numbers, thousandths being taken from thousandth:;, hundredths from hundredths, and teiiths from te7iths. The decimal point in the answer will fall exactly beneath the decimal points in the subtrahend and minuend, cyphers may be added (or supposed to be added) to the right of the decimal figures in the minuend, as this will not alter the value (see page 42), and the subtraction may proceed as in whole numbers. We have, therefore, the following rule for subtraction : — EULE XII. 1°. Place the quantities so that their decimal points shall he in the same vertical li^ie. 2°. Next proceed as in subtraction of whole numlers. Examples. Ex. I. Subtract 756 from -897. •897 •756 •141 Here the difference befrsveen 6 thousandths &-ad J thousandths is i thoxisandth, between 9 hundredths -and 5 hundredths is 4 hundredths, between 8 tenths and 7 tenths is i tenth. Ex. 3. From $8765'432i take 99*99. 98765-4321 99'99 98665'442i Ex. 2, From 37-6 take -907. In this instsince 37 "6 may be written 37-600. 37-600 •907 36-69 J Ex. 4. Subtract '97658 from 5'i394. 5"i394 •97658 4-16282 In subtracting 8, o is supposed to occupy the plana above it as 51394° = 5-1394' Ex, 5. Subtract '0000999 ^'0°^ '°^' •0000999 •0099001 Ex. 6. From i take -47712. I •47712 •522J In examples of this kind (Ex. 5, 6, and 8) when the number of decimal figures in the lower line exceeds the number of figures in the upper, it is advisable to mentally supply cyphers to make up the deficiency in the upper line. This may be done without altering the value of the upper line. Ex.7. Subtract 247*258746 from 347"258745- 347-258745 247*258746 99-999999 Ex. 8. From i take 'oooooi. •999999 MuUiplicaUon of Decimals. 47 Examples for Practice, Subtract 1. 3*07 from 6-50 1 ; '79999^01119; 2"9989 from 3 : '999999 from 9. 2. °oo9o8o6 from 39'857~; "00032 from 32 ; '876534 from 8*2 13 14; 364'3i23from456'0546. 3. '99 from I ; '00000099 from 99 ; 'oooooi from 10 ; 3"29ffom999; 25"6o5ofrom j67'392. 4. '9682347 from 65'ooooi ; "79999 from 9; 9'i63 from 8i'68234oi, 5. 'oooooi froid 'oooi ; '000004 from '0004; '00032 from 32 ; '87623 from 24681. 6. From 700 take 7 hundredths; from 'oooi take 'ooooooi. 7. From 42 hundredths take 42 thousandths; 154 millionths fi'om 6231 hundred thou- sandths. 8. From 96 thousandths take 909 ten thousandths ; 92 thousandths from 29 thousand. MULTIPLICATION. 60. We have stated that for every place we shift the decimal point to the right we increase the value of the decimal ten-fold, for every place wo shift it to the left we decrease it ien-foU. Now, in multiplying two decimals together, since the law oi local value hold with regard to the digits comprising the decimals, the process of multiplication will be performed exactly as in ordinary whole numbers ; the only matter requiring consideration will bo the proper posifion of the decimal point. Suppose we have fio multiply 4'935 by 6-28, and let us suppose the decimal point in each case removed to tha extreme right. Then (Art. 57, page 43) we have multiplied the number 4*935 by 1000, and ihe nimiber 6-28 by 100, and we have obtained the numbers 4935' and 628- respectively, Now,4935 X 628=^3099180, but as we increased our original numbers one thousand and 013/5 hundred fold respectively, it is evident our product is increased 1000 X 100, or one hundred thourand fold. Dividing, therefore, the above result, 3099180 by loooooo, or what 5s the saaa* thing (Art. 57, page 43), writing it 3099180- and removin>T the decimal point 5 places to the left, wo get for the product of the number.* 4*9 35 and 6' 2 8 the result 30-99180. It will b-? tJ^en that tho number of decimal places on the product namely, 5, is the sum of the ntrirbers of the decimal figures ia tho two given numbers. We have, therefore, the following rule for multiplication : — EULE XIII. MttUiply the numbers together, as whole numbers, and point off as many decimal places in the product fheginning at the right J as there are decimal places in tho multiplier and multiplicand together. When the decimal places to be pointed off are more in number than the figures of tho product, make up the proper number by prefixing cyphers to tho product. Examples. Ex. I. Multiply 34'xi by 3-72. 34' 1 1 3*72 6822 23877 10233 126-8892 In 34'11 are two decimals; in 3-72 are two; thcre- XOfe four decimal places are pointed oS. Ex. 2. Multiply 236000 by 336000 •48 944 ii328o'oo The product of 236 by -4813 11328; in 236000 are no decimals; in -48 are two decimals ; therefore two places are pointed off in the product. 48 Division of Decimals. Ex. 3. Multiply 56-3 by -08. 56-3 o-o8 4'504 In 56' 3 13 one decimal; in "08 are two; therefore three places are pointed off in the product. Ex. 4. Multiply 5"63 by "00005. 5"63 O'OOOO? 0'00028i5 In 5 "63 are two decimals ; in "00005 ^t6 ^'^^ > there- fore throe cyphers must be prefixed to the product 2S1;, and seven decimals marked off. Ex. Multiply "0048 by "000012. •0048 •0000C00576 The product of 4^ by 12 is 576; in •0048 are 4 deci- mals; in "00001 i are six decimals; therefore the pioduct must contain ten decimals (four and six) and seven cyphers are prefixed to 576, whence the product is "0000000576, as above. Ex. 6. Find the valuo of i"005 X 'oo.? X "0064. i"005 "005 •005025 "0064 20100 300150 •0000321600 Examples foe Practice. Pind the value of 2"5 X 4; "25 X 40; 2"5 X 476; *'5 X 47<5; '0025 X A-l^; -025 X 0476. •0002 X 'ooioi ; 90"oi X 0-034; 'oooS X •00014; ^.nd "6005 X ■oo35. 2i"56 X "0035; 24"35 X "074; 35-85 X 2-09; and "004716 X "22240656. ioo'ooo8 X "000306; 7535060 X 62-3906; and 3r503oi X i7"°352- 25067823 X 'ooooooi ; 394"2003 X "00000003; and '834567834 X -00000008. 47-83 by 10, 100, 1000, iV, Totr, roT)-^; *5 X 1000; -75 X looooo. 22-5 X "0241 X '0024; -0003 X -oi X 500000; "006 X -00012, 2-7 X "27 X '027 X 270; -2 X "04 X 'coS X 64000; 8-004 X "004. i-i X 'Oil X 101 X Old ; 'oij X 16 X "007 X 305 J 1003 X 6-12. DIVISION. 61. Let it be required to divide 37-015 by 6-73. By shifting the decimal point to the right of the dividend and divisor so as to turn both into -whole numbers, we increase the number 1000-fold, and the divisor 100-fold. The former of these alterations will have the same eflect as multiplying the quotient by 1000, the latter the same as dividing by 100 ; so that the quotient will be 10 times too great, and must be further divided by 10, i.e., one decimal place must be pointed ofif to give a correct result. Had it been required to divide 370-15 by 6-73 where there is the same number of decimal places in both dividend and divisor, by shifting the decimal points so as to make both whole numbers, we should increase the dividend 100-fold, and the divisor 100-fold; this would not affect the value of the result, and the quotient would bo a whole number requiring no decimal point at all. If the given quantities had been 370-15 by "673, so that there had been fewer decimals places in the dividend thin in the divisor, by converting both into whole numbers we should have increased the dividend 100-fold and the divisor 1000-fold. This would have decreased the divisor 10-fold, and to obtain the correct result we should hava had to multiply the quotient by 10. Division of Decimals. 49 We can hence determine the following practical rule for the division of decimals : — ■ EULE XIV. Divide as in whole numbers. The rule for placing the decimal point is, that the quotient must have as many decimal figures as the decimal places in the dividend exceed those in the divisor ; that is, the quotient and divisor together must contain as many decimals as the dividend. - Examples. Ex I. Divide 640-87458 by ()i,'^. "Wg here proceed as in ordinary division, except that a cypher is annexed to the dividend for carrying in the division. The dividend 640- 874580 contains six decimal places, the divisor 64"5 contains one, therefore the quotient 993604 must contain (6 — i) five decimal places; then five decimal places counted from the right to the left gives 9-93604, the quotient required. Ex. 2. Divide •239316 by 3256. •3256)'2393i6(735 22792 1 1 396 97^8 16280 16280 Here the number of decimal places in the dividend -2393160 (including the cypher supplied to carry on the division) is seven ; the divisor (-3256) contains five decimal places, and/o?y "ooofS; 9'6i4 by -0000019; "25 ty 31-25; 8-92 by 237'6567. „ -03679 by 2-83; 165-434 by 36-2; -027472 by 3-434; 61000 by '825. „ 17-171717 by 343-4; 1255 by 1-004; 12-55 by 1004; 7-231068 by -12, » -012550 by 1004000; 12-55 by -01004: 1255 by 10-04; 4 by -ooooi. „ -001255 by 1004; -01 by 1000; 6821091-97627 by 88-03. „ -6 by 6: 6 by -6; -06 by 60; -006 by -6 ; 600 by -6; 600 by -06; and -006 by 600. „ -00636056 by -86; 6100 by -825; 23by-ooo579; 37-69416. by -156 ; -0016 by -000008 ; 6 by -000000 1 ; -8 by -0000002; -000054 ^Y 9 ; 4000 by -000125. „ 36 by loooo ; 9-3 by 10 ; 52-306 by 100 ; 8 by loooo ; 2-0076364 by loooooo. „ f/' to 10 decimal places ; \\^^ to 12 decimal places ; ir,r^-, to 25 decimal places. EEDUOTION. 63. The great convenience of decimals makes it often desirable to reduce vulgar fractions to a decimal form. ' To reduce a vulgar fraction to a decimal. "We have to change the fraction to another equivalent fraction whose denominator is of the form 10, 100, 1000, &c. To do this we multiply the numerator and denominator of the fraction by 10, 100, 1000, &:c., as may bo necessary, i e., we add a certain number of cyphers (the same to each) ; we then divide the numerator and denominator by the original denomi- nator. These operations will not alter the value of the fraction. If the numerator by the addition of the cyptiers becomes divisible by the denominator, without remainder the required decimal is found ; if not, a circulating or recurring decimal is produced as is shown in the following examples : — Ex. I. To reduce § to a decimal. 5 — so on _• o r.ir) ,r, , B — »uuo -TO — rool) — 025. Here we multiply numerator and denominator by looo and divide them by 8. The resulting fraction rbVo" represented as a decimal is -625. 5* Redudion of Decmah. Ex. t. To reduce ^f| to a decimal. issoono 1 n e a lo'ooo' — 'IS Here we multiply numerator and denominator by loooo, and then divide them by '625. The resulting fraction -rooirs represented as a decimal is -1968. 625)i23oooo(-i968 625 6050 5625 4250 3750 5000 5000 The work is shortened thus : — "We put down the numerator 123 as divi- dend, and denominator 625. as a divisor, and adding cyphers as often as required, we obtain as a quotient the significant digits of the decimals ; and the number of cyphers added to the dividend will be the number of places to be marked ofif in the question. Hence to convert vulgar fractions into decimals we proceed by the following rule. EULE XV. Annex a cypher to the numerator, and then divide hj the denominator ; if there be a remainder, annex another cypher, and continue the division, still annexing a cypher, either till the division terminates without a remainder, or till as many decimals as are considered necessary are obtained : the quotient, with a decimal point before it, will be the value of the fraction in decimals. Examples. Ex. I. Reduce ^ to a decimal fraction. 5)i'o Dividing 10 by 5 (the cypher being added) wo find that i is = 0-2. That i = 0-2 is easily proved, for 1- = ig; consequently, by divi(ij'Jig both tho numerator and denomi- nator by 5, we hn7e i = vo = '2. Ex. 3. Reduce |^ to a decimal fraction. 3)foooo •3333) &°- Dividing 10 by 3 gives 3, the next cypher Added gives another 3, and so on con- tinually. Ex. 5. Reduce H to a decimal. (4)25-00 36 = 4 X 9 36 ^ • 1 9) 6-25 Ex. 2. Convert f into a decimal fraction. 8)3*000 •375 That f = -37.5 is proved thus, | = fa§§ ; consequently, dividing the numerator and denominator by 8, (the denominator of the fraction), we have |- = f^^^ = -375. Ex. 4. Reduce ^x to a decimal. ii)6-o •694444, &c. Hence f| ■=■ "694 which is called a mixed recurring or circulating decimal, consisting of the non-recurring part 69, and _ the recurring part 444, and usually written with a point or dot above the figure which is repeated. •5454 It is plain from the remainder that 54 would recur continually, so that -f^ is equal to a recurring decimal ; 54 being the recurring period. Ex. 6. Reduce ^^5- to a decimal. ( 2)3*000 7)1-500 •2142857 Hence y^j = -2 142857 ; the recurring part 142857 having a point above its first and last figures being called its period. If the whole decimals recurs, it is called a 2^Hre circulator. Reduction of Denniatk. S3 Ex, 7. Convert -^ into a decimal. ii3)8"oo(o'o7079, &c. 791 900 791 1090 1017 73 &°« - When the o is annexed to the 3, the divisor 113 will go no time; therefore the first decimal place is to be occupied with a cypher. Annexing now a second o, the next decimal figure is 7, and the work proceeds as above. The quotient shows that yI^ =: "07079, &c. ; the decimals may ^^e carried out to any extent. Ex. 8. Reduce -^ to a decimal. i28)i"ooo('oo78i25 1040 1024 160 128 320 256 640 640 Ex. 10. Eeduce -j-fj- to a decimal. ^ = •001953125. Ex. 12. Ex. 9. Eeduce -^{'^^ to a decimal. ^'ftSj = -0123291015625. I Ex. II. Keduce ^-l-j to a decimal. I ^ = -013671875. Reduce tMt to a decimal. •0451206715634837355718782791185729275970619. Examples for Praotiob. r. Change into equivalent decimals iV, H. if- \h -i^y ^> oh> and i^. ■z. Reduce to decimals -5;^, vf) ih -ir) -rlaj irf> f ffj and afa-j carrying interminanta to Bev=,i:a. decimal places. 64. It becomes important to observe what fractions will produce termi- lidting decimals. Suppose a fraction in its lowest terms; then in reducing it to a decimal we multiply the numerator by 10, 100, looo, &c. Now the numerator contains no factor common to the denominator, and by this multi- plication we introduce the factors 2 and 5 as often as we please and no others. Unless, then, the denominator contains no other factors except twos and fives, this multiplication cannot render the numerator divisible by it. Hence the only fractions which will produce terminating decimals are those whose denominators contain only a and 5 as prime factors. All other fractions will produce circulating decimals, though in- many cases the period is so long that it would be tedious to find it. 65. Decimals are most frequently used to make calculations oa numbers that have been obtained by observations of some kind, by measuring, for instance, or weighing ; and it is very seldom indeed that the accuracy of these observations can be relied on to within one five-thousandth part of the unit employed. Now if we cannot rely on the measurement beyond three decimal places, it is needless to carry the result derived from it any farther. In all operations with decimals, then, whether terminating or repeating, we may usually stop at the third or fourth place, and need very rarely go beyond the fifth or sixth. We may, however, attain any degree of exactness that may fee required, by carrying the decimals far enough. 54 Reduction of Decimah. 66. With respect to repeating decimals, if perfect accuracy be necessary, they must in most cases be reduced to vulgar fractions before they are added, subtracted, multiplied, or divided. In almost all the applications of decimals, however, an approach to accuracy is sufficient, and this is attained by carrying the decimal only to a moderate number of digits, and omitting the rest. If, in converting a vulgar fraction into a decimal, we stop after the third digit, for instance, adding unity to that digit, if the next be 5 or upwards, it does not differ from its exact value by more than one five-thousandth part of the unit employed. Thus, -172 differs from 172437 by '0004372, which is less than -0005. Similarly, 983 differs from -98276 by -000^317, which is also less than -0005. 67. To reduce any quantity or fraction of one denomination to the decimal of another denomination. EULE XVI. Reduce the number of the lowest denomiyiaiion to a decimal of the next higher denomination, prefix to this decimal the number of its denomination given in the question, if any, atid reduce this also to a decimal of the next higher order, and so on till all the numbers of the given denomination are exhausted, and the decimal of the required denomination has been obtained : the last result will be the answer. Examples. Ex. I. Let it b9 required to express 173. 5id. as a decimal of ;^r. The process will be first to express the fractional part of a penny as a decimnl of a penny ; placing the 5 as a whole number before this decimal, to divide that result by 12, in order to reduce it to the decimnl of a shilling; placing the 17 as a whole number before this decimal, to divide that result by 20 in order to reduce it to the decimal of a pound. This will be written as follows : — 12) 5-25 pence 20) 1 7 '43 7 5 shillings •871875 of a pound. It will be seen from this, that whatever we should divide by in whole numbers in order to bring penceinto shillings, or shillings into pounds, that we must likewise divide by in this case, only marking off correctly the decimal results. Ex. 3. Find what decimal of an hour is 40'". There are 60 minutes in an hour; _h™co I minute is ',;V of an hour, and 40 minutes is 13 of I hour, which gives o-66 of i hour. 6o)40"oo o'66 of an hour. Ex. 2. Express as decimals of a degree 27° 18' 35". 60 35" 60 18-5833 27-30972 Here for convenience of arrangement we write the 35" uppermost, and the 1 8' and 27° directly under it, and draw a vertical line to the left of the lino opposite these numbers ; write for divisors the numberof that denomi- nation which makes one of the next higher — namely, 60 opposite the seconds, since 60" z=z i', and 60 opposite the minutes, because 60' = 1°. Then dividing 35 by 60 we get •5 S3, which we write af:er the minutes, which gives i8''583 ; this again divided by 60, the number of minutes in a degree, gives the quotient '30972, which being annexed to the degrees, 27°, gives the answer 27°-30972. Ex. 4. Find what decimal of an hour is 15™. Hero I minute is -cV of i hour, 15 minutes is -J;} of I hour; hence Jj gives 0-25 of l hour. 60)15-00 0-25 of an hour. Reduction of Decimals. 55 Ex. 5. Find what decimal of i degree is 8' 37" 37" are ^'^ of i', or o-6i of i' ; then i' is -a\s of 1° ; hence 8'-6i are -/u- of 1°, or o°-i43. 60)37" 60) 8-6 1 6 0-143 of a degree. Ex. 7. Find what decimal of i mile (nautical) is 700 feet. There are 60S0 feet nearly in a nautical mile ; hence r foot is ^7j\— of a mile, and 700 feet are -iis»o of i mile, which gives o'ii5 of I mile nearly. 6o8o)7oo-o(o"ir5 6080 9200 6080 31200 30400 Ex. 6. Find what decimal of i day is 3I. 42". 42"^ are ^5- of r hour, or oi>-7 ; and 1^ is T.V of I day ; hence 3''7 is --il of 1 day, or' o'i'i54i66, &c. 60)42" 24) 3'7,' OT54166 Ex. 8. Find what decimal of i foot is 8| inches. First, J is 0-75 of I inch; hence 8 J inches are 8-75 inches. Then, i inch is -A" of r foot; hence 875 inches are -'1", or 0729 of I foot. 12)875 0729 68, Or, reduce the given quantities to the lowest denominations when there are more than one, and also the integer to ivhich it is referred, to the same denomination; then divide the given quantity hy the integer thus reduced. Ex. I. (Ex. 7 above.) The given quantity 700 feet, being all of one denomi- nator requires no further reduction. The integer i mile, reduced to the same de- nomination, is 6080 feet; then 700 divided by 6080 gives 0-115. Ex. 2. (Ex. 8 above.) 8 inches and 3 quarters are 35 quarters, and i foot reduced to the same denomination is 48 quarters ; then 35 divided by 48 gives 0729. Examples for Practice. Express as decimals of an hour 17™; 29™; 42™; ?5™; 48™; and 58'". Eeduce 5'' iz^ 25™ 39«'92 to decimals of a week; 18^'^ to decimals of a year. Express as decimals of a day the following quantities: — ii^ 14'^ 13™ 12'; 29<' i-]^ 11™ 45'; 15*^ ^t 48"^ 54'; and 119^ 5'' ig'" I5^ Express as decimals of a degree the following quantities: — 8' 11' 15"; 19" 40' 45"; 104° 16' 7j"; and 82° 19' 30". "What decimal is 3 qrs. 2 1 lbs. ; 12 cwt. i qr. i61bs. ; i4cwt. iqr. ; 10 cwt. i qr. r4lbs. ; 13 tons 6 cwt. 7 lbs. ; i ton 13 cwt. ; 17 tons 18 cwt. 3^ lbs. of a ton. ? 63. 4d.; 8s. 8^d. ; 4s. 7id. ; 15s. ii|d. ; 19s. lo^d.; ;,{'i it,%. iil» 10 = 10^ 2 » 100, » 100 = lO' 3 ). lOOO, » 1000 = 10- 4 „ 1 0000, )) lOOOO ■=! lO' &c. &c. Now, if the above Tables were amplified by the insertion of the logarithms of all the numbers between i and lo, between lo and loo, &c., we should have a Table of logarithms of all numbers from i to loooo; and whatever may be the difficulty of determining the intermediate logarithms, it is at once easily seen that the logarithms of all numbers between i and lo, i.e.., between io° and lo^ must lie between o and i, and will be o + a fraction, that is, a decimal less than i ; of all numbers between lo and loo, i.e., between lo^ and lo^ must lie between i and 2, and will be i + a fraction, or a decimal between i and 2 ; of all between 1 00 and 1 000 will be 2 + a fraction, and so fortt ; or the integral part of each intermediate logarithm will be one less than tho anmber of integral figures in the quantity of which it is the logarithm. Thus, the logarithms of 2, 3, 4, &c., to 9, have o as the integral part; those of 10, 1 1, 12, &c., to 99, have i as the integer; those of 100, loi, 102, &c., to 999, have 2 as the integer; and so forth. Hence Tables of logarithms usually supply only the fractional or decimal part 5 the integral part is always known from the number of integers in the value whose logarithm is wanted. Very few logs, can be expressed in terminating decimals, but this causes little inconvenience, since a log, carried to six or seven decimal places is sufficiently exact for all common purposes. 79. The integers 1,2, 3, 4, &:c., which are the logarithms of 10 and its powers (see 78), are chief indices, and the logarithms intermediate to these, as for instance 1-778151 (which is the logarithm of 60) cosisting of an integer and a decimal fraction, though they are also indices, are usually referred to as consisting of au inde.v^' and mantissa], the integral part being specially termed the index or characteristic, because it indicates, by being one less, how many integral places are in the corresponding natural number, and the annexed decimal being called the mantissa. Example. — In the lo^. 4'6i63';9, the figure (4) standing to the left of the decimal point 13 the characteristic or index, and tho remaining portion ('616339) is the mantissa or decimal ])art. * In order to avoid confusion from the use of the word " index" to signify two things, we shall throughout this work employ the term characteristic when speaking of logarithms, and index when speaking of roots or powers. t Mantista, a Latin word signifying an additional handful; something over and above an exact quantity. On Logarithm. 6i So. To find the characteristic of the logarithm of any number greater than unity we have, therefore, the following rule : — EULE XVIII. The cJiaracteristic of the logarithm of a number greater thart unitg, i.e., of a whole or mixed number, is one less than the number of the digits of its integer part Thus : the characteristic of the logarithm of 849 is 2 ; for the number S49 is an integer consisting of three digits (that is the number between too and looo) and i less than 3 is 2. Also, the index of the log. of 264-^6 (which is a mixed number) is 2, since the integral part of the number, namely 264, is a number between 100 and 1000, or consists of 3 digits, and one less than 4 is 3. Again, 3 is the characteristic of the logarithm of 3847'2i6, s-ince this number has 4 integral digits ; while o is the characteristic of the logarithm of 3-847216, since this number has one integral digit. Again, the characteristic of the log. of a number of o»e place of integers (such as 5 or 5"o8, or 5-0801) is o. Again, every number with tivo places of integers (such as 50, or 50-8, or 50-813) is I. Again, every number -with three places of integers (such as 508, or 508*2, or 50825) has for its characteristic 2, and so on. Examples for Practice. Write down the characteristics of the logarithm of the following numbers : — I. 365 6. 69710 2. 4-8 7- 45-82 3- 64375 8. 8640 4- 28-9 9- 75 5- 6 10. 7-265 II. 474000 16. 473-908 12. 4256-45 17- 54793000 n- 3-9 18. 21256-8 14. 8 19. 2 -14006 15- iS 20. 50-7406 10" = I =Z I 10-^ = -A- = o-i 10-" = Toff = 0"0I ro-=' = 10"00 = O'OOI io-* = toooiy = o-ooor &c 81. It has been shown that in the common system of logarithms (Briggs') the log. of I is o ; consequently, if we wish to extend the application of logs. to fractions, we must establish a convention by which the logs, of numbers wholly decimal, i.e., less than unity, may be represented by numbers less than zero, i.e., by negative numhers. Extending, therefore, the above principles to negative exponents, since o T is the logarithm of -r in this system. ^ }, "01 „ 3 »> '001 „ 4 „ 'oooi „ &C. It follows from this, that when the number is a decimal with all its digits significant, in value between i and -^, its log. is negative, yet not so small as the log. of ^, which is — i . Its log. therefore will be something between o and — • I , or — i with some positive decimal added. Hence — i is its characteristic. When the number is a decimal with zero as its first digit, in value therefore below ^V ^^* ^^^ so low as y^> its log. is less than — i, but not so small as — 2, and so will be — 2 with some positive decimal attached. Thus 2 is the characteristic. The log. of a decimal between 'oi and •001 is some number between — 2 and — 3, and its characteristic is — 3 ; of a number between "ooi and -oooi its log. is between — 3 and — 4, and its characteristic is — 4; and generally, following this reasoning, it will appear that the ^^ On ZogdritTims. cliaraeteristio of a decimal fraction is negative, and may be known from its denoting the place of the first significant figure of the decimal, as being the ist, 2nd, 3rd, &c., place after the point; hence, 82. To find the characteristic of any number less than unity, i.e., of a decimal, we have the following RULE XIX. The characteristic of the logarithm of a number less than unity, and reduced to the decimal form, is negative and one more than the number of cyphers following the decimal point. A negative characteristic is denoted by writing over it the negative sign ( — ), thus 7, 2, 3, &c.* Thus the chatacteristic of the logarithm of '00521 is 3, since the number of cyphers following the decimal point increased hy i is 3. Similarly the index of log. of -156 is T „ „ •or56 is i ,, ,, '00046 is 4 „ „ '0000C0721 is 7 83. But in order to avoid the confusion that might arise by the addition and subtraction of negatim indices, the following rule is frequently used. EULE XX. Add I to the number of cyphers between the decimal point and the first significant figure, and subtract from 10 ; the remainder is the index required. Thus the characteristic of the log. of '04 is I, or 8, since i added to the number of cyphers following the decimal point is 2, then 2 from 10 is 8. Similarly the index of log. of '140 is 9. „ „ . -0149 is 8. „ „ '00064 is 6. „ „ '000000721 is 3. (a) If the characteristic of a vulgar fraction is required, it must first be reduced to an equivalent decimal fraction, and then the index is found by the rule. Thus, the index of log. \, or of log. -125 is I or 9. „ of log. -^, or of log. '04 is 2 or 8. „ oflog;24f, or 24'4 i. Examples fob Praoticb. "Write down the characteristics of the logarithms of the following decimal fractions : — I. •045 6. •OOOOOOf II '4537 16. •037299 3. •9 7- '01 12 •009 17- '00000052018 3- •0004 8. •0003127 13 •0000008 18. •000000205379 4- •6798 9- •02803 14 ' 000064 19. •J 5- •0062 10. •7007 15- •000485 20. •0000000000382 * The negative sign ( — ) is written above the characteristic, thus 2, instead of before it, to show that it affects only the characteristic and not the mantissa, which remains positive. If it were written in front of the complete logarithm it would signify that the entire logarithm was negative, but such logarithms are never employed in the operations connected with navigation. On Logarithms. 63 84. The characteristic may also be found as follows : — EULE XXI. Place your pen hetween the first and second figure, (not cypher), and count one for each figure or cypher., until you come to the decimal point ; the number thus given will he the characteristic : hut observe that if you count to the left you inmt sub- tract the number found from 10, and consider the remainder as the characteristic. Thus, in finding the log. of 4'6oi7, if you placo tho pen between the first figure (4) and second (6), it falls on the decimal point ; in this case the characteristic is o. Next, in tho case of log. of 4601-7, place your pen between 4 and 6, and count I ' , ' ; the characteristic is 3. Next, in the case 4601700, here the decimal point falls behiud the last cj^phor (No. 5). Hence, counting as before, we have |j2-jI<;6 ^^^ ^^^ characteristic is 6. Again, in the case of log. '00046017 the first significant figure is 4. Hence, counting, we have .L^ ^ , but here we count to the left, so that the characteristic is negative, or 4, which taken from xo is 6. Again, in the case of log. of •46017, we have || °'^, and the characteristic is i, or 9. 85. The mantissa of the logarithm depends entirely on the relative valua of the figures composing the quantity whose logarithm it is, and not at all upon the numerical value of that quantity : thus, the mantissa of the log. of 13 is '1 13943, which is also the mantissa of 1*3, or 130, or 1300, for in each case the i and the 3 have the same relative value. So the mantissa of a logarithm is always the same, if the significant figures remain the same, and is not altered by the addition of cyphers to the right or left of these figures, or what is equivalent, by the multiplication or division of the quantity by 10 or any power of 1 ; it is only the characteristic which alters its value by an alteration in the position of the decimal point, i being added to the character- istic for every place the decimal point is removed to the right, that is, for every 10 by which the quantity is multiplied; or, i is subtracted from the characteristic for every place the decimal point is removed to the left, that is, divided by 10. The logarithm of 745S00 bein 5-872622 that of 74580 is 4-872622 » 7458 » 3-872622 » 745-8 >> 2-872622 » 74-58 » 1-872622 tt 7-458 » 0-872622 » •7458 )> 01 1-872622 • 9-872622 » •07458 » or 2-872622 '8-872622 n •007458 >> or 3-872622 7-872622 » •0007458 >» or 4-872622 6-872622 64 ' On Logarithms. ON TABLES OF LOGAPJTHMS OF NUMBEES. 86. In Eaper's, Norie's, and the collection of nautical tables intended to accompany this work, the Tables of the Logarithms of Numbers are arranged so as to give the mantissse of the natural numbers from i to loooo. If the reader will open an ordinary table of logarithms, such as is contained in the above-mentioned works, he will find a short table of logs, from i to 100 immediately preceding the general table, and giving the entire logarithm ; following which is the general table, on opening which he will find a vertical column on the left side of the page containing three digits and ten columns of logarithms headed by the digits o, i, 2, .... 9. These last are fourth digits to be attached to the former three, so that the table thus embraces numbers from 1000 up to 9999. Opposite to every such number is a number with six places of figures. This is a decimal, though to save printing the decimal point is not printed, and it is the decimal part, or mantissa, of the logarithm of the number to which it corresponds. The characteristics are never printed, but are prefixed according to the Rules XVIII, XIX, and XX. Hence from such a table we can take out the logarithm of any number with any four significant digits. In the margin of all tables of logarithms the difi'erenoe of the successive logarithms in that part of the table is set down. In some tables, also, there are little tables in the margin called talles of proportional parts. These are placed under every successive difference, and contains for that difference the number to be added in respect of each unit's digit, so as to form the logarithm of numbers of six digits. These numbers are found as directed by the Eule XXV, page 66. 87. If the number be given, its logarithm may be found as follows : — To find the logarithm of a number consisting of not more than two digits, i.e., which does not exceed 100, using the short table of logarithms, from i to 100, preceding the general table. EULE XXII. Seelc for the number proposed, considered as a whole mimier, in the column at the top of which is No., and the logarithm will he found opposite to it in the next column to the right hand. Prefix the proper characteristic (by changing it if necessary'' to the mantissa (see Eules XVIII and XIX, pages 61 and 62). The result is the logarithm sought. Note. — It may be observed here, once for all, that the proposed number must be considered a whole number, and in case a decimal point occurs in the given number, no notice is taken of it till we come to the insertion of the characteristic; and should cj'phers occur heiwesn the decimal point and the first sigyiificmt figure, these also, are disregarded in entering the table, being only taken into account when determining the characteristic. Ex. I. Required the logarithm of 21, 2T, '21, and •ozr. In the first page of the Table, and in one of the vertical columns marked JS'o., we find 21 against which stands 1-322219, the logarithm sought. Since the mantissa of the logarithm On Logarithms. 65 of any number consisting of the same figures is the same whether the number be integral, fractional, or mixed, the logarithms of the numbers 2T, •21, and '021 will have the same decimal part as 21, the characteristic only being changed, consequently the logarithm of fi is 0-322219, the logarithm of -21 is 9'3222i9, and the logarithm of '021 is 8'3222i9. Ex. 2. To find the logarithm of 52, 5-2, -52, and -00052 : — In the Tables we find the log. of 52 is i'7 16003, and, therefore, simply changing the index, the log. of 5-2 is 0-716003, "52 is 9-716003, and the log. of -00052 is 4 6716003 or 6-716003. No. Nat. No. No. Nat. No. No. Nat. No. No. Nat. No I. 5 7- 41 13- 94 19. •0091 2. 9 8. •004 14. Jor '5 20. 25-0 3- •009 9- 2-4 15- 1 or -75 21. •024 4- •01 10. 24 16, 2 '5 22. •000035 5- •0001 II. •24 17- I or -25 23- •000057 6. 14 12. •0021 18. -09 24. ■ToVo 88. To find the logarithm of a number consisting of not more than three places of figui-es (from 100 to 1000). EULE XXIII. Find the given number in the left-hand column of the Table, and opposite it, in the next column, will stand the mantissa or decimal part of the logarithm. Prefix the characteristic according to Eules XVIII and XIX. The result is the logarithm sought. Note. — "When we say "three figures " we mean independently of cyphers either to the right or to the left. Thus we should include 6340, 73200, and -00265 under the head of thi» problem. Note. — If the number is less than three figures make up three by placing cyphers, if not already present (or by supposing them placed), on the right of the number, cyphers so added being regarded as decimal ; then proceed as directed in the above Rule XXIII. Thus the logarithm of 75 is the same as that of 75-0 ; the logarithm of 8 is the same as of 8"oo; and that of -035 is the same as of -0350. Example. Ex. I. Required the log. of 476, 4-76, and -00476. We seek in the left-hand column of the Table for 476, against which in the column marked at the top, stands the mantissa corresponding thereto ; and this part by the rule is the same for each of the above numbers. Now prefixing the index according to the number of integral figures in the natural number, we find the log. of 476 is 2-677607; of 4-76 is 0-677607; and of -00476 is 7-677607. Again, the logarithm of 576 is 2-760422; that of 39*4 is 1-595496; that of -0253 is 5-403121. No. Nat. No. No. Nat. No. No. Nat. No. No. Nat. No. 1. 100 5. 673 9. 896 13. -0147 2. 145 6. 794 10. 1-47 14. 434 3. 2-94 7. 982 II. •147 15. •0000448 4. 361 8. 4-80 12. •901 16. 44^00 89. If the number contains four places of figures, exclusive of final cyphers, or cyphers included between the decimal point and the first signifi- cant figure. EULE XXIV. Find the first three figures in the vertical column on the lefi marked Nq., (tnd the fourth in the horizontal column at the toj> of the page. Under thia last, and opposite K 66 On Logarithms. the three Jigiires, will he found the mantissa of the logarithm sought. Prefix the index according to Eules XYIII and XIX. The result is the logarithm Bought. Example. Ex. I. Eequired the logaritlims of 4587 and of 0-0004587. The first three figures (viz., 458) being found in the column to the left marked No., and the fourth (7) in the line of digits at the top of the page, the decimal part of logarithm (•661529) is found in the same horizontal line as the three first figures of the given numher, and in the same column as the fourth. The characteristic is 3, being one less than the number of integers in the whole number ; whence the completed logarithm is 3-661529. The logarithm of -0004587 is 4-661529, the characteristic being negative, and one more than the number of prefixed cyphers. Again, the logarithm of 3470 is 3*54t33o ; that of 3*492 is 0-543074 ; and that of 0-3468 ia 1-540079; that of 74-39 is 1-871515; that of 325600 is 5-512648, in which case the mantissa of 3256 is taken out, since it is the same as the mantissa of 3256000. No. Nat. No. No. Nat. No. No. Nat. No. No. Nat. No. 1. 1000 4. 5432 7. '01012 JO. 987-6 2. 1234 5. 26-06 8. 94*87 II. '06843 3. 25-65 6. 2-606 9. 7'777 i2. '002784 Note. — The foregoing rule may be used not only in the case of numbers consisting of four places of figures, but may be made to include all numbers consisting of less than four significant digits, and so enable us to dispense with the Eules XXII and XXIII. Thus, if the number consists of less than four figures, make up four by placing cyphers, if not already placed (or by supposing them placed), on the right of the number ; cyphers so added being regarded as decimals. Then proceed to find the mantissa of the log. by the foregoing rule. Thus, the log. 75 is the same as the log. of 75"oo ; the log. of 8 is the same as the log. of 8-000; and that of -035 the same as that of '03500 (3500 in the tables). 90. Although the tables in Eaper, Norib, and the "Nautical Tables" accompanying this work are constructed so that the mantissas corresponding to more than four figures cannot be taken out directly, yet the mantissse of numbers containing five or six figures can be found from them without much trouble by means of the tabular difference taken out of the extreme right hand column of the page (see 86). If the number consists of more than four figures other than final cyphers, or if the number be a decimal fraction, cyphers immediately following the decimal point, we use EULE XXV. 1°. Cut off the first four figures and consider the rest as a decimal. 2°. Find the mantissa corresponding to the first four figures (Eule XXIV). 3°. Multipl'j the tabular difference hy the decimal cut off, i.e., by the remain- ing figures of the given number, and cut off from the right-hand as many figures as there are in the multiplier, but at the same time adding unity if the highest figure thus cut off is not less than 5. 4.°. Add the integer part of this product to the figures of the mantissa just found. These proportional parts are thus compiled on the supposition that the difference between the numbers ^nearly equal to each other J is proportional to On LogariiJimJ. 67 the difference between their logarithms. This proportion can be shown to be approximately true. The result is the mantissa of the recxuired logarithm. The characteristic or index is found by Eules XVIII and XIX, pages 6 1 and 62. Examples. Ex. I. Required the logarithm of 28434. Tab. difif. Mantissa of 2843 = '453777 153 Tab. diff. 153 X 4 = 6i-2 = + 61 X 4 Characteristic = 4 61,2 or 6r The log. of 28434 = 4-453838 We seek in the left-hand column of the Table for 284 (the first three digits) and also at the top of the page in one of the horizontal columns we find 3 (the fourth figure), then in a line with the lormcr and in the column with the latter at the top we have 453777. which is the mantissa of 2843. In a line with the quantity in the right-hand column marked Difi"., stands tab. diff. 153; which multiplied by 4, the remaining digit of the given number, produces 612 ; then cutting off one digit from this (since we have multiplied b)' only ox^ digit) it becomes 61, which being added to ^Sllll (the mantissa of 2843) makes 453838, and, with the char icteristic, 4-453838, the required logarithm. The logarithm of 284-34 is 2-453838, and the log of -028434 is 2-453838 or 8-453838. Ex. 2. Eequircd the logarithm of 12806. Tab. diff. Mantissa of 12S0 r= -107210 338 Tab. diff. 338 X 6 = 202-8 = -f 203 X 6 Characteristic = 4 202,8 or 203 The log. of 12806 ^ 4-107413 Ex.3. Find the logarithm of 873457. Tab. diff. Mantissa of 87 34 = "941213 50 Tab. diff. 50 X 57 = 28-50 = -j- 29 -j- 57 Characteristic = 5 28,50 or 29 The log. of 873457 — 5-941 242 The mantissa of the first four figures is found thus :— Opposite the 873 and under 4 stands 941213 ; then in the right-hand column in a line with this stands the diff. 50, which being multiplied by 57, the remaining digits of the given number, makes 2580 ; from this we cut off two digits to the right (^inco wo havo multiplied by tico digits), when it becomes 18 ; but as the highest digit cut off is 5, we add unity, whic!i makes 29. Then 5-941212 (the log- arithm of 8734) -j- 29 =: 5-941242 is the required logarithm. Ex. 4, Required the logarithm of 62S007. Mantissa of 6280 = "797960 Tab. diff. Tab. diff. 69 X 07 = 4-83 = + 5 69 Characteristic =5 X 07 The log. of 628007 = 5-797965 4,83 or 5 The log. of 628-067 is 2-798006, and the log. of -00628067 i^ 3'7 9^006 or 7-798006. The mantissa of the log. of each of these numbers being the same, the index only being varied. (See Rules XYIII and XIX.) I. 38475 7- 435 '60 13- 200000 19. 365152 2. 384-75 8. 78-604 14. -056214 20. 997-1370 3- 12345 9- 2-2055 15- -0098563 21. 32-1908 4- 543'2i 10. 0-7S362 16. 643786 22. 1-032764 5- 66666 II. 1 0000 17- 1129-06 23- loooii-y 6. 9244-8 12. •000800073 18. ■998095 24. 596-423 6^ On Logarithms. gi. To find the natural number corresponding to a given logarithm. — If the logarithm be given, tho number which corresponds to it may be found by the following rules, which are the converse of those last given for finding the logarithm when the number is given. Since the characteristic denotes how many places the first significant figure stands to the right or left of the unit's place ; converselj^, therefore, if logs. be given having for characteristics i, 2, 3, .... 7, 2, 3, ... . there are in the integral parts of the number to which these logs, belong, 2, 3, 4, .... o, T, 2, .... digits respectively. In illustration of these remarks take the following: — Log. 4*589950 (in -which characteristic 4) gives 38900 3'sS995o 3 •• 3890 ^^^S^95° 2 .. 389 i"58995o J .. 38*9 2'589950 o .. y^ 019-589950} ■^°'^9 .. -389 or 8-589950 j ""■ ■> -^ &c, &c. In which it will be observed that the first answer must consist of five integers, because the index of the given logarithm is 4; that the second answer must contain four integers, because the index of the given logarithm is 3 ; that the third answer must contain three integers, because the index of the logarithm is 2, &c., &c. ; and that the sixth answer must be a decimal fraction having the first significant figure in the place of tenths, because the logarithmic index is T ; and lastly, that the seventh answer must be a decimal fraction having the first significant figure in the place of hundredths, because the logarithmic index is 2. 92. From the foregoing it is evident that when the figures of che natural number have been found, we must place the decimal point so that the nujaber of integral figures may be one more than the characteristic denotes. Cyphers must be supplied to the right, if necessary, to make up the number, hence EULE XXVI. Add I to the charaeteristic of the given logarithm, and mark off to the left the number of figures for whole numbers; the rest (if any ) loill he decimals. If the characteristic is negative place the decimal point to the left of the natural number found, along with as many cyphers as may remove the first significant figure to that place of decimals which the index expresses; that is, one cypher fewer than the number denoted by the characteristic, whence, to find the place of the decimal, we have the following EULE XXVII. Tlie number corresponding to a logarithm xcith a negative index is wholly decimal, and the number of cyphers foUotving the decimal point is one less than the character- istic of the logarithm. On LogarUhms. 69 But Instead of the negative characteristic its arithmetical complement is sometimes used, in which case we proceed by EULE xxvni. Add I to the index, and subtract the mimber thus found from 10; the remainder is the member of cyphers to be prefixed to the figures taken out of the Tables. Place the dot before the first cypher. 93. To find the natural number corresponding to any given logarithm. When the mantissa or decimal part of the logarithm can be found exactly ia the Table, we proceed by EULE XXIX. 1°. Seeh out the mantissa, and take from the column No. the three figures in the same horizontal roto. 2°. From the head of the column take the fourth figure. 3°. Fro}}i the characteristic fnd by the rules already given the ^^osition of the decimal point, and so adjust the local value of the figures. (Eules XXVI, XXVII, and XXVIII, No. 92, page 68). {a) When the characteristic of the given logarithm requires a greater number of digits to the left of the decimal point than there are in the number found by the above rule, the deficiency is made up by adding a sufficient number of cyphers to the right. ib) If the natural number is a decimal fraction, and the final figure or figures are cyphers, they need not be written down. Examples. Ex. I. Given the logarithm 2'6()%()'jo to find the natural number. Entering the Table with the decimal part "698970, we find the natural number corres- ponding to it to be 5, or 50, or 500, or 5000, &c., but as the index of the logarithm is 2, the natural number must contain three integral figures. Hence the natural number of 2-698970 is 500. Ex. 2. Given the logarithm 3*539954 or 7'539954: find the number. Entering the Table with the decimal part, we find the corresponding number is 3467 ; to this we prefix two cyphers, since the index is 3 ; or adding i to 7 (8), and subtract 8 from 10, we have 2, the number of cyphers to be prefixed, and then the decimal point ; hence the number corresponding to 7*539954 is "0034567. Ex. 3. "What number corresponds to the logarithm 4'2i43i4. The decimal part of the log. being found opposite 163 and under the figure 8 at the top of the page : therefore the digits of the required number are 1638. But as the characteristic is 4, there must be in it 5 places of integers. A cypher is annexed (see Rule XXIX, (a). Hence the required number is 16380. Ex. 4. Required the natural numbers corresponding to logs. 0*176091 and 4*i76o9r. (i) The mantissa -176091 stands in the Table opposite 150, and the column with at the top; and the] characteristic shows that one of these is integral, whence the number sought ia 1*500 or 1*5 (see Rule XXVI, page 68). (2) The mantissa of second log. being the same as that of the first, the corresponding number will consist of the same significant figures, but the characteristic 4 shows that the first significaut figure (i) must occupy the fourth place to the right of the decimal point, ■whence the number sought is -00015. (See Rule XXVII or XXVIII, pages 68 and 69.) 70 On Logarithms. Ex. 5, Eequired the natural nnmber ■whose logarithms are respectively i -8 135 14, 0"3034i2, 4-996993, 2-299943 or 8-299943, 4-000000, 4-000000, 7-816109, we shall find them to be as follows: — 1-813514 = log. of 65-09 •303412 = 2-OII 4-996993 = 99310 2-299943 1_ or 8-299943 J •01995 4-000000 = lOOOO 4-000000 = •0001 7-816109 = 65480000 "Where it will be observed that the first answer must contain only two integers, as the index of the given logarithm is i ; that the second must contain only one integer as the characteristic is o ; that the third must consist of five integers, because the index of the given logarithm is 4, and therefore to 9931, the number found in the Table, a cypher is annexed, (see Rule XXIX, {a) ; and that the fourth answer must bo a decimal, having the first significant figure two places to the right of the decimal point because the characteristic is z ; the fifth answer must consist of five integral figures (a cypher being annexed to make up the number) since the characteristic is 4; the mantissa of the sixth log., or -000000, gives the corresponding natural number 1000, but adjusting the decimal punctuation, or the local value of the figures, the characteristic 4 denotes that the first significant figure (i) must stand in the fourth decimal place, and, therefore, three cyphers must be prefixed, and the natural number will be -0001 — the three final cyphers not being written down. Finally, the mantissa of last log. being found in the table gives the natural number corresponding as 6548, to which annex four cyphers; the characteristic 7 determines the number to consist of 8 integral figures. No. Log. No. Log. No. Log. No. Log. I. 0-47712X 9 3-898506 17- 2-990561 25- 7-991093 2, 0-041393 10 2-538574 18. 4-541579 26. 7-903524 3- 0-973128 II. 1-394977 19. 1-744058 27. 2-621488 4- 1-161368 12 3-845098 20. 1-501196 28, 9-901349 5- 0-812245 13 7-000000 21. 7-875061 29. 3-662758 6. 2-767898 14 5-825426 22. 6-6o2o6o 30. 4-851258 7- 0-39445^ 15 4-698970 23- g-845098 3r- 6-913761 8. 1-478422 16. 5-000000 24. 3-605197 32- 5-868527 94. When, as usually happens, the mantissa cannot be found exactly in the Tables, but lies between two successive records in the Tables, and it is proposed to find the corresponding number correct to six places of figures, . other than final cyphers immediately following a decimal point, the number is to be found by the method of proportional parts, on the supposition that, between two successive records in the table, the number advances in pro- portion to the increase of the logarithm. 95. To find the natural number corresponding to a given logarithm, when more than four figures are required. We proceed by EULE XXX. i"^. Having found the next loiver mantissa in the Tables, note the four figures which correspond to it. 2°. From the given logarithm subtract that taken out of the Tables, divide the remainder (annexing as many cyphers as there are digits required above four) by the tabular difference, and reduce the quotient to the form of a decimal. On Logarithms. 71 3°. To the four figures already found, add this decimal, and shift the decimal point to suit the characteristic of proposed logarithm. The result will be the required number. Note. — It is needless to annex many cyphers to the dividend. We cannot witli safety carry the natural number to more than six figures when the tabular difference contains three, or to more than five when the tabular difForence contains only two. Examples. Ex. I. Given the logarithm 3-543027 to find the natural number. Given logarithm 3'543027 Mantissa nest lower in Table •542950 which corresponds to 349r. Tab. diff. = i24)'77oo(-62 744 260 24S Attaching this (•62) to the four figures, we have 349162, Sec. The decimal punctuation or local value of the figures of the number can now bo adjusted, and as the index is 3, we obtain, by pointing off four figures to the left, 3491-62, the natural number sought. Ex. 2. Given the logarithms 5-654329 and 2-65427 3 to find the natural number j. Given logarithm 5'654329 Mantissa next less in Table '754^73 -which corrsponds to 4; ri. Tab, diff. = 96)5600(58 480 800 768 Ans. 451158. 654273, which corresponds with the natural number 451 1, is the logarithm next less than the given one ; therefore the first /&!ofore ; thus 9 from 9, o from g, 8 from 9, 6 from g, 5 from q, and as the next figure, 4, is the lowest sUjnificant figure (see Rule), we take it from 10, which leaves 6; lastly, the cypher at the end is repeated. The ar. co. of 2-170630 is 7-829370 7-217034 „ 10-782966 . 3-178680 „ ij-821320 97. A subtractive quantity is, by this means, made additive. The process is equivalent to subtracting the number from lo, and the reason of it is evident on considering that to add 3 and subtract 10 is the same as to subtract 7. In like manner, instead of subtracting 42"' 10' for example, -we may add 17™ 50' (the complement of 60"), provided we subtract i'' (or 60'"); and thus any number of quantities, of which some are additive and some subtractive, may be rendered all additive, provided that the larger numbers ■which are employed in taking the complements be themselves subtractive. MISCELLANEOUS. 98. We here insert a collection of numbers, the logarithms of which are io bo taken out of the Tables. I. 8 II. 63-5 21. 844-4 31- 93'7654 41 I 0000000 2. o-i 12 6390 22. •92096 32. 5*793 42 -000000062 3. 4-9 13 •1463 33- -0899 33- 50000 43 30000-9 4- 3^ 14 3-874 24. 1 0000 34- 700090 44 10000*9 5- 380 I.? 6754 25- 4800 35- 264000 45 594500 6. 100 16 •0876 26. 9080-8 36. 404007 46 8S590000 7. '0001 17 ■3467 27. '00058 37- 500909 47 287-642 8. 24-6 18. 1-083 28. •035871 38- 48-627 48 0-003564 9. 3-88 19. 0-125 29. •000448 39- 93'5U 49- -oodS56i)»3-6 0. 900 20. 0-0009 3°- 4480000 40. ■032764 50 65480000 * A very curious and valuable artifice, discovered by Gunter about 1614. Un Logarithms, 73 I. 2"30963o 10. 0-565021 19. 2. 3-676968 IX. 0-778441 20. 3- 0-95424.1 12. 2-769504 21. 4- 1-698970 13- 5'774i5J 22. 5. oooooo 14. 5-421604 23- 6. 2-000000 15- 3-000000 24- 7- 2*564494 16. 6-39445* 25- 8. 3'563362 17- 1-415674 26. 9- 2-621754 18. 1-188591 27. 99. Required the natural number of the following logarithms : — 2-309630 10. 0-565021 19. 2-954243 28. 5606389 37. 1-883030 3-676968 IX. 0-778441 20. 3-959041 29. 5-000000 38. 3-625343 0-954243 J2- 2-769504 21. 4-705864 30. 2-881955 39- ''725364 1-69S970 13. 5-774152 22. 0-415974 31. 1-I673I7 40. 5-627407 0000000 14. 5-421604 23. i-oooooo 32. 7-875061 41. 3-686216 3-954243 33- 0-000186 42. 0-400573 2-716003 34. 6-947385 43. 5002559 5-654243 35- 2-9630S1 44. 4-321547 0-434294 36- 0-763947 45. 0-875061 100. Finally, we recommend the student to commit to memory the follow- ing table of logarithms to two places : — No. Log. No. Log. No Log. 1. 00 4. 60 7. 85 2. 30 5. 70 8. 90 3. 48 6. 78 9. 95 MULTIPLICATION BY LOGARITHMS, loi. In multiplication we proceed by RULE XXXII. 1°. Find the logarithms of the nianhcrs, the product of which is required. (For the method of taking out tho log. of a number see pagea 64 to 6-.) Note. — If any of the quantities is a decimal, eith<^r the nepalive cbaraotoristic of that quantity or ita arithmetical complement is to be used (see Eules XIX and XX, page 62.) 2°. Add these together, the sum mil he the logarithm of the product. 3°. Find from the Tables the corresponding member. (For the method of finding the corresponding number to a log., see pages 68 to 71.) This will be reqiiired product. KoTE r. — "When the characteristics are negative and subtract jd from ro (see Rule XX. page 62), if the sum of such characteristic exceeds the sum of tens boriuwed, the product, ■will be a -whole nuioV -^j othir-wise it -will be a decimal. (See Ek. 15, paj?rt 75.) Note 2. — When the char-^cterLstics of the logarithms to be added are all positive, it id evident that their sum will bo poaiti^e. Note 3. — If the characterist'cs arc all negative, their fium diminished by the figure — if any — carried from the sum of the mantiss© or positive decimal parts will be negative. (Ex, 9.) Note 4. — If some characteristm are positive and tha^ others negative, find the sum of the positive characteristics together with any figure which may be carried from the decimal part of the logarithm; also add the negative characteristics together; subtract the less of these quantities from tho greater and prefi.K to the difference the sign belonging to the greater. But if a positive and a negative characteristic are exactly equal to each other, cancel both ; this is done in practice by simply drawing the pea through them. (Ex. 13.) Examples.* I. Multiply 77 bj' 100. The log. of 77 and 100 being taken from the table, wo have 77 log. 1-886491 100 log. 2-OO0O0O 7700 log. 3-886491 We hive here added the loprs. of the piven factors, and having sough", in the Table for the mantissa •886491, we have found tlie figure.s of the iiai.. no. corresponding to he 770c ; the index 3 determines /our of thesp to he integ-al ; hence the predict is 7700 (Rule XXVI, page 68). 2. Multiply 97 by 83. The log. of 97 and 83 being taken from the Table, we have 97 log. 1-986773 83 log. I -9 1 907 8 8051 log. 3-905850 We add the logs, of the given factors, and then seek in the Table fur the mantissi '905850, whici corre.sponds to the natural number 8051 ; the index 3 determines /oM/- of these to he integral; hence tho proiuct is 8051 (Rule XXVI, page 68). * In these examples, and for several of the subjoined Exercises, the logarithmic is more tedious than the ordinary method of calculation ; the purpose here intended being simply to make the student familiar with the process of finding products logarithmically. It must be remembered too, that by the logarithmic process, we generally obtain only an approxi- mate value of the required result. L 74 On Loga/rithms. 3. Multiply 378 by 50. 378 log. 2-577492 50 log. i'698970 18900 log. 4-276462 Themantissaof log., viz., '276462, is found crnc^?;/ in the Table in a line with 189, ami under o ; but as the characteristic 4 rnquircs 5 digits in the integer part, we therefore add a cypher (o) , which gives iSgco as the nat. no. corresponding to the proposed log. This is according to Rule XXIX (a), page 69. 4, Multiply 3456 by 500- 3456 log. 3-538574 500 log. 2-698970 1728000 log. 6-237544 The characteristic 6 requires 7 digits in the integer part of product, we therefore annex 3 cyphers which gives 17280^0 as the nat. no. required. (See Rule XXIX [a), page 69. 5. Multiply 963 by 48-9 by common logarithms. The log. of 963 and 48-9 being taken from the Table, wo have 963 log. 2-983626 48-9 log. 1-689309 log. 4-672935 _ (next lower in Table) -672929 gives 4709 Product 47090-7 92)6-00(06 552 48 We have here added the logs, of the given factors together, and having sought for the given mantissa •672935, which is not to bo exactly found in the Tables, we obtain the next less mantissa -672929, ■which we subtract from the given mantissa ; the difference is 6, to which two cyphers are annexed, and then we divide by the tabular difference 92, whence we obtain 07 nearly ; the remainder, 48, beins more than half the divisor, i is added to the last figure in the quotiant (6) ; attaching these to the four figures obtained previou.sly, we have 470907; the charact ristic 4 determines five of these to be in- tegral; hence the product is 470907 (Rule XXVI, page 68). The multiplier containing one decimal place, the product is worked out to one place of decimals. 8. Multiply 29-42 by 8-6 by common logarithms. 29-42 log. 1-468643 8-6 log. 0-934498 2-403141 (next lower in Table) 403 1 20* gives 2530 Product 253-012 171)2100(12-^- In this instance the characteristic of the log. of the product is 2, hence the integral part of the natural number must contain 3 figures ; but since there are decimals in both factors, there must be decimals in the product — as many decimal places as there are in both the multiplier and multiplicand together. In 29-42 are two decimal places, and in 8'6 one ; hence in the product three decimal places are required, making, with the three integral figures, in all six places. Now the nest lower man ti8sa_ found in tho table gives the four corresponding figures 2=;3^, leav- ing two figures to be foimd. (See Rule XXX, page 70.) * This log. is taken from Norie, and is incorrect in the last decimal figure, which ought to te i, as given in Raper's table; the true log. being -403 1205 2. 6. Multiply 734 by 23. 734 log. 2-865696 23 log. 1-361728 log. 4*227424 (next lower mantissa) 227372 corresponds to 1688. Product 16882 258)520(2 — 7. Multiply 498 by 376. 498 log. 2-697229 376 log. 2-575188 187248 log. 5-272417 306 DifiF. 232)11100(48 nearly 9. Multiply -0567 by -00339. Both multiplier and multiplicand being decimalsi the characteristics of these factors will be negative! but instead wo use their arithmetical complementsj thus :— •0567 log. 8-753583 •00339 log. 7 "530200 •0001922 log. 6-283783 Here 10 is borrowed to find the characteristic both of the multiplicand -0567, and the multiplier -00339 (see Rule XX, page 62). The sum of the charac- teristics, including the 1 carried from the decimal part of the log., amounts to 16; reject 10 and write down 6 for the index of the log. of product. Then, seeking in the Table for the decimal part, viz., -28 J 783, the natural number corresponding to it is found to be 1922 ; and since the sum of the indices 16 is 4 less than the 20 borrowed, (see Rule XXXII, Note 1, page 73) the product is a decimal fraction, and ttiQ first significant digit must stand in the fourth decimal place ; hence the product is -0001922. Or thus — using negative indices :— •0567 log. 2-753583 •00339 log. 3"53O20o •0001922 log. 4*283783 In adding, when we come to the place of tenths, the process is 5 and 7 are 12, 2 to put down and 1 to carry, and since the characteristics are^both negative ("S") and (T), we diminish their_sum ( 5 ) by the num- ber carried (i), which leaves 4 for the index (see Rule XXXII, Note 3, page 73). "We prefix 3 cvphers because the index being i- the first significant figure of product must stand in the fourth place from the decimal point. On Logarithns. 1<, 10. Multiply 99'9 by 8-63. 99'9 los- i'999565 863 log. o'936oii 862-136 log. 2-935576 558 5,0)180,0 36 Multiply 436 by i^"j. 436 log. 2-639486 19-7 log. 1-294466 8589-2 log. 3*933952 43 51)90(2 nearly 12. Find the product of -073 by -00028 by logarithms. •073 log. 8-863323 •00028 log. 6-447158 •00002044 log. 5-310481 Or, using the negative characteristic, thus : — •073 log. 2-863323 •00028 log. 4-447158 •00002044 log. 5-310481 In adding, when ■we come to the place of tenths, the process is <, and 8 are 13, 3 to put down and i to carry; and this 1 being a positive quantity. Hence in the above, +1, — 2, and — 4 arc to be algcbrically added together to form the new characteristic. The sum of the two characteristics (both negative) viz., — 2 and — 4 is — 6, which diminished by + 1 leaves — 1; for the new characteristic. We prefix/o;«r oyiihors, because the characteristic being 5 shows that the first significant figure must stand in the fiflli decimal place. (Rule XXVII, page 68.) 13. Multiply 24000 by "000783. 24000 log. 4-3802 1 r •000783 log. 6-893762 18-7919 + log. 1-273973 Here 10 was borrowed in detonnining the index of the log. of '000783, and since the sum of the indio<'s (including i carried from the dfcimal part of log.) is elcfen, we reject or jxnj hack the 10 borrowed, which leaves 1 for the index, and the nat. number corres- ponding is found to be 18791Q, and we mark oflf to the right two figures (i more than the characteristic) whence the answer is 1879194-. Or thus — using negative indices: Ex. 13. 24000 log. 4-380211 •000783 log. 4-893762 18-7919 -flog. 1-273973 Here the 1 which is carried after adding i, 8, and 3 (in the place of tenths), instead of increasing the T leaves 3. This is according to Rule XXXII, Note 4, page 73. 14. Multiply ^0172 by •00214. •0172 log. 8-235528 •00214 log. 7-330414 •000036808 log. 5-565942 In this instance 10 is borrowed, in finding the in- dex of the log. both of the multiplier and multipli- cand, and 10 is rejected from the sura, which sum (15) being ^ /Msthan the amount borrowed (20), indicates that the product must be a decimal fraction, and the fir.it significant digit stands in the fifth decimal place ; hence the product is •000036808. This is according to Rule XXXII, Note i, page 73. Or thus — using negative indices : Ex. 14. •0172 log. 1-235528 •00214 log. 3-330414 •000036808 log. 5-565942 The characteristics of both logs, being negative, the sum of them is taken, and this, with the necrative sign over it, is put down us the characteristic of the log. of product. We prefix four cyphers to the num- ber taken out of the Table, since the characteristic being 5, the first significiint figure of the product must stand in the fifth place from the decimal point. 1$. Required the product of i7'25, 0-82, and 0^065. 17-25 log. i^236789 •82 log. 9-913814 •065 log. 8-812913 0-919425 log. 9-963516 Here 10 is borrowed, to find the characteristics of log. of both of the second and third factors, and sub- tracting the sum of the indices, lo from 20 leaves i ; the ,sum being less than the number borrowed, the proiluct is a decimal, and hence the first .significant figure must occupy the first place to the tight of tho decimal point. (See Rule XXXII, Note 1, page 73.) Or thus : — 17-25 log. 1-236789 -82 log. 1-913814 •065 log. 2-812913 0-919425 log. T-963516 Here we have i to carry from tho mantissa, which added to tho positive characteristic 1 (characteristic of log. I7'25, .see above) makes positive 2. Now the sum of the negative indices is "s (negative 3), and, therefore, since where one is positive and the other is negative, the_differcncc is the characteristic ; we have + 2 from "3" leaves T for the characteristic, (see Rule XX XII, Note 4, page 73) and tlie first signifi- cant figure of the quotient must occupy the first place to the rio-ht of the decimal point. (Rule XXVI, pagetSJ. • ^6 On ZogarWhms. Examples fob Practice. Multiply by logs. 85 by 70 ; 39 by 27 ; 100 by 10 ; and 369 by 9. „ 53S by 1-74; 601 by 18;. 250 by 12-5 ; and 3964 by 7. „ 20-42 by 0-5 ; 3-646 by 0-75 ; 2-745 by 0-24 ; and 5-792 by G-^. „ 5671 by 4-7; 517 by 6591; 60-609 by 72; 1-955 by 10-04; and 758S75 by 8. ,, 127 by 304; 476 by loa; 8o'o8 by 5-98 ; 5760 by 30; and 970 by 630. )> 3.7"6by249; 44"4by22'i; 182-7 by 250; 2807 by 200; and 63-055 by 84. „ 280054 by5o; 30967 by 90; 23716 by 350 ; and 45670 by 690. ,, 81-33 by 15-3; 47-6 by 6-82; 1 0000 by 10; and 4-02674 by •0123456. „ 789^0 by 400; 756-875 by 8; 94-055 by 74; and 1975 by 10-76. j» 732 by 543; 587 by 66-4; 3000 by 100-14; and 60060 by 700. „ 543-29 by 3800-62; 90-43 by 712-2; 87-305 by 4-09 ; an I 209-36 by 46. „ 348-25 by 7-125; 498-256 by 41-2467; 56-3426by -023579; -123456 by 26813-9. „ -0001468 by -000395 ; o-ooo6 by 10-0004; -605 by'00000091; and 35'69i by -0048. „ -00146 by -039 ; 5900 by -00071 ; 4-189 by -00071 ; and 247-55 by 56-72. „ 527"45 ty 1-6938; 10-5526 by ZiTJAS; '007461 by •3^5^']6^ ; and -0700397 by '0086752. 16. „ -iby'i; -ooor by -oooM ; -on by i-oi and 'ooioi ; and 1000 by 100. DIVISION BY LOaAEITHMS. 102. In division we proceed by EULE XXXIII. 1". Find the logarithms of the numbers the quotient of which is required. KoTE. — If the dividend or divisor, or both, are decimaln, ihe negative characteristic of that quantity, or its arithmetical complement, is to boused. 2°. Sultract the logarithm of the divisor from that of the dividend, f adding 10 to the characteristic of this last, if required) ; the difference will he the logarithm of the quotient. 3°. Find from the Tables the corresponding number. This will be the required quotient. Note r. — When the divisor is greater than the dividend, the characteristic of the logarithm of the quotient will come out negative — the quotient itself being, evidently, a decimal ; but if we wish to avoid the use of negative characteristics it will be necessary to add 10 to the characteristic of the dividend when subtracting the logarithm of the divi^:or, and tho characteristic of the remainder is the arithmetical complement of the negative characteristic of the quotient. (See Ex. 4, 5.) Note 2.— If, for the sake of convenience, the line containing the quantity to be subtracted, when the quantities have been written down one under th^ other, is called the take line and the quantity from which it is to be subtracted the from line, then subtracting in the usual way until we come to the characteristics ; if their signs aro alike take the diflference of them, and if the from line is the greater, prefix to the remainder the ffiven ngn ; but if the tal^e line is the greater prefix the contrary of the given sign. If the nigns are different, take the sum of the characteristics and prefix ihe aign of the f-r/m line. The figure borro-wed when Subtracting the decimal part of the logarithm, when carried to the characteristic, is always to be added, and therefore make a nogativo characteristic less, thus 2 carried to 5 makes it 3. Note 3. — Otherwise, if one or both of tho given terms are decimals, remove the decimal points till the factors contain whole numbers, and the dividend the greatest; then if the dividend he more places removed than the divisor, remove the decimal point of the quotient as many places to the left hand, but if the divisor be more places removed, then remove the decimal point of the quotient as many places to the right hand, If the dividend and divisor be equally removed, the quotient is not to be altered. Vn ZogafWhrns. 11 I. Dhvide 3192 by 76. The loGf. of 3192 is taken out according to Rule XXIV, page 6$, and the log. of 76 by Kule XXII, page 64. 3192 log. 3-504063 76 log. i'S8o8i4 Quotient 42*0 log. 1-633249 3. Divide 579416 by 4324. Log. of 5794 r:^ 762978 Tab. d iff. 75 Farts foE 16 -|- 1 ^ X 1 6 Log. of 579416=5-762990 450 579416 log. 5-762990 4324 log. 3'635886 134-0 log. 2*127104 a. Divide 830772 by 982. The log. of 830772 is taken out by Rule XXV> page 66. "We seek in the Iclt-hand column of the Table (No.) for 830 (the first three digrits), and also at the topofthepagcinoneof the horizontal columns for the fourth figure 7, then in a line with the first and under the latter wo have 919444. In a line with this quantity and in the right-hand column marked Z'»/f.stands52, which multiplied by the remaining fig- ures of the nat. number, viz. 7.-'., produces 3744; then cutting off two discits from these (since we multiplied by tivo digits) it becomes 37, which being added to 519444, the mantissa of 8307, makes 910481, and with the characteristio 5, is 5-919481. The work wil' stand thus :— Log. 8307=919444 Diff. for 72 -f- 37 Tub diff. 52 X 7» 9194S1 830772 log. 5-919481 982 log. 2-9921 ir Quotient 846-0 log. 2-927370 37,44 4. Divide 34 by 582. 34 log. 1-531479 582 log. 2-764933 Quotient -05842 log, 8-766556 In this instanoo 10 is added to the characteristic bf the dividend to enabla the subtraction of the log. or divisor to be made, and to avoid negative charac- f eristics ; the divisor is greater than the dividend, the quotient therefore is a decimal. (Sec Note i, p 76.) Or thus, using negative Characteristics : — 34 log. i'53i479 583 log. 3-764923 •05842 log. 3-766556 In this example, when carrying 1 to the character- istic) 2, we have to subtract 3 from 1 which gives —2 (ncgativo 2) ; or according to Note 2, the character- istics having liko signs ( + ) their difference is taken, and the take line being the greater, prefix to the remainder a contrary sign (— ) to the given one. 5. Divide 3672 by 51000, 3672 log. 3-564903 51000 log. 4'70757o Quotient -072 log. 8-857333 Here 10 is added to the characteristic of the divi- dend before subtracting log. of divisor. The divisor being greater than the dividend the quotient is evidently a decimal. (See Note i, page 76.) Or thus, using negative characteristics :— 3672 log. 3'564903 51000 log. 4-707570 •072 log. 2-857333 In this instance, when carrying 1 to the character- istic 4, we have to subtract 5 from 3, which gives —2 (negative 2) ; or by Note 2.— Take the difference of characteristics, as they are of the same sign (+), and prefix a contrary sign (— ) to the remainder, tho take line being tho greater. 6. Find the quotient of -09983 -i- -67. •09983 log. 8-999261 •67 log. 9-826075 •i49log. 9-173186 Before -subtracting the log. of divisor from that of tho dividend 10 must be added to the characteristic of the dividend; the quotient is therefore a decimal. Or using negative character.sticsthe work will stand thus :— •09983 l0£ -67 log 2-999261 T-826075 •149 log. 1-173186 In this instance both characteristics are of the same sigp ( — ), the from line tho greater ; the character- istic of log. of quotient is marked witU sign (— ). 7. Divide "01958 by •4828. •01958 log. 8-291813 •4828 log. 9-683767 •04056 log. 8-608046 Hero 10 has to be borrowed in subtracting thelog. of divisor from that of dividend ; or using negative characteristics the work will stand thus :— •01958 log. 2-291813 •4828 log. 1-683767 •04056 log, 2-608046 To obtain the characteristic of the quotient (Tithe I that Ls carried, and which is positive, is added to tho 1 j)roducing o, which has then to be subtracted from a , leaving a , 78 0)1 Loganthms. 8. Divide 1 8-792 by -000783. Log. 1879 = 273927 Diff. 232 Difl'. for 2 = -f 46 2 Log. 18-792 = 1-273973 46,4 Log. Log. Log. 18-792 = 1-273973 •000783 = 6-89^762 24000 r= 4-350211 The divisor here is less than the dividend, the for- mer (iS-792) being amixednumber, whilst the latter is a decimal {'000783) ; the quotient, therefore, is an integer. Or, using the negative characteristic of divisor. Log, Log. 18-792 = 1-273973 •003783 = 4-893762 Log. 24000 = 4-380211 In the subtraction it will be seen thjit carrpng 1 to the T we say 1 and "4 make "3, and 3 taken from 1 leaves + 4. The characteristics being of different names, + and — , their sum is taken, and the re- mainder takes the same sign as the fi'om line — in this case it is positive {+). In writing down the result the + is left out. 9. Divide 26843 by -03010. Log. 2684 =428782* Diff. for 3=4" 49 Tab. diflf. 162 X 3 Log. 26843 =428831 48,6 or 49 * Eaperj. Or thus:- 26843 log. 4"42883i •03010 log. 8-478566 891794 log. 5-950265 219 49)4600(94 nearly. 26843 log. 4'42883i •03010 log. 5-478566 891794 log. 5-950265 Ifcre the characteristic of the second log. is -2, but following the rule, we have changed it to posi- tive 2. The characteristic of first log. beine positive 4, the two are added, the sum being positive 6, but having borrowed 1, the correct characteristic is 5, and being positive, the quotient will contain 6 integral figures. 10. Divide -8 by -0000002. -8 log. 9-903090 •0000002 log. 3-301030 4000000 log. 6-602060 The divisor being less than the dividend, the quo- tient is eWdently an integer, and the characteristic denotes that it is a whole number consisting of seven places of figures ; cyphers are therefore annexed to make up the required number. Or, Log. '8 = 7-903090 Log. -0000002 r= 7-301030 Log. 4000000 = 6-602060 The characteristics are both negative, fake their difference, and prefix to the remainder the contrary sign to the given one, as the take one is the greater. II. Divide -00815 by -000275. 0-00815 log. 7-9x1158 0-000275 log. 6-439333 Quotient 29-6364 log. 1-471825 Or, -00815 ^oR- 3"9m5^ •000275 log. 4-439333 29-6364 log. 1-471825 The index of the divisor T being supposed changed to positive /,, the difference between which and ■3 leaves positiv-e 1 for index of quotient. Or, proceed- ing according to Note 2 — Since the characteristica have like signs, take their difference ; the remainder takes a positive sign, or a contrary sign to the take line, which is the greater. • 12. Divide 469-76 by 0-937. 469-76 log. 2-671877 0-937 log. 9-971740 Quotient 501-345 log. 2-700137 Or, 469-76 log. 2-671877 0-937 log. 7-971740 501-345 log. 2-700137 13. Divide 6 by 'ooooooi. 6 log. 0-778151 -OOOOOOI log. 3-000000 60000000 log. 7-778151 The divisor is li'i.'< than the divilend, the quotient, therefore, is a whole number, and the characteristic 7 indicates that it consists of 8 places of figures; annex cyphers to make up the number. Or, 6 1og. 0-77S151 •OOOOOOI log. 7- 60000000 log. 7-778151 On Logarithms. 19 14. Divide '012550 by 1004000. •012550 log. 8-098644 1004000 log. 6'ooi734 •0000000125 log. 2'0969io The divisor bcine: greater than the dividend, the j'uoticnt is a decimal. Or, "012550 log. 2'098644 1004000 log. 6'ooi734 •0000000125 ^^o- 8*0969 10 The characteristic of the dividend is T, that of the divisor positive 6 ; then according to Note 2, the signs being unlike, tako the sum of the characteristics, prefixing the sign of the//om lino [—). 15. Divide -027472 ty 3-434. Log. 2747 =: -438859 Diff. 158 Ditr. for 2 z= + 32 X 2 Lof 2747 = -438891 31.6 Log. -02747 = 8-438891 Log. 3-434 = 0-535800 Log. -008000 = 7-903091 Or, using negative characteristics, thus: — Log. -02747 :zr 2-438891 Log. 3-434 = 0-535800 Lo" -008 = 3-903091 Examples for Practice. Divide 6391 by 77 ; 21636 by 36; 6384 by 76; and 93750 by 750. „ 9504000 by 98; 45000 by 9; 6071000 by 8; and 58469 by 981. „ 382-746 by 593; 218432 by 495; 300360 by 100-12 ; and 365-55 by 5-5. » 783254 ty 250689; -79632 by -019354; -0092852 by -0003461 ; and -654831 by -474586. „ -0008464 by -0002852; -05826 by -95381 ; '019354 by -79632 ; -0003461 by '0092852 ; -00005 by 2-5, by 25, and by "0000025. „ 77000000 by 9999 ; 680300 by 681500; 100-002 by 1-0012 ; and 75759-6 by 13062. „ 1-32704 by -0358; '7i56by2-6S878; 87-641 by "003368; and '563426 by -023574. „ 999999 by loior; 57634-1 by 276-4; 69-7565 by -97564; and 352740 by 56780. „ 40048000 by 800 ; 11123100 by 340 ; i8692ioby9o; and 1875000 by 15000. „ 75-2484 by 8-59; 147392 by 440; 1962820 by 10-04; and 888888 by 8S000. „ 248-25 by 364-87 ; -235316 by 293-864; 5"6949 by 53'058; 3876000 by 1200; and 42 by -00007. „ o63i4by '0007241 ; '004728 by ^2382 ; 36'49 by 192-24; '048S69 by -oo7^698• I9 -^ 72 ; 19 -f- '72 ; '19 -7- '72 ; 19 -1- -0072 ; 6 -f- '0000003 ; and 9 -f- -0000003. ■01237-^-10846; 28-7642-^-083456; -oiooii -f- 0993 ; and -048869 -f- '007 1698. •I -7- -0004572 ; I -7--0011636 ; 11-2221 -J- I II ; 4000-^-000125; and '562625 -f-52643 •0001 -j- 'oooi ; loooooo — -ooooooi ; 10 •—- 100; -oooooooi -|- -oooooi ; 1000 -f- -j\,-. 103. When it is proposed to find the value of an expression in which both multiplication and division are sigmfiod, the sum of the logarithms of the factors of the dividend, diminished by the sum of the logarithms of the factors of the divisor, will be the logarithm of the value required. Thus : to find the value of Example. 209 X 573 X 63 287 X 2101 287 log. 2-4578S2 2101 log. 3-322426 209 log. 2-320146 573 log. 2-758155 63 log. 1-799341 5780308 6-877642 5-780308 Ans,: 12-5x22 log. 1-097334 8o On Logarithms. 1 04. It is very often expedient to transform the logarithm of a divisor into that of a multiplier, and it is customary, in such calculations, to avoid not only negative logarithms, but negative indices also, by substituting for a subtraction logarithm its arithmetical complement (See No. 96, page 72^. This makes the operation consist of a single addition, only we must diminish the result by subtracting 10 for every arithmetical complement that has been used. By this means the process of division is less open to error from mistakes when logarithms with negative characteristics would be subtracted.* To apply this method to the example ahove : — Having found in the Table the log. of the divisor 287, -we may at once transform it into the addition logarithm 7-542118, and similarly, for the log. of 2101 we may writo 6'677574, and then the calculation -will proceed con- tinuously as follows : — ■ 209 log. a'32or46 573 log. 2-758155 63 log. 1-799341 287 ar. CO. 7-542118 — 10 3IOI ar. CO. 6-677574 — 10 1-097334 ./4ms. ; 12-5122 Examples for Practice. r. 7 X 8-73 84 X -00769 X 683 8-4 X -0769 X '006^3 ■54963 598 X •0000146 X '039 59-8 X -CK500146 X '0039 2. 67-038 X -010705 X 4-1525 28-045 X 1-3564 X -0942537 -7854 X 3-1416 X -086725 48-375 X 2-71828 X 52359 3, Divide -06314 X -7438 X '102367 by -007241 X 12-9476 X -496523) and compare theresult with the product of 8-71979 X -057447 X -0206168. 105. Degree of Dependence. — The number of places of figures which may be obtained in a result derived from any table of logarithms is the same, usually rejecting prefixed cyphers, as the number of decimals to which the lof arithms are carried. JBut towards the end of the Table the last place thus obtained cannot always be depended upon within a unit, that is, provided the mantissa of log. is greater than -9388. Thus, for instance, the log. 3-7575 corresponds to the no. 5721 and the log. 3-7576 to 5722, nearly. It will moreover be noticed that the log. tables are, in fact, useless for dealing with numbers consisting of more than eight places of figures. Thus, for instance, we should not find any difference between the log. of 23-47832 and the log. of 23-4783297. This remark should be kept in mind, because it is mere waste of time to employ more figures than are required to insure a certain degree of precision in the result. * To divide by any number n is the same in effect as to multiply \)y its reciprocal \ (that is, the quotient of unity divided by that number, and is so called from an exchange of places between a nnmerator and denominator : thus, the reciprocal of -| is 3, that of 6, or f, is i). Therefore to subtract log. n is the same in effect as to add log. \-=.o — log. n. 8i TRIGONOMETRICAL TABLES. 1 06. There are two kinds of trigonometrical tables; the first, called the Tahle of Natural Sines, Cosines, Sj'c, contains the numerical values of the sines, cosines, tangents, &c., that is, of the trigonometrical ratios for each given value of the angle; the second, called the Table of Logarithmic Sines, ^c, contains the logarithms of the numbers in the first Table.* TABLE OF NATUEAL SINES, &c. 107. The trigonometrical functionsf or ratios are numbers which are capable of being calculated from geometrical principles, and accordingly certain series have been investigated, and certain algebraic expedients devised for the general purpose of determining the trigonometrical ratios. With such aid the sines, cosines, &c., of all angles from 0° to 90° (i.e., for all values of A, from A = o up to A = 90) have been computed to several places of decimals and arranged in tables called Tables of Natural Sines, Cosines, Sfc. In some tables the angles succeed each other at intervals of i", in others at intervals of 10"; but in ordinary tables (as Table XXVI, Noeie) at intervals of i', and to the last mentioned we shall refer. 108. The statement of the method by which such tables are constructed is unsuitable to the pages of the present work. The mode of using them in computation we shall now proceed to explain. 109. The arrangement of this table will be understood from a simple inspection. It contains the sines, cosines, &c., of angles between zero and 90°, generally for every minute, and the fluctuations of angles containing a number of degrees, minutes, and seconds, have to be found by interpolation similar in their nature to those that are required to be used in tables of logarithms of numbers. This interpolation is based upon the supposition that the diflfer- ences of the sines, &c., are proportional to the difi'erences of the angles, and this proportion, though theoretically inexact, gives, in general, a sufiicient approximation, provided the difference of the angles of the table are suffi- ciently small. 1 1 o. Eeferring to the Tables (Table XXVI, Nome) it will be seen that the degrees are given at the top of the column, and the minutes down the left hand side of the page for the sines. And, for the cosines, the degrees are given at the bottom of the page, and the minutes up the right hand side of the page. * The usual trigonometrical tables are given in conjunction with tables of logarithms, and they more frequently give logarithms only than sines, cosines, &c., themselves. When logarithms were invented they were called artificial numbers, and the originals for which logarithms were computed, were accordingly called natural numbers. Thus, in speaking of a table of sines, to express that it is not the logarithms of the sines which are given, but sines themselves, that table would be called a table of natural sines, and the logarithms of these would be called not logarithms of sines but logarithmic sines, ^c. t By the functions of angles (sometimes called their trigonometrical or geometrical functions) are meant their sines, tangents, secants, versed sines, and chords ; the word function signi- fying any quantity that is depeniit«it on another changing as it changes. M 82 Trigonometrical Tables. The difference of the trigonometrical ratios for loo" are given at the foot of each column. 111. In using these Tables, we have either to find the sine, cosine, &c., of an angle whose value is given in degi'ees (°), minutes ('), and seconds (") ; or to find the corresponding angle in degrees, minutes, and seconds. 112. If the value of the angle be given in degrees and minutes only, the sine, cosine, &c., is found directly from the Tables, in which are registered the values of the trigonometrical ratios. All the numbers contained in such Tables as Nome's Table XXVI must be understood as decimals. Thus, nat sine 7° 7' = •123890 „ sine 59 40 = •863102 „ cosine 15 30 r= •963630 „ cosine 71 12 = *322266 113. As the sines, cosines, &c., pass through all their possible numerical values while the angle varies from 0° to 90°, the tables are not extended beyond 90° ; such computations would be superfluous, for the sine or cosine of an angle between one and two right angles, viz., of an angle greater than 90° is the same in numerical value as the sine, cosine, &c., of an angle as much below 90°, and is known from the recorded sine or cosine of its supplement.* Whence also Nat. sine 156° 42' = sine 23° 1 8' := -395546 ,, cosine 108 48 = cosine 71 12 = "322266 „ sine 140 16 = sine 39 44 = "639215 „ cosine 140 16 = cosine 39 44 = "769028 114. If the angle contains seconds, we must proceed by the method of proportional parts, as in the following examples : — EULE XXXIV. 1°. Find from the Table the nat. sine, cosine, Sfc, which corresponds to the degrees and minutes. (Norib, Table XXVI.) 2°. Multiply the difference ly the seconds, and divide hj 1 00. Note. — To divide by 100 we have merely to cut off the two right-hand figures. 3'^. If the required quantity he a nat. sine, tangent, or secant, add the result to the last figures obtained in 1° ; if it be a cosine, cotangent, or cosecant, subtract. The result will be the required sine, cosine, &c. Note i. — The reason of this rule is founded on the principle that for a small interval, such as one minute, the increase of the sine is proportional to the increase of the angle. Note 2. — It is necessary to bear in mind that the sine, tangent, and secant (under 90°) for which the tables are constructed increase as the arc increases, whilst the cosine, co- tangent, and cosecant decrease as the arc increases. This will require the corrections connected with a sine, a tangent, or a secant to be added, and those connected with a cosine, a cotangent, or a cosecant to be subtracted whether arcs or their functions be sought from the tables. * Def. — The supplement of an angle is the result when the angle is subtracted from 1 80°. In other words, an angle and its supplement together make 1 80°, or two right angles, thus, 23" 19' is the supplement of 156° 41', and 156° 41' is the supplement of 23° 19'. Trigonometrical Tables. 83 Examples. Ex. I. Find the nat. sine of 12' 44' 27" Nat. sine 12" 44' = 220414 437 X 27 Tab. diff. =4-128 100 Am. 220542 Nat. sine 12° 44' 27* = 220542 To obtain the parts for the second we multiply the tabular difference by the number of seconds and divide by 100, thus; — Tab. diff. 473 No. of seconds X 27 33" 946 Ex. 2. Find the nat. cosine of 31° 28' 42". Nat. cosine 31° 28' := "852944 253 X 42 Tab. diff. = — 106 127,71 128 nearly. Tab. diff. 253 Seconds X 42 506 roi2 Ans. •852838 Nat. cosine 31° 28' 42" = "852838 106,26 106 Examples for Practice. To find the nat. sine of 1. 34° 48' 15" 3. 71" 20' 43" 2. 60 7 18 4. 21 44 21 To find the nat. cosine of 1. 14° 15' 3" 3. 80° 22' 22" 2. 70 47 40 4. 5 22 10 5" 6. 46° 22' 37" 76 57 49 5- 6. 46° 31' 41' 29 40 48 53° 7' 49" 86 3 17 38° 31' 10" 8 19 17 115. If the value of the sine, cosine, &c., be given, and it is required to find the angle, we use the following rule : — EULE XXXV. 1°. IHnd in the Tables the next lower nat. sine, nat. cosine, Sfc, and note the corresponding degrees and minutes. 2°. Subtract this from the given sine, cosine, ^c, multiplying the difference hy 100; divide by the tabular difference, and consider the result as seconds. 3°. If the given value he that of a sine, tangent, or secant, add these seconds to the degrees and minutes found in 1° ; ij it he that of a cosine, cotangent, Sfc, subtract. The result will be the required angle. Note. — In taking out the angle for a natural cosine we may take out the next greater natural cosine, and subtract the given natural cosine from it ; and having found the seconds ("), as above, they are additive. The trigonometrical ratio corresponding to the 7iext less angle being written down in every case, confusion will be avoided, as the additional seconds •will always be additive. Examples. Ex. I. Given the natural sine = ©"732156 : find the angle. Given nat. sine 732156 Sine 47° 4' = 732147 next lower in Table XXVI, NoRiE. Tab. diff. := 327 327)900(3" nearly (additional seconds for nat. sine.) 981 Ans. : 47° 4' 3". Tables of Logmithms of Trigonometrical Ratios. The log. 732156 is sought for in Table XXVI, Noeie, but as it cannot be found exactly, the next leas is taken which corresponds to 47° 4'. The difference of the logs, is then found, two cyphers added (which is equivalent to multiplying by 100), and the product divided by the tabular difference ; the quotient is the additional seconds. Ex. 2. Given the natural cosine 853267 : find the angle. Given nat. cosine 853267 Cosine 31° 26' =: 853248 next lower in Table XXVI, Norie. 31° 26' o" Tab. difif. = 253 253)1900(7" (to be subtracted), — 7 1771 31 25 33 129 Ans.: ■^1° 2S' 5l"- Ex. 3. Find the angle whose natural cosine is 728713. Proceeding according to Note, page 83. Here nat. cosine of required angle = -728713 Nat. cosine of next less angle, or 43° 13'= 728769 Tab. diff. = 334 334)5600(17" nearly, to bo 334 added. 2260 2338 . ' . angle required = 43° 13' 17". Examples for PRAcrnc:E. Given the nat. sines, to find the angle. 1. -898002 3. -8 5. -444 7. -740912 9. -75214 2. -370383 4. -920411 6. -20389 8. -^ft or -529221 10. -96 Q-iven the nat. cosine, to find the angle. 1. -448807 3. -726998 5. -51484.1 7. -769388 9. -817726 II. 999000 2. '948397 4. -702017 6. -914237 8. -974822 10. -215515 12, -6 TABLES OF LOGARITHMS OF TRiaONOMETRICAL RATIOS. 116. The Trigonometrical Ratios being numbers, have logarithms that correspond to them. In practice the logarithmic are generally far more useful than the natural sines, &c., though the latter are often necessary, or in some simple kinds of calculation, preferable. 117. As the sines and cosines of all angles, and the tangents of angles less than 45^, are less than radius or unity, being proper fractions, the logarithms of the value of these quantities, properly, have negative characteristics. In order to avoid the inconvenience of printing negative logarithms, and for other reasons, 10 is added to the characteristic before it is registered in the table of logarithmic sines, &c., so that we find the characteristic g instead of I, 8 instead of 2, &c. Thus, on referring to the Table of Natural Sines (Table XXVI, Noiiie), we find natural sine of 16° = -275637. If we cilculute the logarithm of -275637, we find its value is 7-440338 ; if to this 10 is added we find that Log. sine 16° ^ 9-440338. Tables of Logarithms of Trigonometrical Ratios. !? To preserve uniformity, the characteristics of the logarithms of all the other ratios, namely, of the log. tangents, cotangents, secants, and cosecants are increased by lo. In trigonoraeti'ical operations this is convenient, but principally because the extraction of roots very seldom occurs. It may be observed here that the uniform addition of lo to the charac- teristic gives the logarithm of loooo million times the natural number. Thus, 9"J99327 is the log. of 3979486000, and this latter number is the natural sine cor- responding to a radius of loooo millions, instead of a radius of unity. 118. TJsual arrangement of Tables of Logarithmic Sines, Cosines, &c. — The table of logarithmic sines, cosines, tangents, cotangents, secants, and cosecants, contain all arcs from zero (0°) through all magnitudes up to 90°, the log. of radius, as just stated, being 10. At the top of the page is placed the number of degrees, and in the left-hand column each minute of the degree, opposite to which are arranged the numerical values of the log. sine, cosine, &c., of the corresponding angle in those columns, at the top of which those terms are placed. The headings of the columns run along the top, thus, as far as 44°. The degrees from 45° to 90° are placed at the bottom of the page, and the minutes of the degree arranged in a right-hand column, so that the angles read off on the right-hand side are complemental to those read off at the points exactly opposite on the left-hand side, the values of the sines, cosines, tangents, &c., being found in the columns at the bottom of which those terms are found. This arrangement is rendered practicable by the circumstance of every angle between 45° and 90° being the complement of another between 45° and o'^, every sine of an angle less than 45° is the cosine of another greater than 45°, every tangent is a cotangent, &c. ; the sines, tangents, &c., of angles being respectively equal to the cosines, cotangents, &c., of the complements of the same angle. The following shows the usual arrangement of such tables :— - M Sin. D. Oosec. ■ Tan. D. Cot. Sec. D. Cos. M M M Cos. D. Sec. Got. D. Tan. Cosec. D. Sin. Besides the columns headed " sine, tangent," &c., are three smaller columns headed ** Diff." They contain, in most tables, the differences between the values of the consecutive logarithms in the contiguous columns on either side, but corresponding to a change of 100" in the arc (not the difference corres- ponding to 60" of arc or angle) ; and it must be kept in mind that the same difference is common to the sine and cosecant, to the tangent and cotangent, and to the secant and cosine. They are inserted for the convenience of finding the values of the sines and cosines, &c., of angles which are expressed in degrees, minutes, and seconds. 1 19. The above, as just stated, is the usual arrangement of most tables, but in the earlier editions of Norie and some other works the arrangement is somewhat different. 86 Tahlea of Logarithms of Trigonometrical Ratios. The columns are arranged thus: — M Sine. Diff. Cosine. Diff. Tangent. Diff. Cotangent Secant. Cosecant. M M Cosine. Diff. Sine. Diff. Cotangent Diff. Tangent. Cosecant. Secant. M Since the same difference is common to the sine and cosecant, to the tangent and cotangent, in this arrangement, then, it must be particularly borne in mind, that the first "Diff." column (from the left) belongs to the first column of logarithms on the left hand of the page, and is also the "Diff." for the first column on the right of the page; that the second column of "Diff." (from the left) belongs to the second column of logarithms from either the right or left of the page; and that the third column of "Diff." belongs to the third column from either the right or the left, which may be otherwise expressed, thus: — A cosecant takes a sine "diff." ; a secant takes a cosine " diff." ; and cotangent takes a tangent ** diff." 1 20. In the use of these Tables, as in that of the natural sines, two questions present themselves : — First, having given the angle in degrees, minutes, and seconds, required the log. sine, log. cosine, &c. Second, having given the log. sine, log. cosine, &c., required the value of the angle in degrees, minutes, and seconds. 121. "When an angle is presented in degrees and minutes only, the tabular logarithm of its sine, tangent, &c., will be found (Table XXV, Noeie, or Table 68, Eaper) simply by inspection, according to the following : — EULE XXXVI. 1°. If the angle or arc is less than 45°. Find the page having the given degrees at the top, and in the left-hand marginal column find the minutes, then opposite the minutes, and in the column which is marked at the top with the name of the ratio, will be found the logarithm sought. 2°. If the angle be greater than 45°. Look for the page having the given degrees at the bottom, and find the minutes in the right-hand column ; the logarithm of the proposed function of the angle will be found opposite the minutes in the column marked at the foot with the name of the ratio whose logarithm is sought. Examples. Ex. I. Find the log. sine of 37° 47'. As the arc is less than 45°, by looking at the top of the table for the degrees (37°), and in \hQ first column on the lelt for the minutes (47), we find in the column having at its top the •word sine, the fiirures 9-787232, which is the log. sine of the arc required. Ex. 2. Find the log. tang, of 75° 34'. Here, as the arc is greater than 45°, looking at the bottom of the table for the degrees (75°), and in the last or right-hand column for the minutes (34'), we find in the column having tang, at the bottom io"58943i, which is the log. tangent of 75° 34'. Log. sine of 40° 4' is 9-808669 Log. sine of 57° 5 IS 9-924001 Log. cosine of 21 38 „ 9-968278 Log. cosine of 79 51 .. 9-246069 Log. tangent of 84 13 „ 10-994466 Log. tangent of 21 50 » 9-602761 Log. cotangent of 55 58 „ 9'829532 Log. cotangent of 27 45 .. 10-278911 Log. secant of 70 20 „ 10-472954 Log. secant of 44 59 » 10-150389 Log. cosecant of 8 35 „ 10-826092 Log. cosecant of 69 54 » 10-027291 13 Log. sine 11° 20' 14 Log. cosec. 35 41 15 Log. cosine 23 14 i6 Lo^'. sec. 47 54 17 Log. cotang. 70 39 18. Log. sme 57 12 Tables of Logarithms of Trigonometrical Ratios. 87 Examples for Praotioe. Take out the logarithms of the following trigonometrical ratios. 1. Log. sine 9° 10' 7. Log. cos. 53" 28' 2. Log. cosec. 40 40 8. Log. sine 51 49 3. Log. cosine 12 48 9. Log. sec. 60 34 4. Log. tang. 37 26 10. Log. cotang. 79 19 5. Log. cotang. 8 25 11. Log. cosec. 45 45 6. Log. sec. 43 I 12. Log. sine 53 56 122. If the value of the angle be given in degrees, miniites, and seconds, we proceed by EULE XXXVII. 1°. Find from the table the sine, tangent, secant, cosine, Sfc, which corresponds to the degrees and minutes; also take out the number in the contiguous column headed "Diff." on the same line (See Nos. 1 18 and 1 19, page 85.) 2°. Multiply the tabular difference (" Difif.") by the seconds, reject the last two figures (always two) of the product for the division by 100, and the remaining figures will furnish the proper correction for seconds. Note i. — If the value of the two figures cut off is not less than fifty, one must be added to the first right-hand figure left. 3°. If the required quantity be a sine, tangent, or secant, add the result to the last figures obtained in 1° ; if it be a cosine, cotangent, or cosecant, subtract.* The result will be the required sine, tangent, secant, cosine, &c. Note 2. — The process above is sufficiently accurate, unless for the sines and tangents of very small angles, and for the tangents and secants of angles very near 90". When an angle of degrees, minutes, and seconds, and of less magnitude than 3°, occurs in calculation, neither the logarithmic sine nor the logarithmic tangent will be found very accurately from the ordinary Tables. In some books, as Hutton's ''Mathematical Tables," a special Table is given, containing the logarithmic sines and tangents to every second in the first two degrees of the quadrant. By that Table we should find the correct log. tang, of 1° 25' 45" to be 2-3970503, whereas, by using the tab. diff. for r^ 25' and 1° 26' in the ordinary Table, we should get the less accurate result, 2-3970448, because for such small auL^lcs, the succes- sive tabular diS'erences for one minute shows too rapidly a wide departure from equality. When an angle of degrees, minutes, and seconds, and within less than 3" of 90° occurs in calculation, we cannot, for the reason just stated, obtain very accurately from the ordinary Tables either the logarithmic or the natural tangent. Thus, the true log. tang, of 88° 4' 15* is 1-6029497 ; but by the ordinary Tables we would get for the last three figures 552. Norib gives the log. sin. and log. tang, to every ten seconds of the first two degrees of the quadrant, and Raper gives the log. sines to every second up to 1° 30', and to every ten seconds up to 4° 30'- * In some Tables these differences are those due to r minute, or 60 seconds, and are got by simply subtracting the greater of the logarithms from the less. The difference d, due to any smaller number («) of seconds is found from such Tables by the projiortion 60 : « : : D : d, so that d=: — But as before observed the differences usually given in the Tables are those due not to 60 seconds but to 100 seconds, so that in these Tables, d=:^~ : and thug d is found somewhat more readily. 88 Tables of Loga/rithms of Trigonometrical Ratios. Examples. Ex. I. Find the log. sine of 6° 36' 27*. Here the given number of degrees (6") being less than 45°, look for them in the head line at the top of the page, turning over the leaves till the proper page is found, then in that page look in the second line for the name of the column wanted, viz., the sine ; and in the left-hand vertical column marked M at the top, find the number of minutes (36') ; having found the minutes, then in the same line and under sine is found 9-060460, which is the log. sine corresponding to 6° 36'. Now this log. being found in the first column on the left, the tabular difi'erence must be taken out of the first "diff." column from the left. It will be noticed that there is no diff. exactly opposite to 36', but between 36' and 37' will be found the diff. 1817, which multiplied by the seconds (27") gives 49059, and rejecting the two last figures from this product (for the division by 100) gives quotient 490, which being increased by I, since the figures cut off exceed 50 (see Note i, page 87) gives 491 as the correction of the logarithm for the seconds. The work will stand thus : — Log. sine 6° 36' ^ 9'o6o46o Tab. diff. 1817 27" gives +491 X 27 9"o6o95r 12719 3634 490,59 = 491 Ans. : Log. sine 6° 36' 27' =z 9-06095 1. ^ Ex.2. Find the log. cosine of 13" 5' 32". ^ The log. cosine of 13° 5' is 9-988578, and the tabular difference corresponding to the log. cosine of the given degrees and minutes is 50; this being multiplied by 32 (the given number of seconds), and pointing off two figures to the right, is 16 to be subtracted, because the cosine is a decreasing log. ; therefore — Log. cosine 13° 5' = 9-988578 Tab. diff. 50 32" gives — 16 X 32 9-988562 100 150 Ans. : Log. cosine 13" 5' 32" = 9' 16,00 or 16 The parts for the seconds are subtracted in this instance, being a colog. (See Rule XXXVII, 3°. Ex. 3. Find the log. tangent of 72° 59' 8". The log. tangent of 72° 59' is 10-514209, and the tab. diff. corresponding to the given degrees and minutes is 753 ; this being multiplied by 8 (the number of seconds), and point- ing off two figures to the right is 60, which is additive ; thus : — Log. tang. 72° 59' o" =: 10-514209 Tab. diff. 753 Parts for 8" = + 60 8 Log. tang. 72° 59' 8" = 10-514269 60,34 Ex. 4. Find the log. cotangent of 73° 21' 7". The log. cotangent of 73° 21' is 9-475763, and the tab. diff. corresponding to the cotangent of the given degrees and minutes is 767 : this being multiplied by 7 (the given number of seconds), and pointing off two figures to the right is 54, which is to be subtracted in this instance, being a colog. Log. cotang. 73° 21' o" =: 9-475763 (Tab, diff. 767) X 7 g^ Parts for l" — — 54 100 ^|^' '"^ Log. cotang. 73=' 21' 7" = 9-475709 1 The parts for the seconds are subtracted in this instance, being a colog. (See Rule XXXVII, f. Tables of Logm-ithms of Trigonometrical Ratios. 89 Ex. 5. Take out log. sine 1° 5' 34". Here tte angle whose log. sine is sought bting lei?fl than 2°, it must, therefore, he taken out of the special part of the Table (see Table XXV, page 107, Norie). The next less angle to be found in the Table is 1° 5' 30", the log. sine of which 8-279941, and the corres- ponding tabular "DiflP." (for 10" in this part of the Table) is 1104, which multiplied by 4, the seconds over 30, gives 4416, and cutting off one figure from the right, for tha division by 10, gives the correction 442, to be added'to the logarithm taken out of the Table ; thus the work stands as follows : — Ex.6. Log. sine 1° 5' 30" ■=. 8'a7994i Parts for 4=4" 44^ Tab. diff. 1 1 04 4 Log. sine i 5 34 := 8-280383 441,6 or 442 nearly. Required the cosecant of 3" 7' 21 '. Log. cosecant 3° 7' 0" = 1 1 -264646 Parts for 21 = — 810 Tab. diff. 3857 21 Log. cosecant 3 7 21 = ii'263836 3857 7714 809,97 Ex. 7. Take out the cosine of 88° 20' 46". (See Table XXV, Norie, page 108). In the special part of the Table, at the bottom part, we get 88° 20' 40" (the next less angle), and the cosine opposite is 8-460761, and the corresponding tabular "Diff." (for 10" in this part of the Table) is 729, which multiplied by 6, the seconds over 40', gives 4374, and cutting off one figure from the right, for the division bj' 10, gives the correction 437 to be subtracted rom the logarithm taken out of the Table ; thus the work stands as follows : — Log. cosine 88° 20' 40" = 8-460761 Tab. diff. 729 Parts for 6 = — 437 6 [] / ? C Log. cosine 88 20 46 = 8-460324 437>4 or 437 123. For the functions of an angle between 90° and 180° we may take the same functions of its supplement ; hence, To find the logarithm of a trigonometrical ratio of an angle greater than 90°, we have the following EULE xxxvin. Subtract the angle from 1 80° and look for the remainder, which is called its supplement in the Tables. Examples. Ex. r. Find the log. sine of 110° 24'. Subtract it from 180°. From 1 80° 00' Subtract 1 10 24 Remainder 69 36 (Supplement). Look for the log. sine of remainder (namely 69° 36'), which is 9-971870; or log. eine HQ° 24' := 9-971870. go Tables of Logo/rithms of Trigonometrical Ratios. Ex. 2. Find the log. secant of 95" 43' ; also the log. cosecant of the same. Subtracting 95° 43' from 180° o' gives remainder 84° 17", and look for the log. secant of 84° 17', which is 1 1 001701 ; . ' . log. secant of 95" 43' is 11-001701. Again, look for the cosecant of 84° 17', which is 1 0*002 165 ; . ' . log. cosecant of 95° 43' is io-ooai65. Ex. 3. Find the log. tangent of 128° 55' 47". From 180° o' o" Subtract 128° 55' 47" (Eemainder) = 51° 4' 15" (Supplement). . ' . Supplement of the given angle = 51° 4' 13", Log. tangent 51° 4' o" = io"092664 Tab. di£F. 431 Parts for 13 = + 56 X 13 Log. tangent 5 1 4 13 = io'09272o 1293 431 56,03 or 56 . Log. tangent 128° 55' 47" = 1 0*09 27 20. Am. 124. But a readier way, and the better practical method, is to proceed as follows: — EULE XXXIX. Diminish the given angle hy go°, and look out the remainder in the tables, observing that if the trigonometrical ratio have " co " prefixed to it drop the " co," but if it have not " co," prefix it, then find the logarithm corresponding to the new ratio and angle. Or, If A denote any angle less than 90°, then For sine ........ (90 -\- A) take out cosine A „ tangent (90 -\- A) cotangent A „ secant ...... (90 -\- A) cosecant A „ cosine (90 + A) , sine A „ cosecant .... (90 + A) secant A ,, cotangent .... (90 + A) tangent A Obs. — This rule may easily be remembered by observing that to the sine, tangent, and secant, co is prefixed, while from the cosine, cosecant, and cotangent, the co is dropped, and in each case the excess of 90° of the angle is used. Examples. Ex. I. Find the log. cosine of 1 10° To find the log. cosine of 110°, or log. cosine (90 -\- 20), take out the log. sine 20°, which is 9-534052- Ex. 2. To find the log. secant of 160° 12', take out the cosecant 70° 12', which is 10-026465. Log. cosine of 143° 24' := Log. sine 53° 24' is 9*904617 Log. cosecant of 99 37 = Log. secant 9 37 „ 10-006146 Log. sine of io§ 2 = Log. cosine 19 2 „ 9*975583 Tables of Logarithms of Trigonormtrical Ratios. 9 1 Ex. 3. Find the log. cosecant of 131° 45' 19". Subtracting 90° from 131° 45' 19" = 41" 45' 19". Log. cosecant 131° 45' 19" = log. secant 41° 45' 19". Log. secant 41° 45' o" =: io-i27228 Tab. diflf. 188 Parts for 4" '9 =^ + 3^ X 19 Log. secant 41 45 19 =: io'i27264 1692 188 35,72 In tbis instance "co" is prefixed to the given trigonometrical ratio, then, according to rule, "co" is dropped, and the log. corresponding to the new ratio is taken out for the remainder resulting from the given angle when diminished by 90°. Ex. 4, Required the log. tangent 99° 32' 58". Log. tangent 99° 32' 58" = cotangent 9° 32' 58". Log. cotangent 9° 32' o" = 10-774844 Tab. diff. 1288 Parts for + 5^ = — 747 58 Log. cotangent 9 32 58 =: 10-774097 10304 6440 747,04 J 25. In Eaper the required logs, are given to every half minute, and, therefore, the required log. is to be taken out to the nearest less arc to that given; and adjoining it will be seen a column of "Parts," from which the correction for the remaining seconds is to be taken, and this correction is to be added if the log. taken out be a sine, tangent, or secant, but subtracted if it be a cosine, cosecant, or cotangent (that is, if the log. have co prefixed). See Note 2, page 82. In most cases the parts are given in the adjoining column for every second up to 30", but when the angle is small, or large, some are given for each second up to lo". In this case, if the parts for a larger number of odd seconds than 10 are required, take them out in instalments. When the angle i.s very small, or very large, the parts are not calculated at all, but the difference for a half minute is given opposite to each logarithm. In this case multiply the given dififerenco by the number of the odd seconds and divide by 30. The result will be the parts required. Examples. Ex. I. Find log. sine 37° 19' 51". Log. sine 37° 19' 30" = 9-782713 Diff. for 21 = + 58 Log. sine 37 19 51 = 978277 1 Ex. 2. Find log. cotang. 64° 53' 39". Log. cotang. 64° 53' 30" = 9-67o8i3 Diff. for 9 = — 49 Log. cotang. 64 53 39 = 9-670764 Ex. 3. Find log. tang. 8° 32' 18'. Log;, tang. 8° 32' o' = 9-176224 Diff. for 10 =; 143 8 = 115 Log. tang, 8 32 18 := 9-176482 9^ Tahles of Loga/rithms of Trigonometrical Ratios. Ex. 4. Find log. cosine 83° 41' 57". Log. cosine 83° 41' 30" = 9'0409i5 Diff. for 10" = 191 Diff. for 27 = — 515 DiflF. for 10 = 191 DiflF. for 7 = 133 Log. cosine 83 41 57 = 9*040399 Ex. 5. Find log. cosec. 3° 7' 21' Diff. for 27 = 515 Log. cosec. 3° 7' o" ■=. 11*264646 Diff. for 30" = 1160 Diff. for 21 = 31 Log. cosec. 3 7 21 = 11*263834 1160 3320 3,0)24360 Diff. for 21" = 813 1 26. But for the purpose of lessening the labour of finding the log. sines and cosines in the case where the parts are not given, two other tables have been constructed (Tables 66 and 67). T«?ble 66 gives the logarithms of sines of small angles from 0° to 1° 30', and the logarithms of cosines of large angles from 88° 30' to 90° to each second. Table 67 gives the logarithms of sines of angles from 1° 30' to 4° 31' and the log. cosines of angles from 85° 29' to 88° 30' to every ten seconds. In Table 67 there are columns of "parts" by which the log. of the sine or cosine of an angle containing odd seconds may be found, and conversely. Each page of this Table is divided into six spaces by horizontal lines, and there are two columns of parts in each space. The left-hand column of parts is to be used when the angle is in the upper half of the space, and the right-hand column when it is in the lower. Examples. Ex. I. Take out log. sine 1° 5' 34". Log. sine 1° 5' 34" = 8 '280383. Ans. Ex. 2. Take out log. ooeine 88° 43' 57". Log. cosine 88° 43' 57" = 8*344790. Ans. Ex. 3. Find log. sine 4° 26' 18". Log. sine 4° 26' 10" =: 8*888446 Diff. for 8 = 216 Log. sine 4 26 18 =8-888662 lu this instance the parts are taken from the right-hand column, because the angle, 4° 26' 10", ia situated in the lower half of the space. Examples for Praotioe. Eequired the log. sine, tangent, secant, cosine, cotangent, and cosecant corresponding to the following arcs : — I- 6°53'56" 4- 56°54'i7" 7- i''49'47" 10. 115° 34' 41' 2. 29 9 30 5. 10 10 6 8. 87 28 45 II. 119 40 48 3. 37 49 14 6. 70 47 40 9. I o 40 12. loi 40 19 Tables of Logarithms of Trigonometrical Ratios. 93 Take out of the Table the following : — 13- Log. sine 2° 40' 10" 22. Log. cosec. 1 27° 30' 40" 14. Log. sine 170 30 39 23- Log. cosec. 141 16 51 ij- Log. sine i 49 47 24. Log. sine 3 53* 1 6. Log. eosine 89 59 19 2J- Log. cotang. 89 23 37 17. Log. cosine 88 40 56 26. Log. cosine 87 33 27 18. Log. cosine 108 40 60 27. Log. cosine 88 50 29 19. Log. tang. I 8 7 28. Log. tang. I 2 18 30. Log. cotang. 378 29. Log. 860. loi 8 7 ar. Log. tang. 1 14 9 30 3°- Log. sine no II 18 127. If the value of the log. sine, log. cosine, &c., i.e., the logarithm of a trigonometrical ratio, be given, and it is required to find the corresponding angle in degrees and minutes, we use EULE XL. Looh for the logarithm in the several columns of the table marked at the top or bottom with the name of the given trigonometrical ratio, which being found exactly or the nearest, whether larger or smaller, to it, will give the degrees and minuted answering to the given logarithm, being careful to observe that when the name of the given ratio is found at the top of the table, then the degrees are to be taken from the top and the minutes from the left-haQd marginal column ; but if the name of the ratio is found at the bottom of the table, take the degrees /ro»i the bottom and the minutes /rom the right-hand side of the page. Note. — In using the Table inversely, for example, in searching for the angle which has 9'6ii294 for the logarithm of its sine, the student must not distinguish sine from cosine, nor tangent from cotaagent, but must consider sines and cosines as one table, tangents and cotangents as one table, and must cast an eye on both, and get to 9'6ii294 as fast as he can. For want of this caution some beginners will turn over page after page until they come to 4j°, and then back again to the very page that was first opened. ExAJiPLE. Ex. I. Required the angle corresponding to the log. sine 9'729223. In page 142, Table XXV, Nouib, under the word "Sine," and opposite 25' in left-hand marginal column, are the exact figures, the degree (being sought at the head of the page, because the column in which the figures are found is named at the head) is 32° ; therefore, the angle is 33° 25'. If the angles for the cosine of the same logarithm be required, the degrees are found at the bottom, and the minutes in the right-hand column, and is 57° 35' accordingly. Log. sine 9-731009 = 32=34' Log. cosine 9-995555= 8»ii' Log. sine 9-871073 = 48 o Log. rcsc. 10-030580 = 68 45 Log. tang. 9-787036 = 31 29 Log. cotang. 10-508820 = 17 13 Log. tang. 10-047850 = 48 9 Log. cosine 9-718497 = 58 28 Log. sec. 10-043673 =: 25 16 Log. cosec. 10-307885 = 29 29 Log. sec. 10-566325 = 74 15 Log. cotang. 11-197235=: 3 38 * When the tabular difference is considerable, as in this instance, the log. is easier reduced from the log. of the nearest minute. 94 Tables of Logarithms of Trigonometrical Ratios. 128. If the value of the log. sine, log. cosine, &c., be given, and it is required to find the corresponding angle, in degrees, minutes, and seconds, we use RULE XLI. 1°. Find in the Tables (XXV, Norie) the next lower log. sine, log. cosine, Sfc, and note the corresponding degrees and minutes ; also, take the number from the corresponding part of the adjoining column of '■'■ Biff.^'' 2°, Subtract this from the given log. sine, log. cosine, 8fc., multiply the difference by 100, i.e., annex two cyphers, divide by the tabular difference, and consider the result as seconds. 3°. If the given value be that of a log. sine, log. tangent, or log. secant, add these seconds to the degrees and minutes found in 1° ; if it be that of a log. cosine, log. tangent, or log. cosecant, i.e., if the log. have CO prefixed, subtract. The result will be the required angle. Note. — If the given log. be a cosine, cosecant, or cotangent, we may seek out the next greater to the given lo^. : then proceed by 2° to find the seconds, which add to the degrees and minutes as found by 1°.* Examples. Ex. I. Given log. sine = 9*422195 (or 7-422195) : find the angle. We take out 9-421 857, the log. sine of ij"^ 19', as it is the logarithm next less than the given one, which we take, as the logarithms in the columns increase with the angle. The difference of these logarithms is 338, and if two cyphers be aiBxed to the difference, and the number then divided by 768, taken from the column of Diff. in the Table, we have 44 for the number of seconds to be added to the degrees and minutes before taken out. The work will stand thus : — Given log. sine 9-422195 Tab. log. sine next less 9-421857 = log. sine 15° 19' Tab. diff. for 100" = 768)-338oo(44" additional seconds. 3072 3080 3072 Therefore 9-422195 = log. sine of 15° 19' 44''' Ex. 2. Given log. cosine = 9-873242 (or 1-873242) : find the angle. Here we take out 9-873223, the log. cosine of 41° 41', as it is the log. cosine in the Table next less than 9-873242. The difference between these two logarithms is 19 ; and if two cyphers be affixed to the difference we get 1900 ; whence 1900 divided by 187, the number from the column of " Diff." gives lo for the number of seconds to be subtracted. Hence the required angle is 41° 40' 50". The work will stand thus : — Given log. cosine 9-873242 Tab. log. cosine next less 9-873223 = log. cosine 41° 40' 19 X 100 ^ 10" subtractive. 19 187 Tab. diff. for 100" = 187. Therefore, 9-873242 = log. cosine of 41° 40' 50" * By writing down the log. corresponding to the next less angle in every ease the seconds are always additive, thus avoiding confusion as to their mode of application. (See Note, Eule XXXV, page 83.) Tables of Logarithms of Trigonometrical Ratio$. 95 Proceeding according to Note, Rule XXXV, page 83. Given log. cosine 9'873242 Tab. log. cosine next greater 9"873335 = 41° 40' (next less angle). Tab. diff. for 100" =r 187)9300(50", nearly, additive. . ' . angle required = 41° 40' 50". Examples fob Praotiob. Eequired the Angles (to the nearest second), the 1 Log. Sine of which is : — I. 9"74i279 4. 8-600700 7. 9*500000 10. 7-456430 13* 9*900000 2. 9*926ioo 5. 9*518317 8. 9-800000 II. 9-999631 14. 8-846217 3- 8*707654 6. 9-929638 9. 9*909176 Find the Arc to the Log. Cosine of 12. 9'974538 15- 8*462167 I. 9*787140 4. 9-995637 7. 9-517232 10. 9932338 13- 7*799520 2. 9*750333 5- 9'i79726 8. 9-212036 II. 9-998970 14. 9-000000 3- 8*134758 6. 9*273216 9. 8-361861 Find the Arc to the Log. Secant of 12. 8-281485 15- 9*013628 I. 10*013839 3. 10*000765 5. 10*746129 7- 10-022719 9* 10*315400 2. 10*205665 4. 10-048398 6. 11-005231 Find the Arc to the Log. Cosecant of 8. 11*642535 10. 1 1*200000 I. 10*347194 4. 10-974476 7. 11-000873 10. 10-070362 13- 10009000 2. 10*252208 5. 10-121000 8. 11-467931 II. 10-900000 14. 10*061462 3- 11-005231 6. 11-442539 9. 11*166007 Find the Arc to the Log. Tangent of 12. 11*079003 15- 11*290123 I. 10*636863 4. 10-827204 7. 11-276400 10. 9-642876 13* 10-060431 2. 10-000100 5. 10-150328 8. 8-297036 II. 9-846175 14. 8-668612 3- 10-287342 6. 8-961007 9. 9-716135 Find the Arc to the Log. Cotangent of 12. 11*282456 15- 8-258262 I. 9-742961 4. 10-060431 7. 8-327691 10. 9*100100 13* 8-460000 2. 10-876432 5. 10-710880 8. lo-oroioi II. 10*825001 14. 9*374611 3- 10-287632 6. 1 1*197568 9. 8-781464 12. 8*272775 '5- 12*069844 MISCELLANEOUS. 1. If Bine A = '432651, find log. sine A. 2. If tang. A = 3, find log. tang. A. 3. Given log. cos. A = 9-236713, find uat. cos. A. 4. Given log. tang. 35° 20' = 9*850593, find log. cotang. 35° 20' without using any tables at all. 5. Find the log. cosec. 68^^ 45' 24" from the table of natural sines only. 6. Given log. sec. A := 11*024680, find.nnt. 00s. A. 7. Given log. cosine A = 9*450981, find A (i) from a table of log. cosines, and (2) from a table of nat. cosines. 8. Given nat. sec. A = 2*005263, find A (i) from a table of nat. sines and cosines, and (2) secant from a table of log. secants. 9. Sine 36° X tang. 54° =: "654. 10. Find by the tables the angle whose sine is v^tV- 11. Given log. cot. A =: 11-015627, find nat. cot. A. 12. Find nat. cot. 45° 18' 17" from the table of cotangents. 13. Find to the nearest second the angle whose sine is 5^, \, i, and \^. H* „ „ tang, is fj, ^, and ifj. 96 Tables of Loga/rithms of Trigonometrical Ratios. 129. It is also necessary to have a distinct conception of the limits to which the Trigonometrical Ratios tend when the angles become right-angles. The following are the Trigonometrical Eatios for the angles 0° and 90° : — Sin. 0° = o Cos. 0° =: I Tang. 0° = o Got. o» = 00 * Sec. o» = I Gosec. 0° = oe And the following, therefore, are the Logarithms of their Trigonometrical Eatios : — Sin. 90° =: I Cos. 90"* = Tang. 90" := 00 Cot. 90° =: Sec. 90° := oc C086C. 90'' = I Log. sin. Log. COS. Log. tang. Log. cot. Log. sec. Log. cosec. Log. sin. Log. cos. Log. tang. Log. cot. Log. see. Log. cosec. 90" 90° 90° 90' 90° 90° 130. When these values occur amongst others requiring to be added to or subtracted from them, the learner must be careful to remember that the addition to or subtraction from them of finite numbers cannot alter them. Hence the explanation of the results in the following : Examples. Ex. I. Add together log. cot. 0° and log. sine 20°. Log. cot. 0° = 00 Log. sine 20° = 9'5 34032 Am. 00 Ex. 1. Add together log. cos. 90° and log. tang. 43°. Log. cos. 90° = — 00 Log. tang. 45° = lo'oooooo An*. 00 Ex. 3. From log. cos. 0° take log. sine 62° 48'. Log. cos. 0° = — 00 Log. sine 62° 48' = 9-949105 Ans. — 00 Ex. 4. From log. tang. 21° 48' 30* take log. cot. 90°. Log. tang. 21° 48' 30' = 9*602212 Log. cot. 90° = — 00 Ana. 131. In the event of a bad or obliterated figure in the table, it may be convenient to know that the tangents are found by subtracting the cosines from the sines, adding always 10, or the radius ; the cotangents are found by subtracting the tangents from 20, or the double radius, and the secants are found by subtracting the cosines from 20, the diameter of a circle whose radius is 10. * This mathematical symbol is called infinity. 97 NAVIGATION. DEFINITIONS. 132. Navigation is a general term denoting that science which treats of the determination of the place of a ship on the sea, and which furnishes the knowledge requisite for taking a vessel from one place to another. The two fundamental problems of navigation are, therefore, the finding at sea the present position of the ship, and the determining the future course. 133. The place of a ship is determined by either of two methods, which are independent of each other: — ist. By referring it to some other place, as a fixed point of land, or a previous defined place of the ship herself. 2nd. By astronomical observations. 1 34. It has been customary to employ the term Navigation in a restricted sense to the first of these methods ; the second is usually treated of under the head of Nautical Astkonomt. Navigation and Nautical Astronomy are the two great co-ordinate divisions of the " Art of Sailinff on the Sea," as the old writers quaintly worded it. The first branch of the art is accomplished by means of the Mariner's Compass, which shows the direction of the ship's track ; the Log, which, with the help of sand-glasses for measuring small intervals of time, gives the velocity or the rate of sailing, and thence the distance run in any interval ; and also a Chart of appropriate construction ; in short, this branch of the art relates to the directing the ship's course under the varying forces of widds and currents, and the estima- tion of her change of place. The second division is that branch of practical astronomy by which the situation of the observer on the globe is ascertained hy a comparison of the position of his Zenith with relation to the heavens with the known position of the Zenith of a known place at the same moment. The principal instruments are the sextant for measuring the altitudes and taking the distances of heavenly bodies ; and a chronometer to tell us the diff'-rence in time between the meridian of the ship and the first meridian ; also a pre-calculated astro- nomical register, such as the Nautical Almanac, the Connaissance de Temps of France, &c. The solution of problems in nautical astronomy requires the use of spherical trigonometry, which is therefore characteristic of this method of navigation. 135. A Sphere is a solid body bounded by a surface, every point of which is equally distant from a fixed point within it ; this fi^ed point is called the centre ; the constant distance is called the radius. Every section of a sphere by a plane is a circle. 136. A Great Circle of a sphere is a section of the surface by a plane which passes through its centre. A Small Circle of a sphere is a section of the surface by a plane which does not pass through its centre. Or, a great circle is the circle of a sphere having for its centre the centre of a sphere, thus dividing the sphere into two equal parts; no greater circle can be traced upon its surface. All other circles are called small circles. All great circles of a sphere have the same radius. All great circles bisect each other. 137. The Axis of any circle of a sphere is that diameter of the sphere which is perpendicular to the plane of the circle. The extremities of the axis axe called the poles of the circle. 98 Navigation — Definitiom. 138. The extremities of that diameter of a sphere which is perpendicular to the plane of a circle are called the poles of that circle. In the case of a small circle, the poles are distinguished as the adjacent and remote pole. All parallel circles have the same poles. The distance of every point in the circumference of a circle from either of its poles is the same. The poles of a great circle are 90"^ distant from every point of the circle. 139. Regarding any great circle as a primary circle, all great circles which pass through its poles are called its secondaries. All secondaries cut their primary at right-angles. The arc of a great circle is measured by the angle subtended by it at the centre of the sphere, which is also the same as the angle of inclination, at its pole, of two secondaries drawn through its extremities. 140. The earth is nearly a globe or sphere. The ordinary proofs of this are of the following nature : — ist. When a vessel is seen at a considerable distance on the sea, in any part of the world, the hull is entirely or partly concealed by the water, though the masts are visible. 2nd. Ships have actually and repeatedly made the circuit of the globe ; that is, by sailing from a port in a westerly direction they have returned to it in an easterly direction. 3rd. When we travel a con- siderable distance from north to south, a number of new stars appear, successively, in the heavens, in the quarter to which we are adv^ancing, and many of those in the opposite quarter gradually disappear, which would not happen if the earth were a plane in that direction. 4th. In an eclipse of the moon, which is caused by the intervention of the body of the earth between the sun and moon, the shadow of the earth thrown on the moon ia found in all cases, and in every position of the earth, to be a circular figure ; the earth therefore, which casts that shadow, must be a round body. 141. The earth, however, is not a perfect sphere, but of the figure of an oblate spheroid very nearly, that is, a figure traced out by an ellipse revolving round its shortest axis, being flattened in at the poles, and bulging out in a corresponding degree at the equatorial regions — the curvature being less as we recede from the equator to the poles ; such a figure, in fact, as would be produced if a hoop were slightly flattened by pressure, and then made to revolve about the shortest diameter thus produced. The shortest diameter (that which joins the poles) being 7899 statute miles, and that of the fullest parts (about the equator) being nearly 26^ more. We can, of course, in a work like this, give no intelligible account of the refined mathe- matical processes by which the most probable values of the flattening in, and of the absolute dimensions have been obtained. It is sufficient to say that from a combination of the measurements of ten arcs of the meridian, Bessel has deduced the following results : — * Greater, or equatorial diameter 41,847,192 feet = 7925-604 miles. Lesser, or polar diameter 41,707,324 „ = 7899-114 „ Difi'erence of diameter, or polar compression . 139,768 „ = 26-471 „ Proportion of diameters, as 299-15 to 298-15. And from the result it follows that the polar diameter is shorter than the equatorial by about ^^ (one three hundredth) part. This quantity is technically called the compression.^ * Astrononische Nachrichten, No. 438. j- The best values for its dimensions, however, appear to be those given by Capt. Clakkb. Equatorial diameter 41847662 feet = 12754937 metres. Polar axis ......... 41707536 „ =12712227 ,, Nmigation — Definitions. 99 142. The Axis of the Earth is that diameter about which it is supposed to tvirn round once in twenty-four hours. The direction of this rotation is from west to east, thus causing all the heavenly bodies to have an apparent motion from east to west. 1 4-3. Poles. — The two extremities of the axis of the earth are^called the poles of the earth, distinguished respectively as the North Pole and South Pole — N S (see Fig.) The former being that to which we in Europe are nearest. As they are the extremities of a diameter they are 1 80° apart. 144. Equator (from Latin cequare, to divide into equal parts), called also by seamen the Line, is a great circle civcumscribing the earth, every point of which is equally distant from the poles, being 90° from each, as WM'E ; and dividing the globe into two equal parts called hemispheres ; that towards the north pole is called the northern hemisphere, as N "VV E, and the other the southern hemisphere, as S WE. (See Figure above). If a plane be supposed to pass through the centre of the earth at right- angles to its axis, it will intersect its surface in a great circle called the Equator. The equator is chosen as the primary circle for co-ordinates. At all places on this circle the sun rises at 6'' a.m., and sets at 6'' p.m., all the year round ; the days and nights are therefore equal, being 12'' each. 1 45. The Meridian of any place is a semi-circle passing through that place and the poles, and therefore cutting the equator at right-angles, as NM'S, NWS, NZS. (See Figure.) The other half of the circle is called the opposite meridian. Every point on the surface of the earth may be conceived lOO Navigation — Definitiom. to have a meridian passing through it ; hence there may be as many meri- dians as there are points in the equator. Of all these innumerable meridians one is always selected as the Initial Circle of Longitude, or, as it is commonly called, the First Meridian. It is a matter of arbitrary choice amongst different nations ; thus the first meridian with us is that of Greenwich, whilst the French refer to Paris, &c. Meridians (i. Meridies, from medius dies, mid-day) are so called because they mark all places which mark noon at the same instant, for when any one of the meridians is exactly opposite the sun it is mid-day with all places situated on that meridian ; and with the places situated on the opposite meridian it is consequently midnight. They are secondaries to the Equator, and on them Latitudes are reckoned North and South from their primitive. They also mark out all places which have the same longitude, and are hence called " Circles of Longitude." Every portion of the meridian lies north and south ; and places lying north and south of each other are said to be on the same meridian. The direction of the meridian towards the north pole is called north, and marked N. ; the opposite direction is called south, marked S. Directions at right-angles to the meridians are called east and west ; the right hand looking to the north east, the left hand west : they are marked E. and W. 146. Latitude is the distance from the equator, measured in degrees (°), minutes ('), and seconds ("),* on the meridian of the place, or its angular distance from the equator measured by the arc of the meridian intercepted (cut off), between the place and'the equator, or by the corresponding angle at the centre of the sphere : it is marked north (N.), or south (S.), according as the place is to the north or south of the equator. Thus the arc A' M' (Eig., page 99), is the latitude of a place A' (supposed Greenwich), and is marked N., because A' is to the north of W M' E ; and the latitude of B' is M' B', and marked S., because the place B' is to the south of the equator, whilst U, or its equal E Z, is the latitude of 0, or of F. As the latitude begins at the equator (lat. 0°), and is reckoned thence to the poles (lat. 90°), where it terminates, therefore the greatest latitude a place can have is 90°, and all other places must have their latitude inter- mediate between 0° and 90°. 147. Parallels of Latitude are small circles of the sphere parallel to the equator, that is, equidistant from it in every point, and hence all the places of the same latitude being at the same distance from the equator, are said to be on the same parallel ; thus (Fig., page 99) AN, T S, OF, and h B' are portions of parallels of latitude, and all places on F, and b B', &c., have the same latitude, being on the same parallel. 148. Co-Latitude is the complement of the latitude to 90°; thus the co- latitude of A' (Fig., page 99) is A N, of B' is B' S. 149. The Difference of Latitude (abbreviated diff. lat.) between two places, or of the parallels E and T S, or of any places on these parallels, is the arc of a meridian included between their parallels of latitude, showing how far one of them is to the northward or southward of the other ; thus * All circles, great or small, are supposed to be divided into 360 equal parts called degrees (°) 60' (minutes) make one degree, and 60" (seconds) make one minute. Nwvigation — Definitions . i o i (Fig., page 99) A'^ is the difference of latitude of the two places A' and B' ; F S between the places F and T, or T. The difference of latitude between two places can never exceed 1 80^. The difference of latitude of the ship is therefore the distance made good in a north or south direction. This is also called her '* northing " or " south- ing ^^ these names being indicated by the initials N. and 8. 150. It is evident that when two places are on the same side of the equator, their diff. lat. is found by subtracting the less latitude from the greater ; and that when they are on opposite sides of the equator, that is, when one place is in north latitude and the other in south latitude, the sum of their latitudes is the diff. lat. Thus the diff. of lat. of A' and B', which is A' b, is the sum of the north lat. A' M', and of the south lat. Z B', or M' B'. 151. Meridional Parts. — At the equator a degree of longitude is equal to a degree of latitude ; but as we approach the poles, while (upon the sup- position that tho earth is a sphere) the degrees of latitude remain the same the degrees of longitude become less and less. In the chart, on Mercator's projection, the degrees of longitude are made everywhere of the same length, and, therefore, to preserve the proportion that exists at every part of the earth's surface between the degrees of latitude and the degrees of longitude, the former must be increased from their natural lengths, more and more as we recede from the equator. The lengths of small portions of the meridian thus increased, expressed in minutes of the equator, are called meridional parts ; and the meridional parts for any latitude is the line, expressed in minutes (of the equator), into which the latitude is thus expanded. The meridional parts computed for every minute of latitude from 0° to 90°, from the Table of Meridional Parts, which is chiefly used for finding the meridional difference of latitude in solving problems in Mercator's sailing, and for con- structing charts on Mercator's projection. 152. The Meridional difference of Latitude is the quantity which bears the same ratio to the difference of latitude that the difference of longitude bears to the departure. It is the projection of the difference of latitude on the Mercator's chart, and takes its name from the meridional parts, by the use of a table of which parts it is found. 153. Middle Latitude. — When the two places are situated on the same side of the equator, the middle latitude is the latitude of the parallel passing mid- way between them ; its value is therefore half the sum of the latitudes of the two places. When the two places are situated on opposite sides of the equator the simple "middle latitude" is replaced by the two half latitudes of each of the places. (See Eaper's Navigation, page 98.) 154. Longitude is the arc of the equator iutercepted between the first meridian and the meridian of the place, and is, therefore, the measure of the angle between the two meridians; thus, (Fig., page 99) take N A' M' d S as the meridian of Q-reenwich, then, the longitude of A', or of any place on the meridian N A' M' J S is 0, and taking N U as the meridian of T, then the arc of the equator M'U, reckoned in degrees (°), minutes ('), and seconds("), or the angle M' N U, which M' U measures, is the longitude of T from M', the Nmigation — Definitions . meridian of Greenwich ; the arc M' Z, or the angle M' N Z is the longitude of the points Z, N, S, and F, or of any place on the meridian N Z ; the arc W M', or the angle W N M', which W M measures, is the longitude of the meridian NWS, or of any place on that meridian. Longitude is reckoned from the first meridian, both eastward and westward, tUl it meets at the opposite point of the equator, therefore the longitude can never exceed 1 80'^. It will be evident that the latitude alone will be insuflicient for the determination of the position of a place. If we state that a certaia place is 45° north of the equator, it will be impossible to ascertain certainly the place in question, inasmuch as there is a circle of points on the earth, all of which are 45° north of the equator. If we suppose a circle drawn round the surface of the northern hemisphere parallel to the equator, at the distance from the equator of 45°, every point of such circle will be equally characterised by the latitude of 45°. But if we state its latitude and longitude, wo can fix at once and unequivocally, the position of the place. Thu--, let us suppose that its latitude is 50° north, its longitude 30° east of Q-reenwich ; its position will be found by imagining a circle parallel to the equator, drawn upon the northern hemisphere at a distance of 50° from the equator ; then supposing a meridian drawn through Greenwich intersecting this parallel, and another drawn so as to cross the equator at a point 30^ east of the former ; the place in question will be upon the line parallel to the equator first drawn, inasmuch as it will be 50° north of the equator, and it will also be in the meridian last drawn, inasmuch as it will be 30° east of Greenwich. Since, then, it will be at the same time upon both these lines, it will necessarily be at the point where they cross each other at the east of the standard meridian of Greenwich. The place of a ship on the apparently indefinable and trackless face of the ocean can, in this manner, be as accurately marked down and discussed as any known and visible spot on the stable land. 155. Difference of Longitude between two places is the arc or portion of the equator included between their meridians, or, which is the same thing, the corresponding angle at the pole. To measure, therefore, the diff. of lojigitude of two places, we must follow down their meridians to the equator, and then take the included portion of the equator itself. It is named East or West, according to the direction in which the ship is proceeding ; thus, if we take A and F (Fig., page 99) to represent two places on the surface of the globe, the arc U Z, or the angle U N F, is diff. of long, between A and F, and is East, the arc U Z being the difference of H K, and H is the difference of the longitudes of P K and P 0, or of any two places on those meridians, and WZ, the sum of WM', and M'Z is the difference of the longitudes of the meridians NWS and N Z B' S. 156. Horizon. — The remote bounding circle which, to an eye elevated above the surface of the ocean, appears to unite sea and sky, is called the visible or sea-horizon. A plane conceived to touch the surface of the earth at any place, and to be extended to the heavens, is called the sensible horizon of that place. And a plane parallel to this, but passing through the centre of the earth, is called the rational horizon of that place. 157. When a ship, in sailing from one place to another, preserves the same angle with the meridians, as she crosses them in succession, she is said to sail on a Jblhumb Line. Thus, a ship in sailing from A to F (Fig., page 99) is supposed to describe on the sea a curve A F, which cuts the meridians N A, NB, NO, &o., at the same angle; that is, the angles N A F, NBF, NCF, are supposed to be equal. The rhumb line coincides with the meridian when Na/oigation — Definitions. i o 3 the course is due N. or S., or with a parallel of latitude when the course is due E. or W. On any other course but these the rhumb line is a spiral, approaching nearer and nearer to one of the poles at every convolution, but never reaching it. Sucb a curve is appropriately called the Equiangular Spiral, and the Loxodrnmte Curve; and also because, in sailing on it, we keep on the same rhumb or point of the compass, it is ci^Ued the rhumb curve. That such a curve may bo drawn through any two given points ■will appear from this consideration, — that from one of the points an infinite number of these curves can bo drawn, making diflFerent angles with the meridian, and on some one of these the second point must lie. It is evident also that only one of these curves can pass through the two points. It is the track used ordinarily in navigation, for when out of sight of land the compass determines the ship's track, and hence the selection of that track which makes a constant angle with tho meridian, the advantages of such a selection being that the sea- man is not required to alter his course. It would seem desirable to take tho shortest route on the voyage, and this is the arc of a great circle ; but the great circle drawn botwoon two places— except it happens to be on tho equator, or a meridian itself — cuts successive meridians at different angles, as a little consideration will show. When in sight of his port the compass is no longer needed, and the rhumb line is given up, and the port is made for on the great circle. When accurately following the compass course, we are, in strictness, only approximating, though very closely approximating, to a rhumb line, on account of the continuous change in tho variation, due to tho magnetic pole and the pole of the earth not being coincident. 158. The Course, from one point of the earth's surface to another, is the constant angle which the rhumb curve, joining the two points, makes with the meridians, or it is the direction in which a ship sails from one place to another, this direction being referred to the meridian, which lies truly north or south, or to the north or south line of the compass by which the ship is steered. The former is distinguished as the True Course, the latter as the Compaxs Course. The course steered is the angle between the meridian and the ship's head. The course made good is the angle between the meridian and the ship's real track on the surface of the sphere. The course is reckoned from the north towards the east or west, when the ship's head is less than eight points from tlie north ; and similarly from the south point. The coiirse is measured in points of 1 1° 1 5' each, or in degrees and minutes. 1 59. The Distance between two places is the arc of the rhumb line joining them, expressed in nautical miles of 60 to the degree of latitude. Thus (Fig., page 99) the length of line AF, expressed in minutes of a great circle of the earth, is called the distance.* It must never be lost sight of that the distance is not necessarily nor generally the shortest distance between the two places, that is, tho distance as the "crow flies." On a Mercator's Chart the rhumb curve is represented by a straight line, but it must be borne in mind, that oquil parts of any such line do not represent equal distances on the earth. The Meridian Distance between two places is the are of a parallel of lati- tude between them. * Minutes of a great circle are usually called nautical miles, or simply miles. I04 Navigation — Definitions. 1 60. Departure is the sum of all the intermediate meridian distances made in going from ono place to another, computed on the supposition that the distance is divided into indefinitely small equal parts. It is the distance, in nautical miles, made good towards the east or west, and such departure is expressed in miles, and not like the longitude, in arc. When the two places are on the same parallel, the departure is identical with the distance. When the places do not differ much in latitude, and are on the same side of the equator, an approximation to the departure is found in the arc of the parallel of middle latitude included between the meridians of the two places. If the subjoined right-angled plane triangles be taken to illustrate the terms defined above (Nos. 158 to 160), AB will represent the distance sailed, that is, the length of A F on the globe (see Fig. i) ; AC drawn N. and S., or in the meridian, shows the angle CAB, the course ; A C will represent A (Fig. I ), while B C drawn E. and W. will represent the sum of the small departures H B, I C, K D, &c., from the successive meridians which it crosses. 161. If a ship's course be due north or south, she sails on a meridian, and therefore makes no departure ; hence the distance sailed will be equal to the difference of latitude. If a ship sails either due east or due west, she sails on a parallel of latitude ; in which case she makes no difference of latitude, and the departure is identical with the distance. When the course is 4 points, or 45 degrees, the difference of latitude and departure are equal. When the course is less than 4 points, or 45 degrees, the difference of lati- tude exceeds the departure ; but when it is more than 4 points, or 45 degrees, the departure exceeds the difference of latitude. 162. Magnetic Course is the angle which the ship's track makes with the magnetic meridian ; such an angle can only be shown by a compass not affected with deviation ; but since the compasses of all iron ships have more or less deviation, and any course steered by such compass is magnetic in a certain sense, it has been deemed necessary to distinguish these when corrected for deviation as correct magnetic courses. 163. Compass Course is the angle which the track of the ship makes with the north and south line of the compass card ; such a course is affected with deviation and variation ; applying the deviation the result is the correct mag- netic course ; applying both the deviation and the variation it becomes the true course. Navigation — Definitions. loc 164. The True Bearing* of an object or place is the angle contained between the meridian and the direction of the object, and ia the same thing as the course towards it. It is thus qualified to distinguish it from the "Compass" and " Correct Magnetic Bearing." 165. Correct Magnetic Bearing. — The " correct magnetic bearing " of an object is the angle which its direction makes with the magnetic meridian; such an angle can only be found by a compass not affected with deviation. It is the bearing observed with the azimuth compass after being corrected for local deviation. A magnetic bearing, as given in "sailing directions" and on charts, is the correct magnetic bearing — in respect to a compass affected with deviation. 166. Compass Bearing. — The bearing of an object as taken by the com- pass. It is the angle between the direction of the needle of the standard compass on board the ship of the observer and the direction of the object ; it is therefore affected by the deviation and variation of the compass ; but the deviation to be applied in this case is that due to the azimuth (direction) of the ship's head, not that on the point of hearing ; when this correction is applied it becomes the correct magnetic bearing, and if, further, the correction for variation be applied, the true bearing or azimuth is deduced ; E. deviation and variation to the right, W. deviation and variation to the left. Taking a bearing of an object is called setting it. The bearings of two objects taken from the same place constitute Cross Bearings, the lines of direction of the two objects intersocting or crossing each other at the place of the observer. 167. The Tropics of Cancer and Capricorn are the parallels of latitude 23° 28' N. and S. The Sun is vertical at noon twice in the year to every place between the tropics, and never to any place outside of them. The space between the tropics is called the Torrid Zone. 168. The parallel of latitude which is 23° 28' from the north polo is called the Arctic Circle ; and that which is at the same distance from the South pole is called the Antarctic Circle. Within these circles the sun does not set during part of the summer, nor rise during part of the winter. The spaces within these circles are called the Frigid Zones. The spaces between the tropics and the polar circles are called the Temperate Zones. * Before the introduction of iron in such large quantities into the construction and equip- ment of steamers and iron sailing ships, bearings and courses were deemed to be sufficiently- well defined when spoken of as true and magnetic — the latter qualifying term being used simply to indicate the direction by compass as affected by variation only, according to the locality. But on board an iron ship compass bearings and compass courses though magnetic — inasmuch as they are the indications of a magnetic needle — are no longer such in the old sense of tho term, since thej' are affected by deviation. Under these circumstances it has been found necessary, especially in respect to this class of ships, to adopt a modification of the old nomenclature, and so "bearing" or "course" admits of an additional qualifying term not previously recognised, viz., correct magnetic. io6 PRELIMINARY RULES IN NAVIGATION. 169. I>EF. — The latitude and longitude of the place left are called the latitude from and longitude from ; the latitude and longitude of the place arrived at are called the latitude in and longitude in. 170. Given the latitude from and latitude in or to, to find the true differenes of latitude. To find the difference of latitude. (For definition, &c., see Nos. 149 and 150, pages 100 and loi.) EULE XLn. 1°, When the latitudes have like names — Subtract the less latitude from the greater, and multiply the degrees in the remainder by 60, adding in the minutes. The result is the true difference of latitude. z°. When the latitudes have unlike names — Take the sum of the two latitudes^ redMce it to minutes. The result is the true difference of latitude. 3°. To name the diff. lat. — If the latitude to is North of the latitude from, mark the diff. of latitude North (N.) ; but «/ latitude to is South o/" latitude from, mark diff. latitude South (S.) Latitudes are reckoned north and south of the equator. If these different directions are considered the one positive and the other negative, the difference of latitude of two places is always found by taking the algebraic difference of their latitudes. Examples. Ex. I. Find the diff. of lat. between Tynemouth Light, in lat. 55° i' N., and the Naze of Norway, in lat. 57° 58' N. Lat. Tynemouth 55° i' N. Lat. Naze 57 58 N. 2 57 60 177 N. The lat. from (Tynemouth) and lat. to (Naze) being of the same, name., that is, both North, the dijfferetice of them is taken for the diff. lat., and since we have to pas^; from the lower North lat. to a higher, the diff. lat. is marked North (N.) Ex. 3. A ship from lat. 32° 40' N., sails to lat. 20° 47' N. : what is the diff. of lat. made? Ex. 2. Kequired the diff. of lat. between Cape Formosa, in lat. 4° 15' N., and St. Helena, in lat. 15° 55' S. Lat. C. Formosa 4" 15' N. Lat. St. Helena 15 55 S. 20 10 60 Lat. from 32° 40' _ Lat. to 20 47 N N. " 53 60 D. lat. 713 S. The ship here passes from a higher N. lat. to a ower N. lat., and to do so must evidently sail S. ; whence we mivtlt diff. lat. S. D. lat. 1210 S. The lat. from (C. Formosa) is North, and the lat. to (St. Helena) is South, it is evident that the ship must sail South in order to pass from North lat. into South; whence we put South (S.) to the diff. of lat. Ex. 4. Eequired the diff. of lat. between Port Natal, in lat. 29° 53' S., and Akyab, in lat. 20° 8' N. Lat. Port Natal 29° 53' S. Lat. Akyab 20 8 N. 5° I 60 D. lat. 3001 N. As the ship (Port Natal) is in S. hemisphere and Akyab is in the N. hemisphere, to pass from the former into the latter the ship must sail N. Preliminary Rules in Navigation. 107 Ex. 5. A ship from lat. 50° S. arrives in lat. n$° 29' S. : what is the diff. of lat. P Lat. from 50° o' S. 45 29 S. Lat. in D. lat. 4 31 = 271 N. Here the ship passes from a higher to a lower S. lat., and to do so must evidently sail N. ; whenco th« ditf. lat. is marked N. Ex. 6. A ship from lat. 13° 45' S., arrives in lat. 26° 15' S. : required diff. lat. Lat. from 1 3° 45' S. Lat. in 26 15 S. 12 30 = 750 S. Here the ship passes from a lower to a higher S. lat., and to do so must evidently soil 8, : whence S. is marked against the diff. of lat. (a) When one of the places has no latitude, or is on the Equator, the latitude of the other place is equal to the difference of latitude. Ex. 7. A ship from a place A, lat. o, is bound to a place B, lat. 25° 8. ; required the diff. lat. Since lat. is reckoned from the Equator lat. 0° (N. or S.), to pass from o' to 25° S., the ship must evidently sail S. ; -whence the lat. of B (25°) is diff. of lat., and is marked S. Ex. 8. A ship from a place A, in lat. 10° N., arrives at a place B, in lat. 0°: re- quired the diff. lat. made. One place being on the Equator, and the other in 10° N., the diff. of lat. is evidently 10° or 600', and is named S., because it is evident the ship must sail South to pass irom 10° N. to 0° N. Examples for Praotioe. Required the difference of latitude between the place A and the place B in each of the following examples : — I. L*t. A 55° o' N. 2. Lat. A 50" 38' N. 3. B 58 23 N. B 42 48 N. 4. Lat. A 3 42 8. 5. Lat. A 13 15 8. " I 48 N. Lat. B A 10 lo N. Boo B Lat. A 49 53 8. B 42 13 8. Lat. A 58' 34' 8. B 63 17 8. Lat. A o o B 2 37 S. Lat. A o 17 S. B I 17 N. 171. To find the meridional difference of latitude, having given the lati- tude from and latitude in. (For definition, see page loi, Nos. 151 and 152). EULE XIHI. Take the meridional parts for the two latitudes from the Table of meridional parts ; take the difference if the latitudes are of the same name, but their sum if the names a/re unlike. The result is the meridional difference of latitude. Examples. Ex. 3. Lat. left 49° 58' 8., and lat. bound to 32° 42' S. : find me». diff. of lat. Lat. left. 49" 58' 8. M. parts 3471 Lat. to 32 43 8. yg 2078 Ex. r. Lat. A 49° 10' N., lat. B. 27" 40' N. : find the mer. diff. of lat. Lat. A 49'' 10' N. B 27 40 N. M. parts 3397 » '729 Mer. d. lat. 1668 Mer. d. lat. 1 393 Ex. 3 20° 8' N Lat. left 29' 53' 8., and lat. required mer. diff. of lat. Lat. left 29° 53' 8. M. parts 1880 Lat. to 20 8 N. „ 1234 to Ex. 4. Lat. from 46° 40' N ., and lat. to 34° 32' S. : find the mer. diff. of lat. Lat. left. 46° 40' N. M. parts 3173 Lat. to 34 22 8. ,, 2198 Mer. d. lat. 5371 Mer. d. lat. 31 14 Examples for Practice. Find the meridional difference of latitude in each of the following examples : — I. Lat. from 34° 40' N. Lat. in. 33^ 20' N. 4. Lat. from 15'' 44' N. Lat. in 4° 20' S. a. „ 24 12 8. 49 10 8. 15 18 N. S' t 60 20 8. f> 6? 10 8. 52 47 S. 6. n >i 4 20 N. 1 68 Preliminary Rules in Nmigaiion. 172. To find the latitude in, having given the latitude from and true difference of latitude. EULE XLIV. 1°. When the latitude from and true difference of latitude have a lilie name — To the latihicle from add the true difference oj latitude (turned into degrees^ minutes, and seconds, if necessary) : the sum will he the latitude in, of the same name as the latitude from. 2°. When the latitude from and true difference of latitude have unlike names — Under the latitude from, put the true difference of latitude fin degrees and minutes, if necessary) : the remainder marked with the name of the greater is the latitude in. Examples. Ex. I. A ship from lat. 59° 27' S., sails South, until the diff. lat. is 374 miles : re- quired the lat. come to. 6jO)37j4 I^at. from 59° 27' S. D. lat. 6 14 S. 6" 14' S. Lat. in 65 41 S. Ex. 3. A ship from lat. 55° i' N. sails North, 94 miles : find the lat. in. 6,0)9,4 Lat. from 55° I'N. D. lat. I 34 N. 1° 34' Lat. in. 56 35 N. Ex. 5. A ship from lat. 0° 49' S. sails North, 83 miles: required the lat. in. 6,0)8,3 i'»3' Lat. from 0° 49' S. D. lat. I 23 N. Lat. in o 34 N. Ex. 2. A ship from lat. 2° 25' N. sails South, 1 80 miles : what lat. is she in P 6,0)18,0 Lat. from 2° 25' N. D. lat. 3 o S. 3O0' Lat. in o 35 S. la this example it is evident that as the diff. lat. is more than the lat. left, the ship must have crossed the Equator, and consequently has come into South lat. Ex. 4. A ship from lat. 28° 39' N. sails South, 131 miles: required the lat. in. 6,0)13,1 Lat. from 28° 39' N. D. lat. 2 II S. 2° 11' Lat. in 26 28 N. Ex. 6. A ship from lat. 3° 12' N. saUs South, 192 miles: required the lat. arrived at. Lat. from 3° 12' N. D.lat. 3 12 S. On the Equator o o 6,0)19,2 Examples fob Practice. Find the latitude in, in each of the following examples : — Lat. from 31° 10' N. D. lat. 172' N. „ 29 38 N. „ 104 S. „ 3 2 S. „ 190 N. „ 2 56 S. „ 357 N. „ o o „ 168 S. 6. Lat. from 0° 8' N. D lat. 182' S. 7- )j 39 N. » 59 S. 8. j» 358N. J) 238 S. 9- >) 4 48 S. >» 288 N. 0. I) 35 >5 S. ») 229 S. 173. To find the middle latitude, having given the latitude from and latitude in. (For definition see No. 153, page loi.) RULE XLV. The name being supposed alike, that is, both North or both South — Add together the true latitudes, and take half the sum ; the result is the middle latitude. Note. — When the names are unlike, the middle latitude (which is seldom required hut for obtaining the departure) should be found by means of a table ; but in this case it may perhaps be as well to avoid the use of the middle latitude in any of the common problems of navigation. Preliminary Rules in Naviffafion. 109 Examples Find the mid. lat., having given Ex Ex. 1. the lat. from 50° 25' N,, and lat. in 47"^ 1 2' N Lat. from 50° 25' N. Lat. in 47 12 N. 2)97 37 Lat. from 6' 28' S., lat. in 14° 50' S. required the mid. lat. Lat. from 6' 28' S. Lat. in 14 jo S. 2)21 li Mid. lat. 39 Mid. lat. 48 48 Examples for Praotioe. Required the middle latitude in each of the following examples : — I. Lat. from 16" 10' S. D. lat. 138' S. 4. Lat. A 63° 53' S. Lat. B 59" 10' S. 2- „ I 40 S. „ 61 8. 5. „ 56 10 N. „ 50 15 N. 3- ). 36 22 N. „ 90 S. 6. „ 67 20 S. „ 61 42 S. 1 74. To find the difference of longitude, having given the longitude from and longitude to. (For definition see No. 155, page 102.) EULE XLYI. 1°. When the longitudes are of the same name — Take their difference and reduce the same to minutes, place E. or W. against the remainder, according as the longitude to is East or West of longitude from. 2°. When the longitudes are of contrary names — Take the sum of the two longs., which sum, if less than 180°, is the diff. of long., and attach E. or W., according as the long, to is East or West of long, from ; but when the sum exceeds 180° subtract it from 360°, for the diff. of long., and reduce the remainder thus found to minutes, attaching to it the contrary name to that found in the tisual way. Longitudes are reckoned East or West of the first meridian. If these differeat directions are considered one positive and the other negative, the difference of longitude of two places is always found by taking the algebraic difference of their longitudes. Examples. Ex. I. Find the difi". of long., having given the long, from 89° 42' W., and long, in 79° 42' W. Long, from 89° 42' W. Long, in 79 42 W. 60 D. long. 600 E. The ship here passes from a high W. long, to a lower, and diff. long, must be E. to do so. Ex. 3. A ship from Cape Bajoli, long. 3° 48' E., is bound to Cape Sicie, in long. 5° 51' E. : required the diflf. of long. Long. Cape Bajoli 3° 48' E. Long. Cape Sicie 5 5 1 E. 60 Ex. 2. Required the diff. of long., having given the long, from 12° 20' E., and long, in 2° 45' W. Long, from 1 2° 20' E. Long, in 2 45 W. 15 5 60 D. long. 123 E. The long, to Cape Sicie is E. of long, from Cape Bajoli, therefore, diff. of long, is marked E. The ship must evidently sail E. D. long. 905 W. The ship here passes from E. long, to W. long., and in order to do so diff, long, must be W. Ex. 4. A ship from Tynemouth, in long, i" 25' W., is bound to long. 7° 12' E. ; re- quired the diff. of long. Long, from i^ 25' W. Long, to 7 12 E. 8 37 60 D. log. 517 F. The ship here is about to cross the meridian of Greenwich (long 0°) and pass from W. long, to E. long., whence the diff. of long, must be E. to do .so. no Prelimina/ry Rules in Navigation. Ex. 5. Find the difF. long, between Aca- pulco, long. 99° 54' W., and Pellew Island, long. 134° 21' E. Long. Acapulco 99° 54' W. Long. Pellew Island 134 21 E. Being greater than 180° it is subtracted from Diff. of long, is 234 15 E. 360 o Ex. 6. A ship from long. 177° 50' E. arrives in long. 178° 10' W. : what diflF. of long, has she made ? Long, left 177° 50' E. Long, in 178 10 W. 125 45 "W. 60 D.long. 7545 W. By gaing E. and W. from Greenwich, the two places in this example will be found to be 234° 15' asunder, but as both places are for our purpose upon one circle, the smaller arc of the circle must be taken to find how far apart the places Acapulco and Pellew Island are separated; so that the sum 234° 15' is subtracted from 360", the whole oircumferenoe of a circle, for the required answer. Being greater than 180" it is subtracted from Diff. of long, is 356 360 W. o E. D. long. 240 E. Ex. 7. A ship from long. 5° bound to a port in long. 90° W. : of long, must she make P Long, from 5° 12' W. Long, to 90 o W. 12' W. is what diff'. 60 D. long. 5088 W. The ship here passes from a less to a greater "W. long. ; and therefore the diff. of long, must bo W. to do so. Ex. 8. A ship from long. 165'^ bound to a place in long. 72° 12' E. ; diff. of long, must she make P Long, left 165° o' E. Long, to 72 12 E. E. is what 92 48 60 D. long. 5568 W. The ship in this example sails from a greater to a less long. (E. long.), the diff. long, is therefore of a different name to the long. left. 9^ 29' w. Long. B 4° 29' W. 7- Long. A 0° 55' E. Long. B 7° 3'E. I 25 w. 7 a E. 8. „ 40 10 E. » 33 lo E. 6 II E. 5 45 W. 9- „ 178 30 W. » 178 30 E. 4 20 W. 10. „ 176 34 E. „ 176 34 W. 4 20 W. 10 E. II. „ 38 32 W. 8 43 E. 7 2 E. 12. „ 5 12 W. » 25 12 W. Examples fob, Praotioe. Required the difference of longitude between a place A and a place B in each, of the following examples : — Long. A 9 2. 3 4 5 175. To find the longitude in, having given the longitude from and the diflterence of longitude. EULE XLYII. 1°. When the longitude from and the difference of longitude have like names — To the longitude from add difference of longitude (turned into degrees, if necessary) : the sum, if not more than 180°, will be the longitude in, of the same name as the longitude from; hut if the sum exceed 180°, subtract it from 360°, and the remainder is the long, in and of a contrary name to long. from. 2°. When the longitude left and difi'erence of longitude have unlike names — Under longitude from, put difference of longitude fin degrees and minutes, if necessa/ryj ; take the less from the greater ; the remainder, marked with the name of the greater, is the longitude in. PreUmina/ry Rules in Navigation. Examples. Ex. I. A ship from long. 5° 12' W. makes diff. long. 113' W: required the long. in. Long, from 5° 12' W. 6,0)11,3 D. long. r 53 W. 53' W. Lonf. 7 5 W. Ex. 2. A ship from long. 1° 25' W. sails E. until her diff. of long, is 177': required her long. in. Long, from 1° 25' W. 6,0)17,7 D. long. 2 57 E. Ex. 3. A ship from long. 0° 57' E. sails W. until her diff. of lonjf. is 201' : find the long. in. Long, from o''57' E. 6,0)20,1 D. long. 321 W. 3" 21 W. Long, in 2 24 W. 2° 57' E. Long. in. i 32 E. Ex. 4. Let the long, left be 174° 4' W., and the diff. of long. 797' "W. : required the long. in. 6,0)79,7 Long, from 174° 4' W. D. long. 13 17 W. 13° 17' W. Being greater than 180" 187 21 W. subtract from 360 o Ex. 5. liong. from 3° 40' W., diff. of long. 220' E. : required the long. in. Long, from 3° 40' W. 6,0)22,0 D.long. 3 40 K. 3° 40' E. Long, in o c On the meridian of Greenwich. Long, in 172 39 E. Ex. 6. A ship from long. 177° 40' "W. makes 140' diff. of long, to the W. : required the lonsf. arrived at. 6,0)14,0 Long, from i77''4o'W. D. long. 2 20 W. 2° 20' W. Long, in 1 80 o W. or, 180 o E. Examples for Pkaoticb. Required the longitude in, or arrived at, in each of the following examples Long, from 5° 48' W. D. long, no' W. 7. „ o 59 W. „ 137 E. 8. „ 29 10 E. ,, 114 E. 9. „ 3 10 E. „ 220 W. 10. „ 2 47 W. „ 242 E. II. „ 3 12 E. „ 237 W. 12. Long, from 41° 29' W. I), long. 139' E. 3- 4- J. 6. 13. Define meridian of the earth, equator, parallel of latitude. Which of these are great circles, and why ? 98 54 E. , 115 >v. 302 E. 178 13 E. „ 201 E. 177 6 W. » 237 w 179 59 W. 2 w 112 THE COMPASS, 176. The Compass* is simply an instrument which utilises the directive power of the magnet. A magnetised bar of steel, apart from disturbing forces and free to move, points in a definite direction, and to this direction a 1 others may be referred, and a ship guided on any desired course. There are various adaptations of the instrument, according to the use it is specially intended for. The compass intended for use on board ship is called the "Mariner's Compass," and according to the purpose it is intended for it is named the Steering Compass, the Standard Compass, and the Azimuth Compass. 177. The Mariner's Compass consists of a circular card, vrhich represents the horizon of the observer ; the circumference or edge of the card being divided according to two systems of notation into points and degrees. * The origin of the compass is very obscure. The ancients were aware that the loadstone attracted iron, but were ignorant of its directing property. The instrument came into use in Europe sometime in the course of the thirteenth century. The Compass. 1 1 3 (i). By Points. — There ai-e 32 points ; and each of those divisions is again sub-divided into four parts called quarter points. A point of the compass being therefore the 32nd part of the circumference of a circle is equal to 11° 15'. The four principal points, or, as they are called, the cardinal points, are the North (represented by N.), South (S.), East (E.), West (W.), the East being to the right, and West to the left, when facing the North. All the points of the compass are called by names composed of these four terms. Thus, the points half-way between the cardinal points are called after the two adjacent cardinal points; hence the point midway between the North and East is called North-east, and represented by N.E. ; so midway between South and East is called South-east (written S.E.); in like manner we get South-west (written S.W.), and North-west (written N.W.)* A point half-way between one of these last and a cardinal point is called, in like manner, by a name composed of the nearest cardinal point and the adjacent points, N.E., N.W., S.E., and S.W. Thus, the point half-way between N. and N.E. is called North-north-east (written N.N.E) ; the point between E. and N.E. is called East-north-east (written E.N.E.); and so we have E.S.E., S.S.E., S.S.W., W.8.W., W.N.W., and N.N.W. The points next the eight principal points, namely, N., N.E., E., S.E., S., S.W., and N.W., are named by placing hj between the letter representing the point to which it is adjacent and the next cardinal point in the same direction. Thus, the point next to N., on the east side, is called North by East, i.e., North in the direction towards East (written N. by E.) ; that next N.E., towards the North, is called North-east by North (N.E. by N.), i.e., North-east in the direction towards North; and so we have N.E. by E., E. by N., E. by S., S.E. by E., S.E. by S., S. by E., S. by W., S.W. by 8., S.W. by W., W. by S., W. by N., N.W. by W., N.W. by N., N. by W. ; in this manner we get other sixteen points. We have thus got names to all the thirty-two points of the compass. Each point is again sub-divided into half points and quarter points. A half point, which is the middle division between two points, is called after that one of its adjacent points which is either a cardinal point or is the nearest to a cardinal point. Thus, the middle division between N. and N. by E. is called North-Aa^-east (written N. ^ E.) Half points near N.E., N.W., S.E., and S.W., take thoir name from these points. Thus we say N.E. i N., N.E. by E. ^ E.f * These new directions also give names to the four quarters of the compass, as, when we say that " ihe wind is in the S.W. quarter," meaning thereby not exactly S.W., but some- where between S. and W. t In naming the half and quarter points it is advisable in some cases to sacrifice system to simplicitJ^ Thus, for example, Beamea commonly say N.N.E. \ E. instead of N.E. by N. ^ N. ; we do not, however, siy E.N.E. \ E., though this is simpler than E. by N. |- N., since it is at once seen to be 6| points. It would of course be more systematic, as a matter of geometry, to reckon the half points always from N. or S., because the ship's course is reckoned from the meridian ; but on the other hand, as a m ttter of names, regard will be had to the whole points between which it falls, and to the order in which these are taken. 114 The Compass. The same holds for a quarter and for three-quarters as for a half point, all of -which^are named upon the same principle as the subordinate points. In chosing the name to use we must be guided by circumstances. In some problems it is convenient always to reckon uniformly from North or South, but generally the simpler name will be the preferable one ; and similarly for quarters and three-quarters of a point. (2). By Degrees. — The whole circumference is divided into three hundred and sixty degrees (360°), each degree into sixty minutes (60'). This furnishes a notation for the compass more minute than points, half points, and quarter points. We still reckon from the cardinal points : thus, to indicate a division which has 72° 48' to the East of North we write N. 72° 48' E. 178. The name of the opposite point to any proposed point is known at once without referring to the compass, by simply reversing the name or the letters which compose it — thus, the opposite of N. being S. and of E. being "W., the opposite point of N.E. by N. is at once known to be S.W. by S., the opposite of W. \ S. is E. | N., and so on. 179. Repeating the points in any order is called boxing the compass ; to do this is, of course, one of the first things a seamen learns. 180. As the ship's course, which is sometimes expressed in points and sometimes in degrees, is always reckoned from the North or South point, the seaman has to refer at once, in using the Tables, to the number of points or degrees in any course given by name. The following table, which exhibits the degrees, minutes, and seconds in each quarter point of the compass, will be convenient for reference. A TABLE OF THE ANGLES, which every Point and Quarter Point of the Compass makes with the Meridian, NOETH N. by E. N.E. by N. N.E. N.E. by E. E.N.E. E. by N. East. N. by W. N.N.W. N.W. by N. N.W. by W. W.N.W. W. by N. West. Points , „ Points 2 48 45 i 5 37 30 1 8 26 15 1 11 15 1 14 3 45 I 16 S2 30 I 1 19 41 15 1 1 22 ^0 2 2S 18 4-; 2 28 7 30 2 : 30 5t> IS 2 : ?.^ 45 3 3t> 33 45 3 ?9 22 30 3 ? 42 11 15 3 1 45 4 47 48 45 '^ ? ,So 37 30 '^ i 5S 26 15 4 1 5t) 15 5 M 3 45 5 ^ 61 52 30 5 § 64 41 15 5 1 b^ 30 6 70 18 45 6 ^ 1 n 7 30 75 50 15 6 i 78 45 81 33 45 7 i 84 22 30 7 f «7 11 15 I * 90 8 SOUTH S. by E. S.S.E. S.E. by S. S.E. by E. E. by S. S. by W. S.S.W. S.W. by S. S.W. S.W. by W. W.S.W. W. by S. West. The Compass. 115 181. The card for practical use is generally made of mica covered with paper, so as to be as light as possible. Two or more magnetic needles,* which are small steel bars magnetised, are fixed below the circular compass card, but parallel with its meridional line, so that the N. ends of the needles shall coincide (in direction) with the N. end of that line, and the 8. ends of the needles with the S. end of the same line. An inverted conical brass socket, called a cap, with a hard stone in its centre, is passed through a hole in the centre of the card, and the whole is then accurately balanced on a sharp centre or pivot rising from the middle of a brass or copper bowl, and sufficiently large to admit of the card moving freely within it : the cover of the bowl is glass, which, while protecting the card from wind and weather, admits of its indications being distinctly seen. There is also a vertical line drawn inside the bowl which is called the lubber's line. The bowl, having a weight fixed to it below, is placed in gimhals, which are brass hoops or rings so arranged as to admit of motion about two horizontal axis at right-angles to one another, i.e., each turning upon two pivots at opposite points of the hoop next greater in size; by this means the loaded bowl remains nearly horizontal during the confused and irregular motion of the ship. To the deck, in front of the helmsman's position, a stand called a Binnacle is firmly fixed, which may be of any shape — octagonal, square, or pillar-like — sometimes of wood, sometimes of brass : within it are supports or bearings into which the pivots or outer rings of the compass bowl fit, and its movable top or cover is fitted with a glass front and a lamp or lamps to cast a light on the compass card by night. This constitutes the Steering Compass. 182. The helmsman steers the ship so that a line parallel to the keel passes over the centre of the card, and the point prescribed as the course. Care is taken to place the box so that the lubber's point in the bowl and the centre of the card are in a line fore-and-aft, or parallel to the keel ; but as the lubber's point deviates a little from its proper position when the ship is heeled over, seamen do not implicitly depend upon it, as, indeed, the name implies. 183. The Azimuth Compass is a compass of superior construction, parti- cularly adapted to observe bearings. It is mounted on a stand, and is fitted with two small frames carrying vertical wires, called sight-vanes, for the purpose of observing objects elevated above the horizon. In one of these vanes there is a long and very narrow slit, and in the other is an opening of the same kind, but wider, and having a wire up and down the middle of it, exactly opposite the slit. 1 84. In the best modern instruments, a horizontal ring is expressly pro- vided to carry the vertical wire frame, and instead of having a wire next the eye, a glass prism, acting by internal reflection, is placed there, so arranged that one-half of the pupil of the eye can observe the wire on the further side of the horizontal ring and the distant object, and the other half of the pupil can see the graduations of the compass card by internal reflection in the * The object of using several magnets i:j to increase the magnetic moment of a given weight of steel. ii6 The Cmypoii. prism. This prism, is a solid piece of glass, whose sides are parallelograms and ends triangles. The compass card is very carefully and minutely grad- uated ; besides the points and quarter points being marked, the circumference over which the prism passes is graduated in degrees, and usually cut to every 20', and this graduation is arranged so that we may read off the bearing at once, and is reckoned in more ways than one, for facilitating taking bearings from different cardinal points. The card can be brought to rest by a stop. There is also a contrivance for throwing the card off its centre when the instrument is not in use, to prevent the fine pivot being worn, and the sensi- bility of the compass impaired. This instrument is known as the Prismatic Azimuth Compass. 185. In observing bearings on board ship the card should never be stopped, but two or more bearings being read off as quickly as convenient, the mean should be used ; for, as the vessel, and consequently the compass card, have always some motion, the card may not therefore be stopped exactly in the middle of its vibration, which, as it may be supposed to vibrate equally on both sides of the line of direction of the object, is essential to the true result. 1 86. The Standard Compass on board ship is the one placed in a particular spot on deck, or above it. It should be placed in the middle of the ship, and fixed on a permanent and secure pillar or support, raised at such a height (not less than 5 feet) as to permit amplitudes of the sun and bearings of the land to be conveniently observed by it. In the Eoyal Navy it is used as an azimuth compass, being fitted with an azimuth circle, which is graduated so as to show the angle between the ship's head and any heavenly body, as measured on the horizon (thus acting as a dumh card) ; the sight vanes and reading prism being fitted to the azimuth circle in such a way as to turn freely in azimuth without moving the compass or disturbing the card. The card of the azimuth compass should not exceed 7 to 7^ inches diameter. It should also be in a position as far as possible removed from any con- siderable mass of iron — at least 5 feet from iron deck beams — and should not be within i o feet of the extremity of any elongated iron mass, especially if vertical, such as funnels, stanchions, or the spindle of the wheel ; and it should be received as a general rule that no iron, subject to occasional removal, is to be placed within 1 5 feet of this compass, either on the same deck, or that below it. But in the mercantile marine the practice prevails of taking these bearings by means of a dumb card, many of which are in use, and answer the purpose intended.* They can be placed in any part for observing, and the true bearing of the ship's head is determined, so that by comparing it with the bearing shown by the Standard Compass the deviation of the latter can be ascertained. Under these circumstances we recommend that the only con- sideration for determining tho selection of a place for the Standard Compass should be favourable conditions connected with its compensation. * Perhaps the best and most useful of these is the one known as Bain & Ainslet's " Oompass Corrector." The (hmpass. 1 1 7 187. Steering Compasses being placed according to the requirements of the slaip, the modei'ato and uniform amount of deviation generally attainable at the Standard Compass by soleetion of position, cannot always be secured. Still we should do the best we can, for if. as frequently happens, the steering wheel is placed near an iron stern-post or rudder-head, and further fitted with an iron spindle — near which, of necessitv, the steering compass is fitted — then large and perplexing deviations may be expected, defying oven approximate correction by magnets, causing much inconvenience to the helmsman, and possibly a total loss of the services of the compass on the ship proceeding into southern latitudes.* The following rules to avoid the inconvenience and even danger just pointed out, have been recommended in selecting a place for steering com- passes : — "Not to be within half the width of the ship from the stern-post or rudder-head ; the spindle of the steering wheel and the foremost support on which the wheel works not to be of iron ; avoid vertical iron." The needle should be at least 3 ft. 6 in. from iron deck beams, and as much higher as can be made convenient to the helmsman. In addition to the rules already given for the guidance of seamen, the following (given by Capt. Evans, Superintendent of the Admiralty Compass * Bridge Compass. — Those who arrant;;e for the construction and equipment of iron steamers ought not to lose sight of tho fact that the hridge compass is the most important of all. It is on tho bridi^e where the officer of the watch is stationed on all occasions when caution is required, and if the compass before him is a reliable one he is better enabled to n'lvigate tho ship safely than by a compass situated in any other part. Yet many errors are committed in placing this compass. Take an instance, " where the bridge compass was rendered untrustworthy from the ventilator of the engine-room being close in front of it. Two reasons were assigned for this. The one was that the bell-mouthed top required to be turned round in order to regul ite the amount of ventilation, it was convenient to have it within reach of the bridge. The other was, that in placing the bridge farther from the ventilator :i greater evil would be incurred, since it would require that the compass should either be brought nearer to the iron mast or nearer to the funnel. It was not understood that this ventilator, small as it was compared with the mast or the funnel, produced five times as much error as these two combined, if placed equally near. The head of the ventilator was a blue pole by induction, and was on a level with the compass, and when the bell-mouth, or cowl, was turned towards the stern, not two feet from the compass on the bridge, it caused two points of deviation, but when the cowl was turned towards the bows only a half point of error was produced, and when turned to the starboard or port side other complications resulted rendering the compass useless, because the red pole of the ventilator was so far distant as to render no appreciable amount of repulsion to compensate the attrac- tion of the upper pole. But the blue and red poles of the mast and funnel were situ 1 ted one above the compass end the other below, so as nearly, if not altogether, to compensate each other. Thus, being ignorant of the laws which regulate deviation, in attempting to avoid an imagined evil, a s^erious real error was committed. This compass being found unworthy of reliance was removed, and the captain in ascending the Gulf of St. Lawrence had continuallj' to refer to the Standard Compass on the deck, which in hazy weather rendered the navigation more difficult than it would have been had there been a trustworthy bridge compuss." Again, it not unfrequently happen? ^^it *he bridge on which the compass stands is placed directly over the engine and close against the central iron stanchion which supports the rail running along the bridge, while according to another arrangement the compass stands on a semicircular piece of planking projected from the bridge forward with the railing carried round it, with perhaps, as in some instances, live and in others four iron stanchions in close proximity to the compass, while in otlier cases where the circular space is just large enough for the binnacle there may be only three iron stanchions, but in all these cases these iron stanchions are only a very few inches from the card, and the card itself within si.x. inches ot the plane of the top of the stanchions. It cannot be a matter of surprise that no dependence can be placed on compasses so placed. ii8 - The Compass. Department) are worthy the attention of the Naval Architect and those superintending the equipment of the ship : — ( I ) . In all designs for the construction of iron ships, a place to be prepared for the Standard Compass, and to be shown in the plan. (2). The Standard Compass not to be within half the breadth of the ship from the rudder-head and stern-post or iron- cased screw well, not to be nearer an iron deck or iron deck beams than five feet. (3). In ships built near North, the Standard Compass to be as far forward as the requirements of the ship will permit. In ships built head near South, to be as far aft as the requirements of the ship will permit, subject to Eule 2. In ships built nearly JEast or West, the Standard Compass not to be near either extremity of the vessel. (^). To be as far as possible from transverse iron bulkheads. (5). As far as possible, no masses of iron — as boilers, engines, bulkheads, or stanchions — should be placed below the compass, or within 55° of the vertical line through the centre, the angle being drawn from the compass as centre to the centre of the mass in question.* (6). To the above we would add, not to be nearer the break of the poop, either before or aft, than half the breadth of the vessel. Note.— " Comparative Merits of Large and Small Compasses.— Of late years much diversity in practice has pravailed as to the size of compasses for use on board ship. The Admiralty Standard Card, for example, is fitted with needles, the maximum lengths of which are 7^ inches, while ia large passenger steam vessels the needles are frequently 12 to 15 inches, and even longer. The chief object in the employment of large compasses is to enable the helmsman to steer to degrees, and a more accurate course is presumed to be preserved. " With reference to this increased size it must be observed that competent authorities limit the length of efficient compass needles to 5 or 6 inches; beyond this limit an increase of length is alone accompanied by an increase of directive power in the same proportion, and if the thickness of the needle be preserved, the weight, and consequently the friction, increases in the same ratio. No advantage of directive power is therefore gained by increase of length, but with the increased weight of the card and appendages, the increase of friction probably far exceeds the increase of directive force ; sluggishness is the result, which is further exaggerated by the extreme slowness of oscillntion of long needles compared with short ones. " Large cards, however convenient in practice, are therefore not without danger, for the course steered m;ty deceive the seaman by seeming right to the fraction of a degree, but which avails little if the card itself is wrong half a point, and the ship in consequence hazarded. In the opinion of the writer, the present Admiralty Standard Card is as large as should be used for the purposes of navigation ; and that as regards safety in the long, steady, * Investigation has shown that the effect of a sphere of iron within this cone is prejudicial by diminishing the directive forco and incre^ising the heeling error to windward — when without the cono it would be beneficial in both respects. Hence the recommendation. With reference to the magnetic character of boilers, or tanks, it has been stated that the tififect is the s;une as if they were solid bodies, on the assumption that magnetism exists entirely on the surface of iron masses. Tliis is not the case ; it is, however, true that the effect of hollow masses of iron incronses very rnpidly with the increase of the thickness of the iron, so that the limit of thiciknosH is speediy reached when the effect of the body is sensibly the same as if it were solid ; for example, in a tank 4 feet in diameter and i-ioth of an inch thick, the efiect is about ^ of a solid mass of the same size; in a similar sized tank ^ of an inch thick, the effect would be about half that of a solid mass. See a valuable investigation by Mr. Abchlbald iSmixh, in the Phil. Trans, for 1865, pages 304 — 31S. The Compass. i%g and fast ship, the choice is really between the Admiralty Card and a smaller one. In short, the question may be thus stated : the smaller a card the more correctly it points, the larger a card the more accurately it ia read." — Manual of the Deviation of the Compass, by Capt. Evans, R.N. 1 88. There is no advantage in having a large number of compasses in a ship : since, unlike the mean results of a number of chronometers, for example, the mean results of any number of compasses need not necessarily be near the truth, as they may all be largely in error, and that error may be all in one direction. Hence the necessity of depending upon one compass alone, but that compass should be in the best position in the ship, of the best manufacture, and the constant attention of the navigator should bo devoted to ascertain its errors. ADJUSTMENTS OF THE COMPASS. 189. (i). The direction of the magnetism of the needle or the " magnetic axis" should be in a line along the middle of the needle itself, otherwise the needle will not point with exactness to the magnetic North and South. To examine whether this is the case reverse the needle on the card. If after this reversion the N. and S. points of the card are also found to be reversed, the adjustment is good. As this error obviously affects all points of the compass alike, it may be included in the total variation of the particular compass as found by observa- tion, and therefore need not be made the subject of special examination. (2). The pivot must he in the centre of the graduated circumference of the card. If it is not, the difference of bearing of two objects will not be the same when measured on different parts of the edge. This adjustment is generally good. (3). The line joining the eye-vane and the ohject-vane, called the "Zme of sighV^ of the Azimuth Compass, must pass directly over the pivot. This condition is examined by noting carefully the bearing of a distant object, and then turning the compass half round so as to reverse the vane and the slit, and then repeating the observation with an object eight points from the first. The bearings taken directly should be identical with those taken by reversion. The effects of this error, if any, may be eliminated by taking the mean of the direct and reversed bearings every time the instrument is used. (4). The sight-vanes mitst he vertical, i.e., the eye-vane and the ohject-vane must each be vertical. This can be examined only on shore, by observing whether the wires coin- cide through their length with a plumb line, or any vertical edge. When this adjustment is not perfect, or when the bowl is not maintained in a strictly horizontal position, bearings are most correctly obtained when the object is low. CORRECTING COURSES. 190. The corrections of the compass are those quantities which must be applied to the indications of the instrument to obtain the reading that would be given if the north point of the compass card always corresponded to the north point of the horizon. Three corrections are sometimes necessary to be applied to the course steered by compass, to reduce it to the true course ; and the converse. These are called 1 . The Leeway. 2. The Variation of the Compass. 3. The Deviation of the Compass. I. LEEWAY. igi. The angle included between the direction of the fore-and-aft line or keel of a ship, and that in which she moves through the water, as indicated by her wake, is called the Leeway, A ship is said to be on the port tack when the wind is on her port side, that is, on the left-hand side of a person looking forward ; and on the star- board tack when the wind is on her starboard side, that is, on the right-han I side of a person looking forward.* When the ship is not going before the wind, she w^ill not only be forced forward in the direction of her head, but in consequence of the wind pressing against her sideways, her actual course will be to " leeward " of the apparent course she is lying. The amount of leeway differs in different t^hips ; depend- ing on their construction, on the sails sets, the velocity forward, and other circunistances. Experience and observation are required to judge what amount of leeway to allow in each case. The correction for leeway is necessary to deduce the course made good from the course steered, and it is one of the corrections to be applied in reducing the compass course to the true course in the day's work ; the correction being allowed according to RULE XLVIIL When the ship is on the port tack, allow the leeway to the right of the course steered ; but when on the starboard tack, allow it to the left, the observer looking from the centre of the compass towards the point the ship is sailing upon. Examples. Ex. I. The course steered ia N.W. by 1 Ex. 2. Course by Compass S. by E., W., the wind N. by E., leeway \\ points. wind E. by S., leeway 2|- points. The ship has the starboard tacks aboard ; The ship is on the port tack, then 2| therefore, the leeway {i\ points) allowed to points allowed to the right of S. by E., is the left of N.W. by W., gives corrected [ fc>. by W. f W., the Course corrected for Course W. by N. f N. | leeway. * A ship is said to be on the tack of the side from which the wind comos, even if it be on the quarter. Correcting CowBes. Ex. 3. Course N.E. by N., the wind i Ex. 4. Course steered West, the wind N.W. by N., the leeway i point. - N.W. by N., leeway 3^ points. The ship being on the port tack, i point ' The ship is on the s^rtriosrc? tack, 3J points to the right of N.E. by N is N.E., the cor- ! to the left of West is S.W. \ W., the com- rected Course. ' pass course made good. 192. The points of the compass are frequently spoken of in calculation with reference to their position to the right or left of the cardinal point towards which the spectator, who is supposed placed in the centre of the compass, is looking. Supposing the given point of the compass have Worth in it, then looking from the centre of the card over the cardinal point North, he has the quad- rant from North to East on his right hand, and the quadrant from North to West on his left hand. Hence any point between North and East is said to be to the right of North, and every point between North and West is said to be to the left of North; thus, N.N.E. is said to be " two points" to the right of North (for shortness usually written 2 pts. E. of N.), and W.N.W. "six points" left of North (written 6 pts. L. of N.) Again, suppose the given point of the compass to have South in it, then the observer, looking from the centre of the card and facing South, has the quad- rant from South to East on the left hand, and the quadrant from South to West on the right hand. Hence any point between the S. and E. is said to be to the left of South, and any point between S. and W. is said to be to the right of South ; thus S.E. by S. is said to be " three points " to the left of S. (for shortness written 3 pts. L. of S.), and W. by S. is " seven points " right of S. (usually written 7 points R. of S.) Adopting this notation the work in the above examples will stand thus : — Examples. Ex. I. Ex. 2. Course steered N.W. by W. is 5 pts. L of N Leeway carries ship i^ ,, LofN Sum is corrected course 6^ „ L of N or W. by N. | N. Ex.3. Course steered N.E. by N. is 3 pts. R of N Leeway carries ship i „ R of N Course steered S. by E. is i pt. L of S Leeway carries ship if „ R of S The difference is if „ R of S S. by W. i W. Ex.4. Course steered West is 8 pts. R of S Leeway carries ship 3^ „ L of S Sum is corrected course 4 „ R of N The diff. is corrected course 4^ „ R of S or N.E. I or S.W. j W. Examples for PaAonoE. Correct the following courses for leeway. Course Steered. Wind. Leeway. Course Steered. Wind. Leeway. 1. S.S.W. S.E. 4 5, E. |N. N. byE. i* 2. S.W. ^W. W.N.W. 2i 6. N.W. |N. N.E. by E. i| 3. N. by E. E. by N. | , 7. S.W. by W. S. by E. il 4. N.N.E. ^E N.W. iN. 2 8. N.E. i E. N. ijr W. i| 122 Correcting Cowrses. (a) WJien the ship is hove-to, take the middle point between that to which she comes up and that to which she falls off for the compass course, and correct this for leeway. Examples. ' Ex. I. A ship lying- to under her main- sail, with her starboard tacks aboard, comes up E. by S., and falls ofi' to N.E. by E., making 5 points leeway. What compass course does she make good ? The middle point between E. by S. and N.E. by E. is E. by N., then 5 points to the left hand gives X.N.E., the compass course made good. Ex. 3. A ship lying-to comes up S. by E. and falls off to S.E. by E., the wind being S.W., making 5 points leeway : required the compass course. The middle point between S. by E. and S.E. by E. is S.E. by 8., then 5 points to the left hand (the ship having starboard tacks on board) is East, the compass course made good. Ex. 2. A ship lying-to under a close- reefed main topsail, with her port (larboard) tacks on board, comes up to S.S.W. and falls off to S.W. by W., making 2^ points leeway. What compass course does she make? The middle point between S.S.W. and S.W. by W. is S.W. ^ S., then 2^ points to the riff hi hand is W.S.W. Ex. 4. A ship lying-to with port tacks on board, comes up W. by S. and falls off N.W. by W., making 5 points leeway. What course does she make good ? The middle point between W. by S. and N.W. by W. is W. by N., then 5 points to the riffht hand is N.N.W., the compass course made good. 2. THE VAEIATION OF THE COMPASS. 193. The needle points to the magnetic North, which in few parts of the world agrees with the true North ; the diiference between them is called the Variation of the Compass.* The variation is said to be easterly when the North end of the needle is drawn to the eastward, and loesterly when drawn to the westward of the true North ; thus, when the North end of the needle points to that part of the horizon which is true N.N.W. \ W., the variation is said to be 2^ points West ; but when it points to the N. by E. part of the horizon, the variation is said to be i point East. 194 The variation is different in different places,f and it is also subject to a slow change in the same place, and becomes alternately East and West.;}: * This is the term commonly employed by nautical men ; but among men of science the term "Magnetic Declination" is usually substituted for " Magnetic Variation." t At G-reenwich, at the present time, the variation is 20° W., or the North end of the magnetic needle does not point exactly North, but 20° W. of North. In the West Indies the variation is o; ut Cape Farewell, 53° W. ; at Cape Horn, 23° E. ; at Hobart Town, 10° E.; at Canton 1° E. ; and Cape of Good Hope, 29^" W. Generally in Europe, Africa, and the Atlantic, the variation is- westerly, while in America and the Pacific it is easterly. \ " The system of Magnetic Meridians has undergone considerable changes in the times of modern accurate science. The southern point of Africa received from the Portuguese voyagers in the fifteenth century the name of L'Agulhas (the needle), because the direction of the compass needle, or the Local Magnetic ILeridian, coincided with the Geographical Meridian : it now makes with it an angle of about 30° W. In the sixteenth century, the compass needle in Britain pointed East of North : it I'oints from 20° to 30° (in different parts of the British Isles) West of North. At the present time, a change of the opposite character is goins^ on: in 1819 the westerly declination at Greenwich was about 24'' 23', which was probably its maximum ; in the last 30 years it has diminished from 23^° to 20° nearly. It is believed that the magnetic poles are rotating round the geographical poles from East to West." — A Treatise on Magnetism, designed for the use of Students in the University. By George Biddell Airy, M.A., L.L.D., D.O.L. Correcting Cotirses. 123 It also changes slightly at diflFerent times of the day.* Its value for each locality is indicated on charts, and always to be found by easy methods. 195. Variation is one of the "corrections" in deducing the true course and bearing from the course and bearing observed with the compass. It is given on the charts used in navigation. The method of correcting Compass Courses or Bearings for Variation will be readily understood by means of an example. Suppose the variation of the compass is found to be two points East — that is, the neeJle is directed two points to the right of the North point of the heavens — that is, points N.N.E. instead of N. ; then the N.N.W. point of the compass card will evidently point to the true north, and every other point on the card will be shifted round two points. If, therefore, a ship is sailing bi/ compass N.N.W., or, as it is usually expressed, hLr compass course is N.N.W., her true course will be North ; that is, two points to the right of the compass course. In a similar manner it may be shown that when the variation is two points westerly, the true course will be two points to the left of the compass course. 196. To find the true com'se, the compass course being given. EULE XLIX. Allow easterly variation to the right of the compass course. ,, westerly ,, left ,, looking from the centre of the card over the point to be corrected.] Examples. Taking the courses between North and South round by E. Ex. I. Course steered N.E. by E., vari- ation 2| points West, to find the true course. Here tiie compass course is X. 5 points E., and the variation is westerly, and hence must be applied to the left, thereby bringing it 2J points nearer to the North (X. 5 E — 2:| = N. 2.| E.), that is, within 2\ points of North : the true course is therefore N.N.E. Ex. 2. Course steered the same, viz., N.E. by E., variation if points East. Here the compass course is N. 5 points E., and the variation easterly, and hence must be applied to the right, thereby carrying the course aicay from the JVorth townrds the East, that is, 6\ points to the eastward of North (N. 5 E. + if E. = N. 6J E.) ; the true course is therefore E. by N. J N. * Besides the gradual changes which occur in terrestrial magnetism, both as regards direction and intensity of force, in the course of long periods of time, there are minute fiuctu- ations continually traceable. To a certiin extent these are dependent on the varying positions of the sun, and, to a much smaller extent, of the moon, with respoct to the place of observation ; but over and above all regular and periodic changes, there is a large amount of irregular fluctuations, which occasionally bfcorno so great as to constitute what is called a mngxetic storm. These variations occur with great rapidity, causing deflections to the right and left comparable m their rate or period of alternation with ordinary telegraphic signalling; accidental variations of 70' have been observed. "Magnetic Storms " are not connected with thunder storms, or any other known disturbance of the atmosphere, but are invariably connected with exhibitions of aurora borealis, and with spontaneous galvanic currents in the telegraph wires, and this connection is found to bo so certain, that upon remarking the display of one of the three classes of phc-nomeua, wo can at once assert that the other two are observable (the aurora borealis sometimes not visible h( re, but certainly visible in a more northern latitude). t The learner must be careful to remember when correcting his courses that he is to suppose himself looking from the centre of the card over the point to be corrected. When he places the compass card before him, mistakes vory frequently occur in the application of the variation between the East and West points round by South ; thus, taking the compass with the North point placed before or from the observer, while an error could scarcely arise when correcting courses in the N.E. and N. W. quadrants, it would be diSercni, with the S.E. and S.W. quadrants, unles.s he bore in mind that in the latter instance the compass card should be placed before him, as if he were facing the South. From what has been said it will be seen that in correcting courses, the significance of Right on the face of a compass card is as tlie hands of a watch tnove over the dial, and Left the contrary direction. 1 24 Correcting Courses. Ex. 3. Course by compass N.N.E., vari- ation 2^ points West, the true course 2\ points to the left hand of N.N.E., or N. J W. Ex. 5. Compass course S.E., variation I5 points Eist, then the true course (allowinar the variation to the right) will be S.S.E. | E., or S. 2| points E. Ex. 7. Compass course East, variation 2 points West, then allowing 2 points to the left gives true course E.N.E. Now proceeding to the courses between North and South round by West. Ex. 9. Course by compass N.W. | W., variation 2 points West, then the tru£ course (allowing the variation to the left) will be W. by N. i N., or N. C\ points W. Ex. II. Again, compass course S.W. by S., variation 1^ points West, the trite course (allowing variation to the left) will be S. i W. Ex. 13. Compass course S.S.W., varia- tion 3^ West, then allowing 3^ W. to the left of S.S.W. gives S. by E. J E., or i\ points E. Ex. 15. But with compass course West, and variation 3 J West, then allowing 3^ points to the left of W., the true course is S.W. I W., or S. 4f points W. Ex. 4. Compass course S. by E. varia- tion 25 East, 2j points allowed to right of S. by E. is S. by W. I W., or S. 4 W. Ex. 6. But compass course S.E., varia- tion 2\ points West, then the true course (allowing the variation to the left) will be E. by S. \ S., or S. 6\ points E. Ex. 8. Compass course E., variation 2f points East, then the true course (allowing the variation to the right hand) is S.E. by E. iE. Ex. 10. Taking the same compass course, viz , N.W. \ W., when the variation is i| points East, the true course (allowing the variation to the right) will be N.W. by N., or N. 3 points W. Ex. 12. Compass course S.W. by S. (as before) variation if East, the true course (allowing the variation to the right) will be S.W. f W., or S. 4| points W. Ex. 14. Compass course W., variation 2J E., then the tru,e course (allowing 2^ points to the right) is N.W. by W. \ W., or N. 5I points W. Ex. 16. Compass course N.N.W. | W., variation 3^ points East, then 35 points to the right of N.N.W. \ W., is N. f E. 197. The learner should so familiarise himself with the compass card as to be able entirely to dispense with its use in correcting courses, and when he has acquired such knowledge, he will find the following rule serviceable, ia which the points of the compass are treated numerically. EULE L. 1°. Put down the points and quarter points which the compass course is to the right or left of North or South, marking them R. or L. accordingly. 2°. Underneath put the variation, marhing it also R. or L., accordingly as it «s E. or W. l^. If th0 names are alike, take the sum, with that name, for the true course. (a) When the sum amounts to 8 points, it is either E. or W. (b) When the sum exceeds 8 points, take it from 1 6 points ; the remainder is the true course to he reckoned from the opposite point to that which the com- pass course is reckoned from . That is, it is to be reckoned from the North if it had previously been reckoned from S., but murked S. if previously marked N. ; also, if marked L (left) change to R (right) ; but if marked R change to L. 4°. If the names are unlike, take the difPerence, and mark it the same name as the greater. Correcting Courses. »^5 (c) I/the variation, heitig subtractive, exceeds the amount from which it is to be subtracted, take the points of the cowse from the variation, and name it the course towards West if it had previously been Easterly, but towards the East if it had been Westerly. (d) Also bear in mind that o points is either North or South as the case may be. The following are examples of this method of applying the variation, and the numbers and letters in brackets refer to the rule as given above : — I. Compass Courses :— S.S.W. ; N. by E. ^ E. ; W.S.W. ; and E. by N. Variation 3^ points Easterly. Required the True Courses. S.S.W. S.S.W. = 2 R. ofS. Var. 3^R.[3l Sura 5^ R. of S 8.W. by W. \ W Here the sum is taken for true course, the names bei; alike. N. by E. \ E. i\ R. of N. 34 R. [3°] N.E. by E. Here the names being alike the sum is taken. No. 3°). W.S.W. 6 R. ofS. 3^R- [b] — 9^ R. of S. 16 6| L. of N. W. by N. \ N. E. by N. 7 R. ofN. _34R- [b] — loj R. of N. 16 5iL.ofS. S.E. by E. \ E. 2. Compass Courses : — N.N."\^^ S. by E. ; W. 5 N. ; and E. by S. Variation 2\ W. N.N.W. 2 L. ofN. ^L. [3°] 4^ L. of N. N.W. \ W. S. by E. I L. of S. ^L- [3°] 3^ L. of S. S.E. J S. W. *N. E. by S. 7|L.ofN. ^^L. [b] 7 L. of S. 2iL. [b] 10 L. ofN. 16 — 9|L. ofS. 16 6 R. of S. 6^ R. of N. W.S.W. E. by N. 4 N. 3. Compass Courses:— N.E. \ E. ; S.W. f W. ; N. by E. ; and S. by W. Variation 2^ points West. N.E. \ E. S.W. f W. N. by E. S. by W. 4* R. of N. 2|L. [4°] 4f R. of S. 2|l. [4°] I R. OfN. 2iL. [c] I R. ofS. 2iL. [c] 2iR. ofN. 2^ R. of S. i^L. ofN. liL. ofS. N.N.E. i E. S.S.W. \ w. N. by W. i W. S. by E. i E. 4. Compass Courses :— N.W. by W.; S.E. by E. ; N. by W. Variation i\ points East. W. ; and S. by E. N.W. by W. S.E. by E. N. by W. \ W. 8. by E. 5 L. ofN. 3iR- M 5 L. ofS. 3iR. [4°] I* L. of N. 3iR. [c] r L. ofS. 3iR- [c] if L. OfN. if L. of S. i| R. of N. 2IR. ofS. N. by W. 1 W. 8. by E. i E. N. by E. i E. S.S.W. i W. 126 Correeting Counes. 5. N.N.E., Variation 2 points W. ; S. by E., Variation i point E. ; W. by S., Variation I point E. ; and E.S.E., Vnriatioa 2 points W. N.N.E., Var. 2 W. S. by B., Var. i E. W. by S., Var. i E. E.S.E., Var. 2 W. 2 R. of N. I L. of S. 7 R. of S. 6 L. of S. 2 L. I R. I R. 2 L. o [d] o [d] 8 R. of S. [a] 8 L. of S. [a] N. S. W. E. 6. North, Variation 2 points E. ; South, Variation 2 points W. ; West, Variation 2 points W. ; and East, Variation 2 points E. N., Var. 2 E. S., Var. 2 W. W., Var. 2 W. E., Var. 2 E. o = N. o = S. 8 R. of S. 8 L. of S. 2 R. of N. 2 L. of S. 2 L. 2 R. 2 R. of N. [3°] 2 L. of S. [3°] 6 R. of S. [4°] 6 L. of S. [4°] N.N.E. S.S.E. W.S.W. E.S.E. 198. If the learner has carefully gone through the preceding examples, he will have noticed that JEasterly variation in its application to Compass Courses increases them in the N.E. and S.W. quarters of the compass, and decreases them in the N. W. and S.E. quarters. Westerly variation decreases the courses in the N.E. and S. W. quadrants, and increases it in the N. W. and S.E. ; we have, therefore, EULE LI. Westerly variation is — from all points between N. and E S. and W. Easterly variation is + to all points between N. and E S. and W. Westerly variation is + to all points between N. and W S. and E. Easterly variation is — from all points between N. and W S. and E. We shall now proceed to illustrate the foregoing rule, which is very generally used in the correcting of courses. 1. Compass Courses :— N.N.E. ; S. by W. \ W. ; W.N.W. ; S.E. i E. Var. 3^ E. N. 2 E. S. I* W. N. 6 W. S. 4| E. + 3i E. + 3I E. - 3^ E. - 3i E. N. 5^ E. S. 4f W. N. 2f W. S. i^ E. N.E. by E. I E. S.W. f W. N.N.W. f W. S. by E. i E. 2. Compass Courses :— E.N.E. ; W. by S. ; N.N. W. ; S. by E. Var. 3^ E. N. 6 E. S. 7 W. N. 2 W. S. I E. + 3i E. + 3i E. - 3i E. - 3i E. N. 9i E. S. loi W. N. il E. S. 2J W. 16 16 — — — N. by E. \ E. S.S.W. J W. S. 6| E. N. 5| W. E. by S. \ S. N.W. by W. | W. 3. Compass Courses :— N.E. ; S.W. \ S. ; N.W. | N. ; S.E. \ S. Var. 2^ W. N. 4 E. S. 2,1 W. N. z\ W. S. 3| E. _ 2^ W. — 2t W. + 2? W. + 2^ W. N. ij E. 8. i\ W. N. 5I W. S. 6 E. N. by E. I E. S. by W. ^ W. Correcting Courses. ttj 4. Compass Courses : -N. by E. ; S. by w. k W.; W. ^N.; ; E. by S. Var. 2iW. N. I E. — 2iW. S. ik w. — 2^W. N. 7^ W. + 2iW. S. 7 E. -i-24W. N. if W. S. I E. N.^W. 16 S. 9I E. 16 8. 61 W. N, 6J E. 5. Compass Courses :— N.N.W. | W. ; S.S.E. f E. ; N.E. by E. J E. ; S.W. by W . | W. Variation zf E. N. 2f W. S. 2f E. N. 5i E. S. si W. — 2I E. — 2I E. + 2| E. + 2I E. o o N. 8 E. S, 8 W. North. South. East. West. 199. Sometimes it may be desirable to express the Variation in degrees, in which case we proceed as follows : — EULE LII. 1°. Correct the compass courses for leeway as hefore directed, and convert the numher of points thus found into degrees, marking them R or L, according as they are right or left of N. or S. 2°. Underneath write the variation, marking it E or L, according as it is E. or W. Take the sum with the common name, if the names are alike, and the difference with the name of the greater, if the nam s are unlike. The result will be the number of degrees the true course is from N. or S. according as the course, corrected for leeway, is reckoned from the N. or S. (a) If, in taking the sum the numher of degrees exceed 90° take the supplement to 1 80°, and reckon the true course from the opposite point to that from which the course corrected for leeway is reckoned; also change the letter E or L. Examples. Compass Course. S.W. \ s. N. by E. W.JN. Winds. Leeway. W. by N. ^ N. f E. by N. 3j S.W. by S. ■ I Var. 23° W. 20° E. 25^ W. True Course. S. 8»W. N. 5° W. S. 850 w. S.W. i S. = 3| R. of S N. by E. = I pt. R. ofN. W. iN. = 7^L.o: 35 R. of S. Leeway | L. I R. ofN. 3iL. 7iL. OfN. I R. 2|R. 2iL. ei or 31° R. of S. Var. 23 L. or 25° L. of N. 20 or 70° L. 25 L. 8 R. of S. S. 8° VV. 5 L. ofN. N. 5° W. — 95 L. ofN. 180 85 R. ofS. S. 85° W. iz8 Correcting Courses. EULE LIII. To find the compass course, the true course and variation being given. Easterly variation is allowed to the left. Westerly „ ,, right. Examples. Taking the courses between North and South round by East. Ex. I. Let the true course be N.E. by E., where the variation is i|: points West, the compass course (allowing westerly variation to the right) will be E.N.E. \ E. Ex. 3. Suppose the true course to be S.E. by E., where the variation is 2J points West, the compass course (allowing variation to the right) will be S.S.E. J E. Taking the courses between North and South round by West. Ex. 5. Let the true course be N.W. by W., where the variation is 2J points West, then the oompass course (allowing westerly variation to the right) will be N.N.W. J W. Ex. 7. With the true course West, and the variation 2 points West, then the compass course (allowing 2 points to the right) will be W.N.W. Ex. 9. With the true course S.W. | S., where the variation is i\ points West, then the compass course (allowing westerly varia- tion to the right) will be S.W. by W. I W. Ex. 2. Taking the same course, viz. N.E. by E., where the variation is i\ points East, and then the compass course (allowing easterly variation to the left) will be N.E. i N. Ex. 4. But the same course, viz., S.E. by S., where the variation is 2^ points easterly, will give the compass course (allow- ing easterly variation to the left) S.E. by E. I E. Ex. 6. Suppose the course to be the same, viz., N.W. by W., where the varia- tion is i\ points easterly, the compass course (allowing easterly variation to the left) will be W. \ N. Ex. 8. Taking the same course West, suppose the variation (o be 2 points East, then the compass course (allowing 2 points to the left) is W.S.W. Ex. 10. But with the same course, viz., S.W. I S., where the variation is i\ points East, the compass course (allowing easterly variation to the left) will be S.S.W. 201. Treating the points of the compass numerically, we proceed according to the following EULE LIV. Proceed according to Eule L, page 1 24, in every particular^ except that the variation is to be allowed the opposite way to that of correcting compass courses, viz., Westerly variation is to be allowed to the right and marked E, and Easterly variation is to be allowed to the left and marked L. Examples. I. True Courses: —N.N.E. ; S. by W. | W. ; E. by N. \ N. W. by S. Var. 2^ W N.N.E. S. by W. \ W. E. by N. ^ N. W. by S. 2 R. ofN. a^R. il R. of S. 6^ R. of N. 2^ R. 2| R. 7 R. ofS. 2iR. 4^ R. of N. N.E. \ E. 4 R. of S. 9 R. of N. — 16 S.W. 7 L. ofS. 9^ R. of S. 16 6i L. of N. B. by S. W . by N. J N. Correcting Courses. 129 2. True Courses :-N.E. by E. | E. ; S.W. by W. ; E. by S. J S. ; N.W. by W. Variation 3I E. N.E. by E. J E. S.W. by W. E. by S. J S. N.W. by W. 5i R. of N. 5 R. of S. 6| L. of S. 5 L. of N. 3iL. 3^L. sIl. 3iL. 2 R. ofN. ifR. ofS. 10 L. ofS. 8iL. ofN. — — 16 16 N.N.E. S. byW. |W. — — 6 R. ofN. 7iR. ofS. E.N.E. W. i S. 3. True Courses:— N, by W. | W. and S. by E.; Variation 3^ W. S. by W. and N. by E. ; Variation 3I E. N. byW. JW. S. byE. S. by W. N. by E. I* L. of N. I L. of S. I R. of S. I R. of N. 3|R. 3iR- 3fL- 3|L. if R. ofN. 2iR. ofS. 2|L. ofS. 2fL. ofN. N.byE. |E. S.S.W. fW. S.S.E. | E. N.N.W. | W. 202. To convert true course into compass course, we may proceed accord- ing to the following EULE LV. Westerly variation is + to all points between N. and E S. and W. Easterly variation is — from all points between N. and E S. and W. Westerly variation is — from all points between N. and W S. and E. Easterly variation is + to all points between N, and W S. and E. 130 DEVIATION OF THE COMPASS. GENEEAL STATEMENT OF EACTS AND LAWS OF MAGNETISM. 203. Magnets, Natural and Artificial. — Natural magnets, or loadstones^ are exceedingly rare, although a closely allied ore of iron, capable of being strongly acted upon by magnetic forces, and hence called magnetic iron ore, is found in large quantities in Sweden and elsewhere. Artificial magnets are usually pieces of steel which have been permanently endowed with magnetism by the action of other magnets. The needle, or bar of steel, in the Mariner's Compass is an artificial magnet. 204. Poles, Neutral Lines, and Axis. — The property of attracting iron is very unequally manifested at different points of the surface of a magnet. If, for example, an ordinary bar-magnet be plunged into iron filings, these become arranged round the ends of the bar in feathery tufts, which decrease towards the middle of the bar where there are none. The name poles is used, in a somewhat loose sense, to denote the two terminal portions of a magnet, or to denote two points, not very accurately defined, situated in these portions. The middle position to which these filings refuse to adhere is called the neutral line. Every magnet, whether natural or artificial, has two poles and a neutral line. The shortest line joining the two poles is termed the axis of the magnet. 205. Magnetic Meridian at any station is best defined as the direction of the declination needle. 206. The Magnetic Equator or aclinic line is the line which joins all those places of the earth where the needle remains quite horizontal, or where there is no dip. This line does not coincide with the geographical equator, nor is it a great circle, but a somewhat irregular curve crossing the geographical equator at two points almost exactly opposite each other, one near the West coast of Africa, in the Atlantic, and the other in the middle of the Pacific Ocean, and never receding from it further than 1 2° ; the position of the two being nearly coincident in that part of the Pacific where there are few islands, and most divergent when traversing the African and American continents. 207. Magnetic Poles. — At two points, or rather small linear spaces on the earth's surface, the needle assumes a position perpendicular to the horizon, or the dip is 90°. These two spots are called Magnetic Poles. At the North magnetic pole, the North pole of the needle dips; at the South magnetic pole, the south pole of the needle dips. The terrestrial magnetic poles do not coincide with the geographical ones, nor are these points diametrically opposite. The position of these poles are latitude 70' N., long. 97° W., and lat. 73^° S., long. 147° E. The line of no variation passes through these poles, and the lines of equal variation converge towards them. Deviation of the Compass. l^i 208. Magnetic Elements. — A. knowledge of terrestrial magnetism implies a knowledge of (i) Declination or Variation, (2) Inclination, (3) Intensity. These are called the magnetic elements of the place at which they are observed. 209. Magnetic Needle. — Magnetic Declination or Variation. — Any magnet freely suspended near its centre is usually called a magnetic needle, or more properly a itiagnetised needle. When a magnetised needle is bo suspended or mounted that it can vibrate in the horizontal plane, it will take a definite direction, to which it always comes back after displacement. In this position of stable equilibrium, one of its ends points to the direction called magnetic north, and the other magnetic south, which differ, in general, by several degrees from geographical (or true) north and south. This is the principle on which compasses are constructed. The angle between the mag- netic meridian and the geographical meridian is called the Variation of the Compass. Imaginary lines on the surface of the eartli, passing through all points where the needle points due north and south, are called Lines of no Variation, and lines passing through all points where the needle is deflected from the geographical meridian are called Lines of Equal Variation. These are extremely irregular curves, and form two closed systems surrounding two points, which may be called Centres of Variation. One of these points is in Eastern Siberia, the other in the Pacific Ocean, in the vicinity of the Marquesas. 2 1 o. Lip, or Inclination. — When a needle is prepared in the unmagnetised state for mounting in a compass, with its centre of gravity very little below its point of support, and is adjusted to horizontality, on being magnetised it will place itself in a particular vertical plane called the magnetic meridian, and will take a particular direction in that plane. This direction is not hori- zontal, except at the equatorial regions of the earth, but inclined generally at a considerable angle to the horizon, and this angle is called dip, or inclination. Its value at Q-reenwich, at present, is about 67°, the end which points to the north pointing at the same time downwards. In the northern hemisphere, generally, it is the north end of the needle which dips, and in the southern hemisphere it is the end which points south. The value of the dip, like that of the variation, difi'ers in difi'erent localities. It is greatest in the polar regions, and decreases with the latitude to the equator, where it is approximately zero. Dip, like the variation, varies greatly, not only from place to place, but also from time to time. In 1843 the dip at Q-reenwich was about 69° i', it has diminished, with a rate continually accelerating, till in 1868 it was 67° 56'. It is also subject to slight annual and diurnal variations, being about 15' greater in summer than in winter. Intermediate to the poles and equator lines are drawn through all points where the needle makes the same angle with the horizon. These are called Lines of Equal Inclination or Dip. T32 Bmation of the Compass. To help the seaman to understand the above remarks, let him proceed as follows : — Having provided a little unspun silk, by means of a bit of wax, or otherwise, attach the silk fibre to the magnetic needle by a single point at its middle. Place a magnet on the table, and hold the needle over the equator of the magnet. The needle sets horizontal. Move it towards the north end of the magnet, the south end of the needle dips, the dip augmenting as the north pole is approached, over which the needle, if free to move, will set itself exactly vertical. Move it back to the centre, it resumes horizontality ; l^ass it towards the south pole, its north end now dips, and directly over the south pole the needle becomes vertical, its north end being now turned down- ward. Thus we learn that on one side of the magnetic equator the north end of the needle dips ; on the other side the south end dips, the dip varying from nothing to ninety degrees. If we go to the equatorial regions of the earth with a suitably suspended needle, we shall find the position of the needle horizontal. If we sail north, one end of the needle dips ; if we sail south, the opposite end dips ; and over the north or south terrestrial magnetic pole the needle sets vertical. In this manner we establish a complete parallelism between the action of the earth and that of an ordinary magnet. Note.— The dip is of importance to the navigator, as it appears to regulate the local deviation of the compass. It also renders necessary an adjustment to secure the horizon- tality of the compass card. 2 11. The horizontal position of the needle and card is preserved by a sliding brass weight fitted for the purpose, or by dropping sealing wax on one end of the needle. This adjustment will often require to be repeated after a considerable change of place. 212. Mutual Action of Poles. — On presenting one end of a magnet to one end of a needle thus balanced, we obtain either repulsion or attraction, according as the pole which is presented is similar or dissimilar to that to which it is presented. Poles of contrary names attract one another ; poles of the same name repel one another. This property furnishes the means of distinguishing a body which is merely magnetic (that is, capable of temporary magnetization) from a permanent magnet. The former, a piece of soft iron, for example, is always attracted by either pole of a permanent magnet, while a body which has received permanent magnetization has, in ordinary cases, two poles, of which one is attracted where the other is repelled. Magnetic attractions and repulsions are exerted without modification through any body which may be interposed, provided it be not magnetic. 213. Names of Poles. — The phenomena of variation and dip above described evidently require us to regard the earth, in a broad sense, as a magnet, having one pole in the northern and the other in the southern hemisphere. Now, since poles which attract one another are dissimilar, it follows that the magnetic pole of the earth, which is situated in the northern hemisphere, is dissimilar to that end of a magnetised needle which points to the north. Hence, great confusion of nomenclature has arisen, the usage of the best Deviation of the Con^ase. 133 writers being opposite to that which generally prevails. Popular usage in this country, however, calls that end or pole of a needle which points to the north the north pole, and that which points to the south the south pole* 214. Magnetic Induction. — When a piece of iron is in contact with a magnet, or even when a magnet is simply brought near it, it becomes itself, for the time, a magnet with two poles and a neutral portion between them. If we scatter filings over the iron they will adhere to its ends, as shown (204). If we take away the influencing magnet the filings will fall off, and the iron will retain either no traces at all, or only very faint ones of its magnetization. If we apply similar treatment to a piece of steel, we obtain a result similar in some respects, but with very important differences in degree. The steel, while under the influence of the magnet, exhibits much weaker efi'ects than the iron ; it is much more difiicult to magnetise than iron, and does not admit of being so powerfully magnetised ; but, on the other hand, it retains its magnetization after the influencing magnet has been withdrawn. This property of retaining magnetism, when once imparted, has been named coercive force. Steel, especially when very liard, possesses great coercive force ; iron, especially when very pure and soft, scarcely any. In magnetization by influence, which is also called magnetic induction, it will be found on examination that the pole which is next the inducing pole is of contrary name to it ; and it is on account of the mutual attraction of dissimilar poles that the iron is attracted by the magnet. The iron can in its turn support a second piece of iron, this again can support a third, and so on through many steps. A magnetic chain can thus be formed, having two poles. An action of this kind takes place in the clusters of filings which attach themselves to one end of a magnetised bar, these clusters being composed of numerous chains of filings. 215. Magnetization by the action of the Earth. — The action of the earth on magnetic substances resembles that of a huge permanent magnet, and hence the terrestrial magnetism will induce magnetism precisely as explained in 214. All soft or cast iron rods or bars, or other elongated forms of soft or cast iron, unless the position of their length is at a right-angle to the line of the direction of the earth's magnetic force, are immediately rendered mag- netic by induction from the earth, and the nearer the iron is in direction to the line of force or dip the greater will be the amount of induction. When a bar of soft iron is held on the magnetic meridian and parallel to the dip, it becomes immediately endowed with feeble magnetic polarity. The lower * Sir W. Thomson calls the north-seeking pole the south pole, and the other the north pole, because the former is similar to the south and the latter to the north pole of the earth. In like manner most French writers call the north-iet Icing pole of a needle the austral, and the other the boreal pole. Fakaday, to avoid the ambiguity which has attached itself to the names north and south pole, calls the north-seeking end the marked, and the other the un- marked pole. AiRT, for a similar reason, employs in his recent Treatise on Magnetism, the distinctive names red and bhte to denote respectively the north-seeking and south-seekin"- ends ; these names, as woil as those employed by Fakaday, being purely conventional, and founded on the custom of marking the north-seeking end of a magnet with a transverse notch or a spot of red paint. Maxwell and Jenkin, in a report to the British A»sociation, call the south-seeking pole of a needle positive, and the north-seeking pole negative. 134 Deviation of the Compass. extremity is a north pole, and if the north pole of a small magnetic needle be approached, it will be repelled. If the bar is held vertically the lower end will still be a north pole, but of less intensity ; the upper end a south pole, also of leas intensity. If the bar is held horizontally north and south, the north end will be a north pole, but of still lesser intensity ; the south end a south pole, also of lesser intensity. If we now turn the bar in the same horizontal plane its magnetism will diminish, and if placed in an east and west direction, it will lose its polarity, and if we turn it still further until its position is reversed, the magnetic poles of the bar will be reversed. While the bar is held with its length in the direction of the dip, if it be struck repeatedly with an iron hammer, it will be found, on removing it, to be a true magnet, the end which was lowest being charged with north mag- netism, and this magnetism is not transient like the induced magnetism of soft iron, changing its place in the bar with every change in the position of the bar, but is constant like that of a steel bar, retaining the same mag- netism whatever the position of the bar. By reversing the position of the bar and striking it a few blows with the hammer, its magnetism is reversed. The magnetism of the bar so struck resembles that of a steel magnet in all respects but this, that while, perhaps, no change can be remarked in hours or days, it infallably diminishes in a long time. To express this partially permanent character, the term Subpermanent Magnetism has been adopted. 216. A sphere of soft iron will be magnetised in the same way, however held. The diameter in the line of dip will be the axis of magnetism, and the lower and north half of the surface will be north, the upper and south half south. In bodies of any other shape the effects will be similar. In an iron ship on the stocks, intense magnetism is developed by the pro- cess of hammering ; N. magnetism being developed in the part of a ship which is below and towards the north, and S. magnetism in the part which is above and towards the south. 217. In the northern hemisphere all vertical or upright bars, such as stanchions and angle-irons composing the frames of ships, are magnetised by induction, their lower ends being north poles, the upper ends south poles, the upper end attracting the north pole of the needle held near them. On the other hand, in the southern hemisphere, these conditions are reversed ; the upper ends of vertical iron are north poles, repelling the north pole of a compass needle and attracting the south pole. On the magnetic equator, where there is no dip, vertical soft iron has no polarity, because its position is at right-angles to the earth's line of force or dip. It is different with hori- zontal pieces of soft iron ; they exert the same influence on a compass needle in both hemispheres, and in all latitudes. 218. The hull of an iron ship acts as a permanent magnet on compasses placed outside the vessel as well as those placed inside ; an iron ship must therefore be viewed in its effect on a properly placed magnet rather as one great magnet, than as an aggregation of smaller magnets. Deviation of the Compass. 135 Keeping in view that the inductive effect from the earth's magnetism is greatest in the line of the dip, and the existence of a neutral equatorial plane at right-angles to the line of dip in spherical bodies, we are prepared to see that each iron ship must have a distinct distribution of magnetism depending on the place of building, and the direction of the head and keel while building; the ship's polar axis and equatorial plane conforming more or less to the line of dip of the earth at the place where built, and a plane at right-angles to that line ; abundant observation and experiment have proved this important general principle. 219. To illustrate this principle : let us suppose, as in the following Figs. 3, 4, 5, and 6, that four iron ships, or four composite-built ships, with ribs, beams, stanchions, and deck girders of iron, are building on the cardinal points of the compass, in a port in England where the line of the earth's total magnetic force is inclined 70° to the earth's horizontal magnetic force, or in other words, where the dip of the needle is 70°. Fig. 3 shows the magnetic state of a ship built head North magnetic. The line marked Dip passes through the centre of the ship ; it shows the direction of the line of the earth's magnetic force. The line marked Equa- torial or Neutral line is the line of no deviation, and runs at right-angles to the Dip. The after body of the ship, or the portion which is shaded, has Fig. 3. Head North while building. S. (hlue) polarity, and the fore body, or white portion of the figure, N. {red) polarity; the upper part of the stern would have the S. {Hue) polarity developed in a high degree ; the lower part of the bows would have the N. {red) polarity equally developed. At the stern the north end of a compass needle would be strongly attracted ; at the bow the south end of the needle would be strongly attracted ; while a compass placed outside of the ship's topsides, above the line of no deviation, the north end of the needle will be attracted ; if it be placed below that line the north end of the needle will be repelled and the south end attracted, in accordance with the law of magnetism. (No. 205.) 136 Deviation of the Compass. Fig. 4. Head South while building. Fig. 4 represents the magnetic condition of a ship built head south. It will be seen by comparing Fig. 4 with Fig. 3 that the conditions are reversed; in Fig. 3 the magnetism of the after body of the ship is soutb {blue), while in Fig. 4 the after part of the ship possesses north (red) polarity ; now the fore body of the ship has S. {blue) polarity, while in Fig. 3 it has N. {red) polarity ; the upper part of the bow has S. {blue) polarity developed in a high degree, and the lower part of the stern N. {red) polarity equally developed. The N. {red) polarity of the stern repels the north end of the compass needle, and the S. {blue) polartiy of the bow attracts it. The dotted line crossing the equa- torial line in Figs. 3 and 4 shows the probable position of the neutral line after the ship has been some time afloat, with her head in an opposite direc- tion to that in which she was built, or after she had made a voyage. The place of little or no deviation in a ship built head north is towards the bow, but in a ship built head south, towards the stern. Fig. 5. Head East while building. Deviaiion of the Comjjass. »37 Fig. 5 is intended to show the magnetic state of a ship built head East. The whole of the upper part of the ship has S. {blue) polarity ; the whole of the lower part has N. (red) polarity ; but the S. {blue) polarity predominates on the starboard side, and the north end of a compass needle, if carried at the usual height of a compass along the amidship line of the upper dock from end to end, is attracted to the starboard side. Fig;. 6, Head West while buildin"-. ^ 3 In Fig. 6, ship built head "West, the magnetic conditions of Fig. 5, head East, are reversed ; the whole of the upper part of the ship has still S. {blue) polarity, and the lower N. {red) polarity ; but tho magnetism of the port side of the upper works is developed in a higher degree than the starboard side, and the N. end of a compass needle, if carried along the upper deck from end to end, would be attracted to the port side. In other words, in these ships the whole of their decks have a S. {blue) polarity, yet in that part which was North while the ship was being built, this S. {blue) polarity is developed in a less degree than on the opposite side, consequently, the N. point of tho compass is drawn towards that part of the ship in which the S. {blue) polarity is developed in the highest degree. The deviation in both cases is rarely largo, but less regular than in ships built head South. 138 Deviation of the Compass. Theoretically, there should be no spot of no deviation on the deck of ships built East or West.* Ex. 7. Head North at Australia. Fig. 7 represents an iron ship built head North in Australia, with a dip of about 68° South. In this ship the shaded part showing S. polarity lies below the equatorial line. It will be useful to compare this figure with Fig. 3, and mark the difference in the magnetic state of the two ships. 220. A little attention to the above diagrams will give the seaman a rough idea of the distribution of magnetism in iron ships ; but it must be borne in mind that all large detached pieces of iron in a ship, such as iron masts, funnels, cylinders, and other masses of vertical iron, are independent magnets ; in north magnetic latitude, their lower ends being north poles, their upper ends south poles. 221. The compasses of composite ships with iron frames and iron deck beams, are affected in the same way as those of ships built wholly of iron. * From the special maptnetic properties developed in a ship according to her position when building, it follows that a compass aft, in the usual place of the steering binnacle, tho character of the deviation — though not the amount— ma.y be approximately represented in a tabular form, as follows : — Approximate maornetic direction Approximate easterly deviation occurs Maximum westerly deviation of ship's hiead while building, when ship's head hy compass is near when ship's head by compass is near N. W. E. N.E. N.W. S.E. E. N. S. S.B. N.E. S.W. S. E. W. S.W. S.B. N.W. W. S. N. N.W. S.W. N.E. Deviation of the Compass. 1 39 DEVIATION OF THE COMPASS. 222. The Deviation of the Compass is the angle through which the mag- netic needle is deflected from its natural position by the disturbing force of iron near it, that is, the angle included between the magnetic meridian and a plane passing through the poles of a compass needlo. The deviation is named East or West according as the north point of the compass so disturbed is to the east or west of its natural position. Deviation consists of two principal parts, the Semicircular and the Quad- rantal, following different laws, and requiring two different kinds of compen- sation ; there is sometimes a third part of small amount called the Constant. 223. In the case of iron ships, as in that of iron bars (215), percussion and vibration, by hammoriug in rivotting, render the iron of which the vessel is constructed more susceptible to the inductive force of the earth, and causes the magnetism, which the iron of the sliip thus acquires, to partake more of the character of permanent magnetism. Still this subpermanent magnetism undergoes a considerable diminution by being submitted to percussion, with the ship's head in a different position to that in which it was when she was being built, and especially if in a contrary direction. But the iron of which a ship is constructed always retains a large amount of this subpermanent magnetism as long as it remains in the form of a ship. The deviation arising from subpermanent magnetism is greater than that which is tho result of transient induced magnetism. The polarity of the ship's magnetism, while she remains on the stocks, takes the direction of the earth's line of force or dip, and its effects on compasses will evidently depend on the direction the ship's head was whilst being built. Taking the case of a ship built head north (Fig. 3, page 135), tho fore part of the ship has acquired north magnetism, and its action will be precisely the same as that of the north pole of a magnet ; hence, on northerly courses, the north end of the compass needle will be repelled, and tho directive power of the needle will be diminished. On southerly courses tho north end of the needle points towards the stern, which has acquired subpermanent south magnetism, then the directive power of tho needle is increased. On easterly and westerly courses the effects on the compass are greatest, since the force acts at right-angles to the needle ; and on all intermediate positions of the ship's head the dis- turbances due to such positions are intermediate. As the ship's head is brought east of north, repulsion of the north end of the needle takes place, and westerly deviation is the result, and it reaches its maximum value when the fore-and-aft lino of the ship is at right-angles to the needle ; beyond that position the fore part of the ship attracCs the south end of the needle, and westerly deviation is still the Tesult. This attraction continues imtil the ship's head reaches south, when the 1 ae of action of the ship lies in the same direction as the needle, and no disturbance occurs, but the directive power of the needle is greater. On bi 'Hging the ship''s head round west of south, the south polo of the needle still continues to be attracted, which causes easterly deviation, and it again attains its maximum when the fore- 140 Deviation of the Compa»s. and-aft line of the ship is at right-angles to the disturbed needle ; this must occur to the north of west. After that point has heen reached by the ship's head, the fore part of the ship repels the north end of the needle, easterly deviation still being the result until the ship's head is again at north. Thus we find that in an iron ship the disturbance of the compass is little or nothing when her head is on or near the points to which her head or stern were directed while building, and is greatest when the ship's head is directed to the points of the compass that were abeam while on the building slip ; and, moreover, that easterly deviation is caused when the ship's head is in one half of the compass, and westerly deviation in the other. The deviation caused ,by subpermanent magnetism, and the effects of magnetism induced in vertical iron, has received the name of Semicircular Deviation, from producing opposite effects when the ship's head is on opposite semicircles of the compass, as the ship's head moves round a complete circle of azimuth. This error is caused by the subpermanent magnetism acquired in building, and the mag- netism induced in vertical ii'On. The part due to subpermanent magnetism remains the same in hind, though different in amount, in all latitudes, unless the ship be subjected to strains or other mechanical violence. The part caused by the magnetism induced in vertical iron changes with a change of geographical position, or more correctlj', as the dip changes, and is of con- trary names on opposite sides of the magnetic equator, that is, if westerly deviation be produced on one side, easterly will be produced on the other. At the magnetic equator the earth's magnetism acts horizontally, and vertical soft iron will have no magnetism, and the semicircular deviation arising therefrom will disappear. As a general rule the magnetism producing semicircular deviation, in a ship built in north magnetic latitude, attracts the north end of a compass needle to that part of a ship which was south from the compass while build- ing ; hence, the semicircular deviation in iron ships is generally represented by the effect of a magnet at the part of the ship which was south in building, with its south end towards the compass. Thus, in a ship built head north, the north end of the needle is drawn towards the stern. The following table will show the part of a ship towards which the north end of a needle is generally drawn, that is, the position of the permanent south pole developed in the process of construction. Ship's head while building. '^^^ "^^^^^ ^^"^ f ^ ^J^ «o™P^«^ ^,'^f^ ^° the '^ ° poop or quarter deck is usually drawn. North . towards the stern. N.E ,, starboard quarter. East ,, starboard side. S.E ,, starboard bow. South ,, bows or right ahead. S.W ,, port bow. West ,, port side. N.W ,, port quarter. Deviation of the Com^iass. 141 SOUTM Fig. 8 will further illustrate the way in which the permanent magnetism, and the inductive magnetism of vertical iron acts upon the compass to produce semicircular deviation. Lot it be supposed that the whole of the south polarity or attractive power of the above magnetism is concentrated in the point P on the port quarter of a ship built with her head near N.W. The ship is supposed to be swung round the compass, beginning at the-N.W. point. The small circles represent the compass, the thick lines N' S the compass needle, the dotted lines the magnetic meridian or the direction of the needle when free from deviation. Beginning at N.W., and noting the position of the point P, it will be observed that t"here can be no semicircular deviation i44 Deviation of the Compass. with ship's head in that direction, because the attractive force of the ship's magnetism at the point P is in a lino with the compass needle N S. As the ship's head swings round tow'ards the west, the relative positions of the point P and the compass needle will alter, and P will exert a pulling force upon the north end of the needle, causing it to deviate to the right from N to N', shown in the figure at "West. The easterly deviation will increase until the ship's head swings to S.W., where it attains its greatest or maximum amount. After passing S.W. it gradually decreases past South until the ship's head reaches S.E., the opposite direction to that in which her head was built, where it is again zero or nothing. The point P is now on the opposite side of the compass to what it was when her head was at N.W., but it will be observed that it is in a line with the needle, and can exert no deviating influence over it. As the ship swings with her head towards the East, the needle will gradu- ally be drawn to the left hand until the westerly deviation attains its maximum at N.E. After passing N.E. the westerly deviation will decrease past North until the ship's head again reaches N.W., at which point there is no deviation. A very slight inspection of the figure will show that in the semicircle from N.W. round by the West to S.E., the deviation is easterly ; while in the semicircle, or half the compass, from S.E. round by the East, the deviation is westerly. The above is merely given for the sake of illustration, but it must be remembered that no two ships are alike in their influence on the compass, nor will the ship's magnetism have the same effect on two compasses placed on different parts of the deck. 224. QiUadrantal Deviation is so named from its being easterly and westerly, alternately, in the four quadrants as the ship moves round a complete circle of azimuth. It is caused by the transient or inductive mag- netism of horizontal soft iron, such as iron deck beams, the iron spindle of the wheel, &c. It is zero or nothing when a ship's head is near the North, South, East, or West points, and greatest on the quadrantal points. It is generally easterly in the N.E. and S.W. quadrants, and westerly in the N.W. and S.E. quadrants of the compass. Quadrantal deviation remains unchanged in all magnetic latitudes, and provided that the iron in the ship be of good quality, the quadrantal deviation will be little, if at all, altered by the lapse of time. To illustrate the way in which horizontal soft iron produces quadrantal deviation, let us suppose the whole of the induced magnetism in a ship to be represented by the soft iron bar B in Eig. 9. This cannot be so in actual practice, because the athwartship horizontal iron produces quadrantal devia- tion as well as the fore-and-aft iron, but we may suppose it may for the sake of clearness. The small circles represent the compass, the thick lines within the small circles the compass needle, the dotted lines within the compass the magnetic meridian. Beginning at north, it will be observed that the bar B is parallel with the magnetic meridian, and will therefore be an inductive magnet while it is in or near that position (215), its after end, marked S, being a south pole ; but as the bar B is in a line with the compass needle N, it cannot exert any deviating power upon the needle, either to the right or left. As the ship's head swings towards the N.W., the relative positions of Deviation of the Compass. H3 Fig. 9. Magnetic. North South the bar B and the needle N are altered, and the south end of the bar draws the north end of the needle to the left from N to N'. As the ship's head approaches the west, the bar B loses its polarity, and at west it is at right- angles to the magnetic meridian, and ceases to exert any influence on the compass. The ships head now swings towards the S.W., and the bar B, as it turns towards the south pole, again becomes an inductive magnet ; its after end being a north pole, and drawing the south end of the compass needle from S to S'. When the ship's head reaches south there is no quadrantal devia- tion, because the bar B is in a line with the compass needle. As her head swings towards the S.E., the needle is drawn from S to S', causing westerly deviation. At east there is no deviation, for the same reason that there was none at west. After passing east, the after end of the bar B becomes a south 144 Deviation of the Compass, pole, and draws the north, end of the needle to the right-hand in the N.E. quadrant. As the ship's head approaches the north, the quadrantal deviation gradually decreases until it becomes nothing at north. The reader will observe that the bar B in this case produces easterly deviation in the N.E. and S.W. quadrants, and westerly deviation in the N.W. and S.E. quadrants. Cases may arise where the deviation is westerly in the N.E. and S.W. quad- rants, but they are very rare. 225. The constant part of the deviation is generally very small, and is the same for every point of the compass, it often arises from defects in the compass itself. An error in the correct magnetic bearing of a distant object used to ascertain the deviation, will give an apparent constant deviation : for example, if the correct magnetic bearing of a lighthouse be S. 46*^ E., and the observer assumes it to be S. 44° E., and finds the deviation by it, there will be an error of 2° in the deviation thus found on every point of the compass ; or, in other words, the westerly deviation will be 2° less, and the easterly deviation will be 2° moro than it ought to be. When a ship is swung hurriedly, and her head is not allowed to remain for a minute or two on any point before observations are made, there is a temporary constant deviation produced ; and this temporary deviation is easterly when the ship is swung to the left, as from East to North, and is westerly when the ship is swung to the right, as from North to East. 226. Mechanical Compensation or Correction of the Compass by means of Magnets and Soft iron. — These adjustments were first proposed by Mr. Airy, the Astronomer Royal, and are now universal in the merchant service. 227. Correction of the Semicircular Deviation. — As this error is caused by magnetism induced in vertical iron, and by subpermanent magnetism acquired in building, it is very diificult in practice to separate the semicircular deviation caused by vertical iron from that caused by subpermanent magnetism. Could this be easily efi'ected, a natural inference would be to compensate the former by vertical soft iron, and the latter by a magnet, each compensator placed so as to produce opposite efiects to those of the ship. An arrangement of this kind was adapted to several ships by Mr. Rundell, the Secretary of the Liverpool Compass Committee, who placed a vertical iron bar before the compass, leading from the keel (when possible) to a point some height above the level of the card. The distance from the compass and height to which it must be carried to be determined by experiment, but a diificulty has been experienced in adjusting this method so as to ensure success. The order of proceeding for compensation is as follows : — The ship must be upright, or on an even beam, with all her iron stores on board, in the positions which they are intended to occupy while at sea. The position of the binnacle being decided on, draw a line upon the deck, fore-and-aft, through the centre of the place where the binnacle is to stand. Draw another line across the deck, at right-angles to the former, through the same centre. . . Bwiation of the Convpaa. 145 Provide two or more powerful magnets from 1 8 inches to 2 feet in length.* Let the ship's head be swung to the North or South, correct magnetic — either of these points will do. When the ship's head is steady at one of these points, observe whether there is any deviation ; if there is any, lay one of the magnets on the deck athwartship, with its centre exactly on the fore- and-aft line drawn on the deck at some distance from the binnacle ; move it gradually (not hurriedly) to or from the foot of the binnacle until the compass points correctly. The magnet may be placed either before or abaft the binnacle, whichever is most convenient, but its centre must always be over the fore-and-aft line drawn on the deck, and it must be kept at right-angles to the ship's keel. If the compass needle deviate to the left, the north end of the magnet must be placed to the left, and conversely. After the compass has been made to point correctly at either the north or south points, swing her head round to the east or west correct magnetic (either will do), and steady her head on one of these points. If there be any deviation, place the other magnet fore-and-aft, either on the port or starboard side of the binnacle, with its centre on the athwartship line drawn on the deck ; move it to or from the foot of the binnacle until the compass points correctly. If the ship was built with her head nearly north or south, two magnets may be required. These may either be placed one on each side of the compass, or both on the same side, as may be convenient ; if placed on the same side of the binnacle, lay them an inch apart, and under all circumstances parallel, but always similar ends (N. or S.) directed to the same part of the ship. The adjuster should be careful to see that the centre of the magnet is kept on the fore-and-aft line, so that one of the poles of the magnet be no nearer the binnacle than the other. It must be remembered that this correction will only hold good for a small rdnge of latitude, and while the ship's magnetism continues in the same state as when the correction was made. 228. Correction of the Q,uadrantal Deviation. — As the quadrantal devia- tion is caused by the action of horizontal soft iron, a natural inference is, that soft iron should be used for compensating this error, so placed as to cause opposite effects to those of the ship. The compensations for semicircular deviation being complete, the ship's head should be swung to one of the intercardinal points, N.E., N.W., S.E., or S.W., correct magnetic; and the binnacle being fitted with two small brass boxes, one on each side of and on a level with the compass ; if there is any deviation place a quantity of small iron chain in the boxes until the compass points correctly. For greater certainty swing the ship to each of the other quadrantal points. * Compensating magnets should be from lo to i8 or 24 inches in length, their breadth one- tenth of their length, and their thickness one-fourth their breadth. 146 Deviation of the Compass. For the correction of large quadrantal deviations, cast iron correctors from 9 to 1 2 inches long, and 3 to 3 J inches diameter, are preferred by the Liver- pool Compass Committee. The ship's head is to be steadied on one of the quadrantal points, and the correctors, one on each side of the compass, and on the same level as the needle, are to be moved to or from the compass until the quadrantal deviation is corrected, but not nearer than ij times the length of the needles from the centre of the card. It is only in very rare instances that the deviation is westerly on the N.E. and S.W. points, or easterly on the S.E. and N.W. ; but if so, the correctors or chain-boxes are required to be placed on the fore- and-aft ends of the binnacle. The adjustment for the quadrantal deviation should always be made, as it tends to reduce the heeling error.* In some cases there is a small amount of quadrantal deviation produced by horizontal soft iron running from the quarters to the opposite bows ; iron in this position produces a quadrantal deviation, which is greatest when the ship's head is at N., S., E., and W., and least with the ship's head at N.E., S.E., S.W., and N.W. ; it is, however, generally of so small amount that it may, in ordinary cases, be disregarded. Although the correctors will compensate for a greater amount of deviation than the chain-boxes, they have in many instances been found to be insufficient, especially in cases of ships with iron decks. Heeling Error. — Although a ship's compasses may be corrected by the above methods, they can only be depended upon so long as she remains upright. Besides the ordinary deviation of the compass there is a deviation caused by the heeling of iron ships, which may increase or decrease the deviation observed when the ship is upright. Cases have been observed in which the deviation from heeling has amounted to as much as two degrees for each degree of heel of the ship, that is, without altering the real direction of the ship's head, the apparent alteration in direction has amounted to 40° by heeling the ship from 10° to starboard to 10° to port. The effect is very serious in those parts where the wind is steady, and the ship inclined in the same direction for many days or weeks in succession.! 229. To ascertain the amount of Heel. — The instrument specially adapted to indicate the amount of heel is the clinometer. It consists of a brass semi- circle, graduated at the edge to degrees, beginning at the middle of the arc and continued both ways ; and to the centre a plumb line is attached. The instrument is fixed at right-angles to a fore-and-aft section of the ship, as a beam, or athwartship bulkhead, with the diameter placed upwards, and parallel to the deck. When the index points to 0, the vessel is upright, but * These correctors are too frequently absent, and it should be remembered that they very essentially improve the action of the compass — not only diminishing the deviation, but increasing; the directive force. f- " Usually, in an iron ship, when her head is placed north or south, the ship's inclination through an angle of n degrees disturbs the compass through an angle of n degrees ; but in some particular instances it has been known to disturb the compass as much as 2 w degrees. — A Treatise on Navigation, by G. B. Airy, M.A., LL.D., D.L., page 182. Deviation of the Compass. 147 when she heels either way, the plumb line being free to move on its centre is always vertical, and the point at which it cuts the graduated edge shows the number of degrees that the vessel deviates from the perpendicular, that is, the heel of the ship. A compass card with the needle detached will answer the purpose, aud au index may be made with a thread and plummet depending from the end. 230. How the Deviation from Heeling is caused. — The heeling error depends partly ou vertical induction in transverse iron, and partly on vertical force arising from subpermanent magnetism in the ship, combined with that from vertical induction in vertical soft iron. The fore-and-aft iron is not disturbed from its horizontal position by heeling, consequently the athwart- ship beams then produce their full influence in disturbing the compass. When an iron ship heels over, forces, which before acted vertically, and did not disturb the horizontal compass needle, now act to one side and produce deviation ; while transverse iron which was previously horizontal, becoming inclined, acquires magnetism by induction (215). In north magnetic latitude the upper or weather ends of athwartship beams, for example, become south poles, and the lower ends north poles ; hence, from both these causes, the north end of the needle is drawn to windward. But, if the iron does not extend entirely across, as when a sky-light or hatchway is fitted, the opposite effects are produced ; for then the end of the iron nearest the compass on the weather side is a north pole, and that nearest it on the lee side a south pole j and under these conditions the north end of the needle is drawn to leeward. In vertical iron the force acting on the needle is no longer directly under it, but is shifted to the weather side of the ship, and thus in north magnetic latitude, as a general rule, the tendency of both horizontal and vertical iron is to draw the north end of the needle to windward. The vertical action of subpermanent magnetism modifies the result of these causes, and may either cause an increase or a diminution of the error so produced. If a ship has acquired subpermanent magnetism by having been built with her head north, there is a strong vertical force acting downwards (see Fig. 3, page 135) from the whole after body of tlie ship having south magnetism or polarity ; this would conspire with the vertical induction in transverse iron, in attracting the north end of the needle to the weather side, as the ship heels over, and thereby increasing the change of deviation from other causes. On the other hand, if a ship be built with her head south, the vertical force acts upwards (see Fig. 4, page 1 36), the after part of the ship has acquired north mag- netism, or polarity, and the north end of the needle, as the ship heels over, is repelled by it to the lee side, the vertical force acting in antagonism, in this case, to the transverse force, thus decreasing the error caused by soft iron. Thus is shown why in England the deviation of ships built there, with their heads northerly, are most affected by heeling. In the ordinary position of the compass on the quarter-deck, we may, in most cases, if we know the direction in which the ship's head was built, anticipate the direction of the heeling error, and form an approximate estimate of its amount. Ships built with theii- heads from about S.W. to 148 Deviation of the Compass. S.E. by way of north, the upper parts have south polarity ; and in those of this group built with their heads from N.W. to N.E., this south polarity is strongly developed near the position of the compass. In all these ships the north end of the compass needle will be drawn to windward, and forcibly so in the last named group. In the ships built with their heads from about S.W. to S.E. by way of south, their upper parts near the position of the compass have N. polarity, and hence the heeling error may be to leeward or to windward — and in either case small in amount — according as the vertical force, or force from transverse iron predominates. 2 3 T . Position of ship's head for greatest and least change of Deviation from Heeling. — There appears to be no deviation from heeling when the ship's head by compass is east or west, but it increases as the ship's head is moved from these points, and is greatest when the ship's head by disturbed compass is north or south. When the ship's head by compass is either east or west, the disturbing force, from the ship's heeling, acting at right-angles to the fore-and-aft midship line, tends to bring the needle into the magnetic meridian, and consequently no change in deviation can be produced from heeling. On the other hand, when the ship's head by compass is either north or south, the disturbing force acts at right-angles to the needle ; hence the greatest change of deviation resulting from a vessel's heeling takes place when her fore-and-aft line is in the magnetic meridian. 232. In north latitude, in ships built with their heads to th© north, with their compasses in the usual position, the deviation from heeling is much larger than in ships built with head to the south. In north latitude the north end of the needle is generally drawn towards the weather side of a ship, yet a small deviation to leeward has also been observed in north latitude, in some ships which were built in a southerly direction. In high south latitudes, where the dip is south, the north end of the needle has been observed to deviate to leeward. Compasses which are least affected by heeling in the northern hemisphere have generally the greatest amount when south of the equator, and vice versa.* * This should be particularly considered by masters of iron ships about to proceed to a port south of the equator. The heeling error not only causes deviation, but also unsteadiness, and in some cases when a ship has rolled with her head north or south by compass, the card has spun, and this has rendered it impossible to keep the ship on any required point. It is not, however, always found that compasses having the largest heeling error are most productive of this incon- venience. " This unsteadiness depends on two conditions : first, the amount of the change of the deviation arising from any number of degrees of heeling ; and second, the existence or non-existence of isonchronism between any number of the periods of the vibrations of the card and any number of the ship's rolling. Taking this latter condition into consideration when inconvenience has been experienced, arising from the unsteadiness of the compass, the card has been changed, since either a more sluggish or more active needle would be more or less in unison with the ship's roll." With a compensated card, while the ship remains in the same latitude, if we choose a card which is steady when the ship rolls with her head north or south, it will also be steady when her head is in any other direction. Still cases have arisen in which a compass that is steady in one latitude has vibrated, and even spun, when by changing the ship's place the earth's horizontal force has changed. To obviate this difficulty, washers, made to rest on the card with the cap passing through the middle, have been employed with advantage. Deviation of the Compass. 149 233. Effects of Heeling.— The eflFect of the heeling error, when the north end of the needle is drawn to windward, is to throw the ship to windward of her supposed position when steering on northerly courses ; and to throw her to leeward when steering on southerly courses. Therefore, to make a straight course, when heeling, a ship should be kept away by compass on either tack on northerly courses ; and she should be luffed up on either tack on southerly courses. The effect in the few cases in which the compass needle is drawn to leeward is the reverse, and in the southern hemisphere, also, the reverse of these rules holds good ; but this is a point which can only be ascertained for each ship. The heeling error may be expressed in terms of the deviation when upright, and the following are the results : — On Northerly Courses . Stirboard tack — E. dev. is increased, W. dev is decreased. Port tack — W. dev. is increased, E. dev. is decreased. On Southerly Courses : — Starboard tack — W. dev. is increased, E. dev. is decreased. Port tack — W. dev. i^ decreased, E. dev. is increased. And when the deviation when the ship is upright is small in amount and decreases by heeling, it may become reversed in name. In the few cases in which the North end of the compass needle is drawn to lee- ward, the rule above is of course reversed. 23+. Correction of the Heeling Error. — The mechanical correction of the heeling en-or is made by a magnet placed in a vertical position immediately below the centre of the compass card. The ship's head is to be placed north find south, correct magnetic ; she is then heeled over to port and to starboard, and the magnet raised or lowered until the compass points correctly. In most cases the north end of the vertical magnet should be uppermost. The deviation arising from a ship heeling being semicircular, this cor- rection holds good only while a ship continues near the magnetic latitude where the adjustment was made : hence, arrangements must be made for sliding the magnet along as different latitudes are reached, and for removing it, and even reversing it in high latitudes of opposite names. 235. After all the compensations have been accurately made, there will still remain small residual errors ; for these the ship must be swung, and a table of deviations made for use. When an arrangement of magnets is employed to neutralise those large deviations occasionally found, and caused by the iron ship's magnetism, the compass so corrected can never be con- sidered as entirely compensated, and the deviation must be expected to change on change of latitude, and from other causes. It will thus be seen that the seaman can have no absolutely safe guide, except in the system of actual and unceasing observation. 1 50 Deviation of the Compass. METHODS OF FINDINa THE AMOUNT OF THE DEVIATION. 236. When in port, there are two principal methods in general use for finding the deviation, viz. :— Method I, by tlie known correct magnetic bearing of a distant object, and Method II, by reciprocal simultaneous bearings, i.e., with a compass on board and a compass on shore. 237. Method I. — By the known bearing of a distant object, — The requisite warps being prepared, the ship is to be gradually swung round so as to bring her head successively upon each of the 32 points of the Standard Compass ; and when the ship and the compass card are perfectly steady, and her head exactly on any one point, the direct bearing of some well-defined object is to be observed with the Standard Compass, and registered. The ship's head is to be gently warped round in the same manner to the next point, and when duly stopped and steadied there, the bearing of the same object is to be again set, and again recorded; and so on, point after point, till the exact bearing of the one object has been ascertained with the ship's head on every separate point of the compass. 238. The object selected for this purpose should be at such a distance that the diameter of the space through which the ship revolves shall make no sensible difi'erence in its real bearing, and should not exceed the one- hundredth part of the distance of the object. The distance must depend on the range the ship takes when swingiug ; if she be at anchor, in a tide-way, from 6 to 8 miles is not too much ; brought up by the middle (in a dock) 2 miles will suffice. 239. The next step is to determine the correct magnetic bearing of the selected object from the ship ; or in other words, the compass bearing it would have from on board if it were not disturbed by the attraction of the iron in the ship. This is efi"ected by taking the compass to some place on shore (avoiding local influence) from which the part of the ship where the compass stood and the object of which the bearings had been observed shall be in one with the observer's eye, or else in the exactly opposite direction. The bearing of the object from that spot will evidently be the correct magnetic bearing from the ship by the compass. The difi'erence between the correct magnetic bearing of the object and the successive bearings which were ob- served with the compass on board, when the ship's head was on the several points, will show the error of each of these points which was caused by the ship's iron; or, in other words, the Deviation of the Standard Compass according to the direction in which the ship's head was placed. (b) The correct magnetic bearing of the distant object will be the mean value of all the observed bearings, if observed on equidistant points ; or of four or more compass bearings, if taken also, on equidistant compass points. 240. II. — By reciprocal bearings. — Should there be no suitable object visible from the ship, and at the requisite distance as stated above, the deviations must be ascertained by the process of reciprocal bearings. A second compass is placed on shore where it will be entirely beyond the influence of iron of any description and where it can be distinctly Deviation of the Compass. 1 5 i seen from the Standard Compass on board. Then take, simultaneously (known by pre-concerted signal), the bearing from each other of the compass on shore and the compass in the binnacle, as the sliip is wai'ped round so as to bring her head successively upon each of the thirty-two points of the Standard Compass on board, or on each alternate point. To ensure the success of this operation, the compass on shore should not be more distant from the ship than is consistent with the most distinct visibility with the naked eye, of both compasses from each other. The observations should be made as strictly simultaneous as possible, the time at which each bearing is taken being noted both on shore and on board. It will be found convenient in practice, for the shore observer to chalk each observation on a black board, to be read at once from the ship, in order that the observation may be repeated if any apparent inconsistency presents itself. Before this process is complete, the Standard Compass should be carried on shore, in order to be compared with the compass used there, by means of the bearing of some distant object, and the difference, if any, is to be recorded ; and in all cases, when compasses are compared, the caps, pivots, &c., should be first carefully examined. The shore compass gives correct magnetic bearings. The difference between the correct magnetic bearing of the Standard Compass as observed from the shore, and the bearing of the shore compass as observed from the ship, with her head in any particular point, reversed, i.e., with J 80° added or subtracted, will show the error on that point which was caused by the ship's iron ; in other words the deviation of the Standard Compass according to the direction in which the ship's head was placed. 241. III.— By Marks on the Dock'Wall. — This is a very convenient method where it can be practised. At Liverpool the correct magnetic bearings of the Yauxhall chimney, from various points of the dock walls, are painted in large figures on the walls, so that the bearing of the same chimney may be observed as the ship swings with the wind and tide ; and at the same time that bearing marked on the wall, which is on a line between the Standard Compass and the chimney, is noted. The difference between those bearings is the deviation for the point on which the ship's head is at the time. In a similar manner, at Cronstadt, the correct magnetic bearings of a conspicuous point on a public building are painted on the mole. 242. If, during the operation of swinging, a haze obscures the shore com- pass, while the sun at the time is shining brightly, a number of points may be secured by time-azimuths, which otherwise might be lost. Time-azimuths are also advantageous where the second of the above methods cannot be used for want of an assistant observer for the shore compass ; and when the first of the above methods is not available owing to the length of the ship and the scope of the moorings, combined with the most distant objects in sight, not being sufficiently far off to render the difference of their bearings in- sensible as the ship swings round to the tide. In such cases Oodfrayh 152 Deviation of the Compass. Azimuth Diagram, as also Azimuth or Sun's True Bearing Tables, computed for intervals of four minutes, by Staff-Commander J. Burdwood, E.N., published by the Admiralty, will be found useful as superseding the calcu- lation for the determination of the True Azimuth. 243. Commander Walker, E.N., has shown* that the deviation may be ascertained with suflB.cient accuracy by selecting a distant object, as before, "and as the ship swings by wind or tide from one point to another, write down the compass bearings of the distant object opposite the direction of the ship's head. As the ship swings round there will be two nearly opposite points of the compass on which the bearings of the distant object agree, and this should be the correct magnetic bearing of the object." The deviation is then found as in the first method. 244. The Dumb Card. — The difficulty of finding the correct magnetic bearing of the ship's head may be obviated, however, by using the dumb-card, i.e., a compass card without the needle, slung in gimbals, with its centre over a fore-and-aft line of the vessel, and as near to its middle as possible. The card is fitted with sight vanes, similar to an azimuth compass. Having obtained the correct magnetic bearing of a distant object, place the card so that it shall point out that direction, and screw the sight vanes to the card, so as to cut the object with the thread. Then, as the ship is swung, the card must still be kept pointing out the correct magnetic bearing of the object by means of the sight vanes, and where the fore-and-aft line meets the edge of the card, must then be the correct magnetic bearing of the ship's head. 245. To name the Deviation. — Rule. — When the reading by the shore compass (reversed), or the correct magnetic bearing of the distant object, is to the right of the reading by the compass on board, the deviation is easterly ; when to the left, westerly. Thus, suppose the correct magnetic hearing from tho shore compass, with ship's head at N.W., is N. 15° E., and the bearing of shore compass from the ship is S. 11" W. ; to find deviation proceed thus : — Reverse of the bearing by shore compass j g x c° W or correct magnetic bearing j ' ^ Bearing from ship S. 1 1 W. Deviation 4 E. When the ship's head lies N.N.E., let the binnacle compass bearing of the shore object or compass be N. 19° 30' E., and the bearing of the binnacle compass from the shore com- pass be S. 27° o' W. : required the deviation. The opposite point to S. 27° o' W. is N. 27° o' E , which is 7° 30' to the right of N. 19° 30' E. Hence the deviation is 7° 30' E. 246. The directions of the ship's head having been taken by the compass in the ship, are therefore afiected by the local attraction, and the apparent compass bearing of the ship's head differs from the correct magnetic bearing by the amount of the local deviation dne to the position of the ship. For Magnetism of Ships and the Mariner's Compass. Deviation of the Compass. 153 instance, when the ship is apparently lying with her head east, it is not the true magnetic cast ; but supposing the local deviation to be one point easterly, the east point of the compass card will be drawn to E. by S., and the true magnetic direction of the ship's head will be E. by S. The observations and tabulated results are incomplete until the correct magnetic bearing of the ship's head at each observation is found. 247. The following shows the arrangement of tabular forms for finding the deviation by the several processes described. I. By bearing of a distant object. Correct magnetic bearing of distant object from ship N, 63° o' W., distant II miles. Ship's Head by the Standard Compass. East E. by S. . E.S.E. ... S.E. by E. S.E Bearin,^ of Distant Object by the Standard Compass, N. 83° 20' W. N. 82 15 W. N. 81 5 W. N. 72 30 W. N. 77 40 W. Deviation of Standard Compass. 20° 20' E. 19 15 E. 18 5 E. 16 30 E. 14 40 E. And similarly at all points of the compass. II. By reciprocal bearings. Time.* gh 10" A.M 9 H 17 21 26 32 Ship's Head by the Standard Compass. North . . . . N. by E. . . N.N.E N.E. by N.. N.E N.E. by E. . SmULTANEOTJS BEABING8. From Standard Compass on board. S. 37° 50' E. S. 45 o E. S. 51 40 E. S. 57 20 E. S. 61 50 E. S. 65 30 E. From the shore Compass. N. 41° o'W. N. 42 25 W. N.4330W. N.44 10 W. N.45 o W. N.46 oW. Deviation of Standard Compass. 3° 10' W. 2 35 E. 8 10 E. 13 10 E. 16 50 E. 19 30 E. And 80 on through all the points of the compass. 248. The seaman must remember that the corrections thus obtained belong to the compass by which the observations are made, and to that com- pass while it is in its proper place, and that these corrections will furnish no guide whatever to the efifects of the iron on a compass placed in any other part of the ship ; but if, while swinging, the direction of the ship's head by the other compasses is noted and tabulated, the deviation of all the compasses can be found. * The time — as taken by compared watches — may be omitted if the shore observations can be clearly made out by being chalked on a black board. '54 Deviation of the Compass. The following is a Table of Deviations to which, reference is to be made in working the following examples. TABLE OF DEVIATIONS. ship'b head. DEVIATION. ship's head. DEVIATION, North % 22 W. I 46 E. 3 20 E. 5 14 E. 7 14 E. 8 54 E. 10 44 E. 11 40 E. 10 44 E. 9 54 E. 9 8 E. 7 20 E. 6 18 E. 5 E. 3 24 E. I 42 E. South 1(5 E. 1 50 W. 3 16 W. 4 48 W. 6 16 W. 7 40 W. 9 18 w. 10 34 W. 11 50 W. II 10 w. 10 16 w. 9 18 W. 7 5^ W. 6 18 W. 5 3 W. 3 10 W. N. by E S. by W N.N.E N.E. by N N.E N.E. by E. E.N.E E. by N S.S.W S.W. by S s.w S.W. by W w.s.w W. by S East „ E. by S E.S.E S.E. by E S.E "West W.byN W.N.W N.W. by W N.W S.E. by S S.S.E S. by E N.W. byN N.N.W N.byW 249. The purposes for which a Table of Deviations so formed are : — I st. — To correct the course steered by the compass, in order that the correct magnetic course actually made good may be used in the calculation of the ship's reckoning, or to lay it down on the chart. 2nd. — If one or more bearings of the land are taken, to correct these bear- ings by the amount of deviation due to the direction of the ship's head at the time. 3rd. — If we wish to shape a course for a port, and having, either by calculation, or as taken from the chart, the correct magnetic course to be made good, so to apply the deviation as to obtain the compass course to be steered. EULE LYI. To find the correct magnetic course, having given the compass course and deviation. Express the compass course in degrees, 8fc. ; look in the Table of Deviations /or the deviation opposite the given course, then, Easterly deviation allow to the right. Westerly „ ,, left. Examples. Correct the following compass courses for deviation, as given in Table above : — I. E.S.E. = 6 points L. of S. 6 points L. of S. = 67° 30' L. of S. Deviation (Table) 9 8 R. Cor. mag. course 58 22 L. of S. or S. 58 22 E. In this instance, the deviation being Easterly, ajlow to the right. N.N.W. = 2 points L. of N. 2 points L. of N. = 22° 30' L. of N. Deviation (Table) Cor. mag. course 2 L. 27 32 L. of N. or N. 27 32 W. The deviation in this instance being Westerly allow to the left. Deviation of the Compass. S^ S.W. = 4 points R. of S. 4 points R. of S. =z 45° o' R. of S. Deviation (Table) 6 16 L. Cor. wwy. course 38 44 R. of S. or S. 38 44 W. 4. W. = 8 points R. of S. 8 points R. of S. = go' o' R. of S. Deviation (Table) u 50 L. Cot. ma^. course 78 10 R. ofS. or S. 78 10 W. 5. W.iN.= pts. L. of N. = 87" 1 1' L. of N. Deviation = 1 1 40 L. 180 51 L. ofN. Cor. maff. course 8i 9 R. of S. or S. 81 9 W. N.W. by W. I W. = si L. of N. = 61° 41' L. of N. Deviation =: 10 2 L. West W. by N. Dev. for J pt. Dev. for W. = ii''5o' W. = II 10 W. 4)0 40 = o 10 = II 50 Dev. for W. J N. = 11 40 W. N.W. byW. = 9°i8'W. W.N.W. = 10 16 W. N. 71 43 W. Change for i pt. = 58 The deviation (see Table) for N.W. by "W. and W-N-W. is found, and the difference of these quantities is the correction for i point, which divided by 4 gives the cor- rection lor i point = 0° 14'. Now compass course is J point from W.N.W., therefore, apply correction 0° 14' to the deviation for W.N.W., the result is the deviation for N.W. by W. J W., and it is to be subtracted, because the deviation fur N.W. by W. is less than for W.N.W. 7. N. i E. = ^ pt. R. of N. = 5" 38' R. of N. Deviation =: o 42 R. 4)58 Change of dev. for J pt. = 14 Dev. for W.N.W. = 10 16 Dev. for N.W. byW. f W. = 10 2 W. Deviation at N. = 0° 22' W. N. by E. I 46 E. a) I 24 o 42 E. Deviation at N. =: 0° 22' W. N. by W. 3 10 W. Cor. moff. course 6 20 R. of N. or N. 6 20 E. One deviation being W., and the other E., half the dif- ference of the two is taken for the deviation. 8. N. J W. = J pt, L. of N. = 5° 38' L. of N. Deviation =1 46 L. Cor. maff. course N. 7 24 W. Half the sum of the deviations for N. and N. by W. is taken for deviation on N. i W., both deviations bein^of the same name Or proceed thus — take the deviations from Table for North and for N. by W. ; take the difference and half it, apply this to the devia- tion for North, adding because the deviation is greater for N. by "W. than for North. Examples for Praotioe. Correct the following courses steered for deviation, as given in the Table^ page 154. Dev. on f pt. 2)3 32 = I 46 W. I. N.E. by N. 8. S.E. by E. i.S- S.W. by S. 22. West. 2. North 9- N. by E. ^ E. 16. S.^E. 23- N.W. i W. 3- N. by W. ^ W. 10. South 17- W.iS. 24. S. |W. 4- B.W. ^ W. II. W. JN. 18. N.N.E. ^ E. 25- W. by S. ^ S ,■;• W. ^N. 12. S.E. i S. 19. N. JE. 26. E. JS. 6. S.S.W. n- E. :} N. 20. W. by N. 27. East 7- E. by N. 14. N.W. ^ W. 21. S. by E. 28. W.N.W. 250. Proceeding to correct the courses for the deviations given in Table I, a second Table, arranged like the following, may be made for all the points of the compass. 156 Deviation of the Compass. TABLE II. Courses steered by Compass. Deviations. Correct Magnetic Courses. North or o° 0'22' W. N. o°22' W. or North. N. by E >i N. II 15' E. I 46 E. N. 13 I E. „ N. byE. IE. N.N.E jj N. 22 30 E. 320E. N. 25 50 E. „ N.N.E. i E. N.E. byN. .. j» N. 33 45 E. 5 '4E. N. 38 59 E. „ N.E. 1 N. n.h: jj N. 45 E. 7 14 E. N. 52 14 E. „ N.E. f E. N.E. by E. .. jj N. 56 15 E. 854E. N. 65 9 E. „ N.E. by E. f E. E.N.E >i N. 67 30 E. 10 44 E. N. 78 14 E. „ E. by N., nearly. E. byN N. 78 45 E. II 40 E. S. 89 35 E. „ East. East )> 90 10 44 El. S. 79 16 E. „ E. by S. E. byS » S. 78 45 E. 954E. S. 6851E. „ E.S.E. IE. E.S.E )j S. 67 30 E. 9 8E. S. 58 22 E. „ S.E. by E. iE. S.E. byE. .. )> S. 56 15 E. 7 20 E. S. 48 55 E. „ S.E. f E. S.E >> S. 45 E. 6 'i 8 E. S. 38 44 E. „ S.E. A S. S.E. by S. . . )j S. 33 45 E. 5 oE. S. 28 45 E. „ S.S.E. i E. S.S.E )) S. 22 30 E. 324E. S. 19 6 E. „ S. by E. f E. S. by E )j S. II 15 E. I 42 E. S. 9 33 E. „ S. 1 E. South yy 16 E. S. 16 W.„ South. S. by W )1 S. II 15 w. I 50 W. S. 9 25 W. „ S. 1 w. S.S.W )5 S. 22 30 W. 3 16 W. S. 19 14W.,, S. by W. 1 W. S.W. byS. .. j> S- 33 45 W. 448 W. S. 28 57 W. „ S.S.W. i W. s.w S. 45 w. S. 56 15 w. 6 16 W. 7 40 W. s. 3844W.,, s.w. IS. S. 48 35 W. „ S.W. f W. S.W. by W... W.S.W y) S. 67 30 w. 9 18 W. S. 58 12 W. „ S.W. by W. ^ W. W. by S yy S. 78 45 W. 10 34 w. S. 68 II W.„ W.S.W. iW. West yy 90 II 50 w. S. 78 10 W. „ W. by S. W.byN. .... yy N. 78 45 W. II 10 w. N. 89 5-; W. „ West. W.N.W >) N. 67 30 W. 10 16 w. N. 77 46 W. „ W.N.W. 1 W. N.W. by W. )> N. 56 15 W. 9 18 W. N. 65 33 W. „ N.W. by W. | W. N.W )5 N. 45 W. 752 W. N. 52 52 W. „ N.W. f W. N.W. byN. .. n N. 33 45 W. 6 18 W. N. 40 3 W. „ N.W. by N. f W. N.N.W jT N. 22 30 W. 5 2W. N. 27 32 W.„ N.N.W. iW. N. by W )> N. II 15 W. 3 10 W. N. 1425 W.„ N. by W. i W. To obtain from the Table above the correct magnetic course of the ship from the course shown by the Standard Compass, look in the first column of the Table for the latter ; the second column gives the deviation when her head is on that point ; and in the third column (the deviation having been applied as directed in Rule LVI) the seaman will find the correct magnetic course given there, by inspection. This Table will be found more useful than the common Table of Deviations, as it shortens the calculation, when it is required to fractions of a point, as a halt, a quarter, &c., and when steering on a whole point, the correct magnetic course is known at sight. The following examples will show the use of the Table. Examples. Ex. I. The ship's head by Standard Compass is N.E. ^ E. : what is the correct magnetic course ? (Using the Table above.) Oorr. mag. course for N.E. = N. 52° 14' E. „ „ N.E. by E. = N. 65 9E. 2)117 23 Oorr. mag. co. for N.E. * E. = N. 58 41 E. Here the courses taken from the Table are of the sa7ne name, therefore, half the sum is evidently the correct magnetic course corresponding to N.E. | E. Ex. 2. Find correct magnetic course when Standard Compass course is N. ^ E. (using the above Table). Corr. mag. course for North ^ 0° 22'W. N.byE. = i3 l E. 2)12 39 Corr. mag. course for N. ^ E. =: 6 19 E. Here the courses corresponding to North and N. by E. are of contrary names; hence, for a Stan- dard Compass course, midway between N. and N. by E., we use half the difference of the correct magnetic courses corresponding to these points. Deviation of the Compass. 157 Ex. 3. Ship's head by Standard Compass is N. I E. ; find, by moans of the Tabic, the corresponding correct magnetic course. Corr. mag. course for North =: 0° 22' W. N. iE. = i3 I E. 4)13 23 DifiFerence for \ point = 321 Corr. maof. course for North :rz o 22 W. Corr. mag. course for N. ;| E. = 259 E. Ex. 4. Required the correct magnetic course when ship's head by Standard Com- pass is W. I N. Corr. mag. course for West =: S. 78°io' W. „ W.byN. ) s „o ,w isN.89<'55'W.orS.90°5'W. 1 ^'9° -5 vv . " 55 3 4)35 45 Difference for f point -\- 8 56 Corr. mag. course for West = S. 78 10 W. Corr. mag. course for W.fN. = S. 87 6 W. 251. Correction of Compass Bearings. — In order to correct the bearing of the object as taken by the Standard Compass, note the direction of the ship's head by that compass while taking the observation, then enter the first column of either Table I or II with that, and in the second column will be found the deviation to be applied to the bearing of the object. (See 164-166, page 105). Easterly deviation to be allowed to the right, and westerly to the left, as in Eule LYI. Cavtion. — Be careful to remember that the deviation to be applied is that due to the compass course, not that on the point of bearing ; and the consequence of a misapplication of the deviation, by applying that for the point of bearing instead of the deviation for the compass course, may lead into danger, if not loss. Examples. Ex. I. The bearing by Standard Compass of the South Foreland is N.N.W., the course by the same is E.N.E. : required the correct magnetic bearing. Taking out the deviation from the Table for the direction the ship's head was on at the moment the bearing was taken, we have Bearing by Standard Compass of South Foreland N. 22° 30' W. or 22' 30' L. of N. Deviation by Table for E.N.E (applied to right) 10 44 E. „ 10 44 R. Correct magnetic bearing II 46 W. „ II 46 L. of N. or N. 11" 46' W. Ex. 2. Ship's head E.S.E. by compass, the bearing by the same compass of the Start Point is N. 20° W. : required the correct magnetic bearing. Bearing of Start Point by Standard Compass N. 20° W. or 20° L. of N. Deviation by Table (applied to the left) Correct magnetic bearing Ex. 3. Two islands bear S.E. and W.S.W. ; correct magnetic bearing of each. Bearing by Standard Compass S.E. 14 W. „ 14 L. N. 34 W. „ 34 L.ofN. or N. 34° W. the ship's head is N.E. : required the = S. 45° E. or 45" L. of S. 7 E. „ 7 R. Deviation by Table (applied to the right) =: Correct magnetic bearing =: S. 38 E. ,, 38 L. of S. or S. 38° E. Bearing by Standard Compass W.S.W. =: S. 67° 30' W. or 67° 30' R. of S. Deviation by Table (applied to the right) = 7 14 E. „ 7 14 R. Correct magnetic bearing = S. 74 44 W. ,, 74 44 R. of S. or S. 75° W. 158 Deviation of the Compass. Examples for Praotiob. In the following examples the ship's compass course and the bearing of the object by compass are both given, and it is required to find the magnetic bearings of the objects, using the same deviation table (Table of Deviations, page 154). No. sup's Head by Compass. Compass Bearing. No. Ship's Head by Compass. Compass Bearing. I 2 3 4 5 6 West East. E.byS. iS. North. South. E.JS. S. by W. i W. 7 8 9 10 II 12 E. J N N.i W. W.fS. W. by S. i S. S. iE. W. by N. 1 N. N. by W. i W. S.S.E N.E.byN W.N.W W.byN. iN... S.byW. 1 W... N.E.JE E. |S N. J E S.iE E.JN 252. Q-iven a correct magnetic course by the chart between two points of land, to find the course that must be steered by compass. EULE LVn. Easterly deviation is allowed to the left, Westerly ,, ,, riglit, taking care that the deviation applied is that of the correct magnetic course. Note. — In this case it is important to remember, not only is the general rule of applying the deviation reversed, but the correction to be applied is the deviation due to the given magnetic course, not that due to a compass course, as in Rule LVI ; that is, to the correct magnetic course as found from the chart, or by calculation, the deviation, as due to that course, must be applied as directed above, in order to find the course to be steered by compass approximately. It will be observed that on those courses near which the deviation is considerable, and rapidly changing, the deviation on a given magnetic course is considerably different from that on the compass course of the same name. In such cases it will be necessary to again enter the table with the approximate course and get the corresponding deviation and apphj it to the correct magnetic course ; the result will be the compass course to be steered to make good the given correct magnetic course. Example. Ex. I. Required the Compass Course that shall make correct magnetic W. by S. Entering the first column of Table I, or II, with W. by S., the deviation on that point is found to be io^° W., which allowed to the right would be about West ; and since the devia- tion for this last does not difi"er from the deviation used, it may be considered that to make correct magnetic W. by S., the course to be steered is about West. 253. By comparing the first and third columns of Table II, the seaman may also by inspection, or a single interpolation, determine what course he will have to steer by the Standard Compass, in order to take up any given correct magnetic course. For example, let the given correct magnetic course be N.E., or N. 45° E. ; on referring to column 3, it will be found that N. 45° E. lies nearly midway between N. 38" 59' E. and N. 52° 14' E., the Deviation of the Compass. 159 Standard Compass courses corresponding to which are N. E. by N. and N.E. ; the course to be steered is consequently N.E. ^ N. If great accuracy be required, it will be necessary to find the exact proportion between the actual changes of the ship's head with reference to the horizon. Referring to the same example, he will find that the ship's head by Standard Compass between N.E. by N. and N.E., the actual angular change, is 13° 15', represented by 1 1° 1 5' of the compass. In shaping a course, therefore, between these points, the value of the half point is about 6^ 37', the quarter is 3° 18', and similarly for smaller divisions of the rhumb. To prevent, however, the possibility of error in such an important opera- tion as that of shaping a course, a separate Table, may be advantageously constructed expressly for that object. See Table III, where the desired course being sought in the first column is immediately followed by the course to be steered by the Standard Compass, and given in degrees and minutes, as well as points and fractional parts. TABLE III. Correct Magnetic Course proposed to bo steered. Course that must be steered by the Standard Compass in order to make the Correct Magnetic Course. North North or nearly North. N. 10° E. „ N. 1 E. N. 19 E. „ N. byE. f E. N. 29 E. „ N.N.E. f E. N. 38^E. „ N.E. JN. N. 48I E. „ N.E. i E. N. 59 E, „ N.E. by E. i E. N. 68 E. „ E.N.E. N. 79 E. „ E. by N. East „ East. S. 78 E. „ E. byS. S. 65 E. „ S.E. by E. | E. S. 52^10. „ S.E. |E. S. 39|E. „ S.E. iS. S. 26iE. „ S.S.E. |E. S. 13" E. „ S. byE. |E. South „ South . S. i2iW. „ S. by W. i W. S. 25IW. „ S.S.W. iW. S. 39" W. „ S.W. ^ s. s. 52J w. „ s.w. i w. S. 65hW. „ S.W. by W. J W. S. 78 W. „ W. by S. i 8. N. 89 W. „ West. N. 78iW. „ W. byN. N. 68 W. „ W. byN.JN. N. 58 W. „ N.W. byW. iW. N. 48 W. „ N.W. 1 W. N. 38 W. „ N.W. i N. N. 28 W. „ N.N.W. ^ W. N. 18 W. „ N. by W.|W. N. ^w. „ N.fW. North. North. N. by E N.N.E N.E. byN N.E N.E. by E E.N.E E. by N East E. by S E.S.E ,.. S.E. by E S.E S.E. by S S.S.E S. by E South S. by W S.S.W S.W. by S S.W s.w. by W W.S.W W. by S West W. by N W.N.W N.W. by W N.W N.W. byN N.N.W N.byW North i6o Deviation of the Compass. 254. The following examples are designed to show the method of correct- ing courses for leeway, variation, and deviation. Examples. Ex. I. Course steered E.N. E. ; wind S.E. ; leeway 2 1 points ; variation 17° E. ; devia- tion 21° E. : required the true course. Here the course by compass is E.N.E., or 6 points right of North. The ship being on the starboard tack the leeway is applied to the left, and hence 2 J „ left. The difference is the course corrected for leeway . . 3^ ,, right of North. Which expressed in degrees, &c.„ is 39" 22' right of North. The variation and deviation are of same name, their sum, viz., (21° E. -|- 17° E.) 38° E., is 38 o right. True course 77 22 right of North. Or N. 77" 22' E. Ex. 2. Course by compass N.N.W. ; wind N.E, : leeway 2J points ; variation 45° W. ; deviation 16° 52' E. ; find the true course. Here the ship's course is N.N.W 2 points left of North. The ship being on the starboard tack the leeway is applied to the left, and hence is 25 ,, left. Therefore the sum is the course corrected for leeway 4J „ left. Which expressed in degrees, &c., is 50° 38' left of North. The variation and deviation are of contrary names, their difference, viz., (45° W. — 16° 52' E.) 28° 8' W., is 28 8 left. Sum 78 46 left of North. True Course N. 7 8° 46' W., or W. by N. In this example the compass course and leeway are given in points, we therefore take thg course and allow the leeway from the wind, which gives the course corrected for leeway, viz., 4I points left of North, which expressed in degrees, &c., is 50° 38' L. of N. ; then the difference of the variation and deviation is taken, as they are of contrary names, the remainder — which takes the name of the greater— is then applied ; the result is the true course. Ex. 3. Course by compass W. by S. ; wind S. by W. ; leeway i| points; variation 36° 34' E. ; deviation 13° 50' W. : find the true course. Compass course W. by S. or 7 points right of South. Leeway, port tack allowed to the right if ,, right. Sum exceeds 8 points 8| „ right of South. Subtract from 16 7I „ left of North. or 81° 34' left of North. Variation and deviation are of contrary names, the difference, viz., (36° 34' E. — 13"^ 50' W.) = 22" 44' E 22 44 right. True course 58 50 left of North. or N. 58" 50' W. Deviation of the Compass. i6i Examples fob Pkaotiob. From the following Compass Courses find the True Courses : — No. Compass Courses I. N.E. by E. 2. North 3- N.N.W. 4- West 5- S.8.E. h E. 6. E.f S. 7- South 8. 9- E. fS. W.N.W. lO. N.E. by E. II. W. by S. 12. South '3- West 14. S.S.W. i W. 15- N.W. by W. 16. E. by N. 17. W. by S. ^ S. ig. E. iS. 19. S.TV.byS. 20. South 21. S.W. \ S. 22. E. ^S. 23- East 24. W. IN. 35- N. iW. 26. E. |N. 27. N. by W. i W 28. up S.E. \ offE. byN. / up S.W. \ W. 1 29. off W. byN. J up N.W. \ N. ) 3°' off W. byN. j "Winds. N. by W. E.N.E. N.E. S.S.W. S.W. \ s. N.E. by N. w.s.w. S. by E. North N. by W. S. by W. E.S.E. N.N.W. S.E. by S. N. by E. S.E. by S. S.by W. N.N.E. \ E. W. by N. E.S.E. S.S.E. \ E. S. by E. i E. S.S.E. N.N.W. W. by N. N.N.E. N.E. ^ E. South S.JE. N. by E. Leeway. 2|- pts. Deviation. 33 pts- E. I „ E. ij „ E. li „ E. NAPIEE'S DIAGEAM. 255. It is often of the utmost importance in various branches of physical science to represent tables of related numbers by means of curve lines, or other figures that show to the eye the nature of the relations or laws expressed, or rather concealed, within the mass of figures constituting the tables. Not only does such a mode of representation at once manifest these laws — almost rendering them palpable — but it further points out in what cases natural laws are not represented, and therefore what the cases are that require a greater amount of observation. These modes of representation are com- monly known as Graphic Methods. Various "graphic methods" of delineating the deviation have been devised ;* but the method introduced here is due to J. E. Napier, Esq., F.E.S., and is one peculiarly adapted for this purpose, as it is equally appU- * Graphic methods for correcting the ship's course for the Deviation of the Compass have also been designed by Rear Admiral Ryder, i\Ir. Archibald Smith, F.R.S., and Mr. W. W. RuNDELL. Admiral Ryder's which is an extension of Napier's diagram, is published by the Admiralty. Mr. Smith's, known as the straight line method, is published by the Board of Trade, and also furnished to H.M. ships for fleet tactics, for which it is well adapted. Mr. Rundell's is known as the circular method. They are all useful in practice. NAPIER'S DIAGRAM, Showing Curve of Deviation, Ship West Deviation, East Deviation. -i^'2. Deviation of the Compass. 161 Examples fob Pbaotiob. From the following Compass Courses find the True Courses : — No. Compass Courses N.E. by E. North N.N.W. West S.8.E. i E. E.JS. South E.f S. W.N.W. N.E. by E. W. by S. South West s.s.w. i w. N.W. by W. E. by N. W. by S. i S. E. ^S. S.W.byS. South S.W. ^ S. E. ^S. East W. iN. N. A W. E. |n. N. by W. ^ W. up S.E. > off E. by N. / up S.W. ^ W. I offW. byN. j up N.W. ^ N. \ ofiF W. by N. I "Winds. N. by W. E.N.E. N.E. S.S.W. S.W. i s. N.E. by N. w.s.w. S. by E. North N. by W. S. by W. E.S.E. N.N.W. S.E. by S. N. by E. S.E. by S. S. by W. N.N.E. ^ E. W. by N. E.S.E. S.S.E. i E. S. by E. i E. S.S.E. N.N.W. W. by N. N.N.E. N.E. ^ E. South S.iE. N. by E. Leeway. a| pts. NAPIEE'S DIAGEAM. 255. It is often of the utmost importance in various branches of physical science to represent tables of related numbers by means of curve lines, or other figures that show to the eye the nature of the relations or laws expressed, or rather concealed, within the mass of figures constituting the tables. Not only does such a mode of representation at once manifest these laws — almost rendering them palpable — but it further points out in what cases natiiral laws are not represented, and therefore what the cases are that require a greater amount of observation. These modes of representation are com- monly known as Graphic Methods. Various "graphic methods" of delineating the deviation have been devised;* but the method introduced here is due to J. R. Napiee, Esq., F.R.S., and is one peculiarly adapted for this purpose, as it is equally appU- * Graphic methods for correcting the ship's course for the Deviation of the Compass have also been designed by Rear Admiral Ryder, Mr. Archibald Smith, F.R.S., and Mr. W. W. RuNDELL. Admiral Ryder's which is an extension of Napier's diagram, is published by the Admiralty. Mr. Smith's, known as the straight line method, is published by the Board of Trade, and also furnished to H.M. ships for fleet tactics, for which it is well adapted. Mr. Rundbll's is known as the circular method. They are all useful in practice. l6z Deviation of the Compass. cable whether the points on which the observations have been made are or are not precisely equidistant. It requires no calculation, and only a moderate degree of neat-handedness. The method consists of two parts, the diagram and the curve. The diagram is the same for all vessels. 256. Construction of the Diagram. — In this method the diagram consists of a central or vertical line of convenient length — say 1 8 inches — which may be considered as representing the margin of the compass card cut at the north point, and straightened and extended in the following way : — N ' E S W N This line, which may be taken to represent no deviation, is divided into 32 equal parts, representing the 32 points of the compass, commencing at the top with North, and ranging in the order of N. by E., N.N.E., &c., and ending with North at the bottom. The central line is then intersected at each of the 32 points by two straight lines, one a, plain line and the other a dotted line. The plain and dotted lines make an angle of 60° with the central line and with each other, and so forming a set of equilateral triangles with the central line, and converting the diagram into a simple addition and subtraction table. On the right side of the central line the dotted lines incline downwards, and the plain lines upwards. The reverse is the case on the left. The central line is further divided into 360 equal parts, representing degrees, and these divisions are numbered from 0° at the top to 360° at the bottom. They are also numbered, according to the usual mode of dividing the compass card, from 0° at North and South, up to 90° at East and "West. This division of the central line into degrees serves also as a scale by which the deviations are laid off. 257. Requisite Observations to be made. — The least number of observed deviations available for obtaining a complete curve are the deviations on 4 points distributed equally, or nearly so, round the compass ; but, if possible, the deviations should be observed on 8 or more points. If the observations are observed on 4 points only, these should be at or near N.E., S.E., S.W., and N.W., and from these it is possible to form a fairly approximate curve. The points next in importance are North, East, South, and "West. If the deviations have been observed at or near the eight principal points, a curve can be drawn which will give the deviation on every point of the compass within very small limits of error.* 258. Cases may also occur in which by the ship swinging round at her anchors in a tide-way or to the wind, or by the aid of a steam-tug, the devia- tion may be observed on various directions of the ship's head, not being necessarily exact points of the compass ; or similarlj'-, whilst under steam or sail at sea, a number of azimuths of the sun may be observed, and hence the deviation obtained. * The examination of a collection of curves made from actual observations, as in the report of the Liverpool Compass Committee, &c., will show that there is so much regularity that these interpolated deviations may generally be relied upon, although certain cases, such as the U.S.S- Roanoke (Keport of Nautical Academy of Sciences, 1 863), the irregularities are considerable. Deviation of the Compass. 163 In these cases the Graphic method here described furnishes a ready and effectual mode of obtaining a result on which the errors of individual observa- tions are as far as possible compensated and any egregious errors eliminated. 259. Construction of the Curve of Deviations. — Easterly deviations are laid down to the right of the central line, westerly deviations to the left.* The amount of the deviation is taken from the scale of degrees on the central line ; then, if the deviation has been determined with the ship's head on an exact compass point, lay off the amount of the deviation on the dotted] line which passes through that point ; but if not observed on the exact point, then on a line parallel to the dotted line, the compass course or direction of the ship's head being still taken from the central line, and mark the point so determined with a cross, or dot encircled in ink. Perform the same operation for each observed deviation. Then with a pencil and a light hand draw a flowing curve, passing as nearly as possible through all the crosses, or dots encircled ; and when satisfied that the curve is good, draw it in ink. This is the curve of deviations. If any of the pencil marks be out of the fair curve, it may be assumed that an error has been made in the observation for that point. Note. — When the curve alters its form considerably near the North point as in several Curves, Examples for Practice, it will he advisable to bring the North points of the dia- gram together in the form of a drum and then draw the curve. The process will be best understood by explaining the projection corres- ponding to the observations as given in the following table : — Ship's Head by Standard Compass. Deviation. Ship's Head by Standard Compass. Deviation. North N.E East S.E 6° 30' W. 13 W. 22 15 W. 23 30 w. South S.W West N.W 5'3o'E. 28 35 E. 19 15 E. 3 E. Deviation Curve in Diagram. I. The first compass course on which an observation has been made is North, and the observed deviation is 6° 30' W. With a pair of dividers take from the central line a distance equal to deviation 6^*^, and from North on the central line lay oflF the deviation on the dotted line which passes through that point towards the left — the deviation being W. ; at the extremity of the distance make a dot or cross, * In Rear Admiral Ryder's plan, the central line is the diagonal of a square and the other lines make angles of 45° with it, and at right-angles to each other and to the sides of the square, which sides are divided into 360' ; the top and bottom representing correct magnetic courses, the sides compass courses. By this method the correct magnetic course correspond- ing to a given compass course, or the compass course corresponding to a given corrt^ct mag- netic course, is found as by a table of double entry. The two methods, it will be seen, are the same in principle. Mr. Napier's will perhaps be found more convenient in construction by the expert; Admiral Ryder's more simple in use by the inexpert. t If the table of deviations are given for the correct magnetic coursos and not the compass courses or direction of the ship's head, the same process is gone through, except that the deviations are in that case laid off on the plain lines. It is, however, now generally under- stood that this procedure is contrary to practice and may lead to error. 164 Deviation of the Compass. 2. The second compass course on which an ohservation has been made is N.E., and the observed deviation is 13° o' W. With dividers take from the central line a distance equal to deviation 1 3", and from N.E. on the central line lay it off on the dotted to the left — deviation being W. ; and the point so determined mark with a cross or dot. 3. The third compass course on which an observation has been made is East, and the observed deviation is 22° 15' W. Take from central line 221°, and from E. on central line and on the dotted line passing through it, lay off the observed deviation to the left — devia- tion being W. ; and mark the point so determined with a dot or cross. 4. Compass course S.E., observed deviation 23° 30' W. Take from the central line a distance equal to deviation 23^°, and from S.E. on the central line lay off on the dotted line passing through the same point the amount of deviation to the left — the deviation being W. ; make a dot. 5. Compass course South, deviation 5° 30' E. Measure on the central line a distance equal to deviation 5!°, and having found compass course South on the central line, lay off the amount of deviation on the dotted line which passes through it towards the right — deviation being E. ; and make a dot or cross. 6. Compass course S.W., deviation 28° 35' E. From the central line take a distance equal to observed deviation 28^°, and having found S.W. on the central line, lay off on the dotted line passing through that point the amount of deviation to the right — deviation being E. ; make a dot or cross. 7. Compass course West, observed deviation 19° 15' E. Measure on the central line a distance equal to deviation 195", and from West lay off on the dotted line passing through that course the amount of deviation towards the right — deviation E. ; and make a dot or cross. 8. Compass course N.W., deviation 3° o' E. Take from central line a distance equal to 3°, and lay off on the dotted line passing through N.W., the amount of deviation (3°) towards the right — deviation E. ; and make a dot. 9. Then, with a pencil and a light hand, draw a flowing curve, passing as nearly as possible through all the crosses or dots, and if satisfied with the curve in pencil, draw it in ink. Note. — The learner should take a pair of dividers and go through the ahove process on the diagrams here given (see Plate I). He should then take the blank diagram (see Plate II), and make the curve on it. Ex. 2. Construct a curve of deviations, using for the purpose the following observations : — (See Deviation Curve A in Diagram.) Ship's Head by Standard Compass. Deviation. Ship's Head by Standard Compass. Deviation. North N.E East 1° 15' W. 22 30 E. 26 50 E. 17 E. South S.W 1° 50' E. 15 W. 26 W. 27 W. West N.W S.E The following describes the process of construction : — 1. With a pair of dividers take from the central line a distance equal to deviation i° 15', or \\°, and from North on the central line, lay the distance off on the dotted line passing through that point and towards the left— being W. ; at the extremity of the distance make a dot or cross. 2. Take from the central line a distance equal to 22^° (22° 30'), and lay it off on the dotted line, from N.E. towards the right — being E. ; make a dot or cross. Deviation of the Compass. 165 3. Take from the central line a distance equal to ^6\° (26° 50'), and lay it off on the dotted line, from Eaat towards the right — being E. ; make a dot or cross. 4. Take from the central line a distance equal to 17" (17=' o'), and lay it off on the dotted line, from S.E. towards the right — being E. ; make a dot or cross. 5. From the central line take a distance equal to i|* (i' 50*), and 1 ly it off on the dotted line, from South towards the right— the deviation being E. ; make a dot or cross. 6. From the central line take a distance equal to 15°, and lay it off on the dotted line, from S.W. towards the left — the deviation being W. ; make a dot. 7. From the central line take a distance equal to 26°, and lay it off on the dotted line, from N.W. towards the left — deviation being W. ; make a dot. t. From the central line take a distance equal to 37°, and lay it off on the dotted line, from W. towards the left — deviation being W. ; make a dot. 9. Repeat the admeasurement first made, from North, at the lower end of the central line. 10. Then, with a pencil and a light hand, draw a flowing curve, passing as nearly as possible through all the dots or crosses ; when satisfied that the curve is good, draw it in ink. This is the curve of deviation. Ex. 3. Construct a curve of deviations, using for the purpose the following observations : — (See Deviation Curve B in Diagram). Ship's Head by Standard Compass. Deviation. Ship's Head by standard Compass. Deviation. North N.E East 0" 22' W, 7 14 E. 10 44 E. 6 18 E, South S.W 0° 16' W. 6 16 W. 11 50 W. 7 52 W. West S.E N.W Selecting those dotted lines which pass through the points representing the different directions of the ship's head, the deviations are laid off: thus, at North, we mark off oi° {0° 22') to the left on the dotted line passing through North — because deviation is West ; at N.E., we take 7^° from the central line, and lay it off to the right (deviation being E.) on the dotted line passing through N.E. ; at East, io| is taken from the central line and laid off to the right on the dotted line passing through East ; and so with the others, being careful to remember that the known deviations must be laid down on the dotted lines, easterly to the right and westerly to the left of the central line. The curve is then drawn neatly through all the points so laid down, and it will be found that the deviation for any other point taken from the curve corresponds with that taken from the Table of Deviations given at page 154 ; and the curve thus drawn can be used instead of the Table. 260. How the Curve is used. — The curve of deviations having been com- pleted, the diagram affords a ready and convenient method of applying the deviation to the ship's course. This correction may be required as follows : — ist, from the compass course which has been steered, it may be required to find the correct magnetic course to be laid down on the chart ; 2nd, from the correct magnetic course given by the chart, it may be required to find the compass course on which the ship's head ought to be kept ; 3rd, if one or more bearings of the land are taken, to correct these bearings by the amount of the deviation due to the direction of the ship's head at the time. The correc- tions are given by the following rules : — 1 66 Deviation of the Compass. 261. To find the Deviation on any Compass Course. EULE LVIII. On the central line find the given course ; then, with a pair of dividers, measure the distance /rowi that point to where the curve cuts the dotted line passing through the course ; but if no dotted line proceeds from the course, then tneasure from the course on the central line to the curve in a direction exactly parallel to the nearest dotted lines : that distance measured on any part of the central line will give the deviation in degrees. Examples. Ex. I. What is the deviation on Compass Course N.E. by N. («) for the deviation curve B ; {h) using the deviation curve A f (a) Having found the given course on the central line, with a pair of dividers measure the distance from N.E. by N. to where the curve cuts the dotted line proceeding from that point; this distance taken to the central line gives 6'' E. (b) Measuring with a pair of dividers the (iistance from N.E. by N. to where the curve cuts the dotted line proceeding from that point, the deviation measured on the central line is found to be 19° E. Ex. 2. Required deviation in compass course W.S.W., using deviation A. Find W.S.W. on the central line, measure the distance from that point to where the curve cuts the dotted line proceeding from it ; this distance taken to the central line gives deviation 21^° W. Ex. 3. What is the deviation for compass course N.E. | N., using the curve C ? Place one leg of a pair of dividers at N.E. \ N. on the central line, and from thence measure the distance to the curve in a direction exactly parallel to the nearest dotted lines ; this distance taken to the central line gives the deviation 12^° W., the deviation for N.E. \ N. Ex. 4. Find the deviation for Standard Compass Course N. 84° W., using deviation curve A. Find N. 84" W. on the central line, and placing one foot of a pair of dividers on that point, from thence measure the distance in the direction of an imaginary line drawn parallel to the nearest dotted lines ; apply this distance to the central line, which shows the deviation for the ship's course is 26 J'^ W. Examples for Practice. Eequired the deviation for each of the following compass courses, using (a) the curve A, and (5) the curve : — S.S.E. ; N.E. by N. ; N.W. ; N.W. by N. ; S.W. by W. ; W.N.W. ; South ; N. 48° E. ; S. 52° W. ; E. by S. ^ S. ; N. 41° W. ; and S. 38° E. Note. — Persons may differ one or two degrees in their estimate of what constitutes a fair curve ; it is therefore quite likely that students may find their answers differ a degree or two from those given in this work. For further exercises in this matter the learner may take the Table of Deviations given on page 154, and using the curve B, find the deviations for each of the 32 points of the compass ; the results ought to agree pretty nearly with those given in the Table. 162. We have now the following easily applied solution of the two following problems : — Problem I.— From a Compass Course, to find the corresponding Correct Magnetic Course. Deviation of the Compass. 167 EULE LIX. On the central line find the given Standard Compass Course, and move on the dotted line draicn from it, or in a direction parallel to the dotted lines till you reach the curve, and then move on a plain line, or in a direction parallel to the plain lines, till you get hack to the central line. The point on the central line at which you arrive is the correct magnetic course required.* Mote. — The directions in the above rule are easiest done by means of a pair of dividers. To move on the dotted line, or in a direction parallel to it, place one leg of a pair of dividers on the course, and the other leg at that point on the curve -which is intersected by the dotted line proceeding from the course, or a point on the curve where a line included between the leg of the dividers on the central line and the le;; on the curve shall be exactly parallel to the nearest dotted lines, then to return to the central line — keep the first leg of the dividers fixed and lift the other ofiF ihe curve, move in a direction parallel to the plain lines until you reach the central line, the point wher?'. the dividers cut the central line shows the correct magnetic course; or keep the leg of the dividers which is on the curve fixed, and move the other leg ofi" the central line parallel to the plain line until it again cuts the central line, which indicates the correct magnetic course required. Examples. Ex. I. The course steered by standard compass is N.N.E. ; what is the correct magnetic course to lay down on the chart (using the curve in the diagram, Plate I.) Find the given compass course N.N.E. on the central line, then take a pair of dividers, put one leg of the dividers on N.N.E., from which extend the other leg along the dotted line passing through the point till the curve is reached, then keeping the leg on the central line fixed, move the one ofi' the curve, and then return to the central line in a direction parallel to the plain line; it will be found to intersect it at N. 13^° E., or N. by E. \ E., nearly, the required correct magnetic course. Ex. 2. The course steered by compass is N.E. by N. : required the correct magnetic course (using Curve A, Plate I). Follow the dotted line extending from N.E. by N. to where the curve cuts it, by placing one leg of a pair of dividers on the course found on the central line and the other at the point where the dotted passing through N.E. by N. cuts the curve, then keeping the leg on the central lino fixed, lift, the other from the curve and move in the direction of the nearest plain lines till the central line is reached ; then thj correct magnetic course will be found to be N. 52' E., or N.E. f E., nearly. Ex. 3. The course steered is S.S.W. : required the correct magnetic course (using curve C, Plate I). Place one leg of the dividers on the compass course S.S.W. on the central line and the other on the place where the dotted line proceeding from it is intersected by the curve, then keeping the leg on the central line fixed, r(;turn with the other leg to the central line in a direction parallel to ih.o plain line, it will be seen that the correct magnetic course is S.W.J W Ex. 4. The course steered by compass is S.E. \ E. : required the corresponding correct magnetic course (using curve C, Plate I.) Place one leg of the dividers on S.E. | E. on the central line, and the other leg on the curve, being careful to keep the two points of the dividers exactly parallel to the nearest dotted lines, then lift the leg off the curve and return to the central line, the place where this last line intersects the central line, shows the correct magnetic course to be S. 75^° E., or E. by S. J S. • It will be observed that this is merely the addition or subtraction, as the curve is to thi right or left of the central line, of the deviation on the course, since the three sides of the triangle ABO passed over by the pencil or leg of dividers are all equal. 1 68 Deviation of the Compass. Ex. 5. Given the standard compass courses N. 38° E. and S. 49° W. : required the correct magnetic courses (using the A deviation curve, Plate I.) Place one leg of the dividers oa the standard compass course N. 38° E. on the central line, and move the other leg out in a direction parallel to the nearest dotted line until it meets the curve; then, keeping the leg which is on the central line fixed, move the other leg in the direction of the plain lines until it returns to the central line. The point arrived at shows the correct magnetic course is N. 58° E., nearly, or N.E. by E. \ E. In a similar manner the correct magnetic course is found to be S. 33° W. Examples for Praotiob. In each of the following examples the compass course is given to find the corresponding correct magnetic course, using curve A and curve C Curve A.-N. 41° W. ; N. 65° 30' E. ; S. 38= E.; S. 79" W.; West; N.E. ; S.E. \ E. Curve C.-N. 39° W. ; N. 48° E.; S. 50° E. ; N>78°W.; N. 70' W. ; N. 34°E.; S. 76" E. ; N. 70^° W. ; N. 26° W. ; and S. 47° W. Problem II.— From a given Correct Magnetic Course to find the corres- ponding Compass Course. RULE LX. On the central line take the given correct magnetic course, and move on the plain line drawn from that point or in a direction parallel to the plain lines till you arrive at the curve ; and then move on a dotted line, or in a direction parallel to the dotted lir^s till you get hack to the central line. The point on the central line at which you arrive is the compass course required.* See Note to Rule LIX, page 167. Examples. Ex. I. Given correct magnetic course N.N.E., to find the corresponding compass course (using curve C). Place one leg of the dividers on N.N.E. on the central line, extend the other leg to the spot where the plain line proceeding from N.N.E. meets the curve ; then keeping the leg on the central line fixed, lift the one on the curve and return to the central line in a direction parallel to the dotted line, it will be found to intersect at N.E. by N., the required course by standard compass. Ex. 2. What compass course will make correct magnetic S.E. (using curve A). Find S.E. on the central line ; place one leg of the dividers on the spot and the other leg on the curve where the plain line that passes through S.E. cuts the curve ; then keep the leg that is on the central line fixed, lift the other leg ofi' the curve and move it in the direc- tion of the nearest dotted line till it again touches the central line ; the compass course that makes correct magnetic S.E. is shown on the central line to be S. 67 1° E., or E.S.E. Ex. 3. It is found from the chart that the correct magnetic course from the ship's position at noon to the Start Point is N. 86'* E. What course must be steered by Standard Compass (using curve C) ? Find the correct magnetic course N. 86° E. on the vertical line ; place one foot of the dividers on the spot, then follow thence with the other leg in a direction parallel to the nearest ^/atV. line until it meets the curve, and then return with the other leg to the central line in a direction parallel to the dotted line ; the compass course required is S. 69^° E. * To assist the memory the following rhyme is given in the Admiralty Manual :— I. " From compass course magnetic course to gain, Depart by dotted and return by plain." II. " But if you wish to steer a course allotted. Take plain from chart and keep her head on dotted." Deviation of the Compass. 169 Ex. 4. Required the compass course to make correct magnetic W. by N. (using curve A). Place one leg of the dividers on W. by N. and the other on the curve where i\\eplai>i line that passes through "W. by N. cuts it ; then keep the leg that is on the central line fixed, but lift the other off the curve and move it in the direction of the dotted line till it again touches the central line ; it will then be seen that the compass course that makes correct magnetic W. by N. is N. 51° W., or N.W. \ W. Ex. 5. Given the correct magnetic courses N. 64° E. and N. 85° W., to find the Standard Compass course (using the C deviation curve, Plate I). Find N. 64° E. on the central line ; place one leg of the dividers on the point and the other on the curve, being careful to keep both legs of the dividers exactly parallel to the nearest plain line ; keep the leg of the dividers that is on the central line fixed and move the other in the direction of the nearest dotted line till it meets the central line ; the point of intersection in this instance is N. 85^° E. In a similar manner the Standard Compass Course corresponding to correct magnetic course N. 85° W. is found to be S. 70" W. Examples for Practioe. In each of the following examples the correct magnetic course is given, to find the compass course (using curve A and curve C). Curve A.— S. 73" 30' W. ; N. 42° 15' E. ; S. 15° 30' W.; N. 14° 15' E. ; N. 62=45' E.; E. 15° S. ; W. 45° S. ; N.E. by N. ; W.S.W. Curve C— S. 44° W. ; N. sH° E. ; S. 5° W. ; N. 24" E. ; N. 83° E. ; S. 50° E. ; S. 22° W. ; N.E. \ E. ; and S.W. \ S. 263. We shall now proceed to show the application of the foregoing rules to Questions 7, 8, g, and 10, of List B, which contains the questions on the Deviation of the Compass required of Candidates for Certificates as Master Ordinary. Question 7. List B. 264. Given the bearings of a distant object ly Standard Compass on eight equidistant* points, to find the Correct Magnetic Bearing of the distant objectf and thence the Deviation. EULE LXI. 1°. If the Compass Bearings are all of the same name, i.e., if they are all reckoned/ro»» N. or S. towards E. or W. : Take the sum of the Bearings in each column, then add these sums together, and divide hy 8 ; the result is the Correct Magnetic Bearing of the distant object of the same name as the Standard Compass Bearings. Note. — A bearing North or South must be expressed as o**, whilst East and West bearing must be expressed as 90°, reckoned either from North or South (as most convenient) towards East or West ; thus East may be written as N. 90° E., or as S. 90° E. ; similarly West may be written either as N. 90" W. or S. 90° W. * The 8 equidistant points are the 4 cardinal points and the four quadrantal points, viz . N.W., S.W., S.E., and N.E. t The Correct Magnetic Bearing thus found is not strictly accurate ; it will difier from the correct quantity by what is called the co-efficient A. The co-efficient A is found by adding the Deviations (algebraically),— that is, add together the Westerly deviations, also add the Easterly deviations together, and take the less from the greater, and mark the dif- ference of the same name as the greater, — and divide by 8. This, however, is not required to be understood by Masters Ordinary, although it is required for Masters Extra. lyo Deviation of the Compass. 2°. If the Compass Bearing be of different names : (a) If some of the Bearings are reckoned from North and the others from South : Take either set/rom i8o°, and they will all he reckoned from the same point North or South ; the name as to East or West remains unaltered ; add all the hearings together and divide by 8, the result is the correct magnetic bearing of the same name as the unaltered bearings. (b) If some of the Bearings are towards the East and others towards the West: Find the sum of those which are reckoned towards East ; and also the sum of those which are towards the West ; then take the less/rom the greater, and mark the difference of the same name as the greater ; the result divided by 8 gives the Correct Magnetic Bearing of the distant object, which is of the same name as the difference. Note. — On the form given at the Marine Board Examinations there is not sufficient space to perform the last mentioned addition and division : there is only room to find the first two Bums ; the rest, however, can he finished in the margin. 3°. To find the Deviation for each of the given Coiirses. — {a) If the Correct Magnetic and Compass bearings are of like names : Take their difference. {b) If one is reckoned from North and the other from South, first take the Correct Magnetic Bearing from 1 8o°, and the remainder will have the same name as the compass bearing, then take the difference between the Correct Magnetic and Compass Bearings. ((?) If both bearings are from North or both from South, but one is towards East and the other towards West, take their sum : The difference or sum wiU be the deviation. 4°. To name the Deviations. — If the Correct Magnetic Bearing is to the right of the Compass Bearing the deviation is East, but if to the left it is West. EXAMPLE I. In the following Table give the correct magnetic bearing of the distant object, and thence the Deviation. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S.4I°W. S. 59 W. S.64 w. S. 60 W. South .... S.W West N.W S.46°W. S. 25 W. S. 18 W. S.23 W. N©TK. — The above is the form in which the Table is given at the Local Marine Board Examinations. The Table is printed, excepting the columns of Bearings, which the Examiner fills up in writing. The following is the above example worked ;— t>eviation of the Compass. 171 Ship's Head by Standard Compass. North N.E. . East . S.E. . Bearing of Distant Olijpot by Standard Compass. S. 41" W. S.59 W. S. 64 W. S. 60 W. I Ship's Head Deviation : by Kequired. jj Standard Compass. i°E. 17 W. 22 W. 18 W. South s.w. . West. N.W. Bearing of Distant Object by Standard Compass. S. 46° W. S.25 W. S. 18 w. S.23 w. Deviation Required. 4''W. 17 E. 24 E. 19 E. S. 224°W. S. 112 w. 8)336 S. II2°W. S. 42° W. = Correct magnetic bearing of distant object. Here we first add together the eight given bearings of the distant object, making 336% and divide the sum by 8, giving as the result S. 42° W., the correct magnetic bearing of the distant object. We next take the difiFerence between the correct magnetic bearing thus obtained and each bearing given in the table ; thus, with the ship's head at North, the bearing by com- pass is S. 41" W., and the difference between this and the correct magnetic bearing S. 42° W. is 1°, and because the correct magnetic bearing is to the right hand of the standard compass bearing the deviation is East. Again, with the ship's head at N.E., the compass bearing is S. 59° W., and the difference between this and the correct magnetic bearing S. 42° W. is 17°, but now the correct magnetic bearing is to the left of the compass bearing, hence the devia- tion is West, and so on with the remaining bearings, and the work will stand as follows: — N. S. 42MV. S. 41 W. r E. S. S. 42°W. S. 46 W. N.E. S. 42° W. S. 59 W. 17 W. S.W. S. 42° w. S. 25 w. East. S. 42° W. S. 64 w. 22 W. West. S. 42°W. S. 18 w. S.E s 42° W. s 60 w 18 w N.W. s 42° w. j^ 23 w. Ship's Head. Correct mag. bear. Standard com. bear. Deviation Ship's Head. Correct mag. bear. Standard com. bear. Deviation 4 W. 17 E. 24 E. 19 E. NoTB. — In the above work, the deviations are obtained by subtracting the less bearing from the greater, because they are all of the same name. (See LXI, 3^ and 4°, page 170.) Q.UE8TI0X 8. List C. (See Eule LX, Problem II, page 168.) From the above table construct a Napier's Curve, and give the courses you would steer by standard compass to make the following courses correct magnetic : — (i.) S.E. (2.) N.E. IE. (3.) S. io°W. (4.) E.^N. To Construct the Curve from the Table. — With a pair of dividers take from the central line 1°, the deviation for ship's head North ; and lay it off from North, on the central line along the dotted line passing through the given point, and towards the right — being East ; at the extremity of the distance make a dot or cross. Next take 17° from the central line, and with one foot of the dividers on N.E. on the central line, lay off this distance on the dotted line passing through the given point and to the left, because the deviation is West. Proceed in like manner with the deviation 22° W. at East; with 18= W. the deviation at S.E. ; and with 4° W. the deviation at ^^outh. The deviation at S.W., West., and N.W. being easterly, must be applied to the right of the central line along the ditted lines pro- ceeding from those points. Now draw with a pencil a curve passing as nearly as possible through the pointa found, and when satisfied with its uniformity, draw it in ink, lyi Deviation of the Compass. To find the Standard Compass Course. — (i.) Place one foot of a pair of dividers on S.E. on tlie central line, and the other foot on the point where the plain line extending from S.E. ia cut by the curve ; then keeping the first foot fixed and lift that on the curve, moving it in a direction parallel to the nearest dotted line, towards the central line ; the compass course that makes correct magnetic S.E. is shown on the central line to be S. 29° E., or S.S.E. | E., (easterly). (2.) Take N.E. \ E. on the central line, then placing one foot of the dividers on that spot extend the other foot from this point and parallel to the nearest plain line until it cuts the curve ; then keeping the first foot fixed, move the one on the curve parallel to the nearest dotted line and towards the central line ; the compass course that makes correct magnetic N.E. 5 E. is shown on the central line to be E. by N. f N. (3.) Place one foot of the dividers on S. 10° W. on the central line, and extend the other foot parallel to the nearest plain line, and to the right until it meets the curve ; then keeping the foot on the central line fixed, move the foot oa the curve thence parallel to the nearest dotted line until it arrives at the central line, which shows that the compass course to be steered is S. 9' W., nearly, in order to make S. 10° W. correct magnetic. (4.) One foot of the dividers being placed on E. \ N., move the other foot to the left from that spot and parallel to the nearest ^^a«w line until it is cut by the curve ; from thence, keeping the first foot fixed on the central line, move parallel to the nearest dotted line till the central line is reached ; the compass course that makes correct magnetic E. J N. is shoAxm on the central line to be S. 72° E. Question 9. List B. Suppose you steer the following courses by the standard compass, find the correct mag- netic courses from the curve drawn: — (See Problem I, Rule LIX, page 167.) (i.) S.S.E.|E. (2.) S.fW. (3.) E. byN.fN. (4.) N. i W. (i.) To find the Correct Magnetic Course. —With one foot of the dividers on S.S.E. | E. on the central line, move the other foot on a line to the left and parallel to the nearest dotted line until it cuts the curve ; then keeping the first foot fixed, move the foot on the curve from thence parallel to the nearest j9/am one and towards the central line ; the correct mag- netic course is at once seen to be S. 455° E. (2.) Placing one foot of the dividers on S. f W. on the central line, move the other foot on a line to the right, parallel to the nearest dotted line and cutting the curve ; from thence, keeping the first foot fixed on the central line, move in a direction parallel to the nearest plain line until the central line is reached ; the correct magnetic course is thus shown to be S. 10° W. (3.) Take the point on the central line representing E. by N. | N., place a leg of the dividers on that point and move the other leg in a direction parallel to the dotted lines, and after meeting the curve, return parallel to the plain lines, until the central line is again reached ; the correct magnetic course is thus found to be N. 48f ° E. (4.) One leg of the dividers being fixed on N. \ W. found on the central line, move the other leg in a direction parallel to the dotted lines till the curve is reached ; and from thence returning to the central line in a direction parallel to the plain lines, we find correct magnetic course is North. Question 10. List B. 265. Q-iven the Bearings of two (or more) distant objects by the Standard Compass and also the Azimuth of the Ship's Head : required the correct magnetic bearing of these objects. Deviation of the Compass. '73 EULE LXn. 1°. Find the Deviation corresponding to the direction of the ship's head, taking it from the Napier'' s Deviation Curve with a pair of dividers. 2°. Apply the Deviation thm found (keeping the legs of the dividers the same distance apart) to the Bearing of distant object hy Standard Compass, thus — Place one leg of the dividers on the part of the central line which represents the Standard Compass Bearing, and lay the other leg on the central line. Upwards if deviation for ship's head is W. Downwards if deviation for ship's head is E. The number of degrees there indicated will be the correct magnetic bearing from N. or S. towards the E. or W. The following bearings of distant objects have been taken by the Standard Compass as above ; with the ship's head as given, find the correct magnetic bearing. 1 Ship's Head. Compass Bearing. SMp'8 Head. OompasB Bearing. I. 2. West S.S.E East. E. by S. i S. 3- 4- E. |N N.E. ^E N. f W. W.iS. On the central line find the given course, West, and with a pair of dividers measure the distance from the point to where the curve cuts the dotttd line proceeding from the course ; this distance taken to the central line gives deviation 24° E. for ship's head at West. We next apply the deviation to East — the bearing of distant object by standard compass — by placing one leg of dividers on central line at East and laying the other leg on the central line downwards (the deviation being East), the number of degrees there indicated ; whence the correct magnetic bearing of distant object is S. 66° E, Again, ship's course S.S.E, is found on the central line ; with dividers measure from thence to where the dotted line proceeding from given course is cut by the deviation curve ; this distance, taken from the central line, gives deviation 13° W. The standard compass bearing of distant object taken when ship's head was at S.S.E. is E. by S. \ S., or S. 76° E., to which apply deviation, found as above, by placing one leg of dividers on S. 76° E. on the central line, and applying the other leg upwards on the central line (deviation being West) ; the result is the correct magnetic bearing S. 89° E. The deviation due to E. | N. — the direction of the ship's head — is found on deviation curve to be 22^° W. Apply this deviation to compass bearing N. J W., or N. 8° W., to upwards on the central line (deviation being West), which gives correct magnetic bearing of distant object N. 30^° W. N.E. J E. is the next given direction of the ship's head, and applied to the central line, the deviation for which on curve is found to be 1 8" W. The deviation thus found being applied upwards from W. ^ S., or S. 87" W., gives the correct magnetic bearing of distant object S. 69° W. '74 Deviation of the Compass. EXAMPLE II. I. In the following Table give the correct magnetic bearing of the distant object, and thence the Deviation : — (Correct Magnetic Bearing of distant object = N. 89° 4' W.) Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Bequired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Kequired. North .... S. 87°W. 3»E. South .... N. 87»W. 2°W. N.E S. 70 W. 31 E. s.w N. 75 W. 14 w. East S. 71 W. 20 E. West N. 68 W. 21 W, S.E S. 81 w. 10 E. N.W N. 72 W. 17 w. S. 310 w. N. 302 W. S. 418 w. (180° X 4) = 720 8)728 S.418 W. S. 91 w. = 180 Correct Magr letic Bearing. or N. 89 W. In this example the bearings given in the left hand column are reckoned from S. towards W., while the bearings in the right hand column are reckoned from N. towards "W. ; there- fore, before adding up the latter column, each bearing must be subtracted from 180°, and the remainder, in each case, is the bearing reckoned from S. towards W. Subtracting N. 87° W. from 180°= S. 93° W. which put in the place of N. 87° W. N. 75W. „ i8o=S. 105W. „ „ N. 75W. N. 68 W. „ i8o=:S. 112W. „ „ N. 68 W. N. 72 W. „ 180 = 8. 108 W. „ „ N. 72 W. We may, however, proceed as above, viz. : — add up the Bearings reckoned from N. towards W. ; and since there are four bearings so reckoned (from N. towards W.) ; take the sum from 720 (180 X 4) ; the remainder is the sum of the bearings to be reckoned from S. towards W. It is evident that to subtract the sum of the four bearings from 720° is the same thing as to subtract each bearing from 180° and to add the remainders. N. 91° W. 88 W. 3E. Ship's Head. Correct magnetic bearing S. Compass bearing S, Deviation Ship's Head. South. Correct magnetic bearing N. 89° W. Compass bearing N. 87 W. Deviation 2 W. N.E. S. 91- W. S. 70 w. 21 E. S.W. N. -Sg" W. N. 75 W. East. S. 91° W. 71 w. 20 E. S West. N. 89° W. N. 68 W. 21 W. S.E. S. 9i=>W. 8. 81 W. 10 E. N.W. N. 89° W. N. 72 W. 17 W. 14 w. II. From the above Table construct a Napier's curve, and give the courses you would steer by standard compass to make the following courses correct magnetic : — (I.) W.byS.fS. (2.) N.^E. (3.) E. |N. (4.) S.E. | S. Answers:— (i.) West. (2.) N. 1° E. (3.) N. 60^^ E. (4.) S. 49° E. Note. — For the method of constructing the curve, see No. 259, page 163, and for Rule to find standard compass course to steer, see Rule LX, page 168. Deviation of the Compass. »75 III. Suppose you steer the following courses by standard compass, find the correct magnetic courses from the curve drawn : — (i.) North. (2.) S.S.W. i W. (3.) E. by S. \ S. (4). N.E. \ E. Answers:— {\). N. 3" E. (2.) S. 15° W. (3.) S. 59= E. (4.) N. yi'E. IV. You have taken the following bearings of a distant object by your standard compass as above ; with the ship's head as given, find the correct magnetic bearing. Ship's Head. Compass Bearinf s. Ship'i Head. Compass Bearinga. S.fW N.W. by W. . . South. E. f N. N.N.E.JE. .. S.E S.W. i s. E.^S. By the curve the deviation for ship's head S. f W. is s\° W. ; then placing one leg of the dividers on compass bearing South on central line, and laying the other leg upwards (devia- tion being West) the correct magnetic bearing of distant object is found to be S. ^\° E. The deviation by the curve for ship's head N.W. by W. is 1 7° W. ; then placing one leg of the dividers on the part of central line representing compass bearing E. f N., the other leg applied upwards on the central line intersects it at N. 63° E. : what is the correct mag- netic bearing ? For position of ship's head N.N.E. f E. the deviation by the curve is i8^° E. ; placing one leg of the dividers on the compass bearing S.W. \ S., and laying the other leg down- wards (deviation being East) it shows the correct magnetic bearing to be S. 60^' W. Ship's head S.E. gives on curve deviation 10^° E. ; one leg of dividers placed E. \ S. and the other leg applied downwards (deviation being East) gives on central line correct magnetic bearing S. 76^° E. EXAMPLE III. Question 7. List B. I. From the following Table find the correct magnetic bearing of the distant object, and thence the deviation : — Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. 1 Deviation Reqmred. 1 Ship's Head by Standard Compais. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E N. i°E. N. 13 W. N. 15 W. N. 8 W. South .... S.W West .... N.W N. s'W. N. I E. N. 9 E. N. 12 E. In this example some of the bearings are towards East and the others towards West ; we therefore write down in one column those which are reckoned towards East, and add these up; we likewise write down in another column those which are reckoned towards Wegt, and add up this column ; the less sum being subtracted from the greater gives as a remainder N. 16^ o' W., which divided by 8 gives N. 2"^ W. for the correct magnetic bearing: the work will stand thus : — Ship's Head. Compass Bearings. Ship's Head. Compass Bearings. North N. i=>E. N.E. N. i3'=W. S.W. N. I E. East N. 15 W. West N. 9 E. S.E. N. 8 W. N.W. N. 12 E. South N. 3 W. N. 23 E. N. 39 W. N. 23 E. 8) N. 16 W. Correct magnetic bearing is N. 2 W. 176 Deviation of the Compass. The deviation for each position of the ship's head is next to he found, and we proceed as follows : — Ship's Head. North. Correct magnetic bearing N. 2° W. Compass bearing N. i E. Deviation 3 W. Ship's Head. South. Correct magnetic hearing N. 2" W. Compass bearing N. 3 W. N.E. N. 2=W. N. 13 W. East. N. 2" W. N. IS W. S.E. N. 2° W. N. 8 W. II E. 13 E. 6 E. S.W. N. 2°W. N. I E. West. N. 2' W. N. 9 E. N.W. N. 2'W. N. 12 E. Deviation I E. 3 W. ri W. 14 W. Question 8. List B, From the above Table construct a Napier's curve, and give the courses you will steer hy standard compass to make the following courses, correct magnetic. X. N.E. ^E. 5. N.N.E. 2. W.S.W. 6. S. i8|°W. 3. N.N.W. 7. N. 4° E. 4. S.S.E. 8. S. ez^'^E. NoTB. — For the method of constructing the curve, see No. 259, page 163, and for Rule for finding standard compass courses to steer, see Problem II, Rule LX, page 168. . — I. N. 40° E. 5. N. 18 E. 2. S. 76" W. 6. S. 19 W. 3. N. i4i»W. 7- N. 5 E. 4. 8. 26" E. 8. S. 73 B. Question 9. List B. Suppose you steer the following courses by the standard compass, find the correct mag- netic courses from the curve drawn : — I. E. byS. JS. 2. E. JN. 3. S. 85° W. 4. N. 1° E. For the Rule for working this Question, see Problem I, Rule LIX, page 167. Answer.— 1. S. 6s\ E. 2. N. 84J E. 3. N. 74^ W. 4. N. ij W. Question 10. List B. You have taken the following bearings of distant objects by your standard compass as above, with the ship's head as given, find the correct magnetic bearings. (See Rule LVIII, page 166). Ship's Head. S.E. by E. W. i N. . , Compass Bearing. N. 68° E. S. 54 W. Ship's Head. S.S.W. i W. N. fE Compass Bearing. N. 4» E. South. The deviation by curve for ship's head S.E. by E. is 7^° E., and the correct magnetic bearing corresponding to N. 68" E. is found to be N. 75^° E. Ship's head W. ^ N. gives on curve the deviation 11° W., whence the correct magnetic bearing corresponding to compass bearing S. 54° W. is found to be S. 43° W. Ship's head S.S.W. J W. gives deviation i^° W. on the curve, and the correct magnetic bearing corresponding to compass bearing N. 4° E. is found to be N. 2j° E. Ship's head N. f E. gives on curve the deviation o, whence the correct magnetic bearing is South— the same as the compass bearing. Dwiation of the Compass. m Examples for Praotioe. Example I. I. Prom the following Table find the correct magnetic bearing of the distant object, and thance the deviation : — (S.S. B- Heeling io° to Port). Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East .... S.E N. ']^°W. W. I s. W. 19 s. S. 62 w. South .... s.w West .... N.W S. 76°W. W.12N. W.19N. N. 72 W. Answer : — Correct magnetic bearing, West, or S. 90° W. In such a case as this, where the bearings are near the East or West points, and are given from different points, they must all be expressed from either the North or South point by taking their equivalents as fallows, where they are all expressed for the North point. The first bearing is expressed from the North towards the West. The second bearing is expressed from the West towards the South, but by adding 90° we get the equivalent bear- ing N. 91° W.; similarly, the third bearing becomes N. 109' W. The fourth bearing 5. 62^ W. is taken from 180", and this becomes N. 118° W., and the fifth bearing is also taken from 180°, and the equivalent bearing is thus found to bo N. 104° W. The sixth and seventh bearings being reckoned from W. to N., subtracting each from 90°, whence we get N. 78° W. and N. 71° W. as the respective equivalent bearings. The last bearing being N. 72° W., is written down as it stands, unchanged. Answer: — Deviations— i. 13° W. 2. 1° E. 3. 19° E. 4. 28° E. 5. 14° E. 6. 12" W. 7. 19° W. 8. i8»W. II. From the above Table construct a Napier's curve, and give the courses you would steer by the standard compass to make the following courses correct magnetic: — I. N.E.^N. 2. E.byS. JS. 3. S.S.W. | W. 4. N.N.W. i W. Answer:—!. N. 42° E. 3. N. 87° E. 3. S. 28" W. 4. N. 10° W. III. Given the following courses steered by standard compass to find the correct mag- netic courses from curve drawn : — I. N. byE. ^E. 2. S.S.E. ^E. 3. W. ^ S. 4. W. by N. | N. Answer:— I. N. 6^° E. 2. S. 2° E. 3. S. 69° W. 4. N. 89° W. IV. You have taken the following bearings of distant objects by your standard compass with the ship's head S.E. by E. J E. : find the correct magnetic bearing : — Standard compass bearings: — W. by N. \ N. and N. f W. Answer. — Deviations: — i. 26' E. 2. 20° W. Correct magnetic bearings : — i. N. 51° W. 2. N. i8°W. AA 178 Deviation of the ConytoHs. Example II. I. From the following table find the correct magnetic bearing of the distant object and thence the deviation : — Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. SMp's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.B East S.E N. 42° W. N. 17 W. N. 9 W. N. 17 W. South .... S.W West .... N.AV N. 44°W. N. 68 W. N. 76 W. N. 64 W. Answer : — Correct magnetic bearing N, 42° W. Deviation o» ; 25" W. ; 33° W.; 25" W.; 2° E. ; 26° E.; 34° E. ; 22° E. II. From the above table construct a Napier's curve and give the courses you would eteer by standard compass to make the following courses correct magnefic : — r. N.W. iW. 2. S.E. ^S. 3. E.iS. 4- W. | S. Answer:— I. N. 84° W. 2. S. 24° E. 3. S. iS^E. 4. S. 53^° W. III. Suppose you steer the following courses by standard compass, find the correct mag- netic courses from the curve drawn : — I. W. byN. fN. 2. South. 3. E. by N. ^ N. 4. N. by E. | E. Answer:—!. N. 39° W. 2. S. if° W. 3. N. 43^" E. 4. N. 7' B. rv. You have taken the following bearings of distant objects by your standard compass as above; with the ship's head as given, find the correct magnetic bearing; — Ship's Head. Bearing of Distant Object by Standard Compass. Ship's Head. Bearing of Distant Object by Standard Compass, S.E E. by S N. by E. 1 E. N. by W. f W. W. by S. 1 S. . . N.E.JE E. JN. S.W. 1 S. Answer : — Deviations for S.E. is 25° W. ; for E. by S. is 32° W. ; for W. by S. f S. is 32° E. ; N.E. \ E. is 26|° W. Correct magnetic bearings N. 80° W. ; N. 52° W. ; S. 64=' E. ; S. 10° W. Example III. I. From the following table find the correct magnetic bearing of the distant object, and thence the deviation : — Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Reqiiired. North .... N.E East S.E N. 13'^W. N. 35 W. N.41 w. N. 35 W. South .... S.W We.9t N.W N. 23°W. N. 2 W. N. 5 E. North. Answer : — Correct magnetic bearing N. 1 8° 6' W. Deviations :— 5° W. ; 17° E.; 23° E.; 17° E.; 5° E. ; 16° W.; 23° W.; 18° W. Deviation of the Compass. {•fg II. From the above table construct a Napier's curve, and give the courses you would steer by standard compass to make the following courses correct magnetic : — I. N. ^W. 2. E. iS. 3. N.W. by W. J W. 4. W. | S. Answ0r:—i. North. 2. N. 73° E. 3. N. 44° W. 4. N. 71° W. III. You have steered the following courses by standard compass : find the correct mag- netic courses from the curve drawn : — I. E. ^N. Answer : — i. S. 72^° K . S.E. |E. 3. N. fW. 4- S.W. is. 2. S. 35° E. 3. N. i6^° W. 4. S. 27" W. IV. You have taken the following bearings of distant objects by your standard compass ; with the ship's head as given below, find the correct magnetic bearing : — Ship's Head. Bearing of Distant Object by Standard Compass, Ship's Head. Bearing of Distant Object by Standard Compass. S.W.JS North E. iS. S.E. i S. N.E. i N E. byS. ^S. .. W. JS. E. iS. Answer .-—Deviations :— 14° W. ; 5° W. ; 16° E.; 22° E. Correct magnetic bearings, N. 84° E. ; S. 42° E. ; N. 77° W. ; S. 65° E. Example IY. I. From the following Table find the correct magnetic bearing of the distant object, and thence the deviation : — (S.S. Royal Charter, before compensation.) Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E N. 8i=E. N. 79 E. N. 68 E. N. 48 E. South .... S.W West N.W N. i6°E. N. 2 E. N. 8 E. N. 51 E. 4° W. Answer : — Correct magnetic bearing N. 44° E. Answer. — Deviations: — i. 37° W. 2. 35" W. 3. 24° W. 5. 28' E. 6. 42° E. 7. 36" E. 8. 7^W. II. From the above Table construct a Napier's curve, and give the courses you would steer by standard compass to make the following courses correct magnetic : — I. E. JN. 2. N. |W. 3. S. byW. iW. 4. W. ^ N. Answer:—!. S. 77° E. 2. N. 29^ E. 3. S. 7° E. 4. 8. 53° W. m. Suppose you steer the following courses by standard compass, find the correct mag- netic courses from the curve drawn : — I. N.E. \ E. 2. S.E. \ S. 3. South. 4. W. \ S. Answer:—!. N. i6|° E. 2. S. 39^' E. 3. S. 28° W. 4. N. 58^° W. IV. You have taken the following bearings of distant objects by your standard compass as above ; with the ship's head ^J. by E. ^ E., find the correct magnetic bearings. Standard compass bearings : — S.W. ^ W. and S. f E. Answer. — Deviation : — 34° W. Correct magnetic bearings : — S. 6" W. and S. 43° E. I So Deviation of the Compass. Example V. I. From the following table find the correct magnetic bearings of the distant object, and thence the deviation. (S.S. Eoyal Charter, after compensation.) Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Recniired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 88" E. South. S. 88 E. S. 83 E. South .... s.w West N.W S. 82" E. S. 86 E. S. 88 E. S. 84 E. Answer : — Correct magnetic bearing S. 86° E. Answer.— UsVidliona:— I. 2° E. 2. 4° E. 3. 2= E. 4. 3° W. 5. 4° W. 6. 0°. 7. 2° E. 8. 2° E. II. Given the correct magnetic cotirse, to find the courses to steer by standard compass : — I. W. JN. 2. S.W. iS. 3. E. byS. JS. 4. N.N.E. J E. Answer.— 1. N. 89° W. 2. S. 42° W. 3. S. 73°E. 4. N. 22^° E. m. Given courses by standard compass, to find correct magnetic : — I. N.byW.fW. 2. W. byN. fN. 3. S. J E. 4. E. by N. J N. Answer.— I. N. 20° W. 2. N. 70° W. 3. S. 7^° W. 4. N. 78° E. IV. You have taken the following bearings of distant objects by standard compass with ship's head as given below, find the correct magnetic bearing ; — Ship's head S.S.E. \ E., compass bearings S. \ W. and N.E. by E. \ E. Correct magnetic bearings : — S. 1° W. and N. 57^° E. Example VI. I. From the following table find the correct magnetic bearing of the distant object, and thence the deviation. (S.S. Royal Charter, at Melbourne.) Answer : — Correct magnetic bearing S. 83|-° W. = S. 84° W. Deviations:— 22° E.; 18° E.; 1° E. ; 19° W.; 21° W.; 15° W. ; 1° W. ; 17" E. IE. From the above table construct a Napier's curve, and give the course you would steer by standard compass to make the following courses correct magnetic: — I. N. ^ E. 2. S. i W. 3. E. J S. 4. N.E. ^ E. Answer.— I. N. 16° W. 2. S. 25° W. 3. S. 86° E. 4. N. 29^° E. III. You have steered the following courses by standard compass ; find the correct magnetic courses from the curve drawn : — I. N.E. i N. 2. S.E. ^ S. 3. W. I N. 4. N.W. ^ N. Answer:—!. N. 59'' E. 1. S. 59° B. 3. N. 87" W. N. 21° W. Dwiation of the Compass. i8i IV. You have taken the following bearings of distant objects by your standard compass as above ; with the ship's head by standard compass N. \ W., find the correct magnetic bearing. Bearings of distant object by standard compass S.S.E. \ E. and E. ^ N. Answer : — Deviation 23° E. Correct magnetic bearings : — S. 2^° E. and S. 73" E. Example VII. I. In the following table give the correct magnetic bearing of the distant object, and thence the deviation. (S.S. Royal Charter, after return to Liverpool from Melbourne.) Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S.i7°E. S. 14 E. South S. II W. South .... S.W West N.W S. 12»W. S. 3 W. S. 7 E. S. 12 E. Answer : — Correct magnetic bearing, S. 16° E. Deviations:— 14° E.; 11° E.; 3° W. ; 14' W.; 15° W.; 6° W. ; 4° E. ; 9° E. II. From the above table construct a Napier's curve, and give the course you would steer by standard compass to make the following courses correct magnetic : — I. N.E. |N. 2. E. fN. 3. S. 8^°W. 4. N.W. ^ W. Answer.— I. N. 29^° E. 2. N. 8i^° E. 3. S. 20^" W. 4. N. 58° W. III. Given the following courses by standard compass, to find the correct magnetic courses from the curve drawn : — I. N.E. IE. 2. N.N.W. iW. 3. W. byS. |S. 4. S.E. | E. Answer.— I. N. 60° E. 2. N. 13° W. 3. S. 70^° E. 4. S. 66^° E. rV. You have taken the following bearings of two distant objects by your standard compass, with the ship's head S.S.E. | E. St. Catherine's Point, N.E. ^ N. Needles Light, N.N.W. J W. Answer. — Deviation for ship's head, N.N.W. ^ W. is 153° W. Correct magnetic bearings : — N. 24° E. and N. 43° W. EXAMPLB Vin. I. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required- Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East .... S.E N.6o»W. N. 80 W. N. 84 W. N. 78 W. South .... S.W West .... N.W N. 66°W. N.53 w. N. 42 W. N. 41 W. Answer : — Correct magnetic bearing of distant object is N. 63° W. Deviations :— 3° W. ; 17° E.; 21" E.; 15° B. ; 3° E. ; 10° W.; 21° W. ; 22° W. l82 Deviation of the Compass. II. From the Table construct a Napier's curve (a), and give the courses you would steer by the standard compass to make the following courses correct magnetic: — I. N. by E. 2. S.W. \ W. 3. E. J N. 4. N.N.W. 5. E.S.E. 6. S.W. \ S. 7. N. JE. 8. W. |S. 9. N. JW. 10. S. iW. II. E. iS. 12. E.N.E. For the method of constructing the curve see No. 259, page 163 ; and for the Rule for finding the correct magnetic course see Problem II, Rule LX, page 168. Answer:— 1. N. 8|° E. 2. S. 65!° W. 3. E.N.E. 4. N. 12" W. 5. S. 88° E. 6. S. 50° W. 7. N. 3f° E. 8. N. 75° W. 9. North. 10. S. 1° E. II. N. 72^''E. 12. N. 49^''E. III. Suppose you have steered the following courses by standard compass, find the correct magnetic courses from the curve drawn : — (See Problem I, page 167.) I. N.E. by E. 2. E.S.E. 3. S.W. by S. 4. South. 5. S.E. J S. 6. W. byN. fN. 7. S. i W. 8. E. | N. 9. W. | S. 10. N.E. by E. J E. II. N.W. JN. 12. S.W. JW. Answer :—i. N. 75!" E. 2. S. 49^° E. 3. 8. 27 J° W. 4. S. 4° W. 5. S. 26° E. 6. S. 86" W. 7. S. 5^° W. 8. S. 77^° E. 9. S. 62° W. 10. N. 82° E. II. N.6o°W. 12. S. 4i|''W. IV. You have taken the following bearings by your standard compass as above ; with the ship's head as given, find the correct magnetic bearings: — No. Ship's Head. W.N.W. ... S.E. by E. N.E. by N. E. by S. I S. Bearing of Distant Object by Standard Compass. South. N.W. by N. North. N.N.W. Answer : — i. Corr. mag. bear. S. 24° E. 5. S. 3^° E. 6. N. 72° E. 7. S. 72 No. Ship's Head. N.iE W. byN.^N.. E. i N E. " fS. Bearing of Distant Object by Standard Compass. S. iE. E. 4S. E. iN. W. by S. \ S. 2. N. 17' W. E. .. 3. N. 14° E. N. 84° W. 4. N. 3° W. 266. From the definitions and principles given in pages 84 and 90, are deduced the following formulae or equations, and these expressed in words constitute the common rules of navigation for finding the place of a ship, that is, its latitude and longitude. FUNDAMENTAL FORMULA OF NAViaATION. Departure = distance x sine course, [i] . * . log. dep. = log. dist. + log. sine course — 10. True diff. lat. = distance x cosine course, [2] . ' . long, true difi". lat. = log. dist. X log. cos. course — 10. DifF. long. = meridional diff. lat. X tang, course, [3] . ' . log. diff. long. = log. mer. diff. lat. + log. tang, course — 10. In parallel sailing. Distance = diff. long, x cos. lat., [4] . * . log. dist. = log. diff. long. + log. cos. lat. — i o. In middle latitude sailing. Dep. (nearly) = diff. long. X cos. mid. lat, [5] . ■ . log. dep. (nearly) = log. diff. long. + log. cos. mid. lat. — 10. NAPIER'S DIAGRAM, Showing" Curve of Deviation. Ship West Deviation. East Deviation, N.bW. i83 THE TRAVERSE TABLE. 267. In all collections of tables for the use of navigators there is inserted a table containing the true dijference of latitude and departure, corresponding to certain distances (at intervals of one mile) up to 300 nautical miles, for every course, at intervals of a quarter point, and also of degrees, from 0° to a right-angle (90°). Tables I and II (Eaper or Norie). This table is constructed by solving a right-angled triangle, of which one angle represents the course, and the hypothenuse the distance; by giving these different and successive values, the corresponding values of the other two sides are found, which sides represent the true difference of latitude and departure. Thus, in the triangle CAB, right-angled at C : if the angle CAB represent a given course, and A B a given distance, the side A will be the true diff. of lat., and B the dep. corresponding to that course and distance. Given course 30°, and distance 25 miles, compute cor- responding true diff. lat. and dep. In the triangle CAB, let A =: 30°, and AB = 26°; then true diff. lat. = A .= A B X cos. A = 25 X cos. 30°, .'. true diff. lat. = 2i'-65, and dep. = BC = AB X sine A = 25 X sine 30°, .* . dep. = i2'"5. Inasmuch as the sine of an angle is the cosine of its complement, it is evident that the difference of latitude and departure for any course are the dep. and diff. lat. for the comple- ment of that course. Thus, let it be required to find the diff. lat. and dep. for course 60° and dist. 25 miles: — diff. lat. for 60° = 25 X cos. 60*^ = 25 X sine 30° = dep. for 30°, and dep. for 60° = 25 X sine 60° = 25 X cos. 30°= diff. lat. for 30°. "When the diff. lat_ and dep. are computed in this manner up to 45°, the diff. lat. and dep. for course above 450 may be found by changing the titles to the columns, and hence the table is compactly arranged by interchanging the headings of the columns containing these elements at the top and bottom of the page, and reading the top reading for courses from 0° to 45^, and the bottom reading for courses from 45° to 90°. This table may be used for a great number of problems depending for their solution on the relation of the several parts of a right- angled triangle, and since all the relations between any two quantities may be expressed as functions of some angle in terms of the sine, cosine, or tangent ; it may be used, in fact, as a general proportional table. In these Tables the course is found at the top of the Table, when under 4 points or 45° ; but at the bottom of the Table, when it exceeds 4 points or 45°. The first column contains the distance to 60 miles, the second column contains the difference of latitude, expressed in minutes and tenths, and the third column, similarly expressed, contains the departure ; but if the coxirse exceeds 4 points, or 45°, the second column contains the departure, and the third column the difference of latitude. The other columns are a continuation of the former, exactly upon the same principle, and extending to 300 miles of distance. (See Tables I and II, Norie or Eaper.) i83 THE TRAVERSE TABLE. 267. In all collections of tables for the use of navigators there is inserted a table containing the true difierence of latitude and departure, corresponding to certain distances (at intervals of ono mile) up to 300 nautical miles, for every course, at intervals of a quarter point, and also of degrees, from 0° to a right-angle (90"). Tables I and II (Eaper or Nokib). This table is constructed by solving a right-angled triangle, of which one angle represents the course, and the hypothenuse the distance ; by giving these diflPerent and successive values, the corresponding values of the other two sides are found, which sides represent the true diflference of latitude and departure. Thus, in the triangle CAB, right-angled at C : if the angle CAB represent a given course, and A B a given distance, the side AC will be the true diff. of lat., and B the dep. corresponding to that course and distance. Given course 30°, and distance 25 miles, compute cor- responding true difif. lat. and dep. In the triangle C A B, let A = 30°, and A B = 26° ; then true diff. lat. = A =: A B X cos. A = 2 5 X cos. 30°, .*. true diff. lat. = 21 ''65, and dep. = BC = AB X sine A = 25 X sine 30°, .■ . dep. = i2'*5. Inasmuch as the sine of an angle is the cosine of its complement, it is evident that the difference of latitude and departure for any course are the dep. and diff. lat. for the comple- ment of that course. Thus, let it be required to find the diff. lat. and dep. for course 60° and dist. 25 miles: — diff. lat. for 60° = 25 X cos. 60*^ = 25 X sine 30° = dep. for 30", and dep. for 60" = 25 X sine 60° = 25 X cos. 30°:= diff. lat. for 30°. "When the diff. lat_ and dep. are computed in this manner up to 45°, the diff. lat. and dep. for course above 450 may be found by changing the titles to the columns, and hence the table is compactly arranged by interchanging the headings of the columns containing these elements at the top and bottom of the page, and reading the top reading for courses from 0° to 45°, and the bottom reading for courses from 45° to 90°. This table may be used for a great number of problems depending for their solution on the relation of the several parts of a right- angled triangle, and since all the relations between any two quantities may be expressed as functions of some angle in terms of the sine, cosine, or tangent ; it may be used, in fact, as a general proportional table. In these Tables the course is found at the top of the Table, when under 4 points or 45° ; but at the bottom of the Table, when it exceeds 4 points or 45°. The first column contains the distance to 60 miles, the second column contains the difference of latitude, expressed in minutes and tenths, and the third column, similarly expressed, contains the departure ; but if the course exceeds 4 points, or 45°, the second column contains the departure, and the third column the difference of latitude. The other columns are a continuation of the former, exactly upon the same principle, and extending to 300 miles of distance. (See Tables I and II, Noriis or Eaper.) 184 Trmer%e Talh. USE OF THE TABLE. 268. Q-iven the courBe and distance, to find the difference of latitude and departure. EULE LXin. With the Course open the Tables, and under or above the proper number of points {or degrees), and opposite the distance, will be found the difference of latitude and departure. Ohs. — When the course is found at the bottom of the page, care must be taken to see that the diff. of lat. and the dep. are taken from the proper column above the words departure and diff. lat. It must be carefully remembered that when the course is less than 4 points or 45°, the diff. lat, exceeds the dep. ; but when it is more than 4 points or 45° the dep. exceeds the diff. lat. Examples. Ex. I . A ship sails N.W. ^ N. a distance of 7 8 miles : required the diflference of latitude and departure by inspection. The given course is 3^ points ; and referring to Table I we find the page assigned to this course to be page 14, Norie, or page 436, Kaper's Navigation, in which against 78, in column headed dist., stands 60*3 under the head lat., and 49*5 under the head dep. We conclude, therefore, that for the given course and distance, the difference of latitude is 6o'3 miles, and the departure 49-5 miles. Ex. 2. Suppose the course to be 5^ points, and the distance 98 miles. Then, since the course here exceeds four points, we look for it at the foot of the page (page 10, Norie, or 432, Raper), and against 98 in the distance column we find in the adjacent column (marked at the bottom dep. and diff. lat.) dep. 86-4, and diff. lat. 46-2, so that the difference of latitude made is 46'2, and the departure 86-4. Ex. 3. Course N.E. by N., distance 129 miles: find diff. lat. and dep. Enter Table I, and find 3 points at the top, and in one of the columns marked dist. find the distance 129, then in the columns opposite to this, marked lat. and dep. at the top, stands the difference of latitude 107 -3, and departure 71 •7. Ex. 4. Course E. by N. |- N., distance 264 miles : find diff. of lat. and dep. Open Table I at 6^ points, found at the bottom, and opposite the distance 264 stands departure 252'6, and difference of latitude 76-6. Ex. 5. A ship sails N. 40° E., 50 miles : required the diff. of lat. and departure. The course being less than 45°, is found at the top (Table II), and the distance being under 60 miles, is found in the left hand column ; therefore, on the page (56 Nouie) is 40° at the top, and opposite to 50 in the distance column (marked Dist.) is 38-3 under Lat., and 32" I under Dep., the difference of latitude and departure required. Ex. 6. A ship sails N. 64° W,, 175 miles : required the diff. of lat. and departure. The course being more than 45°, is found at the bottom, in page 42, and opposite to the distance 175 miles, is 76'7 over Lat., and i57'3 over Dep., which was required. {a) To find diff. lat. and dep. when there are tenths in the distance. Take the distance as an entire number of miles, i.e., as a whole number, and find the corresponding diff. lat. and dep., /row each of which cut off the right hand figure, or tenths, and remove the decimal point one place to the left hand, which will give the required diff. lat. and dep. in miles and tenths of a mile.* The tenths, however, must be increased by i, if the figure cut off is 5, or upwards. * The reason of this rule is that the Traverse Table is entered with a distance ten times as great as the given distance, and the resulting diff. lat. and dep. is divided by ten. This is done by merely imagining the decimal point to be removed one place to the right before entering the table, and then one place to the left after taking out the results (see p. 26, (10). Traverse Table. 185 Examples. Ex. I. Course 3^ points, distance 20-3 ; required the diflf. lat. and dep. corresponding thereto. With course 3^ points, and dist. 20'3, taken as 20*3, wo get tho diff. of lat. i56'9, dep, i28"8; now cut off the right hand fii;uro of each (the 9 and 8), and shifting the decimal point one place to the left, we have diff. lat. i5"7, and dep. 12-9. It will be observed that the tenths are increased by i, in each case, as the figure cut off in both cases exceeds 5. Ex. 2. Required the diff. lat. and dep. corresponding to course 4^ points, and dist. 24"3 miles. With course 4^ points, and dist. 24-3 (as 243 miles), we find diff. lat. i54"2, and dep. i87"7 : hence we obtain, after dropping the tenths, and removing the decimal point in each one place to tho left, i5'4 and i8"8, for the required quantities. The tenths in the dep., it will be observed, are increased by i, since the figure dropped exceeds 5. Ex. 3. A ship sails N. 67° E., distance 29'*5 : find diff. of Ut. and dep. corresponding. In this case take out for distance 295. Thus, for 67° and distance — 295 =: ii5"3 •i'ff' l8.t., 27r-5 dep. . ■ . 295 = 11-5 diff. lat., 27'-2 dep. After dropping the tenths, and removing the decimal point one place to the left, we have diff. lat. 1 1-5 and dep. 27-2 (the tenths of dep. is increased by i, as the figure cut off is 5.) Ex.4. N. 34" W., and dist. 20-6 miles (as 206), give diff. lat. 170-8, and dep. ii5'2 dropping the tenths in each case (the 8 and the 2), and shifting the decimal point one place to the left, we get diff. lat. 17-1 N., and dep. 11-5 W. The tenths of diff. lat. must be increased by i, as the figure cut off (8) exceeds 5. Ex. 5. N. 65" E., and dist. 21-5 (as 215), give diff. lat. 90*9, and dep. i94'9, which is diff. lat. 9'i N., and i9'5 E. It will be observed that tho tenths are increased by i, in each case, as the figure dropped exceeds 5. {h) If the distance exceeds the limits of the Traverse Table. Take the half, the third, &c., so as to bring it toithin the limits, taking care to multiply the corresponding quantities by 2, 3, &c. Ex. 6. Let the course be 37°, and distance 435 : required the corresponding diff. lat. and dep. 435 divided by 3 gives 145. Course 37°, and dist. 145, give diff. lat. ii5"8 and dnp. 87-3 X 3 X 3 Diff. lat. 347*4 Dep. 261 9 If the distance had been 43^5, the diff. lit. would have been 34*7, and the dep. 261. (c) But when the distance is between 300 and 600 we may proceed as follows : — Take out diff. of lat. and dep. for 300, and for the excess of 300 ; take the sum of tho quantities thus found, cut off the last figure, and remove the decimal point as before. BB i86 Traverse Table. Ex. 7. Course 65°, and distance 526 -. rrquired the correspon'Hng diff. of lat. and dep. Course 65°, and dist. 300, give diff. lat. 126 8, and dep. 271*9 226 955 204-8 526 222-3 476-7 If the distance were 52-6 we should proceed as above, and then cutting off the last figure of each, and removing the decimal point one place to the left, the diff. of lat. is 22-2, and dep. 47*8. The tenths are increased by i in the last case of dep., as the figure cut off in one exceeds 5. Ex.8. •I as 6 Course S. ri). 53' E., dist. 6ri (take Ex. 9. dist. 68-7 Course S. as 687). 44° E., dist. 68-7 (take Course. 53° Dist. 300 300 II Diflf. Lat. 180-5 180-5 6-6 Dep. 239-6 239-6 8-8 Course. 44° Dist. 300 300 87 Diff. Lat. 215-8 2151 62-6 Dep. 208-4 208-4 60-4 53 611 367-6 488-0 44 687 494-2 477-2 .'. S. 53° E., dist. 6i-i, give diff. lat. 36 8 S., dep. 48-8 E. . Course S. 44° E., dist. 68-7, give diff. lat. 49*4 S., and dep. 47-7 E. Ex. 10. Find from the table the diff. lat. and dep. for 483''7 on a N. 37° W. course. Course N. 37* W. Diff. Lat. 239-6 Dist. 300 185 ■7 485-7 i47'7 •6 387-9 Dep. 180-5 111-3 •4 292-2 The decimal 7 we take out as 7, which gives diff. lat. 5-6, and dep. 4-2, and shifting the decimal point in each one place to the left we have for diff. lat. 0-6, and for the dep. 0*4. Examples for Practice. In each of the following examples find the difference of latitude and departure corresponding to the given course and distance : — No. Given Course. S.S.E. E. by S. S.W. i S. W.|N. S.E. by E. \ E. S.W. \ W. E. by N. \ N. Dist. 30 48 136 84 56 225 183 Given No. Course. Dist. 8. S. 72° W. 35 9- N. 21 W. 24'5 10. S. 65 w. 257 II. S. 80 w. 14-7 12. N. 27 W. 30-6 13- W. 10 S. 42-8 14. N. 18 W. 34-9 269. Given the difference of latitude and departure, to find the conrse and distance. KULE LXiy. Seek in, the Traverse Table (Table II) '"ill the diff. lat. and dep., or the quantities agreeing most nearly with them, are found ' .pposite eaeh other in the proper columns ; to the left or abreast of the quantities thi s found, and under "Dist," will he the distance made good; and the Course fin degrees) must be taken from the top of the page when diff. lat. is greater than the dep. ; but from the bottom of the page when the dep. is greater than the diff. lat, Trmerse Table. 187 The Course will have the same name (from N, or S. towards E. or W.) as the diff. lat. and dep. made good. Note. — Always seek for the larger of tho two given numbers in the column next the distance, viz., the column marked " Diff. L^i ." at the top, and examine page after page, until the smaller number is found by its side in the column marked "Dep." at the top; being careful to remember that when diff. lat. is greater than the dep. the course will be at the top ; otherwise it is to be found at the bottom. {a) When the diff. lat. and dep. ou two successive pages of the Traverse Table, appear to be equally near the given diff. lat. and dep., neither page giving values actually corresponding ro them, take a course midway between those on the two successive pages as tl^e course actually made good. {b) If the difference of latitude anl departure, or any of the sides of the proposed triangle, should exceed the limits of the Traverse Table, they may be divided by any number that will bring them within these limits, and then the results from the table multiplied by the same number will give the required parts of the proposed triang-'O ; observing that the angle or course must in no case be multiplied or divided, because the course will be the same, whether determined to the whole difference of latitude or departure, or by using an aliquot part of the same. {c) In certain extreme cases, as when diff. lat. and dep. are very small, the course cannot be found correctly as above, since the diff. lat. and dep. may be found to agree in their resjiective columns for several successive pages of the table. In such cases, the diff. lat. and dep. may be multiplied by 10, and the products, when found to agree in their respective columns, will give the correct course; but the distance from the table, divided by 10, will give the correct distance. Examples. Ex. I. A ship having sailed between the N. and E., until her difference of latitude is 199 miles, and the departure i44'6 : required her course and distance. In page 52, Norie, or page 474, Raper, these quantities will be found to correspond with 246 in the distance column, and with the aigle 36'' found at the top of the table (the diff. lat. being greater than dep.) ; the course is tiioreforo N. 36' E., and distance 246 miles. Ex. 2. A ship has made upon one cours; 36 miles diff. lat. to the northward, and 58 miles dep. to the wesiward: required the cou.se and distance run. Look for 58 and 36 in two iidjoining co uf is marked "diff. lat." and "dep." at the top. In the table will be found 57*7 and 160 abre -at of each other. In the same line at the dist. column will be found 68, the distance soughl. As 58, the dep., is greater than diff. lat. 36 the course is taken from the bottom of the p ;ge. It is 58°. As the ship has gone to tho N. and W., the course she has made is N. 58^ W. Ex. 3. A ship having sailed between S. md W. until her difference of latitude is 40 miles, and her departure i39"4 milos : requir 1 the course and distance. In page 32, Norie, or page 454, Rapkr, tl'^ course answering to diff. latitude 40 miles, and departure i39'4 miles, corresponds with t.ie angle 74^^, at the bottom of the table, and opposite the distance 145 miles; the course is therefore S. 74° W., and distance 145 miles, which were required. 1 88 Traverse Table. Ex. 4. Given the diflF. lat. 24o'-o S., and dep. 2o8'-6 E. ; required the corresponding course and distance. (See Rule LXIV {b), page 186.) Aa both diflF. lat. and dep. exceed the limits of the table, divide them by 2, so as to bring them within these limits, thus : — Diff. lat. Dep. 2)240-2 2)208-8 I20-I 1 04*4 Then seeking for 1201 and 104-4 in two adjoining columns marked "diflf. lat." and "dep." at the top. In the Table are found 1 2o''o and 104-3 abreast of each other. In the same line in the distance column will be found 159, which must be multiplied by 2 (the number the diflF. lat. and dep. were divided by), the product 318 is the distance sought. As 120-1, the diflF. lat., is greater than the dep, the course is taken from the top of the page. It is 41°. As the ship has gone to the S. and E., the course she has made is S. 41° E. Ex. 5. Given diflF. lat, I'-o and dep. o'-i ; to find the corresponding course and distance. Here diflF. lat. I'-o and dep. o'-i are found in Tables to j^ive as course 3°, 4", 5°, 6°, 7°, and 8° ; now which is the correct answer in this case. Multiplying difi. lat. and dep., say by 10, we obtain diff. lat. lo'o and dep. i''o ; these are found in the tables to correspond to course 6° and distance 10 miles. Dividing the distance by 10 gives i (dividing by 10 as the tables were entered with ten times the diflf. lat. and dep.), hence course 6° and distance i mile. Examples for Praotioe. In each of the following examples the difference of latitude and departure are given, to find the corresponding course and distance. Given No. Diflf. Lat. Dep. 72-7 S. 25-0 E. 72*3 N. i7i'7 E. 64-0 N. i46'9 W. 98-6 S. 37-5 E. 415-6 N. 240-0 W. No. 6. 7- 8. Diflf. Lat 37-9 N. 53-3 S. 160-7 s. Given Dep. 36-4 E. 76-0 W. 1 6-5 W. 9- 0. 172-6 S. 164-2 N. 7-9 W. 262-8 E. TRAVERSE SAILING. 270. Traverse Sailing is the case in plane sailing when the ship makes several courses in succession, the track being zigzag, and the direction of its several parts " traversing," or lying more or less athwart of each other. For all these actual courses and distances run on each, a single equivalent imaginary course and distance may be found, which the ship would have described had she sailed direct for the place of destination. Finding this course is called " Working a Traverse." In order to do this, the difference of latitude and departure for each distinct course must be found, and the aggregate of the several differences of latitude and departure taken for the single difference of latitude and departure which would be made by sailing from the place left to that reached on a single course. The determination of this course, and the corresponding distance, is then to be effected.* 271. In resolving a traverse, it is usual to take the diff. lat. and the dep. due to each of the component courses from the Traverse Table ; hence we proceed by the following EULE LXV. 1°. Draw out a form similar to that given in the example following . 2'^. In the column headed Courses, enter each course in succession; and in column Dist., enter the distance run on each course. In entering the courses in the appropriate column, reckon, in each case, the points, and fractional parts of a point, from the North or South, whichever is nearest, and write them down as in the following example. 3°. Take out of the Traverse Tables (Table I or II, Eapeb or Norie), the difference of latitude and departure to each course and distance, and enter the lati- tude in column N. or S., and the departure in column E. or W., according to the name of the course. Thus, if the course is S.E. by S., the difference of latitude must be entered in the column S., and the departure in the column E. ; if the course is W. \ N., the difference of latitude must be entered in the column N., and the departure in the column W. 3a. When the co^trse is exactly North or South, the distance and diff. lat. are the same, there is no departure, and the whole distance is entered as difference of latitude in the corresponding column N. or S., as the case may be ; so also when the course is due East or West, the departure is indentical with the distance, there is no difference of latitude, and the whole distance run is entered as departure in the E. or W. column. (See pages 191 — 192, Exs. 2, 3, and 5). * The plane sailing formula — Dep. =. Dist. X Sine course (i) D. Lat. = Dist. X Cos. course (2) give for each course and distance the corresponding departure and difference of latitude ; and taking the algebraic sum of all the diff. lats. and dep., we get the course from formula Tang course = ^g^ and then the distance from formulae (i) and (2). The Trwerae Table is used to obviate the necessity of computations. 1 90 Traverse Sailing. 4°. Add the diff. oflaU. in each column, and write the sum at the bottom of each, write the less of the two sums under the greater, and take their difference. Do the same with the departure. 5°. These differences are the diff. lat. and dep. made good on the whole, and each takes the name of the column it stands in. 6°. The course and distance are then found by Eule LXIV, page 1 86. Note. — (a). When there is no resulting departure the Traverse Table need not be referred to, as the ship has returned to the same meiidian, and the course made good is North (N.) or South (S.), according as the diff. lat. is North or South, and the distance is equal to the diff. !at. (See Example 5, page 192.) (b) Similarly, when there is no resulting difference of latitude, the course made good will be either East (E.) or West (W.) as the departure made good is East or West, and the distance will be of equal value with the differunce of departures. See Ex. 4, page 192. (c) Should the difference of latitudes and also the departures balance each other, in which case the ship will have made good neither difference of latitude nor departure, the vessel must be considered to have returned to the place from which she set out. It may be advisable for a beginner, before he proceeds to take out the quantities from the Traverse Tables, to write a dash in all the places not to be occupied by a diflerence of lati- tude or departure, in order to avoid writing a quantity in the wrong column. Such helps, however, are useless to an expert computer. Examples. Ex. r. A ship from the Dudgeon Light, in lat. 53° 19' N., sails S.S.E. \ E., 8 miles; E.N.E., 23 miles; N.W. by W. \ W., 36 miles; E. | N., 48 miles; and N.W. J W., 46 miles: required the latitude arrived at, also the course and distance made good. Courses. Dist. Diff. Lat. I Departure. N. S. E. W. S.S.E. i E =S. 2i=E.* E.N.E = N. 6 E. N.W.byW.iW.=:N.5i W. E. f N = N. 7i E. N.W.^W =N. 4iW. 8 36 48 46 8-8 170 7-0 292 7*2 3*4 21-3 47 '5 31-8 35-6 • The courses are given in this double form merely as an illustra- tion of the method of using them. 62-0 7-2 7-2 72-2 67-4 67-4 54-8 4-8 5w Explanation. — The courses and distances are entered in their proper columns, in the same order as they stand in the question; then in Traverse Table I, Nokie or Raper, the diff. lat. and dep. to each course and distance is found and entered in their proper column. (See Rule LXV, 3°.) The sum of the respective columns, N., S., E., and W., is next found, and set down at the bottom of each column, and the difference between the Northing, viz., 62-0, and the Southing, viz., 7-2, is taken, which leaves 54-8 N., which is the DLff. lat. made ; the diflference between the Easting 72-2, and Westing 67-4, leaves 4*8 E., departure. We proceed in the next place to find the course and distance made good, thus : — T m rr VI TT ( Diff. lat. 54*8 N. \ • ^ / Course N. s" ^- \ ^„^^ „^„^ In Traverse Table II. ( ^^^ jj^^-S^.g ^ J give { pj^^ ^^ J^^^^ J made good. This is an illustration of the remark, that when the diff. lat. is more than the dep., the course is less than 4 points, or 45° (see No. 161, page 104), and it is named from the N. towards the E., since the diff. lat. is N. and the dep. E. Lat. left (or sailed from) Dudgeon Light 53° 19' N. \ Diff. lat. 54*8 = 55 N. f Lat, in is found according to i Rule XLLV, 1°, page 108. Lat. in (or arrived at) 54 14 N. / Traverse Sailing. 191 Ex. 2. A ship from Cape Espicheli, in lat. 38'^ 25' N., sails as follows : S.W. by W., 28 miles; W. by N., 55 miles; West, 47 miles; S.E. | S., 25 miles; South, loi miles; W. J S., 72 miles : required the latitude in, also the conrse and distance made good. Courses. Dist. DiflF. Lat. Departure. N. S. E. W. S. i'W 28 S5 47 25 lOI 72 10-7 20'I loro io'6 14-9 23-3 53-9 47-0 71-2 N. 7 W W.* S. -ji E S.* 8. 7i W • See Rule LXV, 3a, page 189. 10-7 147-3 • 10-7 149 1 95 '4 14-9 136-6 1805 We seek in the Traverse Table till the diflF. of lat. 136-6, and dep. 1 80-5, are found opposite each other, in their respective columns; the nearest to these are i8o'5 and 136-0, which give the course (at the bottom of page, dep. being the most) S. 53° W., and distance 226'. This is an illustration of the remark. No. 161, page 104, that when the departure exceeds the difference of latitude, the course is more than 45°. Lat. left Diff. lat. 136-6 38° 25' N. = 2 17 S. Lat. in (or arrived at) 36 8 N. The lat. in is found according to Rule XLIV, 2°, page 108. Ex. 3. A ship from lat. 37° 24' S., sails the following true courses :— S.W. by S., 20 miles; West, 16 miles; N.W. by W., 28 miles; S.S.E., 32 miles; E.N.E., 14 miles; S.W., 36 miles : required the lat. in, also the course and distance made good. Courses. Dist Diff. Lat. Departure. N. S. E. W. S. 3'» W 20 16 28 32 14 36 15-6 5 "4 16-6 29-6 25-5 12-3 12-9 ii-r 16-0 23-3 ^5-5 W.* N. 5 W S. 2 E N. 6 E S. 4 W • See Rule LXV, la, page 189. 2I-0 71-7 21-0 25-2 75-9 25-2 JO-7 50-7 We seek in the several pages of the Traverse Table TI, for the diff. lat. 50-7 ; and dep. 50-7 ; the nearest found to these are diff. lat. 50-9, dep. 50-9, give course S. 45° W., distance 72 miles. The diff. lat. and dep. being of equal amount, the course is 45°, or 4 points, which illustrates the remark. No. 161, page 104. Lat. left 37° 24' S. \ The lat. sailed from being South, and the Diff bit 1:0-7 1= cr S / **^P having siiled South, the ship has ovi- ■ ' ■ -^ ' __L ' } <^6nt]y increased her South lat., whence the . i sum of lat. from and diff. lat. i.s taken to obtain Lat. arrived at 38 15 S. / lat. in.— (See Rule XLIY, 1% page xo8. 192 Trmerse Sailing. Ex. 4. A ship from lat. 46° 20' N. sails (all tnie courses) N. 72° E., 21 miles ; N. 38° E., 17 miles; S. 26° W., 13 miles; S. 73° E., 19 miles ; S. 1° W., 19 miles; S. 65° E., 48 miles ; N. 76° E., 19 miles ; N. 48" E., 48 miles : required the lat. in, also the course and distance made good. Courses. Dist. DifiF. Lat. Departure. N. S. E. W. N. 72°E 21 17 13 19 19 48 19 48 6-5 i3'4 4-6 3^"i 11-7 5-6 19-0 20'3 20'0 10-5 i8-2 43-5 18-4 35-7 5'7 0-3 N. 38 E S 26 W. .. S 7-! E S. I w S. 6c E N. 76 E N. 48 E 56-6 56-6 56-6 146-3 6-0 6-0 i4o"3 Course due East, and dist. 140-3, the same as the departure. (See No. 161, page 104.) The Traverse Table being filled up, the sums of the Northings and Southings are both 56-6, and being of contrary directions, show that the ship has returned to the same parallel of latitude which she sailed from. The sum of the Eastings is 146-3, and that of the Westings 6-o ; their difference, 140-3, shows that tho ship has gained so much to the Eastward, that being the greater. Consequently the Course is due East, and the Distance 140-3, the same as the departure. Ex. 5. A ship from a place in lat. 1° 5' S., sails the following true courses : — N. 17° E., 13 miles; North, 38 miles; N. 27° E., 18 miles; N. 79° E., 25 miles; S. 83° W., 23 miles; S. 48° E., 25-2 miles; N. 48° W., 27-1 miles; N. 36° W., 21 miles: required the latitude in, also the course and distance made good. Courses, N. 17° E.. North* . N, 27 E., N, 79 E,. S. 83 W. S. 48 E.. N. 48 W, N, 36 W, See Bule XLV, 3a, page 189. Dist, 13 38 18 25 23 25-2 27-1 21 DiflF. Lat. N. I2"4 38-0 i6-o 4-8 i8-i 17-1 106-3 19-7 86-6 19-7 Departure. E. 55*2 55'^ W, 22-8 20-1 123 55-2 The Traverse Table being completeil, tho sum of the Northings is 106-3, ^^^ t^° ^um of the Southings is 19-7, the difference 86-6, and to that amount the ship has altered her latitude. The miles of departure in the East are 52-2, and those in the "West column are also 52-2 ; but as the East and West depai-tures destroy one another, there is no resulting departure, and therefore it is not necessary to refer to the Traverse Table. The ship is under the same meridian as she sailed from; consequently, the course is due North, and the dis- tance sailed is equal to the diff. lat., viz., 86-6. This is according to No. 161, page 104. Latitude left Diff. lat. 6,0)8,6-6 15 'S. Latitude in I 266 = I 27 N. o 22 N. The ship being 1° 5', or 65 miles, S. of the equator, must evidently be in N. lat. after making 87 miles of Northing. Thus, in subtracting one of the quantities from the other, the difference takes the name of the greater. Rule XLIV, page 108. The course is North, and dist. 86*6, the same as diff. lat. Traverse Sailing. 193 Ex. 6. A ship from latitude 46° 10' N., sails as follows : — S. 48° E., 25 miles ; S. 51° E., 18-9 miles; N. 87° E., 12-4 miles; S. 70^ E., 14-5 miles; S. 68° E., 21 -6 miles; N. 25° W., 16-4 miles; N. 8° E., 78 miles; N. 19^ E., 13-7 miles; N. 76° E., 39"6 miles: required the lat. in, also the course and distance made good. Courses. Dist. S. 48'' E. S. 51 E. N. 87 E. S. 70 E. S. 68 E. N. 25 W N. 8 E. N. 19 E. N. 76 E. 25 18-9 12"4 21'6 i6'4 7-8 i3'7 396 Diff. Lat. N. 0-7 14*9 7-7 130 9-6 45-9 41-7 4-2 S. Departure. W. E, 16 7 II-9 5"o 4i'7 186 '4*7 I2"4 13-6 20'O I"I 4-5 38-4 6-9 123-3 6-9 1164 6-9 For the method of taking out the course and distance from the Traverse Table when the distance is given in tenths, see Rule LXIII, page 1 84. Examples for Practice. 1. A ship from the Texel in lat. 52° 58' N., sails W. by N., 44 miles ; S. by E., 45 miles ; W. by S., 35 miles ; S.S.E., 44 miles; W.S.W. ^ W., 42 miles; find diff. lat. and dep., the course and dist. made good, also the lat. arrived at. 2. A ship from Heligoland, lat. 54° 12' N., sails W.S.W., 12 miles; N.W., 24 miles ; S. by W., 20 miles ; N.W. by W., 32 miles ; S. by E., 36 miles ; W. by N. 2 N., 42 miles ; S.S.E. ^ E., 16 miles ; W. f N., 45 miles : required diff. lat. and dep., course and dist. made good, also the lat. arrived at 3. A ship sails from lat. 3' 50' N., sails S.S.W., 112 miles ; S. by E., 86 miles ; S.S.E., 112 miles ; S. by W., 86 miles : find diff. lat. and dep., the course and dist. made good, also the lat. arrived at. 4. Yesterday we were in lat. 19° S., and since then have sailed S.E. ^ S., 13 miles; S. by E., 19 miles; S.E. by E., 22 miles; E. by S. J S., 32 miles; N.N.E., 20 miles; N. by W. i W., 27 miles ; N.E. by E. | E., 24 miles ; S.W. ^ S., 10 miles. 5. A ship from lat. 1° N., sails East, 8 miles ; E. J N., 20 miles ; S.E. by E., 33 miles ; 8. I W., 31 miles; N.E. ^ N., 43 miles; South, 28 miles; S. f E., 21 miles; S.byW.^W., 12 miles : required diff. lat. and dep., course and dist. made good, and also the lat. in. 6. A ship from lat. 1° 10' N., sails N. 40° W., 20 miles ; S. 56° W., 51 miles ; S. 19° W., 19 miles; S. 48' W., 16 miles; N. 85° E., 28 miles; S. 44° E., 15 miles ; N. 22° "W., 25 miles ; S. 9° E., 54 miles ; find diff. lat. and dep., course and dist. made good, also the lat. in. 7. A ship from lat. 47° 12' N., sails S. 31° W., 16 miles; N. 72° E., i3'-i ; S. 52°'W., 15' ; S. 44° E., i5'-i ; N. 44° W., i9'-7 ; N. 77° E., ii'-4; S. 40° W., 16' ; S. 14° E., 6' : required the course and dist. made, the lat. arrived at, and the dep. made. 8. Since leaving lat. 34° 11' N., we have sailed the following courses : — N. 36° W., 27'; N. 24° E., 30'; S. 75^ W., 47'; S. 8o» W., 29'; N. 72° W., 42'; N. 78° W., 34'; S. 12° E., 28'; required the course and dist. made, the lat. arrived at, and the dep made. 9. Since leaving lat. 36° 35' S., the ship has sailtd N. 84° W., 18'; N. 89° W., 3o''4; N. 67^ W., 29'-9; N. 39»\V., 33'-9; N. 8-^ W., 25'-9 ; N. 73^ W., 34'-9 ; N. 86^ W., 44'-7 ; S. 65° E., 56' ; required the lat. arrived at, and the course and dist. made good. 10. A ship sails from lat. 1° 46' N., on the following compass courses, viz., N. 84° W., 23 miles; S. 6° W., 48 miles; S. 61' W., 37 miles; S. 24° W., 30 miles;' N. 75° W., 44 miles ; N. 69° W., 37 miles ; N. 20'^ W., 38 miles ; and N. 33° W., 36 miles : required the lat. arrived at, and the course and dist. made good, the variation of the compass being 21^° W. 194 PARALLEL SAILING. 272. When two places lie on the same parallel of latitude, or due East or West of each other, the distance between them estimated along a parallel, or E. and W. (which is all departure) is converted into difference of longitude ; or, on the other hand, the difference of longitude is converted into distance by Parallel Sailing. Since the meridians are all parallel at the equator and meet at the poles, the distance between any two meridians, measured East and West, is less as the latter is greater — that is, the absolute number of miles, or of feet, in a degree of longitude, is less as the latitude in which they are measured is greater. Hence, also, a given number of miles between two meridians corresponds to a greater difference of longitude, as the latitude in which they are measured is greater. For example, two places in lat. 10°, and distant 60 miles East and West from each other, have 60' "9 diff. long. In lat. 60° N. or S., two places similarly situated have 2° o' diff. long., while at 73° the diff. long, is 3° 25'. Questions of this kind are solved by Parallel Sailing. 273. Given the departure made good on a given parallel of latitude, to find the diff. of long, corresponding thereto. EULE LXVI. 1°. Take out of the Tables the log. secant of latitude (rejecting 10 from index), and the log. of the departure made good. 2°. Add these logs, together, and find the nat. number corresponding thereto. The result is the difference of longitude required. 274. In parallel sailing, the latitude being constant, the difference of longitude bears a constant ratio to the distance, and all problems may be completely solved by the solution of a right-angled plane triangle, and there- fore by inspection of the Traverse Table by EULE LXVII. With the latitude of the parallel as a course, and the distance sailed on it as difference of latitude, the corresponding distance, in the Traverse Table, is the difference of longitude. Examples. Ex. I. In lat. 29° 51' S., the dep. made good 161 miles: required the diff. of long. Lat. 29° 51' Secant 0-061815 Dep. 161 Log. 2'2o6826 Log. 2*268641 Diff. of long. 185-6. i^«/ Inspection. In Traverse Table II, lat. 30° as course, dep. 161 -i in lat. column, give diff. long. 186 miles in dist. col. Ex. 2. A ship sailed 94-6 miles on the parallel 64° 38' N. : required the diff. long. Lat. 64° 38' Secant 0-368141 Dep. 94-6 Log. 1-975891 Log. 2-344032 Diff. of long. 220-8. i?y Inspection. In Traverse Table II, lat. 64° as course and dep., 94'7 give diff. lat. in dist. column 216 miles; and course 65°, and dop. 94-7, give diff. long, in dist. column 224 miles ; therefore, the diff. long, for 64^° will equal 216 + 224 -j- 2 = 220 miles. I*(Mrallel Sailing. '95 Ex. 3. From long. 0° 59' W., the dep. made was 125 East, on the parallel of 52° : required the long. in. Lat. 52° Secant 0-210658 Dep. 125 Ijog. 2'0969io Log. DifF. long. 6,0)20,3 2-307568 3 23 = 3° 23' E. Long, left o 59 W. Long, in 2 24 E. Ex. 5. In lat. 71° 25' N., the dep. made good was 71^ miles: required the difif. of long. Lat. 7 i^ 25' Secant 0-496640 Dep. 71-25 Log. 1-852785 Ex. 4. A ship from long. 179° 20' W. sails 109 miles West, on the parallel of 61° 25' : what is the long, in ? Lat. 61° 25' Secant 0-320176 Dep. 109 Log. 2-037426 Log. 2-357602 Diff. of long. 227-8 W. 6,0)22,8 3»48'W. 179 20 W, 3 48 Long, in 183 360 8 W. Log. 2-349425 Diff. of long. 223-6 nearly. or 176 52 E. Ex. 6. In lat. 80°, the dep. made good was 80 miles : required the diff. of long. Lat. 80° Secant 0-760330 Dep. 80 Log. 1-903090 Log. 2-663420 Diff. of long. 460-7. Examples for Practice. In each of the following examples the difference of longitude is required: — No. Lat. in. Dop. No. Lat. in. Dep. 1. 6" 7'N. 249' W. 5. 46^^37' S. 352' E. 2. 19 48 S. 324 E. 6. 64 16 N. ^65-7 W. 3- 39 57 N. 398 W. 7. 51 28 S. 70-9 E. 4. 60 o N. 74 W. 8. 60 o S. 204 E. 275. The method of parallel sailing will apply correctly enough for all practical purposes to cases where the course is nearly East and West (true). In latitudes not higher than 5°, when the distance does not exceed 300 miles, the departure may be used at once for the difference of longitude, the result- ing error scarcely exceeding one mile. 276. Given the difference of longitude of two places on the same parallel, to find their distance as measured along the parallel. EULE LXVIII. To the log. of the diff. of long, add the cosine of lat. ; the sum (neglecting i o) is log. of the distance required. Example. Ex. I. Required the distance between St. Abb's Head, in latitude 55° 55' N., longitude 2° 10' W., and Uraniberg in the same latitude, but in longitude 12° 52' E. Longitude of St. Abb's Head 2° 10' W. Longitude of Uraniberg 12 52 E. 15 2 60 Difference of longitude 902 miles. Log. distance =: log. diff. long. -\- log. cosine lat. Log. diff. long. 902 = 2*955207 Log. cos. lat. 55° 55' = 9-748497 Log. distance 505'-5 =: 2703704 196 Parallel Sailing. 277. Given the meridian distance and difference of longitude, to find the latitude. EULE LXIX. From the log. of meridian distance (adding 10 to the index) subtract the log. of diff. long. ; the remainder is the log. cosine of the latitude. Example. Ex. I. From a place in longitude 3° 12' W., a ship sails due East 246 miles, and by observation is found to be in longitude 4° 8' E. : required the latitude of the parallel on which she sailed. By Calculation. mer. dist. Long, left 3° 12' W. Cos. lat. _ ^MnoHg. Long, in 4 8 E. D. long. 7 20 = 440 miles. Mer. dist. 246 log. {-\- 10) 12*390935 Diff. long. 440 log. 2-643453 Lat. 56° o' cos. 9*747482 By Inspection. Since the diff. long, given, 440, exceeds the distance given in the Traverse Table, its half is taken, and also the half of the meridian distance 246, these are 220 and 123 respectively. Entering the Tables with 220 as distance, and 123 as diff. lat., we find, on searching the Table, these quantities, in their respective columns, on the page with 56° at the bottom ; hence the latitude sought is 56°. Examples for Practice. 1. Required the compass course and distance from A to B. Given lat. A 52° 15' S. ; var. i\ points W. ; long. A 37° 30' 'W. B52 15 S.; dev. 8°5o' W. ; B 48 18 W. 2. A and B lie on the parallel of 58° 30' N. Given long. A 15° 12' E. B 13 18 W. What is the distance between them in nautical miles ? 3. Define a great circle and a small circle of a sphere, giving an example of each. What connection is there between the tropic of Cancer and the Arctic Circle ? 4. Required the compass course and distance from A to B. Lat. A 28° 40' N. ; var. i| points W. ; long. A 2° 20' E. B 28 40 N.; dev. 8° 50' E.; B 4 10 E. 5. In what latitudes are the lengths of a degree of longitude 30 and 20 miles respectively ? 6. In travelling 35 nautical miles on the parallel of 55" 25' N., how much do I change my longitude ? 7. Find the true course and distance from A to B. Lat. A 54° 25' S. ; long. A 15° 30' E. B54 2j S.; B 9 15 W. 197 MIDDLE LATITUDE SAILING. 278. Middle Latitude Sailing is a method founded on the principle of parallel sailing, converting Departure into Difference of Longitude, and the Difference of Longitude into Departure, when the ehip's course lies obliquely across the meridian, that is, when besides departure she makes difference of latitude. Suppose a ship, in going on the same course, from latitude 40" to latitude 44°, make 100 miles departure: this departure, if made good altogether in latitude 40°, would give 130-5 difference of longitude by Rule LXVI, page 194 ; and again, if made good in latitude 44°, it would give 139 difference of longitude. Now, since the ship has sailed between these two parallels, and not on either of them exclusively, her real difference of longitude must be between i30'5 and 139, and therefore we may conclude it to be not far from that which would result from a departure made good altogether in the middle parallel ; hence the name Middle Latitude Sailing. Middle latitude sailing, then, is founded on the consideration that the arc of the parallel of middle latitude of two places intercejited between their meridians is nearly equal to the departure. If we conceive the ship to sail along this middle parallel, we may apply the principle of parallel sailing to the cases in point. In parallel sailing, as has been shown, the departure (or distance) and difference of longitude are connected by the relation, dep. = diff. of long. X cos. lat. When the ship's course lies obliquely across the meridian, making good a difference of latitude, a modification of this formula gives the formula for middle latitude sailing, dep. (nearly) = diff. of long. X cos. mid. lat.; or in logarithms, log. dep. ■=. log. diff. of long. -|- log. cos. mid. lat. — 10. Middle latitude sailing has thus the same two cases as parallel sailing, and accordingly the rules for inspection and computation already given, Rule LXVI, page 194, apply equally to this sailing, observing merely to read middle latitude for latitude. 279. To find the latitude and longitude in, the course and distance from a known place being given, by Traverse Table and Middle Latitude. By working a Traverse the difference of latitude nnd departure are obtained. Hence, by applying the difference of latitude to the latitude from, we have the latitude in. The middle latitude is then found, and the solution of the problem completed by the aid of the formula above. No. 278, note, viz. : — Diff. long. = dep. X sec. mid. lat. the diff. of long, applied to the long, from giving the long, in, hence— EULE LXX. 1°. With the given course and distance enter the Traverse Table^ and take out true difference of latitude and departure (see Eule LXIII, page 1 84). 2°, With difference of latitude atid latitude from, Jind latitude in (see Rule XTilY, page 108). 3°. Get the middle latitude, as directed, Rule XLV, page 108. 4°. With the middle latitude as course, look in the difference of latitude column for the departure, the corresponding distance at the top is the difference of longitude. 5"^. With difference of longitude and longitude from get longitude in, as in Rule XLYII, page 1 1 o. Note.— When the departure to be looked for as difference of latitude at the middle latitude, is beyond the limits of the Table, one-half, one-third, &c., must be used, and the resulting difference of longitude multiplied by the divisor, in order to get the whole difference of longitude. 198 Middle Latitude Sailing. Examples. Ex. I. A ship from lat. 52° 6' N., long. 35° 6' W., sailed S.W. by W., 256 miles : required her latitude and longitude in. 6,0)14, Diff. lat. 2° Lat. from 52 Lat. in 49 44 N. DifF.long.=dep.xsec.mid.lat. 2)101 50 Mid. lat. Mid. lat. Dep. 50 55 By Calculation. 50° 55' sec. 0*200349 212-9 ^°S- 2"328i76 2)2I2'9 \ dep. io6'4 ^y Inspection. Mid. lat. 5 1° as course (Table II), and half dep. io6"4, in diflF. lat. column, give in dist. column 169 miles, the half the diff. of long. Then 169 X 2 = diff. long. 338. 6,0)33,8 Diff. long. 337-7 log. 2-528525 5 38 Long, from Long, in 40 44 "W. 5°38'W.\.2„.^ 35 6W. (,^^S Explanation.— ThQ difference of latitude and departure is found as described in Rule LXIII, page 184. The latitude in is found by Rule XLIV, page 108; and thence the middle latitude, by adding the latitude from and latitude in together, and divided by 2 (see Rule XLV, page io3). The departure exceeding the limits of the Tables, the half ig taken. Then with middle latitude as a course, and half the departure, in difference of latitude column, half the difference of longitude is found in the distance column. This being doubled (as half the departure was taken) and divided by 60, gives the difference of longitude expressed in dt'Tees and minutes. The ship is in West longitude, sailing West, add difference of longitude to longitude left to obtain longitude in (Rule XLVII, page no). This Method by Inspection is the usual case at sea of Working the Day's Work. Ex. 2. A ship from lat. 48° 27' S., and long. 29° 12' W., sails S.E. by S., 22-5 miles: required the latitude in, also the longitude in. Course S.E. by S. = 3 pts. ; then 3 pts. and dist. 22*5 give diff. lat. 18-7, and dep. 12-5 (see Rule LXIII, page 184). By Inspection. ^Mid. lat. 48^° as course, and dep. as diff. lat., give in dist. column 19 miles, which is the diff. of long. Diff. long, o" 19' E. Long, left 29 12 "W. Diff Lat. lat. from in 0° 48 19' 27 S. S. s. ^d8 Lat. 48 46 2)97 13 5 c § Mid. lat. 48 36 J- Mid. lat, Dep. I I Long, in 28 53 W. (The long, in is found by Rule XLVII, page no). By Calculation. 48" 36' sec. 0-179594 12-5 log. 1-096910 Diff. long. 18-9 log. 1-276504 Ex. 3. A ship from the Lizard, in lat. 49' 57' N., sails between the South and West until diff. lat. is 126-7, ^°d "^^P- io^"6 : required the latitude come to, and difference of longitude. 6,0)12,6-7 2 6-7 or 2° 7' S. ^ <« Lat. left 49 57 N. Lat. in 2 47 50 N. )97 47 — Sm Mid lat. 48 53 J By Inspection. Then mid . lat. 48° 54', say 49°, as a course, and dep. 102-3 (nearest), found in the lat. column, opposite to which, in the dist. column, is 156, nearest, the difference of longitude. By Calculation. Mid. lat. 48° 53' sec. 0-182042 Dep. 102-6 log. 2-oiri47 Diff. long. 1560 log. 2-i93i5 Middle Latitude Sailing. '99 Ex. 4. Lat. from 59° o' N., long, from 3" 33' E., the ship sailed between S. and E., making difF. lat. 81-7 and dep. 17 2*7. B. Lat. 6,0)8,1-7 I 21-7 or i"»22'S. I^at. from 59 N. Lat. in 57 38 N. 116 38 Mid. lat. 58 19 By Inspection. 58° Dep. 159-0 give D. long. 300 13-8 „ 26 172-8 „ 326 By Calculation Mid. lat. 58° 19' sec. 0-279655 Dep. 172-7 log. 2*237292 log. 2-516947 D. long. 328-8 6,0)32,8-8 5°28'-8 59'. Dep. 154*5 give D. long. 300 i»o „ 35 i72"5 335 D. long, for mid. lat. 58° is 326 »> »> 59 >» 335 DifiF. for i°ofmid. lat. 9 60' (or i") 19 9 6,0)17,1 2-85 D. long, for mid. lat. 58° is 326 Corr. for 19' (over 58°) + 2-8 D. long, for mid. lat. 58" 19' is 328-8 Remark. — When the mid. lat. is high and between two whole degrees, and also the dep. great as in this example, the diff. long, is best found by calculation. Ex. 5. Sailed from A, in lat. 50' 48' M., long. 1° 10' W., S. 41° E., 275 miles. Entering Traverse Table II with dist. 275 miles, and course 41°, the true diff. lat. is 207'*5, or 3° 27'-5 iS. ; applying this to lat. from, the lat. in is 47° 2o'-5 N. The corresponding dep. is taken out at the same opening, which is i8o'-4. The mid. lat., or half sum of lat. from and lat. in, is 49° to the nearest degree. The dist. corresponding to 49° as a course, and 1 80' '4 in diff. lat. column, is found to be 275', in degrees 4'^ 35' E., which is the diff. long. Applying this to the long, from, 1° 10' W., we have the long, in 3° 25' E. Examples for Praotioe. In each of the examples following, the latitude and longitude arrived at are required to be found, having given the latitude and longitude from, with the course and distance sailed. I. 2. Lat. from. 25° 35' N. 32 30 N. 3- 4- 5* 6. 39 30 S. 46 24 S. 20 29 N. 56 N. Long, fi om. 60° 0' w. 25 24 w. 74 20 E. 178 28 E, 179 10 W. 29 50 w. Course. Dist. E.N.E. 296 N.W. by W. f W. 212 S.W. by W. I W. 2IO S.E. f E. 278 W. by S. J S. 333 S. 47° E. 168 200 MERCATOR'S SAILING. 280. Mercator's Sailing, like middle latitude sailing, relates to finding the difference of longitude a ship makes when sailing on any oblique rhumb, and is a perfectly general and rigorously true method, which the other is not. Mercator's sailing is characterised by the use of the Table of Meridional Parts, and the chart constructed by means of it called Mercator's Chart. With the assistance of this Table, the rules of plane trigonometry su£B.ce for the solution of all the problems. In the triangle ACB let A be the course, AB the distance, AC the true difference of latitude, CB the departure; then corres- ponding to AC, the Table of Meridional Parts gives AC, the meridional difference of latitude, and completing the right- angled triangle AO'B', C'B will be the difference of longitude. In addition, then, to the three canons of plane sailing which can be deduced from the triangle ACB, the triangle ACB' gives the characteristic canon of Mercator's Sailing (since C'B' =: AC tang. A) diff. long. = mer. diff. lat. X tang, course. 281. Q-iven the latitudes and longitudes of two places, to find the course and distance between them. EULE LXXI. 1°. Find the true difference of latitude, according to Rule XLII, page 106. 2°. Find the meridional difference of latitude, Eule XLIII, page 107. 3°. Next find the difference of longitude, Eule XL VI, page 109. 4.°.* To find the course. — From the log. of diff. of longitude (increasing its index hy 10 J, subtract the log. of meridional diff. of lat. : the remainder is the tangent of course, which take out of the tables, and place before it the letter of diff. of lat., and after it the letter of diff. of long. Note. — Be careful to remember that when the sum of the longitudes exceeds 180°, it must be taken from 360", and the course must be named the same as the longitude left. 5°. To find the distance. — To the secant of course (rejecting lofrom the index J, add the log. of diff\ of lat. : the sum is the log. of distance, the natural number corresponding to which find in the tables. * From the formula : — Diff. long. Tang, course = M^^rdrffTTaX ^ True diff. lat. Dist. =: -7^ — ~ — ' Cos. course. log. tang, course — 10 = log. diff. long. log. mer. diff. lat. . log. dist. =: log. true diff. lat. -|- log. sec. course — 10. Mercator^s Sailing. 201 Examples. Ex. r. Required the course and distance from Tynemouth Light to the Naze of Norway. Lat. Tynemouth 55' i' N. Lat. Naze 57 58 N. 2 57 60 Mer. parts 3970 Mer. parts 4291 Mer. diff. lat. 321 DifF. oflat. 177 N. To ^d the Course. I DifiT. longf. 507 Log. (4- 10) 12-705008 I Course 57° 40' Mer. diff. lat. 321 Log. 2-506505 Diff. of lat. 177 Long. Tynemouth 1° 25' W. Long. Naze 7 2 E. 8 27 60 Diff. of long. 507 E. To find the Distance. Secant o"27i773 Log. 2-247973 Tang. 10-198503 Course N. 57"' 40' E. Distance 331 Log. 2-519746 Ex. 2. Required the course and distance from A to B. Lat. A Lat. B 5i''a3'N. 48 23 N. 3 60 Mer. parts. 3606 Mer. parts. 3326 Mer. diff. lat 280 Long. A Loner. B 9° 29' "W. 4 29 W. 5 60 Diff. oflat. 180 S. Diff. long. 300 Log. (+ 10) 12-477121 Mer. diff. lat. 280 Log. 2-447158 Course 46° 58^' Diff. lat. 180 " Tang. 10-029963 Courses. 46° 58J'E. Diff. of long. 300 E. Secant 0-166014 Log. 2-255273 Log. 2-421287 Distance 263-8. Ex, 3. Required the course and distance from Cape Bajoli to Cape Sicie. Lat. Cape Bajoli 40° i' N. Lat. Cape Sicie 43 3 N. 3 2 60 Diff. of. lat. 182 N. Mer. parts 2624 Mer. parts 2867 Mer. diff. lat. 243 Long. Cape Bajoli 3=48' E. Long. Cape Sicie 5 51 E. Diff. long. 123 Log. (+ 10) 12-089905 Mer. diff. lat. 243 Log. 2-385606 Tang. 9-704299 Course N. 26'' 51' E. Course 26° 51' Diff. lat, 182 60 Diff. of long. 123 E. Secant 0-049542 Log. 2-260071 Distance 204. Log. 2-309613 Ex. 4. Required the course and distance from Cape Formosa to St. Helena. Lat. Cape Formosa 4° 15' N, Lat. St. Helena 15 55 S. 20 10 60 Diff. oflat. 1210S. Mer. parts 255 Mer. parts 968 Mer. diff. lat. 1223 Long. Cape Formosa 6" 1 1' E. Long. St. Helena 5 45 W. II 56 60 Diff. long. 716 Log. (+ 10) 12-854913 Mer. diff. lat. 1223 Log. 3-087426 Course 30° 21' Diff. of lat. 1 2 10 Tang. 9-767487 Course S. 30° 21' W. Vii Diff. of long. 716 W. Secant 0-064012 Log. 3-082785 Log. 3-146797 Distance 1402. I'd Mercator^s Sailing. Ex. 5. Required the course and distance from Bahia to Fernando Po. Lat. Bahia 13° i' S. Mer. parts 788 Long. Bahia 38° 32' W. Lat. Fernando Po 3 48 N. Mer. parts 228 Long. Fernando Po 8 43 E. 16 49 60 Mer. diflf. lat. 1016 47 15 60 Diff. of lat. 1009 N. Diff. long. 2835 Log. (+ 10) 13-452553 Mer. diff. lat 1016 Log. 3'oo6894 Diff. of long. 2835 E. Course 70° 17' Secant 0-471895 Diff. of lat. 1 009 Log. 3"oo389i Tang, 10-445659 Course N. 70° 17' E. Required the course and distance from A to B. Mer. parts 3007 Mer. parts 4065 Mer. diff. lat. 7072 Ex.6. Lat. A Lat. B Required t 44° 44' S. 55 55 N 100 39 60 Diff. of lat. 6039 N. Distance Log 2991. 3-475786 Long. Long. A B 148° 44 39' 44 W. E. 193 360 23 E. 166 37 60 Diff. long. 9997 Log. (4- 10) 13*999870 Mer. diff. lat. 7072 Log. 3-849542 54° 43 Tang. io'r50328 210 446)11800(26 2676 Diff. of long. 9997 "W. Course 54° 43' o" Secant 0*238358 Parts for 26" 77 Diff. of lat. 6039 Log. 3-780965 10457 I^og- 4-019400 116 416)2840(7 Distance 10457 nearly. Course N. 54° 43' 26" W. This question worked to the nearest minute of arc gives course N. 54° 43' W., and distance 10455 miles. Ex. 7. Required the course and distance from Cape East, Ne-w Zealand, to Cape Horn. Lat. Cape East 37° 42' 8. Lat. Cape Horn 55 59 S. 18 17 60 Mer. parts 2445 Mer. parts 4072 Mer. diff. lat. 1627 Long. Cape East 178° 40' E. Long. Cape Horn 67 16 W. Diff. of lat. 1097 S. 245 56 "W". 360 o 114 4 60 Diff. long. 6844 Log. (+ 10) 13-835310 Mer. diff. lat. 1627 Log. 3-211388 76° 37' Tang. 10-623922 558 935)36400(39 2805 8350 8415 Course 8. 76° 37' 39" E. Diff. of long. 6844 E. Course 76° 37' o" Secant 0*635515 Parts for 39" 345 Diff. of lat. 1097 Log. 3-040207 3-676067 53 Distance 4743-2 nearly, 92)140(2 Mereator^s Sailing, 203 Examples for Practice. Eequired the course and distance from A to B in each of the following examples : — A 38° 14' B39 51 N. N. A 2° v'E. B 4 18 E. A 49 53 B48 28 N. N. A 6 19 W. B 5 3 W. A 64 30 B 60 40 N. N. A 4 20 W. B 10 E. A 54 54 B 34 22 S. S. A 60 28 W. B 18 24 E. A 45 '5 B 47 10 N. N. A 35 26 W. B 32 15 W. A 34 22 B15 55 S. S. A 18 29 E, B 5 43 W. A 49 57 B36 58 N. N. A 5 12 W. B 25 12 W. A 35 14 B 18 23 S. S. A 75 30 E. B 12 2 E. A 4° 24' N. A 7»46'W. B 8 48 S. B 13 8 E. A 57 43 S. A 10 37 E. B55 35 s. B I 28 W. A 6 11 N. A 80 15 W. B 6 S. B 39 16 W. A 55 28 N. A I 9 E. B57 58 N. B 7 3 E. A 35 51 S. A 138 54 E. B38 5^ N. B 165 53 W. A 15 30 N. A 176 34 E. B 15 30 S. B 176 34 w. A 22 22 S. A 122 22 w. B33 33 N. B in II E. A 17 N. A 180 E. B 20 N. B 161 E. '3- 14. 15- 16. 282. To find the latitude aud longitude in, having given the latitude from, the longitude from, and the course and distance between the two places by Traverse Table and meridional parts.* RULE LXXII. 1°. With given course and distance enter the Traverse Table and take out the corresponding true difference of latitude, Rule LXIII, page 1 84, from which and latitude from find latitude in, as in Rule XLIV, page 108, and then meridional difference of latitude, as in Rule XLIII, page 107. 2°. At the given course look in the column of the true difference of latitude for the meridional difference of latitude ; the corresponding departure wi,ll be the difference of longitude, from which and the longitude from find the longitude in, as in Rule XL VII, page 1 ] o. Examples. Ex. I, A ship from lat. 55'^ i' N., long. 1° 25' W., sails S.S.E. ^ E., 246 miles : required the lat. in and long. in. Entering Traverse Table II, with course S. 2^ points E., and distance 246, we obtain difF. lat. 2i7"o, and dep. ii6'o. 6,0)21,7 Lat. left 55° I'N. \ _ Mer. parts 3970 D. lat. 3 37 S. ] >^ Mer. parts 3607 3° 37' _ . - ' bd - { 3 - ■^ a s p. Lat. in 51 24 N. ^ ^ §, Mer. diff. lat. 363 \ mer. diff. lat. i^i's ) ^ The general method of solution bj- '' meridional parts," is from the formula :— True diff. lat. = dist. X cos. course. . ■ . log. true diff. lat. = log. dist. -|- log. cos. course — 10. Diff. long. =: mer. difi. lat. X tang, course. . ' . log. diff. lout/. = log. mer. diff. lat. -\- log. tang, course — 10, 2b4 Meroator^s Sailing. The course 2^ points, and half mer. diff. lat. 181 '5 (in diff. lat. column), the nearest fonild in the Tahlc is iSry, the coiTespondinp; departure is 97"i, which multiplied hy 2 (having divided mer. diff. lat. by 2) gives diff. long. i94"2 miles. 6,0)19,4-2 Long, left 1° 2s' W. \ <^ xhe ship being i" 25' W., or 85' West of D. long. 3 14 E. f^^ii Greenwich, must evidently be in East longi- •?' I a' .^__ i ^^ fr tude, after having sailed 194 miles to the Long, in i 49 E. ) ^ S Ea-stward (see Rule XLTII, page no). Ex. 2. A ship from lat. 42"^ 36' S., long. 178° 43' E., makes diff. lat. i78''i S., and dep. 240' '2 E. : find lat. in and long. in. Course 4f points, and dist. 299, diff. lat. i78'i, dep. 240-2. 6,0)17, 8-1 Lat. left 42° 36' S. \ i> Mer. parts 2830 D. lat. 258 i:^. Mer. parts 3078 jp^ 2»58' f X 2 (M2 Lat. in 45 34 S. ^ | §, Mer. diff. lat. 2)248 / -3 g, \ M p- 124 / s Course 4f points, and half mer. diff. lat. 124 (in diff. lat. column), give in dep. column i67'i, which doubled is 334*2, the diff. long. 6,0)33,4-2 Long, left i78''43' E. \ M- D. long. 5 34 E. 1 > . 5° 34' f^2 184 17 E. ^2. 360 o Long, in 175 43 W. Ex. 3. From lat. 50° 48' N., and long. 1° 10' W., sailed S. 41° E., 275 miles: required the lat. in and long. in. In the Traverse Table at the distance 275, and course 41°, the corresponding t?'iie diff. lat. is 20TS, or 3°27'"5, which being subtracted fromso''48'N.,the lat. in is 470 2o''s N. ; taking out the mer. p arts for ^o°4S', and 47"^ 2o''5, the yner. diff. lat. is found to bo 317, to half which as a true diff. lat., and the course 14°, the dep. is i37'8, twice which is 275-6, — that is, the diff. long, is 4° 36' E. : hence the long, in is 3° 26' E. Ex. 4. From lat. 50° 30' N., and long. 37° 55' W., sailed S.W. | S., until arrived at lat. 52° 15' N. Lat. from 50° 30' N. Mer. parts 3521 Course 3J points, and mer. diff. lat. Lat. in 52 ic N. Mer. parts 3690 in diff. lat. oolumn, give in f J y 125-4, which is the din. long. Mer. diff. lat. 169 Long, left 37° 55' W. 6,0)12,5-4 D. long. 2 5 W. 2° 5' Long, in 40 o W. Examples fob Practiob. For examples for practice in this problem take those given in middle latitude sailing at page 199. REMAEKS ON MIDDLE LATITUDE AND MERCATOR'S SAILINGS. 283. "The difference of longitude found by middle latitude is true at the equator, and very nearly true for short distances in all latitudes, especially when the course is E. or W. In high latitudes, when the distance is great and the course oblique, the error becomes considerable : but the result may be made as accurate as we please by sub-dividing the distance run into small portions, and finding the diflference of longitude for each portion separately. The difference of longitude deduced by middle latitude sailing is too small : M0rcator'8~_Sailinff. 205 an estimate of the error for places on the same side of the equator may be formed by the help of a few cases. Suppose the course 4 points or 45°, and the difference of latitude 10'' or 600'; then if this difference of latitude is made good in any latitude below 30"^, the error of the difference of longitude will not exceed 2' ; if made good between the parallels of 40° and 50'^', the error will be about 3'; and between 60° and 70° about 19', or J of a degree. For smaller distances the errors will be much less, and for greater distances much greater, as they vary in much more rapid proportion than the distances. It has been observed before that when the course is large, the difference of longitude should be found by middle latitude in preference to Mercator's sailing ; because, although the latter is mathematically correct in principle, yet a small error in the course may, when the course is large, produce a considerable error in the difference of longitude. The reason of this is easily shown. In middle latitude sailing we convert the departure into difference of longitude. The process increases the departure iu a proportion which is less than 2 to i in all latitudes below 60° ; and exceeds 3 to i in all latitudes beyond 70°. The error of the departure, increased in the same proportion, becomes thus the error of difference of longitude. Now when the course is nearly E. or W., the departure is nearly the same as the distance, and an error of some degrees in the course does not affect the departure sensibly ; hence in this case the error of the difference of longitude depends on that of the distance alone. But in Mercator's sailing, on the other hand, we convert the meridional difference of latitude into difference of longitude, and the process, when the course is large, converts a given meridional difference of latitude into a difference of longitude much greater than itself; and thus increases the error of the meridional difference of latitude in the same proportion. Thus, for example, at the course 80°, the difference of longitude exceeds the meridional difference of latitude in the proportion of 6 to i ; at the course 85° this proportion is 11 to i . Now, when the course is large, a slight change in it sensibly affects the difference of latitude, and also the meridional difference of latitude, which is deduced directly from it. In high latitudes the meridional parts vary rapidly, and the error of the difference of longitude is increased accordingly ; heuce the precept more especially demands attention in high latitudes." — Roper'' s Practice of Navigation, pp. 103, 104. io6 THE DAY'S WORK. 284. This is the process oi finding the ship's place at noon — that is, its lati- tude and longitude, having given the latitude and longitude at noon preceding, or a departure taken since, the compass courses and distances run in the interval, the leeway (if any), variation and deviation (if any), direction and rate of current (if any), &c., &c. EULE LXXIII * \^. Correct each course for leeway, variation, and deviation (see Rules XLYIII to LV, pages 120 to 160, which arrange in the tabular form as in the example following . Add together the hourly distances sailed on each course, and insert the same in the Table, opposite the true course. Note. — Allow the leeway in points before expressing the course in degrees. Departure Course, — When a departure has been taken, consider the opposite to the bearing as a course, ivhich correct for variation, and the deviation due to the direction of the ship's head when the bearing was taken, and insert in the Table as an actual course, with the distance of the object as a distance. The departure course is generally put down iti the Tables as the first course. See No- 251, page 157. As the ship leaves the land, the bearing (by oompass) of some prominent object or known headland is taken, and its distance is generally estimated by the eye ; this process is called " taking a departure." The latitude and longitude of the landmark are known ; and thence^ by supposing the ship to have sailed on a course the opposite to the bearing of the object, through the distance that object is off, we thus obtain, on commencing a voyage, a deter- minate starting p(jint, from whence lo reckon the subsequent courses and distances. Thus, supposing, for example, a ship leaving the Tyae observes Tynemouth Light dipping, and setting it, finds its bearing to be W. by N., distant (by estimation) 20 miles. Now in sailing from Tynemouth Light to the jTesent position of the ship, she would have to sail in the opposite direction to the bearing of the light, viz., E. by S., 20 miles. At the end of the day, the Day's Work give? us a change of the ship's place as referred to the landmark, and not the su2)j)osed posilion. I'or methods of determining the distance, see Raper's Practice of Navi- gation, on Taking Departures, ch., IV, pp. 114 — 122. Current Course. — The set of a current is to be corrected for variation only (being correct magnetic^, and inserted in the Table as a course ; the drift being taken as a distance. The current course is generally inserted in the Table as the last course. 2°. Take out of the Traverse Tables (Table I or II, Raper or Norie) the difference of latitude and departure to each course and distance, (see Rule LXIII, page 184), and proceed to find the difference of latitude and departure made good as directed in Rule LXV, page 1 89, Traverse Sailing. 3°. Find the course and distance made good (see Rule LXIV, page 186.) 4°. Find the latitude in by applying the difference of latitude to the latitude from (see Rule XLIV, page 108). If a departure has been taken, the diflference of latitude is to be applied to latitude of the point of land ; if otherwise, to yesterday's latitude. * Nearly the entire process of computing the Day's Work has already been given, and if the learner has thoroughly mastered the rules laid down in the preceding pages, he will lind uo difficulty in working the Day's Work without reference to them. The Day's Work. iof Note. — "When tho course is less than 5 points or 56°, the difference of longitude may be found by either or both Middle Latitude or Mercator's method, but if the course exceeds 5 points the method of Middle Latitude should bo used in preference to Mercator's (see Remarks in pages 204—205). 5°. To find the difference of longitude. — By Middle Latitude Sailing. (a) Find the middle latitude as directed, Rule XLV, page 108. (b) Next at the page of Traverse Table on which the degrees fat top or bottom J correspond to middle latitude, fifid the departure in a difference of latitude column, then the corresponding distance is the difference of longitude of the same name as the departure (see Rule LXX, 4°, and note, page 197.) (c) Or thus, by calculation : — To log. sec. of middle latitude add log. of dep., the sum f rejecting 10 from the index J is the log. of diff. of long. When the latitude left and latitude in are of contrary names, that is, in low latitudes, no sensible error can arise from taking the departure itself as the difference of longitude. 6°. If the ship has made a due E. or due W. course good, the difference of longitude is found by Parallel Sailing, thus: — With the latitude as a course and the departure in a difference of latitude column, then the corresponding distance is the difference of longitude (see Rule LXVII, page 194). 7°. To find the difference of longitude. — By Mercator's Sailing. (a) Find meridional difference of latitude (see Rule XLIII, page 107). (b) Then with course and meridional difference of latitude fin a latitude column), find the corresponding departure, which is the difference of longitude (see Rule LXXII, page 203). (c) To find the long. in. — With the longitude left and difference of longi- tude ^w(/ the longitude in (see Rule XLYII, page no). When a departure has been taken the longitude left is that of the point of land ; otherwise that of yesterday. EXAMPLE I. H. Courses. K. 1^0- Winds. Lee- way. Devia- tion. Remabks, &c. pts. I S.S.E. J E. 4 2 East. 2 7°E. A point of land in 2 4 3 lat. 42° 12' S., long. 3 5 42" 58'W., bearing by 4 5 2 compass E. by N. | N. 5 4 dist. 21 miles. Ship's 6 N.N.E. 4 I East. 2i 9°E. head S.S.E. \ E. ; de- 7 3 8 viation as per log. 8 3 5 9 3 2 10 S.W. J w. 3 5 W.N.W. 'f 7^°W. II 3 6 12 4 I 2 N. iE. 4 4 2 3 W.N.W. 2i i°W. Variation 20° W. 3 4 4 4 4 5 5 s.s.w. 6 2 West. I i^^W. 6 6 4 7 6 2 8 6 5 AcurrentsetW.S.W. 9 N. by W. ^ W. 6 2 West. I 10° W. correct magnetic 26 10 5 7 miles from the time the II 5 3 departure was taken 12 5 4 to the end of the day. 208 The Bay'i Work. The Departure Course. The opposite point to E. by N. ^ N. is W. by S. ^ S., and the ship's head being S.S.E. \ E., the deviation is same as given in log. for S.S.E. \ E., viz., 7° E. W. by S. ^ S. = 6^ pts. R. of S. Deviation 7 Variation 20 L. or 73° 8'R. of S. - j.3 L. True course 60 8 E. of S. or S. 60° W., distance 21 miles. This is inserted in the Traverse Table as 1st course. tst Course, S.S.E. iE. H. K. S.S.E. ^ E. = 2^ pts L. of S. I 4-2 Leeway 2 ,, R. L. of S. 2 3 4 4" 3 5"2 /^- ,° -,«' T. r^f « 5 4-0 Dev. 7° R, Var. 20 L ;} 13 L. 22*7 True course 18 38 L. of S. or S. 19° E., distance 22*7 miles. The distance 22'7, is found by adding up the hourly distances sailed, until the course is altered at 6 o'olodi. Insert this course and distance as 2nd course. znd Course, N.N.E. The deviation for N.N.E. is q" E. N.N.E. = 2 pts. R. of N. Leeway aj » L- oi„ L. ofN. or N. 2° 49' E. of N., L. of N. Dev. g-R-Jix L. Var. 20 L. ) H. K. 6 4' I 7 3-8 8 3-5 9 3'2 i4'6 True course 13 49 L. of N. or N. 14° W., distance 14-6 miles. The distance, 14-6, is found by adding up the hourly distances from 6 o'clock until the course is changed at 10 o'clock. ^rd Course, S.W. ^ W. The deviation for S.W. J W. is -j^" W. S.W. J W. = 4| pts. R. of S. Leeway i| „ L. 2f R. of S. or 3o°56'R. of S. Dev. fio'h) L. Var. 20 o L j ' ■' K. 3-5 3-6 4-0 I 4*2 15*3 H. 10 II 12 True course 3 26 R. of S. or S. 3° W., distance i5'3 miles. Distance, i5''3, is found by adding up hourly dis- tances from 10 o'clock until 2. 42 Previous to opening the Traverse Table to take out the diflferenco of latitude and departure to each course and distance in the above table, fill up the columns not wanted : thus, in the first course, S. 60° W., the S. and W. will be wanted, and the N. and E. will not bo wanted ; fill up these last two columns by drawing a dash under N. and E. Proceed in the same manner with the other courses. 2. lb find the difference of latitude and departure to each course and distance by the Traverse Table. Enter Traverse Table, and take out the difference of latitude and departure corresponding to 60° and distance 21'. Insert them in the columns S. and W. The second course is S. 19° E., and the distance 22*7. Then, 19 degrees and distance 227 (omitting the decimal point) give difference of latitude 2i4-6, departure 73*9 ; now dropping the tenths in each, namely, the 6 and the 9, and shifting the decimal point one place to the left, we have difference of latitude 21 '5, departure 7-4, which insert in columns S. and E., the course being marked S. and E. The third course is N. 14° W., and distance 14-6. Look for 14 degrees and distance r4*6, which give difference of latitude 1417, departure 35-3 ; now dropping the tenths, the 7 and the 3, and increasing the preceding figure by i, in the first case, as the tenths exceed 5, we have, by removing the decimal point one figure to the left, the difference of latitude 142, and departure 3^5. Proceed in this way with the remaining courses, KE 2IO lite Dmfi Work. Next we find the sum of the four columns, when it appears the ship has sailed 45*0 N., and 89'9 S. ; therefore, upon the whole, the difference of latitude is 44*9 S. The sum of the eastings is 12-3, of the westings sS'li ''iid the departure made good is 43*4 W. 3. To find the Goiirse and Distance made good.~^\iQ difference of latitude is 44-9 and departure 43*4 found to correspond in their columns, give course S. 44° W., distance (i^\ miles (see Rule LXIV, page 186). 4. We next apply the diflFerence of latitude, 45' S. (44"9), to the latitude left, 42° 12' S., (the latitude of point of land), taking the sum, as they are of the same name, and the latitude 42° 57' S., takes the name of either (Rule XLIV, page 108). 5. To find the Difference of Longitude. — Take out the meridional parts for latitude left, 42° 12', and also for latitude in, 42° 57', and take the less from the greater, as the latitudes are of one name. The remainder is meridional difference of latitude (XLIII, page 107). Or, find middle latitude by adding together latitude left and latitude in, and divide the sum by 2 ; the quotient is the middle latitude (Rule XLV, page 108). Then the course 44°, in Table II, and meridional difference of latitude 61', found in difference of latitude column, gives in departure column 59', or difierence of longitude 59' (Rule LXXII, 2°, page 203). Or, the middle latitude 42^° in Table II, and departure 43-4 in difference of latitude column, gives in distance column 59', the difference of longitude (Rule LXX, 4°, page 197). Thus:— Mid. lat. 42° and dep. 43'"4 give in dist. column 58^ and „ 43 „ 43*4 ,. 59^ 2)118 Diff. long. 59 The difference of longitude 59' W. (that found by Mercator's sailing), added to longitude left 42° 58' W., gives longitude in 43'' 57' W. (Rule XLVII, page no). EXAMPLE II. H. Courses. K. i-6 Winds. Lee- way. Deviation. Remarks, &c. pts. I S. by W. 4 I W. by S. ^\ 0° A point of land in 2 3 9 lat. 62° 18' K, long. 3 4 85' 17' E., bearing by 4 4 compass N. by E. \ E., 5 S.W. 1 w. 3 5 S. by E. z\ 8°W. 16 miles. Ship's head 6 3 4 S. by W. Deviation 7 3 I as per log. 8 3 9 E. f S. 5 4 S. by E. If 15° E. 10 5 6 II 5 4 12 5 6 I W.N.W. 4 4 North. 3 i8|°W. 2 4 4 Variation 42° E. 3 4 2 4 5 5 N.W. \ N. 9 6 S. by W. i6rw. 6 10 2 7 II 4 8 11 8 9 E. |N. 3 4 N. by E. 3i 171° E. 10 3 2 A current set the II 3 ship W.S.W. (correct 13 2 4 magnetic), 23 miles. The Day's Worh 211 W. N. E. S. W. N. The Departure Course. The opposite point to N. 't>y E. \ E. is S. by W. ^ W.. and the ship's heiid being S. by "W., the deviation is the same as given in the log. for S. by W. S. by W. \ W. = li pts. R. of S. or 14° 4' R. of S. Var. 42'^ R. ) , ^ ^ True Course _^6 o R. ofS. or S. 56° W., distance 16 miles. This is iusertcd in the Traverse Table as ist course. ist Course, S. by "W. The deviation for S. by "W. is o. S. by W. = Leeway pt. R. of S. „ L. : ., L. ofS. Dev. o Var. 42" R or 14° 4' L. of S. I 42 o R. E. 4' I 3"9 4 4 16 True CoursR 28 o R. ofS. or S. 28° W., distance r6'. The distance i6''6 is found by adding up the hourly distances until the course is chansfd at 5 o'clock. Insert this course and distance in Traverse Table as the 2nd course. 2nd Course, S.W. | W. The deviation for S.W. f W. is 8° W. S.W. I W. = 4| pts. R. of S. H or 7f „ L. ofN. 13 „. 87° 11' L. of N. Dev. 8" L. ) T, Var. 42 R. 1 34 ° ^■ True Courso 53 o L. of N. or N. 53° W., distance 13'. The distances is obtained by adding up the hourly distances sailed from 5 o'clock until the course is changed at 9 o'clock. Leeway Sum exc. 8 pts. Subtract fr"'m 3i . 8^ , 16 , R. , R. ofS. 5 3"5 6 3-4 7 3-1 8 3 T,rd Course, E. f S. The deviation for E. f S. is 15° E. E. I S. = 7^ pts. L. of S. H. Leeway i| Sum exc. 8 pts. 9 Subtract from 16 L. L. of S. R. ofN. K. 5*4 5-6 5 '4 5-6 or 79° R. ofN. Dov. IC° R. ) T> .r XT Var. 42 R.) 57 R-ofN. f^um exc. 90° 136 R. ofN. Subtract from 180 True Course 44 L. of S. or S. 44° E., distance 22'. Hourly distances sailed from 9 o'clock until course is changed at i o'clock being added up, give the dis- tance to enter in the Traverse Table. ^th Course, W.N.W. The deviation for W.N.W. is 18^" W. w.x.w. = 6 pts. L. of N. H. K. Leeway 3 ., L. I 4*4 — 2 4-4 Sum exc. 8 pts. 9 „ L. ofN. 3 4-2 Subtract from 16 4 5 Dev. i8°3o'L. Var. 42 R. R. of S. or 7 8° 45' R. of S. 23 30 R. R. of S. Sum. exc. 90° 102 Subtract from 1 80 True Course 78 L. of N. or N. 78" W., distance 18'. The distance is found by adding up hourly dis- tances sailed from 1 o'clock until the course is changed at 5 o'clock. ^th Course, N.^\^ ^ N. The deviation for N.W. i N. is i6^° W, N. W. i N. =r 3| pts. L. of N. H. Leeway o 3i L. of N. 5 6 9-6 102 7 8 1 1-4 II-8 43'o or 39" 22' L. of N. Dev.i6''3o'L. ) T> Var.42 R. h^ 30 R. True Course 13 52 L. ofN. or N, 14° W., distance 43'. Add up hourly distances saUed from 5 o'clock until course is changed at 9 o'clock. 212 The Drn^'s Work. 6th Course, E. | N. The deviation for E. |- N. is 17^° E. E. |N. = vipts.R. ofN. 11. K. Leeway 3^ „ R. 9 3-4 10 32 Sum exc. 8 pts. loj ,, R. of N. 11 3'o Subtract from 16 1224 5i L. of S. Current Course, W.S.W. W.S.W. = 6 pts. R. of S. or 67- 30' R. of S. Variation 42 R. Sum exc. 90° 109 30 R. of S. Subtract from 180 True Course 70 30 L. of N. or N. 71° W., distance 23'- or 61° 53' L. of S. Dev. i7°i<'R) T, True Course 2 38 L. ofS. or S. 3° E., distance 12'. The corrected courses are written down to the nearest degree and the work will stand as follows : — Courses. Dist. N. E. W. S. 56° w. S. 28 w. N. 53 W. S. 44 E. N. 78 W. N. 14 W. S. 3 E. N. 71 W. 16 16 13 22 18 43 12 23 7-8 3-7 41-7 7-5 8-9 141 15-8 15-3 0-6 133 7-5 ro'4 17-6 io"4 21-7 60-7 50-8 50-8 15-9 80-9 15-9 9-9 65-0 62 28 N. ^^ Course N. 81° W., distance 66 miles. Diff. lat. 9'"9 and dep. 65'-o being found to correspond in their columns, give course N. 81° W., distance 66 miles. Course exceeds 56°, diff. long, must be found by middle latitude sailing. 62° 18' N.^ "?«; The mid. lat. is high and between two whole degrees, 10 N. I ^° therefore we proceed thus: — Mid. lat. 62° as course (in Table II), and dep. 64*8 (nearest to 65'o) as diff. lat., give in dist. column 138 ; and mid. lat. 63° and dep. 64^9 (nearest to 6 ^'•6) as diff. lat., 3.3 give in dist. column 143 : whence it is evident that for 1° . i^rt (or 60') of mid. lat., the diff. long, increases j' : we next 62 23 J J'^ make the proportion 60' : 23' : : 5 : a: 5 6,0)11,5 2 nearly, the correction for 23', Lat. left Diff. lat. Lat. in Sum Mid. lat. 124 46 Mid. lat. 62^^ gives D. long. 138 Correction for 23' 2 . ■ . Mid. lat. 62° 23' gives D. long. 140 By Calculation. Mid. lat. 62° 23' sec. 0-333900 Dep. 65' log. 1-812913 Diff. long. 140'* 2 log or 2^ 2o''2. 2-146813 Long, left 85° 17' E. \ %i Diff. long. 2 20 W. Ma"- — (rt^l Long, in 82 57 E. ; >^ p- The Day's Work. 213 Dev. Var. V. y. E. S. W. N. Departure Course. The opposite point to bearing N. by W. A W. is S. by E. ^ E. S. by E. J E. = i.i pts. L. of S. or i6°53' L. of S. 4 R. I ii_ ^• i5 7 R. ofS. or S. 160 W., clistanco n'. 1** Chiirsc, E. by S , dev. ig" E. E. by S. = 7 pts. L. of S. Leeway ■= i >> L. 7i „ L. ofS. ov8io34'L. of 8r 10° R. ) -n 14 R. I iL^ ^- True course 48 34 L. of S^ or S. 49° E., distance 56'-6. 2>id Onitrse, E. J S., dev. zc^ E. E. J 8. =- 7I pts. U of S. Leeway = J 1. I^'- Dev Vai 7| „ L.ofS. Dev. Var. or870 1l'L. of S. S"R:h4 OK. 01 ird 53 U L. of S. S. 53' E., distance 6i''i. Course, E.S.E., E., distance 68' o. Sth CbMr^e.S.E. iE.,dev. 15° E. S.E. iE. =.4ipts. L. ofS. Leeway ™ J „ L. 5 „ L. of S. 0»s6°l5'L. of 8. Dev. X5° B. ) .„ ^ „ Var. 14 R. j '9 ° ^• True course tj 15 L. of S. or S. 27° E., distance n'-y. 6th Course, N. by E., dev. 2° E. N. by E. « 1 pt. R. OfN. Leeway «= si » L. 4J „ L. OfN. or S3°2«'L. ofN. Dev. 2° R. > ,, ^ „ Var. H R. ( ^^ ° ^• 37 26 L. ofN. or N. 37° W., distance 4'7. Currcut Course. N.W. by N. = 3 pts. L. ofN. or 33" 45' L. ofN. ■^(r. 14 o R. 19 45 L. ofN. or N. 4o° W., distance ij'. EXAMPLE III. H Courses. K T^ Winds. r ii Remarks, &c. i3 ^■43 I E. by S. 10 6 4 6 S. by E. pts 19° E. A point of land in lat. 47° 44' 8., long. 179° 7' E., bearing by i 1 1 compass N.bW.JW. 4 II 4 dist. II miles. Ship's 5 6 I? head E. by 8. Dev. E.|8. 12 S.byE.p. i 20" E. as per log. 7 12 3 8 12 4 ' 9 12 10 12 3 I1 E.S.E. 13 4 Soutb . i iS'E. 12 I '3 '3 4 6 Variation 14° E. 2 14 3 3 14 3 4 N.E.byE. 13 8 N.byW. * 19" E. 5 13 8 6 '3 5 7 13 5 8 '3 4 A current sot the 9 S.E. i E. 12 1 SSW 1 W 1 M°h:. shipN.W. by N. cor- 5 4 rect magnetic 13 mis. II N. by E. 2 E.N.E. 5¥ 2»E. from the time the dc- partiu-e was taken to 12 2 6 the end of the day. Courses. Dist. II N. S. E. W. S. i6°W. .. io"6 3*o S. 49° E. . . 57 37-1 42-7 S.53"E. .. 61 36-8 48-8 S. 44° E. . . 69 49'4 477 S. 83° E. . . 68 J'9 67-7 8. 27° E. . . 35 21-8 ii-i N. 37" W. . . 5 3-8 2-8 N. 20° W. . . 13 12*2 4-4 i6*o i6i-6 2i8-o IO"2 i6"o lo-a '456 207-8 Diff. lat. i45'6 and dep. 207'8, found to oorrespend in the columns, give course 8. 55° E., and distance 254 miles (see B«ilc LXTV, page 186). Lat. left Diff. lat. 47°44'S. 2 26 S. 50 10 S Sum 2)97 54 Mid. lat. 48 57 I2 Mid. lat. 49'' as course in Table II, and half of the dep. 103 ■9(the whole dep. being tuo large a number to be found in the Table) gives in distance column isSwhich multiplied by a (as only half the dep. was used in entering the Table) gives diff. long. 316 miles. Mid lat. 48°57' sec. o'l8262i Dep. 207-8 log. 2-317645 D. long. 3i6'-4 or 5° 16' 4 log. 2-500266 Long, left 1790 7'E. D. long. 326' 5 16 E. %im eijfi. i! £nibt. front a^ »i E. Long, in 175 37 W. > 2 ft 214 The Day's Work. "W. N. E. S. W. N. Departure Course. Opposite to bearing E. by S. ^ S. 64pts. L.of S. or Tf 8' L. of S. Dev. i7°45'R-j 3 30 L. Yar. 21 IS L. ( _J_2_ 75 38 L. of S. or S. 77° E., distance 15'. 1st Course, E. by N. 7 pts. R. of N. Leeway \ „ I<. 6| „ R. ofN. or 75'' 56' R. of N. Dev. i7°45'R-> 3 30 L. Var. 21 15 L. I ^ ^" -^ 72 26 R. of N. or N. 720 E., distance 49'. 2»d Course, E.S.E. 6 pts. L. ofS. Leeway | „ L, 6.i „ L. of S. or 73° 8' L. of S. Dev. 13" 30' R. I 75 L. Var. 21 15 L. f ' ^^ 80 53 L. of S. or S. 81° E., distance 42'. p-d Course, N.E. by E. 5 pts. R. of N. Leeway 1 ,, L. Dev. Var. I7°15'R. 21 15 L. 4 .. or 45° R. } 4 1" R. of N. OfN. orN.4i°E 41 R. OfN. ., distance ^3'. 4«A Course, S.S.E. s pts. L. of S Leeway ij „ R. Dev. Var. 5°3o'R. 21 IS L. I „ L. ofS. or 8° 26' L. of S. 1 15 45 L. 24 II L. of S. or S. 24° E., distance 29'. ith Course, S.E. by S. 3 pts. L. of S. Leeway 2 „ R. 1 „ L. of S. or 11°15'L. of S. Dev. 8°3o'R. 1245L. Var. 21 15 L. ) 2- 24 o L. of S. or S. 24° E., distance 22'. 6th Course, E.S.E. 6 pts. L. of S. Leeway 2I „ R. 3} „ L. ofS. or 42°ii'L. of S. Dev. 13°30'R. t 7 ^r l Var. 21 15 L- ) ^ ^^ 49 56 L. of S. or S. 50° E., di.stancc 19'. Current Course, S.S.W. J W. 2^ pts. R. of S. or 28° 7'R. of S. Var. 21 15 L. 6 52 R. of 3." or S. 7° W., distance 18'. EXAMPLE IV. H Courses. KA- Winds. 1 Devia- tion. Eemarks, &c. I 2 3 4 5 6 E. byN. E.S.E. 12 4 12 2 12 2 12 2 10 6 •0 5 S.E. by S. South. pi8 1 i7|''E. Apoint.Tynemouth inlat. 55°i'N., long. 1° 25' W., bearing by compassW. bN.^N. di.st. 15 miles. Ship's head E. by N. Dev. as per log. 7 8 9 N.E.byE. 10 10 8 4 5 2 S.E.iby E. I i7i"E. 10 8 3 II 8 3 12 8 2 1 S.S.E. 7 4 East. l^ .S^E. Variation 2ii° W. 2 7 2 3 7 2 4 5 6 S.E. by S. 7 5 5 2 8 6 E. by N. 2 8fE. 7 5 4 8 9 10 1 1 12 E.S.E. 5 5 4 4 4 2 4 6 5 5 N.E. ^i n^E. A current set the shipS.S.W.iW. cor- rect magnetic 18 mis. from the time the de- parture was taken to the end of the day. Courses. Dist. N. .8. E. W. S. 77° E. .. N. 72 E. .. S. 81 E. .. N. 41 E. .. S. 24 E. .. S. 24 E. .. S. 50 E. .. S. 7 w. .. 15 49 42 33 29 22 19 18 151 24-9 3'4 6-6 26-5 20* r 12-2 17-9 i4'6 46-6 4f5 21-6 11-8 8-9 i4'6 2-2 40*0 86-7 40-0 159-6 2-2 2-2 467 i57'4 Diff.lat. 46';7| give in Table II !S'?^'^^%°^• Departure i57'4 J ° { Dist. 164^ Lat. left. 55' i' N. -d . DifiF. lat- 47 S. ] =* 2 Lat. in 54 14 N. 2)109 15 \ _!> To find D.long.(i) By Calculation Lat. 54° 37' sec. o'237288 Dep. 157-4 log- 2"i97°o5 D. long. 271 '8 log, 2*434293 Mid. lat. 54 37 Mid. lat. 54° and dep. 1S7'"5 in diff. lat. column (the nearest in Table to 157'4), gives in dist. column 268 for diff. long. ; and mid, lat. 55° and dep, 157'7 in diff, lat, column give in dist, column diff. long. 275 ; whence it is evident that for 1° change of mid, lat, we have (275 — 268) = 7' change in diff. long,, thus:— 60 : 37 : : 7 : 4'3. Mid, lat. 54°, dep. 157 give D. long. 268 Long, loft i°25'"W. • Correction for 37' + 4*3 D, long, 272' or 4 32 E, i Mid. lat. 54° 37', dep. 157-4 give 272*3 Long, in 3 7 E. 1^2 The Dm/8 Work. 215 W. N. E. 8. -W. N. Departure Course. The opposite po i ut to N.E. by N. isS.W. by S. S.W. by S. =3 pts. R. of S. or 33° 45' R- of S. Dev. ii°L. I , T Var. 25L-I ^^ ° ^^ True Course 2 1 5 L. of S. or S. 2° E., distance 17 miles. 1st Course. W. byN. = 78" 45' L. of N. Leeway o Dev. 11° L. ) .s „ T Var. 25 L. 36 o L. Sum exc. 90° 115 o Subtract from 180 o True Course 65 o R. of S. or S. 65° W., distance 25 miles. 2nd Course, W.S.W. W.S.W. =6 pts. R. ofS. Leeway(port tack) ^ „ R. 6i „ R. of S. or 73° R. of S. Dev. 9°L. I ^ Var. 25 L. ] 34 L- True Course 39 R. of S. or S. 39° W., distance 22 miles. ird Course, W.N.W. W.N.W. = 6 pts. L. of N. Leeway(port tack) i „ R. 5 ,, L. of N or 56° L. of N. Dev. Var. 90 25 L. L. i» L. True Course 90 L. of N. or West, distance 19 miles. ^th Cotirse, S."W. S.W. - 4 pts. R. of S. Leeway (star, tack) li „ L. 2j „ R.ofS. or 31" R. of S. Dev. 6° L. > T Var. 25 L. ) 3i L. or South, distance 16 miles. Sth Course, S.W. by W. S.W. by W. = 5 pts. R. ofS. Leeway{star. t;ick)ij „ L. 3i „ R. ofS. or 37" R. of S. 33 L. Dev. 8° L. ( Var. 25 L. ) True Course 4 R. of S. or S, 4° W., distance 13 miles, tth Course, South. South = o pts. Leeway (star, tack) 2J „ L. 2i „ L. of S or 25^ L. of S, Dev. 0° 1 Var. 25 L. 25 L. True Course 50 L. of S. or S. 50^ E., distance 13 mUes. Current Course, N.W. J W. N.W. J W. = sf L. of N. Variation 25 L. True Course 78 L. of N. or N. 780 W., distance 6 miles. EXAMPLE V. H Courses. K ^ Winds. Devia- tion. Remarks, &c. I 2 3 4 5 6 W. by N. W.S.W. 6 6 6 6 5 5 3 3 4 6 5 E.S.E. S. pts. 11° W. 9° W. A point, Lizard, in lal. 40° 58' N, long. 5° 12' W., bearing by compass N.E. by N., distance 17 miles. (Ship's ).eadW. 6 N) Dev. as per log. 7 8 9 W.N.W. 5 5 5 5 4 S.W. I 9°W. 10 4 8 II 4 6 12 I S.W. 4 4 6 W.N.W. 'i 6°W. Variation 25° W. 2 4 3 4 <; S.W.5W. 4 3 3 2 8 6 N.W.JW. '1 8°W. 6 3 4 7 3 8 9 10 II 12 s. 3 3 3 3 3 3 3 2 2 W.S.W. 4 0° A current set (cor- rect majinetic) N.W. f W., 6 miles, from the time the depar- ture was taken to the end of the day. Courses. Dist. N. S. E. W. S. 2° E. . . S. 65 W. . . S. 39 W. .. West South S. 4 w. .. y. 50 E. .. N. 78 W. .. 17 25 22 19 16 13 13 6 1-2 17-0 10-6 171 i6'o 13-0 8-4 0-6 lo-o 22-7 13-8 19-0 0-9 59 1-2 821 1-2 io'6 623 106 51-7 80-9 Diff. lat. 8o'-9 S. > .„„ ■ t,^. ,„ ^j ( Course S. 32J0 W. Departure 517 W. / ^ive in Table 11 J ^^^^^^^^^^^ ^^ ^^iigg. Lat. left 40^ 58' N. \ ^joo Diir. lat. 1 21 s, Kj 2 Lat. in. 48 37 N. ^ S I' Mer. parts 3471 ' m Sum 2)93 35 Middle lat. 40 17 Course S. 32^° W, Mer. parts 3347 Mer. diff. lat. 124 , 3^ Si p.'ia'ti'l^-oh) • ""^\^"" ^ " f (In depa?iJr'e column). Mid. lat. 49° Dep5i''8(asditf.lat. ; give in Table II {gf, 1-^,^^,1,^,3 Longitude left Diff. longitude Longitude in 5°12'W. \ M 2 6 31 V.)^>^E 2l6 The Layh Work. EXAMPLE VI. H. Courses. K. -h Winds. Lee- way. Deviation. Remarks, &c. I N.E. 1 E. 6 N.N.W. ^i i6|^ E. A point of land in 2 6 lat. 47° 35' y., long. 3 6 6 1 79'^.26' E., bearing by 4 6 4 compass S.S.fB., dist. 5 N.E.byE.JE. <; 7 S.E. 1 E. 4 i7r E. 14 miles. Ship's head 6 9 8 N.E. ^ E. Dev. as 7 6 3 per log. 8 6 2 9 S.E. f E. 12 N.E. by N. o ii«E. lO 12 4 II 12 12 ir 6 Variation 25° B. I South. 4 6 E.S.E. 2 2i° W. 2 4 6 3 4 6 4 •J a 5 N.W.bW.iW. 4 4 S.W. ^ W. I* 1 8^° W. 6 4 ^ A current set the shfp 7 4 <; N.E. by E. 1 E., cor- 8 4 6 rect magnetic, 36 9 N.W. J W. 12 6 S.W.bW.^W. i i8'W. miles, from the tjmo lO 12 S the departure was 1 1 T7 4 taken to the •nd of 12 12 5 the day. Coursefl. Dist. 14 25 24 48 19 18 50 36 N. S. E. W. N 13° W 13-7 4-6 14-6 37*7 11-3 45-9 13-4 223 236 i4'o 36-0 2-9 13-4 io'6 328 59'7 8 63 E N 'TO E g 17 E R AC "W N 36 W N 41 W. East 706 70-6 ^©•^ 95-9 59"7 36-2 Having filled up the Traverse Table, the sum of the Northings and Southings are equal, consequently the latitude remains unaltered, or, the ship, after sailing the foregoing courses and distances, has returned to the same parallel. Altogetjier, the vessel has sailed 95''9 towards the east on four courses, while she has made .59'-7 Wd^ting on the other fonr, leaving 36''2 of progress towards the East ; hence The Course is East, distance 36'-2 (see No. 161, page 104). To find the Latitude and Longitude in. The ship not having altered her latitude, the latitude arrived at is the same as the latitude left, viz., 47° 35' S., and consequently the difiF. of long, made good is to be found by Parallel Sailing, Rule LXVII, page 194, thus:— give in Table II Lat. 47 1° (as course) and Dep. (in diff. lat. column) By Calculation. Lat. 47° 35 Dep. 36'-2 sec. log. 0*171007 i'558709 &; long. 5^7 log. 1*7297 16 Long, left DiflF. long. Long, in Subtract from Long, in Diff. loDg. 53I'. (in distance column). 179° 26' E. + 53-5 E. 180 19*5 E. 360 o ^2 !» 4P-5 W. The Bay's Work. 217 Examples for Praotioe. EXAMPLE I. H. Courses. K. ^ Winds. Lee- way. Deviation. Remarks, &c. pts. I S.E. by S. 12 5 E. by N. i 3"^ A point of land in 2 12 5 lat. 35° 15' N., long. 3 12 6 75"" 30' W., bearing by 4 12 4 compass W. by N., 5 East. 9 8 N.N.E. I 23' E. dist. 19 miles. Ship's 6 9 4 headVS.E. by 8. De- 7 9 4 viation as per log. 8 9 4 9 N.E. 10 4 N.N.W. ^ 17° E. 10 10 6 II 10 4 12 10 6 1 2 North. II 10 4 W.N.W. 1 4'E. Variation 15° W. 3 10 2 4 10 4 5 N.N.E. II Eait. * 13" E. 6 10 4 7 10 4 8 10 2 A current set N.E. by 9 E.N.E. 9 •S North. I l^° E. E. correct magnetic 52 10 8 8 miles from the time the II 9 4 departure was taken 12 9 3 to the end of the day. EXAMPLE II. H. Courses. K. 1^ Winds. Lee- way. Deviation. Remarks, &c. pts. I East. 9 4 S.S.E. f 16° E. A point, Flambro' 2 9 6 Head, lat. 54° 7' N., 3 9 long. 0° 5' W., bearing 4 S.E. by E. 10 4 S. by W. ^ I2»E. by compass N.W. *W. 5 10 2 dist. 17 miles. Ship's 6 10 4 head E.S.E. Devia- 7 E.^S. 6 7 S. by E. ^ E. li 15° E. tion 13° E. 8 6 6 9 6 7 10 S. by W. 5 ! S.E. by E. 2 0° II 4 8 12 I 4 4 6 6 Variation 25° W. 2 South. 4 4 E.S.E. ^ 2»E. 3 4 4 4 4 2 5 S.E. by S. 3 -; E. byN. 2* 8<'E. 6 3 5 7 1 A current set (correct 8 E.N.E. 3 S.E. 3i 18° E. magnetic) N.N.E. , 6 9 3 miles, from the time 10 3 the departure was II 3 taken to the end of 12 3 the day. fp 3l8 The Day'ti Work. EXAMPLE III. H. Course 3. K. T^ Winds. Lee- way. Deviation. Remarks, &c. pts. I N.N.E. 9 5 East. i 8°E. A point of land in 2 9 5 ' lat. 43° 47' N., long. 3 9 6 7° 51' W., bearing by 4 9 4 compass S.W. by S., 5 E.N.E. 3 6 S.E. 2i iSl° E. dist. 13 miles. Ship's 6 4 4 head N.N.E. Devia- 7 g 5 5 6 tion as per log. 9 E.S.E. South. 2 • i3^°E. lO 7 II 6 4 12 5 6 I W. by N. I N. 6 s.w. ^ s. I| i5f°W. Variation 25° W. 2 5 6 3 5 4 4 6 5 S.S.E. 5 6 s.w. i 5FE. 6 6 2 7 6 4 A current set the ship 8 7 N. by W. correct mag- 9 N.N.W. 6 5 West. I 12° W. netic, 21 miles, from lO 6 4 the time the departure II 7 was taken to the end 12 7 2 of the day. EXAMPLE IV. H. Courses. K. T^o Winds. Lee- way. Deviation. Remarks, &c. pts. I S.W. i w. 6 5 S. by E. i E. 2 i4r w. A point of land in lat. 2 6 2 46°i2'S., long. 2"'io'W. 3 6 6 bearing by compass 4 6 7 E. by S. ^ S., 20 mis. 5 N. |E. 8 2 E.N.E. 1 20" E. (Ship's head W. by 6 8 2 compass). Deviation 7 S 9 8 6 g'E. S. by E. ^ E. 9 6 3 S.W. J W. 2* 20° W. 10 .? .5 II 5 6 12 4 6 I W. by S. 6 4 S. by W. 2^ s°w. 2 6 S Variation 14° E. 3 S 6 4 6 -; 5 E.N.E. 6 S.E. 2i i3rE. 6 6 4 7 7 6 8 6 9 S.S.W. i W. 6 7 S.E. I^ 18° W. A current set the ship 10 6 S.W. J W. by compass II .") 6 22^ miles these last 5 12 5 7 hours. The Bay's JFork. 219 EXAMPLE V. H. Courses. K. ^ Winds. Lee- way. Deviation. Remarks, &c. pts. I W.S.W. 9 4 N.W. ■ ^ 10° W. A point, lat. 35° 1 0' N. 2 9 6 ^'^^g- 5° 36' VV., bear- 3 9 4 inj^bycompassE.byS. 4 9 6 Ship's head N.N.E., 5 North. II 4 W.N.W. • i 3»W. dist. 9 miles. Devia- 6 II tion 9° E. 7 II 2 8 II 4 9 N.W. 8 4 W.S.W. ^ 17° W. 10 8 6 II 8 4 12 8 6 I S.W. by S. II 7 W. by N. 1 IS 5°W. 2 II 5 Variation 23° W. 3 II 4 4 II 4 5 W.S.W. 9 6 N.W. ^ lo'W. 6 9 5 7 8 9 9 4 East. I 4 S.S.E. I 15° E. 10 6 4 A current set the ship II 6 (correct magnetic) 12 6 S.E. by E., 15 miles. EXAMPLE VI. H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. I N.N.W. i W. 3 5 N.E. pts. If 2° W. A point, lat. 29° 59' N. 2 4 2 long. 32°54'E..beiiiing 3 4! 3 bycompassN.N.E.^E. 4 E.S.E. 2 i 7 N.E. 2 7°E. dist. 15 miles. Ship's 5 3 i head N.W. by W. 6 3 3 Deviation 6° W. 7 4 8 S.fE. S 4 E.S.E. H 2°W. 9 5 10 5 5 II 4 5 12 4 i 6 I N.E. J N. ■ 4 7 E.S.E. i^ 8^E. Variation 25° W. 2 4 2 3 4 4 4 3 7 5 3 6 W.iN. 3 ■5 s.s.w. ^ w. li 9'^W. 7 4 3 A current set the ship 8 3 6 (correct magnetic) 9 3 6 N.E., 30 miles, from 10 N. by E. 8 S E. by N. i 4 6»E. the time the departure 11 9 3 was taken to the end 12 9 2 of the day. 220 The Day's JFork. EXAMPLE VII. H. Courses. K. ^ Winds. Lee- way. Deviation. Remarks, &c. I N.N.W. lO 2 West. pts. f i9i°E. Apoint,lat.44°2o'S., 2 9 4 1 long. 1 7 6* 49' W. , bear- 3 4 c Q 4. ing by compass E. by West. 9 8 6 N.N.W. I • ii°W. N. i N., distance 18 miles. Ship's head 6 8 4 N.N.W. Deviation 7 8 4 as per log. g 8 6 9 W.by S. ir 6 N.W. by N. 1 4rw. lO 12 2 II II 8 12 12 4 I S.S.W. i W. 6 ■? West. li i7i"W. Variation 15° E. 2 6 3 1 3 6 6 9 4 4 5 South. 6 E.S.E. 1 21° W. 6 q 4 7 9 ■; A current set the ship 8 q ^ (correct magnetic) 9 S. by E. 1 E. T2 "J E. by S. i 20° W. N. by E., 18 miles, lO 12 6 from the time the de- II 12 5 parture was taken to 12 12 4 the end of the day. EXAMPLE VIII. H. Courses. K. ^ Winds. Lee- ! way. Deviation. Remarks, &c. pts. I E.S.E. 12 4 N.E. ^ 13° E. Apoint,lat.62'' i8'N. 2 12 <; long. 63° 17' W., bear- 3 12 <; ing by compass 4 5 12 6 W.N.W. , distance 21 E.JN. 4 N.N.E. 3i i7»E. miles. Ship's head 6 4 4 E.S.E. Deviation as 7 4 3 1 per log. 8 4 3 9 E.|S. 8 4 S.S.E. x4 13° E. 10 8 5 II 8 6 12 I S.W. f w. & 3 5 5 S. by E. 3i 8" W. Variation 60° W. 2 3 5 3 3 4 4 3 6 5 S. by W. 5 3 W. by S. 2i 0° 6 7 8 5 3 4 A current set the Bhip(correctmagnetic) 9 10 W.N.W. /| 2 North. 3 17" W. E. by S. 1 S., 49 miles, 4 2 from the time the de- 1 1 ,1 • 2 parture was taken to 12 4 4 the end of the day. The Day's Work. 221 EXAMPLE IX. H. Courses. K. -1^0 Winds. Lee- way. Deviation. Remarks, &c. pts. I 2 South. 5 4 3 8 E.S.E. 'i 2''E. Apoint, lat.^°49' N. long. 43° 54' W., bear- 3 4 i; ing by compass 4 4 4 N.E. \ N., distance 14 5 N.E.I J N. 6 6 E. by S. ^ S. I 14^ E. miles. Ship's head 6 6 4 South. Deviation as 7 6 per log. 8 6 9 S.S.W. i w. 5 .S S.E. ^ S. '^ s°w. lO 5 6 II 5 4 12 '; f I E.^S. 8 S. by E. i E. i 17° E. Variation 53° W. 2 8 4 3 8 4 4 8 2 5 s.w. \ s. 4 6 S.S.E. ^ E. 2 5°W. 6 "> 4 7 4 4 A current set the 8 4 6 ship(correct magnetic) 9 S.E. ^ S. 6 4 E. by N. ^ N. I 10' E. S.E. \ E., 41 miles, lO ■ 6 3 from the time the de- II 6 parture was taken to 12 6 3 end of the day. EXAMPLE X. H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pts. I S.W. i W. 4 8 S. by E. ^i 3'FE. A point of land, lat. 2 <; 2 36" 10' S., long, no* 10' 3 5 2 W.,' bearing by com- 4 .<; 3 pass E. by N., dist. 14 5 W. by S. J S. 4 3 S. by W. A ^i^E. miles. Ship's head 6 4 3 S.W. ^ W. Devia- 7 4 3 tion as per log. 8 4 3 9 W. by N. 1 N. 6 S.W. 2 17^ E. 10 5 4 II 6 2 12 6 4 I N.W. \ W. <; 6 W. by S. 1 S. A 7rW. 1 <; 4 Variation 20° E. 3 .s S 4 <; 8 S W. by S. 7 4 S.^W. '* 24° E. 6 7 4 7 8 2 8 8 2 9 S.AV. % 2 S. by E. H 33° E. A current set the 10 4 • ship the last 8 hours II S 7 E. 2 S. (coirect mag- 12 5 4 netic) 2 miles an hour. lit The Bay's Work EXAMPLE XI. H. Courses. K. -h- Winds. Lee- way. Deviation. Bemarks, &c. pts. I N.W. by W. 8 4 N. by E. * 15° E. A point of land in 2 8 4 lat. 55° 59' 8., long. 3 8 4 67"" i6'W., bearing by 4 8 4 compass! E.S.E., dist. 5 North. 6 E.N.E. I 22° E. 17 miles. Ship's head 6 S 6 N.W. by W. Devia- 7 a 6 tion ij" E. 8 6 9 N.W. by N. 4 N.E. by N. 1 4 if E. lO II a 12 Variation 23° W. I West. 4 6 s.s.w. 4 4°w. 3 12 4 4 12 4 5 N.N.E. 7 3 East. I 19^ E. 6 7 4 7 7 4 A current set the ship 8 7 4 N. by W. correct mag- 9 S.S.E. 9 S East. i 20° W. netic 27mile8,from the lb 9 S time the departure was ir 9 4 taken to the end of 12 9 4 the day. EXAMPLE XII. H. 3 4 5 6 7 8 9 10 II Courses. W. by N. W.^S. w.s.w. s.w. i w. s.^w. West. K. -^ 30™ have elapsed since that noon. But 5I' 30"> a.m. civil time on May loth is 17'' 30^ astronomical time on the 9th of May, for the 9th day of the month, according to the astronomical reckon- ing, commences at noon of the 9th civil time, and ends at noon of the loth civil day (the hours being reckoned up to 24), and 5^ 3o'»> a.m. of the loth is 17^ 30™ from noon of the 9th. Ex. 2. October 7th, at 3*> 20"^ p.m., civil time, is October 7th, at 3'^ 20" astronomical time. (See 1° of Rule LXXIV.) Ex. 3. October 7th, at 3^ lo™ a.m., civil date, h Octobor 6'^ 15'' 20™ aRtronomical date; since 7* less i^ is 6^, and i2'> added t j 3'' 20"" is ij^ 2o"\ (See 2° of Rule.) Ex. 4. January 3i8t, at 7'> 20™ p.m., civil time, is January 3i8t, at y*" 20™ astronomical time. (Rule LXXIV, i°.) 2 24 Longitude in Arc and Longitude in Time. Ex. 5. February ist, at 6^ 18™ A.M., civil date, is January 31'* 18^ ig"" astronomical date; since February i-^, diminished by i"*, gives January 31'*, and 12^ added to 6^ 18™ is 18^ 18". (Rule LXXIV, 2°). Ex. 6. What is the astronomical date corresponding to 1873, January ist, 8^ a.m. The corresponding astronomical date is 1872, December 31"* 20''. In this case the year is diminished by r, since in diminishing the day of the month by i, the reckoning throws us back into the last month of the previous year, i.e., the day before January ist, 1873, ^l^o 12'' added to S^ is 2oh. Examples for Practice. Express the following dates in astronomical time. I Jan. 2nd, 4h 381" 9«A.M. 7- Dec. 31st, 2 Feb. 27th, 8 12 P.M. 8. July ist, 3 Aug. 14th, 6 28 40 P.M. 9- July ist, 4 April ist. 7 54 19 A.M. 10. Oct. ist, • I June 4th, 4 18 3 A.M. II. 1872, Jan. 1st, 6 Sept. ist. 8 10 52 A.M. 12. i«73, Jan. ist. 6h 1%^ 34«p.M. 8 3 24 P.M. I 30 10 A.M. 10 12 P.M. 8 9 50 A.M. 44 12 A.M. 288. Given the astronomical date, to find the corresponding civil date. RULE LXXV. If the hours of astronomical time he less than iz^ write p.m. after it, and it will he the required civil time ; hut if the astronomical time he greater than 1 2**, add I to the days, diminish the hours hy i 2 and write a.m. after it : the result will be the required civil time. Express the following astronomical dates in civil time : — I. Jan. loth, i6'» 31'" 15^ 2. Oct. 14th, i5'> 17™ 13s 28 56 Dec. 3rd, 5 16 12 15 II 4. Mar. 3 ist, 23 10 16 15 7 Mar. 2 ist, 7 24 12 10 54 6. 1872, Jan. ist, 9 50 41 10 54 1872, Dec. 31st, 22 41 56 Feb. 3rd, II May 17 th, 7 Mar. 13th, 23 Sept. ist 8 Aug. 3 1 St, 20 LONGITUDE IN ARC AND LONGITUDE IN TIME. 289. The earth rotates uniformly on her axis once in twenty-four hours, and thus every spot on her surface describes a complete circle, or 360°, in that space of time ; hence the longitude of any place is proportional to the time the earth takes to revolve through the angle between the first meridian and the meridian of the place, and thus the longitude of a place may.be expressed either in arc or in time.* Longitude in arc and longitude in time are easily convertable, for since 360° is equivalent to 24*" (360 -r- 24 = 15°), 15° is equivalent to i*", 15' to i'", and 15" to i"; whence 1° is equivalent to 4"" {i.e., the 15 th part of i hour or 60°) i' ,, 48 {i.e., the 15th part of i minute or 6o») I" „ 4t {i.e., the 15th part of i second or 6ot)f and the following rules are sufficiently clear. * In reckoning by arc, each degree is divided into sixty minutes, and each minute into sixty seconds. In reckoning by time, each hour is also divided into sixty minutes, and the minutes into sixty seconds. But a distinct notation for each of these has been adopted, degrees, minutes, and seconds being represented by ° ' ", and hours, minutes, and seconds |)y h m s J and care should be observed not to use the same marks for both, great confusion arising from so doing. t A third is the name given to the sixtieth part of a second. Longitude in Arc and Longittide in Time. 22$ 290. To convert arc (or longitude) into time. EULE LXXVI. Multiply the degrees, minutes, Sec, by 4; this turns the degrees (°) into minutes (") of time, minutes (') into seconds (') of time, and the seconds (") into thirds (t) of time; or in other words, mark the resulting figures thus : — Those under seconds (") thirds (t), those under minutes (') seconds ('), those under degrees (°) minutes ("), and those to the left of the latter, hours C"). Note. — Instead of thirds it is customttry to uf e tenths of seconds, in which case the thirds must be reduced to tenths by dividing by 60 (see Rule XVI, page 54). Examples. Ex. I. Convert 12° 18' 15" into time. Ex. 2. Convert 25" 15' 16" into time. 12° 18' 15" 4 49™ 13* ot Four times 15" are 60", which contains &o once and o over ; write this remainder do^vn under the seconds (") and mark it thirds (t) as directed in the Rule, carrying the i. Again, 4 times 18' are 72, and the 1' carried makes 7.^ ; 60 goes in 73 once and 13 over; ■write this remainder (13) under the minuti s (') and call them seconds (•) and carry the i. Again, 4 times 12 are 48, and i carried makes 49 ; write this imder degrees (") and mark it minutes (") : whence the time corresponding to arc 12° 18' 15" is 49™ 13' ot. Ex. 3. Turn 77° 2' 10" into time. 77" 2' 10" 4 6o)4ot-o ^h gm g. 4ot •66 or, 5h 8™ 8"66 4ot66 Ex. 5. "What time is equivalent 15° 47' 58"? 15° 47' 58" 4 25° 15' 16" 4 ih^im js^t F.jur times 16" are 64", which contains 60 once and 4 over, and according to Rule this remainder placed under seconds {'') becomes thirds (t), and the i is to be carried. Again, feur times 15' are 60 and i carried makes 61, which contains 60 once and 1 over ; write the remainder i under minutes {'), and carry 1 : four times 25 are 100 and 1 carried gives 101, and 60 into 101 goes once and 41 remainder, which remain- der being placed under degrees (°) gives minutes (") and the 1 carried on being placed to the left of the latter is maiked hours {^) ; whence i"" 41"' i» 4t is the time corresponding to the arc 25° 15' 16". Ex. 4. What time corresponds to 127° 32' 40" P 127° 32' 40* 60)40^-0 -66 gh .jom lo' 4ot jh jm ijB ^2t or, 8^" 30" io"66 178° 45 53" 4 or, !•> 3" ii»"86 Examples for Practice. Reduce the following arcs into time : — "'"55" 3*32* or, I ih 55™ 3«-53 6o)32t-oo •53 1. i8''54'; i2''4o'45"; 137° 27'; 96' 10' 45' ; and 89° 16'. 2. 67° 42'; 76° 20' 30*; 1° 25' ; 140° 32' 10" ; and 69° 29'. 3. 0° 58''6; 49'' 4' 20"; 0° 26'-8 ; 14° 2' 30"; and 130' 19'. 4. 9° 14'; 163° 2' 48"; 0° 37' 4"; 2° 18' 12"; and 170° 15'. 5. 108^ 37'; 10° 27' 14*; 2° 29'; 84° 42' 30*; and 0° 34^'. 6. o" i3'-5; 51° 10' 12"; 156° 52': i78°49' 45"; ando''4i'-7. TO CONVEET TIME INTO LONGITUDE. 2gi. It has been shown (No. 289, page 224) that 4" of time are equivalent to 1° of arc ; hence it is evident that if we bring any given time into minutes, and divide by 4, we shall have the corresponding arc in degrees, minutes, and seconds. This is the reverse of the last process, oa 226 Longitude in Arc and Longitude in Time. EULE LXXVII. Reduce the hours and minutes into minutes, after which place the seconds, ^'c, then divide all hy 4, and the quotient will he the degrees, minutes, Sfc, of the corres- ponding arc; or, in other words, after dividing by 4, mark the resulting figures thus : — Those under minutes (") degrees (°), those under seconds (^) minutes ('), those under thirds (*) seconds ("). Examples. Ex. I. Turn i'' 5" 12*' into arc 60 4)65"" I2«ot 16° 18' O" Multiply x^ by 60, add the minutes (5) and divide by 4, the quotient is 16' with remainder i. Multiply this remainder by 60, and to the product add the 12 seconds; the sum is 72. Again, the quotient of 72, divided by 4, is 18, which is minutes (') ; whence the arc corresponding to the time i*" s™ i2» is 16° i8'. Ex. 2. Reduce 6'' 24™ 43^ into arc. 6"^ 24'n 43» 60 4)384" 43« ot 96" 10' 45" Multiplying 6'' by 60, and adding the 24" to the product, gives 384 as the sum; the quotient of this, divided by 4, is 96", with no remainder. 43'" divided by 4 gives quotient 10' with remaiufler 3 : remainder 3 multiplied by 60 gives 180, which divided by 4 gives quotient 4s" : therefore, 96" lo' 4j" is the arc which corresponds to 6'> 24™ 43'. Ex. 3. What arc corresponding to oh ^ym jgs p 4)oh 47"" 36* II" 54' In this instance it is not necessary to multiply by 60, as there are no hours to reduce into minutes : we divide 47"° at once by 4. Ex. 5. Convert 8*> 17"" 2>S^'5 ^°to arc. gh 17m 3^8.^ 60 4)497" 35" 30* 124° 23' 52''-5 30 Multiply the hours (Sh) by 60, and adding the minutes (17™) to the product gives 497"'; divide the result by 4, the quotient is 124°, with remainder 1. Again, multiply the remainder just obtained (1°) by 60, and to the product add the seconds of time, viz., 35», the sum is 95 ; then divided by 4, the quotient is 23' (minutes of arc) with remainder 3. Next multi- ply this last remainder by 60, the product is 180, to which add the 3ot ; and the sum 210, divided by 4, gives 52" of arc, and remainder 2, to which annex a cypher and divide by 4, the quotient is •$ of seconds of arc. Ex. 4. What is the equivalent arc to 9'' 25" 37'? 9'' 25" 37' 60 Ex.6. 174° 57' 40"-5 Multiply ll*" by 60, and to product 660 add jg™, dividing the simi, viz., 699 by 4 gives 174°, with re- mainder 3 ; this remainder (3) multiplied by 5o, and 50* added to product gives 230 ; this sum divided by 4 gives 57% with remainder 2 ; remainder 2 multiplied by 60, and 42' added, gives sum 162, which divided by 4 gives 40", and remainder 2; remainder 2, with a cypher annexed and divided by 4, gives quotient •5 ; whence the arc corre.'iponding to ii*- 39™ so"7 is i74°57'4°"'5. 4)565""- 37' 0* 141° 24' 15" Convert ii** 39" 50^-7 into arc III' 39m5o''-7 60 •7' 6 4)699™ 50»42t 42 Examples for Pbactioe Convert the following times into arc : — I, ii> 13" 52B 6. 9h 49™ 38s ri. oh 21™ 30^-9 16. oh 20™ 41 2. 3 52 4 7- 34 58-2 12. II 41 666 17- 8 36 56 3. 42 12 8. I 41 1-6 13. 3 52 18. 5 51 4. II 15 21 9. 5 59 4 14. 9 56 19. " 59 57 5. 4 29 5 10. 8 17 6 15- 52 20. I 52 -227 GREENWICH DATE. EEDUCTION OF GREENWICH DATE. 292. Def. — The Greenwich. Date is the day and time (reckoned astro- nomically) at Greenwich corresponding to a given day and time elsewhere. It is necessary to find the Greenwich date before the information contained in the Nautical Almanac can be made available, because all the elements there tabulated are given for time at the meridian of Greenwich. As in almost every computation of nautical astronomy we are dependent for some data upon the Nautical Almanac — and these are commonly given for Greenwich — it is generally the first step iu such a computation to deduce an exact or, at least, an approximate value of the Greenwich astronomical time. It need hardly be added that the Greenwich time should never be otherwise expressed than astronomically. The Greenwich Date is found at once from a chronometer, the error and the rate of which is known ; but it can also be found by means of the approximate time at place and the approximate longitude. 293. To find the Greenwich Date, the time at any other place and the longitude being given. EULE LXXVIII. 1°. Express the ship time astronomically (Rule LXXIV, page 223). 2". Convert the longitude into ^me (Rule LXXYI, page 225). 3°. In West longitude. — Add longitude in time to ship time ; the sum, if less than 24 hours, is the corresponding Greenwich date on the same dag with the ship date ; «/" greater thati 24 hours, reject the 24 hours, and put the day one forward. 4°. In East longitude. — From ship astronomical time subtract longitude in time, i/less than the hours, minufes, &fc., of the ship date; the remainder is the corresponding Greemoich date of the same day as the ship date ; if the longitude in time he greater than the hours, minutes, i^-c, of ship astronomical, add 24 hours to the latter, and put the day one back bejore the subtraction is made. 5°. When it is noon at the place. — The longitude in time, if West, is the Greenwich date {apparent time) ; but z/East, subtract the longitude in time from 24 hours : the remainder is the Greenwich date (apparent time) after noon of the preceding day. (a) Erom this last it is evident that when the sun is on a meridian in West longitude, the Greenwich apparent time is precisely equal to the longi- tude, that is, the Greenwich apparent time is after the noon of the same date with the ship date, by a number of hours, &c., equal to longitude. When the sun is on a meridian in East longitude, the Greenwich apparent time is before the noon of the same date as the ship date, by a number of hours equal to the longitude in time. 228 Greenwich Date. Note. — A bad habit prevails in -writino; datos, of separating tbc month and day from the hours, minutes, and seconds. The day of the month should always precede the minor divisions of time which give the precise instant of the day intended. Examples. Ex. I. November 9th, at 4*" 10™ p.m., apparent time at ship, longitude 32° 45' W. : required the corresponding time at Greenwich, or the Greenwich date. Ship date (A.T.) Nov. 9"^ 4'' lo"^ Longitude 32" 45' Long, in time . -\- x 11 4 Greenwich date, Nov. 9621 Ex. 2. June 5th, at 7'' 15"" a.m., app. time at ship, longitude 140° 30' E. : find corresponding Greenwich date. Ship date (A.T.) June 4'^ 19^ 15" Longitude in time — 9 22 6,0)13,1 o 2h i|m qb Ex. 3. January 3rd, at 8^ 1 2™ p.m., mean time at ship, long. 50° 45' E. : find Green- wich date. Green, date (A.T.) June 4 9 53 Ship date (M.T.) Longitude in time Green, date (M.T.) Jan. 3<' Sh 12™ — 3 *3 Jan. 3 4 49 Ex. 4. April 27th, at 5'^ 35'" 45^ a.m., app. time at ship, long. 122° 13' W. : what is corresponding Greenwich date ? Ship date (A.T.) April 26^ 17*' 35™ 45" Longitude 122° 13' W. -\- 8 8 52 26 25 44 37 — 24 Ex. 5. July 20th, at 3'' 35™ 7" p.m., mean time at ship, long. 85° 24' E. : find corres- ponding Greenwich date. Ship date (M.T.) July 20^ 3^ 35'" 7' + 24 or 19 27 35 7 Longitude 85° 24' E. — 5 41 36 Green, date (A.T.) April 27 i 44 37 Green, date (M.T.) 19 21 53 31 In Ex. 4, the added longitude advances the day of the month. (This illustrates latter part of 3° of the Bule). In Ex. 5, a day (or 24 hours) is borrowed before the subtraction is made, since the longitude in time exceeds the astronomical ship date, thus making the days of the month at Greenwich one less than at the place. (This illustrates the latter part of 4° of the Rule). Ex. 6. 1 87 3, January I st, 3^40™ 20' P.M., mean time at ship, long. 95° 7' E. : find the Greenwich date. Ship date (M.T.) 1873, Jan. i^ 3ii4on'20» Longitude 95° 7' E. — 6 20 28 Ex. 7. 1872, January ist, 2*1 i™ a.m., mean time at ship, long. 107° 4' W. : find the Greenwich date. Ship date (M.T.) 187 1, Dec. -^1^11^1^ o» Longitude 107° 4' W. -4-7816 Green.date(M.T.)i872, Dec. 31 21 19 52 i Green, date (M.T.) 1872, Jan. i 4 9 16 Ex.9. ^872. October ist, long. 2° "W., ih^i sun on meridian : required Greenwich I date (app. time). June ii<'i8''4o"o* I Ship date (A.T.) October !<• o'' o™ + 2 49 4 I Longitude 2° W. +8 Ex. 8. 1872, June 12th, 6'' 40" a.m., app. time at ship, long. 42° 16' W. : find the Greenwich date. Ship date (A.T.) Longitude 42° 16' W. Green, date (A.T.) June n 21 29 4 Ex. 10, Required the Greenwich date when the sun is on the meridian of a place in long. 80° 44' E., on January 12th. The sun being on the meridian, it is app. noon : hence Ship date (A.T.) Jan. i2<' o'' o™ o» Longitude 80° 44' E. — 5 22 56 Green, date (A.T.) Jan. 11 18 37 4 Green, date (A.T.) October i o 8 Ex. II. What is the Greenwich date wiitn the sun is on the meridian of a place in long. 155° 19' W., on March 3i8t? Ship date (A.T.) March 31'i o** o™ o» Longitude 155^^ 19' W. + 1021 16 Green, date (A.T.) March 31 10 21 16 In Ex. 10, the hours, "cc, of longitude to be subtracted are to be taken from a borrowed day, or 24 hours, thus making the day 0! . ..e month at Greeuwicli oue less than at the place. (See 5° of Bule). Reduction of Elements from Nautical Almanac. izg Ex. 12. 1882, February I st, long. 135° E.: find the Greenwich date when the sun is on the meridian. Ex. 13. 1883, January ist, the ship in long. 160° 30' E. : required the Greenwich date when the sun is on the meridian. Ship date (A.T.) February Longitude 135^ E. — 9 Ship date (A.T.) 1883 Longitude 160° 30' E. Jan. 1*^ o'> o" — 10 42 Green, date (A.T.) January 31 15 o Green, date (A.T.) 1882 Dec. 31 13 18 Examples for Practice. Required the Greenwich date in each of the following examples : I. 1882, January 6th !it 3h4o"i6»p.M. apparent time, long. 66-^56' o"W 2. February 13th at 8 40 3 A.M. apparent time, long. 21 4 W 3- February ist at 5 10 50 A.M. mean time, long. 14s 20 30 E. 4- March 15 th at 9 16 22 P.M. apparent time, long. 17 4 E. 5- May 15th at 8 38 35 A.M. apparent time, long. 141 51 ijW. 6. Novemijer 1st at 5 010 P.M. mean time. long. 114 30 oE. 7- December ist at 8 5 A.M. mean time, l')ng. 158 10 W. 8. July ist at 4 033 P.M. apparent time, long. 170 S5 15 E. 9- August 4th at 6 31 32 P.M. apparent lime, long. 100 17 30 E. 10. September ist at 8 29 I A.M. mean time. long. 148 47 30 W. II. December 28th at 2 42 10 P.M. mean time, long. 50 40 oE. 12. July 8th fit 4 36 A.M. apparent time, long. 178 51 W. 13- February ist at noon apparent time, long. 153 40 E. 14. Juno 1st at noon apparent time, long. 83 50 oE. 15- March 2nd at noon apparent time, long. I 25 w 16. 1883, January ist at noon apparent time, long. 149 10 E. REDUCTION OF ELEMENTS FROM NAUTICAL ALMANAC. 294. The Nautical Almanac* or Astronomical Ephemeris contains the riglit ascension, declination, &c., of the principal heavenly bodies for given equi- distant instants of Greenwich time ; the right ascension and declination of the sun and planets, for example, being given for every day at noon (o'' o™ o") at Greenwich, while for the moon these elements are given for every hour. Before we can find from the Almanac the value of any of these quantities for a given local or ship time, we must find the corresponding Greenwich date (Rule LXXVIII, page 227). Where this time is exactly one of the instants for which the required quantity is put down in the Ephemeris, nothing more is necessary than to transcribe the quantity as there put down. But when, as is mostly the case, the time falls between two of the times in the Ephemeris, we must obtain the required quantity by interpolation, it being requisite to apply a correction to that taken from the Almanac, in order to reduce it to its value at the given instant. To facilitate this interpolation the Almanac * The French Ephemeris, La Connaissance des Temps, is computed for the meridian of Paris, the German Beliner Astronomisches Jahrbuen for the meridian of Berlin. All these works are published annually several years in advance. 2 30 Hediiction of EhmenU from Nautical Almanae. contains the rate of change, or difference of each of the quantities in some unit of time, or, which is in general the simplest method, we may make use of certain tables computed for the purpose, called Tables of Proportional Logarithms. To use the difference columns with advantage, the Greenwich time should be expressed in that unit of time for which the difference is given : thus, when the difference is for one hour, the time must be expressed in hours and decimals of an hour ; when the difference is for one minute of time, the time should be expressed in minutes and decimals of a minute. 295. Simple Interpolation. — In the greater number of cases in practice, it is sufficiently exact to obtain the requisite quantities by simple interpolation ; that is, by assuming that the difference of the quantities are proportional to the differences of the times, which is equivalent to assuming that the differ- ences in the Ephemeris are constant. This, however, is never the case ; for example, referring to the Nautical Almanac for 1882, the variation in decli- nation for I hour for ist January is 12 "•66 2nd „ 1 3 "-So Srd „ i4''-94 &c. But the error arising from the assumption will be smaller the less the interval between the times in the Ephemeris ; hence, those quantities which vary most irregularly, as the Moon's Eight Ascension and Declination, are given for every hour of Greenwich time ; others, as the Moon's Parallax and Semidiameters, for every twelfth hour, or for noon and midnight ; others, as the Sun's Eight Ascension, &c., for each noon ; others, as the right ascensions and declinations of the fixed stars, for every tenth day of the year. TO EEDUCE SUN'S DECLINATION. 296. The declination of the sun is given in the "Nautical Almanac," pages I and II of each month, for every day both for apparent and mean noon at Greenwich. The difference of declination for one hour f'Var. in i hour' ^ J is always annexed, and is intended to facilitate the reduction of the quantities from noon to any other time. In general it is necessary to take out the required quantities for the nearest Greenwich time to the given time, and interpolate in either direction to the given instant of Greenwich time. Method I. — By hourly difference.* EULE LXXIX. 1°. Get a Greenwich date hy means of ship time, expressed astronomically, and longitude (see Eule LXXVIII, page 227), or iy means of chronometer. To express the Greenwich time in hours and decimals of an hour. Annex a cypher to the minutes and divide by 60, or divide the minutes by 6, and consider the quotient as tenths of an hour, and to this prefix the hours. For example, let it be required to express 7^ iS" in hours and decimals of an hour. Then 6 is contained in 18 three times; to this prefix the hours (7) and we have 7-3 hours. — (See Ex. 3, page 54). * This method of reducing the sun's declination is the one required to be used at the Local Marine Board Examinations. Redwtion of EhmenU from Nautical Almanao. 2 3 r 2°. Take out of Nautical Almanac the declination for the nearest noon to the given Greenwich date, noting whether the declination is increasing — in which case affix the sign + to it — or decreasing — when the sign — must be affixed — and a little to the right place the " Var. in i hour'' ^ found in page I of the month in the Nautical Almanac. {a) When Greenwich date u given in apparent time, use page I of the month, but for mean time use page II of the month. {b) The tenths of seconds {") of declination as given in the Nautical Almanac may bo rejected when less than five, but call them i" when thoy amount to five or above — thus, 6""4 is put down 6", but 42"-7 will be put down 43" ; and it may bo here observed, that whenever a decimal is rejected in a final result if the first decimal figure be 5, or above it, add I to the last figure of the result. {e) When the seconds of time (in Greenwich date) are less than 30*, they may be rejected; but if above 30', increase the minutes of time by i""; thus, Greenwich time 2>' 35™ 408 would be called 2'' 36"". 3°. Multiply the '^Var. in i hour^' hy the hours, and fractional parts of an hour, that have elapsed since, or must elapse before that noon, as the case may be; the product reduced to minutes and seconds is the cha7ige of declitiation in the time from noon. 4°. Apply this corrc'ctio7i to the declination for the nearest noon to the given time, i.e., to the declination of the same noon as that for which the "Var. in i hour " has been taken as follows : — (a) When the Decl. is increasing for has -f affixed), the correction for the time elapsed since noon is additive, but the correction for the time that must elapse is subtraetive. (b) When the Bed. is decreasing for has the sign — affixed), the eorrectioti for the time elapsed since noon it subtraetive, but the correction for the time that must elapse before noon is additive. The result is the declination sought. NoTB. — It must be remembered that when the declination is taken out of the Nautical Almanac for the noon of the day folloiving that of given Green, date, the correction is applied the contrary way to the sign affixed. Examples. Ex. I. Greenwich date, Jan. lo'' 6'' ; in this case take 22"o8 the Diff. for hour on the 20th, which multiplied by 6 gives the correction of the Decl. for tlio loth day — to be sub- tracted because the Declination is decreasing, and we have multiplied by the number of hours that have elapsed since noon. Ex. 2. Greenwich date, Jan. lo'' 19''; in this case take 23"-i4, tho "Var. in i hour" on the nth, which multiplied by the difference between 24^' and 19'' gives the correction of tho Decl. for the nth day — to be added because the Decl. is decreasin//, and we have multiplied by the number of hours that must elapse before noon of the i ith day. Ex. 3. Greenwich date, April 2"^ 6|'' ; in this case take 57""49, the "Var. in i hour" on the 2nd, which multiplied by 6|>', gives the correction of the declination for the 2nd April — to be added because the Decl. is increasing, and we have multiplied by 6^ hours the time that has elapsed since noon, April 2nd. 232 Reduction of Elements from Nautical Almanac. Ex. 4. Greenwich date, April 2^ i']\'" ; in this case take 57"'25, the " Var. in i hour " on the 3rd, which multiplied hy 6| (the diff renfe between 24'' an i 17!^) gives the correction of the Decl. for the 3rd day — to he subtracted because the Decl. is increasing, and we have multiplied by the number of hours that must elapse before noon, 3rd. 5°. If the correction when subtractive exceeds the declination itself subtract the declination from the proportional part ; the remainder is the declination of the contrary name. In March, when the declination changes from South to North, and in September, when it changes from North to South, if the correction, by being subtractive, exceed the declination, subtract the declination from the correction, and call the remainder N. in March, but S. in September. (See Ex. 3.) Method II.— By proportional logarithms. EULE LXXX. 1°. Find a Greenwich date. 2°. Take out of the Nautical Almanac the declination for the noon at Green- wich, and that following it. 3°. When the declinations are of like names, take their differeyice ; but when of different names, take the sum : this is the daily variation of declination. (a) When the declination is increasing, place the sign of addition (4-) hefore the daily variation; hut whn the declination is decreasing, place the sign of subtraction ( — ) hefore it. 4°. Under the daily variation place the hours and minutes of Greenwich time, and take from the tahle (Table XXI A, Eaper, or XXXIII, Norie) log. of change of declination in 24 hours and log. of hours and minutes of Greenwich time ; the sum of these logs, found in the table will give the proportional part of daily change of declination. In using Table XXI A, Kaper, or Noeie XXXIII, minutes (') of declination, and hours of time (h), are found at the top of the columns ; seconds (*) of declination, and minutes (") of time at the side columns. 5°. -Apply the proportional part to the declination at the first noon, adding when the declination is increasing, but subtracting when the declination is decreasing. The result is the declination at the time required. Examples. Ex. 1. 1882, January 13th, at i^ 54" 16^ p.m., app. time at ship, long. 30° 4' E. : find the sun's declination. Ship date (A.T.) January 13'' 3^54™! 6' Longitude 30° 4' Longitude (30' 4' E.) in time — 2 o 16 4 c Am 5_4 Green, date (A.T.) January 13'^ ih54'>' o^ 60)54-0 6,0)12,0 16 or, i^*9 "9 2'»o™i6' Reduction of Elements from Nautical Almanac. '-31 Method I. Decl., page I, N.A., for January 13th, app. noon, is 21° 26' 44" S., decreasing ( — ), and var. for i hour is 25"" 82. Var. for i^", 13th, noon 25"-82 Green, time i'' 54"" = 1^-9 x i"9 Method II. Decl. app. noon, page I, N.A. Jan. 13th, 2i°26'44'S. 14th, 21 16 12 S. Daily var. Green, time 10 32 Correction 23238 2582 49-058 Correction — o 50 r3th, at noon 21 26 44 S. log- 3576 log. 1-1015 log. I -459 1 Red. decl. 21 25 54 S. Decl., noon, Jan. 13th 2 1^26' 44' S. ( — ) Correction — 49 Red. docl. 21 25 55 S. The correction 49", which is for a time elapsed since the noon for which the decli- nation is taken out, is subtracted from docliaation at noon, because the declina- tion is docroasing. Having found the Qroonwich date, the sun's declination is taken from the Nautical Almanac, where it is found in page I of the month (the Greenwich date being in app. time), and on the same page and in colunm headed " Var. in 1 hour " is found the change of declination for 1 hour past noon ; nest observe that the decli- nation is decreasing, and make a note of it. Now, since the declination changes 25"-82 in 1 hour past noon, how much does it change in the Greenwich time past noon, viz., ib 54'" ? First annex a cypher to the minutes (54") and divide by 60 ; thus 60 is contained in 541 nine times and nothing over. To this we prefix the hour, and we then have the Greenwich time i^ S4'" = i'''9 expressed in hoiurs and decimals of an hour. (See Rule XVI, page 54.) Set this under the hourly di£f. and then proceed as in multiplication of decimals, the result- ing figures are 49058, but as we have two decimals in the multiplicand and one in the multiplier, in all three places, three figures are to be marked off from the right hand, leaving 49" (see Kule XIII, page 47). Ex. 2. 1882, May 21st, at 7'' 50'» a.m., mean time at ship, long. 149" 30' E. sun's declination. required the Ship date (M.T.) May 2o<' 19*150"" Longitude 149° 30' E. — 9 58 Green, date (M.T.) May 20 9 52 Decl., page II, N.A., May 20th, at noon, is 20° 1' i" N. (mereasingj , var. for i^ at noon, May 20th, 31"- 18. Var. for i*" gh ^2"" = 9''*87 nearly By Hourly Diff. 52™ = M)52o 866 or -87 nearly. Decl. mean noon, page II, N.A. May 20th, 20^ i' i"N. 2I8t, 20 13 19 N. ly X 9-87 21826 Daily variation Green, time Correction Decl. 20th, noon Red. decl. + r2 18 9*'52'" log. log. log. 2903 3860 24944 28062 6,0)30,7-7466 ■f 20 20 5' 3' I I N. 6 4 N. 6763 Correction -f 5'8 nearly. Decl., May 20th, noon 20° i' i"N. 4- Corr. for 9^ 52™ = _j_ ^ g The correction 5' 3", which is for the time elapsed since noon, is added to the declination at noon, because the declination is increasing. Red. decl. 20 6 9 N. JSxplanation.—H&sin^ found the Greenwich date ; with this date the sun's decUnation is taken out of the Nautical Almanac, page IT for May (the Greenwich date being mean time), and in page I of the month, in the column headed "diff. for 1 hour" is found the chanp:e for i hour past noon ; next observe whether the declination is increasing or decreasing. In this instance it is increasing, and we note th's. Now, since the declination increases 3i"-i8 in i', what will be the change in Green, time past noon, viz., 9!" 52" ? We have now to express this in hours and d-eimal parts of an hour. Then 52™ = I o of an hour, and annexing cyphers to 52 and dividing 60, we have 60 in 520, or 6 in 52 goes 8 times and 4 over; 6 is contained in 40 (the remain- der and a cypher annexed) six times and 4 over, or seven times nearly— two places of decimals only being used. Set the '87 under the hourly dificrence to which prefix the gi", and then proceed as in common multipli- cation. The resultmg fi-ures are 3077466, but as we have two decimal places in the multiplicand, (ind two decimal places in the multiplier, in all four, four flgiu-e s are to be marked off from the right hand, leaving 307". but since the first decimal figure (7) exceeds 5, we increase the seconds by i" in consequence, and the cor- rection is 3o>i", which divided by 60 gives + 5' 8", the correction of declination. 11 U 234 Re&ucUon of Elements from Nautical Almanac. Ex. 3. 1882, Greenwich date, Marcli 13"^ i4>> 24™ i8«, mean lime: required the sun's declination. Green, date (M.T.) March Subtract from Time before noon March 14th ijdi^hi^migs 35m 42» or 36" = M)36o ~ -6 9 35 4> or 9''*6 Decl. page II, N.A., March 14th, at noon is 2° 27' 14 S. ( — ), and var. in 1^ = 59"'i6. Var. in I*" =: Time before noon 59*-i6 9^ 35496 53244 6,0)56,7 "936 Correction + 9' 28' Decl., March 14th, noon 2° 27' 14" S. ( — ) By Aliquot parts. Var. in i** = 59"'i6 9 30" 6 ¥' 6,o)56,-793 Correction 9 28 Red. decl. 3 36 42 S. In this example the Greenwich noon of March 14th is nearer the given time (which exceeds 12'') than the Greenwich noon, March 13th; therefore take out tlie declination from page II of the month, Nautical Almanac (because mean time), for noon March 14'', and from page I of the month, also the "Var.in 1 hour" corrpsponding to this declination, vIh., S9"'i6. Next subtract the hours and minutes of Greenwich time from 24'', the remainder g'^ 35" 42' or q^ 36" is the time tha' must elapse b ofore noon 14th. Divide tlie minutes of this last by 60 to get decimals of an hour ; thus 6 is uontained in 36 six times, hence we have '6 (see Rule XVI, page 54), to this we prefix the hours (g) and we then have g^-e. Next multiply the hourly difference by this, and the resulting figures are 567936, then three figures marked off from the right hand, leaves 567", which being increased by i in consequence of the first figure on the right of the decimal jioint exceeding 5, gives for the correction 568", which divided by 6c gives 9' 28". And since the declination at noon 14th is decreasing, it is evident that the declination at g"" t,6<^ before that noon will be more than at noon 14th, and the correction 9^ 28" is, therefore, to be added ; whence the reduced declination is 2° 36' 42" S. Ex. 4. 1882, March 20th, 6'' 39™, app. time at Greenwich : required the sun's declination. Green, date (A.T.), March 20^ 6'>39™ 39" = Decl., page I, N.A., March 20th, at noon, is 0° 4' 52"-6 B. ( — ), and var. in i^ = 59"'25. Var. in i*> = 59''"25 6-65 •65 By Aliquot parts. Var. in ih = 6,0)39,4-0125 Correction — 6" 34 March 20th, at noon Correction for 6h-65 5. (-) Eed. decl. o 1 41 N. In this example the correction 6' 3,^, which is that due to the time elapsed since the noon for which the declination is taken out, Is suhtr active and the d clination decreasing ; but since the correction exceeds the declination itself, subtract the declination from the correction ; the remainder is the reduced declination, and of a contrary name to that at noon. We may consid r the sun moving northward, and therefore the correc- tion may be marked N. ; and since it is greater than the decl. it shows that the sun has crossed the equator, and has now North decl. Hence the difference between the decl. and the correction must be taken, and marked with the name of tie greater. Reckiction of Memmts from Nautical Almanac. 235 Ex. 5. 1882, February nth, at 8^' 54"> 47" p.m., apparent time, long. n° 4' W. : find the declination. Ship date (A.T.) February 1 1'' S*' 54'" 47' Longitude in time -|- 44 16 Longitude ii*^ Green, date (A.T.) February 1 1** 9 39 3 or 965 Hourly Difif., page I, N.A. Feb. nth at noon 49 "48 965 44"! 6* Bi/ Aliquot Parts. 49^-48 9 6,0)47,7-4820 Correction 7"57 Decl., p.ige I, N.A. Feb. nth, at noon 13° 56' 59" S. ( — ) Corr. for 9'' 39'" — 7 57 30 f 44532 6 ft 2474 3 i 494 247 5,o)47.7"47 Reduced decl. 13 49 2 y- 7-57 297. To find the declination of the sun at the time of its transit over a g^ven meridian. When the sun is on a meridian in West longitude, the Greenwich apparent time is precisely equal to the longitude ; that is, the Greenwich apparent time is after the noon of the same dat') with the ship date by a number of hours, equal t > the lougitude in time. When the sun is on a meridian in East longitude, the Greenwich apparent time is before the noon of the same date as the ship date by a number of hours, equal to the longitude in time. Hence, to obtain the sun's declination for apparent noon at any meridian we have RULE LXXXI. Take the declination from the Nautic al Almanac (page I of the month) for Greenwich apparent noon of the same duie as the ship date, and apply a correction equal to the hourly difference multiplied by the longitude, observing to add or subtract this correction according as the numbers in the Nautical Almanac may indicate for a time before or after noon. ExAJktPLES. Ex. I. 1882, September loth, the sun on tho meridian, long. 100^ 35' E. sun's declination. required the Longitude 100° 35' E. 4 6,0)40,2 20 6*'42'"20'. Sun's decl., page I, N.A. Sept. loth, noon 4° 53' 8'N. ( — ) Corr. for 6*" 42" -4- 6 22 The lonsrituile being e"" ^2'° 20* East, the Green. A.T. is 6^ 42™ bffore the noon of September loth— the same date as the ship date. The decl. is taken out of the Nautical Almaruic, page I of the month ; also take out tit the same time the hourly ditf. ; the work will stand thus : — Var. in i'', page I, N-A. Sept. loth, noon 56*-95 Time from noon loth, 6'' 42'" ^ 6-7 Reduced decl. 6 59 30 N. 6,0) 3», I -565 As the declination is decreasini/, the declination at (,h ^2"> before noon will be greater than that for noon. Correction 622 236 Reduction of Elements from Nautical Almanac. Ex. 2. 1882, June ist, the sun on the meridian, long. 75° W. : required the sun's decl. I^ong. 75° W. 4 The longitude being s^ "West, the Greenwich A.T. is sh after the noon of June ist — the same date as the ship date. The deel. 6 olio o ^® taken out of the Nautical Almanac, page I of the month; '■'•'' also take out at the same time the var. in i""; the work will stand thus : — Sun's decl., page I, N.A. Var. in !*>, page I, N.A. June ist, at noon 22° 4' 57 " N. -\- 2o"'25 Correction -\- i 4.1 5 Red. decl. 22 6 38 N. 6,0) 10,1 "25 Correction i'4i As the decl. is increasing, the decl. at s"" after noon will be greater than that for noon. Ex. 3. In the last question suppose the longitude to be 75° E. The longitude being s^ E., the Green, A.T. is s^ before the noon of June ist — the sama date as the ship date. The decl. is taken out of the Nautical Almanac, page I of the month ; also the var. in 1^, and the work is as follows : — Sun's decl., page I, N.A. June ist, at noon 22° 4' 57' N. + Var. in !*> = 20'"25 Correction — i 41 5 Ked. decl. 22 3 16 N. 6,0)10,1-25 As the decl. is increasing, the decl. at s"" before noon will be less than that for noon. 298. Interpolation by Second Diflferences. — The differences between the successive values — given in the Nautical Almanac as functions of time — are called the^rs^ differences; the differences between these successive differences are called the second differences ; the differences of the second differences are called the third differences, &c. In simple interpolation we assume the function to vary uniformly ; that is, we regard the first difference as constant, neglect- ing the second difference, which, is, consequently, assumed to be zero. In interpolation by second differences we take into account the variation in the first difference, but we assume its variations to be constant ; that is, we assume the second difference to be constant, and the third difference to be constant. When the Nautical Almanac is employed we can take the second differences into account in a very simple manner. In this work, since the year 1863, the difference given for a unit of time is always the difference belonging to the instant of Greenwich time against which it stands, and it expresses, therefore, the rate at which the function is changing at that instant. This difference, which we may here call the first difference, varies with the Greenwich time, and (the second difference being constant) it varies uniformly, so that its value for any intermediate time may be found by simple interpolation, using the second differences as first differences. Now, in computing a correction for a given interval of Greenwich time, we should employ the mean, or average value, of the first difference for the interval, and this mean value, when we regard the second differences as constant, is that which belongs to the middle of the interval. Hence, to take into account the second differences, we have only to observe the very simple rule — employ that (interpolated J value of the first difference which corresponds to the middle of the interval for xohich the correction is to he computed. Redttction of Elciaents from Nautical Almanac. 237 299. Degree of Dependence. — The sun's declination changes nearly i' an hour, or 1" in 1™, in March and September; hence to insure it to i" in the extreme case, the Greenwich date must be true to i". Examples for Practice. Required the sun's declination in each of the following examples : — [These are prop;iratory to working Amplitudes, Azimuths, &c.] January jth, February 2nd, March 31 at, March 26th, May 1 6th, April 29th, Juno loth, November ist, September ist, October ist, December i6th, November 14th, ghjjmjjs A.M. 390 P.M. 6 2 13 P.M. 7 8 22 A.M. 9 17 20 A.M. 2 26 52 P.M. 8 45 O P.M. 10 20 16 A.M. 8 20 40 A.M. 6 II 30 A.M. 4 35 32 A.M. 6 45 8 P.M. app. time at ship app. time at ship app, time at ship mean time at ship moan time at ship mean time at ship app. time at ship moan time at ship app. time at ship mean time at ship app. time at ship mean time at ship long. 108" 7' W. long. 52 45 W. long. 156 3 E. long. 72 47 E. long. 45 40 W. long, no 57 W. long. 129 30 E. long. II 17 E. long. 172 9 E. long. 68 15 W. long. 4 8 E. long. 100 2 E. In each of the following examples it is required to find the sun's declination when the sun is on the meridian (at apparent noon) : — '3- 14. 15- 16. 17- 18. >2, Jan. 19th, Feb. 1 6th, Mar. 2olh, May 8th, June 21st, Mar. 2olh, long, long, long, long, long, lonij. 86°57' W. 72 59 E. 168 3 W. 10 35 W. 167 IS E. 129 o W. 19. 20. 23- 24. 1882, July 28th, „ Sept. 23rd, „ Oct. ist, ,, Dec. 22nd, 1883, Jan. ist, 1882, Sept. 23rd, long. 2" o' W. long. 156 o E. long. 170 58 E. long. 179 52 E. long. 156 48 E. long. 174 15 E. 300. The Polar Distance of a heavenly body is its angular distance from the elevated pole of the heavens ; it is measured by the intercepted arc of the hour circle passing through it, or by the corresponding angle at the centre of the sphere. According as the North or South pole is elevated, we have the North Polar Distance, or the South Polar Distance. 301. To find the Polar distance of a celestial object, proceed according to the following rule : — RULE LXXXII. When the latitude of the place, and declination of the object, are of the same name subtract the declination from 90° ; but when the latitude and declination are of contrary names, add the declination to 90° ; the result in either case is the polar distance. When the latitude is o, the declination, either added to or taken from 90"^, is the polar distance. Examples. Declination. Polar distance. 8' 12' 18" S 98^ 12' 18" 22 30 o N 67 30 o Lat. N. N. S. N. S. S. 31 '5 S 87 28 45 30 23 15 S. 7 22 32 N. 26 42 12 S. 120 2- 12 48 N. 15 ... 97 22 32 ... 63 17 48 ( 102 48 2 \or 77 II 58 2j8 Rechiction of Elements from Nautioal Almanac. TO FIND THE EQUATION OF TIME. 302. Apparent Solar Day is the interval between two successive transits of the actual sun's centre over the same meridian ; it begins when that point is on the meridian. The apparent solar day is variable in length from two causes ; first, the sun does not move uniformly in the ecliptic — its apparent path sometimes describing an arc of 57', and at other times an arc of 61' in a day ; second, the ecliptic twice crosses the equinoctial — the great circle whose plane is perpendicular to the axis of rotation — and hence is inclined to it in its different parts ; at the point of intersection the inclination is about 23° 27', •at two other limiting points they are parallel. A uniform measure of time Is obtained by the invention of the Mean Solar Day. 303. Mean Solar Day is the interval between two successive transits of the mean sun over the same meridian ; it begins when the mean sun is on the meridian. This fictitious body is conceived to move in the equinoctial with the mean motion of the actual sun in the ecliptic. The length of the mean solar day is the average length of the apparent solar days for the space of a solar year. 304. Eq^uation of Time is the difference between apparent and mean time. It is measured by the angle at the pole of the heavens between two circles passing, the one through the apparent sun's centre, the other through the mean sun. The Equation of Time is so called because it enables us to reduce apparent to mean, or mean to apparent time. In consequence of the motion of the sun in the ecliptic being variable, and the ecliptic not being perpen- dicular to the axis of the earth's rotation, apparent time is variable, and this fluctuation is considerable, amouutiug to upwards of half an hour— apparent noon sometimes taking place as much iis 16" before mean noon, and at others as much as 14^"' after. These are the greatest values of the equation of time ; it vanishes altogether four times a year — this occurring about April 15th, June 15th, September ist, and December 24th. It is calculated and inserted in the Nautical Almanac for every day in the year. On page I of each month the equation of time given is that to be used in deducing mean from apparent time ; that on page II is to be used in deducing apparent from mean time. The difference in the value of the two arises from the one being that at apparent noon, and ihe other that at mean noon. As these may be separated by an interval of more than a quarter of an hour, the equation of time given in pages I and II may differ by a quarter of the " Var. in i hour" given in the adjoining column. The equation of time is Itself a portion of mean time. 305. To Reduce Equation of Time to Greenwich date. — The method of correcting the equation of time for the G^reenwich date is similar to that for correcting the sun's declination, and the " Variation in i hour " may be used for the purpose. EULE LXXXIII. 1°. Get a Oreenwich date, as before. NofE.— TUb time by chronometer when error and rate are applied to it, gives Mean Time at Greenwich. Reduction of Elements from Nautical Almanac. 23.9 2°- Take out of Nautical Almanac (page II of the month) the Equation of Time for the noon of Greenwich date, a7i,d mark it additive or subtractive, accordinij to the heading of Equation of Time at the top of the column in page I of the month ; aho note whether it is increasing — ivhen affix the sign + ; or decreasing — affixing the sign — ; at the name time take from the column in page I the " Var. in 1 hour.^^* Note. —It sometimes happens that the precept for applying the Eq. of Time changes hi the course of the month. Thus in April, 1882, a hlack lino is placed bt;tween the E(i. T. for the 14th and that for the 15th, indicating that a change of precept occurs between those days. The Equations above the line, page I, have to be added, those below have to be subtracted. 3°. Multiply the '^Var. in i hour" b>/ the howrs 0/ Greenwich time, and when great precision is necessary, by the fractional parts of an hour also. The result is the correction to be applied to the equation of time taken from, the Nautical Almanac, and is to be added when equation of time is increasing, but subtracted when equation of time is decreasing ; the result is the Equation of Time sought. Note. — "We may, as in reducing the decimation (see preceding Rule LXXXIII), take the Eq. T. and "Var. in !•> " from the Nautical Almanac for the nearest noon to the Green- wich time, and multiplying; the " Var. in i''" by the time that must elapse before noon ; the correction thus o^itainod must be applied to the Eq. of T. taken out of Nautical Almanac in a contuxry way to that directe*! above, that is to iay, when correcting backwards the rule is Eq. T. increasing, subtract — Eq. T. decreasing, add. (See Exu. 3, 4, and 5). (a) When the correction, being subtractive, exceeds the equation of time itself subtract the equation of time from the correction ; the remainder is the reduced equation of time sought — and it is to be subtracted /row apparent time when equa- tion of time at noon is directed to be added, bid added to apparent time when equation of time at noon is directed to be subtracted ; i.e., the Equation lias to be applied to A.T. according to the precept /or the day following the given day. EXA^EPLES. Ex. I. 1882, January 29th, 6'' 53"" 49' mean time at Greenwich ; find Equation of time to be applied to apps-rent time in working the chronometer. Eq. of Time, page II, N.A. Hourly DiflF., page I, N.A. Jan. 29th, add i3'"24''6 (-j-) Jan. 29th, at noon o''4a3 Corr. for 6*'"9 + 2-9 6^ 54"^ is 6'>*9 69 Red. Eq. Time 13 27-5 3807 2538 (To be added to app. time.) Correction 2-9187 or, 2«'9 In working this example the "Var. for i hour" is t;iken from the Nautical Almanac from the column ki page I of the month, and against the given day. The Greonwioh date being mean time, take the oquati jn o time from page II of the month, ;ind mark it ndditi /■ to iipp. time as directed at the top of the column la page I ; also note that the equation is increasing. The Green, time being d^ 54"' or 6*" 9 ; hourly difference is nultiplied by 6-9 giving thi product 29187 ; and s nee there are three decimal figures in U.U. (•425) and one in Green, time (-9) in all four, four decimal place- are marked off from the right hand of the produot, the result a'-giSy or z'-g is the correction to be applied to Wfi Eq. of time at noon, and is to be added to it because it is that due to time elapsed since noon while the Eq. T. is increasing. * As the equation of time is not a uniformly varying quantity, it is not quite accurate to compute its correction iis abovo, by multiplying the givf^n hou'ly tlifference by the number of hours in the Greenwich lime ; for that process assumes that this hourly difl'erence in the same for each hour. The variations in the hourly difference are, liowever, so small that it is only when extreme precision is r- (luired that recourse must be had to the more exapt method of interpolation for second differences. 2^0 Reduction of Elements from Nautical Almanac. Ex.2. 1882, September 30th, lo^ 15™ mean time at Greenwich: find the Equation of time to be applied to app. time in working the chronometer. Eq. of Time, page II, N.A. Diff. for i^, page I, N.A. Sept. 30th, noon, subt. lo" z^-o (-j-) Sept. 30th, at noon 0^-8 10 Corr. for lolh + ^'3 i°i Red. Eq. T. 10 10*3 Diff. for 10 hours 8100 Diff. for \ hour 202 (To be subtracted from A,T.) Correction 8*302 or, 88-3 Ex. 3. 1882, December 24th, loh 54'" mean time at Greenwich: find the Equation of time to be applieil to apparent time in working the chronometer. Greenwich date (M.T.) Dec. 24th, lo^ 54™ = Dec. 24^ io^>-9. Eq. of Time, page II, N.A. Dec. 24th, noon, suht. o"" g^T ( — ) Var. i hour =: i*"243 Corr. for io'"9 — 13-5 10-9 Red. Eq. T. 04-4 11 187 1243 (To be added to A.T.) r3'5487 The Eq. T. is taken out for noon of Deo. 24tli, page II, N.A., the sign — being affixed because Eq. T. is decrcasmg, and accor.ling to the precept at the top of the column in page I the Eq. T. at the Green, noon is suht. from A.T. The "Var. i;i 1 hour" multiplied by the time from noon — expre<;sed in hours and decimals of an hour — gives the correction, which is suhtractive from the Eq. T. at noon, because the latter is decreasing ; but since the correction exceeds the Eq. T. itself, take this latter from the correction, the result is the reduced Eq. T., and is to be applied to A.T. the contrary way to the Eq. T. at noon, i.e., according to the precept for the day following the given day, and is therefore additive to apparent time. Ex. 4. 1882, August 3i8t, 15'^ 42™ 15% mean time at Greenwich : find Equation of time to be applied to apparent time. Eq. of Time, page II, N.A. Hourly Diff., page I, N.A. Aug. 31st, at noon, add o^ios-7 (— ) Aug. 3i8t, noon 08-775 Corr. for 1 5^ 42" — 12-2 15^42" = 15-7 Red. Eq. T., subt. o 1-5 5425 3875 (To be subtracted from A.T.) 775 Correction 12" 1675 or, I2*'2 In this case the correction is suhtractive, and exceeds in amount the equation of time at noon, therefore the equation of time is taken from the correction, and the remainder is the reduced equation of time to be suhtracted from A.T., according to the precept for the day following the given day — a change of precept occurring between Aug, 31st and Aug. 32nd (Sept. 1st) — which change is shown by means of a black line drawn between the equations for the two named days. Ex. 5. 1883, June 14th, 22^ 25™ 21', mean time at Greenwich: find Equation of time to be applied to apparent time in working the chronometer. Greenwich date, June 14th, 22^ 25'" Hourly Diff., page I, N.A. ■ T June 13th, noon o«c3o or, 22^.4 ^^^,^ Eq. T., page TI, N.A. ~^^ June 14th, nooD, subt. o™ 3^'55 ( — ) 1060 Correction for 2 2*'"4 — irSy 1060 Tied.'Eq.T., add o 8*32 Correction 11-8720 {Toheaddedto A.T.) or, ii'-87 Correction of the Observed Altitude. 241 In this case also, the correction is siihtrartive, and exceeds the equation itself, therefore, the equation is subtracted from the correction and the difference is the deduced Eq. T., which is to be applied to apparent time according to the precept for the day following the given day. By using the Eq. T. corresponding to the nearest Greenwich noon, viz., that for June 15th, the work will stand thus : — Green, date, June 14th, Subtract from 24 Time from noon, June 15 th i"35 or, i''-6nly. Eq. T. page II, N.A. June 15th, at noon, add Correction for i^'6 Hotirly DiflF., page I, N.A. June 15th, at noon, o*"535 Time from noon 15th i'6 3210 535 — 0-86 Correction or, •8560 os-86 Red. Eq. T. add 0839 (To be addid to A.T.) The Eq. T. would be test at i^e before noon than what it is at noon, the oorrootion is therefore subtracted from the noon Eq. of Time. EXAKTPLES FOR PrAOTIOE. In each of the following examples it is required to find the equation of time corresponding to the given Greenwich date : — I. [882, Jan. 5th, at ^hjjmosM.T. I [ 1882, June 14th, at I2h52™ o«M.T. 2. „ Feb. 1 8th, at 8 20 M.T. 12 ,, Aug. 3i8t, at 15 54 A.T. 3- „ Mar. 24th, at 348 M.T. 13 ,, May 14th, at 9 36 A.T. 4- „ April 14th, at 22 30 10 M.T. 14 ,, April 14th, at 22 36 53 M.T. 5- ,, May 19th, at 6 56 M.T. 15 ,, Nov. 14th, at 21 35 A.T. 6. ,, Jxme 14th, at II 49 50 M.T. 16 „ July 20th, at 20 57 16 M.T. 7- ,, July 1 6th, at I 14 A.T. 17 „ Dec. 24th, at 18 2 54 M.T, 8. „ Aug. 3 1 at, at 21 14 40 A.T. 18 „ Oct. 26th, at 7 56 21 M.T. 9- „ Sept. 1 8th, at 53 10 M.T. 19 ,, Dec. 24th, at 7 18 A.T. 0. „ Oct. 5th, at 19 19 2 A.T. 20 „ June 14th, at 6 41 20 M.T. CORRECTION OF THE OBSERVED ALTITUDE. 306. Def. — The Altitude of a celestial body is the angular distance of the body from the horizon. It is measured by the arc of a circle of Azimuth (which is hence generally called a circle of altitude) passing through the plane of the body, or by the corresponding angle at the centre of the sphere. 307. The corrections necessary to reduce an altitude observed from the sea-horizon with a quadrant or sextant, &c., to the true altitude, consist of the index correction, the dip, the correction of altitude, or the joint effect of refraction and parallax, and, in certain cases, of the semi-diameter. The altitudes of heavenly bodies are observed from the deck of a ship at sea, with the sextant, for the purpose of finding latitude, longitude, &c. Such an altitude is called the "observed altitude." There are certain instrumental and ciroumstnntial sources of error by which this is affected :— (a) The sextant (supposed otherwise to be in adjustment) may have an index error ; (*) The eye of the observer being elevated above the surface of the sea, the horizon will appear to be depressed, and the consequent altitude in reality too great; and (c) One of the limbs of the body may be observed instead of its centre. When the correction H 242 Correction of the Observed Altitude, for these errors and method of observing are applied— "the index correction," "correction" for dip, and " semi-diameter," — the observed is reduced to the apparent altitude. But, again, for the sake of comp irison and computation, fill observations must be transformed into what they would have been had the bodies been viewed through a uniform medium, and from one common centre — the centre of the earth. Th j altitude supposed to be so taken is called the 'Hrue altitude ;" it may be deduced from the apparent altitude by applying the corrections called "corrections for refraction" (Table V, Norie, or XXXI, Raper), and "correction for parallax" (Table VI, Norie, or XXXIV, Raper), which, however, are sometimes given in tables combined under the names "correction of altitude" (Table XVIII, Norie). («') "Correction for refraction;" when a body is viewed through the atmosphere, refraction will cause the apparent to be greater than the true altitude ; hence the correction for refrac- tion is subtractive in finding the true from the apparent altitude, [b') " Correction for parallax ;" the position of the observer on the surface, especially for near bodies, will cause the apparent to be less than the true altitude ; hence the correction for parallax is additive in finding the true from the apparent altitude. TO COBEEOT THE SUN'S ALTITUDE. EULE LXXXIY. 1°. Correct the observed altitude of the sun for index error, if any. 2°. Subtract the dip answering to height of eye (Table V, Nokie, and Table XXX, Rapes) ; the remainder is the apparent altitude of the limb observed. 3°. Subtract the refraction (Table lY, Norie, and XXXI, Rapek), add the parallax (Table VI, Norie, and XXXIY, Raper) ; or take out the " correction in altitude ofsun^^ (Table XVIII, Norie), and subtract it; the remainder is the true altitude of the observed limb. 4°. Take from page 11 of the month in the Nautical Almanac the sun's semi- diameter, adding it when the sun's lower limb (l.l.) is observed, but subtracting it when the sun's upper limb (u.l.) is observed ; the result thus obtained is the true altitude of the sun's centre. Table 9, Norie, and Table 38, Raper, contain the gross correction of altitude, or the corrections for dip, refraction, sun's semi-diameter, and parallax — exclusive of index error — which are sometimes used in solving questions in nautical astronomy when great precision is not necessary. Examples. Ex. I. 1882, January 6th, the observed altitude sun's l.l. 39° 8' 30", index correction 4- 33", height of eye 19 feet : required the true altitude. Saper. Obs. alt. sun's l.l. Index correction Dip. (Table 30) 39 8' 30" 33 9 3 4 15 Norie. Obs. alt. sun's l.l. Index correction Dip (Table 5) App. alt. sun's l.l. Corr. alt. (Table 18) True alt. sun's l.l. Semi-diameter True altitude 39° + 39 8' 30" 33 9 3 4 " App. alt. sun's l.l. Ref. (Table 31) \ —Par. (Table 34) / 39 4 48 I 5 39 4 52 I 3 39 + 39 3 49 16 18 20 7 True alt. sun's l.l. Semi-diameter True altitude 39 + 39 3 43 16 18 20 I Correction of the Observed Altitude. 243 Ex. 2. 1882, June 18th, the obserwd altitude sun's l.l. 71° 19' 20', index correction -|- 3' 46", height of eye 18 feet: r.-quired the true altitude. Obs. alt. sun's l.l. Index correction Dip. (Table 30) Ref. — o' 20' ) Par.+ 3 ] Semid., p. II, N.A. True altitude 71° 19 20" + 346 71 23 6 — 4 10 71 18 56 — 17 71 18 39 + 15 46 71 34 25 Obs. alt. sun's l.l. Index correction Dip (Table 5) Corr. ofalt. (Table 18) Semi-diameter True altitude 71° 19' 20'' + 346 71 23 6 — 4 4 71 19 2 — 17 71 18 45 + 15 46 71 34 31 Ex. 3. 1882, October 8th, the observed altitude sun's l.l. 19' 50' 10", index correction 4" 50", height of eye 16 feet. Obs. alt. sun's l.l. Index correction Dip Ref. — 2' 41* Par. + 8 Semi-diameter True altitude 19" 50' 10* + 50 19 51 o — 40 19 47 o — 2 33 19 44 27 + 16 3 20 o 30 Obs. alt. sun's l.l. Index correction Dip Correction of altitude Semi-diameter True altitude 19° 50' lo" + 50 19 51 o — 3 50 19 47 10 — 2 29 19 44 41 + 16 3 20 o 44 Ex. 4. 1882, August 8th, observed altitude sun's u.l. 12° 52' 30", index correction -\- 3' 10", height of eye 17 feet. Obs. alt. sun's U.L. Index correction Dip 17 feet Ref, — 4' iiM Par. + 8 / Semi-diameter True altitude •*°52' 30" + 3 'o 12 55 40 — 4 5 12 51 35 — 4 3 12 47 32 — 15 49 Obs. alt sun's u.l. Index correction Dip Correction altitude Semi-diameter True altitude 12° 5 2' 30" + 3 JO 12 55 40 — 3 57 " 51 43 — 3 56 " 47 47 — '5 49 12 31 5S 12 31 43 ! Examples fob Praotiob. Jan. 29th, Obs. alt. sun's l.l. 17" 44' 30" Index corr. — i' 25" Eye 16 feet. Feb. 1 8th, Mar. 24th, April 20th, May 8th, June 19th, July 1 6th, Aug. 7 th, Sept. 2nd, Oct. nth, Nov. i5tb, Dec. 14th, 4 10 29 50 30 76 3 o 58 38 20 24 48 30 65 r o 85 13 20 u.l. 28 16 20 U.L. 67 44 o U.L. 14 3 40 U.L. 12 10 5 + 55 + ' 3 — I 27 — I 10 — I 14 + 017 — 2 10 -4 8 — • 38 + 4 I — o 49 17 10 18 20 18 144 TO FIND THE LATITUDE BY A MERIDIAN ALTITUDE OF THE SUN. EULE LXXXV. 1°. With the ship's date and hngitiide in time, find the Qreemoich date in apparent time (Rule LXXVIII, 5°, page 227). 2°. Take the sun's declination from Nautical Almanac, (page I of the month), and correct it for the Greenwich date (Rule LXXX, page 233), Instead of proceeding according to 1° and 3° the declination may be found thus : — i. Take the sun's declination from the Nautical Almanac,. tor apparent noon, page I; and also the corresponding hourly difference. 2. Multiply the hourlj' diff. by long, in time, expressed in hours and decimals of an hour. 3. When the declination is increasing the correction is to be added in West, but subtracted in East longitude ; but when the declination is decreasing, subtract in West but add in East longitude. See Rule LXXXI, page 235. 3°. Correct the observed altitude for index error, dip, semi-diameter, and refrac- tion and parallax, and thus get the true altitude (Rule LXXXIY, page 242); subtract true altitude from 90° : the result will be the true zenith distance^ 4°. Call the zenith distance N. when the observer is North of sun, or when the sun bears South; call zenith distance S. when, the observer is Souik of sun, or when it bears North. 5°. Add together the declination and zenith distance, when they have the same name (see Exs. i and 3) ; but take the difference if their names be unlike (see Exs. 2. 5, and 6) ; the latitude is N. or S., as the greater ia. 6°. When the declination is 0°, the zenith distance is the latitude, and of the same name as the zenith distance (see Ex. 7) ; and when the zenith distance is 0°, the declination is the latitude, which is of the same name as the declination (see Ex. 4). Examples. Ex. I. 1882, January 15th, in longitude 72° 42' W., the observed meridian altitude of the sun's l.l. (lower limb) was 59° 42' 10", bearing North; index error -J- 2' 10", height of eye 1 4 feet : required the latitude. The observation was made when the sun was on the meridian, that is, at apparent noon; the date, therefore, at the place of observation is January 15th, o"" o"" o». But the meridian of the place of observation is 72° 42' W. of meridian of Greenwich, and therefore the sun is 72° 42' W. of meridian of Greenwich; or, in time 4'' 50™ 48% since 72° 42' is equivalent to 4'» 50™ 48s (see below). It is, therefore, 4'^ ^o™ i\%'> past apparent noon at Greenwich, and the Greenwich date is found by adding 4^ 50'" 48' to the time of apparent noon at ship, January 15th, thus: — Ship date, January 15"' 6^ o"^ o» 72° 42' Longitude 72° 42' W. -f- 4 50 48 4 Greenwich date Jan. 15 4 50 48 4'' 50™ 48* With this date the sun's declination must be taken out of Nautical Almanac, where it will be found in page I for January. It may be reduced to Greenwich date by means of the Tables, or by "hourly difi.," thus: — * When true altitude exceeds 90°, subtract 90° from it. Latitude hy Sun's Meridians-Altitude. 245 Decl., page I, N.A. Jan. 15th, at noon 21° 5' 15" S. (— ) Corr. for 4'' 51"" 2 15 Eed. decl. S. In working this example the H. diff. for the 7Won of the day is taken. We divide the minutes of Green- wich timeby6; thus,6i3 contained in 51 eight times and three over, 6 is contained in 30 (the remainder 3 with a c added) five times ; hence wo have the decimal •85, to this wo prefix the hours (4), and wo then have 4''"85 to multiply by. As the Greenwich date w;ints lo" of 5 hours, wo might have multiplied the hourly diff. by 5, and deducted one-sixth of hourly diff. from the product. Hourly diff., page I, N.A. Jan. 15th — 27"-87 4t'5i'" = 4''-85 X 4-85 13935 22296 1114S 6,0)13,^-1695 Correction — 2 15 Latitude 51 6 32 S. Oba. alt. sun's l.l. Index error Dip (Table 30) 59° 42' 10* N. -|- 2 10 59 44 20 — 3 40 Norte. Obs. alt. sun's l.l. Index error Dip (Table 5) Corr. alt. (Table 18) Semi-diameter True altitude Zenith distance Declination Latitude 59° 4- 59 59 42' io"N 2 10 44 20 3 36 App. alt. sun's l.l. Refraction (Table 31) 59 40 40 — 34 40 44 29 Parallax (Table 34) 59 40 6 + 4 59 40 10 4- 16 18 59 56 28 90 59 + 59 90 40 15 16 18 True alt. sun's l.l. Semi-diameter 56 33 True altitude 30 21 3 27 s. 3 oS. Zenith distance Declination 30 3 32 s. 21 3 S. 51 6 27 S. Hinn'Hr.T, ni added, because they are of the same name. (This is according to No. 5° of the Bule). Ex. 2. 1882, February 3rd, in longitude 139° 42' W., the observed meridian altitude of the sun's l.l. 56° 56' 56", bearing South ; indeK correction — 3' 4" ; height of eye 14 feet. Ship date, February Long, 139° 42' W. 3d o*" o^" o' 4- 9 18 48 Green, date, February 3 9 18 48 Decl., page I, N.A., Feb. 3rd =: 16° 27' 20" 8^ deer. Hourly diff. 44""3i. Hourly diff. Feb. 3rd, noon 44"'3i Time from noon 9'' 1 8"* X 9'3 13293 39879 6,0)41,2-083 Correction Decl., noon, Feb. 3rd Red. decl. — 6 52 j 16 27 20S — I 16 20 28 S. By Raper : index, corr. — 3' 4' ; dip — 3' 40" ; refr. — o' 38* ; par. -\- 4' ; semid. + r6' 16" ; truo alt. 57° 5' 54' ; latitude 16° 26' 46" N. Norie. Obs. alt. sun's l.l. Index correction 56° 56' 56" S. — 3 4 Dip (Table 5, Norie) 56 53 52 3 36 App. alt. sun's l.l. Corr. alt. (Table 1 8) 56 50 16 32 True alt. sun's l.l. Semi-diameter 56 + 49 44 16 16 True altitude 57 6 90 Zenith distance Declination 32 16 54 oN. 20 28 S. Latitude 16 33 32 N. The diffurence of zenith distance and declination is taken because they are of contrary names. (See No. s^ofKule). 246 Latitude ly Sun^s Meridian Altitude. Ex. 3. 1882, March 20th, longitude 158° 5' W., observed meridian altitude of the sun's L.L. 52° 52' 50", bearing South ; index correction + i' 5" ; height of eye 1 2 feet. Green, date, March 20'' g** 12™ or, 9'"2 Hourly diff., noon, 20th 59"'^5 T. from noon, 20th, at g*' 12™ X 92 1 1 850 53325 6,0)54.5* 100 Correction — 95 Decl., March 20th, noon 0° 4' 53" S., deer. Correction — 95 Red. decl. o 4 12 N. By Raper : dip — 3' 20" ; refr. — o' 44" ; par. + 5"; semid. + ^6' 5'. True alt. 53° 6' i", and latitude 36° 58' 11" N. Obs. alt. sun's l.l. Index correction Dip CNorieJ App. alt. sun's i.L. Corr. of alt. True alt. sun's l.l. Semi-diameter True altitude Zenith distance Declination Latitude 52° 52' 50' S. + I 5 52 53 S5 — 3 19 52 50 36 - 38 52 49 58 + 16 5 53 6 3 90 o o 36 53 57 N. o 4 12 N. 36 58 9 N. Ex.4. '882, April i6tb, longitude 139' 50' E., observed meridian altitude sun's l.l. j° 46' 10", bearing North; index correction + i' 56"; height of eye 18 feet. Ship date (A.T.), April 16^ d^ o™ o» Long, in time — 9 19 20 Green, date, April 15^ 14 40 40 Time from noon, April 16'' 9 19 20 Decl., page I, N.A. H. diff., noon, April 1 6th 53"'" Green, time 9'' 19™ = X 9"3 15933 47799 Correction Decl., noon, i6th Correction Hed. decl. 6,0)49,3-123 — 8 14 10" 11' 7"N. iner. — 8 14 10 2 53 N. By Raper : index corr. -\- i' 56" ; dip — 4' 10" ; ref., &c., o' ; semid. + 15' 58', True alt. 89° 59' 54', lat. 10° 2' 47" N. Obs. alt. sun's l.l. Index correction Dip (Norte) Corr. alt. Semi-diameter Zenith distance Declination Latitude 89° 46' + I 10" N. 56 89 48 — 4 6 4 89 44 2 $ 44 15 2 58 90 90 10 2 53 N. 10 2 53 N. Instead of finding Green, date as above, we may proceed as follows : — Since the longitude is g"" 19™ 2o' East, the Greenwich date is therefore that amount before the noon of April loth (the noon of the ship date), then the decl. and hourly diff. is taUen out of the Nautical Altnanar, page I, for the nearest noon to Greenwich date, viz., noon ot April i6th, and hourly diff. 53"il is multiplied by 9''-3 ; the resulting figures are 49J"'i23, or 8' 14", the correction. The declination at noon Increasing will evidently be less gh ig™ before noon} therefore the correction is 8' 14" to subtract (see Rule LXXXV, 2°, note, page 444). Latitude by Sun's Meridian Altitude. J47 Ex. 5. 1882, July 13th, longitude 100° W., observed meridian altitude eun'a l.l. 68° 2'o', bearing North; index correction — 25", height of eye 17 feet. Ship date, July 1 3<' o*" o"> o» Long, in time 6 40 o Green, date, July 13'* 6 40 o Decl., p.I., N.A., July 13th, 2i'49'22*N. — Hourly difT., July 13th, noon 2i''*93 Time from noon 6'» 40™ 30" Correction Decl. noon, 13th Jh ,3158 1096 365 6,0)14,6-19 — 2 26 21 49 22 N. Bed. deol. 31 46 56 N. By Raper : Index corr., — 25"; dip, — 4*5"; corr.alt., — 21"; semid., -|- i5'46"; true alt., 68° 12' 55*; latitude o*^ o' 9" S. Norie. Obs. alt. sun's l.l. Index correction Dip (Table 5) Corr.alt. (Table 18) Semi-diameter (N.A.) True alt. Zenith distance Declination 68" 2' o"N. — 25 68 I 3S — 3 57 67 57 38 — 20 67 57 18 + 15 46 68 13 4 90 o o 21 46 56 S. 21 46 56 N. Latitude o The ship is on the equator. "When the zenith distance and declination are numerically equal, and of contrary names, the ship is on the equator. Ex. 6. 1882, December 17th, longitude 175° 45' E., observed meridian altitude sun's l.l. ■)° 54' 20" bearing North ; index correction -\- 4' 4", height of eye 24 feet. Ship date Dec. Long, in time jiyd qS\ Qin o» II 43 o Green, date (A.T.) Dec. 1 6'' 12 17 o T. from noon Dec. 17"* 11 43 o or ri'>-7 Decl., page I, N.A., Dec. 17th, at noon, is 23° 23' 34" S. (4-). Var. in i"* 5" '9 T. from noon, 17th 11 -7 3633 5709 6,o)6,o'733 Correction i i Decl., noon, Dec. 17th 23° 32' 34" S. + Corr. for II "i 43™ — 11 Red. dec]. 23 21 33 S. The declination is taken out for the nearest noon to Green, date, viz., Dec, 17th, and corrected for the interval between it and the Green, time, which is e^ual to the long;itude in time, viz., iii> 43'o (= ix*";}. Obs. alt. sun's L.L. 89°54'2o"N Index correction + 44 Dip (Table 5) 24 feet 89 58 24 — 4 42 Corr of alt. (Table 18) 89 53 42 Semi-diameter 89 53 42 + 16 18 True altitude 90 10 Zenith distance 10 N. Declination 23 21 33 S. Latitude 23 II 33 s. The true altitude by Raper's Tables is 90° 9' 52", zenith dist. 0° 9' 53", latitude 23° 11' 41" y. 90° is subtracted from the true altitude ; the re« mainder is the zenith distance, North. 248 Latitude hy Sun^s Meridian Altitude. Ex. 7. 1882, September 23rd, long. 123° 45' E., observed meridian altitude sun's L.t. 40° 9', bearing North; index correction + 20", height of eye 18 feet. Green, date (A.T.) Sept. 22nd, 15*' 45'" Time from noon 23rd, 815 Decl., page I, N.A., Sept. 23rd, at noon is 0° 8' 2' S., increasing, hourly diff. 58"'47. H. difiF. Sept. 23rd, noon 58"'47 Time from noon, Sept. 23rd 8*25 29235 1 1694 46776 6,0)48,2-3775 — 8 2 Decl., Sept. 23rd 8 2S. Red. decl. 000 Obs. alt. sun's l.l. Index correction Dip 18 feet (Table 5) Corr. alt. (Table 18) Semi-diameter, N.A. True altitude Zenith distance Declination Latitude 40° + 40 9' o"N. 20 9 20 4 4 40 5 i6 I I 40 4 15 15 59 40 90 20 14 49 39 46 S. 49 39 4<5 S. By Raper : index correction -\- 20" ; dip — 4' 10" ; refr. — i' 9" ; par. -\- 7" ; semid. + '5' 59'; true alt. 49° 39' 54'; latitude 49° 39' 54' S. Ex. 8. 1882, June 25th, longitude 59° 15' E., observed meridian altitude sun's u.i. 60° 24' 10" (zenith South of observer) ; index correction — 3' 17" ; height of eye 30 feet. June 25"* o'' o" — 3 57 Ship date (A.T.) Long. 59° 15' E. Green, date (A.T.) June 24th 20 3 Time from noon, June 25th 3 57 or, 3h-95 Decl., p. I, N.A., 25th, 23° 23' 29" N. deer. H. diff., June 25th, noon 4"-o7 T. from noon, 25th, is f" sT^ = 3'95 2035 3663 1221 Correction i6'0765 Decl., June 25th, at noon 23° 23' 57"N. deer. Correction + 1 6 Reduced declination 23 24 13 N. The decl. is taken out for the nearest noon to Green date, viz., June 25th, at noon, and corrected for the interval between it and the Greenwich time, which is equal to the long, in time, viz., ^^ 57" (= 3-95 hrs.) We might have found the correction for 4'', and taken from this result the change for 3™, or one-twentieth of the hourly difference. Obs. alt. sun's u.l. Index correction 60° 24' ro*N. — 3 17 Dip 30 feet (Table 5) 60 20 53 — 5 »5 Corr. of alt. (Table 18) 60 15 38 — 29 Semi-diameter 60 15 9 — 15 46 True altitude 59 59 23 90 Zenith distance Declination 30 37 S- «3 24 13 N, Latittide By Raper : Index corr. - — 5' 20"; refr. — 33''-4; Semid. — 15' 46"; true alt latitude 6" 36' 29 " S. 6 36 24 S. - 3' 17"} dip par. + 4'-2; • 59° 59' 18"; Latitude hy Sun's Meridian Altitude. 249 Ex. 9. 1882, Autjust 23rd, longitude 168° 25' W., observed meridiMii nltitude sun's L.I.. 40' 5' 30", observer N. of sun ; index corr. — 54' ; height of eye i a feet. Green, date, Aug. 23rd, ii*" 13™ 40'. Decl., page T, N.A., August 23rd, at noon, 11° 24' 12" N., decreasing, hourly diff. S°^'q(> X ii'"23 nearly := 572""28o8 or 9' 32*, tho corr. to bo subtracted ; whence rod. de«l. = 11° 14' 40" N. By Norie : index corr. — 54' ; dip — 3' 19"; corr. of alt. — i' i'; semid. 4" '5' 52* ; true alt. 40° 16' 8". True altitude 40° 1 6' 8" 90 o o Zenith distance Declination Latitude 49 43 52 N. II 14 40 N. 60 58 32 N. By Raper : index corr. — 54" ; dip — 3' 20" ; refr. — i' 9"-5 ; par. -|- 7 " ; semid. -|- 15' 52"; true alt. 40" 16' 5'; latitude 60° 58' 35" N. Ex. 10. 1883, January ist, longitude 150' E., observed meridian altitude sun's L.L. 70° 20' (zenith N. of sun) ; index corr. — 30" ; height of eye 19 feet. Green, date, 1882, Dec. 3i8t, 14*^ o". Time from noon, Jan. ist, 1883, or Dec. 32nd =z longitude in time 10'' o". Decl., 1882, December 32nd, 23" o' 50* S., decreasing, hourly difF. i2"'39 X 10'' (long, in time E.) = 1 23''"9o or 2' 4" ; whence red. decl. 23° 2' 54" S. By Norie: index corr. — 30"; dip — 4' 1 1'; corr. alt. — 18'; semid. -^ i6' i8"{ true altitude 70* 31' 19". True altitude 70° 31' 19* 90 o o Zenith distance Declination Latitude 19 28 41 N. 23 2 54 8. 3 34 13 S. Examples for Praotioe. In each of the follow ing examph )8 the latitude is required : — No Civil date. Longitude. Obs. alt. Bun's l.l. Index corr. Eye. I 1882 Jan. loih, 49°5i'W. 68' 39' 40' N. -f 5' 10" 13 ft. 2 )) Feb. ist. 39 51 E. 72 43 50 S. + I 42 n 3 » March 8th, 89 48 E. 51 49 30 s. — 3 17 15 4 )> April 28th, 165 23 W. U.L 82 51 10 N. 4-4 10 li 5 ■ )> May 2nd, 32 3 E. U.L 46 18 S. 20 6 )> June nth, 62 57 E. L.L 42 24 45 N. + 2 15 21 7 1) July 20th, 156 38 W. 51 58 30 N. — 2 39 16 8 »> Aug. 19th, 82 30 W. 57 41 S. — I 3 22 9 )i Sept. 23rd, 166 30 E. 41 36 10 8. — 4 41 17 10 ») Oct. 23rd, 90 12 W. 54 40 40 S. — 49 18 II >) Nov. 15th, 80 II E. 67 43 S. + 1 38 15 12 » Dec. loth, 55 20 E. 25 52 15 8. + 2 n n )» Sept. 2 ist, 60 I E. 56 26 N. 20 14 It March 20th, 103 30 W. 61 49 30 S. — 3 17 15 »5 >i April 7 th, 139 45 W. 89 55 50 8. 4-5 10 12 16 >) Sept. 23rd, 123 45 E. 83 40 30 8. 18 «7- » Nov. 3rd, 106 E. 70 29 45 N. 4-1 22 '9 18 )» Sept. 23rd, '73 58 E. 71 19 20 S. + 3 40 18 19 ,, Feb. 1 2th, 8 12 W. 29 55 20 S. — r 10 »9 20 »t March 20th, 77 45 E. 76 58 15 N. — 2 20 21 21 1883 Jan. ist. 125 32 E. U.L 54 57 20 8. 4- 2 10 22 22 1882, Oct. ist, 71 20 E. U.L 82 15 N. — 3 15 14 KK 250 ON AMPLITUDES, 308. The Correction or Error of Compass is found by comparing the bearing of the sun or other celestial body, as shown by the compass, with the true bearing, as found by calculation. 309. The True Amplitude is the bearing of a celestial body at rising or setting {i.e., when its centre is on the rational horizon), from the true East or West point, found by calculation, from the latitude of the place and declina- tion of the body, or taken by inspection from a table, of which these quanti- ties are the arguments (Table XLII, Norie, or LIX, Raper). 310. The Magnetic Amplitude is the bearing of a celestial body at rising or setting from the compass East or West points, found by direct observation with an instrument fitted with a magnetic needle, as the Azimuth Compass. The magnetic amplitude is distinguished as observed, or apparent, and corrected. The observed or apparent magnetic amplitude of a celestial body is its bearing from the compass East or West point, when it appears in the sea-horizon of an observer standing on the deck of a ship. The corrected magnetic amplitude is the bearing of the body from the compass East or West point, when on the rational horizon, as it would appear to a spectator at the centre of the sphere through an uniform medium. The diurnal circles of the celestial bodies being, except at the equator, inclined to the horizon, and more and more the higher the latitude, any cause which affects the time of rising will affect the apparent amplitude, and in a greater degree as the latitude increases. The following are the causes ; — i. The eleva- tion of the observer depresses the sea-horizon, while it does not affect the place of the celestial body — hence by reason of the dip the body appears to rise before it is truly on the sensible horizon. 2. The great refraction at the horizon causes the body to appear to rise considerably before it comes to the sensible horizon. 3. When a body is in the sensible horizon, to an eye at the centre of the sphere it has already passed the rational horizon- This being the effect oi parallax, is only of importance in the case of the moon. These cor- rections will be found in Table 59 A, Raper. RULE LXXXVI. 1°. With the ship date and longitude in time, find the Greenwich date (see Rule LXXVm, page 227. The time of sunrise and sunset is generally given in apparent time. 2°. Take out of Nautical Almanac, page I, the sunh declination and correct it for this date (see Rule LXXTX, page 230). 3°. Take from the Table the log. sine of declination, and log. secant of latitude (rejecting 10 from the index J ; the sum of these is log. sine of true amplitude, which take out of Tables. (Table XXV, Norie, or LXVIII, Raper). 4°. To name the True Amplitude. — If the body is rising, or a.m., mark true amplitude East ; if it is setting, or p.m., mark it West ; mark it also North when declination is North ; or South when declination is South. The time of sun rising is always a.m., and of sun setting p.m. (a) When the declination is o, the true amplitude is o ; that is, it is East if the object is rising. West if it is setting. (b) When the latitude is o, the true amplitude is of the same amount as the declination. On Amplitudes. 251 5°. Correction or Error of the Compass for the Position of Ship's Head. — Under the true amplitude write the observed amplitude ; then — (a) If loth amplitudes are North or both South, take their difference. (b) When one is North and the other South, take their sum. (c) If one is reckoned from East and the other from West, take the True Amplitude /row 180°, and change the name /row East to West, or from West to East ; the name as to North or South remains unaltered ; then take their difference. The sum or difference fas the case may be) is the entire correction, or error of the compass.* The observed amplitude must be reckoned from East or West towards the North or South, and then expressed in degrees and minutes before it is placed underneath the true. Thus, the magnetic amplitude S.E. by E. \ E. isE. 2^ points S., or E. 28° 7' 30" S. 6°. To name the Error of Compass. — The correction is named East when the true amplitude is to the right of observed amplitude ; West when true is to the left of magnetic : the observer being supposed looking from the centre of the compass in the direction of the observed amplitude. Note. — The learner will find it very useful to draw a figure, thus : — Make a rough sketch of the compass by drawing two lines crossing at right-angles, the ends of which will represent the four cardinal points, which mark N., S., E., W., (see Fig. Ex. i) ; then to name the error of the compass proceed as follows: — Consider the cardinal point from which the amplitude is reckoned as the origin, and draw two straight lines from the centre to represent the true and magnetic amplitudes, and mark their extremities T and M respectively — taking care to place the line T further from the origin if the true be greater than the observed (or magnetic) amplitude, but nearer the origin if the true is les-i. The arc between M and T is the error which will be East when T is to the right of M, but West if to the left. It is easily seen whether the error of the compass is the sum or difierenco of T and M. 7°. To find the Deviation. — Under the error of the compass place the variation : then (a) If they a/re of like names, i.e., are both East or both West, take their difference. (b) But if they have unlike names, i.e., if one is East and the other is West, take the sum. The sum or difference fas the case may be J is the deviation. (c) If the variation is o, the error of the compass is also the deviation. (d) If the error of the compass is o, the deviation is of the same amoiint as the variation. 8°. To name the Deviation. — The deviation is of the same name as the error, unless the error has been subtracted from the variation, in which case the deviation is of a contrary name to the error, i.e., the deviation is E. when the error is W., but W. when error is E. • The result as deduced above is generally called the variation, but the effects of the iron in the ship modify the bearing by compass. Every error determined on board ship is com- pounded of variation proper and deviation, and is the entire correction necessary to be applied to every bearing taken, and course steered, but will vary with the position of the ship's head and heel of the ship. If the iron of the vessel exorcise no influence on the com- pass, the result obtained is only variation, and ought to agree with that registered on the , chart. 25* On Amplitudes. Also, when the error is o, the deviation is of opposite name to variation ; when variation is o, the deviation is of same name as error : thus— Error 14" lo'W. Var. 2 25 E. Error i4°io'W. Var. 2 25 W. Error 14° 10' W. Error 0° o'W. Var. 22 25 W. Var. 22 35 W. Error 14° lo'W. Var. o o Dev. 16 35 W. Dev. 11 45 W. Dev. 8 15 E. Dev. 22 25 E. Uev. 14 10 W, 9°. Otherwise the observer must suppose himself in the centre of the com- pass, looking in the direction of the variation, — then the deviation is East when the error of compass is to the right of the variation ; West when the error of compass is to the left of the variation — both the error of the compass and the variation being reckoned from the ^oxth point of the compass. Note.— It will be convenient for beginners to draw a figure for the deviation, thus : — (See Fig. 2, Ex. i.) Make a rough sketch of the compass ; the upper part, of the vertical line being taken to represent the origin, which mark N., and mark the extremities of the horizontal line W. and E. respectively. Then from the centre of the compass draw two lines to represent the error of compass and the variation, calling them E and V respectively. The line E must be drawn to the right of N. if the error of compass is E., but to the left of N. if the error be W. ; similarly, the line V must be to the right of N. when the variation is E., but to the left of N. if the variation is W. Take care to draw E further from N. than V if the error of compass is greater than the variation, but nearer to N. if the error is the less. The deviation is the distance from V to E, and is East when E is to the right of V, but West when E is to the left of V. It is easily seen whether the deviation is the sum or difference of E and V. Note. — In the following examples the seconds of declination are rejected. When the seconds are 30, or above, i is added to the minutes ; but when they are below 30 nothing is twided to the minutes. Examples. Ex. I. 1882, January 6th, at 4'^ 44™ 27' a.m., apparent time at ship, lat. 37° 59' S., long. 36" 24' W., the sun's magnetic amplitude was S.E. by E. ^ E. : required the true amplitude and error of compass ; and supposing the variation to be 3° 40' E., required the deviation for the position of the ship's head at the time of observation. Ship date (A.T.) Jan. s^ i^h 44m 27^ Long, in time + * 25 36 Ghreen. date (A.T.) Jan. 5 19 10 3 Time from noon Jan. 6th 4 50 = 4'»*83 Or thus, H.D. i8"-3o 5 H. diff., noon, Jan. Time from noon, 4'^ 6th 50"" — i8*-30 X 4-83 noon 6,( + 22 5490 14640 7320 Correction DecL, 6th, 3)8,8-3890 I 28 28 38 S. (- -) Bed. decl. 22 30 6 S. lO" I ^h 6,0 91-50 8,8-45 I 28 The declination is here taken for the nearest noon, viz., the 6th, and since the Green, time wants only 4'' sc" of being noon of 6th, ^24'' o<^ — \g^ lo™ = 4'' SC"), multiply the hourly difif. by this quantity, and apply the resulting correction the opposite way, since the declination is de- creasinn the decUnation at 4I' 50" before noon will be more than it is at noon, hence we add the correction. Or, multiply by st, then since si" is 10" in excess of 4'' 50", deducting i-6th of H.D. from the product above, the result ia correction. On Amplitudes. 2S3 Decl. Lat. 37 59 sine 9"582840 aecant 0-103369 sine 9'686209 Kg. 1. N. W. (a.m. and S. decl.) True amp. E. 29° 3' S. (S.E. by E. ^ E.) Mag. amp. E. 28 7^ S. = E. 2^ polnii. Error of compass o 55^7 E., because true amplitude is to the rty/isi of wa^we^jc amplitude. '*■ To find the Deviation. Error of compass 0^55^ E. Error and var. same Fig- 2- N Variation by chart 3 40 E. name, take the <^^^f we. -E. Deviation 2 44^ W., because the error of compass is to the left of the variation. "W. Make a rough sketch of the compass as in Fig. i in the above example. In this examjile the magnetic amplitude is reckoned from E. towrrds S. (S.E. bj' E. | E. ^ E. 2^ pts. S. = E. 28° 7I' S.) To represent this, draw a line from the centre of the compass to a point M, somewhere between E. or S. Again, the true amplitude is reckoned from E. towards iS. To represent this, draw a line from the centre of the compass to point T, further from E than M is from E, because the true amplitude is greater than the magnetic amplitude. Then it is evident that the line T, or the true amplitude, is to the right of the line M, or tlie magnetic amplitude. Hence, by Rule, 6°, the error of the compass is East. Again, to name the deviation : — Draw a figure (see Fig. 2 above) and mark the end of the vertical line N, to represent the true meridian (or true North point), and the extremities of the horizontal line W and E respectively, to represent West and East. Next, from the centre of the compass draw a line E (see Fig. 2) to the right of North, to represent the Error of the Compass, which is E. ; and since the variation is also East, draw another line V to the right of North, but further from N than E is, because the variation is greater than tha error. (See Fig. 2.) It is evident that the deviation is the angle included between E and V, and is East because E, the error, is to the right of V, the variation (Rule, 7^^ and 9°). It is evident too that in this instance the deviation is the difference of E and V. Otherwise : — By 8* of Rule, the error being subtracted from the variation, the deviation is of the opposite name to the error, i.e., the error being E., the deviation is named W. Ex. 2. 1882, February i6th, at 4'' 58™ p.m., apparent time at ship, latitude 51° 9' N., longitude 15^^ W., sun's observed amplitude W. | N. : required the true amplitude and error of the compass ; and supposing the vnriation to be 28" 30' W. : required the deviation for the position of the ship's head at the time of observation^ H. diff,, noon, Feb. i6th — 52''"i5 J"97 Ship date (A.T.), Feb. i6'i Long, 15' 0' W. + I Green, date (A.T.), Feb. 16 5 58 H.D.= or, = 52'" 15 6h 5''-97 2- \ J-oh 31290 — 1-74 6,0 31,1-16 CWT. 5 " 36505 46935 26075 6,0)31,1-3355 5 "J Correction — 5 " Decl., noon, Feb. i6th 1* 15 17 S. rf47ni20* — 7 II 44 Green, date (A.T.), April i2<*io 35 36 Declination 8° 54^' Latitude 20 2 or, i6^-6 sine 9' 1 89922 secant 0-027 106 sine 9*217028 (a.m. and N. decl.) True amp. E. 9° 29' N. (E. \ pt. N.) Mag. amp. E. 2 49 N. H. diff., April 12th, noon •\- 54"'67 io"6 32802 5467 6,o)57.9'502 9 39'S Correction Decl., 12th, noon + 9' 39*"5 8° 44 51-4 N. Red. decl. 8 54 31 N. N. Error of compass 6 40 W, the true amplitude being to tho left of magnetic. On Amplitudes. 255 To find the Deviation. Error of compass 6° 40' W. \ Different names, Variation i 40 E. ] take the sum. Deviation 8 20 W., because the error is to the left of the variation. B N.v W. -E. Ex. 4. 1882, June loth, at 4'^ 45™ r.M., apparent time at ship, latitude 36° 42' S., longi- tude 109° 30' E., magnetic amplitude W. 29° 12' N., variation 7° 20' W. : required the deviation for the position of the ship's head at the time of observation. Ship date (A.T.), June \o^ 4'' 45™ or June 9 28 45 Long. 109° 30' E. — 7 i8 Green, date (A. T.), June 9 21 27 Time from noon, June 10 2 33 =: ^^'5S Declination 23° i^' sine 9-592324 Latitude 36 42 secant o'095947 sine 9-68827 1 H. diff., noon, June loth, -\- 11 "•35 T. from noon, 2'' 33"" ^ 2''"5S ^'SS 5675 2270 Correction Decl., loth, noon Red. decl. 28-9425 — o' 29" 23° r 59 N. 23 I 30 N. (p.m. and N. decl.) True amp. W. 29° 12' N. ) Same name, Mag. amp. W. 29 12 N. ) take their difference. Error of compass o o Obs. — In this instance the error of com- Variation 7 20 pass is o, and the dev. is equal in amount W. to the variation but of an opposite name. Deviation 7 20 E., because the error is to the riffht of the variation. Ex. 5. 1882, July 3i8t, at 4'' 26™ a.m., apparent time at ship, latitude 46° 3' N., longi. tude 165° 58' W., sun's magnetic amplitude N.E. by E., variation 13° o' W., ship's head E. by N. Ship date (A.T.) Long, in time Green, date Declination 18° 12^' Latitude 46 3 July 30'' i6*'26'" o^ + " 3 52 H. diff., July 3i8t, noon 37"ir ^h 30m := 3.^5 July 31 3 29 52 or 3t'-5 sine 9-494813 secant 0-158622 sine 9-653435 18555 "133 6,0)12,9-885 Correction — 2' 10" Decl., 31st, noon 18" 14 47 N deer (a.m. and N. decl.) True amp. E. 26" 45^' N. (E. af pts. N.) Mag. amp. E. 30 56 N. Red. decl. 18 12 37 N. Error of compass 4 10^ E., the true amplitude being to the riffht of Variation 13 o W. the magnetic. Deviation 17 10^ E., because the error is to the riffht of variation. Ex. 6. 1882, Sept. 2ind, at 6** o"" p.m., apparent time at ship, latitude 24° 40' S., longi- tude 146^ i^' W., sun's magnetic amplitude W. 2° 50' N., variation 7° 40' E, 256 On Amplitudes. Ship date, Sept. 2tA 6^ o™ c^ H. diflF., noon, Sept. 22nd — 58"*47 Long. 146° i5'W.,intime+ 9 45 o 8Jh Green, date, Sept. 22 15 45 o 46776 1462 Time from Sept. 23rd 8 15 6,0)48,2-38 The decl. being; 0° the true amplitude is o", or Correction — 82 W. o°o', whence the en or of compass is 2° 50' \V., Decl., 22nd, noon o 6 zS.incr because the trw amplitude is to left of magnetic. Red. decl. 000 lb Find the Deviation. Error of compass 2° 50' W. Variation 7 40 E. Deviation 10 30 W., because the error of compass is to the left of variation. Ex. 7. 1882, December 9th, at 8^ 27"" a.m., apparent time at ship, latitude 54° 35' N., longitude 53° 15' W., sun's magnetic amplitude S.E. | E., varialion 36° 20' W., ship's head S.W. by W. Ship date (A.T.), Dec. 8^2o'i27'n o^ Decl. at noon, Dec. 9th, 22° 51' 6" S. Long, in time + 3 30 o The Green, date being noon, Dec. loth, one of the instants for which the decl. is Green, date, Dec. 8 24 o o put down in the Almanac, nothing more is necessary than to transcribe the quan- or Dec. 9000 tity as there put down. Declination 22''5i' sine 9'589i90 Latitude 54 35 secant 0-236933 sine 9-826123 (a.m. and S. decl.) True amp. E. 42° 4' S. (E. 3I pts. S.) Mag. amp. E. 39 22 S. Error of compass 2 42 E., the true amplitude being to Variation 36 20 W. the right of magnetic Deviation 39 2 E., because the error is to the right of variation. Ex. S. 1882, December 2i8t, at 4*^ 31™ p.m., apparent time at ship, latitude 41° 12' N., longitude no" 45' E,, sun's setting amplitude S.W. j W., variation o. Green, date, Dec. 20'^ 21^ S"". Decl., noon, Dec. 2i8t, 23° 27' 7" S., incr. Declination 23° 27' sine 9-599827 H. difl'., noon, 21st -j- o"-49 X time Latitude 41 12 secant 0-123543 from noon, 2^ 52"^ (=: 2''-87) = 1-41 = corr. — i"; red. decl. 23*^ 27' 6". sine 9*723370 (pjn. and S. decL) True amp. "W. 3i°56' S. (W. 3| pts. S.) Mag. amp. W. 42 1 1 S. Error of compass 10 15 E., the true amplitude being to the Variation o o right of magnetic Deviation 10 15 E. The Variation being o, the Error of the compass is also the Deviation. On Amplitudes. a 57 Ex. 9. 1882, June 19th, at 9'' 40™ p.m., apparent time at ship, lat. 62° 31' N., long. 60° 24' W., sun's magaotic amplitude, N.N.E. ; and supposing tho variation of the compass is 57° 50' W., required the deviation for the position of the ship's head at the time the ohservation was taken. Ship date (A.T.), June 19'' 9*' 40™ o» H. diff., 19th 2''i2 Long. (60° 24' W.) hi time -^- 4 i 36 T. from noon X 137 Green, date (A.T.) June 19th 13 41 36 Corr. of decl. 29-044 i3*''7 Decl., Juno 19th, noon 23° »6' i8*N. wkT. Oorr. 29 Declination 23° 27' sine 9599827 Latitude 6j 31 secant 0-335837 Red. decl. 23 26 47 N. sine 9-935664 (p.m. and N. decl.) True amplitude W. 59" 35' S. 180 o E. 120 25 N. (E. 6 points N.) Mag. amplitude E. 67 30 N. = N.N.E. Error of compass 52 55 W., because true amplitude is to the Variation 57 50 W. left of mag. amplitude. Deviation 4 55 E., because error is to tho riffht of variation. Tho true and observed amplitudes must both be reckoned /row the same point of the com- pass, E. or W., buf- in this instance ono is reckoned from W. and tho other from E; there- fore, by taking either of them from 180°, they would both be reckoned from the same point — the true amplitude, in this example, is taken from 1 80°, and it is then reckoned from E. instead of W. Next take tho diflferonco of the amplitudes, as they are both marked N. ; and since the true amplitude is to the kft of the magnetic — looking from the centre of the com- pass in the direction of the magnetic — the error of compass is W. The error of compass and variation being of the same name, take their difiference for the deviation, which mark E., because the error of the compass is to the right of variation, looking from the centre of the compass in tho direction of the variation. Ex. 10. 1882, July ist, at 8*' 36™ p.m., lat. 56' 4' N., long. 64° 50' W., sun's magnetic amplitude North, variation 36° o' W. Green, date, July r^ 12'' 55™ 20* Decl., page I, N.A., July ist, at noon, or i2'>-92 23" 6' 49' N. rf«<^., H.D. io"-i9 X 12-92 Declination 23° 4^' sine 9-593215 Latitude 56 4 secant 0-153188 sine 9-846403 gives correction — 2' 12", whence JRed. Decl. is 33° 4' 37" N. (p.m. and N. decl.) True amplitude W. 44° 36' N. (N., or W. 8 pts. N.) Mag. amplitude W. 90 o N. (8 pts. = 90°) Error of oompase 45 ^4 W. Variation 36 o W. Deviatioti 9 24 W. 258 Finding the Time of High Water. Examples for Practice. In each of the following examples the Error of Compass and Deviation are required for the position of the ship's head at the time of observation. Civil Date. 1882. Jan. 27th,, Feb. 17th March 29th April 5th . . Nov. 7 th . . May 26th . . June 2nd . , July 14th Aug. 27th Sept. 8th . . Oct. ist .. Sept. 23rd Nov. 3rd . . Dec. 4th . . March 20th Sept. 23rd June 9th . , Feb. 26th . . April 30th May 27th. . June 1 8th Mar. 6th . . April loth Dec. 14th. , App. Time. h 6 6 5 6 5 7 8 6 50 5 44 5 47 5 48 6 o 6 34 7 56 6 o 6 o 6 o 7 49 6 28 7 40 I 47 6 14 6 45 4 35 40 A.M. O P.M. O A.M. O P.M. O A.M. O A.M. 2 P.M. 58 A.M. O P.M. O A.M. 50 A.M. O A.M. O P.M. 48 P.M. O P.M. O A.M. O A.M. O A.M. 12 P.M. O P.M. O A.M. O P.M. O P.M. O A.M. Latitude. 35 42 N. 34 57 N. 25 50 s. 20 20 S. 27 41 s. 51 22 S. 52 30 N. 28 59 S. 21 4 S. 24 22 N. 42 44 S. 56 41 s. 29 20 s. 49 59 S. 55 10 N. 60 I S. o o 62 5 N. 24 58 N. 47 40 N. 63 54 N. 31 24 S. 53 58 N. 42 o S. Longitude. : W. IE. 12 52 40 8 _. 127 35 W. "55 30 E. 70 2 W. 48 o E. 27 6 W. Ill II W. 36 19 E. 57 30 W. 175 15 W. 179 42 E. 136 35 E. 160 45 E. 15 54 E. 33 45 W. 10 21 AV. 12 52 W. 138 52 w. 148 3W. 174 20 w. 2 10 E. 178 33 E. 74 56 E. Sun's Mag. Amp. S.E. by S. . . S.W. by W... E.S.E W. 6° 40' N. E. JN. , E. fS N.N.W. i W. N.E. IN. .. N.W. |W. .. East E. f N E. AS w.s.w S.W. by W. .. West East E.iN S S F W.'by N."i'N, W. by N N. by W. i W W. 16" 52' N. W.f s South Variation. 21 50 w. 7 40 E. 23 40 W, 6 40 E. n 50 E. 3'? 20 W. 37 20 W. II 40 E. 23 10 W. 18 "(O E. II E. 2 50 W. 16 E. M E. 21 50 W. 20 11 W. 3'; 45 W. 10 E. 20 I-; E. 2"; E. 17 ■JO W. 16 10 E. 19 20 W. ON FINDING THE TIME OF HIGH WATER. "BY THE ADMIEALTY TIDE TABLES." 311. These Tide Tables, published annually, give the time (a.m. and p.m.) of high water, and the height for every day in the year, at the following places, viz. : — Brest, Devonport, Portsmouth, Dover, Sheerness, London, Harwich, Hull, Sunderland, North Shields, Leith, Thurso, Q-reenock, Liverpool, Pembroke, Weston-super-mare, Holyhead, Kingston, Belfast, Londonderry, Sligo Bay, Galway, Queenstown, and "Waterford. 312. To find the times of high water from the Tide Tables if the place is one of the Standard Ports, proceed by EULE LXXXVII. Turn to the month in the Tide Tables and find the given place ; then opposite the given date will stand the morning (a.m.) and afternoon (p.m.) times of high water required. Note. — When the mark — occurs it shows that there is but one tide during that day ; no high water, therefore, takes place in the morning or afternoon in which the mark appears. Thus, wishing to know the time of high water at North Shields on the 9th of February, 1880— on turning to February under the head of North Shields (see page 13), it is seen at a Find/ing the Time of High Water. 259" glance that high water takes place at 2'^ 26™ a.m., and that the height of tide is 12 ft. i in. above the mean low tvater level of spring tides, and that the time of high water on the afternoon of same day is 2h 51'", while the height of tide ahove the low water level of spring tides is 12 ft. 6 in. Similarly, desiring to know the particulars of the tide at Brest on the morning of April 20th, 1880 (see page 26), the mark — shows that no tido occurs on the morning of that day; there will be a high water at ii^ 32™ p.m. on the 19th, and again at o** 7™ p.m. {i.e., 7" past noon) of the 20th April, but none in the interval. Again, if it be required to know the times of high water on May ist, 1880, at Weston- Buper-mare — on turning to May, and under Weston -super-mare (see page 39), and opposite the ist we find that the times of high water are 11'' 24™ a.m., and 11'' 57™ p.m. respectively. 313. If the place at which the time of high water is required be not a Standard Port, it is to be referred (if in the west of Europe) to a Standard Port, by adding or subtracting a certain constant to the time of that Standard Port, as directed in the Tables. In pages 103 to 108 of the Admiralty Tables, 1880, will be found upwards of two hundred ports on the coasts of the United Kingdom, and in Europe, for which Standard Ports of Reference are given, and the time which is to be added to or subtracted from the time of high water at such Standard Port. 314. To find the times of high water by the Tidal Constants. RULE LXXXVin. 1°. Seeh in the '■'■Tide Tables,''^ pages 104 — 108, in the left-hand column for the given place, and in the column headed " Standard Port for Reference," will be found the Standard Port for the given place ; also, from the column headed "Time," and opposite the given place, take out the "Constant," being careful to note whether it is additive (marked -f ), or subtractive (marked — ). 2°. Take out of ^^ Admiralty Tide Tables,^'' pages i — 97, the morning (a.m.) and afternoon (p.m.) times of high water at the " Standard Port for Reference," being careful to annex the letters a.m. or p.m. to the tides so taken out. (a) If a blank ( ) occurs in either morning (a.m.) or afternoon (p.m.) column, use the preceding time of high water instead when the Constant is marked additive ( + ), but use the time of high water following the blank ( ) when the difference is marked subtractive ( — ). 3°. To the times of high water at the Standard Port just taken out, apply the Constant (No. 2°), adding or subtracting said Constant according as it is marked -\- or — ; the result in each case, if less than 1 2'', is respectivehj the morning (a.m.) and afternoon (p.m.) times of high water required. (See Exs. i and 10). (a) When the sum of the Constant and the morning (a.m.) time of high water at the Standard Port exceeds 12^, deduct 12**, the remainder is the afternoon (p.m.) time of high water at the given place. To obtain the morning (a.m.) time of high water at the given place, if any, add the Constant to the preceding afternoon (p.m.) time of high water at the Standard Port, and if the sum exceeds 1 2'', deduct I 2'', the remainder is the morning (a.m.) tide sought (Ex. 3), but if the sum be less than 12'', it is the afternoon (p.m.) tide of the day before, and there is no morning (^a.m.) tide that day at the given place. (Ex. 4). »6o Finding the Tinie of High Water. (b) When the Constant added to the morning (a.m.) time of high water at the Standard Port is less than li^ (i-©-, gives morniag (a.m.) tide at given place), hut when added to the afternoon (p.m.) tide at the Standard Port is greater than 11^, there is only a morning (a.m.) tide at the given place on that day. (Ex. 5). Note. — When the sum of the Conetant and the tide taken from the Tables is less than 12'', it remains a tide of the same name as that used, but when the sum exceeds 12^, the time over 1 2'' will be a tide of the name following that taken out ; consequently, in such a case take from the Tables the tide immediately preceding the one you require. (c) When the Constant is subtractive, and exceeds the morning (a.m.) time of high water at the Standard Port, reject this last and use the following after- noon (p.m.) tide at the Standard Port. If the subtractive Constant exceeds the afternoon (p.m.) tide at Standard Port, \-£^must he added to this last before subtraction is made, the remainder will he the morning (a.m.) tide at the given place. For the afternoon (p.m.) tide use the following tide at Standard Port, that is, the morning (a.m.) time of high water next day, borrowing iz*" if Con- stant exceeds it, the remainder is afternoon (p.m.) tide at the given place. (d) If Constant being subtractive, exceeds the Standard morning (a.m.) high water, but is less than the Standard afternoon (p.m.) tide, there is only an afternoon (p.m.) tide at the given place on that day. (e) If when the Constant is subtractive, the Standard afternoon (p.m.) tide has to be increased i z^, but Constant is less than the Standard morning (a.m.) tide following ; there is only a morning (a.m.) tide at the given place that day. Examples. Ex. I. 1 880, January lath : find the times of high water at Scarborough. Turning to Admiralty Tide Tables for 1880, at page 106, in the left-hand oohimn, and under the head of "Porta of Great Britain," we find Scarborough, and in the right-hand column, immediately abreast, we find that the Standard Port for Reference in this instance is Sunderland, and in the column under Time we have the Constant + o"" 49™, that is, we have to a-dd c^ 49™ to the time of high water at Sunderland on any day in order to obtain the corresponding time of high water at Scarborough. The wosk wtU stand as follows : — Port for Eeference— Sunderland, Jan. 12th 3'' 18" a.m. Jan. 12th ■^^z'^ p.m. Constant for Scarborough -J- o 49 4- o 49 Time of H.W. Scarborough, Jan. 12th 4 7 a.m. Jan. 12th 4 31 p.m. Ex. a. 1880, Feb. 17th : find times of high water a.m. and p.m. at Bordeaux. Turning to page 107, Admiralty Tide Tables, it is seen that the Port for Reference for Bordeaux is Brest, and the Constant is + i*" 3", that is, the Bordeaux tides are j*" 3™ later than the Brest tides, and consequently jh ya must bc added to the time of high water at Brest on any day, to obtain the corresponding time of high water at Bordeaux. Port for Reference— Brest, H.W., Feb. 17th 7^45'" a.m. Feb. 17th 'i^ e^^s.^. Constant for Bordeaux +33 +33 Times H.W. Bordeaux, Feb. 17th 1048 a.m. Feb. 17th ir 9 p.m. It may be here remarked, that on adding a Constant to the time at the Standard Port for Reference, a morning tide frequently becomes an afternoon tide, and an afternoon tide may become a morning tide for the next day, in which case the afternoon tide of the previous day must be employed to find the morning tide at the given port. (See 3" (a) of Bule). Mnding the Time of Sigh Water. Ex. 3. 1880, May 19th: find times of high water, a.m. and p.m. at Cherbourg. Tho Standard Port for Reference fur Cherbourg (see page lo8, Afhniralty Tide Tables) is Brest, and the Constant is + 4'' 2™, that is, for the times of high water at Cherbourg we must always add 4'' a"" to the times of high water at Brest. In this instance, high water at Brest, May 19th, occurs at iih 8™ a.m. {i.e., 52'" before noon) ; consequently, ^h jm (the Cherbourg Constant) added to that time must evidently give a i'.m. tide at Cherbourg ; the a.m. high water at Cherbourg must, therefore, be sought from tho previous (p.m.) tide at Brest. In thi.s example it will bo seen that when the additive constant is applied to the preceding afternoon (r.M.) tide tho sum oxcoeds la"", consequently the tide flows past midnight — the excess of 12I' being evidently the morning (a.m.) tide at Cherbourg. Tho work stands as follows : — Port for Reference— Brest, H.W., May 19th uh8">A.M. I\lay i8th ioh35"'i'.M. Constant for Cherbourg -4-42 +42 15 10 14 37 12 — 12 Times H.W. Cherbourg, May 19th 3 10 r.M. May 19th 2 37 a.m. If the morning tide, by adding a Constant, is more than 12'', and thus becomes an afternoon tide, but the afternoon tide of the day before remains hss than 12'' when the Constant is added, then there is no morning high water at the required port, thus : — Ex. 4. 1880, June 22nd : find a.m. and p.m. tides at Flashing. Tho Standard Port for Beference in this case is Dover, and the Constant + i"" 42"!. In this case it is high water at Dover, June 22nd, at lo'' 24"" a.m. {i.e., i^^^S'^'betorQ noon), and the Constant ih 42"' added to that will evidently give a p.m. tide at Flushing. The preceding time of high water at Dover i.e., tho time of high water in the afternoon (p.m.) of the previou.s day must bu employed to obtain the morn- ing (or A.M.) tide at Flushing— if any. In this example it will be seen that when the additive constant is applied to the preceding afternoon tide at Dover, the sum is less than 12'>, consequently, the tide does not flow p»st midni^ki — the result being p.m. tide of June 21st. There is, therefore, no a.m. tide on tho 22nd of June a* Flushing. Time H.W. Dover, June 22nd io''24'" a.m. June 2i8t 9'' 54™ p.m. Constant 4" ^ 4^ + i 42 June 2 1 fat II 36 P.M. Time H.W. Flushing, June 22nd o 6 p.m. (No a.m. tide.) So, also, if, when the Constant be added to the morning (a.m.) time of high water, the time is less than 1 2^, but when added to the afternoon time is greater than i z^, there is only a morning time of high water at the given port. Ex. 5. 1880, April 1 8th : find the times of high water a.m. and p.m. at Lyme Regis. Turning to page 107 of the Admiralty Tide Tables for 1880, it will be seen that tho Standard Port for Heferenoo in this instance is Devonport, and the Constant is + oi' 38™, i.e., in order to obtain the time of high water on any given day we must add o>< 38" to the times of high water at Devonport on that day. Port for Reference— Devonport, April i8th io*>48'" a.m. April i8th ii^ 27'" p.m. GonBtant -\- o 3S Constant + o 38 H.W. Lyme Regis, April 18th 11 26 a.m. April i 8th 12 5 p.m. or April 19th o 5 a.m. U.W. Lyme Regis, April 18th, 11'' 26'" a.m. ; no p.m. It also frequently occurs that when there is hut one high loater at the Standard Port — a blank ( — ) occurring in one of the columns — there may be two at the given po^rt. 262 Finding the Time of Sigh Water. (a) When there is only a morning time of high water at the Standard Port, i.e., a blank occurs in p.m. column, and the Constant is additive, the morning tide may become an afternoon tide, and the afternoon tide of the day before may become the morning tide required, thus : — Ex. 6. 1880, August ist: find times of high water a.m. and p.m. at Filey Bay. Eeferring to page 106, Admiralty Tide Tables for i88o, it will be seen that the Standard Port for Eeference in this case is Sunderland, and the Constant is + oi" 58™. There is only one high water at the Standard Port, which occurs at 11'' 43"^ a.m., i.e., cf' ij"^ before noon. The constant + o^ 58'" being added to this morning time evidently gives the afternoon (p.m.) tide (see Rule y (a); and the morning (a.m.) tide is sought by applying the constant to fkQ previous afternoon (p.m.) tide, which occurs on July 31st, (the last day of the previous month) at 11'' 5™ p.m., i.e., o'' 55° before midnight; the result is July 31st, 12'' 3™ p.m., which is equivalent to August lat, o'' 3"° a.m. Sunderland, H.W., August I st ii'>43mA.M. Sunderland, July 31 st ii** 5™ p.m. Constant for Filey + ° i^ Constant + ° 5^ 12 41 12 3 P.M. — 12 — 12 Filey Bay, H.W., August ist o 41 p.m. H.W., August ist o 3 a.m. This example shows the method of proceeding when the first tide has to be taken from the first day of one month and the second tide from the last day of the preceding month. (b) When there is only an afternoon (p.m.) time of high water at the Standard Port, i.e., when a blank ( — ) occurs in the morning (a.m.) column at Standard Port, and the Constant is additive, the previous afternoon (p.m.) time of high water at the Standard port may become the morning (a.m.) time of high water at the given port, while the afternoon (p.m.) time of high water at the Standard Port may give the afternoon (p.m.) time of high water required, thus: — Ex. 7. 1880, August 12th : find a.m. and p.m. times of high water at Stromness. The standard Port for Reference is Thurso, and the constant is + o'' 32" (page 106, Admiralty Tide Tables, 1880). A blank occurring in the morning (a.m.) column of the Standard Port for Reference, shows that there ig no morning tide at that place, only an afternoon time of high water. The Constant being additive, we must take out the high water for the previou^s afternoon (p.m.), which is iii> 46™ p.m. on the nth, i.e., oh 14™ before midnight on the nth; the Stromness Constant, + o^ 32", added, evidently gives the next morning (a.m.) tide at Stromness, since 12^ iS" past noon August nth is evidently o'' 18™ a.m. on August 12th. The afternoon (p.m.) time of high water at Stromness is found by adding the Stromness Constant to the afternoon (p.m.) time of high water at Thurso. The work stands thus :— Standard Port for Reference— Thurso ; Constant + o'' 32°'. Thurso, H.W., August nth ii^^6"'¥.yL. H.W., August 12th o^ 9'np.M. Constant + o 32 Constant + o 32 August nth 12 18 P.M. H.W., August 12th o 41 p.m. 12 Stromness, August 12th o 18 a.m. It sometimes happens that when there is but one high water at the Standard Port, there may be but one at the given port. (a) When a blank ( ) occurs in the morning column for the given day at the Standard Port, i.e., when there is no morning tide at that place, and the Constant being additive, the previous afternoon (p.m.) time of high water at the Standard Port may give the afternoon (p.m.) tide at the given port for the previous day, and the afternoon (p.m.) time of high water at the Standard Port for the given day gives the afternoon (p.m.) time of high water for thd same day at the given port. Fmding the Time of High Water. 263 Ex. 8. 1880, July 23rd : find times of high water a.m. and p.m. at Boulogne. The Standard Port for EoferGnco for ISouloprne is Dover, and the Constant is + o'' 13™ (see Admiralty Tide Tables, i88o, page 108). A blank occiirrino: in the morning (a.m.) coliunn of Port for Reference (Dover), and the Constant being additive (-f ), the preceding time of high water, i.e., that for the afternoon (p.m.) of the previous day is used to determine the morning tide at Boulogne — if any. In this example it will be seen that on applying the Constant to the previous p.m. tide at Dover, the sum (11'' 52"") is less than lai", consequently it remains a p.m. tide of the preceding day — the 22nd. There is no a.m. tide. The Constant added to the afternoon tide of given day at the Standard Port givee the required afternoon tide at Boulogne. Time H.W. Dover, July 22nd ii*>39°'p.m. Dover, July 23rd o'> 5™ p.m. Constant for Boulogne + 013 Constant + ° ^ 3 TimeH.W. Boulogne, July 22nd 11 52 p.m. Boulogne, July 23rd o 18 p.m. No A.M. tide; o*" 18"' p.m. Ex. 9. 1 880, January 8th : find a.m. and p.m. times of high water at Whitby. The Standard Port for Reference for Whitby is Sunderland, and the Constant -|- oi" 23"". There is tio after- noon tide on Jan, 8th, at Standard Port. The morning lime of high water at Sunderland is ii'' 37™ a.m., or o'' 23" before noon, and the Constant o'' 23'" added gives noon as the time of high water at the given port. Time of H.W. Sunderland, January 8th 11'' 37" a.m. Constant for Whitby -\- o 23 12 o A.M. or Noon Ex. 10. 1880, January 23rd: find the times of high water a.m. and p.m. at the Needles Point. Turning to the "Admiralty Tide Table" for 1880, at page 107, in the left-hand column, wo find Needles Point, and in the right-hand column, immediately abreast, we find that the Standard Port for Reference which in this instance is Portsmouth, and in the column under Time we have the Constant — i*" sS"'t that is, we have to subtract i^ 55'" from the time of high water at Portsmouth on any day in order to obtain the cor- responding time of high water at Needles Point. The work will stand as follows : — Port for Reference— Portsmouth, Jan. 23rd, 8*> 26™ a.m. Jan. 23rd, 9^ 4"" p.m. Constant for Needles — i 55 — r 55 Times H.W. Needles, Jan. 23rd, 6 31 a.m. Jan. 23rd, 7 9 p.m. When the Constant is subtractive, the morning tide at the Standard Port frequently becomes an afternoon tide of the day before, and the afternoon tide of the given day becomes a morning tide, in which case the morning tide of the succeeding day must be employed to find the afternoon tide at the given port, as in the example following : — Ex. II. 1880, May i6th : find a.m. and p.m. tides at Portland Breakwater. In this case the Standard Port for Reference is Portsmouth, and the first tide at Portsmouth occurs at 3'' 34'n A.M. (i.e., 3'' 34™ after midnight), consequently, since Portland Constant shows that high water occurs there 4i> 40m earlier than at Portsmouth, and since that quantity, subtracted from May 16th, 3'' 34n> a.m., would give a P.M. tide on the 15th at Portland ; wo therefore use the Portsmouth tide of the 16th p.m., and of the 17th A.M. thus : — (See Rule). Time H.W. Portsmouth, May 1 6th 3'' 58™ p.m. lilay 17th 4'' 25"^ a.m. -j- 12 +12 15 58 i6 25 Constant for Portland — 4 40 — 4 40 Times H.W. Portland B'kwater, May 1 6th 11 18 a.m. ii 45 p.m. When at the Standard Port there is only an afternoon high water, and the Constant is subtractive, and greater than the given time, the afternoon tide for 3j64 FinMng the Time of High Water. the given day will give the morning tide required, and the morning time of high water for the succeeding day must be employed to determine the after- noon time, thus : — Ex. 12. 1880, July 17th: find a.m. and p.m. tides at Falmouth. The Standard Port for Reference is Devonport, and the Constant is — oi" 46"'. A blank ( ) occurs In tlie morning column of the 17th, we therefore use the next tide (as the Constant is auhtractwe), viz., the p.m. tidft, laii being added to make the subtraction, thus : — Time H.W. Devonport, July 17th o^ 16" p.m. (next tide) July i8th o^ 57" a,m. + 12 Constant — o 46 12 16 July 18th O II A.M. Constant for Falmouth — o 46 Times H.W. Falmouth, July 17th 11 30 a.m. Here there is no p.m. tide on July 17th at Falmouth. The following example shows the mode of procedure when the first tide is taken out for the last day of one month, and the second tide on the first day of the succeeding month. Ex. 13. 1880, February 29th: find a.m. and p.m. times of high water at Aberystwyth. The Standard Port for Reference is Holyhead, and the Constant is — 2t' 40™. A blank (— ) occurs in the morning column of February 29th at the Standard Port ; there is, therefore, only an afternoon tide at that port on the 29th, and the Constant being subtractive, we use the afternoot), time of high water at Holyhead (increased by 12'') to obtain the morning (a.m.) time of high water at Aberystwyth; and the next time of high water at Holyhead, i.e., the time of high water at Holyhead on the morning (a.m.) of March »et must be employed to determine the p.m. tide at Aberystwyth. The work will stand as below : — H.W. Holyhead, Feb. 29th, o>> ^^ p.m. (next tide) H.W., March ist, o*' 25'" a.m. 12 + 12 12 5 12 25 Constant — 2 40 Constant — 2 40 H.W. Aberystwyth, Feb. 29th 9 25 a.m. H.W. February 29th, 9 45 p.m. Ex. 14. 1880, July ist: find a.m. and p.m. times of high water at Milford Hav«i (entrance). The Standard Port for Reference is Pembroke, and the Constant — oi- 2on> (see page 105, Adnviralty Tide Tables). Constant exceeds Standard a.m. tide, therefore reject it ; but it is less than p.m. ; ttiere Is only a P.M. tide at Milford Haven. Time H.W. Pembroke, July ist o^ 6"" a.m. o^ 35'" p.m. Constant — 20 — 20 No A.M. tide on July 1st at Milford Haven. 15 P.M. Ex. 15. 1880, July 17th: find a.m. and p.m. times of high water at Skull. The Port for Reference for Skull is Queenstown, and the Constant is — oh 59m (see page 104, AdmtraJty Tide Tables, 1880). Here the first tide at Queenstown occurs at 11'' 50™ a.m., and there is no afternoon (p.m.) tide that day, as a blank occurs in that column ; but the next tide occurs July 18th, oh 32"! a.m., and since the Constant for Skull shows that the time of high water there takes place o^ 59™ earlier than at Queenstown, that quantity subtracted from July iS^ oh 32" a.m., will give a p.m. tide at Skull ; therefore use times of high water at Queenstown on the morning (a.m.) of July 17th, and the morning (a.m.) of July 18th ; thus r— Queenstown, H.W., July 17th 1 1^50™ A.M. H.W., July 1 8th o^^^'^a.k. Constant — o 59 +12 Skull, H.W., July 17th 10 51 a.m. 12 yi. Constant — o 59 Skull, H.W., July 17th II 33 p.». Mndinff the Time of Sigh Water. 265 Ex. 16. 1880, October 28th : find a.m. and p.m. times of high water at Ballycottin. The Standard Port for Reference w Waterford, and the C mstant — o'' 20'°. In this example there is a blank in the morning column at Standard Port, wo therefore take the afternoon tide, and the Constant being of the same value precisely as the time of high water, the result is a noon tide at IJallycottin. (Only tide). Time H.W. Waterford, Oct. 28th o>> 26™ p.m. Constant — 26 Time H.W. Ballycottin, Oct. 28th o o There is only one high water on the 28th October, and this occurs at Noon. Examples for Praotioe. In each of the following examples it is required to find the time of high water a.m. and p.m. I. 1880 , Jan. 8th, Mary port. 16. 1880 , Jan. 24th, Southampton. 2. Feb. 29th, Cardigan. 17- June 2nd, Abervrach. 3- March i8th, Aberystwyth. 18. June 17 th, Ballycottin. 4- April 14th, Lerwick. 19. Juno 19th, Valentia Harbour 5- May 30th, Portland Bk' water. 20. Jan. 22nd, Bayonne. 6. June 14th, Ballycastle. 21. May 23rd, Aberdeen. 7- July 8th, Boulogne 22. Dec. 29th, Tay Bar. 8. August 13th Wexford. 23. June 16th, Bordeaux. 9- Sept. ist, Cadiz. 24. June loth, Douglas. 10. Oct. 12 th, Torbay. 25- June loth, Wicklow. II. Nov. loth, Stromness. 26. March ist, Poole. 12. Dee. 12th, Aberdeen. 27- Nov. nth Ilfracombe. 13- Jan. 22nd, Gibraltar. 28. Dec. 7th, Port Rush. 14. Feb. 7th, Blyth. 29. Feb. 1 8th, Coleraine. 15- March 25th, Peterhead. 30. Feb. 5th, Bantry Harbour. 315. In pages 151 to 232 of Admiralty Tide Tables for 1880, are given the times of high water at full and change of a great number of ports, by which we are enabled to calculate approximately the time of high water on each day. The Constant is found by taking Brest as the Standard Port, at which place the time of high water, full and change, is 3'' 47". The difference between the full and change at the given port and Brest will be the Constant to be employed (as in the preceding Eules), except there be a great difference of longitude, in which case the correction for the moon's meridian passage must be employed, since for the greatest longitude this correction may amount to half an hour. Shovdd the longitude, however, not exceed 5*^, it may be neglected, as doing so will scarcely make more than a difference of one minute. It must also be observed that the longitude of Brest is 4^° W. of Greenwich, and in strictness, therefore, in determining this correction 4° should be subtracted, if the longitude of the place be East, or added if it be "West. The correction is found in Table XVI, Norie, or Table XXVIII, Paper. Hence : 316. To find the time of high water at Foreign Ports whose Constants are not given in the Tide Tables. EULE LXXXIX. 1°. To find the Constant.— /« the Alphabetical List of Ports at the end of the Admiralty Tide Tables for 1880, page 189 — 232, find the time of high water Full and Change, at Brest, and also that corresponding to the given port ; sub- tract the less from the greater of these two times, and the remainder will he the Constant, additive if the full and change (F. SiQ.) at the given port is greater than that of Brest, but subtractive if less. ^^^ Mndinff the Time of Sigh Waier. 2°. Take out the times of high water at Brest for the given day, and apply the Constant as directed in the preceding Eule LXXXYIII, pages 259 — 260; the result is the time of high water (nearly) at the given place. 3°. Take out the longitude of Brest and also of the given place ; talce the sum if the names are alike, hut take the difference if the names are unlike. 4°. Take out (from the column to the left of those containing the times of high water at Brest) the moonh transit for the proposed day and the following one, if the long, is West; but for the given day and the preceding one if the long, is East. Their difference, in either case, is the Daily Variation, or Eetardation. 5°. Take from Table 28, Raper, or Table 16, Nome, the correction corres- ponding to the daily variation and longitude. 6°. Apply this correction by addition in West longitude, but by subtraction in East longitude, to the approximate times of high water already found, the result is the times of high water on the proposed day at the given place. Examples. Ex. I. 1880, March 19th: required the time of high water at Victoria Elver, Turtle Point (N.W. coast of Australia), longitude 130° E. Time of H.W. full and change, Victoria Kiver 7''i5'" (p- 231) „ „ Brest 3 47 (p. 196) Constant + 3 28 ]) 'b transit, March 19th, 6h 43m P.M. Long. Victoria River 130° E. rSth, 5 53 „ Brest 4 W. 50 126 Under 50^ and against 126° longitude, in Raper, Table 28, or Norie, 16, we find 17"= to be subtracted, because the longitude is E. Time H.W. Brest, March 19th 8^ 52'" a.m. Constant + 3 ^^ Correction for longitude Time H.W. at Victoria > River, March 19th ) 12 20 — 17 12 3 — 3 Time H.W. Brest, March i8th 8^ 22'n p.m. Constant -)- 3 ^8 II 50 P.M. — o 17 Time H.W. at Victoria ) i^ 33 P-M. River, March i8th ] No A.M. tide. Ex. 2. 1882, October 9th : find the times of high water at Sandy Hook, long. 74° W. Time of H.W. full and change, Sandy Hook 7h 29™ (p. 225) „ „ Brest 3 47 (p. 196) Constant + 3 42 ]) 'b transit, Oct. 9th 4*> 42™ Long. Sandy Hook, 74" W. loth 5 42 „ Brest + 4 W. I o 78 60'" Under 6on> and against 78° in longitude, in Raper, Table 28, or Noeie 16, we find 13™ to be added, because the longitude is West. Time H.W. Brest, Oct. 9th 6h 46" a.m. Constant + 3 4^ 10 28 A.M. Correction for longitude +13 T. H.W. Sandy Hook, Oct. 9th 10 41 a.m. Time H.W. Brest, Oct. 9th 7'> 13™ p.m. Constant -}" 3 4^ 10 55 P.M. Correction for longitude + ^3 T. H.W. Sandy Hook, Oct. 20th ii 8 p.]n. Filling the Time of High Water. 267 Ex. 3. 1882, May 25th: required the times of high water at Nelson, New Zealand, longitude 173° E. Time of H.W. full and change, at Nelson 9'' jc" (p. 217) „ „ Brest 3 47 (P- 196) Constant +63 J) 's transit, May 25th, o'»4i"a.m. 23rd, II 36 P.M. 65" Under 65" and opposite 169" (173° — 4°) in Table 16, Nokie, or 28, Kapeb, stands the correction 30™ to be subtracted. Time H.W. Brest, May 25th 4'' 3'" a.m. Constant -1- 6 3 10 6 A.M. Correction for longitude — 30 Time H.W. Nelson, May 25th 9 36 a.m. Time H.W. Brest, May 25th ^^ 28™ p.m. Constant +63 10 31 P.M. Correction for longitude — 30 Time H.W. Nelson, May 25th 10 i p.m. Ex.4. 1882, August 8th: find the times of high water at Cape Virgin, Straits of Magellan, longitude 68° W. Time of H.W. full and change, Cape Virgin 8^ 30™ (p. 232J „ „ Brest 3 47 (P- 196) Constant + 4 43 ]) 's transit, August 8th i^^G"^ 44'° and long. 72° "W. (68<> + 4^) give corr. + 8™. 9th 2 30 44 Time H.W. Brest, Aug. 8th, 41'56'»a.m. ; Time H.W. Brest, Aug. 8th 5^ 12" p.m. Constant -|- 4 43 ; Constant -^ 4 43 9 39 Correction for longitude + 8 Time H.W. C. Virgin, Aug. 8th 9 47 a.m. 9 55 Correction for longitude + g Time H.W.C. Virgin, Aug. 8th 10 3 p.m. Examples fob Pbaotiob. On the dates given, find the times of high water at the undermentioned places. 1882, January 12th: Rio Janeiro, longitude 43° 9' W. „ August ist: Caracus River, Ecuador, longitude 67° W. 3. „ September 26th: Auckland, New Zealand, longitude 175° E. 4. „ May 20th : Point de Gallo, Ceylon, longitude 80° E. 5. „ February 28th: San Francisco Bay, longitude 122° W. 6. „ September 27th: Malacca Fort, longitude 102" 15' E. 7. „ July 12th: Port Jackson, North Head, longitude 151° 16' E. „ July 31st: St. Julian, longitude 67° 38' W. 9. „ February 4th: Awatska Bay, longitude 158" 47' E. „ February loth: Cape Cod, longitude 70° 6' W. „ January 2nd : Point de Galle, longitude 80° E. ,, August 24th: Macao, longitude 113° 34' E. 268 GREENWICH DATE BY CHRONOMETER. 317. The Error of Chronometer on Mean Time at any place is the dif- ference between the time indicated by the chronometer and the mean time at that place. The error of chronometer on Mean Time at Greenwich is the dif- ference between the time indicated by the chronometer and the mean time at Greenwich. The error is said to be fast or slow as the chronometer is in advance of or behind the mean time at Greenwich. 318. Rate of Chronometer is the daily change in its error, or the in- terval it shows more or less than twenty-four hours in a mean solar day. If the instrument is going too fast, the rate is called gaining ; if too slow, losing. 319. To find the Rate. — The rate of a chronometer is determined by com- paring its errors for mean time, as found by observation at a given place, on different days. Thus, if by observation a chronometer is found 20' slow, and at the end of ten days is found to be 50' slow for mean time at the same place, it has evidently lost 30' in ten days, whence its mean daily rate is 3^ losing. If on a given day, chronometer be 12^ fast, and at the end of thirteen days z^-^'fast for mean time at any place, it must have gained 45' in thirteen days, or its rate is about 3'- 5 a day gaining. Hence the amount of the daily rate (supposed uniform) is found by the following EULE XC. Write one error under the other, then Both errors fast, or both slow, take their difference. One error fast and the other slow, take their sum. Bring the sum or remainder into seconds, and divide by the number of days between the dates of the two errors ; the result will be the daily rate in seconds and tenths, or (perhaps) tenths only. 320. To name the Rate. — When the chronometer is fast either on Green- wich mean time, or on the time at place, if the error is increasing, the rate is gaining ; if decreasing, the rate is losing. When the chronometer is slow, if the error is increasing, it is losing; if decreasing, it is gaining. When the chronometer is fast, and the error changes to slow, the rate is losing ; if the error changes from slow to fast, the rate is gaining. Examples. Ex I. A chronometer was 25" 20^ slow for mean time at Greenwich on Nov. 20th, and on November 30th was 24™ 45" slow on Greenwich mean time : required the daily rate. November 20th, chronometer slow 25«n2o« November 30th, „ slow 24 45 Change of error in 10 days 35 Bate for i day 3*5 gaining. In this example the chronometer is slow oa November 20th, and the error is decreasing, therefore the chronometer is gaining. Greenwich Date hy Chronometer. 269 Ex. 2. A chronometer was slow 28™ 5' on mean time at Greenwich, Feb. 27th, 1880, and on March i ith was slow 29^" 36' on mean time at Greenwich : find daily rate. 1880, February 27th, chronometer slow 28™ 5' Feb. 29 (leap year) 1880, March nth, ,, slow 19 36 Feb. 27 Change of error in 13 days i 31 March i3)9i(7'"-o — 91 Int. 13'* The error of chronometer, which is slow, is increasing, it is therefore losing 7*"o. Ex. 3. A chronometer was fast i™ 23* on mean time at Greenwich, Juno 2nd, and on July <8t was/aai i" 37''5 on mean time at Greenwich : find daily rate. June 2nd, chronometer / a day. ) gains 2''6 in a 2'! l6h? ( 6)i6"oo ( 4)2-66 32-66 3-6 6, 19596 6532 0)8,4-916 i'"24«-9 Ex. 2. If a chronometer loses 9''"4 in a day, what is its loss in i2'i 9'^ 34™ = 12'' lo*" (nearly) ? 24 Prefixing the days to the decimals of a day. 6)10 4)1-66 13*41 9'4 4964 1 1 169 6,0)11,6-654 in'56""7 or I ""57' The result obtained by this rule in these examples is a little more than by the previous one of aliquot parts, as we have taken g^ 34™ as icA in this, while in the other it was reckoned g^ 30". 323. Before going to sea, the error of the chronometer on Greenwich mean time, and its daily rate, are supposed to have been accurately deter- mined, either at an observatory by means of daily comparison with an astronomical clock, or by observations taken by a sextant at a place whose longitude is known. 324. When the error of a chronometer on Greenwich mean time, and also its daily rate, are known, we may determine Greenwich mean time at some other instant, as when an observation is taken, by the following EULE XCII. 1°. To the time hy chronometer apply the original error, adding it if the chronometer was slow, rejecting 24'* if greater than 24'', and putting the day one forward ; hut if the chronometer is fast, subtract original error, increasing time shown hy chronometer hy 24^, if necessary, and puttitig the day one hack. 2°. Find the numher of days and parts of a day, to the nearest hour, elapsed since the original error was ascertained. 3°. Multiply the daily rate of chronometer hy the elapsed time, and add thereto the proportionate part for the fraction of a day, found hy proportion or otherwise; the result is the accumulated rate in the interval. 4°. To the result found hy i ° add the accumulated rate, if chronometer is losing ; hut subtract if gaining ; the result will be mean time at Greenwich, at the instant of observatipn. 272 Oreenwich Date "by Chronometer. Examples. Ex. I. 1882, Jan. 30th, P.M. at ship, time by a chronometer, Jan. 29'' \^ 47™ 48»*3, which was g"" i9'-6 slow for Greenwich mean time Doa ist, i88i, and on Jan. ist, 1882, was lo™ 24»'7 slow on Greenwich mean time. 1 88 1, Dec. ist, slwo 1882, Jan. ist, slow Change of error in 31 days 9"i9«-6 10 24-7 hiin^. Deo. yi^ Dec. I 60 T ^° Jan. I i)65-i(2-i 62 31 31 Int. 3i<» The chronometer being slow and the error increasing, the rate must be marked losing, 'Daily rate Time by chron., Jan. Original error Jan. Accumulated rate Greenwich date, Jan. 29di5h47'"48>*-3 4- 10 24' 7 Interval from January isk 29 15 58 13-0 to January -\- I o-i a9th i5*> 58"" I8 2 8<*i6hnly. 29 15 59 13-1 28-1 28 468 42 58-8 i*o •3 6,0)6,0-1 ^ Ace. rate i^o'"! Ex. 2. 1880, March 20th, p.m. at ship, an observation was made when the time by chronometer was March %o^ o^ 7"! 539, which was 50™ 5i« fast on Greenwich mean time, November 22nd, 1879, ^"^^ on December 21st, 1879, was /««< 47™ 33''8 for mean noon at Greenwich : required the Greenwich date by chronometer. November 22nd, chron. fast December 21st, fast Change of rate in 29'* 47 33-8 Nov. Nov. Dec. Int. 3od 22 3 17-2 60 8 21 29)i97'2(6'8 losing. 174 232 23a 29* The chronometer is fast and the error decreasing ^\hQ rate is therefore losing. Time by chron., March 20'' o*" 7"'55* Original error Accumulated rate Greenwich date, March or 19 24 7 55 — 47 33 8 23 20 21 2 + 10 II 7 Dec. Jan. 3 1 Feb. 29 March 19 23 'Rate 19 23 30 32-9 Intr. 89 23 In finding accumulated rate, as the interval is within half an hour of 90 days (23I), we might have multiplied by 90, and deducted i-48th \\^ is i-48th of a day) from the daily rate. 6»-8 89 612 544 605-2 3'4 23 6,0)61,1-7 t. Ace rate 10 11-7- Greenwich Date hy Chronometer. z-j^ Ex. 3. Time by a chronometer, Sept. -j"^ 23^' i6"> 28", which wa8 57'" 479 slow on Green- wich mean time, June 30th, and on July 1 2th, was 56"" 53* slow on mean time at Greenwich. June 30th, chron. slow 57'"47* June 30^ Jul}' 12th, „ slow 56 53 30 Change in rate in 1 2 days 54 Daily rate 4*5 gaining. July Int. In this instance the chronometer is slow and the orror decreasing, the rate 4>"5 is therefore to be marked gaining. Time hy chron. Sopt. •]^it,'^i6"'i%^ July 31 fRate 4»*5 Original error + 5^ 53 Sept. 8 o 13 21 19 58 360 Accumulated rate — 421 Aug. 31 ^ 225 Sept. Greenwich date, Sept. 8090 Int. Examples for Praotioe. 6,0)26, 1 "O l^Acc. rate 4 2ro 1. 1882, February i6th, a.m. at ship, an observation was taken, when the corresponding time by a chronometer was 1 'eb. i6'' S^ 59™ 25", which was i'' 20" 23'"*4/rt«< on Greenwich mean time, December ist, i88r, and on January 23rd, 1882, was i^ 14'" ^y fast on Green- wich mean time : required the Greenwich date by chronometer. 2. A chronometer showed April z()^ 5^ o'" 0% which was fast 33"> 309-3 on Greenwich mean time, March 19th, and on March 26th was 34™ 20^ fast for mean time at Greenwich : required the Greenwich date by chronometer. 3. A chronometer showed May 7^ 6^ 9'" 48% which was slow 11'" q'-^. on Greenwich mean time, February i6th, and on February 26th was ii"" 4i''6 slow for Greenwich mean time: required the Greenwich date by chronometer. 4. The chronometer showed June 25'' 2i'» 29'» 53', which was 30"" ii'fast on Greenwich mean time, March 31st, and on April 15th was 30™ 45* fast for mean time at Greenwich : required the Greenwich date by chronometer. 5. 1882, October 25th, p.m. at ship, time by chronometer Oct. 25'^ 8'' 31'" lo^, which was 12™ 9«"2 slow on Greenwich mean time, July 20th, and on August 13th was 10™ 2^ slow for Greenwich mean time : required the Greenwich date by chronometer. 6. Time by chronometer January 19^ 13^ 21'" 2^^, which was 53™ 47' fast on mean time at Greenwich, October 24th, and on October 31st was 53™ 199 fast for mean time at Green- wich : required the Greenwich date by chronometer. 7. Time by chronometer November 8"* 16'' 2"' 3', which was 33'" o' slow on mean time at Greenwich, July 3i8t, and on August 12th was 32™ 2«*4 slow on mean time at Greenwich. 8. Time by chronometer August i*! o*" 3™ 0% which was 6"^ 4" fast on mean noon at Greenwich, May 31st, and on June 14th was 4™ 2<>-2 fast for Greenwich mean time. 9. Time by chronometer May i"* 13^ 23™ 10', chronometer slow 3™ 23^ on mean time at Greenwich, February 2nd, and on February 28th was 3" 49''o slow on Greenwich mean time. 10. Time by chronometer January 20*^ o^ 4™ 21% which was 20" fast on mean time at Greenwich, November 20th, i88r, and on December loth, i88t, was 4' fast on mean time at Greenwich : required the Greenwich date by chronometer. 11. Time by chronometer September 27'' i6'» 34™ 31*, which was o^" 20^ fast on mean time at Greenwich, April 19th, and on May 9th was o™ 18' slow for Greenwich mean time: required the Greenwich date by chronometer. 12. Time by chronometer April 16'' 5^" 36"" i2», which was i" 2' slow for mean time at Greenwich, January 24th, and on February 28th was 2^^ fast for Greenwich mean time. 274 Greenwich Date hy Chronometer. 325. When the "chronometer question" is given in a form similar to that below, we have to determine for ourselves the day of the month at Greenwich, that is, if the time shown by chronometer was i'', z**, 3'', &c., on the civil or on the astronomical day ; for, a frequent source of embarrassment in interpreting the indications of a chronometer arises from the division of its face into twelve instead of twenty-four parts, so that the same position of the pointer represents two periods of the day twelve hours distant. Thus, at z^ past noon, and again at 14'' past noon the hands are in the same place, and it is necessary to determine whether it should be read as z^ or 14'', 5'' or ly'', 6'' or iS*" past noon, and so on. To determine this point proceed according to this rule : — EULE xcin. 1°. Get an approximate Greenwich date by means of ship mean time nearly and the longitude hy account (Eule LXXVIII, page 227). 2°. Proceed, as directed in Rule XCII, page 271, to apply the original error and accumulated rate to the time hy chronometer. If the diflference between Greenwich dates thus found by the two methods is nearly 1 2^, then the Greenwich date by chronometer, found as above, must be increased by 12'', and the day put one bach, so as to make the two dates agree both in the day and hour nearly. Examples. Ex. I. August 3rd, at about -^^ p.m., longitude by acct. 75' W., the chronometer marks gh urn .^B^ and is 6™ 10^ fast on Greenwich mean time : what is the Greenwich astronomical date? Approx. T. at ship, Aug. Longitude 75° W. 3^ + Time by chron. Error of chron. fast ghiim .yB — 6 10 Approx. Green, date, Aug. 380 Green, date, Aug. 3rd » 4 57 In this example the approximate Greenwich time is S'', it is evident that the chronometer must have shown S*" from noon also. Ex. 2. June i8th, at 10^ 52" p.m., me^n time at ship nearly, long. 60° W., an observation was taken, when a chronometer showed 2'' 48™ 40*, on June 6th its error was known to be jm J08-2 fast on Greenwich mean time, and its mean daily rate 3^-5 gaining: required the mBan time at Greenwich when the observation was taken. Approx. Ship date, June i8 58'" io», and diff. 7 gives V to add, whence the term correspondiner to 9-15731 is ?!> 58" Il». 56^ 3»' o" Parts for 1 3 799"74 sine 9-921 190 Difi. 139 13 + 18 S6°3i'i3" sine 9-921208 417 ^39 18,07 fj6 To Find the Sov/r-emgle. Ex. 2. Given the true altitude 17' 16' 12", latitude 50° 42' S., reduced declination 20° 6' 17" iS. (when polar dist. is 69° 53' 43") : find the hour-anarle. Altitude Latitude Polar dist. Sum Half sum \ sum— alt. Hour-angle 17" 16' 12" 50 42 o 69 53 43 137 51 5S sec. oi9»335 cosec. 0-027304 68 55 57 COS. 9-555660 51 39 45 sine 9-894521 5*1 48m 6« log. 9-67582,0 79 i« = 3 In NoRTK, Table 31, we seek for the nearest log. to 9-67582, the nearest to which is 9-67579, which corresponds to s"" 48™ 5' ; then in column jjrop. part we seek for 3 , which gives !• to add, whence hour- angle is s"" 48'" 6'. Ex. 4. Latitude 0°, declination 0°, true altitude 30' : required the hour-angle. True altitude 30° o' 90 o Zenith distance 60 o 4 6,0)24,0 o Hour-angle 4I1 o"" OS Ex. 3. Given the true altitude 1 3° 28' 42", latitude 10° 35' S., reduced declination 23° 2 3'54''N. (or polar distance 113° 23' 54"): find the hour-angle. Altitude Latitude Polar dist. Sum Half sum \ sum— alt. Hour-angle 13° 28' 42" 10 35 o "3 23 54 sec. 0-007451 cosec. 0-037268 cos. 9-559623 sine 9-914694 4h4o'n4i« log. 9-51903,6 137 27 36 68 43 48 55 15 6 18 = 6 The nearest log. to 9-51904 is 9-51898, which gives ^h ^em ^o", the diff. 6 found at right hand in prop, parts gives l», whence hour-angle is 4'' 40™ 41'. Ex. 5. Given true altitude 75°, latitude 0°, declination 0° : find the hour-angle. True altitude 75° o' 90 o Zenith distance 15 o 6,0)6,0 Hour- angle i h o™ o» Examples for Praotioe. Eequired the hour-angle or meridian distance in each of the following examples : — Latitude 30° 15' S. Decl „ 39 27 S. I. True altitude 11° 21' 29' 2. 30 2 4 3- 27 48 22 4- 34 49 46 5- 25 38 II 6. 15 59 13 7- 29 2 27 8. 20 34 4 40 10 N. 39 20 S. o 29 N. 60 5 N. nation 15° 21' 4"N. 5 48 23 N. 23 26 44 N. 21 15 7 S. 23 I 55 N. 7 25 38 S. 000 23 27 21 N. 277 LONGITUDE BY CHRONOMETER, FROM AN OBSERVED ALTITUDE OF THE SUN. In the followinfj problem two chronometer errors are given, which, by means of the elapsed time, give a daily rate : hence proceed as follows : — RULE XCV. 1°. Write down the time by chronometer, apply its second error, adding if it is slow, subtracting if it is fast ; then apply the accumulated rate, adding if losing, subtracting if gaining (see Rule XCII) : the result is the Greenwich date at the instant of observation. 2°. Take out of Nautical Almanac, page II, the sun's declination and the equation of time for the noon of Greenwich date, and tlie corresponding hourly difference for each : also take out the sun's semi-diameter. 1°. Reduce the sun's declination and equation of time to the Greenwich time (Rules LXXIX and LXXXIII). 4.°" For the Polar Distance. — Subtract the reduced declination from 90°, if latitude and declination of same name ; hut if of different names add 90° to decUnation. 5°. For the True Altitude. — Correct observed altitude for index error, dip, refraction and parallax, or correction in altitude, and semi-diameter, and thzis get the true altitude (Rule LXXXIV). 6°. Find the hour-angle or meridian distance by Rule XCII.* 7°. When the observation is made in the afternoon, the hour-angle is apparent time past noon of the given day at ship — before which write t/te date at ship, but if the observation is made in the marning, take tlie hour-angle from 24'', the remaim,der in apparent time at ship reckoned from noon of the preceding day, the time at place in both instances being expressed in astronomical time. Examples. Ex. I. January 6th, p.m. at ship; sup- i E>. :. T muary 6th, a.m. at ship; sup- pose the sun's hour-angle to be 3'' 40™ i8» : pose the sun's hour-angle to bo 3'^ 40'" i8« : what is the apparent time at ship P , what is the apparent time at ship ? Here the time being p.m., we have the Here the hour-angle is 3^'4o™i8* ship date app. time, January 6"* 3'' 40" i8». 24 o o Ship date app. T., Jan. 5th 20 19 42 * In finding longitude by chronometer the logs, used in finding the hour-aAgle are required to be taken out for seconds of arc. 278 Longitude % Ohromrmter. Ex. 3. June ist, p.m. at ship ; suppose ' Ex. 4. June ist., a.m. at ship ; suppose the hour-angle to be 3'' 54™ 39' : required the apparent time at ship. Here the time being p.m, we have the app. time at ship, June i<^ 3^ 54™ 39^ the hour-angle to be 3'^ 54™ 39^ : what is the apparent time at ship ? Hour-angle 3^54"" 39' 24 o o App. T. at ship, May 31st 20 5 21 On comparing these examples with paragraph 7°, which they are intended to illustrate, the seaman will have no difficulty in understanding that, since the sun's Hour-angle is the Distance (in time) of the object from the meridian, if the observation is made in the after- noon (p.m.), as in Ex. i , the time will be 3'' 40'!"' 1 8^ past noon of the 6th day ; that is, the ship date (astronomical time) is January 644'n36* Original error ■=. — 8 42 II 6 35 54 Accumuliitod rate — 97 (a) Interval from Oct. I st to Jan. nth 6^ is 103'' 6'>. Daily rate Green, date, Jan. 11 6 26 47 Sept. 30 30 103 159 530 Oct. Nov. Dec. Jan. 31 30 31 103d iV''::\ Obs. alt. O's L.L. Dip Corr. alt. Semi-diameter True alt. 12 20 30 — 44 12 16 26 4 9 12 13 17 + 16 18 12 28 35 6,0)54,7-3 By Raper's Tables : dip — 4' 10", refr. — 4' 23,' par. -|- 9", semid. + 16' 18", and true alt. 12° 28' 24". Aco. rate 9 7 '3 (a) Tho chronometer having been rated, September ^th, there arc no days loft in September. Decl., page II, N.A. H.D. Jan. nth, noon 2i°46' 37*8. <^«w. a3'72 Correction — 3 33 6*45 Red. decl. Polar dist. Eq. T. page II, N.A. II.D. Jan. nth, noon 8'"i4'-i inor. -980 Correction + 6-3 21 44 48. Ill 44 4 n86o 9488 14233 6,0)15,3-9940 Altitude Latitude i2"28 35' 49 30 Polar dist. 1 1 1 44 4 Corr. — 3 33 sec. o" 1 87456 cosec. o'033026 n3 42 39 86 51 19 74 22 44 Red. Eq. T. 8 20-4 To be added to A.T. App. T. at ship, Jan. Eq. time 6-45 490 392 588 6-32,0 n 17'" 42^ (see Rule XCV, 7°, page 277). Ex. 2. 1882, May 20th, p.m. at ship, latitude 50° 43' N., obs. alt. sun's l.l. 17° 10", index corr. — i' 39*, height of eye 28 feet, time by a chronometer May 2o50597 Hour-anglo 65 34 56 3^.36-58^ 9"95873i 9-317646 App. Time ship, July 2nd Eq. Time Mean time ship, July 2nd Mean time Oreen., July 2nd Longitude in time 20*'23"' 2' + 3 51 20 26 53 16 18 48 4 8 5 Longitude 62°i'i5*E. By Raper : Log. sine sq. 9'3i7694 gives hour-angle 3'' 36™ 58", longitude 62" i' 15* E. A.T. ship, July 2'' 20 23 2 ' In this example the time of observation ia a.m. at ship, the hour-angle Is therefore the time before noon of July ^rd, anil as all calculations are made in the astronomical ilay, we take the hour-anglo from 24'', the remainder is A.T. at ship, reokone 1 from noon of the proccdini/ day, viz., that of July 2nd. Since the ob.ser- vation ia made in the forenoon at Rhip we may tako the hour-angle IVom the bottom of tho table, and tho time .so found is A.T. at ship, reckoned from noon of tho precoding day. (See Rule XCV, page 277). Ex. 4. 1882, April 14th, A.M. at ship, latitude 52° 10' N., obsotvtd altitude sun's l.l. 18" 20' 25", index corr. -\- 55", height of eye 12 foot, time by chron. 14'* 5^ 5™ 5" (Ixing p.m. at Greenwich), which vimfast 5'" 52«-4 for moan noon at Greenwich, February 14th, and on February 26th vi&sfaat 6™ 38" for mean noon at Greenwich. February 14th, chronometer /(M< 5" 52" February 26th, „ fast 6 38 T. by chron., Ap. i4<* Original error — Int. 5"^ 5"" 5' 6 38 4 58 27 3 3 12 12) Interval from Feb. 26th to April 14th jh^ is 48d s^. Rate 38-8 Interval 48 46 3-8 gaining. Ohs. alt. 0's ij.i. Index corr. Dip Corr. alt. Semid. t%° 20' 25" 4- ss 14 Accumulated rate — 18 21 20 — 3 19 Green, date, April 14 4 55 24 orr. -1- 55", a' 55". par. !*. true alt. 4 I i B 182-4 -6 18 18 I — 3 42 By Rapw : Index c Dip. — 3' 20", refr. — -4- 8'. semid. 4- ic' cf 6,0 18 15 19 + 15 58 )'8,3-i 18° 31 II". Decl., page II, N.A. Ap. i4tli, noon 9° 29' 18' N. incr. Correction •\- 4 24 3 3'' True alt. 18 31 17 H.D. Eq. T., page II, N.A. H.D. + 53"92 r4th, noon, rtrfrfo"ii39-98 <*for. o«-629 X 4-9 Correction — 3-08 X 4-9 Re J. decl. Polar dist. Altitude Latitude 9 3^ 42 N. 80 37 18 18^31' 17" 52 10 Polar dist. 80 27 18 sec. cosec. 6,0)26,4-208 ReJ. Eq. T. o 10-90 (To be added to A.T.) -\- 4 24 corr. App. T. ship, April Eq. Time 3-0821 _^5i 8 35 75 34 17 COS. 57 3 sine log. 0-212280 0-006055 9-396501 9-923827 9"53867 3 r3d i9>'ii~55s + II 13 19 12 6 14 4 55 24 9 43 18 Mean T. ship, April Mean T. Green., April Longitude in time Longitude 145° 49' 30"^". By Jiaper : Log. sine sq. 9-538703 gives hour-angle 4'' 48"" 6», long. 145° 49' 45" W. Hour-angle 4^ 48"" 5' 24 A.T.S.Ap.i3'ir9 11 55 The observation ha\-ing been made a.m. at .ship, the hour-anglo Is tho time before noon at ship viz. that ot April 14th ; therefore take the hour-angle from 24I', and the remainder is apparent time at ship', reckoned from the preceding noon, April 13th ; or if the hour-amjlo is tak.n from the bottom of the table (it being a m at ship) it is the apparent time— April 13'! igi" ii" 54"— reckoning from tlie preceding noon at ship. The Greenwich and ship dates being different (the Greenwich date being tho greater, or in advance of ship date) therefore subtract the latter from the former (borrowing 24i> to enable the subtraction to be made). 00 282 Longitude hy Chronometer. Ex. 5. 1882, March 6th, p.m. at ship, latitude 40° 20' S., ohs. alt. sun's l.l. 16° 20' index corr. + 30", height of eye 18 feet, time by a chronometer 5^ 20^ 10™ (being 6^ 8^ lo™ a.m. at Greenwich), which was 6™ 14^ fast for mean noon at Greenwich, January 30th, and on February 13th was/asi 4™ 29^ for mean time at Greenwich. January 30th, chronometer /as< 6™ 14' Feb. 13th, Change in 14 days. T. by chron., March 5<> 20^ lo™ o^ 4 29 Original error Accumulated rate Green, date, March T. from noon, 6th fast 4 29 I 45 60 or 105' Interval from Feb. 1 3th to March 5th 20**, is 2 1"^ 20*". 14 2)105" 7) 52-5 5 20 5 31 + 2 44 5 20 8 15 3'' 52" Rate TS 7 '5 losing, because fast and less fast. Ob-J. alt. 0*8 L.L. 16° 20' o" Index corr. Dip + 30 16 20 30 — 4 4 By Raper : Index corr. + 30", dip — 4' 10', refr. 3' 1 8", par. + 8", Bemid.-}- 16' 9', true alt. i6°29'i9''. Decl., page II, N.A. 6th noon 5° 35' 1 3"S. deer. Correction + 3 45 H.D. 58-14 387 157-5 3-7 Corr. alt. Semid. True alt. page II, N.A. + II'»24« n +2 •7 •4 16 16 26 — 3 5 2 5 16,3-7 16 13 21 + 16 9 2 43-7 Eq. time, 6th, noon Correctio 16 29 30 H.D. deer. 0-598 X 4 Red. Eq. time 11 27-1 To be added to A.T.) 2-392 Red. decl. 5 38 58 S. 6,0)22,5-0018 Polar dist. 8421 2 Corr. + 3 45 Declination and equation of time are both taken out for the nearest noon at Greenwich, viz., 6th, and corrected for the time wanting to noon, that is, for 3*» 52" or 3''-87. Altitude Latitude Polar dist. 16^29' 30* 40 20 o 84 21 2 sec. o'l 17879 cosec. 0-0021 15 141 10 32 70 35 16 54 5 46 App. T. at ship, March Eq. time M.T. ship, March M.T. Green., Jan. 6<* 4'^52™3is + ir 27 6 5 5 20 3 58 8 '5 Long, in time Longitude 133° 55' 45* E. 8 55 43 By Raper: log. sine sq. 9'550i34 gives hour-angle 4*> 52™ 32% long. 133° 56' o" E. 10s. 9-52161 1 ine 9-908486 A.T. S. Mar. 6d4h52'"3i» log. 9-550091 The observation having been made p.m. at ship, the hour-angle is the app. time at ship, before which write the date at ship, viz., March 6th ; then mean time at ship being one day in advance of mean time at Green- wich, we subtract the latter from mean time at ship (borrowing 24'') to enable us to complete the subtraction. Ex. 6. 1882, June 15th, p.m. at ship, latitude 13° 54' S., obs. alt. sun's l.l. 16° 16' 16", index corr. 4- o' 16", height of eye 16 feet, time by a chronometer 15"' o^ 16"" 16" (or 6^ 16"' 16' P.M.) which was 2^ 13™ 37' fast for mean noon at Greenwich, April ist, and on April i6th was 2^ 16"" 16^ fast on mean time at Greenwich. April ist, chron. fast 1 6th, „ ■Change in 15 days Daily rate 2 39 ios-6 gaining. 2 39 60 i5)i59(io"6 15 90 90 Longitude by Chronometer. 2«3 T. by chron., June 15'^ o'>i6"i6« Original error — 2 16 16 14 22 o o Accumulated rate — i o 35 Green, date, June 14 21 49 25 T. from noon June 15 2 11 By Raper : Index corr. + o' 16', dip — 4*0", rei'r. — 3' 18", par. -j- 8", semid. -\- 15' 47*, true alt. 16° 35' 9". Dec!., page II, N.A. June 15th, noon, 23° 19' 36" N. + Interval from April 1 6th to June 14th 22'', is 59<' 22'>. Daily rate io*-6 Interval 59 954 530 625-4 5'3 3*5 ■9 2 i 8 2 Obs. alt. O's L.L. 16° 16' 16" Index corr. -\- 016 16 feet Corr. alt. Semi-diameter True alt. 16 16 32 3 50 16 12 42 3 5 16 + 9 37 15 47 16 25 24 6,0)63,5-1 Correction Red. decl. Polar dist. Altittide Latitude — 14 23 19 22 N. 113 19 22 1 6° 23' 24" 13 54 o Polar dis t. 113 19 22 143 38 46 Half sum alt. 7 1 49 23 55 23 59 Hour-angle 4'' 19" 41' sec. cosec. COS. sin. log. H.D. 6-24 2-2 13-728 0-012905 0*037021 9-494089 9'9i547i 9-459489 10 35-1 Eq. T., page II, N.A. 15th, noon, add o™ 9"2 -\- Correction — i - 1 Eed. Eq. T. o 8-i (To be added to A.T.) H.D. + o*535 X :t'^ 1-070 l^i 4h 1901419 -f 08 App. T. ship, June Eq. Time Mean T. ship, June Mean T. Green., June Long, in time Longitude 97° 36' o" E. By Raper : Log. sine sq. 9-459550, hour- angle 4'' i9"> 42% long. 97° 36' 15" E. 15 4 19 49 14 21 49 25 6 30 24 In. this example the observation is made p.m. : hence the hour angle is apparent time at ship, before which we write the date at ship, viz., June 15th (see head of question) ; and since the mean time at ship is June isth 4'' 19™ 52% which is in advance of mean time at Greenwich, the latter being June 14'' aii" 49'" 25% we subtract mean time at ship from mean time at Greenwich, and 24'' is borrowed in subtracting. Ex. 7. 1882, June 23rd, p.m. at ship, latitude 0°, observed altitude sun's l.l. 20° 25', height of eye 20 feet, time by chronometer June 23^' 6'' 4™ 40', which was fast 13™ n^ on Greenwich mean time, April 6th, and on May ist was 12"^ 1'* fast for mean noon at Green. Interval from April 6th to May ist is 25 days ; the change in rate in that time is 70" : then 70' -j- 25 = 2^-8 losing. T. by chron., Jan. 23<> 6'' 4"'4o» Original error — 12 i Accumulated rate n 5 52 39 + 2 29 Interval from May ist to June 23rd 6'', is 53'* 6h. Daily rate 29-8 Interval 53 Green, date, June 23 5 55 8 By Raper : dip — 4' 20", refr. — 2' 36" par. -\- 8*, semid. -|- 15' 46', and true alt. 20° 33' 58". 84 140 6 1 i 1 148-4 •7 6,0)14,9-1 Obs. alt. O's L.L. 2o» 25' o" Dip — 4 »7 20 : Corr. alt. Semi-diameter True altitude 20 34 4 20 20 43 2 25 20 18 18 15 46 19*1 Longitude hy Chronometer. Deol.,page II, N.A. or 113 26 ti H.D. June 13rd, Correction noon 23° 26' 23" N. — 12- 2-OI 5 9 Ked. deol. 23 26 11 N. 1809 Polar dist. 66 33 49 1005 [1-859 Eq. T., page II, N.A. 23rd, noon i"'53^'4 + Correction + 3-2 Red. Eq. T. i 56-6 {.Added to A.T.) H.D. •538 59 4842 2690 3"i742 Altitude Latitude Tolar dist. Hour-angle Eq. Time 20- 34 4" O O O 860. O'OOOOOO 66 33 49 eosec. 0-037393 -87 7 53 43 33 56"5 «08. 9-860089 -22 59 52-5 sine 9-591840 ^hj^m^^s log. sr 48932,2 •+• » 57 . . M.T. ship 23a 4 .^i 54 iLT. a. £/i 5 55 8 Lon^. in T. i 23 J4 Longitude 2cf' 48' jp' W. iBy !Raper the answer Altitude Latitude Polar dist. Hour-angle Eq. Time ao° 34 4" 000 sec. 113 26 II cosec. J34 o 15 67 o 7-5 cos. 46 26 3-5 sine 4h29">578 log. -f I 57 M.T. ship 23'* 4 31 54 M.T. G. ^z<' 5 55 8 Long, in Time i 23 14 Longitude 20° 48' 30" W. oomes out the same. 0-000000 0-037393 9-591840 9-860089 9-48932,2 Ex. 8. 18^, October loth, p.m. at ship, latitude at noon 20° 41' S., ship had sailed N.E. (truo) 54 miles eiiioe noon, obs. alt. sun's l.l. i 8° 45', height of eye 15 feet, time by chrono- meter Octobur 9'' i6'> 28™ 42', which was slow ii~ 44^ on mean time at Greenwich, August 26th, and on September loth was slow lo™ 26': required the longitude at time of observa- tion and aleo at noon. Tojind tlie difference of latitude and difference of longitude. — The course 4 points and distance 54 miles give diff. lat. 38'-2 and dep. 38'-2. The diff. of lat. is marked North, because the ship at the time of aights was to the North of the position at noon ; and is subtracted from the lat. at noon^ viz., 20^ 41' S. to get the lat. at sights, the result is 20° 2' 48" S., and the dep. is named East, because the ship at the time of sights was to the East of the position at noon. The mid. lat. 20° 22' as a course, and dep. 38'-2 as a diff. lat., give the distance 41' as diff. of long, and is named East, because the ship at time of sights is east of her position at noon. The daily rate is 5»-2 gaining ; the interval 29"! 16'' X 5^*2 gives accumulated rate 2™ 34»-3 ; Greenwich date October 9"* 16" 36" 34*; polar distance 83° 24' 13", red. eq. T. i2"> 53^-9 snbL from A.T.-; true alt. (Nosib) 18° 54' 43" ; latitude in at sights 20° 2' 48" S. ; hour-angle 4^ 48™ 56»; mean time at ship Oct. lo"* 4^ 36™ 2" ; long, at^time of observation 179° 52' o" E. ; di& long. 4.1 '.4 also longitude at noon 179° 27' o' W. Examples fob Praotiob. I. 1882, January 2nd, a.m. at ship, latitude 36° 59' S., observed altitude sun's l.l. 49° 10', index correction — 2' 40", height of eye 14 feet, time by a chronometer i^ 19'' 8™ 50' (being 7*" 8" 50' a.m. at Green'wich), which was slow 18™ 2' for mean noon at Greenwich, November 30th, 1881, and on December 7 th, 1881, was 19" 10^-6 alow for mean time at Greenwich : required the longitude. Longitude by Chnmcmet&r. 285 2. 1882, February 19th, a.m. at ahip, latitude 38° 18' S., observed altitude sun's l.l. 21° 30' 40", index correction — 6' 45", height of eye 14 feet, time by a chronometer 18'' 19'' 53™ 37»'6 (being 7'' 53" 37"6 a.m. at Greenwich), which was 4™ i6»*6/a*^ for mean noon at Greenwich, January 23rd, and on January 30th was 5"" 9»"8 fast for mean time at Greenwich. 3. 1882, March 28th, p.m. at ship, latitude 20° 19' S., observed altitude sun's l.l. 30° 14', index correction — 2' 10", height of eye 30 feet, lime by chronometer 28'' o'' 10™ (being oh iQtn p inL lit Greenwich), which was 54'" 48" /a«^ for mean noon at Greenwich, October 2oth, 1881, and on December and, 1881, was 51"^ ^6^ font for mean uoou at Greenwich. 4. 1882, April 6th, A.M. at ship, latitude 53° 5' N., observed altitude sun's i,.i- 16° 8' 40", index correction — 40', height of eye ij feet, time by a chronometer ^^ 19'' iS'" 49" (being 7I' 1 8'n 49" A.M. at Greenwich), which was o'" 4»'4 slow for mean noon at Greenwich, February nth, and on March nth was 2" i%*fmt for mean noon at Greenwich. 5. 1882, May 19th, P.M. at ship, latitude 2° 58' S., observed altitude sun's l.l. 30° 30'^ index correction -\- 52", height of eye 19 feet, time by chronometer 19'' o'' 23™ 58*, which was 2%' fast for mean noon at Greenwich, January 3rd, and on January 3131 was 43" slow on mean time at Greenwich. 6. 1882, June 15th, A.M. at ship, latitude 12° 11' N., observed altitude sun's l.l. 39° 39' 40", index correction + 20", height of eye 17 feet, time by a chronometer i4<* 17'' 59'" 30' (being 5^' 59'" 30' A.M. at Greenwich) which was slow 5"^ 56'"3 for mean timo ^t Greenwich, April 2otb, and on May 12th was 2"> 29»-5 slow for mean noon at Greenwich. y. 1882, July 5th, A.M. at ship, latitude 23° 48' N., observed altitude sun's lj« 48° 36' 50", index correction — 50', height of eye 17 feet, time by chronometer 5"^ o'' 42™ 38'* (being oil 42M 388 P.M. at Greenwich), which was fast 4'" 47«"8 for mean noon at Greenwich, May 6th, and on June ist w&sfMt 6™ 50' for mean noon at Greenwich. 8s. 1882, August 13th, A.M. at ship, latitude 30° 46' S., observed altitude sun's l.l. 27° 15', inde.K Correction — 1' 15*, height of eye 21 feet, time by a chronometer 13'' 2'' o" (being 2'' o™ P.M. at Greenwich), which was slow 26"" 7'"6for mean noon at Greenwich, April loth, a«d on May ist was slow 25"" 13" for mean noon at Greenwich. 9. 1882, September ist, p.m. at ship, latitude 35*^ 49' N., observed altitude sun's l.l. 44° 32' lo", index correction -\- 1' 46", height of eye 20 feet, timo by chronometer August ji 578-4 for mean noon at Greenwich, July 3rd, and on July 3 ist w&sfast 12™ 17" for mean noon at Greenwich. 10. 1882, October 25th, p.m. at ship, latitude 51'^ 30' S., observed altitude sun's l.l. 40° 22', iude.v. correction — i' 50", eye 20 fe«t, time by chronometer 25'* 8'' 22" i" (or gh 22™ 19 p.m.), which was slow 24"" S^'2 for mean noon at Greenwich, June 14th, and on July 20th was slow 21™ 19* for mean noon at Greenwich. 1 1- 1882, November 27th, a.m. at ship, latitude 39*^ 20' S., observed altitude eun's l.l. 34° 37' J5*> index correction -J- »' 'S'j eye 18 feet, time by a chronometer 27<' 7'" 41™ 30" (being p.m. at Greenwich), which was fast 31"' 54^ for mean noon at Greenwich, October 2ott(, and on November 9th was 29'" ^q^ fast on mean noon at Greenwich. T2. 1882, December 25th, a.m. at ship, latitude 9' 59' fcj., observed altitude sun's l.l. 10"^ 38' 45", index correction — 3' 12", eye i8 feet, time by a chronometer 24'' 11^ 36"'- o» (being a.m. at Greenwich), which was slow 34" 198*1 for mean noon at Greenwich, July ist, and on July 29th was slow 38"' 39*"5 for mean noon at Greenwich. 13. 1882, January ist, p.m. at ship, latitude 38° 28' S., observed altitude sun's l.l 39° o', index correction — a' 25*, eye 12 feet, time by chronometer i"* 11^ 58™ 398 (being I'.M. at Greenwich), which was shw i^ 49™ 19* fbr mean nooii at Greenwich, September 12th, i88i, ivnd oa October i3tb was i** i2°> 53' slovf for mean noon at Grueuwich. 2 86 Longitude hy Chronometer. 14. 1882, February nth, a.m. at ship, latitude 53° 12' N., observed altitude sun's l.l. 12° 10', index correction — 49", eye 12 feet, time by chronometer io 31^ on mean time at Greenwich, August ist, and on Sept. 4th viaafast 30™ 6» for mean noon at Greenwich. 16. 1882, February 6th, p.m. at ship, latitude 6° 58' N., observed altitude sun's u.l. 21° 43' 40", index correction o", eye 18 feet, time by a chronometer iii^ 40™ 26' (being a.m. at Greenwich), which was slow 16™ 4''8 on mean noon at Greenwich, January 2nd, and on January 20th was slow 17™ 42* on mean noon at Greenwich. 17. 1882, May ist, P.M. at ship, latitude 21° 8' N., observed altitude sun's l.l. 28° 5' 30", index correction + 2' 50", height of eye 16 feet, time by a chronometer April 30'' iS^ 50™ i^^-^. (being 6^ ^o"' 29^*4 a.m. at Greenwich), which was 10"" 128 slow for mean noon at Greenwich, December 31st, 1881, and on February 17th, 1882, was 7'" 33^*6 slow for mean noon at Greenwich. 18. 1882, April 2ist, P.M. at ship, latitude at noon 0° 20' N., observed altitude sun's u.l. 32° 21' 10", index correction — i' 10", eye 12 feet, time by a chronometer 3^ 44™ i^ (being a.m. at Greenwich) which was slow 9"^ ']^ioT mean noon at Greenwich, November 14th, 1881, and on January nth, 1882, was slow 7™ 34^*2 for mean noon at Greenwich, course sincd noon S.W. by W. (true), distance 36 miles : required the longitude at the time of observation, and also at noon. 19. 1882, August 2ist, A.M. at ship, latitude at noon 0° 20' S., observed altitude sun's L.L. 33° 49', index correction -\- 2' 10", eye 15 feet, time by chronometer 8^ 14"! o^ (being P.M. at Greenwich), which was slow 4" 40^ for mean noon at Greenwich, March 13th, and on April 30th was slow 5"" 40* for mean noon at Greenwich, course till noon S.W. by W., distance 36 miles : required the longitude at time of sights, and also at noon. 20. 1882, March 20th, a.m. at ship, latitude o", observed altitude sun's l.l. 28° 50' 10", index correction + 1', eye 23 feet, time by chronometer 20 20™ 48' ; therefore there is no chronometer time to correct for error and rate, but from the Nautical Almanac, page II, the Decl. and Eq. T. is taken and corrected in the usual way. Greenwich date, August 28'^ 3'>20™48» Obs. alt. 0*8 l.l. 35° 4a' 40" Dip — 3 50 or 3»'-35 6,0)21" y"zs Oorr. of alt. Semi-diameter True alt. 35 38 50 I 11 35 + 37 38 15 53 35 53 31 Sumner^s Method. 291 Decl., page II, N.A. Aug. 28th, noon 9° 29' 58" N. deer. Corr. for 3'» 21™ — 2 58 H.D. 5 3"' 26 3"-35 Eq. T., page II, N.A. 2-47 H.D. 0-736 3-35 Bed. decl. Polar dist. 9 27 o N. 80 33 o 26630 Red.Eq.T. o 53"3i 15978 (\!o hQ added to K.T .) 15978 3680 2208 2208 6,0)17, 8"42io Corr. 2'4656o ^ 58 It will be further noticed that at the end of the question two assumed latitudes are given — in this case 49° o' N. and 50° N. For each lat,, with the true alt. and pol. dist. (as above) next proceed in the usual way to find the longitudes. True alt. Latitude Polar dist. 35° 53' 31" 49 80 33 165 26 31 sec. cosec o' 1 83057 0-005934 Altitude Latitude Polar dist. 35° 53' 31" 50 80 33 166 26 31 sec. cosec 0-191933 0005934 82 43 15 COS. 9-102790 . 83 13 15 COS. 9-072040 46 49 44 sine 9-862914 47 19 44 sine 9-866438 A.T. S. Aug. 2 Bq. time + 53 log. 9''54<595 A.T. S. Aug. Eq. time 2 85 3^438 + 53 log. 9" 1 36345 M.T. S. Au^. 2 M.T. G.Aug, a 82oh28™32» 24 o Time from noon, April 24^ 331 3 5 Decl., page II, N.A. H.D. Eq. T., page II, N.A. April 24^, noon 13" 5' o"N. iwcr. 49"-02 April 24^, noon 2>" o»-q Corr. for 3h-5 _ 252 3-5 Corr. for 3h -5 _ 1-6 H.D. 0-454 3'5 Red. decl. Polar dist. 13 90 8 N. 24510 Red. Eq. time i 59-3 2270 14706 (To be «Miif. from A.T.) 1362 76 57 52 6,0)17,1-570 1-5890 Obs. alt. O's L.L. Dip Corr. 30" 24' 40' — 4 4 30 20 36 — I 30 Semi-diameter True alt. 30 19 6 + 15 56 30 35 2 2 516 To find point A, latitude 49° 30' N. Altitude 30° 35' 2* Latitude 49 30 o sec. 0-187456 Polar dist. 76 57 52 cosec. 0-011338 157 2 54 78 31 27 COS. 9-298754 47 5^ 25 sine 9-870666 Hour-angle 24 o o App. T. ship, April 23^ 20 851 Eq. time — 1^9 log. 9-368214 800 gives 4" Altitude Latitude Polar dist M.T. ship, April 23d 20 6 52 M.T. Green., April 23d 20 28 32 Longitude in time 21 40 = 5° 25' W. (A) To find point A' in latitude 49'^ 50' N. 30° 35' Hour-aDgle 2" 49 50 o 76 57 52 157 22 54 78 41 27 48 6 25 3''5o'"3i' 24 sec. 0-190431 cosec. 0-011338 cos. 9-292484 sine 9-871802 log. 9-366055 App. T. ship, April 23'' 20 9 29 Eq. time — i 59 M.T. ship, April 23'' 20 7 30 M.T. Green., April 23d 20 28 32 Longitude in time 21 2 = 5» 15^' W. (A') 1^^ Sumner's Method. On Mercator's chart mark the point A in latitude 49° 30' N., long. 5° 25' W. ; also, mark the point A' in latitude 49° 50' N., long. 5° 15^' W. Join A and A' by a straight line and the ship will be on this line somewhere, it being the projection of the line of position of the ship, or as it is also sometimes called, the parallel of equal altitude of the ship. The line AA' being produosd until it meets the land it will be found to intersect the land near the Lizard Point ; and when referred to the compass on the chart it will be foun'l to bear N. by E. | E. (true). Consequently, although the exact position of the ship is not known, it is certain that by steering N. by E. ^ E. the ship will make the Lizard. For the Sun's True Atimuth. — From any point on AA' draw a perpendicular to AA' on the side of it the sun is known to be at the time of observation, and the sun's bearing will be found to be 8.E. \ E. (nearly). Problem II. 336. To find the latitude and longitude of a ship by circles of position projected on Mercator's chart. Having given two altitudes of the sun, the times of observation by chrono- meter (whose error for Greenwich mean time is known), also the sun's decli- nation at both times, and the latitude by dead reckoning, to find the true latitude and longitude of the ship. EULE XOVII. 1°. Find the Greenwich date of each observation by correcting the timst shown by chronometer for error and accumulated rate. 2°. Find the declination and polar distance for each of the Greenwich dates. 3". Find the equation of time corresponding to each Greenwich date. 4°. Correct each observed altitude for index error (if any), dip^ refraction^ pa/rallax, and semi-diameter. 5°. Assume two latitudes, differing about a degree or less, and embracing the supposed latitude of the ship ; thus it may be convenient to assume the latitudes from id to 30' on each side of the supposed latitude of the ship.* Compute the longitude (by the usual method) with each assumed latitude, first from one altitude and then with the other, as follows : — (a) With the ist altitude, the less assumed latitude, and the sun's polar distance, corresponding to the time of the first observation: call this A. (b) With the ist altitude, the greater assumed latitude, and the sun's polar distance, corresponding to the first observation; call this A'. (c) With the 2nd altitude, the less assumed latitude, and the sun's polar dis- tance, at the time of taking the second observation : call this B. (d) With the znd altitude, the greater assumed latitude, and the sun's polar distance, corresponding to the time of the second observation : call this B'. Thus, four longitudes will be found which may bo designated as A, A', B, and B' respectively, to facilitate reference to them. * It is not, however, essential that the same assumed latitudes should be used in com- puting both lines of position ; it is only more convenient to do so, as it saves some logarithms. In the case of two altitudes of the same object, as of the sun in this case, where a course and distance have been made in the interval, if the course has been nearly North or South, it would be bebtor to assume two latitudes diflferiug from those used for the first observation, and such that they may be more in accordance with the altered position of the ship. Sumner's Method. 295 7°. On Mercator's chart project A A', the ship's line of position, or parallel of equal altitude, corresponding to the frst altitude. Thus, (a) On the less assumed parallel of latitude marh the longitude A, and in the greater assumed parallel of latitude mark the longitude A'. (b) Join the points A and A' by a straight litie, which will be the projection of the ship's line of position at the first observation, and on or near this or this produced the ship is situated. 8°. Project, in exactly the same manner, B B', the ship's line of position, corresponding to the second altitude. That is, (a) On the less assumed latitude marlc the spot corresponding to longitude B, and on the greater assumed latitude mark the corresponding longitude B'. (b) And the line joining the points B B', will he a second ship's line of posi- tion, and on or near this or this produced the ship is also situated. The position of the ship is at the intersection of the two lines thus produced,* from which its latitude and longitude may be easily found. Or proceed thus : — 9°, On the less assumed latitude marh the longitudes A and B ; on the greater assumed latitude mark the longitudes A' and B' ; draw a line through the points A and A', cutting both parallels ; also draw another line through the points B and B' : the position of the ship is at the intersection of these lines. The following example will illustrate the last paragraph of the rule : — Ex. On the ist day of August, 1880, at about 10'' apparent time at ship, an altitude of the sun was observed, the latitude by dead reckoning being 50° 16' North. The longitudes an ascertained by working with the assumed latitudes 50° N. and 51° N. are as follows : — Longitude corresponding to 50° N. is 7° 5' W. (A). 51° N. is 8' 15' W. (A'). Proceed as in the preceding rule to project the ship's line of position. Mark on the chart the spot (A) corresponding to latitude 50° N. and longitude 7° 5' W. (see plate). Again, mark the spot corresponding to latitude 51° N. and longitude 8'^ 15' W. on the chart (A'). Draw a line connecting these two spots, A A'. At 3'' P.M. of the same day an altitude of the sun was observed, the ship having remained stationary between the observations, workinrj with the same assumed latitudes. Longitude corresponding to 50° N. is 8° 30' W. (B). 5i°N.is5°i5'W. (B'). Project, in exactly the same manner as above, the ship's line of position corresponding to the second altitude by marking off on the parallel of 50" N., the longitude of B = 8^ 30' W., and the longitude B B' = 5° 15' W. on the parallel of 51° N. Connect the two points B and B' by a straight line. Then A A' in the diagram (see plate) shows the two longitudes corresponding to ihefirit observation, and B B' the longitudes corresponding to the second observation. The intersection of these lines at shows the position of the ship. The latitude is 50° 19' N. instead of 50' 14' N,, as by dead reckoning ; the longitude la 7" 27I' W. The line B B' prolonged until it meets the land, gives the bearing of Lundy Island N.E. by E. I E. (true). * If the lines of position when computed and projected intersect considerably beyond one or other of the assumptions, then take another latitud ^ a little beyond that of the inter- secting point, compute anew for this, and so project again. The position wiU be more accurately determined in this manner. 296 Sumner^ s Method. 10°. If the ship has changed her station between the observations. — Set off from any part of the line AA' the distance sailed between the observations in the direction of the true course made good. Through the point thus found draw a parallel to AA', and produce it till it meets BB' ; and the intersection of its new position will give the place of the ship at the second observation. Note. — When the ship is steering a straight course, the line indicating the track can, from any point in AA', be drawn, and the distance laid off, on the completion of the second observation. This evidently accomplishes the same result as correcting the altitudes. It possesses the advantage of being simple, and when the chart has the magnetic compass plotted upon it, the compass course can be laid off between the observations. The con- venience of allowing in this manner for the run of the ship in the interval is worthy of special attention, as one-half the computation is performed without waiting until the second observation is taken. The following examples will illustrate this part of the rule : — Examples. Ex. I. Suppose in the forenoon (see preceding example) an altitude of the sun was observed, the latitude by dead reckoning being 50° 14' F. The longitudes, as ascertained by working with the assumed latitudes 50° N. and 51° N., are as follows: — Longitude corresponding to 50° N. is 7° 5' W. (A). 51° N. is 5° 15' W. (A'). Tn the afternoon of the same day a second altitude of the sun was observed, the ship in the interval having sailed N.E. (true) 20 miles. Working with the same assumed latitudes as before, the Longitude corresponding to 50° N. is 8° 30' W. (B). 51° N. is 5° 15' W. (B'). Project A A', the s'lip's line of position, or line of equal altitudes, corresponding to the first altitude. The ship having moved in the interval, the projection of the first line of equnl altitudes, must now be moved parallel to itself in the direction of and through the distance of the run. Set off from any part of the line A A', 20 miles due N.E. Draw- through this point a line a a! parallel to A A', this will be the line of equal altitude at the time of the first observation, corrected for change of station. (See Plate.) Next project B B', the line of equal altitude corresponding to the second observation. The intersection at C of the line B B' with a a' or that corresponding to the first observation as corrected for change of station, shows the position of the ship. The latitude is 50° 28' N., instead of 50° 14' N., as by dead reckoning ; the longitude is 6° 58' W. It will be evident that the second observation will not give the same altitude as if the ship had remained st^ionary, but will give a line of position or line of equal altitude 20 geo- graphical miles further to N.E. Be carefal not to confound these with longitudinal miles, as in this lat. 20 geographical miles equal 31 miles of longitude. Ex. 2. It is required to project two observations, showing by the intersection of lines of position the place of the ship, supposing her to have sailed in the interval between the first and second observations 13 miles N.W. \ W. (true). The longitudes at the first observation, as ascertained by working with the assumed lati- tudes 49° N. and 50° N., were as follows: — Longitude corresponding to latitude 49° N. is 4° 33^' W. (A). 5°° N. is 3» 49' W. (A'.) Sumner's Method. 297 Mark on the chart the spot A in latitude 49° N. and long. 4° 33^' W., and also the point A' in lat. 50° N. and loniij 3° 49' W. Connect these two points A and A' by a straight lino, this line lies N.E. by N. and S.W. by S. and proJuced to meet the land it will be found to cut the Eddystone, bearing N.E. by N. Next project this first line of position for the run of the ship, by setting off from any point in A A' the distance, 13 miles N.W. \ W. (true), and draw a line a a through this point parallel to A A'; this will bo the line of position at the time of the first observation, corrected for change of station. After a suitable change of bearing of the sun an altitade of the sun is observed, and the longitudes as ascertained by working with the same assumed latitudes, as before, were as follows : — Longitude corresponding to 49° N. is 3° 42^' W. (B). 50°N.is5°35'W. (B'). Mark the point B on the chart in lat. 49° N. and long. 3° 42^' W., also the point B' in lat. 50° N. and long. 5° 35' "W. Connect thrse two points B and B' by a straight line ; this line is the ship's line of position at the second observation, and its intorsection at C of the line a a' the position of the ship at the second observation, in lat. 49° 27^' N., long. 4° 33^' W. The line of position at the second observation, prolonged to meet the land, cuts the Land's End, bearing N.W. i W. Ex. 3. It is required to project two observations of tlie sun, showing by the intersection of their lines of position the place of the ship, supposing the ship to have sailed N.E. by E. ^ E. (true), 19 miles, in the interval between the observations. The longitudes at the first observation, by working with the assumed latitudes 49° 40' N. and 50° o' N., were as follows : — Longitude corresponding to latitude 49° 40' N. is 177° 56I' E. (A). „ n „ 50° o'N. is 179° 27i'W. (A'.) Working with the same assumed latitudes, the corresponding longitudes at the second observation were as follows : — Longitude corresponding to lat. 49° 40' N. is 179° 48^' W. (B). 50° o' N. is 179° 48' W. (B'). Mark on the chart the spot corresponding to A, lat. 49" 40' N., long. 177° 56|-' E. ; also the point A', lat. 50' o' N., long. 179° 27^^' W. Connect these two points by a straight line ; this is the ship's line of position at the first observation. Set off from the line A A' 19 miles in the direction of the course, N.E. by E. | E. (true), and draw a line a a' through this point parallel to AA', which will be a projection of the line of position at the first observation for the run of the ship. Again, mark the spot B in lat. 49° 40' N., long. 179" 48^ W., and the spot B' in lat. 50' o'N., long. 179° 48' W. ; these two points being joined by u straight line are a projection of the ship's line of position at the second observation, and the intersection of this line B B' with the line a a (or the line of position at the first observation corrected for change of station) gives the place of the ship at the second observation, lat. C 50° 4' N., long. 179'^ 48' W. The first line of position trends E. by N. and W. by S., and the sun's true bearing is S. by E. We shall now proceed to work out in full detail an example showing how to determine the ship's position by the intersection of lines of position corres- ponding to two altitudes of the sun, having given the ship's course and dis- tance in the interval. Ex. 4. 1880, February 29th, a.m. at ship, and uncertain of my position, when the chron. showed February 28'* 23^ 16"" 55* M.T.G. ; suppose the obs. alt. sun's l.l. 29° 46' 50", and again p.m. same day, when the chronometer showed I'ebruary 29'^ 4^ 13" 50^, the obs. alt. 298 Sumner^ s Method. Bun's L.L. 15' 47' 10% the ship having made 36 miles on a true E. by S. course, height of eye 20 feet : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was taken. Suppose the ship's position at the time of taking the second altitude to be as follows : — B = lat. 49= o' N., long. 5° 30' W. ; B' = lat. 50° o' N., long. 6° 30^' W. First Observation. Green. d:ite, Feb. 2i^2;^'^i6'"S5^ '1 imo from noon, Space for minor corrections. coseo. 97042' 0"= 0-003934 Pts. for 50 = + 14 97 42 50 0-003948 Tab. diff. = 28 50 Corr. 14,00 58° 22' 0"= 9-930145 Pts. for 49 = + 64 29° o 44 oh-73 Decl., page II, N.A. H.D. Eq. T., page II, N.A. H.D. Feb. 29'^, noon 7" 42' f)" '6. deer. 56"-8S Feb. 29'^, 12"' 37'*-7«?«c/- 0-490 Corr. for o''-7 3 -f- o 41 0-73 Corr. for o'' -7 + 0-3 "j Red. decl. 7 42 50 S. Polar dist. 97 42 50 17064 Red. Eq. time 12 38-0 0^-3430 39816 (To be added to A.T.) 58 22 49 diff 9-930209 Tab. 130 - 49 1170 520 Corr. 63,70 Bine. S8°52 Pte. for 0"= = 9'932457 49 = = + 62 = 9"9325i9 88 52 49 Tab. diflf 127 49 1143" 508 Corr 62,23 Correction of observed altitude. Obs. alt. O's L.L. 29' 46' 50' Dip — 4 17 Corr. Semi-diameter True alt. 29 42 33 — I 32 29 41 I + 16 10 41-5224 Computation of time and lon2:itude, with assumed latitude 49'' N. Altitude 29° 57' 11" Latitude 49 o o sec. 0*183057 Polar dist. 97 42 50 cosec. 0-003948 COS. 8-463665 29 57 Hour-angle ih-jo™ 3' log. Altitude Latitude Polar dist. App. T. ship, Feb. 2 Eq. time M.T. ship, Feb. 28'! M.T. Green., Feb. 2S Longitude in time Long. A 8° 35' W. With assumed latitude 50° N. 29° 57' 11" 50 o o sec. 0-191933 97 42 50 coseo. 0-003948 76 40 I 88 20 58 22 49 1^ 30m 3. 22 + 29 57 12 38 22 23 42 35 16 55 34 20 8-580879 47 41 gives 3' 177 40 I cos. sine 88 50 8-308794 58 52 49 9932519 Hour-angle i''i6'"ii« log. 8-437194 10 App. T. ship, Feb. 28'' 22 43 49 Eq. time + 12 38 9 = M.T. ship, Feb. 28^ 22 56 27 M.T. Green., Feb. 28d 23 16 55 Longitude in time 2028 Long. A' 5° 7'W. Sumner^ s Method. 299 Project A A', the ship's line of position, or line of equal altitudes, corresponding to the Jirst altitude. Next, project B B', the line of equal altitudes corresponding to the second observation. The ship havinpf moved in the interval between the observations, the projec- tion of the first line of equal altitudes must now bo moved parallel to itself in the direction of and through the distance of the run. Set off from any part of A A', 20 miles due N.E., as follows : — From the graduated meridian opposite the line extending from A to A' take off the dis- tance 20 miles with the dividers ; lay tho dividers down, and taking the parallel rules, place them on the compass in the corner of the chart, over N.E. ani S.W. ; then work the parallels (strictly preserving the direction) towards the line of the first position A to A' ; having reached that line, draw another extending from any part of it in the direction of the course N.E., and on this last line lay off the 20 miles of distance already taken with your dividers. Lay the edge of the parallels on the line AA', and working them (preserving the direction) to the point just laid off; driw through this point another lino a a' parallel to A A'. The intersection at C of the line B B' with a a', or that of the first observation as corrected for change of station, shows the position of the ship. The latitude is 50° 28' N., the longitude is 6^^ 58' W. Examples for Practice. 1. 1882. If at sea, March ist, a.m. at ship, and uncertain of my position, when the chronometer showed February 28'' 10'' 50'" 54" M.T.G. ; suppose the obs. alt. sun's l.l. 29° 46' 50": and again, p.m. on the same day, when the chronometer showed February 28"^ i5i> 50™ 49» M.T.G., obs. alt. sun's l.l. 15' 47' 10", the ship having made 32 miles on a true E. by N. course, height of eye 19 feet. Kequired the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was taken, assuming the ship to be between latitude 49° 10' N. and 49° 40' N. On looking at the question it will be noticed that there is a second Greenwich date mean time; also a second observed altitude; in this case February 28"^ 15*1 50"" 49" M.T.G., and obs. alt. 15° 47' 10" ; these are the basis of the two positions which will be furnished by the Examiners, and respecting which tho circular says — " Candidates will not be for the present obliged to perform the calculations."* Suppose the ship's calculated position at the time of taking the second altitude to he as follows : — B = Lat. 49° 10' N. Long. 179° 50' W. B' = Lat. 49° 40' N. Long. 179° 35' E. 2. 1882, May 19th, A.M. at ship and uncertain of my position, when the chronometer showed May 18^ 22h i" 39' M.T.G., obs. alt. sun's l.l. 48° 57', and again, p.m. the same day, when the chronometer showed May 19'' i^ 37™ 18' M.T.G. , obs. alt. sun's l.l. 40° 45' 15" the ship having made between the observations 25 miles on a true N. 45° E. course, height of eye 18 feet : required the lino of bearing when the first altitmle was taken, and the posi- tion of the ship by Sumner's Method when tho second altitude was observed, assuming lati- tudes 48° 30' N. and 49° o' N. Suppose the ship's calculated position at the time of taking the second altitude to be as follows: — B = Lat. 48° 30' N. Long. 5" 18^' W. B'=: Lat. 49° o' N. Long. 5° 33' W. 3. 1882. If at sea, November 8th, a.m. at ship, and uncertain of my position, when the chronometer showed November 7'' 22*' 29"" 2' ; suppose obs. alt. sun's l.l. 19° 49' ; and ao-ain P.M. on the same day, when the chronometer showed November S"" 3'' 48™ 57*^ M.T.G., obs. alt. sun's L.L. 10° 52' 30", the ship having made 41 miles on a true E. ^ N. course in the * For the sake of practice it may be advisable to verify the positions of B and B'. 300 Sumner^ s Method. interval, height of eye 19 feet : reqiiirod the line of hearing when the first altitude was taken, assuming the ship to be between lats. 48° 10' N. and 48° 40' N. Suppose the ship's calculated position at the time of taking the second altitude to be as follows : — B = Lat. 48° 10' N. Long. lo" 33I' W. B' = Lat. 48° 40' N. Long. 11° 13^' W. 4. 1882, February 3rd, a.m. at ship, and uncertain of my position, when the chronometer showed February 2^ 22'' 17" i6« M.T.G., obs. alt. sun's l.l. 19° 49' 50"; and again, p.m. ■same day, when the chronometer showed February 3'* 3^ 37"" 11^ M.T.G., obs. alt. sun's l.l. 10" 53' 30", the ship having made 39 miles on a true E. by N. ^ N. course in the interval, height of eye 19 feet : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming the ship to be between lats. 48° 10' N. and 48° 40' N. Suppose the ship's calculated position at the time of taking the second altitude to be as follows ; — B = Lat. 48" 10' N. Long. 0° 21J' E. B' = Lat. 48^ 40' N. Long. 0° 17I' W. 5. 1882, January 24th, a.m. at ship, and uncertain of my position, when the chronometer showed January 23'' 22*> 19"" i« M.T.G., obs. alt. sun's l.l. 9° 40' 15'; and again, p.m. same day, when the chronometer showed January 24'^ 2^ 19°' 5^ M.T.G., obs. alt. sun's l.l. 18° 20' lo", the ship having sailed 18 miles on a (trun) S.E. by E. ^ E. course in the interval between the observations, height of eye 21 feet, required the time of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was taken, assuming the ship to be between latitudes 51° 15' N. and 50° 45' N. Suppose the ship's position (by calculation) at the time of second observation to be as follows : — B = Lat. 51° 15' N. Long. 17" 16J' W. B' ^ Lat. 50° 45' N. Long. 14° 17J' W. 6. 1882, March 27th, a.m. at ship, and uncertain of my position, when chronometer showed March 26^ lo'' 26™ 24^ G.M.T., obs. alt. sun's l.l. 36'^ 38' 30" ; again, p.m. same day, when chronometer showed March 36d i6'> 37'" io», obs. alt. sun's l.l. 15° 15' 20", height of eye 22 feet, course and distance sailed in tho interval between the observations S. by E. f E. (true), distance 37 miles : required the line of bearing when the first altitude was taken, and fhe position of the ship by Sumner's ' Tethod when the second altitude was observed, assuming the ship to be between latitudes 51° 40' N. and 51° 10' N. Suppose the ship's position (by calculation) at the time of second observation to be as follows: — B = Lat. 51° 40' N. Long. 179° 45' W. B' = Lat. 51° 10' N. Long. 178° 31^' W. 7. 1882, March ist, a.m. at ship, and uncertain of my position, when the chronometer showed February 28'^ 22'' 48"^ 15^ M.T.G., the obs. alt. sun's l,l. 29° 44' 30" ; and again, p.m. same day, when the chronometer showed March i"* 3'' 46" 50^ M.T.G., obs. alt. sun's l.l. 15° 50' 10", the ship having made 42 miles on a true E. by N. ^ N. course in the interval, height of eye 15 feet : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 49° 15' N. and 49° 45' N. ' Suppose the ship's jiosition (by calculation) at the time of taking the second altitude to be as follows: — B = Lat. 49° 15' N. Long. 1° 9' E. B' = Lat. 49° 45' N, Long, o'' 39I' E. Sumner's Method. 8. 1882, February 3rd, a.m. at ship, being uncertain of my position, when the chrono- meter showed February 2^ 16'' 33"" 36^ M.T.G., obs. alt. sun's l.l. 20° i' 23"; and again, P.M. on the same day, the chronometer showed February 2^ 21*1 55™ 54% obs. alt. sun's l.l. 1 1° 10' 30*, the ship having made 40 miles on a true E.N.E. course in the interval, height of eye 19 feet : required the line of bearing when the first altitude was taken, and the posi- tion of the ship by Sumner's Method when the second altitude was observed, assuming the latitudes 47° 10' N. and 47° 40' N. Suppose the ship's position (by calculation) at the time of taking the second altitude to bo as follows: — B =: Lat. 47° 10' N. Long. 86° 14!' E. B' := Lat. 47° 40' N. Long. 85° 37^' E. 9. 1882, January 24th, a.m. at ship, being uncertain of my position, when the chrono- meter showed January 24'* 9'' 14"" 44', obs. alt. sun's l.l. 9° 40' 10"; and again, p.m. same day, when the chronometer showed January 24"^ 13^ 14"" 30^ M.T.G., obs. alt. sun's l.l. 18° 31' 15', the ship having made 23 miles on a true S.E. ^ S. course in the interval, height of eye 22 feet: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming the ship to be between the latitudes 51° 15' N. and 50° 45' N. S'appose that by calculation the ship's position at the time of the second observation is as follows : — B = Lat. 51° 15'N. Long. 178° 26' E. B':= Lat. 50° 45' N. Long. 178° 40^' W. 10. 1882, March 2i8t, a.m. at ship, being uncertain of my position, when the chronometer showed March 2o<' 2oi> 38" 58^ obs. alt. sun's l.l. 22° 49' 50" ; again, p.m. same day, when chronometer showed March 21^ 2*' iS" 50* M.T.G., obs. alt. sun's l.l. 32° 29' 50", the ship having made 29 miles on a true N. 52*^ W. course in the interval, height of eye 21 feet: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming the ship to be between the latitudes 50° 10' N. and 50° 50' N. Suppose that by calculation the ship's position at the time of the second observation is as follows : — B = Lat. 50° 10' N. Long. 0° 26^' E. B' = Lat. 50" 50' N. Long, o" 48^' W. 11. 1882, March 15th, a.m. at ship, being uncertain of my position, when chronometer showed March i4> 7 15 55 — „ 12 14 38 S. 51 2 N. 6. ,, 13 47 28 — 17 50 57 N. 42 36 S. 7- ») 45 30 — „ 23 2 S. ,, 00 8. J) 25 40 10 -j- ,, 000 ,, 00 9- >j 40 7 21 — 17 4 3 S. 33 51 S. 345. Q-iven the true bearing and compass bearing, to find the error of the compass. EULE XCIX. 1°. To find the amount of the Error of the Compass. — Reckon the True and Magnetic Azimtiths from the same point of the compass — North or South. Caution. — Be careful when taking the truB bearing from 180°, not to change the East or West name ; only the North or South. (a) If one of the azimuths be expressed from the North and the other /rom the South, take either of them from 1 80°, and it will then be reckoned /ro/w the same point as the other. (b) If the bearing by compass be reckoned from East or West, towards North or South, take it from 90°, «w^ reverse the position of the letters; or, add 90°, and it will then be expressed from the opposite point to that from which it is reckoned when taken from 90°. Example. Ex. Suppose magnetic azimuth to be W. 18° 30' N, ; then subtract the magnetic azimuth from 90° thus: — Or add 90*^ to the magnetic azimuth, thua: — 90 o W. 18 30 N. N. 71 30 W. The azimutli is thus reckoned from the North Pole. W. 18'30'N. 90 o S. 108 30 W. The azimuth is thus reckoned from the South Pole. (c) When the magnetic azimuth is either East or West, it is to be reckoned as go'^from North or South, according as the true azimuth is North or South. 2°. Take the difference of the true and magnetic azimuths when measured towards the same point of the compass, East or West ; but lohen measured towards different pnnts, i.e., ivhen one is reckoned towards East and the other towards West, take the sum ; the result is the error of the compass or correction. 3°. To name the Correction of Compass. — Let the observer look at the two azimuths for bearings] from the centre of the compass — then if the true azimuth is to the right of the magnetic azimuth^ the correction is East ; but if the true azimuth is to the left of the magnetic azimuth, the error is West. Variation hy an Azimuth. 307 Examples. Ex. I. Given true azimuth N. 44° 20' E. and the sun's bearing by compass (or mag- netic azimuth) N. 17" 10' E. : required the error of compass. True az. N. 44°2o'E. n m.a. Mag. az. N. 17 10 E. Error 17 10 E. Tho observer being supposed looking from the centre of the compa.ss in the direction qJ', the maenotic azimuth, then the true azimuth lies to the right hand of the magnetic azimuth, whence the error of compass is to be marked East. Ex. 3. Given true azimuth kS. 69° W., magnetic azimuth S. 47° W. : required the error of compass. True az. Mag. az. Error S. 69=- W. S. 47 W. 22 E. The observer being supposed looking from the centre of the compass in the direction of tho magnetic azimuth C M, then the true azimuth, T, lies to the right hand of the magnetic azimuth, whence the error of the com- pass is East. Ex. 5. The true azimuth S. 62° 41' E., and magnetic azimuth E.y.E.: required the error of compass. True az. S. 62' 41' E. S. 6 pts. E. = Mag. az. S. 67 30 E. Error of compass 4 49 E. Here the error of compass is East, since the true fiximtith is on the right of tho magnetic azimuth, thi' observer looking fiom the centre of the compass in the direction of tho magnetic azimuth. Ex .7. TrueazimuthN.72''E., magnetic azimuth East. True aaimuth N. 72" E. Mag. azimuth East = N. 90 E. Error of compass 18 W. Ex. 9. The true azimuth S. 90' 33' E., and magnetic azimuth N. 81° 20' E. : find the error ot compass. True azimuth IS. 90° 33' E. 180 o or N. 89 27 E. Mag. azimuth N. 8i 20 E. Error of compass 8 7 E. The t7-ue azimuth beina; reckoned from S., while the magnetic azimuth is expressed if dm N., the true is subtracted frtnn 180°, in order to rei kuii it Ironi the same point as the magnetic iizimutb, viz., from N. Ex. 2. Given true azimuth S. 70=' 57' E. the magnetic azimuth S.E. by E. f E. : required the error of compass. Mag. az. S.E. by E. | E. = S. 64-^ 41' 15" E. True az. S. 70*57' o"E. m.a. s Mag. az. S. 64 41 15 E. x | Error 6 15 45W. E^ The error of compass is in this instance West, because when I looking from the centre of the ■•■ compass in the direction of the ** magnetic azimuth, the true azimuth is on the left hand of the magnetic. Ex. 4. True azimuth N. 50° 12' E., and the magnetic azimuth N. 61" 50' E. : re- quired the correction of compass. True azimuth N. 50' 1 2' E. Magnetic azimuth N. 61 50 E. Error of compass 11 38 W. The error of compass is here West, because the true azimuth is to the Jcft hand of the magnetic azimuth, tho observer being supjiosed to look from the ceutro of the compass in the direction of the magnetic azimuth. Ex. 6. The true azimuth S. 82'^ 50' W., and magnetic azimuth W. 15'^ N. Trueaz. S. 82=50' W. W. 15'' N. =. Mag. az. S. 105 W. Error of compass 22 10 W. The error of compass is West, the triic azimuth being to the left of maqnetic, 90° is added to the compass bearing in order to reckon it from the same point as the true azimuth ; ihus, from S. to W. is 90°, and from W. to W. 15° N. is 15° more; hence mag- netic azimuth is S. 105° W. Ex. 8. Tho true azimuth is S. 76-^ W., and the magnetic azimuth West. True azimuth S. 76" o' W, Slag, azimuth S. 90 o W. Error of compass 14 o W. The magnetic azimuth West is reckoned as 90° from S., because the true azimuth is reckoned from S. Ex. 10. The true azimuth N. 69° 39' W., and magnetic azimuth 8. 93*^ 30' VV. : find tho error of compass. True azimuth N. 69° 39' W. 180 o or IS. no 21 W. azimuth S. 93 30 W. Error of compass 16 51 E. Tlie true azimuth is here taken from i8o°,inoi-.Ier to reckon it fiom the same point as the magnetic azimuth. 3o8 Variation hy an Azimuth. Ex. II. True azimuth S. 36° W., mag- netic azimuth S. g'^ E. True azimuth S. 36° W. Mag. azimuth S. 9 E. Error of compass 45 E. Ex. 1 3. True azimuth N. 49° E., mag- netic azimuth N. 3° W. True azimuth N. 49° E. Mag. azimuth -N. 3 W. Error of compass 52 E. Ex. 12. True azimuth N. 68° "W., mag- netic azimuth N. 5° E. True azimuth N. 68° W. Mag. azimuth N. 5 E. Error of compass 7 3 W. Ex. 14. True azimuth S. 50° E., mag- netic azimuth S. 8^ W. True azimuth S. 50° E. Mag. azimuth S. 8 W. Error of compass 58 W. RULE C. 1°. To find Greenwich Date. — With ship time and longitude in time find the Greenwich date (Eule LXXYIII, page 227). 2°. To find Polar Distance. — Take from page II, Nautical Almanac, the sun^s declination and reduce it to Greenwich date (Rule LXXIX, page 2 30) ; also tahe out surHs semi- diameter . If apparent time is given, use Nautical Almanac, page I. 3°. To find True Altitude. — Correct observed altitude for index error, dip, refraction, parallax, and semi-diameter, and thus get the true altitude (Rule LXXXIY, page 242). 4°. To find True Azimuth. — Proceed according to Rule XCVIII, page 304, to find the true azimuth. 5°. To find Error of Compass. — Having found the true azimuth, proceed hy Rule XCIX, page 306, to find the entire correction or error of the compass. 6°. To find Deviation. —Next place the variation behw the error of compass and proceed as in the amplitude (7° and 8° of Rule LXXXVI, page 245), the result is the deviation for the position of the ship's head at the time of obgervation. Examples. Ex. I. 1882, May 19th, 3'' 7™ 44^ p.m., mean time at ship, latitude 41° 53' N.j longitude 60° 19' W., sun's hearing by compass S. 104° 40' W., observed altitude sun's l.l. 43° 56' 7", height of eye 1 8 feet, index correction o' : required the true azimuth and error of the com- pass ; and supposing the variation to be 17° 10' W. : required the deviation of the compass for the position of the ship's head at the time of observation. Ship date (M.T.) May ig^ ^^ 7'"44' Long. 60° 19' W. in time -+-4 116 Green, date (M.T.) May 19790 By Baper : Dip — 4' 10", rofr. — i' i", par. -\- 6", semid. -\- 15' 50". True alt. 44" 6' 54". Obs. alt. O's L.L. 43=56' 7" Dip — 44 Corr. altitude Semi-diameter True altitude 43 52 3 53 43 51 + 15 10 50 44 7 o Variation hy 48™22s Watch /««< — o 29 16 Diff. long. 3^:4^14 60 Obs. alt. 0'8 L.L. Index correction"' Dip for 20 feet Correction of alt. Semi-diameter True altitude 2 19 6 — - 2 10 2 16 56 60° 20' 30 + 2 5« 60 n 28 4 17 60 19 II — o 28 60 18 43 + 16 18 60 35 I App. time at ship, Jan. z^ d'^iS^^6* Long, in time -4" 4 45 3^ Green, date (A.T.), Jan. 2 5 228 Decl., page I, N. A., January and, at noon 22° 54' 20" S. (decreasing) . H. diflf., Jan. 2nd, noon i3''-8o Greenwich date 5'' 2"", or X 5 6,o)6,9*oo Correction — i' 9" Decl., Jan. 2nd, noon 22° 54' 20' S. dev. Correction — 19 Bed. decl. 22 53 II S. By Rapgr : Index corr. + 2' 58", dip — 4' 20", refr. — o' 34", par. + 4", semid. -\- 16' 18", true alt. 60° 45' 53'. Method I. Time from noon i6'^56^ rising* 3-435880 Latitude- 52° 6' cos. 9-788370 Declination 22 53 cos. 9-964400 1544 nat.no. 3' [88650 1544 T. alt. 60^35' i" nat. sine 871073 Z. diet. 29 14 9 S. nat. cos. 872617 Decl. 22 53 ir S. (next greater) 638 Lat. 52 7 20 S. 239)2100(9" • The index of log. rising is increased by 1. See Note to 7°, page 314. Method II. Constant log. Lat. by D.K. 52° 6' S. Declination 22 53 S. Mer. zen. dist. 29 13 Time from noon i6"'56' Reduction True altitude Meridian alt. Zenith distance Declination 5"6i546 cos. 9-78837 COS. 9-96440 cosec. 0-3 1 148 log. 7 13486 29 14 7 S. 22 53 II S. Latitude 18 S. 6,0)65,2 log. 2-81457 + 10' 52" 60 35 I 60 45 53 90 By Baper : Lat. 52° 7' 7^" S. Method III. — By Towson's E»-Meridian Tables. 0*8 red. declination Aug. Table i, Index 30 22°53' ii^'S. + 3 21 Augmented declination 22 56 32 S. This is the method required of Candidates at Liverpool. True altitude Aug. Table 2, Index 30 Meridian zen. dist. Declination 60" 35' I" S. + 14 4 60 49 5 29 10 55 S. aa 56 32 S. latitude 52 7 27 S. Reduction to Meridian. 3'7 Ex. 2. 1882, February 6th, a.m. at ship, lat. acct. 51° 58' N., long. 105° 41' W., obs. alt. sun's L.L. South of observer 22° 10' 30", index corr. -\- 56", height of eye 22 feet, time by watch 6'! oh 4™ 4% found to be aS" 4-]' fast on app. time at ship, difF. of long, mado to East 298 miles since error of watch on app. time al ship was determined : required the latitude by reduction to meridian. Time by watch, Feb. "Watch /a*i; Diff. long. ^9-8x4 60 App. time at ship, Feb. Time from noon, Feb. Obs. alt. 0*8 L.L. Index correction Dip Corr. of alt. Semi-diameter True altitude gd oh 4m 48 — 28 47 5 23 35 17 + I 59 5 23 37 16 24 6 22 44 22» + 10' 30" 56 22 II 26 4 30 [22 6 56 2 II 22 4 45 16 15 App. time at ship, Feb. Long. 105° 41' "W. Greenwich date, Feb. Hourly diff. 6h 40™ = jdjjhj^mjfi* + 7 2 44 6 6 40 o or, 6'' 67 — 46" '02 X 6-67 32214 27612 27612 6,0)30,6-9534 5' T Declination, page I, N.A. Feb. 6th, noon, 15° 43' 7" S. deor. Correction — 5 7 Bed. decl. 15 38 Method I. 260 Time from noon 22"i44' rising 3-689030 Latitude 5'° 50' cos. 9-789665 Declination 15 28 cos. 9-983981 2919 nat. no. 3-465276 1919 True altitude 22^21' o" nat. sine 380263 Zen. distance 67 28 9 N. nat. cos. 383182 Declination 15 27 46 S. Latitude 52 o 23 N. The nat. sine being worked to six places of figures, I is added to index of log. rising. Constant log. Latitude Declination Mer. zen. dist. T, from noon Method II. 5'°58'N. 15 28 S. 67 26 22™44« cos. cos. cosec log. 5 "6 1 5455 9-789665 9-983981 0-034594 7-390540 6,0)65,2 log. 2-8 14235 Reduction True altitude Mer. altitude Zenith distance Declination + ^o'52' 22° 21 N. S. 22 31 52 67 28 8 15 27 46 Latitude 52 22 N. Method III.— £^ Towson's Ex- Meridian Tables. O's red. declination 15'* 27' 46" S. Aug. Table i, Index 57 -f- 4 23 Augmented declination 15 32 9 True nltitude 22*21' o* Aug. Table 2, Index 57 + 6 35 22 27 35 Meridian zen. dist. 67 32 25 N. Augmented declination '5 31 9 S. Latitude 52 o 16 N« 3.8 Reduction to Meridian. Ex. 3. 1882, August 7th, A.M. at ship, lat. acct. 40° 52' N., long. 36" 47' W., obs. alt. Bun's L.L. South of observer 65° i', index corr. ■\- 17*, eye 14 feet, time by watch 11'' 15'" 46% found to be 26™ i6» slow of app. time at ship, the diff. of long, made to East was 17 miles after the error on app. time at ship was determined : required the latitude. Time by watch, Aug. "Watch alow 6d2 3'ii5™46" -f 26 16 Diff. of long. 6 23 42 + I 2 8 App, time at ship, Aug. 6 23 43 10 24 Time from noon, Aug. 7 16 SO Obs. alt. O's L.L. 65° i' 0" Index correction + 17 Dip 65 I 17 - 3 36 Corr. altitude 64 57 41 — 23 Semi-diameter 64 57 18 + 15 49 True altitude 65 13 7 Method I. Time from noon 1 6'"5o» rising 3-430750 Latitude 40°52' cos. 9-878656 Declination 16 22 cos. 9 982035 1956 nat. no. 3-291441 nat. no. 1956 True altitude 65* 13' 7" nat. sine 907913 Zen. distance 24 30 46 N. nat, cos. 909869 Declination 16 22 14 N. Latitude 40 53 o N. App. time at ship, Aug. 6d 23*^4 3'»io« Long. 36° 47' "W. -|- 2 27 S Greenwich date, Aug. 7 2 10 18 By Raper : True altitude 65° 1 3' 3" H.D., 7th noon, 42"-07 X 2-2 8414 8414 6,0)9,2-554 i' 33" Decl., Aug. 7 th Correction page I, 16° N.A. 23'47"N. deer I 33 Red. decl. 16 22 14 N. Method II. Constant log. Latitude acct. 4o°52'N. cos. Declination 16 22 N. cos. Mer. zen. dist. 24 30 cosec. Time from noon 1 6'"50« log. 5"6i54^5 9-878656 9-982035 0-382273 7-129720 6,0)97,3 log. 2-988139 Reduction True altitude + 16' 13" 65° 13 7 Meridian alt. 65 29 20 Mer. zen. dist. Declination Latitude Bv Haver 24 30 40 N. 16 22 14 N. 40 52 54 N. .' Lat. Ao C2 ii8 I I. Method III. — By Towson's Ex- Meridiem Tables. O's red. declination Aug. Table i, Index 33 16° 22' 14" N. + 2 30 Augmented declination 16 24 44 N. True altitude 65' 1 3' 7" Aug. Table 2, Index 33 -f- 19 2 Augmented altitude 65 3* 9 Zenith distance 24 27 51 N. Augmented declination 16 24 44 N. Latitude 40 52 35 N. Seduction to Meridian. 5»9 Ex. 4. 1882, Sept. 23rd, P.M. at ship, lat. acct. 51° 2' N., long. 117° 8' E., obs. alt. sun's L.I,. South of observer 38== 44' 20", index corr. + i' 8', height of eye 21 feet, time by watch 50" 08 (or 23d o'' 50^"), found to be 39™ 2» fait on app. time at ship, difiF. of long, made to West was 8*2 miles after the error on app. time was determined : required the latitude. Time by watch, Sept. W&tch. fast — 23 o 10 58 DiflF. of longitude — 33 Time from noon, 23rd is Obs. alt. O's L.L. Index correction Dip Corr. altitude Semi-diameter True altitude 23 10 25 10 25 38» + 44' I 20" 8 38 45 4 28 »3 38 41 I 5 4 38 40 I 15 59 App. time at ship, Sept. 2^^ o'>io'n25» Long. 173° 53' E. — 7 48 32 Greenwich date, Sept. 22 16 21 53 By Saper :' True altitude 38' 55' 55" 3856 H.D. I2h 35 — 58"-44 X 16-37 40908 17532 35064 5844 6,0)95,6-6628 15' 56"-7 Deal., page I, Sept. 22nd 0° 1 Correction — i N.A. 5' 2i"N. deer. 5 57 Red. decl. o o 36 S. Method I. Time from noon io™25« Latitude 51° 2' Declination o i rising 3-01399 cos. 9-798560 cos. 0-000000 649 nat. no. 2 8 12550 True altitude 38* 56' nat. no. 649 nat. sine 628416 Mer. zen.dist.5i° Declination o r 8" N. nat. cos. 629065 o 36 S. Latitude 51 32 N. In taking out log. rising for io"> 25% it -will be noticed that the index given at the beginning of the line is j, meaning that the index at the commftnce- ment of the line is 1, but that it changes somewhere along the line, which may easily be known by observing that when the first figfure of the decimal part of the log. changes from 9 to o, the index changes from 1 to 2. Method II. Constant log. 5 Latitude acct. 51° 2'N. cos. 9 Declination o i cos. o Zen.dist.byD.R. 51 3 cosec. o Time from noon io"'25' log. 6 Reduction True altitude Meridian altitude 38 58 52 Mer. zen. dist. 51 i 8 N. Decliniiion o o 36 S. Lttitude 51 o 32 N. 615455 798560 000000 109191 712960 6,0)17,2 log. 2-236166 + 2' 52" 38 56 o Method III. — By Towson's Ex-Meridian Tables. 0'fl red. declination 0° o' 36" S. Aug. Table i, Index 13 -\- o Augmented declination o o 36 S. As the decl. is less than any given in the head of Table I, augmentation is alone required. In this case enter Table I, under least declination, and with givenhotu--angle fiiid corresponding Index number ; with this and the altitude, augmentation II is deter- mined as usual. Latitude True altitude 38^56' o" Aug. Table 2, Index 13 -|- 2 47 Augmented altitude 3S 58 47 Zenith distance 51 i 13 N. Augmented declination o o 36 S. 51 o 37 N. jao Re&uotion to Meriddcm. Ex. 5. 1882, May 5th, p.m. at ship, latitude account 5° 13' N., longitude 61° E., observed altitude sun's l.l. 78° 41' N., eye 17 feet, time by watch 5*" i" 7% which was found fast 4b JO'" 578, diflference of longitude made since, 20^ miles West. App. time at ship, May 5<» d^ 8-"48« Green, date, app. time, May 4,^ 2o'> /^"^ 48^. Time from noon is 8 48 Hourly diff., 5th noon, 42"-66 X Green, time 3''-92 = l6•^"•^1^^ -J- 60 = 2' 47", decl. noon 5th, 16° 18' 6' N. — 2' 47* = red. decl. 16° 15' 19" N. By Norit : True altitude 78° 52 Method I. 47 Time from noon Latitude acct. Declination 8™488 5° 13' 16 15 rising 2"8675io cos. 9'998i97 cos. 9-982294 705 log. 2"848ooo True altitude 78° 52' 47" M. Z. dist. io°54' ji'S. Declination 16 15 19 N. nat. no. 705 nat.sine98i227 nat. COS. 981932 5 20 48 N. Latitude This example cannot be solved by means of Towson's Ex-Meridian Tables, as the altitude exceeds the limits of the Tables. By Eaper : True altitude 78° 52' Method II. 36". Constant log. Latitude D.R. Declination Mer. zen. dist. Time from noon Reduction True altitude 5°i3'N. 16 15 N. 5'6i546 9.99820 9-98229 cosec. o"7i8io log. 6-56649 cos cos 6,0)76,0 log. 2'88o54 + 12' 40" 78° 52 47 Meridian altitude 79 5 27 Mer. zen. dist. Declination 10 54 33 S. 16 15 19 N. 5 20 46 N. Latitude Examples fob Praotioe. 1. 1882, January 4th, a.m. at ship, latitude by account 34° 47' N., long. 27° 12' W., observed altitude sun's l.l. South of the observer was 32" 12' 10", index correction -\- 4' 19", height of eye 28 feet, time by watch o^ 13"" 24% which had been found to be 25"^ 35^ fast of apparent time at ship, difference of longitude made to TFest was 29'-2 after the error on apparent time at ship was determined : required the latitude. 2. 1882, February 28th, p.m. at ship, lat. acct. 43''46'N., long. 12° 31' W., obs. alt. sun's L.L. 38° i' 15" S., index corr. — 5' 10", eye 23 feet, time by watch 22" 3% which had been found to be 8"" 14^^ fast of app. time at ship, diflF. of long, made to JEast was 14' after error on app. time at ship was found : required the latitude. 3. 1882, March 20th, a.m. at ship, lat. acct. 41° 24' 8., long. 105° E., obs. alt. sun's l.l. 47° 46' N., index corr. + 26", eye 22 feet, time by chron. ig'^ i6'> 58'" 12% which had been found to be 6'^ 34'" 34^ slow on app. time at ship, diff. of long, made to Hast was 23' after the error on app. time at ship was determined : required the latitude. 4. 18S2, April 2ist, A.M. at ship, lat. acct. 39° 54' N., long. 6° 6' E., obs. alt. sun's l.l. 61° 26' 35" S., index corr. + i'-, eye 18 feet, time by watch 21^ o'^ 8™ io«, which had been found to be 27™ o^fast on app. time tit ship, diff. of long, made to Hast was 5'aftf r the error on app. time at ship was determined. 5. 1882, May 29th, P.M. at ship, lat. acct. 37° 15' S., lent,'. 107° W., obs. alt. sun's l.l. 30" 22' 30" N., index corr. -|- 49*, eye 22 feet, time by watch 29'i 7*' 9™ 11% which had been found to be 6'' 36"" 56s/fl!«< on app. time at ship, diff. of long, made to iFest was 27' after the error on app. time at ship was determined. 6. 1882, June 19th, A.M. at ship, lat, acct. 44° 24' N., long. 14° 5' W., obs. alt. sun's l.l. 68° 37' 5" South of observer, eye 18 feet, time by watch 11'' 40™ 40% which was found to be 2« 2' ilov on app. time at ship, diff. of long, made to East was 32^' after the error on app. time at ship was determined. 7. 1882, July i6tb, P.M. at ship, lat. acct. 0° 38' S., long. 2° E., obs. alt. sun's l.l. 67° 41' (zen. S.), eye 15 feet, time by watch o^ ii"i 99, found fast on app. time at ship 56^, diff. of long, made since i|' to £ast. Meridiem Altitude of a Fixed Star. 32^ 8. 1882, August 30th, P.M. at ship, lat. acct. 41° 5' N., long. 139° 25' E., obs. alt. sun's L.L. 57" 20' S., index corr. + 2' 21", eye 14 feet, time by watch 22"^ 22', found to be 18' slow of app. time at ship, diff. of long, made to West was 34'. 9. 1882, September 9th, p.m. at ship, lat. acct. 9° 20' N., long. 178° 30' E., obs. alt. sun's L.L. 85° 19' (zen. N.), ej'e 20 feet, time by watch 1 1'' 59"" 40', slow on app. time at ship 9" 21% difiF. of long, made to East was io|'. 10. 1882, October nth, p.m. at ship. lat. acct. 45° 51' N., long. 85° 3' E., obs. alt. sun's L.L. 36° 38' 15" S., index corr. — 5' 15', eye 16 foot, time by watch lo"* 18'' 50™ ro» which was j** 40™ I2S slow on app. time, difiF. of long. 33' W. 11. 1882, November 3rd, p.m. at ship, lat. acct. 32° S., long. 109*^ 39' E., obs. alt. sun's L.L. 71° 50' N., index corr. + 32", eye 18 feet, time by watch 2^ 22'' 22™, which was found 2'' slow, difiF. of long. 28'-7 West. 12. 1882, December 23rd, a.m. at ship, lat. acct. 47'' 22' S., long. 27° 3' W., obs. alt. sun's L.L. 65° 10' 15" N., index corr. -f- 45", eye 12 feet, time by watch 1 1'' 29"! 42^, found to be 18™ 408 slotv, diflF. of long, was 36' East. 13. 1882, January 5th, p.m. at ship, lat. acct. 8^ 50' N., long. 130= 14' W., obs. alt. sun's L.L. 58° 6' 10" S., eye 21 feet, lime by watch o'' 2"' 40', found 13"^ 48^ slow on app. time, diff, of long, made since 16' East. 14. 1882, April 28th, A.M. at ship, lat. acct. 18° 46' S., long. 34° 12' W., obs. alt. sun's L.L. 56° 28' (zen. S.), index corr. -\- i 5", eye 21 feet, time by watch 1 1^^ 49"™ 50^, found fast 2™ 17" on app. time at ship, diff. of long, made since 17^ West. 15. 1882, March 20th, a.m. at ship, lat. acct. 19° S., long. 33° 33' E., obs. alt. sun's l.l. 70° 21' N., index corr. — 2' 10", oye 16 feet, time by wutch 8"' 17% found fast on app. time at ship 26™ ri% diff. of long, made since 14^' East. 16. 1882, April 12th, A.M. at ship, lat. acct. o^, long. 164° 12' W., obs. alt. sun's l.l. 80° 30' N., index corr. — 5' 10", eye 21 feet, time by watch i2<* o^ c" 2', fast on app. time at ship 10™ 51", diff. of long, made to East 7^'. 17. 1882, September i6th, a.m. at ship, lat. acct. 42° 36' S., long. 137° 10' E., obs. alt. sun's l.l. 44'-' 6' N., index corr. ■\- 2' 10", eye 19 feet, time by watch 16'J 8'^ 41™ 43^, which had been found to be 9*^ 2™ ^"j^ fast on npp. time at ship, the diff. of long, made to West was 14' after the error on app. time at ship was determined. 18. 1882, March i6th, a.m. at ship, lat. acct. 37° 42' N., long. 61° 40' E., obs. alt. sun's L.L. 50° o' 30" S., index corr. + 34", eye 15 feet, time by watch lo^^ 53™ 31% found slow on app. time at ship i^ 3™ 22% diff. of long, made since 18' West. 19. 1882, March 5th, p.m. at ship, lat. acct. 33° 35' N., long. 78° E., obs. alt. sun's l.l. 49° 53' '5" ^-i intiex corr. — 3' 15", eye 22 feet, time by watch 4'' ig^ 2™ r2«, found to be ^h jiym 123 slow, diff. of long, was 10' E. 20. 1882, September 22nd, a.m. at ship, lat. acct. 45° 45' S., long. 111° 42' W., obs. alt. sun's L.L. 43"^ 50' N., index corr. — 5' 40", eye 18 feet, time by watch 22'' 7I1 41"! 10% found to be 8'' 4"'- 10* fast, diff. of long, was 13' -5 East. MERIDIAN ALTITUDE OF A FIXED STAR. RULE CII. 1°. Take from Nautical Almanac the starts declination. 2°. To the observed altitude apply the index error, as the s^ign attached directs. 3°. Subtract the dip answering to the height of eye (Table 5, Norie; Table 30, Raper). 4°. Subtract the refraction (Table 4, Norie ; Table 3 1 , Rarer), and thus get the true altitude. 322 Meridian Altitude of a Fixed Star. 5°. Subtract the true altitude from 90; the remainder is zenith distance. 6°. Mark the zenith distance N. or S., according as the observer is North or South, of the star. 1°. Underneath this latt place the declination, and take their sum if they have the same names ; hut take their difference if they have unlike names : the result, in either case, toill be the latitude. The declination of a fixed star changes so slowly that it may be taken out of the Nautical Almanac by inspection, without any prac;ical error resulting; a Greenwich date, therefore, is clearly unnecessary. 8°. When the zenith distance and declination are of the same name, the latitude is of tliat name ; ichen the %enith distance and declination are of different names, the latitude takes the name of the greater. The stars are inserted in the Nautical Almanac in the order of their Right Ascension, from of- to 24^ ; it will, therefore, very much facilitate the finding of the given star in the Nautical Almanac, to turn, in the first instance, to the three pages (290 — 293, Nautical Almanac, 1882, and seek the given star under the head "Mean Places of Stars" for January, and thence obtain the star'd Right Ascension, which find at the top of one of the pages following 311 — 366, Nautical Almanac, 1882), which will give the star, and the declination will be found opposite the day in the side column which is nearest the given day. The degrees (°) and minutes (') are placed at the top of the column (as annexed), and the seconds (") are ranged below, for the sake of economizing space in the second column below the name of the star. If the seconds exceed 60", only take the excess of 60" and increase the minutes (') at the top by i. Thus, on May loth, 1876, (see table annexed) the declination of a Andromedse is 28° 22' 49" N., and on January ist the declination is 28' 23' 3" N., 62"-8 being i' 3", which being added to 28° the head of the column, gives the declination 28= 23' 3". a Andromedse 1 Date. R.A. Decl. N. oh jm 28' 22' Jan. I 45" 1 7 6 2"- 8 II 45*04 6i-8 21 44-90 606 31 44-78 59'2 &c. &c. &c. May 10 45-50 49-2 20 45-80 49'9 21 46-12 509 &c. &c. &c. which stands at Examples. Ex. I. 1882, Dec. 29tb, long. 140" W., the obs. mer. alt. ol the star a Leonis (Regulus) , bearing South, was 52° 7' 30', index corr. 27", height of eye 15 feet : required the latitude. Observed altitude of star 52° 7' 30" S. Index correclion — 27 Dip 15 feet Refraction True altitude Zenith distance Declination (N.A., p. 33J Latitude 52 7 3 3 42 52 3 21 44 52 2 37 90 o o 37 57 23 N. 12 32 6 N. 50 29 29 N. By Raper : Index corr. — 27', dip — 3' 50'', refr. — 46", true alt. 52° 2' 27", lat. 50° 29' 39" N. Ex. 2. 1882, March 12th, long. 10° E., the obs. mer. alt. of the star Pollux, bearing North, was 71° 59' 10", index corr. -\- i' 15", height of eye 1 8 feet : required the latitude. Observed altitude of star 71° 59 io"N. Index correction -\- i 15 Dip 18 feet Refraction True altitude Zenith distance Declination (N.A., p. 334) Latitude By Raper : Index corr. + i' 15", dip — • 3' 10", refr. — 19", true alt. 71° 55' 56", lat. 10° 14' 24" N. 72 4 25 4 71 S^ 21 18 71 90 56 3 18 28 3 18 578- 28 N. 10 14 31 N. Meridian Altitude of a Fixed Star. 3^3 Ex. 3. 1882, March nth, long. 84° W., the ohs. mer. alt. of the star a Argus (CanopusJ, bearing South, was 37° 26', inde.x corr. -\- 47", height of eye 16 feet. Observed altitude of star 37° 26' o' S. Index correction -\- 47 Dip 16 feet Refraction True altitude 37 26 47 — 3 50 37 22 57 — I 15 37 21 42 90 o o Zenith distance 52 38 18 N. Declination (N.A., p. 331) 52 38 18 N. Latitude 000 By Raper : Index corr. + i' i^', dip — 4' o', refr. — i' i6-, true alt. 37° 21' 56", latitude 0° o' 14" N. Ex. 4. 1882, January ist, long. 100° E., the obs. mer. alt. of the star a Canis Majoris (SiriusJ, bearing South, was 59° 59' 50'', in- dex corr. -\- 4' 12", height of eye 24 feet. Observed altitude of star 59° 59' 50" S. Index correction -\- 412 Dip 24 feet Refraction True altitude 60 4 z — 4 42 59 59 20 — 33 59 58 47 90 o o Zenith distance 30 i 13 N. Declination (N.A., p. 332) 16 33 26 S. Latitude 13 27 47 N. Bj^ Raper : Index corr. + 4' 1 2", dip — 4' 50", refr. — 34", true alt. 59° 58' 38", lati- tude 13"^ 28' 38" N. Examples for Praotioe. In each of the following examples it is required to find the latitude :- CIVIL DATE. 1882. I. Nov. 7th, 2. Jan. ist, 3- Aug. 19th, 4- Dec. 22nd, 5- April nth, 6. June loth. 7- Dec. 27th, 8. Nov. 30th, 9- Feb. 2nd, 10. June ist, II. May 22nd, 12. July 17 th, '3- Oct. 17th, 14. March 2nd, 15- April 3rd, 16. Aug. 7 th, 17- May ist, 18. Oct. 29th, 90° W. a Audromedse 75° 10' 3o"S. 27 W. a Aurigag (Gapella) 54 o 15 N. 84 E. a Lyrse (Vega) 50 o 20 N. 82 E. a Persei 51 51 45 N. 142 W. « Virginas fSpicaJ 63 14 30 S. 151 E. a 'E.vid.a.ni ( AcheniarJ 40 10 25 S. 91 W. (Algenih) 78 16 45 S. 24 W. a Arietis 68 23 oN. 76 E. rt Tauri (Aldeharan) , 29 52 10 N. 97 E. a^ Crucis 75 10 30 S. 178 W. a Hydrse 30 28 53 S. 29 E. a Cj'gni 20 13 50 N. 165 E. a Aquilae (AltairJ 60 49 10 N. 154 W. a Canis Majoiis fSiriusJ 58 58 50 N. 1 1 1 E. a Bootis (ArcturusJ 79 49 40 S. 40 W. a Scorpii (Antares) 68 49 30 S. 8 E. a* Centauri 10 2 50 S. 5 W. a Piscis Australis (FomalhautJ .70 6 o N. + 0' + + 3 4-0 + 5 — I — 7 o + + 1 + 27 45 o 40 47 55 25 38 20 40 38 o 55 10 5 54 45 55 25ft. 18 22 26 22 24 24 28 15 14 II 18 17 20 25 21 3H ORDINARY EXAMINATION. EXAMINATION PAPEE 2b ie used by all candidates when appearing for Examination for the first time only. DEFINITIONS. The Candidate is requested to write at least ten of the following dejinitions. The writing should be clear, and the spelling should not be disregarded. 1. The Equator ia a great circle passing round the earth at an equal distance from the two poles. 2. The Poles are the extremities of the axis of the earth. 3. A Meridian is a gre it circle passing through both poles, perpendicular to the equator. 4. The Ecliptic is the great circle of the celestial sphere in which the sun appears to move in consequence of the earth's motion in its orbit. 5. The Tropics of Cancer and Capricorn are the parallels of latitude 23'^ 28' N. and S. 6. Latitude is that portion of the meridian which is contained between the equator and the given place, and is reckoned in degrees, minutes, and seconds. 7. Parallels of Latitude are small circles parallel to the equator. 8. Longitude is an arc of the equator between the "first meridian" and the meridian of the place. 9. The Visible Horizon is the circle bounding the spectator's view at sea. 10. The Sensible Horizon is the plane on which the spectator stands, produced to meet the celestial concave. 11. The Rational Horizon is an imaginary plane parallel to the sensible horizon, and passing through the centre of the earth. 12. Artificial Horizon and its use. It is a small shallow trough, containing quicksilver, or any other fluid, the surface of which affords a reflected im.tge of a heavenly body. It is used for observing altitudes on shore. 13. True course of a ship is the angle which the ship's track makes with the meridian, or N. and S. line of the liorizon. 14. Magnetic Course (correct magnetic) is the angle which the ship's track makes with the magnetic meridian. 15. Compass Course is the angle which the ship's track makes with the N. and S. line of the compass card. i6. Variation of the Compass is the angle which the magnetic needle, under the influence ot terrestrial magnetism only, makes with the meridian. 17. Deviation of the Compass is the angle the compass needle makes with the (correct) magnetic meridian. 18. Tlie Error of the Compass is the angle the compass needle makes with the true meri- dian, being the combined eff'ect of the variation and deviation. 19. Leeway is the angle included between the direction of the ship's keel and the direction of the wake she leaves on the surface of the water. 20. Meridian Altitude of a celestial object is the angular height of that object above the horizon when it is on the meridian of the place of observation. 21. Azimuth of a celestial object is the arc of the horizon between the N. and 8. points, and a vertical cirole drawn through the object. Ordina/ry Mxamination. 325 22. Amplitude is the arc of the horizon between the East point and the centre of the object when rising, or the "West point when setting. 23. Declination of a celestial object is the arc of a circle of declination between the object and the equator. 24. Polar distance is an arc of a circle of declination between the body and the pole (complement of the declination) . 25. Right Ascension of a body is an arc of the equator, or an angle at the pole intercepted between the meridian passing through the first point of Aries, and that over the object. 26. Dip is the angle through which the sea horizon is depressed in consequence of the elevation of the spectator above the surface of the earth. 27. Refraction is the correction to be applied to the place of a heavenly body as actually viewed through the atmosphere, which bends the rays of light which pass through it into a position more nearly vertical, and thus causes the apparent places of the heavenly bodies to be above the true place. 28. Parallax is a correction to reduce an altitude as observed from the surface of the earth, to what it would be if taken from the centre. It is the angle subtended at the object by that radius of the earth which is drawn to ths place of observation. 29. Semi-diameter of a heavenly body is h ilf the angle subtended by the diameter of the visible disc at the eye of the observer. 30. Moon's Augmented Semi-diameter is an increase of the moon's apparent dimension due to increase of altitude, because the Moon's distance from the spectator decreases as the altitude increases. 31. Observed Altitude is the angular distance of a heavenly body from the horizon, as observed witu the sextant or other instrument. 32. Apparent Altitude is the altitude of a celestial body as seen from the surface of the earth ; or, the observed altitude corrected for index error and dip. 33. True Altitude is the altitude of a celestial body as seen from the centre of the earth ; that is, the apparent altitude corrected for refraction, semi-diameter, and parallax. 34. Zenith Distance is an arc of a circle of altitude between the body and the zenith (complement of the altitude). 35. Vertical circles are great circles passing through the z-nith and nadir, perpendicular to the horizon. They are also called Circles of Altitude, because altitudes are measured on them; and Circles of Azimuth, as marking out all the points that have the same azimuth. 36. Prime vertical is a great circle passing through the zenith and nadir, and the East and West (true) points of the horizon. 37. Civil time is the time used in ordinary life to record events. It begins at midnight and ends at the following midnight, and its hours are reckoned through twice 12, from midnight to noon, denoted by a.m. ; and then from noon to midnight, denoted by p.m. 38. Astronomical Time is the time used in all astronomical calculations : it begins at noon and ends at the following noon, its hours being reckoned from o** to 24''. 39. Sidereal Time is the westerly hour-angle of the first point of Aries. 40. Mean Time is the hour-angle which the mean sun is westward of the meridian. 41. Apparent Time is the hour-angle of the apparent or true sun, always reckoning westward. 42. Equation of Time is an angle at the pole between a meridian over the true sun, and one over the mean sun. 43. Hour-angle of a Celestial Object is an angle at the pole included between the meri- dian of the observer and that over the object. 44. Complement of an Arc or Angle is that arc or angle which must be added to it to make a right-angle (90°). 45. Supplement of ditto is that angle which must be added to it to make two right- angles (i8o°). 32< Ordinary Examination. EXAMINATION PAPER— No. I. FOR SECOND MATE. Multiply 7654 by 95, and 950 by 586, by common logarithms. Divide 3654000 by 7308, and 35420 by 323, by common logarithms. H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pts. I W.S.W. 10 8 N.W. i ii°W. A point, lat. 37" 3' N. 2 II 4 long. 9° 0' W., bearing 3 II 4 by compass N.E.^E., 4 II 4 dist. 15 miles. Ship's 5 N.W. \ N. 12 2 w.s.w. i 17° w. head W.S.W. Devia- 6 12 3 tion as per log. 7 13 3 8 12 2 9 N.N.W. 9 6 West. f ii-'W. 10 9 i 5 II 9 5 12 9 4 Variation 22° 30' W. I N.W. by W. 7 8 S.W. by W. li 20»W. 2 7 6 3 7 4 4 8 2 5 S.W. 1 S. 9 3 S.S.E. I 6°W. 6 8 7 7 9 3 A current set the ship 8 8 7 S.W. by W.fW. (cor- 9 W.^S. 10 3 S. by W. 1 2 f4°W. rect magnetic) 8 miles 10 10 3 from the time the de- II 10 2 parture was taken to 12 10 3 the end of the day. Correct the courses for deviation, variation, and leeway, and find the course and distance from the given point, and the latitude and longitude in by inspection. 4. 1882, January ist, in longitude 102° 41' W., the observed meridian altitude of the sun's L.L. was 59" 59' 50", bearing South, index error -|- 50", height of eye 15 feet : required the latitude. 5. In latituJe 37° N., the departure made good was 89*2 miles: required the diflference of longitude by parallel sailing. 6. Required the course and distance from Toulon to Valencia, by Mercator's sailing. Long. Toulon Long. Valencia ' 56' E. 24 W. Lat. Toulon 43° 8' N. Lat. Valencia 39 29 N. ADDITIONAL FOR 0NL5f MATE. 7. 1880, January 14th : find the time of high water, a.m. and p.m., at Cherbourg, Christ- church, and Falmouth. 8. 1882, January ist, at 8^ 4"! a.m. apparent time at ship, in latitude 50° 32' N., longitude 139° 51' W., the sun's magnetic amplitude E. by S. | S. : required the true amplitude and error of the compass ; and supposing the variation to be 23'^ 52' E. : required the deviation of the compass for the position of the ship's head when the observation was taken. 9. 1882, January 29th, p.m. at ship, latitude 42° 26' N., observed altitude sun's l.l. 13° 40', index error — i' 14", height ot eye 16 feet, time by chronometer 29<' 6'» 48™ 40*, which was slow ii"" 22'"3 for mean noon at Greenwich, December 1st, 1881, and on January ist, 1882, was 8"! 7s slow for G-reenwich mean noon : required the longitude by chronometer. ADDITIONAL FOR FIRST MATE, ro. 1882, Jawuary 15th, memi time at ship if' 39™ 44' a.m., latitude 23° 39' S., longitude 127° 52' W., sun's magnetic azimuth S. 103" E., observed altitude sun's l.l. 55° 8' 30', index Ordinary Examination. 327 error — 2' 30', height of eye 12 feet : required the true azimuth and error of the compass ; and supposing the variation be 7° 50' E. : required the deviation of the compass for the position of the ship's head at the time the observation was taken. 11. 1882, January 17th, p.m. at ship, lutituds by account 36" 2' N., longitude 149° 28' E., observed altitude sun's l.l. South of observer was 32'^ 54' 15", inde'^ error ■\- 2' 18", height of eye 22 feet, time by watch 16'' 23'' 59™, which had b^'cn found to be 20" 24" slow on ajip. time at ship, the difference of longitude made to the West since the error of watch on app. time at ship was determined, was 39'"2 : required the latitude by reduction to mt^ridian. 12. 1882, January i6th, a.m. at ship, and uncertain of my position, when a chronometer showed January \^^ 22^ 16™ i^ M.T.G., obs. alt. sun's l.l. 12° 52' 30" ; again, p.m. at ship same day, when chronometer showed January i6<* 2'i 47™ 2», obs. alt. sun's l.l. 14^^ 1' 57", height of eye 17 feet, the ship having made 42 miles on a true X.N.W. course in the interval : required the line of position wh»n the first altitude was observed, and the ship's position by Sumner's Method when the second altitude w.is taken, assuming the ship to be between the latitudes 48° 50' N. and 49° 30' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 48^ 50' N. Long. 5° 47^' W. B' = Lat. 49" 30' N. Long. 7° 28' W. ADDITIONAL FOR MASTER ORDINARY. 13. 18S2, January 24th, the observed meridian altitude of the star a Tauri (Aldelaran) was 92'-' 36', bearing South, index correction — 23", height of eye 20 feet: required the latitude. DEVIATION OF THE COMPASS. N.B. — r/< 24™ 22", or iS'' 2 3>> 24" 22% which had been found to be 5™ slow on apparent time at ship, the difference of longitude made to the East was 30 miles, after the error on apparent time was determined. 12. 1882, April 6th, P.M. at ship, and uncertain of my position, when a chronometer showed April 6'' i^ 17"" 25* M.T.G., obs. alt. sun's l.l. 45° 7' 10"; again, p.m. at ship same day, when chronometer showed April 6"' 4^ 23"^ 24s, obs. alt. sun's l.l. 24° 52' 10", height of eye 18 feet, the ship having made 46 miles on a true N.E. course in the interval: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 50° 20' N. and 50=50' N. ADDITIONAL FOR JMASTER ORDINARY. 13. 1882, April 1 2th, the observed meridian altitude of the star Spica, South of observer, was 20" 58' 40", index correction — 45", height of eye 25 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head ; Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 2°W. S. 5 W. S. 10 W. S. 16 w. 1 South .... S.W 1 West N.W S. 5°E. S. 24 E. S. 17 E. S. 3 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :-E. by N. i N. ; N.W. | W. ; S.W. by S. ^ S. ; S.E. by S. \ S. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass Courses :— N.W. by W. ^ W. ; W. by S. ^ S. ; E. by S. i S. ; N.E. f E. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at S. 24° W., find the bearings, correct magnetic. Compass bearings:— N. 84° W. and W.H.W. Bearings, magnetic : — Orddna/ry Examination. 33? EXAMINATION PAPER— No. Y. FOR SECOND MATE. Multiply 767 by Sg'S, and 10003 by no, by common logarithms. Divide 66889"2 by 99'7, and 3972096 by 144, by common logarithms. 3 — H. Courses. K. T^ Winds. Lee- way. Deviation. Remarks, &c. pts. I S.E. ^ E. 14 S.W. 8^E. A point, lat. 64° 2' S. 2 14 long. i4o°2i'E., bear- 3 14 4 ing by compass 4 '3 6 W. b y.f.S, distance 23 5 E. by S. f S. 4 4 N.E. 3 14° E. miles. .'chip's head 6 4 2 S.E.|E. Deviation as 7 4 2 per log. 8 4 2 9 W.N.W. 4 North. ^i 19° W. 10 3 II 3 12 3 I N.E. i N. 9 2 N.N.W. 1 4 8^E. Variation 37° E. 2 9 2 3 9 2 4 9 4 5 S.W. by S. 8 4 W.N.W. i^ 10° W. 6 8 4 7 7 2 A current set the 8 7 8hip(correctmagnotic) 9 N.N.W. 5 West. 2 11° W. N.E. \ E., 48 miles, 10 4 from the time the de- II 4 parture was taken to 12 4 the end of the day. 4. 1882, May 8th, in longitude 105° 17' W., observed meridian altitude of sun's l.l., bearing North, was 76'^ 3', index correction — i' 27", height of eye 10 feet: required the latitude. 5. In latitude 3° 24' N., the departure made good was 982 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by calculation on Mercator's principle. Latitude A 39° 39' N. Latitude B 27 27 N. Longitude A 51° 51' E. Longitude B 33 33 E. ADDITIONAL FOR ONLY MATE. 7. 1880, May 20th: find the times of high water a.m. and p.m. at Loch Ryan, Tarn Point, Berwick, St. Malo, and Dungeness. 8. 1882, May 2i8t, at 7'> 29™ a.m., apparent time at ship, latitude 45° 53' S., longitude 50° 39' E., sun's magnetic amplitude N.E. \ E. : required the true amplitude and error of the compass ; and supposing the variation to be 31° 50' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, May 22nd, A.M. at ship, latitude 43° 25' N., observed altitude sun's l.l. 32' 8', index correction -\- 47", height of eye 15 feet, time by chronometer 2i<' 2i'> 6" 10% which was slow i2«-6 for mean noon at Greenwich, February 24th, and on April ist was 2"" ^^^ fast for oiean noon at Greenwich : required the longitude. 33^ Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, May 25th, mean time at ship t,'^ 29™ 47^ p.m., latitude 41° 58' N., longitude 96° i' W., sun's bearing by compass N. 118° 30' W., observed altitude sun's l.l. 40° 40' 40", index correction -j- 2' 15", height of eye 12 feet : required the true azimuth and error of the compass; and supposing the variation is 10° 30' E. : required the deviation of the compass for the position of the ship's head at the timu of observation. 11. 1882, May loth, p.m. at ship, latitude account 28° 13' S., longitude 112° 15' W., observed altitude of sun's l.l., North of observer, was 43° 35' 20", index correction — 6' 12", height of eye 19 feet, time by watch 30" 26^ (or lo'^ o** 30" 268), which had been found to be 6'" ^^^ fast on apparent time at ship, the difference of longitude made to the East was 26', after the error on apparent time was determined : required the latitude. 12. 1882, May 30th, A.M. at ship, and uncertain of my position, when a chronometer showed May 29"* 19'' 21™ i5=M.T.G., obs. alt. sun's l.l. 23° 42' 10''; again, p.m. at ship same day, when chronometer showed May 30'^ 2'' 7"" 5', obs. alt. sun's l.l. 56° 45', height of eye 15 feet, the ship having made 49 miles on a true West course in the interval : required the line of bearing when the fir»t altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 50° o' N. and 50° 30' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, May loth, the observed meridian altitude of Spica, bearing North, was 70° 10' 25", index correction + 42", height of eye 22 feet : required the latitude. In the following table give the correci magnetic bearing of the distant object and thence the deviation. Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E East. S. 78 E. S. 7c E. S. 7 1 E. South .... S.W West N.W N. 85° E. N. 63 E. N. 64 E. N. 74 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses : -N.N.E. f E. ; N. 84° W. ; S. 72° E. ; S.W. by W. \ W. Compass courses: — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— N.N.W. ^ W. ; N. 64° E. ; S.E. | E. ; S. 39° W. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at S. 87° E., find the bearings, correct magnetic. Compass bearings: — S. 15° W., and N. 72° W. Bearings, magnetic: — EXAMINATION PAPEE— No. VI. FOR SECOND MATE. 1. Multiply 987 by 543, and 5900 by '00071, by common logarithms. 2. Divide 50800 by 4*835, and 999999 by loioi, by common logarithms. Ordina/ry Examination. 337 3 — H. Courses. K. 10 Winds. Lee- way. Deviit;on. Remarks, &c. pts. I E. by S. 4 3 S. by E. 3i 15° E. Apoint,lat.56''i2'N., 2 - 4 4 long. 1 35°4o'W., bear- 3 4 3 ing by compass WSW 4 4 dist.aijmiies. 8hip'8 5 S.E. by E. 5 6 S. by W. 'i 12° E. head E. by S. Deviu- 6 5 8 tion as per log. 7 5 6 8 •5 1 9 S. by E. 5 7 S.W. by W. '^ 4°E. lO 5 i 8 II 6 I 12 I E. f S. 6 6 5 4 S. by E. '* 15^ E. Variation 25° E. 2 6 5 3 6 4 4 .? 7 5 S.W. f w. 4 8 S. by E. 3i 7° W. 6 4 6 7 4 2 A current set the ship 8 4 4 (correct magnetic) 9 S. by E. 4 J E. by S. 2i 4° E. N.byWJW., i6mls., lO 4 5 from the time the de- II 3 6 parture was taken to 12 4 4 the end of the day. 4. 1882, June ist, in longitude 96° 17' E., the observed meridian altitude of sun's l.l. was 75° 38' 15", bearing North, indc.'c correction -)- 27", height of eye 26 feet : required the latitude. 5. In latitude 35° 54' S., the departure made good was 249 miles : required the difiTerence of longitude. 6. Required the course and distance from A to B, by Mercator's Sailing. Lat. of A 3°i9'N. Long, of A 7i''42'W. Lat. of B 33 2 S. Lour-, of B 122 20 W. ADDITIONAL FOR ONLY MATE. 7. 1880, June 19th : find the time ot high water, a.m. and p.m., ;it Rotterdam, Heligoland, and Rio Janeiro, longitude 43° 12' W. 8. 1882, June 2ist, at 9^ 16"" p.m. apparent time at ship, in latitude 59° 51' N., longitude 64° 42' W., the sun's ma'j;notic amplitude N. | E. : required the true amplitude and error of the compass ; and supposing the variation to be 52° 30' W. : required the deviation of the compass for the position of the ship's head when the observation was taken. 9. 1882, June 14th, P.M. at ship, latitude 2° 2' S., observed altitude sun's l.l. 28* 38', index error -)- 48', height of eye 12 feet, time by chronometer 14"* o*> 3™ i8', which was 2^ 28™ i^^-"] fast for mean noon at Greenwich, April 1st, and on April 30th, was 2^ 24™ 19' fast for Greenwich mean noon : required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1882, June 8th, mean time at ship 7^ 50™ a.m., latitude 15° 45' N., longitude 32° 33' W., sun's magnetic azimuth N. 70° E., observed altitude sun's l.l. 31° 10', index error — i' 22", height of eye 18 feet : required the error of the compass, and supposing the variation be 14'^ 40' W. : required the deviation of the compass for the position of the ship's head at the time the observation was taken. XX 33^ Ordinary Examination. ir. 1882, June 5th, p.m. at ship, latitude by account 61" 58' N., longitude 155° 21' E., observed altitude sun's l.l. South of observer was 49° 50' 30", index error 4* 2' 10", height of eye 2r feet, time by watch 1 1^* 48™ 26', (or 4^ Winds. Lee- way. Deviation. Remarks, &c. pts. I S.E. by S. 6 S.W. by S. 'i 31° W. A ] oint of land in 2 5 6 lat. 51° 25' N., long. 3 5 4 1 7 8° 56' E., bearing by 4 5 compass N.E. by N., 5 South. 5 5 W.S.W. 4 20° W. dist. 17 miles. Ship's 6 5 5 head S.E. by S. De- 7 6 5 viation as per log. 8 6 5 9 West. 6 4 S.S.W. 1 30°E. 10 6 6 II 6 5 12 6 5 I S.S.E. 4 4 S.W. 4 27F w. Variation 28'* W. 2 4 4 3 4 2 4 4 5 S.E. by E. 10 S. by W. \ ' 32^° W. 6 10 6 7 8 9 10 4 A current set the 9 S.S.E. 12 4 East. \ 20^° W. ship N.W. JN. (correct 10 12 6 magnetic), 9 miles, II 12 6 from the time the de- 12 13 4 parture was taken to the end of the day. 4. 1882, July 26th, in longitude 12° 19' W., the observed meridian altitude of the sun's L.L. was 15° 41', bearing North, index correction — 3' 10", height of eye 19 feet: required the latitude. 5. In latitude 25° 20' S., the departure made good was 389 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from Start Point to St. Michael's. Lat. Start Point 50° 1 3' N. Long. Start Point 3° 38' W. Lat. St. Michael's 37 48 N. Long. St. Michael's 25 10 W. ADDITIONAL FOR ONLY MATE. 7. 1880, July i8th: find the time of high water, a.m. and p.m., at Bayonne, lie de Noirmoutier, Port Navalo, Belle Isle, and Bordeaux (by Admiralty Tables). 8. 1882, July 1 2th, at 5'' 9"" p.m. apparent time at ship, in latitude 29° 3' S., longitude 21° 53' W., the sun's magnetic amplitude was W. by N. | N. : required the true amplitude and error of the compass ; and supposing the variation to be 1 1° 20' W. : required the devia- tion for the position of the ship's heal when the observation was taken. 9. 1882, July 17th, P.M. at ship, latitude 31" 32' S., observed altitude sun's L.L. 13° 23' 10", index correction -\- 5", height of eye 16 feet, time by chronometer July 16'' 22'' 3™ 49', which was 9™ y"]* fast for mean noon at Greenwich, June 6th, and on June 14th was fast 8"^ 32»'6 on mean time at Greenwich : required the longitude. 3+0 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, July 4th, mean time at ship 8^ 39™ 2^ a.m., latitude 38° 10' S., longitude 78° 35' W., suq's bearing by compass N. 19° 16' E., observed altitude sun's l.l. 12° 16' 10", index correction — 2' 38'', height of eye 14 feet : required the true azimuth and error of the compass ; and supposing the variation be 17° 20' E. : required the deviation of the compass for the position of the ship's head at the tinae the observation was taken. 11. 1882, July 3i3t, P.M. at ship, latitude by account 45^5' S., longitude 83° 12' E., observed altitude sun's l.l. North of observer was 26° 15' 10", index corr. — 40', height of 63^6 19 feet, time by watch ii^^ 50™, (or 30'' 23'' 50™), which had been found to be 36™ 16' slow on apparent time at ship, the difference of longitude made to the West was 14 miles, after the error ou apparent time was determined : required the latitude by reduction to meridian. 12. 1882, July 5th, P.M. at ship, and uncertain of my position, when a chronometer showed July 5'* ih 4m 28 G.M.T., obs. alt. sun's l.l. 61° 15' ; and again, p.m. at ship same day, when chronometer showed July j'' 6'' 17" 2', obs. alt. sun's l.l. 20^ 18' 50", height of eye 24 feet, the ship having made 19 miles on a true W. by N. \ N. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 50° 40' N. and 5 1 ° o' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 50° 40' N. Long. 8° 12' W. B' = Lat. 51 o N. Long. 8 o W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, July 6th, the observed meridian altitude of the star a Scorpii CAntares), bearing North, 70° 10' 30", height of eye 21 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head by standard Compass. BeariQg of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E West. S.72°W. S. 70 W. S. 82 W. South .... s.w West N.W N.85°W. N. 78 W. N.70 W. N. 73 W, With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— E. by N. | N. ; S.E. by E. | E. ; S. by W. | W. ; N. 1° E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— W. 1° S. ; N. J E. ; N.E. by E. | E. ; S.E. | S. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at N.W. by W., find the bearings, correct magnetic. Compass bearings: — E. \ S., and E. ^ N. Bearings, magnetic ; — Ordina/ry Examination. 3+1 EXAMINATION PAPEE— No. VIII. FOR SECOND MATE. 1. Multiply 777 by 999, and 209-36 by 46, by common logarithms. 2. Divide 1 1 1 1 1 1 by 234, and 1962820 by 10-04, by common logaritlims. 3-— H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pts. I S.S.E. 7 2 S.W. I 6°E. A point of land in 2 6 4 lat. 0° 10' N., long. 3 6 4 ^73° 50' E., bearing by 4 6 4 compass S.W., dist. 5 W.N.W. 7 3 s.w. •i 18° W. 15 miles. Ship's head 6 7 S.S.E. Deviation as 7 6 7 per log. S 6 9 W. \ N. 5 4 S.W. by S. 2 16- W. 10 5 5 II 5 6 12 5 5 Variation 8° E. I S. by E. 1 E. 6 3 S.W. 1^ 5"E. 2 6 3 3 6 2 4 6 2 5 S.S.E. 5 2 s.w. If 5°E. 6 5 3 7 5 5 A current set the ship 8 6 S. by W. correct mag- 9 W. by N. 9 6 s.s.w. 17= W. netic 1 8 miles,from the 10 10 4 time the departure was II II 5 taken to the end of 12 II 5 the day. 4. 1882, August 12th, longitude 92' 12' E., the observed meridian altitude of sun's l.l. bearing North, was 42° 42' 10", index correction — 2' 50", height of eye 17 feet : required the latitude. 5. In latitude 56° 11' S., the departure made good was 356 miles East: required the diiference of longitude by parallel sailing. Required the course and distance from A to B. Latitude A 47° 50' S. Latitude B 40 49 S. Longitude 42° 16' E. Longitude 46 25 E. ADDIl lONAL FOR ONLY MATE. 7. 1880, August ist: find the times of high water, a.m. and p.m., at Ushant, Cadiz, Antwerp, and Penzance. 8. 1882, August 20th, at 5'' 16™ P.M., apparent time at ship, in latitude 42° 5' S., longi- tude 88° 36' W., the sun's magnetic amplitude was W. by S., variation 19" 15' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, August 7th, P.M. at ship, latitude 6"^ 4' N., observed altitude sun's l.l. 24" 5', index correction -j- i' 30", height of eye 12 feet, time by chronometer August 6"^ 20'' 30"" 36', which was ^S^'z fast for Greenwich mean noon, July 14th, and on July 21st was io» slow for Greenwich mean time : required the longitude. 342 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, August 20th, mean time at ship 2'' 35"" 25^ p.m., latitude 52° 2' S., longitude 89° 26' E., sun's bearing by compass N.W. | N., observed altitude sun's l.l. 17° 26', index correction -j- i' 45", height of eye 21 feet: ri^quired the error of compass; and supposing the variation to be 33° 50' E. : find the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, August nth, A.M. at ship, latitude account 39° 3' S., longitude 157° 25' E„ observed altitude of sun's l.l., North of observer, was 34° 37', height of eye 12 feet, time by -watch 7'> 41"' 25" (or lo** i9*> 41'" 25^), which had been found to be 3'> 41™ 8^ slow on apparent time at ship, the difference of longitude made to the East was 33', after the error on apparent time was determined. 12. 1882, August loth, A.M. at ship, and uncertain of my position, when a chronometer showed August 9' zi^ S'" 37* M.T.G., obs. alt. sun's l.l. 38° 17'; again, p.m. at ship same day, when chronometer showed August ic' 3^ ijn> 57% obs. alt. sun's l.l. 40° 10' 15*, height of ej'e 18 feet, the ship having made 2 1 miles on a true N.E. course in the interval : required the line of position when the first altitude was taken, and the ship's position by Sumner's Method when the second altitude was observed, assuming latitudes 49° 40' N. and 50° o' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, August 20th, the observed meridian altitude of the star a Aquilae (AltairJ, bearing North, was 66° 51' 10", index correction + 58", height of 13 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East .... S.E S. i2°E. S. 10 E. S. 6 W. S. 9 W. South S.W West .... N.W S. 4°E. S. 17 E. S. 20 E. S. 16 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses:— E. by N. ^ N. ; S.W. by W. | W. ; N.N.W. | W. ; E.S.E. Compass courses : — Supposing you have steered the foUowiug courses by the Standard Compass, find the correct magnetic courses made, from Iho above deviation table. Compass courses :-N.W. | N. ; W. by S. | S. ; N. i W. ; W. | S. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at W.N.W., find the bearings, correct magnetic. Compass bearings:— S.S.E. iind S.E. by S. . Bearings, magnetic : — ' Ordinary Examination. 343 EXAMINATION PAPEE— No. IX. FOR SECOND MATE. 1. Multiply 247'55 by 56-72, and "03948 by OT959, by common logarithms. 2. Divide (><)"]S^S by 975^4, and 33248100 by 830000, by common logarithms. 3 — H. Courses. K. T^ Winds. Lee- way. Deviation. Remarks, &c. ptB. I S.E. \ E. 10 4 S. by W. \ W. 1 15° E. A point, lat. 54° 7' N. 2 10 2 long. 0° 5' W., bearing 3 10 2 by compass W. f N., 4 10 2 dist. 20 miles. Ship's 5 S.E. by S. 121 4 S.W. by S. \ 5°E. head S.E. ^E. Devia- 6 12; 6 tion as per log. 7 12 5 8 12 5 9 East. II 2 N.N.E. i 21° E. 10 10 6 II 10 1 6 12 10 6 Variation 22° W. I S.E. 1 S. 8 5 S.S.W. ^ w. I 7°E. 2 8 3 3 8 2 4 8 5 S. by E. ^ E. . 7 3 S.W. ^ w. li 3rE. 6 6 8 7 6 6 A current set the ship 8 6 3 E. hy S. \ S. (correct 9 S.E. by E. 8 6 N.E. by E. I i7°E. magnetic) 32 miles, 10 9 2 from the time the de- II 9 2 parture was taken to 12 9 the end of the day. 4. 1882, September 23rd, in longitude 123° 45' E., observed meridian altitude of sun's L.L. bearing North, was 89° 49' 50', index error — 52", height of eye 26 feet: required the latitude. 5. In latitude 20° 15' S., the departure made good was 352 miles W. : required the dif- ference of longitude by parallel sailing. Latitude A 25° 39' N. Latitude B 34 28 S. Longitude A 48° 19' W. Longitude B 18 28 E. ADDITIONAL FOR ONLY MATE. 7. 1880, September nth: find a.m. and p.m. tides at Alderney, Heligoland, Nieuport, Fowey, Hastings, and Dornock Road. 8. 1882, September 30th, at j*'- 45"" p.m., apparent time at ship, latitude 52° 30' N., longi- tude 12° 10' W., sun's magnetic amplitude is.W. | W. : required error of compass; and supposing the variation to be 30° 28' E. : required the deviation of the compass for the posi- tion of the ship's head at the time of observation. 9. 1882, September ist, p.m. at ship, latitude 9" 9' N., observed altitude sun's l.l. 62° 13' 14", index correction -j- 15", height of eye 16 feet, time by chronometer August 3i<* 15^ 34" 28% which was 2>»> io» slow for mean noon at Greenwich, July 28th, and on August 12th was i"" 31' slow on mean noon at Greenwich : required the longitude. 344 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, September i6th, mean time at ship 8^ 3"^ i8« a.m., latitude 4° 22' N., longitude 81° 39' W., sun's bearing by compass N. 93" 20' E., observed altitude sun's l.l. 29° 30' 30", index correction + i'22", height of eye 20 feet: required the true azimuth and error of compjiss; and supposing the variation is 8° 20' E. : required the deviation for the position of the ship's head at the time of observation. ir. 1882, September 23rd, a.m. at ship, latitude acct. 27° 32' S., longitude 168° 51' E. observed altitude sun's l.l., North of observer, was 61° 59' 40", index correction — i' 50", height of eye 18 feet, time by watch 1 1*> 10™ 10' (or 22"* 23^ 10™ 10'), which had been found to be 31™ 31^ slow on apparent time at ship, the difference of longitude made to the East was 24'-4, after the error on apparent time was determined : required the latitude. 12. 1882, September 9th, a.m. at ship, and uncertain of my position, when a chronometer showed September %^ i6'> 23" 50' M.T.G., obs. ;ilt. sun's l.l. xi° 2' 30"; again, a.m. at ship same day, when chronometer showed September %^ 20^ 39"" 25*, obs. alt. sun's l.l. 45° 45' 30'', height of eye 18 feet, the ship having made 19 miles on a true S. 76° E. course in the interval : required the line of position when the first altitude was observed, and the ship's position by Sumner's Method when the second altitude was taken, assuming the ship to be between the latitudes 47° 50' N. and 47° 10' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 47° 50' N. Long. 35° 58^ E. B' = Lat. 47° 10' N. Long. 33° 27J' E. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, September 7th, the observed meridian altitude of star Arcturus was 86° 35' 50*, bearing North, index correction — i' 10", height of eye 12 feet: required the latitude. In the following table give the correct magnetic be.iring of the distant object, and thence the deviation : — Correct magnetic bearing. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses : — W. by S. J S. ; N. J E. ; E. | N. ; S.E. \ E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses : —North ; S.S.W. J W. ; E. by S. J S. ; N.E. \ E. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at N.N.E. | E., find the bearings, correct magnetic. Compass bearings : — N. 79° E. and W. ^ S. Bearings, magnetic ; — Ordinary Examination. 345 EXAMINATION PAPEE— No. X. FOR SECOND MATE. Multiply 560072 by 50, and 10-5526 by 3i7*i45, by common logarithms. Divide 849 1 "9 by 984, and 2064840 by 3800-62, by common logarithms. 3-— H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pts. I w.s.w. II South. 1 11- W. A point, Cape Fare- 2 II well, in lat. 59° 49' N., 3 10 4 long. 43°54'W., bear- 4 10 6 ing by compass 5 West. 5 S.S.W. I i6» W. N. |- E., distance 36 6 5 miles. Ship's head 7 4 5 W.S.W. Deviation as 8 4 5 per log. 9 S.E. 13 S.S.W. h 10^ E. 10 12 2 II 12 4 12 12 4 I S. by E. 6 S.W. by W. 2i 4»E. a 5 5 Variation 70° W. 3 5 4 4 5 5 S.W. by 8. I 5 S.E. by S. 3i 5° W. 6 I 5 7 I 5 A current set the 8 I 5 shipfcorrectmagnetic) 9 s.w. 6 W.N.W. If 6° W. S.S.E., 48 miles, from 10 5 6 the time the departure II 5 4 was taken to the end 12 •5 of the day. 4. 1882, October 20th, in longitude 150° 25' W., observed meridian altitude of sun's l.l., bearing North, was 49° 58' 50", index correction + i' 10'', height of eye 19 feet: required the latitude. 5. In latitude 59° 36' N., the departure made good was 52-9 miles East : required the dijSerence of longitude by parallel sailing. 6. Required the course and distance from A to B, by calculation on Mercator's principle. Latitude A 9° 36' S. Latitude B 7 16 S. Longitude A 2° 10' W. Longitude B i 24 W. ADDITIONAL FOR ONLY MATE. 7. 1880, October ist : find the times of high water a.m. and p.m. at Gibraltar, Ramsgate, Wick, and Berwick. 8. 1882, October 9th, at 5'' 51'" a.m., apparent time at ship, latitude 18° 45' S., longitude 99° 18' E., sun's maijnetic amplitude E. ^ N. : required the true amplitude and error of the compass ; and supposing the variation to be 1° 50' W. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, October 30th, p.m. at ship, latitude 32''45'N., observed altitude sun's l.l. 28° 30', index correction + 2° 30", height of eye 18 feet, time by chronometer Oct. 30"* 11^ 56" 43% which was slow 2™ 28' for mean noon at Greeawich, October ist, and on October 8th was 2"" 44'"8 slow for mean noon at Greenwich: required the longitude. y Y 346 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, October ist, mean time at ship j^ 54™ p.m., latitude 17° 8' S., longitude 152° 33' E., sun's bearing by compass W. \ N., observed altitude sun's l.l. 13° 59', index correction — 22", height of eye 17 feet : required the true azimuth and error of the compass ; and supposing the variation is 7° 40' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, October 2nd, a.m. at ship, latitude account 38° 12' N., longitude 23° 34' W., observed altitude of sun's l.l., South of observer, -was 47° 30', index correction — i' 38", height of eye 17 feet, time by watch !*> 50" (or z^ 1^ 5°")) which had been found to be 2^ 10™ fast on apparent time at ship, the difference of longitude made to the East was 43', after the error on apparent time was determined : required the latitude. 12. 1882, October ist, a.m. at ship, and uncertain of my position, when a chronometer showed October i"* o'> 8™ 56^ M.T.G-., obs. alt. sun's l.l. 23° 19' 50" ; again, p.m. at ship same day, when chronometer showed October i'' 5^ 53™ 52", obs. alt. sun's l.l. 25° 30', height of eye 23 feet, the ship having made 29 miles on a true N.W. f W. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 50° 30' N. and 51° o' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 50° 30' N. Long. 49° 54f ' W. B' =z Lat. 51 o N. Long. 50 40 W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, October 7th, the observed meridian altitude of a Pegasi (MarkabJ bearing South, was 54° 10' 15", height of eye 13 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object and thence the deviation. Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head ^^y standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 37°W. S. 55 w. S. 60 w. S. 57 W. South .... S.W West N.W S. 41' w. S. 20 w. S. 14 w. S. 19 w. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses:— S.E. ; N.E. :^ E. ; S. 10° W. ; E. ^ N. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :-S.S.E. f E. ; S. f W. ; E. by N. | N. ; N. i W. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at S.E. by E. \ E., find the bearings, correct magnetic. Compass bearings:— E. by S. | S., and W.N.W. Bearings, magnetic : — Ordina/ry Examination. 347 EXAMINATION PAPER— No. XI. FOR SECOND MATE. Multiply 45*3 liy 976, and 40*405 by lO'S, by common logarithms. Divide loo'ooz by I'ooiz, and 829440 by 288, by common logarithms. 3 — H. Courses. K. t^ Winds. Lee- way. Deviation. Remarks, &c. pts. I N. by E. 4 2 E. by N. H 3°E. A point of land, lat. 2 3 8 52° N., lonj^. 120" E., 3 4 5 bearinfj^ by compass 4 4 5 N.byE. 1 E.. dist. 16 5 N.E. 1 E. 4 5 N. by W. 3^ 17° E. miles. Ship's head 6 5 N. by E. Deviation 7 5 as per log. 8 4 5 9 W. .^ N. 7 5 N. by W. 'f 17° W. 10 7 5 II 8 12 8 I E.S.E. 4 5 South. 3 13° B. Variation 25° E. 2 4 5 3 4 4 4 5 S.E. i S. 4 6 N. by E. 9'E. 6 4 5 7 4 8 A current set (cor- 8 5 I rect maL;netic)E.N.E., 9 W. f s. 6 N.W. by N. 3i 14" W. 22 miles, from the time 10 6 3 the departure was ir 6 4 taken to the end of 12 6 3 the day. 4. 1882, November 15th, in longitude 80° 11' E., the observed meridian altitude of sun's ii.L. was 67° 44', bearing North, index error -\- i' 38", height of eye 15 feet : required the latitude. 5. In latitude 40° 50' S., the departure made good was 149 miles East : required the dif- ference of longitude by parallel sailing. 6. Required the course and distance from the ship's position to the Lizard, by calculation on Mercator's principle. Latitude of position i7°5o'N. Longitude of position 76^42' "W. Latitude of Lizard 49 58 N. Longitude of Lizard 5 12 "W. ADDITIONAL FOR ONLY MATE. 7. 1880, November 12th : find a.m. and p.m. tides at Newhaven, Torbay, Kiltush, and St. Nazaire. 8. 1882, November loth, at 4*" 3™ 52' a.m., apparent time at ship, latitude 58" 13' S., longi- tude 55° 47' E., sun's magnetic amplitude S. by E. ^ E. : required the true amplitude and error of compass; and supposing the varintion to be 16' 30' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, November 30th, a.m. at s-hip, latitude 40° 40' S., observed altitude sun's l.l. 39° 30', inde.x correction -f 6' 24", height of eye 22 feet, time by chronometer, November 30'' 2'> 58'" 45% which was io"> ^o'-^fast for mean noon at Greenwich, October 13th, and on October 25th, was lo"' ^S'fast for mean noon at Greenwich : required the longitude. 348 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, November 15th, mean time at ship 2^ 46™ 43^ p.m., latitude 45° 31' S., longitude 119° 56' W., sun's magnetic azimuth S. 985° W., observed altitude sun's l.l. 43° 45', index correction — 56", height of eye 20 feet : required the error of compass ; and supposing the variation to be 7° 50' W. : required the deviation for the position of the ship's head at the time of observation. 11. 1882, November i3tb, a.m. at ship, latitude acct. 50° 52' S., longitude 48° 52' W., observed altitude sun's l.l. was 56° N., index correction -\- 23", height of eye 19 feet, time by watch 4"^ 34* (or 13'' d^ 4"> 34^), which had been found to be 43™ 24* fast on apparent time at ship, the difference of longitude made to West was 9 miles after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, November ist, a.m. at ship, and uncertain of my position, when a chronometer showed October 31'* 23^ 19™ lo^ M.T.G., obs. alt. sun's l.l. 19° 8' ; again, p.m. at ship same day, when chronometer showel November i<* 4'' 33'" 29^, obs. alt. sun's l.l. 12° 38', height of eye 15 feet, the ship having made 12 miles on a true S.S.W. course in the interval : required the line of bearing when the first altitude was taken, and the ship's position by Sumner's Method when the second altitude was observed, assuming the ship to be between the latitudes 50° 50' N. and 50° 20' N. For the second observation the positions as given by the Examiner would be as follows :— B = Lat. 50° 50' N. Long. 25° 20' W. B' = Lat. 50 20 N. Long. 24 36 W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, November 7th, the observed meridian altitude of the star a Piscis Australia (FomalhautJ , bearing North, was 59° 40', index correction -\- i' 12", height of eye 23 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation: — Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object bj' Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 88° W. S. 70 w. S. 68 W. S. 80 W. South .... S.W West .... N.W N. 88° W. N. 80 W. N. 72 W. N. 75 W. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— W. by S. ^ S. ; N.W. by W. ^ W. ; E. by S. J S. ; S. :| E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— W. | N. ; N. 42° W. ; S. 64° E. ; N.E. \ N. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at W. by N. \ N., find the bearings, correct magnetic. Compass bearings : — W. ^ N., and S. 36° E. Bearings, magnetic ;— Ordinary Examination. H9 EXAMINATION PAPEE— No. XII. FOR SECOND MATE. 1. Multiply 758900 by 13-5, and 0-006994 by 0-33318, by common logarithms. 2. Divide 999-43 by 67-832, and 1875000 by 15000, by common logarithms. 3-— H. Courses. K. ■^ Winds. Lee- way, Deviation. Remarks, &c. pts. I E. by N. i N. 3 5 N. i E. 2 ii»E. A point of land in 2 3 3 lat. 50° N. long. 40° W. 3 4 beciring by compass 4 4 2 E.N.E. i E., distance 5 W.N.W. 6 5 North. 4 10° W. 1 6 miles. Ship's head 6 6 2 E. by N. 1 N. Devia- 7 5 6 tion as per log. 8 4 7 9 s.s.w. \ w. 4 2 West. ^h 4°W. 10 4 II 3 6 12 3 2 I N.N.W. \ W. 5 6 West. i| 5°W. Variation 36^° W. 2 5 6 3 6 4 4 6 4 5 S.E. f E. 6 3 S.S.W. 'i 7"E. 6 5 6 7 5 2 A current set the 8 5 ship(correct magnetic) 9 S.W. f W. 2 5 S. by E. i E. 3i 6° W. S.S.W., 6 miles, from 10 3 2 the time the departure 11 3 was taken to the end 12 3 3 of the day. 4. 1882, December 3i8t, longitude 123° 45' W., observed meridian altitude of sun's l.l. was 67° 8' 10", bearing South, index correction -\- 9", height of eye 13 feet: required the latitude. 5. In latitude 60=' N., the departure made good was iii miles East: required the dif- ference of longitude by parallel sailing. 6. Required the course and distance from Port San Francisco to Cape Palliser, by Mercator's Sailing. Lat. Port San Francisco 37° 48' N. Long. Port San Francisco 122° 24' W. Lat. Capo Palliser 41 38 S. Long. Cape Palliser 175 21 E. ADDITIONAL FOR ONLY MATE. 7. 1880, December 28th : find the times of high water, a.m. and p.m., at Skull, Westport, Valentia, Limerick, Coleraine, and Tenby. 8. 1882, December 28th, at 4'' 11™ 13' a.m., apparent time at ship, latitude 46° 47' S., longitude 179° 54' W., sun's magnetic amplitude S.E. by E. | E. : required error of compass ; and supposing the variation to be 15° 30' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, December 24th, a.m. at ship, latitude 33° 33' S., observed altitude sun's l.l. 40° 40' 40', index correction -|- 2' 20", height of eye 19 feet, time by chronometer 8'' 7" 37" P.M., which was 6™ 8* slow for Greenwich mean noon, October 3i8t, and on November 12th, was 7™ i6»-2 slow for mean noon at Greenwich : required the longitude by chronometer. 350 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, December 27th, mean time at ship ^^ o" lo' a.m., latitude 15° 12' N., longitude 130° W., sun's bearing by compass E. by S. | S., observed altitude sun's l.l. 20° 15', index correction + 2' 5', height of eye 16 feet : required the error of compass ; and supposing the variation to be 7° 20' E. : required the deviation of the compass for the position of the ship's head when the observation was taken. 11. 1882, December 4th, a.m. at ship, latitude account 51° 54' S., longitude 30" 10' W., observed altitude sun's l.l.. North of observer, 59° 59', index correction — 3' 12", height of eye 20 feet, time by watch i2"> 10' (or 4.^ o^ 12™ lo^), which had been found to be 42'" io» fast on apparent time at ship, the difference of longitude made to the West was 10 miles after the error on apparent time was determined : required the latitude. 12. 1882, December 25th, a.m. at ship, and uncertain of my position, when a chronometer showed December 24'^ 22'^ ii" 2^ M.T.G., oba. alt. sun's l.l. 9° 8'; again, p.m. at ship same day, when chronometer showed December 25"^ i^ 39'", obs. alt. sun's l.l. 15° 3', height of eye 15 feet, the ship having made 35 miles on a true S. 54° E. course in the interval: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 46° 30' N. and 46° o' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1 882, December 2 1 st, the observed meridian altitude of star a Canis Minoris (PtoeyonJ was 52° 51' 50", bearing North, index correction — 49", height of eye 21 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. 1 Ship's Head Bearing of by Distant Object Standard by Standard Compass. Compass. Deviation Required. North .... N.E East S.E S. 2°W. S. 4 W. S. 10 w. S. 16 w. South .... S.W West N.W S. 5°E. S. 24 E. S. 16 E. S. 3 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— N. 78° E. ; E.S.E. ; S.W. by W. J W. ; N. | W. Compass courses: — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass Courses :— S.S.E. ; W. by S. J S. ; S. by W. | W. ; S.E. f S. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at S.W. by S., find the bearings, correct magnetic. Compass bearings:— W. by S. \ S. and N.N.W. Bearings, magnetic; — Ordinm-y Examination. 351 EXAMINATION PAPEE— No. XHI. FOR SECOND MATE. I. Multiply 198400 by &k,, and 448000 by -0000448, by common logarithms. J. Divide 2208000 by 3450, and '085375 by "07425, by common logarithms. 3— H. Courses. K. T^ Winds. Lee- way. Deviation. Remarks, &c. I s.s.w. \ w. 12 2 West. pts. 4 42° W. Apoint,lat.62'»i8'N. 2 12 6 long. 83° 17' E., bear- 3 '3 2 ing by compass 4 '3 N. by E. i E., 5 S.W.fW. II 5 S. by E. 1 5 8°W. dist. 23 miles. Ship's 6 II 4 heiid S.S.W. Devia- 7 II I tion as per log. 8 II 9 E.f S. 5 4 S. by E. If 15° E. 10 5 6 II 5 6 12 5 4 I W.N.W. 4 4 North. 3 19° W. Variation 42° E. 2 4 4 3 4 2 4 5 5 N.W. \ N. 10 6 S. by W. i6°3o'W. 6 10 2 7 II 4 A current set the ship 8 II 8 (correct magnetic) 9 E.f N. 3 2 N. by E. 3i 17° 14' E. S.W. I W., 52 miles, ID 3 2 from the time the de- II 3 2 part\ire was taken to 12 2 4 the end of the day. 4. 1882, August nth, in longitude 92" 12' E., observed meridian altitude sun's l.l. was 42° 42' 10', zenith South of sun, index correction — 2' 50", height of eye 17 feet: required the latitude. 5. In latitude 80° the departure made good was 80 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 51° 30' N. Longitude A 3° 30' 30" W. Latitude B 20 o N. liongitude B 33 4 56 W. ADDITIONAL FOR ONLY MATE. 7. 1880, July 24th : required the times of high water, a.m. and p.m., at Point de Qalle, long. 80° E., St. Nazaire, and Jersey. 8. 1882, October 28th, at %^ 30" a.m., apparent time at ship, in latitude 49° 40' N., longitude 116° 12' W., the sun's magnetic amplitude was E. 10° 40' N. : required the error of compass, and supposing the variation to be 23° 50' E. : required the deviation for the position of the ship's head at the time of observation. 9. 1882, April 1 8th, a.m. at ship, latitndo 50° 48' N., observed altitude sun's l.l. 38° 10' 50", index correction -j- 45", height of eye 16 feet, time by chronometer 9"^ 27™ 2% a.m. at Greenwich, which was o™ 49'*3 slow for mean noon at Greenwich, March 17th, and on April ist was i" ^Z"-"] fast for mean time at Greenwich: required the longitude. 352 Ordinm-y Examination. ADDITION A.L FOR FIRST MATE. 10. 1882, March 9th, mean time at ship 8'' ii"" 42^ a.m., latitude 29° 58' S., longitude 57° 24' E., observed altitude sun's l.l. 28° 23' 15', height of eye 16 feet, sun's azimuth E. 9° 40' S. : required the error of compass ; and supposing the variation to he 17° 10' W. : required the deviation for the position of the ship's head at the time of observation. 1 1. 1882, July 28th, A.M. at ship, latitude account 38° 54' N., longitude 39° W., observed altitude sun's l.l. 69° 10' S., index corr. 4" i' 27", height of eye 23 feet, time by watch iih jm ij8^ slow on apparent time at ship 28™ 45^ the difference of longitude made to East was 32 miles after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, February 28th, a.m. at ship, and uncertain of my position, when a chronometer showed February 28'! 9>» 4™ 128 M.T.G-., obs. alt. sun's l.l. 27" 31'; again, p.m. at ship same day, when chronometer showed February 28"* 12'' 25™ 35^, obs. alt. sun's l.l. 32° 40'^ height of eye 20 feet, the ship having made 27 miles on a true N.E. \ E. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 47° 10' N. and 47" 40' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 47° 10' N. Long. 165° 23' W. B' ^ Lat. 47° 40' N. Long. 167° 23' W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, October 8th, the observed meridian altitude of a Gruis was 50° o', bearing South, index correction — i' 12", height of eye 17 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation. Correct magnetic bearing. SMp's Head ijy Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. Ship's Head ty Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 25" w. S. 21 W. S. 21 W. S. 16 w. South .... S.W West N.W S. 1° w. S. 7 E. S. 6 W. S. 21 W. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— S.W. by W. ; E.N.E. ; S. by W. J W. ; N.N.E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magneiic courses made from the above deviation tablp. Compass courses :— N.E. by E. ; N.W. J N. ; N. ^ E. ; S, by E. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with ship's head at S.S.W. \ W., find the bearings, correct magnetic. Compass bearings : — N.E. by E. and S.W. by W. Bearings, magnetic : — Ordinary 'Examination. 3?3 EXAMINATION PAPER— No. XIY. FOR SECOND MATE. 1. Multiply i73'4 by i'734, and 60030041 by -000273004, by common logarithms. 2. Divide 57634* i by 276, and 471 by 964325, by common logarithms. 3 — H. Courses. K. 10 Winds. Lee- way. Deviation. Remarks, &c. pts. I E.iS. 10 4 S. by E. \ E. 1 4 27F w. A 1 oint of land in 2 10 6 lat. 30° 16' N.,_ Ion-. 3 10 6 179° 52' E., bearin<^ by 4 10 4 compass N. by E. \ E., 5 E. by S. 1 S. 8 6 S. by E. '1 34F W. dist. I ^ miles. Ship's 6 8 4 head E. ^ S. De- 7 7 8 vjation as per log. 8 7 2 9 W. by S. \ S. 9 6 South. 4 7°E. 10 9 4 II 9 5 12 8 5 I S. |W. 9 3 S.E. by E. 'i 46° W. Variation 12° 30' E. 2 9 4 3 4 5 9 3 N. iW. 9 9 5 E.N.E. * 50° E. 6 9 5 7 9 6 8 9 4 A current set the 9 E.JS. 8 4 S.S.E. 2i 28° W. ship N. 1 W. (correct 10 7 6 magnetic), 17 j miles, II 6 4 from the time the de- 12 5 6 parture was taken to the end of the day. 4. 1882, March 20th, in longitude 174° "W., the observed meridian altitude of the sun's L.L. was 89' 56' 10", bearing North, index correction — i' 15", height of eye 15 feet : required the latitude. 5. In latitude 71° 44' N., the departure made good was 164 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by calculation on Mercator's principle. Lat. A 2» 49' N. Long. A 130° 9'E. Lat. B 20 S. Long. B 82 16 W. ADDITIONAL FOR ONLY MATE. 7. 1880, June ist : find the time of high water, a.m. and p.m., at Plymouth Breakwater, Falmouth, Exmouth, and Flambro' Head. 8. 1882, December 6th, at o'' 16™ a.m. apparent time at ship, latitude 67° 19' S., longitude 19° 2' E., the sun's magnetic amplitude was S. | W. : required the true amplitude and error of the compass ; and supposing the variation to be 24° 50' W. : required the deviation lor iho position of the ship's head when the observation was taken. 9. 1882, September ist, a.m. at ship, lat. 47^48' N., observed altitude sun's l.l. 39° 46' 50", index correction -f- 2' 10", height of eye 13 feet, time by chronometer August 31"* 10'' 17°' 20', which was 4™ 50' slow for mean noon at Greenwich, May ist, and on June 30th was slow gm 20' on mean time at Greenwich : required the longitude. Z Z 354 Ordinary Examination. ADDITIONAL FOR FIRST MATE. ro. 1882, March 20th, mean time at ship 9^ 35" a.m., latitude 43° 18' N., longitude 32° 25' W., sun's bearing by compass S. \ E., observed altitude sun's l.l. 35° 2' 50", height of eye 1 2 feet : required the true azimuth and error of the compass ; and supposing the variation be 27° 10' W. : required the deviation of the compass for the position of the ship's head at the time the observation was taken. 11. 1882, September 23rd, a.m. at ship, latitude by acct. 28° 5' S., longitude 170° 57' E.^ observed altitude sun's l.l. North of observer was 61° 40' height of eye 19 feet, time by watch September 21'^ 23^ iif" 4^, which had been found to be 36™ 29* slow on apparent time at ship, the difiFerence of lon2;itude made to the East was 27*5 miles, after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, January 17th, a.m. at ship, and uncertain of my position, when a chronometer showed January 16'' 22^ 10™ Gr.M.T., obs. alt. sun's l.l. 11° 8' 30'; and again, p.m. at ship same day, when chronometer showed January 17"* 2'^ 29™ 24^, obs. alt. sun's l.l. 13° 59' 30", height of eye 18 feet, the ship having made 24 miles on a true N.W. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 50° 50' N. and 51° 20' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 5o» 50' N. Long. 5° 59I' W. B' -^ Lat. 51 20 N. Long. 7 34 W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, December 21st, the observed meridian altitude of the star a Cygni Deneb, bearing North, 56° 18' 10", height of eye 15 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Be quired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. North .... N.E East S.E N. 77°W. N. 88 W. S. 85 W. S. 78 W. South .... S.W West N.W S.62°W. S. 59 W. S.77 w. N.81 w. With the deviation as above, give the courses you would steer by the Standard Oompasa to make the following courses correct magnetic. Correct magnetic courses :— N.E. by E. ; S.W. by S. ; E.S.E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— W. J S. ; N.W. by N. ; N.E. by N. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at N. W., find the bearings, correct magnetic. Compass bearings:— North, and E.S.E. Bearings, magnetic : — Orddnary Examination. 355 EXAMINATION PAPEE— No. XV. FOR SECOND MATE. 1. Multiply looooi by 8, and 160800 by 325, by common logarithms. 2. Divide 37"i49 by 523'76, and 615 by '03075, by common logarithms. 3-— H. Courses. K. -1^ Winds. Lee- way. Deviation. Remarks, &c. pts. I S. by E. J E. 3 3 S.W. i W. 4 5°E. A point, Cape of 2 3 4 Good Hope, latitude 3 3 6 34" 28' S., longitude 4 3 7 18^ 28' E., bearing by 5 N.W. 1 W. 2 i 4 S.W. by W. 3i 18° W. compass N. i W. f W. 6 2 3 dist. 21 miles. iShip's 7 2 3 head S. by E. \ E. 8 2 Deviation as per log. 9 N. by E. J E. 4 6 I^ .w. \ w. -i 6°E. 10 4 4 II 4 7 12 4 3 I S.W.iW. iW. 5 6 N.W. ^ W. ^i 7°W. Variation 25° W. 2 5 7 3 5 4 4 5 3 5 W. by N. i N, 7 5 N. 1 W. h i6° W. 6 7 6 7 7 2 A current set the ship 8 6 7 (correct magnetic) 9 N.E. \ E. J 3 E.S.E. 2 16° E. E. by S. \ S., 14 miles, 10 4 4 from the time the de- II J 5 parture was taken to 12 5 4 the end of the day. 4. 1882, February nth, in longitude 32° 20' E., the observed meridian altitude of sun's L.L. was 30" 25' 10", observer North of sun, index correction — 3' ij", height of eye 12 feet : required the latitude. 5. In latitude 51° 10', the departure made good was 64"3 miles : required the difiFerence of longitude. 6. Required the course and distance from A to B, by Mercator's Sailing. Lat. of A 43° 24' S. Long, of A 65" 39' W. Lat. ofB 26 38 N. Long, of B 15 E. ADDITIONAL FOR ONLiT MATE. 7. 1880, April 2nd : find times of high water at Cape Virgin, longitude 68° W., Water- ford Harbour, and Banff. 8. 1882, March 3i8t, at 6*> i™ 48" a.m. apparent time at ship, in latitude 6° 31' N., longi- tude 155° 10' E., the sun's magnetic amplitude was E. 3° 51' S. : required the true amplitude and error of the compass ; and supposing the variation to be 6° E. : required the deviation of the compass for the position of the ship's head when the observation was taken. 9. 1882, May a7th, a.m. at ship, latitude 55° N., observed altitude sun's l.l. 43° 9' 5", index error — 14", height of eye 14 feet, time by chronometer 9'' 13™ 12* a.m., which was 48«-5 slow for mean noon at Greenwich, AprU 9th, and on April 24th, was/as< 25' : required the longitude. 356 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, July 10th, gh 44m A.M., mean time at ship, latitude 59° 56' N., longitude 40° 20' W., sun's magnetic azimuth S. J W., observed altitude sun's l.l. 44° 49', height of eyo 20 feet: required the error of the compass, and supposing the variation be 51° W. : required the deviation of the compass for the position of the ship's head at the time the observation was taken. 11. 1882, November 8th, p.m. at ship, latitude by account 33° 9' N., longitude 89° 42' E., observed altitude sun's l.l. South of observer was 40° o', index error — 6' 12", height of eye 19 feet, time by watch S*^ 20'^ 20', (or 7^^ 20^ 20™ 20»), which had been found, to be 4'> 8™ 12S slow on app. time at ship, the difference of longitude made to the Hast was 32'-3, after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, April 25th, P.M. at ship, and uncertain of my position, when a chronometer showed April 25"^ 5I1 13™ 20' M.T.G., obs. alt. sun's l.l. 51° 3' 42"; again, p.m. at ship same day, when chronometer showed April 2^^ 9^ 12™ 4% obs. alt. sun's l.l. 16° 20' 3", height of eye 18 feet, the ship having made 21 miles on a true N.E. ^ N. course in the interval : required the line of position when the first altitude was observed, and the ship's position by Sumner's Method when the second altitude was taken, assuming the ship to be between the latitudes 48° 10' N. and 48° 40' N. For the second observation the position as given by the Examiner would be as follows :— B = Lat. 48° 30' N. Long. 58° 26' W. B' = Lat. 49° o' N. Long. 58° 24f' W. ADDITIONAL FOR MASTER ORDINARY. i 3. 1882, July 19th, the observed meridian altitude of the star a Pavonis, bearing South, was 32° 50' 15", index correction + 4' 48", height of eye 23 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation. Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East .... S.E S. 44°E. S. 56 E. S. 39 E. S. 12 E. South .... S.W West .... N.W S. ii°W. S. 13 w. S. 4 W. S. 12 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— N.N.W. ; W.N.W. ; S.W. by W. ; W.S.W. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— E.N.E. ; S.S.E. ; N.W. by W. ; N.E. by E. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at N.E. g E., find the bearings, correct magnetic. Compass bearings : — N. by W. and E. by N. Bearings, magnetic :— Ordinary Examination. 357 EXAMINATION PAPEE— No. XVI. FOR SECOND MATE. 1. Multiply io8'58i by 500, and 964'3204 by "000690041, by common logarithms. 2. Divide 408848 by 202, and '000694321 by •000014798, by common logarithms. 3 — H. Courses. K. -1^ Winds. Lee- Deviation. Remarks, &c. way. I W. by N. \ N. 8 9 N. by W. pts. 21° lo'W. A point of land in 2 8 3 lat. 59° 16' S., long. 3 8 4 179° 42' W., bearing 4 8 4 by compass S.E. ^ E., 5 W. f N. 9 4 N. by W. 2i 19'^50'W. dist. 13 miles. Ship's 6 9 8 head W. by N. i N. 7 8 6 Deviation as per log. 8 8 4 9 W.^N. 9 8 N. by W. ^i i8°i5'W. lO 9 7 rr 9 6 12 9 5 Variation 15° 40' E. I S.S.W. I \v. 8 S W. ^N. 3i 4°3o'W. 2 8 2 3 8 5 4 8 6 5 S.S.E. i E. 3 6 s.w. f 27° 40' E. 6 3 2 7 4 8 A current set the ship 8 5 6 S. 1 E., correct mag- 9 W. 1 S. 8 6 South. ^i 19° 30'W. netic 21 miles, from the 10 8 2 time the departure was II 8 3 taken to the end of 12 8 4 the day. 4. 1882, May i6th, longitude 45° 26' W., the observed meridian altitude of sun's l.l. bearing North, was 86° 34' 20", index correction -j- 4' 16", height of eye 15 feet: required the latitude. 5. In latitude 44° 20' S., the departure made good was 44*2 miles : required the difference of longitude by parallel sailing. 6. Required the courBe and distance from A to B, by calculation on Mercator's principle. Latitude A 9° 2' N. Longitude 171° 19' W. Latitude B i 2 S. Longitude 83 17 E. ADDITIONAL FOR ONLY MATE. 7. 1880, April 20th : find the times of high water, a.m. and p.m., at Abervrach, Morlaix, Gibraltar, and St. Nazaire. 8. 1882, June 24th, at ii*" 5™ p.m., apparent time at ship, in latitude 66° 31' N., longi- tude 9° W., the sun's magnetic am[pliiu le was N. f E. : required the true amplitude and frror of the compass ; and supposing tho variaion to be 33° 30' W. : required ihe deviation cif the compass for the position of the ship's head at the time of observation. 9. 1882, June i5lh, a.m. at ship, latitude 29"^ 10' S., observed altitude sun's l.l. 20" 40', index correction o, heiglit of eye s • feet, time by chronometer June 14"^ 22'' 59™ 20^, which was 4™ 35'/rts< for Greenwioh mean noon, March 20th, and on May 3rd was i"> 17' slow for Q-reenwich mean time : required the longitude. 35« Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, September 22nd, mean time at ship o^ 42'" p.m., latitude 49° 40' S., longitude 146° 56' W., sun's bearing by compass N. by E. \ E., observed altitude sun's l.l. 40° 18', height of eye 14 feet: required the error of compass; and supposing the variation to be 12° 20' E. : find the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, September 23rd, p.m. at ship, latitude account 50° 47' N., longitude 169° 54' E., observed altitude of sun's l.l.. South of observer, was 88° 47', height of eye 12 feet, time by watch September 23"* o'' 58"^ 14% which had been found to be 40" j^y fast on apparent time at ship, the difference of longitude made to the East was 27'-5, after the error on apparent time was determined : required the latitude by reduction to the meridian. 12. 1882, August 28th, A.M. at ship, and uncertain of my position, when a chronometer showed August 27'' 21^ 56"" 4* M.T.G., obs. alt. sun's l.l. 39° 21' 10"; again, p.m. at ship same day, when chronometer showed August 28<* 3^ 20™ 48% obs. alt. sun's l.l. 35° 42' 40", height of eye 16 feet, the ship having made 36 miles on a true N. f W. course in the interval : required the line of bearing when the first altitude was taken, and the ship's position by Sumner's Method when the second altitude was observed, assuming latitudes 48° 50' N. and 49° 30' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, March 3i8t, the observed meridian altitude of the star a Yirginas (SpieaJ, bearing North, was 52° 14', height of 19 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head Bearing of by Distant Object Standard by Standard Compass. Compass. Deviation Required. Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North N.E East .... S.E S. 13 W. 8. 6 W. S. 3 W. South .... S.W West .... N.W S. io°E. S. II E. S. 14 W. s. 37 w. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses: — East; W.S.W. ; S.E. by S. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made, from the above deviation table. Compass courses :— N.W. by W. ^ W. ; North ; E.N.E. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at W.S.W., find the bearings, correct magnetic. Compass bearings : — W. by S. and South. Bearings, magnetic : — Orddna/ry Examination. 359 EXAMINATION PAPEE— No. XYII. FOR SECOND MATE. Multiply 37340 by 1200, and $y62 by o'4i88, by common logarithms. Divide 9145752 by 22-22, and 5*6949 by 53*058, by common logarithms. 3-— H. Courses. K. T^ Winds. Lee- way. Deviation. Remarks, &c. pts. I N.E. 9 N.N.W. f 6|»E. Apoint, lat.37°37'N. 2 8 6 long. 0° 41' W., bear- 3 9 2 ing by compass 4 8 6 N.W. by W. \ W., 5 E.N.E. 12 3 North. i ii°E. dist. 25 miles. Ship's 6 12 3 head N.E. Deviation 7 II 4 as per log. 8 12 9 N.N.W. 10 5 N.E. i 6°W. 10 II I II 10 6 12 10 8 I E.S.E. 6 6 N.E. I* 95" E. Variation 19° W. 2 6 4 3 6 5 4 6 5 5 N.N.E. 4 3 East. 2i 4'E. 6 4 8 7 4 5 A current set the ship 8 4 4 E. by S. (correct mag- 9 S.E. 8 5 E.N.E. li 4°E. netic) 36 miles, from 10 8 7 the time the departure II 7 4 was taken to the end 12 7 4 of the day. 4. 1882, November 2i8t, in longitude 70° 20' E., observed meridian altitude of sun's L.L. bearing North, was 80° 20', index error — 2' 50", height of eye 20 feet : required the latitude. 5. In latitude 35° 39', the departure made good was 66 miles: required the diflference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 6° i' N. Latitude B 6 10 S. Longitude A 60° 14' E. Longitude B 39 15 E. ADDITIONAL FOR ONLY MATE. 7. 1880, September ist : find a.m. and p.m. tides at Lynn Deep, Ramsgate, and Antwerp, and also at Victoria River, longitude 130° E. 8. 1882, July 20th, at 7*'- o"" p.m., apparent time at ship, latitude 43" 4' S., longitude 179° 12' W., sun's magnetic amplitude S.W. by W. \ W. : required error of compass; and supposing the variation to be 9° 40' E. : required the deviation of the compass for the posi- tion of the ship's head at the time of observation. 9. 1882, June 5th, A.M. at ship, latitude 2° 5' S., observed altitude sun's l.l. 28° 4', index correction -\- 4' 25", height of eye 15 feet, time by chronometer June 4'' i2h 28"' 42' which was i™ i^* fast for mean noon at Greenwich, March 6th, and on March 24th was p™ 8» slow on mean noon at Greenwich ; required the longitude. 360 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, November loth, mean time at ship 8^ 45™ 38' a.m., latitude 50° 30' N., longitude 86° 43' E., sun's bearing by compass S. 49° 50' E., observed altitude sun's l.l. 6° 7' 10', height of eye 15 feet: required the true azimuth and error of compass; and supposin-j the variation is 7° 20' E. : required the deviation for the position of the ship's head at the time of observation. 11. 1882, January 8th, a.m. at ship, latitude acct. 35° 10' S., longitude 55° 12' "W. observed altitude sun's l.l.. North of observer, was 76° 44', index correction + i' 18', height of eye 14 feet, time by watch 39" 34' (or 8"^ o^ 39"" 34^), which had been found to b3 50" 3«/a*< on apparent time at ship, the difference of longitude made to the East was 21', after the error on apparent time was determined : required the latitude. 12. 1882, June 14th, P.M. at ship, and uncertain of my position, when a chronometer showed June 14'* 7'' 18™ 50^ M.T.G., obs. alt. sun's l.l. 54° 12' 50"; again, p.m. at ship same day, when chronometer showed June 14"* ii^ i" 13', obs. alt. sun's l.l. ii" 15' 10', height of eye 2 1 feet, the ship having made 25 miles on a true S. W.^W. course in the interval : required the line of position when the first altitude was observed, and the ship's position by Sumner's Method when the second altitude was taken, assuming the ship to be between the latitudes 50° 10' N. and 49° 40' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, February ist, longitude 50° "W., observed meridian altitude of the star a Canis Majoris (Sirius) was 37° 50' 20", bearing South, index correction -|- i' 4", height of eye 19 feet : required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. North .... N.E East S.E North. N. 12° E. N. 29 E. N. 36 E. South .... S.W West N.W N. 24°E. N. 5 E. N. 5 W. N. 5 W. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses:— W.N. W.; W.S.W.; S.E. by E.; S.S.E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :—E. by S. J S. ; S. by E. J E. ; N.W. by W. ; W. J S. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at E. by S. \ S., find the bearings, correct magnetic. Compass b-Rrlp-s :— N.W. by W. \ W. and S. by E. \ E. Bearings, magnetic ; — Orddnary Examination. 361 EXAMINATION PAPEE— No. XVin. FOR SECOND MATE. 1. Multiply 5940 by 530, and -00087214 hy •001963, by common logarithms. 2. Divide 9504000 by 98, and '9649 by 35'0583, by common logarithms. 3-— H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pta. I W.N.W. 12 6 North. i io»W. A point, lat. 36''27'S., 2 12 6 long. 68» 37' W., bear- 3 12 8 ing by compaHs 4 13 E. f S., distance 25 5 S.W. by W. 10 6 N.W. by W. 1 7°W. miles. Ship's hend 6 10 4 W.N.W. Deviationas 7 10 4 per log. 8 10 6 9 N. by E. \ E. 7 3 N.W. I W. 'i ^rw. 10 7 6 II 7 8 12 7 3 I N.W. II 4 N.N.E. i 8°W. Variation 22^° E. 2 II 4 3 II 8 4 II 4 5 S.W. i w. 3 3 W.N.W. 3i 7° W. 6 2 8 7 2 6 A current set the 8 2 3 8hip(corroctmagnetic) 9 N.E. 1 E. 4 7 N. by W. 1 W. 2| 8°E. S.S.W.^W., 32 miles, 10 4 4 from the time the de- II 3 6 parture was taken to 12 3 3 the end of the day. 4. 1883, January ist, in longitude 167° 54' E., observed meridian altitude of sun's l.l., bearing North, was 83° 40', index correction -\- 47', height of eye 23 feet: required the latitude. 5. In latitude 60° 5' S., longitude 179° 17' W., a ship sails due West 96 miles : find the longitude in. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 8° 57' N. Longitude A 79''3i'W. Latitude B 36 50 S. Longitude B 174 49 E. ADDITIONAL FOR ONLY MATE. 7. 1880, March 28th : find the times of high water, a.m. and p.m., at Gibraltar, Port Louis (Mauritius), long. 57^° E., and Halifax, long. 64° W. 8. 1882, November 4th, at 4'' 52"" 42^ a.m., apparent time at ship, latitude 46° 40' S., longitude 8° 57' W., sun's magnetic amplitude S.E. h S. : required the true amplitude and error of the compass ; and supposing the variation to bo 16° 30' W. : required the deviation of the compass for the position of the ship's h' ad at the time of observation. 9. 1882, September ist, a.m. at ship, latitude 15° 31' S., observed altitude sun's l.l. 15* 18' 20', index correction — 20", height of eye 26 feet, time by chronometer, August 3id loh 12™ 40% slow I" 308 on April 15th, and on April 29th was o"" 2^' fast for Greenwich mean time : required the longitude. 362 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, June ist, mean time at ship 8^ 19m a.m., latitude 21° 10' N., longitude 61° 30' E., ohserved altitude sun's l.l. 39° 10', index correction — 15", height of eye 18 feet : sun's magnetic azimuth E. f N. : required the error of compass ; and supposing the variation to be 0° 50' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, April 13th, A.M. at ship, latitude account 0°, longitude 147° 10' E., observed altitude of sun's l.l., North of observer, vras 80° 30', index correction + i' 10", height of eye 16 feet, time by watch o^ o" 12', which had been found to be 11™ i^ fast on apparent time at ship, the difference of longitude made to the East was %\ miles, after the error on apparent time was determined : required the latitude. 12. 1882, April ist, A.M. at ship, and uncertain of my position, when a chronometer showed March 31'' 22^ 22" 13' M. i'.G., obs. alt. sun's l.l. 34° 52*40''; again, p.m. at ship same day, when chronometer showed April i<' 3^ 39m 22", obs. alt. sun's l.l. 31° 20' 30", height of eye 17 feet, the ship having made 19I miles on a true S.E. f S. course in the interval: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 51° o' N. and 50° 30' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, May loth, the observed meridian altitude of a^ Centauri was 10° 4' 15", (zenith North), index correction — 2' 10", height of eye 20 feet : required the latitude. 14. At what time will the star a Aquilse (Altair) pass the meridian of the Land's End on December 8th, 1882, and how far North or South of the zenith. 15. 1882, January 8th, at 2^ 18", what stars will be near the meridian of a place in long. 45° 20' E. In the following table give the correct magnetic bearing of the distant object and thence the deviation. Correct magnetic bearing. Ship's Head by Standard Compass. Bearing; of Distant Object by Standard Compass. Deviation. Bequired. Skip's Head by standard Compass. Bearing of Distant Object Deviation by Standard ; Required. Compass. North .... N.E East , S.E S. 27°E. South. S. 21 w. S. 27 w. s ft South .... S.W West N.W S. 12° W. S. 21 E. S. 37 E. S. 39 B. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— N. by E. f E. ; E. | N. ; S.S.W. \ W. ; N.W. \ N. Compass courses: — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— N. W. \ N. ; S.S.E. 1 E. ; E. by N. ^ N. ; N.N.E. f E. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at S.E. \ S., find the bearings, correct magnetic. Compass bearings :— N.E. \ E., and N. by W. \ W, Bearings, magnetic :— Ordinary Examination. 363 EXAMINATION PAPEE— No. XIX. FOR SECOND MATE. 1. Multiply 2410050 by 5, and 47 by i"405, by common logarithms. 2. Divide •00001400018 by "00000700009, and 2oo4'64 by 34, by common logarithms. 3 — H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. I S. by E. \ E. 12 E.f S. pts. 19^° E- A point, lat. 46" 20' S. 2 II 6 long. i76°44'W.,bear- 3 12 2 ing by compass 4 12 2 E. by N. J N., 5 S. by W. 9 2 S.E. by E. .1 25° E. dist. 13 miles. Ship's 6 9 head S. by E. \ E. 7 9 6 Deviation as per log, 8 9 4 19^° E. 9 S.S.W. \ W. 7 5 West. li 28° E. 10 7 5 II 7 4 12 7 6 I W. by S. II 3 N.W. by N. i 21° E. Variation 15° E. 2 10 8 3 10 8 4 10 6 5 Weat. 9 8 N.N.W. I 20° E. 6 9 6 7 9 4 A current set the ship 8 9 6 (correct magnetic) 9 N.N.W. 9 3 West. i 15F w. N.W. \ W., 36 miles, 10 9 4 from the time the de- II 9 6 parture was taken to 12 9 5 the end of the day. 4. 1882, September 23rd, in longitude 119° 54' E., observed meridian altitude sun's l.l. was 83° 46', zenith South of sun, index correclion — 5' 30", height of eye 18 feet : required the latitude. 5. In latitude 63° 54' N., the departure made good was 63"5 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 9* 59' S. Longitude A 140° 5' W. Latitude B 10 JS. - Longitude B 163 5 E. ADDITIONAL FOR ONLY MATE. 7. 1880, November 30th : required the times of high water, a.m. and p.m., at Margate, Gibraltar, Bordeaux, Ilfracombe, Cromarty, and Batavia, long. 106^ 48' E. 8. 1882, Juno 12th, at 11^ 15"" p.m., apparent time at ship, in latitude 66° 24' N., longitude 4"^ 54' E., the sun's magnetic amplitude was N.N.E. \ E. : required the error of compass, and supposing the variation to be 23° 10' W. : required the deviation for the position of the ship's head at the time of observation. 9. 1882, June 14th, P.M. at ship, latitude 31° 10' N., observed altitude sun's l.l. 48" 59' 10", index corr. -\- i', height of eye 18 feet, time by chronometer June i4<'i2'> 15™ 40*, which wa8/rt«< 4" 30*, January 2nd, for mean noon at Greenwich, and on March 13th was 9" 16* fast lor mean time at Greenwich: required the Ijngitude. 3^4 Ordinary Examination. ADDITION A.L FOR FIRST MATE. 10. 1882, January ist, mean time at ship 4*> lo"" p.m., latitude 31° 50' S., longitude 176° 25' E., observed altitude sun's l.l. 34° 50', height of eye 16 feet, sun's azimuth W. \ N. : required the error of compass ; and supposing the variation to be 20° E. : required the deviation for the position of the ship's head at the time of observation. 11. 1882, December 31st, a.m. at ship, latitude account 52° N., longitude 78° E., observed altitude sun's l.l. 14° 46' S., height of eye 19 feet, time by watch o^ ^6™, fast on app. time at ship i^ 5™ 20', the difference of longitude made to West was 21*4 miles after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, January 24th, a.m. at ship, and uncertain of my position, when a chronometer showed January 21,^ 21^ ig'" 50^ M.T.G., obs. alt. sun's l.l. 9° 34' 50"; again, p.m. at ship same day, when chronometer showed January 24'* i'' 19" 12% obs. alt. sun's l.l. 18° 28' 10"^ height of eye 17 feet, the ship having made 23 miles on a true S.E. by E. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 51° 15' N. and 50° 45' N. For the second observation the positions as given by the Examiner would be as follows :— B = Lat. 51° 15' N. B' = Lat. 50° 45' N. Long. 0° 1 8' W. Long. 3° 37^' W. ADDITIONAL FOR MASTER ORDINARY. 13. 1 882, April I ith, the observed meridian altitude of star Sirius was 61° 3' 50", bearing North, height of eye 16 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation. Correct magnetic bearing. Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East 8.E N.73»W. West. N.88 W. N.80 W. South .... S.W West N.W N. 75" W. N.68 W. N.57 w. N.53 W. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses : — North ; E. by S. ; S. 40° W. ; Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— W. by S. ; E.N.E. ; N.W. by N. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with ship's head N.E. \ E., find the bearings, correct magnetic. Compass bearings : — South and W. by N. Bearings, magnetic ; — Ordina/ry Examination. 365 EXAMINATION PAPEE— No. XX. FOR SECOND MATE. 1. Multiply 30-24 by 12-5, and -034632 by "397302, by commoa logarithms. 2. Divide 8100900 by 900, and -00005 by 2-5, by 25, and by -0000025, ^7 common logs. 3-— H. Courses. K. A Winds. Lee- way. Deviation. Eemarks, &c. pts. I N.E. i E. 13 2 N.byW.^W. i 26- E. A point of land, lat. 2 12 9 50°25'S.,long.i79''4o'E 3 n 5 bearing; by compass 4 13 4 N.by\V.^\V..di9t. i6 5 w.s.w. 3 6 N.W. 2i 4rW. miles. Ship's head 6 4 N.E. i E. Deviation 7 4 as per log. 8 3 4 9 E. by N. 12 2 N. by E. ^ 28° W. 10 12 4 II 12 6 12 12 8 I N.byW.iW. 2 4 N.E. i E. 3 51° E. Variation 14° E. t 2 3 3 2 3 4 2 5 S. |W. 6 9 W. by S. i 50° W. 6 6 8 7 6 8 A current set (cor- 8 7 5 rect niagnetic)E.N.E., 9 E. byN.^N. II 5 N. byE. ^E. * 28» W. 42 miles, from the time 10 12 2 the departure was II II 6 taken to the end of 12 II 7 the day. 4. 1882, September 23rd, in longitude 57° 45' E., the observed meridian altitude of sun's L.L. was 84" 10' 50", bearing North, index error — i' 36", height of eye 16 feet: required the latitude. 5. In latitude 52° S., longitude 0° 40' W., a ship sails 136 miles due £ast : required the longitude in. 6. Required the course and distance from A to B, by calculation on Mercator'e principle. Latitude of A 5°2i'N. Longitude of A 163° i' E. Latitude of B 36 50 S. Longitude of B 73 6 W. ADDITIONAL FOR ONLY MATE. 7. 1880, December 12th: find a.m. and p.m. tides at Aberdeen Bar, Penzance, King's Road (Bristol Channel), Southampton, and Ferrol. 8. 1882, November 5th, at 5*^ 10" p.m., apparent time at ship, latitude 20° 45' N,, longi- tude 116° 45' E., sun's magnetic amplitude S."W. f W. : required the true amplitude and error of compass; and supposing the variation to be 1° E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, August 5th, A.M. at ship, latitude at noon 30" 30' N., observed altitude sun's l.l. 35° 6', height of eye 16 feet, time by chronometer 8^ 39" 22^ p.m., which was/as^ 29™ 32»-4 on Gjeenwich mean noon, July 8th, and on July 20th, was fast 30'" o' on Greenwich mean noon ; course till noon West (true) 48 miles : required the longitude in at noon. 366 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, August 13th, mean time at ship 9*^ 5™ 20^ a.m., latitude 30° 46' S., longitude 78° 50' W., sun's magnetic azimuth N. 25'' E., observed altitude sun's l.l. 27° 12', index correction -\- i 45", height of eye 21 feet : required the true azimuth and error of compass ; and supposing the variation to be 16" 20' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, June X2th, p.m. at ship, latitude account 15° 50' S., longitude 72° 12' E., observed altitude of sun's l.l. 50° 10' 10", zenith South of observer, index correction — 5' 40", height of eye 26 feet, time by watch 28™ 40^ (or 12'' 0*1 28"" 40^), which had been found to be 4™ 44« slow on apparent time at ship, the dijBference of longitude made to West was i6\ miles after the error on apparent time was determined : required the latitude. 12. 1882, November 3rd, a.m. at ship, and uncertain of my position, when a chronometer showed November z^ 2.2^ 38" 45= M.T.G., obs. alt. sun's l.l. 18° 53' 50" ; again, p.m. at ship same day, when chronometer showed November 3"^ 2^ 44'" 15^ obs. alt. sun's l.l. 16° 32' 30", height of eye 19 feet, the ship haviug made 15 miles on a true E. by S. \ S. course in the interval : required the line of bearing when the first altitude was taken, and the ship's position by Sumner's Method when the second altitude was observed, assuming the ship to be between the latitudes 51° 45' N. and 51° 15' N. For the second observation the positions as given by the Examiner would be as follows : — B = Lat. 51° 45' N. Long. 10° 41' "W. B'^Lat. 51 15 N. Long. 9 33 W. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, December 7th, the observed meridian altitude of the star a Arietis was 60° 39' 50", zenith North of star, index correction — 2' 10", height of eye 18 feet : required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head 1 Bearing of by Distant Object Standard ' by Standard Compass. Compass. 1 Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 5i°W. W. 6 S. N. 57 W. W. 64 N. South .... S.W West .... N.W N. 30° W. W. 21 N. W. 8 S. S. 46 w. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. • Correct magnetic courses :— W. f N. ; S.E. ^ E. ; N. by E. J E. ; S. by E. f E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— E. \ N. ; S. by W. ^ W. ; N.W. f W. ; S.W. by W. \ W. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at S.E. \ E., find the bearings, correct magnetic. Compass bearings :— S. ^ W. ; W. by N. ^ N. ; E. i N. Bearings, magnetic : — Ordinary Examination. 367 EXAMINATION PAPEE— No. XXI. FOR SECOND MATE. Multiply 7642 by 7429-5, and o'ooo64 by io"coo4, by common logarithms. Divide "39765 by 25, and loooooo by ooooooi, by common logarithms. 3 — H. Courses. K. tV Winds. Lee- way. Deviation. Kemarks, &c. I N. by W. 6 4 W. by N. pts. 4 8°W. A point of land in 2 6 3 lat. 57° N . long. 40° W. 3 5 6 bearincr bj' compass 4 5 4 N.E.iE.^E., distance 5 S.S.W. 1 w. 4 8 W. i N. 2^ 5°W. 19 miles. Ship's head 6 4 2 N. by W. Deviation 7 3 7 as per log. 8 3 6 9 N.N.E. 1 E. 4 2 N.W. f N. 2 '3°E. 10 4 4 II 4 5 12 4 5 I W. by N. 3 4 N. by W. 2^ nrw. Variation 48° W. 2 3 4 3 3 6 4 3 7 5 S.E. 1 E. 9 5 S.S.W. i 11° E. 6 10 5 7 II 2 8 10 8 A current set the 9 S. ^W. 3 2 w.s.w. 2| 2° W. 8hip(correct magnetic) 10 3 3 W.N.W. for the last II a 8 5 hours, 3 miles an 12 2 7 hour. 4. 1882, June 25th, in longitude 59° 15' E., the observed meridian altitude of sun's u.l. was 60° 23' 15", bearing North, index correc;ion -\- 2' 21", height of eye 30 feet: required the latitude. 5. A ship sailed due West 120 miles from Cape Roca, in latitude 38° 46' N., and longi- tude 9° 30' W. : required the longitude of the ship. 6. Required the compass course and distiince from Cape East, New Zealand, to San Francisco. Variation 14° 20' E., and deviation 5° 40' E. Lat. Cape East 37" 40' S. Long. Cape East 178" 36' E. Lat. San Francisco 37 48*5 N. Long. San Francisco 122° 24' W. ADDITIONAL FOR ONLY MATE. 7. 1880, August 7th: find the times of high water, a.m. and p.m., at Hong Kong, long. 114° E., New York (Sandy Hook), long. 74° \V., and Skull. 8. 1882, June 24th, at 6*> a.m., apparent time at ship, latitude 0° N., longitude 12° 3' W., sun at setting bore by compass S.E. by E. \ E., variation by chart was 21° 40' W. : required the error of compass and the deviation. 9. 1882, September 22nd, a.m. at ship, on the Equator, observed altitude sun's u.l. 17° 20' 40", index correction — i' 18", height rf eye 20 feet, time by chronometer September 22'' 4^ 59" i6», which was 15^ slow for Greenwich mean noon, April 30th, and on June ist was 10*' 6 fast for mean time at Greenwich : required the longitude. 368 Ordinwry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, March 2i8t, mean time at ship 3*' ij" p.m., latitude 9° 7' S., longitude 159° 4' "W., sun's bearing by compass W. \ S., observ^ed altitude sun's l.l. 42° 49' 45*, index correctioTi — 3' 14", height of eye 21 feet, variation by chart 7° 50' E. : required, the error of compass and deviation. 11. 1882, October 4th, a.m. at ship, latitude account 30° 24' S., longitude 140"' 30' E., observed altitude sun's l.l., North of observer, was 63° 37' 10", index correction — i' 15", height of eye 21 feet, time by watch October 3'^ 22'' 37'" 15% which had been found to be ih 10"^ 20' slow on apparent time at ship, the difference of longitude made to the East was 23^ miles after the error on apparent time was determined : required the latitude. 12. 1882, January i6th, a.m. at ship, and uncertain of my position, when a chronometer showed January i6'* 7'" 7"" 58" M.T.Gr., obs. alt. sun's l.l. 13° 10' 30" ; again, p.m. at ship same day, when chronometer showed January 16"* ii*» 22"^ 8', obs. alt. sun's l.l. 15° 17' 30", height of eye 17 feet, the ship having made 24 miles on a true N.E. by N. course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 49° o' N. and 49° 40' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, June loth, longitude 25° "W., the observed meridian altitude of star a Cassiopese, was 85° o' 20", bearing South, index correction -|- 34", height of eye 1 8 feet : required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head Standarl Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard j Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E N. 78°E. E. 22 S. E. 34 S. E. 29 S. South .... s.w West N.W S. 89° E. N. 66 E. E. 23 N. N.79 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses : — N. f E. ; E. ^ S. ; S. ^ E. ; W. \ S. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass Courses :— N.E. \ E. ; S.S.E. J E. ; W. ^ N. ; N. by W. i W. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at S.W. by W. J W., find the bearings, correct magnetic. Compass bearings: — N.E. \ N. and E. | N. Bearings, magnetic ; — Ordinary Examination. 369 EXAMINATION PAPEE— No. XXII. FOR SECOND MATE. Multiply 10003 ^y i'°» ^^^ 12344 by 57, by common logarithms. Divide 3972096 by 144, and 120500 by iio'S, by common lojjaritlims. 3 — H. Courses. K. iV Winds. Lee- way. Deviation. Remarks, &c. pts. I W.N.W. 13 4 S.E. 43rE. A point, 181.51° 8^' N. 2 13 long. 1° 25' E., bear- 3 13 6 ing by compass 4 .10 S.W. 1 S, 5 w. \ s. 9 6 S.S.W. \ 43° E. dist. 25 miles. Ship's 6 9 4 head W.N.W. Devia- 7 9 5 tion as per log. 8 9 5 9 West. 8 4 N.N.W. 1 43FE. 10 8 4 II 8 4 12 8 6 I N.|E. 9 5 N.W. by W. 1 2 i7i°E. Variation 25° W. 2 9 4 3 9 6 4 9 4 5 North. 10 5 W.N.W. 1 4 19^ E. 6 10 6 7 II A current set the ship 8 ID 4 N.W. by W. I W. 9 N.W. JW. 7 4 N. by E. J E. li 44° E. (correct magnetic) 32 10 7 miles, from the time II 7 4 the departurewas taken 12 7 2 to the end of the day. 4. 1882, August 26th, in longitude 92° 3' E., observed meridian altitude of sun's l.l., bearing North, was 35° 35' 20", index error + 2' 17", height of eye 12 feet; required the latitude. 5. In latitude 17° 15' N., the departure made good was 171 '5 miles: required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 55° 40' N. Latitude B 50 25 N. Longitude A 2' 25' W. Longitude B 3 40 E. ADDITIONAL FOR ONLY MATE. 7. 1880, January 8th: find a.m. and p.m. tides at Flambro' Head, Peterhead, Crinan, St. Ives, Cape Wrath, and Whitby. 8. 1882, September 23rd, at 6^- o^ p.m., apparent time at ship, latitude 51° 3' S., longitude 152° 17' E., sun's magnetic amplitude W. 15° S. : required error of compass ; and suppos'ng the variation to be 20° 16' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1882, August 28th, P.M. at ship, latitude 5° S., observed altitude sun's l.l. 38°, index correction + 5' 20", height of eye 21 feet, time by chronometer August 27> 18™ p.m. apparent time at ship, latitude 72° 58' S., longitude 89° "W., the sun's magnetic amplitude was S. | E. : required the true amplitude and error of the compass ; and supposing the variation to be 36° 20' E. : required the deviation for the position of the ship's head when the observation was taken. 9. 1882, August 3i8t, P.M. at ship, lat. o^ N., observed altitude sun's l.l. 45° 5' 30", index correction — 2' 10'', height of eye 15 feet, time by chronometer August 3i<* 9'' ri" 28*, which was 5™ 20' fast for mean noon at G-reenwich, April 15th, and on June i6th was fast 2™ 43' on mean time at Greenwich : required the longitude. 374 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, March 20th, mean time at ship o^ 45m p.m., latitude 36° 18' N., longitude 39° 52' W., suq's bearing by compass S. \ E., observed altitude sun's l.l. 52° 59', height of eye 14 feet : required the true azimuth and error of the compass ; and supposing the varia- tion be 24° 50' W. : required the deviation of the compass for the position of the ship's head at the time the observation was taken. 11. 1882, March 21st, p.m. at ship, latitude by acct. 18° 50' N., longitude 108° 47' E., observed altitude sun's l.l. South of observer was 71° 6' 50", height of eye 18 feet, time by watch March 20'' 23'! 58™ 12*, which had been found to be ii™ 8' slow on apparent time at ship, the difference of longitude made to the East was 18J miles, after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, August 17th, A.M. at ship, and uncertain of my position, when a chronometer showed August 16^ %^ o™ 20' G.M.T., obs. alt. sun's l.l. 28° 14' 10'; again, p.m. at ship same day, when chronometer showed August 16^ 1^ o™ 20', obs. alt. sun's l.l. 46'^ 21', height of eye 12 feet, the ship having made 20 miles on a true East course in the interval : required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 49° 30' N. and 50" o' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, March 31st, the observed meridian altitude of the star a Pegasi (Marhah), bearing North, 33° 20' 50", index correction + i' 20", height of eye 20 feet: required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation;— Correct magnetic bearing. SMp's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Bequired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Bequired. North .... N.E East S.E N. 32°W. N. 13 W. N. 3E. N. n E. • South .... S.W West N.W N. 7°W. N. 26 W. N.26 W. N.18 w. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses : — E. by S. ; West ; North. Compass courses ; — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses : — E. by S. ; West; North. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above, with the ship's head at West, find the bearings, correct magnetic. Compass bearings : — West, and N. 14° W. Bearings, magnetic : — OrcUna/ry Examination. 375 EXAMINATION PAPEE^No. XXV. FOR SECOND MATE. Multiply 6054 by 912, and 2070*5 by 62-0898, by common logarithma. Divide ii7'658 by i46"932, and 167342 by "002, by common logarithms. 3-— H. Courses. K. .1. 10 Winds. Lee- way. Deviation. Remarks, &c. pts. I S.JW. 4 5 W. by S. ^\ 50° W. A point of land in 2 4 2 lat. 62° 20' N., long. 3 4 64° 40' W., bearing 4 3 9 by compass 5 S.W. 1 W. 3 5 S. by E. 3i 9° W. W. bvN.^N., 6 3 4 dist. 21 miles. Ship's 7 3 2 head S. \ W. Devia-. 8 3 3 tion as per log. 9 E.fS. 5 i 4 S. by E. ij 25° W. 10 5 I 3 11 4 4 12 4 3 I W.N.W. 3 6 North. 3 37° E. Variation 59° W. 2 4 5 3 5 3 4 5 7 5 N.W. \ N. 10 2 E.N.E. 28^° E. 6 II 4 7 12 6 A current set the ship 8 13 4 E. by S. 1 S., correct 9 E. |N. 5 .? N. by E. z\ 26|° W. magnetic, 49 miles, 10 5 4 from the time the de- II 5 4 parture was taken to 12 5 the end of the day. 4. 1882, June ist, longitude 44° 40' E., the observrjd meridian altitude of sun's l.l. bearing North, was 72° 14' 10", index correction -\- 3' 45", height of eye 22 feet: required the latitude. 5. In latitude 32° 3'S., longitude 179° 45' W., a ship makes 54 miles "West, then 80 miles North : what is the longitude in, also find the compass course and distance ; variation 1 8' E. : ist deviation 4" 5' E. ; 2nd deviation 3° 10' W. 6. Required the course and distance from Cape Lopatka to Callao. Lat. Cape Lopatka 50° 33' N. Long. Cape Lopatka 156° 46' E. Lat. Callao 12 4 S. Long. Callao 77 14 W, ADDITIONAL FOR ONLY MATE. 7. 1880, May 7th: find the times of high water, a.m. and p.m., at Aberdeen, Montrose, Wick, Fecamp, flellevoetsluis. 8. 1882, December 2i8t, at ii*" 15™ p.m., apparent time at ship, in latitude 66° 25' S., longitude 93° 57* W., the sim's magnetic amplitude was S. ^ E. : required the true amplitude and error of the compass ; and supposing the variation to be 28" 10' E. : required the devia- tion of the compass for the position of the ship's head at the time of observation. 9. 1882, January 29th, p.m. at ship, lati'ude 28^ 45' N., observed altitude sun's fc.L. 17" 46' 30", index correction — 3' 25", height of eye 16 feet, time by chronometer January 28"^ 16'' 31'" 30', which was i"^ iS^'^fast for Greenwich mean noon, December 17th, i88r, and on January ist, 1882, was i'" 3' slow for Greenwich mean time; course since noon N.W. by W. (true), distance 20 miles : required the longitude at the time of observation, and also at noon. 376 Ordinmy Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, July loth, mean time at ship f" 14'" 2' p.m., latitude 38° 2' S., longitud:; 140° 58' E., sun's hearing by compass N. 2° 15' E., observed altitude sun's u.l. 14° 56' 30", index correction -\- 3' 30", height of eye 19 feet, variation by chart 6' 45' E. : required tko deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, November 29th, p.m. at ship, latitude account 6° 20' S., longitude 123° 25' E., observed altitude of sun's l.l. 74°, index correction + 4' o", height of eye 19 feet, time by watch November 28'* 22^ 46™, which had been found to be i^ 27'" slow on apparent time at ship, the difference of longitude made to the West was 12-3 miles after the error on apparent • time was determined : required the latitude. 12. 1882, October 20th, p.m. at ship, and uncertain of my position, when a chronometer showed October 19*^ 16^ 30"" 54' M.T.Gr., obs. alt. sun's l.l. 29° 37' 15"; again, p.m. at ship same day, when chronometer showed October 19"* 19*1 44" 50% obs. alt. sun's l.l. 12° 21', height of eye 20 feet, the ship having made 24 miles on a true N.N.E. \ E. course in the interval : required the line of bearing when the first altitude was taken, and the ship's position by Sumner's Method when the second altitude was observed, assuming latitudes 49° 20' N. and 49° 50' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, May 15th, the observed meridian altitude of the star jS Orionis 52° 20' 30", zenith North of star, index correction — 4' 10", height of eye 15 feet : required the latitude. 14. 1882, September 4th, what bright stars in the Nautical Almanac will pass the meri- dian of a place in longitude 54° 40' E., between the hours of seven and ten. 15. 1882, June 15th, observed meridian altitude of 7; Argus, under the South pole, was 47** 5°' 3°"> index correction + 3' 20", height of eye 20 feet : required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. North .... N.E East .... S.E S. 23°E. S. II E. s. 5 w. S. 20 w. South .... S.W West .... N.W S. 6<=W. S. 18 E. S. 21 E. S. 22 E. With the deviation as above, give the courses you would steer by the Standard Compwa to make the following courses correct magnetic. Correct magnetic courses :— S. J W. ; E. by N. ; S.E. by S. ; W. by N. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find tho correct magnetic courses made, from tho above deviation table. Compass courses :-N.W. by N. ; W.N.W. ; S.E. by E. ; N.N.E. Magnetic courses: — You have taken the following bearings of two distant objects by your Standard Compass as above; with the ship's head at S.E. by S., find the bearings, correct magnetic. Compass bearings:— N. 84" W., and N.W. by W. | W. Bearings, magnetic : — Ordinary Examination. 377 EXAMINATION PAPER— No. XXVI. FOR SECOND MATE. Multiply 6893 by 11300, and '0001468 hy -000395, by common logarithms. Divide 7122 by 8"9596, and 268430 by "00310, by common logarithms. 3 — H. Courses. K. A Winds. Lee- way. Deviation. Remarks, &c. pts. I S.E. \ E. 13 7 N.E.byKiE. i 11° E. A point, lat.59°49'N. 2 14 long. 44" lo'W., bear- 3 13 2 ing by compass 4 13 3 N.W. |- W., distance 5 E.^S. 10 4 S. by E. i E. 1 14° E. 30 miles. Ship's head 6 10 4 S.E. i E. Deviation 7 10 J as per log. 8 10 3 9 N. by W. \ W. 4 N.E. i E. 4 S^W. 10 3 6 II 3 4 12 3 4 I S.S.W. ^ W. II 6 S.E. ^ S. ^ 2rW. Variation 53^° W. 2 II 7 3 II 8 4 II 4 5 N.E. J N. 7 2 E. by S. ^ S. I 14° E. 6 7 3 A current set the 7 7 4 Bhip(correctmagnetic) 8 7 2 S.E. ^E., I -7 knots per 9 S. iE. 12 5 E.S.E. i 3"^ hour, from the time 10 12 the departure was II 12 3 taken to the end of 12 12 4 the day. 4. 1882, October ist, in longitude 84° 40' E., observed meridian altitude of sun's u.l., zenith North, was 57° 20' 30", index correction — 3' 36', height of eye 17 feet: required the latitude. 4.* 1882, July 2nd, in longitude 45' 15' E., observed meridian altitude of the sun's l.l. below the pole, was 10° 19' 45", index correction — i' 15", height of eye 12 feet : required the latitude. 5. A ship from latitude 35° 30' S., longitude 27° 28' W., sailing due East (true) 301 miles : required the compass course steered, and what will be the longitude in, variation if point E., and deviation 8" 50' E. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude A 10° 8' S. Longitude A 175" 18' E. Latitude B 23 12 N. Longitude B 141 15 E. Variation ^ point West, and deviation 7'' 15' West. ADDITIONAL FOR ONLY MATE. 7. 1 880, March 7th : find the times of high water, a.m. and p.m., at Santander, Arcachon, Scarborough, Holy Island, Montrose, and Angra Pequena (S.W. coast of Africa), long. 15' E. 8. 1882, April 25th, at 7*" 22"" 8' p.m., apparent time at ship, latitude 57"^ 18' S., longitude 101° 50' E., sun's setting by compass N, ^ E., variation by chart 35° 50' W. : required the error of the compass and deviation. 9. 1882, August 24th, A.M. at ship, latitu le at noon 37° 59' N., observed altitude sun's L.L. 37° 1 3' 30', index correction + 2' 40", height of eye 1 8 feet, time by chronometer, August 24'* 6'' 13" 24', A.M. at Greenwich, which was i" 5'/««(^ for mean noon at Greenwich, August ist, and on August loth was o"" 42' slow for mean time at Greenwich, course since observation N.N.W., 22' -4 (true) : required the longitude at noon, ceo 378 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1882, November ist, mean time at ship S^ 40™ a.m., latitude 50° 21' N., longitude 23° 56' W., sun's bearing by compass S. \ W., observed altitude sun's l.l. 12° 19', index correction — 3' 20", height of eye 21 feet : required the error of the compass ; and supposing the variation to be 33° 20' W. : required the deviation of the compass for the position of the ship's head at the time of observation. 11. 1882, Slay 29th, A.M. at ship, latitude account 0° 31' S., longitude 150° 40' W., observed altitude of sun's l.l. 67° 41' N., index correction -\- i', height of eye 20 feet, time by watch May 29'! 3^ ■^z'^,fast on apparent time at ship 3'' 38"", the difference of longitude made to East was 26"9 miles, after the error on apparent time was determined : required the latitude by reduction to meridian. 12. 1882, April loth, P.M. at ship, and uncertain of my position, when a chronometer showed April 9"* 16^ 40"^ 24^ M.T.G., obs. alt. sun's l.l. 45° 17' 15'' ; again, p.m. at ship same day, when chronometer showed April 9'' 20^ 31™ 52', obs. alt. sun's l.l. i8° 17', height of eye 18 feet, the ship having made 12 miles on a true W. \ N. course in the interval: required the line of bearing when the first altitude was taken, and the position of the ship by Sumner's Method when the second altitude was observed, assuming latitudes 51° 10' N. and 51° 40' N. ADDITIONAL FOR MASTER ORDINARY. 13. 1882, June x7th, the longitude 98° W., observed meridian altitude of a Serpentis, zenith South of object, was 29° o' 40", index correction -\- 4' 20", height of eye 24 feet : required the latitude. 14. 1882, June 15th, at what time will a Serpentis pass the meridian of a place in lati- tude 37° N., and longitude 15° 30' E. ; what distance N. or S. of the zenith P 15. 1882, May i8th, observed meridian altitude of -q Draconis under the North Pole was 34° 56' 15", index correction — 5' 45", height of eye 22 feet : required the latitude. 16. At the Cape of Good Hope the variation is about 28° W., if the sun at noon bears due North by compass, what is the deviation P In the following table give the correct magnetic bearing of the distant object and thence the deviation. Correct magnetic bearing. SMp's Head by Standard Compass. Bearing; of Distant Object by Standard Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 29°E. S. 33 E. S. 47 E. S. 63 E. South .... S.W West N.W S. 69° E. S. 64 E. S. 48 E. S. 38 E. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses correct magnetic. Correct magnetic courses :— S.W. i W. ; S.S.E. ^ E. ; W. by N. i N. ; N. ^ E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses :— N.E. by N. | N. ; S.W. by W. | W. ; S. i E. ; S.W. Correct magnetic courses : — You have taken the following bearings of two distant objects by your Standard Compass as above ; with the ship's head at N.N.E., find the bearings, correct magnetic. Compass bearings :— S. by W., and W. by N. | N. Bearings, magnetic ; — 379 QUADRANT AND SEXTANT. 348. The ftuadrant and Sextant* are reflecting astronomical instruments for measuring angles, and are the instruments chiefly in use for taking the observations required for the solution of a number of the most useful problems in navigation, such as to find the time, the latitude and longitude of a place. The Quadrant contains an arc of 45° in real extent, and measures a few degrees more than 90°;! it is usually of wood, and the graduated arc, which is ivory, reads to minutes, and sometimes to 30". The Sextant is constructed on the same principles as the Quadrant ; has a gradu- ated limb of more than 60° in real extent ; and furnishes the means of measuring the angle between two objects in whatever direction they may be placed, so that the angle does not exceed 140°. The quadrant serves for common purposes at sea ; but the sextant is used when considerable precision is required, as, for instance, in taking a lunar observation. 349. The form of a sextant, as at present in common use, consists of a single frame of brass, so constructed as to combine strength with lightness ; and in others a double frame connected by pillars (see Fig., page 380). The graduated arc, inlaid in the brass, is usually of silver, sometimes of gold, or platinum. The explanation of the parts of a sextant, and of the adjustments of that instrument, will answer for the quadrant, since the parts and append- ages are common to both. 350. The flat surface of the sextant is called the plane of the sextant; the circular part B C is the arc or limb, which is graduated from right to left from the zero point 0°to about 140*^, and each degree in the best instruments is again sub-divided into six equal parts of 10' each, while the vernier g, used in estimating the sub-divisions of the arc, shows i o". The divisions are also continued a short distance in the opposite direction on the other side of zero (0), towards C, forming what is termed the arc of excess, for the purpose of determining the index error in the manner that will be subsequently explained. The microscope M, and its reflector r, secured at the point d hy a, movable arm dr io the index bar A E, may be adjusted to read ofi' the divisions on the graduated limb and the vernier g. A E is the radius, or index bar, movable along the arc and round a centre, and having a dividing scale (called the vernier) close to the arc, by which the sub-divisions of the arc are read off. The index bar is secured to the arc B C by the intervention of a mill-headed clamp screw « at its back, which must be loosened when the index has to be moved any considerable distance, and when the contact nearly has been made ♦ The first inventor of the sextant (or quadrant) was Newton, among whose papers a description of such an instrument was found after his death, not, however, until after its re-invention by Thomas Godfray, of Philadelphia, in 1730, and perhaps by Hadley in 1731. t This depends on the properties of light, which we cannot consider here. The principle of the sextant is this: — The angle between the first and last direction of a ray which has suffered two reflections in one plane is equal to twice the inclination of the reflecting surfaces to each other. 38o On the Sextant. by hand, the screw is again to be fixed, and a tangent screw s enables the index bar and the vernier* upon it to be moved by a small quantity along the limb, so as to render the contact of the objects observed more perfect than could be effected by moving the index solely by hand ; the other extremity of the index bar has a silvered glass or reflector I fixed perpendicular to the plane of the instrument, with its face parallel to the length of the index bar, and directly over the centre ; another glass h is fixed perpendicular to the plane of the instrument frame H, and facing the index-glass, the lower half only is silvered (being a reflector), and the upper transparent ; it is usually provided with screws, by which its position with respect to the plane of the sextant may be rectified ; the plane of this glass, usually termed the horizon- glass, is made parallel to the plane of index-glass I, when the vernier g is adjusted to zero on the divided arc BC, or if not so made, the want of parallelism constitutes what is termed the index error of the instrument. The telescope t is carried by a ring fastened to a stem E, which can be raised or lowered by a mill-headed screw s" at the back of the frame, for the purpose of so placing the field of the telescope that it may be bisected by the line on the horizon-glass, separating the silvered from the unsilvered part, whereby the brightness of the reflected object and that seen by direct vision may be made equal, and the quality of the observations improved ; the ring and its elevating apparatus are technically known as an " up-and-down piece." It is usual to supply a direct and inverting telescope, of which the latter is to - * Vernier — so called after its inventor, Peter Vernier, of France, who lived about 1630. By some it is called a nonius, after the Portuguese, Nunen or Nonius ; but the invention of the latter (who died in 1577) was quite different. On the Sextant 3$i be preferred, as possessing greater magnifying power, and thus showing a Letter contact of the images of the objects. Two wires parallel to each other, and to the plane of the instrument, are placed in the inverting telescope, within which limit the observation should be made. In the quadrant the telescope is omitted, and the eye is applied to a small circular orifice in a piece of brass, placed in the same position as the telescope in the drawing. Dark glasses of different colours and shades are a necessary accompaniment to the sextant to enable the sun to be observed, and they are usually attached to a hinged joint at K. Four of these glasses or shades are placed at a, between the index and horizon-glasses, so as to admit of one or more of them being interposed between the index and horizon-glass, to moderate the light (if any brilliant object seen by reflection. Three more such glasses, some- times called back shades, are placed behind the horizon-glass at K, any one ( r more of which can also be turned down to moderate the intensity of the light before meeting the eye when observing a bright object, such as the sun. There is also a dark glass which can be placed at the eye-end t of the telescope, which method is preferable to the other, as no error in this is liable to be introduced in the passage of the rays from the index to the horizon- glass.* When observing, the instrument is to be held with one hand by the handle P placed at the back of the frame, while the other hand moves the index. 351. Reading off the Angle. — The following brief directions for reading off will be more readily imderstood by the learner, if he place a sextant before him for reference and examination. It will be seen that the arc (limb) is divided into degrees and parts of a degree, from 0° (zero) to about 140° ; every roth degree is numbered from 0° to 140"^; the space between every 10° is divided into 10 equal parts by straight lines ; consequently every part is 1° ; every fifth line is made a little longer than the others, to represent every fifth degree ; and (in the best instruments) every degree is sub-divided into six equal parts by lines shorter than those which represent the degrees ; those short lines divide every degree into sixths of a degi-ee, or 10', every third line of these short ones being made a little longer to denote 30'. On any part of the arc, therefore, the first short stroke is 10', the second is 20', the third is 30', the fourth is 40', and the fifth is 50'. Wo will suppose it is an instrument of this kind before the learner. The index, up to which an arc is read off, is a line cut in a plate at the end of the movable radius, and is generally distinguished from the other lines on the plate by a diamond-shaped mark, resembling a spear head, und sometimes by 0. Supposing this index to stand exactly at any of the long lines on the arc, that is, so that the two lines are in the same direction ; in such a case the reading oil is easily known, for it must be a certain number * With respect to the darlc glasses, when it is possible (as in observing altitudes of the stin in the mercurial horizon, &c.) to make the obscrvution wilh a single darii glass on the lye-end of the telescope, witliout using anj- shade, this should always be done, for tlie error ( f this dark glass does not aifect the contact at all, and the distortion caused by it is not inagnifi' d, whereas any fault in the dark shade between the index and horizon-glasses iroduces actual error iu the observation, and the distortion is magnified subsequently by the telescope. 382 On the Sextcmt. of divisions, of which the value is seen at once, the reading being degrees and no minutes; it may be lo'^, 12°, 20°, 30°, &c. — any number: but if the index exactly coincides with a short "troke on the arc, in such a case the reading off must be a certain number of divisions and sub-divisions. Thus, if it coincide, for example, with the second line to the left of 40°, then the reading off will be 40"^ 20', or if it coincide with the fifth stroke to the left of 30°, the reading will be 30° 50', since each line on the arc represents 10'. But suppose the index not to stand exactly at any line whatever on the arc, but somewhere between two, as in the above example, between the second and third line from 40°, suppose it appeared to be about half-way between the second and third, lines (the learner may place it in that position). But as this is a rough and imperfect way of estimating the additional minutes and seconds beyond the second division from 40°, the exact value of this small space is known by means of a few divisions on the index plate to the left of the index, and called the Vernier. Tiiese divisions are made less than the arc divisions, so that the line on the pLite immediately to the left of the index is somewhat nearer to the corresponding one on the arc than the small space to be determined. It is nearer thereto, as is manifest by difference of a division on the arc and one on the index plate. In like manner the second line, reckoning from the index, must be nearer to the corresponding line by two differences, the third by three, and so on. At length, therefore, there must be a coincidence of two lines, or nearly so ; that is, they must appear to an eye placed directly over them to lie in the same direction, or nearly so. And since, upon the whole, the lines on the vernier have approached those upon the arc through the small part the index is in advance of 20', this excess must be equal to as many times the difference of two divisions, as there are lines, reckoning from the index, before this coincidence takes place. Hence, if we know the value of a difference, we shall know the value of the small arc to be measured. This difference is known as follows : — By examining the arc of the sextant before us, it will be seen that 60 divisiuns of the vernier just cover or coincide with 59 divisions on the arc, or the difference between a division on the arc and one on the vernier is -^^ of a division of the arc ; if, therefore, a division on the arc is jo', the difference will be J^ of 10', or 10". Every sixth division of the vernier being distinguished by a figure denoting minutes, and the interval between each of these figures is divided into six parts of 10" each. Hence, to read off on a Sextant, ^e proceed thus : — First examine the divisions and sub-divisions on the arc, up to the line which stands before the index. We then move the micro:^cope on the vernier and examine the numbered lines. If any one of these c oincides in direction with the opposite one on the arc, the reading off to be added will be so many minutes ; if not, we observe between which numbered lines the coincidence actually takes place, and then reckon the preceding minutes as numbered, and afterwards the sub-divisions of the vernier, as po many minutes or seconds. Let us now suppose the index to stand betwe'-n the second and third divisions from 40°. In reading off, first 40° 20' is notod on the arc, and then running the microscope farther on the arc, it is observed that a line on the vernier and On the Sextant. 383 an arc line are in the same direction, between tlie lines on the vernier marked 5 and 6. The farther reading off i.-; therefore 5' and some seconds. Ou examining the interval between 5 aid 6, which is divided into six equal parts, the fourth line to the left of 5 is found to be in the same direction with the opposite one on the arc. The remaining reading off is therefore 40". Hence the whole reading off is 40° 25' 40". The sextant supposed under examination is marked to read off to the nearest 10" ; some instruments are graduated to 15" or 30", &c., but the same method of reading off is to be followed as pointed out above. Take a sextant cut to 1 5", then every degree is sub-divided into four equal parts by lines shorter than those whica represent the degrees ; these short strokes divide every degree into fourths of a degree, or 15': then, on any part of the arc the first short stroke is 15', the second 30', and the third 45'. On the vernier the short strokes from minute to minute are each 1 5". If on the vernier is made to exactly coincide with a large stroke on the arc, the reading is degrees and no minutes ; but if on the vernier coincides with a short stroke, then the reading is so many divisions and sub-divisions : thus, for example, if it coincides with the third short stroke to the left of 30°, the reading is 30° 45', since each short stroke to the left of 30° represents 15'. Again, suppose on the vernier to stand somewhere between the first and second strokes to the left of 36°, in the first place the reading would be 36° 1 5', but it must be something more, because the vernier indicates minutes between 1 5' and 30'. Next look along the vernier and see which stroke on it coincides with any stroke on the arc of the sextant : let us say it is the second short stroke to the left of tenth minute stroke, that is, 10' 30", then the reading will be 36° 15', and 10' 30" to add to it, making 36° 25' 30", and so on for any other indication. Once more let us suppose the sextant to be graduated to 20', then on any part of the arc the first short stroke is 20', and the second 40'. If on the vernier exactly coincides with a long mark on the arc, the reading must be a certain number of divisions, that is, degrees and no minutes ; but if on the vernier coincides with any short stroke or sub-division to the left of division, the reading is evidently degrees and minutes (twenties) ; then suppose it stands at the second short stroke to the left of 32°, the reading is 32° 40'. Lastly, suppose on the vernier to stand somewhere between the first and second short strokes to the left of 43°, in the first place the reading will be 43° 20'. Next look along the vernier and see which stroke on it coincides with any stroke on the arc of the sextant : let us suppose it is the second short stroke after the twelfth lon^ stroke, then the reading will be 43° 20', and 12' 40" to add to it, making 43° 32' 40". 352. To read off on the arc of excess. — As has been observed before, the graduation of the arc of the sextant is usually continued to the right of 0, or zero, in which case we have to read oi'' an arc divided from left to right by means of an index which is divided fn'in right to left ; this, however, is easily done if we remember that the line or; the vernier marked 10' must be con- sidered as the commencement of the divisions, 9' must be considered as i', 8' as 2j, 7' as 3', &c. ; or else take the difference between the minutes and 3^4 On the Sextant. seconds denoted by the vernier and i o' ; thus, if the coincidence of lines on the arc and vernier is at 7' 20", we must read this as 2' 40" ; if at 5' 40" -we must read this as 4' 20", and so on. Similarly, for a sgxtant cut to 1 5", if O on the vernier stands between the second and third strokes to the right of on the arc, and the seventh minute stroke of the vernier coincides with a stroke on the arc, then the reading on the arc of excess, that is off, will be 38', since the seventh stroke is 8 when reckoned from the left of the vernier. The preceding remarks relate to the true Quadrant and Sextant, but instru- ments of the first kind are now not unfrequently graduated on the limb to 120°, and the second kind up to 160° ; this arrangement is effected by placing the index-glass at an angle with the index-bar, and so fixing the horizon - glass that it shall be parallel with the index-glass, when 0^ on the vernier coincides with 0° on the limb- ADJUSTMENTS OF THE SEXTANT AND QUADRANT. 353. The adjustments of the sextant and quadrant are: — (i) To set the index-glass and (2) the horizon-glass perpendicular to the plane of the instrument ; (3) to adjust the line of collimation of the telescope, i.e., to set the axis of the telescope parallel to the plane of the instrument ; (4) and to set the horizon-glass parallel with the index-glass, when (zero) on the vernier coincides with (%ero) on the arc; then, if the adjustments cannot be perfected, (5) to find the inde.c error of the instrument : — \st. The index-glass, or central mirror, must be perpendicular to the plane of the instrument. — Place the index near the middle of the arc. Hold the sextant with its face up, the index-glass being placed near the eye, and the limb turned from the observer. Look obliquely down the glass ; then, if the part of the arc to the right, viewed by direct vision, and its image in the mirror, appear as one continued arc of a circle, the adjustment is perfect ; if the reflection seems to droop from the arc itself, the glass leans hach ; if it rises upward, the glass leans forward. The position is rectified by screws at the back. znd. The horizon-glass, or fixed mirror, must be perpendicular to the plane of the instrument. — (a) By the sea horizon — Set on the index to on the arc. Hold the instrument with its face up ; direct the sight to the horizon- glass, give the instrument a small nodding motion ; then if the horizon, as seen through the transparent part of the horizon-glass, and its image, as seen in the silvered part, appear to be in a continued straight line, the adjustment is perfect. For this method of (a) testing there must be no index error, which caution is unnecessary when (h) the sun is used. (b) By the sun. — The instrument being held perpendicular, look at the suu (using the shades) ; sweep the index-glass along the limb, and if the reflected image pass exactly over the object itself, appearing neitlier to the right nor left of the object, then the horizon-glass is perpendicular to the plane of the instrument; if not, tiu-n the adjusting screw, which in some instruments is a mill-headed one at the back of the instrument, while in others it is a small screw behind and near the upper part of the glass itself, whic*h can be turned by placing a capstan-pin into the hole in the head of the screw. On th Sextant. 385 3r<7. The line of collimation,* or in other words, the axis of the telescope , must be parallel to the plane of the instrument.! — Turn the eye-piece of the telescope till two of the parallel wires in its focus appear parallel to the plane of the instrument; then select two objects, as the sun and moon, whose angular distance must not be less than from 100° to 120°, because an error is more easily discovered when the distance is great ; bring the reflected image of the sun exactly in contact with the direct image of the moon, at the wire nearest the plane of the sextant, and iix the index ; then, by altering a little the position of the instrument, make the object appear on the other wire ; if the contact still remains perfect, no adjustment is required ; if they separate, slacken the screw furthest from the instrument in the ring which holds the telescope, and tighten the other, and vice versa if they overlap, ^th. The horizon-glass must be parallel to the index-glass. — Set on the index to on the arc ; screw the tube or telescope into its socket, and turn the screw at the back of the instrument till the line which separates the transparent and silvered parts of the horizon-glass appears in the middle of the tube or telescope. Hold the sextant vertically, that is, with its arc or limb downwards, and direct the sight through the tube or telescope to the horizon ; then, if the reflected and true horizons do not coincide, turn the screw at the back and at the lower part of the horizon-glass till they are made to appear in the same straight line. Then will the horizon-glass be truly parallel to the index-glass. J 354. Def. — Index Error of reflecting instruments, such as the sextant, is the difi'erence between the zero point of the graduated limb, and where tho zero point ought to be, as shown by the index when the index-glass is parallel to the horizon-glass. <^th. To find the Index Correction. — The two objects generally used to determine the index error are (a) the sea horizon, and (b) the sun. (a) JBy the horizon. — Holding the instrument vertically, move the index till the horizon coincides with its image, and the distance of on the index from on the limb is the index error ; suhtractive when on the index is to the left, and additive when it is to the right of on the limb ; but if on the index stands exactly at on the limb, there is no index error. Ex. I. — The horizon and its image being made to coincide, the reading is z on the arc. Then 2' is the Index Correction to he subtracted from every angle observed. Ex. 2. — "When the horizon and its image were made to coincide, the reading was 3' 20" off the arc; the index correction therefore was -\- 3' 20". Note. — The reading may be on the arc proper, or on the arc of excess ; it is on the latter when O on the vernier is to the right of on the limb, and the reading is then said to be off the arc ; when on the vernier is to the left of on the limb, the reading is said to be on the arc. * The line of collimation, i.e., the path of a visual ray passing through the centre of the object-glass, and the middle point between the cross wires. t The error caused by the imperfection of this adjustment is called the "Error of Collimation," and the observed angle is always too great, X Some sextants,' as Troughton's Pillar Sextants, are not provided with the moans for making this adjustment, because it is not absolutely necessary. An allowance, called Index Error, being made for the want of paralleliem of the two glasses when the zeroes coincide. 386. On the Sextant. ( (2). By measuring the smz's semi-diameter. — Fitting the telescope and arranging the shades so that the reflected and direct images of the sun may be viewed clearly and seen of the same brightness, measure the sun's hori- zontal diameter, moving the index forward on the divisions until the images of the true and reflected suns touch at the edges ; read off the measure which will be on the arc ; then cause the images to change sides, by moving the index back ; take the measure again and read off; this reading will (as is generally the case) be off the arc ; half the difference of the two readings is the index correction. When the reading oni\iQ arc is the greater, the correction is suMractive; when the lesser, additive. Examples. Ex. I. On the arc — 33' 10* Off + 30 50 2)2 20 Index corr. sub. i 10 Ex. 2. On the arc — 30' 20" Off + 33 30 2)3 10 Index corr. add i 35 If both readings are on the arc, or both off the arc, half their sum is the index correction — subtractive when both on, additive when both o/the arc. Ex. 3. ist reading on the arc — 6^' 30" 2nd „ „ — I 40 2)67 10 Index corr. sub. 33 35 Ex.4. I st reading off the are + 1' 30" 2nd „ „ + 66 50 2)68 20 Index corr. add. 34 10 One-fourth of the sum of the two readings should be equal to the sun's semi-diameter in the Nautical Almanac for the day ; but if both readings be on or both off the arc, one-fourth their difference should be the sun's semi- diameter. Thus, suppose the ohservations, in Example i, to be made on September 26th, 1882 ; here one-fourth of the sum of the two readings is 16' o", agreeing with the semi-diameter as given in the Nautical Almanac for the given day. This affords a test of the accuracy with which the observation has been made. But in order that the comparison may be a good criterion, we should measure the sun's horizontal diameter which is not sensibly affected by refraction. Ols. — In order to obtain the index correction with the greatest precision, the mean of a number of measurements of the sun's diameter should be taken ; also, the limb should be placed (by hand, before the tangent screw is used) alternately a little open and a little overlapping, so that in making the contact the tangent s(3rew may be turned different ways. Examples for Pbactioe. Ex. I. 1882, April 17th, the reading on the arc 29' 40", the reading off the arc 34' 10* : required the index correction and semi-diameter. Ex. 2. 1882, July 4th, the reading on 33' lo'', off 29' 50"; find index correction antj eemi-diameter, On the Sextant. J87 Ex, 3. 1882, November 13th, on 4' 40", off 60' 10": find index correction and semi- diameter. Ex. 4. 1882, July loth, on 32' 45", off 34' 30": find index correction and semi-diamoter, Ex. 5. 1882, March 21st, off 1° 10' o", off 6' 40" : find index correction and semi-diameter. Ex. 6. 1882, January 17th, on 67'4o', on 2' 30" : find index correction and semi-diameter. 355- The Prismatic Sextant. — In the form of instrument just described, and which is all but universally employed, the angle measureable is limited to 140° ; but we may perha^^s add that Pistor and Martins, of Berlin, have, by an ingenious modification of the horizon-glass (for which they substitute a prism), produced a sextant which will measure any angle up to 180°. This instrument is called the Prisitatio Sextant. The following shows the form of Examination Paper on the Adjustment of the Sextant. EXAMINATION PAPER. Exn. 9a. Fort of ADJUSTMENTS OF THE SEXTANT. Rotation No. The applicant will answer in writing, on a sheet of paper which will he given him ly the Examiner, all the following questions, numbering his answers tvith the numbers corresponding to the questions. I I. — What is the first adjustment of the sextant ? A. — To set the index-glass perpendicular to the plane of the sextant. 2. — How do you make that adjustment ? A. — Place the index near the middle of tho arc, and look into the index-glass so that you can see both tho arc and its reflection ; if they be in one line, the glass is perpendicular, but if not continuous, they are brought so by the screws in tho frame upon which the glass stands. 3. — "What is the second adjustment? A. — To set tho horizon-glass perpendicular to the plane of the sextant. 4. — Describe how you make that adjustment ? A.— Place O of the vernier on on the arc, hold the instrument obliquelj', with its face upwards, and look from the sight vane at tho horizon ; if the reflected part and the direct portions of the horizon are in one line, this aljustment is perfect, but if not, they must be brought in lino by gently moving a screw at the back (top) of the glass. 5. — What id the third adjustment ? A. — To set the index and horizon-glasses parallel when the index is at 0. 6. — How would you make the third adjustment ? A. — Place the index at 0, and holding tho instrument verticallj', look at the horizon ; if the reflected and direct parts are in one line, this adjustment is perfect, but if they are not in one line, move a screw at the back of the horizon-glass until they are. 7. — In the absence of a screw how would you proceed ? A. — I would find the index correction, or as it is called, the index error. 388 On the Sextant. 8. — How would you find the index error by the horizon P A. — Hold the instrument vertically, and, looking at the horizon, move the tangent screw until the horizon in botn parts of the horizon-glass form one line ; the reading is the index error. 9. — How is it to he applied ? A. — To he added when the reading is off the arc, and to subtract when the reading is on the arc. 10.— Place the index at the error of minutes to be added, clamp it, and leave it. The Examiner will see that it is correct. This is a reading off the arc, i.e., on the arc of excess II. — The Examiner will then place the zero of the vernier on the arc, not near any of the marked divisions, and the Candidate will read it. In all cases the Candidate will name or othenvise point out the screws used in the various adjustments. NoTB to 10 and 11. — When the Examiner is satisfied that the Candidate can read the arc of the sextant both on and off the arc, it mil be sufficient to place his initials against 10 and 11 on the paper containing the answer. The above completes the examination of Second and Only Mates. In addition to the above, First Mates and Masters will be required to state in writing : — 12. — How do you find the index error by the sun ? A. — Place the index at about 30' on the arc, and holding the instrument vertically, look at the sun, two suns will be seen ; bring their upper and lower limbs in exact contact, read off and mark down, then place the index at about 30 off the arc, or to the right of O, bring down the upper and lower limbs in contact as before, read off and mark down ; half the difference of these two readings will be the index error. 13. — How is the same applied ? A. — To he added when the greater reading is off the arc, and subtracted when the greater reading is on the arc. 14. — What proof have you that those measurements or angles have been taken with tolerable accuracy ? A. — By adding the two readings together, and dividing the sum by 4 ; if the measure- ments are correct, the result should be nearly equal to the semi-diameter for the day, as given in the Nautical Almanac. If they do not so agree, repeat the observations until they do. 3^9 CURRENT.— SOUNDINGS. TO FIND THE COUESE TO STEER IN ORDER TO MAKE GOOD ANY COURSE IN A KNOWN CURRENT, AND ALSO THE DISTANCE MADE GOOD. 356. Draw a line on a chart to represent the course to be made good; from the ship's place on the chart lay ofT a line in the direction of the set of the current, on which mark off from the ship's place the rate of the current per hour ; then take in the compasses the distance the ship sails in an hour by log, and put one foot on the last-named mark, and from tlie point where the other foot reaches the first line draw a line to the mark on the lino representing the direction of the current. The course to be steered is repre- sented by the line last drawn, and the parallel ruler being placed to it, and moved to the centime of the compass on the chart, will give the course of the ship ; and that portion of the first line drawn, intersected by the last line drawn, will be the distance the ship will make good per hour. On a chart, suppose A to be the place of the ship, B the port of destination ; also A C the set of the current, the rate per hour being taken from the scale of miles and laid off in the direction of the line. Take the distance sailed by the ship per hour from the scale of miles, and with one foot of the dividers at C, make an arc cutting A at D. Join C D, and move the parallel ruler from C D to A, drawing A E parallel to C D ; then A E will be the direction of the ship's head ; and the parallel ruler being moved to the centre of the compass on the chart, will give the course of the ship on the chart ; and A D will be the distance the ship will make good. SOUNDINGS. 357. In the open sea, the tide requires about six hours and a quarter to rise from low to high water, and an equal interval to fall from high to low water. If the rise or fall was an uniform quantity throughout, by simply ^9° Soundings. taking a proportionate part of the rise or fall due to the time of tide, we should at once obtain the quantity required to reduce the soundings to the low water of that day. But the water does not rise in equal proportions, the rise during the first and last hours being very small (about one-sixteenth of the whole range) ; in the second hour there is a considerable increase of rise; in the third and fourth hours a still greater increase of rise ; and then the rise begins to take off in the same proportion as it increased.*' The correct amount for every half-hour, and for various ranges, is given in the "Tide Tables for the English and Irish Ports for 1880," (p. 98, Table B), published by the Hydrographic Office, Admiralty.f 358. As the soundings upon the chart are all referred to or measured downwards from the mean level of low water oi ordinary spring tides, J casts of the lead taken at any other time of the tide, or any other day than full and change, will exceed the depth marked on the chart (except when it happens to be low water of greatest spring tides). It is necessary for the seaman to be able to calculate the difference between the actual depth obtained by means of his lead, and that marked on his chart, in order to the identification of his ship's place, more especially when the range of the tide is considerable, and the depth not great. Also, when about to enter a port in a vessel whose draught of water is nearly equal to the depth, it is necessary to find the height of the tide as exactly as circumstances will permit. 359. Two classes of questions may be proposed in reference to this subject —firstly, to find the depth of water at a given place and time ; secondly, having obtained the actual depth by a cast of the lead, to find the sounding on the chart corresponding thereto, and thence to identify the ship's place. Both these classes of questions require us to know the time of high water and the range of the tide on the given day ; and for this purpose almanacs are published. The most correct, and by far the most useful of all these, are the "Tide Tables" published by the Admiralty, and to which we have already referred. In this book are given the times of high water and the height of the tide for every day in the year, at each of the princij)al ports in Great Britain. * The reader may obtain an idea of this law, suiiiciently exact for practical purposes, in the following manner: — Describe a circle, and divide the circumference into six equal parts on each side, corresponding to the hours of the tide ; then divide the diameter into propor- tional parts, corresponding to a given (assumed) range of tide. Connect the segments of the circle by straight lines drawn across the figure, when it will be perceived that they intersect the diameter at certain divisions of the range. These are the correct quantities respectively due to each hour's rise or fall of such a tide from low to high water, and vice versa. An examination of tliese quantities will show, that in the first hour of the tide the rise is equal to one-sixteenth of the whole range; at two hours from low or high water, the tide has risen or fallen one-fourth of the whole range ; at three hours it has risoa just half iis, range ; at four hours it has risen three-fourths of the whole range ; at five hours to within a sixteenth of the whole range. The above method, which is constructed on principles theoretically correct, will represent with sufficient exactness all that is necessary for practical purposes. t Table XIX, Kaper, which the author, in 1847, computed for Eaper's work, also shows the space through which the surface of the water rises and falls at given intervals from high or low water. X On most charts the soundings expressed are reduced to low water of ordinary spring tides ; but in some charts, however, the soundings are reduced to the low water of extra- ordinary spring tides — such, for example, is the case on the chart of Liverpool, surveyed by Captain Denham, R.N., the soundings on which are reduced to a spring range of thirty feet, while the mean spring range for that place, as dteduced from observations made for two years at the Tide Gauge, St. George's Fier, is 26 feet. Soundings. 391 360. To find how miicli we must subtract from a cast of tlie lead, in order to a comparison with the soundings marked on the chart, proceed by EULE cm. 1°. Open the Admiralty Tide Tables at the proper month; and in the column under the head of the place near your position, and opposite the day of the month, take out the "time " of high water in the morning or afternoon, as the case reqxiires, and also from the adjoining column, under " height,^^ take out the height of the tide. 2°. JV^ext, underneath the time of high loater place the time at ship, and take the diflference and call it "time from high water." 3°. From the height of tide subtract the half mean Spring Range, which stands at the foot of the column. The remainder is the half range of the day. 4°. Unter Table B, page 98, Admiralty Tide Tables, arid under the time from high tvater, and opposite the half range for the given day, take out the correction corresponding thereto, observing whether it is to be added or subtracted. 5°. Add or subtract the correction, as directed, to the mean half Spring Range marked on the chart. The result is the excess of the sounding observed above the sounding recorded on the chart, or is the height of the tide above zero. 6°. Subtract this last from the sounding ihoivn by the lead, the remainder is the sounding shown by the chart. Note T. When it happens to be an extraordinary low ebb tide, the quantity given in Table B will be greater than the half mean spring range, and will be stibtractive. In such cases, subtract the half mean spring range from the correction by Table B, and add tho result to the soundings by lead; the sum will be the sounding on the chart. Examples. Ex. I. 1880, September i6th, at ^^ 41"" p.m., a ship off Liverpool strikes soundings in 8 fathoms: required the corrected soundings to compare with the chart. (Tho half spring range by Captain Denham's chart is 15 foet.) Admiralty Tide Tables (pago 70) ; time of high water at Liver- pool, September 1 6th, 1883 g'' 41'" P.M. Time of sounding 7 41 Time from high water 30 ft. in. Height at Liverpool 26 7 Half mean spring range 13 o Half-range of the day 13 7 In Table B, page 98, under 2'', opposite 13^ ft., stands flf/c^ . 6 9 Half spring range by chart 15 o Correction 3^ fathoms, or 21 9 Depth by lead 8 fathoms. Correction 3I „ Showing the depth by comparison . . . # 4^ „ Whence the depth to compare with the chart is only 47 fathoms instead of 8 fathoms. 392 Soundings. Ex. 2. 1880, October 6th, at 7^' 23™ a.m., a vessel anchored off Weston-super-mare in 6| fathoms; at low water the vessel was "high and dry :" required tho cause of this. (Half spring range by chart 23 feet.) By Table : October i6th, the time of high water at Weston- super-mare 7!^ 47™ A.M. Time of anchoring 7 23 Time before high water o 24 ft. in. Height of tide by Tables 38 8 Half spring range 18 7 Half range . . . . 20 i By Table B, 24"^ and half range 20 feet i inch give add , , 196 By chart: half spring range , .. .. 23 o Correction to low water 42 6 Sounding 6^ fathoms, or 39 o 3 6 Water below the sounding ; or, the ship is found to be 3 feet 6 inches drj' at low water. Ex. 3. 1880, March 2nd, at 4'' 18'" a.m., a vessel has to cross the Victoria Bar, Liverpool : it is required to know what water she will have over the bar. (Depth at low water springs on chart, 11 feet). By Table: March 7 th, time of high water at Liverpool .. 2^ 15™ a.m. Time of crossing the bar •• .. 4 18 Time after high water 2 3 ft. in. Height of tide by Tables 25 7 Half spring range 13 9 Half range for the day 12 10 By Table B: 2i» 3'" and half range 12ft. 7in add 6 5 Half spring range by chart 150 Correction 21 5 Depth on Bar at 2'' 3™ from high water, March 7 th . . . . 11 o By chart : depth on Victoria Bar at low water springs . . 32 5 or 5^ fathoms, nearly. Ex. 4. 1880, August 2i8t, at 2*1 ii"" p.m., offWeston-super-mare, sounded in 4^ fathoms : required the soundings on the chart. Time of high water, Weston-super-mare, August 21 st. . .. 7'' 43™ p.m. Time of sounding 211 Time from high water 5 32 Height of tide, Weston-super-mare, August 2 rst 39ft.2in. Half mean spring range 18 7 Height above half tide , . 20 7 By Table B : 5^ 32n> and half range 20 ft. 7 in subt. 19 10 Half spring range 18 7 Level of tide below zero . . i 3 Soundings by lead 4I fathoms, or 27 o Correction -\- i 3 Soundings on chart 28 3 Or a little less than 5 fathoms. Soundings, 393 Examples for Praotioe. 1. 1880, August 8th, at 9^ 37™ a.m. : required the depth of water on the " Four-fathom Ledge," ofifWeston-super-mare. 2. 1880, June 9th, at i^ 28™ a.m. : off Brest, the depth of water by the lead was loj fathoms : required the soundings on the chart. 3. 188&, August 1 8th, at %^ 8™ p.m., sounded in the Victoria Channel, Liverpool, in 5 fathoms : required the soundings on the chart. 4. 1880, March iSth, at 7*" 55" p.m., a vessel anchored off Weston-super-mare, in 6 fathoms; required the depth at low water (half spring range by chart 23 feet). 5. 1880, March 12th, at s^ 42™ p.m. : required the height of the tide above mean low water of spring tides at Liverpool. 6. 1880, December 13th, at 8'^ 10'" a.m. : going up the Firth of Forth, the load showed 12 fathoms : required the soundings on the chart. 361. The following is the form of the Eule as used at the Liverpool Examinations : — 1°. Take the difference between the tune of high water, full and change, at Liverpool and full and change at ship, and take this difference from the time of high water on the given day at Liverpool ; the result is time of high water at ship. 1°. Next find the time from high water when the " cast " was taken. 3°. Take 1 3 feet, the half mean spring range from the height of tide on given day at Liverpool. 4°. Apply a correction from Table B to the half mean spring range, as directed at the head of the Table ; the result is the Reduction at Liverpool. Lastly. — Find the Reduction at ship (by proportion), thus: — As Spring Range at Liverpool ^ mi. t> j i.- n Is to Spring Range at Ship, ^^^ Reduction or Correction So is the Reduction at Liverpool ( • , t ot foundings To the Reduction at Ship. ) '' ^' ^' taken from the Cast. Er. 1880, September 22nd, at i^ 57™ p.m. at ship, off Holyhead, sounded in 45 fathoms : required the corrected cast to compare with the chart. Full and change at Liverpool iih23ra | Page 152, Admiralty Tide Full and change at Holyhead 10 11 j Tables, 1880. Difference — i 12 Time high water, Liverpool, Sept. rgth i 9 P.M., and Height of tide 25 J ft, Half mean spring range 13 Time high water at ship 11 57 a.m., Time of cast r 57 p.m., Half range for day 12 J Time of cast from high water 2 o ) give in Table B correction -j- 6 Half range 12^ ft. / Half spring range 13 By Proportion. ft. ft. ft. Keduction at Liverpool 19 26 : 19 :: 16 16 fath. ft. 26)304(12 feet =20 26 (nearly) 45 o cast taken. 44 43 o cast corrected. Note.— Tn the above proportion, 26 is the spring range at Liverpool, 16 the spring range at ship, and 20 the reduction at Liverpool. SEE 394 ON THE CHART. 361. A Chart is a map or plan of a sea or coast. It is constructed for the purpose of ascertaining the position of the ship with reference to the land, and of shaping a course to any place. 362. The use to be made of the chart in each case determines the method of projection, and the particulars to be inserted. ([) The chart may be required for coasting purposes, for the use of the pilot, &c., and then only a very small portion of the surface of the globe being represented at once, no practical error results from considering that surface a plane, and a ^' plane chart^' is constructed in which the different headlands, lighthouses, &c., are laid down according to their bearings. The soundings on these charts are marked with great accuracy; the rocks, banks, and shoals, the channels, with their buoys, the local currents, and circumstances connected with the tides, are also noted. (2) Again, for long sea passages the seaman requires a chart on which his course may conveniently laid down. The track of a ship always steering the same course appears as a straight line (and can at once be drawn with a ruler) on the Mercator's chart. Hence the charts used in navigation are Mercator's charts. (3) When great circle sailing is practic- able, and of advantage, a chart on the " central projection,''^ or gnomic, exhibits the track as a straight line, and is therefore convenient.*' ON MEECATOE'S CHAETS. CSee Norie, 2iages 126 — 131 ; or Maper's '■'Practice of Navigation," pages 120 — 127, on this subject) . 363. A chart used at sea for marking down a ship's track and for other purposes, exhibits the surface of the globe on a plane on which the meridians are drawn parallel to each other, and therefore the parts BH, CI, DK, &c. (fig. chap. def. nav.), arcs of parallels of latitude, are increased and become equal to the corresponding parts of the equator UV, VW, &c. Now, in order that every point of this plane may occupy the same relative position with respect to each other that the points corresponding to them do on the surface of the globe, the distance between any points, A and 0, and A and F must be increased in the same proportion as the distance FO has been increased. The true difference of latitude, AO, is thus projected on the * The method lately introduced by Hugh Godfray, Esq., M.A., St. John's College, Cambridge, deserves special mention, as its beauty and simplicity will ultimately lead to its general adoption. A chart on the central projection, as stated above, exhibits the great circle as a straight line, and thus it is seen at once, whether the track between two places is a practicable one ; hence, also, we have by inspection the point of highest latitude. An accompanying diagram then gives the different courses, and distances to be run on each, in order to keep within \ of a point to the great circle. This chart and diagram is fully described in the Transactions of the Cumbridge Fhilosophical Society, vol. X, part II, and is published by J. D. Potter, Poultry. On the Chart. 395 chart into what is called tlio meridional difference of latitude, and the departure 13H -f CI + DK, &c., into the difference of longitude, and the representation is called a Mercator's projection. It is evidently a true representation as to form of every particular small track, but varies greatly as to point of scale in its different regions, each portion being more and more enlarged as it lies farther from the equator, and thus giving an appearance of distortion."* (i.) In charts generally, the upper part as the spectator holds it, is the North, the lower part South, and that towards his right hand the East, that towards the left West, as on the compass card. In a case which sometimes happens when the upper part is not the North, the North part may be known by the North part of the compass. (2.) On Mercator's chart the parallel linos from North to South (from top to bottom) are termed meridians, and they are all perpendicular to the equa- tor ; the meridians on the extreme right and left are the graduated meridians — so called from showing the divisions for degrees and minutes. The latitude is measured on the graduated meridians, and also the distance. (3.) The parallel lines from West to East (from left to right) are called parallels, and they are all parallel to the equator, the parallels at the top and bottom are graduated to degrees and minutes — and longitude is measured on the graduated parallels. Distance cannot be taken from them. (4.) The depth of water is denoted, as also in some places the quality of the bottom. The numerals or figures in harbours, bays, channels, &c., indicate soundings reduced to low water ordinary spring tides. The Roman figures indicate the time of high water at full and change of the moon. Thus : XI hrs. 34™ F & C means that the time of high water is thirty-four minutes past eleven on days of full and new moon. The anchors on the chart denote * It is plain from the principles of Mercator's projection, and from the diagram (page 200) which connects the enlarged meridian with the difl'erence of longitude, that if a ship set out on any point on the globe, and sail on the same oblique rhumb towards the pole, it can reach it only after an infinite number of revolutions round it. For from any point to the pole, the projectci meridian is infinite in length, and so, therefore, is the difference of longitude due to this advance in latitude upon an oblique course. Consequently, this latitude can bo reached only after the ship has circulated round the polo an infinite number of times. These endless revolutions, however, are all performed in a finite time, the entire track of the ship being of limited extent. This, however paradoxical it may appear, is necessarily true from the principles of plane sailing, which shows that any finite advance in latitude is always connected with a finite length of track, this length being '- — ^— cos. course. The apparent paradox of the infinite number of revolutions about the polo being performed in a finite time, becomes explicable when we consider that, whatever be the progressive rate of the ship along its undeviuting course, the times of performiDg the successive revolutions continually diminish as the ship approaches the pole, both the extent of circuit and the time of tracing it tending to zero, the limit actually attained at the polo itself; hence there must ultimately be an infinite number of such circuits to occupy a finite time. When the pole is reached the direction all along preserved may still be continued, and a descending path will be described similar to that jist considered, and which will conduct the ship to the opposite pole, after an infinite number of revolutions round it, as in the former case. In receding from this pnlc the track described will at h^ngth unit's with that at first traced, the point of junction being thut from which the ship originally departed. But for the strict mathematical proof of these I ittor circumstances the student may consult Professor Davies' curious and instructive papers on Spherical Co-ordinates in the Edinburgh Transactions, vol. XII. ^^6 On the Chart. anchorage. The small arrows show the direction of the set of the current, the current going with the arrow. ( 5 . ) Lines called Compasses, similar to those on the compass card, are drawn at convenient intervals on the chart. In charts of large seas, as the Atlantic, these comimsses are generally drawn so that the line from the North to the South point corresponds with the true meridian ; but in coasting charts the same line generally coincides with the correct magnetic meridian. (6.) When the true course between two places is known, it must be remembered that Westerly variation is allowed to the right, and Easterly to the left hand of the true course in order to obtain the compass course. (7.) In " cross hearings,''^ both bearings must be corrected for the deviation due to the direction of the ship's head at the instant of making the obser- vations. (8.) With respect to the method of determining the ship's position by cross bearings, it may be observed that this is the most complete of all methods when the difference of bearings is near 90°; but if the difference is small — as, for example, less than 10° or 20°, or near 180" — the ship's position will be uncertain, because a small error in the bearing will cause a great error in the distance. — (Eapeb, page 120, No. 367.) Exn. 9b. Port of EXAMINATION PAPEE. EXAMINATION IN OHAET. Rotation No. The applicant will be required to answer in writing, on a sheet of pap>er which will be given him by the Examiner, all the following questions according to the grade of Certificate required, numhering his answers with the numbers corresponding ivith those in the question pajjer. I . — A strange chart being placed before you, what should be your special care to determine, before you answer any questions concerning it, or attempt to make use of it ? A.. — Which is the North part of the chart. Note. — If a foreign chart, note wliat meridian it is projected for. 2. — How do you ascertain that in our British charts ? A. — In our British charts there is always at least one compass, the true north point of which is designated by a star or other ornament. 3. — Describe how you would find the course by the chart between any two places, A and B. A. — Lay the edge of a parallel ruler over the two given places, A and B, then taking care to preserve the direction, move one edge of the ruler until it comes over the centre of the nearest compass on the chart ; the circumference of the compass cut by the edge of the ruler would show the course according to the direction the one place is from the other. 4. — Supposing there to be points of variation at the first named place, what would the course be magnetic ? the true course being In answering this question, merely write down the magnetic course corresponding to the true course given. On the Chart. 397 Rule.— To turn true course into a magnetic course, allow Easterly variation to the left hand, Westerly variation to the right hand, thus — True course N.E. by N., with 2 points W. variation, gives N.E. by E. magnetic. „ ,, with 2 points E. ,, N. by E. „ True course S.E. by S., with 2 points W. variation, gives S. by E. magnetic. ,, „ with 2 points E. „ S.E. by E. „ True course S.W. by S., with 2 points W. variation, gives S.W. by W. magnetic. ,, „ with 2 points E. ,, S. by W. ,, True course N.W. by N., with 2 points "VV. variation, gives N. by W. magnetic. „ „ with 2 points E. „ N.W. by W. „ A. — points of variation should bo allowed to the and the magnetic course would bo 5. — How would you measure the distance between those two or any other two places on the chart ? A. — With a pair of dividers measure half the distance on the chart between, then placing one leg of the dividers on the middle latitude on the graduated meridian, measure on each Side of the same, and the number of degrees measured between those two extreme points, brought into miles, will be the distance required. 6. — Why would you measure it in that particular manner ? A. — Because on a Mercator's chart the degrees of latitude increase in length as the lati- tude increases. The above comprises all the questions on the chart that are put to Mates and Only Mates. In addition to the above, the Masters are required to answer : 7. — What do you understand those small numbers to indicate that you see placed about the chart ? A. — Depths of water in fathoms, or feet, as specified on the chart. 8. — At what time of the tide ? A. — At low water ordinary springs. 9. — What are the requisites you should know in order that you may compare the depths obtained by your lead-line on board with the depths marked on the chart ? A. — The time from high water "rise and fall," or as it is now called, the "mean spring range." Note.— The rise of tide at spring and neaps, as well as the range, is given on a chart to facilitate finding the height of tide at different hours between high and low water, 10. — What do the Roman numerals indicate that are occasionally seen near the coast and in harbours ? A. — The time of high water at that place at full and change of the moon. Note. — It is generally expressed thus,— H.'W. at F. and C, VIII'' ja", that is, high water at full and change of the moon occurs at S^ 32". II. — How would you find the time of high water at any place, the Admiralty Tide Tables not being at hand, nor any other special tables available ? A. — To the time of high water at full and change add 49 minutes for every day that has elapsed since the full or change of moon, the sum will be the p.m. tide for the given day approximately; or, to the lime of the moon's meridian passage, corrected for longitude, add the port establishment, the sum will be the p.m. tide required. All the above question'^ should be amwcrcd, but this does not preclude the Examiner from putt iug any other questions of a practical character, or which the local circumstances of the port may require. 39^ On the Chari. PE ACTIO AL EXAMINATION IN THE USE OF THE CHAET. In this problem the chart is that used for Sumner's Method, on which the compass is true : the amount of variation, E. and W., will be written within or near the compass : and the Candidate is supplied with a Table of Devia- tions ; these deviations are given in two columns, one for the s?dp^s head correct magnetic, and the other for the ship^s head ly compass. The form of the question is as follows : — FOE ALL GKADES WHERE THE CHART IS USED. a.— Using Deviation Card No. find the course to steer by compass from to , also the distance. Answer. — Compass Course Distance Variation b. — "With the Ship's head on the above-named Compass Course bore by Compass Distant miles ; find the Lat. and Long, of the Ship when it was on that bearing. Answer. — Latitude Longitude c. — "With the Ship's head as above bore by Compass also bore by the same Compass. Find the Ship's position. Answer. — Latitude Longitude Signature Bate On the different days of Examination, the positions in (Question a), and consequently the directions of the line from one place to the other, the various bearings of the letters (which are taken to be lighthouses or land marks) are changed. The form above, however, indicates the character of the question. 399 DEVIATION OAED— No. I. SHIP'S HEAD WHEN BUILDING. Ship's Head Ship's Head Correct Magnetic Deviations. By Compass Deviations. North iii^W. North 15° W. N.by E 9 W. N.by E 12 W. N.N.E 6 W. N.N.E 7iW. N.E. byN 3 W. N.E. by N 4 W. N.E o N.E N.E. hy E 2^E. N.E. by E 3 E. E.N.E. , 6 E. E.N.E 5iE. E.by N 7 E. E. by N 8 E. East 9 E. East 10 E. E. byS 10 E. E. by S ii|E. ES.E e.. iiJE. E.S.E 12 E. S.E. by E I2|E. S.E. by E 13 E. S.E 13 E. S.E 14 E. S.E. by S 14 E. S.E. by S 15 E. S.S.E 15 E. S.S.E 15 E. S. byE 15 E. S.byE 15 E. South , . , 15 E. South 15 E. S.byW.. 15 E. S.byW 14 E. S.S.W 15 E. S.S.W 12 E. S.W. by S 14 E. S.W. by S 10 E. s.w 10 E. S.W 7 E. S.W. by W 5 E. S.W. by W 4 E. w.s.w jiW. W.S.W I W. W.byS 9IW. W.byS 6 W. West 15 W. West 10 W. W. by N 18 W. W.byN 14 W. W.N.W 20^ w. w.n:w 17 W. N.W. by W 20t W. N.W. by W 19 W. N.W 20 W. N.W 21 W. N.W. byN i8iW. N.W. byN 20J w. N.N.W 16 W. N.N.W 20 w. N.byW 14 w. N.byW 18 w. 400 DEVIATION CAED— No. II. SHIP'S HEAD WHEN BUILDING. Ship's Head Ship's Head Correct Magnetic 1 Deviations. By Compass Deviations. North 6rE. North io» E. N. by E io|E. N.byE 18 E. N.N.E i6 E. N.N.E 25 E. N.E. byN 20 E. N.E. byN.. 29 E. N.K 24 E. N.E 32 E. N.E. by E 28i:E. N.E. by E 33 E. E.N.E 30" E. E.N.E 31 E. E. byN 32 E. E.by N 28JE. East 32 E. j East 24 E. E. byS 30 E. i E. by S 18 E. E.S.E 25 E. E.S.E 10 E. S.E. byE 10^ E. S.E. byE 5 E. S.E I W. S.E I W. S.E. by S 4|W. S.E. by S 4 w. S.S.E 7 W. S.S.E 6 W. S. byE 10 W. S. by E 8 W. South 12 w. South 10 W. S.'ibyW i3|W. S.byW II W. s.s.w i5iW. S.S.W 13 W. S.W. byS i8|w. ; S.W. by S 15 w. s.w 21 w. S.W 17 W. S.W. by W 23 w. s.w. by W 19 W. W.S.W 22iW. W.S.W 2 1 W. W. byS 22 W. W. by S 23 w. West 20^ W. West 23 w. W. by N 19^ w. W. byN 2 2 W. W.N.W 17 w. W.N.W 20 W. N.W.byW 14 w. N.W. by W 18 w. N.W io|W. N.W 15 w. N.W. byN 6iW. N.W. byN 10 W. N.N.W 2 W. N.N.W 3 w. N.byW HE. N.byW 31E, On the Chart. 401 EXPLANATION OF QUESTIONS ON THE CHAET. Given True Course, to find Correct Magnetic and Compass Courses. Allow East variation to the left, West variation to the right ; also, East deviation to the left, West deviation to the right. Given Compass Bearings, to get True Bearings. Allow East variation to the right. West variation to the left. EXAMPLE I. (See chart.) ((?) Using Deviation Card No. T : fiad the course to steer by compass from A to B, also the distance. Lay an edge of a parallel rule over the positions A and B, then move the parallel rule (strictly preserving the direction) until an edge pass threugh the centre of the compass, then the point of the compass coinciding with the edge of the ruler shows the true course from A to B ; suppose it to be S. by E., then proceed as follows : — True course S. by E. = S. 1 1" E. (L. of S.) Variation, suppose ii E. allow to left. Correct magnetic course S. 22 E. = S.S.E. Deviation by card (left side) 15 E. allow to left. Compass course S. 37 E. =: S.E. f S. The variation being 11° E., this allowed to the left makes correct magnetic course S. 22"" E., or S.S.E. ; then referring to Deviation Card No. I, page 399, under Ship's Head Correct Magnetic, the deviation is found opposite S.S.E. to be 15° E. ; this must be allowed to the left on the correct magnetic course, making compasa course S. 37° E., or S.E. f S. "Write down this compass' course, and also the distance from A to B, as measured on the graduated meridian. Answer. — Compass course S.E. :J S. Distance 61 miles. Next, a position C is marked on the chart, and the Candidate is required to answer the following question : — (5) With the ship's head on the above-named compass course, C bore by compass N. 44° E., distant 23 miles : find the latitude and longitude of the ship when it was on that bearing. (See No. i, small chart). Proceed as follows :— Compass bearing N. 44° E. Deviation same as in a 15 E. allow to right. Correct magnetic bearing N. 59 E. Variation same as in (?» 1 1 E. allow to right. True bearing N. 70 E. Place the parallel rules on the compass in the comer of the chart, over N. 70° E. and S. 70° W., then work the parallels (strictly preserving the direction) till an edge pass over C. Draw a line trending S. 70^ W. from C ; also take off the distance from the graduated FFF 'm A/9 m On th Chart. 401 EXPLANATION OF QUESTIONS ON THE OHAET. Given True Course, to find Correct Magnetic and Compass Courses. Allow East variation to the left, West variation to the rigid ; also, East deviation to the left, West deviation to the right. Given Compass Bearings, to get True Bearings. Allow East variation to the right, West variation to the left. EXAMPLE I. (See chart.) (a) Using Deviation Card No. T : find the course to steer by compass from A to B, also the distance. Lay an edge of a parallel rule over the positions A and B, then move the parallel rule (strictly preserving the direction) until an edge pass thr«ui;h the centre of the compass, then the point of the compass coinciding with the edge of the ruler shows the true course from A to B ; suppose it to be S. by E., then proceed as follows : — True course S. by E. = S. 1 1° E. (L. of S.) Variation, suppose 1 1 E. allow to left. Correct magnetic course S. 22 E. = S.S.E. Deviation by card (left side) 15 E. allow to left. Compass course S. 37 E. =: S.E. f S. The variation being 11° E., this allowed to the left makes correct magnetic course S. 22" E., or S.S.E. ; then referring to Deviation Card No. I, page 399, under Ship's Head Correct Magnetic, the deviation is found opposite S.S.E. to be 15° E. ; this must be allowed to the left on the correct magnetic course, making compass course S. 37° E., or S.E. J S. "Write down this compass course, and also the distance from A to B, as measured on the graduated meridian. Answer. ■^Gom.'^Sias, course S.E. J S. Distance 6i miles. Next, a position C is marked on the chart, and the Candidate is required to answer the following question : — {b) With the ship's head on the above-named compass course, C bore by compass N. 44° E., distant 23 miles : find the latitude and longitude of the ship when it was on that bearing. (See No. i, small chart). Proceed as follows :— Compass bearing N. 44° E. Deviation same as in a 15 E. allow to right. Correct magnetic bearing N. 59 E. Variation same as in (^ 1 1 E. allow to right. True beating JJ. 70 E. Place the parallel rules on the compass in the comer of the chart, over N. 70° E. and S. 70° W., then work the parallels (strictly preserving the direction) till an edge pass over C. Draw a line trending S. 70° "W. from C ; also take off the distance from the graduated FFF 402 On the Clio/rt. meridian, so that the middle latitude between C and the ship'a position may bisect the distance contained between the points of tho dividers ; apply the distmce thus found from C in the direction of the line drawn through it, and here is found the ship's position — latitude and longitude — which take from the proper scale and write down. Answer. — Latitude 51° i6|' N. Longitude 178° 2' E. Finally, two positions D and E are marked on the chart, and the Candidate is required to answer the following questions. (See No. r, small chart.) (c) With the ship's head as above, D bore by compass N. 33° E., also E bore N. 74° E. by the same compass : find the ship's position. In this question, the same corrections (variation 25° W. and deviation 11° E.) applied to the compass bearings of D and E, give their true bearings, viz., N. 59° E. and N. 80° E., thus — D E Compass bearings N". 33° E. N. 74° E. Deviation same as in «, allow to right 15 E. 15 E. Correct magnetic bearings N. 48 E. N. 89 E. Variation same as in «, allow to right 1 1 E. 1 1 E. True bearings N. 59 E. =: N.E. by E. \ E. N. 100 E. 180 S. 80 E. = E. I N. Proceed, as in question 5, to draw a line S.W. by W. \ "W. from D, and another line W. \ S. from E ; the point where these two lines intersect is the ship's position : find the latitude and longitude of this point, and write them down. Answer. — Latitude 51° 40' N. Longitude 177° 34^' E. EXAMPLE ir. (See chart). («) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. (Suppose variation 29° E.) True course E. by S. = S. 79° E. Variation, suppose 29 E. allow to left. Exceeds 90°, S. 108 E. Subtract from 1 80 Correct magnetic course N. 72 E. = E.N.E. f E. Deviation by card 6^ E. allow to left. Compass course N. (>^\ E. = N.E. by E. f E. Distance 773 miles. Answer. — Compass course N.E. by E. f E. Distance 775 miles. {b) With the ship's head on the above-named compass course, G bore by compass N. 69!° W., distant 19 miles: find the latitude and longitude of the ship when it was on that bearing. Compass bearing N. 69^° W. Deviation same as in a (>\ E. allow to right. Correct magnetic bearing N. 63 W. =: N.W. by W. f W. Variation same as in a 29 E. allow to right. True bearing N. 34 W. = N.W. by N, .^insw^r.— Latitude 50° 31' N. Longitude 177° 15' E. On ihe CharS. 403 (c) "With the ship's head as above, D bore by compass N. 46^° W., also E boro N. 3^° E. by the same compass : fiad the ship's position. D E Compass bearings N. 463=' W. N. 3^° E. Deviation same as in a, allow to riffhl 6J- E. 6j E. Correct magnetic bearing N. 40 W. N. 10 E. Variation same as in a, allow to riffhf 29 E. 29 E. True bearings N. 11 W. = N. by W. N. 39 E. = N.E. | N. Answer. — Latitude 50° 183 N. Longitude 177° 59' E. EXAMPLE III. (See chart.) (a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Suppose variation 25° W. True course from A to B is E. by N. =. N. 84° E. Variation 25 W. allow to ri(/ht Exceeds 90 N. 109 E. Subtract from 180 Correct magnetic course S. 71 E. = E.S.E. f E. Deviation by card 1 1 E. allow to left. Compass course S. 82 E. = E. | S. Answer. — Compass course E. f S. Distance 52 miles. (i) With the ship's head on the above-named compass course, C bore by compass S.S.W. I W., distant 31 miles. Compass bearing S.S.W. f W. = S. 31° W. Deviation same as in « 1 1 E. allow to rif/ht. S. 42 W. Variation same as in a 25 W. allow to left. True "bearing of C S. 17 W. Answer. — Latitude 49° 33' N. Longitude 178^ 23' E. (c) With the ship's head as above, D bore by compass N.W. f W., also E boro N.N.E. f E. by the same compass : find the ship's position. D E Compass bearing N.W. t W. = N. 53^°W. N.N.E. | E. = N. 3i°E. Deviation same as in a, allow to riffht 1 1 E. 1 1 E. Correct magnetic bearings N. 42^ W. N. 42 E. Variation same as in a, allow to left. 25" W. 25 W. True bearings N. 67^ W. = W.N.W. N. 17 E.=:N. JE.iE. Answer. — Latitude 49° 40' N. Longitude 178° 3' E. EXAMPLE IV. (See chart). {a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Answer, — Compass course S. 68i^ E. Distance 67 miles. Variation 21^ W. 404 On the Chart. (J)) With the ship's head on the ahove-ncamed compass course, C bore hy compass E. hy N. \ N., distant 14 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 51' 29' N. Longitude 179° $$' E. (c) With the ship's head as above, D bore by compass N. 83 1° W., also E bore S. 17° W. by the same compass : find the ship's position. Answer, — Latitude 51^ 23' N. Longitude 179° 42' W. EXAMPLE V. (See chart). {a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Answer. — Compass course S. 9° E. Distance 53 miles. {b) With the ship's head on the above-named compass course C bore by compass S. 47° W., distant 1 8 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 50° 17' N. Longitude 179° 41' W. (c) With the ship's head as above, D bore by compass N. 80° W., also E bore N. 7° W. by the same compass ; find the ship's position. Answer. — Latitude 50° 35' N. Longitude 179° 19I' W. EXAMPLE VI. (See chart.) (fl) Using Deviation Card No. I. : find the course to steer by compass from A to B, also the distance. Answer. — Compass course N. 58° E. Distance 61 miles. Variation 20° E. (J) With the ship's head on the above-named compass course, C bore by compass N. 15° W, distant 19 miles: find the latitude and longitude of the ship when it was on that bearing. Answer,— liSLtituie 49° 26' N. Longitude 179° 195 W. (c) With the ship's head as above, D bore by compass N. 10° E. ; also E bore by the same compass E. by N. |^ N. : find the ship's position. Answer, — Latitude 49° 13' N. Longitude 179° 37' E. EXAMPLE VII. (See chart). (a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Answer.— Gompa,sa eourse S. 66" E. (S.E. by E. | E.) Distance 72 miles. Variation 19° E. (i) With the ship's head on the above-named compass course, C bore by compass S. 57' E, distant 21 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 52" 4' N. Longitude 177° 44' W. On th Chwt. 40; (c) With the ship's head as ahove, D tore by compass N. 71' W., also E bore S. 22' W. by the same compass : find the ship's position. Answer, — Latitude 51° 24^ N. Longitude 178" 11' "W. EXAMPLE VIII. (See chart). {a) Using Deviation Cird No. I : find the course to steer by compass from A to B, also the distance. Answer. — Compass course S. 73° W. (W. by S. ^ S.) Distance 60 miles. Variation 14^° E. (J)) With the ship's head on the above-named compass course, C bore by compass N. 83^° E., distant 17 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 50° 42' N. Longitude 178" 7' W. (c) With the ship's head as above, D bore by compass S.E. \ S., also E bore S.W. \ S. by the same compass : find the ship's position. Answer. — Latitude 50° 5' N. Longitude 177° 58' W. EXAMPLE IX. (See chart.) (a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Answer. — Compass course N. 46'' E. (N.E. ^ E.) Distance 69 miles. Variation 16° W. (J) With the ship's head on the above-named compass course, C bore by compass S. 18° W., distant 17 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 49° 27' N. Longitude 177° 19' W. (c) With the ship's head as above, D bore by compass S. 43° E., also E bore S. 15° W. by the same compass ; find the ship's position. Ansiver. — Latitude 49° 55' N. Longitude 178^^ 50' W. EXAMPLE X. (See chart). (fl) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. Answer. — Compass course N. 84^° E. (E. \ N.) Distance 95 miles. Variation 34° E. (5) With the ship's head on the above-named compass course, C bore by compass N.N.B., distant 35 miles ; find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 51° 45' N. Longitude 175" 51' W. (c) With the ship's head as above, D bore by compass S.S. W., also E bore N. 43° W. by the same compass: find the ship's position. Answer. — Latitude 51° 26^' N. Longitude 175° 30' W. 466 On the Chart. EXAMPLE XI. (See chart). (a) Using Deviation Card No. I : find the course to steer from A to B, also the distance. (See No. ii in small chart). Variation 22° W. Answer. — Compass course S. 56° W., or S.W. by "W. Distance 57 miles. (b) With the ship's head on the above-named compass course, C bore by compass S. 7 1° E., distant 18 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 50' 15' N. Longitude 175° 19' W, (c) With the ship's head as above, D bore by compass W.S.W., also E bore N. by W. | W. by the same compass : find the ship's position. Answer. — Latitude 50° 33' N. Longitude 176° 18' W. EXAMPLE XII. (See chart). (a) Using Deviation Card No. I : find the course to steer by compass from A to B, also the distance. (See No. 12 in chart). Variation 28"^ W. Answer. — Compass course N. 54° W. Distance 50 miles, (5) With the ship's head on the above-named compass course, C bore by compass S. 35° E., distant 22 miles : find the latitude and longitude of the ship when it was on that bearing. Answer. — Latitude 49° 23' N. Longitude 176" 34' W. (c) With the ship's head as above, D bore by compass N.N.E. J E., also E bore N. 64° E. by the same compass : find the ship's position. Answer. — Latitude 49° 32' N. Longitude 176° 37' W. EXERCISES ON THE OHAET. FOR ONLY MATE, FIRST MATE, AND MASTER. f2{o Deviation allowedj. North Sea. Latitude 55° 5' N. 2. Latitude 57°3o'N. Longitude o 5 E. Longitude o 40 E. Required the course and distance to Tynemouth Light. Required the course and distance to Hartlepool. 3. Latitude 6o''2i'N. Longitude o 35 E. Required the course and distance Udsire. to 5. Latitude 55° 40' N. Longitude o 15 W. Required the compass course and the distance to St. Abb's Head Light. 7. Required the true and magnetic bearing and distance between Whitby and the Naze of Norway. 4. Latitude 57° 25' N. Longitude 7 25 E. Required the course and distance to the Naze of Norway. 6. Latitude 58' 25' N. Longitude 2 10 W. Required the compass course and the distance to Duncansby Head. 8 . Required the direct true and magnetic course and distance between Buchanness in Scotland to the entrance of the Texel. 9. A ship from Kinnaird's Head, in Scotland, sailed S.E. by E. (true) 186 miles: required the latitude and longitude she is come to, and the direct course and distance she must sail in order to arrive at Heligoland. On the Chart. 407 10. A ship from Heligoland sailed on a direct course between the North and West 197 miles, and spoke a ship which had run 170 miles on a direct course from Hartlepool : required the latitude and longitude of tho place of meeting, also the course steered by each ship. 11. Sunderland Light, bearing by compass S.W. \ S. Coquet Island „ „ N.W. Required tho latitude and longitude of ship ; also the course and distance to Hartlepool Light. 12. Buchanness Light, N. by W. h W., by compass. Girdleness Light, West. Required the latitude and longitude of ship ; also the course (by compass) and distance to the Staples. 13. The Skerries, North, by compass. Sumburg Head, W. 5 S. „ Required the latitude and longitude in ; also the compass course and distance to Peterhead. 14. Flambro' Head Light, S.W. by S., by compass. Whitby Lights, N.W. by W. f W. „ Required the latitude and longitude in ; also the compass course and distance to Outer Dowsings. 15. The Dudgeon Light, W. by N., by compass. Hasbro' Sand-end Light, S.S.W. „ Required the latitude and longitude of ship ; also the compass course and distance to Flambro' Head. 16. Scarbro' Light was observed to bear S.W. by compass, then sailed E.S.E. 11 miles, and the light then bore West : required the latitude and longitude of the ship at each station, and her distance from the light. 17. Coasting along shore, observed Tynemouth Light to bear W. by S. by compass; then sailed S. by W. 16 miles, and tho light boro N.W. by N. : required the latitude and longitude of the ship, and her distance from the light. English and Bristol Channels, and South Coast of Ireland. 1. Latitude 50° I'N. Longitude 2 4 W. Required the compass course and distance to the Caskets. 3. Latitude 49" 30' N. Longitude 3 30 W. Required the compass course and distance to the Start Point. 5. Latitude 50" 30' N. Longitude o 55 E. Required the compass course and distance to Dungeness. 7. Latitude 5o°5o'N. Longitude 10 35 W. Required the compass course and distance to the Fastnet Rock. 9. Latitude 5o°5o' N. Longitude 7 20 W. Required the compass course and distance to Old Head of Kiusale. 2. Latitude 48°5o'N. Longitude 5 50 W. Required the compass course and distance to Ushant. 4. Latitude fo'io'N. Longitude i 10 W. Required the compass course and distance to St. Catherine's Light. 6. Latitude 48''55'N. Longitude 6 5 W. Required the compass course and distance to tho Lizard. 8. Latitude 5i°52'N. Longitude 6 6 W. Required the compass course and distance tg the Tuskar Light. 10. Latitude 5o°3o'N. Longitude 8 30 W. Required the compass course and diatance to Cape Clear. 4© 8 On ih-e Chart. 11. Longshjps Light, bearing by compass E.N.E. St. Agnes' Light, „ „ N.N.W. ^ W. Required the latitude and longitude in ; also the compass course and distance to the Lizard. 12. Berry Head, bearing by compass N. J E. Start Point „ „ W.byN. JN. Required the compass course and distance to Portland. 13. Bill of Portland, bearing by compass N.W. by W. St. Alban's Head, „ „ N.E. f E. Required the latitude and longitude of ship, and the compass course and the distance to Start Point. 14. Longships Light, bearing by compass S.S.E. Seven Stones Light „ „ W. by S. Required the latitude and longitude of ship ; also the compass course and the distance to Roches Point. 15. Tuskar Rock N.E. by compass. Great Saltees Lightvessel N.W. ^ W. „ Required the latitude and longitude of the ship ; also the course (by compass) and distance to the Smalls. 16. Shipwash Light, bearing by compass W. by N. Galloper „ „ „ S.S.W. Required the latitude and longitude in ; also the compass course and distance to Gorton Lightvessel. 17. Caldy Island Light, bearing by compass E.N.E. Lundy Island Light, ,, „ S. by E. Required the latitude and longitude of ship ; also the compass course and distance to the Smalls. 18. Lizard Lights, bearing by compass E. J S. Longships, „ „ N. J W. Required the latitude and longitude of ship ; also the compass course and distance to St. Agnes' Light. 19. Smalls Light, bearing by compass N. | E. St. Ann's (Milford Haven) „ „ E.S.E. Required the latitude and longitude of ship ; also the compass course and distance to Seal Rock (Lundy Island). 20. Dungeness, bearing by compass N.E. by E. ^ E. BeachyHead „ „ N.W.JW." Required the latitude and longitude of the ship ; and her distance from each place. 21. A ship is bound to Boulogne, being 18 miles distant, and lying directly to wind- ward, the wind being E. by N. (true). It is intended to reach her port on two boards, the first being on the port tack, and the ship can lie within six points of the wind ; required the course and distance upon each tack. 409 ANSWERS. NOTATION, Pages i6 and i-j. I. 100 2. lOI 3. 110 4. 9009 5. 9090 8. 10700 9. 90090 10. 305000 II. 900900 i: 14. 5030049 15. 9900006 16. 58000009 17. 19. 604060005 20. 800003033 21. 900900900 22. 24. 500000000 25. 580245192 26. 707007077 NUMEEATION, Page i3. I. Forty-three. 2. Sixty. 3. Twelve. 4. Twenty-one. hundred and one. 7. One hundred and ten, 8. Five hundred 6. 9909 7. 5074 ". 505550 13. 1003008 70302441 18. 222000035 700000007. 23. 180000000 5. One hundred. 6. One 9. Five hundred and five. 10. Five hundred and fifty. 11. One thousand. 12. Two thousand and twenty. 13. Three thousand three hundred and three. 14. Four thousand and four. 15. Seven thousand seven hundred and seven. 16. Eight thousand eight hundred and eighty. 17. Eighty-seven thousand and fifty-four. 18. Seventy thousand seven hundred and seven. 19. Sixty thousand eight hundred and eighty. 20. Ninety-nine thousand four hundred and four. 21. Nine hundred and three thousand seven hundred and fifty-six. 22, Two hundred and two thousand two hundred and two. 23. Four hundred thousand and four hundred. 24. Five hundred and fifty thousand five hundred and fifty. 25. One million and one. 26. Eight million and forty-seven thousand three hundred and twenty-eight. 27. Four million and ninety thousand and three hundred. 28. Five million two hundred aul ten thousand and seven. 29. Six million and thirty thousand four hundred and five. 30. Nino million nine thousand and nine hundred. 31. Forty-one million forty-one thousand and fourteen. 32. Three million and six. 33. Twenty million eighty-four thousand two hun- dred and sixteen. 34. Five million one thousand eight hundred and sixty. 35. Eight million eighty thousand eight hundred and eight. 36. Fifty-five million seven hundred thousand and five. 37. Seventy-six million and fourteen thousand and fifty-nine. 38. Six million six thousand six hundred and six. 39. Fifty-six million seven hundred thousand five hundred and five. 40. One hundred and twenty million fifteen thousand and fifteen. 41. Two hundred and two million two hundred and two thousand and two hundred. 42. One hundred million one Imndred thousand one hundred and one. 43. Two hundred and seventy-five million eight thousand and five. 44. One hundred million ten thousand and one. 45. Seventy-nine million thirty thousand one hundred and eighty-four. 46. Four hundred and eight million seventj'-six thousand and thirty-two. 47 . Four hundred and one million four hundred thousand and fifty-six. 48. Nine hundred and eight million five hundred thousand and sixty. ADDITION, Page 20. 25 26 I. 1274170 7. 1648127 13. 3312667 19. 8518439 2. 1634607 8. 2067690 14. 3018498 20. 7498159 3. 1659291 9- 33^9175 15- 2797285 2 1. 9560155 4- 2333431 10. 3724599 16. 3519772 22. 5621434 5- 3005313 ir. 4483647 17- 9185198 23. 6524956 6. 1536206 12. 4105670 18. 7485613 24. 8238336 • (0 13788543 (2) 12844819 (3) 14661377 (4) 13937 260 (5) 15878135 (6) 10176 138 (7) 10970368 (8) 13825798 J. 20566726566 SUBTEAOTION, Page 23. I. 621511 9. 681179 17. 2922930923 25. 4244103 a. 539540 10. 507S71 18. 908891C990901 26. 5460813 3- I II. 376099 19. 10020950993 27. 8026758 4- 9 12. 174386 20. 9I089CO9099 28. 98599383 5- 676001 13. 107500 21. 238036793034 29. 983982 6. 554999 14. 222419 22. 5540058 30. 9985268 7- 480895 15. 157406 23. 5866974 31- 743187 8. 590098 16. 8409091 24. 6521913 32. 8457 33. 71880 GQG 4IO Answers. MULTIPLICATION, Pa(/e 27. I. 685295792 5. 550942443156 9. 9876543210 2. 1962965961 6. 45652143474 10. 9803614194 3. 1506172792 7. 3886950304 II. 7774239492 4. 1899328910 8. 5159176101 12. 11031283848 1 }. (2) 476949824 (3) 5432756589 (4) 3052 4788736 (5) 116442828125 (6) 347696421696 (7) 876760466309 (8) 1953586479104 (9) 3960479553381 (11) 13202276674301 (12) 22252570988544 Faffe 30. I. 2684444024 10. 199999929143681 19. 1614054474492415542 2. 5629618680 ir. 2243503727343888 20. 53107710897987 3- 8918232255 12. 355733452311336 21. 40155302248305278754132 4- 61286228934 13. 395130761574453 22. 15232906283422580 5- 15993780666 14. 668094460288461 23. 5060344127 169 150 6. 999999999 15, 8312372968202684 24. 296229611814587191480656 7- 27349835014665 16. 460937797776 25. I 020302907 866668S093030201 8. 32228449759163 17. 1219326311126352690 26. 999400 14998000 149994000 I 9- 770930181732 18. 2872556494008787 27. 9999999500000004000 28. 999400I4998000I49994000I Faffe 31. I. 7239334500 4- 290675534724420 7. 2345457600 2. 10814314500 5. 229262443200000 8, 2315085840 3- 24538039680000 6. 581643960 9- 35333670133890810 10. 12003400820050006000000 I. 10285980 4. 24335360 7. 13085320545 2. 12838608 5. 20146968 8. 1557262880 3. 38114062 6. 2466490572 9. 18532696320 DIVISION, Faffe 35. 1. 67896347-1 5. 66779748-5 9. 25409614-6 2. 194899128-2 6. 39512348-r 10, 100107478-9 3. 99836471 7. 868427625-6 II. 91261430-10 4. 59648952 8. 274473675 12. 4953087942-8 . [2) 96168 (3) 8442-2 (4) 1502-2 (5) 393- ■4 (6) 131-5 (7) 52-2 (8) 23-3 (9) 11-5 (11) 3-5 (12) 2 , (2) 109375 (3) 9602 (4) 1708-3 (5) 448 (6) 150 (7) 59-3 (8) 26-5 (9) 13-1 (") 3-10 (12) 2-4 . [2) 63856 (3) 5606 (4) 997-3 (5) 261-2 (6) 87-3 (7) 34-5 (8) 15-4 (9) 7-6 (11) 2-3 (12) 1-4 . [2) 137828 (3) 12100 (4) 2153-2 (5) 564-2 (6) 189 (7) 74-6 (8) 33-5 (9) ^^-5 (") 4-10 (I2) 2-11 . (2) 31261 (3) 2744-1 (4) 48S-r (5) 128 ( 6) 42-5 (7) 17 (8) 7-5 (9) 3-6 (11) i-i (12) 0-8 . (2) 146406 (3) 12853 (4) 2287-2 (5) 599 -3 (6) 200-4 (7) 79-4 (8) 35-5 (9) 17-5 (11) 5-3 (12) 3-1 (2) 154907 (3) 13599-1 (4) 2420-1 (5) 634-2 (6) 212-2 (7) 84-r (8) 37-6 (9) 18-5 (11) 5-6 (12) 3-3 (2) 73531 (3) 6455-1 (4) 1148-3 (5) 30X (6) 100-5 (7) 40 (8) 17-7 (9) 8-7 (II) 2-7 (12) 1-6 ' Faffo 37. I. 75638-2 10. 8607936214-143 19. 1068392-1 17002 2. 34785-7 n- 576845 20. 544023-195858 3. 542370 12. 245728 21. 70030401 4. 220950-14 13. 567977-3852 22. 730956788-980154321 5. 127474-28 14. 107356-2031 23- 63245553 6. 100045-25 15. loooio-io 24. 9007609600 7. 461002-18 16. 999000-1000 25. 304529878174100474 8. 725983 17. 9000090000-10000 26. 4342944819-49016833 9. 706321-515 18. 8485852-43614 Answers. 411 Pu(/e 39. 1. 463519673763533-5 2. 27201490438560034-10 3. 1582874324701-32 4. 95022741046776-8 5. 14964459409277-63 6. 4489339863279-30 7. 27206980239559-123 8. 34045491087 17 2-1 1. 1S7157296759729-46 2. 962869563912-25 3. 329218107-670 4. 40316322081 10056-69 5. 1039682584-834 6. 3396-5094687 MISCELLANEOUS EXAMPLES, Faye 40. I. 4- 7. 10. 10004; 4004; 44004 2. 474788 3. 280S846363 7398981889800 5. 957610S 6. looiooioo 87846125 8. 99912350214 9. 1000622528890200 2768884-85187 II. 103080207 12. 1202609 71625861494 14. 128721301414200 15. 607862510254-15696883 16. Tho one is larger than the other by forty-nine thousand nine hundred and fifty, i.e., by 49950. 17. 60768396; of 129847 and 40068. 18. 847021,36865365. 19. 6 and 3. 20. 324937594. 31. 300490090, sum ; 275798734, difTurence; 355733812S051336, product. 5555656, sum ; 3086522, difference ; 53346738S3465, product. 22. 372 tons. 23. 127 years. 24. 7852 times. 25. 34 ships. 26. 141. 27. 1002. 28. 65280. 29. 129115. 30. 146, after subtracting it 390 times. 31. 203. 32. 1666350, sum; 1639900, dififerenco; 21862578125, product; 125, quotient. 33. 9843750, sum; 9687500, difference; 762939453125 product; 125, quotient. NOTATION OF DECIMALS, Fu(/e 44. I- •3. ■o3> '003, and 3-3; also -7, 11*7, -33, and 1-015. 2. 'oi, '0021, "0117, '0000003, -I, -53, '007, "ooii, and •00137. 3. 30'i, 40o'oi, 53"oo4i5, 50'oooioi, "441, 33*1, and •000000000501. 4. "9178, 91*78, "09178, '0091, •00009, 520"3, and '90. 5. 3'oi42, 6'728i9, -000672819, and 6728^i9. 6. 7-06, 43-2143, 9-07823457, roooooi, and 35^721341. 7. -073, -0197, •oooooi, -00261, and -oooiooi. 8. i'54, 24-079, 315-008005, -ooooooii, and -00903. 9. "I, '03, -005, -105, -000002, -000060,41-08, looo-ooi, 30-000006, -ooooijand -00002375. 10. Two hundred and eighty-three thousandths; Five thousand three hundred and twenty-one ten thousandths ; Seventy-four thousand eight hundred and ninety- five hundred thousandths ; Eight hundred and twenty-one thousand and fifty-six millionths ; Twenty-seven, and eight thousand three hundred and fifty-four ten thousandths ; Thirty-four, and nine ten thousandths ; Forty-three, and one one hundred and one thousand and seven millionths; Twenty-tbr^e, and seventy- five hundredths ; Two, and throe hundred and seventy-five thousandths ; Two thousand three hundred and seventy-five ten thousandths ; Two thousand three hundred and seventy-five hundred millionths. ir. Six tenths; Ninety-two hundredths; Five thousand four hundred and ninety-eight ten thousandths; Seven, and seven hundredths; Twenty-six, and four hundred and five thousandths ; One millionth ; Thirty-seven hundred thousandths ; Eleven, and one hundred and one thousand one hundred and one millionth ; Four hundred and forty thousand three hundred and eight ten millionths; Eighty-two thousand three hundred and forty-four hundred thousandths ; Thirteen thousand two hundred and thirty-six hundred thousandths. 12. Nine, and four hundred and fifty-sevcu ten thousandths; Four thousand and four, and three hundred and forty-five ten millionths ; Three, and four hundred thousandths ; Five hundred and twenty-four millions six hundred and thirty-four, and eight thousand and thirty-four ten millionths ; Three thousand seven hundred and five thousand millionths ; Twenty-four thousand and fifty-six thousand millionths ; Seven thousand and five, and six hundred and seventy-four thousand millionths ; One hundred thousand, and one ten millionth ; Ten, and one thousandth ; Nine, and twenty-eight millionths ; One, and six thousand and three ton millionths. 4tt Answers. 13. One, and one millionth ; One million and one ten milliontli ; One hundred milliontlis ; One, and thirteen thousand and four hundred thousandths ; Nine, and two hundred and three thousand one hundred and sixtj^-seven millionths; Four, and three million eight thousand and four ten millionths ; Twenty-seven, and four million six hundred and twenty-seven thousand three hundred and fifty ten millionths. ADDITION OF DECIMALS, Page 45. 1. 745-0261 ; 2-919563 5. 538-6422021 2. 886-9326; 1681-679 6. 140-1996; 1408-25559 3. 1437-4179; 330'87552i 7- 5y^1^9' i27*o5034o 4. 4009-0; 501-15998 8. i-iiii ; 42-7162 9- i"2345; 945"5993 SUBTEACTION OF DECIMALS, Page 47. 1. 3'43i; 8-2000X ; -ooii; 8-000001 2. 39-8479194; 31-99968; 7-336606; 91-7423 3. -oi; 98-99999901; 9-999999; 995-710; 541-787 4. 64-0317753; 8-20001 ; 72-5193401 5. -000099; -000396; 31-99968; 24680-12377 6. 699-930 ; -0000999 7. '0378; -062156 8. "00510; 28999-908 MULTIPLICATION OF DECIMALS, Fa(/e 48. 1. lO-o; lo-o; 1190-0; 11-9; -0119; -00119 2. -000000202; 3-06034; -000000112; -00210175 3. -075460; 1-8019; 74-9265; -00104886933696 4. -0306002448; 470116914-4360 ; 536-660075952 5. 2-5067823; -000011826009; -00000006676542672 6. 47-83; 500-0; 75000-0 -001301400; 1-5; -00000072 5-314410; 4-096; '032016 •0001234321 ; '000444080 ; 6138-36 DIVISION OF DECIMALS, Fa^e 51. 1. 19-82421; 14-16015; 11-01345; 2-7533625; i'i2637557 2. -2017386; -1008693; -0672462; -025217325; -0009456496 3. 134-88057; 790-9882353; 59-406396; 24-82661 4. 14-789983; 255-121; 1210-234426; -02 5. -8810891; 908-83768; -1108754; -0174532922 6. 7ir855; 2280-28; 234508 7. '03; 74*84; 43206-7; -000007375; 83671000; -000000000003; "061096. 8. 2681-081081; -0000360074; -0001; 6-578947; -00862. 9. 15000-0; 5060000-0; -008; -0375313. 10. -013; 457; "008; 73939"39- ir. "050005; 1250-0; -0x25; 60-2589. 12. "0000000125; 125-0; 125-0; -00004. 13. "00000125; "ooooi ; 20200; 77485-93. 14. "10; lo-o; -001; "001 ; "looo; loooo ; -ooooi. 15. "0093536; 7393*939; 39723-66; 241-6292; 200-0 ; -60000000; 4000000; "000006; 32000000. 16. -0036; -93; "52306; -0008; -0000020076364. 17. -0882352941; "017256637168 ; -0000999000999000999000999. REDUCTION OF DECIMALS, Page 53. I- •437s; '73; -^H^m; -34375; '1875; -676923; -0112; -275. 2. -5384615; -6470588; -6315789; -iSj; -7167235; -3183098; -4683544; -0104895. Ansioers. 413 Page 55. •2833; -4833; -7; -4166; -8; -9666 •788260449735; -05 3- 12*5925 days; 29-71649305 days; 15-7422916 days; 119-2217013^ days 4. 8°-i875; i9°-679i6; 104^-26875 ; 82°-325 •O46875ton8; -61964285714^03; '7i25ton3; •5i875tons; 13-303125 tons ; i-65toiis; 17-90156251008; -8669642 tons 6. •316; 435416; '23125; '796875; "9927083; 1-697916; and -476196 EXAMPLES FOR PEACTICE, Page 56. 1. I qr. ; 10 lbs. 8 oz. ; 10 oz. ; 19 cwt. 2 qrs. ; 15 cwt. 2 qrs. 4 lbs. o oz. 8-192 drs. ; 4 cwt. 3 qrs. 27 lbs. o oz. 3-6864 drs. ; 7 lbs. 8 oz. 15-36 drs. 2. ii>'47™2*'; 2-^ 9'' 37"! 26f»; i95>i i'' 2™ 385'; iz^ 44™ 2^-868 3. 17' 44"-8o62 ; 7° 51' i5"-3 ; 64° 22' 57"; 10" 52' 21" 4. 17 cub. ft. 2951 1-59847552 cub. in. 5. 106-434785 gallons MISCELLANEOUS EXAMPLES, Pages 56-57. I. -00066287 2. ^ 3. 29'i-53059027 4. 12-175 hours; '0013 5. I N.M. = 1-15202 I.M. ; I I.M. = -86804N.M. ; 1-15:515, and -868421 6. -997269560 day 7. 80-655 degrees, or 80^ 39' 30'; 89-615 grades 8. 1 kilomd'trc = -621382 mile; i mile = 1-609315 kilometre 9- 3 ft- 3'42 in. 10. 34-002; 83100-000831 11. -000038 12. -0434027 eeconds 14. 574-425 ^5- 43'°39783 feet; 6-1575164 yards; 89-03894378 miles 16. 1-240346 17- 7ft-5|i°- 18. 277-2738 cub. in. 19. 3-883 in. 20. 62-321 lbs. ; 35-943 cub. ft. 21. 22-7205 tons; 8639-6805 tons 22. 500 sovereigns 23. -002021; 20210; -1902; 1902000000; 1902000 24. 7970 miles 25. -00000033 CHAEAOTERISTICS OE LOGARITHMS, Page 61. 1. 1 6. 4 "• 5 16. 2 2. 7. 2 12. 3 17. 7 3. 2 8. 3 13. 18. 4 4. I 9. 2 14. 19. 5- 10. 15- I 20. I Page 62. I. 2 or 8 6. 7 or 3 II. T or 9 16. 2 or 8 2. I or 9 7. 2 or 8 12. 3 or 7 17. 7 or 3 3, 4 or 6 8. 4 or 6 13. 7 or 3 18. 7 or 3 4. I or 9 9. 2 or 8 14- 5 or 5 19. T or 9 5- 3 or 7 10. T or 9 15. 4 or 6 20. iT or 9 LOGARITHMS OF NATURAL NUMBERS, Page 65. I. 0-698970 2. 0-954243 3. 3'954243 or 7 "954243 4. 2'O0000O or 8-O0000O 5. 4-000000 6. 1-146128 7. 1-612784 ■ 8. 3-602060 or 6-000000 or 7-602060 9. 0-380211 10. 1-380211 II. 1-380211 or 9-380211 12. 3-322219 or 7-322219 13. 1-973128 14. 7-698970 15. 7-875061 16. 0-397940 or 9-698970 or 9-875061 17- T-397940 18. 2-954243 19- 3"95904i 20. 1-397940 or 9-397940 or 8-954243 or 7-959041 21. 2-380211 22. I'544o68 23- 5-755875 24. 4-698970 or 8-380211 or 5-544068 or 5-755875 or 6-698970 I. 2-000000 2. 2-161368 3. 0-468347 4. 2-557507 5. 2-828015 6. 2-899820 7- 2-992111 8. 0-681241 9. 0952 308 10. 0-167317 II. 1-167317 12. i'954725 13- 21673 17 14. 2-627 366 15. 5-651278 or 9-167317 or 9-954725 or 8-1673 •7 or 5-651278 16. 5651278 4H Page 66. I. 3-000000 2. 3-091315 3. 1-409087 4. 3-734960 5. 1-415974 6. 0-415974 7. 2-005180 or 8-005180 8. 1-977129 9. 0-890812 10. 2-994581 II. 5-835247 or 8-835247 12. 3-444669 or 7-444669 Page 67. I. 4-585178 7. 2-639088 13- 5'30io3o 19- 5-562474 2. 2-585178 8. 1-895445 14. 2-749845 or 8-749845 20. 2-998755 3. 4-O9149I 9- o'343S07 15- 3-993714 or 7-993714 21. 1-507732 4. 2-734968 10. r-894105 or 9-894105 16. 5-808742 22. 0-014001 5. 4'8239°4 II. 4-000000 17. 3-052717 23. 3-000003 6. 3-965898 12. 4-903120 or 6-903120 18. 1-999172 or 9-999172 24- 2-775555 NATUEAL NUMBEE OF LOaAEITHMS, Pages 70—72. I- 3 9- 345'6 17 34800 25. -0000009797 2. 9-4 10. 24-83 18 •5547 26. 80080000 3- H'5 II. 7000 19 •3171 27. •04183 4. 6-49 12. lOOOOOOO 20 . -00000075 28. •000000007968 5- 586 13. 669000 21 . 4000000 29. -0046 6. 248 14. 50000 22 •00000007 3°' •00071 7. 30-09 15. lOOOOO 23 . 4029 31. 8199000 8. 7916 16. 978-5 24 2784 32. 738800 I. 853-52167 8. 543210 15. 678945-3 22. -000290888 2. 4220-3 9. 666660 16. 260418 23- -0174533 3- 71105-9 10. 98765 17. 69500-645 24. 2349632-4 4. 23000-1 II. 84321 18. 12375-426 25. -0000017645 5- 53'i33 12. 123456 19. 1-7 26. -99727 6. 93-8689 13- 342-945 20. 1651374 27- '7854 7. 456780 14- 5555'54 21. -0096532 28. -000856735 29. -000036808 LOaAElTHMS OF NATUEAL NUMBEES, Page 72. 1 . 0-903090 II 1-802774 21. 2-926548 31- 1-972043 41. 7-000000 2 . I -000000 12 3-805501 22. T-964240 32. 4-722552 42. 2-792392 3 . 0-690196 13 1-165244 23- 2-953760 33- 4-698970 43- 4-477134 4 • i'579784 14 0-588160 24. 4-000000 34- 5-845154 44. 4-000039 5 • 2-579784 15 3-829561 25- 4-681241 35. 5-421604 45- 5-774152 6 . 2-000000 16 2-942504 26. 3-958124 36. 5-606388 46. 7-947385 7 . 6-000000 17 ^-539954 27. 4-763428 37- 5-699759 47- 2-458852 8 • ^"390935 18 0-034628 28. 2-554755 38. 1-686877 48. 3-551938 9 . 0-588832 19 I -0969 10 29. 4-651278 39- 1-970876 49. 4-932847 10 • 2-954243 20 3-954243 30- 7-651278 40. 2-515397 50. 7-816109 NA TUE AL NUMBEES OF LOGAEITHMS, Page 73. J. 204 10. y^ii 19. •09 28 404007 37- •763888 2. 4753 II. 6-004 20. -0091 29 I 00000 38. 4220-3 3- 9 12. 588-172 21. 50800 30 -0762 39- 53-1329 4- 50 13- 594500 22. 2-606 31 •147 40. -042404 5- I 14. 264000 23- -I 32 •00000075 41. •0048553 6. 100 15- 1000 24. •009 33 1-00043 42. 2-5152 7- 366-855 16. 2480000 25- •052 34 8859000 43- 100591 t. 3659 17- 26-042 26. 451070 35 0918504 44. - 000209675 9- 418-557 18. 15-438 27. 2 -71828 36 5-80693 45- 7-5 Anstcers. 415 MULTIPLICATION BY LOGAEITHMS, Page 76. 1. 3'7745n =5950; 3"022429= 1053; 3-000000=1000; 3-521269 = 3321. 2. 2-971331 = 936-12 ; 4'034i47 = 'o^iS ; 3-49485° = 3125 ; 4"443232 = 27748. 3. 1-009026 = 10-21; 0-436878 = 2-7345; 1-818753 = -6588; 1-575742 = 37-648. 4- 5"425758 = 266537; 4-532375 = 34070-2; 2-639870 = 436-385; 1-292881 = 19-62826; 6783260 = 6071000. 5. 4-586678 = 38608; 4-677607 = 47600; 2-680225 = 478-878; 5-237543=172800; 5-786113 = 611101. 6. 3-971387 = 9362-39; 2-993736 rrgSi^f^S; 4-659678 = 45675; 5-749272 = 561400; 3-723999 = 5296-6. 7. 7-146212 = 14002718; 6-445142 = 2787032; 6-919110 = 8300615; 7-498480 = 31512319- 8. 3-100249 = 1259-64; 2-511391 = 324-632; 5-000000 = looooo; 8-696466 =r -0497125. 9. 7-499467 = 31583985-5; 3-782115 = 6055; 3-842614 = 6960-08; 4-327379 = 21250-98. lo- 5*59931 1 = 397476-1 ; 3-590806 = 3897-68; 5-477728 = 300419-31; 7-623683 = 42041942. 11. 6-314887 = 2064842-8; 4-808914 = 64404-2; 2-552762 = 357-077; 3-983651 = 9630-555- 12. 3-394677 = 2481-28; 4-312842 = 20551-4; 0-123363=1-32850; 3-519872 = 3310-34. 13. 8-763323 = -000000057986; 3-778168 = -006000236; 7-740796 = -00000055055; 1-233799 = •171317- 14' 5-7554'8 = -00005694; 0-622110 = 4-189; 3-473368 = -00297418; 4-147399 ^ 14041-03. 15. 2-951043 = 893-394; 3-524617 = 3346-7; 3-398070 = -00250075; 4-783612 = -000607593. j6. 2-000000 = -or ; i-oooooo = -00000000 1 ; 5-050035 = -0000112211; 5-000000=: lOOOOO. DIVISION BY LOGAEITHMS, Page 79. 1. 1-919078 = 83 ; 2-778875 = 601 ; 1-924279 = 84; 2-096910 = 125. 2. 2-986680 = 969-8; 3-698970 = 5000; 5-880170 = 758875-4; 2-775257 = 596-015. 3. 1-809855 = -64544; 3-644712=4412-78; 3-477122 = 3000; 1-822584=66-4637. 4. 0-494768 := 3-1244; 1-614317 = 41-145; 1-428589 =: 26-828 ; 0-139814=1-3798. 5. 0-472427 = 2-967762; 2-785908 = -06108; 2-3856S3 ="0243043; 2-571411 =-037274; 5-301030 =; -00002 ; 4-301030=1-000002; 1-301030 = 20. 6. 3-886534 = 7700-764-; 1-999234= -998236; 1-999489 = 99-8825; 3-763429=5800. 7. 1-569001=37-0684; 1-425115 = -266143; 4-376859 = 23815-5; i'378403 = 23-9003. 8. 1-995636 = 99; 1-319142 = 20-85x7; 1-854294=71-498; 0-793259 = 6-2124. 9. 4-699490 = 5006; 4-514747 = 32715; 4-317415 = 20769; 2-096910=125. 10. 0*942505 ^ 8-76; 2-525020 = 334-98; 5-291146=195500; 1-004364=10-1010. 11. 1-832752 = -68038; 4-903504= -C00800763; 1-030734 = -107333; 2-509203 = 323; 5-778151 = 600000. 12. 1-940506 = 87-1978; 2-297735 = -oi98488; r-278331 = -189815 ; 0-833525 = 6-81592. 13. 1-421422=: -265388; 1-421422 =: 26-5388 ; 1-421422 =-265388 ; 3-421422=12653-88; 7-301030= 20000000; 7-477122 = 30000000. 14. T-057101 = -11405; 2-537395 = 344-6631 5-003528 = -0000100816; 0-833525 =2 6-816. '5- 2-339894 = 218-723; 2-934196 = 859-4; 1-004751 = -1011 ; 7-505150=32000000; 5-028878 = -0000106875. 16. o- ::= 1 ; 13" = looooooooooooo ; T- = -1 ; 2- = -oi ; 4- = loooo. 4i6 Ansmrs. NATUEAL SINES AND COSINES, Page 83. Natural Sines. I- 570774 3. 947463 5. 723895 7. 800000 2. 867085 4. 370382 6. 974228 8. 997630 Natural Cosines, Page 83. I. 969227 3. 167237 5. 688000 7. 78 2397 2. 328958 4. 995612 6. 868805 8. 989472 Arcs of Natural Sines, Page 84. I. 63*53' 47" 3- 53° 7'49" 5- 26''2i'34' 7. 47=48' 33" 9- 48=46' 34* 2. 21 44 21 4. 66 59 10 6. II 45 52 8. 31 57 10 Arc of Natural Cosines, Page 84. 10. 73 44 23 I. 63° 19' 58 4. 45° 24' 39" 7. 39° 42' 4" 10. 77° 33' 15" 2. 18 29 12 5. 59 47 8. 12 53 4 II. 2 33 48 3- 43 21 52 6. 23 54 9 9- 35 8 32 12. 53 7 48 LOG. SINES, TANGENTS, SECANTS, ETC., Page 87 . I. 9'20223Z 4- 9"883934 7- 9'774729 ^°- 9' 275658 2. io'i8598i 5. 10-829843 8. 9-895443 II. 10- 144904 3. 9-989071 6. 10-135990 9. 10-308556 12. 9-907590 Pages 92 — 93. NO SINE. TANGENT. SECANT. COSINE. COTANGENT. COSECANT. I 9-079607 9-082763 10-003156 9-996844 10-917237 10-920393 2. 9-611999 9-651805 10-039806 9-960194 10-348195 10-388001 3- 9'787595 9-890004 10-102409 9-897591 IOIO9996 10-212405 4 9-923122 10-185903 10-262781 9-737219 9*814097 10-076878 5 9-246845 9-253720 10-006876 9-993124 10-746280 10-753155 6 9"975i3o 10-457990 10-482859 9-517141 9-542010 10-024870 7 8-504189 8-504410 0-000221 9-999779 11-495590 I-4958II 8 9-999580 11-356298 i'3567i9 8-643281 8-643702 0-000421 9 8-246654 8-246721 o-oooo68 9-999932 11-753279 1-753346 10 9-955206 10-319983 0-364777 9-635223 9-680017 0-044794 II 9-938922 0-244180 0-305259 9"69474i 9"75582o 0-061078 12 9-990926 10-684913 0-693987 9-306013 9-315087 0-009074 n 8-668140 17. 8-361681 21. IO-348I95 25. 8-024643 29. 10-714159 14 9-217118 18. 9-505271 22. 10-100598 26. 8-658227 3°- 9-972464 15 8-504188 19. 9-297036 23. 10-203779 27. 8-305785 i6 6-297326 20. 11-263695 24. 8-546002 28. 8-258261 AECS OF LOG. SINES, Page 95. I. 33' 26' 48" 4. 2° 17' 7" 7. i8''26' 6" 10. 0° 9*50" n- 5 2° 35' 30" 2. 57 30 53 5. 19 15 35 8. 39 7 15 II. 87 38 20 14. 4 I 28 3- 2 55 26 6. 58 15 30 9. 54 13 20 12. 70 34 18 AECS OF LOG. COSINES, Page 95. 15- I 39 39 I. 52° 13' 35" 4. 8' 6' 31" 7. 70° 47' 25" 10. 31° 9' 33" 13- 89° 38' 20" 2. 55 45 8 5. 81 18 8. 80 37 20 11. 3 56 40 14. 84 15 39 3- 89 13 8 6. 79 II 16 9. 88 40 54 12. 88 54 16 AECS OF LOG. SECANTS, Page 95. 15- 84 4 38 I. 14° 23' 15" 3. 3°24' 0" 5. 79°39'5i" 7- 18=22' 17" 9- 61° 4' 15' 2. 51 28 50 4. 26 33 6. 84 19 47 8. 88 41 42 AECS OF LOG. COSECANTS, Page 95. 10, 86 V 57 I. 26' 43' 0' 4. 6° 5' 13" 7. 5° 43' 39" lo- 58° 15' 30" 13- 78=22' 32* 2. 34 I 14 5. 49 119 8. I 57 4 11. 7 13 56 14. 60 13 52 3- 5 40 16 6. 247 9- 3 54 45 "• 4 4^ 5^ 15- 2 56 20 Ansioers. 417 AECS OF LOG. TANGENT, Page 95. 1. 77' o'23" 4. 81° 31' 58" 7. 86'58'i6' 10. 23° 43' 17" n- 48=58' 24" 3. 45 24 5- 54 43 2<5 8. I 8 7 ii. 35 3 32 14. 2 40 10 3- 62 42 21 6. 5 13 23 9. 27 28 54 12. 87 46 AECS OF LOG. COTANGENTS, Page 95. 15- I 2 18 I. 61' 2' 39" 4- 41° i'35" 7- 88'46'54" 10. 82° 49' 23" 13- 88" 20' 53* 2. 7 34 16 5. II 41 8. 44 20 2 II. 8 30 34 14. 76 40 15 3- *7 16 43 6. 3 37 50 9. 86 32 24 12. 88 55 35 15- 29 16 MISCELLANEOUS EXAMPLES. For practice in natural and logarithmic Sines, Tangents, and Secants, Page 95, 1. Nat. sine •432651, its common log. is 9"636i38, which is the log. sine required. 2. Nat. tang. 3, its common log. is io"477i2i, which is the log. tang, required. 3. Loj. 9 '2 367 1 3, its corresponding nat. no. is •172470, the nat. cos. required. 4. The given log. tang. 9"850593, being subtracted from 20, gives io^i49407, the log. cotang. 5. The nat. sine of 68^* 45' 24" is •932050, the log. of which, or 9-969439, is its log. sine, which, being subtracted from 20, leaves, 10030561 for the log. cosec. 6. The log. sec. 11^024680 subtracted from 20, leaves 8*975320, the log. cosine, the nat. no. corresponding to which, or ^094476, is the nat. cosine sought. 7. I. The quantity 9-450981 is found in the tables to be the log. cosine of the arc 73° 35' 31 "• 2. The nat. no. corresponding to the given log. is ^282476, which is the nat. cos. of 73° 35' 31", the arc A sought. 8. I. The square of radius, or i, divided by the nat. sec. 2-005263, gives •498688, the nat. cosine of A, which is found in the tables of nat. cosines to correspond to 60" 5' 12', the value A. 2. The common log. of the nat. sec. 2^005263 is 0-302171, which is found to be the log. sec. of 60° 5' 12*", the arc A sought. DIFFEEENCE OF LATITUDE, Page 107. 1. 203' N. 3. 293' S. 5. 795' N. 7. 610' S. 9. 94' N. 2. 470 S. 4. 330 N. 6. 157 S. 8. 459 N. MEEIDIONAL DIFFEEENCE OP LATITUDE, Page 107. I. 97 2. 2426 3. 345 4. 1216 5. 932 6. 260 LATITUDE IN, Page 108. 1. 34" 2' N. 3. o^ 8'N. 5. 2°48'S. 7. o°2o'S. 9. Equator. 2. 27 54 N. 4. 3 I N. 6. 2 54 S. 8. Equator. 10. 39" 14'S. MIDDLE LATITUDE, Page 109. I. 17° 19' 2. 2' loV 3. 35° 37' 4- 6r3i^' 5. 53° 12^' 6. 64' 31' DIFFEEENCE OF LONGITUDE, Page no. 1. 300'E. 3. 716'W. 5. 270'E. 7. 368'E. 9. i8o'"W. II. 2835'E. 2. 507 E. 4. 260 W. 6. 422 W. 8. 420 "W. 10. 412 E. 12. 1200 W. LONGITUDE IN, Page 1 1 1. I. 7°38'W. 2. i"i8'E. 3. 31° 4'E. 4. o°3o'W. 5- I 15 E. 6. o 45 W. 7. 39 10 W. 8. 92 9 E. 9. 103 56 E. 10. 178 26 W. II. 178 57 E. 12. 179 59 E. Hlin 4i8 LEEWAY— COEEECTED COUESES, Page 121. I. S.W. |S. 2. s.s.w. iw. 3. K IE. 4. N.E. JE. 5- E. bytS. 6. N.W. byW. 7. W.fS. 8. N.E. byE. |E. DEVIATION- TEUE COUESES Page 155- I. N. 38-59' E. 8. S. 48°55' E. i^ S. 28°57' W. 22. S. 78° 10' W. 2. N. 22 W. 9. N. 19 25iE. 16. 8. 4 38iE. 23. S. 56 2iW. 3- N. 20 58! W. 10. S. 16 W. 17. S. 73 io|W. 24. S. 4 34|W. 4- S. 43 39|W. II. S. 84 7iW. 18. 2Sr. 32 24iE. 25. S. 63 ii|W. 5- S. 84 7iW. 12. S. 36 i2|E. 19. N. 3 o'E. 26. S. 74 3JE. 6. S. 24 5IW. 13. S. 87 £. 20. N. 89 55 W, 27. 8. 79 16 E. 7- S, 89 35 E. 14. K 59 i2iW. 21. S. 9 33 E. 28. N. 77 46 W. MAGNETIC BEAEINOS OF OBJECTS, Page 158. I. N. 78-10' E. 5. N. 84°54'E. 9. S. 86° 2'W. 2. S. 72 32 E. 6. S. II 53 W. 10. S. 2 39 E. 3. N. 5 14 E. 7. N. 8 37 E. II. N. 72 33 W. 4. S. 10 16 E. 8. S. 89 13 W. 12. N. 3 6 W. L] 3EWAY, VAEIATION, and DEVIATION— TEUE COUESES, Page 161. I. E. byS. iS. ir. S. 79°W. 21. S. 9°E. 2. N.W. by N. 12. S. 6 E. 22. N. 59 E. 3. N. bvW. 13. S. 73 w. 23. N. 89 E. 4. N.N.W. 14. S. 37 W. 24. S. 60 W. 5. S.E. by E. 15. S. 57iW. 25. N. 10 E. 6. S.E. 16. S. 77 E. 26. S. 6oiE. 7. S. 44° E. 17. N. 84 W. 27. S. 76 W. 8. N. 64 E. 18. S. 81 E. 28. N.E. i E. 9. N. 78 W. 19. S. 32 W. 29. N. fW. 10. S. 85 E. 20. S. 36 E. 30. W. IN. NAPIEE'S DIAGEAM. Deviations, Page 166, (fl) Curve A.— 10° E. ; 19" E.; 24° W.; i8^°W.; 25= W. ; 2° E. ; 23!° E.; 17° W. 24|°E.; 24° W.; 15° E. (c) Curve C— 15° W.; ii|' W. ; oi° W. ; 28i» E. ; io^°E.; 6" E. ; 14*° W.; 28i''E. 241° W. ; 2° E. Correct Magnetic Courses, Page 168. Curve A.— N. 66'' W.; S. 87° E ; S. 23° E.; S. 54° W. ; S. 63!° W. ; N. 68° E. S. 3i|° E. Curve C.-N. 37° W.; N. 33i°E.; S. 7i°E.; N. 62° W. ; N. 58MV. ; N. 20° E. N. 80° E. ; N. 58^° W. ; N. 27^° W. ; S. 76^ W. Compass Courses, Page 169. Curve A.— N. 79° W.; N. 27°E.; S. 24'W.; N. 9' E. ; N. 4i|°E.; N. 78° E. S. 66° W. ; N. 2i\° E. ; N. 86° W. CurveC— S. 2i°W. ; N. 78°E. ; South; N. 35i°E. ; S. 73° E. ; S. 31° E. ; S. 8° W. N. 65!° E.; S. 18° W. DIFFEEENCE OF LATITUDE AND DEPAETUEE, Page 186. No Diff. lat. Dep. I. 27'-7 S. ii'-5 E. 2. 9-48. 47-1 E. 3 100-8 S. 91-3 W. 4 12-3 N. 831 W. 5 28-8 S. 48-0 E. 6 142-7 S. 1739 w. 7 44-5 N. i77'5 E. No. Diff. lat. Dep. 8. 9- io'-8 S. 22-9 N. 33'-3 W. 8-8 W. 10. 109 S. 233 W. II. 12. 2-5 S. 27-3 N. 14-5 W. 13-9 W. 13- 14. 7-4 S. 33-2 N. 42-2 w. 10-8 W, 419 COUESES AND DISTANCES, P<7y. !.AT. in;r. L.\T. IN. COURSE. DIST. I 95-2 S. 92-1 w. 5'°23'N. S. 44" W. 132' 7 20'0 S. 1288 w. 53 52 N. S. 81 W. 130 3 375"6 S. O'O 2 26 S. s. 376 4 o-o 76-8 E. 19 S. E. 77 5 75-1 S- 77-8 E. 15 s. S. 46 E. 108 6 80-4 B. 36-0 W. 10 S. S. 24 W. 88 7 31-0 S. 8-4 W. 46 41 N. S. 15 W. 32 8 24-7 S. 145- 1 W. 34 36 N. N.80 w. 147 9 551 N. 1 29-9 W. 35 39 S. N. 23I W. '39i 10 150-3 S. 56-8 w. 44 S. S. 21 AV. 161 PAEALLEL SAILING, Fat/e 195. I. 25o'3' W. 2. 344-4' E. 3- 5 £9-2 W. 4. 148-0' W. 5- 5I2'5 ^^ 6. 612-0 W. 7. 113-8 E. 8. 408-0 E. Fai/e 196. I. N . 64° 17' 30" W., distance 396-7 m les. 2. 893-4 miles 4. S. 79° 8' 45" E., distance 96-5 miles. 5. 60" and 70° 32'- 6. 6i •6 miles, or 1° i'-6. 7. West, distance 864-1 miles. D. lat. II 3'" 3 » 999 „ 89-8 „ 1656 ,> 967 „ 1 14-6 97 N. 85 s. 230 S. 1232 N. 115 N. 1 107 N. 779 S. ion N. 792 S. 128 N. 731 s. 150 N, 4483 N. i860 S. 3355 N. 180 N. MIDDLE LATITUDE SAILING, Fa^/e 199 D. Ion Dep. 273'-5 „ 187-0 „ 189-8 » 223-3 ), 318-7 „ 122-9 Lat. in 27° 28'N. 34 10 N. 41 o S. 49 10 S. 18 52 N. o 59 S. 305 224 248 334 338 123 Long, in 54° 55' W. 29 8 W. 70 12 E. 175 58 w. 175 12 E. 27 47 W. MEECATOE'S SAILING, Page 203. M. D. LAT. 130 500 1760 166 1230 1080 "39 794 233 733 274 4842 1884 3516 190 D. LONG. 131 E. 76 E. 270 E. 4732 E. 191 E. 1452 w. 1200 W. 3808 w. 1254 E. 725 W. 2459 E. 354 E. 33'3E. 412 E. 7587 w. 1 140 w. COURSK. N. 46''2i' S. 30 19 S. 28 22 N. 69 36 N. 49 o N. 49 44 S. 48 I N.*73 21 E. E. E. E. E. W. W. W. S. 57 39i E. N. 72 II S. 73 24 N.52 16 N. 34 23 S. 12 20 N. 65 8 N. 80 32 W. E. E. E. E. W. W. DIST. 141 98 261 35 34 ns 1713 1 165 35^8 1480 418 2559 245 5432 1904 7978 1094 429 A nsmrs. DAY'S WOEKS COEEECTED FOE LEEWAY, YAEIATION, AND DEVIATION, Pages 217—222. Note. — In the following key, the first line for each day's work is explained by the titles at the top of the page. The second line contains the True Courses. The third line contains the DifF. Lat. and Dep. corresponding to each course ; their names are not given because these are easily seen from the courses in the second line. 1 Courses. Distance. DifF. lat. Departure. Lat. in, Mid. Lat. Diif, Long. long, in. N. 630 E. 227' 102'-2N. zo2'-8 E. 36°S7'N. 36° 6' 251' E. 71° 19' W. N. SgoE.ig' o'-3 ig'-o S. 43°E. 50' 36'-6 34'-i S.7i°E.38' i2'-4 3S''9 , N.S5''E.42' j 24'-i 34'-4 N.30W. 42' 4i'-9 2'-2 N.15<=E.42' 40' '6 lo'-g N.82°E.37' S'-l 36'-5 N.4i°E.52' 39'-2 34'-l S. 77^'= E. 99' 2i'-s S. 6'-6E S. 75°E.3l' I N.72°E.2o' 8'"o 29'9 ' 6''2 i9''o 53" 4!^' N. I 53° 56' 164' E. 2° 39' E. s. es'^E. 17' 6'-4 i5'-8 N. 3°W.6' &'o o''3 N.730E.28' 8'-2 26'-8 S. g-^ W. 19' l8'-8 3'-o S. 5°W. 13' l3'-o I'-i S. 2o''E.io' 9''4 3'"4 N. 24°E.i5' 13'7 6'-i North. 86'-6 86'-6N. 45° 14' N- 70 51' W. N.i7°E.i3' i2'-4 3'-8 North 38° 38' -0 N.27°E.i8' i6'-o 8'-2 N.790E.25' S.83°W. 23' 4'-8 24'-s 2'-8 22'-8 S.48°E.25'-2 l6'-9 l8'7 N.48°W27'-i i8'-l 2o'-i N.360W.21' l7'*o i2''3 4 N44°E 14' 10' '5 N. 9'-8E 46° i.y S 46° 6' 14' E 1 1° 56' "W N5oo"W2o' 12'9 l5'-3 S 7J°"W 26' 7'-6 24'-9 N 40° E 34' 25''0 21'-9 S 54' E 22' l2'-9 i7'-8 N67°E25' 9''8 23'-o N 67° E 26' io'-2 23'-9 S 44° "W 24' 17''3 i6'"7 S62°"W22'-5 io''6 19''9 6 ,S.45°W. 82' 58'-3 S 58'-! W 34°i2'N 34° 41' 71' W 6°47'W S 87° "W 9' o'*5 9'-o S 29° W 38' 33'-2 i8'-4 N23°W4S' 4i'-4 i7'-6 N77°W34' 7'-6 33'-i South 46' 46' -o S29°W37'-5 32'-8 i8'-2 N 7i°E24"8 S 790 E 15' 8'-i 23'-5 1 2'-9 l4'-7 6 N3"W 28'-5 28'-5 N i'-6 W 30° 27'-5 N 30° 13' 2' "W 1 32° 52' E. S3°E15' l5'-o o'-8 N75°Wl2' 3''i ii"6 S63°Ei3' 5'v) ii'-6 S 10° E 2< 24'-6 4'-3 N 8° E 20' i9'-8 2'-8 876° Wis' 3'-6 l4'-6 Nii°W27' 26"s 5''2 N 20° E 30' 28'-2 io"3 7 S5o°W 87' 56'-5S 66'-5 "W 45° i6i' S 44° 48' 94° W |I78°23'W N70o"Wi8' 6''2 l6''9 N2o°E38' 35'-7 i3'-o S 88° W -,4' l'-2 34'-o S 83° W 48' 5'-8 47' 6 S 9° W 25' 24'7 3''9 S 2° W 38' jS'-o i'-3 S 19° E 50' 47'-3 io''3 N 26° E 18' i6''2 7''9 8 N70°E 161' 55'-3 N i5i'-2E 63° 13' N 62° 45' 330' E 1 57°47'W N 66° E 21' 8'-5 i9'-2 N 71° E 50' i6'-3 47'-3 N75°E 17' 4'"4 i6'-4 N 35° E 34' 27''9 i9'"5 S 28° W 14' i2'-4 6'-6 S74°E2i' 5'-8 2o'-2 S2°'Wl7' i7'-o o'-6 N 47°E49' 33'-4 35'-8 9 S77°E 105' 24'-3 S lo2'-5 E 59° 25' N 59° 37' 202'-7E 4o°3l'"W S 12° E 14' 13'7 2'-9 S37°El9' l5'-2 ii'-4 Nli°W25' 24'-5 4'-8 S 13°E 22' 2i'-4 4'-9 N5l°E33' 2o'-8 25'-6 S 4° W 19' i9'-o l'-3 S7i°E25' N79°E4i' 8'-i 23'-6 7''8 4o'-2 iio''iN. 66'7 "W. I 34°2o'S iJ'W. I in°3iJ''W N860W 1 247' l6'-5 N 246'-i "W 44° 3i' S 44° 12' 343' W 177° 28' E. "West 16' 16' -o N 62° W 40' i8'-8 35'-3 N«63°W37' l5'-8 33'-o S 86° W 38' 2' 7 37'-9 S83°W3i' 3'-8 3o'-8 Sso°W3i' ig'-g 23' -7 N8o°'W4i' 7'-3 4i'-4 West 28' 28'0 12 1. Jan. 1' 4 . Mar. 3 1 7. Dec. 31 10. Oct. I ASTEONOMICAL DATES, Page 224. 16^38™ 9" 2. Feb. 27'' Si'iii'o' 3. Aug. 14'^ 6'i28'»4b» 19 54 19 5. June 3 16 18 3 6. Aug. 31 20 10 52 6 18 34 8. July I 8 3 24 9. June 30 23 30 10 o 10 12 II. 1 87 1, Dec. 31 20 9 50 12. 1872, Dec. 31 12 44 12 421 Jan. nth, Oct. 15th, Mil)' 17th, April ist, Sept. ist, CIVIL DATE, Page 224. 4h3I'"I5»A.M., 3 17 13 A.M., 7 15 " PM-, II 10 16 A.M., 8 10 54 P.M., Feb. SriJ, ii>>28""-56'P.M Dec. 3rd, 5 1& 12 P.M Mar. 14th, II 15 7 A.M Mar. 2 ist. 7 24 12 P.M Sept. ist, 8 10 54 A.M 1872, Jan. ist, 9 30 41 P.M., 1873, Jan. ist, 10 48 56 A.M. DEGEEES INTO TIME, Parje 225. i. ihi^m^^t o''-5o"43» gb 9™48' 6''24-43« 5''57'" 4" 2. 4 30 48 S S ^^ 5 40 9 22 8-7 4 37 56 3- 3 54'4 3 '6 17-33 I 47'2 56 10 8 41 16 4- 36 56 10 52 II-2 2 29"6 9 I2'8 II 21 5- 7 '4 28 41 48*9 9 56 5 3S 50 2 18 6. 54 3 24 40 8 10 27 28 II 55 '9 2 46 TIME INTO DEGEEES, Page 226. I. iS°28' 0"' 2. 58' i' 0" 3- io<^33' 0" 4- i68°5o' 15' 5- 67 16 «5 6. 147 24 30 7- 8 44 33 8. 25 15 24 9- 89 46 10. 124 16 30 II. 5 22 43-5 12. 175 16 40 13- 58 14. 2 29 15- 13 16. ■ 5 10 15 17- 9 14 18. 75 12 45 19. 179 59 15 20. 28 GEEENWICH DATES, Page 229. r. Jan. 622"'l03 II. Dec. j'ydj^h igm^os 2. Feb. 12 22 4 19 7- Dec. I 6 32 45 12. July 8 at noon 3- Jan. 31 7 29 28 8. June 30 16 36 52 13- Jan. 31 13 45 20 4- Mar. 15 8 8 6 9- Aug. 3 23 50 22 14. May 31 18 24 40 5- May 15 6 6 16. 10. 1882. Sept. Dec. I 6 24 1 1 31 14 3 20 15- Mar. 2 5 40 SUN'S DECLINATION, Page 237. I. 22' 35' i7"S. 2. i6»4o'4"-5 S. 3- 4° 9' 2 8"-5 N. 4- 2" i i5"-9 N 5- 19 8 39 N, 6. 14 38 43 N. 7- 23 2 I N. 8. 14 28 6 S. 9- 8 27 48 N. 10. 3 13 58 S. II. 23 19 26 S. 12. 18 17 30 S. 13- 20 14 24 S. 14- 12 19 31 S. '5- 6 II N. 16. 17 8 32 N. 17- 23 27 10 N. 18. 3 37 N. 19. 18 57 52-5 N. 20. 2 6N-. 21. 3 3 57 «• 22. 23 27 13 S. 23- 23 2 59-5 S. 24. 3 17 N. EQUATION OF TIME, Page 241. r. + 5'"48'-3 2. + 14™ 6^1 3- 4- 6°>i8«-9 4- 0™ O^'O 5- — 3 45-0 6. + 2-7 7- + 5 46-5 8. + 5-8 9- -5 55-8 10. — II 49"3 II. + 3'2 12. — 1-6 13- -3 52-8 14. O'l 15- — 15 i5"4 16. + 6 7-8 17- + 13-4 18. — 15 58-4 19. + 5*2 20. 0"0 I. 17=52 42 5. 58 48 28 9- 17 51 38 TEUE ALTITUDES, Page 243. 2. 48° 17' 14" 6. 14 56 49 10. 67 22 16 3. 30" 2 9 7- 65 13 4 II- 13 44 33 4. 76° 14' 16" 8. 85 22 51 12. II 45 28 424 MEEIDIAN ALTITUDES , Page 2 4.9. raper's NO. GREEN . DATE. EED. DECL. TRUE ALT. L iTITUDE. TRUE ALT. LATITUDE. I. Jan. lod fiig^24^ 21° 54' 35"!^- 68=57' 22" 42° 57' i3"S. 68° 57' 19" 42° 57' l6"S. 2. Jan. 31 21 20 36 17 4 6S. 72 58 6 2 12 S. 72 58 3 2 9 S. 3- Mar. 7 18 48 4 54 12 S. 51 58 I 33 7 42 N. 51 57 51 33 7 57 N. 4- April 28 II I 32 14 20 59 N. 82 35 15 6 56 14 N. 82 35 10 6 56 9 N. 5- May- I 21 51 48 15 24 8N. 45 57 I 59 27 7N. 45 56 55 59 27 13 N. 6. June 10 19 48 12 23 5 36 N. 42 37 28 24 16 56 S. 42 37 24 24 17 S. 7. July 20 10 26 32 20 34 7N. 52 7 9 17 18 44 S. 52 6 57 17 18 56 s. 8. Aug. 19 5 30 12 39 46 N. 57 50 46 44 49 N. 57 50 45 44 49 I N. 9- Sept. 22 12 54 2 47 S. 41 42 33 48 20 14 JN. 41 42 23 48 20 24 N. 10. Oct. 23 6 48 II 33 38 «. 54 51 19 23 35 3N. 54 51 13 23 35 9 N. II. Nov. 14 18 39 16 18 29 19 S. 67 56 49 3 33 53 N. 67 56 40 3 34 I N. 12. Dec. 9 20 18 40 22 55 49 S. 26 4 46 40 59 25 N. 26 4 36 40 59 35 N. 13- Sept. 20 19 59 56 42 36 N. 5(5 37 9 32 40 15 S. 56 37 5 32 40 19 S. 14. War. 19 18 2 o I 56 N. 61 qS 10 28 3 46 N. 61 58 I 28 3 55 N. 15- Apiil 7 9 19 7 2 33 N. 90 13 41 6 48 52 N. 90 13 40 6 48 53 N. 16. Sept. 22 15 45 83 52 20 6 7 40 N. 83 52 14 6 7 46 N. 17. Nov. 2 16 56 15 2 14-58. 70 42 48 34 19 27 S. 70 42 44 34 19 31 S. 18. Sept. 22 II 35 52 3 16 N. 71 34 38 18 28 38 N. 71 34 32 18 28 44 N. 19. Feb. 12 32 48 13 36 37 «• 30 4 42 46 18 41 N. 30 4 36 46 18 47 N 20. Mar. 19 18 49 14 N. 77 7 26 12 52 20 S. 77 7 23 12 52 23 S. 21. Dec. 31 15 37 52 23 I 34 S. 54 38 7 12 20 19 N, 54 38 6 12 21 20 N. 32. Sept. 30 19 14 40 3 10 24 S. 81 57 15 II 33 9S. 81 37 II II 33 13 S. AMPLITUDES, Page 258. No. Green, date Red. decl. True amp. Sine. I. Jan. 2C''i9''47'" 8» iS°25'46''S E 22°S6'S 9'59074l 2. Feb. 17 4 7 28 11 so 43 S W 14 30JS 9-398865 3- March 29 2 20 20 3 29 26 N E 3 53 N 8 '830356 4- April 4 19 53 6 4 40 N W 6 29 N 9'053M5 5. Nov, 6 22 5 8 16 ig 12 S E 18 30 S 9-501420 6. May- 25 16 44 21 6 23 N E 35 14N 9-761046 7. June 2 9 56 26 22 l5 2 N ■W 38 30 N 9-794098 8. July 14 2 IS 42 21 39 33 N E 24 57 N 9-625221 9- Aug. 27 3 18 44 9 58 22 N "W 13 42 N 9 268637 lo. Sept. 7 21 37 5 40 46 N E 6 13 N 9-036286 11. Oct. I 5 29 5^ 3 20 21 S S 4 33 S 8-899589 12. Sept. 22 6 I 12 9 29 S E 17 N 13' Nov. 2 21 27 40 15 5 46 S "W17 23 S 9-475406 14. Dec. 3 21 13 48 22 15 47 S W 36 7 s 9-770427 15- March 20 4 56 24 000 16. Sept. 22 15 45 000 17- June 8 18 41 24 22 56 9 N E 22 s6 N 18. Feb. 25 20 40 28 8 41 15 S E i3 49 S 9-508481 19. April 30 15 41 40 IS 1 30 N AV i5 37 N 9-456310 20. May 27 17 32 12 21 26 54 N W 32 s^ N 9-734811 21. June 18 1 24 20 23 25 19 N E 64 36 N 9-955851 22. March 6 6 5 20 5 29 7 S W 6 26 S 9-049030 23- April 9 18 50 48 7 56 S N "W 13 34 N 24. Dec. 13 11 35 1^ 23 12 25 S E 32 I S Error of compass. 33°i9''W ig 144E 26 23~W o 11 W 21 19 E 43 4^ W 26 11. jW 22 52 E 25 52 W 6 13 W 12 59 E 3 6 "W 5 7 E 2 22 W o o o o 17 18 "W 48 41 W o 1^.1 W 21 38*'E 3928 E 23 18 W 22 o E 57 58iW Deviation. ii°29''W II 342E 2 43 W 659 W 7 2g E 8 zo W 11 8JE 11 12 E 2 42 W 6 13 W 5 51 W 14 6 AV 7 57 E 18 22 AV 15 ^ w 21 50 E 7 3 E 12 55 "W 10 isiW I 21 E 14 28 E 5 28 -W 5 50 E 38 384W TIDES, Page 265. t ']^I()™A.'iI. ^h^gm P.M. 16. 8h25"'A.M. Shjo^r.-M 2 855 t'^ » 17- II 59 . , No „ 3 II 53 No )> 18. II 12 , , II 41 „ 4 39 58 )» 19. No , 15 „ 5 II 42 No 1) 20. II 58 , No „ 6 10 32 10 57 )i 21. Noon 7 II 52 No >) 22. No , 01,, 8 II 34 II 59 )> 23- No 18 „ 9 II 59 No » 24. 7 , , 24 „ 10 Noon 25- II 43 > , No „ II 2 56 3 39 )> 26. U ^° ' No 12 9 58 10 26 » 27. No , > 36 ), 13 10 10 39 » - 28. 9 54 , , 10 29 „ 14 No 35 » 29. II 36 , No 15 No Noon 30. 10 10 , , 10 58 „ 42 3 TIDES-FOEEIGN POETS, Fa^e 267 I Constant — o''47'" corr. for lonjj. + 6™ 3*^ ^'"a.m. ^hj^mp ji. 2 Constant — 17 corr. for long. -|- 10 Js'o ,, 13 » 3 Constant -}- 3 18 corr. for long. — 23 11 8 „ II 42 „ 4 Constant — I 47 corr. for long. — 10 10 15 „ 10 43 .» 5 Constant — 341 corr. for long. + 15 i 39 ,, I 57 » 6 Constant 4" 3 4'i corr. for long. — 14 016 ,, 56 ,, 7 Constant 4- 4 28 corr. for long. — 18 10 37 ,, 10 56 „ 8 Constant -f 6 58 corr. for long. + 8 5 24 „ 6 3 „ 9 Constant -}- 49 corr. for long. — 24 9 29 ,, 10 5 ,y 10 Constant -f 7 43 corr. for long. + 8 1 1 26 ,, II 47 ., II Constant — I 47 corr. for long. — 10 4 29 „ 4 49 » 12 Constant -|- 6 13 corr. for long. — 15 No „ 02,, FINDING DAILY EATE, Pape 270. I. 22 (lavs i y% gaining 5. 31 days 4'' 2 losing 2. 18 ,; 3-0 losing 6. 22 ,, I -I fo.siwy 3- 17 ,. 4-5 gaining 7. 15 „ 6"4 lonng 4- 15 .. 11-2 gaining 8. 14 „ GEEENWICH DATE, Page 273. 7-0 gaining Daily Rate Ace. Kate Green. Date. Daily Rate. Ace. Rate. Green. Date. 1. b'-i losin •7 2"M5'-S Feb. lo'i 7''47"'47^-5 7. ^'-^iiaining ■;"'$■' 7 Nov. 8<'i6i>26'"59-7 2. 7'l frrt/« ".'7 4 27 April 28 4 21 37 8. 87 losinij 6 57 6 Autj. 105 55 '4 i- 3 "22 losh ,17 3 4"'2 May 7 6 25 16 g. I'o losing 1 2 5 May 1 15 28 1-5 4- 2-2 ^n(/(i /((jf 2 38-1 June 25 20 56 30 10. 08 losing 32 8 J50™42s 3. 4'i5o'"20' 4. 4'' 6n'56' 5- 4 8 45 6. 2 33 42 7-4 3 50'f 8. 4 29 56 CHEONOMETEE, Pages 284—286. 1. Interval 7 da3-s, rate 9'"8 losing, interval 25'* iq*', accumulated rate -j- 4™ i2»'9, Green, date January i"* ig"^ 32™ i3*'5, red. docl. 22° 55' 22" S., rod. eq. T. add 4:^ i6'"25, true alt. 49" 19' 18", sum of logs. 9"i57889, hour-angle z^ 58™ 18', M.T. ship January i"* 21'' 5™ 58'. Longitude 23° 26' 7 "5 E. By Baper : True alt. 49° 19' 13", sine sq. 9'i57923, hour-angle 2'' 58'" 19'. Longitude 23° 25' 52"-5 E. 2. Interval 7 days, rate 75-6 gaining, interval i9 45"" 57^, red. decl. 11° 15' 51", eq. T. 14"^ 3'-35 add, true alt. 21° 34' 14", sum of logs. 9'53i239, hour-angle 4'' 45"'- 15', M.T. ship February iS'^ 19*" 28™ 48". Longitude 4' 17' 15" E. By Saper : True alt. 21° 34' 7', log. sine sq. 9"53i262, hour-angle 4'' 45™ 16'. Longitude A" 17' 30" E. 3. Interval 43 days, rate 4''o, interval 115'' 23^, accumulated rate -\- 7™ 43'"8, red. decl. 3° 4' 16' N., red. eq. T. 5"' 7^*97 add, true alt. 30' 21' 7*, sum of logs. 9'34:946, hour-angle 3'' 43"" 55% ^i-T. ship February 28'^ 3^ 49™ 2"- Longitude 65° 48' 45" E. By Raper : True alt. 30' 21' o', log. sine sq. 9342975, hour-angle 3'' 43™ ^d*. Longitude 6s' 49' E. 4. Interval 28'', rate 5^ 8, interval 25'' 19.P', accumulated rate — 2™ 29^'6, Green, date April 5*1 19'' 13™ 41", red. decl. 6' 26' 44" N., red, eq. T. add 2"" 27'-86, tme alt. i6^ 17' 12"'^ sum of logs. 9-531803, hour-angle 4'* 45"" 28% M.T. ship April 5'' 19*" 17™ o«. Longitude 0° 49' 45" E. By Raper : True alt. 16° 16' 58', log. sine sq. 9"53i873, hour-angle 4'' 45™ 30'. Longitude o" 49' 15" E. 424 Answers. 5. Interval 28'*, rate z^'<, losing, interval IDS'*, accumulated rate -f- 4™ 30% Green, date May i()^ o'l 29™ 10% red. decl. 19° 48' 39" N., red. eq. T. s"" 45''8i subt. from A.T., true alt. 30° 41' 2", suna of logs. 9-340198, hour-angle i'^ 43™ 9^ M.T. ship May 19^ 3^ 39m 23^. Longitude 47° 33' 15" E. By Raper : True alt. 30' 40' 55", log. sine sq. 9*340227, hour-angle 3*^ 43'" io«. Longitude 47° 33' 30" E. 6. Interval 22'', rate ^^'4 gaining, interval 33'' 18'', accumulated rate — 5™ i7*'3, Green, date June 14'! 17'' 56'n 42% red. decl. 23° 20' 14" N., red. eq. T. + o"" 6% sum of logs. 9*277995, hour-angle 3*1 26™ 32^, M.T. ship June 14'^ 201^ 33™ 34'. Longitude 39° 13' E. By Raper : True alt. 39° 50' 18', log. sine sq. 9-278041, hour-angle 3'' 26'" 33'. Longitude 39° 12' 45" E. 7. Interval 26'^, rate 4»-7 gaining, interval 34'^, accumulated rate — 2™ 40', Green, date July 5, accumulated rate -)- 2™ 2^-6, Green, date November 27"^ 7^ 13'" 539, red. decl. 21° 13' 53" S., eq. T. 12™ 53 subt., true alt. 34° 50' 6", sum of logs. 9-420025, hour-angle 4'' 6"" 51% M.T. ship November 26^ 19'' 41™ 4'. Longitude 173' 12' 15" W. By Raper : True alt. 34° 49' 57", log. sine sq. 9-420039, hour-angle 4'' 6" 5I^ Longitude 173° 12' 15" W. 12. Interval 28'^, rate 9^-3 losing, interval 148'' 18^, accumulated rate -\- 23'" 3% Green, date December z^^ 18'' 37™ 42»-8, true alt. 10° 42' 56", sum of logs. 9-638730, hour-angle 5I1 30™ i4«, M.T. ship December 24"* i8h 30'" c. Longitude 1° 55' 45" W. By Raper : True alt. 10" 42*44", log. sine sq. 9-638761, hour-angle 5^ 30™ I5^ Longitude 1° 56' W. 13. Interval 31'^, rate 6^9 losing, interval 80'* 14!^, accumulated rate 9™ i6", Green, date January 1^ i4>i o™ 38% red. decl. 23° 56' 41" S., red. eq. T. + 4" 9^'^7, true alt. 39° 9' 31", sum of logs 9-362235, hour-angle 3^ 49™ 25^ Longitude 151° 45' 45" W. By Raper : True alt, 39" 9' 28", log. sine sq. 9-362253, hour-angle 3'' 49™ 25^ Longitude 151° 45' 45" W, J Answers. 425 14. Interval 3i<^, rate 85'3 gainiiiff, interral 7r' 22'', accumulated rate 9™ 56''9, Green, date February io<* 21'' 33™ 26', red. decl. 13° 59' 12' S., red. eq. T. 14™ zy6 add, true alt. 12' 17' 54', sum of logs. 9'i76958, hour-anglo 3'' 2™ 29'. Longitude 5° 21' 45" W. By Baper : True alt. 12° 17' 48", log. sine sq. 9-177030, hour-angle 3" 2™ 30*. Longitude S" 22' W. 15. Interval 34'', rate 28-5 losing, interval 52 days, accumulated rate + 2™ 10", Green, date October 26'' d^ 26-" ic, red. decl. 12° 31' 15" S., red. eq. T. 15'" 56^-6 subt., true alt. 25" 10' 24', sum of logs. 9'2S6497, hour-angle 3'' 28"" 44". Longitude 62° 42' 45" W. By Eaper : True alt. 25^ 10' 13', log. sine sq. 9-286565, hour-angle 3'' 28™ 45'. Longitude 62° 43' W. i6. Interval iS*!, rate 5^-4 losing, interval 17 days (nearly), accumulated rate i™ 32% Green, date February 5"^ 23'' 59™ 40% red. decl. 15° 33' 6'S., red. eq. T. + 14™ 19^-4, true alt. 21* 21' 7", sura of logs. 9-466313, hour-angle 4'' 21'" 59'. Longitude 69° 9' 30" E. By Raper : True alt. 21° 20' 57', log. sine sq. 9-466348, hour-angle 4'' 22™ o». Longitude 69° 9' 45" E. 17. Interval 48'^, rate 3"'-3, interval 72"' 19'', accumulated rate — 4™ o», Green, dato April 3o> 3™ 37*. Longitude 82' 17' 15" W. 21. Interval 32"', rate o'-8 gaining, interval 113d 15^^, accumulated rate — i™ 3i«, Green, date September 22^^ 15'' 38^' o^, red. decl. 0°, red. eq. T. — 7"" 47s, hour-angle 4'' 3™ 50% M.T. ship September 22d 19'' 48"" 23". Longitude 62° 35' 45" E. By Eaper : True alt. 29° 2' 15", hour-angle 4'' 3'" 51'. Longitude 62° 35' 30" E. SUMNER'S METHOD, Fat/e 299. r. ist red. decl. 7" 42' 55" S; ist red. eq. T. 12" 37»-47 {added to A.T.) ; ist true alt. 29° 57' 17'; lat. 49° 10' gives hour-angle i'' 27™ 53"; long. 178° 27' 30" E. (A). Lat. 49° 40' gives hour-an:;le i^ 21™ y; long. 180° 9' 30" E., or 179° 50' 30" W. (A'). 2nd red. decl. 7° 38' 10' S.; 2nd red. eq. T. 12™ 3584 (rtfWcfHo A.T.) ; 2Dd true alt. 15^^55' 57"; lat. 49° 10' gives hour-angle 3>' 38™ 345; long. 180° 5' o' E., or 179=55' "W. (B). Lat. 49=40' N. gives hour-angle 3^' 36" 34^; long. 179= 35' E. The line of bearing when the first altitude was taken trends E.N.E. and W.S.W. (nearly). The position of ship -when the second altitude was observed was lat. 49° 25' N., long. 179° 49' E. 2. let red. decl. 19° 47' 20" N. ; ist red. eq. T. — 3'" 46'; true alt. 49° 8' 2' ; lat. 48° 30' gives hour-angle 2>' 25™ 6« ; M.T. ship May 18'^ 21^ 31™ 8"; long. 7° 37 j W. (A), Lat. 49°o'giveshour-angle 2h23'n 11'; M.T. ship May i8 Answers. 5° 38' "W. The line of bearing when the first altitude was taken trends N.E. by N. (northerly) and S.W. by S. (southerly). The position of ship at the time of the second observation is lat. 49° sSy N, long. 6° ij' W. 3. ist red. decl. i6» 37' 15" S. ; ist. red. eq. T. 16'" 75-4 sub(. ; ist true alt. 19° 58' 31"; lat. 48° 10' N. gives hour-angle i'> 59™ 6' ; long. 1 1' 3I' W. (A). Lat. 48° 40' N. gives hour- angle ii> 53™ 49S ; long. 9° 44!' W. (A'). 2nd red. decl. 16° 41' 6" S. ; 2nd red. eq. T. 16™ 6^ subt. ; lat. 48^ 10' gives hour-angle 3"^ zz"* 48^ ; long. 10° 33I' W. (B). Lat. 48° 40' N. gives hour-angle 2^ 20™ 9^ ; long. 1 1^ 13!' W. (B'). The line of bearing at first observation trends N.E. by E. 1 E. and S.W. by "W. J W. The ship's position at time of second observation is lat. 48° 5' N., long. 10° 27' W. 4. ist red. decl. 16° 28' 46" S. ; ist red. eq. T. 14'" 48-59 add; ist true alt. 19° 59' 26"; lat. 48° 10' gives hour-angle 2'' o™ 28' ; long. 0° 54^' W. (A). Lat. 48^40' gives hour-angle jh ^^m i^s; long. 0° 23' E. (A'). 2nd true alt. 11° o' 53"; lat. 48° 10' gives hour-angle 3'»24™3os; long. 0° 2ii' E. (B). Lat. 48^40' gives hour-angle 3^ 21 '"54s; long. 0° 17!' W. The line of bearing at first observation runs N.E. by E. i E. and S.W. by W. ^ W. The ship's^position at second observation is lat. 48*^ 22J' N., long. 0° 4^' E. 5. ist red. decl. 19° 10' 24*8.; ist eq. T. 12"' 22^ add; ist true alt. 9° 46' 49"; lat. 51° 15' gives hour-angle 2^ 55"^ 36^ ; long. 15'^ 33|-' W. (A). Lat. 50° 45' gives hour-angle 2h ^gm g3. long. 16° 26I' W. (A'). 2nd red. decl. 19° 7' 58" S. ; 2nd red. eq. T. 12™ 248-77 add; lat. 51° 15' gives hour-angle o^ 57™ 35s ; long. 17° 165' W. (B). Lat. 50° 45' gives hour-angle i'' 8™5i8; long. 14° 27 1' W. (B). The line of bearing at first observation trends N.E. I E. (easterly) and S.W. J W (westerly). The position of ship at second observation is lat. 50° 56' N., long. 15° 26^' W, 6. istred. decl. 2=27' o"N.; istred. eq.T. 5™ 36s-45 « 33"'' 32s; long. 181° 28^' E., or i78°3iJ'W. The line of bearing at first observation trends N.E. by E. i E. and S.W. by W. ^ W. The position of the ship at the second observation is lat. 51° laf N., long. 178° 37J' W. 7. ist red. decl. 7° 31' 34" S. ; ist red. eq. T. iz^" 318-5 add; ist true alt. 29° 55' 26" ; lat. 49° 15'; hour-angle i^^ 29™ 498; long. 1° 23' W. (A). Lat. 49° 45' gives hour-angle jh 23"-- 7s; long. 0° 17' 30' E. (A'). 2nd red. decl. 7° 36' 50"; 2nd red. eq. T. 12™ 298 add; lat. 49° 15' gives hour-angle 3^ 38™ 57^ ; long. 1° 9' E. (B). Lat. 49° 45' gives hour-angle 3^ 36"" 588; long. 0° 39I' E. (B'). The line of bearing at first observation trends N.E. by E. f E. (easterly) and S.W. by W. f W. (westerly). The position of the ship at the second observation is lat. 49° 45' N., long, oo 39^' E. 8. ist red. decl. 16° 33' o" S. ; 1st red. eq. T. 14'" 38-14 s^t?; ist true alt. 20° 11' o" ; li't. 47° 10' gives hour-angle 2^ 701 lo*; long. 83° 19' 15" E. (A). Lat. 47° 40' gives hour-angle jh 2m 258; long. 84^ 30' 30" E. (A'). 2nd red. decl. 16° 29' 2" S. ; 2nd red. eq. T. 14"^ 48-49 ; 2nd true alt. 11° 17' 59"; lat. 47° 10' gives hour-angle 3'* 26'" 498; long. 86" 14' 45' E. (B). Lat. 47° 40' gives hour-angle 3^ 24'" 198; long. 85° 37' 15" E. (B'). The line of bearing at the first observation trends N.E. by E. and S.W. by W. The position of the ship at the second observation is lat. 47° 53|-' N., long. 85° 20' E. 9. ist red. decl. 19° 3' 46" S. ; ist red. eq. T. 12™ 288-88 add; ist true alt. 9° 46' 38"; lat. 51° 15' N. gives hour-angle 2^ 56"" 308; long. 180" i8|' E., or 179° 41^' W. (A). 2nd red. decl. 19° i' 20" S. ; 2nd eq. T. 12"" 31*; 2nd true alt. 18° 40' 21"; lat. 50° 45' gives hour- angle 2>' 59™ 118; long. 179° 38^' E. (A') Lat. 51° 15' N. gives hour-angle o'' 5^"" 438; long. 180° 34' W., or 178° 26' E, (B). Lat. 50° 45' gives hour-angle i^ 7™ 188; long, 178° 40J' W. The line of bearing at first observation trends N.E. J N. (northerly) and S.W. 5 S. (southerly). The position of the ship at the second observation is lat. 50° 53' N., long. 179° 24' W. 10. ist red. decl. 0° 15' 23" N.; ist red. eq. T. 7'" 188-5 add; ist true alt. 22° 59' 25"; lat. 50° 10' gives hour-angle 3'" 31"^ 168; long. 0° 43I' W. (A). Lat. 50° 50' gives hour, angle 3'^ aS-" 488 j Long, 0° 6' E. (A'). 2nd red. decl. 0° 20' 58" N. ; 2nd red. eq. T. -j"' 148-3 Answers. 427 add; 2nd true alt. 32° 40' 10'; lat. 50° 10' gives hour-angle a*" 13"' 22* ; long, o'^ 26^' E. (B). Lat. 50° 50' gives hour-angle 2^ 8™ 22' ; long, o' 485' W. (B'). The line of bearing at the first observation trends N.E. h X. and S.W. | S. The position of the ship at the second observation is Lit. 50" 14' N., long. 0° igY E. I r. ist red. decl. 2" 7' 36" S. ; ist roJ. cq. T. -\- 9'" 4«'4 ; ist true alt. 14° 22' 43" ; lat. 49°°' gives hour-angle 4'' 20'" 17^; long, i^ 16' W. (A). Lat. 49^ 40' gives hour-angle 4I1 iS"" 43S; long. 0° 52^ W. (A'). 2nd red. decl. 2° o' 6" S. ; 2nd eq. T. + 8'" 59'''4 ; 2nd true alt. 23° 27' 13'; lat. 49° o' gives hour-angle 3*" 18'" 40*; long. o''4o|' W. (B). Lat. 49' 40' gives hour-angle 3*^ 15"" 53'; long. 1° 22' W. (B'). Tha line of bearing at the first observation trends N. by E. f E. and S. by W. | W. The position of tho ship at the second observation is lat. 49° 22^ N., long. 1° 4' W. 12. ist red. decl. 16' i' 57" N. ; ist red. eq. T, — 3'n23''-2; ist true alt. 50° 40*55" ; lat- 47^ o' gives hour-angle 1'' 57'' 10'; long. 51° 27 j' W. (A). Lat. 46° 20' gives hour-anglo jh o"" 359; long. 52" 19' W. (A'). Tho line of bearing at the first observation trends N.E. J N. and S.W. ^ S. The position of the ship at the second observation is lat. 46°28'N.,long. 5i°45r W. 13. ist red. decl. 23° 23' 23" N. ; ist red. eq. T. + C" 32*; ist true alt. 52° i' 30" ; lat. 48^ 50' gives hour-angle 2^ 23™ 35*; long. 48" 132^' E. (A). Lat. 48° 10' gives hour-angle 2'' 25™ 455 ; long. 47" 41' E. (A'). 2nd red. decl. 23" 23' 36" N. ; 2nd red. eq. T. -\- o™ 33^-7 ; 2nd tiue alt. 63'' 45' 6'. Lat. 48" 50' gives hour-angle o'' 32™ 44'; long. 47" 29^ E. (B). Lat. 48^ 10' gives hour-angle o'' 43"" 41' ; long. 50° 135 E. (B')* The lino of bearing at tho first observation trends N.E. by N. ^ N. (northerly) and S.W. by S. 5 S. (southerly). Tho position of the ship at the second observation is lat. 48" 30^' N., long. 48° 492-' E. 14. ist rod. decl. 13" 34' 51" S. ; ist red. eq. T. S2(5i;. 16™ 115-69 ; ist true alt. 28° 28' 14' ; lat. 47' 30' gives hour-anglo o'' 33"* ii^; long. 74'^ 38I' E. (A). Assumed lat. 47" 50' {)iot 48°, which would give the sum more than 180°), this gives hour-angle o'' 16'" 55^; long. 70' 34I' E. (A'). Eed. decl. 13" 37' 45 ; eq. T. 16™ 135-6 siibt. ; lat. 47" 30' gives hour-angle jii o"" 379; long. 73" 42;^' W. (B). Lat. 48° o' gives hour-anglo 2'' 57™ 43^; long. 74° 25I' W. (B'). The line of bearing at the first observation trends E. ^ N. (northerly) and W. I S. (southerly). The position of tho ship at the second observation is lat. 47° 46' N., long. 74° 12' W. N.B. — It is scarcely necessary to warn the intelligent navigator against making an assumption of latitude that shall render the computation of the hour-angle impossible : tho sum of the altitude, latitude, and polar distance, can never exceed i8o'\ TEUE AZIMUTHS, Fa^e 306. ' 1. S. 98''43' 3o"E. 4. S. 9o°32' 54"E. 7. 8.56° 4' o"W. 2. S. 41 58 1 8 E. 5. S. 60 9 36 W. 8. East. 3. N. 75 58 4 W. 6. N. 49 7 14 W. 9. N. 84 5 2 W. AZIMUTHS, Faffe 312. No. Green, date. Red. decl. True alt. Sum of logs. True azimuth. EiTor. Deviation. 1. Jan. 23'! 231146™ 7' l9^9'3o"S 38°34'32" i9"73"59 N94°24'44"E i5°3g'44"E ll" 3'44"E 2. Feb. 28 9 43 4 7 44 I S 27 743 19-295153 S 52 44 38 W 10 8 38 E 1 21 22 W 3. Mar. 27 03412 2 40 40 N 2941 3 19703487 S 90 35 56 W 10 39 4 W 7 9 4 "W 4. April 2 19 o 8 518 4 N 11 50 42 19-651753 S K4 5 20 W 5 54 40 W 15 4 40 W 5. May 25 21 9 32 21 18 32 N 43 19 41 19-4181^4 S 61 33 50 E o 11 10 E 15 56 10 E 6. June 20 6 55 44 23 27 4N 16 49 1 19-808282 S 106 37 52 W 30 2 8 \V 7 2 8 "W 7. July 31 1 6 46 18 14 9 N 433548 19-601312 S 78 22 58 E 3 2 58 W 5 47 2 E 8. Aug. 23 2 57 38 II 21 44 N 7 43 28 19759124 S 98 32 30 E 15 7 24 E ij 22 36 W 9. Aug. 31 18 33 2 8 19 2N 30 14 51 19-556169 N 73 43 34 W 7 50 II E II 20 11 E 10. Nov. 25 7 30 28 20 51 43 S 34 2 21 19-696611 N 89 41 22 W 2 30 7 W 11 52 7 W 11. Dec. 16 22 58 2 23 22 2.S S 51 13 o 19712137 N 91 45 50 E 3 54 10 W 5 50 53 E 12. July 21 16 18 50 22 59 26 N 14 19 37 19-259073 N 50 26 36 E 11 33 24 W 5 26 36 E 13. Jan. 6 24930 22 27 49 S 264654 19-732709 N9438 oW 9 5« o W 23 17 o E 14. April 25 14333 13 16 17 N 185257 19-439408 N631544E 12544E i5 5i44E 15. Jan. 29 65347 17 47 45 S 134728 19242075 S 49 24 oW 3126 oW 1056 o "W 16. Eib. I 2 22 23 17 o 41 S 40 7 21 19-651011 N 83 59 54 W 13 9 ':4 "W 16 30 6 E 17. Mar. 26 02958 2 17 17 N 325039 19362660 S 57 23 oE i2 33"o'W' 1327 oE 18. Feb. 25 15 40 o 8 46 8 .S 60 48 32 19-422946 S 61 56 30 "SV o 30 30 E 8 44 30 W 19. Juno 21 7 3 20 23 27 loN 15 46 28 ]984i7;6 S112 54 32'W 75 31 43 W 7 41 43 W 20. Sept. 10 20 16 o 4 33 48 N 42 38 43 18-9^0809 N33 57 20 E 10 52 34 W 1 27 26 E 21. April 24 1432 o 13 850 N 43 634 19-785877 S 102 48 oE 1 26 o E 1 4 o AV 22. Feb. 28 20 15 o 7 34 oS 66 26 29 18-043319 S 12 4 2 AV 14 52 47 E 14 52 47 E 23. Dec. 31 2X 25 30 23 1 23 S 452220 19-891183 N1234948E 11 10 12 AV 9 49 48 E 428 Answers. EEDUCTION TO MEEIDIAN, Fa^es 320—321. No. 3- 4- 5- 6. 7- 8. 9- 10. 13- 14. 15- 16. 17- 18. 19. 20. Green, date. Jan. 4*1 i''34™405 Feb. 28 I 4 49 Mar. 19 16 34 18 April 20 23 17 6 M»j 29 7 38 27 June 19 o 41 12 July 16 o 2 19 Aug. 29 15 2 44 Sept. 8 12 15 43 Oct. 10 18 47 58 Nov. 2 17 I 29 Uec. 22 I 38 58 Jan. 5 8 38 28 April 28 2 3 II ]\Iar. 19 21 28 52 April 12 10 46 29 Sept. 15 14 29 20 15 19 49 I 4 19 8 4 22 7 4 42 Time noon. 14™ 8^ 14 45 25 42 18 30 30 27 15 8 10 19 20 24 9 43 28 10 20 s 9 14 17 32 13 37 16 s6 10 19 Red. docl. 22° 41' 58"S. 7 5' 58 S. o 12 13 S. II 54 31 N. 21 41 51 N. 23 26 21 21 9 5 5 27 7 o 15 2 23 27 33 9N. 22 N. 25 N. 4N. 24 S. 18 S. 4S. 10 S. Mar. Mar. Sept. Reduction. + 5' 52" + 6 30 + 24 36 + 17 54 -{■ 26 10 + 13 44 + 8 40 + 19 8 + 44 48 + 22 30 + 37 2 + 4 .6 + 17 38 + 10 14 + 27 24 -j- 22 10 + 16 22 -j- o 46 + 17 6 + 15 31 Aug.^I. -\- 2' 2 + True alt. (Norie) 32° 26' 19" 38 6 34 47 57 15 61 39 o 30 3-5 8 68 48 28 67 52 44 57 34 6 Exceeds limits 36 44 4 72 2 22 6s 23 36 58 17 34 56 40 4 70 30 47 80 36 17 44 19 4 50 12 46 5° ° 56 43 55 22 14 13 56 N. 7 22 S. 8 55 58 N. 2 44 18 N. 1 43 52 S. 6 2 56 S. o 8 27 N. METHOD II. True alt. (Raper) 32° 26' 12" 38 16 27 47 57 18 6 1 3S 59 30 33 5 68 48 21 67 52 35 57 34 3 True. alt. 32°26' 19" 38 6 34 47 57 15 61 39 o 3° 33 8 68 48 28 67 52 4+ 57 34 6 85 30 34 36 44 4 72 2 22 65 23 36 58 17 34 56 40 4 70 30 47 80 36 17 44 19 4 50 12 46 50 o 56 43 55 22 Nat. no. 1442 1481 4711 2444 6517 1427 942 2945 883 5215 3143 504 2669 1620 2580 lOOI 3385 140 3174 3242 Latitude, 34° 45' 50" 43 55 o 41 50 41 39 57 44 37 18 52 44 24 13 o 37 17 41 12 21 9 14 31 45 53 56 32 24 20 47 59 18 8 5' 45 18 55 50 19 9 42 o 6 13 42 40 20 38 2 57 33 39 6 45 40 41 Latitude (Norie) 34°45'5i"N. 43 54 56 N. 41 50 22 S. 39 57 37 N. 37 18 51 S. 44 24 7 N. o 37 14 S. 41 12 II N. 3- 4- 5- 6. 7- 8. 9- 10. 13- 14. 15- 16. 17- 18. 19. 20. + + + + + 2 19 10 31 2 44 Aug. decl. 22° 44' i8"S. 7 52 56 S. o 12 24 S. II 56 50 N. 21 52 22 N. 23 29 3 N. 21 22 30 N. 9 7 31 N. 36 43 52 72 216 65 22 35 58 17 31 56 37 5t 70 30 37 80 36 16 44 18 59 50 12 37 50 18 2 44 10 45 BY TOWSON : Index. 26 68 41 77 24 II 50 This hour-angle exceeds the limits of the Table. 45 53 2 S. 32 22 54 S. 47 59 12 S. 8 51 38 N. 18 55 46 S. 19 9 II S. o 5 35 S. 42 40 16 S. 38 2 36 N. 33 39 8 S. 45 40 40 S. Aug. II. -f 3' 22" + 5 23 + 24 II 4- 20 6 + 15 38 + 16 37 + 7 12 + 20 52 + + + + + + 19 34 27 7 3 34S. 15 5 37 S. 23 28 5S. 22 36 44 S. 14 15 23 N. o 7 22 S. 76 47 9 32 22 36 + 19 45 + 38 56 + 5 15 + 13 43 + 8 48 + 27 12 This hour-angle exceeds the limits of the Table. + o 42 2 45 oN. 57 +15 31 4" o I I 43 53 S. 2 -j- ° 38 -j- I 23 6 4 19 R. 491 +15 37 4 o o o 8 27 N. 57 -j- 15 18 Latitude (Raper) 34° 45' 5 8 'N. 43 55 5 N. 41 50 19 S. 39 57 38 N. 37 18 54 S. 44 24 14 N. o 37 23 S. 42 12 14 N. 45 53 14 S. 32 23 o S. 47 59 13 S. 8 51 41 N. 18 55 49 S. 19 9 21 S. o 5 36 S. 42 40 21 s. 38 2 45 N. 33 39 2 N. 45 40 48 S. Latitude. 34° 46' i"N. , 43 55 7 N. 41 50 58 S. 39 57 54 N. 37 18 52 S. 44 23 58 N. o 37 34 S. 41 12 33 N. 45 52 37 N. 32 24 19 S. 47 59 14 S. 8 51 59 N. 18 55 46 S. 19 9 23 S. 42 40 25 s. 38 2 43 N. 33 39 8 N. 45 40 S3 S. Ansivers. 429 MERIDIAN ALTITUDE OF STAE, Page 323. No. Doclination. I. 28= ■26'54'N. 2. 45 52 36 ^'• 3- 38 40 50 N. 4- 49 26 44 N. 5- 10 32 59 S. 6. 57 49 47 S- 7- 14 32 9 N. 8. 22 54 3« N- 9- 16 16 14 N. 10. 62 27 12 8. 1 1. 8 9 13 s 12. 44 5' 47 N. 13- 8 33 52 N. 14. 16 33 37 S. 15- 19 47 35 N. 16. 26 10 13 S. 17- 60 21 10 S. 18. 30 14 31 s. True alt. Latitude. True alt. Latitude. 75' 5'55"S- 43" 20' 59' N. 7S° 5'46'S. 43° 21' 8' N. 53 53 45 N. 9 46 21 S. 53 53 38 N. 9 46 14 N. 49 55 2 N. I 24 8 S. 49 55 I N. I 24 9^-; ji 46 49 N. II 13 33 ^• 51 46 39 N. II 13 23 N. 63 13 18 y. 16 13 43 N. 63 13 17 y. 16 13 44 N. 40 5 30 S. 7 55 '7 ^• 40 5 21 s. 7 55 8 S. 78 II 26 S. 26 20 43 N. 78 II 18 y. 26 20 51 N. 68 15 54 N. I 10 32 N. 68 15 49 N. I 10 27 N. 29 52 9 N. 43 51 37 S- 29 51 59 N. 43 51 47 S. 75 4 59 &• 47 32 11 S. 75 4 54 S. 47 32 6 8. 30 i6 28 S. 51 34 «9 N. 30 16 20 y. 51 34 27 N. 20 7 12 N. 25 r I S. 20 7 2 N. 25 I II S. 60 45 36 N. 20 40 32 S. 60 45 28 N. 20 40 40 y. 58 SS 9 N. 47 38 28 S. 58 SS 5 N. 47 38 32 8. 79 42 38 i^- 30 4 57 N. 79 42 29 y. 30 5 6 N. 68 42 51 S. 2 53 4 S- 68 42 48 y. 4 53 I 8. 9 52 32 S. 19 46 18 N. 9 52 24 S. 19 46 26 N. 70 3 15 N. 50 II 16 S. 70 3 14 N. 50 11 17 8. EXAMINATION PAPER— No. I, Pages 326—327. 1. Log. 5-861612 = nat. no. 727130. Log. 5-745622 = nut. no. 556700. 2. Log. 2-698971 =nat. no. 500. By Baper : log. 2-698970 =: 500; log. 2-041393 = nat. no. no. 3. True Courses.— S. 14° W., 15' dap. course; S. 28° W., 45'; N. 76° W., 49'; N. 48MV., 38'; N. 85»"W., 31'; S. 22°W., 36'; S. 54°W., 41'; S. 42° W., 8' current course. Diff. lat. ii'-'] S., dep. i83'"4 W. ; course 8. 67° W. ; dist. 199'. Lat. in 35° 45' N. ; mid. lat. 36° 24'; diff. long. 228'; long, in 12° 48' W. 4. Green, date, Jan. i^' 6*> 50™ 44^ ; red. decl. 22° 58' 1 1" S. ; true alt 60° 12' 47" ; latitude C 49' 2" N. By liaper : True alt. 60" 12' 39'. latitude 6° 49' 10' N. 5. Log. of diff. long. 2-048016 =: diff. long. iii'"j. 6. Diff. lat. 219' y. ; mer. diff. lat. 292' ; diff. long. 3So'W. ; log. tang, of course 10-114401 ; course S. 52"^ 27' 38" W.; log. of distance 2-555608 ; distance 359'-4. 7. Standard, Brest constant + 4'' 2""; g^ 19m a.m., and 9^ 39™ p.m. ,, Portsmouth „ — 2 41 ; 10 36 a.m., and 10 57 p.m. „ Devonport „ — o 46 ; 6 27 a.m., and 6 47 p.m. 8. Green, date, January i^ 5^ 23™ 24"; red. decl. 22° 58' 30* 8.; log. sine true amp. 9-788227. True amp. E. 37° 53' 7' S. Error of compass 21" o' t,"]" ^- Deviation 2° c,i' zf Vif . 9. Interval 31^, rate 6^1 gains. Interval 28'' 7^, ace. rate 2™ 58^-2, Green, date, Jan. 29** 6'' 53°' 49% red. decl. 17° 47' 45" 8., true alt. 13' 47' 28', hour-angle 3° 23' i", red. eq. time add 13™ 27»-5, mean time at ship 29'' 3'' 36'" 29'. Longitude 49° 20' W. Raper : True alt. 13° 47' 12" : hour-angle 3*^ 23" t,'. Longitude 49° 19' 30" W. 10. Green, date, Jan. 15'' 6^ n" 12', red. decl. 21° 2' 27' S., true alt. 55° 18*24", sum of logs. 19-722644, true azimuth N. 93° 12' 42' E. Error of compass 16^ i2'42"E. Deviation 8' 22' 42" E. By Eoper : True alt. 55" 18' 22", sine sq. of azimuth 9-722645, true azimuth N. 93" i2'42"E. Error of compass 16" 12' 42" E. Deviation 8" 22' 42'' E. 11. Time from noon 16"' 47', Green, date, January 16'' 14'' 18"^ 55', red. decl. 20° 46' 59'' 8., true alt. 33° 7' i", nat. no. 2026, nat. cos. mer. zen. dist. 548376, mer. zen. dist. 56' 44' 40" N. Latitude 35° 57' 41' N. Method If. — Raper : True alt. 33° 6' 58", reduction -|- 8' 19". Latitude 35° 57' 43' N. Toicson : Aug. I, 3' 5*, aug. decl. 20" 50' 4", index 30. Latitude 35° 57' 45' N. Aug. II, 5' 10", Latitude 35" 57' 45" N. 430 Answers. 12. ist Red. decl. 20° 54' 49" S. ; eq. T. + 10" z^'S ; true alt. fNorieJ 13° o' 53"; alt. 13° o' 53"; lat. 48° 50'; polar dist. 110° 54' 49"; hour-angle 2*" 26"" 13^, long. 8° 3' W. (A). Alt. 13° 0*53"; lat. 49*30'; polar dist. 110° 54' 49"; hour-angle 2'» 20™ I2^ Longitude 6° 32I' W. (A'). The line of bearing at the first observation trends N.E. | E. (easterlj-) and S.W. f W. (westerly). The position of the ship at the second observation is lat. 49" 42i' N., long. 7° 57' W. 13. Star's decl. 16° 16' 14" N., true alt. 52^^ 30' 36". latitude 53° 45' 38" N. Raper : True alt. 52° 30' 32". Latitude 53° 45' 42*' N. The Curve.— Correct magnetic bearing S. 79° W. Deviations.— ij°W.; 0°; 10° E.; 14° E.; 15° E.; 7°E.; 10° W.; 21° W. Compass courses.— N. 47° W. ; N. 27^° E. ; S. 79" E. ; S. 8^" W. Magnetic courses.— N. 41^° W. ; N. 73° E. ; S. 7° E. ; S. 67° W. Bearings, magnetic. — N. 84° E. ; N. 14° W. EXAMINATION PAPEE— No. II, Pages 329—331. 1. 7"6o232i = 40024074. Log. 6-297979 = nat. no. 1986000. 2. 2'i68747 = i47"485. Log. 5*000004 = nat. no. looooi. 3. True Courses.— N. 51° E., 18' dep. course; S. 73°E., 52'; S. 58° E., 43'; N. 57° E., 36'; N. 38"^ E., 27'; S. 21" E., 24'; S. 40° E., 29'; S. 39° E., 12', current course. Biff. ?«<. 39'-7 S.; «fe;;. i8i'*8 E. ; coe^r^e S. 78° E., 186'. Zfii^. m 46^51' N. ; mid. lat. 4.-]° 11' ; diff- long. (mid. lat. J 267^', or 4° 27 1' E. Longitude in 48° 5 J' W. 4. Green, date, Jan. 31'^ 1%'^ 47"" 4^, or long, in time 5'' 12™ 56^ ; red. decl. 17" 5' 56" S. ; true alt. 78° 17' 52". Latitude 5° 23' 48" S. By Eaper : True alt. 78° 17' 51". Latitude 5° 23' 47" S. 5. Log. of diff. long. 2*232899 = diff. long, i-ji'o. 6. Diff. lat. 2404' S. ; mer. diff. lat. 3104' ; diff. long. 3692' W. ; tang, course io"07534o; coiirse S. 49° 56' 42" W. ; log. of distance 3'i7237o, distance 3736'. 7. Standard, Filey Bay, constant + o^ 5^"" 5 ^o^ 49™ a.m., and 1 1'^ 32™ p.m. „ Milford Haven, ,, — o 16 ; no a.m., and o 7 p.m. „ Cromarty, ,, — 2 21 ; 6 34 a.m , and 7 18 p.m. 8. Green, date, February 19'^ 18'' 10™ 12^; rod. decl 10° 55' 36" S. ; sine 9-287272 ; true amp. W. 11° 10' 21" S. Hrror of con-qyass i ; ace. rate 8'"4i"7; Green, date March 30'i 22'' 53™ i2«-5 ; red. decl. 4° 12' 33" N. ; true alt. 29" 19' 56" ; hour-angle 3° 57' 4" . red. eq. time 0^4"^ 13'; mean time ship March 30'' 20*" 7'" 9". Longitude 41° 30' 52"-5 W. Saper : True alt. 29" 19' 45"; hour-angle 3*> 57™ 5*. Longitude 41° 31' 7''-5 W. 10. Green, date, March 9'' 9'> 43™ 5s ; red. decl. 4' 15' 37" S. ; true alt. 18° 6*51" ; sum of logs. 19-604572, true azimuth N. 78'' 44' 4" E. Correction 7° 21' 34" E. ; deviation 3° 28' 26" W. ir. Time from noon 10™ 14'; Green. date March 24'* 18^ 13™ 34'; red. decl. 1° 47' 40' N. ; true alt. 7 1° 20' 43" ; nat. no. 937 ; nat. cjs. mer. zen. dist. 948400 := 18^ 29' 10' N. Latitude 20° 16' 44" N. Method II. — liaper : True alt. 71= 20' 37" + 10' 17". Latitude 20° 16' 46" N. Towson : Aug. I, -\- o 7"; index no. 13. Aug. II, -|- 10' 14". Latitude 20° 16' 50" N. 12. ist red. decl. 2° 50' 28"-5 N. ; ist oq. T. -|- 5"" iS^-o; true alt. 36° 49' 14* ; lat. 51° 30' gives hour-angle i^ 42"^ 34'; long. iSo° 55' W. or 179° 5' E. (A); lat. 51° o' gives hour- angle ih47™ 245; long. 182° 7V W. or 177° 52^' E. (A'). 2nd red. del. 2° 56' 31" N. ; 2nd red. eq. T. -{- 5™ 13'; true alt. 15° 23' 34"; Ut. 51° 30' gives hour-angle 4*' 35™ i^; long. 179'' 14' "W. ; lat. 51' o' gives hour-angle 4^ 35™ 53'; long. 179° i' W. The line of bearing trends N.E. by E. and S.W. by W. ; the position of ship at the time of second observation, lat. 51° 39' N., long. 179° 18^' W. 13. Arcturus' decl. 19° 47' 33" N. ; true alt. 36° 7' 27". Latitude 34° 5' o" S. Raper : True alt. 36° 7' 22". Latitude 34" 5' 5" S. The Curve.— Correct magnetic bearing S. 70° E. Deviations.— 3° W. ; 20° E.; 25° E.; 230E.; 2° E. ; 24° W.; 25° W. ; 18MV. ' Compass courses.— N. 35' E. ; S. 82° "W. ; N. 85^° W. ; S. 49" E. Magnetic course.".— S.E. by S. ; N. 84^" E. ; N. 86° W. ; N.W. by N. Bearings, magnetic— N. 87I'' W. ; N. 78^' E. 432 Answers. EXAMINATION PAPEE— No. IV, Pages 333—334. 1. Log. 5-II9S63 = i3i78'4 Log. 4-861964 = 72771. 2. Log. 3'5i0784 = 3241-78 nearly. Log. 2-033424 ^ 108. 3. True Courses,— S., 19' dep. course; S. 59° W., 58'; N. 11° W., 15'; S. 28° E., 9'j S. 82° "W., 50'; N. 72° E., 12'; S. 58^ W., 22' ; S. 62° W., 42', current course. Diff. lat. 76'-8 S. ; de2). i42'-3 W. ; course S. 62° W. ; dlst. 162'. Lat. in 51° 29' S. ; diff. long. 225' W. Longitude in \%-^ 25' W., or 176° 35' E. 4. Green, date, April i-^ 5^ 50™ 48^; red. decl. 4° 42' 29" N. ; true alt. 48^ 55' 26". Lat. 45° 47' 3" N. Lla^per : True alt. 48' 55' 19". Latitude 45° 47' 11" N. 5. Log. of difif. long. 2-364667 = Diff. long. 231-6. 6. DifiF. lat. 325' N. ; mer. diflF. lat. 552' ; diff. long. 325' "W. ; tang, course 9-769944; course N. 30° 29' 17' W. ; log. dist, 2-576509 ; dist. 377'-2. 7. Standard, Brest -\- -i^^s^; no a.m., and o^iS™ p.m. ,, Hull + o I ; no a.m., and o 20 p.m. ,, Thurso — I 56 ; d^ 4™ A.M., and o 46 p.m. „ Pembroke +0 4 ; i r 42 a.m., and no p.m. „ Weston-super-mare 4" o 2 ; no a.m., and o 14 p.m. „ Waterford + o 44 ; 11 43 A.M., and no p.m. 8. Green, date, April 27^ 23'' 18"" 50^ red. decl. 14" 11' 48" N. ; log. sine 9-494965 ; true amplitude W. 18° 12' 54* N. Error 15° 32' 6" W..; deviation 3° 37' 54" E. 9. Interval 1 2 days ; rate ii»-3 ^o««^; interval 70'! 22''; ace. rate 13™ 21^-2 ; Green, date, April i4<* 21^ 59™ 3^; red. decl. 9° 48' o" N. ; red. eq. time stibt, o™ o^; true alt. 26° 37' 25"; hour-angle 2^ 59" 9'. Longitude 75° i' 30' E. Raper : True alt. 26" 37*20"; sine sq. 9-161814; hour-angle 2^ 59m 93. Zoh^. 75° i' 30" E. 10. Green, date, April 17'* 2^ 43'" 25^; red. decl. 10° 34' 40' N. ; trae alt. 42° 20' 39''; sum of logs. i9'449652 ; true azimuth S. 64° 6' 18" "W. Error 25° 53' 42" W. ; deviation 6" 3' 42" W. Maper : True alt. 42° 20' 34"; sine sq. 9-449680; true azimuth S. 64° 6' 15" W. Error of compass 25° 53' 42" W. ; deviation 6° 3' 42" W. 11. Time from noon 28™ 38'; Green, date, April i8 38™ 34'' ; red. decl. 1 1" 3'23"N. ; true alt. 54° 20' 16"; nat. no. 5291 ; nat. cos. mer. zcn. dist. 817759 = 35° 8' 20' N. Lat. 46° ii'43"K Method II. — Raper : True alt. 54° 20' 12" ; -|- 31' 33". Latitude 46° 11' 38" N. Towson : Aug. I, -\- 5' 10" ; index no. 76. Aug. II, + 36' 28". Latitude 46° 1 1' 49" N. 12. ist red. decl. 6° 32' 26" N. ; eq. T. -f 2™ 24'; true alt. 45° 18' 15"; lat. 50° 20' givca hour-angle o^ 42™ 45*; long. 7° 34' W. (A) ; lat. 50"' 50' gives hour-angle o'> 28™ 45s ; long. 11° 34' W. (A'). 2nd red. decl. 6" 35' 22' K. ; eq. T. + 2"" 2is-6 ; true alt. 25° 2' 12''; lat. 50° 20' gives hour-angle 3!^ 52™ 308; long. 7° 8' W. ; lat. 50° 50' gives hour-angle 3'' 51™ 15*; long. 7° 265' W. The line of bearing at first observation trends E. by S. and W. by N. ; the position of ship at the time of second observation is lat. 50° 59' N., long. 7° 32' W. 13. Spica's decl. 10° 32' 59" S. Latitude 58° 36' 21" N. Raper : True alt. 20° 50' 30". Latitude 58' 36' 31" l!T. The Curve. — Correct magnetic bearing S. 2° E. Deviations.— 4° W. ; 7° W. ; 12° W.; iS^-W. ; 3° E. ; 22° E.; 15° E.; 1° E. Compass courses.— N. 88" E. ; N. 59° W. ; S. 14° W. ; S. 19" E. Magnetic courses.— N. 56° W.; N. 87^° W. ; East; N.E. Bearings, magnetic— N. 67' W. ; S. 84^° AV. EXAMINATION PAPEE— No. V, Fat/e 335—336. 1. 4-838071 = nat. no. 68876-5. 2. 2-826661 = nat. no. 670-905. >> Liverpool >> — I )> Sunderland )> — I 4 »> Brest >i + 218 1) Dover !> — 27 Green. date, May 20'' i6'> 6"' ' 24'; red. Answers. 433 3. True Courses.— S. 6j» E., 23' dep. course ; S. 6° E., 56' ; S. 14" W., 17' ; N. 80° "W., 13'; E., 37'; S. 44° TV., 31'; N. 26° E., 17'; N. &5° E., 48' current course. Difif. lat. 82'-4 vS., dep. 8o'-6 E. ; course S. 44° E., dist. 115'. Zat. in 65° 24' S., mid. lat. 64° 43', diff. Ion J. 189' E. Long, in 17 8' 30' W. 4. Green, date, May S*" 7'' i™ 8'; red. docl. 17° 12' 48' N.; true alt. 76° 14' 11". Lat. 3° 26' 59" N. Raper : True alt. 76' 14' 3'. Latitude 3° 26' 50' N. 5. Log. of diCF. long. 2-992876 = I>(f. long. 983-7. 6. DiflF. lat. 732' S. ; mer. difF. lat. 881'; difiF. long. 1098' W.; log. tang. 10-095626; course S. 51° 15' 27" \V. ; log. of distance 3-068060; dintancc 1169-6. 7. Standard, Greenock constant — o''56'"; 6''58™ a.m., and 7''28'"r.M. 722 A.M., and 751 P.M. 10 38 A.M., and II 5 P.M. I 59 A.M., and 2 30 P.M. 6 28 A.M., and 6 59 p.m. red. decl. 20° 9' 18" N. ; log. sine 9-694581 ; true amp. E. 29° 40' 5" N. Error of compass 12" 31' 10" E. ; deviation 19° 18' 50" W. 9. Interval 36''; rate 4^-9 gaining; interval ^o^ 21''; ace. rate — 4" 9'; Green, date, May 2 1"* 20'' 59™ 16'; red. dec]. 20^ 23*48'' N. ; red. eq. T. se<5<. 3™ 35^-6 ; true alt. 32"* 19' 31"; sum of logs. 9-452117; hour-angle 4'» 17™ 13'; mean time at ship May 21'^ 19'' 39™ ii». Longitude 20° i' 15" W. Eaper : True alt. 32* 19' 21"; hour-angle 4'' 17™ 14". Longitude 20° i' 30* W. 10. Green, date, May 2^^ g*" 53™ 51'; rod. decl. 21° 3' 26" N. ; true alt. 40° 54*26''; sum of logs. 19-603544; true azimuth S. 81° 57*40" W. or N. 98' 2' 20" W. Error 20° 27'4o''E.; deviation 9" 57' 40" E. Eaper: True alt. 40° 54' 22"; sine sq. 9-633554; true azimuth S. 81° 57' 44" W. or N. 98° 2' 16'' W. Error 20' 27' 40" E. ; deviation 8° 57' 40" E. ir. Time from noon 25™ 25'; Green, date lo** 7'' 54™ 25"; red. decl. 17° 45' 6" N. ; true alt. 43' 39' 55" ; nat no. 5144 ; mer. zen. dist. 45° 55' 34" S. Latitude 28° 10' 28' S. Method II. — Raper : True alt. 43° 39' 49" ; 4" 24' 39". Latitude 28° 10' 26" S. Towson : Aug. I, + 6' 16'; index 61. Aug. II, -j- 18' 17". Latitude 28" 10' 26' S. 12. ist red. decl. 21° 46' 17" N.; istred.eq.T. — 2'"46'*-3; ist true alt. 23° 52' 16" ; lat. 50° o' gives hour-angle 5'> 13™ 24'; long. 9° 21.5' W. (A). Lat. 50' 30' gives liour-angle ^hjjm^jsj long. 9' 26' W. (A'). 2nd red. decl. 21° 48' 46" N. ; 2nd red. eq. T. ««5<. 2™ 448 ; 2nd true alt. 56° ^6' 35". Lat. 50" o' giv3s hour-an^le i'' 27™ 50'; long. 10° 29I' W. (B). Lat. 50° 30' gives hour-angle !*> 23™ 54'; long. 10° 29I' W. (B')- The line of bearing at the first observation trends N. \ W. and S. \ E. The position of the ship at the second observa- tion is lat. 50" 5' N., long. 10° 40^' W. 13. Star's decl. 10° 33' o" S. Latitude 30' 26' 44" S. Raper : True alt. 70' 6' 16'. Latitude 30' 26' 44'' S. The Curve.— Correct magnetrc bearing N. 87' E. Deviations.— 3= W. ; 15^ W.; 23° W. ; 22' W. ; 2°E.; 24° E. ; 230 E. ; 13° E. Compass courses.- N. 45'-' E. ; S. 71^° W. ; S. 49° E. ; S. 39I' W. Magnetic courses.— N. 20V W. ; N. 46° E. ; S. 76^-° E. ; S. 6i^ W. Bearings, magnetic. — S. 8" E. ; S. 85' W. EXAMINATION PAPER— No. VI, Pages 336—338. 1. 0-6221 10 = nat. no. 4-189. (The product). 2. 4-021468 = nat. no. 10506-77. (The quotient). 3. True Courses.— S. 73° E., 23 dep. course; S. 75°E., 17'; S. 33°E., 22'; S. 1° W., 24'; S. 6i^E., 25'; N. 69' W., 18'; S. 43° W., 17'; N. 11° E., 16' current course. DiflF. lat. 55'-9 S., dep. 46'-6 E.; course S. 40° E., dist. 73'. Lat. in 55^ 16' N. ; diff. long. (Mercator) 82'-9 E. ; long, in 134'' 17' W, Diff. long, (by mid. lat.) 83' E. ; long, in 179" 57' W. KICK 434 Answers. 4. Green, date, May 31'^ 17'' 34™ 52=; red. decl. 22" 6' 44" N. ; true alt. 75° 49' 26". Latitude 7° 56' 9" N". 5. Log. of diff. long. 2*487692 = Biff. long. 307'-4. 6. Diff. lat. 2i8i'S. ; mer. diff. lat. 2301'; diff. long. 3038' W. ; tangent course 10-120671 ; coune S. 52° 51' 34" W. ; log. distance 3-880471 ; distance 36 12'- 3. 7. Standard, Dover -r4''33""'; 11'' 46"" a.m., no p.m. „ Harwich — o 33 ; 7 39 a.m., S^^ 12™ p.m. „ Brest — o 47 ; corr. for long. + 7™; 3^ 9™ a.m., -^ 28"^ p.m. 8. Green, date, June 21'* \r^ i\^ 48^; red. decl. 23° 27' ii"N. ; sine of true amplitude 9-898948 ; true amplitude W. 52° 24' 39" N. Error 46° i' 36" "W. ; deviation 6° 28' 24" E. 9. Interval 29'^, daily rate 8^-3 losing, interval 44"^ 22'% accumulated rate 6"^ 13% Green, date June 13'* 21'^ 45™ 12^, red. decl. 23° 16' 38" N., true alt. 28° 49' 40", red. eq. time o™ 58 suit., hour-angle 3^ 49*" 7^ M.T. ship June 13'' 27^1 49™ 2». Longitude 90° 57' 30" E. By Raper : True alt. 31° 18' 45', log. sine sq. 9-813214, true azimuth S. 107° 30' 42" E. Deviation 17° 9' 18" E. 10. Green, date, June 7'^ 22'^ o"" 12s; red. decl. 22° 51' 39" N. ; true alt. 31° 18' 54"; logs. 19-813275; true azimuth S. 107° 30*42" E. Error 2° 29' 18" E.; deviation 17' 9' 18" E. By Raper : True alt. 28" 49' 36" ; hour-angle 3'> 49™ 8^. Longitude 90° 57' 45" E. ir. Time from noon 37™ 26s Green, date, June 4'^ 14^ 16™ 2=; red. decl. 22° 3i'36"N.; true alt. 50° 3' 22"; nat. no. 5778 ; mer. zen. dist. 39° 25' 32" N. Latitude 61° 57' 8" N. Method II. +31' 18". True alt. (Norie) 50' 3' 22"; latitude 61° 56' 58" N. True alt. (Raper) 50° 3' 18''; latitude 61° 57' 2" N. Towson : Beyond the limits of the Table. 12. ist red. decl. 23° 12' 34" N. ; ist red. eq. T. + 3™ 145-3 ; ist true alt. 62° 58' 35" ; lat. 49° 40' gives hour-angle o^ 28™ 3^; long. 177° 45!' E. (A). Lat. 50° o' gives hour-angle oh igm 68 J long. 180° 15' E. or 179° 45' W. (A'). 2nd red. decl. 23° 11' 49" N. ; 2nd eq. T. _|_2mi^sj 2nd true alt. 30° 13' 27" ; lat. 49° 40' gives hour-angle 4*1 40™ 9^ ; long. 180° 15^' E. or i79°44|'W. (B). Lat. 50°©' gives hour-angle 4"^ 40™ 11^; long. 180° 16' E. or 179° 44' W. (B'). The line of bearing at the first observation trends E. by N. (northerly) and W. by S. (southerly). The position of the ship at the second observation is lat. 50° 6' N., long. 179° 44' W. 13. Star's decl. 28° 18' 29" N. ; true alt. {Norie) 48° 33' 45". Latitude ly 7' 46" S. Raper : True alt. 48° 33' 40". Latitude 13° 7' 51" S. The Curve. — Correct magnetic bearing S. 79° W. Deviations.— 12° W.; 0°; 17° E.; 21° E. ; 12^ E.; 4" E. ; 18° W.; 24° W. Compass courses.— N.W. by N. ; S. 70° W.; N. 27 J° E. ; S. i|" E. Magnetic courses.— N.W. by W. ; S. 610 W. ; S. 17° W. ; S. 40^° E. Correct magnetic bearing.— N. 11° E.; N. 87I0W. EXAMINATION PAPEE-No. VII, Pages 339—340. 1. 4-141829 =: 13862-1 ; 5-805908 = 639600. 2. 2-537819 = 345-0; 1-643452 ^44-0. 3. TrueCourses.— S. 25°E., 17'; N. 73° E., 22'; S. 79°E., 24'; N. 80° W., 26' ; N. 85°E., 17'; N. 58" E. 40'; S. 68" E., 51'; N. 62° W., 9' current course. Biff. lat. i'-3 S., dep. ii6'-4 E., course S. 872" E., dist. 116'. Lat. in 51° 24' N., mid. lat. 51° 24', long. 187° E. iow^. w 182° 3' E., or 177° 57' W. 4. Green, date, July 26l^'"]. 6. Diff. lat. 421' N. ; mer. diff. lat. 590' ; diff. long. 249' E. ; tangent 9625347 ; course N. 22° 52' 53" E. ; log. dist. 2-659876 ; distance 457' nearly. 7. Standard, Brest — o''i5"'; no a.m. and o*" 5™ p.m. „ Brest — I 51 ; io''29'" a.m. aud 11 5 p.m. „ Dover -H 5 13 ; no a.m. and o 7 p.m. „ Devonport — 113; no a.m. and o 16 p.m. 4.36 Answers. 8. Green, date, Aug. -2.0^ iV^ lo"! 24'; red. decl. 12° 15' 18" N.; sine 9-456381 ; true amp. W. 16° 37' 8" N. Correction 27° 52' 8" E. ; deviation 8° 37' 8" E. 9. Interval 7'^, rate 6^-6 losing, interval 16^ 20^'', accumulated rate -|- i" 5i'> Green, date, Aug. 6'i6™p.m. „ Sunderland — i 14 ; no a.m., and o 22 p.m. „ Leith — I 17 ; 11'^20'nA.M., and 11 44 p.m. „ Greenock -}- 4 47 ; 2 8 a.m., and 2 27 p.m. 8. Green, date, September 30* 6'' 33™ 408; red. decl. 2° 58' 4" S. ; sine 8-929667 ; true amp. W. 4° 52' 44" S. Error 41° 26' 29' W. ; deviation 10° 58' 29" W. 9. Interval i5<^, rate 2''-6 gaining, interval 19'' isl^, accumulated rate — o™ 51% Green, date August 31"^ 15'^ 35"" 8% red. decl. 8° 21' 44" N., red. eq. T. C" i« suM., true alt. 62° 25' 7", hour-angle i'' 51'" 36^, IM.T. ship August 31'' 25^ 51™ 23'. Long. 154° 6' 45" E. Raper : True alt. 62'^ 24' 56" ; hour-angle i'' 51"" 37'. Longitude 154° 7' o' E. Answers. 437 10. Green, date, September iG^ i^ 29™ 54' ; red. decl. i" 33' 37" N. ; true alt. 29° 41' 59" ; 8um of lo'?3. 19-702438; true azimuth S. 90'' 27' 32" E. Error 3' 47' 32" W. Deviation 12*^ 7' 32" W. liaper : True alt. 29' 41' 53"; sine sq. 9-702440; true azimuth S. 90*^ 27' 34' E. Error 3° 47' 34' W. : deviation 12° 7' 34" W. 11. Time from nooa i6™4i», Green, date, September 22 >> 12'' 27'" 55', red. decl.o°3' i3''S., true air. 62" 9' 19", nat. no. 2348, mer. zon. dist. 27° 33' 19" S. LalHudc 27° 30' 36' S. Method II. -|" '7* 26". Latitude 27° 30' 2'' S. liaper : True alt. 62° 9' 11". Latitude 27° 30' 11''. Toicson : Auj^. I, 4- o, index 35. Aug. II, 4- '7' 29". Latitude 27° 29' 59" S. 12. I at red. decl. 5^23' I "N.; i3t red. eq. T. — 2™ 46'; isttruealt. ii°9'42"; lat.47''5o' gives hour-an2;Ie 5'' 17"" 13'; long. 34° 2^' E. (A). Lat. 47° 10' gives hour-angle 5*" 17™ 30'; long. 33° jSy E. (A') ; 2nd red. decl. 5° 18' 59" N. ; 2nd red. eq. T. — 2"^ 43^'i8 ; 2ud true alt. 45' 56' 32'; lat. 47° 50' hour-angle o^ 5;™ 57'; long. 35'' 58^' E. (B). Lat. 47' 10' gives hour-angle i*» 4™ 2^; long. 33° 27!^' E. (B'). The lino of bearing at first observation trends N. ^ E. and S. 5 W. Tnc position of tho ship at tho second observation is lat. 47^^ 26' N., long. 34° 28' W. 13. Star's decl. 19" 47' 49" N., true alt. 86' 31' 18". Latitude 16° 19' 7' N. (XorieJ. „ „ 86 31 17 „ 16 19 5 N. CRaperJ. The Curve.— Correct magnetic bearing S. 88" "W. Deviations.— 2" E., 20" E., 22° E., 9' E., 4' W., 12° W., 20' W., 17° W. Compass courses. — "West, N. 2° E., N. 595" E., S. 612" E. Magnetic courses. — N. 2° E., S. 17" AV., S. 57^° E., N. 72° E. Bearings, magnetic. — S. 84^° E., N. 76^" W. EXAMINATION PAPER -No. X, Pages 345—346. 1. 7-447213 = 28003548; 3-524617 = 3346-7. 2. 1-936010 = 8630; 2-735031=543-288. 3. True Courses.— S. 75"" E., 36' dep. course; S. 8" E., 43'; S. 15" W., 19'; N. 69° E., 50'; N. 77° E., 21'; S. 5°E., 6'; S. 38" W., 3'; S. 5i°E., 22'; N. 88° E., 48' current course. Biff. lat. 6y% S. ; d-^p. iSV-j E. ; coiirse S. 68^' E. ; did. 181'. Lat. in 58° 43' N. ; mid. lat. 59' 16'; diff, long. 330' E. Long, in 38° 24' W. 4. Green, date, Oct. 20^ io'> i™ 40', red. decl. 10° 33' 31" S., true alt. 50" 11' 14". Lat. 50° 22' 17" S. Raper : True alt. 50° 11' 9'. Latitude 50' 22' 22" S. 5. Log. of diff. long. 2-019277 =: Biff. long. i04'-5. 6. Diff. lat. 140' N., mor. difiF. lat. 142', diff. long. 46' E., tang, course 9'5io47o; course S. i7''57'E. ; log. dist. 2-167799; '^^^^- 147' '• Standard, Brest — 2^ o« „ London — 2 19 „ Leith — 2 55 ,, Sunderland — 14 ii'>54'" A.M. and no p.m. 9 44 A.M. and 10 6 p.m. 9 34 A.M. and 9 54 P.M. no A.M. and 022 p.m. 8. Green, date, Oct. 8"* ii^ 13'" 48", rod. decl. 6' 7' 40' S., true amp. E. 6' 28' 21' S. Error 9' 17' 6" E. ; deviation 11° 7' 6' E. 9. Interval 7'^, rate 2'-4 /o.swy, interval 22^ i2'>, accumulated rate o™ 54", Green, date, October t,o^ 12'' o™ 22*, red. decl. 14° i' o' S., true alt. 28^ 42' 5S', hour-angle 2'' 45"^ 51', red. eq. T. 16™ i6« suht., M.T. ship Oct. 30^ 2'> 29™ 35'. Longitude 142° 41' 45" W. Raper : True alt. 28^ 42' 50" ; hour-angle 2*" 45"^ 52'. Longitude 142° 41' 30" W. 10. Green, date, Sept. 3o<* 18'' 43™ 48'; red. decl. 3° 10' 4" S. ; true alt. 14° 7*4'; sum of logs. 19-691122; true azimuth N. 88"' 58' 26' W. ^/->-or4^ 35'56' W. ; deviation 12° 15*56' W- Raj)er : True alt. 14" 6' 51 " ; sine sq. 9-691132 ; true a-^cimuth N. 88' 58' 30' W. Deviation 12" 16' O" W. 438 Answers. 11. Time from noon 17" 8^; Green, date, Oct. z"* i^ 1711 8'; red. decl. 3° 39' 32" S. ; true alt. 47° 39' 40"; nat. no. 2190; mer. zen. dist. 42° 9' 9' N. Zat. 38° 29' 31" N. Method 11.— Red. + 11' 15": lat. 38" 29' 31'' N. ('NorieJ. 38° 29' 38" N. (Raper) . Towson : Aug. I, -\- 35" index 36. Aug. II, -\- 10' 24". Latitude 38° 29' 49" N. 12. istred. decl. 3° 15' 18" S.; ist red. eq. T. — lo'^ii^-^; ist true alt. 23° 29' 11"; lat. 50° 30' ; hour-angle 3^ 3™ 24a; long. 50° 40J' W. (A). Lat. 51° o' gives hour-angle jh o"-. ^o»; long. 50° if W. (A'). 2nd red. decl. 3° 20' 55" S. ; 2nd red. eq. T. — 10™ 26' ; true alt. 25° 39' 34"; lat. 50° 30' gives hour-angle 2'> 44'« 39^; long. 49° 54I' W. (B). Lat. 51° o' gives hour-angle 2^ 41"" 30'; long. 50° 40' "W. (B'). The line of bearing at first observation trends N.E. ^ N. and S.W. \ S. The position of the ship at the second observa- tion is lat. 51° 7' N., long. 50° 51' W. 13. Star's decl. 14° 34' 46" N., true alt. 54" 6' 7". Latitude 50° 28' 39" N, (Norie) . „ „ „ 54 6 3 „ 50 28 43 N. (RaperJ. The Curve. — Correct magnetic bearing S. 38^ W. Deviations.— 1° E., 17^ W., 22" W., 19° W., 3^ W., iS^'E., 24° E., 19^ E. Compass courses.— S.S.E. | E., E. by N. i N., S. 9" W., S. 72- E. Magnetic courses.— S. 45^° E., S. 10° "W., N. 49° E., North. Bearings, magnetic. — N. 85^^ W., N. 89^ W. EXAMINATION PAPER— No. XI, Pages 347—348. I. 2-645548 = 442-128 ; 5-639859 = 436374- 3. 1-999489 = 99-88; 3-459392 = 2880. 3. True Courses.— S. 42° W., 16' dep. course; N. 14" E., 17'; S. 45' E., 19'; S. 87° W., 31'; S. 63" E., 17'; S. 5" E., 19' ; S. 56' W., 25'; S. 88° E., 22' current course. Diff. lat. 5i'-8 S. ; de}}. 6'-i W. ; course S. 7° W. ; dist. 52'. Lat. in 51° 8' N.; diff. long. 10'. Long. in 119° 50' E. 4. Green, date, November 14'^ 18'^ 39" 16' ; red. decl. 18° 29' 19" S. ; true alt. 67° 57' 49". Latitude 40° 31' 30" S. 5. Log. of diff. long. 2-294311 = diff. long. i96'-9. 6. DiflF. lat. 1928' N. ; mer. diff. lat. 2383'; diff. long. 4290' E. ; tang. 10-255333 ; course N. 60° 56' 56" E. ; log. of distance := 3'598938 ; distance 3971'. 7. Standard, Dover constant + d^i^)"*; 7''42"^a.m., and 8hi4rap.M. ,, Devonport „ — o 17 ; 1 6 a.m., and i 40 p.m. „ Galway „ + ° 7 5 ° 53 ^^•^^•> ^"^^ ^ ^^ '^■^• „ Brest „ — o 7 ; no a.m., and o 13 p.m. 8. Green, date, Nov. g"! i2'» 20'"44S; red. decl, 17° 4' 14" S. ; true amp. E. 33° 52' 16" S.; Correction 42° 3' 59' W. ; deviation 58^ 33' 59' W. 9. Interval 12=^, rate i«-2 losing, interval 36"^ 3'', accumulated rate 43'"2, Green, date Nov. 3od 2^ 48"" 52% red. decl. 21° 42' 33" S, red. eq. T. 11™ 5^-5 suM., true alt. 39° 47' 8", hour-angle 3'' 42™'7% M.T. ship Nov. 29'^ 2011 6™ 47^ Longitude 100° 31' 15" "W. Ra2}er : True alt. 39° 47' 6" ; hour-angle 3'' 42"! 7*. Longitude 100° 31' 7''-5 W. 10. Green, date, Nov. 15'! lo^ 46™ 27s; red. decl. 18° 39' 44"; true alt. 43° 55' 7' ; sum oflogs. 19-514224; true azimuth N. 69° 43' 40" W., or S. 110° 2' 26' W. J'r?-or n" 46' 20" E. Deviation 19° 36' 20'' E. Raper : True alt. 43° 55' 2"; sine sq. 9-514239 ; true azimuth N. 69° 43' 43" W. Deviation 19" 36' 17" E. II. Time from noon 39" 26'; Green, date, Nov. 13'' 2'i 36"" 2«; red. decl. 18" 3' 15" S. ; true alt. 56° 11' 50"; nat. no. 8861 ; mer. zen. dist. 32° 52' 46" S. Latitude 50° 56' i" S. Method II.— Red. + 56' 13", true alt. 56° 11' 50", latitude 50° 55' i2"S., Norie. True alt. 56° 11' 46", latitude jo° 55' i6" S., Ra2}er. Towson : Hour-angle exceeds limits of Table. Amrvers. 439 12. ist red. decl. 14° 29' 29" S. ; ist red. eq. T. — i6™i8»-53; ist true alt. 19° 17' 53"; lat. 50° 50' gives hour-anglo 2'' 3™ 45^ ; long. 24° 48^' W. (A). Lat. 50° 20' gives hour-anglo jh gm ^.2»; long. 26^ 2.f' W. (A'). 2nd red. decl. 14' 33' 41" S. ; 2nd red. eq. T. — i6"i 19'; 2nd true alt. 12° 46' 17'; lat. 50' 50' gives hour-angle 3'> 8"^ 28^; long. 25° 20' W. (B). Lat. 50' 20' gives hour-angle 3*' ii"" 24=*; long. 24° 36' W. (B'). The lino of bearing at the first observation trends N.E. by E. (easterly) and S.W. by W. (westerly). The position of the ship at the second observation is lat. 50" 36^' N., long. 25" o^' W. 13. Star's decl. 30° 14' 33" S., true alt. 59° 36' 2", latitude 60° 38' 23" S., Norie. True alt. 59° 35' 5^"> l(ttitude 60' 38' 35" S., Jiapo: The Curve. — Correct magnetic bearing S. 89° W. Deviations.— I ° E. ; 19"' E.; 21° E.; 9° E. ; 3° W. ; ri°W. ; 19" W.; 16° W. Compass courses.— N. 85° W. ; N. 45^" W. ; N. 86° E. ; South. Magnetic courses.- S. 77° W. ; N. 58' W. ; S. 49° E. ; N. 56" E. Bearings, magnetic. — S. 76" W. ; S. 56° E. EXAMINATION PAPEE— No. XII, Pages 349—350. 1. 6010519 = i0245i6"5 ; 3-367405 = -00233026. 2. 1-168317 = 14-7339; 2-096910=125. 3. True Courses.— S. 45° W., 16' dep. course; N. 73° E., 15'; S. 49° W., 23'; S. 43° E., 15'; N. 47° W., 24'; N. 80° E., 22'; S. 53° W., 12'; S. 14° E., 6', current course. Dif. lat. 25'-8 S. ; dep. 8'-2 W. ; course S. 18° W. ; dist. 27'. Lat. in 49° 34' N. ; dij'. long. 13' W. Longitude in 40"^ 13' W. 4. Green, date, Dec. 31'^ S^ 15'"; rod. decl. 23° 4' i" S. ; true alt. 67° 20' 49"; latitude 0° 24' 50" S. By liaper : True alt. 67° 20' 46". latitude 0° 24' 47' S. 5. Log. of diff. long. 2-346353 =: diff. long. 222'. 6. Diff. lat. 4766' S. ; mer. diff. lat. 5205' ; diff. long. 3735' W. ; tang, course 9-855870 ; course S. 35" 39' 45" W. ; log. of distance 3-768350 ; distance 5866'. 7. Standard, Queenstown, constant — o'' 59" „ Sligo ,, Queenstown „ Galway „ Londonderry „ Pembroke 8. Green, date, Dec. 28*1 4'' 10™ 49'; red. decl. 23° 15' 10" S. ; true amp. E. 35° 12^' S. Correction 9° 53V E. ; deviation 5° 36^' W. 9. Interval 12'^, rate 5'-7 losing; interval 42'' 8'', ace. rate 4™ o''-5. Green, date, Dec. 24'! %^ iS" 44% red. decl. 23° 24' 46" S., red. eq. time add o^ i", true alt. 40° 54' 8", hour- angle 3° 41' 17". Longitude 180" o' o" W. 10. Green, date, Dec. s-i^ 4'' 40™ io», red. decl. 23° 19' 18" S., true alt. 20° 27' 7", sum of logs. 19-362516, true azimuth S. 57° 22' 24" E. Error of compass 12° 56' 21" E. Deviation 5' 36' 21" E. liaper : True alt. 20" 26' 54" ; log. sine sq. 9-362629 = S. 57° 22' 53" E. Hrror of compass 12° 55' 52" E. Deviation 5° 35' 52" E. 11. Time from noon 30'" 40% Green, date, Doc. 4'' i'' 30'", red. decl. 22° 17' 12" S., true alt. 60° 7' 18', nat. no. 5104, zen. dist. 29° 17' 12* S. Latitude 51" 34' 24" S, Method II. -\- 35' 31", true alt. 60° 7' 18", latitudt 51° 34' 23' S., Norie. True alt. 60° 7' 14", latitude 51° 34' 28" S., Raper. Towson : Exceeds limits of Table. 12. ist Green, date, Dec. 24'' 22'' 11™ 2^; decl. 23° 24' 15" S. ; eq. T. -{■ o'" i8«-5i ; true alt. 9° 14' 59"; lat. 46° 30' gives hour-angle 2'> 58"^ 53'; long. 17° 24' W. (A) ; lat. 46° o' gives hour-angle 3'> 2" ii^; long. 18° 13 J' W. (A'). 2nd Green, date, Dec. 25J 3'' 15'" [not x'' 39"!^ ; decl. 23" 23' 54' S. j eq. T. -J- o™ 225-78 j true alt. 15° 12' 14"; lat. 46° 30' gives _o'' 59'"; oh 6"' A.M., , and oh 43" ■ P.M. — 21 ; ; I 30 A.M.; , and 2 3 P.M. — I 19 ; no A.M., , and 23 P.M. + 1 35 ; 2 38 A.M., , and 3 10 P.M. — I 37 ; 3 6 A.M., 1 and 3 32 P.M. — 30 ; I 37 A.M. , and 2 ^^5 P.M. 44© Answers. hour-angle i^ 57'" 39' ; long. 19° 14^' "W. (B) ; lat. 46° o' gives hour-angle 2^ 3™ o" ; long. 17° 54I' W. (B'). The line of bearing at first observation trends N.E. by E. (easterly) and S.W. by W. (westerly). The position of ship at the time of second observation is latitude 45° 50' N., longitude 17' 28' W. 13. Star's decl. s" 31' ^5" N., true alt. 52° 45' 55", latitude 31° 42' 50" S., None. True alt. 52° 45' 52", latitude 31° 42' 53" S., Eaper. The Curve, — Correct magnetic bearing S. 2° E. Deviations.-4° W. ; 6^= W. ; iz'W.; i8°W.; 3° E. ; 22° E.; 14° E. ; 1° E. Compass courses.— East ; S. 50^ E. ; S. 40' W. ; N. 4° W. Magnetic courses.— S. 33° E. ; N. 88|° W. ; S. 30° W. ; S.E. by E. Bearings, magnetic. — N. 86|° "W. ; N. 2° "W. EXAMINATION PAFEE— No. XIII, Pages 351—352. 1. 6-110455 = 1289598. 1-302556 = 20-0704. 2. 2-806180 = 640. 0-060635 = 1-14983. 3. True Courses.— S. 14° "W., 23' dep. course; S. 23°"W.,5i'; N. 87° W., 45'; S. 44° E., 22'; N. 78"^ AV., 18'; N. 14° W., 44'; S. 3°E., 12'; "West 52', current course. Biff.lat. 48'-2 S.; ffe^J. i34''7 W. ; course S. 70!" W., dist. 143'. Lat. in 61° 30' N. ; diff. long. 286' W. Longitude in 78^ 31' E. 4. Green, date, Aug. lo'^ 17'' 51™ 12=; red. decl. 15° i8'58"N.; true alt. 42° 50' 17". Latitude 31° 50' 45" S. By Raper : True alt. 42° 50' 7". Latitude 31° 50' 55" S. 5. Log. of diff. long. 2-663420 =: diff. long. a,6o'-']. 6. Diff. lat. 1890' S. ; mer. diff. lat. 2392' ; diff. long. i774'-43 W. ; co? 22™ 6^; red. decl. 4° 32' 35" S. ; true alt. 28° 33' 55" ; sum of logs. 19-596707, true azimuth N. 77" 53' 22" E. Error 21° 46' 38" W. ; deviation 4° 36' 38" W. 11. Time from noon 25™ 52^; Green, date July 28'' 2'' 10™ %«; red. decl. 18' 56' 42" N. ; true alt. 69° 22' 19' ; nat. no. 4683 ; mer. zen. dist. 19° 51' 6" N. Latitude 38^ 47' 48" N. Method II. + 46' 59"; true alt. 69° 22' 19'. Latitude 38° 41' 25" N. Norie. Towson : Exceeds limits of Table. 12. ist red. decl. 7° 44' 36" S. ; ist red. eq. T. 4- 12™ 38^-4; ist true alt. 27° 41' 12* j lat. 47° 10' gives hour-angle 2^ 12™ 59^; long. 166'' 8 J' W. (A). Lat. 47° 40' gives hour- angle 2'' 9™ 5s ; long. 165° 9I' W. (A'). 2nd red. decl. 7° 41' 25"-5 ; eq. T. -f 12™ 36^; true alt. 32' 50' 32"; lat. 47° 10' gives hour-angle i^ 12™ 21^; long. 165° 23' W. (B). Lat. 47° 40' gives hour-angle i^ 4™ 2i«; long. 167° 23' W. (B'). The line of bearing at the first observation trends N.E. | E. and S.W. f W. The position of the ship at the second observation is lat. 47° 18^' N., long. 165° 57^' W. 13. Star's decl. 47° 31' 39" S. ; true alt. 49° 54' 3". Latitude 7° 25' 42" S. Norie. „ „ .. 49 53 54 ,, 7 25 33 S. Eaper. The Curve. — Correct magnetic bearing S. 1 3° W. Deviations.— 1 2MV.; 8° W. ; 8°W.; 3° W. ; 12° E.; 20° E. ; 7°E.; 8^ W. Compass courses.— S. 36° W. ; N. 75° E. ; S. 4° W. ; N. 31" E. Magnetic courses.— N. 49° E. ; N. 47° W. ; N. 5° W. ; S. 2^° E. Bearings, magnetic— N. 75° E. ; S. 75° W, 11^53'" A.M., and no p.m. II 13 A.M., and ii'>47ni p.m. 4 A.M., and 37 P.M. II 18 A.M., and II 47 P.M. Answers. 441 EXAMINATION PAPER— No. XIV, Pages 353—354. 1. ?'478o98 = nat. no. 300675 ; 4'2r4537 = i6388*38. 2. 2-319771 = nat. no. 208*82 ; 6'6S8798 = -0004884. 3. True Courses.— S. i°E., 15' dep.cour.se; N. 75°E.,42'; N. 62'E., 32'; N. 59° W., 37' ; S.n°E.,37'; N.5i°E.,38'; N.55°E.,28'; N. io°E., i7'-5 current course. Dif.lai. so'-g'i^.; dep. ioo''o'E.; cowrie N. 63° E.; dist. 112. Zal.in ^i°Y^-'y viid. la(. ^o^^i'; diff.long. ii(>''t,; long. VI 181° 48' E., or 178' 12' W. 4. Green, date, March 20^ ii^ 36™ ; red. dec!. 0° 6' 35" N. ; true alt. 90° 7' 18". Lat. o°i3'53''N. Raper : True alt. 90° 7' 10". Latitude o" 13' 45" N. 5. Log. of difF. long. 2-718690 = Diff. long. 523'-2. 6. Diff. lat. 189' S. ; mer. diff. lat. 189'; diff. long. 147° 35' E. =: 8855'; tang, course 1 1*6707 27 ; course S. 88' 46' 38' E. ; log. dist. 3-947270 ; dist. 8857'. 7. Standard, Devonport — o-' 6" „ Devonport — o 46 ,, Devonport -f o 3^ „ Hull — I 59 8. Green, date, Dec. 5^ lo*" 59™ 52'; red. decl. 22° 27' 55" S. ; sine amp. 9-996025 ; true amp. E. 82° 15' 36' S. Error 16° 10' 39" W. ; deviation 8° 39' 21" E. 9. Interval 6o<'; rate 3^-5 losing; interval 62'! lo^*"; ace. rate •\- 3"^ 38^-5; Green, date, Aug. 31 ' 10'' 29™ 185-5; rod. decl. 8^ 26' 20" N. ; eq. timo -\- o'" 2^-5 ; true alt. 40° o' 24"; hour-angle 2'' 25™ 59'; mean time ship Aug. 31'' 21'' 34"' 38-5. Longitude 166° 11' 15' E. Maper : True alt. 40" o' 20"; hour-angle 2'' 25'n 59s. Longitude 166° 11' 15" E. 10. Green, date, March 19^ 23'^ 44"" 40', red. decl. 0° 5' 15" S., true alt. 35° 14' 22'', sum of logs. 19-2 19729 ; true azimuth S. 48' 3' 54' E. Error 45' 15' 9" W. ; deviation 18° 5' 9" W. 11. Time from noon 9"" 37'; Green, date Sept. 22'' 12^ 26™ 35"; red. decl. 0° 3' 13'' N. ; true alt. 61° 51' 21' ; nat. no. 776 ; mer. zen. dist. 28" 2' 58' S. Latitude 27° 59' 45" S. LIethod II. + 5' 40", true alt. 61° 51' 21", latitude 27° 59' 46' S., Norie. True alt. 61° 51' 16", latitule 27° 59' 51" S., Eapcr. Towson : Aug. I, o; Aug. II, 4- s' 27". Latitude 27° 59' 59' S. 12. ist red. decl. 20" 43' 9" S. ; ist. red. eq. T. -\- lo™ 22«-56; ist true alt. 11° 16' 8"; lat. 50° 50' N. gives hour-angle 2'' 29"! 20"; long. 7' 14J' W. (A). Lat. 51° 20' N. gives hour- angle 2'' 24™ 45'; long. 6° 11' W. (A'). 2nd rod. decl. 20° 40' 59"-6 S. ; 2nd red. eq. T. ■\- 10"' 26*; 2nd true alt. 14° 8' 7"; lat. 50^50' gives hour-angle i** 54"* 59'; long. 5° 59!-' W. (B). Lat. 5 1" 20' N. gives hour-angle i^ 48™ 42' ; long. 7° 34' W. (B'). The line of bearing at first ohservation trends N.E. | E. and S.W. f W. The ship's position at time of second observation is lat. 51° 17' N., long. 7° 22' W. 13. Decl. (rt Cygni), Doneb, 44° 52' 3"; true alt. 56° 13' 50". Latitude 11° 5' 53" N. Eaper : True alt. 56° 13' 41". Latitude 1 1° 5' 44" N. The Curve.— Correct magnetic bearing S. 82^ W. Deviations.— 21° "W.; 10' "W.; 3" W. ; 4°E.; 20' E. ; 23° E.; 5^ E. ; 170 W. Compass courses.— N. 64^° E. ; S. 10^° W. ; S. 68^^ E. Magnetic courses. — N. 87^° W. ; N. 54= W. ; N. 22° E. Bearings, magnetic. — N. 17° "\V. ; S. 84^" E. EXAMINATION PAPER-No. XV, Pages 3?5--356. 1. 7718169 =: 52260000. 5-903094=3800007-4. 2. 4-301030:^:20000. 8-850814 = -0709274. 3. True Courses.— S. 40° E., 21' dep. course: S. 65= E., 14': N. 54° W., 9': N. 23= E., 18' : S. 10° W., 22' : S. 60° ^Y., 29' : N. 16^ E., 2o'-6 : N. 79° E., 14' current course. Liff. lat. 1 3'- 8 S., dep. i6'-4 E. : course S. 49^° E., dint. 2ii', Lat. in 34° 42' S., diff. long. 20' E. Long, in 18' 48' E. LLL 442 Answers. 4. Green, date, February lo'* 21^ 50™ 40^ : red. decl. 13° 58' 46" S. : true alt. 30° 33' 20". Latilude 45° 27' 54" N. liaper : True alt. 30° 33' 17". Latitude 45° 27' 57" N. 5. Log. of difT. long. 2"oiO904 ■=. Biff. long. i02'"5. 6. DifF. lat. 4202' N. : mer. difF. lat. 4555' : difi. long. 4847' E. : tang, course 10-026985 : course N. 46^ 46' 44" E. : log. dist. 3-787882 : distance 6136'. 7. Standard, Brest, constant + 4^43™; corr. for long. -j- 10" ; i*^ 6™ a.m., i*'4i'n p.m. ,, Waterford + o 46 ; 9 56 a.m., 10 23 p.m. ,, Leith — I 49 ; 5 28 a.m., 6 7 p.m. 8. Green, date, March 30*1 7'' 41'" 8^ : red. decl. 3° 57' 55" N. : sine 8-842619 : true amp. E. 3° 59' 28" N. Error 7° 50' 28" W. : deviation if 50' 28" W. 9. Interval i^^, rate 4^-9 gaining, interval -^2^ 2i'>, accumulated rate — 2™ 41^, Green, date May 26'^ 21'' 10™ 6^, red. decl. 21° 18' 36' N., red. eq. T. 3"^ 8^ subt., true alt. 43° 20' 9", hour-angle 2^ 53™ 24^. Longitude \° 39' 30" W. 10. Green, date, July \o^ d^ 25™ 20", red. decl. 22° 13' 56" N., true alt. 44° 59' 38", sum of logs. 19-231552, true azimuth S. 48° 46' E. Error 54° 23I' W. ; deviation 3° 23^' W. 11. Time from noon 30'" 41": Green, date, Nov. 7"^ 18*' 31"" 53*: red. decl. 16" 34' 11" S. : true alt. 40" 4' 47''; mer. zen. dist. 49° 22' 53" N. Latitude 32° 48' 42" N. Method II. 4" 3^' 22". True alt. (Norie) 40' 4' 47"; latitude 32° 48' 40" N. Towson : Aug. I, ■\- 8' 30" ; index 81 ; Aug. II, 4" 24' 10'. Latitude 32° 40' 21" N. 12. ist red. decl. i3°i9'6"N.; ist red. eq. T. — 2™io3; ist true alt. 51° 14' 52"; lat. 48° 30' gives hour-angle i'^ 18"" 31=; long. 59° i4f' W. (A). Lat. 49° o' gives hour-angle ihijmjo^; long. 60° 32^ W. (A'). 2nd red. decl. 13° 22' 20" N. ; 2nd eq. T. — 2™ 12'; 2nd true alt. 16" 28' 49"; lat. 48" 30' gives hour-angle 5'' 20"^ 32'^; long. 58° 26' E. (B). Lat. 49° o' gives hour-angle 5'^ 20™ 37*; long. 58° 24I' W. (B'). Tha line of bearing at the first observation trends N.W. by W. \ W. and S.E. by E. \ E. The position of the ship at the second observation is lat. 48° 35J' N., long. 58° 26' W. 13. Star's decl. 57° 6' 22" S. ; true alt. 32° 48' 59". Latitude 00 4' 39" N. Raper : True alt. 32° 48' 53". Latitude o" 4' 45" N. The Curve. — Correct magnetic bearing S. 17° E. Deviations.— 27° E. ; 39° E. ; 22° E.; 5° W. ; 29" W. ; 27° W. ; 23° W. ; 5° W. Compass courses.— N. 28^ W. ; N, 58° W. ; S. 79^° W. ; S. 89" "W. Magnetic courses.— S. 80° E. ; S. 42° E. ; N. 66° W. ; S. 87° E. Correct magnetic bearing. — N. 28° E. ; S. 62° E. EXAMINATION PAPER— No. XVI, Pages 357—358. 1. 4^734724 = 54290-5; 9-823096 = -66542. 2. 3-306211=12024; 1-671357 := 46-92. 3. True Courses.— N. 56" W., 13' dep. course; S. 7o°W., 34'; S. 69''"W., 36'-2 ; S. 59°W., 38'-6; S. 3° "W., 34'-i ; S. 10" W., i7'-2 ; N. 77° W., 33'-5 ; S. 10° W., 21' current course. Eiff.lat.\oi''^ii.; dep. 1^0' -gW.; course S. $6° W.; dist. iSi^/. Lat. in 60° £•]' S. ; diff.long. 302'. Long, in 184° 44' "W., or 175° 16' E. 4. Green, date. Ma}-- 16^ 3'' i"" 44', red. decl. 19° 10' 10" N., true alt. 86° 50' 42'. Lat. 16° o' 52" N. 5. Log. of diff. long. 1-790942 = Biff. long. 6i'"]g. 6. Diff. lat. 604' S., mer. diff. lat. 606', diff. long. 6324' E., log. tang. 11-018519 ; course S. 84° 31' 35" E. ; log. dist. 3-801545 ; dist. 6332'. 7. Standard, Brest, constant -}- 0*127™ ; no a.m. and o''34'" p.m. „ Brest „ -|- I 6 ; 0*138'" a.m. and i 13 p.m. „ Brest „ — 2 o ; 10 7 a.m. and 10 38 p.m. „ Brest „ — 07; no a.m. and noon. AnstoBTS. 443 8. Green, date, June 24'* ii^ 41™, red. decl. 7.y 24' 47" N., sine amp. <)'^()%-]ii, true amp. W. 85° 41' 29' N. Error 12" 44' 46" W. ; deviation 20° 45' 14" E. 9. Interval 44'', rate 8» losing, interval 42'' 23'', accumulated rate + 5™ 43''7> Green, date Juno 14*^ 23*" 6™ 21', red. decl. 23^ 19' 30' N., red. eq. T, 4- o™ 8»-76, true alt. 20° 48' 38", hour-angle 3'' 7"" 27% M.T. ship June 14'' 20'' 52™ 42". Longitude 33° 24' 45" W. 10. Green, date, Sept. 23'' lo^ 29™ 44" ; red. decl. 0° 18' 24" S. ; true alt. 40° 29' 23' ; sum of logs. 17*426341 ; true azimuth N. 5" 55*22" W. Error 20" o' •]" 'W .; deviation 2'^° 20' y'W . 11. Time from noon 19™ 21 », Green, date, September 22^ iz^ 59m 458^ red. decl. 0° 2*41 "N., true alt. 38' 58' 36", nat. no. 2253, mer. zen. dist. 50° 51' 26" N. Latitude $0° 54' 7' N. Method II. — lied. 4- 10' o": lat. 50" 54' 5' N. fNorieJ. Towson : Aug. I, o, index 46. Aug. II, -}- 9' 50". Latitude 50° 54' 15" N. 12. ist red. decl. 9° 41' 59" N. ; ist red. eq. T. + i™ 6^-i ; ist true alt. 39° 32' 10''; lat. 48° 50' gives hour-angle 2'' 32™ 56^ ; long. 6° 58^' W. (A). Lat. 49° 30' gives hour-angle 2*' 29""- 40"; long. 6'^ 9I' W. (A'). 2nd red. decl. 9^^ 37' 11" N. ; 2nd red. eq. T. -\- i™ 2^-6 ; true alt. 35° 53' 31"; lat. 48^ 50' gives hour-angle 2'' 59"^ 14^; long. 5° 7|-' W. (B). Lat. 49° 3° gives hour-angle 2'' 56™ 44'; long. 5^^ 45^' W. (B'). The line of bearing at first observation trends N.E. ^ N. (northerly) and S.W. i S. (southerly). The position of tho ship at the second observation is lat. 50° 6' N., long. 6" 172' W. 13. Star's decl. 10° 32' 59" S. ; true alt. 52° 9' 5". Latitude 48^ 23' 54' S. The Curve. — Correct magnetic bearing S. 11° W. Deviations.— 24" W., 2° W., 5° E., 8^ E., 21" E., 22° E., 3° W., 26° W. Compass courses. — N. 85° E., S. 46^ W., S. 42^° E. Magnetic courses.— N. 81 ^ W., N. 23° "W., N. 70° E. Bearings, magnetic. — (Deviation for ship's head W.S.W. = io^° E.) S, 89^" W., S. io|°W. EXAMINATION PAPER— No. XYII, Pajes 359—360. I- 7'65i355 = 44807938-1; 1-351334 = 22'456i. 2. 5-614476 = 4n6oo-94; 9-030734 = -107333. 3. True Coursea.-S. 72° E., 25' dep. course; N. 4i°E., 35'-4; N. 62° E., 48'; N. 53°"W., 43'; S. 60° E. 26'; N. 18° W., 18'; S. 46° E., 32'; N. 820 E., 36' current course. DiflF. lat. 54'-3 N., dep. i3o'-6 E. ; course N. 67!" E., dist. 141?/. Lat. in 38° 31' N., dij". long. 166' E. Long, in 2° 5' E. 4. Green, date, Nov. 20'' 19'' 18™ 40'; red. decl. 19° 55' 38' S.; true alt. 80' 28' 59". Lat. 29° 26' 39" S. Eaper : True alt. 80' 28' 55' N. Latitude 29° 26' 43' S. 5. Log. of diff. long. 1-909671 = LliJ". long. 81-22. 6. DiflF. lat. 731' S.; mer. difF. lat. 733'; diflF. long. 1259' W.; log. tang. 10-234922; course S. S9° 4iy ^•> log. of distance y 162224.; distance 14^2'. 7. Standard, Hull constant — o''29'"; 3'' 32'" a.m., and 4'' C" p.m. „ London ,, — 2 19 ; 9 30 a.m., and 9 57 p.m. ,, Dover ,, +5 13 ; i 24 a.m., and i 53 p.m. „ Brest „ + 3 28 ; corr. for long. — i6">; 41'37'"a.m., &51'2™p.m. 8. Green, date, July zo'i 1 8^56"' 48'; red. decl. 20^30' i"N. ; true amp. W. 28= 38*4 i"N. Error of compass 59° 34' 56" E. ; deviation 49" 55' E. 9. Interval iS"*; rate 4"-o /os/w^ ; interval 72I 12''; Green, date, June 4'' 12'' 3310 ^o'; red. decl. 22=" 31' 9" N. ; true alt. 28' 18' 5 2"; red. eq. T. 5?" 54^; hour-angle 3'> 52"" 17*; mean time at ship June 4'' 2o'> 5™ 49'. Longitude 113° 2' 15" E. 10. Green, date, Nov. ^^ 14'' 58™ 46'; red. decl. 17° 6' 15* S. ; true alt. 6° 11' 26"; sum of logs. 19-304632 ; true azimuth S. 53° 22' 6" E. Error 3° 32' 6 " W. ; deviation 10° 52' 6" W. 11. Time from noon 9™ 5» ; Green, date Jan. 8'* 3'! 31 "143"; red. decl. 22° 1 1' 54*8. ; truo alt. 76' 57' 49" ; nat no. 594 ; mer. zen. dist. 1 2" 53' 5' S. La/i/wle 35'' 4' 59' S. Method II. -]- 9' 6". Latitude 35° 4' 59" S., Xorie. Towson : Exceeds limits of Table. 444 Answers. 12, ist red. decl. 230 17' 47" N. ; ist red. eq. T. o™ o^-t, ; 1st true alt. 54° 23' 38" ; lat. 50° 10' gives hour-angle z^ o"^ 8'; long. 79° 40^' W. (A). Lat. 49° 40' gives hour-angle jh am 20^; long. 79° 5' W. (A'). 2Rd red. decl. 23° 18' 21" N. ; 2nd red. eq. T. -{- o™ 2^-8 5 2nd tiue alt. 11° 22' o'. Lat. 50" 10' gives hour-angle 6'' 41™ 49^; long. 79° 50^ "W. (B). Lat. 49° 40' gives hour-angle 6^ 40"" 30= ; long. 80° 10' W. (B'). The line of bearing at the first observation trends N.W. ^ N. (northerly) and S.E. \ S. (southerly). The position of the ship at the second observation is lat. 49° 48^' N., long. 80° 4' W. 13. Star's decl. 16° 33' 33" S. ; true alt. 37° 45' 59". Latitude 35° 40' 28" N., Norie. Maper : True alt. 37° 45' 54". Latitude 35° 40' 33" N. The Curve. — Correct magnetic bearing N. 12' E. Deviations.— 12° E.; 0°; 17° W.; 24° "W. ; 12° "W.; 7° E. ; 17° E.; 17° E. Compass courses.— N. 85° W. ; S. 57° W. ; S. 34° E. ; S. 8° E. Magnetic courses.— N. 86° E. ; S.E. by S. ; N. 38^° W. ; N". 79° W. Bearings, magnetic. — (Deviation for ship's head E. by S. f S. ■= 20^° W.) ; N. 827" W. ; S. 37rE. EXAMINATION PAPEE-No. XVIII, Pages 361—362. 1. Log. of product 6'498o62^ product 3148195-6. Log. ^•233506, product -0000017 12. 2. Log. of quotient 4*986680 =: quotient g6<)']g-6. Log. 2-43969I, product •027523. 3. True Courses.— N. 69" W"., 25' dep. course ; N. 58° W., 51'; S. 63°W.,42'; N. 5i°E., 30'; N. 36° W., 46' ; S. 27°W., ri'; S.68°E.,i6; S. 51° \V., 32' current course. Diff. lat, 37*o N., dep. i22'"8 W., course N. 73° W., dist. 128'. Lat. in 36° 50' S., mid. lat. 36° 8', dijf. long. 152' W. Long, in 71° 9' W. 4. Green, date, 1882, December 31'^ i2'> 48"» 24^; red. decl. 23° 3' 9" S. ; true alt. 83° 52' 24". Latitude 16° $2) 'ii' ^• Maper : True alt. 82° 52' 19". Latitude 16° 55' 28" S. 5. Diff. long. i92'5 Y/. Long, in 182° 29'-5 W., or 177° 30^' E. 6. Diff. lat. 2747' S. ; mer. difi. lat. 2919' ; diff. long. 6340' W. ; tang, course io'336855 ; course S. 65° 16' 41" W. ; log. distance 3'8i7459, distance 6568. 7. Standard, Brest constant — 2^ C"; z^^cj^^k.^., 21>59'"p.m. „ Brest ,, — 3 17 ; corr. for long. — 8™ i 14 a.m., i 34 p.m. ,, Brest „ -^42; „ -|- 9™ 8 50 a.m., 9 10 p.m. 8. Green, date, Nov 3'* 17^ 28"' 30=; red. decl. 15° 21' 20" S. ; true amp. E. 22° 41' 24" S. Error 27° 56' 6" W. ; deviation 11° 26' 6" "W. 9. Interval 14"*; rate 8*'5 gaining; interval 124"* 20'^; ace. rate 17"" 41=; Green, date August 31'' i9'> 54"^ 30'; red. decl. 8° 17' 48" N.; eq. T. — o™ 5*; true alt. 15° 25*44"; sum of logs. 9'53i79i ; hour-angle 4'^ 45°" 41^. Longitude 10° o' 45" W. 10. Green, date. May 3i<^ i6'» 13"^; red. decl. 22° 2' 19" N. ; true alt. 39° 20' 26" ; sum of logs. i9"779i99 ; true azimuth S. 101° 41' 8" E. Error 3° 14' 53" W. ; deviation 4° 6' 9'' AV. 11. Green, date, April 12'^ i4'> i™ 5^; time from noon 10"" 15^; red. decl. 8° 57' 37" N. ; true alt. 80° 43' 11" ; nat. no. 988 ; mer. zen. dist. 8' 55' 23" S. Latitude 0° 2' 14" N. Method II. -1- 21' 48". Latitude 0° 2' 36' N., Norie. Towson : The altitude exceeds the limits of the Table. 12. ist red. decl. 4° 35' 13" N.; ist red. eq. T. + 3"" 55^-3 ; rst true alt. 35' 3' 31"; lat. 51° o' gives hour-angle 2^ 21"^ i^; long. 9° 495 W. (A). Lat. 50° 30' gives hour-angle jh 24m 9S ; long. 10° 36J' W. (A'). 2nd red. decl. 4" 40' 16" N. ; 2nd red. eq. T. 4" 3™ 5''; 2nd true alt. 31° 31' 9"; lat. 51° o' gives hour-angle 2*1 51™ 35^; long. 10° 59' W, (B). Lat. 50° 30' gives hour-angle 2'> 53™ 57^; long. 10° 23^' W. (B'). The line of bearing at the first observation trends N.E. and S.W. The position of the ship at the second observa- tion is lat. 50° 19' N., long. 10° 10' W.- 13. Star's decl. 60" 21' 13" S., true alt. 9° 52' 32". Latitude 19° 46' 15" N. The Curve. — Correct magnetic bearing S. 8° E. Deviations.- 1 9° E.; 8° W. ; 29° W. ; 35° W. ; 20° W. ; 13° E.; 29° E. ; 31° E. Compass courses.— N. 2" E. ; S. Coh" E. ; S. 29° W. ; N. 81^° W. Magnetic courses.- N. 11° W. ; S. 56^° E. ; N. 50" E. ; N. 34" E. Bearings, magnetic- N. 153" E. ; N. 49° W. Answers. 445 EXAMINATION PAPER— No. XIX, Faf/es 363—364. 1. 7-080996 = 12050250; i"8i9774 = 66035. 2. 0-301030 =z 2-; 1-770558 = 58-96. 3. True Courses.— N. 70' W., 1 3' d.>p. courso ; S. 20" W., 48' ; S. 60° W., 37'-2 ; S. 54° W., 30' ; N. 7 1"" W., 43'-5 ; N. 66' W., 38'-4 ; N. 15^ W., 37'-S ; N. 36^^ \V., 36' current course. Diff. lat. i8'-5 N., dop. i92'-2 W. ; course N. 84^° VV., dist. 193^'. Lat. in 46' i V S. ; diflF. long. 4*^ 371' W. Long, in iSi° 21^' W., or 178° 38^' E. 4. Green, date, Sept. 22^ 16'' o™ 24s; red. decl. o" o' 15" S. ; true ait. 83' 52' 20". Lai. 6° 7' 25" N. JRqper : True alt. 83° 52' 14". Latitude 6° 7' 31' N. J. Log. of diff. long. 2-159381 = Biff. long. r44''3. 6. Diff. lat. ii99'N.; mer. diff. lat. 1205'; diff. long. 3410' W. ; tangent courso 10-451767; counc N. 70' 32' 17" W. ; log. distance 3-556139 ; distance 3599'. 7. Standard, London constant — 2''i3"'.; 9h53'" a.m., 101117'" p.m. ,, Brest ,, — 20; no a.m., o 4 r.M, „ Brest „ + 3 3 ; 4 43 a.m., 5 7 i-.m. „ Weston-super-maro,, — i 12 ; 3 21 a.m., 3 50 p.m. „ Leith ,, — 2 21 ; 10 16 a.m., io 40 p.m. „ Brest „ -J- 6 13 ; corr. for long. — 16"'; 7h 37"" a.m., and 8'' I°> P.M. 8. Green, date, June 12'^ lo^ 55"" 2\^; red. dec!. 23° 11' 57" N.; sine of true amplitude 9-992978 ; true amplitude W. 79° 44' N. Error 38' 23V W. ; deviation 15*^ 13^ W. 9. Interval 70'', daily rate (4*-oS :=) 4^-1 gaining, interval 93'' 12'', aec rate + 6™ 23^-3, red. decl. 23" 18' 21" N., red. eq. time + o™ 2*-8i, true alt. 49° 1 1' 9", hour-angle 3'^ i™ 29% M.T, ship June 14'^ 3'' i" 32". Longitude 134° 37' 15" "W. 10. Green, date, Dec 31'' 16'' 24'" 2o»; red. decl. 23° i' 15" S. ; true alt. 35° i' 13"; logs. 19-750904; trueazimuthN. 97° 17*52" W. J5^r/-oy i2''55'2 2" W.; rfwia^iow 32^55' 22' W. I r. Time from noon 10™ 46' ; Green, date, Dec. 30*' 18'' 37™ 14s ; red. decl. 23° 6' 34" S. ; nat. no. 625 ; nat. cos. 257950; mer. zen. dist. 75° 3' 6" N. Latitude 51° 56' 32" N. Method II. Eed. -}- 2' H- Latitude 51° 56' 31" N., Norie. Towson : Aug. I, + 1' 21": index 12. Aug. II, 4- 50'. Latitude 51° 36' 34" N. 12. ist red. decl. 19' io'59"S. ; istred.eq.T. -|-i2">228; truealt. 9°4i'48' ; lat. 5i°i5'N'. gives hour-angle 2i> 56™ 19'; long. o°56,f' W. (A). Lat. 50° 45' gives hour-angle 2h59'"499; long, i" 49^-' W. (A') ; 2nd. red. decl. 19° 8' 34" S.; red. eq. T, -j- 12'» 24^; true alt. 18° 37' 49*; long. 3' 17^' \V. (B). Lat. 50°45'gives hour-angle i''5'^36s long, o" 18' W. (B'). Theline of bearing when the first altitude was taken trends N.E. -J- E. and S.W. \ W. The position of Bhij) at the time of the second observation is lat. 51° 19I' N., long, o"^ 15^ E. 13. Star's decl. 10° 33' 38" S., true alt. 60° 59' 28". Latitude 45° 34' 10' S. The Curve.— Correct magnetic bearing N. 73° W. Deviations.— 0° : 17° E. : 15° E. : 7'' E. : 2" E. : 5° W. : 16" W. : 20'' W. Compass courses. — North : N. 85^° E. : S.W. Magnetic courses.— S. 65° W. : N. 841° E. : N. 52" "W. Bearings, magnetic. — S. 19"^ AV., and N. 60" "W. EXAMINATION PAPEE-No. XX, Pa>jes 365—366. r. Log. 2-577492 =: product 378; 2-138598 = -013759. 2. Log. 3-954292 = quotient 9001-0; 5-301030 =: -00002 ; 6"3oio3o = "000002 ; and 1-301030 r= 20. 3. True Courses.— S. 26° W., 16' dep. course; S. 87° E., 53'; S. 52° W., 15'; N. 70" E., 50' ; N. 14" E., 9'; S. 30° E., 28'; N. GZ" E., 47'; N. 82° E., 42' current course. Biff. lat. i'-4 S. ; dejK i82'-5 E. ; course S. 89^° E. ; dist. 182^. Latitude in 50" 26' S. ; diff: long. 286'-5 E. Longitude in 184'' i6\' E., or 175° 33J' W. 446 Answsrs. 4. Green, date, Sept. 7,2^26^^'^; red. decl. o°4' 17" S. ; true alt. 84° 21' 18". Latitude 50 42' 59" S. Raper : True alt. 84° 21' 7". Latitude 5° 43' 10" S. 5. ^(^. ^o?2y. 2 2o'"9 E., or 3° 41' E. Zowi;. in 3° i' E. 6. Diff. lat. 2531' S. ; mar. diff. lat. 2701'; difi. long. 7433' E. ; tang. io'439639; co^irse S. 70° 1' 48" E. ; distance 7411'. 7. Standard, Leith constant — i''i7™; 9*^58™ a.m., and io'i26"'p.m. „ Davenport ,, — J 13 ; o 30 a.m., and i 3 p.m. „ "VVeston-super-mare ,, -|- o 19 ; 3 5 a.m., and 3 40 p.m. „ Portsmouth „ — i 1 1 ; 6 47 a.m., and 7 20 p.m. ,, Brest „ — o 47 ; II 54 a.m., and no p.m. 8. Green, date, November 4 43™ 30'; time from noon 32™ 18^; red. decl. 23° 9' 36" N. ; true alt. 50° 14' 43"; nat. no. 8770; mer. zen. dist. 38^ 57' 47'' S. Lat. 15° 48' 11" S. Method II. •\- 47' 55". Latitude 15° 47' 46'' S. Towson : Aug. I, -{- 12' 16', index 81. Aug. II, -f- 34' 38". Latitude 15° 46' 41" S. 12. ist red. decl. 15' 6' 55" S.; istred.eq.T. — 16™ 19^-3; ist true alt. 19° 3' 11"; lat. 51° 45' gives hour-angle 1^49"" 6'; long. 11° 2^' W. (A). Lat. 51° 15' gives hour-angle i''54'"59'; long, 12° 3of' W. (A') ; 2nd red. decl, 15° 10' 6" S. ; 2nd red. eq. T. — 16"" i9^'i3 ; 2nd true alt. 16" 41' 27"; lat. 51° 45' hour-angle 2^ i']'^ ^o^; long. 10° 41' W. (B). Lat. 51° 15' gives hour-angle 2'' 22™ 22'; long. 9° 33' W. (B'). The line of bearing at first observation trends N.E. by E. |^E. and S. W. by W. J W. The position of the ship at the second observation is lat. 51° 42' N., long. 10° 34' W. 13. Star's decl. 22'' 54' 38" N., true alt. 60" 23' 4". Latitude ^2° li t,^' "iS . fNorieJ . „ „ 60 22 57 „ 52 31 41 N. (Raper). The Curve. — Correct magnetic bearing S. 100° W. or N. 80'^ W. Deviations.— 49° E., le'' E., 23° W., 54° W., 50" W., 11° W., 18" E., 54° E. Compass courses. — S. 8j° W., South, N. 39° W., S. i6J° W. Magnetic courses.— N. (>s° E., S. 27^° E., N. 5" W., S. 620 W. Bearings, magnetic. — S. 46° E., S. 52° W., N. 35" E. EXAMINATION PAPEE— No. XXI, Pages 367—368. 1. 7-754166 = 56776104; 3-806197 = "00640025. 2. 8-201561 = "015906; 13*000000 = lOOOOOOOOOOOOO. 3. True Courses.— S. 6° W., 19' dep. course; N. 50° "W., 23'-? ; S.47''E., i6'"3 ; N.i8»E., i7'-6; S. 8° W., 14-1; N. 87°E., 42'; S. 75° E., 12'; S. 65° W., 15', current course. Biff, lat. i9'"3 S. ; dep. 35' E. ; course S. 61° E. ; dist. 40' E. Lat. in 56° 41' N. ; diff. long. 64' E. Longitude in 38" 56' W. 4. Green, date, June 24J 2oi» 3"^; red. decl. 23' 24' 13" N.; true alt. 60° 4' 7"; latitude 6° 31' 40" S. By Raper : True alt. 60" 4' i" N. Latitude 6° 31' 46" S. 5. Diff. long. i53'"9 W., or 2° 34' W. Longitude in 12° 4' "W. 6. Diff. lat. 4728'-5 N. ; mer. diff. lat. 4896'"5 ; diff. long. 3540' E. ; log. tang. 9-8591 17 ; course N. 35° 52' E. Compass course N. 15° 52' E. ; distance 5588'. Answers. 447 7. standard, Brest, constant 4- 6'' 28"; corr. for long. — 14" ; io''37™a.m. & ioh54t"p.M. „ Brest „ + 3 42 ; „ 4- 9 : 8 14 a.m. & 8 31 p.m. „ Queenstown, constant — c" 59* ; 4^ 41'" a.m. and 4'' ^S"" p.m. 8. Green, date, June 23'> iS^ 48™ 12" ; red. decl. 23° 25' 38" N. ; true amp. 23° 25' 38" N. Hrror of compass 540 22' W. ; deviation 32" 42' W. 9. Interval 32', accumulated rate — i™ 3o«-5, Green, date, Sept. 22 27™ 9" ; red. decl. 4° 15' 31" S. ; true alt. 63' 47' 10' ; nat. no. 963 ; mer. zen. dist. 26° j' 19" S. Latitude 30° 20' 50" S. Method II. 4" 7' 3''- Latitude 30° 20' 50" S. Towson : Aug. I, 4- o' 16" ; index 15". Aug. TI, 4- 8' 6". Latiticde 30^ 20' 54" S. 12. ist red. decl. 20° 20' 33" S. ; ist eq. T. 4- 10™ io»; rst true alt. 13° 18' 58"; lat. 49° o' gives hour-angle 1^ 22'" 12' ; long. 140^ o' W. (A). Lat. 49° 40' gives hour-angle jh i^m ^^s. long. 138= 26}' W. (A'). 2nd red. decl. 20° 48' 31" S.; 2nd red. eq. T. 4- io'"i3'-63; lat. 49° o' gives hour-angle !>> 57m 328; long. 138° 35I' W. (B). Lat. 49° 40' gives hour-angle 1'' 49™ 39'; long. 140° 33I' W. (B'). The line of bearing at first observation trends N.E. by E. and S.W. by W. The position of ship at second observation is lat. 49° 21' N., long. 139" 373' W. 13. Star's decl. 55° 53' 21'' N. ; true alt. 84° 56' 45". Latitude 60° 56' 36" N. The Curve.— Correct magnetic bearing N. 950 E., or S. 85° E. Deviations.— 7° W. ; 17° W.; 29^ W. ; 240 W. ; 4° E. ; 29^ E. ; 28"= E. ; 16' E. Compass courses.-N. \%¥ E. ; S. 600 E. ; S. 5^0 E. ; S. 58' W. Magnetic courses.— N. 32^° E. ; S. 40° E. ; S. 58^° W. ; N. 15" W. Bearings, magnetic. — (Deviation for ship's head S.W. by W. \ W. = 30° E.), N. 69 J E. ; S. 65^' E. EXAMINATION PAPER— No. XXII, Pages 369—370. 1. 6"04i523 = 1100330-8 ; 5-847331 =; 70360806. 2. 4-440657 := 27584 ; 3*000000 = 1000. 3. True Courses.— N. 55° E., 25' dep. course ; N.49°W., 50'; N. 72° W., 38' ; N. 80" W., 33'-8 ; N. 6" E., 37'-9 ; N. 3" W., 42'-5 ; N. 43° W., 29'; West, 32' current course. Liff. lat. i66'-o N. ; dep. ilS'-] W. ; course N. 393" W. ; dist. 215^'. Latitude in 53" 54^' N. ; mid. lat. 52° 31' N. ; diff. long. 224-6. Longitude in 2° 20' W. 4. Green, date, Aug. 2^^ 17'' 51"" 48'; red. decl. 10° 27' 36' N. ; true alt. 35''48'58". Lat. 430 43' 26' S. 5. Log. of diff. long. 2-254252 := diff. long. i79''6. 6. Diff. lat. 315' S. ; mcr. diff. lat. 524'; diff. long. 365'; tang, course 9-842962 ; course S. 34" 51' 35" E. ; log. of distance 2-584204; distance 383'-9. 7. Standard, Hull, constant — i'' 59™ ; o*" 9"" a.m., and o^ 45'" p.m. 9 o A.M., and 9 36 P.M. no A.M., and o 27 p.m. no a.m., and o 39 p.m. 3 45 a.m., and 4 23 p.m. no A.M., and noon. 8. Green, date, Sept. 22'' i9*> 50™ 52' ; red. decl. 0° 3' 59* S. ; sine 7-267198 : true amp. W. 0° 6' 22" S. Hrror 14° 53' 38' E. ; deviation 5° 22' 22" W. 9. Interval lo"!, rate 4»-6 gaining; interval 89'^ 22^'', ace. rate 6"^ 54'. Green, date, Aug. 2^d 22'' 15" 56% red. decl. 9° 41' 40" N., red. eq. time 4" i™ 6'-3, true alt. 38' 15' 44", hour- angle 3° 19' 10", M.T. ship Aug. 28> — I 43 Greenock >> + 4 41 Weston-super-mare )) — 2 10 Thurso »> — 58 Sunderland >» 4-0 23 448 Answers. 10. Green, date, June 15^ 2^ 30" 52', red. decl. 23° 19' 52" N., true alt. 9° 21' 57", sum of logs. I9-86459I, true azimuth S. 117" 39' 46' W. Hrror of compass 65° 9' W. Deviation 14' 29' "W. 11. Time from noon 29™ 55", Green, date, Doc. 235\' ^- (^')- "^^"^ ^'^^ of hearing at first ohservation trends N.E. hy E. \ E. and S.W. hy W. \ W. The position of ship at the time of second observation is latitude 46° i' N., longitude 87" 50J' E. 13. Star's decl. 55" 53' 41" N., true alt. 62° 12' 39", latitude 28° 6' 20" N. The Curve. — Correct magnetic hearing S. 30° W. Deviations.— 14° W.; 15° W.; 15° W.; 10° W.; 13° E.; 30° E. ; 16° E.; 5° W. Compass courses. — S. 2o|° W. ; N. 30° E. ; S. 55° E. Magnetic courses.— S. 75° W. ; N. 74° W. ; N. 62° E. Bearings, magnetic. — N. 14° W. ; N. 70° E. EXAMINATION PAPEE— No. XXIII, Pages i']i—i^i. 1. 5-561769 = 364560 nearly. 7'4i7i39 =: 26130000. 2. o'477i20=r3. 5'903O9O = 800000. 7*903090 = 80000000. 3. True Courses.— S. 24° W., 12' dep. course; S. 43° W., 34'; S. 57° W., 30'; S. 32° W., 32'; S. i'^ W., 34'; S. 3° E., 32'; S. 19° E., 34'; S. 59° "W. 30', current course. Biff. lat. i92'-9 S.; dep. %i-% W. ; course S. 23^° W., dist. 2io|'. Lat. in 58°57'N. ; ndd, lat. 60° 33'; diff. long. i7o'-4 W. Longitude in 181° 42' W., or 178° 18' E. 4. Green, date, Aug. 31*1 i7'> 29™ 12^; red. decl. 8" 19' 59" N. ; true alt. 51° 9' 27". Latitude 30° 30' 34" S. Eaper : True alt. 51° 9' 22". Latitude 30° 30' 39" S. 5. Log. of difF. long. 2-322530 =: diff. long. 2io'-2. 6. Diff. lat. 5968' N. ; mer. diff. lat. 7050' ; diff. long. 2121' W. ; tang. 9-478352 ; course N. 16° 44' 38" W. ; log. distance 3-794644 : distance 6232'. * 7. Standard, Holj'head constant — 3i'io'": 9hi2'"A.M., 9h34'»'p.M. ,, Londonderry ,, — i 37 : 8 30 a.m., 8 49 p.m. „ London ,, — 2 13 : i 55 a.m., 2 17 p.m. „ Weston-super-mare „ — 2 10 : 6 55 a.m., 7 14 p.m. „ Waterford ,, + o 44 : 8 21 a.m., 8 40 p.m. „ Liverpool ,, — o 55 : o 19 a.m., o 39 p.m. „ Brest „ -}- 3 28 : corr. long. — i6'",9h9"> A.M.,9'»29'np.M. 8. Green, date, Dec. 24'! 1^ 14™ 12^; rod. decl 23° 25' 31" S. ; sine 9*972365 : true amp. E. 69° 46' 35" S. Error of compass 25° 50' 55' W. Deviation 27° 10' 55" W. 9. Inteival ii"*; ra,iQ o^-^^ gaining; Interval 92*22''; ace. rate o"5is; Green, date, April i4 28'" 129-5 ; M.T. ship April 14* 19^ 31™ 47=-5. Longitude 44° 30' 22"-5 W. 10. Green, date, Sept. 22'* i8h 27^529; red. decl. o' 2*46" S. ; true alt. 32° 18' 38"; sum of logs. 19-369294, true azimuth S. 57° 51' 54" W. Error ^(y' 30' 36" AV. ; deviation 6° 30' 36" W. ir. Time from noon 1 2'n 183 ; Green, date Sept. 2 2 34'; red. decl. o' 7' 16'' N. ; true alt. 49° 31' 49" ; nat. no. 1094 ; nat. cos. 761843 : mer. zen. dist. 40° 22' 22" N. Latitude 40° 29' 38" N. Method II. Eed. + 5' 48". Latitude 40° 29' 39" N. '[oxvson : Aug. I, o : index 19. Aug. IL + 5' 51". latitude 40' 29' 36" N. Answers, 449 12. ist red. decl. 21" o' 47" S. ; ist red. eq. T. — i2™289-3; ist true alt. 1 1'> i-j' -j" ; lat. 49° o' gives hour-angle 2^ 39™ 59^ ; long. 82'^ 55^' W. (A). Lat. 49° 30' gives hour- angle 2'> 35" 58'; long. 8i°55^' W. (A'). 2nd red. decl. 21' 2'45'S. ; eq.T. — 12™ 25^; true alt. i6'43'47'; lat. 49^0' gives hour-angle i^ 36™ 26'; long. 81° 27' W. (B). Lat. 49° 30' gives hour-angle i'' 29™ 4"; long. 83° 17!' W. (B'). The line of bearing at the first observation trends N.E. | E. and S.W. | W. The position of the ship at the second observation is lat. 49° 19' N., long. 82^ 19' W. 13. Decl. /3 Centauri 59° 48' 36" S. ; true alt. 59' 41' 53". Latitude 29° 30' 29" S. The Curve.— Correct magnetic bearing N. 74° E. DeviatioDB.— 5° W. ; 20° E. ; 25° E.; 18° E. ; 2° "W. ; 16' W. ; 24° W. ; 16^ W. Compass courses.— S. 81^° E. ; N. 72° W. ; N. 482" E. Magnetic courses.— S. Ss" E. ; N. 76° W. ; S. 29° "W. Bearings, magnetic— N. 14° E. ; N. 65° W. EXAMINATION PAPEE— No. XXIV, Pa^es 373—374. 1. 6-996945 = 9929977; 9-337878 = -21771. 2. 0-954242 = 9 ; 5-243038 = 175000. 3. True Courses.— S. 82° E., 42' dep. course ; S. 85° E., 53'; N. 69° E., 47' ; South, 22' ; N. 34' W., 14'; N. 72" E., 37' ; S. 83^ E., 35'; S. 82' E., 52'-5 current course. Dif. lat. 4'*i S. ; dep. 2S^'l E- ; coiirse S. 89° E. ; dist. 252'. Lat. in 37° 41' N. ; diff. long. 319' E. Long, in 20° 19' E. 4. Green, date, November 30' 19'' 28^ 16'; red. decl. 21' 49' 6" S. ; true alt. iS' 54' 37", Latitude 49° 16' 17" N. 5. Log. of diff. long. 2-421932 = diff. long. 264'-2. 6. Diff". lat. 5968' N. ; mer. diff. lat. 7050'; diff". long. 21 21' E. ; tang. 9-478352 ; course N. 16° 44' 38" E. ; log. of distance ^ 3794664 ; distance 6232'. 7. Standard, Brest constant -f '^ 6" „ Brest „ — o 26 „ Brest „ + 2 45 ,, Devonport „ — ' ^3 „ Galvcay „ + 1 35 „ Queenstown „ — 14 ,, Brest ,, 4" 4 28 8. Green, date, February I'l 1 7'» 14™ ; red. decl. 16° 49' 50* ; sine 9-994951 ; true amp. W. 81° 17' S. i:rror 17° 9' E. ; deviation 19° 11' W. 9. Interval 62'', rate 2»-5 losing, interval 76"^ g^, accumulated rate + 3™ ii». Green, date August 31'* 9'' II'" 56', red. decl. 8^ 27' 30" N., red. eq. T. + o™ 3"6o, true alt. 45° 14' 4o"> hour-angle 2^ 56"" 28% M.T. ship Aug. 31'^ 2^ 56™ 32*. Longitude 93° 51' o" "W. 10. Green, date, March 20'' 3'' 24™ 28'; red. decl. o' i' 39' S. ; true alt. 53° 10' 51"; sum oflogs. 17-948571 ; true azimuth S. io°48'58''W. ^n-or 13° 37' 43" E. : deviation 2^" 2^' ^2"^. 11. Time from noon I o™ 34« ; Green, date, March 20'i 16*" 55™ 26"; red. decl. o" 11' 49" N.; true alt. 71'^ i8' 34"; nat. no. 1004 ; mer. zen. dist. 18' 30' 38" N. Latitude 18' 42' 25" N. 12. ist red. decl. 13" 35' 56" N. ; ist eq. T. -f 4™ i'; true alt. 28° 25' 3'; lat. 49° 30' gives hour-angle 4'* 7"^ 41'; long. 179° o' E. (A) ; lat. 50° o' gives hour-angle 4'' 6™ 599; long. 179° 10^' E. (A'). 2nd red. decl. 13'' 31' 10" N. ; 2nd red. eq. T. -{- 3"! 58^-2 ; true alt. 46° 32' 43'; lat. 49° 30' gives hour-angle i''59'" 3"; long. 180' 40 J' E., or 179° ig^'W. (B), Lat. 50' o' gives hour-angle i'' 55™ 50* ; long. 179° 52' E. (B') The line of bearing trends N. by E. and S. by "W. ; the position of ship at the time of second observation ia lat. 50" 3^' N., long. 179" 44' E. 13. Markab's decl. 14° 34' 16 N. : true alt. 33° 16' 27". Latitude 42° 9' 17' S. Norie. )> u w 33 16 31 „ 42 9 13 S. Maper. MMM o''45™A.M., and i''26'"p.m. 1 1 54 A.M., and no p.m. 2 24 A.M., and 3 5 P.M. no A.M., and o i6 p.m. 2 16 a.m., and 2 52 p.m. no a.m., and o 14 p.m. corr. for long. — 19""; 3''48"a.m., 41'29™p.m. 45© Amivers. Tlie Curve. — Correct maguetic bearing N. 12° W. Deviations.— 10° E.: 10° E. : 15° W. : 23' W.: 5° W. : 14" E. : 14° E.: 6° E. Compass courses. — S. 56° E. : S. 75° W. : N. 9° W. Magnetic courses.— N. 83° E. : N. 76° W.: N. 10= E. Bearings, magnetic. — N. 760 W. : North. EXAMINATION PAPER— No. XXV, Pages 375—376. 1. 6-742037 = 5521240-5. 5-109096 = 128557. 2. 1-903505^-800767. 7-922575 = 83670961-5. 3. True Courses.— N. 2° W., 21' dep. course; N. 5i°E., i6'-6 ; S. 25°'W., i3'-4; N.5°W., i9'-3 ; S. 57° W., i9'-i ; N. 70° W., 48' ; N. 33° E., 2i'-3 ; N. 48° E., 49' current course. Diff. lat. 95'-2 N. ; dep. -j'-g ; course N. 6° W. ; dist. 96'. Zat. in 63° 55' N. ; fnid. lat. 63° 7'; diff. long. 18' W. Long, in 64° 57^' W. 4. Green, date, May 31'' 21^ i™ 20% red. decl. 22° 3'56" N., true alt. 72° 28' 57". Lat. ^ 32' 53" N- Raper ■' True alt. 72° 28' 57". Latitude 4° 32' 53" N. 5. Log. of diff. long. 1-804211 = diff. long. Sy-ji. Long, in 179° 11' 17" E. 6. Diflf. lat. 3757' S., mer. diff. lat. 4255', diff. long. 7560' E., log. tangent 10-249622, course S. 60° 38' E., distance 7661'. 7. Standard, Leith — 1^17'": 11^59"! a.m., and no p.m. ,, Leith — o 52 : o 3 a.m., and 0^24™ p.m. „ Leith — 2 55 : 10 21 a.m., and 10 40 p.m. „ Brest + 6 57 : 9 18 a.m., and 9 37 p.m. „ Dover -|" 3 '^ • ° 35 ^•^■1 ^^^ ° 57 ^-^- 8. Green, date, Dec. i\^ 17'' 3o''> 48% red. docl. 23° 27' 5" S., true amp. W. 84° 5' S. Error of compass 14° 21' E. Deviation 13° 49' W. 9. Green, date, Jan. 28^ 16^ 36™ 50% red. decl. 17" 57' 24" S., red. eq. time -j- 13" 22% true alt. 17° 52' 42", D. lat. made since noon 11' 6" added to lat. at noon (28'^ 45') gives lat. at sights 28° 56' 6" N., hour-angle 3^ 47'" 6^ Long, at sight 170° 54' 30" E., diff. long, since man 19' W. Suht. from long, at sights as the ship -was farther East at noon. Long, at noon 171° 13' 27" E. 10. Green, date, July 9"* 11'^ 50" io% red. decl. 22° 15' 58" N., true alt. 14° 36' 40", sum of logs. 19-174552, true azimuth N. 45° 29' r6" W. Error of compass 47° 44' 16" W. Devia- tion 54° 29' 16" W. 11. Green, date, Kov. 28^ 1511 58™ 31''; time from noon 12™ 11''; red. decl 21° 28' S'S.; true alt. 74° 15' 51" ; nat. no. 1306 ; mer. zen. dist. 15° 27' 27" N. ; latitude 6'" o' 41" S. Method II. +17' 12"- Latitude (f o' 11" S. Towson ; Hour-angle exceeds the limits of Table. 12. ist red. decl. 10'^ 18' i" S. ; ist red. eq. T. — 15'" 4^-85 ; ist true alt. 29'' 47' 32'' lat. 49° 20' gives hour-angle o'' 37™ 44^; long. 117° 50^' E. (A). Lat. 49° 50' gives hour- angle o'' 13™ 41% long. 111° 55^' E. (A') 2nd red. decl. 10' 20' 55* S. ; 2nd eq. T. — i^m gs.j ; 2nd true alt. 12° 28' 41 . Lat. 49° 20' gives hour-angle 3'^ 46"' 39'; long, 116° 4of' E. (B). Lat. 49° 50' gives hour-angle 3'' 44™ 39^; long. 116° lof E. (B') The line of bearing at first observation trends E. ^ S. (southerly) and W. \ N. (northerly). The position of the ship at the second observation is lat. 49" 52' N., long. n6° 9' E. 13. Star's decl. 8° 20' 29" S. ; true alt. 52° 11' 54". Latitude 29° 27' 37' N. Norie. Raper : True alt. 52° ii' 45". Latitude 29° 27' 46' N. The Curve.— Correct magnetic bearing S. 8^ E. Deviations.— 15° E. ; 3^ E. ; 13° W.; 28^ W. ; 14° W.; 10" E. ; 13° E.; 14' E. Compass courses.— S. by W. ; S. 87° E. ; S. 13° E. ; S. 85^' W. Magnetic courses.— N. 20° W. ; N. 54° W. ; S. 82= E. ; N. 31= E. Bearings, magnetic— S. 68'' "W. ; West. EXAMINATION PAPEE— No. XXVI, Pages 377—378. 1. 7-891486 = 77890714. 8-763323 =: -000000057986. 2. 2-900314=794-904. 7-950265=89179388. Answers. 45 1 3. True Courses.— East, 30'dep. course; S. 88° E., 54''2; N. 48= E., 4i''6; S. 76° ^Y., i4'-4; S. 22" E., 46'-5; N. 9" W., 29'-i ; S. 51° E., 49'-2 ; N. 79° E., 40'- 8 current course. Biff. lat. i5'*2 S. ; dej). i92'*2 E. : course S. 85!" E. : dist. 193'. Latitude in 59" 34' N. : diff\ long. 380J' E. : longitude in 375 49I' W. 4. Green, date, Sept. 30^ 18'' 21'" 20' : red. dccl. 3'' 9™ 32" S. : true alt. 56° 56' 24'. Lat. 29° 54' 4' N. Raper : True alt. 56' 56' 15' N., latitude 29° 54' 13" N. 4*. Green, date, July 2'' 8*^ 59™ : red. docl. 23° o' 51 ' N. : true alt. 10' 25'59". Latitude 78» 25' 58" N. AVhen the sun is oLserved below the polo (:it midnight), instead of suhtracting the true alt. from 90° add 90° to it ; the lat. will be of the same name as the declination. 5. Difif. long. 369'-7 E., or 6° 10' £1. : long, in 21° 18' W. Compass course N. 61° 29' E., or N.E. by E. \ E., nearly. 6. Diff. lat. 2000' N. : mer. diff. lat. 2045' : diff. long. 2043' W. : log. tang, io'ooqodo ; true course N. 45= "W. Compass course N. 32'' 75 W. List. 2828'"4. 7. Standard, Brest constant — o''i7'n; o''25"'a.m., and i** i™p.m. „ Brest „ -f o 50 ; 132 a.m., and 2 8 p.m. ,, Sunderland „ -}" ° 49 > ° 4^ a.m., and i 26 p.m. ,, Sunderland „ — o 52 ; 11 45 a.m., and no p.m. ,, Leith „ — o 52 ; 10 52 a.m., and 11 26 p.m. ,, Brest „ — 117; corr. for long. — i"; no a.m., and noon. 8. Green, date, April 25<* o*" 34™ 48'; red. decl. 13" 15' 18" N. ; true amp. "W. 25' 6' il. Error 67° 42' W; deviation 31'' 52' W. 9. Green, date, Aug. 23 58™ 288; time from noon 4™ 12*; red. decl. 21° 42' 46' N. ; true alt. 67° 53' 11"; nat.no. 160; mer. zon. dist. 22' 5' 21' S. Latitude cP zz' ■i,^" ^. Method II. + 1' 25". Latitude o'' 22' 38* S. Towson : Aug. I, + o' 12"; index 2. Aug. II, + i 18'. Latitude o" 22' 33" S. 12. ist red. decl. 7° 54' 3" N. ; ist eq. T. + i™ 22«-ii ; ist true alt. 45° 28' 20"; lat. 51° 10' gives hour-angle o^ 50™ 56* ; long. 122° 58^' E. (A). Lat. 51" 40' gives hour-angle o'' 39"" 51'; long. 120" 12^' E. (A') 2nd red. decl. 7° 57' 37" N. ; 2nd eq. T + i" 19'; 2nd true alt. 18' 26' 12"; lat 51' 10' gives hour-angle 4'' 41'" 35'; loag. 122° 45^ E. (B). Lat. 51° 40' gives hour-angle 4*' 40"" 59«; long. 122^ 36^' (B'). The line of bearing at the first observation trends E. by S. ^ S. (southerly) and W. by N. \ N. (northerly). The positioa of the ship at the second observation is lat 51' 10' N., long. 122'' 45V E. 13. Star's decl. 6' 48' 49" N. ; true alt. 28^ 58' 36'. Latitude 54° 12' 35" S. The Curve.— Correct magnetic bearing S. 49° E. Deviations.— 20' W. ; 16' W. ; 2° W. ; 14' E. ; 200 E. ; 15° E. ; i^ W. ; 11° W. Compass courses.— S. 33° W. ; S. 43^ E. ; N. 69" W. ; N. 24' E. Magnetic courses.— N. 9= E. ; S. 74' W. ; S. 16" W. ; S. 60^ W. Bearings, magnetic— S. 7^° E. ; S. 88° W. INDEX EEEOE, Fa^e 386. Semid. 15' 57" 2. — i' 40" Semid. 15' 45" „ 16 12 4. + o 52 „ 16 49 '» ^5 50 6. +35 5 „ 16 17-5 I. + 2' 15' 3- + 27 45 5- + 38 20 45 2 AnBwera. EXEECISES ON THE CHAET. FOR ONLY MATE, FIRST MATE, AND MASTER. North Sea, Pages 406 — 407. X. Course AV.iS. Dist. 49' 2. Course S.W. ^ W. Dist. 163' 3. „ S.E.iS. „ 149 4- „ N. JE. „ 35 5. „ N.W.byW. „ 66 6. „ N.W.^N. „ 30 / True course N.E. h'E. \ „ ( True course S.E. \ S. » 7- \ Mag. do., E. by N: i N. ) 3^9 ». | Mag. do., S. by E. ^ E. j 35 » 9. The sbip is in latitude 55° 59' N., longitude 2^ 40' E., and must sail S.E. \ S. (mag.) 209 miles. 10. The place of meeting was lat. 56= 5' N., lon;^. 3° 11' E. : the course steered by the ship from Heligoland was N.W. f W. (trur), and by the ship from Hartlepool was N.E, by E. \ E. 11. Lat. 55° 153 N. Long. 1° 11' W. 12. ,, 57 16 N. „ I 28 W. 13. „ 60 4 N. „ o 24 W. 14. „ 54 27 N. ,,03 E. 15. „ 53 20 N. „ I 36 E. 16. I st Station — Lat. 54° 24' N., long. 0^20' W., distance 6 miles. 2nd „ „ 54 24 N., „ o 1 W., „ 14 „ 17. ist „ „ 55 8 N., „ I 8 W., „ 12 „ 2nd „ „ 54 53 N., „ I I W., „ 16 „ English and Bristol Channels, and South Coast of Ireland, Pages 407 — 408. Course S.W. by W. Dist. 21' 2. Course S.S.E. \ E. Dist. 37' „ N. byE. |E. „ 44 4- „ N. by E. nearly „ 25 „ N.N.E. IE. „ 25I 6. „ N.E. IE. „ 72I „ E.byN.iN. „ 50 8- » N.byE. iE. „ 21 „ N.byW.fW. „ 6s 10. „ N.fW. „ 67 Lat. 49° 48' N. Long. 6° 72' W. Course E. by S. Distance 37' „ ,, „ E. iN. „ 40 Course S.S.W. Distance 34' „ S.S.W. i W. ,. 96 „ S.W. i S. nearly .. 159 „ S.JE. ). 69 „ N.N.W. f W. 752 „ 50 28 N. „ 2 9 W. „ „ „ „ » N.byW.^W. „ 134 „ 52 2 N. „ 6 19 W. „ S.S.E. I E. „ 30 „ 52 4 N. „ I 58 E. „ N. |E. „ 27 „ 51 25 N. „ 4 59 W. „ N.N.W. \ W. „ 32 „ 49 5i*N. „ 5 35 W. „ N.W. by W. \ W. „ 30 „ 51 41 N. „ 5 39 W. „ S.E.byS. iS. „ 46J / Dist. off Dungeness Light, 21 miles. „ 50 39IN. ,, o 35 E. I ^^ offBeachyHeadLt., 15 „ ist Course— S.E. by S. (true), or S. by E. (mag.), distance 24 miles. 2nd „ N. by E. (true), or N.E. by N. (mag.), distance 24 miles. SOUNDINGS. Depths, &c., Page 393. 1. Time from high water i'' 31™ : half-range for day 18 feet i inch : Table B 4- 12 feet 9 inches : 12 feet 9 inches + 18 feet i inch + 24 feet (4 fathoms). Depth of water required 55 feet 7 inches, or 9^ fathoms. 2. Time before high water i^ 58™ : half-range for day 7 feet 6 inches : Table B + 34 feet 9 inches. Corr. to low water 13 feet 3 inches. Depth 51 feet 3 inches, or 8 J fathoms. 4. Time from high water d^ 15" : half-range for day 20 feet 1 1 inches : Table B -^ 20 feet 7 inches. Corr to low water 43 feet 7 inches. Water below sounding 7 feet 7 inches, or the ship is found to be 7 feet 7 inches dry at low water. 6. Time from high water 3'' 59"^ : half-range for day 5 feet 5 inches : Table B — 2 feet 9 inches; Sounding by chart 6 feet i inch, or 10 fathoms (better). UNIVERSITY OF CALIFORNIA LIBRAR"? THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW "'• -] I VtO mn 30' ti jliL 12 1917 OCT 20 1917 :y A918 rT. r ^ FEB 9 t! 30»n-l,'15 ^9 VD 1566 o .^ 235923 > /4