MMMJk^ TO THE I ' 1 ! f I I I i \ kfff[ J^ UGSB LIBRARY ^ GUIDE Bodfe^J^t^^.^ TO THE LOCAL MAKINE BOAED EXAMINATION. THE OBDINAEY EXAMINATION. BY THOMAS L. AINSLEY, TEACHER OF NAVIGATION. Vsi\'(X'g-%%iml3 (^jtritijcr«, SOUTH SHIELDS: PRINTED AIvD PUBLISHED BY THOMAS L. AINSLEY, 16, MARKET PLACE. LONDON: SIMPKIN, MARSHALL, & Co., STATIONERS' H.VLL COURT; R. H. LAURIE, 53, FLEET STREET ; CHARLES WILSON (L-\TE NoRiE AND WiLSOx), 157, LEADENHALL STREET; J. IMRAY AND SON, 89 AND 102, MINORIES ; J. D. POTTER, 31, POLT^TRY. LIVERPOOL : G. PHILIP jVND SON, CAXTON BUILDINGS. EDINBURGH : J. MENZIES & Co. 1875. SOUTH SHIELDS: THOMAS L. AIXSLEY, PRINTER MARKET PEACE. 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 ot 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. ; S co wage 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, July loih, 1856. ADYEETISEMENT TO THE THIRTY-SECOND 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 come into operation on March ist, 1872. T. L. A. South Shields, Septeinher i^ih, 1875. EERATA ET COREIGENDA. Page 1 8, line 20 from the top (Ex. i), for 490 read 6497, and for 25 read 835. „ 29, Ex. 2, the first divisor should be 9, not 8 ; and the second divisor should 8, not 9. ,, 58, No. 20, for 5'oooo27 read 5'oooooo. „ 23, for 4-722522 read 4-722552. „ 27, for 6-602062 read 6'6o2o6o. „ 65, Ex. 7, the log. of -4828 should be 9"683767, not 9-683687 ; and the log. of quotient "04056 is 8-608046. „ 69, paragraph 96, for nat. sine 136' 42' ::= sine 43° 18' read nat. sine 156" 42' = sine 23° 18' ; and for nat. sine 104° 16' read 140" 16'. ,, 81, paragraph no, line 2, for learned read learner. „ 98, line 17 from bottom, for towards the spectator road towards what the spectator, &c. „ 140, No. 2, the ship's head by compass should be S.S.E. „ 149, line 20 from top, for the central shows read the central line shows. „ 312, Day's "Work, for ship's head S.S.E. i E., read S.E. | E. ,, 319, Ex. 8, line 2 from bottom, the variation should be 30° 28' W., not 30° 28' E. „ 328, Reduction to meridian, after slow on app. time at ship, insert 4'' 8"" 12'. „ 392, Paper XVII, Day's Work, second dist. should be 53', instead of 30'. CONTENTS. Notices of Examinations of Masters and Mates Notices of Alteration in Examination Papers . , Places and Days of Examinations . . Exercises in the Simple Eules of Arithmetic Decimal Fractions On Logarithms Tables of Trigonometrical Ratios . . Tables of Logarithmic Sines, &c. Navigation — Definitions of, &c. .... Preliminary Rules in Navigation — To Find Difference of Latitude To Find Meridional Difference of Latitude To Find the Latitude in To Find Middle Latitude To Find the Difference of Longitude To Find the Longitude in . , The Compass Correcting Courses — Leeway Variation . . Variation and Leeway Deviation of the Compass Napier's Diagram On the Traverse Table Traverse Sailing Parallel Sailing Middle Latitude Sailing Mercator's Sailing . . The Day's Work 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 . . . . 220 On Finding the Greenwich Date 222 To Reduce the Sun's Declination 224 To Find the Equation of Time Correction of Sun's Observed Altitude To Find tlie Latitude by Sun's Meridian Altitude Variation by an Amplitude .. .. ., .. 13 14 31 46 68 71 88 91 92 93 93 94 95 97 104 106 no 114 143 167 171 177 180 183 219 219 232 235 238 244 TIU 005TKWTI. On Finding the lime of High Water, Bv Admiralty Tide To find the rate of a Chronometer Green\\'ich Date by Chronometer To Find the Hour-angle , . On Finding the Longitude by Chronometer 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 Sta Examination Papers Quadrant and Sextant Adjustments of the Quadrant, Sextant, 6cc. On the Chart Mercator's Chart Exercises on the Chart Hon- to Find the Course to Steer in a Known Current . . On the Log Line Marking the Lead Line On Soundings Answers PAOE. bles .. .. 253 259 .. 263 266 .. 268 280 .. 288 r .. 299 • • 302 341 .. 345 349 .. 350 351 •• 352 355 .. 357 357 .. 388 EXAMINATION OF MASTERS AND MATES roR CEETIFIOATES OF COMPETENCY Under " The Merchant Shipping Act, 1854," AND VOLUNTAEY EXAMINATION IN STEAM. 1 . Under the provisions of the " Merchant Shi pping Act, 1 854 , " no " Foreign-going ship' ' * or "Home Trade Passenger Ship"* can ohtain a clearance or transire, or legally proceed to sea, from any port in the United Kingdom unless the Master thereof, and in the case of a Foreign-going Ship the First and Second Mates, or Only Mate (as the case may be), and in the case of a "Home Trade Passenger Ship" the First or Only Mate (as the case may he), have obtained and possess valid Certificates, either of Competency or Service, appro- priate 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 the 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 Mate 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 the 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 he is at the time entitled to and possessed of such Certificate, /or each offence incurs a penalty not exceeding Fifty founds. 2. Every Certificate of Competency for a "Foreign-going Ship " is to be deemed to be of a higher grade than the corresponding Certificate for a " Home Trade Passenger Ship," and entitles the lawful holder to go to sea in the corresponding grade in such last-mentioned Ship; but no Certificate for a ^^ Home Trade Passenger Ship" entitles the holder to go to sea as Master or Mate of a "Foreign-going Ship." 3. Certificates of Competency will be granted to those persons who pass the requisite exami- nations, 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 the Table marked A, page 13. The days for exami- nation 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 Marine Board if the place where they intend to be examined is a port where there is a Local Marine Board, on or before the day of examination (except in the case of Londonf and Liverpool), and must * By a " Foreign-iioing Ship " is meant one which is bound to some place out of the United Kingdom beyond the limits included between the River Elbe and Brest ; and by a " Home Trade Passenger Ship " is meant any Home Trade Ship employed in carrying Passengers ; and it is to be observed that Foreif/n Steam Ships when employed in carryiui/ Fiissciu/crs between plnces in the United Ki>i!/dom are subject ito all the provi- sions of the Act, as regards Certificates of Masters, Mates, and Engineers, to which British Steam Ships are subject : s. 291 of the Merchant Shipping Act, 1854, and s. 5 of the Merchant Shipping Act, &c., Amendment Act, 1862. t At London applications for examination must be made on Fridays from 10 till 4, and on Saturdays from 10 till 3 . At Liverpool applications for examination must be made on Tuesdays, "Wediiesdays, Ilitirsdays, and Saturdays, during office hours. B Examination of Masters and Mates. conform to any regulations in this respect which may be laid down hy the Local Marine Board ; and if this be not done, delay may be occasioned. 5. Testimonials of character, and of sobriety, experience, ability, and good conduct on board of 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 office 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 be verified by the Registrar-General ot 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 other credible person on the spot having personal knowledge of the facts required to be established. Upon application to the Superintendent of the Mercantile Marine Ofiice candidates will be 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 the proper Entries in the Articles of the Ships in which the Candidates have served cannot be counted. Thus, — for instance, a Man will state his Service to have been as Second or Only Mate, and to support this assertion will produce a Certificate of Discharge or 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 Mate. Whenever a Man has, from any cause, 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 page 13, 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 years' 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 TOE CERTIFICATES OF COMPETENCY FOE A ''FOEEIGN-GOINa 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 bearings 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. He must understand the use of the sextant, and be able to observe with it, and read off the arc. (See List A, page 9.) In Seamanship. — He must give satisfactory answers as to the rigging and unrigging of ships, stowing of holds, «S:c. ; must understand the measurement of the log-line, glass, and lead-line ; be conversant with the rule of the road, as regards both steamers and sailing vessels, and the lights and fog signals carried by them, and will also be examined as to his acquaintance with " the Commercial Code of Signals for the use of all Nations." 10. An ONLY MATE must be nineteen years of age, and have been five years at sea. In Navigation. — In addition to the qualification required for a Second Mate, an Only Mate must be able to observe and calculate the amplitude of the sun, and deduce the Examination of Masters and Mates. variation of the compass tlierefrom, and be able to find the longitude by chronometer by the usual methods. He must know how to lay off the place of the ship on the chart, both by bearings of known objects, and by latitude and longitude. He must be able to determine the error of a sextant, and to adjust it, also to find the time of high water from the known time at full and change. (See List A, page 9.) In Seamanship. — In addition to what is required for a Second Mate, 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 the ship's log. He will also be 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 log-book. 11. A FIRST MATE must be nineteen year.s of age, and have served five years at sea, of which one year must have been as either Second or Only Mate, or as both.* In Navigation. — In addition to the qualification required for an Only Mate, he must be able to observe azimuths and compute the variation ; to compare chronometers and keep their rates, and find the longitude by them from an observation of the sun ; to work the latitude by single altitude of the sun off 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 ship in stormy weather, taking in and making sail, shifting yards and masts, &c., and getting heavy weights, anchors, &c., in and out; casting a ship on a lee-shore; and securing the mast in the event of accident to the bowsprit. 12. A MASTEE must be twenty-one years of age, and have been six years at sea, of which at least one year must have been as First or Only Mate, and one year as Second Mate. In addition to the qualification for a First Mate, he must 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 wUl 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 the depths 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 sufficient 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 the measures for preventing and checking the outbreak of scurvy on board ship. He will be questioned as to his knowledge of invoices, charter-party, Lloyd's agent, and as to the nature of bottomry, and he must be acquainted with the leading lights of the channel he has been accustomed to navigate, or which he is going to use. (See List B, page 9.) In cases where an applicant for a certificate as Master Ordinary has only served in a fore-and-aft-rigged vessel, and is ignorant of the management of a square-rigged vessel, he may obtain a certificate on which the words ^'fore-and-aft-rigged vessel" will be written. This certificate does 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. 13. An EXTRA MASTER'S EXAMINATION .is voluntary, and intended for such persons as wish to prove their superior qualification?, and arejdesirous of having certificates for the 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, &c., 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 off the meridian, and also by double altitude of the sun, and to verify the result by Sumner's method. He must be able to calculate the altitudes of the sun or star when they cannot be observed for the purposes of lunars, — to find the error of a watch by the method of equal altitudes, — and to correct the altitudes observed with an artificial horizon. He must understand how to observe and apply the deviation of the compass ; and to * Service iu a superior capacity is in all cases to b& cquiviilent to service in an inferior capacitr, Examination of Ilasters and Mates. deduce the set and rate of the current from the D. R. and observation. He will be required to explain the nature of great circle sailing, and know how to apply practically that know- ledge, but ho will not be required to go into the calculations. He must be acquainted with the law of storms, so far as to know how he may probably best escape those tempests common to the East and West Indies, and known as hurricanes. In Seamanship. — The extra examination will consist of an inquiry into the competency of the 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. QUALIFICATIONS FOE CEETIFIOATES 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 he under- stands the Admiralty regulation as to lights. He must be able to take a bearing by compass, and prick off the ship's course on a chart. He must know the marks in the lead- line, and be able to work and heave the log. 15. A MASTER must have served one year as a Mate in the Foreign or Home Trade. In addition to the qualifications required for a Mate, he must show that he is capable of navigating a ship along any coast, for which purpose he will be required to draw upon a chart produced by the Examiner, the courses and distances he would run along shore from headland to headland, and to give in writing the courses and distances corrected for variation, and the bearings of the headlands and lights, and to show when the courses should be altered either to clear any danger, or to adapt it to the coast. He must under- stand how to make his soundings according to the state of the tide. He will also be 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 Ofiicial Log Book. A first-class Pilot may be examined for a Master's Certificate of Competency for Home Trade Passenger Ships, notwithstanding that he may not have served in the capacity of Mate. GENEEAL 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 be allowed five hours to perform the work ; at the expiration of which time, if they have not finished, they will be declared to have failed, unless the Local Marine Board see fit to extend the time. 17. The fee for examination must be paid to the Superintendent of the Mercantile Marine Office (Shipping Master). If a candidate fail in his examination, half the fee he has paid will be returned to him by the Superintendent of Mercantile Marine Ofiice, on his producing the Form Exn. 17, late HH, which will be given him by the Examiner. The fees are as follow: — "FOR FOREIGN-GOING SHIPS." £ s. d. Second Mate 100 First and Only Mate, if previously possessing an inferior certificate 0100 If not TOO 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 an Extra examination without paijmcnt of any fee, hut if lie fails in his first examination, half a Master's fee will he charged for each siihseqiient examination. FOR "HOME-TRADE PASSENGER SHIPS." .■€ s. d. Mate o 10 o Master 100 Examinaiion of Masters and Mates. 1 8. If the applicant passes he vnM receive the Form Exn. i6, late GG-, from the Examiner, which will entitle him to receive his Certificate of Competency from the Super- intendent of the Mercantile Marine Office, at the port to which he has directed it to he forwarded. If his testimonials have been sent to the Kegistrar to be verified, they will be returned with his Certificate. 19. If an applicant is examined for a higher grade and fails, but passes an examination of a lower grade, he may receive a certificate accordingly, hut no part of the fee will be returned. 20. In every case the Examination, whether for Only Mate, First Mate, or Master, is to commence with the problems for Second Mate. 21. In all cases of failure the candidate must be re-examined de novo. If a candidate fails in Seamanshi2) he will not be re-examined until after a lapse of Six Months, to give him time to gain experience. If he fails three times in Navigation he will not be re-examined until after a lapse of Three Months. 22. As the examinations of Masters and Mates are made compulsory, the qualifications have been kept as low as possible ; but it must be distinctly understood, that it is the inten- tion of the Board of Trade to raise the standard from time to time, whenever, as will no doubt be the case, the general attainments of officers in the merchant service shall render it possible to do so without inconvenience ; and officers 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 Masters will do well to permit apprentices and junior officers to attend schools of instruction, and to aSbrd them as much time for this purpose as possible. EXAMINATION OP MASTEES AND MATES WITH EEFEEENCE TO THE COMMEECIAL CODE OF SIGNALS FOE THE USE OF ALL NATIONS.— INSTEUCTIONS TO EXAMINEES. 23. In transmitting the accompanying copy of the latest edition of the Commercial Code of Signals for the use of the Examiners, the Board of Trade desire to direct attention to the principal points connected with this Code as to which Candidates for examination should be questioned. 24. At the same time, as the subject is probably new to some of the Examiners themselves, the Board recommend to them a perusal of the 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 Booh. The information therein given will be found sufficient to make the Examiners theo- retically acquainted with the characteristics of the New Code, and the advantages it claims to possess over other Codes, and will enable them to appreciate and urge upon Candidates for Examination the facilities which the new System of Signalling affords for easy and rapid communication . 25. The "comprehensiveness" and "distinctness" of the Commercial Code are its chief recommendations . 26. The form of the Hoist generally indicates tho nature of the Signal made, so that an observer can at sight understand the character of the Signal he sees flying. 27. The Examinations should tend to elicit a knowledge of the distinctive features of the Code above alluded to. "With this object the Examiners should make the 2, 3, and 4 Flag Signals on the Frame board which is furnished for the purpose (always taking care first to show the Ensign and the Code Pennant 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. 28. The Candidate ought to know how to find in the Signal Book the communication or the inquiry he desires to make, and how to make the Signal. The Signal to be made should invariably be sought for by the candidate in the Vocabulary and Index, Part II, and never in Part I. 29. The Candidate ought to know how to interpret a Signal. * The object of this is, of course, to distinguish the Signals from those of another Code. Masters' and Mates' Vohmtary Examinations in Steam. The Examiner should place a Signal on the Frame hoard, and vary the Signal by showing a 2 or 3 Flag Signal, or a " Geographical" or a " Vocabulary " Signal, or the name of a Merchant Ship or a Ship of War. The two latter Signals would not of course be found in the Signal Book. The Candidate ought to point thfim out in the Code List of Sh >ps. 30. A Candidate ought to be able to read oflf a Signal at sight, so far as to name the Flags composing the Hoist. 31. lie ought to know the use of the Code Pennant, and of the Pennants C and D, "Yes" and "No." 32. The Candidate should be practised in the use of the Spelling Table, by being made to spell his own name, or some word not in the Vocabulary of the Code. 33. As Ships of War use a diSerent set of Code Flags, the Candidate ought to be aware of the fact, and should know that a plate of the Admiralty Flags is to be found in the Signal Book, as well as plates of the Cude Flags which Foreign Ships of War will use in signalling to Merchant Vessels. He should also know that every OfHcial Log Book contains plates of these Code Flags. 34. A knowledge of the Distant Signals should be required of the Candidate, their object, and the mode of signalling by Distant Code, which will be found at the end of the Signal Book. For the purpose two Black Balls, two Black Square Flags, and two Black Pennants will be furnished with the Frame board, and the Candidate should be required to make one or two Distant Signals, and to read off one or two made by the Examiners. The Ball being the distinguishing symbol of the Distant Signal, any Pennants or Flags of the Code may be employed in conjunction with it, irrespective of colour. The Black Pennants and Flags are merely sent as showing best in the light background of the Frame board. SEMAPHOEES. 35. 'We have as yet no Semaphores on our coasts. The French, however, have upwards of no such stations established on their coasts, at which the Commercial Code of Signals onli/ is used. 36. A plate at the end of the Signal Book explains the method by which the arms of the Semaphore are made to represent by their position (up, down, or horizontal), the three symbols used for distant signalling, viz., a Flag, a Ball, or a Pennant. Before making Signals with the Semaphores, the Black Disc, with the white rim, should be placed on the top of the Semaphore Mast, as it properly forms a part of the Mast itself. 37. The Board of Trade think it of consequence to observe that as the Commercial Code has (in its integrity) been translated into French, and as copies of the Signal Book are furnished to all French Vessels of War and Semaphore stations, any Englishman can now, by this Code, make his wants known to them. Other nations are now negotiating for the adoption of the Commercial Code, and from the favour with which Foreigners seem to have accepted the Code wherever it has been presented to their notice, there is every reason to believe that in a short time the Mercantile Marine of all nations will have the advantage of being able to communicate by an *' Universal Language of Signals." 38. Her Majesty's Government have done all in their power to promote the use of the Commercial Code, and the Government of India and nearly all the Colonial Governments have adopted it, and a large number of Signal Books and Code Lists have already been circulated in the British Possessions abroad. MASTERS' AND MATES' VOLUNTARY EXAMI- NATIONS IN STEAM. 39. Arrangements have been made for giving to those Masters and First and Only Mates who are possessed of or entitled to certificates of competency, an opportunity of undergoing a voluntary examination as to their practical Imowledge of the use and working of the steam Local Marine Board Examinations. engine. These examinations are conducted on the premises, and under^the superintendence of the Local Marine Boards at such times as they may appoint for the purpose ; and the Examiners are selected by the Board of Trade from the Engineer Surveyors appointed under the fourth part of "The Merchant Shipping Act, 1854." 40. Any Master or Mate desiring to be examined in Steam, must deliver to the Superin- tendent of the Mercantile Marine Office, a statement in writing to that effect, upon the form of application (Exn. 2, late EE) ; if the applicant has a Certificate of Competency, such certificate must be delivered to the Shipping Master along with his statement. If he is about to pass an examination for a Certificate of Competency at the same time, the appli-. cations should be sent in together. 41. A fee of one pound must be paid by the applicant for the examination in Steam, and the Superintendent of the Mercantile Marine Office will thereupon inform him of the time and place at which he is to attend to be examined, and the examination will then and there proceed in the same manner as the other examinations. If the applicant fails, and has given in his certificate, it will at once be returned to him, but no part of the fee he has paid will he returned. 42. If he passes, the Report (Exn. 14, late FF) will be sent to the Board of Trade, and the Certificate of Competency with the Form (Exn. 2, late EE) to the Registrar-General of Seamen ; the words " Passed in Steam," with the date and place of examination, will then be entered on the certificate and its counterpart, and the certificate will be sent to the Superin- tendent of the Mercantile Marine Office of the port named in the Application (Exn. 2, late EE) to be delivered to the applicant in the usual manner. 43. The examination is viva voce, and extends to a general knowledge of the practical use and working of the steam-engine, and of the various valves, fittings, and pieces of machinery connected with it. Intricate theoretical questions on calculations of horse-power or areas of cylinders and valves, or any of the more difficult questions which appertain to steam-engines and boilers, will not bo asked. The examination will in fact be confined to what a master of a steam-vessel may be called upon to 'perform in the case of the death, incapacity, or delinquency of the engineer. 44. If the applicant fails to answer some few of the questions, and yet, in the opinion of the Examiner, possesses such a competent knowledge of the parts of the engine generally, and such other practical knowledge of the subject as will enable him to effect the object in view, the Examiner will exercise his discretion as to whether a sufficiently high standard of knowledge has been attained, and pass him or not accordingly. 45. The Examiner will provide drawings and working sections, on a sufficiently large scale, of the various parts of the steam-engine, and of the valves and slides, &c., as may be necessary, and will require the applicant to make use of them in giving his answers to the various questions put to him ; and, if an opportunity offer, the applicant will bo permitted, under the guidance of the Examiner, to start and stop the engine of some vessel which may have her steam up. CERTIFICATES OF SERYICE. 46. A Certificate of Service entitles an Officer who had served as either Master or Mate in a British Foreign-going Ship before the ist January, 1851, or as Master or Mate in a Home Trade Passenger Ship before the ist January, 1854, to serve in those capacities again ; and it also entitles an Officer who has attained or attains the rank of Lieutenant, Master, Passed Mate, or Second Master, or any higher rank in the service of Her Majesty or of the late East India Company, to serve as Master of a British JMerchant Ship, and may be had by application to the Registrar- General of Seamen, Adelaide Place, London Bridge, London, or to any Superintendent of a Mercantile Marine Office in the Outports, on the transmission and verification of the necessary certificates and testimonials. LOCAL MARINE BOARD EXAMINATIONS— NOTICE TO CANDIDATES— OFFICIAL NOTICE. I . Candidates are required to appear at the examination room punctually at the time appointed. Notice of Alteration in Examination Papers. 2. Candidates are prohibited from bringing into the examination room books or papers of any kind whatever. The slightest infringement of this regulation will subject the offender to all the penalties of a failure. 3. In the event of any Candidate being detected in defacing, blotting, writing in, or otherwise injuring any book or books belonging to the Board, the papers of such Candidate will be detained until the book or books so defaced be replaced by him. He will not, how- ever, be at liberty to remove the damaged book, which will still remain the property of the Board. 4. In the event of any Candidate being discovered copying from another, or affording any assistance or giving any information to another, or communicating in any way with another, during the time of examination, he will subject himself to a failure and its consequences. 5. No Candidate will be allowed to work out his problem on a slate or on waste paper. 6. No Candidate will be permitted to leave the room until he has given up the paper on which he is engaged. 7. Candidates will be allowed to work out the various problems by the method and tables they have been accustomed to use, and will be allowed five hours to perform the work. At the expiration of the five hours they will, if they have not finished, be declared to have failed, unless the Local Marine Board or Examiner see fit to lengthen the period in any special case. If however, the period is lengthened in any case, the special circumstances of that case, and the reasons for lengthening the period must be reported to the Board of Trade by the Examiners at the time they send in the report on Form FF. 8. Candidates will find it more convenient, both here and at sea, to correct the decli- nation and other elements from the Nautical Almanac bj' the "hourly differences " which have been given in that work in order to facilitate such calculations ; they will thereby render themselves independent of any proportional or logarithmic table for such purpose. 9. The corrections by inspection from tables given in many works on navigation will not be allowed (see Tables IX, XI, and XXI, in Norie's Epitome; Tables 21 and 38 in Raper's Navigation) ; every correction must appear on the papers of the Candidates. The First-class and Extra Master are referred to page 519 of the Nautical Almanac, 1867, for further information on this subject. 10. Candidates are expected to bring their answers to all problems within, or not to exceed, a margin of one mile of position from a correct result. ir. In finding the longitude by chronometer the logarithms used in finding the hour- angle should be taken out for seconds of arc. 12. In all other problems the logarithms to the nearest minute will be suQiciently correct for all grades, except Extra Master, from whom a degree of precision will be required, both in the work and in the results, beyond what is demanded from the inferior grades. THOMAS GRAY, Assistant Secretary. Board of Trade, Marine Department, Januarxj ist, 1869. NOTICE OF ALTERATION IN EXAMINATION PAPERS. After the rst day of March, 1872, all candidates presenting themselves for examination for Master's and Mate's Certificates for the first time, will be required to give short defi- nitions of so many of the terms contained in the following list (A) as maj' be marked with a cross by the Examiner. These questions are, at the same time, intended to test the can- didate's handwriting and spelling, to both of which special attention should be paid by him. For the " Table of Deviations " which hitherto formed part of Form Exn. 7, the questions contained in the following list (B) have been substituted. Candidates for Certificates of Competency as Masters Ordinary will be required to answer at least twelve of such of these questions as may be marked with a cross by the Examiner. Candidates for First-class Certificates (Master Extra) will be required to answer the whole of these questions. THOMAS GEAY. Notice of Alteration in Examination Papers, List A referred to in Circular. N.B. — The Candidate is to write a short definition against so many of the following tenns as may be imrJced with a cross by the Examines: Tlie Examiner will not 'tnarJi less than lo. The writing should be clear, and the spelling should not be disregarded. -< I. The Equator. 23- Declination. 2. The Poles. 24. Polar Distance. 3. A Meridian. 25- Right Ascension. 4. The Ecliptic. 26. Dip or Depression of the Horizon. 5- The Tropics. 27. Eefraction. 6. Latitude. 28. Parallax. 7- Parallels of Latitude. 29. Semi-diameter. 8. Longitude. 30. Augmentation of Moon's Semi-diameter 9- The Visible Horizon. 31. Observed Altitude. 10. The Sensible Horizon. 32. Apparent Altitude. II. The Eational Horizon, 33- True Altitude. 12. Artificial Horizon and its use. 34- Zenith Distance. 13- True course of a Ship. 35- Vertical Circles. 14. Magnetic Course. .36. Prime Vertical. 15- Compass Course. 37- Civil Time. 16. Variation of the Compass. 38. Astronomical Time, 17- Deviation of the Compass. 39- Sidereal Time. 18. The Error of the Compass. 40. Mean Time. 19. Lee "Way. 41. Apparent Time. 20. Meridian Altitude of a Celestial 42. Equation of Time. Object. 43- Hour Angle of a Celestial Object. 21. Azimuth. 44. Complement of an Arc or Angle. 22. Amplitude. 45- Supplement of Ditto. List B, referred to in Circular. Deviatiok of the Cojipass. The Candidate is to answer correctly at least eight of suei of tJie following questions as are marked with a cross by the Examiner. The Examiner will not mark less than 12. 1. What do you mean by Deviation of the Compass ? 2. How do you determine the deviation (a) when in port, and (i) when at sea ? 3. Having determined the deviation with the Ship's head on the vai'ious points of the Compass, how do you know when it is Easterly and when Westerly ? 4. Why is it necessary in order to ascertain the deviations, to bring the Ship's head in more than one direction ? 5. For accuracy, what is the least number of points to which the Ship's head should be brought ? 6. How would you find the deviation when sailing along a well known coast ? 7. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — ov- . TT J Bearing of w iL^®^^ distant object Deviation by Standard ^ standard required. Compass. 'Compass. Ship's Head by Standard Compass. Bearing of distant object by Standard Compj^s. Deviation required. North. N.E. East. S.E. South. S.W. West. N.W. lo NoUce of Alteration in Examination Papers. 8. With the deviation as above, give the courses you would steer by the Standard Compass to make the following courses, correct magnetic. 9. Supposing you have steered the following courses by the Standard Compass, find the correct magnetic courses made from the above deviation table. 10. You have taken the following bearings of two distant objects by your Standard Compass as above ; with the Ship's head at , find the bearings, correct magnetic. 11. Name some suitable objects by which you could readily obtain the deviation of the Compass when sailing along the coasts of the English Channel. 12. Do you expect the deviation to change ; if so, state under what circumstances ? 13. How often is it advisable to test the accuracy of your table of deviations ? 14. State briefly what you have chiefly to guard against in selecting a position for the Compass. 15. The compasses of iron Ships are more or less affected by what is termed the heeling error ; on what courses does this error vanish, and on what courses is it the greatest ? 16. State to which side of the ship, in the majority of cases, is the North point of the Compass drawn in the Northern hemisphere ; and what effect has it on the assumed position of the Ship when she is steering on Northerly, and also on Southerly courses ? 17. The effect being as you state, on what courses would you keep away, and on what courses would you keep closer to the wind, in (Jrder to make good a given Compass course ? 1 8. Does the same rule hold good in both hemispheres with regard to the heeling error ? EXAMINATIONS OF EXTEA MASTEES, &c. : MINOE ALTEEATIONS IN EXAMINATION PAPEES. Some misunderstanding appears to exist as to the extent of the examination in Compass Deviation which candidates for Extra Masters' Certificates ■will be required to pass. Exami- nation in the Syllabus hitherto voluntary/, became compulsory, so far as Candidates for Extra Master's Certificates were concerned, on the ist January last. It was also announced that, in addition to the questions contained in the afore-mentioned Syllabus, they would, after the ist March, 1872, be required to answer the whole of the elementary questions headed "Deviation of the Compass" on the revised Form Exn. 7. Upon further consideration, these regulations have been slightly modified. Candidates for Extra Master's Certificates will stUl be examined in the whole of the Syllabus, but it will be sufiicient if they answer two-thirds of the questions to the satisfaction of the Exami- ner ; and they will not be required to answer the elementary questions on Exn. 7. Any Master or Mate who wishes to pass a voluntary examination in the Syllabus, can at any time be examined upon payment to the Superintendent of the Mercantile Marine Ofllce of the usual fee of two pounds. If the Candidate passes the examination successfully, an endorsement to that effect will be duly made upon the Master's or JMate's Certificate held by him. If he fail to pass, the fee will not be returned. In addition to the alterations announced, the following minor alterations in the Examination Papers of Masters and Mates came into effect after the ist of March, 1872 : — Bay's Work. — (Exn. 4, No. 3.) The Deviation of the Compass will be given for the several courses in this problem, which latter will therefore require to be corrected for the same. Mercator Sailing. — (Exn. 4, No. 6.) This will in future be required from all candidates for Second Mate's Certificates. Amplitude and Azimuth. — (Exn. 5, No. 2, and Exn. 6, No. i.) From the errors of the Compass as found by these problems, and the Variation which will be given, the Candidates will be required to find the deviation of the Compass for the position or direction of the ship's head at the time the observations were.taken. Longitude hy Chronometer. — (Exn. 5, No. 3.) In this problem the rate of the Chronometer will not be given as heretofore, bul; the Candidate will be required to find the rate from two given errors on different dates. THOMAS GRAY. Notice of Alteration in JExamination JPapers. 1 1 NOTICE OF ALTEEATIONS IN THE EXAMINATIONS OF MASTEES AND MATES. On and after the rst January, 1874, candidates will be required to give \mtten answers to the questions on navigation contained in Lists A. and B., as follows, and to the question marked 0. which has been added to the paper on compass deviation. THOMAS GKAY. List A. Adjustments of the Sextant. The applicant wiU answer in writing, on a sheet of paper which will be given him by the Examiner, all the following questions, numbering his answers with the numbers corresponding to the questions : — Question. 1. What is the first adjustment of the sextant ? 2. How do you make that adjustment ? 3. What is the second adjustment ? 4. Describe how you make that adjustment ? 5. What is the third adjustment ? 6. How would you make the third adjustment ? 7. In the absence of a screw how would you proceed ? 8. How would you find the index error by the horizon ? 9. How is it to be applied ? 10. Place the index at error of minutes to be added, clamp it, and leave it. (Note. — The examiner will see that it is correct.) 11. 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. Note, — In all cases the applicants will name or otherwise point out the screws used in the various adjustments. 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 ? 13. How is the same applied ? 14. What proof have you that those measurements or angles have been taken with tolerable accuracy ? List B, Examination in Chart. The applicant will be required to answer in writing, on a sheet of paper which will be given him by the Examiner, all the following questions according to the grade of Certificate required, numbering his answers with the numbers corresponding with those on the question paper : — 1. 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? 2. How do you ascertain that in our British charts ? 3. Describe how you would find tbe course by the chart between any two places, A and B, I a AppropriaU CeHifiddes. 4. Supposing there to be pomts of variation at the first- named place, what would the course he magnetic P the true course being 5. How would you measure the distance between those two or any other two places on the chart P 6. Why would you measure it in that particular manner ? The above comprises all the questions on the chart that are put to first 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 P 8. At what time of the tide ? 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 ? 10. What do the Eoman numerals indicate that are occasionally seen near the coast, and in harbours P ir. 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 P All the above questions should be answered, but this does not preclude the Examiner from putting any other questions of a practical character, or which the local circumstances of the port may require. The following question has been added to the examination paper on compass deviation :— JQaeetion. 19. Tour steering compass having a large error, how would you proceed to correct that compass by compensating magnets and soft iron, in order to reduce the error within manageable limits ? N.B. — The candidate is required to construct a deviation curve upon a Napier's diagram supplied by the Examiner. APPROPRIATE CERTIFICATE. A PERSON possessing a Master's Certificate, whether of Competency or Service, is eligible to command any vessel of whatsoever tonnage, and either Certificate is sufficient for clearance at the Custom House. But a condition in the Charter-party of vessels taken up by Government for the conveyance of troops, stores, or emigrants, and also the Eegulations of the Principal Steam Packet Companies, require that the Master and principal Officer shall possess Certificates of Competency. The First Mate may engage as Mate of any kind. The Only Mate as Mate when there is no other ; or as Second Mate when there is a First Mate. The Second Mate is not appropriate for any superior station, and must be employed only in cases where a First Mate is also engaged. A Certificate of Competency for a " Foreign-going Ship " is equivalent to a Certificate of equal or lower grade for a " Home Trade Passenger Ship," and entitles the holder to fill the situation of Master or Mate, as the case may be. Certificates of Competency or Service may be either of a grade appropriate to the Stations held for the time being, or of any superior grade. N.B.— Oatjtiom to Opficees peoct^eiko Cbetificates of Chaeactee froji Otvnehs and Captaiks: Ctrtificates of Character from Owners and Captains, must particularly include the word "Sobriety," as tbcy caanot othennse be received by the F^aminers at the Local Maiinc Board. 13 Table A. EXAMINATION DAYS AT DATS. PLACES. I. For Masters and Mates. 2. Aberdeen* » » . » First and third Friday and Saturday in each month. Belfast .... First and third Tuesday in each month. Bristol* .... Tuesday in each week. Cork .... Second and fourth Monday in each month. Dublin .... First and third Thursday in each month. Dundee* ...» Saturday in each week. Glasgow* . . . \ Greenock* . . . / Thursdays and Fridays ; held alternately at each place. Hull* Second and fourth Tuesday in each month. Leith* .... Tuesday in each week. liiverpool* .... Every week — Monday and Tuesday "Foreign Trade;" Thursday and Friday "Homo Trade Passenger" and "Foreign Trade." London .... The examination in Navigation commences every Monday, and the examination in Seamanship takes place as soon as the Navigation examination is finished ; Master's voluntary examination in Steam held on Friday in each week. Shields, South* . Every alternate Thursday. Sunderland* . Every alternate Monday. Plymouth* .... Tuesday in each week. * At theso places Masters' Extra Examinations are held. EXEECISES IN THE SIMPLE RULES OF ARITHMETIC FOE MASTERS AND MATES OP HOME TRADE PASSENGER SHIPS. EXEECISES IN NOTATION AND NUMEEATION. I.— NOTATION. 1. Notation is the art of expressing numbers by figures or symbols, appropriated for that purpose. 2. Unit, or unity, is the name given to that quantity which is to be reckoned as one, when other quantities of the same kind are to be measured. 3. Number is the relation of a quantity to its unit ; the notion of number being suggested by successive repetitions of the individual unit, or number, is the name by which we signify how many objects or things are considered, whether one or more. When, for instance, we speak of one ship, two steamers, three masts, or four yards, the number of things referred to will be one, two, three, or four, according to the case ; and so one, two, three, four, and the rest are called numbers. 4. In the common system of arithmetic all numbers, however large or small, can be expressed by the following symbols or characters, called figures, viz. : — I 2 3 4(;67890 one, two, three, four, five, six, seven, eight, nine, nought. The first nine of these are called significant figures or digits,'^' and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly they denote the simple numbers subjoined to the characters ; where several are placed together the first figure on the right is taken for its simple value, the next, or figure standing in the second place, expresses ten times its simple value, or signifies so many tens ; thus 94 expresses ten times nine units, together with four units more ; the third, or figure standing in the third place, expresses one hundred times its simple value, or signifies so many hundreds ; thus 943 expresses one hundred times nine units, together with four times ten units, and also three units more, and so on by a ten-fold increase for each additional figure that follows it. The value which thus belongs to a figure in consequence of its position or place is called its local * Names frequently throw light on the origin of things. It is interesting to notice that the nfime digits is plainly significant of the early rude method of counting on thejingers; and that the name calculation as plainly refers to the primitive practice of reckoning with pebbles {calcuhis, a pebble). Exermes in the Simple Rules of Arithmetic. 15 100 1,000 value. Therefore, all numbers have a simple or intrinsic value, and also a local value. 5. It appears, then, that in common arithmetic we proceed towards the left from units to tens of units, from tens of units to tens of tens of units, or hundreds of units, from hundreds of units to tens of hundreds of units or thousands of units ; from thousands of units to tens of thousands of units ; from tens of thousands of units to tens of tens of thousands of units, that is to hundreds of thousands of units, thence to tens of hundreds of thousands of units, or millions of units, thence to tens of millions of units, hundreds of millions of units, &c., till we come to millions of millions of units, which are called billions of units, and so on to trillions, quadrillions, &c.'^' The actual scale is as follows : — Units a single one being written as i Teas 10 Hundreds , , , , Thousands , , Tens of Thousands . . . ^ 10 000 Hundreds of Thousands 100,000 Millions 1,000,000 Tens of Millions . , 10,000,000 Hundreds of Millions 100,000,000 Thousands of Millions ,,,,, 1,000,000,000 Tens of Thousands of Millions 10,000,000,000 Hundreds of Thousands of Millions 100,000,000,000 Billions , 1,000,000,000,000 6. When any of the denominators, units, tens, hundreds, &c., is wanting, it becomes necessary to supply its place with the last sign or character, viz., o, which is termed cypher, or nothing — the word cypher in the Arabic signi- fying vacuity. This character, which indicates the absence of all number, is a most important one, inasmuch as its introduction serves to preserve the proper position of the significant figure, thus the number /or^y thousand three hundred and twenty would be expressed in figures by 40320, because the denominations units and thousands are wanting, and the absence of each is indicated by the cypher which occupies its place. EULE I. To write in figures a number expressed in words. — TFrite down a row of noughts, or cyphers, and, as if these blanks were numbers, mark off the periods hy cutting off the last three, then the next three, then the next, and so on ; then * It is worth while to remark that as regards billions there is a difference between the French and English practice ; in French, a billion (or milliard) is one thousand million, in English a billion is a million of millions, and accordingly the word is seldom used in our language, for such large numbers are rarely of any practical use. The old books use a scale of numbers of this kind — A million of millions is a billion, A million of billions is a trillion, and so forth ; but these names are never used in practice, and can hardly be said to belong to the language of arithmetic or to English speech. Ifc may be worth a, passing notice, too, that no distinct ideas are conveyed by any of these terms ; beyond a very moderate extent our notions o',' the value of numbers become confused. The number of ones in a million even, is hard to conceive ; it is a thousand thousand, and would take more than twenty -three days to count through, kept at it for twelve hours a day, and counted one every second. 1 6 Exercises in the Simple Rules of Arithmetic, commencing at the first cypher on the left, put under each the proper figure in the number proposed, taking care that it he in its proper place ; if any vacancies appear under the corresponding cyphers, fill them up with noughts. Thus, let it be required to put into figures the number five hundred and four million, eighty-two thousand and thirty-five. We know the place of millions has six places to the right of it, we therefore put a 000,000,000 nought for the millions, and write six noughts after it, and, as we see, from hundreds being the leading word in the written expression, that the first period will be a complete period, we prefix two noughts more. The requisite number of noughts, divided as proposed, is as in the margin, and under them we now have to write, in their proper places, the figures 5, 4, 8, 2, 3, 5, and then fill up the gaps with noughts; we thus find the number, when written, to be 504,082,035. Examples. Ex. I. Express in figures, five hundred thousand six hundred and four. 000,000 500,604 Ex. 2. Express in figures, eight millions, seven thousand, seven hundred and two. 0,000,000 8,007,702 Ex. 3. Express in figures, sixty-seven million, eight hundred thousand, five hun- dred and six. 00,000,000 67,800,506 Ex. 4. Express in figures, five hundred and twenty millions, three thousand and eleven. 000,000,000 520,003,011 Examples pok Practice. Express the following numbers in Figures : — Sixty-three ; eighty-one ; ninety-nine ; forty ; thirteen. , Two hundred ; three hundred and three ; five hundred and ninety-eight ; eight hundred and eighty-eight. , Four thousand ; one thousand, seven hundred and eighty-three ; six thousand and eighty -three ; seven thousand, nine hundred and thirty ; nine thousand and nine. , Twenty-seven thousand, five hundred and four ; eighty-nine thousand and sixty-four ; thirty-three thousand. . One hundred thousand ; six hundred and seventy-six thousand and fifty ; six hundred and three thousand, two hundred and forty. , Twenty thousand, six hiindred ; ninety thousand and ninety-two ; two hundred and four thousand, six hundred and forty-one; eight hundred thousand and eight hundred. , Three million, six thousand and four ; five million, thirty thousand and forty ; seven million, seven hundred thousand and six ; ten million, ten thousand and ten. . Seven million, three thousand ; eleven million, one hundred and eight thousand, one hundred and six ; fifty-four million, fifty-four thousand and eighty eight ; six hundred and thirteen million, twenty thousand, three hundred and three. , Seventy million, seven hundred and four thousand, and thirty-two ; forty-five million, three hundred and eighty-seven thousand, and twenty -five ; three hundred and forty-nine million, four thousand and sixty-five ; one hundred million, ten thou- sand and one. , Eight hundred and forty-two million, two hundred and forty-eight thousand, four hundred and eighty-four; nine hundred and nine milUon, nine thousand and ninety-nine ; two hundred and twenty-two million, and forty ; three hundred and five million, forty thousand and eight. . Seven hundred million, seven hundred thousand and seven hundred ; two hundred and two million, two hundred and two thousand, two hundred ; nine hundred million, and nine hundred ; one hundred million, ten thousand and one. Exercises in the Simple Rules of Arithmetic. n 2.— NUMERATION. 7. Numeration ia generally applied to the converse process of expressing in words a number which is already expressed in symbols. 8. To express in words the numbers denoted by a line of figures. EULE II. 1°. Divide them into periods of three figures each, beginning at the right hand. z°. Then, commencing at the left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period pronounce the name. Note. — A glance at tho numeration table shows that tho loading figure of each set is hundreds of something; that of the first set, on the right, is hundreds of units, or simply hundreds; that of the next set, hundreds of thousands ; that of the next set, hundreds of millions, and so on. And by thus finding out the local value of the leading figure in each period, the number may be read with ease. When any of the figures is o, a little extra care is, however, necessary. Examples. Ex. I. Express in words 68547329. millions, thousands, units. The number 68547329, when divided into periods as proposed, is 68, 547, 329, pointing to the 3 you say hundreds, and passing to the 5 you say hundreds of thousands ; the incomplete period 68, must, therefore, be 68 millions ; and the entire number 68 millions, 547 thousand, 329, or expressing the whole in words it is, sixty-eight million, five hundred and forty-seven thousand, three hundred and twenty-nine. Ex. 2. Express in words 460305007. millions, thousands, units. The number 460305007 being divided into periods is 460, 305, 007, and is read, four hundred and sixty millions, three hundred and five thousand and seven. Ex. 3. Express in words 999999999. millions, thousand, units. Divided into periods this is 999, 999, 999, and is read, nine hundred and ninety- nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine, Ex. 4. Express in words 561234826479365. billions, thousand, millions, thousand, units. 561, 234, 826, 479, 365 and is read, five hundred and sixty-one billions, two hundred and thirty-four thousand eight hundred and twenty-six million, four hundred and seventy-nine thousand three hundred and sixty-five. Examples for Practice, Express in words :- I. ■43 9- 123 17- 7036 25- 690006 33- 20084216 41. 202202200 2. 60 10. 407 18. 2000 26. 8047328 34- 5001860 42. lOOIOOIOI 3- 88 II. 500 19. 3003 27. 4090300 35- 8080808 43- 275008005 4- 97 12. 999 20. 5505 28. 5210007 36. 55700005 44. 20084216 5- 59 13- 738 21. 37654 29. 6030405 37- 76014059 45- 79030184 6. 12 14. 837 22. 87078 30. 560075 38- 6006606 46. 408076032 7- 21 15- 2760 23- 37003 31- 3000006 39- 56700505 47- 401400056 8. 19 16. 5080 24. 63090 32- 1397475 40. 120015015 48. 908500060 ADDITION. 9. The process of finding a number which shall be equal to the sum of two or more numbers is called addition. The number found, or the answer, is called the sum, and numbers which are added are called addends, i8 Exercises in the Simple Rules of Arithmetic. It is usual in the applications of Arithmetic to express the operation of Addition by signs invented for the purpose : thus, the sum of 4 and 5 is expressed in the form 4 + 5 =: 9, wherein the sign -\- between 4 and 5 denotes the addition of the latter number to the former, and is read ^;/m5 or more by ; and the sign := between 5 and 9 expresses the result of such addition to be 9, or the equality between the sum of the numbers 4 and 5, and the number 9 ; so that the arithmetical expression 4 + 5 = 9 is read 4 plus 5 equals 9. Similar, 3 + 3-j-7r=i2, shows the sum of the three numbers 2, 3, 7, to be 12. 10. The rule for simple addition is as follows : — EULE III. Write the numbers to be added together in vertical columns so that the units of all the numbers may be in one column, the tens in the second, the hundreds 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 be added to the next column, writing down only the units of each column, and under the last column put the entire sum, xohatever it may be. If the sum of any column be an exact number of tens, write o for the units and carry the tens to the next column. Example. Ex. I. Let it be required to find the sum of 26389, 38127, 2815, 490, 25 and 3745. 26389 38127 2815 490 25 3745 Write the numbers as at the side, so that the figures of the same class shall be in the same vertical column ; then taking the sum of each class, we find there are 38 units, 27 tens, 31 hundreds, 25 thousands, and 5 tens of thousands. Now 38 units are 3 tens and 8 units, then writing 8 below the units column, carry the 3 tens to the 27 tens, which together make 30 tens, or 3 hundreds and o tens. Write o below the column of tens and reserve the 3 hundreds to be added to the 31 hundreds; this gives 34 hundreds, or three thousands and 4 hundreds, and 78408 writing 4 below the column of hundreds, carry the 3 thousands to the 25 thousands, and we get 28 thousands, or two 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. II. 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. (I) 321413 452734 1 3042 1 3718 24561 341323 (9) 662593 395266 841923 356627 725983 346783 (2) (3) (4) (5) (6) (7) (8) 543123 536123 123456 761284 657890 692387 876578 234512 453215 234561 612874 278679 4956 49"; 7I3I45 1234 345612 8719 579S 87658 54939 104234 4231 456223 46759 67843 769378 8797 36142 51234 561234 587999 488567 5790 358428 3451 613254 612345 987678 (13) 37429 87958 (15) 768453 (10) (11) (12) (H) (16) 846914 516398 425396 567453 169964 145673 197794 415327 854627 674958 654359 435434 366535 543543 723456 735829 827694 531769 744315 679654 765976 674216 916358 731045 765453 476757 341345 415161 328427 827146 556677 H7954 496059 569765 954131 736259 633289 889900 645679 695969 694313 643167 Exercises in the Simple Rules of Arithmetic. 19 (17) (18) (19) (20) (21) (22) (23) (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 734316 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 526606 842163 946208 161514 453148 555555 •333445 666777 888999 615827 807609 1 3 1549 567963 724483 335445 666777 888999 736846 915827 761346 313499 952637 25. 26. Add together the addends (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. SUBTEACTION. 12. The process of finding a number whicli 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 subtracted, the subtrahend ; and the result of the subtraction, the difference. Thus, then, we have, minuend — subtrahend ^ difference. We may also write this as, minuend =: subtrahend -f difference, which shows the connection between subtraction and addition. The operation of Subtraction is indicated or expressed by the sign — , which is read mmus 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. 1 3. The rule for simple subtraction is as follows : — RULE IV. 1*. Put the smaller numler under the greater, taking care, as in addition, that units shall he under units, tens under tens, hundreds under hundreds, and so on. 2°. Then, leginning at the units, subtract each figure in the lower row from the figure above it, if the loioer figure he not the greater of the two, and put the remainder underneath. (See the operation in Ex. 1). 3°. But if you come to a lower figure ivhich is greater than the figure above it, add 10 to the upper figure and then subtract, putting down the remainder as before, and talcing care to carry i to the next figure of the lower row. (See Ex. 2). 20 Uxercises in the Sm^^Je liules of Arithmetic. From 76594 Subt. 42572 Rem. 34022 From Subt. 86947 29385 Examples. Ex. 1. Let it be required to subtract 42572 from 76594. Ex. 2. Let it be required to subtract 29385 from 86947 ; then placing the former number under the latter (as in the margin) we proceed thus : 5 from 7 and 2 remains ; 8 from (not 4) but 14 and 6 remains, carry i from 8 and 5 remains. In the preceding example we see that 8 cannot be taken from the figure above it, because this is only 4, we therefore add 10 to the 4, converting it into 14 ; but the adding 10 to any figure is simply putting i before it, that is, it is adding i to i'he preceding figure, which i, by carrying it to the next lower or subtractive figure, is taken away again at the next step. In like manner, the 6 in the upper row is converted into 16, and the i thus prefixed to it is afterwards taken away, by i being carried to the next lower figure, and 3 subtracted instead of 2. It is plain that in subtraction the carrying can never amount to more than i. Rem. 57562 4 from 9 and 5 remains ; 9 from 16 and 7 remains, carry i ; 3 Ex. 3. As another example, let 84025506 be subtracted from 1 30741 394, then, having arranged the numbers, as in the margin, we "proceed thus, 6 from 14, 8, carry i ; i from 9, 8 ; 5 from 13, 8, carry 13, 4; therefore the remainder is 46715888 From Subt. I 30741 394 84025506 Rem. 46715888 6 from II, 5, carry i ; 3 from 4, 1 ; o from 7, 7 ; 4 from 10, 6, carry i ; 9 from 14. Verification of Subtraction. — The best verification is to add the subtrahend and difi'erence. This ought to give back the minuend, or original quantity from which the subtraction was made. EXERCISES IN SIMPLE SUBTRACTION. 0) (2) (3) (4) (5) (6) (7) (8) 706205 804601 980001 600501 702001 601002 501001 602004 84694 265061 980000 600492 26000 46003 20106 1 1006 (9) 701628 20449 (ro) 508000 129 (11) 403000 26001 (12) 393436 219050 (13) 321288 213788 (14) 345876 123457 (15) 206011 48605 (16) "3456 65432 (17) 36479236472 28217993216 (22) 32179836472 22222222222 (27) 230962083534589 187524828485771 (18) 3642364231 1284128417 (23) 347986312101 269887360189 (19) 7631026341 5624736794 (24) 7987642062 486428462 (28) lOIOOOIllOlOII lOIIIOOIIOIlO (20) 3462364284 2698768796 (25) lOlIOOIlOIlO lOOIlIOlOII (29) 378219362112 24686762421 (21) 23476212861 17467127437 (26) 479863217896 241826424862 (30) I270106851256I58 II96398779220936 Take each subtrahend 12 times from its minuend in the following examples : — (31) (32) (33) (34) (35) (36) 7432326 6677298 7213545 4362579 6002109 ^100630 157689 67527 57636 9873 45108 6156 Exercises in the Simple Rules of Arithmetic. 21 374 Take two thousand and nine from ten thousand and ninety-six ; three thousand and eight from seven thousand nine hundred and forty-four. 38. From three hundred and two thousand four hundred and sixty-seven take ninety- four thousand six hundred and eighty-one. 39. Take seventy-eight thousand four hundred and one from one hundred and thirty thousand. 40. Find the difference between two hundred and eighteen and one million one hundred. 41. The minuend is one hundred million one hundred and one thousand and ten, and the subtrahend is seventy million seven thousand and seven : find the remainder. 42. The population of Russia is about 67,500,000, that of France 37,050,671 ; how many more people in iiussia than in France ? MULTIPLICATION. 15. Multiplication is the finding the amount of a number repeated any number of times. The number which is repeated is called the multiplicand, the number denoting the repetitions is called the multiplier, and the amount the product. Multiplier x Multiplicand = Product. Multiplicand x Multiplier = Product. The multiplicand and multiplier are termed the factors of the product, because they are factors or makers of the product. The operation of Multiplication is expressed by the sign X) which is read into, or multiplied by ; thus, 5X7 = 35 denotes the result of the multiplication of 5 by 7 to be 35 ; so again, 4 X 5 X 13 = 260 expresses the continued product of 4, 5, and 13. Let it be required to multiply 739 by the single figure 8. Since the product of 739 by 8 is evidently equal to the sum of the products of all its parts, we have the following operation : — Thousands. Hundreds. Tens. Units. o 7 3 9 739 8 7 2 72 = product of 9 by 8 34 240= „ „ 30 „ 8 5 6 5600 = „ „ 700 „ 8 5912 5912= „ „ 739 » 8 In practice, the partial products, 72, 240, and 5600, are not written down, but combined mentally into one sum : thus we say 8 times 9 are 72, write down 2 and reserve the 739 7 tens; then 8 times 3 are 24, and the reserved 7 added thereto gives 31, write 8 down I and carry 3 to the sum of 7 by 8, or to 56 hundreds, and the entire number ' of hundreds is 59, the whole product being 5912. ^^ x6. When the multiplier is not greater than 12. EULE V. Put the multiplier under the multiplicand, units under units, and multiply each figure of the multiplicand, commencing at the unites figure hj the multiplier. Set down the right-hand figure only of the product, when it is a number of more than onefigwe, and carry as in addition. The following worked like the above example require no further explanation. 73826 9073142 531462 8 9 12 jr9o6oS 8165S178 637-7544 22 Exercises in the Simple Rules of Arithmetic. 17. When the multiplier is greater than 12, -we proceed as follows : — ^ EULE YI. 1°. Place the multiplier under the multiplicand so that the units of the former mai/ he under those of the latter, the tens under the tens, ^x. 2°. Write down the product of the whole multiplicand hy the unit's digit of the multiplier. In like manner write down the product of the multiplicand hy each of the remaining figures of the multiplier, observing to place the unit of each line in the column under the figure of the multiplier from which it came. (a) If the multiplicand contain a cypher, treat it as if it were a number, recollecting that ox 1=0, 0x2 = 0, and so on. (b) If one or more of the figures of the multiplier he o, the corresponding partial product, or products, ivill he o, cyphers, and the lines may he entirely omitted, recollecting to give its proper value to the product arising from multiplying hy the next figure. 3°. Then add all these partial products together, and their sum will he the entire product of the two factors. As before, an example will explain this rule. Ex. I. Let it te to multiply 4786 by 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 add the sums together; or, to multiply it by 2000, by 700, by 80, and by 3 times, and add the products together. Ordinary form. Now, 4786 X 3 = 14358 («) 4786 4786 X 80= 382880(3) 2783 4786 X 700 = 3350200 (c) 4786 X 2000 = 9572000 14358 units' product. 38288 tens' „ And the sum of all these is 133 19438 the product required. 33502 hundreds' „ 9572 thousands' „ 1 33 19438 complete ,, "We first multiply 4786 by 3 (a); then by 8, annexing a cypher to the right of the product {b); next by 7, annexing two cyphers to the product {c) ; and, lastly, by 2, annexing three cyphers. If the ordinary method of performing the operation be compared with the detailed process here given, it will be seen that in practice the cj^phers on the right may be omitted, provided care be taken that the first significant figure of each partial product is made to occupy its proper place, i.e., directly under the multiplying figure which supplies that product. Ex. 2. Multiply 7680426 by 500403. ^ 7680426 500403 23041278 = 3 times. 30721704 = 400 „ 38402130 =500000 „ 3843308211678 = 500403 „ Here the first figure (8) of the first partial product is set below the figure 3 in the multiplier, the first figure (4) of the second partial product below 4, the multiplying figure, and the first figure (o) of the third partial product is placed directly under 5, the multiplying figure. (This example illustrates Kule VI, 2°, (a) and (b). Exercises in the Simple Rules of Arithmetic. 23 18. Any number is multiplied by 10 by annexing one cypher, by 100 by annexing ttoo cyphers, by loooo by annexing three cyphers, &c. ; e.g., 85 X 10 = 850, for, by annexing the cypher, the 5 units have become 5 tens, and the 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 Eule V), afterwards the 327600 product is multiplied by 10, 100, or 1000 (as in Art. 18) 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. 19. Hence, if the multiplier to any proposed multiplicand consists of any one or more of the nine digits, followed by a cypher, or any number of cyphers, then multiply according to the following EULE YII. 1°. Place the multiplier under the multiplicand, so that the significant ^_(7Mr« of the multiplier shall stand under the unit's figure of the multiplicand, and multiply the successive figures of the multiplicand ly the significant figure of the multiplier, according to Eule VI. 2°. Then, to the product thus obtained, place to the right the same number of cyphers as are contained in the multiplier. Example. Multiply 123456789 by 80 and 800000. Multiplicand 123456789 Multiplier 80 Product 9876543120 Multiplicand 123456789 Multiplier 800000 Product 98765431200000 In the first of tliese examples we multiply first by 8, according to Rule VI, then annex (i.e., Join to) to the product one cypher, because 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 7?fe cyphers, in order to preserve the product in its proper place as 9 hundred of thousands. 20. If the multiplier or multiplicand, or both, end with cyphers, we may omit them in the working, and proceed according to the following EULE VIII. Multiply the significant ^_^Mr^s of the factors, as directed in Eule V. Then, to the product, aflix as many cyphers as have been omitted from the end of the multiplier or multiplicand, or both. The principle of this annexation has been already explained (No. 18). 24 Exercises in the Simple Rules of Arithmetic. Examples. Thus, if 263 be multiplied by 6200, 570 be multiplied by 3200, and 4076800 by 307000. (0 (^) (3) 263 570 4076800 6200 3200 307000 526 114 285376 1578 171 122304 1630600 1824000 1251577600000 The reason is clear : for in the first case, when ve multiply by 2, we, in fact, multiply by 200 ; and 3 multiplied by 200, gives 600. In the second case, the 7 multiplied by 2 is the same as 70 multiplied by 200 ; and 70 multiplied by 200 gives 14000. In the third the product of the significant figures is 40768 X 307 =■ 12515776, to ihh five cyphers must be annexed, because 100 X 1000=:= looooo; and 12515776 X 1 00000 n: 1251577600000. 21. It is sometimes advantageous to split up a multipliei* which is the pro- duct of two or more 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 Y. 6 ^ 36), or by 4 and 9 (4 X 9 = 36), than to multiply by long multiplication, that is, by 3 tens 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. IJere, 6 X 8 = 48 ; or, 4 X 12 = 48, then, 57894362 6 347366172 2778929376 22. Verification of Multiplication. — I. By casting out nines. — Add to- gether 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 more ; note each result. Multiply the first two 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 figure misplaced.*- Thus, in the annexed example, we say (omitting the 9) 3 and 7 are 10 ; then Multiply 90376.... 7 1 and 6 are 7, which write down. Again, 2 and 8 are i o ; ^ ° ^ "•*''■ then I and 3 are 4, which is also put down near the 271 128 multiplier. Lastly, the product of 4 and 7 are 28, and 723008 2 and 8 are 10, which is i above 9. Write, then, i near '^'^ the product, and cast the nines out of the product thus, 188253208,... r I and 8 are 9 ; 8 and 2 are 10; i and 5 are 6 and 3 are 9 ; 2 and 8 are i o, which being i above 9 shows that the operation most probably is correct. * It is plain, that if any of the figures in the product were made to exchange places, the agreement of the third and fourth results would remain, though the product would be wrong ; as would also be the case if one figure were increased and another diminished by the same number ; all, therefore, that we can safely infer is, that the agreement spoken of must have pl9.ce if the work be correct, so that if it foil, the work is wrong. Exercises in the Simple Rules of Arithmetie. H The truth of all results in multiplication may be proved by using the multiplicand as multiplier, and the multiplier as multiplicand ; if the product thus obtained be the same as the product found at first, the results are in all probability true. 25- 26. 27. 28. 29. 3°- EXERCISES IN SIMPLE MULTIPLICATION. r. 342647896 X 2 2. 654321987 X 3 3- 376543198 X 4 4- 379865782 X 5 13. 58726341 X 23 14. 78954236 X 34 15. 98765240 X 57 16. 93876129 X 95 50014000 X 270 78965430 X 700 43679854 X 806 67869578 X 903 23589647 X 678 86483279 X 567 43- 586371829 X 44. 95400621 X 45- 948375628 X 46. 987654321 X 47. 4771213 X 91823740526 6521734782 485868788 573241789 5832764985 735865000 958866 31622777 378421896 58640987 5906408 6437063 38926392 29362983 X 6 X 7 X 8 X 9 X 4689 X 30700 X 804002 X 6324553 23- 24. 6738579 70030401 8764853 123456789 602059999 X 5928578 X 98067 X 90064 X 5006701 X 77 X 84 48. 98763210 49. 49864023 50. 275361328 51- 5432149 52. 30001000300 987654321 X 10 891237654 X II 647853291 X 12 918273654 X 12 685732 X 15 903421 X 18 356628 X 36 838777 X 48 777838 X 49 434560 X 56 735846 X 64 279819 X 72 356718 X 8r 817938 X 96 X 64038040 X 708600470 X 7462170 X 8705040950 X 400100020000 53. 2793 X 812358 X 857 54. 744615 X 427282^X 15905 cc- 708421 X 930937 X 461762 X 972744 loioioi X 999999 X iiiim X 9090909 9998 X 9999 X loooo X loooi X 10002 9999 X 9999 X 9999 X 9999 X 9999 X 9999 9999 X 99980001 X 999700029999 1234 X 2345 X 3456 X 4567 55 56. 57. 58. 59- 60. DIVISION. 23. The object of division is to find how many times one number is con- tained 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, and what remains over (if any such there be) is called the remainder. Dividend = divisor x quotient + remainder. The operation of Division is expressed by the sign -f-, which is read hy or divide by ; thus, 42 -f- 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. If the divisor be not contained in the dividend an exact number of times, that which remains is called the remainder. 24. The first idea of obtaining the result is to use subtraction and oount the times we have to use it. Thus to find how many times 8 is contained in 34. 26 Exercises in the Simple Rules of Arithmetic. 34 ^ (I) 26 8 (2) (3) (4) We see we can take 8 away from 34 four 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 will 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 "something over" is evidently 34 — 32, or 2. Let it be required to divide 3168 by 27. Here the quotient will con- sist of three digits, and therefore there will be at least 3 separate sub- tractions. Now the figure in the hundreds' place ^i^g cannot be more than i, and if the product 27 hundreds, 27°° = 10° times 27 or 2700, be subtracted from the total product 3168, 'Z the remainder 468 must contain the products of the 270 = 10 times 27 tens and units of the quotient multiplied by the divisor 27. "We now inquire how often 27 is contained \\ _ t\mes 27 ten times in 468, and this is found to be onl}' once ten times; then subtracting the partial product 27 9 tens or 270 from 468, the remainder is 198. Lastly, we have to divide 198 by 27 which gives 7 for a quotient and a remainder 9 ; and, therefore, 3168 contains 27, 100 -1- 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 it will be contained in 3100, or in 3 168 ; and as often as 27 is contained in 46, so many ten times it will be 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. 25. 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 2987618 by 3605. Operation with cyphers in full. 3605)2987618(800 -}- 20 -1- 8 2884000 or 828, Operation without annexing cyphers. 3605)2987618(828 28140 103618 72100 10361 7210 31518 28840 31518 28840 2678 2678 Exercises in the Simple Rules of Arithmetic. 27 Hence we may deduce tlie following rules : — EULE IX. 26. If the divisor be not greater than 12. 1°. Set the divisor at the left hand of the dividend and draw a line leneath which the quotient is to he written. 2°. £1/ the multiplication table find the greatest number of times the divisor is contained in the first figure, or if necessary the first two, or first three figures of the dividend ; set down the quotient and ca/rry the remainder to the next figure of the dividend. 3°. Divide 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 dividend, and so on till all the figures of the dividend are exhausted. The number thus found is the quotient. Examples. Ex. I. Divide 25602 by 3. Placing the dividend and divisor (3) as in the margin, we proceed thus : — 3)256020 3 is contained in 2, no times ; so that nothing is to be placed under the 2 : 3 is ■ contained in 25, 8 times and i over ; 8 and carry i : this r, regarded as S5340 prefixed to the 6, gives the number 16: we therefore say ; 3 in 16, 5 times and i over: 3 in 10, 3 times and i over : 3 in 12, 4 times : 3 in o, o times. Therefore, the quotient is 85340 ; 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 5)7804623 4, o: 5 in 46, 9, an,d i over: 5 in 12, 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 1560924 f it, to the figures of the quotient and call 1560924 f , the complete quotient. EULE X. 27. If the divisor be greater than 12. 1°. At the right hand of the dividend, draw a line for the quotient; at its left hand, and in a line with it, write the divisor. 2°. Mark off a number of fig ares, from the left hand side of the dividend, equal in number to those of the divisor^ or one more if necessary and find the greatest number of times the divisor is contained in this number ; write down this as the f/rst figure of the quotient. 3°. Multiply the divisor by this number, and place the product under the number marked off from the dividend, and subtract. 4°. Bring down to the remainder the next figure of the dividend, and if the remainder thus increased be greater than the divisor, find the greatest number of times the divisor is contained in it, and write this number as the second figure of the quotient, but if not bring down the next figure of the dividetid, or more, until it is greater — recollecting to place a cypher in the quotient for every figure of the dividend so taken except the last : find how often the divisor is contained in this number ; then multiply, subtract, and bring down, Sfc, as before, till all the figures of the dividend are exhausted. The number thus obtained is the quotient required. 28 I^xercises in tJie Simple Rules of Arithmetic. 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 ^'^^3 figures of the dividend, and seeing that 3 is contained in 25 8 ___ times, we should put 8 for the first quotient figure ; but bearing .„ . in mind that when the whole divisor is multiplied by this 8 we 346 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 48 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 the latter will go 4 times, we therefore put 4 for the second quotient figure, and multiplying 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 07ice, 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 94 1 -^i^, which is the 346th part of 256434. Ex. 2. Divide 108419716214 by 5783. 5783) 108419716214(18748005 quotient. Quotient figure (i) 5783 50589 I st remainder with next figure. (8) 46264 43257 2nd „ „ (7) 40481 27761 3rd „ „ (4) 23132 46296 4th „ „ (8) 46264 1^ (o). 322 5th 3221 ,. 6th 32214 7tli „ ,, (5) 28915 3299 final remainder. It must be noticed that if any dividend formed by a remainder and a figure brought down should be less than the 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 the operation annexed. The steps marked l^° are inserted merely to show the principle. In practice we simply put down the two noughts in the quotient, and go at once to 32214 for the divisor. 28. Wheiiever the divisor can be separated into two factors, the division may be effected by the following rule : — EULE XI. 1". Divide by one factor, setting down the quotient and remainder. 2". Divide the quotient hy the other factor, setting down the quotient and re* mwinder ; the smnd qmtient thus obtained is the required quotient* Ux&rcises in the Simjoh Rules of Arithmetic. 29 3°. The proper remainder is found hy multiplying the second remainder hy the first divisor, and to the product adding the first remainder. This rule may be extended to the case of the divisor being divisible into any number of factors, as follows, always setting down as remainder the product of the partial remainder hy all the previous divisors increased by the previous remainder. Examples. Ex. I. Divide 569736869 by 15. Here the remainder 2 in the first quotient is 2 units of the upper line ; but the remainder 4 in the second line consists of 4 15 units of the second line; and as each unit in the second line is three times as great as each unit in the upper line, the remainder 4 is equal to 3 X 4 units of the upper line, i.e., is equal to 12 ordinary units, hence the whole remainder is 2 -4- 12, or is 14. Ex. 2. Divide 8327965 by 72 and 99. 569736869 37982457.... 4 72 l9 8327965 925329- 99 8327965 925329 ii5666.,..i 84120. ,.,9 To deduce the remainders which would have been left had the divisions been performed by 72 and 99 in the usual way, we may observe that the first partial remainder 4 must be units ; but the second dividend being so many collections of 9 units each, the second re- mainder must be regarded as so many collections of 9 units each ; hence the true remainders in these examples are respectively I X 9 + 4 = i3> and 9 X 9 + 4 = 85. Ex. 3. Divide 2671998 by 192. 192 2671998 667999,. ..2 Ans, — Quotient = 139 16 Bemainder =126 "1333. X4+2=6 13916....5 X 6 X 4 + 6 = 126 29. Division may also be abridged where the divisor is terminated by a cypher or cyphers ; we proceed as follows : — EULE XII. 1 °. Cut off the cyphers from the divisor, and as many figures 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 manner (Rule X or XI), and if there are anything remaining after the division annex those figures which are cut off from the dividend; otherwise, the figures cut of will he the remainder. Examples. Ex. I. Divide 3704196 by 20. 2,0)370419,6 185209 \% Ex. 2. Divide 31086901 by 7100. 71,00)310869,01(4378 fi^ 213 ~^ 497 HI 568 31 ^o Exercises in the Simple Rules of Arithmetic. In the first of these examples you mark off with a turned comma the cypher or o in the divisor, and the first figure 6 to the right in the dividend ; this is equivalent to dividing both divisor and dividend by lo. You next divide the remaining figures 370419, to the left in the dividend, by the divisor 2, according to Eule IX ; thus is obtained the quotient 185209, and remainder i ; to this remainder you annex the figure 6, which was cut off, and you have the complete remainder 16. The quotient may now be correctly represented thus, I 85209^1. In the second example you follow the same rule ; that is, you cut off two cyphers in the divisor and two figures in the dividend, and obtain the quotient in the usual way, which is 4378, and remainder 31 ; to this 31 annex the two figures cut off from .the dividend, and you have the complete remainder 31 01. 30. 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 difference 80 obtained hy the quotient. The result should be equal to the divisor, if the working be correct. EXERCISES IN SIMPLE DIVISION. I- 135792^95 "T- 2 5. 400678493 -f- 6 9. 254096146 -7- 10 2. 584697386 -f- 3 6. 276586437 -7- 7 10. 1101182267 -7- II 3- 399345884 -r 4 7- 6947421006 -f- 8 II. 1095137170 -f- 12 4. 298244760 -f- 5 8. 2470263075 -j- 9 12. 59437055312 -j- 12 13. 6489275432689467 -r 14 18. 987654321012345 -f- 66 14- 598432789648320758 -r 22 19- 9357864837986496 -f- 70 15. 56983475689268 -f- 36 20. 483795864973206789 -7- 120 16. 9357864837986496 -f- 50 21. 3591321391621911 -f- 132 17. 5986432685946896 -r- 63 22. 4902550716552769 -f- 144 23. 987654321670 -^ 3000 26. 27410012221749999 -f- 37009 24. 17932810740000 -J- 2600 27. 149778007923526 -j- 618934 25. 147 10962989869 -7- 1709 28. 42243968241835 — 872169 29. 31415926536 -J- 648 34. 6345670000000 -f- 45630425 30, I 000000000000000 31- 4895300478 32. 655650751827 mil 35. 6680943744279021 -f- 467I ^ 5678 36. 4842315713782 — 570634 396218 37. 815240906170 -f- 763054 33. 33201610691892 -j- 7043628 38. 34939053124326 -r 8431072 39. Divide 31415926536 into 100 with as many noughts added as may be necessary to give ten figures in the quotient. t. Express in figures, ten thousand and four. 2. 29483 + 7648 + 32479 + 586 + 298364 + 98765 -f 897 + 789 + 5678 + 99. 3. From 6794006897 take 3985160534. 4. Multiply 94785830 by 78060. 5. Divide 5688208152 by 594. 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. Eind the difference between 1 00000000000 and 87649786. 9. Multiply 326904678 by 3060900. to. Divide 236487698743 by 85409. II. Express in figures, one hundred and three million, eighty thousand, two hundred and seven. Decimal Fractions. 31 12. Add together 69074, 6745, 723, 29, 931648, 9005, 76245, 54267, 47096, and 7777. 13. From 78600070000 take 6974208506. 14. Multiply 167409678 by 768900. 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 farther 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 1 01 001, 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 twentj'-five millions, seven thousand and twenty-one ? 21. Find the sum, difference, and product of 12345678 and 288144412. Find the sum, difference, and product of 1234567 and 4321089. 22. 15996 tons of coal are exported in 43 ships: how many tons does each ship on the average carry ? 23. How many years of 365 days each in 46355 days ? 24. How often can you subtract 6 from 47112 ? 25. How many ships, each carrying 673 men, can transport an army of 22882 men ? 26. By what number must you divide 7460020 in order that the quotient may be 52907 and the remainder 133 ? 27. 2036809 divided by a certain number gives a quotient 2031 with a remainder of 1747 ; find the dividing number ? 28. A ream of paper contains 20 quires of 24 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 15 19 square inches, if there be 85 in the square inch. 30. What will remain after subtracting 213 as often as possible from 83216 ? 31. The product of two numbers is 1270374 and half of one of them is 3129 : what is the other number ? 32. Find the sum, difference, product, and quotient of 1653125 and 13225. 33. Find the sum, difference, product, and quotient of 9765625 and 78125. DECIMAL FRACTIONS. 31. Amthmetioal operations become lengthy and troublesome if tbey 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 i o, i go, i 000, &c. Such fractions are called decimal fractions. 32. 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 Yulgar Fractions. 33. 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 be if it stood one place further to the left ; thus the number 1 1 1 1 denotes one thousand, one 32 Decimal Fractions. hundred, one ten, and one unit, or looo + 100+ 10 + ij 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 vrith 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, the figures decrease in a tenfold ratio from left to right. 34. Now we may evidently extend this principle still further, and on the same plan may represent one-tenth of one, one-tenth of this, or one-hundredth of one, one-thousandth of one, and so on, by simply putting some mark of separation between the integers and these fractions. The mark actually used is a dot or full stop, and is called the decimal point, thus 1 1 1 n 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 10; the next is 100; the next 1000, and so on. In like manner, the unit next the decimal point, on the right, is to, the next tc^, the next i^roo, 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-hundreth of what its value would be if it were in the unit's place, and so on for any number of figures, as in the following table, which may be regarded as an extension of the numeration table. g^S^^^s Si S ^ S ^ B S 7654321-2345678 35. 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 : — 507 — To ^ To77 ~ 1000 — 10 00" 0305 To I 100 > 100 1^ 10000 1 00 0* 10 Z04 — io -T To' TToo i 1000 — 1000* 36. Hence, a decimal is always equivalent to the vulgar fraction whose numerator is the decimal considered as integral, that is, the number itself, tchen the decimal point is suppressed, and whose denominator is i folloxoed by as many cyphers as there are decimal places in it. 37. We generally speak of any figure in a decimal as being in stich a place of decimals ; thus, for instance, in 3-14159, we should say that the 5 is in the fourth place of decimals, the 9 in the fifth place, and so on, reckoning from left to right. 38. 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 * The decimal point ahoiild be put at the top of the line of figures, thus — 5-7, because 5.7 with a stop at the bottom is used in most ^701118 to mean 5 X 7 = 35. Decimal Fractions. 33 say that 2 is a significant digit, becauso it is the first figure which indicatea a number, the cyphers only serving to fix the place in which the 2 occurs. 39. 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 (the 3, the 6, and the 8) are whole numbers or integers, while those which follow the point ("414) are decimals. 40. To read off, or express in words, decimal fractions, read the decimal figures as (/"whole numbers, and to the last figure add the name of the order deter- mined hy 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 unWs place should alwaj^s be made the starting point. It is advisable for the learner to apply to everj'- 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, hundredths, thousandths, &c., pointing to each figure as he pronounces the name of its order. In this way he will be able to read decimals with as much ease as he can whole numbers. 41. 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 significayvb figures. For instance, 07 is the same as 0*70, because the number that expresses the decimal fraction becomes ten times greater while its parts become hundredths, and are therefore diminished ten times, f>in9 -^- — -^2 .700. f.f, bUUS ]^o 100 lOOOj '^''• and hence it is evident that annexing cyphers to the' right hand of decimals does not change their value, for wej only multiply both numerator and denominator by 10, 100, &c., and consequently does not alter their value at all. Again, take a decimal such as -56, which, as already explained, means 5 tenths 6 hundredths, it will follow tliat '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 = ii^ and -560 = J^%«^ = -^. Similarly '23, '230, and "2300, are all of equal value, for expressed as fractions they are respectively ^-io, to¥o, and t¥tjoo. 42. 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 value being diminished ten times for each cypher that is prefixed, thus "7 = I'V, '07 = i-J-jj, -007 = TTnnT> ^^'^ so on. We infer from this, that as the value of a decimal is decreased ten-fold for every cypher added to the left-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 by 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 34 Addition of JDecimah. divided by loo if written -56789, 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 written '723, and is multiplied by 1000 if written 7-23. Examples for Practice. Express as decimals — T 3 3 __3 „„,1 .13. nlari -7 lit 3 3 o^/l 1015 1 21 117 3 1. S3 7. _11 Q-nd 182. 2' Tinr> looou' TT)Tnnr> louOuOuo' loi loo; 1000 ioooo> "'^'^ looooo' -, 3.oi Aoaai .asfcoiis 60009101 ±41 _a3i_ 101 3- 10 » 100 > 100000, mmionths, thousandtlis, tenths? billionths- A ^17fi_ 91X8. nl78 al . I> S203 90 4' 10000) 100 » 100000) 10000) looboo, lo > loo* r .3filA4 6728 1!) 672.8.19 67281900 S' lOOOb) lOOOOO) 10000000001 lOOOO • 6. In the following mixed numbers write the -fractional parts in decimals : — 1 80 A 1 2 145 i-i 78234 5 7 7 lOOOi 4i iTnroTT) y 10000000 tT) „ P, 9 ,- 78 /C 89 ,.7000. . 400637 f^r.^ ^S.Q±0\_ /• 53 To) 47 T!Rr> " ToooO) i 10 00 0' 9 looooooooo) y'-'-' loooooo- 8. Express as decimal fractions the following : — Seventy-three thousandths ; one hundred and ninety-seven ten thousandths; one millionth; two hundred and sixty-one hundred thousandths ; one thousand and one ten millionths. 9. 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. 10. One tenth ; three hundredths ; five thousandths ; one hundred and five thousandths ; two millionths ; sixty millionths ; forty-one and eight hundredths ; one thousand and one thousandth ; thirty and six millionths ; one hundred thousandth ; two thousand three hundred and seventy-five hundred millionths. 11. Express as vulgar fractions — •7, -07, -007, -000007, -327, 3-27, 32-7, -45697, 456-97, -893, -0000893. 12. Express in words the following decimals and mixed numbers : — •283, -5321, -74895, -821056, 27-8354, 34-0009, 43-101007, 23-75, 2-375, -2375, •00002375. 13. Express as vulgar fractions, and reduce to the lowest terms : — -0000001024, -000000000576, -9241, 67-09, '5064919, and -00000000000000065536. 14. Express in words the following : — •6, -92, -5498, 7-07, 26-405, -oooooi, -00037, ii-ioiioi, -0440308, -82344, -13236. 15. Write in words 9-0457; 4004-0000345; 3-400; 524000634-0008034; -000003705; •000024056; 7005-000000674; looooo'ooooooi ; lo-ooi ; 9-000028; 1-0006003. 16. I -00000 1 ; ■ 1 00000 1 ; 'oooooooi ; 1-13004; 9-203167; 4-3008004; 27-4627350. ADDITION. 43. 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 hundredths, or hundredths under thousandths, as the case might be ; and we should be attempting to add together fractions which had not common denominators. 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 them we at once bring the several fractions to a common denomi- nator, 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, whether 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 rules of ordinary addition are immediately applicable. Additiorfof Decimals. 3!? We have, therefore, the following rule for addition :— EULE XIII. 1°. Place the quantities so that their decimal points shall he in the same vertical 'ne ; for then the quantities of the same denomination will stand together. 2°. Then proceed as in addition of whole numbers. Examples. For instance, let it be required to add together -8, -78, and -678. "Where we see that after writing in the answer'8 in the place of thousandths, that 7 hundredths and 8 hundredths added together make 15 hundredths ; but 15 hun- dredths are i tenth and 5 hundredths, writing 5 in the place of hundredths, and carrying one to the place of tenths, we obtain 22 tenths ; but 22 tenths are properly written as 2 integers and 3 tenths. •78 •678 2-258 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 ia the place of thousandths, and carrj'ing i to the place of hundredths, we obtain 10 as the sum 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 46-0045 place of integers, and write 6 in the place of units, and 4 in the place of tens. Ex. 3. Add together 0*35, 47-4, and 9"I2. 0-35 47 '4 9.12 56-87 Ex. 5. Add together i234'6789, 13, 170, -0054, -5, and 87-142. 1234-6789 13 170 •0054 •5 87-142 i505"3263 Ex. 4. Add together 23*628, 4*1056, •0137, and "0042. 23*628 4*1056 •0137 -0042 27'75i5 Ex. 6. Add together 66i99'3226, '301, 54-5, -00632, 1000, -07, and 32745-80008. 66199-3226 -301 54-5 *oo632 1000 •07 32745-80008 Examples fob Pbactice. Find the value of 1. •225, 3-086, 12*17, '005I) ^ii'i 729*54; 2-63, '263, *o263, and -000263. 2. 8-1, 40-652, 98-51, 6957, 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, 360826, 360826, and -22314; 467*3004, 28-78249, 1-29468, and 3-78241. 5- 36"053, '0079' '000952, 417, 85-5803, and -0000501. 6. 87-1 + 0-376 + "0056-1-49 + 3*009 -J- -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, 51-720345, 2-684 and 62-304607. 8. I + -I + -01 + -001 + -0001 ; 4-07 + -6201 + -936 + 29-08 + 1*0101 + 7. 9. I + -2 + -03 + -004 + .0005 ; -7, 5008, 312-907, -4093, 494-5, and 87-003, 36 Subtraction of Decimals. What is the sum of 10. Eighteen hundredths ; seven hundred and forty-five hundred thousandths ; nine thousandths ; lorty-three millionths ; five hundred and eight thousandths ; one hundred and thirty-two thousandths ; one thousand and forty-four ten millionths ; twenty-five hundredths ; five tenths ; and six hundred and five thousandths ? 11. Add together 9 tenths, 92 hundredths, 162 thousandths, 489 thousandths, and 92 millionths. 12. Add together i tenth, 2 hundredths, 16 thousandths, 7 millionths, 26 thousandths, 95 ten millionthsj^and 7 ten thousandths. SUBTEACTION. 44. 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 thousandths, hundredths from hundredths, and tenths from tenths. The decimal point in the answer will fall exactlj' 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 32), and the subtraction may proceed as in whole numbers. "We have, therefore, the following rule for subtraction: — EULE xiy. 1°. Place the quantities so that their decimal points shall he in the same vertical line. 2° Next proceed as in subtraction of whole numbers. Examples. Ex. 2 Ex. I. Subtract '756 from -897. •897 •756 •141 Here the difference between 6 thousandths and 7 thousandths is i thousandth, between 9 hundredths and 5 hundredths is 4 hundredths, between 8 tenths and 7 tenths is i tenth. Ex. 3. From 98765"432i take 99*99. 98765-4321 99'99 From 37'6^take '907. In this instance 37 '6 may be written 37'6oo. 37 600 •907 36-693 98665-4421 Ex. 4. Subtract -97658 from 5'i394, 5'i394 •97658 Ex. 5. Subtract -0000999 from -01. -01 •0000999 4-16282 In subtracting 8, o is supposed to occupy the place above it as 5-13940 ^ 5*1394. Ex. 6. From i take -47712. I •47712 •522i -0099001 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 lino. Multiplication of Decimals. 37 Ex. 7. Subtract 247-258746 from 347-258745. 347-258745 24T25S-J4.6 99-999999 Ex. 8. From i take - 00000 1. •999999 Examples for Pbactioe. Subtract r. 3-07 from 6-501 ; '79999 from 9 ; 2-9989 from 3 ; -999999 from 9. •0090806 from 39-857; -00032 from 32; •876534from 8-21314; 364'3i23 from 456-0546. •99 from I ; -00000099 from 99 ; 'oooooi from 10; 3-29 from 999; 25-6050 from 567-392. •9682347 from 65-00001 ; -79999 from 9; 9'i63 from 81-6823401. •OOOOOI from -oooi ; '000004 from ^0004; ^00032 from 32; ^87623 from 24681. From 700 take 7 hundredths; from •oooi take 'ooooooi. From 42 hundredths take 42 thousandths; 154 millionths from 6231 hundred thou- sandths. From 96 thousandths take 909 ten thousandths; 92 thousandths from 29 thousand. MULTIPLICATION. 45. "We have stated that for every place we shift the decimal point to the riffht we increase the value of the decimal fen-fold, for every place we shift it to the left we decrease it fen-fold. Now, in multiplying two decimals together, since the law of local value hold with regard to the digits comprising the decimals, the process of multiplication will he performed exactly as in ordinary whole numbers ; the only matter requiring consideration will be the proper position of the decimal point. Suppose we have to multiply .4-935 by 6-28, and let us suppose the decimal point in each case removed to the extreme right. Then (Art. 42, page 33) we have multiplied the number 4-935 by 1000, and the number 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 one hundred fold respectively, it is evident our product is increased 1000 X 100, or one hundred thousand fold. Dividing, therefore, the above result, 3099180 by loooooo, or what is the same thing (Art. 42, page 33), writing it 3099180" and removing the decimal point 5 places to the left, we get for the product of the numbers 4-935 and 6-28, the result 30-99180. It will be seen that the number of decimal places on the product, namely, 5, is the sum of the numbers of the decimal figures in the two given numbers. We have, therefore, the following rule for multiplication : — EULE XV. Mkltiply the numbers together, as whole numbers, and point off as many decimal places in the product (beginning at the right) as there are decimal places in the multiplier and multiplicand together. When the decimal places to be pointed off are more in number than the figures of the pjoduct, make up the proper number by prefixing cyphers to the product. Examples. Ex. I. Multiply 34"ii by 3^72. 34'" 3'72 6822 23877 10233 126-8892 In 34*11 are two decimals; in 3-72 are two; therefore four decimal places are pointed off. Ex. Multiply 236000 by -48. 236000 •48 944 113280-00 The product of 236 by -48 is 11 328; in 236000 are no decimals; in -48 are two decimals ; therefore two places are pointed off in the product. 38 l>wmon of Decimals. Ex. 3. Multiply 56'3 by 'oS. 56-3 o-o8 4'504 In 56" 3 is one decimal; in '08 are two; therefore three places are pointed off in the product. Ex. 4. Multiply 5*63 by -00005. 5'63 0-00005 Ex.5. Multiply '0048 by '000012. •0048 •0000000576 The product of 48 by 12 is 576 ; in •0048 are four decimals; in '000012 are six decimals ; therefore the product must contain ten decimals (four and six), and seven cyphers are prefixed to 576, whence the product is '0000000576, as above. 0-0002815 In 5-63 are two decimals; in '00005 are five ; therefore three cyphers must be pre- fixed to tke product 2815, and seven decimals marked off. Ex. 6. Find the value of i"005 X '005 X .0064. 1-005 •005 .005025 .0064 20100 300150 ■0000321600 Examples for Practice. Find the value of 2'5 X 4; "25 X 40; 2"5 X 476; 2*5 X 4*76 ; -0025 X 4-76 ; -025 X 0476. •0002 X "ooioi ; go'oi X 0-034; '0008 X '00014; and '6005 X "0035. •0783 X '461 ; "2764 X 96; -06948 X '0087 ; and -00043 X 4700. 21-56 X -0035; 24-35 X -074; 35-85 X 2-09; and -004716 X -22240656. 1-075 X "oioi ; 8-004 X '004; -0006 X "00012; and -923521 X '28629151. loo'oooS X -000306; 7535060 X 62-3906; and 31-50301 X i7'0352. -000713 X 2-30561 ; 42-10062 X 3-821013; and i'oi42034 X '0620034. 25067823 X -ooooooi ; 394-2003 X -00000003; ^^^ '834567834 X -00000008. 47-83 by 10, 100, 1000, 3L, yio, T-gVo ; "5 X 1000; -75 X looooo. 22-5 X -0241 X -0024; -0003 X '01 X 500000; -006 X -00012. 2.7 X "27 X "027 X 270; -2 X '04 X 'ooS X 64000; 8-004 X -004. I'l X 'Oil X I'oi X -oioi; -013 X i'6 X '007 X Y°5'i ^°°Z X 6-12. DIVISION. 46. 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 looo-fold, and the divisor loo-fold. The former of these alterations will have the same effect 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 off to give a correct result. Had it been required to divide 370'i5 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 loo-fold; this would not affect the value of the result, and the quotient would be 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 decimal places in the dividend than in the divisor, by converting both into whole numbers we should have increased the dividend loo-fold and the divisor 1 000-fold. This would have decreased the divisor lo-fold, and to obtain the correct result we should have had to multiply the quotient by 10. Division of JDecimals. 39 "We can hence determine the following practical rule for the division of decimals : — EULE XVI. Divide as in whole numbers, and point off in the quotient 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. Ex. I. Divide i7'68 by 3-4. 3-4)i7-68(5-2 170 68 68 Here 17 '68 contains two decimals; 3-4 contains one ; therefore 52 must contain the remaining one required, and be written 5'2. Examples. Ex. Divide 547*8 by 66. 66)547-8(8-3 528 19S Here 547*8 contains one decimal; 66 none; hence 83 must contain one, and be written 8" 3. Ex. 3. Divide 4*784 by 9-2. 9-2)4-784(-52 460 184 184 Here 4*784 contains three decimals, and 9*2 one, the remaining two required must therefore be obtained by pointing off both figures, thus, *52. Ex.4. Divide i353'6by 37*6. 37"6)i353'6(36 1128 2256 2256 Here the dividend has one decimal, and the divisor also one, or as many, and the quotient is therefore an integer. (a) When the divide?id has no decimals, cyphers must he annexed, preceded hy the decimal point. Ex. 5. Divide 38 by *o8. •08)38*00 475 Annex two cyphers to 38 ; then the dividend contains two cyphers, and the divisor also two, and the quotient is there- fore an integer. Ex. 6. Divide 132 by 0-7. 0*7)132*00000 1885714 Annexing five cyphers (decimals) gives quotient 1 8857 14. Then the number which added to one decimal in 0*7 to make up five is four. Ans. : 188*5714. (b) When the number of figures in the quotient is not sufficient to make up the required number of decimals, cyphers must be prefixed. Ex. 7. Divide *30285 by 67*3. 67'3)'30285(45 2692 3365 3365 Here *30285 contains five decimals, and 67*3 contains only one; the quotient 45 contains only two, and four are required ; hence two cyphers must be prefixed, and the c^uotient written as *oo45. Ex. 8. Divide 4*784 by 92. 92)4*784(52 460 184 184 Here 4*784 contains three decimals, and 92 none ; the quotient therefore must contain three, and becomes '052. 40 Division of Decimals. (c) When the number of decimal places in the dividend is less than the number in the divisor, annex as many cyphers to the dividend as will make the number equal to the numher in the divisor, and then proceed as in whole numbers. Ex. 9. Divide i4'4 by -0012. The divisor contains four decimal places and the dividend only one, annex three cyphers to the di\adend, so as to make the numher of decimal places in both divisor and dividend equal, and then divide hy 12, and the work will stand thus : — •001 2) 14' 4000 As the numher of decimal figures in the dividend is equal to that of the divisor, we have none to cut off in the quotient. The answer is therefore 12000, an integer. Ex. 10. Divide 3028*5 hy '673. In this instance the divisor contains three decimal places and the dividend only one ; therefore two cyphers are annexed to make the numher of decimal places in each equal. •673)3028-500(4500 2692 3365 3365 The number of decimal figures in the dividend and divisor being equal, we have none to cut off in the quotient. The answer 4500, is therefore an integer. The division may always be carried to any degree of accuracy by annexing cyphers to the dividend, as seen in example 6 [a). 47. The decimal point may be removed altogether from both the divisor and dividend, by continually multiplying each by j o ; for the quotient -will thus remain unaltered. The first decimal in the quotient will then appear only with the first cypher annexed to carry on the division. Example. — Divide 5580 by 0-04. Multiplied by 10 they become 558 and 0-4 multiplied again they become 5580 and 4, the quotient of which is 1395. This easy process furnishes a complete security against wrongly placing the decimal point in the quotient. Examples for Practice. Divide 99-12105 by 5, 7, 9, 36, and 88. „ 6-66816 by 4, 6, 8, 25, and 630. „ -6052158 by 3, 6, 9, 24, and 6400. 6541-234567 by 21; 7461-30765 by 112; 2188-054 by 983. „ 236-041 by 1-75; 67234 by 85; 60-0001 by i-oi ; 300-402 by 12-1. » 325-67543 by 20 02 ; 6842346 by 2682 ; 73-8243 by -061 ; -00006 by -003. „ 34'82725 ty39'5275; 897-25 by -98725 ; 42-9365 by 387-25; and 3-1415926 by 1 80, true to seven decimal places. „ 17-084522 by -024 ; 1237-0519 by -5425 ; 762-151 by -00325. ,, -00033 by -on ; 140-02564 by 1-871 ; 4/32067 by -ooi ; -59 by 80000; 167342 by -002; -001024 by 30517578125; looi by 16384. „ 992 by -37 ; -1599 by 4100; -019 by 190; i by -152 ; -2 by 23-2. „ 1-95 by -00013; 9 614 by -0000019 ; -25 by 31*25; 8-92 by 237-6567. „ -03679 by 283; 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 t>y -01004 j ^"^SS ^y '°'°4) 4 by -ooooi. „ -001255 by ^°°4 > '°i by 1000 ; 202 b}' -oi ; 6821091-97627 hy 88-03. „ 13099 52 by -OOI 1008 ; -005868 by -036, and arrange the divisor, dividend, and quotient in order of magnitude. „ -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; 6iooby-!*25; 23 by -000579 ; 37"694i6 bj' '156 ; -0016 by •000008 ; 6 by -0000001 ; -8 by -0000002 ; -000054 by 9 ; 4000 by •000125, Reduction of Becimah. ±1 19. „ 36 by loooo; 9-3 by 10; 52*306 by 100; 8 by loooo; 2"oo76364 by loooooo. 20. „ — ^ to 10 decimal places; i^ to 12 decimal places; -^^ to 2 c decimal places. 17 1130 lo-oi ^ 21. „ '^''J to 25 decinml places ; °^J^ to 15 decimal places; :2££9840oi_8 ^^ nil _ 531441 159-282 ' decimal places. EEDUCTION. 48. The great convenience of decimals makes it often desirable to reduce vulgar fractions to the decimal form. To reduce a vulgar fraction to a decimal. "W e 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, looj 1000, &c., as may be 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 wiU not alter the value of the fraction. If the numerator by the addition of the cyphers 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 f to a decimal. 5000 _i. 9 885 •f.tp SOOO . " TinriJ "-'J- Here we multiply numerator and denominator by 1000 and divide them by 8. The resulting fraction -{^-^ represented as a decimal is "625. Ex. 2. To reduce ^41 to a decimal. 123 1230000 1868 — 'mfiS 0250000 Here we multiply numerator and denominator by loooo, and then divide them by •625. The resulting fraction xHio represented as a decimal is -625. 625) 1 230ooo(' 1968 625 The work is shortened thus: — we put down the numerator 123 as "°-5° dividend, and denominator 625 as a divisor, and adding cyphers as often as required, we obtain as a quotient the significant digits of the 425° decimals ; and the number of cyphers added to the dividend will be the ^'^ number of places to be marked oflf in the question. 5000 5000 Hence to convert vulgar fractions into decimals we proceed by the following rule. RTJLE XVn. Annex a cypher to the numerator, and then divide hy the denominator ; if there he 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. Q 42 Reduction of Decimals. Examples. Ex. I. Reduce ^ to a decimal fraction. Dividing 101)75 (the cypher being added) we find that i is = 0-2. That \ = 0-2 is easily proved, for | = i§ ; consequently, by dividing both the numerator and denomi- nator by 5, we have i = ^ = '2 Ex.3. Reduced i to a decimal fraction. 3)1*0000 •3333> &c. Dividing 10 by 3 gives 3, the next cypher added gives another 3, and so on, con- tinually. Ex. 5. Reduce -ff to a decimal. '4)25-00 36=4 X 9 36 ■ 9) 6-25 •694444, &c. Hence f f = •694 which is called a mixed recurring or cirenlating 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. Ex. 2. Convert | into a decimal fraction. 8)3-000 •375 That I = -375 is proved thus, f = |^^ ; consequently, dividing the numerator and denominator by 8, (the denominator of the fraction), we have f = J^% = -375. Ex. 4. Reduce ^ to a decimal. ii)6'o ■5454 It is plain from the remainder that 54 would recur continually, so that -^ is equal to a recurring decimal; 54 being the recurring period. Ex. 6. Reduce -5^ to a decimal. [ 2)3-000 14 = 2 X 7 14 — I 7)1' 500 -2142857 Hence ^ = -2 142857 ; the recurring part 142857 having a point above its first and last figures being called its period. If the whole decimal recurs, it is called a pure circulator. Ex. 7. Convert ^fg into a decimal. 113)8-00(0-07079, &c. 791 900 791 1090 1017 73) &c. When the o is annexed to the 8, 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 i^f-3 = "07079, &c. ; the decimals may be carried out to any extent. Ex. 10. Reduce -g\^ to a decimal. 6T2 = •001953125- Ex. 8. Reduce -^^ to a decimal. I28)i-ooo(-oo78i25 1040 1024 160 128 320 256 640 640 Ex. 9. Reduce g\y^ to a decimal. -^^■=. -0123291015625. Ex. 1 1 . Reduce ■g-f^ to a decimal. 5l2 = -013671875. Ex. 12. Reduce -^ to a decimal. = -0451206715634837355718782791185729275970619. Reduction of Decimals. 4^ Examples por Practice. 1. Change into equivalent decimals tk, ii, \%, ii, ^, ^, -^, and \^. 2. Eeduce to decimals ^'3, li, fg, .^^_^ |^^ 1.13^ 3_7^ and -jf^, carrying interminants to seven decimal places. 49. It becomes important to observe what fractions will produce termi- nating decimals. Suppose a fraction in its lowest terms ; then in reducing it to a decimal we multiply the numerator by 10, 100, 1000, &c. Now the numerator contains no factor common to the denominator, and by this multiplication we introduce the factors 2 and 5 as often as we please and no others. Unless, then, the denominator contains no other factor 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 2 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. 50. Decimals are most frequently used to make calculations on numbers that have been obtained by observations of some kind, by measuring, for in- stance, 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 be required, by carrying the decimal far enough. With respect to repeating decimals, if perfect accuracy be necessary, they must in most cases be reduced to vulgar fractions before they are added, sub- tracted, 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 difi'er from its exact value by more than one five-thousandth part of the unit employed. Thus, -172 difi'ers from 172437 by -0004.372, which is less than -0005. Similarly, 983 differs from '98276 ^J •0002317, which is also less than '0005. 51. To reduce any quantity or fraction of one denomination to the decimal of another denomination. EULE xvni. Reduce the numher of the lowest denomination to a decimal of the next higher denomination, prefix to this decimal the numher of its denomination given in the question, if any, and reduce this also to a decimal of the next higher order, and so on till all the numbers of the given denominations are exhausted, and the decimal of the required denomination has been obtained : the last result will be the answer. 44 Reduction vf Decimals. ElLAJMPLES. Ex. I. Let it be required to express 175. s\ii- as the decimal of £1. The process will be first to express the fractional part of a penny as a decimal 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 decimal 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 "4375 shillings •871875 of a pound. It will be seen from this, that whatever we should divide by in whole numbers in order to bringpenceintoshillings,orshillings into pounds, that we must likewise divide by in this case, only marking off correctly the decimal results. 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 18' and 27° directly under it, and draw a vertical line to the left of the line opposite these numbers write for divisors the number of that denomi- nation which makes one of the next higher — namely, 60 opposite the seconds, since 60" =: i', and 60 opposite the minutes, because 60' z^ 1°. Then dividing 35 by 60 we get •583, which we write after the minutes, which gives 18-583 ; 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°-3097i. Ex. 3. Eind what decimal of an hour is 40"^. There are 60 minutes in an hour ; hence I minute is -e^or of an hour, and 40 minutes is %% of I hour, which gives 0-66 of i hour. 60)40-00 0-66 of an hour. Ex. 4. Find what decimal of an hour is i5>". Here i minute is -g-o of ^ hour, 15 minutes is M of I hour; hence \% gives 0-25 of i hour. 60)15-00 0-25 of an hour. Ex. 5. Find what decimal of i degree is 8' 37". 37" are I J of i', or 0-61 of i'; then i' is -^q of 1°; hence 8'-6i are ^'i-^- of i", or o°-i43. 60)37" 6o)8-6i6 0-143 of a degree. Ex. 6. Find what decimal of i day is 3I1 42"". 42™ are f 5 of i hour, or o>>-7 ; and !•> is sV of I day ; hence 3^-7 is -2t of i day, or od- 1 541 66, &c. 60)42™ 24) 3*7 0*154166 Ex. 7. Find what decimal of x mile (nautical) is 700 feet. There are 6080 feet nearly in a nautical mile; hence i foot is ^oVo of a mile, and 700 feet are -i-Q-io of i mile, which gives 0-115 of I mile nearly. 6080)700-0(0-115 6080 9200 6080 31200 30400 Ex. 8. Find what decimal of i foot is 8| inches. First, f is 0-75 of i inch ; hence 8f inches are 8-75 inches. Then, i inch is ^ of i foot; hence 8-75 inches are ^^, or 0-729 of I foot. ^2)8-75 0-729 Reduction of Lecimah. 4.5 52. Or, reduce the given quantities to the lowest denominations when there are more than one, and also the integer to xohich it is referred, to the same denomination ; then divide the given quantity hy the integer thus reduce. Ex. I. (Ex. 7 above). The given 1 Ex. 2. (Ex. 8 above). 8 inches and 3 quantity 700 feet, being all of one denorai- I quarters are 35 quarters, and i foot reduced nator requires no further reduction. The to the same denomination is 48 quarters; integer i mile, reduced to the same de- j then 35 divided by 48 gives 0729. nomination, is 6080 leet ; then 700 divided by. 6080 gives o' 1 15. Examples foe, Pbaotioe. r. Express as decimals of an hour 17™; 29™; 42™; 25™; 48""; and 58"". 2. Reduce ^^ 12^ 25" 39*'92 to decimals of a week. 3. Reduce 29'* 12*^ 44"^ 28-82 to decimals of a day. 4. Add together 2-095 hours, -07 days, "05 weeks, and express the same as the decimal of 365-25 days. 5. A nautical mile is 6082-66 feet, and an imperial mile is 5280 feet ; express each of these miles as decimals of the other. Also find how near the results are to the decimal values of |f and ff . A sidereal day is 23'' 56" 4^-09 ; express this as a decimal of a common day — that is, of 24^ — and give the result to nine decimal places. Express as decimals of a day the following quantities: — 12'' i^ 13™ 12^; 29'' 1711 11™ 45s ; 15'' 17^48™ 543; and 119^ 5h igm i^s^ Express as decimals of a degree the following quantities: — 8° 11' 15"; 19° 40' 45"; 104° 16' 7"; and 82° 19' 30". If 90 degrees correspond to 100 French grades, how many degrees are there in the sum of 41-45 degrees and 41-45 grades. A metre is 39-37079 English inches, a kilometre is 1000 metres; express as decimals of each other a kilometre and an English mile. If the length of a degree of latitude is 365000 feet, and a metre one ten-millionth of 90 degrees : find its length in feet. Express in figures thirty-four and two thousandths, and by it divide 28255662. What alteration must be made in the quotient if the decimal part in the dividend be moved eight places to the left ? 13. The sidereal year being 1,6$^ 6^ 9'" 9^-6, and the tropical year 365'' 5'> 48'" 493-7 : re- duce their difference to the decimal of a tropical year. 14. Supposing the velocity of electricity be 288,000 miles per second, and the earth's cir- cumference to be 25,000 miles : calculate to seven places of decimals the time of transmission of an electric telegraph to the antipodes. 15. A French metre is 39-37 inches nearly: show that a foot is equal to -304 metres nearly. To find the value in a lower denomination of any decimal of a higher denomination, RULE XIX. 1**. Note the number of parts which the unit or integer of the given quantity contains of the next inferior denomination, and multiply the given decimal hy this number; the product is the given quantity expressed in that denomination. 2°. If this product has a decimal part, multiply this decimal hy the number of parts which the unit of the present denomination contains of the next inferior dc' nomination to that Just before employed; this product is the quantity which the given decimal contains of the next denomination. 3°. I'roceed (if there still be decimals), in like manner, to the loweat denomina^ Hon in which the decitmi U required to he expressed. 46 On Loga/rithms. Examples. Ex. I. Find the number of feet in o-i 15 of a mile. 0-115 The next inferior denomination to that of miles is here feet,") ^ go of which the number in one mile is J 230 920 6900 Ans. (in the lowest denomination required) 699-430 feet. Ex. 2. Find the number of seconds in 0-7 of a minute. 0-7 The next inferior denomination to that of minutes is seconds | x 60 of which the number in a minute is ) Ans, 42-0 seconds. Ex. 3. Find the number of inches and eighths in 0*48 of a foot. 0-48 The next inferior denomination to that of feet is inches, of ^ v 12 which the number in a foot is ) 5-76 inches. )n to inches is eighths, ) .^ of which the number in an inch is 6-08 eighths. The next proposed inferior denomination to inches is eighths, \ -v 8 Ex. 4. What is the value of -625 of a cwt. ? The next inferior denomination to that of a cwt. is qrs., of | which the number in a cwt. is j ^ ^ 2500 qrs. The next inferior denomination to qrs. is lbs., of which the | v X number in a quarter is j ^ ^ 14-0 lbs. Examples for Practice. What is tlie value of 1. '7768 tons. 4. "225 tons. 7. '2957795 degrees. 2. 7-85425 degrees. 5. -3625 tons. 8. -165625 tons. 3. 64-3825 degrees. 6. 108725 degrees. 9. "530588715 days. Find the length of a tropical year which contains 365-242218 days. ON LOGARITHMS. 53. Logarithms are numbers arranged in Tables for the purpose of facili- tating arithmetical computations. They are adapted to the natural numbers, I, 2, 3 . . . . iu such a manner that by means of them the operation of Multiplication is changed into that of Addition ; . Division . . ... Subtraction; . Involution . ... Multiplication; . Evolution . ... Division; * * No proof can here be offered that numbers must exist possessing the properties under which we call them logarithms ; neither can any account be here given of the methods of computing such logarithms. The reader will accept the statement that if such numbers exist, bearing the properties aforesaid, they are called logarithms. He must also accept tha On LogmitJvmB. ^.-^ 54- Take any whole numbers, as i8, 813, 6489 ; the first consists of two, the second of three, and the third of four figures or digits. Again, in the mixed number 739-815, the whole number or integral part (739) consists of three digits. 55. By multiplying a number by itself, one, two, three, &c., times succes- sively, we obtain the second, third, fourth, &c., powers of that number; hence, a power of a number is the number arising from successive multiplication by itself. Thus, 3 X 3 = 9 is the square or second power of 3 ; and 5 X 5 X 5 = 125, the cube or third power of 5 ; and so on. These operations are denoted by means of Indices, or small figures placed on the right of the numbers, a little above the line ; thus, 2^=2x2=4, 3^ = 3 X 3 X 3 = 27, and 2^=2X2X2X2X2 = 32, where the Index or exponent denotes the number of factors employed. 56. When there are a series of numbers, such that each is found from the previous one by the addition or subtraction of the same number, they are said to be in arithmetical progression. i, 3, 5, 7, 9, n, &c., are in arithmetical progression. 57. Again, the numbers 3, 6, 12, 24, &c., are in geometrical progression, for each number is formed from the one immediately preceding by multiply- ing by 2. If we take the following series of powers, 31, 3", 33, 3*^ 38^ &c., we find that the exponents proceed in arithmetical progression, and the quantities themselves in geometrical progression. 58. Def. — Logarithms are a series of numbers in arithmetical Y^ogvQ^sion answering to another series in geometrical progression, so taken that o in the former corresponds with i in the latter. Thus, o, I, 2, 3, 4, 5, 6, &c., are the logarithms or arithmetical Bevies, and I, 2, 4, 8, 16, 32, 64, &c., are the numbers or geometrical series, answering thereto — the latter being called the natural number. Or, o, I, 2, 3, 4, 5, the logarithms, and I, 5, 25, 125, 425, 5125, the corresponding numbers. Or, 0, I, 2, 3, 4, 5, the' logarithms, and I, 10, 100, 1000, 1 0000, 1 00000, the corresponding numbers. In which it will be seen, that by altering the common ratio of the geometrical series, the same arithmetical series may be made to serve as tables ■which are published, recording logarithms for the several numbers to which they profess to belong, though he cannot at present verify the computation of these several log- arithms ; and he will be informed how he may use these tables to effect with comparative ease many calculations which would otherwise be most laborious. The truth is, though it requires for its demonstration higher algebra than this work presupposes the reader to be acquainted with, that not only has every number a logarithm, but it has an infinite variety of logarithms, constructed, as the term is, on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unift/ is its logarithm. Any number mit/ht be used as a base ; but in fact there are only two numbers which are ever really used. The one is an unterminating decimal, 2"7i828i8 . . . . , denoted generally by the letter e. This is the base of what is called the natural or Naperian system ; and the advantage of it consists in the ease with which logs, are computed, to this base ; but which we cannot here explain. The other is 10, which is the base in ordinary use, and with this base log. 10 = i. Log- arithms to this base are the only ones which wiU now be considered in their practical use, 48 On, Logmithms. logarithms of any series of numbers. As above, when the common ratio of the geometrical series are 2, 5, and 10 respectively. 59. The common ratio in the geometrical series corresponding to the common difference of i in the arithmetical series is called the base of the system. Thus, the base of the first specimen exhibited is 2, the base of the second is 5, and the base of the third is 10. In the specimens just exhibited we have, in each, taken two ascending progressions, but they might equally well have been two descending pro- gressions, or the one descending and the other ascending. Logarithms, however, as now used in practice, are limited to the case of two progressions, either both ascending or both descending — the former giving the logarithms of integers, the latter of fractional numbers. But a better way of considering logarithms is as follows : — 60. Def. — The logarithm of a number to a given base is the index of the power to which the base must be raised to give the number. For instance, if the base of a system of logarithms be 2, 3 is the logarithm of 8, because ^■=i^^^^y.^X^. And if the base be 5, then 3 is the logarithm of the number 125, because 125 = 53 = 5 X 5 X 5- There may thus be as many different systems of logs, as we please ; but, for practical use, it is necessary to select and adhere to one. That usually employed now is called Briggs' system. 61. We now proceed to describe what is called the common system of logarithms. In the common system of logarithms unity is assumed to be the logarithm of 1 ; that is, i o is the constant base. All the logarithms registered in the Tables commonly used, are indices of the radix or base 10; a Table of logarithms of numbers is in fact nothing more than a Table of the exponents of 10 placed against the several numbers themselves. Accordingly — is the log. of I, because i = 10' I » 10, „ 10 = 10 2 )» 100, „ 100 = 10' 3 )> 1000, „ 1000 = 10' 4 „ &c. 1 0000, „ lOOOO = 10' &c. Now, if the above Tables were amplified by the insertion of the logarithms of all the numbers between i and 10, between 10 and 100, &c., we should have a Table of logarithms of all numbers from i to loooo; and whatever may be the difiiculty of determining the intermediate logarithms, it is at once easily seen that the logarithms of all numbers between i and 10 will be o + a fraction, that is, a decimal less than i ; of all numbers between 10 and 100 will be i + a fraction, or a decimal between i and 2 ; of all be- tween 100 and 1000 will be 2 + a fraction, and so forth; or the integral part of each intermediate logarithm will be one less than the number 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, 11, 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; the integral part is always known from the number of On' Loga/rifhrnB, 49 integers in the value whose logarithm is wanted. Very few logs, can be ex- pressed 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. 62. The integers i, 2, 3, 4, &c., which are the logarithms of 10 and its powers (see 61), are chief indibes, and the logarithms intermediate to these, as for instance i "778 1 5 1 (which is the logarithm of 60) consisting of an integer and a decimal fraction, though they are also indices, are usually referred to as consisting of an index'^ 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 log. 4-616339, the figure (4) standing to the left of the decimal point is the characteristic or index, and the remaining portion ("616339) i^ the mantissa or decimal part. 63. To find the characteristic of the logarithm of any number greater than unity, we have, therefore, the following rule : — EULE XX. The characteristic of the logarithm of a number greater than unity, 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 numher 849 is an integer consisting of three digits (that is a number between roo and 1000) and i less than 3 is 2. Also, the index of the log. of 264-96 (which is a mixed number) is 2, since the integral part of the number, namely 264 is a number between too and 1000, or consists of 3 digits, and one less than 4 is 3. Again, 3 is the characteristic of the logarithm of 3847-216, since 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 one place of integers (such as 5 or 5-08, or 5-0801) is o. Again, everj'^ number with two 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 508-25) has for its characteristic 2, and so on. Examples foe Pbactice. Write down the characteristics of the logarithm of the following numbers : — I- 365 6 2. 4.8 7 3- 643-75 8 4. 28-9 9 5. 6 10 69710 II. 474000 45-82 12. 4256-45 8640 13. 3-9 75 14- 8 7-265 15. 18 16. 473-908 17- 54793000 18. 21256-8 19. 2-14006 20. 50-7406 64. 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 * 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. f Mantissa, a Latin word signifying an additional handful ; something over and above an exact quantitJ^ H 5© On Logarithms. wholly decimal, i.e , less than unity, may be represented by numbers less than zero, i.e., by negative numbers. Extending, therefore, the above principles'to negative exponents, since T^ = 0-or T 2 is the logarithm of ■ i in this system. IO-3 = IO-* = ToW = o-oor Yogoo = O'OOOI 3 4 „ -001 „ „ -0001 „ &c., &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 -j^-, 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 -^ but not 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 "01 and "001 is some number between — 2 and — 3, and its characteristic is — 3 ; of a number between '001 and -oooi its log. is between — 3 and — 4, and its characteristic is 4 ; and generally, following this reasoning it will appear that the charac- teristic of a decimal fraction 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. 65. To find the characteristic of any number less than unity, i.e., of a decimal. The characteristics of the logarithms of all numbers less than unity are negative, and may be found by EULE XXI. 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 characteristic of the logarithm of -00521 is 3, since the number of cyphers following the decimal point increased by i is 3. Similarly the index of log. of •156 » „ "oiS^ „ „ -00046 „ ,, -000000721 is 7 66. But in order to avoid the confusion that might arise by the addition and subtraction of negative indices, the following rule is frequently used. EULE xxn. 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. * 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 Logwrithms. ^i Thus the characteristic of the log. of -04 is 2 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 T or 9. „ of log. ^8, or of log. -04 is 2 or 8. „ oflog.241, or 24-4 I. Examples foe Practice. Write down the characteristics of the logarithms of the following decimal fractions : — I. ■045 6. •ooooooi II. ■4537 16. •037299 2. •9 7- •01 12. •009 17- •00000052018 3- •0004 8. •0003127 13- •0000008 18. •000000205379 4- •6798 9- •02803 14. •000064 19. ■5 5- •0062 10. •7007 15- •000485 20. •0000000000382 67. The characteristic may also be found as follows : — EULE XXIII. Place your pen between 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 toill be the characteristic : hut observe that if you count to the left you must sub- tract the number found from 10, and consider the remainder as the characteristic. Thus, in finding the log. of 4^6oi7, if you place your pen between the first figure (4) and second (6), it falls on the decimal point ; in this case the characteristic is o. Next, in the case of log. of 4601.7, place your pen between 4and 6, and count L^i > the characteristic is 3. Next, in the case 4601700, here the decimal point falls behind the last cypher (No. 5), Hence, counting as before, we have '^li°^]^?6> and the characteristic is 6. Again, in the case of log. -00046017 the first significant figure is 4. Hence, counting, we have ■°i°°4o°i7^ -b^t j^ere we count to the left, so that the characteristic is negative, or 413^^ 4, which taken from 10 is 6. Again, in the case of log. of '46017, we have | °^'' and the characteristic is T, or 9. 68. The mantissa of the logarithm depends entirely on the relative value 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 10 ; it is only the characteristic which alters its value by au <|2 On Loga/rithms. 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 lo 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 lo. The logarithm of 745800 being 5-872622 that of 74580 is 4-872622 ,, 7458 ,, 3"872622 „ 745-8 „ 2-872622 ,, 74-58 ,, 1-872622 „ 7-458 ,, 0-872622 „ -7458 „ 7-872622 or 9-872692 „ '07458 „ 2-872622 or 8-872622 „ '007458 „ 3-872622 or 7-872622 „ "0007458 „ 4-872622 or 6-872622 ON TABLES OF LOGAEITHMS OF NUMBEES. 69. In Raper'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 i ,000 up to 9,999- 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 XX and XXI. Hence, from such a table we can take out the logarithm of any number with any four significant digits. no. 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 XXIV. Seek for the numher proposed, considered as a tohole number, in the column at the top of which is No., and the logarithm will be 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 Eulea XX and XXI, pages 49 and 50. 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 On Loga/rithms. ^^ of it till we come to the insertion of the characteristic ; and should cyphers occur between the decimal point and the first significant figure, these, also, are disregarded in entering the table, being only taken into account when determining the characteristic. Ex. I. Eequired the logarithm of 21, 2'i, -21, and •021. In the first page of the Table, and in one of the vertical columns marked No., we find 21, against which stands 1-322219, the logarithm sought. Since the mantissa of the logarithm of any number consisting of the same figures is the same whether the number be integral, fractional, or mixed, the logarithms of the numbers 2-1, -21, and -021 will have the same decimal part as 21, the characteristic only being changed, consequently the logarithm of 2-1 is 0-322219, the logarithm of -21 is 9-322219, and the logarithm of "021 is 8-322219. Ex. 2. To find the logarithm of 52, 5-2, -52, and -00052 : — In the Tables we find the log. of 52 is 1-716003, and, therefore, simply changing the index, the log. of 5-2 is 0-716003, -52 is 9-716003, and the log. of -00052 is ^-6716003 or 6-716003. Vo. Nat, No. No. Nat. No I. 5 7- 41 2. 9 8. -004 3- -009 9- 2-4 4- ■or lO. 24 5- -0001 II. -24 6. 14 12. •0021 No. Nat. No. 13- 94 14. 4 or -5 15- 1 or -75 16. 2-5 17- i or -25 18. -09 No. Nat. No. 19. •0091 20. 25-0 21. •024 22. -000035 23- •000057 24. 20 off 71. To find the logarithm of a number consisting of not more than three places of figures (from 100 to 1000). EULE xxy. 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 Rules XX and XXI. 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 ot this 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 XXV. Thus the logarithm of 75 is the same as that of 75-0 ; the logarithm of 8 is the same as of 8-00 ; and that of -035 is the same as of '0350. Ex. I. Kequired 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 tho 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 57 6 is 2-760422 ; that of 39-413 1-595496; that of -0253 is 2-403 121. No. Nat. No. No. Nat. No. No. Nat. No. No. Nat. No. 1. 100 5. 673 9. 8-96 13. -0147 2. 145 6. 794 10. 1-47 14. 424 3. 2-94 7. 982 II. -147 15. -0000448 4. 361 8. 4.-8o 12. -901 16. 448000 54- On Loga/rithms. 72. If the number contains four places of figures, exclusive of final cyphers, or cyphers included between the decimal point and the first significant figure. EULE XXVI. Find the first three figures in the vertical column on the left marhed No., and the fourth in the horizontal column at the top of the page. TJuder this last, and opposite the three figures, will he found the mantissa of the logarithm sought. Prefijx the index according to Eules XX and XXI. The result is the logarithm sought. Ex. I. Required the logarithms 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 number, and in the same column as the fourth. The characteristic is 3, being one less than the num- ber of integers in the whole number; whence the completed logarithm is 3'66i529. 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 3-541330; that of 3-492 is 0-543074; and that of 0-3468 is T-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 10. 987-6 2. 1234 5. 26-06 8. 94"87 II. '06843 3. 25-65 6. 2-606 9. 7'777 12. -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 Rules XXIV and XXV. 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 75-00 ; 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). Although the tables in Eaper, Norie, and the "Nautical Tables" accom- panying this work are constructed so that the mantissse 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. 73. If the number consists of more than four figures other than final cyphers, or if the number be a decimal fraction, cyphers immediately follow- ing the decimal point, we use EULE XXYII. 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 XXVI). 3°. Multiply the tabular difference by the decimal cut off, i.e., by the remain- ing figures of the given number, a7id 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 iigure thus cut off is, not less than 5. 4°. Add the integer part of this product to the figures of the mantissa jmt found. The result is the mantissa of the required logarithm. The characteristic or index is found by Eules XX and XXI, pages 49 and 50. On Logarithms. te Ex. I. Eequired the logarithm of 28434. Tah. diff. Mantissa of 2843 = '453777 153 Tab. diff. 153 X 4 = 61-2 = -|- 61 X 4 Characteristic 4 61,2 or 61 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 lino with the former and in the column with the latter at the top we have 453777, which is the mantissa of 2843. I1 ^ 1^^® with the quantity in the right hand column marked Diff., 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 by only one digit) it becomes 61, which being added to 453777 (the mantissa of 2843) makes 453838, and, with the characteristic, 4*453838 the required logarithm. The logarithm of 284*34 is 2-453838, and the log of -028434 is 5-453838 or 8-453838. Ex. 2, Required the logarithm of 12806. Tab. diff. Mantissa of 1280 = *io72io 338 Tab. diff. 338 X 6 = 202-8 = + 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 8734 = '941213 50 Tab. diff. 50 X 57 = 28-50 = + 29 X 57 Characteristic = 5 28,50 or 29 The log. of 873457 = 5-941242 The mantissa of the first four figures is found thus :— opposite the 873 and under 4 stands 9412 1 3 ; 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 2850 ; from this we cut off two digits to the right (since we have multiplied by two digits), when it becomes 18 ; but as the highest digit cut off is 5, we add unity, which makes 29. Then 5-941212 (the log- arithm of 8734) -|- 29 = 5-941242 is the required logarithm. Ex. 4. Required the logarithm of 628007. Mantissa of 6280 = -797960 Tab. diff. Tab. diff. 69 X 07 = 4-83 = -f- 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 is 3-798006 or 7-798006. The Mantissa of the log. of each of these numbers being the same, the index only being varied. — (See Rules XX and XXI.) I. 38475 7 435"6o n- 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 0-78362 16. 643786 22. 1-032764 -;• 66666 II 1 0000 17- 1129-06 23- IOOOj-|-g- 6. 9244-8 12 •000800073 18. •998095 24. 596-423 74. To find the natural number corresponding to a given logaritlim, — If the logarithm be given, the 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. 5 6 On Loga/rithms. Since the characteristic denotes how many places the first significant figure stands to the right or left of the unit's place ; conversely, therefore, if logs, be given having for characteristics i, 2, 3, .... T, 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 , r5h9S° 3 • • 3890 2-589950 2 ■■ 389 i'58995o I •• 38-9 0-589950 o .. 3-89 019-589950) ■^°^9 •• -389 ^■-5^9950) - 8 8 or 8-5899501 ^^'^^ •• °3S9 &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 lastty, 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. 75. From the foregoing it is evident that when the figures of the natural number have been found, we must place the decimal point so that the number 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. 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 fin d the place of the decimal point proceed as follows : — EULE XXYIII. Add I to the characteristic of the given logarithm, and marh off to the left the number of figures for whole numbers ; the rest (if any) loill be decimals. EULE XXIX. The number corresponding to a logarithm with a negative index is wholly decimal^ and the number of cyphers following the decimal point is one less than the character- istic of the logarithm. But instead of the negative characteristic its arithmetical complement is sometimes used, in which case we proceed by EULE XXX. Add I to the index, and subtract the numher thm found from 10; ths remainder is the numher of cyphers to he prefixed to the figures taken out of the Tables. Place the dot before the first cypher. On Loga/rithms. 57 76. To find the natural number corresponding to any given logarithm. When the mantissa or decimal part of the logarithm can be found exactly in the Table, we proceed by EULE XXXI. 1°. Seeh out the mantissa, and take from the column No. the three figures in the same horizontal row. 2°. From the head of the column take the fourth figure. 3°. From the characteristic find hy the rules already given the position of the decimal point, and so adjust the local value of the figures. (Eules XXVIII, XXIX, and XXX, No. 75, page 56.) {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 riile, the deficiency is made up by adding a sufficient number of cyphers to the right. (h) If the natural number is a decimal fraction, and the final figure or figures are cyphers, they need not be written down. Examples. Given the logarithm 2'69897o 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. 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. When number corresponds the logarithm 4-214314. 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 XXXI, («). Hence the required number is 16380. Eequired the natural numbers corresponding to logs. 0-176091 and 4"i76o9i. (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 nimiber sought is 1-500 or 1-5 (see Eule XXVIII, page 56). (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 2f shows that the first significant figure (i) must occupy the fourth place to the right of the decimal point, whence the number sought is -00015. (See Eule XXIX or XXX, page 56.) Eequired the natural numbers whose logarithms are respectively i -81 3514, 0-303412, 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 \ 3^ .. -01995 or 8-299943 j ^-'•^ 4-000000 = ., 1 0000 4-000000 ^ . , -0001 7-816109 = ,, 65480000 S8 On Loga/rithms. 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 onlj' 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 XXXT, {a) ; and that the fourth answer must be a decimal, having the first significant figure two places to the right of the decimal point because the characteristic is 2 ; 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 cj^phers 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. 0-477121 II 3'09i3i5 21. 2-990561 31 7-991093 2. 0*903090 12 3-898506 22. 4-541579 32 7-903524 3- 0-041393 13 2'538574 23- 4-722522 33 2-621488 4- 1-301030 14 I "394977 24. 7-744058 34 9-901349 5- 0-973128 15 3-845098 25- 1-501196 35 3-662758 6. I-161368 16 7-000000 26. 7-875061 36 4-851258 7- 0-812245 17 5-825426 27. 6602062 37 6-778151 8. 2-767898 18 5-602060 28. 8-845098 38 6-913761 9- 0-394452 19 4-698970 29. 3-605 197 39 0-004321 0. 1-478422 20 5-000027 30- 3-444669 40 5-868527 77. 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 proportion to the increase of the logarithm. 78. To find the natural number corresponding to a given logarithm, when more than four figures are required. We proceed by RULE XXXII. 1°. Having found the next lower 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 f annexing as many cyphers as there are digits required above four J by the tabular difference,^' and reduce the qtiotietit to the form of a decimal. 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 with safety carry the natural number to more than six figures when the tabular difference contains three or to more than five when the tabular difference contains only two. C:> * The tabular difference (Diff.) spoken of here is given at the end of every line in the table* and is the difference of the successive logarithms in that part of the table. On Loga/rithms. 59 Given tlie logarithm 3"543027 to find the natural number. Given logarithm 3-543027 Mantissa next lower in Table -542950 which corresponds to 3491. Tab. diff. = i24)*77oo(-62 744 260 248 Attaching this ('62) to the four figures, we have 349162, &c. The decimal punctuation or local value of the figures of the number can now be adjusted, and as the index is 3, we obtain, by pointing off four figures to the left, 349i"62 the natural number sought. Given the logarithms 5-654329 and 2-654273 to find the natural numbers. Given logarithm 5-654329 Mantissa next less in Table -654273 which corresponds to 4511. Tab. difif. = 96)5600(58 480 800 768 Ans. 451 158. 32 654273, which corresponds with the natural number 45 n, is the logarithm next less than the given one ; therefore the first /om^ digits of the required number are 451 r. Adding two cyphers to 56, the diff^ence between 654273 and the given logarithm, it becomes 5600, which being divided by 96, the tahdar difference corresponding- with 45 11, gives 58 as quotient and 32 as remainder. The integers of the required number (one more than 5, the characteristic) are, therefore, 451 158. The mantissa of the second log. being the same as the first one, the natural number will contain the same significant figures, viz., 451 158, but the characteristic 2 shows that the first significant figure of the nat. no. (4) must stand in the second place to the right of the decimal point ; therefore, the nat. no. corresponding to 2-654273 is -0451 158. Let it be required to find the number of which the logarithm is 3-104831. Given logarithm 3 • 1 048 3 1 Mantissa next lower in Table -104828 which corresponds to 1273. Tab. difif. =: 34i)30oo(-oo8 2728 272 Therefore, 3-104831 = log. of 1273-009 nearly. In dividing by tab. difl". we take remainder 3 and a cypher, then 341 in 30 goes no times, which we place down in the quotient, then taking another cypher we have 300, which contains 341 no times, lastly, 341 goes into 3000 eight times with 272 for remainder. The remainder 272 being more than half the quotient the last figure of the quotient (8) is increased by i or unity. No. Log. No. Log. No. Log. No. Log. I. 2-931214 10. 4-994603 19. 0-230449 25- 6-24663 1 2. 3'625343 II. 4*925936 20. 1-217845 26. 1-998813 3- 4-851906 12. 5-091512 21. 3-984671 or 9-998813 4- 4-361730 13- 2-535224 or 7-984671 27. 1-895090 5- 1-725364 14. 3-744726 22. 4-463726 or 9-895090 6. 1-972521 15- 5-831835 or 6-463726 28. 4-932847 7- 5"659707 16. 2-415671 23. 2-241877 or 6-932847 8. 5'7 34968 17- 4841989 or 8-241877 29. 5-565942 9- 5-823904 18. 4092561 24. 6-371000 or 5-565942 6o On Logwithms. 79. To find the arithmetical complemenP^ of the logarithm of a number. The arithmetical complement of a number is the number by which it falls short of the units of the next higher denominator. (If x is any number what- ever, then the arithmetical complement of a; =: 10 — x.) It is abbreviated into or. co. The most expeditious way of finding the arithmetical complement is as follows: — EULE XXXIII. Begin from the left ; subtract every figure from 9 up to the lowest significant figure, which subtract from 10. Repeat the cyphers at the end, if any. [a) When the characteristic is negative it must be added to 9. Examples. Ex. 2. Find the ar. co. of -9086540 Ex. I. Find tlie ar. co. of i '97043 («) Ar. CO. log. required 8*02957 (a) Thus we say, begirmiiig at the left hand, i from 9, 9 from 9, 7 from 9, o Irom 9, 4 from 9, and 3 from 10. The ar. co. of 3'6o72i8 is 6"392782 o"7 14000 .. 9"2ti6ooo 5-631642 .. 4"368358 {b) At. CO. log. required 0-913460 (6) In this example we proceed as before ; thus 9 from 9, o from 9, 8 from 9, 6 from 9, 5 from g, and as the nest figure, 4, is the lotvest siffnificant&guie (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 .. 12-821320 80. A subtractive quantity is, by this means, made additive. The process is equivalent to subtracting the number from i o, 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 subtracted. to MISCELLANEOUS. 8 1 . We here insert a collection of numbers, the logarithms of which are be taken out of the Tables. I. 8 II. 63-5 21. 844-4 31 ?3-7654 41. I 0000000 2. o-i 12. 6390 22. 92096 32 52790 42. -000000062 3- 4-9 13- •1463 23- -0899 33 50000 43- 30000-9 4. 38 14. 3-874 24. 1 0000 34 700090 44. 10000-9 5- 380 ^5- 6754 25- 4800 35 264000 45- 594500 6. 100 16. -0876 26. 9080-8 36 404007 46. 88590000 7- •0001 17- •3467 27. •00058 37 500909 47- 287-642 8. 24-6 18. 1-083 28. -( 335872 38 48-627 48. 0-003564 9- 3-88 19. 0-125 29. -c D00448 39 93-514 49. -000856736 10 900 20. 0-0009 3°- 4480000 40 032764 50. 65480000 82. Eequired the natural number of the following logarithms : I. 2-309630 10 0-565021 19. 2-954243 28 5-606389 37 T-883030 2. 3-676968 II 0-778441 20. 3-959041 29 5-000000 38 3-625343 3- 0*954243 12 2-769504 21. 4-705864 30 2-881955 39 £•725364 4- 1-698970 13 5'774i52 22. 0-415974 31 r-167317 40 2-627407 5- 0-000000 14 5-42x604 23- X -000000 32 7-875061 41 3686216 6. 2-OO0O0O 15 3-000000 24. 3-954243 33 o-oooi86 42 0-400573 7- 2-564494 16 6-394452 25- 2-716003 34 6-947385 43 5-002559 8. 3*563362 17 1-415674 26. 5-654243 35 2-963081 44 4-321547 9- 2-621754 18 1-188591 27. 0-434294 36 0-763947 45 0-875061 A yery curious and valuable artifice, discovered by Gunter about 16 14. On Logwithms. 83. Finally, we recommend the student to commit to memory the follow- ing table of logarithms to two places : — No. 7- 8. 9- STo, Log. No. Log I. 00 4- 60 2. 3- 30 48 5- 6. 70 78 Log. 85 90 95 MULTIPLICATION BY LOaAEITHMS. 84. In multiplication we proceed by EULE XXXIV. 1°. Find the logarithms of the nunibers, the product of which is required. (For the method of taking out the log. of a number see pages 52 to 55.) Note. — If any of the quantities is a decimal, either the negative characteristic of that quantity or its arithmetical complement is to be used (Rules XXXI and XXXII, page 50.) 2°. Add these together, the sum loill be the logarithm of the product. 3°. Find from the Tables the corresponding number. (For the method of finding the corresponding number to a log., see pages 57 to 58.) This will be required product. Note i. When the characteristics are negative and subtracted from 10 (see Rule XXX, page 56), if the sum of such characteristic exceeds the sum of tens borrowed, the product will be a whole number ; otherwise it will be a decimal. Note. 2. "When the characteristics of the logarithms to be added are all positive, it is evident that their sum will be positive. Note 3. If the characteristics are all negative, their sum diminished by the figure — if any — carried from the sum of the mantissa or positive decimal parts will be negative. Note 4. If some characteristics are positive and the 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 the greater and prefix 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 pen through them. Examples.* I. Multiply 77 by 100. The log. of 77 and 100 being taken from the table, we have 77 log. 1-886491 100 log. 2 '000000 7700 log. 3- We have here added tlie logs, of the given factors, and having sought in the Table for the mantissa •886491, we have foimd the flgiires of the nat. no. corresponding to he 7700 ; the index 3 determines four of these to he integral ; hence the product is 7700 (Kule XXVIII, page 56). 3. Multiply 378 by 50. 378 log. 2-577492 50 log. 1-698970 18900 log. 4-276462 The mantissa of log., viz,, -276462, is found exactly in the Table in a line with 189, and imder o ; but as the characteristic 4 requires 5 digits in the integer part, vre therefore add a cypher (o), which gives 18900 as the nat. no. corresponding to the proposed log. This is according to Rule XXXI (a), page 56. 2. Multiply 97 by 83. The log. of 97 and 83 being taken from the Table, we have 97 log. 1-986772 83 log. 1-919078 8051 log. 3-905850 We add the logs, of the given factors, and then seek in the Table for the mantissa •905850, which corresponds to the natural nvimber 8051 ; the index 3 detennines/o!jr of these to be integral; hence the product is 8051 (Rule XXYIII, page 56). 4. Multiply 3456 by 500. 3456 log. 3'53S574 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 ffives 1728000 us the nat. no. required. (See Rule XXXI, (a), page 56). * 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. 62 On Logarithms. J. Multiply 963 by 48-9 by common logarithms. The log. of 963 and 48-9 being taken from the Table, we 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 "We have here added the logs, of the given factors together, and having sought for the given mantissa •672935, which is not to be exactly foimd 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, being more than half the divisor, 1 is added to the last figure in the (juotiL'Ut (5) ; attaching these to the four figui-es obtained previously, we have 470907 ; the characteristic 4 determines five of these to be in- tegral ; hence the product is 47090-7 (Rule XXVIII, page 56). 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 Tab.) 403120* 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, thure must be decimals in the product — as many decimal places as there are in both the multipUer 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 sis places. Now the next lower mantissa found in the table gives the four corresponding figures 2530, leav- ing two figures to he found. (See Rule XXXII, page 58.) * This log. is taken from Norie, and is incorrect in the last decimal figure, which ought to be i, as given in Raper's table ; the true log. being -40312052. 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(2 516 7. Multiply 498 by 376. 498 log. 2-697229 376 log. 2-575188 187248 log. 5-272417 306 Diff. 232) 1 1 100(48 nearly 9. Multiply "0567 by -00339. Both multiplier and multiplicand being decimals, the characteristics of these factors will be negative, but instead we use their arithmetical complements, 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 multipUer -00339 (See Rule XXII, page 50). 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., •283783, 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 XXX TV, Note 1, page 61) the product is a decimal fraction and the first significant digit must stand in the fourth decimal place; hence the product is '0001922. Or thus — using negative indices : •0567 log. 2-7535S3 •00339 log- 3'5302oo •0001922 log. 4"283783 In adding, when we come to the places 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 (a") and ("3), we diminish their 31m ( s ) by the num- ber carried (1), which leaves i for the index (see Rule XXXr\^, Note 3, page 61 ) . "We prefix 3 cyphers because the index being 4 the first significant figure of product must stand in the fourth place from the decimal point. 10. Multiply 99-9 by 8-63. 99-9 log. 1-999565 8-63 log. 0-936011 862-136 log. 2-935576 5,0)180,0 36 Multiply 436 by 19-7. 436 log. 2-639486 19-7 log. i'294466 8589-2 log. 3-933952 43 51)90(2 nearly On Logarithms. 63 12. Find the product of 073 by '00028 by logarithms. •073 log. 8-863323 •00028 log. 6'447i58 •00002044 log. 5'3io48i Or, using negative characteristics, thus : — •O73log. 2^863323 •00028 log. 4^447 158 •00002044 IOq- 5'3I048i Jn adding, when we come to the place of tenths, the process is 5 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 are to be algebraically added together to form the new characteristic. The sum of the two characteristics (both negative) viz., — 2 and — 4 is — 6, which diminished by + i leaves — 5 for the new characteristic. We prefix four cyphers, because the characteristic being 5 shows that the first significant figure must stand in the filth decimal place. (Kule XXIX page 56). 14. Multiply "0172 by •00214. •0172 log. 8-235528 •00214 log. 7'3304i4 •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 sum, which sum (15) being 5 less than the amount borrowed (20), indicates that the product must be a decimal fraction, and the first significant digit stands in the fifth decimal place ; hence the product is -000036808. This is according to Kule XXXIV, Note 1, page 61. Or thus- -using negative indices : Ex. 14. •0172 log. 2-235528 •00214 log. 3-330414 •000036808 log. 5-565942 The characteristics of both logs, being negative, the simi of them is taken, and this, with the negative sign over it, is put down as 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 significant figure of the product must stand in the fifth place from the decimal point. 13. Multiply 24000 by -000783. 24000 log. 4^38o2ii ■000783 log. 6^893762 18-7919 + log- i"273973 Here 10 was boiTowed in determining the index of the log. of -000783, and since the sum of the indices (including 1 carried Itom the decimal part of log.) is elcvt^n, we reject or pat/ bach the 10 boiTOwed, which leaves 1 for the index, and the nat. number corres- ponding is found to be 187910, and we mark off to the right two figures (l more than the characteristic) whence the answer is i8'7gi9+. Or thus — using negative indices: Ex. 13. 24000 log. 4-3802 1 1 •000783 log. 4-893762 18-7919 + log. 1-273973 Here the 1 which is carried after adding 1, 8, and 3 (in the place of tenths), instead of increasing the "4 leaves 3". This is according to Kule XXXIV, Note 4, page 61. 15. Required the product of 17-25, 0-82, and 0-065. 17-25 log. 1-236789 -82 log. 9-913814 •065 log. 8^8i29i3 0-919425 log. 9-963516 Here 10 is borrowed, to find the characteristics of log. both of the second and third factors, and sub- tracting the sum of the indices 19 from 20 leaves 1 ; the sum being less than the number borrowed, the product is a decimal, and hence the first significant figure must occupy the first place to the right of the decimal point. (See Rule XXXIV, Note i, page 61.) Or thus : — 17*25 log. 1-236789 •82 .log. T-913814 •o65log. 2-812913 13- 0-919425 log. 1-963516 Here we have 1 to carry from the mantissa, which added to the positive characteristic i (characteristic of log. 17-25, see above) makes positive 2. Now the sum of the negative indices is 3" (negative 3), and, therefore, since where one is positive and the other is negative thejiitference is the characteristic; we have + 2 from "3 leaves T for the characteristic, (see Rule XXXrV, Note 4, page 61) and the first signifi- cant figure of the quotient must occupy the first place to the right of the decimal point (ilule XXVIII, page 56). Examples fob Practice. Multiply by logs. 85 by 70 ; 39 by 27 ; 100 by 10 ; and 369 by 9. Multiply 538 by 1-74; 601 by 18 ; 250 by 12-5 ; and 3964 by 7. Multiply 20-42 by 0-5; 3-646 by 0-75 ; 2-745 ^7 0-24; and c-792 by 6-5. ^ Multiply 5671 by 4-7 ; 517 by 659 ; 60-609 by 72 ; 1-955 by 10-04 ; and 758875 by 8. Multiply 127 by 304; 476 by 100; 80-08 by 5-98 ; 5760 by 30; and 970 by 630. Multiply 37-6 by 249 ; 44-4 by 22-2 ; 182-7 by 250 ; 2807 by 200 ; and 63-055 by 84. Multiply 280054 by 50; 30967 by 90; 23716 by 350; and 45670 by 690. Multiply 82-33 by 15-3 ; 47-6 by 6-82 ; loooo by 10 ; and 4-02674 by -0123456. Multiply 78960 by 400; 756*875 by 8 ; 94-055 by 74; and 1975 by 10-76. Multiply 732 by 543 ; 58-7 by 66-4; 3000 by 100-14; and 60060 by 700. Multiply 543-29 by 3800-62 ; 90-43 by 712-2 ; 87-305 by 4-09 ; and 209-36 by 46. Multiply 348-25 by 7-125; 498-256 by 41 -2467 ; 56-3426by -023579 ; -123456 by 26813-9. Multiply 0001468 by -000395 ; 0-0006 by 10-0004; '^°S ^7 '00000091; and 35-691 by '0048. 64 On Loga/rithms. 14. Multiply "00146 by "039 ; 5900 Ly '00071 ; 4'i89 hy '00071 ; and 247-55 ^Y S^'l"^- 15. Multiply 527'45 ^7 I'^gS^ ; 10-5526 by 317-145 J -007461 by -3351767 ; and -0700379 by -0086752. 16. Multiply -I by -i ; 'oooi by -ooooi ; -on by i-oi and -ooioi ; and 1000 by 100. 85. DIVISION BY LOGAEITHMS. In division we proceed by EULE XXXV. 1°. Find the logarithms of the numbers the quotient of which is required. Note. If the dividend or divisor, or both, are decimals, the negative characteristic of of that quantity, or its arithmetical complement is to be used. 2°. Subtract the logarithm of the divisor from that of the dividend, f adding 10 to the characteristic of this last, if required^ ; the difference will be the logarithm of the quotient. 3°, Find from the Tables the corresponding number. This will be the required quotient. Note i. When the divisor is greater than the dividend, the characteristic of the log- arithm 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 divisor, and the 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 the 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 are alike take the difference of them, and if the from line is the greater, prefix to the remainder the given sign ; but if the take line is the greater prefix the contrary of the given sign. If the signs are different, take the sum of the characteristics and prefix the sign of the from line. The figure borrowed when subtracting the decimal part of the logarithm, when carried to the characteristic, is always to be added, and therefore make a negative characteristic less, thus 2 carried to 5 makes it 3. Note 3. Otherwise, if one or both of the given terms are decimals, remove the decimal points till the factors contain whole numbers, and the dividend the greatest ; then if the dividend be 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. Examples. I, Divide 3192 by 76. The loff. of 3192 is taken out according to Rule XXVI, page 54, and the log. of 76 by Eule XXIV, page 52. 3192 log. 3-504063 76 log. i-8io8i4 Quotient 42-0 log. 1-633249 3. Divide 579416 by 4324. Log. of 5794 = 762978 Tab. diff. 75 Parts for 16+12 X 16 Log. of 579416 = 5-762990 579416 log. 5-762990 4324 log. 3-635886 12,00 134-0 log. 2-127104 2. Divide 830772 by 982. The log. of 830772 is taken out by Eule XXVII, page 54. We seek in the left hand column of the Table (No.) for 830 (the first three digits), and also at the top of the page in one of the horizontal columns for the fourth figure 7, then in a line with the first and under the latter we have 919444. In a lino with this quantity and in the right hand column marked JJiff. stands 52, which multiplied by the remaining fig- ures of the nat. number, viz. 72, produces 3744; then cutting ofi' two digits from these (since we multiplied by two digits) it becomes 37, which being added to 919444, the mantissa of 8307, makes 9I9481, and with the characteristic 5, is s-gi948i. The work will stand thus : — Log. 8307 = 919444 Tab. diff. 52 Diff. for 72 -|- 37 X 72 919481 830772 log. 5-919481 982 log. 2-992111 Quotient 846-0 log. 2-927370 104 364 37.44 On Logarithms. 65 4, Divide 34 by 582. 34 log. 1-531479 582 log. 2-764923 Quotient "05842 log. 8-766556 In this instance 10 is added to the characteristic of the dividend to enable the subtraction of the log. of di'^isor to be made, and to avoid negative charac- teristics ; the 6 Log. Log. •02747 = 8-438891 3-434 = 0-535800 Log. -008000 =: 7-903091 Or, using negative characteristics, thus : Log. -02747 ^ 2-438891 Log. 3-434 = 0-535800 Log. 3-90309X 1. Divide 2. Divide 3. Divide 4 Divide 5. Divide 6. Divide 7 . Divide 8. Divide 9. Divide 10. Divide Examples foe Pkaotice. 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 ico-12 ; and 365-55 by 5-5. 783254 by 250689; -79632 by -019354; -0092852 by -0003461 ; and -654831 by "474586. •0008464 by -0002852 ; -05826 by '95381; '019354 by -79632; '0003461 by -0092852; 00005 l^y 2"5, by 25, and by -0000025. 77000000 by 9999 ; 680300 by 681500; 100-002 b3'i'ooi2; and 75759'6 by i3'o62. i'32704by -0358 ; '7156 by 2'68878 ; 87'64i by '000368 ; and '563426 by '023574. 999999 by loioi ; 57634'! by 276'4; 69-7565 by -97564; and 352740 by 56780. 40048000 by 800 ; 11123100 bj' 340; i8692ioby9o; and 1875000 by 15000. 75'2484 by 8'59; 147392 by 440; 1962820 by i0'O4; and 888888 by 88000. On Loga/rithms. 67 11. Divide 248-25 by 364-87 ; -235316 bj' 293-864 ; 5-6949 by 53-058 ; 3876000 by 1200; and 42 by -00007. 12. Divide 2064840 by 3800-62; 33248100 by 830000; 13-5056 by -734; and 674-80 by -0763. 13. Divide -06314 by -0007241 ; -004728 by 0-2382 ; 36-49 by 192-24; -048869 by -0071698. 14. 19-^-72; 19-^-72; -i9-f--72; 19 -f- -0072; 6 -f- -0000003 ; and -9 -f- -0000003. 15. -01237-^-10846; 28-7642 -f- 083456; -oiooii -f- 0993; and -048869 -f- -0071698. 16. -I -;- -0004572 ; I -f- -0011636 ; ii-222i-f-iii; 4000-;- -000125 ; and -562625 -j- 52643. 17. 'oooi -7- -oooi ; loooooo -f- -ooooooi ; 10 -f- 100 ; -oooooooi -f- -oooooi ; 1000 -f- ^. 86. When it is proposed to find the value of an expression in which both multiplication and division are signified, 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. 209 X 573 X 63 Thus : to find the value of 209 log. 2-320146 287 X 2101 573 log. 2-758155 63 log. 1-799341 287 log. 2*457882 2101 log. 3-322426 6-877642 5'78o3o8 5-780308 Ans. : 12-5122 log. i "097 3 34 87. It is very often expedient to transform the logarithm of a divisor into that of a midtiplier, 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. 79, page GoJ. This makes the operation consist of a single addition ; only we must diminish the result by subtracting 1 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 above : Having found in the table the log. of the divisor 287, we may at once transform it into the addition logarithm 7*542 1 18, and similarly, for the log. of 21 01 we may write 6-677574, and then the calculation will proceed con- tinuously as follows : — 209 log. 2-320146 573 log. 2-758155 63 log. 8-799341 287 ar. CO. 7-542118 — 10 2101 ar. CO. 6-677574 — 10 1*097334 Ans. : 12*5122. 13- 7 X 8-73 84 X -00769 X 683 8-4 X -0769 X -00683 ■54963 598-00 X •00146 X "039 59-8 X *ooooi46 X -0039 14. 67*038 X *oio705 X 4-1525 28-045 X I -3564 X -0942537 •7854 X 3"i4i6 X -086725 48-375 X 2-71828 X 52359 15. Divide -06314 X -7438 X -102367 by -007241 X 12-9476 X -496523, and compare the result with the product of 87 1979 X '057447 X -0206168. * To divide by any number n is the same in effect as to multiply by its reciprocal ^ (that is, the quotient of unity by that number). Therefore to subtract log. n is the same in effect as to add log. J, = o — log. n. 68 Irigonometrical IMes. 88. 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 logarithms are carried. But 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 gi-eater than '9388. Thus, for instance, the log. 37575 corresponds to the no. 5721 and the log. 37576 to 5722, nearly. 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. TRIGONOMETRICAL TABLES. 89. There are two kinds of trigonometrical tables ; the first, called the Table of Natural Sines, Cosines, Sfc, contains the numerical values of the sines, cosines, tangents, &c., that is, of the trignometrical ratios for each given value of the angle; the second, called the Tahle of Logarithmic Sines, Sfc, contains the logarithms of the numbers in the first Table. ^ TABLE OF NATUEAL SINES, &c. 90. The trigonometrical functions 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, ^c. 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, Norie) at intervals of i', and to the last mentioned we shall refer. 9 1 . 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. 92. The arrangement of this table vdll 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 interpolations 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 differ- * The usual trigonometrical tatles 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, 6[C. Trigdnofnetricat lalhs. 69 ences of the sines, &e., are proportional to the differences of the angles, and this proportion, though theoretically inexact, gives, in general, a sufficient approximation, provided the difference of the angles of the table are sufficiently small. 93. Eeferring to the Tables (Table XXVI, Norie), 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 lottom of the page, and the minutes up the right hand side of the page. The difference of the trigonometrical ratios for 100" are given at the foot 94. In using these Tables, we have either to find the sine, cosine, &c., of an angle whose value is given in degrees (°), minutes ('), and seconds (") ; or to fi,nd the corresponding angle in degrees, minutes, and seconds. of each column. ' 95. 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 Norie's Table XXVI must be understood as decimals. Thus, nat. sine 7° 7' = •123890 , „ sine 59 40 =: •863102 „ cosine 15 30 = •963631 ,, cosine 71 12 = •322266 96. 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 - • • .JSfat. sine 136° 42' = sine 43' 18' = '395546 „ cosine 108 48 = cosine 71 12 = "322266 „ sine 104 16 = sine 39 44 = •639215 „ cosine 140 16 = cosine 39 44 = '769028 97. If the angle contains seconds, we must proceed by the method of proportionl pa/rts, as in the following examples : — EULE XXXVI. 1°. Find from the Table the nat. sine, cosine, Sfc, which corresponds to the degrees and minutes. (Norie, Table XXVI.) ■2°. Multiply the difference hy the seconds, and divide hy 100. Note. — To divide by 100 we have merely to cut off the two right hand figures. 3°. If the required quantity be 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. * Def. — The supplement of an angle is the result when the angle is subtracted from 180". In other words, an angle and its supplement together make 180°, or two right angles, thus, 23° 19' is the supplement of 156" 41', and 156° 41' is the supplement of 23' 19'. 70 HVigonomehical Tables. 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, cotangent, 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. Examples. 1. Find the nat. sine of 12° 44' 27". Nat. sine 12° 44' =: 220414 473 X 27 Tab. diff. = + 128 100 220542 Ans. : Nat. sine 12° 44' 27" = 220542 Find the nat. cosine of 31° 28' 42". Nat. cosine 31" 28' =: "852944 253 X 42 Tab. diff. = — 106 100 To obtain the parts for the seconds we we multiply the tabular difference by the number of seconds and divide by 100, thus : — Tab. diff. 473 No. of seconds X 27 33" 946 127,71 128 nearly. Tab. diff. 253 Seconds X 42 506 1012 Ans. •852838 Nat. cosine 31° 28' 42' = -852838 106,26 106 Examples for Peactice To find the nat. sine of 34° 48' 15" 3. 71° 20' 43" 60718 4. 214421 To find the nat. cosine of 14° 15' 3" 3- 80° 22' 22' 70 47 40 4. 5 22 10 5- 6. 46° 22' 37" 76 57 49 7- 8. 53° 7' 49" 86 3 17 5- 6. 46° 31' 41" 29 40 48 7- 8. 38° 3i'io'' 8 19 17 98. If the value of the sine, cosine, &c., be given, and it is required to find the angle, we use the following rule : — EULE XXXYII. 1°. Find in the Tables the next lower nat. sine, nat. cosine, Sfe., and note the corresponding degrees and minutes. 2°. Subtract this from the given sine, cosine, ^r., multiplying the difference by 100; divide by the tabular difference, and consider the result as seconds. 3°. If the given value be that of a sine, tangent, or secant, add these seconds to the degrees and minutes found in 1° ; if it be that of a cosine, cotangent, ^c, subtract. The result will be the required angle. Note. — In taking out the angle for a natural cosine we may take out the fiext 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 next less angle being written down in every case, confusion will be avoided as the additional seconds will always be additive. Tables of Loga/rithms of Trigonometrical Ratios. ni Examples. 1. Given the natural sine = 0*7 32 156 : find the angle. Given nat. sine 732156 Sine 47' 4' = 732147 next lower in table XXVI, Norie. Tab. (liflF. = 327 327)900(3" nearly (additional seconds for nat. sine). 981 Ans.: 47° 4 3". The log. 732156 is sought for in Table XXVI, Norie, but as it cannot be found exactly, the next less 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. 2. Given ttie natural cosine 853267 : find the angle. Given nat. cosine 853267 Cosine 31° 26' = 853248 next lower in Table XXVI, Norie. 3i°26' o" Tab. jliff. = 253 2 ^■^) I ^oo{j" [to he subtracted), — 7 1771 129 Am.: 31° 25' 53"- 3. Find the angle whose natural cosine is 728713. Proceeding according to Note, page 70. Here nat. cosine of required angle = •728713 Nat. cosine of next less angle or 43° 13' = 728769 31 25 53 Tab. diff. = 334 334)5600(17" nearly, to be 334 added. 2260 2338 Hour angle required =: 43° 13' 17' . Examples for Practice. Q-iven the nat. sines, to find the angle. •898002 3. -8 5. -444 7. ^740912 9. -75214 •370383 4. -920411 6. -20389 8. A^l or -529221 10. -96 G-iven the nat. cosine, to find the angle. •448807 3. '726998 5. •514841 7. ^769388 9. -817726 II. 999000 •948397 4. -702017 6. '914237 8. -974822 10. -215515 12. -6 TABLES OF LOaARITHMS OF TRIGONOMETRICAL RATIOS. 99. The Trigonometrical Eatios 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. 100. As the sines and cosines of all angles, and the tangents of angles less than 45°, are less than radius or unity, the logarithms of the values of these quantities, properly, have negative characteristics. In order to avoid the inconvenience of printing negative logarithms, and for other reasons, i o is added to the characteristic before it is registered in the table of logarithmic sines, &c., so that we find the characteristic 9 instead of T, 8 instead of 2, &c. 7^ Tahles of Logarithms of Trigonometrical Hatios. Thus on referring to the Table of Natural Sines (Table XXVI, Norie), we find natural sine of i6^ = •275637. If we calculate the logarithm of "275637, we find its value is T"440338, if to this 10 is added we find that Log. sine 16° = 9'440338. 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 10. In trigonometrical operations this is convenient, but principally because the extraction of roots very seldom occurs. It may be observed here that the uniform addition of 10 to the charac- teristic gives the logarithm of loooo millioti times the natural number. Thus, 9"599327 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. 1 01 . 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. Cosec. Tan. D. Cot. Sec. D. Cos. M M Cos. n. Sec. Cot. D. Tan. ' Cosec. D. Sin. M 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 con4;iguous columns on either side, but corresponding to a change of 1 00 ' in the arc ; 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. I02. 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 some- what different. Tables of Loga/rithms of Trigonometrical Ratios. 73 The columns are arranged thus : — M Sinfe. Diff. Cosine. Diff. Tangent. Diff. Cotangent Secant. Cosecant. M M Cosine. DifP. 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 to the first column on the right of the page; that the second column of "Diff." belongs the second column -of logarithms from 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. - 102. 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. 103. "When an angle is presented in degrees and minutes only, the tabular logarithm of its siae, tangent, &c., will be found simply by inspection, according to the following : — EULE XXXVIII. 1°. If the angle or arc is less than 45°. Find the degrees at the top of the page, and the minutes in the left-hand marginal column^ 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°. Loohfor the degrees at the bottom of the page, and for 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 thn top of the table for the degrees (37°), and in the first column on the left for the minutes (47'), we find in the column having at its top the word sine, the figures 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 tables 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 10-589431, which is the log. tangent of 75° 34'. Log. sine of Log. cosine of Log. tangent of Log. cotangent of Log. f 'Cant of Lost, cosecant of 4 IS 38 „ 58 „ 8 35 9-* 9-968278 10-99^1466 9'829532 10-472954 io'826o92 Log. sine of 57° 5' is 9-924001 Log. cosine of 79 51 „ 9-246069 Log. tangent of 21 50,, 9-602761 Log. cotangent of 27 45 „ 10-278911 Log. secant of 44 59 „ 10-150389 Log. cosecant of 69 54 „ 10-027291 I. Log. sine 9° 10' 2. Log. cosec. 40 40 3- Log. cosine 12 48 4- Log. tang. 37 26 5- Log. cotang. 8 25 6. Log. sec. 43 I 74 Tables of Logarithms of Trigonometrical Ratios. Examples for Practice. Take out the logarithms of the following trigonometrical ratios. 7. Log. cos. iS'' 28' 8. Log. sine 51 49 9. Log. sec. 60 34 10. Log. cotang 79 19 1 1 . Log. cosec. 45 45 12. Log. sine 53 56 104. If the value of the angle be given in degrees, minutes, and seconds, ■we proceed by EULE XXXIX. 1°. Find from the iaMe tlie sine, tangent, secant, cosine, Sfc, which corresponds to the degrees and mimites ; also taTce out the number in the contiguous column headed "Diff." on the same line (See Nos. loi and 102, page 72.) 2°. Multiply the tabular difference ("Diff.") hj the seconds, reject the last two figures 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 ahove is sufficiently accurate unless for the sines and tangents of very small angles, and for the tangents and secants of angles very n^ar 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 he found very accurately from the ordinary Tables. In some books, as Button'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 i' 25' and 1° 26' in the ordinary Table, we should get the less accurate result, 2-3970448, because for such small angles the successive tabular differences for one minute shows too rapidly a wide departure from equality. "W hen 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. Norie 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 i 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 proportion 60 : a : : 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 thus loo ' d 18 found somewhat more readily. Tables of Loga/rithms of Trigonometrical Ratios. 75 Examples. 1. Find tlie log. sine of 6° 36' 27". Here the given niimber 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, vii!., 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"o6o46o, 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 " difl"." column from the left. It will be noticed that there is no difi". 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 pnoduct (for the division by 100) gives quotient 490, which being increased by I, since the figiires cut ofi" exceed 50 (see Note i, page 74) gives 491 as the correction of the logarithm for the seconds. The work will stand thus : — Log. sine 6" 36' = 9'o6o46o Tab. difi". 1817 27' gives -f 491 X 27 9'o6o95i 12719 3634 490>59 or 491 Arts. : Log. sine 6° 36' 27" =: 9*o6o95i, 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 0^ 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 16,00 or 16 Ans. : Log. cosine 13° 5' 32" = 9-988562. 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 60,24 4. Find the log. cotangent 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 7" = — 54 100 qj. ,/ Log. cotang. 73° 21' 7" = 9-475709 The parts for the seconds are subtracted in this instance being a colog. (See Kule XXXTK, 3^. 76 Tables of Logmithns of Trigonometrical Ratios. 5. Take out log. sine 1° 5' 34". Here the angle whose log. sine is sought being less than 2", it must, therefore, be 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 8'27994i, and the corres- ponding tabular, "Diff" (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 the division by 10 gives the correction 442, to be added to the logarithm taken out of the Table ; thus the work stands as follows : — Log. sine 1° 5' 30" =: 8-279941 Tab. diff. 1104 Parts for 4 = + 442 4 Log. sine i 5 34 = 8-280383 441,6 or 442 nearly. 6. Required the cosecant of 3° 7' 21". Log. cosecant 3° 7' o" z= 11*264646 Tab. diff. 3857 Parts for 21 = — 810 21 Log. cosecant 3 7 21 =11-263836 3857 7714 809,97 105. For the functions* of an angle between 90° and 1 80° we may take tlie same functions of its supplement ; hence to find the logarithm of a trigono- metrical ratio of an angle greater than 90°, i.e., of an obtuse angle, we have the following EULE XL. Subtract the angle from 1 80° and look for the remainder, which is called its supplement in the Tables. Examples. Ex. I. Find the log. sine of 110° 24'. Subtract it from 180°. From 180" 00' Subtract no 24 Eemainder 69 36 (Supplement.) Look for the log. sine of remainder (namely 69° 36'), which is 9-971870; or log. sine 110° 24' = 9-971870. Ex. 2. Find the log. secant of 95° 43' ; also the log. cosecant of the same. From 180° o' Take 95 43 Supplement 84 17 Look for the log. secant of 84° 17', which is 11-001701; . " . log. secant of 95* 43' is ii"ooi7oi. Again, look for the cosecant of 84° 17', which is 10-002165; . ' . log. cosecant of 95° 43' is 10-002165. *Bythe/Mwc) 238 S. „ 4 48 S. „ 288 N. » 35 25 S. >> 229 S. 94 Prelimina/ry Rules in Navigation. Examples. Ex. I. Find the mid. lat., having given 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 Ex. 2. Lat. from 6° 28' S., lat. in 14° 50'S. required the raid lat. Lat. from 6" 28' S. Lat. in 14 50 S. 2)21 18 Mid. lat. 48 48 ' Mid. lat. 10 39 " Examples fob Practice. Required tlie middle latitude in each of the following examples : — 1. 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 S. 5. „ 56 10 N. „ 50 15 N. 3. „ 36 22 N. „ 90 S. 6. ,, 67 20 S. „ 61 42 S. I 54. To find the difference of longitude, having given the longitude from and longitude to. (For definition see No. 134, page 86). EULE XLYIII. 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 ; hut 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 usual way. Longitudes are reckoned East or West of the first meridian. If these different 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 diff. 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 sliip 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 diff. of long. Long. Cape Bajoli 3° 48' E. Long. Cape Sicie 5 5 1 E. 60 Ex. 2 . Kequired the diff. of long. , having given the long, from 12° 20' E., and long, in 2° 45' W. Long, from i2°J2o' E. Long, in 2 45 W. 15 5 60 D. long. 123 E. The lonK. to Capo Sicie is E. of long, from Cape Bajoli, therefore, ditf. 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. 1° 25' W., is bound to long. 7' 12' E. : re- quired the diff. of long. Long from 1° 25' W. Long, to 7 12 E. 8 37 60 D. long. 517 E. Th* fhip here is about to cross the meridian of Greenwich (long, o'^) and pass from W. long, to E. long., whence the dilf. of long, must be E. to do so. Preliminary Rules in Na/vigation. 95 Ex. 5. Find the difT. lonp:. 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 1 80' it is subtracted from DifiF. of Ions:, is 234 15 E. 360 o 125 45 W. 60 D. long. 7545 W. By going E. and W. from Greenwich, the two places in this examplo 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 sub- tracted from 360', the whole circumference of a circle, for the required answer. Ex. 7. A ship from long. 5° 12' W. is bound to a port in long. 90"" W. : what diff. of long, must she make ? Long, from 5° 12' "W. Long, to 90 o "W. Ex. 6. A ship from long. 177' 50' E. arrives in long. 178° 10' W. : what diff. of long, has she made ? Long, left 177" 50' E. Long, in 178 10 W. Being greater than 180° 356 o W. it is subtracted from 360 o DifiF. of long, is 4 o E. 60 D. long. 240 E. 8448 60 D. long 5088 W. The ship here passes from a less to a greater "W. long. : and therefore the diff. of long, must he W. to do so. Ex. 8. A ship from long. 165' E. is bound to a place in long. 72*^ 12' E. : what diff. of ^ong must she make ? Long, left 165° o' E. Long, to 72 12 E. 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. Examples for Pkactioe. Required the clifFerence of longitude between a place A and a place B in each of the following examples : — Long. A 9° 29' W. Long. B. 4'' 29' W. „ I 25 W. „ 72 E. 6 II E. „ 5 45 W. „ 00 „ 4 20 W. „ 4 20 W, „ o 10 E. „ 7 2 E. „ 00 7- Long. A o" 55' E. Long. B7° 3'E. 8. 40 10 E. 33 10 E. 9- 178 30 W. 178 30 E. 10. 176 34 E. 176 34 W. II. 38 32 W. 8 43 E. 12. 5 12 W. 25 12 W i;;;. To find the longitude in, having given the longitude from and the difference of longitude. EULE XLIX. 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 J ; 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 difference of longitude have unlihe names — Under longitude from, put difference of longitude (in 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. 96 Preliminary Rides in Navigation. Examples. Ex. I . A ship from Ion?. 5° 1 2' "W. makes diflF. long. 113' W. : required the long in. Long, from 5° 12' "W. 6,0)11,3 D. long. I 53 W. 1° 53' W. Long, in 7 5 W. Ex. 3. A ship from long, o^ 57' E. sails "W". until her diff. of long, is 201' : find the long. in. Long, from o' 57' E. D. long. 321 W. 6,0)20,1 3° 2 W. Long, in 2 24 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 i" 25' "W. 6,0)17,7 D. long. 2 57 E. 2°57'E. Long, in i 32 E. Ex. 4. Let the long, left be 1 74° 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. i3°i7'W. Being greater than 180° 187 21 W. subtract from 360 o Ex. 5. Long, 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 E. 3° 40' E. Long, in o o 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 long, arrived at. Long, from 177° 40' W. 6,0)14,0 D. long. 2 20 W. 2° 20' W. Long, in 180 o W. or, 180 o E. Examples for Practice. Required tlie longitude in, or arrived at, in each of the following examples r. Long, from 5° 48' W. D. long, no' W. 2. „ o 59 W. „ 137 E. 3. ,, 29 10 E. „ 114 E. 4. „ 3 10 E. „ 220 W. 5. „ 2 47 W. „ 242 E. 6. „ 3 12 E. „ 237 W. 13. Define meridian of the earth, equator, parallel of latitude great circles , and why ? Long, from 41^29'^. D. long. 139'E. 94 4E. „ 115 W. 98 54 E. „ 302 E. 178 13 E. „ 201 E. 177 6 W. „ 237 W. 179 59 W. „ 2 w. Which of these are THE COMPASS. 1 56. 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 all 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 j)urposes it is intended for it is named The Steering Compass, The Standard Compass, and The Azimuth Compass. 157. The Mariner's Compass consists of a circular card, which 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 vnry 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. 98 The Com/pass. I. By Points. — There are 32 points; and each, of those divisions is again sub-divided into four parts called quarter pointn. A point of the compass being therefore the 32nd part of the circumferenee 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 j)oint 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., W.S.W., W.N.W., and N.N.W. The remaining sixteen points are reckoned from the cardinal or secondary point to which each is adjacent, the name of which it takes qualified by the name of the succeeding cardinal point towards which it lies. Thus, the point next to N., on the east side, is called North b}^ East (written N. by E.) ; that next N.E., towards the north, is called North-east by North (N.E. by N.); 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 S., S.W. by W., W. by S., W. by N., N.W. by W., N.W. by N., N. by W. The points of the compass are frequently spoken of with reference to their position to the right or left of the cardinal point towards the spectator is looking ; thus, N.N.E. is said to be " two points " to the right-hand of North ; W.N.W. six points to the left of North. 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-AflsZ/'-east (written N. |- E.) Half-points near N.E., N.W., S.E., and S.W., take their name from these points. Thus, we say, N.E. i N., N.E. by E. i E. 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 I'eckon uniformly from North or South, but generally the simpler name will be the preferable one ; and simi- larly 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 The Compass. 99 a notation for the compass more minute then 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. The name of the opposite point to any proposed point is known at once, without referring to the compass, by simply reversing the names or the letters which compose it — thus, the opposite of N. being S. and of E. being W., the opposite point to N.E. by N. is at once known to be S.W. by S., the opposite of W. i S. is E. f N. and so on. 158. Repeating the points in any order is called boxing the compass; to do this is, of course, one of the first things a seaman learns. 159. 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 na^ne. 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, vrhich every Point and Q,uaxter Point of the Compass makes mth the Meridian, NORTH Points ' „ Points SOUTH 1 \ 2 48 45 -\ 1 1 5 37 30 s 8 26 15 i N. by E. N. by W. 1 II 15 1 S. by E. S. by W. I i 14 3 45 I i I ^ 16 52 30 I g 1 1 I i 19 41 15 N.N.E. N.N.W. 2 22 30 2 S.S.E. S.S.W. 2 i 2 S .25 18 45 ^ f 28 7 30 2 ^ 2 1 30 56 15 2 4 N.E. by N. N.W. by N. 3 33 45 3 S.E. by S. S.W. by S. 3 1 3 i 36 33 45 3 1 3 i 3 it 39 22 30 3 1 42 11 15 N.E. N.W. 4 45 4 , S.E. S.W. 4 ^ 4 i 47 48 45 4 i 50 37 30 4 2 4 i 4 i 53 26 IS N.E. by E. N.W. by W. 5 56 15 5 S.E. by E. S.W. by W. H 59 3 45 5 i 61 52 30 5 ? 5 1 64 41 15 E.N.E. W.N.W. 6 67 30 6 E.S.E. W.S.W. 6 i 70 18 45 6 1 6 s 73 7 30 6 i 75 56 15 6 1 , E. by N. W. by N. 7 78 45 E. by S. W. by S. 7 f 81 33 45 7 |- 7 $ 7 i 84 22 30 7 * 87 ir 15 East. West. ^ 90 8 East. West. 160. The card is laid upon a magnetic needle, which is a small steel bar magnetised, the north end being attached to the north end or pole of the needle."^' The whole is then balanced on a sharp centre or pivot rising from * Some compass cards carry on their lower surface, one, two, four, or more parallel mag- nets with similar poles pointing in similar diiectious. The object of using several magnets is to increase the magnetic moment of a given weight of steel. 100 The Compass. the bottom of a brass bowl, and covererl with glass. The bowl having a weight fixed to it below, is placed in gimbals, which are brass hoops or rings, so arranged as to admit of motion about two independent horizontal axis at right-angles to each other, i.e., each turning upon two pivots at opposite points of the hoop next greater in size ; by this means the loaded bowl re- mains nearly horizontal during the confused and irregular motion of the ship. The pivots of the outer ring fit into bearings in the binnacle (a turret shaped ease fitted with panes of glass and a lamp) and constitutes the Steering Compass. i6i. 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 luhber'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 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. 162. 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. 163. In the best modern instruments, a horizontal ring is expressly pro- vided to carry the vertical wiro 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 prism. This prism is a solid piece of glass, whose sides are parellelograms 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. 1 64. 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 may be supposed to vibrate equally on both sides of the line of direction of the object, is essential to the true result. The Compass. loi 165. Standard Compass. — The Standard Compass on board ship is the one placed on a particular spot on deck, or above it, where the local deviation is nothing, or very small. Such a compass will show magnetic bearings correct, or of ascertained errors, and the deviation of the Steering Compass can at any time be determined by a comparison with it — all other compasses em- ployed being in fact simply considered as auxiliaries to it. It should be fitted with an azimuth circle. This circle should be graduated so as to show the angle between the ship's head and any heavenly body, as measured on the horizon, without using the compass card ; the sight- vanes and reading prism should be fitted to the azimuth circle in such a way as to turn freely in azimuth, without moving the compass bowl or disturbing the card. SELECTION OF BEST POSITION FOE COMPASSES. 166. Standard Compass. — The Standard Compass should be placed in the middle of the ship, and fixed on a permanent and secure pillar or support, raised at such a height as to pel-mit amplitudes of the sun and bear- ings of the land to be conveniently observed by it. It should also be in a position as far as possible removed from any considerable 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 15 feet of this compass, either on the same deck or that below it. 1 67. Steering Compasses being placed according to the requirements of the ship, the moderate and uniform amount of deviation generally attainable at the Standard Compass by selection 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 necessity, the steering compass is fitted — then large and perplexing deviations may be expected, defying even 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 3ft. 6in. 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 Department) are worthy the attention of the Naval Architect and those superintending the equipment of the ship : — 1 . 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 102 The Compass. 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 forwarA 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 Rule 2. In ships built nearly East or West, the Standard Compass not to be near either extremity of the vessel 4. 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 5 5° of the vertical line through the centre, the angle being drawn from the compass as centre to the centre of the mass in question. •■' 168. 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 manu- facture, and the constant attention of the navigator should be devoted to ascertain its errors. Note. — "Comparative Merits of Large and Small Compasses. — Of late years much diversity in practice has prevailed 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 in large passenger steam vessels the needles are frequently 12 to re 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 oscillation of long needles compared with short ones. " Large cards, however convenient in practice, are therefore not without danger, for the course steered may 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 * Investigation has shown that the effect of a sphere of iron within this cone is prejudicial by diminishing the directive force and increasing the heeling error to windward — when without the cone 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 effect is the same as if they were solid bodies, on the assumption that magnetism exists entirely on the surface of iron masses. This is not the case ; it is, however, true that the effect of hollow masses of iron increases very rapidly with the increase of the thickness of the iron so that the limit of thickness is speedily reached when the efi"ect 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 effect is about i of a solid mass of the same size ; in a similiar sized tank |^ ot an inch thick, the effect would be about half that of a solid mass. — See a valuable investigation by Mr. Ai'chibald Smith, in the Phil. Trans, for 1865, pages 304—318. The Compaas. 103 hazarded. In the opinion of the writer, the present Admiralty Standard Card is as large as should bo used for the purposes of navigation, and that as regards safety in the long, steady, 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 is XB^di."— Manual of the Deviation of the Compass, by Capt. Evans, R,N. ADJUSTMENTS OF THE COMPASS. 169. (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 obser- vation and therefore need not be made the subject of special examination. (2,) The pivot must be 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 ^^line of sight " 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 turn- ing 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 hj taking the mean of the direct and reversed bearings every time the instrument is used. (4.) The sight vanes must he vertical, i.e., the eye-vane and the ohjeet-vayie 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. 170. 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. 171. 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 hand side of a person looking forward.'^' When the ship is not going before the wind, she will not only be forced forward in the direction of her head, but, in consequence of the wind pressing againt her sideways, her actual course will be to " leeward'''' of the apparent course she is lying. The amount of leeway differs in different ships ; depend- ing on their construction, on the sails set, the velocity forward, and other circumstances. Experience and observation are requii'ed to judge what amount of leeway to allow in each case. Tlie 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 EULE L. When the ship is on the port tack, allow the leeicay to the right of the course steered ; hut when on the starboard tack, alloiv it to the left, the olserver looking from the centre of the compass towards the point the ship is sailing upon. Examples. Ex. I. The course steered is N.W. by [ Ex. 2. Course by Compass S. by E., W., the wind N. by E., leeway i\ points. wind E. by S., leeway af points. The ship has the starboard tacks on board ; therefore, the leeway {i\ points) allowed to the left of N.W. by W., gives corrected Course W, by N. f N". The ship is on the port tack, then 2f points allowed to the right of S. by E., is S. by W. f W., the Course corrected for leeway. * A ship is said to be on the tack of the side from which the wind comes : even if it be on the quarter. Cwrecting Courses. 105 Ex. 3. Course N.E. by N., the wind N.W. by N., the leeway i point. The ship being on the port tack, i point to the right, of N.E. by N. is N.E., the cor- rected Course, Ex. 4. Course steered W. by S., the wind N.W. by N., leeway 3^ points. The ship is on the starboard tack, 35 points to the left of W. by S. is S.W. ^ S., the Compass Course made good. 172. The points of the compass are frequently treated with reference to their position to the right or left of the cardinal point towards which the spectator is looking, thus N.N.E is said to be "two points to the right of North;" W.N.W. " six points to the left of North." Adopting this notation the work in the above Examples will stand thus : — Ex. I. Course steered N.W. by "W. is 5 pts. L of N Leeway carries ship i\ „ LofN Ex. 2. Course steered S. by E. is r pt. L of S Leeway carries ship if „ R of S Sum is corrected course 6| „ L of N 1 The difference is or W. by N. f N. ! RofS S. by W. I W. Ex.3. Course steered N.E. by N. is 3 pts. R. of N. Leeway carries ship i „ R. of N. Sun is corrected course 4 ,. R. or N.E. Ex.4. Course steered W. by S. is 7 Leeway carries ship 3 pts. R. of S. k ., L. of S. of N. The diff. is corrected course 3^ ,, R. of S. or S.W, i S. Examples for Practice. Correct the following courses for leeway : — 3 Course Steered. Wind. Leeway. Course Steered. Wind. . S.S.W. S.E. i^ -^5■ E. 1 N. N. by E. . S.W. i w. W.N.W. 2i X6. N.W. 1 N. N.E. by E . N. by E. E. by N. 3 -4^1- S.W. bv W. S. by E. .. N.N.E. iE. N.W. ^ N. 2 8. N.E. IE. N. by W. Leeway. (a) When 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 off 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 N.N.E., the compass course made good. Ex. 3. A ship lying-to comes up S. by E. and falls off to S.JE. by E., ths 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 S., then 5 points to the left hand (the ship havinar starboard tacks on board) is East, the con.^ ...s 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. i S., then 2^ points to the right 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. b}^ W. is W. by N., then 5 points to the right hand is N.N.W., the compass course made good. io6 Correcting Cov/rses. 2. THE YAEIATION OF THE COMPASS 173. The needle points to the magnetic north, which in few parts of the world agrees with the true north, the difference 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 westerly 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 z^ points west ; but when it points to the N. by E. part of the horizon, the variation is said to be i point east. 174. 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. J It also changes slightly at different times of the day.§ Its value for each locality is indicated on charts, and always to be found by easy methods. 175. 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 needle is directed two points to the right of the north point of the heavens — that is, points N.N.E. * 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 Greenwich, 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 ; at 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 Ajnerica, and the Pacific, it is easterly. X " 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 Meridian, coincided with the Geographical Meridian : it now makes with it an angle of about 30° W. In the sixteenth century, the the compass-needle in Britain pointed east of north : it now points from 20° to 30= (in differ- ent parts of the British isles) west of north. At the present time, a change of the opposite character is going on: in 18 19 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 rotiting round the geographical poles from East to West." — A Treatise on Magnetmn, designed for the use of Students in the University. By George Biddell Airy, M.A., L.L.D., D.C.L. § Besides the gradual changes which occur in terrestial magnetism, both as regards direction and intensity of force, in the course of long periods of time, there are minute' fluctu- ations continually traceable. To a certain extent these are dependent on the varrying positions of the sun, and, to a much smaller extent, of the moon, with respect to the place of observation ; but over and above all regular and periodic changes, there is a large amount of irregular fluctuations, which occasionally become so great as to constitute what is called a magnetic storm. These variations occur with great rapidity, causing deflections to the right and left comparable in 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 telegraphic wires, and this connection is found to be so certain, that upon remarking the display of one of the three classes of phenomena, we can at once assert that the other two are observable (the aurora borealis sometimes not visible here, but certainly visible in a more northern latitude). Correcting Cowses. 107 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 by compass N.N.W., or, as it is usually expressed, her compass course is N.N.W., her true course will be north ; that is, two points to the right of tJie 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. 176. To find the true course, tlie compass course being given. EULE LI. Allow easterly variation to the right 0/ the compass course, westerly ,, left ,, ,, looking from the centre of the card over the point to he corrected.* Examples. Taking the courses between North and South round by east. Ex. I. Course steered N.E. by E. vari- ation 2| points West, to find the true course. Here the compass course is N. 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 (N. 5 E.— 2|=N. 2^ E.), that is, within 2J points of North ; the true course is therefore N.N.E. iE. 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. I W. Ex. 5. Compass course S.E., variation i\ points East, then the true course (allowing the variation to the right) will be S.S.E. f E., or S. zf 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 true course (allowing the variation to the left) will be W. by N. I N., or N. 6^ points W. Ex. II. Again, compass course S.W. by S., variation 2'^ points West, the true course (allowing variation to the left) will be S. \ W. Ex. 13. Compass course S.S.W., varia- tion 3^ West, then allowing 35 W. to the left of S.S.W. gives S. by E. ^ E., or i^ points 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 away from the North towards the Hast, that is, 6| points to tha Eastward of North (N. 5 E. + i| E. = N. 6? E.) ; the true course is therefore E. by N. i N. Ex. 4. Compass course S. by E. varia- tion 2^ Hast, 2^ points allowed to right of S. by E. is S. by W. ^ W., or S. ij W. Ex. 6. But compass course S.E., varia- tion 2 J points West, then the true course (allowing the variation to the left) will bo E. by S. I S„ or S. 6^ points E. Ex. 8. Compass course E., variation 2^ points Hast, then the true course (allowing the variation to the right hand) is 8.E. by E.iE. Ex. 10. Taking the same compass course, viz., N.W. ^ W., when the variation is ij points Hast, 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 Hast, the trm course (allowing the variation to the right) will be S. W. f W., or S. 4f points W. Ex. 14. Compass course W., variation 2^ E., then the true course (allowing 2^ points to the right) is N.W. by W. ^ W., or N. 5I points W. * 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 he corrected. When he places the compass card before him, mistakes very frequently occur in the application of the varia- tion between the east and wett 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 different with the S.E. and S.W. quadi-ants, unless 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 the hands of a watch move over the dial, and Left the contrary direction. io8 Correcting Courses. Ex.15. But with compass cour.se West, Ex. 16. Compass course N.N.W. | W., and variation 3.^ TFl'.si, then allowing 35 | variation 3J points Hast, then 3^ points to points to the left of W., the true course is the right of N.N.W. i W., is N. f E. S.W. f W., or S. 4| points W. j 177. 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, in which the points^of the compass are treated numerically. EULE LII. i'. Put down the points and quarter points which the compass course is to the right or left of North or South, marhing them R. or L. accordingly. 2". Underneath put the variation, marking it also R. or L., accordingly as it is E. or W. 3°. If the names are alike, talce 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 16 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 marked S. if previously marked N. ; also, if marked L (left) change to R (right) ; but if marked E, change to L. 4°. If the names are unlike, take the diflFerence, and mark it the same name as the greater. (c) If the variation leing subtractive, exceeds the amount from which it is to he subtracted, take the points of the course from the variation, a?id 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 : — 1. Compass Courses :— S.S.W. ; N. by E. ^ E. ; W.S.W. ; and E. by N. Variation 3 J points Easterly. Required the True Courses. S.S.W. S.S.W. = 2 R. of S. Var. 3iE-[3°] N. by E. i E. liR. ofN. 3iE- [3°] W.S.W. 6 R. ofS. 3iE. [b] E. by N. 7 R. ofN. 3iR- [b] Sum 5|E. ofS. 5 — 9I R. of S. 16 — loi R. of N. 16" S.W. by W. i W. Here the sum is taken for the true course, the names being alike. N.E. by E. Here the names ahke the simi is (See No. f) being 6i L. of N. taken. W. by N. i N. 5|L.ofS. SrE. by E. i E. 2. Compass Courses ; ;— N.N.W. ; S. by E. ; W. i N. ; and E. by S. Variation 2| W. N.N.W. 2 L. ofN. 4L. [3°] S. by E. I L. of S. W. 1 N. 71 L. ofN. 2iL. [b] E. by S. 7 L. of S. 4L. [b] 4I L. of N. 3| L. of S. — 10 L. ofN. 16 - 91 L. of S. 16 N.W. \ W. S.E. 1 S. 6 R. ofS. W.S.W. 6^ R.ofN. E^byN.JN. Correcting Courses. i6g 3. Compass Courses :— N.E. i E. ; S.W. | W. : N. by E. ; and S. by W. Variation 2^ points West. N.E. I E. S.W. I W. N. by E. S. by W. 41 R. of N. 4f R. of S. I E. of N. i R. of S. 2^L. [4=] 2iL. [4^] 2iL. [c] 2iL. [c] 2iR. ofX. 2IR. ofS. i^L. ofN. liL. ofS. N.N.E. i E. S.S.W. i W. N. by W. | W. S. by E. i E. 4. Compass Courses:— N.W. by W. ; S.E. by E. ; N. by W. i W. ; and S. by E. Variation 3^ points East. N.W. by W. S.E. by E. N. by W.^ W. S. by E. 5 L. ofN. 5 L. ofS. i|L. ofN, i L. ofS. 3|R. [4°] 3^R. [4=] 3iR. [c] 3iR. [c] ifL. ofN. i|L. ofS. i|R. ofN. 2IR. ofS. N. by W. I W. S. by E. f E. N. by E. f E. S.S.W. i W. 5. N.N.E., Variation 2 points W. ; S. by E., Variation i point E. ; W. by S., Variation 1 point E. ; and E.S.E., Variation 2 points W. N.N.E., Var. 2 W. S. by E., Var. 1 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 E. 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. 178. If the learner has carefully gone through the preceding examples, he will have noticed that Easterly variation in its application to Compass Courses increases them in the N.E. and 8. 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 LHI. 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. iid Correcting Cownes. 1. Compass Courses: N.N.E. ; S. ty W. ^ W. ; W.N.W. ; S.E. ^ E. Var. 3^ E. N. 2 E. S. \\ W. N. 6 W. S. 4^ E. + 3iE. +3iE. -£iE. -3iE. N. 5iE. S. 4fW. I<.2fW. S. iiE. N.E. by E. i E. S.W. | W. N.N.W. f W. S. by E. ^ 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. + 3iE. +3JE. -3_iE. -3_iE. N. li E. S. 2i Vi N. by E. i E. S.S.W. ^ W. N. 91 E. S. loj W. N. li E. S. z^ W. 16 16 S. 6| E. N. 5| W. E. by S. i S. N.W. by W. f W. 3. Compass Courses :— N.E. ; S.W. i S. ; N.W. ^ N. ; S.E. i S. Var. 2i W. N. 4 E. S. 31 W. N. 3^ W. S. 3f E. _ 2i W. — 2i W. + 2i W. + 2i W. N. i| E. S. li ^. N. 5f W. S. 6 E. N. by E. I E. S. by W. i W. 4. Compass Courses :— N. by E. ; S. by W. i W. ; W. i N. ; E. by S. Var. 2^ W. N. I E. S. li W. N. 7^ W. S. 7 E. — 2iW. — 2^W. +2iW. +2|W. N. if W. S. I E. N. 9I W. S. 91 E. 16 16 S. 6iW. N.6fE. 5. Compass Courses :— N.N.W. | W. ; S.S.E. f E. ; N.E. by E. i E. ; S.W. by W. i W. ; Variation 2f E. N. 2fW. S. 2|E. N.fiE. S. 5iW. - 2| E. - 2f E. + 2| E. + 2i E. o o N. 8 E. S. 8 W. North. South. East. West. Examples for Practice. Correct the following Compass Courses for Variation : — COMPASS COURSE. VAR. COMPASS COURSE. VAR. COMPASS COURSE. VAR. I. N.N.E. 2 E. 6. S.S.E. 2 W. 12. N.N.E. 2iW. 2. S.E. ^S. I^E. 7. S.W. iS. if W. 13. S.S.W. fW. 2f E. 3. S.W. i w. ifE. 8. N.N.W. fW. 2i W. 14. E.N.E. 3|E. 4. N.W.byW.i W. i^E. 9. E. JN. 3 E. 15. E. |S. 2jW. 5. N.E.^N. i^W. 10. N.W. by N. 11. S.W. ^S. 3|E. 4iW. 16. W. by N. 2|W. 179. The learner may now proceed to correct the courses steered for the combined effect of leeway and variation, and in doing so we proceed by EULE LIV. WTien they are both to he applied in the same direction, take their sum and applf/ it in the same way ; hut when these corrections are to he applied in opposite Correcting Courses. m directions take their difference, and apply the remainder in the same direction as the greater correction is to be applied ; the result in either case is the true course. Examples. Ex. I. A ship sails N.E. by E. ^ E. on the port tack and makes i| points leeway, the variation is 2^ points East : required the true course. Here the ship's course being N.E. by E. \ E. is 5^ points right of North. The ship being on the port tack the leeway is applied to the right, and is i^ points right of North. The variation being East is applied to the right and hence is 2^ points "right of North. The names being alike we take the sum 9^ points right of North. This being greater than 8 points we take it from 16 points 16 And we get true course 6|- points left of South. or E. by S. i S. Ex. 2. Course by compass S.W. by S., the wind W. by N., leeway 2 points. Variation 3 points East. Here the course steered being S.W. by S. is ... . 3 points right of South. The ship being on the starboard tack the leeway is applied to the left and is 2 points left of South. The names being unlike the difference is taken i point right of South. The variation being 2 points East is applied to riffht and is 3 points right of South. The names being alike the sum is taken and is. . 4 points right of South. True course S.W. 180. Sometimes it may be desirable to express the Variation in degrees, in which case we proceed as follows : — ETJLE LV. 1°. Correct the compass course for leeway as before directed, and convert the number of points thus found in degrees, marking them R or L, according as they are right or left o/N. or S. 2°. Underneath write the variation, marking it E, or L, according as it is E. or W. Take the sum tvith the common name, if the names are alike, and the difference toith the name of the greater, if the names are unlike. The result will be the number of degrees the true course is from N. or S. according as the course, as corrected for leeway, is reckoned from the N. or S. (a) If, in taking the sum the number of degrees exceed 90° take the supplement to 180°, and reckon the true course from the opposite point to that from which the course corrected for leeway is reckoned; also change the letter R or L. 112 Correcting Courses. Examples. COMPASS COURSE. S.W. 4 S. N. by E. W. |N. S.W. i S. = 3i R. Z\ R. of S. Leeway J L. W E. S. ofS. WIND. • ty N. by N. W. by 1 2 S. N LEE-WAT. I . by E. = 1 pt. R. I R. ofN. 3iL. ofN VAH. 23" W. 20° E. 25° W. W. TEUE COURSE. S. 8=W. N. 5°W. S. 85= w. 1 N. = 71 L. of N. 7iL. OfN. I H. 2|R. 2iL, 5 0131° R. of S Var. 23 L. or 25° L. of N 20 or 70° L. 25 L. 8 R. of S. S. 8° W. 5 L. of E. N. 5= W. EUI-E LVI. — 95 L. of N. 180 85 R. of S. S. 85° W. 1 1. 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. £x. 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 2 j points west, the compass course (allowing variation to the right) will be S.S.E. | 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 2\ points icest, then the compass course (allowing westerly variation to the right) will be N.N.W. h 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 2j points west, then the compass course (allowing westerly varia- tion to the right) will be S.W. by W. f W. 182. Treating the points of the compass numerically, we proceed according to the following EULE LVII. Proceed according to Rule LII, page 108, in every particular, except that the variation is to he allowed the opposite way to that of correcting compass courses, viz., Westerly variation is to he allowed to the right and marked E; and Easterly variation is to he allowed to the left and marked L. Ex. 2. Taking the same course, viz., N.E. by E., where the variation is 15- 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. IE. Ex. 6. Suppose the course to be the same, viz., N.W. by W., where the varia- tion is 2^ 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 to 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. Correcting Cowsen. 113 Examples. 1. 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. i W. E. by N. \ N. W. by S. 2 E. ofN. 4R. ofS. 6|R. ofN. 7 R. ofS. 2| R. 2| R. 2^ R. 2^ R. 4j R. of N. 4 E. of S. 9 R. of N. 9 J E. of S. — — 16 16 N.E.^E. S.W. — 7 L. of S. 6\ L. of N. E. by S. W. by N. \ N. 2. True Courses :— N.E. by E. | E. ; S.W. by W. ; E. by S. i S. ; N.W. by W. Variation 3I E. N.E. by E. A E. S.W. by W. E. by S. \ S. N.W. by W. 5iR. ofN. 5 R. ofS. 6|L. ofS. 5 L. ofN. 3i L- 3i L- Ji L. 3i L. 2 R. ofN. i|R. ofS. 10 L. ofS. 8JL. ofN. — — 16 16 N.N.E. S. byW. iW. — — 6 R. of N. 7| R. of S. E.N.E. W. I S. 3. True Courses:— N. by W. i W. and S. by E. ; Variation 3^ W. S. by W. and N. by E. ; Variation 3f E. N. by W. \ W. S. by E. S. by W. N. by E. li L. of N. I L. of S. I R. of S. I R. of N. 3i R- 3J R- 3l L. 3| L. if R. of N. 2i R. of S. 2| L. of S. 2| L. of N. £ N. by E. f E. ^■'^A'TA S.S.W. i W. S.S.E. | E. N.N.W. f W. 183. To convert true coui-se into compass course, we may proceed accord- ing to the following EULE LVIII. 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 i " + to all points between N. and W S. and E. DEVIATION OF THE COMPASS. 184. The large quantity of iron now used in the construction and equip- ment of steamers, iron sailing vessels, and sometimes of wooden sailing vessels, produces a deviation from the magnetic north, which interferes seriously with the navigation of such vessels ; and it is highly important that the officers of the Mercantile Marine should have a thorough acquaintance with the subject of local attraction, and with the correct method of applying to the different points of the compass, the " deviation," which is the effect of that attraction. GENERAL STATEMENT OF FACTS AND LAWS OF MAGNETISM. 185. 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. 186. 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 in 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 portion 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. 187. 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. 18^. 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 Deviation of the Compass. 1 1 5 pole, the south pole of the needle dips. The terrestrial magnetic poles do not. coincide with the geographical ones, nor are these points diameterically opposite. The position of these poles are latitude 70° N., long. 97"^ W., and lat. 73i° S., longitude 147° E. The line of no variation passes through these poles, and the lines of equal variation converge towards them. 189. Magnetic Needle. — Any magaet freely suspended near its centre is usually called a magnetic needle, or more properly a magnetised needle. When a magnetised needle is so 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 con- structed. The angle between the magnetic meridian and the geographical meridian is called the variation. 1 90. Dip, 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 direption in that plane. This direction is not horizontal 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 Grreenwich, 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.* 191. 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, accord- ing 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. 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 * Dip first noticed by Egbert Norman in 1576. ti6 Deviation of the Compass. north end of the magnet, the south end of the needle dips, the dip aug- menting 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 ; pass 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 downwards. 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. The value of the dip, like that of the variation, differs in different 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 Greenwich was about 69° i', it has diminished, with a rate continually accelerating, till in 1868 it was 67° 56'. It is also subject to a slight annual and diurnal variations, being about 1 5' 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. 192. 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. 193. 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 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.^ * Sir "Wm. Thomson calls the north-seeking pole the sotith 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-seeking pole of a needle the austral, and the other the boreal pole. Faraday, 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. Airy, for a similiar reason, employs in his recent Treatise on Magnetism, the distinctive names red and blue to denote respectively the north-seeking and south-seeking ends, these names, as woU those employed by Faraday, 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 pj^int. Maxwell and Jenkin in a report to the British Association call the south-seeking pole of a needle positive, and the north-seeking pole negative. t>eviation of the Compass. \ i i 194. 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 fiKngs over the iron they will adhere to its ends, as shown ( 1 86). If we take away the influencing magnet the filings will fall ofi', 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 effects than the iron ; it is much more difficult 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 hard, 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. 195. 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 194.. 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 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 less 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 wiU 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 ii8 Deviation of the Cotnpass. 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 be 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. 196. 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. 197. 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 ends 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. 198. The hull of an iron ship acts as a permament 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. 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 rioht-angles to that line ; abundant observation and experiment have proved this important general principle. 1 99. To illustrate this principle : let us suppose, as in the following figures 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 dip of the needle is 70". Pig. 5 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 Deviation of the Compact. Fig. 3. Head North while building. S. {him) polarity, and the fore body, or white portion in the figure, N. {red) polarity ; the upper part of the stern would have the S. {blue) 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. igi). Fig. 4. Head Soulih 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 south {Hue), while in fig. 4 the after part of the ship possesses north {red) polarity ; now the fore body of the ship has S. {hlue) 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 equalled developed. At the Deviation of the Compass. stern the N. end of a needle would be repelled, and also attracted to the strong S. {hlue) pole at the bow. The dotted line crossing the equatorial 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 direction to that in which she was built, or after she has made a voyage. Fig. 5. Head East while building. Fig. 5 is intended to show the magnetic state of a ship whose head has been east on the building slip. The whole of the upper part of the ship would have S. [hlue] polarity ; the whole of the lower part would have N. {red) polarity ; but the magnetism of the starboard side of the upper works would be developed in higher degree than the port side, and the N. end of a compass needle, if carried at the usual height of a compass along the amidship line of the upper deck from end to end, would be attracted to the starboard side. Fig. 6. Head West while bviilding. Deviation of the Compass. 121 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. polarity, and the lower N. polarity ; but the 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.* Fig. 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 figure 3, and mark the diiference in the magnetic state of the two ships. 200. 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. 201. The compasses of composite ships with iron frames and iron deck beams, are afi'ected in the same way as those of ships built wholly of iron. * From the special magfnetic 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, the character of the deviation — though not the amount — may be approximately represented in a tabular form, as follows : — Approximate magnetic direction Approximate easterly deviation occurs Maximum westerly deviation of ship's head while building. when ship's head by compass is near when ship's head by compass is near N. W. E. N.E. N.W. S.E. E. N. S. S.E. N.E. S.W. S. E. W. S.-W. S.E. N.W. W. S. N. N.W. S.W. N.E. 122 Deviation of the Compass. DEVIATION OF THE COMPASS. 202. 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 needle. 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. 203. In the case of iron ships, as in that of iron bars (195), percussion and vibration by hammering in rivetting 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 ship thus acquires to partake more of the character of permanent magnetism. Still this sub-permanent 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 sub-permanent magnetism as long as it remains in the form of a ship. The deviation arising from sub-permanent magnetism is greater than that which is the 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 of the ship's head was whilst being built. Taking the case of a ship built head north (fig. 3, page 119), the fore part of the ship has accj^uired 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 the directive power of the needle will be diminished. On southerly courses the north end of the needle points towards the stern, which has acquired sub-permanent south magnetism, then the directive power of the 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 line of the ship is at right-angles to the needle ; bej'ond that position the fore part of the ship attracts the south end of the needle, and westerly deviation is still the result. This attraction continues imtil the ship's head reaches south, when the line 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 bringing the ship's head round west of south, the south pole of the needle stiU continues to be attracted, which causes easterly deviation, and it again attains its maximum when the fore- Deviation of the Compass. 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 been 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 sub-permanent magnetism, and the effects of magnetism induced in vertical iron, has received the name of Semicircular Deviation. Semicircular Deviation, is so called because it is easterly in one semicircle or half of the compass, and westerly in the other half, as the ship's head moves round a complete circle of azimuth. This error is caused by the sub-perma- nent magnetism acquired in building, and the magnetism induced in vertical iron. The part due to sub-permanent magnetism remains the same in kind, though different in amount, in all latitudes, unless the ship be sub- jected 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 correctly, as the dip changes, and is of contrary names on opposite sides of the magnetic equator, that is, if westerly deviation be pro- duced 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 shij) 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. OL- .1. J T_-i 1, -u- The north end of a compass needle on the bhip 8 head while building p^^p ^^ ^^^^^^ ^^^^ j^ ^^^^^^ ^^^^^ 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. 124 Deviation of the Compass. Fig. 8. Magnetic North SCL'Tf-/ Figure 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. Let 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 there can be no semicircular deviation with ship's head in that direction, because the attractive force of the ship's Deviation of the Compass. 125 magnetism at the point P is in a line with a compass needle N S. As the ship's head swings round towards 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 maximum or greatest 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 there influence on the compass, nor will the ship's magnetism have the same effect on two compasses placed on difi;erent parts of the deck. 204. ftuadrantal 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 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 figure 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 (195), 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 126 Deviation of the Compass. Fig. 9. Magnetic. 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 ship's 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 8 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 Deviation of the Compasn. izn 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 ob- serve that the bar B in this case produces easterl}' 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. 205. 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° more 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. Mechanical Compensation or Correction of the Compass by means of Magnets and Soft iron. — These adjustments were first proposed by Mr. Airy, the Astronomer Eoj^al, and are now universal in the merchant service. Correction of the Semicircular Deviation. — As this error is caused by sub- permanent magnetism and by magnetism induced in vertical iron, the same agents must be used to correct it properly, to hold good in all latitudes ; but as it is impracticable to ascertain how much of it is due to the one, and how much to the other, it is customary to correct the whole of it by means of per- manent magnets fixed on the deck, one before or abaft, and another at the side of the foot of the binnacle ; but it must be remembered that this correction will only hold good for a small range of latitude, and while the ship's mag- netism continues in the same state as when the correction was made. 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. Provide two or moi*e 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 or below 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 128 Deviation of the Compass. 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. 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. Correction of the ftuadrantal Deviation. — The semicircular deviation being corrected, and the binnacle being properly fitted with two small brass boxes, one on each side of and on a level with the compass; steady the ship's head on one of the quadrantal points N.E., S.E., S.W., or N.W. ; if there is any deviation, fill one of the chain boxes with a quantity of small chain until the compass points correctly ; if one chain box be not sufficient, fill the other. For greater certainty, swing the ship's head to each of the other quadrantal points. When this adjustment is once properly made, it ought to remain perfect at all times and in all latitudes. Instead of the soft iron chain, the Liverpool Compass Committee prefer cast iron cylinders with hemispherical ends as correctors for the quanrantal deviation. These cylinders are of two sizes, one 9 inches long by 3 inches in diameter, the other 12 inches long by 3^ inches in diameter. 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. It is only in very rare instances that the correctors or chain-boxes are required to be placed on the fore-and-aft ends of the binnacle. The adjust- ment 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. 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 * 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. Deviation of the Compass. izg 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.* 210. 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 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, and an index may be made with a thread and plummet depending from the end. 211. How the Deviation from Heeling is caused. — The heeling error depends partly on 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 athwartship beams then produce their full influence in disturbing the com- pass. 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 (195). In north mag- netic 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 com- pass on the weather side is a north pole, and that nearest it on the lee side a south pole ; and under these conditions the north end of 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 * " 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 particuLir instances it has been known to disturb the compass as much as 2 « degrees. — A Treatise 0.! Navigation, b\ G. B. Airy, M.A., LL.D., D.L., page 182. 130 Deviation of the Compass. 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 119) from the whole after body of the 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 19), the after part of the ship has acquired north magnetism, 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 their heads from about S.W. to 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. 212. 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 compass is north or south. When the ship's head by compass is either east or west, the dis- turbing 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. 213. In north latitude, in ships built with their heads to the 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, Deviation oj the Com/pass. 131 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. '^ 214. Effects of Heeling. — The effect of the heeling error, when the north end of the needle is drawn to windward, is to throw a 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 : — Starboard tack — E. dev. is increased, "W. dev. is decreased. Port tack — W. dev. is increased, E. dev. is decreased. On Southerly courses : — Starboard tack — West dev. is increased, E. dev. is decreased. Port tack — W. dev. is 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. 215. Correction of the Heeling Error. — This correction is made by a vertical magnet placed in the binnacle immediately below the centre of the compass card. The ship's head is to be placed north and 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 name. 2 1 6. After all the compensations have been accurately made, there wiU 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. * This should be particularly considered by masters of iron ships about to proceed to a port south of the equator. ijz Deviation of the Compass. METHODS OF FINDING THE AMOUNT OF THE DEVIATION. 217. When in port, there are two principal methods in general use for jfinding the deviation, viz: — Method I, by the 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. 218. 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. 219. 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 difference 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 swinging ; 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 suf&ce. 220. 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 effected by taking the compass to some place on shore (avoiding local influ9nces) 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 difference 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 aU the observed bearings, if observed on equi-distant points ; or of four or more compass bearings, if taken also, on equi-distant compass points. 221. 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 Com/pa%&. 134 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 ship is warped 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 com- pass 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, «.e., with 180° 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 com- pass according to the direction in which the ship's head was placed. 222. 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 Vauxhall 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. 223. 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 are 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 suflB.ciently far off to render the difference of their bearings in- sensible as the ship swings round to the tide. In such cases God/ray^ a 134 Deviation of the Convpass. 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. 224. Commander Walker, K.N., has shown* that the deviation may be ascertained with sufficient 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. 225. 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 ob- tained 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. 226. To name the Deviation. — Rule. — When the reading hy the shore compass (reversed), or the correct magnetic hearing of the distant ohject, is to the right of the reading hy the compass on hoard, the deviation is easterly ; when to the left, westerly. Thus suppose the correct magnetic bearing from the 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 ) ^ o -rn- or correct magnetic bearing | ' ^ 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. 227. The directions of the ship's head having been taken by the compass in the ship, are therefore affected by the local attraction, and the apparent compass hearing of the ship's head differs from the correct magnetic bearing by the amount of the local deviation due to the position of the ship. For Magnetism of Ships and the Mariner's Compass. Deviation of the Compass. '35 instance, when the ship is apparently lying with her head east, it is not the true magnetic east, 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. 228. 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 1 1 miles. Ship's Head by the Standard Compass. East .... E. by S. E.S.E. .. S.E. by E. S.E Bearing; 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 tbe compass. II, By reciprocal bearings. Time.* 9>» lo'" A.M. 9 H Ship's Head by the Standard Compass. North . . . . N. by E. . . N.N.E N.E. by N. N.E N.E. by E. SIMULTANEOUS BEARINGS From Standard Compass on board. S. 37' S.45 S.5X S.57 S. 61 S. 65 ' 50' E. oE. 40 E. 20 E. 50 E. 30 E. From the shore Compass. N.4I'' o'W. N. 42 25 W. N. 43 30 W. N. 44 10 W. N. 45 oW. 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 so on through all the points of the compass. 229. 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 effects 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. 136 Deviation of the Compass. The following is a Table of Deviations to wluch reference is to be made in working the following examples. TABLE OF DEVIATIONS. ship's head. DEVIATION. ship's head. DEVIATION. North . . N. by E. . N.N.E. . N.E. by N. 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. . 22' w. 1 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 . . S. by W. . S.S.W. . S.W. by S. S.W. . . 16 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 52 W. 6 18 W. 5 2 W. 3 10 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. by N. N.N.W. . N. by W. . 230. The purposes for which a Table of Deviations so formed are : — ist. — 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 bearings 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 LIX. To find the correct magnetic course, having given the compass course and deviation. ^Express the compass course in degrees, Sfc. ; look in the Table 0/ 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 : — 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, al- low to the right. N.N.W. =.= 2 points L. of N. 2 points L. of N. ^ 22° 30' L. of N. Deviation (Table) 5 2 L. Cor. mag. course 27 32 L. of N. or N. 27 32 W. The deviation in this instance being Westerly, al- low to the left. Deviation of the Compass. m 3. S.W. = 4 points R. of S. 4 points K. of S. = 45= o' R. of S. Deviation (Table) 6 16 L. Cot. mag. course 38 44 E. ofS. or S. 38 44 W. 4. W. = 8 points R. of S 8 points R. of S Deviation (Tab! Cor. mag. course 8 points R. of S. = 90° o' R. of S. Deviation (Table) n 50 L. 78 10 E. of S. orS. 78 10 W. 5. W. I N. = 7| pts. L. of N. = 87° 1 1' L. of N. Deviation z=. 11 40 L. 180 51 L. of N. Cor. mag. course 81 9 R. of S. or S. 81 9 W. "West W. by N. Dev. for \ pt. Dev. W. == ii°5o' W. =: II 10 W. 4)0 40 :r: o 10 = II 50 Dev. for W, 1 N. = 11 40 W. 6. N.W. by W. |W. = 5f L. of N. = 6i'^4i'L. ofN. N.W. byW. = 9°i8W. Deviation =: 10 2 L. W.N.W. = 10 16 W. K 71 43 W. Change for i pt. = 58 4)58 The deviation (see Table) forN.'W.by'W.and'W.N.'W. is found, and the difference of these quantities is the correction for 1 point, which divided by 4 gives the cor- rection for J point = o^ 14'. Now compass course is \ point from W.N.W., therefore, apply correction 0° 14' to the deviation for W'.X.'W., the resillt is the deviation for N. W. bv W. ; W., and it is to be subtracted, because the deviation for N."W. by "W. is less than for W.N.W. 7. N. ^ E. = I pt. R. of N. = 5° 38' R. of N. Deviation =0 42 R. Correct magnetic 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. A W. = i pt. L. of N. = 5' 38' L. of N. Deviation := i 46 L. Correct magnetic cotcrse N. 7 24 "W. Half the s>im of the deviations for N. and N. by W. is taken for deviation on N. ^ W. ; both deviations being of the same name. Or proceed thus — take the deviations from Table for North and for N. by W. ; take the ditferencc and half it, apply this to the deviation for North, addiny because the deviation is greater for N. by W. than for North. Examples for Practice. Correct the following courses steered for deviation, as given in the Table, page 136. W. Change of dev. for ^ pt. = 14 Dev. for W.N.W. = 10 16 Dev. for N.W. by W. '^ W. = 10 2 W. Deviation at N. = 0° 22' W. N. by E. I 46 E. 2)1 24 o 42 E. Deviation at N. = o" 22' W. K by W. 3 10 W. 2)3 32 Dev. on ^ pt. =1 46 W. I. N.E. by J^. 8. S.E. by E. 15- S.W. by S. 22. West 2. North y 9- X. by E. h E. 16. S. |E. f-23. N.W. i 3- N. by W. i W. 10. South • 17- W. iS. ^24. S.JW. 4- S.W. 1 W. II. W. ^N. i8. X.N.E. i E. A 25- W. by S. S- W. i N. 12. S.E. j S. . 19- N. i E. \ 26. E. JS. 6. s.s.w. \ w. 13. E. % N. 20. W. by N. . 27- East 7- E. by N. 14- X.W. 1 w. 21. S. byE. 28. W.N.W. 231. Proceeding to correct the courses for the deviations given in Table I, a second Table, arranged like the following, may be made for aU the points of 1 lie compasf. T 138 iation of the Compass. TABLE II. Courses steered by Compass. Deviation. Correct Magnetic Courses. North or o° N. by E „ N. II 15' E. N.N.E „ N. 22 30 E N.E. by N. . . „ N. 33 45 E. N.E „ N. 45 E. N.E. by E „ N. 56 15 E. E.N.E „ N. 67 30 E. E. by N „ N. 78 45 E. East „ 90 E. byS „ S. 78 45 E. E.S.E „ S. 67 30 E. S.E.byE „ S. 56 15 E. S.E „ S. 45 E. S.E. by S „ S. 33 45 E. S.S.E „ S. 22 30 E. S. by E „ S. II 15 E. South , S. by W „ S. II 15 W. S.S.W „ S. 22 30 w. S.W. byS „ S. 33 45 W. S.W „ S. 45 W. S.W. by W. . . „ S. s6 15 W. W.S.W „ S. 67 30 W. W. byS , S. 78 45 W. West „ 90 W. byN „ N. 78 45 W. W.N.W „ N. 67 30 W. N.W. by W. .. „ N. 56 15 W. N.W „ N. 45 W. N.W. by N. . . „ N. 33 45 W. N.N.W „ N. 22 30 W. N. by W „ N. II 15 W. 0° 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. 16 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 52 W. 6 18 W. 5 2 W. 3 10 W. N. 0' 2 2' W. or North . N. 13 I E. „ N. byE. IE. N. 25 50 E. „ N.N.E. i E. N. 38 59 E. „ N.E. 1 N. N.5214E. „ N.E.fE. N. 65 9 E. „ N.E. by E. f E. N, 78 14 E. „ E. by N. nearly. S. 89 35 E. „ East. S. 79 16 E. „ E. bv S. S. 6851 E. „ E.S.E. IE. S. 58 22 E. „ S.E. by E. I E. S. 48 55 E. „ S.E. 1 E. S. 38 44 E. „ S.E. A S. S. 28 45 E. „ S.S.E. i E. S. 19 6 E. „ S. by E. | E. S. 9 33 E. „ S. 1 E. S. 16 W. „ South. S. 9 25 W. „ S. I W. S. 19 14 W. „ S. bv W. \ W. S. 28 57 W. „ S.S.W. A W. S. 3844W. „ S.W. ig. S. 48 35 W. „ S.W. 1 W. S. 58 12 W. „ S W. by W. i W. S. 68 II W. „ W.S.W. |W. S. 78 10 W. „ W. by S. N, 89 55 W. „ West. N. 77 46 W. „ W.N.W. 1 W. N. 65 33 W. „ N.W. by W. § W. N. 52 52 W. „ N.W. 1 W. N. 40 3 W. „ N.W. by N. | W. N. 27 32 W. „ N.N.W. 1 W. N. 14 25 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 ist column of the Table for the latter ; the 2nd column gives the deviation when her head is on that point ; and in the 3rd column ("the deviation having been applied as directed in Rule LIX) 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 half, a quarter, &c., and when steering upon a whole point, the correct mag- netic 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). Corr. mag. course for N.E. =.= N. 52°i4'E. „ N.E.byE. = N. 65 9E. 2)117 23 Corr. mag. co. for N.E. ^ E. = N. 58 41 E. Here the courses taken from the Table are of the same 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.= 13 lE. 2)12 39 Corr. mag. course for N. | E. = 619 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 correspondiog to these points. Deviation of the Compass. 39 Ex. 3. Ship's head by Standard Compass is N. :^ E. ; find, by means of the Table, the corresponding correct magnetic course. Corr. mag. course for North =: o°22'W. N.iE. = 13 lE. 4)13 23 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. ) o TtT J — I S. qo 5 W. Difference for \ point = 321 Corr, mag. course for North =: o 22 W. Corr. mag. course for N. + E. =: 2 59 isN. 89°55'W. orS.90°5'W. 90 II 55 3 4)35 45 Difference for | point -|- 8 56 Corr. mag. course for "West =: S. 78 10 W. Corr. mag. course forW.f N.= S. 87 6 W. 232. 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 144-146, page 90). Easterly deviation to be allowed to the right and westerly to the left, as in Eule LIX. Caution. — 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 H. Correct magnetic bearing 1 1 46 W. „ II 46 L. of N. or N. 46^ 11' 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) 14 W. „ 14 L. Correct magnetic bearing N. 34 W. „ 34 L.ofN. or N. 34° W. Ex. 3. Two islands bear S.E. and W.S.W. ; the ship's head is N.E. : required the correct magnetic bearing of each. Bearing by Standard Compass S.E. Deviation by Table (applied to the right) = S. 4i°E. or45°L. ofS. = 7 E. „ 7 R. Correct magnetic bearing Bearing by Standard Compass W.S.W. Deviation by Table (applied to the right) Correct magnetic bearing — S. 38 E. „ 38 L. of S. or S. 38° E. = S. 67^30' W. or 67° 30' R. of S. = 7 14 E. „ 7 14 R. = S. 77 44 W. „ 74 44 R. ofS. or S. 75^ W. 140 Deviation of the Co'mpass. Examples for Pbactioe. 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 136). No. If^Co^^t Compass Bearing. No. Ship's Head by Compass. Compass Bearing. i 1 x2 3 4 46 West East. •^ 7 E. # N N. AW. s.s.s. t N.E.byN W.N.W W. byN.iN... S. by W. i W. .. E. by S. i S. 8 North. 9 N.E. i E E. fS W.|S. W. by S. I S. S. AE. W. by N. A N. N. by W. A W. South. E. iS. S. by W. i W. 10 11 12 N.iE S. iE E. A N 233. Given a correct magnetic course by the chart betwees two points of land, to find the course that must be steered by compass. RULE LX. Easterly deviation is allowed to the left. "Westerly ,, ,, right taking care that the deviation applied is that of the correct magnetic course. Note. — In this case it is important to remember, not onlj' is the general rule of applying the deviation reversed, but the correction to be applied is the deviation due to the given mag- netic course, not that due to a compass course, as in Rule LIX ; that is, to the correct mag- netic 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 ia 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 apply 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 ioJ° W., which allowed to the right would be about West ; and since the devia- tion for this last does not differ from the deviation used, it may be considered that to make correct magnetic W. by S. the course to be steered is about West. 234. By comparing the first and third columns of this Table H, the sea- man 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 jDeviatioii of the Compass. 141 Standard Compass courses corresponding to which are N.E. by N. and N.E. ; the course to be steered is consequent!}' 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 11° 15' of the compass; in shaping a course therefore, between these points, the value of the half point is about 6° 37', the quarter is f 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 be steered. Course that must be steered by the Standard Compass in order to make the Correct Magnetic Coui'se. North .... N. by E. . . N.N.E. .. N.E. by N. N.E N.E. by E ! N E.N.E. E. by N East E. by S E.S.E S.E. byE. .. 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, byN. .. W.N.W N.W. by W. N.W N.W. by N. N.N.W N. by W. . , North North or N. 10° E. N. 19 E. 29 E. 38IE. 48|E. 59 E. nearly North. N. ^E. N. N. N. N. 68 N. 79 E. E. East S. 78 S. 65 S. 52iE. S. 39|E. S. 26iE. S. 13 E. South S. I2|W. S. 25^ W. S. 39 W. S. 52^ w. s. 651 w. S. 78 w. N. 89 W. N. 78^W. N. 68 W. N. 58 W. N. 48 ~" N. 38 N. 28 N. 18 N. 8^ North W. W. w. w. w. IE. E. N. byE. |E N.N.E. f E. N.E. i N. N.E. \ E. N.E. by E E.N.E. E. by N. East. E. by S. S.E. by E. S.E. I E. SE. i S. S S.E. f E. S. by E. i E. South. S. by W. i W. S.S.W. i W. S.W. \ S. S.W. i W. S.W. by W. f W, W. by S. \ S. West. W. by N. W. by N. i N. N.W. byW. 1 N.W. I W. N.W. i N. N.N.W. \ W. N. by W. f W N. |W. North. W. i^i, Deviation of the Compass. 235. 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^ points ; variation i J W. ; and the deviation zf points E. ; required the true course. Here the course by compass is E.N.E or 6 points right of North. The variation and deviation are of contrary names, their diflerence, viz. (zf E. — i^ W. = i J E.) is i| E. and + i\ points right of North. Therefore the sum is the course corrected for variation and deviation and is • 72 points right of North. The ship being on the starboard tack the leeway is applied to the left, and hence is — i\ left. Difference 5 points right of North. True course N.E. by E. Ex. 2. Course by compass N.N.W. ; wind N.E. ; leeway z\ 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 ^\ „ left. Therefore the sum is the course corrected for leeway 4I ,, 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. 78=46' W., or W. by N. In this example the compass course and leeway are given in points, we therefore take the course and allow the leeway from the wind, which gives the course corrected for leeway, viz., if\ 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. We may, however, if we prefer so to do, express all the quantities in degrees, and the work will then stand as in the following example. Ex. 3. Course by compass W. by S. ; variation 3I E. ; deviation 13° 50' W. ; wind S. by W. ; leeway \\ points : required the true course. Compass course W. by S. ., = 7 8°45' right of South. Variation 3^ points = 36 34 right. 115 19 right of South. Deviation 1350 left. Course corrected for variation and deviation loi 29 right of South. Leeway, port tack i\ points := 19 41 right. Sum exceeds 90° .... 121 10 right of South. Subtract from 180 58 50 left of North. . • . True Course is N. 58° 50' W. Since no course can exceed 90° from either N. or S., when the course as in this example exceeds 90", the result must be taken from 1 80°, and its name changed. (See Kule LV (a), page III.) Deviation of the Compaxn. H3 Examples for Practice. From the following Compass Courses find the True Courses : — No. Compass Courses. ■Winds. Leeway I. N.E. by E. N. bv W. 2^ pts 2. North E.N.E. 2 „ 3- N.N.W. N.E. 3i „ 4- West S.S.W. 2i » 5- S.S.E. |E. S.W. \ S. 32 » 6. E. |S. N.E. by N. 3i » 7- N. by W. W. by N. I „ 8. South W.S.W. 2| „ 9- W. by S. \ S. N.W. 2i » lO. N.E. by E. i E. N. by W. I? „ II. S.W. by W. S. by E.- 2 ,, 12. E.f S. S. by E. If ,. 13- N.E. N.N.W. li „ 14. W.N.W. North 3 j> 15- N.E. by E. N. by W. 1. 16. W. by S. S. by W. 3 4 >' 17- South E.S.E. 3 4, » 18. West N.N.W. ^\ ., 19. S.S.W 1 W. S.E. by S. 2j „ 20. N.W. by W. N. by E. 3 „ 21. E. byN. S.E. by S. ^4 >> 22. W. by S. i S. S. bv W. 2 „ 23. E. IS. N.N.E. i E. 2i „ 24. S.W. by S. W. by N. Ij » 25- South E.S.E. li » 26. S.W. i S. S.S.E. 1 E. 4 » 27. E.iS. S. by E. \ E. * » 28. East S.S.E. 2^ „ 29. W. JN. N.N.W. '1 ,. 30- N. |W. W. bv N. 1 31- E. iN. N.N.E. 2i „ 32- S.E. by E. S. by W. I* „ 33- N. by W. \ W. N.E. J E. 2^ ,. 34- up S.E. •) off E. byN.; South. 4^ „ 35- up S.W. i W. J offW. byN. f S. iE. 04 )> 36. up N.W. i N. \ off W. byN. / N. by E. 4¥ .. Variation. Deviation. 2 pts. W. 32 pts. E. 2 „ w. I „ E. 2f „ E. 4 » E. 2| „ E. 1 „ E. 2i „ E. li ,, w. li „ w. li » E. 2i „ W. li ,. w. i^ „ E. li „ E. 2 „ w. li ,, w. 2i ,. E. li „ E. 3 » E. li „ w. 42^ 0' E. 15° 0' E. 12 W. 16 30 E. 42 E. 18 30 W. 14 E. 19 E. 10 30 E. 19 W. 17 W. 3 E. 18 30 E. 21 W. 17 W. 5 W. 25 W. li w. 32 E. 12 E. 15 E. 15 W. 21 W. 4 w. 25 E. 10 w. 52 W. 2 E. 52 W. 13 w. 52 W. 24 E. 8 30 E. 15 35 E. 8 30 E. 21 30 W. 15 45 E. 6 W. 13 W. 20 E. 18 w. 16 30' E. 53 oW. 8^ W. i\ pts. E. 2 pts. W. 4 ,. w. i\ „ E. 2^ „ E. i| „ E. li „ E. i „ E. li » E. NAPIEE'S DIAGRAM. 236. 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 figui-es that show to the et/e 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, 144 Deviation of the Compasfi. Various " graphic methods " of delineating the deviation have been devised ;••' but the method introduced here is due to J. R. Napier, Esq., F.R.S., and is one peculiarly adapted for this purpose, as it is equally appli- 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. 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 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. The vertical line is then intersected at each of the 32 points by two straight lines inclined to it at an angle of 60° : one of these is a plain line inclined to the right ; the other a dotted line inclined to the left; that is, on the right side of the vertical 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 in 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 cir- cumference of the compass-card, from 0° at North and South, up to 90° at East and West. 237. Requisite Observations to be made. — The least number of ob- served 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 approxi- mate 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. 238. 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 similiarly whilst under steam or sail at sea, a number of azimuths of the sun may be observed, and hence the deviation obtained. * Graphic methods for correcting the ship's course for the Deviation of thft 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 lurnished to H.M. ships for fleet tactics, tor which it is well adapted. Mr. Rundell's is known as the circular method. They are all useful in practice. N.bVV. Deviation of the Compaas. H5 In these cases tlie Graphic method here described furnishes a ready and effectual mode of obtaining a result on which the error of individual observa- tions are as far as possible compensated and any egregious errors eliminated. 239. Construction of the Curve of Deviation. — Easterly deviations are laid down to the right of the central line, Westerly deviations to the left. 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;* the scale on each is the same, and the amount of the deviation may therefore be 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, pass- ing 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 mai'ks be out of the fair curve, it may be assumed that an error has been made in the observation for that point. The process will be best understood by explaining the projection cor- responding to the observations as given in the following table : — Ship's Head by- Standard Compass. Deviation. 1 Ship's Head by Standard Compass. Deviation. North NE 6^30'W. 1 13 W. 1 22 15 w. 23 30 w. South S.W 5" 30' B. 28 35 E. 19 15 E. 3 E. East West N.W S.E Deviation Curve C 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 vertical line lay otf the deviation on the dotted line which passes through that point towards the left— the deviation being West; 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 cornet mig- 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. f If the table of deviations are given for the correct magnetic courses and not tbe compass courses or direction of the ship's head, the same process is gone through, except tliat the deviations are in that case laid ofifon the plain lines. It is, however, now generally under- stood that this procedure is co:.lrary to practice and may lead to error. 146 Deviation of fjip Compnu. 2. The spcond compass course on which an ohservation has heen made is N.E., and the ohserved deviation is 13"" o' W. With dividers take from the vertical line a distance equal to deviation 13°, and from N.E. on the vertical line lay it off on the dotted to the left — deviation heing W. ; and the puint so determined mark with a cross or dot. 3. The third compass course on which an observation has been made is East, and the ohserved deviation is 22° 15' W. Take from vertical line 22^°, and from East on vertical line and on the dotted line passing through it, lay off the observed deviation to the left — deviation 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 vertical line a distance equal to deviation 23^°, and from S.E. on the vertical 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 vertical line a distance equal to deviation 5^°, and having found compass course South on the vertical 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 0^ cross. 6. Compass course S.W., deviation 28^ 35' E. From the vertical line take a distance equal to observed deviation 28 j°, and having found S.W. on the vertical 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 vertical line a distance equal to deviation 19^'', and having found compass course West on the vertical line, lay off on the dotted line passing throuijh 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 vertical line a distance equal to 3°, and having found compass course, N.W., on the vertical line, place one foot of compass on that point and lay off on the dotted line passing through it 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 above 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.) j Ship's Head 1 by Deviation. Standard Compass. Ship's Head by standard Compass. Deviation. North ,.. i°i5'W. N.E 22 30 E. East . . 26 50 E. South y.W 1° 50' E. 15 W. 26 W. 27 w. West S.E 17 E. N.W The following describes the process of construction : — 1. With a pair of dividers take from the central line a distance equal to deviation \° 15', or i^°, and from North on the vertical 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 vertical 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. '47 3. Take from the vertical line a distance equal to 26|° (26° 50'), and lay it ofiP on the dotted line, from East towards the right— being E. ; make a dot or cross. 4. Take from the vertical line a distance equal to 17° (17° o'), and lay it ofif on the dotted line, from S.E. towards the right — being E. ; make a dot or cross. 5. From the vertical line take a distance equal to if° (1° 50'), and lay it off on the dotted line, from South towards the right — the deviation being E. ; make a dot or cross. 6. From the vertical 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 or cross. 7. From the vertical 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. 8. From the vertical line take a distance equal to 27", 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 vertical 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 follovring 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 d= 16' W. 6 16 W. II 50 W. 7 52 W. West S.E N.W Selecting those dotted lines which pass through the points reprtsenting the diflerent directions of the ship's head, the deviations are laid off: thus, at North, we mark off 05° (0° 22') to the left on the dotted line passing through North— because deviation is West; at N.E., we take i\° from the vertical 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 vertical line and laid off to the right on the dotted line passing through East ; and so with the others, being carelul 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 vertical 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 136 ; and the curve thus drawn can be used instead of the Table. 240. 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 ; or, 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 ; or, 3rdly, if one or more bearings of the land are taken, to correct these bearings by the amount of the deviation duo to the direction of the ship's head at the time. The corrections are given by the following rules. Deviation of the Compass. 241. To find the Deviation on any Compass Course. EULE LXI. On the central line find the given course ; then, with a pair of dividers, measure the distance /ro?;* that point to where the curve cuts the dotted line passing through the course ; hut if no dotted line proceeds from the course, then measure 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 pa/rt of the central line will give the deviation in degrees. Examples. Ex. I. "What is the deviation on Compass Course N.E. by N. («) lor the deviation curve B ; {Jj) using the deviation A ? {a) Having found the given course on the central line, with a pair of dividers measure the distance from X.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. {J)) Measuring with a pair of dividers the distance irom N.E. by N. to where the curve cuts the dotted line proceeding from that point, the deviation measured on the vertical line is found to be 19' E. Ex. 2. Eequired deviation in compass course W.y.W., using deviation A. Find W.y.W. on the vertical line, measure the distance from that point to where the curve cuts the dotted line proceeding from it ; this distance taken to the vertical line gives deviation 2i|° W. Ex. 3. What is the deviation on compass course E.S.E., using deviation curve C ? Measure from E.S.E. on the vertical line to the point where the curve cuts the dotted lines proceeding from it ; this distance taken to the central line gives deviation 25° W. Ex. 4. What is the deviation for compass course N.E. ^ N., using the curve C ? Take N.E. \ N. on the vertical line, and draw a faint pencil line parallel to the dotted lines until it meets the curve. The length of this line in degrees taken from the vertical line gives 12^° W., the deviation for N.E. \ N. Or thus:— 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. 5. Find the deviation for Standard Compass Course N. 84° W., using deviation curve A. Through N. 84° W. on the vertical line draw a pencil line parallel to the nearest dotted line, so that it may cross the curve. Or thus : — 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^° W. Examples ion Practice. Required the deviation for each of the following compass courses, using ia) the curve A, {b) the curve B, and (c) the curve C : — 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. i S. ; N. 41° W. ; and S. 38° E. Eor further exercises in this matter the learner may take the Table of Deviations given on page i 56, 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. Deviation of the Com/pass. 149 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. 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. EULE LXII. On the central line find the given Standard Compass Course, and move on the dotted line drawn 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 lack to the central line. The point on the central line at which you arrive is the correct magnetic course required. Note. — 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 leg on the curve shall be cxacthj 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 ofi' the curve, move in a direction parallel to the plain lines until j-ou reach the central line, the point where the dividers cut the central shows the correct magnetic course. 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 C 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 fixed, move the one off 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. bj' N. cuts the curve, then keeping the leg on the central line fixed, lift the other from the curve and move in the direction of the nearest plain lines till the central line is reached ; then the correct magnetic course will be found to be N. 52= E., or N.E. f E., nea^lJ^ Ex. 3. The course steered is S.S.W. : required the correct magnetic course (using curve C, Plate I.) Having found the compass course S.S.W. on the vertical line, place one leg of the dividers on the course and the other on the place where the dotted proceeding from it is intersected by the curve, then keeping the leg on the central line fixed, return with the other leg to the central line in a direction parallel to the plain line, it will be seen that the correct magnetic course is S.W. \ W. Ex, 4. Compass course W.S.W. : what correct magnetic course does this give on the curve A of deviations, Plate I ? From W.S.W. on the central line follow the dotted extending from that point until it reaches the curve, then keeping the leg of the dividers on the central line fixed, move the other that is placed on the curve, in a direction parallel to the plain line, until it cuts the central line, which shows the correct magnetic course S. 47" W., or S.W. \ W. i^o Deviation of the Compass. Ex. 5. 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. i S. Ex. 6. Given the standard compass courses N. 38' E. and S. 49° "W. : required the correct magnetic courses (using the A deviation curve, Plate I.) Having found the standard compass course N. 38' E. on the central line, place one leg of the dividers on that spot 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. i E. In a similar manner the correct magnetic course is found to be S. 33° "W. Examples for Practice. In each of the following examples the compass course is given to find the corresponding correct magnetic course, using curve A, curve B, and curve C. Curve A.-N. 41° W. ; N. 65° 30' E. ; S.' 38^ E. ; S. 79= W. ; West; N.E. ; S.E. f E. ; S. 89° 30' W. ; N.N.W. ; S.S.W. Curve B.— N. 47^° W. ; N. 76° E. ; S. 26|° E. ; S. 6%l° W. ; S. 78° W. ; N. 52^° E. ; S. 43' E. ; S. 78° W. ; N. 27^" E. ; and S. 20° W. Curve C.—N. 39" W. ; N. 48° E. ; S. 50" E. ; N. 78^^ W. ; N. 70' W.; N. 31!° E.; S. 76° E. ; N. 7o|° W. ; N. 26° W. ; and S. 47= W. Problem II. — From a given Correct Magnetic Course to find the cor- responding Compass Course. EULE LXIII. On the vertical 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 lines till you get lack to the vertical line. The point on the central line at which you arrive is the compass course required.* Examples. Ex. I. Given correct magnetic course N.N.E. to find the corresponding compass course (using curve C.) Find N.N.E. on the central line, and placing one leg of the dividers on the spot, extend the other It^g to the spot where the j^/am line proceeding from N.N.E. meets the curve; then keeping the leg on the central 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. * The only difficulty in applying these rules is to remember in each case whether we ought to depart from the central line by a dotted line, and return to it by a ^;/rtm line ; or whether we ought to depart from the central line by a plain line and return to it by a dotted line. The doubt will be removed if the following lines, in which the two lines are versified, are committed to memory : — "From compass course magnetic course to gain, Depart by dotted and return by plain." " But if you wish to steer a course allotted. Take plain irom chart and keep her he^d on dotted." Deviation of the CompnnH. \ r i Ex. 2. "What compass course will make correct magnetic S.E. (using A curve). Find S.E. on the central line, .put 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, liTt the other lei; off the curve and move it in the direction of the nearest dotted till it at,^ain touches the central line ; the compass course that makes correct magnetic S.E. is shown on the central line to he S. 67^° 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 C curve) ? 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 'plain 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. Ex. 4. Required the compass course to make correct magnetic W. by N. (using A curve). Find compass course "W. by N. on the central line, put one leg of the dividers on "W. by N. and the other on the curve where the j!3^«m line that passes through W. by N. cuts the curve ; 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. If the magnetic course required is N."W. by N., required the compass course on the C curve. Find N."W. by N. on the central line; plaice one foot of the dividers on the spot, and the other foot on the place where the plain line proceeding from N. W. by N. meets the curve ; then keeping the foot that is on the central fixed, return to the central line with the other leg in a direction parallel to the dotted line ; it will be found to intersect at N. 32!° W., the required course by Standard Compass. Ex. 6. 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 foot 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 foot of 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 similiar manner the Standard Compass course corresponding to correct magnetic course N. 85° W. is found to be S. 70° "W. Examples for Pkaotioe. In each, of the following examples the correct magnetic course is given to find the compass course, using curve A, curve B, 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 B.-S. 85° W. ; N. ^d)? E. ; S. 18° "W. ; N. 12^° E. ; N. 54^° E. ; S. 85^° E. S. 53° W. ; N. 29^ E. ; and W. by S. Curve C.—S. 44° "W. ; N. 58!° E. ; S. 5° W, ; N. 24" E. ; N. 83° E. ; S.jo^E.; S.22°W. N.E. \ E. ; and S.W. | S. We shall now proceed to s1iow the application of the foregoing rules to Questions 7, 8, 9, and 10 of List B, which contains the questions on the Deviation of the Compass required of Candidates for Certificates as Master Ordinary. 152 Deviation of lie Compms. Question 7. List B. 242. Given the bearings of ;i distant object by Standard Compass on eight equi-diiitanP'- points to find the Oorrect Magnetic Bearing of the distant objectf and thence the Deviation. EULE LXIV. i". If the Compass bearings are all of the same name, i.e., if they are all reckoned //-o^w N. or S. towards E. or W. : Talce the sum of the Bearings in each column then add these sums together, and divide hy 8 ; tlte result is the Correct Magnetic Bearing of the distant object. 2°. If the Compass Bearing be of different names : (a) If some of the Bearings are rechoned from North and the others from South: Talce either set from 180°, and they will all he reckoned from the fiame point North or South ; the name as to East or West remains unaltered. (b) If some of the Bearings are towards the East and others towards the West: Find the sum of these which are reckoned towards East ; and also the sum of those which are towards the West ; then talce the less from the greater, and mark the difference of the same name as the greater ; the result divided ly 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 sums; the rest, however, can be finished in the margin. 3°. To find the Deviation for each of the given Courses. — («) If the Correct Magnetic and Compass bearings are of like names : Tahe their difference. ijb) If one is reckoned from North and the other from South, first take the Correct Magnetic hearing from 180° and the remainder will have the same name as the compass bearing, tfien take the difference hetween the Correct Magnetic and Compass Bearings. (c) i7"both hearings are from North or both from South, hut one is to- •wards East and the other towards West, take their sum : The difference or sum will 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. * The 8 equi-distant points are the 4 cardinal points and the four quadrantal points, viz. : N.W., S.W., S.E., and N.E. + The Correct Magnetic Bearing thus found is not strictly accurate; it will difi'er 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 diflerence of the same name as the greater,— and divide by 8. This, however, is not re- quired to be understood by Masters Ordinary, although it is required for Masters extra. "---/^ Deviation of the Compnss. 153 EXAMPLE I. In the following Table give the correct magnetic bearing of the distant object, and thence the Deviation. Ship's head hy Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. Ship's Head hy 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. Note. — 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 : — Ship's Head hy Standard Compass. Bearing of Distant Object Deviation by Standard Required. Compass. Ship's Head hy Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 4i''W. I'E. S. 59 W. 17 W. S. 64 W. 22 W. S. 60 W. 18 W. South .... S.W West N.W S. 46»W. S. 2^ w. S. i'8 W. S. 23 W. 17 E. 24 E. 19 E. S. 224° W. , S. 112° w. S. 112 W. 8)336 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 difference betv?een 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 : — Ship's Head. N. N.E. East. S.E. Correct mag. bear. S. 42" W. S. 42° W. S. 42° W. S. 42" W. Standard com. bear. S. 41 W. S. 59 W. S. 64 W. S. 60 W. Deviation Ship's head. Correct mag. bear. Standard com. bear. S. 42= W. S. 46 W. 17 W. S.W. S. 42° w. S. 25 w. 22 W. West. S. 42° W. S. 18 w. s 18 w. N.W. . 42° w. .23 w. Deviation 4 W, 17 E. 24 E. 19 E. Note. — 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 Rule LXIV, 3° and 4°, page 162.) 154 Deviation of the Compass. Question 8. List C. (See Eule LXIII, Problem II, page 150.) From the above table r.onstruct 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. i E. (3.) S. 10° W. (4.) E. i N. To Construct the Curve from the Table. — With a pair of dividers talre 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 vertical line, and with one foot of the dividers on N.E. on the vertical line, lay off this dis- tance on the dotted line passing through the given point and to the left, because the devi- ation 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 South. The deviation at S.W., West, and N.W. being easterly, must be applied to the right of the vertical line along the dotted lines proceeding from those point. Now draw vsdth a pencil a curve passing as nearly possible through the points found, and when satisfied with its uniformity, draw it in ink. To find the Standard Compass Course. — (i.) Place one leg of a pair of dividers on S.E. on the central line, and the other leg on the point where the plain line extending from S.E. is cut by the curve ; then keep the first leg 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 leg of the dividers on that spot, extend the other leg from this point and parallel to the nearest j(?/am line until it cuts the curve ; then keeping the first leg 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. \ E. is shown on the central line to be E. by N. | N. (3.) Place one leg of the dividers on S. 10° W. on the central line, and extend the other leg parallel to the nearest plain line, and to the right until it meets the curve ; then keeping the leg on the central line fixed, move the leg on the curve thence parallel to the nearest dotted line until it arrives at the central line, which shows that the compass course to bs steered is S. 9° W., nearly, in order to make S. 10° W. correct magnetic. (4.) E. \ N. is found on the central line, and one leg of the dividers being placed on the point, move the other leg to the left from that spot and parallel to the nearest plain line until it is cut by the curve ; from thence, keeping the first leg 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. \ N. is shown 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 LXII, page 149.) (I.) S.S.E. JE. (2.) S. fW. (3.) E.byN. IN. (4.) N. 1 W. (i.) With one leg of the dividers on S.S.E. f E. on the central line, move the other leg on a line to the left and parallel to the nearest dotted line until it cuts the curve ; then keeping the first leg fixed, move the leg on the curve from thence parallel to the nearest plain one and towards the central line ; the correct magnetic course is at once seen to be S. 45*° E. (2.) Placing one leg of the dividers on S. | W. on the central line, move the other leg on a line to the right, parallel to the nearest dotted line and cutting the curve ; from thence, keeping the first leg 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. Deviation of the Contpass. ^55 (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 jjlain lines, until the vertical line is again reached ; the correct magnetic course is thus found to be N. 48!° E. (4.) One leg of the dividers being fixed on N . j "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. Given the Bearings of two (or more) distant objects by the Standard Com- pass and also the Azimuth of the Ship's Head ; required the correct magnetic bearing of these objects. EULE LXV. 1°. Fmd the Deviation corresponding to the direction of the ship's head, takiny it from the Napier'' s Deviation Curve. (See Rule LXI, page 148). 2°. Apply the Deviation thus found to the Bearing of distant ohject hy Stan- dard Compass, according to Rule LIX, page 1 36, viz. : West Deviation to the left hand, East Deviation to the right hand. 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. Ship's Head. 1 Compass Bearing. Ship's Head. Compass Bearing. I. 2. West S.S.E 1 . I East. J E. by S. i S. 3- 4- E.|N KE.iE N. |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 dotted line proceeding from the course ; this distance taken to the vertical 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 — allowing it to the right hand ; 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. 5 S., or S. 76=' E., to which apply deviation, found as above, to the left hand: the result is the correct magnetic bearing S. 89° E. E. f N. is found on the central line ; then from this spot measure the distance to the curve in a direction parallel to the nearest dotted line ; and apply this to the central line which shows the deviation due to E. f N. — the direction of the ship's head — to be 22^'' W. Apply this deviation to compass bearing N. f W., or N. 8° W., to left hand, which gives correct magnetic bearing of distant object N. 30^° W. N.E. \ E. is the next given direction of the ship's head, which being found on the central line, place one leg of the dividers on the spot and move the other leg out in a direction parallel to the nearest dotted line until it meets the curve ; this distance, taken by the dividers and applied to the central line, shows deviation for ship's head N.E. \ E. to be 18" W. The deviation thus found being applied to the left hand of W. \ S., or S. 87° W., gives the correct magnetic bearing of distant object S. 69" W. tjf6 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 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 .... S. 87=W. 3°E. South .... N. 87° W. 2°W. N.E S. 70 W. 21 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 +) = 720 — - W. ^8)728 S.418 "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 remaider, in each case, is the bearing reckoned from S. toward "W. We maj', however, proceed as above, viz. : — add up the Bearings reckoned from N. towards W. ; and since there are four bearings so reckoned (from X. 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 1 80^ and to add the remainders. Ship's Head. Correct magnetic bearing Compass bearing Deviation Ship's Head. N. 91' W. 88 W. 3 E. 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. 89° W. N. 75 W. 14 W. East. S.E. S. 91° W. S. 71 W. S. 91 w S. 81 w 20 E. 10 E. West N.W. N. 89° W. N. 68 W. N. 89° W N. 72 W 21 w. 17 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. J S. Answers:— {1.) We&t. (2.) N. i'' E. 3. N. 60° E. (4.) S. 49' E. m. Suppose you steer the following courses by standard compass, find the correct mag- netic courses from the curve drawn : — (i.) North. (2.) S.S.W. iW. (3.) E. by S. i S. (4.) N.E. i E. Ansvcers:—{i.) N. 3= E. (2.) S. 13° W. (3.) S. 59° E. (4.) N. 71= E. Deviation of the Compass. i57 IV. You have taken the following hearings of a distant object by your standard compass as above ; with the ship's head as given, find the correct magnetic bearing. SMp's Head. Compass Bearings. Ship's Head. Compas Bearings. S.fW N.W. by W. . . South. E. 1 N. 1 N.N.E. fE. .. S.E S.W. i S. E. iS. E. f N. = 82°R. ofN. N.W. by { J W. gives / '9 ^• 63 E. of N. or, N. 63 E. S.W. i S. = 42° R. of S. gives . . / 2 6o|E. ofS. or, S. 6o|° W. Comp. Bear. S. = S. 0° Dev. for S. f W. is 5^ L. of S. SJL. ofS. Correct Mg. Bear. S. 5^ E. E. iS. = 87°L. ofS. S.E. = loj R. 76|L. ofS. S. 76I 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 : — SMp's Head : Bearing of by Distant Object Standard by Standard Compass. Compass. Deviation Eeqnired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N. I'E. N.E N. 13 W. East N. 15 W. S.E j N. 8 W. i South .... S.W West .... N.W N. 3°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 West, and add up this column ; the less sum being subtracted from the greater gives as a remainder N. 12° o' W., which divided by 8 gives N. 1° 30' W. for the correct magnetic bearing : the work will stand thus : — Ship's Compass Ship's Compass Head. Bearings. Head. Bearings. North N. i»E. N.E. N. 13° 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. The deviation for each position of the ship's head is next to be found, and we proceed as follows : — Ship's Head. North N.E. East. Correct magnetic bearing N. 2°W. N. 2° W. N. 2" W. Compass bearing N. i E. N. 13 W. N. 15 W. S.E. N. 2''W. N. 8 W. Deviation W. E. 13 E. 6 E, 158 Deviation of the Compass. Ship's Head. South. S.W. Correct magnetic bearing N. 2" W. N. 2° W. Compass bearing N. 3 W. N. 1 E. Deviation I E. Question 8. 3 w. List B. West. N. 2°W. N. 9 E. II W. N.W. N. 2°W. N. 12 E. 14 W. From the above Table construct a Napier's curve, and give the courses you will steer by standard compass to make the following courses, correct magnetic. 1. N.E. ^ E. 2. W.S.W. 3. N.N.W. 4. S.S.E. 5. N.N.E. 6. S. i8fW. 7. N. 4° E. 8. S. 62^° E. Note. — For the method of constructing the curve see No. 239, page 145, and for Kiile for finding standard compass courses to steer, see Problem II, Rule LXIII, page 150. Amwers. — 1. N. 40° E. 5, N. 18° E. 2. S. 76°W. 3. N. i4|°W. 4. S. 26° E. 6. S. 19' W. 7. N. 5° E. 8. S. 73° 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 : — 1. E. by S. 1 S. 2. E. f N. 3. S. 85° W. 4. N. 1° E. For the Rule for working this Question, see Problem I, Rule LXII, page 149. Answer.— \. S. 65! E. 2. N. 84^ 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 LXI, page 148). Ship's Head. Compass Bearing. Ship's Head. Compass Bearing. S.E. by E W.iN N. 68=E. S. 54 w. S.S.W. 1 W. . . N. JE N.4''E. South Ship's head ) nlo F S.E. by E. gives i "-' ■^■ Comp. bear. N. 68 E. Corr. mg. br. N. 75^ E. W. i N. gives . . ) S.54 W ^■4^- [c-E. gives . . ) South o South , S.S.W.^W. ) ..^ gives ) ^ N. 4 E. S. 43 W. N. 2\ E. Examples fob Exercise. Example I. 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 N. 60° 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' E.; 3" E. ; 10' W.; 21° W.; 22° W. Deviation of the Compasn. 1 59 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. i W. (3.) E. i N. (4.) N.N.W. (5.) E.S.E (6.) S.W. \ S. (7.) N. iE. (8.) W. |S. (9.)N. iW. (lo.)S. iW. (u.) E. i S. (12.) E.N.E. For the method of constructing the Curve see No. 239, page 145 ; and for the Rule for finding the correct magnetic course see Problem II, Eule LXIII, page 150. Answer :-{i.) N. 8|° E. (2.) S. 65^° W. (3.) E.N.E. (4.) N. i2°W. (5.) S. 88° E. (6.) S. 500W. (7.) N. 3f°E. (8.) N. 75°W. (9.) North. (10.) S. 1° E. (II.) N. 72|°E. (12.) N. 49iE. III. Suppose you have steered the following courses by standard compass, find the correct magnetic courses from the curve drawn : — See Problem I, page 149. (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. iW. (8.) E. | N. (9.) W. | S. (10.) N.E.byE. lE. (II.) N.W. |N. (12.) S.WfW. Answer :-{i.) ^.T si" '^- (2.)S.49|°E. (3.) S. 27^° W. (4.)S.4°W. (5.) S. 26" E. (6.) S. 86°W. (7.) S. 5i°W. (8.) S. 77^ E. (9.) S. 62= W. (10.) N. 82°E. (11.) N. 6o°W. (12.) S. 4if°W. rV. 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 Bearing of Distant Object by Standard Compass. No. Ship's Head. Bearing of Distant Object by Standard Compass. 1 W.N.W 2 S.E. by E 3 N.E. byN 4 E. by S. ^ S . . . . South. N.W. by N. North. N.N.W. 5 6 7 8 N. i E S. iE. E. is. E. ^N. W. by S. i S. W. by N. i N. . . E. i N E. |S Answer:— {i.) Corr. mag. bear. S. 24° E. (2.) N. i7°W. (3.) N. i4°E. (4.) N. 3" W. (5.) S. 3|E. (6.) N. 72°E. (7.) S. 72°E. (8.) N. 84= W. Example II. Ex. I. From the following table find the correct magnetic bearing of the distant object and thence the deviation : — Ship's Head Bearing of by Distant Object Standard , by Standard Compass. Compass. 1 Ship's Head Deviation by Required. Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E N. 42° W. N. 17 W. N. 9 W. N. 17 W. South .... S.W West .... N.W N. 44»W, N. 68 W. N. 76 W. N. 64 W. Answer : — Correct magnetic bearing N. 4.2° W. Deviation 0° ; 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 steer by standard compass to make the following courses correct magnetic: — 1. N.W. I W. 2. S.E. i S. 3. E. i S. 4. W. f S. 4mwer:—l. N. 84° W 2. S. 24° E. 3. S. 58° E. 4, S. 53^° W. i6o Deviatio7i of the Cotnpass. III. Suppose you steer the following courses hy standard compass, find the correct mag- netic course from the curve drawn : — 1. W. byN. fN. 2. South. 3. E. hy N. ^ N. 4. N. by E. | E. Answer:— I. N. 39=^. 2. S. if W. 3. N. 431° E. 4. N. 7° E. IV. You have taken the following hearings 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. j Ship's Head. Bearing of Distant Object by Standard Compass. S.E N. byE. iE. N. by W. 1 W. W. byS. IS... E. |N. E. byS j N.E. iE S.W. f S. Answer .-—Deviations 25° W. 32° W. ; 32° E. ; 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 1 Bearing of by i Distant Object Standard by Standard Compass. , Compass. Deviation Required. Ship's Head Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Eequired. North .... N. 13° W. N.E 1 N. 35 W. East N. 41 W. S.E N. 35 W. South .... S.W West .... N.W N. 23° W. N. 2 W. N. 5 E. North Answer : — Correct magnetic bearing, N. 18° 6' W. Deviations 1-5° W. ; 17° E. ; 23= E. ; 17° E. ; 5° E. ; 16° W. ; 23° W. ; 18^ 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 : — 1. N. I W. 2. E. i S. 3. N.W. by W. J W. 4. W. i S. Answer:— I. North. 2. N. 73° E. 3. N. 44° W. 4. N. 7i°W. III. You have steered the following courses by standard compass; find the correct magnetic courses from the curve drawn : — 1. E. I N. 2. S.E. f E. 3. N. f W. 4. S.W. i S. Answer:-!. S. 72!° E. 2. S. 35° E. 3. N. 16!° 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. 1 1 Bearing of Distant Ship's Head. Object by Standard Compass. 1 S.W.|S North E.iS. S.E. 1 S. N.E. iN ' W. iS. E. by S. i S. . . E. i S. 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. Beviatimi of the Coni^ass. i6i Example IV. I. From the following Table find 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 Eequired. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation llequired. North .... N.E East S.E N. 87° E. N. 71 E. N. 70 E. N. 82 E. South S.W West N.W S. 88" E. S. 81 E. S. 73 E. S. 76 E. Answer : — Correct magnetic bearing, N. Sg"" E. Deviations :— 2° E. ; iS^E.; 19° E.; 7° E. ; 3° W. ; 10° W. ; 18° W.; 15° W. 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 : — 1. S.W. fW. 2. North. 3. S. 45^ E. 4. N. 48|° E. Answer:— I. S. 68° W. 2. N. 1° W. 3. S. 55^ E. 4. N. 33^0 E. III. You have steered the following courses by standard compass; find the correct magnetic courses from the curve drawn : — 1. E. byN. |N. 2. S. 72° E. 3. S. 76° W. 4. N. 26!" W. Answer:— I. S. 89° E. 2. S. 57!° E. 3. S. 6o|° W. 4. N. 38^° W. IV. 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 by Standard Compass. Bearing of Distant Object by Standard Compass. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. East N.N.W N. 87° E. N. 2°E. W. 1 N N.W. i W . . . . N. iW. W. 4S. Answer .•— S. 74° E. ; N. 9° W. ; N. 21° W. ; S. 71° W. Example V. I. In the following Table give the correct magnetic bearing of the distant object, and thence the deviation : — Ship's Head Bearing of by Distant Object Standard by Standard Compass. Compass. Ship's Head Deviation by Required. Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. ■ North .... S. 15° E. N.E S. 34 E. , East ! S. 31 E. S.E i S. 22 E. South .... S.W West .... N.W S. i8°E. S. II E. South S. 3 W. It Answer : — 'irrect magnetic bearing, S. 16° E. Deviatiojib :— 1° W. ; 18° E.; 15° E.; 6°E.; 2°W.; 5°W.; i6°W.j 19*= W. 1 62 Deviation of the Oonipass. 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 : — 1. N. 73" E. 2. N. 68° W. 3. S. 87° E. 4. S. 2° W. Answer :—\. N. 54° E. 2. N. 49°W.. 3. N. 76°E. 4. South. III. Given the following courses by standard compass to find the correct magnetic courses from the curve drawn : — 1. S. W E. 2. N. 77° W. 3. N. 64° E. 4. S. 5" W. Answer:— I. S. 7i°E. 2. S. 85° W. 3. N. 82° E. 4. S. 6° W. IV. You have taken the following bearings of two distant objects by your standard compass as above ; with the ship's head N. 39° E. St. Catherine's Point, N. 82° E. Needles Light, N. 8° W. Answer : — Deviation for ship's head, N. 39° E., is 16" E. Correct magnetic bearing of St. Catherine's Point, S. 82° E. ; and of the Needles Light, N. 8° E. Example VI. I. From the following Table find the bearing of the distant object, and thence the deviation. 4 Ship's Head Bearing of by ' Distant Object Standard , by Standard Compass. Compass. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E North N. 16^ E. N. 24 E. N. 19 E. South .... S.W West -■.,. N.W N. 3»E. N. 16 W. N. 23 W. N. 15 W. Answer : — Correct magnetic bearing, N. 1° E. Deviations:— I °E.; 15° W.; 23^ W.; 18° W.; 2° W. : 17° E.; 24° E. ; 16° E. 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 : — 1. E. by S. 2. W. by S. i S. 3. North. 4. N. 57^° W. Answer .-—1. S. 58° E. 2. S. 56° W. 3. N. 1° W. 4. N. 80° W. III. Suppose you steer the following courses by the standard compass, find the correct magnetic courses from the curve drawn : — 1. W. I N. 2. S.S.W. 3. E. by N. i N. 4. South. Answer .—l. N. s^l" W. 2. S. 3i|° W. 3. N. 52^° E. 4. S. 2° E. IV. 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 : — Ship's Head by Standard. Bearing of Distant Object by Standard Compass. S.E. South N.E. I E. North. Ship's Head by Standard ' Compass. Bearing of Distant Object by Standard Compass. S.W. I S. E. 2 N. .. W. S.J E. Answers:— Deyiations, i%°W.; 2° W. ; 14° E. ; 22° W.; Correct magnetic bearings, N. 33* E. ; N. 2° W. ; N. 73° W. ; S. 25° E. Deviation of the Compass. 163 Example VII. I. From the following Table fiad the correct magnetic bearing of the distant object, and thence the deviation : — Ship's Head Standard Compass. Bearing: of | Distant Object Deviation by Standard Eequired. Compass. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East .... S.E S. 87»W. S. 70 w. S. 72 w. S. 88 W. South .... S.W West .... N.^ N. 84=W. N. 77 W. N. 68 W. N. 72 W. Answer : — Correct magnetic bearing, N. 88° W. ^MSM;er .-—Deviations, 1. 5° E. 2. 22° E. 3. 20° E. 4. 4° E. 5. 4° W. 6. 11° W. 7. 20° W. 8. 16° W. II. From the above Table construct a Napier's curve, and give the Courses you would steer bj the standard compass to make the following courses correct magnetic : — 1. N. 79° E. 2. S. 48' E. 3. N. 84° W. 4. S. 42° W. Answer :—l. N. 55° E. 2. S. 55° E. 3. N. 65° W. 4. S. 55" W. III. Given the following courses steered by standard compass to find the correct magnetic courses from curve drawn : — 1. N. 72° W. 2. N. 84° E. 3. S. 20= W. 4. S, 78" E. Answer .-—1. S. 88° W. 2. S. 74° E. 3. S. 13° W. 4. S. 61° E. 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 : — ' Bearing of Distant Ship's Head. Object by Standard Compass. Ship's Head. Bearing of Distant Object by Standard Compass. S. 68° W N. 82° W. N. 38 E i S. 69 W. 1 N. 78»W S. 9 E S. 3° E. N. 78 E. ^»WM;e>-.-— Deviations, 1. 15° W. 2. 21° E. 3. 20^ W. 4. 3° W. Correct magnetic bearings, 1. S. 83° W. 2. West. 3. S. 23° E. 4. N. 75° E. Example VIII. I. From the following Table find 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 by Standard Compass. Beai-ing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 60° E. S. 44 E. S. 36 E. S. 40 E. South^ .... S.W...... West .... N.W S. 67= E. East N. 87 E. S. 82 E. Answer : — Correct magnetic bearing, S. 64° E. ^««M;e>-.— Deviations:— 1. 4° W. 2. 20' W. 3. 28* W. 4. 24° W. 5 3* E. 6. 36'i; 7. 29= E. 8. 18= E. 164 Deviation of the Compass. 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: — 1. N.N.W. :}W. 2. E. byS. is. 3. S.W. i W. 4. E. J N. Answer:— I. N. 51° W. 2. S. 5o|° E. 3. S. 27^° W. 4. S. 64° E. III. Suppose you have steered the following courses by standard compass; find the correct magnetic courses from the curve drawn : — 1. W. by N. ^ N. 2. N. ^ E. 3. S.E. f S. 4. S. f W. Answer:— I. N. 461° W. 2. N. 2° W. 3. S. 57^° E. 4. S. i8^°W. IV. 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 : — Sliip's Head. Bearing of Distant Object by Standard Compass. Ship's Head. Bearing of Distant Object by Standard , Compass. N.W s.s.w N.E. East. E.iN W. by N. 1 N. . N.| W. S. J W. uimwer.-— Deviations:— 1. 18° E. 2. 18° E. 3. 28' W. 4. 27° E. Correct magnetic bearings :—l. N. 63° E. 2. S. 72°E. 3. N. 36" W. 4. S. 33'W. Example IX. I. From the following Table, find the correct magnetic bearing of the distant object, and thence the deviation : — V 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 Compas% Deviation Required.! North .... N.E East S.E S. 44° E. S. 50 E. S. 58 E. S. 59 E. South .... S.W West .... N.W S. 38° E. S. 16 E. S. 21 E. S. 34 E. 1 1 j 19° E. 5. W. Answer : — Correct magnetic bearing, S. 40° E. ^wjtoer.— Deviations:— 1. 4° E. 2. io» E. 3. 18= E. 4. 6. 24° W. 7. 19° W. 8. 6° 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 . — 1. E. fS. 2. S. |E. 3. W. byS. |S. 4. N. 1 W. Answer:—!. N. 82° E. 2. S. 8|° E. 3. West. 4. N. 5° W. III. Suppose you steer the following courses by standard compass, find the correct magnetic courses from the curve drawn : — 1. S.W. by W. I W. 2. N. 5° W. 3. S. 8" E. 4. N. 8of E. Answer:-!. S. 40° W. 2. N. i W. 3. S. 5° E. 4. S. 83!° E. IV. 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. Ship's Head. Compass Bearing. Ship's Head. Compass Bearing. E.N.E S.W. by W. . . E.S.E. S.E. i S. E.fS W.iN. W. by N. A N. E. by N. | N. ^Mswer.— Deviations:— 1. 13!°^. 2. 25° W. 3. Correct magnetic bearings:—!. S. 54*' B. 2. S. 64" 19^° E. 4. 16° W. E. 3. N. 67^' W. 4. N.54°E. Deviation of the Compass. 165 Example X. I. From the following tahle find the correct magnetic hearings of the distant ohject, and thence the deviation. Ship's head ( Bearing of , ' Ship's Head | Bearing of [ by ] Distant Object Deviation by 1 Distant Object Deviation Standard I by Standard Required. Standard I by Standard Required. C!ompass. | Compass. Compass. Compass. I II I North N.E. . East . S.E. . West. S. 78=^W. S. 77 W. S. 82 W. South .... s. se^w. I S.W N. 89 W. West K. 81 W. 1 N.W N.79 W. I Answer : — Correct magnetic hearing S. 88° W. Answer.— Devmiion:—\. 2° W. 2. 10° E. 3. 11° E. 4. 6. 3°W. 7. 11= W. 8. 13° W. 6°E. 2°E. II. Given the correct magnetic course, to find the courses to steer hy standard compass : — 1. S. 87° E. 2. S. 83" W. 3. N. %° W. 4. N. 57^= E. Answer.— I. N. 8iA° E. N. 85^^ W. 3. N. 4° W. 4. N. 47° E. m. Given courses hy standard compass, to find correct magnetic : — 1. N. 85°E. 2. N. 86=^W. 3. N. 631=' E. 4. S. 9° W. Answer.— I. S. 83I'' E. 2. S. 82' W. 3. N. 75!" E. 4. S. gf W. IV. You have taken the following bearings of distant ohjects by standard compass with ship's head as given below, find the correct magnetic bearing : — Ship's Head. Compass Bearing. Ship's Head. Compass Bearing. N. I2»E N. 86° W S. 75° E. S. 3 W. • S.47°E S. 16° W S. 87° E. S. 88° W. ^mwr.— Deviations:— 1. 2^° E. 2. ii|°W. 3. 5^° E. 4. 0°. Correct magnetic bearings:—!. N. 77|°E. 2. S. 8^ E. 3. S. 8i|°E. 4. S. 88°W. Example XI. I. From the following Table find the correct magnetic bearing of the distant object, and thence the deviation : — r Ship's Head by Standard Compass. Bearing of Distant Object Deviation by Standard Required. Compass. Ship's Head ty Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 17° W. ' South. S. J E. South. 1 1 South .... S. iz-W. S.W S. 31 W. West S. 35 W. N.W S. 30 W. Answer. — Correct magnetic, S. 15° W. Answer. — Deviations: — 1. 2° W. 2. 15* E. 3. 20° E. . ;6°W. 7. so'W. 8. 15-' W. 4. 15° E. 5. 3°E. i66 Deviation of the Cotnpass. II. From the above Table construct a Napier's curve and give the courses j-ou would steer by standard compass to make the following courses correct magnetic : — 1. E. I S. 2. S. I W. 3. N. i4f W. 4. S. 44° E. Answer.— 1. N. 78^° E. 2. S. io°W. 3. N. 9I' W. 4. S. 62^° E. III. Suppose you steer the following courses by standard compass, find the correct magnetic courses from the curve drawn : — 1. W. |N. 2. N.E. IE. 3. N. 78'E. 4 S. iii°W. Amwer.—l. W. by S. 2. N. 67° E. 3. S. 82° E. 4. S. 9° W. IV. 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. Compass Bearing. Ship's Head. Compass Bearing. N. by W. i W. S.W. by W. . . . S. 3°W. S. 86 E. W. by S. f S. E. by S. i S . N. f E. E. by N. 1 N. Answer.— Devieitions:— I. 8= W. 2. i8^°W. 3. 21° W. 4. 21° E. Correct magnetic bearings :—l. S. 5° E. 2. N. 75!° E. 3. N. i2|° W. 4. S. Ss'E. Example XII. I. From the following Table find the correct magnetic bearing of the distant object, and thence the deviation: — \ I ■ 1 Ship's Head Bearing of Ship's Head Bearing of by Distant Object Deviation by ' Distant Object Standard by Standard Required. Standard ' by Standard Compass. Compass. Compass. Compass. Deviation Required. North .... S. 3"W. N.E S. I E. East S. 9 E. S.E S. 12 E. South .... S.W West .... 1 N.W S. I'E. S. 13 W. S. 15 W. S. 8 W. - . '■ Answer : — Correct magnetic bearing S. 2° W. ^nsM^er.-— Deviations:— 1. 1° W. 2. 3^ E. 3. 11° E. 4. 14- E. 5. 3° E. 6. 11° W. 7. 13° W. 8. 6° W. II. Given the correct magnetic courses, to find the courses to steer by standard compass : — 1. N. 41° W. 2. N. 66° E. 3. S. 39° E. 4. S. 79° W. Answer:— I. N. 36!° W. 2. N. 60° E. 3. S. 54° E. 4. N. 88^° W. m. Suppose you steer the following courses by standard compass, find the correct magnetic bearings : — 1. N. 32|°W. 2. S. 70'W. 3. S.E. byE. JE. 4. N. 31° W. Amwer.—l. N. 37' W. 2. S. 55^ W. 3. S. 47I-' E. 4. N. 35° W. rV. You have taken the following bearing of distant objects by standard compass ; with the ship's head as given below, find the correct magnetic bearings : — Ship's Head. Compass Bearing. Ship's Head. Compass Bearing. S. |E Dungeness. N.E. by E. ^ E. Beachy Head. S. 65° W N.W. i W. Answer: Deviations: — 1. 6°E. ; 2. i4°W.; and correct magnetic bearings, 1. N. 68* E.; . N. 64i° W. THE TRAVERSE TABLE. 243. In all collections of tables for the use of navigators there is inserted a table containing the true difference 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). 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, Norie or Eaper.) USE OF THE TABLE. 244. Given the course and distance, to find the difference of latitude and departure. EULE LXVI. With the Course open the Tahles, and under or above the proper number of points [or degrees) and opposite the distance, will be found the difference of latitude and departure. Obs. — When the course is found at the bottom of the page, care must be taken to see that the difif. of lat. and the dep. are taken from the proper column above the words departtirt and diff. lat. It mast 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. 5 N. a distance of 78 miles : required the diflFerence of latitude and departure by inspection. The given course is 35 points ; and referring to Table I we find the page assigned to this course to be page 14, Norie, or page 436, Raper's Navigation, in which against 78, in column headed Dist., stands 60-3 under the head Lat., and 49-5 under the head Bep. We conclude, therefore, that for the given course and distance, the diflerence of latitude is 60-3 miles, and the departure 49-5 miles. * This table is constructed by solving a right-angled triangle, of which one angle repre- sents 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. Inasmuch as the sine of an angle is the cosine of its complement, it is evident that the difference ot latitude and departure for any course are the departure and diflerence of latitude for the complement of that course, and hence the table is compactly arranged by interchanging the headings of the columns containing these elements at the top and at the bottom of the page, and using 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 })etwepn any two quantities may be expressed as functions of some angle in terms of the sine, cosine, or tangent ; it may bq \ued, in fact, as a general proportional table. r68 Tr a/verse 2 able. Ex. 2. Suppose tlic 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, Xorie. 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 dij". lat. 46' 2, so that the difference of latitude made is 46*2, and the departure 864. Ex. 3. Course N.E. by X., distance 129 miles: find diff. lat. and dep. Enter Table T, and find 3 points at the top, and in one of the columns marked Bist. find the distance 129, then in the columns opposite to this, marked lat. and dep. at top, stands the difference of latitude 107-3, ^°'i 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 ~6'6. 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, and the distance being under 60 miles, is found in the left hand column ; therefore, on the page (56 Xorie) is 40' at the top, and opposite to 50 in the distance column (marked Dist.) is 38-3 nnder Lat., and 32-1 ntiderJie^., 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 767 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 bg 1, if the figure cut off is 5, or upwards. Examples. Ex I. Course 3I points, distance 20-3; required the diff. lat. and dep. corresponding thereto. With course 3^ points, and dist. 20-3, taken as 203, we get the diff. of lat. 156-9, dep. 128-8 ; now cut off the right hand figure of each (the 9 and 8), and shifting the decimal point one place to the left, we have diff. lat. 15-7, and dep. 12-9. It will be observed that the tenths are increased by i, in each case, as the figures cut off in both cases exceeds 5. Ex. 2. Required the diff. lat. and dep. corresponding to course 4j points, and dist. 24-3 miles. With course 4^ points, and dist. 24-3 (as 243 miles), we find diff. lat. 154-2, and dep. 187-8 : hence we obtain, after dropping the tenths, and removing the decimal point in each one place to the left, 15-4, and 18-8, 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 E.N.E.. distance 29'-5 ; find diff. of lat. and dep. corrpsponding. La this case take out for distance 295. Thus, for 6 points and distance — 295 = 1 12-9 diff. lat., 272'-5 dep. .' . 29-5 =z 11-3 diff. lat., 27'-3 dep. After dropping the tenths, and removing the decimal point one place to the left, we have diff. lat. 11-3 and dep. 27-3. * The reason of this rule is that the Traverse Table is entered with a distance ten times as great as the given distancp, and the resulting diff. lat. and dep. is divided by ten. This is done by meruly imagining the decimal point to be removed one place to the rifffit before entering the table, and then one place to the kft after taking out the results (see p. 26, (10). Traverse Table. 169 Ex. 4. N. 3 pts. W., and dist, 20-6 miles (as 2c6), give diff. lat. i7i'3, and dep. ii4'4; dropping the tenths in each case (the 3 and the 4), and shifting the decimal point one place to the left, we get diff. lat. 17-1 N., and dep. i r4 W. Ex. 5. N. 6^° E , and dist. 21-5 (as 215), give difif. lat. 90-9, and dep. i94'9, which is difi, lat. 9"i N., and i9"5 E. It will be observed that the tenths are increased by 1, in each case, as the figure dropped exceeds 5. {b) If the distance exceeds the limits of the Traverse Table. Take the half, the third, &c., so as to bring it within the limits, taking care to multiply the corresponding quantities by z, 3, &c. Ex. 6. Let the course be 3^ points, and distance 435 : required the corresponding diff. lat. and dep. 435 divided by 3 gives 145. Course 3^ points, and dist. 145 give diff. lat. 116-5 ^^^ 'i^p. 86-4 X 3 X 3 Diff. lat. 3495 Dep. 259-2 If the distance had been 43-5, the diff. lat, would have been 35-0, and the dep. 25-9. (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 0/300; take the sum of the quantities thus found, cut oft' the last figure, and remove the decimal point as before. Ex. 7. Course 5f points, and distance 526 : required the corresponding diff. of lat. and departure. Course 5^ points, and dist. 300, give diff. lat. 128-3, ^° D. Lat. i36'6 = 2 17 S. f The lat. in is found according to I Eule XLVI, page 93. Lat. in (or arrived at) 36 8 N. ' 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. D18T. DiFF. Lat. Departure. 1 N. S. E. W. S. 3W 20 . 166 1 ii"i w.* 16 16 N. 5"W 28 15-6 23-3 S. 2 E 32 296 i2"3 N. 6E 14 5 '4 129 S. 4 W 36 25-5 25*5 2I"0 71-7 252 75-9 * See Rule LXVIII, :^a, page 172. 2I-0 2C-2 1 i 50'7 ' 50'7 We seek in the several pages of the Traverse Table II, for the diflf. 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 diflf. lat. and dep. being of equal amount, the course is 45", or 4 points, which illustrates the remark, No. 141, page 89. Lat. left. 37° 24' S. ' The lat. sailed from being South, and the T-v-flp 1-i. „„._ rr «5 i sliip having sailed South, the ship has evi- Jjm. lat. jO 7 — 51 ^- . dently increased her South lat., whence the • ( sum of lat. from and diff. lat. is taken to obtain Lat. arrived at 38 15 S. / lat. in.— (See Rule XLVI, 1°, page 93). Ex. 4. A ship from lat. 20° 56' N. sails (all true courses) N.W. by N., 20 miles ; S.W., 40 miles; N.E. by E., 60 miles; S.E , 55 miles; W. by S., 41 miles; E.N.E., 66 miles: required the latitude in, also the course and distance made good. Courses. DiST. DlEF. Lat. Departure. N. S. E. W. N. 3 W 20 i6-6 ii-i S. 4W 40 28-3 28-3 N. 5 E 60 33'3 49"9 S. 4E 55 389 389 S. 7 w 41 80 40*2 N. 6 E 66 25'3 6i-o 75-2 75-2 149-8 79-6 75*2 79-6 70-2 Course due East, and dist. 70-2, the same as the departure. (See No. 141, page 89). The Traverse Table being filled up, the sums of the Northings and Southings are both 75-2, 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 149'8, and that of the Westings 796; their difference, 70-2, shows that the ship has gained so much to the Eastward, that being the greater. Consequently the Course is due East, and the Uttance ^^>■*, the same as the departure. Travene Sailing. I7S Ex. 5. A ship sails from a place in lat. 1° 15' N., the following true courses :— S."W. by W., 45 miles; E.S.E., 50 miles; 8.W., 30 miles; SE, by E., 60 miles; S.W. | S., 63 miles: required the latitude in, also the course and distance made good. COUHSES. DiST. DiFF. Lat. Departure. N. S. E. W. S. 5 "W S. 6E S. 4W S, 5 E S. 31 W 45 50 30 60 63 25-0 19-1 2I"2 33-3 50-6 46-2 49'9 37-4 21-2 37-5 96-1 149-2 96T 96T The Traverse Table being completed, the sum of the Southings is 149-2 miles, and to that amount the ship has altered her latitude. The miles of departure in the East column are 96-1, and those in the West column are also q6i ; but as the East and West departures 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; consetiuontly, the course is due South, and the distance sailed is equal to the diff. of lat., viz., 149'2. This is according to No. 141, page 8g. Latitude left DifiF. lat. 6,0) 14,9-2 29-2 = 2 Latitude in The ship being 1° 15', or 75 miles, N. of the equator, must evidently be in S. lat. after making 149 miles of Southing. Thus, in subtracting one of the quantities from the other, the difference takes the name of the greater. Rule XLVI, page 93. The course is South, and dist. i49'2, the same as diff. lat. Ex. 6. A ship from latitude 46° 10' N., sails as follows: S. 48° E., 25 miles; S. 51° E., 189 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., 137 miles; N. 76° E., 39-6 miles; required the lat. in, also the course and distance made good. Courses. 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. Dist. Diff. Lat. Departure. N. S. E. W. 25 i6-7 i8-6 18-9 12-4 0-7 11-9 H"7 12-4 145 21-6 5-0 8-1 13-6 20-0 16-4 7-8 i3'7 396 14-9 7 '7 13-0 96 45-9 41-7 II 4'5 38-4 6-9 41-7 i23'3 6-9 6-9 4-2 ii6'4 Course N. 88" E. Dist. 11 6§ miles. Exphmation. — The course 48° (found at the bottom of one of the pages in Table I), and dist. 25 (in dist. column), opposite this last stands 16-7 diff. lat., and i8-6 dep., and as the ship is sailing on a S. and E. course, the diff. lat. is -written in the diff. lat. column, and the dep. in the East column. 1^6 Traverse Saih'tiff. To take out the next course and distance wc proceed thus : — 51° and dist. i8'g, taken as 189, give diff. lat. ll8'9, and dep. i46'9, now removing the decimal point in each, one place to the left we have diff. lat. 118-9 = ll'Sg, and dep. 146-9 = i^'Sj; we do not require to use both the decimal places, but if, as in the case with both diff. lat. and dep., the second decimal figure amounts to 5, we add 1 to the first, and the diff. lat. thus becomes ii'g, and the dep. 147. The third course is N. 87° E., and di.stance i2'4; then 87° and dist. 124 (omitting the decimal point), give diff. lat. 06-5, and dep. 123-8 ; now dropping the tenths in each, viz., the 5 and the 8, and increasing the preceding figures by 1 in each case, as the tenths exceed 5, we have, after removing the decimal point one place to the left, diff. lat. 0-7, and dep. 12-4. Proceed in this way with the remaining courses, except the last, in which case the distance being more than 300, we proceed as follows : — Course 76°, and dist. 300 give diff. lat. 72-6 and dep. 291-1 „ „ 96 „ 23-2 „ 93-1 . • , Course 76°, and dist. 396 ,, 95-8 ,, 384-2 Now, cutting off the last figure in each, the 8 and the 2, and removing the decimal point one place to the left, we have diff. lat. 9-6 (not 9-5, as the figure 8 which is dropped exceeds 5, one is added to ihe tenths! , and dep. 38-4. in Traverse Table II } ^^'^^ , 4^2 N. j ^^^ j Couxse N. 88° E. j ^,,, ^,,,. Latitude left 46° 10' N. \ Diff. lat. 4 S. I The latitude in is found according I Eule XL VI, 2°, page 93. Lat. in 46 6 N. / Examples foe 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 difi". 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. J N., 42 miles ; S.S.E. I E., 16 miles ; W. ^ 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 difif. 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. ^ S., 32 miles; N.N.E., 20 miles; N. by W. I W., 27 miles; N.E. by E. | E., 24 miles; S.W. J S., 10 miles. 5. A ship from lat. 1° N., sails East, 8 miles ; E. ^ N., 20 miles ; S.E. by E., 33 miles ; S. I W., 31 miles ; N.E. J N., 43 miles ; South, 28 miles ; S. f E., 21 miles ; S. by W. ^ 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' S., sails E. by N. J N., 56 miles ; N. ^ E., 80 miles; S. by E. I E., 96 miles ; N. ^ E., 68 miles ; E.S.E., 40 miles ; X.N.W. J W., 86 miles ; E. by S., 65 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° 1 1' N., we have sailed the following courses : — N. 36° W., 27 ' ; N. 24° E., 30' ; S. 75° W., 47'; S. 80° 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 sailed N. 84° W., 18'; N. 89° W., 3o''4; N. 67" W., 29-9 ; N. 39° W., 33'-9 ; N. 8» W., 25'-9 ; N. 73° W., 34'-9 ; N. 86° W., 44'-? ; 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., S.W. f W., 62 miles ; S. by W., 16 miles ; W. % S., 40 miles ; S.W. | W., 29 miles ; S. by E., 30 miles ; and S. I E., 14 miles : required the lat. arrived at, and the course and dist. made good, the variation of the compass being 21^° W. PARALLEL SAILING. 248. 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. (wliieh 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. Eor example, two places in lat. 10° and distant 60 miles east and west from each other, have 6o'-() 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. 249. Given the departure made good on a given parallel of latitude, to find the diff. of long, corresponding thereto. EULE LXIX. 1°. Take out of the Tables the log. secant oj latitude (rejecting lofrom index J, and the log. of departxcre made good. 2°. Add these logs, together, and find the nat. number corresponding thereto. The result is the difference of longitude required. 250. 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 therefore by inspection of the Traverse Table by EULE LXX. 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 Ex. 2. A ship sailed 94-6 miles on the good 161 miles: required the diff. of long. parallel 64° 38' K : required the diff. long. Lat. 29° 51' Secant 0-061815 Lat. 64° 38' Secant 0-368141 Dep. 161 Log. 2-206826 Log. 2-268641 Diff. of long. 185-6. Bij Inspection. Dep. 94-6 Log. 1-975891 Log. 2-344032 Diff. of long. 220-8. By Inspection. In Traverse Table II, lat. 30° as course, dep, i6ri '• In Traverse Table II, lat. 64° as course and dep< in lat column, gire cliff, long. i85 mil. < in dist. col. AA 94' 7 give diff. lat. in dist. column 216 miles; and course 65°, and dep. 947, give difif. long, in dist. column 224 miles ; therefore the diff. long, for 64^" wiU = 216 + 224 -j- 2 ■= 22° nules. 178 Pwrallel Sailing. 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 Log. 2 -0969 10 Diff. long. 6,0)20,3 Log. 2-307568 3 23 = 3°23'E. Long, left o 59 "W. Long, in 2 24 E. 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. 2-037236 Log. 2-357602 Diff. of long. 227-8 W. 6,0)22,8 3'48'W. 179 20 W. 3 48 Long. in. 183 8 W. 360 o Ex. 5. In lat. 71° good was 71^ miles; long. Lat. 71° 25' Dep. 71-25 25' N., the dep. made required the diff. of Secant 0-496640 Log. 1-852785 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. I 903090 Log. 2-663420 Diff. of long. 460-7. Lat. in. I. 6° 7'N. 2. 19 48 S. 3- 39 57 N. 4- 51 17 N. <;• 60 N. 6. 46 37 s. Lat. in. Dep. . 64°i6'N. 265'-? W. . 51 28 S. 709 E. • 37 N. 94 W. 60 S. 204 E. II 15 N. 365 W. • 54 53 «• 342 E. Log. 2-349425 ' Diff. of long. 223-6 nearly. Examples for Practice. In each of the following examples the difference of longitude is required : — Dep. 249' W. 324 E. 398 W. 294-8 W. 74 W. 352 B- 251. 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 be used at once for the difference of longitude, the resulting error scarcely exceeding one mile. 252. Griven the difference of longitude of two places on the same parallel, to find their distance as measured along the parallel. EULE LXXI. To the log. of the diff. of long, add the cosine of lat. ; the sum (neglecting 10) 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 60 Difference of longitude 902 miles. Log. distance = log. diff. long. + log. cosine lat. Log. diff. long. 902 = 2-955207 Log. COS. lat. 55° SS = 9'748497 Log. distance 505'-5 = 2-703704 Pa/rallel Sailing. 1 79 253. Given the meridian distance and diflference of longitude to find the latitude. RULE LXXn. 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. ElCAMTLE. 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 tbe latitude of the parallel on which she sailed. By Calculation. Cos lat = ™^^- ^^*- ^°°§^- ^®^* 3° 12' W. diff. long. Long, in 4 8 E. D. long. 7 20 =r 440 miles. Mer. dist. 246 log. (-j- 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 25 S.; B 9 15 W. MIDDLE LATITUDE SAILING. 254. Middle Latitude Sailing is a method founded on tlie principle of parallel sailing, converting Departure into Difference of Longitude, and the Difference of Longitude into Departure, when the ship'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°, makes 100 miles departure: this departure, if made good altogether in latitude 40'', would give i30"5 difference of longitude by Rule LXIX, page 177 ; 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 130-5 and 139, and therefore we may conclude it to bo 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 intercepted between their meridians, is nearly equal to the departure. If we conceive the ship to sail along this middle parallel, we may apply the princi]>le of parallel sailing to the cases in point. In parallel sailing the departure (or distance) and difference of longitude are connected by the relation, dep. = diff. of long. X COS. lat. ^hen 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. Jliddle latitude sailing has thus the same two cases as parallel sailing, and accordingly the rules for inspection and computation already given, Eule LXIX, page 177, apply equally to this sailing, observing merely to read middle latitude for latitude. 255. To find the latitude and longitude in, the course and distance from a known place being given, by Traverse Table and Middle Latitude. EULE LXXni. 1°. With the given course and distance enter the Traverse Table, and take out true difference of latitude and departure (see Eule LXVI, page 167). 2°. With difference of latitude and latitude from,- Jind latitude in (see Eule XLYI, page 93). 3°. Get the middle latitude, as directed, Eule XLYII, page 93. 4°. With the middle latitude as course, look in the difference of latitude column for the departure, the corresponding distance at the top ts the difference of longitude. 5°. With difference of longitude and longitude from get longitude in, as in Eule XLIX, page 95. 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 diffetence qf longitude. Middle Latitude Sailing. i8i 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. Di^ance "256* miles! } ^'^^ '^'^- ^^*- '42-2, and dep. 212-9 (see Kule LXVI, page 167). 6,0)14,2 Diff. lat. 2°22'S. 2)212-9 Lat. from 52 6 N. 2 22 ^ i dep. 106-4 Lat. in 49 44 N. Mid. lat. 51° as eoiirse (Table II), and half dep. ^/lO't ,5° 1064, in diff. lat. column, give in dist, column 169 _^^^_ miles, the half the diff. of long. Then 169 x 2 = llid. lat. 50 55 ^' i°"S. 338. 6,0)33,8 5°38'W. Long, from 35 6 W. 5 38 Long, in 40 44 W. Explanation.— T\iQ difference of latitude and departure are found as described in Rule LXVI, page 167. The latitude in is found by Rule XL VI, page 93 ; and thence the middle latitude, by adding the latitude from and latitude in together, and divided by 2 (see Rule XLVII, page 93). The departure exceeding the limits of the Tables, the half is 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 degrees and minutes. The ship is in West longitude, sailing West, add difference of longitude to longitude left to obtain longitude in (Rule XLIX, page 95). This 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 dijBF. lat. 18-7, and dep. 12-5 (see Rule LXVI, page 167). Mid. lat. 48^0 as course, and dep. as diff. lat. give in dist. column 19 miles, -which is the diff. of long. Diff. lat. 0° 19' S. > Lat. from 48 27 S. Lat. in 48 46 S. 2)97 13 Mid. lat. 48 36 J Diff. long. 0° 19' E. Long, left 29 12 W. Long, in 28 53 W. (The long, in is found by Rule XLIX, page 95). Ex. 3. A ship from the Lizard, in lat. 49° 57' N., sails W.S.W., 163 miles, variation z\ points "W. : required the latitude come to, and difference of longitude. W.S.W. by compass is (allowing 2^ points westerly variation) S.AV. \ S. true, which in Table II, and dist. 163, gives diff. lat. 126, and dep. 103*4. 6,0) 12,6 6 or 2° 6' S. Lat. left 49 57 N. Then mid. lat. 48° 54', say 49°, as a course, and T i • ^, "NT and dep. 103-4, found in the lat. column, opposite ijai. m 47 51 j.>i . ^.Q .^iiicii^ i^ tiie ^gt. column, is 158, nearest, the difference of longitude. 2)97 48 Mid. lat. 48 54 1»2 Middle Latitude Sailing. Ex. 4. Lat. from 59° o'N., long, from 3' 33' E., course S.E. by E.fE., distance 191 miles. Course 5f points, distance 191 miles (in Table I) give diff. lat. 8i"7 and dep. i72"7. D. Lat. 6,0)8,1-7 I 21 '7 or i°22'S. Lat. from 59 o N. By Calculation. Mid. lat. 58° 19' sec. 0-279655 Dep. i72'7 log. 2*237292 log. 2-516947 Lat. in 57 38 N. 116 38 D. long. 328-8 6,0)32,8-8 Mid. lat. 58 19 58° 5°2i By Inspection. Dep. 159-0 give D. long. 300 172-8 60' (or 1°) -. 19' 9 26 326 59" Dep. 154-5 give D. long. 300 18-0 172-5 35 335 D. long, for middle lat. 58° is 326 ). 59 .. 335 Diff. for 1° of mid. lat. q 6,0)17,1 2-85 T>. 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 mid. lat. Ex. 5. Sailed from A, in lat. 50° 48' N., long, i'^ 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 S. ; 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 i8o'-4 in diff, lat. column, is found to be 275', in degrees 4° 35' E., which is the diff. long. Applj-ing this to the long, from, 1° 10' W., we have the long, in 3° 25' E. Examples for Praotice. In each of tlie 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. Lat. from. Long. from. Course. Dist. I. 25°35'N. 60° o'W. E.N.E. 296 2. 32 30 N. 25 24 W. N.W. by W. I W. 212 3- 39 30 S. 74 20 E. S.W. by W. f W. 210 4- 46 24 S. 178 28 E. S.E. f E. 278 5- 20 29 N. 179 10 W. W. by S. I S. 333 6. 56 N. 29 50 W. S. 47° E. j68 MERCATOR'S SAILING. 256. 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 partSy and the chart constructed by means of it called Mercator's chart. With the assistance of this Table, the rules of plane trigonometry suffice 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 ACB', 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 CB' ■=: AC tang. A) diff. long. = mer. diff. lat. X tang, course. 257. Given the latitudes and longitudes of two places, to find the course and distance between them. EULE LXXIV. 1°. Find the trice difference of latitude, according to Eule XLIV, page 91. 2°. Find the meridional difference of latitude, Rule XLV, page 92. 3°. Next find the difference of longitude, Rule XL VIII, page 94. 4°.* To find the course. — From the log. of diff. of longitude {increasing its index by 10 J, subtract the log. of meridional diff'. of lat.: the remainder is the tangent of course, xohich take out of the tables, and place before it the letter of diff. of lat., and after it the letter of diff. of long. 5°. To find the distance. — To the secant of course f rejecting 1 o from the index), 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 = Mer. diff lat: _ Tru e diff. lat. (jpg^ course . log. tang, course — 10 =z log. diff. long, log. mer. diff. lat. . • . log. dist. ■=. log. true diff. lat. + log. sec. course — 10. 184 Mereator^s Sailing. Examples. Ex. I. Eequired 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 Diff. of lat. 177 N. Mer. parts 3970 Mer. parts 4291 Mer. difiF. lat. 321 Long. Tj-nemouth i''25'W. Long. Naze 7 2 E. 8 27 60 DifF. of long. 507 E. To find the Course. Diff. long. 507 Log. (-4- 10) 1 2-705008 Mer. diff. lat. 321 Log. 2-506505 Tang. IO-I98503 Course N. 57° 40' E. To find the Distance. Course 57° 40' Secant 0-271773 Diff. of lat. 177 Log. 2*247973 Distance 331 Log. 2-519746 Ex. 2. Eequired the course and distance from A to B. Mer. parts 3606 Mer. parts 3326 Mer. diff. lat. 280 Lat. A Lat. B 5i°23'N 48 23 N. 3 60 Diff. of lat. 180 S. Diff. long. 300 Log. (+ 10) 12-477121 I Course 46° 5 8 1' Mer. diff. lat. 280 Log. 2-447158 Diff. lat. 180 Tang. 10-029963 Course S. 46" 58^' E. Long. A Long. B 9° 29' W. 4 29 W. 5 60 Diff. of long. 300 E. Secant 0-166014 Log. 2-255273 Log. Distance 263-8. 2-421287 Ex. 3. Eequired the course and distance from Cape Bajoli to Cape Sicie. Lat. Cape Bajoli 40° 1' 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- 60 Diff. of long. 123 E. Diff. long. 123 Log. (-4- 10) 12-089905 I Course 26° 51' Mer. diff. lat. 243 Log. 2-385606 ] Diff. lat. 182 Secant 0-049542 Log. 2-260071 Tang. 9-704299 Course N. 26' 51' E. Distance 204. Log. 2-309613 k Ex. 4. Eequired 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 Mer. parts 255 Mer, parts 968 Mer. diff. lat. 1223 Diff. of lat. 1210 S. Long. Cape Formosa 6° 1 1' E. Long. St. Helena 5 45 "W. II 56 60 Diff. of long. 716 W. Diff. long. 716 Log. (+ 10) 12-854913 I Course 30° 21' Secant 0-064012 Mer. diff. lat. 1223 Log. 3-087426 | Diff. of lat. 1210 Log. 3-082785 Tang. 9-767487 Course S. 30° 21' W. Distance 1402 Log. 3' 146797 Mereator^s SaiUny. 185 Ex. 5. Required the course and distance from Bahia to Fernando Po. Lat. Bahia 13° i' S. Mer. parts 788 Lat. Fernando Po 3 48 N. Mer. parts 228 16 49 Mer. diff. lat. ioi6 60 Diff. of lat. 1009 N. Diff. long. 2835 Log. (-|- 10) 1 3*45255 3 Mer. diff. lat. 1016 Log. 3"oo6894 Long. Bahia 38''32'W. Long. Fernando Po 8 43 E. 47 15 60 Diff. of long. 2835 E. Tang. 10-445659 Course N. 70" 17' E. Course 70° 17' Diff. of lat. 1009 Secant 0-47 1895 Log. 3-003891 Distance 299 1 Log. 3'475786 Ex. 6. Required the course and distance from A to B. Lat. A Lat. B 44° 44' S. 55 55 N. 100 39 60 Mer. parts 3007 Mer. parts 4065 Mer. diff. lat. 7072 Long. A Long. B 148° 39' W. 44 44 E. Diff. of lat. 6039 N. Diff. long. 9997 Log. (-{- 10) 13-999870 Mer. diff. lat. 7072 Log. 3-849542 54° 43 Tang. 10-150328 210 446)11800(26 2880 2676 Course N. 54° 43' 26" W. 193 360 23 166 60 37 Course 54° 43' o" Parts for 26" Diff. of lat. 6039 Diff. of long. 9997 W. Secant 0-238358 77 Log. 3'7 80965 10457 Log. 4-019400 116 Distance 10457 nearly, 416)2840(7 Ex. 7. Required the course and distance from Cape East, New Zealand, to Cape Horn. Lat. Cape East 37° 42' S. Lat. Cape Horn 55 59 S. 18 17 60 Diff. of lat. 1097 S. Mer. parts 2445 Mer. parts 4072 Mer. diff. lat. 1627 Long. Cape East 178° 40' E. Long. Cape Horn 67 16 W. Diff. long. 6844 Log. (-4- 10) 13-835310 Mer. diff. lat. 1627 Log. 3-211388 76° 37' Tang. 10-623922 558 935)36400(39 2805 8350 8415 Course S. 76° 37' 39" E. 245 360 56 114 60 4 Course 76' 37' o' Parts for 39" Diff. of lat. 1097 Log Diff. of long. 6844 E. ' Secant 0-635515 Distance 4743*2 nearly 345 3-040207 3-676067 53 92)140(2 1 86 Mercator'i Sailing. Examples for Pkaotice. Required the course and distance from A to B in each of the following examples. LATITUDE. LONGITUDE. LATITUDE. LONGITUDE. f '■ A 38° 14' N. B 39 51 N. A B 2° 7'E. 4 18 E. ^'ii. A 35° B 18 14' S. 23 s. A B 75°3o'E. 12 2 E. ^ 2. A 49 53 N. B 48 28 N. A B 6 19 W. 5 3 W. *• 12. A 4 B 8 24 N. 48 S. A B 7 46 w. 13 8 E. / 3- A 53 18 N. B 57 58 N. A B 055 E. 7 3 E. y 13- A 57 B55 43 S. 35 S. A B 10 37 E. I 28 W. ^ 4- A 50 4 N. B 51 25 N. A B 5 42 W. 9 29 W. /14. A 55 B 50 40 N. 25 N. A B 2 25 W. 3 40 E. \^- A 64 30 N. B 60 40 N. A B 4 20 w. 10 E. ' 15- A 6 B 6 11 N. S. A B 80 15 W. 39 16 W. + 6. A 22 55 S. B 34 22 S. A B 43 9 E. 18 29 W. V16. A 55 B57 28 N. 58 N. A B I 9 E. 7 3 E. V 7- A 54 54 S. B 34 22 S. A 60 28 W. B 18 24 E. ^n- A 35 B 38 51 S. 52 N. A B 138 54 E. 165 53 W. i 8- A 45 15 N. B 47 10 N. A B 35 26 W. 32 15 W. r 18. Ax5 B15 30 N. 30 S. A B 176 34 E. 176 34 W. > 9- A 34 22 S. B 15 55 S. A B 18 29 E. 5 43 W. * 19- A 22 B33 22 S. 33 N. A B 122 22 W. Ill II E. * 10. A 49 57 N. B 36 58 N. A B 5 12 W. 25 12 w. > 20. A 17 B 20 N. N. A B 180 E. 161 E. 2 1 . Required the compass '- 1 i% f ^ Lat. A 33" 1 ^^rtl*- ,'^^; B 42 course and distance from A to B. [8'S. ;long. A 72° 0' W. ; var. 16° 0' E. 3 S. ; B 173 30 E. ; dev. 9 25 E. hf 258. To find the latitude and 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 LXXV. 1°. With given course and distance enter the Traverse Table and tahe out the corresponding true difi'erence of latitude. Rule LXYT, page 167, fro7n which and latitude from find latitude in, as in Rule XL VI, page 93, and then meridional difference of latitude, as in Rule XLV, page 92. 2°. — At the given course look in the column of the true difference of latitude for the meridional difference latitude; the corresponding departure ivill he the difference of longitude, from tvhich atid the longitude from find the longitude in, as in Rule XLIX, page 95. * The general method of solution by "meridional parts," is from the formula: — True diff. lat. z=: dist. X cos. course. . • . log. true diff". lat. = log. dist. -\- log. cos. course — 10. DiflF. long. =: mer. diflF. lat. X tang, course. . • . log. dif, long, z=. log. mer. diflf. lat. + log- tang, course — 10. Mercator^s SadUng. i go Examples. Ex. r. A ship from lat. S5° i' N., long, i^ 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. 2 1 7*0, and dep. iiG'o. 6,0)21,7 Lat. left 55° I'N.^ „ Mer. parts 3970-) D. lat. 3 37 S. I >: ^ Mer. parts 3607 j l> . 3° 37' ' ^ "^ Lat. in 5124N. [^1 Mer. diff. lat. 363 f* o | J '^ J mer. diff. lat. 181-5]^ The course 2^ points, and half mer. diff. lat. 181-5 (in longitude, after having sailed 194 nules to Long, in i 49 E. j ^ ^^^ Eastward (see Rule XLIX, page 95). Ex. 2. A ship from lat. 42° 36' S., long. 178° 43' E., sails S.E. f E., 299 miles ; find lat. in and long. in. Course 4f points, and dist. 299, give diff. lat. 178-1, dep. 240-2. 6,0)17,81 Lat. left 42°36'S.^^ Mer. parts 2830^ k;- T). lat. 2 58 S. I 5 • Mer. parts 3078 | J . 2° 58' \>< ^ I M g> Lat. in 45 34 S. f 1 1 Mer. diff. lat. 2)248 f = | J I 124J « Course 4I points, and half mer. diff. lat. 124 (in diff. lat. column), give in dep. column i67-i, which doubled is 334-9, the diff. long. 6,0)33,4-2 Long, left i78°43'E, 1 - D. long. 5 34 E. I G 5" 34 I « g 184 17 E. )> o o 360 o Long, in 175 43 W.J « 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 4i=>, the corresponding true diff. lat. is 207-5, or 3° 27-5, which being subtracted from 50° 48' N., the tat. hi is47°2o'-sN. ; taking out the mer. parts for 50° 48', and 47^ 2o''s, the mer. diff. lat. is found to be 317, to half which as a true diff. lat., and the course 14°, the dep. is 1378, 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. f S., until arrived at lat. 52° 15' N- Lat. from 50^ 30' N. Mer. parts 3521 Course 3} points, and mer. diff. lat, Lat. in 52 15 N. Mer. parts 3690 in ditf. lat. column, give in rfcp. column 125-4, w'hich is the diff. 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 for Practice. For examples for practice in this problem take those given in middle latitude sailing at page 182. 1 88 Mercator's Sailing. EEMAEKS ON MIDDLE LATITUDE AND MEROATOE'S SAILINGS. 25Q. "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 difference of longitude for each portion separately. The difference of longitude deduced by middle latitude sailing is too small : 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 i o" 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 ^ 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 in 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 tlae course 85° this proportion is 1 1 to 1 . 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 ; hence the precept more especially demands attention in high latitudes." — Roper's Practice of Navigation, pp. /03, 104. THE DAY'S WORK 260. This is the process o{ finding the ship^s place at woo;;— that is, its latitude 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 LXXVI * i". Correct each course for leeway, variation a7ul deviation (see Eules L to LY, pages 104 to in), 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. 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 in the Table as the first course. See No. 232, page 139. As the ship leaves the land, the hearing (by compass) 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 point, from whence to reckon the subsequent courses and distances. Thus, supposing for example a ship leaving the Tyne 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 present position of the ship, she whould have to sail in the opposite direction to the bearing of the lights, viz., E. by S., 20 miles. At the end of the day, the Day's Work gives us a change of the ship's place as referred to the landmark, and not the supposed position. For methods of determining the distance, see Raper's Practice of Navigation, 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, Eaper or Norie) the difference of latitude and departure to each course and distance (see Eule LXVI, page 167), and proceed to find the difference of latitude and departure made good as directed in Eule LXVIII, page 172, Traverse Sailing. 3°. Find the course and distance made good (see Eule LXVII, page 1 70). 4°. Find the latitude in by applying the difference of latitude to the latitude from (see Eule XLVI, page 93). If a departure has been taken, the difference 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 find no difficulty in working the Day's Work without referwice to them. 190 The Day's Wort Note. — When the 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 be used in preference to Mercator's (see Remarks in page 188). 5°. To find the difference of longitude. — By Middle Latitude Sailing. (a) Find the middle latitude as directed, Eule XL VII, page 93. (b) Next at the page of Traverse Table on which the degrees fat top or lottomj correspond to middle latitude, find the departure in a difference of latitude column, then the corresponding distance is the difference of longitude (see Rule LXXIII, 4°, page 180). 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 hy 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 LXX, page 177). 7°. To find the difference of longitude. — By Mercator''s Sailing. (a) Find meridional difference of latitude (see Rule XLV, page 92.) (b) Then with course a^id meridional difference of latitude (in a latitude column), find the corresponding departure, which is the difference of longitude (see Rule LXXV, page 186). (c) With the longitude left and difference of longitude find the longitude in (see Rule XLIX, page 95). When a departure has been taken the longitude left is that of the point of land ; other- wise that of yesterday. EXAMPLE I. H. Courses. K. -A- Winds. Lee- way. Devia- tion. Remarks, &c. pts. I S.S.E. 1 E. 4 2 East. 2 7» E. A point oi land* in 2 4 3 lat. 42° 12' S., long. 3 5 42° 58° W., bearing by 4 5 2 compass E. by N. 3 N. 5 4 dist. 21 miles. IShip's 6 N.N.E. 4 I East. 2.i 9°E. head S.S.E. | E. ; de- 7 3 8 viation as per log. 8 3 5 9 3 2 10 S.W. 1 w. 3 5 W.N.W. If 7rW. II 3 6 12 4 I 4 2 Variation 20° W. 2 N. iE. 4 3 W.N.W. 2I i»W. 3 4 4 4 4 5 5 s.s.w. 6 2 West. 1 2 2^''W. 6 6 4 7 6 2 8' 6 5 A current set W. S.W. 9 lO N. by W. A W. 6 2 West. i 10=" W. correct magnetic 26 5 7 miles from the time the II 5 3 departure was taken 12 5 4 to the end of the day. The Bafs Work. 191 The Departure Course. The opposite point to E. by N. h N. is "W. by S. 5 S., and the ship's head being S.S.E.1 E., the deviation is same as given in log. for S.S.E. \ E., viz., 7° E. W. by S. I S. = 6i pts. R. of S. Deviation 7 Variation 20 L or 73^ 8' R. of S. K; }.3 L. True course 60 8 R. of S. or 8. 60° W., distance 2 1 miles. This is inserted in the Traverse Table as 1st course. IS!! Course, S.S.E. % E. Leeway Dev. Var. ^ 11j. = 24 pts. L. 01 b. 2 „ R. 0^ „ L. of S. or 5° 38' L. of S. £{■3 ^- I 4-2 3 5-0 4 5'2 5 4'o 22-7 True Course 18 38 L. ofS. or S. 19° E., distance 22*7 miles. The distance 22'* 7, is found by adding up the hoiirly distances sailed, until the course is altered at 6 o'clock. Insert this course and distance as 2nd course. ind Course, N.N.E. The deviation for N.N.E. is 9° E. h. N.N.E. = 2 pts. E. of N. 6 Leeway i\ „ L. 7 — 8 o\.. L. of N. q orN. 2°49'E.ofN., L.ofN. Var. 20 L. 1 4' I 3-8 3-5 3*2 i4'6 True course 13 49 L. of N. or N. 14° W., distance i4"6 miles. The distance, 14''5, is found by adding up the hourly distances from 6 o'clock until the course is changed at 10 o'clock. ird Course, S.W. | "W. The deviation for S.W. | W. is 7§o W. S.W. I S. = 4^ pts. R. of S. H. Leeway L. R, of S. Dev. 7'30'Li Var. 20 o L / ' or 30° 56' R. of S. 30 3'5 3-6 4-0 4-2 15-3 True Course 3 26 R. of S. or S. 3° W., distance r5"3 miles. Distance, 15'"3, is found by adding up hourly dis- tances from 10 o'clock until 3. 4^7* Course, N. \ E. The deviation for N. \ E. is 1° W. N. i E. = \ pt. R. of N. H. Leeway R. 2I „ R. of N. or 30° 56' R. of N. Dev. 1° L. ) T ■XT T ^ 21 O L. Var. 20 L. True Course 9 56 R. of N. or N. \o° E., distance 13-2 miles. K. 4' 3 4'4 4'5 [3-2 ^th Course. S.S.W. The deviation for S.S.W. is i.\° W. S.S.W. = 2 pts. R. of S. H. Leeway | „ L. 5 Dev. Var. or 16" 53' 30' L. \ 20 o L 22 30 R. of S. R. of S. L. True Course 5° 37' L. of S. or S. 6° E., distance 25-3 miles. K. 6-2 6-4 6-2 6-5 25'3 eth Course N. by W. \ W. The dev. for N. by W. \ W. is 10° W. h. N. by W. ^ W. = i|L. ofN. 9 |R. 10 |L. ofN. 12 or 8° 26' L. of N. Dev. 10° L. I -r Var. 20 L. I 3 6-2 57 53 5"4 22"6 True course 38 26 L. of N. or, N. 38° W., distance 22'-6. Current CourseW.S.W. W.S.W. == 6 pts. R. of S. or 67° 30' R. of S. Deviation 20 o L. 47 30 R. of S. or, S. 48" W., distance 26' 192 Tht Bay's Work The corrected courses are written down to the nearest degree, and the work will stand as follows : — Courses. D18T. N. S. E. W. S. 60° w. - . - • 21 xo-5 i8-2 S. 19° E. - - - 1 22*7 21'5 7 "4 N. 14° W. . i4"6 14-2 3"5 S. 3° w. - . i5'3 i5'3 0-8 N. 10° E. - . - - 13-2 13-0 2*3 S. 6° E. - . - - 25-3 25-2 2-6 N. 38' W.- . 22-6 17-8 i3'9 S. 48° W. - - 26 17-4 19*3 45-0 899 45*o 12-3 55*7 12-3 44'9 43*4 Difference latitude 44"9"1 Departure 43'4/ Latitude left 42° 1 2' S. Diff. latitude 45 S. Latitude in 42 57 8. Sum 2)85 9 Middle lat. 42 34 Course S. 44' Mer. diff. lat, give in Table II (■Course S. 44° "W.* (Distance 62^ miles. . 6vi] give in S Meridional parts 2798") ^ 1^ Meridional parts 2859 | ^ g, p^l> ^!^ o ^ Mer. D. lat. 6i||l" Jrt * The course being less than 56°, the difference of longitude may he found both hy middle latitude and Mercator's method, T hie II /Difference of longitude 59-0 ^ t (in departure column). Mid. latitude 42 J" \ Dep. 43-4 (as d. lat.) J Longitude left 4.2° 58' W Diff. longitude o 59 W • rr vi TT fDifference of longitude cq' give m Table II | ^j^ ^j^^^^^^ column). Var. 42 R / -59 ^S ^■ True Course 2 38 L. of S. or S. 3° E., distance i2'"4. The corrected courses are -written down to the nearest degree and the work will stand as follows : — Courses. S. 56° W. S. 28» W. N. 53» W. 8. 44° E. . N. 78° W. N. 14° W. S. 3° E. . N. 71° W. DiST. 16 166 217 178 43*1* I2'4 22-5 N. 81 3'7 41-8 7-3 6o*9 9'3 8-9 14-7 12'4 51-6 i5'i 0-7 15-8 W. i3'3 7-8 io'7 17-4 io'4 21-3 80-9 iS-8 65- r y_^ N. 14° W., distance 43'* i. Course 14° Dist. 300 D. lat. 291-1 Dep. 72*6 „ 14 131 127-1 31-7 „ 14 431 418-2 104-3 Diff. lat. 9'-3 and dep- 65'-! being found to correspond in their columns, give course 'N. 82° W., distance 66 miles. The mid. lat. is high and between two whole degrees, therefore, we proceed thus: — Mid. lat. 62° as course (in Table II), and dep. 65-3 (nearest to 65-1) as did. lat., give in dist column 139 ; and mid. lat. 63*^ and dep. 649 (nearest to 65'- 1) as ditf. lat., give in dist. column 143 : whence it is evident that for 1° (or 60') of mid. lat., the di£F. long, increases 4' : we next make the proportion 60' : 22' : : 4 : a; 4 Lat. left DiflF. lat. Lat. in Snm 9 N. p,ji 62 27 N. ^|g 124 45 1 » Mid. lat. 62 22 J ^^ Mid. lat. 62° gives D. long. 1 39 Correction for 22' 1-4 Mid. lat. 62° 22' gives D. long. 140-4 Mid. lat. 62° 22' sec. 0-333658 Dep. 65'-! log. 1-813581 DiflF. long. 140-4 log. or 2° 20' -4 2-147239 6,0)8,8 1*4 the correction for 22' Long, left 85° 17' E. >■ « . DiflF. long. 2 20 W. j^ 2^ ( gJ Long, in 82 57 E. ; | ^ * To take out the diflf. lat. and dep. for course N. 14° W., dist. 41'- 3. (See Rule LXVI, (c) page 169. ^^- 196 3 he Bay^t Work. EXAMPLE III. H. COUHSES. K. iV Winds. Lee- way. Devia- tion. Remarks, &c. pts. I E. by S. 10 6 S. by E. i 4 19° E. A point of land in 2 II 4 lat. 47° 44' S., long. 3 II 2 179° 7' E., bearing by 4 II 4 compass N. by W.JW. 5 12 dist. II miles. (Ship's 6 E.iS. 12 S. by E. i E. 4 20° E. head E. by S.). Dev. 7 12 3 as per log. 8 12 4 9 12 10 12 4 II E.S.E. 13 4 South. f 18° E. 12 I 13 13 4 6 Variation 14° E. 2 14 3 14 3 4 N.E. by E. 13 8 N. by W. 1 2 19° E. 5 13 8 6 13 5 7 13 5 A current set the ship 8 13 4 N.W. by N., correct 9 S.E. i E. 12 5 S.S.W. i w. f 15° E. magnetici 3 miles from 10 12 2 the time the departure II N. by E. 2 4 E.N.E. si 2°E. was taken to the end 12 2 3 of the day. W. N. E. S. W. N. Departure Course. The point of land from which the depar- ture is taken bears N. by W. | W., and the opposite point to this bearing is S. by E. 5 E. The ship's head when departure was taken is E. by S., the deviation for which is 19'^ E. S. by E. i E = i\ pts. L. of S. Dev. Var. or 16° 53' L. of S. r:}33 ^- 16 7 R. of S. or S. 16° W., distance 11' is^ Course. The dev. for E. by S. is 19° E. H. K. E. by S. = 7 pts. L. of S. I 10-6 Leeway ^= J ,, L. 2 11*4 — 3 II-2 7i „ L.ofS. 4 11-4 — • <; 12 or 81° 34' L.ofS. Dev. 19° R."l ^^ ^ T) Var. 14 R.I 33 ° ^' 56-6 True course 48 34 L. of S. or S. 49° E., distance 56''6. The distance 55''6 is foiind by adding up the hourly distances until the course is changed at 6 o'clock. 7.nd Course. The dev. for E. J S. is 20° E. E.f S = Leeway = 75 pts. L. of S. 71 L. of S. Dev. 20° R or 87° 11' L.ofS. Var. 14 T? f 34 R.-I R.; R. 53 II L. of S. or S. 53° E., distance 6i'*i. 12*3 12*4 12 12*4 6ri 2,rd Course. The dev. for E.S.E. is 18° E. E.S.E. = 6 pts. L. of S. Leeway = J „ L. Dev. Var. i8» R. 14 R- I „ L. ofS. or 75° 56' L. of S. 32 o R. 43 56 L. of S. or S. 44° E., distance 68''7 i3'4 13-4 136 14 H-3 687 The Day^g Work 197 \th Course. The dev. for N.E. by E. is 19° E. N.E. byE. = 5 pts. R. ofN. Leeway = | ,, R. 5i R. of N. or 61° 53' R. of N. Dev. 19° R.-l „ _ T> Var. 14 RJ 33 ° ^• Sum exc. 90° 94 53 R. of N. Subt, from 180 o H. K. 4 13-8 5 i3-» 6 i3"5 7 13-5 8 13-4 True course 85 7 L. of S. or S. 85° E., distance 68'-o. 68-0 5 Var. 14 R.r9 ° K- True Course 27 15 L. ofS. or S. 27° E., distance 24'-7. I2'5 I2'2 24-7 6th Course. The dev. for N. by E. is 2° E. N. by E. =: i pt. R. of N. H. Leeway = 5f ,, L. 11 2-4 2"3 4r„ L.ofN. Dev. 2° R Var. 14 R or 53° 26' L. of N 4-7 16 o R. or :}_ 37 26 L. of N. N. 37° "W., distance 4'- 7 Current Course. N.W. by N. = 3 pts. L. of N. or 33° 45' L. of N. , Var. 14 o R. 19 45 L. of N. or N. 20" W., distance 13'. The corrected courses are written down to tlie nearest degree, and the work will stand thus : — Courses. S. 16° W S. 49° E. («) S.53°E-(*) S. 44° E. (c) S. 85° E s. 27° E. . . . ■. a^/O N. 37° W N. 20° W DiST. N. S. E. W. II io'6 3"o S6-6 611 37"i 368 42-7 48-8 68-7 68 49'4 5-9 47'7 67-7 24'5 47 3-8 21-8 ii-i /' 1 2-8 13 I2"2 i6-o 4*4 i6i-6 2i8-o IO*2 160 10-2 i45'6 207-8 To take out of the Traverse Table the courses marked («), {h), and {e). {a). Course S. 49° E,, distance 56'-6. Taking dist. 56''6 as ^66 we have Course. Dist. D. Lat. Dep. 49° 300 196-8 226-4 49 266 i74"5 200-8 49 566 37i"3 427*2 . - . S. 49° E. and dist. 56''6 gives diff. lat. 42-7 S., and dep. 37-1 E, 19^ The Day's Work. (b) Course S. 53° E., dist. 6ii. (Take dist. 6i-i as 6ii.) Course. Dist. D. Lat. Dep. 53° 300 1805 239-6 „ 300 180-5 239-6 II 6-6 88 53 611 367-6 488-0 . - . S. 53° E., dist. 6i-i gives diff. lat. 36-8 S., and dep. 48*8 E. (c) Course S. 44° E., dist. 68'-7. (Take dist. 68'-7 as 687.) Course. Dist. D. Lat. Dep. 44" 300 215-8 208-4 „ 300 2I5'8 208-4 „ 87 62-6 60-4 44 687 494-2 477-2 . • . Course S. 44° E., dist. 68'-7, gives difif. lat. 49'-4, and dep. 47'-?. Diff. lat. 145-6 and dep. 207-8, found to correspond in the columns, give course S. 55' E., and distance 254 miles (see Rule LXVII, page 170). Lat. left Difl. lat. Lat. in Sum Mid. lat. Mid. lat. Dep. D. long. 47 '44' S.^ 26 S. ! So P. I P.H 50 10 S. 'v,m'> 2)97 54 I J -3« sec. 0-182621 log. 2-317645 48 57 48° 57' 2078 3i6"4 log. 2-500266 Mid. lat. 49° as course in Table II, and half of the dep. 103-9 ('^® whole dep. being too large a number to be found in the Table) gives in distance column 158, which multi- plied by 2 (as only half the dep. was used in entering the Table) gives diff. long. 316 miles. or 5° 16 -4 Long, left D. long. 326' Sum exc. 180° Subt. from Long, in 175 37 W.J ^ EXAMPLE IV. H. Courses. K. tV Winds. Lee- Devia- Kemarks, &c. way. tion. I N.E. J E. 6 N.N.W. 4 1 61° E. A point of land in 2 6 2 lat. 47° 35' S., long. 3 5 6 179° 26' E., bearing by 4 6 4 compass S.E. | E., dist. 5 N.E.byE.iE. 5 7 S.E. i E. 2i i7r E. 15 miles. (Ship's head 6 5 8 N.E. ^ E.) Dev. as 7 5 9 per log. 8 5 9 9 S.E. 1 E. 12 6 N.E. by N. 11° E. 10 12 4 • II 12 3 12 I South. II 4 I 7 E.S.E. 2 2i°W. Variation 25° E. 2 4 8 3 4 9 4 5 N.W.bW.^W. 4 4 9 2 S.W. J W. If 1 8^° W. 6 4 3 A current set the ship 7 4 5 N.E. by E. f E., cor- 8 4 6 rect magnetic, 36^ 9 N.W. i W. 12 6 S.W.bW.^W. i 18° W. miles, from the time 10 12 5 the departure was II 12 4 taken to the end of 12 12 3 the day. 4 The Lay's Work. 199 Departure Course. The opposite point to S.E. | E. is N.W. J W. 4I pts. L. of N. or 53° 26' L. of N. Dev. i6°i5'E.\ „ ,, p Var. 25 R.| 41 ^5 R- True Course 12 11 L. ofN. or N. 12° W., distance 15'. 7.nd Courie, N.E. by E. \ E. 5J pts, R. of N. H. K. Leeway 2^ ,, L. 5 5-7 — 6 58 si „ R.ofN. 7 5-9 — 8 5-9 or 36° 34' R. of N. Dev. i7»45'R.^ xj 23-3 Var. 25 o R. ) 4^ 45 ■«• True Course 79 19 R.ofN. or N. 79° E., distance 2 3'" 3. 4^A Course, South. O pts. H. K. Leeway (port tack) 2 ,, R. i 47 — 2 4-8 2 „ R. of S. 3 4-9 — 4 4'9 or 22° 30' R. of S. True Course 45 15 R. of S. or S. 45° E., distance i9'*3. 6th Course, N.W. ^ W. 4^ pts. L. of N. H. K. Leeway J „ R. 9 12-6 — 10 125 4^ „ L. ofN. II 12-4 — 12 12-3 or 47°49' L. of N. Dev. i8°L. i jj 498 Var. 25 R. / 7 on. True Course 40 49 L. of N. or N. 41° W., distance 49'"8. Courses. N. 12° W S. 63° E N. 79° E S. i7°E.(a) S. 45''W N. 36° W N. 41° W. (J) East \st Course, N.E iE. 4^ pts R. of N. H. K. Leeway (port tack) 2\ „ E. I 6 — 2 6-2 6| „ R. of N. 3 S6 — 4 6-4 or75°56'R. ofN. Dev. i6°i5'R. 1 t, Var. 25 oR.) 4^ ^^ R- 24*2 Exceeds 90° 117 11 R.ofN. Subtract from 180 o True Course 62 49 L. of S. or S. 63° E., distance 24'*2. yd Course, S.E. | E. 4I pts. L. of S. (Leeway o) or 53° 26' L. of S. Dev. 11° R. I ^ -D Var. 25 R. ; 36 o R. True Course 17 26 L. of S. or S. 17° E., distance 48'"4. K. 12-6 I2'4 12-3 IIT 48-4 5th Course, N.W. by W. J W. 5i pts. L. of N. Leeway i| „ R. Vi „ L.ofN. or 42° 11' L. ofN. Dev. i8°3o'L. 1 <- ,„ -r Var. 25 R. 1 ^ 30 R- H. 5 6 7 8 4-2 4'3 4*5 4-6 176 True Course 35 41 L. of N. or N. 36' E., distance i-j'-S. Current Course, N.E. by E. f E. 5I pts. R. of N. Variation or 64°4i'R. ofN. 25 o R. True Course 89 41 R. of N. or East, distance 36'"5. DiST. N. S. E. W. 15 14-7 31 24-2 ii-o 21-6 23-3 4-5 22-9 48-4 46-3 14-2 193 137 13-7 17-6 14-2 10-4 49"8 37-6 32-7 36-5 36-5 71-0 71-0 95-2 59'9 71-0 59'9 35-3 200 The Lay's Work. For method of taking out the courses in the Traverse Table marked [a) and {b) respec- tively, see page 169 (4) and (c). 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. Altogether, the vessel has sailed 95' -2 towards the east on four courses, while she has made 59'-9 westing on the other four, leaving 35'"3 of progress towards the east: hence The Course is East, distance 35'-3 (see No. 141, page 89). 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 diff. of long, made good is to be found by Parallel Sailing, Rule LXX, page 177, thus:— £tS£) ) give, in Table II. { (i„ Slice^coLtn.) 1 79° 26' E. ^ 52-5 E. \^ Long, left Diff. long. Long, in 180 i8'5E. ^M « Subtract from 360 o "3 p. \a Long, in 179 41-5 W. J EXAMPLE V. -^ H. Courses. K. tV Winds. Lee- way. Deviation. Eemarks, &c. I E. by N. 12 4 S.E. by S. 1 4 i7r E. A point, Tynemouth 2 12 2 in lat. 55° i' N., long. 3 12 2 i" 25' W., bearing by 4 12 2 compass W. byN. \ N. 5 E.S.E. 10 6 South. I n¥ E. dist. 15 miles. (Ship's 6 10 5 head E. by N.) Dev. 7 10 4 as per log. 8 10 5 9 N.E. by E. 8 2 S.E. by E. 1 17^ E. 10 8 3 II 8 3 12 8 2 I S.S.E. 7 4 East. li s¥^- Variation 2iJ° W. 2 7 2 3 7 2 4 7 2 5 S.E. by S. 5 8 E. by N. 2 8|°E. 6 5 6 7 5 4 A current set the ship 8 5 2 S.S.W. i W. correct Q E.S.E. 5 4 N.E. 2i n^E. magnetic, 18 miles 7 10 4 6 from the time the de- II 4 5 parture was taken to 12 4 5 the end of the day. The Bmi's Work. 20I W. N. E. S. W. K Departure Course. Opposite to bearing E. by S. | S. 6^ pta. L. of S. or 73 8'L. ofS. Dev. 17° 45' R.\ ; j^ Var. 21 15 L.J ^ ^ 76 38 L. of S. or S. 77° E., distance 15'. ist Course, E. by N". 7 pts. E. of N. Leeway \ „ L. 6^ „ R. ofN. or 75° 56' R. of N. Dev. 17-^45' R.-| j^ Var. 21 15 L.J -* -^ 72 26 R. of N. or N. 72° E., distance 49'. ■ 2nd Course, E.S.E. 6 pts. L. of S. Leeway- J „ L. 6i „ L. ofS. or 73° 8' L. of S. Var. 21 ,5 L.J 7 +5 L- 80 53 L. of S. or S. 81° E., distance 42'. ^rd Course N.E. by E. 5 pts. R. of N. Leeway i „ L. 4 „ R. of N. or 45° R. of N. Dev. i7»i5'R.1 ^ Var. 21 15 L.J ^ 41 R. of N. or N. 41° E., distance 33'. 4.th Course, S.S.E. 2 pts. L. of S, Leeway i^ ,, R. I „ L. of S. or 8' 26' L. of S. Dev. 5°3o'R.-l j. ^. t Var. 21 15 L.J ^^ ^^-5 ^• 24 II L. of S. or S. 24° E., distance 29'. Leeway Sth Course, S.E. by S. 3 pts. L. of S. 2 » R. r „ L. of S. or ii°i5'L. of S. Dev. 8°3o'R.l ^^ -^ Var. 21 15 L.J ^^ 24 o L. of S. or S. 24° E., distance 22'. 6th Course, E.S.E. 6 pts. L. of S. Leeway 2J ,. R. 3f „ L.ofS. or 42° ii'L. of S. Dev. 13'^30'R.) , Var. 21 15 L.} 7 45 L. 49 56 L. of S. or S. 50° E., distance 19'. Current Course, S.S.W. 1 W. 2J pts. E. of S. or 28° 7' R. of S. Var. 21 15 L. 6 52 R. of S. or S. 7° W., distance 18'. DP 202 The. Day\ Work. Courses. R. 77° E. N. 72^ E. S. 8i- E. N. 41*^ E. S. 24° E. S. 24° E. S. 50° K S. 7° W. DiST. J5 '9 4.2 33 29 22 19 18 N. S. 1 3-4 , I5-I 6 6 24-9 1 265 ; 20*1 12-2 17-9 40-0 86-7 40'o 467 E. W. 14 6 46-6 4'"5 21'6 11-8 8-9 i4"6 1596 2"2 157 '4 Diff. lat. 46''7 \ ■ • m vi tt Departure 157-4) S^^^ ^° Table II I Course S. 73!° E. \ Distance 164^'. Lat. left DifF. lat. 55° I'N.^-g . 47 S. n.5^ Lat. in k bo 54 14 N. \^^ % 2)109 15 |g Mid. lat 54 37 To find Diff. long, {i) By Calculation. Lat. 54° 37' sec. 0-237288 Dep. 157-4 log. 2-197005 Diff. long. 271-8 log. 2-434293 To find Diff. long. (2) By Inspection. Mid. lat. 54° and dep. i57''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. i57'-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 — 268J ■=! 7' change in diff. long., thus: — 60 : 37 : : 7 : a; 7 6,0)25,9 4*3 Mid. lat. 54° and dep. 157 give D. Long. 268 Correction for 37' + 4'3 Mid. lat. 54° 37' and dep. 157-4 give 272-3 Long, left 1° 25' W. « D. long. 272' or 4 32 E. Long, in 3 7 E. The BoAfh Work. 203 Departure Cotirse. The opposite point to N.E. by N. is S W. by S. S.W. by S. = 3pts. R. ofS. or 33° 45' B. ofS. Dev. ii°L. ( g L. Var. 25 L. \ ■^ True Course 2 15 L. of S. or S. 2°E., distance 17 miles. 1st Course. ■W. byN. = 78'45'L. of N. Leeway o Dey. 11" L. ( f- Var. 25 L. I 3° o L. Sum exc. go° 115 o Subtract from 180 o True Course 65 o E. of S. or S. 65° W., distance 25 miles. 2nd Course, "W.S.W. ■W.S.W. =6 pts. E. ofS. Leeway (port tack) g ,, R. Dev. Var. L. ) 6i or 73= „ R. of S. R. of S. 34 L. 90 25 True Course 39 R. of S. or S. 390 W., distance 22 miles. ird Course, W.N.W. W.N.W. = opts. L. of N, Leeway (port tack) 1 „ R. 5 „ L. ofN. or 56° L. of N. Dev. 9° L. Var. 25 L. ( 34 L- True Course 90 L. ofN. or West, distance 19 miles. 4th Course, S.W. S.W. = 4 pts.R.ofS. Leeway (starb. tack) li ,, L. Dev. 6° L. Var. 25 L. 2i or 3i< 31 L „ R.ofS. R. of S. or South, distance 16 miles. 5th Course, S.W. by W. S.W. by W. =5 pts.R.ofS. Leeway (starb. tack) i| „ L. 3i „ R.ofS. or 37° R. of S. Dev. 8° L. Var. 25 L. 33 L- True Course 4 R. of S. or S. 4° W., distance 13 miles. 6th Course South. South = o pts. Leeway (starb. tack) 2J ,, L. Dev. o" Var. 25 L. z\ „ L.ofS. or 25° L. of S. 25 L. True Course 50 L. of S. or S. 50° E., distance 13 miles. Current Course, N.W. J W. N.W. fW. 53<'L. ofN. Variation 25 L. True Course 78 L. of N. or N. 78° W., distance 6 miles. EXAMPLE VI. H Courses. k' tV Winds. 1-1 ^ D i;viA- TION. Remarks, &c. pts. r W. by N. 6 3 3 E.S.E. I 1° W. A point, Lizard, in 2 6 lilt 49^58' N.,lonp;. 5^12' W., bearing by 3 ^ 4 compass N.E. by N., 4 6 distance 17 miles. 5 6 W.S.W. 5 6 5 S. 1 2 9°W. (Ship's head W. 6 N) Dev, as per log. 7 .9 .? 8 <; 4 9 W.N.W. .I S.W. I 9°W. ID 4 8 II 4 6 12 I S.W. 4 4 6 W.N.W. a 6»W. Variation 25° W. 2 4 3 4 2 4 3 8 <; S.W.iW. 3 6 N.W.iW. if 8°W. 6 3 4 7 3 A current set 8 3 (correct magnetic) q s. ■\ 3 W.S.W. 2T o-" N.W. JW., 6 miles, 3 3 from the time the departure was taken 3 to the end of the 12 3 2 day. Courses. DiST. 17 25 22 19 16 13 13 6 N. S. E. W. S. 2° E. . S. 65° W. S. 39'= W. West . . South. . S. 4" W. S. 50° E. N. 78°W. I "2 I'2 17-0 106 171 i6-o 13-0 8-4 82-1 1-2 809 0-6 I0"0 22-7 138 19*0 0-9 10-6 51-7 . IO-6 Diff. lat. 8o'-9 S. ) . . T, , 1 jj ( Course S. 32^° W. Departure 517 W. ( ^^^^ ^^ ^^^^^ ^^ \ Distance 96 miles. Mer. parts 48 37 N. Lat. left Diff. lat. Lat. in Sum Middle lat. 49 17 Course S. 32^° W. Mer. diflf. lat. 124 (D. lat. col.) 49° 58' N. \ ^ j5 1 21 S. * ^ "^ 2)98 35 b. ho 3471 Mer. parts 3347 Mer. diff. lat. 124 } gives in Table II | fi^i^^S^Jcolumn.) Mid. lat. 49° ) ■ . ,,, , I YT ) ^^^- long. 79' Dep 5i'-8(a8diff.lat) i ^^^* ^'^ ^^^^^ ^^ \ (In distance column.) Longitude left s^i2'W. \ -lA Diff. longitude 1 19 W. I .2M ^ Longitude in 6 31 W. / M g ^ Z04 The BoAf'ii Work. EXAMPLE VII. H. COUKSES. K. iV Winds. Lee- way. Devia- tion. Remarks, &c. pts. I N.W. 1 W. 3 4 N. by E. i E. i| i8»W. A point, Cape Swaine 2 3 6 in lat. 52"^ 15 N., long. 3 3 i28''3o'W., bearing by 4 E. IS. 4 2 N.N.E. 1 E. 2 15^ E. compass E. by N. 5 N. 5 4 2 dist. 16 miles. (Ship's 6 4 5 head W.X.W. ; devia- 7 4 5 tion as per log. 8 4 6 9 S. by E. 1 E. .-J 2 S.W. 2^ 6»E. lO 5 II 5 2 12 f 6 • I W. by N. 1 N. 3 6 S.W. 2|- i7rw. Variation 25° E. 2 3 4 3 3 6 4 3 4 5 S. IE. 4 6 S.W.byW.|W. 2i 3=E. 6 ■ 4 7 7 5 5 8 t 2 AcurrentsetE.JS.fS 9 N.W. 6 '? w.s.w. I^ 17° W. correct magnetic 20 lO 1 8 miles from the time the II 5 4 departure was taken 12 5 3 to the end of the day. C0UE8ES. S. 80° w. N. 63° W. S. 24" E. . S. 17° E. . N. 32' W. South . . N. 23° W. S. 45° E. . D18T. 16 10 22 21 14 2C 23 20 N". 4-5 2 :"2 37-6 201 20-I 14-1 77*1 376 39'5 8-9 61 14-1 29-1 W. 15-8 8-9 7 '4 9-0 41-1 29*1 DifiF. ot lat. 39"5, and dep. i2'o, give course S. 17" W., and distance 41 miles.* (Rule LXVII, page 170.) Lat. left 52^i,5'-oN.^ ^ Ditf. lat. — 39\5 S. I ,,- Lat. in 51 35-5 N. Mid. lat. 51 55 Mer. parts 3690 Mer. parts 3626 M. diff. lat. 64 • The course being less than 56°, the diff. of long, may b9 found both by Middle L«titud« aad M«ra«toir't gailiog. The course S. 17° W., and mer. diff. lat. 64 in lat. column, give dep. 19' '6, which is the required diff. of long., see Rule LXXV, page 186. Or mid. lat. 52° as course, and dep. i2'-o, give diff. of long. 1 9 1' in dist. column, Rule LXXin (4°), page 180. Long. Cape Swaine 128° 30' W. Diff. of long. 4- 20 W. Loii§. ia i»% je W. / I t«5 . M in h3 on S) to The Ba/y's Work. 205 i^ EXAMPLE YTII. H, Courses. K. A Winds. Lee- way. Dktia- TION. Remarks, Acc. pt8. r ^.iW. <; 4 W. by N. 4 2°W. A point of land in , 2 ,') 4 lat. 54-^ 30' N., long. 1 3 <; 3 60° 0' W., bearing by 1 4 <; compass W. by N . 1 N . 5 S.W. 1 S. 2 1 W. by N. 2* 6»W. 14 miles. (Sbip'sbead , 6 2 <) N.N.W.) Deviation i 7 2 7 6-W. i 8 N.N.E. 1 E. 3 6 N.W. i N. 4 4-E. 1 9 3 5 1 10 3 4 1 II 3 7 12 3 X I W. ^ N. 4 2 N.N.W. H 11° W. Variation 42° West. 2 4 4 3 4 5 4 4 6 5 S.E. ^ E. 6 S S.byW.^W. 1 8=E. 6 6 4 7 6 A current set the 8 <; 8 8hip(correct magnetic) 9 S.iW. 2 4 S.E. by E. 3 i°W. N.W. 1 W., 23 miles, 10 2 3 from the time the II 2 3 departure was taken 12 2 to the end of the day. Courses. N. 59° E. N. 30° W. S. 40° E. . N. 18° E. S. 17° W. East . . S. 4°E. . S. 87° W. DiST. 14 2I'I 7*7 18 17-7 24-7 9 23 N. 7-2 i8-3 17-1 42'6 33"o 9-6 S. 5"9 169 90 1*2 33"o 5-6 24-7 0-6 47'9 38-8 9'i W. io*6 23*0 38-8 DifF. lat. 9-6, dep. 9'i, give course N. 43° E., dist. 13 mUes.* (Rule LXVil, page 170). Lat. left 54° 30' N . ^ ^ Diff. lat. 10 N. ] " S > Lat. in 54 40 N. ! »^ I. 2)109 10 I IP Mid. lat. 54 35 J ^^ Mer. parts 3916^ I >■ ■ Mer. parts 3933 (s g" 17J • The diff. of lon^. may be found both by Middle Latitude and liferoator'8 method — tke oouree being l»u tUaa 56*. Course 43°, and mer. difiF. lat. 17, give in dep. column the diff. of long. 16 miles (Rule LXXV, page 186); or mid. lat. 54^° as course, and dep. 9" I in diff. lat. column, give in dist. column 16, the diff. long. (Rule LXXTII, 4% page 180). Long, left 60° o' W. ^ « Diff. long. 16 E. / ^ « L«Bg. ia 59 44 W. / ^ ^ 2o6 The Lay's Work. y EXAMPLE IX. H. Courses. K. tV "Winds. Lee- Devia- Remarks, &c. way. tion, I A point of land in 2 N.N.W. 8 N.E. I 12° W. Int. 0" 15' S.. long. 3 8 4 170'' 44' E.. bearing by 4 7 6 compass N . E. 1 E. dist. 5 E.S.E. 6 2 N.E. li 13^ E. 17 miles. (Ship's head 6 5 8 N.N.W.) Dev. as 7 5 4 per log. 8 5 2 9 5 4 lO E. by N. 3 4 N. by E. 2 i8°E. II 4 12 4 t Variation 8-^ E. I 5 8 2 S 4 3 N.byW.f W. 5 3 N.E. 4 I of W. 4 .S 7 5 6 6 7 7 6 A current set tbe ship 8 N.W. \ N. i; 6 N.X.E. It 17' W. N. by E., correct mag- 9 <; 4 netic, 23 miles, from lO <; 7 the time the depar- II s 3 ture was taken to the 12 5 4 end of the day. Courses. DiST. S. 47" W ' 17 N. 38= W 24 S. 29° E I 28 S. 53° E I 23 N. 39° W 30 N. 7i°W 27-4 N. 19° E 23 N. 23*3 8-9 21-7 II-6 24'5 13-8 49'9 E. 13-6 18-4 7-5 12-4 14-8 18-9 25-9 39'5 720 39 5 32-5 Diff. lat. 22'-9 Dep. 32-5 Lat. left Diff. lat. 0° 15' S. \ 5 . Lat. in 8 N.j|S. rr,V« ,-r, ToVI^ TT I COUrSB N. C C° W. give in 1 able 11. ', ,. ^ -'-' , ° \ distance 40 . Long, left Diff. long. 170= 44' E. Long, in 170 11-5 E. 32-5W. (^c< o to Since the latitude left and latitude in are of contrary names, the ship has sailed near the equator, and the departure itself may be taken as the difference of longitude. (See No. 251, paga 178, and 5° (i) note, page 190.) The Boy'K WorL 207 EXAMPLE X. Vr H. Courses. K tV Winds. Lee- Devia- Remarks, &c. way. tion. pts. I N.N.E. q <; Eust. 4 8° E. A point of land in 2 9 <; lat. 4-5° 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. ^1 i5i° E. dist. 13 miles. (Ship's 6 4 4 head N.N.E.) Dev. 7 8 9 5 as per log. E.S.E. 5 6 South. 2 iS^E. 10 7 II 6 4 12 I W. by N. i N. 5-' 6 6 S.W. i S. ^h i5|° W. Variation 25° W. 2 5 6 3 5 6 4 4 5 S.S.E. 6 s.w. h 5rE. 6 6 2 7 6 4 A current set the ship 8 7 N. by W., correct mag- 9 N.N.W. 6 1; West. I 12° W. netic, 21 miles, from 10 6 4 the time the departure II 7 was taken to the end 12 7 2 of the day. Courses. N. 17° E North N. 27° E N. 79° E S. 83^ W S. 48° E N. 48='W N. 36° W Dist. 13 N. S. E. W. 12-4 3-8 38 38-0 18 i6-o 8-2 25 4-8 24"5 23 2-8 22-8 25*2 16-9 i8-7 27-1 i8-i 20-1 21 17-0 io6-3 12-3 19-7 55-2 55-2 19-7 5S'2 86-6 Course, North, distance 86''6. The Traverse Table being completed, the sum of the northings is io6''3, while that of the southings is i9''7. The departure in the east column amounts to 55''2, and that in the west column, also, to 55''2 ; but as the east and west departures destroy one another, there is no reaulting 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 distance sailed is equal to the diflF. lat., viz., 86''6. This is according to No. 141, page 89. Latitude left 43° 47' N, Diff. latitude i 27 N Latitude in 45 14 N. To find the Latitude and Longitude in. Longitude left 7° 51' W. Longitude in 751 W. X 208 Th- T)ay\ Work. Examples foe Praotioe. EXAMPLE I. H. Courses. K. 1 Winds. Lee- way. Deviation. Remarks, &c. I S.E. by S. 12 <; E. by N. 1 3°E. Apointjat. 35°i5'N. 2 12 <; long. 75" 30' W., bear- 3 12 6 ing by compass W. by 4 12 6 N., dist. 19 miles. ? East. q 8 N.N.E. I 23° E. (Ship's head S.E. byS. 6 9 4 Dev. as per log. 7 9 7 8 9 4 9 N.E. lO 4 N.N.W. * 17° E. lO lO 6 11 lO 4 12 lO 4 Variation 15" W. I North. II W.N.W. i 4°E. 2 lO 4 3 9 8 4 lO 4 5 N.N.E. II East. i 13" E. 6 lO 4 7 lO 4 A current set the ship 8 lO 4 N.E. by E. correct 9 E.N.E. q 2 North. I i8i'' E. magnetic, 52 miles lO 8 8 from the time the de- II q 4 parture was taken to 12 9 3 the end of the day. EXAMPLE II. H. Courses. K. iV Winds. Lee- way. Deviation. Kemaeks, &c. I East. q 4 S.S.E. pts. 16° E. ■ A point, Flambro' 2 q 6 Head, lat. 54° 7' N., 3 9 long. o°5' W., bearing 4 S.E. by E. 10 4 S. bv W. i 12-E. by compass N. W. bW. 5 10 2 dist. 17 miles. (Ship's 6 10 4 head E.S.E.) Devia- 7 E. |S. 6 7 S. by E. i E. xi 15° E. tion 13° E. 8 6 6 9 6 7 10 S. by W. S S.E. bv E. 2 0° II 4 8 12 4 6 1 4 6 Variation 25" W. 2 South. 4 4 E.S.E. H 2°E. 3 4 4 4 4 2 5 S.E. by S. 3 5 E. by N. 4 8°E. 6 3 <; 7 3 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 11 3 taken to the end of 12 3 the day. The Lay's Work. 209 EXAMPLE IIL H. COUKBES. K. 1 IS Winds. Lee- way. Detia- TION. Remakks, &c. pts. I w.s.w. 10 6 N.W. ^ II' W. A point, lat. 37° 3 N. 2 10 4 long. 9° W., bearing 3 10 <; by compass E.i N.^ N. 4 10 <; dist. 14 miles. (Ship's 5 W. by N. 8 N. by W. li 17° w. head W,S.W.) De- 6 7 5 viation as per log. 7 8 9 7 7 12 5 N.N.W. 4 N.E. 1 5 12° W. 10 12 6 II 12 12 I North. 12 10 W.N.W. f 2°W. Variation 22° W. 2 10 6 3 10 4 4 10 5 S.W. 6 2 W.N.W. i| 6°W. 6 6 6 7 6 2 A current set (cor- 8 6 rect magnetic W. b N. 9 W. by S. 7 4 S. by W. ^i 14° w. 30 milesfrom the time 10 7 4 the departure was II 8 taken to the end of 12 8 2 the day. EXAMPLE IV. H. 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 COTJBSES. K. t\t S.W. \ w. N. |E. S. by E. ^ E. W. by S. E.N.E. S.S.W. i w. Winds. 6 5 6 ! 2 6 I 6 6 I 7 Lee- way. 2| pts. S. by E. i E. , 2 E.N.E. S.W. ^ w. S. by W. S.E. S.E. Devia- TIOX. 6" W. 6°E. 8- W. i°E. 6°W. Remarks, &c. A point of land in lat. 46'^i2'S.,long.2°io'W. bearing bj- compass E. by S. ^ S., 20 miles. (Ship's head W. by compass.) Deviation 9°E. Variation 14° E. A current pet the ship S.W. A W. by compass 22^ miles these last 5 hours. T}ie Bwy's Work. ^^ EXAlitPLE V. H. Courses. K. tV Winds. Lee- way. pts. Devia- tion. Remarks, &c. 1 W.S.W. 9 4 N.W. \ io°W. Apoint,lat. 35°io'N. 2 9 3 long. 5° 36 W., bear- 3 I 9 4 ing by compassE. by S. 4 1 9 A (Ship's head N.N.E.) 5 North. II 4 W.N.W. \ 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. i 17° W. lO 8 3 II 8 4 12 8 4 I S.W.byS. II W. by N. 1 2 s°w. 2 II .S Variation 23° W. 3 11 4 4 II 4 5 W.S.W. 9 6 N.W. ^ 10° w. 6 9 5 7 S 9 9 4 East. 9 6 4 S.S.E. I 15° E. lO 6 4 A current set the ship II 6 (correct magnetic) 12 6 S.E. by E., 15 miles. EXAMPLE VI. H. Courses. K. .V Winds. Lee- Devia- Eemabks, &c. way. tion. I N.N.W. i W. 3 5 N.E. pts. 2°W. Apoinf-.lat 29°59'N. 2 4 2 lonti.32'54' K..b< ariiig 3 4 3 bycompassN.N E.^E. 4 E.S.E. 2 7 N.E. 2 7°E. dist. 15 miles. (Ship's 5 3 head N.W. by W.) 6 3 3 Deviation 6'' W. 7 4 8 S. |E. 5 4 E.S.E. H 2°W. 9 5 10 5 5 II 4 5 12 4 6 I N.E. i N. 4 7 E.S.E. 4 8°E. Variation 25° W. 2 4 2 3 4 4 4 3 3 3 7 5 6 W. iN. 5 S.S.W. 1 W. i| 9° W. 7 4 3 Acurrrnf set tbe ship 8 3 6 (correct ma gnetic) 9 3 6 N'E., 30 milps, from 10 N. by E. 8 5 E. by N. i 6^E. the lime Iht- dnp irture 1 1 9 3 w,i8 ImIc 11 to the tiid 12 9 2 of the day. The Boat's Work. EXAMPLE VII. H. Courses. K. tV Winds. Lee- Devfa- Eemarks, &c. way. TION. pts. I N.X.W. lO West. 3 4 I'E. Apoint,lat. 44''2o'S., 2 9 4 loDj? i76°49'W., hear- 3 9 4 ini^ bv compass E. by 4 9 N. 1 N., di.slance i8 5 West. 8 4 N.N.W. I 8= W. milns. (iShip'fi head 6 8 4 N.N.W.) Deviation 7 8 4 as per log. 8 8 4 9 W. by S. II 6 N.W. by N. i 9" W. ro 12 2 II II 8 12 12 I S.S.W.iW. 6 3 West. I^ IO° w. Variation 15° E, 2 ^6 3 6 4 4 6 5 South. 9 3 E.S.E. ^ 8' W. 6 9 4 7 9 5 A current set the ship 8 9 4 (correct masjnetic) 9 l^.byE. ^E. 12 5 E. by S. i 9' W. N. by £., 18 miles, lO 12 6 from the time the de- 11 12 ,-) parture was taken to 12 12 * the end of the day. EXAMPLE Vni. H. COTJKSES. K. Winds. E.S.E. E.|N. E.|S. S.W. f w. S. by W. W.N.W. 12 J 12 •5 1 12 3 1 4 4 4 3 1 4 3 1 4- 8 8 8 3 4 5 6 8 5 - 3 S 3 S 3 3 3 2 5 3 5 3 5 3 5 2 4 2 4 2 4 2 4 2 N.E. N.N.E. S.S.E. S. by E. W. by S. North. Lee- way. pts. 2k Detia- TION. Remarks, &c. i^ 3t 13 E. 17° E. 13° E- 8° W. I?'- W. Apoint,lat.62''i8'N. long. 63° 17' W., bear- ing by compass W.N.W. (Ship's head E.S.E.), dist. 21 miles. Deviation as per log. Variation 60° W. A current set the ship (correct magnetic) E. Ijy S^. I S., 49 miles, from the time the de- parture vfas taken to the end ot the day. / 212 The DoAi's Wwk. EXAMPLE IX. ■ H. Courses. K. tV Winds. Lee- way. Devia- tion. Eemarks, &c. pfs. I South. <; E.S.E. 4 2''E. A point, l>it.59"'49' N. 2 4 8 loiif?. 43° 54 W., hear- 3 4 <; ing: It y compass 4 4 4 N.E. i N., distKiice 14 5 N.E. 1 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 2 9 S.S.W. i w. 5 5 S.E. ^ S. 4 i'W. lO 5 4 II 5 4 12 ,<; 4 I 2 E. ^S. 8 8 3 4 S. by E. \ E. * 17° E. Variation 53° W. 3 8 4 4 8 2 5 5 S.W. ^ S. 4 5 4 6 S.S.E. \ E. 2 5=W. 7 4 8 4 4 A current set the 9 S.E. \ S. 6 4 E.byN.JN. I 10° E. ship (correctmagnetic) lO 6 2 S.E. \ E., 1-7 knots II 6 per hour during the 12 6 3 whole of the day. EXAMPLE X. H. Courses. K, -At Winds. Lee- Devia- Remarks, &c. way. tion. pts. I S.W. \ W. 4 8 S. by E. ^k 6°W. A point of land, lat. 2 <; 2 36° 10 iS.,long. 110° 10' 3 ■? 2 W., bearing by com- 4 <; 3 pass E. by N., dist 14 5 W. by S. \ S. 4 3 S. by W. 4 10° W. miles. (Ship's head 6 4 3 S.W. \ W.) Devia- 7 4 3 tion as per log. 8 4 3 9 W.byN.|N. 6 S.W. 2 8^» W. 10 5 4 II 6 2 12 6 4 I N.W. J W. <; 6 W. by S. f S. 4 srw. 2 ,1 4 Variation 20° E. 3 5 5 4 ^ ,1 8 5 W. by S. 7 4 . S. ^ W. li 9= W. 6 7 4 7 8 2 8 8 2 9 S.W. <; 2 S. by E. 2^ 5°^. A current set the 10 4 ship the last 8 hours II 5 ' 7 E. \ S. (correct mag- 12 5 1 4 netic) 2 miles an hour. The Lay's Work. 213 ^ EXAMPLE XI. H. COUKSES. K. tV Winds. Lee- way. Devia- tion. Kemakks, &c. I 2 3 4 5 N.W. by W. North. 8 8 8 6 4 4 4 4 N. hy E. E.N.E. 4 I 4°W. 3-2. A point of land in lat. 55° 59' S.. Ions;. 67° 16' W., bearing by compaiis E.S.E., dist. 17 miles. (Shiji'shead 6 t 6 W.) Uev. 8^ W. 7 8 5 6 6 9 N.W. by N. 4 N.E. by N. 1 4 i°W. 10 II 2 12 I West. 4 S.S.W. \ 9° w. Variation 23° W. 2 6 3 12 4 4 5 N.N.E. 12 7 4 3 East. I 15° E. 6 7 8 9 S.S.E. 7 7 7 9 4 4 4 <; East. * 5°W. A current set tbe ship N. by W. correct mag- netic, 27 miles, from 10 9 <; the time the depar- II 9 4 ture was taken to the 12 9 4 end of the day. EXAMPLE Xn. H. CotJKSES. K. IT Winds. Lee- Devia- Eemakks, &c. 6 3 way. tion. I W.N.W. S.W. pts. I 18° W. A point, Butt of Lewis 2 6 3 inlat.58'^29'N.,loni,'. 3 6 2 6" 12' W., bearing by 4 6 2 compass S.E. by S., 5 S.W. by W. 6 2 N.W. by W. i\ 9° W. dist. 15 miles. (Ship's 6 <> 8 head We^t) Devia- 7 <; 6 tion 16° W. 8 5 4 9 W. by N. 5 2 N. by W. 4 17° W. 10 4 8 II 4 6 12 4 4 I N. by W. 3 2 W. by N. H 7° w. Variation 31° W. 2 2 6 3 3 2 4 W. by S. 5 N.W. by N. If 14° w. 5 4 6 6 4 4 7 ') A current 8et(correct 8 W.N.W. <) 3 North. i\ 18° w. magnetic) E S.E.. 9 9 s 3 miles, from the time 10 <; 4 the departure was II North. 6 <; W.N.W- h 2°W. taken to the end of 12 6 5 the day. 214 The DoAfh Work. EXAMPLE XIII. H. Courses. K. l\ Winds. Lee- Devia- Remarks, &c. way. tion. pts. I S.E. by E. 3 2 S. hy W. 2h 8' E. A point of land in 2 3 4 lat 62° 9' S., long. 3 3 4 140° 17 E.,bearin^bv 4 W. by S. .S S. by W. If 11° W. compass S.S.W. ^ W. 5 4 4 dist. 25 miles. (Ship's 6 4 6 head S.E. i E.) Dev. 7 S.S.W. 4 5 West. 2 3°W. 70 E. 8 4 3 9 4 2 10 N.N.W. 3 6 West. 4 5°W. II 3 4 12 3 2 8 4 I 3 N.N.E. i E. 2 East. 3^ 5°E. Variation 31° E. 3 2 2 4 I 8 5 I 6 6 S.S.E. 2 4 East. 3 5='E. 7 2 2 8 2 4 9 lO East. I 6 N.N.E. 3l 11° E. I 4 A currpnt set by com- II N.W. I <; N.N.E. 4 8° W. pass S. 1 W., 3 miles 12 I 5 an hour for 7^ hours. EXAMPLE XIV. H. COUKSES. K. ^ Winds. Lee- way. Devia- TIO.V. Remarks, &c. pts. r S.S.W. i w. 3 6 West. H 11° W. A point, lat. jo'20' S. 2 3 <; long. 2o''io'E..bearinf^ 4 by compass N.E. ^ N. 4 5 6 4 2 dist. 15 miles. Devia- S.S.W. 6 4 W. |N. If 10° W. tion 25° E. (Ship's 6 1 head E. ^ N.) 7 6 4 8 6 3 9 N.N.W. ^ W. 6 W.^S. 3 13° w. 10 6 2 II 6 12 1 8 I E. by S. f S. 5 4 S. |E. 2i 21° E. Variation 2| pts. W. 2 3 4 5 5 6 5 5 * If/f 5 S.W. by W. 5 4 Ditto. 2^ 18° W. 6 5 4 7 8 6 3 6 A current set theship 9 10 S.^E. 10 6 West. i°W. by compass (correct 1 1 4 mngnetic) E.S.E , 2 1 1 12 4 miles an hour during 12 12 6 the whole day. The Bay's Wm-k. 215 EXAMPLE XV. H. Courses. K. 1 "10 Winds. Lee- way. Devia- tion. Remarks, &c. pts. I S. by E. \ E. 12 E.^S. 1 4 5"E. A point of land in 2 I I 6 lat. 46^ 26 S., long. 3 12 2 i76'44'W.,bearingby 4 IZ 3 compass E. by N. ^ N. 5 S. by W. P 2 S.E. byE. 1 26° E. dist. 23 miles. (Ship's 6 P head S. by E. \ E.) 7 9 6 Deviation as per log. 8 9 4 9 S.S.W. k W. 7 5 W. ^i ^'W. lO 7 5 II 7 4 12 7 6 I W. by S. 1 1 3 N.W. by N. 1 a i6rE. Variation 15° E. 2 lO 8 3 lO 8 4 lO 6 5 W. q 8 N.N.W. I 17° E. 6 q 6 A current set the 7 9 4 ship N.W. 1 W. (cor- g q 6 rect magnetic) i\ 9 N.N.W. q 3 W. f 20j» W. knots per hour, from 10 q 4 the time the departure II q 6 was taken to the end 12 9 5 of the day. EXAMPLE XVI. H. Courses. K. tV Winds. Lee- Devia- Remarks, &c. way. tion. pts. I E. 1 J^. 4 8 S.S.E. i E. ik 12° E. A point of land in 2 ■) 2 lat. 46-" 37 N., long. 3 <) 2 53" 30' VV., bearing l.y 4 4 8 compass S.W. f W. 5 N. by E. f E. 7 E. iN. h zo'^E. dist. 19 miles. (Ship's 6 7 6 head E. ^ N.) Devia- 7 8 8 tion as per log. 8 9 6 9 S.E. ^ S. 10 N.E.byE.§E. \ 9°W. 10 9 6 II 9 4 12 10 Variation 31' West. I W.byN. iN. 9 S.W. i W. k 6°W. 2 8 6 3 9 4 8 4 5 N.E. 1 E. 10 4 N.N.W. X. 20° E. 6 10 6 A current set the 7 q 8 ship(('(jrrect magnetic) 8 9 2 E. by N. 1 N., 17 9 S.E.byE.fE. 8 ■; N.E. \ E. i 2° W. miles, from ihe time 10 1 1 8 7 5 8 the departure was taken to the end of 12 7 2 the day. 2l6 Tilt Day'K JFork. f EXAMPLE XVII. H. Courses. W.N.W. W. |S. West. N. |E. North. N.W. i: W. K. tV n 4 13 '3 6 10 9 6 9 4 9 5 9 8 8 8 8 5 4 4 4 6 9 5 9 9 4 6 9 4 10 10 5 6 II 10 4 7 4 7 7 4 7 4 Winds. S.E. s.s.w. N.N.W. N.W. by W. W.N.W. N.byE.^E. Lee- way. pts. Devia- tion. 23j° W. 23° W. 23r w. of E. 20° W. Eemarks, &c. Apoint,lnt.5i^8|'N. lon<^. 123' E., hearing by compHss IS.W. | 8., dist. 25 miles. cShip's head W.N.W. Dev. as per log. Variation 25° W. A current set the ship N.W. by W. I W., correct magnetic, 32 miles from the time the df^parture was taken to the end of the day. EXAMPLE XVIII. H. Courses. K. tV Winds. Lee- way. Devia- tion. Remarks, &c. pts. I E.iS. 12 4 North. 20° E. A point, lat. 37=45'N. 2 13 long. 15° 0' E., bear- 3 13 6 ing by compass W.|N. 4 14 dist. 42 miles. (Ship's 5 E.fS. 12 N.N.E. ^ E. i 17" E. head E. ;| S. Devia- 6 II 4 as per log. 7 II 6 g 12 9 S.S.E. 5 4 East. 2i 2°E. 10 5 5 11 S 6 12 <; <; I N.N.E. 2 4 East. 3i 11° E. Variation 14"" W. 2 2 3 I 6 4 2 5 E. ^S. 9 5 N.N.E. |E. i i7»E. 6 9 5 7 9 8 q 9 E.S.E. 8 4 N.E. * 14° E. A current set E.S.E. 10 8 6 (correct magnetic) II 8 s 2| miles per hour for 12 9 the last 21 hours. Thi Dwy's Work. 217 Y- + No. Courses. DiST. Winds. Lee- way. Devia- tion. Remarks, tfec. 19. S. iW. 9 S.E. by E. pts. ■1 I'W. A point, in lat. 34° 50' S., E.N.E. 1 1 SE 18° E. long. 20' 1' E., bearing by S.S.Br 19 East. 60E. compass N. J "W., distance 15 miles. Ship's head E.S.E. 13 N.E. 4 13° E. S. by E. J E. Dev. 5° E. S.E. 26 E.N.E. 1 10° E. Variation 28° "West. E.N.E^ 7 N. 2i 18° E. ^V current set by compas.s N.W.bvN. 10 N.E. by N. I 16° W. W. by S., 2^ miles an hour S.E. i E. N.E. 22 13 N.E. by E. N.N.W. * io»E. 15° E. from the time the departure was taken to the end of the of the day. 20. S.W. 10-7 ■ W.N.W. !• 6'W. A point, in lat. 25° 39' S., 1 W.N.W. 8-7 S.W. Ij 10° W. long. 45'^ 7' E., bearing by W. by S. 27-9 S.by W. 11° W. compass N.E. by E. J E., distance 17 miles. (Ship's S.E. by S. 28 S.W. by S. i 5^E. head S.W.) Dev. a-* per log. S.W. by W. E. by S. 8-7 122 S. by E. S. by E. 15 8° W. 10° E. Variation 22° "West. S.S.W. \ W. S. bv E. 11-4 S.E. 4 S. S.W. by W. 4 4" W. A current set bv compass ri-2 4 2^E. S.W. I "W., 35 miles, from the time the departure was S.W.' ^ S. 6-7 W. by N. ^ N. 2 6" W. taken to the end of the day. 21. East. 529 S.S.E. i 11° E. A point. Cape East, in lat. Soutb. IO-5 23'4 E.S.E. 3 1° E. 37° 42' S., long. 178'' 40' E,, 1 N.N.E. East. 4 4»E. bearing by compass "W. J S. dist.23miles. (Ship's h«ad S.E. ^ E. 12 N.E.byE.iE. 2 70 E. East.) Dev. as per log. E.S.E. 25'I N.E. I 8^E. Variation 14° East. E. by N. \ N. 13 N. by E. 2i n°E. A current set by compass East. 22-2 N.N.E. li 11° E. E. by N.^N., 48 miles, from E.^N. 37-6 S.E. by S. 4 1 10' E. the time the departure was taken to the end of the day. 22. W.N.W. 30 North. , 10° W. A point, lat. 56° 27' S., S.W. by W. N. by E. A E. i4'8 i6-2 N.W. bv W. N.W. \ W. li 8° W. long. 68= 3 7' W. , bearing by compass E. J S., distance 15 miles. (Ship's head N.W. 166 N.N.E. \ 8^ W. "W.N."W.) Dev. as per log. , N.JW. 8-3 W.N.W. 2^ 2°W. Variation 22^° East. ^ A current set the ship < S.W. \ w. II W.N.W. 3 7°W. W.N.W. S.W. by W. 12-4 30 S.W. N.W. by W. 2; 10° W. I 8° W. S.S."W. I W. {con-ect mag- netic), 19 miles, from the time the departure was 1 N.byW.iW. 25-9 N.E. 1 E. ■' 4= W. taken to the end of the day. A point, in lat. 16= 5' S., 23. W. f N. 3S N.byW.AW. ^\ 12° w. tlong. 179"' 36' W., bearing W. 26 N.N.W. l| l| 13° w. by compass N.E., dist. N.N.W. 34 N.E. 5'W. ,15 miles. (Ship's head 'S."W. by "W.) Dev. 8° "W. S.W. bv W. 56 N.E. 8° W. Variation 37° West. S.S.W. 45 S.E. 1 4 s^-w. A current .set the ship S. by E. 26 S.W. by W. 4 2=E. during the day S.W. j S. (correct magnetic) 24 miles. ¥F PRELIMINARY RULES IN NAUTICAL ASTRONOMY. CIYIL AND ASTEONOMICAL DAY. 261. The Civil Day, or common method of reckoning time, begins at midnight, and ends the following midnight, the interval being divided into two periods of 1 2 hours each ; the fii-st twelve hours, from midnight to noon, are denoted by a.m. {ante meridian) ; the latter, from noon to midnight are styled P.M. (post meridian) ; thus we say 10 a.m. when an event occured at ID o'clock in the morning, and 10 p.m. when it occured at 10 o'clock in the evening. 262. The Astronomical Day begins at noon and ends at the following noon, and is later than the civil day by twelve hours. The hours are reckoned throughout, or continuously from o** to 24''. The distinction of a.m. and p.m. is not recognised in astronomical time. Thus, 1 1 o'clock in the forenoon of the second of January ja the civil reckoning of time corres- ponds to January i day 23 hours in the astronomical reckoning ; and 1 o'clock in the afternoon of the former to January 2 days i hour of the latter reckoning. 263. Since the civil day commences at the midnight preceding the noon which commences the astronomical day, it is evident that the civil mode of reckoning is always twelve houi's in advance of the civil reckoning, and heace we have the following Rule for converting civil into astronomical time. Given civil time at ship, to reduce it to astronomical time. / EULE LXXVII. ' 1°. If the civil time at ship be p.m., it will also be astronomical time, p.m. being omitted. 2°. If the civil time be a.m., add twelve to the hours and subtract one from the days of the month; also omit a.m. The result in each case is the Astronomical )ate. Examples. E.\. I. May loth. at 5'' 30™ p.m., civil time is 5*' 30"^ astronomical time of the same date ; because the loth astronomical day begins at noon of the loth civil day, and 5^ 30™ have elapsed since that noon. But 5'' 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 j** 30™ a.m. of the 10th is 17'' 30"' from noon of the 9th. Ex. 2. October 7th. at 3'' 20"" p.m., civil time, is October 7tli, at 3'' 20™ astronomical time. (See 1° of Rule LXXVII.) Ex. 3. October 7th, at 3'' 20*" a.m., civil date, is October 6' i5*> 20'" astronomical date; since 7"* less i^ is 6'*, and 12'' addedto 3*" 20" is 15** 20™. (See 2*^ of Rule). Ex. 4. January 3i8t, at 7'' 20™ p.m., civil time, is January 3i8t. at 7'' 20™ astronomical time. (Rule LXXVII, i°.) Longitude in Arc cmd Longitude in Time. 219 Ex. 5. February ist, at 6h 18"" a.m., civil date, is January 31'' i8'' 18'" astronomical date; since February i 15* Feb. 3rd, II 28 36 (3.) May •17th, 7 15 II Mar. 13th, 23 15 7 (5.) Sep. ist, 8 10 54 Aug. 31st, 20 10 54 (2.) Oct. 14th, iS» 17™ I3« Dec. 3rd, s 16 12 (4.) Mar. 3 1 St, 23 10 16 Mar. 2 1 St, 7 24 12 (6.) 1872, Jan. iSt, 9 .lo 41 1872, Dec. 31st, 22 48 56 LONGITUDE IN ARC AND LONGITUDE IN TIME. 265. The earth rotates uniformly on her axis once in twenty -four hours, and thus every spot on her sm'face 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 lohgitude in time aj-e easily convertable, for since 360^ is equivalent to 24*" (360 -;- 24 =z i5°)> 15° is equivalent to i*" ; 15' to i"", and 15" to i^; whence 1° is equivalent to 4" (i.e., the 15th part of i hour or 60™) i' „ 4« {i.e., the 15th part of i minute or 60') I" „ 4t {i.e., the 15th part of i second or 6ot^t 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 hour.s, minutes, and seconds, by ''«>»; and care should bo 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. 220 Longitude in Arc and Longitude in Time. To convert arc (or longitude) into time. EULE LXXIX. Multiply the degrees, minutes, Sfc-, ly 4, this turns the degrees (°) into minutes (") "/ time, minutes (') into seconds (') of time, and the seconds (") into thirds (*) of time ; or, in other words mark the resultiiicj figures thus : — Those under seconds (") thirds (*), those under minutes (') seconds ('), those under degrees (°) minutes (") and those to the left of the latter, hours (•"). Note. — Instead of thirds it is customary to use tenths of seconds, in which case the thirds must be reduced to tenths by dividing by 60, (See Eule XVIII, page 43). EXAHFLES. Ex. I. Convert 12° 18' 15' into time. Ex. 2. Convert 25° 15' 16' into time. 15" 25" 15' 16" 4 4 49'"i3» ot Four times 15" are 60", ■which contains 60 once and o over, write this remainder down \inder the seconds (") and mark it thirds (t) as directed in the rule, can-yinj; the 1 ; Again 4 times j8' are 72, and the 1' carried makes 73 ; 60 goes in 73 nnce, and 13 over ; write this remainder (13) under the minutes (') and call them seconds («) and carry the 1 ; Again, 4 times 12 are 48, and l carried makes 49 write this under 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. 6o)40fo •66 4ot-66 77" 2 10" 4 jh 8" 8^ 4ot or, s^ ^-n 8^-66 Ex. 5. What time is equivalent to ii°47'58"? IS" 47' 58 4 ih 41m J. 4t Four 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 1 is to be carried : Again, four times 15' are 60 and 1 carried makes 61 ; which contains 60 once and i over, write the remainder 1 under minutes ('), and carry 1 ; tour times 25 are loo and i earned gives 101, and 60 into 101 goes once and 41 remainder, which remainder being placed under degrees (') gives minutes (■») and the i carried on being placed to the left of the latter is marked hours C") ; whence i>'4i°' i" 4t is the time corresponding to the arc 25° 15' 16". Ex. 4. What time corresponds to 127° 32' 40 ? 127= 32' 40" 4 30™ 10^ 40t 6o)40fo •66 or, ih 3" ii«-86 or, 8*^ 30"" io*-66 Ex. 6. Convert 178° 45' 53" into time. 178" 45' 53" 4 6o)32t'oo •53 "■"SS" 3' 32t or, ijh 55™ 35-53 Examples for Pkactice. Heduce the following arcs into time : — 18° 54 ; 12" 40' 45" ; 137'^ 27' ; ^6"- 10' 45 " ; and 89° 16'. 67" 42' ; 76° 20' 30"; i^ 25'; 140° 32' 10" ; and 69'^ 29'. o" 58'"6; 49° 4' 20"; 0° 26'8; 14° 2' 30"; and 130° 19'. 9° 14' ; 163° 2' 48"; 0° 37' 4" ; 2° 18' 12"; and 170° 15'. 108° 37' ; 10° 27' 14" ; 2" 29' ; 84° 42' 30" ; and 0° 34^'. o" i3''5 ; 5^° 10' ""; 156° 52'; 178° 49' 45"; ^nd o"^ 4i'-; TO CONVERT TIME INTO LONGITUDE. It has been shown (No. 2b^, page 219,) that 4" of time are equivalent to 1° of arc ; hence it is evident that if wo 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 procebS. Longitude in Are and Longitude in Time. 221 EULE LXXX. Reduce the hours and minutes into minutes^ after which place the seconds, Sfe., then divide all by 4, and the quotient will be the degrees, minutes, Sfc., of the corres- ponding arc; or, in other words, after dividing by 4, mark the resulting ilgures thus : — Those under minutes (") degreess (°) ; those under seconds (*) minutes ('), those under thirds (*) seconds ("). Examples. Ex. I. Turn i'' 5'" i2» into arc. l" 5™ I2« 60 4)65>"i2'ot 16" 18' o" Multiply ii" by 60 add the minutes (5) and divide by 4, the quotient is i6» with reniaindei- 1. Multiply this remainder by 60, and to the product add the 12 SFcuuds ; the sura is 72 : Agaiu, the quotient of 72 divided by 4 is 18, which is minutes (') ; whence the arc corresponding to the time 1'' 5™ 12» is 16° 18'. Ex. 3. Reduce 6'' 24™ 43' into arc. 6h 24™ 43» 60 4)384"" 43' ot 96° 10' 45" Multiplying 6^ by 60, and adding the 24™ to fhe product, gives 384 as the sum ; the quotient of Ihis divided by 4 is ge'', with no remainder. 43'" divided by 4 gives quotient lo' witli remainder 3 : remainder 3 multiplied by 60 gives 180 which divided by 4 gives quotient 45" : therefore g6° lo' 45 ' is the arc which, corresponds to 6^ 24™ 45", Ex. 3. What arc corresponding to oh 47™ 36" ? 4)0'' 47"" 36" "°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. 4. What is the equivalent arc to 9" iJ"" 37' 60 4)565™ 37' o* 141 24 15 Ex. 5. Convert %^ 17™ 35»*5 into arc. Sh 17" 60 35"5 4>497"' 35' 3°* 30 124-23 52-5 Multiply the hoaMs (Sh) by 60, and adding the minutes (17™) to the product gives ^g?"; divide the result by 4 ; the quotient is 124°, with remainder i. Agam, miiltiply the remainder just obtamed (i") by 60, and to the product add the seconds of time, viz., 35» ; the sum is g5, thou divided by 4, the quotient is 23' (minutes of arc) with remainder 3. Next multi- plythis last remainder by 60, the product is 180, to which add the jot ; 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 -5 of seconds of arc. Ex. 6. Convert n'^ 39™ 5o''7 into arc. Ilh 39m ^o«-7 60 4)699" 50' 42t 42 174" 57 40-5 Multiply Ilh i)y 60 and to product 660 add jg", dividing the sum, viz., 6gg by 4 gives 174° with re- mainder 3 ; this remainder {3) multiplied by 60, 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 corresponding to 11'' 3g" ^o'■^ is i74°S7'4o"-5. Examples for Praotiob. Convert the following times into arc : I, ih 13m 52' 6. 9.. 49m 38B II. oh 2 1™ 30<'*9 16. oh 20"" 418 2. 3 52 4 7- 34 58-2 12 II 41 6-66 17- 8 36 56 3- 42 12 8. I 41 1-6 13 3 52 i8- 5 51 4. II *5 21 9- S 59* 4 14 9 56 J9- " 5^ 57 5- 4 29 5 lo. 8 17 6 15 52 20. I 52 GREENWICH DATE. EEDUCTION OF GREENWICH DATE. 266. 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 infoi'mation 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 in 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. To find the Greenwich date, the time at any other place and the longitude being given. EULE LXXXI. 1°. Express the ship time astronomically (Rule LXXVII, page 218). 2°. Convert the longitude into time (Rule LXXIX, page 220). 3°. In West longitude. — Kjyojongittde in time to ship time ; the sum, ^less than 24 hours, is the corresponding Greenwich date on the same day with the ship date ; //"greater than 24 hours, reject the 24 hours, and put the day one forward. 4°. In East longitude. — From ship astronomical time subtract longitude in time, if less than the hours, minutes, Sfc, of ship date ; the remainder is the corresponding Greenwich date of the same day as the ship date; ij the longitude in time he greater than the hours, minutes, Sfc, of ship astronomical, add 24 hours to the latter, and put the day one back before 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 ij east, subtract the longitude in time from 24 hours ; the remainder is the Greenwich date (apparent time) after noon of the preceding day. (a). From 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, i&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. Gremwich Date. 223 Note. — A bad habit prevails in writing dates, of separating the 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 gtb, at 4'' ic" 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. ^^ 4'' 10™ Longitude 32" 45' Long, in time -1-2 11 ' ' . 4 Greenwich date, Nov. 9 621 6,0)13,1 o Ex. 2, June 5th, at v*" 15" a.m., app. tilne at ship, longitude 140*^ 30' E.: find cor- responding Greenwich date. Ship date (A.T.) June 4'^ 19'' 15" Longitude in time — 922 Green- date (A.T.) June 4 9 53 Ex. 3. January 3rd, at 8^ 12"- p.m., mean time at ship, long. 50° 45' E. : find Green- wich date. Ship date (M.T.) Longitado in time Green, date (M.T.) January 3'! 8'' 12" — 3 23 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^ i7''35"4S' Longitude 122° 13' W. -\- 8 8 52 26 25 44 37 — 24 Green, date (A.T.) April 27 i 44 37 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 Longitude 85° 24' E. Green, date (M.T.) or 19 27 35 7 — 5 41 36 ^9 21 53 31 In example 4, the added longitude advances the day of the month. (This illustrate.? latter pnrt of 3^ of the Eule.) In example 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. 1873, January rst, 3^40'" 20* p.m., mean time at ship, long. 95° 7' E. : find the Greenwich date. Ship date (M.T.) 1873, Longitude 95° 7' E. Jan. I'' 311401120* — 6 20 28 Ex. 7. 1872, January ist, <)^ i™ a.m., meati time at ship, long. 107° 4' W. : find the Greenwich date. Ship date (M.T.) 1871, Dec. 31^ 2 1^ i™ o* Longitude 107* 4' W. -\- 7 816 Green, date (M.T.) 1872, Dec. 31 21 19 52 | Green, date (M.T.) 1872, Jan. i 4 9 16 Ex. 8. 1872, June 12th, 6'' 40'" a.m. app. time at ship, long. 42° i6' W. : find the Greenwich date. Ship date (A.T.) June i v^ i8h4o"i o^ Longitude 42° 16' W. 4- 2 49 4 Green, date (A.T.) June 11 21 29 4 Ex. 10. Ef^quired the Greenwich date when the sun is on the meridian of a jilace in long. 80" 44' E., on January 12th. The sun being on th3 meridian, it is apparent noon : hence Ship date (A.T.) Jan 12^ o*' o"^ o^ Longitude 80' 44' E. — 5 22 56 Green, date (A.T.) Jan. n 18 37 4 Ex.9. iS72> October ist, long. 2° W., the sun on meridian : required Greenwich date (app. time). Ship date (A.T.) October i^ o'l o™ Longitude 2" W, -\- 8 Green, date (A.T.) October 108 Ex. II. What is the Greenwich date when the sun is on the meridian of a place in lor.g. 155^ 19' W., on March 31st .^ Ship date (A.T.) March n^ o'' o™ qb Longitude 155° 19' W. 4- 10 21 16 Green, date (A.T.) March 31 10 21 16 In example 10, the hours, &c., of longitude to be subtracted are to be taken from a bonowed day, or-z^ hours, thus making the day of the month at Greenwich one less than at the place. (See 5° of Eule.) 224 Reductmt of Elements from Nautical Almanac. Ex. 12. 1872, February ist, long. 135° E.: find the Green\vich date when the sun is on the meridian. Ship date (A.T.) Longitude 1 35" E. Green, date (A.T.) February Ex. 13. 1873, January ist, the ship in long. 160° 30' E. : required the Greenwich date when the sun is on the meridian. Ship date (A.T.) 1873, Jan. i"* o^ o" Longitude 160^ 30' E. — 10 42 January 31 15 o ' Green, date (A.T.) 1872, Dec. 31 13 18 Examples for Practice. Eequired the Grreenwich date in each of the following examples :- — I 1876, January 6lh at 3'i4o'"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. 145 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 15 w. 6 November ist at 5 010 P.M. mean time, long. 114 30 E. 7 December ist at 8 5 A.M. mean time, long. 158 10 W. 8 July ist at 4 033 P.M. apparent time, long. 170 55 15 E. 9 August 4th at 6 31 32 P.M. apparent time, long. 100 17 30 E. 10 September ist at 8 29 I A.M. mean time, long. 148 47 30 W ir December 28th at 2 42 10 P.M. mean time, long. 50 40 E. 12 „• July 8th at 4 36 A.M. apparent time, long. 178 51 W 13 February ist at noon, apparent time, long. 153 40 E. 14 June ist at noon, apparent time. long. 83 50 E. 15 March 2nd at noon, apparent time, long. I 25 W 16 1877, January ist at noon, apparent time, long. 149 10 olE. REDUCTION OF ELEMENTS FROM NAUTICAL ALMANAC. The N'autieal Almanac^ or Astronomical Ephemeris contains the right ascen- sion, declination, &c., of the principal heavenly bodies for given equidistant 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 everj' hour. Before ire 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 LXXXI, page 222). Where this time is exactly one of the instants for which the required quantity is put down in the Ephemeris, nothing more ie 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 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 oiproportional\o^Sixiih.ms. * The French Ephemeris, Lo Connaissance des Temps, is computed for the meridian of Paris, the German Berliner A$tronamisc}Hs Jahrbucn for the meridian of Berlin. All these work* are published annually several years in advance. Reduction of Elements from Nautical Almanac. 225 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. 267. 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 differences in the Ephemeris are constant. This, however, is never the case ; 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 Right 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. 268. 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 Oreenwich. The difference of declination for one hour /"* Var. in i how'''' ) 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 hoiirly difference.* RULE LXXXII. 1°. Get a Oreenwich date hy means of ship time, expressed astronomically, and longitude (see Rule LXXXI, page 222), or by means of chronometer. To express the Greenwich time in hours and decimals of an hour. Annex a cipher 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 ^^ 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 44.) 2°. Take out 0/ Nautical Almanac the declination for the nearest noon to the given Greenwich date, noting whether the declination is increasing or decreasing ; and a little to the right place the " difference for i hour,'''' found in page I, N.A. (a). When Greenwich date is given in apparent time, use page I of the month, but call them I" when they amount to five, or above — thus 42""7 would be 43". (J). The tenths of seconds (") of declination may be rejected when less than five, but for mean time, use page II of the month. (c). 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 i^ ^^^ 408 would be called 2^ 36"". * This method of reducing the sun's declination is required to be used by the Board of Trade at the Local Marine^Board Examinations, GQ 226 Red/uction of Elements from Nautical Almanac. 3°. Multiply the ^^ Hourly Biff.^^ by 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 change of declination in the time from noon. 4°. Apply this correction to the declination for the nearest noon to the given time i.e., the declination of the same noon as that for which the "Var. in i hour" has been taken as follows : — (a) When the Bed. is increasing, the correction for the time elapsed since noon is additive, but the correction for the time that must elapse is subtractive. (b) When the Bed. in decreasing, the correction for the time elapsed since noon is subtractive, but the correction for the time that must elapse before noon is additive. The result is declination sought. Examples. Ex. I. Greenwich Date, Jan. lo* 6^ ; in this case take 22"-o8 the Diff. for i hour on the loth, which multiplied bj' 6 gives the correction of the Decl. for the 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"- 14 the "Var. in 1 hour" on the nth, which multiplied by the diflerence between 24^ and i^^ gives the correction of the Decl. for the i ith day — to be added because the Decl. is decreasing and we have multiplied by the number of hours that must elapse before noon the nth day. Ex. 3. Greenwich Date, April 2'' 6|h ; 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 and we have multiplied by 6| hours the time that has elapsed since noon, April 2nd. Ex. 4. Greenwich Date, April 2'' 17I'' ; in this case take 51" "^Si ^^^ " Var. in i hour'' on the 3rd ; which multiplied by 6| (the difference between 24'! and 175'') gives the correc- tion of the Decl. for the 3rd daj'^ — to be subtracted because the Decl. is increasing and we have multiplied by the number of hours that must elapse before noon, 3rd. ( c°). If the correction when subtractive exceeds the declination itself, suhstract the declination /row 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 X. in March, but S. in September. — (See example 3.) Method II. — By proportional logarithms. EULE LXXXin. 1°. Find a Oreenwich date. 2°. Take out of the Nautical Almanac the declination for the noon at Gfreev^ wich, and that following it. Mechiction of Elements from Nautical Almanac. 3°. When the decimations are o/like names, take their difference; 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 ( + ) before the daily variation ; but when the declination is decreasing, place the sign of subtrac- tion ( — ) before it. 4°. Under the daily variation place the hours and minutes of Greenwich time, and take from the table (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, Raper, or Norie XXXIII, minutes (') of declination, and hours of time (*>), are found at the top of the columns ; seconds (") of declination, and minutes {'") of time, at the side columns. 5°. Apply the proportional pa/rt to the declination at the ftrst 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. I. 1876, January 13th, at 3^ 54" sun's declination. 16^ P.M., app. time at ship, long. 30' 4' E. : find the Ship date (A.T.) January 13'' Longitude (30° 4' E.) in time — Green, date (A.T.) January 13'' 3''54"i6* Longitude 30° 4' 2 o 16 4 54m ■= 54 i'>54'" o' 60)54-0 6,0)12,0 16 or, 1^-9 ■9 2h o'ni6' Method I. Decl., page I, N.A., for January 13th, app. noon, is 21*^ 32' 34" S., decreasing, and Hourly diff. is Method II. Decl. app. noon, page I, N.A. Jan. 13th, 31*^32' 33-6 S. 14th, 21 22 15-2 8. H. diff., 13th, noon Green, time i*" 54™ =: 25"-24 i''-9 X 19 22716 2524 S., s. deer. Daily var. — 10 18 Green, time i''54'" log. 3674 log. liorj Correction — 49 13th, at noon 21 32 34 S. log. 1-4689 Correction — 47*956 21° 32' 24" - 48 21 31 46 Red. decl. 21 31 45 S. Decl., noon, 13th, Correction Bed. decl. The correction 49" is subtnicted from declination at noon, because the decli- nation is decreasing. Having found the Greenwich date, the STin'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 column headed " Var. m i hour" is found the change of declination for 1 hour past noon ; next observe that the decli- nation is decreasing, and make a note of it. Now, since the declination changes 25"'24 in 1 hour past noon how much does it change in the Greenwich time past noon, vii!., ii" 54"! ! First annex a cypher to the minutes (54™) and divide by 60 ; thus 60 is contained in 540 nine times and nothing over. To this we prefix the hour, and we then have the Greenwich time i'' 54"' = \^-g expressed in hours and decimals of an hour. (See Rvile XVIII, page 43). Set this under the hourly difl'. and then proceed as in multiplication of decimals, the resulting figures are 47956, but as we have two decimals in the multiplicand an is the time that must elapse before notn •zoth. We di^-ide the minutes of this last by 60 to get decimals of an hour, thus, 6 is contained in 21 three times and three over ; a cypher being annexed to the remainder 3 makes 30, then 6 is contained in 30 five times, hence we have -65 (see Rule XVIII, page 43), to this we prefix the hours (1) and we then have i^-^s- Kext multiply the hourly ditlcrcncc by this, and the residtiug figuxes arc 799740, then since we have two Heduction of Elements from NauUcal .Almanac. 229 decimal fignres in the multiplicand ('24). and two in the multiplier, (-35) in all four,/oMr figures must he marked off from the right hand, leaving 79, which being increased by 1 in coasequence of the first figure on the right of the decimal point exceeding 5, gives for the correction 80", which divided by 60 gives 1' 20". And since the declination at noon, 20th, is increasiiuj, it is evident that the declination at i^ 21'" before that noon will be less than at noon, and the correction 1' 20" is therefore to be subtracted ; whence the Reduced Declination is 0° 4* 33" N. (see No. 5° (a) of the Rule LXXXII). Ex. 4. 1876, February nth, at 8'' 54"^ 47^ p.m., app. time, long. 11° 4' W. : find the declination. Ship date (A.T.) February 11^ 8i^54'"47« Longitude ri" 4' Longitude in time ^- 44 16 4 Green, date (A.T.) February 11^ 9 39 3 44'ni6» 9'65 Decl. page I, N.A. Hourly diff., page I, N.A. Feb. nth at noon 14° 'j' ^2"S, fdecr.J Feb. nth at noon 49'" ^4 Corr. for g^ 39"" — 7 45 9*65 Red. decl. 13 59 58 S. 6,0)47,4-2010 Correction 7-54 269. To jfind the declination of the sun at the time of its transit over a given meridian. When the sun is on a meridian in West longitude, the Grreenwich apparent time is precisely equal to the longitude ; that is, the Grreenwich apparent time is ajter the noon of the same date with the ship date by a number of hours, equal to the longitude 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 hoiu:s, equal to the longitude in time. Hence, to obtain the sun's declination for apparent noon at any meridian we have. EULE LXXXrV. Take the declination from the Nautical Almanac (page I of the month) for Greenwich apparent noon of the same date 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. Examples. Ex. I. 1876, September loth, the sun on the meridian, long. 100^ 35' E. : required the sun's declination. Longitude 100° 35' E. . The longitude being Cf' /]2"' 2o» East, the Green. ^ A.T. is 61' 42™ before the noon of September 10th — the same date as the ship date. The decl. is taken 6,0)40,2 20 out of the Nautical Almanac, page I of the month ; also take out at the same time the hourly ditf. ; the ^t „ , work will stand thus ; — Sun's decl., page I, JJ.A. H.D., page I, N.A. Sept. I oth, noon 4''43' 8"N., «fe«'. Sept. loth noon 57" 04 Corr. for 6^' 42"" -j- 6 22 Time from noon loth, 6'' 42'" z=. 6-7 Red. decl. 6 49 30 N. 6,0)38,2 168 Aa tfce declination is rf«c/«««jf>(7, the declination at 6'' 42'" . — — _ before noon will be greater than that for noon. ' Correction fyzi ajo Reduction of Elements from Ncmtical Almanac. Ex. 2. 1876, June ist, long. 75° W. : Ex. 3. In the last question suppose the required the declination when the sun is on long. 75" E. the meridian Ship date 1 Long, in time -{- 5 Ship date (A.T.) June it* ©•' o"* Ship date (A.T.) June i<* o'' o" Long, in time — 5 o Green, date (A.T.) May 31 19 o Green, date (A.T.) June i 5 c H. diflF., noon, June ist 19 "-81 Time from noon X 5 6,0)9,905 Correction + i 39 Decl., June ist, noon 22 8 51 N., incr. Red. decl. 22 10 30 N. H. diif., June ist, noon 19"" 85 Time from noon, June ist 5 6,o)9.9'35 — I 39 Decl. June ist, noon 22 8 51 N. incr. Red. decl. 22 7 12 N. 270. Interpolation by second differences. — 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, neglecting 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 Grreenwich 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 which the correction ts to be computed. ill. Degree of dependence. — The sun's declination changes nearly i' an hour, or i" m i™, in March and September ; hence to insure it to i" in the extreme case, the Greenwich dat^^ u\n'. I'! hiu ' > i™. Rechiction of Elements from NoMtical Almanac. 23t Examples for Practice. Eequired the sun's declination in each of the following example? :— [These are preparatory to working Amplitudes, Azimuths, &c.] I 1876, January 5 th, 6>>23"32« A.M. app. time at ship long. 108° 7'W. 2 )> February 2nd, 390 P.M. app. time at ship long. 52 45 W. 3 >) March 31st, 6 2 12 P.M. app. time at ship long. 156 3 E. 4 » March 26th, 7 8 22 A.M. mean time at ship long. 72 47 E. 5 )> May 1 6th, 9 17 20 A.M. mean time at ship long. 45 40 W. 6. !» April 29th, 2 26 52 P.M. mean time at ship long. no 57 W. 7- >» June loth, 8 45 P.M. app. time at ship long. 129 30 E. 8. » November ist. 10 20 16 A.M. mean time at ship long. II 17 E. 9- » September ist, 8 20 40 A.M. app. time at ship long. 172 9 E. to. „ October ist, 6 H 50 A.M. mean time at ship long. 68 15 W. [I. » December i6th, 4 35 32 A.M. app. time at ship long. 4 8 E. t2. )> November 14th, 6 45 8 P.M. mean time at ship 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) : — 13- 1876 Jan. 19th, long. 86°57'W. 19. 1876, July 28th, long. 2° o'W 14. Feb. 1 6th, long. 72 59 E. 20. )) Sept. 22nd, long. 156 w. 15- Mar. 2 ist, long. 168 3 E. 21. ,, Oct. ist, long. 170 58 E. 16. May 8 th, long. 10 35 W. 22. ,, Dec. 22nd, long. 179 52 E. 17- June 2 ist, long. 167 15 E. 23- 1877, Jan. ist, long. 156 48 E. 18. Mar. 20th, long. 129 W. 24. 1876, Sept. 23rd, long. 174 15 E. 272. 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. 273. To find the polar distance of a celestial object, proceed according to the following rule : — EULE LXXXV. When the latitude of the place, and declination of the object, are of the same name subtract the declination from 90° ; hut ivhen the latitude and declination are 0/ 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. N. N. S. N. S. S. Polar DisUnce. 8' 12' 18" S 98° 12' 18" 22 30 o N 67 30 o 2 31 15 S 87 28 45 30 23 15 S 120 23 15 7 22 32 N 97 22 32 26 42 12 S 63 17 48 12 48 2 N ( ^°^ 48 2 I or 77 II 58 232 Jteductior). of Elements from Nautical Almanac. TO FIND THE EQUATION OF TIME. 274. 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 some- times 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 points 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 Bay. 275. 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. 276. Equation 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, amounting to upwards of half an hour — apparent noon sometimes taking place as much as 1 6"" before mean noon, and at others as much as 1 4^™ 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 the 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. 277. To reduce equation of time to Greenwich date. — The method of correcting the equation of time for the Grreenwich date is similar to that for correcting the sun's declination, and the "Variation in i hour" may be used for the purpose. EULE LXXXVI. 1". Get a Greenwich date, as hef&re. Note. — The time by clironometer when error and rate are applied to it, gives Mean Tirne at Greenwich. ReckicUon of JElements from Nautical Almanac. 233 2°. Tahe out of Nautical Almanac (page II of the month) the equation of time for the noon of Greenwich date, and mark it additive or subtractive, accord- ing to the heading of equation of time at the top of the column in page I of the month ; also take from the column in page I, the '^Var. in i hour.'^* Note. — It sometimes happens that the precept for applying the Eq. of Time changes in the course of the month. Thus in April 1876, a hlack line is placed between the Eq. T. for the 14th and that for the 15th, indicating that a change of precept occurs between those days. The Equations above the line has to be added, those below have to be subtracted. 3°. Multiply the ''Var. in i hour^' hj the hours, and when great precision is necessary, hy the fractional parts of an hour also. The result is the correction to he 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. — When correcting backwards from the following noon, the rule will be Equation increasing, subtract, and decreasing, add. (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 from apparent time when equa- tion of time at noon is directed to be added, but added to apparent time when equation of time at noon is directed to be subtracted; i.e. the Equation has to be applied to A.T. according to the precept for the day following the given day. Examples. Ex. I. 1876, January 29th, 6^ 53"" 49' mean time at Greenwich ; find Equation of time to be applied to apparent time (in working the chronometer.) Eq. of Time, page II, N.A. Hourly Diff. page I, N.A. Jan. 29th, add i3'"2o'"r incr. Jan. 29th, at noon o''447 Corr. for 6h-9 -\- yi 6'' 54™ is 6i>-9 6-9 Bed. Eq. Time 13 23-2 4023 (To be added to app. time.) 2682 Correction 3*0843 or, 3-1 In working this example the "Diff. for i hour" is taken from the Nautical Almanac from the column in page I of the month, and against the given clay. The Greenwich date being mean time, take the equation o time from page II of the month, and mark it additwe to app. time as directed at the top of the column in page I; also note that the equation is increasing. The Green, time being 61" 54"" or eii'g ; hourly difference is multiplied by 6-9 giving the product 30843 ; and since there are three decimals figures in H. D. ('447) and one in Green, time (q) in all four, four decimal places are marked off from the right hand of the product, the result 3'o843 or yi is the correction to be appUed to the Eq. of time at noon, and is to be added to it be- cause it ia that due to a 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 as above, by multiplying the given hourly difference by the number of hours in the Greenwich time ; for that process assumes that this hourly difference in the same for each hour. The variations in the hourly difference are, however, so small that it is only when extreme precision is rtiinired that recourse must be had to the more exact method of interpolation for second differences. HH 2 34 Reduction of Elements from Nautical Almanac. Ex, 2. 1876, September 30th, io'> i5"» 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. Sept. 30th, noon, suht. 10^ iv-^ incr. Corr. for loi'' + 8-3 Diff. for ih, page I, N.A. Red. Eq. T. 10 19-8 (To be subtracted from A.T.) Sept. 30th, at noon, Diff. for 10 hours Diff. for \ hour Correction 08-805 8050 2or 8-251 8-3 Ex. 3. 1876, December 23rd, 12^ 56'", mean timo at Grreenwich: find equation of time to be applied to apparent time. Green, date, Dec. 23rd, Subtract from 22*^56" 24 o Time from noon, Dec. 24th, i 4 Eq. of time, page II, N .A. Dec. 24th, suht. o 6^6% incr. Corr. for jh — 1-25 Hourly diff. Time from noon Correction is-246 1-246 Eed. eq. of time o 5-43 (To be subtracted from A.T.) In this example the equation of time is taken for the nearest noon to Greenwich date, viz., Dec. 24th. To obtain the correction we go back 1 hour, and since the equation of time is increas'tny at noon, Dec. 24th, it was less at one hour earlier, therefore the correction is subtractive. Ex. 4. 1876, August 31st, 5*' 42™ 15', mean time at Greenwich : find equation of time to be applied to apparent time. Eq. of time, page II, N.A. Aug. 31st, at noon, add o" i«-4i deer. Corr. for 5'' 42"^ — 4-45 Bed. eq. of time, subt. o 3-04 (To be subtracted from A.T.) Hourly diff., page I, N.A. Aug. 31st, noon, o«-78,o 5" 42'° = 5*7 546 39° Correction 4'446 or, 4'"4S In this case the correction is subtractire, 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 subtracted from A.T., according to the precept for the day following the given day— a change of precept occuring 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. 1876, June 13th, 22i» 25'^ 2i» mean time at Greenwich; find equation of time to be applied to apparent time, in working the chronometer. Hourly Diff. page I, N.A. Greenwich date, Jane 13th, 22'> 25™ 22'>-4 June 13th, noon, Eq. T. page II, N.A. June 13th, noon, subt. o™io»-57 deer. Correction for 22'>-4 — 1165 BM. Eq.T. {add) o 1-08 (To be added to A.T.) 0«-520 22-4 2080 1040 1040 Correction 1 1-6480 iis-65 Correction of the Observed Altitude. • 235 In this case also, the correction is subtractive, and exceeds the Equation itself, therefore, the equation is subtracted from the correction and a change of precept is made i.e., the equation of time at noon being 4m6- iractive, after it has been subtracted from the correction ; to the remainder prefix the precept add to A.T. By using the Eq. T. of time corresponding to the nearest Greenwich noon, viz., that for June 14th the work ■will stand thus : Green, date, June 13th, 1,2^2^"' Hourly DiflF. page I, N.A. Subtract from 24 June i4tli, at noon, 0-526 Time from noon, June 14th 1*35 or, I '''6 nly. Eq. T. page II, N.A. June 14th, at noon, add o'ni^'gg incr. Correction for i'''6 — 0*84 Time from noon 14th i"6 3156 526 Correction '8416 or, o*-84 Red. Eq. T. add o rij (To be added to A.T.) The Eq. Time would be less 'dXx^'b before noon than what it is at noon, the correction is therefore subtracted from the noon Eq. of Time. Examples for Practice. In each of the following examples it is required to find the equation of time corresponding to the given Grreenwich date : — I. 1876, Jan. 5th, at 4^ 33'" 0^ M.T. n. 1876, June 13th, at 2 2i'52n> os;M.T. 2. Feb. 1 8th, at 8 20 M.T. 12. „ Aug. 3 1 at, at 15 54 A.T. 3- Mar. 24th, at 3 4 8 M.T. 13- „ May 14th, at 9 36 A.T. 4- April 14th, at 16 8 10 M.T. 14. „ April 13th, at 21 36 53 M.T. 5- May 19th, at 6 56 M.T. 15- „ Nov. 14th, at 21 35 A.T. 6. June 13th, at 22 49 50 M.T. 16. „ July 20th, at 20 57 16 M.T. 7. July i6tli, at I 14 A.T. »7- „ Dec. 23rd, at 18 2 54 M.T. 8. Aug. 31st, at 21 14 40 A.T. 18. „ Oct. 26th, at 7 56 21 M.T. 9- 8ept. 1 8th at 53 10 M.T. 19. ,, Dec. 24th, at I 30 A.T. 0. Oct. 5th, at 19 19 2 A.T. 20. „ Aug. 3 1 at, at I 48 M.T. CORRECTION OF THE OBSERVED ALTITUDE. 278. 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. 279. 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 circumstantiul sources of error by which this is affected : — («) The sextant (supposed otherwise to be in adjustment) may have an index error : {b) 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 236 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 altitruU. But, again, for the sake of comparison and computation, all 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. The altitude supposed to be so taken is called the " true 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 XVIU, 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 refraction 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 COEEECT THE SUN'S ALTITUDE. EULE LXXXVII. 1°. Correct the olserved altitude of th« sun for index error, if ciny. 2°. Subtract the dip answering to height of eye (Table V, Norie, and Table XXX, Eaper) ; the remainder is the appa/rent altitude of the limb observed. 3°. Subtract the refraction (Table IV, Norie, and XXXI Eaper), add the parallax (Table VI, Norie, XXXIV, Eaper) ; or take out the " correction in altitude of sun ^^ (Table XVIII, Norie), and subtract it; the remainder is the true altitude of the observed limb. 4°. Take from page II of the month in the Nautical Almanac the sun's semi- diameter, adding it when the sunh lower limb (l.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 son etimes used in solving questions in nau'ical astronomy when great precision is not necessary. Examples. Ex. I. 1876, January 6th, the observed altitude sun's l.l. 39° 8' 30", index correction •\- 33", height of eye 19 feet : required the true altitude. Maper. 1 Norie. Obs. alt. sun's l.l. 39° 8' 30" j Obs. alt. srm's l.l. 39° 8' 30" Index correction -j- 33 j Index correction -f 33 39 9 3 I Dip (Table 30.) —415 I Dip (Table 5.) App. alt. sun's l.l. 39 4 48 ! App. alt. sun's l.l. Ref. (Table 31.) ^ ,. -^ ,, . —Par. (Table 34 True alt. sun's l.l. Semi-diameter 39 9 3 4 15 39 4 48 I 5 39 3 43 16 18 ReJ. (Table 31.) \ _ i c Corr. alt. (Table 18.) True altitude 39 20 i True alt. sun's l.l. Semi-diameter 39 9 3 4 II 39 4 52 I 3 ¥ 3 49 16 18 True altitude 39 20 7 Correction of the Observed Altitude. ^yj Ex. 2, 1876, June i8th, tlie observed altitude sun's l.l. 71° 19' 20", index correction + 3' 46", height of eye 1 8 feet : required the true altitude. Obs. alt. sun's l.l. Index correction Dip (Table 30.) Eef. — o' 20' ^ Par. + 3 j Semid., p. II, N.A. True altitude 71° 19 20' + 3 46 71 23 4 6 10 71 18 56 17 71 + 18 15 39 46 Obs. alt. sun's l.l. Index correction Dip (Table 5) Corr. of alt. (Table 18) Semi -diameter True altitude 71' 19' 20" + 3 46 71 23 4 6 4 71 19 2 17 71 + 18 15 45 46 71 34 31 71 34 25 Ex. 3. 1876, October 8th, the observed altitude siin's l.l. 19° 50' 10", index correction -)- 50', height of eye 16 feet. Obs. alt. sun's l.l. Index correction Dip Eef. — Par. + i' 41" \ 8 ] Semi-diameter True altitude 19° 50 10 50 19 51 — 4 19 47 — 2 33 19 44 + 16 27 3 Obs. alt. sun's l.l. Index correction Dip 20 o 30 Correction of altitude — Semi-diameter True altitude 19- + 50 10' 50 19 51 3 50 19 47 2 10 29 19 44 41 + 16 3 o 44 Ex. 4. 1876, 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 Kef. — 4 Par. + ■"") 12" + 52 3 30 10 12 55 4 40 5 12 51 4 35 3 3 4 5 6 7 8 9 10 II. tz. Semi-diameter True altitude 1876, Jan. 29th, Feb. 1 8th, Mar. 24th, April 20th May 8th, June 19th, July 1 6th, Aug. 7th, Sept. 2nd, Oct. nth, Nov. i5tb> Dec. 14th, 12 47 32 — 15 49 Obs. alt sun's u.l. Index correction Dip Correction altitude Semi-diameter True altitude 12 52 30 + 3 10 12 55 40 3 57 12 51 43 3 56 12 47 47 15 49 12 31 58 12 31 43 Examples for Praotice. Obs. alt. sun's l.l. 17° 44' 30' Index corr. 48 4 10 „ 29 50 30 » 76 3 o 58 38 20 ,, 24 48 30 „ 65 I o » 85 13 20 „ U.L. 28 16 20 „ U.L. 67 44 O „ V.L. 14 3 40 „ U.L. 13 10 5 „ -,'25' + 55 + 1 3 Eye » 16 feet 12 „ 17 „ — I 27 » 10 „ — I 10 J) 18 „ — r 14 + 17 20 „ 14 ,, — 2 10 )> 18 „ -4 8 -I 38 + 4 I If 10 „ 15 » — 49 n " »> TO FIND THE LATITUDE BY A MERIDIAN ALTITUDE OF THE SUN. \ EULE LXXXVHL i"^. With the ship's date and longitude in time, find the Greenwich date in apparent time (Rule LXXXI, 5°, page 222). 2°. Take the sun's declination from Nautical Almanac (page I of the month), and correct it for the Greenwich date (Rule LXXXIII, page 225). Instead of proceeding according to i~ and 2° the declination may be found thus : — (i) Take the sun's declination from the Nautical Almanac, for apparent noon, page I ; and also the corresponding hourly difference. (2) Multiply the hourly 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 LXXXIV, page 229. 3'^. Correct the observed altitude for index error, dip, semi-diameter, and refrac- tion and parallax, and thus get the true altitude (Rule LXXXVH, page 236); 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 South of sun, or when it bears North. 5°. Add together the declination and zenith distance, when they have the same name (see examples i and 3) ; but take the difference if their names be unlike (see examples 2, 5, and 6) ; the latitude is N. or S., as the greater is. 6°. When the declination is 0°, the zenith distance is the latitude, and of the same name as the zenith distance (see example 7) ; and when the zenith distance is 0°, the declination is the latitude, which is of the same name as the declination (see example 4). Examples. Ex. I. 1876, January 15th, in longitude 72° 42' W., the observed meridian altitude of the sun's l.l. (lower limb) was ^9° 42' 10', bearing North ; index error -|- 2' 10', height of eye 14 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^. 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™ 48* (see below). It is, therefore, 4*^ jo"" ^%« past apparent noon at Greenwich, and the Greenwich date is foimd by adding 4*^ jC" 48* to the time of apparent noon at ship, January 15th, thus: — Ship date, Januarj" i^'^ ©h o™ o« 72° 42' Longitude 72° 42' W. + 4 50 48 4 Greenwich date Jan. 15 4 50 4g 4'^ 50'" 48' With this date the sun's declination must be taken out of Nautical Almanac, where it will be foimd in page I for January. It may be reduced to Greenwich date by means of the Tables, or by " hourly diff.," thus : — * When tnte altitude exceeJo 90', subtract 90' firom it. Latitude hy Sun' ft Meriilian Altitude. ^59 Decl., page I., N.A. Jan. 15th at noon 21° 11' 32"-2 S. (dem-.J Corr. for 4** 51™ 2 124 Hourly diff., page I, N.A. Jan. 15th — 27 "'3° 4i'5in--4h-85 X 4-85 Red. decl. 21 9 20 S. 13650 21840 10920 6,0)132-4050 Correction — 2 12-4 In working this example the H. diff. for the noon of the day is taken without correcting it for the middle time as explained in No. 270 page 230. We divide the minutes of Greenwich time by 6 ; thus, 6 is contained in 51 eight times and three over, 6 is contained in 30 (the remainder 3 with a o added) five times; hence we have the decimal -85, to this we prefix the hours (4), and we then have 4'>'85 to multiply by. As the Green, date wants ic" of 5 hours, we might have multiplied the hourly diff. by 5, and deducted one-sixth of hourly diff. from the product. Eaper. Norie. Obs. alt. sun's l.l. Index error Dip (Table 30) App. alt. sun's l.l. Refraction (Table 31) Parallax (Table 34) True alt. sun's l.l. Semi-diameter True altitude Zenith distance Declination Latitude 59° 42' 10" N + 2 10 59 44 20 — 3 40 59 40 40 — 34 59 40 6 + 4 59 40 10 + 16 18 59 56 28 90 30 3 32 S. 21 9 208. Obs. alt. sun's l.l. Index error 59° 42' io''N -j- 2 10 Dip (Table 5) 59 44 20 3 36 Corr. alt. (Table 18) 59 40 44 29 Semi-diameter 59 + 40 15 16 18 True altitude 59 56 33 90 Zenith distance Declination 30 21 3 27 S. 9 20 S. Latitude 51 12 47 S. 51 12 52 S. The meridian zenith distance and declination are added, because they are of the same name. (This is according to No. 5° of the Rule). Ex. 2. 1876, February 3rd, in longitude 139° 42' W., the obaeryed meridian altitude of the sun's l.l. 56° 56' 56", bearing South ; index correction — 3' 4" ; height of eye 14 feet. Ship date, February Long. 139° 42' W. 3"^ + O" o™ o^ 9 18 48 Greenwich date, Feb. 3 9 18 Decl. page I, N.A., Feb. 3rd =: 16° 37' 7" S., deer. Hourly diiF. 43"-95. Hourly diff. Feb. 3rd noon 43"'95 T. from noon, c/^ 18" + 9"3 13185 39555 Corr. Decl. noon, Feb. 3rd 6,0)40,8-735 — 6 49 16 37 7 S. Red. decl. 16 30 18 S. Norie. Obs. alt. sun's l.l. Index correction Dip (Table. 5, Norie) App. alt. sun's l.l. Corr. Alt. (Table 18) True alt. sun's l.l. Semi-diameter True altitude Zenith distance Declination Latitude 56=56' 56" S. — 34 S^ 53 3 52 36 56 50 16 32 56 + 49 44 16 16 57 90 6 32 16 54 30 oN. 18 S. By Raper : index corr. — 3' 4" ; dip — 3*40"; refr. — o' 38"; par. + 4'; semid. -\- 16' 16" ; true alt. 57° 5' 54" ; latitude 16° 23' 12" N. 16 23 42 N. The difference of zenith distance and declination i> taken because they are of contrary names. See No. 5° of Rule. 240 Zattitcde hy SurCs Meridian Altitude. ^ Ex. 3 1876, March 20th, longitude 37° 45' E. LI. 52° 52' 50", bearing South : index correction Long, in time 2h 30% or 2-5. Hourly diff., noon, 20th, 59"'24 T. from noon, 20th, at 2'' 31" X 2-5 29620 Correction Decl. 20th, noon Correction Red. decl. 6,o)i4,8'ioo — 2 28 0° 5' 53' N., «■««-. — 2 28 o 2 25 N. By Eaper : dip — 3' 20" ; ref. — o' 44" ; par. + 5" ; semid. -|- 16' 5". True alt. 53° 6' I", and latitude 36° 56' 38" N. , observed meridian altitude of the sun's -\- i' 5" ; height of eye 12 feet. Obs. alt. sun's l.l. Index correction Dip (Norie) App. alt. sun's l.l. Corr. of alt. True alt. sun's l.l. Semi-diameter True altitude Zenith distance Declination Latitude 5 2° + 52' 50" s. I 5 52 53 55 — 3 19 52 50 36 38 52 + 49 58 16 5 53 6 3 90 36 53 57 N. 2 25 N. 36 56 22 N. ^(^yj.'^J --V The longitude 37° 45' in time is 2'' 31"", or 2'>'5, and being east, the Greenwich date is •!> 31" before the noon of March 20th (the noon of the ship date), then the decl, and hourly diff. is taken out of the iVaK^/coi^imanac, page I, for the nearest noon to Greenwich date, viz., noon of March 20th, and hoittly diflf. is multiplied by 2'''5 ; the resulting figures are i48"'ioo, or 2^ 28°>, the correction. The decl. at noon 20th, increasing, would evidently be less, 2^ 31"° earlier, therefore the correction — z' 28" is subtractive. See Rule liXXXrv, page 229. ^ Ex. 4. 1876, April 1 6th, longitude 139° 50' E., observed meridian altitude sun's l.l. '»' 89° 46' 10", bearing North ; index correction -\- i 56"; height of eye 18 feet. Long, in time (139° 50' E.) ^ 9^ 19™ 2o», or 9''-3. Decl. page I, N.A. Hourly diff., noon, i6th 52"'9i Green, time 9'' 19™ = X 9' 3 15873 47619 6,o)49,2'o63 Correction — 8 12 Decl. noon, i6th Correction 10° 20' 52"N. incr. — 8 12 Eed. decl. 10 12 40 N. By Eaper: index corr. + i' 56"; dip — 4' 10"; ref., &c., o' ; semid. + i5'58". "True alt. 89° 59* 54", latitude 10° 12' 34' N. Obs. alt. sun's l.l. Index correction Dip (Norie) Corr. alt. Semi-diameter Zenith distance Declination Latitude 89' 46' io"N. + ^ 56 89 48 6 — 44 89 44 2 ¥ 44 15 2 58 90 90 10 12 40 JN 10 12 40 N. -f Latitude by Sun's Meridian Altitude. 241 Ex. 5. 1876, July 13th, longitude 50° o' E., observed meridian altitude sun's l.l. 68* 2' o*, bearing north ; index correction, — 25 ", height of eye 17 feet. / Long, in time Hourly diff., 1 3th, noon Time from noon 3^ 20"" •jh 20™ O' si" 22"-32 3 Correction Decl. noon, 1 3th Ked. decl. 20"^ I ^^ I 6696 ! I 744 6,0)7,4-40 + 1 14 21 45 42 N. 21 46 56 N. By Eaper : index corr., — 25" ; dip, — 4' 5"; corr. alt., — 21"; semid., -|- 15' 46"; true alt., 68° 12' 55"; latitude 0° o' 9" S. Norte. Obs. alt. 8\m's i.L. Index correction Dip (Table 5) Corr. alt. (Table 18) Semi-diameter (N.A.) True alt. Zenith distance Declination Latitude 68'» 2' o"N. — 25 68 I 35 — 3 57 67 57 38 — 20 67 57 i8 + 15 46 68 13 4 90 o o 21 46 56 S. 21 46 56 N. The ship on the Equator. When the zenith distance and declination are numerically equal, and of contrary names, the ship is on the Equator. Ex. 6. 1876, December 17th, longitude 175° 45' W., observed meridian altitude sun's 1..I1. 89° 54' 20 " bearing north, index correction + 4' 4", height of eye 24 feet. Green, date (A.T.) Dec. 17th, ii^ 43"" or, iih'7 Decl. page I, N.A. Dec, 17th, 23° 23*46" S., iner easing. H. diff. Dec. 17th, noon Time from noon 4"-66 "■7 3262 466 466 54-522 Correction 55" Decl. Dec. 17th, noon 23° 23' 46" S., incr, Corr. for ii*" 43" -j- 55 Eed, decl. 23 24 41 S. Obs. alt. sun's l.l. Index correction Dip (Table 5) 24 feet Corr. of alt. (Table 18) Semi-diameter True altitude Zenith distance Declination Latitude 890 + 54' 20" N 4 4 89 58 24 — 4 42 89 53 42 1 90 53 42 16 18 10 23 10 N. 24 41 S. 23 14 41 S. The true altitude by Raper's Tables is 90° 9' 52", Zenith dist. 0° 9' 52*, latitude 23° 14' 49* S. 90° is subtracted from the true altitude, the re- mainder is zenith distance, North. II 242 Latitude hy Sun^s Meridian Altitude. -t Ex. 7. 1876, September 22nd, long. 76° 30' W., observed meridian altitude sun's l.l. 40° 9', bearing North, index correction -j- 20", height of eye 1 8 feet. Green, date (A.T.) Sept. 22nd, 5'' 6"'. Decl. page I, N.A., Sept. 22nd, at noon 0° 4' 58" N., decreasing, Hourly diff. 58"-50. H. diff. Sept. 22nd, noon 58''-5o Time from noon 5'i 5850 29250 6,o)29,8'35o Decl. Sept. 22nd - 4' 58" 4 58N 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" 9' o"N. + 20 40 9 20 — 4 4 40 5 16 — I r 40 4 15 + 15 59 40 20 14 90 49 39 46 S. 49 39 468.- By Raper : index corr. + 20" ; dip — 4' 10'' ; refr. — 19"; par. -\- 7 ; semid. + 15' 59" ; t^U6 alt. 49° 39' 54" ; latitude 49° 39' 54" S. Ex. 8. 1876, June 25th, longitude 59° 15' E., observed meridian altitude sun's u.l. 60° 24' 10" (zenith South of observer) ; index correction — 3' 17" ; height of eye 30 feet. Ship's date (A.T.) June 2$^ °^ °" Long. 59° 15' E. — 3 57 Green, date (A.T.) June 24th 20 3 Decl. p. I, N.A., 25th 23° 23' 29* N. deer. H. diff'. June 25th noon 4"'54 T. from noon 25th is 3i»57n>=3-95 2270 4086 1362 Correction 17-9330 Decl. June 25th at noon 23° 23' 29"N. deer. Correction -\- 18 Reduced declination 23 23 47 N. The decl. is taken out for the nearp.it noon to Green, date, \i'/.., June 25th at noon, and con-fcted for the interval between it and the Green, time, which is equal to the long, in time, \-iz., j*' 57'" (= 305 hrs.) Wo might have found the correction for 4I', and taJien from this result the change for 3«> or one-twentieth of the hourly diflference. Obs. alt. sun's u.l. 60° 24 10" N Index correction — 3 17 Dip 30 feet (Table 5) 60 20 53 — 5 15 Corr. of alt. (Table 18) 60 15 38 — 29 Semi-diameter 60 15 9 — 15 46 True altitude 59 59 23 90 Zenith distance 30 37 S. Declination 23 23 47 N. Latitude 6 36 50 S. By Raper : Ind. corr. — 3' 17" ; dip — 5' 20"; refr. — 33"'4; par. -\- 4"'2 ; Semid. — 15' 46"; true alt. 59° 59' 18"; latitude 6° 36' 55' S. Latitude hy Sun's Meridian Altitude. H3 ^ Ex. 9. 1876, August 23rd, longitude 168° 25' W., observed meridian altitude sun's L.L. 40° 5' 30", observer N. of sun ; index corr. — 54" ; height of eye 12 ft. Green, date, Aug. 23rd, ii"* 13"" 40*. Decl. page I, N.A., August 23rd, at noon, 11° 15' 17" N., decreasing, Hourly dift'. 5i"'2o X ii'"23 nearly ■=z 574"-976 or 9' 35', the eorr. to be subtracted ; whence red. decl. = 11° 5' 42" N. By Norie : index corr. — 54" ; dip — 3' 19"; corr. of alt. — i' i" ; semid. + 15' 52"; trm alt. 40° 16' 8". True altitude 40= 16' 8' 90 o o Zenith distance Declination Latitude 49 43 52 N. II 5 42 N. Ex. 10. 1877, 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, 1876, Dec. 31st, 1^^ o^. Time from noon, Jan. ist, 1877, or Dec. 32ud := longitude in time 10'' o"". Decl. 1876, Dec. 32nd, 22° 58' 47" S., increasing, Hourly diff. i2''-90 X loi^ (long. in time E.) = i29"-oo or 2 9" ; whence red. decl. 23° o' 56' S. By Norie: index corr. — 30'; dip — 4' 11" ; corr. alt. — 18" ; semid. + 16' 18" ; true altitude 70^ 31' 19". True altitude 70" 31' 19" 90 o o 60 49 34 N. Zenith distance Declination Latitude By Baper : index corr. — 54' ; dip — 3' 20" ; refr. — i' ^"'5 ; par. + 6''5 ; semid. -\- 15' 52"; true alt. 40° 16' 5"; latitude 60° 49' 37" N. 19 28 41 N. 23 o 56 S. 3 32 li N. Examples FOR Pkactioe. In each of the following examples the latitude is required :— No. Civil date. Longitude. Obs. alt. sun's l.l. Index COIT. Eye. -J. I 1876, Jan. loth. 49^51' W. 68° 39' 40" N. + 5' 10" 13 ft. y 2 )> Feb. ist, 39 51 E. 72 43 50 S. + 1 42 13 -♦ 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 10 18 _ 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 X.7 )> 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 „ Aug. 26th, 92 3 E. 35 35 20 N. + 2 17 12 _ro >> Sept. 23rd, 166 30 E. 41 36 10 S. — 4 41 17 •- II i> Oct. 23rd, 90 12 W. 54 40 40 s. — 49 i8 ~ 12 « Nov. 15th, 80 II E. 67 43 S. + 1 38 15 - 13 „ Dec. loth. 5S 20 E. 25 52 15 s. + 2 17 ^14 J) Sept. 2 ist, 60 I E. 56 26 N. 20 -15 )) March 20th, 89 30 E. 61 49 30 S. — 3 17 15 ^16 „ April 7th, 139 45 W. 89 ss 50 s- .+ 5 10 12 -S17 » May 1 6th, 45 26 W. 86 34 19 N. -f 4 16 15 Nl8 » Sept. 23rd, 90 45 E. 83 40 30 S. 18 + 19 )> Nov. 3rd, 106 E. 70 29 45 N. + 1 22 '9 / 'f 20 )> Sept. 22nd, 173 58 W. 71 19 20 S. + 3 40 18 *^ 421 ji Feb. 12th, 8 12 W. 29 55 20 S. — I 10 21 tt- 22 )) March 20th, 77 45 E. 76 58 15 N. — 2 20 L 23 1877, Jan. ist. 125 32 E. U.L 54 57 20 S. + 2 10 22 y 24 1876, Oct. ist, 71 20 E. U.L 82 15 N. — 3 15 14 25^ ON AMPLITUDES 281. — 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. 282. — 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 quan- tities are the arguments (Table XLII, Norie, or LIX, Eaper). 283. — 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, whUe it does not effect 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 corrections will be found in Table 59 A, Raper. EULE LXXXIX. 1°. With the ship date and longitude in time, find the Greenwich date (see Eule LXXXI, page 222. The time of sunrise and sunset is generally given in apparent time. 2°. Tahe out of Nautical Almanac (page I) the sun's declination and correct it for this date (see Eule LXXXTE, page 225). 3°. Take from the Table the log. s%ne of declination, and log. secant of latitude (r^ecting i o from the index) ; the sum of these is log. sine of true a^nplitude, which take out of Tables. (Table XXV, Norie, or Table LXVIII, Eaper). 4°. To name the True Amplitude. — ^ the bodg is rising, or a.m., mark true amplitude East ; if it is setting, or p.m., mark it West. : mark it also North, tolien 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 0, the true amplitude is o ; that is, it is East if ths object is rising — West, if it is setting. (b) When the latitude is o, the true amplitude is of the same amount as the dtclinatioH. On AmpUtudeB. 245 ij°. Correction or Error of the Compass for the Position of Ship's Head. Under the true amplitude write the magnetic amplitude ; then — (a) If both amplitudes are North or both South, talie 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 from 180°, and change the name from 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 /'as the case may he J is the entire correction, or error of the compass. The magnetic amplitude must be reckoned from East or "West towards the North or South, before it is placed underneath the true. Thus : the magnetic amplitude S.E. by E. ^ E. ia E. 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 magnetic 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 magnetic 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 less. 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 diflference of T. and M. 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 deter- mined on board ship' is compounded 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 the heel of the ship. If the iron of the vessel exercise no influence on the com- pass, the result obtained is only variation, and ought to agree with that registered on the chart. The deviation is found as follows : — 7°. To find the Deviation. — Under the error of the compass place the variation; then (a). If they are 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 {as the case may be) 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 amount as the variation. 8°. To name the Deviation. — The observer must suppose himself in the centre of the compass, looking in the direction of the variation, — then the deviation ia East wh,en the error of compass is to the right of the variation j i^6 On AmpUtudtis. West when the error of compass is to the left of the variation — loth the error of the compass and the variation being reckoned from the ^ovXh. point of the convpass. Note. — It will be convenient for beginners to draw a figure for the deviation thus : — (See Fig. 2, Example 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 difiference of E and V. I^oTE. — Persons who understand the algebraic signs will find it easier to use them in find- ing the deviation, which is always the algebraic difference between the error of compass and the variation, and which may be thus expressed, — when of contrary names, add ; but when of the same name, subtract. 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 added to the minutes. Examples. Ex. I. 1876, 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. 5^ i6>^ 44™ 27^ Long, in time + 2 25 36 Green, date (A.T.) Jan. 5 19 10 3 H. diff. noon, Jan. 6th Time from noon, 4'' 50" — 17 "71 X 4-83 Correction Decl. 6th, noon Ked. declination Decl. 22° 34' Lat. 37 59 5313 14168 7084 6,0)8,55393 4- I 26 22 32 49 s. 0] U" 6,0 r thus, H.D i7"-7i 5*^ 10" 88-55 — 2-95 8,5-60 I 26 22 34 15 The decl. is here taken for the nearest noon, viz., the 6th, and since the Green, time wants only 4I' 50"> of being noon of 6th (241' o°> — \g^ lo" = 41" so"), "we multiply the hourly diff. hy this quantity, and apply the resulting correction the opposite way to throw it back, for since the declination is decreasing the declination at 4'' 50=1 before noon will be more than it is at noon, hence we add the correction. Or, multiply by s"", then since s"- is ion> in excess of 4'' 50"-, deducting i-6th of H.D. fpom the product above, the result is correction. sine 9-584058 secant 0-103369 Fig. 1. N. sine 9-687427 (a.m. and S. decl.) True amp. E. 29* 8' S. (S.E. by E. ^ E.) Mag. amp. E. a8 7^ S. = E. 2^ points. ISaofxc of comg>aai i o^ E.. because true amplitude is to the right of fnagnetic amplitude. On Atnplitudes. 247 To find the Deviation. Error of compass + 1° o^' E. Variation by chart -j- 3 40 E. Deviation — 2 39^ W., because the error of compass is to the left of the variation. Fig. 2. "W.- Make a rough sketch of the compass as in Fig. i in the above example. In this example the magnetic amplitude is reckoned from E. towards S. (S.E. by E. ^ E. = E. i\ pts. S. =: E. 28° 7^' 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 S. 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 the 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 the 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. It is evident too that in this instance the deviation is the difference of E and V. Ex. 2. 1876, February i6th, at 4'' 58™ p.m., apparent time at ship, latitude 51° 9' N., longitude 15° W., sun's magnetic amplitude W. \ N. : required the true amplitude and error of the compass ; and supposing the variation to be 28° 30' W. : required the deviation for the position of the ship's head at the time of observation. Ship date (A.T.), Feb. Long. 15° o' W H.D. 2UV 6,0 31116 — 1-73 30.9'43 5 9"4 Feb. led 4''58'n + I H. diff., noon, Feb. i6th — 5i"-86 5-97 , Feb. 16 5 58 5i"-86 6 36302 46674 25930 Correction Decl., noon, i6th Bed. decl. 6,0)30,9-6042 5 9"5 — 5 10 12 26 48 S. 12 21 38 Declination 12° 22' Latitude 51 9 sine 9-330753 secant 0-202536 sine 9-533289 (P.M. and 8. decl.) True amp. W. i9°58' S. (W. \ pt. N.) Mag. amp. W. 2 49 N. Error of compass 22 47 W., the trtie amplitude being to the Uft of magnttie. 248 On Amplitudes. To find the Deviation. Error of compass — 22° 47' W. Variation — 28 30 W. Deviation + 5 43 E., because the error is to the right of the variation. ■w.- Make a rough sketch of the compass by drawing two lines crossing at right-angles ; and since the magnetic amplitude is reckoned from W. and N., draw a line M somewhere between W. and N. to represent it. Again, the true amplitude is reckoned from W. towards S. ; draw another line T somewhere between W. and S. to represent the true amplitude. It is easily seen that the error of the compass is the angle included between M and T, i.e., the sum of the true and magnetic amplitudes ; and it is evident T, the true amplitude, is to the left of the magnetic amplitude, the observer being supposed looking from the centre of the compass in the direction of magnetic amplitude ; whence, according to Rule, 6°, the error of the compass is marked West. To name tlie Deviation. — Draw another compass, and taking N. as the origin, and to repre- sent the error of compass, draw a line E from the centre of the compass, but to the left of N., because the error is West. Again, the variation also being West, draw another line V to the left of N., but further than E is from N., because the variation is greater than error. It is easily seen that, in this instance, the deviation is the difference of V and E, and the deviation is named East, because the error is to the right of the variation ; the observer being supposed looking from the centre of the compass, in the direction of the variation. Ex. 3. 1876, April 13th, at 5'' 47"^ 20^ a.m., apparent time at ship, latitude 20° 2' N., longitude 107° 56' E., sun's magnetic amplitude E. \ N., variation 1° 40' E. Ship date, (A.T.) April Long. 107° 56' E. 12'^ i7"47'"20« — 7 II 44 H. diff., April 12th, noon -{- 54*48 io"6 Green, date, (A.T.) April 1 2th 10 35 36 32688 54480 , Declination 9° 4^' sine Latitude 20 2 secant 9-167907 o"027io6 57,7-488 9-37-5 (a.m. and N. decl.) True amp. E. 9° 40' N. (E. \ point N.) Mag. amp. E. 2 49 N. Error of compass 6 51 W. the true amplitude being to the left oitnagnetic. Correction Decl. 12th noon Red. decl. + 9' 37" 8° 54 55 4 32 To find the Deviation. Error of compass 6° 51' W. Variation i 40 E. Deviation 8 31 W. the error is to the left of the variation. E N.T W. Ex. 4. 1876, June loth, at 4'^ 45™ p.m., apparent time at ship, latitude 36° 42' S., longi- tude 120° 30' E., magnetic amplitude W. 29° 15' N., variation 7° 20' W. : required the deviation for the position of the ship's head at the time of observation. On Amplitudes. 249 Ship date (A.T.), June loti 4^45" or June 9 28 45 Long. 120° 30' E. — 82 Green, date (A.T.), June 9 20 43 Time from noon June 10 317 Declination 23" 4' sine 9'593o67 Latitude 36 42 secant o'095947 ?0H H. diff., noon, June loth, -\- io"-<)o T. from noon, 3^ 17m z=i 3'"3 3-3 Correction Dec!-, loth, noou Red. decl. (p.m. and N. decl.) True amp. W. 290 15' N. Mag, amp. "W. 29 15 N. Error of compass Variation 7 20 W. Deviation 7 ,40 E., because the error is to the right of the variation. 3270 3270 23= 35'97o 0' 36' ' 4 16 N. 23 3 40 N. In this instance the error of compass is o, and the deviation is equal in amount to the variation, but of an opposite name. Ex. 5. 1876, July 3i8t, at 4'' 26"^ a.m., apparent time at ship, latitude 460 3' N., longi- tude 165° 58' "W., sun's magnetic amplitnie N.E. by E., variation 13^ o' W., ship's head E. by N. Ship date, (A.T.) liong. in time Green, date, J.uly 30'' 1 6'! 2')^ qs July 3i-ro?' is to the right of variation. Ex. 6. 1876, Sept. 22nd, at 6*> o™ p.m., apparent time at ship, latitude 24° 40' S., longi- tude 13° 30' E., sun's magnetic amplitude W. 2° 50' N., variation 10° 40' W. Ship date, Sept. 2 2d 6^ cm o^ Long. 13° 30' in time — 54 o Green, date, Sept. 22nd 560 or, <-i'' The decl. being 0° the true amplitude is o", or W. o"' o', whence the error of compass is 2° 5 o' W., because the true amplitude is to left of magnetic. H. diff., noon, Sept. 22nd, — 58"-5o 51 5850 29250 6,0)29,8-350 Correction — 4 58 Decl., 22nd noon o 4 58 N. Red. decl. KK Ijo On Amplitudes. To find the Deviation. Error of compass — 2=50' W. Variation — 10 40 W. Deviation + 7 50 E., because the error of compass is to the rujht of variation. Ex. 7. 1876, December loth, at 8" 27'»> a.m., apparent time at ship, latitude 54" 35' N., longitude 53" 15' W., sun's magnetic amplitude S.E. \ E., variation 36° 20' W., ship's head S.W. by W. Green, date, Dec. i)^%^ C" o» Decl. at noon, Dec. loth, 22° 59' 18' S. The Green, date being noon, Dec. loth, or Dec. xo o o o one of the instants for which the declination is put down in the Almanac, nothing more is necessary than to transcribe the quantity as there put down. Declination 22° 59' siuo 9-591580 Latitude 54 35 secant o"236933 sine 9'8285i3 (a.m. and S. decl.) True amp. E. 42^22' S. (E. 3^ pts. S.) Mag. amp. E. 39 22 S. Error of compass + 3 o E., the true amplitude being to the Variation — 36 20 W. /-f;*//:; of magnetic. Deviation + 39 20 E., because the error is to the right of variation. Ex. 8. 1876, December 20th, at 4'^ 3i°> p.m., apparent time at ship, latitude 41^ 12' N., longitude 110° 45' E., sun's setting amplitude S.W. \ W., variation o. Green, date, Dec. 19'' 21^ 8™ Decl. noon, Dec. 20th, 23° 27' 15" S., incr. Declination 23° 27' sine 9*599827 H. diff., noon, 20th, + i"-i3 X time Latitude 41 12 secant 0-123543 from noon, 2i» 52™ (= 2^-87) = 3-2431 = corr. — 3"; Red. decl. 23^ 27' 12". sine 9-723370 (p.m. and S. decl.) True amp. W. 31" 56' S. (W. 3^ pts. S.) Mag. amp. "W. 42 1 1 S. Error of compass •\- 10 15 E., the i>-i«3 amplitude being to the Variation -j- o right of magnetic. Deviation 10 15 E. The Variation being o, the error of the compass is also the deviation. Ex. 9. 1876, November 15th, at 6^ 45" p.m., apparent time at ship, latitude 31° 56' N., longitude 75^^ 30' W., sun's setting amplitude "W. by S. f S., variation 6=" 30' W., ship's head N.N.E. Green, date, November 15'' 11'' 47" H. diff. noon, 15th + 37"'83 X time from Declination 18=" 47' sine 9-507843 noon \i^ 47^" (= \i>-l) := 446"- 394 = corr. Latitude 31 56 secant 0-071264 -|"7"'26; Red. decl. 18° 47' 14" S. sine 9-579107 ^p.M. and S. decl.) True amplitude W. 22° 18' 8. (W. if pts. S.) Mag. amplitude W. 19 41 S. = W. by S. f S. Error of compass — 2 37 W. the true amplitude being to left Variation — 6 30 W. [of magnetic. Deviation + 3 53 E. the error being to the right of [variation. On Amplitudes. 251 Ex. 10. 1876, June igtli, at 9^ 40™ p.m., apparent time at ship, lat. 62° 31' N., lonqf. 60° 24' W., sun's magnetic amplitude, N.N.E. ; and supposing the variation of the comptiss is 57° 50' W., required the deviation for the position of the ship's head at the time the ob- servation was taken. Ship date (A.T.), Juno ig^ gh 40m o» H. difi"., 19th i"-66 Long. (60'' 24' W.) in time -|- 4 i 36 T. from noon X 137 Green, date (A.T.) June 19 13 41 36 Corr. ofdecl. 22742 }^ 13^ '7 Decl. June 19th, noon 23° 26' 56" Declination 23° 27' sine 9-599827 corr. 23 Latitude . 62 31 secant 0335837 sine 9'935664 (p.m. and N. decl.) True amplitude W. 59° 35' S. 180 o Red. decl. 23 27 19 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 left Variation 57 50 W. of mag. amplitude. Deviation 4 55 E., because error is to the ri^ht of variation. Before comparing the true and magnetic amplitudes, they must both be reckoned /ro»« the same point of the compass, E. or W., but in this instance one is reckoned from W. and the other from E. ; therefore, by taking either of them from 1 80°, they would both be reckoned from the same point — the true amplitude, in this example, is taken from 180°, and it is then reckoned from E. instead of W. Next take the difference of the amplitudes, as they are both marked N. ; and since the true amplitude is to the left of the magnetic — looking from the centre of the compass 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 difference 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 the direction of the variation. Ex. II. 1876, July 20th, at 7'^ o™ p.m., apparent time at ship, lat. 34° 51' S., long. 172° 28' E., sun's magnetic amplitude W. f N., variation 8*^ 30' E. Green, date, July 19J 19'' 30™ 8' H. diflf. noon, 20th — 28'63 X time from noon Declination 20^ 36^-' sine 9546515 4*^ 30"' (4''"5) = i28"'835 =: corr. -\- 2 9"; Latitude 34 51 secant 0-085842 Red. decl. 20° 36' 24" N. f sine 9-632357 (p.m. and N. decl.) True amplitude W. 25° 24' N. (W f N.) Mag. amplitude W. 8 26 N. Error of Compass + 1658 E., the true amplitude being to Variation 4" 8 30 E. [the riffht of the magnetic. Deviation -\- 8 28 E., because error is to the n^A< of variation. Ex. 12. 1876, March 24'' 5^ 58™ p.m. apparent time at 'ship, latitude 22° 15' S., longitude J^ 179° 12' W., sun's magnetic amplitude S.W. by W. \ W. variation 9" 40' E. The Green, date is March 2^^ 17'' 54™ 48". Then 24'' — 17*' 55™ = 6'' 5"', or 6''-i ncarij', the time from noon, March 25th. Thfi dfcl , noon, Mnroh 25:h is 2"^ 4' 4 N., iticrensing \ H. diff., 29th. noon. 58"-88 X 6'-i = 359-168. or 5' 59" 8ubtr,ii.:t.ivt;; the decl. iiii;i-e.i5Hiig, Will bt! less 6'- 5" bcloiu iiooii Uiuu at noon. Red. decl. 1° 58' 5" N. 2^2 On Amplitudes. Declination i^'fS' Latitude 22 15 sine 8-535523 secant 0*033605 sine 8"569i28 (p.m. and N. decl.) True amp. W. 2° 7^" N. (W. 2i pts. S.) Mag. amp. W. 25 19 S. Corr. of compass -j- 27 26^ E. Variation -{- ^ 40 E. Deviation -\- i"] 46^ E. Ex. 13. 1876, July ist, at $^ 36"' p.m., lat. 56° 4' N., long. 64° 50' W., sun's magnetic amplitude North, variation 36° o' W. Green, date, July i<* 1 2'' 55"" 2o» or, i2*»*92 Declination 23'^ 3' sine 9-592770 Latitude 56 4 sec. 0-253188 Decl. Page I, N.A. July 1st, at noon, 23° 5' i2"-5 N. deer., H. D. io"-66 : H. D. io"-66 X 12-92 gives correction — 2' i7"-7, whence Jied. Decl. is 23° 2' 54"-8 S. sine 9-845958 (p.m. and N. decl.) True amplitude W. 44° 32' N. (N., or W. 8 pts. N.) Mag. amplitude W. 90 o N. (8 pis. =: 90°) Error of compass Variation — 45 28 W. — 36 o W. Deviation — 9 28 W. 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. No. Civil date. App. time. Latitude. Longitude. Sun's Mag. Amp. Variation. 1876. h m s „ -, I Jan. 27th 6 55 40 A.M. 35 42 N. 12 52 W. S.E. by S. 21 50 W. a " 2 Feb. 17th.... 6 48 P.M. 34 57 N. 40 8E. S.W. bv W. 7 40 E. % 3 March 29th . . 5 50 A.M. 25 50 ^. 127 35 W. E.S.E. 23 40 W. y 4 April 5 th .... 6 15 P.M. 20 20 S. iSS 30 E. W. 6° 40' N. 6 40 E. *. 5 Nov. 7th .... 5 25 A.M. 27 41 S. 70 2 w. E. iN. 13 50 E. y- 6 May 26th 7 56 A.M. 51 22 s. 48 oE. E. f S. 35 20 W. > 7 June 2nd .... 882 P.M. 52 30 N. 27 6 W. N.N.W.iW. 37 20 w. 8 July 14th.... 6 50 58 A.M. 28 59 s. HI II W. N.E. i N. II 40 E. i 9 Aug. 27th . . 5 44 P.M. 21 4 s. 36 19 E. N.W. 1 W. 23 10 W. •n 10 Sept. 8th .... 5 47 A.M. 24 22 N. 57 30 W. E. 4 II Oct. ist > 48 50 A.M. 42 44 s. 175 15 w. E. 1 N. 18 50 E. -» 12 Sept. 23rd . . 600 A.M. 56 41 s. 179 42 E. E. \ S. 11 oE. f- '3 Nov. 3rd 6 34 P.M. 29 20 s. 136 35 E. W.S.W. 2 50 W. y 14 Dec. 4th .... 7 56 48 P.M. 49 59 S. 160 45 E. S.W. by W. i6 oE. f 15 March 20th . . 600 P.M. 55 10 ^• 179 24 E. W. ^"N. 15 E. -f 4 16 Sept. 22nd . . 600 P.M. 60 I S. 13 54 E West. 21 50 W. 17 June 9th 600 A.M. 10 21 W. E. iN. 20 15 w. 18 Feb. 26th .... 7 49 A.M. 62 5 N. 12 52 W. S.S.E. 35 45 W. 19 April 30th . . 6 28 12 P.M. 24 58 X. .3852W. W. by N. i N. 10 oE. V , 20 May 27th 7 40 P.M. 47 40 N. 148 3 w. W. bv N. 20 15 E. + 21 June 1 8th .. I 47 A.M. 63 54 N. 174 20 w. N. bv W. 1 W. 25 oE. i 22 March 6th . . 6 14 P.M. 3, 24 s. 2 10 E. W."i6°52'N. 17 50 W. 23 April loth .. 6 45 P.M. 5.^ 58 N. 178 33 E. W. f S. 16 10 E. + 1 24 Dec. 14th 4 35 A.M. 42 .S. 74 56 E. South. 19 20 W. ON FINDING THE TIME OF HIGH WATER. " BY THE ADMIRALTY TIDE TABLES." 284. 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, Grreenock, Liverpool, Pembroke, Weston-super-mare, Holyhead, Kingston, Belfast, Londonderry, Sligo Bay, G-alway, Queenstown, and Waterford. 285. To find the times of high water from the Tide Tables if the place is one of the Standard Ports, proceed by EULE XC. 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 i.s 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 7th of February, 1875 — 0^ turning to February under the head of North Shields (see page 13), it is seen at a glance that high water takes place at 4^' 8"^ a.m., and that the height of tide is 12ft. loin. above the mean low water level of spring tides, and that the time of high water on the afternoon of same day is 4** 27™, while the height of tide above the low water level of spring tides is 13ft. 2in. Similarly, desiring to know the particulars of the tide at Brest on the morning of March 3rd, 1875 (^^^ P^ge 18), the mark — shows that no tide occurs in the morning of that day ; there will be a high water at ii*> 41™ p.m. on the 2nd, and again at o^ 27"" p.m. {i.e., 27™ ]>ast noon) of the 3rd, but none in the interval. Again, if it be required to know the times of high water on May ist, 1875, at Weston- super-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 2^ 24"! a.m., and 2'' 57™ p.m. respectively. 286 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, 1875, 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. 287. To find the times of high water by the Tidal Constants. EULE XCI. 1°. Seeli in the " Tide TaUen,^'' pages 104 — 108, in the left hand column /or the given place, and in the column headed "Standard Port of Eeference" will he found the .Standard Port for the given place ; also, from the column headed "Time," and opposite the given place, take out the "Constant," bev g careful to note ^ohother it is additive (markwd -\-\ w subtractive (mai'ked — j. i^^ Finding the lime of Sigh Water. 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 of 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 tJie preceding tide instead when the Constant is marked additive (+), hut use. the tide following the blank ( ) when the difference is marked sub- tractive ( — ). 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 marked + or — ; the result in each case, if less than iz"", is respectively the morning (a.m.) and afternoon (p.m.) times of high water required. (a) When the sum of the Constant and the morning (a.m.) time of high water at the Standard Port exceeds 12^, deduct iz'', 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 a given place, if any, add the Constant to the preceding afternoon (p.m.) time of high water at the Standard Poi*t, and if the sum exceeds 12'', deduct 12'', the remainder is the morning (a.m.) tide sought, but if the sum he less than iz^, 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. (b) When the Constant added to the morning (a.m.) tide at the Standard Port is less than \i^ (i.e., gives morning (a.m.) tide at given place) ; hut when added to the afternoon (p.m.) tide at the Standard Port is greater than iz^, there is only a morning (a.m.) tide at the given place on that day. jSj'oTE. When the sum of the constant 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 iz^, the time over 12^ will be a tide of the namefolloiviruj that taken out. (c) When the Constant is sub tractive, and exceeds the morning (a.m.) tide at the Standard Port, reject this lad and use the following afternoon (p.m.) tide at the Standard Port. If the subtractive Constant exceeds the afternoon (p.m.) tide at Standard Poit, 1 1^ 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 sdiex- noon {i^ 'n.) tide use the following tide at Standai'd Port, that is, the morning tide of next day, borrowing 1 2'' if Constant exceeds it ; the remainder is after- noon (p.m.) tide at the given place. (d) If Constant being subtractive, exceeds the Standard morning (a.m.) tide, hut is less than the Standard afternoon (p.m.) tide, there is only an after- noon (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 he increased 12'', 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. 18751 January 3rd: find the times of high water a.m. and p.m. at the Needles Point. Turning to the "Admiralty Tide Table " for 1875, at page 107, in the left hand column, we find Needles Point and in the right hand column, immediately abreast, we find that the Standard Port of Reference which in this instance is Portsmouth, and in the column under Time we have the Constant — i"" 55", that is, we have to siibtinit i'' 55™ from tlie time of high water at Purtsiuouth on any day in order to obtam, the cenespuading ^ir^n of high water at N«ecU« Point. The wirk w >V ^rtnn 1 »s follows :— Finding th» times of High Water. 255 Port of reference— Portsmouth 8'' i"^ a.m. 8'i34"«p.m. Constant for Needles — i 55 — i 55 Time H.W, Needles, Jan. 3rd 6 6 a.m. 6 39 p.m. Ex. 2. 1875, Feb. 12th :. find times of high water a.m. and p.m. at Bordeaux. Turning to page 107, A'dmiralty Tide Tables, it is seen that the Port of Reference for Bordeaux is Brest, and the constant is 4- 3'' y^, that is, thG Bordeaux tides are 3'' 3™ later than the Brest tides, and conseiiuently 3'' 3" must be added to the time of high water at Brest on any day, to obtain the corresponding time of high water at Bordeawx, Port of reference — Brest, Feb. 12th, 7h 47m a.m. 8''io'"p.m. Constant for Bordeaux +33 +33 Times H.W. Bordeaux 10 50 a.m. ii 13 p.m. It may be here remarked, that on adding a comtant to the standard, a morning tide frequently becomes an afternoon tide, and an afternoon tide may beeome a morning tide /or the next dag. (See 3° (a) of Eule.) Ex. 3. 1875, March i6th : find times of high water, a.m. and p.m. at Cherbourg. The Standard Port of Keference for Cherbourg (gee page io8, Admiralty Tide Tables) is Brest, and the Con- stant is 4- 41' 2'", that is, for the times of high water at Cherbourg, we must always add 4'' 2"" to the times of high water at Brest. In this instance, high water at Brest, March 16th, occurs atii'ig™ a.m. (i.e., 51"" before noon) ; consequently 4'' 2>ii (the Cherbourg constant) added to that time must evidently give a p.m. tide at Cherbourg ; the a.m. high water at Ch^rboug must, therefore, be sought from the previous (p.m.) tide at Brest, thus : — Port of reference —Brest, H.W., Constant for Cherbourg Ilh g« + 4 2 ' A.M. P.M. loh I4'» P.M. + 4 ^ 15 " — 12 14 16 — 12 3 II : 216 A.M. Times H.W. Cherbourg, March i6th, If the morning tide, by adding a Constant, becomes an afternoon tide, and the afternoon tide of the day before remain less than 12^ when the Constant is added there is no morning high water at the required port thus : — Ex. 4. 1875, April 19th : find a.m. and p.m. tides at Flushing. The Standard Port of Reference in this case is Dover, and the Constant + i'^ 42™. In this case it is high water at Dover, April 19th, at io'> 21™ a.m. {i.e., i^ 39« before 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., the time of high water in the afternoon (p.m.) of the previous day must be 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 iz^, consequently, the tide does not flow past midnight — the result being p.m. tide of April i8th. There is, therefore, no a.m. tide on the 19th of April at Flushing. Time H.W. Dover, April 19th 10^21"" a.m. April i8th io>> 4™ p.m. Constant -j- i 4^ + i 42 12 3 April i8th 11 46 p.m. Time H.W. Flushing, April 19th o 3 p.m. (No a.m. tide.) 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 : — 256 Finding the Time of Sigh Water. Ex. 5. 1875, May nth: find a.m. and p.m. tides at Portland Breakwater. In this case the Standard Port of Reference is Portsmouth, and the first tide at Portsmouth occurs at 3'' 30"° A.M. [i.e., 3'' 30m after midnight), consequently, since Portland Constant shows that high water occurs there 4i> 40™ earlier than at Portsmouth, and since that quantity, subtracted from May nth, t,^ so™ a.m., -would give a P.M. tide of the icth at Portland ; -we therefore use the Portsmouth tide of the nth p..m., and of the 12th A.M. thus : — Time H.W. Portsmouth, May nth s'^jS™ p.m. + i^ Constant for Portland — 4 4° May 12th 4*' 30™ A.M. + 12 Time H.W. Portland B'kwater,May nth 11 18 a.m. Ex. 6. 1875, Jiine 28th: find a.m. and p.m. tides at Falmouth. 16 30 _ 4 40 1 1 50 P.M. The Standard Port of Reference is Devonport, and the Constant is — o'' 46"'. A blank ( ) occurs in the morning column of the 28th, we therefore use the next tide (as the constant is suhtractive),yiz., the p.m. tide, 12'' being added to make the subtraction, thus : — Time H.W. Devonport, June 28th oh2i»np.M. (next tide) June 29th o'>57" a,m. 12 Constant — o 46 12 21 Constant for Falmouth — o 46 June 29th on a.m. Times H.W. Falmouth, June 28th n 35 a.m. Here there is no p.m. tide on June 28th at Falmouth. Ex. 7, 1875, July 12th: find a.m. and p.m. times of high water at Milford Haven (entrance). The Standard Port of Reference is Pembroke, and the Constant — o'' oo™ (see page 105, Admiralty Tide Tables). Constant exceeds Standard a.m. tide, therefore reject it ; but it is less than p.m. ; there is only a P.M. tide at Milford Haven. Time H.W. Pembroke, July 12th o^ 5" a.m. 0I133™ p.m. Constant — 20 — 20 013 P.M. No A.M. tide on 12th at Milford Haven. Ex. S. 1875, Fehruary 15th : find a.m. and p.m. times of high water at Ballycottin. The Standard Port of Reference is Waterford, and the Constant — o'' 20™. (Only tide) Time H.W. Waterford, Feb. 15th oh26"> p.m. Constant — 26 Time H.W. Ballycottin, Feb. 15th o o There is only one high water on the 15th, and this occurs at Noon. Examples tor PRAoncE. In each of the following examples it is required to find the times of high ■water, a.m. and p.m. : — No Civil Date. Place. No. Civil Bate. Place. I. 1875, Dec. 14th, Cherbourg. 12. 1875 Jan. 19th, Cadiz. 2. Dec. 23rd, Aberdeen. 13- » Sept. 24th Lundy Island. 3. Nov. 8th Stromness. 14. )) Oct. 22nd, Havre. 4- May 14th, St. Nazaire. 15- )> Oct. 8th, CardiflP, 5- April 1 8th, Ushant. 16. )) Sept, 17 th, Ramsey, 6. May 12th, Portland B'kwater.|^ 17. )> July 2 1 St, Maryport. 7- June 27th, Dartmouth. 18. )> March i6th, Wexford. 8. Oct. 23rd, Ballycottin, 19. >i May 1 6th, Gibraltar. 9- Dec. 6th, King Road. 20. » May 22nd, Dublin Bay, 10. Oct. 9th, Falmouth. 21. » Aug. nth. Milford Haven II. Sept. nth, Ferrol. 22. >> Sept. 17 th, Nieuport. MnMng the lime of High Water. 257 288. In pages 151 to 232 of Admiralty Tide Tables for 1875, 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 Rules, 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 amotmt to half an hour. Should 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° shoid'd be sub- tracted, 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, Eaper. Hence : 289. To find the time of high water at Foreign Ports whose constants are not given in the Tide Tables. EULE XCII. 1°. To find the Constant — In the Alphabetical List of Ports at the end of the Admiralty Tide Tables (for 1875 p. 189 — 232), jind the time of high water., Full and Change, at Brest, and also that corresponding to the given port ; subtract the less from the greater of these two times, and the remainder will he the Constant, additive if the full and^change (F. & Q.) at the given port is greater than that of Brest, hut sub tractive if less. 2°. Take out the times of high loater at Brest for the given day, and apply the constant as directed in the preceding Rule, XCI pages 253 — 254; the result is the time of high water (nearly) at the given place. 3°. Talce out the longitude of Brest and also of the given place ; take the sum if the names are alike, but 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 moon^s transit for the proposed day and the following one, if the long, is west ; hut for the given day and the preceding one if the long, is east. Their difference, in either case, is the Daily Variation, or Retardation. 5°. Take from Table 28, Raper, or Table 16, None, the correction corres- ponding to the daily variation and longitude. 6". Apply this correction by addition in west longitude, hut hy subtraction in east longitude, to the approximate times of high toater already found, the result ie the times of high water on the proposed day at the given place. Examples. Ex. I. 1875, March 30th: required the time of high water at Victoria River, Turtle Point (N.W. coast of Australia), longitude 130° E. Time of H.TV. full and change, Victoria River ■j'>^ 15'" (p. 229) Brest 3 47 (P- i94) Constant +3 28 J) 's transit, March 30th, 6^ 16" p.m. Long. Victoria River, 130* E. 29*^1,5 23 » Brest 4 W. 53 126 Under sj" and aarainet 196* loootntude, in Raper, Table 28, or Norie, 16, we find ifl" to be subtracted liManM tbs longitaiie is £. LI 258 Finding the Time of High Water. Time H.W. Brest, March 30th 'i^ii,^"'KM Constant -|" 3 ^^ 12 13 Correction for longitude — 018 Time H.W., Brest, March 30th 9''25'»p.m. Constant -|- 3 2J 12 53 - o 18 Time H.W. at Victoria \ I Time H.W. at Victorial 12 35 p.m. River, March 30th, | ii55^^-m- Eiver, March 30th .. J" No P.M. tide. ! on March 3i8t o 35 A.M. Ex. 2. 1875, October 20th: find the times of high water at Sandy Hook, longitude 74° W. Time of H.W. full and change, Sandy Hook .... 7^ icf^ (p. 222) » „ Brest 3 47 (p- 194) Constant 4-3 42 ]) 's transit 20th, 4^ 44" Long. Sandy Hook, 74= W. 2ist, 5 46 ,, Brest +4 W. I 2 78 62'n Under 62"' and against 78° in longitude, in Raper, Table 28, or Norie, 16, we find 13"' to b« added, because the longitude is West. Time H.W., Brest, Oct. 20th 7hio'"A.M. 1 Time H.W., Brest, Oct. 20th 7^39" p.m. Constant -}- 3 42 10 52 A.M. Correction for longitude + 13 Constant + 3 42 II 21 P.M. Correction for longitude -}- 13 Time H.W. Sandy Hook, 1 Time H.W. Sandy Hook, \ ,^ , „ Oct.2oth .1 " 5A.M. 0ct.2oth \ .} "34 P.M. Ex. 3. 1875, May 2ist: required the times of high water at Nelson, New Zealand, longitude 173° E. Time of H.W. full and changq, at Nelson 9'' 50'" (p. 215) » » Brest 3 47 (p. 194) Constant + 6 3 ]) 's transit, May 2 ist, o^ 23™ a.m. 19th, II 36 P.M. 47 Under 47°' and opposite 169° (173° — 4°) in Table 16, Norie, or 28, Raper, stands the correction ai™ to be subtracted. Time H.W., Brest, May 21st 4^ 8"" a.m. Time H.W., Brest, May 21st 4^24"' p.m. Constant + 6 3 Constant + 6 3 10 II A.w. 10 27 P.M. Correction for longitude — 2 1 Correction for longitude — 2 1 Time H.W. Nelson, May 21 st 9 50 a.m. Time H.W., Nelson, May 2 ist 10 6 p.m. Ex. 4. 1875, August 3rd : 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. 229) Brest 3 47 (p. 194) Constant + 4 43 5 -8 transit, August 3rd, i" 51™ ^^^ ^ ^ ^ ^^^ ,,„. ^ g-. August 4th, 2 36 ^ 45 Chremmch Bate hj Chrmomei&r. 259 Time H.W. Brest. Aug. 3rd 4'' 51" a.m. Time H.W. Brest, Aug. 3rd jhiimp.si. Constant -j- 4 43 ' Constant + 4 43 9 34 Correction for longitude .... -j- 8 Time H.W.C. Virgin Aug. 3rd 9 42 a.m. 9 54 Correction for longitude .... + 8 TimeH.W.C.Virgin, Aug. 3rdio 2 p.m. Examples for Praotiob. Ex. I. 1875, -A-^g- 12th: find the times of high water at Caracus River, Ecuador, longi- tude 67° W. Ex. 1. 1875, September 22nd: find the times of high water at Auckland, New Zealand, longitude 175° E. Ex. 3. 1875, M^ay 15th: find the times of high water at Point de Galle, Ceyloe, longi- tude 80° E. Ex. 4. 1875, February 22nd : find the times of high water at San Francisco Bay, longi- tude 122° W. Ex. 5. 1875, Sept«mber 23rd: find the times of high water at Malacca Fort, longi- tude 102" 15' E. Ex. 6. 1875, ^^Y 22nd : find times of high water at Port Jackson, North Head, longi- tude 151° 16' E. Ex. 7. 1875, July 27th : find times of high water at St. Julian, longitude 67' 38' W. Ex. 8. 1875, July 26th : find times of high water at Awatska Bay, longitude 158' 47' E. Ex. 9. 1875, J^y ^^t^ '• fi^d times of high water at Cape Cod, longitude 70° 6' W. Ex. 10. 1875, June 3rd: find times of high water at Point de Galle, longitude 80° E. GREENWICH DATE BY CHRONOMETER. 290. 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. 291. Rate of Chronometer is the daUy 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. 292. 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" sloto, 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 iz" fast, and at the end of thirteen days ^f fast for mean time at any place, it must have gained 4^" 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 dividing the change of the error by the number of days in the interval between observations. 26o Grumiwich Date by Chronometer. 293. To name the rate. — When the chronometer is J ant either on Green- wich mean time, or on the time at place, if the error is inereasing, the rate is gaining ; if deereasing, the rate is losing. When the chronometer is slow^ if the error is increaning, 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'" 2o« November 30th, „ slow 24 45 Change of error in 10 days 35 Rate for i day 3*5 gaining. In this example the chronometer is slow on November 20th, and the error is decreasing, therefore the chronometer is gaining. Ex. 2. A chronometer was slow 28™ 5^ on mean time at Greenwich, Feb. 27th, 1876, and on March i ith was slow 29™ 36^* on mean time at Greenwich : find daily rate. 1876, February 27th, chronometer slow 28'" 5* Feb. 29 (leap year). 1876, March nth, „ slow 29 36 Feb. 27 Change of error in 13 days 131 March 11 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, June 2nd, and on July ist, was/as< i"" 37*'5 on mean time at Greenwich : find daily rate. June 2nd, chronometer /««< July ist, „ fast Change in 29 days I 37'5 June 30 June 2 H'S i8 July I Int. 29 29)i4-5(o"5 i4'5 The error of chronometer is fast and increasing, hence the daily rate is 08-5 gaining. Ex. 4. A chronometer was fast i™ 51* on mean time at Greenwich, May ist, and on May i5th was 41* fast on mean time at Greenwich: find daily rate. May ist, chronometer /asi i"5i' May 1 May 15th, „ fast o 41 May 15 Change in 14 days i 10 Int. 14 70 35 14 5 In thin example the chronomoU^i iafaai and Iho qu-qi dtcrtnuii/g, the rate therefore is losing^ Qreenwich Date by Chronometer. 261 294.. When the error is found to have changed from fast to slow, or from slow to fast, the rate is the sum of the errors divided by the number of days elapised. Examples. Ex. I. July 28tli, at 3'' p.m., the chronometer was o™ 6s'o fast, and on August 4th at same time, it was o°> i7'-i slow : required the daUy rate. July 28th, at 3*^ P.M., chronometer /««< o™ 6»*o August 4th, „ „ slow o 17-1 Change of error in 7 days 23 'i 3 '3 In this example the error has changed from fast to slow, the chronometer therefore is losing. Ex. 2. A chronometer was slow !>" 4' on mean time at Greenwich, March ist, and on March 23rd, was o™ i^^Sfast on mean time at Greenwich : required the rate of chronometer. March ist, chronometer slow i™ 4' March 23 March 23rd, „ fast o 19-6 March r Change of error in 22 days i 23'6 Int. 22 (nearly) ? 24 Prefixing the days \ to the decimals of \ a day. ) . . 32-66 2-6 19596 6532 6,0)8,4-916 jm 24»*9 Prefixing the daj-s j to the decimals of a day. 6)10 4)1-66 . . . 12*41 94 4964 11169 6,0)11,6-654 i"° 56«-7 or I™ ST 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 io'> in this, while in the otlior it was reckoned g*" 30". Greenwich Date by Chronometer. 263 297. 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. 298. 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 XCIV. 1°. To the time hy chronometer apply the original error, adding it if the chronometer tvas slow, rejecting 24.'' «/ greater than 24'', and putting the day one forward ; but if the chronometer is fast, subtract original error, increasing time shown by chronometer by 24'', if necessary, and putting the day one back. 2°. Find the number 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 by the elapsed time, and add thereto the proportionate part for the fraction of a day, found by proportion or otherwise ; the result is the accumulated rate in the interval. 4°. To the result found by 1°, add the accumulated rate, if chronometer is losing ; but subtract if gaining ; the result will be mean time at Greenwich, at the instant of observation. Examples. Ex. I. 1876, Jan. 30th, r.M. at ship, time by a chronometer, Jan. zcf- 15'' 47"" 48*"3, which was 9™ i9"6 slow for Greenwich mean time Dec. ist, 1875, and on January ist, 1876, was 10™ 248-7 slow on Greenwich mean time. 1875, Dec. ist slow 1876, Jan. ist slow i9»-6 24-7 Change of error in 31 days = i $'i 60 3i)65"i(2-i losing 62 Dec. 31'^ Dec. I Jan. I Int. 31"^ 31 31 The chronometer being slow and the error increasing, the rate muit be marked losing. Time by chron., Jan. Original error Accumulated rate Greenwich date, Jan. 29" i5''47'"48''3 -\- 10 24-7 Interval from January ist Jan. 29 15 58 130 -|- I O'l 29 15 59 i3'i f Daily rate to January 29th is^ 58"" is 28J 16'' nearly. 28 168 42 58-8 i*o ■3 6,o)6,oT .Ace. rate i™o»'i 264 Gremwicli Date hy C1ironometm\ Ex. 2. 1876, March 20th, p.m. at sliip, an observation was made when the time by chronometer was March 20'' o^ 7'" 55^, which was 50™ 51" fast on Greenwich mean time, November 22n(i, 1871;, and on December 2i8t, 1875, was /«■?< 47"" 33'-8 for mean noon at Greenwich : required the Greenwich date by chronometer. November 22nd, chron./as< 50"" 5i'"o Nov. 30'* December 21st, fast 47 33-8 Nor. 22 Change of rate in 291^ = 3 17*2 60 Dec. Int. 29'* 29)i97*2(6*8 losing 232 232 The chronometer is fast and the error decreasing, the rate is therefore losing. Time by chron. March 20'^ o^' 7'"55* Dec. 31 fRate Original error Accumulated rate or 19 24 7 55 — 47 33"8 23 20 2I"2 + 10 ii'7 Dec. 31 f 21 10 1 Jan. 31 Feb. 29 March 19 23^ Greenwich date, March 19 23 30 329 Intr. 89 23 In tinding accumulated rate, as the interval is within half an hour of 90 days (23^''), we might have multiplied by 90 and deducted i-48th (!•» is i-48th of a day) from the daily rate. 6»-8 612 544 605-2 3"4 2'3 6,o)6i,i'7 l^Acc.rate 10 11 "7 Ex. 3. Time by a chronometer, Sept. 7'' 23'^ 16"^ 28*, which was 57" 47*" slow on Green- wich mean time, June 30th, and on July 12th, was 56"' <,3» slow on mean time at Greenwich. June 30th, chron. s^tt! 57"H7' July 12th, „ slow 56 53 Change in rate in 12 days =: 54 Daily rate June 3o"28' + 5653 8 13 21 — 4 21 July Aug. Sept. Int. 31 12 19 31 < 8 58 'Rate 4'-5 58 360 225 Greenwich date, Sept. 8090 6,0)26,1-0 l^ Ace. rate 421-0 Examples for Pbactioe. Ex. I. 1876, February i6th, a.m. at ship, an observation was taken, when the corres- ponding time by a chronometer was Feb. i6<* 8'* 59™ 2^*, which was i*> 20" 2 2'' 4- fast on Greenwich mean time, December ist, 1875, ^^^ ^^ January 23rd, 1876, was i^ r4"> 2^^ fast on Greenwich mean time : required the Greenwich date by chronometer. Ex. 2. A chronometer showed April 29"* 5^ o"" o», which w&afast 33" 30^-3 on GreMiwich mean time, March 19th, and on March 26th was 34™ 20' fast for mean time at Greenwich: required the Greenwich date by ohronoraeter. Greenwich Date hy Chronometer. 265 Ex. 3. A chronometer showed May 7^ 6^ gm 48% which was slow ii"" 9»-4 on Greenwich mean time, February i6th, and on February 26th was 11™ 418-6 slow for Greenwich mean time : required the Greenwich date by chronometer. Ex. 4 The chronometer showed .Tune 25'* 21^ 29™ 53^,^which was 30"' 12^ fast on Green- wich mean time, March 3i8t, and on April 15th, was 30"' 4.^^ fast, for mean time at Green- wich : required the Greenwich date by chronometer. Ex. 5. 1876, October 25th, p.m. at ship, time by chronometer Oct. 2^^ ^^ 31'"' 10% which was 12" 9*"2 slotv on Greenwich mean time, July 20th, and on August 13th was 10™ 2^ slow for Greenwich mean time : required the Greenwich date by chronometer. Ex. 6. Time by chronometer January ig** 13'! ai"" 25*, which was 53™ 4"]" fast on mean time at Greenwich, October 24th, and on October 31st was 53"" \.<)^fast for mean time at Greenwich : required the Greenwich date by chronometer. Ex. 7. Time by chronometer November 8<' i6'» 2" 3*, which was 33" o« slow on mean time at Greenwich, July 31st, and on August 12th was 32™ 2^*4 slow on mean time at Greenwich. Ex. 8. Time by chronometer August 1*1 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. Ex. 9. Time by chronometer May !<• 13'' 23n"' 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. Ex. 10. Time by chronometer January 20'^ o'^ 4m 21s, which was 20^ fast on mean time at Greenwich, November 20th, 1875, and on December loth, 1875, ^^^ 4.' fast on mean time at Greenwich : required the Greenwich date by chronometer. Ex. II. Time by chronometer September 27'' 16'' 34™ 31*, which was C" 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. Ex. 12. Time by chronometer April 16'^ 5'' 36"" 12^ which was i™ 2' slow for mean time at Greenwich, January 24tb, and on February 28th was 2^^ fast for Greenwich mean time. 299 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'', &e., on the civil or on the astronomical day ; for, a frequent source of embarrass- ment 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 2^ 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 2^ or 14^ 5** or 17'', 6'' or 18'' past noon, and so on. To determine this point proceed according to this rule : — RULE XCV. 1°. Get an approximate Greenwich date hy means of ship mean time nearly and the longitude hy account (Rule LXXXI, page 222). 2°. Proceed, as directed in Rule XCIV, page 263, to apply the original error and accumulated rate to the time hy chronometer. If the difference between Greer --ich dates thus found by the two methods, is nearly 12'', then the Greenwich date by chrononuter found as above, must be increased by 12^1, and the day put one back, go as to make the two dates agree both in the day and hour nearly. 266 To Find, the Sour-angle. Examples. Ex. I. August 3rd, at about 3*" p.m., longitude by acct. 75° W., the cbronometer marks gh urn .ys^ and is 6™ 10^ fast on Greenwich meati time : what is the Greenwich astronomical date? Approx. T. at ship, Aug. 2^ 3^ o" Longitude 75° W. + 5 o Approx. Green, date, Aug. 380 Time by chron. 8'- 11" Error of chron. fast — 6 Green, date, Aug. 3rd 8 4 57 In this example the approximate Greenwich time is 8'', it is evident that the chronometer must have shown 8*" from noon also. Ex. 2. June i8th, at lo^ 52™ p.m., mean time at ship nearly, long. 60° "W., an observa- tion was taken, when a chronometer showed 2^ 48™ 40=, on June 6th its error was known to be 3" lo^- 7. fast on Greenwich mean time, and its mean daily rate 3^*5 gaining : required the mean time at Greenwich when the observation was taken. Approx. Ship date June i8iis log. 9'i573i,3 15724 7 = 1" Note. — In Norie, Table XXXI, the next less log. to 15731 is 15724, which, gives 2I' 58"' 10*, and diff. 7 gives 1' to add, whence the term corresponding to 9-15731 is 2^ 58™ 11'. The pol. dist. being greater than 90°, take the secant of decl. for the cosec. of pol. dist., and add the prop, part for 8". cosec. 0-019753 + 5 Diflf. 65 107= 9' 8" cosec. 0-019758 5,20 Observation. — A-lways cut off two figures when working for seconds of arc. 81° 54' o" cos. 9-148915 DifF. 1481 Parts for 54 — 800 54 81° 54 54" cos. 9-1481 15 sine 9-921 190 + 18 sine 9-921208 5924 7405 56° 31' 0" Parts for 13" 799'74 DifF. 139 13 56" 31' 13' 417 139 18,07 Ex. 2. Given the true altitude 17° 16' 12 ', latitude 50° 42' S., reduced declination 20° 6' 17' S. (when polar dist. is 69" 53' 43"): find the hour-angle. Altitude 17° 16' 12" Ex. 3. Given the true altitude 1 3° 28' 42", latitude 10" 35' S., reduced declination 23 23' 54" N. (or polar distance 113^ 23' 54"): find the hour-angle. Altitude 13° 28' 42" Latitude 50 Polar dist. 69 42 53 51 55 43 55 57 sec. cosec. COS. sine log. o'i98335 0-027304 9'55566o 9-894521 9-67582,0 79 Latitude 10 35 Polar dist. 113 23 54 sec. cosec. COS. sine log. 0-007451 0-037268 Sum 137 Sum 137 27 36 Half sum 68 Half sum 68 43 48 1 sum — alt. 55 15 6 Hour-angle 4'^ 40'" 41' 9*559623 ^sum — alt. 5 1 39 45 9-914694 Hour-angle 5'' 48'n 6^ 9'5i903>6 898 i»=3 In Norie, Table 31, we seek for the nearest log. to 967582, the nearest to which is 9-67579, which corresponds to 5I' 48"' 5* ; thenia column prop, part we aeek for 3, which gives i> to add, whence hour-angle is sh 48" 6". 1^ = 6 The nearest log. to 9-51904 is 9-51898, which gives 4" 40"' 40', the diff. 6 found at right hand m prop. parts gives i», whence hom-augle is 4'' 40" 41". 268 Longitude hy Chronometer. Ex. 4. Latitude o', declination 0°, true altitude 30° : required the hour-angle. True altitude 30° o' 90 o Zenith distance 60 6,0)24,0 o Hour-angle ^h Qin o» Ex. 5. Given true altitude 75°, latitude 0°, declination 0° : find the hour angle. True altitude Zenith distance Hour-angle 75" 90 15 6,0)6,0 jh omo» Examples for Practice. Required the hour-angle or meridian distance in each of the following examples. : — I. True altitude, ii°2i' 29' Latitude 30'15'S. Declination i5°2i' 4'N. 2. 30 2 4 39 ^7 S. 5 48 23 N. 3- 27 48 22 40 ro N. 23 26 44 N. 4- 34 49 46 39 20 S. 21 15 7 S. 5- 25 38 II 29 N. 23 I 55 N- 6. 15 59 13 60 5 N. 7 25 38 S. 7- 29 2 27 000 8. 20 34 4 23 27 21 N. 9- 37 40 000 LONGITUDE BY CHRONOMETER, FROM AN OBSERVED ALTITUDE OF THE SUN. RULE xcvn. 1°. To the time hy Chronometer apply its original error and accumulated rate, as directed in Rule XCIV ; the result is the Grre^nwich 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 the corresponding hourly difference for each : also take out the surCs semi-diameter. 3°. Reduce the surCs declination and equation of time to the Greenwich time (Rules LXXXII and LXXXVI) ; also find the polar distance (Rule LXXXV). 4°. Correct observed altitude for index error, dip, correction in altitude, and semi-diameter, and thus get the true altitude. (Rule LXXXVH). 5°. Find the hour-angle or meridian distance by Rule XCVI.* 6°. 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 the date at ship, but if the observation is made in the morning, take the hour-angle from 24'', the remainder is apparent time at ship reckoned from noon of the preceding day, the time at place in both instances being expressed in astronomical time. * In finding longitude by chronometer the logs, used in finding the hour-angle are required to be taken out for aecoudt> of arc. Longitude hy Chronometer. 269 Examples. Ex. I. January 6th, p.m. at ship; sup- pose the sun's hour-angle to he 3h 40" 1 8^ : what is the apparent time at ship ? Here the time heing p.m., we have the ship date app. time, January 6'^ 3^ 4o"> i8». Ex. 2. January 6th, a.m. at ship; sup- pose the sun's hour-angle to be 3^ 40™ i8»: what is the apparent time at ship ? Here the Hour-anglo is ^ 40™ 1 8» 24 o o Ship date app. T., Jan. 5th 20 19 42 Ex. 3. June ist, p.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. T. at ship, June i"^ 3^ 54m 39s. Ex. 4. June ist, a.m. at ship; suppose the hour-angle to be 3'' 54"" 39': what is the apparent time at slup ? Hour-angle 3'' 54" 24 o 39' App. T. at ship. May 3 ist 20 5 On comparing these examples with paragraph 6°, which they are intended to illustrate, the seamen 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 afternoon (p.m.), as in Ex. i, the time will be 3'' 40"' 10^ past noon of the 5th day; that is, the ship date (astronomical time) is January 6<* 3*" 40" 10^ — the astronomical day commences always at noon ; but if the observation be made in the morning (a.m.), the hour-angle will be the time before noon of the 6th day ; or, as shown in Ex. 2, 20^ 19™ 42^* past noon of the day before, — that is, January 5'^ 20'' 19'" 42^ In Ex. 3, similarly, the observation being p.m., the time will be 3'' 54'" 39^ past noon of June ist, while in Ex. 4, the observation being a.m., the time will be 3*^ 54"" 39* before noon of June ist, i.e., 10^ 5™ 2i» past noon, May 31st. In the new edition of Norie's Tables the hour-angle is so arranged that when the obser- vation is made p.m. at ship it is read from the top of the page ; when a.m. from the bottom ; in which case the necessity of deducting from 24h (as explained in paragraph 6°) is obviated. 7°. To apparent time apply the reduced equation of time, adding or subtracting as directed in page I, Nautical Almanac, and so get mean time. 8°. Under ship mean time put Greenwich mean time — not forgetting the day in each, case: — subtract the less from the greater ; the remainder is longitude in time, which convert into arc °'" ; see Rule LXXX, or Table 17, Eaper, or Table 19, Norie. In taking the difference of Greenwich mean time and ship mean time, if the days of the month be different, it will be necessary to add 24 to the hours of the more advanced (that is the one whose days are most), in order to enable the subtraction to be made. 9°. Call the longitude West when Greenwich time is greater than ship mean time ; but East when Ghreenwich mean time is least. Note. — "When the latitude at noon is given, the latitude in at the time of observation must be found by means of the course steered and distance sailed. The diff. of lat. from noon is to be named iS'orth and South, according as the ship at the time of observation is north or south of her latitude at noon. When the longitude is found as in examples i to 10 (or according to paragraph No. 8° and 9°, page 269), the diff. of long, between the ship at the time of observation and noon must be applied to find the longitude at noon. The diff. of long, is to be named East or West, according aa the ship is east or west of its position at noon. l']0 Longitude hy Chronometer. Examples. Ex. I. 1876, January nth, p.m. at ship, latitude 49" 30' N., the observed altitude sun's L.L. was 12' 20' 30", height of eye 18 feet, time by a chronometer January 11'' 6'' 44™ 36" (being p.m. at Greenwich), which was 6™ 8*'3 fast for mean noon at Greenwich, September ist, 1875, ^^^ 0° September 30th, 1875, was 8'" ^2^ fast on Greenwich mean time; required the longitude bj' chronometer. September ist, chronometer /«s< 6™ %^'i 30th, „ fast 8 42-0 Change in 29 days 2 33*7 29)i53'7(5"3 T. by chron., Jan. ii<* 6^'44™36« Original error = o 8 42 Accumulated rate Green, date, Jan. " 6 35 54 9 7 (a) Interval from Oct. ist to Jan. nth 6|'^ is ro3 19™ 53* (or o^ 19™ 53* P.M.), which was 33'/a«< on mean noon at Greenwich, March 20th, and on April ist, was 23«-4/as< on Greenwich mean time. March 20th chronometer / 33™ 2 2^ (being 3^* 4*" 33™ 22* A.M. at Greenwich), which was i7'» i6*'4 fast for Greenwich mean noon May i6th, and on June ist was i6"> 22^ fast for mean time at Greenwich : required the longitude. May i6th. Chronometer /ax< 17™ i6*'-4 June ist, „ fast 16 22-0 Change in 16 days Daily rate 54'4 T. by chron., July Original error Accumulated rate Green, date, July 2d igii 33m 22^ — 16 22 2 16 17 o + I 48 2 16 18 48 Interval from June ist to July 2nd, i6*>, is 3i dip — 4' 10", refr. 3' 18', par. + 8", semid. ■{• 16' 9", true alt. 16*29' 19". Decl., page II, N.A. H.D. 6th noon, 5° 24' 35"S. — 58-23 Correction + 3 45 3' 87 21 Dip i i57'5 i 3-7 2-5 Corr. alt. 6,0)16,3-7 Semid. 2 43'7 True alt. + 30 16 20 3«o 4 4 16 16 26 3 5 16 + 13 21 16 9 Eq. time, page II, N.A. 6th noon -|- ii'»20«'4 Correction + 2-4- B«d. decl. Polar dist. 5 28 20 S. 6,0)22,5-3501 31 40 Corr -f 3 45 Red. Eq. time 11 22*5 (To be added to A.T.) 16 29 30 H.D. 0-604 X 4 2-416 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' 40 20 84 31 30" 40 sec. cosec. cosin. sine log. 0-117879 0-001984 141 21 10 A.T.S.Mar 70 40 54 " 35 5 ^54' 9-519701 9-908972 9"548536 A.T. ship, March Equation time M.T. ship, March M.T. Green., March Longitude in time 6d 4h5im543 + II 23 6 5 3 17 5 20 8 15 8 55 2 Longitude 133° 45' 30" E. By Raper : Log. sin. sq. 9-548579 gives .hour-angle 4^ 51" 55', long. 133" 45' 45" E. 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 24I') to enable us to complete the subtraction. \' Ex. 6. 1 876, November 19th, a.m. at ship, latitude 39° 20' 9., obs. alt. sun's l.l. 34° 37' 55"^ index corr. -f i' ic', height of cyo 14 feet, time by chronometer iS^.z^^ 49"" 32* (or 19'' 11'' 49™ 32* A.M.), which was slow 56'" 57*'i for mean noon at Greenwich, September ist, and on September 19th was 58'" 52»-3 slow on Greenwich mean time. 274 Longitude by Chronometer. Sept. I st, chronometer s^i^ 56"" 57"! Sept. 19th, „ alow 58 52'3 Change in 1 8 days 5i' 60 i8)ii5-2(6»-4 lOg A- T. by chron. Nov. Original error or Accumulated rate Green, date Nov. i8J23h49™32' + 58 5^-3 18 24 48 34-3 19 o 48 24-3 + 6 306 19 o 54 54'9 Sept. 30 days 19 Sept. 1 1 Oct. 31 Nov. 19 I*! nearly Intr. 61 I Daily rate Interval 6-4 61 By Eaper: index cor. -\- 1' 10", dip 390'4 — 3' 40", ref. — i' 25", par. -4- 7", Prop. par. i*" '% semid. + 16' 14", true alt. 34° 50' 21", 6,0)39,0-6 72 72 Obs. alt. O's L.L. 34' 37' 55" Index error + i 10 Dip for 14 feet Corr. altitude Semi-diameter True altitude 34 39 — 3 5 36 34 35 — I 29 15 34 34 + 16 H H 34 50 28 Decl. page II, N.A. Nov. 19th noon, 19° 37' 44" S. Correction -j" 3^ Red. decl. Ace. rate 6 306 H.D. -f 34"' 39 55""= '9 Eq. T. page II, N.A. Nov. 19th noon sub. i4"i95'57 Correction — -53 H.D. — o"594 X -9 19 38 15 S. 30"95i Polar dist. Altitude Latitude Polar dist. 70 21 45 34* 50' 28" 39 20 o sec. OTI1556 70 21 45 cosec. 0-026025 Red. Eq. T. 14 19-04 (To be subt. from A.T.) Hour-angle o'5346 4" 24 9' 144 32 13 72 16 6 COS. 37 25 38 sine 9-483672 9-783727 Hour-angle j^ 2'" 9" log. 9-404980 App. T. ship, Nov. 18^ Eq. Time Mean T. ship, Nov. 1 8'^ Mean T. Green, Nov. 19"* Longitude in time 19 57 14 51 19 19 43 32 54 55 5 II 23 Longitude 77=50' 45''W. i By Raper : Log. sin. sq. 9-405015, gives Hour-angle 4'' 2'" 10% Long. 77° 51' o" W. In this example, the observation is made a.m. at ship, the hour-angle is therefore the time before noon, Nov. 19^, and since all our computations are made in astronomical time — which dates from noon — we take the hour-angle from 24'', which gives apparent time at ship reckoned from noon of the preceding day. The mean time at ship and mean time at Greenwich are of different dates, mean time at Greenwich being more advanced, take the mean time at ship from mean time at Greenwich, borrowing 24'' to enable the subtraction to be made. Ex. 7. 1876, June 15th, p.m. at ship, latitude 13° 54' S., obs. alt. sun's l.l. 16'' 16' 16", index corr. -j- o' 16", height of eye 16 feet, time by a chronometer 15'^ o*> i6"> 16' (or c*" 16™ i6* P.M.) which was 2'' 13"* t,"]^ fast for mean noon at Greenwich, April ist, and on April 1 6th was 2'' 16"" 16^ fast on mean time at Greenwich. April ist, chronometer /flw< 2'»i3'"37'' i6th, „ fast 2 16 16 Change in 15 days Daily rate 2 39 io*-6 gaming. 2 39 60 '5)159(1°"^ 15 90 90 Longitude hy Chronometer. 275 T. by chron., June i^"* o^i6<^iG^ Original error — 2 1 6 1 6 14 22 o o Accumulated rate — ^o 35 Green, date, June 14 21 49 25 By Kaper : Index corr. -\- o' 1 6", dip — 4' o', refr. — 3' 18", par. -j- 8", semid. + 15' 47", true alt. 16° 25' 9". Interv al from Obs. alt.0's L. April 1 6th to June Index corr. 14th 22^, is 59'* 22''. Daily rate io-6 Interval 59 i6 feet 954 530 Corr. altitude 12 8 1 6254 5-3 Semi-diameter 2 i 4 3'5 ■9 True altitude + o 16 16 16 32 3 50 16 12 42 3 5 i6 4- 9 37 15 47 16 25 24 6,0)63,5- 10 351 Decl., page II, N.A. June 15th, noon 23'^ 20' 58' N. Correction — 13 Red. decl. Polar dist. Altitude Latitude Polar dist. 23 20 45 N. 113 20 45 1 6° 25' 24' 13 54 o 113 20 45 H.D. 5'79 1158 1158 12-738 sec. 0-012908 cosec. 0-037096 Eq. T., page II, N.A. 15th, noon, add o"i4*-7 Correction — i-i Red. Eq. T. 13-6 H.D. + o'532 X 2*^ 1-064 (To be added to A.T.) App. time at ship, June 15"* 4'' i9™38* Equation time -\- o 14 43 40 9 71 50 4 55 24 40 Mean time at ship, June 15 4 19 52 Mean time at Green. June 14 ai 49 25 cos. 9493825 Longitude in time 6 30 27 sine 9-915530 I Longitude 97^36' 45" E. By Raper : Log. sinfi sq. 9-459415, Hour- Hour-angle 4''i9'"38s log. 9'459359 angle 4'' 19"" 39". Longitude 97"^ 37' o" E. 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 15th ^h igm ^2; 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 2411 is borrowed in subtr^tijtig. Ex. 8. 1876, September 23rd, a.m. at ship, latitude 59° 30' N., observed altitude sun's h/ L.L. 10° 50' 10", index cbrreotion -)- 6' 10', height of eye 18 feet, time by chronometer ■LiA ii'> 44™ 20' (or 22^ II" 44™ 20S P.M.) which was 35"^ i(f fast for mean noon at Green- wich, July 14th, and on August 13th was 30" t^' fast on Greenwich mean time. July 14th fast 35™ 19' July 31 August 13th fast 30 4 14 Change in 30 days Daily rate T. by chron. Sept. 22'' ii'>44™2os Original error — 3° 4 Accumulated rate Green, date, Sept. 22 II 14 16 + 7 5 5 15 io'5 Interval from Aug. 13th to Sept. 22nd, III", is 40'' II >> Daily rate 10-5 40 22 n 21 21 By Raper: Index corr. -j- 6' 10", dip — 4' 10', refr. — 4' 56", par. + 8", semid. -}- 15*59', truealt. 1 1° 3' 21" r 420-0 t I 3'5 13 6,0)42,4-8 7 4-8 17 Aug. 13 Interval 30 Obs. alt. 0*8 L.L. 10° 50' 10" Index corr. -}- 6 lo Dip Corr. alt. Semi-diameter Trtie altitude i x 333 10 56 20 -^4 10 52 16 4 42 10 + 47 34 15 59 276 Longitude by Chronomd&r. Deel. page II, N.A. H.D. Sept. 22nd, noon 0° 4'5i"N. — 58''5o Correction — "4 ri'>2o"rz:ii-35 Red. decl. o 6 13 S. 6,o)66,3"975o Polar dist. Altitude Latitude Polar dist. 90 6 13 "" 3' 33" 59 30 o sec. 90 6 13 coseo. 160 39 46 80 19 53 69 16 20 sine o"29453i 0"00000I ;9-2a5i78 9-970938 Hour-angle 4*' 30™ 25* log. 9-490648 Eq. T. page II, N.A. 22nd, noon 7™30»2 Correction -j- 98 Red. Eq. T. 7 40-0 (To be subt. from A.T.) Hour-angle App. T. at ship, Sept. Equation of Time H.D. + o»-866 "•35 982910 4h 30>n25' 24 22'>I9 29 35 — 7 40 Mean T. at ship, Sept., 22 19 21 55 Mean T. at Green., Sept., 22 11 21 21 Longitude in time 8 o 34 Longitude 120° 8' 30" E. Ex. 9. 1876, 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'" ii^ on Green- wich mean time, April 6th, and on May ist, was 12™ 1* fast for mean noon at Greenwich. 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. Interval from May Obs. alt. ©'a l.l. ist to June 23rd, 6^, Dip is 53"! 6*^. 23 5 52 39 Daily rate i^-% Ace. rate + 2 29 Interval 53 T. of chron., Jan. 2 3<' 6^ 4™40« Original error — 12 i Green, date, June 23 5 55 8 84 140 By Raper : dip — 4' 20', refr. — 2' 36" par. -\- 8", semid. -|- 15' 46", and true 6 | | | 148-4 Corr. alt. Semi-diameter True altitude 20" 2J 4 17 20 20 43 » 25 20 + 18 18 15 46 alt. 20' 33' 58' Decl. page II, N.A. 23rd noon 23° 26' 17 "N. Correction — 15 7 6,0)14,9-1 Red. decl. Polar dist. Altitude Latitude Polar dist. 23 26 2 N. 66 33 58 20" 34' 4" 000 66 33 58 H.D. 2-48 5-9 2232 1240 14-632 2 29-1 Eq. T. page 11, N.A. 23rd noon i'»58«-8 Correction 3-2 Rf 1. Eq. T. sec. 0-000000 cosec. 0-037385 {Added to A.T.) Altitude 20° 34' 4' Latitude 000* Polar dist. 113 26 2 87 8 Hour-angle Eq.T. 43 34. I 22 59 57 4h29™57» + 2 2 cos. sine log. 9-860080 9'59i863 9-489328 M.T.S. 23d 4 31 59 M.T. G. 23d 5 55 8 Long, in T. i 23 9 Longitude 20° 47' 15" W. '34 6 67 3 gle 46 25 59 Hour-an Eq. T. 4'' + 29" 2 '57 2 M.T.S. M. T. G. 23d 23« T. 4 5 31 55 59 8 Long* in I 23 9 sec. cosec. cos. sine log. 20 34 4 H.D. •540 5-9 4860 2700 3-1860 0-000000 o"037385 9'59i863 9-860080 9-489328 Longitude 20° 47' 15" W. By Kaper the answer comes out the same. Longitude hy Chronometer. '■11 X. Ex. 10. 1876, Sept. 22nd, P.M. fit ship, lat. 0° o', obs. alt. sun's l.l. 28° 52', height of eye 17 feet, time by chronometer Sept. 22'' 4*' 59™ 41', which was slow 15^ on Greenwich mean time, April 30th, and on June ist -w&sfast ios-6 on Greenwich mean time. April 30th, chronometer slow ©'"15s June ist, „ fast o io'6 Change in 32 days Daily rate T. by chronometer, Sept. 22^ ^^^^"'j^i^ Original error — . io-6 25-6 o"8 gaining. June April May June 30 30 31 Accumulated rate Greenwich date, Sept. 22" 4 59 30'4 — I 30-4 22 4 58 o July Aug. Sept. 30 I 29 31 31 Interval 32 Interval 1 1 3 days. Rate o"8 6,o)9,o'4 Ace. rate i 30-4 4h-97 Interval 113 Decl., page II, N.A. Sept. 22nd, 0° 4' 5o""5 N. deer. Corr. for 5''"89 — 4 50-7 H.D. - 58"-5o 497 Eq. T. Sept. 22nd, sub. 7'"3o»"2i Corr. 5h-9 + 4-33 Red. decl. o Obs. alt. sun's l.l. Dip Corr. altitude Semi-diameter True alt. Zenith distance Do. in time 6,0)29,0-7450 4 50'7 lied. Eq. T. 7 34-54 (To be subt. from A.T.) H.D. 0^-866 4" 3 30 28 52 3 57 28 48 I 3 35 28 + 46 15 28 59 29 2 27 90 O O 60 57 33* 4h 3m^o«-2 App. T. ship, Sept. £q. T. 4 3'"50''2 — 7 34-5 Mean T. ship, Sept. 22'^ 3 56 15*7 Mean T. Green., Sept. 22'! 4 58 o-o Long, in time Longitude I I 44'3 15° 26' 4" W. By Raper : True alt. 29° 2' 16", whence H.A. (or zen. dist.) in time =: 411 3™ 50»-9. Long. 15° 25' 54" W.k Ex. II. 1876, October loth, p.m. at ship, latitude at noon 20° 41' S., ship had sailed N.E. (true) 54 miles since noon, obs. alt. sun's l.l. 18° 45', height of eye 15 feet, time by chro- nometer October ()^ i6'» 28" 42% which was slow 11™ 44' on mean time at Greenwich, August 26th, and on September loth was slow 10™ 26' : required the longitude at time of observation, and also at noon. To find the 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 named North, because the ship at the time of sights 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 difif. 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. * Both latitude and declination being o, the zenith distance converted into tiiAe is the hour-uigle. 278 Longitude hy Chronometer. The daily rate is 5''2 gaining; the interval 29<*. iG^ X 5''2 gives accumulated rate 2™ 34*'3; Greenwich date October 9<* 16'' 36"" 34''; polar distance 83° 14' 2"; red. eq. T. 13" 2«-25 subt. from A.T. ; true alt. (Norie) 18° 54 43 " ; latitude in at sights 20' 2' 48' S. ; hour-apgle 4*'49'n 10'; mean time at ship October io'*4*' 36"' 8'; long, at time of observation 179° 53' 30" E.; difF. long. 41 ; also longitude at noon 179° 25' 30" W. Examples for Practice. * Ex. I. 1876, 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** S"" 50* (being 7'^ 8" 50' a.m. at Greenwich), which was slow 18" 2' for mean noon at Greenwich, November 30th, 1875, and on December 7th was 19"" io»-6 slow for mean time at Greenwich: required the longitude. ^ Ex. 2. 1876, February 19th, a.m. at ship, latitude 38° 18' S., observed altitude siin's l.l. 21° 30' 40", index correction — 6' 45", height of eye 14 feet, time by a chronometer i8<* 19'' 53™ 37*"6 (being -^^ 53™ 37*"6 a.m. at Greenwich), which was 4"" i6^-6 fast for mean noon at Greenwich, January 23rd, and on January 30th was 5" 9^*8 fast for mean time at Greenwich. Ex. 3. 1876, 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, time by chronometer 28<* o'> 10" ■^ (being o^ 10'" p.m. at Greenwich), which was 54" 48* fast for mean noon at Greenwich, October 20th, 1875, and on December 2nd, 1875, was ji" 56* fast for mean noon at Greenwich. V Ex. 4. 1876, April 6th, a.m. at ship, latitude 53° 5' N., observed altitude sun's l.l. 16° 8' 40", index correction — 40", height of eye 15 feet, time by a chronometer 5'^ 19'' 18" 49' (being 7'' 18™ 49' a.m. at Greenwich), which was o"" 4'-4 slow for mean uoon at Greenwich, Febru£u:y nth, and on March i ith was 2"^ "38» fast for mean noon at Greenwich. Ex. 5. 1876, May 19th, p.m. at ship, latitude 2" 58' S., obs. alt. sun's l.l. 30' 30', index correction -|- 52*, height of eye 19 feet, time by chronometer 19'* o^ 23™ 58', which was 28' fast for mean noon at Greenwich, January 3rd, and on January 31st was 42" slow on mean time at Greenwich. Ex. 6. 1876, 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 1 7 feet, time by a chronometer 14'^ 1 7'^ 59" 30^ (being 5*' 59™ 30^ a.m. at Greenwich), which was slow 5"» s^'^'l ^or mean time at Greenwich, April 20th, and on May 12th was 2" 29»'5 slow for mean noon at Greenwich. ^ Ex. 7. 1876, July 5th, a.m. at ship, latitude 23^^ 48' N., observed altitude sun's l.l. 48° 36' 50", index correction — 50", height of eye 17 feet, time by chronometer 5^o'»42™ 38^ (being o*^ 42™ 38^ p.m. at Greenwich), which was/««<^4'" 47^*8 for mean noon at Greenwich, May 6th, and on June ist was/as^ 6°" 50* for mean noon at Greenwich. \ Ex. 8. 1876, August 13th, A.M. at ship, latitude 30° 46' S., observed altitude sun's l.l. 27° 15', index correction — i' 15", height of eye 21 feet, time by a chronometer i3"i z^ o" (being 2^> o™ p.m. at Greenwich), which was sloio 26™ 7''6 for mean noon at Greenwich, April loth, and on May ist, was slow 25" 13'' for mean noon at Greenwich. > Ex. 9. 1876, September ist, p.m. at ship, latitude 35° 49' N., observed altitude sun's l.l. 44° 32' 10", index correction ■\- 1' 46", height of eye 20 feet, time by chronometer August 31'* igh 24'" 57' (being 7'' 24" 57^ a.m. at Greenwich), which vf&ifast 1 1"" ^7*'4 for mean noon at Greenwich, July 3rd, and on July 3 ist was/as< 12"" 17* for mean noon at Greenwich. Ex. 10. 1876, October 25th, p.m. at ship, latitude 51° 30' S., observed altitude sxm's l.l, 40° 22', index correction — i' 50", eye 20 feet, time by chronometer 25'' %^ 22™ i» (or gh 22"" i« P.M.), which was slow 24™ 8"a for moan noon at Greenwich, June 14th, and on July 20th was slow ii" 19' fbr mean noon at Greenwich. Longitude hy Chronometer. 270 Ex. II. 1876, November 27tli, a.m. at ship, latitude 39° 20' S., observed altitude sun's l.l. 34° 37' 55". index corr. + i' 15", eye 18 feet, time by a chronometer 27^ 7^ 41"-- 30^ (being P.M. at Greenwich), which vr-dafast 31"' 54' for mean noon at Greenwich, October 20th, and on November 9th was 29"^ 40^ fast on mean noon at Greenwich. Ex. 12. 1876, December 24th, a.m. at ship, latitude 9° 59' S., observed altitude sun's l.l. 10° 38' 45". index correction — 3' 12", eye 18 feet, time by a chronometer 23^ i7h ^6'^ o» (being a.m. at Greenwich), which was slow 34™ 19'- 1 for mean noon at Greenwich, July ist, and on July 29th was slow 38™ 39^-5 for mean noon at Greenwich. Ex. 13. 1876, January ist, p.m. at ship, latitude 38° 28' S., observed altitude sun's l.l. 39° o', index correction — 2' 25', eye 12 feet, time by chronometer i-^ ii^ 58™ 29^ (being P.M. at Greenwich), which was slow 1^ 49>n 19s for mean noon at Greenwich, on September 12th, 1875, ''nd on October 13th was i^ 52" 538 slow for mean noon at Greenwich. Ex. 14. 1876, February nth, a.m. at ship, latitude 53° 12' N., observed altitude sun'a L.L. 12° 10', index corr. — 49", eye 12 feet, time by chronometer 10*1 22h 22™ 22* (being a.m. at Greenwich), which wdsfist 34"! 4i3'7 for mean noon at Greenwich, October 31st, and on December ist, 1875, was/«s< 38™ 59^ for mean noon at Greenwich. Ex. 15. 1876, October 26th, a.m. at ship, latitude 28° 10' N., observed altitude sun's u.l. 25° 32' 20", index correction o", eye 17 feet, time,by chronometer o'^ 54m 6* (being p.m. at Greenwich), which was fust 31™ 31' on mean time at Greenwich, August ist, and on Sept. 4th wda fast 30" 6^ for mean noon at Greenwich. Ex. 16. 1876, February 6lh, p.m. at ship, latitude 6° 58' N., observed altitude sun's u.l. 21" 43' 40", index corr. o', eye 18 feet, time by a chronometer ii^ 40™ 26* (being a.m. at Greenwich), which was slow 16™ 48-8 on mean noon at Greenwich, January 2nd, and on January 20th was slow 17"" 42' on mean noon at Greenwich. Ex. 17. 1876, May ist, p.m. at ship, latitude 21° 8' N., observed altitude sun's l.l. 28° 5' 30", index corr. + 2' 50", height of eye 16 feet, time by a chronometer April 30^ 18'' 50" 29«'4 (being 6'' 50™ 29''4 a.m. at Greenwich), which was 10"^ 12^ slotv for mean noon at Grecnwiih, December 3i3t, 1875, and on February 17th, 1876, was 7™ 33''6 slow for mean noon at Greenwich. Ex. 18. 1876, April 2ist, P.M. at ship, latitude at noon 0° 20' N., observed altitude sun's U.L. 32° 2i' 10', index correction — i' 10", eye 12 feet, time by a chronometer 3^ 44m i3 (being a.m. at Greenwich) which was slow 9"" 7^ for mean noon at Greenwich, November 14th, 1875, and on January nth, 1876, was slow 7™ 34'*'2 tor mean noon at Greenwich, course since noon S.W. by W. (true), distance 36 miles: required the longitude at the time" of observation, and also at noon. Ex. 19. 1876, August 2i8t, A.M. at ship, latitude at noon 0° 20' S., observed altitude sun'a L.L. 33° 49', index correction + 2' 10", ej'e 15 feet, time by chronometer ^^ 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. Ex. 20. 1876, March 20th, a.m. at sliip, latitude 0=^, observed altitude sun's l.l. 28" 50' 10", index corr, -}" ''» ^J'^ ^3 ^6®*, time by chronometer 2o» (being p.m. on 14th at Greenwich), which -was fast 4"^ 35' for mean noon at Greenwich, March 20th, and on May 3rd was slow I" 17' for mean noon at Greenwich. VARIATION BY AN AZIMUTH. In this problem the Error of the Compass is required by computing the true bearing of the sun, and taking the difference between the true bearing and the bearing by an Azimuth Compass. 300. The Azimuth of a heavenly body is the arc of the horizon inter- cepted between the cardinal point adjacent to the elevated pole, and the circle of altitude passing through the body, or it is the angle at the Zenith contained between the vertical circle passing through the elevated pole (the meridian) and tlie vertical circle passing through the object. Azimuth is usually reckoned from the north or south point, eastward and westward from 0° to I 80°. 301. True Azimuth is the bearing of an object from the true north or south point, and is the azimuth found by calculation from the observed alti- tude or hour-angle of the body. It is in general simply called The "Azimuth'^ but it is thus qualified as the True Azimuth to distinguish it from the Mag- netic Azimuth, which is the bearing of the object from the compass North or South point, and which is found by direct observation with an instrument carrying a magnetic needle. The difference between the true and magnetic azimuth gives the entire correction of the compass — variation and deviation combined. 3or"-\ Given the latitude, altitude, and declination of an object, to find the true azimuth. EULE XCVIII. 1°. Add together the polar distance, the latitude, and the altitude* take half the sum, and take the difference between the half sum and the polar dist. Note, — When the latitude is o, suppose it to be of contrary name to the declination when finding the polar distance. 2°. Add together the log. sec. of latitude, the log. sec. of altitude, (rejecting tens J, the log. cosines of the half sum and remainder ; the sum (rejecting tens J is log. sine scfuare of true azimuth (Table 69 Eaper). Or half the sumof the four logs, is the log. sine of half of the true azimuth, which take out of the table (Table 24 Norie), and double it ; the result is the true azimuth. 3°. Reckon the true azimuth from S., when the latitude is N., but from the N. when the latitude is S. ; towards E. when it is a.m., or when the altitude is in- creasing, but towards W. when it is p.m., or when the altitude is decreasing. (a) When latitude is 0°, if declination is N. reckon the azimuth from the South; «/ declination is S., reckon the azimuth from the North. (b) When both latitude and declination are 0°, the object moves on the prime vertical, or is E. while the altitude is increasing and W. while the altitude is decreasing. Note. — The logs, are taken out in these examples to the nearest second. * The learner will observe that in this formulje the pol. dist., lat., and alt. occur in the reverse order of that in Eule XCVI (finding hour angle) in which the initials form the word alp. In finding the azimuth the initials form pla. The 2nd and 3rd terms, take secants ; the last two, cosines. By this arrangement the term which has to be taken for the half sum is always on the top. Variation hj mi Azimuth. Ex. r. Given the latitude 47° 46' S. ; declination 22° 27' 22' (or polar distance 67° 32' 38") : true altitude 26"^ 44' decremimj or (being p.m.) Polar dist. 67° 32' 38" Latitude 47 46 o sec. Altitude 26 44 o sec. Examples. Ex. 2. Latitude 37° 15' N., declination 22*^ 22' 58' N., true altitude 39"^ 20' 8' — (p.m.) Polar dist. 67° 37' 2" 37 15 o sec. o'099o86 39 20 8 sec. o* 1 1 1570 0172533 0-049095 Sum Half sum 142 2 38 71 I 19 Half sum — p.d. 3 28 41 cos. 9-512159 cos. 9-999200 2)19-732987 True az. N. 47° 20 12- sine 2 94 40 W. 9-866493 470 12" = 23 The true azimuth is here marked N. because the latitude is S., andW. because the altitude is decreas- ing, it being p.m. Ex. 3. Latitude zS'' 3' N., declination 12° 39*50" S., true altitude 25° 12' ^'-\- (a.m.) Polar dist. 102° 39' 50 " 28 3 o 25 12 4 Latitude Altitude sec. sec. 0-054267 0-043438 Sum 155 54 54 Half sum 77 57 27 cos. 9-319392 P. dist. — 2 sum 24 42 23 cos. 9'958307 2)i9'375404 Half azimuth 29° 10' 22" sine 9-687702 True az. S. 58 19 E. The sum of the four logs., rejecting 10 from the index is the log. sine square of true azimuth ; seek for log. in Table 69, Raper, and the corresponding arc is 58° 19', whence true azimuth is S. 58° 19' E., the same as by Norie's Table. Ex. 5. Latitude 34" 19' S., declination 7° 5' 27" S., true altitude 40° 55' 57 ' (+ p.m.) Polar dist. 82° 54' 33" Latitude 34 19 o sec. 0-083054 Altitude 40 55 57 sec. 0-121776 Sum Half sum Remainder 158 9 30 79 4 45 cos. 3 49 48 cos. 9-277500 9999029 2)19-481359 33 23 40 sine (, ^40679 True az. X. 6^ 47 20 E. I Latitude ! Altitude Sum Half sum 144 12 10 72 6 5 Half sum— p.d. 4 29 3 44° 51' 58" sine 9-848467 cos. 9-487610 COS. 9-998668 2)19-696934 True azimuth S. 89 44 W. The sum of logs, (less 10 in the index) being found in the log. sine square (Table 69, Raper), gives true azimuth as above — without the trouble of halving the sum of logs, and multiplying the arc correspond- ing thereto by 2. Ex. 4. Latitude 38' 46' N., declination 7° 41' 56" S., true altitude 27^ 16' 8"— (p.m.) Polar dist. 97° 41 '56" Latitude 38 46 o sec. 0-108071 Altitude 27 16 8 sec. 0-05 11 64 Sum Half sum Eemainder 163 44 4 81 52 2 cos. 15 49 54 cos. 9" 150657 9-983206 2)19-293098 26 18^ sine 9-646549 True azimuth S. 52 36^ W. Ex.6. Latitude o' declination 1 5° 2' 2 7 "N. True altitude 24^* 12' lo" — Polar dist. Latitude Altitude Sum Half sum Remainder 105° 2 27" 000 24 12 10 sec. o"039957 129 14 37 64 37 18 COS. 9-632045 40 25 9 cos. 9-881568 2)i9'553570 36 44 sine 9-776785 S. 73 38 W. 282 Variation by an Animuth. Examples fok Pkaotioe. In each of the following examples it is required to find the true azimuth : (The sign + means a.m., and the sign — means p.m.) +r. True altitude 7° 43' 27'' + Declination ii»28' 32" N. Latitude 51° jo' N. <^- » 28 30 S3 + 21 56 45 S. 26 20 N, 1-3- )> 12 50 46 — 9 36 51 N. 15 47 S. Hr 4- i> 29 41 59 + 2 38 14 N. 4 22 N. JhS- „ 7 15 55 — 12 14 38 S. 51 2 N. ^6. ,1 13 47 28 — 17 50 57 N. 42 36 S. ^7. >> 45 30 — 23 2 S. -^ 8. „ 25 40 10 + 000 9- ») 40 7 21 — 17 4 3 «• 33 51 S. 301. Given 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 Azimuths from the same point of the compass — North or South. (a) If one of the azimuths be expressed from the North and the other from the South, take either of them from 180°, and it will then be reckoned /row the same ^oint as the other. (b) Jf the bearing by compass be reckoned from East or West, towards North or South, take it from 90°, and reverse the position of the letters ; or, add 90°, and it will then be expressed from the opposite ^om^ to that from lohich it is reckoned when taken from 90°. Example. Ex. Suppose magnetic azimuth to beW. 78^ 30' N ; then subtract the magnetic azimuth from 90° thus ; — 90° 00' W. 78 30 N. The azimuth is thus reckoned N. 11 30 W. from the north pole. (c) When the magnetic azimuth is either East or West, it is to he reckoned as ^0° from North or South, according as the true azimuth is North or South. 2°. Take the difi'erence of the true and magnetic azitnuths when ifieaszired to- wards the same point of the com pass, East or West ; but when measured toivards different joo/w^s, i.e., when one is reckomd towards East and the other towards West, take the sum ; the result is the ei-ror of the compass or correctiou. 3°. To name the Correction of Compass. — Let the obsercer look at the two azimuths for bearings J from thu centre of the compass — then if tfie true azimuth is to the right of the magnetic azimuth, the correctiou is East j but if tJie true fdiimuth IS to the left of the magi.etic azmuth, the error is West. Variation hy an Aiimuih. 283 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 27 10 E. The observer being supposed looking from the centre of the compass in the direction of the s magnetic azimuth, then the trttc azimuth lies to the rij/ht hand of the magnetic azimuth, whence the error of compass is to be marked £(ist. Ex. 3. Given true azimuth S. 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. E The observer being supposed looking from the centre of the compass in the diri'ction of the magnetic azimuth C M, then the true azmmth, C T, lies to the rir/ht hand of the magnetic azimuth, whence the eiTor of the com- pass is Hast. Ex. 5. The true azimuth S. 62° 41' E., and magnetic azimuth E.S.E. : required the error of compass. True az. S. 62° 41' E. S. 6 pts. E. = Mag. az. «. 67 30 E. Error of compass 4 49 ^• Here the error of compass is East, since the true azimuth is on the right of the magnetic azimuth, the observer looking from the centre of the compas^s in the direction of the magnetic azimuth. Ex. 7. True azimuth N. 72° E., mas netic azimuth East. True azimuth 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 of compass. True azimuth S. 90°33'E. 180 o or, N". 89 27 E. Mag. azimuth N. 81 20 E. Error of compass 8 7 E. The truf azimuth being reckoned from S., while the inau'iKtic a/ciniuth is e:. jires-eii t)om N-, the t;ae is subtracted horn 180^, in order to reckon it Ukjil. the same jfaint as the magne^c azimuth, viz. , from N. Ex. 2. Given true azimuth S. 70" 57' E. the magnetic azimuth S,E. hy E. f E. : required the error ot 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. \ : Error 6 15 45W. E. The error of compass is in this | in'itance 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 nzimulh N. 61" 50' E. : re- quired the correction of compass. True azimuth N. 50^ 12' E. Magnetic azimuth N. 61 50 E. Error of compass 11 38 W. The error of compass is here West, because the tnie azimuth is to the left hand of the magnetic azimuth, the observe r being supposed to Ljuk from the centre (if the compass in the dii'ection of the magnetic azimuth. Ex. 6. The true azimuth S. 82' 50' W., and magnetic azimuth W. 15"^ N. True az. S. 82° 50' W. 15° N. = Mag. az. S. 105 W. W. Error of compass 22 10 W. The error of compass is West, the true azimuth being to the li'ft of magnetic, 90° is added to the compass bearing in order to reckon it from the same point as the true azimuth ; thus, 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. The true azimuth is S. 76'^ W., and the magnetic azimuth West. True azimuth S. 76' o' W. Mag. 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 S. 93° 30' W. : find the error of compass. True azimuth N. 69^^39' W. 180 o or, S. no 21 W. Mag. azimuth S. 93 30 W, Error of compass 16 51 E. The true azimuth is here talcen from 180", in order to rei kuu it Irum the t>ame point as the magnt:tio assimu'th. 284 Variation by an Azimuth. Ex. II. True azimuth S. 36° W., mag- netic azimuth S. 9" E. True azimuth S. 36° W. Mag. azimuth S. 9 E. Error of compass 45 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. 13. True azimuth N. 49° E., mag- netic azimuth N. 3' W. True azimuth N". 49° E. Mag. azimuth N. 3 "W. Error of compass 5 a E. Ex. 14. True azimuth S. 50° E., mag- netic azimuth S. 8° W. True azimuth S. 50° E. azimuth S. 8 W. Error of compass 58 W. EULE C. i^. With ship time and longitude in time find the Greenwich date (Rule LXXXI, page 222.) 2°. Talce from page II, Nautical Almanac, the sun's declination and reduce it to Greenwich date (Rule LXXXII, page 222); also take out sun's semi- diameter. If apparent time is given, use Nautical Almanac, page I. 3°. Correct observed altitude for index error, dip, refraction, parallax, and semi-diameter, and thus get the true altitude (Rule LXXXVII, page 236). 4°. Proceed according to Rule XCVIII, page 280, to find the true azimuth. 5°. Having found the true azimuth, proceed by Rule XCIX, page 282, to find the entire correction or error of the compass. 6°. Next apply the variation to the error of compass according to Rules 7° and 8° of Rule LXXXIX, page 245, the result is the deviation /or the position of the shipi's head at the time of observation. Examples. \ Ex. I. 1876, May 19th, 3'^ 7'" 44' p.m., mean time at ship, latitude 41° 53' N., longitude 60° 19' W., sun's bearing 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 19'' 3^ 7 '"44" Long. 60° 1 9"W. in time +4 i 16 Green, date (M.T.) May 19790 By Raper : Dip — 4' 10", ref. — i' 1 ", par. ■\- 6', semid. -\- 15' 50". True altitude 44" 6' 54*. Obs. alt. O's L.L. Dip 43° 56' 7" — 4 4 Corr. altitude 43 52 3 — 53 Semi-diameter 43 51 I® + 15 50 True altitude 44 7 Variation hy an Aaimuth. 285 Polar dist. Latitude Altitude 70° I '52" 41 53 44 7 sec. sec. COS. COS. 0-128132 0-143922 9-317284 9"99577i Decl. page II, N.A. 19th, noon, 1 9° 54' 22 "N. Correction +3 4^ Eed, decl. 19 58 8 90 Polar dist. 70 i 52 Correction H.D. + 3 1 "'65 7"i5 Sum Half sum 156 I 52 78 56 7 59 4 15825 3165 22155 Remainder 6,0)22,6-2975 2)19-585109 + 3' 46" Half azimuth True azimuth S. 76 40 W. . azimuth S. 104 40 W. Error of compass 28 o W. Variation 17 10 W. Deviation 10 50 W. V N. The true azimuth being to the left of the magnetic, the errm- of compass being to the left of the variation. Ex. 2. 1876, September 2nd, mean time at ship 8'» 59"" a.m., latitude 39° 31' S., longitude 127° 45' W., sun's bearing by compass N. 29° 50' E., observed altitude sun's l.l., 26° 40' 37", height of eye 1 8 feet : required the true azimuth and error of the compass : and supposing the variation to be 9° 50' E. : required the deviation of the compass for the position of the ship's head at the time of observation. Ship date (M.T.) Sept. 1^20^59" Long. 127° 45' W. +831 Grn. date (M.T.) Sept. 2 5 30 By Eaper: Dip — 4' 10", ref. — i' 56', par. + 8", semid. + 15' 54". True alti- tude 26° 50' 33'. Decl. 2nd, p. II, N.A. 7° 42' 31" N. — 5 2 Correction Reduced declination Polar distance Obs. alt O's L.L. Dip. (T. 4) Corr. altitude Semi-diameter True altitude Hourly diflf., noon, 5^ 30"' = 7 37 29 N. 90 o o 97 37 29 ; 26° 40' 37* — 4 4 26 36 33 — I 45 26 34 48 + 15 54 26 50 42 54"-95 X55 27475 27475 6,0)30,2-225 5' 2" Polar dist. Latitude Altitude Sum Half sum Remainder Half azimuth 97 37 29' 39 31 o 26 50 42 163 59 II Half true azimuth 26° 11' 50" 0-112690 0-049523 81 59 35 COS. 9-143931 15 37 54' COS. 9-983633 2)19-289785 26° 1 1' 50" sine 9-644892 True azimuth Magnetic azimuth Error of compass N. 52 24 E. N. 29 50 E. 22 34 E. The true azimuth being to the rij/ht of magnetic. To find the Deviation. Error of compass -}" 32°34'E. Variation 9 50 W. i Deviation 4- T3 44E. 296 Variation hy an Azimuth. > Ex. 3. 1876, July 5th, mean time at ship 6^ 55" 51' p.m., latitude 50° 53' N., longitude 119° 8' E., sun's bearing by compass N. 69° o' W., observed altitude sun's l.l. 9° 40', index correction + 3' 50", height of eye iS feet, variation 4° o' W. Ship date (M.T.) July s<^ S^ss'Si^ Long. 119'' 8' E. — 7 56 32 Green, date (M.T.) July 4 22 59 19 Hourly diff. Time from noon i'' Correction Decl., noon, July 5th, Red. decl. Polar dist. i4"-64 i4'-64 or + 15" 22 44 58 N. Obs. alt. O's L.L. Index correction Dip Corr. altitude Semi-diameter True altitude 9' 40' o" + 3 JO 9 43 4 JO 4 9 39 5 46 17 9 + 34 15 29 46 22 45 13 N. 90 67 14 47 9 JO iJ By Raper : Index corr. 4" 3' 50", dip — 4' 10', ref. — 5' 31", par. -\- 8", semid. -|- iS 46". True alt. 9° 50' 3", Polar dist Latitude Altitude Sum Half sum Remainder Half azimuth 67° 14' 47" 50 53 o 9 JO ij 127 58 2 63 J9 I 3 15 46 True azimuth sec. 0*200038 Magnetic azimuth sec. 0006434 Error of compass N. 65° 51' 40" "W. N. 69 o o W. 3 8 20 E. COS. 9-642097 The true and magnetic azimuths being recUonfd Irom opposite points, the true is cos. 9-999295 taken from 180°, aiid the remainder reckoned from the oppoaite point, whence true azimuth 2)19-847864 is N. 65" 52' W. The true azimuth being to the right of magnetic, the error of compass is East. 57° 4 10" sine 9-923932 True azimuth S. 114 8 20 "W. 180 o o or, N. 65 51 40 W. To find the Deviation. Error of compass 4- 3° 8' E. Variation — 40 W. Deviation +7 8 E. Ex. 4. 1876, February loth, at 8'' 2™ a.m., mean time at ship, latitude 50° 48' N., longi- tude 77^ 30' W., sun's bearing by compass S.E. by E. ^ E., observed altitude sun's l.l. 7° 10' 40", index correction — i' 6", height of eye 15 feet: required the true azimuth and error of compass ; variation 14° o' W. Ship date, M.T., Feb. Long. 77° 30' W. gd 20^" 2' + 5 10 Feb. 9 25 12 — 24 Green, date, M.T., Feb. 10 I 12 Hourly diff. decl. noon Time from noon i** i2'» - 48"-55 X 12 Correction Decl. loth, noon 14 — 58"-26o 27 36 «. Red. declination 14 26 38 S. 90 Ptolar difittmce 104 *6 38 Obs. alt. O's L.L. Index correction Dip for 15 feet Correction of altitude Semi-diameter True altitude 7° 10 40" - I 6 7 9 34 3 42 7 5 J2 7 6 6 J8 46 16 14 7 IJ Bv Bnper; Indf-x corr. — i' 4'', dip — 3' 50', rtri. — 719, par. -^ 9', seiuid. -|- 16 14", true alt. 7^ 14 50 . Vcvriatmi hy im Azv/nutk. 287 Polar dist. Latitude Altitude Sum Half sum Kemainder Half azimuth 104° 26' 38" 50 48 o sec. 7 15 o sec. 0199263 0-003486 162 29 38 28" II' True azimuth S. 56 22 E. Mag. az. (S.E. hy E. ^ E.) S. 59 4 E. il 14 49 COS. 23 II 49 COS. 9-182347 9"963389 2)19-348485 Error of compass Variation + 2 42 E. — 14 o W. Deviation + 16 42 E. 28° 11' 7" sine 9-674242 The error is East, the true azimuth heing to the riffht of the magnetic. Ex. 5. 1876, January 2i8t, at 10'' 14'" a.m. app. time at ship, lat. 39° 3' S., longitude 96° 28' E., sun's bearing hy compass E. 2° 30' S., observed altitude sun's u.l. 46° 15', index correction — 2' 43", height of eye 19 feet : required the true azimuth and error of compass ; variation 17° o' W. : find the deviation. Ship date, M.T. Jan. Long. <)&" 28' E. 20'' 2 2*" 14'" O* — 6 25 52 Green, date, M.T. Jan. 20 15 Hourly difF. 2i8t, noon Correction Decl. 2i8t, noon, page I Red. decl. Polar distance Polar dist. 69° 56' 30" Latitude 39 3 Altitude 45 51 6634 26536 6,0)27,1-994 + 4' 32" 19 58 58 S. Obs. alt. 0*8 u.L. Index correction Dip for 19 feet Correction of altitude Semi-diameter True altitude 46' 15' 0' — 2 43 46 12 17 — 4 ir 46 8 6 — 49 46 7 17 — 16 17 45 51 o 20 3 30 b. 69 56 30 sec. 0-109805 sec. 0-157054 By Raper : Index corr. — 2' 43", dip - 4' 15", ref. — 56", par. + 6'', semid. - 16' 17", true alt. 45° 50' 55". True azimuth N, 78''22'E. Mag. azimuth N. 92 30 E. Sum Half sum 154 50 30 rror of compass Variation — 14 8 W. — 17 o W. 77 25 15 cos. 9-338035 Polar dist. — do. 7 28 45 cos. 9-996289 2)19-601183 39° 11' 4' sine 9800591 Deviation + 2 52 E. In this example the best way is to reckon the magnetic azimuth from the North, the same as the true ; thus from N. to E. is 90°, and from E. to E. 2° 30' S., is 2° 30' ; therefore, the magnetic azimuth is N. 92" 30' E. True azimuth 78 22 Ex. 6. 1876, June ist, at 9'> 40"" a.m. mean time at ship, latitude 60" N., longitude 40'^ 20' W., observed altitude sun's l.l. 44° 48' 50", index correction + 3' 17", height of eye 18 feet, sun's bearing by compass S. h W. : required the true azimuth and error of compass ; variation 51° 15' W. The Greenwich date is June !<• oh 21" 20^ True altitude (Norie) 45° 3' o", hourly difiF. of decl. i9'-8i X .^^ = 7", which, addid to docJ. June istat noon, viz., 22° 8' 52" N., gives he red. decl. 22^ 8' 59" N., polar distance 67^ 51' i", sum of logs. 19-220455, true aaimuth S. 48=" 6' E. 288 Vernation by an Azimuth. True azimuth S. 48° 6' 26" E. S. ^ point W. ^ Mag. azimuth S. 5 37 30 "W. Error of compaas — 53 43 56 W. true azimuth left of way. Variation — 5' ^5 ° W. Deviation — 2 28 56 W. because cnw is left of variation Examples foe Pkaotice. In each of the following examples it is required to find the true azimuth, also the error of compass and deviation for the position of the ship's head at the time of observation. No.' Civil date. 1876. M.T. ship. I Latitude. ' Longitude. 10 • •11 1-12 -14 •15 16 17 18 19 -H 20 -t 21 22 23 Jan. 24th ! 8''22"'3s»A.M. Feb. 28th ! 3 14 o P.M. March 27tli ' 4 6 40 p.m. April 3rd 6 20 o P.M. May 27th ' 9 3 20 A.M. June 20th 6 10 o P.M. July 31st ' 8 46 30 A.M. Aug. 23rd S 54 58 A.M. Sept. 1st 3 47 50 P.M. Nov. 25th 470 P.M. Dec. 17th i 9 10 30 A.M. July 3rd I 8 26 50 A.M. Jan. 6th ' 5 2 14 p.m. April 2sth ! 7 56 41 a.m. Jan. 29th I 3 36 35 P.M. Feb. 1st I 3 44 51 P.M. March 26th , 9 5 50 a.m. Feb. 26th 2 48 o P.M. June 21st • 3 22 o P.M. Sept. 11th j 700 A.M. April 25th I 8 50 o A.M. March ist I o 35 o p.m. 1877, Jan. 1st ...j 9 27 10 A.M. 26° s'S. 38 46 N. 4 22 N. 49 59 N. 55 oN. 43 45 N. 38 18 N. 51 ID N. 10 40 S. 39 58 S. 29 10 S. 32 10 s. 47 46 S. 27 20 S. 42 26 N. 33 51 S. 43 6N. 5 o N. 66 40 N. 37 o S. 5 35 N. IS 30 N. 5o°53'W. 97 16 W, 53 7 E. 169 58 E. I 33 W. 11 26 "W. 65 4 w. 135 40 "W. 138 42 E. 50 52 "W. 26 53 W. 62 o E. 33 II E. 86 43 W. 49 18 W. 20 37 E. 51 2 "W. 167 o E- 55 20 "W. 19 o "W. 94 30 E. 6s o E. o 25 E. Sun's Obs. I Ht. I bearing by alt. sun's of I Variation, compass. l.l. .eye.] E. by N. S. 42° 36' W. W. by N. West. S. 61 45 E. N. 43 20 W. S. 75 20 E. N. 66 20 E. W. JN. W. iN. S. 84 20 E. N. 62 o E. N. 84 40 W. N. 61 s° E. W. 9 10 S. N. 70 50 W. S. 44 50 E. N. 118 34 W. N. J E. N. 44 50 E. N. 75 46 E. S. iE. S.E. 38° 23' 10 26 57 14 29 30 so 11 43 o 43 8 51 16 40 20 43 24 58 7 38 o 30 4 10 33 51 o 51 I 13 14 II 37 26 37 27 18 44 55 13 38 46 39 56 10 32 40 o 60 37 o 15 38 o 42 28 o 42 55 o 66 14 o 45 10 50 4°36'E. 11 30 E. 3 30 W. 9 10 E. 15 45 W. 23 o W. 8 so W. 25 30 E. 3 30 W. 9 22 E. 9 45 W. 17 o W. 3i 15 W. 14 30 W. 20 30 w. 29 40 w. 26 o W. 9 15 E. 67 so W. 12 20 W. 2 30 E. 17 I 21 o W. ON FINDING THE LATITUDE BY REDUCTION TO THE MERIDIAN. 302. The latitude of a place is most simply determined by observation of the meridian altitude of a known heavenly body. When such an observation cannot be obtained by reason of the state of the weather, the altitude of the body may often be obtained a little before or a little after its meridian pass- age. And if at the time of observing such an altitude near the meridian, the hour- angle of the body is known, we may find by computation very nearly the difference of altitude by which to reduce the observed to the Meridian altitude. The correction is called the "Reduction to the meridian." This method, in point of simplicity, is little inferior to the meridian altitude, to which it is next in importance. The altitude may also be determined by a direct process, deduced from spherical trigonometry. The former is the method used in the following pages. The Term "near the meridian" implies a meridian distance limited according to the latitude and declination, and also the degree of precision T^ith which the time is known (see Eaper, Table 47). Re^AMit'icn to Meridiem. 289 EULE CI. I °. To the time shown by the toatch, expressed astronomically, apply the error of the watch for apparent time,'^' adding when the watch is slow frejecting i^*" when the sum exceeds 24*" and putting the day one forward), subtracting when the toatch is fast f increasing the time shown by watch by 24."*, if necessary, and putting the day one back. J 2°. Wext turti into titne the difference of longitude made since the error of the watch was determined ; adding when the difference of longitude is East, subtracting when difference of longitude it West ; the result is apparent time at ship when the observation was naade.f 3°. If apparent time at ship is p.m., it is the time from noon; when it is a.m., f reckoning from the preceding noon J subtract it from 24*", the remainder is the time from noon. Examples. Ex. I. Suppose it is p.m. at ship, and Ex. 2. Again, suppose it is a.m. at ship, the watch when corrected shows January and the watch when corrected indicates Feb. 2d Qii i6«> 56* (see example i following): ^^ aj*" 37" i6» (see example 2 following), then the time from noon is i6'» 56' past , then we have noon of the 2nd. 24'' o™ o" \ In this instance it is 23 37 16 / 32" 44« before noon of I the 6th. 22 44 ; 4°. With apparent time at ship and longitude, find Greenwich date in apparent time (Rule LXXXI, page 222.) 5°. Take out of Nautical Almanac, page I, the declination, and reduce it to the Greenwich date (Rule LXXXII, page 225.) 6°. Correct the observed altitude of tun's upper or lower limb, and so get the true altitude of sun^s centre (Rule LXXXVII, page 236). Method I. 7°. Take out log. rising of time from noon (Table 29, Norie), log. cos. declination (Table 25, Norie), and log. cos. of latitude (Table 25, Norie). Note. — In using the natural sines and cosines to six places, it will be necessary to add i to the index of the log. rising, because, as given in the Table, it is only adapted to five places of figures. ♦ The error of chronometer for apparent ti'ne at place, should be noted when the morning sights are taken for determining the longitide. This with the diff. long, made in the interval between this last time and the time 01 observing the ex-meridian altitude, will give the apparent time at ship. If the ship has not changed her meridian since the time of morning sights the result obtained by applying the error of chronometer is, of course, the apparent time at ship. t The reason for this rule will appi^ar on considering that if a watch is set to the time at any given meridian, it will be iloio for any meridian to the eastward, but fast for any meridian to the westward, at the i-rtte of i"" for 15' diff. long., since the sun comes to the easterly meridian earlier, and to the westerly meridina later. 290 Hedudion to Meridimi. Caution. — In the use of the table of log. rising (XXIX, Norie), care must be taken that the correct indices are used when the minutes of the time from noon are i, 3, 10, or 32. It is necessary to notice, that the imlicos in the tabic sometimes change in the column whore they could not be inserted for want of room ; this may, however, be easily known by observing that the tirst figure of the decimal part of the log. changes from 9 to o. Thus the log. rising of o'' i™ o' is 9-97860 but the log. rising of o^ i"" 5' is 0-0481 3. The index, as given in the table, is in the form %, which means that it changes from 9 to o somewhere in that line. Similarly, opposite ic", the index is in the form ^, and the numerator i is the index of the log. rising of lo"" 0% lo"" 5'*, lo™ 10% and of 10™ 15', and changes to 2 somewhere between lo™ ij« and 10" 20*. 8°. Take the sum of these and ^nd the natural number corresponding thereto. (Table 24, Norie). 9°. To the natural number just found add the natural sine of the true altitude, (Table 26, Norie) ; the sum is natural cosine of meridian zenith distance, which take out of the Table, and name it North or South, according as the observer is North or South of the sun. See Rule LXXXVIII, 4°, page 238. I o°- -Apply the reduced declination to the %enith distance, taking their sum if they are of the same name, but their difference if of contrary names ; the result, in either case, is the latitude of the same name as the greater. Note. — The foregoing Method (Method I) is only convenient when the computer is pro- vided with a table of natural sines and cosines, as well as a table of log. versed sines, or the logarithmic value of 2 sine'' \ t. 303. We may also compute directly the reduction of the observed altitude to the meridian altitude by the following : — = Method II. 1°. Add together the following logarithms : — Constant log. 5-615451; ; (this is the log. of - . ^ J .* Log. cosine of latitude by ac'count (Table 25, Norie). Log. cosine of declination (Table 25, Norie). Log. cosecant of meridian senith distance as deduced from, latitude by D.R. and declination (Table 25, Norie). The log. of time from noon ; (this is twice the log. sine of half the hour- angley. — (Table 31, Norie, and 69, Raper). The sum of these logs., rejecting bens from the index, will be the log. of the reduction in seconds ("). — (Table 24, Norie). The zenith distance from latitude by D.K. is found as follows : — "When the latit-ude and declination are both of the same name, take their difference ; when latitude! and declination have different names, take their sum : the result in either case will be zenith distance by D.R. * If we use the constant log. 0-301030 (this is log. of 2) instead of that given above, viz., 5-615455, the sum of logs., rejecting tenw from index, will be log. sine of reduction in minutes (') and seconds ('). Table 66, fiai..er, or Table 25, Norie. Red/uction to Meridian. 291 2°. Add the reduction to the true altitude : the result is the meridian altitude.* 3°. Hewing the meridian altitude : find the latitude m hy the method of meri- dian altitudes (Eule LXXXVIII, page 238). NoTB. — This Method (Method II) does not approximate so rapidly as the preceding (Method I), but the objection is of little weight when tlie observations are very near the meridian. On the other hand,. it has the great advantage of not requiring the use of the table of Natural sines. 304. At the Liverpool Local Marine Board Examinations the Candidate is expected to solve this problem by means of Towson's ex-meridian Tables : hence we have Method III. 1°. Enter TeM-Q I (Toweon) v/nder nearest declination and find nearest hour- angle, against which stands Augmentation I, which add to declination, at the same time take out corresponding index number in the mwrgin. 2°. Enter Table II under true altitude and opposite index number, find Aug- mentation II, which add to true altitude, and thence find latitude as in meridian altitude. 305. In the following method the latitude is obtained by a dii-ect process, deduced from spherical trigonometry, and wholly independent of the latitude by account. Method IV. 1°. To find Arc I. — To the secant of the hour-angle (or time from noon) add the tangent of the declination ; the sum (rejecting \ o in the index) will be the tangent of Arc I, which is named North or South according to the declination. 2°. To find Arc II. — Add together cosecant of declination, sine of Arc I, and sine of true altitude ; the sum is the cosine of Arc II, to be marked of a con- trary name to the bearing of the sun. 3°. If Arcs I and II are of the same name take their sum ; but their dif- ference if of contrary names ; the result in either case is the latitude of the same name as the greater. Note. — As a check against any gross mistake, it should be borne in mind that Arc I is always a little greater than the declination, and Arc IT is (nearly) the complement of the altitude when the hour-angle is small. The tang, of decl. and the cosec, of the same are, of course, fovmd on the same page, and generally, also tang. Arc I and the corresponding sin. of same will be found on that page ; while the sin. of alt. and coa. Arc II are found in another page. In example I, page 292, the logs, which are found at the same opening of the Tables are marked with the same letter. * This is only an approximate mcriaaau altitude, in stiictuess a 8c6ond reduction should be computed. tgt BeducUon to Meridiem. Examples. Ex. I. 1876, January 2nd p.m. at ship, latitude by account 52° 6' S., longitude 71° 23' W., observed altitude sun's l.l. North of observer 60° 20' 30", index correction -\- 2' 58', height of eye 20 feet, time by watch, January 2'* o*^ 48" 22", which was found to be ag'" 16* fast on apparent time at ship, difference of longitude 32-4 miles to West: required the latitude by reduction to meridian. Time by watch, Jan. Watch /««< Diff. long. '-^^ Time from noon, Jan. Obs. alt. O's L.L. Index correction — ■ o 29 16 2 o 19 6 — 2 10 2 o 16 56 60° 20' 30" App. time at ship, Jan. Long, in time Greenwich date, Jan. jd o*»i6"'56* + 4 45 32 2 c 2 28 Decl., page I, N. A., January 2nd, at noon, 22° 57' 34" 8., (deereaaingj . H. diff., Jan. 2nd, noon, i3"'20 Greenwich date f*" a™, or X 5 Dip for 20 feet Correction of alt. Semi -diameter True altitude Method I. Time from noon i6'"56« rising* 3 '435 8 80 Latitude 52° 6' cos. 9-788370 Declination 22 56^ cos. 996421 1 1542 nat. no. 31 60 23 28 — 4 17 60 19 II — o a8 60 18 43 + 16 18 60 35 I 6,o)6,6'oo Correction — i' 6* Decl., Jan. 2nd, noon, 22" 57' 34*8. deer. Correction — 16 Red. decl. 22 56 28 S. T. alt. 60" 35' i' 1542 nat. sine 871073 Constant log. Lat. by D. R. Declination Mer. zen. dist. Time from noon i6'"36 Method II. 561546 52° 6' S. COS. 9'78837 22 56^ 8. COS. 9'9642i 9 9g cosec. o'3i227 log. 7' 1 3485 Z. dist. 29 14 10 S. nat. cos. 87261? Decl. 22 56 28 S. (next greater) 638 Lat. 52 10 38 S. 239)2300(10 nearly. Reduction True altitude Meridian alt. 6,0)65,3 log. 2-81516 + 10' 53" 60 35 I (• The index of log. rising is increased by 1. See Note to 7° page 289). 60 45 54 90 Zenifii distance 29 14 6 S. Declination 22 56 28 8. Latitude 52 10 34 8. Method III. — By TowsorCs Ex-Meridian Tables. 0*8 red. declination Aug. Table r, Index 30 Augmented declination 22°56' 28" 8. + 3 21 22 59 49 8. True altitude 60° 35' o"S. Aug. Table 2, Index 30 -|- 14 4 Thi» is the method required of Candidates at Liverpool. Meridian zen. dist. Declination 60 49 4 29 10 56 8. 22 59 49 8. 52 10 45 S. Time from noon Declination Arc I Arc IT Latitude Latitude Method IV. 4° 14' o" sec. 0-OOII87 22 56 28 S. tang. 9-626609 cosec. 0-409165 22 59 50 8. tang. 9-627796 sine 9-591828 True alt. 60' 35' i* sine 9-940055 29 TO 50 8. ..#.»....... « I . « i « . . . . cos, 9-941048 5a 10 40 S. Seduction to Meridian. 293 Ex 2. 1876, February 6th, a.m. at ship, lat. acct. 51* 50' N., long. 105° 41' "W., obs. alt. sun's L.L. South of observer 22*^ lo' 30", index corr. -j- 56', height of eye 22 feet, time by watch 6'^ Qh 4'" 4», found to be 28™ ^T^fast on app. time at ship, diflF. of long, made to East 29-8 miles siuce error of watch on app. time at ship was determined : required the latitude by reduction to meridian. Time by watch, February 6'' d^ 4'" 4' Watch /«s< — 28 47 App. time at ship, February 5<'23i'37'"i6» Long. 105° 41' W. +7 2 44 5 23 35 17 + I 59 DifF. long. — 6^- App. time at ship, February 5 33 37 16 24 o o Greenwich date, February 6 6 40 o or, 6'i-67 Time from noon, February 6 o 22 44 Hourly diff. 6*' 40"" = — 46"'02 X 6-67 Obs. alt. O's L.L. Index correction Dip Correction ef alt. Semi-diameter True altitude Method I. 22" 10 30" + 56 22 II 26 — 4 30 22 6 56 — 211 22 4 45 + 16 15 260 32214 27612 27612 6,0)30,6-9534 5' 7" Declination, page I, N.A. Feb. 6th, noon, 15° 43' 7" S. deer. Correction — 57 Time from noon 32"44' rising 3-689030 Latitude 5'° 5°' coa. 9-790954 Declination 15 38 cos. 9983629 3926 nat. no. 3-466213 2926 True altitude 21* 31' o" nat. sine 380263 Zen. distance 67*28'7" N. nat. cos. 3831! Declii»ation 15 38 o 8. Latitude 51 50 7 N. (The nat. sine being worked to six places of figures, 1 is added to index of log. rising). Red decl. 15 38 oS. Constant log. Latitude Declination Mer. zen. dist. T. from noon Method II. 51° 50' N. COS. 15 38 S. COS. 67 28 cosec 22™44' log. 5-615455 9-790954 9-983629 0-034489 7-390540 6,0)65,3 log. 2-815067 Reduction True altitude + 10' 53" 22° 21 Mer. altitude 22 31 53 Zenith distance 67 28 7 N. Declination 15 38 o 8. Latitude 51 50 7 N. Method III. — By Towspn's Ex-Meridian Tables. 0's Red. declination Aug. Table i. Index 57 Augmented declination 15° 38' 0*8. -4- 4 23 15 42 23 True altitude Aug. Table 2, Index 57 Meridian zen. dist. Augmented declination Latitude 22* 21' 0" + 6 35 22 27 35 67 32 25 N 15 42 33 8. 51 50 2N Time from noon 3 3™ 44' : Declination 15° 38' Method IV. 5° 41' seC. 0-002140 8. tang. 9-446898 cosec. 0-569473 Arc I Arc II Latitnda 15 42 34''S. 67 3? 33 N.. 51 50 8N. tang. 9-449038 . , sine 9-43250S True alt. 22° 21' sine 9580085 COS. 9-5S2066 *94 Red/uction to Meridian. Ex. 3. 1876, April 7th, A.M. at ship, lat. by acct. 58° 50' N., long. 51° 42' W., obs. alt. sun's L.L. South of observer 37*^ 42' 15", index corr. — i' 6", height of eye 10 feet, time by watch 7** oh 59'" 50", found to bo i'' 22"^ fast on app. time at ship, the diff. of long, made to East was 20*7 miles after the error on apparent time at ship was determined : required the latitude by reduction to meridian. Time by watch, April Watch fast April Diff. log. ??^ App. time at ship, April Time from noon, April Obs. alt. 0's L.L. Index correction Dip Corr. altitnde Semi-diameter True altitude T oi'59'"5o« — I 22 o App. time at ship, April 6'*23''39™i3» Long. 51° 42' W. -|- 3 26 48 6 23 + 37 I 50 23 6 23 24 39 13 7 20 47 37" 42' I 15 6 37 41 3 9 2 37 38 I 7 7 37 37 16 Greenwich date, April 7 3 6 jh.j By Raper : True altitude 37° 53' 2" H.D. 31 5615 16845 6,0)17,4-065 2 54 Decl. apparent noon 7th 7" 4' 14' N. Correction -|- 2 54 37 53 o Method I. 1300 Time from noon 20'° 47' rising 2'6i2340 _ Latitude acct. 58° 50' cos. 9-713935 ' Declination Declination 7 7 cos. 9-996639 Mer. zen. dist 2110 nat. no. 2-324214 True altitude 37° 53' nat. no. 21 10 nat. sine 614056 Mer. zen. dist. 5 1 57 48"N. nat. cos. 616166 Declination 7 7 8 N. Latitude 59 4 56 N. Red. decl. 7 7 8 N. Method 11. Constant log. Latitude D.R. 58°5o'N. cos. ^ " 7 7 N. cos 51 43 Time txom noon 2o"'47 5"6 15455 9"7i3935 9996639 cosec. 0-105 167 log. 7-312710 Reduction True altitude 6,0)55,5 log. 2-743906 ■4- 9' 15" 37''53 ° Meridian altitude 38 215 Zenith distance Declination Latitude 51 57 45 N. 7 7 8 N. 59 4 53 N. Method III.— By Towson'a Ex- Meridian Tables. 0's red. declination Aug. Table i. Index 53 Augmented declination 7^ 7' 8"N. 4- I 44 7 8 52 N. True altitude Aug. Table 2, Index 53 Mer. zen. dist. Augmented declination 37'"53' o"N. + 11 3 N. Latitude Method IV. sec. 0-001788 38 4 3 51 55 57 N. 7 8 52 N. 59 4 49 N. Time fromnoon 20"47« = 5''ii 45 Declination 7° 7' 8 " N. tang. 9-096532 , cosec. 0-906828 Arc I Arc II Latitude 7 8 53 N. 51 56 5 N. 59 4 58 N. tang. 9-098320 sine 9-094938 True alt. 57° 53' sine 9-788208 COS. 9-789974 Reduction to Meridian. 295 Ex. 4. 1876, August 7th, A.M. at ship, lafc. acct. 40° 52' N., long. 36° 47' W., obs. alt. Bun's L.r. South of ohserver 65° i', index corr. + 17 ', eye 14 feet, time by watch 1 1*^ 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 ajip. time at shij) was determined : required the latitude. Time by watch, Aug. 6^27,^15^/^6* App. time at ship, Aug. 6''2 3''43'nio» "Watch slow + 26 16 Diff. of long. 6 23 42 2 + I 8 App. time at ship, Aug. 6 23 43 10 24 Time from noon, Aug. 7 16 50 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 Long. 36" 47' W. Greenwich date, Aug. -f 2 27 By Raper: True altitude 65° 13' 3" H.D. 42"-34 X 2-2 8468 6,o)9>3"i48 I 33 True altitude 65 13 7 Method I. Time from noon i6"'5o' rising 3*430750 Latitude 4°° 52' cos. 9"878656 Declination 16 15 cos. 9-982297 1958 nat. no. 3'29i703 Decl. apparent noon. Aug. 7th 16° 16' 28" N. Correction — 1 33 Red. decl. 16 14 55 N. Method II. Constant log. 5 "6 15455 Latitude acct. 40° 52' N. cos. 9-878656 Declination 16 15 N. cos. 9-982297 Mer. zen. dist. 24 37 cosec. 0-380325 Time from noon 1 6™5o« log. 7-129720 nat. no. 1958 True altitude 65" 1 3' 7" nat. sine 907913 Zen. distance 24 30 45 N. nat. cos. 909871 Declination 16 14 55 N. Latitude 40 45 40 N. Reduction True altitude Meridian alt. Mer. zen. dist. Declination Latitude 6,0)96,9 log. 2-986453 + 16' 9" 65° 13 7 65 29 16 24 30 44 N. 16 14 55 N. 40 45 39 N. Method III. — By Towson's Ex-Meridian Tables. Q's red. declination 16° 14' 55 ' True altitude Aug. Table i, Index 33 -j- 2 30 Augmented declination 16 17 25 N. Aug. Table 2, Index 33 Augmented altitude Zenith distance Augmented declination 65° i 3' r -f 19 2 65 32 9 Latitude 24 27 51 N. 16 17 25 N. 40 45 16 N. Method IV. Time from noon Declination Arc I Arc II Latitude 4° 12 30 sec. 0-001173 16 1455 N. tang. 9-464560 cosec. 0-553143 16 17 25 N. tang. 9-465733 ,. jsine 9-447940 True alt. 65° 13' 7" eine 9958045 24 28 lo N. 40 45 '35 N. cos. 9-959128 2g6 deduction to MericUmi. Ex. 5. 1876, March gth, a.m. at ship, lat. aoct. 30° 21' 8., long. 16° 45' W., obs. alt. sun's L.L. North of observer was 63° 37', index corr. — i' 57", eye 21 feet, time by watch, March S"" 21^^ 49™ 25' (being g*' 49"^ 25' a.m. on 9th), found to be i** 59"" iC slow of apparent time at ship, the difF. of long, made to East was 23^ miles after the error on apparent time at ship was determined : required the latitude. Time by watch, March 8'^2i'i49'"258 Watch slow I 59 10 Diff. long. '3'-5 X 4 00 App. time at ship, Marob Time from noon, March Obs. alt O's L.L. Index correction Dip Corr. of alt. Semi-diameter True altitude 63 46 24 Method I. Time from noon 9"5i' rising 2*9654io Latitude 3o°2i' cos. 9'935988 Declination 4 13 cos. 9*998820 Ship date (A.T.) March Long, in time '2 3h5o"' 9* 170 ¥ 48 I 35 34 8 ^3 50 9 9 9 51 63° 37' I 0" 57 3 23 N 63 35 4 63 30 40 24 63 + 30 16 16 8 Green, date (A.T.) March 9 o 57 9 Hourly diff. or, oh'95 58'-69 oh 57na = 0-95 Correction 29345 52821 55'7555 or, 56" Decl. app. noon, page I, N.A. March 9th 4° 14' 14" S- deor. Correction — 56 Red. decl. 4 13 18 S. Method II. Constant log 5"6i5455 Latitude 30°2i' o'S. cos. 9*935988 Declination 4 13 18 S. cos. g-ggSSao nat. no. 795 log. 2*9002 18 nat. no. 795 True altitude 63° 46' 24" nat. sine 897052 Mer. Z. dist. 26 7 42 T. from noon 9 ""51^ coaec. o*356i7o log. 6664380 6,0)37,2" log. 2-570813 Mer. Z. dist. 26° 7' 25"S. nat. cosine 897847 Declination 4 13 18 S. Reduction True altitude + 6' 12" 63 46 24 (By Norie). Latitude 30 20 43 S. Mer. altitude 63 52 36 Zen. mer. dist. Declination Latitude 26 7 24 S. 4 13 18 S. 30 20 42 S. Method III. — £1/ Toicson's Ex-Meridian Tables O's red. declination 4°i3' i8"S. True altitude Aug. Table i, Lidex 12 -J- 13 0*8 Augmented declination 4 13 31 S. Aug. Table 2, Index 13 Augmented altitude Zenith distance O's Augmented decl. 63=46' 24" + 6 27 63 52 51 26 7 9 S. 4 13 31 S. Latitude Method IV. Time from noon 9" 51" ^ 1° 27' 45" sec. 0-000400 Declination 4 13 18 tang. 8'868i5o..,, Arc I 30 20 40 S, , cosec. I 133031 4 13 32 S. tang. 8-868550 , . .sine 8367367 True alt. 63° 46' 24' sine 9952818 Arc II LftUtudo 26 7 12 S cos. 9-953216 30 20 44 IS. Reduction to Meridian. 297 Ex. 6. 1876, Sept. 23rd, P.M. at ship, lat. acct. 51° 2' N., long. 173° 53' E., obs. alt. sun's L.L. South of observer 38° 44 20", index corr. -|- i' 8', height of eye 21 feet, time by watch 50™ 0% (or 23d o'' 50'") found to bo 39™ T.'^fasi on app. time at ship, diff. of long, made to West was 8'2 miles after the error on app. time was determined: required the latitude. Time by watch, Sept. Watch fast 23a o''5o" 39 23 o 10 58 DifT. of longitude — 33 App. timo at ship, Sept. Leng. 173^ 53' E. Greenwich date, Sept, 23" 35 32 22 12 34 53 App. time at ship, Sept. 23 o 10 25 Time from noon 23rd is 10 25 By Raper : True altitude 38° 55' 55'' H.D. 12I' 35>' X 12-6 Obs. alt. O's L L. 38^44' 20" Index correction +18 Dip Corr. altitude Semi-diameter True altitude Method I. Time from noon io'«25s Latitude 51^ 2' Declination o 7 38 45 28 — 4 23 35100 11700 5850 6,0)73,7-100 38 41 12' 17 ' 38 40 I + 15 59 38 56 o Decl. page I, N.A. Sept. 22nd 0° 4' 58" N. de^:r. Correction — 12 17 Red. decl. o 7 19 S. Method II. rising cos. COS. 3"oi399 9-798560 9-99999,9 650 nat. no. 2-812549 Constant log. Latitude acct. 51° 2' Declination o 7 Zen.dst. by D.R. 51 9 Time from noon io""25' N. 5"6i546 cos. 9-79856 COS. 9'99999 cosec. 0-10855 log. 671296 True altitude 38^ 56' nat. no. 650 nat. sine 628416 6,0)17,2 log. 2-23551 Mer. zen. dist. Declination i' 7"N. nat. co?. 629066 7 19 S. Latitude 50 5348 N. Reduction True altitude Meridian altitude Mer. zen. dist. Declination In taking out log. rising for 10"' 25-, it will be noticed that the inrlex given at the beginning oi the line is J, meaning that the iuilex at the commonoe- ment oT the line is 1, but that it changes somewhere t o(,-f,,/lp along the line, wliich may easily be knoTvn by J-iaiilu observing that when the first figure of the decimal part of the log. changes from 9 to o, the index changes from i to 2. Method III. — By Towso7i's Ex- Meridian Tables. 0's Red. declination 0° 7' 19" S. True altitude Aug. Table I, Index 13 + o Aug. Table 2, Index 13 + 2' 38-56 52" 3858 52 51 I 7 8 19 S. 50 53 49 ^• Augmented declination o 7 19 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 given hour-angle find corresponding Index number; ■with this and the altitude, augmentation II is deter- mined as is other case. Augmented altitude Zenith distance Augmented declinadon Latitude 3 8° 56' o" + 2 47 38 5 8 47 51 I 13 N. o 7 19 S. 50 53 54 N, Ex. 7. 1876, 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 5I' i'^ 7", which was found /rt»^ ^h JO"" 57% difference of longitude made since, 20j miles West. App. time at ship. May 5'' o^ 8'»48* Green, date app. time, May 4^20'' 4ni4^3 Time from noon ifl 8 48 igS Mednction to Meridian. U) Hourly diff. 5th noon, 42"'34 X Green, time f'-<)2 z=l i65"-9728 -|- 60 = 2' 46", decl. noon 5th, 16° 26' 4" N. — 2' 46' = red. docl. 16" 23' 18" N. By Norio : True altitude 78^ 52' 47". Time from noon Latitude acct. Declination Method I. S"i48i' .5' 13' .6 23 rising 2"8675io 9998197 9'98i986 COS. COS. 704 log. 2-847693 By Raper: True altitude 78'^ 52' Melhrd TL Constant log. Latitude DR. Declination Mer. zen. dist. Time from noon 8">48* 16 23 N. COS. COS. C03CC 10-. nat. no. 704 True altitude 78^52' 47" nat. sine 981227 M. Z. dist. Declination Latitude 10 54 31S. nat. cos. 981931 j6 23 18 N. 5 28 45 N. This example cannot be solved by means of Towson's Ex-moridian Tables, as the altitude exceeds the limits of the Tables. 36". 56 1546 9-99820 ^•98199 0'1l2-]6 6-56649 Reduction True altitude 6,0)75,0 log. 2-87489 -|- 12' 30" l^'S^ 47 Meridian altitude 79 5 17 Mer. zen. dist. Declination Latitude Examples tor Practioe. 10 54 43 s. 16 23 18 N. 5 28 35 N. ■• Ex. I. 1876, 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 corr -{- 4' 19", height of eye 28 feet, time by watch o'> 13™ 24% which had been found to be 25"" 35s/(m< of app. 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. Ex. 2. 1S76, February 28th, p.m. at ship, lat. acct. 43° 46' N., long. 12° 31' W., obs. alt. sun's L.L. 38" 1' 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, diff. of long, made to East was 14' after error on app. time at ship was found : required the latitude. Ex. 3. 1876, March 20th, a m. at ship, lat. acct. 41° 24' S., long. 105° E., obs. alt. sun's L.L. 47° 46' N., index corr. -\- 26", eye 22^feet, time by chron. 19'^ i6'> 58'" i2«, which had been found to be 6'' 34™ 34^ slow on Mpp. time at ship, diff. of long, made to East was 23' after the error on app. time at ship was determined : required the latitude. Ex. 4. 1876, 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 2i 40™ 128 slow on app. time, diff. of long. 33' W. ■^ Ex. ir. 1876, Nov. 3rd, P.M. at ship, lat. acct. 32" S., long. 109" 39' E., obs. alt. sun's li.L. 71° 50' N., index corr. -|- 32", eye 18 feet, time by watch ^^ 22*' 22"", which was found 2^ slow, diff. of long. 2 8'"7 West. J, Ex. 12. 1876, Dec. 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. -\- 45', eye 12 feet, time by watch n^ 29™ 42', found to be 1 8" 40* slow, diff. of long, was 36' East. -f Ex. 13. 1876, Jan. 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, time by watch d^ 2'" 40% found 13™ 48^ slow on ajDp. time, diff. of long, made since 16' East. J Ex. 14. 1876, 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 17I' West. Ex. 15. 1876, July 13th, A.M. at ship, lat. acct. 54° 35' S., long. 152° 20' W., obs. alt. ■ sun's L.L. 13° 17' N., index corr. -\- 47", eye 12 feet, time by watch it,^ 71' 54™ i2», which had been found to be 8^ 1411 I'j^ fast on app. time at ship, diff. of long, made to West was 34' after error on app. time was determined. "^ Ex. 16. 1876, March 20th, a.m. at shii?, lat. acct.. 19° S., long. 33° 33' E., obs. alt. sun's L.L. 70° 21' N., index corr. — 2' 10', eye 16 feet, time by watch 8"' 17% found fast on app. timo at ship 26'" 11% diff. long, made since 14-y East. \ Ex. 17. 1876, 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 12'^ o" o™ 2% fast on app. time at ship 10'" 51% diff. of long, made to East 7^'. Ex. 18. 1876, Sept. 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^ 8'' 41™ 43% which had been found to be c/^ 2™ 4-]»fast on app. time at ship, the diff. of long, made to West was 14' after the error on app. time at ship was determined. , Ex. 19. 1876, March i6th, a.m. at ship, hit. acct. 37° 42' N., long. 61" 40' E., obs. alt. sun's L.L. 50° o' 30" S., index corr. + 34'> eye 15 feet, timo by watch 10'' 53'" 3P, found sloiv on app. time at ship i'» 3™ 22', diff. of long, made since 18' West. J- Ex. 20. 1876, December 3i8t, a.m. at ship, lat. acct. 52° N., long. 12^ 53' W., obs. alt. sun's L.L. 14° 46' S., eye 19 feet, time by watch o'' ^6"^, which -was fast on app. time at ship jh ^m 2o», diff. of long. 2i'-4 West. Ex. 21. 1876, March 5th, p.m. at ship, lat. acct. 33" 35' N., long. 78'' E., obs. alt. sun's L.L. 49° 53' 15" S., index corr. — 3' 15", eye 22 feet, time by watch 4'' ig^ 2"' 12% found to be 5'' 17™ 12' slow, diff. of long, was 10' E. ,f- Ex. 22. 1876, September 22nd, a.m. at ship, lat. acct. 45° 45' S., long, in" 42' \V., obs. alt. sun's L.L. 43° 50' N., index corr. — 5' 40", height of eye 18 feet, time by watch 22'' 7^ 41'" io«, found to be 8'' 4™ 10^ fast, diff. of long, was i3'-5 East. tEx. 23. 1876, December 23rd, p.m. at ship, lat. acct. 42° 16' N., long. 4" 39' W., obs. alt. sun's u.L, 24° 14' 10" S., eye 11 feet, time by watch o*> 50"' 5^', fast on app. time at ship i9« 38', diff. of long. 2i''3 West. MERIDIAN ALTITUDE OF A FIXED STAR. EULE CII. 1°. Take from Nautical Almanac the starts declination. 2°. To the observed altitude apply the index error, a^t the aign attached directs. 3°. Subtract tJie dip answering to tha height of eye (Table 5, Norie ; Table 30, Eaper). 3P0 Meridian AUitudt of a Fixed 8im\ 4°. Subtract the refraction (Table 4., Norie; Table 31, Eaper), and thus get the true altitude. 5°. Subtract the true altitude from 90; the remainder is the zenith distance. 6°. Marh the zenith distance N. or S., according as the observer is North or South of the star. 7°. Underneath this last place the declination, and take their sum if they have the same names ; but take their difference if they have unlike names ; the result, in either case, will be the latitude. The declination of a fixed star changes so slowly that it may be taken out of the Nautical Almanac by i»spectio)i, without any practical 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 that name; when the zenith distance and declination are q/" 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 o'^ 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 (297 — 299, Nautical Almanac, 1876, and seek the given star under the head "Mean Places of stars" for January, and thence obtain the star's Right Ascension, which find at the top of one of the pages following 317 — 373, Nautical Almanac, 1876), which will give the star, and the declina- tion 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 in- crease the minutes (') at the top by i. Thus, on May loth, (see table annexed) the declination of a Andromedae is 28^ 22' 49" N., and on January 1st, the declination is 28^ 23' 3" N., 62"-8 being i' 3", which being added to 28° 22', which stands at the head of the column, gives the declination 28° 23' 3". Examples. a Andromedas | Date. R.A. Decl. N. oh im 28" 22' Jan. I 45'- 1 7 62"- 8 II 45"04 618 21 44-90 6o-6 31 44-78 59-2 &c. &c. &c. May 10 45*50 1 49-2 20 45-80 49'9 21 4612 509 &c. &c. &c. Ex. I. 1876, Dec. 29th, long. 140' W., theobs.mer. alt. of the star«Leonis {Reguha), 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 correction — 27 Dip ij feet Refraction True altitude Zenith distance Declination (il^.A., p. 345) 52 7 3 3 42 52 3 21 44 52 90 2 37 37 12 57 23 34 I Ex. 2. 1876, March 12th, long. 10° _E., obs. mer. alt. of the star Follux, bearing North, was 71° 59' 10", index corr. + i' i5''» height of eye 18 feet : required the latitude. Observed altitude of star Index correction Dip 18 feet Refraction True altitude N. N. 71° + 59' I io"N 15 72 25 — 4 4 71 S^ 21 18 71 56 3 90 18 28 19 33 N Latitude 5° 3^ 24 N. ByRaper: Index corr. — 27", Dip — 3' .50*, ref. — 46", true alt. 52^^ 2' 27 , latitude 50° 31' 34" N. Zenith distance Declination (N.A., p. 341) Latitude 10 15 36 N. By Raper : Index corr. -j- i' 15', dip — 3' 10', ref. — 19", true alt. 71° 55' 56", latitude 10'^ 15' 29' N. Meridian Altitude of a Fixed Star. 301 Ex. 3. 1876, March nth, long. 84° W., the ohs. mer. alt. of the star a Argus fCanopusJ, hearing South, was 37° 26' index corr. + i' 12', height of eye 16 feet. Observed altitude of star 37^26' o"S. Index correction -\- 112 Dip 16 feet Refraction True altitude Zenith distance Declination {N.A., p. 338) Latitude 000 By Eaper: Index corr. -{■ i' 12", dip — 4' o", ref. — i' i6", true alt. 37° 21' 56", latitude 0° o' 11" N. Ex. 4. 1876, January ist, long. 100° E., the ohs. mer, alt. of the star a Canis Majoris CSiriusJ, hearing South, was 59° 59' 50", in- dex corr. 4-41 2 ", height of eye 24 feet. Observed altitude of star 59" 59' 5o"S. Index correction -j- 412 37 27 12 — 3 50 Dip 24 feet Refraction True altitude Zenith distance Declination (N.A. P- 339) 60 4 2 — 4 42 37 23 22 — I 15 59 59 20 — 33 37 22 7 90 59 58 47 90 52 37 53 N. 52 37 53 S. 30 I 13 N 16 32 44 S. Latitude 13 28 29 N. By Eaper : Index corr. + 4' ' 2", dip — 4' 50", ref. — 34", true alt. 59° 58' 38", lati- tude 13° 28' 38' N. In •* I. t 2. •^ 3- •^ S- 4 6. 4 8. •f 10. fir. / 12. ^•14. •i- 15- 416. /x8. f 19- '^20. Examples for Pkaotice. each, of the following examples it is required to find I'lVIl. DA'IK. I.ONt;. STAK. 0I3S. ALT 1876. Nov. 7th, 90° W. « Andromedic 75° 10' 30' Jan. ist, 27 "W. n Aurigdi ("CapellaJ 54 o 15 Aug. 19th, 84 E. a liyim fVeffaJ .- 50 o 20 Dec. 22nd, 82 E. « Persei 51 51 45 April nth, 142 W. « Virginas ^^S^JUffi^ .......... 63 14 30 June loth, 151 E. Eridani (Achernar) 40 10 25 Dec. 27th, 91 W. (Algenib) 78 16 45 Nov. 30th, 24 W. « Arietis 6823 o Feb. 2nd, 76 E. aTsLMTi f"Aldebara>i) 29 52 10 June ist, 97 E. a^ Crucis 75 10 30 May 22nd, 178 W. a Hydrse 30 28 53 July 17th, 29 E. a Cygni , 20 13 50 Oct. 17th, 165 E. « Aquilas f'AlfairJ 60 49 10 March 2nd, 154 W. « Canis Majoris ^-Strm^ .... 58 58 50 April 3rd, III E. fl! Bootis (ArcturusJ 79 49 40 Aug. 7th, 40 W. a Scorpii (Antares) 68 49 30 May ist, 8 E. «- Centauri 10 250 Oct. 29th, 5 W. a Piscis Australia (FomalhautJ 70 6 o March 31st, 36 E. a Pegasi (MarkabJ 33 20 50 Sept. nth, 12 W. a Cassiopefc 62 24 50 the latitude : — COKR. KY£ 'S. + 27" 25f N. I 45 18 N. 22 N. + 40 26 S. + 3 47 22 S. + 55 24 s. — 25 24 N. — I 38 28 N. + 5 20 15 s. — I 40 14 s. — 7 38 II N. 18 N. + ° 55 17 N. + 1 10 20 S. — 2 5 25 S. — I 54 21 S. — 45 20 N. + 55 12 N. + r 20 20 N. — 7 30 19 ORDINARY EXAMINATION. EXAMINATION PAPER To he used hy all Candidates when appearing for Examination for the first time only. DEFINITIONS. The Candidate is requested to ivrite at least ten of the folloiving definitions. The writitng aliould he clear, and the s2}cUing shoidd not be disregarded, I. — The Equator is 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 great circle passing through hoth 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 ia the plane on which the spectator stands, i)roduced 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 afi"ord3 a reflected image 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 horizon. 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. 16. Variation of the Compass ie the angle which the magnetic needle, under the influence of 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. The Error of the Compass is the angle the compass needle makes with the true meridian, being the combined eS'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 obsorvatimi. 21. Aaimuth of a celestial obj«ot is the arc of the borinon between the N. and 8. points, and a vertical circle drawn through the object. Ordinary ExaiMnatio)). 303 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. 37. 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 the place of observation. 29. Semi-diameter of a heavenly body is half 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 with 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 zenith 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 oi' to 20^^. 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. Eqiiation 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 sight-" angles (180°). 304 Ordinary Examination. EXAMINATION PAPER.— No. I. FOR SECOND MATE. Multiply 7654 by 95 by common logarithms. Divide 3654000 by 7308 by common logarithms. 3 — H. COT'RSES. K. tV Wrans. Lee- Devia- Remarks, ifcc. 1 way. TIOX. pts. I w.s.w. 1 '° 8 N.W. \ 11° 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 (list. 15 miles. (Ship's 5 - N.W. ^ N. 12 2 W.S.W. \ if W. head AV.S.W.) Dev. 6 12 3 as per log. 7 12 3 8 12 2 9 N.N.W. 9 6 West. 1 11° w. 10 9 5 II 12 9 9 5 4 Var. ^x' 30' W. I N.W. by W. 7 8 S.W. by W. •i 20= W. 2 7 6 3 7 4 4 8 2 5 S."W.^S. 9 3 S.S.E. I 6°W. 6 8 7 7 9 2 A current set the ship 8 8 7 S.W. by W.|-W. (cor- 9 W. |S. 10 3 S. by W. 1 a 14^ w. rect magnetic) 8 miles 10 10 2 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. 1876, 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 latitude 37° N., the departure made good was 89-2 milos : required the difference of longitude by parallel sailing. 6. Required the course and distance from Toulon to Valencia, by Mercator's sailing. Lat. Toulon 43° 8' N, Long. Toulon 5° 56' E. Lat. Valencia 39 29 N. Long. Valencia o 24 W. ADDITIONAL FOR ONLY MATE. 7. 1875, Januarj' 6th: find the time of high water, a.m. and r.M., at Cherboui-g and Portland Breakwater. 8. 1876, January ist, at S"" 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. 1876, January 29th, p.m. at ship, latitude 42° 26' N., observed altitude sun's l.l. 13" 40', index error — i' 14", height eye 16 feet, time by chronometer 29*^ 6h 48"" 40% which was slow II™ 22^'3 for mean noon at Greenwich, December ist, 1875, ^'^^ <^^ January ist, 1876, was 8" 7« slow for Greenwich mean noon : required the longitude by chronometer. Orddim^y Exammation. y^e ADDITIONAL FOR FIRST MATE. 10. 1876, Jinuary 15th, mean time at ship i)^ 39™ 445 a.m., latitude 23° 39' S., longitude 127° 52' "W., sun's maj^netic azimuth S. 103° E„ observed altitude sun's l.l. 55° 8' 30", index 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. 1876, January 17th, p.m. at ship, latitude by account 36° 2' N., longitude 149" 28' E., observed altitude sun's l.l. South of observer was 32° 54' 15", index error -|- 2' 18', height of eye 22 feet, time by watch 1 1'> 59"!, which had been found to be 20'" 24' slow on apparent 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 meridian, ADDITIONAL FOR MASTER ORDINARY. 12. 1876, January 24th, the observed meridian altitude of the star a Tauri (AJdebaran) was 52™ 36' bearing South, index correction — 23", height of eys 20 feet: required the latitude. DEVIATION OF THE COMPASS. N.B. — The Candidate is to answer correcthj at least eight of such of thefoUowing questions as are marked with a cross hy the Examiner. The Ej-aminer nill not >nark less than twetre. 1 . What do you mean by Deviation of the Compass ? A. The deflection of the compass needle from the magnetic meridian caused by the attraction of the iron of the ship. 2. How do you determine the deviation («) when in port, (and b) when at sea? A. By bringing the ship's head successively upon each of the thirty two points of the Standard Compass, or on each alternate point, and then (a) by taking reciprocal simul- taneous bearings ; or by the observer on board taking the bearings of a distant object whose correct magnetic bearing is known, or of some conspicuous object in a line with figures on a dock wall. (b) At sea, by bearings of well known conspicuous objects in a line on the the coast, or b)' amplitudes and azimuths and the known variation at the place of the ship. 3. Having determined the deviation with the ship's head on the various points of tho compass, how do you know when it is Easterly and when Westerly ? A. When the correct magnetic bearing of the distant object is to the right of the reading of the compass on board, the deviation is easterly, when to the left, westerly. 4. Why is it necessary, in order to ascertain the deviations, to bring the ship's head in more than one direction ? A. Because the deviation alters as the direction of the ship's head is changed. 5. For accuracy, what is the least number of points to which the ship's head should be brought r A. Eight; although, if the deviations be known on the four quadrantal points, N.E., S.E., S.W., and N.W., with the aid of Napier's diagram a good deviation curve may be formed. 6. How would j-ou find the deviation when sailing along a well known coast .'^ A. By taking with the Standard Compass the bearing of two well defined objects in a line, as for instance, the bearings of two beacons, two lights, two points of land, not too near one another, and whose correct magnetic bearing is known, from the chart or otherwise; then the difference between the correct magnetic bearing and the compass bearing is the deviation for the direction of the ship's head when the bearing was taken. 3o6 Ordinary Examination. 7. In the following table give the correct magnetic bearing of the distant object, and thenco the deviation. Correct magnetic bearing. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. i Shin's Head Bearin;? of fyltantod Distant Object Compass. | %„SUnd-^ Deviation Required. North .... N.E East S.E N. 86° W. S. 79 W. S. 69 W. S. 6s W. t South .... S. 64° W. S.W ' S. 72 W. West .... S. 89 W. N.W N. 80 W. 1 8. 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. ; N.N.E. ; E.S.E.; S.S.W. Compass courses : — 9. 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.; E.N.E. ; S.S.E.; W.S.W. Magnetic courses : — 10. 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 AV., find the bearings, correct magnetic. Compass bearings: — E. by S. \ S. and N. \ E. Bearings, magnetic : — 11. Name some suitable objects by which you could readily obtain the deviation of the Compass when sailing along the coasts of the English Channel. The South Foreland lighthouses in one, bearing W. by N., correct magnetic : Beachy Head lighthouse just open of the cliffs to the eastward, bearing N.W. by W., correct mag- netic ; Portland lights in one N.N.W. f W. ; Prawl Point and Start lighthouse in one and Lizard lights in one. \ 11. Do you expect the deviation to change ; if so, state under what circumstances ? ] A. Yes, it changes rapidly for several months after the ship is launched, an alteration ' also takes place by change of magnetic latitude, and in ships running long upon one course and then changing the course, bj'^ the heeling of the ship, and by taking in a cargo of iron. 13. How often is it advisable to test the accuracy of j'our table of deviations ? A. Frequently, in a new vessel ; when nearing land ; and under the circumstances stated in last question. 14. State briefly what you have chiefly to guard against in selecting a position for the compass. A. Elongated iron, especially if vertical, such as stanchions, davits, capstan, spindles, funnels, ventilating shafts, &c., and the compass should be as fiir removed as possible from transverse bulk heads. 15. The Compasses of iron Ships are more or less affected by what is termed the heeling error ; on what courses does this error vanish, and on what courses is it the greatest ? A. It vanishes when the Ship's head is East or West by compass, and is greatest when the Ship's head, by compass, is North or South. 16. State to which side of the ship, in the majority of cases, is the North point of the Compass drawn in the Northern hemisphere ; and what effect has it on the assumed position of the Ship when she is steering on Northerlj', and also on Southerly courses ? A. The North point of the Compass is drawn to the weather side in the majority of cases. The effect of this is to throw the Ship to windward on northerly courses, and to leeward on southerly courses. 17. The effect being as you slate, on what courses would you keep away, and on what courses would you keep closer to the wind, in order to make good a given Compass course ? Ordinary Examination. 307 A. I would keep her away on either tack on northerly courses, but on either tack on southerly I would keep her close to the wind. 18. Does the same rule hold good in both hemispheres with regard to the heeling error ? A. No, ships which have a large heeling error to windward in northern latitudes, will probablj' have as large a heeling error to leeward in high southern latitudes ; but it is recommended in order to determine it, that observations be made in every ship. 19. Your steering compass having a large eri-or, how would you proceed to correct that compass by compensating magnets and soft iron in order to reduce the error within manage- able limits. A. Draw a line upon the deck, fore-and-aft, through the centre of the binnacle. Draw another line across the deck at right-angles to the former, through the same centre. Bring the ship's head either North or South (correct magnetic), and placing a magnet on the deck athwartship, with its centre exactly on a fore-and-aft line drawn on the deck at some distance from the binnacle; move it gradually to or from the foot of the binftade until the compass points correctly. If the compass needle deviates to the left, the north end of the magnet must be placed to the left, and conversely. Next swing the ship's head round to the East or West (correct magnetic) and steady her on one of these points, and place a 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. A gain : the semicircular deviation having thus been corrected, and the binnacle being properly fitted with two small brass boxes, one on each side of, and on a level with the compass ; steady the ship's head on one of the quadrantal points, N.E., S.E., S.W., or N.W. : if there is any deviation fill one of the chain boxes with a quantit)' of small chain until the compass points correctly ; if one chain box be not sufiicient, fill the other. For greater certainty, swing the ship's head to each of the other quadrantal points. EXAMINATION PAPEE.— No. n. 3-— FOR SECOND MATE. Multiply 50030 by 800 by common logarithms. Divide 9999-46 by 6t% by common logarithms. H. Courses. K. ■h Winds. Lee- WAY. Devi.\- TIOX. Remarks, &c. pts. r S.E. by E. 13 2 N. 11= E. A point of land in 2 13 3 lat. 47^ 31' N., long. 3 12 5 52° 33' W., bearing 4 13 by compass W.S.W., 5 S.E. II E.N.E. J g'E. dist. 18 miles. (Ship's 6 10 5 head S.E. by E.) De- 7 10 4 viation as per log. 8 II I 9 E. by N. 8 8 S.E. by S. I 17° E. 10 8 4 II 9 4 12 9 I E.N.E. 6 8 S.E. 4 i5»E. Variation 28= W. 2 6 7 3 6 5 4 7 5 S.S.E. 5 8 East. 2 7'»E. 6 5 8 7 6 4 A current set (cor- 8 6 rect magnetic) S. *E., 9 S.E. by S. 7 E. by N. li 8»E. 1 2 miles, from the time 10 7 3 the departure was II 7 4 taken to the end of the 12 1 1 7 3 day. 3o8 Ordino/ry JExmmnatwn. 4. 1876, February ist, in longitude 78° 14' E., the observed meridian altitude of sun's L.L. was 78° 4' 10", bearing South, index error -j- 55', height of eye 12 feet: required the latitude. 5. In latitude 47° 30' N., the departure made good was i i^'s miles : required the difference of longitude by parallel sailing. 6. Eequired the course and distance from St. Helena to Cape Horn, by calculation on Mercator's principle. Latitude St. Helena 15" 55' S. Latitude Cape Horn 55 59 S. Longitude St. Helena 5° 44' W. Longitude Cape Horn 67 16 W. ADDITIONAL FOR ONLY MATE. 7. 1875, February 5th: find a.m. and p.m. tides at Filey Bay, Milford Haven, and Cromarty. 8. 1876, February 20th, at 6'' 9™ p.m., apparent time at ship, latitude 1 1° 58' S., longitude 179° 42' E., sun's magnetic amplitude S.W. by W. ^ W. : required the true amplitude and error of compass; and supposing the variation to be 10° 20' E., required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1876, February loth, a.m. at ship, latitude 50° 48' N., observed altitude sun's l.l. 9'' 10' 50', index correction — 3' 20', height of eye 18 feet, time by chronometer February gi gh ^gm 253^ which was 37" 58^-8 /«i< for mean noon at Greenwich, December 20th, 1875, and on January loth, 1876, was 34>" iz^ fast for mean noon at Greenwich: required the longitude. ADDITIONAL FOR FIRST MATE. to. 1876, February i6th, mean time at ship 8^ 7"> 35^ a.m., latitude 51'^ 2' N., longitude 140° 34' W., sun's magnetic azimuth S. 36° 20' E., observed altitude sun's l.l. 7° 16' 40", index correction — 6' 10", height of eye 15 feet: required the error of compass; and supposing the variation to be 25° W. : required the deviation for the position of the ship's head at the time of observation. ir. 1876, February 15th, a.m. at ship, latitude acct. 55° 59' S., longitude 54° 18' E., observed altitude sun's l.l. North of observer was 46° 22' 10'', index correction — i' 50", height of eye 19 feet, time by watch o™ 5^, which had been found to be 2,0™ fa^t on apparent time at ship, the difference of longitude made to the East was i6'-8 : required the latitude. ADDITIONAL FOR MASTER ORDINARY. 13. 1876, February 12th, the observed meridian altitude of star Procyon, South of observer, was 77° 18' 10", index correction + 19", 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 by Standard Compass. Bearing of Distant Object Deviation by Standard ; Eequired. Compass. 1 Ship's Head by Standard Compass. Bearing of ' Distant Object Deviation by Standard Eequired. Compass. 1 North .... N.E East S.E S. 34' E. S. 58 E. S. 62 E. S. 52 E. South .... S.W West .... N.W S. 43° E. S. 31 E. S. 17 E. S. 15 E. Ordinary Meamination. if>9 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.E. by E. ^ E. ; W. i S. ; W. f N. ; E. by S. \ S. Compass courses : — Supposing you have steered the following courses bj- the Standard Compass, find the correct magnetic courses made from the above deviation table. Compass courses:— W. by N. ^ N. ; N.E. | N. ; S.W. \ S. ; S.E. by 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. \ S., find the bearings, correct magnetic. Compass bearings: — N. 79° W., and S. 19° W. Bearings, magnetic : — EXAMINATION PAPEE.— No. III. FOR SECOND MATE. 1. Multiply 84'8 by 62-8, by common logarithms, and prove the result. 2. Divide 666'666 by 8'88, by common logarithms, and prove the result by decimals. 3-— H. Courses. K. i\ Winds. Lee- way. Devia- tion. Remarks, &c. pts. I S.S.E. 5 6 \ East. ^k 3°E. A point of land in 2 5 6 lat. 62°N. long. i50°E. 3 5 bearing by compass 4 4 8 W. by S.|S., distance 5 S.S.W. \ W. 4 7 West. 2J 4°W. 17 miles. (Ship'shead 6 4 8 S.S.E.) Dev. as per 7 5 2 log. 8 5 3 9 w.s.w. 5 South. 4 9°W. 10 6 II 6 5 12 6 c I W. iN. 6 6 N. by E. 11° W. Variation 31° E. 2 3 7 6 6 4 4 4 5 6 7 East. 6 S.S.E. 4 10° E. 5 4 8 A current set the ship 8 4 6 (correct magnetic) 9 E.S.E. 4 S. by W. 9''E. N.N.E., 21 miles from 10 4 5 the time the departure II 4 5 was taken to ^e end 12 5 of the day. 4. 1876, March 20th, longitude 173° 18' E., the observed meridian altitude of sun's l.l. was 89=" 38' 10" bearing North, index correction + 4' 27", height of eye 18 feet: required the latitude. $. In latitude 34" 28' S., the departure made good was 394*2 miles : required the dif- ference of longitude by parallel sailing. 6. Required the course and distance from the Cape of Good Hope to Cape Frio. Lat. Cape of Good H«p© 34° 28' S. Long. Cape of Go«d Hope 18' 28' E. Lat. Oape Frio 33 o 8. Long. Oajre Frio 41 57 W. 3IO Ordinary Examination. ADDITIONAL FOR ONLY MATE. 7. 1875, March nth: find the times of high water, a.m. and p.m. (hy Admiralty Tables) at Quillebaeuf, Havre, Poole, Yarmouth Roads, Lerwick, and Beaumaris. 8. 1876, March 6th, at 5'^ 31"^ 52^ i>.m. apparent time at ship, in latitude 52" 12' N, longitude 138' 54' W., the sun's magnetic amplitude was W. by S. ^ S. : required the error of compass, and supposing the variation to be 24° E. : required the deviation for the position of the ship's head at the time of observation. 9. 1876, March 31st, a.m. at ship, latitude 26=' 9' N., observed altitude sun's l.l* 29° 10' 20", height of eye 26 feet, time by chronometer 31'' o'' 4™ 50% which was 58" ^%^ fast for mean noon at Grecnwicb, November 20th, 1875, and on December 3i8t, 1875, was jh 2'n55'-8/(T.?< for mean time at Greenwich: required the longitude. ADDITIONAL FOR FIRST MATE. ID. 1876, Jrarch loth, mean time at ship, 7'' 35"' 25' a.m., latitude 42° 41' S., longitude 148^ 5' E., sun's bearing by compass S. 108= 37' 30" E., observed altitude sun's l.l. 17° 57'4o", height of eye 19 feet : required the error of the compass ; and supposing the variation to be 10" 50' E. : required the deviation for the position of the ship's head at the time of observation. 11. 1876, March 25th, p.m. at ship, latitude acct. 20' i' N., longitude 89' 10' E., observed altitude sun's l.l. South of observer was 71' 9', height of eye 18 feet, time by watch iih 38" 12= (or 24'' 23I' 38-" 128), which had been found to be 31™ 8^ slow on apparent time at ship, the difference of longitude made to East was it,\ miles after the error on apparent time was determined : required the latitude by reduction to meridian. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, March 19th, the observed meridian altitude of a Bootis (ArcturiisJ, 36° 10' 20", bearing North, index correction -|- 2' 42 ", 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. cv,- , IT J ' Bearing of Ship's Head Distant Object by Standard by standard Compass. -fcompass. Deviation required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation required. North .... S. 67° E. N.E East. East i N. 85' E. S.E N. 87 E. South .... S.W West .... N.W S. 72°E. S. 46 E. S. 45 E. S. 52 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.E. ^ E. ; S.W. by W. ; W. by S. f S. ; S.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 :-S.E. by E. ^ E. ; N.E. by E. ^ E. ; N.W. by W. f W. ; N.N.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. 24° E., find the bearings, correct magnetic. Compass bearings :~W. | S. and E.N.E. Bearings, magnetic : — EXAMINATION PAPER.— No. IV. FOR SECOND MATE. 1. Multiply, by common logarithms, 456 by 28'9. 2. Divide, by oommon logarithms, 462927 by i42'8. Ordinary JExaMhiation. 311 3-— H. Courses. K. 1 "Winds. Lee- Devia- Eemarks, &c. — way. tion. pts. I S.W. i W. 14 5 S.E. 6^W. Apoint,lat. 50'12'S. 2 14 2 long. 1 79°4o'\V., bear- 3 14 6 ing by compass N.fW. 4 14 7 dist. 19 miles. (Sbip'a 5 N. |E. 4 E.N.E. 3i 3°E. head S.W. \ W.) De- 6 3 6 viation as per log. 7 3 6 8 3 8 9 S. by E. i E. 2 4 S.W. ^ W. ^k 6°E. 10 2 3 II 2 3 12 2 Variation 14'' East. I W. by S. 12 2 S. by W. 1 14° W. 2 12 4 3 12 6 4 12 8 5 E.N.E. 3 S.E. 4 19° E. 6 2 3 7 3 4 A current set (cor- 8 3 3 rect mag.) S.W. i W., 9 S.S.W. i w. 5 6 S.E. If 4°W. 42 miles, from the time 10 5 7 the departure was 11 5 3 taken to the end of the 12 5 4 day. 4. 1876, April ist, in longitude 87° 42' W., observed meridian altitude sun's l.l. South of observer was 48^ 42' 30", index correction -\- 1' 42', height of eye 18 feet : required the latitude. 5. In latitude 49° 57' N., the departure made good was 149 miles : required the difference of longitude by parallel sailing. 6. Required the course and distance from A to B. Latitude A 56° 35' S. Longitude A 2° 15' E. Latitude B 51 10 S. Longitude B 3 10 W. ADDITIONAL FOE ONLY MATE. 7. 1875, April 13th: required the times of high water, a.m. and p.m., at Ecrehous, Blakeney, Portree, Llanelly, Cardiff, and New Eoss. 8. 1876, April 28th, at 5'' 14™ 2^ p.m. apparent time at ship, latitude 38^ 19' S., longitude 88° 48' E., sun's magnetic amplitude N.W. by W., variation 19° 10' W. : required the deviation. 9. 1876, April 15th, P.M. at ship, latitude 37^ 49' S., observed altitude sun's l.l. was 26° 27' 30', index correction — 49", height of eye i3feet, time by chronometer 14'' 21 '» 48™ 17% which was 4™ 51^ fast for Greenwich mean noon, January 22nd, and on February 3rd, was 2"» 35^"4/««< lor mean time at Greenwich : required the longitude. ADDITIONAL FOE FIEST MATE. 10. 1876, April 17th, mean time at ship 2^ 49'" 45' p.m., latitude 39° 50' N., longitude 1° 35' E., sun's bearing by compass West, observed altitude sun's l.l. 42° 10', index corr. — 45", height of eye 14 feet : required the error of compass ; and supposing the variation to be 19° 50' W. : required the deviation of the compass for the position of the ship's head when the observation was taken. 11. 1876, April 19th, A.M. at ship, latitude account 46*^ 15' N., longitude 178° 12' E., observed altitude of sun's l.l. South of observer 54° 7', index correction + 2' 12", height of eye 20 feet, time by watch ii'' 24"" 22% or 18'' 23*^ 24™ 22", which had been found to be 5"" slow on apparent time at ship, the difference of longitude made to the £ait was 30 miles, after the error on apparent time was determined. 3'« Qrdinm'y Exnmwntion. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, April 12th, the ohserved meridian altitude of the star Spica, South of ohserver, was 20" 58' 40", index correction — 45'', height of eye 25 feet : required the latitude. In the following table give the correct magnetic hearing of the distant object, and thence the deviation: — Correct magnetic hearing. Ship Head jyi^i^^t Object Deviation by Standard by Standard ' Required. Compass. fcompass. i Ship's Ilead 1 by 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. South .... S.W.' .... 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. i S. Compass courses : — Supposing you have steered the following courses by the Standard Compass, tind the correct magnetic courses made from the above deviation table. Compass Courses :-N.W. by W. i W. ; W. by S. J S. ; E. by S. \ S. ; N.E. f E. Correct magnetic courses. You have taken the following bearing 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.S.W. Bearings, magnetic: — EXAMINATION PAPER— No. V. FOR SECOND .MATE. 1. Multiply 767 by 89-8, bj- common logarithms. 2. Divide 66889"2 by 99'7, by common logarithms. 3 — H. Courses. K. ItV Winds. Lee- way. Devia- tion. Remarks, &c. pts. I S.E. \ E. H 1 S.W. 8^E. Apoint, lat. 64°2'S., 2 H 1 long. 140" 21 'E., bear- 3 H 4 ing by compass W. by 4 n 6 S. f S., distance 23 5 E. by S. f S. 3 4 N.E. 3 14° E. miles. (Ship's head 6 3 3 S.S.E. i E.) Devia- 7 3 4 tion as per log. 8 3 5 9 10 W.N.W. 3 4 4 3 North. 2f 19" W. II 4 3 12 4 4 I N.E. \ N. 12 2 N.N.W. 1 8^E. Variation 37' E. 2 12 .S 3 12 3 4 W.S.W. 3 6 N.W. 3i 19° W. 5 3 4 6 3 7 N.N.W. <; 7 West. 2 11° W. A current set the ship 8 <; 6 1 (correct magnetic) 9 10 '? 7 N.E. i E., 48 miles, s.s.w. 6 1 West. I* 3'W. J from the time the de- 1 1 6 7 parture was taken to 12 7 8 the end of the day. Ordmary ExaminaUon. 313 4. 1876, May 8th, in longitude 105 017' W., observed meridian altitude of sun's l.l., hearing 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 37 N. Longitude A 5i°5i' E. Longitude B 33 33 E. ADDITIONAL FOR ONLY MATE. 7. 1S75, May 2ist: find the times of high water a.m. and p.m. at Loch Ryan, Tarn Point, Berwick, St. Malo, and Dungeness. 8. 1876, May 21st, at 7'' 29'n a.m. apparent time at ship, latitude 45^ si 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 3 1° 50' E. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1876, 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 chron. 21^ 2ih 6'" 10% which was slow i2'-6 for mean noon at Greenwich, Februar)- 24th, and on April ist, was 2^ ^^^ fast for mean noon at Greenwich ; required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1876, May 25th, mean time at ship, 3^ 29™ 47s p.m., latitude 4t° 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 + 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 compasa for the position of the ship's head at the time the observation. 11. 1876, 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 io 45™ p.m., apparent time at ship, latitude 52= 30' N., longitude 12° 10' W., sun's magnetic amplitude N.W. | W. : required error of compass ; and supposing the variation to be 30' 28' E., required the deviation of the compass for the position of the ship's head at the time of observation. 320 Ordinary ^Examination. 9. 1876, Sept. ist, P.M. at ship, latitude 9" 9' N., observed altitude sun's l.l. 62" 13' 14', index correction + 15") height of eye 16 feet, time by chronometer August 31"' ij"^ 34™ 28', •which was 2™ lo'* sloiv for moan iioou ut Greiinwich, July 28th, and on August 12th was I™ 31' slow on mean noon at Greenwich : required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1876, Sept. i6th, mean time at ship S*" 3"^ 18^ 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 -\- 1' 22", height of eye 20 feet: required the true azimuth and error of compass ; and supposing the variation is 8° 20' E. : required the deviation for the position of the ship's head at the time of observation. 11. 1876, Sept. 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 bj' watch 11'' 10" lo^ (or 22'* 2311 10™ io»), 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. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, Sept. 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 bearing of the distant object, and thence the deviation: — Correct magnetic bearing. Ship 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 Kequired. North N.E East S.E S. 86° W. S. 68 W. S. 66 W. S. 79 W. South ....' N. 88° W. S.W.- .... N. 80 W. West .... N. 72 W. N.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. ^ E.; E. | N. ; S.E, I E. Compass courses : — Supposing you have steered the following courses by the Standard Compass, lind the correct magnetic courses made from the above deviation table. Compass Courses:— North; S.S.W. \ W. ; E. by S. f S. ; N.E. \ E. Correct magnetic courses : — You have taken the following bearing of two distant objects by j'our Standard Compass as above, with the ship's head at N.N.E. f E., find the bearings, correct magnetic. Compass bearings : — N. 79° E. and W. \ S. Bearings, magnetic : — EXAMINATION PAPER.— No. X. FOR SECOND MATE. 1. Multiply 560072 by 50, by common logarithms. 2. Divide 8491-9 by 98-4, by common logarithms. Ordinary Exmnination. 32» 3-— H. Courses. K. tV Winds. Lee- way. Devia- tion. Remarks, &c. pts. I W.S.W. II South. i 11° W. A point, Cape Fare- 2 II well, in lat. 59*^ 49' N., 3 lO 4 long. 43° 54 W., bear- 4 10 6 ing by compass N. ^E. 5 West. 5 s.s.w. I i6*W. dist. 36 miles. (Ship's 6 5 head W.S.W.) Dev. 7 4 5 as per log. 8 4 5 9 S.E. 13 s.s.w. h 10° E. lO 13 1 II 12 4 12 12 4 I S. by E. 6 s.w. by W. 4 4''E. 2 5 5 Variation 70° West. 3 5 4 4 5 5 S.W.byS. I 5 S.E. by S. 3^ 5°W, 6 I 5 7 8 9 Calm. •• ., Calm. W. ^N. 3 S.S.W. \ W. 2i 16° W. A current set (cor- lO s.w. 7 6 W.N.W. if 6° W. rect magnetic) S.S.E., II 7 4 48 miles during the 12 7 24 hours. 4. 1876, October 2otb, in longitude 150' 25' W., obierved meridian altitude of sun's L.i. bearing North, was 49* 58' 50', index correction -J- i' ^°'i height of eye 19 feet: required the latitude. 5. In latitude 59° 36' N., the departure made good was 5a'9 miles East : required the diflference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's Sailing. Latitude of A 9"36'S. Longitude of A 2°io'W. Latitude of B 7 16 8. Longitude of B i 24 E. ADDITIONAL TOR ONLY MATE. 7. 1875, October loth: find a.m. and p.m. times of high water at Calcutta, longitude 88° E., Falmouth, and Scarborough. 8. 1876, October 9th, at 5'> 51'" a.m. apparent time at ship, latitude 18' 45' S., longitude 99° 18' E., sun's magnetic amplitude E. ;| N. : required the error of compass ; and supposing the variation to be i' 50' W. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1876, October 30th, p.m. at ship, latitude 32° 45' N., observed altitude sun's l.i,. 28' 30', index correction -|- 2' 30", eye 18 feet, time by chronometer, October 30'' n** 56™ 43% which was 2™ 28' slow for Greenwich mean noon, October ist, and on October 8th, was a™ 44»-8 altw for mean time at Greenwich : required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1876, October ist, mean time at ship 4^^ 54™ p.m., latitude 17° 8' S., longitude ij2° 33' E., sun's bearing by compass W. ^ N., observed altitude sun's l.l, 13° 59', index corr. — 22", oyo 17 feet: required the error of compass; and supposing the variation to be 7° 40' E. : required the deviation for the position of the ship's head at the time of observation. 11. 1876, October 2nd, a.m. at ship, latitude account 38^ 12' N., longitude 23' 34' W., observed altitude sun's l.l., South of observer, 47° 30', index correction — i' 38'', eye 17 feet, time by watch i*" 5o-», (or 2** i^ 50™,) which had been found to be 2^ 10"^ fast on apparent time at ship, the difference of longitude made to Ea-zt was 43 mileis : required the latitude. XX 322 Ordinary Examination. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, October 7th, the observed meridian altitude of the star a Pegasi (Marhah) was 54° 10' 15", bearing Souih, height of eye 13 feet: required the hititude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation: — Correct magnetic bearing. Ship 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. 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. ; Y..\ 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. | W. ; E. by N. | N. ; N. J W. Correct magnetic courses : — You have taken the following bearing 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 : — EXAMINATION PAPEE.— No. XI. FOR SECOND MATE. 1. Multiply 45*3 by 9*76, by common logarithms. 2. Divide 100-002 by i'ooi2, by common logarithms. 3-— H. Courses. K. -h Winds. Lee- way. Devia- tion. Remarks, &c. pts. I N. by E. 4 2 E. by N. ■^k 3°E. A point of land in 2 3 8 lat. 52°N.,long. 1 2o°E. 3 4 5 bearing by compass 4 4 <; N. byE.iE., dist. 16 5 N.E. f E. 4 5 N. by W. ■\\ 17° E, miles. (Ship's head 6 5 N. by E.) Deviation 7 5 as per log. 8 4 "J 9 W. f N. S 6 N. by W. 4 17° w. 10 6 2 11 6 3 12 6 4 I 6 12™ lo^), which had been found to be 42'" 10' fast on apparent time at ship, the diflference of longitude made to the West was 10 miles after the error on apparent time was determined : required the latitude. ADDITIONAL FOR MASTER ORDINARY. jj.. 1876, Dec. 2ist, the observed meridian altitude of star a Canis Minoris (ProcyonJ 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 by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Esquired. Ship's Head by Standard Compass. Bearing of i Distant Object | Deviation by Standard | Eequired. Compass. \ North .... N.E East S.E S. 2» W. S. 4 W. S. ro W. S. 16 W. South .... S.W West .... N.W S. i'-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. ^ 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. ^ S. ; S. by W. i W. ; S.E. ^ S. Magnetic courses:— You have taken the following bearings of two distant objects by your Standard Com- pass 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 : — EXAMINATION PAPER— No. XIII. FOR SECOND MATE. 1. Multiply 448000 by "0000448, by common logarithms, 2. Divide •o853y5 by '0742^;, by onrnxaan logsarithmB. 326 Ordinary Examination. 3 — H. Courses. K. 1 Winds. Lee- Devia- Remarks, &c. 10 way. tion. pts. I S.S.W. \ W. 12 2 West. JL 4 42° W. Apoint,lat.62°i8'N. 2 12 9 long. 83° 17' E., bear. 3 12 5 by compass N.i E. ;| E. 4 13 o dist. 2 3miles. (Ship's 5 S.W. 1 w. " 5 S. by E. \ 8° W. head S.S.W.) Devia- 6 II 4 tion'as per log. 7 II 2 8 II 3 9 E.f S. 5 4 S. by E. l| 15° E. lO 5 6 II 5 4 12 5 3 I W.N.W. 4 4 North. 3 19^ W. Variation 42' E. 2 4 2 3 4 2 4 5 O 5 N.W. i N. JO 7 S. by W. o 16= 30' W. 6 lO 2 7 II 4 8 II 8 9 E. f N. 3 4 N. by E. 3i 17° 14' E. lO 3 2 A current set the ship II 3 O S.W. i W. (correct 12 2 8 magnetic) 52 miles. 4. 1876, August nth, in longitude 92"* 12' E., the observed meridian altitude of 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. Eequired the course and distance from A to B, by Mercator's Sailing. Latitude of A 51° 30' N. Longitude of A 3° 30' 30" W. Latitude of B 20 o N. Longitude of B 33 4 56 W. ADDITIONAL FOR ONLY MATE. 7. 1875, July 24th : find a.m. and p.m. tides at Point de Galle, long. 80° E., St. Nazaire, and Jersey. 8. 1876, October 28th, at 8'' 30™ a.m. apparent time at ship, latitude 49° 40' N., longi- tude 116° 12' W., sun's bearing by compass E. io°4o' N. : required the true amplitude and 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. 1876, April i8th, a.m. at ship, latitude lo' 48' N., observed altitude sun's l.l. 38'io'5o", index correction + 45", height of eye 16 feet, time by chronometer 9"^ 27™ 2^, a.m. at Green- wich, which was o"" 49^*3 doio for mean noon at Greenwich, March 17th, and on April ist was I™ 58^'7 fast for mean time at Greenwich : required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1876, March 9th, mean time at ship ?,^ 11™ 42' a.m., latitude 29° 58' S., longitude 57" 24' E., observed altitude sun's l.l. 28" 23' xj", height of eye 16 feet, sun's azimuth E. 9° 40' S. : required the error of compass; and supposing the variation to be 17^ xo' W. : required the deviation for the position of the ship's head at the time of observation. IX. 1876, July 28th, A.M. at ship, latitude account 38° 54' N., longitude 39° W., ob- served altitude sun's l.l. 69' 10' S., index corr. -j- i' 27", height of eye 23 feet, time by wtitch 11'' 3™ 15', slow on apparent time at ship 28™ 45', the difference of longitude made to Emt was 32 miles after the error on apparent time was determined : required the latitude by rsduction to meridian. Ordinary JExmiinattoth. 327 ADDITIONAL FOR MASTER ORDINARY. 12. 1876, October 8th, the observed meridian altitude of a Oruis was 50° o' S., index correction — i' 12", height of 636 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. «« Head ! i,Sr6,°L Compass. -bompass. Deviation required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation required. North .... S. 25° W. N.E S. 21 W. East S. 2 1 W. S.E S. 16 W. South .... S.W West .... N.W S. I'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. i W. ; 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 E. ; N.W. i N. ; N. ^ E. ; S. by E. Correct magnetic courses : — You have taken ihe following bearings of two distant objects by your Standard Compass as above ; with the 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 : — EXAMINATION PAPER.— No. XIV. FOR SECOND MATE. I. Multiply icoooi by 8, by common logarithms. 2. Divide 37' 149 by 3-— 523 76, by common logarithms. H. Courses. K. -iV Winds. Lee- way. Devia- tion. Remarks, &c. I 2 3 4 5 6 7 8 S. by E. J E. N.W. ^ W. 3 3 3 3 2 2 2 2 3 4 6 7 4 3 3 S.W. i W. S.W. by W. pts. 3i 5°E. 18° W. A point. Cape of GoodHope,in latitude 34° 28' S., longitude 18" 28' E., bearing by compass N. i W. f W., dist. 2imiles. (Ship's head S. by E. ^ E.) Deviation as per log. 9 N. by E. 1 E. 4 6 N.W. 1 W. ^i 6°E. 10 4 4 II 4 7 12 I 2 S.W.5W.I W. 4 5 5 3 6 7 N.W. ^ W. If 7°W. Variation 25° W. 3 5 4 4 5 6 7 8 9 W. by N. ^ N. N.E. i E. 5 7 7 7 6 3 5 6 2 7 N. ^W. E.S.E. 1 2 16° W. 16° E. A current set (correct magnetic) E. by S.iS., 14 miles, from the time 10 II 4 4 s the departure was taken to the end of 12 5 ' day. 328 Oi'dinary Examination. 4. 1876, Feb. nth, in longitude 32° 20' E., the observed meridian altitude of the sun's L.L. was 30° 25' 10", observer North of sun, index correction — 3' 15", height of eye 12 feet: required the latitude. 5. In latitude 51" 10' the departure made good was 64-3 milys : re(iuiied the difference of longitude by parallel sailing. 6. Required the course and distance from A to B, by Mercator's sailing. Latitude A 43° 24' S. Longitude A 65° 39' W. Latitude B 26 38 N. Longitude B 15 8 E. ADDITIONAL FOR ONLY MATE. 7. 1875, April 2nd : find times of high water at Cape Virgin, longitude 68^ W., Water- ford Harbour, and Banff. 8. 1876, March 31st, 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 error of com- pass ; and supposing the variation to be 6° E. : required the deviation for the position of the ship's head at the time of observation. 9. 1876, May 27th, A.M. at ship, latitude 55° N., observed altitude sun's l.l. 43° 9' 5", index correction — 14", height of eye 14 feet, time by chronometer 9'^ 13" 12^ a.m., which was slow 485'5 for mean noon at Greenwich, April 9th, and on April 24th was fast o™ 25" : required the longitude. ADDITIONAL FOR FIRST MATE. 10. 1876, July loth, 9'" 44'" A.M., mean time at ship, latitude 59° 56' N., longitude 40° 20' W., observed altitude sun's l.l. 44° 49', sun's magnetic azimuth S. \ W., height of eye 20 feet : required the error of the compass : and supposing the variation be 51° W. : required the deviation for the position of the ship's head at the time of observation. 1 1. 1876, November 8th, p.m. at ship, latitude by account 33° 9' N., longitude 89° 42' E., observed altitude sun's l.l. 40° o', South of observer, index corr. — 6' 12", height of eye 19 feet, time by watch 8^ 20™ 20«, (or 7'' 20'' 20"" 20'), slow on apparent time at ship, the dif- ference of longitude made to the East was 32-3 miles: required the latitude by reduction to meridian, ADDITIONAL FOR MASTER ORDINARY. 12. 1876, July 19th, the observed meridian altitude of a Pavonis 32" 50' 15", bearing South, 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 Bequired. 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, AVith 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.M.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. \ E., find the bearings, correct magnetic. Compass bearings: — N. by W., and E. by N. Bearings, magnetic : — Ordinmy Exanmiation. 329 EXAMINATION PAPEE.— No. XV. FOR SECOND MATE. 1. Multiply 5362 by o'4i88, by common logarithms. 2. Divide 5*6949 by 53'058, by common logarithms. 3-— H. Courses. K. 1 IT! Winds. Lee- way. Devia- tion. Remarks, &c. pts. I N.E. 9 N.N.W. 1 61° E. Apointlat. 37"'37'N. 2 8 6 long. 0° 41' W. bear- 3 9 2 ing by compass 4 8 6 N.W. by W. i W. 5 E.N.E. 12 3 North. i 4 11° E. dist. 25 miles. 6 12 3 (Ship's head N.E.) 7 II 4 Dev. as per log. 8 12 9 N.N.W. 10 5 N.E. 1 2 6°W. 10 II I II 10 6 12 10 8 I E.S.E. 6 6 N.E. ij 92° E. 2 6 4 Variation 19° W. 3 6 5 4 6 5 5 N.N.E. 4 3 East. 2i 4°E. 6 4 8 7 8 4 4 5 4 A current set by com- 9 S.E. 8 5 E.N.E. 4 4°E. pass E. by S., 36 miles 10 8 7 from the time the de- II 7 4 parture was taken to 12 7 4 the end of the day. 4. 1876, November 21st, in longitude 70° 20' E., observed meridian altitude of the sun'a L.L. was 80° 20', bearing North, index correction — 2' 50", height of eye 20 feet : required the latitude. 5. In latitude 35° 39', the departure made good 66 miles. 6. Required the course and distance from A to B, by Mercator's sailing. Latitude A 6° i' N. Longitude A 60° 14' E. Latitude B 6 10 S. Longitude B 39 15 E, ADDITIONAL FOR ONLY MATE. 7. 1875, September ist: find the times of high water, a.m. and p.m., at Victoria River, longitude 130° E., and also at Beachy Head and Antwerp. 8. 1876, January i6th, at 7^ 22™ p.m., apparent time at ship, latitude 43° 4' S., longitude 10° 6' W., sun's magnetic amplitude W. 15'^ 56' S. : required the error of compass; and gupposing the variation to be 23° 20' W. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1876, June 5th, a.m. at ship, latitude 2° 5' S., observed altitude sun's l.l. 28° 4', index correction -\- 4' 25", eye 15 feet, time by chronometer, June 4'' i^^ 28-^ 42', which was jm ^3 fast for mean noon at Greenwich, March 6th, and on March 24th was o™ 8^ slow on mean time at Greenwich : required the longitude. UU 330 Or&ino/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1876, November 10th, S** 45™ 38^ a.m., mean time at ship, latitude 50' 30' N., longitude 86° 43' E., observed altitude sun's l.l. 6° 7' 10'', height of eye 15 feet, sun's magnetic azimuth S. 49° 50' E. : 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 at the time of observation. 11. 1876, January 8th, a.m. at ship, latitude account 35° 10' S., longitude 55° 12' W., observed altitude of sun's l.l. 76° 44', N., index correction -J- i' 18", height of eye 14 feet, time b}' watch 39™ 34^ (or 8=1 o*" 39"" 34*), which was 50™ 3' fast on apparent time at ship, the difference of longitude made to the East 21', after tha error on apparent time was determined : required the latitude, ADDITIONAL FOR MASTER ORDINARY. 12. 1876, February ist, longitude 50° "W., observed meridian altitude of the star a Canis Majoris fSiriusJ 37° 50' 20" S., height of eye 19 feet, index correction + i' 4' : required the latitude. In the following table give the correct magnetic bearing of the distant object, and thence the deviation : — Correct magnetic bearing. ^^P^*^^ij by standard Compass. j ^compass. Deviation required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation required. 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. § S. ; S. by E. ^ E ; N.W. by 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 E. by S. \ S. find the bearings, correct magnetic. Compass bearings :— N. W. by W \ W., and S. by E. \ E. Bearings, magnetic : — EXAMINATION PAPER— No. XVI. FOR SECOND MATE. 1. Multiply 5940 by 530, and •00087214 by -001963, by common logarithms. 2. Divide 9504000 by 98, and '9649 by 35*0583, by common logarithm^. Ordmiry Examination. 331 H. Courses. K. tV WiNBS. Lee- Devia- Remarks, &c. — way. tion: pts. I W.N.W. 12 6 North. 1 4 10' W. Apoint, lat. 36° 27'S., 2 12 6 long. 68° 37' W., bear- 3 12 8 ing by coiupass E. f S. 4 13 dist. 25 miles. (Ship's 5 ' S.AV. by W. 10 6 N.W. by W. 1 7°W. head W.N.W.) De- 6 10 4 viation as per log. 7 10 4 8 10 6 9 N. by E. 1 E. 7 3 N.W. 1 W. ^\ 2|°W. 10 7 6 II 7 8 12 I N.W. 7 II 3 4 N.N.E. 4 8'W. Variation 22\° East. 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 (correct 8 2 3 maguetic)S.S.W.|W. 9 N.E. ^ E. 4 7 N. by W. \ W. 2| 8'E. 32 miles, from the time 10 4 4 the departure was II 3 6 taken to the end of the 12 3 3 day. 4. 1877, January ist, in longitude 167° 54' E., the observed meridian altitude of sun'a I.L., 83° 40', zenith North of sun, 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 of A 8° 57' N. Latitude of B 36 50 S. Longitude of A 79° 31' W. Longitude of B 174 49 E. ADDITIONAL FOR ONLY MATE. 7. 1 875, March 28th : find the times of high water at Gibraltar, Port Louis (Mauritius), long. 57^° E., and Halifax, long. 64° W. 8. 1876, November 4th, at 4'^ 52'" 42« a.m., apparent time at ship, latitude 46° 40' S., longitude 8° 57' W., sun'a magnetic amplitude S.E. | S. : required the error of compass; and supposing the variation to be 16^ 30' W. : required the deviation of the compass for the position of the ship's head at the time of observation. 9. 1876, Sept. 1st, A.M. at ship, latitude 15* 31' S., observed altitude sun's l.l. was 15° 18' 20", index correction — 20", height of eye 26 feet, time by chronometer, Auq:'ist •jjd 20'' 12™ 40% slow 1" JO' on April 15th, and on April 29th was o™ z^^ fast on Greenwich mean time : required the longitude. 33« Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1876, June ist, 8'' ig"" a.m., mean time at ship, latitude 21° 10' N., longitude 61° 30' E., observed altitude sun's l.l. 39° 10', index correction — 15", hei|?ht 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. 1876, April 13th, A.M. at ship, latitude account 0°, longitude 147° 10' E., observed altitude of sun's l.l. 80' 30', North of observer, index correction -\- 1' 10', height of eye 16 feet, time by Avatch o'' o™ 12^ which had been found to be 11™ V fast on apparent time at ship, the difference of longitude made to the East was %\ miles, after the error on apparent time was determined. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, May loth, the observed meridian altitude of a- Centuri was 10° 4' 15", (zenith North), index correction — 2' 10", height of eye 20 feet : required the latitude. 13. At what time will the star a Aquila; (AltairJ pass the meridian of the Land's End, on December 8th, 1876, and how far North or South of the Zenith. 14. 1876, 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 Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard Compass. Deviation Required. North .... N.E East S.E S. 34° w. S. 30 W. S. 18 w. S. 2 E. South .... S.W West .... N.W S. 6°E. South. S. 13 W. S. 25 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. by W. | W. ; E. | N. ; N. by E. | E. ; 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. by E. ^ E. ; E. ^ S. ; S.W. by S. ; N. ^ 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. by W. J W., find the bearings, correct magnetic. Compass bearings:— W. | S. and E. by N. \ N. Bearings, magnetic : — Ordina/ry Examination. 333 EXAMINATION PAPEE.— No. XVII. FOR SECOND MATE. Multiply 30*24 by 12*5, and 034632 by -397302, bj'^ common logarithms. Divide 8100900 by 900, and -00005 ^7 ^'5i by 25, and by -0000025 ^y common logs 3 — H. Courses. K. .1. Winds. Lee- Devia- EeMARIvS, &c. way. tion. pts. I N.E. i E. 13 2 N. by W. i W. i 4°E. Apointoflandinlat. 2 12 9 5o''25'S.,long.i79°4o'E 3 13 5 bearing bj' compass 4 13 4 N.byW.iW.dist. 16 5 W.S.W. 3 5 N.W. 2i 9rw. miles. ((Ship's head 6 4 N.E. I E.) Devia- 7 4 I tion as per log. 8 3 8 9 E. by N. 12 2 N. by E. i 11" E. 10 12 4 II 12 6 12 12 8 I N.byW.^W. 2 4 N.E. i E. 3 4°W. Variation 14° East. 2 2 3 3 2 3 4 2 5 S. |W. 6 9 W. by S. i 2'' W. 6 6 8 7 6 8 A current set by 8 7 5 compass E.N.E., 42 9 E. by N. i N. II 5 N. by E. \ E. f 8*E. miles, from the time 10 12 2 the departure was II II 6 taken to the end of the 12 II 7 day. 4. 1876, September 23rd, in longitude 57°45'E., observed meridian altitude of sun's L.L., 84° 10' 50", bearing North, index correction — i' 36", height of eye 16 feet: required the latitude. 5. In latitude 52* S., longitude o" 40' "W., a ship sails 136 miles due East : required the longitude in. 6. Required the course and distance from A to B. 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. 1875, December 12th: find a.m. and p.m. tides at Aberdeen Bar, Penzance, King's Road (Bristol Channel), and Southampton. 8. 1876, November 5th, at 5^^ 10™ p.m. apparent time at ship, in latitude 20° 45' N., longitude 116° 45' E., sun's magnetic amplitude was S.W. f W. : required the 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. 1876, 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 chron. %^ 39*" 22' p.m., which was fast 29™ 32'-4 on Greenwich mean noon, July 8th, and on July 20th, vfa.afast 30™ o» on Greenwich mean noon ; course till noon West (true) 48 mil«s : required the longitude in at noon. 334 Ordina/ry Examination. ADDITIONAL FOR FIRST MATE. 10. 1876, August 13th, moan time at ship 9'' 5™ 20' a.m., latitude 30' 46' S., longitude 78" 50' W., sun's bearinu; by compass 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 th(i 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. 1876, June 12th, P.M. at ship, latitude account 15° 50' S., longitude 72° 12' E., observed altitude of sun's l.l., jo"" 10' 10", zenith South of observer, index correction — 5' 40", height of ej'e 26 feet, time by watch 28'" 40' (or 12"^ o'' 28'" 40*), which had been found to he slow 4™ 44' on ajjparent time at ship, the difference of longitude made to West was 16 j, after the error on apparent time was determined : required the latitude. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, December 7th, the observed meridian altitude of the star a Arietis was 60° 29' 50", zenith North of star, index correction — 2' lo'', 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 hearing. Deviation Required. Ship's Head by Standard Compass. Bearing of Distant Object by Standard "Compass. Deviation Eequired. North .... N.E East S.E S. 2° W. S. 10 E. S. 23 E. S. 34 E. South .... S.W West .... N.W S. 35° E. S. 30 E. S. 20 E. S. 10 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.W. \ N. ; S.AY. by S. | S. ; N.N.E. ; 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. f W. ; S.E. f E. ; W.S.W. ; S. by E. \ E. Magnetic courses : — You have taken the following bearings of two distant objects by your Standard Com- pass as above, with the ship's head at N. \ E., find the bearings, correct magnetic. Compass hearings: — W. by N. and N.W. | W. Bearings, magnetic :— EXAMINATION PAPER— No. XVIII. FOR SECOND MATE. 1. Multiply 7642 by 7429*5, and 000064 ^y 100004, by common logarithms. 2. Divide "39765 by 25, and loooooo by '0000001, by common logarithms. Ordinary Examination . 335 3-— H. Courses. K. 1 "Winds. Lee- Devia- Remarks, &c. way. tion. pts. I N. by W. 6 4 W. by N. i^ 8°W. Apoint, lat. 57'0'N. 2 6 3 long. 40" 0' W., bear- 3 5 6 ing by compass 4 5 4 N.E. by E. i E. 5 S.S.W. 1 W. 4 8 W. ^ N. ^i 5= W. dist. 19 miles. (Ship's 6 4 2 headN.^iW.) Devia- 7 3 7 tion as per log. 8 3 6 9 N.N.E. 1 E. 4 2 N.W. 1 N. 2 13° E. lO 4 4 II 4 5 12 4 5 I W. by N. 3 4 N. by W. 4 17F w. 2 3 4 Variation 48" W. 3 3 6 4 3 7 5 S.E. a E. 9 8 S.S.W. 1 4 11° E. 6 lO 5 7 II 2 8 lO 8 9 S. |W. 3 2 w.s.w. 2| 2° W. A current set correct lO 3 3 magnetic W.N. W. for II 2 8 the last 5 hours, 3 12 2 7 miles an hour. 4. 1876, June 25th, in longitude 59"* 15' E., the observed meridian altitude of sun's u.L. bearing North, was 60" 23' 15", index correction -|- 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 distance from Cape East, New Zealand, to San Francisco. A'^arialion 14° 20' E., and deviation 5° 40' E. Latitude Cape East 37° 40' S. Longitude Cape East 178° 36' E. Latitude San Francisco 37 48 5 N. Longitude San Francisco 122 24 W. ADDITIONAL FOR ONLY MATE. 7. 1875, August 7th : find times of high water a.m. and p.m. at Hong Kong, long. 1 14° E., New York (Sandy Hook), long. 74° W., and Skull. 8. 1876, 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. 1876, September 22nd, a.m. at ship, on the Equator, observed altitude sun's u.L. 17° 20*40", index correction — i' iS", height of eye 20 feet, time by chronometer September 22'^ 4*^ 59^" j6', which was slotv 15^ for Greenwich mean noon, April 30th, and on June ist yf&afast io^"6 for mean time at Greenwich : required the longitude. 336 Ordinary ExmnitiaUon. ADDITIONAL FOR FIRST MATE. lo. 1876, March list, mean time at ship 3^ 15'" p.m., latitude 9=7' S., longitude i59''4'"W., sun's bearing by compass W. \ S., the observed altitude sun's l.l. 42= 49' 45', index correction — 3' 14", height of eye 21 feet, variation by chart 7' 50' E. : required the error of the compass and deviation. If. 1876, October 4th, a.m. at ship, latitude account 30^ 24' S., longitude 140° 30' E., observed altitude sun's i..l. North of observer was 63" 37' 10', index corr. — i' 15", height of eye 21 feet, time by watch October 3-^ 22^ 37™ 15% which had been found to be i*> lO™ 20» slow on apparent time at ship, the difference of longitude made to East was 23^ miles after the error on apparent time was determined : required the latitude. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, June loth, longitude 25° "W., the observed meridian altitude of the star a Cassiopege, bearing South, was 85° o' 20, index correction + 34", 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. ov -cT A ' Bearing of iT c^ T:ttt^ i Distant Object Deviation by Standard by Standard Keauired. Compass. ^Compass. ^^.^.. ^v Standard Compass. Compass. Deviation Required. North N. i8°W. N.E 1 N. 17 W. East N. 13 W. S.E N. 8 W. South N. 2° W. S.W N. I E. West .... N. I W. N.W I N. 6 W. i 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.S.W.; N.E. by E.; S. by E. ; W. ^ S. Compass courses : — Supposing you have steered the following courses by the Standard Compass, lind the correct magnetic courses made from the above deviation table. Compass Courses :-S.W. | W. ; S. | W. ; E. J N. ; E.S.E. Correct magnetic courses : — You have taken the following bearing of two distant objects by your Standard Compass as above, with the ship's head at N.E. | E., find the bearings, correct magnetic. Compass bearings : — W. I S. and N.E. by E. Bearings, magnetic: — EXAMINATION PAPER.— No. XIX. FOR SECOND MATE. Multiply 6054 by 912, and 2070-5 by 62-0898, by common logarithms. Divide 117-658 by 146-932, and 167342 by '002, by common logarithms. Or dinar t/ Exmnination. 337 3-— H. Courses. K. rtr "Winds. Lee- Devia- Eemarks, &c. — way. tion. pts. I S. 1 W. 4 5 W. by S. 4 5°E. Apoint.lat. 62'»2o''N'. 2 4 2 long. 64° 40' W., bear- 3 4 ing by compass 4 3 9 W.byN.JN., 5 S.W. ^ W. 3 5 S. by E. z\ 9" W. dist. 21 miles. (Ship's 6 3 4 bcadS.iW.) Devia- 7 3 2 tion as per log. 8 3 3 9 E. 1 S. 5 4 S. by E. ^1 15° E. 10 5 3 II 4 4 12 4 2 I W.N.AV. 3 6 Nortb. 3 19° W. Variation 59° West. 2 4 5 3 5 3 4 5 7 5 N.W. i N. 10 2 E.N.E. a 17° w. 6 II 4 7 12 6 A current set by 8 13 4 compass E. by S. \ S., 9 E. |N. 5 5 N. by E. 3* 18° E. 49 miles, from the time lO 5 4 the departure was II 5 4 taken to the end of the 12 ^ 5 day. 4. 1876, June ist, in longitude 44° 40' E., observed meridian altitude of sun's l.l. was 72° 14' 10", zenith North of sun, 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. 1875, May 4th : find the a.m. and p.m. tides at Aberdeen, Wick, Fecamp. 8. 1876, December 28th, at 4'> 35" a.m., apparent time at ship, latitude 40° 10' S., longi- tude 75° E., sun rose by compass South: required error of the compass; and supposing the variation to be 19' 10' W. : required the deviation of the compass for the position of the ship's head when the observation was taken. 9. 1876, January 29th, p.m. at ship, latitude at noon 28° 45' N., observed altitude sun's L.L. 17° 46' 30', index correction — 3' 25", height of eye 16 feet, time by a chronometer, Jan. 28'' 16'^ 31"" 30'-, which was i™ i6^-^fast for mean time at Greenwich, December 17th, 1875, and on January ist, 1876, was i™ 3* sloxv for mean time at Greenwich; course since noon N.W. by W. (true), distance 20 miles: required the longitude at the time of observation, and also at noon. 338 Ordinary Examination. ADDITIONAL FOR FIRST MATE. 10. 1876, July loth, mean time at ship 3»" 14™ 2' p m., Lttitude 38^ 2 S., longitude 140° 58' E., sun's bearing hy compass N. 2° 15' E., ohserved altitude sun's v.l. 14° 56' 30", index correction -J- 3' 30", height of eye 19 feet, variation Ly chart 6' 45' E. : required the deviation of the compass for the position of the ship's head. 11. 1876, 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", eye 19 feet, time by watch November 28"^ 22'' 46", v?hich had been found to be i"^ 27™ alow on apparent time at ship, the difference of longitude m;ide to the West was 12-3 miles after the error on apparent time was determined : required the latitude. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, May r5th, thfi observpd meridian altitude of starts Orionis 52° 20' 30", zenith North of star, iudex correction — 4' 10", height of eye 15 feet : required the latitude. 13. 1876, September 4th, what bright stais in the Nautical Almanac will pass the meridian of a place in longitude 54° 40' E., between the hours of seven and ten. 14. 1876, June 15th, observed meridian altitude of t) Argus, under the South pole, was 47^ 50' 30", 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. „, . , , , Bearing of Ship s head Distant Object by Standard tv Standard Compass. j Compass. T, • ij Ship's head Deviafaon ^ standard required. -Compass. Bearing of Distant Object by Standard Compass. Deviation required. North .... S. 23»E. N.E S. II E. East S. 5 W. S.E S. 20 W. : South S.W AVest 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 Compass to make the following courses correct magnetic. Correct magnetic courses :—S. I W.; E. by F. ; S.E. by S. ; W. by 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. 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 :— EXAMINATION PAPEE.^No. XX. FOR SECOND MATE. 1. Multiply 6893 by 1130c, and -0001468 by -000395. by common logarithms. 2. Divide 7123 by 8"9596, and 268430 by '003010. by common logarithms. Ordinary JSxamination. 339 3-— H. Courses. K. 1 17 Winds, Lee- way. Devia- tion. Remarks, &c. pts. I S.E. iE. '3 7 N.E.byE.iE. 1 4 11° E. Apomt,lat.59°49'N. 2 '4 lon^. 44" 10' W., bear- 3 13 2 ing by compass 4 13 3 N.W. i W., dist. 5 E.|S. 10 4 S. by E. 1 E. 3 14° E. 30 miles. (Ship's head 6 10 4 S.E. iE.) Deviation 7 10 5 as per log. 8 10 3 9 N.byW.iW. 4 N.E. IE. 2i 8i° W. lO 3 6 II 3 4 12 3 4 I S.S.W.iW. II 6 S.E. i S. i 2l=W. Variation 53i^ W. 2 " 7 3 II 8 4 II 4 5 X.E. i N. 7 2 E. by S. I S. I 14° E. 6 7 3 7 7 4 A current set by com- 8 7 2 pasaS.E. iE. rvknots 9 S. iE. 12 5 E.S.E. i 3°E. per hour trom the time 10 12 the departure was II 12 3 taken to the end of 12 12 4 day. 4. 1876, October ist, longitude 84° 40' E., the observed meridian altitude of sun's u.l., zenith North, was 57' 20' 30", index corrrection — 3' 36", height of eye 17 feet: required the latitude. 4.* 1876, July 2nd, in longitude 45° 15' E., observed meridian altitude of the sun's l.l., below the pole, was 10° 19' 45', index con-ection — 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. Latitude of A 10° 8' S. Longitude of A 17^' 18' E. Latitude of B 23 12 N. Longitude of B 141 15 E. Variation \ point Wesf, and deviation 7° 15' West. ADDITIONAL FOR ONLY MATE. 7. 1 875, March 1 8th : find the times of high water, a.m. and p.m., at Cadiz, Southampton, Angra i'equena (S.W. coast of Africa), longitude 15° E. 8. 1876, April 25th, at 7'' 22"^ 8« p.m. apparent time at ship, latitude 57" 18' S., longitude loi' 50' E., sun's setting by compass N. 5 E., variation by chart 35° 50' W. : required the error of the compass and deviation. 9. 1876, August 24th, A.M. at ship, latitude at noon 37° 59' N., observed altitude sun's L.L. 37° 13' 30', index corr. •\- 2' 40'', eye 18 feet, time by chronometer, August 24<* 6'' 13"" 24% a.m. at Greenwich, which was i"" 5'/«4' for mean noon at Greenwich, August ist, and on August loth was o"^ /[z^ slow for mean time at Greenwich, course since observation N.N.W., 2 2' 4 (true) : required the longitude at noon. 3+0 Ordinary JExamination. ADDITIONAL FOR FIRST MATE. 10. 1876, Novemher 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 corr. — 3' 20", eye 21 feet : required the error of the compass ; and supposing the variation to be 33° 20' W. : required the deviation for the position of the ship's head at the time of observation. 11. 1876, May 29th, A.M. at ship, latitude account o' 31' S., longitude 150' 40' W., ob- served altitude sun's l.l. 67^ 41' N., index correction + 1', height of eye 20 feet, time by •watch May 29"^ 3*" 32"", fast on apparent time at ship 3*^ 38"^, the difference of longitude made to E'lst was 269 miles, after the error on apparent time was determined : required the lati- tude by reduction to meridian. ADDITIONAL FOR MASTER ORDINARY. 12. 1876, June 17th, 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. 13. 1876, 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 ? 14. 1876, 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. 15. 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 ? In the following table give the correct magnetic beating 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 Recjuired. 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 . \ W. ; S.S.E. | E. ; W. by N. i N. ; N. i 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. i N. ; S.W. by W. f 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 : — QUADRANT AND SEXTANT. 307. The Quadrant 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. 308. 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 ; 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 instru- ment, will answer for the quadrant, since the parts and appendages are common to both. 309. 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 the zero point 0° to about 140°, and each degree in the best instruments is again sub-divided into six equal parts of i o' each, while the vernier ff used in estimating the sub-divisions of the arc shows xo". The divisions are also continued a short distance in the opposite direction on the other side of zero (o), 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 micro- scope M, and its reflector r secured at the point <^ by a moveable arm dr to the index bar A E, may be adjusted to read off the divisions on the graduated limb and the vernier ff. The index bar A E is secured to the arc B C by the intervention of a mill-headed clamp screw s at its back, which must be loosened when the index has to be moved any considerable distance, and when * 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 Godfkay of Philadelphia, in 1730, and, perhaps, by Hadley, in t This depends on the propetties of ligb^, which w» cannot consideT here. 342 On the Fiextant. the contact nearly has been made b}^ 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 and directly over the centre ; I^>>£ik I another glass b is fixed perpendicular to the plane of the instrument frame H, of which the lower half only is silvered and the upper transjjarent ; 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 B C, 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 tu 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 Bome it is called a nonius, after the Portuguese, Munen or Nonius; but the invention of the latter (who died in 1577) was quite different. On the Sextant. 343 be preferred, as possessing greater magnifying power, and thus showing a better 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. Pour 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 of any brilliant object seen by leflection. Three more such glasses, some- times called back shades, are placed behind the horizon-glass at K, any one or 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. 3 1 o. Reading off the Angle. — The following brief directions for reading off will be more readily understood by the learner, if he place a sextant before him for reference and examination. It will be seen that the arc is divided into degrees from o° to about 140°; every loth degree is numbered from o 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 degree, or 10'. We will suppose it is an insti-ument 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 moveable radius, and is generally di^tinguished from the other line on the plate by a diamond-shaped mark, resembling a spear-head. Supposing this index to stand exactly at any of the lines on the arc, that is, so that the two lines are in the same direction ; in such a case the reading off is easily known, for it must be a certain number of divisions and sub-divisions, of which the value is seen at once. Thus, if it coincide, for example, with the second line to the left of 40°, then the reading off will be 40° 20', since each line on the arc represents 10'. * With respect to the dnrk j^lasses, when it is possihle (as in observing: altitudes of the sun in the mercurial horizon, &c.) to make thn observation with a single dark glass on the ej-e-end of the telescope, without using any shade, this should always be done, for the error of this dark glass does not all'ect the contact at all, and the distortion caused by it is not magnified, whereas any fault in the dark shade between the index and horizon -glasses produces actual error in the observation, and the distortion is magnified subsequently by the telescope. 344 ^^' ^^'^ Sextant. 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. These divisions are made less than the arc divisons, so that the line on the plate 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 vei-nier 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 divisions of the vernier just cover or coincide with 59 divisions on the arc, or the difference between a divison on the arc and one on the vernier is qL of a division of the arc ; if therefore a division on the arc is 10', the difference will be -g^ of 10' or 10". Every sixth division of the vernier being distinguished by a figure denoting minutes, and the in- terval between each of these figures is divided into six parts of 10" each. 311. To read off on a Sextant. — First examine the divisions and sub- divisons on the arc, up to the line which stands before the index. We then move the microscope on the vernier and examine the numbered lines. If any one of these coincides in direction with the opposite one on the arc, the read- ing 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 so many minutes or seconds. Let us now suppose the index to stand between the second and third divisions from 40°. In reading off, first 40° 20' is noted on the arc, and then running the microscope farther on the arc, it is observed that a line on the vernier and an arc line are in the same direction, between the lines on the vernier marked 5 and 6. The farther reading off is therefore 5' and some seconds. On examining the interval between 5 and 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". On the Sextant. 34; The sextant supposed under examination is marked to read off to the nearest 10" ; some instruments are graduated to 1 5" or 30", &c., but the same method of reading ofF is to be followed as pointed out above. 312. 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 o^ or zero, in which case we have to read off an arc divided from left to right by means of an index which is divided from right to left; this, however, is easily done if we remember that the line on the vernier marked 10' must be con- sidered as the commencement of the divisons, 9' must be considered as i', 8' as 2', 7 as 3', &c. ; or else take the difference between the minutes and seconds denoted by the vernier and 10' ; 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. ADJUSTMENTS OF THE SEXTANT AND QUADEANT. 313. The adjustments of the sextant and quadrant are: — (i) To set the mdex-glass and (2) the horizon-glass perpendicular to the plane of the instrument; (3) to adjust the line of coUimation 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 o (zero J on the vernier coincides ivith o (zero) on the arc; then, if the adjustments cannot be perfected, (5) ^0 find the index error of the instrument : — \st. The index-glass, or central mirror, must be perpendicular to the plane of the instrument. — Place the index to about the middle of the arCi 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 lach ; if it rises upward, the glass \Q2iTx% forward. The position is rectified by screws at the back. ^nd. 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 ii^ 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 (b) the sun is used. (b) By the sun. — The instrument being held perpendicular, look at the sun; sweep the index-glass along the limb, and if the reflected image pass exactly over the object itself, appearing neither to the right nor left of the object, then the horizon-glass is perpendicular to the plane of the instrument ; if not, turn 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, which can be turned by placing a capstan-pin into the hole in the head of the screw. y Y 34^ On the Sextant. ■^rd. 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 neai-est the plane of the sextant, and fix 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 he parallel to the index-glass. — Set o on the index to o on the arc ; screw the tube or telescope into its socket, and turn tlie 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 tangent screw at the back 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. f 314. Def. — Index Error of reflecting instruments such as the sextant, is the difi'erence between the zero point of the graduated limb, and where the zero point ought to be as shown by the index when the index-glass is parallel to the horizon-glass. ^ih. 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) By the horizon. — Move the index till the horizon, or any distant object, coincides with its image, and the distance of o on the index from o on the limb is the index error ; suhtractive when o on the index is to the left, and additive when it is to the right of o on the limb. Example i . — The horizon and its image being made to coincide, the reading is 2' on the arc. Then 2' is the Index Correction to be subtracted fron every angle observed. Example 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". (2.) Or measure the sun's horizontal 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 be off the arc; half the difference of the two readings is the index correction. * The error caused by the imperfection of this adj ustment is called the "Error of Collimation' and the observed angle is alwa)'s too great. t Some sexttints, as Tronghton's PilLir Soxtan^s, are not provided with the menns for making this adjustment ; because it is not absolutely necessary. An allowance, called Index Error, being made for the want of parallelism of the two glasses when the zeroes coincide. On the Sextant. 347 When the reading on the arc is the greater, the correction is suhtractive ; when the lesser, additive. Examples. Ex. I. On the arc — 33' 10" Off + 30 50 2)2 20 Index Cohr. sub. i 10 Ex. 2. On the arc — 30' 20" Off + 33 3 ° 2)3 10 Index Corr. add 1 35 If both readings are on the arc, or both off the arc, half their sum is the index correction — suhtractive when both on, additive when both q^'the arc. Ex. 3. ist reading on the arc — 65' 30" Ex. 4. ist reading off the arc + i' 30" 2nd do. on the arc — i 40 2nd do. off the arc 4" 66 50 2)67 10 Index Corr. sub. 33 35 2)68 20 Index Corr. add. 34 10 One-fourth of the sum of the two readings shotild 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 diffirence should be the sun's semi- diameter. Thus, suppose the ohservations, in Example r, to he made on Septemher 26th, 1876 ; 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. In order to obtain the index correction with the greatest precision, the mean of a number of measures of the sun's diameter should be taken. Examples for Pbactice. Ex. I. 1876, 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. 1876, July 4th, the reading on 33' 10", off 29' 50': find index correction and semi-diameter. Ex. 3. 1876, Novemher 13th, on 4' 40 , off 60' lO' ; find index correction and semi- diameter. Ex. 4. 1876, July 10th, on 32' 45", off 34' 30": find index correction and semi-diamoter. Ex. 5. 1 876, March 2 ist, off i' 10' o", off 6' 40' : find index correction and semi-diameter. Ex. 6. 1876, January 17th, on 67' 40", on 2' 30" : find index correction and semi-diamctcr. 315. The Prismatic Sextant. — In the form of instrument just described, and which is all but universally employed, the angle measurable is limited to 140"; but we may perhaps add that Pistor and Martins, of Berlin, have, by an ingenious miidifi'jation 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 Fribmaxio Sbxtakx. 34^ On the Sexitmt. The following shows the form of Examination Paper on the Adjustment of the Sextant. EXAMINATION PAPER. Ezn. 9a. Port of ADJUSTMENTS OF THE SEXTANT. Rotation No. The applicant will answer in writing, on a sheet of paper which will be given him by the examiner, all the following questions, numbering his answers with the numbers corresponding to the questions. I. — What is the first adjustment of the sextant ? A. — The index-glass must be perpendicular to the plane of the sextant. 2. — How do you make that adjustment ? A. — Place the index near the middle of the arc, and look into the index-glass so that you can see both the arc and its reflection ; if they be in one line, the glass is perpendicular, but if not in one line, they are brought so by gently moving the screws in the frame upon which the glass stands. 3. — What is the second adjustment? A. — The horizon-glass must be perpendicular to the plane of the sextant. 4. — Describe how you make that adjustment ? A. — Place of the vernier on on the arc, hold the instrument obliquelj^, with its face upwards, and look at the horizon ; if the reflected part and the direct portions of the horizon are in one line, this adjustment is perfect, but if not, they must be brought in line by gently moving a screw at the back of the glass. 5. — What is the third adjustment? A. — The index and horizon-glasses must be parallel when the index is at 0. 6.— How would you make the third adjustment P A. — Place the index at 0, and holding the instrument vertically, look at the horizon ; if the reflfcted 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. 8. — How would you find the index error by the horizon ? A. — Hold the instrument vertically, and, looking at the horizon, I would move the tangent screw until the horizon in both parts of the horizon-glass form one line ; the reading is the index error. 9. — How is it to be applied ? A. — To be 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. Note. — The examiner will see it is correct. 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. Note. — In all cases the candidate will name or otherwise point out the screws used in the various adjust- ments. Note to 10 and 11. — When the examiner is sati.=fied that tie candidate can read the arc of the sextant both on and off the arc, it will be sufficient to place his initials against 10 and 11 on the paper containing the answ«r. The above eomphtei the exmminatien (^ Seeottd and Onh/ Mates, On the Chart. 349 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-— I would place the index at about 32' on the arc, and, looking at the sun, two suns will be seen ; bring their upper and lower limbs in exact contact, read oflf and mark down, then place the index at about 32' off the arc, or to the right of 0, 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.— It is to be added when the greatest reading is off the arc, and subtracted when the greatest reading is on the arc. 14. What proof have you that those measurements or angles have been taken with tolerable accuracy ? A. — I would add the two readings together, and divide the sum by 4 ; if the measurements 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, I would repeat my observations until they do. ON THE CHART. 316. 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 wilh reference to the land, and of shaping a course to any place. 317. The use to be made of the chart in each case determines the method of projection, and the particulars to be inserted, (i) 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 charV 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 be 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 practicable, and of advantage, a chart on the *' central projection,'^ or gnomic, exhibits the track as a straight line, and is therefore convenient."' * The method lately introduced by Hugh Godfray. Esq., M.A., St. John's Collpge, Cambridge, deserves special mention, as its beauty and simjilicity will ultimately lead to its general adoption. A ch irt, on the central projection, as stated above, exhibits the great circle as a straight line, and thus it is seen at once, whether the tra( k between two places is a practicable one; hence, also, we have by inspection the point of highest latitude. An accouipanving dia^rRm then gives the different courses, and distances to be run on earh, in ord<'r to keep within ^ of a point to the jjreat circle. 'I'his chart and dintiram is fully de- scribed in the Tramaclions of the Cambridge Fhiloiophisal Society, vol. X, part 11, and is published by J. D. Potter, Poultry. 350 On the Cha/rt. ON MEECATOE'S CHAETS. (See Norie, pages 126 — 131 ; or Raper's '■'Practice of Navigation," pages 120 — 127, on this subject.) 318. A chart used at sea for marking down a ship's track and for other purposes, exhibits the surface of the ^lobe 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 chart into what is called the meridional difere^ice of latitude, and the departure BH + 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 lines 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 * It is plain from the principles of Mercator's projection, and from the diagram (page 183) which connects the enlarged meridian with the difference of longitude, that if a ship set out on any point on tl)e 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 projected 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 be reached only after the ship has circulated round the pole 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 p tradoxical 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 '- ^ — The apparent paradox of the infinite number of revolutions about the pole being performed in a finite time, becomes explicable- when we consiiler that, whatever be the progressive rate of the ship alonii iis undeviating course, the times of perlorming the successive rf volutions continually diniin'sh as the ship avproacties the pole, both the extent of circuit and the time of tracin" it tending; to zero, the limit actually attained at the })ole 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 just 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 recedinij from this pole the track desiribed will at length unite with that at lirst traced, the point ot juii'tion bein^ thnt from which the ship orii;inailj t->)j...rted. But for the strict riiMtheniatical proof i--f these latter circunistances tlie student m,-.;- ctnsult Professor Davieb' curious and instructive papers on tSpIi^rical C«-ordinat«» in (ke Edmburgit TiwuiRCtionB, vol, XII. On the Chart. 351 — 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 tf)p and bottom are graduated to degrees and minutes — and longitude is measured on the graduated parallels. (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 eleveu on days of full and new moon. The anchors on the chart denote anchorage. The small arrows show the direction of the set of the current, the current going with the arrow. (5.) Lines called Compasaes, similar to those on the compass card, are drawn at convenient intervals on the chart. In charts of large seas, as the Atlantic, these compasses 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 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 diiference 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. — (Raper, page 120, No. 367.) EXERCISES ON THE CHART. FOR ONLY MATE, FIRST MATE, AND MASTER. North Sea. (i.) Latitude 55° 5' N. Longitude o 5 E. Required the course and distance to Hartlepool. (3.) Latitude 53° 35'^- Longitude o 55 E. Required the course and distance to the Dudgeon Light. (5.) Latitude 6o°2i'N. Longitude o 35 E. Required the course and distance to Udsire. (2.) Latitude J7°3o' N. Longitude o 40 E. Required the course and distance to Tjnemouth Light. (4.) Latitude 55°io'N. Longitude o 35 E. Required the course and distance to Flambro' Head. (6.) Latitude 57° 25' N. Longitude 7 25 E. Required the course and distance to the Naze of Norway. 352 On the Chart. (7.) Latitude 55''28'N. I (8.) Latitude 58°5o'N. Longitude o 30 W. Longitude 4 33 E. Required the course and distance to Required the course and distance to Tj'nemouth Light. ' Sundevoieg (9.) Latitude 55°4o'N. Longitude o 15 W. Required the compass course and the distance to St. Abb's Head Light. (10.) Latitude 58°25'N. Longitude 2 10 W. Required the compass course and the distance to Duncansby Head. (12.) Required the direct true and magnetic Course and Distance between Bachanness in Scotland to the entrance of the Texel. (11.) Required the true and magnetic Bearing and Distance between Whitby and the Naze of Norway. (13.) A ship from Kinnaird's Head, in Scotland, sailed S.E. by E. (true) 186 miles: required the latitude and lonsritude she is come to, and the direct course and distance she must sail, in order to arrive at Heligoland. (14 ) A ship from Helii^oland 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 the place of meeting ; also the course steered by each ship. (15.) Sunderland Light, bearing by compass S.W. \ S. Coquet Island ,, „ N.W. Required the latitude and longitude of ship ; also the course and distance to Hartlepool Light. (16.) Buchanness Light, N. by "W. \ W., by compass. Girdleness Light, W^est. Required the latitude and longitude of ship ; also the course (by compass) and distance to the Staples. (17.) The Skerries North by compass. Sumburg Head, W. \ S. „ Required the latitude and longitude in ; also the compass course and distance to Peterhead. (18.) Flambro' Head Light, S.W. by S. by compass. Whitby Lights, N.W. by W. | W. „ Required the latitude and longitude in ; also the compass course and distance to Outer Dowsinga. (19.) Farn Lights, S."W. by S., by compass. Berwick Lights, W. by N. ,, Required the latitude and longitude ; also the distance from each light. (20.) 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. (21.) Scarbro' light was observed to bear S.W. by compass, then sailed E.S.E. ir miles, and the light then bore West : required the latitude and longitude of the ship at each station, and her distance from the light. (22.) Coasting along shore, I observed Tynemouth light to bear W. by S, by compass; I then sailed S. by W. 16 miles, and the light bore 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. (i.) Latitude 50" i' N. j (2.) Latitude 48^50'N. Longitude 2 4 W. [ Longitude 5 50 W. Required the compass course and distance Required the compass course and distance to the Caskets. to Ushant. (3.) Latitude 49° 30' N. Longitude 3 30 W. Required the compass course and distance to the Start Point. (4.) Latitude 50° 10' N. Longitude i 10 W. Required the compass course and distance to St. Catherine's Light. On the Chmt. 353 (5.) Latitude 50°3o'N. Longitude o 55 E. Required the compass course and distance to Dungeness. (7-) Latitude 50° 10' N. Longitude 3 10 W. Required the compass course and distance to Portland. (9,) Latitude 50° 50' N. Longitude 10 35 W. Required the compass course and distance to the Fastnet Rock. (11.) Latitude 50^18' N. Longitude o 10 E. Required the compass course and the dis- tance to Beachy Head. (13.) Latitude 51° 6' N. Longitude 6 12 W. Required the compass course and the dis- tance to St. Anne's Head Light. (15.) Latitude 50°5o'N. Longitude 7 20 W, Required the compass course and distance to Old Head of Kinsale. (17.) Latitude 5o°4o' N. Longitude 6 30 W. Required the compass course and distance to Lundy Island. (I9-) (6.) Latitude 48°55'N. Longitude 6 5 W. • Required the compass course and distance to the Lizard. (8.) Latitude 49°55'N. Longitude 3 55 W. Required the compass course and distance to the Eddystone. (10.) Latitude 50'55'N. Longitude 6 55 W. Required the compass course and distance to Trevose Head. (12.) Latitude 51° 16' N. Longitude 10 38 W. Required the compass course and the dis- tance to the Fastnet Light. (14.) Latitude 51^52' N. Longitude 6 6 W. Required the compass course and the dis- tance to the Tuskar Light. (16.) Latitude 50^30' N. Longitude 8 30 W. Required the compass course and distance to Cape Clear. (18.) Latitude 5i°28'N. Longitude 6 30 W. Required the compass course and distance to Smalls Rock. Longships Light, hearing 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. (20.) Cape Barfieur, bearing by compass N.W. St. Marcouf, „ „ S.W. Required the latitude and.longitude of ship ; also the compass course and distance to Cape de la Heve. (21.) Berry Head, bearing by compass N. \ E. Start Point, „ „ W. by N. | N. Required the compass course and the distance to Portland. (22.) Bill of Portland, bearing by compass N.W. by W. St. Alban's Head „ „ N.E. \ E. Required the latitude and longitude of ship, and the compass course and the distance to Start Point. (23.) 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. (24.) Tuskar Rock N.E. by compass. Great Saltees Lightvessel N.W ^ W. „ Required the latitude and longitude of ship ; also the course (by compass) and distance to the Smalls. 354 On the Chart. (25.) 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. (26.) Bembridge Lightvessel, bearing by compass N. \ W. Owers Lightvessel, „ „ East. Required the latitude and longitude of ship ; also the compass course and distance to St. Catherine's Point. (27.) Needles Light, bearing by compass N. ^ E. St. Catherine's Light, „ „ E. | S. Eequired the latitude and longitude of ship ; also the compass course and distance to St. Alban's Head. (28.) 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. (29.) Lizard Lights, bearing by compass E. ^ S. Longships, „ ,, N. \ W. Eequired the latitude and longitude of ship ; also the compass course and distance to St. Agnes' Light. (30.) Mine Head Light, bearing by compass N.E. | N. Ballycottin Light, „ „ N.W. Required the latitude and longitude of ship ; also the compass course and distance to Old Head of Kinsale. (31.) 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.) (32.) Dungeness, bearing by compass N.E. by E. | E. BeachyHead „ „ N.W. J W. Required the latitude and longitude of the ship ; and her distance from each place. (33.) 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 lier 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. EXAMINATION PAPER. Ezn. 9b. Port of EXAT^nNATION IN CHART. Rotation No. The applicant will be required to answer in writing, on a sheet of paper which will he given him by the Examiner, all the following questions according to the grade of Certificate required, numbering his ansivers with the numbers correspo7tding with those in the question paper. I. — A stransre 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, On the Chart. 355 2. — How do you ascertain that in our British Charts ? A. — In our British charts there is alwaj's 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.— I would lay the edge of a parallel ruler over the two given places, A and B, then taking care to preserve the direction, I would move one edge of the ruler until it came 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 A. — points of variation should be allowed to the and the magnetic course would be 5. — How would you measure the distance between those two or any other two places on the chart ? A. — I would measure one-half the distance on the chart by my dividers, then placing one leg of the dividers on the middle latitude, I would measure on each side of the same, and the distance measured between those two extreme points would be the distance. 6. — Why would you measure it in that particular manner r A. — Because on a Mercator's chart the degrees of latitude increase as you approach the poles. The above comprises all the qicestions 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. 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 ol'tained by your lead-line on board with the depths marked on the chart ? A. —The time of the tide and the "rise and fall," or as it is now called the "mean spring range." 10. — What do the Homan 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. ir. — 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 I would add 49 minutes for every day that has elapsed since the full or change of moon, the sum would be the p.m. tide for the given day approximately ; or, to the time of the moon's meridian passage, corrected lor longitude, add the port establishment, the sum would be the p.m. tide required. All the above questions should be answered, but this does not preclude the Examiner from putting any other questions of a practical character, or which tJi^ local circumstances of the port may require. TO FIND THE COUSSE TO STEER IN OEDER TO MAKE GOOD ANY COURSE IN A KNOWN CURRENT, AND ALSO THE DISTANCE MADE GOOD. Draw a line on a chart to represent the course to be made good ; from the ship's place on the chart lay off 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; 356 Ijog Line. 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 the point where the other foot i*eaches the first line draw a line to the mark on the line representing the direction of the current. The course to be steered is represented by the lino last drawn, and the parallel ruler being placed to it, and moved to the centre 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. LOa LINE. 3 1 9, The length of the stray-line should be sufficient to allow the log-chip to be clear of the eddies of the vessel's wake. The distance between the knots should bear the same proportion to the number of seconds run by the glass intended to be used, as the number of feet in a nautical mile bears to the number of seconds in an hour. The number of feet in a nautical mile is 6080. The number of seconds in an hour is 3600. McvrUng the Lead Line. — Soundings. 357 Therefore, to find the length of a knot corresponding to a 28 seconds glass, we proceed as follows : — -^ 3600 ; 6080 ; : 28 28 48640 12160 ft. in. 360,0)17024,0(47 3J 1440 2624 2520 104 12 360)1248(3^ 1080 We have for glasses running 30 seconds and 32 seconds the following proportions : — 3600 : 6080 : : 30 : 50 feet 8 inches. 3600 ; 6080 : : 32 : 54 feet o\ inch. MARKINa THE LEAD LINE. 320. In nautical phrase the lead line has '' nine marks and eleven deeps." At two fathoms, the mark is leather ; at three fathoms, leather ; at five, white rag ; at seven, red rag ; at ten, a piece of leather with a hole in it ; at thirteen, blue rag ; fifteen, white rag ; seventeen, the same as at seven ; at twenty fathoms, a piece of cord with two knots. 321. Deep-sea lead lines are marked the same as far as twenty fathoms ; then add a piece of cord with an additional knot for every ten fathoms, and a strip of leather for every five fathoms. SOUNDINaS. 322. 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 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; * The Eule for Proportion is— Multiply the second and third terms together and divide this product by the first term, the quotient will be the fourth term required. 358 Soundings. 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 1875," (p. 98, Table B], published by the Hydrographic Office, Admiralty. f 323. As the soundings upon the chart are all referred to or measured downwards from the mean level of low water of ordinary spring tides, ^ 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. 324. 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 principal ports in Great Britain. * The reader may obtain an idea of this law, sufficiently 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 these 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 risen just half its 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 upon principles theoretically correct, will represent with sufficient exactness all that is necessary for practical purposes. t Table XIX, Raper, which the author, in 1847, computed for Raper'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 deduced from observations made for two years at the Tide Grauge, St. George's Pier, is a6 feet. Soundings, 359 325. To find how much we must subtract from cast of the 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 headof the place near your position, and opposite the day of the month, take out the " time " of high tvater in the morning or afternoon, as the case requires, and also from the adjoining column, under ^^ height,'''' take out the height of the tide. 2°. Next, underneath the time of high water 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, whic'h stands at the foot of the column. The remainder is the half range of the day. 4°. Enter Table B, page 98, Admiralty Tide Tables, and under the time from high water, 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 shown by the lead, the remainder is the sounding shown by the chart. Note I. 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 subtractive. In such cases, subtract the half mean spring range from the correction by Table B, and add the result to the soundings by lead ; the sum will be the sounding on the chart. Examples. Ex. I. 1875, September r5th, at 9^ 19"^ p.m., a ship off Liverpool strikes soundings in 8 fathoms: required the corrected soundings to compare with the chart. (The half spring range by Captain Denham's chart is 15 feet.) Admiralty Tide Tables (page 70) ; time of high water at Liver- pool, September 15th, 1875 Il''I9'"P.M. Time of sounding 9^9 Time from high water 20 ft. in. Height at Liverpool 26 6 Half mean spring range 13 o Half-range of the day .. 13 6 In Table B, page 98, under 2^, opposite 13I ft., stands add, , 6 9 Half spring range by chart ^S ° Correction 3^ fathoms, or 21 9 Depth by lead 8 fathoms. Correction 3 J „ Showing the depth by comparison . . . . 4^ ,, Whence the depth to compare with the chart is only 4^ fathoms instead of 8 fathoms. Ex. 2. 1875, October i6th, at 7'^ 42*" a.m., a vessel anchored off Weston-super-mare in 6\ fathoms ; at low water the vessel was " high and dry :" required the cause of this. (Half spring range by chart 23 feet.) ,6o Soundings. By Table: October i6th, the time of high water at 'Westoii- super-mare .. .. , VIZIS'" a.m. Time of anchoring 7 4^ Time before high water o 24 Height of tide by Tables 40ft. o in. Half spring range 187 Half range 21 5 By Table B, 24" and half range 21 feet 5 inches give add 20 10 By chart; half spring range 23 ° Correction to low water 43 1° Sounding 6| fathoms, or 39 ° 4 10 Water below the sounding ; or, the ship is found to be 4 feet 10 inches dry at low water. Ex. 3. 1875, March 7th, at g*" i''^ 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, 1 1 feet) . By Tables : March 7th, time of high water at Liverpool . . 11^4™ a.m. Time of crossing the bar 9 i Time from high water 23 Height of Tide by Tables 25ft. 7in. Half spring range 13 o Half range for the day .. 12 7 By Table B : 2'' 3" and half range 12ft. 7in add 6 3 Half spring range by chart 13 o Add for Liverpool chart 20 Correction .... 21 3 Depth on Bar at 2^ 3"* from high water, March 7th . . . . no By Chart : depth on Victoria Bar at low water springs . . 323 or 5 1 fathoms, nearly. Ex. 4. 1875, September 17th, at 2^ 34" p.m., off "Weston-super-mare, sounded in 4^ fathoms : required the soundings on the chart. Time of high water, Weston-super-mare, September 17th S*" 6" p.m. Time of Sounding 2 34 Time from high water 5 32 Height of Tide, Weston-super-mare, September 1 8th . . .. 39ft. 6in'. Half mean spring range 187 Height above half tide 20 1 1 By Table B: 5i> 32'" and half range 21 ft subt. 20 3 Half spring range 18 7 Level of tide below zero 18 Soundings by lead 4^ fathoms, or 27 o Correction , -|-i 8 Soundings on chart 28 8 Or a little less than 5 fathoms. Sounchn^s. 361 Examples for Practice. Ex. I. 1875, August i8th, at ^^ 12™ a.m.: required the depth of water on the "Four- fathom Ledge," off 'Weston-super-maro. Ex. 2. 1875, June i8th, at s^ 17" p.m. ; off Brest, the depth of water by the lead was io| fathoms : required the soundings on the chart. Ex. 3. 1875, August 1 6th, at g** 24'" p.m., sounded in the Victoria Channel, Liverpool, in 5 fathoms : required the soundings on the chart. Ex. 4. 1875, September i8th, at 8'' 38" a.m., a vessel anchored off "Weston -super-mare in 6 fathoms : required the depth at low water. Ex. 5. 1875, March 9th, at 5'' 42™ a.m. : required the height of the tide above mean low water of spring tides at Liverpool. Ex. 6. 1875, December 25th, at 9'' 39'" a.m. : going up the Firth of Forth, the lead showed 12 fathoms : required the soundings on the chart. 326. The following is the form of the Rule as used at the Liverpool Examinations : — 1°. Take the difference between the time of high water, full and change, at Liverpool and full and change at ship, and tahe this difference /ro«j the time of high water on the given day at Liverpool ; the result is time of high water at ship. t°. 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, a« 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 \ mu t> j i.- n T * ^^ -D i oi- I i he Reduction or Correction Is to bprmg Range at Ship, { f Q rl " So is the Reduction at Liverpool i ■ , i . t ^ j- ^i ^ , To the Reduction at Ship. ) '' *' ^' taken from the Cast. 1875, September 19th, at i'' 57™ p.m. at ship, off Holyhead, sounded in 45 fathoms: re- quired the corrected cast to compare with the chart. Full and change at Liverpool 11*123'" ( Page 152 Admiralty Tide Full and change at Holyhead 10 11 ) Tables, 1875. Difference — i 12 Time high water, Liverpool, Sept. 19th i 9 P.M., and Height of tide 27ft. Half mean spring range .13 Time high water at ship 11 57 A.M. — Time of cast 1057 P.M. Half range for day 14 Time of cast from high water 2 o) give in Table B correction -\- 7 Half range 14ft. J Half spring range 13 By Proportion. — ft. ft. ft. Reduction at Liverpool 20 26 : 20 : : 16 16 fath. ft. 26)320(12 feet = 20 26 45 o cast taken. 60 43 o cast corrected. 52 _ Note. — In the above proportion, 26 is the spring range at j{ Liverpool, 16 the spring range at ship, and 20 the reduction lit Liverpool. AAA ANSWERS. NOTATION, page i6. 1. 63; 8r; 99; 40; 13. 2. 200; 303 ; 598 ; 888. 3. 4000; 1783; 6083; 7930; 9009, 4. 27504; 89064; 33000. 5. looooo; 676050; 603240. 6. 20600; 90092; 204641; 800800. 7. 3006004; 5030040; 7700006; loorooio. 8. 7003000; 11108106; 54054088; 613020303. 9. 70704032; 45387025; 349004065; lOOOIOOOI. 10. 842248484; 909009099; 222000040; 305040008. 11. 700700700; 2O23027.00 ; 900000900; lOOOIOOOI. NUMEEATION, page 17. I. Forty-three. 2. Sixty. 3. Eighty-eight. 4. Ninety-seven. 5. Fifty-nine. 6. Twelve. 7. Twenty-one. 8. Nineteen. 9. One hundred and twenty-three. 10. Four hundred and seven, ir. Five hundred. 12. Nine hundred and ninety-nine. 13. Seven hundred and thirty-eight. 14. Eight hundred and thirty-seven. 15. Two thousand seven hundred and sixty. 16. Five thousand and eighty. 17. Seven thousand and thirty-six. 18. Two thousand. 19. Three thousand and three. 20. Five thousand five hundred and five. 21. Thirty-seven thousand six hundred and fifty-four. 22. Eighty- seven thousand and seventy-eight. 23. Thirty-seven thousand and three. 24. Sixty- three thousand and ninety. 25. Six hundred and ninety thousand and six. 26. Eight million, forty-seven thousHnd three hundred and twenty-eight. 27. Four million, ninety thousand and three hundred. 28. Five million, two hundred and ten thousand and seven. 29. Six million, thirty thousand four hundred and five. 30. Five hundred and sixty thousand and seventy-five. 31. Three million and six. 32. One million, three hundred and ninety-seven thousand four hundred and seventy-five. 33. Twenty million, eighty- four thousand two hundred and sixteen. 34. Five million, one thousand eight hundred and sixtv. 35. Eiaht million, eiuhty thou.sand eitjht hundred and eight. ^6. Fifty-five million, seven hun(ired thousnnd and five. 37. Seventy-six million, 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 hundr d and two million, two hundred and two thousand two hundred. 42. One hundred million, one hundred thousand one hundred and one. 43. Two hundred and seventy-five million, eight thousand and five. 44. Twenty million, eighty-four thousand two hundred and sixteen. 45. Seventy-nine million, thirty thousand one hundred and eighty-four. 46. Four hundred and eight million, seventy-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. SIMPLE ADDITION, pagen 18 — 19. I. 1274170 2 1634607 3. 1659291 4- 2333431 5- 3005313 6. 1536206 7 1 648 1 27 8. 2067690 9- 3329'75 10. 3724599 II. 4483647 12 4105670 13. 3312667 14. 3018498 15- 2797285 16. 35'9772 17 9185198 18. 7485613 19. 8518439 20. 7498 '59 21. 956011; 1; 22 56?I4U 23. 6524956 24. 82383^,6 25- (i) i37«8543 (2) 12844819 (3) 14661377 (4) 13937260 (5) 15878135 (6) 10176138 (7) 10970368 (8) 13825798. 26. 20566726566 Answers. 363 SIMPLE SUBTRACTION, pn,,, 20. I. 621511 5. 676001 9. 681179 13. 107500 17- 21. 25- 29. 33- 37- 41- 8261243256 6009085424 91089009099 353532599691 65Z'9'3 8087 ; 4936 30094003 2- 539540 6- 554999 10. 507871 14. 222419 18. 2358235814 22. 9957614250 26. 238036793034 30. 73708072035222 34. 4244'03 38. 207786 42. 30449329 4. 9 8. -590998 7. 480895 II. 376999 12. 174386 15. 157406 16. 58024 19. 2006289547 20. 763595488 23. 78098951912 24. 7501213600 27. 43437255048818 28. 9088910990901 31- 5540058 32- 5866974 35. 5460813 36. 8026758 39- 55599 4o. 999882 SIMPLE MULTIPLICATION, page 25. I. 685295792 5. 550942443156 9- 9876543210 13. 1350705843 17. 27349835014665 20. 199999929143681 23. 12838608 26. 55275801000 29- 15993780666 32.5750745672129 35- 2997332184 38- 24335360 41. 28894158 44. 6680943744279021 47. 2872556494008787 50. 2054793040961760 53- 1944460921158 14. 1962965961 45652143474 9803614194 2684444024 15- 1506172792 3886950304 7774239492 5629618680 4. 1899328910 8- 5159176101 12. 11019283848 16. 8918232255 18. 22591055500000 21. 10285980 24. 40261296 27- 35205962324 30. 490360 9193 33- 531954730112 36. 2466490572 39. 47094144 42. 78522048 45. 8312372968202684 48. 6324602392508400 51. 47287079491501550 54. 50603441 27 169150 56. 10203029078666688093030201 57. 99999995000000040000 58. 999400149980001499940001 59. 999400149980001499940001 19. 770930181732 22. 16261578 25- 1350^80000 28. 61286228934 31. 2243503727343888 34. 32228449759163 37. 38114062 40. 20146968 43- 39513' 2893090991 46. 121932631112635269 49- 35333670133890810 52. 12C03400820050006000000 55. 29622961181458719J480656 60. 45673337928960 SIMPLE DIVISION, page 30. 15- 17- 19. t. 67896347-1 2. 194899128-2 5. 66779748-5 6. 39512348-1 9. 25409614-6 10. 100107478-9 13. 463519673763533-5 1582874324701-32 95022741046776-8 133683783399807-6 27206980239559-123 329218107-670 8607936214-143 24'993504-5'8790 48481368-72 862152-1422 4713708 14278693864-88877 1068392-1 17002 3183098861-2599872450 23- 25- 27. 29. 31- 33- 35- 37- 39- 3. 99836471 4- 59648952 7. 868427625-6 8. 274473675 II. 91261430-10 12. 4953087942-8 14. 27201490438560034-10 16. 187157296759729-46 18. 14964459409277-63 20. 40316322081 10056-69 22. 34045491087172-1 24, 6897234900 26. 740630987644-3^203 28. 48435S30-477265 30. 90000900009-1 32. 1654772 34. 139066-29316950 36. 8485852-43614 38. 4144081-7839494 364 Answ^ru. MISCELLANEOUS EXAMPLES, page 30. I. 10004 2. 474788 3. 2808846363 4. 7398981889800 5. 9576108 6. looiooioo 7. 87846125 8. 99912350214 9. 1000622528890200 10. 2768884-85187 II. 103080207 12. 1202609 13. 71625861494 14. 128721301414200 15. 607862510254-15696883 16. The 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. (21.) 300490090, sum ; 275798734, difi"erence; 3557 338 12805 1336, product, 5555656, sum ; 3086522, difference; 5334673883465, product. (22.) 372 tons. (23.) 127 years. (24.) 7852 times. (25.) 34 ships. (26.) 141. (27.) 1002. (28.) 65280. (29.) 129115. (30.) i46after subtracting it 390 times. (31.) 203. (32.) 1666350, sum; 1639900 difference; 21862578125, product; 1 25, quotient. (33.) 984375o,sum; 9687500, difference; 762939453125 product; 125, quotient. NOTATION OF DECIMALS, page 34. •3, -03, -003, and 3-3; also -7, 11-7, -33, and 1-015. •01, '0021, •0117, "0000003, '^t '53) 'oo?! *ooii, and •00137. 30"i, 4oo'oi, 53*00415, 50"oooioi, '441, 33'i, and "000000000501. •9178, 9i"78, 09178, "0091, "00009, 520-3, and "90. 3"oi42, 6-72819, "000672819, and 6728-19. 7"o6, 43-2143, 9"07823457, roooooi, and 35-721341. S3'9> 47'73. 6-0069, 3-7, 9-000400537, and 902-030401. •073, -0197, "oooooi, "00261, and -oooiooi. i"54, 24-079, 315-008005, -ooooooii, and -00903. "I, "03, -005, -105, -000002, -000060, 41-08, looo'ooi, 30-000006, *ooooi, and "00002375. 7 . 7 7 7 32 7 ^27 -,-. 7 * 5 aa 7 Arf,_2J^ 893 a-nA 883 TJ5> i0 0> 1000> lOOOOOOl lOOOl JXOO) J-'lOl 100000> nO"lOO> 1000' ""^ lOOOOOOO' 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 hundred and one thousand and seven mLUionths ; Twenty-three, and seventy- five hundredths ; Two, and three hundred and seventy-five thousandths ; Two thousand three hundred and seventy-five ten thousandths ; Two thousand three hundred and seventy-five hundred millionths. *3" ^TSTtTaT' X 5625O000OO> TOTJOO' "/ 100> 10000000' ""^ 152587 89062 50OOO" 14. 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. 15. Nine, and four hundred and fifty-seven 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 oae thousandth ; Nine, tuid tW9aty-eight roillionthi ; One, and six thouMiud aad tlur«g t*u milMwuth*. Antwers. 365 16. One, and one millionth ; One million and one ten millionth ; One hundred millionths ; One, and thirteen thousand and four hundred thousandths ; Nine, and two hundred and three thousand one hundred and sixty-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 35. 7450261 ; 2"9i9563. 886-9326; 1681-679. i43V4ii79; 330"87552i. 4009-0; 501-15998. 538-6422021. 140-1996; 1408-25559. 53-6769; 127-050340. I'liii ; 42-7162. i'2345; 9455993. 2-9291474. 2-471092. 0-1627165. SUBTEACTION OF DECIMALS, page 37. I. 3-431; 8-20001; -oon ; 8-0O0O0I. 2- 39'H79'94; 3i'99968; 7'3366o6; 9I-7423- 3. -01; 9899999901; 9-999999; 995-710; 54i'787- 4. 64-0317753; 8-20001; 72-5193401. 5. -000099; '°°°39^ i 3i'99968; 24680-12377. 6. 699-930; -0000999. 7. '0378; -062156; -00510; 28999-908. MULTIPLICATION OF DECIMALS, page 38. 1. lo-o; lo-o; 1190-0; ii'9; -0119; •00119. 2. -000000202; 3-06034; -000000112; -00210175. 3. -0360963; 26-5344; -000604476; 2-02100. 4. -075460; 1-8019; 74-9265; •00104886933696. 5. -0108575; '032016; -000000072; •26439622160671. 6. -0306002448; 470116914-4360; 536-66oo75952, 7. -00164389993; 160-86701632806; -06288405909156. 8. 2-5067823; -000011826009 ; -00000006676542672. 9- 47'83; S°°'°\ 750oo'o. 10. "coi 301400; 1-5; -00000072. 11. 5-314410; 4-096; -032016 12. '0001234321; '000444080; 6r38'36. DIVISION OF DECIMALS, page 40. i. 19-82421; 14-16015; 11-01345; 2-7533625; m6nsS1- 2. 1-66704; 1-H136; 0-83352; -2667264; "01058438. 3. -2017386; -1008693; -0672462; -025217325; -0009456496. 4- 3"'48736o333; 66-6188183; 2-2258942. 5. 134-88057; 790-9882353; 59-406396; 24-82661. 6. 14-789983; 255-121; 1210-234426; "02. 7. •8810891J 90883768; •uo87j;4; •0174^33332. 8. Tivla; %iU-%%\ 3i4jo8. 366 Answers. •03; 7484; 432o6'7 ; '000007375; 83671000; "000000000003; '061096. 268i"o8io8i ; -0000360074; -oooi ; 6'578947 ; •C0862. 150000; 5060000-0; "ooS ; 0375313. ■013; 4'57; 'ooS; 73939 39- ■050005; 1250-0; -0125; 602589. ■0000000125; 125-0; 125-0; -00004 -00000125; 'ooooi ; 20200; 77485-93. 1 1900000; •163. ■10; lo'o; ■ooi ; "ooi ; "looo; loooo; "ooooi. •0093536; 7393'939; 39723'66; 24r6292; 200-0; -60000000; 4000000; "000006; 32000000. 0036; '93; 52306; -0008; -0000020076364. ■0882352941; -017256637168; -0000999000999000999000999. •0000000900090009000900090; -000000123456790; -00000618. REDUCTION OF DECIMALS, page 43. i- "4375: "73; •3142857; "34375; "1875; -676923; -0112; -275. 2. -5384615; -6470588; -6315789; -185; -7167235; -3183098; ^4683544; -0104895. paffe 45. ■2833; -4833; ■?; "4166; -8; -9666. ■788260449735. 3- 29'530588i94- 4. 12-175 hours; '0013. 5. I N.M. = 1-15202 I.M. ; I I.M. = -86804 N.M., 1-1515, ■868421, 6. '997269560 day. i2 44'^ 2'-86 CHAEACTERISTICS OF LOGARITHMS, page 49. 6. 4 II. 5 16. 2 7. 2 12. 3 17. 7 18. 4 19. o 20. I II. 5 12. 3 13- 14. 15- I CHARACTERISTICS OF LOGARITHMS, page 51. 1. 2 0r8 6. 7 or 3 11. r or 9 16. 2 or 8 2. 7 or 9 7. 2 or 8 12. 3 or 7 17. 7 or 3 3. 2for6 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. I or 9 5. 3 or 7 to. I or 9 15. 4 or 6 20, IT or 9 3^7 LOaAEITHMS OF NATUEAL NUMBEES, paffes 53— ?5- I. 0-698970 2. 0-954243 3- 3-954243 or 7-954243 4. 2-OO000O or 8-000000 5- 4-000000 6. 1-146128 7. 1-612784 8. 3-602060 or 6*oooooo or 7-602060 9- 0-38021 1 10. 1-380211 II. 7-3802x1 or 9 3802x1 12. 3322219 or 7-322219 13- 1-973128 14. 7-698970 Dr 9-69S970 15. 7-875061 or 9-875061 16. 0-397940 17- 1-397940 18. 2-954243 19. ■3-9sqo4i 20. 1-397940 or 9-397940 or 8-954243 or 7-959°4i 21. 2-380211 22. 5-544068 23- 5-755875 24. 4-698970 r 8-380211 2-000000 2. c 2-16 )r 5-544068 or 5-755875 7507 or 6-698970 I. 1368 3- 0-468347 4. 2-55 5. 2-82*8015 6. 2-899820 7. 2-99 2111 8. 0-68x241 9. 0-95 S308 xo. 0-167317 II. T167317 12. 1-954725 13. 2-1673x7 14. 2-62' 7366 15. 5-651278 or 9" 1 67317 or 9"954725 or 8-1673x7 015-651278 16. 5-651278 I. 3-000000 4 2. 3-091315 3. 1-409087 • 3*734960 5. 1-415974 6. 0-415974 7. 2-005180 or 8-005180 8 ■ 1-977129 9. 0-890812 10. 2-994581 II. 1-835247 12 • 3-444669 I. 4"585i78 7- or 8-835247 or 7-444669 2639088 13. 5-30x030 19. 5-562474 2. 2-585x78 8. 1-895445 14. 2-749845 or 8-7 9845 20. 2-998755 3. 4091491 9- 0-343507 15- 3993714 or 7-993714 21. 1-507732 4. 2-734968 10 i- 894 1 05 16. 5-808742 22. 0-014001 or 9-894105 5. 4823904 II. 4-000000 17. 3-052717 23. 3-000003 6. 3965898 NATURAL 12. c NU 4-903120 )r 6-903120 x8. I 999172 or 9-999172 mges 24- 2-775555 MBERS OF LOGARITHMS, ^ 58-59. I- 3 II. 1234 21. 978-5 3X -0000009797 2. 8 12. 7916 22. 34800 32 80080000 3- II n- 345 -6 23- 52790 33 ■04x83 4. 2 14. 24-83 24- -5547 34 •000000007968 5- 9'4 15- 7000 25- -3171 35 -oc 546 6. 14-5 16. I 0000000 26. -00000075 36 -00071 7. 6-49 17- 669000 27. 4000000 37 -000006 8. 586 18. 400000 28. ..-00000007 38 81 99000 9. 2-48 19. 50000 29. 4029 39 I'OIO 10. 30-09 I. 853-52167 20. 8. I 00000 30. 2784 40 73 22. 8800 543210 15- 678945-3 •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- 53-133 12. 123456 19. 1-7 26. ■99727 6. 938689 13- 342-945 20. 1651374 27. •7854 7. 456780 14. 5555-54 29. 2X. -0096532 •000036808 28. •000856735 368 AmufBTi. LOGAEITHMS OF NATUEAL NUMBEES, page 60. I. 0*903090 11 1-802774 21. 2-926548 31 1-972043 41 7^000000 2. F-oooooo 12 • 3'8o55oi 22. 1-964240 32 4-722552 42 2-792392 3- o'690i96 13 7-165244 23- 2-953760 33 4-698970 43 4-477134 4- i"579784 14 . 0-588160 24. 4-000000 34 5^845'54 44 4-000039 5- 2-579784 15 . 3-829561 25- 4-681 241 35 5-421604 45 5-774152 6. 2"O0O0OO 16 • 2-942504 26. 3^958124 36 5-606388 46 7-947385 7- 6'oooooo 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 . 1096910 29. 4-651278 39 1-970876 49 • 4-932847 lO 2*954243 20 NATUE • 3*954243 30- AL NUMBEES 7-651278 40. 2-515397 MEITHMS, page 50 60 . 7-816109 OF L0( I. 204 10. 3-673 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. 2-6o6 31 •147 40 •042404 S I 14. 264000 23- T 32 •00000075 41 •0048553 6 100 15- 1000 24. •009 33 1-00043 42 2-5152 7 366855 16. 2480000 25- •052 34 8859000 43 10059 1 8 3659 17- 26-042 26. 451070 35 •0918504 44 •000209675 9 418-557 18. i5'438 27. 271828 36 5-80693 45 7-5 MULTIPLICATION BY LOOAEITHMS, page 63. I- 3*774517 = 5950; 3-022429=1053; 3-000000=1000; 3-521269 = 3321. 2. 2^971331 = 936-12; 4-034147 = 10818; 3^494850= 3125; 4-443232 = 27748. 3. roo9026 = IO-2I ; 0-436878 = 2-734^; 1-818753 = -6588 ; 1-575742 = 37-648. _4, 5-425758 = 266537; 4-532375 = 340702; 2-639870 = 436-385; 1-292881 = 19-62826. 6783260 = 6071000. 5. 4-586678=38608; 4-677607 =; 47600; 2-680225 = 478878; 5-237543=1172800; 5 786113 = 6iiior. 6- 3-97>387 = 9362-39; 2-993736 = 985-68; 4-659678 = 45675; 5-749272 = 561400; 3-723909 = 52966. 7. 7-I462I2 = 14002718; 6-445142 = 2787032; 6-9I9IIO = 8300615; 7-498480 = 31512319- 8. 3'ioo249 = 1259-64; 2*511391 = 324632; 5-0C0000 = looooo; 8-696466 = -0497125. 9. 7-499467 = 31583985-5; 3-782115 = 6055; 3-842614 = 6960-08; 4*327379 = 21250-98. 10. 5*5993" = 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 = 33io-34. 13. 8-763323 = -000000057986; 3-778168 ^ -006000236; 7-740796 = -00000055055; 1-233799 = -171317- 14. 5-755418 ::.= -00005694; 0-622110 =: 4-189; 3-473368 = -00297418; 4*147399 =: i404ro3. 15. 2-951043 = 893394; 3-524617 = 3346-7; 3-398070 = -00250075; 4-783612 = •000607593. 16. 2*000000 = -oi ; T-oooooo =: -ooooooooi ; 5-050035 = *ooooii22ii ; 5*000000 = I 00000, Answers. 369 DIVISION BY LOGAEITHMS, pages (^G-b-j. 1. i"9i9078 =z 83 ; 2778875 = 601 ; 1-924279 =: 84; 2'0969io = 125. 2. 2-986680 = 969-8; 3-698970 = 5000; 5-880170 = 758875-4; 2-775257=596-015. 3. r 809855 = 64544; 3644712=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; 2-78590S = -0610S; 2-385683 = -0243043; 2-571411 = •037274; 5-301030 = -00002 ; 4-301030 z= -000002 ; i-30io:;o =z 20. 6. 3-886534 = 7700-76+; 1-999234 = -998236; 1-999489 = 99-8825; 3-763429 = 5800. 7. 1-569001 = 37-0684; i-425ii5 = -266i43; 4-376859 = 23815-5; 1-378403 = 23-9003. 8. 1-995636 = 99; 1-319142 = 20-8517; 1-854294=71-498; 0-793259 = 6-2124. 9- 4 699490 = 5006 ; 4'5i4747 = 32715 ; 4'3'74i5 = 20769 ; 2-096910 = 125. 10. 0942505 r= 8-76 : 2-525020=1:334-98; 5-291146=195500; 1-004364=10-1010. 11. 1-832752 = -68038; 4-903504 = -000S00763; i'030734 = -107333; 2-509203 = 323; J'778i5i =600000. 12. 2-735031=543-288; 1-602689 = 40-058; 2-264818=184; 3-946651 =: 8844-04. 13. 1-940506 = 87-1978; i-297735 = -0198488; T-278331 = -189815; 0-833525 = 6-81592. 14. 1*421422 = "26^: 265388; 7-301030 15. 1-057101 = -1 1405; 2'537395 = 344-663; J-003528 = -0000100816; 0-833525 = 6-816. 16. 2-339894=218-723; 2-934196=859-4; 1004751 = -101 1 ; 7-505150=32000000; 5-028878 ^ -0000106875. 17. o- = 1 ; 13- ^= 1 0000000000000 : r- = -I ; z' = -oi ; 4- rr 10000. ..J388; 1-421422 = 26-5388; 1-421422 = -265388; 20000000; 7-477122 = 30000000. '4:1422 = NATUEAL SINES AND COSINES, page 70. Natural Siuei. 570774 867085 947463 370382 723895 974228 7. 800000 8. 997630 1. 969227 2. 32895S 63' 53' 47" 21 44 21 Natural Cosines, page 70. 3. 167237 5. 688000 99561; 6. 868805 Arcs of Natural Siues, page 7 1 . 53' 7' 49 5- 26^21' 34" 7. 47^48' 33" 66 59 10 6. II 45 52 8. 31 57 10 7- 782397 8. 989472 9. 48' 46' 34' ^o. 73 44 23 1. 63° 19-58' 2. 18 29 12 3- 43 21 52 Arcs of Natural Cosines, ^r/.i/e 71, 4- 45''24'39" 7- 5- 59 o 47 8. 6- 23 54 9 9- LOG. SINES, TANGENTS, SECANTS, ETC., page 74. 1. 9-202234 2. 10-185981 3. 9-989071 B15B 4. 9-883934 5. 10-829843 6- 10-135990 77 33 15 2 33 48 53 7 48 JO. y-275658 11. 10-144904 12. 9-907590 37© Answers, LOG. SINES, TANGENTS SECANTS, ETC., page 78. NO. SINE. TANOF.NT. SKCANr. COSINE. COTANGENT. COSECANT. I 9-079607 9-082763 10-003156 9-996844 10-917237 10-920393 2 9-6.1999 9-65.805 10-039806 9-960194 10-348195 10-388001 3 9'787595 9-890004 10-102409 9'89759i 10-109996 10-212405 4 9-923122 10-185903 10-262781 9"7372i9 9-814097 10-076878 5 9-246845 9 253720 10-006876 9-993124 10-746280 10-753155 6 9'975'3o 10-457990 10-48^859 9'5'7'4i 9-542010 10-024870 7 8-504189 8-504410 000022 1 9-999779 "•495590 1-495811 8 9-999580 I 1-356298 1-356719 8-643281 8-643702 0-000421 9 8-246654 8-246721 o-oooo68 9-999932 "•753279 ^•753346 lO 9-955206 10-319983 0-364777 9-635223 9-680017 0-044794 II 9-938922 0-244180 0-305259 9-694741 9*755820 061078 12 9-990926 10-684913 0693987 9-306013 9'3i5o87 0-009074 n 8 668140 17. 8-36168I 21. 10 348195 25. 8-024643 29. IO7I4I59 14 9-217118 18. 9-505271 22. 10 100598 26. 8-658227 30- 9-972464 15 8-504188 19. 8-297036 23. 10 203779 27. 8-305785 16 6297326 20. 11-263695 AECS 24. 8 546002 28. 8-258261 1$ 80. OF LOG. SINES, pa^ I. 33° 26' 48" 4. 2° 17' 7" 7. 18^26' 6" 10. 0° 9' 50" n- 52° 35' 30" 2, 57 30 53 5- 19 15 35 8. 39 7 15 "• 87 38 20 14. 4 I 28 3- 2 SS 26 6. 58 15 30 AECS 0: 9- 54 13 20 12. 70 34 18 age 80. 15- I 39 39 F LOG. COSINES, p I. 52° 13' 35" 4. 8^ 6' 31" 7. 70^47' -5" 10. 31' 9' 33" 13- 89= 38' 20" 2. 55 45 8 5. 81 18 8. 80 37 20 II. 3 56 40 14. 84 15 39 3- 89 13 8 6. 79 II 16 AECS 0] 9. 88 40 54 12. 88 54 16 age 80. 15- 84 4 38 ^ LOG. SECANTS, p I. 14° 23' 15" 3. 3»24 0" 5- 79 '39' 51" 7- i8°22' 17" 9- 61° 4' 15" 2. 51 28 50 4. 26 33 6. 84 19 47 8. 88 41 42 10. 86 22 57 AECS OF LOG. COSECANTS, page 80. 26= 43' 0" 4- 6° 5' 13" 7- 5 =43' 39" 10. 58^ 15' 30" 13- 78= 22' 32 34 I 14 5- 49 II 9 8. I 57 4 II. 7 13 56 14. 60 13 52 5 40 16 6. 2 4 7 9- 3 54 45 12. 4 46 56 15- 2 56 20 AECS OF LOG. TANGENT, page 80. I. 2. 3- 77° o'23" 45 24 62 42 21 61° 2' 39" 7 34 16 27 16 43 4. 8i"'3i'58" 7. 5- 54 43 26 8. 6. 5 13 23 9. AECS OF LOG. 4. 41° i'35" 7. 5. II 41 8. 6. 3 37 50 9- 86^58' 16' 10. 187 II. 27 28 54 12. 23° 43' '7" 35 3 32 87 46 , page 80. 8 2° 49' 23" 8 30 34 88 55 35 13- 14. 15- 14. 15- 48= 58' 24' 2 40 10 I 2 18 I. 2. 3- COTANGENTS 88° 46' 54" 10. 44 20 2 II. 86 32 24 12. 88=20' 53' 76 40 15 29 16 Answers. 371 MISCELLANEOUS EXAMPLES. For practice in natural and logarithmic Sines, Tangents, and SecsLuta, paae 80. (i.) Nat. sine -432651, its common log. is 9 636138, which is the! log. sine required. (2.) Nat. tang. 3, its common log. is 10-477121, -which is the log. tang, required. (3.) Log. 9-236713, its corresponding nat. no. is -172470, the nat. cos. required. (4.) The given log. tang. 9-850593, being subtracted from 20, gives 10-149407, 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 ro-030561 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' 3^"' 2- 'r^^ ^^^- ^°- 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. DIFFERENCE OF LATITUDE, pa^e 92. 7. 610' S. 9. 94' N. 8. 459 N. I. 203' N. 3- 293' S. 5- 795' 'N. 2. 470 S. 4- 330 N. 6. 157 S. MERIDIONAL DIFFERENCE OF LATITUDE, pa^e 92. I. 97 2. 2426 3. 345 4. 1216 5. 932 6. 260 LATITUDE IN, page 93. r. 34' 2' N. 3. 0° 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 94. I. 17" 19' 2. 2" 10^' 3. 35= 37' 4. 61" 34' 5. 53" 12^' 6. 64° 31' DIFFERENCE OF LONGITUDE, page 95. 1. 300' E. 3. 716' W. 5. 270' E. 7. 368' E. 9. i8o'W. ir. 2835' E. 2. 507 E. 4. 260 W. 6. 422 W. 8. 420 W. 10. 412 E. 12. 1200 W. LONGITUDE IN, page 96. I. 7''38'W. 2. i°i8'E. 3. 31° 4' E. 4. o" 30' 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. LEEWAY— CORRECTED COURSES, page 105. X 1. S.W. I S. -^2. S.S.W. i W. i 3. N. A E. > 4. N.E. i E. .J. E. byS. f 6. N.W. byW. ^ ^. W. i; S. >. 8. N.E. byE.JE. 37^ Anstvers. VAEIA.TION— TEUE COUESES, par/e I 10. I. N.E. 2. S.S.E. 3. W.S.W. 4- N.W. 5- N.N.E. 6. S.E. 7- S.S.W. 6. N.VV\ bv AV. 9. S.li ;. by E. ^ E. 10. N. |E. 11. S. 'Ie/ ,11. N. ^W. 13- S.W. bv W. i W. 14. E. by S. i 8. 15. E.N.E. 1 6. W. by k 1 S. 137- DEVIATION— TEUE COUESES, paffe I N. 38°59' E. 8. S. 48=55' E. 15. S. 28°57' W. 22. S. 7!^^ 10' W. ., j;> 2 3 N. N. 022 W. ^9. N. i9 25^E. ^16. S. 4 38iE. 2058IW. 10. S. 16 W. J, in. S. 73 io|W. i 23- '24. s. 56 2|w. ^i/; ^ S. 4 34|W.;/.$^ 4 S. 4339IAV. ri. S. 84 7^W. .,18. N.32 24iE. S. 63 ii|W. 5 S. 84 7^W. -412. S. 36 i2iE. 19. X. 3 E.^, S. 74 3|K-'^ 6 s. 24 5^W. 13. S. 87 E. 20. ¥.89 55 W. 27. S. 79 16 E. > 7 s. 89 35 E. .^ 14. N. 59 i2|W. 2 1. S. 9 33 E.. 28. N. 77 46 W. MAGNETIC BEAEING-S OF OBJECTS, page 140. 41. N. 78°io'E. 5. J N. 84'54'E. 9. S. 86° 2' W. V 2. S. 72 32 E. 6.->- S. II 53 W.fJJ'i !',io. S. 2 39 E. N. 5 14 E. S. 10 16 E. 7.J: N. 8 37 E. S. 89 13 W. 6 W. I. E. by S. i S. 2. N.W. by N. 3- N. by W. 4- N.N.W. 5- S.E. by E. 6. S.E. 7- N.W. 1 N. 8. South 9- S. by W. i W. 10, S.E. by E. f E. II. W. iN. 12. S. 440 E. EY lATION- -TEUE COUESES i?«^e 143 13- N. 64' E. 2,5- S. 36' E. 14. N. 78 W. 26. S. 9 E. 15- S. 85 E. 27. N.59 E. 16. S. 79 "W. 28. N. 89 E. n- S. 6 E. 29. S. 60 W. 18. s. 73 w. 3°- N. 10 E. 19. S. 37 W. 31- S. 6oiE. 20. S. 57iW. 32- S. 77 E. 21. S. 77 B. 33- S. 76 w. 22. N. 84 W. 34- N.E. i E. 23- S. 81 E. 35- N. f W. 24. S. 32 W. 36. W.iN. NAPIEE'S DIAGEAM. Deviations, jf^r/yc 148. (a) CunvE A.— lo^E.; 19' E.; 24* W. ; iSi»W. ; 25* W.; 2» E. ; 23FE.; 17° W. 24|° E. ; 24» W. ; 15^ E. (5) Curve B.— 3° E. ; si" E. ; 5-^ W. ; ^° W. ; io|^ W. ; o^ 8^ E. ; 7|° W. ; 10° E. 6»W.; 5i°E. (c) Curve C.—i5°W.; n^'W.; o^°W.; 28i° E.; io^°E.; 6=E.; 14F W.; 28^° E. 24r W. ; 2" E. Correct Magnetic Courses, 2>(^ff<^ 150- Curve A.— N. 66° W. ; S. 87° E. ; S. 23° E. ; S. 54' W. ; S. 63^° W. ; N. 68° E. S. 3iJ° E. ; S. 63° W. ; N. 4i|° W. ; S. 16° W. Curve B.— N. 56° W. ; N. 86° E. ; S. 22|» E. ; S. 58^° W. ; S. 67' W. ; N. 61° E, S. 37' E. ; S. 67° W. ; N. 31^= E. ; S. 18° W. " Curve C— N. 37° W. ; N. S3i° E. ; S. 75° E. ; Jf. 6i° W. ; N. 58° W. ; N. 20° E. N. 80° E. ; N. 58^° W. ; N. 27^° W. ; S. 76° W. Answers. 373 Compass Courses, pciffe 151. Curve A.— N. 79° W. ; N. 27' E. ; S. 21^- W. ; N. 9^ E. ; N. 4if E. ; N. 78° E. ; S. 66" W. ; N. 21^" E. ; N. 86= W. Curve B.—N. 83° W. ; N. 3ri°E.; S. 20^ W. ; N. loi^E. ; N. 47° E. ; N. 83^^^ E. ; S. 64° W. ; N. 2f E. ; N. 89° W. Curve C.-S. 21° W. ; N. 78^ E. ; South ; N. 35^^ E. ; S. 73' E. ; S. 31° E. ; S. 8' W. ; N. 651' E. ; S. 18° W. DIFFERENCE OF LATITUDE AND DEPARTUEE , paffe 169. NO. Bill' J AT. DEr. NO. DIFF LAT. BEP. I. 27'-7 S. ii'-5 E. 8. io'-8 S. 33'-3 W. 2. 9-4 s. 47-1 E. 9- 22-9 N. 8-8 W. 3- loo'S s. 9. -3 W. 10. 10-9 S. 23-3 W. 4- '2-3 N. 83-1 W. 1 1. 2-5 S. 14-5 W. 5- 28-8 s. 48-0 E. 12. 27-3 N. 13-9 W. 6. 142-7 s. 173-9 w. 13- 7-4 S. 42-2 w. 7. 44-5 N. 177-5 E. COURSES AND 14. 33-2 N. ES, pa^^ 171. 10-8 W. DISTANC NO. COURSE:^. DIST, NO, COWRSES. nisT, I. S. 19 " E. 77' 6. N. 44° E. 52^ 2. N. 67 E. 186^ 7- S. 55 W. 93 3- 'i!f.66 iW 161 8. S. 6 W. 161J 4. S. 21 E. IO5I 9. S. 2i W. 173 5- "N. 30 W 480 TRAVERSE [o, N. 58 paffe 176. E. 310 SAILING, NO. ]). LAT. DEr, LAT. IN COURSE. DIST. I. 95"2 S. 92-1 w. 5i°23'N. S. 44° W. 132' 2. 20-0 s. 128-8 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. 120-0 N. 149-0 E. 50 N. N,5i E. 192 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. 129-9 ^• 35 39 S. ^•23i W. i39i 10. i50"3 S. 56-8 W. 44 S. S. 21 W. 161 PARALLEL SAILING-, pac/e 178. I. 250-3' W 2. 344-4' E. 3- 519-2' W. 4. 471-3' W 5. 148-0 w. 6. 512-5 E. 7- 612-0 W. 8. 113-8 E. 9. 117-7 w 10. 408-0 E. II. 372-2 E. 12- 594*5 E. 2Mffe 179. I, N, 64° 17' 30" "W., distance 396-7 miles. 4, S. 79 8 45 E., distance 96-5 miles. 6. 61-6 miles, or 1° i''6 2. 893-4 miles. 5. 60° and 70° 32', 7. "West, distance 864-1 miles. 374 MIDDLE LATITUDE SAILING, page 182. D. lat. ii3''3 „ 99'9 89-8 ,, 1656 967 „ 114-6 Dep. 273'-5 „ 187-0 „ 189-8 ,, 223"3 „ 3ii^7 „ 122-9 Lat. in 27° 28' N. „ 34 10 N. „ 41 o S. „ 49 10 S. „ 18 52 N. „ o 59 S. D. long. 305' » 224 „ 248 334 » 338 ,1 123 Long, in 54° 55' W, „ 29 8 W. „ 70 12 E. » 175 58 w. „ 175 12 E. „ 27 47 W. MERCATOE'S SAILING-, page 186. ^ 3 -*- 4 * 5 *6 ^7 ^8 i- 9 ^10 -< II 4 13 >(- 14 ^ 15 \ 16 + 17 ^ 18 -V 19 .^ 20 D. LAT. 97 N. 85 S. 280 N. 81 N. 230 S. 687 s. 1232 N. iijN. 1 107 N. 779 S. loii N, 792 S. 128 N. 31S s. 731 s. 150 N. 4483 N. i860 S. 3355 N. 180 N. M. D. LAT. 125 130, 497 128 500 785 1760 1230 1080 "39 794 233 524 733 274 4842 1884 3516 190 1>. LONG. 131 E. 76 E. 368 E. 227 W. 270 E. 3698 W. 4732 E. 191 E. 1452 w. 1200 W. 3808 "W. 1254 E. 725 W. 365 E. 2459 E. 354 E. 3313 E. 412 E. 7587 W. 1 140 W. COIRSF.. DIST. N.46»2i' E. 141 S. 30 19 E. 98 N.36 31 E. 348 N. 60 35 W. 165 S. 28 22 E. 261 S. 78 I W. 3309 N.69 36 E. 3534 N. 49 E. 175 N.49 44 W. 1713 S. 48 I W. 1165 N.73 21 W. 3528 S. 57 39I E. ■ 1480 N. 72 II W. 418 S. 34 52 E. 384 S. 73 24 E. 2559 N.52 16 E. 245 N. 34 23 E. 5432 S. 12 20 E. 1904 N. 65 8 W. 7978 N. 80 32 W. 1094 DAY'S WOEKS COEEECTED FOE LEEWAY, YAEIATION, AND DEVIATION, page 208. •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 Difif. 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, ' ' : 1 DifF, Lat. Departure. Lat. in. , Mid. lat. Diff, long, j Long, in, i N. 63|» E. 227' ioi'-7N. 202'-7E. j 36°S7'N. 36°6' j 2So|'E. 71° 19^ W. N. 89° E. 19' , S.43oE.so'-2 S7l=>E38'-3 'Ns5='E41'-8 N3°"W4i'-6 N i5°E42'-2 iN82°E36'7 j N4i°Es2' i2'-5 36'-2 j 24-0 34'-2 4l'-5 2'-2 i 4o'-8 lo'-g | 3'-i 36'-3 : 39'-2 34'-! 2 S. 77i°E. ' 99' 2i'-sS- 96'-6E. 53°45j'N. 53° 56' 1 164' E. 2<'39'E. S.68°E.l7' N.73°E. 28 1 S. ys^E-ai' N.72°E.2o' 6'-4 15' 8 ' 8'-2 26'-8 8'-o 29'-9 6'-2 Ig'-o N.3^W.6'| 6'-o o'-3 1 { S. 9°^. 19' i8'-8 3-0 S.5°W.t3' i3'-o I'-i S. 200E. 10' 9'-4 3''4 N.24°E.15' l3'-7 6'-i 3 1 S 71° "W. 167' 53'-7 S- 158'-4'W. I 36° 9' N. 36° 36' 197^' "W. (i2'l7i'W. S.4o''"W.l4' S.26^'W.42' S.48'W.3o'lN.62''W.49' N.i6^W.4i'| S. 3° E. 25' S-sy'W.ji' I lo'-7 9'-o 1 37'-7 l8'-4 j 2o'-i 22'-3 i 23'-o 43'-3 | 39'-4 ll'-J I 25'-© l'-3 ( l6'-9 26'-o S. 79^ "W. 30' I 5'7 29''4 Answers. 375 Courses, N. 71° W. N. so^W. 20' 12'-9 l5'-3 Distance, I DifF. Lat. 55r 4'-i 2S'-7 i l7'-7 N. 3o''4 14' 8 Departure. 52's W. S34°E2i'-4 17'7 i2'o Lat. in. 45° 54' S. N70°W24'j 8'-3 22'8 Mid. lat. 46"' 3' N64<'E26'-3 ii'S 23' 6 Diff. long. I Long, in, ISh' W. 3° 25h' W. Ss6°'W"23'-2 ij'o i9'-2 S62°W22'-5 lo''6 ig'-g S. 45° W. S 87° "W. 9' o'-5 9'o S29°W37'7 33'-o i8'-3 S7'-5 S. N. 23°W. 45' 4l'-4 ]7''6 57--5 W. N77°W33'-s 7''S 32''6 34'^ 12^' N. South. 45'"3 70' "W. S29°W37'-5 32 -8 i8'-2 N7i°E24'-8 8'-i 23'-s 6° 46' W. S. 79°E. 15' 2'-9 14'7 N. 30 W. 28' s 28'-s N. i'-6 W. 30= 27'- 5 N. 30° 13' z'W. 32° 52' E. S.3°E.i5' l5'*o o'-8 N.7S°W.12' S.63°E.i3' 3'-i il'-6 5'-9 ii'-6 S. io°E. 25' 24'6 4'-3 N.8<'E.2o' i9'-8 2' 8 S.76°-W. 15' 3'd i4'-6 N. 11°W.27' 26'-5 5''2 N. 20" E. 30' 28'-2 io'-3 •u -f- S. 56° w. 119' 65'-3 S. 98'- 5 w. 45° 26' S. 44° 53' 139' W. 179° 8' W. N.88°W.i8' o'-Q i8'-o N.2°E.37'-8 S86°W-33'-6 37'-8 l'-3 2'-3 33'-5 S79°W47'-6 g'-i 46'-7 S i6°'W24'-7 23'7 6' 8 Si5°W37'-6 36'-3 9'-7 S. 8° E. 50' 49'-S 7'o N.26°E. 18' i6'-2 7'-9 N. 70° W.^l 162' 56'-oN. 151' 6 E. 63° 14' N. 62° 46' 33i'-4 E. 57° 46' W. N.66°E. 21' N7i°E49'7 8'-s i9'-2 i6'-2 47'-o N75°El7'-3 N.35-E.34' 4-5 i6'7 27'-9 ig'-s S28°"W"i3'-5 ii'-g 6'-3 S74°E2l'l 5'-8 2o'-3 S 20"W 16' -8 l6'-8 o'-6 N.47°E. 49' 33"4 3S'-8 r S. 77^° E. 105' 22'7 S. lo2'-s E. 59° 26' N. 59° 37' 202'-sE. 4o°3l''S'W. S. 12°E. 14' i3'7 2'-9 S37°Ei8'7 Nii°W25'-2 14''9 ll'-3 24-7 4'-8 Si3°E2i'7 2i'i 4'-9 N5i°E33'-3 2l''0 2S''9 S4°"Wi8'-4 i8''4 l'-3 S.71°E24'-9 N79°E4o'-8 8'-i 23'-5 7'-8 4o'-i 10 I N. 64° W. N87°Wi4' o'7 i4''o "5 "West 2o''5 49'-8 N. lo3'-5 "W. N69°Wi7'-2 6'-2 i6'-i N 36°W24' 19''4 i4''i 35° 20' S. N.8°W.22'-3 22'-l 3'-l 35° 45' N73°W3l'-2 g'u 29'-8 127^' "W. S88°W2o'-3 o'7 2o''3 li2°i7j'"W. S. 64°E. 16' 7'o i4'-4 11 I N. 73^ "W. S. 82°W. 17' I 2'-4 i6'-8 136' S88oW33'-6 39'"9 N. i88oW33'-6 N3i°W23 l'-2 33'-6 19''9 12' N3i°"W"23'-2 i3o'-iW. I 5S°19'S- N6i°W44'-6| 2i'-6 39'-o S6i°W47'-8 23'-2 4i'-8 231' W. 710 7' "W. N3^E29'-s 29'-5 i'-5 S45°E37'-8 26' 7 26' '7 N 34^^27' 22'-4 I5'l 12 S. 53r W. ^■81°^. 15' 2"3 i4'-8 N. 82=E.9' l'-3 8'-9 84^ S.75»W.25' 6"5 24'-l 5o'-2 S. S. 2°"W. 23' 23''o o''8 67'-8-W. I 57°39'N. 58° 4' S. 36W. 19' i5''4 "'"z N. 213W.9' 8'-4 3'-2 S. 14°W.19' i8''4 4''6 128' "W. S. 490 W. 16' lo'-5 I2'-l 8° 20' W. N. 27°"W. 13' n'-6 5'-9 13 8.3rE. N.66°E.25' I0''2 22'-8 N.67°W.3' l'-2 2'-8 S.45°E. 10' 7'-i 7'-i S39°'W22''5 i7''5 i4''2 7'-8 S. N62""W. 14' 6''6 i2''4 o'-s E. S. 283"W. 13' ii'-5 6'-i 62° 17' S. N.34°E.i3' ioi-8 , 7''3 62^13' N. 25" E. 8' 7''3 3''4 S. 47' "W". 7' 4'-8 5"l 140° 18' E. S. 9° E. 3' 3''o o''5 14 S. 45°E. S. 39°"W. 15' 11'7 9'-4 110' S42°Ei5'-3 Il''4 lo''2 77'-3 S. S33°E25'-6 2i'-5 i3'-9 78'-3E. I 5i°37'S. 50° 58' N.44°W. 24' 17''3 i6'7 N.8o°E.22' 3'-8 2i'7 S4i°W2i'7 l6''4 I4''2 124^' E. S. 32'E.47' 39'-g 24'-9 22" 14^' E. N.87°E.48' 2'-5 47'-9 J- 15 N. 84°"W. 23' 2''4 22'-9 183' S.6''W.48''i 47''8 s'-o i83'-o S6i='W37'-2 iB'-o 32''S S. 240W. 30' 27'-4 I2'-2 46°2o'S. N75"W43'-5 ii'-3 42'-o 46° 20' 265' "W. N69=W38'-4 i3'-8 35'-9 N2ooW37'-8 35'5 "'-g i78°5i'E. N. 330W. 36' 3o''2 i9''6 16 I N. 5o^°E. 135' 85'-8 N. N. 34°E. 19' I i5''8 io''6 N. si°E. 20' i2'6 15' 5 N 3°E.33' 33'-o i'7 io4'-3 E. I 48° 3' N. 47° 20' S-79°E. 39' 7''4 i^'-i i S.730W.35' io'-2 33'-5 I N. 45°E. 40' 28'-3 28'-3 154' E. N.88°E.32' I'l 32''0 50^ 56' W. N.42°E. 17' i2''6 ii''4 17 N. 86° W. 171' N.i2°W.25' 24''5 5'"2 S. 640 W. 40' i7'-S 36'o n'-8N. S.42°W. 48' 35'7 32'-i i7o''S W. |5i°2o'i8"N| 51° 14' 24" I 272''9'W. S33°W33'-8 28'-4 i8'-4 I N6°"W37'-9 I 37'-7 4'o N2ioW42''5 39'7 i5'-2 S73°'W29''2 8'-5 27'-9 3° 10' -w. West 32' 18 S. 74° E. S. 78°E. 42' 8'-7 4i'-i 269' S.8IOE.53' 8'-3 52 -3 74'- 1 S. 258'-i E. [ 36°3i'N. S. 790E.47' 9''o 46''i S. 9° E. 22' 2i'7. 3''4 N. 17°^. 8' 7''7 2'-3 37° 8' S. 76°E.37' g'o 35'-9 323'-8 E. I 20° 24' E. S59°E34''5 i7'-8 29'-6 S820Es2'-5 I 7'"3 52''° 37^ Answeii'8. 19 Courses. Distance, Diff. lat. Departure. Lat. in. Mid. lat. Diff. long. Long. in. S. 24° K. 92' 83'-8 S. 37'-9 -I- 36° 14' S. .S66'E. 13 5"3 11-9 35'' 32' S 550 E 26 14-9 21-3 46' E. N83^E7 09 6'9 20= 47' E S. 29''E. 15' I3''i 7''3 S.63°E.22' lo'-o 19' 6 S. 40 E. 9' 9''o 0' 6 N. 4o°E. 13' lo'o 8''4 N. p-'E. 11' 9''-3 5'-8 S. 5i°"W.6o' 37'8 4''^'-6 S. 42' E. 19 141 127 N89=>Wio 0'2 lO'O / 20 SnJMV 116' n3'-6S 23' 3 W 27' 33' S. 26' 36' 26' W 44' 41' E S34^Wi7 141 95 851 E 11-2 7'I 87 S6 "W 107 10-6 i-l Sli E67 6-6 1-3 ]S'8s°W87 0-8 87 S 29 W 35 30'6 17-0 S51 Wzyg 176 217 S 59'^ E 28 14-4 24-0 S43W87 6-4 5-9 N75E 122 3-2 n-8 S27^"Wir4 I0'2 5'2 Sr 21 Syo^E 219' 75'-o S. 205'-4 E. 38'57'S 38-- 10' 262 E. 176^ 58 w S 73' 1^23 67 220 S77''E37-6 8-5 366 S68°E52-9 19-8 49'o N 87' E 48 2-5 47 '9 S2j°Wio-s 97 4'i N 26° £23-4 21-0 103 S7°E 12 11-9 1-5 S 34° K 25-1 208 14-0 S57'E 13 7-1 30-9 .S5IE22-2 14-0 17-3 k 22 X 65= AV no JU7 X 99-3 w 55' 43' S 5'^' 3' 178 W 71= 35 W N69='AVis 5-4 14-0 S68MY30 ii-z 27'8 N58=W3o 15'9 25-4 X4-W259 25-8 1-8 S 62- W 14-8 7-0 13-1 Ssi'T^ig 12-0 14'8 N56^Ei6-2 9"i i3"4 N36=',Wi6-6 13-4 0-8 X43°E8-3 6-1 57 S3o=>Wii 9"5 5-5 X 30^^12-4 107 6-2 23 S120W 204 1997 S 43-0 W 19° 25' S 17' 45 45 W 179° 39 E South 15 S33°W38 31*9 207 S23^W26 23'() IO"2 N. 790 "W 34 6-5 33-1 Sii°"\V50 55-0 107 Si5'E45 43-5 ii-e S 60" E 26 13-0 22-5 •S 5° W 24 23*9 2-1 ASTEONOMICAL DATES, page 219. I. Jau. I'l 16^38'" 95 4. Mar. 31 19 54 19 7. Dec. 31 6 18 34 10, Oct. I o 10 12 2. Feb. 27^ gh 12™ 0= 5. June 3 16 18 3 8. July I S 3 24 II. 1871, Dec. 31 20 9 50 3. Aug. 14'! 6'»28n'40' 6. Aug. 31 20 10 52 9. June 30 23 30 10 12. 1872, Dec. 31 12 44 12 CIVIL DATE, page zig. I. Jan. nth, 4'>3i™i5'A.M. Feb. 3rd, Ill>28' "56^ P.M. 2. Oct. ijth, 3 17 13 A.M. Dec. 3rd, 5 16 12 P.M. 3. May 17th, 7 15 II P.M. Mar. 14th, II 15 7 A.M. 4. April ISt, II 10 16 A.M. Mar. 2 ist, 7 24 12 P.M. 5. Sept. ist, 8 10 54 P.M. Sept. ist, 8 10 54 A.M. 6. 1872, Jan. ist, 9 50 41 P.M., DEGREES I^ 1873, Jan. ist E, I) a go 220 10 48 s(> A.M. no TIM I. jhi^mjgs oh50"H3' 9i> 9»48» 61' 24° 43' 5''57"^ 4' 2. 4 30 48 5 5" 5 40 9 22 8-7 4 37 56 3- 3 54"4 3 16 17-33 I 47-2 56 ID 8 41 16 4- 36 56 10 52 II '2 2 29-6 9 12-8 II 21 .?• 7 14 28 41 48*9 9 56 5 38 50 2 18 6. 54 3 24 40-8 TIME INTO 10 27 28 S, page 221 19 2 46-8 DEGREE I. 1 8° 28' 0' 2. 58° i' 0' 3- 10=33' 0" A- i68''5o' 15" 5- 67 16 15 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 n- 58 14. 2 29 15- 13 16. 5 10 15 17- 9 14 18. 75 12 45 19. J79 59 '5 20. 28 377 I Jan. 6d 8'' 8" > 2 Feb. 12 22 4 It) 3 Jan. 31 7 29 28 4 Mar. 15 8 8 6 5 May 15 6 6 GREENWICH DATES, page 224 6. Oct. 3lJ2l'>22'"IO* 7. Dec. I 6 32 45 June 30 16 36 52 Aug. 3 23 50 22 fc'ept. I 6 24 II Dec. 31 14 3 20 8. 9- 10. 16. 1872. 11. Dec. 27''23i' 19™ 30* 12. July 8 at noon. n- Jan. 31 13 45 20 14. May 31 18 24 40 •5- Mar. 2 5 40 SUN'S DECLINATION, page 231. I. 5- 9- 13- 17- 2 2° 39' 14' S. 19 J5 5 N. 8 18 12 N. 20 21 38 S. 23 27 26 N. 2. 16^49' 4s"S. 6. 14 47 22 N. 10. 3 24 25 S. 14. 12 31 S. 18. • 14 22 N. 3. 4' 20' 9'N. 7. 23 4 17 N. II. 23 20 56 S. 15. 18 31 N. 19. 18 51 51 N. 4- 8. 12. 16. 20. 2° 18' I'N. 14 36 54 S. 18 24 43 S. 17 16 6 N. 5 1 1 S. 21. 3 14 24 S. \ 22. 23 27 20 S. 23. 23 I 2 S. 34. 7 7 s. EQUATION OF TIME, page 235. I 5 9 13 17 + 5"'34^-8 — 3 43'5 -6 7'i + 3 52"4 — 0'7 2. 4" i4™io»'3 6. + 1-4 10. — II 58'2 14. 4" ° i°'° 18. — 16 1-4 3. 4- 6'°iis-8 7- + 5 47'8 II. +0 I '4 IJ- + 15 "'3 19. — 8-6 4- 8. 12. 16. 20. _ o-n lS-5 + 15-3 4" iro 4-6 7-9 O'O NO. /l. •4 2. -*3- — 5- - 6. -7- -8. «- 9- _ II. - 13- -N 14- - 15- -16. -17. ^8. +• 19. •+ 20. f 21. •4 22. 23- 24. TRUE ALTITUDES, page 237. 17^52' 42" 2. 48° 17' 14' 3. 30° 2' 9" 4. 76° 14' 16" 58 48 28 6. 24 56 49 7. 65 13 4 8. 85 22 51 27 51 38 10. 67 22 16 II. 13 44 33 12. II 45 28 MERIDIAN ALTITUDES, page 243. ORF.F.N. DATE. Jan. ioi9">24s Jan. 31 21 20 36 Mar. 7 18 o 48 j4.pril 28 II I 32 May I 21 51 48 June 10 19 48 12 July 20 10 26 32V ■A-ug. 19 5 30 o Aug. 25 17 51 48 Sep. 22 12 54 o Oct. 23 6 o 48 RED. DECL. 12. Nov. 14 18 39 16 Dec. 9 20 18 40 Sept. 20 19 59 56 Mar. 19 18 2 o April 7 9 19 o May 16 3 I 44 Sept. 22 17 57 o Nov. 2 16 56 o Sept. 22 II 35 52 Feb. 12 o 32 48 Mar. 19 18 49 o Dec. 31 15 37 52 Sept. 30 19 14 40 ocq 21=59 17 13 4 43 14 29 15 32 23 7 20 29 12 31 10 18 o 7 11 43 18 36 22 58 o 32 o o 7 12 19 16 012 IJ 10 o 6 13 47 o o 23 o 3 20 44"S. 36 S. 32 s. 41 N. 27 N. 44 N. 16 N. 13 N. 26 N. 37 S. 10 S. 26 S. 31 s. 15 N. o 57 N. 34 N. 33 S. 50 S. 21 s. 39 S- 46 N. 35 S. 51 S. TRUE ALT. 68°57' 22" 72 58 6 Ji J8 I 15 82 3J 45 57 I 42 37 28 52 7 9 57 50 46 35 48 58 41 4^ 33 54 51 19 67 56 49 26 4 46 56 37 9 61 58 10 90 13 41 86 50 41 83 52 20 70 42 48 71 34 38 30 4 42 77 7 26 54 38 7 81 37 15 L.\TITUDE. 43° 2' 22" S. o II 42 S. 33 18 27 N. 7 456N. 59 35 26 N. 24 14 48 S. 17 23 35 S. 44 40 27 N. 43 J 2 36 N. 48 9 50 Jf. 23 25 31 N. 3 26 45 N. 40 56 43 N. 32 50 36 S. 28 I 6 59 50 N. 16 N. 16 7 15 N. 5 55 7 N. 34 28 18 19 2S. I N. 46 7 39 N. 12 51 48 S. 12 21 '^8 N. J I 43 36 S. TRUE 68° 57 72 58 51 57 82 35 45 56 42 37 52 6 57 50 35 48 41 42 54 51 67 56 26 4 56 37 61 58 90 13 86 50 83 52 70 42 71 34 30 4 77 7 54 38 81 37 raper's alt. latitude. '19" 3 51 10 55 24 57 45 54 23 13 40 36 5 I 40 34 14 44 32 36 23 6 II 43" 2 25" S.** o II 39 S. "* 33 18 37 N. ■♦• 7 4 51 N. 7 . 59 35 32 N. ' 24 14 52 S. 17 23 47 s. y. 0^'. 44 40 28 N 43 52 40 N./^ J 48 10 o N. 23 25 37 N,~ 3 26 54 N. 40 56 53 N. 32 50 40 S. 28 I 59 N. - 6 59 17 N. '.^ 16 7 8 N. "* 5 jj 13 N. ^ s;^ 34 28 6 S. 18 19 9 N. "f 46 7 45 N. ^ 12 51 51 S. " 12 21 19 N. » II 43 40 S. - 1%, 378 Answers. AMPLITUDES, piige 252. GHEBN. DATE. nF.D. DKCL. TRVE AMP. EHEOR OF COMPASS DEVIATION I Jan. 26'' 19^47™ 8« 18° 35'27"«- E. 23 = 7' S. 33° 8' W. 11° 18' W. 2 Feb. 17 4 7 28 12 2 21 S. \7. 14 45 s. 19 E. II 20 E. 3 JMnrch 29 2 20 20 3 40 8 N. K. 4 4.^ N. 26 34^ W. 2 54^ w. 4 April 4 '9 53 6 15 10 N. W. 6 40 N. 6 40 W, ^ 5 Nov. 6 22 5 8 16 27 19 S. E, 18 39l S. 21 28 E. i 38 E. / 6 May- 25 16 44 21 II 21 N. E. 35 23 N. 43 49 W. 8 29 W. •* 7 June 2 9 56 26 22 19 42 N. W. 3« 37| N. 26 4 W. II 16 E. ^ 8 July 14 2 15 42 21 35 42 N. E. 24 53 N. 22 56 E. II 16 E. -^ 9 AuiT. 27 3 18 44 9 49 6 N. ^V. 10 3M N. 26 2 W. 2 52 W. ^ lO Sej)t, 7 21 37 "0 5 30 50 N. E. 6 3^ N. 6 3h W. 6 3^ W. *^ II Oct. I 5 29 50 3 30 47 S. E. 4 47 S. • 13 13 E. 5 37 W. 1^ 12 Snpt. 22 6 I 12 54 S. E. 2 s. 2 47 W. 12 47 w. ? 13 Nov. 2 21 27 40 15 14 22 s. T/. 17 33 s. 4 57 E. 7 47 E. -/ 14 1)<'C. 3 21 13 48 22 19 39 S. XL 36 13 s. 2 28 W. 18 28 w.y w. -^ '5 Mitrch 19 18 2 24 5 37^ W. 20 37l i6 Sept. 22 5 4 24 21 50 E. V '7 June 8 18 41 24 22 58 39 ^• E. 22 sH N. 17 21 w. 2 54 E. ■/ i8 19 20 21 22 23 24 Feb. April Mny June MMPch April Dec. 25 30 27 18 6 9 13 20 15 »7 I 6 18 1 1 40 28 43 40 32 12 24 20 5 20 50 48 35 '6 8 5 21 23 5 8 23 53 33 ■^• 19 58 N. 3' 32 N- 26 8 N. 18 29 S. 6 18 N. 14 23 S. E. W . W. E. W. F. 19 16 33 64 6 '3 32 i6i 46] 1 41 13I 52i 4 s. N. N. N. s. N. S. 48 21 39 23 22 57 '3l 6 46 23 5^ 19 56 w. w. E. E. vv. E. W. 12 28^ 10 6 I 31 14 23 5 i5i 6 9 38 36 W. «f W. -» E. f w.~f TIDES, ;;6fe 256. I. 8^27- A.M. S^if 2"' P.M. 12. II"58'" A M. 2 3 10 7 4 30 10 33 » 56,, 13- 14. No 2 56 4 5 " 33 2 12 „ No „ 22 8 !! 15- 16. No( n No )) 6 7 8 No II 46 No ,, 02 No ,, 3 >, 2 „ 17- 18. 19. 24 1 37 No )> 9 No » 12 „ 20. No „ 'I 5 ,> II 54 „ 21. No >> I No ,, 01 TIDES.- V 22. 56 3, page J) 259. -rOEEIGN PORTJ I. Constant — o*" 17' "corr., for long. + 9- no A...M , ^ 4 2. Constant -j- 3 18 corr., for long. - 29- II' 4™ )) I I 42 3- Constant — i 47 corr., for long. — 8-^ 10 44 , J I I 10 4- Constant — 3 41 corr., for long. + 14- I 37 )) I 53 5- Constant -)- 4 13 corr., lor long — 17™ 49 >) I 37 6. Constant + 4 28 corr., for long. - 19- 10 13 )> 10 33 7- Constant + 6 58 corr., for long. + 11" 4 41 )) 5 14 8. Constant — 17 corr., for long. — 20™ 8 26 >> 8 SS 9- Constant + 7 43 corr., for long -\- 10™ II 27 ») II 47 10. Constant — i 47 corr., for long. — 12"" 41 )) I 6 No P.M 0^'20'" 3 44 I 43 2 26 I 7 21 I 15 FINDING DAILY RATE, pa.je 261. 22 days 3''8 gaining 5. 31 d »ya 4'"2 losing 18 ,, 3'o Inshiq 6. 22 ,, ri Inamg 17 „ 4*5 yaii.ing 7. 15 ,, 6-4 losing 15 „ ii"2 gaining 8. 14 „ 7*0 gaining Answers. 379 GREENWICH DATE, pn^e 264. DAILY SATE. . 6«8 losing . yi gaininy . y22losinff 2'2 gaining 5-3 gaining J' Jo yum till 6. 4-0 losing I. 4^26"'24-' 6. 2 33 42 4 27 3 46-2 2 381 6 28-9 5 22 GR.EN". DATK. Peb. 16* 7'>47'"47*' April 28 4 2t 37 May 7 6 25 16 June 25 20 56 30 Oct 25 8 34 43 Jan. 19 12 33 28 DAILY HATE. . 4' 8 uaining . 87 losing . 10 losing K 08 losing . 1 g losing . 2 '6 gaining T"S' 7 6 57-6 1 25 o 32-8 4 29-2 2 2-8 V I i ^ 3 ^ 4 5 ^ 6 /' ^" V 12, — 13 ^14 -.IS -16 -18 - 19 -f 20, 21 HOUR- ANGLE, page 268. 2" 50"" 42" 4 3 50"i 3. 4'' 50" 20' 8. 4 29 56 RATE, -9-8 — 4-0 + 56 — 2-5 + 9-4 + 47 + 2-6 + 07 + 47 -67 — 9-3 -69 + 8-3 — 25 — 54 + 3-3 + 1-6 — 1-25 + 0-3 — 80 GRF.EN. DATE, Jan. 1^ i9i>32™i3' 6 Feb. 18 19 45 57 Mar. 27 23 25 52 April 5 19 13 47 May 19 o 29 12 5 June 14 17 56 42 July 5 ° 33 8 Aug. 13 2 20 42 Aug. 31 19 12 18 Oct. 25 8 35 42 Nov. 27 7 13 53 Dec. 23 18 37 335 Jan. 1 14 o 38 Feb. 10 21 33 26 Oct. 26 o 26 10 Feb. 5 23 59 40 April 30 18 53 59 April 20 15 48 54 Aug. 21 8 22 2 Mar. 20 1 32 46 June 13 22 6 12 CHRONOMETZ RED. DECL, 22° 58' 3 3" S. 11 27 38 S. 3 15 .S2 N. 6 37 11 K. 19 54 38 X. 23 20 23 N. 22 44 50 X. 14 28 33 X. 8 8 49 X. 12 27 6 S. 21 19 I S. 23 25 35 S. 22 59 46 S. 14 10 5 s. 12 40 36 S. 15 43 18 S. IS 12 21 X. II 57 34 X. II 48 55 N. o 7 17 X. 23 18 14 N. RS, paffes TRUE ALT. 4-j°19' 18' 21 34 15 30 21 7 16 17 12 30 41 2 39 50 48 48 47 5 27 23 29 44 44 41 40 31 I 34 50 6 10 42 56 39 9 31 12 17 54 25 10 24 21 21 7 28 18 45 31 59 22 34 2 1 29 I 4 30 49 32 4- 9- 278 Eft. + + + + + + 3 29 20 GREEN. DATE. Xov. Si le*" 26'" 59»"7 Aug. 1 o 5 55-4 May 1 13 28 15 Jan. 20 o 4 50 Sept. 27 i5 39 18 April 16 5 33 40 5. 4h 8"^ 45. — 279. TIME. HOUR-ANGLE, 4m 2« 2'>58'"27' 14 8 4 45 51 51 3 43 37 2 22 4 46 25 3 44 3 43 9 on 3 26 31 4 21 3 o 10 4 32 309 o 14 2 37 16 15 56 2 30 49 II 55 4 7 5 00 5 30 16 3 55 3 49 33 14 29-5 3 o 58-5 16 o 3 28 12 14 19 4 21 49 33 4 20 12 1 23 3 48 51 2 45 3 40 30 7 26 4 3 56 ° 1 I 53 44 LONG. 23 '20' 15' 4 25 O 65 41 30 32 30 47 33 7" 39 14 30 52 14 30 79 4 45 111 II o 95 12 15 173 13 15 1 57 30 151 47 30 4 58 45 62 33 30 69 7 o 140 47 15 179 52 30 179 33 21 82 19 o »6s 1 15 K 2 3,2 ^ w. E. E. ^^'- »« J^ "W. w. E. r*-l \Q,Ci ^O y/ 98°43' 30" E. C; 0,' - ■ ■ 4. 41 58 18 E. A ^^-''i- 75 58 4 W... ,\:' 6. TRUE AZIMUTHS, page z^z. 90 3 54" E. r^. -■^v. s. 56' 4' 9 36 W. ci^i- 8. Eust. 7 14 W. S'*'^^'^- N. 84 5 o'W. 3 W. \/<^'r,ru ^ /^ AZIMUTHS, i?flye 288. GREEK. DATE. 23'' 23'' 46™ 7» 28 9 43 4 27 O 34 12 2 19 O 8 26 21 9 32 20 6 55 44 31 1 6 46 23 2 57 58 31 18 33 2 25 7 30 28 16 22 58 2 2 16 18 50 6 2 49 30 25 1 43 33 29 6 53 47 1 2 22 23 26 o 29 58 25 15 40 o 21 7 3 20 10 20 16 o 24 14 3- o 29 20 15 o 31 21 25 30 RED. DECL. 19° 1 7' 40" s. 7 56 26 S. 2 51 33 N. S 28 40 N. 21 23 18 X. 23 27 28 X. 18 7 41 N. It 12 47 N. 8 9 25 N. 20 57 11 S. 23 23 42 S. 22 57 31 N. 22 32 o S. 13 25 16 X. 17 56 49 S. 17 10 11 s. 2 28 O X. 8 58 26 S. 23 27 23 N. 4 23 47 N. 13 16 12 N. 7 23 39 S. 22 59 21 S. TRUE ALT. 38°34'32" 27 7 44 29 4» 3 II 50 42 43 19 41 16 49 1 43 35 48 7 43 28 30 14 51 34 2 21 50 13 o 14 19 37 26 46 54 j8 52 56 13 47 28 40 7 21 32 50 38 6o 48 32 15 4> 28 42 38 43 43 6 34 66 26 29 45 22 20 LOGS. 19732443 19-289433 19705031 19-654122 19-420714 19-808333 19'599598 15757568 10-558129 19-697632 19-712392 19-259818 19733459 19-437017 15-237697 ic 653024 15 367425 15 416939 10-841808 15-943200 i;.- 786896 15-173709 19-890968 TRUE N. 94' S. 52 S. 90 S. 84 S. 61 S.lc6 S. 78 S 98 N.73 X. 89 X. 91 N. 50 X. 94 N.^3 S. 49 N. 84 S. 57 S. 61 S.112 N. 34 S.102 S. 14 S. 56 AZIMUTH. '35'44"E. 22 16 W. 48 18 "W. 22 10 w. 45 56 E. 38 26 "W. II 54 E. 18 20 E. 55 14 W. 49 26 W. 47 56 E. 29 24 E. 44 28 W. 4 6 E. 8 6 W. 14 18 AV. 43 44 E. 28 6 W. 55 10 W. 27 36 E. 58 6 E. I 52 W. 13 24 E. COURECTIOX. l5°5o'44"E. 9 46 16 E. 10 26 42 "W. 5 37 50 "W-. o 56 W. 30 1 34 W. 2 51 54 w. 15 21 40 E. 7 38 31 E. 2 38 11 "W. 3 52 4 "W. 11 30 35 "W. 10 4 28 W. 1 14 6 E. 31 41 54 W. 13 24 18 W. 12 53 44 W. 2 6 E. 75 31 5 W. 10 22 24 W. 1 15 54 E. 16 50 37 E. 11 13 24 "W. DEV1.AT10N. li°i4'44"E. I 43 44 "W. 6 56 42 W. 14 47 50 "W. 15 44 4 E. 7 1 .34 "W. 5 58 6 E. 10 8 20 "W. 11 8 31 E. 12 on W. 5 52 56 E. 5 29 24 E. 23 10 32 E. 15 44 6 E. II II 54 W. 16 15 42 E. 13 6 16 E. 9 12 54 W. 7 4' 5 W. I 57 36 E. 1 14 6 W. 16 50 37 E. 9 46 36 E. K. fr / r 38o Answers. ii NO. W I. 4- 1: - 9- Yio. -f- 15- I 16. ■:< 17- 18. -/- 20. -V 21. 4 22. 3- 4- 5- 6. 7- 8. 9- 10. Jan. Feb. Mar. April May June July Aug. Sept. Oct. Nov. Dec. Jan. April July Mar. April EEDUCTION TO MEEIDIAN, pa^es 298—299. GUEES". J)ATE. time noon. HKD. BECL. TUVK ALT. XAT. NO. ^d jh^^ni^^o' 14'" 8« 22°45' 4l"S. 32' 26' I9" 144! 8 4 23 8. 38 6 34 1480 o I 4 S. 47 57 15 47 1 1 12 3 52 N. 61 39 o 2443 21 46 14 X. 30 33 8 6514 23 26 58 N. 68 48 28 1427 21 17 14 N. 67 52 44 944 8 55 57 N. Mar. Dec. Mar. Sept. Dec. 28 I 4 49 19 16 34 18 20 23 17 6 29 7 38 27 19 O 41 X2 16 O 2 19 29 15 2 44 8 12 15 43 10 18 47 58 2 17 I 29 22 I 38 58 5 8 58 28 28 2 3 II 13 9 46 59 19 21 2,8 52 12 10 46 29 15 14 29 20 15 19 49 r 31 o 40 46 4 19 8 4 22 7 4 42 23 o 48 31 TIME NOON. 14'" 8« 14 45 25 42 18 30 30 27 15 8 10 19 20 24 9 43 28 10 20 5 9 14 17 32 13 37 22 21 16 56 10 19 22 O 4 19 10 46 20 4 22 6 29 55 5 17 I N. 7 10 31 S. 15 10 55 S. 23 26 ^4 S. 22 37 12 S. 14 22 40 N. 21 42 6 N. 3 24 N. 9 4 -41 N. 2 34 8 N. 1 33 5 S. 23 3 35 S. 5 52 22 S. o I 56 S. 23 26 26 S. 32' 26' 19" 38 6 34 47 57 15 61 39 o 30 33 8 68 48 28 67 52 44 57 34 6 85 30 34 36 44 4 72 2 22 65 23 36 58 17 34 56 40 4 13 26 24 70 30 47 So 36 17 44 19 4 50 12 46 14 54 41 50 o 56 43 55 22 23 52 44 5213 3140 504 2668 1619 2558 2580 1 001 3386 140 624 3175 3242 5775 34°42 7 N. f . ; . 43 42 35N. , "l.: «'7 41 39 32 s. /, , ,'y 40 7 6 N. j/{) ,'r " 37 14 36 S. ^y ni I, 44 24 51 N. i^Z,!i^y ;, ■ o 41 24 S. ^/ '^ ■'/'•/ 41 2 52 N.^./ g -. 9 428N. f^>^n 32 32 59 S. '\' ' ^ , 47 58 388- ^- >^'^' 8 47 43 i\ ' 47 37 45 ^■Lin.si'' 30 6 48N.y^. ^^ 1^8 ^^^ ^'^"^ A- N, 19 48 3 Amwtrt. 381 EXAMINATION PAPEE— No. I, pages 304—305. 1. Log. 5-861612 = nat. no. 727130. (The product by numbers = 727130.) 2. Log. 2-698971 = nat. no. 500. (The quotient.) By Raper : log. 2-698970 = 500. 3. True Courses. — S. 14° W., 15' dep. course; S. 28^ W., 45' ; N. 76° "W., 49'; N. 48" W., 38'; N. 85°W., 31'; S. 22° W., 35'-9 ; S. 54°W., 41'; S. 42" W., 8' current course. Biff. lat. 77'-6 S., dep. i83'-4 "W. ; course S. 67° W. ; dist. 199'. Lat. in 35° 45' N. ; diff'. long. 228'; long, in 12° 48' W. 4. Green, date, Jan. i*! 6^ 50"^ 44"; red. decl. 23' i' 13" S. ; true alt. 60° 12*47'; latitude 6°46'o"N. By Raper : True alt. 60' 12' 38". Latitude 6° 46' 9' N. 5. Log. of diff. long. 2-048016 =: diff. long. iii'-7. 6. Diff. lat. 219' S.; mer. diff. lat. 292'; diff. long. 380' W.; log. tang, of course 10- 11 4401 ; course S. 52° 27' 38' AV. ; log. of distance 2*555608 ; distance 359''4. 7. Standard, Brest constant + 4*^ 2™; 6'^ 47" a.m., and 7'» 7''i p.m. ,, Portsmouth „ — 4*140™; 6'^ c™ a.m., and 6^ 21'" p.m. 8. Green, date, January 1^ 5'' 23™ 24"; red. decl. 23° i' 31" S.; log. sine true amp. 9V789r25. True amp. E. 37° 58' 40'- S. Error of compass 21° 6' 10" E. Deviation 2° 45' 5o"W. 9. Interval 31'^; rate 6«-3 gains. Interval 28'* 7** ; ace. rate 2™ ^%^-z ; Green, date Jan. 29'' 6*» 53"» 49'; red. decl. 17'' 56' 49* S. ; true alt. 13° 47' 28' ; hour-angle -^^ 22'" 9'; red. eq. time add 13™ 235-2, mean time at ship 29'' 3^ 35" 32". Longitude 49° 34' 15" W. Raper: True alt. 13° 47' 12": hour-angle 3'' 32™ 9'. Longitude 49° 34' 15'' "W. 10. Green, date, Jan. 15'' 6^ 11™ i2»; red. decl. 21° 8' 48" S. ; true alt. 55° 18' 24"; sum of logs. 19-723998 ; true azimuth N. 93' 24' 4" E. Error of compass 16" 24' 4" E, Deviation %- 34' 4" E. By. Raper : True alt 55* 1 8' 22" ; sine sq. of azimuth 9*723998 ; true azimuth N. 93° 24' 4' E. Error of compass 16° 24' 4" E. ; Deviation 8' 34' 4" E. 11. Time from noon 16™ 47^; Green, date, January 16"^ 14'' 18"" 55*; red. decl. 20°53'37''S.; true alt. 33" 7' i" ; nat. number 2025, nat. cos. mer. zen. dist. 548375 ; mer. zen. dist. 56° 44' 40" N". Latitude 35° 51' 3" N. Method II. — Reduction + 8' 18'. Latitude 35° 51' 7" N. Towson : Aug. I, 3' 5' ; index 30 ; Aug. II, 5' 10'. Latitude 35" 51' 7'' N. 12. Star's decl. 16° 15' 38' N. ; true alt. 52° 30' 36'. Latitude 53' 45' 2" N. Raper: True alt. 52° 30' 32'. Latitude 53° 45' 6" N. The Curve. — Correct magnetic bearing S. 79° W. Deviations.— 15° W. ; o" ; 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. 4i|° W. ; N. 73' E. ; S. f E. ; S. 67^ W. Bearings, magnetic. — N. 84° E. ; N", 14" W. EXAMINATION PAPER— No. II, pages 307—309. 1. 7*602321 = 40024074. (The product.) 2. 2-168747 = i47'485. (The quotient.) 3. True Courses.— N. 51° E., 18' dep. course ; S. 73° E., 52' ; S. 58° E., 43' ; N. 57' E., 35'-6; N. 38° E., 27'; S. 21° E., 24'; S. 40* E., 29' ; S. 39° E., 12'; current course. Diff. lat. 39'-9 S. ; dep. i8i'-5 E. ; course S. 77^° E., 186'. Lat. in 46" 51' N. ; diff. long. fmid. lat.) 267', or 4"' 27' E. Longitude in 48° 6' W. 4. Green.date, January 3i p.m. ,, Leith ,, — 2^'2i™; ii^ 47™ a.m., and no p.m. S. Grfien. date, February 19'' 18'' 10™ 12*; red. decl. 11° 7' 29" S. ; true amp. W. 11' 22' 30' S. Error of compass 19° 33' 45" E. Deviation 9" 13' 45" E. 9. Interval 21^; rate io*-8 losing; Interval 30'^ 92'> ; ace. rate 5™ 28*; Green, date, Feb. 9 4™ ; no a.m. and o** 4" p.m. „ Portsmouth ,, — 2*' 31™; ii*" 30™ a.m., and ii'' 50™ p.m. „ Harwich „ — 2^51™; iii^ 37™ a.m., and iih 57m p.m. ,, Thurso ,, -^2^ 2"" ; o^ 26™ a.m., and o'" 46"^ p.m. „ Liverpool ,, — o"5i'"; o'> 27™ a.m., and c^ 46'" p.m. 8. Green, date, March 6^ 14'' 47"" 28"; red. decl. 5° lo' 2" S. ; true amp. W. 8° 27' S. Correction 8° 25' 30" E. ; deviation 15° 34 30" W. 9. Interval 41^; rate 5«-8 ^amm^ ; interval 90'! 2 3^; ace. rate 8™ 47^-5 ; Green, date, March 30'' 22'' 53"" 7* ; red. decl. 4° 23' 14" N. ; true alt. 29° 19 56' ; hour-angle 3*' 57™ 26'; red. eq. time add 4"' 7» mean time ship March 30'' 2o'> 6™ 41'. Longitude 41° 36' 30' W. Raper: True alt. 29" 19' 45' ; hour-angle 3'' 57"^ 27^ Longitude 41° 36' 45" W. Ansivers. 383 10. Green, date, March 9'^ 9*" 43™ 5«; red. decl. 4° 4' 54" S. ; true alt. 18° 6' 51"; 8um of logs. 19-602167 ; true azimuth N. 78° 28' 28" E. Correction 7° 5' 58" E. ; deviation 3° 44' 2" W. 11. Time from noon ic" 14'; Green, date, March 24^ 18'' 13™ 34^; red. decl. i°58' 24"N ; true alt. 71° 20' 43"; nat. no. 936; nat. cos. mer. zen. dist. 948399 =: 18^ 29' 11" N. Latitude 20° 27' 35" N. Method II. 1- 10' 23". Latitude 20° 27' 24" N. Towson: Aug. I, + o' 7"; index no. 13. Aug. II, -\- 10' 14". JjUt. 20° 27' 34" N. 12. Arcturus' decl. 19° 49' 24" N. ; true alt. 36° 7' 27". Latitudt 34° 3' 9" S. Eaper : True alt. 36" 7' 22'. Latitude 34° 3' 14" S. The Curve. — Correct masnetic bearing S. 70° E. Deviatioiis.-3° W ; 20== E ; 25° E. ; 23° E. ; 2° E. ; 24° W. ; 25° W. ; 18° W. Compass courses.— N. 3<;° E. ; S, 82" W. ; N. 85 1" W. ; S. 49° E. Magnetic courses.— S. E. by S. ; N. 84^ E. ; N. 86" W.; N.W. by N. Bearings, magnetic. — N. 872'' W. ; N. 78!° E. EXAMINATION PAPEE-No. IV, pages 310-312. 1. 4*ii9863 =:: I3I78-4. (The product.) 2. 3'5i0784 = 3241-78 nearly. (The quotient.) 3. True Courses.— S., 19' dep. course; S. 59° W., 58'; N. n° W., 15'; S. 28' E., 9'; S. 82*^ W., 50'; N. 72° E., 12'; S. 58^ W., 22' ; S. 62 W., 42', current course. Diff. lat. 76'-8 iS. ; dep. l^^'^1 W. ; course S. 62^" \V. ; dist. 162'. Lat. in 51° 29' 8, ; diff. long, 225' W. Longitude in 183' 25' W., or 176° 35' E. 4. Green, date, April 1'^ 5'' 50'" 48^* ; red. decl. 4° 53' 7" N. ; true alt. 48° 55' 26". Lati- tude 45° 57' 41" N. Eaper: True alt. 48° 51;' 18". Latitude 45° 57' 49" N. 5. Log, of diff. long. 2-364667 = Liff. long. 231-6. 6. Diff. lat. 325' N. ; mir. diff. lat. 552'; diff. long 325' \V. ; tang, course 9-769944; course N. 30" 29' 17" W.; distance lll'"2'. no A.M., and o*" 8™ P.M. no A.M., and o" 4'n p.m. no A.M., and o^ 34'" p.m. I ,h 23"! A.M., and no p m. no A.M., and o*> 5™ p.m. ijh 23™ A.M. no p.m. 7. Standard, Brest, -|- 2'> 45"^ „ Hull, -f o" 1" „ Thuiso — i'' 56" „ Pembroke -|- o'^ 4" „ Weston-supf^r-maro + o*" z" „ Waterfbid -)- o^ 44™ 8. Gret'Ti. da(e, April 27'' 23'' 18'" 50'; red. decl, 14° 20' 32" N. ; true amplitude W. 18° 24' 15" N. Error 15' 20' 45" VV. ; deviation 3° 49' 15'' E. 9. Interval 12 days; rate ii'*-3 losing; interval 7i 58" 19'. Longitude 74° 45" E. 10. Green, date, April 17'' 2'' 43™ 25' ; red. decl. 10° 44' 20' N. ; true alt. 42° 20' 39" ; sum of logs. 19-453401 true azimuth IS. 64° 24' 44" \V. Error 25° 35' 16" W. ; deviation S" 45' 16" W. 11. Time from noon 28™ 38' ; Green, date, April 18'' 11'' 38™ 34"; red. decl. 11" 12' 57" N. ; true alt. 54° 20' 16'; nat. no. 5288; nat. cos. mer. zen. dist. 817756 = 35° 8' 17" N. Lati- tude 46" 21' 13- N. Method II. h 3'' 4°'- Latitude 46" 21' 4" N. Towson : Aug. I, -\- 5' 10'; index no. 76. Aug. II, -\- 36' 28'. Latitude 46° 21' 23" N. 12. Spica's decl. 10° 31' 5" S. Latitude 58'^ 38' 15" N. Eaper; True alt. 20" 50' 30". Latitude 58° 38' 25" N. 384 Aniu>0rs. The Curve. — Correct magnetic bearing S. 2° E. Deviations.— 4° W. ; 7° W. ; 12° W.; 18° 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 i" W. ; East ; N.E. Bearings, magnetic. — N. 67° W. ; S. 84^° W. EXAMINATION PAPER.— No. V, pa^es 312—314. 1. 4'838o7i ^ nat. no. 68876'5. 2. 2 826661 = nat. no. 670"905. 3. True Courses.-S. 65° E., 23' dep. course ; S. 6° E., 56' ; S. 14' W., 17' ; N. 80° W., 13'; E., 37'; S. 49° W., 10; N. 26'' E., 17'; S. 40° W., 21; N. S^^'E., 48' current course. Ditf. lat. 82'-8 S., dep. 8i'*i E. ; course S. 44° E., dist. 116'. Zat. in 65" 25' S., dij^. long. 190' E. Long, in 143° 31' E. 4. Green, date, May %^ 7'' i" 8'; red. decl. 17° 20' 20" N. ; true alt. 76° 14' 11". Lati- tude 3° 34' 31" N. 5. Log. of diff. long. 2'992876 :=. Biff. long. ()%y]. 6. Diff. lat. 732' S. ; mer. diff. lat. 88i'; diff. long. 1098' W. ; log. tang. 10095626; coxirse S. 51° 15' 27" W. ; distance ii69'6. 7. Standard, Greenock, constant — o'^ ^(>^ ; ii*^ 35°^ a.m., and ii'' 52™ p.m. ,, Liverpool, ,, — o^ i™ ; ii^ 42=" a.m., and 11'' 59°» p.m. „ Sunderland, ,, — i'' 4™; 2*1 42™ a.m., and 2*1 58™ p.m. ,, Brest, „ -\--£^i%^; 6'' 26"" a.m., and 6'> 42™ p.m. „ Dover, „ — oh 27""; ii*> 9"^ a.m., and ii"^ 27™ p.m. 8. Green, date. May 20'^ i6'> 6'" 24'; red. deol. 20° 15' 3" N. ; true amp. E. 29° 49' N. JError of compass 12° 22' E. deviation 19° 28' W. 9. Interval 37"^ ; x&ie 4^-^ gaining ; interval 50'' 2 1^; ace. rate — 4"^ 4*^-2 ; Green, date, May 21"* 2o'> 59™ 21^; red. decl. 20° 29' 22" N. ; red. eq. time sub. 3'" 33*"5 ; true alt. 32' 19' 31"; hour-angle ^ 17"" 32^-5 ; mean time at ship May 21'^ 19'' 38™ 54'. Longitude 20° 6' 45' W. 10. Green, date. May 25^ 91^ 53" 51' ; red. decl. 21° 8' 27' N. ; true alt. 40° 54' 26" ; logs. i9'634762 ; true azimuth S. 82° 6' 4'' W. or N. 97° 53' 56' W. Error 20' 36' 4" E. ; deviation 10° 6' 4" E. 11. Time from noon 25"" 25^; Green, date io'*7''54™ 25'; red. decl. 17° 52'2i" N. ; true alt. 43° 39' 55" ; nat. no. 5152 ; mer. zen. dist. 45° 55' 31" S. Latitude 28° 3' 10" S. Method II. \- 24' 35". Latitude 28° 3' 15" S. Towson: Aug. I, -\- 6' 16"; index 63. Aug. II, -j- 18' 17". Latitude 28^ 3' ii" S. 12. Star's decl. 10° 31' 5" S. ; zen. dist. 19° 53' 44" S. Latitude 30° 24' 49" S. The Curve. — Correct magnetic bearing N. 87° E. Deviations.— 3° W. ; 15= W. ; 23° W; 22° W. ; 2° E. ; 24' E. ; 23° E.; 13° E. Compass courses.— N. 451° E. ; S. 71^° W. ; S. 49° E. ; S. 39^° W. Magnetic courses.— N. 20^° W ; N. 46° E. ; S. 76!° E. ; S. 61" W. Bearings magnetic— S. 8° E. ; S. 85° W. EXAMINATION PAPER.~No. YI, pages 314—315. 1. 0-622110 = nat. no. 4"i89. (The product). 2. 4-021468 = nat. no. io5o6'77. (The quotient). 3. True Courses. — S. 73° E., 22'-5 dep. course; S.75°E., 17'; S. 33° E., 22'; S. i°W., 24'; S. 61" E., 25'; N. 69^ W., 18'; S. 43° W., 17' ; N. 11° E., 16' current course. Diff. lat. 55'-8 S., dep. 46''i E. ; course S. 39I' E., dist. 72J'. Lat. in 55° 16' N. ; diff. long. (Mercator) 82'-9 E., long, in 134° 17' W. Diff. long, (by mid. lat.) 82'E., long, in 134° 18' W. 4. Green, date, May 31^ 17'' 34™ 52'; red. decl. 32° 6' 44" N. ; true alt. 75^^ 49' 26"; Latitude 7" 56' 9" N. 5. Log. of diff. long. 2-487692 == Diff'. long. 307'-4. Answers. 385 6. Diff. lat. 2i8i'S. ; mer. diflf. lat. 2301'; difiF. long. 3038'"W. ; tangentcouree io'i2o67i; cottrse S. 52" 51' 34" "W. ; distance 36i2''3. 7. Standard, Dover, + 4** 33™ ! 3*" 28"" a.m., 3'' 49™ p.m. ,, Harwich — o'' 33""- ; 1 1^ 43™ a.m., no p.m. „ Brest, — o'> 47™ ; corr. for long. -|- 7"" ; 3'' 9™ A.M., 3*' 28'" p.m. 8. Green, date, June 21 >' 13'' 34'" 48*; red.decl. 23'' 27' 19" N. ; sine oftrueamp. 9'898987 ; true amp. 52° 25' N. Error 46° i' W. ; deviation 6" 29' E. 9. Interval 29^; daily rate 8'' 3 ; interval 44'' 22''; ace. rate 6"M 3' ; Green, date June 13'! 2i*> 45"" 12'; red. decl. 23° 18' 12" N. ; true alt. 28^49' A°" '■> ^^d. eq. time o'" i^ additive ; hour-angle 3'' 49™ 6° ; mean time ship June 13'^ 27'' 49"^ 7^ longitude 90' 58' 45" E. 10. Green, date, June 7'* 22'' o'» 12^; red. decl. 22° 54' 18' N. : true alt. 31° 18' 54"; logs. 19-813501 ; true azimuth S. 107' 33' 48" E. Error 2° 26' 12" E. ; deviation 17° 6' 12" E. 11. Time from noon 37™ 26" ; Green, date, June ^^ 14'' 16'" 2^ ; red. decl. 22'^ 34' 51" N. ; true alt. 50° 3' 22"; nat. no. 5776 : mer. zen. diet. 39° 25' 32" N. Lat. 62° o' 23" N. Method II.4- 31' 18". Latitude 62° o' 25" N. Towson : Bej'ond the limits of the Tahle. 12. Star's decl. 28° 19' 34" N. ; true alt. 48'^ 33' 45". Latitude 13° 6' 41* S. The Curve.— Correct magnetic tearing S. 79' W. Deviations 12° W.; o; 17° E.; 21° E.; 12° E.; 4" E. ; 18= W.; 24" W. Compass courses.— N.W. by N. ; S. 70° W. ; jS^. 27!" E. ; S. i|» E. Magnetic courses.— N.W. by W. ; S. 61° W. ; S. 17° W. ; S. 4o|° E. Correct magnetic bearing. — N. 11° E. ; N. 87^° W. EXAMINATION PAPEE.— No. YII, pages 315—317. 1. 4"i4i829 ■=: I3862-I. 2. 2-537818 = 345-0. 3. TrueCourses.-S. i4nV., 17'; S.68°E.,22'; S.57''E.,7'; N.53°W.,6'; S.54'W., 26'; S. 6i=E., 17'; S. 25'W., 27'; S. 42°E., 15'; S. 26^E., 18'; N. 62° W., 9' current course. Diff. lat. 96'-o S., dep. 9'-9 E. ; course S. 6' E., dist. 96^'. Lat. in 49° 49' N., diff. long. 1 6' E. Long, in 9° 13' W. 4. Green, date, July 26'' o'' 49'" 16'; red. decl. 19° 19' o" N. ; true alt. 15° 46' 12'. Latitude 54^ 54' 48" S. 5. Log. of diff. long. 2-633861 ■=z Diff. long. 430-4 nearl}'. 6. Diff. lat. 745' S. ; mer. diff. lat. 1042'; diff. long. 1292' W. ; log. tang. 10-093395; course S. 51° 6' 50" "W. ; distance ii86'-7. 7. Standard, Brest, constant — o'l 2™; S^ 9™ a.m., 8^ 34n-- p.m. „ Brest, ,, — 0^45™; 7'» 26'" A.M., 7!^ 51™ p.m. „ Brest, „ — o'' 5'"; %'» 6™ a.m., S** 31'" p.m. „ Brest, ,, — oh 29^" ; 7'^ 42™ a.m., 8'' 7"" p.m. „ Brest, ,, +3"^ 3""; ii'' 14"" a.m., ii'' 39™ p.m. 8. Green, date, July 12'' 6^ 36™ 32^ ; red. decl. 21' 52' 5' N. ; true amp. "W. 25° 13' N. Correction 5° 32' E. ; deviation 16° 52' E. 9. Interval %^; rate 5*-55 losing; interval 32'' 22'' ; ace. rate + 3™ 2*-7 ; Green, date July i6'i 2ih 58™ 19'; red. decl. 21° 7' 57" N. ; true alt. 13° 31*23*; red. eq. time 5>" 52' additive; hour-angle 3'' 51"^ 38*; mean time ship 16'^ 27'' 57™ 30*. Longitude 89°47'45'_E. 10. Green, date, July 4"^ i'' 53'" 22^; red. decl. 22° 50' 12" N. ; true alt. 12^ 21' 31'; sum of logs. 19-207424 ; true azimuth N. 47" 20' 50" E. Error iS' 4' 50' E. ; deviation 16" 44*50" E. 11. Time from noon 25"" 20' ; Green, date July 30'' iS"" 52'" 32* ; red. decl. 18° 11' 31" N. ; true alt. 26"^ 24' 19'- ; nat. no. 4094; mcr. zen. dist. 63' 19' 57' S. Latitude 45° 8' 26" S. Method II. -\- 15' 45'. Latitude 45" 8' 32 S. Towson : Aug. I, -|- 6' 17 " ; index 63. Aug. II, + 9' 27". Latitxide 45° 8' 26" S. 12. Star's decl. 26° 9' 34' S. ; true alt. 70^ 5' 46". Latitude 46° 3' 48" S. DDD 386 Answers. The Curve. — Correct magnetic bearing N. 89° AV. Deviations.— 1° E.; 19° E. ; 21° E. ; 9" E. ; 4° W. ; ri-'W. ; 19' W. ; 16° W. Compass courses.— N. 51° E.; S. 81^° E. ; S. 23^° W. ; North. Magnetic courses.— S. 70" W. ; N. 6° E. ; N. Sof" E. ; S. 33^ E. Bearings magnetic. — N. 75° E. ; N. 66| E. EXAMINATION PAPER— No. VIII, pa^es 317—318—319. 1. 5*889986 = 776221-4. 2. 2-676541=474833. 3. True Courses.— N. 59° E., 15' dep. course ; S. 20° E., 26' ; N. 61' "W., 27' ; N. 70° W., 22'; S. 24° E., 25'; S. 29° E., 22'; N. 88° W., 43'; S. 19'' W., 18' current course. Dif. lat. 53'-6 S ; dep. ^o'-^ W. ; course S. 43^° W. ; dist. 73J'. LatitwJe in 0° 44' S. ; diff. long. 50^' W. Longitude in 172° 59^' E. 4. Green, date, Aug. ii<'i7''5im 12'; red. docl. 14° 53' 15" N.; true alt. 42° 50' 18". Latitude 32" 16' 27" S. 5. Log. of difF. long. 2-805956 ^ L)'ff long. 6t,<)'"]. 6. DiflF. lat. 421' N. ; mer. diff. lat. 590' ; diff. long. 249' E. ; course N. 22° 52' 53" E., distance 457' neaily. 7. Standard, Brest — i^i']'^; no a.m. and o'> 5'" p.m. „ Brest — 22; 1 1*" 20" A.M. and 1 1 50 p.m. ,, Leith 4" ° 5 ! ^^ 5^ am. and no p.m. 8. Green, date, Aug. 20'' ii^ lo"" 24^; red. deol. 12° 6' 38" N. ; true amp. "W. 16° 25' N. Correction 27° 40' E. ; deviation 8° 25' E. 9. Interval 7'^; daily rate 6'*6 fosjH^ ; interval 1 6'' 20P; ace. rate + i" 51'; Green, date, Aug. 6^ 26^ 32™ 37'; red. decl. 16° 18' 58" N. ; true alt. 24° 17' i"; red. eq. time 5™ 29^ additive; hour-angle 4'' 25™ 44'; mean time ship 6'^ 28^ 3i"> 13^. Longitude 119° 39' E. 10. Green, date, Au}?. i9<^ 20'' 37™ 41^; red. decl. 12° 18' 45"^.; true alt. 17° 36' 21"; sum of logs. 19-060115; true azimuth N. 39° 37' 6'' W. Error 3° 3' 21" W. ; deviation 36° 53' 21" W. Raper: True alt. 17° 36' 15"; sine sq. logs. 9-060205 = N. 39^ 37' 22" W. Error 3° 3' 37" ^- > (feviation 36° 53' 37" W. 11. Time from noon 35"^ 15*; Green, date, Aug ist ic^ ii^ 55"^ 5'*; red. decl. 15° 14*58' N. ; true alt. 34° 48' 15" ; nat. no. 8845' ; mer. zen. dist. 54° 34' 36 S. Latitude 39° 19' 38" S. Method II. — Reduction -\- 37' 26". Latitude 39° 19' 26" S. Towson: Aug. I, + 10' 26"; index 90; Aug. ]!, -j- 26' 35". Latitude 39° 19' 46" S. 12. Star's decl. 8^ 32' 38" N. ; true alt. 66^ 48 17'. Latitude 14° 39' 5" S. (Norie). „ „ „ 66 48 13 „ 14 39 9 S. (RaperJ. The Curve. — Correct magnetic bearing S. 8° E. Deviations.- 4"^ E. ; 2 E. ; 14° W. ; 17° W. ; .^.° W. ; 9° E. ; 12° E.; 8° E. Compass courses.— N. 85!== E. ; S. 51^ W. ; F. 38^ W. ; S. 50° E. Magnetic cour.ses.— N. 32^^ W. ; S. 85° W. ; N. i'^ W. ; N. 86|° W. Bearings, magnetic. — S. 12° E. ; S. 233° E. EXAMINATION PAPER— Xo. IX, pages 319—320. 1. 7-888411 =: -007734. (The product.) 2. 6 854294 = -00071498. (The quotient.) 3. True Courses.— S. 89° E., 20' dep. course ; !1 66" E , 41 '-2 ; S. 56° E., 49'-8 ; S. 83° E., 42'-6; S. 66° E., 32'-7 ; S. 49° E., 26'-7 ; S. 50" E., 36'- 1 ; N. 85" E., 32' current course. Diff. lat. ioi'-4 S. ; dep. i25'-4 E. ; course S. 68'' 1\ ; dist. 271'. Latitule in 52° 26' N. ; diff. long. 420' E. Longitude in 6° 55' E. - .. 4. Green, date, Sept. 22"* 8'' 15™; red. decl. 0' 3' 5" S. ; true alt. 90=" i' 49'. Latitude 0° i' 16" S. Answers. 387 5. Log;, of diff. long. 2"574252 = dif. long. 375''2. 6. DifF. lat. 3607' S. ; mer. diff. lat. 3798' ; diff. long. 4007' E. ; tang. 10-023264; course S. 46" 32' E. ; distance 5243'. 7. Standard, Brest + 2'' 59"^ ; 3h"i7'n a.m. and 31136™?.!!. ,, Brest 4- 4 '3 ; corr. for long. — 14™; 4 17 a.m. and 4 56 P.M. „ Brest 4- 3 47 ; >. — " ! 3 54 a.m. and 4 33 p.m. 8. Green, date, Sf>ptemher 3c'' 6^ 33™ 40^; red. decl. 3° 8' 32" S. ; true amp. W. 5' 10" S. Err»r 41'' 44' W. Deviation 11° 16 W. 9. Interval 15'^ ; rate 2^6 yatVis ; interval I9''i5|h; ace. rate — o"'5i«; Green, date, August 31^ 15'^ 35"" 8'; red. decl. 8° 12' 7' N. ; red eq. time 0"^ 11' subtractivc ; true alt. 62° 25' 7"; hour-angle i'' 51"' 34^; mean time at ship, August 31'! 25'' 51'" 23'. Longitude 154° 3' 45" E. xo. Green, date, Septemher 16^ i'' 29™ 54'; red. decl. 2° 23' 23" N". ; true a1t. 29° 41' 59"; sum of logs. 19-700958 ; true azimuth 6. 90' 15' 48" E. Error 3° 35' 48" W. Deviation 11° 55' 48' W. 11. Time from noon i6" 41s; Green, dafe, September 22'^ 12'' 27™ 55'; red. decl. 0° 7' 11" S. ; true alt. 62" 9' 19'; nat. no. 234S ; mer. zen. dist, 27° 33' 19" S. Latitude 27" 40' 3°" S. Method II. +17' 32". Latitude 27° 40' 27" S. Towson: Aug. I, -)- i", inlex 35 ; Aug II, + 17' 29". Latitude 27° 40' 23" S. 12. Star's decl. 19" 49' 38' N. ; true alt. 86° 31' 1 8'. Latitude 16° 20 56" N. 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. 59!° E ; S. 6i^° E. Magnetic courses.— ISl. z" E. ; S. 17° W. ; S. 571" E. ; N. 72° E. Bearings, magnetic— S. 84^° E. ; N. 76^° W. EXAMINATION PAPEE— No. X, pages 320—321—322. 1. 7"4472i3 =: 28003548. (The product.) 2. i-936oio = 86-30. (The quotient.) 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., 3'; S. 3SMV., 3'; S. 5i°E., 22'; N. 88^ E., 48' current course. Dif'. lat. 6^''2 S. ; dep. i66'-7 E. ; course S. 69!° E. ; dist. 179'. Lat. in 58'-^ 44' N. ; diff. long. 326^' E. L.ong. in 38° 27^' W. 4. Green, date, Oct. 20'' io'» i" 40'; red. decl. 10° 43' 13' S. ; true alt. 50" 11' 14?. Lat. 50° 31' 38" S. Eaper : True alt. 50° 11' 9 . Latitude 50' 32' 3" S. 5. Log. of diff. long. 2-019277 = Liff. long. I04*5. 6. Diff. lat. 140' N.; mer. diff. lat. 142'; diff. long. 214' E. ; tang, course 10-178126; course N. 56' 26' E. ; distance 253'-2. 7. Standard, Brest — i^ 17™; corr. for long. — 11™; loiiji" a.m. and ii''23™p.m. ,, Devenport — o 46 ; no a.m. and o 38 p.m. „ Sunderland -|- o 49 ; no a.m. and o 31 p.m. 8. Green, date, Oct. 8'' ii'» 13™ 48"; red. decl. 6'^ 17' 52' S. ; true amp. E. 6° 39' 8" S. Error 9° 28' E. Deviation 11° 18' E. 9. Interval 1^; rate 2»-4 losing ; interval 22"* 12''; ace. rate o™ 54'; Green, date, Oct. 30'' 12'' o™ 22^ ; red. decl. 14"^ 9' 57" S. ; tvue alt. 28' 42' 59' ; hour-angle 2^ 45'" 6» ; red-, eq. time 16"" 17= subt. ; mean time at ship, Oct. 30'* 2^ 28"" 49^ Longitude 142° 53' 15' W. 10. Green, date, Sept. 30'' 18'' 43™ 48*; red. decl. 3° 20' 31" S. ; true alt. 14^ 7' 3"; sum of log. 19-692568 ; true azimuth N. 89"^ 9' 42' W. Correction 4° 47' 12" W. Deviation 12° 27' 12" W. 11. Time from noon 17'" 8^ ; Green, djte Oct. 2^ i^ 17™ 8= ; red. decl. 3'' 49' 58' S. ; tru« alt. 47* 39' 40'; nat. no. 2190; mer. zen. dist. 42° 9' 9" N. Latitude 38' 19' 11" N, 388 Answers. Method IL—Bed. + 11' 15"; lat. 38-^ 19' 7" N fNorieJ. ; 38° 19' 16" N. ('Eai)erJ. Towson : Aug. I, + 38", index 36; Aug. II, -f- 10' 28". Latitude 38^ 19' 16" N. 12. Star's decl. 14° 32' 42" N. ; true alt. 54" 6' 7". Latitude 50° 26' 35" N". The Curve.— Correct magnetic bearing S. 38' AV. Deviations.— I ^E.; 17" W.; 22° W. ; 19MV. ; 3" W. ; 18' E.; 24." E. ; h/ E. Compass cour.scs.— S.S.E. | E. ; E. by N. -| N. ; S. 9° W. ; S. 72' E. Magnetic courses. — S. ^s¥ E. ; S. 10" W. ; N. 49° E. ; North. Bearings, magnetic.— N. 85^" W. ; N. 89^ W. EXAMINATION PAPEE— No. XI, pages 322—323—324. 1. 2-645548 = 442-128. (The product.) 2. 1-999489 = 99-88. (The quotient.) 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. Liff lat. 5i'-8 S. ; dep. 6'-i W. ; course S. 7° W. ; disi. 52'. Lat. in ^1° 8' N. ; diff. long. 10'. Longi- tude in 119° 50' E. 4. Green, date, Nov. 14^ 18'' 39™ 16^; red. decl. 18" 36' 25" S.; true alt. 67' 57' 49". Latitude 40° 38' 36" S. 5. Log. of diff. long. 2-2943 11 :=.L>iff. long. 196-9. 6. Diff. lat, 1928' N. ; mer. diff. lat. 2383' ; diff. long. 4290' E. ; tangent course 10-255333 ; course "ii. 6o°57'E. ; distance ^^ji'. 7. Standard, Brest 4* 7'' 53™ ; corr. for long. — io"> ; io'> o™ a.m., 10'' 21"^ v.u. „ Dover -j- o 39.; 10 16 a.m., 10 40 p.m. ,, Devonport 4"° ^7; 429 a.m., 4 52 p.m. 8. "Green, date, November 9'^ i2>» 20"" 44^; red. decl. 17° 12' i"S. ; true amp. E. 34° 9' S. Correction 41° 47' W. Deviation 58° 17' "W. 9. Interval 1 2'' ; rate Vz losing; Interval 36'' 3'> ; ace. rate 43^-2; Green, date, Nov. 30^ 2'' 48"" 52^; red. decl. 21" 47' 11" S. ; red. eq. time 1 0^548 suhtractivc; true alt. 39° 47' 8"; hour-angle t,^ 42™ 21=; mean time ship Nov. 2()^ 20^ 6'" 45^ Longitude lod^ 31' 45" W. 10. Green, date, Nov. i5<^' io'> 46"" 275 ; red. decl. 18° 46' 44" true alt. 43° 55' 7" ; sum of logs. 19-516742 true azimuth N. 69^ 57' 34' W., or S. 110° 2' 26" S. Correction 1 1° 32' 26" E. Deviation 19° 22' 26" E. 11. Time from noon 39'" 26*; Green, date, November i3'^2h 36™ 2*; red. decl. 18° 10*35 'S.; true alt. 56° 11' 50" ; nat. no. 8855 ; mer. zen. dist. 32' 52' 45" S. Latitude ^1° 3' 20" S. Towson : Hour-angle exceeds limits of Table. 12. Star's decl. 30° 16' 35" S. ; true alt. 59" 36' 2". Latitude 60° 40' 33' S. The Curve.— Correct magnetic bearing S. 89° W. Deviations.— I °E.; 19° E.; 21" E.; 9^ E. ; 3' "W. ; ii^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 324—325- 1. 3'364038 = '00231227. 2. 1-168317 = 14-7339. 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. Diff. lat. 25'-8 S. ; «fe^;. 8'-2 W. ; comz-scS. 18" W. ; dist,2f. Z«/. w 49^^ 34' N. ; diff. long, iz'^- Long, in 40*^ 1 3' W. Answers. S^g 4. Green, date, Dec. 3i ; corr. for long. 14'° ; 9"' 58"' a.m., 10'' i7"> p.m. ,, Queenstown, constant — i'' 19"^ ; 3*" 56'" a.m., 4^ i^^ p.m. „ Galway, + o^ 7'" ; 4*" 56"" a.m., 5^ 15°^ p.m. 8. Green, date, Dec. 2%'^ 4'' lo™ 49^; red. decl. 23° 15' 10" S. ; true amp. E. 35° 12^ S. Correction 9° 533' E. ; deviation 5° 36^' W. 9. Interval 12^; rate 5^-7 losing; interval 42'! 8''; ace. rate 4™ o'^-5. Green, date, Dec. 24J 8'> 18™ 54« ; red. decl. 23° 24' 46' S. ; red. eq. time additive o'« 17^ ; true alt. 40'' 54' 8" ; hour-angle 3'' 41'° 17^ longitude 178° 58' 30" W. 10. Green, date, Dec. 27J 4'' 40™ io»; red. decl. 23° 18' 17" S. ; true alt. 20° 27' 7"; sum of logs 19-362795; true azimuths. 57^23' 36" E. Co/->-ecn i2» ; red. decl. 15° 11' 16" N. ; true altitude 420 50' 17 ". Latitude 31° 58' 27 S. 5. Log. of diff. long. 2-663420 = Diff'. long. 460-7. 6. Diff. lat. 1890' S. ; mer. diff. lat. 2392' ; diff. long. i774'*43 W. ; course S. 36^ 34' W. ; distance 2353'. 7. Standard, Brest — i" 47'"; corr. for long. — 9'"; 5'^ 30™ a.m., and 5'' 52"" p.m. „ Brest — 07: 7 19 A.M., and 7 41 p.m. „ Brest + 2 38 : 10 4 a.m., and 10 26 p.m. 8. Green, date, Oct. 2 8. 60° W., 29': N. 16^ E , 20' -6: N. 79' E., 14' current course. Liff. lat. 1 3'- 8 S., dej}. i6'-4 E. : course S. 492° E., dist. 21^', Lat. in 34° 42' S., dij^. long. 20' E. Long, in 1 8° 48' E. 4. Green, date, Feb. lo"^ 21'^ 50™ 40' : red. decl. 14° 9' 38" S. : true a't. 30^ 33' 20'. Lati- tude 45° 17' 2" N. 5. Log. of difF. long. 2 010904 = L'ff. Long. i02"5. 6. DiflF. lat. 4202' N : mer. difiF. lat. 4555': diff. long. 4847' E. : tang, course 10026985 : course N. 46° 47' E. : distance 6136'. 7. Standard, Brest, constant + 4'^ 43™ ; corr. for long. + 9"!; 5^ 26™ A.M., 6'^ o™ p.m. „ Waterford, i*" 42™ a.m., a^ 20™ p.m. „ Leith, — i*^ 49"" ; 9*> 48'" a.m., 10'' 21"" p.m. 8. Green, date, March 30<* 711 41"" 8«; red. decl. 4° 8' 37" N. ; true amp. E. 4° 10* N. Correction 8" i' W. ; deviation 14'^ i' W. 9. Interval 1^^ ; rate 4^-9 gaining ; interval 32"^ 21^ ; ace. rate 2™ 41'; Green, date, May 26^1 21'' 10™ 6«; red. decl. 21° 23' 19" N. ; red. eq. time 3"" 4' subt. ; true alt. 43° 20' 9-'; hour-angle 2^ 53"" 55% Longitude 1° 46' 15" W. 10. Green, date, July lo"* o»» 25™ 20*; red. decl. 22° 10' 41" N. ; true alt. 44° 59' 38' ; sum of logs. 19228412 ; true azimuth S. 48^ 34' 44' E. Correction 54° 12' 14" W. ; deviation 3° 12' 14" W. 11. Time from noon 30™ 41*; Green, date, November 7'' 18'' 31'" 53'; red. decl. 16" 42' 12' S. ; true alt. 40° 4' 47"; mcr. zen. dist. 49° 22' 51" N. Zoie^Mrfe 32° 40' 39" N. Method II. j- 32' 17''. Latitude 32° 40' 49" N. Towson: Aug. I, + 8' 30"; index 81 ; Aug. II, -4- 24' 10". Latitude 32° 40' 21' N. 12. Star's decl. 57° 7' 43" S. ; true alt. 32'^ 48' 59". Latitude o"" 3' 18" N. The Curve. — Correct maa:netic 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. 79I" W. ; S. 89° W. Magnetic courses.- S. 80° E. ; S. 42^^ E. ; N. 66° W. ; S. 87° E. Bearings magnetic. — N. 28^ E. ; S. 62° E. EXAMINATION PAPER— No. XV, pages 329—330. 1. f3si334 = 22-4561. a. 9030734= -107333. 3. True Courses. —8. 72^ E., 25' ddp. course; N. 4i°E., 35'-4; N. 62° E.,48'; N. 53'W., 43'; S. 60° E., 26 ; N. 18^ W., i8'; S. 46" E., 32'; N. 82=^ E., 36'. Biff. lat. S'^'Z N., dep. 130' 6 E. ; course N. 67^^ E., dist. i4ij. Lat. in 38' 31' N., diff. long. 166' E. Long, in i" S' E. Answers. 391 4. Green, date, Nov. 2o<^ 19'' 18™ 40*; red. decl. 20° i' 50" S. ; true alt. 80' 28' 59": latitude 29° 32' 51" S. 5. Log. of diff. long. i'90967i = dif. Ung. 8i'22. 6. Diff. lat. 731' S. ; mer. diff. lat. 733' ; diff. long. 1259' W. ; tang, course io"234922 ; course S. 59° 47g' W. ; distance 1453'. 7. Standard, Brest, -4- 5'>,i3'" ; corr. for long. — if"" ; 9^ 26™ a.m. and 9'' 42™ p.m. ,, Dover, -j- o^ 8'" ; no a.m. and o^ 7" p.m. „ Dover, + f*" 13™ ; 4*^ 53"" a.m. and 5'' 12"" p m, 8. Green, date, Jan. 16^ S*' 2" 24'; red. decl. 20° 56' 37" S. ; true amp. W. 29° 17^' S. Cortection 13° 21 j' W. ; deviation 9° 385' E. 9. Interval 1 S'' ; r-dte 4^ o losing ; internal 7 z"! 1 2*^ ; Greenwich date, June 4'! 12^ 33™ 40'; red. defl. 22° 34' 25" N. ; true alt. 28' i8'52" ; red. eq. time subt, i"'48^; hour-angle 3'' 52"" i3«. Longitude 113"^ 4' 45' E. 10. Green, date, Nov. 9^ 14'' 38™ 46'; red. decl. 17° 14' 2" S. ; true alt. 6° 11' 26'; sum of logs. I9-300927'; true azimuth S. 53^ 7' 24" E. Correction 3° 17' 24" W. ; deviation 10° 37' 24" W. 11. Time from noon 9"' 5^; Green, date, Jan. 8<^ 3'' 31'" 43^; red. decl. 22° 16' 35" S. ; true alt. 76' 57' 49"; nat. no. 5(^4; mer. zen. dist. 12° 53' 5" S. Latitude 35° 9' 40" S. Method II. \- <)' 9". Latitude 35° 9' 42" S. Towson : True alt. exceeds the limits of the Table. 12. Star's decl. 16" 32' 50" S. ; true alt. 37^ 45' 59". Latitude 35° 41' 11" 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 headE. by 8.^8.:= 205° W.) ; N. 82^° W. ; S. 37l° E. . EXAMINATION PAPEE-No. XVT, pages 330—331—332. 1. Log. of profluct 6-498o62 z=: product 3i48i95'6 Log 6-23^506, product 0000017 12. 2. Log. of quotient 4 28™ 30*; red. decl. 15" 29' 49" 8. ; true amp. E. 22" 55' S. Correction 27° 423' W. : deviation 11° i2g' W. 9. Interval 14'*; rate %■"■$ gaining ; interval 124'' 20*"; acc. rate 17™ 41^; Green, date, August 31'^ 19^ 54™ 30'; red. 'feci. 8=' 8' 11" N.; eq. time — C" 14'; true alt. 15° 25' 44"; hour-an^le 4'^ 45™ 41". Longitude 10° 6' 15' W. 10. Green, date. May 3i<* 16^ 13""; red. decl. 22" 6' 17" N. ; true alt. 39^ 20' 26' ; sum of logs. i9'77973o; true azimuth 8. 101° 47' 34" E. Gonection 3° 21' 19' W. ; deviation 40 11' 19" W. 11. Green, date, April iz^ 14^ i"^ 5^ ; time from noon 10™ 15'; red. decl. 90 7' 38" N. true alt. 80° 43' 11" ; nat. no. 987 ; mer. zen. dist. 8° 55' 22" S. Lalittide 0° 12' 15" N. Method II. \- 11' 24". Latitude 0° 12' 2" N. 392 Answers. Towson : The altitude excefids the limits of the Table. 12. Star's decl. 6o-> 19' 27" S. ; true alt. 9° 52' 32". Latitude 1,9° 48' i" N. The Curve. — Correct miipnetic bearing S. 14° W. Deviation.— 20° W.; 16^ W.; 4" W. ; 16^ E.; 20° E. ; 14" E. ; I'E. ; n" W. Compass courses.— 8. 4° E. ; S. 89' E. ; N. 340 E. ; N. 13^° E. Magnetic courses. — S. 7' W. ; East; S. 500 W. ; N. 150 W. Bearing magnetic. — N. 79^"" W. ; S. 85= E. EXAMINATION PAPER— No. XVII, paf/es 333—334. 1. Log. of product 2-577492 = product 378 ; 2T38598 = "013759 and i'30io3o = 20. 2. Lo 13'"; 311 18'" a.m., and 3'' 45'" p.m. „ Weston-super-mare, ,, ■\-d"i<)"'; 6'^ 2"^ a.m., and 6^ 31m p.ji^ „ Portsmouth ,, — i^ 11™; 9^ 24™ a.m., and 9^ 50™ p.m. 8. Green, date, Nov, 4^ 21I1 23™ ; red. decl. 15° 51' 4" 8. : true amp. W. 16° 59' S. Cor- rection 19° 35' E. ; deviation 18" 35' E. 9. Green, date, Aug. 5"^ P^ 8™ 44^ ; red. decl. 16' 44' 18' N. ; red. eq. time + 5"^ 4°': true alt. 35" 16' 45'; hour-angle 3'" 53'" 55=. Longitude at sight 179° 14' 45" W. : diff. long, since sight 55' 42". Longitude at noon 180° 10' 27" W., or 179° 49' 33' E. 10. Green, date, Aug. 13'! 2^ 20™ 40''; red. decl. 14° 28' 33' N. ; true alt. 27* 23' 29"; Bum of logs. 19-259920; true azimuth N. 50° 29' 48' E. Error 250 29' 48" E. Deviation 9° 9' 48'' E. 11. Green, date, June II'' 19^43™ 30s ; time from noon 32™ 18'' ; red. decl. 23M1' 32"N.; true alt. 50° 14' 43"; nat. no. 8768. Latitude 15° 46' 14" S. Method II. -f- 47' 52'. Latitude 15° 46' 3" S. Towson: Aug. I, 4" i^' ^^' > index 81 ; Aug. II, -\- 34' 38". Latitude 15° 46' 41' S. 12. Star's decl. 22° 52' 59'' N. ; true alt. 60° 23' 4". Latitude 52° 29' 55" N. The Curve. — Correct magnetic bearing S. 20" E. Deviations. — 22" W. ; 10° W.; 3° E. ; 14° E. ; 15° E. ; 10° E. ; O"; 10^ W. Compass courses.— N. 24° W. ; S. 140 W. ; N. 35^° E. ; N. 13° E. Magnetic courses.— S.S.W. ; S. 41° E. ; S. 72^"^^. ; S. 1° E. Bearings, magnetic. — S. 8o|° W. ; N. 6%h° W. EXAMINATION PAPER--No. XVIII, i?^//^.!? 334—335—336. 1. 7-754166 = 56776104. 3-806197 = -00640025. 2. 8 201561 z= -015906. 13-000000=10000000000000. 3. TTrue Courses.— S. 6'' W-, 19' dep. course; N. 50'' W., 23'-7 ; S. 47° E., i6'-3; N. 180 E., i7'-6 ; S. %" W., I4'i ; N. 87° E , 42'-3 ; S. 75° E., 12' ; 8. 65° W., 15' current course. Biff. lat. k)''-^ S. ; dep. 35'-3 E. ; course S. 6i|"E. ; dist. 405 E. Lat. in ^6° ^i' i^.: Biff. long. 65' E. Long, in 38° 55° \V. 4. Green, date, June 24'' 20>> 3™ : red. decl. 23° 23' 46' N. : true alt. 60° 4' 7' . Latitude 6° 32' 7" S. Answers. 393 Eaper: True alt. 60° 4' i" N. Latitude 6^ 32' 13' S. 5. Difif. long. i53'"9 "W., or 2" 34' W. Longitude in 12° 4' W. 6. DiflF. lat. 4728'-5 N. : mer. difi. lat. 4896'-5 : diflF. long. 3540' E. : log. tang. 9'859ii7 ; -.se N. 35° 52' E. Conqyass cottrse N. i5''52'E.: r/ZsYrwce 5588'. 7. Standard, Brest 4"6''28"': corr. for long. — 12"': i'mj^a.m. and i1'33'»p.m. „ Brest -|- 3 42 : corr. for long. + 9 : 11 8 a.m. and 11 26 p.m. „ Queenstown — o 59 : : 7 33 a.m. and 7 49 p.m. 8. Green, date, June 23* 18'' 48"" 12" : red. decl. 23" 25' 23" N. : true amp. 23" 25^' N. : EtTor of compass 54' 21 j W. Deviation 32° 41^' "W. 9. Interval 32'!: rate o'-8: interval 113'' 5'': ace. rate i'" 3C5. Green, date, Sept. 22'' 4'' 57"" 35«: red. decl. o" o' o"; red. eq. time — 7" 34-'5 : true alt. 16' 56' 12 : hour- angle 4'^ 52™ i5^'2. Longitude 149° 21' 9" "W. 10. Green, date, March 21'' 13'' 51'" i6«: red. decl. 0° 43' 8" N. : true alt. 42° 57' 18"; sum of logs. 19-619736: true azimuth N. 80° 24' W. Error of compass i^° 1 2,^ 'E. Deviation 70 23I' E. ir. Green, date, Oct. 3'' 14'' 27'" 9** : time from noon 10™ 51' : red. decl. 4° 25' 54' S, true alt. 63° 47' 10" : nat. no. 963. Latitude 30' 31' 12' S. Method II. -j- 7' 34". Latitude 30° 31' 12' S. Towson ; Aug. I, + o' 16" : index 15". Aug. II, + 8' 6'. Latitude 30° 30' 54" S. 12. Star's decl. 55° 51' 19" N. : true alt. 84" 56' 45'. Latitude 60" 54' 34" N. The Curve,— Correct magnetic bearing N. 8' W. » Deviations.— 10° E. : 9^ E. : 5' E. : 0°: 6° W. : 9" W. : 7° W. : 2' W. Compass courses.— S. 76^ W. : N. 48° E. : S. 6° E. : N. 88^" W. Magnetic courses.— S. 40^= W. : S. 2" E. : S. 88"^ E. : S. 65° E. Bearings, magnetic. — N. 84!^ "W. : N. 64^^ E. EXAMINATION PAPER— No. XIX, pai/es 336—337—338. 1. 6-742037 = 5521240-5. 5-109096=128557. 2. 1-903505 = -800767. 7-922575 = 83670961-5. True Courses. — N. 53^ E., 21' dep. course: S. 74=" E., i6'-6 : S. 250 W., 13-4: N. 35* E., i9'-3 : S. i' W., i9'-i : S 65° W., 47'-6 : N. 77° E., 21-3 : N. 48° E., 49' current course. Diff. lat. lo'-i N. : dep. ^z'-o E. : course N. 79=' E. : dist. 53'. Lat. in 62° 30' N. : dij'. long. 112 E. Long, in 62° 48' "W. 4. Green, datp. May 31'^ 21'' i*" 20«: red. decl. 22° 7' 52" N. : true alt. 72° 28' 57 '. Lat. 39° 38' 55' N. 5. Log. of diff. long. 1-7477 19 = dif. long, ss'-g^. 6. Diff. 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 — i^i-]™: iih59'n a.m. and no p.m. ,, Leith — 2 55 : 10 21 a.m. and 1 0^42"" p.m. „ Brest + 6 57 : 918 a.m. and 9 39 p.m. 8. Green, date, Dec. 27^ 11'' 35™: red. decl. 23" 17' 27" S. : true amp. E. 3P gi'S. £)ror .of compass 58" 505' W. Deviation 39' 405 W. 9. Green, date, Jan. 2^^ 16^ 36™ 50' : red. decl. 18° 6' 23" S. : red. eq. time + 13"" 17' : true alt. 17° 52' 42* : hour-angle 3'' 46"" 35* : ' long, at sight 170° 45' 30" E. : diff. long, since sight 19' W. Long, at noon 171° 4' 30". E. 10. Green, date, July 9^ i7h 50'" i.o': red. decl. 22^ 12' 50' N. : true alt. 14° 36' 40": sum of logs. 19-176158 : true azimuth N. 45° 34' 36" W. Error of comjjass 47"" 49' 36" W. Deviation 54° 34' 36" W. £ £ £ 394 ^' 11. Green, date, Nov., 28'' 15'' 58"" 31': time from noon 12™ ii«: red. decl. 2i'>33'o"S. true alt. 74° 15' 51": rat. no. 1306': mer. zen. did. 15° 27' 27" N. : latitude 6' 5' 33" S. Method IT. + ^7' 6- Latitude 6" 5' 53" S. 12. Star's decl. 8° 20' 44" S. : true alt. 52° ii- 54": latitttde 29° 27' 22" N. Towson : Hour-angle exceeds the limits of Tal le. The Curve.— Correct magnetic bearing S. 8° E Deviations. — 15° E. : 3' E. : 130 W.: 28° W. : 14'' W. : 10° E. : 13^ E. : 140 E. Compa.ss courses. -S. by W. : S. 87" E. : S. 13- E. : S. 85.^" W. Magnetic courses.— N. 20° W. : N. 54° W. : S. .32" E. : N. 32' E. Bearings, magnetic— S. 68° W: West. EXAMINATION PAPEE— No. XX, pages 338—339—340. 1. 7'89i486 = 77890714. 8'763323 ^ •oococ 0057986. 2. 2-900314= 794-904. 7-950265 = 891793F8. 3. True Courses. — East, 30' dep. course: S. ;!8° E., 54'-2 : N. 48° E., 4i'-6 : S. 76^ "W., 14' 4 : S. 22" E., 46-5: N. 9° W., 29'-! : S. 51° H., 49'-2 ; N. 79° E.. 4o'-8 current course. Diff. lat. i5''2 8. : dep. \f)^'■^ E. : course S. 855° ?:j. : dist. 193'. Latitude in 59° 34' N, : diff. long. 3805° E. : longitude in 37° 49 .|' W. 4. Green, date, Sept. 30'' i8*> 21™ 20^ : red. d( cl. 3" 19' 58'' S. : true alt. 56° 56' 24" : lat. 29° 43' 38" N. 4.* Green, date, July 2*^ 8^ 59>n^ red, decl. .2° 59' o" N., true alt. 100 25' 59", latitude 77° 26' 59" N. 5. Diff. long. 369'-7 E., or 6° 10' E. : longitude 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. 2043' : t iff. long. 2043' W. : log. tang. lo-oooooo; true course N. 45° W. Compass course N. 32" 7I' ''7. Dist. 2828'-4. 7. Standard, Brest — 2'' 2™ 11*^52™ a.m. and no p.m. „ Portsmouth — i 11 84 a.m. and 8'" 35™ p.m. „ Brest — I 17 corr. for Img. — i"" 06 a.m. and o 36 p.m. 8. Green, date, April, 25^ o^ 34" 48^: red. de;l. 13° 24' 19" N. : true amp. W. 25^25' N. Error of compass 67° 24' W. Deviation 31° 34' W 9. Green, date, Aug. 23'' 18'' 16™ 50^: red. d' ol. to" 59' 40' X. : red. eq. time + 2™ 7^: true alt. 37° 26' 49", hour-angle 3'* 23™ 6^ Long at sight T,f -7,1 .^^" 'E.: di^f. long. 10 45" W. longitude at noon 35° 22' o" E. 10. Green, date, Oct. 31'' 22'' 15™ 44^: red. rjecl. 14° 37' 27" S. : true alt. 12° 23' 17": sum of logs. 19-217276: true azimuth S. 47° ^5 H" ^- Error 50° 43' 59" W. Deviation 17° 23' 59" W. 11. Green, date, May 29"^ 9*^ 58"" 28^ : time f om noon 4™ 12' : red. decl. 21° 47' 7" N. : true alt. 670 53' 11": nat. no. 160: mer. zen. dis. 22° 5' 21" S. Latitude 0° 18' 14" S. Method II. -\- 1' 29". Latitude 0° 18' 17" S. Towson: Aug. I, + o' 12" : index 2. Aug. II, + i' 18": latitude 0° 18' 12" S. 12. Star's decl. 6° 48' 49" N. : true alt. 28° 5E 36" : latitude 54° 12' 35" S. The Curve. — Correct magnetic bearing S. 49^ E. Deviations.— 20° W. : 16° W.: 2° W. : 14° E. ; 20° E. : 15° E. : 1° W. : 11° W. Compass courses. — S. 33° W. : S. 43° E. : N. 6)° W. : N. 24° E. Magnetic courses. — N. 9" E. : S. 74° "W. : S. lo' W. : S, 60° W. Bearings, magnetic. — S. 2° E. : S. 88° W. INDEX EEEOJ I, page 347. ^' I. + 2-15" Semid. i5'57" J^ 2. — 1-40" Semid. 15' 45" . 3- +2745 » 16 12 ^_4. +052 „ 1649 "\-S + 38 20 „ 15 50 ^ 6. + 35 5 „ 16 17-5 Amwers. 395 EXEECISES ON THE CHAET. FOE ONLY MATE, FIRST MATE, AND MASTER North Sea, pages i^\ — 352. Dist. 49' 2 » 19 4 „ 149 6 „ 41 8 W. Course W. \ S. „ S. by W „ S.E.iS. „ W. by S. i S. „ N.W. by W. /True course N.E. i E. i Mag. do., E. by N. a N. 66 329 Course S.W. i \V. Dist. i6r „ S.W. i S. >> 67 „ N. i E. II 3.S „ S.E. Ny S. )> ,?o „ N.W. ^ N. )» 30 / True coursfi S E. | ( Mag. do , S. by E. S. ) ^E 3i' 13. The ship is in latitude 55° 59' N., longitude 2° 40' E., and must sail S.E. \ S. (mag.) 209 miles. 14. The place of meeting was lat. 56° 5' N., long. 3° 11' E. : the course steered by the ship from Heligoland was N.W. f W. (true), and by the ship from Hartlepool was N.E. by E. I E. Long. 1° 1 1' W. I 28 W. o 24 W. 3 E. 1 34 W. I 36 E. ist Station — Lat. 54° 24' N., long. 0° 20' W,, distance 6 miles, and „ „ 54 24 N., „ o i W., „ 14 „ ist „ „ 55 8 N., „ I 8 W., ,, 12 „ 2nd „ „ 54 53 N., „ r I W., „ 16 „ 15- Lat. 55°i5rN. 16. )) 57 16 N. 17- 11 60 4 N. 18. » 54 27 N. 19. » 55 49 N- 20. C3 20 N. Course S.S.W. Distance 34' „ S.S.W. 1 w. 96 „ S.W. i S. (nrly) 159 ,, S. 1 E. 69 Earn Lights . Berwick Light 12^ 14^ Course N.N.W. | W. ISh English and Bristol Channels, and South Coast of Ireland, pp. — 352- -3?4- I. Course S.W. by W. Dist. 21' 2. Course S.S.E. J E. Dist. 37' 3- ,, N. by E. IE. » 44J 4- i> N. by E. (nrly.) 25 5- ,, N.N.E. \ E. .» 251 6. )) N.E. 1 E. 72^ 7. Portland E. by N. IN. ,. 36 8. )» N. by W. i W. 21 9- Course E. by N. ^N. „ 50 10. » S.E \ E. 77 II. )> N.N.E \ E. „ 26 12. )' E. by S. A S. 40 13- )) E. by N. IN. » 53 14. )) N. by E. i E. 21 15- )) N. by W .f W » 65 16 )» N. 1 W. 67 17- )) E. I S. „ 76 18. ,, E.iS. 355 19. Lat. 49^48' N. Long 6^ 7i'W. Course E. by S. Distance 37' 20. » 49 40 N. I 3 W. ,, S.E. 1 E. 45 21. „ .... )> E. iN. 40 CO 28 N. 2 9 w. 23- » t> >> N. by W. ^ W. »34 24. » 52 '"2 N. 6 19 W. ,, S.S.E. 1 E. 30 25- >> 52 4 N. I 58 E. 1> N. |E. 27 26. )> 50 35 N. 55 W. „ N.W. by W. 1 W. 15 27. )» 50 32 N. I 3o|W. „ N.W. by W. 21 28. )> 51 25 N. 4 59 W. ,, N.N.W. \ W. 32 29. >i 49 515 N. 5 35 W. )> N.W. by W. \ W. 30 30- jj 51 47 N. 7 42| W. >> W. by N. 32i 31- jj 51 41 N. 5 39 W. ,, S.E. by S. 1 S. 46^ 32. „ 50 39I N. >, 35 E. fDisi . off Dungeness Light, 21 miles, off Beachy Head Lt., 15 ,, 33- First Course— S.E. by S. (true), or S. by E . (mag.), distance 24 miles. 34- Second „ N . by E . (true), or N.E. by N. (mag.), distance 24 miles. 396 Answern. SOUNDINGS. Depths, &c., page 361. 1. Time from high water i*> 31"" ; half-range for day 18 feet 4 inches; tabk B 4- 13 feet. Depth of water required 55 feet 7 inches, or 9'^ fathoms. 2. Time before high water i"^ 47™ ; half-range for day 7 feet 2 inches ; table B -f 4 feet 2 inches. Depth 60 feet 4 inches, or 10 fathoms. 3. Time before high water i'' 46™ ; half-range for day 11 feet 3 inches ; table B -|- 6 feet 9 inches. Depth by chart 8 feet 3 inches. 4. Time from hi^h water o^^ 15™ ; half-range for day 20 feet 9 inches ; table B -|- 20 feet 5 inches. Corr. to low water 43 feet 5 inches. Water below sounding 7 feet 5 inches. 5. Time from high water ^'^ 41'" ; half-range for day 14 feet 5 inches ; corr., table B, sub. 14 feet 3 inches ; height of water i foot 3 inches below zero. 6. Time from high water 3^^ 59" ; half-range for day 5 feet 5 inches ; table B sub. o feet 10 inches. Sounding by chart 71 feet 2 inches, or 12 fathoms nearly. THOMAS I.. AINSI.KT, PRINTER, MARKET PLACE SOUTH »HIT:LDS. gi^se IIBRARJ UC SOUTHERN REGIONAL LIBRARY FACILITY A 000 605 192 4